Download Beams, Bursts, Bubbles, and Bullets: Relativistic Outflows in

Beams, Bursts, Bubbles, and Bullets:
Relativistic Outflows in Astrophysics
by
Sebastian Heinz
M.S., University of Colorado, 1997
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Department of Astrophysical and Planetary Sciences
2000
This thesis entitled:
Beams, Bursts, Bubbles, and Bullets:
Relativistic Outflows in Astrophysics
written by Sebastian Heinz
has been approved for the Department of Astrophysical and Planetary Sciences
Prof. Mitchell C. Begelman
Prof. Ellen G. Zweibel
Date
The final copy of this thesis has been examined by the signatories, and we find that both the
content and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.
Heinz, Sebastian (Ph.D.)
Beams, Bursts, Bubbles, and Bullets:
Relativistic Outflows in Astrophysics
Thesis directed by Prof. Mitchell C. Begelman
This thesis investigates the physics of relativistic outflows in astrophysical scenarios.
We focus on collimated flows, known as jets, that occur in Active Galactic Nuclei (AGN) and
Gamma-Ray Bursts (GRBs).
The production and early propagation of relativistic jets is examined in the case of acceleration by dynamically dominant tangled magnetic fields. It is shown that such a configuration
behaves in a similar fashion as a pure particle pressure dominated jet, however, it can avoid radiative losses and inverse Compton drag, which hamper particle dominated flows. The radiative
signatures of such jets and several complications, such as dissipation of magnetic energy and
radiation drag, are considered.
To investigate the energetics and propagation of the kpc scale jet, we study the jet in the
nearby galaxy M87. We find that the complex spectral behavior and the surprising correlation
between radio brightness and the optical spectral index can be explained by simple adiabatic
acceleration of the radiating particles. The lack of significant spectral evolution is consistent
with magnetic fields below equipartition by a factor of a few.
The impact of the enormous energy flux in the AGN jets on the large scale intergalactic
environment is studied on the basis of a simple dynamical model. In general, the flow will
displace the surrounding medium, leading to a depression in the X-ray surface brightness. We
show how a grid of models can be used to infer important physical parameters, such as the
average kinetic jet power from Chandra observations of radio galaxies embedded in clusters.
Relativistic flows also occur in Gamma-Ray Bursts. It has generally been accepted that
only models in which the gamma rays originate from internal variations in the flow (internal
iv
shocks) can explain the complex temporal signatures of GRBs. We present a new model, based
on external dissipation of kinetic energy stored in dense bullets, that can also explain the millisecond variability seen in some bursts. The basic observational characteristics of such a model
are presented, along with a preliminary analysis of the requirements that the viability of such a
flow imposes on the central engines of bursts.
v
Acknowledgements
This work was supported by a two year Fulbright grant and NSF grants AST 91-20599,
AST 95-29170, AST 95-29175, and AST 98-76887.
Contents
Chapter
1 Introduction
1.1
1.2
1
AGNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.1.1
AGN Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.1.2
Accretion in AGNs and Unification of Radio Quiet AGNs . . . . . . .
5
1.1.3
Radio Loud AGNs . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.1.4
Relativistic Jets in AGNs . . . . . . . . . . . . . . . . . . . . . . . . .
10
GRBs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.2.1
Lightcurves and Timing . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.2.2
Spectra and Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
1.2.3
Afterglows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
1.2.4
GRBs as Relativistic Flows . . . . . . . . . . . . . . . . . . . . . . . .
18
2 Jet Acceleration by Tangled Magnetic Fields
2.1
2.2
21
Tangled Fields as an Alternative . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.1.1
The Twin Exhaust Model . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.1.2
Compton Rockets . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.1.3
Magneto-Centrifugal Acceleration . . . . . . . . . . . . . . . . . . . .
25
2.1.4
Tangled Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
vii
2.3
2.4
2.5
2.2.1
Treatment of Magnetic Field . . . . . . . . . . . . . . . . . . . . . . .
30
2.2.2
Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
Dynamical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
2.3.1
Critical Points
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
2.3.2
The Effects of Gravity on the Sonic Transition . . . . . . . . . . . . .
40
Solutions in the Self-Similar Regime and Asymptotic Solutions . . . . . . . . .
44
2.4.1
Opening Angles and Causal Contact . . . . . . . . . . . . . . . . . . .
48
2.4.2
Equipartition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
2.4.3
Full Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
2.4.4
Radiation Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
2.5.1
2.6
Tradeoff Between Dissipation and Acceleration and Synchrotron Brightness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
2.5.2
Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
2.5.3
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
3 Energetics of Jets and the M87 Jet
3.1
66
The M87 Jet After 79 Years of Surveillance . . . . . . . . . . . . . . . . . . .
66
3.1.1
The Spectral Aging Problem . . . . . . . . . . . . . . . . . . . . . . .
70
3.1.2
Proposed Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
3.2
Adiabatic Effects on Synchrotron Emission . . . . . . . . . . . . . . . . . . .
75
3.3
Relativistic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
3.4
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
3.4.1
Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
3.4.2
Comparison with Earlier Models . . . . . . . . . . . . . . . . . . . . .
92
3.4.3
X-ray Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
viii
3.5
3.4.4
Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
3.4.5
Particle Acceleration Radius . . . . . . . . . . . . . . . . . . . . . . .
97
Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
4 Evolutionary Signatures of Radio Galaxies
100
4.1
The Expansion of a Radio Galaxy: Connection Between CSOs and FRIIs . . . 101
4.2
Description of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.3
4.4
4.5
4.2.1
The Dynamical Model . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.2.2
Calculation of the X–Ray Brightness . . . . . . . . . . . . . . . . . . 106
4.2.3
Observational Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . 107
Applications to Existing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.3.1
Perseus A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.3.2
Other Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Predictions for (Chandra) Observations . . . . . . . . . . . . . . . . . . . . . 115
4.4.1
Detectability of Young Sources . . . . . . . . . . . . . . . . . . . . . 115
4.4.2
Extended Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.5.1
Complications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.5.2
Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5 A Shotgun Model for Gamma-Ray Bursts
5.1
127
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.1.1
External Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.1.2
Internal Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.2
The Shotgun Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.3
GRB Efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.4
Simulating Lightcurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.5
Afterglows in the Shotgun Model . . . . . . . . . . . . . . . . . . . . . . . . . 138
ix
5.6
5.7
Making Bullets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.6.1
Why Bullets Must be Contained by Adiabatic Compression . . . . . . 141
5.6.2
Rayleigh-Taylor Driven Bullet Formation . . . . . . . . . . . . . . . . 143
5.6.3
Ram Pressure Confinement and Acceleration . . . . . . . . . . . . . . 145
Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6 Conclusions
154
Bibliography
159
Appendix
A A More Realistic Dissipation Law for the Tangled Field Model
168
B Details of Synchrotron Emission
170
B.1 Synchrotron radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
B.1.1
Synchrotron Losses: . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
B.1.2
Emitted Synchrotron Spectrum: . . . . . . . . . . . . . . . . . . . . . 171
B.1.3
Equipartition: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
B.2 The Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
B.3 Spectral Cutoff Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
C Derivation of the Dispersion Relation for the Relativistic Rayleigh-Taylor Instability
176
Figures
Figure
1.1
Approximate Composite Spectra for AGNs Compared to Normal Galaxies . . .
4
1.2
Cartoon of the Standard Accretion Picture in AGNs . . . . . . . . . . . . . . .
7
1.3
Image of Cyg A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.4
Cartoon of Cocoon-Shell Morphology . . . . . . . . . . . . . . . . . . . . . .
11
1.5
Lightcurves of a Sample of GRBs . . . . . . . . . . . . . . . . . . . . . . . .
20
2.1
Cartoon of Magneto-Centrifugal Acceleration . . . . . . . . . . . . . . . . . .
27
2.2
Cartoon of Tangled Field Model . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.3
Solutions Around Critical Points . . . . . . . . . . . . . . . . . . . . . . . . .
41
2.4
Map of Tangled Field Parameter Space . . . . . . . . . . . . . . . . . . . . . .
46
2.5
Acceleration Efficiency for Dissipative Jets . . . . . . . . . . . . . . . . . .
47
2.6
Equipartition
for Dissipative Jet . . . . . . . . . . . . . . . . . . . . . . . .
50
2.7
Analytic Solutions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
2.8
Radiative Efficiency of Dissipative Jets . . . . . . . . . . . . . . . . . . . . . .
55
2.9
Projected Brightness Profiles for Radiative Jets . . . . . . . . . . . . . . . . .
57
2.10 Projected Brightness Profiles for Non-Radiative Jets . . . . . . . . . . . . . . .
59
2.11 Synchrotron Polarization for Optically thin Jets . . . . . . . . . . . . . . . . .
61
3.1
Radio Image of Virgo A (Small Scale) . . . . . . . . . . . . . . . . . . . . . .
68
3.2
Optical and Radio Jet in M87 . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
xi
3.3
The M87 Jet Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
3.4
Emissivity Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
3.5
Cutoff-Frequency Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
3.6
Chi square–Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
3.7
Best–Fit B-Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
3.8
Best–Fit vs. Equipartition Field . . . . . . . . . . . . . . . . . . . . . . . . . .
86
3.9
Estimated Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
3.10 Best–Fit Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
3.11 Best–Fit vs. Equipartition Pressure . . . . . . . . . . . . . . . . . . . . . . . .
91
3.12 Particle Acceleration Radius . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
4.1
Radial Density Profile for a Radio Galaxy Expanding Into a King Atmosphere . 109
4.2
ROSAT HRI Image of NGC 1275 (Perseus A) . . . . . . . . . . . . . . . . . . 111
4.3
Brightness Profiles for Slices shown in Figure 4.2 . . . . . . . . . . . . . . . . 112
4.4
Contours of Allowed Parameter Space for Figure 4.2 . . . . . . . . . . . . . . 114
4.5
Diagnostics for Small Sources . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.6
Detectability Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.7
Diagnostics for Large Sources . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.1
Illustration of the Angular Spreading Problem . . . . . . . . . . . . . . . . . . 130
5.2
Illustration of the Shotgun Model . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.3
Constraints Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.4
Real and Simulated Lightcurves . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.5
Bullet masses from Rayleigh-Taylor Instability in GRBs . . . . . . . . . . . . 145
5.6
Cartoon of Ram Pressure Acceleration of Bullets . . . . . . . . . . . . . . . . 146
B.1 Theoretical Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Chapter 1
Introduction
Astrophysical scenarios often offer unique settings to test and investigate extreme conditions which cannot be reproduced in a laboratory. An important class of objects fulfilling such
a role are ultra-relativistic flows encountered in relativistic jets and gamma-ray bursts (GRBs).
A thorough understanding of these flows will shed light on the complex astrophysical systems
that they are part of, as well as enhance our understanding of the princ pal physical mechanisms
behind the jet phenomenon.
The conditions in these flows are extreme in two respects: first, they move at relativistic
bulk speeds, with Lorentz factors1 of order
Lorentz factors of order
10 in the case of AGN jets, and even higher
100 1000 in the case of GRBs. Second, the internal energy of
these flows is dominated by relativistic forms of energy (rather than rest mass energy), at least
in parts of the flow, like the injection region and the shocks found in knots and hot spots. This
internal energy can be in the form of relativistic particle pressure, magnetic fields, or photons
(or even neutrinos, as might be the case in GRBs). Whichever form of energy is chosen can
have important consequences on the flow, as we will discuss in the following chapters.
The result of these extreme conditions is an equally extreme display of power: AGNs
continuously release amounts of energy that can outshine their host galaxies by orders of magnitude, and GRBs are so bright that they can briefly become the brightest objects in the universe,
as they release the isotropic equivalent2 of more than a solar rest mass energy in only tens of
1
2
p
is defined as
1= 1 v 2 =c2 , where v is the velocity of the object in question and c the speed of light.
The isotropic luminosity is derived from the observed flux by assuming that the source radiates isotropically. If
2
seconds in gamma rays alone.
This thesis will investigate several aspects of the relativistic flows that occur in radio
galaxies and GRBs. Each chapter will contain a brief introduction to the specifics of the problem
addressed in the chapter, while the remainder of this chapter will present a broader introduction
of the field of AGNs and GRBs. The reader familiar with these topics can skip directly to x2.
1.1
AGNs
Defining an AGN is not as simple as it may appear at first. While all AGNs have some
outstanding characteristic, there is no one single observational marker that is shared by all types
of AGNs (though some are very common). AGNs are best defined from a theoretical standpoint,
since there exists a model that seems to apply to all types of AGNs:
An Active Galactic Nucleus (AGN) is any galactic nucleus that harbors a massive accreting
black hole.
While this model has not yet been proven conclusively, we can use it as a working definition of AGNs for the time being, keeping in mind that, however unlikely, the theory might be
wrong.
The reasoning behind the black hole paradigm is indirect. Imaging black holes directly
is one of the Holy Grails of astrophysics and many exotic proposals have been made for how to
achieve this. The main difficulty is, of course, their small angular size: Even in M87 the event
horizon of the hole has an angular size of only 2 micro arcseconds.
The best resolution that
can be achieved to date is roughly 100 times worse than that (Junor and Biretta 1995). We thus
depend on circumstantial evidence. Luckily, such evidence has accumulated in recent years.
The strongest arguments for the black hole paradigm include:
Relativistic Iron K lines: Detections of extremely broad, double peaked lines with signatures of
both gravitational redshifts and Doppler shifts corresponding to v
>
c=3 indicate the presence
the radiation is beamed, the total energy released is reduced by the inverse of the covering fraction of the beam.
3
of strong gravity at work (Fabian et al. 1995). While the mass of the black hole cannot be
determined by this method, the observations are inconsistent with any compact object other
than a black hole, and are thus the best evidence for the existence of black holes yet.
Water Masers: Water masers have been detected from several AGNs, where they occur in the
outer (molecular) part of the accretion disk and can thus be resolved by VLBI. Very accurate
redshift measurements can be used to map out the orbits of the material. The best candidates
for this method are NGC 1068 and NGC 4258 (e.g., Kartje et al. 1999). Material has been
observed to orbit in Keplerian motion around central masses of 3:5 107 M and >
107 M
respectively, at radii of
0:13 pc and 0:65
pc respectively. The associated mass densities are
difficult to explain with anything but a massive black hole.
Variability: Causality constraints based on the observed strong variability on time scales of
order days to years imply that the emission region must be no larger than the corresponding
light crossing length. The enormous luminosities of AGNs suggest that, even with the most
efficient mechanism known to release energy (per unit mass), millions to billions of solar masses
are required to produce the energy emitted by some AGNs. The tight constraints on the size
of the emitter, together with the mass requirements, strongly argue in favor of the black hole
hypothesis (Rees 1978a).
From the maser measurements and from the dynamics of the central stellar population,
we know that the masses of these black holes range from 106 to 109 M .
1.1.1
AGN Classification
In the following we will list the main observational characteristics of AGNs.
Luminosity: Many AGNs emit at luminosities comparable to or surpassing the luminosity of
their hosts. Estimates of AGN bolometric luminosities range from
1048 ergs s
Lbol
1042 ergs s
1044 ergs s 1 .
1 , compared to the typical galaxy luminosity of L
1 to
4
1010
108
νFν
106
104
102
100
1010
1012
1014
ν
1016
1018
Figure 1.1: Composite spectra (Elvis et al. 1994) for radio loud AGNs (dotted line, stars), radio
quiet AGNs (dash-dotted line, diamonds) compared to a dusty galaxy (dash-triple-dotted line,
triangles) and an elliptical (dashed line, squares). Units are arbitrary
Broad band continuum spectra: Most AGNs have very broad band spectra compared to normal
galaxies, which feature spectra dominated by starlight (i.e., predominantly in the optical, or
IR in the presence of dust). Most of the energy in normal galaxies is emitted over one or two
decades in frequency. In contrast, most AGNs emit roughly equal amounts of energy over many
decades in frequency, ranging from the IR, or even the radio in the case of radio galaxies, all the
way to X-ray/gamma ray wavelengths. We can generally group AGNs into radio loud and radio
quiet objects, with the obvious definition for both. To visualize the difference, Fig. 1.1 shows
schematic composite spectra of the two classes.
Variability: Since the light from normal galaxies is dominated by starlight, variability in normal
galaxies is basically non-existent. On the other hand, AGNs can show variability on extremely
short time scales: flux changes by a factor of two or more have been observed on time scales
as short as hundreds of seconds at X-ray energies. At lower energies, the variability timescales
tend to grow longer (months to years in the optical), and less pronounced. More luminous
5
AGNs tend to vary on longer time scales (consistent with larger black hole sizes). Very strong
optical variability on time scales of days can be seen only in a few extreme classes of AGNs,
namely blazars.
Polarization: Most AGNs show slightly enhanced levels of polarization. In the optical, strong
polarization in AGNs is generally associated with strong variability, while in principle every
AGN that has resolved radio emission shows polarization at the few 10% level. Strong polarization generally indicates non-thermal emission mechanisms, most likely synchrotron emission
(see Appendix B).
Emission lines: Strong nebular lines characterize the spectra of most AGNs. These lines tend
to be broader than those found in starburst galaxies, and in some classes of AGNs they can
reach FWHM of
103
104 km s
1 . Even more extreme are the X-ray lines found in some
Seyfert galaxies: they display FWHMs of c=3 and line shapes that can best be interpreted as
relativistic distortions.
Small angular sizes: Most of the IR/optical/X-ray emission is released very close to the horizon
of the black hole, which makes it unresolvable with techniques available today.
Apart from these characteristics, AGNs can share many more features. Given these commonalities among some AGNs, it has to be said that there is remarkable diversity among any
random sample of AGNs. It is now believed that these differences can be resolved into a few
basic groups of AGNs, coupled with orientation effects and changes in luminosity of their different components.
1.1.2
Accretion in AGNs and Unification of Radio Quiet AGNs
The energy in AGNs is most likely released by matter falling down the potential well of
the central black hole. This matter will carry angular momentum and thus cannot simply fall
inward. Instead, the gas will settle into a disk. Viscous stresses (most likely due to magnetic
6
fields) will then slowly transport angular momentum outward, thus moving material inward.
At the same time, orbital kinetic energy is released into internal energy. The disks tend to be
the hottest closest to the black hole, which is why X-ray observations are the best candidates
to probe the innermost regions of AGNs and the strong gravity of the central black hole. If a
fraction of this energy is subsequently released in the form of isotropic radiation, we can place
a fundamental upper limit on the luminosity and the associated accretion rate. If the radiative
force on the flow exceeds the gravitational force, accretion is no longer possible. This limit is
set by
Ledd = 4GMmp c=T 1:3 1038 M=M ergs 1 ;
(1.1)
where M=M is the mass of the central object in units of solar masses, mp is the proton mass,
c is the speed of light, and T is the Thomson cross section (e.g., Frank, King, and Raine 1990).
The limit on the accretion rate then turns out to be
M_ edd 4mp Rc=T :
(1.2)
Here, R is the radius of the central object. This Eddington luminosity is the natural value against
which we much compare the emission from AGNs. This limit is independent of the geometry
of the accretion disk.
Several modes of accretion can occur, depending on the micro-physics, (most significantly, the cooling rate in the disk). If the disk can cool efficiently (i.e., most of the released
energy is radiated) the disk will be relatively cold and geometrically thin. If instead most of
the energy is trapped inside the disk (either due to opacity or radiative inefficiency), the disk
is hotter and geometrically thick. The latter scenario has been dubbed “Advection Dominated
Accretion Flow” (e.g., Narayan and Yi 1994), since most of the energy is advected through the
horizon and never leaves the flow. Whether these ADAFs exist is still a topic of hot debate
(e.g. Blandford and Begelman 1999). However, we are relatively certain that both radiatively
efficient and inefficient flows do occur in AGNs.
7
τ >>1
Figure 1.2: Cartoon of the standard accretion picture in AGNs. Shown are the molecular torus
(green) , the accretion disk (pink), the narrow line clouds (brown), the broad line clouds (yellow), a “corona” (red), the black hole, and an obscured and an un-obscured line of sight.
Observations also suggest that at larger distances, where the angular momentum of the
infalling matter is less severe of a constraint, matter can form a toroidal structure. At those
distances, the temperatures in the accreted matter are sufficiently low for most of the material
to be molecular and dusty. Figure 1.2 illustrates the general picture. Radio quiet AGNs can
be explained on the basis of this picture alone: Broad emission lines are produced by clouds
orbiting relatively close in (the broad line region), while narrow lines are produced farther out,
where densities are lower (thus we can see both forbidden as well as permitted lines). The torus
can obscure the central source (and the broad line region) and, at the same time, scatter some
of the light back into the observer’s line of sight. The orientation of the observer with respect
to the axis of the torus therefore determines what is seen by the observer (Antonucci and Miller
1985). In this picture, Seyfert 1s are seen more face on, while Seyfert 2s are seen edge on. The
luminosity of a typical AGN will roughly scale with the mass of the central black hole (equation
1.1). Together, orientation and luminosity can account for most of the variation seen in radio
quiet AGNs.
8
1.1.3
Radio Loud AGNs
About 10% of all AGNs show enhanced radio emission (see Fig. 1.1), at about a tenth
of a percent of the bolometric luminosity of the source. The radio emission generally exhibits
high linear polarization, indicative of its synchrotron origin. Furthermore, in many cases the
radio emission can be spatially resolved, unlike the rest of the AGN spectrum. Thus the radio
brightness must be due to a separate mechanism from the nuclear emission. In general, radio
loud AGNs are referred to as radio galaxies (radio loud quasars are often classified separately
as radio quasars).
We can divide the class of radio loud AGNs according to their brightness distribution:
sources that emit most of their radio power on larger scales (kpc, well resolved) are called lobedominated sources, and those that emit on small scales (pc scales, mostly unresolved with VLA)
are called compact or core-dominated objects.
In the lobe-dominated class, the morphology of the resolved structure shows three distinct components: On the largest scales we see diffuse emission from two oblong regions on
opposite sides of the nucleus, roughly in the shape of balloons. These structures are called the
radio lobes (thus the classification of the sources as lobe-dominated) and can be hundreds of
kiloparsec in size, larger even than their host galaxies. The nucleus itself can generally also be
seen. Finally, connecting nucleus and lobes are two narrow beams of radio emission, which,
though not visible in all cases, are generally believed to exist in all sources, as they are thought
to feed the lobes. These beams are called the radio jets and their properties shall occupy most
of this thesis. Figure 1.3 shows some examples of radio galaxies. Both jets and lobes show
significant amounts of linear polarization.
The morphology and radio power of radio galaxies are generally correlated. Low power
sources with radio power below Lrad
1042 ergs s
1 show rather curvy jets on both sides of
the nucleus. Their radio lobes tend to be brightest closest to the nucleus and fade towards their
edges (which are consequently not very well defined). Radio Galaxies falling into this category
9
Figure 1.3: VLA map of prototypical FR 2 radio galaxy Cyg A.
are classified as FR 1 sources (Fanaroff and Riley 1974). Members of the more powerful FR 2
class, with radio power in excess of Lrad
1042 ergs s
1 , tend to show a straight jet on one side
of the nucleus only, while the lobes are edge-brightened and well defined, showing very bright
“hot spots” at the end of the jet-lobe structure. The cores of both core- and lobe-dominated
sources show elongated structure when resolved with VLBI.
The radio spectra of these sources generally exhibit power-law behavior with fluxes obeying the relation F
/
. For the lobe-dominated sources, the power-law indices
fall into
the range from 0.5 to 1, while compact sources tend to be flatter, with 0. Together with the
polarization information, the spectra point towards synchrotron radiation as the mechanism for
the radio emission (see Appendix B for a brief review). The flatness of compact sources can be
understood as a superposition of self-absorbed spectra. In these sources, we can also see significant contribution of synchrotron radiation at IR and even optical/UV wavelengths, sometimes
overpowering the thermal emission and line emission from the “ordinary” AGN. These sources
also show polarization and strong variability in the optical band and are referred to as blazars.
In these sources, it is believed that relativistic effects enhance the jet emission so much that it
10
dominates the overall appearance of the source.
Among the compact sources, the Compact Steep Spectrum sources (CSS sources Fanti
et al. 1990) and Gigahertz Peaked Sources (GPS sources O’Dea et al. 1991) show distinctly nonflat spectra. Resolved images expose them to be intrinsically small radio galaxies. A related
class of sources are the compact symmetric objects (CSOs, smaller than 500 pc, Wilkinson et al.
1994) and medium symmetric objects (MSOs, 0.5 to 15 kpc, Fanti et al. 1995), which show a
symmetric but miniature jet-lobe structure.
1.1.4
Relativistic Jets in AGNs
Jets are at the heart of radio loudness in AGNs. They are very fast collimated outflows of
highly energetic plasma. The projected opening angle of AGN jets is often only a few degrees
and they can be very straight (see, for example, the image of Cyg A in Fig. 1.3). The jets are
collimated and collinear over several orders of magnitude in length and often show clear signs
of elongation on the smallest angular scales resolved (e.g., NGC 6251 Perley et al. 1984).
The surface brightness of radio jets tends to be concentrated into (often relatively evenlyspaced) smaller knots rather than having smooth surface brightness distribution. Far downstream the originally straight jets often become more curvy (possibly due to their decreasing
speed and interaction with ambient material - called the “dentist drill” effect, Scheuer 1982)
and terminate in the hot spots of the lobes. This suggests that the lobes are fed with relativistic
plasma by the jets. Both analytic models and numerical simulations show that the interaction
of the head of the jet with the ambient material results in a strong shock and a backflow of
material, which forms a cocoon-like structure around the jet, called the radio lobes. This picture, outlined in Fig.1.4, will be built on in x4 to develop a powerful method of analyzing X-ray
images of radio galaxies. Some jets are also visible at optical and UV frequencies, and several
jets have even been detected in soft X-ray, most notably M87 (Biretta et al. 1991), 3C273, and
most recently PKS 0637-752 (Chartas et al. 2000). As with the lobes, we know that the radio
emission from the jets is of synchrotron origin. The same is true for the rest of the spectrum
11
shell expansion
n
coconosion
expa
back
flo
w
AGN
"dentist drill" jet
hot spot
shocked ISM shell
cocoon (relativistic plasma)
contact discontinuity
Figure 1.4: Cartoon of the morphology of an intact radio galaxy. The radio plasma in the cocoon
is shown in blue, the IGM, which follows a King profile, is shown in pink, the shocked shell in
red, and the jet in green. The arrows indicate motion (not to scale). According to the “dentist
drill” effect, the jet is shown to wiggle, causing the effective area of the hot spot (the termination
shock of the jet) to be much larger than the area of the jet.
with the possible exception of the X-ray emission, which could also be due to inverse Compton
scattering.
As is true for radio lobes, radio jets often show a significant amount of polarization,
reaching levels of a few tens of percent (compared to the maximum of 70% for typical radio
spectral indices 0:5). While this indicates the presence of magnetic fields, it does not imply
that this field is well ordered over large scales (see x2). The field orientation can be parallel or
perpendicular to the direction of the jet and can change abruptly, as can be seen, for example,
in the knots of M87.
Proper motion measurements of the knots sometimes show apparent velocities in excess
12
of the speed of light. Superluminal motion is caused by the apparent retardation of light emitted
by the moving object; a radiation source moving at a small angle towards the observer at close
to the speed of light will show an apparent motion app
the sky (Rees 1966). The inferred actual
vapp =c faster than 1 on the plane of
of the emission region (which can either be a bulk
velocity of an emitting plasma blob or a pattern speed of a bright region in the flow, as would be
encountered in a shock) must be comparable or larger than app , while the viewing angle must
be smaller than, or comparable to 1=app .
A stronger argument for relativistic bulk motion in jets is the fact that we often only see
one jet where there are two radio lobes and the two associated hot-spots. Clearly, if the idea of
hot spots and lobes being created and fed by the jets is correct, a second jet should be present
yet hidden from our view. This is exactly what we would expect if the bulk speed of the jet
plasma were relativistic and if the angle between the approaching jet and the line of sight were
small. This effect is due to relativistic beaming and will be extensively referred to in x3.
A third argument for relativistic motion is the strong variability seen in blazars. As with
GRBs, the only reasonable explanation for the fast variability combined with such large fluxes
is relativistic bulk motion.
While the above arguments only supply approximate lower limits to , the most plausible
range of Lorentz factors compiled from all these arguments falls in the range
we adopt a fiducial value of
10.
3<
<
100;
For a jet to be visible at all at such large , it has to be
pointing roughly in our direction, which immediately introduces a strong angular dependence
of the observational signatures of the jet. Coupled with variations in the kinetic power and
emissivity in the jet, we can explain the differences in FR 1 and FR 2 as due to FR 2s having
consistently larger radio power, while blazars and BL Lacs are oriented so closely towards the
observer that relativistic beaming affects the whole spectrum, swamping out emission lines and
introducing variability and polarization even at optical frequencies.
While it is now widely accepted that jets move at relativistic bulk speeds, the physical
conditions inside the jet are far less well known. Estimates of magnetic field strength and
13
particle pressure depend on the assumption of equipartition between the two. While providing
us with an estimate that is probably accurate to within an order of magnitude, it does not resolve
the question which form of pressure dominates the internal energetics of the source. This very
important question in terms of jet dynamics and jet emission will be discussed in x2 and x3
respectively.
Furthermore, the composition of the jet plasma is unknown: it can be made up of either
pairs or of electrons and ions. The nature of the particles also has important consequences for
jet dynamics, since the baryon density in the jet will determine the terminal Lorentz factor that
can be achieved in the acceleration phase. Both of these issues can in principle be resolved with
high resolution broadband imaging.
Finally, the most pressing open issue is the question of jet production and dynamics.
A number of promising models have been proposed, which we will briefly mention here and
describe in more detail in x2. There are a number of channels other than radiation through
which the accretion energy can be released. Most notably, both Poynting flux and simple kinetic
energy flux (in the form of a wind) can remove energy from the flow. If the luminosity in these
channels is comparable with the
Ledd , we can expect to see a dramatic effect on the overall
structure of the flow. The amount of energy available could easily explain the energy released
in jets. The ubiquitous presence of accretion disks in astrophysics and the abundance of energy
released by them strongly points towards black-hole-accretion disk systems as the origins of
AGN jets, especially since we can see the signatures of both jets and disks in some cases. For
the rest of this thesis we shall assume that jets are produced in the inner regions of the accretion
disk/black hole system.
Any jet model must explain two things: the acceleration of the plasma to Lorentz factors
of order 10, and the collimation of that flow to narrow opening angles. Acceleration can be
provided by several means:
Internal energy of relativistic particles (BR74). In this case, the terminal Lorentz factor of the
14
flow is given by the average random particle Lorentz factor
at the base of the flow, modulo
losses of energy through other channels like radiation. Collimation must be provided by an
external agent, such as static pressure in the surrounding medium.
Inverse Compton enhanced radiation pressure (O’Dell 1981). This mechanism is based on
the fact that a region containing relativistic particles in the presence of a strongly anisotropic
radiation source will be accelerated away from that source, which also provides the collimation.
Poynting flux in large scale organized fields (Blandford and Payne 1982). This is the standard
model for jet production. Acceleration is provided by centrifugal forces along inclined field
lines, electric fields induced by rotation of the field in the rotating frame of the accretion disk
or a spinning Kerr black hole, and simple pressure gradients of particle and field pressure.
Collimation is provided by tension in the toroidal field and stresses communicated along the jet
from the inner disk.
All of these models have shortcomings and, while the latter model has been embraced by
most researchers in the field, it has not been proven conclusively. In x2 we will outline where
these models run into difficulty and we will offer a complementary approach to the organized
field model.
1.2
GRBs
While we do not have any direct observations of jets in Gamma Ray Bursts (GRBs), there
is now increasing evidence that they too exhibit collimated relativistic outflows (e.g. Stanek et al.
1999). The physics behind GRBs is even less clear than in the case of AGN jets, thus we are
limited to very crude models and collimation of the flow is mostly treated as an afterthought.
GRBs were discovered in the 1960s by a system of satellites monitoring circum-terrestrial
space for nuclear explosions, which had been banned by the “Outer Space Treaty”. It was soon
discovered that they were extra-terrestrial in origin, and rather mysterious in nature. Since
15
their discovery we have learned a good deal about their origin, yet a large part of their mystery
remains.
A GRB is a very bright flash in gamma rays that lasts anywhere from less than one second
to a few hundred seconds. BATSE, the most prolific GRB monitor, detected about one burst per
day, with high sensitivity and almost perfect sky coverage. This number has to be compared to
the estimated rate of supernovae in the entire universe of about
1s
1 (the reason will become
apparent when we discuss GRB progenitors). GRB light curves show variation on time scales
as short as 1 millisecond and for the time they emit, they can outshine the entire gamma ray sky.
Their positions are distributed isotropically, which implies that they must originate either in the
outer solar system, the halo of the Milky Way, or on cosmological scales. Until the discovery of
GRB afterglows in galaxies at cosmological redshifts, the distance scale of GRBs was an open
question, and for short duration GRBs, for which there have not been any afterglow detections,
it still is unknown.
1.2.1
Lightcurves and Timing
Figure 1.5 shows light curves from a small sample of GRBs. Above all, they show the
intrinsic diversity in GRBs: light curves can be smooth and single humped or very spiky, with
no apparent pattern in the peaks.
Upon closer investigation, the envelopes of many bursts do show a commonality: they
seem to follow a fast-rise exponential-decay (FRED) profile (seen, for example, in burst 971208
in Fig. 1.5). Zooming in on the individual spikes reveals that they also appear to follow a FRED
profile. FREDs are not uncommon and can be produced by a number of processes, which has
led to literally hundreds of proposed models for GRBs, most of which can be classified as rather
unorthodox (or even perverse, Rees, private communication).
GRBs are detected by all-sky gamma ray monitors that have built-in triggers. They
continuously monitor the gamma-ray sky and whenever a significant rise in flux over the background is observed, they send an envelope of data bracketing the trigger event to the ground.
16
The positional information can be used by other instruments to look for signals in other wavelengths from the GRB, or simply to improve the accuracy of the GRB position with follow-up
observations.
GRB durations are often measured in cumulative fluxes: t50 is the time between the
points where 25% and 75% of the total flux have been received, while t90 is the time between
the points when 5% and 95% of the flux have been received.
Grouping GRBs by their duration reveals a clearly bimodal distribution. GRBs fall into
two categories: short duration bursts, which last less than about a second, and long duration
bursts, which last longer than roughly 1 second. Due to their brief existence, it has been impossible to observe signals from short duration bursts at longer wavelengths. As a result, we have
very limited information about their nature. Most research has thus been concentrating on long
duration bursts, and for the rest of the thesis we too shall only be concerned with long duration bursts. That is not to say that the theory presented cannot be applicable to short duration
bursts as well, but the existing observational body is too small to support or rule out any such
statement.
The power spectrum of a large sample of long duration bursts has been studied by Beloborodov et al. (1998). It shows a rise at low frequencies corresponding to the distribution of
burst durations, a portion that follows a power law with slope
5=3,
and a steep drop of on
time scales shortward of 1 Hz. The power-law part of the PSD is indicative of the distribution
of individual spikes and should included in any GRB theory.
1.2.2
Spectra and Fluxes
The only spectral information we have about GRBs is from the gamma rays, with one
exception, GRB990123, which was simultaneously detected in the optical, but only in one filter.
