UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO
PROGRAMA DE POSGRADO EN ASTROFÍSICA
ASTROFÍSICA TEÓRICA
AN ANALYSIS OF THE SSC (SYNCHROTRON SELF-COMPTON) MODEL FOR
X-RAY AND GAMMA-RAY EMISSION IN BLAZARS AND ITS IMPLICATIONS
TESIS
QUE PARA OPTAR POR EL GRADO DE:
MAESTRO EN CIENCIAS (ASTROFÍSICA)
PRESENTA:
JAMIE MARLEN AGUILAR MURILLO
DIRECTORES DE TESIS:
DRA. MARÍA MAGDALENA GONZÁLEZ SÁNCHEZ
DR. NISSIM ILLICH FRAIJA CABRERA
INSTITUTO DE ASTRONOMÍA, UNAM, CAMPUS C.U.
MIEMBROS DEL COMITÉ TUTOR
DRA. MARIANA CANO DÍAZ
DR. ANTONIO PEIMBERT TORRES
DR. ALDO ARMANDO RODRÍGUEZ PUEBLA
INSTITUTO DE ASTRONOMÍA, UNAM, CAMPUS C.U.
CIUDAD UNIVERSITARIA, CD. MX, ENERO 2022
 
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Acknowledgements
This research was financially supported by Programa de Apoyo a Proyectos de In-
vestigación e Innovación Tecnológica (PAPIIT) AG-100317, IN106521 and
IG101320 .
I would like to acknowledge and give my thanks to my supervisors Magda and Nissim who
made this work possible. I would also like to thank the examining committee for all the com-
ments and suggestions. I would like to thank the Consejo Nacional de Ciencia y Tecnoloǵıa
(CONACyT) for the financial aid, the Instituto de Astronomı́a and the UNAM, for all the
resources and opportunities granted, that allowed the culmination of this work.
I would like to thank my parents for all the advices and all the sacrifices they made to
give me a great education. A special thanks to my mom for all the support and love. I
would like to thank all the friends that handed me a helping hand through the whole under-
grad and grad experience and specially through life’s ups and downs.
I would like to give special thanks to my spouse Lloret Gironés for supporting me through all
the hardships of life and always believing in me. Thank you for taking care of me through all
those sleepless and stressful nights of research. Your unconditional support and understand-
ing held me up when I couldn’t, without your kind words of love and support I wouldn’t
have made it this far. I couldn’t have asked for a better partner in life.
i
ii
Resumen
En esta tesis abordamos el estudio de la emisión de rayos-X y rayos-γ de los blazares BL
Lacs. Proponiendo un modelo leptónico del jet con factor de Lorentz Γ, que contiene un
campo magnético, B, constante y una región de emisión llena de electrones relativistas, la
cual asumimos que es esférica de radio rd. Se propone que el jet esté produciendo dichas
emisiones mediante un proceso de sincrotrón self Compton (SSC), lo que quiere decir que los
electrones que emiten radiación por proceso sincrotrón, correponden a la misma población
que por efecto de compton inverso sede enerǵıa a los fotones de sincrotrón. Tomamos como
fuente de estudio Mkr 421, consideramos el estudio hecho por [González, M. M 2019] en el
cual concluyen que las emisiones de rayos X y rayos γ están correlacionados y que dicha
correlación es congruente con un model SSC y es capaz de reproducir los flujos observados
[Acciari, V. A., 2014] mediante el incremento de la densidad de electrones, sin un cambio en
el valor de B. Tomamos dicha conclusión y la ponemos a prueba con el modelo explicado
en esta tesis. Se prueba que para correlaciones bajas el espacio de parámetros (B y la
densidad de electrones) que es capaz de reproducir dichos flujos es amplio y a medida que la
correlación se vuelve mas pronunciada los valores de campo magnético que describen los flujos
altos disminuye. Se prueba que la conclusión de [González, M. M 2019] no se mantiene cierta
ya que para correlaciones mas pronunciadas el valor del campo magnético debe disminuir
para ser capaz de reproducir los flujos altos. Para futuros trabajos se espera poder usar el
modelo presentado en esta tesis para estudiar un grupo de BL Lacs y de tal manera entender
mejor la f́ısica de los BL Lacs y probar si cada blazar tiene un campo magnético dado o si
ii
estos mostrarán correlación siempre.
iii
iv
Abbrevations and Acronyms
AGN Active Galactic Nucleus
AGNs Active Galactic Nuclei
BH Black Hole
BL Lac BL Lacertae
BLR Broad Line Region
CMB Cosmic Microwave Background
EC External Compton
EGRET Energetic Gamma-Ray Experiment Telescope
FSRQ Flat Spectrum Radio Quasars
HBL High-frequency Peaked BL Lacs
LBL Low-frequency Peaked BL Lacs
Mkr Markarian
NLR Narrow Line Region
SC Self-Compton
SPB Synchrotron Proton Blazar
v
SSC Synchrotron Self-Compton
VHE Very High Energy
vi
Contents
Acknowledgements i
Resumen ii
Abbrevations and Acronyms v
1 Introduction 1
1.1 Main components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Unified Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Blazars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Relativistic Blazar Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Hadronic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
vii
1.6 Leptonic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.7 Blazars with Correlation of the X-ray and γ-ray Emission . . . . . . . . . . 16
2 Synchrotron Self-Compton Model 19
2.1 Synchrotron Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.1 Minimum Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.2 Cooling Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.3 Maximum Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.4 Synchrotron Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Inverse Compton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Discussion and Conclusions 29
4 Appendix A 33
4.1 Emission and Absorption Lines . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 Broad and Narrow Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5 Appendix B: Physics background 35
viii
5.1 Synchrotron Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.1.1 Power Emitted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.1.2 Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Inverse Compton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2.1 Energy Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2.2 IC Power Emitted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2.3 IC Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Bibliography 55
ix
x
List of Figures
1.1 Representation of the gravitational potential of a non rotating BH, taken from
[Beckmann V. 2012 ]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Proton(p) photon (γ) interaction, neutral pion (π0) decay. . . . . . . . . . . 8
1.3 Proton photon interaction, charged pion decay and the following primary
decay channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 A visual representation of the jet model. . . . . . . . . . . . . . . . . . . . . 20
3.1 Theoretical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.1 Helical motion of a particle in an uniform magnetic field. . . . . . . . . . . . 36
5.2 Emission cones at different points of an accelerated particle’s trajectory. . . . 40
5.3 Geometry of the scattering of a photon by an electron initially at rest. . . . . 45
xi
5.4 Scattering geometries in the observer’s frame K and in the electron rest frame
K ′. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
xii
List of Tables
1.1 Types of AGNs [Malizia, A. 2020] . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Lepton flavors [Bettini, A. 2008]. . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Hadron types [Bettini, A. 2008]. . . . . . . . . . . . . . . . . . . . . . . . . . 8
xiii
Chapter 1
Introduction
In this chapter I will review some concepts about Active Galactic Nucleus (AGN) needed to
later understand the model presented in this thesis. Active Galactic Nuclei (AGNs) are ener-
getic astrophysical sources powered by accretion onto black holes (BH) in galaxies, and they
radiate across the full electromagnetic spectrum, from radio to γ-rays.[Padovani, P. 2017 ]
1.1 Main components
In this section I will introduce some of the main components of AGNs so it serves as tool for
understanding of the main interest of this thesis.
1
Black Hole
Figure 1.1: Representation of the gravitational potential of a non rotating BH, taken from
[Beckmann V. 2012 ].
Black holes posses such high masses so that their gravitational pull is so strong to al-
low the escape of light. One of the basic characteristics of a BH is the presence of an
event horizon, which is a boundary where matter and light fall inward towards the BH,
but never come back out.[Salpeter, E., 1964, Zeldovich, Ya. B., 1964, Lynden-Bell, D. 1969]
[Schwarzchild, K. 1916] , described the gravitational field near a non rotating BH in the
context of Einstein’s theory of General relativity. His solution described the event horizon as
a spherical surface centered on the BH. This surface is determined by Schwarzschild radius,
as seen in figure 1.1. Any event within the boundary defined by Schwarzschild radius (Rs),
will not be able to communicate with an outside observer. [Schwarzchild, K. 1916]
Rs =
2GMBH
c2
, (1.1)
where G is the gravitational constant, c the speed of light in the vacuum, and MBH the BH’s
mass. [Schwarzchild, K. 1916]
The Kerr metric describes the space time geometry in the vicinity of a rotating
massive object such as the central BH of an AGN. Rotating black holes are formed in the
gravitational collapse of a collection of stars or gas, which would most likely have nonzero
angular momentum.[Blandford, Znajek 1977]
2
Accretion disk
Accretion disks in galactic centers are formed naturally by in-falling gas that sinks into the
central plane of the galaxy, while retaining most of its angular momentum. The assumption is
that the viscosity in the disk is enough to provide the needed mechanism to transfer outward
angular momentum of the gas to allow it to spiral into the center, losing a considerable
fraction of its gravitational energy on the way. The energy lost in the process can be converted
into electromagnetic radiation, with extremely high efficiency.[Netzer, H. 2013].
Narrow Line Region
The Narrow line Region (NLR) is located at 100 pc up to kpc scales from the BH. It is biconial
shaped region with low velocity gas (<900km/s). The radiation that passes through the NLR
produces permitted and forbidden narrow lines in the optical to UV band [Malizia, A. 2020].
Broad Line Region
The Broad line Region (BLR) is about the 0.1-1 pc scale from the BH. This region is made up
of high velocity gas (1000-5000km/s). The radiation that passes through the BLR produces
permitted broad emission lines in the optical spectra.[Malizia, A. 2020]
Torus
The torus located at a distance of 1 to 100 pc from the BH, is made up of gas and dust. It
is responsible for the absorption of the the continuum. It may be as thick to be capable of
3
hiding the emission up to several keVs. [Malizia, A. 2020]
Jet
The interaction between the BH’s magnetic field and the accretion disc may create powerful
magnetic jets that eject material perpendicular to the disc at relativistic speeds and extend
for hundreds to thousands of parsecs.[Malizia, A. 2020] We will discuss this component with
more detail later on.
