Time Dependent Leptonic and Lepto-Hadronic Modeling of Blazar Emission

A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University

In partial fulﬁllment of the requirements for the degree Doctor of Philosophy

Christopher S. Diltz April 2016

This dissertation titled Time Dependent Leptonic and Lepto-Hadronic Modeling of Blazar Emission

by CHRISTOPHER S. DILTZ

has been approved for the Department of Physics and Astronomy and the College of Arts and Sciences by

Markus Bottcher¨ Adjunct Professor of Physics and Astronomy

Robert Frank Dean, College of Arts and Sciences 3 Abstract

DILTZ, CHRISTOPHER S., Ph.D., April 2016, Physics Time Dependent Leptonic and Lepto-Hadronic Modeling of Blazar Emission (149 pp.) Director of Dissertation: Markus Bottcher¨ Active galactic nuclei (AGN) are known to exhibit multi-wavelength variability across the whole electromagnetic spectrum. In the context of blazars, the variability timescale can be as short as a few minutes. Correlated variability has been seen in different bands of the electromagnetic spectrum: from radio wavelengths to high energy γ-rays. This correlated variability in different wavelength bands can put constraints on the particle content, acceleration mechanisms and radiative properties of the relativistic jets that produce blazar emission. Two models are typically invoked to explain the origin of the broadband emission across the electromagnetic spectrum: Leptonic and Hadronic Modeling. Both models have had success in reproducing the broadband spectral energy distributions (SEDs) of blazar emission with different input parameters, making the origin of the emission difficult to determine. However, flaring events cause the spectral components that produce the SED to evolve on different timescales, producing different light curve behavior for both models. My Ph.D. research involves developing one-zone time dependent leptonic and lepto-hadronic codes to reproduce the broadband SEDs of blazars and then model flaring scenarios in order to find distinct differences between the two models. My lepto-hadronic code also considers the time dependent evolution of the radiation emitted by secondary particles (pions and muons) generated from photo-hadronic interactions between the photons and protons in the emission region. I present fits to the broadband SEDs of the flat spectrum radio quasars (FSRQs) 3C 273 and 3C 279 using my one-zone leptonic and lepto-hadronic model, respectively. I showed that by considering perturbations of any one of the selected input parameters for both models: magnetic field, particle injection 4 luminosity, particle spectral index, and stochastic acceleration time scale, distinct differences arise in the light curves for the optical, X-ray and γ-ray bandpasses that can separate leptonic and lepto-hadronic models. I find that decreasing the stochastic acceleration timescale for a one-zone leptonic model will result in a decrease in flux in the X-ray band as opposed to the lepto-hadronic model, in which an increase is seen. I also find that increasing the magnetic field produces a drop in the X-ray and γ-ray bands while for the lepto-hadronic model, an increase is observed in both bands. In the final part of my Ph.D. research, I use my leptonic and lepto-hadronic codes to reproduce the SED of the FSRQ 3C 454.3. The SED fits are then used to model a large multi-wavelength flare that 3C 454.3 exhibited in November 2010. I find that a combination of parameter changes to the magnetic field, particle injection luminosity, stochastic acceleration timescale and particle spectral index is needed to model the November flare. Using both codes to model the flare, I found that the lepto-hadronic model can reproduce both the broadband spectral energy distribution of 3C 454.3 in its quiescent and flaring states and can reproduce the integrated light curves in three bandpasses; optical R, Swift XRT and Fermi γ-rays. I also found that the fits to the SED of 3C 454.3 in its flaring state could not be reproduced by our one-zone leptonic model. 5

This thesis is dedicated to my mother, Victoria Diltz, who encouraged me to follow my dreams. 6 Acknowledgments

I would like to express my gratitude to my doctoral advisor, Dr. Markus Bottcher¨ for his guidance, assistance, patience and support. His extensive knowledge has helped me overcome many obstacles in completing this work. His discussion on many matters has provided resolutions to many aspects of my thesis work. I also owe my gratitude to Professors Joseph Shields, Douglas Clowe and Daniel Phillips who have indirectly assisted me with their extensive knowledge in their respective ﬁelds and helpful assistance in furthering my understanding of science and astronomy. I would like to thank the professors of my dissertation committee; Dr. Markus Bottcher,¨ Dr. Joseph Shields, Dr. Douglas Clowe, Dr. Daniel Phillips and Dr. Tatiana Savin for taking time oﬀ their busy schedule to read and evaluate my work and to make the defense of this thesis possible. During my research, I have developed collaborations with many colleagues for whom I grant my sincerest thanks for their brilliant insight and invaluable assistance to my doctoral work. I would like to thank Dr Svetlana Jorstad and Dr Alan Marscher for providing the SED and light curve data for 3C 454.3. I would like to thank Dr. Matthew Baring for his assistance in providing references to papers useful for my work. I also extend my gratitude to Vaidehi Paliya for his fruitful collaboration on my work. I would also like to thank Dr. Daniel Phillips for his service as my local advisor at Ohio University. I express my apologies that I could not mention all others who supported me either directly or indirectly with this work. Finally, I give my loving thanks to my sisters: Teri Cooperrider, Mirisa Reed, Lindsay Medina, Alisha Applegate, my brother Jason Bousquet, my father Bruce Diltz for their endless support and encouragement. I also give my thanks to the friends I have met during my time at Ohio University who have also been a source of support and inspiration. Most of all, I give my thanks to my mother and grandmothers whom I’ve lost over the 7 course of my doctoral work. I would not have made it through to the end without their encouragement and tireless support. 8 Table of Contents

Page

Abstract...... 3

Dedication...... 5

Acknowledgments...... 6

List of Tables...... 10

List of Figures...... 11

List of Symbols...... 16

List of Acronyms...... 18

1 Introduction...... 21 1.1 Active Galactic Nuclei...... 21 1.2 Blazars...... 26 1.3 Multi-Wavelength Telescopes...... 30 1.3.1 Optical Telescopes...... 30 1.3.2 X-Ray Telescopes (Swift XRT and NuSTAR)...... 33 1.3.3 γ ray Telescopes (Fermi and IACTs)...... 34

2 Modeling Blazar Emission...... 38 2.1 General Considerations...... 38 2.2 Leptonic Modeling of Blazars...... 42 2.2.1 Synchrotron Emission...... 42 2.2.2 Compton Scattering...... 44 2.2.3 Synchrotron Self Compton...... 51 2.2.4 External Compton Scattering...... 53 2.3 Lepto-Hadronic Modeling of Blazars...... 58 2.3.1 Photo-Hadronic Interactions and Pion Production...... 60 2.3.2 Neutrino Production and Emission...... 67 2.4 γγ Pair Production in Blazar Jets...... 69

3 Modeling the Broadband Emission of 3C 273, 3C 279 and 3C 454.3...... 73 3.1 General Considerations...... 73 3.2 Modeling the SED of 3C 273...... 74 3.3 Modeling the SED of 3C 279...... 78 3.4 Modeling the SED of 3C 454.3...... 84 9

3.4.1 Leptonic Model Fits...... 86 3.4.2 Hadronic Model Fits...... 89

4 Light Curve Modeling of Blazar Emission...... 92 4.1 General Considerations...... 92 4.2 Light Curve Modeling of Leptonic Models (3C 273)...... 95 4.3 Light Curve Modeling of Lepto-Hadronic Models (3C 279)...... 101 4.4 Discrete Correlation Analysis...... 109 4.5 Modeling the Exceptional Nov. 2010 Flare of 3C 454.3...... 113 4.5.1 Leptonic Light Curve Modeling of 3C 454.3...... 114 4.5.2 Lepto-Hadronic Light Curve Modeling of 3C 454.3...... 118

5 Discussion...... 124

6 Outlook...... 129

Appendix: Fokker-Planck Equation and Discrete Correlation Analysis...... 140 10 List of Tables

Table Page

2.1 Laboratory-frame electron and muon neutrino distribution functions [15].... 68

3.1 Parameter values used for the equilibrium fit to the SED of 3C 273...... 77 3.2 Parameter values used for the equilibrium fit to the SED of 3C 279...... 81 3.3 Parameter values used for the leptonic and lepto-hadronic equilibrium fit to the SED of 3C 454.3...... 88

4.1 Best-fit parameters of the Gaussian fit to the discrete correlation functions between selected bands of light curves for the leptonic fit of 3C 273. Negative values for the normalization indicate an anti-correlation between bands. Negative values for the peak time indicate a lag between the first and second bands...... 111 4.2 Best-fit parameters of the Gaussian fit to the discrete correlation functions between selected bands of light curves for the lepto-hadronic fit of 3C 279. Negative values for the normalization indicate an anti-correlation between bands. Negative values for the peak time indicate a lag between the first and second bands...... 111 4.3 3C 454.3 model light curve fit parameters for the leptonic and lepto-hadronic models. The negative value for the perturbation of the particle spectral index indicates spectral hardening. Conversely, a positive value indicates a spectral softening...... 118 11 List of Figures

Figure Page

1.1 6 cm radio image of the FR II radio galaxy Cygnus A [84]...... 23 1.2 Artist’s conception of an AGN [105]...... 24 1.3 Uniﬁcation model of AGN [77]...... 26 1.4 SED of diﬀerent blazar types demonstrating the blazar sequence [42]...... 28 1.5 Image of the MDM observatory at Kitt Peak (credit: M. Bottcher).¨ ...... 31 1.6 Schematic of the Swift Telescope [78]...... 33 1.7 Layout of the Swift XRT [79]...... 34 1.8 Artist’s rendering of NuSTAR in orbit [80]...... 35 1.9 Artist’s rendering of the Fermi Space Telescope [81]...... 36 1.10 The VERITAS Gamma-Ray Telescope Array on Mt. Hopkins [108]...... 37

2.1 Sample output of synchrotron emission from a power law distribution (q = 2.5) 4 of electrons with cutoffs γmin = 10 and γmax = 10 for a magnetic field of B = 1 G and Doppler factor δ = 15 with and without SSA given by equation 2.2.1. The dashed green curve denotes the optically thick, low frequency part 5/2 of the synchrotron spectrum, S ν ∝ ν . The blue dashed curve denotes the 1/3 optically thin, low frequency part of the synchrotron spectrum, S ν ∝ ν . The magenta dashed curve is the power law spectral fit α = (q − 1)/2 for the synchrotron spectrum...... 45 2.2 Geometry of the Compton scattering event in the rest frame of the electron [29]. 46 2.3 Total Compton cross section as a function of the photon energy, 0 in the rest frame of the electron [29]...... 47 2.4 Geometry of a Compton scattering event in a.) the lab frame and b) the rest frame of the electron [29]...... 48 2.5 Definition of the angles and their corresponding cosines for the transformation between the laboratory and electron rest frames [29]...... 49 2.6 Sample output of SSC emission with Doppler factor δ = 15 from a power law 3 5 distribution (q = 3.6) of electrons with cutoffs γmin = 10 and γmax = 10 interacting with its own synchrotron radiation in an emission region with a magnetic field of B = 0.5 G moving with Doppler factor δ = 15...... 52 2.7 Geometry of the accretion disk/jet system with the formation of magnetic field lines in the accretion disk and black hole magnetosphere [36]...... 53 2.8 Geometry of the accretion disk relative to the rotation axis. Rmin denotes the innermost stable circular orbit [36]...... 56 12