Within the limits of the spectral window of BATSE (sensitive from about 20 keV to 600 keV)
and OSSE, which observed some bursts simultaneously with BATSE, GRBs show a broad, nonthermal spectrum that typically peaks at around 200 keV. While the flux drops off to both sides
17
of the peak,
F can be almost constant at the high energy end of the spectrum, while the one
optical point available lies far below the gamma rays, indicating that
F drop off quickly at
lower energies. Keeping in mind that the spectral shape above the peak is somewhat uncertain
and that a significant amount of energy might be hidden at unobservably high energies, the
fluence of the GRB can be estimated by F;peak .
If afterglow observations (see next paragraph) provide the distance to a given burst, the
isotropic energy it radiated can be calculated by multiplying the fluence with
4d2 , where d
contains the necessary cosmological corrections. The inherent assumption in calculating this
value is that the burst radiated isotropically, which, while convenient, might well be wrong.
For the
O(30) GRBs for which redshifts have been obtained,
the inferred isotropic energies
are staggering: within only a few tens of seconds GRBs can release anywhere from
few 1054 ergs s
1051
to
1 , more than a solar rest mass, in gamma rays. Coupled with their fast
variability, this places severe constraints on the physics of GRBs (see 1.2.4).
1.2.3
Afterglows
The launch of Beppo-SAX, with an onboard burst monitor and a high resolution X-ray
imager, led to the discovery of emission from GRBs at other wavelengths. This is accomplished
by imaging the location of the burst at high enough X-ray spatial resolution that the location
of the fading X-ray source that is sometimes associated with the GRB can be determined. The
X-ray emission is generally detected after the GRB is over and is qualitatively different from
the GRB in that it does not show rapid variability. Any emission that is discovered after the
GRB itself has ended is called afterglow.
With accurate positions, other instruments can look for emission from the GRB successor
and indeed, afterglows have been found at optical and radio wavelengths. Optical images reveal
that GRBs are generally associated with galaxies, and redshift information from these galaxies
shows them to lie at cosmological distances.
Afterglows generally exhibit powerlaw spectra that can be explained by synchrotron
18
emission that is self-absorbed at low energies and shows a cooling break that moves to lower
energies as the afterglow ages. In some bursts, the afterglow seems to show additional breaks
late into its evolution. As the peak of the emission moves to lower energies with increasing age
of the afterglow, the brightness slowly fades, also following a powerlaw behavior.
1.2.4
GRBs as Relativistic Flows
The fact that GRBs are so bright and at the same time release most of their energy in
gamma rays within a few tens of seconds puts severe constraints on their physical nature. If
GRBs were non-relativistic, the standard causality argument (used above as an argument in favor of black holes present in AGNs) would limit their typical size scales to only few 1011 cm.
The fast variability on time scales as short as a few milliseconds reduces these limits by another
few orders of magnitude. The presence of gamma rays implies that photon-photon interactions
will lead to copious pair production. The compactness parameter, which measures the importance of photon-photon interactions, greatly exceeds unity for a non-relativistic GRB: l 108 .
As a result, pair production would quickly raise the optical depth in GRBs to LTE values (many
orders of magnitude larger than unity), which would render the bursts completely opaque, making the escaping spectrum thermal (in contradiction to the observed non-thermal gamma ray
spectrum) and prolonging the burst duration to lengths inconsistent with the observations.
The solution to this dilemma is the same that has been found to work in blazars: relativistic bulk motion with
1 can explain the fast variability and the non-thermal GRB
spectrum without running into difficulty with the extreme luminosity of the burst. This is because the observed variability time and burst duration in a relativistic flow are shorter than the
corresponding comoving time scales by a factor of order
1=
2 . This is due to retardation of
the radiation emitted in the comoving frame, which shortens the observed emission time scale
obs by a factor of order 1=
3 , and time dilation, which stretches obs by a factor of . The
implied size of the emission region is correspondingly larger. In addition, a Doppler shift in the
observed frequency implies that the average photon energy in the comoving frame is lower by a
19
factor of order 1= , reducing the number of photons able to pair-produce by a factor / 1= 2 ,
where
need
is the spectral index defined above.
In order to avoid the compactness problem, we
>
100, larger than typical AGN jet Lorentz factors by an order of magnitude or more.
This makes GRBs the most extreme large scale test-beds for special relativity known to date.
The question then arises as to what causes the gamma-ray emission. If the energy radiated away in the GRB were simply carried along as internal energy, the ratio of kinetic to radiated energy would be large and the efficiency of the burst would in turn be very low. The energy
requirements for such a burst would be staggering. It is thus most commonly assumed that the
internal energy in the burst is recovered from the kinetic energy in the flow by shocks slowing
the material down to only “moderately” relativistic speeds (i.e.,
10, roughly corresponding
to braking from ludicrous to ridiculous speed, Brooks 1987). The cause of these shocks is the
defining feature for the two mainstream GRB models. In the external shock model, interaction
with the external ISM leads to a strong shock that produces the gamma rays, whereas in the
internal shock model the flow shocks with itself due to variation in the outflow speed of the
material. These two models will be outlined in more detail in x5.1.
20
250
GRB980923
14000
GRB980425
GRB990123
4000
200
12000
150
3000
8000
100
2000
6000
50
10000
1000
4000
2000
0
0
10
20
30
40
50
60
0
20
40
60
80
0
0
50
100
150
500
2000
GRB971209
GRB971214
GRB980124
3000
400
2500
1500
300
1000
200
500
100
2000
1500
1000
500
0
0
0
2
4
6
8
10
12
0
20
40
60
80
1500
1400
0
0
20
40
60
80
100
500
GRB910711
GRB970925
1200
GRB971208
400
1000
1000
300
800
600
200
500
400
100
200
0
0
0
0
2
4
6
8
10
12
0
20
40
60
80
0
200
400
600
800
Figure 1.5: Archival lightcurves for a sample of GRBs as seen by BATSE. Vertical axis:
counts/sec in all four channels; horizontal axis: time in seconds. Note that the change in the
variability/noise level in GRB971208 at 600 sec is due to a change in the binning factor. Remarkable bursts in this sample are GRB990123: the most energetic burst seen and the only burst
for which a contemporaneous optical counterpart has been detected; GRB910711: the the burst
with the shortest time scale variation ever observed ( 3ms); GRB971208: the longest and
smoothest burst ever observed.
Chapter 2
Jet Acceleration by Tangled Magnetic Fields
2.1
Tangled Fields as an Alternative
The key unsolved issue in jet physics is the question of jet creation. In other words: what
mechanism provides the large amounts of energy-momentum flux we see in relativistic jets and
at the same time restricts this flow to an opening angle of <
10Æ ? We know from independent
arguments that black holes can most likely be found at the centers of both AGNs and GRBs (all
serious GRB engine models ultimately form a black hole at their center). We also know that the
most efficient and readily available way of energy release for ordinary matter is conversion of
gravitational energy into other forms of energy when falling down the gravitational potential of
a compact object. This mechanism is well known to operate both in Galactic compact objects
(specifically, X-ray binaries) and AGNs. The introduction mentioned the three most probable
models for jet acceleration, which we will investigate now in a little more detail.
2.1.1
The Twin Exhaust Model
The first model that is a viable candidate for the large-scale dynamics of extragalactic
radio jets is the ‘twin exhaust’ model (Blandford and Rees 1974, BR74 throughout the rest of
this thesis). In this model, ordinary relativistic particle pressure provides the bulk acceleration
via conversion of internal to kinetic energy, i.e., simple adiabatic expansion. On a superficial
level, this model corresponds to the standard polytropic solar wind model.
The underlying assumption in this model is that the pressure of the external medium
22
surrounding the jet is stratified. This is not an unreasonable assumption, since the central black
hole as well as the host galaxy and (if the AGN is located in a cluster) the cluster provide a
strong gravitational potential in which any quasi-static medium would necessarily develop a
strong pressure gradient. An even steeper pressure gradient could be expected if the medium
were not stationary but in the form of a slow (i.e., non-relativistic) wind. In any case, we can
expect the pressure of the ISM/IGM to decrease with distance from the center.
Since the wall in a stationary jet does not expand sideways, the jet must be in pressure
equilibrium with its surroundings. Since the external pressure is stratified, the jet will also be
stratified, and if the jet is non-radiative, it will expand adiabatically to satisfy this requirement.
Since the symmetry of this problem is broken by the geometry of the accretion disk, we can
expect most of the energy released in the central region to escape preferentially along the symmetry axis. The equations of motion then dictate that the flow accelerate along that axis and
that its cross section contract for the sub-sonic portion of the flow and expand for the supersonic portion. In other words, the flow goes through a self-imposed De Laval Nozzle. In the
ultra-relativistic part of the flow (i.e., where
jet (r )
Rjet (r)
where
r
/ pext (r)
/
1) we simply have (see BR74):
1=4
jet ;
(2.1)
is the distance from the center, pext is the external pressure, and
the jet. The jet is collimated as long as d2 R=dr 2
Rjet
the radius of
< 0, i.e., as long as d ln pext =d ln r < 4, a
reasonable assumption. This does not guarantee that the flow will stay in causal contact with
itself and its environment, however. That question will be addressed in x2.4.1.
It has to be noted that the acceleration in this model is rather gradual, since the Lorentz
factor only varies as the
1=4 power of the external pressure.
Since the relativistic core of the
flow must be injected close to the horizon of the black hole (say at 10 gravitational radii), there
are many decades in radius to provide enough acceleration to get up to Lorentz factors of order
10 and higher. There are, however, more serious problems this model must face up to:
23
First, cooling processes for particles with highly relativistic random motions (necessary
to produce outflows with large bulk Lorentz factors ) are very efficient (both synchrotron and
inverse-Compton cooling go as 2 ). These loss processes compete with and possibly disable
bulk acceleration.
Second, the external pressure required to contain the flow must be equal to the internal
pressure of the jet. For standard ISM/IGM densities, this would imply higher pressures than
observed by X-ray observations of AGN environments. This problem could be circumvented
by a slow wind surrounding the relativistic core of the jet; Compton drag can also hamper the
efficiency of this process since a significant fraction of the particles are relativistic (see x2.4.4
below).
2.1.2
Compton Rockets
Since relativistic jets are known to contain ultra-relativistic particles (i.e., particles with
relativistic random motions), Compton scattering off of those particles must be very efficient.
Compton up-scattering is isotropic in the frame of the relativistic particle as long as the photon
energy in that frame is small compared to the particle rest mass. If the relativistic particle
encounters a photon, it will preferentially be scattered away from that photon, because the
energy and momentum transfer from electron to photon peaks for head-on collisions, and even
though the photon direction in the particle frame will be beamed towards the direction of motion
of the particle, statistically, the angle of incidence in the observer’s frame will lead to a stronger
momentum transfer for head-on scattering. Thus, a cloud of ultra-relativistic particles in the
presence of a highly non-isotropic radiation source (the best example here would, of course, be
a point source) will accelerate away from that source very quickly. The accelerative force on the
cloud will be proportional to h 2 i=h i, since the energy transfer by inverse-Compton scattering
for a particle goes as 2 while the inertia of the particle is proportional to . This process was
originally proposed as a mechanism for jet acceleration by O’Dell (1981) and has been dubbed
the “Compton Rocket”.
24
While this mechanism does not in itself provide a means of collimation, the fact that
matter will be propelled radially away from the radiation source can lead to jet formation if the
symmetry of the system is broken. In particular, a geometrically thick accretion disk can funnel
and shield the flow. In such a model, relativistic plasma is released in the evacuated funnel
along the rotation axis, with most of the radiation intercepted by the accelerated plasma coming
from the center of the funnel.
However, very soon after the Compton rocket was proposed, it was realized that the
same process that causes this strong acceleration also puts severe limits on the efficiency of the
Compton Rocket. Radiative losses in this scenario will limit the terminal bulk Lorentz factor of
the flow to
1 3 (Phinney 1982, which also provides a sound derivation of IC acceleration).
One could imagine a scenario where relativistic particles in the jet are constantly replenished
(an assumption also employed in other contexts, see x3.2, also x2.2.1.2). There is, however, an
even more severe problem: The radiation field encountered by the jet, while likely anisotropic,
is not going to correspond to a pure point source. As soon as the jet reaches a bulk Lorentz
factor large enough to make the radiation field in the rest frame of the plasma isotropic by
relativistic aberration, the acceleration will terminate and the jet will simply cool radiatively.
In other words, the Lorentz factor is limited by the opening angle O of the radiation source,
1=O. In fact, radiation drag can actually decelerate the jet if the external radiation field is
isotropic in the lab frame, which leads to both random energy losses and kinetic energy losses
by the same process.
Nonetheless, it is worth investigating the effects of radiation drag in jet physics and research on radiative acceleration is currently being pursued by several groups. Some recent publications in the field include Renaud and Henri (1998); Sikora et al. (1996); Luo and Protheroe
(1999) and references therein.
25
2.1.3
Magneto-Centrifugal Acceleration
The most widely accepted model for jet acceleration is magneto-centrifugal acceleration,
as proposed by Blandford and Payne (1982). Acceleration and collimation are provided by
organized magnetic fields and since the energy is initially stored in the field, radiative losses
are much less limiting in this model. Except for the Blandford-Znajek effect (which will be
discussed below) all of the flavors of this model derive the energy that drives the flow directly
from the orbital energy of the accretion disk.
The particle densities and ionization fraction in the disk will likely be high enough to
justify the magneto-hydrodynamic approximation and thus any magnetic field present in the
disk will be frozen into the plasma by virtue of the induction equation. If a large scale poloidal
field (any field perpendicular to the toroidal, i.e.
component of the field) is present, it will be
dragged along by the orbital motion of the plasma. This will wind up the field (i.e., introduce a
toroidal component) and it will also act as a slingshot: a blob of plasma sitting above the disk
(e.g., in a corona) will be accelerated along the field line it is sitting on by the centrifugal force
due to the rotation of that field line as long as the angle between the field line and the disk surface
is less than
60Æ and the field points away from the disk center.
Once the outflow is launched
by this slingshot mechanism, tension in the field will collimate the flow (two effects contribute
here: tension due to the toroidal component as well as tension in the poloidal component that is
communicated downstream from the disk, where the field lines are anchored and cannot move
outward). Gradients in the magnetic and particle pressure will also act to accelerate the flow.
Figure 2.1 demonstrates the different ingredients of this picture.
Depending on the uniformity of the field, the wound up field lines imply that a current
must flow along the jet due to Ampere’s law. In order to conserve charge, this current must
return to the disk/black hole. In order to close the current loop, the assumption of zero resistivity
must break down somewhere downstream. While this is a good assumption, the details of
this process are critical to determining the actual power of the jet. Since we have no good
26
understanding of this aspect of the flow, a critical piece of the puzzle is still missing even if the
jet is accelerated by these MHD effects.
Many different modifications of this model have been investigated in a large body of
research, a cross section of which can be found in Blandford and Payne (1982); Heyvaerts and
Norman (1989); Li et al. (1992); Camenzind (1986); Okamoto (1979) and references therein
(the latter three references deal with the relativistic complications encountered in real jets).
The magneto-centrifugal model has a number of advantages over the previous two approaches. As mentioned, radiative losses are not as important. Furthermore, we know from
observations that jets do contain magnetic fields. How much energy is present in those fields is
an open question (see x3), but even if the field is below equipartition (as defined in xB.1.3), it is
lower by only an order of magnitude or so. Finally, the model provides a simple explanation for
both acceleration and collimation by the same process, which means less stubble for Ockham’s
razor.
There are, however, a number of shortcomings. First, it is not at all clear if a large scale
ordered field can be maintained above a disk. Such a field would either have to be dragged in
with the accreted matter, or it would have to be generated in the disk via dynamo-processes.
If the accretion is based on angular momentum transport via magneto-turbulent viscosity, it is
unlikely that the disk can drag in sufficient poloidal flux to support a jet, as has been argued by
Lubow, Papaloizou, and Pringle (1994a). If instead angular momentum transport occurs via a
disk wind, it might be possible to support the proper field geometry simply by field dragging,
however, such disks seem to be unstable (Lubow, Papaloizou, and Pringle 1994b). This leaves
field generation by a disk dynamo, as has been investigated in a first attempt by Tout and Pringle
(1996). In this case, turbulence in the disk due to magneto-rotational instabilities (MRI Balbus
and Hawley 1991) produces a predominantly toroidal field, although it might be possible to
generate poloidal field along the way via a turbulent cascade. This question has yet to be
properly investigated, preferentially with the help of high-resolution numerical simulations.
Any force transfer through the magnetic field, which ultimately helps to collimate and
27
wou
nd-u
p fie
magnetic pressure
gradient
ld
magnetic tension
b
blo
cence
ac
plas
ma
tri
ler fuga
a ti l
on
Disk rotation
Figure 2.1: Cartoon of the general mechanism in magneto-centrifugal acceleration. Plasma is
flung out along field lines anchored in the accretion disk. For this to work the angle between
disk and the field line must be 60Æ . The flow is collimated by magnetic tension. Further
acceleration is provided by pressure gradients in the wound-up field and in particle pressure.
accelerate the jet, can only occur within the Alfven surface (the surface where the fluid becomes
super-Alfvenic). Beyond this surface, the fluid is causally disconnected from the disk and will
essentially act as a free fluid and it is therefore likely that another form of collimation would
have to come into play (i.e., external pressure confinement, Begelman 1995, B95 hereafter).
28
Furthermore, the rate of acceleration is rather slow, roughly logarithmic, which would make it
difficult to convert a large fraction of the Poynting flux into kinetic energy flux (Begelman and
Li 1994).
Finally, magnetically collimated flows are violently unstable to pinch and kink instabilities (Begelman 1998; Eichler 1993). Any such flow will be disrupted quickly and depart from
any self-similar structure it might have had, and thus from the analytic solutions found in the
above-mentioned publications. If collimation of jets by ordered fields is possible, it must look
somewhat different from any of these approaches.
The Blandford-Znajek Mechanism (Blandford and Znajek 1977) is, in a sense, a subcategory of magneto-centrifugal mechanisms. The energy flux in this picture is provided by
conversion of the spin-energy of a Kerr black hole into Poynting flux. For this mechanism to
work, a sufficient amount of magnetic field must thread the horizon of the black hole, which acts
as a fly wheel that drags the magnetic field along due to its finite resistivity. If this assumption
is satisfied, a very large energy reservoir can be tapped into to power the jet at rapid rates.
Collimation must be provided by the usual means (either magnetic self-collimation or external
pressure). Recently a vigorous debate has arisen regarding the viability of producing sufficient
field strength in the vicinity of the black hole (Livio et al. 1999; Krolik 1999).
2.1.4
Tangled Fields
Polarization measurements show that the magnetic field in jets is probably not well or-
ganized — the polarization is generally well below the maximal value of 70%; only in knots
does the polarization tend towards this value (note, however, that interpretations of polarization measurements are often ambiguous, since different field geometries can sometimes lead to
the same net polarization). This argues for the presence of largely unorganized, chaotic fields,
which could easily account for the high polarization measured in the knots if they are interpreted as shocks, compressing the field in the shock plane (Laing 1980, B95). This goes hand
in hand with the fact that the field produced in the disk by dynamo processes is expected to
29
be highly chaotic. Since the conditions in the jet will likely be controlled by disk physics, we
should expect the same statement to be true for the magnetic field in the jet at least close to the
disk. These arguments led us to investigate the dynamics of jets containing large amounts of
such disorganized magnetic fields.
The rate of acceleration in a jet propelled by internal (isotropic) particle pressure in an
external pressure gradient is limited to
to bulk Lorentz factors of
/ pext
1=4 (BR74), which means that the acceleration
10 100 would occur over length scales > (1000) rg for external
pressure gradients p / z 2 . One might think that an anisotropic pressure in the form of chaotic
magnetic fields could increase the rate at which the jet is accelerated, if the excess momentum
flux is oriented along the direction of the jet. However, we will show that under a given set of
simple assumptions the rate of acceleration is actually the same as in the classic case considered
by BR74, i.e.,
/ pext
1=4 .
It is unlikely that the magnetic field evolves without some form of dissipation, especially
if it is highly unorganized (reconnection is a diffusive process, so strong gradients in the field, as
are present if the field is highly tangled on small scales, will likely lead to increased dissipation).
These loss processes can compete with the efficiency of bulk acceleration by removing energy
from the flow reservoir. We will investigate the effects that such a tradeoff might have on the
dynamics and appearance of jets.
2.2
The Model
The model we are employing here is closely related to (and an extension of) the ‘twin-
exhaust’. We adopt a similar scenario under which the jet is launched into a stratified external
medium. In the case we are considering, turbulent and highly disordered magnetic fields dominate the internal energetics of the jet. See also Heinz and Begelman (2000).
We have illustrated the overall picture of the model in Fig. 2.2: Interstellar magnetic field
is advected inward by the accretion disk. Turbulent shear then amplifies the field and tangles it
up (dynamo action). This effect will grow stronger with decreasing distance to the black hole.
30
Eventually, regions of very high field strength will develop. Due to their buoyancy they will
accelerate away from the black hole, forming an initial outflow. This outflow is then collimated
by the pressure of the external medium. The jet channel is constrained by pressure balance, i.e.,
the jet will expand or contract in such a way that an equilibrium solution is set up for which
the flow is stationary. As the flow expands, we assume that micro instabilities and turbulence
constantly rearrange the field. As in the pure particle pressure case, the flow can go through
a critical point, where the radius
R has a minimum and beyond which the flow will become
self-similar (if the external pressure itself behaves self-similarly with distance to the black hole)
before the rest mass energy starts dominating the inertia, at which point the jet will reach a
terminal Lorentz factor
1 . Along the way, the field might dissipate energy via reconnection-
like processes, and radiation drag might alter the dynamics. We assume that no energy or
particles are exchanged between jet and environment, except for radiative losses. However, the
momentum discharge (the jet thrust
quantities like 0 , U 0 i
Q) need not be conserved along the jet.
By assuming that
hB 0i2i=8 (where a prime denotes that the quantity is measured in the
comoving frame) do not vary significantly across the jet we simplify the analysis to a quasi1D solution. We ignore effects of shear at the jet boundaries. We assume throughout most of
this chapter that the advected matter is cold (i.e., enthalpy density
h0
n0mparticlec2 ).
We
are looking for stationary flow along the jet (i.e., far from the terminal shock), enabling us to
drop time derivatives. Finally, to make this quasi 1D treatment possible, we will need to make
the assumption that the jet is narrow, which in our case implies that the opening angle is small
compared to the beaming angle, i.e.,
dR=dz
1=
. As we will later show, this also implies
that the jet is in causal contact with its environment (as required by the assumption that the jet
is in pressure equilibrium with the surrounding medium).
2.2.1
Treatment of Magnetic Field
We will use cylindrical coordinates
(r; ; z ) with the z -axis oriented along the jet axis.
The flow velocity is not aligned with the z -axis for r
> 0 (the jet expands). We assume that the
31
Figure 2.2: Cartoon of the general picture employed in this chapter. Tangled magnetic
field is generated in the disk, advected inward by the disk flow, and accelerated away
from the black hole to form an initial outflow. Under suitable conditions, this outflow
is then collimated and accelerates away from the core, keeping pressure balance with
the thermal pressure provided by the jet environment.
32
sideways velocity is small compared to vz but non-vanishing, i.e., the flow is well collimated.
e
e
The magnetic field is expressed in a different basis, since the standard basis vectors r and z
6= 0. One axis of this new basis is aligned with
the local velocity vector, ek . The second unit vector e is coincident with the -unit vector of
the lab coordinate system. The third unit vector e$ is obtained from the cross product of the
are not orthogonal in the comoving frame for r
other two. In the comoving frame we have
B0 = B 0$ e$ + B 0e + B 0k ek:
(2.2)
As mentioned before, the 1D approximation is only possible if the opening angle is small compared to the beaming angle. This is because the Lorentz factor will not be nearly uniform
across the jet otherwise. The assumption that dR=dz
1=
simplifies the equations of motion
significantly.
Following B95, who investigated similar jets in the non-relativistic limit, we assume that
the magnetic field is highly disorganized. In the comoving frame, averages over the individual
components and cross terms vanish while the energy density in the individual components is
not zero:
hB 0iB 0ji = 0; for i 6= j;
h B 0 i i = 0;
hB 0i2i 8Ui0 6= 0:
(2.3)
Lorentz transformation of the field to the lab frame (and to the cylindrical coordinate system
aligned with the jet axis) yields
B=
vz 0
v
v
B $ + r B 0k ^er + B 0^e + z B 0 k
v
v
v
vr 0 B $ ^ez :
v
(2.4)
Some of the components are now correlated. The electric field in the lab frame is
E = vz
2.2.1.1
B 0^er
p
vr 2 + vz 2 B 0 $^r vr B 0^ez :
(2.5)
Magnetic Equation of State
Without the presence of turbulent rearrangement of the field, flux freezing would govern
the behavior of the individual components. If we assume the presence of turbulent mixing
33
between the different field components, we might expect the field to follow a modified evolution
according to
dB 0 i =
X
j
X
@B 0 j @B 0 j d
(
v
)
+
dR;
ij
ij
@ v @R j
(2.6)
where the subscript ff denotes the value the derivative would take under flux freezing, and ij
and ij are arbitrary mixing coefficients. Note that we assume that large scale field produced
by shear (for example due to a boundary layer) will be tangled and cascaded to small scale
turbulent field. We also assume that the effects of turbulent kinetic energy are negligible (an
assumption that should be lifted in future, more realistic prescriptions of this model). Based on
this picture we therefore choose the following convenient ad-hoc parametrization of the field
evolution with Lorentz factor
B 0 $ 2 / B 0 2 / (v )
and jet radius R, including rearrangement:
2+1 R 2+2 ;
B 0 k 2 / (v )3 R
4+4 :
This is the magnetic equation of state we use. In the case of pure flux freezing, i
In the case of a completely isotropic field we have 1
(2.7)
= 0 for all i.
= 2=3, 2 = 2=3, 3 = 4=3, = 4=3.
Note that this prescription is still fully general [until we make some limiting assumptions about
the i (r; z )]. Since the re-arrangement process mixes the perpendicular and parallel components
of the field, we would expect that the field behavior is changed from flux freezing in such a way
that the coefficients i are bracketed by the values they take in the case of flux freezing.1 Since
the case of a purely isotropic field must be included in our analysis, it is clear that this condition
requires that 2
< 0 < 1 and 3 < 0 < 2 .
We define two quantities to characterize the anisotropy of the magnetic pressure:
U 0 U 0?
0k
;
U k + U 0?
1
U 0 U 0$
Æ 0
;
U + U 0$
(2.8)
Fully developed, compressible MHD turbulence is still an open and very difficult problem. Sridhar and Goldreich (1990); Goldreich and Sridhar (1995, 1997) have provided advanced treatment of this problem in the context of
anisotropic incompressible MHD turbulence. They do find that turbulence tends to be anisotropic in the sense that
the turbulent cascade might propagate a preferred field directionality to smaller scales. The situation considered in
these papers is different from the one considered here in that we assume that no ordered large scale field is present
and the fact that the plasma is internally relativistic and compressible. It would be an important step forward to
generalize the results by Goldreich and Sridhar to the parameter choices considered here.
34
where
U 0?
U 0$ + U 0.
purely parallel for toroidal for
Æ = 1.
Thus, the magnetic field is purely perpendicular for
= 1, and
= 1. The perpendicular component is purely radial for Æ = 1 and purely
The field is perfectly isotropic for
= 1=3 and Æ = 0.
It is obvious
from equation (2.7) that Æ is constant for any combination of parameters, since U 0 $
/ U 0 by
assumption.
While this parametrization alone is rather unrestrictive, we can limit it to a one parameter
family by assuming that the i are constants under any possible variation of
and R, and that
the rearrangement process does not change the total comoving energy density in the magnetic
field. (Otherwise the same process would have to act as an energy sink, since we assume that
the magnetic field is the dominant term in the internal energy budget. We would therefore be
dealing with a dissipative process, which we will address in x2.2.1.2.) We can solve for i in
terms of
by fixing either
or
R and demanding that the total energy density U 0
P U 0i
behave the same as it would following flux freezing:
dU 0 = U 0 ? [(1 2) d( v) + (2 2) dR] + U 0 k [3 d( v) + (4
=
for arbitrary d(
2U 0 ? (d( v) + dR) 4U 0 k dR
4) dR]
(2.9)
v) and dR. Constancy of any of the i then implies constancy of and substi-
tution of from equation (2.8) yields
Ui
/ ( v ) 1 R
3 1 = 1 + ; 2 = 1 ;
3 = 1; 4 = 1 (2.10)
which includes the isotropic case, where the magnetic field behaves like a relativistic gas, for
which = 1=3, Æ = 0.
It turns out that one can find special analytic solutions with constant i that satisfy equation (2.9) without the requirement that
be constant (see x 2.4).
For these solutions the rear-
rangement process conserves the comoving magnetic energy density only under the variations
35
in
and
R allowed by the Bernoulli equation [i.e., d( v) and dR in equation
(2.9) are not
arbitrary]. The only condition on the parameters i for such a solution is that 1 =2
= 3 =4 .
These solutions are limited to the self-similar range, where the jet is dominated by magnetic
pressure. Once they approach the terminal phase (i.e.,
0 >
U 0), the parameters i must vary
with z . For the rest of this chapter we will assume that equation (2.10) holds unless indicated
otherwise.
2.2.1.2
Dissipation of Magnetic Energy
It is unlikely that the tangled magnetic field evolves without any dissipation of its energy (e.g., via reconnection). We thus include a simple, ad hoc prescription of magnetic energy
losses. We base this parametrization on the idea that the magnetic field is always in a nearly
force-free equilibrium. However, it is impossible to maintain perfect force-free conditions everywhere and as the jet expands in either direction, the field responds by rearrangement between
the different components (eq. [2.10]) and by dissipation of some of its energy. We therefore assume that the dissipation rate is roughly proportional to the divergence of the velocity in the
comoving frame:
@U 0 i U 0 i r v0 @ diss
(2.11)
or, in the lab frame
@
@U 0 i U 0 i ln vR2 :
@z diss
@z
(2.12)
This ansatz can easily be generalized to different i for different field components (e.g., if the
Alfvén velocity factors into ). For a more realistic dissipation model see the Appendix.
We will assume that the dissipated energy goes into isotropic particle pressure, which is
then either (a) radiated away immediately as isotropic radiation in the comoving frame or (b)
accumulated until the jet reaches a state of equipartition between particle pressure and magnetic
field.
36
2.2.2
Equations of Motion
Let us write the relativistic continuity equation as
0 vz R2 = const:
(2.13)
The energy and momentum equations are given by
T ; = 0, (T
is the stress-energy
tensor, separable into a matter and an electromagnetic part). In the absence of gravity, this
T ; = 0, which will be sufficient for the analysis through most of this chapter
reduces to
since most of the acceleration will likely take place at distance z
rg GM=c2 , where M is
the mass of the central black hole. It turns out, however, that gravity is important in discussing
the critical points of the jet, in which case we will approximate the covariant derivative by a
Newtonian potential (
0
30
=
3
33
3
=
00
rg =z2 ). We will comment on the accuracy of
this approximation in x2.3.1.
Since there is no energy exchange between the jet and the environment, using the expression for the electromagnetic field measured in the lab frame from x2.2.1, we write the
energy equation as
T 0 ; = 0
(neglecting gravity). We then integrate the equation over a
cross-sectional volume of the jet and convert it to a surface integral using Gauss’s law. The
contribution from the sidewall is zero, giving
2
0 c2 + 4p0 vz R2 +
(where the
1
v
4 z
2
B 0 2 + B 0 $ 2 R2 L = const:
B 0 i 2 are now averaged quantities).
(2.14)
Dividing equation (2.13) into equation (2.14)
gives the relativistic Bernoulli equation. In the more general case including radiative losses and
gravity we have
!
B0 2
d 2 2 02
vz R c + 4p0 + 2 ?
dz
8
rg 2 2 0 2
+2 2 vz R c + 4p0 + 2U 0 ? + Srad = 0;
z
(2.15)
where Srad is the energy lost to radiation leaving the jet. We have to make some assumption
about the form of Srad , i.e., the amount of energy radiated away [cases (a) and (b) from x2.2.1.2].
37
The z -momentum flux Q can be calculated in much the same way (integrating T 33 across
a jet cross-section). Since the jet can exchange z -momentum with the environment, the momentum discharge need not be conserved, however. Dropping terms of order vr 2 , the integration
yields
Q
Z
A
2 v 2 R2 ( + 4p0 =c2 ) + R2 p0
dT 33 + R2
h
2
1 + v2 U 0 ?
i
U 0k :
(2.16)
The sideways momentum equation is given by T 1 ;
the solution be stationary (i.e.,
vr T 0 ; = 0. The condition that
@R(z )=@t = 0) gives the pressure balance condition between
the jet and its environment. We assume that the internal structure of the field adjusts to maintain
the given cross-section. Since we assume that vr
@
small, @r
vz , the internal variation will be sufficiently
vr @
vz @z , to justify the assumption of uniformity (note: this assumption is only
satisfied if the jet is in causal contact). We are thus only interested in the pressure balance
condition at the jet walls, r
= R, which gives
pext = U0 + Uk0 U$0 + p0 :
Note that
(2.17)
U$0 = 0 directly at the jet boundaries,
since the magnetic field is assumed not to
penetrate the contact discontinuity. However, since interior pressure balance demands that U0 +
Uk0
U$0 + p0 be constant, we can set U0
U$0 = const: and substitute it for U0 at r = R,
which gives equation (2.17).
2.3
Dynamical Solutions
Before we start analyzing the equations presented above, it is worth noting that in the
case of a cold (p0
= 0) jet and a magnetic field following pure flux freezing (1 = 2 = 0) the
only possible solution to equation (2.14) far away from the core (i.e., z
rg ) is (z) = const:,
i.e., the jet expands sideways to satisfy pressure balance, without accelerating. This is because
both the kinetic energy flux and the Poynting flux do not vary with R, while they do vary with
38
v, so that equation (2.14) becomes an equation of
only. Thus, fixing the total jet power
L
fixes . While a scenario like this might explain the coasting phase of the jet (where no more
acceleration occurs), it cannot account for the initial bulk acceleration we are looking for.
Note that this is different than the case of anisotropic, relativistic particle pressure in the
absence of isotropization (i.e., simply under adiabatic behavior of the individual components).
In that case, the components scale like pz
/ ( v) 2 R
2,
pr
/ p / ( v) 1 R
3 . We might
expect a behavior like this for a relativistic turbulent pressure term. The sideways pressure is
simply pr . At relativistic speeds, the solution approaches the one found by BR74,
p
/R/
1=4 . Thus, unlike in the magnetic case described in this chapter, it is generally possible to
accelerate a jet with anisotropic particle pressure without making any arbitrary assumptions
about the randomization process. This is simply because only the perpendicular component
of the field,
U 0 ?, contributes to the Poynting flux, while all components of the pressure enter
equation (2.14), which introduces a dependence on R, making a solution
6= const: possible.
For a magnetically dominated solution to exist, on the other hand, we need a field rearrangement
process at work, such as was described in x2.2.1.1. But even under such favorable conditions, a
proper, accelerating solution is not always guaranteed.
