Download c Copyright by Jonathan C. McKinney, 2004

c Copyright by Jonathan C. McKinney, 2004
°
BLACK HOLE ACCRETION SYSTEMS
BY
JONATHAN C. MCKINNEY
B.S., Texas A&M University of College Station, 1996
M.S., University of Illinois at Urbana-Champaign, 1999
DISSERTATION
Submitted in partial fulfillment of the requirements
for the degree of Doctor of Philosophy in Physics
in the Graduate College of the
University of Illinois at Urbana-Champaign, 2004
Urbana, Illinois
Abstract
Accretion onto a black hole is the most efficient known process to convert gravitational energy
into radiation. Some gamma-ray bursts (GRBs), some X-ray binaries, and all active galactic
nuclei (AGN) are likely powered by accretion onto a rotating black hole. I model the global, timedependent accretion flow around a black hole using the nonradiative viscous hydrodynamic (VHD),
Newtonian magnetohydrodynamic (MHD), and general relativistic MHD (GRMHD) equations of
motion, which are integrated numerically. I wrote the VHD and nonrelativistic MHD numerical
codes, and coauthored the GRMHD code (Gammie et al., 2003).
I first studied VHD accretion disk models, such as those studied by Igumenshchev et al. (1999,
2000) and Stone et al. (1999), hereafter IA and SPB. IA and SPB use the VHD model of accretion,
but they gave qualitatively different measurements of the energy per baryon accreted, angular
momentum per baryon accreted, and the radial scaling law for various quantities. While there
was concern in the community that the different results were due to numerical error, I found
that the different results could be reproduced using my single VHD code to model the accretion
flow (McKinney and Gammie, 2002). The differences in their results were due to differences in
their experimental designs. Seemingly small changes to the VHD model introduced nonnegligible
changes to the results. This suggests that a self-consistent MHD, rather than phenomenological
VHD, model for turbulence is required to study accretion flow.
First discovered during the VHD study described above, I found that VHD, nonrelativistic
MHD, and GRMHD numerical accretion disk models can produce significant numerical artifacts
unless the flow near the inner radial boundary condition, at rin , is out of causal contact with the
flow at r > rin . For the VHD model, this corresponds to setting rin so the flow there is always
ingoing at supersonic speeds. For the MHD models, this corresponds to setting rin so the flow there
is always ingoing at superfast speeds. For the GRMHD model, this is easily constructed by using
Kerr-Schild (horizon-penetrating) coordinates. In this case, rin is chosen to be inside the horizon,
where all waves are ingoing.
I next studied MHD and GRMHD numerical accretion disk models to test the results of previously studied phenomenological disk models. These models are based on the Shakura & Sunyaev
α-disk model and suggest the disk should terminate at the innermost stable circular orbit (ISCO) of
a black hole. Simplified GRMHD models predict super-efficient accretion due to energy extraction
from a rotating black hole. I found that MHD and GRMHD numerical models show that the disk
does not terminate at the ISCO, and magnetic fields continue to exert a torque on the disk inside
the ISCO. The disk will likely continue to emit radiation inside the ISCO, altering the predicted
spectra of accretion disks. GRMHD numerical models of thick and thin disks show that the energy
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per baryon accreted closely follows the thin disk efficiency, so super-efficient accretion does not seem
to be a generic property of thick magnetized relativistic disks (McKinney and Gammie, 2004).
The Blandford-Znajek (BZ) effect, describing the extraction of spin energy of a rotating black
hole by the magnetosphere, plausibly powers the jet in some GRBs, some microquasars, and all
AGN. I found that GRMHD numerical models of thick disks around a rotating black hole show
that an evacuated, nearly force-free magnetosphere develops as predicted by BZ (McKinney and
Gammie, 2004). The BZ solution for the energy extracted is remarkably accurate in this region
for a black hole with a/M . 0.5 and qualitatively accurate for all a/M , where a is the Kerr spin
parameter and M is the mass of the black hole. GRMHD numerical models with a/M & 0.5 show
a mildly relativistic (Lorentz factor Γ ∼ 1.5 − 3) collimated Poynting jet around the polar axis.
Currently, no self-consistent MHD model of the accretion flow around a black hole shows a jet with
Γ & 3. Additional physics is likely required to obtain Γ ∼ 100 as models predict in GRBs, and to
obtain Γ ∼ 3 − 10 as seen in some microquasars and AGN.
I studied the VHD, MHD, and GRMHD accretion models by performing numerical simulations
on our group’s Beowulf computer clusters, which I designed and constructed. For about $50,000,
one can buy a private cluster of computers that will provide as much computing power as today’s
typical time-shared “supercomputer.” I give an account of the procedures necessary to design,
build, and test a Beowulf cluster. The main conclusion is obvious: test one’s code on test nodes
before purchasing the entire cluster in order to confirm the performance and reliability of the chosen
components (CPUs, motherboard, network, etc.).
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To my mom.
v
Acknowledgments
I strongly express my gratitude to my supervisor, Prof. Charles Forbes Gammie, whose expertise,
understanding, and patience added considerably to my graduate experience. Charles demonstrates
a broad-minded, razor-sharp knowledge of astrophysics, and his example has driven me to become
a better scientist. Charles has tirelessly reviewed many of my written works, and with his help, I
have gradually improved my skills as a writer.
I express a special thanks to Scott Noble and Ruben Krasnopolsky, who both proofread the
unpublished parts of my thesis. They gave excellent comments that greatly improved the focus,
clarity, and readability of the thesis. Throughout their careers as postdocs for Charles, they have
provided me with many stimulating conversations and uncountable helpful pointers. I thank Paul
Ricker for excellent comments about the Beowulf cluster appendix.
I thank the members of my thesis committee: Charles Gammie, Stu Shapiro, Susan Lamb, and
Jen-Chieh Peng. Their time is precious and I am grateful for all the comments on my written
thesis and oral defense. I particularly thank Stu Shapiro for being a great source of inspiration and
guidance in the study of relativity and compact objects.
I thank Charles Gammie, Stu Shapiro, Bill Watson, David Campbell, and Peter Anninos for
writing recommendation letters for my future career as a postdoc at Harvard with Ramesh Narayan,
who I thank for hiring me and being patient while I finish my thesis.
I thank and love my family for the support they provided me through my entire life and in particular, I must acknowledge my fiancé and best friend, Elena, without whose love, encouragement,
and editing assistance I would not have finished this thesis.
My financial support was largely provided by a NASA GSRP Fellowship Grant NGT5-50343
(S01-GSRP-044) and partially supported by a GE fellowship. During my thesis work, Charles
was supported by an NCSA Faculty Fellowship, the UIUC Research Board, NSF ITR grant PHY
02-05155, and NSF PECASE grant AST 00-93091. Computations were done in part under NCSA
grants AST010012N and AST010009N using the Origin 2000 and Posic Linux cluster at NCSA.
Some computations were performed on Platinum.
vi
Table of Contents
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
List of Abbreviations and Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
1 Observations, Theory, and Models . . . . . . . . . . . . . . . . . . .
1.1 Summary of Introduction . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Introduction to Black Hole Systems . . . . . . . . . . . . . . . . . . .
1.2.1 Formation of Black Holes . . . . . . . . . . . . . . . . . . . .
1.2.2 Gamma-Ray Bursts . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Black Hole X-ray Binaries . . . . . . . . . . . . . . . . . . . .
1.2.4 Normal and Active Galactic Nuclei and Quasars . . . . . . .
1.3 Basic Accretion Disk Theory . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Accretion Luminosity and Mass of the Compact Object . . .
1.3.2 Some Accretion-Based Arguments . . . . . . . . . . . . . . .
1.4 Models of Accretion Disks and GRBs . . . . . . . . . . . . . . . . . .
1.4.1 Angular Momentum Transport Models . . . . . . . . . . . . .
1.4.2 Radiative Disk Models . . . . . . . . . . . . . . . . . . . . . .
1.4.3 Gamma-Ray Bursts Models . . . . . . . . . . . . . . . . . . .
1.5 Characteristic Quantities and Model Validity Estimates . . . . . . .
1.5.1 Estimated State and Structure of Accretion Flow . . . . . . .
1.5.2 Validity of the Fluid, MHD, and ideal MHD Approximations
1.6 Summary of Motivation for a GRMHD Model and Open Questions .
1.7 Summary of Dissertation Results . . . . . . . . . . . . . . . . . . . .
1.7.1 Viscous Hydrodynamics Summary . . . . . . . . . . . . . . .
1.7.2 Global 2D/3D MHD Summary . . . . . . . . . . . . . . . . .
1.7.3 HARM / GRMHD Summary . . . . . . . . . . . . . . . . . .
1.7.4 BZ Effect Summary . . . . . . . . . . . . . . . . . . . . . . .
1.7.5 BZ/Inflow Solution Comparison Summary . . . . . . . . . . .
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2 Numerical Models of Viscous Accretion Flows Near Black Holes
2.1 Summary of Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Numerical Treatment of Low Density Regions . . . . . . . . .
2.4.2 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.5
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2.4.3 Code Tests . . . . . . . . . . . . . . . . . . .
Results . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Fiducial Model Evolution . . . . . . . . . . .
2.5.2 Dependence on Inner Boundary Location and
2.5.3 Comparison of Torus and Injection Models .
2.5.4 Other Parameters . . . . . . . . . . . . . . .
VHD Summary . . . . . . . . . . . . . . . . . . . . .
Global 2D MHD Simulations . . . . . . . . . . . . .
Global 3D MHD Simulation . . . . . . . . . . . . . .
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3 HARM: A Numerical Scheme for General Relativistic
3.1 Summary of Chapter . . . . . . . . . . . . . . . . . . . .
3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 A GRMHD Primer . . . . . . . . . . . . . . . . . . . . .
3.4 Numerical Scheme . . . . . . . . . . . . . . . . . . . . .
3.4.1 Constrained Transport . . . . . . . . . . . . . . .
3.4.2 Wave Speeds . . . . . . . . . . . . . . . . . . . .
3.4.3 Implementation Notes . . . . . . . . . . . . . . .
3.5 Code Verification . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Linear Modes . . . . . . . . . . . . . . . . . . . .
3.5.2 Nonlinear Waves . . . . . . . . . . . . . . . . . .
3.5.3 Transport . . . . . . . . . . . . . . . . . . . . . .
3.5.4 Orszag-Tang Vortex . . . . . . . . . . . . . . . .
3.5.5 Bondi Flow in Schwarzschild Geometry . . . . .
3.5.6 Magnetized Bondi Flow . . . . . . . . . . . . . .
3.5.7 Magnetized Equatorial Inflow in Kerr Geometry
3.5.8 Equilibrium Torus . . . . . . . . . . . . . . . . .
3.6 Magnetized Torus Near Rotating Black Hole . . . . . . .
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .
Magnetohydrodynamics 82
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4 A Measurement of the Electromagnetic Luminosity of
4.1 Summary of Chapter . . . . . . . . . . . . . . . . . . . .
4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Review of Analytic Models . . . . . . . . . . . . . . . .
4.3.1 Coordinates . . . . . . . . . . . . . . . . . . . . .
4.3.2 Governing Equations . . . . . . . . . . . . . . . .
4.3.3 Blandford-Znajek Model . . . . . . . . . . . . . .
4.3.4 Equatorial MHD Inflow . . . . . . . . . . . . . .
4.4 Numerical Experiments . . . . . . . . . . . . . . . . . .
4.4.1 Fiducial Model . . . . . . . . . . . . . . . . . . .
4.4.2 Comparison with BZ . . . . . . . . . . . . . . . .
4.4.3 Comparison to Inflow Solution . . . . . . . . . .
4.5 Parameter Study . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Black Hole Spin . . . . . . . . . . . . . . . . . .
4.5.2 Field Geometry and Strength . . . . . . . . . . .
4.5.3 Numerical Parameters . . . . . . . . . . . . . . .
4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . .
a Kerr
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Black
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5 Future Studies . . . . . . . . . . . . . . . . . . . . . .
5.1 Thin Disks . . . . . . . . . . . . . . . . . . . . . . .
5.2 Relativistic Jet-Disk Connection . . . . . . . . . . .
5.3 Gamma-Ray Bursts: GRMHD Collapsar Model . . .
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A Model of Plasmas . . . . . . . . . . . . . . . . . . . .
A.1 Equation of State . . . . . . . . . . . . . . . . . . . .
A.1.1 EOS for GRBs . . . . . . . . . . . . . . . . .
A.1.2 EOS for X-ray binaries and AGN . . . . . . .
A.2 Validity of the Fluid Approximation . . . . . . . . .
A.3 Validity of the Ideal Fluid Approximation . . . . . .
A.4 Validity of the Plasma and MHD Approximations . .
A.5 Validity of the Ideal MHD Approximation . . . . . .
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B Beowulf cluster . . . . . . . . . . . . . . . . . . . . .
B.1 History of Design Decisions for our Clusters . . . . .
B.2 Cluster Performance and Advanced Network Drivers
B.3 Cluster access to Internet . . . . . . . . . . . . . . .
B.4 Hardware . . . . . . . . . . . . . . . . . . . . . . . .
B.5 Software . . . . . . . . . . . . . . . . . . . . . . . . .
B.5.1 Choices for OS . . . . . . . . . . . . . . . . .
B.5.2 OS Installation . . . . . . . . . . . . . . . . .
B.5.3 Software Installation and Usage Notes . . . .
B.5.4 MPI Implementation in Fluid Codes . . . . .
B.5.5 Running MPI Jobs . . . . . . . . . . . . . . .
B.6 Testing Cluster Reliability and Performance . . . . .
B.6.1 Reliability and Performance Issues . . . . . .
B.6.2 Bandwidth and Latency of Network . . . . .
B.6.3 Code Performance . . . . . . . . . . . . . . .
B.7 Beowulf Cluster Summary . . . . . . . . . . . . . . .
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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
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List of Figures
1.1
Cartoon plot of one hemisphere of the axisymmetric accretion disk, corona, funnel/wind, and plunging regions. The outer radius of the GRMHD simulation is
40GM/c2 . The fiducial radius (r = 12L, where L = GM/c2 ) is where all “state”
quantities are evaluated. These state quantities are then used to estimate the validity
of the fluid, MHD, and ideal MHD approximation at this location. . . . . . . . . . . 39
2.1
Time-averaged spatial structure of fiducial run (Run A; α = 0.1, rin = 2.7GM/c2 ,
and rout = 600GM/c2 ). Shown are the density (upper left), Bernoulli parameter (Be = (1/2)v 2 + c2s /(γ − 1) + Ψ) (upper right; dotted line is a negative contour), scaled mass flux r2 sin θ(ρ0 v) (lower left), and scaled angular momentum flux
r3 sin2 θ(ρ0 vvφ + Π · φ̂) (lower right). The flow is not symmetric about the equator because the flow exhibits long timescale antisymmetric variations. Convective
bubbles form at the interface between positive and negative Bernoulli parameter
(i.e. unbound and bound matter). . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The evolution of Ṁ /Ṁinj , e = Ė/(Ṁ c2 ), and l = L̇ c/(GM Ṁ ) in the fiducial run
(Run A). The dotted line indicates the thin disk value. The run has clearly entered
a quasi-steady state. The evolution is relatively smooth with a small variation on a
timescale τ ≈ 4 × 104 . This is the timescale for convective bubble formation (the low
point in rest-mass accretion rate is when bubble forms). For this model the bubble
forms at alternate poles. A full cycle requires of order one rotation period at the
injection radius. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The radial run of θ and time averaged quantities from the fiducial run (Run A).
Shown are the density (upper left), squared sound speed (upper right), radial velocity
(lower left), specific angular momentum (lower right; solid line), and circular orbit
specific angular momentum (lower right; dashed line). Crudely speaking, the inner
flow is consistent with a radial power law. The best fits to a power law are: ρ0 ∝ r−0.6 ,
cs ∝ r−0.5 , |vr | ∝ r−2 , and vφ ∝ r−0.8 . The plots are averaged over θ = π/2 ± π/6.
The effect of moving the inner boundary on the accretion rates of mass, angular
momentum, and energy (Run A vs. Run B). The top panel shows Ṁ /Ṁinj , the
middle panel Ė/(Ṁ c2 ), and the bottom panel L̇ c/(GM Ṁ ). The solid curve is Run
A, which has rin = 2.7GM/c2 . The dashed curve is Run B, which has rin = 6GM/c2 .
Evidently Run B has a different variability structure and different time averaged
values for the accretion rates. The relatively rapid and high-amplitude variations in
Run B are due to nonphysical interactions with the inner radial boundary. Only by
ensuring a supersonic flow (as in Run A) can one avoid these nonphysical effects.
Field line snapshot after 8 orbits (at the t = 0 torus density maximum at r =
9.4GM/c2 ) for a global 2D MHD simulation. Full non-linear turbulence drives the
accretion process. Note that the polar field is essentially radial while the accretion
disk is dominated by turbulence generated by the MRI. . . . . . . . . . . . . . . .
2.2
2.3
2.4
2.5
x
. 70
. 72
. 73
. 75
. 80
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
The L1 norm of the error in u for a slow wave as a function of Nx for both the
monotonized central (MC) and minmod limiter. The straight lines show the slope
expected for second order convergence. . . . . . . . . . . . . . . . . . . . . . . . . .
The L1 norm of the error in the single nonzero component of the velocity for an
Alfvén wave as a function of Nx for both the monotonized central (MC) and minmod
limiter. The straight lines show the slope expected for second order convergence. . .
The L1 norm of the error in u for a fast wave as a function of Nx for both the
monotonized central (MC) and minmod limiter. The straight lines show the slope
expected for second order convergence. . . . . . . . . . . . . . . . . . . . . . . . . .
The run of density in the Komissarov nonlinear wave tests. . . . . . . . . . . . . . .
The run of ux in the Komissarov nonlinear wave tests. . . . . . . . . . . . . . . . .
Snapshot of the final state in HARM’s integration of Ryu & Jones test 5A (a version
of the Brio & Wu shock tube) but with c = 100. The figure shows primitive variable
values at t = 0.15. Quantitative agreement is found to within ≈ 1%, as expected. .
Snapshot of the final state in HARM’s integration of Ryu & Jones test 2A, with
c = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Convergence results for the transport test. . . . . . . . . . . . . . . . . . . . . . . .
A cut through the density in the nonrelativistic Orszag-Tang vortex solution from
HARM (solid line, with c = 100), from VAC (dashed line), and 4× the difference
(lower solid line) at a resolution of 6402 . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of results from HARM and the nonrelativistic MHD code VAC for the
Orszag-Tang vortex. The plot shows the L1 norm of the difference between the
two results as a function of resolution for the primitive variables ρ0 (squares) and u
(triangles). The straight line shows the slope expected for first order convergence.
The errors are large because they are an integral over an area of (2π)2 . . . . . . . .
Convergence results for the unmagnetized Bondi accretion test onto a Schwarzschild
black hole. The straight line shows the slope expected for second order convergence.
Convergence results for the magnetized Bondi accretion test onto a Schwarzschild
black hole. The straight line shows the slope expected for second order convergence.
The equatorial inflow solution in the Kerr metric for a/M = 0.5 and magnetization
parameter Fθφ = 0.5. The panels show density, radial component of the four-velocity
in Boyer-Lindquist coordinates (with the square showing the location of the fast
point), the φ component of the four-velocity, and the toroidal magnetic field B φ =
F φt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Convergence results for the magnetized inflow solution in a Kerr metric with a/M =
0.5. Parameters for the initial, quasi-analytic solution are given in the text. The
straight line shows the slope expected for second order convergence. The L1 error
norm for each of the nontrivial variables are shown. The small deviation from second
order convergence at high resolution is due to numerical errors in the quasi-analytic
solution used to initialize the solution. . . . . . . . . . . . . . . . . . . . . . . . . .
Convergence results for the Fishbone and Moncrief equilibrium disk around an
a/M = 0.95 black hole. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Density field, for a magnetized torus around a Kerr black hole with a/M = 0.5 at
t = 0 (left) and at t = 2000M (right). The color is mapped from the logarithm of
the density; black is low and dark red is high. The resolution is 3002 . . . . . . . . .
xi
93
94
95
97
98
99
101
102
103
104
106
108
109
110
112
113
3.17 Evolution of the rest-mass accretion rate (top), the specific energy of the accreted
matter (middle), and the specific angular momentum of the accreted matter (bottom)
for a black hole with a/M = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
Initial (left) and final (right) distribution of log ρ0 in the fiducial model on the r sin θ−
r cos θ plane. At t = 0 black corresponds to ρ0 ≈ 4 × 10−7 and dark red corresponds
to ρ0 = 1. For t = 2000, black corresponds to ρ0 ≈ 4×10−7 and dark red corresponds
to ρ0 = 0.57. The black half circle at the left edge is the black hole. . . . . . . . . .
(a) The distribution of β, b2 /ρ0 , and ut in the fiducial run, based on time and hemispherically averaged data. Starting from the axis and moving toward the equator:
(1) ut = −1 contour shown as a solid black line; (2) b2 /ρ0 = 1 contour shown as a
red line; (3) β = 1 contour shown as a magenta line that nearly matches part of the
ut = −1 contour line; and (4) β = 3 contour is shown as cyan line. (b) Motivated
by the left panel, the right panel indicates the location of the five main subregions
of the black hole magnetosphere. They are (1) the disk: a matter dominated region
where b2 /ρ0 ¿ 1; (2) the funnel: a magnetically dominated region around the poles
where b2 /ρ0 À 1 where the magnetic field is collimated and twists around and up
the axis into an outflow; (3) the corona: a region in the relatively low density upper
layers of the disk with weak time-averaged poloidal field; (4) the plunging region;
and (5) the wind, which straddles the corona-funnel boundary. See Section 4.4.1 for
a discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Initial (left) and final (right) distribution of Aφ . Level surfaces coincide with magnetic field lines and field line density corresponds to poloidal field strength. In the
initial state field lines follow density contours if ρ0 > 0.2ρ0,max . . . . . . . . . . . . .
Contour plot of the time and hemispheric average of Aφ . Level surfaces coincide
with magnetic field lines and field line density corresponds to poloidal field strength.
Evolution of rest-mass, energy, and angular momentum accretion rate for our fiducial
run of a weakly magnetized tori around a black hole with spin a = 0.938. For
500 < t < 2000 the time average of these values is Ṁ0 ' 0.35, Ė/Ṁ0 ' 0.87, and
L̇/Ṁ0 ' 1.46 as shown by the dashed lines. The dotted lines show the classical thin
disk values (Ė/Ṁ0 ' 0.82 and L̇/Ṁ0 ' 1.95). See Section 4.4.1 for a discussion. . . .
(EM )
Electromagnetic energy flux density FE
(θ) on the horizon for the fiducial run,
based on time and hemisphere averaged data. The mean electromagnetic energy flux
is directed outward. See Section 4.4.1 for a discussion. . . . . . . . . . . . . . . . . .
The run of the force-free parameter ζ for the a = 0.5 run; when ζ ¿ 1 the field is
approximately force-free. The parameter has been time and hemisphere averaged.
The contours show (beginning from the pole and moving toward the equator) ζ =
10−3 , 10−2 , 10−1 . The small closed contours at large radius and close to the axis have
ζ = 10−2 . The small closed contours from the equator to θ ∼ π/4 have ζ = 10−1 .
See Section 4.4.2 for a discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Left panel: Magnetic field angular frequency on the horizon relative to black hole
rotation ω(θ)/ΩH . The solid line indicates time and hemisphere averaged data from
our a = 0.5 MHD integration. The middle dotted line is the prediction of the BZ
model (ω/ΩH = 1/2). The dashed line (top) is the value predicted by the inflow
model. Right panel: the run of field rotation frequency ω with radius along a single
field line that intersects the horizon at θ = 0.2. ω is constant to within 3%, as
expected for a steady flow. See sections 4.4.2 and 4.4.3 for a discussion. . . . . . . .
xii
130
131
133
134
135
137
138
139
(a) Square of radial field ((B r (θ))2 ) on the horizon in the a = 0.5 MHD integration,
from time and hemisphere averaged data. Solid line is the field for our numerical
model. The dotted line shows the Blandford and Znajek (1977) perturbed monopole
solution with the field strength normalized to the numerical solution at the pole.
(EM )
The dashed line is the inflow solution. (b) Electromagnetic energy flux FE
(θ) on
the horizon in the a = 0.5 MHD integration, from time and hemisphere averaged
data. The solid line shows the numerical model, the dotted line shows BZ’s spun-up
monopole solution, and the dashed line shows the inflow solution. See sections 4.4.2
and 4.4.3 for a discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.10 A comparison of the time-averaged fiducial model near the equator (within θ =
π/2 ± 0.3) with the inflow solution of Gammie (1999). In the right two panels the
black dotted line is the thin disk value. In all cases the red vertical line is the location
of the ISCO. The black line for the upper left panel is the numerical result. For the
other three panels, the particle term is shown in cyan, the internal energy term is
shown in magenta, and the electromagnetic term is shown in green. The blue line
in each plot represents the inflow model result. Notice that the run of density with
radius shows no feature at the ISCO. See the Section 4.4.3 for discussion. . . . . . . 142
4.11 The ratio of electromagnetic to matter energy flux on the horizon. The solid line
indicates numerical data while the dotted line indicates a best fit of Ė (EM ) /Ė (M A) =
−0.068(2 − r+ )2 . See Section 4.5.1 for a discussion. . . . . . . . . . . . . . . . . . . . 145
4.9
B.1 Astronomy Building computer room electrical layout as of Feb, 2004. BH is located
north and center. This excludes the addition of another new node for BH and the
collaborating group’s cluster of 12 nodes. . . . . . . . . . . . . . . . . . . . . . . . .
B.2 BH Beowulf cluster primary elements. . . . . . . . . . . . . . . . . . . . . . . . . . .
B.3 Digital photograph of BH cluster. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.4 Cluster block diagram of BH cluster. Figure shows bandwidth between elements on
motherboard and switch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.5 Bandwidth (left) and latency (right) for gigabit Ethernet connection on BH cluster.
xiii
178
180
181
185
200
List of Tables
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Fiducial Black Hole Accretion Systems
Fiducial Black Hole Accretion Systems
Fiducial Black Hole Accretion Systems
Accretion Flow Regions . . . . . . . .
Accretion Disk State . . . . . . . . .
Validity of Fluid Approximation . . .
Validity of MHD Approximation . . .
Validity of Ideal MHD Approximation
2.1
2.2
Parameter List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Results List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1
4.2
4.3
4.4
Commonly used symbols . . . . . .
Black Hole Spin Study . . . . . . .
Field Strength and Geometry Study
Resolution Study . . . . . . . . . . .
B.1
B.2
B.3
B.4
B.5
B.6
Single CPU performance in ZCPS . . . . . . . . . . . . . . . . . . . . .
Origin 2000 MPI performance in kZCPS for 2D MHD code . . . . . . .
NCSA Platinum MPI performance in kZCPS for 2D & 3D MHD code .
BH Xeon Cluster MPI performance in kZCPS for ZEUS-based 2D MHD
BH Xeon Cluster MPI performance in kZCPS for 3D MHD code . . . .
BH Xeon BH Cluster MPI performance in kZCPS for 2D HARM code .
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List of Abbreviations and Acronyms
ADAF Advection Dominated Accretion Flow.
AGN Active Galactic Nuclei.
ASCA Advanced Satellite for Cosmology and Astrophysics.
BATSE Burst And Transient Source Experiment.
BeppoSAX Beppo Satellite per Astronomia X, in honor of Giuseppe “Beppo” Occhialini.
BC Boundary Condition.
BH Black Hole.
BHC Black Hole Candidate.
BL Lac BL Lacertae.
BTU British Thermal Unit.
BZ Blandford & Znajek (1977).
CDM Cold Dark Matter.
CPU Central Processing Unit.
CV Cataclysmic Variable.
EM ElectroMagnetic.
EMF Electromotive Force.
EOS Equation Of State.
GM Glenn’s Messages (Myrinet message protocol).
GR General Relativity.
GRB Gamma-Ray Burst.
GRMHD General Relativistic MagnetoHydroDynamics.
GRS Galactic Radio Source.
xv
HARM High Accuracy Relativistic Magnetohydrodynamics.
HD HydroDynamics.
HDE Henry Draper Extension (catalogue).
HETE High Energy Transient Explorer
HLL Harten-Lax-van Leer
HMXB High-Mass X-ray Binary.
HST Hubble Space Telescope.
IA Igumenshchev & Abramowicz (2000,2001).
IC Index Catalogue (of nebulae).
IMBH Intermediate-Mass Black Hole.
IMF Initial Mass Function.
IR InfraRed.
ISCO Inner-most Stable Circular Orbit.
KS Kerr-Schild.
LF Lax-Friedrich.
LIGO Laser Interferometer Gravitational Wave Observatory.
LISA Laser Interferometer Space Antenna.
LMC Large Magellanic Cloud.
LMXB Low-Mass X-ray Binary.
MC Monotonized Central.
MGC Millennium Galaxy Catalogue.
MHD MagnetoHydroDyanmics.
MKS Modified Kerr-Schild.
MPI Message Passing Interface.
MRI MagnetoRotational Instability.
MTW Misner, Thorne, & Wheeler (1973).
xvi
NGC New General Catalogue.
NS Neutron Star.
NSE Nuclear Statistical Equilibrium.
OS Operating System.
PWF Popham, Woosley, & Fryer (1999).
QED Quantum ElectroDynamics.
QPO Quasi-Periodic Oscillation.
RXTE Rossi X-ray Timing Explorer.
RJ Ryu & Jones (1995).
RMR Relativistic MagnetoRotators.
ROSAT Röntgen Satellite.
SGI Silicon Graphics Incorporated.
SLE Shapiro, Lightman, & Eardley (1976).
SDSS Sloan Digital Sky Survey.
SMBH Super-Massive Black Hole.
SN Supernova.
SPB Stone, Pringle, & Begelman (1999).
SRMHD Special Relativistic MagnetoHydroDynamics.
SS/SS73 Shakura & Sunyaev (1973)
SXT Soft X-ray Transient.
ULX UltraLuminous X-ray (object).
URCA The URCA process involves scattering of a neutrino, proton, neutron, and electron. Coined
by Gamow & Schoenberg in 1941 in humorous reference to the money drain they experienced
gambling in Urca, Brazil.
UV UltraViolet.
VHD Viscous HydroDynamics.
xvii
VLA Very Large Array.
VLBI Very Long Base Interferometry.
VLBA Very Long Baseline Array.
XMM-Newton X-ray Multi-Mirror - Newton.
XSPEC X-Ray Spectral fitting package.
YSO Young Stellar Object.
xviii
Summary
About twice per day, gamma-ray observatories detect a gamma-ray burst (GRB) with a typical
duration of seconds (for a review see Piran 2004). Some GRBs have an associated lower-energy
afterglow. Redshift measurements of the afterglow, together with the fireball model of GRBs,
suggest GRBs have a luminosity of 1045 − 1053 erg/s. A high luminosity GRB is likely the most
powerful event in the universe. The known duration of GRBs allows one to estimate that most
GRBs have a total energy of ∼ 1051 erg, which is comparable to the energy of a supernova. The
so-called fireball model suggests a GRB comes from an outflow with a Lorentz factor of Γ ∼ 100,
so they have one of the fastest bulk flow velocities in the universe. A GRB likely originates from
the collapse of a massive star into a neutron star or black hole. The duration of a GRB is much
longer than the radial free-fall time for the stellar envelope, which suggests the stellar matter has
a nonnegligible angular momentum and an accretion disk likely forms. The GRB is thought to be
generated by a jet from the accretion of stellar matter onto a rapidly spinning neutron star or black
hole. The jet is suspected to penetrate the stellar surface and, according to the standard model,
the observed gamma-rays are produced in relativistic shocks within the jet.
About half of all star systems have only one star. Some single-star systems contain a massive
star that collapses into a neutron star or black hole, which eventually accretes all the stellar matter
that could form a disk or material that falls back from the supernova. Some of these may continue
to accrete interstellar matter or roam the universe undetected except by gravitational lensing and
faint X-ray emission. However, about half of all star systems are binaries, and the neutron star
or black hole can continue to accrete by capturing matter from the companion star. Such systems
have a high X-ray luminosity, so are called X-ray binaries (for a review see Lewin et al. 1995).
With a persistent power output up to 1036 − 1038 erg/s, X-ray binaries are the brightest persistent
sources of X-rays in the sky. By comparison, the Crab pulsar, otherwise the brightest X-ray source
in the sky, has an X-ray luminosity of ∼ 1037 erg/s. Many X-ray binaries, called microquasars, have
a relativistic jet with Γ ∼ 3 − 10. Some neutron star X-ray binaries unstably burn hydrogen and
helium leading to thermal X-ray bursts with a power output of 1036 −1039 erg/s, and some unstably
burn carbon leading to thermal X-ray bursts with a power output of 1043 erg/s. The compact object
in some X-ray binaries has a mass M ∼ 3 − 20 M¯ , which suggests that they cannot be neutron
stars, so they are dubbed black hole candidates (BHCs). None of these BHC X-ray binaries exhibit
thermal X-ray bursts. This suggests they each have no surface and so likely each contain a black
hole with an event horizon. An example black hole X-ray binary is Cygnus X-1, which harbors a
compact object with M ∼ 7 − 13 M¯ (Webster and Murdin, 1972; Bolton, 1972; Gies and Bolton,
1986; Herrero et al., 1995).
xix
Observations of a galaxy’s stellar dynamics, gas kinematics, or the galactic mass to light ratio
suggest that in most galaxies the center harbors a supermassive (M ∼ 105 − 109 M¯ ) black hole
(for a review see Krolik 1999a). Such supermassive black holes likely formed from accretion onto,
or merging of, intermediate-mass black holes with M ∼ 102 − 104 M¯ . The motions of stars near
our galactic nucleus, SgrA*, provide the best evidence for a black hole with M ∼ 2.6 × 106 M¯
(Eckart and Genzel, 1997; Ghez et al., 1998; Schödel et al., 2002). An active galaxy is a galaxy
from which a significant fraction of the energy output is emitted by the nucleus rather than by the
stars, dust, and interstellar gas. Such nuclei are called active galactic nuclei (AGN). AGN have
a luminosity of 1042 − 1048 erg/s, which has long been considered to be generated by accretion
onto the supermassive black hole. Observations of the active galaxy MCG-6-30-15 show a highly
redshifted Fe Kα line emission profile with significantly different strengths for the red and blue
wing of the profile. These features of the Fe Kα line profile are consistent with emission from a
rapidly rotating v/c ∼ 0.2 accretion disk within a few gravitational radii (GM/c2 ) of a (possibly
rotating) black hole (Pariev and Bromley, 1998; Tanaka et al., 1995).
Accretion onto a black hole is the most efficient known process to convert gravitational energy
into radiation. Some GRBs, some X-ray binaries, and all AGN are likely powered by accretion of
plasma onto a rotating black hole. Models of the accreting plasma typically use the hydrodynamic
(HD) approximation, or when including a magnetic field, the magnetohydrodynamic (MHD) approximation. Since these systems involve relativistic flow around a rotating black hole, the MHD
approximation is solved using general relativity (GR) theory, which gives the GRMHD equations
of motion. This thesis mostly involves the study of nonradiative accretion disks using the GRMHD
approximation. Despite some useful time-independent and simplified HD, MHD, and GRMHD analytic solutions, a numerical model is likely required to study the global, time-dependent accretion
flow. Typically, the time-dependent solution of accretion flow is only a quasi-steady state, where
flow quantities can have large deviations from the time-averaged value. I developed a viscous HD
code, a nonrelativistic MHD code, and a GRMHD code.
The thin disk solution of Shakura & Sunyaev (SS) uses an unmagnetized viscous model for
angular momentum transport, which is likely due to MHD turbulence (Balbus and Hawley, 1991).
The SS model predicts that the accretion disk terminates at the innermost stable circular orbit
(ISCO) of a black hole. The SS solution is often used to interpret observations of spectra from
accretion disks, and the termination of the disk leads to a significant spectral feature. Some studies
assume the SS solution to be accurate, and properties such as black hole spin, disk orientation,
and the location of the disk inner-edge are derived. I used the nonrelativistic MHD and GRMHD
codes to study magnetized accretion flow. Like others at the time, I found that the disk does not
terminate at the ISCO, and magnetic fields continue to exert a torque on the disk inside the ISCO.
The disk will continue to emit radiation inside the ISCO, altering the predicted spectra of accretion
disks and altering those derived quantities (Reynolds and Begelman, 1997). For X-ray binaries and
AGN, a more accurate spectral fit may suggest that the Fe Kα line emission is from farther out in
the disk and that the black hole is not necessarily rotating.
xx
Blandford and Znajek (1977) (BZ) argued that accretion flow around a black hole leads to
evacuation of the polar regions and the development of a force-free magnetosphere. They found a
solution, called the BZ-effect, which describes the extraction of spin energy of a rotating black hole
by the magnetosphere. I used the GRMHD code to numerically study thick disks around a rotating
black hole and found that an evacuated, nearly force-free magnetosphere develops as predicted by
BZ. The BZ solution is remarkably accurate in this region for a black hole with a/M . 0.5, where
a is the Kerr spin parameter. For models with a/M & 0.5, I find a mildly relativistic (Γ ∼ 1.5 − 3)
collimated Poynting jet around the polar axis. However, additional physics is likely required to
obtain Γ ∼ 100 as models predict in GRBs, and to obtain Γ ∼ 3 − 10 as seen in some microquasars
and AGN.
The Gammie (1999) inflow solution describes a cold, thin accretion flow around a rotating black
hole. The Gammie solution predicts the radial dependence of comoving mass density, comoving
magnetic energy density, and the accretion rate of energy and angular momentum per baryon. The
Gammie solution also predicts accretion could occur super-efficiently (i.e. more energy is released
than rest mass accreted), and that likely the efficiency is greater than that in a thin disk due
to the magnetic field in the plunging region. I found the numerical GRMHD solution for the
radial dependence of those quantities to be marginally consistent with the Gammie inflow solution.
Discrepancies are likely due to the numerical solution consisting of hot disk, while the Gammie
solution is for a cold disk. None of the numerical models I studied showed super-efficient accretion.
In fact, for most thick and thin disks, the energy per baryon accreted closely followed the thin disk
efficiency found by Bardeen (1970).
I designed and constructed a Beowulf cluster of computers to help develop the VHD, MHD,
and GRMHD codes and use these codes to perform numerical simulations of accretion flow. I
describe how to design, build, and test a Beowulf cluster. The main conclusion is that one should
test one’s own code on test nodes and construct a test cluster before purchasing the entire cluster.
Performance and reliability vary greatly between personal computers (PCs) and server computers
(SCs). The main performance difference is due to the PCI bus, on which both add-on and builtin network chips operate. The network chip and the PCI bus are typically the bottleneck in
the performance of parallel calculations. Today’s PCs are all composed of 32-bit 33Mhz-based
motherboards, while SCs are composed of 64-bit 66/133Mhz-based motherboards. A PC is about
1/2 the price of an SC, so PCs are an attractive option. However, for a cluster with more than
only about 4 nodes, a cluster of PCs costs more per unit performance than a cluster of SCs.
Since some or all black hole accretion systems could harbor a thin disk, I plan to study thin
disks and to determine how the BZ luminosity depends on disk thickness. A few groups have
studied GRMHD models of jets, and all fail to show Γ factors as high as in observed jets. I plan
to study other mechanisms for jet acceleration and collimation, and I plan to study the connection
between the disk and jet. I also plan to include the relevant microphysics to the GRMHD model
to study the accretion disk that likely forms during a GRB.
xxi
1
Observations, Theory, and Models
This chapter summarizes the observations, theory, and models of black hole accretion disk systems.
Section 1.1 outlines this introduction.
1.1
Summary of Introduction
Against the force of gravity, a spheroidal star is mostly supported by pressure forces, while a thin
disk is mostly supported by centrifugal, or even magnetic, forces. The formation of a disk typically
occurs because radiative energy is released faster than angular momentum is transported. The disk
can be described as a result of the conservation of mass, energy, and momentum in the gravitational
potential. As the disk matter falls in the gravitational potential, collisions allow the gravitational
potential energy to be converted into radiative energy (instead of kinetic energy). Such a disk is
known as an accretion disk. The entire structure of the disk and surrounding dynamically-coupled
medium is called the accretion flow. The gravitating body in some accretion disk systems is a
planet (held up by gas or matter pressure), young stellar object (YSO) (held up by gas or radiation
pressure), white dwarf (star held up by electron degeneracy pressure), or neutron star (NS) (star
held up by neutron degeneracy pressure). This thesis is primarily concerned with those accretion
disk systems that contain a black hole. For a review, see Shapiro and Teukolsky (1983).
Accretion disk models often use the fluid approximation to describe the underlying microphysics
of ionized media. The fluid approximation assumes that the smallest region of interest contains
a large number of particles and that particle collisions are frequent enough to sustain a statistical equilibrium state. A magnetized fluid is often modeled by the magnetohydrodynamic (MHD)
approximation. The typical MHD approximation assumes the fluid is ionized, can be treated as
a so-called weakly-coupled plasma, and particles have negligible gyration radii. A weakly-coupled
plasma is one that contains both positive and negative charges, but is effectively neutral over a
distance that spans many particles. A simplified form of the MHD approximation is the ideal
MHD approximation, which assumes that the plasma is perfectly conducting, currents arise instantaneously from fields, the Hall effect is negligible, electronic pressure gradients are negligible,
and the pressure is isotropic (see, e.g., Krall and Trivelpiece 1973). A further simplification is the
single-component ideal MHD approximation, which assumes all species are in statistical equilibrium within a fluid element and that there is a well-defined average density, velocity, etc. for the
fluid element. For a gas of protons and electrons, the single-component approximation involves
neglecting the electron inertia. For some fluids, some of these approximations are coupled to the
same underlying approximation. For example, if the electron inertia is negligible, then currents
1
arise instantaneously from fields for a gas of protons and electrons. Throughout the thesis, the
accretion flow is assumed to be well-described by the single-component ideal MHD approximation,
but this will be tested near the end of this chapter. This thesis considers both the nonrelativistic
and general relativistic ideal MHD approximations to describe the black hole accretion flow.
To provide context for the accretion disk theory to be presented later, Section 1.2 gives a
summary of some astrophysical objects that probably contain an accreting black hole, such as
gamma-ray bursts (GRBs), X-ray binaries, a source in the Galactic nucleus called SgrA*, and
active galactic nuclei (AGN). That section introduces the basic observed and implied features of
these objects and also introduces 6 fiducial black hole accretion disk systems: 2 GRBs, 1 X-ray
binary, SgrA*, and 2 AGN. The purpose of the discussion is to provide context for the accretion disk
theory discussion that follows and to motivate general relativistic magnetohydrodynamic (GRMHD)
models of these systems by summarizing the evidence for black holes and magnetized relativistic
motion. Later in this chapter, accretion disk models are developed for these 6 fiducial systems.
These models are used to determine some accretion flow properties, such as the mass density and
temperature of the disk. Also later, the validity of the fluid, MHD, and ideal MHD approximations
as a model for the accretion flow in these 6 fiducial systems is tested.
Since the inner region of the accretion disk is typically not resolved by current telescopes, the
theory of this part of the accretion disk often rests on a spatially-unresolved flux of photons. This
flux can be decomposed into a luminosity, time variability, and spectra of the disk system. Section 1.3 outlines the basic theory of disks around compact stars (such as black holes) by discussing
the luminosity, time variability, and some spectral features of accretion disks. The section shows
how the luminosity and time variability independently provide an estimate for the mass of the
accreting compact object. The section discusses how some spectral features, such as the Fe Kα line
profile, may even provide details about the spatial structure of the inner disk in some X-ray binaries
and some AGN (Karas et al., 1992; Tanaka et al., 1995; Pariev and Bromley, 1998; Miller et al.,
2004). The purpose of this section is to provide accretion theory-based arguments that suggest
some of the 6 fiducial objects, and those like them, are likely powered by an accretion disk that
extends close to a black hole.
An element of a disk accretes only by losing angular momentum. Since angular momentum is
conserved, other elements must gain angular momentum and so move outward. Thus, for a disk
to accrete, angular momentum must be transported outward. The first models of accretion disks
assumed some form of magnetic or hydrodynamic turbulence drives angular momentum transport
(Shakura and Sunyaev, 1973). The turbulence is often phenomenologically modeled as an effective
shear viscosity that causes differentially-rotating layers to exchange angular momentum (Novikov
and Thorne, 1973; Eardley and Lightman, 1975; Pringle, 1981). While a hydrodynamic instability
may generate turbulence in a disk, no known hydrodynamic instability drives sufficiently vigorous
angular momentum transport to account for observations of astrophysical accretion disks (for a
review, see Balbus and Hawley 1998). Later, it was realized that magnetized accretion disks are
naturally unstable, and that the so-called magnetorotational instability (MRI) could drive angular
2
momentum transport and account for accretion in astrophysical disks (Balbus and Hawley, 1991).
The first part of Section 1.4 summarizes the effective viscosity and MRI models for the transport of
angular momentum in a highly ionized accretion disk. The purpose of that section is to motivate
the use of a magnetic, rather than viscous, model of the angular momentum transport.
Radiative processes are self-consistently included in accretion disk models that seek to account
for the observed spectra and luminosity of accretion disk systems. Some radiative processes, such
as Compton scattering or bremsstrahlung, are involved in determining the structure of the disk
and in producing a significant observational feature. Other radiative processes, such as Fe Kα line
fluorescence, are weakly involved in determining the disk structure, but are important observational
features from disks. Some radiative processes, such as synchrotron radiation from non-thermal particles, require physics beyond the thermodynamic equilibrium approximation. The second part of
Section 1.4 discusses radiative accretion disk models that self-consistently determine disk structure.
The purpose of that section is to provide a summary of previous work and to provide estimates of
the rest-mass accretion rate for the 6 fiducial systems.
Ultimately, radiative models are required to compare with observations, however a nonradiative
GRMHD model may be sufficient to describe the mechanism of jets in black hole accretion systems.
This thesis focuses on testing the plausibility of the Blandford-Znajek (BZ) effect (Blandford and
Znajek, 1977) as a mechanism to generate relativistic jets, rather than focusing on the mechanisms
involved in determining observational spectra from the disk. The last part of Section 1.4 discusses
GRB models and the likelihood of the BZ-effect powering a GRB. GRBs are discussed more extensively than X-ray binaries and AGN since the nonradiative GRMHD models, as developed in
this thesis, more readily apply to the accretion disk that forms in the so-called “collapsar” model
of a GRB (Woosley, 1993; Paczyński, 1998; MacFadyen and Woosley, 1999). That is, it is later
shown that the collapsar GRB model likely does not need a detailed radiative transport model for
photons and neutrinos.
Section 1.5 estimates various quantities, such as mass density, temperature, velocity, magnetic
field, and other quantities for the 6 fiducial systems. No radiative GRMHD model, with a realistic
equation of state, has been developed or studied numerically. Therefore, a combination of models
is used and checked for consistency. Section 1.5 uses the results of nonradiative GRMHD numerical
models (as in Chapter 4), unmagnetized radiative neutrino-cooled analytic models (as in Popham
et al. 1999), and unmagnetized photon-cooled radiative analytic models (as described by Shakura
and Sunyaev 1973; Novikov and Thorne 1973) of accretion disks to make all the estimates. GRB
systems are approximated by a radiative ideal Fermi gas equation of state, while X-ray binary and
AGN systems are approximated by an ideal gas + photon radiation pressure equation of state. As
a consistency check of the estimates for the 2 GRB systems, the results of the nonradiative, ideal
gas GRMHD numerical model are compared to the radiative, ideal Fermi gas model. A summary of
the radiative Fermi model is provided in the first part of Appendix A. The purpose of the section is
to provide estimates of the various quantities that help to provide an intuition about these systems.
Section 1.5 also estimates the validity of the fluid, MHD, and ideal MHD approximation for the
3
6 fiducial systems. These estimates are as discussed in any plasma physics book, and summarized
in Appendix A. The purpose of the section is to show that the single-component ideal MHD
approximation is an excellent model1 for the 6 fiducial (and likely all) black hole accretion systems.
Section 1.6 summarizes the properties of GRBs, X-ray binaries, SgrA*, and AGN that suggest a
GRMHD model is required to study the accretion disk near the black hole in these systems. Some
open questions are posed about the source of the luminosity and jets in these systems, which may
be resolved by a GRMHD model.
Section 1.7 reviews the main results of chapters 2-5 in this thesis.
1.2
Introduction to Black Hole Systems
Einstein (1916) formulated his general relativity (GR) theory, which has stationary solutions called
black holes (for a discussion see, e.g., Shapiro and Teukolsky 1983, Chapt. 12). Einstein’s GR theory
today remains the standard and most fundamental theory of gravity (see, e.g., Misner et al. 1973,
and for a historical perspective see Thorne 1994). Schwarzschild (1916) derived an exact, static,
and spherically symmetric solution to Einstein’s equations. A generalization of the Schwarzschild
solution for a stationary, charged2 , rotating, space-time is the Kerr-Newman metric (see, e.g.,
Shapiro and Teukolsky 1983, Chapt. 12, for a discussion). The Schwarzschild and Kerr-Newman
solutions to Einstein’s equations define the properties of stationary, classical black holes as studied
in this thesis. This thesis focuses on black holes as the object at the center of the accretion disk.
The first discussion in this section summarizes the plausible methods for forming a black hole
as a remnant of the collapse of a massive gas cloud or a massive star, critical mass build-up on
a degenerate star due to accretion, or the collision between two compact objects. The types of
astrophysical systems involved in such black hole formation scenarios include 1) gamma-ray bursts,
which each likely contain a black hole with M ∼ 3 M¯ ; 2) black hole X-ray binaries, which each
likely contain a black hole with M ∼ 3 − 20 M¯ ; and 3) galaxies, some of which contain one or
more super-massive black holes (SMBHs) with M ∼ 106 − 109 M¯ , where M¯ = 1.989 × 1033 g is
the mass of the Sun.
The second discussion describes the properties of gamma-ray bursts, X-ray binaries, and active
galactic nuclei, and introduces 6 fiducial black hole accretion disk systems: 2 gamma-ray bursts, 1
X-ray binary, a radio source at the center of our galaxy called SgrA*, and 2 active galactic nuclei.
These 6 fiducial systems are used as examples of black hole accretion systems for several calculations,
such as those in Section 1.5. Table 1.1 summarizes the estimated distance to, black hole mass of,
and bolometric luminosity of the 6 fiducial objects described in this section. Throughout this thesis,
similar tables are presented that cumulate additional quantities as relevant to the discussion in each
section. A summary description of these objects/events is given below.
1
This is shown to be true everywhere except in current sheets.
Charged black holes are likely not astrophysically relevant. Any buildup of charge is quickly neutralized by
accretion and pair production.
2
4
Table 1.1.
Fiducial Black Hole Accretion Systems 1
Parameter
NS-BH
GRB 030329
LMC X-3
dL
M [M¯ ]
Lbol [erg/s]
?
3
1053
804 Mpc
3
3 × 1052
55 kpc
10
1038
SgrA*
8 kpc
2.6 × 106
1037
NGC4258
7 Mpc
4 × 107
3 × 1043
M87
18 Mpc
3 × 109
2.3 × 1042
Note. — dL is the luminosity distance, M is the mass in solar mass units, and Lbol is the
bolometric luminosity. These quantities are shown for the 5 fiducial black hole accretion
disk systems. See text for references.
Some of the objects discussed in this section have their distance from the Sun determined by
the observed redshift (z = λobs /λemit − 1, where λ is the photon wavelength) of spectral lines. For
these objects, the distance is found by integrating the Friedman-Robertson-Walker (FRW) metric to
determine the comoving distance (dC ) that gives a luminosity distance of dL = (1+z)dC . The power
law ΛCDM model and WMAP (and related) data give a Hubble constant h ∼ 0.71 km s−1 Mpc−1 ,
Ωm ∼ 0.27, and ΩΛ ∼ 0.73 for a flat (k = 0) FRW metric model of the cosmos (Spergel et al.,
2003).
1.2.1
Formation of Black Holes
Subrahmanyan Chandrasekhar found that if a white dwarf (an accepted object at the time) is
sufficiently massive, then the degenerate electrons will become highly relativistic and can drive
the star to collapse (Chandrasekhar, 1931b,a). He predicted that a white dwarf would undergo
collapse if the mass reaches Mch ∼ 1.4 M¯ . The collapse of a white dwarf can lead to a neutron
star supported by neutron degeneracy pressure. Eddington (1935) realized that if Chandrasekhar
was correct, then a star with a mass much larger than 1.4 M¯ should collapse to a black hole3 .
Modern estimates of the upper limit to the mass of a neutron star give M ∼ 1.8−2.2 M¯ (Akmal
et al., 1998). If one only requires GR and causality to hold, then the upper limit is M ∼ 3.4 M¯
(Rhoades and Ruffini, 1974; Hartle, 1978), but see (Yuan et al., 2004; Abramowicz et al., 2002).
Regardless of the equation of state, once an object is contained within a light trapping surface, GR
theory predicts that the result must be a black hole (Hawking and Ellis, 1973). A classical black
hole is a simple, conservative model, while alternatives invoke exotic neutron star physics (Bahcall
et al., 1990) or modifications to GR in the strong-field limit (see, e.g., Babak and Grishchuk 2003;
DeDeo and Psaltis 2003). This thesis considers a compact object with a mass greater than about
M ∼ 3.4 M¯ to be a black hole candidate (BHC).
3
The term black hole was coined later by Wheeler in 1968.
5
There are two likely ways of creating a “stellar mass black hole” with M ∼ 3 − 20 M¯ : 1) stellar
core-collapse; and 2) compact object collisions. Collapse of the iron core of a massive star can lead
to a black hole or neutron star remnant and the generation of a supernova (for a review see Woosley
et al. 1993; Wheeler et al. 2000). Black hole formation may also occur during the collision of two
neutron stars (Eichler et al., 1989), or during other collisions involving a compact object (Popham
et al., 1999). Isolated solar-mass black holes have been indirectly detected by micro-lensing events
(Bennett et al., 2002; Agol et al., 2002; Agol and Kamionkowski, 2002). Black holes are found with
normal (non-compact) stellar companions in binary systems (for a review see, e.g., Lewin et al.
1995; McClintock and Remillard 2003), which are called black hole X-ray binaries due to their
powerful X-ray emission. The black hole in the binary likely results from the collapse of a massive
star, while the other star remains intact.
The initial mass function (IMF) ξ(M ) specifies the distribution in mass of newly formed stellar
populations, and ξ(M ) can be used to estimate the mass and number of black holes compared to
other stars. From the observed dependence of bolometric luminosity on star mass, Salpeter (1955)
suggested that ξ(M ) ∝ M −(1+x) , where x = 1.35 and the distribution extends from a lower bound
of M1 ∼ 0.1 M¯ to an upper bound of M2 ∼ 125 M¯ (see also Miller and Scalo 1979; Scalo 1986).
The present-day stellar IMF is now thought to extend up to M ∼ 200 M¯ (Larson, 2002) and studies
of population III (low-metallicity) stars suggest the IMF is probably top-heavy (Schneider et al.,
2002). Stars with zero metallicity have little mass loss, so massive low-metallicity stars may collapse
to massive black holes. Numerical simulations predict there should exist stars with M > 100 M¯
(Abel et al., 2000, 2002). Other studies suggest the first stars with M < 140 M¯ should have
mass loss like normal metallicity stars (Fryer, 1999). For stars with 140 M¯ < M < 260 M¯ , an
electron-positron pair instability during oxygen burning leaves no remnant. For M > 260 M¯ , the
star collapses directly to a black hole with a remnant mass about half the original stellar mass. For
M ∼ 105 M¯ , no stable hydrogen burning occurs and the protostar collapses directly to a black
hole (Baumgarte and Shapiro, 1999; Shibata and Shapiro, 2002).
Cold dark matter (CDM) models suggest the first generation baryonic clouds have M ∼ 106 M¯
(Bromm et al., 2002). Such clouds are suggested to fragment into stars. However, a small fraction
may suffer direct collapse to a black hole. If one thousandth of the initial cloud mass collapses
directly to a black hole, then there would exist intermediate-mass black holes (IMBHs) with masses
M ∼ 102 − 104 M¯ (Balberg and Shapiro, 2002; Shapiro and Shibata, 2002). This is consistent
with the observed correlation between the mass of the central SMBH and the mass of the bulge of
a galaxy, which is MBH /Mbuldge ∼ 10−3 (Kormendy and Gebhardt, 2001). Hierarchical growth of
galaxies through accretion of satellite galaxies leads to IMBHs in off-center positions, as confirmed
by numerical simulations of dense star clusters (Portegies Zwart et al., 1999; Portegies Zwart and
McMillan, 2002; McCrady et al., 2003).
There are so-called ultraluminous X-ray sources (ULXs) that are observationally similar to
black hole X-ray binaries, but ULXs have a higher luminosity of L ∼ 1039 − 1041 erg/s. ROSAT,
ASCA, and Chandra observations show that about 50% of spiral galaxies have ULXs that are
6
more compact and variable than AGN (Fabbiano et al., 2003), off-center from the galactic center,
and associated with star-forming regions. ULXs are also observed in globular clusters in elliptical
galaxies (Kaaret et al., 2001). A ULX is found in, for example, the center of irregular galaxy M82
(Cigar galaxy) (Matsumoto et al., 2001). ULXs could correspond to accreting IMBHs. Simple
luminosity constraints suggest M & 102 − 103 M¯ , dynamical friction constraints imply M <
106 M¯ , and the velocity cusps in globular clusters imply M ∼ 103 − 104 M¯ (Van Der Marel,
2004). However, if the emission is beamed or time-dependent, these ULX sources could be stellarmass black holes (King et al., 2001). ULXs may be explained as IMBHs confined to the disk
of the host galaxy that accrete interstellar matter (Krolik, 2004), although see Rappaport et al.
(2004). Whether ULXs are actually accreting IMBHs or stellar mass black holes is an active area
of research.
The formation of a SMBH is a likely result of star formation (Rees, 1984; Genzel et al., 1997).
The formation of SMBHs could be due to one or more IMBHs that fall to the galaxy center due
to dynamical friction (Haiman and Loeb, 2001). A scenario for creating a SMBH involving the
collision of multiple IMBHs may be unable to account for all SMBHs today (Islam et al., 2003).
This scenario would result in little black hole spin to power radio jets (Hughes and Blandford,
2003; Gammie et al., 2004), so if the scenario is accurate then black hole spin would not power
jets. A single IMBH seed could be driven to a supermassive mass by accretion of gas. This
accretion scenario generates the observed correlations (Kormendy and Gebhardt, 2001) between
black hole mass and bulge mass or bulge velocity dispersion (Van Der Marel, 2004). However,
recent evidence of a SMBH binary in radio galaxy 3C66B is consistent with the hierarchical growth
of galaxies (Sudou et al., 2003). Thus, both hierarchical (merger dominated) and anti-hierarchical
(gas accretion dominated) scenarios may have occurred.
The preceding discussion summarized how black holes form with different masses, ranging from
stellar mass black holes with M ∼ 3 − 20 M¯ to SMBHs with M ∼ 106 − 109 M¯ , and some
basic observational evidence was given for the existence of these black holes. The following discussion summarizes the evidence for the existence of black holes and accretion disks in specific
events/objects. The purpose of the following discussion is to suggest that some gamma-ray bursts,
some X-ray binaries, the centers of some normal galaxies, active galactic nuclei, and quasars are
likely powered by an accreting black hole.
1.2.2
Gamma-Ray Bursts
Gamma-ray bursts (GRBs) are brief bursts of X-rays or gamma-rays that are observed a couple
times a day, last from seconds to hours, appear randomly distributed in the sky, and do not
repeat (for a review see Piran 1999, 2004). The Vela satellites detected the first GRB in 1967.
Not until 1973 were reports of GRBs declassified and published (Klebesadel et al., 1973). In
1990, the Burst and Transient Source Experiment (BATSE), onboard the Compton Gamma-Ray
Observatory, localized several hundreds of bursts. Their random distribution on the sky suggested a
7
cosmological origin (Paczyński, 1986; Meegan et al., 1992). However, not until 1997 did BeppoSAX
detect GRB970508 with an optical counterpart (afterglow) with emission lines showing a redshift of
z = 0.835 (dL = 5300 Mpc) (Metzger et al., 1997; Frail et al., 1997), demonstrating a cosmological
origin.
Approximately two GRB detections occur per day. Possibly related events called X-ray flashes
(XRFs) are detected several times a year and exhibit no gamma-ray emission (Heise et al., 2001;
Yamazaki et al., 2002). GRB durations form a bimodal distribution of short-duration GRBs lasting
an average of δT ∼ 0.3 s and of long-duration GRBs lasting on average of δT ∼ 35 s (Kouveliotou
et al., 1993). The shortest GRB currently ever observed lasted for δT ∼ 6ms (Bhat et al., 1992) and
had 200µ s temporal structure, while the longest GRB (actually an XRF) currently ever observed
lasted for 2550 s (0.7 hr) (see review by Zand et al. 2003). Short-duration GRBs are observed to have
a harder spectrum compared to long-duration GRBs (Paciesas et al., 1999). Some long-duration
GRBs have an associated X-ray to radio afterglow emission that contains spectral lines that allow
a redshift measurement, which indicates these GRBs occur at cosmological distances. The typical
redshift of a GRB is z ≈ 1 (dL = 6600 Mpc) (Mészáros, 2001) and has been measured up to z ∼ 4.5
(dL = 43000 Mpc) (Andersen et al., 2000). No afterglow, and hence redshift, has been observed for
short-duration GRBs.
The GRB emission is likely relativistically beamed, but the afterglow redshift can be used to
estimate the distance and isotropic rest-frame energy Eiso (γ) = 4πFγ d2L (1 + z)−1 and isotropic
rest-frame luminosity Liso (γ) = Eiso (1 + z)/δT = 4πFγ d2L , where Fγ is the observed fluence of
gamma-rays (see, e.g., Frail et al. 2001). For example, GRB 990510 had a 50 − 300 keV fluence
of Fγ = 3 × 10−5 erg/ cm2 over a period of δT ∼ 33 s (Kippen, 1999) and an associated optical
afterglow with FeII and MgII absorption lines with redshift z = 1.619 (Vreeswijk et al., 1999). This
gives dL ∼ 7600 Mpc. Assuming an isotropic emission, the total energy is Eiso (γ) ∼ 5 × 1052 erg
and total luminosity is Liso (γ) ∼ 4.2 × 1051 erg/s. “Cosmic GRBs” at redshift z & 0.5 are possibly
standard candles with an actual total energy production of about Eγ ∼ 1051 erg (Sari et al., 1999;
Frail et al., 2001; Panaitescu and Kumar, 2001; Bloom et al., 2003), although there are significant
outliers with Eγ ∼ 1048 − 1050 erg (Sazonov et al., 2004; Soderberg et al., 2004).
In most GRB models, the GRB is suspected to be produced from a relativistic jet that was
emitted, by some mechanism involving accretion, near the polar axis of a black hole accretion
disk system. The “collapsar” model scenario, used to explain long-duration GRBs, entails the
collapse of a massive star and formation of an accretion disk around a newly-formed black hole of
mass M ∼ 3 M¯ (Woosley, 1993; Paczyński, 1998; MacFadyen and Woosley, 1999). Short-duration
GRBs may be generated by the collision of a neutron star and black hole, which subsequently forms
an accretion disk around the rotating black hole (Narayan et al., 1991, 1992).
This introduction later estimates the properties of a disk that could form during a neutron
star - black hole (NS-BH) collision and during the collapse of a massive star, such as what likely
occurred in GRB 030329.
8
NS-BH Collision Model
There are no afterglows, and so no redshift measurements, associated with short-duration GRBs.
Thus, these events have no estimated isotropic luminosity that could provide a clue about the nature
of the progenitor. One model for short-duration GRBs is a NS-BH collision, which is estimated to
have an isotropic luminosity of about 1053 erg/s and last less than a second (Narayan et al., 1992;
Popham et al., 1999).
GRB 030329
The prompt gamma-ray emission from GRB 030329 lasted for about δT = 25 s (Vanderspek et al.,
2003), and within hours was followed by an afterglow observed from X-ray, UV, optical, infrared,
and radio emissions. Observations of a supernova light curve (SN2003dh) in the afterglow of
GRB 030329 confirmed that the collapse of a massive star is spatially and temporally connected
to the GRB (Stanek et al., 2003; Kawabata et al., 2003; Uemura et al., 2003; Mészáros, 2003;
Hjorth et al., 2003). Emission lines in the afterglow have a redshift of z = 0.169 (dL = 804 Mpc)
(Greiner et al., 2003b). The GRB fluence within 30 − 400 keV was Fγ ∼ 10−4 erg cm−2 , which
gives Eiso (γ) ∼ 6 × 1051 erg and Liso (γ) ∼ 3 × 1050 erg/s. The likely total beamed energy is
Eγ ∼ 2 × 1049 erg in a beam with angular width ∼ 3.5◦ (see, e.g., Greiner et al. 2003a), which is
an outlier in a GRB Eγ histogram.
1.2.3
Black Hole X-ray Binaries
Observations indicate that 50-80% of stars are in multiple star systems, with most of these being
binary stellar systems (for a review see Evans 1999; Tohline 2002). About half of all stars are
in binary systems, and some contain an accreting compact object (likely a neutron star or black
hole) and have strong emissions of X-rays that likely originates from an accretion disk (for a review
see, e.g., Lewin et al. 1995; McClintock and Remillard 2003). X-rays are emitted mostly from
the inner region of the accretion disk near the collapsed star (see, e.g., Shapiro and Teukolsky
1983, Chapt. 13 for discussion). X-ray binaries have a persistent X-ray luminosity in the range
of L ∼ 1032 − 1038 erg/s, where the companion star has a bolometric luminosity in the range of
1032 − 1038 erg/s.