2.9 Sample output of EC emission located at a distance along the jet of r = 0.12 pc produced from a power law distribution (q = 3.6) of electrons with cutoffs 3 5 γmin = 10 and γmax = 10 interacting with an accretion disk of luminosity 45 ∗ −5 LD ≈ 7.0 × 10 erg/s peaking at an energy of 0 ≈ 3.5 × 10 surrounding a 8 supermassive black hole of mass MBH ≈ 3.0 × 10 M . The red dashed curve denotes the EC spectral fit from scattering in the Thomson regime α = (q − 1)/2. 58 2.10 Sample output of EC emission produced from a power law distribution (q = 3 5 3.6) of electrons with cutoffs γmin = 10 and γmax = 10 interacting with a 44 isotropic radiation field of luminosity Lext ≈ 2.0 × 10 erg/s peaking at an ∗ −5 energy of 0 ≈ 2.0 × 10 ...... 59 2.11 The total pγ cross-section as a function of the photon’s energy in the rest frame −30 2 of the proton, r,[49] (1µbarn = 10 cm ). The contributions of different baryon resonances (red, dashed), the direct channel (green, dotted), and multi- pion production (brown) are shown individually...... 61 2.12 The energy distribution of the charged pion production rates generated from the interaction between a power law distribution of protons (q = 2.4) with cutoffs 8 γmin = 1 and γmax = 10 and a power law photon field (α = 1.7) with cutoffs −4 5 min = 10 and max = 10 ...... 66 2.13 Cross Section for γγ pair production as a function of the invariant interaction energy 12(1 − µ)[29]...... 71 2.14 Sample output of the γγ opacity from an isotropic power law distribution of 0 5 photons with normalization constant nph = 4.0 × 10 , spectral index (α = 1.8), −4 4 16 cutoffs min = 10 and max = 10 in an emission region of size R = 2.0×10 cm. 71

3.1 Image of the FSRQ 3C 273 and its jet from Chandra [82]...... 75 3.2 The fit between our one-zone leptonic model and the multi-wavelength data set (red data points) [3] for 3C 273. Green data points represent archival data. Black solid = total SED fit; Red dashed = electron/positron synchrotron; green dashed = SSC; blue dashed = accretion disk; magenta dashed = EC (disk); indigo dashed = EC (BLR)...... 78 3.3 Time series of the superluminal motion exhibited by 3C 279 from the VLBA [85]...... 79 3.4 The fit between our one-zone lepto-hadronic model and the multi-wavelength data set (red data points) [3] for 3C 279. Green data points represent archival data. Black solid = total SED fit; Red dashed = proton synchrotron; Green dashed = electron/positron synchrotron; blue dashed = accretion disk; magenta dashed = muon synchrotron; indigo dashed = pion synchrotron...... 83 3.5 All sky image of the galactic plane and the location of FSRQ 3C 454.3 from the Fermi Space Telescope [83]...... 85 13

3.6 The fit between our one-zone leptonic model and the multi-wavelength data set [3] for 3C 454.3. Black solid = total SED fit; Red dashed = electron/positron synchrotron; green dashed = SSC; blue dashed = accretion disk; magenta dashed = EC (disk); indigo dashed = EC (BLR)...... 87 3.7 The fit between our one-zone lepto-hadronic model and the multi-wavelength data set [3] for 3C 454.3. Black Solid = Total SED fit; Red dashed = proton synchrotron; Green dashed = electron/positron synchrotron; blue dashed = accretion disk; magneta dashed = muon synchrotron; indigo dashed = pion synchrotron...... 90

4.1 Electron energy distribution in the quiescent state and at the peak of the magnetic field perturbation in the leptonic model scenario...... 97 4.2 Normalized light curves for the magnetic field perturbation in the leptonic model scenario. Black dashed = Radio light curve; Red dashed = X-ray light curve; Green dashed = R Band light curve; Blue dashed = HE γ-ray light curve. 97 4.3 Normalized light curves for the electron injection luminosity perturbation in the leptonic model scenario. Black dashed = Radio light curve; Red dashed = X-ray light curve; Green dashed = R Band light curve; Blue dashed = HE γ-ray light curve...... 98 4.4 Electron energy distribution in the quiescent state and at the peak of the stochastic acceleration timescale perturbation in the leptonic model scenario.. 100 4.5 Normalized light curves for the stochastic acceleration timescale perturbation in the leptonic model scenario. Black dashed = Radio light curve; Red dashed = X-ray light curve; Green dashed = R Band light curve; Blue dashed = HE γ-ray light curve...... 101 4.6 Normalized light curves for the magnetic field perturbation in the lepto- hadronic scenario. Black dashed = R Band light curve; Red dashed = X-ray light curve; Green dashed = HE γ-ray light curve; Blue dashed = VHE γ-ray light curve; Magenta dashed = Electron neutrino light curve...... 103 4.7 Normalized light curves for the proton injection luminosity perturbation in the lepto-hadronic scenario. Black dashed = R Band light curve; Red dashed = X-ray light curve; Green dashed = HE γ-ray light curve; Blue dashed = VHE γ-ray light curve; Magenta dashed = Electron neutrino light curve...... 104 4.8 Proton energy distribution in the quiescent state and at the peak of the stochastic acceleration timescale perturbation in the lepto-hadronic scenario... 106 4.9 Normalized light curves for the stochastic acceleration time scale perturbation in the lepto-hadronic scenario. Black dashed = R Band light curve; Red dashed = X-ray light curve; Green dashed = HE γ-ray light curve; Blue dashed = VHE γ-ray light curve; Magenta dashed = Electron neutrino light curve...... 107 4.10 Proton energy distribution in the quiescent state and at the peak of the proton spectral index perturbation in the lepto-hadronic scenario...... 108 14

4.11 Normalized light curves for the proton spectral index perturbation in the lept- hadronic scenario. Black dashed = R Band light curve; Red dashed = X-ray light curve; Green dashed = HE γ-ray light curve; Blue dashed = VHE γ-ray light curve; Magenta dashed = Electron neutrino light curve...... 109 4.12 Broadband fit to the SED of 3C 454.3 using our leptonic model during the quiescent and flaring states. The quiescent-state data points included in the fit are plotted in red [3]; additional, archival data are plotted in grey. Broadband data during the Nov 2010 flare are plotted in cyan [113]. The model curves are: Black solid = total spectrum; Red dashed = synchrotron emission from electrons/positrons; Green dashed = synchrotron self Compton emission; Blue dashed = thermal emission from accretion disk; Magneta dashed = EC emission from accretion disk; Indigo dashed = EC emission from BLR..... 117 4.13 Broadband fit to the SED of 3C 454.3 using our lepto-hadronic model during the quiescent and flaring states. The model curves are: black solid = total spectrum; red dashed = proton synchrotron emission; green dashed = synchrotron emission from electrons/positron; blue dashed = thermal emission from accretion disk; magenta dashed = pion synchrotron; violet dashed = pion synchrotron...... 120 4.14 Light curve fits between the data [113], and our lepto-hadronic model in the Fermi bandpass (20 − 300) GeV...... 122 4.15 Light curve fit between data and our lepto-hadronic model in the Swift XRT bandpass (0.2 − 10) keV...... 123 4.16 Light curve fit between the data and the lepto-hadronic model in the R band... 123

A.1 Discrete correlation function between the optical R and X-ray bands for the magnetic field perturbation in the leptonic model scenario. The red curve represents the Gaussian fit to the DCF using equation 4.5...... 142 A.2 Discrete correlation function between the optical R and HE γ-ray bands for the magnetic field perturbation in the leptonic model scenario. The red curve represents the Gaussian fit to the DCF using equation 4.5...... 143 A.3 Discrete correlation function between the optical R and X-ray bands for the electron injection luminosity perturbation in the leptonic model scenario. The red curve represents the Gaussian fit to the DCF using equation 4.5...... 143 A.4 Discrete correlation function between the optical R and HE γ-ray bands for the electron injection luminosity perturbation in the leptonic model scenario. The red curve represents the Gaussian fit to the DCF using equation 4.5...... 144 A.5 Discrete correlation function between the optical R and X-ray bands for the stochastic acceleration timescale perturbation in the leptoic model scenario. The red curve represents the Gaussian fit to the DCF using equation 4.5..... 144 A.6 Discrete correlation function between the optical R and HE γ-ray bands for the stochastic acceleration timescale perturbation in the leptonic model scenario. The red curve represents the Gaussian fit to the DCF using equation 4.5..... 145 15

A.7 Discrete correlation function between the X-ray and HE γ-ray bands for the magnetic field perturbation in the lepto-hadronic model scenario. The red curve represents the Gaussian fit to the DCF using equation 4.5...... 145 A.8 Discrete correlation function between the HE and VHE γ-ray bands for the magnetic field perturbation in the lepto-hadronic model scenario. The red curve represents the Gaussian fit to the DCF using equation 4.5...... 146 A.9 Discrete correlation function between the X-ray and HE γ-ray bands for the proton injection luminosity perturbation in the lepto-hadronic model scenario. The red curve represents the Gaussian fit to the DCF using equation 4.5..... 146 A.10 Discrete correlation function between the HE and VHE γ-ray bands for the proton injection luminosity perturbation in the lept-hadronic model scenario. The red curve represents the Gaussian fit to the DCF using equation 4.5..... 147 A.11 Discrete correlation function between the X-ray and HE γ-ray bands for the stochastic acceleration timescale perturbation in the lepto-hadronic model scenario. The red curve represents the Gaussian fit to the DCF using equation 4.5...... 147 A.12 Discrete correlation function between the HE and VHE γ-ray bands for the stochastic acceleration timescale perturbation in the lepto-hadronic model scenario. The red curve represents the Gaussian fit to the DCF using equation 4.5...... 148 A.13 Discrete correlation function between the X-ray and HE γ-ray bands for the proton spectral index perturbation in the lepto-hadronic model scenario. The red curve represents the Gaussian fit to the DCF using equation 4.5...... 148 A.14 Discrete correlation function between the HE and VHE γ-ray bands for the proton spectral index perturbation in the lepto-hadronic model scenario. The red curve represents the Gaussian fit to the DCF using equation 4.5...... 149 16 List of Symbols

LD Accretion disk luminosity

θobs Angle between the observer and the jet axis

θ Angle between two photons kB Boltzmann constant c Speed of light

µ Cosine of interaction angle θ

2 Dimensionless photon energy (Eph/(mec )) e± Electron/Positron

µ± Muon/Anti-Muon

π± Pion/Anti-Pion

α Energy spectral index

M Mass of the SMBH m˙ Mass accretion rate of the SMBH

RBLR Extent of broad line region nph Photon density uBLR Energy density of the broad line region

ν Photon Frequency rg Larmor Radius

γ Particle Lorentz factor

β Speed of the particle normalized to c 17

γcm Lorentz factor in the center-of-momentum frame uB Energy density of the magnetic ﬁeld

B Magnetic ﬁeld

κ Opacity

τγγ Optical depth

2 cm Photon energy in the center-of-momentum frame/(mec )