2.3.1
Critical Points
Since the jet will likely be injected with sub-relativistic speed, the question arises as to
where the jet crosses possible critical points and at what velocity. If the jet is injected at large
distances from the central black hole, we can neglect gravity; if it is injected close to the hole
we will have to include at least a phenomenological gravity term.
As a first step, we will set M
= 0 artificially (still assuming the presence of an external
pressure gradient) and neglect dissipation (Srad
= 0). Equation (2.15) gives
i dv
0 c2 + 4p0 + 2U 0 ? v2 + 4 (1 ad ) p0 + 2U 0 ?
vdz
dR
= 0;
+ 4 (2 ad ) p0 2 (1 + ) U 0 ?
Rdz
2
h
(2.18)
39
where ad is the adiabatic index of the particles, ad
d ln p0=d ln 0.
This equation has a
critical point when the expression in square brackets vanishes. At such a point, the jet crosssection must satisfy
dR=dz = 0, i.e., the jet must go through a nozzle, the position and cross
section of which are determined by the dynamics of the flow. Following the notation of BR74,
the velocity at which that happens is
s
c? =
4p0 ? (ad 1) 2U 0 ?? ;
0 ? c2 + 4p0 ? + 2U 0 ??
where the subscript
(2.19)
? indicates that the quantity is evaluated at the critical point z? .
Since for
a magnetically dominated jet
lim !0 c? = 0, the critical point exists only for < 0.
purely isotropic field, where = 1=3, c? reduces to the sound speed of a relativistic gas with
For a
p
ad = 4=3, c? = 1=3.
Locally we can always write pext
/z
, thus we define
d ln pext =d ln z:
(2.20)
We can then substitute the pressure balance condition (2.17) into equation (2.18) in the limit
0 c2 + 4p0 U 0 and eliminate R, which yields
2
h
i
(3 + ) v2 + (1 + 3 )
dv
= (1 + ) :
vdz
z
(2.21)
This equation also has a critical point with a critical speed of
cy s
1 + 3
:
3 (2.22)
Unlike equation (2.18), solutions cannot cross this critical point, since there dv=dz
! 1 (but
see x2.3.2).
We expect dpext =dz
Since
< 0, so solutions always accelerate (decelerate) for v > cy (v < cy ).
cy only exists for < 1=3, solutions with > 1=3 always accelerate.
equation (2.18) implies that for
v > c? (v < c? ) the
In that case
jet is expanding (contracting) in the
r-direction. Since c? only exists for < 0, solutions with > 0 always expand sideways.
If, on the other hand, injected with v
< 1=3, two branches of solutions exist: (a) solutions which are
> cy , which always accelerate and go through a nozzle at v = c? cy , and (b)
40
solutions which are injected with
v < cy , which always decelerate.
Thus, at sufficiently large
distances from the core for gravity to be negligible (see x2.3.2), highly anisotropic solutions
with
1 have to be injected at relativistic velocities to be accelerating, since cy ! 1 as
! 1. This corresponds to the right branch of the dashed solutions plotted in Fig. 2.3 (which
includes the effects of gravity, see x2.3.2).
It is instructive to look at the case of pure anisotropic relativistic particle pressure again.
(pk p?)=(p? + pk). If we fix p by some rearrangement process as we did for the magnetic field in x 2.2.1.1 (which might occur, for example, if
We define the pressure anisotropy as p
there is coupling between magnetic field and turbulent pressure as suggested by B95), the behavior is very similar in the sense that accelerating solutions for p
have to be injected at super-critical velocity v
If we simply let the components of
arrive at a different critical velocity,
cyp =
in this case), which exists only for p
q
(2 + 2p )=(3 + p).
solutions injected with
Since c?p
v > cyp
q
> cyp = (1 + 3p)=(5 p ).
p evolve adiabatically (without rearrangement), we
q
(5 + 7p )=(9 + 3p ) (note that is not constant
> 5=7.
> cyp
> 1=3 (i.e., for p? > 2pk )
for p
The solution once again has a nozzle at c?p
=
> 5=7, and since dR=dz < 0 for v < c?p ,
must accelerate to satisfy pressure balance, which means that
p increases with z , reducing cyp and thus making the flow more super-critical (i.e., once above
the critical point, the solution moves away from it).
2.3.2
The Effects of Gravity on the Sonic Transition
As seen in the previous section, a solution for > 1=3 that starts out with v < cy will
always decelerate in the absence of gravity. However, as is the case in the solar wind, gravity
can actually help a flow go through a critical point. We thus consider M
> 0 in this section. A
good overview of MHD effects in general relativity can be found in Phinney (1983).
We can go through the same arguments as in x2.3.1. The critical speeds are still given by
equations (2.19) and (2.22), but now the critical conditions are different. At c? , equation (2.15)
41
Γ†v† Γ*v*
Γv
10
1
10
z†
100
z/rg
1000
Γv
10
1
z†
10
100
z/rg
Figure 2.3: Inner solutions for = 2 in the vicinity of the critical point zy in the absence of
dissipation but including gravity for the two possible cases, < 1=3 and > 1=3. In the
first case (panel a), shown for = 0:9, the transonic solutions are thick black lines, suband super-critical solutions are thin black lines. Dashed lines show double-valued solutions (of
interest is only the upper right quadrant for z > zy , v > cy ). The critical values cy , zy are shown
as dash-dotted lines, c? is plotted as a grey dotted line. The condition dR=dz = 0 is shown
as a dash-triple-dotted line and regions where dR=dz < 0 are shown as hatched areas. In the
second case (panel b, for = 0) only super-critical solutions are possible. All solutions are
expanding sideways and all possible initial values lead to acceptable solutions. We have once
again indicated the location of zy .
42
gives
2r 2
dR
4p0 ? (2 2ad ) 2 (1 + ) U 0 ??
+ 0 ? c2 + 4p0 ? + 2U 0 ?? g = 0
Rdz
z?
instead of
dR=dz = 0.
Since
(2.23)
1 and ad > 1, we can infer that dR=dz > 0 at z? , i.e.,
there is no ‘geometric’ nozzle at z? anymore. The solution can always adjust dR=dz to satisfy
this condition, thus the first critical point c? becomes irrelevant.
Inclusion of the gravity term changes equation (2.21) to
2
h
i
(3 + ) v2 + (1 + 3 )
dv
= (1 + )
vdz
z
(3 + )
2rg
:
z2
(2.24)
Now solutions can cross the critical point cy , since the right hand side of equation (2.24)
vanishes at
zy 3 + 2rg
:
1+ v 6= cy
If
at
zy ,
(2.25)
dv=dz = 0
the solution must follow
at that point. This is true for
all . Since that is the only zero of equation (2.24), we can therefore conclude that solutions
accelerating at any z
> zy will accelerate for all z > zy .
Solving the equations for dR=dz instead gives
h
2 1
(3 + ) v2 + which also has a critical point at
i
cy .
dR
=
Rdz
For
v2 + v = cy ,
+ (
z
1)
2rg
;
z2
the right hand side of this equation only
vanishes at zy . In that case, dR=dz remains finite. For all other solutions (i.e., if z
v = cy ), we must have singularities
in both
dv=dz
(2.26)
and
dR=dz .
6= zy when
The singularity in
dv=dz
evident from Fig. 2.3 and from equation (2.24); pressure balance then requires that
must have a singularity of opposite sign, since
dpext =dz
is assumed to be finite, i.e.
is
dR=dz
pext
is
continuous.
We have numerically integrated equation (2.24) for two representative cases (
and
= 0, = 2 )
= 0:9
and plotted them in Fig. 2.3. Solutions are qualitatively different for
< 1=3 and > 1=3:
43
For < 1=3, there is one accelerating transonic solution, given by the condition in
equation (2.25), shown in the upper panel of Fig. 2.3 as a thick solid black curve. This
is also the only solution accelerating for all z . As in the case of a regular adiabatic
wind (Parker 1958), there also exists a decelerating transonic solution. Regions where
solutions contract in the r -direction (i.e., where
dR=dz < 0) are shown as hatched
areas. There are four more branches of solutions. Two branches are double-valued
(shown as dashed curves in Fig. 2.3). The left branch of those solutions can be rejected
since those solutions only exist for
z < zy.
branch to exist, it must be injected with
For an accelerating solution on the right
v > cy .
This corresponds to the solutions
discussed in x2.3.1 for which gravity can be neglected.
The remaining two branches are solutions that are always sub- or supercritical (plotted
as thin solid black curves in Fig. 2.3). The sub-critical solutions decelerate for large
z
and always stay sub-relativistic. They are uninteresting as possible candidates for
relativistic jets. The supersonic solutions decelerate for
z < zy
and accelerate for
z > zy. These solutions correspond to the super-critical solutions mentioned in x2.3.1.
! 1, zy ! 1. This is not
necessarily an indication that no solution is possible for 1, since for those cases
They always expand in the sideways direction. As we let the critical speed is very close to 1, thus the solution can attain large . Furthermore,
as we saw above, the solution is expanding even before it goes through zy . We can thus
have a regular (though sub-critical) accelerating jet even for For 1.
> 1=3, there is only one branch of solutions, all of which start out decelerating,
shown in the bottom panel of Fig. 2.3. As the solutions reach zy they begin to accelerate
and behave the same way as described in x2.3.1. Since zy moves inward for increasing
, this is no handicap.
For
> 1=3 we have zy < 8rg =
from equation (2.25),
which, for reasonable values of , is well in the regime where relativistic corrections
become important and inside the region where we expect the injection to occur. All of
44
these solutions have positive sideways expansion dR=dz
The transonic solution for > 0 for all z .
< 1=3 has some additionally nice features: Since we know
zy , we can relate the jet cross section to the total jet power Ly at zy . Assuming that the jet is still
magnetically dominated at zy , the kinetic luminosity of the jet is
Ly = Ry2 y2 vy2U 0 ?y :
(2.27)
The external pressure at zy is pext y and so
Ry =
v
u
u
u Ly Æ
t
+ 1+
1 (1 + )
p
:
pext y (2.28)
While we do not know pext y , for most parameter choices Ly is very nearly equal to L1 ,
which we have a reasonably good handle on from an observational point of view. Furthermore,
we can estimate the jet width at observable distances and scale the solution back to
gives us an estimate of
py and thus U 0 ?y .
This in turn will allow us to determine the original
matter loading of the jet from estimates of the terminal Lorentz factor
2.4
zy , which
1.
Solutions in the Self-Similar Regime and Asymptotic Solutions
For an already relativistic jet in the ‘self-similar’ range zy
is the location where 0 c2
(2.14) give
/R
z zinertia (where zinertia
= 2U 0 ?) a self-similar solution can be found.
2 =1 and with equation (2.10) we have
Equations (2.10) and
/ R.
Under the conditions of equation (2.10), the pressure balance condition gives
/ R / pext
1=4 ;
(2.29)
the same as in the case of isotropic particle pressure considered by BR74. For future reference
we define the acceleration efficiency
d ln
;
d ln pext
thus for this simple case (2.30)
= 1=4.
45
If we adopt the less limiting restriction 1 =2
= 3 =4 (see x2.2.1.1) instead of equation
(2.10), we can find powerlaw solutions in three limiting cases:
/ R 2 =1 / pext 2 =41 , which can be very
efficient for 1 2 . As pointed out in x2.2.1.1, one would generally expect that
(a) For
Æ = 0, the solution is given by
2 < 0 < 1 .
1, i.e., highly anisotropic, perpendicular fields, the solution is given by
/ R 2 =1 / pext 1=(21 =2 2) , which has a limiting efficiency of 1=2.
(b) For
(c) For If Æ
1 the solution is approximately the same as case (a).
> 0, the solution will in general approach solutions (b) or (c) (i.e., ! 1). For Æ < 0 it
is possible that the solution approaches a finite terminal Lorentz factor and zero opening angle
if radial tension cancels the pressure due to
Uk and U .
Note once again that these solutions exist only for
the coefficients
that 1
1 , 2 , 3 ,
and
4
Fig. 2.4 shows the different regimes.
1 and c2 U?. For all other cases
are not constant. Note that we would generally expect
2, since otherwise the re-arrangement process would be acting preferentially for
changes in geometry in one specific direction, which seems arbitrary. Thus, these results reduce
to the well known 1=4.
We can look for solutions in the presence of dissipation of magnetic energy. We now
have to consider equation (2.15). We assume that the energy goes completely into relativistic
particles, thus energy conservation implies
dp0 =
dz dissipation
1 dU 0 :
3 dz dissipation
(2.31)
The particle energy can subsequently be radiated away as isotropic radiation. As long as p0
U 0 ?, the radiative case is no different from the non-radiative case, since the adiabatic term in
the particle pressure does not contribute to the dynamics.
A powerlaw type solution is once again possible only if the enthalpy is negligible compared to the magnetic energy density (see x2.4.2). In that case the solution in the self-similar
46
1.0
ζ → (−1)
η → µ1/(2µ2 - 2µ1)
δ
0.5
ζ → 1
η → −µ2/4µ1
0.0
ζ→(1+δ)/(1-δ)
η → 0
-0.5
-1.0
0.1
1.0
-µ2/µ1
10.0
Figure 2.4: Regions in parameter space for the most general choice of parameters i possible
in the self-similar region. We have indicated the various limits of at sufficiently low external pressure. Along with 1 we have indicated the limiting efficiency achievable when has
reached is limiting value. For 2 =1 > 1, the efficiency can become very large. For Æ < 0,
it is possible that the jet stalls due to radial tension. Note also that for Æ < 0 there is a minimum
value for at the injection below which no solution is possible, given by min > (1+ Æ )=(Æ 1).
Even if 0 > min , the solution will contract and decelerate if Æ is below a certain value. We
have plotted this parameter range for a jet that starts out isotropically (i.e., with = 1=3) as a
grey area. For smaller , this area will become larger.
range is given by
/ pext
= 1 at = 1.
We have plotted
with = 1=[4 + 6(3 + 5 )=(3 3 2
as a function of and
2 6 )] 1, with
in Fig.
the efficiency is larger than in the case without dissipation, for
2.5. For
> 3=5 it is smaller.
limiting efficiency that can be achieved in such a jet under the condition that
given by < 3=5
The
dR=dz > 0 is
1=2, as ! 0. This happens because the dissipative process can convert energy
in the parallel field component Uk (which does not enter eq. [2.14]) into particle pressure, which
must be taken into account in the energy balance. Also shown are areas in --space where the
1
2
5
-4
.3
0
47
0.25
0.40 0.50
1.00
0.1
0
0.2
0
0.30
Λ
0.10
dΓ/dz < 0
dR/dz < 0
d(αo/αb)/dz > 1 for:
3<ξ<4
2<ξ<3
ξ<2
0.01
-0.5
0.0
ζ
0.5
Figure 2.5: Contour plot of acceleration efficiency d ln =d ln pext for radiative dissipation and different values of and . The grey areas indicate parameter values for which the jet
is either not collimated (light grey with dark grey border) or decelerates (dark grey with light
grey border). The hatched areas indicate regions where the ratio o =b of opening angle to
beaming angle grows with z for given values of 0 < < 4.
conditions d
=dz > 0, dR=dz > 0 are not satisfied.
Note that the actual acceleration is not limited by the dissipative process: highly anisotropic
fields with strong dissipation can produce faster acceleration, however, such cases are actually
contracting in
R, thus we do not consider them here.
While dissipation can produce strong
magnetic pressure gradients, the inherent assumption of a stationary jet, which leads to the condition of sideways pressure balance, restricts the run of pressure in the jet. If the energy is
dissipated too fast, the jet will fall out of causal contact with itself and its environment and time
independence would no longer be a good assumption.
48
2.4.1
Opening Angles and Causal Contact
Since a jet is generally defined as a collimated outflow, we can ask under which condi-
tions the solutions from above are actually collimated. The collimation condition
dR=dz < 1
d ln pext =d ln z = < 4 as long as the jet is magnetically
translates to a pressure gradient
dominated, the same as in the particle pressure dominated case. The presence of dissipation
can alter this value. Generally, the collimation is increased by dissipation, since the sideways
pressure is reduced, thus the jet does not need to expand as much. In the coasting phase, where
the jet is no longer accelerating, this condition changes to < 3 + < 4.
Given the solutions from above, we can investigate the ratio of the opening angle o to
the beaming angle b
o =b =
dR
dz
1 . In the absence of dissipation we can write
/ z=2 1 ;
(2.32)
independent of . Thus, for steep pressure gradients
> 2, the opening angle will grow faster
than the beaming angle and will thus always become larger even if it starts out being smaller.
For shallow pressure gradients
< 2,
the situation is reversed, i.e., the beaming angle will
eventually become larger than the opening angle. The presence of dissipation changes this
behavior qualitatively: the ratio o =b now depends on both and , as illustrated in Fig. 2.5.
This has consequences for the appearance of the jet, since the effective beaming angle is
given by the larger of the two angles. Under the assumption that the jet is always collimated,
the opening angle in the coasting phase will always become smaller than the beaming angle,
since the jet does not accelerate anymore. This could have important consequences for the
morphology of superluminal sources: if the beaming angle were smaller than the opening angle,
one might expect to see larger jet misalignments, or lose the jet morphology altogether. The
appearance would become sensitive to the emissivity and local Lorentz factor as a function of
position across the jet cross section.
Jets that expand too fast will eventually lose causal contact with their environment. This
happens when the Alfven crossing time of the jet becomes larger than the expansion time (in
49
the comoving frame), i.e., when
R
p0
z
> 0 exp =
0
vAlfven c
vdp =dz
c
(where we approximated vAlfven c) or
R
0A =
R>
(2.33)
z
:
(2.34)
This corresponds (up to the factor ) to the criterion when o
> b .
Thus, for
> 2 (in the
absence of dissipation) the jet will eventually lose causal contact with its surroundings (see Fig.
2.5 for values of
and
where this is the case).
As mentioned in x2.2, a quasi 1D treatment
is no longer possible, since the internal pressure balance is now regulated by shocks traveling
inward from the jet walls. After the jet reaches the terminal phase, it will re-gain causal contact,
since the opening angle will continually decrease (assuming the jet is still collimated).
2.4.2
Equipartition
Constant pumping of magnetic energy into particle pressure can lead to equipartition
between U 0 and p. We can use the self-similar solution to estimate eq , where the accumulated
particle pressure surpasses the magnetic energy density (including effects of adiabatic cooling
on the accumulated particle pressure, where we assume that it behaves as a relativistic gas,
1>
eq the solution might
be altered. For some parameter values p never reaches the level of U 0 ? . In that case we estimate
which gives an upper limit on the resulting pressure). Thus, for
the asymptotic ratio
(p0 =U 0 ?)1.
Figure 2.6 shows the results of those estimates. For large ,
equipartition can be reached quickly, thus, unless the energy going into particles is subsequently
radiated away, the assumption that the particle pressure be negligible compared to the magnetic
field energy density will be violated beyond eq .
We define the energy distribution function as
f ( ) F0 s;
Z 2
1
f ( )d = n0 ;
(2.35)
4
e6
5
3
-2
.1
0
50
1.00
0.5
0.3
Λ
0.2
0.10
2
4
10
1.0
0.1
100
1e
6
1e4
0.01
-0.5
0.0
ζ
0.5
Figure 2.6: Value of for which the pressure accumulated by non-radiative dissipation reaches
equipartition with U 0 ? . Neglecting p is no longer justified beyond that . For <
0:6 the
pressure never catches up with U 0 ? , in that case we plotted contours of the limiting ratio p0 =U 0 ?
(dashed lines). The hatched and grey regions correspond to the regions in Fig. 2.5.
where is the Lorentz factor according to a particle’s random motion, measured in the comoving frame, and 1
2 are the lower and upper spectral cutoffs.
Since we assume that the
magnetic field is dominating the internal energy budget, synchrotron losses can be very strong,
provided the particle energy spectrum is flat enough so that most of the energy is at high particle energies (i.e.,
s < 2).
In that case we can expect most of the energy to be radiated away
immediately and the corresponding electrons will lose most of the inertia, thus the dissipated
energy will not lead to a build-up of particle pressure. If, however, synchrotron losses are weak
compared to dissipation (e.g., if the spectrum is too steep, or if synchrotron self-absorption
traps most of the radiation to inhibit cooling), the effects of particle pressure can become important, as demonstrated in Fig. 2.6. For a discussion of the observational effects of the different
51
radiative scenarios see x2.5.1
2.4.3
Full Solutions
We can solve the full equation (2.15) in the regime
1, i.e., for relativistic jets.
As mentioned before, the pressure balance condition leads to a simple algebraic equation in
and
R.
In the absence of dissipation and gravity, equation (2.14) is in fact another algebraic
equation relating
and R, thus, the two equations can be solved for
(pext ) using a numerical
root finder. Apart from reproducing the scaling behaviors established in x2.3, this will enable
us to determine the terminal Lorentz factors and the length scales over which the transitions
between different dynamical phases occur. Furthermore, we can use the full dynamical model
to investigate the evolution of such observational quantities as polarization and synchrotron
brightness.
In the absence of dissipation, the terminal Lorentz factor that can be reached with such a
jet is simply
lim!0 =
1 pext
!
0
0 0 c2 + 2U?0 0 + 4p0 0
;
0 0 c2
(2.36)
where subscripts 0 denote quantities evaluated at some arbitrary upstream point.
This simple solution is no longer possible in the presence of dissipation, which introduces a sink term into equation (2.14). As a result, we have to use equation (2.15) instead.
Once the energy has been converted into particle pressure, it can be radiated away as isotropic
radiation, which will not affect the dynamics of the jet any further (assuming that p is dynamically unimportant). If the energy is stored as particle pressure, the pressure could eventually
become dynamically important (see x2.4.2). Until that happens, though, the two solutions are
identical. The terminal Lorentz factor is always reduced (see x2.5.1), but the acceleration efficiency can be increased for < 3=5 (see x2.4). We have plotted the solution for the radiative
case (the one case solvable analytically) in Fig. 2.7.
52
Γ(z)/Γ0
Terminal Γ for Λ=0
10
pure magnetic jet (self similar case)
cold matter (2Uperp=40ρ0c2)
cold matter, dissipation for Λ=0.05
cold matter, dissipation, radiation drag
1
1
10
100
103
(z/z0)(ξ/2)
104
105
106
Figure 2.7: Analytic solutions in the limit v 1 for = 0: a) self similar limit, valid for
very large U?0 0 c2 ; b) solution with cold matter, U?0 =0 c2 = 20, i.e., 1 = 41; c)
radiative dissipation with = 0:05, otherwise same parameters as b); d) same parameters as
c) but including radiation drag. Note that the dissipation of energy alters both the acceleration
efficiency and the terminal Lorentz factor 1 .
2.4.4
Radiation Drag
The presence of ultra-high-energy particles in AGN jets suggests that inverse Compton
(IC) enhanced radiation drag might be dynamically important. While O’Dell (1981) initially
suggested that pair jets might be accelerated by the Compton rocket effect, Phinney (1982)
showed that it is hard to accelerate a plasma beyond fairly modest Lorentz factors by radiation
pressure without a continuous source of particle acceleration to offset the strong IC cooling of
the plasma. Furthermore, if the radiation source is not point-like, the terminal Lorentz factor is
limited by the solid angle
the radiation source subtends.
On the other hand, radiation drag
can hamper the bulk acceleration of plasmas containing relativistic particles in the presence of
53
a radiation field, if those particles are continuously reheated to overcome the IC losses. The
dissipation mechanism discussed above could provide such reheating. Here we will consider
the effect of radiation drag in the simplest possible prescription.
We assume that the IC cooling and the dissipational heating time scales are short compared to the adiabatic timescale. If this is not satisfied, the influence of radiation drag will be
reduced. We can then expect dissipational heating to nearly balance IC losses in a near equilibrium situation. Thus the amount of IC drag is controlled by how much dissipation there is.
For this approximation to be valid, IC losses must dominate the loss processes of the particles,
i.e., the radiation energy density
magnetic field energy density
U 0 rad in the comoving frame must be large compared to the
U 0 (for large enough
this is always going to be the case, since
the external field will be Doppler boosted). Finally, we assume that h 2 2 i 1, where is the
particle Lorentz factor in the comoving frame. This sets an upper limit of
U 0 rad 6 103 ergs cm
3
"
U 0 d ln R2 1014 cm
02
c d ln z
z
#
(2.37)
on the comoving radiation energy density (otherwise IC cooling would have lowered the upper
spectral cutoff to 2
1). These assumptions allow us to eliminate U 0rad from the equations,
since the drag term and the cooling term are both proportional to U 0 rad . In a sense, then, we are
presenting an upper limit on the importance of IC radiation drag over large length scales. It has
to be kept in mind, though, that drag can be much more important in non-stationary situations
(like, for example, in shocks), which are beyond the scope of this thesis.
We assume the jet is moving through a radiation field that is locally isotropic in the
lab frame. Following Phinney’s treatment (1982), we can calculate the loss rate and the force
density due to IC scattering in the comoving frame and then transform back to lab frame to find
the additional term for equation (2.14). We find that radiation drag always decreases both the
acceleration efficiency and the terminal Lorentz factor
1 by moderate amounts. It does not,
however, introduce qualitatively new features. To demonstrate this, we have plotted a solution
including radiation drag for otherwise identical parameters in Fig. 2.7.
54
2.5
Discussion
2.5.1
Tradeoff Between Dissipation and Acceleration and Synchrotron Brightness
In the following section we will investigate the observational effects of the jets we have
introduced in this chapter. A highly dissipative jet will radiate away a large fraction of its
internal energy along the way before reaching the terminal Lorentz factor
1 , while a non-
dissipative jet will convert all its internal energy into kinetic energy flux. Since the jet will
ultimately terminate and reconvert its kinetic energy flux into random particle energy when it
slams into the surrounding medium, the ratio of kinetic luminosity (which could be estimated
based on the energy input into the lobes, based on the source size and its age) to the radiative
luminosity Ldiss (z ) (i.e., the integrated luminosity of the jet before reaching the terminal shock)
should give us some indication of the importance of dissipation.
We have already seen in
x 2.4.3 that the presence of dissipation can lower
lowering the kinetic energy flux at the terminal shock,
1, thus
L1 (dominated by cold kinetic energy
flux), and the produced hot-spot luminosity. A given fraction
b
of the terminal luminosity
bL1 will be radiated away, which can be estimated from the hot-spot and cocoon luminosity
(calculating
b is,
of course, a highly non-trivial matter), giving us a handle on
.
We have
plotted the ratio
Ldiss
L1
(2.38)
in Fig. 2.8. To make that plot, we chose parameters such that in the absence of dissipation the
jet would reach
1( = 0) = 2U 0 ?0 =(0 0 c2 ) = 100. If we had chosen a larger (smaller) value
of this parameter, the lines in the plot would move down (up), since
1 depends non-linearly
on both and 2U 0 0 =(0 0 c2 ), so this plot is only a representative one of a family of similar plots
for different
1 ( = 0).
Another question is what the spatial distribution of the jet emission is, since there are several competing effects: the optical depth to self absorption (and inverse Compton up-scattering),
.4
0
-1
5
2
6
55
100
25
5
625
1
25
10-1
Λ
1
0.2
10-2
0.04
10-3
-1.0
-0.5
0.0
ζ
0.5
1.0
Figure 2.8: Plot of the ratio Ldiss =L1 , indicating how much of the energy carried in the jet is
radiated away and how much will reach the terminal shock (this energy can be used to heat the
hot spot and to inflate the radio lobes). This plot was constructed for 10 1 ( = 0) = 100.
Lower values of 10 will shift the lines in this plot upwards. Note that this plot was constructed
using the assumption that all the dissipated energy is radiated away on the spot.
Doppler boosting and opening angle (the larger of which will determine the opening angle of the
cone into which most of the radiation goes), and of course the dissipative power of the jet itself
[depending on and
(z ), R(z )]. This question can be asked with respect to the frequency in-
tegrated brightness I or the spectral brightness I . If we assume that all the dissipated energy is
radiated away on the spot, we can calculate the local dissipation rate, which must then be equal
to the local frequency integrated jet emissivity
j 0 in the comoving frame.
This assumption de-
pends on the injected particle energy spectrum. If the spectrum is flatter than s = 2, most of the
energy is in the high energy particles. In that case, synchrotron cooling can be efficient enough
for our assumption of on the spot radiation to be effective. If, on the other hand, s > 2, most of
the energy is in the low energy particles, synchrotron radiation will not be efficient (unless 1
56
is very high, in which case the injected spectrum would rapidly cool to a quasi mono-energetic
distribution), and the jet will accumulate particle pressure or radiate by other means (note that
Compton cooling would be equally insufficient to balance heating in this case).
To investigate the first case we will set
s < 2.
Given a viewing angle we can then
determine the observed total intensity I , given by
I/
where
dU 0
dz
diss
D2
R
;
sin (2.39)
is the angle between line of sight and jet axis and
[ (1 v cos )]
D
1 is the
Doppler factor. This expression takes relativistic beaming and the relativistic corrections to
foreshortening into account. One might expect that the integrated brightness peaks at a certain
distance from the core, since D is strongly peaked at
1=. However, in our prescription the
dissipation drops off too fast for this effect to be important. The main difference in the brightness evolution is that a dissipative jet has a different efficiency
and generally expands less
rapidly in the sideways direction (and will reach a smaller terminal Lorentz factor
effect will become important once the jet has reached
1 ). This
1 and only for large enough to signif-
icantly alter the dynamics ( >
0:1). We have plotted I as a function of z for different values of
, = 0, U 0 0 = 20 0 0 c2 , and a viewing angle of = 10Æ in Fig. 2.9, arbitrarily normalized to
I ( = 0:01) to increase dynamic range (the brightness decreases by many orders of magnitude
along the jet). As is obvious from the plot, for small
, only the overall normalization
of
I
varies with , whereas for large enough , the brightness distribution itself changes shape due
to the altered dynamics. Also shown are the Doppler factors
D for the different parameters,
which are primarily responsible for the different shapes.
The situation can become more difficult in the opposite case, i.e., if most of the radiation
is trapped (e.g., by synchrotron self-absorption, which implies a steep spectrum, s > 3) or if the
deposited energy is simply not efficiently radiated (e.g., if 2 < s < 3). In that case the emission
of dissipated energy could be delayed, leading to a relative brightness peak downstream. To
briefly investigate this possibility, we assume the latter case, i.e.,
2 < s < 3, which is not an
57
10
D
I(Λ)/I(Λ=0.01)
100
1
Λ=.004
Λ=.010
Λ=.027
0.1
1
10
100
Λ=.074
Λ=.200
Λ=.541
103
z/z0
104
105
106
Figure 2.9: Plot of the frequency integrated brightness for a radiative jet (i.e., all the dissipated
energy is radiated away on the spot) as a function of z for different values of , = 0, = 2,
U 0 ?0 =00 = 20, and = 10Æ , arbitrarily normalized to the intensity curve for = 0:01 to
increase the contrast (thick black curves). Also shown are the corresponding Doppler factors D
as thin grey curves. The shapes of the individual intensity curves are mainly determined by the
variation of D . Note that a significant observable effect is achievable only for >
0:1.
unreasonable choice for AGNs (see x2.5.3, for example). We assume that the energy flux F
peaks at high energies p : either at the spectral break of 1=2 expected in a scenario in
which high energy particles are constantly re-injected, where the spectral index is given by
d ln I
;
d ln (2.40)
or at the spectral cutoff produced by synchrotron and IC cooling (in the absence of a strong
break). The position of p depends on adiabatic effects, radiative cooling, and heating due to
dissipation. The spectrum will be self-absorbed at low frequencies, which generally leads to
an observed spectral index of
5=2 (for an exact treatment of the spectral shape at the
self absorption turnover see De Kool, Begelman, and Sikora 1989). If we take
2 < s < 3 or
58
1=2 < < 1, the self absorbed part contributes a negligible fraction to the total brightness and
most of the energy is emitted at the high end of the spectrum.
In this case it is impossible to calculate the brightness analytically as a function only of
, R, and z . Rather, one can numerically integrate the evolution equation of the peak frequency
under adiabatic cooling, dissipative heating [we assume a self-similar transfer of energy from
magnetic field to the particles such that
ing on the basis of the solution
(d=dz )jdiss / (dU 0 =dz )jdiss], and synchrotron cool-
(z ) given above.
We can estimate the run of
I
by scaling it
with the brightness at p , taking account of relativistic beaming and aberration. If we choose
to normalize the intensity curves as we did in the previous case, we can get around fixing the
absolute normalization of U 0 , since it only enters linearly into the intensity and will thus cancel
out upon normalization. We used
U 0 ?0 = 20 0 0 c2 , along with s = 5=2 and plotted the fre-
quency integrated intensity in Fig. 2.10 with otherwise the same parameter values as in Fig. 2.9,
once again normalized to I for = 0:01.
Note that for large values of our assumption that
the jet be magnetically dominated and that particle pressure be negligible can break down (see
Fig. 2.6), so curves with high
are to be taken with a grain of salt.
Nevertheless, it is inter-
esting to note that the intensity drops less rapidly with z for larger values of , corresponding
to the delayed emission mentioned above. The shapes of these curves depend only weakly on
s.
The different bends in the curves stem from the evolution of the Doppler factor (shown in
Fig. 2.9), and the evolution of the peak frequency. The slopes of the curves are produced mostly
by the evolution of the lower cutoff frequency 1 (see eq. [2.35]), which enters the expression
for the synchrotron brightness through the powerlaw normalization of the particle distribution,
and by the evolution of Lorentz factor and jet radius (entering through the particle density and
the integration of the emissivity across the jet).
2.5.2
Polarization
While the degree of polarization is highest for homogeneous magnetic fields, jets with
tangled or disorganized field can exhibit a net polarization if there is a net anisotropy in the field
59
104
Λ=.004
Λ=.010
Λ=.027
Λ=.074
Λ=.200
Λ=.541
I(Λ)/I(Λ=0.01)
103
100
10
1
1
10
103
100
104
105
z/z0
Figure 2.10: Plot of the frequency integrated brightness for a marginally non-radiative jet (only
a dynamically small amount of the dissipated energy is radiated away on the spot) as a function
of z for different values of , = 0, U 0 0 =00 c2 = 20, = 2, and = 10Æ , arbitrarily normalized
to the intensity curve for = 0:01 to increase the contrast (the absolute intensity drops by
many orders of magnitude). Note that these curves do not depend on the absolute value of the
magnetic field, which cancels out due to the normalization.
(Laing 1980). The measured polarization will depend not only on , but also on the viewing
angle and the bulk Lorentz factor . The polarization of radiation from a powerlaw distribution
of electrons with index s in a region of homogeneous field is given by
I I
s+1
? k =
I? + Ik s + 7=3
(2.41)
where I is the intensity at a given optically thin wavelength. To calculate the integrated polarization, we average across the jet. To do this we decompose the radiation into polarization along
the jet axis and perpendicular to it. Furthermore, we assume that field directions are distributed
among all solid angles and introduce a weighting function that distributes the field orientations
60
to the required anisotropy,
w(#) = sin #
(2.42)
where # is the angle between the field and the z -axis and is determined by the anisotropy. We
can solve for under the condition that hBk2 i = (1 + )=(1
=
)hB?2 i:
1 + 3
:
1+
(2.43)
We correct the viewing angle for relativistic aberration, which has a significant impact on the
observed polarization, since the average polarization will go to zero for a jet seen head on.