An X-ray binary is a true binary system, which is resolved as a binary by observing the motion
of the normal star due to the gravitational influence of the compact object. The motion is detected
by 1) observing the wobble motion of the normal star on the celestial sphere ; 2) spectroscopic
observations of the light-of-sight velocity ; or 3) the eclipsing of one object by another. The mass
of the compact object in X-ray binaries is often determined by the mass function
f (Mc , Mn , i) =
Mc sin3 i
Porb vn3
=
,
(1 + Mn /Mc )2
2πG
(1.1)
where Mc is the mass of the compact object, Mn is the mass of the normal star, i is the inclination
9
of the orbit with respect to the line of sight, Porb is the orbital period of the binary, and vn is the
line of sight velocity. A lower limit to Mc can be found by setting Mn = 0 and i = π/2, while
often other secondary measurements allow higher lower limits on the mass of the compact object.
The distribution of the compact object’s estimated mass in X-ray binaries shows a clear bimodal
distribution with about 22 objects with mass 1.1 − 2 M¯ and a distribution of about 18 objects
with masses from 3 − 16 M¯ with a mean and peak of 10 M¯ (see, e.g., Zand et al. 2004; Postnov
and Cherepashchuk 2004, figure 1, and see McClintock and Remillard 2003).
Classes of X-ray Binaries
There are approximately 240 known X-ray binaries that are classified according to their supposed
underlying physics (Zand et al. 2004 and references therein). These classifications include high-mass
X-ray binaries (HMXBs), low-mass X-ray binaries (LMXBs), Be X-ray binaries, X-ray bursters,
X-ray pulsars, soft X-ray transients (SXTs), and microquasars. A neutron star is the compact
object in some HMXBs, some LMXBs, all Be X-ray binaries, all X-ray bursters, all X-ray pulsars,
some SXTs, and some microquasars. A black hole is the compact object in some HMXBs, some
LMXBs, some SXTs, and some microquasars. A single object can be in multiple classes.
X-ray binaries are likely powered by the accretion of matter from the normal star onto the
compact star. The formation of the accretion disk can be described by two limiting scenarios associated with either HMXBs or LMXBs. HMXBs are X-ray binaries with a massive M ∼ 10 − 30 M¯
stellar companion, such as a Be (B-type with prominent emission lines) star or blue supergiant. An
accretion disk likely forms from the companion’s stellar wind material. LMXBs are X-ray binaries
with a low-mass M . 5 M¯ stellar companion, which is typically a main sequence solar mass star.
In some cases the companion star is degenerate or evolved (subgiant or red giant). An accretion
disk forms as the companion overfills its Roche lobe (Lagrange gravitational equipotential surface
between the compact and normal star) and matter plunges onto the compact object. For a catalogue of LMXBs see Liu et al. (2001). The spatial distribution in our galaxy of X-ray binaries
shows about 90 LMXBs with a slight concentration in the center, while HMXBs are more evenly
distributed (Grimm et al., 2002).
SXTs are LMXBs that are only discovered after undergoing a so-called accretion outburst
(also called an X-ray nova) (Tanaka and Shibazaki, 1996). Typical SXTs are usually faint or
unobservable in X-rays during a “quiescent” state. Typical SXTs contain a K-type subgiant or
dwarf that is transferring mass to a black hole through an accretion disk. It is thought that during
“quiescence,” mass is accumulating in an accretion disk. When an outburst occurs, most of disk
falls into the compact object. The accretion outburst likely occurs due to some type of temperature
dependent disk instability. A similar mechanism operates in dwarf novae (Cataclysmic Variables
(CVs)) (Warner, 1995).
Galactic microquasars (Mirabel et al., 1992) are X-ray binaries that are probably powered by
a black hole accretion disk system or an exotic mechanism for energy emission from a neutron
star. Galactic microquasars are named for their strikingly similar features to quasars or AGN
10
(Mirabel and Rodrı́guez, 1994, 1999; Fender and Belloni, 2004), such as apparently superluminal
(v/c ∼ 3 − 10) jets, which plausibly originate from a relativistic system with a well-defined axis
of rotation. Microquasars have a higher X-ray luminosity than other X-ray binaries. Unlike other
X-ray binaries, microquasars might also be sources of gamma-rays (Romero, 2004). AGN and
microquasars are likely more than simply morphologically similar (Rees, 1998), since they may
both contain an accreting rotating black hole.
This thesis later estimates the properties of the disk around the black hole in LMC X-3 described
below. A description of the classic galactic microquasar GRS1915+105 and the classic black hole
X-ray binary Cygnus X-1 is also given below.
LMC X-3
For a review of LMC X-3 see Cowley (1992). Large Magellanic Cloud (LMC) X-3 is in the LMC
about 55 kpc from the Sun. LMC X-3 likely contains a B5 subgiant with mass 4 − 8 M¯ in a 1.7day
orbit around a compact object, so LMC X-3 is often classified as a HMXB. The normal companion
star’s shape is severely distorted by the compact object, which is likely a ∼ 10 M¯ mass black hole
(Cowley et al., 1983). The X-ray luminosity of LMC X-3 is Lx ∼ 1038 erg/s (White and Marshall,
1984; Treves et al., 1988). Despite being classified as a HMXB, recent XMM-Newton observations
suggest that the disk is likely formed by Roche-lobe overflow (Soria et al., 2001). As with many
X-ray binaries, the observed disk flux of hard or soft X-rays shows periodic behavior (Cowley et al.,
1991; Ebisawa et al., 1993) and transitions between hard and soft emission (Wilms et al., 2001a).
GRS1915+105
A classic galactic microquasar is GRS1915+105 (V1487 Aql), which is about 12.5 kpc away from
the Sun. The companion star is a late-type giant with M ∼ 0.8 ± 0.5 M¯ , so is also classified
as a LMXB (Greiner et al., 2001b; Harlaftis and Greiner, 2004). GRS 1915+105 is the most
luminous of all known X-ray binaries (Done et al., 2004), with a highly variably X-ray output of
∼ 1038 − 1040 erg/s and shows emission from apparently superluminal jets (Mirabel and Rodrı́guez,
1994, 1999; Fender and Belloni, 2004). Radio synchrotron emission from the jet shows plasma blobs
moving at apparently superluminal speeds with an estimated true speed of v/c ∼ 0.92. The black
hole mass is suspected to be about 14 M¯ (Greiner et al., 2001a).
Cygnus X-1
Cygnus X-1 (Cyg X-1) is a source about 2.6 kpc away in the Cygnus constellation. The optical
counterpart is a supergiant called HDE 226868 which is an O9-B0 supergiant with a surface temperature of 3.1 × 104 K and a mass of 20 − 30 M¯ , so Cygnus X-1 is classified as a HMXB. Cyg X-1
is the first object that was broadly agreed to require the existence of a black hole. The compact
object is believed to be a black hole since it has a mass of 7 − 13 M¯ (Webster and Murdin, 1972;
Bolton, 1972; Gies and Bolton, 1986; Herrero et al., 1995).
11
1.2.4
Normal and Active Galactic Nuclei and Quasars
All the stars from a normal (non-active) galaxy together have a typical luminosity of 1044 erg/s.
Normal galaxies derive their luminosity from an ensemble of stars and so are, to zeroth order,
related to a black body spectrum with a small range of strong emission. An active galaxy is a
galaxy from which a significant fraction of the energy output is emitted by the nucleus rather
than by the stars, dust, and interstellar gas. The luminosity of such nuclei is typically between
1042 − 1048 erg/s. Some nuclei outshine their host galaxy by 4 orders of magnitude. The nuclei
in such active galaxies are called active galactic nuclei (AGN) (see review by Krolik 1999a). The
distance from the Sun to a typical active galaxy is less than a Gpc (z < 0.2), which is close enough
to resolve the stellar distribution and sometimes the individual stars. AGN come in many different
types, which are largely a result of viewing angle (for a discussion of AGN unification models see,
e.g. Krolik 1999a).
Unlike normal galaxies, most AGN emit in a broad-band spectrum. For example, NGC 4151
is observed to have a flat spectrum from the mid-infrared (1013 Hz, 0.04 eV) to the hardest X-rays
observed (1019 Hz, 41 keV). The active galaxy Markarian 421, known as a blazar, shows dramatic
variability with flares reaching into the TeV range (Punch et al., 1992). AGN are observed to
have spectra with emission lines such as the Lyα, Balmer lines, and the X-ray Fe Kα line near
6.4 keV. These lines are Doppler-broadened from relatively broad lines (several thousand km s−1 )
to relatively narrow lines (few hundred km s−1 ). For example, MCG-6-30-15, known as a Seyfert
I, exhibits Fe Kα line fluorescence, whose broad, blue-shifted profile likely indicates emission from
deep within the gravitational potential of a supermassive black hole (Tanaka et al., 1995). See
Krolik (1999a) for a discussion of AGN spectra features and their likely origins.
Within AGN, there are many intermediate absorbers and emitters between our line of sight
and the accretion disk that significantly affect the observations. For example, the broad-line and
narrow-line regions in AGN are an interesting study alone (see, e.g., Krolik 1999a; Elvis 2000, and
references therein).
Some of the first observations of AGN were of so-called radio galaxies with jets emerging from
a point-like source. A typical jet has lobes that extend a few hundred kiloparsecs, a distance that
is comparable to or larger than that of the host galaxy size (see, e.g., the classic review by Bridle
and Perley 1984). AGN jets are observed to be relativistic with some having variable structure
moving at an apparent speed of v/c ∼ 3 − 10 (see, e.g., West et al. 1998). AGN jets are sources of
synchrotron radiation (Baade, 1956), indicating the existence of a strong magnetic field. The radio
synchrotron radiation is likely due to internal shocks that produce high-energy electrons in the jet
magnetic field. Recent results also find that X-rays can be produced as the AGN jet collides with
external matter and produces knots within the jet (Hardcastle et al., 2003; Kraft et al., 2003). These
structures are often tracked to measure the apparent velocity, which is typically superluminal. Jet
motion in quasar 3C279 has been followed over a period of several years (Wehrle et al., 2001), and
the jet has apparent velocities of 4.8 − 7.5c. The quasar 3C273 has an apparently superluminal
12
v/c ∼ 10. HST images of the jet in M87 show well-defined knotty structures (Bridle et al., 1994),
which move at an apparently superluminal v/c ∼ 6.
AGN are likely powered by accreting SMBHs of mass M ∼ 106 − 109 M¯ , as estimated from
1) the AGN’s compactness and luminosity ; 2) kinematic models of stellar motion (rotation profile
and velocity dispersion) near nuclei ; 3) observations of Doppler shifts or maser emission from disk
(see review by Kormendy and Richstone 1995; Richstone et al. 1998) ; 4) reverberation mapping
(Blandford and McKee, 1982; Peterson, 1993; Peterson et al., 2004) ; and 5) relativistic effects for
some AGN (Fabian and Vaughan, 2003). Using Hubble Space Telescope (HST) photometry and
ground-based observations of kinematics, Magorrian et al. (1998) apply dynamical models to 36
nearby galaxies. They find that 97% of early-type galaxies have SMBHs, and this suggests SMBHs
could be common in (active) galactic nuclei. Similar results are found by van der Marel (1999);
Tremaine et al. (2002).
The galactic material near the nucleus provides a source of matter to form an accretion disk.
Observations of NGC 4261 show a radio source (3C 720) and jet that likely originate from nearby
a black hole, which is estimated to have M ∼ 5 × 108 M¯ (Ferrarese et al., 1996). Around the radio
source is a dusty torus with a radius 240 pc (∼ 106 GM/c2 ) and mass of Mdisk ∼ 5 × 105 M¯ . The
mass is determined by assuming NGC 4261 has the same ratio of surface density of hydrogen to
color excess E(B-V) as our Galaxy, as provided by Bohlin et al. (1978). The torus mass distribution
is perpendicular to the orientation of the jet (Jaffe et al., 1993), which suggests the disk is feeding
the black hole and producing the jet. The centers of the galaxy, nucleus, and disk do not coincide.
This offset between the disk and galaxy suggests that the torus may have formed from an interacting
dwarf galaxy, but it must have been captured > 107 years ago (Ferrarese et al., 1996). The offset
between the nucleus and galaxy could be accounted for by recoil from a time-dependent jet, as
suggested by Shklovski (1982); Rudnick and Edgar (1984). Despite recent advances in observational
technology, only instruments currently in the concept phase will have sufficient angular resolution
to spatially resolve the inner (R . 100GM/c2 ) accretion disk in AGN (Rees, 2001). Therefore,
there remain fundamental questions that can only be answered by folding observations through
models of AGN disk structure.
A quasi-stellar object (QSO, or quasar) is a distant unresolved luminous object, which is likely
the nucleus of an active galaxy with an unresolved distribution of stars or protostellar gas. The
redshift of quasars ranges from z = 0.06 − 6.4 (dL = 265 Mpc − 64000 Mpc), where most quasars are
at z ∼ 1.6 (dL = 12000 Mpc) (see Spinard review article in Mason 2004 and see the Sloan Digital
Sky Survey (SDSS) derived quasar-redshift histogram in Oguri et al. 2004). The luminosity of
quasars and high-luminosity AGN are comparable. Estimates for the luminosity and compactness
of quasars suggests they are likely powered by accretion onto a SMBH with M ∼ 106 − 109 M¯
(Zeldovich, 1964; Salpeter, 1964). High redshift quasars are likely newly forming galaxies, where
all that can be seen is the bright nucleus. Quasars likely evolve into AGN (Lynden-Bell, 1969).
For the purposes of this thesis, it is sufficient for AGN and quasars to be referred to collectively as
AGN.
13
This thesis later focuses on estimating disk properties within the nucleus of three galaxies that
contain a SMBH: 1) the normal galaxy SgrA* ; 2) the active galaxy NCG4258 ; and 3) the active
galaxy M87.
SgrA*
The Milky Way is a spiral galaxy about 31 kpc across and has a total luminosity of about 1042 erg/s
(from about 400 billion stars) (Mihalas and Binney, 1981). The Galactic nucleus contains a compact
variable radio source called Sgr A* that has a bolometric luminosity of 4 × 1037 erg/s (for reviews
see Genzel and Townes 1987; Blitz et al. 1993; Morris and Serabyn 1996; Melia and Falcke 2001)
and is about 8 kpc away from the Sun (Reid, 1993). SgrA* may be an X-ray source (Baganoff et al.,
2003) and a source for high-energy (∼TeV) gamma-rays (see, e.g., Aharonian and Neronov 2004).
As similarly observed in the nucleus of NGC 4261, within 3 pc of SgrA* there is a circumnuclear
disk, with T ∼ 7000 K, ρ ∼ 107 cm−3 , and M ∼ 3 × 105 M¯ (Shukla et al., 2004), where T is the
temperature and ρ is the rest-mass density. This disk is the likely source of matter for the inner
accretion disk.
The strongest case for a SMBH is in our own galaxy. Infra-red imaging of stellar orbits (e.g.
S2 with pericenter of 6 × 10−4 pc, 124AU, 2100GM/c2 ) in the central region of our galaxy suggest
a dark object lies near the center with M ∼ 2.6 × 106 M¯ (Eckart and Genzel, 1997; Ghez et al.,
1998; Schödel et al., 2002). A study of the accelerations and velocities of stars in the central region
shows the central mass must be extremely compact (Ghez et al., 2000; Eckart et al., 2002). The
only known object that could remain stable for a sufficient amount of time is a SMBH. Alternatives,
such as a compact cluster of neutron stars, will collapse on short times scales compared to the age
of a nucleus (Genzel et al., 1997; Maoz, 1998).
NGC4258
The spiral galaxy NGC4258 (M106) is about 7 Mpc (Greenhill et al., 1995b; Herrnstein et al.,
1999) away in the constellation Canes Venatici, has a bulge (galaxy without spiral disk) bolometric
luminosity of 5 × 1042 erg/s, has a nucleus with a 2 − 10 keV X-ray luminosity of 4 × 1040 erg/s
(Makishima et al., 1994), and has an estimated nuclear bolometric luminosity of 3 × 1043 erg/s
(see, e.g., Wilkes et al. 1995; Herrnstein et al. 1998; Yuan et al. 2002; Woo and Urry 2002). VLBI
observations of the nucleus show water maser emission from a rotating disk between 0.13 pc−0.26 pc
(Greenhill et al., 1995a). The acceleration of the maser spots or the proper motion of the spots
allows a geometric distance measurement (Greenhill et al., 1995a; Herrnstein et al., 1999). The
kinematics of the rotation suggest the nucleus likely harbors a 3.9×107 M¯ mass black hole (Watson
and Wallin, 1994; Miyoshi et al., 1995).
14
M87
The elliptical galaxy M87 (NGC4486,3C 274) is about 30 kpc across, 18 Mpc away in the Virgo
cluster (Whitmore et al., 1995), has a total luminosity of 2×1044 erg/s, and has a nuclear bolometric
luminosity 2.3×1042 erg/s (Ho, 1999). Within 18 pc of the central region of the nucleus is an ionized
(T ∼ 104 K) accretion disk whose velocity profile suggests it harbors a 3 × 109 M¯ mass black hole
(Sargent et al., 1978; Young et al., 1978; Ford et al., 1994; Harms et al., 1994; Macchetto et al.,
1997). M87 also shows a well-collimated jet at large distances (Biretta et al., 1991, 1999) and a
broad jet near an accretion disk (Junor et al., 1999) with “jet formation” at about .015 pc (3100AU,
100GM/c2 ) from the black hole. HST images of the jet in M87 show a well-defined knotty structure
(Bridle et al., 1994), which moves at an apparent speed of v/c ∼ 6.
1.3
Basic Accretion Disk Theory
The previous section introduced GRBs, X-ray binaries, and AGN, and introduced 6 fiducial objects
that are a reference point for some following discussions. As was discussed, many of these systems
likely harbor a black hole. A black hole by itself possesses no astrophysically significant means to
generate energy. However, a black hole coupled with an accretion disk is possibly the most efficient
producer of large amounts of energy in the universe. Some GRBs, some X-ray binaries, and all
AGN are likely powered by gas accretion onto a black hole.
Through dissipative processes, gas tends to bind gravitationally to a compact object. As for a
spiral galaxy, the rings of Saturn, protoplanetary/protostar disks, and similar objects, this matter
eventually forms a disk if the energy of the matter is lost faster than its angular momentum. In the
case of a highly ionized plasma, the angular momentum is then transported outward by magnetic
instabilities and matter is accreted onto the compact object (for a review see Balbus and Hawley
1998). For typical astrophysical systems, the disk is centrifugally supported and rotates many times
before finally plunging into (or onto) the compact object. During this process a significant amount
of energy can be released per baryon rest-mass. The basic theory for how this works is described
in this section.
Below is a description of some basic theoretical concepts used to estimate the mass of the
compact object and the luminosity of the accretion disk. This basic theory is then applied to argue
that most AGN likely harbor SMBHs, GRBs likely originate from stellar objects rather than an
AGN or quasars, and many X-ray binaries likely each harbor a black hole.
1.3.1
Accretion Luminosity and Mass of the Compact Object
A black hole is the most compact object known to exist, and this allows black hole accretion
disk systems to release large amounts of gravitational binding energy per baryon accreted. The
compactness of a star can be estimated as C = GM/Rc2 , where R is the surface radius, M is
the mass of the black hole, and G is Newton’s constant. The nonrelativistic gravitational binding
15
energy is proportional to Epot ∼ GM m/R, where m is the baryon mass. The nonrelativistic
efficiency (energy released per baryon rest-mass energy) can be estimated as η ∼ Epot /(mc2 ) = C,
which is largest for a black hole at η ∼ 0.5−1 for R at the horizon. A general relativistic calculation
defines this efficiency more precisely.
Nominal Accretion Luminosity
A reasonable estimate for an upper limit to the luminosity of an accretion disk can be found for a
disk that is always in Keplerian motion. The matter is assumed to accrete in a tightly wound spiral
around the compact object. The accreting matter slowly drifts inward due to angular momentum
exchange between successive rings of matter. The gravitational binding energy is assumed to be
released as radiation. The final inner radial ring is at the innermost stable circular orbit (ISCO)
for a black hole and the surface for other stars (Bardeen, 1970). For the case of a nonrotating black
hole surrounded by a thin disk terminating at the ISCO, the energy per baryon released is η ∼ 6%,
while for a maximally rotating black hole η ∼ 42%. Some disk models suggest η . 10−4 near the
black hole (see, e.g., Narayan and Yi 1995), while others suggest η & 1 (see, e.g., Gammie 1999).
These efficiencies set an upper limit Lacc , referred to as the nominal accretion luminosity, to the
actual luminosity L, with
L . Lacc ≡ η Ṁ0 c2 ,
(1.2)
where Ṁ0 is the accretion rate of rest-mass. For a star with surface radius R, all the remaining
kinetic energy is released as radiation at the surface, where L = 1/2Ṁ0 vf2f = GṀ0 M/R is released
by a rest-mass with free fall velocity vf f and accretion rate Ṁ0 at the surface. The surface efficiency
is thus η = L/Ṁ0 c2 = GM/Rc2 = C, which is η ∼ 0.01% for a white dwarf and η ∼ 10% for a
neutron star. These efficiencies can be compared to η ∼ 0.7% for hydrogen burning in stars.
Eddington Luminosity and Minimum Mass
Another upper bound to the luminosity of an accretion disk can be estimated by considering
spherical accretion onto a compact object (Bondi, 1952). Assume that the gas flow is composed of
mostly ionized hydrogen, spherical, in steady-state, nonrelativistic, and optically thin (τγ ¿ 1) to
photons. If the gas has a thickness H, then the optical depth for a uniform mass density is
τγ ∼ κρ0 H,
(1.3)
where κ is the opacity for the appropriate density and temperature regime (Bell and Lin, 1994).
For a typical temperature and density of the accretion flow, electron scattering dominates and
then κ ∼ σT Ye /mp , where Ye is the number of electrons per baryon and σT =
6.65 ×
10−25 cm2
8π 2
2 2
3 (e /me c )
∼
is the Thomson scattering cross section. For such an accreting gas, the radial
force balance between radiative scattering of electrons and the gravitational force on protons gives
an estimate for the maximum luminosity of the system and the minimum mass of the gravitating
16
object.
First, the force of radiation on the gas is found. The radiative energy flux is F = L/(4πr2 ),
where L is the luminosity of a radially-directed photons from a compact region. From a photon
momentum of p = E/c, the radiative momentum flux is then Prad = L/4πr2 c. This is the pressure
exerted by photons on a completely absorbing surface. The radiation force on a gas depends on
the opacity, where the outward force on a single free electron is Frad = σT Prad = LσT /4πr2 c for a
gas of completely ionized hydrogen. Second, the inward force of nonrelativistic gravity on the gas
is Fgrav = GM (mp + me )/r2 ≈ GM mp /r2 . The protons and electrons are assumed to maintain the
neutrality of the plasma by electrostatic coupling.
In this spherical, optically thin, nonrelativistic approximation, the force balance between radiation and gravity results in a maximum luminosity, called the Eddington luminosity LE , where
L . LE ≡
4πGM c
= 1.3 × 1038 (M/ M¯ ) erg/s = 3.2 × 104 (M/ M¯ ) L¯ ,
κ
(1.4)
where L¯ = 3.89 × 1033 erg/s is the luminosity of the Sun. A system with mass M with L > LE
would stop accreting due to outward radiative forces. Setting equation 1.2 equal to equation 1.4
gives the Eddington rest-mass accretion rate of
Ṁ0,E ≡ LE /ηc2 .
(1.5)
The Eddington argument can be inverted to give the minimum mass (Eddington mass)
M & ME = 3 × 10−5 (L/ L¯ ) M¯ ∼
= (L/L39 )(10 M¯ )
(1.6)
for the gravitating body in a system that radiates at the Eddington luminosity, where L39 ≡
1039 erg/s. So an object with L = L39 must have a mass M > 10 M¯ to sustain such a luminosity.
The estimate for maximum luminosity LE and minimum mass ME for a system undergoing
steady spherical uniform accretion of an optically thin medium works well to explain the upper
limit to several non-spherical accretion systems (see, e.g., Margon and Ostriker 1973). However,
there are magnetized super-Eddington atmospheres that are unstable, but that are quasi-steady
in a time-averaged (statistical) sense (Begelman, 2001). Also, the Eddington luminosity does not
generally apply to all optically thick mediums. As shown later, the disk in GRB systems is very
optically thick to photons, so the Eddington luminosity is not expected to be an upper limit. The
luminosity is so high in GRB systems that if the disk were optically thin, then outward radiative
forces would essentially instantaneously halt accretion. Table 1.2 includes the data from Table 1.1
with the addition of the ratio of luminosity to Eddington luminosity for the 6 fiducial objects.
Time Variability: Maximum Size and Reverberation Mapping
The maximum size of the emitting region can be estimated from the fluctuations in the frequency integrated luminosity with time scale δT . Assume that the variations in the luminosity of background
17
Table 1.2.
Fiducial Black Hole Accretion Systems 2
Parameter
NS-BH
GRB 030329
LMC X-3
dL
M [M¯ ]
Lbol [erg/s]
Lbol /LE
?
3
1053
3 × 1014
804 Mpc
3
3 × 1052
8 × 1013
55 kpc
10
1038
0.08
SgrA*
8 kpc
2.6 × 106
1037
3 × 10−8
NGC4258
7 Mpc
4 × 107
3 × 1043
0.006
M87
18 Mpc
3 × 109
2.3 × 1042
6 × 10−6
objects can be accounted for, and that the variations are from the system of interest. Suppose that
the entire emitting region is contained within a single point of observation, then fluctuations in the
luminosity from causally connected regions must be within R . cδT . There could be historically
inevitable or coincidental fluctuations over larger scales, but R is the largest causally connected
emitting size due to the limited speed of light.
More information can be gathered from the time-variability of spectral features than the variability of the frequency integrated luminosity. Close to AGN there are many intermediate optically
thick emitting and absorbing clouds (see, e.g., Krolik 1999a; Elvis 2000, and references therein).
One can time the emission response of so-called broad-line regions (BLRs) in comparison to the
continuum emission. It was realized that the time delay between the emission-line variations and
continuum are due to light travel-time effects within the BLR. The emission lines echo or “reverberate” the changes in the continuum from the accretion disk. This delay can be used to constrain
the proper motion and size of the BLRs, and then the BLR kinematics can be used to estimate the
mass of the black hole (Blandford and McKee, 1982; Peterson, 1993; Peterson et al., 2004).
Fe Kα line Profile
An accretion disk is sometimes modeled as having a hot corona (see, e.g., Ostriker 1976), which is
suspected to be generated similarly as the Sun’s. In such corona models, the disk is often treated as
a cold slab of material. The corona of an accretion disk can generate X-rays, which either Compton
scatter free electrons in the accretion disk or get photoelectrically absorbed by a neutral atom.
Photons above the threshold energy for transition cause the ejection of an electron from a neutral
atom in the disk. The largest cross section for cosmic abundances is associated with the excitation
of a K-shell (n=1) electron into the L-shell (n=2). The de-excitation leads either to fluorescence of
a K-α photon or to the electron exciting another electron (autoionization) that carries the energy
away. The resulting spectrum is a combination of the fluorescent lines, that result from cosmic
abundances, and the incident flux that is typically modeled as an X-ray power law (George and
Fabian, 1991). The fluorescence also includes the Compton reflection bump at higher energies
(E ∼ 30 keV) due to reprocessing (Compton scattering or reflection) of the X-ray power law in the
optically thick slab. The observed intensity of this feature in AGN is consistent with the presence
18
of an accretion disk that subtends a solid angle of 2π (see, e.g., Nandra et al. 1991). The Kα line
from iron has the largest fluorescent emission and has a rest-frame energy of E = 6.40 keV. AGN
observations show that this feature has a much wider profile than expected from simply absorption
along the line of sight (Makishima, 1986). The true X-ray reflection processes in an accretion disk
are typically more complicated than that described above (Ballantyne et al., 2001).
Often reported is the equivalent width (EE.W. ) of the emission line. This is equal to the energy
at which the cumulative flux at E < EE.W. in the continuum emission has the same energy as that
contained within the line emission. When viewing the incident plus reflected spectrum, the Fe Kα
line has an equivalent width of E ∼ 180 eV. Other lines have much smaller equivalent widths (Matt
et al., 1997).
In summary, in the corona model where the disk is a cold slab, the Fe Kα line emission is
produced by fluorescence when hard X-rays from a hot corona illuminate the cold, optically thick
component of an accretion disk (George and Fabian, 1991). However, this model has come under fire
since observations show the X-ray continuum and Fe Kα line profile are not temporally correlated
in MCG-6-30-15 (see, e.g., Matsumoto et al. 2003). This suggests the basic model may be wrong
or that a detailed model is required.
An important question is whether the properties of the disk and black hole can be measured
without directly resolving the accretion disk. Fe Kα line profiles are important since they may
allow a reconstruction of disk properties such as disk radii, the spin of a black hole, inclination, and
emissivity (Chen and Halpern, 1989; Fabian et al., 1989; Laor, 1991; Pariev and Bromley, 1998).
Observations of the Fe Kα line profile from black hole accretion system are discussed below.
1.3.2
Some Accretion-Based Arguments
The above basic theory shows that an accretion disk could be an efficient producer of large amounts
of radiation. It is interesting to give rough arguments that invoke an accretion disk to explain the
luminosity and some spectral features in astrophysical systems. The following discussion argues
that most quasars likely have supermassive black holes, that some X-ray binaries likely each contain
a black hole, that GRBs are likely related to stars rather than AGN or quasars, that the extraction
of rotational energy from a rotating black hole could lead to a large fraction of the luminosity from
GRBs, X-ray binaries, and AGN (or quasars), and that Fe Kα line emission from AGN and X-ray
binaries probes the inner radial region of an accretion disk around a (likely rotating) black hole.
Soltan Argument for Rotating Super-Massive Black Holes in Quasars
Black hole accretion has long been considered the most likely power source of quasars (Zeldovich,
1964; Salpeter, 1964). The ratio of quasar radiative energy density to supermassive black hole mass
density is ∼ 0.2 (Yu and Tremaine, 2002; Elvis et al., 2002). If the accretion proceeds as a thin
disk (Bardeen, 1970) with radiative efficiency η(a) around a black hole with spin parameter a, then
η & 0.2. This suggests that the average black hole spin is a ∼ 0.96. Even an efficiency of η & 0.1
19
gives a & 0.67. This result also applies to nonradiative magnetized thick disk models, where the
thick disk efficiency is found to be comparable to the thin disk efficiency (McKinney and Gammie,
2004).
Assume that accretion proceeds with a radiative efficiency of ∼ 10%, where the black hole is the
spent fuel from the accretion process. One can estimate the remnant black hole mass (Soltan, 1982;
Rees, 1984; Cavaliere and Padovani, 1988) from the efficiency η ∼ 0.1, average quasar luminosity
L ∼ 1045 erg/s, timescale of quasar activity (age of universe: T ≈ 4.3 × 1017 sec), typical distance
to a quasar (redshift z ∼ 1.6 → dC ≈ 1.4 × 1028 cm), observed number of quasars per square degree
(nQ ∼ 100 deg−2 → NQ ≈ 4×106 ), and average number density of L? galaxies (n? ∼ 2.4×106 Gpc−3 ,
where Gpc = 109 pc) that are capable of harboring a SMBH (Krolik, 1999a). The average accreted
mass per L? galaxy is then
µ
M∼
NQ
n?
¶Ã
LT
¡4 3 ¢
2
ηc 3 πdC
!
∼ 107 M¯ .
(1.7)
The left fraction is the number of quasars per galaxy for a given volume of space. The right fraction
is the mass accreted per quasar per unit volume of space. The product is the average mass accreted
into the nucleus per galaxy. This suggests that the typical remnant object from accretion should
be a SMBH, and the mass is in basic agreement with observations. For a more recent version of
this argument, see Merloni (2004).
Black Hole Mass of GRBs and AGN from Variability
As mentioned above, the shortest GRB currently ever observed lasted for 6ms (Bhat et al., 1992)
with 200µ s temporal structure. This gives a maximum size of 60 km and a maximum black hole
mass of 40 M¯ for the emitting source, so likely this GRB is (and likely others are) related to a
compact star rather than to AGN or quasars.
Intra-day variability is typically observed in all quasars and AGNs, while in MGC-6-30-15 the Xray luminosity shows rapid variability on order of 100 s (Reynolds et al., 1995; Yaqoob et al., 1997).
MCG-6-30-15 in an active galaxy in the constellation of Centaurus with a redshift of z = 0.0078
(dL = 37 Mpc). For MCG-6-30-15, R . cδT ∼ 170 astronomical units (AU). The compactness
gives M . 2 × 107 M¯ for a black hole. The luminosity of 4 × 1043 erg/s (Reynolds et al., 1997)
is limited by the Eddington luminosity, and so M & 3.2 × 105 for a black hole. More advanced
time-series based arguments can be made that estimate the black hole mass to be M ∼ 106 M¯
(Reynolds, 2000; Lee et al., 2000; Vaughan et al., 2003).
Within AGN there must exist an object capable of generating power that is comparable to or
even larger than that of an entire galaxy of stars, yet it must be compact enough to fit within
∼ 100AU. The current standard model is that the central engine of AGN is an accreting SMBH
(for an introductory review see, e.g., Armitage 2004).
20
Dimensional Argument for Blandford-Znajek Effect
The mechanism of jet production is an area of active research. One mechanism that may generate a
jet is the extraction of rotational energy from a rotating compact star by the surrounding magnetosphere or magnetized disk. Compact stars have the largest magnetic fields due to the approximate
advection of magnetic flux during collapse. Compact stars, such as black holes and neutron stars,
can hold significant rotational energy that can be released through magnetic braking between the
star and the magnetized part of the accretion flow (such as a magnetosphere) (Goldreich and Julian 1969; Blandford and Znajek 1977; Kim et al. 2004 and references therein). The process of
magnetically extracting rotational energy from a rotating star is called the Blandford-Znajek (BZ)
effect.
For a black hole, the power generated by the BZ-effect can be estimated dimensionally from
3 ), and the
the local energy density of the magnetic field (∼ B 2 ), the volume of the black hole (∼ r+
light crossing time (∼ r+ /c), giving
µ
LBZ ∼
2
B 2 r+
c
∼ 10
45
B
105 G
¶2 µ
M
107 M¯
¶2
µ
erg/s ∼ LE
B
105 G
¶2 µ
M
107 M¯
¶
(1.8)
´
√
¡
¢³
where B is the magnetic field strength near the black hole, r+ ≡ GM/c2 1 + 1 − a2 is the
radius of the event horizon, a ≡ J/M 2 , and J is the angular momentum of the black hole. This
dimensional estimate does not behave properly for arbitrary a, but it is accurate within an order
of magnitude for a & 0.5 − 1. A more accurate estimate is obtained by invoking the black hole spin
angular frequency ΩH = ac/2r+ (see MTW §33.4), which gives LBZ ∼ B 2 (GM/c2 )4 Ω2H /c that has
the correct a dependence (R. Krasnopolsky, private communication). The density scale can be set
by assuming the accretion rate is at the Eddington rate Ṁ0,E (e.g., with efficiency, η = 10%) or by
estimating the rest-mass accretion rate for a particular object from observations of the luminosity
and independent estimates for the mass of the compact object. The magnetic field near the black
hole can then be estimated from MHD simulations or arguments for equipartition between the gas
and magnetic pressure near the black hole (i.e. β = pg /(b2 /2) ∼ 1), where the pressure is estimated
from some model of accretion. For a typical AGN or quasars, MHD simulations give B ∼ 105 G and
LBZ ∼ 1045 erg/s, which is comparable to the luminosity of jets and radio lobes in the strongest of
AGN. Renormalizing the BZ luminosity for GRBs and X-ray binaries shows that the jets in these
objects may also be powered by the BZ-effect.
More advanced analytic estimates suggest that the BZ process is too inefficient to account for a
large fraction of power output except in a handful of AGN with low accretion rates and relatively low
efficiency (see, e.g. Ghosh and Abramowicz 1997; Armitage and Natarajan 1999; Livio et al. 1999).
However, recently the BZ effect has been determined self-consistently using numerical models of
force-free models (Komissarov, 2001) and, as performed in this thesis, GRMHD models of disks
(McKinney and Gammie, 2004). It is found that the BZ luminosity is indeed lower than nominal
accretion luminosity, but that the luminosity due to the BZ effect is focused in a collimated jet at
21
the axis of the black hole accretion disk system. Thus, the BZ effect remains a plausible source of
jet energy.
A similar total luminosity is estimated for electromagnetic disk winds (Blandford and Payne,
1982; Krasnopolsky et al., 1999, 2003) and for super-Eddington radiation-driven winds (Lucy and
Solomon, 1970; Castor et al., 1975; Abbott, 1982; Vitello and Shlosman, 1988; Warner, 1995; Murray
et al., 1995; Proga et al., 1998). Likely all these effects are important to some degree for some
systems.
X-ray Binary QPOs as Probe of Accretion Disk and Space-Time
Quasi-periodic oscillations (QPOs) with kHz frequencies are observed in the time series of the X-ray
emission from some neutron star accretion systems and some black hole accretion systems, such as
GRS1915+105. These kHz QPOs provide a plausible means to probe the inner part of an accretion
disk and map the space-time in the vicinity of a black hole. For those black hole systems that have
kHz QPOs, the kHz QPOs exhibit a scaling of frequency with black hole mass (f ∝ 1/M ) that is
consistent with the kHz QPO being produced in an accretion disk near the black hole (see, e.g.,
Abramowicz et al. 2004, and references therein). For a discussion see di Matteo and Psaltis (1999);
Strohmayer (2001); Abramowicz et al. (2004); Kato (2004) and references therein.
Event Horizon Argument for Compact Objects in X-ray Binaries
Of the ∼ 240 known X-ray binaries, ∼ 100 show time-dependent phenomena such as pulsations and
strong X-ray bursts that do not correspond to accretion outbursts. X-ray bursters have frequent
X-ray bursts with quiescent (lower, persistent) X-ray flux between. The quiescent luminosity is
order 1032 − 1034 erg/s (the companion star has a bolometric luminosity 1032 − 1039 erg/s). X-ray
bursts are classified as Type I and II, where only Type I bursts are discussed since only they are
relevant to this discussion. Type I X-ray bursts are observed as a burst of thermal emission with a
luminosity of 1036 − 1039 erg/s that last for seconds and recurs in hours to days. The burst energy
is found to be proportional to the duration of the preceding inactivity period. There are about
80 known Type-I X-ray bursts. There are about 8 known so-called superbursts with luminosity
1043 erg/s that lasts for hours and recurs in years (Zand et al., 2004). Type-I bursts are thought
to occur due to unstable nuclear burning of hydrogen or helium, and in the case of superbursts, of
carbon (see, e.g., Woosley et al. 2004, and references therein).
Type I bursts are believed to be due to unstable nuclear burning on the surface of neutron
stars, while the persistent background flux from such objects comes from conversion of gravitational energy into radiation. For standard Type-I X-ray bursts, the unstable burning comes from
4H →
4 He,
3 4 He →
12 C,
and 5 4 He + 84 H →
104 Pd,
which releases 6.7 MeV, 0.6 MeV, and
6.9 MeV per baryon, respectively. The latter two processes are the “triple alpha” and the rapid
proton (rp) process. The persistent release of gravitational energy is E ∼ GM m/R ∼ 200 MeV per
baryon. Thus, the ratio of gravitational to thermonuclear energy is about 30 − 40.
22
Evidence for the thermonuclear origin for Type I bursts includes that the ratio of persistent
energy to burst energy is 30 to 40 and one observes Type I behavior: the longer the preceding fuel
accumulation, the more intense the burst. The burning site is argued to be a neutron star since
4 gives a typical
only one normal star is seen in the optical, the Stefan-Boltzmann law L = σ4πR2 Teff
neutron star radius, and the maximum luminosity is consistent with the Eddington luminosity for
a neutron star.
Type I X-ray bursts may provide a useful tool to study X-ray binaries with BHCs. No BHC
has ever been observed to have a Type I X-ray burst, which is consistent with the fact that such
bursts can only occur on a surface. Assuming this is correct, observations of Type I bursts on BHCs
would reject the event horizon hypothesis and the existence of classical black holes. Thus refining
models of Type I X-ray bursts is important in order to predict their occurrence (Narayan and
Heyl, 2002, 2003; Woosley et al., 2004). One must be able to predict not only why some neutron
stars have Type I bursts, but also why some neutron stars do not have Type I bursts. Preliminary
studies suggest that there is evidence for event horizons (Narayan, 2003a,b; Cornelisse et al., 2003;
Tournear et al., 2003; McClintock et al., 2004), but see Abramowicz et al. (2002).
Black Hole Mass/Rotation from Fe Kα line Profiles
Observations of line emission from AGN and X-ray binaries show a broadened profile that may be
due to relativistic effects in the accretion disk near a rotating black hole. One can generate line
profiles as observed on Earth by mapping photon trajectories in Kerr space-time either analytically
(Karas et al., 1992) or numerically (Bromley et al., 1997; Cunningham, 1975). Evidence of an
accretion disk in AGN includes the ASCA observations of the Fe Kα line profile from MCG-630-15. The Fe Kα line profile in MCG-6-30-15 is shifted to the red, likely demonstrating general
relativistic redshifting within a few GM/c2 of the black hole. The red and blue wings of the Fe
Kα line profile have different intensities, which are consistent with the emission coming from parts
of the accretion disk that move towards and away from the point of observation. The red and
blue wing intensities suggest the accretion disk is moving relativistically with speeds approaching
0.2c (Pariev and Bromley, 1998; Tanaka et al., 1995; Fabian et al., 2002; Vaughan and Fabian,
2004). The emission profile is consistent with line emission from within a few Schwarzschild radii
(2GM/c2 ) of the black hole (Tanaka et al., 1995; Fabian et al., 2002; Vaughan and Fabian, 2004).
X-ray binary Fe Kα line profiles have also been observed and studied (Miller et al., 2004), and the
profiles are consistent with the Fe Kα line being emitted within a few Schwarzschild radii of the
black hole. Thus, the Fe Kα line apparently allows a probe of the innermost part of the accretion
disk.
With Chandra’s and XMM-Newton’s high resolution X-ray imaging, it may be possible to use
Fe Kα line profiles to test specific aspects of general relativity such as frame dragging (Bromley
et al., 1997). Models of the Fe Kα line from MCG-6-30-15 suggest the emission is ≈ 3 times more
likely to originate from a disk around a rotating black hole than a non-rotating black hole (Bromley
et al., 1997; Dabrowski et al., 1997). Also, observations of the Fe Kα line are consistent with the
23
Table 1.3.
Fiducial Black Hole Accretion Systems 3
Parameter
NS-BH
GRB 030329
dL
M [M¯ ]
Lbol [erg/s]
Lbol /LE
Ṁ0
?
3
1053
3 × 1014
5 M¯ / s
804 Mpc
3
3 × 1052
8 × 1013
0.1 M¯ / s
LMC X-3
SgrA*
NGC4258
M87
55 kpc
10
1038
0.08
10−8 M¯ / yr
8 kpc
2.6 × 106
1037
3 × 10−8
10−5 M¯ / yr
7 Mpc
4 × 107
3 × 1043
0.006
10−2 M¯ / yr
18 Mpc
3 × 109
2.3 × 1042
6 × 10−6
10−2 M¯ / yr
presence of energy extraction from a rotating black hole by the BZ-effect (Wilms et al., 2001b;
Miller et al., 2002; Maraschi and Tavecchio, 2003).
Despite these suggestions about black hole rotation and energy extraction, the typical Fe Kα
line profile calculation, such as performed by XSPEC (Speith et al., 1995), assumes the accretion
disk is unmagnetized, viscous, and thin. As is discussed in the next section, such thin disk models
typically assume that the accretion disk terminates at the ISCO, which marks a sharp transition
where matter plunges into the black hole. This sharp transition would lead to specific spectral
features in the observations. As discussed below, magnetic fields can torque the disk within the
plunging region and keep the matter from simply falling into the black hole at the ISCO. Indeed,
numerical GRMHD studies of accretion disks (see, e.g., McKinney and Gammie 2004), suggest that
no sharp transition occurs at the ISCO. Inside the ISCO, the disk should produce weak Fe Kα line
emission that the typical thin disk model might be incorrectly interpreting as evidence for black
hole rotation and energy extraction (Reynolds and Begelman, 1997).
1.4
Models of Accretion Disks and GRBs
The previous section discussed the most basic principles of accretion disk theory and gave some
accretion-based estimates for the properties of these systems. This section goes into more detail
by discussing mechanisms for angular momentum transport, some radiative models of accretion,
and models for GRBs. First, a discussion of angular momentum transport summarizes why a selfconsistent magnetic model is required to study accretion disks. Next, a summary of the thin α-disk
model (Shakura and Sunyaev, 1973) derivation is given as an example of a radiative viscous HD
model. The purpose of presenting the derivation summary is to show how such disk solutions are
found and to show the basic structure of the equations. Next, there is a summary of other radiative
disk models. Finally, the section ends with a discussion of models of GRBs. The purpose of the
discussion of GRB models is to motivate a GRMHD model for the study of GRBs and core-collapse
supernovae in general.
The GRB and radiative disk models discussed in this section have been used by others to
estimate the rest-mass accretion rate of the 6 fiducial systems. Table 1.3 includes the data from
24
Table 1.2 and adds an estimate for the rest-mass accretion rate. The rest-mass accretion rates for
the NS-BH GRB and Collapsar GRB (GRB 030329) disks come from simulations and estimates in
MacFadyen and Woosley (1999); Popham et al. (1999). Estimates for the rest-mass accretion rate
for LMC X-3 are discussed in Ebisawa et al. (1993); Shimura and Takahara (1995); Wilms et al.
(2001a); Brocksopp et al. (2001). The rest-mass accretion rate for SgrA* comes from Quataert
et al. (1999). The rest-mass accretion rate for NGC4258 comes from Gammie et al. (1999). The
rest-mass accretion rate for M87 is from Reynolds et al. (1996); Ho (1999).
1.4.1
Angular Momentum Transport Models
The purpose of this discussion is to summarize why an MHD, rather than HD or viscous HD (VHD),
model is required to study ionized accretion disks. First, there is a short discussion of turbulence,
which has long believed to be the driver of angular momentum transport in accretion disks. Next,
the α-disk model is summarized. Finally, the MHD model is discussed. The discussion of the
MHD model includes a summary of the magnetorotational instability (MRI) that likely dominates
hydrodynamic instabilities in the generation of turbulence and angular momentum transport in
accretion disks (for a review see Balbus and Hawley 1998). Now, a general discussion of turbulence
is presented.
Turbulent flow can be qualitatively defined as a complex pattern of flow in which microscopic
perturbations are enhanced to macroscopic scales as an expression of the flow’s internal degrees of
freedom (Kadomtsev, 1965; Tennekes and Lumley, 1972; Eckman, 1981; She and Leveque, 1994;
Frisch, 1995). The small-scale pattern of turbulent flow is typically sensitive to the initial conditions
(the flow is chaotic), but the average pattern of turbulent flow is insensitive to the initial conditions
(the flow is ergodic). Turbulence is 1) stationary if the average values do not vary with time; 2)
uniform if the average values do not depend on position; and 3) isotropic if the average values at a
specific point in space do depend on the direction. For a finite system undergoing fully developed
turbulence, there is a cascade of energy from large to small scales, at which point the energy is
dissipated by kinematic viscosity or resistivity into heat (Tatarski, 1961; Kolmogorov, 1941b,a).
Kolmogorov found that an incompressible fluid demonstrating stationary, uniform, isotropic turbulence has an “inertial range” of cascading
energy, with a wave energy Wk = k −5/3 , where k is the
q
magnitude of the wave vector k = kx2 + ky2 + kz2 . The inertial range is between the largest dimensions of a physical system and the dissipation scale. A magnetized, anisotropic, compressible fluid
might not be expected to be described by this simple Kolmogorov spectrum, but paradoxically, astrophysical measurements are consistent with the Kolmogorov spectra. Two example systems that
follows the Kolmogorov spectra are the magnetized solar wind (Belcher and Davis, 1971; Goldstein
et al., 1995; Leamon et al., 1998) and the interstellar electron density spectrum on small scales
(108 cm − 1015 cm) (Armstrong et al., 1995) and large scales ( pc) (Lazarian et al., 2001).
To study turbulence in accretion disks, one must first identify the source of turbulence and then
determine the mechanism of dissipation. Since both of these effects have not been well understood
25
for an accretion disk, models of turbulence have been developed. Generally, all black hole accretion
models assume the approximation of hydrodynamics (HD), or when including a magnetic field,
of MHD. Many accretion models have been developed since the pioneering work of Bondi who
described a model for inviscid (perfect, ideal) HD spherical accretion (Bondi, 1952). A summary of
some of the early models by Shakura and Sunyaev (1973); Novikov and Thorne (1973); Pringle and
Rees (1972); Pringle (1981); Frank et al. (1992) and others can be found in chapter 14 of Shapiro
and Teukolsky (1983).
Magnetically-driven winds off the disk are also likely important for angular momentum transport
(Blandford and Payne, 1982; Krasnopolsky et al., 1999, 2003) (and see references therein). However,
this discussion only considers the process of angular momentum transport in the interior of the disk.
Anomalous Viscosity
See Section 2.3 for the viscous HD (VHD) equations of motion. The so-called α-disk model (Shakura
and Sunyaev, 1973) introduces into the (spherical polar) HD equations of motion an r-φ stress (Πrφ )
that is proportional to αP , where α ≡ Πrφ /P is a dimensionless constant and P is the pressure.
The addition of a shear stress allows for angular momentum transport by driving an exchange
of momentum between differentially rotating layers of the disk. This anomalous viscosity is an
attempt to model the effect that turbulence generates locally within the disk.
While there are global HD instabilities that could initiate the required turbulence, nonlinear
studies show that known HD instabilities saturate at low levels or do not apply to an accretion
disk near a black hole. No local HD linear or nonlinear instabilities are known to exist in Keplerian
disks (Balbus and Hawley, 1998), but this is not a settled issue (see, e.g., Bisnovatyi-Kogan 2004;
Kuznetsov et al. 2004). The α-based shear viscosity model can be considered as a zeroth-order
approximation to what is mostly likely a magnetic effect, which is described in the next section.
The thin disk model of black hole accretion based upon viscosity requires that the inner edge of
the accretion disk be at the ISCO. Fluid flows through the ISCO and plunges into the black hole
without any effect on the accretion disk or the polar region, as proposed by Page and Thorne (1974).
They also realized early on, however, that a magnetic field could significantly alter this picture of
the plunging region. The magnetic field within the plunging region may torque the accretion
disk outside the ISCO (Krolik, 1999b; Gammie, 1999; Agol and Krolik, 2000). Before discussing
magnetic torques inside the ISCO, a summary of the MHD equations of motion is presented.
MHD Equations of Motion
The governing equations for nonrelativistic ideal MHD are the same as those for viscous HD without
viscous heating, except for the addition of the Lorentz force to the momentum equation:
ρ0
Dv
= −∇P − ρ0 ∇φ + (∇ × B) × B
Dt
26
(1.9)
and the addition of the induction equation:
∂B
= ∇ × (v × B)
(1.10)
∂t
√
where B is the magnetic field vector (a factor of 4π is absorbed into the definition of B, as in
Heaviside-Lorentz units). The general dispersion relation for linear waves in a uniform background
is:
(vp2 )(vp2 − (k̂ · va )2 )(vp4 − (cm vp )2 + (cs (k̂ · va ))2 ) = 0
(1.11)
where vp = ω/k is the phase velocity, k is the wave vector, k̂ = k/k is the wave unit vector, ω is the
p
√
frequency of oscillation, va = B/ ρ0 is the Alfvén velocity, cm = c2s + va2 is the magnetosonic
p
speed, cs = ∂P/∂ρ0 |S is the sound speed, and S is the entropy. There are clearly 8 wave solutions.
The first term gives two modes: the entropy mode that corresponds to entropy perturbations
at constant pressure ; and the monopole mode that is removed by the solenoidal condition of
∇ · B = 0. The second term gives left- and right-going Alfvén waves, which are transverse and
move at the speed va cos θ, where θ is the angle between k and va . The third term gives leftand right-going fast and slow magnetosonic waves. For this term, if va · k = 0 the fast wave is
a purely compressive,
q longitudinal wave with phase speed cm . For waves of arbitrary va · k the
1 2
1 4
2 2
2 1/2 , where the +(−) corresponds to the fast(slow)
phase velocity is
2 cm ± 2 [cm − 4cs va cos θ]
magnetosonic wave.
Clearly, the limitation of the above nonrelativistic MHD model is the inability to model relativistic flows. While a pseudo-Newtonian potential (Paczyński and Wiita, 1980) reproduces the
positions of the ISCO, the marginally stable circular orbit, and approximately the binding energy
of the last stable orbit, the potential is unable to model a rotating black hole. See Chapters 3
and 4 for the general relativistic coordinates, governing equations, and the characteristics of the
general relativistic MHD (GRMHD) equations. Both nonrelativistic and relativistic MHD models
find similar results that deviate from expectations built on the α-disk model.
MHD Turbulence
Early accretion disk theory considered magnetic stresses as insignificant (Shakura, 1972). Later
accretion disk studies realized that a magnetic field may provide a type of viscosity (Novikov
and Thorne, 1973; Eardley and Lightman, 1975; Pringle, 1981). Indeed, previous experimental
(Velikhov, 1959) and theoretical (Chandrasekhar, 1961) work found instabilities in differentially
rotating magnetized fluids. Balbus and Hawley (1991) rediscovered this magneto-rotational instability (MRI) (or Balbus-Hawley instability) by performing a local stability analysis of a magnetic
fluid in a differentially rotating Newtonian disk. The MRI operates similarly near a Kerr black hole
(Gammie, 2004). The MRI generates angular momentum transport even when the field is weak.
A local stability analysis shows that the MRI grows on a dynamical time of Ω−1 and that the
wavelength of the fastest growing mode is λc ∼ va /Ω (Balbus and Hawley, 1991), where Ω is the
27
angular frequency of rotation. Including a magnetic field (and thus the MRI) solves the question
of “what is α” in α-disk models by self-consistently generating the turbulence from the magnetic
field’s interaction with the fluid. It is still useful, however, to use α as a dimensionless measure of
the stress that transports angular momentum. In MHD, α ≡ −δvr δvφ /P and αmag ≡ −δBr δBφ /P
are parameters one can measure anywhere in space (δ is the difference from the mean flow, so α
measures the turbulent part of the flow).
Development of the MRI can be understood qualitatively by considering the stability of a
magnetized fluid element in a differentially rotating disk. If a fluid element at radius r0 is threaded
by a field line that is parallel to the axis of rotation, then a radial disturbance in the fluid can cause
the field line to bend. The initial disturbance causes the fluid element at r < r0 to gain rotational
speed by conservation of angular momentum. The fluid element at r > r0 loses speed for the same
reason. The field line simply flexes back and resists the separation if the field strength is high.
However, if the field strength is relatively low, the field line pulls slightly back on the fluid elements
and reduces the angular momentum of the lower fluid element at r < r0 and raises the angular
momentum of the fluid element at r > r0 (i.e. the field transfers angular momentum). This in
turn increases their speed and thus separation from the original position resulting in an instability
(Balbus and Hawley, 2002). In general, a magnetized accretion disk is unstable to the MRI when
dΩ2 /dr < 0, where Ω is the frequency of rotation. This criterion is easily satisfied for an accretion
disk around a compact object. By comparison, the Rayleigh (hydrodynamic) criterion is dl/dr < 0
for instability, where l = r2 Ω is the specific angular momentum. While no accretion disk model
satisfied the Rayleigh criterion for instability, both theoretical and observed accretion disks have
an angular velocity that decreases outwards.
Numerical simulations of the MRI confirm the linear stability analysis (Balbus and Hawley,
1991). Subsequent numerical work showed that the MRI is capable of sustaining turbulence in the
nonlinear regime (Hawley and Balbus, 1991). The MRI was shown to generate sufficient angular
momentum transport to sustain accretion, and the MRI develops a dynamo in the nonlinear regime.
The rediscovery of the MRI renewed hope that accretion disks could be understood by a direct
calculation, rather than by using the α-disk model.
One simplified form of the HD or MHD equations of motion is called the local shearing box
approximation, which is a result of an expansion of the equations at a specific radius with a box size
δr. The inner and outer “radial” boundaries are treated as periodic with an additional shearing term
that describes the differential rotation. Simulations of accretion disks have been performed in the
local shearing box approximation (Hawley and Balbus, 1991; Hawley et al., 1995; Stone et al., 1996),
which determined the relevance of the MRI instability to, and the vertical magnetized structure
of, accretion disks. Accretion disk simulations with 2D axisymmetric or cylindrical symmetries,
and full 3D simulations, have been performed to study the global structure of the accretion disk
near the black hole (Armitage, 1998; Hawley, 2000, 2001). For an initially 2D symmetric accretion
disk, 3D geometry appears to be required only to sustain a magnetic dynamo. All these simulations
have shown that the MRI self-consistently accounts for the transport of angular in ionized accretion
28
disks.
MHD Effects Inside ISCO and Ergosphere
The previous discussion showed how the MHD model goes beyond the predictive capability of the
α-disk model by self-consistently describing the generation of turbulence and angular momentum
transport. Another new result is that the MHD model shows that the disk is magnetically torqued
inside the ISCO, as seen in both pseudo-Newtonian numerical models (Hawley, 2000; Hawley and
Krolik, 2001, 2002) and GRMHD numerical models (De Villiers et al., 2003a; De Villiers and
Hawley, 2003b; De Villiers et al., 2003b; McKinney and Gammie, 2004). Although, how applicable
these simulations are to, say, CVs, YSOs, and protoplanetary disks is not certain (Armitage et al.,
2001; Reynolds and Armitage, 2001). The α-disk model assumes that the accretion disk terminates
at the ISCO, which marks a sharp transition where matter plunges into the black hole. This
sharp transition leads to specific spectral features in the observations. As mentioned above in the
discussion of the Fe Kα line profile, the presence of a magnetic torques inside the ISCO can affect the
conclusions about evidence for black hole rotation and energy extraction. Models of the Fe Kα line
profile using an α-disk model seemed to be consistent with the presence of an accretion disk around
a rotating black hole (Bromley et al., 1997; Dabrowski et al., 1997; Wilms et al., 2001b; Miller
et al., 2002; Maraschi and Tavecchio, 2003). However, inside the ISCO the disk should produce
weak Fe Kα line emission that the α-disk model might be incorrectly interpreting as evidence for
black hole rotation and energy extraction (Reynolds and Begelman, 1997).
The magnetic field may also extract angular momentum and energy from a rotating black
hole into the accretion disk through an MHD version of the BZ effect (Blandford and Znajek, 1977;
McKinney and Gammie, 2004). Relativistic magnetized steady state accretion disks postulated that
a magnetic field might allow a higher accretion efficiency than the thin disk efficiency (including
super-efficient η > 1) due to magnetic torques in the plunging region (Gammie, 1999). Remarkably,
GRMHD simulations of both thick and thin disks show an efficiency similar to that of a thin disk
(McKinney and Gammie, 2004).
1.4.2
Radiative Disk Models
The previous section established that magnetic fields are required to self-consistently describe the
generation of turbulence and process of angular momentum transport within an accretion disk.
MHD models also showed that the disk is magnetically torqued inside the ISCO, and that energy
can be magnetically extracted from a rotating black hole. Radiative processes in an accretion disk
are also dynamically important because they cool the disk, introduce radiative instabilities, and
introduce a radiative pressure. Ultimately, of course, several different radiative processes may be
involved in determining observations, but the focus of this section is on those radiative processes
that determine disk structure.
29
Shakura and Sunyaev (1973) constructed an accretion disk solution (SS73 model) that describes
the structure of a disk that steadily accretes due to a local anomalous viscous transport of angular momentum. The anomalous viscosity is parameterized by the α model that was described
in Section 1.4.1. By approximating the thin disk by a surface density using a height-integrated
approximation, including optically thick thermal bremsstrahlung absorption and optically thick
electron scattering, they find a complete solution to the accretion flow and spectrum emitted from
the disk surface. This model is a useful starting point for estimating the properties of accretion
disks. A relativistic version of the thin disk model is given by Novikov and Thorne (1973); Eardley and Lightman (1975), but a pseudo-Newtonian approximation of the gravitational potential
Φ = GM/(r − 2GM/c2 ) (Paczyński and Wiita, 1980) is sufficient to grasp the salient aspects of
the solution. A short derivation of this SS73 model is provided in Shapiro and Teukolsky (1983),
§14.5, where below is a summary of the main points. Following the SS73 derivation summary is a
discussion of more advanced radiative disk models.
The SS73 model assumes that 1) the disk scale height is much less than the radius (H/R ¿ 1) ;
2) an anomalous fluid viscosity introduces a shear stress (Πrφ ∼ αP ) ; 3) rest-mass is conserved ; 4)
angular momentum is conserved, where there is a balance between the angular momentum accreted
and angular momentum removed by the viscous stress ; 5) vertical momentum is conserved and
vertical equilibrium is established, where H/R ≈ cs /vK ; 6) energy is conserved, where there is
a balance between kinetic energy, viscous heat generation, and cooling by photon radiation that
diffuses through the optically thick disk or cools directly in an optically thin disk. The pressure is
assumed to be due to radiation and the gas. In the innermost radial region where the temperature is
highest, radiation pressure typically dominates gas pressure. In this thesis, disks in X-ray binaries
and AGN are modeled with this photon-cooling model. For such disks, the opacity is typically
dominated by electron scattering, and this is assumed here. These assumptions allow one to derive
the complete state of the disk, where the midplane rest-mass density, midplane temperature, and
dimensionless disk height to radius ratio are found to be
ρ0 = (1.5 × 10−5 g cm−3 )η 2 (αM )−1 ṁ−2 r3/2 Ξ−2
T
= (5 × 107 K)(αM )−1/4 r−3/8
(1.12)
h/r = 0.95η −1 ṁr−1 Ξ,
where Ξ ≡ 1 − (6/r)1/2 , ṁ ≡ Ṁ0 /Ṁ0,E , Ṁ0,E is the Eddington mass accretion rate given in
Equation 1.5, M is measured in units of M¯ , and r, h are both measured in units of GM/c2
(i.e. h/r is dimensionless). Notice that the assumption that the disk is thin (h/r ¿ 1) is not valid
in the inner-radial region when the luminosity is near the Eddington limit.
The standard SS73 disk model predicts a disk midplane temperature of T ∝ M −1/4 , and thus
AGN would be expected to have much lower temperatures than X-ray binaries. The accretion disk
in AGN and X-ray binaries, as described by the SS73 model of accretion, produces a soft quasithermal spectrum dominated by UV and optical emission in AGN, and dominated by soft X-rays
30
in black hole X-ray binaries. However, there is greater uniformity in the observed X-ray spectra
in these systems, which have hard X-ray emissions in a power law out to E ∼ 0.3 MeV. Also, the
SS73 model is unstable in the radiation-dominated region of the inner accretion disk, thus violating
the steady-state assumption (Lightman and Eardley, 1974; Piran, 1978; Abramowicz, 1981).
Since the SS73 model is unstable and does not account for hard X-ray emission, other models
have been invoked to explain the X-ray emission. These models often introduce a temperature
difference between the ions and electrons in the flow, known as a two-temperature model. This is
plausible since the average energy released per accreted particle is
Ẽ ≡
Ė
∼ ηc2
Ṁ0
µ
GM/c2
R
¶
,
(1.13)
where Ė is the total energy accretion rate. Since R ∝ GM/c2 , Ẽ is independent of the mass of the
compact object. For η ∼ 10%, this available energy is up to ∼ 51 keV (∼ 6 × 109 K) for electrons
and ∼ 94 MeV (1012 K) for protons. Thus, it is plausible that the electrons and protons could have
different temperatures, with inverse Comptonization cooling the electrons and generating hard Xrays (Thorne and Price, 1975; Sunyaev and Truemper, 1979). Depending upon how thermally
coupled the electrons and protons are, there may be more or less energy available to the electrons
to power inverse Comptonization. This process has been considered as a plausible mechanism for
generating hot coronae in accretion disks (Stern et al., 1995). A discussion of Comptonization can
be found in Rybicki and Lightman (1979); Pozdniakov et al. (1983) and is summarized below.
Compton scattering involves a photon with energy Eγ and an electron with mass me and gas
temperature Te . If me c2 À Eγ À 4kb Te , then the kinetic energy of the electron is negligible and
the photon loses energy. However, if Eγ ¿ 4kb Te , then it can be shown that in the lab frame the
photon gains an energy proportional to Γ2e , where Γe is the electron’s relativistic Lorentz factor.
As long as Eγ ¿ 4kb Te , the photon gains energy until the photon reaches thermal equilibrium with
the electrons. For low enough photon energy and high enough electron energy the photon can be
highly energized, a process referred to as unsaturated inverse Comptonization. If there is little
amplification the process is referred to as saturated inverse Comptonization.
The “hot disk” SLE model (Shapiro et al., 1976) is a two temperature model. In this model,
in the inner-radial accretion disk, ions reach T ∼ 1011 − 1012 K, while electrons reach T ∼ 109 K.
Hard X-rays are produced by inverse Comptonization of soft photons from electrons in the cooler
parts of the disk. The two-temperature models suggest that far away from the compact object,
where the disk is relatively cold, the disk follows the SS73 model. The disk there is thin, generating
the UV and optical spectra observed. In the inner radial regions a transition to a two-temperature
thick disk may occur, and then the ions remain hot and heat electrons, which continuously cool to
generate the hard X-rays observed. The transition radius is inversely proportional to the rest-mass
accretion rate. At low mass accretion rates, the disk is in the “quiescent” state with much of the
inner radial region of the disk forming a thick disk generating low-hard (low luminosity, hard X-ray)
emission. At the highest rest-mass accretion rates, the thin disk reaches close to the compact object
31
and disk generates high-soft (high luminosity, soft X-ray or UV) emission.