σγγ Photon-Photon cross section

Rs Schwarzschild radius

T(R) Thermal temperature of accretion disk at radius R

TBLR Thermal temperature of the Broad Line Region R

σT Thomson cross section 18 List of Acronyms

AGN Active Galactic Nucleus

BAT Burst Area Telescope

BL Lac BL Lacertae

BBB Big Blue Bump

BH Black Hole

BLR Broad Line Region

BZ Blandford-Znajek Mechanism

CGRO Compton Gamma-Ray Obsrvatory

CN Crank-Nichelson

CMB Cosmic Microwave Background

DCF Discrete Correlation Function

EC External Compton

EBL Extra Galactic Background Light

EC External Compton

EGRET Energetic Gamma Ray Experiment Telescope

EHT Event Horizon Telescope

ERF Electron Rest Frame

FR I(II) Fanaroﬀ-Riley Type I(II)

FSRQ Flat Spectrum Radio Quasar

GBM GLAST Burst Monitor 19

GRB Gamma-Ray Burst

HBL High Frequency Peaked BL Lac

HE High Energy

HESS High Energy Stereoscopic System

HSP High Synchrotron Peaked Blazar

IACTs Imaging Atmospheric Cherenkov Telescopes

IBL Intermediate Frequency Peaked BL Lac

IC Inverse Compton

IR Infra Red

IRAF Image Reduction and Analysis Facility

ISP Intermediate Synchrotron Peaked Blazar

KN Klein Nishina

LAT Large Area Telescope

LSP Low Synchrotron Peaked Blazar

MAGIC Major Atmospheric Gamma-ray Imaging Cherenkov Telescope

MHD Magneto Hydrodynamics

Near-IR Near Infra Red

NLR Narrow Line Region

NuSTAR Nuclear Spectroscopic Telescope Array

SSA Synchrotron Self Absorption

SED Spectral Energy Distribution

SMBH Super Massive Black Hole 20

SSC Synchrotron Self Compton

UHECR Ultra High Energy Cosmic Ray

UV Ultra Violet

UVOT UV-Optical Telescope

VERITAS Very Energetic Radiation Imaging Telescope Array System

VHE Very High Energy

VLA Very Large Array

VLBA Very Large Baseline Array

WEBT Whole Earth Blazar Telescope

XMM X-Ray Multi-Mirror Mission

XRT X-Ray Telescope 21 1 Introduction

A galaxy is a gravitationally bound collection of stars, interstellar gas, dust and dark matter. It is widely accepted that galaxies harbor supermassive black holes (SMBH) in their centers. Matter falling into the gravitational well of the SMBH produces an accretion disk, which represents the source of power for an active galactic nucleus at its core.

1.1 Active Galactic Nuclei

An active galactic nucleus is a bright compact region at the center of a galaxy. The galaxy that harbors an active galactic nucleus is called an active galaxy. The region emits radiation over the whole of the electromagnetic spectrum, from radio wavelengths to high energy γ-rays, often outshining the broadband emission of the stars that make up the galaxy. Active galaxies are usually characterized by their high bolometric luminosities, broadband thermal and non-thermal emission, prominent broad and narrow emission lines. It is widely believed that every galaxy harbors a supermassive black hole in their centers, ranging in masses between 106 to 1010 times the mass of the sun. The black hole collects nearby stars, dust, gas and other matter and accretes it onto itself forming an accretion disk. The material falls into the gravitational well that makes up the black hole. Gravitational and frictional forces heat up the material as it spirals towards the center, with

−3/4 a characteristic thermal temperature of T(r) ∝ (r/Rs) , where T(r) is the disk

2GM temperature at radius r from the center and Rs = c2 represents the Schwarzschild radius of the supermassive black hole [90]. Accretion processes are extremely eﬃcient at mass energy conversion, in some instances converting 10 to 40 percent of the mass of material around the black hole to radiative energy. The spectrum of an accretion disk depends on the mass of the black hole. For a supermassive black hole, the electromagnetic radiation peaks in the optical-ultraviolet wavelength bandpass. The diﬀerent components of the AGN give rise to the observational features seen. These features include: 22

1) A super-massive black hole at the center, and release of gravitational potential energy of material accreted onto the black hole represents the main source of energy that the AGN emits. 2) An accretion disk that orbits around the black hole; its thermal spectrum peaks at optical- ultraviolet wavelengths (T ≈ 105 K). 3) The Broad Line Region (BLR): gas clouds around 0.1 parsecs from the black hole and disk, traveling with high speed around 104 km/s, with a line dominated spectrum with strong features at UV wavelengths. 4) The Narrow Line Region (NLR): gas clouds located much further out from the black hole traveling with low speeds around 102 km/s, with a line dominated spectrum with strong features at UV wavelengths. 5) A Dusty Torus: a torus of dust, extending parsecs outside the SMBH with a thermal spectrum, peaking at IR wavelengths (T ≈ 103 K). 6) Corona: hot gas in the vicinity of the inner edge of the accretion disk that upscatters soft disk photons up to X-ray energies and reflects emission off of the accretion disk. 7) Relativistic Jets: high velocity outflows along the spin axis of an AGN that emit broadband radio to optical/UV emission due to synchrotron radiation from electrons/positrons and X-ray to high energy γ-rays through inverse Compton scattering or hadronic processes. A fraction of active galactic nuclei possess relativistic jets that are produced along the axis of rotation of the accretion disk. The jets are composed of high energy particles, such as electrons, positrons, hadrons and photons that interact with magnetic fields and each other to produce the broadband emission across the electromagnetic spectrum. The jets can carry significant amounts of electromagnetic and radiative power and can extend hundreds of kiloparsecs into the intergalactic medium, (see Figure 1.1). The mechanism 23

Figure 1.1: 6 cm radio image of the FR II radio galaxy Cygnus A [84].

through which the jet is powered and launched is, however, not well understood. In 1977, Roger Blandford & Roman Znajek developed a model in which electromagnetic energy could be extracted from the rotational energy of Kerr black holes [19]. Their model showed that the curved space time around a rotating black hole acts as a medium through which an electromagnetic field could be generated. A net positive flux of electromagnetic energy is then extracted from the rotation, which thereby powers the jets [29]. Their model produced electromagnetic luminosities that were comparable to the jet luminosity of supermassive black holes. However, In 1982, Blandford and David Payne developed a new model in which electromagnetic energy could be extracted from the magnetic field anchored in accretion discs rather than the black hole [20]. Both models serve as viable sources through which electromagnetic energy could be extracted from black holes. Computational magnetohydrodynamics have demonstrated the Blandford-Znajek and Blandford-Payne mechanisms as viable modes through which relativistic jets could be powered [54, 55, 68]. Magnetohydrodynamical simulations have also been performed to gain insight into the jet outflows of Sgr A* at the center of the Milky Way [74] and the 24 large elliptical galaxy, M87 [75]. Observations of the base of the jets, however, can not distinguish which model can best explain how jets are formed because of the poor angular resolution of the current radio interferometers used. Large, sub millimeter interferometers, such as the Event Horizons Telescope (EHT), can probe jet outflows in compact regions with angular resolution on the order of few Schwarzchild radii, giving insight on the physics of accretion disks and the mechanisms through which relativistic jets are powered [93].

Figure 1.2: Artist’s conception of an AGN [105]. 25

AGNs can be divided into separate subclasses based on their radio emission. These subclasses are referred to as radio-quiet and radio-loud AGNs. Radio-quiet AGNs are galaxies that do not produce strong radio emission from the jets. Seyfert 1, Seyfert 2 galaxies and radio-quiet quasars are examples of radio-quiet AGN. Seyfert 1 show strong broad emission lines in their optical/UV spectra while Seyfert 2 galaxies show much narrower emission lines. Radio quiet quasars are typically more luminous than Seyfert 1 galaxies. Radio quiet quasars exhibit strong optical and X-ray continuum emission, along with broad and narrow emission lines. Radio-loud AGNs generate substantial radio emission from both the relativistic jet and the lobes that are produced in the intergalactic medium. Radio-loud AGNs include radio galaxies, radio-loud quasars and blazars. Radio galaxies can be separated into two classes known as Fanaroff-Riley I and II (FR I and FR II) galaxies [40]. The distinction depends on the morphology of the radio emission, with FR I galaxies being brighter towards the center while FR II galaxies being brighter near the edges, see Figure 1.1. The class distinction also shows a separation in luminosity, with FR II galaxies being more luminous than FR I galaxies [40]. Radio loud quasars exhibit the same features as radio quiet quasars with the addition of a jet component in the emission. Blazars are a class of AGN in which the relativistic jet is point directly towards the Earth. The unification model suggests that the distinguishing characteristics between the different classes of radio quiet and radio loud AGNs is due to the angle of the observer relative to the axis of rotation of the accretion disk. The type of AGN can be determined based on the angle between the observer and the jet axis. (see Figure 1.3)[90]. In the unification model, Seyfert 1 galaxies are radio quiet galaxies viewed at a shallow angle between the line of sight of the observer and the rotational axis of the galaxy. Observers have a direct view of the active nucleus in the center of an AGN. The accretion disk, broad and narrow emission lines, and soft X-ray emission from a hot corona are visible [105]. 26

Figure 1.3: Uniﬁcation model of AGN [77].

For Seyfert 2 galaxies, the viewing angle relative to the AGN rotation axis is larger, causing the active nucleus to be more obscured by a dusty torus, which prevents the optical, X-ray and broad emission lines from being observed. The narrow emission lines, which are much further out from the active nucleus, can still be seen. For radio-loud uniﬁcation, radio loud AGNs viewed with a shallow angle are seen as radio loud quasars while radio loud AGNs that are viewed edge on appear as radio galaxies. Radio loud AGNs that are viewed directly down the axis of the jet are referred to as blazars.

1.2 Blazars

Within the category of AGNs, blazars are one of the most luminous and variable objects seen in the observable universe. Blazars are AGNs that have their relativistic jets pointed directly towards the earth. As a result, the beamed emission is strongly Doppler enhanced and extremely variable across all bands of the electromagnetic spectrum. 27

Blazars are also strong sources of high energy γ-rays, seen in energies in excess of 100 MeV or more. Blazars also exhibit strong polarization from radio to optical/UV wavelengths, in some cases as high as 20 − 30 percent. The spectral energy distribution of blazars consists of two broadband peaks. The first, which extends from radio wavelengths to soft X-rays, is generally accepted to be due to synchrotron emission from high energy electrons/positrons moving at relativistic speeds in a magnetic field [94]. The second peak extends from soft/intermediate X-rays to high energy γ-rays. The physical nature of the second peak is less understood. Two models have been proposed to explain the origin of the second peak; these are referred to as leptonic and hadronic models. In the framework of leptonic models, the second peak is due to inverse Compton scattering of the seed photon fields by the electrons/positrons within the emission region. The low-energy target photon fields can be the synchrotron photons within the emission region (SSC = synchrotron self Compton) [50], or external photons (EC = external Compton), which can include the accretion disk [33, 34], the broad line region [BLR; 21, 98], or infra-red emitting, warm dust [18]. One-zone leptonic models have had great success in reproducing the spectral energy distributions of blazars [30, 42] and in investigating their variability properties in different wavelength bands [26, 44, 58]. In the framework of hadronic models, the second broadband peak is due to proton synchrotron emission. The relativistic protons interact with the radiation fields within the emission region, producing high energy pions, which then decay to produce muons, electrons, positrons, and neutrinos in a high energy cascade. One zone lepto-hadronic models have also had success in modeling the emission of different blazars. [e.g., 30, 59, 60, 64, 65]. Blazars can be divided into two categories, based on the appearance of broad emission lines in their optical-UV spectra. Blazars that exhibit strong emission lines are classified as flat spectrum radio quasars (FSRQs) [63]. Blazars that display weak or no emission lines are called BL Lacs [62]. BL Lacs can be subdivided further into three 28 categories, based on the location of the peak energy of their synchrotron emission in their SED: 1- LSP: low synchrotron peaked blazars; the synchrotron peak energy is at low

s 14 energies, the far IR or IR band, with νpeak . 10 Hz. 2- ISP: intermediate synchrotron peaked blazars; the synchrotron peak energy is at

14 s 15 intermediate energies, with 10 Hz. νpeak . 10 Hz. 3- HSP: high synchrotron peaked blazars; the synchrotron peak energy is at high

s 15 energies, with νpeak & 10 Hz [87]. Figure 1.4 illustrates the SEDs of diﬀerent types of blazars based on the location of their synchrotron peak frequencies.