The components mentioned added up over the years to contribute to the understand-
ing of a large variety of AGN classes at all wavebands and led to the Unified Model of
AGN.[Malizia, A. 2020][Urry, M.; Padovani, P. 1995]
1.2 Unified Model
The unified model hypothesises that the diversity of AGN can be explained as a viewing
angle effect. An AGN have been classified as radio loud or radio quiet, this according
to its emission at radio frequencies, which may be strong or weak [Malizia, A. 2020]. Yet
[Padovani, P. 2017 ] considers this radio loudness classification as obsolete, misleading and
inappropriate and instead divides them in terms of jetted and non-jetted, referring to the
presence of the jet or lack of it. In this thesis I will consider the radio loudness classification.
Blazars considering the radio loudness classification [Malizia, A. 2020] are considered
radio loud AGNs and jetted [Padovani, P. 2017 ].
4
Radio loudness AGN type Subtype
Radio loud
Radio Galaxy FR I
FR II
Quasar Type 1
Type 2
Blazar
FSRQ
BL Lacs
Radio quiet
Seyfert Type 1
Type 2
Table 1.1: Types of AGNs [Malizia, A. 2020]
1.3 Blazars
Blazars host a jet oriented at a small angle (≲ 15 - 20°) with respect to the line of sight.Blazars
spectra extend over a broad range of energies, from radio to TeV γ-rays. The blazar type
is made up of Flat Spectrum Radio Quasars (FSRQs) and BL Lacs. The main difference of
these two sub-classes lies in their optical spectra. FSQRs show strong broad emission lines.
BL Lacs show at most weak emission lines, sometimes they show absorption features, and
in many situations they are completely featureless. To understand the nature of blazars we
must understand the nature of their jet. [Urry, M.; Padovani, P. 1995] [Padovani, P. 2017 ]
The spectral energy distribution (SED) of blazars is characterized by a non-thermal
continuum that extends from radio to very high energy γ-rays. They also show a thermal
emission in optical/UV which is associated to the accretion disk, and emission lines from the
BLR. The dominant emissions are non thermal which are due to the relativist jet. The SED
is made up of two distinguishable radiative components due to the jet: the first one peaks
5
from infrared to X-rays, the second one peaks in the γ-ray band, from MeVs to TeVs. This
implies a population of particles that can radiate photons over the electromagnetic spectrum
[Cerruti, M. 2020].
1.4 Relativistic Blazar Jet
Blazars have their jet pointing close to our line of sight. The relatavistic jet is moving
towards the observer with speed βc, Lorentz factor Γ, therefore it will appear in the ob-
server’s frame as showing a projected superluminal motion speed βprojected > 1. Also
the emission from the jet is boosted in the direction of the movement which mean that
the flux density in the observer’s frame scales as Fv = δ3DF
′
v′ where δD is the Doppler
factor. [Hovatta, T., Linfords, E. 2020, Cerruti, M. 2020] According to theoretical models
jets are magnetically dominated [Blandford, Znajek 1977, Hovatta, T., Linfords, E. 2020].
For blazar γ-ray jets, there are two general ideas, the leptonic and hadronic jet models
[Netzer, H. 2013]. Leptonic radiative processes are the ones associated with electrons/positrons
while hadronic radiative processes are the ones associated with protons and nuclei. We con-
sider that there exists an efficient particle accelerator in the jet [Cerruti, M. 2020].
Leptonic blazar jet models describe that they have a accretion disk around a BH
which powers the jet. Occasionally emitting regions (also known as blobs) of ultrarelativis-
tic electrons are ejected at relativistic bulk velocity. Synchrotron radiation is produced by
charged particles moving in a magnetic field. For simplicity it is assumed that the emitting
region or blob is a sphere of radius R in the jet, moving with bulk Lorentz factor Γ, and
with a homogeneous magnetic field B [Böttcher, M., 2001, Hovatta, T., Linfords, E. 2020,
Cerruti, M. 2020, Böttcher, M. 2013]. Leptonic models explain the SEDs high energy com-
ponent as originated by inverse Compton emission of high energy electrons that scatter off
lower energy photons. Some possible target photon fields for Compton scattering are the
synchrotron photons produced within the jet, Synchrotron Self-Compton (SSC) mechanisms
6
[Bloom, S. 1996, Marscher, A. P. 1985, Maraschi, L. 1992, Tavecchio, F. 1998 , Lindfors, E. J 2005,
Graff, P. 2008], external photons from the disk(through external Compton (EC) [Dermer, C. D. 1992,
Dermer, C. 1993]), EC from the cosmic microwave background (CMB) [Harris, D. E. 2002]
and other EC sources [Blazéjowski M. 2000, Arbeiter, C 2002].
For the hadronic jet models we consider, a jet with a blob in which there are also
protons.[Sikora, M., 2011, Böttcher, M. 2013] Several works explain the SED of blazars through
hadronic models. Protons may contribute through the radiative output of jets in blazars
through proton synchrotron emission [Mannheim, K. 1993, Mücke, A. P., Protheroe, R.J. 2001,
Aharonian, A. 2000, Mücke, A., Protheroe, R. 2003][Böttcher, M. 2013] or through photo-
pion production [Mannheim, K. 1989] [Böttcher, M. 2013, Sikora, M., 2011]. The emission
by protons is suppressed and needs to be compensated by a higher magnetic field values, the
particle density or both. [Sikora, M., 2011, Hovatta, T., Linfords, E. 2020, Cerruti, M. 2020]
1.5 Hadronic Models
Hadronic models rely on the existence of relativistic hadron populations, located in an emit-
ting magnetized blob, which moves at relativistic speeds along the jet axis. Electron and pro-
ton populations are being co-accelerated. The emissions are related to proton-photon inter-
action (pγ), either with photons due to external sources, or photons of the same blob. Other
processes to consider, depending on the model, are synchrotron due to the co-accelerated
electrons or/and protons and other decay mechanisms due to different hadron’s decay inter-
actions.
I will review a few concepts for a better understanding of the models described in
this section and the following ones. Let’s start by defining which particles are hadrons and
which are leptons as shown in table 1.2 and 1.3. As shown in figures 1.2 and 1.3, when
7
a proton interacts with a high energy photon, the interaction (known as pγ) may lead to
the production of additional hadrons. Perhaps to neutral pion, as seen in figure 1.2, which
may in turn decay. Its primary or dominant decay channel is into two photons. Photons
primarily interact with matter via pair production, i.e. electron e− and positron e+ pairs.
This production will continue as long as the photon interacts with a nucleus, and has an
energy above the total rest mass energy of the two particles produced. In turn, electron-
positron annihilation may occur.[Bettini, A. 2008]
Leptons
Charged electron e muon µ tau τ
Neutral electron neutrino νe muon neutrino νµ tau neutrino ντ
Table 1.2: Lepton flavors [Bettini, A. 2008].
Hadrons
Byrons Protons p neutrons n
Mesons Pions π kaons K
Table 1.3: Hadron types [Bettini, A. 2008].
p
γ
p
π0
γ
γ
Figure 1.2: Proton(p) photon (γ) interaction, neutral pion (π0) decay.
Through the deceleration of a charged particle, the loss of kinetic energy is converted
into radiation. pγ interactions produce a charged pion, as seen in figure 1.3, and the primary
decay channel of a charged pion, π±, is into a muon, µ±, and muon neutrino, νµ, (π
+ −→
8
µ++νµ, π
− −→ µ−+ ν̄µ). Which, in turn, the primary decay channel of charged muons, µ±,
is into an electron, e−, electron neutrino, νe, and a muon neutrino, νµ, (µ
− −→ e−+ ν̄e+ νµ,
µ+ −→ e+ + νe + ν̄µ).[Bettini, A. 2008]
p
γ
n
e−
ν̄e
p
π±
µ±
νµ
e±
νe, ν̄e
νµ, ν̄µ
Figure 1.3: Proton photon interaction, charged pion decay and the following primary decay
channels.
[Mücke, A. P., Protheroe, R.J. 2001] proposed a synchrotron proton blazar (SPB)
model where the high-energy bump of the SED is due to accelerated protons, while the
low energy bump is due to synchrotron radiation by co-accelerated electrons. It is assumed
that the relativistic e− radiate synchrotron photons, which serve as target radiation field
for pγ interactions, and for the subsequent pair-synchrotron cascade, which develops due to
photon-photon pair production. Using Monte Carlo technique to simulate the interactions
and subsequent cascades, it was possible to fit the high-energy portion of the observed SED
of Markarian (Mrk) 501, during the April 1997 flare. In the model for Mkr 501, synchrotron
radiation from protons in a highly magnetized emission region are responsible for the TeV
bump, while the X-ray bump mainly by the synchrotron radiation from the accelerated elec-
trons. If the optical depth of TeV photons is significantly higher, a corresponding amount
may be redistributed to X-ray energies.
[Mücke, A., Protheroe, R. 2003] presented a parameter study of the SPB to explain γ-
ray loud objects, LBLs (Low-frequency Peaked BL Lacs ) and HBLs (High-frequency Peaked
9
BL Lacs ). It was stated that the model needs strong magnetic fields together with proton,
muon, pion synchrotron radiation to produce the double bumped SED observed during active
phases of γ-ray emission. HBLs’ radiation is dominated by proton synchrotron radiation at
MeV-TeV energies. In LBLs, besides the proton synchrotron radiation, there is a significant
contribution from muon synchrotron radiation at MeV-TeV energies.
Some of the characteristics that hadronic models share is that they mostly need large
magnetic fields. Evidence that hadrons contribute to the γ-ray emission would be confirmed
with associated neutrino observations, which would rule out purely electromagnetic processes
and require for the decay of hadrons. It is possible to have high energy SED component in
blazars and proton synchrotron peak frequency at around 10-100 GeV, assuming, magnetic
field, B ≃ 10 − 100 G, maximum proton Lorentz factor, γp,max ≃ 109, and Doppler factor,
δ ≃ 10− 50. [Sikora, M., 2011, Hovatta, T., Linfords, E. 2020, Cerruti, M. 2020]
1.6 Leptonic Models
Leptonic models explain the high energy component as originated by inverse Compton emis-
sion of high energy electrons that scatter off lower energy photons. Several radiation fields or
photons that work as targets for the high energy electrons to scatter have been proposed. For
instance, photons produced through synchrotron emission by the same population of elec-
trons, optical, UV, X-ray emission from the central accretion disk, emission from the BLR
and the NLR, infrared emission from the torus, and radiation from the cosmic microwave
background, just to mention a few of the possible radiation fields that may serve as targets
for inverse Compton processes. Other considerations are: a magnetic field with no preferred
orientation in the co-moving frame of the emission region, and a spherical emitting region
filled up with hot plasma.