While the SLE model is unstable (Pringle, 1976; Piran, 1978), the two basic considerations may
still be viable. First, accretion flow may be two-temperature. Second, Comptonization of photons
may provide the hard X-rays even for a single temperature flow. For example, magnetic dissipation
has long been considered a mechanism for heating a hot coronal region (Galeev et al., 1979), and
this is seen in most global simulations of accretion disks. This is consistent with the so-called
“hot corona” model (see, e.g., Ostriker 1976), which also models hard X-rays production as due to
Comptonization.
A possibly stable version of the SLE model is the advection dominated accretion flow (ADAF)
model. As with the SLE model, the ions remain hot, are unable to cool efficiently, and hold most of
the accretion energy (Ichimaru, 1977; Rees et al., 1982). The fiducial nonrelativistic ADAF model
(Narayan and Yi, 1995) includes synchrotron and bremsstraulung cooling, and Comptonization and
synchrotron self-absorption, where a relativistic version is given by Gammie and Popham (1998);
Popham and Gammie (1998). The key result is that the flow is highly inefficient (η ∼ 10−4 ).
This is due to the hot ions holding most of the energy down to the neutron star or black hole,
while electrons cool and generate the observed spectra. A black hole absorbs the ADAF with little
radiation from the accretion disk (Narayan and Yi, 1994, 1995; Abramowicz et al., 1995). Such
models have been applied fairly successfully to X-ray binaries (Ichimaru, 1977), to AGN (Rees
et al., 1982), and to our galactic nucleus (Narayan et al., 1995, 1998; Quataert et al., 1999).
ADAF models have been used to study whether the compact object in an X-ray binary is a
neutron star or black hole. The primary difference between these objects is that the neutron star
has a surface, while a black hole does not. A possible method to determine the existence of the
event horizon (or the lack of a surface on a BHC) is by comparing the quiescent luminosity of X-ray
binaries with neutron stars to X-ray binaries with BHCs. It is found that X-ray binaries with BHCs
are substantially underluminous compared to their neutron star counterparts per unit Eddington
luminosity, which suggests that the excess luminosity in neutron stars is due to emission from the
surface and that the BHCs have no surface (Narayan et al., 1997; Menou et al., 1999; Garcia et al.,
2001; McClintock et al., 2004). In order to confirm this hypothesis one needs to accurately model
the accretion disk. The radiative models of Narayan et al. basically agree with observations by
assuming that the accretion disk takes on either a highly radiative, thin disk state or a nonradiative,
thick disk state. In either case, the matter releases its energy upon impacting the surface of the
neutron star. If BHCs had a surface, then they too would exhibit this surface luminosity. Since
they do not, this suggests they have no surface, and given their large mass suggests they are likely
classical black holes.
The stability of the ADAF model is not certain, and the flow could be unstable to developing
outflows near the black hole (Blandford and Begelman, 1999) or be convectively unstable (Quataert
and Gruzinov, 2000). The success of the two-temperatures or hot-corona models is that they
predict the X-ray spectrum better than the SS73 model due to the inclusion of electronic cooling
by Comptonization.
32
All of these radiative models assume a viscous model to describe the transport of angular
momentum. The existence of the Balbus & Hawley instability (Balbus and Hawley, 1991) suggests
one should study magnetized accretion disks rather than viscous accretion disks. Only simplified
analytic models have appeared that include a magnetic field (Gammie, 1999; Merloni, 2003; Li,
2004). Radiative disks have been studied using time-dependent, radiative MHD numerical models
of accretion flow (see, e.g., Agol et al. 2001; Turner and Stone 2001; Turner et al. 2002; Turner
2004). These are effectively SS73 models with magnetic fields. Unlike the SS73 model, these
simulated disks have a layered vertical structure and are less radiation-dominated for otherwise
similar parameters as the SS73 model. No numerical or analytic models have been constructed
that include a magnetic field with otherwise similar physics to, say, the two-temperature model
with ADAF-type radiative processes. No radiative GRMHD models of accretion disks have yet
been studied numerically.
1.4.3
Gamma-Ray Bursts Models
The standard model for the GRB engine is a hyper-accreting black hole or neutron star (for a
review see, e.g., Piran 1999; Mészáros 2002; Piran 2004). The accretion flow is expected to produce large amounts of energy from annihilation of neutrinos or from spin energy extracted from a
rotating magnetized compact object. Respectively, these two processes are believed to produce an
ultrarelativistically moving pair plasma that internally shocks to produce gamma-rays, or to produce a Poynting flux jet that develops hydromagnetic instabilities that shocks internally to produce
gamma-rays. Eventually, this outflow can interact with the surrounding media to produce so-called
external shocks.
This section first discusses the prompt gamma-ray and afterglow emission (for a review see
Piran 1999, 2004). The purpose of this discussion is to suggest that the GRB progenitor must be
relativistic, compact, and likely sometimes occurs during a supernova. Next, this section discusses
the “collapsar” model of the accretion system for long-duration GRBs (Woosley, 1993; Paczyński,
1998; MacFadyen and Woosley, 1999). The purpose of this discussion is to suggest that a GRMHD
model is required to study the collapsar model. Later in Section 1.5, it is shown that the nonradiative GRMHD models used in this thesis (in Chapters 3 and 4) are a good model for the study
of collapsars.
Models for GRB Emission and Afterglow Emission
The prompt gamma-ray emission from a GRB is found to be non-thermal and highly variable
in time. The temporal variability of GRB light curves suggests the GRB progenitor has a size
R . 60 km for some short-duration bursts, and R . 300 km for some long-duration bursts. However,
an isotropic estimate for the number density of gamma-rays nγ ≈ (1053 erg)/((500 keV)R3 ) is much
larger than what is required to generate electron-positron pairs (Ee− e+ ∼ 2me c2 ). This implies the
emitting media would have an optical depth of τ ∼ 1015 , and suggests that the emission should
33
be thermal. The prompt gamma-rays are actually non-thermal, and this paradox is called the
“compactness problem” (Piran, 1999, 2004). One resolution to the compactness problem is to
assume that the flow is moving ultrarelativistically. By assuming the observed radiation is from
a source moving toward us with Lorentz factor Γ, blue-shifting of photons and modifications to
the probability of photon collisions require Γ & 100 to obtain τ . 1. However, there is no wellunderstood mechanism for jet collimation or large Lorentz factors.
If geometric or relativistic beaming occurs, then an estimate of the actual luminosity can be
obtained by appropriately reducing the estimate for the isotropic luminosity. The value of the
actual luminosity can be useful to determine the nature of the progenitor. If beaming occurs
geometrically, then the isotropic luminosity estimate must be reduced by a beaming fraction of
∼ θj2 /2 for θj . 1, where θj is the opening angle of one side of, an assumed, bipolar jet (Rhoads,
1999; Sari, 1999). If beamed relativistically, a local observer sees an enhancement of Γ2 , so one
must reduce the estimate by 1/Γ2 . One expects 3 possible phases if the initial Lorentz factor is
large, and these phases lead to approximately 3 different domains in the light curve (Sari et al.,
1999): 1) early relativistic beaming with ∼ (θj Γ)2 causally disconnected patches; 2) a slowdown
leads to Γ ∼ θj and thus rapid sideways expansion that leads to a break in the light curve; and 3)
at late time Γ ∼ 1 and one observes an exponentially decaying emission.
In this ultrarelativistic fireball model, internal shocks are produced because the outflow has
a range of velocities. These relativistic internal shocks produce gamma rays via synchrotron or
synchrotron self-Comptonization emission. Such a fireball model has a “baryonic contamination”
problem (see, e.g., Piran 2004). A flow contaminated with baryons can absorb the radiative energy
into bulk kinetic energy, producing too few photons compared to observations. Internal shocks
allow this kinetic energy to be converted back into radiation, but too much conversion too quickly
leads to sub-relativistic bulk motion leading again to a compactness problem.
Another solution to the compactness and baryonic contamination problems is to assume the
jet carries nonradiative energy in the form of a Poynting flux, and the flow only creates gammarays at sufficiently large distances. For example, a Poynting flux could be converted into radiation
far from the collapsing star by internal dissipation (see, e.g., Thompson 1994; Mészáros and Rees
1997; Spruit et al. 2001; Drenkhahn 2002; Drenkhahn and Spruit 2002; Sikora et al. 2003; Lyutikov
et al. 2003) such as magnetic reconnection. Preliminary (controversial) evidence for a magnetic
dominated outflow has been found in GRB 021206 (Coburn and Boggs, 2003), consistent with a
magnetic outflow directly from the inner engine (Lyutikov et al., 2003). As discussed below, a
Poynting flux jet may be primarily responsible for GRBs.
The GRB afterglow is suspected to be generated by interaction with the interstellar medium
in external shocks (see, e.g., Piran 2004). Optical, radio, and X-ray afterglows have energies of
1050 − 1052 erg, generally about 1/10th of the energy in gamma-rays. GRBs, such as GRB 990123
and GRB 990510, show breaks in the optical light curve, which is consistent with an external shock
model. An analysis of the light breaks using an adiabatic synchrotron model for cooling (see, e.g.
Piran 1999, 2004), similar to the Blandford-McKee self-similar solution (Blandford and McKee,
34
1976), shows the breaks are consistent with Γ ∼ 100 − 500. There is no temporal scaling between
the GRB and afterglow lightcurves, and this lack of scaling is consistent with the GRB as generated
in internal shocks and the afterglow as generated in external shocks.
A GRB likely develops from a compact central engine (Piran, 1999; Mészáros, 2002). If one
assumes that the GRB emitting region is causally connected by light propagation effects, then the
variability from GRB light curves gives an estimate of the maximum emitting region (R ∼ cδT ).
The typical GRB variability is 1 ms − 1 s, giving a maximum emitting region of R ∼ 300 km.
The shortest GRB currently ever observed lasted for 6ms (Bhat et al., 1992) with 200µ s temporal
structure. This gives a maximum size of R = 60 km for the emitting source. Energetic and beaming
considerations suggest a GRB is a collimated jet with an energy of ∼ 1051 erg and a Lorentz factor
of 102 − 103 (Sari et al., 1999; Frail et al., 2001; Panaitescu and Kumar, 2001; Bloom et al., 2003).
Such supernova-like energies suggest a subclass of massive stars undergoing core-collapse could
generate GRBs. There is no clear evidence for whether the GRB is a “standard candle” with
standard energy output and jet beaming characteristics (e.g. see references in Liang et al. 2004).
Recent observations suggest that at least some GRBs are not standard candles. For example, GRB
031203 had a very low luminosity from an otherwise normally beamed GRB (Sazonov et al., 2004;
Soderberg et al., 2004).
The estimated GRB event rate of ∼ 1/(107 yr) per galaxy, or ∼ 1/(105 yr) if there is beaming,
also suggests a GRB-SN connection. HST observations show an association between optical afterglows and host galaxies (Sahu et al., 1997; Akerlof et al., 1999), and show a correlation between
GRBs and star-forming regions in galaxies (Fruchter et al., 1999). Core-collapse supernovae models
have suggested a GRB-SN connection since GRB 980425 and SN 1998bw at z = 0.008 (Galama
et al., 1998; Iwamoto et al., 1998). The afterglow from long-duration GRB 030329 and supernova
SN2003dh at redshift z = 0.169 occurred essentially simultaneous in time and overlap in space
confirming a supernova connection to long duration GRBs (Uemura et al., 2003; Mészáros, 2003;
Hjorth et al., 2003; Kawabata et al., 2003; Stanek et al., 2003). Chandra observations of supernova
remnant W49B in our own Milky-Way show the likely remnants of a GRB (Keohane et al., 2004),
so a forensic study may reveal details of the GRB event.
Models for the Engine of Long/Short Duration GRBs
The temporal variability of GRB light curves suggests the GRB progenitor is a system with a
size R . 60 km, and M . 40 M¯ if a compact object, for some short-duration bursts; and with
a size R . 300 km, and M . 200 M¯ if a compact object, for some long-duration bursts. The
progenitor must release the binding energy of a solar mass object over an extended period (many
light crossing times), rather than in a single explosion. This suggests an accretion disk origin is
plausible. Accretion in other systems is known to produce relativistic jets. The two likely scenarios
for generating a GRB are 1) accretion of a 0.001 − 1 M¯ disk around a few solar mass black hole
; or 2) gamma-ray dipole or spin-down emission from a highly magnetized neutron star called a
magnetar.
35
The matter that forms the accretion disk around the solar-mass black hole or neutron star
could be from core-collapse material, supernova fallback material, a neutron star, a white dwarf,
or a Wolf-Rayet star. GRB events have a duration that follows a bimodal distribution, which
could result from an accretion disk system formed from these different types of secondary material
and their associated angular momentum (and thus circularization radius and accretion time scale)
(Popham et al., 1999).
The current leading explanation for a long-duration GRB is the collapse of a massive star
(M ∼ 25 − 40 M¯ ). The most attractive model for long-duration GRBs is the collapsar model
(Woosley, 1993; Paczyński, 1998; MacFadyen and Woosley, 1999) of which there are two types. A
Type I collapsar results from a collapsing massive star that has too weak a shock wave to generate
a supernova. The collapse of the iron core leads to a neutron star, which then quickly leads to a
black hole and accretion disk from the fall back of the envelope. A Type II collapsar is a successful
supernova of a massive star that blows away all helium and some heavy elements outside a neutron
star core. Some material fails to reach escape velocity, falls onto the neutron star, and slowly forms
a black hole and accretion disk.
The collapsar model as developed by Woosley (1993); MacFadyen and Woosley (1999) includes
several pieces of microphysics, such as 1) a realistic EOS (Lattimer and Swesty, 1991; Lattimer,
1996; Blinnikov et al., 1996; Lattimer and Prakash, 2000); 2) the URCA process for nuclear burning
(Bodenheimer and Woosley, 1983); and 3) neutrino cooling and annihilation in the optically thin
parts of the flow (Itoh et al., 1989, 1996; MacFadyen and Woosley, 1999). For a discussion of
collapsar-like GRB accretion disk models, see for example Popham et al. (1999); Narayan et al.
(2001); Kohri and Mineshige (2002); Di Matteo et al. (2002). For a discussion of neutrino-driven
explosions, see for example Fryer and Mészáros (2003). For a discussion of optically thick neutrino
accretion disks, see for example Lee et al. (2004).
The collapsar model has been studied numerically, including 1) viscous HD models with the
above microphysics, following the evolution of a pre-supernova star to collapse, accretion disk
formation, and jet formation (MacFadyen and Woosley, 1999); 2) special relativistic MHD models
of the jet (MacFadyen et al., 2001); 3) nonrelativistic MHD models with the above microphysics,
following the formation of a disk and jet (Proga et al., 2003); and 4) stationary space-time general
relativistic models, with no microphysics, of a black hole with a pseudo-post supernova star (Mizuno
et al., 2004a,b). Numerical simulations show that the penetration of a jet through a stellar surface
and evolution of the jet through a surrounding pressure gradient can lead to large Lorentz factors of
Γ ∼ 44, close to that required by the standard fireball model (see, e.g., Aloy et al. 2000; Zhang et al.
2003). Numerical models have yet to include numerical relativity and MHD to study the collapse
of a rotating magnetized massive star to a black hole (see review by Stergioulas 2003, §4.3).
Despite extremely detailed neutrino physics, typically the magnetic field is neglected in corecollapse supernovae calculations that seek to follow the entire collapse of a massive star (for the
latest typical code see, e.g., Liebendörfer et al. 2004). In unraveling the mechanism by which corecollapse supernovae explode, the simulation of a successful supernova has been found to be sensitive
36
to the accuracy of the neutrino transport (Messer et al., 1998; Yamada et al., 1999; Burrows et al.,
2000; Rampp and Janka, 2000; Liebendörfer et al., 2001; Mezzacappa et al., 2001). For some time
this has been regarded as implying one requires highly accurate neutrino physics to model corecollapse supernovae (Liebendoerfer, 2004), however this could also be interpreted as suggesting
other physics (say, a magnetic field) is required to model core-collapse supernovae.
The GRB is suspected to be due to either (or both) 1) neutrino and anti-neutrino annihilation
that develops into an electron-positron pair fireball at the polar axis of the disk or black hole
and eventually produces gamma-rays; or 2) an electromagnetic Poynting flux from the disk or
black hole following field lines at the poles and eventually produces gamma-rays. As discussed
above, Poynting flux generating GRB models solve both the compactness and baryon-contamination
problems. The energy extracted from a rotating black hole by a magnetic field can easily exceed
the energy for neutrino-mediated energy transport (Mészáros and Rees, 1997). Indeed, all corecollapse events may be powered by MHD processes rather than neutrino processes (Leblanc and
Wilson, 1970; Bisnovatyi-Kogan and Ruzmaikin, 1974; Bisnovatyi-Kogan et al., 1976; Symbalisty,
1984; Woosley and Weaver, 1986; Duncan and Thompson, 1992; Khokhlov et al., 1999; Akiyama
et al., 2003; Thompson et al., 2004a). Core collapse explosions are observed to be significantly
polarized, asymmetric, and often bi-polar, indicating a strong role of rotation and a magnetic field
(Wang and Wheeler, 1996; Wheeler et al., 2000; Wang et al., 2001, 2002, 2003). Core-collapse also
involves shearing subject to the Balbus-Hawley instability as in accretion disks (Akiyama et al.,
2003). Thus, there is sufficient evidence that long-duration GRB models should include a magnetic
field.
There are fewer observational phenomena associated with short-duration GRBs, and so it is
more difficult to model short-duration GRBs than to model long-duration GRBs. Without an
afterglow counterpart to the GRB, a redshift measurement is not possible, and correlating a short
GRB with a host galaxy is difficult. The short-duration GRBs are observed to have harder gammarays than long-duration bursts. This is believed to be due to a lack of baryons obstructing gammaray emission.
Short-duration GRBs could be due to binary NS mergers (Narayan et al., 1992) or NS-BH mergers (Narayan et al., 1991). The NS-NS or NS-BH collision processes are currently the best model
for short-duration GRBs, since these systems produce harder gamma-rays as seen in observations.
However, short-duration GRBs could be produced by a relative of the collapsar model, which has
no supernova and little baryon-loading that also leads to harder emission. During NS-NS or NS-BH
collisions, gravitational wave emission is likely significant, and if the GRB is close enough to Earth,
gravity waves may be detectable by LIGOII/LISA (van Putten et al., 2004). Thus, gravitational
waves may help determine the origin of short-duration GRBs.
Some GRBs may also be due to spin energy extraction from a highly magnetized neutron star
called a magnetar (Wheeler et al., 2000; Rees and Mészáros, 2000; Thompson et al., 2004b). In
the so-called cannon-ball model, the asymmetric stellar collapse leads to a magnetar moving at
∼ 1000 km s−1 (Dado et al., 2002; Huang et al., 2003), where a pair plasma fireball is ejected
37
highly anisotropically (∼ 3 − 10◦ ), but otherwise as described by Usov (1992).
1.5
Characteristic Quantities and Model Validity Estimates
In Section 1.2, 6 fiducial black hole accretion systems were introduced to establish their distance
from the Sun, black hole mass, and bolometric luminosity. It was argued that these objects likely
have accreting black holes. In Section 1.3, the basic theory of accretion was introduced, and
allowed a calculation of the ratio of the bolometric to Eddington luminosity. It was found that
the luminosity, variability, and spectra could be used to deduce basic and precise properties of
black hole accretion disk systems. In Section 1.4, magnetic and radiative models of accretion disks
were introduced, which have allowed researchers to estimate the rest-mass accretion rate of the 6
fiducial systems. It was found that a magnetic model was required for self-consistent generation of
turbulence to drive angular momentum transport, and that radiation is crucial to determine the
thickness and structure of an accretion disk. It was also discussed how GRMHD models are likely
required to study GRBs and core-collapse supernovae. This section uses this cumulative knowledge
to estimate the state of the accretion disk in the 6 fiducial systems by using an approximate radiative
GRMHD model of the accretion flow.
First, the “state” of the accretion flow is determined. The “state” refers to quantities such as
the density, temperature, and magnetic field strength. In principle, the state should be determined
by a completely self-consistent numerical or analytic solution. However, no such solution has
been derived or numerically constructed for a radiating magnetized disk in general relativity. A
compromise is made by using radiative unmagnetized analytic models together with nonradiative
GRMHD numerical solutions of accretion disks as studied in chapter 4. A consistency check is
performed to verify that, indeed, this is a reasonable treatment. Once the complete magnetized
state of the disk is determined, the validity of the fluid, MHD, and ideal MHD approximations is
tested for the 6 fiducial systems.
Even though the accretion disk may extend to R & 1000GM/c2 , the focus of this study is
on the state of the accretion flow, and the validity of the approximations, near the black hole
within R . 40GM/c2 where relativistic effects are important. The density and temperature vary
widely within and between systems. While the conditions for the MHD approximation are plausibly
satisfied in the bulk of the disk, they may not be satisfied, for example, in the more vacuous corona
(Novikov and Thorne, 1973). While I have tested the validity of the approximations for the entire
global accretion flow, only the state of the disk and the validity of the approximations in the disk
are shown. The disk results shown are representative for the non-disk part of the global solution,
and any deviations are discussed.
1.5.1
Estimated State and Structure of Accretion Flow
In order to estimate the state of the GRB disk, and the magnetic field in the other disks, we use
results of nonradiative GRMHD numerical models of accretion flow described in chapter 4. For
38
FUNNEL
WIND
P
0
L
E
CORONA
BLACK HOLE
DISK
PLUNGING REGION
r=12L
EQUATOR
r=40L
Figure 1.1 Cartoon plot of one hemisphere of the axisymmetric accretion disk, corona, funnel/wind,
and plunging regions. The outer radius of the GRMHD simulation is 40GM/c2 . The fiducial radius
(r = 12L, where L = GM/c2 ) is where all “state” quantities are evaluated. These state quantities
are then used to estimate the validity of the fluid, MHD, and ideal MHD approximation at this
location.
those simulations, the mass density, pressure, and magnetic field are determined self-consistently
for a thick disk with scale height to radius ratio of H/R ∼ 0.26. In that study, it was found that the
global accretion flow can be understood basically as having different regions based upon the ratio of
magnetic energy to mass energy (B 2 /ρ0 c2 ) and the ratio of gas to magnetic pressure (β ≡ pgas /pb ,
where pb = B 2 /2). Figure 1.1 shows the regions of the accretion flow. The disk is defined to be
where β < 1/3, the plunging region where β > 1/3 within a disk scale height, the corona region
where 1/3 < β < 1, and the funnel region where B 2 /ρ0 > 1. The figure shows the fiducial radius
r = rf id = 12GM/c2 and the outer radius r = rout = 40GM/c2 used in the simulation. The fiducial
radius rf id is the location where all “state” quantities are evaluated and all “validity parameters”
are evaluated that test the validity of the fluid, MHD, and ideal MHD approximations.
These regions can be treated approximately based upon their average properties. We take a
spatiotemporal average of quantities during the turbulent period of a global GRMHD simulation.
The time average is performed over t = 1000GM/c3 to t = 2000GM/c3 , and a volume integral is
performed over each region for each quantity in the table. Table 1.4 shows the average value of
the dimensionless rest-mass density ρ0 /ρ? , where ρ? is some fiducial rest-mass density; the value
of B 2 /ρ0 ; and the value of β for each of the 4 regions. I have determined the state (density,
temperature, magnetic field, optical depth, etc.) for all of these regions for each of the 6 fiducial
objects. I have also determined the validity of the fluid, MHD, and ideal MHD approximations for
all these regions for each of the 6 fiducial systems. For brevity, this section only presents the details
39
Table 1.4.
Accretion Flow Regions
Region
ρ0 /ρ?
B 2 /ρ0
β = pgas /pb
Disk
Plunge
Corona
Funnel/Jet
0.072
0.18
0.0068
0.00023
0.0004
0.018
0.003
7.4
9.5
1.6
1.5
0.0047
Note. — Dimensionless quantities for regions
as described in Figure 4.2 for H/R = 0.26
GRMHD simulation. See Section 1.5 for a detailed description.
for the accretion disk itself.
From Table 1.4 one can obtain the dimensionless ideal gas temperature
θ≡
pgas
β/2
≡
,
2
B /ρ0
ρ0 c2
(1.14)
which can be converted to kelvin for an ideal gas of protons (where pgas = ρ0 kb Tp [K, ideal]/mp ) to
obtain
µ
Tp [K, ideal] ∼
=θ
mp c2
kb
¶
.
(1.15)
The GRMHD simulation of a thick disk, with an initially fixed H/R = 0.26, has a rest-mass
accretion rate of
Ã
Ṁ0,numerical = AM
µ
ρ? c
GM
c2
¶2 !
,
(1.16)
where AM = 0.2 from the GRMHD numerical simulation. By setting the numerical mass accretion
rate equal to the true rest-mass accretion rate (Ṁ0,numerical = Ṁ0 ), the rest-mass density scale (ρ? )
and then rest-mass density (ρ0 ) can be determined from Table 1.4. The true mass accretion rate is
determined from Table 1.3.
Our GRMHD model is a nonradiative model with an initial (and approximately always) height
to radius ratio of H/R ∼ 0.26. By using the ideal gas law relating density, temperature, and
40
pressure, H/R ∼ cs /vK , and B ∝
√
ρ0 c, one can derive the following proportionalities
µ
ρ0 ∝
P
∝
T [K] ∝
B ∝
¶
H −1
R
µ ¶
H
R
µ ¶2
H
R
µ ¶−1/2
H
,
R
(1.17)
and from numerical simulations one can estimate that
µ
Ṁ0 ∝
H
R
¶−3/5
.
(1.18)
These proportionalities are used to scale the simulated solution with H/R = 0.26 to another solution
with arbitrary H/R. For the sake of consistency, the true rest-mass accretion rate is assumed to
vary with H/R similarly as the numerical rest-mass accretion rate. It is assumed that the physical
rest-mass accretion rates are no different for thin or thick disks. The rest-mass accretion rate is
determined by a few GRMHD simulations that show Ṁ0 /ρ0 ∝ (H/R)2/5 . No detailed GRMHD
studies of thin disks (H/R . 0.01) have been performed that generally verify these scaling laws,
but they should be roughly accurate.
The state parameters are estimated for a characteristic length, time, and rest-mass density
GM
c2
GM
=
3
!µ
Ãc
¶
1
Ṁ0
=
,
AM
cL20
L0 =
T0
ρ?
(1.19)
where L0 and T0 are the natural length and time scales for a black hole, and ρ? is found from Ṁ0 =
Ṁ0,numerical . The characteristic value of any other quantity can be found from these definitions of
mass, length, and time.
GRB Disk State Determination
Since the GRMHD simulations are nonradiative, the value of H/R is estimated from the results of an
unmagnetized, viscous, neutrino-dominated accretion disk model (Popham et al., 1999) (hereafter
PWF). PWF determine the balance between the energy loss due to radiation of neutrinos and
energy gain due to turbulent dissipation. The PWF model provides the disk scale height per unit
radius (H/R) at a particular radius for a given rest-mass accretion rate. This value of H/R is used
41
to scale, as described above, the state resulting from the nonradiative GRMHD simulation with
H/R ∼ 0.26 to the PWF solution for H/R.
X-ray Binary and AGN Disks State Determination
For the X-ray binary and AGN disks, a thin disk state is self-consistently found by solving the
equations of mass, angular momentum, and energy conservation for a radiative disk as in the socalled α-disk model (Shakura and Sunyaev, 1973). This typically results in H/R . 0.05. A thick
disk state (studied, but not shown below) is assumed to be an ADAF with H/R ∼ 1 (Narayan
and Yi, 1995). The only additional component required for these unmagnetized models is the field
strength, which is estimated using the H/R scaling law above. The non-disk regions are assumed
to follow Table 1.4 for both thin and thick disks.
The State of the Accretion Disk
The “state” of the accretion disk for the 6 fiducial black hole accretion systems is shown in Table 1.5.
The columns of the table correspond to one of the 6 fiducial systems, and the rows give estimates
for state parameters such as density and field strength. Only the state of the disk region is shown.
The first 5 rows of Table 1.5 are from Table 1.3 and shown here for convenience.
State parameters in Table 1.5 are
1. rf id : Characteristic fiducial radius in centimeters in a typical numerical study. Used to
determine accretion and rotation time scales.
2. rres : Characteristic smallest radius in centimeters in a typical numerical study. Used to
determine characteristic smallest scale L.
3. L: Characteristic smallest length scale in centimeters in a typical numerical study. The radial
range is resolved logarithmically dr/r ∝ const. and the height range as H/rres ∝ const..
For Nr radial zones dr = rres log (rout /rin )/Nr , where rin and rout are the inner and outer
radial ranges of interest. For this study typically rin ∼ GM/c2 , near the event horizon, and
rout ∼ 40 − 400GM/c2 .
4. T : Characteristic smallest time scale in seconds in a typical numerical study (T = L/c).
5. ρ0 : Characteristic rest-mass density in g cm−3 .
6. B: Characteristic magnetic field in Gaussian units.
7. B/Bcrit,e : Fraction of QED electronic critical field for which the magnetic field exhibits
particle modes and photons behave nonlinearly. Bcrit,e ≡ m2 c3 /(e~) ∼
= 4 × 1013 G. See for
e
example Heyl and Hernquist (1999). Only GRB disk has a field around the critical value.
8. Tp : Temperature of protons (ionized hydrogen) and other particles in kelvin.
42
Table 1.5.
Parameter
NS-BH
dL
M [M¯ ]
Lbol [erg/s]
Lbol /LE
Ṁ0
?
3
1053
3 × 1014
5 M¯ / s
rf id [cm]
rres [cm]
L[cm]
T [s]
ρ0 [g cm−3 ]
B[G]
B/Bcrit,e
Tp [K]
xp
xe
nn /ndegen,n
ne /ndegen,e
Ye
Xnuc
T /tN SE
Xion,H
tacc
trot
H/rf id
prad /pgas
pgas /pgas,0
τγ
τν
5.3 × 106
4.4 × 105
5.9 × 103
2 × 10−7
6.1 × 1011
1.7 × 1015
38
2.1 × 1010
0.0019
3.5
0.32
25
0.01
0.17
1.3 × 104
0.043
0.091 s
3.9 ms
0.26
0.0031
0.39
3.4 × 1015
1.2
GRB 030329
Accretion Disk State
LMC X-3
SgrA*
NGC4258
M87
804 Mpc
3
3 × 1052
8 × 1013
0.1 M¯ / s
55 kpc
10
1038
0.08
10−8 M¯ / yr
8 kpc
2.6 × 106
1037
3 × 10−8
10−5 M¯ / yr
7 Mpc
4 × 107
3 × 1043
0.006
10−2 M¯ / yr
18 Mpc
3 × 109
2.3 × 1042
6 × 10−6
10−2 M¯ / yr
5.3 × 106
4.4 × 105
5.9 × 103
2 × 10−7
1.6 × 1010
2.7 × 1014
6.1
1.2 × 1010
0.0011
2
0.0095
1.6 × 102
0.5
0.17
48
0.026
0.15 s
3.9 ms
0.2
0.0057
1.6
3.3 × 1015
0.0084
1.8 × 107
1.5 × 106
2 × 104
6.6 × 10−7
0.0072
1.1 × 106
−
1.8 × 107
1.7 × 10−6
0.0031
−
−
0.5
−
−
1
32 s
13 ms
0.025
18
−
6.3 × 102
−
4.6 × 1012
3.8 × 1011
5.1 × 109
0.17
1.5 × 10−6
1.4 × 102
−
3.5 × 105
3.2 × 10−8
6 × 10−5
−
−
0.5
−
−
1
1.4 × 102 yr
0.93 hr
0.0011
0.62
−
1.5 × 103
−
7.1 × 1013
5.9 × 1012
7.9 × 1010
2.6
2.4 × 10−7
2.8 × 102
−
4.8 × 105
4.4 × 10−8
8.2 × 10−5
−
−
0.5
−
−
1
2.7 × 102 yr
0.6 day
0.0031
9.7
−
1.1 × 104
−
5.3 × 1015
4.4 × 1014
5.9 × 1012
2 × 102
1.3 × 10−8
3.7
−
7.1 × 104
6.5 × 10−9
1.2 × 10−5
−
−
0.5
−
−
1
8.4 × 105 yr
0.12 yr
0.00048
0.57
−
6.6 × 103
−
Note. — First 4 parameters are previously discussed and are based upon astrophysical observations.
The value of Ṁ0 is based upon both observations and models of the accretion disk (see Section 1.4). The
remaining parameters describe the state of the plasma in the accretion disk for various astrophysical
systems at a radius of r = rf id = 12GM/c2 , close to the black hole. The GRB-type disks (first
2 columns) are modeled using a radiative VHD PWF model of the disk thickness (Popham et al.,
1999) and nonradiative GRMHD simulations to set all other quantities. The other disks (remaining 4
columns) are determined by the SS73 model of a thin accretion disk (Shakura and Sunyaev, 1973) for
all quantities except the magnetic field strength, which is determined by scaling the GRMHD solution
to the appropriate H/R.
43
9. xp : Proton relativity parameter, xp = kb Tp /(mp c2 ). Protons are relativistic if greater than 1,
otherwise nonrelativistic.
10. xe : Electron relativity parameter, xe = kb Te /(me c2 ). Electrons are relativistic if greater than
1, otherwise nonrelativistic. It is assumed that Te ∼ Tp .
11. nn /ndegen,n : Ratio of actual number density of neutrons to number density of neutrons if
degenerate. Neutrons are degenerate if greater than 1, otherwise nondegenerate. See Kohri
and Mineshige (2002). Only the GRB disk has nonnegligible neutron/proton degeneracy.
12. ne /ndegen,e : Ratio of actual number density of electrons to number density of electrons if
degenerate. Electrons are degenerate if greater than 1, otherwise nondegenerate. See Kohri
and Mineshige (2002). Only the GRB disk has nonnegligible electron degeneracy.
13. Ye : Electron fraction: Ye = ne /(np + nn ). Assumes charge neutrality and that the positron
number density is small (true for all the cases studied here). See Kohri and Mineshige (2002).
14. Xnuc : Mass fraction in free nucleons. Rest are mostly α particles. Assumes nuclear statistical
equilibrium. See Woosley and Baron (1992). Not relevant for SS73 model of disk for X-ray
binary and AGN. A hot thick disk model may require this.
15. T /τnse : Ratio of the characteristic smallest time to the time to achieve nuclear statistical
equilibrium. See Khokhlov (1989). Not relevant for SS73 model of disk for X-ray binary and
AGN. A hot thick disk model may require this.
16. Xion,H : Ionization fraction of electrons in a hydrogen gas. This assumes thermal statistical
equilibrium, and that pressure and radiative ionization are negligible. In GRB disks, the
interparticle spacing ∼ the electron orbit radius, so pressure ionization may increase the
ionization fraction. See Equation A.10 in Appendix A.
q
17. τacc : Accretion time scale: τacc ∼ (1/α) rf3 id /(GM )/(H/rf id )2 . Assumes α ∼ 0.1, and the
α model approximately holds.
√
3/2
18. τrot : Rotation time scale: τrot ∼ 2π(rf id + a) GM . Assumes a ∼ 0.9375, although at rf id
this is not a significant effect.
19. H/rf id : Scale height of the disk per unit radius at the fiducial radius. Determined selfconsistently without a magnetic field for the thin disk state. The SS73 solution is used for
AGN and X-ray binaries, and the PWF solution is used for GRBs.
20. prad /pgas : Ratio of radiation to gas pressure, where prad = 13 aT 4 for an optically thick medium
(τγ À 1), where a = 4σ/c and σ = 2π 5 kb4 /(15h3 c2 ) is the Stefan-Boltzmann constant. The gas
pressure for X-ray binary or AGN disks is pgas = (kb /µ)ρ0 T , where µ is the mean molecular
weight per free particle. For a hot fully ionized plasma of cosmic abundances µ ∼
= 0.62, while
44
for a cold cloud of molecular hydrogen µ ∼
= 2.4. The GRB disk pressure is determined as in
Kohri and Mineshige (2002).
21. pgas /pgas,0 : Ratio of GRMHD simulation estimated gas pressure to actual gas pressure as
described above. Error results if radiation pressure, degeneracy pressure, or other non-ideal
gas pressures exceeds the ideal gas pressure used in the GRMHD simulation. Not applicable
to X-ray binary and AGN thin disk solution, which is constructed self-consistently as in SS73.
22. τγ : Disk optical depth to photons τγ ∼ κρ0 H, where κ is the mean opacity for the given temperature and density (see Bell and Lin 1994 for photon scattering and absorption opacities).
23. τν : Disk optical depth to neutrinos. Since this is negligible for accretion flow near X-ray
binaries and AGN, this is only shown for GRB disks.
GRB Disk State
See Appendix A for a discussion of GRB equations of state. The field in GRB accretion disks
is slightly above the critical magnetic field where QED effects, such as photon splitting γ → γγ
and electron/positron/photon pair generation γ → e+ e− γ, occur (see, e.g., Erber 1966). Notice
that the thin GRB disks have sub-relativistic protons and neutrons but have marginally relativistic
electrons. The NS-BH and collapsar disks have degenerate electrons, but marginally non-degenerate
protons and neutrons. For the NS-BH disk, the degeneracy of neutrons leads to a small electron
fraction (i.e. high neutronization). The GRB disks are marginally in nuclear statistical equilibrium.
At this particular radius, alpha particles dominate free nucleons. The PWF solution at slightly
larger radii gives a disk that is hotter (thicker) and so has more free nucleons.
Gas pressure dominates the thin GRB disks. The ratio of actual gas pressure to simulation
gas pressure shows the NS-BH and collapsar disks are well approximated by the simulation gas
pressure. Thus, these estimates are self-consistent. Since the GRB disks are very optically thick,
the radiation in the disk can be approximated by a radiation pressure term. The collapsar disk has
a low neutrino optical depth, so the optically thin approximation is valid. Thus, a nonradiative
GRMHD model can be easily modified to include optically thin neutrino emission in order to study
radiative GRMHD collapsar models of long-duration GRBs. The NS-BH disk is estimated to have
an optical depth of τ ∼ 1, but the effects of electron degeneracy may lower this value (Kohri and
Mineshige, 2002).
The states of the plunging and coronal regions are similar to the state of the disk. The funnel
region within the NS-BH GRB disk has a magnetic field strength ∼ 300× that of the critical field
strength, while the collapsar disk field strength is ∼ 40× the critical field strength. This indicates
a need to account for QED effects in the funnel region.
45
Table 1.6. Validity of Fluid Approximation
Parameter
NS-BH
L/λmf p,e−p
T νc,e−p
Re
4.5 × 1011
1.9 × 1010
1.5 × 1014
GRB 030329
LMC X-3
SgrA*
NGC4258
M87
1.7 × 1012
8.2 × 1010
2.5 × 1014
1.1 × 106
2.7 × 103
6.9 × 106
1.6 × 1011
6.3 × 107
7.2 × 1012
2.1 × 1011
8.7 × 107
8.3 × 1012
4 × 1013
7.1 × 109
4 × 1015
Note. — If a system has all values much greater than unity, then the ideal fluid approximation is valid. If all but Re are much greater than one, then the viscous fluid approximation
is valid. The ideal fluid approximation is valid for all these systems in the thin disk state
shown. In a thick disk state (not shown) the X-ray binary and AGN systems do not explicitly
behave as a fluid, but may be forced to behave like a fluid due to plasma instabilities, as
occurs in the solar wind.
X-ray binary and AGN State
In the thin disk state, the X-ray binary and AGN have low relativity parameters and essentially
no degeneracy. Treating the fluid as a nonrelativistic ideal gas is a good approximation as long
as either 1) the gas pressure dominates over radiation pressure in the optically thin regime or 2)
the disk is optically thick. Notice that all the disks are optically thick at the fiducial radius, so a
radiative diffusion treatment can be used to study these disks at this location. Indeed, treating the
radiation as simply a pressure may be sufficient.
The thick disk in the X-ray binary or AGN is estimated to be H/R ∼ 1. These disks are either
marginally optically thick or completely optically thin. The thick disk (ADAF) models of LMC X-3,
Sgr A*, and NGC4258 likely require radiative transport to understand their radiative properties at
the fiducial radius. However, the thick disk (ADAF) model of M87 is very optically thin and the
radiation flux may be modeled by an optically thin emissivity. The states of the (thick) plunging
and (thick) coronal regions are similar to the state of the thick disk.
1.5.2
Validity of the Fluid, MHD, and ideal MHD Approximations
Table 1.6 shows the estimates that test the validity of the fluid approximation. See Section A.2 for
a discussion of the role of these dimensionless parameters. The value of L/λmf p,A−B is the number
of mean free paths, within a distance L, for particle A hitting particle B. The value of T νc,A−B
is the number of collisions in a characteristic time. The conditions L/λmf p À 1 and T νc À 1
were tested for all colliding species, but for brevity only the electron-proton values are shown. The
value of Re is the Reynolds number. If a system has all values much greater than unity, then the
ideal fluid approximation is valid. If all but Re are much greater than one, then the viscous fluid
approximation is valid.
46
Table 1.7. Validity of MHD Approximation
Parameter
T ωp
L/λD
Λ
NS-BH
2.1 × 1014
5.3 × 1013
3.7 × 105
GRB 030329
LMC X-3
SgrA*
NGC4258
M87
3.1 × 1014
8.2 × 1013
1.3 × 105
1.7 × 109
5.4 × 109
7.6 × 106
6.4 × 1012
1.5 × 1014
1.4 × 106
4 × 1013
7.8 × 1014
5.6 × 106
6.9 × 1014
3.5 × 1016
1.4 × 106
Note. — If all rows are much greater than unity, then the plasma approximation is valid
for that system. The plasma approximation is valid for all these systems.
All species are highly coupled in the GRB accretion disks. This includes the photons, which
cannot escape the disk. For the hotter regions, the temperature reaches the point where nuclear
interactions dominate over Coulomb interactions. The values shown for electron-proton collisions
are within a few orders of magnitude of the values for the proton-proton and neutron-proton
collisions. The weakest coupling is between protons and photons, but even these together act as a
fluid.
For the X-ray binary and AGN, the values are shown for the thin disk. The thin X-ray binary
and AGN disks have photons well-coupled to electrons, making the disk optically thick. All species
are well-coupled except photons and protons.
For the X-ray binary and AGN, thick (ADAF) disk models of the disks may not be treatable
by the fluid approximation without further consideration. The most coupled pair in these systems
is the weak coupling between protons and electrons. Despite the possible necessity of Boltzmann
transport in AGN and X-ray binaries with thick disks, likely plasma instabilities operate as in the
solar wind and the plasma can be treated effectively as a fluid. The solar wind too cannot be
explicitly treated as a fluid, yet the MHD approximation is a reasonable approximation (see, e.g.,
Usmanov et al. 2000). Plasma instabilities likely force the solar wind to behave as a fluid (Feldman
and Marsch, 1997), and similar phenomena may occur in thick disks. In any event, the singlecomponent fluid approximation serves as an interesting starting point for more realistic plasma
calculations. The plunging, corona, and funnel region regimes are similar.
Table 1.7 shows the estimates that test the validity of the plasma/MHD approximation. See
Section A.4 for a discussion of the role of these dimensionless parameters. From top to bottom,
these parameters are associated with 1) charge separation and a finite current rise time ; 2) charge
separation; and 3) the coupling of plasma (> 1 implies weakly coupled). If all rows are much greater
than unity, then the plasma approximation is valid for that system. The plasma approximation is
valid for all the states and regions of all objects. Thus, quasi-neutrality holds.
Table 1.8 shows the estimates that test the validity of the single-component ideal MHD approximation. See Section A.5 for a discussion of the role of these dimensionless parameters. In summary,
47
Table 1.8.
Parameter
NS-BH
T ωg,p
L/λg,p
T ωp2 /ωg,e
Lωp2 dv/(ωg,e c2 )
2
Ldvωg,e /vtherm,e
Lωg,p /vtherm,p
(L/λg,p )2 /(T ωg,p )
RM
dv/vdrif t,p−n
dv/vdrif t,e−n
dv/vdrif t,B
3.1 × 1012
5.1 × 1013
3.4 × 1013
6.5 × 1010
2.6 × 1012
5.1 × 1013
8.2 × 1014
6.7 × 1013
1.5 × 1011
Validity of Ideal MHD Approximation
GRB 030329
LMC X-3
SgrA*
NGC4258
M87
5 × 1011
1.1 × 1013
3.1 × 1014
6 × 1011
6.6 × 1011
1.1 × 1013
2.3 × 1014
1.1 × 1014
1.2 × 1012
7 × 103
3.8 × 106
2.3 × 1011
4.4 × 108
4 × 106
3.8 × 106
2.1 × 109
7.6 × 1011
2.3 × 105
1.9 × 105
4.4 × 108
2.2 × 105
8.7 × 108
1 × 1017
1.9 × 1014
6.6 × 109
8.7 × 108
3.4 × 1012
5.3 × 1014
2.4 × 1010
1.1 × 1017
1.9 × 1014
7 × 106
2.4 × 1010
1.2 × 1017
2.4 × 1014
1.5 × 1011
2.4 × 1010
7.9 × 1013
1.3 × 1016
2.7 × 109
1.9 × 1016
2.4 × 1014
7 × 106
6.2 × 1010
3.7 × 1019
7.1 × 1016
1 × 1012
6.2 × 1010
5.4 × 1014
5.4 × 1016
1.2 × 1012
7.3 × 1021
7.1 × 1016
Note. — If the term is much greater than unity, then the associated effect is negligible and further
motivates the ideal-MHD approximation. If all terms are much greater than unity, then the singlecomponent ideal-MHD approximation is valid for that system. The single-component ideal-MHD
approximation is valid for all these systems.
in order from top to bottom, the parameters correspond to the importance of 1) the gyration time
scale ; 2) gyration length scale ; 3) charge separation ; 4) the Hall effect (J × B) ; 5) the electron
pressure effect ; 6) an anisotropic pressure ; 7) finite gyration radii (FLR vs. MHD order, where > 1
implies MHD ordering) ; 8) Ohmic dissipation (RM ≡magnetic Reynolds number) ; 9) ambipolar
diffusion drift between protons and neutrons ; 10) ambipolar diffusion drift between electrons and
neutrons ; and 11) the drift between electrons and protons that generates the current to sustain the
magnetic field. If the term is much greater than unity, then the associated effect is negligible and
further motivates the ideal-MHD approximation. If all terms are much greater than unity, then
the ideal-MHD approximation is valid for that system.
The GRB disks (disk thickness set by PWF solution) and both the thin (SS73) and thick
(H/R ∼ 1) disk solution for X-ray binary or AGN disks suggest the ideal MHD approximation is
valid. The plunging, corona, and funnel regions are qualitatively similar.
In summary, the fluid approximation is valid for the GRB disks and thin X-ray binary and AGN
disks. However, to maintain fluidity in thick disks in the X-ray binary or AGN, plasma instabilities
are required to maintain fluidity. This is likely to occur, as in the solar wind (see, e.g., Usmanov
et al. 2000; Feldman and Marsch 1997). The plasma approximation is valid for all disks in all
regions. If fluidity is sustained, then the plasma and single-component ideal-MHD approximations
are valid for all systems in all regions.
48
1.6
Summary of Motivation for a GRMHD Model and Open
Questions
HST, Chandra, and ground-based observations of GRB counterparts/afterglows, such as associated
with GRB 020813 and GRB 030329, suggest a supernova connection to GRBs (see, e.g., Uemura
et al. 2003; Mészáros 2003). These results support the black hole accretion disk model for generating
GRB jets (see, e.g. Piran 1999; MacFadyen and Woosley 1999; van Putten and Ostriker 2001; Butler
et al. 2003; Mészáros 2003). Observations of GRBs will always be folded through models, since the
accretion disk will not be resolved in the foreseeable future. The current most attractive model
for long duration GRBs is the collapsar model (Woosley, 1993; Paczyński, 1998; MacFadyen and
Woosley, 1999). In order to determine the flow’s Lorentz factors and the importance of a magnetic
field to jet collimation (Cameron, 2001; Wheeler et al., 2002), a GRMHD model is likely required.
There are currently about 22 known X-ray binaries with a BHC (McClintock and Remillard,
2003; Zand et al., 2004), where about 18 of those are confirmed (dynamical) black hole X-ray
binaries. Estimates of the mass of the black hole come from observations of oscillations in the
motion of the stellar companion. For example, the X-ray binary system Cygnus X-1 was found
to contain a compact object with a mass of M ∼ 7 − 13 M¯ , which is in excess of any possible
neutron star (Webster and Murdin, 1972; Bolton, 1972; Gies and Bolton, 1986; Herrero et al., 1995).
Some of these X-ray binaries, called microquasars since comparable to quasars and related AGN
(Mirabel and Rodrı́guez, 1994, 1999; Fender and Belloni, 2004), exhibit apparently superluminal
(v/c ∼ 3 − 10) jets, which likely originate from a relativistic system with a well-defined axis of
rotation. Microquasars likely form similarly to the jets that form in GRBs, and also likely require
a GRMHD model.
Astronomers have long thought that quasars are likely powered by the accretion of matter onto
a SMBH (Zeldovich, 1964; Salpeter, 1964; Blandford, 1984), and AGN have long been considered to
be related to quasars (Lynden-Bell, 1969). RXTE X-ray observations of AGN show rapid variability
on the order of days. Observations of MCG-6-15-30 show variability on timescales of order 100 s
(Reynolds et al., 1995; Yaqoob et al., 1997), suggesting that the very inner region of an accretion
disk is being probed (Edelson and Vaughan, 2000). Radio interferometry has been able to probe
within a few hundred gravitational radii of a central black hole within Sgr A* and M87 (see, e.g., Lo
et al. 1998; Junor et al. 1999; Doeleman et al. 2001). Fe Kα line emission shows different intensities
for the red and blue wings, which suggests the accretion disk is moving relativistically with speeds
approaching 0.2c (Pariev and Bromley, 1998; Tanaka et al., 1995). Thus, a general relativistic
model of the accretion disk is required.
Some of the first observations of AGN were of so-called radio galaxies with jets extending a few
hundred kiloparsecs out of a point-like source. AGN jets are observed to be relativistic with some
jets having variable structure moving at an apparent speed of 10 times the speed of light, (see,
e.g., West et al. 1998). AGN jets are sources of synchrotron radiation (Baade, 1956), indicating
the existence of a strong magnetic field. HST, VLA, and VLBI all provide high-resolution images
49
of jets, including M87’s jet showing detailed large and small scale structure with apparent speeds
of v/c ∼ 6 (see, e.g., Biretta et al. 1999). However, only future observatories have any hope of
resolving the inner accretion disk at the source of the jet in AGN (Rees, 2001). AGN jets are likely
produced similarly to jets in GRBs and microquasars, and also likely require a GRMHD model.
Some GRBs, some X-ray binaries, and all AGN, are likely powered by a black hole accretion
disk system. The single-component ideal-MHD approximation applies to the disk and surrounding
regions near the black hole for all these objects. A GRMHD model is likely required to study this
general relativistic, highly magnetized plasma flow. Since the GRMHD equations are non-trivial to
study analytically, it is necessary to perform numerical studies of these systems. In particular, selfconsistent, time-dependent studies of the global solution (i.e. disk-corona or disk-jet connection)
require a numerical model.
It has yet to be determined what fraction of the observed luminosity from AGN or X-ray binaries
is due to (a) a radiative accretion disk, (b) black hole spin energy extraction, or (c) a disk wind. For
GRBs, photon production due to neutrino annihilation should also be considered as a component
of the luminosity. A radiative GRMHD model is required to determine the disk contribution to the
luminosity due to photons or neutrinos, and has yet to be studied self-consistently for the global
accretion flow. To determine the contribution of energy from black hole spin energy extraction or
the disk near the black hole, one requires a GRMHD model.
Unanswered questions about AGN jets, microquasar jets, and GRB jets include: 1) what is
the energy source for jets? ; 2) what is the connection between the disk and jet? ; 3) what is
the mechanism for jet production, collimation, and variability? ; and 4) is the jet composed of
ion/electron or positron/electron pair plasma?
1.7
Summary of Dissertation Results
The introduction summarized the evidence for black hole accretion disk systems and discussed
how black hole accretion disk systems exhibit relativistic magnetized flow near a (likely rotating)
black hole. It was discussed how a magnetic field self-consistently generates turbulence and angular
momentum transport in a disk, while unmagnetized models using an α-viscosity model introduce
ad hoc parameters. The observational evidence for relativistic magnetized flow, and the lack of selfconsistency in unmagnetized models, demonstrates that a GRMHD model is required. A discussion
was presented of radiative accretion disk models, which find the structure of an accretion disk by
solving the equations of mass, energy, and momentum conservation. At the end of the introduction,
the state of the accretion disk in 6 fiducial systems was discussed, and the single-component ideal
MHD approximation was found to be valid for these 6 fiducial black hole accretion disk systems.
It was shown that an optically thin radiative GRMHD collapsar model can accurately and selfconsistently model some long-duration GRBs. Finally, a summary was provided for why a GRMHD
numerical model is required to study accretion flow in GRBs, X-ray binaries, and AGN.
Some isolated regions of an accretion flow may be well-described by simplified analytic solutions,
50
such as the Blandford-Znajek (BZ) solution in the nearly force-free funnel region around a rotating black hole (Blandford and Znajek, 1977) and the Gammie inflow solution for the high-density
plasma flow in the equatorial plunging region (Gammie, 1999). Typically, in deriving analytic solutions, the simplifying assumptions do not allow an understanding of the connection between, and
a self-consistent solution for, the black hole, disk, corona, and jet. This is referred to as describing
the “global” accretion flow, rather than the “local” accretion flow. Despite some successful simplified analytic solutions, the VHD, MHD, and GRMHD equations are sufficiently complicated that
numerical studies would likely yield results not easily (or possibly) derived analytically. Numerical
studies are likely required to understand the global, time-dependent structure of accretion flows
around rapidly rotating black holes.
In order to study black hole accretion disk systems, I first developed a code that integrates
the 2D global Newtonian VHD equations of motion with a pseudo-Newtonian potential (McKinney
and Gammie, 2002); then I developed a code that integrates the 2D and 3D global nonrelativistic
MHD equations of motion with a partially relativistic treatment of Alfvén and sound waves and an
artificial resistivity to capture heating in current sheets (unpublished); and finally I helped develop
a code that integrates the GRMHD equations of motion in a stationary space-time (Gammie et al.,
2003).
The main results of this thesis are that 1) the inner radial boundary condition must stay
out of causal contact with the rest of the flow in order to avoid nonphysical outflows and other
numerically-induced artifacts; and the viscosity models are compromised by the arbitrariness of
model parameters (McKinney and Gammie, 2002); 2) the BZ-effect is likely an important source
of energy due to the development of a strong magnetic field in the funnel region; 3) the BZ model
accurately predicts the electromagnetic luminosity in the nearly force-free funnel region near black
holes with spin parameter a . 0.5 ; 4) the BZ model is qualitatively accurate for all black hole
spins; 5) the BZ-effect contributes significantly to the energy content of jets in systems with a thick
disk around a black hole with a spin parameter of a & 0.5; 6) for thick or thin disks, the nominal
accretion efficiency is typically close to the classical thin disk value and generally less than unity
(McKinney and Gammie, 2004), while the Gammie inflow model suggested the efficiency may be
larger than thin disk (including super-efficient) due to magnetic torques in the plunging region
(Gammie, 1999) ; and 7) for thick and thin disks, the angular momentum per baryon accreted
is consistent with the predictions of the Gammie inflow model, which suggested that the angular
momentum per baryon accreted should be less than predicted by viscous, unmagnetized thin disk
models. The following discussion summarizes the results of this thesis to be described in detail in
chapters 2 to chapters 4. Following these chapters, in chapter 5, is a summary of possible future
studies.
51
1.7.1
Viscous Hydrodynamics Summary
The turbulent transport of angular momentum has long been suspected as the cause of accretion in
disks around stars. Viscosity is a natural model to capture the effective friction due to the mixing
action of turbulent flow (Shakura and Sunyaev, 1973). This viscosity is called an anomalous
viscosity, since the true molecular viscosity is too small to drive angular momentum transport.
Typically, the anomalous viscosity is estimated as generating a stress proportional to αP , where α
is a constant and P is the pressure. Such a model is referred to as an α-disk model. The actual
mechanism of angular momentum transport was unknown until the rediscovery of the MRI (Balbus
and Hawley, 1991). The MRI likely dominates the generation of turbulence in ionized accretion
disks (Balbus and Hawley, 1998).
I decided to develop a viscous hydrodynamics (VHD) code to study black hole accretion. The
VHD code uses a numerical method based on ZEUS-2D (Stone and Norman, 1992) with the addition
of an explicit scheme for the viscosity. ZEUS uses an operator-split, finite-difference algorithm on
a staggered mesh that uses an “artificial viscosity” to capture shocks. This artificial viscosity is
only activated in shocked regions. I was particularly interested in measuring the mass, energy,
and angular momentum flux through the inner radial boundary. These diagnostics are useful in a
discussion of accretion disk luminosity and variability.
I used the VHD code to study black hole accretion models, which were similar to previously
studied models using various experimental designs (Igumenshchev and Abramowicz 1999, 2000;
Igumenshchev et al. 2000; all hereafter IA), (Stone et al. 1999, hereafter SPB). IA and SPB used
different models for the accretion disk, anomalous viscosity, and gravitational potential. Unfortunately, these authors found quite different results using otherwise similar methods. The primary
differences between IA’s and SPB’s results are in the energy per baryon and angular momentum
per baryon accreted. IA and SPB also found different results for the radial scaling power laws of
density, pressure, energy per baryon accreted, and angular momentum per baryon accreted. It was
uncertain if these differences were due to the experimental design or due to bugs in the numerical
code. Our goal was to form a bridge between these different results by using an identical code to
study both types of experiments.
IA chose to form an accretion disk by injecting marginally bound matter (Bernoulli parameter <
0) far from the hole at the rate Ṁ0,inj . SPB choose to start with a torus in hydrostatic equilibrium.
IA include all components of the (spherical polar) Navier-Stokes viscous stress tensor, while SPB
only include the r − φ and θ − φ components. SPB only include the toroidal shear components.
They suggest the differential rotation leading to the MRI causes negligible poloidal shear compared
to the toroidal shear. Also, IA and SPB use different forms for the viscosity coefficient ν (units of
p
length2 /time). IA use ν ∝ c2s /Ωk , where cs = ∂P/∂ρ0 |S is the sound speed, S is the entropy,
p
and Ωk = GM/r3 is the Keplerian angular velocity. SPB typically use ν ∝ ρ0 , where ρ0 is the
rest-mass density. IA’s viscosity prescription is chosen to concentrate the stress where the disk is
hot, while SPB’s choice is meant to confine the stress to the bulk of the mass. Any dimensionally
52
reasonable form of the viscosity coefficient can be used, but the goal is to choose a form that mimics
the MRI driven angular momentum transport.
I used a single VHD code to study both IA’s and SPB’s experimental designs, and I was able
to reproduce the results of both IA and SPB. This confirms that the differences in their results
were not due to a bug, but due to differences in their experimental design. Model choices that
introduced the most significant differences include the choice of viscosity prescription and how to
model the source of matter. However, I also found that both IA’s and SPB’s choices for other
model parameters can lead to numerical artifacts. The model parameters in question are the radius
of the inner boundary condition (rin ) and the form of the black hole’s gravitational potential.
The gravitational potential can be either purely Newtonian (φ = GM/r) or pseudo-Newtonian
(φ = GM/(r − 2GM/c2 )) (Paczyński and Wiita, 1980).
There are two competing factors in deciding where to place the inner boundary: 1) the desire to
produce a simulation free of numerical artifacts; and 2) computational cost. The inner boundary’s
location, rin , must be chosen so the boundary condition applied there does not affect the flow at
r > rin . In modeling a black hole, one must choose rin so that as rin → 2GM/c2 there are negligible
changes in the solution. Thus, one must convergence test a solution as rin → 2GM/c2 .
Of course, a Newtonian potential has no intrinsic length scale. However, as in Bondi flow (Bondi,
1952), the sonic point (the radius for supersonic flow at which −vr /cs = 1) can determine a length
scale. For inviscid flow, the sonic point determines the physical point at which information cannot
travel upstream. In more complicated flows, this supersonic boundary may be time-dependent, but
one can define a minimum distance away from rin that always contains a supersonic flow. Indeed,
near a black hole there must exist a surface that is always supersonic (Shapiro and Teukolsky,
1983).
Moving rin → 2GM/c2 in a nonrelativistic code is expensive (time step, dt ∝ (r − 2GM/c2 )).
Thus, the inner boundary is usually placed close to the horizon, but far enough away, so the
simulation can be completed in a reasonable time. However, if rin is chosen such that the flow is
subsonic at the inner boundary, information can travel back into the flow. Clearly, this violates
the physical model of a black hole and may lead to spurious measurements for the accretion rate
of mass, angular momentum, and energy.
For the VHD numerical models with a pseudo-Newtonian potential, only by choosing rin .
2.7GM/c2 was the flow supersonic at rin . In comparison, a numerical model with a pseudoNewtonian potential with rin = 6GM/c2 has subsonic flow at rin . As studied by IA, a numerical
model with a Newtonian potential has subsonic flow at all rin .
I found that the accretion of energy and angular momentum per baryon are affected by numerical
artifacts for models with subsonic flow at rin . Models with subsonic flow at rin show up to 5 times
lower rest-mass accretion rates, and such models show spurious high-frequency oscillations in the
accretion rates of mass, energy, and angular momentum. Models with a Newtonian potential
never develop consistent supersonic flow at rin , and the angular momentum per baryon accreted
oscillates around zero instead of taking a definite value. Models with subsonic flow at rin showed
53
more complicated outflows of matter and energy in the polar regions. Thus, significant numerical
artifacts can be avoided by 1) using a pseudo-Newtonian potential; and 2) choosing a suitable rin ,
such that the flow is supersonic there.
These VHD simulations are limited by the 2D axisymmetric condition, the use of viscosity as
a model of turbulence (as opposed to direction simulations of MHD turbulence), and use of the
pseudo-Newtonian potential with no relativistic corrections.
1.7.2
Global 2D/3D MHD Summary
The α-disk viscosity model continues, to this day, to be used as a model for the turbulent transport
of angular momentum in accretion disks. The thin α-disk model of accretion, based upon this
turbulent viscosity, predicts that the disk plunges radially into the black hole in a sharp transition
at the innermost stable circular orbit (ISCO) (Bardeen, 1970; Shakura and Sunyaev, 1973; Page
and Thorne, 1974; Thorne, 1974; Abramowicz et al., 1978). However, the true kinematic viscosity
is much lower than the anomalous viscosity used in α-disk models.
A magnetic field was discovered to develop a magneto-rotational instability (MRI) in accretion
disks with a high ionization fraction (Balbus and Hawley, 1991; Hawley and Balbus, 1991). The
MRI dominates kinematic viscosity as a mechanism for the generation of turbulence and angular
momentum transport in accretion disks (Balbus and Hawley, 1998). The magnetic field may be
dynamically important inside the ISCO and generate a non-negligible torque on the disk, and the
magnetic field could allow for a super-efficient nominal accretion luminosity (Gammie, 1999; Krolik,
1999b). A magnetic field also likely plays a central role in the generation and collimation of a jet
near a compact object (see, e.g., Blandford and Znajek 1977; Blandford and Payne 1982; Begelman
et al. 1984). In particular, the BZ effect may be responsible for producing a jet in gamma-ray
bursts (GRB), microquasars, and active galactic nuclei (AGN) (Blandford and Znajek, 1977). The
magnetic field could generate and heat corona (see, e.g., Stern et al. 1995).
I was interested in solving the 2D or 3D MHD equations of motion as applied to black hole
accretion systems to determine 1) whether the torque on the disk diminishes near the ISCO ; 2)
whether a magnetic field can launch and collimate jets; and 3) whether the polar field is strong
enough to make the BZ-effect comparable in magnitude to the accretion disk luminosity. I was
also interested in whether the polar BZ-effect provides the energy source for jets, but a direct
measurement of the BZ-effect requires a general relativistic calculation.
I developed a ZEUS-type code (Stone and Norman, 1992) that evolves the equations of nonrelativistic MHD. This MHD code uses an artificial resistivity (Stone and Pringle, 2001) to avoid
the generation of a class of numerical artifacts called “point shocks” and poor energy conservation
in the dissipation of current sheets. The MHD code also includes relativistic corrections that limit
the Alfvén and sound speed to the speed of light (Miller and Stone, 2000), as described below.
Typically, MHD numerical studies of a black hole accretion system show the development of
an evacuated funnel region around the poles. In unpublished simulations, I found that this funnel
54
√
region had Alfvén speed va = B/ ρ0 ∼ 100 − 1000c. Since the stability criteria on the time step is
dt < dx/va (dx being some grid size), a large Alfvén speed sharply reduces the allowed time step. I
modified the equations of motion to limit the speed of Alfvén and sound waves to the speed of light ;
this is referred to as a wave speed limiter. The method involves including a partial correction for the
p
displacement current. The modified equations of motion have Alfvén speed va = B/ ρ0 + B 2 /c2 ,
so as ρ0 → 0, va → c. The sound speed is similarly limited.
The axisymmetric approximation may lead to unrealistic accretion flow geometries. Global
3D simulations of nonaxisymmetric accretion flow can give rise to a self-sustained dynamo, while
axisymmetric flow cannot (Cowling, 1934). A numerical method that uses 3D spherical polar coordinates, rather than 3D Cartesian coordinates, allows more accurate (less diffusive) simulations
of spherical or cylindrical accretion flow. However, the coordinate singularity in such a coordinate
system presents at least two problems. First, the coordinate singularity makes it difficult to accurately model non-radial flow near the axis. Second, the Alfvén wave speed stability criteria on
the time step near the polar axis in 3D spherical polar coordinates gives dt ∝ r sin θ/va for motion
in the φ-direction. For a standard rectangular-based grid, the flow is always over-resolved in the
φ-direction near the outer radial region of the polar axis, and so simulations require many more
time steps to complete an evolution.