Figure 1.4: SED of diﬀerent blazar types demonstrating the blazar sequence [42]. 29

Most FSRQs belong to the LSP category of blazars and display synchrotron peaks

s 14 less than νpeak . 10 Hz. Most BL Lac objects, in contrast, have synchrotron peak s 14 s 17 frequencies above νpeak > 10 Hz and can be as high as νpeak ≈ 10 Hz. LSPs tend to exhibit more power in the γ-ray bandpass in comparison to HSPs. In the context of the leptonic model, the high γ-ray power is due to the presence of external radiation fields around the emission region, which leads to strong external Compton emission that produces the second peak in the SED. The increased emission from external photon sources causes stronger radiative cooling on the electrons/positrons, causing the peak of the synchrotron emission to be located at lower energies [42]. For HSPs, the contribution of external radiation fields is considerably smaller. This causes the γ-ray emission to be reduced and weaker radiative cooling on the electrons/positrons. The peak of the synchrotron spectrum is then located at much higher energies. In the unification model, FR II galaxies are the parent radio galaxies for FSRQs while FR I galaxies are the parent radio galaxies of BL Lacs. The stronger luminosities exhibited by FR II galaxies would then correlate with the stronger γ-ray luminosities seen in FSRQs. Conversely, the weaker luminosity seen in FR I galaxies would then correlate with the weaker γ-ray luminosity of a BL Lac. The observed correlation between the location of the synchrotron peak and the total bolometric luminosity of a blazar is referred to as the blazar sequence. Leptonic and lepto-hadronic models have had success in explaining and reproducing many features in the broadband SEDs of blazars with different sets of input parameters, making it difficult to determine the dominant energy carriers in the jets. Determining which sets of particles, electron/positrons or protons carry the energy in the jet can put tighter constraints on how the jets are powered. Determining the composition of the jets can also give clues as to whether blazars are a source of ultra high energy cosmic rays (UHCERs). The composition of the jets can be found through time dependent simulations of one-zone leptonic and lepto-hadronic models. Different broadband spectral components 30 make up the SEDs for both models. Simulating flaring scenarios for both models causes the spectral components to change in unique ways and evolve different timescales. The focus of my Ph.D. work is to simulate a set of flaring scenarios for both the one-zone leptonic and lepto-hadronic models in order to find distinct differences between them. I apply these time dependent models to a particular flare of a blazar in order to determine whether the leptonic or lepto-hadronic model can best explain all features, both the SED and the light curves of the blazar.

1.3 Multi-Wavelength Telescopes

Blazars produce broadband emission over the whole electromagnetic spectrum. Dedicated multi-wavelength campaigns are essential for obtaining simultaneous observations to piece together the broadband spectral energy distributions of blazars. Ground-based and space-based telescopes that collect light from radio waves to high energy γ-rays are utilized in these multi-wavelength campaigns.

1.3.1 Optical Telescopes

Ground-based and spaced-based optical observations serve as an important tool in understanding emission from blazars. Ground-based telescopes, such as MDM, have provided important observations of broadband optical spectra. The MDM Observatory is an optical astronomical Observatory located at the Kitt Peak National observatory, west of Tucson, Arizona. The MDM consists of two reflecting telescopes, the 2.4 m Hiltner Telescope, and the 1.3 m McGraw-Hill Telescope. It is owned and operated by the University of Michigan, Dartmouth College, Ohio State University, Columbia University, and Ohio University. I participated in two observing runs at the MDM observatory using the 1.3 m telescope. I monitored the variability of several different blazars in five different optical bands, ultraviolet (U), blue (B), visual (V) red (R), and the near infra red (I). The blazars monitored include 3C 66A, AO 0235+164, BL Lacertae, and AO 0235+16. Flat 31

ﬁelds and bias frames were collected in the evening along with the raw images of our target blazars. The images collected at the observatory were reduced using the software packages from the Image Reduction and Analysis Facility (IRAF) 1. The objective of my observations at the MDM observatory was to identify features related to the short term spectral variability and ﬂux exhibited by blazars. The variability behavior seen in the optical band of blazars reveals clues to the evolution of the underlying electron/positron population. Understanding the evolution of the electron population gives clues to the acceleration and cooling processes taking place in the emission region.

Figure 1.5: Image of the MDM observatory at Kitt Peak (credit: M. Bottcher).¨

The observing runs done at the 1.3 m telescope gave me valuable experience in working with a research telescope, the process of collecting images, reducing the images and collecting photometry using the IRAF software packages. A majority of the observations that I participated in at the MDM Observatory were part of much larger collaborative projects with other optical telescopes around the world, known as the Whole Earth Blazar Telescope (WEBT). The optical observations were also coordinated with

1 IRAF is distributed by the National Optical Astronomy Observatories, which are operated by Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. 32 observations in different wavelength bands, such as X-ray observations from the X-ray Multi-Mirror Mission (XMM) and Chandra and VHE γ-ray observations from the Very Energetic Radiation Imaging Telescope Array System (VERITAS) in Arizona. Participating in these observing runs has led to my co-authorship on 6 publications [4–6,8, 91, 111]. Space-based multi-wavelength telescopes, such as Swift, are also valuable tools for understanding blazars. Since its launch in Nov 2004, Swift has explored many different properties regarding gamma-ray bursts (GRBs), including their radiative properties, their origin, and their role in answering questions on the evolution of the early universe. Swift possesses three different instruments for the study of GRBs: the Burst Alert Telescope (BAT), the X-ray Telescope (XRT) and the Ultraviolet/Optical Telescope (UVOT). The BAT detects a given GRB event and determines its coordinates in the sky. The BAT covers a large fraction of the sky (2 sr) in the energy range of 15 keV ≤ E ≤ 150 keV and is able to locate the position of a given event with an accuracy of 1 to 4 arc-minutes in 10 to 15 seconds. The position is relayed to the ground where ground based telescopes can catch the GRB with the information provided by BAT. Immediately following a GRB event detection by the BAT, the spacecraft slews to the position of the source and points both the UVOT and the XRT at the GRB location. The spacecraft’s rapid targeting capabilities (20-70 secs) allows about 100 GRBs to be detected each year by Swift. In between GRB observations, Swift can be used for multi-wavelength observations of other objects in space, such as: AGNs, supernova remnants and flaring activity in the Galactic center. The UVOT is a diffraction-limited 30 cm (12” aperture) Ritchey-Chretien reflector, sensitive to magnitude 24 in a 17 minute exposure, which provides optical coverage (170-650 nm) in a 17’ x 17’ field of view. 33

Figure 1.6: Schematic of the Swift Telescope [78].

1.3.2 X-Ray Telescopes (Swift XRT and NuSTAR)

The other main instrument on the Swift satellite is the XRT. The XRT is designed to monitor the X-ray afterglows after the BAT has detected and located the position of the GRB in the sky with a 110 cm2 eﬀective area at 1.5 keV and a 23.6 x 23.6 arcmin ﬁeld of view. With its rapid acquisition of the target GRB, (20-70 s), the XRT is able to obtain positional accuracy of the source to within 5 arcsec. The Swift XRT is a class 1 Wolter X-ray telescope that operates in an energy range of 0.2 keV ≤ E ≤ 10 keV. With its grazing optics, it is able to obtain an angular resolution of 18 arsec. The NuSTAR (Nuclear Spectroscopic Telescope Array) is a space based telescope that focuses high energy X-rays from astrophysical sources, operating in an energy range of 3 keV ≤ E ≤ 79 keV. NuSTAR utilizes grazing incidence optics to focus high energy X-rays. For this, NuSTAR uses a class 1 Wolter X-ray telescope with a 10.15 meter focal length that are held at the end of a long deployable mast. A built in laser system is used to determine the locations of the optics and the focal plane at all times. This ensures that if 34

Figure 1.7: Layout of the Swift XRT [79].

the optics and the focal plane move relative to each other during a given exposure, each photon detected can be mapped back to the correct location on the sky. NuSTAR is able to achieve an angular resolution of 9.5 arcsec and an eﬀective area of 850 cm2 at 9 keV and 60 cm2 at 78 keV. Since the launch of NuSTAR, new discoveries are appearing in many diﬀerent areas of astrophysical research. NuSTAR has proven vital to measuring the spin parameter of black holes in galactic centers to tracing the decay of 44Ti in the remnants of supernovae. The latter has given astronomers new clues on how large stars are able to shed their outer layers when a supernova explodes.

1.3.3 γ ray Telescopes (Fermi and IACTs)

The Fermi γ-ray Space Telescope has given astronomers an unprecedented view of the high energy sky since its launch into orbit on June 11, 2008. The telescope carries two instruments: the Large Area Telescope (LAT) and the GLAST Burst Monitor (GBM). Its primary instrument, the LAT, detects gamma rays with energies between 20 MeV to 300 35

Figure 1.8: Artist’s rendering of NuSTAR in orbit [80].

GeV and an angular resolution of < 0.1 degree at 1 GeV. It has a field of view of almost 20 percent of the sky and an effective area similar to its predecessor, the Energetic Gamma Ray Experiment Telescope (EGRET) aboard the Compton Gamma-Ray Observatory (CGRO) during its operational run from 1991 to 2000. The second instrument, the GBM, detects gamma-rays with energies between 8 keV to 30 MeV in order to detect and observe Gamma-ray bursts (GRB) and their subsequent afterglows. Fermi has a nominal five year lifetime with prospects of 10 years worth of operation. Since the seven years after its launch, Fermi has detected large γ-ray bubbles coming from the center of our galaxy, shown that supernova remnants can accelerate cosmic particles to extremely high energies and detected one of the brightest GRBs ever recorded, GRB 130427A. The Fermi mission represents an international collaboration to observe the γ-ray sky with high quality resolution with contributions from the United States, France, Spain, Iceland, Italy, Japan, Germany and Sweden. For VHE γ-rays, astronomers use Imaging Atmospheric Cherenkov Telescopes (IACT). IACTs work by detecting and imaging the flashes of Chernkov radiation 36

Figure 1.9: Artist’s rendering of the Fermi Space Telescope [81]

produced by cascades of relativistic particles that are created when VHE γ-rays interact with the upper atmosphere. An incoming γ-ray produces high energy electron/positron pairs from its interaction with an atmospheric particle. The high energy electron/positron pairs undergo Bremsstrahlung radiation that generates more high energy photons. The secondary photons continue to initiate pair-production, generating a shower of charged particles that produce short (∼ 10 − 20 ns) bursts of light. The shower spreads out laterally as they approach the surface of the Earth, producing a large effective area for IACTs. IACTs, such as the High Energy Stereoscopic System (HESS), the Very Energetic Radiation Imaging Telescope Array System (VERITAS), and the Major Atmospheric Gamma Imaging Cherenkov Telescopes (MAGIC) use large segmented mirrors, separated up to 120 meters apart, to detect the faint Chernknov light. The mirrors reflect the light onto photomultiplier tubes that reconstruct the image of the shower. IACTs, such as VERITAS, HESS and MAGIC have detected supernova remnants, blazars, and pulsars 37 from GeV to TeV γ-ray energies. IACTs require large collaborative efforts. VERITAS, for example, is currently run by a collaboration of scientists from different institutions, such as: NASA and the Harvard Smithsonian Center for Astrophysics.

