Download The Magnetic Field of the Milky Way

The Magnetic Field of the Milky Way
by
Ronnie Jansson
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Department of Physics
New York University
January, 2010
——————————–
Prof. Glennys R. Farrar
Acknowledgments
This thesis would not have been possible if not for the keen insights and constant
support of my advisor, Prof. Glennys Farrar. I have benefited greatly from her
sage advice and owe her my deepest gratitude for guiding me through the thorny
jungle of physics research these last five years. It has been – and continues to be
– a pleasure working together with Glennys.
I am indebted to many of the professors at the CCPP for always being willing to
explain things to me: Andrew MacFadyen, David Hogg, Andrei Gruzinov, Patrick
Huggins, Roman Scoccimarro and Michael Blanton. Particular thanks to Andrew
and David for savvy career advice and writing recommendations. Special thanks
to Prof. Paul Chaikin for enlisting in my thesis committee.
Many thanks to my collaborators, Andre Waelkens and Torsten Ensslin. I am
very grateful to Yosi Gelfand for his insights on the interstellar medium; Jo-Anne
Brown, Bryan Gaensler and Ilana Feain for giving me access to their non-public
data; and Mulin Ding for handling all my computer worries and always increasing
my disk quota.
I also want to thank Christine Waite for making sure I got paid.
I owe a great deal of gratitude to my fellow graduate students: Ronin, Rakib,
Phil, Grant, Rachel, Abhishek, Seba, Guangtun, Lisa, Jeff, Eyal, Jo and Morad.
Finally, my warmest thanks to my family and friends – you didn’t help me with
this thesis, but I like you anyway.
iii
Abstract
The magnetic field of the Milky Way is a significant component of our Galaxy,
and impacts a great variety of Galactic processes. For example, it regulates star
formation, accelerates cosmic rays, transports energy and momentum, acts as a
source of pressure, and obfuscates the arrival directions of ultrahigh energy cosmic
rays (UHECRs). This thesis is mainly concerned with the large scale Galactic
magnetic field (GMF), and the effect it has on UHECRs.
In Chapter 1 we review what is known about Galactic and extragalactic magnetic fields, their origin, the different observables of the GMF, and the ancillary
data that is necessary to constrain astrophysical magnetic fields.
Chapter 2 introduces a method to quantify the quality-of-fit between data
and observables sensitive to the large scale Galactic magnetic field. We combine
WMAP5 polarized synchrotron data and rotation measures of extragalactic sources
in a joint analysis to obtain best-fit parameters and confidence levels for GMF
models common in the literature. None of the existing models provide a good
fit in both the disk and halo regions, and in many instances best-fit parameters
are quite different than the original values. We introduce a simple model of the
magnetic field in the halo that provides a much improved fit to the data. We show
that some characteristics of the electron densities can already be constrained using
our method and with future data it may be possible to carry out a self-consistent
analysis in which models of the GMF and electron densities are simultaneously
iv
optimized.
Chapter 3 investigates the observed excess of UHECRs in the region of the
sky close to the nearby radio galaxy Centaurus A. We constrain the large-scale
Galactic magnetic field and the small-scale random magnetic field in the direction
of Cen A, and estimate the deflection of the observed UHECRs and predict their
source positions on the sky. We find that the deflection due to random fields are
small compared to deflections due to the regular field. Assuming the UHECRs are
protons we find that 4 of the published Auger events above 57 EeV are consistent
with coming from Cen A. We conclude that the proposed scenarios in which most of
the events within approximately 20◦ of Cen A come from it are unlikely, regardless
of the composition of the UHECRs.
v
Contents
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
Introduction
1
1 Galactic magnetism
5
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2
Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.2.1
Faraday rotation measures . . . . . . . . . . . . . . . . . . .
6
1.2.2
Synchrotron emission . . . . . . . . . . . . . . . . . . . . . .
12
1.2.3
Starlight polarization . . . . . . . . . . . . . . . . . . . . . .
15
1.2.4
Zeeman effect . . . . . . . . . . . . . . . . . . . . . . . . . .
16
1.2.5
Ultrahigh energy cosmic rays
. . . . . . . . . . . . . . . . .
18
Ancillary data for magnetic field observables . . . . . . . . . . . . .
20
1.3
vi
1.3.1
Relativistic electron density . . . . . . . . . . . . . . . . . .
20
1.3.2
Thermal electron density . . . . . . . . . . . . . . . . . . . .
23
The magnetic field of the Milky Way . . . . . . . . . . . . . . . . .
25
1.4.1
The regular field . . . . . . . . . . . . . . . . . . . . . . . .
27
1.4.2
Turbulent fields . . . . . . . . . . . . . . . . . . . . . . . . .
28
1.4.3
Fields in the halo . . . . . . . . . . . . . . . . . . . . . . . .
28
1.5
Magnetic fields in external galaxies . . . . . . . . . . . . . . . . . .
30
1.6
Cosmological magnetic fields . . . . . . . . . . . . . . . . . . . . . .
31
1.7
The origin of cosmic magnetic fields . . . . . . . . . . . . . . . . . .
34
1.7.1
Generation of seed fields . . . . . . . . . . . . . . . . . . . .
35
1.7.2
The α − Ω dynamo . . . . . . . . . . . . . . . . . . . . . . .
36
1.7.3
Numerical simulations . . . . . . . . . . . . . . . . . . . . .
37
1.4
2 Constraining models of the large-scale Galactic magnetic field
with WMAP5 polarization data and extragalactic rotation measure sources
42
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
2.2
Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
2.3
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
2.3.1
Quality-of-fit . . . . . . . . . . . . . . . . . . . . . . . . . .
48
2.3.2
Parameter estimation . . . . . . . . . . . . . . . . . . . . . .
48
2.3.3
Producing simulated observables from models . . . . . . . .
49
vii
2.3.4
2.4
2.5
2.6
2.7
Disk - halo separation . . . . . . . . . . . . . . . . . . . . .
50
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
2.4.1
Synchrotron radiation . . . . . . . . . . . . . . . . . . . . .
51
2.4.2
Extragalactic RM sources . . . . . . . . . . . . . . . . . . .
55
3D models of the magnetized interstellar medium . . . . . . . . . .
63
2.5.1
large-scale magnetic field . . . . . . . . . . . . . . . . . . . .
63
Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . .
73
2.6.1
Fitting models to entire data set . . . . . . . . . . . . . . . .
75
2.6.2
Fitting models to disk data . . . . . . . . . . . . . . . . . .
77
2.6.3
Fitting models to halo data . . . . . . . . . . . . . . . . . .
81
2.6.4
Scale heights of electron distributions . . . . . . . . . . . . .
83
2.6.5
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . .
88
3 Magnetic deflection of UHECRs in the direction of Cen A
92
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
3.2
Centaurus A - an overview . . . . . . . . . . . . . . . . . . . . . . .
96
3.3
Observables of the magnetic field toward Cen A . . . . . . . . . . .
98
3.3.1
Synchrotron emission . . . . . . . . . . . . . . . . . . . . . .
98
3.3.2
Extragalactic rotation measure sources . . . . . . . . . . . . 106
3.3.3
Other potentially useful data sets . . . . . . . . . . . . . . . 106
3.4
Magnetic field modeling . . . . . . . . . . . . . . . . . . . . . . . . 108
viii
3.4.1
Results of the fit . . . . . . . . . . . . . . . . . . . . . . . . 113
3.4.2
Purely random fields . . . . . . . . . . . . . . . . . . . . . . 114
3.5
Estimated deflections . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.6
Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 119
Conclusion
120
A Maximum Likelihood Method for Cross Correlations with Astrophysical Sources
126
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
A.2 Maximum Likelihood Approach for the Cross-Correlation Problem . 129
A.2.1 The HiRes Maximum Likelihood method . . . . . . . . . . . 129
A.2.2 Extending method to differing luminosities . . . . . . . . . . 131
A.3 Simulation trials
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
A.3.1 Dilute source and UHECR data sets . . . . . . . . . . . . . 135
A.3.2 Dispersion in results . . . . . . . . . . . . . . . . . . . . . . 136
A.3.3 High source and UHECR densities . . . . . . . . . . . . . . 136
A.3.4 Sensitivity to experimental resolution . . . . . . . . . . . . . 138
A.3.5 Clustering of sources . . . . . . . . . . . . . . . . . . . . . . 141
A.4 Application to BLLacs and x-ray clusters . . . . . . . . . . . . . . . 142
A.4.1 X-ray clusters . . . . . . . . . . . . . . . . . . . . . . . . . . 142
A.4.2 BLLacs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
ix
A.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 148
Bibliography
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
x
List of Figures
1.1
A hypothetical UHECR multiplet with deflections from various
GMF models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
1.2
Comparison of two relativistic electron distributions. . . . . . . . .
24
1.3
The NE2001 thermal electron density model. . . . . . . . . . . . . .
26
1.4
The vertical distribution of the total magnetic field strength. . . . .
29
1.5
Magnetic field orientations of M51 and NGC 5775. . . . . . . . . . .
32
1.6
A schematic view of the α − Ω dynamo. . . . . . . . . . . . . . . .
38
1.7
A simulated galactic disk field. . . . . . . . . . . . . . . . . . . . . .
40
2.1
Overview of the implemented analysis. . . . . . . . . . . . . . . . .
47
2.2
Polarized synchrotron intensity, Mollweide projection. . . . . . . . .
52
2.3
Sky maps of Stokes Q and U. . . . . . . . . . . . . . . . . . . . . .
53
2.4
Synchrotron mask. . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
2.5
Extragalactic RM sources. . . . . . . . . . . . . . . . . . . . . . . .
58
2.6
Total variance of the RM sources. . . . . . . . . . . . . . . . . . . .
59
xi
2.7
Histogram of RM values as a function of galactic latitude. . . . . .
60
2.8
Field model examples. . . . . . . . . . . . . . . . . . . . . . . . . .
67
2.9
Field model examples, continued. . . . . . . . . . . . . . . . . . . .
68
2.10 The reduced χ2 for a selection of GMF models. . . . . . . . . . . .
75
2.11 The best-fit disk and halo models. . . . . . . . . . . . . . . . . . . .
79
2.12 The 1σ and 2σ confidence levels for the best-fit parameters of the
Sun08D model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
2.13 The 1σ and 2σ confidence levels for the best-fit parameters of the
E2No model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
2.14 The best-fit vertical magnetic scale height vs. the vertical scale
height of the NE2001 thermal electron density of the thick disk. . .
86
3.1
The published Auger UHECR events above 57 EeV around Cen A.
95
3.2
Centaurus A in total intensity (Stokes I) at 408 MHz (Haslam et al.
1982). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
97
Polarized synchrotron radiation at 22 GHz with texture following
the magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
3.4
Polarized synchrotron intensity around Cen A. . . . . . . . . . . . . 102
3.5
PI with polarization angles added. . . . . . . . . . . . . . . . . . . . 103
3.6
Circled low-emission regions near Cen A. . . . . . . . . . . . . . . . 104
3.7
WMAP 22 GHz total synchrotron radiation. . . . . . . . . . . . . . 105
3.8
188 extragalactic sources with lines-of-sight outside Cen A. . . . . . 107
xii
3.9
The 7 pulsars within an angular distance of 10◦ from Cen A with
measured RM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.10 Polarized synchrotron intensity and starlight polarization bars. . . . 110
3.11 Estimates of UHECR source locations near Cen A . . . . . . . . . . 118
A.1 The number of correlations vs. the detector resolution. . . . . . . . 137
A.2 Comparison of the extracted number of correlations for specific realizations using the two ML methods. . . . . . . . . . . . . . . . . . 138
A.3 Correlations found by the HiRes method for the case of very high
event densities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
A.4 Average number of found correlations when resolution is incorrectly
estimated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
A.5 Sensitivity to clustering in source dataset. . . . . . . . . . . . . . . 142
xiii
List of Tables
1.1
Observables of the Galactic magnetic field. . . . . . . . . . . . . . .
2.1
Best-fit parameters and 1σ confidence levels for the best-fitting models. 78
3.1
Best fit GMF parameters . . . . . . . . . . . . . . . . . . . . . . . . 114
20
A.1 Correlations between X-ray clusters and UHECRs. . . . . . . . . . . 142
A.2 Correlations between BLLacs and UHECRs. . . . . . . . . . . . . . 146
A.3 HiRes UHECR cross correlations with BL Lacs using the noninteger
implementation of the generalized Maximum Likelihood method. . . 147
xiv
Introduction
Magnetic fields are ubiquitous in the universe: they are present in planets, stars,
galaxies, clusters of galaxies, and – though yet not measured – are likely to exist
in the very voids separating galaxies and clusters. The effects of cosmic magnetic
fields are diverse. They align the spinning micron-sized dust particles in the interstellar medium, accelerate and deflect cosmic rays, and slow the gravitational
collapse of matter into galaxies and stars.
In the Milky Way, typical field strengths are of order micro-Gauss, but up
to three orders of magnitude greater in filaments in the Galactic Center. While
the magnetic field exhibits regular large-scale structure reminiscent of the matter distribution in the spiral arms, small-scale random fields are comparable in
strength. The average energy density of the Galactic magnetic field (GMF) is about
1 eV/cm3 , which is similar to the energy density of starlight and cosmic rays, and
the kinetic energy density due to thermal motion of particles in the Galaxy. Thus
we expect the GMF to have non-negligible effects on processes occurring throughout the interstellar medium, such as star formation, energy transport, cosmic ray
1
propagation, etc.
The history of the study of Galactic magnetism started with the discovery of the
polarization of starlight by Hiltner (1949). The phenomenon was presumed to be
caused by magnetic fields permeating interstellar space, and was soon given a plausible explanation by Davis & Greenstein (1951) in terms of magnetically aligned
dust grains. After the discovery of synchrotron radiation (Schwinger 1949) and
the development of radio astronomy in the early 1950s several non-thermal radioemitting sources were argued to consist of energetic electrons spiraling in magnetic
fields, emitting synchrotron radiation (Alfvén & Herlofson 1950; Shklovsky 1953).
These were indirect observations of interstellar magnetic fields. Bolton & Wild
(1957) suggested that magnetic fields could be studied directly by the Zeeman effect. Due to technical challenges, the discovery of Zeeman splitting in interstellar
gas of neutral hydrogen did not occur until a decade later (Davies et al. 1968). At
the same time, the first measurements of the Galactic magnetic field using Faraday rotation of the polarized radio emission from pulsars were made by Lyne &
Smith (1968). Faraday rotation applied to polarized extragalactic radio sources
led Davies (1968) to conclude that a large-scale, regular magnetic field permeated
the Milky Way.
The thesis is organized as follows. Details of the above observables, how they
are related to the magnetic field, and a summary of what has been learned about
the Galactic magnetic field up to now is covered in the next Chapter, and is
2
intended to provide the necessary background information and context for the
subsequent chapters.
Chapter 2 is the central piece of the thesis, where we constrain models of the
large-scale Galactic magnetic field common in the literature by a joint analysis
of the polarization data in the WMAP5 22 GHz band and rotation measures of
extragalactic radio sources. The synergistic advantage of combining these two data
sets goes beyond the benefit of achieving a larger number of data points; the two
observables probe mutually orthogonal components (perpendicular and parallel
to line-of-sight) of the magnetic field. We develop estimators of the variance in
the two data sets due to turbulent and other small scale or intrinsic effects. We
calculate the χ2 for a particular choice of GMF model and parameter choice, and
use a Markov Chain Monte Carlo algorithm to optimize the parameters of each field
under consideration. Thus we can quantitatively compare the validity of different
models with each other.
In Chapter 3 we extend the GMF modeling of Chapter 2 by adding more data,
and combining separate field models for the disk and halo and simultaneously fit
them to the combined RM and polarized synchrotron data sets. We predict the
UHECR deflection in a particularly interesting region of the sky, in the direction
of the nearby galaxy Centaurus A, to confirm or refute the claim that the galaxy
is the source of numerous UHECR observed in its general direction on the sky.
With models of the Galactic magnetic field improving, it may become possible
3
to correlate UHECR arrival directions – corrected for magnetic deflection – with a
data set of UHECR source candidates. In appendix A, based on Jansson & Farrar
(2008), we present a Maximum Likelihood method to evaluate the quality of the
correlation between two such data sets.
4
Chapter 1
Galactic magnetism
1.1
Introduction
This Chapter is intended to provide the necessary backround – theoretical and
observational – to the subsequent chapters. Since the Galactic magnetic field was
discovered 60 years ago (Hiltner 1949) the field (of research) has grown immensely.
Hence, we will concentrate on issues directly relating to the results obtained in the
later chapters, and lightly touch on more general topics that provide context for
the research presented in this thesis. For a more in-depth treatment of galactic
and cosmic magnetic fields, many excellent reviews exist, e.g. work by Beck et al.
(1996), Widrow (2002) and Kulsrud & Zweibel (2008).
The Chapter is organized as follows: section 1.2 discusses the relevant observables and their connection to the magnetic field; section 1.3 treats the important
5
ancillary data of Galactic electron distributions; sections 1.4 and 1.6 briefly review
the knowledge of Galactic and extragalactic magnetic fields, respectively; the final
section covers popular theories of the origin of cosmic magnetic fields.
1.2
Observables
While direct measurements of the Galactic magnetic field are not possible, a host
of indirect methods exists. In this section we briefly review the most common
observables. Table 1.1 gives a summary of the currently available data sets.
1.2.1
Faraday rotation measures
The derivation below follow Harwit (2006) and Rybicki, G. B. and Lightman, A. P.
(1986).
Preliminaries
Under the assumption that charges are stationary and in vacuum, the electric field
P
P
in the presence of a collection of charges i qi is E = F/q = i rq3i ri . If charges
i
are moving, they experience a magnetic force; and if not in vacuum but a dielectric
medium, that medium cancels some of the electric field by rearranging itself. We
can define D, the dielectric displacement, which is equal to the electric field caused
6
by the charge distribution if it were in vacuum. For a uniform dielectric,
F=
q X qi
ri
i ri3
and
D = E.
(1.1)
Next, define the polarization field as
P=
D−E
( − 1)E
=
,
4π
4π
(1.2)
which is the field obtained from the rearrangement of charges in the dielectric
(the 4π is just a convention), i.e., the field caused by the “bound” charges. The
polarization field is also equal to the electric dipole moment per unit volume,
P = nqd,
(1.3)
where n is the density of dipoles of charge q and separation d.
Dispersion measures
In an ionized medium, assuming electron-ion collisions are rare, an electromagnetic
wave E(r, t) = E0 (r) cos ωt accelerates an electron according to
mr̈ = −eE = −eE0 (r) cos ωt,
7
(1.4)
with m and −e the mass and charge of the electron. Solving for r, we get
r=
e
E.
mω 2
(1.5)
Following equations 1.2 and 1.3 we find,
ne2
( − 1)E
E=
,
P = −ner = −
2
mω
4π
(1.6)
from which we obtain the dielectric constant of the ionized medium,
=1−
4πne2
.
mω 2
(1.7)
For a wave propagating along the x-direction, its electric field amplitude can be
expressed as E = E0 cos(kx ± ωt), where
k 2 c2
k 2 c2
ω =
=
2 ,
1 − 4πne
mω 2
2
(1.8)
which gives the frequency,
ω 2 = k 2 c2 +
4πne2
≡ k 2 c2 + ωp2 ,
m
8
(1.9)
where we have defined the plasma frequency,
r
ωp ≡
√
4πne2
∼ 5.6 × 104 n rad s−1 .
m
(1.10)
When ω < ωp , the wave cannot propagate (which happens for instance at ω . 1
MHz in Earth’s ionosphere). When ω > ωp , the group velocity becomes
vg =
dω
c
=q
,
dk
1 + ωp2 /c2 k 2
(1.11)
i.e., the group velocity is frequency dependent. A pulse traveling a distance D will
arrive in tD = D/vg . Assuming ω ωp we can approximate tD as
D
D
tD =
'
vg
c
ωp2
1− 2
ω
− 12
D
'
c
ωp2
1+ 2
2ω
4πe2
D
=
+
c
2mcω 2
Z
D
n ds,
(1.12)
0
where the dispersion measure is defined as
Z
DM ≡
D
n ds.