10
In the case where the radiation field that serves as target for inverse Compton scat-
tering is the radiation emitted through synchrotron by the same population of electrons we
have the called SSC (Synchrotron Self-Compton) mechanisms.
[Marscher, A. P. 1985] presented models for the variability of compact radio sources,
with special interest on the applications for the flare of 3C 273. It was presented that the
emissions observed were to be expected if the relativistic electrons responsible for the flare
are continuously injected with the same energy distribution as for the quiescent emission
and subsequently suffer significant energy losses to synchrotron radiation or Compton scat-
tering. They presented a model based on an uniform expanding source, allowing a variable
injection of relativistic electrons and magnetic field and including synchrotron, Compton,
and expansion losses which could reproduce the observed behavior of 3C 273, if the injection
is allowed to vary with radius.
Since the nature of the uniform expanding source model did not seem to fill the
expectations, they tried a different approach. They adopted a physical model of the source
and determined whether the temporal behavior of the model fulfills the observations. So
they chose a model in which the jet is conical, directed nearly along the line of sight, with
constant opening and with relativistic electrons and a magnetic field injected at a point
which lies at an axial distance from the vertex. Any increase in the pressure of the jet
flow will result in the formation of a shock. The electrons gain energy at a constant rate
as they cross the shock. They found that an important consequence of a shocked region
in a relativistic jet is a variable self-Compton X-ray and γ-ray emission. Nevertheless, the
approach is an approximate. The Compton spectrum is quite sensitive to uncertainties in
the physical parameters. They stated that Compton losses during the initial stages of the
flare indicates that Compton scattering may produce the observed X-ray and γ-rays. The
model produces several observationally testable predictions, such as that of 1983 flare in 3C
273.
[Maraschi, L. 1992] studied the blazar 3C 279 through a model in which they assumed
11
that X-rays and γ-rays are produced in the same region by relativistic electrons that inter-
acting with a magnetic field radiate photons that through inverse Compton produce γ-rays.
In this model, γ-ray luminosity is higher than the emitted at lower frequencies, produced by
synchrotron processes, therefore, the radiation energy density is higher than the magnetic
energy density. From the maximum observed radiation on blazar 3C 279, they inferred that
synchrotron processes would produce the observed UV and soft-X-ray photons. The shape
of the jet proposed is parabolic in the inner part and conical in the outer parts. For sim-
plicity, the emitting region is homogeneous and spherical and in the source frame photons
are isotropically distributed. The considered size of the γ-ray emitting region is inconsistent
with the variability time scale of 2 days. Bulk acceleration of the plasma occurs in the inner
parabolic part of the jet, where the bulk Lorentz factor increases with distance. In the hard
X-ray and γ-ray band, only the inner portions of the jet contribute. In the ‘high-state’,
the synchrotron emission steepens from infrared to UV band and the inverse Compton from
the hard X-ray to the γ-ray band. In the observed ‘low-state’, the Compton component,
predicted with the model, falls above the lowest X-ray, data which may be due to the lack
of simultaneous observations. They concluded that for emitting regions of up to 2× 1017 cm
and Lorentz factors between 5.5 and 14, the model is more accurate.
[Bloom, S. 1996], due to detection of γ-ray energies > 100 MeV, stated that it was
necessary to reconsider models to link the radio to optical emission to X-ray and γ-ray emis-
sion. In their model, they presented a component due to synchrotron emission and a first
and second order self-Compton component, the component due to inverse scattering. Their
main goal was to explore the dependence of the broadband spectra on physical parameters,
in order understand the observed variability in terms of parameters such as magnetic field,
B, and electron density, Ne, and other parameters. They considered a simple model for
the synchrotron radiation, and did not considered expansion, energy losses, relativistic time
delay, particle acceleration, etc.
Without modeling the causes for variability, they examined how synchrotron and
12
self-Compton components vary with the magnetic field, source size, electron density and
electron energies using the common expressions to determine the values of B and Ne in
terms of the observable parameters. They developed a numerical program that generates
synchrotron and inverse Compton spectra, finding that, when the magnetic field changes,
both the synchrotron and the self-Compton component flux densities increase. When varying
R in the interval [6.5×1017, 1.3×1018, 2.6×1018]cm, there is an increase in the scattered flux,
as expected since it increases with volume as R3. Varying electron upper cutoff energy, γ2 ,
causes synchrotron and scattered spectra to extend to higher frequencies. Varying electron
lower cutoff energy γ1, does not have much effect on the X-ray emission but the self Compton
flux is higher at optical frequencies. Keeping the product of Ne and R constant, has an effect
similar to that of an increase in B, except that the upper cutoff energies remain constant. Ne
causes X-ray and γ-ray flux densities to vary by larger fractions. This contrasts with models
where γ results from photons external to the jet. According to EGRET-detected blazar, the
second order SCC is not an important contributor, but there is a contribution seen in some
observations. Their SSC model and their analytical considerations provide a formulation to
relate the variability of the high energy spectra to that of the radio-IR emission. The second
order self-Compton scattering tends to dominate the γ-ray emission in sources with steeper
spectral indices, first order SSC emission remains as the main candidate to explain the hard
γ-rays observed in sources.
[Tavecchio, F. 1998 ] proposed a model assuming a spherical, filled with relativistic
electrons and a magnetic field with no preferred direction and the energy distribution of the
electrons following as broken power law. With the source transparent to γ-rays, the high
energy photons may interact with the low energy photons, producing pairs explaining the
time lags with the hypothesis that they are associated to the time necessary for freshly high
energy electrons to cool. The break energy and the two spectral indices are determined
through observed spectra. The cooling is considered as dominated by synchrotron processes
and the break in energy results from a balance between the cooling time and the escape
time. The soft photon lags observed in some sources are explained by radiative cooling of
high energy particles. It was showed that the constraints lead to equations that can be
13
expressed in terms of the magnetic field and the Doppler factor. The value found for the
magnetic field intensity and Doppler factor for Mkr 501 are ∼ 0.3 G and ∼ 10, respectively.
For two sources, Mrk 421 and PKS 2155-304, the derived parameters were a Doppler factor
∼ 25 and a magnetic field B ∼ 0.2 G.
[Lindfors, E. J 2005] took into account a SSC model for the observed SEDs of 3C 279,
considering the inverse Compton maximum to be in the γ-ray energy range. The Doppler
boosting factor has to be high for inverse Compton and, in the external Compton mecha-
nisms, the magnetic fields need to be strong, and also the luminosity of the external photon
field has to be high. So, they present a multi component scenario, where the synchrotron
emission originates from both the jet and from the shocks propagating downstream in the
jet. From observed time delays from radio to γ-ray flares, they estimated that the shock are
at distances of a several parsec from the jet apex. Beyond the BLR, they may not be enough
external photons to produce a significant external Compton radiation. They found that with
the size, Doppler boosting and the variability time scale constraints, the SSC model cannot
reproduce all the high energy flux in quiescent or high states, even though the fits show that
X-ray emission may be plausible to be reproduced by the shocks.
[Dermer, C. D. 1992] proposed an EC model with Compton scattering of accretion
disk photons by relativistic non-thermal electrons in the jet. The accretion disk source and
the core of the AGN emit target photons isotropically, photons passing through the blob
follow trajectories parallel to the jet axis. [Dermer, C. 1993] proposed an EC from the disk
as well and angle dependence. Photons that enter the jet from the side, can make more
important contributions to the loss rate than luminous emission emitted from the innermost
regions of the supermassive BH. Photons that interact at large angles with respect to the
jet axis, dominate the electron energy loss rate in the comoving fluid frame. The Compton
scattering energy loss rate of the relativistic electrons in the blob is mainly due to accretion
disk photons, rather than scattered photons at distances ≲ 0.01 − 0.1pc from the central
source. This eliminates the need to call upon different source models for quasars and BL
Lac objects, for this to be true BL Lacs would be viewed as weak quasars, with no emission
14
lines.
[Blazéjowski M. 2000] assumed relativistic electrons/positrons enclosed within a very
thin shell that propagates along the jet where energy losses are due to synchrotron, Comp-
ton scattering of synchrotron radiation and Compton scattering of external radiation. Three
cases are considered, one, where the γ-rays are produced by Comptonization of near-IR radi-
ation, other, due to Comptonization of the broad line region and the last, as a combination
of both. An electron/positron injection function as a power law injected at a constant rate.
It is assumed that the shell propagates at a constant Lorentz factor. The model approxi-
mates a situation in which the shell containing relativistic plasma is formed as a result of
the collision of two perturbations moving down the jet at different speeds. Near-infrared
Comptonization (model A) yields spectra smoother and closer to the observed ones, than
the spectra produced by BLR Comptonization (model B). Lorentz factors for electrons where
the cooling break occurs need to be larger for model A than in model B, causing the X-ray
portion of the SSC component to be harder in model A, compared to model B. In the γ-
ray spectral band, the spectral component, due to IR, has a break at ∼1 GeV, whereas in
observations from EGRET/CGRO blazar spectra extend to at least 5 GeV. Therefore, the
EGRET spectra can be produced only by the combination of IR and BLR components. This
also provides an interesting explanation for γ-ray spectra being steeper than the X-ray spec-
tra. According to Blazéjowski, Comptonization of infrared radiation produced by hot dust
should be taken into account in all radiation models of blazars, since such radiation is likely
to be present in quasar cores at parsec scales, where it is sufficiently dense to compete with
BLR flux, as the source of Compton cooling. If blazar radiation is produced at distances
larger than the distance at which energy density of the BLR light peaks, the IR component
is expected to dominate over the BLR component. EC models predict the general steepening
of the γ-ray and synchrotron spectra with energy in qualitative agreement with observations.
However, the lower observed variability amplitude of synchrotron flux compared with γ-ray
flux suggests contamination of synchrotron radiation produced at larger distances or station-
ary shocks. The X-ray spectrum would be a superposition by the SSC component and the
harder EC component.
15
[Arbeiter, C ] considers EC from both the dust torus and the accretion disk, stating
that for small angles to the jet axis and distances of more than a few accretion disk radii the
dust surrounding the AGN is more important as target photon source for inverse-Compton
scattering. [Harris, D. E. 2002] suggested an EC contribution from cosmic microwave back-
ground (CMB) photons. Inverse Compton emission from the relativistic electrons scattering
off the CMB is incapable of producing the observed intensity of X-rays, unless relativistic ef-
fects that modify the apparent luminosity are considered. Finding that this model is favored
by higher redshifts and steeper radio spectra, while local sources with flatter radio spectra
are most likely dominated by synchrotron emission.