The MHD code was extended to include a type of 3D Cartesian grid in order to avoid these two
problems. In this case, there is no coordinate singularity. In order to maintain decent spherical
symmetry and avoid introducing a strong m = 4 mode due to the Cartesian grid, I model the inner
and outer boundaries of the grid as pseudo-spherical (i.e. Cartesian grid approximation of an inner
and outer sphere).
Using this MHD code, I reproduced work by Hawley (2000) and Stone and Pringle (2001) who
numerically model a thick disk of magnetized fluid using the MHD approximation. I found that
there is no sharp transition at the ISCO in any fluid quantities, and the torque does not vanish
near the ISCO. Thin disks studies are required to compare with the thin disk model, but resolving
a thin disk requires much more computer power and has not yet been attempted.
Nonrelativistic or pseudo-Newtonian (Paczyński and Wiita, 1980) MHD models cannot model
black hole rotation, hence cannot show black hole energy extraction by the BZ-effect. However,
2 c, where B is the
dimensional analysis allows one to estimate the BZ luminosity as LBZ ∼ B 2 r+
magnetic field and r+ is the radius of the horizon. Since jets form around the polar axis of an
accretion disk near the black hole, I determine this dimensionally estimated BZ luminosity in the
polar region. The numerical model of an accretion disk thus provides the ratio of the BZ luminosity
per unit rest-mass accretion rate. If I assume an Eddington rest-mass accretion rate, then I find
that the BZ luminosity is comparable to the energy generation rate observed in jets and radio lobes
in the most luminous AGN.
Simulations also show a collimated Poynting flux dominated jet in the funnel region along the
polar axis. However, while the flow appears to be relativistic, there is no way to determine this
accurately with a (mostly) nonrelativistic numerical model. The modified Newtonian code was
55
unable to properly evolve the relativistic flow in the funnel, since the Alfvén and sound speeds
reached near the speed of light. For example, I found it impossible to remove the inner radial
boundary condition from causal contact with the rest of the flow (i.e. one cannot keep the inner
radial boundary inside the fast point in MHD). Also, while I found a large Poynting flux from
the horizon, the Newtonian model should not be capable of producing such an effect. Most of the
Poynting flux energy could be artificially generated by contact between the inner radial boundary
and the flow at larger radii. Thus, confidence was lost in the numerical evolution of the funnel
region. The Lorentz factor of the jet and the dependence of the jet and BZ luminosity on black
hole spin remain undetermined until a general relativistic calculation is performed. Ultimately,
these problems with the (unpublished) nonrelativistic MHD results pushed us to develop a general
relativistic code.
1.7.3
HARM / GRMHD Summary
In collaboration with Charles Gammie and Gábor Tóth, I helped developed the HARM (High
Accuracy Relativistic Magnetohydrodynamics) code. HARM directly solves the equations of ideal
MHD in conservative form using a Godunov method that solves for the conserved fluxes using a
simplified Lax-Friedrich (LF) or Harten-Lax-van Leer (HLL) scheme. Conservative methods can
accurately resolve shocks. However, if the various types of energy (rest-mass, internal energy,
kinetic energy, and magnetic energy) in the problem become disparate in magnitude, then due to
the form of the conservative equations of motion (the energy scales are sometimes summed together)
truncation error in one type of energy can lead to large errors in other energies.
HARM uses 8 primitive quantities: 1) comoving mass density; 2) comoving internal energy
density ; 3,4,5) three velocity components; and 6,7,8) three magnetic field components. HARM uses
the primitive quantities to compute 8 conserved quantities: 1) rest-mass density or, equivalently,
particle number ; 2) total energy; 3,4,5) three momentums (one is angular momentum); and 6,7,8)
three magnetic fluxes. The physically conserved quantities are total rest-mass (or total particle
number), energy, angular momentum, and magnetic flux. HARM computes the conserved quantities
from primitive quantities ; evolves the conservative quantities ; and inverts the conserved quantities
to primitive quantities using a multidimensional Newton-Raphson method.
At the same time as HARM was being written, De Villiers and Hawley (2003a) developed a fully
relativistic code using a different numerical method. Their code solves the nonconservative form of
the equations of motion where primitive variables are directly evolved. This has the advantage of
being faster than HARM, since it avoids the inversion of conserved to primitive variables. It is also
less diffusive compared to LF or HLL methods. However, their method has problems with accuracy
in strong shocks and it is uncertain whether loss of conservation of particle number, energy, and
angular momentum significantly affects the solution.
In practice, HARM and the method by De Villiers and Hawley (2003a) give comparable results
when applied in astrophysical contexts. Indeed, their method is better able to handle regions with
56
b2 /ρ0 À 1 (magnetic energy density to mass energy density ratio much greater than one), such
as the funnel region. However, both codes have difficulty in this funnel region when evolving an
accretion disk around a rapidly spinning black hole. Their current method uses Boyer-Lindquist
coordinates that requires a high-resolution grid near the event horizon for an accurate measurement
of the accretion rate of energy and angular momentum per unit mass. HARM uses Kerr-Schild
(horizon-penetrating) coordinates, and this allows HARM to find reasonable accretion rates of
energy and angular momentum at lower resolution.
HARM could be improved to handle flows with larger b2 /ρ0 by using methods that reduce the
truncation error. HARM is limited by the exclusion of dynamic space-time, radiation, and other
microphysics. Some additional physics will likely continue to be modeled phenomenologically. One
useful phenomenological model is a cooling function that maintains a constant disk scale height to
radius ratio (H/R). This can be used to study disks with different H/R independent of the origin
of the scale height, which would naturally be determined by balancing radiative and gas pressure
effects. Other physical processes that do not dynamically couple to the accretion flow, such as the
synchrotron cooling in a jet, can be accurately modeled without significant modification to HARM.
1.7.4
BZ Effect Summary
Some GRBs, microquasars, and AGN may be powered by the electromagnetic braking of a rapidly
rotating black hole. The BZ-effect (here, broadly defined as any electromagnetic means of extracting
energy from a rotating black hole) is the most likely astrophysical means of extracting energy from a
rapidly rotating black hole. Estimates for the nominal black hole spin in astrophysical environments
give a rapid black hole spin of about a ∼ 0.92 (Gammie et al., 2004). Phenomenological estimates
determined that the BZ luminosity is likely small compared to the disk luminosity (Ghosh and
Abramowicz, 1997; Armitage and Natarajan, 1999; Livio et al., 1999). However, phenomenological
models have only been worked out for simple field geometries. Given the complicated nature of the
accretion flow and its connection to the jet, their estimates may not apply to physical accretion
flow or to a global solution of the disk + jet (McKinney and Gammie, 2004).
I investigated the BZ-effect using HARM via axisymmetric numerical simulations of a rapidly
rotating black hole surrounded by a magnetized plasma. The plasma is described by the equations
of GRMHD, and the effects of radiation are neglected. The evolution is followed for 2000GM/c3 ,
and the computational domain extends from inside the event horizon to typically 40GM/c2 . The
initial conditions are similar to the nonrelativistic MHD study. The numerical model starts with a
plasma torus, in hydrostatic equilibrium, with an overlaid weak magnetic field. The torus is at an
initially fixed H/R ∼ 0.26, and typically remains at that H/R since no radiation was included.
I found that all models with a & 0.5 have an outward Poynting flux on the horizon in the KerrSchild frame, which means energy is extracted from the black hole. None of the models have net
outward energy flux on the horizon. One model, with a net magnetic flux through the disk, shows
a net outward angular momentum flux on the horizon. The model with a net magnetic flux in the
57
disk also has the largest black hole energy extraction rate per unit nominal accretion luminosity of
80%. Thus, the BZ luminosity can be comparable to the disk luminosity.
The limitations of the numerical models presented include the assumption of axisymmetry,
an ideal gas equation of state, and a nonradiative gas. The assumption of axisymmetry is likely
not problematic since I find quantitative agreement with 3D results for the energy and angular
momentum per baryon accreted through the horizon (De Villiers and Hawley, 2003a; De Villiers
et al., 2003a) and the ratio of electromagnetic energy flux to matter energy flux on the horizon
(Krolik, private communication). Models in axisymmetry may overestimate the BZ effect by, for
example, allowing the presence of axisymmetric sheets of magnetic field. In 3D, these sheets
would be disrupted by MHD turbulence. Also, a compressed or tangled axisymmetrically-forced
magnetic field is less likely to diffuse than the same magnetic field in 3D. Radiative effects are
crucial for comparing with observations, but are only easy to implement in limited form, e.g. weak
synchrotron cooling in jets. Radiative effects are likely dynamically important in accretion disks in
X-ray binaries and AGN, such as demonstrated by the photon bubble instability (Gammie, 1998).
1.7.5
BZ/Inflow Solution Comparison Summary
I compared the GRMHD numerical results of black hole accretion to two analytic steady state
models: 1) the force-free magnetosphere solution of Blandford & Znajek (Blandford and Znajek,
1977) and 2) the equatorial inflow solution of Gammie (Gammie, 1999). The BZ solution applies to
the funnel region near the polar axis of the rotating black hole, while the Gammie inflow solution
applies to the equatorial region inside the ISCO.
I presented a self-contained rederivation of the Blandford-Znajek model in Kerr-Schild (horizon
penetrating) coordinates. Unlike the original BZ derivation, this solution does not require the use
of a physical observer to determine the boundary condition on the coordinate singularity at the
horizon, as required in Boyer-Lindquist coordinates. The solution for the far-field radial dependence
of the energy flux only relies on the solution’s separability, rather than requiring the solution to
match Michel’s (1973) solution, as done by BZ.
I used HARM to evolve an accretion disk around a black hole with a spin parameter of a = 0.5.
A nearly force-free region developed in an evacuated funnel region around the poles of the black
hole. Thus, in this region the BZ solution could be compared to the solution from the numerical
model. In this force-free region, on the horizon, I measured 1) the radial magnetic field (B r ), 2) the
electromagnetic rotation frequency (ω), and 3) the electromagnetic energy extracted (Ė (EM ) ). The
analytic BZ solution of these quantities was found to be in excellent agreement with the GRMHD
numerical model solution. Since the BZ solution is a perturbative solution, valid for a ¿ 1, rapidly
rotating black holes are not expected to follow the BZ solution. However, all black hole spins
(−0.938 ≤ a ≤ 0.969) and field geometries tested show a nearly maximally efficient BZ-process
with ω ≈ ΩH /2 on the horizon, as expected (Thorne and MacDonald, 1982), where ΩH ≡ a/(2r+ )
is the rotation frequency of the black hole. Models with arbitrarily large black hole spin show a
58
nearly force-free evacuated funnel region with a BZ-luminosity that is qualitatively similar to the
BZ solution. As in the BZ solution, the numerical solution contains field lines that extend to large
radius in the nearly force-free funnel region. As predicted by the BZ solution, along such field lines
the value of ω is nearly constant.
The Gammie inflow solution determines the fully relativistic MHD flow quantities for a stationary, cold (H/R = 0) disk. The solution is completely determined by the rest-mass accretion
rate and a parameter that sets the magnetization of the flow. The Gammie inflow solution does
fairly well at predicting the space-time average of many flow quantities. In particular, the Gammie
inflow agrees fairly with the numerical solution for the accretion rate of particle energy vs. radius,
accretion rate of electromagnetic energy vs. radius, and comoving magnetic energy vs. radius. The
radial velocity and mass density vs. radius do not fit well. This is partially because the Gammie
inflow solution does not model hot flow, while the numerical solution is of a hot disk. Future analytic studies may investigate a hot inflow solution, and relax other assumptions (see, e.g., Li 2004),
to compare with numerical results. The Gammie inflow model suggests that magnetic torques in
the plunging region may increase the nominal accretion efficiency η (even beyond unity). I found
all models have η < 1. Remarkably, the efficiency closely follows the thin disk efficiency of Bardeen
(1970).
59
2
Numerical Models of Viscous
Accretion Flows Near Black Holes
2.1
Summary of Chapter
We report on a numerical study of viscous fluid accretion onto a black hole. The flow is axisymmetric
and uses a pseudo-Newtonian potential to model relativistic effects near the event horizon. The
numerical method is a variant of the ZEUS code. As a test of our numerical scheme, we are able
to reproduce results from earlier, similar work by Igumenshchev and Abramowicz and Stone et
al. We consider models in which mass is injected onto the grid as well as models in which an
initial equilibrium torus is accreted. In each model we measure three “eigenvalues” of the flow: the
accretion rate of mass, angular momentum, and energy. We find that the eigenvalues are sensitive
to rin , the location of the inner radial boundary. Only when the flow is always supersonic on
the inner boundary are the eigenvalues insensitive to small changes in rin . We also report on the
sensitivity of the results to other numerical parameters.1
2.2
Introduction
Black hole accretion flows are the most likely central engine for quasars and active galactic nuclei
(AGN) (Zeldovich, 1964; Salpeter, 1964). As such they are the subject of intense astrophysical
interest and speculation. Recent observations from XMM-Newton, Chandra, Hubble, VLBA, and
other ground- and space-based observatories have expanded our understanding of the time variability, spectra, and spatial structure of AGN. Radio interferometry, in particular, has been able
to probe within a few hundred gravitational radii (GM/c2 ) of the central black hole, e.g. Lo et al.
(1998); Junor et al. (1999); Doeleman et al. (2001). Despite these observational advances, only
instruments now in the concept phase will have sufficient angular resolution to spatially resolve the
inner accretion disk (Rees, 2001). And so there remain fundamental questions that we can only
answer by folding observations through models of AGN structure.
All black hole accretion flow models require that angular momentum be removed from the flow
in some way so that material can flow inwards. In one group of models, angular momentum is
removed directly from the inflow by, e.g., a magneto-centrifugal wind (Blandford and Payne, 1982).
Here we will focus on the other group of models in which angular momentum is diffused outward
1
Published in ApJ Volume 573, Issue 2, pp. 728-737. Reproduction for this thesis is authorized by the copyright
holder.
60
through the accretion flow.
It has long been suspected that the diffusion of angular momentum through an accretion flow is
driven by turbulence. The α model (Shakura and Sunyaev, 1973) introduced a phenomenological
shear stress into the equations of motion to model the effects of this turbulence. This shear stress
is proportional to αP , where α is a dimensionless constant and P is the (gas or gas + radiation)
pressure. This shear stress permits an exchange of angular momentum between neighboring, differentially rotating layers in an accretion disk. In this sense it is analogous to a viscosity (see also
Lynden-Bell and Pringle (1974)) and is often referred to as the “anomalous viscosity.”
The α model artfully avoids the question of the origin and nature of turbulence in accretion
disks. This allows useful estimates to be made absent the solution to a difficult, perhaps intractable, problem. Recently, however, significant progress has been made in understanding the
origin of turbulence in accretion flows. It is now known that, in the magnetohydrodynamic (MHD)
approximation, an accreting, differentially rotating plasma is destabilized by a weak magnetic field
(Balbus and Hawley, 1991; Hawley and Balbus, 1991). This magneto-rotational instability (MRI)
generates angular momentum transport under a broad range of conditions. Numerical work has
shown that in a plasma that is fully ionized, which is likely the case for the inner regions of most
black hole accretion flows, the MRI is capable of sustaining turbulence in the nonlinear regime
(Hawley and Balbus, 1991; Hawley et al., 1995; Hawley, 2000; Hawley and Krolik, 2001).
Studies of unmagnetized disks have greatly reduced the probability that a linear or nonlinear
hydrodynamic instability drives disk turbulence. While there are known global hydrodynamic
instabilities that could in principle initiate turbulence, these have turned out to saturate at low
levels or require conditions that are not relevant to an accretion disk near a black hole. As of this
writing, no local, linear or nonlinear hydrodynamic instabilities that transport angular momentum
outwards are known to exist in Keplerian disks (Balbus and Hawley, 1998).
Work on magnetized disks has now turned to global numerical models. These are possible
thanks to advances in computer hardware and algorithms. Recent work by Hawley (2000); Hawley
and Krolik (2001), Stone and Pringle (2001), and Hawley and Krolik (2001) considers the evolution
of inviscid, nonrelativistic MHD accretion flows in two or three dimensions. Some of this work
uses a pseudo-Newtonian, or Paczyński and Wiita (1980), potential as a model for the effects of
strong-field gravity near the event horizon.
Other work on global models has considered the equations of viscous, compressible fluid dynamics as a model for the accreting plasma (Igumenshchev and Abramowicz, 1999; Stone et al., 1999;
Igumenshchev and Abramowicz, 2000; Igumenshchev et al., 2000). The viscosity is meant to mock
up the effect of small scale turbulence, presumably generated by magnetic fields, on the large scale
flow. In light of work on numerical MHD models, this may seem like a step backwards. The MHD
models, however, are computationally expensive and introduce new problems with respect to initial
and boundary conditions. It therefore seems reasonable to investigate the less expensive α based
viscosity models. In this paper we investigate axisymmetric, numerical, viscous inflow models.
This work was motivated by the earlier work of Igumenshchev and Abramowicz (1999, 2000) and
61
Stone et al. (1999), hereafter referred to as IA99, IA00, and SPB99, respectively (IA99 and IA00
are collectively referred to as IA, in which case SPB99 is simply referred to as SPB). These authors
studied similar viscous inflow models yet found different radial scaling laws for radial velocity,
density, and angular momentum. They also found different values for the accretion rate of mass
and angular momentum. They used different experimental designs, however. We set out to discover
whether the results from these authors differed due to numerical methods or model parameters.
Along the way, we took a systematic approach to studying numerical parameters and boundary
conditions. One particular point of concern, which will be described in greater detail below, is the
inner boundary condition. This lies in the energetically dominant portion of the flow, so errors there
can potentially corrupt the entire model. In this paper we show that aspects of results presented by
other researchers are sensitive to model and numerical parameters. These results should be useful
to others contemplating large-scale numerical models of black hole accretion.
The paper is organized as follows. In § 2.3 we discuss our models. In § 2.4 we discuss numerical
methods. In § 2.5 we discuss a fiducial solution and results from a survey of other solutions. In
§ 2.6 we summarize our results.
2.3
Model
We are interested in modeling the plasma within a few hundred GM/c2 of a black hole. We will
consider only axisymmetric models (the work of Igumenshchev et al. (2000) suggests that 2D and
3D viscous models give similar results). Throughout we use standard spherical polar coordinates
r, θ, and φ.
We solve numerically the axisymmetric, nonrelativistic equations of compressible hydrodynamics in the presence of an anomalous stress Π, which is meant to model the effects of small-scale
turbulence on the mean flow. The governing equations then express the conservation of mass
Dρ0
+ ρ0 (∇ · v) = 0,
Dt
momentum,
ρ0
and energy,
(2.1)
Dv
= −∇P − ρ0 ∇Ψ − ∇ · Π,
Dt
(2.2)
Du
= −(P + u)(∇ · v) + Φ.
Dt
(2.3)
Here, as usual, D/Dt ≡ ∂/∂t + v · ∇ is the Lagrangian time derivative, ρ0 is the rest-mass density,
v is the velocity, u is the internal energy density, P is the pressure, and Ψ is the gravitational
potential. The dissipation function Φ is given by the product of the anomalous stress tensor Π
with the rate-of-strain tensor e
Φ = Πij eij ,
62
(2.4)
(sum over indices) where the anomalous stress tensor is the term-by-term product
Πij = −2ρ0 νeij Sij ,
(2.5)
(no sum over indices) where Sij is a symmetric matrix filled with 0 or 1 that serves as a switch for
each component of the anomalous stress. The rate of strain tensor e is a symmetric tensor that in
spherical polar coordinates has
1
∂vr
− (∇ · v),
∂r
3
(2.6)
1 ∂vθ
vr
1
+
− (∇ · v),
r ∂θ
r
3
(2.7)
vr
vθ
1
1 ∂vφ
+
cot θ − (∇ · v) +
,
r
r
3
r sin θ ∂φ
(2.8)
err =
eθθ =
eφφ =
1 ∂ vθ
1 ∂vr
erθ = (r ( ) +
),
2 ∂r r
r ∂θ
1 ∂ vφ
1 ∂vr
erφ = (r ( ) +
),
2 ∂r r
r sin θ ∂φ
and
(2.9)
(2.10)
1 ∂vθ
1 sin θ ∂ vφ
(
)+
).
eθφ = (
2 r ∂θ sin θ
r sin θ ∂φ
(2.11)
P = (γ − 1)u.
(2.12)
The equation of state is
For the gravitational potential we use the pseudo-Newtonian potential of Paczyński and Wiita
(1980): Ψ = −GM/(r − rg ) (here rg ≡ 2GM/c2 ). This potential reproduces features of the orbital
structure of a Schwarzschild spacetime, including an innermost stable circular orbit (ISCO) located
at r = 6GM/c2 . In a few cases we use the Newtonian potential Ψ = −GM/r for comparison with
others’ work. IA and SPB describe results exclusively from a Newtonian potential, although IA00
report in a footnote that otherwise identical experiments in a pseudo-Newtonian potential show
significant changes in the flow structure for r ∼ 2GM/c2 .
We must now make some choices for the anomalous stress tensor. One might argue on very
general grounds for a Navier-Stokes prescription, and indeed IA and we use a prescription where
all elements of S are 1 (the “IA prescription”). SPB, on the other hand, use the Navier-Stokes
prescription with all components zero except Srφ , Sθφ , Sφr , and Sφθ (the “SPB prescription”). SPB
justify this choice by arguing that it more appropriately models MHD turbulence; this was later
supported by results presented in Stone and Pringle (2001).
We must also choose a viscosity coefficient. We consider three different prescriptions: one
similar to those chosen by IA; a second viscosity coefficient similar to that chosen by SPB; and a
third, similar form that vanishes rapidly near the poles. Explicitly,
ν = α(c2s /ΩK ),
63
(2.13)
ν = α(ρ0 /ρ? )Ω0 r02 ,
(2.14)
ν = α(c2s /ΩK )sin3/2 (θ),
(2.15)
are the IA, SPB, and MG prescriptions, respectively, where cs =
Ω2K ≡
p
γP/ρ0 is the sound speed, and
1 ∂Ψ
GM
=
r ∂r
r(r − rg )2
(2.16)
is the “Keplerian” angular velocity. Here ρ? and Ω0 are values of ρ0 and Ω at a fiducial radius r0 .
The choice of viscosity coefficient for IA and MG is based, as usual, on dimensional arguments.
SPB99’s choice focuses the viscosity where most of the matter is, a numerical convenience. Our MG
prescription is a small modification of the IA prescription to concentrate the viscosity toward the
equator. These choices are to a large extent arbitrary, although one might attempt to motivate the
choice by comparison with MHD simulations, as do Stone and Pringle (2001). Nevertheless, some
dynamical properties of MHD turbulence, such as the elastic properties that produce magnetic
tension and hence Alfven waves, can never be modeled with a viscosity.
The model has boundaries at θ = ±π/2 and at r = rin , rout . At the θ boundaries we use
the usual polar axis boundary conditions. At the radial boundaries we use “outflow” boundary
conditions; ideally these boundary conditions should be completely transparent to outgoing waves.
Fuel for the accretion flow must be provided either in the initial conditions or continuously over
the model evolution. Some global numerical accretion flow models have started with an equilibrium
torus. Examples include Hawley (1991), Hawley et al. (1995), Hawley (2000), Hawley and Krolik
(2001), Hawley et al. (2001), and Hawley and Krolik (2001). Others have started with an initial
configuration of matter that is not in equilibrium. For example, matter may be placed in orbit
about the black hole, but with sub-Keplerian angular momentum, so that once the simulation
commences it immediately falls toward the hole. Examples of this approach include Hawley et al.
(1984), Koide et al. (1999), Koide et al. (2000), and Meier et al. (2001). This approach may enhance
transients associated with the choice of initial conditions, although it can also be physically well
motivated, as in studies of core-collapse supernovae. One can also inject fluid continuously onto
the computational grid over the course of the evolution. Examples of this include IA99 and IA00.
One might also use an inflow boundary condition at the outer radial edge of the grid, as in Blondin
et al. (2001). The main advantage of injection models is that they allow one to achieve a steady,
or statistically steady, state. In this paper we will consider only the equilibrium tori and on-grid
injection models.
The equilibrium tori (Papaloizou and Pringle, 1984; Fishbone and Moncrief, 1976; Jaroszynski
et al., 1980) are steady-state solutions to the equations of inviscid hydrodynamics. They assume a
polytropic distribution of mass and internal energy, P = Kρ0 γ , and a power-law rotation profile,
Ω ∝ (r sin θ)−q . The radial and meridional components of the velocity vanish. There are 5 parameters that describe the torus: (1) the location of the torus pressure maximum r0 ; (2) the location
of the innermost edge of the torus, rt,in < r0 ; (3) the maximum value of the density, ρ? = ρ0 (r0 );
64
(4) the angular velocity gradient q = d ln Ω/d ln R; and (5) the entropy constant K.
On-grid injection adds matter to the model at a constant rate. The matter is injected with
a non-zero specific angular momentum and with zero radial or meridional momentum in a steady
pattern ρ˙0 (r, θ) which is typically symmetric about the equator. Parameters for this scheme include:
(1) a characteristic radius for injection rinj ; (2) the rate of mass injection Ṁinj ; (3) the specific
angular momentum of the injected fluid, vφ = f1 rΩK . We usually set f1 = 0.95, so that the
fluid circularizes near rinj . This restricts transients associated with circularization to the outer
portions of the computational domain; (4) the internal energy of the injected fluid, u = f2 ρ0 Ψ.
We always set f2 = 0.2 so that the fluid is marginally bound, i.e. has Bernoulli parameter Be ≡
(1/2)v 2 + c2s /(γ − 1) + Ψ < 0.
One must also choose the injection pattern ρ˙0 (r, θ). IA99 choose a radially narrow region, but
do not explicitly give ρ˙0 (r, θ). We use a Gaussian, but found that none of the results are sensitive
to the precise profile. The accretion rate of mass, energy, and angular momentum are insensitive
to large changes in the size of the injection region except for the extreme cases of filling the entire
θ width or injecting in 2 locations. These extreme cases are sufficiently different to be referred
to as a completely different model; they lead to a qualitative change in the flow. For example, a
full range θ injection region has matter that will collide with any outflow at the poles. A bipolar
injection leads to an equatorial outflow. Our models have radial width σr = 0.05(rin + rout )/2 and
σθ = π/8.
2.4
Numerical Methods
Our numerical method is based on ZEUS-2D (Stone and Norman, 1992) with the addition of
an explicit scheme for the viscosity. ZEUS is an operator-split, finite-difference algorithm on a
staggered mesh that uses an artificial viscosity to capture shocks (in addition to the anomalous
viscosity in equations [2.2]). This algorithm guarantees that momentum and mass are conserved to
machine precision. Total energy is conserved only to truncation error, so total energy conservation
is useful in assessing the accuracy of the evolution.
The inner and outer radial boundary conditions are implemented by copying primitive variable
values (ρ0 , u, and v) from the last zone on the grid into a set of “ghost zones” immediately
outside the grid. Inflow from outside the grid is forbidden; we set vr (rin ) = 0 if vr (rin ) > 0 and
vr (rout ) = 0 if vr (rout ) < 0. Since we expect inflow on the inner boundary, this switch should
seldom be activated. We have found that frequent activation of the switch is usually an indication
of a numerical problem.
We use a radial grid uniform in log(r − rg ). We require that dr(r)/(rin − rg ) ≤ 1/4 so that the
structure of the pseudo-Newtonian potential is well resolved. The grid is uniform in θ. The grid
has Nr × Nθ zones.
65
2.4.1
Numerical Treatment of Low Density Regions
Like many schemes for numerical hydrodynamics, ZEUS can tolerate only a limited dynamic range
in density. It is therefore necessary to impose a density minimum ρ0 f l to avoid small or negative
densities. Our procedure for imposing the floor is equivalent to adding a small amount of mass to
the grid every time the floor is invoked. Mass is added in such a way that momentum is conserved.
To monitor the effect of the density floor, we track the rate of change of total mass and total energy
(from kinetic energy change) due to this procedure, Ṁf l and Ėf l .
We set ρ0 f l = 10−10 Ṁinj c3 /(GM )2 for injection runs and ρ0 f l = 10−5 ρ? for torus runs. Lower
values for ρ0 f l do not lead to a significant change in the solution. Larger values of ρ0 f l give
Ṁf l ∼ Ṁ , the accretion rate through the inner boundary. The atmosphere also becomes more
massive and begins to affect torus stability– vertical oscillations are excited in the inner disk by a
Kelvin-Helmholtz like instability. This should be avoided.
We must also surround the torus in a low density atmosphere in the initial conditions. The density of the atmosphere is ρ0 f l and the internal energy density is u = Uo ρ0 Ψ, where Uo is a constant
fraction of order unity (e.g. IA and we choose Uo = 0.2). The addition of the atmosphere has no
effect on the solution since the mass source’s evolution eventually dominates the flow everywhere.
SPB99 choose a different method of constructing the initial atmosphere but obtain late-time results
that are similar to ours.
It is also necessary to impose a floor uf l on the internal energy density. This we take to be the
minimum value of u in the initial atmosphere. As for the mass, we track the rate of change of total
energy due to the internal energy floor, that along with the kinetic energy is included in Ėf l .
2.4.2
Diagnostics
Global numerical simulations of accretion flows are complicated; it is possible to measure many
quantities associated with the flow. Some are astrophysically relevant, and some are not. In our
view particular interest attaches to the time-averaged flux of mass, energy, and angular momentum
through the inner boundary. Physically, these are directly related to the luminosity of the accretion
flow and the rate of change of mass and angular momentum of the central black hole. As described
by Narayan and Popham (1993), these are in a sense the nonlinear “eigenvalues” of the model.
The rest-mass accretion rate is
Z
Ṁ =
ρ0 v · dS,
(2.17)
S
where S is the inner radial surface of the computational domain. The total energy accretion rate is
Z
Ė =
1
(( v 2 + h + Ψ)ρ0 v + Π · v) · dS,
2
(2.18)
S
where h = (u + P )/ρ0 = γu/ρ0 is the specific enthalpy with our equation of state. The angular
66
momentum accretion rate is
Z
r sin θ(ρ0 vvφ + Π · φ̂) · dS.
L̇ =
(2.19)
S
It is also sometimes useful to focus on the reduced eigenvalues l = L̇/Ṁ and e = Ė/Ṁ . These
value of mass, energy, and angular momentum are recorded at about 2 grid zones away from rin .
This avoids any error that may occur when evaluating directly on the boundary where the inflow
boundary condition is applied.
We also track volume-integrated quantities, the flux of mass, energy, and angular momentum
across all boundaries, and floor added quantities in order to evaluate the consistency of the results.
Mass and angular momentum are conserved to machine precision, although “machine precision”
implies a surprisingly large random walk in the integrated quantities over the full integration because
the calculation requires millions of timesteps.
Total energy is conserved to truncation error, not machine precision, and thus is a useful check
on the quality of the simulation. Total energy conservation implies
Ėerr = Ėvol + Ė + Ėout − Ėf l ,
(2.20)
where Ėvol is the rate of change of the volume integrated total energy, Ėout is the flux of total
energy through the outer radial boundary, and Ėf l is the rate of total energy added due to the
kinetic energy change (because of the mass density floor) and internal energy density floor. Ideally,
Ėerr = 0. Truncation errors can (and do) lead to cumulative, rather than random, changes in the
total energy. A useful gauge of the magnitude of these errors is Ėerr /E. For all runs we performed
the error rate is within 10% of 10−5 c3 /GM for a torus run and within 10% of 10−4 c3 /GM for a
viscous injection run.
2.4.3
Code Tests
Our version of ZEUS reproduces all hydrodynamic test results from Stone and Norman (1992),
including their spherical advection and Sod shock tests. We also find excellent agreement with
steady spherical accretion solutions, i.e. Bondi flow (Bondi, 1952). An inviscid equilibrium torus
run also persists for many dynamical times with insignificant deviations from the initial conditions.
We have parallelized our code using the MPI message passing library. On the Origin 2000 at
NCSA we are able to achieve about 2.5 × 107 zone updates per second using 240 CPUs, or about 35
GFLOPs. This is 159 times faster than the single CPU speed, which represents a parallel efficiency
of 66%.
67
2.5
Results
The initial motivation for undertaking this calculation was to understand differences between results
reported in IA and SPB. Using our code, which is based on the same algorithm used in SPB’s
calculations, we ran a series of tests attempting to reproduce SPB’s results. These test calculations
used all of SPB’s model choices, including SPB’s viscosity prescription, a Newtonian potential,
and a torus for the mass source. We were able to reproduce most quantitative results reported in
SPB99’s torus calculations. This includes their radial scaling laws. For example, in a model that
is identical to SPB99’s Run B, we find Ṁ ∝ r, ρ0 ∝ r0 , c2s ∝ r−1 , vφ ∝ r−1/2 , and |vr | ∝ r−1 .
These power law slopes are identical to those reported by SPB99. As another example, we found
Ṁ = 1.23 × 10−3 torus masses per torus orbit at the pressure maximum for a model identical to
SPB’s Model A (their fiducial model); SPB report Ṁ = 1.0 × 10−3 in the same units. Given the
fluctuations in rest-mass accretion rate, our value and SPB’s value are fully consistent. We were
even able to reproduce certain numerical artifacts associated with the inner radial boundary, such
as a density drop and temperature spike near the inner boundary.
Recall that SPB evolve an initial torus and allow it to accrete; IA use a different experimental
design in which matter is steadily injected onto the grid. They also use a different viscosity prescription. We ran a second series of test calculations attempting to reproduce IA’s results. These
test calculations used all of IA’s model choices, including viscosity prescription, Newtonian potential, etc. We were able to reproduce all of IA99’s calculations except those that include thermal
conduction (which we did not attempt to reproduce). In each case we found that the qualitative
nature of the flow is similar to that described in IA99. In particular, we agree on which models are
stable and unstable and which models exhibit outflows. We also find qualitative agreement with
their contour plots of, e.g., density pressure, mass flux, and Mach number. We also find qualitative
agreement with their radial run of cs /VK and specific angular momentum. Our results do not agree
precisely, but this is likely due to small differences in mass injection scheme (because IA99 do not
give their ρ˙0 (r, θ)). Finally, we can also reproduce the radial scalings given in IA00 for their model
A.
The most significant difference between the results of IA and SPB was due to the choice of
anomalous stress prescription, as might have been anticipated. Qualitatively, the stress components
that are included in IA and not SPB tend to smooth the flow and suppress turbulence. Thus
SPB99’s simulations result in more vigorous convection than simulations performed by IA. The
choice of mass supply (torus vs. injection) also leads to a significant difference between IA and
SPB’s results; this is discussed in more detail below.
The fact that we can reproduce both SPB’s and IA’s results using a single code is consistent
with the hypothesis that differences between their reported results (e.g. the lower degree of convection reported in IA than SPB, and the differences in radial scaling laws) is due to differences in
experimental design and viscosity prescription rather than numerical methods. While we cannot
completely rule out the possibility that SPB, IA, and we have made identical experimental errors,
68
Table 2.1.
Run
Nr
A
B
C
D
E
F
108
80
80
64
128
128
Nθ
50
50
50
40
80
80
Visc.
IA
IA
IA
MG
MG
MG
Potential
Parameter List
rin /rg
PN
PN
Newt.
PN
PN
PN
rout /rg
1.35
3
3
1.2
1.4
1.4
Rinj /rg
300
300
300
76
21
81
248
248
248
62
17
21
γ
α
3/2
3/2
3/2
3/2
5/3
5/3
0.1
0.1
0.1
1.0
0.01
0.01
tf (c3 /GM )
7.3 × 105
4.4 × 105
7.3 × 105
3.3 × 103
3.0 × 104
6.8 × 104
Note. — IA and MG are viscosity prescription described in equations 7-9. PN is the pseudo-Newtonian
potential of Paczyński & Wiita (1980). Run B uses Run C as initial conditions. Here rg = 2GM/c2 , and in
run F Rinj is the position of the torus density peak ρ0 .
Table 2.2.
Run
A
B
C
D
E
F
Results List
Steady State Time (GM/c3 )
Max. Mach at rin
Ṁ0
−(Ė/(Ṁ0 c2 )) × 10−2
L̇ c/(GM Ṁ0 )
4.3 × 105
2.4 × 105
2.4 × 105
≥ 3.3 × 103
5.5 × 103
2.0 × 104
-1.4
+0.0
+0.0
-0.4
-3.0
-3.4
5.96 × 10−2
1.95 × 10−2
9.46 × 10−3
≥ 3.59 × 10−2
3.48 × 10−2
5.03 × 10−1
2.06
3.72
6.77
4.64
3.01
3.11
1.75
1.29
.0746
−0.167
3.41
3.35
Note. — Runs A-E are injection runs with rest-mass accretion rate units in Ṁ0,inj and Run F is a torus run with rest-mass
accretion rate unit in ρ0 (GM )2 /c3 . Run C’s angular momentum fluctuations are 10 times the average value shown.
this seems unlikely. This comparison thus lends credibility to SPB, IA, and our numerical results.
2.5.1
Fiducial Model Evolution
We now turn from reproducing earlier viscosity models to considering new aspects of our own
models. First, consider the evolution of a “fiducial” model (Run A in Table 2.1 and Table 2.2).
The fiducial model has rin = 2.7GM/c2 , rout = 600GM/c2 , rinj = 495GM/c2 , γ = 3/2, α = 0.1,
Nr = 108, Nθ = 50. It uses a pseudo-Newtonian potential, mass is supplied by injection, and the
viscosity prescription follows IA. It was run from t = 0 to t = 7.3 × 105 GM/c3 .
Run A is similar to IA99’s “Model 5”, except that it uses a pseudo-Newtonian potential. In a
statistically steady state the flow is characterized by a quasi-periodic outflow. Hot bubbles form
at the interface between bound (Bernoulli parameter Be = (1/2)v 2 + c2s /(γ − 1) + Ψ < 0) and
unbound (Be > 0) material. These hot bubbles are buoyant and move away from the black hole.
This appears to be a low-frequency, low wavenumber convective mode (IA refer to it as a “unipolar
outflow”). Higher wavenumber convective modes are evidently suppressed by the viscosity.
Figure 2.1 shows time-averaged plots of various quantities in the fiducial run. The time average
69
Figure 2.1 Time-averaged spatial structure of fiducial run (Run A; α = 0.1, rin = 2.7GM/c2 , and
rout = 600GM/c2 ). Shown are the density (upper left), Bernoulli parameter (Be = (1/2)v 2 +c2s /(γ−
1) + Ψ) (upper right; dotted line is a negative contour), scaled mass flux r2 sin θ(ρ0 v) (lower left),
and scaled angular momentum flux r3 sin2 θ(ρ0 vvφ + Π · φ̂) (lower right). The flow is not symmetric
about the equator because the flow exhibits long timescale antisymmetric variations. Convective
bubbles form at the interface between positive and negative Bernoulli parameter (i.e. unbound and
bound matter).
70
is performed from 200 equally spaced data dumps from t = 4.3 × 105 GM/c3 to t = 7.3 × 105 GM/c3 .
We show only the region rin < r < 30GM/c2 , whereas the computational domain is much larger:
rin < r < 600GM/c2 . The injection point is located far outside the plotted domain at r =
496GM/c2 . Because of the strong time-dependence of the flow in the fiducial run, the flow at any
instant may look very different from these time averaged plots.
The upper left panel in Figure 1 shows the average density; notice that the density is not symmetric about the equator. This is because the flow involves long-timescale quasi-periodic variations
which are not quite averaged out over the course of the run. The upper right corner shows the
Bernoulli parameter Be. Dotted lines are negative; notice that there is a substantial amount of
fluid near the equator that is unbound in the sense that Be > 0. Nevertheless, this material is still
flowing inward in a nearly laminar fashion. Near the poles, the time-averaged Be < 0, but this
region experiences large fluctuations. Polar outflows are typically associated with positive fluctuations in Be. The lower left panel shows the scaled mass flux r2 sin θ(ρ0 v). Notice that much of the
mass flux is along the surfaces of the inflow rather than along the equator. The lower right panel
shows the scaled angular momentum flux r3 sin2 θ(ρ0 vvφ + Π · φ̂). As for the mass flux, most of
the activity is along the surface of the flow.
Figure 2.2 shows the time series of the reduced eigenvalues: Ṁ /Ṁinj , e = Ė/(Ṁ c2 ), and
˙
l = Lc/(GM
Ṁ ). Also shown as dashed lines are the thin disk values for e and l. These assume
a thin, cold disk terminating at r = 6GM/c2 . The low value of l is due to two effects. First, the
disk is already sub-Keplerian by the time the flow reaches the innermost stable circular orbit. In
addition, there are residual viscous torques in the plunging region that lower the specific angular
momentum of the accreted material (see Figure 2.3, below). Notice that Figure 2.2 shows a smooth
evolution that varies on a timescale τ ≈ 4 × 104 at late time. The largest variations in rest-mass
accretion rate are related to the appearance of large convective bubbles.
Figure 2.3 shows the θ and time averaged run of several quantities with radius. The averages
are taken over 4.3 × 105 GM/c3 < t < 7.3 × 105 GM/c3 and |θ − π/2| < π/6 2 . The upper left plot
shows the run of density. Notice that here, as for the other quantities, there is a spike near rinj ,
an intermediate region, and then an inner, roughly power-law region. The upper right plot shows
(cs /c)2 ; the lower left shows |vr |/c. Notice that the radial velocity exceeds the speed of light at the
inner boundary. Similarly the azimuthal velocity vφ /c shown in the lower right panel approaches
the speed of light. Also shown in that panel is the circular velocity (dashed line). Evidently the
flow is slightly sub-Keplerian at most radii.
The radial run of flow quantities in the inner regions can be fit by power laws, as done by IA
and SPB. Our best fit power laws for the fiducial model (Run A) over 2.7GM/c2 < r < 20GM/c2
are ρ0 ∝ r−0.6 , cs ∝ r−0.5 , |vr | ∝ r−2 , and vφ ∝ r−0.8 . Between 2.7GM/c2 < r < 6GM/c2 , vφ is
best fit by vφ ∝ r−0.9 , which is nearly, but not exactly, consistent with conservation of fluid specific
angular momentum (vφ ∝ r−1 ). Angular momentum is not exactly conserved at r < 6GM/c2
because of viscous torques.
2
Averaging over |θ − π| < π/36 produces nearly identical results, but we have chosen to use IA’s range in θ.
71
Figure 2.2 The evolution of Ṁ /Ṁinj , e = Ė/(Ṁ c2 ), and l = L̇ c/(GM Ṁ ) in the fiducial run (Run
A). The dotted line indicates the thin disk value. The run has clearly entered a quasi-steady state.
The evolution is relatively smooth with a small variation on a timescale τ ≈ 4 × 104 . This is the
timescale for convective bubble formation (the low point in rest-mass accretion rate is when bubble
forms). For this model the bubble forms at alternate poles. A full cycle requires of order one
rotation period at the injection radius.
72
Figure 2.3 The radial run of θ and time averaged quantities from the fiducial run (Run A). Shown
are the density (upper left), squared sound speed (upper right), radial velocity (lower left), specific
angular momentum (lower right; solid line), and circular orbit specific angular momentum (lower
right; dashed line). Crudely speaking, the inner flow is consistent with a radial power law. The
best fits to a power law are: ρ0 ∝ r−0.6 , cs ∝ r−0.5 , |vr | ∝ r−2 , and vφ ∝ r−0.8 . The plots are
averaged over θ = π/2 ± π/6.
73
The careful reader may notice that the power law slopes quoted in the last paragraph are not
consistent with a constant rest-mass accretion rate. This is because the power laws are derived
from averages over |θ − π/2| < π/6, following SPB. If one averages over all θ (following IA) and
time, then the resulting profiles are consistent with constant mass, energy, and angular momentum
accretion rates, as they must be for a flow that is steady when averaged over large times.
2.5.2
Dependence on Inner Boundary Location and Gravitational Potential
Having established that the differences between SPB and IA’s models are due to model choices
rather than numerics, we were also interested in studying whether any features of global viscous
accretion models are strongly dependent on numerical parameters. The first parameter we considered was the location of the inner boundary.
In models that use a Newtonian gravitational potential (such as IA and SPB) the location of the
inner boundary is not an interesting parameter in the sense that there is no physical lengthscale that
one can compare rin to: it is simply a scaling parameter. In models that use a pseudo-Newtonian
potential, however, there is a feature (a “pit”) in the potential on a lengthscale GM/c2 . Starting
with our fiducial model, then, what is the effect of shifting rin ?
Our fiducial run has rin = 2.7GM/c2 . This may be compared with Run B, which has rin =
6GM/c2 . Figure 2.4 compares the accretion rates in the two runs. Evidently there are two changes
in the solution. First, the time-averaged accretion rates differ by a large factor. The mean restmass accretion rate is factor of 3 lower in Run B than Run A. The reduced eigenvalues e and l
also differ by about 50% (see Table 2). Second, the time variation of the accretion rates differs,
with Run B showing far more short-timescale variations. The short-timescale variations are due to
the interaction of unstable convective modes with the boundary conditions. Inspection of the runs
reveals an enhancement of convection and turbulence near rin in Run B.
The differences between Run B and Run A are caused by the boundary location. Gradual
variation of rin (in models not discussed in detail here) reveals that if the flow on the inner boundary
is everywhere and always supersonic, then the solution is similar to Run A. If the flow is subsonic,
then the solution exhibits artifacts like those seen in Run B.
Evidently forcing the flow to be supersonic on the inner boundary causally disconnects the flow
from the boundary.
3
This eliminates nonphysical reflection of linear and nonlinear waves from the
boundary and renders the precise implementation of the numerical boundary conditions irrelevant.
We do not want the reader to think that this problem arises because we happened to choose the
wrong numerical implementation of the boundary conditions. Our implementation is the standard
ZEUS outflow boundary condition, and it is widely used in astrophysical problems. While it may
be possible to implement more transparent boundary conditions in the context of other numerical
schemes, a survey of the numerical literature shows that in multiple dimensions this is an area of
3
Although the viscous fluid equations of motion are not hyperbolic, and the flow in the supersonic region is in
principle in causal contact with the rest of the flow, the coupling is exponentially weak.
74
Figure 2.4 The effect of moving the inner boundary on the accretion rates of mass, angular momentum, and energy (Run A vs. Run B). The top panel shows Ṁ /Ṁinj , the middle panel Ė/(Ṁ c2 ),
and the bottom panel L̇ c/(GM Ṁ ). The solid curve is Run A, which has rin = 2.7GM/c2 . The
dashed curve is Run B, which has rin = 6GM/c2 . Evidently Run B has a different variability
structure and different time averaged values for the accretion rates. The relatively rapid and highamplitude variations in Run B are due to nonphysical interactions with the inner radial boundary.
Only by ensuring a supersonic flow (as in Run A) can one avoid these nonphysical effects.
75
active research (Roe, 1989; Karni, 1991; Dedner et al., 2001; Bruneau and Creus, 2001), and that
no general solution to the problem has been found.
Furthermore, a simple example shows that no local extrapolation scheme can work for all
accretion problems. Consider a numerical model of a steady spherical inflow (Bondi flow) in a
gravitational potential Ψ(r). Let us suppose that we are primarily interested in accurately measuring Ṁ . We know from the analytic solution of the problem that Ṁ depends on the shape of
the potential everywhere outside the sonic point. If we place the inner boundary rin outside the
sonic point and use a local extrapolation scheme, we won’t always get the correct answer because
the local extrapolation doesn’t have any information about the shape of the potential between rin
and the sonic point. Put differently, one can’t determine a global solution from local extrapolation
at the boundary. The key point is that, while aspects of the solution may be accurate, Ṁ (and L̇
and Ė) are sensitive to the boundary conditions.
It is worth noting that the outer boundary is always in causal contact with the flow, but does
not cause the same type of artifacts as the inner boundary. Experiments show that the flow is
qualitatively insensitive to the location and implementation of the outer boundary condition. The
time averaged Ṁ , however, is sensitive to both rout and rinj /rout . The time averaged l and e scale
out this mass dependence and so are qualitatively and quantitatively insensitive to both rout and
rinj /rout .
It is also worth noting the effects of changing the gravitational potential. Run C (identical to
IA99 Model 5) is identical to Run B except that the potential is now Newtonian. It is qualitatively
similar to Run B, but Ṁ is now a factor of 5 lower than Run A. Run C also has the property that
l oscillates about 0.0. This is a problem if the focus of the simulation is measuring Ṁ or L̇.
To summarize: the location of the inner radial boundary can determine the character of the flow.
If the flow is everywhere and always supersonic (or super-fast-magnetosonic in MHD) on the inner
boundary then boundary-related corruption of the flow is impossible. Since it is computationally
expensive to place the inner boundary very deep in the potential (for our model, the time step
dt ∼ (rin − 2GM/c2 )), the optimal location for the inner boundary is just inside the radius where
the radial Mach number always exceeds 1.
The results of IA and SPB do not focus on the time-dependence of the accretion values, so
much of their discussion is unaffected by their treatment of the inner radial boundary. As discussed
below, there are small changes related to the appearance of outflows.
2.5.3
Comparison of Torus and Injection Models
The torus and injection methods represent sharply different approaches to studying accretion flows.
The equilibrium torus presents a physically well-posed problem, but the accretion flow is transient:
no steady state can be achieved. The injection method reaches a quasi-steady state, but much of
the computational domain is wasted on evolving the injection region, which has no astrophysical
analog: it is nonphysical. It is natural to ask whether these two widely used schemes for supplying
76
mass can be made comparable or used to measure any of the same quantities.
We selected two runs, E (torus) and F (injection), that had similar mass distributions in an
evolved state. The torus run was studied at a time when Ṁ was close to its maximum. Run F’s
Ṁ is a factor of 10 larger than Run E’s. This difference might have been anticipated from the
sensitivity of the injection run to rout and rinj /rout : runs in which mass is concentrated closer to
the outer boundary tend to have lower accretion rates because more of the mass escapes through
the outer boundary. The time averaged rest-mass accretion rate is therefore strongly dependent on
the method of mass supply.
The energy and angular momentum accretion rates per unit mass are, however, insensitive to
the experimental design. We find that e and l differ by less than 3% in Runs E and F (see Table
2.1 and Table 2.2). These quantities are apparently set by conditions near the inner boundary (the
ISCO for the Pseudo-Newtonian potential), and can be measured in either type of experiment.
2.5.4
Other Parameters
We have varied rinj and rout /rinj , and as reported above, these strongly affect the time-averaged
value of Ṁ . The sense of the effect is that a simulation with a larger rout /rinj loses less matter
through the outer boundary, and this results in more matter streaming back into the black hole
(by up to a factor of 10). The qualitative nature of the flow, however, is roughly independent of
rout /rinj in that, e.g., the temporal power spectrum of Ṁ is similar. The qualitative nature of the
flow is dependent on rinj . If one fixes rout /rinj and all other parameters, the range in α where
unipolar outflows are observed tends to become smaller and disappears altogether for rinj as small
as 40GM/c2 .
The dependence of accretion models similar to ours on α has already been investigated by IA.
They find that the flow changes from turbulent to laminar as α is increased and the higher viscosity
damps modes of increasing lengthscale. IA find that α . 0.03 the flow is turbulent, and for α & 0.3
the flow is laminar. For 0.03 < α < 0.3 the flow exhibits a “unipolar” outflow. Our results are in
agreement with IA. However, our models with a pseudo-Newtonian potential and super-sonic flow
at the inner radial boundary show a slight shift in the values of α that exhibit unipolar outflows.
We did find a critical value of α ≈ 0.5 above which no supersonic flow at rin could be achieved
due to viscous heating, at least for γ = 3/2 and γ = 5/3 and for rin ≥ 2.1GM/c2 . Smaller values of
rin were not computationally practical. This high α is typically associated with a bipolar outflow,
as seen by IA. Even in this case, however, a choice of rin = 2.1GM/c2 instead of rin = 6GM/c2
leads to a qualitatively different profile for the flow. The flow with smaller rin = 2.1GM/c2 has a
bipolar outflow starting at larger radius (10GM/c2 ) rather than immediately on the boundary as
with rin = 6GM/c2 , and the rest-mass accretion rate increases by a factor of 3.
Finally, we studied the dependence of the results on numerical resolution. We find that Nr ×
Nθ = 108 × 50 is sufficient at α = 0.1 to resolve the shortest wavelength convective mode. Also, we
chose our value of r0 to agree with SPB99’s torus models. We experimented with varying r0 and
77
find that, all things being equal, smaller r0 gives more laminar flow.
2.6
VHD Summary
Work in this field will shortly focus on global MHD models in pseudo-Newtonian potentials and
in full general relativity. In our view it is useful to understand the solution space for physically
and numerically simpler viscous models before turning to MHD. It is even possible, as Stone and
Pringle (2001) have claimed, that viscous hydrodynamics provides a crude approximation to the
MHD results. In any event, this investigation provides a preview of some of the experimental issues
that will play a role in most future global numerical investigations of accretion flows.
This investigation was initially motivated by a desire to understand whether the differences
between earlier global viscous hydrodynamics simulations performed by IA and SPB were caused
by differences in experimental design or numerical method. IA and SPB reported different degrees
of convective turbulence in their models and found different radial scalings for vertically averaged
quantities such as temperature and density. Using a single code, we were able to reproduce both
sets of results. We conclude that the differences are due to experimental design.
We also found, while reproducing IA and SPB’s results, that some aspects of our solutions were
sensitive to the numerical treatment of the region close to the inner boundary in models that use
a pseudo-Newtonian potential. In particular, l and e, the specific angular momentum and energy
of accreted material, are strongly dependent on rin , the location of the inner boundary. When the
flow is supersonic at rin the location of the boundary does not affect l and e. But when the flow is
subsonic at rin the flow interacts strongly with the numerical boundary condition. This produces
spurious outflow events and makes l and e dependent on rin . Evidently for accurate measurement
of these quantities it is necessary to isolate the numerical boundary condition behind a sonic
transition that is located within the computational domain; one must place the inner boundary
condition inside a “sound horizon”.
We are not saying that all models that lack a sonic transition in the computational domain
are fatally flawed. Whether the treatment of the inner boundary condition is problematic or not
depends on what is being measured. For the nonlinear eigenvalues L̇, Ė, and Ṁ that we have
focused on here, however, the treatment of the inner boundary condition is crucial. Furthermore,
the only guarantee that the inner boundary condition is not governing the solution is to isolate it
behind a sonic transition; this is the only completely safe choice.
The location of the inner boundary may prove even more crucial in MHD models. Much of
the character of the flow is determined in the turbulent, energetically important region of the flow
just outside the fast magnetosonic transition, just as the region immediately outside the sonic
transition determines the nature of the viscous flows described in this paper. We have performed
some preliminary numerical tests and find that, as in the viscous flow, l and e for an MHD flow
are sensitive to the treatment of the inner boundary. For various reasons it may prove difficult
to achieve a fast magnetosonic transition in the computational domain; accurate treatment of the
78
inner boundary condition may require fully (general) relativistic MHD.
We have also compared two commonly used experimental designs for black hole accretion flow
studies: models that begin with an equilibrium torus, and models that continuously inject fluid
onto the grid. The choice between these models is to some degree a matter of taste. We find the
equilibrium torus slightly easier to initialize and analyze. Remarkably, the two different approaches
produce indistinguishable measurements for l and e, the specific angular momentum and energy of
the accreted material.
A parallel, viscous, axisymmetric hydrodynamics code based on that used in this paper can be
found at http://kerr.physics.uiuc.edu.
2.7
Global 2D MHD Simulations
Prior analytic calculations find that the magnetic stress −Br Bφ → 0 near the innermost stable
circular orbit (ISCO) of a black hole. This has bearing on whether magnetic stresses can exert a
torque between the hole and disk. Thus, at least two interesting questions regarding a magnetic
disk are: 1) does the magnetic stress approach 0 near the ISCO? and 2) is the polar field strong
enough for the BZ effect to be relevant?
With a global 2D MHD code we have been able to reproduce prior work by Hawley (2000)
(hereafter H00) and Stone and Pringle (2001) (hereafter SP01). This work involves an equilibrium
torus (in HD) threaded by a field where the vector potential component Aφ ∝ ρ, so that the
field resides totally within the torus and is axisymmetric. H00, SP01, and we all find there is
nothing special about the ISCO, contrary to most prior analytic calculations (except in, for example,
Gammie (1999)). The magnetic stress continues to grows as r → 2GM/c2 directly through the
ISCO, and is relatively large compared to the Reynolds stress −vr vφ there as well.
To discover whether the polar field is substantial enough to make the BZ effect a significant
source of energy, we need to simulate both the pole and the disk. This is not possible for H00 since
his computational grid excises the polar region, and SP01 do not report on this. We find that the
polar field surrounding the black hole is relatively strong compared to that in the disk. This field
is supported by the disk and hole interaction as the disk continually pumps and compresses field
into the black hole.
Figure 3 shows the field lines of a 2D axisymmetric, spherical polar coordinate simulation after
about 8 orbits (at the t = 0 torus density maximum) once full non-linear MRI turbulence has
developed. We measure the polar magnetic field strength to be 2-4× that of the average disk field
at the inner radial edge, contrary to estimates by Ghosh and Abramowicz (1997); Livio et al. (1999).
With such a strong polar field, it is plausible that the BZ effect generates a significant fraction of
AGN power output.
One can compute the power output of the BZ effect using equation 1.8. I estimate the field
strength B by assuming my computer simulation’s mass accretion rate Ṁ is equal to the Eddington
¡ ¢−1 M
value (Ṁ ∼ 1.3 × 1038 ηc2
( M¯ ) erg s−1 ), where η is order 10%. In the simulation the accretion
79
Figure 2.5 Field line snapshot after 8 orbits (at the t = 0 torus density maximum at r = 9.4GM/c2 )
for a global 2D MHD simulation. Full non-linear turbulence drives the accretion process. Note that
the polar field is essentially radial while the accretion disk is dominated by turbulence generated
by the MRI.
80
rate scales like Ṁ ∝
ρ0 (rh /2)3
(GM/c3 )
and the field strength scales like B ∝
p
ρ0 c2 , where ρ0 is the typical
density of the initial mass source. Thus, Ṁ can be used to solve for ρ0 in cgs units which is then
used to solve for B in cgs units. This gives P ∼ 4 × 1045 erg s−1 .
By limiting the calculation to 2D, the field is forced to avoid twisting around the pole or diffusing
around the pole, so we may be overestimating the polar field. Also, as stated by the anti-dynamo
theorem (Cowling, 1934), an isolated 2D axisymmetric system cannot sustain a magnetic dynamo.
Thus, the next step in modeling the black hole environment is to perform a global 3D calculation.
2.8
Global 3D MHD Simulation
A global 3D calculation involves about twice the number of computational operations per grid zone,
and requires N 3 zones instead of N 2 , so a 3D calculation can be quite expensive. One would prefer
to use spherical polar coordinates since the black hole is spherical, and because in these coordinates
the ZEUS algorithm conserves angular momentum well.
We performed global 3D MHD calculations in spherical polar coordinates and found that the
singularity at the poles leads to several problems. The most significant problem is that grid zones
near the polar axis become smaller since zones have a side with arc length of r sin θdφ. In order
to guarantee a stable algorithm, the Courant condition requires dt ∝ dx/v for a grid size dx and
fluid speed v. Since the grid size is dx ∝ θ near the θ = 0 pole, and the polar region is nearly a
vacuum with potentially strong fields (hence large va ), this leads to a requirement that dt ∝ θ/va
and severely limits the time step.
We therefore decided to use a Cartesian grid. This presents two problems: 1) angular momentum
is no longer well conserved by the algorithm ; and 2) The grid geometry forces an m = 4 mode
(spherical mode for the φ direction) in the solution. We try to diminish both of these problems
by cutting an approximation to a sphere out of the Cartesian mesh at the outer boundary and
around the black hole. Preliminary results show that the time step is well behaved and the m = 4
mode is minimal. Angular momentum conservation appears acceptable, but quantitative analysis
is pending.
Questions we will attempt to answer with this code are: 1) does an accretion flow generate a
magnetized outflow? ; 2) how does Bremsstrahlung cooling affect a magnetized accretion disk? and
3) what is the power output due to the BZ effect?
81
3
HARM: A Numerical Scheme for
General Relativistic
Magnetohydrodynamics
3.1
Summary of Chapter
We describe a conservative, shock-capturing scheme for evolving the equations of general relativistic
magnetohydrodynamics. The fluxes are calculated using the Harten, Lax, and van Leer scheme. A
variant of constrained transport, proposed earlier by Tóth, is used to maintain a divergence free
magnetic field. Only the covariant form of the metric in a coordinate basis is required to specify
the geometry. We describe code performance on a full suite of test problems in both special and
general relativity. On smooth flows we show that it converges at second order. We conclude by
showing some results from the evolution of a magnetized torus near a rotating black hole.1
3.2
Introduction
Quasars, active galactic nuclei (AGN), X-ray binaries, gamma-ray bursts, and core-collapse supernovae are all likely powered by a central engine subject to strong gravity, strong electromagnetic
fields, and rotation. For convenience, we will refer to this class of objects as relativistic magnetorotators (RMRs). RMRs are among the most luminous objects in the universe and are therefore the
center of considerable theoretical attention. Unfortunately the governing physical laws for RMRs,
while well known, are nonlinear, time-dependent, and intrinsically multidimensional. This has
stymied development of a first-principles theory for their evolution and observational appearance
and strongly motivates a numerical approach.
To fully understand RMR structure, one must be able to follow the interaction of a nonMaxwellian plasma with a relativistic gravitational field, a strong electromagnetic field, and possibly
a strong radiation field as well. This general problem remains beyond the reach of today’s algorithms
and computers. A useful first step, however, might be to study these objects in a nonradiating
magnetohydrodynamic (MHD) model. In this case the plasma can be treated as a fluid, greatly
reducing the number of degrees of freedom, and the radiation field can be ignored. The relevance
of this approximation must be evaluated in astrophysical context and will not be considered here.
1
Published in ApJ Volume 589, Issue 1, pp. 444-457. Reproduction for this thesis is authorized by the copyright
holder. Initial HARM work as in paper primarily performed by CFG, later efforts primarily by JCM, and GT provided
numerical advice.
82
We were motivated, therefore, to develop a method for integrating the equations of ideal, general
relativistic MHD (GRMHD), and that method is described in this paper. This is in a sense welltrodden ground: many schemes have already been developed for relativistic fluid dynamics. What
has not existed until recently is a scheme that: (1) includes magnetic fields; (2) has been fully verified
and convergence tested; (3) is stable and capable of integrating a flow over many dynamical times.
A pioneering GRMHD code has been developed by Koide and collaborators (see, e.g. Koide et al.
2002, Koide et al. 1999, Koide et al. 2000, Meier et al. 2001). Our code differs in that we have
subjected it to a fuller series of tests (described in this paper), we can perform longer integrations
than the rather brief simulations described in the published work of Koide’s group, and our code
explicitly maintains the divergence-free constraint on the magnetic field. A GRMHD Godunov
scheme based on a Roe-type approximate Riemann solver has been developed by Komissarov and
described in a conference proceeding (Komissarov, 2001).
A rather complete review of numerical approaches to relativistic fluid dynamics is given by
Martı́ and Müller (2003) and Font (2003). The first numerical GRMHD scheme that we are aware
of is by Wilson (1977), who integrated the GRMHD equations in axisymmetry near a Kerr black
hole. While it was recognized throughout the 1970s that relativistic MHD would be relevant to
problems related to black hole accretion (particularly following the seminal work of Blandford
and Znajek (1977) and Phinney (1983); see also Punsly (2001)), no further work appeared until
Yokosawa (1993). The discovery by Balbus and Hawley (1991) that magnetic fields play a crucial
role in regulating accretion disk evolution (reviewed in Balbus and Hawley (1998)) and the absence
of purely hydrodynamic means for driving accretion in disks (Balbus and Hawley, 1998) further
motivated the development of relativistic MHD schemes. More recently there have been several
efforts to develop GRMHD codes, including the already-mentioned work by Koide and collaborators
and by Komissarov. A ZEUS-like scheme for GRMHD has also been developed and is described in
a companion paper (De Villiers and Hawley, 2002).
Special relativistic MHD (SRMHD) is the foundation for any GRMHD scheme, although there
are nontrivial problems in making the transition to full general relativity. SRMHD schemes have
been developed by vanPutten (1993); Balsara (2001); Koldoba et al. (2002); Komissarov (1999)
and Del Zanna et al. (2003). We were particularly influenced by the clear development of the
fundamental equations in Komissarov (1999) for his Godunov SRMHD scheme based on a Roetype approximate Riemann solver, and by the work of Del Zanna and Bucciantini (2002) and Del
Zanna et al. (2003) who chose to use the simple approximate Riemann solver of Harten et al. (1983)
in their special relativistic hydrodynamics and SRMHD schemes, respectively.
Our numerical scheme is called HARM, for High Accuracy Relativistic Magnetohydrodynamics.2
In the next section we develop the basic equations in the form used for numerical integration in
HARM (§2). In §3 we describe the basic algorithm. In §4 we describe the performance of the code
on a series of test problems. In §5 we describe a sample evolution of a magnetized torus near a
rotating black hole.
2
Also named in honor of R. Härm, who with M. Schwarzschild was a pioneer of numerical astrophysics.
83
3.3
A GRMHD Primer
The equations of general relativistic MHD are well known, but for clarity we will develop them here
in the same form used in numerical integration. Unless otherwise noted c = 1 and we follow the
notational conventions of Misner et al. (1973), hereafter MTW. The reader may also find it useful
to consult Anile (1989).