(1.13)
0
The dispersion measure can be used to measure the distance to pulsars. The
spectrum of each individual pulse contains a wide band of frequencies. This allows
measurement of the arrival-time delay for different frequencies of the pulse, and
hence the dispersion measure. By assuming an electron density (typical value
9
∼ 0.03 cm−3 ) we can calculate the distance to the pulsar.
Rotation measures
A circularly polarized electromagnetic wave incident on a free electron induces
circular motion of the electron. If an external magnetic field B is also present
parallel to the direction of propagation of the radiation, the electron is subject to
a Lorentz force F = (−e/c)v × B. Let r denote the displacement of the electron.
Depending on whether the incident radiation has a right or left handed circular
polarization, the Lorentz force will be directed along r or −r, respectively. The
centripetal force experienced by the electron is then,
−mω 2 r = −eE ±
eωBr
.
c
(1.14)
Solving for r yields,
eE
r=
m
eωB
mω ±
mc
2
−1
,
(1.15)
which together with equation 1.2 and 1.3 we can use to identify the dielectric
constant
L,R
4πne2
=1−
m
eωB
ω ±
mc
2
−1
=1−
4πne2
,
mω(ω ± ωc )
(1.16)
where the gyrofrequency ωc ≡ eB/mc, is the frequency a non-relativistic electron
spirals in a magnetic field of strength B. For B in µG , we obtain ωc ≈ 17B
s−1 . Due to L 6= R , left- and right-handed polarized waves propagate at different
10
speeds. Since any linearly polarized wave can be seen as a sum of two circularly
polarized waves with equal amplitude but opposite helicity, it follows that a linearly
polarized wave has its direction of polarization rotated when propagating through
√
a magnetized plasma. The refractive index of a medium is n = , so we can write
L − R = n2L − n2R = 2nω ∆n,
(1.17)
where we approximate, assuming nω − 1 1,
nω ≈ 1 −
2πne2
.
mω 2
(1.18)
Inserting this into equation 1.17, assuming ω ωc and ne2 /mω 2 1, we obtain
∆n =
4πne2 ωc
.
mω 3
(1.19)
Using v = c/n, the difference in velocity of the left and right handed polarized
waves are ∆v = c∆n/n2ω ≈ c∆n. Thus the lag in phase per unit time is ω∆n,
with the direction of linear polarization rotating at half this rate, ∆θ ≈
ω∆n
.
2
Substituting for ∆n and ωc , using ω = 2πc/λ, and identifying the magnetic field
strength in our derivation to be the component of an arbitrary magnetic field
parallel to the direction of propagation of the electromagnetic wave, we arrive at
11
the expression for Faraday rotation:
e3 λ2
∆θ ≈
2πm2 c4
Z
D
nBk ds ≡ RM λ2 ,
(1.20)
0
which defines the Rotation Measure, RM . Thus, for a source with intrinsic linear
polarization direction θ0 , the observed polarization direction will be
θ = θ0 + RM λ2 .
(1.21)
For astrophysically interesting situations, we can write in units of cm−2 ,
◦
Z
RM ' 0.5
0
L
n
0.1 cm−3
Bk
µG
ds
kpc
.
(1.22)
For example, at λ = 21 cm the net rotation is ∆θ ∼ 200◦ after traversing 1 kpc
of a medium with n ∼ 0.1 cm−3 and Bk ∼ 1 µG. For the same medium, K-band
photons (22 GHz, λ = 1.4 cm) is rotated ∼ 1◦ only.
1.2.2
Synchrotron emission
Relativistic electrons spiralling along magnetic field lines radiate synchrotron radiation . For a power law distribution of relativistic electrons – commonly called
“cosmic ray” electrons – the electron density ncre is characterized by a spectral
12
index s,
ncre (E)dE ∝ ncre,0 E −s dE,
(1.23)
where ncre,0 is a normalization factor. The synchrotron emissivity (Rybicki, G. B.
and Lightman, A. P. 1986) is
1+s
jν ∝ ncre,0 B⊥2 ν
1−s
2
.
(1.24)
For a regular magnetic field and the above relativistic electron distribution, the
emitted synchrotron radiation has a large degree of linear polarization, around
75% for a power law distribution of electron with spectral index s = 3. Observationally, the percentage of polarization is typically much lower than this due
to depolarizing effects such as Faraday depolarization and the presence of turbulent or otherwise irregular magnetic fields which depolarize the radiation through
line-of-sight averaging.
The equipartition argument
A widely used, and widely debated (see Beck & Krause (2005), and references
therein) assumption about the interstellar medium (ISM) in the Miky Way and
other galaxies is the energy equipartition between magnetic fields and cosmic rays.
Equipartition assumes an equality between the magnetic and cosmic ray energy
densities, B = cr . When this extremely useful relation is assumed to hold, it
13
enables the calculation of magnetic field strengths of other galaxies, interstellar
clouds, or any other astrophysical object based on observable radio (synchrotron)
emission.
An illustrative example of equipartion is the estimation of the vertical scale
height of the Galactic magnetic field. The observed Galactic synchrotron emissivity
has an exponential scale height of z0syn ≈ 1.5 kpc (Beck 2001). Since B ∝ B 2
and cr ∝ ncre , equipartition applied to equation (1.24) yield for the synchrotron
emissivity,
j ∝ ncre B
1+s
2
∝ B2B
1+s
2
∝B
5+s
2
.
(1.25)
If then, j = j0 exp(−z/1.5 kpc), it implies that
z
2
B = B0 e− 1.5 kpc 5+s
(1.26)
If we assume a synchrotron spectral index s = 3 (Bennett et al. 2003), it follows
that the vertical exponential scale height of the Galactic magnetic field is z0 ≈ 6
kpc.
The instances, and to what degree, the equipartition argument can be trusted
are still under debate. Duric (1990) provide general arguments why equipartition
should be reliable to at least within an order of magnitude.
14
1.2.3
Starlight polarization
The polarization of optical starlight was first discovered by Hiltner (1949). As
light from stars in the same vicinity were found to have similar polarization directions it was concluded that the interstellar medium was the cause of this effect,
not the stars themselves. Soon after, Davis & Greenstein (1951) argued that the
polarization was caused by the alignment of spinning, non-spherical grains through
the mechanism of paramagnetic relaxation. This effect causes the long axis of the
grains to align perpendicular to the ambient magnetic field and preferentially absorb light that is polarized in the direction of the grain’s long axis, and hence gives
a net polarization of the unabsorbed light parallel to the magnetic field. However,
since the discovery of the Davis-Greenstein effect, ten additional processes that
affect the alignment of interstellar grains have been discovered (see Draine (2003)
for a review). The various processes include effects due to, e.g., electron spin, nuclear spin, dust-gas temperature differences, radiative torques, and H2 formation
on the grains. Each process may be the dominant cause of grain alignment for a
particular grain size, temperature or shape, or depend on the ambient radiation
field or magnetic field. It is humbling to note that of all those processes – even
those discovered during the last ten years – none required any physics beyond what
was known in 1950.
Beyond the formidable challenges and incomplete nature of grain alignment
physics, additional drawbacks of using starlight polarization to study the large
15
scale GMF exist: First, since individual stars must be observed, only the nearby
(. 3 kpc) part of the Galaxy can be probed. Second, starlight polarization is a
self-obscuring effect based on the extinction of light, and about 3% polarization
corresponds to roughly one visual magnitude of extinction. For these reasons
starlight polarization is best used as a probe of small scale magnetic structures,
such as interstellar clouds, and not the large scale GMF.
Currently, about 104 measurements of starlight polarization exist. This number
is growing quickly through the use of automated surveys (e.g., Clemens et al.
(2009)) and is approaching 106 within about a year.
1.2.4
Zeeman effect
An electron orbiting a nucleus acquires a magnetic moment
Z
µ=
i · da =
L
eme vr
eL
−ev 2
πr = −
=−
≡ −µB
2πr
2me
2me
~
(1.27)
where the Bohr magneton is defined as µB = e~/2me . The normal Zeeman effect is
due to the magnetic moment caused by the electron’s orbital angular momentum.
The interaction energy between a magnetic field and a magnetic dipole is ∆E =
−µ · B. If we let the direction of the magnetic field define the z-axis, the orbital
angular momentum of the electron projected on to the z-axis takes on values
Lz = ml ~, with ml = 0, ±1, . . . , ±l. Measuring this splitting of atomic levels in
16
stars yield typical values of ∼ 1 G, but with fields greater than 3×104 G having
been measured in some A stars. For the Sun, sunspots of 1000s G are common. In
the interstellar medium, one typically measures Zeeman splitting in OH and water
masers, or the 21 cm line for neutral clouds. In the latter case, three different
transitions of the hyperfine splitting of the ground state are possible, with ∆ml =
0, ±1. The unshifted frequency, ν0 = 1.42 GHz, corresponds the case ∆ml = 0,
and the frequencies
ν± = ν0 ±
eB
4πme c
(1.28)
correspond to ∆ml = ∓1. For a magnetic field of order 10 µG the difference in
frequency is ∆ν = ν+ − ν− ∼ 30 Hz. This splitting in frequency is tiny compared
to the Doppler broadening of the lines. Per km/s of random motion of the gas,
the broadening is ∆ν = ν0 v/c ∼ 5 kHz. While it is thus impossible to actually
measure the splitting of the 21 cm line directly, partial information of the magnetic field can be recovered due to the transitions having different polarization
properties. Viewed along the magnetic field direction the two shifted components
are left/right circularly polarized, and the unshifted component missing. Viewed
perpendicular to the field, the shifted components are linearly polarized normal to
the field line, and the unshifted component is linearly polarized in the direction
of the field. In theory, measuring the Stokes Q, U (linear polarization) and V
(circular polarization) is enough to determine Btot . In practice, however, Stokes
Q and U are extremely weak (they are proportional to the second derivative of
17
the line profile) and only the Stokes V (the difference between the two circular
polarizations, and proportional to the first derivative of the line profile) is possible
to measure, yielding constaints on Blos only.
1.2.5
Ultrahigh energy cosmic rays
Ultrahigh energy cosmic rays (UHECRs) are most likely atomic nuclei of extragalactic origin, accelerated in magnetic shocks to kinetic energies ∼ 1018−20 eV.
The ultrarelativistic nuclei travel at the speed of light and have their trajectories
deflected when traversing cosmic magnetic fields. The larmor radius for a UHECR
is
R = 110 kpc ×
ZB1µG
,
E100 EeV
(1.29)
which translates to a deflection angle of a few degrees for a 100 EeV proton using
standard estimates of the Galactic magnetic field.
One of the major unanswered questions in astrophysics is what astrophysical
objects act as sources of cosmic rays at the highest energies. A multitude of sources
has been proposed (e.g., AGNs, BLLacs, GRBs, radio galaxies, galaxy clusters),
but the obfuscating nature of cosmic magnetic fields has so far left all cases of
claimed correlations unconvincing. An accurate characterization of the large scale
GMF will allow calculations of UHECR deflection angles if the composition of
UHECRs are known. Even a modest improvement in correcting the source direc-
18
tions will significantly aid in the search for UHECR sources.
A unique tool in studying the cosmic magnetic fields between us and the
UHECR source is available if the composition and true source of a UHECR are
known. Given its charge Z, the deflection angle of a cosmic ray is a direct meaR
surement of B⊥ ds along its trajectory. Specifically, it is not dependent on any
ancillary data, such as the electron density, As will be discussed in §1.3, electron
densities are poorly known and introduce a large uncertainty to the best-fit parameters of models of the Galactic magnetic field inferred from synchrotron emission
and rotation measures. Indeed, reliable measurements of B⊥ from UHECR observations would allow excellent constaints on, e.g., the distribution of relativistic
electrons by measurements of synchrotron radiation.
Until recently the highest energy cosmic rays have usually been presumed to
be protons. However, in the analysis of air showers (The Pierre Auger Collaboration: J. Abraham et al. 2009a) evidence have been found that the composition of
UHECRs becomes heavier at the highest energies. If this is true, the expected magnetic deflections become much larger than previously assumed, and the prospects
of UHECR astronomy may become bleak.
Multiplets
A powerful way to break the degeneracy between symmetries in GMF models
would be to use future UHECR multiplets. As the deflection angle of a UHECR
19
Table 1.1: Observables of the Galactic magnetic field.
Dataset
Synchrotron emission
RM: pulsars
RM: X-Galactic
Starlight polarization
Zeeman splitting
UHECR multiplets
measures what?
B⊥ orientation
Bk
Bk
B⊥ orientation
Bk in situ
B⊥
ancillary data
ncre
ne
ne
grain physics
none
CR charge
data points
3×50k (WMAP)
529
∼1500
∼10k
∼100s
none yet
region covered
full sky
mainly disk; . 10 kpc
roughly uniform
mainly disk; . 3 kpc
near quadrant
-
is inversely proportional to its energy and proportional to the transverse magnetic
field, the arrival directions of an UHECR multiplet will roughly form a string on the
sky; the highest rapidity (proportional to E/Z) cosmic ray closest to the direction
of the source, and the lower rapidity cosmic rays further removed according to their
energy. From their energies and angular position on the sky, their common source
can be estimated. A hypothetical multiplet placed in the southern hemisphere is
shown in figure 1.1, together with the arrival directions of the multiplet as predicted
for an example spiral field with the four different GMF symmetries. It is clear that
if a number of such multiplets would be discovered in future experiments they
would be very useful in breaking the degeneracy of the model symmetries.
1.3
1.3.1
Ancillary data for magnetic field observables
Relativistic electron density
The Galactic density of relativistic electrons, also known as cosmic-ray electrons,
is a poorly known quantity. Relativistic electrons and positrons (both contribute
equally to synchrotron emission) are assumed to be accelerated in shocks of su20
−10
Source
BSSS
−20
BSSA
DSSS
−30
b
DSSA
−40
−50
−60
290
300
310
l
320
330
Figure 1.1: A hypothetical UHECR multiplet source along with the UHECR locations as predicted from bisymmetric (BSS, see §2.5.1) and disymmetric (DSS)
spiral GMF models. UHECR are protons with energies selected in the 1019−20 EeV
range, with the size of the markers proportional to the event energies.
21
pernova remnants (SNRs). Direct measurements have been made in the energy
range 106 − 1012 eV, but at the lower end solar modulation completely obscures
the Galactic contribution.
Following Page et al. (2007) (who were influenced by Drimmel & Spergel (2001))
we use a simple exponential model for the spatial distribution of cosmic ray electrons,
Ccre (r, z) = Ccre, 0 exp(−r/hr ) sech2 (z/hz ),
(1.30)
with the scale heights hr = 5 kpc and hz = 1 kpc. The quantity Ccre (r, z) is defined
by
N (γ, r, z)dγ = Ccre (r, z)γ p dγ,
(1.31)
where N is the number density. The normalization factor Ccre, 0 is such that for
10 GeV electrons, Ccre (Earth) = 4.0 × 10−5 cm−3 , the observed value for 10 GeV
electrons at Earth (Strong et al. 2007). The number density for other energies is
calculated assuming a power law distribution with spectral index p = −3 (Bennett
et al. 2003).
A numerical tool for cosmic ray propagation, GALPROP (Strong et al. 2009),
could potentially be used to make better predictions of the 3D distribution of
electrons. GALPROP uses observational data for nuclei, electrons and positrons,
gamma rays and synchrotron radiation etc., and solves the transport equations
taking into account effects such as diffusion, nuclear spallation, and various energy
22
losses. However, many estimates and assumptions of the sources of cosmic rays
must be made in order to calculate the final distribution. In the case of relativistic
electrons, GALPROP assumes that the production of electrons follows the distribution of supernova remnants. However, since it is difficult to determine the
Galactic distribution of supernova remnants from observations, GALPROP uses a
modeled pulsar distribution (Strong et al. 2004; Lorimer 2004), with the assumption that pulsars trace supernova remnants. Unfortunately, it is not clear if the
dearth of observed pulsars close to the Galactic center is due to a lack of pulsars
or selection effects. The modeled pulsar distribution used by GALPROP assumes
that the number density of pulsars in the Galactic center is zero. How well this
corresponds to reality is not clear (Bailes & Kniffen 1992), and is still under debate. Figure 1.2 compares the spatial term C(r, z) predicted by GALPROP with
the the Drimmel-Spergel model used throughout this work.
1.3.2
Thermal electron density
The best available 3D model of the Galactic distribution of thermal electrons
is the NE2001 model (Cordes & Lazio 2002, 2003), based on pulsar dispersion
measures, measurements of radio-wave scattering, emission measures, and multiple
wavelength characterizations of Galactic structure. The model has four different
components: a thin disk, a thick disk, spiral arms, and some local over/underdense
regions (e.g., supernova remnants). The model is shown in figure 1.3. The thick
23
Figure 1.2: Top: Spatial term for relativistic electron density, C(r, z) in cm−3 ,
used in this work. Botton: Spatial term from GALPROP code (Andy Strong,
2009, private communication).
24
disk has a vertical scale height of 0.97 kpc. We adopt NE2001 as our baseline
model for the thermal electron density. We note, however, that the vertical scale
height in this model is poorly constrained due to uncertainties in pulsar distances.
Gaensler et al. (2008) estimate the thick disk scale height to be 1.83+1.2
−2.5 kpc by
only using pulsars with independently measured distances, and Sun et al. (2008)
also suggest an increase in the scale height by a factor of two in order to avoid an
excessively large magnetic field in the halo, when fitting to RM data. In section
2.6.4 we investigate the sensitivity of the best-fit magnetic field parameters on the
choice of electron density scale height.
1.4
The magnetic field of the Milky Way
Magnetic fields are ubiquitous in the Galaxy; they permeate the diffuse interstellar
medium and extend beyond the Galactic disk, and are present in stars, supernova
remnants, pulsars and interstellar clouds. The magnetic field in the diffuse ISM
has a large scale regular component and also a small scale turbulent component.
A standard estimate of the strength of the total Galactic magnetic field near the
Sun is 6 ± 2 µG (Beck 2001). The ratio between regular and random field strength
is estimated from starlight and synchrotron data to be 0.6 − 1.0, but is expected
to vary throughout the Galaxy: it is believed that the total field in optical arms is
the strongest, and mainly turbulent; in the inter-arm regions the regular field may
25
Figure 1.3: The NE2001 thermal electron density model. The figure shows a
30 × 30 kpc Galactic disk at z = 0. The grayscale is logarithmic. The light colored
features in the solar neighborhood are a number of underdense regions. The small
dark speck in one of the ellipsoidal underdensities is the nearby Gum nebula and
Vela supernova remnant.
26
dominate, possibly forming “magnetic arms” that extend farther than the optical
arms. Within (∼ 200 pc) of the Galactic center Ferriere (2009) estimates the field
strength to be ∼ 10 µG and roughly poloidal in shape in the diffuse medium, and
finds ∼ 1 mG fields in filaments and dense clouds.
1.4.1
The regular field
Mapping the large scale geometry of the regular field in our own Galaxy is extremely challenging and require extensive modeling. That is the main topic of this
thesis and covered in Chapter 2. Specific examples of models for the large scale
field is given in §2.5.1.
As seen in external galaxies, the regular field tends to have a spiral shape,
reminiscent of the matter distribution in the disk. The local pitch angle is difficult
to measure (Vallée 2005), but is estimated to p ' 10◦ (where p = 0◦ correspond
to a completely azimuthal field). The pitch angle is likely spatially varying.
The Sun is located between the Perseus and the Sagittarius spiral arms. Rotation measures indicate that the magnetic field is clockwise in the Perseus arm
(located outside the solar circle), and counter-clockwise in the Sagittarius arm
(Wielebinski & Beck 2005). The nature and number of large scale field reversals
are still an open question, and several other reversals have been suggested (Han
et al. 1999).
27
1.4.2
Turbulent fields
The interstellar medium is turbulent over a very large range of scales. Armstrong
et al. (1995) showed that the power spectrum of the interstellar thermal electron
density was consistent with a power law with a Kolmogorov spectral index of
5/3 (Kolmogorov 1941) from scales of 107 to 1015 cm (∼ 10−11 to 10−3 pc). At
the largest scales, Haverkorn et al. (2008) found that stellar winds or protostellar
outflows dominate the energy injection of turbulent energy on parsec scales in
spiral arms, while supernova and superbubble expansions are the main sources of
energy in the interarm regions and occur on scales of 100 pc.