Furthermore, the angle between line of sight and magnetic field is important in determining the
relative brightness of a region. Figure 2.11 shows the predicted polarization for the cases shown
in Fig. 2.7, a spectral index of = 0:5, and a viewing angle of = 10Æ . Since the anisotropy of
the jet is fixed, the variation in the polarization ( ) is solely caused by changes in the viewing
angle due to relativistic aberration. Generally, the polarization will be perpendicular to the jet
axis if
< 1=3 and parallel if > 1=3.
As long as equation (2.10) holds, an extremum
() will be present and it should indicate the position where = 1= sin , i.e., where the
viewing angle corrected for aberration is 0 = 90Æ . Note that this polarization is averaged
in
across the jet. In order to compare these predictions to actual measurements a relatively small
correction for the emission weighted averaging across the jet at different angles must be made.
The qualitative predictions of this section should be unaffected by that caveat.
2.5.3
Applications
Finally, we sketch out some applications for this model. The best studied AGN jet is that
in M87, since it is the closest unobscured source (though by no means a particularly powerful
one). The properties of the M87 jet are discussed at length in x3, here we will only list the
properties important for this analysis. The central black hole in M87 has a relatively well
2:4 109 M or rg 3:5 1014 cm. The kinetic power of M87
is estimated to be of order Lkin 1043 44 ergs sec 1 (Reynolds et al. 1996b, BB). Taking the
determined mass of
M
61
pure magnetic jet (self similar case)
cold matter (2Uperp=40ρ0c2)
cold matter, dissipation for Λ=0.05
cold matter, dissipation, radiation drag
Π[%]
30
20
10
0
1
10
100
103
z/z0
104
105
106
Figure 2.11: Optically thin polarization as a function of z for the same parameter values as in
Fig. 2.7 and a viewing angle of = 10Æ . We chose a spectral index of = 0:5. The variation
in is solely due to changes in the aberrated viewing angle along the jet.
proper motion measurements based on HST data (Sparks, Biretta, and Macchetto 1996, SBM
hereafter) at face value, the terminal bulk Lorentz factor likely falls into the range2
10 with a viewing angle
of
20Æ .
The average polarization at kpc distances (where the
jet has likely reached terminal velocity) is roughly
corresponds to 6<
1<
10% parallel to the jet axis , which
0:3 with the numbers given above.
Reynolds et al. (1996b) showed that the jet probably consists of pairs rather than ionized
gas, so the jet might actually still be accelerating (if indeed a large fraction of the particles has
> 10. Note, however, that the presence of shocks, clearly visible as knots in all wavelengths,
calls for a more sophisticated model which takes time-dependence and MHD instability effects
2
Note that VLA measurement show slower proper motions (Biretta, Zhou, and Owen 1995) and the relationship
between the observed pattern speeds and is not known. There are, however, no direct measurements of bulk
motions, so for lack of better knowledge we will use the larger values obtained from the optical data.
62
into account. For a discussion of the nature of the shocks observed in M87 see BB.) Furthermore, there is evidence that the jet is not magnetically dominated on large scales (z
> 100 pc)
and that the magnetic field is actually somewhat below equipartition (Heinz and Begelman
1997, see x3), which means in this context that the magnetic field must have been dissipated
non-radiatively, accounting for the particle pressure at larger distances. The observed spec-
0:6, which is at least in the correct regime for
synchrotron radiation to be inefficient at radiating away the dissipated energy (see x2.5.1).
At a viewing angle of 20Æ the jet radius at knot A (z 3 kpc) is R 35 pc with an
approximate opening angle of o 0:7Æ , much smaller than the beaming angle of b >
6Æ ,
tral index at optically thin wavelengths is
thus the jet is narrow at least at VLA resolution. Note that entrainment of ambient material may
be important before the jet reaches knot A (Bicknell and Begelman 1996, BB hereafter). Using
knot D as a reference point does not change this analysis significantly, however. There are no
good estimates of the pressure gradient in the innermost regions of the M87 X-ray atmosphere
(this situation should change, though, with upcoming Chandra observations of M87). The
pressure at large distances is approximately pISM
10
10 dyn cm 2 , but BB argue that the
pressure in the radio lobes is significantly higher than this, pext
1:5 10 9 dyn cm
2 . For
lack of better information we will assume the latter value and a smooth pressure gradient with
= const.
The radiative luminosity of the jet itself is Lrad
>
3 1042 ergs sec
1 (Reynolds
et al. 1996b), which argues for > 4 10 3 . If (as we speculated above) much of the particle
pressure at large distances was indeed produced by dissipation, could be much larger.
The jet probably reaches terminal
1 10 somewhere between VLBI scales and VLA
scales, 2 103 rg
0:2 pc < z1 < 200 pc 2106 rg . For small enough to be dynamically
unimportant we can assume that 1=4 in the self similar regime. Arbitrarily setting z0 10 rg gives 0:8 <
<
1:74. The central pressure p0 is then 2 10 4 dyn cm 2 < p0 <
200 dyn cm
2 , which requires an rms magnetic field of
balance (using an assumed z0
0:09 G < B 0 0 < 90 G for pressure
0:3). The initial jet width strongly depends on : 8 rg < R0 < 9000 rg for
= 10rg .
The latter value is unrealistic (and inconsistent with the limits put on
63
the jet width by VLBI observations, Junor and Biretta 1995), since most of the energy output of
the disk into the jet will be provided close to the black hole. It is therefore most reasonable to
2 103 rg and 1:7. The total jet power implied by the numbers given
is of order L 2 1044 ergs sec 1 , consistent with the estimate by BB. Overall it seems that
assume that z1
this model is consistent with the observed properties of M87 to first order if we adopt a pressure
gradient following 1:7.
The total isotropic energy output of E
>
1054 ergs of GRB 990123 (Akerlof et al. 1999;
Bloom et al. 1999) argues strongly in favor of non-isotropic gamma-ray burst (GRB) scenarios,
so jet models explaining the apparently beamed nature of these sources (e.g., Meszaros and
Rees 1997; Sari, Piran, and Halpern 1999) enjoy newly enhanced popularity. One of the standard scenarios for the energy sources of GRBs is a massive accretion event (either a neutron star
ski,
- neutron star merger or the accretion of a neutron star by a black hole: Narayan, Paczyn
and Piran 1992), which leads to the formation of a disk after tidal disruption of one of the objects. Since neutron stars already display large magnetic fields, one might expect strong shear
amplification of this field in the disk, leading to a scenario similar to what we described in
x2.2.
Similarly, the hypernova approach (Paczyn
ski 1998; Woosley 1993) can produce a pre-
collimated outflow, which might then evolve into a jet. Application of our model to GRBs is,
however, not as straightforward as in the case of mature radio galaxies. This is because GRBs
are highly time dependent (corresponding to the adolescent stages of radio galaxies): the lifetime of the jet (roughly of the order of the light travel time of the material) is shorter than the
sound crossing time of the bubble the jet blows into the environment, so a pressure balanced
solution as described above (which will be set up after the jet and the ambient material have
equilibrated, i.e., after the jet has existed for a few sound crossing times) might not be a good
approximation. The investigation of jet dynamics in the context of GRBs and in the framework
of tangled fields as introduced above will therefore be the subject of future work. Here we simply wish to point out the benefits of jet models in general and an approach based on our model
in particular:
64
Acceleration of GRB outflows by Poynting flux has the advantage that collimation can
be provided not only by the external medium, but also by the field geometry itself
(note that for Æ
= 0 the perpendicular component of the field does not contribute to the
sideways pressure, so for 1 the jet can have a large Poynting flux yet orders of
magnitude smaller sideways pressure than a particle dominated jet would have for the
same energy flux).
One would not need to invoke shocks to produce emission in this context, since the
internal dissipation of magnetic energy would provide a natural source of high energy
particles and photons to produce gamma rays (see Thompson 1994 for an example of
how internal dissipation can power GRBs). The short term variability seen in GRBs
could then be explained by inhomogeneities (e.g., variations in the inhomogeneity of
the field) imprinted on the outflow by variability in the central engine itself, which is
expected to have time scales of the same order as the ones observed.
2.6
Chapter Summary
In this chapter we presented an alternative jet acceleration model that utilizes tangled
magnetic fields rather than organized fields. Collimation must be provided by an external agent,
but magnetic tension can still reduce the amount of pressure needed to collimate the jet. Unlike jets accelerated by particle pressure or radiation pressure, radiative losses are not a severe
constraint on this model, and it avoids MHD instabilities encountered in magneto-centrifugal
jets.
This model is similar to the original ‘Twin Exhaust’ model (BR74). Analytic solutions
show that
/
pext
1=4 . Dissipation of magnetic energy can both increase or decrease the
acceleration rate, while radiation drag will always slow down the jet. We calculated the change
in the terminal Lorentz factor
1 due to dissipation and produced surface brightness, as well
as polarization predictions for small scale jets. Finally, we demonstrated that the model is
65
consistent with the observations of the M87 jet.
Future research is necessary to include the effects of turbulent kinetic energy, time dependence, 2D effects, such as shear layers, and the presence of large scale ordered magnetic
fields. It seems most likely that a realistic jet contains both ordered and chaotic field and a
combination of the magneto-centrifugal model with the tangled field model would be a more
realistic approach.
Chapter 3
Energetics of Jets and the M87 Jet
It should be evident from the previous chapter that knowledge of the internal makeup of
jets is critical to understanding the origin and acceleration mechanism of jets. Since resolution
limits prohibit us from studying the innermost regions of jets directly, the only way to learn
about the physical conditions in the plasma is from circumstantial evidence gathered at larger
radii. If we can determine the magnetic field strength, its level of uniformity, and the nature of
the particles (i.e., pairs vs. ions) are at large radii, we can then try to extrapolate backwards to
pin down which of the acceleration mechanisms mentioned in x2 is at work.
As it turns out, determining even such fundamental properties as the magnetic field
strength in the jet is far from trivial. The most promising way to measure this particular (and
particularly important) quantity is via measurements of particle evolution under the influence
of synchrotron losses. To do this, resolving the jet at high frequencies (optical and preferably
X-ray) is essential, which limits us to the most nearby sources. The nearest and best-studied
AGN jet accessible at all frequencies is that in the galaxy Messier 87. In this chapter we will
use the example of M87 to demonstrate how we can learn more about the energetics of AGN
jets. The following work has been published in Heinz and Begelman (1997) and Heinz (1997).
3.1
The M87 Jet After 79 Years of Surveillance
M87 is the central cluster galaxy and brightest member of the Virgo cluster. It shows
the typical characteristics of a large elliptical galaxy (it has been classified as an E0 pec galaxy,
67
Sandage 1961) with a total luminosity of about 30 L . Recent distance measurements put it at
a distance of 16 Mpc (Nielsen et al. 1999). Surface brightness measurements reveal a strong
central peak in density (Young, Westphal, Kristian, Wilson, and Landauer 1978), suggesting the
presence of a 109 M black hole at the center of M87. This hypothesis has been strengthened
by HST observations of what appears to be an accretion disk around the central compact object,
which reveals Doppler motion consistent with Keplerian rotation around a 3
4 109 M black
hole (Ford et al. 1994; Macchetto, Marconi, Axon, Capetti, Sparks, and Crane 1997).
The noticeable feature of M87 is its bright jet, visible all the way from radio to X-ray
wavelengths. First observed by Curtis (1918), it has become one of the prototypical radio jets
(mostly due to its proximity, since the radio galaxy itself does not particularly stand out from
the crowd of modest power radio sources).
The large scale structure of the radio source shows strong signs of interaction with
the environment or non-steady behavior of the central engine, with size scales ranging from
5 kpc (VLA radio jet and lobes) to the large scale radio source (about 100 kpc). M87 has
been classified as an FR I. Indeed, the total radio power of Virgo A is not overwhelming:
Lradio
1042 ergs sec
4 1044 ergs sec
2 . The kinetic power, estimated to be roughly between
1043
and
1 (BB, Reynolds et al. 1996b), is similarly small compared to the Eddington
luminosity of LEdd
1047 ergs sec
1 . This has led to the hypothesis that the central object in
M87 is fed by an advection dominated flow (Reynolds et al. 1996a).
Even though the jet in M87 has been observed in every possible wavelength band (Figures 3.1 and 3.2 show M87 in the radio, and optical bands), some puzzles remain about the
nature of its emissivity and composition. Polarization observations at both optical and radio
wavelengths have shown that the emission from the jet at wavelengths longward of
100 Å
is most likely of synchrotron origin (Baade 1956; Owen, Hardee, and Cornwell 1989). Collimated structure can be traced back to
terminates
0:01pc from the core (Junor and Biretta 1995), and
2500 away from the core (corresponding to a projected length of about 2 kpc at the
assumed distance of 16 Mpc) in the western radio lobe, with optical emission still detectable at
68
Figure 3.1: Combined high dynamic range VLA map of the jet and lobes at 15 GHz and 0:00 15
resolution (Biretta 1993).
this distance. The jet looks very similar at optical and radio wavelengths, although SBM show
that differences (e.g., in the transverse brightness profiles) do exist. Reynolds et al. (1996b)
argue that the jet plasma is very likely composed of electrons and positrons rather than heavier
particles such as protons, but more conclusive evidence is needed to settle this question.
Far from having a smooth appearance, the jet exhibits a series of bright knots at intervals
roughly
2:00 5 apart (see Figure 3.1).
The nature of the knots is uncertain, but they are usually
attributed to internal shocks from either flow instabilities (BB) or variable outflow at the source
(Rees 1978b). Figure 3.2 shows the sharp increase in brightness in the knot and its filamentary
appearance, arguing against variable outflow as a source of the brightness variations observed.
The lack of any emission or absorption features in the spectrum of the jet makes a direct
determination of the bulk velocity of the jet plasma impossible. However, the onesidedness of
the M87 jet has been interpreted as the result of relativistic beaming. Adopting a lower limit
on the jet-to-counterjet radio brightness ratio of 150 (Biretta 1993), one can set a lower limit of
jet
2 on the bulk Lorentz factor of the plasma with line of sight inclinations of LOS 35Æ .
69
Figure 3.2: HST FOC image (upper panel) and 2cm VLA map of the jet (lower panel), adopted
from SBM. The bottom panel shows the distance from the core and the knot labels.
Even though the dynamic range of optical HST images is larger than that achieved by Owen,
Hardee, and Cornwell (1989), the constraints on jet and LOS (the dynamical parameters of the
jet) set by the radio observations are stronger than what follows from the optical non-detections.
This is because the brightness ratio goes as [(1
cos LOS ) = (1 + cos LOS )]3+ and R 0:5, the radio spectral index, is smaller than O 1:2, the optical spectral index. Doppler beaming of the counter jet emission away from the observer would explain the existence of the radio
lobe on the eastern side of the core, for which no jet is visible. New infrared observations that
detect emission from the counterjet region at the 3 level (Stiavelli, Peletier, and Carollo 1997)
seem to confirm the hypothesis of relativistic beaming. The Doppler beaming interpretation is
also bolstered by proper motion measurements of the knots (Biretta et al. 1995), which show
characteristic velocities vproper
0:5 c in knots A and B, with some features exhibiting much
larger proper velocities (a subfeature of knot D appears to show superluminal motion at a level
of 6c, Biretta, Sparks, and Macchetto 1999).
These motions broadly support the interpreta-
tion of the knots as relatively weak, oblique shocks moving down the jet with pattern speeds
70
significantly smaller than the bulk speed of the flow (BB).
Figure 3.3 shows the integrated spectrum for the knot A-B region (Meisenheimer et al.
1996), which is typical for the spectrum observed along the jet. Both the knots and the interknot
regions exhibit a featureless power law spectrum of index R
the optical data with a power law of index RO
1:8 in the optical.
0:5 in the radio, connecting to
0:65, and steepening to O between 1:2 and
This steepening trend is also found in observations at infrared (e.g., Stocke
et al. 1981) and ultraviolet (Perola and Tarenghi 1980) wavelengths, as confirmed recently by
HST observations (Boksenberg et al. 1992, , SBM). X-ray observations made with Einstein
Observatory (Biretta et al. 1991) and ROSAT (Neumann et al. 1996; Reynolds et al. 1996a)
reveal X-ray emission from several spots along the jet (mainly the core and knot A, possibly
also from knots D and B). However, the origin of the X-ray emission is unknown and it is
not clear whether the spectrum breaks between optical and X-ray wavelengths to a spectral
index of OX
1:4, with the X-ray emission still being of synchrotron origin, or whether the
X-ray emission is produced by a different mechanism, such as inverse Compton scattering or
bremsstrahlung.
3.1.1
The Spectral Aging Problem
One might hope to detect the effects of synchrotron cooling and relativistic particle ac-
celeration by studying the spectrum as a function of position along the jet. Such measurements
(SBM) show that the radio–to–optical spectral index,
optical spectral index,
RO , is very nearly uniform, while the
O , is anti–correlated with the brightness,
i.e., the optical spectrum is
flatter in regions of higher intensity. Modeling the optical steepening as a high-energy cutoff
imposed on a power law spectrum, Meisenheimer (1999, hereafter M99) and Meisenheimer
et al. (1996) find a corresponding correlation between brightness and cutoff frequency, i.e., a
higher cutoff frequency at higher intensities1 . Both of these results are striking in the lack of a
strong secular decline in the cutoff frequency with distance from the core, as would be expected
1
Appendix B reviews basic properties of synchrotron radiation and details of the model fit by M99.
71
Figure 3.3: Top panel: radio-to-optical spectrum for the integrated knot AB region. The radio-to-optical spectral index is RO 0:65, the spectrum
cuts off at 1015 Hz. Bottom panel: optical-to-X-ray spectrum for the
integrated knot A-B region. The origin of the X-ray emission is unknown.
The spectrum could either exhibit a cutoff in the UV and have a separate
component accounting for the X-ray emission or (if the X-ray emission is
of synchrotron origin) it might break in the optical and connect to the X-ray
with a OX 1:4 power law. From Meisenheimer et al. (1996).
72
naively if the steepening were due to synchrotron cooling.
Indeed, these observational results do not compare well with simple quantitative models
of synchrotron cooling in the M87 jet. The usual assumption of an equipartition between magnetic and particle energy density1 in the jet plasma leads to estimates of the magnetic field of
order
300 G , with values up to 500 G in knot A. The synchrotron lifetime2
with Lorentz factors of
only
for electrons
106 needed to produce the optical emission in a 300 G field, is
2:3 1010 sec, which for mildly relativistic bulk velocities (e.g., 0:5 c) implies a travel
distance of less than
120 pc.
But the projected length of the jet is about
2 kpc, and even the
distances between the most prominent knots are longer than the estimated cooling length. Yet
the spectrum between radio and optical bands remains remarkably constant along the jet, with
only minor variations in the optical spectral index.
This presents a paradox: After a travel distance of
2 kpc one
would expect the high
energy particles to have cooled in such a way that the spectrum would drop very steeply above
1012 Hz.
This cutoff frequency is a factor of
103
smaller than what is actually observed.
The discrepancy becomes worse if the magnetic field is stronger than the equipartition value, as
suggested by Owen et al. (1989) in order to explain the confinement of the overpressured jet via
magnetic tension force. (An alternate explanation for the confinement — that the radio cocoon
surrounding the jet is overpressured with respect to its surroundings — has been proposed by
BB.) Three explanations for the discrepancy between the expected and the observed amount of
cooling have appeared in the literature:
(1) First-order Fermi re–acceleration3 in the knots, interpreted as shocks, could produce
high energy electrons (and possibly positrons) from a synchrotron-cooled distribution,
with a power law of roughly the index observed. It could also explain the observed Xray emission. However, since the power-law index produced by Fermi acceleration is
a strong function of the compression ratio one would need some fine tuning to explain
2
3
For a definition of this term see Appendix B.1.1
For a short review on Fermi acceleration see Kirk 1994, for example.
73
the observed constancy of the radio–to–optical spectral index, which does not equal the
limiting value for a strong adiabatic shock. Furthermore, there does not seem to be a
significant amount of cooling between the knots even at the highest optical frequencies
observed, which would be expected for the assumed B -fields and interknot distances.
(2) Particles could be transported in a loss-free channel in the interior of the jet, with the
bulk of the emission produced in a thin outer layer of high magnetic field strength
(Owen et al. 1989). In this picture, the knots and filaments would be interpreted as
instabilities with greatly increased magnetic field strength wrapped around the jet. The
emission would then be fed by particles from this channel. The radio brightness profiles
across the jet seem to suggest a limb-brightened emission, but a reinvestigation of the
HST observations (SBM) shows that the optical emission is more concentrated to the
inner regions of the jet. Also, a new radio volume emissivity map by SBM (see Figure
3.4), produced by deconvolving a processed 2 cm VLA image of the jet, places the
brightest spots in the jet interior. This argues against a field-free zone in the jet interior.
(3) On–the–spot reacceleration by a yet unknown process could maintain the cut-off particle momentum at the observed level, as has been proposed by Meisenheimer et al.
(1996) in a model similar to ours (see x3.4.2). This has the advantage of explaining
all the observed features, but invokes unknown physics to explain the apparent lack of
cooling.
3.1.2
Proposed Solution
Inspired by the observed correlation between the emissivity variations and the cut-off
frequency, by the newly deconvolved volume emissivity (SBM), and by the new evidence for
relativistic bulk velocities (Stiavelli et al. 1997; Biretta et al. 1995, 1999), we propose a simple
way of explaining the observations. The only standard assumption we give up is the assumption
of equipartition, which does not seem to have a very firm physical foundation anyway. Magnetic
74
fields smaller than equipartition by a factor of 1:5
in excess of
2
3, coupled with bulk Lorentz factors
3, can readily explain the lack of strong synchrotron cooling.
jet
In our model
the fluctuations in the cutoff frequency are produced by weak shocks, so that the influence of
the compressions on the plasma distribution function can be considered to be adiabatic4 . As a
result, we are able to explain the general behavior of the cutoff reasonably well; the magnetic
fields we derive from our own fits to the data are below equipartition with a convincing level of
confidence, but the inferred total pressures do not necessarily need to exceed the equipartition
values. Additionally, for relativistic jets the equipartition values for
B -field and pressure are
less than in the nonrelativistic limit, so the pressures we derive can fall below the equipartition
value in the nonrelativistic case.
The importance of this model therefore lies in its ability to explain the observations of
the jet (i.e., the lack of cooling and the amplitude of the fluctuations in emissivity and cutoff
frequency) in a self consistent fashion without invoking unknown physics. Since independent
determination of magnetic fields in optically thin synchrotron emitting environments is otherwise impossible, this method provides a unique opportunity to learn more about the physical
conditions in extragalactic jets. The fact that our model requires relativistic effects to be at work
is a further step in the chain of evidence for relativistic bulk velocities of the jet plasma in large
scale radio jets.
This chapter is organized as follows. In x 3.2 we present a nonrelativistic treatment of the
synchrotron emissivity, taking into account cooling and assuming that the particles and fields
respond adiabatically to changes in the flow density. Once relativistic effects are incorporated
into the treatment in
x 3.3, we use the data of SBM and M99 to constrain the magnetic field
strength of the jet. Chapter 3.4 discusses confinement and stability of the jet in the light of
x 3.3, the production of X-ray emission in knot A,
polarization, and limits on the particle acceleration site; and x 3.5 gives a brief summary of
the pressures derived from the results of
4
The energy of the accelerated particle is proportional to its initial energy and the compression ratio of the
plasma to the 4=3 power. Thus, if multiple species or populations of particles exist in the plasma, the fraction of
internal energy in each species remains constant throughout the acceleration.
75
the results and future prospects. The Appendix gives a brief review of major properties of
synchrotron radiation, shows the solution to equation (3.4) explicitly, and comments on the
fitting procedure by M99.
3.2
Adiabatic Effects on Synchrotron Emission
Our model rests on the hypothesis that Fermi acceleration is unnecessary to explain the
fluctuations of radio–to–optical emissivity and cutoff frequency along the M87 jet3 . Given
certain assumptions about the orientation and degree of disorder in the magnetic field, and the
degree of anisotropy permitted in the relativistic electron distribution, we can relate changes in
both the emissivity and the cutoff frequency uniquely to changes in the density of the jet fluid.
These adiabatic effects are readily combined with the effects of synchrotron cooling (Coleman
and Bicknell 1988). In effect, given the emissivity map of SBM, we can predict the run of
the cutoff frequency along the jet, and vice-versa. Since we also have M99’s observations of
the cutoff frequency as a function of position, our adiabatic model is subject to a powerful
self-consistency check.
Observationally, the main changes in emissivity and cutoff frequency are rather localized, and associated with the positions of the knots. These small-scale fluctuations therefore
provide the strongest check on our assumption of adiabaticity. The large-scale trends then
determine the best fit to the magnetic field strength. We have already seen that the apparent
lack of a large-scale synchrotron cooling trend is incompatible with a magnetic field strength
as large as the mean equipartition value (see Fig. 3.5). Neglecting relativistic and projection
effects, it would require a magnetic field as low as
25 G to obtain a cooling length of 2 kpc
at optical frequencies. This is an order of magnitude smaller than the mean equipartition field
and would require a total (particle + magnetic) pressure of
1:2 10
7
dyn cm
2 to produce
the observed average amount of synchrotron emission, compared to an equipartition value of
4:0 10 9 dyn cm
2.
As noted earlier, the physical basis of equipartition is weak. Estimates of
B -fields in
76
radio hot spots based on the synchrotron cooling time indicate that in some cases equipartition
might be correct up to a factor of
2 (Meisenheimer et al. 1989), but the conditions in the
jet might very well be different from those in the lobes. We are therefore free to consider the
magnetic field strength to be a free parameter. Applying our adiabatic model to the observational
data, we can derive an estimate for the magnetic field strength. In the case of the M87 jet, this
estimate lies below equipartition, even when relativistic effects are taken into account (x 3.3).
Our model for the evolution of the particle distribution function follows that of Coleman
and Bicknell (1988). We assume that pitch angle scattering due to plasma micro–instabilities
keeps the particles close to an isotropic distribution in the fluid rest frame. In the absence
of cooling, this would imply that the relativistic electrons respond to compressions like a
adiabatic = 4=3 (i.e., ultrarelativistic) fluid, but this behavior will be modified by synchrotron
cooling. Because the magnetic field is frozen into the plasma, its strength should change as the
plasma density fluctuates along the jet.
Depending on the orientation and the degree of disorder of the field, its variation will
depend roughly on the density change to some power :
B (r) / %(r) , where % is the proper
particle density and r is the distance from the core. For a completely disordered magnetic field,
=
2 , whereas for a homogeneous field the power depends on the orientation of the field with
3
respect to the compression normal, with = 1 for an orthogonal orientation, = 0 for parallel
orientation. Since the polarization of the jet is of order 10%-20% in the interknot regions,
compared to the maximum polarization of 70% for a homogeneous field, it is likely that the
magnetic field has a disordered component, so it is reasonable to assume to be of order 23 (but
see x3.4.4). The ordered
B -component is aligned with the jet axis almost everywhere except
in the brightest knots, as can be seen from polarization measurements (Biretta 1993). We do
not know its orientation with respect to the compression normal, because the orientations of the
(presumably oblique: see BB) shocks are unknown. However, our results are not very sensitive
to what the actual value of
is, as we will show later. We therefore make the simplifying
assumption of a single exponent describing the field variations. Using the scaling relation for
77
the synchrotron emissivity of a power law momentum distribution
f (p) d3 p N0
p a d3 p = N0 p
4
(corresponding to a spectral index of
=
a+2 dp;
(3.1)
a 3 ) under adiabatic compression (e.g., Coleman
2
and Bicknell 1988, see equation [B.5]):
2+3
j / (B sin #)1+ % 3 ;
(3.2)
where # is the angle between magnetic field and line of sight, we can express the field relative
to its value at r0
= 000 :5, the (arbitrary) injection point at which we start the calculation, as a
function of the emissivity ratio j=j (r0 ):
B = B (r0 )(j=j (r0 ))
where (3.3)
[1 + + (2 + 3)=(3 )]
1 . A necessary condition for this approach to be valid (in
addition to the assumed isotropy and the absence of Fermi acceleration) is the assumed steadystate injection of relativistic particles and fields by the central engine, which allows us to relate
densities and fields at each
r to the corresponding
values at the injection point r0 at a given
instant of time.
Is it plausible to neglect Fermi acceleration in the shocks that comprise the knots? Except for knot A, the brightness changes along the jet are moderate. In knot A, the brightest
feature, the emissivity changes by a factor of order
10 (SBM), which, if entirely due to a sud-
den compression of the plasma, can be produced by a proper compression ratio of r
measured in the respective rest frames of the plasma if we take infer smaller density contrasts,
r
1:5.
=
2:7 as
2
3 . For the other knots we
Consistent with this observation, we will henceforth
take the knots to be weak shocks, in accordance with the suggestion by BB that the knots are
highly oblique (and therefore weak) shocks. Thus, because Fermi acceleration leaves the spectral index unchanged if the shock is weak enough, we will henceforth neglect its effect on the
cutoff frequency. Section 3.4 discusses Fermi acceleration in more detail, with particular focus
on the possibility of Fermi acceleration occurring in knot A.
78
Furthermore, because the shocks are believed to be oblique, we take the fluid velocity to
be constant to first order, both in magnitude and direction. For the shock jump conditions in the
non–relativistic limit (which we consider in this section) the velocity component perpendicular
to the shock plane
v? is inversely proportional to the density,
thus for a proper compression
ratio of 2:7, as seen in knot A, the perpendicular velocity component should change by a factor
of 0:37. For highly oblique shocks, v? is small compared to vk and the velocity will not change
significantly. Moderate changes in velocity would be easy to incorporate in principle; yet with
our current ignorance of the velocity field and shock parameters, such a level of detail is unwarranted. We will comment on the validity of this assumption in
x 3.3.
We also postpone a
treatment of the motion of the knots until x 3.3, and assume them to be stationary for the rest of
this section.
With these assumptions we are ready to calculate the downstream cutoff frequency for a
given initial cutoff momentum in the injected particle distribution and a given B -field at r0 . We
use the transport equation as presented by Coleman and Bicknell (1988):
df 1 d% @f
@
+
p = Ap 2 (p4 f )
dt 3% dt @p
@p
written in the rest frame of the fluid. Here,
A=
(3.4)
f = f (p) is the electron distribution function and
4e4
2
9m4e c6 B is the synchrotron loss term. The equation is valid for the assumed case of
isotropy and negligible Compton losses (for a brief discussion of Compton losses see x 3.4.1).
The solution of this equation is
0
1 14
3
p %
f (p(r)) = f0 (p0 ) @ 0 1 A
p %3
(3.5)
0
where f0 is the injected momentum distribution and
1
(%=%0 ) 3 p0
p(r) =
R t(r)
1
(1 + p0 t(r0 ) A(t0 )(%(t0 )=%0 ) 3 dt0 )
(3.6)
(see Coleman and Bicknell 1988 and Appendix B.2) where the subscript 0 denotes the values at
injection point r0 . Even for arbitrarily large p0 , p(t 6= 0) can only reach a limiting value of
pmax (t) (%=%0 )1=3
1 0
0
0
t(r0 ) A(t )(%(t )=%0 ) 3 dt
R t(r)
(3.7)
79
10-2
j(2cm) [Jy/pixel3]
10-3
10-4
10-5
10-6
10-7
10-8
200
400
600
800
1000
pixel
Figure 3.4: Emissivity map provided by SBM (top panel), produced by symmetrifying the 2cm
VLA map of the jet. The scaling is logarithmic in Jy/pixel3 for 0:00 0223 pixels. The bottom
panel shows a slice along the jet axis to demonstrate the scaling of the emissivity as a function
of distance from the core (solid line) and the emissivity averaged across the jet (thick line). Note
that the variations along the central slice are much higher than for the averaged j . This is very
likely due to the fact that the deprojection method is least accurate towards the jet axis.
Thus, any momentum distribution subject to synchrotron cooling without replenishment of high
energy particles must develop a cutoff at high energies.
Equation (3.6) describes how the momentum of a given particle changes along a streamline. Thus, if the distribution initially cuts off at pc;0 , we can calculate the cutoff momentum
1
pc (r) downstream. Because in our model the density %(r) is proportional to B (r) and because
we know the scaling of B with r from equation (3.3), we can eliminate % and B=B0 from equation (3.6). The remaining parameters are B0 , pc;0 , and j (r )=j0 , the latter being provided by the
Sparks et al. data, shown in Figure 3.4.
The cutoff momentum pc is related to the observed cutoff frequency c by the expression
c =
3e 2
p B sin #:
4m3e c3 c
(3.8)
Equation (3.8) contains another parameter, #, the angle between the line of sight and the mag-
80
netic field. For now we shall set the factor sin # 1, which is valid for disordered fields, since
1) the regions in which the field is perpendicular to the line of sight have the highest emissivity,
and 2) assuming randomly oriented fields, half of the field orientations lie in the range from 60Æ
to 90Æ to the line of sight, i.e., sin #
0:866. Thus the cutoff frequency is mainly determined
by field orientations close to 90Æ or sin # 1.
We can now determine the free parameters
B0 and pc;0 by applying a least chi-squared
method to fit the observed cutoff frequency c;obs with the value determined from equations
(3.6) and (3.8), using the emissivity map j (r )=j0 provided by SBM. We prefer to average the
emissivity across the jet (we averaged j over a disk with jet diameter at every location along the
jet), which minimizes small scale variations probably due to the deprojection procedure, which
is most unreliable towards the jet axis (we will comment on the possible uncertainty introduced
by this step in x3.3). Because M99 also averaged across the jet, this seems to be the most
appropriate way of calculating the cutoff frequency. Figure 3.5 (calculated for a bulk Lorentz
factor of jet
= 1:1,
a radio spectral index of R
= 0:5, LOS = 90Æ ,
and
=
2
3 ) shows
the observed cutoff frequency c;obs (vertical bars) with error bars and the best fit curve (solid
line), which seems to reproduce the scaling of the cutoff frequency reasonably well. (The radio
spectral index seems to break to
0:65 at 10 GHz, so we have used both = 0:5 and
0:65 in our fits with insignificant differences in the average parameters but smaller chi-squared
for
0:65; see x3.3.)
equipartition
For comparison the plot also shows the best fit cutoff frequency for
B -fields (dashed line).
The mean
B -field is of order 10 G, even smaller than
the zeroth order estimate made at the beginning of this section. Assuming (arbitrarily) a lower
cutoff at = 107 Hz and the observed high-frequency cutoff at 1015 Hz yields an average
total pressure of
value of
p
8 10
8
dyn cm
3 10 9 dyn cm
2 for the given parameters, compared to an equipartition
2 . In calculating absolute values for both pressure and
B-
field, projection (i.e., foreshortening and length scale) effects must be taken into account, since
the emissivity was derived for a side-on view of the jet. This introduces a factor of
in intrinsic emissivity and pressure for a given magnetic field and a factor of
sin LOS
4
(sin LOS ) 7
in
81
1016
νcutoff [Hz]
1015
1014
1013
1012
0
5
10
15
distance in arcsec
20
Figure 3.5: Measured cutoff frequency along the jet from M99 (vertical lines with error bars).