Figure 1.10: The VERITAS Gamma-Ray Telescope Array on Mt. Hopkins [108]. 38 2 Modeling Blazar Emission

The results of this chapter and parts of the text have been published in the Journal of High Energy Astrophysics and the Astrophysical Journal: Diltz, C., & Bottcher,¨ M., 2014, Journal of High Energy Astrophysics, 1, 63D: Time dependent leptonic modeling of Fermi II processes in the jets of ﬂat spectrum radio quasars. Diltz, C., Bottcher,¨ M., & Fossati, G., 2015, The Astrophysical Journal, 802, 133D: Time Dependent Hadronic Modeling of Flat Spectrum Radio Quasars

2.1 General Considerations

In this chapter, we describe the general features for one-zone models used to study the broadband emission from blazars. In the sections that follow, we go into the specific details for one-zone leptonic and lepto-hadronic models and how we compute the individual spectral components. In modeling the broadband emission from blazars, there are different frames that are utilized to compute the different spectral components that make up the SED for the observer. Throughout this study, we consider 4 different frames of reference: we denote quantities in the comoving frame with no primes, quantities in the AGN frame with a star ∗, quantities in the observer’s frame with a subscript/superscript ”obs”, and quantities in the electron rest frame (ERF) with primes, see section 2.2.2. For both models, we consider a homogeneous, one zone model, where a power law distribution of ultra-relativistic particles (electrons/positrons, hadrons),

−q Q(γ, t) = Q0γ H(γ; γmin, γmax) is continuously injected into a spherical region of size R, moving along the jet with a bulk Lorentz factor Γ, embedded in a homogeneous, randomly oriented magnetic ﬁeld of strength B. Here, H(x; a, b) is the Heaviside function deﬁned as H = 1 if a ≤ x ≤ b and H = 0 otherwise. The normalization factor for the injection spectrum is determined through the particle injection luminosity, Lin j, through the relation: 39

  Lin j 2−q  2 2−q 2−q if q , 2,  Vbmic −  γmax γmin Q0 =  (2.1)  Lin j  2 γmax if q = 2.  Vbmic ln( ) γmin

where Vb denotes the comoving volume of the emission region and mi denotes the rest mass of the injected particle. The size of the emission region is constrained through the

observed variability time scale, ∆tvar, using the relation R ≤ δc∆tvar/(1 + z), where z denotes the redshift to the source. It moves relativistically with Lorentz factor Γ, resulting in Doppler boosting characterized by the Doppler factor δ = 1/Γ(1 − βΓcosθ). The emission is Doppler boosted into a viewing angle of size θobs ≈ 1/Γ, enhancing the bolometric luminosity by a factor of δ4 and reducing the variability time scale in the comoving frame by a factor of δ−1. With the bulk Lorentz factor, the location of the

2 emission zone can be constrained through the relation Raxis = 2Γ c∆tvar/(1 + z)[36]. Both models use second order Fokker-Planck equations to track the time evolution of the particle distributions [37, 38]. We assume that the pitch angle scattering timescale is much smaller than the dynamic timescales acting on the particle distributions. As a result, we treat the particle distributions to be isotropic in the co-moving frame. For an arbitrary particle distribution, (i = e, p, π±, µ±), the Fokker-Planck equation is given by the relation:

2 ∂ni(γ, t) ∂ γ ∂ni(γ, t) ∂ ni(γ, t) ni(γ, t) = [ ] − (˙γ · ni(γ, t)) + Qi(γ, t) − − (2.2) ∂t ∂γ (a + 2)tacc ∂γ ∂γ tesc γtdecay

2 2 where tacc denotes the stochastic acceleration timescale, a = vs/vA gives the square of the ratio between the shock and the Alfven wave velocities, tesc denotes the dynamical escape time scale for the particles which we parametrize as a multiple of the light crossing time: tesc = ηR/c where η ≥ 1. The value of η is kept as a free parameter. The termγ ˙ denotes the combined loss rate for the particle distribution. Particles, such as charged pions and muons decay over a given decay time scale. The final term in the Fokker-Planck equation 40 represents the loss of the particles due to decay in the laboratory frame of the emission region. We utilize the implicit Cranck-Nichelson (CN) scheme to numerically solve the second order Fokker-Planck equation, see Appendix. Given the unconditional stability of the CN scheme, we can set the time step to any size to ensure quick convergence to the equilibrium solution to the Fokker-Planck equation. This quick convergence to the equilibrium solution is motivated to reproduce the spectral components that make up the SEDs of blazars. We set the size of the time step in our code initially to ≈ 106−7 s. This time step size is considerably longer than the time scales for all loss terms, acceleration terms and escape terms in all particle Fokker-Planck equations. However, choosing an arbitrarily large time step allows quick convergence to the equilibrium solution but fails to probe evolutionary processes of the particle spectra that can occur on smaller timescales; such as radiative, adiabatic cooling and acceleration processes. Different photon fields contribute to the overall broadband emission for both models. The combined photon field includes synchrotron emission from different particle species, synchrotron-self-Compton, external-Compton radiation of the electrons/positrons (Compton emission from the heavier particle species is strongly suppressed due to their much higher masses) as well as the radiation fields produced by the decay of neutral pions. We then solve a separate evolution equation for the combined photon field, nph(ν, t):

∂nph(ν, t) 4π X 1 1 = · j (ν, t) − n (ν, t) · ( + ) (2.3) ∂t hν k ph t t k esc,ph abs

where jk(ν, t) denotes the emission coeﬃcient for a given spectral component, the sum, k,

is over the all radiation components and all particle species, tesc,ph = 4R/3c denotes the

photon escape time scale and tabs denotes the absorption time scale due to synchrotron self absorption (SSA) by electrons/positrons and γγ absorption, see section 2.2.1 and 2.4. The absorption time scale can be deﬁned through the opacity as 41

R tabs = (2.4) c · (τSSA + τγγ) where τSSA and τγγ denote the synchrotron-self-absorption and γγ absorption opacities. The diﬀerential equations that track the time evolution of the photon ﬁelds is solved simultaneously with the set of Fokker-Planck equations until equilibrium is reached. The observed spectral components of the SED are evaluated through the relation:

2 4 obs obs obs obs h · ν · nph(ν, t) · δ · Vb ν Fν (ν , t ) = 2 (2.5) 4πdL · tesc,ph where the frequency and time in the comoving and observer’s frame are connected

obs obs through the relations: ν = δν and ∆t = ∆t/δ, and dL represents the luminosity distance to the source. In both models, the particles interact with magnetohydrodynamic Alfven waves in the plasma. If the Doppler-shifted wave frequency is a constant multiple of the particle gyrofrequency in the particle guiding center frame, then a resonant interaction between the particle and the transverse component of the electric field of the MHD wave will occur [35, 36]. The particle will experience either an accelerating or decelerating electric field in the transverse direction of motion over a fraction of the cyclotron period, resulting in an increase or decrease in energy. The accelerating or decelerating electric field causes the particle distributions to diffuse in energy, pushing particles to higher or lower energies in a diffusion pattern. This energy diffusion typically causes the particle distributions to have a pronounced curvature in the energy spectrum, resembling a log parabola [96]. Second order Fermi acceleration is therefore a viable mechanism for producing log-parabola particle spectra which may be hard enough to reproduce the hard spectra of γ-ray emission observed in several TeV blazars [12, 57]. The strength of the particle diffusion depends on the spectral index of the MHD turbulence, q, which defines how turbulence cascades onto smaller length scales. A Kolomogorov, (p = 5/3), or a Kraichnan, 42

(p = 3/2), spectrum are most often used to model MHD turbulence. In this study, we restrict the spectral turbulence to p = 2 to simulate hard sphere scattering between the MHD waves and the particle spectra.

2.2 Leptonic Modeling of Blazars

In the leptonic scenario, electrons/positrons are responsible for the broadband emission, from radio to high energy γ-rays. The radio to UV/X-ray is due to synchrotron emission from electrons. The high-energy (X-ray – γ-ray) emission is due to inverse Compton scattering of low-energy photons off the same electrons/positrons. The seed photon fields can include the synchrotron emission that the electrons/positrons emit or photons from external sources, such as the accretion disk surrounding the SMBH, the broad line region or an infrared dusty torus. In this section, we describe a one zone time-dependent leptonic model that incorporates Fermi acceleration and self-consistent radiative losses, including synchrotron and Compton scattering on internal (SSC) and external (EC) radiation fields as well as synchrotron self-absorption and γγ absorption and pair production.

2.2.1 Synchrotron Emission

We consider a continuous injection of relativistic electrons into the emission region with a randomly tangled magnetic field of strength B. With the distribution of electrons/positrons, we compute the synchrotron emission coefficient based on the energy density of the magnetic field using the relation:

1 Z ∞ jsyn(ν, t) = dγne(γ, t) · P(ν, γ) (2.6) 4π 1 where the term P(ν, γ) denotes the spectral synchrotron power of a given electron/positron. The synchrotron power is approximated through the relation [29]: 43

πc m ν1/3 32 2 e 2 2 −ν/νc P(ν, γ) = re ( ) uB γ 4/3 e (2.7) 9Γ(4/3) mi νc 2 where i denotes the particle emitting the synchrotron radiation, uB = B /8π denotes the energy density of the magnetic ﬁeld and re denotes the classical electron radius. The critical frequency for the synchrotron spectrum is given by the relation:

obs 6 2 νsyn = 4.2 × 10 B(G)(me/mi)δ γ Hz. For illustrative purposes, we can approximate the power per unit frequency of a single electron due to synchrotron emission by the Dirac delta function, see [29] for a detailed review:

32πc 2 me 2 2 P(ν, γ) = re ( ) uB γ δ(ν − νc) (2.8) 9 mi Placing equation 2.8 into equation 2.6 then yields the emission coeﬃcient:

r 4c 2 me 2 −3/2 1/2 ν jδ(ν) = re ( ) uB ν0 ν ne( ) (2.9) 9 mi ν0

2 where ν0 is deﬁned through νc = ν0γ . We see that a power law distribution of electrons with a spectral index of q, we recover a power law distribution of photons with spectral

index αsy = (q − 1)/2, see Figure 2.1. As the electrons emit synchrotron radiation, they lose energy in the process. For an isotropic distribution of particles emitting synchrotron radiation, the loss rate is given by the equation:

4 u m γ − c σ B e 3 γ2 ˙ syn = T 2 ( ) (2.10) 3 mec mi

where σT denotes the Thomson cross section of the particle emitting the radiation. Synchrotron emission can also be absorbed by the same electrons that are emitting the photons. Under the assumption that the energy of the photon absorbed is much smaller than the energy of the interacting electron, the SSA coeﬃcient is determined using the 44 relation: 1 Z ∞ ∂ n (γ, t) α ν, t − dγ P ν, γ γ2 e ( ) = 2 ( ) ( 2 ) (2.11) 8πmeν 1 ∂γ γ For a power law distribution of electrons, equation simpliﬁes to the expression:

(q + 2) Z ∞ n (γ) αPL ν − dγ P ν, γ e ( ) = 2 ( ) (2.12) 8πmeν 1 γ To illustrate the functional dependence of the photon frequency and magnetic ﬁeld on the SSA absorption coeﬃcient, we use equation 2.8 for the delta approximation of the power per unit frequency of a radiating electron to obtain [29]:

(q + 2) 3e q−2 q+2 q+4 2 2 2 − 2 αδ(ν) = ( ) re B ν (2.13) 18πme 4πmec − q+4 The dependence of the absorption coeﬃcient on the frequency α(ν) ∝ ν 2 shows that for lower frequencies, the absorption increases substantially. The absorption coeﬃcient

(q+2) dependence on the magnetic ﬁeld α(ν) ∝ B 2 also shows that for large magnetic ﬁelds, the SSA opacity also increases substantially. Figure 2.1 gives the synchrotron spectrum

from a power law distribution of electrons with index q = 2.5 and cutoﬀs γmin = 10 and

4 γmax = 10 in a magnetic ﬁeld of B = 1 G with and without SSA correction. The dependence of the SSA opacity on frequency shows that synchrotron emission from a distribution of electrons becomes optically thick at lower frequencies, in accordance with equation 2.13.

2.2.2 Compton Scattering

In the leptonic modeling scenario, the second broadband spectral component that extends from X-rays to high energy γ-rays is believed to be due to inverse Compton scattering of seed photon ﬁelds within the emission region with relativistic electrons. Compton scattering represents a critical element to explaining the high energy emission from blazars. 45 Syn. Emission w/wo SSA Absorption:

15 10

Syn. Emission with SSA Syn. Emission without SSA

14 10

[Jy Hz] 10 ν F ν

12 10

11 10 8 9 10 11 12 13 14 15 16 17 18 10 10 10 10 10 10 10 10 10 10 10 ν [Hz] Figure 2.1: Sample output of synchrotron emission from a power law distribution (q = 2.5) 4 of electrons with cutoffs γmin = 10 and γmax = 10 for a magnetic field of B = 1 G and Doppler factor δ = 15 with and without SSA given by equation 2.2.1. The dashed green curve denotes the optically thick, low frequency part of the synchrotron spectrum, 5/2 S ν ∝ ν . The blue dashed curve denotes the optically thin, low frequency part of the 1/3 synchrotron spectrum, S ν ∝ ν . The magenta dashed curve is the power law spectral fit α = (q − 1)/2 for the synchrotron spectrum.

We begin by normalizing the photon energies in units of the electron rest mass energy

2 by the definition, = hν/mec . This definition will be utilized as our photon energy variable through this discussion, as well as in section 2.4 regarding γγ opacity and pair-production. The Compton scattering between an electron of energy γ and a photon of energy is determined through the Compton cross section, also referred to as the Klein-Nishina (KN) cross section. From Figure 2.2, we can define the scattered photon

0 0 energy as s and the scattered photon angle χ . The KN cross section is then given by the equation [29]: 46

Figure 2.2: Geometry of the Compton scattering event in the rest frame of the electron [29].

dσ r2 0 0 0 0 e s 2 s − 2 0 0 − s 0 0 = ( 0 ) ( 0 + 0 sin χ ) δ(s 0 0 ) (2.14) dΩsds 2 s 1 + (1 − cosχ ) 0 0 0 where dΩs = dcosχ dφs denotes the solid angle element of the direction of motion of scattered photons. Integrating the diﬀerential KN cross section over all scattered photon

0 0 energies, s and directions corresponding to the solid angles, dΩs, gives the total Compton cross section:

πr2 202 (1 + 0) 02 − 20 − 2 σ (0) = e (4 + + ln(1 + 20)) (2.15) C 02 (1 + 20)2 0 The dependence of the total Compton cross section as a function of photon energy in the electron rest frame is given in Figure 2.3. For small values of photon energy in the ERF, 0 1, the cross section assumes a constant value, equal to the Thomson cross section. This energy regime is referred to as the Thomson regime. In the Thomson regime, the

0 0 scattered photon energy is identical to the original photon energy, s ≈ in the electron rest frame. This implies that Compton scattering in the Thomson regime can be considered to be elastic, causing little energy loss to the scattering electron. However, for larger values of the photon energy in the ERF, the transfer of energy from the photon to the electron can be more profound. Larger photon energies result in a 47 reduction of the total cross section. The energy regime 0 1 is referred to as the Klein-Nishina regime. From equation 2.15, we ﬁnd that for larger photon energies, the cross section reduces to the relation:

3 ln(20) σ (0) ≈ σ (2.16) C 8 T 0

Figure 2.3: Total Compton cross section as a function of the photon energy, 0 in the rest frame of the electron [29].

Figure 2.3 illustrates the total cross section with the limits in both the Thomson and the Klein-Nishina regimes. For computing the Compton emission coeﬃcients in our model, it is most convenient to consider the scattering event in the laboratory frame. In these applications, the laboratory frame is the rest frame of the emission region of the relativistic jet. A diagram of the angles and their corresponding cosines between the ERF and the laboratory frame is given in Figure 2.4. From the diagrams, we ﬁnd that the photon energy in the ERF and the laboratory frame are related by the transformation: 48

Figure 2.4: Geometry of a Compton scattering event in a.) the lab frame and b) the rest frame of the electron [29].

0 = γ(1 − βµ) (2.17) where the scattered photon energy in the ERF and the laboratory frame are related by the transformation:

0 0 s = sγ(1 + βµs) (2.18)

In the Thomson regime, γ 1, the scattering will be elastic between the electron and the photon. As a result, the original and scattered photon energies in the ERF will be the

0 0 same, s = . This implies that in the laboratory frame, the scattered photon energy and

2 the original photon energy are related by the equation, s ∼ γ , producing higher energy emission. This phenomenon is referred to as inverse Compton scattering (IC). We next transform the diﬀerential KN cross section from the ERF to the laboratory frame. 49

Given that the cross section is an area that is perpendicular to the direction of motion of the electron, it remains invariant under transformations between frames in the direction

0 0 of motion. The other two terms, dΩs and ds, are related to the laboratory frames by the

0 2 0 −1 transformations, dΩs/dΩs = δs and ds/ds = δs , where the Doppler factor is given by the relation δs = 1/(γ(1 − βµs)). This implies that the diﬀerential KN cross section is related between the ERF and the laboratory frame by the equation:

dσC s dσC = 0 0 0 (2.19) dΩsds s dΩsds Now that we have the cross section in the laboratory frame, we need to consider the kinematics and the direction of the scattered radiation in the laboratory frame. We consider an electron and an incoming photon moving in arbitrary directions, speciﬁed by

the angles with respect to a z-axis, ψe and ψph, see Figure 2.5. We can then compute the

angle cosines, namely µ, µs and κ in both the laboratory frame and the electron rest frame. Using spherical trigonometry, we ﬁnd that:

Figure 2.5: Deﬁnition of the angles and their corresponding cosines for the transformation between the laboratory and electron rest frames [29]. 50

µ = ηph ηe + sin(ψph) sin(ψe) cos(φ) (2.20)

0 0 0 0 0 0 µs = κ µ + sin(χ ) sin(θ ) cos(φ ) (2.21)

For Compton scattering due to ultra-relativistic electrons (γ 1, β → 1), extreme relativistic aberration occurs for the photon paths between the laboratory and electron rest frame. An electron traveling with a large Lorentz factor will only see a narrow cone of emission (of size 1/γ) from an isotropic photon ﬁeld. In the electron rest frame, however, µ0 ≈ −1, for nearly all the photons in the direction of motion. This approximation is referred to as the ”Head-on approximation” to the Compton cross section. The resulting photon distribution will be strongly peaked along the direction of motion of the electron before scattering. As a result, all scattered photons will scatter in the direction of the incoming electrons, Ωs ≈ Ωe. The Compton cross section using the Head-on approximation can then be recast by the relation:

dσC dσC |Head−on = δ(Ωs − Ωe) (2.22) dΩsds ds

where dσC/ds is given by the relation:

dσ πr2 1 2 0 2γ0 C e { − s s 2} = 0 y + 0 + ( 0 ) H(s : , 0 ) (2.23) ds γ y γ y γ y 2γ 1 + 2 0 where y = 1 − (s/γ), = γ(1 − βµ), and H(x : a, b) denotes the Heaviside function defined at the beginning of the section. Equations 2.22 and 2.23 represent the differential scattering cross section in the comoving frame used to compute the inverse Compton emission coefficients for the different photon fields present in the jet. With the differential scattering cross section, the inverse Compton emission coefficient for a distribution of 51 photons upscattered by a distribution of particles in the comoving frame of the emission region is given by the relation [36]:

I Z ∞ I Z ∞ dσC j(s, Ωs) = hcs dΩph dnph(, Ωph) dΩe dγne(γ, Ωe)(1 − βe cos ψ) 0 1 dsdΩs (2.24) where nph(, Ωph) denotes the photon distribution being upscattered. The angle ψ denotes the angle between the electron and the target photon directions and is given by the relation:

q p 2 2 cosψ = µµe + 1 − µ 1 − µecos(φ − φs) (2.25)

where µ represents the angle of the photon/electron relative to the axis of the jet in the comoving frame.

2.2.3 Synchrotron Self Compton

In the jet environment, diﬀerent photon ﬁelds can be upscattered to produce the high energy emission observed from blazars. The photons created from synchrotron emission can be upscattered to produce SSC emission. The distribution of synchrotron photons is considered to be isotropic in the jet frame since the radiating electrons are also isotropic.

For an isotropic distribution of photons and electrons, nph(, Ωph) = nph()/4π,

ne(γ, Ωe) = ne(γ)/4π, and using the Head-on approximation, equation 2.24 reduces to the double integral [50]:

Z ∞ Z ∞ hcs j(s) = dγne(γ) dnph()g(s, , γ) (2.26) 4π 1 0 where the integration kernel g(s, , γ) is given by the relation:

πr2 4γ2 g( , , γ) = e ( s − 1) (2.27) s 2γ4 52

2 if /4γ ≤ s ≤ and:

2πr2 (4γq)2 (1 − q) g( , , γ) = e {2q ln(q) + (1 + 2q)(1 − q) + } (2.28) s γ2 1 + 4γq 2 2 ≤ ≤ 4γ if s 1+4γ , where q is deﬁne by the relation:

q = s (2.29) 4γ2(1 − /γ)

Synchrotron + SSC:

15 10

Total Emission Synchrotron 14 SSC 10

13 10 [Jy Hz] ν F

ν 12 10

11 10

10 10 8 10 12 14 16 18 20 22 24 10 10 10 10 10 10 10 10 10 ν [Hz] Figure 2.6: Sample output of SSC emission with Doppler factor δ = 15 from a power law 3 5 distribution (q = 3.6) of electrons with cutoﬀs γmin = 10 and γmax = 10 interacting with its own synchrotron radiation in an emission region with a magnetic ﬁeld of B = 0.5 G moving with Doppler factor δ = 15.