1.4.3
Fields in the halo
The extent of the Galactic magnetic field away from the disk is very difficult to
determine. Han & Qiao (1994) used pulsar RMs to determine the scale height of the
regular field and found z0 ' 1.5 kpc. Another approach is to assume equipartition
between cosmic rays and magnetic fields, and use the observed vertical distribution
of synchrotron emissivity to estimate the vertical distribution of the magnetic field
(see figure 1.4). This leads to estimates of the scale height to z0 ' 5 − 6 kpc. A
third approach of Boulares & Cox (1990) is to assume that there is an approximate
equipartition between the non-thermal pressure forms: magnetic, cosmic ray, and
dynamic pressure. By estimating the non-thermal pressure as a function of z,
and assuming that the magnetic field is aligned parallel to the plane, the vertical
28
Figure 1.4: The vertical distribution of the (total) magnetic field strength, taken
from Cox (2005). The solid blue curve is derived from the vertical distribution of
the synchrotron emissivity. The dashed red curve is derived under the assumption
that a third of the non-thermal pressure is magnetic.
29
distribution of magnetic field strength can be calculated, as shown in figure 1.4.
The field geometry is unlikely to be the same in the halo as in the disk, and rotation
measure maps of the sky show less variation at larger latitudes, implying a simpler
structure than the disk. Furthermore, as described below, external edge-on galaxies
often exhibit significant vertical fields far from the disk.
1.5
Magnetic fields in external galaxies
Niklas (1995) used synchrotron data and equipartition arguments to estimate the
total magnetic field strength for a sample of 74 spiral galaxies and found the
average value to be 9 ± 3 µG , although local values within the spiral arms could
reach 20 µG . Turbulent fields tend to be strongest in the optical spiral arms,
while regular fields dominate the interarm regions. Typical pitch angles of the
magnetic spiral field are 10◦ to 40◦ , and are similar to the pitch angles of the
corresponding optical arms. It is worth noting that if the large scale field was
completely frozen into the gas – whose particles follow almost circular orbits –
the magnetic field would be more tightly wound, with a smaller pitch angle than
is observed. Flocculent and irregular galaxies exhibit magnetic field strengths of
similar magnitude as spiral galaxies. More interestingly, these galaxies often have
regular spiral magnetic fields.
No large scale field reversal such as is thought to be present in the Milky Way
30
has ever been observed in an external galaxy. Only three large reversals have been
observed (Beck 2008), and they are limited in extent (not an entire spiral arm) and
could be explained by the magnetic field being a superposition of dynamo modes
that for a localized region yield a change in the field direction. It has also been
suggested by Beck (2008) that the Milky Way reversals may not be coherent over
several kpc, or that they are restricted to a thin region near the plane only.
Observations of edge-on galaxies reveal a magnetic field in the galactic disk that
is mainly parallel to the disk. In the halo, however, a vertical component becomes
more prominent (see figure 1.5), giving the halo field an “X”-shape if viewed edgeon. Numerous galaxies of this kind have been observed (Beck 2009). Golla &
Hummel (1994) studied the specific case of NGC 4631 and found that the vertical
field lines are connected to regions of star formation in the disk. Conversely, in
regions of weaker star formation the magnetic fields were observed to be parallel
to the disk. Presumably galactic winds and outflows drag the the magnetic field
out from the disk, creating the X-shaped halo fields, and may also be the origin of
intergalactic magnetic fields (Beck 2009).
1.6
Cosmological magnetic fields
The nature of magnetic fields on cosmic scales is an important topic in cosmology.
Primordial magnetic fields can have had significant effects on structure formation,
31
Figure 1.5: Left: M51 observed at 6cm, with color indicating total intensity and
polarization bars rotated 90◦ to show the orientation of magnetic field lines. Copyright MPIfR Bonn (Beck, Horellou & Neininger). Right: NGC 5775, an edge-on
spiral galaxy with contours of total emission at 6cm wavelength and bars oriented
with the magnetic field. The contours are overlaid on an optical Hα image. Copyright: Cracow Observatory.
32
as well as big bang nucleosynthesis and neutrino physics (Enqvist et al. 1993).
Indeed, cosmological observations can provide limits on certain large scale magnetic
field scenarios (see Grasso & Rubinstein (2001) for a review). For instance, a
uniform cosmological magnetic field with a coherence length equal to the Hubble
length would introduce anisotropic pressure and break the observed isotropy seen
in COBE data if the field strength is greater than ≈ 3 nG.
No detection of magnetic fields in the intergalactic medium (IGM) has been
reported. A widely quoted upper limit on magnetic fields in the IGM was derived
by Kronberg (1994), who found BIGM ≤ 10−9 G for fields of coherence length of
a Mpc from RM observations of QSOs. However, this limit was later found to be
severely underestimated, and Farrar & Piran (2000) found RMs to instead yield
the limit BIGM ≤ 10−6 G.
Magnetic fields in clusters of galaxies have been measured. Recent examples
are Govoni et al. (2005), who found ordered magnetic fields on the scale of ∼ 400
kpc in the galaxy cluster Abell 2255. Govoni & Feretti (2004) found µG strength
fields in galaxy clusters, and fields reaching tens of µG at the center of cooling core
clusters.
33
1.7
The origin of cosmic magnetic fields
One of the major open questions in astrophysics is how cosmic magnetic fields are
generated at all. For a recent review of this topic, see e.g., Kulsrud & Zweibel
(2008). This section will only briefly cover the main issues of this important
subject.
While stellar magnetic fields are easily explained by a dynamo mechanism because of the short rotation period of the star, galactic magnetic fields are difficult
to explain since a galactic dynamo would only have had ∼ 50 rotations and needs
a strong seed field (∼ 10−21 G), whose origin is still unclear. Different theories
of galactic magnetogenesis predict different topologies of the magnetic field and
clues to the origin of galactic scale magnetic fields can be found by studying the
symmetries of the large scale field, i.e., the existence of field reversals in or between
spiral arms, or if the field orientation in the south of the galaxy is different from
the north, etc.
It was first noted by Fermi (1949) that the enormous inductance of the Galaxy
would make the characteristic decay time of the Galactic magnetic field orders of
magnitude longer than the age of the universe. The problem is thus how to explain
what mechanism could oppose the inductive voltages and generate such a magnetic
field to begin with.
The standard explanation of generating a galactic magnetic field has two steps.
First, the generation of a tiny seed field where previously no magnetic field existed
34
at all, followed by a process of which the seed field is magnified in strength to the
observed values and made partially coherent on larger scales. Numerous processes
that can supply a seed field has been suggested (Widrow 2002), some of which are
covered below. The favored mechanism to achieve the amplification of seed fields
is a galactic dynamo and is discussed in §1.7.2.
1.7.1
Generation of seed fields
A way to generate a magnetic field from a nonexistent initial field was suggested
by Biermann (1950). In a plasma with fluctuations in the electron pressure Pe
(e.g., due to turbulence), electrons flow to low pressure regions which leads to an
electric field,
E=−
∇Pe
.
ne e
(1.32)
Applying Faraday’s law we get,
∂B
c
= −c∇ × E = ∇ ×
∂t
e
∇Pe
ne
.
(1.33)
By assuming the ideal equation of state for the electrons, Pe = ne kB Te , and using
the identity ∇ × (f ∇g) = ∇f × ∇g, equation (1.33) becomes
ckB
∂B
=−
∇ne × ∇Te .
∂t
ne e
35
(1.34)
This is the essential mechanism of the so-called Biermann battery: if the electron
density and temperature gradients are not parallel, the right-hand side of equation (1.34) is non-zero and a magnetic field is produced where there previously
was none. Non-parallel density and temperature gradients are expected in astrophysical plasmas (Widrow 2002), and the generation of seed fields of order 10−21 G
through the Biermann battery mechanism has been done in cosmological computer
simulations by Kulsrud et al. (1997).
Magnetic fields are thus expected to be generated during structure formation
through the Biermann battery process. However, even if a star would be born with
no magnetic field, a stellar Biermann battery will produce an internal seed field
that will quickly be amplified by a stellar dynamo. Stellar winds or other ejecta
could then propel magnetic fields into the ISM and act as a seed field for a galactic
dynamo.
Other scenarios for the creation of seed fields through, e.g., AGNs and early
universe phase transitions, are described by Widrow (2002).
1.7.2
The α − Ω dynamo
When a sufficient seed field is present in a galaxy it is believed that a dynamo mechanism can act on the seed field increasing its coherence length and exponentiating
the field strength by transferring mechanical energy into magnetic energy.
A pictorial view of the α − Ω dynamo powered by supernova explosions is
36
presented in figure 1.6. In panel (a) an initial seed field is directed in the forward
motion of galactic rotation, with a star about to go supernova located below it. In
the next panel the supernova creates a bubble in the ISM and stretches the field
line in a (vertical) loop. Because of the expansion of the shell the moment of inertia
of the shell has been greatly increased. By Conservation of angular momentum
the bubble slows down in a fixed frame, and rotates opposite the galactic rotation
in the galactic frame. This coriolis effect is called the α-effect, and is shown from
above in panel (c). The points A and B of the field line, at the base of the bubble,
lie at slightly different radii and are now stretched farther apart due to differential
rotation, shown in panel (d-f). This is the Ω-effect, and will continue to amplify the
field strength linearly in time. The flux lines above the plane are assumed to rise
indefinitely, which will thus remove the negative magnetic flux, leaving the flux in
the disk increased but with flux conservation upheld globally. This is a large-scale
version of the mean-field α −Ω dynamo driven by turbulence and acting on smaller
scales. For an in-depth review of the dynamo mechanism, see Kulsrud (1999).
1.7.3
Numerical simulations
Past attempts to use numerical simulations to study magnetic fields were mainly
for cosmological fields (Ryu et al. 1998; Dolag et al. 2002). Recently (Dubois &
Teyssier 2009) used numerical MHD to follow the evolution of the magnetic field
during formation of a dwarf galaxy, and the eventual formation of a galactic disk
37
Figure 1.6: A schematic view of the α − Ω dynamo, taken from Kulsrud & Zweibel
(2008). In panels (a) and (b) a supernova creates a bubble in the ISM, stretching a
magnetic field line into a loop. From the top view in (c), where dashed lines show
the upper parts of the field line, coriolis forces twist the loop into the poloidal
plane. Differential rotation then stretches the field lines, increasing the magnetic
flux in the disk (d). Eventually, the upper part of the field line is expelled from
the disk, shown in (e) and (f).
38
and the ejection of magnetic field lines into the IGM by a supernova driven wind.
The authors provide an explanation for an IGM magnetic field of 0.1 nG, assuming
equipartition magnetic fields in dwarf galaxies. These galactic fields were put in
by hand, however, and need a separate explanation.
Kotarba et al. (2009) use N-body and smooth particle hydrodynamics (SPH)
simulations of the evolution of magnetic fields and gas in a galactic disk. The
authors find that the homogeneous seed field is amplified by an order of magnitude
after five rotations of the disk. This amplification is not sufficient to explain
observations of high-redshift galaxies with µG magnetic fields reported by. e.g.,
Kronberg et al. (2008). The simulations do not include small-scale turbulent effects
that lead to the α-effect in an α − Ω dynamo, and hence are not expected to
cause the necessary exponentiating of the magnetic field strength. However, the
simulations result in a disk field geometry closely resembling observed galaxies, as
can be seen in figure 1.7, with magnetic field lines aligned with the spiral pattern
of the gas.
The most promising attempt to simulate a fully functioning dynamo leading
to observed field strengths is the work by Hanasz et al. (2009). The authors
simulate a “cosmic ray driven dynamo”, where the magnetized interstellar medium
is dynamically coupled with the cosmic ray gas. The initial seed fields are randomly
distributed magnetic dipoles associated with magnetic supernova ejecta. Hanasz et
al. find that the seed fields get amplified exponentially, with an average e-folding
39
Figure 1.7: A simulated galactic disk field, taken from Kotarba et al. (2009).
Color shows gas density, and vectors show the magnetic field (length of vectors
scale logarithmically with field strength).
40
time of 270 Myr (approximately the galactic rotation period). The growth of the
magnetic field saturates at approximately 4 Gyr with field strengths of 3-5 µG in
the disk. The initially random seed field have developed into a large scale field with
the shape of a tightly wound spiral. Moreover, viewed edge-on a thick halo field
with an X-shaped structure becomes apparent, reminiscent of radio observations of
external galaxies seen in figure 1.5. This recent development adds strong support
to the dynamo theory of the origin of galactic magnetic fields.
41
42
Chapter 2
Constraining models of the
large-scale Galactic magnetic field
with WMAP5 polarization data
and extragalactic rotation
measure sources
Chapter Abstract
We introduce a method to quantify the quality-of-fit between data and observables depending on the large-scale Galactic magnetic field. We combine WMAP5
43
polarized synchrotron data and rotation measures of extragalactic sources in a
joint analysis to obtain best-fit parameters and confidence levels for GMF models
common in the literature. None of the existing models provide a good fit in both
the disk and halo regions, and in many instances best-fit parameters are quite
different than the original values. We note that probing a very large parameter
space is necessary to avoid false likelihood maxima. The thermal and relativistic
electron densities are critical for determining the GMF from the observables but
they are not well constrained. We show that some characteristics of the electron
densities can already be constrained using our method and with future data it may
be possible to carry out a self-consistent analysis in which models of the GMF and
electron densities are simultaneously optimized.
2.1
Introduction
Among the probes of Galactic magnetism, Faraday rotation measures (RMs) and
synchrotron radiation are among the best-suited for studying the large-scale Galactic magnetic field. Other probes are either mostly applicable to the nearby part of
the Galaxy (starlight polarization) or mainly used to study small-scale structures
like dense clouds (the Zeeman effect). While synchrotron radiation has been used
extensively for studying external galaxies, Faraday rotation measure has been the
method of choice for probing the magnetic field of our own Galaxy (e.g., Han et al.
44
2006; Brown et al. 2007). To most simply probe the large-scale Galactic magnetic
field (GMF) with synchrotron radiation, a full-sky polarization survey is needed,
at frequencies where Faraday depolarization effects are negligible and where other
sources of polarized radiation (e.g., dust) are either negligible compared to synchrotron radiation or possible to exclude from the data. The 22 GHz K band of
the Wilkinson Microwave Anisotropy Probe (WMAP) provides such a data set.
In this Chapter we constrain models of the large-scale Galactic magnetic field
common in the literature by a joint analysis of the polarization data in the WMAP5
K band and rotation measures of extragalactic radio sources. The synergistic
advantage of combining these two data sets goes beyond the benefit of achieving
a larger number of data points; the two observables probe mutually orthogonal
components (perpendicular and parallel to line-of-sight) of the magnetic field. We
calculate the χ2 for a particular choice of GMF model and parameter choice, and
use a Markov Chain Monte Carlo algorithm to optimize the parameters of each
field under consideration. Thus we can quantitatively measure the capability of
different models to reproduce the observed data.
The content of this Chapter is structured as follows: we briefly comment on the
connection synchrotron radiation and Faraday rotation has with magnetic fields
in section 2.2. Section 2.3 explains our general approach to constrain models of
the GMF. Section 2.4 describes the two data sets used in the analysis. Section
2.5 details the various 3D models of the magnetized interstellar medium we have
45
studied. Section 2.6 give our results and a discussion, and is followed by a summary
and conclusions.
2.2
Input
As is clear from sections §1.2.2 and §1.2.1, the observables we are using are not
direct measures of the Galactic magnetic field itself, but convolutions of the various
~ ne and ncre ). The ideal
components of the magnetized interstellar medium (B,
way to constrain models of the GMF would thus be to simultaneously and selfconsistently model the thermal and relativistic electron distributions together with
the magnetic field. This is beyond the scope of the present work, but should become
feasible when even larger data sets become available. Instead we use 3D models
of ne and ncre present in the literature (detailed in section 2.5) and do not vary
their parameters, except in a separate study of the magnetic field scale height’s
dependence on the scale height of the thermal electron distribution in section 2.6.4.
2.3
Method
Our strategy to model the GMF is straightforward. As illustrated in figure 2.1,
starting from 3D models of the distribution of relativistic electrons (ncre ), thermal
electrons (ne ) and a model of the large-scale GMF parametrized by a set of numbers
p~0 we produce simulated data sets of polarized synchrotron radiation and Faraday
46
Figure 2.1: Overview of the implemented analysis.
rotation measures. Generating the simulated data is done using the Hammurabi
code (Waelkens et al. 2009). We compare the simulated data sets to the observational data and calculate χ2 , as a measure of the quality-of-fit. This value together
with p~0 is passed to a Markov Chain Monte Carlo (MCMC) sampler that generates
a new set of GMF parameters, p~1 , as described in section 2.3.2. This scheme is
then iterated until we obtain a Markov Chain that has sufficiently sampled the
parameter space. In this way we find the best-fit parameters and confidence levels
for each GMF model, and can compare and quantify the capability of different
models to reproduce the observed data.
47
2.3.1
Quality-of-fit
We define χ2 in the usual way,
χ2 =
N
X
(data − model)2
σi2
i
,
(2.1)
with i labelling the data points (pixels for synchrotron data, point sources for RMs).
The variance, σi2 , is of utmost importance and obtaining it is a key accomplishment of this work. The variance is primarily not experimental or observational
uncertainty (which will typically be a negligible contribution), but astrophysical
variance stemming from random magnetic fields and inhomogeneities in the magnetized ISM. In the case of extragalactic RMs there is also a contribution to the
variance from Faraday rotation in the intergalactic medium and the source host
galaxy. The method we have developed to measure σ 2 from the data is described
in §2.4.1 and §2.4.2. This variance map contains valuable information about the
random magnetic fields. Indeed, when studying turbulent and small-scale magnetic fields this variance map is itself one of the observables that a given model of
the random field should be able to reproduce.
2.3.2
Parameter estimation
To minimize χ2 for a GMF model and find the best-fit parameters we implement
a Metropolis Markov Chain Monte Carlo algorithm (Metropolis et al. 1953) to
48
explore the likelihood of the observed data. We use the approach outlined in, e.g.,
Verde et al. (2003). For a set of parameters p~j we calculate the (unnormalized)
1
2 (~
pj )
likelihood function L(~pj ) = e− 2 χ
. We then take a step in parameter space
to p~j+1 = p~j + ∆~p, where ∆~p is set of Gaussianly distributed random numbers
with zero mean and standard deviation ~s (the “step length” of the MCMC). If
L(~pj+1 ) > L(p~j ) this new set of parameters is accepted, and if L(~pj+1 ) < L(p~j )
the new step is accepted with a probability P = L(~pj+1 )/L(~pj ). If a step is not
accepted, a new p~j+1 is generated, and tested. This process is continued until the
parameter space has been sufficiently sampled. To decide when this has happened
we use the Gelman-Rubin convergence and mixing statistic, R̂ (Gelman & Rubin
1992). In the vein of Verde et al. (2003) we terminate the Markov Chain when the
condition R̂ < 1.03 is satisfied for all parameters. To achieve a smooth likelihood
surface we run all Markov Chains to at least 50k steps.
2.3.3
Producing simulated observables from models
We use the Hammurabi code developed by Waelkens et al. (2009) to simulate fullsky maps of polarized synchrotron emission and rotation measures from 3D models
~ The produced polarized synchrotron sky maps (Stokes Q and
of ne , ncre and B.
U parameters) are in HEALPix1 format (an equal-area pixelation scheme of the
sphere, developed by Górski et al. (2005)) and are calculated taking Faraday depo1
http://healpix.jpl.nasa.gov
49
larization into account. The Galactic contribution to the rotation measure (RM)
of extragalactic point sources can also be calculated. For details, see Waelkens
et al. (2009).
2.3.4
Disk - halo separation
Significant magnetic fields in galactic halos are observed in external galaxies (Beck
2008). However, the magnetic field in our own Milky Way galaxy has proven
very difficult to study, and a long-standing question in Galactic astrophysics is the
nature and extent of a magnetic Galactic halo. For instance it is unclear whether
the Galactic halo field may be completely distinct from the magnetic field in the
disk, in the sense that their topologies are different. In other words, are the disk
and halo fields best modeled by the same GMF model (with perhaps different
best-fit parameters) or distinctly different GMF models? To address this question
we separate the sky into a disk part and a halo part, and optimize each GMF
model under consideration using only data from one part at a time.