The first governing equation describes the conservation of particle number:
(nuµ );µ = 0.
(3.1)
Here n is the particle number density and uµ is the four-velocity. For numerical purposes we rewrite
this in a coordinate basis, replacing n with the “rest-mass density” ρ0 = mn, where m is the mean
rest-mass per particle:
√
1
√ ∂µ ( −g ρ0 uµ ) = 0.
−g
(3.2)
Here g ≡ Det(gµν ).
The next four equations express conservation of energy-momentum:
T µ ν;µ = 0,
(3.3)
where T µ ν is the stress-energy tensor. In a coordinate basis,
∂t
¡√
¡√
¢
¢ √
−g T t ν = −∂i
−g T i ν + −g T κ λ Γλ νκ ,
(3.4)
where i denotes a spatial index and Γλ νκ is the connection.
The energy-momentum equations have been written with the free index down for a reason.
Symmetries of the metric give rise to conserved currents. In the Kerr metric, for example, the
axisymmetry and stationary nature of the metric give rise to conserved angular momentum and
energy currents. In general, for metrics with an ignorable coordinate xµ the source term on the
right hand side of the evolution equation for Tµt vanish. These source terms do not vanish when
the equation is written with both indices up.
The stress-energy tensor for a system containing only a perfect fluid and an electromagnetic
field is the sum of a fluid part,
µν
Tfluid
= (ρ0 + u + p)uµ uν + pg µν ,
(3.5)
(here u ≡ internal energy and p ≡ pressure) and an electromagnetic part,
1
µν
TEM
= F µα Fαν − g µν Fαβ F αβ .
4
(3.6)
Here F µν is the electromagnetic field tensor (MTW: “Faraday”), and for convenience we have
84
absorbed a factor of
√
4π into the definition of F .
The electromagnetic portion of the stress-energy tensor simplifies if we adopt the ideal MHD
approximation, in which the electric field vanishes in the fluid rest frame due to the high conductivity
of the plasma (“E + v × B = 0”). Equivalently the Lorentz force on a charged particle vanishes in
the fluid frame:
uµ F µν = 0.
(3.7)
It is convenient to define the magnetic field four-vector
1
bµ ≡ ²µνκλ uν Fλκ ,
2
(3.8)
where ² is the Levi-Civita tensor. Recall that (following the notation of MTW) ²µνλδ = − √1−g [µνλδ],
where [µνλδ] is the completely antisymmetric symbol and = 0, 1, or −1. These can be combined
(with the aid of identity 3.50h of MTW):
F µν = ²µνκλ uκ bλ .
(3.9)
Substitution and some manipulation (using identities 3.50 of MTW and bµ uµ = 0; the latter follows
from the definition of bµ and the antisymmetry of F ) yields
1
µν
TEM
= b2 uµ uν + b2 g µν − bµ bν .
2
(3.10)
Notice that the last two terms are nearly identical to the nonrelativistic MHD stress tensor, while
the first term is higher order in v/c. To sum up,
1
µν
TMHD
= (ρ0 + u + p + b2 )uµ uν + (p + b2 )g µν − bµ bν
2
(3.11)
is the MHD stress-energy tensor.
The electromagnetic field evolution is given by the source-free part of Maxwell’s equations
Fµν,λ + Fλµ,ν + Fνλ,µ = 0.
(3.12)
The rest of Maxwell’s equations determine the current
J µ = F µν ;ν ,
(3.13)
and are not needed for the evolution, as in nonrelativistic MHD.
Maxwell’s equations can be written in conservative form by taking the dual of eq.(3.12):
F ∗µν ;ν = 0.
(3.14)
∗ = 1²
κλ is the dual of the electromagnetic field tensor (MTW: “Maxwell”). In ideal
Here Fµν
2 µνκλ F
85
MHD
F ∗µν = bµ uν − bν uµ ,
(3.15)
which can be proved by taking the dual of eq.(3.9).
The components of bµ are not independent, since bµ uµ = 0. Following, e.g., Komissarov (1999),
it is useful to define the magnetic field three-vector B i = F ∗it . In terms of B i ,
bt = B i uµ giµ ,
(3.16)
bi = (B i + bt ui )/ut .
(3.17)
The space components of the induction equation then reduce to
√
√
∂t ( −gB i ) = −∂j ( −g (bj ui − bi uj ))
(3.18)
and the time component reduces to
√
1
√ ∂i ( −g B i ) = 0,
−g
(3.19)
which is the no-monopoles constraint. The appearance of B i in these last two equations are what
motivates the introduction of the field three-vector in the first place.
To sum up, the fundamental equations as used in HARM are: the particle number conservation
equation (3.2); the four energy-momentum equations (3.4), written in a coordinate basis and using
the MHD stress-energy tensor of equation (3.11); and the induction equation (3.18), subject to the
constraint (3.19). These hyperbolic
3
equations are written in conservation form, and so can be
solved numerically by well-known techniques.
3.4
Numerical Scheme
There are many possible ways to numerically integrate the GRMHD equations. A first, zerothorder choice is between conservative and nonconservative schemes. Nonconservative schemes such
as ZEUS (Stone and Norman, 1992) have enjoyed wide use in numerical astrophysics. They permit
the integration of an internal energy equation rather than a total energy equation. This can be
advantageous in regions of a flow where the internal energy is small compared to the total energy
(highly supersonic flows), which is a common situation in astrophysics. A nonconservative scheme
for GRMHD following a ZEUS-like approach has been developed and is described in a companion
paper (De Villiers and Hawley, 2002).
We have decided to write a conservative scheme. One advantage of this choice is that in
one dimension, total variation stable schemes are guaranteed to converge to a weak solution of the
equations by the Lax-Wendroff theorem (Lax and Wendroff, 1960) and by a theorem due to LeVeque
3
The GRMHD equations exhibit the same degeneracies as the nonrelativistic MHD equations.
86
(1998). While no such guarantee is available for multidimensional flows, this is a reassuring starting
point. Furthermore, one is guaranteed that a conservative scheme in any number of dimensions
will satisfy the jump conditions at discontinuities. This is not true in artificial viscosity based
nonconservative schemes, which are also known to have trouble in relativistic shocks (Norman and
Winkler, 1986). Conservative schemes for GRMHD have also been developed by Komissarov (2001)
and by Koide et al. (1999).
A conservative scheme updates a set of “conserved” variables at each timestep. Our vector of
conserved variables is
U≡
√
−g(ρ0 ut , Ttt , Tit , B i ).
(3.20)
These are updated using fluxes F. We must also choose a set of “primitive” variables, which are
interpolated to model the flow within zones. We use variables with a simple physical interpretation:
P = (ρ0 , u, v i , B i ).
(3.21)
Here v i = ui /ut is the 3-velocity.4 The functions U(P) and F(P) are analytic, but the inverse
operations (so far as we can determine) are not.
To evaluate U(P) and F(P) one must find
5
ut
There is also no simple expression for F(U).
and bµ from v i and B i . To find ut , solve the
quadratic equation gµν uµ uν = −1. Next use equations (3.16) and (3.17) to find bµ ; these require
only multiplications and additions. The remainder of the calculation of U(P) and F(P) requires
raising and lowering of indices followed by direct substitution in equation (3.11) to find the components of the MHD stress-energy tensor.
Since we update U rather than P, we must solve for P(U) at the end of each timestep. We
use a multidimensional Newton-Raphson routine with the value of P from the last timestep as
an initial guess. Since B i can be obtained analytically, only 5 equations need to be solved. The
Newton-Raphson method requires an expensive evaluation of the Jacobian ∂U/∂P. In practice we
evaluate the Jacobian analytically. It is possible to evaluate the Jacobian by numerical derivatives,
but this is both expensive and a source of numerical noise.
The evaluation of P(U) is at the heart of our numerical scheme; the procedure must be robust.
We have found that it is crucial that the errors (differences between the current and target values
of U) used to evaluate convergence in the Newton-Raphson scheme be properly normalized. We
√
normalized the errors with −gρ0 ut .
To evaluate F we use a MUSCL type scheme with “HLL” fluxes (Harten et al., 1983). The
fluxes are defined at zone faces. A slope-limited linear extrapolation from the zone center gives
PR and PL , the primitive variables at the right and left side of each zone interface. We have
implemented the monotonized central (“Woodward”, or “MC”) limiter, the van Leer limiter, and
the minmod limiter; unless otherwise stated, the tests described here use the MC limiter, which is
4
We initially used ui as primitive variables, but the inversion ut (ui ) is not always single-valued, e.g. inside the
ergosphere of a black hole. That is, there are physical flows with the same value of ui but different values of ut .
5
Del Zanna et al. (2003) have found that the inversion P(U) can be reduced analytically to the solution of a
single, nonlinear equation.
87
the least diffusive of the three.
From PR , PL , calculate the maximum left and right-going wave speeds c±,R , c±,L , and the fluxes
FR = F(PR ) and FL = F(PL ). Let cmax ≡ MAX(0, c+,R , c+,L ) and cmin ≡ −MIN(0, c−,R , c−,L ),
then the HLL flux is
F=
cmin FR + cmax FL − cmax cmin (UR − UL )
cmax + cmin
(3.22)
If cmax = cmin , the HLL flux becomes the so-called local Lax-Friedrichs flux.
3.4.1
Constrained Transport
The pure HLL scheme will not preserve any numerical representation of ∇·B = 0. An incomplete list
of options for handling this constraint numerically includes: (1) ignore the production of monopoles
by truncation error and hope for the best (in our experience this causes the scheme to fail in any
complex flow); (2) introduce a divergence-cleaning step (this entails solving an elliptic equation
at each timestep); (3) use an Evans and Hawley (1988) type constrained transport scheme (this
requires a staggered mesh, so that the magnetic field components are zone face centered); (4)
introduce a diffusion term that causes numerically generated monopoles to diffuse away (Marder
(1987); this typically leaves a monopole field with rms value somewhat larger than the truncation
error).
We have chosen a fifth option, a version of constrained transport that can be used with a
zone-centered scheme. This idea was introduced by one of us in Toth (2000), where it is called
the flux-interpolated constrained transport (or “flux-CT”) scheme 6 . It preserves a numerical
representation of ∇ · B = 0 by smoothing the fluxes with a special operator. The disadvantage of
this method is that it is more diffusive than the “bare”, unconstrained scheme. The advantage is
that it is extremely simple.
To clarify how zone-centered constrained transport works we now give a specific example for
a special relativistic problem in Cartesian coordinates t, x, y. To fix notation, write the induction
equation as
∂t B i = −∂j Fij ,
where we have used
(3.23)
√
−g = 1 and the fluxes Fij are
Fxx = 0
Fyy = 0
Fxy = by ux − bx uy
(3.24)
Fyx = bx uy − by ux = −Fxy
Notice that the fluxes are centered at different locations on the grid: F x fluxes live on the x face of
each zone at grid location i − 1/2, j (we use i, j to denote the center of each zone), while F y fluxes
6
We have also experimented with Tóth’s “flux-CD”, which preserves a different representation of ∇ · B = 0. This
appears to be slightly less robust. It also has a larger effective stencil.
88
live on the y face at i, j − 1/2. The smoothing operator replaces the numerical (HLL-derived) Fij
with F̃ij , defined by
F̃xx (i − 1/2, j) = 0
F̃yy (i, j − 1/2) = 0 h
F̃xy (i, j − 1/2) = 81 2Fxy (i, j − 1/2)
+Fxy (i + 1, j − 1/2) + Fxy (i − 1, j − 1/2)
−Fyx (i − 1/2, j) − Fyx (i + 1/2, j)
−Fyx (i − 1/2, j − 1) − Fyx (i + 1/2, j − 1)
h
F̃yx (i − 1/2, j) = 18 2Fyx (i − 1/2, j)
i
(3.25)
+Fyx (i − 1/2, j + 1) + Fyx (i − 1/2, j − 1)
−Fxy (i, j − 1/2) − Fxy (i, j + 1/2)
−Fxy (i − 1, j − 1/2) − Fxy (i − 1, j + 1/2)
i
It is a straightforward but tedious exercise to verify that this preserves the following corner-centered
numerical representation of ∇ · B:
∇ · B = (B x (i, j) + B x (i, j − 1) − B x (i − 1, j) − B x (i − 1, j − 1)) /(2∆x)
+ (B y (i, j) + B y (i − 1, j) − B y (i, j − 1) − B y (i − 1, j − 1)) /(2∆y),
(3.26)
where ∆x and ∆y are the grid spacing.
3.4.2
Wave Speeds
The HLL approximate Riemann solver does not require eigenvectors of the characteristic matrix
(as would a Roe-type scheme), but it does require the maximum and minimum wave speed (eigenvalues). These wave speeds are also required to fix the timestep via the Courant conditions. The
relevant speed is the phase speed “ω/k” of the wave, and it turns out that only speeds for waves
with wavevectors aligned along coordinate axes are required. Suppose, for example, that one needs
to know how rapidly signals propagate in the fluid along the x1 direction. First, find a wavevector
kµ = (−ω, k1 , 0, 0), that satisfies the dispersion relation for the mode in question: D(kµ ) = 0. Then
the wave speed is simply ω/k1 .
The dispersion relation D(kµ ) = 0 for MHD waves has a simple form in a comoving frame.
= γp/w, (the last holds only if
In terms of the relativistic sound speed c2s = (∂(ρ0 + u)/∂p)−1
s
√
p = (γ − 1)u) and the relativistic Alfvén velocity vA = B/ E, where E = b2 + w and w ≡ ρ0 + u + p,
the dispersion relation is
¡
¢
ω ω 2 − (k · vA )2 ×
¢
¢
¡ 4
¡
2 + c2 (1 − v2 /c2 )) + c2 (k · v )2 /c2 + k 2 c2 (k · v )2 = 0,
ω − ω 2 k 2 (vA
A
A
s
s
s
A
(3.27)
Here c is the (temporarily reintroduced) speed of light. The first term is the zero frequency entropy
89
mode, the second is the Alfvén mode, and the third contains the fast and slow modes. The eighth
mode is eliminated by the no-monopoles constraint.
√
√
The relativistic sound speed asymptotes to c γ − 1 = c/ 3 for γ = 4/3, and the Alfvén
speed asymptotes to c. In the limit that B 2 /ρ0 À 1 and p/ρ0 6À 1, the GRMHD equations
are a superset of the time-dependent, force-free electrodynamics equations recently discussed by
Komissarov (2002c); these contain fast modes and Alfvén modes that move with the speed of
light. They are indistinguishable from vacuum electromagnetic modes only when their wavevector
is oriented along the magnetic field.
To find the maximum wave speeds we need to evaluate the comoving-frame dispersion relation
for the fast wave branch from coordinate frame quantities. This is straightforward because the
dispersion relation depends on scalars, which can be evaluated in any frame: ω = kµ uµ ; k 2 = Kµ K µ ,
2 = b bµ /E;
where Kµ = (gµν +uµ uν )k ν is the part of the wavevector normal to the fluid 4-velocity; vA
µ
√
µ
(k · vA ) = kµ b / E. The relevant portion of the dispersion relation (for fast and slow modes) is
thus a fourth order polynomial in the components of kµ . This can be solved either analytically or
by standard numerical methods. The two fast mode speeds are then used as cmax and cmin in the
HLL fluxes.
We have found it convenient to replace the full dispersion relation by an approximation:
2
2
ω 2 = (vA
+ c2s (1 − vA
/c2 ))k 2 .
(3.28)
This overestimates the maximum wavespeed by a factor ≤ 2 in the comoving frame. The maximum
error occurs for k k vA , vA /cs = 1, and vA ¿ c, and it is usually much less, particularly if the fluid
is moving super-Alfvénically with respect to the grid. This approximation is convenient because it
is quadratic in kµ , and so can be solved more easily.
3.4.3
Implementation Notes
For completeness we now give some details of the implementation of the algorithm.
Time Stepping. Our scheme is made second order in time by taking a half-step from tn to
tn+1/2 , evaluating F(P(tn+1/2 )), and using that to update U(tn ) to U(tn+1 ).
Modification of Energy Equation. A direct implementation of the energy equation can be
inaccurate because the magnetic and internal energy density can be orders of magnitude smaller
than the rest mass density. To avoid this we subtract the particle number conservation equation
from the energy equation, i.e., we evolve
√
√
√
∂t ( −g(Ttt + ρ0 ut )) = −∂i ( −g(Tti + ρ0 ui )) + −gTλκ Γλtκ .
(3.29)
In the nonrelativistic limit, this procedure subtracts the rest mass energy density from the total
energy density.
90
Specification of Geometric Quantities. In two dimensions we need to evaluate gµν , g µν ,
√
and −g at four points in every grid zone (the center, two faces, and one corner) and Γµνλ at the
zone center. It would be difficult to accurately encode analytic expressions for all these quantities.
HARM is coded so that an analytic expression need only be provided for gµν ; all other geometric quantities are calculated numerically. The connection, for example, is obtained to sufficient
accuracy by numerical differentiation of the metric. This minimizes the risk of coding errors in
specifying the geometry. It also minimizes coordinate dependent code, making it relatively easy to
change coordinate systems. Minimal coordinate dependence, besides following the spirit of general
relativity, enables one to perform a sort of fixed mesh refinement by adapting the coordinates to
the problem at hand. For example, near a Kerr black hole we use log(r) as the radial coordinate instead of the usual Boyer-Lindquist r, and this concentrates numerical resolution toward the
horizon, where it is needed.
Density and Internal Energy Floors. Negative densities and internal energies are forbidden
by the GRMHD equations, but numerically nothing prevents their appearance. In fact, negative
internal energies are common in numerical integrations with large density or pressure contrast.
Following common practice, we prevent this by introducing “floor” values for the density and
internal energy. These floors are enforced after the half-step and the full step. They preserve
velocity but do not conserve rest mass or energy-momentum.
Outflow Boundary Conditions. In the rotating black hole calculations described below we
use outflow boundary conditions at the inner and outer radial boundaries. The usual implementation of outflow boundary conditions is to simply copy the primitive variables from the boundary
zones into the ghost zones. This can result in unphysical values of the primitive variables in the
ghost zones– for example, velocities that lie outside the light cone– because of variations in the
metric between the boundary and ghost zones.
We have experimented with a variety of schemes for projecting variables into the ghost zones in
the context of black hole accretion flow calculations (described in §5). We find that some are more
robust than others. The most robust extrapolates the density, internal energy, and radial magnetic
field according to
√
√
P (ghost) = P (boundary) −g(boundary)/ −g(ghost),
(3.30)
the θ and φ components of the velocity and magnetic field according to
P (ghost) = P (boundary)(1. − ∆r/r),
(3.31)
and the radial velocity according to
P (ghost) = P (boundary)(1. + ∆r/r).
(3.32)
The extrapolation of θ and φ components of magnetic field and velocity results in weak damping
91
of these components near the boundary. Slightly different choices of the extrapolation coefficients
(i.e. (1. − 2∆r/r)) are much less robust.
Performance. We have implemented both serial and parallel versions of the code. In serial
mode the code integrates the black hole accretion problem (described in §5) at ≈ 54, 000 zone cycles
per second on a 2.4 GHz Intel Pentium 4, when compiled using the Intel C compiler. The parallel
code was implemented using MPI.
3.5
Code Verification
Here we present a test suite for verifying a GRMHD code. The tests are nonrelativistic, special
and general relativistic, and one and two dimensional. The list of problems for which there are
known, exact solution is short, since exact solutions of multidimensional GRMHD problems are
algebraically complicated. This list of test problems was developed in collaboration with J. Hawley
and J.-P. de Villiers. Unless otherwise stated we set γ = 4/3 and c = 1.
3.5.1
Linear Modes
This first test considers the evolution of a small amplitude wave in two dimensions. The unperturbed
state is ρ0 = 1, p = 1, ui = 0, B y = B z = 0, B x = B0x . The basic state is parametrized
by α = (B0x )2 /(ρ0 c2 ); our fiducial test runs have α = 1. Onto this basic state we introduce a
perturbation of the form exp(ik · x − iω(k)t), where (kx , ky ) = (2π, 2π), and the amplitude is fixed
by δB y = 10−4 B0x . The computational domain is x, y ∈ [0, 1), [0, 1), and the boundary conditions
are periodic. The wave is either slow, Alfvénic, or fast.
This test exercises almost all terms in the governing equations. The numerical resolution is
(Nx , Ny ) ≡ N (5, 4) zones, and the integration runs for a single wave period 2π/ω, so that a perfect
scheme would return the simulation to its original state. We measure the L1 norm of the difference
between the final state and the initial state for each primitive variable. For example, we measure
Z
L1 (δρ0 ) =
dxdy|ρ0 (t = 0) − ρ0 (t = 2π/ω)|.
(3.33)
for the density.
All primitive variables exhibit similar convergence properties (as they must, since with the
exception of the magnetic field, they are tightly coupled together). In Figures 3.1, 3.2, and 3.3
we present the L1 norm of the error for runs using the monotonized central limiter and a Courant
number of 0.8, in addition to the results for the minmod limiter. These runs have α = 1. Figure
3.1 shows the results for the slow wave, Figure 3.2 for the Alfvén wave, and Figure 3.3 for the fast
wave. Evidently the convergence rate asymptotes to second order, although more slowly for the
minmod limiter.
The code performs similarly well over a range of α, provided only that δB 2 /2 ¿ p, which is
necessary for the wave to be in the linear regime. We have been unable to find a value of B0x where
92
Figure 3.1 The L1 norm of the error in u for a slow wave as a function of Nx for both the monotonized
central (MC) and minmod limiter. The straight lines show the slope expected for second order
convergence.
93
Figure 3.2 The L1 norm of the error in the single nonzero component of the velocity for an Alfvén
wave as a function of Nx for both the monotonized central (MC) and minmod limiter. The straight
lines show the slope expected for second order convergence.
94
Figure 3.3 The L1 norm of the error in u for a fast wave as a function of Nx for both the monotonized
central (MC) and minmod limiter. The straight lines show the slope expected for second order
convergence.
95
the code fails completely for a linear amplitude disturbance, although for very large values of B0x
the evolution becomes inaccurate because of numerical noise in the evaluation of P(U).
3.5.2
Nonlinear Waves
Komissarov (1999) has proposed a suite of one-dimensional nonlinear tests for special relativistic
MHD. Komissarov presents a total of 9 tests (see his Table 1). The nonlinear Alfvén wave (test
5), and the compound wave (test 6) cannot be reconstructed without a separate derivation of the
exact analytic solution, and we will not provide that here. For the remaining tests Komissarov’s
Table 1 contains several misprints that are corrected in Komissarov (2002b). Our code is able
to integrate each of Komissarov’s remaining 7 tests, although in some cases we must reduce the
Courant number (usually 0.8) or resort to the slightly more robust van Leer slope limiter. Tests
that required special treatment are: fast shock (Courant number = 0.5); shock tube 1 (Courant
number = 0.3, van Leer limiter); shock tube 2 (Courant number = 0.5); collision (Courant number
= 0.3; van Leer limiter).
Figures 3.4 and 3.5 show the run of ρ0 and ux , respectively, for all 7 tests. These may be
compared with Komissarov’s figures. Notice that, unlike Komissarov, we have in all cases set
Nx = 400 and x ∈ (−2, 2). We have not obtained the exact solutions used by Komissarov, but the
solutions can still be checked quantitatively. For example, the slow shock wave speed is 0.5; since
the calculation ends at t = 2 the slow shock front should be, and is, located at x ≈ 1.0. The fast
shock speed is 0.2, so at t = 2.5 the fast shock wave front should be, and is, located at x ≈ 0.5.
There are artifacts evident in the figures. In particular there is ringing near the base of the
switch-on and switch-off rarefaction waves. This is common and is seen in Komissarov’s results as
well. In addition the narrow, Lorentz-contracted shell of material behind the shock in shock tube
1 is poorly resolved; the correct shell density is ≈ 0.88 but a resolution of Nx > 800 is required to
find this result to an accuracy of a few percent. There is also a transient associated with the fast
shock that propagates off the grid and so is not visible in Figures 3.4 and 3.5.
A complete set of nonlinear wave tests for one dimensional nonrelativistic MHD was developed
by Ryu and Jones (1995) (hereafter RJ). We can run these under HARM by rescaling the speed of
light to c = 102 in code units, where all velocities in the tests are O(1). This should lead to results
that agree with RJ to O(v/c) ≈ 1%. The results can be checked quantitatively by comparison to
the tables provided by RJ.
Figure 3.6 shows our results for RJ test 5A, which is a version of the familiar Brio and Wu
(1988) magnetized shock tube test. Like RJ we use 512 zones between x = 0 and x = 1, and
we measure the results at t = 0.15. To compare to RJ quantitatively, consider ux in the region
behind the fast rarefaction wave, near x = 0.7. RJ report ux = −0.277 here, while we measure
ux = −0.273, which differs by 1%, as expected. Similar agreement is found for the other variables.
The most unsatisfactory feature of the solution is the visible post-shock oscillations. The amplitude
of these features varies, depending on the Courant number (here 0.9) and the choice of slope limiter
96
Figure 3.4 The run of density in the Komissarov nonlinear wave tests.
97
Figure 3.5 The run of ux in the Komissarov nonlinear wave tests.
98
Figure 3.6 Snapshot of the final state in HARM’s integration of Ryu & Jones test 5A (a version
of the Brio & Wu shock tube) but with c = 100. The figure shows primitive variable values at
t = 0.15. Quantitative agreement is found to within ≈ 1%, as expected.
99
(here MC).
Figure 3.7 shows our results for RJ test 2A. In the region near x = 0.6, RJ report that By =
1.4126, and we find essentially exact agreement (By = 1.41262) after averaging over a small region
near x = 0.6. This test has two pairs of closely spaced slow shocks and rotational discontinuities
that are difficult to resolve, and our scheme barely obtains the correct peak values of uy and By ,
even though there are about 20 zones inside the “horns” visible in the By panel of the figure.
3.5.3
Transport
This special relativistic test evolves a disk of enhanced density moving at an angle to the grid
until it returns to its original position.
The computation is carried out in a domain x, y ∈
[−0.5, 0.5), [−0.5, 0.5) and the boundary conditions are periodic. The initial state has v x = v y = 0.7,
or ux = uy ≈ 4.95, corresponding to ut ≈ 7.07. The initial density ρ0 = 1 except in a disk at
r < rs = 0.45, where ρ0 = 3/2 + cos(2πr/rs ). The initial pressure p = 1, and the initial magnetic
field is zero. The test is run until t = 10/7, when the system should return exactly to its initial
state.
Numerically, we use the monotonized central limiter and set the Courant number to 0.8. The
resolution is fixed so that Nx = 5Ny /4. Figure 3.8 shows the L1 norm of the error in ρ0 as a
function of x resolution. The convergence rate asymptotes to second order.
3.5.4
Orszag-Tang Vortex
The Orszag-Tang vortex (OTV) is a classic nonlinear MHD problem (Orszag and Tang, 1979). Here
we compare our code, with the speed of light set to 100 so that that it is effectively nonrelativistic,
to the output of VAC (Toth and Odstrcil, 1996), an independent nonrelativistic code developed by
one of us. The version of VAC used here is TVD-MUSCL using the monotonized central limiter.
It is dimensionally unsplit and uses a scheme similar to HARM to control ∇ · B. The problem is
integrated in the periodic domain x ∈ (−π, π], y ∈ (−π, π] from t = 0 to t = π. Our version of the
OTV has γ = 4/3, but is otherwise identical to the standard problem.
Results are shown in Figure 3.9, which shows ρ0 along a cut through the model at y = π/2 and
t = π. The resolution is 6402 . The solid line shows the results from HARM; the dashed line shows
the results from VAC. The lower solid line shows the difference between the two multiplied by 4.
Evidently our code behaves similarly to VAC on this problem.
We can quantify this by asking how the difference between the HARM and VAC solutions
changes as a function of resolution. Figure 3.10 shows the variation in the L1 norm of the difference
between the two solutions. Thus the line marked ρ0 shows
Z
dxdy|ρ0 (HARM; N2 ) − ρ0 (VAC; N2 )|
(3.34)
evaluated at t = π. The codes converge to one another approximately linearly, as expected for a
100
Figure 3.7 Snapshot of the final state in HARM’s integration of Ryu & Jones test 2A, with c = 100.
101
Figure 3.8 Convergence results for the transport test.
102
Figure 3.9 A cut through the density in the nonrelativistic Orszag-Tang vortex solution from HARM
(solid line, with c = 100), from VAC (dashed line), and 4× the difference (lower solid line) at a
resolution of 6402 .
103
Figure 3.10 Comparison of results from HARM and the nonrelativistic MHD code VAC for the
Orszag-Tang vortex. The plot shows the L1 norm of the difference between the two results as a
function of resolution for the primitive variables ρ0 (squares) and u (triangles). The straight line
shows the slope expected for first order convergence. The errors are large because they are an
integral over an area of (2π)2 .
104
flow containing discontinuities. If this study were extended to higher resolution convergence would
eventually cease because the HARM solution would differ from the VAC solution due to finite
relativistic corrections.
3.5.5
Bondi Flow in Schwarzschild Geometry
Spherically symmetric accretion (Bondi flow) in the Schwarzschild geometry has an analytic solution
(see, e.g., Shapiro and Teukolsky 1983) that can be compared with the output of our code. This
appears to be a one-dimensional test, but for HARM it is actually two dimensional. Although
the pressure is independent of the Boyer-Lindquist coordinate θ, the θ acceleration does not vanish
identically. This is because pressure enters the momentum equations through a flux (−∂θ (p sin θ) in
the Newtonian limit) and a source term (p cos θ in the Newtonian limit). Analytically these terms
cancel; numerically they produce an acceleration that is of order the truncation error.
Our test problem follows that set out in Hawley et al. (1984): we fix the sonic point rs =
8GM/c2 , Ṁ = 4πr2 ρ0 ur = −1, and γ = 4/3. The problem is integrated in the domain r ∈
(1.9, 20)GM/c2 for ∆t = 100GM/c3 . We use coordinates based on the Kerr-Schild system, whose
line element is
ds2 = −(1 − 2r/ρ0 2 )dt2 + (4r/ρ0 2 )drdt + (1 + 2r/ρ0 2 )dr2 + ρ0 2 dθ2 +
¡
¢
sin2 θ ρ0 2 + a2 (1 + 2r/ρ0 2 ) sin2 θ dφ2
2
−(4ar sin θ/ρ0
2 )dtdφ
− 2a(1 + 2r/ρ0
(3.35)
2 ) sin2 θdrdφ,
where we have set GM = c = 1. In (3.35) only ρ0 2 = r2 + a2 cos2 (θ); elsewhere ρ0 is density. In this
test, a = 0. We modify these coordinates by replacing r by x1 = log(r). The new coordinates are
implemented by changing the metric rather than changing the spacing of grid zones. We measure
the L1 norm of the difference between the initial conditions (exact analytic solution) and the final
state. The difference is taken over the inner 3/4 of the grid in each direction, thus excluding
boundary zones where errors may scale differently. This test exercises many terms in the code
because in Kerr-Schild coordinates only three of the ten independent components of the metric are
zero.
The L1 norm of the error in internal energy for the Bondi test is shown in Figure 3.11. Similar
results obtain for the other independent variables. The solution converges at second order.
3.5.6
Magnetized Bondi Flow
The next test considers a Bondi flow containing a spherically symmetric, radial magnetic field. The
solution to this problem is identical to the Bondi flow described above because the flow is along
the magnetic field, so all magnetic forces cancel exactly. This is a difficult test, however, because
numerically the magnetic terms cancel only to truncation error. This causes problems at high
magnetic field strength.
We use the same Bondi solution as in the last subsection and parameterize the magnetic field
105
Figure 3.11 Convergence results for the unmagnetized Bondi accretion test onto a Schwarzschild
black hole. The straight line shows the slope expected for second order convergence.
106
strength by b2 /ρ0 at the inner boundary. Our fiducial run has (b2 /ρ0 )(r = 1.9GM/c2 ) = 10.56.
The L1 norm of the error in the internal energy is shown in Figure 3.12. Similar results obtain for
the other independent variables. The solution converges at second order.
We have considered models with a range of (b2 /ρ0 )(rin ). Lowering (b2 /ρ0 )(rin ) produces results
similar to those in our fiducial test run. Raising (b2 /ρ0 )(rin ) first causes the code to produce
inaccurate results (at ∼ 103 , where the radial velocity profile is smoothly distorted from the true
solution) and then to fail (at ∼ 104 ). This is an example of a general problem with conservative
schemes when the basic energy density scales (rest mass, magnetic, and internal) differ by many
orders of magnitude.
3.5.7
Magnetized Equatorial Inflow in Kerr Geometry
This test considers the steady-state, magnetized inflow solutions found by Takahashi et al. (1990), as
specialized to the case of inflow inside the marginally stable orbit by Gammie (1999). This solution
exercises many of the important terms in the governing equations, in particular the interaction of
the magnetized fluid with the Kerr geometry.
We use Boyer-Lindquist coordinates to specify this problem, but the solution is integrated in the
modified Kerr-Schild coordinates described above. The flow is assumed to lie in the neighborhood
of the black hole’s equatorial plane and is thus one dimensional, much like the Weber-Davis model
for the solar wind. As above, we set GM = c = 1.
The particular inflow solution we consider is for a black hole with spin parameter a/M = 0.5.
The model has an accretion rate FM = −1 = 2πρ0 r2 ur (adopting the notation of Gammie 1999).
The magnetization parameter Fθφ = r2 B r = 0.5. The flow is constrained to match to a circular
orbit at the marginally stable orbit. This is enough to uniquely specify the flow. It follows that
(see Gammie 1999) Ftθ = ΩFθφ , where Ω is the orbital frequency at the marginally stable orbit.
For a/M = 0.5, Ω ≈ 0.10859. The solution that is regular at the fast point has angular momentum
flux FL = 2πr2 (ur uφ − br bφ ) ≈ −2.8153 and energy flux FE = 2πr2 (ur ut − br bt ) ≈ −0.90838.
The fast point is located at r ≈ 3.6167, and the radial component of the four-velocity there is
ur = −0.040547. Figure 3.13 shows the radial run of the solution.
We initialize the flow with a numerical solution that is subject to roundoff error. The nearequatorial nature of the solution is mimicked by using a single zone in the θ direction centered on
θ = π/2. The computational domain runs from 1.02× the horizon radius rh to 0.98× the radius of
the marginally stable orbit rmso . For a/M = 0.5, rh = 1.866, and rmso = 4.233. The analytic flow
model is cold (zero temperature) but we set the initial internal energy in the code equal to a small
value instead. The model is run for ∆t = 15.
Figure 3.14 shows the L1 norm of the error in ρ0 , ur , uφ , and B φ and a function of the total
number of radial gridpoints N . The straight line shows the slope expected for second order convergence. The small deviation from second order convergence at high N in several of the variables
is due to numerical errors in the initial solution, which relies on numerical derivatives (Gammie,
107
Figure 3.12 Convergence results for the magnetized Bondi accretion test onto a Schwarzschild black
hole. The straight line shows the slope expected for second order convergence.
108
Figure 3.13 The equatorial inflow solution in the Kerr metric for a/M = 0.5 and magnetization
parameter Fθφ = 0.5. The panels show density, radial component of the four-velocity in BoyerLindquist coordinates (with the square showing the location of the fast point), the φ component of
the four-velocity, and the toroidal magnetic field B φ = F φt .
109
Figure 3.14 Convergence results for the magnetized inflow solution in a Kerr metric with a/M = 0.5.
Parameters for the initial, quasi-analytic solution are given in the text. The straight line shows
the slope expected for second order convergence. The L1 error norm for each of the nontrivial
variables are shown. The small deviation from second order convergence at high resolution is due
to numerical errors in the quasi-analytic solution used to initialize the solution.
110
1999).
3.5.8
Equilibrium Torus
Our next test concerns an equilibrium torus. This class of equilibria, found originally by Fishbone
and Moncrief (1976) and Abramowicz et al. (1978), consist of a “donut” of plasma surrounding a
black hole. The donut is supported by both centrifugal forces and pressure and is embedded in a
vacuum. Here we consider a particular instance of the Fishbone & Moncrief solution.
A practical problem with this test is that HARM abhors a vacuum. We have therefore introduced floors on the density and internal energy that limit how small these quantities can be. The
floors are dependent on radius, with ρ0 min = 10−4 (r/rin )−3/2 and umin = 10−6 (r/rin )−5/2 . This
means that the torus is surrounded by an insubstantial, but dynamic, accreting atmosphere that
interacts with the torus surface. To minimize the influence of the atmosphere on our convergence
test, we take the L1 norm of the change in variables only over that region where ρ0 > 0.02ρ0 max .
The problem is integrated in modified Kerr-Schild coordinates. The Kerr-Schild radius r has
been replaced by the logarithmic radial coordinate x1 = ln(r), and the Kerr-Schild latitude θ
has been replaced by x2 such that θ = πx2 + (1/2)(1 − h) sin(2πx2 ). Clearly 0 ≤ x2 ≤ 1 maps
to 0 ≤ θ ≤ π. This coordinate transformation has a single adjustable parameter h; for h = 1
we recover the original coordinate system (the θ coordinate is simply rescaled by π). As h → 0
numerical resolution is concentrated near the midplane.
We have integrated a Fishbone-Moncrief disk around a black hole with a/M = 0.95, to maximize
general relativistic effects. We set ut uφ = const. = 3.85 (this is the defining feature of the FishboneMoncrief equilibria) and rin = 3.7. The grid extends radially from rin = 0.98rh = to rout = 20.
The coordinate parameter h described in the last paragraph is set to 0.2. The numerical resolution
is N × N , where N = 8, 16, 32, . . . , 512, and the solution is integrated for ∆t = 10. Figure 3.15
shows the L1 norm of the error for each variable as a function of N . Second order convergence is
obtained.
The sum of the evidence presented in this section strongly suggests that we are solving the
equations of GRMHD without significant, compromising errors.
3.6
Magnetized Torus Near Rotating Black Hole
Finally we offer an example of how HARM can be applied to a real astrophysical problem: the
evolution of a magnetized torus near a rotating black hole. Again we set GM = c = 1.
The initial conditions contain a Fishbone-Moncrief torus with a/M = 0.5, r(pmax ) = 12, and
rin = 6. Superposed on this equilibrium is a purely poloidal magnetic field with vector potential
Aφ ∝ MAX(ρ0 /ρ0 max −0.2, 0), where ρ0 max is the peak density in the torus. The field is normalized
so that the minimum value of pgas /pmag = 102 . The orbital period at the pressure maximum
(r = 12), is 264 as measured by an observer at infinity.
111
Figure 3.15 Convergence results for the Fishbone and Moncrief equilibrium disk around an a/M =
0.95 black hole.
112
Figure 3.16 Density field, for a magnetized torus around a Kerr black hole with a/M = 0.5 at t = 0
(left) and at t = 2000M (right). The color is mapped from the logarithm of the density; black is
low and dark red is high. The resolution is 3002 .
113
The integration extends for ∆t = 2000, or about 7.6 orbital periods at the pressure maximum.
Figure 3.16 shows the initial and final density states projected on the R = r sin(θ), Z = r cos(θ)
plane. Color represents log(ρ0 ). The coordinate parameter h, which concentrates zones toward the
midplane, is set to 0.2. The torus atmosphere is set to the floor values (see above), and the MC
limiter is used. The numerical resolution is 3002 .
The flux of mass, energy, and angular momentum through the inner boundary are described
in Figure 3.17. Initially the fluxes are small because the initial conditions are near an (unstable)
equilibrium. The magnetorotational instability (Balbus and Hawley, 1991) e-folds for just over an
orbital period, after which the magnetic field has reached sufficient strength to distort the original
torus and drop material into the black hole. Later, the torus is turbulent and accretion occurs at
a more or less steady rate.
3.7
Conclusion
Like all hydrodynamics codes, HARM has failure modes. We will discuss one that is likely to
be relevant to future astrophysical simulations. When B 2 /ρ0 À 1 and B 2 À u, the magnetic
energy is the dominant term in the total energy equation. Because the fields are evolved separately,
truncation error in the field evolution can lead to large fractional errors in the velocity and internal
energy. An example of this was discussed in §3.5.6, where the magnetized Bondi flow test fails for
large values of B 2 /ρ0 .
Another example can be found in the strong cylindrical explosion problem of Komissarov (1999),
where an overpressured region embedded in a uniform magnetic field produces a relativistic blast
wave. HARM fails on the strong-field version of this problem unless we turn the Courant number
down to 0.1, use the minmod limiter, and sharply increase the accuracy parameter used in the
P(U) inverter. This is a particularly difficult problem, with B 2 /ρ0 as large as 104 . The problems
caused by magnetically dominated regions appears to be generic to conservative relativistic MHD
schemes, where small errors in magnetic energy density lead to fractionally large errors in other
components of the total energy. At present this is unavoidable, and has motivated the development
of schemes for the evolution of the electromagnetic field in the force-free limit (Komissarov, 2002c).
Finally, to sum up: we have described and tested a code that evolves the equations of general
relativistic magnetohydrodynamics. This code, together with the code described in a companion
paper by De Villiers and Hawley (2002), are the first that stably evolve a relativistic plasma in
a Kerr spacetime for many light crossing times. The advent of practical, stable GRMHD codes
opens the door for the study of many problems in the theory of RMRs. For example, it may be
possible to directly evaluate the importance of magnetic energy extraction from rotating black holes
and the importance of black hole spin in determining jet parameters. It may also be possible to
couple these schemes to numerical relativity codes and use them to study dynamical spacetimes
with electromagnetic sources.
114
Figure 3.17 Evolution of the rest-mass accretion rate (top), the specific energy of the accreted
matter (middle), and the specific angular momentum of the accreted matter (bottom) for a black
hole with a/M = 0.5.
115
4
A Measurement of the
Electromagnetic Luminosity of a Kerr
Black Hole
4.1
Summary of Chapter
Some active galactic nuclei, microquasars, and gamma ray bursts may be powered by the electromagnetic braking of a rapidly rotating black hole. We investigate this possibility via axisymmetric
numerical simulations of a black hole surrounded by a magnetized plasma. The plasma is described
by the equations of general relativistic magnetohydrodynamics, and the effects of radiation are
neglected. The evolution is followed for 2000GM/c3 , and the computational domain extends from
inside the event horizon to typically 40GM/c2 . We compare our results to two analytic steady state
models, including the force-free magnetosphere of Blandford & Znajek. Along the way we present
a self-contained rederivation of the Blandford-Znajek model in Kerr-Schild (horizon penetrating)
coordinates. We find that (1) low density polar regions of the numerical models agree well with the
Blandford-Znajek model; (2) many of our models have an outward Poynting flux on the horizon in
the Kerr-Schild frame; (3) none of our models have a net outward energy flux on the horizon; and
(4) one of our models, in which the initial disk has net magnetic flux, shows a net outward angular
momentum flux on the horizon. We conclude with a discussion of the limitations of our model,
astrophysical implications, and problems to be addressed by future numerical experiments.1
4.2
Introduction
A black hole of mass M and angular momentum J = aGM/c, 0 ≤ a/M < 1 has a free energy
associated with its angular momentum (or “spin”). This energy can, in principle, be tapped by
manipulating particle orbits so that negative energy particles are accreted (Penrose, 1969). Spin
energy can also be tapped by superradiant scattering of vacuum electromagnetic waves (Press
and Teukolsky, 1972), gravity waves (Hawking and Hartle, 1972; Teukolsky and Press, 1974), or
magnetohydrodynamic (MHD) waves (Uchida, 1997). It can also be tapped through the action of
force-free electromagnetic fields (Blandford and Znajek, 1977).
The Blandford-Znajek (BZ) effect– broadly used here to mean the extraction of energy from
rotating holes via a magnetized plasma– appears to be the most astrophysically plausible exploita1
Published in ApJ 20 August 2004 issue. Reproduction for this thesis is authorized by the copyright holder.
116
tion of black hole spin energy. Relativistic jets in active galactic nuclei, galactic microquasars, and
gamma-ray bursts (GRBs) may well be powered by the BZ effect. Despite some hints (see, e.g.,
Wilms et al. 2001b, Miller et al. 2002, Maraschi and Tavecchio 2003) and the general consistency
of this idea with the data, however, there is no direct observational evidence for black hole energy
extraction. In this paper we take an experimental approach and study the BZ effect through direct
numerical simulation of a magnetized plasma accreting onto a black hole.
The energy stored in black hole spin is potentially large. If Mirr is the “irreducible mass” of
the black hole where, in units such that G = c = 1,
1
2
Mirr
= M r+ ,
2
and r+ = M (1 +
(4.1)
p
1 − (a/M )2 ) is the horizon radius, then the free energy is
µ
61
Espin = M − Mirr < 5.3 × 10
M
8
10 M¯
¶
erg.
(4.2)
or ≈ 30% of the gravitational mass of a maximally rotating hole. This corresponds to a luminosity
of . 4 × 1010 (M/108 M¯ ) L¯ if released over a Hubble time.
Estimates suggest that black hole accretion is surprisingly efficient, in the sense that the ratio of
quasar radiative energy density to supermassive black hole mass density is ∼ 0.2 (Yu and Tremaine,
2002; Elvis et al., 2002). During the accretion process some mass-energy is radiated away and the
rest is incorporated into the black hole. Through electromagnetic spindown this energy gets a
second chance to escape. A combination of efficient thin disk accretion (in which all radiation
is somehow permitted to escape) followed by the Penrose process can in principle extract up to
√
(1 − 1/ 6)c2 = 0.59c2 per gram of accreted rest-mass. In practice, of course, much less energy is
likely to be available. One goal of our investigation is to discover how much less. Part of the answer
may lie with the calculations already described in Gammie et al. (2004): if black hole spins are
limited by the equilibrium value found there (a/M ≈ 0.92) then the nominal thin disk efficiency of
the accretion phase is about ≈ 17%, much less than the 42% expected at a/M = 1.
In this paper we consider the self-consistent evolution of a weakly magnetized torus surrounding
a rotating black hole. The evolution is carried out numerically in the axisymmetric ideal MHD
approximation. As the evolution progresses the computational domain develops matter dominated
regions near the equator and electromagnetic field dominated regions near the poles. To fix expectations for the structure of these regions we review two analytic models for the interaction of a
magnetized plasma with a black hole in § 4.3. Along the way we develop the relevant notation and
coordinate systems. In § 4.4 we describe our numerical model and give a summary of numerical
results for a high resolution fiducial model. In § 4.5 we consider the dependence of our results on
model parameters. A discussion and summary may be found in § 4.6. From here on we adopt units
such that GM = c = 1. Table 4.1 gives a list of commonly used symbols.
117
Table 4.1.
Symbol
Fiducial Value
a
r+
risco
redge
rmax
ΩH
Rin
Rout
β
γ
Model Parameters
0.938
1.347
2.044
6
12
≈ 0.3477
0.98r+
40
100
4/3
Ṁ0
Ė
Ė (EM )
Ė (M A)
L̇
L̇(EM )
L̇(M A)
L̃
Diagnostics
see sections
see sections
see sections
see sections
see sections
see sections
see sections
see sections
b2 /2
B r ,B θ ,B φ
Aφ
ṽ r
ω
Ω
Commonly used symbols
4.3.2
4.3.2
4.3.2
4.3.2
4.3.2
4.3.2
4.3.2
4.4.1
&
&
&
&
&
&
&
&
Description
black hole spin (J/M 2 )
√
radius of the event horizon (r+ = 1 + 1 − a2 )
radius of the ISCO (innermost stable circular orbit)
radius of inner edge of torus
radius of the pressure maximum
spin frequency of zero angular momentum observer at r+
inner radial grid location
outer radial grid location
pgas,max
ratio of gas to magnetic pressure (initially pmag,max
)
pgas = (γ − 1)u
4.4
4.4
4.4
4.4
4.4
4.4
4.4
4.5
Variables
see section 4.4.3
see section 4.3.2
see section 4.4
see section 4.4.1
see sections 4.4.2 & 4.4.3
see section 4.4.3
rest-mass flux into the black hole
energy flux into the black hole
electromagnetic energy flux
matter energy flux
angular momentum flux into the black hole
electromagnetic angular momentum flux
matter angular momentum flux
L̃ = Ė (EM ) /(−²Ṁ0 ) ; ² = 1 − Ė/Ṁ0
electromagnetic energy density in the fluid frame
∗ it
magnetic field components. B i = F
azimuthal component of electromagnetic vector potential
asymptotic radial velocity (i.e. v r at r = ∞)
spin frequency of electromagnetic field
spin frequency of fluid (Ω = uφ /ut )
118
4.3
Review of Analytic Models
In this section we review two quasi-analytic, steady state models for the interaction of a black
hole with the surrounding plasma. The purpose of this review is to introduce our coordinate
system and notation and to describe the models in a form suitable for later comparison with
numerical results. Along the way, we give a self-contained derivation of the BZ effect in Kerr-Schild
(horizon penetrating) coordinates. To the extent that the analytic and numerical models agree, the
comparison also builds confidence in the numerical models.
4.3.1
Coordinates
Before proceeding it is useful to define three coordinate bases for the Kerr metric.
Boyer-Lindquist (BL) coordinates. These are the most familiar coordinates for the Kerr metric.
In BL coordinates t, r, θ, φ
µ
¶
2r
Σ
A sin2 θ 2 4 a r sin2 θ
ds = − 1 −
dt2 + dr2 + Σ dθ2 +
dφ −
dφ dt
Σ
∆
Σ
Σ
2
(4.3)
where Σ ≡ r2 + a2 cos2 θ, ∆ ≡ r2 − 2r + a2 and A ≡ (r2 + a2 )2 − a2 ∆ sin2 θ. The determinant of the
metric g ≡ Det(gµν ) = −Σ2 sin2 θ. In BL coordinates the metric is singular on the event horizon
at r = r+ where ∆ = 0.
Kerr-Schild (KS) coordinates. The Kerr-Schild coordinates t, r, θ, φ are regular on the horizon.
They are closely related to BL coordinates: r[KS] = r[BL] and θ[KS] = θ[BL]. The line element is
µ
¶
µ ¶
µ
¶
2r
4r
2r
2
ds = − 1 −
dt +
dr dt + 1 +
dr2 + Σ dθ2
Σ
Σ
Σ
2
¶
¶
µ
µ
2r
2
2
sin θ dφ2
+sin θ Σ + a
1+
Σ
µ
¶
µ
¶
4 a r sin2 θ
2r
−
dφ dt − 2 a 1 +
sin2 θ dφ dr,
Σ
Σ
2
(4.4)
and g = −Σ2 sin2 θ.
The transformation matrix from BL to KS is
and
∂t[KS]
2r
= ,
∂r[BL]
∆
(4.5)
∂φ[KS]
a
= ;
∂r[BL]
∆
(4.6)
all other off-diagonal components are 0 and all diagonal components are 1. The inverse transformation matrix is identical, with the signs of the off-diagonal components reversed.
Modified Kerr-Schild (MKS) coordinates. Our numerical integrations are carried out in a mod119
ified KS coordinates x0 , x1 , x2 , x3 , where x0 = t[KS], x3 = φ[KS], and
r = ex1 ,
(4.7)
1
θ = πx2 + (1 − h) sin(2πx2 ).
2
(4.8)
Here h is an adjustable parameter that can be used to concentrate grid zones toward the equator
as h is decreased from 1 to 0. The transformation matrix from KS to MKS is diagonal and trivially
constructed from the explicit expressions for r and θ in equations 4.7 and 4.8.
4.3.2
Governing Equations
For a magnetized plasma the equations of motion are
¡ µν
µν ¢
T µν ;ν = TMA
+ TEM
= 0.
;ν
(4.9)
where T µν is the stress-energy tensor, which can be split into a matter (MA) and electromagnetic
(EM) part. In the fluid approximation
µν
TMA
= (ρ0 + ε + p)uµ uν + pg µν ,
(4.10)
where ρ0 ≡ rest-mass density, ε ≡ internal energy, p ≡ pressure, uµ is the fluid four-velocity, and
we assume throughout an ideal gas equation of state
p = (γ − 1)ε.
(4.11)
In terms of F µν , the Faraday (or electromagnetic field) tensor,
1
µν
(4.12)
TEM
= F µγ F ν γ − g µν F αβ Fαβ ,
4
√
where we have absorbed a factor of 4π into the definition of F µν . We assume that particle number
is conserved:
(ρ0 uµ );µ = 0.
(4.13)
The evolution of the electromagnetic field is given by the space components of the source-free
Maxwell equations
∗
F
µν
;ν
= 0,
(4.14)
∗
where F is the dual of the Faraday, and the time component gives the no-monopoles constraint.
The inhomogeneous Maxwell equations
J µ = F µν ;ν
120
(4.15)
define the current density J µ but are otherwise not required here. We adopt the ideal MHD
approximation, where
uµ F µν = 0,
(4.16)
which implies that the electric field vanishes in the rest frame of the fluid.
In our numerical models the fundamental (or “primitive”) variables that describe the state of
∗
it
the plasma are ρ0 , ε, B i ≡ F , plus three variables which describe the motion of the plasma. In
Gammie et al. (2003) we used the plasma three-velocity. Here we use
ũi ≡ ui +
where γ ≡
γβ i
,
α
(4.17)
p
1 + q 2 , q 2 ≡ gij ũi ũj , β i ≡ g ti α2 is the shift, and α2 = −1/g tt is the lapse. We made
this change to improve numerical stability. Because the three velocity components have a finite
range, truncation error can move the plasma velocity outside the light cone. The variables ũi have
the important property that they range over −∞ to ∞, and this makes it impossible for the plasma
to step outside the light cone.
To write the electromagnetic quantities in terms of the primitive variables, define the four-vector
bµ
with bt ≡ giµ B i uµ and bi ≡ (B i + ui bt )/ut . With some manipulation one finds
µν
TEM
= b2 uµ uν +
b2 µν
g − bµ bν ,
2
(4.18)
and
∗
F
µν
= bµ uν − bν uµ .
(4.19)
The no-monopoles constraint becomes
√
( −gB i ),i = 0.
(4.20)
A more complete account of the relativistic MHD equations can be found in Gammie et al. (2003)
or Anile (1989).
4.3.3
Blandford-Znajek Model
BZ studied a rotating black hole surrounded by a stationary, axisymmetric, force-free, magnetized
plasma. They obtain an expression for the energy flux through the event horizon and, given a
solution for the field geometry when a = 0, find a perturbative solution when a ¿ 1. Here we
present a self-contained rederivation, which will be compared to numerical models in Section 4.4.2.
Those not interested in the derivation may find a summary set of equations in 4.3.3. A comparison
of the analytic BZ model to our numerical models can be found in Section 4.4.2.
We follow an approach that differs slightly from BZ. We solve T µν ;ν = 0 directly rather than
using Jµ F µν = 0, which is equivalent in the force-free approximation. Also, because our solution
121
is developed in KS coordinates, which are regular on the horizon, we obtain the BZ solution by
applying a regularity condition on the horizon and at large radius, rather than the physically
equivalent approach of applying a regularity condition on the horizon in the Carter tetrad (Znajek,
1977) and then applying the result as a boundary condition in BL coordinates. Finally, if we assume
separability of the solution then we do not need to require that the solution match the flat-space
force-free solution of Michel (1973).
Derivation in KS coordinates
Over the poles of the black hole it is reasonable to expect that the density is low, but the field
strength is comparable to that at the equator. In the limit that
b2 À ρ0 + ε + p,
(4.21)
where b2 is the field strength in the fluid frame, one may assume that the matter contribution to
the stress energy tensor can be ignored and
µν
T µν ≈ TEM
.
(4.22)
This is the force-free limit.
The ideal MHD condition uµ Fµν = 0 implies that the electric field vanishes in the rest frame of
∗
the fluid. Therefore the invariant E · B = 0, or in covariant form F
µν
Fµν = 0. The electromagnetic
field is then said to be degenerate.
In the force-free limit the governing equations are then
µν
TEM;ν
=0
(4.23)
and
∗
F
µν
;ν
= 0.
(4.24)
As BZ point out, the same basic set of equations can be derived without assuming that the plasma
obeys the fluid equations.
We now specialize to KS coordinates and write down the Faraday tensor in terms of a vector
potential Aµ , Fµν = Aν,µ − Aµ,ν . We assume that the field is axisymmetric (∂φ → 0) and stationary
∗
(∂t → 0). Evaluating the condition F
It follows that one may write
µν
Fµν = 0, one finds
Aφ,θ At,r − At,θ Aφ,r = 0.
(4.25)
At,θ
At,r
=
≡ −ω(r, θ)
Aφ,θ
Aφ,r
(4.26)
where ω(r, θ) is an as-yet-unspecified function. It is usually interpreted as the “rotation frequency”
122
of the electromagnetic field (this is Ferraro’s law of isorotation; see e.g. Frank et al. 2002, §9.7 in a
nonrelativistic context). This yields Fµν in terms of the free functions ω, Aφ , and B φ , the toroidal
magnetic field:
Ftr = −Frt = ωAφ,r
(4.27)
Ftθ = −Fθt = ωAφ,θ
√
Frθ = −Fθr = −gB φ
(4.28)
Frφ = −Fφr = Aφ,r
(4.30)
Fθφ = −Fφθ = Aφ,θ
(4.31)
(4.29)
with all other components zero. Written in this form, the electromagnetic field automatically
√
√
satisfies the source-free Maxwell equations. Notice that Aφ,θ = −gB r and Aφ,r = − −gB θ .
We want to evaluate the radial energy flux
Z
π
Ė ≡ 2π
dθ
0
√
−gFE
(M A)
where FE ≡ −Ttr . This can be subdivided into a matter FE
(4.32)
(EM )
and electromagnetic FE
part,
although in the force-free limit the matter part vanishes. Similar expressions can be written for
the angular momentum flux L̇ and angular momentum flux density FL , and for the mass flux Ṁ0
and mass flux density FM . In the limit of a steady flow these conserved quantities correspond to
the radial flux measured by a stationary observer at large distance from the black hole.
Using the definition of the electromagnetic stress-energy tensor (4.12) and the relations (4.27)(4.31), it is a straightforward exercise to evaluate
(EM )
FE
= −2(B r )2 ωr(ω −
a
) sin2 θ − B r B φ ω∆ sin2 θ.
2r
(EM )
The radial angular momentum flux density is FL
(4.33)
(EM )
= FE
/ω. One can verify by direct trans√
formation that FE [KS] = FE [BL] and FL [KS] = FL [BL]. On the horizon r = r+ = 1 + 1 − a2 and
∆ = 0, so the horizon energy flux is
(EM )
FE
|r=r+ = 2(B r )2 ωr+ (ΩH − ω) sin2 θ
(4.34)
where ΩH ≡ a/(2r+ ) is the rotation frequency of the black hole (see MTW §33.4). This result,
which is identical to BZ’s result, implies that if 0 < ω < ΩH and (B r )2 > 0 then there is an outward
directed energy flux at the horizon. Because the flux was evaluated in KS coordinates the horizon
did not require special treatment as in Znajek (1977).
To finish evaluating Ė (EM ) we need to find Aφ , ω, and B φ . This requires solving the equations
of motion (4.9). They can be evaluated directly or in the reduced form J µ Fµν = 0 (as in BZ), in
which case one must also evaluate the currents using Maxwell’s equations. In either form this is a
123
difficult, nonlinear problem which probably cannot be solved in any general way.
To make progress, BZ find solutions to the equations of motion when a = 0, then perturb
them by allowing the black hole to spin slowly with a ¿ 1. If we assume that the initial field has
ω = B φ = 0, then we may expand the vector potential
(0)
(2)
Aφ = Aφ (r, θ) + a2 Aφ (r, θ) + O(a4 ),
(4.35)
(1)
where Aφ = 0 by symmetry (Aφ should be even in a). The field rotation frequency vanishes in
the unperturbed solution, and ω (2) = 0 because ω should be odd in a, so
ω = aω (1) (r, θ) + O(a3 )
(4.36)
B φ = aB φ(1) (r, θ) + O(a3 ).
(4.37)
and similarly for the toroidal field
(2)
(0)
We are now in a position to find the free functions Aφ , ω (1) , and B φ(1) , given an initial field Aφ
that satisfies the basic equations when a = 0.
(0)
BZ consider two forms for Aφ : a monopole field and a paraboloidal field. Here we review only
(0)
the (possibly split) monopole, where Aφ = −C cos θ and C is an arbitrary constant. One may
obtain the perturbed solution by making the following sequence of deductions.
(1) The t and φ components of equation (4.9), expanded to lowest nontrivial order in a, require
(EM )
that FL
(EM )
and FE
be independent of radius. Therefore they are functions of θ alone. Since
(EM )
FE
(EM )
= aω (1) FL
,
(4.38)
we conclude that ω (1) is a function of θ alone.
(2) The r component of equation (4.9), together with the requirement that B φ(1) be finite on
the horizon (all components of Fµν are well-behaved on the horizon in KS coordinates), yields a
single nontrivial solution:
B
φ(1)
C
=− 2
4r
µ
¶
2
(1)
1 − 4ω +
r
(4.39)
This solution is well behaved at the horizon and at large radius as long as ω (1) is finite on the
horizon and grows less rapidly than r2 at large r.
(3) The θ component of equation (4.9), which is the trans-field force balance equation, can now
(2)
(2)
be reduced to an equation involving Aφ and ω (1) . If we require that Aφ = Cf (r)g(θ), then one
may deduce that (a) ∂θ ω (1) = 0, i.e. ω (1) = const.; (b) g(θ) = cos θ sin2 θ. Then f (r) must satisfy
2f 0
6f
f +
−
+
r(r − 2) r(r − 2)
00
Ã
r+2
(ω (1) − 1/8)(r2 + 2r + 4)
−
r3 (r − 2)
r(r − 2)
124
!
=0
(4.40)
which is equivalent to BZ’s equation (6.7). This has an exact solution with two constants of
integration. One of the constants of integration is set by requiring that the solution be finite on
the horizon. Part of the solution can be regularized at large r by fixing the other constant of
integration, but the remaining divergence can only be zeroed by setting ω (1) = 1/8; this is already
suggested by the form of the preceding equation. For r > 2 the regular solution is
µ
¶
2
2
r r2 (2r − 3) 1 + 3r − 6r2 r 11
1
r r2
f (r) = Li2 ( ) − ln(1 − ) ln
+
ln +
+
+ − ,
r
r
2
8
12
2 72 3r 2
2
(4.41)
where Li2 is the second polylogarithm function:
Z
Li2 (x) = −
1
dt
0
ln(1 − tx)
.
t
(4.42)
For r < 2 the solution is given by the real part of equation (4.41). In the limit of large r
1
f (r) ∼
+O
4r
µ
ln r
r2
¶
,
(4.43)
which agrees with BZ.
(2)
To sum up, using only the assumption of separability of Aφ
and the regularity of physical
quantities in Kerr-Schild coordinates on the horizon and at infinity, we find
ω (1) =
B φ(1) = −
1
8
C
4
(1 + )
2
8r
r
(4.44)
(4.45)
and
(2)
Aφ = Cf (r) cos θ sin2 θ.
(4.46)
with f (r) given by equation (4.41). Our solution is identical to BZ’s after transforming to BoyerLindquist coordinates and transforming from our B φ to BZ’s BT , although BZ’s expression for f (r)
contains some unclosed parentheses.
BZ Derivation Summary
In Kerr-Schild coordinates, then, the magnetic field components are
Br =
¢
C ¡
C
+ a2 4 −2 cos θ + r2 (1 + 3 cos 2θ)f (r) ,
2
r
2r
B θ = −a2
C
cos θ sin θf 0 ,
r2
125
(4.47)
(4.48)
both accurate through second order in a, and
B φ = −a
C
4
(1 + ),
2
8r
r
(4.49)
accurate through first order in a. In Boyer-Lindquist coordinates,
B r [BL] = B r [KS],
(4.50)
B θ [BL] = B θ [KS],
(4.51)
B φ [BL] = B φ [KS] − B r [KS]
(a − 2rω)
,
∆
(4.52)
and BZ’s toroidal field
BT = ∆ sin2 θB φ [BL]
(4.53)
(which is different from BZ’s Bφ ).
There has been some concern about causality in the application of the force-free approximation
(see, e.g., Punsly 2003, see also Komissarov 2002a, 2004a). The MHD equations are hyperbolic
and causal (as are the equations of force-free electrodynamics). Below we show that a numerical
evolution of the MHD equations agrees well with the BZ solution in those regions where b2 /ρ0 À 1.
This is either a remarkable coincidence or else the BZ solution is an accurate representation of the
strong-field limit of ideal MHD.
For comparison with computational models, the most relevant aspects of the BZ theory are that:
(1) the field is force-free; (2) the field rotation frequency ω = a/8+O(a3 ) in the monopole geometry
case and ω = a/8 + O(a3 ) at the poles (θ = 0, π/2) in the paraboloidal field case considered by
BZ;2 (3) if the field geometry is nearly monopolar and a is small enough that the expansion to
lowest order in a is accurate, then B r (θ) is given by equation (4.47); and (4) if the field geometry
is monopolar and a is small, then the energy flux density FE ∝ sin2 θ on the horizon. We compare
this analytic BZ model to our numerical models in Section 4.4.2.
4.3.4
Equatorial MHD Inflow
Gammie (1999) considered a stationary, axisymmetric MHD inflow in the “plunging region”, between the innermost stable circular orbit (ISCO) and the event horizon. The flow was assumed to
be cold (zero pressure), nearly equatorial, and to proceed along lines of constant latitude θ. The
latter assumption ignores the requirement of cross-field force-balance. This model is analogous to
the Weber and Davis (1967) model for the solar wind, only turned inside out so that the wind flows
from the disk into the black hole. The model builds on earlier work by Takahashi et al. (1990),
Phinney (1983), and Camenzind (1986). The analytic model derived here will be used to compare
to numerical models in Section 4.4.3.
2
According to the numerical results of Komissarov (2001) and the argument of MacDonald and Thorne (1982), ω
adjusts to ≈ ΩH /2 hole even at large a.