Equation 2.26 represents the most common expression for the evaluation of SSC emission in the environments of AGNs jets. Figure 2.6 gives a sample output of the SSC spectral

3 component from a power law (q = 3.2) distribution of electrons with cutoﬀs γmin = 10

5 and γmax = 10 interacting with its own synchrotron emission to produce SSC at X-ray and γ-ray energies. The electrons emit synchrotron emission in a magnetic ﬁeld of 53

B = 0.5 G in an emission region with a Doppler factor of δ = 15. This corresponds to an

pk 6 2 13 observed synchrotron frequency of νsyn = 4.2 × 10 B δγmin Hz ≈ 3.15 × 10 Hz. The Compton scattering falls within the Thomson regime for the low energy cutoﬀ of the

pk −5 electron spectrum, synγmin = 8.1 × 10 1. This produces an observed peak SSC

pk pk 2 19 frequency located at νSSC = νsynγmin ≈ 3.15 × 10 Hz, which is consistent with the output shown in Figure 2.6.

2.2.4 External Compton Scattering

Figure 2.7: Geometry of the accretion disk/jet system with the formation of magnetic ﬁeld lines in the accretion disk and black hole magnetosphere [36].

Radiation from the accretion disk, or the BLR and dusty torus surrounding the SMBH are extremely luminous and serves as a source of target photons that can be upscattered by electrons in the jet to produce high energy γ-rays. For the emission spectrum of the accretion disk surrounding the SMBH, we consider the geometrically thin, optically thick, blackbody solution of Shakura and Sunyaev [97], where the energy is derived from the 54 viscous dissipation of gravitational potential energy of the matter accreted onto the SMBH. The radiative energy ﬂux in the AGN frame is given by the equation:

d2E 3GMm˙ = ϕ(R) (2.30) dAdt 8πR3 where M andm ˙ represent the mass and mass accretion rate of the SMBH, R is the disk radius and ϕ(R) is deﬁned by the relation:

1/2 ϕ(R) = 1 − βi(Ri/R) (2.31)

where Ri represents the innermost stable circular orbit around the SMBH and βi represents

the fraction of angular momentum captured by the black hole at the radius Ri. Integrating equation 2.30 over the area of the accretion disk, gives a total bolometric luminosity of the disk:

Z 2 d E 3GMm˙ 2βi Ld = dA = · (1 − ) (2.32) dAdt 2Ri 3 Equating the power per unit area given by equation 2.30 with the Stefan-Boltzmann law, we ﬁnd that the characteristic thermal temperature of the accretion disk is given by the equation:

3GMm˙ T R ϕ R 1/4 ( ) = ( 3 ( )) (2.33) 8πR σSB so that the mean photon energy for the optically thick disk at a given radius, R, is then

∗ 2.7kBT(R) (R) = 2 . For the power per unit area, the corresponding intensity of the disk 0 mec ∗ ∗ concentrated at the peak energy ≈ 0(R) can then be approximated as [36]:

∗ ∗ 3GMm˙ ∗ ∗ I ∗ (Ω , R) = ϕ(R) δ( − (R)) (2.34) 16π2R3 0 55

With the intensity of the disk, the observed νFν ﬂux for the accretion disk integrated over all radii is then found to be:

l L SS Edd Edd s 4/3 − f = 2 ( ) exp( s/max) (2.35) 2πdLη f max With the intensity spectrum of the accretion disk, we can compute the corresponding energy and number density of the disk as a function of the distance and angle in the AGN and comoving frames. The energy density of the disk is found by the relation

∗ ∗ ∗ ∗ u∗ (Ω , R) = I∗ (Ω , R)/c. The number density and energy density of the photon ﬁeld are

∗ ∗ ∗ ∗ 2 related by n∗ (Ω , R) = u∗ (Ω∗, R)/ mec . The number density of the accretion disk photon ﬁeld per unit normalized photon energy, solid angle and disk radius in the AGN frame is then given by the equation:

∗ ∗ ∗ ∗ 3GMm˙ δ( − 0(R)) n ∗ , R ϕ R (Ω ) = 2 3 3 ( ) ∗ (2.36) 16π mec R To evaluate the inverse Compton emission coefficient, equation 2.24, for the accretion disk, we must transform the number density of the accretion disk photon field from the AGN frame to the comoving frame. External radiation fields will typically not be isotropic in the frame of the emission region because of strong relativistic aberration. We use the transformation between the stationary and comoving frames for a number density:

n∗(∗, Ω∗) n(, Ω) = (2.37) (Γ(1 + βµ))2 Rewriting the normalized photon energies from the AGN frame to the comoving frame gives the following expression for the number density of photons for the accretion disk in the comoving frame:

3GMm˙ ϕ(R) δ( − (R)) n , R 0 (Ω ) = 2 3 3 4 4 (2.38) 16π mec R Γ (1 + βµ) 56

Figure 2.8: Geometry of the accretion disk relative to the rotation axis. Rmin denotes the innermost stable circular orbit [36].

We must also rewrite the number density in terms of the angle between the jet axis and the direction of the photon in the AGN frame, µ∗, and then transform the angle into the comoving frame to carry out the integration over the photon direction. From the geometry of the disk, see Figure 2.8, we find that we can rewrite the disk radius in terms of the cosine of the angle between the axis of the jet and direction of the photon in the AGN p frame, R = r (µ∗−2 − 1), where r denotes the location of the emission region along the jet axis. Using equation 2.24 and 2.38 and the representation of the radius in terms of the directional cosine in the comoving frame, we can compute the corresponding emission coefficient in the comoving frame. For radiation fields more extended than the accretion disk, the angular dependence of the photon fields is weaker. The angular characteristics of the photon fields in the comoving frame are more attributed to the relativistic aberration from the bulk motion of the emission region. For the broad line region or the dusty torus, we approximate the photon field as an isotropic spherical blackbody of temperature Text and size Rext surrounding the SMBH. This approximation has had success at modeling the external Compton emission from the BLR for different blazars [30]. With these considerations, we model the energy density of the radiation field assuming the emission is concentrated at 57

∗ 2 the peak of the blackbody emission in the AGN frame, 0 = 2.7kBTet/mec . The energy density of the isotropic radiation ﬁeld in the AGN frame can be written in the form:

L u∗(∗, Ω∗) = ext δ(∗ − ∗) (2.39) 2 2 0 8π Rextc

where Lext represents the observed bolometric luminosity of the isotropic radiation ﬁeld

and Rext represents its corresponding radius. The number density of the isotropic radiation ﬁeld in the AGN frame can then be found through equation 2.37. Transforming the number density into the comoving frame is then:

L δ( − ) n (, Ω) = ext 0 (2.40) ph 2 2 3 4 4 8π Rextmec Γ (1 + βµ) From equations 2.38 and 2.40, we then use equation 2.24 to evaluate the inverse Compton emission coefficients from the accretion disk and an isotropic radiation field surrounding the black hole that characterizes a BLR or dusty torus. Figures 2.9 and 2.10 give a sample output for the external Compton spectral components for an accretion disk and an isotropic radiation field resembling a BLR with a power law distribution of electrons (q = 3.6) with

3 5 cutoﬀs γmin = 10 and γmax = 10 moving with a bulk Lorentz factor of Γ = 15. With a

8 black hole mass of MBH ≈ 3.0 × 10 M , the emission region is at a large enough distance

2 along the jet (d Γ Rdisk)[29] to assume that most accretion disk photons enter the emission region from behind. The disk photons will be deboosted by a factor of 1/Γ. We

∗ −5 have assumed that the accretion disk peaks at 0 ≈ 3.0 × 10 . The scattering between accretion disk photons and electrons in the comoving frame falls within the Thomson

∗ −3 regime, 0γ/Γ ≈ 2.0 × 10 . The scattering in the Thomson regime results in a EC spectral index of α = (q − 1)/2, see Figure 2.9. The peak of the accretion disk spectral component

pk ∗ 2 21 in the observer’s frame is then ECD ≈ 0(δ/Γ)γpk ≈ 4.32 × 10 Hz, consistent with the output given in Figure 2.9. For the isotropic radiation ﬁeld, the beaming of the emission in the forward direction will cause the ﬂux to be boosted by a factor Γ. For a peak energy of 58

−5 ∗ 0 ≈ 2.0 × 10 , the scattering is also in the Thomson regime, 0γΓ ≈ 0.3. For this reason, pk 2 ∗ 23 the EC spectral component should peak at EC = δΓγpk0 ≈ 5.62 × 10 Hz, which is consistent with the output, see Figure 2.10. However, for electrons with energies higher

∗ 3 than γ ∼ 1/Γ0 ≈ 3.0 × 10 , the scattering is dominated by Klein-Nishina eﬀects, resulting in a steeper slope than the Thomson approximation.

EC (Accretion Disk):

13 10

Total Emission Accretion Disk

12 10 [Jy Hz] ν F ν

11 10

10 10 19 20 21 22 23 24 25 26 10 10 10 10 10 10 10 10 ν [Hz] Figure 2.9: Sample output of EC emission located at a distance along the jet of r = 0.12 pc 3 produced from a power law distribution (q = 3.6) of electrons with cutoﬀs γmin = 10 5 45 and γmax = 10 interacting with an accretion disk of luminosity LD ≈ 7.0 × 10 erg/s ∗ −5 peaking at an energy of 0 ≈ 3.5 × 10 surrounding a supermassive black hole of mass 8 MBH ≈ 3.0 × 10 M . The red dashed curve denotes the EC spectral ﬁt from scattering in the Thomson regime α = (q − 1)/2.

2.3 Lepto-Hadronic Modeling of Blazars

In the lepto-hadronic model, the high-energy emission originates from the synchrotron emission by ultra-relativistic protons. The relativistic protons then interact with the radiation ﬁelds in the emission region. The photo-hadronic interactions produce 59

EC (Isotropic Radiation Field):

14 10

Total Emission Isotropic Radiation Field 13 10

12 10

11 10 [Jy Hz] ν 10 F

ν 10

9 10

8 10

7 10 19 20 21 22 23 24 25 26 27 28 10 10 10 10 10 10 10 10 10 10 ν [Hz] Figure 2.10: Sample output of EC emission produced from a power law distribution 3 5 (q = 3.6) of electrons with cutoﬀs γmin = 10 and γmax = 10 interacting with a isotropic 44 ∗ −5 radiation ﬁeld of luminosity Lext ≈ 2.0×10 erg/s peaking at an energy of 0 ≈ 2.0×10 .

high energy pions, which then decay to produce muons, electrons, positrons, and neutrinos [10, 76]. Recently, a time dependent hadronic model that considered a Fokker-Planck equation with the incorporation of radiative losses, second order Fermi acceleration and the emission produced from the final decay products of the photo-hadronic interactions was utilized to explain blazar emission [114]. The production rates of final decay products were derived by analytical parametrizations of the energy distributions for the neutrino, electron/positron and photon distributions from the interactions of relativistic protons with the photon fields, [53]. However, in order for this approach to be viable, the synchrotron cooling time scales of the intermediate decay products (pions and muons) must be significantly longer than their decay time scale (in the co-moving frame of the emission region), which restricts the combination of maximum proton Lorentz factor, γp,max, and

10 magnetic ﬁeld B to B γp,max 5.6 × 10 G[30]. If blazar jets are the sites of the 60

19 acceleration of ultra-high-energy cosmic rays (Ep & 10 eV), then such models are only applicable in the range of magnetic ﬁelds of B . 5 G, substantially lower than usually found in hadronic modeling of blazars. For higher magnetic ﬁelds or maximum proton energies, the synchrotron emission from muons and possibly also charged pions becomes non-negligible. In this section, we describe a new, time dependent lepto-hadronic model that considers the radiation emitted by all secondary products and incorporates Fermi acceleration and self-consistent radiative losses for all particle species (including photo-pion production losses for protons). Our time dependent model allows us to explore parameter regimes where muon and pion synchrotron emission is no longer negligible.