1.7 Blazars with Correlation of the X-ray and γ-ray
Emission
Several multiwavelength observations for several blazars have been performed to identify the
emission mechanisms and to constrain the parameters describing the emission in blazar jets.
In one zone leptonic models, we would expect that the emission at different wavelength to
be correlated when produced essentially by the same electron population. V. Acciari (2014)
[Acciari, V. A., 2014] studied observations of Markarian 421 over a span of 14 years with
TeV energies observations from Whipple (December 1995 to May 2009) and from VERITAS.
They correlated Whipple observations with data taken at other wavelengths for part or all
of the epoch 1995-2009. The most complete overlap was with Rossi X-ray Timing Explorer
(RXTE) database for the entire 14 years. RXTE had three non-imaging X-ray detectors, one
of which is the All-Sky Monitor (ASM). The RXTE ASM data for the epoch show evidence
for emission correlated with γ-ray data on monthly and yearly timescales. The correlation
coefficients for monthly-binned data is rc=0.75 and for yearly-binned date is rc=0.89. The
strong correlation suggests that the X-rays and γ-rays are emitted by the same population
of electrons in the same region of the jet. Yet other wavelengths (radio, optical, infrared)
16
are not strongly correlated.
[Rachen Jörg P. 2001] found from simultaneous multiwavelength observations of Mrk
421 and Mrk 501, that their variability in the X-ray and γ-ray regime is largely correlated
and stronger than in other bands. Which would serve as an argument against hadronic
models. [Gutierrez, K 2006] studied the multiwavelength observation campaign of blazar
1ES 1959+650, May 2003 to June 7 and compared the source characteristics with those
measured during observations taken during previous years. They found correlation between
the simultaneously measured TeV γ-ray and X-ray fluxes during the 2002 and 2003 obser-
vation campaigns. Both, the fluxes and photon indices, appear to be correlated for the
2002 data set. The lack of a correlation between the high energy (X-ray and γ-ray) and
low energy (radio and optical) flux levels observed in the years 2002 and 2003 suggests that
the high energy radiation is produced close to the central engine and that the low energy
radiation is produced farther downstream of the jet. [Tachibana, Y. 2014] studied the data
in optical, X-ray and γ-ray bands for blazar 3C 454.3. Variation, especially activities like
flares, are seen to be completely correlated across the three energy bands. In terms of the
flare amplitudes the X-ray band seems to show lower variabilities than the optical and γ-ray
bands. [González, M. M 2019] used a maximum likelihood approach to estimate the monthly
robustness in Whipple-RXTE/ASM correlation for Mkn 421.
Considering the correlations found on several blazars, it is plausible to consider that
mechanisms behind the X-ray and γ-ray emission on blazars are due to one zone leptonic
models. So far, different models have been constructed to explain the observed emissions,
both leptonic and hadronic. Within these models that were mentioned earlier, the TeV
orphan flare remains either as an open question or explained as a whole new phenomena.
Using [González, M. M 2019] considerations for the correlation for Mkn 421, a model is
proposed to explain the X-ray and γ-ray emission of blazars as one physical process dependent
of parameters, whether they are in quiescent, flaring state or presenting Tev orphan flares.
17
18
Chapter 2
Synchrotron Self-Compton Model
As an attempt for a better understanding of blazars hence their physical properties, in
this thesis we propose a SSC model. Before taking this task we undertook the task of
studying the existing models mentioned in the previous chapter. Mkr 421 [Sikora, M. 2000],
is one of the closest and brightest therefore serves as a good study source, observations
[Macomb, D. J 1995, Fossati, G. 2008, Acciari, V. A 2011, Aleksić, J. 2015, Aleksić, J 2015,
Baloković, M. 2016, Acciari, V. A., 2014] suggest correlation between the X-ray and γ-ray
emissions, this may be interpreted well in a SSC model.
In this chapter, I will discuss the a one zone synchrotron self-Compton model proposed
in this thesis. We consider a leptonic jet model which will depend on four parameters,
electron density Ne,constant magnetic field B,a bulk Lorentz factor Γ and radius rd. We
consider an emitting region (blob) as an sphere for simplicity with a radius rd and it is filled
with a relativistic electron density Ne, which is moving towards the observer with a bulk
Lorentz factor Γ, and filled with a homogeneous, constant magnetic field B, as shown in
figure 2.1 . Here on we define the universal constants c = ℏ = 1 in natural units.
19
Accretion disk
BH
Magnetic field
(B)
Emitting region with
relativistic electrons (Ne)
rd
Γ
Observer
Figure 2.1: A visual representation of the jet model.
2.1 Synchrotron Emission
As we reviewed in (Appendix) 5.1 synchrotron radiation is produced by the interaction of
charged particles moving in a magnetic field. If there is a particle with charge qe and Lorentz
factor γe = E/me in the region it would radiate a synchrotron power in he form of equation
5.23. We assume that the blob is made up of more than a single electron, instead is made
up of a population of electrons described with a broken power law distribution of Lorentz
factor γe, defined between γe,m and γe,max as [Longair, 1994],
Ne(γe) = N0,e





γ−α
e γe,m < γe ≤ γe,c
γe,cγ
−(α+1)
e γe,c < γe < γe,max.
(2.1)
Where N0,e is the normalization factor,which is associated to the number density of the
electrons in the emission region; α is the power index of the electron distribution; the Lorentz
cooling factor ,γe,c, which gives the cooling energy, break in energy, of the electron where the
slope changes; the minimum and the maximum energy of the electron population given by
their respective electron Lorentz factors γe,m and γe,max.
20
2.1.1 Minimum Energy
We assume an equipartition of energy, we have that magnetic energy density is given by
UB = B2/8π and the total electrons energy density is described as,
Ue = me
∫
γeNe(γe)dγe, (2.2)
where me is the rest electron mass, γe is the electron’s Lorentz factor and Ne(γe) is given by
equation 2.1. If we consider that a fraction of the total energy density is used to accelerate
the electrons we may proceed to find the minimum electron Lorentz factor. We integrate our
electron distribution Ne(γe) =
∫
Ne(γ
′
e)dγ
′
e defined between γe,m and γe,max and for α > 2
we find:
Ne ≃
N0,e
α− 1
γ−α+1
e,m . (2.3)
Solving for the electron energy density given by equation 2.2 for the electron distribution
given by equation 2.1 we find:
Ue ≃
N0,eme
α− 2
γ−α+2
e,m . (2.4)
Using 2.3 and 2.4, we may write the minimum electron Lorentz factor in terms of the energy
density and the distribution of the electrons. As follows,
γe,m =
α− 2
me(α− 1)
Ue
Ne
. (2.5)
2.1.2 Cooling Energy
As electrons radiate synchrotron photons, they lose energy (cooling energy). The cooling
time, tcool, is the time related to the energy loss due to synchrotron radiation. The energy loss
per unit of time or the power emitted due to synchrotron is found earlier given by equation
5.14 and keeping in mind the energy of an electron is given by ϵe = γeme, we can obtain the
21
cooling time as [Inoue, S. 1996],
tcool =
ϵe
dϵe/dt
=
γeme
4
3
σTγ2
eUB
=
3me
4σTγeUB
. (2.6)
The blob moves at relativistic velocities with Doppler factor δD along the jet in an outwards
direction. We consider the dynamic time scale associated with the expansion of the emitting
region as it travels along the jet, may also be interpreted from a mathematical point as
an energy independent particle escape from the blob , which is defined as [Inoue, S. 1996,
Cerruti, M. 2020]:
td ≃
rd
δD
. (2.7)
The electron’s cooling Lorentz factor, γe,c satisfies tcool ≃ td [Inoue, S. 1996],
3me
4σTγe,cUB
=
rd
δD
. (2.8)
Doing so, we get the cooling electron Lorentz factor as
γe,c =
3me
4σT
δDr
−1
d U−1
B . (2.9)
2.1.3 Maximum Energy
The blob’s electrons must have an energy limit reflecting the acceleration mechanisms taking
place. The maximum energy of the electrons is estimated by equating tcool with tacc which
is given as [Inoue, S. 1996],
tacc =
√
π
2
me
qe
γeU
−1/2
B =
√
π
2
U
−1/2
B
ϵe
qe
. (2.10)
tacc ≃ tcool (2.11)
√
π
2
U
−1/2
B
ϵe,max
qe
=
3m2
e
4σTUB
1
ϵe,max
(2.12)
ϵe,max =
[
9m4
eq
2
e
8πσT
]
1
4
U
−1/4
B . (2.13)
Now we can write the maximum electron Lorentz factor as,
γe,max =
[
√
π
2
3qe
4σT
U
−1/2
B
]1/2
. (2.14)
22
2.1.4 Synchrotron Spectrum
We have that the energy of a photon is given by ϵ = hν, photons emitted by synchrotron
have ν ≃ γ2qeB
me
, as discussed in Appen. B equation 5.19. The energy of the emitted photons
by synchrotron is given by,
ϵsynγ =
δDqeU
1/2
B
me
√
8πγ2
e . (2.15)
To find the minimum, cooling, and maximum energy of the emitted photons we take the
above equation and substitute the corresponding Lorentz factor.