126
The MHD inflow model is stationary (∂t → 0), axisymmetric (∂φ → 0) and nearly equatorial
(θ ≈ π/2) so ∂θ → 0 by symmetry. In addition flow proceeds along lines of constant θ. As a
result the model is one dimensional with a single independent variable r. The nontrivial dependent
variables are the radial and azimuthal four-velocity ur and uφ , the radial and azimuthal magnetic
field B r and B φ , and the rest-mass density ρ0 .
With these assumptions the equations of general relativistic MHD can be integrated completely.
The constancy of energy flux
and angular momentum flux
√
− −gT r t = const.,
(4.54)
√
−gT r φ = const.,
(4.55)
follow from T µ ν;µ = 0. The source-free Maxwell equations imply
√
−gB r = const.,
(4.56)
which expresses the constraint ∇ · B = 0, and the relativistic “isorotation law”,
√
√
∗ rφ
−g F = −g(ur bφ − uφ br ) = const.
(4.57)
where bµ is the magnetic field four-vector (defined above). Finally, conservation of particle number
implies
√
−gρ0 ur = const.
(4.58)
These five constants yield five constraints on the five nontrivial fundamental variables ur , uφ , B r ,
B φ , and ρ0 . Given the constants, and using the constitutive relations that relate the constants and
fundamental variables, one can solve the resulting set of nonlinear equations for the fundamental
variables at each radius.
The next step is to determine the constants. The radial magnetic flux and the rest-mass
flux are determined by conditions in the disk and can be left as free parameters. The remaining
three degrees of freedom are fixed by imposing boundary conditions. Gammie (1999) imposed the
following conditions: (1) the flow is regular at the fast point (the flow is automatically regular at
the Alfvén point– see Phinney (1983) for a discussion– and the slow point is absent because the
flow is cold) ; and (2,3) the four-velocity components ur and uφ match onto a cold disk at the
ISCO.
Energy can be extracted from the black hole if the Alfvén point lies inside the ergosphere
(Takahashi et al., 1990). Gammie (1999) calculated Ė and L̇ as a function of a and B r and showed
that for even modest magnetic field strength these were modified from the values anticipated in
classical thin disk theory. The implications of these modified fluxes for the structure– particularly
the surface brightness– of a thin disk were explored by Agol and Krolik (2000).
For comparison with numerical models, the key predictions of the inflow model are: (1) the
127
constancy of the conserved quantities with radius; (2) matching of the flow velocity to circular
orbits at the ISCO; (3) modification of the angular momentum and energy fluxes from their thin
disk values; and (4) the run of all the fluid variables with radius in the plunging region.
4.4
Numerical Experiments
All our experiments evolve a weakly magnetized torus around a Kerr black hole in axisymmetry.
The focus of our numerical investigation is to study a high resolution model (4.4.1), compare with
the BZ model (4.4.2), and compare to the Gammie inflow model (4.4.3). In § 4.5 we investigate how
various parameters affect the results. Any dimensional quantity can be recovered from a numerical
quantity since we set GM = c = ρ? = 1, where ρ? is some rest mass density set to unity in a
simulation.
The initial conditions consist of an equilibrium torus (Fishbone and Moncrief 1976 ; Abramowicz
et al. 1978) which is a “donut” of plasma with a black hole at the center. The donut is supported
against gravity by centrifugal and pressure forces, and is embedded in a vacuum. We consider
a particular instance of the Fishbone and Moncrief (1976) solutions, which are defined by the
condition ut uφ = const. We normalize the peak density ρ0,max = ρ? to 1 and fix the inner edge of
the torus at redge = 6. We also set γ = 4/3.3 Absent a magnetic field, the initial torus is a stable
equilibrium.4
Into the initial torus we introduce a purely poloidal magnetic field. The field can be described
using a vector potential with a single nonzero component Aφ ∝ MAX(ρ0 /ρ0,max − 0.2, 0) The field
is therefore restricted to regions with ρ0 /ρ0,max > 0.2. The field is normalized so that the minimum
ratio of gas to magnetic pressure is 100. The equilibrium is therefore only weakly perturbed by the
magnetic field. It is, however, no longer stable (Balbus and Hawley, 1991; Gammie, 2004).
Our numerical scheme is HARM (Gammie et al., 2003), a conservative, shock-capturing scheme
for evolving the equations of general relativistic MHD. HARM uses constrained transport to maintain a divergence-free magnetic field (Evans and Hawley, 1988; Toth, 2000). The inversion of conserved quantities to primitive variables is performed by solving a single non-linear equation (Del
Zanna and Bucciantini, 2002) or by a slower but more robust multi-dimensional Newton-Raphson
method. Unless otherwise stated we use modified Kerr-Schild (MKS) coordinates with h = 0.3.
The computational domain is axisymmetric, with a grid that typically extends from rin = 0.98r+
to rout = 40, and from θ = 0 to θ = π/2.
HARM is unable to evolve a vacuum, so we are forced to introduce “floors” on the density and
internal energy. When the density or internal energy drop below these values they are immediately reset. This sacrifices exact conservation of energy, particle number, and angular momentum,
although it is reasonable to assume that when the floors are small enough the true solution is
recovered. The floors are position dependent, with ρ0,min = 10−4 r−3/2 and εmin = 10−6 r−5/2 . We
3
4
We have run a limited number of γ = 5/3 models and find results essentially identical to those discussed below.
In axisymmetry. The torus is unstable to global nonaxisymmetric modes (Papaloizou and Pringle, 1983).
128
discuss the effect of varying the floor in Section 4.5.3.
At the outer boundary we use an “outflow” boundary condition. This means we project all
primitive variables into the ghost zones while forbidding inflow. The inner boundary condition is
identical except that, because the boundary is inside the event horizon, we never need to worry
about backflow into the computational domain. At the poles we use a reflection boundary condition
where we impose appropriate symmetries for each variable across the axis.
4.4.1
Fiducial Model
First consider the evolution of a high resolution fiducial model with a = 0.938. This is close to the
spin equilibrium value (where d(a/M )/dt = 0) found by Gammie et al. (2004) for a series of similar
Fishbone-Moncrief tori.
The fiducial model has ut uφ = 4.281, the pressure maximum is located at rmax = 12, the inner
edge at (r, θ) = (6, π/2), and the outer edge at (r, θ) = (42, π/2). The orbital period at the pressure
3/2
maximum 2π(a + rmax ) ' 267, as measured by an observer at infinity.
The numerical resolution of the fiducial model is 4562 . The zones are equally spaced in modified
Kerr-Schild coordinates x1 and x2 , with coordinate parameters h = 0.3. Small perturbations are
introduced in the velocity field, and the model is run for ∆t = 2000, or about 7.6 orbital periods
at the pressure maximum.
The initial state is Balbus-Hawley unstable. The inner edge of the disk quickly makes a transition to turbulence. Transport of angular momentum by the magnetic field causes material to
plunge from the inner edge of the disk into the black hole. The turbulent region gradually expands
outward to involve the entire disk. The disk relaxes toward a “Keplerian” velocity profile, meaning
that the orbital frequency along the equator is close to the circular orbit frequency. The disk enters a long, quasi-steady phase in which the accretion rates of rest-mass, angular momentum, and
energy onto the black hole fluctuate around a well-defined mean.
Figure 4.1 shows the initial and final density states projected on the (R, z = r sin θ, r cos θ)plane. Color represents log(ρ0 ). The initial density maximum is 1 and the minimum is ≈ 4 × 10−7 .
The final state contains shocks driven by the interaction with the magnetic field, outflows near the
surface of the disk, and an evacuated “funnel” region near the poles.
The left panel in Figure 4.2 indicates the relative densities of internal, magnetic, and restmass energy. The magenta and cyan contours show the ratio of the average pressure to average
magnetic pressure, β ≡ 2p̄/b¯2 . The overbar indicates an average taken over 1000 < t < 2000 and
over both hemispheres. The cyan contour indicates β = 3 and encircles most of the high density,
approximately Keplerian disk. The magenta contour indicates β = 1. The red contour indicates
where b¯2 /ρ¯0 = 1. Between the pole and this contour the magnetic energy density exceeds the
internal and rest-mass energy density. The black contour surrounds a region, extending to large
radius, where −ut > 1 and the flow is directed outward (at large radius −ut asymptotes to the
Lorentz factor). That is, the particle energy-at-infinity is larger than the rest-mass density: so
129
Figure 4.1 Initial (left) and final (right) distribution of log ρ0 in the fiducial model on the r sin θ −
r cos θ plane. At t = 0 black corresponds to ρ0 ≈ 4 × 10−7 and dark red corresponds to ρ0 = 1. For
t = 2000, black corresponds to ρ0 ≈ 4 × 10−7 and dark red corresponds to ρ0 = 0.57. The black
half circle at the left edge is the black hole.
130
WIND
FUNNEL
CORONA
BLACK HOLE
DISK
PLUNGING REGION
Figure 4.2 (a) The distribution of β, b2 /ρ0 , and ut in the fiducial run, based on time and hemispherically averaged data. Starting from the axis and moving toward the equator: (1) ut = −1
contour shown as a solid black line; (2) b2 /ρ0 = 1 contour shown as a red line; (3) β = 1 contour
shown as a magenta line that nearly matches part of the ut = −1 contour line; and (4) β = 3
contour is shown as cyan line. (b) Motivated by the left panel, the right panel indicates the location of the five main subregions of the black hole magnetosphere. They are (1) the disk: a matter
dominated region where b2 /ρ0 ¿ 1; (2) the funnel: a magnetically dominated region around the
poles where b2 /ρ0 À 1 where the magnetic field is collimated and twists around and up the axis
into an outflow; (3) the corona: a region in the relatively low density upper layers of the disk with
weak time-averaged poloidal field; (4) the plunging region; and (5) the wind, which straddles the
corona-funnel boundary. See Section 4.4.1 for a discussion.
131
the fluid is in a sense, unbound. We use the value of ut to estimate the radial component of the
3-velocity at infinity (ṽ r ), which is independent of the coordinate system.
The right panel in Figure 4.2 defines some useful terminology inspired by the left panel, following
De Villiers and Hawley (2003a) and Hirose et al. (2003). Moving from the axis to the equator, the
“funnel” is the nearly evacuated, strongly magnetized region (b2 À ρ0 + ε + p), that develops over
the poles. The “wind” consists of a cone of material near the edge of the funnel that is flowing
outward with an asymptotic radial velocity of ṽ r ∼ 0.75c. Near the outer edge of our computational
domain the wind becomes marginally superfast. The “corona” lies between the funnel and the disk
and has b2 /2 ∼ p except in strongly magnetized filaments. In the “disk” b2 /2 < p and the plasma
follows nearly Keplerian orbits. Finally, the “plunging” region, which lies between the disk and
the event horizon, contains accreting material moving on magnetic field and pressure modified
geodesics.
Figure 4.3 shows the evolution of the poloidal magnetic field. The panels show contours of
constant Aφ , so the density of contours is directly related to the poloidal field strength, and the
contours follow magnetic field lines. The contours are projected on the (R = r sin θ, z = r cos θ)plane, and show the initial and final state. The initial field is confined to a region much smaller
than the torus as a whole because field is introduced only in those portions of the disk that have
ρ0 > 0.2ρ0,max . Notice that by the end of the simulation the field has mixed in to the funnel region
and has a regular geometry there. In the disk and at the surface of the disk the field is curved on
the scale of the disk scale height. The field strengths and geometries we see are consistent with
Hirose et al. (2003). This includes the absence of disk to disk field loops, and that the funnel
field collimates instead of connecting back into the disk (thus providing a means for the outflow to
escape to large radii).
Figure 4.4 shows contours of time and hemisphere averaged Aφ . The time averaged field is even
more regular in the funnel than the snapshot in Figure 4.3. Time averaging tends to sharply reduce
the field strength in the corona and disk because the field fluctuates in magnitude and direction
there.
Figure 4.5 shows the accretion rate of rest-mass (Ṁ0 ), energy per unit rest-mass (Ė/Ṁ0 ), and
angular momentum per unit rest-mass (Ė/Ṁ0 ) evaluated inside the horizon at the inner boundary
of the computational domain. For 500 < t < 2000 the time average values are Ṁ0 ≈ 0.35,
Ė/Ṁ0 ≈ 0.87, and L̇/Ṁ0 ≈ 1.46. These average values are shown as dashed lines. The dotted
lines show the classical thin disk values Ė/Ṁ0 ≈ 0.82 and L̇/Ṁ0 ≈ 1.95 obtained by setting these
ratios equal to respectively the specific energy and angular momentum of particles on the ISCO.
The energy per baryon is therefore slightly above the thin disk value, but the angular momentum
per baryon is significantly below the thin disk value.
It may be useful to recast the energy flux in terms of a nominal “radiative efficiency”5 ² =
1 − Ė/Ṁ0 . For the fiducial run ² = 13%, which is slightly lower than the thin disk with ² = 18%.
This is likely due to the high temperature of the flow. On the horizon about 20% of the energy flux
5
Our evolution is nonradiative, so the true radiative efficiency is zero.
132
Figure 4.3 Initial (left) and final (right) distribution of Aφ . Level surfaces coincide with magnetic
field lines and field line density corresponds to poloidal field strength. In the initial state field lines
follow density contours if ρ0 > 0.2ρ0,max .
133
Figure 4.4 Contour plot of the time and hemispheric average of Aφ . Level surfaces coincide with
magnetic field lines and field line density corresponds to poloidal field strength.
134
Figure 4.5 Evolution of rest-mass, energy, and angular momentum accretion rate for our fiducial
run of a weakly magnetized tori around a black hole with spin a = 0.938. For 500 < t < 2000 the
time average of these values is Ṁ0 ' 0.35, Ė/Ṁ0 ' 0.87, and L̇/Ṁ0 ' 1.46 as shown by the dashed
lines. The dotted lines show the classical thin disk values (Ė/Ṁ0 ' 0.82 and L̇/Ṁ0 ' 1.95). See
Section 4.4.1 for a discussion.
135
would vanish if we set the internal energy to zero. The corresponding zero-temperature efficiency
(1 + ut ) would be 32%.
The chief object of our study is to measure the electromagnetic luminosity of the hole. The time
and hemisphere averaged electromagnetic energy flux on the horizon is shown in Figure 4.6. In the
funnel region the energy flux density is outward, as predicted by the force-free model of BZ. We
compute other interesting quantities by integrating over the horizon and taking a time average (for
technical reasons we are using a less resolved time sampling here than used to make Figure 4.5, but
the time averages have fractional differences of only 10%). We find Ė (EM ) /Ė (M A) = −2.3%, where
the energies per baryon are Ė (EM ) /Ṁ0 = −0.018 and Ė (M A) /Ṁ0 = 0.77. It is useful to define the
ratio of electromagnetic luminosity to nominal accretion luminosity L̃ = Ė (EM ) /(−²Ṁ0 ). We find
L̃ = 16%. Thus while the electromagnetic energy flux is outward, it is a small fraction of the inward
material energy flux and the BZ luminosity is small compared to the nominal accretion luminosity.
The BZ luminosity from the polar regions of the horizon provides about 1/2 of the radial
electromagnetic energy flux (Ė (EM ) ) in the wind, while another 1/2 comes from the surface of the
corona but is mostly dissipated into kinetic energy of the wind. Therefore, the BZ-effect is an
important source of Poynting flux in the wind at large distances since it flows freely along highly
ordered poloidal field lines connected to unbound plasma.
A control calculation at a = 0 and a resolution of 2562 gives Ė (EM ) /Ė (M A) = 0.33% and
L̃ = −6.5%, where the energies per baryon are Ė (EM ) /Ṁ0 = 0.0032 and Ė (M A) /Ṁ0 = 0.95.
Ė (EM ) /Ṁ0 > 0 and L̃ < 0 are as expected, since the outward energy flux must vanish for a
nonrotating hole (i.e. the BZ effect is not operating). For our sequence of models the BZ effect
does not operate for a . 0.5 (see Section 4.5.1). The matter energy flux ratio may be compared to
the thin disk value of Ė (M A) /Ṁ0 = 0.94.
4.4.2
Comparison with BZ
The BZ solution was reviewed in Section 4.3.3. BZ were able to find steady force-free field solutions
in the limit that a ¿ 1. Since the fiducial run has a = 0.938, we ran a special a = 0.5 model for
comparison with BZ.
The BZ solution was found in the force-free limit, so the first question one might ask is whether
any region of the model is force-free. To measure this we recall that in the force-free limit
T µν ;ν = F µν Jν = 0.
(4.59)
So when the field is force-free the parameter
¯
¯ µν
¯ F Jν Fµκ J κ ¯
¯
¯
ζ=¯
Jµ J µ Fκλ F κλ ¯
(4.60)
is small compared to 1.
Figure 4.7 shows the time and hemispherical averaged ζ(r, θ) from t = 1000 to t = 2000 for
136
(EM )
Figure 4.6 Electromagnetic energy flux density FE
(θ) on the horizon for the fiducial run, based
on time and hemisphere averaged data. The mean electromagnetic energy flux is directed outward.
See Section 4.4.1 for a discussion.
137
Figure 4.7 The run of the force-free parameter ζ for the a = 0.5 run; when ζ ¿ 1 the field is
approximately force-free. The parameter has been time and hemisphere averaged. The contours
show (beginning from the pole and moving toward the equator) ζ = 10−3 , 10−2 , 10−1 . The small
closed contours at large radius and close to the axis have ζ = 10−2 . The small closed contours from
the equator to θ ∼ π/4 have ζ = 10−1 . See Section 4.4.2 for a discussion.
138
Figure 4.8 Left panel: Magnetic field angular frequency on the horizon relative to black hole
rotation ω(θ)/ΩH . The solid line indicates time and hemisphere averaged data from our a = 0.5
MHD integration. The middle dotted line is the prediction of the BZ model (ω/ΩH = 1/2). The
dashed line (top) is the value predicted by the inflow model. Right panel: the run of field rotation
frequency ω with radius along a single field line that intersects the horizon at θ = 0.2. ω is constant
to within 3%, as expected for a steady flow. See sections 4.4.2 and 4.4.3 for a discussion.
the a = 0.5 model. The contours show (beginning from the pole and moving toward the equator)
ζ = 10−3 , 10−2 , 10−1 . The entire funnel region has ζ < 10−2 and is therefore effectively force-free.
This is true in both a time-averaged and instantaneous sense in the funnel for all our runs. This
opens the possibility that the BZ solution describes the funnel.
A key feature of the BZ model is that the field rotation frequency ω ≈ ΩH /2 for a ¿ 1 if the
field has a monopole geometry. Figure 4.8a shows the ratio ω/ΩH on the horizon. Within the
force-free region, which runs from 0 < θ < 0.4 on the horizon, the average ω/ΩH ≈ 0.45. The small
difference from the BZ could be due to higher order terms in the expansion in a, but Komissarov
(2001) has integrated the equations of force-free electrodynamics for a monopolar field geometry
and at a = 0.5 finds that ω rises from ≈ 0.495ΩH at the pole to ≈ 0.51ΩH at the equator, so this
seems unlikely. The difference is more likely due to small deviations from force-free behavior (mass
loading of field lines by the numerical “floor” on the density).
In an axisymmetric steady state both the force-free equations and the MHD equations predict
that the rotation frequency ω (and other quantities) are constant along field lines. Figure 4.8b
shows the variation of ω with radius along a field line that intersects the horizon at θ = 0.33. As
expected ω ≈ const., with a variation of less than 3% from maximum to minimum.
BZ’s spun-up monopole model makes definite predictions about the variation of B r and FE on
139
Figure 4.9 (a) Square of radial field ((B r (θ))2 ) on the horizon in the a = 0.5 MHD integration, from
time and hemisphere averaged data. Solid line is the field for our numerical model. The dotted
line shows the Blandford and Znajek (1977) perturbed monopole solution with the field strength
normalized to the numerical solution at the pole. The dashed line is the inflow solution. (b)
(EM )
Electromagnetic energy flux FE
(θ) on the horizon in the a = 0.5 MHD integration, from time
and hemisphere averaged data. The solid line shows the numerical model, the dotted line shows
BZ’s spun-up monopole solution, and the dashed line shows the inflow solution. See sections 4.4.2
and 4.4.3 for a discussion.
140
the horizon. Figure 4.9a shows the variation in time and hemisphere averaged (B r )2 and compares
to BZ’s monopole field calculation. The single adjustable parameter of the model normalizes the
field strength. We have set this normalization by requiring that (B r )2 match at the pole. Evidently
the variation matches the BZ prediction closely even well outside the force-free region at θ ≈ 1.1.
Figure 4.9b shows the variation in radial energy flux on the horizon as predicted by the BZ model
using the pole-normalized field. Here the match is quite close out to θ ≈ π/4. It is slightly surprising
that the BZ solution does so well even in regions that are not force-free. This is likely a result of
trans-field force balance and geometry controlling the distribution of field on the horizon and hence
the radial energy flux.
To summarize: in our low spin numerical experiment the funnel is approximately force-free
within the funnel. It is approximately in a steady state and hence ω is approximately constant
along field lines. Furthermore, ω, B r , and the radial electromagnetic energy flux are all in good
agreement with the spun-up monopole force-free model on the horizon. We have not compared
the entire funnel region with the monopole model because the field is collimated there and not
well-described by the monopole solution.
4.4.3
Comparison to Inflow Solution
The inflow solution of Gammie (1999) considers a near-equatorial stationary MHD inflow in the
plunging region, reviewed in Section 4.3.4. Here we compare the inflow models with the fiducial
model. Unlike the funnel, the plunging region is rapidly fluctuating, so we expect the inflow model
to match only the time-averaged data from the simulation.
The inflow model has two free parameters: the field strength and the accretion rate. The field
strength we match by finding the parameter that gives the best fit to the mean magnetic energy
density between the ISCO and the event horizon. The rest-mass flux is chosen to agree with the
time-averaged data from the simulation. The ratio of the field strength to the square root of the
accretion rate is a dimensionless parameter that controls the solution; in the units of Gammie
√
(1999), where 2πρ0 ur −g = −1, we use Fθφ = 1.09 for the comparison model.
Figure 4.10 shows a comparison of ur , L̇/Ṁ0 , comoving energy densities (ρ0 , b2 /2, and ε), and
energy fluxes (Ė/Ṁ0 ) in the inflow solution. The comparison data from the fiducial run has been
averaged over |θ − π/2| < 0.3 and 500 < t < 2000. Each panel in the figure contains a vertical line
at the ISCO.
The upper left panel compares the radial component of the four-velocity (in KS and BL coordinates) in the inflow and numerical solutions. The substantial differences are due to the finite
temperature of the flow; the inflow solution is cold by assumption. Radial pressure gradients in
the numerical model (which are absent in the inflow solution) begin to accelerate material inward
outside the ISCO, and the flow becomes supersonic near the ISCO.
The upper right and lower right panels show components of the energy and angular momentum
flux from the simulation and inflow solutions. The dashed horizontal line in each case shows the
141
Figure 4.10 A comparison of the time-averaged fiducial model near the equator (within θ = π/2 ±
0.3) with the inflow solution of Gammie (1999). In the right two panels the black dotted line is
the thin disk value. In all cases the red vertical line is the location of the ISCO. The black line for
the upper left panel is the numerical result. For the other three panels, the particle term is shown
in cyan, the internal energy term is shown in magenta, and the electromagnetic term is shown in
green. The blue line in each plot represents the inflow model result. Notice that the run of density
with radius shows no feature at the ISCO. See the Section 4.4.3 for discussion.
142
values expected for a thin disk; the inflow solution is constrained to match the thin disk at the
ISCO. The cyan lines show uφ (upper panel) and ut (lower panel) from the simulation, while
the blue lines show the prediction from the inflow model. The energy flux matches rather well
(although notice that this is only a small fraction of the energy flux), while the angular momentum
is overestimated; in the simulation the plasma has sub-Keplerian angular momentum by the time
it reaches the ISCO.
The electromagnetic components of the per unit mass flux of angular momentum flux (b2 uφ /ρ0 −
br bφ /(ρ0 ur )) and energy flux (b2 ut /ρ0 − br bt /(ρ0 ur )) are also shown in the upper and lower right
panels of Figure 4.10 (green line ≡ simulation, blue line ≡ inflow solution). The inflow solution
matches well, although it tends to overestimate the magnitude of the outward directed energy flux.
The magenta lines in the upper and lower right panels show the internal energy component
of the normalized angular momentum flux ((ε + p)uφ /ρ0 ) and energy flux (−(ε + p)ut /ρ0 ). This
component of the fluxes is zero by assumption in the inflow solution, and it is evidently an important
component of the fluxes in our thick disk simulations. This leads to large corrections to the angular
momentum and energy fluxes; the total normalized angular momentum flux is significantly smaller
than the thin disk prediction, while the energy flux is, seemingly by conspiracy, very close to the
thin disk.
The lower left panel shows the rest-mass density from the inflow solution (upper blue line)
and from the simulation (cyan line). The mass flux in the inflow solution is normalized so that it
matches the simulation mass flux. Since mass flux is approximately constant with radius, the run
of density is directly related to the run of ur . What is remarkable here is that there is no feature
in the simulation ρ0 near the ISCO. In fact it is nearly constant from well outside the ISCO in
to the event horizon. The surface density varies smoothly as well. This confirms the point made
by Krolik and Hawley (2002) in their pseudo-Newtonian solution: there is no sharp feature at the
ISCO. This has implications for iron line profiles, as discussed by Reynolds and Begelman (1997).
The lower left panel also shows the run of internal energy density in the simulation (it is zero by
assumption in the inflow solution). Again, there is no sharp feature at the ISCO, just a gentle rise
inward toward the event horizon. Because the density is nearly constant with radius this implies
that entropy is increasing inward. Therefore there is some dissipation of kinetic or magnetic energy
into internal energy in the inflow region.
The lower left panel of Figure 4.10 shows the run of magnetic energy density b2 /2 in the inflow
solution (lower blue line) and simulation (green line). The normalization of the inflow magnetic
energy is a parameter, but its radial slope is not.
Finally, the inflow solution predicts that ω = ΩISCO . Figure 4.8a shows the run of ω/ΩH on
the horizon for the a = 0.5 model. The dashed line shows the ISCO value of ω/ΩH . At the equator
the time-averaged numerical value lies within about 10% of the ISCO value: the numerical average
ω/ΩH = 0.685, while the ΩISCO /ΩH = 0.8136 at the ISCO. In the a = 0.938 run the numerical
average ω/ΩH = 0.681, while ΩISCO /ΩH = 0.745 at the ISCO.
To sum up, the inflow model does a surprisingly good job of matching some aspects of the
143
time-averaged simulation. It does not match the profile or boundary condition at the ISCO for the
radial velocity or the total angular momentum and energy fluxes, because the simulation flow is
hot, while the inflow solution has zero temperature by assumption.
What is most surprising is that the energy per baryon accreted in the numerical model matches
the thin disk prediction. The inflow model predicts that the energy per baryon accreted should
be lower than the thin disk prediction, enhancing the nominal accretion efficiency (Gammie, 1999;
Krolik, 1999a; Agol and Krolik, 2000). The difference is apparently due to the finite temperature
of the numerical model and the consequent change in boundary conditions at the ISCO. These
boundary conditions evidently adjust themselves to maintain the energy flux at the thin disk value.
The angular momentum flux is affected by the field, however, with the specific angular momentum
of the accreted material in the fiducial run about 25% lower than the thin disk.
4.5
Parameter Study
Our numerical model has a number of physical and numerical parameters. Here we check the
sensitivity of the model to: (1) black hole spin parameter a; (2) initial magnetic field geometry
and initial magnetic field strength; and (3) numerical parameters such as (a) location of the inner boundary (rin ); (b) outer radial (rout ) boundary; (c) radial and θ resolution, including the
coordinate parameter h; and (d) parameters describing the density and internal energy floors.
4.5.1
Black Hole Spin
The fiducial run has a rather low outgoing electromagnetic energy flux compared to the ingoing
matter energy flux. It is possible that this varies sharply with black hole spin and that more rapidly
rotating holes exhibit much larger electromagnetic luminosity. We have performed a survey over a,
keeping all parameters identical to those in the fiducial run, except that the resolution is lowered
to 2562 and the location of the pressure maximum is adjusted to keep H/R ≈ const.
The results are shown in Figure 4.11 and described in Table 4.2. The figure shows the measured
ratio of electromagnetic to rest-mass energy flux; the dashed line shows a fit
Ė (EM )
≈ −0.068(2 − r+ )2 .
Ė (M A)
(4.61)
This fit applies only to this particular sequence of models; models with different initial field geometries give different results, as we shall see below. For all a > 0 we find Ė (EM ) > 0 in the funnel.
For a < 0.5 this outward funnel flux is balanced by an inward electromagnetic energy flux near the
equator. For our most extreme run with a = 0.969 the outward electromagnetic flux is still dominated by the inward particle flux. The ratio of electromagnetic luminosity to nominal accretion
luminosity is L̃ = 27%, so the nominal accretion luminosity dominates over the BZ luminosity.
The accretion rate of angular momentum is also a strong function of spin. As discussed in
Gammie et al. (2004), accretion flows around rapidly spinning holes have da/dt < 0. Our fiducial
144
Figure 4.11 The ratio of electromagnetic to matter energy flux on the horizon. The solid line
indicates numerical data while the dotted line indicates a best fit of Ė (EM ) /Ė (M A) = −0.068(2 −
r+ )2 . See Section 4.5.1 for a discussion.
145
Table 4.2.
a
-0.938
0.000
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0.500
0.600
0.750
0.875
0.895
0.900
0.938
0.969
Black Hole Spin Study
104 × Ė (EM ) /Ė (M A)
Ė/Ṁ0
L̇/Ṁ0
ȧ/Ṁ0
Ṁ0
L̃
105
34.4
31.2
35.8
29.7
26.9
9.17
3.30
1.32
1.15
-9.85
-28.5
-81.8
-291
-254
-315
-318
-410
0.958
0.950
0.952
0.948
0.949
0.948
0.946
0.937
0.933
0.937
0.933
0.929
0.908
0.852
0.891
0.882
0.856
0.869
3.806
3.068
3.025
2.896
2.881
2.817
2.749
2.759
2.605
2.763
2.583
2.489
2.150
1.440
1.723
1.674
1.396
1.374
-5.583
-3.049
-2.921
-2.713
-2.597
-2.439
-2.302
-2.217
-1.975
-1.986
-1.665
-1.347
-0.808
-0.152
-0.204
-0.118
0.067
0.217
-0.908
-0.870
-0.709
-0.767
-0.796
-0.776
-0.747
-0.571
-0.620
-0.241
-0.252
-0.318
-0.276
-0.170
-0.215
-0.193
-0.203
-0.172
-0.24
-0.065
-0.062
-0.066
-0.055
-0.050
-0.016
-0.0049
-0.0018
-0.0017
0.014
0.037
0.083
0.17
0.20
0.24
0.23
0.27
Note. — All models same as fiducial except at a resolution of 2562 and rmax
is used to keep H/R ∼ constant. These values can be compared to Tables 3
and 4. The efficiency is 1− Ė/Ṁ0 . A positive ȧ/Ṁ0 corresponds to a spindown
of the black hole since Ṁ0 < 0.
146
model, in fact, is spinning down. Previous estimates suggested that spin equilibrium is reached at
a ∼ 0.998 (Thorne, 1974). Our models reach spin equilibrium at a ∼ 0.92.
The variation of field strength and geometry with black hole spin is also of interest. To measure
variation of field strength, we probe the flow near four locations: 1) in the funnel near the horizon
(“funnel/horizon”); 2) in the plunging region near the horizon (“plunging/horizon”); 3) at the
ISCO; and 4) at the pressure maximum. We then take a time and spatial average of the comoving
electromagnetic energy density b2 /2 over a small region near each of these locations. The ratio of
b2 (funnel/horizon) to b2 (plunging/horizon) changes from 0.43 at a = 0 to 0.74 at a = 0.938. The
ratio of b2 (funnel/horizon) to b2 (ISCO) varies from 2.53 at a = 0 to 2.14 at a = 0.938. The ratio
b2 (funnel/horizon) to pressure maximum varies from 4.8 at a = 0 to 15.7 at a = 0.938. In summary,
the field strength increases from the ISCO to the horizon by a factor of ∼ 3 at a = 0 and by a
factor of ∼ 6 at a = 0.938, and on the horizon is slightly larger at the equator than at the poles by
a factor of ∼ 2. Only the ratio of b2 (pressure maximum) to other locations in the plunging region
or at the horizon depends strongly on black hole spin.
Our observed increase in horizon field strength with black hole spin agrees with results reported
by De Villiers et al. (2003a). (Livio et al., 1999) suggest that 1) there is no reason the field strength
near the black hole horizon should be stronger than in the central regions of the disk, which they
suggest implies the electromagnetic output from the disk (corona) dominates the BZ luminosity ;
and 2) the spin of the hole is probably irrelevant to the electromagnetic output in the wind. We have
found that the field strength from the pressure maximum of the disk to the horizon increases by a
factor ∼ 5 − 16 depending on the black hole spin. Also, for a = 0.5 we find that the BZ luminosity
contributes about 1/5 to the total Ė (EM ) in the wind, and for a = 0.938 about 1/2. The rest of
Ė (EM ) in the wind is provided by the disk corona, which is mostly dissipated into kinetic energy
of the wind. This suggests that black hole spin is important in determining the electromagnetic
energy flux in the wind, and that the BZ luminosity in the funnel can be a significant contribution
to Poynting flux at large distances.
We see no sign of the expulsion of flux from the horizon reported by Bicak and Janis (1985), who
find that the flux through one hemisphere of the horizon, due to external sources and calculated in
axisymmetry using vacuum electrodynamics, vanishes when the spin of the hole is maximal. It is
possible that we have not gone close enough to a = 1 to observe this effect.
To investigate the variation of field geometry in the funnel region with a we trace field lines
from θin on the horizon to θout on the outer boundary and define a collimation factor θin /θout . The
collimation factor is similar for all field lines in the funnel region. It reaches a minimum of ≈ 5/2
for the fiducial run, and rises to nearly 2 for a = 0 and again to nearly 2 for a ∼ 1. The collimation
factor depends on the location of the outer boundary; for models with rout = 400 the collimation
factor is 10 and the field lines are nearly cylindrical at the outer boundary.
We have also studied the variation of the field rotation frequency ω in the funnel. ω/ΩH varies
weakly with a, from 0.53 at a = 0.25 to 0.45 at a = 0.938, consistent with the hypothesis advanced
by Thorne et al. (1986) that ω/ΩH ≈ 1/2.
147
Table 4.3.
Field Strength and Geometry Study
Field Geometry Aφ
β
Ė (EM ) /Ė (M A)
Ė/Ṁ0
L̇/Ṁ0
ȧ/Ṁ0
Ṁ0
L̃
A0φ
A0φ
A0φ sin (log (r/h))
A0φ | sin(2θ)|
r sin θ
r sin θ
100
500
100
100
100
400
−0.0312
−0.0115
−0.0355
−0.0112
−0.147
−0.157
0.856
0.879
0.892
0.888
0.773
0.813
1.40
1.94
1.24
1.91
−0.997
0.0617
0.0674
−0.293
0.278
−0.299
1.807
1.184
−0.203
−0.0474
−0.541
−0.0746
−1.769
−0.715
0.21
0.085
0.42
0.083
0.79
0.67
Note. — A0φ is the fiducial model field geometry and β = 100 is the fiducial ratio of
gas to magnetic pressure. The r sin θ field geometry is a uniform vertical field model with β
set by disk values at the equator. All other model and numerical parameters are as in the
fiducial model except that the resolution is 2562 . The efficiency is 1 − Ė/Ṁ0 . A positive ȧ/Ṁ0
corresponds to a spindown of the black hole because Ṁ0 < 0.
4.5.2
Field Geometry and Strength
The outcome of the simulation may also depend on the field geometry and strength in the initial
conditions. This seems more likely for axisymmetric models such as ours where the evolution may
retain a stronger memory of the initial conditions than comparable three dimensional models.
We begin by investigating the dependence of outcome on initial field strength, parameterized
by β ≡ pgas,max /pmag,max (notice that the two maxima never occur at the same location in space,
so this ratio varies over a wide range when evaluated at individual locations in the disk). We
consider models with β = (100, 500) and find a weak dependence on β. For the β = 100 model
(the fiducial model at a resolution of 2562 ) we find ω/ΩH ≈ 0.45, Ė (EM ) /Ė (M A) ≈ −3.1%, and
L̃ = 21%. β = 500 leads to ω/ΩH ≈ 0.42, Ė (EM ) /Ė (M A) = −1.2%, and L̃ = 8.5%. Notice that
a higher spatial resolution is required to fully resolve weak field models, although all runs in this
comparison were done at 2562 ; the decrease in electromagnetic energy extracted at β = 500 may
therefore be due to resolution.
We also vary the field geometry from the single loop used in our fiducial model, which has
vector potential Aφ ∝ M AX(P/Pmax − 0.2, 0). We do this by multiplying the vector potential by
sin(log(r/h)) or | sin(2θ)|. The former decompresses the field lines at the inner radial edge giving a
field strength that is more uniform around the loop (for an extended disk this would yield a sequence
of field loops centered at the midplane with alternating sense of circulation). The latter yields two
loops, one centered above the equator and the other below, with the same sense of circulation.
The sin(log(r/h)) modulation gives ω/ΩH ≈ 0.44, Ė (EM ) /Ė (M A) ≈ −3.6%, and L̃ = 42%. The
| sin(2θ)| modulation gives ω/ΩH ≈ 0.40, Ė (EM ) /Ė (M A) ≈ −1.1%, and L̃ = 8.3%. Increasing the
148
number of initial field loops therefore leads to a weak (factor of 2 − 3) decrease in Ė (EM ) /Ė (M A) ,
while making the field strength more uniform around the loop increases L̃ by a factor of 2 with a
nearly constant Ė (EM ) /Ė (M A) . Higher resolution studies may better resolve these simulations and
show weaker dependence on field geometry.
We have also considered a purely vertical field geometry: Aφ ∝ r sin θ. In a Newtonian context
this would correspond to a uniform z field in cylindrical coordinates. The field is normalized so
that β = pgas,max /pmag,max = 100 and 400 in the equator of the torus. The outcome is different
from any of the other models.
The funnel field in the vertical field run is strong compared to the disk field. The accretion rate
is larger, by a factor of 5, than the fiducial run. In the early stages there is a brief net outflow of
energy from the black hole (although the total energy released from the hole is negligible compared
to the energy gained at later times). The β = 100 model has a high mean efficiency; Ė/Ṁ0 = 0.77,
compared to 0.82 expected for a thin disk. There is also a net outflow of angular momentum from
the black hole, with L̇/Ṁ0 = −1.00, compared to 1.95 expected for a thin disk. The wind has a peak
asymptotic radial velocity ṽ r = 0.94c, attained near the outer boundary, compared to ṽ r = 0.75c
for the fiducial run. Finally, the model has ω/ΩH ≈ 0.41, Ė (EM ) /Ė (M A) ≈ −15%, and L̃ = 79%.
The β = 400 vertical field model has very similar properties, which suggests that we are resolving
the β = 100 model. Table 4.3 summarizes measurements from the varying field geometry models.
The models with net vertical field exhibit markedly different behavior from the fiducial model.
It seems likely that some of this difference is due to the axisymmetric nature of the model; in 3D
matter can accrete between the vertical field lines without having to push them into the hole. That
is, in 3D, it would be easier for the hole to rid itself of the dipole moment that it acquires in the
net vertical field calculation. But we cannot say with any confidence what the outcome is until a
full 3D experiment on a disk with nonnegligible magnetic dipole moment.
4.5.3
Numerical Parameters
We have run the fiducial model at resolutions of 642 , 1282 , 128 × 64, 2562 , and 4562 . There is a
weak dependence on resolution in the sense that Ė (EM ) /Ė (M A) is smaller at higher resolutions.
Lower resolution models do not sustain turbulence for as long as high resolution models, so we
average over 500 < t < 1000, when all models are turbulent. Table 4.4 gives a summary of results
from the resolution study. In every case the nominal radiative efficiency is close to the thin disk
value.
Resolution of the near-horizon region, where the energy density is large, is also a concern,
because our accretion rates are measured there. We have checked dependence on radial numerical
resolution of the near-horizon region by modifying the coordinate definition in equation (7) to read
r = R0 + ex1 rather than r = ex1 . Increasing R0 from 0 to the horizon radius increases the number
of grid zones located near the horizon. We ran a model with R0 = 0.5 and found no significant
difference from a comparable model with R0 = 0. This suggests that we are adequately resolving
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Table 4.4.
Resolution Study
Resolution
Ė (EM ) /Ė (M A)
Ė/Ṁ0
L̇/Ṁ0
ȧ/Ṁ0
Ṁ0
L̃
642
128 × 64
1282
2562
4562
-0.0528
-0.0438
-0.0447
-0.0316
-0.0261
0.914
0.841
0.887
0.874
0.865
1.630
1.420
1.518
1.274
1.381
0.036
0.121
0.087
0.198
0.216
-0.159
-0.165
-0.167
-0.186
-0.299
0.55
0.23
0.38
0.27
0.18
Note. — Numerator and denominators are separately time averaged
from 500 < t < 1000 at the horizon. This interval is chosen so that all
models are turbulent (in the lowest resolution model turbulence decays
shortly after t = 1000). The 4562 model is the fiducial model. The nominal radiative efficiency is 1 − Ė/Ṁ0 . A positive ȧ/Ṁ0 corresponds to a
spindown of the black hole because Ṁ0 < 0.
the near-horizon region.
We also varied rin and rout and found no measurable difference in Ė (EM ) , Ė (M A) , (B r )2 , and ω
on the horizon. We have moved rout from 40 to 400 and rin from 0.7r+ to 0.98r+ and find negligible
differences in these quantities on the horizon. The solution is not sensitive to the location of the
inner or outer boundary. Moving the inner boundary of the computational domain outside the
horizon (e.g. 1.05r+ ) leads to strong reflections from the boundary conditions and, ultimately,
failure of the run. It is possible that better inner boundary conditions or higher resolutions could
overcome this difficulty, but it seems cleaner to simply leave the boundary inside the event horizon
at r = 0.98r+ , out of causal contact with the rest of the simulation.
The model with larger rout = 400 does exhibit some new features. The magnetic field lines in
the funnel region have a collimation factor of 10 by the time they reach the outer boundary. At
R = 40, however, both the rout = 400 model and the fiducial model have a collimation factor of
5/2. By rout = 400 the field lines are nearly cylindrical. The peak of the radial component of the
asymptotic 3-velocity in the wind is identical to the fiducial run with ṽ r = 0.75c, indicating little
acceleration between R = 40 and R = 400.
The main numerical uncertainty in our experiments arise from the floor on the density and
internal energy. We varied the floor scaling from ρ0,min = 10−4 r−3/2 and umin = 10−6 r−5/2 to
ρ0,min = 10−4 r−2.7 and umin = 10−6 r−3.7 (we chose these scalings so that b2 /ρ0 would be nearly
constant with radius in the funnel). While this significantly affects b2 /ρ0 , it does not otherwise
affect Ė (EM ) and Ė (M A) or the mean values of B r and ω measured on the horizon.
We varied the floor normalization at r = 1 from the fiducial values (ρ0,min , εmin ) = (10−4 , 10−6 )
to (10−5 , 10−7 ), and (10−6 , 10−8 ). This causes almost no change in the flow near the horizon. In
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the funnel, however, we are at the limit of our ability to integrate the MHD equations (b2 /ρ0 À 1).
Our integration fails when we attempt to use a mass density floor ρ0,min (r = 1) . 10−5 when
rout À the outer edge of the initial torus. Lower floors lead to faster outflows ṽ r À 0.99c in the
funnel region, which are more likely to be numerically unstable. These results hint that low density
models will produce fast outflows, but a confirmation awaits a more stable GRMHD algorithm.
The funnel region is difficult to integrate reliably, because when b2 /ρ0 À 1 small fractional errors
in field evolution lead to large fractional errors in the evolution of other flow variables. This is a
consequence of our conservative scheme, in which all the dependent variables are coupled together
by the interconversion of primitive and conserved variables. Evolution of the MHD equations in
nonconservative form (e.g. using an internal, rather than total, energy equation), as in De Villiers
and Hawley (2003a), may be slightly more robust, although De Villiers and Hawley eventually
experience similar problems in the funnel. In any event, the close correspondence between the
numerical experiment and the BZ model raises confidence in the results and suggests that the
magnetic field, if not the mass density and internal energy density, is being evolved reliably.
4.6
Discussion
We have used a general relativistic MHD code, HARM, to evolve a weakly magnetized thick disk
around a Kerr black hole. Our main result is that we find an outward electromagnetic energy flux on
the event horizon, as anticipated by Blandford and Znajek (1977). The funnel region near the polar
axis of the black hole is consistent with the Blandford-Znajek model. The outward electromagnetic
energy flux is, however, overwhelmed by the inward flux of energy associated with the rest-mass
and internal energy of the accreting plasma. This result essentially confirms work by Ghosh and
Abramowicz (1997) that suggested the BZ luminosity should be small or comparable to the nominal
accretion luminosity (L̃ . 1).
One of our models discussed here, however, begins with a vertical field threading the torus,
exhibits a brief episode of outward net energy flux. This appears to be a transient associated with
the initial conditions. The same model exhibits a steady net outflow of angular momentum from the
black hole. Of all our models, the vertical field model has the largest negative −Ė (EM ) /Ė (M A) ≈
15% (ratio of the electromagnetic energy flux to ingoing matter energy flux) and largest L̃ =
Ė (EM ) /(−²Ṁ0 ) ≈ 80% (ratio of electromagnetic luminosity to nominal accretion luminosity). This
suggests that the BZ effect could play a significant role if the disk has a net dipole moment and
accumulates magnetic flux that crosses the horizon. This possibility will be considered in future
work.
Consistent with the results found earlier by De Villiers et al. (2003a), we find that our models
can be divided into four regions: (1) a “funnel” region with b2 /ρ0 & 1 and β ¿ 1; (2) a corona
with 1 . β . 3; (3); an equatorial disk with β > 3; and (4) a plunging region between the disk
and event horizon with β ∼ 1 and a nearly laminar inflow from the disk to the black hole. We also
find no feature in the surface density at or near the ISCO (see Figure 4.10), which agrees with the
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results by Krolik and Hawley (2002); De Villiers et al. (2003a) and consistent with Reynolds and
Begelman (1997). This is contrary to the sharp transition predicted by thin disk models and used
by XSPEC to fit X-ray spectra.
We have shown that the funnel region is nearly force-free, and is well-described by the stationary
force-free magnetosphere model of Blandford and Znajek (1977), for which we have presented a selfcontained derivation in Kerr-Schild coordinates. We find agreement between the BZ model and our
simulations in measurements of energy flux, magnetic field, efficiency of accretion, and spindown
power output. In all cases we find that in the force-free region the field rotation frequency is about
half the black hole spin frequency, ΩH ≡ a/(2r+ ). This spin frequency maximizes electromagnetic
energy output from the hole. This result is consistent with expectations of MacDonald and Thorne
(1982) and the force-free numerical results of Komissarov (2001).
We have also compared the time-average of the plunging region in our fiducial model with the
stationary MHD inflow model of Gammie (1999), which assumes that the flow matches a cold disk at
the ISCO. The inflow model matches the simulated rest-mass flux and electromagnetic flux of energy
and angular momentum surprisingly well, particularly considering the strongly variable nature of
the simulated flow in the plunging region. The inflow model fails to match other aspects of the
flow, such as the radial component of the four-velocity. This is mainly due to the finite temperature
of the simulated flow; the inflow solution assumes zero temperature. It is slightly surprising that
the total angular momentum flux is close to the value predicted by the zero temperature inflow
solution, and 20% less than what is predicted by the thin disk, yet the total energy flux is almost
exactly what is predicted by the thin disk. It is as yet unclear whether this is due to coincidence
or conspiracy.
For a set of models similar to the fiducial model, the ratio of electromagnetic to matter energy
fluxes is sensitive to the black hole spin, reaching −7% for a ∼ 1. The evolution is sensitive to the
initial field geometry. Models with a net vertical field are more efficient, and more electromagnetically active than models with comparable field strength but zero net vertical field. Our models have
a weak dependence on resolution in the sense that as resolution increases the relative importance
of electromagnetic energy fluxes on the horizon diminishes.
With an rout = 400 model we demonstrate that an outgoing electromagnetic energy flux can
reach large radii. The field in the funnel region does not connect back into the disk. Rather the
poloidal components lie parallel to the polar axis. The field lines are collimated by a factor of 5/2
at r = 40 and by a factor of 10 at r = 400. An outflow along the boundaries of the funnel reaches
a maximum ṽ r ≈ 0.75c, but this is sensitive to the value of our artificial density “floor”: a model
with lower density reaches even larger radial velocities at the outer boundary of the computational
domain.
Koide et al. (2002) have evolved a cold, highly magnetized uniform density plasma (ρ0 /p =
0.06, b2 /ρ0 = 10) as it falls into a rapidly spinning (a = 0.99995) black hole in Boyer-Lindquist
coordinates for a time ≈ 14GM/c3 using the MHD approximation. This initial state does not
correspond to an accretion disk system. They demonstrated, however, that a transient net energy
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extraction is possible from a spinning black hole. Because of the short evolution time they are
unable to say whether the energy extraction process is possible in steady state. Koide (2003) gives
an expanded discussion of the above system.
In contrast to the results of Koide et al. (2002) and Koide (2003), we model a disk with an
initially hydrodynamic equilibrium fluid that is weakly magnetized. We also use a Kerr-Schild (horizon penetrating) coordinate system that avoids potential problems associated with the treatment
of inner boundary condition in Boyer-Lindquist coordinates. In our simulation the Balbus-Hawley
instability drives turbulence and accretion in a steady state where we evolve for a time 2000GM/c3 .
We measure a sustained outward electromagnetic energy flux that is smaller than the inward matter
energy flux (i.e. net inward energy flux). Their model is evolved for too short a time to observe the
unbound mass outflow in the funnel region as seen by us and De Villiers et al. (2003a, 2004).
De Villiers et al. (2004)(hereafter DH) have also considered the numerical evolution of weakly
magnetized tori around rotating black holes. Their models are quite similar to ours in many
respects, although they differ in that: (1) their models are three dimensional while our models are
axisymmetric; (2) they use a nonconservative numerical method (De Villiers and Hawley, 2003a);
(3) DH use Boyer-Lindquist while we use Kerr-Schild coordinates; (4) DH choose γ = 5/3 while
we use γ = 4/3; (5) DH’s initial pressure maximum is located at 25M , while ours are typically at
12M . Our results for the energy and angular momentum per baryon accreted from Table 4.2 can be
compared to Table 1 of DH by computing Ė/Ṁ0 = ∆Ei /∆Mi and L̇/Ṁ0 = ∆Li /∆Mi . For models
with a = (0, 0.5, 0.9) DH find Ė/Ṁ0 = (0.91, 0.91, 0.84) while we find Ė/Ṁ0 = (0.96, 0.93, 0.88).
For the same models DH find L̇/Ṁ0 = (3.1, 2.6, 1.9), while we find L̇/Ṁ0 = (3.1, 2.6, 1.7). Given
the differences in the models and numerical methods, this quantitative agreement is remarkable.
Our models and De Villier and Hawley’s models also agree qualitatively in the sense that both show
a similar geometry of disk, corona, and funnel and both imply that spin equilibrium is achieved at
a ∼ 0.9 (see Gammie et al. 2004).
Komissarov (2001) finds the BZ solution to be stable in force-free electrodynamics, and Komissarov (2002a, 2004a) find the BZ solution to be causal, but inconsistent with the membrane
paradigm. We find our numerical solutions to be consistent with the BZ solution in the low-density
funnel region around the black hole. A numerical general relativistic MHD study of strongly magnetized (monopole magnetic field) accretion by Komissarov (2004b) is also consistent with the BZ
solution. For the strong field chosen he finds a considerably faster outflow (Lorentz factors of ≈ 14)
than found in our models (Lorentz factors of ≈ 1.5 − 3.0). Komissarov’s model does not contain a
disk.
The limitations of the numerical models presented here include the assumption of axisymmetry
and a nonradiative gas. The effect of axisymmetry can be tested by comparing our models with
the three dimensional models of De Villiers et al. (2004); the angular momentum and energy per
accreted baryon in the two models differs by only a few percent. In addition the jet structure
observed in De Villiers et al. (2004) is nearly axisymmetric. This is encouraging, although it is
unlikely that an axisymmetric calculation can capture the full range of possible dynamical behavior
153
in the accretion flow.
The radiation field, which we have completely neglected here, is likely to play a significant role
in the flow dynamics, through radiation force on the outflowing plasma in the wind and through
photon bubbles in the disk Gammie (1998); Socrates and Blaes (2002). It will also, of course, play
a significant role in heating and cooling the plasma. This is clearly the most significant limitation
of our calculation– particularly from the standpoint of comparison with observations– and clearly
the most numerically difficult problem to overcome.
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5
Future Studies
For future work, I plan to perform simulations that will 1) establish the power of the BZ effect as a
function of disk thickness ; 2) determine the connection between the disk and the jet; 3) determine
the relationship between the black hole spin and jet; and 4) establish the requirement of GRMHD
to jet speed, collimation, and stability. I also plan to specifically study the collapsar model of GRBs
using a GRMHD model with additional GRB-type microphysics.
5.1
Thin Disks
Chapter 4 presented a study of the BZ luminosity of a rotating black hole surrounded by a thick
disk, for which the scale height (H) is comparable to the radius (R). I plan to extend those
results to the thin disk regime, which may be applicable to some black hole accretion disk systems.
Prior analytic work suggests that thin disks may have an insignificant BZ luminosity (Ghosh and
Abramowicz, 1997; Livio et al., 1999). If thin disks have a weak BZ-effect and if the energy from
the BZ-effect dominates the energy content of a jet in black hole systems, then those systems with
a black hole and jet most certainly have a thick accretion disk near a rapidly rotating black hole.
A self-consistent treatment of radiation generally leads to a disk with a radially dependent
H/R, especially for disks near the black hole event horizon. While radiative effects are important,
a general relativistic treatment is non-trivial (see, e.g., Cardall and Mezzacappa 2003; Cardall
2004). I plan to model thin disks with an ad hoc cooling function that sustains a constant H/R
for all radii. The proposed H/R study may be sufficient to obtain an estimate for the BZ power
output as a function of H/R without resorting to radiative transport.
Compared to an otherwise equivalent thick disk simulation, the planned thin disk simulation
requires a larger radial numerical resolution in order to resolve the MRI. Resolving the MRI, with
about 6 zones per fastest growing wavelength, is crucial in sustaining turbulence for a sufficient
period of time to reach steady state. For example, a typical thick (H/R ∼ 0.26) disk simulation
requires a resolution of 256 × 256. A thin disk model with a = 0 and H/R = 0.01 requires a
resolution of 6500 × 1200 and a thin disk model with a = 0.938 and H/R = 0.02 requires a
resolution of 9800 × 920. Also, in order to avoid artificially loading the (larger) funnel region with
baryons, the thin disk model requires a lower density floor than the thick disk model. Also, a lower
density floor is required for convergence of the magnetic field in the funnel region. Using a lower
density floor (and thus increasing b2 /ρ0 ) may require improving the algorithms used in HARM.
155
5.2
Relativistic Jet-Disk Connection
GRMHD computational studies are on the verge of understanding the mechanism of jet formation
from an accretion disk around a rotating black hole (McKinney and Gammie, 2004; De Villiers et al.,
2004). These numerical models are beginning to identify 1) the mechanism of jet acceleration ; 2)
the relative contributions of electromagnetic and kinetic energy to the jet; and 3) the baryon loading
in the jet – i.e. the amount of rest-mass in the jet. The resolution of these issues are one focus of
my future study, and some work has begun as described below.
The source of energy generating a jet could arise from the BZ mechanism (an electromagnetic
black hole wind), from the Blandford-Payne mechanism (a matter disk wind), from an electromagnetic disk-corona wind, radiation driving, or some combination of all these processes. I performed
GRMHD simulations of rapidly rotating (a & 0.5) black holes that are spun-down by the BZ-effect,
which generates an outgoing electromagnetic energy flux in a mildly relativistic (v/c ∼ 0.3), collimated outflow that reaches large distances. The black hole spin energy is extracted directly from
the rotating black hole and flows along collimated field lines frozen within unbound, baryon-poor
matter. For the particular parameter space I explored, the Poynting flux always dominates the
matter-energy flux in the jet.
There is a small matter-energy jet at the edge between the corona and funnel in our models with
high black hole spin. The electromagnetic component of this “disk wind” appears to only provide
a source of acceleration for the outgoing matter in the funnel region, since the fluid is tied-up in
the magnetic field within the corona. Hawley’s group at Virginia, and our group, have performed
simulations that show that black hole spin plays a significant role in determining the existence and
power of a jet, but the details are only beginning to be studied (De Villiers et al., 2004).
5.3
Gamma-Ray Bursts: GRMHD Collapsar Model
I plan to use a GRMHD model to study the accretion disk that likely forms during a GRB, in
the framework of the collapsar model (Woosley, 1993; Paczyński, 1998; MacFadyen and Woosley,
1999). Some interesting unresolved questions include 1) what fraction of the observed luminosity of
GRBs is due to power by black hole spin energy extraction, accretion disk luminosity, or neutrino
annihilation, 2) what is the mechanism for jet production, collimation, and GRB variability ; 3)
what is the efficiency of neutrino annihilation and magnetic field reconnection ; and 4) is the jet
composed of ion/electron or positron/electron pair plasma and what particle/radiation physics
dominates GRB radiation.
Numerical studies of the Type I collapsar model have been performed using Newtonian hydrodynamic (HD) (MacFadyen and Woosley, 1999) and special relativistic HD models (Aloy et al.,
2000; Zhang et al., 2003). Type II collapsars have been studied using Newtonian HD numerical
models (MacFadyen et al., 2001). All of these models include a realistic EOS, neutrino cooling, and
some estimates of neutrino annihilation. These unmagnetized numerical models include an ad hoc
156
alpha-viscosity model of angular momentum transport (Shakura and Sunyaev, 1973). However, the
magnetic field is likely explicitly required to launch and collimate jets (Cameron, 2001; Wheeler
et al., 2002), and the angular momentum transport is likely driven by the MRI (Balbus and Hawley,
1991).
Recent numerical studies of non-self-gravitating accretion disks use the nonrelativistic MHD
approximation (Proga et al., 2003) and use a code based on ZEUS (Stone and Norman, 1992).
They include a realistic EOS, photodisintegration of helium, and neutrino cooling. They observe
a neutrino luminosity of ∼ 1052 erg/s, which is close to the expected GRB luminosity. However,
their model does not include neutrino annihilation, so their jet contains no energy from annihilating
neutrinos that may alter the jet structure. Also, they cannot estimate what fraction of the energy
in the jet comes from neutrino annihilation compared to electromagnetic luminosity from the disk.
Neutrino-generated jets are likely important (see, e.g., Fryer and Mészáros 2003). They observe a
jet that is dominated by Poynting flux, rather than matter-energy flux. However, since their model
is Newtonian, they are unable to measure the Lorentz factor of the jet and cannot model a spinning
black hole, which may be required to generate a Poynting flux via the BZ-effect.
As discussed above, I used a similar ZEUS-based MHD code with a pseudo-Newtonian potential
(Paczyński and Wiita, 1980) to study similar flows without microphysics. My results suggest that
their studies of the jet could be dominated by numerical artifacts. For example, I found it impossible
to remove the inner radial boundary condition from causal contact with the rest of the flow (i.e. one
cannot keep the inner radial boundary inside the fast point in MHD). Also, while I found a large
Poynting flux from the horizon, the Newtonian model should not be capable of producing such an
effect. The results of Proga et al. (2003) are likely similar. Most of the Poynting flux energy could
be artificially generated by contact between the inner radial boundary and the flow at larger radii.
Indeed, GRMHD simulations show that accreting black holes with spin parameter a = 0 generate
weak outflows and much smaller Poynting flux at large distances (McKinney and Gammie, 2004).
GRMHD numerical models suggest Poynting flux jets are a natural result of accretion around
a rapidly rotating black hole (De Villiers et al., 2003a; Koide, 2003; McKinney and Gammie,
2004). However, such simulations ignore possibly important sources of jet energy, such as neutrino
annihilation, and lack accurate models of reconnection (see, e.g., Sikora et al. 2003). Recent
GRMHD collapsar simulations have been performed that fail to generate the estimated observed
GRB energy and Lorentz factors (Mizuno et al., 2004a,b). However, their model includes no
microphysics; they use a shock model, rather than the core of a massive star, as their initial
conditions; and the simulation lasts for only a small number of dynamical times (∼ 200GM/c3 ).
The black hole accretion disk system itself may not generate the large Lorentz factors. For example,
a radial pressure gradient in the star and external atmosphere can accelerate the jet to Γ ∼ 44,
close to that required by observations (see, e.g., Aloy et al. 2000; Zhang et al. 2003).
The current GRMHD numerical model has an ideal-gas equation of state and includes no
radiation of neutrinos. To the current GRMHD numerical model, I plan to add a realistic equation
of state (EOS) and neutrino cooling and annihilation in the optically thin parts of the flow. I plan to
157
first include an approximate EOS (Popham et al., 1999; Kohri and Mineshige, 2002), which includes
electrons, positrons, protons, neutrons, alpha particles, and photons as a single-component fluid. All
particles are assumed to be in thermodynamic and nuclear statistical equilibrium. This approximate
EOS allows the electrons, positrons, protons, and neutrons to have (fairly) arbitrary relativity and
degeneracy. I later plan to include an more accurate EOS solver that allows for any relativity
and degeneracy (Lattimer and Swesty, 1991; Lattimer, 1996; Blinnikov et al., 1996; Lattimer and
Prakash, 2000). The disk is optically thick to all particles except potentially neutrinos, so the
single-component fluid approximation is not applicable to neutrinos. Since photons and positrons
are in thermal equilibrium, they will be treated as an additional source of radiative pressure. The
photodisintegration of alpha particles and recombination of free nuclei into alpha particles will also
be included.
Neutrinos are to be included as part of the fluid for parts of the disk that are optically thick
to neutrinos and otherwise stream freely for the optically thin parts. The neutrinos can radiate in
the neutrino “optically” thin parts of the disk (see, e.g., Itoh et al. 1996; MacFadyen and Woosley
1999), while neutrinos provide a pressure in the neutrino “optically” thick parts of the flow. This
neutrino pressure can be estimated as if optically thick, but with an extinction factor in the radiative
neutrino emissivity. As a result, the pressure is reduced by a corresponding factor (Lee et al., 2004).
A diffusion scheme for the radiation of neutrinos in the optically thick regime will then be applied.
If required, a full Boltzmann transport scheme will be used (see, e.g., Cardall and Mezzacappa 2003;
Cardall 2004). Initially, I do not plan to include nuclear burning, since it is likely only relevant to
the supernova component and not to the GRB jet.
158
A
Model of Plasmas
This appendix discusses the assumptions of the fluid, MHD, and ideal MHD approximations of
a plasma (for a classic review see, e.g. Landau and Lifshitz 1959, 1980). The purpose of this
appendix is to present an overview of the parameters estimated in Section 1.5, which shows that
the single-component ideal MHD approximation is valid for the accretion flow likely to be present in
GRBs, X-ray binaries, and AGN. As described there, only thick accretion disks, with their diffuse
plasmas, are not explicitly treatable by the fluid approximation. However, the solar wind too
cannot be explicitly treated as a fluid, yet the MHD approximation is a reasonable approximation
(see, e.g., Usmanov et al. 2000). Plasma instabilities likely force the solar wind to behave as a fluid
(Feldman and Marsch, 1997), and a similar phenomena may occur in thick disks. A discussion of
plasma instabilities is beyond the scope of this thesis, so the discussion below is restricted to the
fluid, MHD, and ideal MHD approximations. First, a general overview of these approximations is
given.
If a gas predominantly collides elastically and contains many particles moving randomly within
a characteristic length scale (L), then the gas can be approximated as conforming to a space-velocity
distribution function. Liouville’s theorem and expressions for the body-forces are used to derive
the distribution evolution equation. This results in the so-called Vlasov or collisionless Boltzmann
equation. Collisions can be introduced as a source or sink of particles within the distribution,
resulting in the collisional Boltzmann equation. The fluid approximation further assumes that
these collisions drive the system through a series of statistical equilibrium states.
The fluid approximation can be derived as a series of mass-weighted velocity moments of the
Boltzmann equation. The MHD approximation can be derived by including a series of chargeweighted velocity moments of the Boltzmann equation. In general, the fluid has many species,
but these can often be reduced into a single-component fluid or MHD approximation. The singlecomponent approximation is obtained by summing each of the equations over species and by appropriately defining the average quantities. For perfect (ideal) fluids, collisions only drive the system
to an equilibrium state, and other effects on the distribution function are ignored. For imperfect
fluids, a closure scheme, such as the asymptotic closure scheme of Chapman-Enskog (Chapman and
Cowling, 1953), is used to relate higher order (say, in ² = λmf p /L) collisional terms as functions
of fluid quantities obtained from the velocity moments, where λmf p is the mean free path between
particle collisions.
The discussion that follows does not present a text-book derivation of the fluid, MHD, and ideal
MHD approximations, but rather presents a summary of the salient features of the approximations
and gives parameters for estimating the validity of the approximations. Note, however, a general
159
plasma is sufficiently complicated that no set of dimensionless parameters can absolutely determine
whether each part of the approximations, described below, are valid. Specific configurations of
matter and magnetic field can develop macroscopic instabilities, for example as with the magnetorotational instability (Balbus and Hawley, 1991). Also, microscopic particle effects can build up
to become macroscopically relevant, for example during reconnection (see, e.g., Parker 1963, 1979;
Syrovatskii 1981; Biskamp 1986; Taylor 1986; Parker 1988; Masuda et al. 1994; Low 1996) or due
to plasma instabilities (see reviews by, e.g., Melrose 1986; Hasegawa 1975; Begelman and Chiueh
1988). Only a full nonlinear study can verify the dimensional arguments presented below.