In the case of polarized synchrotron emission it is very difficult to separate the
contribution of the large-scale field from that of smaller, more nearby structures
simply because the flux from a particular structure scales as 1/r2 . Thus we apply
a mask on the synchrotron data that covers the disk and clearly defined nearby
structures such as the Northern Spur (see section 2.4.1). Extragalactic rotation
measures are not subject to the inverse square effect since the relevent sources are
50
point-like and the RM contribution of some volume of the ISM is independent of
its distance from us (see Eq. 1.22).
In this Chapter we will refer to the ‘disk’ and ‘halo’ as distinct regions on
the sky, where the dividing line between the two is given by the set of directions
pointing towards z = ±2 kpc at Galactocentric radii R = 20 kpc, i.e., |b| . 4◦ at
l = 0◦ and |b| . 10◦ at l = 180◦ . This division is plotted in, e.g., figure 2.5.
2.4
2.4.1
Data
Synchrotron radiation
In the WMAP K-band (22 GHz) the observed polarized radiation is dominated by
Galactic synchrotron emission. In the WMAP five-year data release the observed
emission is further analyzed and separated into foreground components caused by
synchrotron, dust and free-free emission (Gold et al. 2008). Throughout our analysis we will use this synchrotron component as our data set for polarized Galactic
synchrotron emission. The polarized WMAP component was first included, and
analyzed in the context of the GMF, in the three-year data release (Page et al.
2007); subsequent GMF modeling of that data set was done by, e.g., Jansson et al.
(2008) and Miville-Deschênes et al. (2008).
The WMAP polarization data is in the form of two HEALPix maps of the
Stokes Q and U parameters. The resolution is ∼1◦ , with about 50k pixels on
51
Figure 2.2: Polarized synchrotron intensity (color) in a Mollweide projection with
galactic longitude l = 0◦ at the center and increasing to the left, overlaid with a
texture showing the projected magnetic field directions (i.e., the observed polarization angle rotated by 90◦ ). Image created using the line integral convolution
code, ALICE, written by David Larson (private communication).
the sky. To get a more easily interpreted map we can instead plot the polarized
p
intensity P = Q2 + U 2 and with the polarization angle rotated 90◦ to depict the
projected magnetic field direction overlaid on the polarized intensity (see figure
2.2).
It is clear from figure 2.2 that emission from the Galactic disk is strong, with
polarization angles consistent with a magnetic field parallel to the disk. Also
present is the prominent Northern Spur, a highly coherent structure stretching
from the disk to high latitudes, and likely the result of the shock between two
52
Figure 2.3: Upper panel: sky maps of Stokes Q (left) and U (right), in 8◦ resolution.
Lower panel: sky maps of σQ and σU , respectively.
expanding nearby supernova shells (Wolleben 2007).
While structures like the Northern Spur can be masked out when studying
the large-scale GMF, smaller features and irregularities in the polarized emission
abound due to the turbulent nature of the magnetized interstellar medium. Since
we are interested in the large-scale regular magnetic field and want to avoid the
random field in this study we smooth the synchrotron data to 8◦ and use maps with
768 pixels, as shown in figure 2.3.
53
Estimating the variance
To calculate χ2 for the synchrotron data an estimate of the variance is needed
that takes into account not just experimental uncertainties but fluctuations in the
measured emission due to random magnetic fields. The variance should de-weight
the χ2 of pixels in directions of strong random fields or other localized sources of
polarized emission.
We estimate the variance in Q (and similarly in U ) by
2
σQ,i
=
N
X
(Qi − qj )2
N
j
.
(2.2)
2
Here σQ,i
is the variance of the ith pixel in the 8◦ -pixel Q map, whose pixel has
the value Qi . The quantity qj is a pixel value from the 1◦ -pixel Q map, where j
enumerates the N = 69 1◦ -pixels that are within a 4 degree radius of the coordinate
on the sky corresponding to the center of the 8◦ -pixel Qi . The lower panels of figure
2.3 show sky maps of σQ and σU .
Masking
As pointed out in section 2.3.4; because the synchrotron flux from structures such
as supernova remnants scale as the inverse square of the distance to the object,
flux from nearby structures in the disk can exceed emission caused by the largescale GMF in the diffuse ISM. However not all parts of the disk may be polluted
54
by bright nearby synchrotron emitters. For example, the large “Fan region”, a
strongly polarized part of the sky around l ∼ 120◦ to 160◦ and |b| . 15◦ has been
argued to be caused by an ordered, large-scale (∼kpc) magnetic field oriented
transverse to our line-of-sight and with minimal Faraday depolarization (Wolleben
et al. 2006). Other such regions may exist, but it would require more extensive
analysis of polarized emission at lower frequencies that probe predominantly the
nearby part of the Galaxy (due to Faraday depolarization obscuring more distant
emission) in order to use them in the current analysis.
In this work we take the simplest approach and mask out the disk and strongly
polarized local structures such as the Northern Spur. We use the polarization
mask discussed in Gold et al. (2008), degraded from 4◦ pixels to the 8◦ resolution
maps we use for the Stokes Q and U data. In cases where an 8◦ pixel covers a
region of both masked and unmasked 4◦ pixels, the larger pixel is made part of the
mask. This mask covers 33.5% of the sky, shown in figure 2.4, and leaves 511 pixels
unmasked. Because only a single of these pixels lies inside the ‘disk’, as defined
in section 2.3.4, we only use polarized synchrotron data when fitting data in the
‘halo’, hence using 510 data points for each of the Q and U maps.
2.4.2
Extragalactic RM sources
We use 380 RMs of extragalactic sources from the Canadian Galactic Plane Survey
(CGPS, Brown et al. (2003)), 148 RMs from the Southern Galactic Plane Survey
55
Figure 2.4: The mask used for the polarized synchrotron data. The mask covers
33.5% of the sky.
(SGPS, Brown et al. (2007)) and 905 RMs from various other observational efforts
(Broten et al. 1988; Clegg et al. 1992; Oren & Wolfe 1995; Minter & Spangler
1996; Gaensler et al. 2001), for a total of 1433 extragalactic RM sources. The data
set contains Galactic longitude and latitude (l◦ and b◦ ), rotation measure (RM)
and observational uncertainty (σobs ). Typical values of RMs are ± a few hundred
radians/m2 in the disk and ± a few tens of radians/m2 at large angles from the
disk. The observational uncertainty reported for individual sources tends to be
roughly a factor of ten smaller than the typical magnitude of the RM in a given
direction.
Some of these 1433 RMs are multiple measurements of the same extended
source. Including multiple measurements of one source as if they were different
sources would lead us to underestimate the total variance in the rotation measure of
that region. To avoid this we round the l◦ and b◦ of each source to one decimal, and
56
if there are multiple RMs with identical coordinates we replace them with a single
2
2
RM, whose variance (σobs
) is the mean of the σobs
’s of the individual measurements;
the rationale for this is discussed in §2.4.2. This procedure removes 103 ‘sources’,
leaving 1330.
Removing outliers
The measured RM of an extragalactic source is the sum of contributions from the
diffuse Galactic interstellar medium (the medium we wish to study), the intergalactic medium and the RM intrinsic to the source. In addition, sightlines that
pass through specific structures such as supernova remnants or H II regions can
receive very large contributions to their measured RM due the high electron density and possibly strong magnetic fields pervading these structures (Mitra et al.
2003). Thus it is expected that the variance in the distribution of RM values is
significantly larger for sightlines close to the Galactic plane compared to those at
large angles to the plane, and this is observed (see figure 2.7).
In order to remove the sources whose rotation measure is likely dominated
by either highly localized Galactic or purely non-Galactic contributions, whose
variances are intrinsically different, we divide the RMs into a ‘disk’ component and
a ‘halo’ component, in identical fashion to the division described in section 2.3.4.
For each of these we calculate the mean and the standard deviation of the RM
values. We use a modified z-scores method (Barnett & Lewis 1984) and exclude
57
Figure 2.5: Top: The full sample of 1433 extragalactic RMs, in Galactic coordinates (l = 0◦ at the center and increasing to the left). Blue squares represent
negative RM values, corresponding to a magnetic field pointing away from the
observer. Red circles show positive RM values. The area of the markers scale as
the magnitude of the rotation measure. The black dotted lines marks our division
between the disk and halo. bottom: The final sample of 1090 extragalactic RMs
(559 in the disk, 531 in the halo) used in the quality-of-fit analysis.
58
Figure 2.6: The total variance, σEGS , of the 1090 RMs used in the data fitting.
The area of the markers scale as σEGS and uses the same scale as the RM plots in
figure 2.5.
any extragalactic sources (EGS) with an RM value three standard deviations away
from the mean. These two steps are then repeated until no values of RM are more
than three standard deviations away from the mean of the remaining sample. This
process removes 21 EGS from the disk and 72 EGS from the halo.
Estimating the variance
2
As our objective is to study the large-scale regular field, the variance σEGS,
i for the
ith EGS should ideally account for rotation measure contributions of non-Galactic
origin and contributions due to small-scale random magnetic fields. That is, when
2
these contributions to the rotation measure are large, σEGS,
i should be large, so
that the ith EGS has a small weight in the calculation of χ2 . There is more than
one way to calculate this variance, and we settle on the following scheme to obtain
59
40
N
30
20
10
0
−2000
−1500
−1000
−500
0
500
RM (rad m−2)
1000
1500
2000
−2000
−1500
−1000
−500
0
500
RM (rad m−2)
1000
1500
2000
200
N
150
100
50
0
Figure 2.7: Histogram of RMs for the disk region (upper panel) and halo region
(lower panel). Dashed lines demarcate RM values excluded as outliers (see section
2.4.2).
60
an estimate of the variance:
2
σEGS,
i
=
N
X
(RMi − RMj )2
N
j
2
+ σobs,
i,
(2.3)
2
where j labels the sources closest to the ith source, and σobs,
i is the observational
variance. To get a decent estimate of the variance we require a minimum of N = 6
sources. For a source in the disk, we use all neighboring sources within θmin = 3◦ of
the ith source. If less than 6 sources are found within θmin we increase the angular
search radius until 6 sources are included or we reach θmax = 6◦ . If we still have
fewer than 6 sources, we exclude the ith source from our sample. For EGS in the
halo, where significant changes in the large-scale GMF are not expected to occur at
as small angular scales as in the disk, we instead use θmin = 6◦ and θmax = 12◦ . 46
sources in the disk and 101 in the halo have an insufficient number of neighboring
sources to give a satisfactory estimate of the variance and are excluded. The
remaining sample, which will be used in the quality-of-fit calculation, consists of
1090 extragalactic sources and are shown in figure 2.5. The corresponding map of
σEGS is shown in figure 2.6.
Having explained the procedure for measuring σEGS , we now return to the issue
of how to treat multiple observations of the same source. The correct procedure
depends on the particular situation. If the source were pointlike and the observations perfectly aligned, then the actual variance in the results of the measurements
61
2
should in principle be the same as the mean value of the σobs
’s for the set of ob-
servations. However if the source is extended and the measurements are not at
exactly the same position, their variance probes the σEGS from the foreground
turbulent ISM as well as from the RM in the source, which is characteristically
much larger than the observational uncertainty in each individual measurement.
Therefore assigning that variance to the combined observation in making the fit
would incorrectly reduce its pull in the fit compared to sources with a single measurement. With the present dataset of RMs, there are only a limited number of
cases of multiple measurements and it is possible to examine them individually. In
only one case is the source extended enough that the variance in the observations
2
2
is significantly larger than the mean of the σobs
, and for that case the σEGS
value
2
we derive is nearly the same with either σobs
assignment, vindicating a posteriori
our procedure, described in section 2.4.2. For future large datasets this issue will
require deeper analysis. If the prescription used here is not adequate, an automatic
method will need to be developed to handle all cases, which is applicable for both
extremes.
62
2.5
3D models of the magnetized interstellar
medium
To calculate simulated data sets of rotation measures and synchrotron emission
we need 3D models of the thermal and relativistic electron densities as well as the
Galactic magnetic field. For relativistic electrons we adopt the simple exponential
model detailed in §1.3.1. For thermal electrons we adopt the NE2001 model of
Cordes & Lazio (2002), also detailed in §1.3.2.
2.5.1
large-scale magnetic field
Numerous models of the large-scale GMF have been proposed in the literature over
the last couple of decades. In this Chapter we test a representative selection of
models. Most of the considered models were originally proposed to describe either
the entire GMF or just the disk field.
All fields considered here are truncated at Galactocentric radius R = 20 kpc.
For all models we set the distance from the Sun to the Galactic center to R = 8.5
kpc. This is the most commonly used distance in the original published forms of
the models used in this thesis. We note that after a few years of confusion on this
topic, the most reliable estimate of this distance is now 8.4 ± 0.6 kpc (Reid et al.
2009).
63
Logarithmic spirals
This class of models contains the most common models in the literature (see, e.g.,
Sofue & Fujimoto (1983); Han & Qiao (1994); Stanev (1997); Harari et al. (1999);
Tinyakov & Tkachev (2002)).
We define the radial and azimuthal field components as
Bθ = −B(r, θ, z) cos p,
Br = B(r, θ, z) sin p,
(2.4)
where p is the pitch angle of the spiral. The function B is defined as
r
B(r, θ, z) = −B0 (r) cos θ + β ln
r0
e−|z|/z0 ,
(2.5)
with β = 1/ tan p. The pitch angle p is positive if the clockwise tangent to the
spiral is outside a circle of radius r centered on the Galactic center. At the point
(r0 , θ = 180◦ ), the field reaches the first maximum in the direction l = 180◦ outside
the solar circle. We set the magnetic field amplitude B0 (r) to a constant value,
b0 , for r < rc , and B0 (r) ∝ 1/r for r > rc . With the vertical scale height, z0 , the
model has five parameters: b0 , rc , z0 , p and r0 .
The above parametrization is usually named bisymmetric spiral (BSS) and
exhibits field reversals between magnetic arms (see figure 2.8). By taking the
absolute value of the cosine in equation (2.5) we get the disymmetric spiral (DSS).
64
This latter model is often referred to as an axisymmetric spiral in the literature.
However, we prefer to reserve the term ‘axisymmetric’ to mean “independent of
azimuthal angle”. A DSS (BSS) field is a logarithmic spiral field that is symmetric
(antisymmetric) under the transformation θ → θ + π.
Another distinction that can be made is a model’s symmetry properties under
z → −z, i.e., reflection through the disk plane. We denote a field as symmetric (S)
with respect to the Galactic plane if B(r, θ, −z) = B(r, θ, z), and antisymmetric
(A) if B(r, θ, −z) = −B(r, θ, z). This notation agrees with, e.g., Tinyakov &
Tkachev (2002), Harari et al. (1999), and Kachelrieß et al. (2007); however it
conflicts with Stanev (1997) and Prouza & Smida (2003).
We thus distinguish between four different sub-models of the logarithmic spiral
on the basis of two symmetries; BSSS , BSSA , DSSS and DSSA .
Sun et al. logarithmic spiral with ring
Sun et al. (2008) test a variety of GMF models for the Galactic disk using extragalactic RMs. The authors find the model best conforming with data to be an
axisymmetric field with a number of field reversals inside the solar circle. Following
equation (2.4), Sun et al. defines B(r, θ, z) = D1 (r, z)D2 (r), with




B0 exp − r−R
−
R0
D1 (r, z) =



Bc exp − |z|
z0
65
|z|
z0
if r > Rc ,
(2.6)
if r ≤ Rc .
and
D2 (r) =




+1 r > 7.5 kpc








−1 6 kpc < r ≤ 7.5 kpc
(2.7)




+1 5 kpc < r ≤ 6 kpc







−1 r ≤ 5 kpc,
where +1 corresponds to a clockwise magnetic field direction when viewed from
the north pole.
Since B(r, θ, z) does not depend on θ, this is an axisymmetric field in the true
sense of the word. Sun et al. take the pitch angle to be the average for the spiral
arms in the NE2001 model, p = −12◦ ; the other parameters are B0 = Bc = 2 µG,
Rc = 5 kpc, R0 = 10 kpc and z0 = 1 kpc. We take these six quantities as
parameters to vary in our fit. This field model (hereafter named Sun08D ) is plotted
in figure 2.8.
Sun et al. (2008) also include a separate halo component, detailed in §2.5.1. In
section 2.6 we consider the full, composite Sun08 model, as well as the disk and
halo components separately.
Prouza-Smida halo field
The halo component of the Sun08 model is a slightly modified version of the
toroidal halo component proposed by Prouza & Smida (2003). The model (PS03
hereafter) consists of two torus-shaped fields, above and below the Galactic plane,
66
Figure 2.8: Top left: An example of a bisymmetric spiral (BSS) field model for
p = −10◦ , rc = 10 kpc and r0 = 10 kpc. The corresponding disymmetric spiral
(DSS) looks the same except that there is no field reversal between magnetic spiral
arms. Top right: Sun08D , the favored disk model of Sun et al. (2008). Bottom
left: The field proposed by Brown et al. (2007), based on the NE2001 thermal
electron density model. Bottom right: Halo field model proposed by Prouza &
Smida (2003), and the halo part of the Sun08 composite model.
67
Figure 2.9: Left: Toroidal disk field proposed by Vallée (2008). Right: The magnetic field model by the WMAP team (Page et al. 2007).
with opposite field direction (i.e., antisymmetric in z). The magnitude of the field
is given by
BφH (r, z)
=
r − r0H
r
.
2 H exp − H
|z|−z0H
r0
r0
1
B0H
1+
(2.8)
z1H
The parameters used in Sun et al. (2008) are B0H = 10 µG , r0H = 4 kpc, z0H = 1.5
H
H
kpc, z1H = z1a
= 0.2 kpc for |z| < z0H and z1H = z1b
= 0.4 kpc otherwise. This
model is plotted in figure 2.8.
68
Brown et al. field
In Brown et al. (2007) the authors propose a modified logarithmic spiral model2
(hereafter Brown07) influenced by the structure of the NE2001 thermal electron
density model with the aim to explain the SGPS rotation measure data only (i.e.,
the fourth quadrant region, 253◦ < l < 357◦ ). The model has zero field strength
for Galactocentric radii r < 3 kpc and r > 20 kpc. Between 3 ≤ r ≤ 5 kpc
(the “molecular ring”) the field is purely toroidal (i.e., zero pitch angle). For
r > 5 kpc, eight magnetic spiral regions with pitch angle 11.5◦ are defined with
individual field strength bj . The field in the molecular ring and the spiral region
corresponding to the Scutum-Crux spiral arm is oriented counterclockwise, and the
remaining regions clockwise. The field strength in region j has a radial dependence
~ j | = bj /r, with the Galactocentric radius, r, in kpc. The vertical extent of the
|B
field was not considered, as the model was proposed to explain the measured RMs
in the Galactic disk. The model field is shown in figure 2.8.
In our analysis we generalize this model by introducing three free parameters:
β, which scales the overall magnetic field strengths (bj ) used in the original model;
~ j | ∝ bj /r for r > rc ;
rc , such that the field strength is constant for r < rc and |B
and an exponential vertical scale height z0 .
2
Details of the model parametrization are not given in Brown et al. (2007), but obtained
through private communication with the authors, and not reproduced here.
69
Vallée field
In Vallée (2008) and Vallée (2005) the author models the magnetic field in the disk
as a perfectly toroidal field consisting of concentric rings of width 1 kpc. The model
we consider here, Vallée08, has nine rings between 1 kpc and 10 kpc Galactocentric
radii, each with a constant magnetic field strength (see Vallée (2008) for details).
The field is clockwise as seen from the North Galactic pole, except between 5 kpc
and 7 kpc where the field is reversed. In the published model the distance between
the Galactic center and the Sun is set to 7.6 kpc. For this reason we rescale the
radial location of the boundaries between the magnetic rings by 8.5/7.6. We note
that this rescaling still does not allow an entirely fair comparison of the model.
The model is shown in figure 2.9. The only two parameters we vary in the fit is
a single, overall scaling factor for the field strengths and an exponential vertical
scale height.
WMAP field
In Page et al. (2007) the authors cite Sofue et al. (1986) and Han & Wielebinski
(2002) as reason to model the regular GMF with a bisymmetric spiral arm pattern.