Solid line: best fit curve as calculated from the emissivity measured by SBM for jet = 1:1,
= 23 , and LOS = 90Æ ; dashed line: best fit curve for the case of equipartition calculated from
SBM data for the same set of parameters.
equipartition pressure.
It seems that the proposed modest compressions can account for the fluctuations seen in
the spectral cutoff, and the large scale decrease in c is well reproduced. It would be helpful to
determine both the emissivity and the spectral index maps from the same method and data, thus
eliminating errors due to different reduction procedures. We comment on the deviations and
uncertainties in this fit in x 3.3. It is important to note that this technique should be independent
of what the actual shape of the particle distribution is, because it simply tracks the behavior of
a single feature in the spectrum, which could be identified with either a break or a cutoff.
As seen in this section, one runs into problems with the jet pressure for nonrelativistic
bulk velocities. Also, the observed mildly relativistic proper motion of the knots and the jet’s
onesidedness favor a relativistic interpretation, as do the knot spacing and morphology (BB).
The results indicate that Lorentz factors of order
2
5 fit the observations
best. In the next
section we will investigate the effects of these suggested relativistic bulk velocities.
82
3.3
Relativistic Effects
Relativistic motions not only explain the onesidedness of the M87 jet, but also help to
solve the synchrotron cooling problem mentioned in the introduction. The travel time in the
electron rest frame is reduced by a factor jet due to time dilation, the intrinsic emissivity is
reduced by a factor
D2+
(where
D
is the Doppler factor
D=[
jet (1
and the intrinsic cutoff frequency is Doppler shifted downward by a factor
cos LOS )]
D.
1 ),
As a result, the
apparent synchrotron lifetime can be a significant underestimate of the intrinsic value.
Biretta (1993) estimates the lower limit on the jet–to–counterjet radio brightness ratio to
150 380 (the higher value corresponds to the assumption that jet and counterjet have
identical appearance). Based on this limit we adopt line–of–sight angles 35Æ and Lorentz
factors 2. Note that even Doppler factors smaller than unity can lead to a large jet–to–
be
counterjet brightness ratio, as the counterjet brightness is severely reduced for large jet .
We repeat the analysis of x 3.2 using the emissivity profile of SBM, this time corrected for
Doppler boosting and projection effects. Again, for fitting the cutoff frequency only emissivity
ratios are important, so these corrections do not change the fitting procedure as long as changes
in the bulk velocity can be neglected. We are confident that at least the direction of the flow is not
changed significantly before knot C. The only knot for which such effects could be important is
knot A, because it displays a jump in emissivity of 11, whereas in the other knots the emissivity
is increased by a factor of order 3 only, which implies very moderate compressions. Using the
relativistic continuity equation for an oblique shock we have estimated the post–knot A Lorentz
factor to be jet;A+
3 for a pre–knot A
jet;A
= 5, a compression ratio of 3 and intrinsic
obliquities 60Æ . Although the change in jet might seem large at first glance, the impact such
a velocity change in the shock at knot A as on our fits is not large, as we will explain below.
We modify equation (3.4) to follow the electron distribution in the fluid rest frame by replacing
t with d with d , where
and dt
d
is the proper time. The same changes apply to equation
(3.6). Strictly speaking, the fluid frame is not an inertial frame and we would have to include
83
accelerational terms into the equation, introducing an anisotropy. But our assumption should be
adequate, provided that isotropization takes place over short enough scales.
Treating the response of the distribution function to compressions as adiabatic and assuming isotropy (i.e., an adiabatic index of 43 ) we calculate the changes in
B and cutoff mo-
mentum (measured in the fluid frame) from the emissivity changes in the fluid frame. The
substitution t ! takes care of the time dilation effects.
In order to incorporate the observed motion of the knots, we must correct
A( 0 ) and
%( 0 ) in equation (3.6) for light travel time effects between the source and the observer.
This
is because the knots move during the time it takes a particle to travel from r0 to r . We need
to know the ratios B=B0 and %=%0 experienced by a particle as a function of proper time 0 in
order to be able to do the integration in equation (3.6). Because we infer the relative values of
B and % at a given position from the emissivity ratio j (ruid )=j0 , it is important to know the
velocity of the emissivity pattern.
We assume that the pattern of density and field fluctuations retains its shape and moves
along the jet at a fixed speed vpattern , taken to be smaller than vuid and set equal to
0:55 c
everywhere in our calculations for simplicity. If the present field distribution (i.e., at a given
time t
= 0 in our frame, corrected for light travel time effects) is expressed by B (r), then the
field at time t0 and position r 0 is given by B (r 0 + vpattern t0 ). Now, for a particle currently
at r , the equation of motion is
r0 = r
by the particle as a function of time is
vuid t0 .
B [r
Therefore, the field distribution experienced
(vuid
vpattern )t0 ].
Appropriate modifications
to equation (3.6) are straightforward. The effect of the pattern speed on the result is not very
dramatic, reducing 2min by about 6%.
With this set of assumptions we can once again proceed to integrate the modified equation
(3.6) for various LOS and jet . Using a minimum chi-square routine we can determine the
best–fit values for B0 , and p0 . Relation (3.3) then yields B (r ).
Figure 3.6 shows a typical chi-square plot calculated for jet
=
2 . The equipartition value for the average
3
= 3, LOS = 25Æ ,
and
B -field is shown as a shaded area at 89 G.
84
1.6
1.4
equipartition B-field
P0 × 1010[g cm sec-1]
1.8
4
1
1.2
3
2
5
1.0
0.8
20
40
60
Bmean[µG]
80
Figure 3.6: Contour plot of the 2 values as a function of Bmean and the injection cutoff momentum P0 for = 25Æ and jet = 3:0. Shown are the contours corresponding to a reduced
2 of 45 and integer multiples of this value. The shaded line on the right of the plot shows the
equipartition B -field for this set of parameters.
The upper limit on Bmean , set by 22min contours, lies at 49 G, 75% above the best fit value,
Bmean = 28 G.
The lower limit set by
The average equipartition field of
Bmean
22min
is
5 G, 80% below the best fit value.
89 G lies above even the 52min contour.
The
lower limit on B is not nearly as strict, due to the fact that cooling is not dominant, i.e., we can
produce a similar spectral behavior by reducing the magnetic field and increasing the particle
energy, which produces the tear-shaped appearance of the contours. Thus, strictly speaking, the
best fit values for the B -field should be regarded as upper limits.
The reduced 2min values (i.e., 2min divided by the number of degrees of freedom) fall
above 44, which is uncomfortably high. However, because we do not have formal errors for the
emissivity deprojection by SBM, which will introduce a significant uncertainty, a high value
for 2min is not all that discouraging. Estimating the average uncertainty in the emissivity by
comparing the averaged emissivity to that derived from taking only a slice along the jet yields
an uncertainty of order 50%, which leads to uncertainties in the predicted cutoff frequency of
85
roughly 20%. This is significantly higher than the formal error in M99’s data and will reduce
the 2min by a factor of approximately 10.
The 2min values are dominated by the region beyond knot A. The post–knot A residuals
in our fit are not larger than the residuals in the pre–knot–A region, but because the post knot A
region is brighter, the error bars on the measured cutoff frequency are smaller, which increases
the 2min . The deprojection procedure, which assumed an axially symmetric flow, breaks down
beyond knot A, which will introduce significant uncertainty. Also, non–uniformities in the
emissivity could lead to large errors if the optical emission peaks at different locations than does
the radio emission. Field orientation effects and changes in jet and LOS might also contribute
to the error. We performed the same procedure just out to knot A and found that, with the same
parameters, the reduced 2min shrinks to 13. Leaving B and pc as free parameters reduces 2min
to 10, but also reduces the B -field significantly. Because in this case the algorithm mainly fits
the region around knot A (where the error bars are smallest), we cannot expect the global run of
c to have significant impact on the fit, which would be necessary to extract information about
the average magnetic field. We conclude that the reproduction of fine detail is not satisfactory
in the region beyond knot A. However, the gross run of c , which is principally responsible for
constraining our parameters, is reasonably well reproduced.
The best–fit average magnetic field Bmean , as a function of jet and LOS , is plotted in
Figure 3.7. Figure 3.8 shows the average ratio hB=Beq i1 as a function of jet for LOS
= 15Æ to
30Æ in increments of 5Æ , and the area corresponding to the limiting jet–to–counterjet brightness
ratio of 150 - 380. Note that the equipartition magnetic field has to be corrected by a factor of
2+
1
D 3+ (sin LOS ) 3+ for projection and Doppler boosting of the emissivity; this has already
been taken into account in the figure. Clearly, for jet in the range
3
5 and LOS <
25Æ the
departure from equipartition is not very large (roughly a factor of 0:2 < hB=Beq i < 0:6).
In order to test the dependence of the best fit B0 on the parameter
the same curves for
= 1 and =
B1 B2
from the 1
B1
=
we have calculated
1
15 . Figure 3.9 shows the fractional deviation
2 curve for models with
3
15Æ
LOS 30Æ .
B
B The deviation is small
86
45
Bmean [µG]
40
35
30
brightness ratio limit
15˚
20˚
25˚
30˚
25
20
15
2
3
4
5
Γjet
6
7
8
Figure 3.7: B -field averaged along th jet between 000 :5 and 2200 as a function of jet for LOS =
15Æ (solid line), LOS = 20Æ (dotted line), LOS = 25Æ (dashed line), and LOS = 30Æ (dasheddotted). The grey region indicates the jet–to–counterjet brightness ratio limit of 150 (dashed
grey boundary) - 380 (solid grey boundary, Biretta 1993).
0.70
brightness ratio limit
15˚
20˚
25˚
30˚
〈 B / Bequipartition 〉
0.60
0.50
0.40
0.30
0.20
2
3
4
5
Γjet
6
7
8
Figure 3.8: Ratio of the best–fit B -field to the respective equipartition B -field Beq averaged
between 000 :5 and 2200 for = 23 as a function of jet . The different values of LOS are labeled
similarly to Fig. (3.7). The grey region shows the jet–to–counterjet brightness limit (dashed
grey boundary: 150, solid grey boundary: 380).
87
Γjet,AΓjet,A+
ζ = 1 / 15
ζ=1
α2cm= 0.65
(∆B/B)ζ,Γ,α
0.20
0.15
0.10
0.05
0.00
2
3
4
5
Γjet
6
7
8
Figure 3.9: Fractional deviation BB of the best fit B -field (averaged between 000 :5 and 2200 )
jB1 B2 j , where = 2 and either = 1
for a. two different values of : BB
1
2
B1
3
15
(hatched, solid boundary) or 2 = 1(hatched, dotted boundary); b. a uniform jet, compared to
Bbreak j , where B
jBuniform
a jet slowing down at knot A: BB
break is the best fit average
Buniform
B -field for a jet slowing down from jet;A to jet;A+ at knot A, and Buniform is the best fit
average field for a uniform jet with jet = jet;A (dark grey region, short dashed boundary)
The latter is plotted versus
and jet = jet;A+ (light grey region, long dashed boundary).
jB1 B2 j for our standard
B
jet;A+ ; c. two different 2cm radio spectral indices:
B B1
value 1 = 0:5 and 2 = 0:65, shown as the black area. (The width of the band corresponds to
LOS between 15Æ and 30Æ in each case.)
compared to the expected errors introduced by the simplifications we made and to the range in
B allowed by our minimum chi-square procedure, at most 12% for 2 =
1
15 and small jet . This
is not a very reasonable value for in any case, because the field has a random component, thus
should be higher, and the probability of the field being in the shock plane (thus having = 1)
is twice as high as for the field being normal to the shock. We conclude that our ignorance of
the precise behavior of B under compression is not a serious obstacle to the application of our
model.
We also tested the impact a change in jet at knot A might have on our results. As we
mentioned earlier, the best fit B -field values we derive are upper limits. This is the reason why
88
a change in jet at knot A does not change our results significantly: generally, lower Lorentz
factors require lower fields to explain the observed lack of cooling. If the jet is slowed down
beyond knot A, we will need lower average fields to fit this region. However, lowering the field
does not change the quality of the fit much (the 2 is essentially unchanged), so the global field
strength is simply set by the region with the lower Lorentz factor, jet;A+ (the relative scaling
of B is still determined from equation 3.3, taking relativistic beaming into account). We have
introduced by hand a change of jet at knot A into our model (we solved the continuity equation
at the shock, assuming an obliquity of
60Æ , for the velocity change that would reproduce the
observed emissivity jump of 11, including relativistic beaming and adiabatic compression), and
calculated the fractional deviation BB
Bbreak j of the derived averaged B -field.
jBuniform
Buniform
Here, Buniform is the best fit average B -field derived for uniform jet and Bbreak is the best fit
field for a jet slowing down from jet;A to jet;A+ at knot A. Figure 3.9 shows BB
for
uniform jet models with jet set to either jet;A or jet;A+ (filled light and dark grey regions,
respectively). The latter is always less than 18% for the parameter range we used. Note that for
post–knot–A jet;A+ ’s above 5, the pre–knot–A jet;A exceeds 8.5, thus jet;A+
>
7 can be
ruled out on the basis of gross energy balance arguments (see next section).
For completeness we have shown the deviation of the best fit average B -field BB
jB1 B2 j for a 2 cm radio spectral index of = 0:65 instead of = 0:5 as the black region
2
1
B1
in Figure 3.9. One can see that the difference is negligible compared to other uncertainties.
3.4
Discussion
In the preceding sections, we have demonstrated that a) magnetic fields slightly below
equipartition and b) moderately relativistic effects are able to explain the general behavior of
the spectrum in M87. In this section we will examine the confinement properties of the jet
and compare our model with a previous model by Meisenheimer et al. (1996). We will also
comment on the production of X-ray emission in knot A, and on the consistency of our model
with polarization measurements, and we will estimate the minimum distance from the core at
89
which particle acceleration must occur.
3.4.1
Confinement
Naturally the question arises whether the jet can be confined under the conditions we
proposed above. The usual assumption for a jet to be confined is that it is in pressure equilibrium
with its surroundings. Alternatively, one could imagine the jet to be freely expanding into an
underpressured surrounding medium.
BB argue that in order to produce shocks via Kelvin-Helmholtz instability, some interaction between jet and surrounding medium has to take place, as opposed to a free expansion
scenario. They also show that the minimum Lorentz factor jet for a freely expanding jet with
no cold matter content is at least 13, much higher than the values we have used above. Because
we would most certainly fall out of the beaming cone for such a high jet jet, the intrinsic emissivity would be much higher than the observed value. As BB point out, the energy flux of the
jet would far exceed the estimates made on the basis of the expanding bubble the M87 jet blows
into the ISM. A jet that high would also raise questions about the location at which the jet is
decelerated to nonrelativistic velocities, and seems inconsistent with the claimed detection of
IR counterjet emission by Stiavelli et al. (1997). We can therefore rule out the picture of a freely
expanding jet. As a consequence we need a mechanism to provide confinement, i.e., we need
to set the jet pressure in relation to the ambient pressure.
The ambient gas pressure in the center of M87 has been derived by White and Sarazin
(1988) from fitting cooling flow models to the Einstein X-ray observations. The values they find
fall into the range pISM
= 1 10
10 dyn cm 2 to p
ISM
= 4 10
10 dyn cm 2 . It is important
to note that the pressure of the interstellar medium in M87 might not be representative of the
pressure of the immediate environment of the jet. In fact, BB’s analysis of the helical KelvinHelmholtz instability leads to the conclusion that the ambient medium of the jet is significantly
overpressured with respect to the interstellar medium in M87. Note also that the pressure in
the knots might well exceed the ambient pressure without losing confinement, as long as the
pmean [dyn cm-2]
90
90˚ non relativistic equipartition pressure
brightness ratio limit
15˚
20˚
25˚
30˚
10-8
10-9
ISM pressure
ISM
pressure
2
3
4
5
Γjet
6
7
8
Figure 3.10: Total pressure pmean averaged along the jet between 000 :5 and 2200 as a function of
jet . Labels according to Fig. (3.7). For comparison, the dashed-triple-dotted line shows the
equipartition pressure for a non-relativistic jet seen edge on (i.e., LOS = 90Æ ). The hatched
area shows the estimated ISM pressure (White and Sarazin 1988) in M87.
average pressure does not.
We have calculated the average total pressure pmean in the jet from the averaged emissivity and the best–fit B -field for various angles and Lorentz factors, as shown in Figure 3.10.
The assumptions we have made in constructing this plot are analogous to those of SBM, who
(arbitrarily) assumed a lower cutoff at 107 Hz, a high energy cutoff at 1015 Hz, a spectral index
of RO
= 0:5, and equipartition between heavy–particle and electron energy.
the spectrum is steeper than R
(Note: because
= 0:5 above 10 GHz, our estimate of the pressure is likely to be
an overestimate.) We have also assumed isotropic emission in the plasma rest frame by using
an average value of # = 54Æ for the term sin1+ # in the emissivity equation (3.2).
For comparison we have also calculated the equipartition pressure and plotted the ratio of
pressure to equipartition pressure in Figure 3.11. It is obvious that we are far above equipartition
for small values of jet and large LOS , but as we approach the favored range of
and LOS
<
30Æ , ptotal
jet
3
approaches the equipartition value. The exact value of the pressure
91
brightness ratio limit
15˚
20˚
25˚
30˚
〈 p / pequipartition 〉
4
3
2
1
2
3
4
5
Γjet
6
7
8
Figure 3.11: The ratio of the best–fit particle pressure to the respective equipartition value averaged along the jet as a function of jet . Labels according to Fig. (3.7).
depends critically on the details we put into the model spectrum. For a jet composed entirely of
electrons and positrons, the pressure would go down by a factor of 12 , whereas the equipartition
4
pressure would only decrease by a factor of ( 21 ) 7 = 0:67. The lack of information about
the low–frequency spectrum inhibits any statements about the low–energy particle distribution.
However, it is safe to assume that the power law does not continue down to non-relativistic
energies.
It is obvious that a magnetic field far below equipartition alone cannot explain the behavior of the jet — it might account for the spectral changes but it requires the pressure to be much
higher than that of the surrounding medium. Field orientations close to the line of sight will
lead to an underestimate in emissivity, pressure, and intrinsic cutoff frequency and will require
even lower magnetic fields and even higher pressures in order to prevent significant cooling.
This changes as we increase jet : the inferred pressures are close to the value for the interstellar
medium in M87 as derived by White and Sarazin (1988), for jet
3 5 and LOS <
25Æ .
This, combined with the possible overpressure of the jet’s immediate environment relative to
92
the ISM, leads to the conclusion that there is no confinement problem.
For small values of the magnetic field, one might ask if inverse Compton losses become
dominant. A simple order of magnitude estimate shows that this is not the case. The ratio of
synchrotron to inverse Compton loss timescales is equal to the ratio of photon energy density
to magnetic field energy density (Rybicki and Lightman 1979). The magnetic field strength
has been estimated above. To derive an estimate of the photon energy density produced by the
synchrotron emission, we normalize to the radio luminosity corrected for beaming and projection effects and integrate over an RO
= 0:65 radio–to–optical power law that cuts off at 1015
Hz. This shows that the inverse Compton lifetime due to just the synchrotron radiation field of
the jet is roughly an order of magnitude longer than the synchrotron lifetime for the parameter
range we suggested above. The starlight background at the center of M87 also contributes to the
photon energy density. Using an isothermal sphere profile, normalized to the total luminosity of
M87, we arrive at a central photon energy density roughly an order of magnitude smaller than
that of the magnetic field, small enough to justify the assumption of negligible Compton losses.
3.4.2
Comparison with Earlier Models
It is instructive to compare our model to an earlier ad-hoc model by Meisenheimer et al.
(1996, see also Biretta 1993), which bears a lot of similarity to our model. They start from the
same assumption that the spectral changes along the jet can be explained by simple compressions and assume that the cutoff momentum c is almost constant along the jet, parameterizing
it as a function only of the transverse jet radius (measured from the 2 cm radio map): c
/R
1
3
2
— note that for an adiabatic compression of the plasma transverse to the flow c goes as R 3 .
They take the
B -field
to consist predominantly of a toroidal component,
B ,
and hold the
poloidal component Bz fixed. They determine the longitudinal compression ratio of B and of
the particle density
n from their fit to the cutoff frequency c with equation 3.8.
However, in
1
an adiabatic compression, the cutoff momentum varies as n 3 and will therefore be affected by
longitudinal compressions as well (here is where our assumption of a disordered field allows
93
us to determine a relation between density and magnetic field, so we can solve equation 3.8
uniquely for B ). They neglect the fact that the synchrotron emissivity is enhanced in adiabatic
2
compressions by n1+ 3 B 1+ (equation 3.2) rather than n B 1+.
Since the shocks might
well be oblique, their assumptions that B scales as the longitudinal compression ratio and that
Bz is constant might also not be valid.
Meisenheimer et al. (1996) favor an intrinsically onesided, subrelativistic jet, viewed
close to perpendicular (LOS
90Æ ). Knot A would be a head–on shock in this scenario. As we
have mentioned above, for this set of parameters additional acceleration has to be provided to
maintain the optical emission out to large distances from the core. Meisenheimer et al. (1996)
favor an unknown global acceleration process to explain the constancy of the cutoff momentum. With these assumptions, their model yields similar results to ours in that it reproduces the
small scale brightness variations on the basis of the changes in cutoff frequency. Our model
could thus be regarded as an extension of their approach, putting it on the theoretical basis of
adiabatic compressions, with a different mechanism for providing the large scale constancy of
the spectrum.
3.4.3
X-ray Emission
An important result from the analysis above is that cooling longward of UV energies is
not important over the length of the jet — the proper time is reduced by a significant factor and
the magnetic fields are small enough to leave the spectral shape unchanged. This conclusion
begs the question of the origin of the X-ray emission detected by the Einstein Observatory and
ROSAT. Both observations show emission from the core/knot D region and knot A.
3.4.3.1
The Role of Particle Advection:
Ultra-high-energy particles, capable of radiating in the X-ray regime, could be carried
out from knot D, where X-ray emission is observed, to knot A, and reaccelerated in the shock
by the adiabatic compression mechanism discussed above. For this to happen the spectrum
94
would have to break rather than cut off in the optical. The presence of a break instead of a
cutoff would not change our fits, as long as the break is located above the frequency we fitted.
We have calculated the behavior of particles with X-ray emitting energies along the jet and
found that for our best fit
at
B -fields cooling out to knot A will have produced a spectral cutoff
1017 Hz – which is where most of the Einstein HRI’s sensitivity lies. Since the B -fields
we derived are upper limits, a lower field could leave the distribution function unchanged even
at such high energies. Therefore this mechanism of producing X–rays is marginally consistent
with our model. It might also account for at least part of the X–rays. Note that, as suggested
by various authors (e.g., Biretta and Meisenheimer 1993), the X-ray emission could also be of
non–synchrotron origin altogether.
3.4.3.2
Fermi Acceleration at Knot A:
The compression of a factor
3 inferred from the analysis above indicates that Fermi
acceleration might be present in knot A, although the moderate change in emissivity and the
constancy in radio and optical spectral index suggests that it might not be very efficient. It
is possible that Fermi acceleration occurs at parts of the shock only, resulting in a particle
distribution composed of a compressed and a Fermi-accelerated preshock distribution.
In order for Fermi acceleration to take place at all the shock has to be subluminal, i.e.,
the intersection point of a given magnetic field line and the shock front has to move with a speed
smaller than the speed of light, in which case we can find a frame in which the magnetic field
is perpendicular to the shock front. This is the case for fields not too closely aligned with the
shock plane. In the nonrelativistic case the field orientations leading to a superluminal shock are
rare, and subluminal shocks are the rule rather than the exception, so one would expect Fermi
acceleration to take place.
Because in relativistic shocks the percentage of superluminal field orientations rises
sharply with shock , Fermi acceleration should become less important. In this limit, most of the
particle acceleration would occur through the mechanism of “shock drift acceleration”. Begel-
95
man and Kirk (1990) presented a theory of this process valid in the relativistic case. They show
that the adiabatic approximation is still accurate in the limit of p p
<
1, where
p is the up-
stream Lorentz factor in the perpendicular shock frame and p the corresponding velocity. We
have calculated this quantity for various obliquities and field orientations appropriate for knot
A and it seems to fall into the desired range. For a disordered magnetic field we would have
to average the resulting spectrum over all possible field orientations. The more superluminal
the shock the less important would effects of Fermi acceleration be. Relativistic corrections to
shock drift acceleration are only important for high p p , which, again, depends on the field orientation. Averaging over all possible field orientations would probably render these corrections
unimportant.
Fermi acceleration both changes the shape of an incoming power law spectrum and amplifies it. For an incoming electron spectrum of the form
f (p) = A0 p
a in the test particle
limit (i.e., the pressure provided by the accelerated particles is negligible), and a nonrelativistic
shock, the change of the spectrum depends on
s
3r
r 1 , where
r is the compression ratio.
If
s < a the spectral index is changed to s. If s > a the slope remains unchanged but the spectrum
is still amplified by a factor s s a (Kirk 1994). The radio spectral index is plying a 4
0:5 0:65, im-
4:3. To provide a boost in emissivity by a factor of 11, s has to be 5:1 5:5,
implying a compression ratio of 2:2
2:5 (assuming =
2 ). Note that this is very close to
3
the compression ratio one derives for an adiabatic compression. The spectral index produced by
such a shock is 1 1:25, consistent with the observed optical 1:2. Drury et al. (1982)
showed that the produced power law softens as one departs from the test particle limit. Also, the
simple treatment stated in this paragraph breaks down in the case of relativistic shocks, where
the spectral index is no longer a simple function of the compression ratio.
Recent investigations by Ballard and Heavens (1992) have shown that oblique relativistic shocks can produce rather steep spectra, but other results indicate that they might be more
efficient in accelerating particles (i.e., producing flatter spectra) than their nonrelativistic counterparts (Kirk and Heavens 1989). SBM find an optical–to–X-ray spectral index of OX
1:4
96
for the knot A region, which seems to be consistent with low–efficiency acceleration.
If Fermi acceleration were present at the shock and effective enough to change the spectral shape in the optical, it would no longer be feasible to use the data from the whole jet to
determine the magnetic field. Rather, the same analysis could simply be carried out separately
for the pre– and post–knot A regions of the jet. We would then have to make an estimate of
the shock strength based on the known parameters in order to determine the ratio of pre– to
post–shock Lorentz factor . Even in this case we would need Lorentz factors of order jet
3
to solve the cooling problem.
However, based on the observed mild spectral changes, and the assumption that pitch
angle scattering is strong, we conclude that Fermi acceleration, if present, will not be efficient
enough at accelerating electrons to affect the spectrum below the cutoff. Inefficient Fermi acceleration might very well be present in knot A, producing the X-ray emitting particles observed.
Prediction of the produced high–energy spectral index has to wait for more conclusive results on
Fermi acceleration at relativistic oblique shocks. As is well known, shocks are more efficient
at acceleration protons than electrons by a factor corresponding to the mass ratio of the two
particle species. In order to explain efficient electron acceleration, an efficient energy transfer
mechanism from protons to electrons is commonly invoked (Coulomb scattering is insufficient).
Such a leap of faith is unnecessary in M87, as we have shown that no strong particle acceleration
is needed to explain the observations. Slow energy transfer from protons to electrons can then
be helpful in explaining the presence of high energy electrons and optical synchrotron emission
in longer jets (such as 3C15).
3.4.4
Polarization
An important complication to the treatment above is the fact that the magnetic field
will not be completely disorganized — polarization measurements show that in some regions
a homogeneous component is present. In fact, it is possible that cancellations between regions
with homogeneous fields but different orientations occur along the line of sight (Meisenheimer
97
1992). In such a case, the assumption of disorganized fields, leading to
justified. However, since the impact that the parameter
2 , is no longer
3
has on the fit is minor (Fig. 3.9), we
feel this caveat is not very severe and merely mention this complication here.
In addition to the unknown orientation of the jet itself, the field orientation is also unknown. Since synchrotron emission depends on the magnetic field orientation to the line of sight
# as sin1+ #, it can be strongly peaked away from the field direction. The cutoff frequency also
depends on sin #. Because the cutoff frequency is determined by sampling all regions along the
line of sight, it is not obvious which value to choose for #. Fortunately, in a domain of disordered field the regions with the field oriented close to perpendicular will contribute most of the
flux, thus the error we make by setting sin # to 1 will not be too large. Note that we used a value
of # = 53Æ in calculating the pressure, the appropriate average of sin1:5 # over 4 steradian.
Knot A shows a polarization of order 35% and a field orientation close to the shock plane.
Neglecting relativistic effects, the small amount of upstream polarization, and the fact that shear
might reduce the compression of the field, we can use the approximate formula (Hughes and
Miller 1995)
where
+ 1 (1 r 2 ) cos2 ;
+ 35 2 (1 r 2) cos2 is the fractional polarization, r is the proper compression ratio,
(3.9)
and
is the line of
sight angle from the shock plane, to obtain a lower limit on the compression ratio in knot A of
r 2, valid for the inferred range of viewing angles with respect to the shock plane (BB). This
is consistent with the compression ratios inferred above. Equation (3.9) also provides an upper
limit of 35Æ on ; however, the shock is assumed to be oblique, so we cannot use this result to
constrain LOS .
3.4.5
Particle Acceleration Radius
Having an estimate of both magnetic field and cutoff momentum at the injection radius
r0 , we can now try to determine where the actual particle acceleration has to occur. We assume
98
some radial dependence for the magnetic field in the inner portion of the jet (i.e., smaller than
0:500 ), for example a power law: B
/r
. Furthermore we make the simplifying assumption
of a constant jet . By also taking B to be proportional to % , which determines the radial dependence of %, we can invert equation (3.6) and solve for the radius at which the cutoff momentum
approaches infinity, in other words, the minimum radius inside of which acceleration has to
occur:
racc = r0 1 +
1
A r0
4p0 B02 e4 9 jet c7 m4e , and
where
A=
A r0
! 1.
(3.10)
1
(2+ 31 ) 1 . The acceleration radius racc approaches zero if
It is obvious that the estimate of racc depends critically on the value of . In
Figure 3.12 we have plotted racc as a function of for LOS
= 25Æ and
jet
= 3. In the same
figure we have also plotted racc for the case in which B no longer scales like % — in this case
we have taken % / r 2 and used the same values for LOS and jet (dashed line).
If the jet expands at constant opening angle and with uniform jet , the decline of B with
radius should correspond to 2, since the density scales like r
2 and dissipative effects will
probably limit the rate of decline. Adopting an upper limit on the magnetic field at 0:01 pc of
B 0:1 G (Reynolds et al. 1996b) limits to values smaller than 1. For
=
2 , this constrains the acceleration radius to be r
acc
3
jet
= 3, = 1, and
10 pc, or 000 :06. On the other hand,
the radius of acceleration cannot lie inside the Schwarzschild radius of the central black hole,
which is of order 10 4
pc for a 109 M black hole.
The dependence of racc on , , and the core magnetic field is too strong to make any
detailed predictions about where the acceleration actually has to occur. However, the racc –curve
is rather flat throughout most of the possible range for , which suggests that the most plausible
value for racc falls between 1 and 10 pc. This is intriguingly far away from the central engine.
99
102
injection radius
racc [pc]
10
1
10-1
10-2
10-3
10-4
0.2
Limited by Schwarzschild radius
0.4
0.6
0.8
σ
1.0
1.2
Figure 3.12: The minimum acceleration radius racc , as a function of (B / r ) for jet = 3
and LOS = 25Æ in the case of B / % (solid line). The dashed line shows racc for the case
of % / r 2 and the same values of jet and LOS . The hatched area indicates the limit set by
10 Schwarzschild radii for the 109 M central black hole. The dashed-dotted line shows the
(arbitrary) injection radius r0 = 000 :5 or 80 pc.
3.5
Chapter Summary
We used the example of the M87 jet to demonstrate how broad band observations can
be used to improve estimates of the physical conditions inside AGN jets. We showed that
equipartition is not necessarily a good assumption in these jets. We also showed that the detailed
radio surface brightness distribution and the spatial variation in the optical spectral index can
both be explained by the same process: adiabatic compression of the jet plasma in relatively
weak shocks. Fermi acceleration is not necessary (though not excluded) to explain the lack of
cooling observed in the jet if we allow the jet to be moderately relativistic and the magnetic
field to be below equipartition by a factor of a few. The external pressure required to confine
the jet turns out to be comparable to what is inferred from X-ray measurements of the cluster
environment.
Chapter 4
Evolutionary Signatures of Radio Galaxies
The previous chapter demonstrated how, given a few simple assumptions, we can measure the field strength in the resolved portion of a radio jet. There is a second crucial parameter
we need to know in order to pin down the conditions within the jet: the jet power L. A relatively
robust method is in place to estimate this quantity. Once again, though, this method relies on
estimate of the magnetic field with the usual assumption of equipartition, and it is not clear
how reliable it really is. In this chapter, we will propose a different method to complement
measurements based solely on the total radio power in the lobes.
Another motivation for the work in this chapter is that the evolution of radio galaxies
has not yet been fully understood. Since radio galaxies pump large amounts of energy into
the surrounding medium, we need to understand how the cluster properties change with time if
they host a radio galaxy. This is important for cosmological arguments based on cluster masses
through X-ray measurement and on the SZ-effect, but it is also crucial for our understanding of
AGNs, since they must constantly be fed by the ambient material, and any disturbance of that
medium will ultimately lead to a change in that feeding behavior. In other words, AGNs could
be subject to strong feedback. The method discussed below could also be used to search for
very young radio galaxies that are heavily obscured at long wavelengths by free-free absorption
and would thus be hard to find in radio surveys, through X-ray observations. In this chapter we
will discuss the X-ray signatures of expanding powerful radio sources. The following work has
been published in Heinz, Reynolds, and Begelman (1998).
101
4.1
The Expansion of a Radio Galaxy: Connection Between CSOs and FRIIs
In recent years there has been some progress in understanding how various classes of
powerful extragalactic radio sources can be described in the context of an evolutionary picture.
Recent radio surveys have identified classes of powerful sources which are morphologically
similar to FR II radio galaxies but appreciably smaller. Sources less than 500 pc in extent have
been termed Compact Symmetric Objects (CSOs; Wilkinson, Polatidis, Readhead, Xu, and
Pearson 1994), whereas those in the size range 0.5–15 kpc are often referred to as Medium
Symmetric Objects (MSOs; Fanti, Fanti, Dallacasa, Schlizzi, Spencer, and Stanghellini 1995).
These classes of small sources, which form approximately one quarter of current flux–limited
radio surveys, are thought to correspond to the early stages of full-sized FR-II radio galaxies
(Begelman 1996; Readhead, Taylor, and Pearson 1996). Given this age template, we can attempt
to build a self-consistent model of radio galaxy evolution.