A.1
Equation of State
An equation of state (EOS) is needed to close the fluid or MHD equations of motion. The EOS
of matter near the black hole depends mostly on the density and temperature if the system is
in nuclear and thermodynamic statistical equilibrium. Thermodynamic statistical equilibrium is
maintained if all species are in thermal contact through collisions. Thus, one needs to know the
elastic and inelastic scattering and absorption cross sections.
The elastic scattering cross sections can be used to estimate not only the validity of the fluid
approximation, but also to estimate the thermodynamic coupling between species. If some species
are thermally coupled faster than the accretion time scale, then they will equilibrate to similar
temperatures. However, if they do not transfer energy by collisions faster than they are accreted,
then they could decouple, resulting in a two-temperature flow for a gas of protons and electrons. For
example, it is plausible that in X-ray binaries and AGN a hot two-temperature disk forms (Shapiro
et al., 1976; Narayan and Yi, 1995). In two-temperature or disk-corona models, the electrons can
cool by Comptonization, while the protons and nuclei have no mechanism to cool efficiently before
being accreted by the black hole. This effect is neglected for the EOS discussed here.
The particle distributions need not be thermal, nor are observations constrained to only observe
the most average electrons within the thermal distribution. Low-frequency radio observations of
SgrA* could be explained if a small fraction of electrons reside in a power-law tail of energies (Yuan
et al., 2003). Non-thermal electrons may be generated in reconnection events, such as has been
observed in tokamak experiments of high density plasmas generated by MHD modes (Savrukhin,
2002) or in solar flares (Priest, 1981; Kahler, 1992; Zank and Gaisser, 1992; Miller et al., 1997;
Priest and Forbes, 2002).
A.1.1
EOS for GRBs
Assuming that GRBs arise from a stellar core-collapse scenario (Woosley, 1993; Paczyński, 1998;
MacFadyen and Woosley, 1999), the accretion disk formed during a GRB likely has mostly free
neutrons, protons, alpha particles, electrons, positrons, various heavy nuclei, and neutrino and
photon radiation (see, e.g., MacFadyen and Woosley 1999). In thermodynamic and nuclear statistical equilibrium, the GRB disk state can be treated using an ideal Fermi EOS with arbitrary
160
degeneracy for all species and arbitrary relativity parameter (x) for all species, where x ≡ kb T /mc2
for a species with mass m (Blinnikov et al., 1996). Specifically, the approximations of Kohri and
Mineshige (2002) are used in this thesis. A modified form of equation (26) in Kohri and Mineshige
(2002) is used to improve the accuracy of the results near the transition between degenerate and
non-degenerate electrons (Q = mn − me is reintroduced into that equation, as required for consistency). A Fermi EOS leads to important deviations from an ideal gas EOS due to neutronization
in thin disks that may form during a GRB. The Fermi EOS determines the fraction of electrons
per baryon (Ye ).
Fully ionized helium (alpha particles) can be dynamically important in the accretion shock that
develops during GRB disk formation. The accretion shock, where nuclei are photodisintegrated
into free nucleons, is located at r ∼ 120GM/c2 in the collapsar model (MacFadyen and Woosley,
1999). Also, within the neutrino-cooled disk or after the free nucleons are ejected out in a wind,
they recombine to form heavier nuclei (e.g., alpha particles and
56Ni).
Indeed, nucleosynthesis
is important for understanding the generation of supernova and the GRB-supernova connection.
These estimates ignore the supernova component and focus on the GRB event itself, which likely
does not require following nucleosynthesis (MacFadyen and Woosley, 1999). This will at most
require understanding the photodisintegration and recombination of alpha particles. If nuclear
statistical equilibrium (NSE) holds (Khokhlov, 1989), then a Saha-like equation can be used to
determine the fraction of free nucleons from
µ
Xnuc = 30.97
T
1010
¶9/8 ³
ρ ´−3/4
exp
1010
Ã
−6.096
( 10T10 )
!
(A.1)
(Woosley and Baron, 1992). The remainder of nucleons are bound in α particles.
The gas pressure (pgas ) is defined as the sum of each species’ (proton, electron, neutron, alpha
particles, nuclei, radiation, and neutrino) partial pressure, with baryons or electrons being either
completely nondegenerate or completely degenerate, and completely relativistic or completely nonrelativistic (Kohri and Mineshige, 2002). It has been found that such approximate equations of state
agree to within 10% (Popham et al., 1999) of more general calculations by Blinnikov et al. (1996).
In this approximation, if the disk is optically-thick to photons, then the photons are treated as a
radiation pressure. If the disk is optically thin to photons, then photons stream out with a pressure
extinction factor of about 1 − exp(−τγ ), where τγ is the photon optical depth given in equation 1.3.
The local cooling rate would be multiplied by exp(−τγ ). This does not model radiative transport
processes that may lead to radiative instabilities, such as the photon-bubble instability (Gammie,
1998).
The same radiation pressure extinction prescription is applied to the neutrinos (Lee et al., 2004)
with a neutrino optical depth of
τν ∼ τs,ν + τa,eN + τa,ν ν̄ ,
where τs,ν = 2.7 × 10−7
³
Tp
1011
´2 ¡
ρ ¢
H,
1010
τa,eN = 4.5 × 10−7 Xnuc
161
(A.2)
³
Tp
1011
´2 ¡
ρ ¢
H,
1010
τa,ν ν̄ = 2.5 × 10−7
³
Tp
1011
´5
H, Tp is the proton temperature, and H is a typical disk scale height
(Di Matteo et al., 2002). These three terms respectively correspond to 1) neutrino-neutron/proton
scattering ; 2) neutrino absorption onto free protons or neutrons ; and 3) inverse electron-positron
pair annihilation (i.e. inverse URCA). These processes dominate the inverses of the bremsstrahlung
and plasmon neutrino emission processes, which are thus neglected. For an overview, see also
Shapiro and Teukolsky (1983) §18.5. In the electron degeneracy regime, the neutrino emission
is slightly reduced due to a reduction of Ye (Kohri and Mineshige, 2002). The effect of electron
degeneracy on the optical depth is not accounted for in this thesis, so the above estimated τν should
be considered a reasonable estimate, but strictly only an upper bound.
Some numerical studies of simplistic pseudo-analytic viscous (unmagnetized) Fermi gas models
of GRB disks have been performed (Popham et al., 1999; Narayan et al., 2001; Kohri and Mineshige,
2002; Di Matteo et al., 2002). GRMHD (or even magnetized) numerical models of accretion disks
that incorporate a self-consistent Fermi gas EOS have yet to be performed by anyone. In Section 1.5,
where Ye , pgas , and other quantities are reported for the GRB disks, the relation between mass
density, pressure, and magnetic field is actually from the GRMHD numerical model, rather than
from a self-consistent model that includes a Fermi EOS. Section 1.5 reports the ratio of Fermi
gas pressure to ideal gas pressure to check the consistency of this actually inconsistent approach.
It turns out that this approach is fairly consistent, and certainly good enough for estimating the
validity of the fluid, MHD, and ideal MHD approximations.
A.1.2
EOS for X-ray binaries and AGN
The species’ abundances in accretion disks in X-ray binary and AGN are most likely similar to
cosmic abundances. A disk with cosmic abundances would be composed of mostly ionized hydrogen,
some ionized helium, few ionized metals, electrons, and photon radiation (Fabian, 1998). For the
temperatures and densities of these accretion disks, the appropriate approximate EOS is simply
an ideal gamma-law gas + radiation pressure (which is simply a reduced form of the GRB EOS
described above). This EOS is used to compute the actual structure of the accretion disk for these
objects using SS73 thin disk (Shakura and Sunyaev, 1973) or ADAF thick disk (Narayan and Yi,
1995) models.
A.2
Validity of the Fluid Approximation
This section summarizes the parameters used to estimate the validity of the fluid approximation.
The fluid approximation is valid only if the characteristic time (T ) is much longer than the effective
collision time τ . Also, the effective mean free path for particles λmf p must be much less than
the length scale (L) of the problem. An “effective” collision is one that has any nonzero cross
section but generates a 90◦ change in the direction of motion. Essentially all particles interact at
some density and energy, but this section only considers the dominant, typically elastic, scattering
162
mechanisms. Most elastic scattering processes dominate inelastic scattering when energies are less
than the mass energy of a particle.
In order to determine the mean free path, we should consider the total scattering cross section
σ = σ(E), where E is the center of mass energy. The effective mean free path for particle type A
to collide with particle type B is
p
µAB /mA
,
σAB nB
λmf p,A−B =
(A.3)
where nB is the number density of particle type B, and µAB = mA mB /(mA + mB ) is the reduced
mass. Notice that the mean free path for particle A to hit particle B is not necessarily the same as
the mean free path for particle B to hit particle A.
To determine the collisional rate between each species, the relativistic relative velocity
vA + vB
1 + (vA vB /c2 )
vrel,A−B =
(A.4)
is used to estimate the mean relative velocity within a thermal distribution. This expression for the
relative velocity assumes that each species is an ideal gas moving with relativistic thermal velocity
p
x(2 + x)
,
1+x
vtherm = c
(A.5)
where x = kb T [K]/(mc2 ) is the relativity parameter. This is simply the relativistic extension of
p
the nonrelativistic thermal velocity vtherm,non−rel. ∼ KE/m. The relative velocity can be used
to find the binary collision rate
νc,A−B =
vrel,A−B
.
λmf p,A−B
(A.6)
The proton-electron scattering cross section σpe is determined by the Coulomb interaction. The
nonrelativistic Coulomb energy U (r, v) = 12 mv 2 − Ze2 /r of one charged particle in the electrostatic
field of another vanishes U (r, v) = 0 at a radius
λc =
Ze2
,
kb Tp
(A.7)
where Z is the atomic number. This is the characteristic radius for interaction and thus σZe ∼ λ2c .
³
´2
1+x
Relativistic corrections, of order unity, give σZe ∼ λ2c x(2+x)
csc4 (θ/2), where x = kb Tp /(me c2 )
and θ is the scattering angle. There are additional corrections of order unity not considered (see,
e.g., McKinley and Feshbach 1948). A more accurate cross section accounting for Debye screening
is discussed below. The Thomson scattering cross section for photon-electron scattering is σT,e =
0.665 × 10−24 cm2 . For photon-proton scattering σT,p = 1.967 × 10−31 cm2 .
163
Additional Scattering Cross Sections for GRB Disks
The proton-proton elastic scattering cross section is also dominated by the Coulomb interaction,
except at high energies that occur in hot (thick) disks. For example, strong nuclear forces at
50 MeV, corresponding to Tp ∼ 6 × 1011 K, give σpp ∼ 0.110−24 cm2 (Berdoz et al., 1986). Notice
that at such energies the Coulomb cross section is only σc,pp = 8 × 10−30 cm2 , which is much
smaller than the total scattering cross section. The Coulomb term dominates the nuclear term
at low energies. At low energies the nuclear neutron-proton scattering cross section reaches up to
σpp,nuc.max = 36 × 10−24 cm2 (see Table 3-3 in Wong 1990). The transition between Coulomb and
strong interaction domination is at Tp ∼ 108 K.
Proton-proton inelastic scattering becomes important above energy production for mesons such
as pions. The lightest meson is π ± with E ∼ 140 MeV, which corresponds to a temperature of
Tp ∼ 1.6 × 1012 K. This corresponds to twice the upper limit of the typical temperatures in a
GRB accretion disk, although such processes may be important in short moments when/where the
plasma may be significantly hotter. The cross section for pion production in pp-scattering above
E ∼ 140 MeV is σpp,π . 0.0110−24 cm2 (see fig 3-4 in Wong 1990). Our own galactic nucleus may
emit gamma-rays due to proton-proton collisions in a thick disk due to neutral pions decaying into
∼ 70 MeV gamma-rays (Mahadevan et al., 1997).
Nucleons in GRB disks likely have kinetic energies of order 9 MeV. Neutron-proton cross sections for 9 MeV are σnp ∼ 0.4 × 10−24 cm2 (see, e.g., Peterson et al. 1960, and references therein).
At Tn,p ∼ 1013 K ∼ 860 MeV, σnp ∼ 0.4 × 10−25 cm2 . As energies reach zero, the nuclear neutronproton scattering cross section reaches up to σnp,nuc.max = 70 × 10−24 cm2 (see Table 3-3 in Wong
1990). Scattering cross sections between the dipole moment of a neutron and electron can be found
in Yakovlev and Shalybkov (1990).
A.3
Validity of the Ideal Fluid Approximation
This section summarizes the parameters used to estimate the validity of the ideal fluid approximation. In the approximation of the Boltzmann equation, higher order (² = λmf p /L ¿ 1) terms
are associated with, for example, shear viscosity, kinematic viscosity, and thermal conductivity. In
the MHD approximation, higher order terms are associated with Ohmic losses and the diffusion
of magnetic field. All these effects are generally non-adiabatic and so alter the entropy of a fluid
element. See Shu (1992) for more detailed discussions.
A standard measure of the importance of viscosity is the ratio (Re), known as the Reynolds
number, of the inertial term to the viscous term in the momentum equation, where
Re =
vL
νvisc
(A.8)
and νvisc ≡ µ/ρ is the kinematic viscosity associated with diffusion of vorticity, v is the relative shear
velocity, L is the characteristic length scale, and the shear viscosity coefficient is µ ∼ mvthermal /σ.
164
Thus νvisc ∼ vthermal λmf p gives Re ∼ vdr/(vthermal λmf p ). The Reynolds number can also be
considered as the ratio of the viscous drag time on the largest scales (L2 /νvisc ) to the eddy turnover
time of a fluid element (L/v). The Reynolds number should be measured relative to the comoving
frame of the fluid to be consistent with the scattering cross section measurement, and so the only
velocity that matters is a relative (or differential) velocity. If the flow is supersonic and λmf p ¿ L,
then Re À 1 and so viscous forces are much less important than inertial effects.
In order to motivate an inviscid (ideal) approximation for the fluid, the Reynolds number must
satisfy Re À 1. As shown in Equation A.8, the limit to such an approximation is the smallest
characteristic differential velocity that needs to be resolved over a characteristic length scale. In an
accretion disk, one smallest characteristic differential velocity desired to be resolved is the radial
differential shear velocity
¶¯
µ
dvK ¯¯
,
v = dv ≡ dr
dr ¯(r=rf id )
(A.9)
where dr = L is the smallest differential distance to be resolved at r = rf id , vK ≡ rΩK is the
Keplerian velocity, and dv is evaluated at fiducial outer radius of rf id = 40GM/c2 , within which
p
this thesis is focused. The nonrelativistic Keplerian velocity is vK = GM/r, while the general
√
relativistic Keplerian velocity is vK = r GM /(r3/2 + a) for black hole spin parameter a for orbits
beyond the ISCO. Another useful characteristic velocity is the sound speed v = cs and characteristic
length scale corresponding to the disk scale height L = H or L = (H/R)R.
A.4
Validity of the Plasma and MHD Approximations
This section summarizes the quantities used to estimate the validity of the plasma approximation.
This discussion of the plasma, MHD, and ideal MHD approximations follows any plasma (Chapman and Cowling, 1953; Spitzer, 1956; Stix, 1962; Boyd and Sanderson, 1969; Biskamp, 1993;
Baumjohann and Treumann, 1996; Treumann and Baumjohann, 1997), astrophysics (Shu, 1992),
or electrodynamics (Jackson, 1962) book. The standard plasma approximation assumes that 1)
the fluid is ionized; 2) the time scale of interest (T ) is larger than the time scale for electric field
oscillations, and the smallest length scale of interest (L) is larger than the electric field screening
length (λD ) ; and 3) the plasma is weakly coupled. The MHD approximation in addition assumes
that the particles behave like a fluid.
A plasma is assumed to be at least partially ionized, since otherwise the fluid approximation
is sufficient. In thermal statistical equilibrium (and ignoring radiative ionization), the number of
ionized atoms can be determined from the fractional ionization by the Saha equation. The thermal
ionization fraction (X) for nonrelativistic (i.e. x ¿ 1) hydrogen (H) is found from the Saha equation
µ
¶
X2
1
IH
3/2
=
(2πme kb TH ) exp −
,
1−X
nh3
kb TH
(A.10)
where IH ∼ 13.6 eV is the ionization potential energy. Metals, such as iron, have an ionization
165
energy of Ipe ∼ 7.9024 eV for the outermost electron. At the temperatures of typical accretion
disks, the resulting ionization fractions are similar for hydrogen and metals. The disks in all of the
systems studied in this thesis are estimated to have high ionization fractions.
Macroscopic electric fields are present in otherwise neutral plasmas due to charge separation,
which can lead to plasma oscillations. The square of the plasma oscillation frequency is given by
√
2
2
2
2
ωp2 = ωpe
+ ωpp
= ωpe
1 + R ≈ ωpe
,
(A.11)
2 = 4πe2 n /m , and ω 2 = 4πe2 n /m . The oscillation frequency of electrons
where R = me /mp , ωpe
e
e
e
p
pp
and protons is identical, but the oscillation (velocity) amplitude of the protons is R times that of
the electrons. Thus, the protons are relatively immobile for plasma oscillations. Note that only
a warm plasma with proton-acoustic modes oscillates at ωpp . Different geometries for the plasma
oscillations (such as a spherical charge compression oscillation) give a plasma frequency to within
factors of 10 of ωp . Relativistic corrections are difficult to derive analytically, but a numerical
solution is easily found. For electronic oscillations reaching Lorentz factor Γ ∼ 18 (estimated for a
GRB-type accretion disk), the fully relativistic plasma frequency is ∼ ωp /4. If T ωp /(2π) À 1, then
plasma oscillations can be ignored on time scales longer than T , the shortest time scale of interest
by definition. This is estimated to be true for all the black hole accretion systems studied in this
thesis.
While ωp is the plasma frequency for a cold plasma, a hot plasma has a range of allowed
frequencies, with a lower limit of ωp . No electric waves can propagate with frequency ω < ωp ,
and waves with wavelength λ < λD are Landau damped (see below). A relativistic treatment of
the plasma waves is necessary if the maximum nonrelativistic velocity of the electrons (vmax,e =
√
vmax,p /R = x0 2πωpe / 1 + R ∼ x0 2πωpe ∼ x0 2πωp ) gives ∼ x0 2πωp & c. One might expect that
the typical displacement of x0 would be the Debye length that is described below, which then simply
suggests that a relativistic correction is needed if vtherm,e & c or kb Te & me c2 . In the ultrarelativistic
limit of a hot plasma of electrons and nonrelativistic protons, ωp ∼ 4πe2 ne c2 /(3kb Te ) (see, e.g.
Medvedev 1999). Thus, relativistic corrections always lead to a lower plasma frequency. Since
the plasma approximation is only valid for large ωp , relativistic corrections should be included
to validate whether T ωp /(2π) À 1. Even for the most relativistic flows studied in this thesis,
T ωp /(2π) À 1, so the plasma approximation is valid.
For a hot plasma with electrons moving at speed vtherm,e , a characteristic length scale is the
Debye length
λD =
ωp
vtherm,e
.
(A.12)
If there are many electrons within the Debye length scale of a proton, then one can show that
the electric field of a proton is screened by the electrons on a length scale of λD (see, e.g., Shu
1992). This sets the outer scale for the Coulomb interaction, while the inner scale is set by the
radius, at r = λc , at which particle thermal energy is equal to the potential energy. By symmetry,
if electrons screen the field of protons, then protons have screened the field of electrons. At scales
166
much larger than λD the plasma acts collectively, while at scales much smaller than λD the plasma
acts as individual charges. For scales less than λD , plasma oscillations are Landau damped due
to resonant wave-particle interactions (for a physical interpretation of Landau damping see, e.g.,
Shu 1992, p.401). Thus, if λD ¿ L, then the plasma acts collectively on scales larger than L, the
smallest scale of interest by definition. All the black hole accretion systems studied in this thesis
are estimated to have λD ¿ L, so the plasmas in these systems act collectively.
The plasma parameter is defined as the reciprocal of the Coulomb logarithm, which is the
number of particles Λ with number density ne = np within the screening sphere of radius r = 2λD .
This gives Λ = 4/3πne (2λD )3 ∼ 10πne λ3D . A more careful derivation for a hydrogen plasma gives
Λ = 24πne λ3D .
(A.13)
One can show that 4πΛ ∼ (λn /λc )3/2 ∼ (λD /λc ) ∼ ωp /νc , where λn = n−1/3 is the interparticle
spacing and νc is the Coulomb collision frequency described below. A weakly coupled plasma is
defined as having Λ À 1, such that Debye screening is effective and charge neutrality (ne− −
ne+ ∼ np ) holds. In a weakly coupled plasma, the particle kinetic energies are large compared
to potential energies. This is what is typically meant by a plasma. A strongly coupled plasma is
defined as Λ ¿ 1, which implies the particle potential energies are large compared to the kinetic
energies. A strongly coupled plasma is irrelevant for the plasmas studied in this thesis. Notice
from Λ = (3/(4πn))1/2 (1/e3 )(kb Tp )3/2 that strongly coupled plasmas tend to be cold and dense,
whereas weakly coupled plasmas are diffuse and hot. All black hole systems studied in this thesis
have Λ À 1, so they are weakly coupled plasmas.
For weakly coupled plasmas in which the Coulomb interaction is screened on the Debye length,
binary collisions are well-defined. Since Coulomb collisions involve many small angle scattering
events, νc must account for the inner and outer Coulomb interaction scales. The typical collision
frequency due to binary Coulomb interactions is νc ∼ ν0 ∼ vthermal nλ2c . A more accurate accounting
of Debye screening gives
µ
νc = ν0 log
¶
λD
.
λc
(A.14)
In summary, a medium is a weakly coupled plasma if Xion ∼ 1, λD ¿ L, T ωp /(2π) À 1, and
Λ À 1. As first discussed for the fluid approximation, the MHD approximation of a plasma assumes
that collisions sustain a statistical equilibrium space-velocity distribution function of particles on
time scales much shorter than T , the shortest time scale of interest by definition (λmf p ¿ L and
T νc À 1).
A.5
Validity of the Ideal MHD Approximation
This section discusses non-ideal MHD effects and the parameters used to estimate whether they are
negligible. Non-ideal MHD effects are due to particle collisions. Particle collisions can be modeled
167
via plasma transport coefficients, which result from higher-order expansions of the Boltzmann
equation and from a closure relationship between the higher-order terms and the lower-order terms.
It is assumed that any higher-order Boltzmann transport properties can be treated as diffusive terms
in a relaxation theory (see, e.g., §19.31 in Chapman and Cowling 1953). Below we discuss some of
these higher-order effects and the validity of approximations that neglect them.
The general multi-component MHD approximation makes no explicit assumptions about the
nature of the electromagnetic field or current and charge sources on scales of L or T for L À λD
and T À 2π/ωp . However, for a specific derivation of the MHD approximation, the current (J) and
charge (ρe ) densities must be chosen to be consistent with the electric and magnetic fields as defined
by Maxwell’s equations. Several reasonable simplifications can be made, such as to 1) model the
protons and electrons (or any other particles) as one component — thereby ignoring drift among
species and the inertia of electrons ; 2) treat the plasma as ideal – thereby ignoring the effects of
collisions (apart from the fluid approximation itself) ; and 3) neglect the displacement current (as
done in the nonrelativistic approximation). These are referred to as the single-component, ideal,
and nonrelativistic MHD approximations, respectively.
If np = ne , then the current distribution for protons and electrons is J = enp up − ene ue ≡
−ene ve , where ve is the drift velocity of electrons relative to the protons. Ampere’s law then gives
that the drift velocity of electrons relative to protons to sustain the magnetic field is
vdrif t,B ∼
cB
4πene L
(A.15)
(Shu, 1992). For all the black hole accretion flows considered in this thesis, the drift velocity is
much smaller than any shear velocity. This further supports the single-component ideal MHD
approximation.
In weakly coupled plasmas, the magnetic field can effectively play the role of collisions by
constraining charged particles to magnetic field lines within a Larmor radius. Indeed, there is such
a thing as a “collisionless shock” (Tidman and Krall, 1971). A charged particle gyrates around a
field line until interrupted by collisions. The gyration frequency for a constant speed around a field
line is
ωg =
eB
,
Γmc
(A.16)
which assumes that the field is created by an external current. The gyration (Larmor) radius is
λg =
ωg
vthermal
.
(A.17)
The ideal MHD approximation assumes that the gyration radii and gyration time scale of particles
are negligible, such that L À λg and T À 2π/ωg . If the magnetic field is nonuniform over the
length . λg , then the gyration itself produces a net current and the electrons can drift significantly
relative to the protons. One can show that gyration only produces a net current if the electron
inertia is non-negligible, which leads to a break-down in the single-component approximation. In
168
current sheets, the gyration of particles and electron inertia must be accounted for. Ignoring
current sheets, the gyration must be considered if T ωgp À (L/λgp )2 À 1, which is the so-called
“finite Larmor radius” (FLR) ordering associated with the so-called FLR-MHD approximation.
The finite size of the gyration can be ignored if (L/λgp )2 À T ωgp À 1, which is the so-called
MHD ordering associated with the common MHD approximation. For the black hole accretion
flows studied in this thesis, the MHD ordering is estimated to hold. This further supports the
single-component ideal MHD approximation.
In the following it is assumed that charge neutrality holds, the flow can be modeled as a singlecomponent, and the gyration of particles can be ignored. All that remains to close the equation
is a charge current density J. In the most trivial form J = σ0 E, as given by Ohm’s law. A socalled generalized nonrelativistic Ohm’s law (see Krall and Trivelpiece 1973, Eq. 3.5.9) includes
the effects of a finite current-rise time (∂J/∂t term), Hall-effect (J × B term), and an anisotropic
charge pressure – the so-called pressure effect (∇ · P term, where P is the pressure tensor). A
general relativistic generalized Ohm’s law in addition accounts for relativistic thermal velocities,
relativistic beaming of particles, and is covariant (Meier, 2004). The MHD approximation is often
used in a form that ignores these effects, where J×B can be neglected if Lωp2 /ωg,e dv/c2 À 1, ∂J/∂t
can be neglected if T ωp À 1, off-diagonal pressure terms can be neglected if Lωg,p /vtherm,p À 1,
2
and ∇pe can be neglected if Ldvωg,e /vthermal,e
À 1, where pe is the electron pressure (Krall and
Trivelpiece, 1973). By ignoring ∂J/∂t, currents are generated by fields instantly, which is acausal
(if only negligibly so in practice). The off-diagonal pressure terms are related to the neglect of
the finite Larmor radius, since the protons and electrons feel a different average force due to their
different orbital radii. All these effects are estimated to be negligible in the black hole accretion
systems studied in this thesis, so the rest of this section discusses a less generalized Ohm’s law.
From a single-component negligible-drift mostly-ideal (ideal except for standard Ohmic dissipation) nonrelativistic MHD approximation, one can show that the conduction electron current in
the ion rest frame (primed) quantities is
J0 = −ene ve0 = σ0 E0 ,
(A.18)
or in terms of lab frame (unprimed) quantities is
J = σ0 (E + v × B/c),
(A.19)
where σ0 = ne e2 /(me νc ) is the electrical conductivity. One can show that the evolution of the
magnetic field, given by Faraday’s law of induction, contains a purely inductive term called the
Lorentz effect and a purely diffusive term associated with an electrical resistivity η = c2 /(4πσ0 ) =
c2 νc /ωp2 . Associated with the resistivity is the conversion of fluid motion into heat energy called
Joule or Ohmic dissipation, which is a similar phenomena as found for viscous fluids. The time
scale for diffusion is tD ∼ L2 /η, so typically in astrophysical (large L) systems, with an otherwise
similar resistivity, tD À t. The resistive term ηJ in the generalized Ohm’s law can be neglected if
169
the so-called magnetic Reynolds number
RM ≡
Ldv
η
(A.20)
gives RM À 1, where RM is derived as the ratio of the EMF (Lorentz) to diffusion terms in the
induction equation. The magnetic Reynolds number can also be considered as the ratio of the
magnetic field decay time (L2 /η) to the eddy turnover time (L/v). The black hole accretion flows
studied in this thesis have an estimated RM À 1. This further supports the single-component ideal
MHD approximation.
This simple picture of resistivity is inaccurate in the presence of current sheets, which are
considered to be present in, for example, solar flares. In current sheets, the resistivity is suspected
to be driven by more complicated processes, such as by plasma or MHD instabilities. These
processes are sometimes modeled by a larger anomalous resistivity, generically referred to as Bohm
2
diffusion. Some Bohm models suggest an effective resistivity with ηef f ∼ vtherm
/ωg . Some self-
consistent mechanisms have been proposed (see, e.g. Drake et al. 1994), but the study of current
sheets is an active area of research.
A nonuniform magnetic field leads to E × B-like curvature, gradient, and pitch drifts. The
pitch drift can lead to the reflection of charged particles due to magnetic mirrors, such as in a
magnetic bottle or the Earth’s van Allen belts. Gravity acts as an effective electric field to produce
a gravitational drift. The curvature and gradient drift speeds are . vtherm (λg /L), which can be
compared to typical flow speed v or typical differential flow speed dv. One can derive so-called
gyrokinetic-MHD hybrid models to allow for such effects. These magnetic drift speeds are small in
all systems of interest in this thesis since λg /L ¿ 1 and vtherm ∼ v, dv. This further supports the
single-component ideal MHD approximation.
A partially ionized plasma has neutrals. The drift velocity between neutrals and charged particles due to ambipolar diffusion can be compared to the characteristic smallest velocity (Draine
et al., 1983), or it can be estimated by considering the ratio of the ambipolar diffusion to the
inductive term in the induction equation (Balbus and Terquem, 2001). By setting the drag force
fd = γdrag,i−n ρn ρi vdrif t,i−n equal to the nonrelativistic Lorentz force fl =
velocity can be estimated as
vdrif t,i−n ∼
B2
4πγdrag,i−n ρn ρi L
,
1
4π (∇ × B) × B,
the drift
(A.21)
where γdrag,i−n = hvrel,i−n σin i/(mi + µ), vrel,i−n is the speed of the charged particle i as seen in
the rest frame of the neutrals n, σin is the scattering cross section between i and n, µ = ρ/nn
is the mean mass per particle, ρ is the mass density, and nn is the number density of neutral
particles (see, e.g., Shu 1992). For slow velocities, neutral-ion or neutral-electron coupling oc−1
curs due to an induced dipole moment in the neutral particle, and then σin ∝ vrel,i−n
, so that
γdrag,i−n ≈ Const. ∼ 3 × 1013 cm3 g−1 s−1 . At large enough drift velocities this effect is diminished and eventually the scattering cross section is dominated by the particle geometry, giving
170
³
σin ∼ 4π
(rn +ri )2
mn +mi
´
(|vdrif t,i−n |). The drift of electrons relative to neutrals can also be estimated
(see, e.g., Balbus and Terquem 2001). Despite the fact that some black hole accretion disks studied
in this thesis have low ionization fractions, most have negligible drift velocities compared to the
characteristic smallest velocity, dv. Only thick (hot) disks modeled for an X-ray binary and some
AGN give nonnegligible drift velocities between protons and neutrals, but they are fully ionized so
this is not relevant. This further supports the single-component ideal MHD approximation.
At Tn,p & 1011 K, which may occur in a GRB disk, the nuclear interactions dominate the
proton-neutron scattering cross section. At all relevant temperatures, the GRB disk has a large
neutron-proton collision frequency and so neutrals do not likely drift. Estimating the GRB disk
neutral drift is nontrivial, so this is not estimated precisely in this thesis. Rough estimates show
that the protons and neutrons do not drift, although direct electron-neutron drift could occur were
it not for the proton-electron electrostatic attraction. Some relevant estimates are provided in
Goldreich and Reisenegger (1992), where the neutron star matter cross sections are taken from
Yakovlev and Shalybkov (1990).
For (Re, RM ) → ∞, Ohm’s law gives
E + v × B/c = 0,
(A.22)
which if combined with the equations of motion is referred to as the ideal MHD approximation.
The ideal MHD approximation assumes there is no electric field in the frame comoving with the
protons. In this case, dissipation of magnetic energy does not occur, and the magnetic field lines are
frozen into the fluid (so-called “flux-freezing”). The single-component ideal MHD approximation
is estimated to be valid for the black hole accretion disk systems studied in this thesis.
171
B
Beowulf cluster
In this appendix, we discuss clustering of computers to achieve an efficient, low cost way of doing
massive calculations through parallel processing. Such a computing cluster is referred to as a
Beowulf cluster. A Beowulf cluster consists of otherwise independent computers, called nodes,
connected by a network. In the simplest setup, all nodes connect to a single switch. For the
purposes of access from the outside, another switch with Internet access is used. The parallel
computing power of a Beowulf cluster falls somewhere between that of a massively parallel machine,
such as the Cray T3D or SGI Origin 2000, and a simple network of workstations with no collective
processing.
We give a synoptic history of our cluster design in § B.1, discuss performance issues in § B.2,
and Internet accessibility in § B.3. In § B.4, we discuss the hardware that comprises a cluster,
such as the motherboard, CPU, memory, network, and hard drive (HD). In § B.5 we discuss the
operating system (OS), message passing interface (MPI), and how to make a hyperbolic physics
code a parallel program. In § B.6 we discuss important tests of a cluster, such as testing CPU
performance, reliability, code compiler, memory, network interface, and how our code performs.
Finally in § B.7 we summarize the results and conclusions of this appendix.
B.1
History of Design Decisions for our Clusters
A short history of Beowulf clusters can be found at the Beowulf website1 . Several factors influenced
the desire to build clusters of computers for parallel computation using commercial off-the-shelf
(COTS) products, which includes: 1) in the early 1990s, the performance per unit price of individual
computers became comparable to classical supercomputers, for which the CPUs obtain each other’s
data via proprietary communication technology rather than what is classified as network technology;
2) in the mid-1990s, computer hardware and the Linux OS became highly reliable; 3) in the mid1990s, committees formed to promote and develop parallel communication technology, such as the
MPI2 (Message Passing Interface) Forum; and 4) developers at Argonne National Laboratory3
(ANL) and Mississippi State University (MSU) developed MPICH4 , a portable implementation of
MPI.
Parallel processing on a cluster of computers became both practical and more affordable than
supercomputers. Computer centers such as NCSA began experimenting with Beowulf clusters in
1
http://www.beowulf.org/beowulf/history.html
http://www.mpi-forum.org/
3
http://www.anl.gov/
4
http://www-unix.mcs.anl.gov/mpi/mpich/
2
172
the year 2000 with a cluster called Posic consisting of 550Mhz Intel Pentium III systems. In 2001,
NCSA created a production cluster called Platinum consisting of 1Ghz Intel Pentium III systems
(now retired by Tungsten)5 . The typical lifetime of a Beowulf cluster has been about 2 years.
NCSA offers a large amount of resources that can be obtained by submitting a proposal through
an allocations board6 . Based upon one’s request for computer time, they decide how much computer
time one should be given. For example, in 2002 we obtained 100,000 CPU-hours of computer time
on NCSA’s Platinum cluster of 1GHz Intel P3s, and in 2003-2004 we obtained 300,000 CPUhours on NCSA’s Tungsten cluster of 3Ghz P4 Xeons (each Tungsten time unit is worth about 5X
that of a Platinum time unit). An NCSA allocation expires after about a year from the point of
activation. Using an NCSA cluster removes the technical hassle associated with computer hardware
and OS details. The major problem with supercomputer centers such as NCSA is they often explore
bleeding edge computer science and make it difficult to perform bleeding edge science. For example,
the NCSA Tungsten cluster was scheduled to be ready October 2003, but it was not in production
mode until June 2004. The primary reason for the delay was their attempt to use a new file system
called Lustre7 , which turned out to have bugs. The purpose of Lustre is to replace NFS (network
file system)-type technologies for distributed file systems.
Since 1999 we have investigated Beowulf clusters as platforms for high-resolution astrophysical
simulations. Our ultimate goal was to perform general relativistic accretion flow simulations using
the to-be developed HARM (see chapter 3). In comparison to computer time on an NCSA cluster, a
local cluster is more easily accessible, allows us complete control over what programs are installed,
and is self-managed. Self-managing a cluster gives the user “root” privileges, allowing the user to
control the basic Linux environment, such as the version of compiler, MPICH, and other utilities
that cannot be installed by a user on an NCSA cluster. We also considered a local cluster useful
for test simulations before using NCSA computer time to perform production simulations. NCSA
computer time can be used rather quickly when using many CPUs, and mistakes can be costly.
With a small budget one can create a cluster that rivals a large NCSA cluster in the quantity
of CPU-hours over the period of a year available for our private use. The number of hours in a
year of computing time as allocated by NCSA is 8760. For an equivalent of 100,000 CPU-hours on
Platinum, we would require 6 dual-CPU 1Ghz P3s. This is quite inexpensive. For an equivalent of
300,000 CPU-hours on Tungsten, we would require about 17 3.06Ghz dual-CPU nodes.
The largest expense of a Beowulf cluster is the initial cost of hardware and associated environmental expenses (such as cooling units and power circuits). Much time can be spent installing
the OS. As we discuss below, some Linux distributions essentially eliminate the complications and
time needed to install and upgrade the OS. The primary expense of money and time after the
cluster is built is replacing failed node components. In particular, HD failure is the limiting factor
in determining the number of nodes to purchase. In our experience, HDs fail the fastest of any
5
http://www.ncsa.uiuc.edu/UserInfo/Resources/
http://www.ncsa.uiuc.edu/UserInfo/Allocations/
7
http://www.lustre.org/
6
173
hardware component, at a rate of about 1%8 per year for mechanical failure, and based on our
experience, the failure rate is about 5% per year for partial data loss requiring reinstalling the OS.
If we assume it would be reasonable to reinstall a node’s hard drive about once per month and each
node has one HD, then the largest cluster one should purchase is 240 nodes. If one is only limited
by one mechanical failure per month, then the largest cluster one should purchase is 1200 nodes.
This gives an operational cost of about $200/month for the new HD, plus time and effort to install
the new HD.
Below we discuss the purchase of a 16 node 2.4Ghz P4 Xeon cluster, that together with our
collaborators will soon (as of this writing) expand into a 28 node Xeon cluster. We should expect
about one HD failure per year resulting in loss of data that requires reinstalling the OS. The current
cost per node is about $1600 and network equipment about $5000. This will give us a total number
of Tungsten-equivalent CPU-hours of about 192,000 for about $50,000 of NSF funds (equivalent
to about 3 graduate students for one year). This equates to $0.26 per Tungsten CPU-hour. We
would require about $78,000 to obtain about 46 nodes to have 300,000 Tungsten CPU-hours per
year. Before discussing our cluster in more detail, we first discuss the history of our older clusters.
We then discuss the history of our decision making for our Xeon-based cluster.
Our first cluster was based on 2 dual-CPU Alpha-based nodes. At the time, Alpha CPUs were
considerably faster than Intel CPUs (Pentium III at 550Mhz). Despite the high price of the Alpha
systems, 2 nodes are much easier to manage than the equivalent 6-8 dual-CPU Intel nodes needed.
Fewer nodes also allows for higher performance for network intense operations. At the time we
chose a Myrinet9 network technology running at 1000Mbit/s, which was significantly faster than the
relatively inexpensive Ethernet technology that ran at 100Mbit/s. The cluster operated as expected,
so we added an additional node when funding became available. By choosing the most expensive
cluster components, the cluster was shielded from the performance uncertainties associated with
low-end technology. However, the Alphas were sufficiently high-end to be cumbersome in Linux.
For example, they were difficult to upgrade to new Linux versions and incompatible with several
programs. This 3-node Alpha cluster served us for about 2 years.
In 2001, we received additional funding to purchase a new cluster consisting of 3 2.0Ghz dualCPU Intel nodes connected by Myrinet and gigabit Ethernet, where the Myrinet boards from the
Alphas were used in the Intel systems. The nodes were named Alphadog, Gravitas, and Horizon,
so the cluster was called the AGH cluster. The Intel-based systems were easier to manage than the
Alpha systems due to the lack of full Linux support for Alphas, although the 2-year old Alphas
had comparable performance. This system served us for about a year as a 3-node cluster, and for
the past 1.5 years has been the backbone of our Xeon cluster described below.
In 2002, our group and collaborators received $50,000 in NSF funds to purchase a computer
cluster to perform simulations of relativistic magnetized gravitating flows. Given our group’s prior
cluster experience we decided to split the $50,000 between the 2 groups. Our group forged ahead
8
9
Typical manufacturer reported HD annualized failure rate
http://www.myrinet.com/
174
and developed a cluster while our collaborators researched how their code performs on such systems.
This gave us $25,000 to purchase a computer cluster. We originally considered simply purchasing
systems as in the AGH cluster, but we first decided to study what systems would give the highest
performance per unit cost. We thought it plausible to test whether single-CPU Intel Pentium 4
(P4) systems with either gigabit or Myrinet would perform as well as the AGH nodes. This would
be advantageous since the P4 nodes were about 2-3X lower in cost than each node in the AGH
cluster. The P4 test cluster could also be tested against the AGH cluster, and any gigabit or
Myrinet board can be swapped between the P4 and AGH cluster for comparison.
We purchased a test-cluster of 4 P4 nodes with the hope that they would eventually form part of
a larger cluster, where in the worst case scenario these nodes would become new workstations. We
tested the P4 nodes with 32-bit 33Mhz 100Mbit Ethernet onboard (82801 BA/BAM/CA/CAM),
the AGH cluster nodes with 64-bit 133Mhz gigabit Ethernet onboard (82544EI rev2), P4 and AGH
nodes with Myrinet boards (LANai 7 version PCI64A and M2L-PCI64B-2, 2MB boards, 66Mhz),
and 4-5 different gigabit boards provided by our supplier, SWT10 . These gigabit boards included
the 32-bit 33Mhz Intel Pro/1000 MT Desktop (Intel 82545EM rev01), 64-bit 66Mhz Intel Pro/1000
MT Server (Intel 82540EM rev02), and a 64-bit 66Mhz 3com 3C996B-T (Broadcom NetXtreme
BCM5701 rev 21). We did not really consider buying expensive gigabit boards (>$200), but we
wanted to understand how such boards performed.
Chips on the motherboard are generally much less expensive than add-on boards. Onboard chips
can often provide increased performance due to an efficient design and because they are integrated
with the latest features included on the motherboard. Myrinet at the time cost $1200 per board,
and a Myrinet switch for 16 hosts cost $5000. Today the Myrinet boards are 50% cheaper, but the
switches cost about the same. The gigabit boards range from $50 for the Intel Desktop to $500 for
the 3COM board. We purchased an 8-port non-blocking switch for $700 in order to test gigabit, a
Linksys EG000811 .
We compared all permutations of boards, systems, and number of processors involved in a
calculation using MPI bandwidth/latency benchmarks, a ZEUS-based code, and HARM in order
to test system+network performance. We found that the AGH cluster using its onboard gigabit
performed the best. The AGH cluster using onboard gigabit even outperformed Myrinet in dualCPU HARM calculations using all 3 nodes and 6 CPUs. We found that the P4 cluster with any
board gives about 45MB/s peak bandwidth, whereas the AGH cluster gives about 90MB/s with any
64-bit board. The P4 systems with any gigabit board obtained a latency of ∼ 70µ s for transferring
small message sizes, Myrinet obtained ∼ 9µ s, the add-on 32/64-bit 66Mhz gigabit obtained ∼ 50µ s
on AGH, and the AGH onboard gigabit obtained ∼ 50µ s. HARM performed best on the AGH
onboard gigabit by passing relatively large blocks per unit of computation. In such a case the
latency of ∼ 50µ s vs. ∼ 9µ s for Myrinet is indistinguishable. The ZEUS-based code with many
more small communications per computation performed best on Myrinet. Note that MPI tests
10
11
http://www.swt.com/
http://www.linksys.com/products/product.asp?grid=29&prid=407
175
on 64-bit 66Mhz systems show year 2004 Myrinet technology to achieve about 225MB/s and small
message size latency of about 8µ s12 , so its plausible that Myrinet could offer a significant advantage
over gigabit today. However, today’s 10Gigabit technology may counter that peak bandwidth13 .
Despite the likely higher peak bandwidth of 10Gigabit, the latency will still be bounded by the
TCP overhead of standard Linux drivers.
The lower P4 performance is likely due to the 32-bit 33Mhz PCI bus14,15 . As of this writing no
single-CPU Intel workstation motherboard has a faster bus than 32-bit 33Mhz. The 32-bit 33Mhz
Intel Desktop gigabit performed poorly in an AGH node compared to 64-bit 66Mhz gigabit boards,
indicating that the 32-bit board, rather than the rest of the system, was limiting the communication.
The onboard AGH gigabit likely performed better than the add-on 64-bit 66Mhz gigabit boards
because the AGH cluster nodes have a 64-bit 133Mhz PCI-X bus connection to the gigabit, and
the built-in design allows motherboard manufactures to design an optimal interface that allows the
onboard gigabit to operate as efficiently as possible at the full speed of PCI-X.
Based upon these results we considered purchasing systems similar to AGH as we had originally
intended. However, around this time in early December 2002, we learned of a soon-to-be-released
type of motherboard that would have all the technical benefits of a server type motherboard that
is typically an ATX-E (extended) form factor16 , but would be compact in ATX form factor like
workstation motherboards. The so-called Value Server motherboard17 (at the time the Tyan Tiger
i7501) costs about the same as a high-end workstation motherboard ($250), while costing about
1/2-1/3 as much as the typical server motherboard18 (at the time the Tyan Thunder i7501). There
is always an uncertainty about performance for untested systems, so we had doubts that this
was a proper choice. The AGH motherboard was from Tyan and worked great, the Value Server
motherboard was also from Tyan, the Value Server motherboard is actually a similar model to the
full server motherboard, and both contain very similar components. In particular, the onboard
Intel gigabit chip was only slightly different and supposedly improved from the AGH gigabit chip.
Thus, we decided to take the risk. This allowed us to obtain two more nodes due to the lower cost
per node. In the worst case scenario these new nodes would be turned into a computer farm19 . We
decided to obtain about 12 nodes to add to the AGH cluster for a total of 15 nodes20 . The delivery
time for providing the systems was delayed a few times due to the lack of product availability, but
we finally had the systems by the beginning of February 2003.
A gigabit network switch, as opposed to a network HUB, is required for fast communication
between nodes. A switch communicates to a network board only that information meant for it
12
http://www.myrinet.com/myrinet/performance/MPICH-GM/index.html
http://www.10gea.org/index.htm
14
http://www.compute-aid.com/64bitpci.html
15
http://www.intel.com/design/bridge/
16
“Form factor” denotes the style and size of the way in which the motherboard attaches to the computer case.
17
http://www.tyan.com/products/html/tigeri7501.html
18
http://www.tyan.com/products/html/thunderi7501.html
19
Most of the cluster’s computer cycles have been used for single CPU processes since its activation.
20
At the time (Aug 2004) of this writing we have added 3 more new nodes, and we have removed 1 older AGH-type
node to serve as a web server
13
176
based upon its IP address. A HUB performs no such routing and every network board receives
data meant for all nodes. High-end switches are typically non-blocking, which means that the
communication is never limited by the switch and all network boards can achieve maximum speed.
We originally planned on purchasing an HP 4108gl 21 , but we required a gigabit switch that followed
the always changing University of Illinois regulations as determined by the Network Design Office 22
(NDO). The typical function of the NDO is to design a network that suites the needs of University
staff that fits with the NDO’s idea of what constitutes appropriate networking equipment. They
knew of problems with the HP 4108gl that could cause network failures for certain devices on the
University network. After providing the NDO with the desired purpose and design of our network,
the NDO agreed on the HP 5308xl
23
switch, which we ultimately purchased for the new cluster.
This switch is composed of a chassis and up to 8 modules. The chassis holds the modules, contains
the technology to allow all the modules to communicate, and powers and cools the entire switch.
The modules contain the technology to handle a specific type of network connector, such as copper
or optical fiber. The cost of the switch chassis (model J4819A) was $2036. The gigabit copper
modules with 4 ports each (HP Procurve Switch XL 100/1000-T module (J4821A)) were $746 each.
For 15 nodes and one Internet connection we required 4 modules, but purchased 5 modules since
we would likely add more nodes later.
The final cluster element was the rack to hold the system. The purpose of the rack is to save
floor space by stacking vertically while still allowing easy access to each node. We decided to buy a
metal rack from Bed Bath and Beyond from InterMETRO24 . Each unit supports 800lbs and each
of the 4 shelves (including top) supports about 200lbs. We purchased a unit that would contain 12
new nodes and the switch on one shelf, while the Alpha cluster and 3 older AGH nodes would be
on another shelf. Each new node is about 30lbs, each Alpha about 70lbs, old AGH nodes about
50lbs, and the switch is about 30lbs. This gives 360lbs for each shelf, so the shelving is more than
sufficient.
With the equipment to be purchased settled, we set out to determine whether there was sufficient
power and cooling for the new cluster in the Astronomy building’s computer room. We needed to
support our old Alpha cluster as well. One can determine the peak power required for a given
type of equipment using the APC calculators25 online or estimate it from individual specification
sheets. A typical new or AGH-type node consumes 270Watts of power, the HP 5308xl gigabit
switch consumes about 620Watts, and the 3 old Alpha systems consume about 370 Watts each.
With an efficiency factor of 1.33 for Watts to VA, we obtain ∼77A for a 120V line. Thus we required
a total of about 4 20A electrical circuits.
During this investigation it was important to determine the electrical layout and total power
available for our system. The electrical outlet and hardware components in the Astronomy Building
21
http://www.hp.com/rnd/products/switches/switch4100glseries/overview.htm
http://www.cites.uiuc.edu/commtech/ndo.html
23
http://www.hp.com/rnd/products/switches/switch5300xlseries/overview.htm
24
http://www.metro.com/consumer/index.cfm
25
http://www.apcc.com/template/size/apc/index.cfm
22
177
Figure B.1 Astronomy Building computer room electrical layout as of Feb, 2004. BH is located
north and center. This excludes the addition of another new node for BH and the collaborating
group’s cluster of 12 nodes.
computer room as of early year 2004 are shown in Figure B.1. We determined that the addition
of our computers required rewiring 2 240V 30A circuits into 3 120V 20A circuits. Often older
computers used 240V since higher power consumption is more efficient at higher voltages. Before
making these rewiring changes we tried putting the cluster into operation by using any available
outlets. In particular, 1 20A outlet was connected to a set of 8 nodes, and we experienced power
failures when fully using the cluster. The peak requirement for 8 nodes is 24A, so we were simply
underestimating the likelihood of the peak power use. This was temporarily corrected by further
distributing the power over the outlets with remaining capacity until the 240V 30A outlets were
converted to 120V 20A.
The power in Watts consumed also gives the heat generated, where the typical cooling unit is
quantified in how many tons of cooling it provides, where 1 ton is 12000 BTU/hr and 1 Watt is
3.413 BTU/hr. Thus we required about 1.6 tons of cooling. The items already in the machine room
included 7 300Watt machines, 1 switch at 1600 Watts, and lighting at about 300 Watts for a total
of 3700 Watts. Thus the total peak cooling required was about 2.7 tons, while the cooling unit at
the time was only 2 tons provided by a Liebert Challenger 226 . Thus in principle we did not have
sufficient cooling. In practice these are peak cooling requirements not often necessary except for
26
http://www.liebert.com/support/training/env/manuals_obsolete.asp
178
extended periods of maximum use of all systems. Essentially the heating of electrical components
is much more extended in time than the electrical demand. Thus, while electrical requirements are
strict for any moment in time, only prolonged periods of maximum use require maximum cooling.
The cooling unit handled our needs, and we have never had an overheating incident since the cluster
was installed.
Later additions by us and our collaborators in mid-2004 increased the need for power and
cooling. After accounting for the addition of 3 new machines by us and a future addition of 12
nodes (attached to our switch) by our collaborating group, the total required cooling is 4 tons for
the entire machine room. This required the purchase of an additional 5 tons of cooling provided
by a Liebert Challenger 300027 . The new 5 ton unit will supply the entire computer room, while
the older 2 ton unit will operate only when the 5 ton unit fails. This allows time to shut down
computers properly. These additional computers also required 3 more 20A circuits. The total
estimated cost for this cooling+electric upgrade is $28,500.
Our new cluster is called BH, an acronym for Black Hole. A basic schematic of the BH cluster
is shown in Figure B.2. A digital photograph taken soon after the OS, software, and MPI were
installed is shown in Figure B.3. BH currently has 17 nodes where one is used primarily for
compilation of code, initiating parallel programs, and services such as a web server.
B.2
Cluster Performance and Advanced Network Drivers
Soon after BH was set up, we measured the latency and peak bandwidth for the new nodes and were
disappointed that the peak bandwidth was only 60MB/s and the latency was 60µ s. This realized
our worst case scenario that the nodes would be turned into a computer farm. This outcome was a
result of impatience and the desire to avoiding wasting money by purchasing more test units that
we would not use in a cluster.
We considered methods to optimize the network performance. We hypothesized that the default
Intel drivers were not properly dealing with the new chip in the new BH nodes. We could use
specialized network drivers or specialized Linux OSs with modified kernels to improve performance.
Some of the projects or companies out there that offer related services include GAMMA28 , EMP29 ,
M-VIA30 , Scyld31 , Score32 , MPI/Pro33 , and Scali34 . Some of these projects or companies offer
specific Linux distributions with modified kernels, some offer highly specific Linux distributions
with management software and no modified kernel or drivers, and others offer specialized Linux
drivers to be used with any Linux distribution. As far as speed is concerned, the specialized Linux
27
http://www.liebert.com/dynamic/displayproduct.asp?id=542&cycles=60Hz
http://www.disi.unige.it/project/gamma/index.html
29
http://www.osc.edu/~pw/emp/
30
http://www.nersc.gov/research/ftg/via/
31
http://www.scyld.com/
32
http://www.pccluster.org/
33
http://www.mpi-softtech.com/
34
http://www.scali.com
28
179
NODES
BH00-BH16
17 NETWORK CABLES
1 NETWORK CABLE
CLUSTER SWITCH
INTERNET SWITCH
Figure B.2 BH Beowulf cluster primary elements.
180
Figure B.3 Digital photograph of BH cluster.
181
drivers offer improvements by including an OS-bypass or user-level communication drivers. The
typical Linux network driver requires support of TCP and other archaic OS-level protocols that are
not optimal for intensive communications. These OS-bypass drivers separate their activity from the
Linux kernel and only provide special services such as a protocol for MPI communication. Another
feature that improves network efficiency is called zero-copy. This feature allows the network to
run without interaction with the CPU in processing memory buffers. Today, many companies,
such as Intel, are including the zero-copy feature in Linux network drivers. These features are
included in the commercial product Myrinet when using their GM drivers. The most interesting
driver modification is GAMMA, which obtains a latency of 11µ s and 113MB/s on a Server-based
motherboard 1Ghz P3 cluster, exceeding the performance of Myrinet at the time. Donald Becker,
master of the networking universe, father of the Beowulf cluster, and chief technical officer &
founder of Scyld, considered GAMMA the most promising technology at the time.
However, this seemed an unreliable option since many projects such as GAMMA have a very
short life (e.g. Giganet-VIA, ServernetII-VIA, InfiniBand, U-Net, AM II, LPC, PM, FM, GigaEPM, and BIP are all dead) and were no longer being worked on at the time of our investigations.
Specifically, GAMMA supported different devices and obtained quite excellent performance on an
optical-based Netgear GA621, but by the time of this investigation we already purchased a copperbased gigabit switch. The successor to the GA621 with a copper connection called Netgear GA622
actually includes an unrelated chip that performs poorly by comparison. The M-VIA project only
supported one still-available expensive $500 board (SysKonnect35 ). The EMP project seemed to
be quite interesting from a performance perspective, however few chips were supported and the
network boards were unavailable or expensive.
The typical problem for these programmers is that by the time their specialized driver is written
for a chip, the manufacturer no longer produces the board with that chip. Often a single board
will have many revisions of the same chip with entirely different, incompatible functions that have
to be reprogrammed (or re-hacked). Support from manufacturers for Linux driver developers has
always typically been difficult to obtain due to the secrecy associated with corporate technology.
Typically, chips are hacked by programmers until the drivers work, but driver development has
not kept up pace with chip revisions. Clearly this process leads to unreliable support, where the
obsolescence of a chip can mean the death of an associated project. Only companies that develop
both the hardware and drivers with OS-bypass (such as Myricom) can offer reasonable reliability
and stability to the user.
Another problem is that these OS-bypass related drivers and OSs are generally non-trivial
to install and only come with obscure instructions. Drivers such as GAMMA come with many
restrictions on how the network can be used due to limitations in the state of the development
of the specialized driver or OS. Discussions with Donald Becker regarding OS-bypass provided a
historical insight. He says it took about 20 years to “get TCP right”, so likely it will take a while
before OS-bypass or a related technology is cheap and reliable.
35
http://www.syskonnect.com/syskonnect/performance/gig-over-copper.htm
182
We did communicate with Marc Ehlert who designed GAMMA and he used our ZEUS-based
Newtonian 2D MHD code to test GAMMA on his 1Ghz P3 cluster. GAMMA obtained no better
performance than default Linux drivers with our code for a typical problem and resolution. It
is possible that a faster CPU would make better use of the higher bandwidth and lower latency
GAMMA offers. Ultimately, the problem with GAMMA was that it only supported single-CPU
nodes. Donald Becker, who has been a network driver programmer for several network chips, reports
to us that the SMP (Shared Memory Processor) issue is a critical reason why OS-bypass has not
been successfully implemented (except at the time by Myricom with Myrinet). Apparently the
driver must be much more sophisticated to be compatible with SMP. Since most Beowulf clusters
take advantage of SMP nodes, this is a large conflict. At the time of this writing the GAMMA
project does support dual-CPU nodes, but still only supports the NetGear GA621 for gigabit.
Ultimately we got “lucky” and found a large increase in performance on BH by using the advice
of Intel and Tyan regarding our performance problems. Tyan explained that the newer gigabit
chips were actually not malfunctioning, but the Intel driver was flawed for the newer chips, which
was a known issue to Intel. They suggested 2 trivial changes to the “/etc/modules.conf” file to pass
parameters to the Intel driver. This includes the “InterruptThrottleRate=0” and “TxIntDelay=0”
lines. We are currently using Intel driver “e1000-4.4.19”. After incorporating this change the AGHtype node latency improved from 50µ s to 30µ s, and the peak bandwidth changed from 90MB/s
to 83MB/s. The new Value Server system latency improved from 60µ s to 27µ s, and the peak
bandwidth improved from 60MB/s to 100MB/s.
B.3
Cluster access to Internet
The University of Illinois at Urbana-Champaign (UIUC) provided BH with a range of private IP
addresses. A private IP address can access the UIUC network, but not the Internet outside. Private
addresses were provided due to the scarcity of addresses available on the UIUC domain. Private
addresses also enhance security by not providing “Internet intruders” with any means to connect.
To use the private IP address we connect our gigabit switch to the Astronomy Department network
switch, which connects to the UIUC network.
In order for part of BH to be visible to the Internet we also obtained a single public IP address. One computer (BH00) is connected to the Internet by directly connecting one of its network
interfaces to the Astronomy Department network switch. This single computer can thus provide
web, ftp, ssh, and any other service to computers on the Internet. Actually, the Internet related
services are not required to run on BH and certain programs can be moved to other nodes to obtain
a higher level of security for BH. Security must be maintained by updating the OS when patches
that remove vulnerabilities are available.
The final basic set of connections made in a cluster is between every node in the cluster and
the cluster switch. For BH, we have chosen a simple star-type network topology that provides
maximum performance per unit cost.
183
B.4
Hardware
The hardware in a cluster must be reliable, relatively stable against new products, high performance,
and all the equipment must work together seamlessly. A typical system primarily consists of a
motherboard, CPU, memory, network interface, and hard drive. Figure B.4 is a cluster block
diagram for the BH cluster and includes all the important elements of a cluster that we discuss
in this section, along with the size and bandwidth of the pipe between each chip and the network
components.
The motherboard forms the backbone of each node by providing a means for communication
between various types of hardware. Chips on the motherboard communicate through many different
protocols at various speeds to other chips on the motherboard. A motherboard typically contains
those hardware chips that are relatively inexpensive (compared to additional hardware), commonly
in use, and have no need to be upgraded. The primary chips on the motherboard are called the
chipset. This is comprised of some number of chips that coordinate activity between any secondary
chips on the motherboard, the CPU, memory, and other additional hardware. The BH nodes use
the E7501 chipset36 .
The chipset is primarily composed of 3 chips: 1) Memory Controller Hub (MCH) ; 2) I/O
Controller Hub (ICH) ; and 3) PCI Controller Hub (PCH). The MCH is the spinal cord of the
nervous system and coordinates activity between the CPU, ICH, and PCH. Secondary chips could
include hard drive controllers, hardware monitoring, external I/O controller, PCI controller, and
BIOS. Today’s motherboards often incorporate other chips that would otherwise be added as a
PCI board. This includes a video controller or one or more SCSI (advanced hard drive) controllers.
The motherboard used for the new BH nodes is the Tyan Tiger 750137 .
The brain of a computer is the CPU, which interprets and executes program instructions that
can be operated on program information. The new BH nodes have 2.4Ghz Intel P4 Xeon CPUs38
with a 512KB L2 cache. In today’s typical system, the set of CPUs communicate on a single bus
called the front side bus (FSB). Each CPU in a dual-CPU system communicates to other CPUs
or the MCH using this bus that operates at 400 - 533Mhz for the BH cluster systems. The FSB
protocol is 64-bit and thus the peak bandwidth is 3.2GB/s - 4.27 GB/s, respectively (each period
of the cycle contains 64-bits of data).
The CPU itself has memory units called L1, L2, L3 caches (L stands for level). The L1 cache
is the smallest and fastest cache. Progressively higher numbered L caches are larger and slower.
The purpose of a CPU cache is to temporarily store program instructions and information. In
the BH cluster the L1 cache is a mere 8K, but operates at 16.4GB/s. The L2 cache is 512K and
operates at 14GB/s. The size and speed of the CPU caches are determined by cost effectiveness
for a given amount of performance as predicted by market prices for memory. There are several
advanced features included in various CPUs. The Intel and AMD lines of CPUs include SIMD (sin36
http://www.intel.com/design/chipsets/e7501/index.htm
http://www.tyan.com/products/html/tigeri7501_spec.html
38
http://www.intel.com/products/server/processors/server/xeon/index.htm
37
184
Figure B.4 Cluster block diagram of BH cluster. Figure shows bandwidth between elements on
motherboard and switch.
185
gle instruction, multiple data) type instructions that allow for a single instruction, which executes
every fixed number of CPU clock cycles, to operate on multiple data. This reduces the overhead
associated with instructions and allows integer and floating point operations to be performed 2X
faster for instructions operating on 2X the standard data size. Another feature of relevance for
clusters is Hyper-Threading. This feature is meant to increase CPU efficiency by having a single
physical processor support up to 2 thread (or processes). This improves performance when multiple programs compete for CPU time in a multi-tasking environment. However, typically in a
cluster environment one only wants to have 1 process per CPU, so this is not typically beneficial.
Some benchmarks with 1 process per CPU show that Hyper-Threading actually slows down performance39 . We find no difference in performance for our codes running 1 process per CPU with
or without Hyper-Threading.
The main memory of a computer serves as a relatively fast repository of program data. The
main memory is essentially an extension of the CPU caches but operates over the FSB bus through
the MCH. A typical Intel server based motherboard has dual-channel memory40 , which means the
bus connecting the memory to the MCH has 2 independent channels for 2 sets of memory chips.
This allows the 2 CPUs (or any other device) to independently operate on those 2 sets of memory
chips. Each channel of memory in each node of the BH cluster has memory that communicates to
the MCH in 64-bit chunks at 266Mhz for an effective total of 128-bit memory. The bus between
the MCH and memory is actually 144-bit, which includes 16 bits of error correction encoding to
guarantee the data is accurate. The memory used in the BH nodes is called PC2100 DDR. The PC
refers to personal computer and the 2100 refers to 2100MB/s, which is the approximate effective
peak bandwidth in MB/s per channel.
Memory will continue to increase in bandwidth, but historically the large increases in memory
bandwidth often do little for overall computational performance. Most scientific codes do well to
make optimal use of the CPU caches.
The network provides the means to transfer data between all the nodes. Some classes of problems
solvable by a computer can be solved on a cluster of networked nodes in order to model the function
of a single computer. Any problem that can be broken into a collection of localized problems, which
can each be efficiently solved on one node, can benefit from a cluster of nodes in a network.
While there are many ways of connecting a network that can be specialized for some classes of
problems, the most general topology for one connection per node is simply to connect the node to a
network switch. In this case it is best if the network switch is non-blocking. A non-blocking switch
allows every port to communicate to every other port simultaneously, bidirectionally at maximum
speed. Current network technology is fastest per unit cost for gigabit networks. Gigabit network
chips are often incorporated into server-type motherboards which greatly lowers the cost. A gigabit
network chip has a peak bidirectional bandwidth of 0.25GB/s.
One can potentially increase the bandwidth between any 2 nodes by giving a node another
39
40
http://www.Linuxclustersinstitute.org/Linux-HPC-Revolution/Archive/PDF02/11-Leng_T.pdf
http://www.cpuplanet.com/features/article.php/1587771
186
network connection. However, one has to determine if the CPU, memory, and PCI bus have
sufficient bandwidth to make this worthwhile. Generically, the CPU and memory are much faster
than the PCI bus, and thus the PCI bus is the bottleneck. However, the true measured bandwidth
can significantly differ from theoretical estimates, so in practice one should do experiments on
various hardware setups. For example, Intel P4 motherboards with a 32-bit 33Mhz PCI bus have
a peak bandwidth of 133MB/s, while on that bus the gigabit network performance is typically
45MB/s independent of the hardware. Older server systems have a 33Mhz 64-bit PCI bus which
gives a peak bandwidth of 264MB/s, while peak gigabit performance is about 90MB/s. Newer
Myrinet and 10Gigabit technologies can reach near the theoretical limits of a 66Mhz 64-bit bus
operating at a bidirectional total of 440MB/s.