They choose the model
B(r, φ, z) = B0 [sin ψ(r) cos χ(z)r̂ +
cos ψ(r) cos χ(z)φ̂ + sin χ(z)ẑ]
70
(2.9)
where ψ(r) = ψ0 + ψ1 ln(r/8 kpc) and χ(z) = χ0 tanh(z/1 kpc), and let r range
from 3 kpc to 20 kpc. In Page et al. (2007) the distance from the Sun to the
center of the Galaxy is taken to be 8 kpc. By fitting model predictions of the
polarization angle γ to the WMAP K-band data by using the correlation coefficient
rc = cos(2(γmodel − γdata )), they report a best fit for χ0 = 25◦ , ψ0 = 27◦ and
ψ1 = 0.9◦ . With these parameters rc, rms =0.76, the rms average over the unmasked
sky. This loosely wound spiral is plotted in figure 2.9.
A few comments about the WMAP model: Firstly, Page et al. (2007) characterize their model as a bisymmetric spiral, while in fact it is not (for a fixed radius, |B|
has the same value at all azimuths, and thus is an axisymmetric field). Secondly,
using the rms average (as opposed to the arithmetic mean) of rc leads to an incorrect estimate of the best-fit parameters, since, e.g., a pixel with rc = −1 (worst
possible fit) is indistinguishable to rc = 1 (best possible fit). Finally, equation 2.9
and the above quoted best fit parameters differ from the published quantities due
to typos3 in Page et al. (2007).
Because the authors of Page et al. (2007) only used the polarization angle
in their analysis no constraints on B0 were found. When performing the model
fitting with the WMAP field model in this Chapter we use a four parameter
(B0 , χ0 , ψ0 , ψ1 ) model. In accordance with Page et al. (2007) no vertical scale
height is used.
3
Errata: http://lambda.gsfc.nasa.gov/product/map/dr2/pub papers/threeyear/polarization/
errata.cfm
71
Exponential fields
To investigate the efficacy of specific model features in lowering the total χ2 for
the optimized model we test a very simple class of GMF models, and iteratively
add some complexity. We start with a basic axisymmetric model defined by an
overall field strength B0 , radial and vertical exponential scale heights r0 and z0 ,
and pitch angle p. As in the case of the logarithmic spirals we also test for symmetry/antisymmetry under the transformation z → −z. We denote these models
“ES ” and “EA ”.
We also test a somewhat more refined model, by introducing a single field
reversal. We let B0 → −B0 for r < rc , and also allow the pitch angle to be
different for r < rc . Again allowing for different symmetry under reflection in the
Galactic plane, we label these six-parameter models “E2S ” and “E2A ”.
The antisymmetry with respect to the Galactic plane of the signs of Extragalactic RMs at |b| & 15◦ has been used to support the idea that the magnetic
field in the halo is antisymmetric with respect to reflection through the plane (Han
et al. 1997). However, the antisymmetry of the signs of RMs seem to hold only for
the inner part of the Galaxy, i.e., |l| . 100◦ , which may imply that the GMF in
the halo is not globally antisymmetric. To quantify this, we also investigate a third
pair of models: “E2No ” and “E2So ”. These are identical to E2S, A , except that for
E2No the only field reversal occurs at r < rc , z > 0 (North), and at r < rc , z < 0
(South) for E2So .
72
2.6
Results and discussion
To quantitatively compare the different GMF models introduced in section 2.5.1
we use the χ2 per degree of freedom, also known as the reduced χ2 ,
χ2dof = χ2tot /(N − P ),
where N is the number of data points and P is the number of free parameters
used in the minimization. For each GMF model we find χ2dof for three separate
cases: using the complete data set, using only the data in the disk region, and
finally using only the halo region data. Each model will thus have three separate
sets of best-fit parameters. There is no obvious way to best present the results
graphically; in figure 2.10 we put the reduced χ2 of the “single” field models (i.e.,
not the Sun08 composite model, but see §2.6.1) for the three cases on three parallel
axes, and connect the points for each model. For comparison, we also calculate
~ = 0), which serves as a upper bound on the
the reduced χ2 for the null case (B
possible range of χ2dof , and is marked by a red bar in figure 2.10.
It should be noted that the relative value of the reduced χ2 for different models
is of greater importance than the absolute value for a given model, since the latter
is sensitive to somewhat arbitrary choices in the calculation of σEGS and σQ,U . We
also recall that in this Chapter we are not aiming to find the best possible model of
the GMF, as that would require including random fields and testing a larger set of
73
models. Our purpose is to present a new method of constraining GMF models and
draw broad conclusions regarding GMF models currently used in the literature.
From figure 2.10, a few general things are clear.
• There is no overall tendency of field topologies to simultaneously be a good
(or bad) fit to both the disk and halo data. This may imply that the Galactic
magnetic field in the halo has a topologically distinct structure compared to
the disk, i.e., that the GMF is not a “single” field with the same form but
slightly different parameters in the halo.
• The spread of minimized χ2dof for disk data is significant; the models that
are antisymmetric under z → −z are all strongly disfavored.
• The spread of χ2dof between models when fitting to the halo data alone is
much less than when fitting to disk data alone. This is to be expected since
~
polarized synchrotron data is only used in the halo and is identical for B
~ making for instance a symmetric model indistinguishable from an
and −B,
antisymmetric model. The synchrotron data set also has roughly twice the
number of data points as the RM data set in the halo.
• For the class of axisymmetric models “E2”, the quality of fit to halo data
is remarkably sensitive to the model’s specific symmetry with respect to the
z = 0 plane. The case where only the inner region of the Galactic magnetic
field is antisymmetric is strongly favored over the completely symmetric or
74
E2S E2No Brown07 Sun08D BSSS E2A E2So WMAP DSSS PS03 Vallee08 BSSA ES DSSA EA
Full sky
Disk
1
1.2
1.4
1.6
1.8
χ2
2
2.2
2.4
2.6
Halo
Figure 2.10: The reduced χ2 for a selection of GMF models. Each model has been
optimized to fit the data for the full data set, the disk data, and the halo data,
respectively. In each case a separate set of best-fit parameters has been obtained.
The model names are ordered according to the reduced χ2 for the full sky data
~ = 0.
sample. The red bars mark the reduced χ2 for the null case, B
antisymmetric model.
2.6.1
Fitting models to entire data set
As noted above, figure 2.10 suggests that the magnetic structure of the Galactic
disk is significantly different from that of the halo. To test this hypothesis further
we investigate the GMF models with the best performance in the disk and the
halo. For these two models, Sun08D and E2No , we note in table 2.1 the reduced χ2
for the models when fitted to the region (disk or halo) it performs best in, as well
as the reduced χ2 for the same model when fitted to the other regions (disk, halo,
75
all sky) when the same parameters are used as for the best performing region. It
is evident that there is a strong tendency for a model to do poorly in regions not
used in the optimization. Indeed, this effect is so pronounced that when fitted to
the disk (halo) data the Sun08D (E2No ) model predictions for the halo (disk) is
~ = 0. We thus conclude that the disk and halo
even worse than the null case, B
magnetic fields should be studied separately, and later combined into a complete
model of the Galactic magnetic field; as done in, e.g., Prouza & Smida (2003)
and Sun et al. (2008). Imposing continuity and flux conservation is in general a
non-trivial challenge.
Sun08 composite model
The Sun08 model warrants some additional comments, as it is unique among those
considered in this Chapter in that it has separate components for the disk and the
halo. The full model is not included in figure 2.10, since it only makes sense to
fit a “composite” model to the full sky data set. Instead we note the reduced χ2
for two interesting cases in table 2.1. When the full 11 parameter Sun08 model
is optimized to the full data set, the lowest χ2 for any model is achieved. Note
however that both the reduced χ2disk and reduced χ2halo are 0.2 larger than the
lowest χ2 achieved when fit by separate components (Sun08D and E2No ). From
figure 2.10 it is clear that the PS03 halo component is a relatively poor model of the
magnetic field in the halo and is thus forcing the disk component to depart from its
76
preferred parameters to improve the fit in the halo. Because of the large number
of parameters (11) we refrain from calculating the confidence levels for the best-fit
parameters of the full model. Instead we consider the case when the parameters of
the disk component are fixed to their optimized values (obtained using disk data
only), except for the vertical scale height, z0 , which is allowed to vary together
with the halo field parameters. The best-fit parameters are summarized in table
2.1. We note that in combination with the halo field the preferred scale height
of the disk component is relatively small (∼ 0.5 kpc). The preferred halo field
strength is considerable, but as discussed in Sun et al. (2008) this is most likely
due to an underestimation of the thermal electron scale height (see also §2.6.4 for
further discussion).
2.6.2
Fitting models to disk data
77
Table 2.1: best-fit parameters and 1σ confidence levels for the bestfitting models. The ‘Region’ column refer to what data set was used
to optimize parameters. The set of reduced χ2 has been calculated
from the model predictions of the listed best-fit parameters.
a
b
The
full
Sun08
model
with
11
parameters.
The full Sun08 model when keeping the parameters of the
disk component (except the vertical scale height) fixed to their
best-fit values, and varying the halo parameters only.
?
Note: the
likelihood function of rc for the E2No model has a complicated
shape, and thus has a less well defined error range (see figure
2.13).
Model
best-fit Parameters
Sun08D
B0 = 1.1 ± 0.1 µG,
Bc = 0.16 ± 0.20 µG
p = 35 ± 3◦ ,
rc = 5.7 ± 0.2 kpc
Region
χ2f ull
χ2disk
χ2halo
disk
2.27
1.35
2.61
halo
1.86
2.94
1.48
all
1.63
1.55
1.68
all
1.68
1.42
1.78
R0 = 5.1 ± 0.9 kpc
E2No
B0 = 2.3 ± 0.1 µG,
rc = 8.72 kpc?
p1 = −2 ± 2◦ ,
p2 = −30 ± 1◦
Sun08a
Sun08b
z0disk = 0.44 ± 0.05 kpc,
B0H = 4.9 ± 0.7 µG
H
z1a
= 0.12 ± 0.05 kpc
H
z1b
= 8.5 ± 1.5 kpc
r0H = 18 ± 4 kpc
z0H = 1.4 ± 0.1 kpc
The best-fit field model for the disk data is Sun08D , the axisymmetric field with
multiple field reversals presented in Sun et al. (2008). The best-fit parameters
78
Figure 2.11: Left: The best-fit Sun08D -type model for the magnetic field in disk
of the Milky Way. Right: The best-fit GMF field for the halo data, E2No . The
plot shows the field for z > 0. In the south (z < 0), the field direction in the inner
region (r < 8.7 kpc) is flipped. The relative normalization is arbitrary.
are summarized in table 2.1 and figure 2.12. Note that some of these best-fit
parameters are very different from the choices in Sun et al. (2008). Some differences
are expected since the parameter values were arrived at using slightly different
data and a significantly different method of analysis which probes a much larger
parameter space. If, e.g., the pitch angle is enforced to the originally proposed
value of p = −12◦ the reduced χ2 optimized for the disk data increases from 1.35
to 1.44, causing the model to lose its lead position.
The runner-up for the best-fit disk field is the Brown07 model. It is not surprising that these two models perform the best, since in addition to the parameters
allowed to vary in the fit (three parameters for Brown07, six for Sun08D ) both
79
45
40
p (deg)
rc (kpc)
6
5.8
5.6
30
25
5.4
1
1.2
1.4
b0 (µ G)
1
45
1.2
1.4
b0 (µ G)
0.8
0.6
bc (µ G)
40
p (deg)
35
35
30
0.4
0.2
0
−0.2
25
−0.4
5.4
5.6
5.8
rc (kpc)
6
2
4
6
r0 (kpc)
8
10
Figure 2.12: The 1σ and 2σ confidence levels for the best-fit parameters of the
Sun08D model when fitted to disk data.
80
models have several fixed parameters that were originally obtained using extragalactic RM data sets similar to the one used in the present analysis. These fixed
parameters make a fair comparison of the two models difficult.
2.6.3
Fitting models to halo data
The best-fit field model for the halo data is ’E2No ’, by a significant margin. bestfit parameters are summarized in table 2.1 and confidence levels plotted in figure
2.13, while the field itself is plotted in figure 2.11.
Because of the limited radial and vertical extent of the electron distributions,
the contribution to the simulated synchrotron emission and rotation measures from
regions far from the center of the disk are effectively zero, regardless of the magnetic field in that region. We allow the radial and vertical scale height parameters
of the E2No magnetic field to vary in the MCMC up to 30 kpc and 10 kpc, respectively. Figure 2.13 makes it clear that very large magnetic scale heights are
preferred. If the assumed ne , ncre are correct, we interpret this to mean that a fairly
constant field strength is preferred (the spatial extent of which we cannot infer).
Alternatively, it can be a sign that the scale heights of the electron distributions
are underestimated.
81
8.76
2
0
p1 (deg)
rc (kpc)
8.74
8.72
8.7
−2
−4
−6
8.68
2.2
2.4
b0 (µ G)
2.6
2.8
−34
8
28
6
26
r0 (kpc)
z0 (kpc)
2
−8
4
−30
−28
p2 (deg)
−26
24
22
2
0
2
−32
2.2
2.4
b0 (µ G)
2.6
2.8
20
2
2.2
2.4
b0 (µ G)
2.6
2.8
Figure 2.13: The 1σ and 2σ confidence levels for the best-fit parameters of the
E2No model when fitted to halo data.
82
2.6.4
Scale heights of electron distributions
It should be emphasized that synchrotron emission and rotation measures are not
observables of the Galactic magnetic field alone, but the magnetized interstellar
~ ne , ncre ). To properly model the GMF a comprehensive and selfmedium (B,
consistent treatment – where parameters of all three components of the magnetized
ISM are optimized simultaneously – would be desirable, which falls beyond the
scope of this Chapter.
In this section we briefly investigate how changes in the scale height of electron
distributions affect the best-fit parameters of some GMF models. This is by no
means an exhaustive study, and is mainly intended to test the versatility of our
general method of analysis, and to be considered a precursor to future work.
The vertical scale height of thermal electrons
A result common for all GMF models under consideration is that constraints on
their vertical scale height are extremely weak. For the disk, the majority of RMs
lie within |b| < 1.5◦ and no useful constraint in the vertical magnetic scale height
can be obtained. For the halo, constraints on the scale height are also weak. An
illustration of this is that a field like the WMAP model which lacks a vertical scale
height altogether still can achieve a fairly good χ2 in the halo.
The reason the magnetic scale height is poorly constrained is because the observables depend on the product of the magnetic field and an electron density (ne
83
for RM and ncre for synchrotron emission). The models of these electron density
distributions have their own vertical scale heights and thus significantly reduce the
RM and synchrotron emission at larger vertical distances from the disk independently of whether a significant magnetic field exists there or not.
Recent work by Gaensler et al. (2008) has shown that the vertical scale height
of thermal electrons may be significantly underestimated in the NE2001 model,
possibly by a factor of two. It is thus important to consider the impact of a change
in the thermal electron scale height on the best-fit vertical magnetic scale height.
To investigate this we modify the vertical scale height, h0 , of the thick disk in the
Cordes & Lazio (2002) NE2001 model (keeping fixed the product of the mid-plane
density and the vertical scale height, which is constrained by pulsar DMs), and
fit the E2No model to the RM and polarized synchrotron data in the halo. All
magnetic field parameters except B0 and z0 are kept fixed to their best-fit values
with the original NE2001 scale height. As seen from the results, which are plotted
in figure 2.14, the preferred vertical magnetic scale height and its uncertainty
decrease as the electron density scale height is increased. For a large electron
density scale height, h0 > 1.5 kpc, the preferred vertical magnetic scale height is
z0 ≈ 1.3−2 kpc. Additionally, if the vertical scale height of the relativistic electron
density tracks h0 and this should be increased as well, the resulting best-fit value
of z0 would be smaller still. When in a future analysis both random and regular
magnetic fields are considered together, equipartition arguments could be invoked
84
to relate the magnetic field and cosmic ray electron density scale heights.
Radial scale length of relativistic electrons
The analysis method introduced in this Chapter can readily be extended from a
formal point of view, to include varying parameters of the electron densities as
well as of the GMF models. We test the feasibility of this by letting the galactocentric radial scale length of the relativistic electron model vary (but enforcing the
overall normalization such that ncre (Earth) is at its measured value). Redoing the
parameter estimation for the halo model E2No with this added parameter, we find
the new best-fit parameters to be within the previously noted confidence levels,
and the best-fit radial scale length becomes rcre = 6.6 ± 1.5 kpc, which is close to
the fixed value rcre = 5 kpc.
2.6.5
Discussion
An obvious but important comment is that a best-fit model is not necessarily a
‘good’, or correct, model. And best-fit parameters and confidence levels have little
relevance if the underlying model is a poor depiction of reality. For example, most
of the models under consideration in this Chapter have a pitch angle as a free
parameter, but the best-fit pitch angles differ by a factor of a few between models.
Hence, if we lack confidence in the veracity of a given model we cannot trust the
optimized value of the pitch angle.
85
5
4.5
4
z0 (kpc)
3.5
3
2.5
2
1.5
1
0.5
0
0.5
1
h0 (kpc)
1.5
2
Figure 2.14: The best-fit vertical magnetic scale height vs. the vertical scale height
of the NE2001 thermal electron density of the thick disk when fit to halo RMs and
polarized synchrotron data. The E2No field model was used, keeping all parameters
except B0 and z0 fixed to their best-fit values. The error bars contain 90% of the
MCMC sampled points.
86
So how do we achieve confidence in a GMF model? More data, in particular
RM sources in “empty” regions of figure 2.5, will strongly reduce the number of
GMF models that can reproduce the data well. Of special importance are the large
gaps of RM data in the disk at 0◦ . l . 60◦ and 150◦ . l . 250◦ . At present, a
few topologically distinct GMF models considered in this Chapter reproduce the
RM data almost equally well, while at the same time making significantly different
predictions for the rotation measure in the regions of the disk lacking data. For
example, in the disk the Sun08D , Brown07 and E2s models fit the data best, yet
the latter model has only one field reversal and the others three; thus even the sign
of RM will differ between the models for part of the disk. Filling these gaps in the
data is thus crucial. In addition, more data at high Galactic latitude will enable
regions of the sky to be constrained where now there are too few RMs to measure
σ. Pulsar RMs are abundant in the disk, and future work will incorporate this
data set as well. In addition, polarized synchrotron data in the disk can be used if
the contamination due to local synchrotron features are modeled and subtracted.
A complimentary approach is to introduce a χ2theory term that incorporates
~ = 0) and information gained through
our understanding of physics (e.g., ∇ · B
observations of magnetic fields in external galaxies similar to the Milky Way (e.g.,
that pitch angles only vary within some specific range). Adding χ2theory to χ2tot would
be a first step toward a fully Bayesian analysis. It would act to disfavor unphysical
models that, at present, are able to reproduce the current data well. Similarly,
87
using predictions from a theory of galactic magnetogenesis, such as dynamo theory,
could possibly provide powerful constraints on our freedom to build models of the
GMF. However, with the genesis of galactic magnetism still not fully understood,
we believe it prudent to study the Galactic magnetic field in the most empirical
way possible. Hopefully, a firm observational understanding of the magnetic field
of the Milky Way will be a significant aid in resolving the theoretical question of
the formation of such a magnetic field.
Model-building
When building a model of the Galactic magnetic field it is essential to be able
to quantify the improvement of the model by the inclusion or modification of
a particular model feature. This is especially true when models become more
complex than what can be described by a handful of parameters, as we have shown
in this work is necessary. The reduced χ2 provides such a tool as outlined above.
In combination with codes to simulate mock data sets and perform parameter
estimation, the reduced χ2 creates a simulation laboratory where one can easily
assess and compare the pull on χ2 of any proposed model feature.
2.7
Summary and conclusion
In this Chapter we combined the two main probes of the large-scale Galactic magnetic field – Faraday rotation measures and polarized synchrotron emission – to
88
test the capability of a variety of 3D GMF models common in the literature to account for the two data sets. We used 1433 extragalactic rotation measures and the
polarized synchrotron component of the 22 GHz band in WMAP’s five year data
release. To avoid polarized local features we applied a mask on the synchrotron
data, covering the disk and some prominent structures such as the North Polar
Spur.
We developed estimators of the variance in the two data sets due to turbulent
and other small-scale or intrinsic effects. Together with a numerical code to simulate mock RM and synchrotron data (the Hammurabi code, by Waelkens et al.