ϵsynγ,m =
δDqeU
1/2
B
me
√
8πγ2
e,m (2.16)
=
δDqeU
1/2
B
me
√
8π
[
(α− 2)
me(α− 1)
Ue
Ne
]2
. (2.17)
Then the photon’s minimum energy is given by:
ϵsynγ,m =
√
8π(α− 2)2qeδDU
1/2
B U2
e
(α− 1)2m3
eN
2
e
. (2.18)
Similarly the photon’s cooling energy
ϵsynγ,c =
δDqeU
1/2
B
√
8π
me
γ2
e,c (2.19)
=
δDqeU
1/2
B
√
8π
me
(
3me
4σT
δDr
−1
d U−1
B
)2
. (2.20)
rearranging, we get
ϵsynγ,c =
9qeme
√
8π
16σ2
T
δ3Dr
−2
d U
−3/2
B , (2.21)
When electrons and photons coexist in a region the repeated scattering of photons will
modify their spectrum. When an IC is considered, we include the correction Compton
parameter,
(
1 + Ue
UB
)
in γe,c [Rybicki, G. 1986, Sari, R. 2001]
γe,c =
3me
4σT
δDr
−1
d U−1
B
(
1 +
Ue
UB
)
. (2.22)
So then our photon’s cooling energy is given by
ϵsynγ,c =
9qeme
√
8π
16σ2
T
(
1 +
Ue
UB
)−2
δ3Dr
−2
d U
−3/2
B . (2.23)
23
Now, calculating the photon’s maximum energy
ϵsynγ,max =
δDqeU
1/2
B
√
8π
m3
e
γ2
e,max , (2.24)
ϵsynγ,max =
δDqeU
1/2
B
√
8π
m3
e
√
2
π
3m2
eqe
4σT
U
1/2
B . (2.25)
Thus photon’s maximum energy is given by
ϵsynγ,max =
3q2e
σTme
δD(1 + z)−1 . (2.26)
The observed synchrotron spectrum is obtained by the shape of the electron distribution, we
may estimate the photon spectrum through emissivity ϵγNγ(ϵγ)dϵγ; as ϵγ ≃ ⟨ϵγ⟩ δ(ω − ωc)
doing as [Rybicki, G. 1986, Longair, 1994] we show that for spectral indices of the electron
distribution α and (α−1), we obtain a photon distribution with spectral indices p = (α−1)/2
and p = α/2, respectively. Therefore photon spectrum, can be written as
[
ϵ2γN(ϵγ)
]
γ,syn
= Aγ,syn













(
ϵγ
ϵsynγ,m
)4/3
ϵγ < ϵsynγ,m ,
(
ϵγ
ϵsynγ,m
)−α−3
2
ϵsynγ,m < ϵγ < ϵsynγ,c ,
(
ϵsynγ,c
ϵsynγ,m
)−α−3
2
(
ϵγ
ϵγ,c
)−α−2
2
ϵsynγ,c < ϵγ < ϵsynγ,max ,
(2.27)
where
Aγ,syn =
4σT
9
d2zδ
3
DUBr
3
dNeγ
2
e,m . (2.28)
Aγ,syn is the proportionality constant estimated by the product of the total number of radiat-
ing electrons in the volume and the maximum radiated power, where dz is the distance from
the source. Equation 2.27 describes the synchrotron spectrum as function of the magnetic
field B, the electron density Ne, the size of the emission region rd, the bulk Lorentz factor Γ
and the spectral index α.
2.2 Inverse Compton
Now, we will calculate our photons due to inverse Compton. From now on the primed values
signify the electrons’ rest frame. In the Appen. B we find the incident photon’s energy given
24
by the equation 5.40 (ϵ′ = ϵγ(1 − β cos θ)), where β = v
c
which is the β factor from the
Doppler shift and θ is the angle between the electrons trajectory and the photon as seen in
the lab frame. If we suppose that photons (which entered at angle θ′ in the electrons frame,
as seen in figure 5.4, the primed values signify the electrons rest frame) rebound at an angle
θ′1. We say θ′1 − ϕ = θ′. Using equation 5.34 in the electrons frame:
ϵ′1 =
ϵ′
1 + ϵ′
mc2
(1− cosϕ)
. (2.29)
Going back to the lab frame, we have using 5.41 (ϵ1 = ϵ′1γ(1 + β cos θ′1)). Replacing the
ϵ′1(equation 2.29) on equation 5.41, we have the scatter photon energy as:
ϵ1 =
ϵ′γ(1 + β cos θ′1)
1 + ϵ′
mc2
(1− cosϕ)
. (2.30)
Now replacing 5.40 on 2.30 we have:
ϵ1 =
ϵγ2(1− β cos θ)(1 + β cos θ′1)
1 + ϵ′
mc2
(1− cosϕ)
. (2.31)
On equation 2.31, I left ϵ′ in the denominator, just so it doesn’t look so crowded, also we
want the limit where ϵ′ ≪ mec
2 and leaving it as it is seems convenient. So, for ϵ′ ≪ mec
2
equation 2.31, turns into ϵ1 ≈ ϵγ2(1−β cos θ)(1+β cos θ′1). In a ‘typical’ collision θ ∼ θ′1 ∼ π
2
so, we get:
ϵ1 ≈ ϵγ2. (2.32)
To find the energies of the photons due to inverse Compton, we are going to use equation
2.32. In 2.32 ϵ is the incident energy which, in our model will correspond to the energy
of photon emitted by synchrotron. Here we denote as ϵγ,i where i denotes whether if it is
cooling, maximum or minimum. So, the inverse Compton photons energy, here we will call
it SSC since the seed electrons of both the synchrotron and inverse Compton are from the
same population , is given by:
ϵSSCγ = γ2
e,iϵ
syn
γi , (2.33)
where γ2
e,i is the corresponding Lorentz factor and ϵsynγi is the incident energy of the photon,
which is given by the synchrotron photon energies we found earlier. We call it ϵSSCγ , since
the same electrons that are synchrotron radiating, scatter up their own radiation, which
25
is known as SSC (sycnrotron self Compton). Using 2.33, we find the energies of radiated
photons through inverse Compton. The minimum energy is given by
ϵSSCγ,m = γ2
e,mϵ
syn
γ,m . (2.34)
The minimum energy is given by
ϵSSCγ,m =
√
8π(α− 2)4
(α− 1)2
qeδD
m5
e
U4
eU
1/2
B
N4
e
(1 + z)−1 . (2.35)
The maximum energy is given by
ϵSSCγ,max =
9
4
√
2
π
q3e
σ2
Tme
δDU
−1/2
B (1 + z)−1 . (2.36)
The cooling energy is given by
ϵSSCγ,c =
81qem
3
e
√
8π
256σ4
T
δ5Dr
−4
d
(
1 +
Ue
UB
)−4
U
−7/2
B (1 + z)−1. (2.37)
The Compton scattering spectrum obtained as a function of the synchrotron spectrum
is
[
ϵ2γN(ϵγ)
]
γ,SSC
= Aγ,SSC













(
ϵγ
ϵSSC
γ,m
)4/3
ϵγ < ϵSSCγ,m ,
(
ϵγ
ϵSSC
γ,m
)−α−3
2
ϵSSCγ,m < ϵγ < ϵSSCγ,c ,
(
ϵSSC
γ,c
ϵSSC
γ,m
)−α−3
2
(
ϵγ
ϵSSC
γ,c
)−α−2
2
ϵSSCγ,c < ϵγ < ϵSSCγ,max ,
(2.38)
where Aγ,SSC is the proportionality constant,
Aγ,SSC =
4σT
9
d2zδ
3
DUe r
3
dNeγ
2
e,m . (2.39)
Equations 2.38 and 2.39 describe the inverse Compton spectrum as function of the
magnetic field B, the electron density Ne, the size of the emission region rd, the bulk Lorentz
factor Γ and the spectral index α. As observed, the inverse Compton spectra is strongly
related to the synchrotron spectrum. In fact, the dependency on the parameters is the same
for both spectra. A linear correlation between both emission is expected.
26
For gamma emission in the energy range of hundreds of GeV to TeVs, the main
spectral component is the third power law in equation 2.38. For instance, the value for
ϵSSCγ,c is 836.8GeV for standard values of rd = 1.8 × 1016 cm, Γ = 22, B = 6.1 × 10−2 G and
Ne = 104.2 cm−3 . It is not understood what causes a blazar to flare, but if a correlation
between gamma-ray emission and X-ray emission is measured, it is interesting to find the
parameter space that could reproduce the measured fluxes. For purposes of this thesis, the
spectral index is considered fixed and the parameter space is found for different correlations
for a blazar similar to Mrk 421.
27
28
Chapter 3
Discussion and Conclusions
Blazar exhibits challenges for a leptonic or hadronic model with one-zone emission. These
sources demand sometimes atypical parameters for explaining the hard spectrum shape,
especially in the TeV gamma-ray band. We propose a one-zone synchrotron self-Compton
model to explain the TeV and X-ray observations and its possible correlation during flaring
or quiescent states. An electron population with two power-law functions was required to
describe these observations. The full set of parameters derived is: Doppler factor, magnetic
field, emitting radius and electron density. It is worth noting that although a two emitting
region model (more than one electron population) has been proposed for several blazars
in order to describe the TeV and X-ray observations, we only require a one emitting zone.
Similarly, in our analysis we do not consider accelerated protons in the emitting region, which
could require a high intensity of the magnetic field. Although annihilation-line photons may
attenuate the TeV gamma-ray spectrum produced in the emitting region, this effect was not
analyzed in this thesis. It is relevant to say that our one-zone SSC model was applied to a
high peaked synchrotron blazar, and not to a low or intermediate peaked synchrotron blazar,
which do not show a lineal correlation as discussed in the literature.
29
The peak energies of the synchrotron and inverse Compton emissions are at 0.2 keV
and 0.8 GeV, respectively. Thus, it makes sense to test the existence of a correlation between
fluxes in the energy ranges of tens of keV and hundreds of GeV. One of the most studied
blazars at all energies is Mrk 421. In fact, several authors have looked for correlations between
TeV γ- and X-ray fluxes during high- and low- activity. For instance, [Macomb, D. J 1995],
[Fossati, G. 2008], [Acciari, V. A 2011] and [Aleksić, J. 2015] reported TeV flares in coinci-
dence with flaring activities in X-rays occurred in 1994 May, 2001 March, 2008 May and 2010
March, respectively. They reported strong correlations with no lags. Furthermore, Mrk421
was monitored in γ-rays and X-rays in 2009, between January and June, [Aleksić, J 2015]
and in 2013 January - March [Baloković, M. 2016]. In 2009, [Aleksić, J 2015] reported a
harder-when-brighter behavior in the X-ray flux, measuring a positive correlation between
TeV and X-ray fluxes with zero time lag. In 2013, [Baloković, M. 2016] reported a significant
TeV/X-ray correlation on a timescale of about one week. Finally, a robust study in different
time scales and flux level of the correlations with a data set for 14 years [Acciari, V. A., 2014]
with Whipple was performed by [González, M. M 2019]. The correlation is reproduced by
increasing the electron density. They argue that it is possible to describe all flux levels
without varying the magnetic field.
In order to test the conclusion, in this thesis we consider a blazar like Mrk 421 and
assume different linear correlations. We use our model, described earlier, to reproduce the
fluxes within the different hypothetical correlations. This to find the different values of
magnetic field and electron density that reproduce the fluxes. To prove the hypothesis.