Central to our understanding of these sources is the following theoretical picture (first
proposed by Scheuer 1974). Relativistic plasma flows from the central AGN in the form of
collimated jets, passes through terminal shocks corresponding to the radio hot-spots, and inflates
a ‘cocoon’ which envelops the whole source. This cocoon becomes highly overpressured with
respect to the surrounding interstellar/intracluster medium and, hence, drives a strong shock
into this material. The swept-up material forms a dense shell separated from the cocoon by a
contact discontinuity (see, e.g., Begelman and Cioffi 1989). In the late stages of evolution, the
expansion of the cocoon/shell becomes subsonic and the cocoon disrupts and mixes with the
ambient medium. A cartoon of the general morphology of an intact source is shown in Fig.1.4.
Although low-frequency radio observations do reveal well-defined synchrotron emitting
cocoons (e.g., Cygnus-A; Carilli, Perley, and Harris 1994), there is relatively little direct observational evidence for the shocked shell. In principle, there are at least two methods of detecting
the shell. Firstly, one can search for the optical line emission that is excited near the shock
front (see, e.g., BB for an explanation of the H line emission in M87). Such line emission
102
is very sensitive to unknown parameters such as the fraction of cold material in the surrounding ISM/ICM and the ionization state of that material. Secondly, one can search for the X-ray
emission from the shocked ISM/ICM and the associated cavity in the ambient material: Since
the diffuse X-ray emission in clusters comes from the hot IGM, and since the cocoon material
is dominated by relativistic plasma, which does not emit any significant amount of X-rays, the
displacement of IGM by cocoon material will necessarily lead to a depression in X-ray flux.
Cavities in the ICM have been observed in Cygnus A (Carilli et al. 1994), Hydra A (McNamara
et al. 2000), Abell 4059 (for which we have been granted observing time on Chandra), and
Perseus A (Böhringer et al. 1993). We argue below that at least in the case of Perseus A not
only the cavity but also the shocked shell can be seen.
Once we have confirmed that our model is a valid (be it crude) description of radio galaxy
morphology and evolution, we can then apply this it to existing X-ray data and make predictions
for future observations, specifically with Chandra.
4.2
Description of the Model
4.2.1
The Dynamical Model
We first outline the basic assumptions used in our model to find a simple, robust descrip-
tion of the early stages of radio galaxy evolution into a surrounding hot medium. Our model is
based on the analysis by Reynolds and Begelman (1997). Following this work, we make several
simplifying assumptions:
(1) Spherical symmetry. For the purpose of this chapter it is sufficient to neglect the prolate
structure observed in most radio sources, since more detailed hydrodynamics would be
required in order to determine the shape of the cocoon beyond a self–similar form (see,
e.g., Clarke, Harris, and Carilli 1997). The level of detail required in such simulations and the amount of computing power necessary to explore parameter space in the
desired manner would defeat the scope of this chapter. The observed elongations are
103
moderate (axial ratios of order 3 in FR II sources, see Carilli et al. 1994). The dependence of our results on the source radius is relatively weak, which makes us confident
that the application of our model to non-spherical sources will introduce minor errors
only.1
(2) Purely relativistic gas inside the cocoon (i.e., the adiabatic index in the cocoon is c
4 ) and non-relativistic gas in the swept up shell (
s
3
=
=
5
3 ). The latter assumption is
valid in all but the early stages of the most luminous sources, in which the electrons
become relativistic.
(3) Uniform pressure. We take the pressure in cocoon and shell to be uniform and equal.
This is a reasonable approximation in the context of the previous assumption, as the
p
sound speed inside the cocoon (c=
3) will be significantly higher than the expansion
velocity of the shell. The radio hot spots will be overpressured, but we will neglect this
complication in the following.
(4) A King-model X–ray atmosphere provided by either the host galaxy or the cluster in
which the AGN is embedded. The density profile thus behaves as:
"
r 2
(r) = 0 1 +
rc
#
3
2
;
(4.1)
where 0 is the central density and rc is the core radius.
can take any positive value;
however, for the interface between cocoon and shell to be stable against Rayleigh–
Taylor instability we need to assume
<
2 (see below).
3
is observationally deter-
mined by the ratio of the velocity dispersion of the cluster galaxies to the temperature
of the cluster gas.
(5) Non-radiative shocks. We neglect energy loss due to radiative losses in the equations
below. This is justified as long as the cooling time is long compared to the source
1
A self-similar, non-spherical model has been advocated recently by Kaiser and Alexander (1999), however, the
model is not self-consistent in that it assumes uniform interior pressure yet non-spherical expansion.
104
lifetime, a condition satisfied in the parameter range we consider.
The system is well defined by energy conservation within the cocoon and the shell:
c
1
Vc p_ + c V_c p = L(t)
1
(4.2)
and
s
1
4
Vs p_ + s V_s p = rs2 (rs )r_s 3 ;
1
2
(4.3)
and the ram pressure condition at the shock:
pc;s (t) = (rs )r_s 2 :
(4.4)
Here, Vs and Vc are the shell and cocoon volumes, respectively, rs is the shock radius, and
pc;s = p(t) is the (uniform) interior pressure, which is a function of time. L(t) is the kinetic
luminosity of the jets feeding the cocoon. A dot indicates a time derivative, i.e.,
p_ = dp=dt.
Equation (4.4) holds only in the case of supersonic expansion, a condition well satisfied in the
early evolutionary stages of our models but which is violated as sources pass a characteristic
size. Once a source has decelerated below the ambient sound speed (typically of the order of
csound <
1000 km sec
1 ) the evolution will resemble an expansion wave rather than a shock
wave. The shell will thin out and eventually blend into the ambient medium; the cocoon–
shell interface will become unstable and collapse on timescales of order the free fall time. The
kinetic luminosity
L(t)
in equation (4.2) can in general be time dependent to allow for the
intermittency suggested by Reynolds and Begelman (1997, see x4.5.1.1). For now, we will take
it to be constant. In a sense, this can be interpreted as a time averaged luminosity LhL(t)i.
To explore parameter space we have integrated equations (4.1) to (4.4) numerically over
a time span of 108 years, assuming an initially small source. Our models were calculated over
a grid of input parameters L and rc . We used the following parameter values:
Luminosities ranging from
of
L = 1046 ergs sec
L = 1042 to 1052 ergs sec
1 . We use a fiducial value
1 throughout the chapter except where indicated. Note that
105
luminosities in excess of L 1048 ergs sec 1 can be considered unphysical, since they
correspond to Eddington luminosities for black hole masses >
1010 M . However, as
L=0 is relevant to the dynamics, so we
chose to hold 0 fixed and explore a wide range of L. Unphysically high values of L
will be shown below, only the combination
can be interpreted as relevant to sources in low density environments.
Core radii in the range of 50 pc rc
500 kpc, with a fiducial value of 500 pc, typical
of elliptical galaxies.
We set
1 throughout the rest of the chapter, corresponding to
2
/r
1:5 for
r rc .
We fixed the central density to be 0
For power-law density distributions
= 1:7 10
/r
25 g cm 3 or n
e;0
= 0:1 cm
3.
and constant, non–zero kinetic luminosity,
a self-similar solution to the equations is possible. This solution is a good indicator of how the
more general solution scales with the input quantities L, 0 , and rc . A necessary condition for
a self-similar solution is that the cocoon radius rc be a fixed fraction
of the shell radius, i.e.,
rc rs . Under these assumptions, equations (4.2) to (4.4) yield
1
9s 2 + 1 3
=
18s 2 8
rs (t) = r0
t 5
t0
3
(4.5)
;
(4.6)
where r0 is the shell radius at time t0 , which is defined by
!1
0 r0 5 3
t0 = C1 L
(4.7)
with
C1 "
363
(5 )3 (c
1)
3c
4+
3
# 31
(4.8)
It follows from equation (4.4) and the assumed pressure equilibrium between cocoon and
shell that the shell temperature (assuming an ideal, non–relativistic gas) is proportional to the
106
square of the expansion velocity, r_s 2 . Thus, for a given
r the temperature in the self-similar
solution goes as
T
/ r_s2 / 1=t0 2 / (L=0 )2=3 :
Equation (4.6) reveals that solutions for
(4.9)
2 (i.e., 2 in the limit
3
r
rc ) are
Rayleigh–Taylor unstable, since for those values the cocoon–shell interface is always accelerated and the shell is very dense compared to the cocoon gas.
We can see from the basic set of equations that the only two parameters entering the
solution are a radial scale factor (rc in the King-profile case, r0 in the self-similar case) and
L=0 — this statement holds even for arbitrary density profiles. While rc and 0 can in principle
be determined by direct observation, L can only be inferred theoretically from observed radio
brightnesses. However, this conversion is not trivial and it would be very useful to constrain the
kinetic luminosity directly.
4.2.2
Calculation of the X–Ray Brightness
Because equations (4.1) to (4.4) do not specify the density and temperature structure
inside the shell itself, we have to make additional assumptions about the radial dependences of
and T . In this chapter we take both variables to be uniform within the shell and use the ideal
gas law and mass conservation of the swept up material to convert from the pressure given by
equation (4.4) to the temperature Tshell . The shock jump conditions dictate the values of shock
and Tshock immediately behind the shock. In the case of a strong shock, the density jumps to
shock
4preshock , and the temperature jumps to kTshock 2 (ss+1)1 2 r_s2 where is the
molecular weight. Comparing shock and Tshock to the average shell values reveals that the shell
is overdense in most cases, i.e., on average the shell is colder than the most recently shocked
material, as is expected due to adiabatic expansion. The density ratio is
6 over most of the
parameter range, close enough to the strong shock jump value of 4 to assume near uniformity
of within the shell.
107
We calculate the specific X–ray emissivity due to thermal bremsstrahlung, taking the
material to be composed of fully ionized hydrogen only (Rybicki and Lightman 1979):
25 e6 2 1=2
dW
=
T
dV dt d 3mc3 3km
1=2 n2 e h=kT g :
e
(4.10)
Since we are only interested in supersonic cases, we will have to discard parameter values for
which the temperature drops below the ambient temperature Tambient
1 10 keV at a given
radius. Because all high resolution imaging X-ray facilities available in the near future have
bandwidths not far above this range, we can assume kT
> h 13:6 eV, thus the Gaunt factor
is given by the small angle uncertainty principle approximation:
p
3
4 kT
g =
ln
;
h
where (4.11)
= 1:78 is Gauss’s number.
We also use equations (4.10) and (4.11) to compute the thermal bremsstrahlung emission
from the hot galaxy/cluster gas, which we assume to be isothermal throughout the rest of this
chapter. Clearly, the existence of a massive cold component to the ISM/ICM (as expected if a
cooling flow operates) will alter the properties of the solution, in particular the emissivities, as
relatively cold, dense material will radiate more efficiently. This aspect will be commented on
in x4.5.1.2.
4.2.3
Observational Diagnostics
Figure 4.1 shows the surface brightness results (i.e., the emissivity integrated along the
line of sight) of integrating the model with our fiducial parameters. Shown are radial profiles at
different times as indicated in the figure. Three basic features are identifiable from the figure,
indicated by shadowed regions on the bottom of the plot:
a) The flat part inside the shell, steepening into the bright shell. This component includes
all lines of sight penetrating the cocoon, i.e., r
< rc .
b) The shell. We define this part as all lines of sight outside the cocoon but still penetrating
the outer shell, i.e., rc
< r < rs .
108
c) The undisturbed cluster emission, i.e., r
> rs . This part simply tracks the King-profile
atmosphere. In our case, since we used an index of
=
1 , the surface brightness in
2
the power law part goes as r 2 and flattens into the core.
We identify the following readily–measurable diagnostics which will be used in the rest of this
chapter to investigate source parameters:
i) shell-cluster ratio: the ratio of the surface brightness at the line of sight tangential to
the cocoon (at rc ) to the surface brightness at the line of sight tangential to the shell (at
rs ). The emission from the shell (without any contribution from the X–ray atmosphere)
is brightest along the former line of sight. The latter is the brightest line of sight outside
the region of shell emission. A high contrast is important for the detectability of the
L=0 , we can see that brightness ratios are also going to depend on L and 0 only in the combination L=0 , since
source. Since the dynamical solution depends only on
the density normalization 0 cancels from equation (4.10). For a purely self-similar
solution, we can thus expect this ratio to be proportional to T 1=2
/ (L=0 )
1=3 [see
equation (4.9)].
ii) center-cluster ratio: the ratio of the surface brightness at the central line of sight to
the surface brightness at the line of sight tangential to the shell. This ratio indicates if
the central lines of sight are brightness enhanced or depressed compared to the cluster
emission, i.e., if there is an ‘X–ray hole’. Again, since it is a brightness ratio, the
center-cluster ratio should depend only on
it should go as T 1=2
/ (L=0 )
L=0 and rc, and for the self-similar case
1=3 . The presence of a strong point-like AGN X-ray
component will swamp the cluster emission at the very center. Because the brightness
profile is very flat at central lines of sight (as can be seen from Fig. 4.1), we can avoid
the contamination by taking an off-center value for the central surface brightness and
will only make a small error.
109
0.15
0.61
2.33
8.76
nomenclature
brightness [arbitrary units]
0.03
rc
rs
a
0.1
b
1.0
c
10.0
r [kpc]
Figure 4.1: Source evolution, seen through a flat 1 5 keV bandpass for our fiducial parameters
(see x4.2). The x-axis is in units of kpc, the y-axis in arbitrary flux units. The different curves
correspond to radial profiles at different times as labeled in the figure (in units of 106 years).
The thick grey curve corresponds to t = 0, i.e., the undisturbed cluster profile.
iii) shell count rate : the integrated count rate from all lines of sight penetrating the shell
(in other words: all of areas a and b in Fig. 4.1), including back– and foreground emission from the X–ray atmosphere. This quantity is easier to determine than the background subtracted emission from just the shell. Notice, however, that since it depends
on absolute normalization, both the distance to the object
d and the density normal-
ization 0 factor into the shell count rate , thus we cannot express it as a function of
L=0 and rc only, rather, a factor of (0 =d)2 remains.
The particular values of these diagnostics for a given source will depend on the assumed
instrumental response, as the cluster and shell have different temperatures and thus different
110
spectra. In the following we will use both the ROSAT HRI band and the Chandra ACIS-S band
as indicated.
4.3
Applications to Existing Data
4.3.1
Perseus A
Figure 4.2 shows a 50 ksec ROSAT HRI exposure of Perseus A2 . Per A is a radio galaxy
with an estimated kinetic power of
>
1043 ergs sec
1 (Pedlar et al. 1990), at a redshift of
0:02 or 80 Mpc h75 1. It is located in a dense cluster environment with core densities
of n0 0:02 0:1 cm 3 , a core radius of rc >
50 kpc, and a temperature of 7 keV (see
z
White and Sarazin 1988). The elliptical shell structure is readily visible from the plot and has
been the subject of a paper by Böhringer, Voges, Fabian, Edge, and Neumann (1993). The shell
semi–minor and semi–major axes are approximately
to
12 and 17 kpc for a Hubble constant
of
3000 and 4500 , respectively, corresponding
H = 75 km sec 1 Mpc
1 . Because the shell is
so well–defined over a significant angle, we are confident that the source is still in supersonic
expansion or has only recently crossed the sound barrier. It is also obvious from the image that
our assumption of a spherically symmetric, stationary cluster medium is idealized — the bright
feature to the east indicates that “cluster weather” has probably had a significant impact on the
appearance of the structures. However, the brightness changes are moderate and, keeping in
mind those caveats, we feel justified in applying our model to this source with some caution.
We have computed our model for a grid of various rc and L. The cluster gas was assumed
to have a temperature of kT
7 keV and a central density of n0 = 0:1 cm
3 . The integration
was stopped at a size of 16 kpc, the approximate size of the source. We computed the three
diagnostics described in x4.2.3 assuming the ROSAT passband. We also calculated the region in
rc –L space for which the source is still supersonic.
L=n0 must exceed 5 1046 ergs cm3 sec
2
We find that for a core radius rc
1 , and for a density of n
0
0:02 cm
These data were obtained from the LEGACY public archive situated at GSFC (NASA).
>
50 kpc,
3 , the mean
111
Declination (2000)
ile
3
e2
of
pr
of
il
pr
30″
profile 4
32′ 00″
31′ 00″
30″
30′ 00″
e5
il
of
pr
profile 1
41° 29′ 30″
55s
50s
03h 19m 45s
Right Ascension (2000)
Figure 4.2: A 50 ksec ROSAT HRI exposure of Per A, smoothed with a 2 arcsec Gaussian beam.
We have chosen the contrast to emphasize the shell structure. These data were downloaded from
the LEGACY public archive at GSFC (NASA). The black lines show the paths along which we
chose to take brightness profiles for our diagnostics.
kinetic luminosity must be L > 1045 ergs sec 1 to satisfy the supersonic condition. The shaded
region in Fig. 4.4 shows the region in parameter space which is forbidden if we insist that the
source be supersonic.
To compare these models to the data for Perseus A, we took radial brightness profiles
at several selected locations (shown in Fig. 4.2 and Fig. 4.3). We decided to hand-pick these
112
profile 1
2.00
1.35
1.00
profile 2
1.6
count rate [arbitrary units]
1.1
1.0
profile 3
1.30
1.12
1.00
profile 4
1.35
1.00
profile 5
1.9
1.4
1.0
Figure 4.3: Brightness profiles 1 through 5 according to Fig. 4.2 in arbitrary units.
locations rather than assign them randomly due to the complications caused by the “cluster
weather” we pointed out earlier. The enhancement at the shell compared to the brightness just
outside (i.e., the shell-cluster ratio) is roughly3 a factor of
1:3
2.
The region correspond-
ing to these values in parameter space is shown in Fig. 4.4 as a vertically hatched area with a
dashed border. Approximating the core–subtracted central surface brightness by the brightness
3
The HRI resolution is of order 4 arcsec, which is roughly the expected width of the shell at a radius of 40 arcsec
(see equation 4.5), thus we expect the shell to be just marginally resolved. The brightness ratios we extract from the
image will thus be lower limits. The core is the dominant feature and will have to be removed.
113
minimum between core and shell, we find a brightness ratio of the interior against the cluster
immediately outside the shell (i.e., the center-cluster ratio) of 1
1:3.
The region in rc – L0
space compatible with this condition is also shown as a hatched area (horizontal lines) with a
dotted border in Fig. 4.4. We also estimated the total count rate for the shell area (i.e., the shell
count rate from x4.2.3) to be 0:64 s
1 from taking an elliptical ring aperture. To display
the predicted total count rate in the same plot we will have to remove a factor of (0 =d)2 from
this value (see section 4.2.3). Using the assumed density from White and Sarazin (1988) of
0 0:02 0:1 cm
3 and d = 80 Mpc, we can calculate the corresponding diagonally hatched
area in Fig. 4.4.
White and Sarazin (1988) provide an estimate for the core radius of rc
10 kpc, whereas
White, Jones, and Forman (1997) chose a core radius of 150 kpc. These values for rc are shown
as two thick, dash–dotted lines in Fig. 4.4.4 Together with the other constraints in Fig. 4.4, these
estimates allow solutions in a range of L46 =n0:1 from 5 10
1 to 5
102 . The lower end
of this range can be ruled out by the requirement of supersonic expansion. Keeping in mind the
rather large uncertainty introduced by our measurements and in the input parameters we used
we find that the different areas in Fig. 4.4 match up in a self–consistent fashion.
The fact that the average kinetic luminosity is L
> few 1045 ergs sec
1 is in itself a
very interesting result, as it suggests that the total power output of Per A is significantly higher
than the simple estimates based on the equipartition energy content of the cocoons (Pedlar et al.
1990). A possible conclusion might be that equipartition is not a good approximation in this
case (the particle pressure will most likely exceed the magnetic pressure). It is interesting to note
that a kinetic power estimate based on the conversion factors by Bicknell, Dopita, and O’Dea
(1997) suggests that the instantaneous kinetic luminosity is
L<
1044 ergs sec
1 . Hence this
may be evidence that Perseus A is in a relatively quiescent state (maybe corresponding to the
“off” state of Reynolds and Begelman 1997).
4
Note that, even though there is more than an order of magnitude discrepancy between those values, the models
constraints are virtually the same for both values. Formally, the larger value provides a better fit.
114
rc[kpc]
100.0
c)
10.0
1.0
b)
a)
0.1
10-4
10-2
100
102
L46/n0.1
104
106
Figure 4.4: Model contours in parameter space for Perseus A, assuming isothermal cluster gas
at 7 keV and a shell size of 16 kpc, seen through the ROSAT HRI response. The three hatched
regions correspond to the observational diagnostics described in x4.2.3: a) the observed shellcluster ratio of 2 (hatched vertically), b) the observed center-cluster ratio of 1:1 (hatched
horizontally), and c) the limits set by the observed shell count rate of >
0:64 counts sec 1 and
3
the estimated central density of n0 <
0:1 cm (White and Sarazin 1988, , hatched diagonally).
Keep in mind that the predicted count rates for a given L=0 and rc still scale with an additional
n0 2 , thus the count rate only allows us to set limits in this plot. The grey area shows the subsonic
region in parameter space. The two thick dash–dotted lines correspond to the estimated core
radius of Per A (White et al. 1997; White and Sarazin 1988).
It should be noted here that an as of this date unpublished Chandra observation of Perseus
A exists (Fabian, private communication), that seems to indicate that the shocked shell temperature is in fact below the ambient temperature. This result, if it holds up under scrutiny, is clearly
inconsistent with our model, as the shocked shell must always be hotter than the un-shocked environment. An obvious explanation for this discrepancy would be that the external medium is in
115
fact non-homogeneous, containing clouds of cold material, which, upon being shocked, would
radiate very efficiently, thus lowering the shock temperature. Efforts are underway to investigate
the effects of a two-phase IGM on the observational effects discussed in this chapter.
4.3.2
Other Examples
Another well known source which exhibits an X–ray structure similar to the one we pro-
pose here is Cygnus A. This source has been object of an extensive study by Clarke et al. (1997),
who ran a 3–dimensional simulation of a jet advancing into a surrounding cluster medium. Even
though there are cavities and brightness excesses visible at the center and the rim of the cocoon,
respectively, the structure is not nearly as reminiscent of a shocked shell as the one in Per A.
It is more than twice as distant as Perseus A and also exhibits a much stronger cooling flow,
both of which will tend to make the analysis we suggest harder. We therefore decided not to
apply our method to this source, although this would be an excellent target for application of
our model once future high resolution data are available.
There are a number of other sources for which suspicious holes in the X–ray morphology
have been detected, for example in Abell 4059 and Hydra A (McNamara et al. 2000). Hydra A
is intrinsically less powerful and spherical symmetry is clearly broken in this object, possibly
due to the source being subsonic. More detailed modelling of such sources, with the help of
numerical simulations, is necessary to understand this source. We have been granted Chandra
time to observe Abell 4059 scheduled for late 2000.
4.4
4.4.1
Predictions for (Chandra) Observations
Detectability of Young Sources
Young sources are generally hard to detect in all wavelength bands, because they are
small and evolve quickly, so we expect the local density of young sources to be very small,
whereas distant sources will be faint and unresolved. They will be particularly hard to detect if
116
they are in an “off-state” (in the sense of Reynolds and Begelman 1997), in which case radio
surveys will most likely select against them, since the radio hot spots fade away quickly and
the spectrum becomes very steep. In such cases, optical observations of shock-excited H
emission, and the X–ray emission that we are investigating here could provide ways of finding
such candidates. The main issue with these sources is detectability.
The best spatial resolution available in X–ray imagers today is 0.5 arcsec (Chandra HRC)
to 1 arcsec (Chandra ACIS, taking account of proper oversampling). Thus, in order to resolve a
source, it has to subtend more than 2 arcsec on the sky (this assumes that the core, which is by
far the dominant feature, will not contaminate more than the central resolution element). But
even if a source can be resolved, we shall show that extremely long observing times will be
necessary to achieve significant signal–to–noise: even though the surface brightness is higher
for small sources, the cluster brightness also rises toward the center.
We have calculated the three observational diagnostics from x4.2.3 for a source of size
500 pc over a grid of 2100 parameter combinations of rc between 50 pc and 500 kpc and
L
between 1042 to 1052 ergs sec 1 , assuming the Chandra ACIS-S passband. The result is shown
in Fig. 4.5.
The shell temperature is generally higher for small sources. Fig. 4.5a shows the temperature as a function of
L=0 and rc for a shell radius of 500 pc. The hatched region shows the
parameter values for which a source of this size at the fiducial cluster temperature of 4 keV has
become subsonic. Even at such a young age, we see that low power sources or sources in a very
dense environment have already stalled. For higher luminosities the solution is well behaved
and basically follows the scaling relations (4.6) - (4.8). For very high luminosities/low densities the temperature formally reaches values in excess of 100 keV, beyond which our model is
certainly not valid, as we assumed the shell gas to be strictly non–relativistic.
Figure 4.5b shows the calculated shell count rate for a source at a distance of 100 Mpc,
3 2
. Since
normalized to a central density of n0 = 0:1 cm 3 , so what is plotted is 0:1 ncm
0
increases with n20 , sources in dense cluster environments will be much easier to detect. The
117
figure shows that for big and dense clusters we might hope to detect a sufficient number of the
photons from the shell.
In addition to a high count rate a second issue for detectability is the contrast of the shell
structure (i.e., the shell-cluster ratio). Fig. 4.5c shows that the shell-cluster ratio is significant
over a large range of luminosities and core radii. However, for sources still within the core of
the cluster profile the ratio drops and approaches one (i.e. undetectability), since in such cases
the line of sight passes through a deep fore- and background screen of cluster gas, which tends
to swamp out the shell emission. Thus we have a detection dilemma: For sources with large
core radii, the count rates are high but the contrast is small. For sources which have broken
out of the flat part of the cluster gas, the contrast is high, but the count rates drop. There is an
optimal range in core radius and luminosity, which is the larger, the higher the density of the
host environment is.
Figure 4.5d shows the center-cluster ratio to indicate whether we see the cocoon as an
X–ray hole or whether it shows a brightness enhancement relative to the cluster gas. In the
presence of a bright core, it will presumably be easier to detect a source which shows a central
brightness depression against the cluster, since in such a case we will see a dark ring between
core and cluster.
We conclude that the redshift constraints and the fact that bright cores will tend to render
the shells of barely resolved sources invisible (i.e., the core will swamp out the shell) poses a
serious problem for the detectability of young sources. Generally, though, it should be possible
to detect close sources in sufficiently dense environments. The detectability constraints have
been compiled in Fig. 4.6. This figure has been constructed as follows. For a given set of
source parameters and distance, we calculate the count rate in the ‘observed’ shell (i.e. the
annulus between rc and rs ),
Cshell .
We also compute the count rate in an annulus of the same
area lying just outside the observed shell,
Ccluster .
The shell is deemed to have been detected
if the brightness profile is seen to possess a jump at rs . Thus, we calculate the exposure time
needed to measure
Æ = Cshell
Ccluster to within 30 per cent (i.e.
the exposure time required
118
to demonstrate the non-zero value of delta to 3 sigma confidence). We have used L46 =n0:1
and rc
= 500 pc and rc = 5 kpc.
=1
In terms of detectability this is a more meaningful quantity
than the actual brightness ratio of the two rings, since a source is only detectable if there is a
visible jump in brightness across the shell.
4.4.2
Extended Sources
As was explicitly shown in x4.3.1 for Perseus A, application of our observational diag-
nostics to a well resolved source can place constraints on the source parameters, especially if
the core radius of the cluster is known.
We have computed our three observational diagnostics from x4.2.3 for a much more
extended source of size 16 kpc, as shown in Fig. 4.7. At this size, only powerful sources will
have maintained a supersonic coasting speed, and we can expect the transition between super–
and subsonic sources to be a strong indicator of the overall source power, since the break in the
data happens at a well defined value of
Tcluster .
L=0 for given core radius rc and cluster temperature
The temperature shows the dependence on core radius and power we expect from the
scaling relations in equations (4.6)-(4.8).
Because large sources (by definition) subtend a larger solid angle, their count rates can
be much higher than those for small sources, even though the peak shell surface brightness
decreases. Since we include the fore– and background cluster emission in the shell count rate
(shown in Fig. 4.7b), this effect becomes even more pronounced. Also, since the shell will likely
be resolved and separated from the core, the recognition of the structure itself and the reduction
of shell parameters will be simpler than for small sources. The strong dependence of the shell
count rate on the core radius for rc
<
16 kpc is introduced because we fixed the central density,
thus the mass enclosed in a sphere radius rs (i.e., the swept up mass) increases with the core
radius, and so does the shell density. For rc
>
16 kpc the dependence is not as strong.
In this
case, it is produced by the fact that we integrate over a longer line of sight of undisturbed cluster
gas within the core for larger rc . As in the case of Perseus A, we can see that the total count
119
rates are in a comfortable regime to achieve good statistics over a wide parameter range for a
cluster with our fiducial density.
If the core radius can be measured by other means (e.g., from optical or X-ray data) and
if the redshift of the source is known, the brightness ratios in Fig. 4.7c and 4.7d can be used
to constrain
L=0 to no more than two possible values in the L=0 –rc –plane, independently
from the total count rate (if the supersonic condition is applicable we can fully constrain this
parameter in many cases). In this case, the total count rate can be used to determine 0 and L
separately, and thus measure the age of the source. Density, age, and in particular the kinetic
luminosity will be useful input into jet models.
With Chandra ACIS-S it is feasible to obtain imaging spectra from resolved sources.
In the case of ACIS with a spatial resolution of about 1 arcsec, the source has to be larger
than 2 arcsec in radius, thus the redshift restriction will be
z < h75 =8.
interested in the bremsstrahlung emissivity goes as T 1=2 e h=kT
shell temperature is mainly dependent on
In the regime we are
ln [kT=(h )], and since the
L=0 , spectra are very useful to separate out this
parameter. A color–color–image would emphasize regions of different temperature (i.e., the
shell versus the undisturbed X-ray atmosphere). For high enough count rates it will even be
possible to obtain information about the foreground absorption. Spectra can also give a handle
on the composition of the cluster gas from spectral line analysis.
4.5
4.5.1
Discussion
Complications
The model we have discussed in the previous sections offers a simple way of probing
the radio source structure and determining important parameters. However, since it is a gross
oversimplification of reality, we have to discuss several complications.
120
4.5.1.1
Intermittency
As recently proposed by Reynolds and Begelman (1997), a large population of sources
might be intermittent (i.e., strongly time variable in their kinetic energy output), thereby explaining the observed size distribution of CSOs and MSOs, which shows a previously not understood
flattening at small sizes. This could be due to the fact that sources spend a significant fraction of
their lifetime in a quiescent state, in which the jets are turned “off”. In such a case, the shell and
the cocoon are still expanding, but they will slow down and the shell will thicken. Since a large
part of the radio luminosity comes from the radio hot spots, which fade away rapidly, the radio
luminosity also decreases. Whereas for large sources the cocoons filled with relativistic plasma
provide enough emission to be detected in the radio, young dormant sources will be very hard
to detect with radio observations. Thus we might hope to detect the X–ray signatures of such
sources.
The major difference that has to be incorporated into our analysis is a time variable
L(t) instead of L in equation (4.2). Since we integrated the equations numerically, this change
presents no difficulty. For simplicity we followed Reynolds and Begelman (1997) in using
a “picket-fence function” for
L.
We chose a duty cycle of 10%, i.e., the source is “on” for
10,000 years and “off” for 90,000 years. Depending on the average source power, this happens
at different evolutionary stages of the source, thus we should expect to see changes from the
previous figures. However, as the source grows, the influence of the intermittency becomes
smaller and the behavior approaches the solution for constant luminosity, corresponding to an
average power of L
= hL(t)i.
essentially unaltered.
For large sources, the results we presented above are therefore
121
4.5.1.2
Cold Material and Mixing
An important unknown we have neglected to include in our treatment is the possible
multi–phase nature of the host ISM/ICM. A cold or warm phase5 would be hard to detect in an
X–ray observation, although it can sometimes be seen via X-ray absorption (Allen et al. 1995;
Fabian 1994). Spectra could help in finding multiple temperature components in the continuum
emission and abundances of low ionization states from spectral lines and edges. The presence
of a cold component could severely alter the dynamics of our expanding shell if both filling
factor and mass residing in the cold phase were high enough.
The general picture is that, as the shell reaches the cold blobs of material, a shock is
driven into the cold matter with the same strength (i.e., same Mach number) as the shock into
the ISM/ICM, since the cold and hot phases are assumed to be in pressure equilibrium. The
cold material will therefore be heated and radiate either in the X–ray regime (if it is hot enough)
or in the UV/optical. Depending on their sizes, the blobs could either be completely evaporated
or their remnants might remain inside the cocoon. It might get shredded by hydrodynamical
instabilities, in which case it would mix with the shocked gas and the cocoon plasma. The
cold material could cool very efficiently and might radiate away a lot of the shell energy. Since
recent Chandra observations of Perseus A seem to indicate the presence of cold material inside
the shocked shell, it is of key importance to include these effects in future applications of this
model. Future work will also concentrate on the optical line emission we would expect to see
from such filaments (see, e.g., BB).
A related question is the possibility that dynamical instabilities (e.g., Rayleigh-Taylor,
Kelvin-Helmholtz) could mix material from the shell and the cocoon, producing pockets of
non–relativistic material in the relativistic cocoon. If mixing is strong, a curtain of intermediate
temperature material could form and absorb radio emission from the cocoon, thus producing
the characteristic spectral shape of Gigahertz Peaked Sources (GPS), as described by Bicknell,
5
There is now mounting evidence for the existence of large amounts of warm, T 105 K gas in clusters, Buote
2000
122
Dopita, and O’Dea (1997). We are currently investigating this effect with the help of numerical simulations, guided by the dispersion relation of relativistic, cylindrical Kelvin-Helmholtz
instabilities.
Foreground absorption might also affect our observational diagnostics. Even though the
column density of Galactic material is well known in most directions, the presence of a cold
component in the host cluster could have a significant effect on the detected signal. As long as
the covering factor and the filling factor of the cold component are small, this effect should be
negligible. If, however, the cold matter covers a large cross section of the source with sufficient
column density, absorption can change the spectral shape and therefore alter not only the total
expected count rates but also the brightness ratios, as they depend on the temperature difference
between the shell and the cluster. Current data suggest that typical values for the column density
of NH of cold material intrinsic to such sources are of the order
NH
1021 cm
2 , too low to
affect the Chandra band significantly.
4.5.1.3
Density Profile
Clearly, an isothermal King profile is a gross oversimplification of reality, but it should
at least provide us with a first order approximation. Introducing two separate King profiles (one
for the cluster, one for the Galaxy itself) might provide a better description of reality, but it
would also increase the number of free parameters by three (rc ; 0 , and ), all of which would
have to be determined by other means to improve this analysis. We decided that the increase in
realism would not justify the necessary computing time and the uncertainty as to which values
we should chose for the new parameters.
4.5.2
Chapter Summary
We presented a model for radio galaxy evolution and the interaction of radio galaxies
with their environment. This model can be used to calculate the observational signatures of this
interaction, manifesting itself in the form of cavities blown into the ISM by the radio lobes and
123
in a strong shock surrounding these cavities. We showed how this model can be used to infer
physically important parameters of the radio galaxy, such as the average total kinetic power.
Our analysis of Perseus A demonstrates that the power in its jets might be much larger than that
inferred from traditional estimates based on equipartition arguments. Furthermore, our model
suggests how Chandra observations can be used to detect very young radio galaxies which
would be obscured at longer wavelengths. Our scheduled Chandra observations of the cluster
Abell 4059 and its strong central radio galaxy will be analyzed on the basis of this model.