For motherboards that have PCI-X technology, which is a standard 64-bit bus at 133Mhz, the
peak bandwidth is about 1GB/s. Thus raw estimates would conclude that the PCI-X bus has
sufficient capability to handle up to 4 gigabit network chips. The actual measured bandwidth of
a gigabit network chip on the PCI-X bus is not measurably different from the 64-bit PCI bus at
33Mhz at 90MB/s. Actual hardware never operates at peak bandwidth for a variety of reasons,
such as network drivers and true hardware operating conditions.
As described in Section B.1, we did consider advanced drivers that eliminate the overhead
associated with the CPU interaction and OS. The GAMMA based driver can achieve a much lower
latency and a bit more peak bandwidth than default Linux drivers. However, we found this option
to be unreliable and impractical due to several limitations of the current technology and scarce
support for existing hardware.
A gigabit switch connects all the nodes in a cluster. A network switch should be non-blocking
for maximum performance. A non-blocking switch never limits the traffic between each node by
processing all routing at sufficiently high rate so data is never stalled. Other features include hardware support for jumbo-frames. The typical mean transfer unit (MTU), or smallest message size,
is no more than 1500 bytes. Jumbo-frames support typically allows up to 9000 bytes. This value
is the maximum at which the standard TCP 32-bit CRC error correction check loses effectiveness
against data loss. However, this is only useful for very long message size data transfers not typical
of all computing done on clusters. The BH cluster uses an HP 5308xl non-blocking switch and
obtains up to 9.6GB/s bandwidth on the so-called “backplane”. This describes the connections
behind all the ports. Each port input can go to 1 port output, and each pair of input/outputs
uses 0.125GB/s of bandwidth. Thus, theoretically the switch could handle up to 76 ports before
performance is inhibited. Currently the BH cluster uses 17 ports and our collaborating group will
soon require 12 more ports, yet this will still be far from over-utilizing the switch. Unlike computer
OSs, network switches are very efficient at network processing due to their proprietary, optimized
OSs.
Older forms of the gigabit network technology exist, such as the 10Mbps Ethernet that theoretically has a peak bidirectional bandwidth of 2.5MB/s, but actually only obtains (under any OS or
driver known to the author) about 0.5MB/s. Likewise, a 100Mbps Ethernet network theoretically
187
could obtain a bandwidth of 25MB/s, but achieves about a peak of 10MB/s under a broad number
of OSs, hardware, and drivers.
We use hard drives for both immediate simulation diagnostics and storage of old simulations.
Each node has the capability to hold 4 HDs on the EIDE channel, and by distributing the HDs
among the many nodes the effective bandwidth is parallelized and the reliability is enhanced compared to a single large hard drive system.
The hard drive used in computer clusters can be a weak point in terms of reliability, and can be
critical since failure can mean loss of data – generally the most important item in any cluster. The
typical HD manufacturer specification sheets report a rate of failure using the mean time between
failure (MTBF), the number of bits before 1 bit is read or written with errors, the annualized failure
rate (AFR), and the “minimum” design lifetime of a drive. At the time of this writing, the typical
high performance EIDE HD has a MTBF of 500,000 to 1,000,000 hours, a bit error rate of 1 out of
1013 − 1015 , an annualized failure rate of about 1%, and a “minimum” design lifetime of 5 years.
The MTBF and bit error rate only indicate HD error in reading or writing and assumes the drive
does not otherwise fail catastrophically, such as from bearing failure (often associated with clicking
sounds). The AFR is most representative in that it reports the actual rate of return due to failure
by customers.
Over the 4 years of dealing with approximately 30 HDs (in an arbitrary number of computers),
we have had about 6 HD failures, where about 2 have been terminal (i.e. not restorable after
attempting a reformat). If this were representative of HD failure rates, this represents a 5% chance
that in a year a HD will fail with loss of data. Our experience of terminal failures is a ∼2% annual
failure rate. The reported failure rate by manufacturers for actual used drives and returned due
to any problem is 1% per year. Perhaps our luck is 2-5X worse than average, but more likely our
problems are associated with obtaining the latest technology since we are interested in purchasing
the new, larger hard drives. Also, we likely accessed our HDs more frequently than the average user.
In summary, we suggest replacing drives that are still alive every 5 years and expect about 2-5%
to die each year. Making backups to other HDs or having HDs perform some form of redundancy
of data is critical.
We have often considered getting a tape backup system, but have always ended up buying a
HD due to their simplicity, general robustness, and low price. Tape backups are generally more
robust than hard drives, but are of course sensitive to a magnetic field. The author has had
personal experience with consumer tape backups before CDRs became available, and often had
reliability problems with only 2 year old tape backups that were thought to be in a safe location
and environment.
We do have a CD burner and recently added a DVD burner to store small data sets or home
directories containing code and other source material. CDs and DVDs are not immune to data
loss problems. Typical CDs or DVDs, especially those one burns using consumer devices, only
have a guaranteed lifetime of about 5-10 years, with some perhaps having as little as 2 years of
188
life 41,42 . CD-R manufacturers performing media longevity studies using industry defined tests and
mathematical modeling, claim an theoretical longevity from 70 years to over 200 years. We suggest
that any data on CDs or CDRs, for which one needs to guarantee reliability, be refreshed every 5
years.
B.5
Software
In this section we discuss the software required to build, operate, and maintain a cluster using
Linux. We discuss the MPI library and how a fluid code uses MPI to perform parallel calculations.
B.5.1
Choices for OS
Linux43 offers a reliable, stable, and high performance OS for a cluster of computers. Compared to
Windows and other OSs that have come and gone (such as IBM’s OS/2), Linux is a programmer’s
dream. Linux is highly configurable, with open source44 access to the entire system45 . Since
the author started using Linux in 1994 several versions of Linux have come and gone. Today
RedHat Linux dominates the commercial market, with the free version of RedHat no longer being
supported except through the Fedora project. Other versions worth mentioning are Mandrake,
Knoppix, Debian, SUSE, Gentoo, and Slackware. The author has used all of these versions and
has found RedHat to be the easiest to deal with. RedHat keeps to the simplicity of the old
Linux versions, thus avoiding confusing configuration issues introduced by versions such as SUSE.
Essentially SUSE adds all the problems of a Windows-like configuration without the benefits of
Windows’ seamlessness. For example, several parts of SUSE’s configuration program “lock-up”
and can corrupt the configuration files. The SUSE configuration files are more complicated than
other distributions, and are generally not human readable. Most problems are due to lack of broad
support by hardware companies for Linux. Since the demise of a free RedHat version, Fedora
and Mandrake are the biggest contenders that are vying for the non-commercial market, which is
applicable to private university computer clusters. We are currently evaluating whether Fedora is
a reasonable migration path from our current installation of RedHat.
Also, there are several “cluster” Linux distributions46 that include the means to install a cluster
from one master node by simply having the slave nodes boot off a CD or floppy. The master node
distributes and installs Linux on the entire cluster. Probably the most promising is the Scyld47 distribution that provides a single system image cluster using specially developed tools and libraries.
Scyld not only makes installation trivial, but the entire cluster can be managed and operated like
41
http://news.independent.co.uk/world/science_technology/story.jsp?story=513486
http://www.pctechguide.com/09cdr-rw.htm
43
http://www.Linux.org/info/
44
http://www.gnu.org/
45
Several commercial products and drivers exist for Linux that are closed source, but these have never been used
for this cluster.
46
http://lcic.org/distros.html\#cluster
47
http://www.scyld.com/
42
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a single stand-alone computer. Scyld also modifies the kernel to achieve optimal performance for
a broad range of network hardware. The price is typically $200 per node. Another commercial
product, Scali48 , competes with Scyld without the installation and management benefits, but with
the performance benefits. The performance of Scyld and Scali can be better than of a standard
Linux distribution, although generally they require advanced network hardware to achieve higher
performance than standard Linux distributions. Scali and Scyld on Myrinet achieve a drop in
latency from 7-8µ s to 4µ s and an increase in peak bandwidth from 220MB/s to 260MB/s. OSCAR49,50 is a free cluster distribution that offers the same installation and management benefits
as Scyld without the performance enhancements. Note that none of these products attempts to
develop OS-bypass network drivers for specific hardware, such as the GAMMA project discussed
in Section B.2.
B.5.2
OS Installation
For our relatively small 15-17 node BH cluster, we decided to install RedHat and manage the
cluster ourselves. We wanted to avoid the complications associated with learning a new OS, specific
configuration issues, and we had a fear of losing support due to the death of several cluster projects.
The performance of our cluster using standard Intel drivers with modified parameters was excellent
compared to any cluster with such cheap technology. Below we describe the steps we followed to
install the cluster by “hand”, after the cluster hardware installation has been completed. During
this discussion we mention places where we learned of problems with the system.
First, download the distribution. If you plan to install from CD, the best advice we can give
is download an ISO51 distribution from the fastest server you can find. The availability of online
ISO images to burn a CD to install Linux is certainly becoming more common today. In the past,
most distributions online were set up for direct installation. This is generally an unreliable means
to install Linux due to network slowdowns or outages.
If you plan to do a network installation, download the distribution in any form (e.g. extract the
ISO images) and place the distribution on a local directory that has an ftp server. All computers
can use ftp to do a network installation from your own ftp server. For RedHat the floppy is used
to boot each node to install in network mode.
Second, install Linux on each node if you are not using a second-generation Beowulf Linux
distribution that installs the OS automatically. We simply installed Linux on each node manually,
fairly simultaneously, by doing a network installation on each node. It takes about 5 minutes
to start the installation off the floppy. This requires switching a keyboard and monitor to each
node, but is fairly efficient. For RedHat we chose a “custom installation” with “install everything”
selected so that all libraries and applications are installed. Each BH node took about 30 minutes
48
http://www.scali.com/
http://sourceforge.net/projects/oscar/
50
http://www.openclustergroup.org/
51
http://www.Linuxiso.org/
49
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to install Linux.
The time it takes for each computer to install Linux is a good first measure of whether the
systems are independently performing properly. We had 3 systems take 45-60 minutes to install. We
found them to have slower than average hard drive performance by using “hdparm -tT /dev/hda”.
In particular the fast computers showed a buffered read bandwidth of 45MB/s, buffered-cache read
of ∼ 700MB/s, while the slower computers had a large variation in HD speeds from 15MB/s to
35MB/s and no different buffered-cache read performance. This result was based upon repeated
tests52 . Swapping hard drives, cables, CPU or chipset heat sinks, etc. lead to no differences. Only
after replacing the motherboard did the slow systems reach the normal performance of all the other
nodes.
Third, recompile the kernel for a streamlined Linux. We compiled our own kernel to trim
down unnecessary modules and features. The only clear general statement that can be made about
compiling a kernel is to avoid unneeded hardware drivers. One should go through every option
available and attempt to fully understand whether it is needed or not. Generally new kernels
change only slightly so this is not a difficult process when learned once. We installed Linux 2.4.20,
and the kernel config file53 is available online. Linux 2.654 will likely be more desirable than Linux
2.4, since it includes many network related optimizations that may reduce network latency and
increase peak bandwidth.
B.5.3
Software Installation and Usage Notes
Here we discuss the installation of MPI, Kerberos, Nagios, the Intel compiler, and group software.
We discuss how we installed this on one node and then distributed that single modified state of the
file system to the other nodes.
The installation procedure for MPI is well documented55 . We found that compiling with the
SMP option “-comm=shared” leads to significant loss in performance, despite the CPUs apparently
being fully utilized. Using the standard options results in good performance, even if the CPUs are
under-utilized due to communication time taking longer than CPU time for a given calculation.
Kerberized56 versions of several programs are available. Kerberos offers the same benefits as
ssh but with higher performance. The default settings for Kerberos connection are encrypted
password exchange, but unencrypted connections. This results in faster communications. For
buffered transfers, we find that a typical “scp” copy using OpenSSH, or any other ssh program,
uses 80-130% of the CPU(s) on the client side and < 1% on the server side for transfers on gigabit,
while only obtaining about 13MB/s. Unencrypted or Kerberized versions of “ftp” or “rcp” give
about 100MB/s transfer bandwidth and 30% CPU utilization on the server side and 10% on the
client side.
52
http://rainman.astro.uiuc.edu/cluster/scripts/hdparm.sh
http://rainman.astro.uiuc.edu/cluster/scripts/jon2.4.20smp
54
http://www.kernel.org/
55
http://www-unix.mcs.anl.gov/mpi/mpich/
56
http://web.mit.edu/kerberos/www/
53
191
We have also found Kerberos is very useful for scripting between nodes. We find that ssh
commands can timeout after generating greater than about 10 simultaneous ssh commands from
a single node. This means that a scripted run through many computers may or may not reach
all computers, an intolerable unreliability. We have never encountered such problems with “rsh”,
so we use this for scripting when generating simultaneous commands to the cluster. The “rsh”
program is also much faster than ssh for starting MPI jobs.
On a related note, we found “ncftp” on RedHat 8.0 to be slow. We obtain about 4-6MB/s on
gigabit and the server side uses 100% CPU (“wu-ftpd” or “in.ftpd”) and 13-20% CPU on the client
side. Using normal ftp gives 90-100MB/s at 100% CPU. For unknown reasons “ncftp” is causing
the CPU to be under-utilized. No such problems were encountered under SUSE.
We found it useful to install a cluster monitoring program called Nagios57 . Nagios can be used
for many advanced purposes and is easily and highly configurable. We use it to monitor each node
in the BH cluster and all our workstations for 1) ping, 2) network services availability (SMTP, ftp,
ssh, HTTP), 3) number of users, 4) total processes, 5) too high CPU load, temperature problems
using the lm sensors58 package (currently version “lm sensors-2.7.0” which requires “i2c-2.7.0”),
and 6) too small hard drive space available. It “nags” a specified user or group about problems via
email, which is why SMTP (sendmail) is run on the cluster. It reports the status of the computers
and allows changes to be made through a well designed web interface, which is why HTTP (apache)
is run on the cluster. Nagios can be run on any computer that can access the relevant services
on the remote computers. Other Nagios-like programs are available, such as Big Brother59 and
Ganglia60 .
The compiler of choice for Intel-compatible nodes is “icc” for C and C++ and “ifc” for Fortran.
These offer about 2-3X faster performance than “gcc” for C or g77 (f77) for Fortran, for our codes.
A free non-commercial license is available online for both the C/C++ compiler61 and the Fortran
compiler62 . The standard “gcc” remains necessary for several programs that are incompatible with
“icc” due to Intel’s lack of full Linux support. For scientific codes this is often a moot point
because they make limited use of complicated libraries. However, several programs and libraries
cannot be compiled with “icc”. Attempts will either generate compiler errors (that can sometimes
be “fixed”), or the code will simply not function properly (sometimes generating insidious failures
of apparently unrelated origin). We suggest only compiling something with “icc” if the code does
not use complicated routines or external libraries. Intel’s “icc” packages include a debugger and
one can purchase a profiler, but the standard gnu debugger “gdb” and gnu profiler “gprof” can be
used with “icc”. The most accurate profile report from gprof on a code compiled with “icc” is with
no optimizations, which can alter the profile significantly compared to an optimized compilation.
57
http://www.nagios.org/
http://secure.netroedge.com/~lm78/
59
http://www.bb4.org/
60
http://ganglia.sourceforge.net/
61
http://www.intel.com/software/products/compilers/clin/noncom.htm
62
http://www.intel.com/software/products/compilers/flin/noncom.htm
58
192
When compiling with optimizations, gprof can fail to create a reasonable report, so we only use
gprof as a guide.
Compiling MPI, especially the Myrinet version of MPICH over GM, is often problematic with
“icc”. GM is Myrinet’s network device driver for accessing Myrinet hardware using MPI functions.
Also, compiling Fortran 90 support for MPI with simultaneous support for C/C++ can be tricky
and has unresolved problems due to Intel’s incomplete C and Fortran support. A script we used to
compile the standard MPI is available online63 .
Group software should be placed in “/usr/local” so the software can be easily copied to any
computer. Any additional software that is installed from source (in “/usr/src”) should be installed
to “/usr/local” and “encapped” with EPKG64 or a similar program. EPKG allows one to easily
install and uninstall any installed package without worrying about corrupting the “/usr/local” path
with old programs. One simply makes a link from your program directory to “/usr/local/encap”
and run inside that directory “epkg -i <dirname>”.
Many of the installation issues, such as what services to install, can be worked out on one
computer and distributed to all other computers. See the script online for a guide to how this
was done for BH65 . All BH nodes have ssh, Kerberized (ftp,telnet,rsh,klogin,kshell,eklogin), while
currently the master node (BH00) has also “httpd”, “smtp”, and “kerberos-adm” installed.
We found it useful to create a cluster equivalent to Linux programs “top” and “df” that would
post to a website. This required updating the procps version to “procps-3.1.6”, which includes,
for example, improved “top” and “ps” programs. The new “top” program has many options, and
we set these options by including a “.toprc”66 file in the home directory of root. These options
determine the final output of running “top” as root. We then run a cron job every 3 minutes for
the cluster “top” and cluster “df” by adding the line to the “/etc/crontab/” list67 . We also run
a script every hour that shows the user usage of disk space. The scripts for the cluster “top” and
“df” are available online68 . The list of available top and df websites is available online69 . As an
aside, we suggest using the C language for complicated scripts. C scripts can be powerful, fast, and
incorporated with pure scripting languages like bash. C scripts have the advantage of being easy to
understand, easy to make, and are easy to read. Bash70 , perl, and other scripting languages have
many exceptions to the basic rules of syntax, and this syntax complexity makes it difficult to write
marginally complicated tasks.
Prior to the BH cluster we had been using NFS (network file system) to allow every computer
to see the disk of every computer with a large hard drive for general storage. While this may
have security concerns, the primary practical problem with this is a lack of stability. We found
63
http://rainman.astro.uiuc.edu/cluster/scripts/mpich.make.intel.p4
http://www.encap.org/epkg/
65
http://rainman.astro.uiuc.edu/cluster/scripts/clusterinstall.sh
66
http://rainman.astro.uiuc.edu/cluster/scripts/.toprc
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http://rainman.astro.uiuc.edu/cluster/cronstuff/crontab
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http://rainman.astro.uiuc.edu/cluster/cronstuff
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http://rainman.astro.uiuc.edu/cluster/computers.html
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http://www.tldp.org/LDP/abs/html/
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that if any single computer with a storage drive went down or became unavailable, all computers
mapping that drive locally would lockup. This destroys the usefulness of NFS by eliminating the
distributed nature of a cluster of workstations. For this reason we do not use NFS, and we simply
assume that each person can copy information using “scp” or “rcp”, and that MPI programs will
manage their diagnostics to write to disk in parallel or collect the data to one drive manually. That
is, the MPI program should not assume that the disk is one large disk for all nodes. This loses
the transparency of disk storage provided by NFS, but is worth the increased stability and parallel
performance. Lustre71 may provide an NFS-like distributed file system, but with improved stability,
performance, and security. NFS is fairly insecure and has stability and performance problems that
Lustre seeks to eliminate. Lustre removes the NFS stability problems by avoiding any single point
that can bring total failure of the distributed file system. Lustre allows up to 10,000 nodes with
nearly 90% efficiency of total hard drive bandwidth for parallel file I/O.
We maintain security by periodically updating packages from RedHat. Since the demise of
the free RedHat, we may attempt to move to Fedora, Mandrake, or some commercial cluster
distribution like Scyld or Scali. We maintain security by using “tcp wrappers”. A “tcp wrapper” is
a program that checks all incoming connections for certain services and only allows computers with
pre-specified IP addresses to connect. This is controlled by the “hosts.allow” and “hosts.deny” files
in “/etc/”. For the website we control access either through standard entries in “httpd.conf”, or by
using “.htaccess” files and assigned passwords. The former is used to control access to somewhat
sensitive web pages by computers outside our group, and the latter is used to control Nagios access.
Mistakes can be made in setting up which services are running. One can check what services are
running on a computer using “nmap <hostname>”.
B.5.4
MPI Implementation in Fluid Codes
MPI is a library specification for message-passing, proposed as a standard by a committee of
vendors, implementers, and users. MPICH72 is a freely available, portable implementation of MPI.
Historically, for those who develop their own physics codes, there have been 2 distinct approaches
to parallelizing a code. One approach is to use a pre-existing library suite that permits a simplified,
but prescribed method of parallelization. The other approach involves directly using an MPI library,
such as MPICH, and developing an interface. Interfaces to the MPI libraries, such as DAGH73,74 ,
Kelp75 , and CHOMBO76 , invite the programmer to ignore the details of MPI and write good physics
code. They often offer very complicated services that only take a handful of functions to use and
so are easy to implement, such as parallel adaptive mesh support in CHOMBO.
Problems with such MPI interfaces include: (1) a lack of continued support (DAGH, Kelp), (2)
71
http://www.Lustre.org/documentation.html
http://www-unix.mcs.anl.gov/mpi/mpich/
73
http://www.caip.rutgers.edu/~parashar/DAGH/
74
http://www.cs.utexas.edu/users/dagh/
75
http://www.cs.ucsd.edu/groups/hpcl/scg/kelp/
76
http://seesar.lbl.gov/anag/chombo/
72
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over-generalization and slowness (DAGH), and (3) integration into code is fixed and very dependent
on that library. As with any Linux project, support is often only guaranteed when supported by a
large set of corporations or universities. Projects like DAGH and Kelp were only supported by a
few universities and later collapsed as funding was not available and key programmers moved on.
We wished to avoid performance problems, being dependent on a high-level library, and stability
uncertainties. We decided to write our own MPI interfaces for intra-nodal setup, all inter-nodal
communications, and for parallel file I/O using the MPICH library. We only use about 10 MPI
functions to do all our parallel processing related tasks. These functions include 1) initialization
type functions: MPI Init, MPI Comm size, MPI Comm rank, MPI Get processor name, which are
trivially used and involve about 5 lines of code; 2) management functions: MPI Barrier, used to
fully synchronize all CPUs at 1 point in the code, which is rarely used; 3) Transfer functions:
MPI Bcast, MPI Reduce, MPI Allreduce, MPI Wait, MPI Irecv, MPI Isend; and 4) finishing functions: MPI Abort, MPI Finalize, which are trivially used at the end of either an aborted or successful simulation, using about 5 lines of code. The transfer functions involve the most complicated
part of MPI by 1) setting up the communications between nodes (about 40 lines of code); 2) the
communication of boundary values shared to nodes, which is about 100 lines of code; and 3) the
diagnostics of parallel file output and other collective operations to obtain integrated quantities,
which involves about 2000 lines of code.
Clearly the diagnostics are the most difficult and time consuming part of making a code parallel
using MPI, despite our effort to modularize the procedures. Once the modularization is done it is
trivial to output new diagnostics by following examples in the code. Several MPI-277 (MPI version
2) parallel I/O functions can be used, as we have done. However, support for MPI-2 features
was non-existent in 1999 when we were developing our code to use MPI. Even in 2002, NCSA’s
Platinum cluster had no support for parallel I/O. In order to use NCSA clusters as soon as possible
we implemented our own file I/O procedures. Our file I/O routines use a fixed small memory
storage buffer regardless of the number of nodes or number of columns of data per zone. These
routines can write to disk on a single master node or write in parallel to each node’s hard drive.
The second most complicated MPI coding involves the transfer functions.
Ideal fluid codes operate on a grid of zones with typically hyperbolic equations of motion. Such
a code can be decomposed (“domain decomposition”) into smaller per-CPU grids and the boundary
values on each CPU can be exchanged to the neighboring CPUs when necessary. The routine that
transfers the boundary values between nodes is quite trivial and was developed and debugged in
a week by the author for a ZEUS-based code in 1999. There are at least 2 methods to improve
transfer efficiency for a simple homogeneous domain decomposition: 1) use non-blocking routines;
and 2) use multi-layered boundary value zones. A non-blocking routine allows the CPU to continue
processing while the network hardware is transferring data to other CPUs. This compared to
“standard” blocking routines that halt all involved CPUs until the communication is complete.
Blocking routines are a bit more trivial to implement, which is why they are more common. Non77
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blocking routines alone might allow a small gain in efficiency by parallelizing transfers between all
nodes when transferring boundary values between all computers each time step. Optimal use of
non-blocking routines is obtained by using 2 layers of boundary zones per CPU. One boundary
layer is the standard boundary conditions layer, while the next layer is a 2-zone wide ring of
non-boundary zones just inside the boundary conditions layer. This assumes the stencil size is
2-zone each direction which is typical of ZEUS and HARM. The procedure to follow is simple:
1) start computations on all zones; 2) exchange boundary values using non-blocking routine; 3)
continue calculation on the remaining subset of non-boundary layer zones not including the 2zone wide ring; 4) once the subset calculation is complete, complete the non-blocking call with
an MPI Wait() call; and 5) calculate the 2-zone ring that required the updated boundary values.
This procedure continues indefinitely. This procedure allows the CPU to compute while boundary
values were being transferred, allowing maximum CPU utilization.
The new time step is computed by first computing the minimum time step required for each
CPU, then using an MPI Allreduce() call. This communicates to every CPU the minimum of all
the CPUs values for the time step. When a algorithmic failure occurs in MPI, its often difficult
to track down the error without careful MPI considerations. Failure checks can be organized by
having every function that could return a 1 on failure and 0 on success, and checking every function
call’s return value. Upon success the code would continue. Upon failure the code would return a 1
for failure. As long as the failure checks are placed before each MPI-type call, then all the CPUs
will fail synchronously. Ignoring this issue will leave hung jobs and no clear record of the cause of
the failure.
B.5.5
Running MPI Jobs
For a large cluster, or for a large number of users for a relatively small cluster, preexisting schedulers
and batch processing applications are necessary. The most widely used program to help organize
all the requests of users to perform computations is the OpenPBS78,79 batch system (PBS stands
for portable batch system). This system accepts user requests to run MPI programs on a cluster,
runs the program, and can even schedule when the program should run. However, the most widely
used scheduler for determining when someone’s program should run is the Maui80 scheduler. These
are popular programs, and NCSA uses both PBS and Maui. These programs are not trivial to
set up. This author set up these systems on the BH cluster, but we decided that they were too
restrictive for our small group.
For relatively small clusters or for clusters with few people, its easy to rely on the users to check
what processors are used and how much hard drive space is available to ultimately decide what
nodes the parallel job will run on. In order to facilitate this process scripts are useful to generate
easy to access lists of this information. As described previously, we have a script that generates a
78
http://www.openpbs.org/
http://www-unix.mcs.anl.gov/openpbs/
80
http://www.supercluster.org/maui/
79
196
webpage showing the “cluster top” and “cluster df”. These show available CPUs, what processes
are on each CPU, CPU usage, and hard drive space available on each node. This is useful for both
parallel and single CPU runs.
We leave it up to the user to run MPI programs or single-CPU programs. The standard
“mpirun” script that is included with MPICH has several seemingly complicated restrictions on
starting an MPI program. For example, the job must be started on the first node that will perform
the calculations. There are some ways around this particular problem, but other problems remain.
The author has written a simple script that makes it easier to start a batch job from one node and
then run the job on other nodes. This “mympirun.sh” script will also create a working directory on
every node and copy the binary file to that directory on every node. This is useful in a non-NFS file
system environment where each hard drive space on each node is invisible to the other nodes. The
script uses either “ssh” or “rsh” and “scp” or “rcp” to perform the startup tasks. The programs
“rsh” and “rcp” are much faster for starting jobs on many nodes. The script is available online81 .
B.6
Testing Cluster Reliability and Performance
Benchmarks can provide insight into how different software, hardware, and configurations of each of
these modify performance. Ultimately one should test the code to be run on a system, but synthetic
benchmarks can be useful for more general performance characteristics. A synthetic benchmark
is simply a test that measures an isolated hardware component or feature. Synthetic benchmarks
also allow one to compare all nodes to each other to verify that each node’s hardware components
operate nominally.
B.6.1
Reliability and Performance Issues
Chips on the motherboard are unlikely to fail if cooled properly. However, the heat sinks on the
CPUs can be unreliable. The fans can stop working or the thermal contact between the CPU
and the heat sink can be poor. One can use an IR (infrared) thermometer (available at, e.g.,
Radio Shack) to obtain a precise temperature measurement for all the chips. We obtained an IR
thermometer to measure the CPU temperature to be sure the “lm sensors” package was measuring
the temperature accurately since some CPU temperatures were reported as being quite high. We
found those temperatures to be accurate, and so we decided to remove the sides from the cases.
This dropped the CPU temperatures from 50C to 25-40C depending upon the vertical location of
the computer in the computer room (higher systems are hotter). During this process we found that
the gigabit chip runs at about 70C, the hottest of all the chips on the motherboard. Tyan discussed
this with Intel and it was determined that this chip is known to run this hot.
An efficient way to find problems in hardware and software is to stress test, or burn-in, the
cluster. Burning-in a computer cluster involves stress-testing the cluster to maximum levels of
81
http://rainman.astro.uiuc.edu/cluster/scripts/mympirun.sh
197
CPU, memory use, hard drive use, and network communication for long periods of time. This tests
whether there is sufficient power and cooling available and tests whether any hardware components
might easily malfunction. CPUs from AMD and Intel are extremely reliable and generally operate
as specified. We test CPU performance using lapack, specfp, stream82 , our own codes, and a few
other interesting benchmarks. The Benchmark HQ website83 and sourceforge84 are good resources
for Linux benchmarking software.
Despite the reliability of AMD and Intel CPUs, the author’s experience is that motherboards
made for AMD CPUs (AMD does not make motherboards), and chipsets not made by AMD or
Intel for AMD or Intel CPUs, are much less reliable than those chipsets and motherboards designed
directly by AMD or Intel. There are several computer companies85,86,87 that do not offer anything
except Intel products for chipsets, motherboards, and CPUs due to reliability problems with AMD
motherboards and chipsets or non-Intel chipsets for Intel CPUs. Assuring compatibility with the
CPU and other motherboard components is a complex task. The CPU has generally been the most
complex piece of equipment on a motherboard88 , so a motherboard and chipset designed by a CPU
manufacturer is more likely to be reliable and compatible with existing technology. We chose an
Intel 2.4Ghz Xeon CPU with an Intel chipset and a Tyan motherboard. Tyan is a well-respected
motherboard manufacturer for Intel CPUs. We chose a CPU speed that was not overpriced by
Intel, so was comparably priced to AMD processors of the same speed. It is possible that over the
past couple years that AMD related chipsets and motherboards have become more robust. It is yet
to be seen how reliable the AMD Opteron motherboards and chipsets are.
Memory performance can often be a bottleneck in physics computations by having (1) limited
bandwidth and high latency to/from the CPU, (2) limited bandwidth and high latency to/from the
system bus, (3) limited bandwidth to/from the interface boards. Typically ECC (error correction
coding) is used in memory for cluster computers to prevent errors in memory affecting the rest of
the system. Most server-based motherboards require ECC memory. Generally memory in today’s
market is reliable assuming one chooses a memory module that is a brand-name rather than generic.
A special memory feature of server motherboards is dual-channel memory. Dual-channel memory
allows interleaved access to the memory by the 2 CPUs, which allows each CPU in dual-CPU
nodes to access the memory at nearly the peak memory bandwidth. Dual-channel memory reduces
so-called “memory contention” that can occur when both CPUs access the memory simultaneously.
We have found that specific memory speeds have little impact on the speed of our codes. That
is, within the range of memory speeds available, the fastest memory seems to be little different from
the standard memory. We purchased DDR PC2100 memory, where DDR stands for double data
82
http://www.cs.virginia.edu/stream/
http://www.benchmarkhq.ru/english.html
84
http://lbs.sourceforge.net/
85
http://www.dell.com/
86
http://www.gateway.com
87
http://www.simplifiedcomputers.com/
88
Actually, for the past 2 years 3D graphics chips have been more complicated than even CPUs
83
198
rate, and the memory operates at 2100MB/s peak bandwidth. We use “memtest86”89 to check
single-CPU memory speed and check for memory errors during a reboot of a system. Memtest86
acts as an entire OS that loads like any Linux kernel. One simply adds another entry to the
“/etc/grub.conf” or “/etc/lilo.conf” files. “Grub” and “lilo” are programs that run at boot time
that load the primary OS component and pass a few key arguments to the OS. One can also test
the memory using “stream”.
B.6.2
Bandwidth and Latency of Network
Two basic measures of network performance are (1) peak bandwidth for large messages and (2)
latency for small messages. These 2 measures characterize a full analysis of the amount of time it
takes to transfer a message of arbitrary size. In comparing various onboard gigabit chips and PCI
boards in 64-bit 66Mhz and 133Mhz PCI slots, we realized that the onboard chips operate faster or
equal to add-on boards. In sections B.1 and B.2 we gave the performance results for all our tests
on gigabit and Myrinet. These benchmarks used the “mpptest” program included with MPICH
in the “examples/perftest” directory, and an example is available online90 . Figure B.5 shows the
1-way bandwidth and latency for MPI communications between 2 2.4Ghz BH nodes for a range
of message sizes. Notice that gigabit only achieves peak bandwidth for message sizes greater than
about 20kB reaching a peak bandwidth of about 100MB/s. Also, notice that the latency for small
message sizes is about 27µ s. These performance figures can be compared to similar tests with
modern Myrinet boards on MPICH over GM91 that achieve a peak bandwidth of about 250MB/s
and a latency of about 9µ s. NetPIPE92 can also be used to measure network performance, but it
does not account for overhead due to MPI.
One can measure the data overhead associated with MPI on the network by comparing the
“tcpdump” results for the number of bytes transferred on a device with the known number of
transfers in, say, a fluid code. From a fluid code one can count the total number of bytes transferred
as boundary values per time step as N = Nv NBC Nx Nb/v Nt , where Nv is the number of named
single unit variables transferred, NBC is the number of boundary zones per named variable, Nx
is the typical dimension of the edge between CPUs, Nb/v is the number of bytes per variable,
and Nt is the number of unidirectional transfers per time step per CPU to CPU communication.
For our ZEUS-based MHD code, we have Nv = 36, NBC = 2, Nb/v = 8, and Nt = 2. This
gives N = 1152Nx , or about 1kB per zone edge. The typical message passed has Nv = 1 and
Nt = 1, so gives N = 16Nx . For a typical value of Nx = 128 the typical message size is 2kB per
time step. This is a small message size, and from Figure B.5 we see that the bandwidth is only
20MB/s. Often when running on hundreds of CPUs (e.g. 200 = 25 × 8) Nx = 64 leads to a more
practical total resolution. In this case the bandwidth on gigabit is only about 10MB/s, 1/10 the
89
http://www.memtest86.com/
http://rainman.astro.uiuc.edu/cluster/scripts/mpptest.sh
91
http://www.myri.com/myrinet/performance/MPICH-GM/index.html
92
http://www.scl.ameslab.gov/netpipe/
90
199
Figure B.5 Bandwidth (left) and latency (right) for gigabit Ethernet connection on BH cluster.
peak bandwidth. For a 3D simulation no such limits occur since then Nx = N1 N2 . A practical
resolution is N1 = N2 = 32. In this case the smallest message size is N = 16kB giving 85MB/s on
gigabit, near the peak bandwidth. HARM operates by always transferring Nv = 8, so N = 128Nx ,
and for Nx = 64 we have the smallest message size is N = 8kB for a bandwidth of 60MB/s, which
is reasonably near the peak bandwidth. HARM in 3D will easily reach the peak bandwidth of
gigabit, and consequently require a faster network.
One can also check for whether the communication is completing faster or slower than the CPU
processing by checking the percent of CPU usage in “top” (assuming MPICH is compiled without
“-comm=shared”). For example, our HARM code with 1282 zones per node achieves about 80%
CPU efficiency when using 16 CPUs. We advise not running single processes on the nodes running
MPI jobs with lower than 100% efficiency, since we find that the actual speed of the MPI processes
slows down nonlinearly in proportion to their CPU usage when the CPU is oversubscribed. This
could be due to memory contention. Simply using “nice” to “renice” the other processes to a nice
level of 19 does not completely remove the cache inefficiency produced by oversubscribed CPUs,
but leads to performance more proportional to the CPU usage shown by “top”.
B.6.3
Code Performance
Older NCSA systems, such as the Origin 2000 (now replaced by Copper), included utilities such as
“perfex” that gave detailed information about cache use (percent miss), MFLOPs, and other vital
statistics useful for profiling a code. There are some dead open source projects with “perfex”-like
200
utilities93 . The current prevailing and actively supported “perfex”-like utility is PAPI94 . For 32-bit
Intel-compatible CPUs, PAPI requires the kernel be recompiled after a patch is installed.
We assume all our codes are well designed for the cache and follow high performance coding
etiquette95,96 . We use the well-known performance metric of “zone cycles per second” (ZCPS) for
measuring fluid code performance. This refers to the number of grid elements that are calculated
per real time second on a given system. The code efficiency for a given number of CPUs is defined
as % Eff= ZCPS/(NCP U ∗ ZCPS1CP U ) ∗ 100, where NCP U is the number of CPUs and ZCPS1CP U is
the ZCPS for 1 CPU. In general, testing of a fluid code should be performed both with and without
diagnostics that are written to file. This allows one to test both raw processing speed and normal
processing speed, which allows us to evaluate how the HD is limiting raw performance. Below we
only report the raw processing speed in ZCPS.
We consider the performance of our viscous hydrodynamics (VHD), 2D magnetohydrodynamics
(MHD), 3D MHD, and HARM codes on some of the systems available to our group over the years.
All these codes can be run in parallel using MPI. The systems tested include the NCSA SGI Origin
2000 composed of 256 R10000 250Mhz CPUs networked internally by SGI’s proprietary hardware
connecting each CPU’s memory, the NCSA Platinum cluster composed of 1024 1Ghz Intel Pentium
3 (P3) CPUs on 512 dual-CPU nodes networked with Myrinet, the NCSA Itanium cluster composed
of 268 Intel 800Mhz Itanium I CPUs on 134 dual-CPU nodes networked with Myrinet, and our
BH cluster composed of 28 2.4Ghz P4 Xeon CPUs on 14 dual-CPU nodes networked with gigabit
(we exclude the 2.0Ghz P4 Xeons from these tests). We expect NCSA’s Tungsten 3.06Ghz P4
Xeon cluster to show similar performance to BH, with Tungsten using Myrinet showing improved
efficiency for a large number of processes. We provide tables of the number of ZCPS and efficiency
of some of our codes on each system. We run all codes in double precision. Single precision offers
some speed advantages, but often is insufficient to preserve important mathematical constraints.
For example, after the number of time steps needed to study accretion flows, a single-precision
based solenoidal constraint leads to errors larger than truncation error. All the codes are run with
similar initial conditions that model the accretion flow near a black hole.
As a baseline for parallel performance comparison, the number of ZCPS for each of our codes for
a single CPU run is shown in Table B.1. The HARM code tests shown are from our original HARM
code that had no failure checks, so is faster than our latest version of HARM with many failure
checks. These checks slow our new HARM down by about a factor of 2X. We plan to profile the
new HARM to optimize these failure checks. Notice that for our codes the Intel Itanium I processor
is about a factor of 2X slower than the older 1Ghz Intel P3 processor, which was unexpected based
upon synthetic benchmarks. Our collaborating group performing numerical relativity simulations
found Itanium I CPUs to be about 2X faster than the 1Ghz Intel P3 processor. This demonstrates
93
http://www.osc.edu/~troy/lperfex/
http://icl.cs.utk.edu/papi/
95
http://www.intel.com/cd/ids/developer/asmo-na/eng/microprocessors/ia32/pentium4/optimization/
43896.htm
96
http://rainman.astro.uiuc.edu/cluster/sgiperf.pdf
94
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Table B.1. Single CPU performance in ZCPS
CPU
R10000
Alpha 21264
1GHz P3
800Mhz Itanium I
2.4Ghz P4 Xeon
VHD
2D MHD
3D MHD
HARM
110
210
155
82
300
100
204
141
75
282
110
70
30
144
30
17
8
40
Note. — ZCPS from actual measurements. The resolution per
CPU is 642 for 2D and 403 for 3D. Tungsten performance can be
estimated by simply multiplying the Xeon performance by ∼ 1.3.
the necessity to profile one’s own code rather than relying on anecdotes or synthetic benchmarks.
Notice also that the 4-5 year old Alpha CPUs are nearly as fast as the 2.4Ghz Intel P4 Xeon CPUs.
We briefly mention our Alpha cluster with Myrinet that has only 6 CPUs. For a 4-CPU MPI
test of the 2D MHD we find an efficiency of 80%, while for a 6-CPU test we find an efficiency of
70%.
When the NCSA Origin 2000 was available, we performed MPI tests using our 2D MHD code
with artificial resistivity and the Alfvén -limiter enabled. File writing of diagnostics was disabled.
The results of these tests are shown in Table B.2. We found the Origin 2000 to have disk performance
problems for more than about 64 CPUs, which showed up when turning on file writing of our
diagnostics. File writing with all diagnostics using 256 CPUs drops the efficiency to 37%, while
with half the diagnostics drops to 55-65%. Otherwise the performance levels off at about 70%
efficiency even up to 256 CPUs, which is excellent for the 2D MHD code.
The tables of 2D & 3D MHD and HARM performance on Platinum and BH show columns
for 1) the number of cpus; 2) the number of CPUs per node used; 3) the number of nodes used;
4) the CPU tile geometry; 5) the number of ZCPS per unit 1000; and 6) the parallel efficiency.
Note that the 2D and 3D MHD code’s inner loop stride is the “i” or “x”-direction. Thus, the
CPU tile geometry is denoted as Tx × Ty , where Tx is the number of tiles in the x-direction and
Ty is the number of tiles in the y-direction. Alternatively, HARM’s inner loop stride is the “j” or
“y”-direction. For comparison purposes the CPU tile geometry in the table is flipped to show “y
× x”. This allows direct table-to-table comparisons.
We performed nonrelativistic MHD simulations using NCSA’s Platinum cluster using the 2D
and 3D MHD codes. The results of the 2D and 3D MHD tests are shown in Table B.3. Notice
that Platinum operates about 20% faster on 1cpu/node than 2cpus/node. Our collaborating group
performing numerical relativity simulations found a decrease in performance between 30-60% when
202
Table B.2.
Origin 2000 MPI performance in kZCPS for 2D MHD code
# CPUs
Speedup
% Eff
kZCPS
1
4
9
16
36
49
64
121
256
1.0
3.7
8.7
11.2
24.2
34.1
36.5
92.7
173.9
100
93
90
70
67
70
57
77
72
110
410
955
1235
2662
3751
4015
10201
19129
Note. — The resolution per CPU is 642
in 2D.
using 2cpus/node compared to 1cpu/node, so they avoid using 2cpus/node. NCSA jobs are charged
per node rather than per CPU, so a 30% performance decrease is reasonable to avoid the factor
of 2X charge in CPU-hours. We have underlined those tests that show notably low efficiencies
compared to similar runs. No simple reason could be found for these low efficiencies.
See Table B.4 for MPI tests of the 2D MHD code on our BH cluster. All these tests are run on
the 2.4Ghz Xeon nodes only, thus we test up to only 14 nodes (rather than 17 – the total number
of nodes). These simulations are in 2D spherical polar coordinates. We test both 642 and 2562 tile
sizes per CPU. Artificial resistivity and the Alfvén -limiter are enabled. File writing is disabled.
Timing is based upon the wall clock time with nodes otherwise free of processes. The code runs
for a fixed number of time steps (about 1500 for 642 and about 90 for 2562 ). We have underlined
those tests that show notably low efficiencies compared to similar runs. No simple reason could be
found for these low efficiencies.
See Table B.5 for MPI tests of the 3D MHD code. These simulations use a tile size per
CPU of 403 . The grid is based upon a 3D Cartesian coordinates with inner and outer Cartesian
approximations to spherical shells. Artificial resistivity and the Alfvén -limiter are enabled. File
writing is disabled. Timing is based upon the wall clock time with nodes otherwise free of processes.
The code is run for a fixed number of time steps (about 120).
See Table B.6 for MPI tests of HARM. These simulations are in 2D spherical polar coordinates
in Kerr-Schild coordinates. We test both 642 and 2562 tile sizes per CPU. The timing is based
upon the wall clock time with nodes otherwise free of processes. The code runs for a fixed number
of time steps (about 100 for 642 tile and about 10 for 2562 ). We use the latest HARM code running
a fiducial simulation as in McKinney and Gammie (2004) with a fixed 3 iterations per zone for
203
Table B.3.
NCSA Platinum MPI performance in kZCPS for 2D & 3D MHD code
# CPUs
2D MHD
1
2
2
3
4
4
6
6
32
32
3D MHD
1
2
2
4
4
6
6
8
8
16
16
24
24
32
32
cpus/node
# nodes
1
2
1
1
2
1
2
1
2
1
1
1
2
3
2
4
3
6
16
32
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
1
1
2
2
4
3
6
4
8
8
16
12
24
16
32
geom
kZCPS
% Eff
1×1
1×2
1×2
1×3
2×2
2×2
2×3
2×3
4×8
4×8
141
218
246
351
190
437
267
589
2146
2800
100
77
87
83
34
77
32
70
48
62
1×1×1
1×1×2
1×1×2
1×2×2
1×2×2
1×2×3
1×2×3
2×2×2
2×2×2
2×2×4
2×2×4
2×3×4
2×3×4
2×4×4
2×4×4
70
117
137
220
265
308
377
418
514
775
964
985
1212
1321
1582
100
84
98
79
95
73
90
75
92
69
86
59
72
59
70
Note. — The resolution per CPU is 642 in 2D and 403 in 3D.
Using 2562 gives a bit lower per CPU performance and a bit better
MPI efficiency.
204
Table B.4.
BH Xeon Cluster MPI performance in kZCPS for ZEUS-based 2D MHD code
# CPUs
cpus/node
# nodes
geom
1 kZCPS
1 %Eff
2 kZCPS
2 %Eff
1
2
2
3
4
4
4
4
6
6
6
6
12
12
12
12
14
14
14
14
14
28
28
28
1
2
1
1
2
1
1
1
2
1
1
1
2
1
1
1
2
1
1
1
1
2
2
2
1
1
2
3
2
4
4
4
3
6
6
6
6
12
12
12
7
14
14
14
14
14
14
14
1×1
1×2
1×2
1×3
2×2
2×2
4×1
1×4
2×3
1×6
6×1
2×3
2×6
2×6
1 × 12
12 × 1
2×7
2×7
7×2
14 × 1
1 × 14
4×7
1 × 28
28 × 1
282
511
513
696
720
787
638
650
872
1297
954
976
1587
1626
2069
1990
1767
1762
1951
2352
2354
2245
3849
4191
100
91
91
82
64
70
57
58
52
77
56
57
47
48
61
59
45
45
50
60
60
28
49
53
305
513
572
753
843
950
848
1007
1110
1428
1195
1218
2176
2108
2375
2000
2493
2297
2335
2359
2680
4431
5094
4187
100
84
94
82
70
78
70
83
61
78
65
66
60
58
65
55
58
54
55
55
63
52
60
49
Note. — (1 )642 per CPU tile size. (2 )2562 per CPU tile size.
205
Table B.5.
BH Xeon Cluster MPI performance in kZCPS for 3D MHD code
# CPUs
cpus/node
# nodes
1
2
2
4
4
6
6
8
8
12
12
16
16
16
16
24
28
1
2
1
2
1
2
1
2
1
2
1
1
1
1
1
2
2
1
1
2
2
4
3
6
4
8
6
12
16
16
16
16
12
14
geom
1×1×1
1×1×2
1×1×2
1×2×2
1×2×2
1×2×3
1×2×3
2×2×2
2×2×2
2×2×3
2×2×3
2×2×4
1×4×4
1 × 1 × 16
16 × 1 × 1
2×3×4
2×2×7
Note. — Per CPU tile size is 403 .
206
kZCPS
% Eff
161
282
307
528
600
750
854
931
1117
1317
1323
1673
1603
1784
1609
2247
2740
100
88
95
82
93
78
88
72
87
68
68
65
62
69
66
58
61
Table B.6.
BH Xeon BH Cluster MPI performance in kZCPS for 2D HARM code
# CPUs
cpus/node
# nodes
geom
1 kZCPS
1 %Eff
2 kZCPS
2 %Eff
1
2
2
2
2
3
4
4
4
4
6
6
6
6
6
12
12
12
12
12
14
14
14
14
14
14
28
28
28
28
1
2
2
1
1
1
2
1
1
1
2
2
1
1
1
2
2
1
1
1
2
2
1
1
1
1
2
2
2
2
1
1
1
2
2
3
2
4
4
4
3
3
6
6
6
6
6
12
12
12
7
7
14
14
14
14
14
14
14
14
1×1
1×2
2×1
2×1
1×2
3×1
2×2
2×2
1×4
4×1
2×3
3×2
6×1
1×6
3×2
2×6
6×2
6×2
12 × 1
1 × 12
2×7
7×2
7×2
2×7
1 × 14
14 × 1
7×4
4×7
1 × 28
28 × 1
17
33
33
35
35
46
61
61
57
61
91
86
88
86
92
175
174
164
169
172
209
207
198
200
203
209
348
369
335
396
100
97
97
100
100
90
90
90
84
90
89
84
86
84
90
86
85
80
83
84
88
87
83
84
85
88
73
78
70
83
18
32
33
34
34
49
67
62
62
62
96
97
93
101
100
152
192
184
185
184
169
219
191
168
205
213
430
377
427
408
100
89
92
94
94
91
93
86
86
86
89
90
86
94
93
70
89
85
86
85
67
87
76
67
81
85
85
75
85
81
Note. — (1 ) 642 per CPU tile size. (2 ) 2562 per CPU tile size. Tile geometry is
flipped for direct comparison to the 2D & 3D MHD code tables. Actual tile geometry
shown is Ty × Tx , where Tx is the number of CPU tiles in the “x-direction” and Ty is
the number of CPU tiles in the “y-direction”.
207
variable inversion, which is typical. Some simulations with many failure corrections use up to 15
iterations per zone, reducing per CPU performance by about 50%. The results of this table may
differ from other results obtained using older versions of HARM. These differences are due to how
the code was configured when tested and a lack of optimizations in the latest version of HARM. We
disable file writing for these tests. All these tests are run on the 2.4Ghz Xeon nodes (i.e. not the
2.0Ghz Xeon nodes). We have underlined those tests that show notably low efficiencies compared
to similar runs with different tile geometries. In these poorly performing cases, the CPU geometry
is extended in the inner loop stride direction.
Similar tests performed on the BH cluster using 100Mbps Ethernet show similar performance
to gigabit for a small number (≤ 8) of CPUs for HARM and 3D MHD tests, but 2D MHD test
results for any CPU number & 4 results in poor performance due to the higher latency of Ethernet.
HARM is not functional on the Tungsten cluster due to issues that are not yet understood.
Tungsten started operating in production mode with Myrinet on June 23, 2004, about the time
this appendix was written. We expect Tungsten to operate better than BH due to the Myrinet
interface, and we expect Tungsten using gigabit will operate similarly to BH after taking the CPU
speed difference into account. We expect that for a large number of CPUs (say 200 operating on
25 × 8 CPU tile geometry), Tungsten with Myrinet will achieve an efficiency of about 70%.
We find that for HARM on the BH cluster, using only 1cpu/node gives faster performance
than 2cpus/node by 10-17%. The older Alpha cluster shows no change in performance between
1cpu/node and 2cpus/node. Some Platinum results show 2cpu/node performance problems, especially for 2D MHD simulations that show up to 50% performance degradation on 2cpus/node
compared to 1cpu/node. HARM performs quite similarly on both 1 and 2 cpus/node and on any
tile geometries. Our collaborating group performing numerical relativity calculations using their
latest codes finds an even larger performance drop of 50-70% using 2cpus/node from 1cpu/node.
Despite this effect they still plan to purchase dual-CPU nodes due to the cost effectiveness of dualCPU nodes. They might plausibly modify their code or develop new code to improve dual-CPU
efficiency. Our collaborating group finds overall higher MPI efficiencies due to a large number of
computations per zone, although they may improve MPI performance by changing from DAGH
(an antiquated MPI library) to directly using MPI functions.
Notice also that the CPU tile geometry can significantly impact performance. For a fixed
tile size, the number of bytes transferred is smallest for an equal number of tiles per dimension.
However, for codes that require many small messages such as the 2D MHD code, extended CPU
geometries can be more efficient by increasing the bandwidth of message passing and making the
CPU to do more work per unit of communication. Otherwise, the CPU is under-utilized due to
the system waiting for the messages to complete passing. For example, the 2D MHD code using
28 CPUs and 2cpus/node with 4 × 7 CPU tile geometry achieves only 40% CPU utilization, while
with 1 × 28 CPU tile geometry achieves 80% CPU utilization. Assuming the code uses the CPU
cache efficiently on the inner loop stride, elongating along the inner loop stride has no benefits. In
fact, we find a slightly larger performance for tile geometries elongated along the outer loop stride
208
for 28 CPUs with 2cpus/node for the 2D MHD code. However, for 4 CPUs and 1cpus/node a 2 × 2
geometry is optimal.
Increasing the tile size can also affect performance for codes that operate on small message sizes.
By comparing the 642 and 2562 cases in Table B.4, notice that the larger 2562 per CPU tile size
typically runs faster per CPU and is typically more efficient in MPI due to larger message sizes.
B.7
Beowulf Cluster Summary
We have described the method used to design, build, test, and use a Beowulf cluster from start
to finish. The design process involves testing plausible configurations of hardware and software by
setting up small test clusters. This process is critical in verifying a specific configuration will be
successful. Building a Beowulf cluster involves mounting the nodes on a rack or shelf, considering
cooling and electrical issues, and setting up the physical network connections and physical components of the network switch. We found that a simple, cheap rack suffices for the number of nodes
(15-28) we purchased. Modifying the electric outlets and circuits in the computer room took time
to authorize and complete, and in the meantime we learned that stress-testing a system is key to
verifying that the power supplied to the cluster is sufficient. The network has been transparent
and working at 100MB/s peak bandwidth and 27µ s latency using standard Intel drivers with some
modified parameters.
We found that synthetic benchmarks tell only a small part of the performance story, and that
testing codes expected to run on the cluster (or similar test cluster) is the only clean way to estimate
cluster performance. We generally perform hyperbolic, fluid dynamic simulations of accretion
flows around black holes. Our collaborators perform numerical relativistic dynamical space-time
simulations. They find very different results on nearly every system we tested, including the final
BH cluster. For example, they find their code performs 2X faster on Itanium I CPUs than on
Platinum 1Ghz Intel P3 processors, while we find our code performs 2X slower.
We have provided tables of single CPU and parallel performance of all our accretion flow codes,
including our VHD, 2D MHD, 3D MHD, and GRMHD (HARM) codes. We have tested these codes
on several systems, including the SGI Origin 2000, the NCSA Platinum cluster, the NCSA Itanium,
and our BH cluster. We expect NCSA’s latest cluster, Tungsten, to show similar performance to
BH. Tungsten using Myrinet should show improved efficiency for a large number of processes.
We found that tile size, tile geometry, and whether one uses 1cpu/node or 2cpus/node can
significantly affect parallel performance. Our HARM code is least affected by such considerations,
with 10X the operations per CPU per zone compared to the 2D MHD code. This results in a much
longer time between communications, leading naturally to a higher parallel code efficiency.
HARM and the 2D MHD codes perform marginally better at 2562 tile size than at 642 tile size.
HARM typically performs best with a larger number of tiles along the outer loop stride, especially
for the 2562 tile size per CPU. However, the 2D and 3D MHD codes show no consistent preference
for tile geometry for their worst case efficiencies. HARM and the 3D MHD codes essentially run
209
the same on 1cpu/node and 2cpus/node, while the 2D MHD code clearly operates more efficiently
when using 1cpu/node.
210
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Curriculum Vitae
Jonathan C. McKinney
Address
Physics Department
University of Illinois at Urbana-Champaign
1110 West Green Street
Urbana, IL 61801
Phone: (217) 766-6555
fax: (217) 244-7638
email: [email protected]
web: rainman.astro.uiuc.edu/ jon/
Personal
Birth November 22, 1973 in Methuen, MA, USA
Education
2004 Ph.D. in Physics, Theoretical Astrophysics, UIUC
Dissertation: Black Hole Accretion Disk, Jet, and Corona (Prof. Gammie, advisor)
1999 M.S. in Physics, UIUC
1996 B.S. in Physics (Magna Cum Laude), Texas A&M Univ., College Station
Sr. Thesis: 2-D Wavefunction Time Evolution using Wavelets/Fourier transforms investigating
Periodic Potentials (Prof. Siu Ah Chin, advisor)
Jr. Thesis: Atmospheric Polarization due to Incoherent Light from Sun: A Layer Model of the
Atmosphere (Prof. George W. Kattawar, advisor)
250
Fellowships, Awards, & Contributions
2004 Institute for Theory and Computation (ITC) fellowship, CfA, Harvard College Observatory
2003 Coauthored NCSA NRAC proposal awarded 300,000 Tungsten SUs, with a significant portion
for my research. PI: Charles F. Gammie
2002 Contributed to NSF ITR Program Award 0205155, MHD Simulations in Full General Relativity . PI: Charles F. Gammie
2001 Contributed to NSF Award 0093091, Theory of Black Hole Accretion Flows . PI: Charles F.
Gammie.
2001-2004 NASA GSRP Fellow (S01-GSRP-044), annual proposals, sole author, source of current
funding
1997, 2000 General Electric Fellow (for scholastic excellence as graduate)
1996-1997 Faculty Achievement Award (for leadership in the College of Science at Texas A&M)
1996 Summer Research Fellowship, University of Illinios at U-C
1994-2003 Golden Key National Honor Award: National Honor Society
Employment and Training
2004- ITC Postdoctoral Fellow, CfA, Harvard College Observatory, Harvard University
2001-2004 NASA Fellow, UIUC
1999-2000 Research Assistant, UIUC
Computational and theoretical study of black hole accretion disks.
Advisor: Prof. Charles Gammie
Molecular Clouds, Galactic Dynamics, Accretion Disks
2000 Teaching Assistant, UIUC, Graduate level, The Physics of Compact Objects , Prof. Stuart L.
Shapiro
1998 Research Assistant, UIUC
General relativistic hydrodynamic processes involving shocks as applied to cosmological sheets.
Advisor: Prof. Mike Norman, Sr. res. scientist, NCSA (now UCSD)
Numerical methods to model astrophysical fluid dynamical systems
1997-1999 Teaching Assistant, UIUC, ENGR. level: Q.M., E&M, Mech., and Stat. Mech.
251
1996-1997 Teaching Assistant, Texas A&M, Graduate level: Stat. Mech.
1997Sum Electrical Engineer Assistant for Turnkey in Plano, TX for Mr. Steve Williams
1996Sum Research Assistant (REU), UIUC
Symmetry properties and chaos in electron transport in semiconductor superlattices.
Advisor: Prof. David K. Campbell (now Dean Boston University)
Nonlinear Dynamics of Electrons in Mesoscopic Nanostructures
1996 Research Assistant, Texas A&M University
1-D and 2-D wavefunction time evolution using wavelet/Fourier transform. Periodic reflectionless
quantum waveguides.
Advisor: Prof. Siu Ah Chin
Theoretical nuclear physics; high-density matter; lattice calculations; Monte Carlo methods
1995Sum Repair and Maintenance Technician for Dallas Semiconductor in Dallas, TX for Mr.
David Massey
1995 Research Assistant, Texas A&M University
Atmospheric polarization due to incoherent light from Sun interacting with the layers of the sky.
Light from Moon and Sun that create the green and blue shock on sunsets and sunrises.
Advisor: Prof. George W. Kattawar
Atmospheric/oceanic optics; radiative transfer w/ elastic and inelastic scattering in the atmosphereocean system
1994 Computer Technician and Autocad design for B.L. & P. Engineers in Dallas, TX for Scott
Brady
Publications
McKinney, J.C. and Gammie, C.F., A measurement of the hydromagnetic luminosity of a Kerr
black hole , 2004, ApJ, 611 , 977M
Watson, W. D., Wiebe, D. S., McKinney, J. C., and Gammie, C. F., Anisotropy of magnetohydrodynamic turbulence and the polarized spectra of OH masers , 2004, ApJ, 604 , 707W
Gammie, C.F., Shapiro, S.L., and McKinney, J.C., Black hole spin evolution , 2004, ApJ, 602 ,
312G
Gammie, Charles F., McKinney, Jonathan C., and Tóth, Gábor, HARM: A numerical scheme for
general relativistic magnetohydrodynamics , 2003, ApJ, 589 , 444G
McKinney, J. C. and Gammie, C. F., Numerical models of viscous accretion flows near black holes
, 2002, ApJ, 573 , 728M
252
Anninos, P. and McKinney, J. Relativistic hydrodynamics of cosmological sheets , 1999, Phys. Rev.
D 60 , 064011
K. N. Alekseev, E. H. Cannon, J. C. McKinney, F. V. Kusmartsev, and D. K. Campbell. Symmetrybreaking and chaos in electron transport in semiconductor superlattices , 1998, Physica D. 113 ,
129-133
K. N. Alekseev, E. H. Cannon, J. C. McKinney, F. V. Kuzmartsev, and D. K. Campbell. Spontaneous dc current generation in a resistively shunted semiconductor superlattice driven by a terahertz
field , 1998, Phys. Rev. Lett. 80 , 2669-2672
McKinney, J. C., Alekseev, K. N., Cannon, E. H., and Campbell, D. K., Dissipative chaos and
symmetry-breaking in semiconductor superlattices, 1996, REU Thesis, U. of Illinois, Urbana
Publications in Preparation
McKinney, J.C. and Gammie, C.F., General relativistic MHD simulations of thin disks , 2004, in
preparation
Refereed Journals
2002 - Astrophysical Journal, Astrophysical Journal Letters
Invited Talks
CTA Seminar on Theoretical Astrophysics & General Relativity
http://www.physics.uiuc.edu/Research/CTA/seminars/
2003Spr Intermediate-Mass Black Holes: Formation Theories & Observational Constraints
2002Spr Efficient Acceleration and Radiation in Poynting Flux Powered GRB Outflows
2002Fal High-Energy Gamma Rays from AGN, GRBs, and Plerions
2001Fal Black Hole accretion in Active Galactic Nuclei (also prelim. exam)
2001Spr Bar-Driven Dark Matter Halo Evolution: A Resolution of the Cusp-Core Controversy
2000Fal Gamma-Ray bursts: Magnetized Collapsars and duration of GRBs
253
2000Spr Planet Formation: The Effects of Thermal Energetics on 3-D Hydrodynamic Instabilities
in Massive Protostellar Disks
1999Fal Discussion of Numerical Methods on the study of AGN: GRMHD
1999Spr Global Magnetohydrodynamical Simulations of Accretion Tori
REU (Research Experience for Undergraduates)
http://www.physics.uiuc.edu/education/undergrad/reu/
1996Sum Semiconductor Superlattices
Professional Memberships
1999- American Astronomical Society (AAS)
1999- American Physical Society (APS)
Undergraduate Memberships
1994-1997 Society of Physics students (SPS)
1992-1996 Texas A&M Physics Club
Meets every Tuesday with guest speakers
Ongoing Projects such as the high school laser show
Participated in all these and weekend shows to the public
1993-1997 Texas A&M Astronomy Club
Meets every Friday Night
Observe and photograph different phenomena, such as the Shoemaker-Levy Comet impact on
Jupiter
http://www.physics.sfasu.edu/astro/sl9.html
Computational Experience
Beowulf Cluster Principle designer, builder, and manager of a 32 CPU Linux gigabit & Myrinet
cluster for testing, development, and up to medium scale simulations. Design and associated publications:
http://rainman.astro.uiuc.edu/cluster/
254
Digital Demo Room Helped create a web portal for astrophysical simulations: http://ddr.astro.
uiuc.edu
Software Written Developed ZEUS-like parallel 3D MHD code from scratch
Contributed significantly to writing HARM
Parallelized all our groups codes using MPI
Workshops NCSA Microprocessor Performance Tuning, Jan 2002
NCSA Linux Clusters Institute Workshop, Oct 2001
NCSA MPI Workshop, Mar 2001
Science Applications Expert: Mathematica, Supermongo, Matlab
Basic: Maple
Operating Systems Expert: All forms of Linux, DOS, and Windows
Basic: VMS, Solaris, SUN
Programming Expert: C, FORTRAN77 & 90, C++, Visual C++, Bash
References
Prof. Charles F. Gammie
Dept. of Physics, MC-704
University of Illinois
at Urbana-Champaign
1110 West Green Street
Urbana, IL 61801-3080
(217) 333-8646 (office)
(217) 244-7638 (fax)
[email protected]
Prof. Stuart L. Shapiro
Dept of Physics, MC-704
University of Illinois
at Urbana-Champaign
1110 West Green Street
Urbana, IL 61801-3080
(217) 333-5427 (office)
(217) 333-9783 (lab)
[email protected]
Prof. William D. Watson
Dept. of Physics, MC-704
University of Illinois
255
at Urbana-Champaign
1110 West Green Street
Urbana, IL 61801-3080
(217) 333-7240 (office)
[email protected]
Prof. & Dean David K. Campbell
Boston University
College of Engineering
44 Cummington Street
Boston, MA 02215
(617) 353-2800 (office)
(617) 353-5929 (fax)
[email protected]
Dr. Peter Anninos
University of California
Lawrence Livermore
National Laboratory
Livermore, CA 94550
[email protected]
256