Results are shown in Figure 3.1.
Figure 3.1 shows in lower panels four different correlations, becoming steeper from
panel (a) to (d) and, in upper panels, the magnetic field and electron density that reproduce
the fluxes in the lower panels. As observed, for a weak correlation, panel (a), the parameter
space is broad. As the correlation is steeper, the values of the magnetic field that describes
the highest fluxes are narrower. The flaring states are obtained by increasing the electron
density. The conclusion by [González, M. M 2019] does not hold for the steepest correlation
30
when a decrease in the magnetic field is required in order to obtain the highest fluxes.
Observations of the polarization may confirm this decrease.
In this thesis, a theoretical model was developed with the minimum number of as-
sumptions such as one electron population described by the simplest function (a broken
power law) with a constant spectral index for all levels and duration of activity of the
blazar. We do not consider neither hadronic scenarios and its electromagnetic cascades due
to photohadronic interactions nor external inverse Compton emission with broadline and
dust radiation as seed photons. This allows to perform a model-independent study of the
correlation of the VHE γ-ray and the X-ray emission.
For future work, this model could be used to study a sample of high-energy peak
BL Lacs, as shown in this thesis. In particular, it is not clear if there is a characteristic
magnetic field for each blazar or all blazars will show a correlation between TeV γ- and
X-rays. Consequences in the parameter space when a spectral evolution is considered have
to be studied. In particular, when the spectrum varies from soft to hard as the flux increases.
Here, a theoretical description of the spectrum has been considered, but also a study of the
correlation as function of the empirical model fitting the spectra has to be performed. It has
been observed a degree of dispersion on the correlation that has to be understood, if it is
intrinsic to the blazar, for example, for the existence of more than one electron population or
if this is related to systematic in the observations. We want to emphasize that this modeling
will be applied to a sample of high-energy peak BL Lacs.
Further studies in the dependencies of the parameters analyzed need to be done to
understand better the acceleration mechanisms, radiation processes and jet composition,
among others in blazars.
31
(a) (b) 
(c) (d) 
Figure 3.1: Theoretical Results.
Panel (a): 2D parameter space (upper) allowed by hypothetical correlation of emission in VHE
γ-rays and X-rays for a source as Mrk 421 (lower). The 2D parameter space corresponds to the
magnetic field and electron density, and the hypothetical correlation are for a slope of 0.10± 0.02.
The regions in yellow, purple, pink and violet represent levels of flux activity of the source within
the correlation, from the very-low, low, high and very-high states, respectively. The green region
corresponds to highest gamma ray fluxes observed for Mrk 421, some of them not following the
correlation and associated to orphan flares. The “very low”, “low” and “high” states of Mrk 421,
as measured by VERITAS, are displayed as black dotted, dot-dashed and solid lines, respectively.
Panel (b), (c) and (d) are the same as panel (a) but the hypothetical correlations with slopes of
1.0± 0.2, 2.0± 0.4 and 3.0± 0.6.
32
Chapter 4
Appendix A
4.1 Emission and Absorption Lines
An emission line appears in a spectrum, if the source emits specific wavelength of radiation.
An emission takes place when electrons in an atom, which are excited (e.g. being heated,...),
make a transition from a high energy state to a lower energy state. When the electron
falls back and leave the excited state, the energy is re-emitted in the form of a photon.
[COSMOS, Osterbrock, D. 2006]
An absorption line appears in a spectrum, if an absorbing material is located between
the source and the observer. Such material may be outer layers of a star, a cloud of interstellar
gas or a cloud of dust. An atom, atomic nuclei or molecule may absorb photons with a given
energy if such energy is equal to the difference between two of their energy states. This lines
are seen usually as dark lines or lines of reduced intensity. [COSMOS, Osterbrock, D. 2006]
33
4.2 Broad and Narrow Lines
The broad lines of AGNs come from material close to the BH, they are broadened because the
emitting material is revolving around the BH with high velocities causing Doppler shifts of
the emitted radiation. The narrow lines come from an emitting region farther from the BH,
moving slower, which causes the, to narrower than the broad lines. [Osterbrock, D. 2006]
34
Chapter 5
Appendix B: Physics background
5.1 Synchrotron Radiation
Synchrotron radiation may be defined as the radiation emitted by charged relativistic parti-
cles in interaction with a magnetic field. The following discussion is based on the concepts
presented in Rybicki’s Radiative Processes in Astrophysics textbook [Rybicki, G. 1986].
5.1.1 Power Emitted
So as mentioned before, and in a bold manner, we may say synchrotron radiation is due to a
charged particle moving at relativistic velocities in interaction with a magnetic field, which
proceeds to move in a helical motion, as shown in figure 5.1. Keeping this in mind, we will
start by finding the motion of a particle of mass, m, and charge, q, in a magnetic field, to
proceed on calculating the magnetic field at hand and, therefore, be able to find our total
power emitted, this to allow us to understand the processes it takes for this radiation to be
35
emitted and the energy that it conveys. Perhaps, to do so, we will first study the case of one
particle opposed to the distribution of charged particles known to participate in this process,
this for a better understanding. Taking in consideration all the above mentioned, when we
speak of finding the equation of motion of a particle, we imply the forces that act upon this
particle, in this case, we have the Lorentz force acting upon it due to the magnetic field (we
will use Lorentz four-force notation as seen on equation 5.1).
v
v⊥
v∥
B
Figure 5.1: Helical motion of a particle in an uniform magnetic field.
F ν =
q
c
F ν
µU
µ. (5.1)
Where q is the charge of the particle, c is the speed of light and (Fνµ) Maxwell’s tensor,
which represents the interaction between electromagnetic forces and mechanical momentum,
is given by
Fνµ =








0 −Ex −Ey −Ez
Ex 0 Bz −By
Ey −Bz 0 Bx
Ez By −Bx 0








,
where Ei are the components of the electric field, Bi are the components of the magnetic
field and ν and µ are the subscript and upperscript of the tensor and the four-velocity vector
is
Uµ =








c
vx
vy
vz








.
36
Using the above mentioned, we find that
d
dt
(γmv⃗) =
q
c
v⃗ × B⃗, (5.2)
d
dt
(γmc2) = qv⃗ · E⃗ = 0. (5.3)
As shown in figure 5.1, when a charged particle enters into an uniform magnetic
field it has a helical motion where the Lorentz force acting upon the particle acts as a
centripetal force, making the particle go in circles, changing the velocity’s direction, but not
its magnitude. The combination of this circular motion and the uniform motion along the
field is a helical motion of the particle. This change is perpendicular to the magnetic field.
Separating the velocity components along the field, we have the following
d
dt
v∥ = 0,
d
dt
v⊥ =
q
γmc
v⊥B sinαs.
Where αs is the angle between v and B. The frequency of the rotation, or gyration is
ωB =
qB
γmc
. (5.4)
Finding the equation of motion allowed us to find the acceleration of the particle and, in
turn, its gyration, which will be needed in the following derivations.
Now to, find total power emitted, we are going to consider the retarded potentials
(the Liénard-Wiechart potentials). We are going to find the magnetic field, which in turn,
will lead as to the derivation of the power emitted. Considering a particle of charge, q, that
moves along a trajectory, r⃗ = r0(t), and its velocity at any time is v⃗(t) = ṙ0(t). The charge
and current densities are given by
ρ(r⃗′, t) = qδ3(r⃗′ − r⃗0(tret)),
j⃗(r⃗′, t) = qv⃗(t)δ3(r⃗ − r⃗0(tret)).
37
We need the charge density at a retarded time, to which we can write as the integral over
time of the charge density multiplied, by another delta function
ρ(r⃗, t) = qδ3(r⃗′ − r⃗0(t
′))
∫
dt′δ(t′ − tret).
To find the scalar potential, we have
ϕ =
∫
d3r⃗′δ3(r⃗′ − r⃗0(tret)
|r⃗ − r⃗′|
∫
dt′δ
(
t′ −
(
t− r⃗ − r⃗′
c
))
,
doing the spatial integration, the delta function is nonzero only where its argument is zero,
so all contributions to the integral come where r⃗′ = r0(t
′). With R(t′) = r⃗ − r⃗0(t
′) and
R = |R(t′)|. We then have
ϕ = q
∫
R−1(t′)δ(t′ − t+ R(t′)/c)dt′,
A =
q
c
∫
v⃗(t′)R−1(t′)δ(t′ − t+ R(t′)/c)dt′.
Delta function vanishes for t′ = tret. Let us change variables t′′ = t′, so we have dt′′ =
dt′+ 1
c
Ṙ(t′)dt′ = (1+ 1
c
Ṙ(t′))dt′ also we will define the unit vector, η = R
R
, and we have Ṙ(t′) =
−ηv⃗(t′). So, the potentials in terms of η and t′′ allow us to easily solve the integrals with
the delta function setting in t′′ = 0 ≡ tret. So, we obtain the Liénard-Wiechart potentials,
where the brackets denote retarded times.
ϕ =
[
q
κR
]
, (5.5)
A =
[
qv⃗
cκR
]
. (5.6)
Using the above, we solve for |E⃗rad| = |B⃗rad| = q ˙⃗v
Rc2
sinΘ and the Poynting vector is in the
direction of η and has a magnitude
S =
c
4π
E2
rad =
c
4π
q2 ˙⃗v2
R2c4
sin2 Θ. (5.7)
This corresponds to an outward flow of energy, along the direction η⃗. We can put this into
the form of an emission coefficient. The energy dW⃗ emitted per unit time into solid angle
dΩ about η⃗ can be evaluated by multiplying the Poynting vector by the area dA = R2dΩ
represented by Ω at the field point:
dW⃗
dtdΩ
=
q2 ˙⃗v2
4πc3
sin2 Θ. (5.8)
38
We may obtain the total power emitted into all angles by integrating this over the solid
angle:
P =
dW⃗
dt
=
q2 ˙⃗v2
4πc3
∫
sin2 ΘdΩ. (5.9)
So we obtain the Larmor’s formula
P =
2q2 ˙⃗v2
3c3
=
2q2a⃗2
3c3
. (5.10)
Expressing P in terms of the three-vector acceleration
P =
2q2
3c3
γ4(a2⊥ + γ2a2∥). (5.11)
The acceleration is perpendicular to the velocity, with magnitude a⊥ = ωBv⊥, so that
the total emitted radiation is,
P =
2q2
3c3
γ4 q2B2
γ2m2c2
v2⊥. (5.12)
For an isotropic distribution of velocities, it is necessary to average this formula over all
angles for a given speed β. Let α be the pitch angle, which is the angle between the field
and the velocity. Then we obtain
⟨β⟩ = β2
4π
∫
sin2 αdΩ =
2β2
3
. (5.13)
The total power emitted can also be written in terms σT = 8πr20/3, which is the Thomson
cross section, UB = B2/8π is the magnetic energy density and the classical electron radius
r0 = q2/mc2.