Future research will concentrate on the effects of non-uniform external media, nonsphericity, and dynamical instabilities occurring at the cocoon-shock interface.
124
b) shell counts
100.0
10.0
10.0
rc [kpc]
100.0
10 0
rc [kpc]
a) Temperature
0.1
10-4
-3
10
10-2
10 4
10 6
1.0
10 2
1.0
10-1
-5
10
0.1
100
102
L46/n0.1
104
106
10-4
c) shell-cluster ratio
10-4
0
4.0
.
8 00
10.0
1.0
0
2.0
4.0
16.0
10-2
106
1.00
rc [kpc]
rc [kpc]
2.00
0.1
104
1.00
100.0
10.0
1.0
100
102
L46/n0.1
d) center-cluster ratio
1.00
100.0
10-2
0.1
100
102
L46/n0.1
104
106
10-4
10-2
100
102
L46/n0.1
104
106
Figure 4.5: Contour plots of the shell temperature and our three diagnostics (see x4.2.3) as
functions of the source parameters rc and L=0 for a source with shell radius 500 pc at a distance
of p
100 Mpc. The contour levels are separated by factors of 10 in plots a) and b) and by factors
of 2 in plots c) and d). We assumed the Chandra ACIS-S response. The grey area on the left
side of the plot indicates parameter values for which a source of this size has turned subsonic
for a cluster temperature of 4 keV. The following quantities are shown: a) Temperature in keV,
2
b) shell count rate , normalized to n0 = 0:1 cm 3 , i.e., 0:1 cm 3 =n0 . At large core radii,
cluster emission dominates, producing horizontal lines in the figure. c) shell-cluster ratio, d)
center-cluster ratio.
125
1.000
rc=500 pc
L=1046ergs sec-1
z
0.100
0.010
64
4k
0.001
101
1.000
pc
kp
c
p
1k
c
50
0p
c
25
102
103
104
t [sec]
0p
c
105
106
rc=5 kpc
L=1046ergs sec-1
64
kp
c
z
0.100
4k
pc
0.010
2k
pc
1k
pc
50
0.001
101
102
103
0p
104
t [sec]
c
25
105
0p
c
106
Figure 4.6: Detectability constraints. This figure shows the exposure times needed to measure
Æ, the difference in the count rate from the annulus between rc and rs to the count rate from an
annulus of equal area lying just outside rs , to within 30 per cent. This demonstrates the non-zero
value of this difference to 3 sigma confidence. The lines correspond to different source sizes as
indicated in the figure. They are terminated when the spatial resolution limit of Chandra ACISS is reached, i.e., when the source becomes smaller than 4 arcsec across. Since it is properly
oversampled, the Chandra HRC has twice the resolution, but a lower effective area and no
energy resolution. The symbols correspond to different sources we picked to demonstrate what
can be achieved. Star: VII Zw485, square: Per A, diamond: 4C34.09, triangle: NGC1052.
126
b) shell counts
100.0
100.0
10.0
10.0
rc [kpc]
-1
10
1.0
-3
10
10 4
10 2
10 0
1.0
1
10
0.1
-5
10
0.1
10-2
100
102
L46/n0.1
104
106
10-4
10.0
0
1.0
1.0
2.0
4.0
8.0
0
1.0
1.0
0
10.0
rc [kpc]
100.0
16.
104
106
d) center-cluster ratio
100.0
0.1
10-4
100
102
L46/n0.1
1.00
rc [kpc]
c) shell-cluster ratio
10-2
2.0
10-4
4.0
rc [kpc]
a) Temperature
0.1
10-2
100
102
L46/n0.1
104
106
10-4
10-2
100
102
L46/n0.1
Figure 4.7: Same as Fig. 4.5 for a source of 16 kpc radius.
104
106
Chapter 5
A Shotgun Model for Gamma-Ray Bursts
5.1
Introduction
While it is not known if GRBs are in fact collimated outflows, they are certainly ultra-
relativistic (even more so than AGN jets). Their interaction with the environment is well established and key to our understanding of GRBs, as afterglows have revolutionized the science of
GRBs. In fact, it has yet to be proven that the GRB is not itself due to the interaction of a relativistic outflow with its environment. In this chapter we will argue that just such an interaction,
in a configuration not previously considered, might be the key to solving the GRB puzzle. The
main observational features of GRBs were introduced in x1.2, including the two main features
for any GRB theory to explain: the enormous energies released in only a few tens of seconds,
as well as GRB variability.
As established in x1.2, GRBs are probably produced by shocks in a relativistic flow.
This flow must be launched before it can shock. As in the case of AGNs, the most efficient
energy production mechanism available is the release of gravitational potential energy of matter
falling down the gravitational well of a compact object. The short timescales and the absence
of nuclear activity in typical GRB host galaxies suggest that the central engine of GRBs is built
around stellar mass black holes rather than massive ones. Typical GRB energies are comparable
to the gravitational energy released by about a solar mass of material accreting onto a compact
object, supporting the notion of stellar central engines. The two most popular mechanisms for
rapid release of such large amounts of energy are mergers of compact objects, with at most one
128
of them being a black hole (otherwise the energy would be released exclusively in the form of
gravitational energy) and extremely energetic supernovae, dubbed hypernovae. In the case of a
merger of two non-singular objects (i.e., N.S. — N.S., N.S. — W.D., W.D. — W.D.), a black
hole forms quickly, around which an accretion disk of residual material releases its energy over
a viscous timescale, roughly comparable to the length of the observed burst. Similarly, in a
hypernova, believed to occur in massive stars with M
>
35M , the initial collapse produces a
black hole of mass 10M , while the stellar mantle collapses to a thick accretion disk.
While the exact mechanism of energy release and GRB formation is still unknown, the
result of this energy release is, in most cases, rather similar: a large amount of radiation released
in a small volume (comparable to the gravitational radius of the central object) will immediately
reach a state of pair equilibrium, with very large opacities and any radiation present will be unable to escape. The specific internal energy of this material is large enough for it to be unbound
and expand freely, essentially like a bomb. The internal energy in the plasma is dominated by
relativistic forms of energy, shared by radiation, magnetic fields, and pairs, and the flow will
accelerate to relativistic bulk speeds very quickly. Such a scenario is called a fireball, since the
(extremely hot) gas will expand spherically in the absence of any external means of collimation.
Acceleration will continue until the internal energy has been converted adiabatically into kinetic
energy, with the terminal Lorentz factor approaching the ratio of the initial internal energy density to rest mass energy density in baryons (see also x2). Independently of GRBs, this scenario
has been investigated by Blandford and McKee (1976). Once it has reached its terminal
1,
the material will form a shell of thickness comparable to the size of the initial fireball at rest (as
seen from an observer stationary in the frame of the central engine). This shell is the starting
point for most GRB models.
5.1.1
External Shocks
In the first serious GRB model (Rees and Meszaros 1992; Katz 1994b; Sari and Piran
1995), the coasting shell sweeps up ambient material in a double-shock structure. This shock
129
is the origin of the observed radiation, which is why this model has been called the ”external
shock model”. The shell will have transferred most of its kinetic energy into internal energy
of both the forward shock into the ISM and the reverse shock into the shell when it has swept
up a fraction of
1=
of its own mass. This is because the swept-up material is shocked, with
post-shock particles reaching random Lorentz factors comparable to the shock Lorentz factor
, boosting the inertia of the swept up material by a factor of order
energies,
100, and an ambient density of n 1 cm
. For typical GRB
3 , this occurs at a distance of order
1016 cm from the center, leading to an observed emission time scale of order 10 100 sec,
comparable to the light crossing time of the shell itself (which is not compressed by relativistic
effects).
The energies and spectra from such a shock match the observations well, however, the
model cannot reproduce the observed millisecond timescale variability, since the produced radiation is emitted in a single, broad spike of length . In order to explain the fast variability,
it has been argued that the external medium might not be uniform (e.g., Shaviv and Dar 1995;
Fenimore et al. 1996), thus leading to spikes in the emission whenever the shell encounters a
cloud. There is ongoing discussion about the validity of this argument, due to what Sari and
Piran (1997) call the angular spreading problem, illustrated in Fig. 5.1: the observer will only
receive radiation originating from an area of the shell with viewing angles LOS
< 1=
with
respect to the motion vector of that material due to relativistic beaming. A shocked cloud of
radius
r at a viewing angle 1=
will have a projected depth of order
Æ
r= < R
2,
where R is the radius of the shell. For typical ISM densities, these clouds must be so small in
order to explain the short term variability that they can only cover a very small fraction of the
shell surface, and the efficiency of the conversion of shell-kinetic energy into radiation must
be correspondingly small (Sari and Piran 1997). Dermer and Mitman (1999) argue that clouds
very close to the line of sight produce much shorter spikes while at the same time contributing
a relatively large fraction of the flux. However, such a model implies a trend for spikes in the
lightcurve to become broader at later times in the burst, which is not seen in the observational
130
r
1/Γ
r
r
Γ
Γ
central
engine
clouds
Figure 5.1: Illustration of the angular spreading problem in the dirty fireball model (Dermer and
Mitman 1999), which employs clumpy external medium in the attempt to rescue the external
shock model.
data.
Fenimore et al. (1996, 1999a, see also Woods and Loeb 1995) recently suggested that an
external shock scenario could give rise to the observed variability if the spherical symmetry of
the outflow were broken, still in the context of what Sari and Piran (1997) call a ‘Type I’ model,
i.e., the burst duration is set by the slowing-down time of the ejecta. Once again, though, the
observed temporal constancy of the pulse width in individual spikes of GRB990123 seems to
rule out such a model (Fenimore et al. 1999b).
5.1.2
Internal Shocks
The difficulties of the external shock model to explain the short term variability prompted
Sari and Piran (1997) to postulate that the gamma ray emission must instead be produced by
the internal shock scenario (Narayan et al. 1992; Rees and Meszaros 1994). The emission in
this model is produced by internal dissipation of kinetic energy without the need for an external
agent. While other means of dissipation are possible (e.g., inverse Compton scattering losses
of turbulent field energy, as proposed by Thompson 1994), the most popular incarnation of the
internal dissipation scenario simply involves shocks of ultra relativistic shells moving at different bulk Lorentz factors . Each time two shells collide, a spike of radiation is released. In this
131
picture, the duration of the burst T is set by the time scale over which the central engine operates, while the substructure in the bursts on time scales
is produced by the inhomogeneities
in the outflow. As in the external shock model, the signal received by the observer is spread out
due to the curvature of the shock front. Assuming spherical shocks, this spread in time is of
order Æ
R=(c 2 ), where R is the radius of the shell, as evident from Fig. 5.1. For inter-shell
distances corresponding to the observed spike distribution and for
for the shocks to catch up with each other is of order
>
100, the typical radius
1012 cm, and the angular spreading time
is of order a few milliseconds, shorter than the observed burst substructure. Thus, unlike the
external shock model, the internal shock model can explain the variability in GRBs.
However, recent estimates of the energy conversion efficiency
indicate that, at best, a
few percent of the bulk kinetic energy carried by the outflow can be converted into gamma
rays in internal shocks, which leads to uncomfortably high requirements on GRB energies
(Panaitescu et al. 1999; Kumar 1999). Larger efficiencies can be achieved if very large dispersion in
is assumed(Katz 1997). A non-spherical geometry can also reduce the required
energy, however, a very small opening angle of the outflow implies a high rate of unobserved
GRBs, which is hard to reconcile with the number of observed supernovae (which are believed
to produce GRB precursors - either compact objects or hypernovae, as explained in x1.2).
5.2
The Shotgun Model
In this chapter we propose a different way by which substructure in the outflow can
produce a GRB, also via the interaction with the external medium, but in a ‘Type II’ scenario,
i.e., the duration of the burst is set by the lifetime of the central engine (Heinz and Begelman
1999, see also Chiang and Dermer 1999; Blackman et al. 1996). In our model, the outflow itself
is very clumpy, with most of the energy concentrated in small blobs, which are sprayed out with
high
over a small opening angle. These bullets then slam into the surrounding medium (not
unlike a meteor shower or a shotgun blast), where they release their kinetic energy and produce
gamma ray emission via external shocks. This scenario is illustrated in Fig. 5.2.
132
Γ
1/
θ
LO
S
low
erg
Aft
B
GR
g
stin
Coa
n
atio
eler
Acc
Figure 5.2: Illustration of the shotgun model.
In the context of our model, the spikes in BATSE lightcurves are produced by individual
bullets of cold ejecta slamming into the surrounding medium. As we will show, a distribution of
masses and/or Lorentz factors of these bullets can reproduce the observed signatures of GRBs
reasonably well.
In the following we assume that the central engine of the burst releases a number of
bullets N 0 distributed over a fan of opening angle of 10Æ with Lorentz factor 1000
3
10 100 s. Each bullet is assumed to expand sideways
with a velocity of v? = = 10 2 2 = 1= , measured in the observer’s frame ( is the
sideways velocity in the comoving frame). The assumption that 1 implies that the internal
and released over a time period of T
expansion speed is very sub-relativistic.
Since
1=
, we only see a fraction of the total released energy,
Eobs = 1047
133
E53 =
3
2 ergs (where E is the inferred isotropic energy in units of 1053 ergs) and an observed
53
number of bullets N
100 N100 = N 0 =(2 2). It is essential in our model that the covering
fraction be less than unity (otherwise it would turn into an internal shock model), thus we require
p
< 1= N .
A bullet of mass
Mb = 5:5 10
3
13 E M =N
53 100 3 will have converted half of its
kinetic energy into internal energy (which can subsequently be radiated away as gamma rays,
see below) after it has swept up or ploughed through a column of interstellar gas of mass Ms
10 3 3 s, then the length over which the
material is swept up must be of the order of R = 3 1013 3 3 2 cm. If the ambient density
is namb 108 n8 cm 3 , the required Lorentz factor is
Mb =
. If the duration of the observed spike is s
= 2300 E53
2
2 N100 1=8
3 3 n8
(5.1)
This model fails for low ambient densities, as has already been discussed in the literature
(e.g., Sari and Piran 1997). However, if the surrounding medium is very dense, n 108 cm 3 ,
Lorentz factors of
1000 can explain the observed short term variability.
The immediate
conclusion is that in this model GRBs are not caused by mergers of naked compact objects.
ski
Rather, the required high ambient densities tie this model to the hypernova picture (Paczyn
1998; Woosley 1993), which predicts that the material surrounding the blast is dense because
of the pre-hypernova stellar wind. (Dense circum-GRB matter was also suggested in a different
context by Katz 1994a).1
The fate of the outer layers of a hypernova precursor is unknown. If the bullets have to
travel through a significant fraction of the star’s mantle (which is optically thick and thus not
the site where the gamma rays are produced), their opening angle must be extremely narrow:
p
< 5 10 4 E53 =( N100 3 MM ), where MM is the mass of the mantle in units of M . Since
hypernovae are believed to originate from rapidly rotating massive stars collapsing into compact
1
A merger of two compact objects could produce a bullet GRB if the two compact objects were shrouded by a
red-giant envelope in a common envelope context. In order to produce the required energies, at least one of these
compact objects should be a neutron star or black hole, reducing the likelihood of such an event.
134
objects, it is possible that the material along the rotation axis has collapsed before the GRB, in
which case the bullets would travel freely until they hit the circumstellar material.
Similarly, little is known about the conditions of the pre-hypernova circumstellar material other than that it must be dense. Massive stars are known to have strong winds with mass
10 6 10 4 Myr 1 and wind velocities from vw 20 km s 1 (red supergiant) to vw 1000 km s 1 (blue supergiant). These winds must still be present after the
loss rates of
M_
star collapses. In the following, we will assume that the GRB is produced in this leftover wind.
The density profile in the ambient matter, then, roughly goes as n
/r
2 outside some radius
R0 . It is natural to assume that R0 is of the order of the stellar radius, R0 1012 cm for a blue
supergiant and R0
1014 cm for a red supergiant. As a conservative estimate we assume that
the sphere inside R0 is evacuated.
If we parameterize the density as
namb(r > R0 ) =
where
M_
1:5 1036 cm
r2
3 M_
6
(5.2)
v20
6 is the mass loss rate in units of
10 6 M yr
1,
r
is in cm, and v20 is the wind
velocity in units of 20 km s 1 , the observed slowing-down time scale is given by
= 0:05 s
independent of R0 .
E53 v20
4 _ ;
2
2 N100 3 M
6
Thus
(5.3)
can be of the order of a few milliseconds for both red supergiant
and blue supergiant winds if
1000.
However, the observed time scale could conceivably
be longer than this. The angular smearing time scale ang is defined as the spread in light travel
time to the observer across the emitting surface. For a bullet at a viewing angle of
gives ang
(R0 + R)=
E53 v20 =(
3
2
2
2 , which is longer than
if
1=
this
R0 >
R= = 1:5 1017 cm _ 6 ). This is only of concern for very dense red supergiant winds, and
100 M
3N
only if the region interior to R0 is evacuated.
In Fig. 5.3 we show various limits on namb and
for a fixed opening angle of 2
= 1:
a) Each bullet is expected to plough through undisturbed medium. Thus, the covering
fraction
N2 of all the bullets together must be smaller than 1.
For a slowing down
135
time of 3
= 1, this gives the dashed line in the plot, to the left of which the covering
fraction is larger than unity.
b) The material between the location where the bullets release their energy and the observer must be optically thin. For
3
= 1 and for R0 = 0 (the most conservative
limits) this constraint produces the dash-dotted line to the left of which the optical
depth is larger than unity.
c) The forward shock must be radiative (see x5.3). This constraint is shown as a light grey
region inside of which the shock is not radiative.
d) The angular smearing time ang must be smaller than the observed stopping time .
This limit is shown as a dotted line for
R0 = 1014 cm.
To the right of this line, the
smearing time is longer than the observed slowing down time.
The hatched region in the plot shows how the allowed region of parameter space opens up if we
relax the time scale requirements to 3
= 10.
It is worth noting that this model makes an exception to the rule that external shocks
cannot produce ‘Type II’ behavior (Sari and Piran 1997). This is for two reasons: First, the
opening angle of the ejecta is so small that the angular smearing time is short compared to .
Second, the ambient density is so high that the observed slowing-down time is 10 3 s. As a
result, the total duration of the burst is determined by the time the central engine operates, while
the short term variability is determined by the mass of the bullets and the statistical properties of
the outflow. This is an important difference from the internal shock model, where the variability
timescale is set by the intrinsic time scale of the central engine (e.g., the orbital time in a merger
scenario).
5.3
GRB Efficiencies
In order for the efficiency of the burst to be reasonable, most of the internal energy pro-
duced in the shock must be radiated away immediately (this requirement holds for all GRB
136
104
101
100
10−1
10−2
O Star
Wind
M˙ −6/v20
102
Red Giant
Wind
103
10−3
100
1000
Γ
10000
Figure 5.3: Constraints on the ambient density and for E53 = 1, N100 = 1, and = 0:01.
a) covering factor larger than unity: left of dashed line for 3 = 1, dark grey area for 3 =
10. b) optical depth between bullet and observer larger than unity: left of dash-dotted line for
3 = 1, medium grey area for 3 = 10. c) forward shock not radiative: light grey area (we
assumed B = 1). d) angular smearing time longer than slowing down time: right of dotted
line. The hatched area shows how the allowed region in parameter space expands if we relax
the requirement on from 3 = 1 to 3 = 10.
models). Electron synchrotron radiation is the only mechanism remotely efficient enough to
produce the gamma rays. While the efficiency also depends on the transfer of energy from protons to electrons, we assume here that this process is efficient. Since we know the observed peak
frequency of the gamma rays (of order
500 keV , Piran 1999), we can then estimate the radia-
tive efficiency under the assumption that the gamma rays are produced by electron synchrotron
radiation. It is usually assumed that the magnetic field in the shocked gas is in equipartition
with the energy density in relativistic particles. We therefore parameterize the magnetic field
strength as UB
BUB;eq , where UBeq is the equipartition magnetic field energy density.
Since the shocked wind material is likely flowing around the bullets at close to the speed
137
of light (like a cocoon surrounding a radio jet), for efficient cooling we require that the cooling
time scale in the comoving frame be smaller than the light travel time across the surface of the
bullet (R0 + R)=
c. If that were not the case, the material pushed aside by the bullet would
cool adiabatically rather than radiatively. Independent of
R0 , this translates to the condition
p
1430 [M_ 6 22 B=v20 ]1=4 E =500 keV (to the left of the light grey area in Fig. 5.3),
where E is the observed peak energy of the gamma rays. This is not a strict condition, however,
since we do not know what the efficiency of the GRB is.
5.4
Simulating Lightcurves
Since our model assumes a central engine at work (essentially a black box shooting out
bullets at a rate R(t)), any distribution of spikes could be reproduced, since R(t) is arbitrary. It
is, however, surprisingly simple to reproduce the main features seen in different burst profiles
by varying only a few parameters in our model. For simplicity, we assume that all the bullets
have the same initial
and the same sideways expansion rate, i.e., constant . We are left with
two parameters — the number of bullets N100 and the average slowing-down time (eq. [5.3]) —
and two unknown functions: the mass distribution of the bullets
N (Mb ) and the rate at which
they are released R(t). We assume that N (Mb ) / Mb 1=3 , chosen to give the observed power
spectrum of P ( )
/
1=3 (Beloborodov et al. 1998). For R(t) we assume (for lack of better
knowledge) that the bullets are released randomly over a time interval of 15 sec.
To produce synthetic GRB lightcurves, we calculated the time dependence of
and the
associated dissipation rate. Assuming the bullets are radiating efficiently and correcting for
Doppler boosting and frequency shifts, we then computed the composite lightcurve for each
bullet. We have plotted two simulated light curves in Fig. 5.4 for 3
c) and
N = 104
(panel d). It seems that simply by varying
N100
= 10 and N = 100 (panel
and
3 we can produce a
wide range in light curve shapes. More complex features (like the gaps seen in panel a) must be
related to the activity of the central engine and cannot be reproduced by a random spike rate as
assumed above. We have also plotted the light curve produced by the deceleration of a single
138
BATSE count rate [arbitrary units]
a)
0
b)
10
20
30
40
0
c)
0
5
10
15
5
10
15
20
d)
5
10
15
20
0
t [sec]
20
Figure 5.4: BATSE light curves for GRB920627 (panel a) and GRB980329 (panel b) and two
synthetic light curves. These curves were calculated for a burst duration of 15 s, a mass distribution of N (Mb ) / Mb 1=3 , an average slowing down time of = 0:01 s, and N100 = 1
(panel c), N100 = 100 (panel d). The insert in panel c shows a template light curve for a single
shot.
bullet along the line of sight (Fig. 5.4c, insert). Note that this profile is very similar to a true
FRED (fast rise, exponential decay) profile. The rise is instantaneous and the decay follows a
steep power-law (to first order). While indicative, these calculations are still rather crude and
simplistic. A more careful analysis of shotgun GRB lightcurves should be carried out in the
near future.
5.5
Afterglows in the Shotgun Model
Afterglows are an important test for any GRB model. How can we understand an after-
glow in the context of the shotgun model? It is not immediately obvious why our model should
produce an afterglow at all. This is because the bullets are assumed to spread sideways. As
mentioned above, the bullets will have lost half of their kinetic energy at R, where they have
swept up
1=
of their own mass. If we simply followed the dynamics of an individual bullet
139
further in time, it would lose the rest of its energy exponentially quickly (Rhoads 1997). This
is because the sideways velocity in the lab frame goes as 1= , so that when
starts decreasing,
the sideways velocity increases, which in turn increases the cross-sectional area of the bullet.
As a result, the bullet can sweep up more mass, which leads to a run-away process. This would
imply that the ejecta would come to a complete stop not far away from
R and the observed
afterglow would last less than a day.
However, there are many bullets traveling together. As they expand, they start increasing
the covering factor of the blast. Once the collection of bullets reaches unit covering factor, they
stop slowing down exponentially, since further sideways expansion does not lead to an increase
in swept up mass. As a result, the bullets start traveling collectively, resembling a collimated
blast wave with opening angle
rather than a meteor shower. The only possible difference
between our model and the standard afterglow models is that in our model, the external density
follows a power-law behavior instead of being constant, which has been discussed by Dai and Lu
(1998). Since the opening angle of the merged blast is much larger than the opening angle of the
individual bullets, the sideways expansion does not affect the dynamics until much later, when
the blast has spread by an angle of order (Rhoads 1997). This transition from constant opening
angle to rapid sideways expansion has been used in other models to explain the temporal break
seen in the afterglow lightcurve of GRB990510 (Stanek et al. 1999; Harrison et al. 1999; Sari
et al. 1999).
5.6
Making Bullets
Having established that the bullet model can indeed explain the observational signatures
of GRBs, we now turn to the question of how these bullets are produced. This is a far more
complicated problem to solve and here we will only outline avenues that might be taken in more
depth in future research. The typical bullet mass that must be produced by any such model is
given by M
10 13 ME53=( 4 N100 ), while the size of the bullets is only 1010 cm.
Both of the approaches outlined here start from a common fireball scenario. It is therefore
140
worthwhile to review the properties of a stationary fireball type wind. Such a wind corresponds
to a simple adiabatic spherical wind. For the sake of simplicity, we assume that the flow is
ultra-relativistic in the observer’s frame. The conservation of energy and momentum flux (here,
no sideways transport of momentum can occur, since the flow fills 4 sterrad) read
2v
c2 + 4p r2 = const:
(5.4)
and
2 v2
c2 + 4p r2 + 2 pr2 = const:
In the limit of v
2 r2 p
(5.5)
c, p c2 , this translates to
const:
(5.6)
The equation of state can now be written (see x2)
p/
4=3 r 8=3
(5.7)
and the solution is simply
/ r;
p / r 4:
(5.8)
Thus, in an idealized fireball, the Lorentz factor grows linearly with
r until the flow reaches
c2 p, at which point the flow turns cold and no more internal energy is available for acceleration. At this point the flow has reached its terminal Lorentz factor,
1
3p0=0 c2 .
The
subscript 0 denotes quantities evaluated at the injection point. It is clear from this that the ratio
of rest mass energy density to pressure determines
1. Since the rest mass density is domi-
nated by baryons, it is the baryon fraction rather than the total particle density that determines
the Lorentz factor of the flow. Since GRBs are assumed to have
1 100
1000, the baryon
fraction in the flow must be small. Since the problem of achieving low baryon loading concerns
every GRB model and is the topic of ongoing research, we will simply assume that such a low
baryon fraction can be established and proceed from there.
141
The acceleration measured by a comoving observer (after a little algebra) is simply given
by
a = c3
@
v@r
c2=r0 ;
(5.9)
where r0 is the injection radius. Thus, the comoving observer feels constant acceleration, which
makes treatment conveniently simple. Having established these key relationships, we can look
at the first possibility for bullet production.
5.6.1
Why Bullets Must be Contained by Adiabatic Compression
One might ask if it is possible to start out with a flow that has uniform density but non-
uniform types of pressure, e.g., mostly photons in one region and magnetic fields in another.
In such a case, it might be possible for the radiation to leak out of the radiation-dominated
region in a photon diffusion time. The exterior magnetic pressure would then squeeze the
now underpressured region, creating a density contrast, which can essentially be described as
a bullet. However, for a bullet of mass
pairs, is T
Mb
10 13 M the optical depth, even neglecting
1010 =R52 , where R5 is the bullet radius in units of 105 cm so the bullets are
still optically thick even when they slow down, and the radiative diffusion time is much longer
than the travel time of the bullet. Unless the conditions in the flow are different than what is
typically assumed to make the flow optically thin, it is not possible to create bullets by squeezing
radiation out of them. Thus, any compression of these bullets must be adiabatic.
There is one exception, though: if the fireball is hot enough, the rapid establishment of
pair equilibrium will lead to a high rate of neutrino production from pair annihilation. While
the entire flow is highly optically thick to photons when it is launched (i.e., a few tens of gravitational radii away from the central engine), under certain conditions it is still optically thin to
neutrinos. The optical depth of the entire fireball to neutrinos is
2 10
7
E53
r7 t10
(5.10)
142
while the neutrino emissivity is given by
7:82 1025
E53 9=4
ergs cm 3 s
t10 r7 2
1
(5.11)
Note that pair annihilation dominates over pair capture neutrinos for the conditions encountered
in GRBs, which is why we can neglect the baryon density in this expression.
Thus, if a region exists that is hotter than its environment, most likely due to higher photon pressure and lower magnetic field pressure, it can cool rapidly and compress significantly
before cooling adiabatically due to the expansion of the flow if the neutrino cooling rate satisfies
U
< r=c = 3:3 10 4 r7 ;
(5.12)
where U is the energy density in the fireball. This can be translated into an energy requirement:
E53 >
1:6 103 r76=5 t10
(5.13)
It is interesting to note that fireballs are neutrino cooled for isotropic energies in the range
1:6 103 r7 6=5 t10 <
E53 < 5 106 r7 t10
(5.14)
where the upper limit is given by the condition that the fireball be optically thin to neutrinos.
Thus, if the total burst energy falls within that range, the fireball will cool rapidly until the
energy falls below that threshold, i.e., the total isotropic energy of the actual GRB is limited to
1:6 1056 ergs s 1 r7 6=5 t10 .
Note that this argument is independent on whether the burst is isotropic or not because
it only depends on the thermodynamic state of the plasma. In a sense, then, GRBs could be
regarded as standard candles since their luminosities seem to be limited from above, and determining the luminosity envelope as a function of redshift could be used to complement SN 1a
observations. In reverse, we can use a large observed burst energy to set a lower limit on the
injection radius. Since GRB efficiencies are poorly understood, these limits are so far uninterestingly low.
143
5.6.2
Rayleigh-Taylor Driven Bullet Formation
The inherent assumption about the internal shock model is that the flow is not stationary,
but rather highly variable, with different epochs of the fireball having different baryon loading.
Here we wish to investigate the effects of Rayleigh Taylor instability on a fireball with abrupt
changes in rest mass density, as might occur in an internal shock scenario. All the groundwork
for this has been laid in the previous paragraphs and Appendix C.
Once again we write the isotropic energy released by the GRB as
and the time scale over which this release takes place as
t = 10 t10 sec.
E
E53 1053 ergs
Without loss of gen-
erality we can assume that the burst is isotropic, as long as the opening angle of the entire
outflow is larger than the beaming angle of the flow after it reaches terminal velocity. We
parameterize the injection radius as
10M black hole is Rgrav
r = 107 r7 cm.
1:5 106 cm.
p 1026 ergs cm 3 E53 t10 1 r7
Note that the gravitational radius of a
The pressure inside the fireball is then given by
2.
We now assume that a contact discontinuity exists in the flow, with the rest mass density
higher on the side further away from the central engine. Taking the surface to be perpendicular
to the direction of motion, and keeping in mind that the flow is accelerating, this corresponds
to a heavier fluid sitting on top of a lighter fluid, which is the classical condition for RayleighTaylor instability to occur. For the sake of simplicity, we will neglect the curvature of the
interface, thus the analysis from Appendix C directly applies. The densities on either side of
the interface are the last parameter necessary. We know that GRBs reach
100 1000, so
0 3p=100 3p=1000 and we assume a density contrast of order 10 between the two shells.
It is clear from the dispersion relation that Rayleigh-Taylor instability is weakened in
relativistic gases, since the inertia is given by the enthalpy rather than just the rest mass density,
but since both sides of the interface must be in pressure balance, no additional buoyancy effects
arise. In other words: the stabilizing inertia is increased, while the de-stabilizing force is not.
Why, then, one might ask, should Rayleigh-Taylor instability be of importance in GRBs, where
144
the plasma starts out highly relativistic. The answer lies in the enormous acceleration felt by
the plasma: in the comoving frame, the acceleration is given by
a
9 1013
cm sec 2 ;
r7
(5.15)
quite large compared to any acceleration encountered in most other places in the universe, larger
than the typical acceleration at the base of AGN jets by the inverse ratio of the black hole masses
involved, roughly 105
108 .
It is well known that the fastest growth rates for non-magnetic Rayleigh-Taylor instability
without surface tension are found at the shortest wavelengths. However, we are interested in the
modes that contain the most mass and can still grow during the time the flow accelerates. An
important requirement on any growing mode is that it stay in causal contact during the entire
acceleration phase, which will put an upper limit on the wavelength.
To calculate the growth time for a given mode, we integrate the inverse growth rate over
the acceleration time of the flow. In order to take account of the sideways expansion of the flow,
we scaled the wavelength with r, in other words,
k(r) = k0 (r0 =r).
The integration was done
numerically and the results are shown in Fig.5.5.
We can see that the most massive bullets fit well with the masses we inferred in the
previous sections.
Once the instability has become non-linear, the heavy fluid will fall behind in the form
of fingers, which would be rather similar to the bullets we are looking for. As we can see, the
most massive bullets are contained in the largest wavelengths. Since the flow keeps expanding
sideways even after acceleration terminates, we still have to contain the bullets, otherwise they
would spread and be much larger than what is needed for the shotgun model to work. The
next section will outline a method how confinement can be established. However, this method
can work to produce bullets even in the absence of Rayleigh-Taylor instability. The purpose of
the previous analysis was simply to show that spherical symmetry in a standard internal shock
scenario is likely to be broken, producing clumps of matter that contain mass on the scales
145
Mb [g]
1020
1019
10
z/z0
Figure 5.5: Mass enclosed in the longest wavelength mode that has grown non-linear as a function of distance from the core for r7 = 1 and 1 = 1000.
expected in GRBs.
5.6.3
Ram Pressure Confinement and Acceleration
The so called plasmon model was the first approach at explaining radio galaxies. It
was envisioned that the radio lobes were blobs of relativistic plasma that were ejected from
the nucleus in opposite directions and were subsequently interacting with the ISM. The ISM
would then decelerate the blobs and at the same time confine them through ram pressure (see
Christiansen 1969, for details). Even though this picture has long since been discarded in favor
of the jet/lobe paradigm, it might be of interest in the context of GRB bullets, as we will show
in the following.
If a blob of gas of a given mass
Mb is embedded in a stream of gas that has a relative
velocity vrel with respect to it, the blob will experience a ram pressure from that flow that will
act to accelerate it in order to decrease the relative velocity of the blob with respect to the flow.
146
background scale height
bullet radius
background flow
contact
discontinuity
bullet
scale height
bullet
acceleration
α
bow shock
static pressure
+ ram pressure
Figure 5.6: Cartoon of bullet acceleration by ram pressure. The bullet is shown in red, the
background flow in blue, and the shocked material confining the bullet is shown in yellow. Also
shown are bullet radius and bullet scale height (roughly equal) and the background scale height
(not necessarily the same).
At any given point at the surface the ram pressure will be given by
pram = hf
2 2 2
rel vrel cos ;
(5.16)
where is the angle of incidence of the flow, i.e., the angle between v and the surface normal
(see Figure 5.6 for illustration), rel is the relative Lorentz factor between blob and background
flow, measured in the frame of the blob, and hf the enthalpy in the background flow. Force
balance across the blob surface will lead the shape of the blob to change until it reaches pressure equilibrium between ram pressure, internal bullet pressure, and the static pressure in the
background flow.