(2009)) this allows us to calculate the total χ2 for a given GMF model, and then
find best-fit parameters and confidence levels for a given model. This enables us
to quantitatively compare the validity of different models with each other.
From applying our method on a selection of GMF models in the literature we
conclude the following:
• None of the models under consideration are able to reproduce the data well.
• The large-scale magnetic field in the Galactic disk is fundamentally different
than the magnetic field in the halo.
• Antisymmetry (with respect to z → −z) for the magnetic field in the disk is
strongly disfavored. For the halo, antisymmetry of the magnetic field in the
inner part of the Galaxy is substantially preferred to both the completely
89
symmetric or antisymmetric case.
We find the existence of several false maxima in likelihood space to be a generic
trait in the models considered. Specifically, for the best-fit disk model in this work,
the true maximum is far removed from the configuration considered in its original
form. However, we caution that this optimized configuration should not be trusted
as its structure is clearly unphysical. We posit a combination of two explanations
for this fact: firstly, the underlying model may not be sufficiently detailed to
explain the GMF or may be otherwise incorrect. Secondly, the lack of RM data in
large sections of the disk may create both local and global likelihood maxima that
would not otherwise exist.
From this and other observations we recommend the following regarding future
GMF model-building:
• It is necessary to probe a large parameter space to avoid false likelihood
maxima.
• A more complete sky coverage of RM data is necessary to derive a trustworthy
GMF model.
• Optimized parameters are sensitive to the underlying electron distributions, and these quantities should be included in the optimization for selfconsistency and to achieve correct estimates of confidence levels.
90
• When possible, combining separate data sets should be done to maximize
the number of data points in the analysis, to probe orthogonal components
of the magnetic field, and to ensure the final model’s consistency with the
most observables.
We also emphasize that by having a well-defined measure of the quality-of-fit, such
as provided in this work, the efficacy of any given model feature (i.e., symmetry,
extra free parameter, etc.) can be quantified, which is essential when building the
next generation of models of the GMF.
In the comparatively near future the number of measured extragalactic RMs
will increase by more than an order of magnitude (see e.g., Taylor (2009); Gaensler
et al. (2004); Gaensler (2009)) and Planck data will become available, which together should allow large parts of the currently valid model and parameter space
to be excluded.
91
Chapter 3
Magnetic deflection of UHECRs
in the direction of Cen A
Chapter Abstract
The Pierre Auger Observatory has observed an excess of Ultrahigh Energy Cosmic
Rays (UHECRs) in the region of the sky close to the nearby radio galaxy Centaurus A. We constrain the large-scale Galactic magnetic field and the small-scale
random magnetic field in the direction of Cen A, using WMAP 22 GHz data and
extragalactic rotation measure sources, including 188 sources around Cen A. From
the best fit models of the GMF we estimate the deflection of the observed UHECRs
and predict their source positions on the sky. We find that the deflection due to
random fields are small compared to deflections due to the regular field. Assuming
92
the UHECRs are protons we find that 4 of the published Auger events above 57
EeV are consistent with coming from Cen A. We conclude that the proposed scenarios in which most of the events within approximately 20◦ of Cen A come from
it are unlikely, regardless of the composition of the UHECRs.
3.1
Introduction
Centaurus A, our nearest active radio galaxy, was first considered as a possible
source of UHECRs by Cavallo (1978). Farrar & Piran (2000) considered a scenario
where most UHECRs originate from Cen A. With the recently completed Pierre
Auger Observatory, numerous events have been observed close to Cen A (see figure
3.1) and interest in Cen A as a possible UHECR source has been revived. Wibig
& Wolfendale (2007) consider Cen A to be one of three sources that combined are
responsible for all observed UHECRs. An analysis of the significance of correlation
between UHECRs and Cen A was performed by Gorbunov et al. (2008). Hardcastle
et al. (2009) and Croston et al. (2009) investigate the plausibility of the giant radio
lobes of Cen A as acceleration sites of UHECRs. Kachelrieß et al. (2009) consider
the case of the radio jet at the core of Cen A as an accelerator. Rachen (2008)
considers various mechanisms that could accelerate UHECRs in radio galaxies such
as Cen A. Fargion (2008, 2009) argues that Cen A is the source of the dozen or
so events in the region of the sky surrounding Cen A; he suggests that the CRs
93
consist of light nuclei (He, Be, B, C, O) that are smeared by Galactic random
magnetic fields oriented parallel to the Galactic plane, causing the observed CRs
to be aligned along a “string” perpendicular to the plane, as seen in figure 3.1.
Whether Cen A can be the source of several or of many of the highest energy
Cosmic Rays depends on the nature of the Galactic and extragalactic magnetic
fields along the line of sight in the general direction of Cen A. If a large-scale
coherent field dominates the deflection and this field is known, then the magnitude
and direction of the deflection of each CR can be calculated assuming they are
protons, or for any other charge assignment. In order for many of the highest
energy UHECRs to be produced by Cen A, deflections in turbulent fields must
dominate coherent fields – otherwise, UHECRs from Cen A would be arranged
according to decreasing rigidity (E/Z) along some (one-sided) arc originating at
Cen A which is manifestly not observed.
In this chapter, we constrain models of the magnetic fields between Cen A and
the solar system, using extragalactic Faraday rotation measures and synchrotron
radiation observed by WMAP. By constraining both the regular and random magnetic fields, we predict the direction to the sources of the observed UHECRs in the
vicinity of Cen A and assess the plausibility of Cen A as a source of a subset of
the observed UHECRs.
The chapter is organized as follows: in section 3.2 we give a brief overview of
the relevant facts that are known about Centaurus A as an astrophysical object.
94
Figure 3.1: The published Auger UHECR events above 57 EeV around Cen A.
Energies are marked in EeV, with contours from radio data (Haslam et al. 1982)
outlining Cen A (center) and parts of the Galactic plane.
Section 3.3 discusses the available observational data that can constrain the magnetic field in the direction of Cen A. Section 3.4 describes our best model of the
regular and random magnetic field. Section 3.5 discusses the expected deflection
and source location of observed UHECRs. We end with section 3.6, a summary
and discussion.
95
3.2
Centaurus A - an overview
Centaurus A (NGC 5128) is a Fanaroff-Riley Class I (FR-I) radio galaxy (Israel
1998). The massive elliptical host galaxy is located at (l, b) = (309.5◦ , 19.4◦ ).
Thanks to its proximity and size, its enormous radio lobes combine into the largest
extragalactic object on the sky, with an angular size of 8◦ -by-4◦ , corresponding to
a physical size of 500×250 kpc at the distance of 3.4 Mpc. Figure 3.2 show Cen A
in total intensity at 408 MHz (Haslam et al. 1982).
About 5 kpc from the central galaxy, jets from the accretion disk surrounding
the central supermassive black hole expand into plumes as they plow into the
ambient intergalactic medium. These plumes are called the inner radio lobes.
Some material goes farther, creating the northern middle lobe, which extends to
30 kpc and lacks a southern counterpart. The giant outer radio lobes extend 250
kpc in projection both in the north and the south. The 3D orientation of the lobes
is not well-known, but the northern lobe is believed to be closer to us than the
southern lobe. It has often been suggested that Cen A is in fact a “misaligned”
BL Lac1 .
1
A sub-class of active galactic nuclei (AGN), with a relativistic jet closely aligned with the
observer’s line-of-sight.
96
Figure 3.2: Centaurus A in total intensity (Stokes I) at 408 MHz (Haslam et al.
1982).
97
3.3
Observables of the magnetic field toward
Cen A
Estimating the magnetic field structure of the Milky Way (and beyond) in a particular direction is very challenging. From studies of external galaxies (see chapter 1)
we know that even galaxies that exhibit highly regular large-scale magnetic fields
often have smaller irregularities that can make predictions of the magnetic field in
a given direction based on the correct ‘global’ GMF model incorrect. However, in
the absence of extensive observational data in the direction of interest, relying on
global GMF modeling is a necessary crutch.
The study of the magnetic field in the direction of Cen A serves as a useful
case study in how to estimate Galactic magnetic fields in any given direction.
While we will end up not including starlight polarization data or pulsar rotation
measures, these data sets could in general be extremely useful in constraining the
GMF in a given direction. A detailed description of the observables below and
their connection to magnetic fields is given in chapter 1. In this section we discuss
only that part of the analysis that specifically relates to Cen A.
3.3.1
Synchrotron emission
The low frequency radio image of Cen A in figure 3.2 clearly shows the giant
lobes as well as the northern middle lobe. Galactic foreground emission – mainly
98
Figure 3.3: Polarized synchrotron radiation at 22 GHz (Page et al. 2007), capped
at 0.1 mK. The texture follows the magnetic field (the measured polarization angle
rotated 90◦ ). The inset show a zoomed view of Cen A.
99
synchrotron radiation – is also visible in this image at lower galactic latitude (b).
To extract information about the large-scale regular magnetic field, however, it is
preferable to study the polarized intensity and polarization angle at a frequency not
strongly affected by Faraday rotation, which acts to obfuscate polarized radiation
emitted from larger distances. The WMAP5 22 GHz synchrotron maps (Gold et al.
2008) provide such a data set. Figure 3.3 shows a large part of the sky, with color
scale indicating polarization intensity and texture tracing the projected magnetic
field (i.e., the polarization angle rotated by 90◦ ). The texture gets more prominent
in regions with strong PI. A blow-up of the Cen A region is shown in the inset. It
is clear from figure 3.3 that Cen A is located in a special position on the sky: at
the very edge of the highly polarized region that is part of the nearby North Polar
Spur (NPS) or radio loop 1 (Wolleben 2007). Figure 3.4 make this point clearer
by plotting the PI along the meridian passing through Cen A. The significant
polarization intensity bordering Cen A to the left is hence most likely coming from
very nearby (. 200 pc) and is not a good indicator of the magnetic field structure
in the diffuse interstellar medium, which presumably has the greatest impact on
UHECR deflections because of its great extent. Fortunately, Cen A lies at the edge
of this local synchrotron emission, and to the right of Cen A the radiation is much
weaker and probably dominated by emission from the diffuse ISM. Figure 3.5 show
the same WMAP data in greater detail, with bars indicating the polarization angle.
The low emission region to the right of Cen A could theoretically contain significant
100
information about the magnetic field: the polarized intensity can be used to infer
the strength of the transverse magnetic field (also the relevant quantity for the
deflection of UHECRs), and the polarization angle to infer the orientation of the
transverse magnetic field and hence also the UHECR deflection. Unfortunately,
the signal-to-noise is poor in the low emission region, and a reliable estimation
of the polarization angle is not possible with the WMAP data. The estimated
1σ errors in the polarized synchrotron map in this region is approximately 2 µK
(Dunkley et al. 2009).
To compare simulated PI with observational data it is necessary to first estimate the PI due to Galactic synchrotron emission and its variance due to random
magnetic fields. We use the same method described in §2.4 and average the WMAP
Stokes Q and U parameters over a circle of radius 4◦ to estimate the PI in the
circled region. The variance of the PI is calculated from the 1◦ pixels within the
circle. The measured PI in the direction of Cen A is obviously dominated by emission from Cen A itself, and cannot be used. Local features dominate much of the
surrounding sky. While the area adjacent Cen A to the right appear mostly uncontaminated by local emission, the PI averaged over a 4◦ circle is quite sensitive to
the exact placement of the circle. For this reason we manually pick four locations
(see figure 3.6) to place 4◦ circles and compute the average PI and variance of these
regions. In this way we estimate the polarized intensity to be PI = 0.006 ± 0.009
mK in the general direction of Cen A. The total intensity is less sensitive to local
101
Figure 3.4: Polarized synchrotron intensity (WMAP, 22 GHz) of the b=19.4◦
meridian after averaging Stokes Q and U over 4 degree radii circles. The Cen
A value marked is the average PI for the Cen A region, as estimated in §3.3.1. The
North Polar Spur and Fan region are marked.
102
Figure 3.5: Polarized synchrotron radiation at 22 GHz around Cen A, with bars
showing the polarization angle. The central contour mark the boundary of Cen A.
103
Figure 3.6: WMAP 22 GHz polarized synchrotron radiation and circles marking
the regions used to estimate the Galactic contribution to the PI in the direction of
Cen A.
variations, but more sensitive to Galactic latitude (see figure 3.7). Thus we only
use the 4◦ circle adjacent to Cen A, which yields the estimate I = 0.10 ± 0.07 mK.
This data will be crucial in estimating the presence of random fields, done below
in §3.4.
104
Figure 3.7: WMAP 22 GHz total synchrotron radiation.
105
3.3.2
Extragalactic rotation measure sources
In addition to synchrotron data, Faraday rotation measures yield complementary
constraints on the large-scale GMF. Feain et al. (2009) recently measured the
RMs of 188 extragalactic sources with lines-of-sight near, but outside, Cen A (see
figure 3.8). This data set can provide an excellent constraint on the line-of-sight
component of the intervening magnetic field. The average RM of the sources is
−54 rad m−2 , with a standard deviation of 32 rad m−2 .
3.3.3
Other potentially useful data sets
Pulsars
Pulsar rotation measures could conceivably provide powerful constraints on the
line-of-sight component magnetic field toward Cen A. The benefit of pulsars RMs
compared to EGS RMs is that the former only probe the Galactic RM contribution
between the observer and the pulsar, while the latter may be contaminated by RM
contributions of extragalactic origin or RM intrinsic to the source. However, as seen
in figure 3.9, only 7 pulsars within 10◦ of Cen A have measured RMs. Moreover,
the distances of the pulsars can only be estimated by their dispersion measure
and assuming a 3D model of the thermal electron distribution (for details, see
§1.2.1). The distances labeled in the figure have been estimated using the NE2001
thermal electron model (Cordes & Lazio 2002, 2003), with modifications according
106
Figure 3.8: 188 extragalactic sources with line-of-sights outside Cen A (Feain et al.
2009). Red squares denote positive rotation measures (corresponding to a line-ofsight electron-density-weighted average magnetic field toward the observer), blue
circles denote negative rotation measures. The size of the markers is proportional
to the magnitude of the rotation measure.
107
to Gaensler et al. (2008), i.e., a thick disk mid-plane density of 0.014 cm−3 with a
vertical scale height of 1.83 kpc. It is clear from figure 3.9 that there is significant
differences in the measured pulsar RMs, with the pulsar probing the farthest even
having a different sign than the EGS RMs in figure 3.8. An explanation of this
is the significance of random fields and the somewhat lower latitude of the pulsar
compared to the EGS. Because the number of EGS available, and the relatively
low variance of their RMs, we do not include any pulsar RMs in our analysis
and interpret the average EGS RM as the total Galactic contribution of rotation
measure in the direction of Cen A.
starlight polarization
Starlight polarization (see §1.2.3) provides mainly directional information on the
transverse magnetic field. As such, it could be very useful in predicting the direction of the deflection of UHECRs. Unfortunately, as seen in figure 3.10, the
numerous starlight polarization measurements that exist are all from very nearby
stars, and thus probe only a small fraction of the magnetic field between Earth and
Cen A. Hence we will not include the starlight polarization data in our analysis.
3.4
Magnetic field modeling
The above estimates of the RM, I and PI are not sufficient to fix the Galactic
magnetic field structure in the direction of Cen A because they are given by line108
Figure 3.9: The 7 pulsars within an angular distance of 10◦ from Cen A with measured RM. Red squares (blue circles) denote positive (negative) rotation measures
with area proportional to the magnitude of the RM. Distances in kpc are given in
black rectangles, inferred from the pulsar dispersion measures using the NE2001
thermal electron model (with modifications according to Gaensler et al. (2008)).
109
Figure 3.10: Polarized synchrotron intensity (WMAP, 22 GHz) and starlight polarization bars. Length of bars indicate polarization fraction. Distances are labeled
in kpc.
110
of-sight integrals along which the field reverses. Thus, we determine the parameters
of the best-fit global GMF model by using all-sky PI and RM data as in Chapter
2 and Jansson et al. (2009), then test the quality of the resultant total field in
the Cen A direction by comparing predicted and observed RM and PI for that
direction in particular. At a later stage we use the observed total intensity to
estimate the strength of random fields in the Cen A direction.
In Chapter 2 we investigated the quality of most of the large-scale GMF models
proposed in the literature to explain polarized synchrotron and extragalactic RM
data. The magnetic field in the disk was found to have a fundamentally different
form to the field in the halo. In this paper we combine the best models of the disk
and the halo from Chapter 2 into a global GMF model. For the halo this is the
E2No model, while for the disk, we choose the Brown07 model (Brown et al. 2007)
instead of the optimized Sun08 disk model (Sun et al. 2008), which had unphysical
best-fit parameters; see Chapter 2 for details.
We generalize the analysis of Jansson et al. (2009) by replacing the vertical scale
height of the disk field with a parameter, hdisk , that defines the height of the sharp
transition between the disk and halo field. The model of the Galactic halo field
is further generalized to include an out-of-plane component, as suggested by radio
observations of edge-on external galaxies (Beck 2009). The out-of-plane component
of the halo field is taken to be axisymmetric and specified by its field strength just
X
X
X
above (below) the Galactic disk, B0,N
orth (B0,South ), pitch angle p , elevation angle
111
θX , and radial and vertical exponential scale lengths r0X and z0X . The B0X ’s can
be negative, to allow for various symmetries (magnetic field lines directed toward
or away from the disk, ‘quadrupole-like’; or having the same direction both above
and below the disk, ‘dipole-like’). Hereafter, we will refer to the out-of-plane halo
component as the “X-field” component, since it is partially motivated by the Xshaped field structures observed in external, edge-on galaxies (Beck 2009).
To constrain the random dispersion of UHECRs about their coherent deflection it is crucial to put robust limits on random fields. We include two types of
random fields in our analysis: purely random fields and “striated” random fields.
Supernovae and other outflows are expected to produce randomly-oriented fields
with a coherence length λ of order 100 pc or less. Differential rotation will shear
these random fields, producing a striated configuration whose predominant orientation is plausibly aligned with the local coherent field. Striated fields produce
a net polarized intensity emission even if the average field vanishes, but do not
contribute in the leading order to rotation measures. We include the effects of
striated magnetic fields by adding a multiplicative factor to the calculation of PI,
such that if this factor is equal to unity we recover the original expression. We let
the factor be a free parameter in the large-scale GMF model. There is an obvious
degeneracy between the strength of striated magnetic fields and the relativistic
electron density: if we write the multiplicative factor as α(1 + β), we can interpret
2
2
α as being the scaling of the relativistic electron density, with Bstri
= βBreg
. The
112
distribution of relativistic electrons in the Galaxy is not well enough known to
permit this degeneracy to be disentangled at present. Of course, since β > 0 if
α(1 + β) is found to be less than unity we can conclude that α < 1, and that
ncre has been underestimated. To avoid underestimating the random deflection of
UHECRs, we here assume α = 1 if α(1 + β) > 1.
3.4.1
Results of the fit
We apply the method detailed in Chapter 2 to find the field parameters which
best fit the WMAP 22 GHz polarized synchrotron emission and all published extragalactic RMs (including the recent addition of 188 RMs in the Cen A direction,
from Feain et al. (2009)). The reduced χ2 of the best fit is 1.24. This is a substantial improvement over the previous-best global GMF model, with χ2 = 1.63,
although an exact comparison is not possible due to the different number of data
points and parameters. The predicted PI and RM in the Cen A direction for the
best-fit field parameters are 0.013 mK and -49 rad m−2 , respectively, and are thus
in good agreement with the observed values.
The best-fit GMF parameters are summarized in table 3.1. The transition
between disk and halo field occurs at a relatively low height, 0.14 kpc, i.e., more
akin to a thin disk than a thick disk. Most other best-fit parameters are similar to
those found in Chapter 2, and we refer to that Chapter for parameter definitions.
The out-of-plane halo component is found to be contained in the inner part of
113
Table 3.1: Best fit GMF parameters. Due to the large number of parameters,
those whose error ranges are small (. 10%), or have negligible affect on the final
estimation of UHECR deflections have been set to their best-fit values in the final
MCMC analysis and thus lack error estimates. For the vertical scale height of
the out-of-plane field only a lower bound is possible with the current data. See
Chapter 2 for parameter definitions.