P =
4
3
σT cβ
2γ2UB. (5.14)
5.1.2 Spectrum
Now that we have the total power emitted, we will discuss the spectrum of synchrotron
radiation, which must be related to the detailed variation of the electric field as seen by the
observer. Due to beaming effects, the emitted fields appear to be concentrated in a narrow
39
set of directions about the particle’s velocity. The observer will see a pulse of radiation
confined to a time interval much smaller than the gyration period. The spectrum will be
spread over a much broader region than one of order ωB/2π.
p1 p2
ra
∆θ
1
γ
Observer
Figure 5.2: Emission cones at different points of an accelerated particle’s trajectory.
In figure 5.2, we can tell that the observer will see the pulse from points p1 and p2
along the particle’s path, where these points are such that the cone of emission of angular
width ∼ 1/γ includes the direction of observation. The distance ∆s, along the path, can be
computed from the radius of curvature of the path, ra = ∆s/∆θ, and from geometry, we
know ∆θ = 2/γ, so that ∆s = 2ra/γ. Using the equation of motion we found earlier, since
|∆v⃗| = v∆θ and ∆ = v∆t, we may rewrite the equation is terms of the radius of curvature
of the path and the gyration as
ra =
v
ωB sinα
. (5.15)
In turn, ∆s is given by
∆s ≈ 2v
γωB sinα
. (5.16)
The time tp1 and tp2 ( time at which the particle passes through point p1 and p2 respectively
) are given by ∆s = v(tp2 − tp1), so that
∆tem = tp2 − tp1 ≈
2
γωB sinα
. (5.17)
This duration of emission (or emission time) is given by the arclength traversed given a
gyration radius ra, combined with the velocity v, of the particle while the time it takes the
40
radiation to move a distance ∆s (the time it takes the light to travel from the beginning
and the end of the emission), we will call it traveled time and is given by ∆ttrav = ∆s/c =
2v/γcωB sinα. The arrival time is the difference of the time of the emission and the time it
takes for it to travel from the beginning and the end of the emission. So the traveled time
is given by
∆tA =
2
γωB sinα
(
1− v
c
)
≈ 1
γ3ωB sinα
. (5.18)
We expect that the spectrum will be fairly broad, cutting off at frequencies like 1/∆tA,
defining a critical frequency, which is where we have the maximum power emitted in the
spectrum, then we expect the spectrum to extend to something of order ωc before falling
away, where ωc follows
ωc ≡
3
2
γ3ωB sinα. (5.19)
or
νc =
3
4π
γ3ωB sinα. (5.20)
We can obtain a lot of information about the spectrum, by using the fact that the electric
field is a function of θ just through the combination γθ, where θ is a polar angle about the
direction of motion. So we have E(t) ∝ F (γθ), where t is the time measured in the observer’s
frame. We set the zero of time and the path length s to be when the pulse is centered
on the observer. Using arguments similar to those used to find ∆s, we find θ ≈ s/a and
t ≈ (s/v)(l−v/c). Then, the relationship of θ to t is found to be γθ ≈ 2γ(γ2ωB sinα)t ∝ ωct.
This way, we can derive the dependence of the spectrum on ω. The Fourier transform
of the electric field is Ê(ω)
∫∞
−∞
g(ωct)e
iωtdt. The spectrum dW/dω is proportional to the
square of Ê(ω). Integrating this over solid angle and dividing by the orbital period, both
independent of frequency, gives the time averaged power per unit frequency, dW/dtdω =
T−1dW/dω ≡ P (ω) = C1F (ω/ωc) where F is a dimensionless function and C1 is a constant
of proportionality. In turn, we have
P =
2q4B2γ2β2 sin2 α
3m2c3
. (5.21)
and
ωc =
3γ2qB sinα
2mc
(5.22)
41
We find that, for the highly relativistic case (β ≈ 1), the power per unit frequency emitted
by each electron is
P (ω) =
√
3
2π
q3B sinα
mc2
F
(
ω
ωc
)
. (5.23)
In formula for P (ω) given above, we can easily tell there is no factor of γ appears, except for
that contained in ωc. This fact is enough to derive an extremely important result concerning
synchrotron spectra. Often, the spectrum can be approximated by a power law over a limited
range of frequencies. When this is so, we may define the spectral index as the constant α in
the following expression
P (ω) ∝ ω−α. (5.24)
This is the negative slope on a log− log plot. Usually, the spectra of astronomical radiation
has a spectral index that is constant over a fairly wide range of frequencies: for example,
the Rayleigh-Jeans portion of the blackbody law has α = −2. A similar result sometimes
holds for the particle distribution law of relativistic electrons. Often, the number density
of particles with energies between E and E + dE (or γ and γ + dγ) can be approximately
expressed in the form
N(E)dE = CE−pdE. (5.25)
or
N(γ)dγ = Cγ−pdγ (5.26)
The total power radiated per unit volume per unit frequency by such a distribution is given
by the integral of N(γ)dγ times the single particle radiation formula over all energies or γ.
P (ω) = C
∫ γ2
γ1
P (ω)γ−pdγ ∝
∫ γ2
γ1
F
(
ω
ωc
)
γ−pdγ (5.27)
Integrating, we find that our spectral index is given by
α =
p− 1
2
(5.28)
42
5.2 Inverse Compton
The following discussion, just as the previous section is based on knowledge acquired from
Rybicki’s Radiative Processes in Astrophysics textbook [Rybicki, G. 1986].
5.2.1 Energy Transfer
First, we are going to discuss some previous concepts to understand Inverse Compton scatter-
ing, starting with the scattering from electrons at rest. For low photon energies, hν ≪ mec
2,
the scattering of radiation from free charges process is Thomson scattering which describes
photon scattering by free electrons in the low energy limit, the incident photons are ap-
proximated as a continuous electromagnetic wave. We assume an electron at rest, and an
electromagnetic wave of frequency ν ≪ mec
2/h, also that the incoming wave is completely
linearly polarized. In order to neglect the magnetic force (q/c)(v⃗ × B⃗), we require that the
oscillation velocity to be v ≪ c. This, in turn, implies that the incoming wave has a suffi-
ciently low amplitude. The electron starts to oscillate (E⃗ = E0 sinωtẑ) in response to the
varying electric force qE (me
d2z
dt
= qeE0 sinωt = mea), and the average square acceleration
(⟨a⟩ =
√
a2) is
⟨a2⟩ = e2E2
0
2m2
. (5.29)
The emitted power per unit solid angle is given by the Larmor formula dP/dΩ = q2a2 sin2 Θ/(4πc3)
(equation 5.9, before integrating over the solid angle), where Θ is the angle between the ac-
celeration vector and the propagation vector of the emitted radiation. Note that Θ is not
the scattering angle, which is the angle between the incoming and the scattered wave (or
photon). We have
dP
dΩ
=
q2E2
0
8πm2c3
sin2 Θ. (5.30)
The scattered radiation is completely linearly polarized in the plane defined by the
43
incident polarization vector and the scattering direction. The flux of the incoming wave is
dP
dσ
= Si = cE2
0/(8π). The differential cross section of the process is then
dσ
dΩ
=
dP/dΩ
Si
= r20 sin
2 Θ. (5.31)
The total cross section can be derived in a similar way, but considering the Larmor formula
integrated over the solid angle (equation 5.10). The total cross section is
σT =
P
Si
=
8π
3
r20. (5.32)
To find the scattering of an unpolarized incoming wave, we will assume that the incoming
radiation is the sum of two orthogonal completely linearly polarized waves, and then adding
the associated scattering patterns. Since we have the freedom to chose the orientations of
the two polarization planes, it is convenient to chose one of these planes as the one defined
by the incident and scattered directions, and the other one perpendicular to this plane. The
scattering can be then considered as the sum of two independent scattering processes, one
with emission angle Θ, the other with π/2. The scattering angle (the angle between the
scattered wave and the incident wave) is θ = π/2−Θ, we have
dσT
dΩ
=
1
2
r20(1 + cos2 θ). (5.33)
We have ϵ and ϵ1 are the incident and scattered photon energy, respectively, dσT/dΩ is the
differential Thomson cross section for unpolarized incident radiation, and r0 is the classical
electron radius, when ϵ = ϵ1 the scattering is called elastic. Here, we have review Thomson
scattering, including Thomson cross section, now we will proceed to find the energy of
a scattered photon. We assumed that the electron is at rest, the initial and final four-
momenta of the photon are P⃗γi = (ϵ/c)(1, ni) and P⃗γf = (ϵ1/c)(1, nf ) , and the initial and
final momenta of the electron are Pei = (mc, 0) and Pef = (E/c, p) ,where ni and nf are the
initial and final directions of the photons as shown in figure 5.3. Conservation of momentum
and energy is expressed by P⃗ei + P⃗γi = P⃗ef + P⃗γf . Rearranging terms and squaring gives
|P⃗ef |2 = |P⃗ei + P⃗γi − Pγf |2 which eliminates the final electron momentum. We obtain
ϵ1 =
ϵ
1 + ϵ
mc2
(1− cos θ)
. (5.34)
44
In terms of wavelength, this can be written as
λ1 − λ = λc(1− cos θ), (5.35)
where the Compton wavelength is defined by
λc =
h
mc
. (5.36)
P⃗γi
ϵ = hν
P⃗ef
P⃗γf
ϵ1 =
hν1
θ
Figure 5.3: Geometry of the scattering of a photon by an electron initially at rest.
We found that change of the wavelength is of the order of λc, upon scattering. For
long wavelengths, λ ≫ λc (hv ≪ mc2), the scattering is closely elastic. When this condition
is met, we can assume that there is no change in the photon energy in the rest frame of
the electron. The differential cross section for unpolarized radiation is shown in quantum
electrodynamics (Heitler, 1954) to be given by the Klein-Nishina formula
dσ
dΩ
=
r20
2
ϵ21
ϵ2
(
ϵ
ϵ1
+
ϵ1
ϵ
− sin2 θ
)
. (5.37)
In the non relativistic regime, we have approximately(x ≪ 1)
σ ≈ σT
(
1− 2x+
26x2
5
+ ...