Since the blob is accelerated, its internal pressure must be stratified, with a scale height
147
of b
= pb =(gb hb ),
d ln pb (z )
= 1=b
dz
(5.17)
where cb is the internal sound speed and gb its acceleration. If the background flow is accelerating, it too must be stratified, with a different scale height than the blob, f .
The pressure balance condition then reads:
pb (z ) = pf (z ) + hf (z ) 2 v2 cos2 (5.18)
This situation is illustrated in Fig.5.6. Using
"
dR
dR
cos =
1+
dz
dz
2 # 1=2
(5.19)
we can write equation 5.18 as
v
dR u
1
t 2 2
=u
v
h
f
dz
p p
b
1
f
:
(5.20)
While equation 5.20 cannot be solved analytically, one can show that in the limit pram
pf ;0 (i.e., negligible static pressure in the external flow) an approximate solution can be found
(Christiansen 1969) with
R
b f
f b
b;
(5.21)
in the case of an internally isothermal bullet, or
R 2:4
b f
f b
pb
2:4b 2:4 ah
;
b
for an adiabatic bullet. This is of the same order as the blob scale height (we expect f
(5.22)
> b
since the blob is compressed by ram pressure). Thus, as it turns out, the blob will be roughly
spherical in shape. Even if pf is not negligible, the sideways dimension of the blob will be
comparable to the scale height as long as pram is comparable to pf . For the case where pf
is much larger than pram the blob shape will not be altered by ram pressure effects from its
original shape, but in this case the blob acceleration will also be small. If the background flow
is accelerating itself, pram will eventually catch up with pf . Since the surface area of the blob
148
is known, the acceleration of the blob can be calculated from the momentum flux through that
surface. This determines the dynamics of the blob completely: the acceleration of the bullet (in
the ultra-relativistic limit) is given by
d b pramRb 2
=
;
dr
hb Vb
(5.23)
where Vb is the bullet volume and hb Vb is the bullet inertia. Since
Rb
and Vb both depend
on the acceleration through equation 5.20, this is a system of coupled equations, which can, in
general, only be solved numerically.
5.6.3.1
Bullet Acceleration
In the standard fireball model it is inherently assumed that the initial fireball is homogeneous, i.e., the baryon density is the same everywhere. This assumption is relaxed in the internal
shock model, where the baryon density varies as a function of time, but is constant on shells of
constant r at least within areas larger than the beaming angle. It is no surprise that such a setup
cannot give rise to bullets.
There is, however, no reason to assume that the isotropy of the fireball cannot be broken
on smaller scales. It is quite plausible to assume that a distribution of baryon densities exists in
regions of different sizes. For simplicity, we will assume in the following that the initial fireball
contains overdense roughly spherical regions of a given size
RB , not unlike raisins in a cake
batter.
In such a case, a hierarchical acceleration scenario will unfold: As long as the entire flow
is relativistic, it will accelerate in unison following the linear acceleration phase outlined above.
The overdense regions will become non-relativistic first and will start lagging behind with respect to the background flow. This will induce ram pressure acceleration and confinement. This
ram pressure acceleration will continue until the bullets reach a significant fraction of the speed
in the background flow or until they encounter the stationary ISM. Since the pressure inside the
blob is enhanced over the static pressure in the background flow, they expand at a slower rate
149
than the background flow. This provides confinement.
We write the initial radius of the overdense region at the base of the flow as
R5 105 cm.
This is the initial bullet size at the base of the flow, i.e., at
r = r0 .
R0 =
After
accelerating with the flow, the bullet will become rest-mass dominated when b c2
>
3pb .
the initial bullet density is b;0 and the initial bullet and background pressure is pb;0
= p0 , this
happens at a distance rc
If
3p0=0 r0 , where = rc =r0 . At this point the background flow is
still relativistic, and the size of the bullet is now Rc
= R0 rc =r0 .
At the base of the flow the scale height in bullet and background gas are the same,
b = f .
Throughout the linear acceleration phase of the bullet, this will continue to be the
case, since during this phase the rest mass density is dynamically unimportant. The scale height
of the background flow measured in the frame of the bullet is f
= b dppff =dr = r0 =4. The bullet
Lorentz factor enters through the Lorentz contraction from observer’s scale height to bullet scale
height. The bullet starts out much smaller than its own scale height (otherwise it would be out
of casual contact) and when it reaches rc it should still be smaller or of the same order as b
(again, it must still be in causal contact).
If the bullet size Rc at rc is similar to b then the ram-pressure acceleration model discussed above directly applies and we can simply integrate the equations to the point where the
bullets hit the ISM.
If the bullet size is smaller than b when it slows down, it will have to adjust to the
Christiansen solution discussed above. The bullet has to stay in pressure equilibrium with its
environment at all times, which (by equation 5.18) translates into an intermediate relation between bullet size
Rb and its scale height, which will be valid for R < b .
For an internally
isothermal bullet, this can be written as
b =
Rb
:
ln (1 + pram=pf ) + Rfb
(5.24)
The bullet behaves adiabatically, implying a relation between bullet pressure and radius
R. The bullet acceleration is determined through it scale height simply by a = cs 2 =, where a
150
is the bullet acceleration measured in the bullet frame and cs 2
= ad p=h is the sound speed in
the bullet. Once the bullet size reaches b the Christiansen model can be used.
As stated above, these equations will have to be integrated numerically, but in the case
where Rc
b we can do the integration analytically. Ram pressure confinement gives
Rb =
where ad
p
cs 2
= 2 ad b ;
2
c d b =dr b c d b =dr
(5.25)
= 4=3, since the bullet is optically thick and thus radiation pressure dominated. The
bullet behaves adiabatically, thus
pb / R 4 ;
b / R 3 :
(5.26)
Together with equation 5.25 this gives
s
2:4pb;c
;
b;c c2 d b =dr
(5.27)
2:4pb pb;c
R;
Mb c2 b;c c2 c
(5.28)
R = Rc
and
d b
=
dr
s
where Mb is the bullet mass.
We assume that pram
> pf
so that pb
pram.
The ram pressure in the bullet frame is
given by
2 2
rel vrel :
pram = hf
(5.29)
We assume that the relative velocity between bullet and background flow is already relativistic,
so that rel
pram /
f
b
1. In that case, we have the proportionality relation
1
2 :
b r2
(5.30)
Note that this relation is valid both for accelerating and coasting background flow: in the accelerating phase, hf
f
4pf =c2 / r
4 and
f
/ r2, while in the coasting case hf f / r
2 and
= const. With this the equation of motion becomes
d b
=
dr
r
3 2:4 pb;c rc
4 b;c c2 Rc r
b;c
;
b
(5.31)
151
which has the solution
b
=
v
u
u
t2
s
b;c
3 2:4
4
1=2
3 2:4
4
pc rc r ln
b;c c2 Rc
rc
r7 r =
100 ln
:
b;c
R5
rc
The ram pressure as a function of r is simply pram pram;c(rc
2
3
1=2
bullet behaves adiabatically and pb
Rb = Rb;c
pram;c
pram
1=4
(5.32)
2
b;c =r b ) , and since the
pram, we have
s
= Rb;c
r
rc
b
:
b ;c
(5.33)
Since b only grows logarithmically beyond rc , the bullet is well collimated and even initially
very large bullets will be small enough to satisfy the size constraints in the bullet model.
As was mentioned above, a key constraint on this model is that the bullet stay in causal
contact throughout acceleration and collimation phase. Since the bullet expansion is very slow
beyond rc , the strongest constraint on the bullet size is that it be in causal contact at rc , i.e.,
Rc <
rc=
b;c , which translates to
R5 <
r7100=
b;c
= r7 100
b;0 c2
:
3 p0
(5.34)
If the bullet were to become causally disconnected, the ram pressure collimation mechanism
would not be able to act on the entire bullet. Instead, the bullet would be compressed in the
direction of motion to become oblate and then break apart into units roughly the size of a
causally connected patch. The longitudinal collapse of the bullet would presumably occur in a
shock, since the bullet is not causally connected longitudinally either when it reaches rc . This
would correspond to an internal shock, since most of the energy goes into heating the bullet
rather than accelerating it. It is therefore possible that the internal shock model and the shotgun
model co-exist.
The causality requirement can be translated into an efficiency estimate. Equation 5.34
tells us that the largest bullet possible satisfies b;c
such a bullet boosts b to
b;ISM 3:2
s
c
ln
100r7 =R5 . Ram pressure acceleration for
rISM 109 cm
:
1014 cm rc
(5.35)
152
where rISM is the location where the ISM starts slowing down the bullets. The mass in the bullet
is
Mb = 4=3R0 3 b;0 = 4=3R5 3 1015 cm3 3p0 = c c2
and its kinetic energy when it reaches the ISM is Ekin
Mb
(5.36)
2
b;ISM c . If the number of bullets
within the beaming angle is 100N100 , then the total energy in the bullets is
Ebullets = 400N100
b
3 M c2 ;
b
(5.37)
while the total energy in the fireball is
Ereball 12 1015 c p0 r7 2 t10 cm2 s = 1053
p0
r 2 t ergs:
26
10 dyn 7 10
(5.38)
The total efficiency of the burst, defined as the kinetic energy in bullets divided by the total
energy in the fireball, is
burst Ekin;bullets
Ereball
1 N100 R5
680
:
t
(5.39)
10
We note that the efficiency of the burst decreases if the bullets are smaller than the maximally
causal size assumed in this analysis. Since the energy efficiency is mostly determined by the
filling factor of the bullets, a smaller bullet size would imply a larger number of bullets to satisfy
reasonable efficiency limits. Note that the acceleration is somewhat more efficient in this case,
since the bullet must adjust to the Christiansen solution first. Detailed numerical simulations
are needed to solve this problem self-consistently.
1.
Equation 5.39 shows that for the bullet efficiency to be high we need N100 R5 =t10
While r7 does not enter into the efficiency, it does determine b;c , and thus the terminal
the bullets. Since we require the bullets to have
spikes, increasing
of
1000 in order to produce millisecond type
R5 implies increasing r7 if the bullets are to stay in causal contact.
Large
injection radii are difficult to reconcile with the idea that the GRB is produced close to the
horizon of the central black hole. Increasing the number of bullets much beyond N100
also difficult to justify.
10 is
153
It thus seems that the region of parameter space where ram pressure acceleration can
produce GRBs is very restricted. We note, however, that not all GRBs display millisecond
timescale variability, which relaxes the requirements on
and
N100 somewhat.
Furthermore,
some GRBs show both smooth, relatively long lasting sub-structure and spikes (e.g., GRB
980425 and 990123, see Figure 1.5). In such cases, the smooth sub-structure might be produced
by the background flow in an ordinary external shock scenario while the spikes are produced
by bullets in the flow. This would lower the required efficiency for the burst, since most of the
energy could reside in the background flow.
Finally, these estimates are still very crude. Since the terminal bullet Lorentz factor b
enters into the efficiency to the third power, estimates of b are crucial in determining
and
more detailed calculations are needed before any definite conclusions about the viability of ram
pressure acceleration can be drawn. Another important issue to consider is the effect of KelvinHelmholtz instability, which might act to disrupt the bullet before it reaches the ISM. Future
work will focus on these questions.
5.7
Chapter Summary
This Chapter presented a GRB model based on the shocks that small blobs of rela-
tivistically moving material drive into the dense atmosphere of hypernova progenitors. The
lightcurves produced by such bursts are consistent with the shape and variability seen in BATSE
lightcurves. Afterglows fit naturally within this picture, as bullets merge at the end of the GRB
phase and form a common shock front.
We find that bursts with satisfactory efficiencies (of order
injection radius is larger than generally assumed, i.e., r7
imply b;c
0:1) can be achieved
if the
>
10, in which case R5 > 5 would not
100. Bullet numbers in the range of N100 10 are also within reasonable limits.
We have shown that bullet production via ram pressure acceleration is possible, though rather
restrictive in the parameter ranges possible. More research is necessary to evaluate the effects
of magnetic fields and to solve the equations of motion completely self-consistently.
Chapter 6
Conclusions
This thesis investigated the physics of relativistic outflows encountered in AGNs and
GRBs. Jets are a ubiquitous phenomenon in astrophysical scenarios, yet the production and
collimation of jets are still open issues and the interal makeup of jets is unknown. Due to their
cosmological distances and small angular sizes, GRBs are even less well understood.
The second chapter presented a simple analytic model of jet acceleration. The jet is
accelerated by tangled magnetic fields, with collimation being provided by pressure from an
external medium. Using tangled fields as a jet acceleration mechanism is a new approach,
based on B95, as previous models of MHD jet acceleration concentrated mostly on large scale
organized fields (though tangled field evolution in a more general context and under somewhat
different conditions has been considered extensively, e.g. Goldreich and Sridhar 1997). Our
analytical quasi-1D approach is limited to narrow jets, for which the opening angle is smaller
than the beaming angle, coincident with the condition that the jet be in causal contact with its
environment. We introduced an ad-hoc process that redistributes energy between perpendicular
and parallel field to facilitate efficient conversion of internal energy to kinetic energy. Without
such a process, stationary jet acceleration by disorganized fields is impossible. In the absence
of dissipation, the rate of acceleration achieved under this scenario is the same as in the case
of relativistic particle pressure,
/ pext
1=4 . We also found analytic solutions beyond the
self-similar region, which enable us to calculate the terminal bulk Lorentz factors of such jets.
In order for these jets to reach the observed
> 10, they must be magnetically dominated at
155
early stages.
We estimated the impact of dissipation of magnetic energy on the dynamics of the flow
by considering a simple, phenomenological prescription of the loss process. The presence of
dissipation lowers the terminal Lorentz factor
1 and generally changes the rate at which the
jet is accelerated (the latter effect is noticeable only if the dissipation rate is comparable to the
adiabatic expansion rate). We also included the effects of radiation drag in the simplest scenario,
which always lowers the efficiency
and 1.
The amount of radiation drag in our model is
controlled by the amount of dissipation replenishing the high energy particle pool but seems to
be dynamically unimportant.
We calculated the frequency integrated surface brightness for the extreme cases where
all or very little of the dissipated energy is radiated away on the spot and found that, while the
brightness drops off very rapidly in all considered cases due to the expansion of the jet, values of
the dissipation efficiency >
0:05 can have a significant impact on the intensity as a function
of z . In the marginally non-radiative case, the build-up of particle energy leads to a slower
decline in intensity with z for larger . Finally, we applied this model to the prototypical radio
galaxy M87 and found that it is consistent with the observed properties.
It should be noted again that this model is a gross oversimplication. In reality, it is
likely that the field will contain both ordered and chaotic regions, and a combination of the two
associated models seems most appropriate. This work should thus be regarded as only a first
step towards an integrated picture.
In the third chapter, we investigated the energetics of jets, concentrating on the best
studied example available, the jet in M87. We proposed that the apparent lack of synchrotron
cooling in the M87 jet most likely indicates the presence of a sub-equipartition magnetic field.
While the total (particle + magnetic) pressure needed to explain the observed synchrotron emissivity is uncomfortably high in the nonrelativistic limit, Doppler beaming effects consistent with
bulk Lorentz factors in the range
2
5 lower the pressure requirements considerably.
Fluctu-
ations in the synchrotron emissivity and spectral cutoff frequency are consistent with adiabatic
156
changes in the magnetic field strength and particle energies that accompany compressions and
rarefactions along the flow, and Fermi acceleration along the flow is not necessary to explain
the observations.
The knots are identified with relatively weak shocks, as inferred from other data by BB.
The first-order Fermi acceleration expected to occur at such shocks, if any, would generate a particle energy distribution steeper than the n(E )
/ E
2 needed to produce the radio–to–optical
synchrotron spectrum. Thus, effects of particle acceleration along the jet might be apparent
only shortward of the cutoff frequency, e.g., in the X-ray band. However, a disordered magnetic
field or a “superluminal” field orientation with respect to the shock front (Begelman and Kirk
1990) could further reduce the efficiency of Fermi acceleration, hence we should continue to
regard the origin of the X-ray emission as unknown.
The pressure estimates we derive are consistent with the assumption that the M87 jet is
embedded in a moderately overpressured bubble, as suggested by BB. As a result, it seems that
the set of parameters we have suggested above can solve the cooling problem of the M87 jet,
as sub-equipartition fields are able to explain both the behavior of the cutoff frequency and the
confinement of the jet.
Using derived values for the magnetic field and the cutoff momentum at r0 we can put an
upper limit of 10 pc on the radius at which most of the particle acceleration occurs. Explaining
why the acceleration is confined to a particular scale (which may be quite large compared to the
size of the central black hole) poses an interesting problem for future work.
The analysis of the M87 jet we carried out could be improved by better deprojection
models of the jet. Higher quality spectral data both in the near infrared and the X-ray will
help to verify the assumption of constancy of the underlying spectral shape and decrease the
uncertainty in the location of spectral features (the knowledge of which is essential for this
technique to work). The community is still awaiting the publication of the HST/NICMOS and
the Chandra results.
Given the fact that M87 is a relatively weak source, it would be very interesting to in-
157
vestigate other sources to see if the results derived above carry over. Such an effort is now
underway: we are participating in a broad band radio/optical/UV campaign of several nearby
AGN jets, which will hopefully be extended to X-ray energies in Chandra Cycle 2. The techniques outlined in chapter 2 should be very useful in analyzing these jets.
Chapter 4 presented a simple model for the evolution of powerful radio galaxies into a
surrounding ISM/ICM in order to make predictions about the detectability and appearance of
such sources for future X–ray missions. We assumed uniform pressure, spherical symmetry,
and a King profile density distribution in the ambient medium to describe the cocoon and the
shell of swept-up material and have calculated a grid of models for various source parameters
to provide observational diagnostics for high resolution X–ray observations.
We are particularly interested in young sources, since signatures of intermittency are
most pronounced in the early stages of source evolution. However, there is only a rather limited
range of parameters for which we might hope to detect the shell of such sources, since the
expected count rates are low for all but the most dense environments. Also, contamination by a
bright core could make such detections impossible.
Larger sources offer more chances not only for detection but also for analysis and application of our model grid. For a source with known redshift and core radius, it should be
possible, at least in principle, to determine the central density of the King profile, the average
kinetic source luminosity, and the source age. The knowledge of these parameters could help
a great deal in understanding the process of jet formation. As an example we apply our model
to a 50 ksec ROSAT HRI observation of Perseus A, and find that the time averaged power most
likely exceeds 1045 ergs sec 1 .
We have shown that GRBs can be produced by a shower of cold, heavy bullets shot at
bulk Lorentz factor
1000 into a dense medium. The required densities are consistent with a
stellar wind from either a blue or red supergiant, which ties our model to the hypernova scenario.
The gamma rays are produced by the shocks these bullets drive into the ambient gas. The total
duration of the burst is then determined by the time the central engine operates rather than the
158
slowing-down time of the bullets, while the latter produces the short-term variability seen in
many bursts. After the gamma ray phase (when the bullets have lost half of their kinetic energy
to radiation) the blast waves of the individual bullets merge into a single collimated shock front,
which produces a standard afterglow in a declining external density profile.
One possible creation mechanism for such bullets is ram pressure acceleration and collimation. The seeds for the bullets can either be provided by an inherently inhomogeneous,
non-isotropic fireball or, by Rayleigh-Taylor instability of onion-shell like internal shock fireballs. Both mechanisms could occur in the standard hypernova scenarios commonly envisioned
as possible central engines for ordinary internal shock GRB models.
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Appendix A
A More Realistic Dissipation Law for the Tangled Field Model
The dissipation law we assumed in x2.2.1.2 was only one of many plausible ad hoc
models. Since (for finite conductivity, i.e., beyond the limit of perfect MHD) reconnection will
occur whenever there is field reversal on sufficiently small scales (which would certainly be the
case in a highly tangled geometry), we would expect dissipation to occur even if there were
no change in field geometry due to expansion or acceleration. In that case we might expect
that the dissipation timescale is proportional to the time it takes a disturbance to travel a given
characteristic length (e.g., the jet width) in the comoving frame, i.e.,
dU 0 i v
U 0 i Alfven ;
dz diss
R
(A.1)
absorbs
the effects of resistivity and all the unknown physics of the
where the parameter
reconnection process. Once again, it is straightforward to generalize to the case of different i
for different components of the field.
It is possible to solve the set of equations in the magnetically dominated case (i.e., setting
0 + 4p0 = 0), under the assumption that = const: and in the relativistic limit,
limit vAlfven
= 1, which simplifies the treatment significantly.
1. In this
We assume that the dissipated
energy is radiated away immediately, which leads to a modified equation (2.15)
2 + 6 d
+ (2 + 2 )
R 3 3
dz
(2 + 2 )
dR
= 0:
Rdz
(A.2)
The pressure balance equation (2.17) is also modified:
(
1)
d
dz
(3 + )
dR
Rdz
= =z
R
(A.3)
169
where we used U 0
(z ) =
/ pext and equation (2.20). This set of equations can be solved to give
n
h
=4 1 A (z=z )1
0 (z=z0 )
0
=2
1
io(3+5 )=(4+4 )
;
(A.4)
with
A
1
:
R
(1
=
2)(1
+ 3 )
0 0
(A.5)
0 denote quantities evaluated at some arbitrary upstream point z0 .
In the limit of
! 0 this solution appropriately reduces to the result without dissipation, i.e.,
/ z=4. The
Subscripts
solution can essentially take on three different behaviors: If > 2, the solution will asymptot-
/ z=4, i.e., = 1=4. If < 2, on the other hand, two different scenarios
6 1=4 (but still constant)
can occur: the flow can either approach a self-similar behavior with =
in the limit of z z0 , or, if A > 0 (i.e., > 1=3), the flow can actually stall, i.e., ! 0
for z ! z0 [(1 =2) (1 + 3 ) =A 1]1=(1 =2) . In this limit, the magnetic field dissipates
ically approach
away too quickly to satisfy pressure balance and the jet must contract and decelerate to increase
its internal pressure, thereby increasing its dissipation rate, which leads to a run-away process.
Eventually, the massless approximation will break down, in which case the Alfven velocity will
drop, lowering the dissipation rate (furthermore, the particle pressure will gain in importance,
eventually stabilizing the jet against external pressure, in which case the jet would behave as
described by BR74). This process would produce an observable hot-spot (and possibly a shock)
at a fixed distance.
Appendix B
Details of Synchrotron Emission
B.1
Synchrotron radiation
In this appendix, we will give a brief overview of the properties of synchrotron radiation
that are important in this paper. We will mainly follow the review by Rybicki and Lightman
(1979).
B.1.1
Synchrotron Losses:
Synchrotron radiation is produced by relativistic particles moving in a magnetic field,
thus it is the relativistic equivalent to cyclotron radiation, the crucial distinction being time dilation and Doppler beaming effects. Consider a charged particle moving trough a homogeneous
magnetic field
B with velocity v c.
The Lorentz force perpendicular to the magnetic field
will cause it to perform circular motion around the field line with frequency !B
where
qB=(mc),
q is the particle charge, its Lorentz factor, m its mass, and c the speed of light.
This
circular motion is superposed on the linear motion of the particle along the field, thus it will
travel on a helical trajectory.
In its rest frame the particle will experience an acceleration
v
velocity . The magnitude of this acceleration is given by
a = a? perpendicular to its
a? = !B v?.
Using the Larmor
formula for the power an accelerated particle emits (which is covariant if the radiation process
171
is forward-backward symmetric), we arrive at a loss of
dW
=
dt
2q2 jaj2
2q 4 2 B 2 2
=
v :
3c3
3c5 m2 ?
Averaging over an isotropic distribution of pitch angles we arrive at an average loss rate per
particle of
d
=
dt
4q4 B 2 2
;
9c5 m3
or, expressed for the particle momentum:
dp
4q 4 B 2 p2
=
= Ap2 ;
dt
9c6 m4
(B.1)
where A from chapter 3.2 and vparticle
c have been used. From this we can easily derive the
synchrotron loss time of a particle with an initial momentum p0 , which we define as the time
it takes the particle to lose half of its energy due to synchrotron losses in a constant magnetic
field:
tloss (p0 A)
B.1.2
1
(B.2)
Emitted Synchrotron Spectrum:
To derive the expression for the spectral emissivity we need to recall the expression for
the energy in the electric field of an accelerated particle moving at velocity
Z
dW
2
2
2
2
= (q ! =4 c ) d!d
n (n v) exp i!(t0
v(t)
2
n r0(t0 )=c)dt0 ;
(B.3)
n is the line of sight normal at an angle # to v0 , the particles velocity at retarded time
t0 = 0, and r0 (t0 ) the particles retarded position. Using the small angle approximation for sin
and cos to leading order, v c, and (1 v=c) (2 2 ) 1 , the factor (t0 n r (t0 )=c) can
where
approximately be written as
t0
cos # sin !B sin (t0 )
!B sin (2 2 ) 1
"
(sin )
(1 + 2 #2 )t0 + !B2 2
3
2 t03
#
:
172
Accordingly, we can write
n (n v)=c
=
ek cos (!B sin t0 )sin# e? sin (!B sin t0)
ek # e?!B sin t0
The small angle approximation is valid here because the integrand in equation (B.3) becomes
highly oscillatory for large arguments, thus the integral itself goes to zero.
The integral in equation (B.3) can be expressed in terms of the modified Bessel function
K 53 .
We can define the critical frequency, the frequency at which the emissivity for a particle
with given energy mc2 peaks, to be (see equation [3.8]):
2
sin #
3 qB
2mc
!crit ( )
and, after averaging over solid angle, arrive at the formula for the power radiated per particle:
p
3q3 B sin # ! Z 1
K 5 (y)dy
P (!) =
2mc2 !crit !=!crit 3
(B.4)
For a power law momentum distribution as defined in equation (3.1), equation (B.4) yields a
spectral emissivity of
j (!) = N0
(mc)(a 5)=2
C1 N0 (B sin #)(a
B.1.3
p
3q3 B sin #
2c(a 1)
3a + 13
12
1)=2 ! ( a+3)=2
3a 7
12
!
3qB sin #
(a 3)=2
(B.5)
Equipartition:
With equation (B.5) and the assumption of equipartition between energy density in rela-
tivistic particles Uparticle and magnetic field energy density
netic field
Beq .
B 2 =8 we can solve for the mag-
Its value depends on the spectral slope and the cutoff in the momentum dis-
tribution at low (pmin ) and high energies (pmax ), since we have to integrate the momentum
distribution from equation (3.1) to find the total energy density
Uparticle =
Z pmax
pmin
N0 cp3 a dp
=
8
h
>
>
< N0 c p4 a
4 a
>
>
:
max
N0 c log ppmax
min
N0 C 2
p4mina
i
(for a 6= 4)
(for a = 4)
9
>
>
=
173
>
>
;
Setting this equal to the magnetic field energy density we can solve for N0 in terms of Beq and
insert this into equation (B.5), which enables us to solve for Beq :
!
3)=2 2=(a+3)
8C2 jobserved !(a
Beq =
C1
B.2
1 a
sin #( a+3 )
(B.6)
The Transport Equation
Equation (3.4), the transport equation, is of Fokker-Planck type, where the left hand side
represents the total advective derivative in phase space and the right hand side corresponds
to a scattering term due to radiative losses. Following Coleman and Bicknell (1988) we can
introduce a new variable q
p%
1=3 and rewrite equation (3.4) as
df p d% @f
df
+
=
+
dt 3% dt @p
dt
Df
=
Dt
dp Dq p @f
dt Dt q @p
Dq @f
Dt @q
@ 4 = Aq 2 %1=3
qf :
@q
Here, f
= f (t; x; p) is the momentum distribution function for assumed anisotropy. We define
D
Dt
@
@t@ + v r + dp
dt @p
to be the total advective time derivative in phase space. It follows that
D4 Dq
q f 4q3 f
Dt
Dt
q4
Dq @f
D 4 Dq @ 4 =
qf
qf
Dt @q
Dt
Dt @q
@ 4 = Aq2 %1=3
qf
@q
or, in the fluid rest frame
@4 @ 4 Dq
qf =
qf
+ Aq2 %1=3 :
@t
@q
Dt
174
Using the method of characteristics, we can solve this equation by finding the streamlines in
q t space, along which the quantity q4 f is conserved, i.e., we solve the characteristic equation
Dq
=
Dt
Aq2 %1=3 ;
which yields
q =
q(t0 )
R
1 + q(t0 ) tt0 A%1=3 dt0
q(t0 )
1 + q(t0 )(%0 )1=3 with the obvious definition for . This leads directly to equation (3.5).
From the fact that q 4 f is conserved along characteristics, we can easily see that the
expected spectral shape of a cooled power law is given by
f (p) = f (p0 )
B.3
% 4=3 p0 4
=p
%0
p
a
% 4=3
(1 p )a
%0
4
(B.7)
Spectral Cutoff Shapes
M99 used the 2 cm VLA maps by Owen et al. (1989) and ground based high resolution
data in the B, R, I, H, and K bands to fit a model spectrum along the jet. They assumed a
power law with sharp cutoff in the electron momentum distribution that drops immediately to
zero at the cutoff momentum and they left the power law index and the cutoff momentum as
free parameters. However, in Appendix B.2 we showed that the cutoff that naturally develops
due to synchrotron losses upstream tends to have a different spectral shape around the cutoff,
depending on the power law index a of the injected particle distribution. For a
4:3, i.e., for
RO 0:65, the expected cutoff in the synchrotron spectrum will be less sharp. Both cases are
shown in Figure B.1. As diffusion processes will also wash out the sharpness of the cutoff, we
should theoretically expect the spectrum to fall off less steeply than what was assumed in the
fit.
This demonstrates that the error estimate by M99 is rather optimistic, as it does not take
uncertainties in the exact shape of the cutoff into account (which could potentially have an
175
100
cutoff momentum pc
f(p) [arbitrary units]
102
10-2
10-4
10-6
10-8
10
101
100
j(ν) [arbitrary units]
1000
cutoff frequency νc
100
momentum p [arbitrary units]
10-1
10-2
10-3
sharp cutoff
natural cutoff
10-4
10-5
102
103
104
105
ν [arbitrary units]
106
107
Figure B.1: Electron momentum distributions and corresponding synchrotron spectra calculated for two different cutoff shapes. solid line: steep cutoff; dashed line: cutoff according to
synchrotron cooled distribution. All units are arbitrary.
effect on the amplitude of variations seen in the best fit c ). With increased error bars, our
model would yield more acceptable values for 2min , especially in the post knot A region.
Appendix C
Derivation of the Dispersion Relation for the Relativistic Rayleigh-Taylor
Instability
The derivation of the Rayleigh Taylor instability for a relativistic plasma by Allen and
Hughes (1984) is based on the formalism used in Chandrasekhar (1961), using Eulerian rather
than Lagrangian first order variables, which is not a particularly transparent derivation and is
formally not self-consistent (thought is does produce the correct results). Since the dispersion
relation given by Allen and Hughes (1984) contains typographical errors we found it appropriate
to derive the fully relativistic dispersion relation for any ideal gas (i.e., for arbitrary adiabatic
indices ad and arbitrary ratios of random internal energy to rest mass energy density). It should
also be noted that the incompressible analysis commonly used is only valid for wavelengths
smaller compared to the scale height in the fluid, even in the non-relativistic case. The analysis
by Mathews and Blumenthal (1977) assumes isothermal perturbations, which is not a realistic
assumption in the cases we are interested in.
The equation of motion is given by
T i ;
ui T 0 ; = 0,
where we assumed the
Einstein summation convention, i runs from 1 to 3. u is the four-velocity,
Here, w
the stress-energy
the fluid Lorentz factor. After a little algebra, this can be written as
tensor, and
2w
T
@v
@p
2
+ (v r) v v=c
+ rp = 0:
@t
@t
(C.1)
= e + p is the enthalpy, e is the internal energy, p is the pressure, and the rest mass
density, all measured in the fluid rest frame. For polytropic gases, e = 1=(ad
is the ratio of specific heats;
v is the three-velocity, vi = ui=
, i from 1 to 3.
1)p where ad
177
The continuity equation for a conserved particle number density n can be written as
@
@
+r n+n
+ r( v) = 0;
@t
@t
(C.2)
while the equation of state is
@
@
+ r p + p
+ r( v) = 0:
@t
@t
(C.3)
All of these equations are fully general within the limits of special relativity (i.e., weak gravity).
Specializing to Rayleigh-Taylor instability, we assume that a gravitational field is present
equivalent to the acceleration of the plasma in the GRB. We will assume that a planar contact
discontinuity exists in the fluid, placed at
z = 0.
Gravity is assumed to act perpendicularly
to that surface. The background flow is compressible, so for self-consistency we must assume
that the density and pressure of the background flow are stratified. For mathematical simplicity
we will assume that the gas has an exponential dependence and is isothermal on either side of
the discontinuity. Without loss of generality, we can also specify that the x-axis lie along the
k
direction of the wave vector .
Assuming small perturbations in the dynamical quantities, we can linearize the equations.
We can then Fourier transform the linearized equations in x and t and transform to a coordinate
system in which the perturbed interface is at rest. The new coordinates are (Mathews and
Blumenthal 1977):
n
x i t
k
z = z 0 exp (ikx + nt);
(C.4)
where is the (first order) displacement of the interface.
The perturbed equations then take the form
@w
@
) + (w n ) = 0
@
@
@p
@w
np~ + ad p(ikv + ) + (w n ) = 0
@
@
@p
nhv + ikp~ ik = 0
@
@ p~
nhw + + gh~ = 0
@
n~ + (ikv +
(C.5)
178
Here, a ~ indicates a first order quantity,
v
and
w are the (first order) velocity in the x and z
directions, and n i! , where ! is the temporal frequency.
The equations also indicate that the dependence of all first order quantities is the same.
Furthermore, the zeroth order quantities also have the same dependence (by assumption). A
little algebra then gives
@2w
@ 2
g
"
h
p
1
1 @w
c2 @
!#
hn2
1 k2 g2 2
+
+k
w=0
ad n2 p
pad
(C.6)
This equation has the well known solution
w = C+ exp K+ + C exp K (C.7)
where C+ and C are constants to be determined later and
K =
hg
2p
g
2c2
s
g 2 hg
+ 1
2p 2c2
(C.8)
1 k2 g2 hn2
+
+ k2
2
ad n p
pad
Since the energy contained in the perturbation must be finite, we must chose the constants
C such that the solution decays exponentially away from the interface. In other words, the real
part of K must be negative (positive) for positive (negative) .
The equations of motion dictate that the pressure be continuous across the discontinuity.
Equating the first order pressure on both sides finally leads to the dispersion relation. A little
algebra reveals
p~ =
k2 p + n2 nw @p
h ad
@
2
pk
ad
2
n + h
ad np @w
@
:
(C.9)
The dynamic boundary condition is simply
1
w
= (w vk ) = ;
n
n
(C.10)
179
dropping the second order term. This is derived from requiring that the fluid on both sides of the
interface move parallel to each other, in other words, no fluid crosses the interface. Together,
these equations yield
k2 gh1 + n2 K1 h1 k2 gh2 + n2 K2 h2
= n2 h2
:
n2 h1 + k 2 2
ad;1
p1
p2 + k ad;2
Here the subscripts 1 and 2 denominate the two sides of the shock.
(C.11)