Model
Best fit Parameters
Brown07 disk αD = 0.2,
rc = 15 kpc,
hdisk = 0.14 kpc
E2No halo
B0H = 1.7 ± 0.2 µG,
z0H = 4.4 ± 2.6 kpc
rcH = 9 kpc
p1 = 11 ± 3◦ ,
p2 = −33 ± 2◦ ,
r0H = 29 kpc
X
X
X halo
B0,N orth = 15 ± 9 µG, B0,South = 20 ± 6 µG, θX = 55 ± 5◦ ,
pX = −77 ± 6◦ ,
r0X = 2.2 ± 0.2 kpc
z0X & 4 kpc
Striated fields β = 1.4 ± 0.3
the Galaxy (r0X =2.2 kpc) with a significant vertical extent (z0X & 4 kpc). This
halo component is very strong (15-20 µG ), with a prominent vertical component
X
X
X
upward (B0,N
= 55◦ elevation), and a weak azimuthal
orth , B0,South > 0, and a θ
component (pitch angle pX ≈ −80◦ ). We find evidence of striated magnetic fields
(or underestimated relativistic electron density), with β = 1.4 ± 0.3 if the standard
ncre is accepted.
3.4.2
Purely random fields
We can now estimate the strength of purely random fields in the particular direction
of Cen A, by comparing the observed total intensity, Iobs , of synchrotron emission,
to the simulated total intensity, Isim , from the modeled global and striated fields
alone. If Iobs > Isim , the difference can be attributed to a random component
of the GMF. Taking the magnetic field energy in the purely random component
114
2
to be proportional locally to the energy in the global component, i.e., B⊥,rand
=
2
γB⊥,regular
, and assuming a spectral index p = −3 for the relativistic electron
γ
2
. Fitting I in the direction of
distribution so that I ∝ B⊥
, gives Iobs /Isim ≈ 1 + 1+β
Cen A, with β = 1.4, gives γ = 11 ± 9. With the best-fit values for β and γ, the
magnetic energy division comes to roughly 10% − 10% − 80% for regular, striated,
and purely random fields, respectively. An analysis by Jaffe et al. (2009) of the
magnetic field in the Galactic disk only, found the corresponding values 10% −
40% − 50%, but this is not a worrying discrepancy. One expects the proportion of
random field to differ in the disk and halo (e.g., toward Cen A) due to differences
in structure of the turbulence and the effects of differential rotation; moreover both
estimates have large uncertainties.
3.5
Estimated deflections
In the limit of small-angle deflections, the deflection angle is inversely proportional
to the cosmic ray rigidity (the ratio of CR energy and charge). Using our best-fit
global GMF model, we find that the deflection of a UHECR in the regular magnetic
field in the direction of Cen A is δθreg ≈ 4◦ ± 0.5◦ (Z/E100 ), where Z is the charge
and E100 is the energy of the CR in units of 100 EeV.
Given β and γ, we can estimate the magnetic dispersion due to striated and
random fields as follows. Striated fields are oriented with the regular field, and
115
cause dispersion along the direction of the deflection due to the regular field. The
random fields cause dispersion in all directions. Because deflection is linear in the
magnetic field strength we can treat striated and random fields separately. To
estimate the magnetic dispersion we propagate each CR through domains of size
λ through the Galaxy, taking the field in the ith domain to be the sum of the
~ i , and a random part: B
~ i + √γ|Bi |n̂ for random fields and
local model GMF, B
√
~ i + β|Bi |n̂ for striated fields, where n̂ is a unit vector chosen to have a different
B
random direction in each step (n̂ = ±1 in the case of striated fields). The domain
size λ corresponds to the maximum coherence length of the turbulent field; this
is uncertain but is plausibly of order ≈ 100 pc. Repeating the propagation many
times for a given CR rigidity, the smearing angle σrand is defined to be the opening
angle of a cone containing 68% of the arrival directions. As one would expect,
√
σrand ∼ λ. In the direction of Cen A, we find
p
λ100 ,
σrand = 1.6◦ +0.8
(Z/E
)
100
−1.2
p
σstri = 0.4◦ ± 0.2◦ (Z/E100 ) λ100 .
(3.1)
(3.2)
where λ100 is the coherence length of the turbulent field in units of 100 pc. The
quoted uncertainties are obtained by adding in quadrature the standard deviation
of the arrival directions and the uncertainty in σ due to the uncertainty in β and
γ. We note that a more sophisticated treatment with a Kolmogorov spectrum
116
would be expected to give a somewhat lower amount of dispersion, since in that
case power is shared over a range of scales rather than being concentrated in the
largest coherence length which is most effective at deflecting UHECRs, so Eq. 3.1,
3.2 give maximum estimates of the magnetic dispersion from Galactic fields.
Given a model of the GMF, the observed UHECR arrival directions can be backtracked through the field assuming the CRs are protons, to find the true source
direction in that case. Auger estimates an ≈ 14% statistical and ≈ 22% systematic uncertainty on the CR energies (The Pierre Auger Collaboration: J. Abraham
et al. 2009b). Adding them in quadrature produces a ≈ 26% energy uncertainty
for each event. Figure 3.11 shows the inferred source directions of the published
Auger UHECR’s in the vicinity of Cen A, assuming the CRs are protons deflected
by our best-fit GMF. The source region shown for each CR is obtained as follows.
Repeatedly backtracking a CR using global field parameters randomly drawn from
the (converged and well-mixed) Markov chain used in the GMF parameter optimization, and only selecting from the subset of parameters contained in the 68%
of the Markov chain with lowest χ2 , as well as accounting for the 26% energy uncertainty and event resolution, yield the dashed contours. Including as well the
dispersion due to striated and random fields yields the solid contours.
117
Figure 3.11: Estimates of source locations (black contours) for individual Auger
events using backtracking through the best-fit GMF model. Dashed contours show
the spread in the predicted UHECR source locations due to the uncertainty in the
best-fit GMF parameters, the angular resolution of the Auger observatory, and
energy uncertainty of the observed cosmic rays. The solid contours include the
dispersion due to random and striated magnetic fields. Gray arrows point from
a source region to the corresponding UHECR, originating at the position of the
source for the best-fit GMF and no magnetic dispersion. Energies are labeled
in EeV. Colored regions indicate the regions where UHECRs are expected to be
observed if the source is the Cen A central galaxy. The green, red and blue regions
are for UHECRs of rigidity 200, 60 and 40 EeV/Z, respectively. The regions take
into account the event resolution of the Pierre Auger Observatory, the dispersion
due to random and striated magnetic fields, and the spread due to the uncertainty
of the best fit GMF parameters.
118
3.6
Results and discussion
In our best-fit model, a 60 EeV proton with Cen A as its source is deflected by
approximately 7 degrees, predominantly toward the Galactic plane. Smearing due
to random magnetic fields is sub-dominant to the deflection due to the regular
field. The dispersion can be seen in Fig. 3.11. Taking a generous view of the
possible deflections and assuming the UHECRs are protons, we conclude that 4 of
the 27 published Auger UHECRs are consistent with originating from Cen A or
its radio lobes.
UHECRs coming from Cen A must lie on an arc or swath originating within its
giant radio lobes, with events of lower rigidity further from the source – unless the
intergalactic medium surrounding the Milky Way in the hemisphere toward Cen
A has a much-larger-than-expected random magnetic field Farrar & Piran (2000).
Fig. 3.11 shows the expected swath of CRs for three different rigidities, E/Z = 40,
60, and 200 EeV, with Cen A as the source. The opening angle of the swath is
approximately given by 2σrand /|δθreg | ≈ 50◦ . For small deflections, the numerator
and denominator in this expression are both inversely proportional to the rigidity
of the cosmic rays. Thus the opening angle of the swath is independent of the
charge, or composition, of the UHECRs. Hence, regardless of their composition,
it is unlikely that significantly more than 4 of the 27 Auger CRs can be attributed
to Cen A, so that those scenarios in which the majority of UHECRs surrounding
Cen A originate from Cen A can be excluded.
119
Summary
In this thesis we have investigated, optimized, and developed models of the Galactic magnetic field; and constrained the deflection of UHECRs towards a specific
potential source, Centaurus A.
In chapter 2 we combined the two main probes of the large scale Galactic
magnetic field – Faraday rotation measures and polarized synchrotron emission –
to test the capability of a variety of 3D GMF models common in the literature
to account for the two data sets. We used 1433 extragalactic rotation measures
and the polarized synchrotron component of the 22 GHz band in WMAP’s five
year data release. To avoid polarized local features we applied a mask on the
synchrotron data, covering the disk and some prominent structures such as the
North Polar Spur.
We developed estimators of the variance in the two data sets due to turbulent
and other small scale or intrinsic effects. Together with a numerical code to simulate mock RM and synchrotron data (the Hammurabi code, by Waelkens et al.
(2009)) this allowed us to calculate the total χ2 for a given GMF model, and then
120
find best fit parameters and confidence levels for a given model. This enabled us
to quantitatively compare the validity of different models with each other.
From applying our method on a selection of GMF models in the literature we
concluded that none of the models under consideration are able to reproduce the
data well. We noted that the large scale magnetic field in the Galactic disk is
fundamentally different than the magnetic field in the halo. Antisymmetry (with
respect to z → −z) for the magnetic field in the disk is strongly disfavored. For
the halo, antisymmetry of the magnetic field in the inner part of the Galaxy is
substantially preferred to both the completely symmetric or antisymmetric case.
We found the existence of several false maxima in likelihood space to be a
generic trait in the models considered. Specifically, for the best-fit disk model in
this work, the true maximum is far removed from the configuration considered in
its original form. However, we caution that this optimized configuration should
not be trusted as its structure is clearly unphysical. We posit a combination of
two explanations for this fact: firstly, the underlying model may not be sufficiently
detailed to explain the GMF or may be otherwise incorrect. Secondly, the lack of
RM data in large sections of the disk may create both local and global likelihood
maxima that would not otherwise exist.
We made several recommendations for future GMF model-building. To avoid
false likelihood maxima two steps are important: first, it is necessary probe a large
parameter space; and second, to use a more complete sky coverage of the RM data.
121
To achieve correct estimates of confidence levels it is necessary to self-consistently
include the underlying electron distributions in the optimization. Finally, when
possible, combining separate data sets should be done to maximize the number of
data points in the analysis, to probe orthogonal components of the magnetic field,
and to ensure the final model’s consistency with the most observables.
We also emphasized that by having a well-defined measure of the quality-offit, such as provided in this work, the efficacy of any given model feature (i.e.,
symmetry, extra free parameter, etc.) can be quantified, which is essential when
building the next generation of models of the GMF.
In chapter 3 we investigated the plausability of the nearby radio galaxy Cen A
to have been the source of several UHECRs observed in the vicinity of Cen A. We
extended the work done in chapter 2, by combining two models of the disk and
halo GMF, and simultaneously fit them both to the combined RM and polarized
synchrotron data sets, now also extended to include 188 extragalactic RM sources
near Cen A. We found the best-fit transition between the disk and halo field to
occur at ' 140 pc above and below the mid-plane of the Galaxy. The predicted
RM and polarized synchrotron intensity in the direction of Cen A by our GMF
model, when optimized to the full sky data set, was found consistent with the
observed data in that direction. With this model, we estimated that a 60 EeV
proton arriving from the direction of Cen A is deflected by approximately 3 − 8
degrees, predominantly toward the Galactic plane. We found that deflections are
122
consistent with four of the 27 published Auger UHECRs being protons originating
from Cen A.
Using WMAP total synchrotron intensity, we constrained the strength of the
turbulent field of the Galaxy, along the sightline toward Cen A. This allowed us
to estimate the smearing of UHECR arrival directions due to propagation through
this random field: ≈ 0.7◦ Z/E100 . This smearing is sub-dominant to the deflection
due to the regular field, and makes it unlikely that significantly more of the 27
Auger CRs can be attributed to Cen A, even assuming some of them are iron.
This smearing is too low to allow significantly more of the 27 Auger CRs to be
attributed to Cen A, even assuming some of them are iron.
In all cases, UHECRs coming from Cen A must lie on an arc or swath originating within its giant radio lobes, with events of lower rigidity further from the
source. The width of the swath is small and dominated by measurement uncertainty if the UHECRs are protons and Galactic random fields are more important
than extragalactic ones, or it may be large if some UHECRs are iron and the extragalactic deflection is maximal. However in every case there will be a residual
asymmetry in the distribution of CR arrival directions, about Cen A, in the direction of the GMF deflection. The presence or absence of such asymmetry can help
answer the question of how many UHECRs are produced by Cen A.
123
Conclusion
There are two common – and extreme – perceptions of the state of Galactic magnetic field research among astrophysicists: those who believe that realistic GMF
models exist and the magnetic field is well understood, and those who believe
modeling the GMF is a hopeless enterprise, bound to fail. In this thesis, it is
demonstrated that both these views are wrong.
Over the last 40 years, progress in understanding, and modeling, the GMF has
been slow. The interstellar medium is a mess; it is a turbulent, out-of-equilibrium
system with many utterly different components (gas, dust, cosmic rays, magnetic
fields, radiation). The interconnectedness of these components make understanding one without the others difficult. However, as knowledge and data on one
component increase – so does our understanding of the others. There are good
reasons to believe that our understanding of the GMF can be greatly improved in
the near future.
In this thesis we present a systematic approach to GMF modeling that incorporates multiple observables and can self-consistently include underlying variables
124
such as electron densities in the model fitting. We show that new insights are
possible with the current data.
In future work, we will extend the analysis of the random magnetic fields,
include the synchrotron total intensity, and self-consistetly include the relativistic
electron density (e.g., using the GALPROP code). We will go further with modelbuilding, and use, e.g., cross-validation to find the optimal model complexity.
The available extragalactic RM data will increase an order-of-magnitude within
a year, which will enable the use of more complex models, and greatly reduce
degeneracies in the current set of models. Synchrotron data from, e.g., PLANCK
will provide excellent additional constraints. A much improved thermal electron
density model is scheduled to appear within the next couple of years. Combined,
the prospects for significant improvements of Galactic magnetism are good indeed.
125
Appendix A
Maximum Likelihood Method for
Cross Correlations with
Astrophysical Sources
Appendix abstract
We generalize the Maximum Likelihood-type method used to study cross correlations between a catalog of candidate astrophysical sources and Ultrahigh Energy Cosmic Rays (UHECRs), to allow for differing source luminosities. The new
method is applicable to any sparse data set such as UHE gamma rays or astrophysical neutrinos. Performance of the original and generalized techniques is evaluated
in simulations of various scenarios. Applying the new technique to data, we find an
126
excess correlation of about 9 events between HiRes UHECRs and known BLLacs,
with a 6 × 10−5 probability of such a correlation arising by chance.
A.1
Introduction
Correlation studies play a fundamental role in establishing or ruling out candidate
sources for rare events such as the highest energy cosmic rays and UHE gamma
rays. When the number of events is low, it is necessary to use sensitive methods
to be able to identify the sources and quantitatively assess the possibility of an
incorrect identification. The purpose of the present work is to evaluate and improve
these methods. If the hypothetical sources are commonplace rather than rare, so
that the typical separation between candidate sources is not large compared to
the directional uncertainty of individual events, then an entirely different approach
must be used, requiring much larger datasets. Fortunately, the GZK horizon allows
the surface density of source candidates to be reduced by increasing the energy
threshold; this played an essential role in the recent Auger discovery (Abraham
et al. 2007) of a correlation between UHECRs and galaxies in the Veron-Cetty
Veron Catalog of Quasars and AGNs. Furthermore, it is plausible that only unusual
astrophysical objects can produce very rare events, in which case the candidate
source catalog may naturally be sparse.
Hints of excess correlations were reported earlier between ultrahigh energy cos-
127
mic rays (UHECRs) and BLLacs by Tinyakov & Tkachev (2001), Gorbunov et al.
(2004), Abbasi et al. (2006) and between UHECRs and x-ray clusters by Pierpaoli
& Farrar (2005). The analyses in Tinyakov & Tkachev (2001), Gorbunov et al.
(2004), and Pierpaoli & Farrar (2005) determine the number of correlated events
within some given angular separation between UHECR and source, and calculate
the “chance probability” of finding a correlation at the observed level by doing a
large number of simulations with no true correlations. The (Abraham et al. 2007)
analysis in addition “scans” to find the optimal UHECR energy threshold and
maximum source redshift.
In order to incorporate the experimental resolution on an event-by-event basis,
Abbasi et al. (2006) proposed a Maximum Likelihood-type procedure and applied it
to studying correlations between UHECRs and BLLacs. This procedure (denoted
the HiRes procedure, below) is motivated under the unphysical assumption that
every candidate BLLac source has the same apparent luminosity. Even if BLLacs
were standard candles with respect to UHECR emission, the BLLacs in the catalog
have a large range of distances which would imply an even larger range of apparent
luminosities, so one does not want to rely on such an assumption. Furthermore
the validity of the method was not demonstrated via simulations.
In this work we introduce a ML prescription which avoids the assumption of
equal apparent source luminosity and allows the potential sources to be ranked
according to the probability that they have emitted the correlated UHECRs. We
128
test and compare both methods with simulations in a variety of situations. We
find that the HiRes method gives the correct total number of correlated events
even when the sources do not have equal apparent luminosities, as long as the
numbers of events are sufficiently low and the candidate sources are not too dense
or clustered themselves. We find that our new method performs better in these
more challenging cases. In a final section, we apply the new procedure to BLLacs
and x-ray clusters.
A.2
Maximum Likelihood Approach for the
Cross-Correlation Problem
A.2.1
The HiRes Maximum Likelihood method
In the HiRes Maximum Likelihood method of Abbasi et al. (2006), the aim is
to find, among N cosmic ray events, the number of events, ns , that are truly
correlated with some sources, of which the total number is M . Hence, there are
N −ns background events whose arrival directions are given by a probability density
R(x), which is simply the detector exposure to the sky as a function of angular
position, x. For a true event with arrival direction s, the observed arrival direction
is displaced from s according to a probability distribution Qi (x, s). Note that Qi
is governed only by the detector resolution. For the analysis given in Abbasi et al.
129
(2006), Qi is taken to be a 2d symmetric Gaussian of width equal to the resolution
σi , of the ith event. (Note that, throughout, the parameter σ is related to σ68 , the
radius containing 68% of the cases, by σ68 = 1.51σ.) Dispersion due to random
magnetic fields can be incorporated into the Qi by generalizing the variance to
2
2
2
σef
f ective = σdetector + σmagnetic , where σmagnetic may vary with direction and event
energy. In practice the magnetic dispersion is not well known and has to be treated
as an unknown or argued to be smaller than σdetector .
The probability density of observing the ith event in direction xi is
Pi (xi ) =
ns
PM
Q(xi , sj )R(sj ) N − ns
+
R(xi ),
PM
N
N k=1 R(sk )
j=1
and the likelihood for a set of N events is defined to be L(ns ) =
(A.1)
QN
i=1
Pi (xi ), which
is maximized when ns is the true number of correlated events. Since L is a very
small number, which depends on the number of events, it is more useful to divide
L by the likelihood of the null hypothesis, i.e., ns = 0, to form the likelihood ratio
R(ns ) = L(ns )/L(0). The logarithm of this ratio is then maximized to obtain the
number of correlated events, ns . The significance of the correlation is evaluated by
measuring the fraction F of simulated isotropic data sets with as large or larger
value of ln R.
130
A.2.2
Extending method to differing luminosities
If the source candidates for UHECRs were standard candles with known distances,
or if the relative fluxes from different sources were known, we could generalize the
above method by simply attaching the appropriate relative weight for each source
and maximizing the likelihood ratio to obtain the number of correlations, ns . However, for the applications we have in mind we do not know the relative luminosities
of the putative sources, nor do we expect them to be standard candles. Thus, to
generalize the HiRes method to allow for sources with differing luminosities we
assign an a apriori unknown number of cosmic rays, nj , separately for each source,
P
with ntot = M
j=1 nj . The probability density generalizes to
PM
Pi (xi ) =
where R̄s =
PM
j=1
j=1
nj Q(xi , sj )R(sj )
N R̄s
+
N−
PM
j=1
nj
N
R(xi ),
(A.2)
R(sj )/M . As above, we divide the likelihood L by the likelihood
of the null hypothesis, L({nj } = 0), to obtain the likelihood ratio
R=
N
Y
i=1
"
1
N
PM
j=1 nj Q(xi , sj )R(sj )
R(xi )R̄s
−
M
X
!
nj
#
+1 .