)
, (5.38)
for the extreme relativistic regime we have (x ≫ 1)
σ =
3
8
σTx
−1
(
ln 2x+
1
2
)
, (5.39)
where x ≡ hν/mc2.
45
When the electron is not at rest, but has an energy greater that the typical photon
energy, there can be a transfer of energy from the electron to the photon. This process is
called inverse Compton. Let us call K the lab or observer’s frame, and let K’ be the rest
frame of the electron. Note that the previous expressions for scattering of electrons at rest
should now be primed. The scattering event as seen in each frame is given in figure 5.4
x
ϵ
ϵ1
θ
θ1
K
x′
ϵ′1
θ′1
ϵ′
θ′
K ′
Figure 5.4: Scattering geometries in the observer’s frame K and in the electron rest frame
K ′.
So we obtain the incident and scattered photon energy
ϵ′ = ϵγ(1− β cos θ), (5.40)
ϵ1 = ϵ′1γ(1 + β cos θ′1). (5.41)
We also know
ϵ′1 ≈ ϵ′
(
1− ϵ′
mc2
(1− cosΘ)
)
, (5.42)
cosΘ = cos θ′1 cos θ
′ + sin θ′ sin θ′1 cos(ϕ
′ − ϕ′
1), (5.43)
where ϕ′
1 and ϕ′are the azimuthal angles of the scattered photon and incident photon in
the rest frame. In the case of relativistic electrons, γ2 − 1 ≫ hν/mc2, the energies of the
photon before scattering, in the rest frame of the electron, and after scattering are in the
approximate ratios
1 : γ : γ2 (5.44)
providing that the condition for Thomson scattering in the rest frame γϵ ≪ mc2 is met.
This process, therefore, turns a low-energy photon to a high-energy one by a factor of order
46
γ2. Since the intermediate photon energy can be as high, perhaps up to 100 keV, and still
be in the Thomson limit, it can be seen that photons of high energies (γ × 100keV ) can be
produced. From the conservation of energy, we can write ϵ′1 < γmc2+ ϵ. Fixing ϵ and letting
γ become large, we see that photon energies larger than ∼ γmc2 cannot be obtained. So, we
found the energy transmitted from an electron at rest and from electrons in motion.
5.2.2 IC Power Emitted
Now, we want to find the formulas for the case of a given isotropic distribution of photons
scattering off a given isotropic distribution of electrons. Here, we have the photon phase
space distribution function (n(p)), which is a Lorentz invariant and vdϵ is the density of
photons having energy in range dϵ. Then v and n are related by
vdϵ = nd3p, (5.45)
in turn
vdϵ
ϵ
=
v′dϵ′
ϵ′
. (5.46)
The total power emitted (scattered) in the electron’s rest frame can be found from
dE ′
1
dt′
= cσT
∫
ϵ′1v
′dϵ′, (5.47)
where v′dϵ′ is the number density of incident photons. Now we will assume that the change
in energy of the photon in the rest frame is negligible compared to the energy change in the
lab frame, γ2 − 1 >> ϵ/mc2; thus we can equate ϵ′1 = ϵ′. Now,
dE1
dt
=
dE ′
1
dt
, (5.48)
by the invariance of emitted power. So, using Lorentz invariants we have
dE1
dt
= cσT
∫
ϵ′2
v′dϵ′
ϵ′
= cσT
∫
ϵ′2
vdϵ
ϵ
. (5.49)
Here we assumed that γϵ ≪ mc2, so that the Thomson cross section is applicable. Consid-
ering equation 5.40, we can say
dE1
dt
= cσTγ
∫
(1− β cos θ)ϵvdϵ, (5.50)
47
this in the K frame. Considering an isotropic distribution of photons we have
dE1
dt
= cσTγ
2
(
1 +
1
3
β2
)
Uph, (5.51)
where
Uph ≡
∫
ϵvdϵ, (5.52)
which is the initial photon energy density. The rate of decrease of the total initial photon
energy is
dE1
dt
= −cσT
∫
ϵvdϵ = −σT cUph. (5.53)
Therefore, the net power lost by the electron, and, thereby, converted into increased radiation
is
dErad
dt
= cσTUph
[
γ2
(
1 +
1
3
β2
)
− 1
]
. (5.54)
So we find that the power is
Pcomp =
dErad
dt
=
4
3
σT cγ
2β2Uph. (5.55)
When the energy transfer in the electron rest frame is not neglected, equation 5.55 becomes
Pcomp =
dErad
dt
=
4
3
σT cγ
2β2Uph
[
1− 63
10
γ⟨ϵ2⟩
mc2⟨ϵ⟩
]
, (5.56)
where ⟨ϵ2⟩ and ⟨ϵ⟩ are the mean values integrated over Uph. Note that equation 5.56 allows
energy to be either given or taken from the photons. From our discussion on synchrotron
radiation, we found that the power emitted by each electron is given by equation 5.14. Using
this, we find
Psynch
Pcomp
=
UB
Uph
, (5.57)
the radiation losses due to synchrotron emission and to inverse Compton effect are in the
same ratio as the magnetic field energy density and photon energy density. Note that this
result also remains true for arbitrary values of the electron’s velocity, not just for ultra
relativistic values. It does, however, depend on the validity of Thomson scattering in the
rest frame so that γϵ ≪ mc2.
48
Let N(γ)dγ be the number of electrons per unit volume with γ in the range γ to
γ + dγ. So then, the total power emitted is of the form of
Ptot =
∫
PcompN(γ)dγ, (5.58)
if we consider
Ne(γe) =





Cγ−P γmin < γ < γmax
0 otherwise,
(5.59)
then, with β ∼ 1 we have
Ptot =
4
3
σTUphC(3− p)−1
(
γ3−p
max − γ3−p
min
)
. (5.60)
If we, instead, find the total power from a thermal distribution of nonrelativistic electrons
of number density ne and let γ ≈ 1 and ⟨β2⟩ = 3kT/mc2 we have
Ptot =
4kT
mc2
cσTneUph, (5.61)
where 4kT
mc2
is the fractional photon energy gain per scattering when ϵ ≪ 4kT .
5.2.3 IC Spectrum
Now that we found the total power emitted, let’s discuss its spectrum, which here it depends
on both the incident spectrum and the energy distribution of the electrons. Though this is
true we can only determine the spectrum for the scattering of photons of a given energy ϵ0 off
electrons of a given energy γmc2, since the general spectrum can be found by averaging over
the actual distribution of photons and electrons. We will consider isotropic distributions for
both the electrons and photons, therefore, the scattered photons are then also isotropically
distributed, so we will just need to find their energy spectrum.
Here, we will consider that the scattering in the rest frame is isotropic we assume that
dσ′
dΩ
= 1
4π
σT = 2
3
r20, also it is convenient to refer to intensity, I, based on the photon number
49
rather than energy. The number of photons crossing area dA in time dt within solid angle
dΩ and energy range dϵ, IdAdtdΩdϵ. This intensity can be found from the monochromatic
specific intensity by dividing by the energy. Suppose that the isotropic incident photon field
is monoenergetic, I(ϵ) = F0δ(ϵ − ϵ0) where F0 is the number of photons per unit area, per
unit time per steradian. So, now we will determine the scattering of a beam of electrons of
density N and energy γmc2 traveling along the x axis (as shown in figure 5.4). The incident
intensity field in the rest frame K’ is
I ′(ϵ′, µ′) = F0
(
ϵ′
ϵ
)2
δ(ϵ− ϵ0). (5.62)
We find that the number of emitted photons per unit volume per unit steradian, for elastic
scattering (assuming ϵ the incident energy equals ϵ′1 the scattered photon energy), So it
follows
j′(ϵ′1) =





N ′σT ϵ′
1
F0
2ϵ2
0
γβ
ϵ0
γ(1+β)
< ϵ′1 <
ϵ0
γ(1−β)
0 otherwise,
(5.63)
The emission function in frame K can be found from Lorentz invariant jν
ν2
,
j(ϵ1, µ1) =













ϵ1
ϵ′
1
j′(ϵ′1),
NσT ϵ1F0
2ϵ2
0
γ2β
ϵ0
γ2(1+β)(1−βµ1)
< ϵ1 <
ϵ0
γ(1−β)(1−βµ1)
,
0 otherwise.
(5.64)
The above results hold for a beam of electrons. To obtain the results for an isotropic
distribution of electrons, we shall average over the angle between the electron and emitted
photon we obtain
j(ϵ1) =
NσTF0
4ϵ20γ
2β2













(1 + β) ϵ1
ϵ0
− (1− β) 1−β
1+β
< ϵ1
ϵ0
< 1
(1 + β)− ϵ1
ϵ0
(1− β) 1 < ϵ1
ϵ0
< 1+β
1−β
0 otherwise,
(5.65)
We can easily check that
∫ ∞
0
j(ϵ1)dϵ1 = NσTF0,
∫ ∞
0
j(ϵ1)(ϵ1 − ϵ0)dϵ1 = NσT
4
3
γ2β2ϵ0F0.
50
Since NσTFo is the rate of photon scattering per unit volume, per unit solid angle, the first
of these shows the conservation of number of photons upon scattering. The second shows
the average increase in photon energy per scattering.
The spectrum resulting from the scattering of an arbitrary initial spectrum with a
power law distribution (as in equation 5.59) of relativistic electrons can now be found, we
will use v(ϵ), the initial photon number density (from equation 5.45) which is related to the
isotropic intensity by v(ϵ) = 4πc−1I(ϵ). The total scattered power per volume per energy is
dE
dV dtdϵ1
= 4πϵ1j(ϵ1) =
3
4
cσTC
∫
dϵ
(ϵ1
ϵ
)
v(ϵ)
∫ γ2
γ1
dγγ−p−2f
(
ϵ
4γ2ϵ
)
. (5.66)
The spectral index that we find is
α =
p− 1
2
, (5.67)
just as the one found for synchrotron emission.
51
52
Acknowledgements
This research was financially supported by Programa de Apoyo a Proyectos de In-
vestigación e Innovación Tecnológica (PAPIIT) AG-100317, IN106521 and
IG101320 .
53
54
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