(A.3)
j=1
Maximizing ln R, with respect to the set {nj } of M numbers, determines the most
probable values of the set {nj } of M numbers as well as the set {(ln R)i } of N
numbers. The numbers {(ln R)i } are figures of merit, providing information about
131
how strongly correlated the individual cosmic ray events are to the catalog of
sources, and allow us to rank the cosmic rays in order of their likelihood of being
correlated to the source data set (similar information can be obtained in the HiRes
approach from the contributions of individual cosmic rays to the sum in ln R(ns )).
A crucial difference between the generalized method and the HiRes method
P
is that in the new method ntot =
nj provides an estimate of all correlations,
i.e., both true and random correlations, whereas the HiRes method yields only
an estimate of the number of true correlations. In the generalized method one
obtains nj > 0 if any cosmic ray event is close enough to any source, regardless
of the degree of correlation between the two data sets elsewhere. For the HiRes
method, the single optimized parameter, ns , will be greater than zero only if the
degree of correlation between the two data sets are greater than what is expected
for a random sample of cosmic rays (weighted by the detector exposure).
We now turn to the estimation of ns with the new method. A crude estimate
of the number of true correlations for the new method is ntot − n̄rand , where n̄rand is
the average number of correlations obtained when cosmic rays are uncorrelated to
the data set of potential sources, i.e., cosmic rays drawn from an exposure-weighted
isotropic distribution.
A better measure of ns is summarized by the following equations, where the
superscript in parentheses labels the “order” of refinement, giving successively
132
better approximations to ns :
n
(0)
= ntot =
M
X
nj
(A.4)
j
n(1) = f¯−1 (n(0) − n̄rand (N ))
(A.5)
n(2) = f¯−1 (n(0) − n̄rand (N − n(1) )),
(A.6)
where n̄rand (N ) is the average total number of correlations found with N events
drawn from an isotropic distribution weighted by the detector exposure, and where
f¯ is the average fraction of true events that are recovered as correlated. This
fraction will typically be slightly smaller than unity since under the assumption
that true correlations are separated according to a Gaussian distribution, some
events are separated too far from their sources to be accepted as correlated events.
We measure f¯ by simulations.
Up to this point, we have not specified whether {nj } are real numbers or
integers. Indeed, the method allows either choice. A non-integer implementation
has the virtue of encoding more information into the set of numbers, {nj }, while in
the integer case it is necessary to also study {(ln R)i } in order to rank the sources in
terms of correlation quality to the set of cosmic rays. An integer approach may be
preferred as it provides a more concise answer as to which are the likely correlated
sources. Moreover, in the case of strong correlations and ample statistics, the
integer implementation is likely to offer a more intuitive result in terms of singlets,
133
doublets, etc.. In the following sections we will use the integer implementation,
unless noted otherwise.
The procedure to maximize ln R may depend on whether {nj } are non-integers
or integers. In the present work, we use a commercial non-linear optimization
package for the former case. One might expect to encounter problems when the
number parameters to be fitted (M ) exceeds the number of data points (N ). How2
ever, in practice this never becomes an issue as long as M πσmean
is small compared
to the solid angle observed so the number of random correlations is not too large.
Then, most potential sources are well separated from most UHECR events, and
thus have nj = 0, thanks to the rapid cutoff of Q for s σ, where s is the separation between cosmic ray and source. For the integer case we first note that
the exponential form of Q and the diluteness of the source data set will make it
extremely rare for a correlated event to be ascribed to the wrong source. In other
words, a cosmic ray event must be at almost the exact same (small) angular distance from two potential sources for there to be any ambiguity as to from which
source the event originated. For the ith cosmic ray, this allows us to consider only
the source j that maximizes the quantity Q(xi , sj )R(sj ), and put that quantity to
zero for all other sources. The maximization of equation A.3 is then a matter of
testing, for each source, whether ln R decreases when nj is increased from 0 to 1.
If it does, then nj = 0. If ln R increases, then we repeatedly increase nj by unity,
until ln R decreases and we find the correct number of correlations, nj , for that
134
source.
A.3
Simulation trials
In vetting the two methods with simulations, we test their ability to correctly
reproduce the number of true correlations on mock data sets. This allows us to
explore the effect of large event and source densities, the effect of anisotropy in the
source distribution, and the consequences of having incorrect event resolutions.
A.3.1
Dilute source and UHECR data sets
We begin by testing the ability to reproduce the number of true correlations in the
simplest case of dilute, random sources. A first simulation is done using half the
sky, uniform detector exposure R(x), 156 randomly distributed sources and 271
cosmic rays (the numbers relevant to the BLLac studies of Gorbunov et al. (2004)
and Abbasi et al. (2006)). Ten of the cosmic rays are Gaussianly aligned (a cosmic
ray paired to a source, with angular separation according to the probability density
Q, taken to be a 2d Gaussian of width σ), for various event resolutions. A second
simulation uses the actual BLLacs source positions; these are more clustered than
the random case. As shown in figure A.1, both methods reproduce well on average
the correct number of correlations. The error bars shown include 90% of the 10k
realizations, and are similar for both methods. As σ increases, the variance in the
135
number of found correlations increases rapidly.
A.3.2
Dispersion in results
To test the extent of agreement between the two methods in individual realizations,
we generate 156 source positions at random on one hemisphere, align 10 cosmic
rays to 10 of the sources and distribute 161 cosmic rays at random, then determine
the number of correlations identified, with both procedures. Figure A.2 shows the
results of 1000 repetitions, for σ = 0.4◦ and for 0.8◦ . From this exercise we conclude
that the two ML methods do not generally agree for individual realizations, in spite
of the fact that both methods reproduce the correct value in the mean.
A.3.3
High source and UHECR densities
When there are very large numbers of events and of potential sources, we expect the
generalized ML method to perform worse than the HiRes method, beacause n(2) is
obtained by taking the difference of two very large numbers, n(0) and n̄rand . Moreover, as the source density becomes very large it becomes impossible to reliably
distinguish the “contributing” sources. The total number of correlations becomes
the only interesting quantity to calculate. Thus, only the HiRes method should
be used for the case of very high event densities. However, the HiRes method also
deteriorates at high densities, as shown in figure A.3.
136
25
HiRes
n(2)
20
n
15
10
5
0.1
0.4
0.7
σ [°]
1
1.3
1.6
0
25
HiRes
n(2)
20
n
15
10
5
0.1
0.4
0.7
σ [°]
1
1.3
1.6
0
Figure A.1: The number of correlations vs. the detector resolution. Error bars
(slighly separated horizontally for readability) contain 90% of the cases. Top:
Random sources; Bottom: actual BLLac positions −30◦ ≤ δ ≤ 90◦ .
137
25
20
20
nHiRes
nHiRes
25
15
10
10
5
5
0
15
0
5
10
15
20
0
25
(2)
n
0
5
10
15
n(2)
20
25
Figure A.2: Comparison of the extracted number of correlations for specific realizations using the two methods. 1000 realizations of ten cosmic rays aligned to
sources. Left: For cosmic rays with event resolution σ = 0.4◦ . The banding is due
to using the integer implementation of the generalized method. Right: σ = 0.8◦ .
A.3.4
Sensitivity to experimental resolution
If the resolution of cosmic ray events are consistently over- or underestimated in a
given data set, the extracted correlations will be incorrect. To test the sensitivity
of the two methods to this problem we repeat the first type of simulations, but
rescale the event resolution when aligning a cosmic ray to a source. Figure A.4
shows the average number of correlations found, as a function of the amount by
which σ is rescaled, for the two methods. The new method is far less sensitive to
incorrectly estimated resolution than is the HiRes method.
138
12
10
8
N=20
n
N=60
6
N=150
N=300
4
N=500
2
0
0
100
200
300
400
number of sources, M
500
600
Figure A.3: Correlations found by the HiRes method with 10 true correlations,
M sources and N events in 100 square degrees. The method’s prediction becomes
increasingly inaccurate, as the dilute approximation becomes less valid.
139
13
HiRes method
n(2) method
12
11
found correlations
10
9
8
7
6
5
4
0
0.5
1
1.5
rescaling factor of event resolution
2
Figure A.4: Average number of found correlations as a function of the factor by
which σ is rescaled by when “Gaussianly aligning” sources.
140
A.3.5
Clustering of sources
As seen in the lower panel of figure A.1, spatial correlations within the data set of
potential sources may skew the number of UHECR correlations found. In figure
A.5 we show the results of a simulation with clustering of potential sources. The
figure shows the mean for 10k realization of 271 cosmic ray events with σ =
0.4◦ , and two different scenarios for the correlation with source clustering, for
156 candidate sources. In both cases, candidate sources and CRs are distributed
over one hemisphere; 100 sources are placed in ten randomly positioned clusters
with ten sources distributed around each cluster center according to a 2d Gaussian
of width d, and the remaining 56 candidate sources are placed at random in the
hemisphere. In the first case, a randomly selected source in each cluster has one
cosmic ray event Gaussianly aligned to it and the remaining 261 cosmic rays are
placed at random. In the second case, ten CRs are Gaussianly aligned with ten of
the randomly placed source candidates not in any cluster and the remaining CRs
are placed at random. As figure A.5 demonstrates, the HiRes method significantly
overestimates the true number of correlations if the sources are in clustered regions
and underestimates it when the candidate sources show significant clustering but
the UHECRs do not come from the clustered regions. By contrast the new method
performs well in this test.
141
11
n
10
9
HiRes: case A
HiRes: case B
8
n(2): case A
n(2): case B
7
0
1
2
3
4
5
d [°]
Figure A.5: Sensitivity to clustering in source dataset, for UHECRs from (A) dense
or (B) sparse regions.
A.4
Application to BLLacs and x-ray clusters
In this section we apply both the HiRes method and the generalized method on
two previously investigated data sets of suggested UHECR sources.
A.4.1
X-ray clusters
Table A.1: Correlations between X-ray clusters and UHECRs. The
integer implementation has been used for the generalized method.
For comparison, using the non-integer implementation we find
n(2) = 3.3 and n(2) = 9.6 for the HiRes and AGASA event sets,
respectively.
Generalized ML Method
HiRes Method
142
UHECR
HiRes
AGASA
Both
N
n(0)
n(2)
F
nHR
FHR
143
22
5.2
0.3
1.5
0.3
36
24
9.9
6 10−3
9.2
9 10−3
179
37
12.7
0.06
4.6
0.1
The possibility that X-ray clusters are sources for UHECRs was investigated by
Pierpaoli & Farrar (2005) using a binned analysis; a correlation was found between
AGASA events and x-ray clusters taken from the NORAS catalog (Bohringer et al.
2000). Here we study the correlation between X-ray clusters and two samples of
UHECRs, both separately and combined. We use the 375 X-ray clusters (Bohringer
et al. 2000) in the region 0◦ < dec < 80◦ , and |b| > 20◦ . For cosmic rays, we use
the 143 HiRes events with E > 1019 eV, and 36 AGASA events with E > 4 × 1019
eV, in the same angular region. Note that in Pierpaoli & Farrar (2005) the authors restricted the correlation search to x-ray sources within the estimated GZK
maximum distance for each UHECR event. Imposing the GZK restriction reduces
the number of found correlations in simulated random samples, but Pierpaoli &
Farrar (2005) found that it does not significantly decrease the number of correlations between UHECRs and x-ray clusters so that it increases the significance of
the correlation. In this work we have not taken the GZK distance restriction into
account, as our focus is on comparing the two ML methods.
When applying the Maximum Likelihood methods to x-ray clusters, we need
to take the angular size of the clusters into account, as they can be of the same
143
order as the cosmic ray event resolution. Therefore we introduce an effective event
√
resolution to be used in the calculation of Q, σef f = σ 2 + r2 , where r is the
angular radius of the cluster. When combining the data sets we calculate the
P
P
detector exposure by R(x) =
i Ni Ri (x)/
i Ni , with i labelling the different
data sets. For the generalized method we use the integer implementation.
The results of the various analyses are shown in table A.1. We see that the two
Maximum Likelihood methods are in fair agreement in all cases about the significance of the correlation, measured by F. However, the number of correlations inferred differs strikingly between the two methods for the combined HiRes-AGASA
dataset. Perhaps surprisingly, the HiRes method finds more correlations in the
AGASA sample alone than in the combined dataset. This is a hazard of using the
HiRes method that we have probed by simulations. Consider the case of combining
two sets of events, one with poor event resolutions with a number of events aligned
to the set of sources, and the other set with good event resolutions but all events
uncorrelated to the set of sources. In such a case the HiRes method significantly
underestimates the number of correlated events. Conversely, if the aligned events
all are taken from the set with good resolutions, the HiRes method overestimates
the correct number of correlations. This explains the discrepancy between the
methods for the combined data in table A.1, since the AGASA events have poor
event resolutions relative to the HiRes events. In effect, the HiRes method imposes
a sort of internal consistency on the distribution of correlations, which could be
144
inappropriate, if one data set had systematically incorrect σ values for instance.
A.4.2
BLLacs
The binned analysis performed in Gorbunov et al. (2004) on the sample of 156
BLLacs with optical magnitude V < 18 from the Veron 10th Catalog (VeronCetty & Veron 2001) and the 271 HiRes events with E > 1019 eV showed a
correlation at the 10−3 level. Applying the HiRes method on this sample we
find ns = 8.5 correlations, and the fraction F of Monte Carlo runs with greater
likelihood than the real data to be 2 × 10−41 . Analyzing the same UHECR-BLLac
data with the generalized ML method we find the number of correlations is n(2) =
9.2 with the integer implementation and 9.3 with the non-integer implementation,
and F = 6 × 10−5 for both. In fact from figure A.1, we see that the HiRes method
underestimates ns by 0.7 on average under these conditions, so on average this is
the expected discrepancy. However as seen from figure A.2, the difference between
the estimated number of correlations is also consistent with the variance from
one realization to the next. With 1σ error bars, our best estimate is 9.2 ± 2.5
correlations.
Now including AGASA events too, in Table A.2 we list the results for the
generalized method (integer implementation) and the HiRes method when applied
1
This ns differs from the value (ns = 8.0) quoted in Abbasi et al. (2006) probably due to
the HiRes exposure map provided to us by HiRes (S. Westerhoff, private communication) being
slightly different than the one used in the published HiRes analysis. For purposes of comparison
of the two methods we do not require a perfect exposure map.
145
to HiRes and AGASA data, both separately and combined. We note that while
the calculated number of correlations differ between the methods, the difference is
within the expected dispersion.
One of the advertised benefits of the generalized ML method is that it provides
a way to rank sources according to the likelihood of them being correlated with
cosmic ray events. To exemplify this we list in Table A.3 the individual BLLacs
and correlated HiRes cosmic rays ranked according to the non-integer number of
correlations found per source (cf. Table 1 in Gorbunov et al. (2004)). The list
includes all cosmic rays with positive ln R, which brings the number of cosmic
rays listed to 16. Note that the sixth listed cosmic ray is not deemed a correlation
by the generalized method, but is kept in the list to clarify why the corresponding
source is found to have nj > 1 by the non-integer implementation of the method.
Finally, we note that it would be desirable to impose restrictions on the sample
of BLLacs to include only those within an estimated GZK maximum distance for
each cosmic ray event, but unfortunately the redshift is known for only a fraction
of the BLLacs.
Table A.2: Correlations between BLLacs and UHECRs. The integer implementation has been used for the generalized method.
Generalized ML Method
UHECR
HiRes
HiRes Method
N
n(0)
n(2)
271
15
9.2
146
F
nHR
FHR
6 10−5
8.5
2 10−4
AGASA
Both
57
13
6.4
328
21
12.0
0.04
2.6
0.1
3 10−5
10.8
1 10−4
Table A.3: HiRes UHECR cross correlations with BL Lacs using
the noninteger implementation of the generalized Maximum Likelihood method.
BLLacs
UHECRs
Name
z
V
nj
∆θ (◦ )
ln R
E (EeV)
RA (◦ )
dec (◦ )
SBS 1508+561
...
17.3
1.905
0.83
3.18
21.7
226.56
56.61
0.89
2.90
11.0
228.86
56.36
0.60
4.20
64.7
162.61
49.22
1.04
2.08
24.1
165.00
49.29
0.32
4.33
12.0
118.77
48.08
1.37
0.21
15.7
120.10
47.40
MS 10507+4946
GB 0751+485
0.140
...
16.9
17.1
1.858
1.268
1ES 1959+650
0.047
12.8
0.991
0.12
4.61
16.4
300.28
65.13
FIRST J11176+2548
0.360
17.92
0.990
0.12
4.60
30.2
169.31
25.89
Ton 1015
0.354
16.50
0.986
0.37
4.24
13.3
137.22
33.54
RXS J01108-1254
0.234
17.9
0.986
0.37
4.24
10.5
17.79
12.56
RXS J03143+0620
...
17.9
0.980
0.50
3.89
27.1
48.53
5.84
RXS J13598+5911
...
17.9
0.970
0.62
3.47
24.4
210.02
59.80
RGB J1652+403
...
17.3
0.958
0.70
3.15
16.5
253.62
39.76
RXS J08163+5739
...
17.34
0.950
0.74
2.99
20.7
123.78
56.93
OT 465
...
17.5
0.849
0.95
1.98
10.5
265.37
46.72
1ES 2326+174
0.213
16.8
0.527
1.12
1.08
38.1
352.40
18.84
147
A.5
Summary and Conclusions
We have introduced a generalization of the HiRes Maximum Likelihood method,
which allows the most likely sources of individual events to be identified and ranked.
Using simulations we have tested the two Maximum Likelihood methods and find
that they complement each other well: the HiRes method allows a fast way to estimate the number of true correlated events, while the new method gives the quality
of correlation between individual sources and cosmic rays rather than just the total number of correlated events. Furthermore, the new method is less sensitive to
the validity of the estimated angular resolution, and has less systematic bias when
candidate sources are clustered (as astrophysical sources such as BLLacs are). Of
course, for any given data set, mere statistical fluctuations can result in conclusions
that would not be borne out with a larger data set. We conclude that both methods should be used: if they disagree markedly on the total number of correlations
the data may have some statistical anomaly and be difficult to interpret.
Although we have tested both Maximum Likelihood methods’ ability to reproduce the true number of correlations under many different conditions, it is impossible to test for every conceivable statistical property of future data sets to which
these methods could be applied. Thus we recommend, when applying these methods on new data samples, that the methods first be tested by simulations using
the candidate source positions but randomly generated cosmic ray directions with
some fraction of them Gaussianly aligned to some randomly chosen sources. By
148
repeating this procedure, the variance and possible systematic bias in the extracted
number of correlations can be determined for that particular detector exposure,
source and event statistics, resolution, etc.
We applied the ML methods to find correlations between x-ray clusters and
AGASA events with E > 4 × 1019 eV, as done in Pierpaoli & Farrar (2005) but
without imposing the GZK horizon. We find ≈ 10 excess correlations, with a
chance probability of 6 × 10−3 . The same analysis for HiRes data with E > 1019
indicates 3 − 5 excess correlations, which is expected by chance about 30% of the
time. Imposing the restriction that the correlated x-ray cluster not be farther
than the GZK distance cutoff was found in Pierpaoli & Farrar (2005) to reduce
the number of chance correlations without significantly decreasing the number of
true correlations. Since in this work we did not impose the GZK restriction, we
cannot compare the present analysis with that of Pierpaoli & Farrar (2005).
Finally, we applied the generalized Maximum Likelihood method to check previously claimed correlations between UHECRs and BLLacs. We corroborate that
there is a significant correlation between BLLacs of the Veron 10th catalog and
HiRes cosmic rays with E > 1019 eV (Abbasi et al. 2006). The generalized ML
method yields a slightly greater number (≈ 9 instead of ≈ 8) excess correlations
and a lower chance probability (6 × 10−5 instead of 2 × 10−4 ). Simulations show
that the HiRes ML method systematically underestimates the number of correlations by 0.7 when the source candidates are clustered like the BLLacs, suggesting
149
that the present result is the more reliable. The striking difference between the apparently significant correlation observed in the Northern sky (HiRes) and Auger’s
null result (Harari 2007) is a mystery. It may be due to greater average magnetic
deflection in the Southern sky, or greater incompleteness in the Southern BLLac
catalog, or a statistical fluctuation or a combination of the above.
150
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