2.3.1 Photo-Hadronic Interactions and Pion Production

We consider a continuous injection of relativistic electrons and protons into the emission region with a magnetic field of strength B. Large magnetic fields, B ≥ 10 G are desired to ensure that the proton Larmor radius is confined to within the size of the emission region, R ≈ 1015 − 1016 cm. High magnetic fields along the jet are also desired if the relativistic jets of blazars are powered through the extraction of electromagnetic energy from the rotation of Kerr black holes via the Blandford-Znajek (BZ) mechanism. Through the BZ mechanism, the base of the jet will possess extremely large magnetic fields, producing electromagnetic luminosities of

47 9 4 9 LBZ ≈ 10 (MBH/10 M )(B/10 G) erg/s. For an FSRQ with a 10 M solar mass black

49 hole that produces a bolometric luminosity in excess of Lγ ≥ 10 erg/s, magnetic fields of around B ≈ 100 G can exist along the jet at distances in excess of z ≈ 1017 cm. Following the initial injection, the electrons and protons give off synchrotron radiation from the radio to high energy γ-rays in the form of equations 2.6 and 2.7. With the proton distribution and the seed photon fields generated from the synchrotron emission, we evaluate the pion production rates based on the photohadronic 61

Figure 2.11: The total pγ cross-section as a function of the photon’s energy in the rest −30 2 frame of the proton, r,[49] (1µbarn = 10 cm ). The contributions of diﬀerent baryon resonances (red, dashed), the direct channel (green, dotted), and multi-pion production (brown) are shown individually.

interaction cross section between protons and photons, see Figure 2.11. The total proton-photon cross section is divided into separate components, corresponding to different reaction channels through which the neutral and charged pions are produced: direct resonances (such as the ∆ resonance), higher resonances, direct single-pion production and multi-pion production. We assume the target photon field for photo pion production is isotropic in the emission region. This limits the model to consider only photon fields that are produced in the emission region (synchrotron radiation). Incorporating external photon fields in the pion production rates would require an additional integration of the differential cross section against the proton and photon angular distributions. We use the prescription developed by [49] for the photo production rate of pions from the interaction between isotropic proton and photon distributions:

Z ∞ Z ∞ dEp Qb(Eb) = Np(Ep) dnγ()Rb(x, y) (2.41) m c2 E Ep th p b 2Ep 62 where Eb represents the energy of the secondary particle created (pion), Np(Ep) represents the proton distribution as a function of the proton energy, Ep, n() represents the photon ﬁeld that the protons interact with as a function of the normalized photon energy

2 = hν/(mec ), and th = 294 (corresponding to an energy of 150 MeV) represents the threshold below which the cross sections are zero. The dimensionless variables x and y are given by:

E x = b (2.42) Ep

E y p = 2 (2.43) mpc The response function R(x, y) in the photo production rate of pions is given by [49]:

X X 1 Z 2y R (x, y) = RIT (x, y) = d σIT ( )MIT ( )δ(x − χIT ( )) (2.44) b 2y2 r r r b r r IT IT th and is summed over all interaction channels that produce the proton-photon cross section

IT as a function of photon energy, r, in the parent nucleus rest frame, σ (r). The functions

IT IT Mb (r) and χ (r) represent the multiplicity of daughter particles and the mean energy fraction that is deposited into the daughter particles for a given interaction channel, respectively. Computationally evaluating these integrals turns out to be very tedious. For this reason, we utilize a simpliﬁed prescription [49] of the pion production rates in which the interactions are split up into separate components that take into account the resonances, direct production and multi-production channels and assumes that the multiplicity and deposited mean energy fractions are independent of the interaction energy, r. The response function then simpliﬁes to:

IT IT IT IT R (x, y) = δ(x − χ )Mb f (y) (2.45) 63 where f IT (y) is given by the relation:

1 Z 2y f IT y d σIT ( ) = 2 r r ( r) (2.46) 2y th With the simpliﬁed response function, the photo-production rate of pions can then be written in the more compact form:

E m c2 Z ∞ m c2 yχIT IT b · p p IT IT Qb = Np( IT ) dynγ( )Mb f (y) (2.47) χ Eb th/2 Eb This single integral is easy to evaluate numerically. The photo-production rate of pions

IT IT now depends on the response function, f (y), the multiplicities, Mb , and the mean energy fraction deposited into the secondary particles, χIT , for the dominant interaction types. The values of the multiplicities and the mean energy deposited for the resonance, direct production and multi-pion production as well as the response functions used, are tabulated in [49]. Figure 2.12 gives a sample output of secondary pions generated from a

8 power law distribution of protons (q = 2.4) with cutoﬀs γmin = 1 and γmax = 10

−4 interacting with a power law distribution of photons α = 1.7 with cutoﬀs min = 10 and

5 max = 10 . For Compton scattering of photons by protons, the Compton cross section is a

2 −7 factor (me/mp) ≈ 2.98 × 10 times smaller in comparison to electron Compton scattering cross section. The proton Compton cross section is of order

2 −31 2 σC ∼ (me/mp) σT ≈ 1.98 × 10 cm , which is about three orders of magnitude smaller than the proton-photon interaction cross section for pion production. The delta resonance enhances the scattering cross section by an order of magnitude above

−20 2 σC ∼ 8.99 × 10 cm , but this is still far short of what would be needed for proton Compton to have a significant impact on the flare dynamics. Similarly, for pion Compton scattering, if the pion differential scattering cross section scales in the same form as the electron cross section, then the Compton emissivity of pions upscattering photons would

2 −30 2 scale as σC ∼ (me/mπ) σT ≈ 8.99 × 10 cm . This is about one order of magnitude 64 below the interaction cross section through any other channel of pion production, in which

−28 2 the cross section is on the order of hundreds of micro barns, σpγ→π ∼ 10 cm . Compton scattering of photons with pions also produces high energy γ-rays beyond ν ≈ 1028 Hz, which is already optically thick due to electron/positron pair production. As a result, we neglect Compton scattering of photons due to protons and pions. As protons interact with the photon ﬁelds to generate pions, they lose energy in the process. With the formalism adopted for the pion production rates, the cooling time scale for the proton distribution from the production of pions can then be given by [49]:

X IT IT IT −1 tcool(Ep) = ( Mp Γ (Ep)K ) (2.48) IT IT IT where K and Γ (Ep) represent the inelasticity and interaction rate of the protons with the photons through a speciﬁc interaction channel. The interaction rate is given by:

Z ∞ Ep ΓIT (E ) = dn () f IT ( ) (2.49) p 2 γ 2 thmpc mpc 2Ep The energy-loss term in the proton Fokker-Planck equation due to pion production is then:

Ep X γ˙ = − = −γ · MIT ΓIT (E )KIT (2.50) pγ t (E )m c2 p p p cool p p IT Protons can also be lost through neutron conversion due to photohadronic interactions. The rate at which primary protons are lost due to conversion into neutrons can be given by the expression [49]:

X IT IT −1 tesc,n(Ep) = ( Mp0 Γ (Ep)) (2.51) IT,p0,p where p0 denotes the new nucleon created in the photohadronic reaction. In the proton Fokker-Planck equation, an additional loss term due to neutron conversion is considered with a loss time scale given by equation 2.51. For protons, the radiative cooling time 65 scales can be longer than the typical dynamical time scale of the expansion of the emission region. As a result, we also include adiabatic losses in our model. Assuming a conical jet with an opening angle of θ ∼ 1/Γ, the adiabatic cooling rate is given by the relation:

3cθ γ γ˙ = − obs (2.52) ad R For low energy protons, adiabatic losses usually dominate. Only for the highest energy protons do synchrotron losses start to become signiﬁcant. With the pion production subroutines, we track the time evolution of the charged pions through a separate Fokker-Planck equation. The pions are subjected to synchrotron losses, escape and stochastic acceleration. There is an additional loss term in the pion Fokker-Planck equation due to pion decay. The pion decay timescale in the particle rest frame is

−8 −17 tdecay ≈ 10 s. Because of the short decay time for the neutral pions, tdecay ≈ 10 s, we assume the neutral pions decay instantaneously into γ-rays. Charged pions decay into charged muons, producing an injection term that is used for a separate muon Fokker-Planck equation. The muons lose energy from synchrotron losses and gain energy from stochastic acceleration. The muon decay time scale in the laboratory

−6 frame is longer than for charged pions, tdecay ≈ 10 s. As a result, charged muons can produce more synchrotron radiation before they decay when compared to charged pions. The high energy protons must meet certain constraints with the magnetic ﬁeld before pions and muons can be generated that produce synchrotron emission comparable to proton synchrotron. If the muon and pion synchrotron radiation timescales are shorter than the decay time scale in the comoving frame and the photo-pion losses are comparable to the proton synchrotron losses, then pion and muon synchrotron emission can no longer be neglected [30]. The code of [38] allows us to choose arbitrary values of the maximum proton Lorentz factor and magnetic ﬁeld, including the regime: 66

Secondary Particle Production Rates:

-2 10

+ π Production: - π Production: -4 10

] -6 -1 10 s -3 [cm 2 γ ) γ -8

Q( 10

-10 10

-12 10 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 γ

Figure 2.12: The energy distribution of the charged pion production rates generated from the interaction between a power law distribution of protons (q = 2.4) with cutoffs γmin = 1 8 −4 and γmax = 10 and a power law photon field (α = 1.7) with cutoffs min = 10 and 5 max = 10 .

  11  7.8 × 10 G for pions, Bγ ≥  (2.53) p   5.6 × 1010 G for muons. in which muons and pions produce substantial synchrotron emission. Muons have a loss term in their Fokker-Planck equation due to charged muon decay (µ+/µ−) into positrons and electrons. The muon decay term serves as an additional injection term for the electron/positron distribution. The electrons/positrons generated from pair production, muon decay and from the original power law injection represent the source term for the electron/positron Fokker-Planck equation. The four coupled Fokker-Planck equations are solved simultaneously with the differential equations that model the time evolution of the synchrotron photon fields for each particle species: protons, charged pions, charged muons and electrons/positrons [38]. Bethe-Heitler pair production is not expected to play 67 an important role in our model since the pion-production rate dominates over the loss rate from BH pair production, so it is therefore neglected. The large magnetic field strengths used for lepto-hadronic fits produce strong particle cooling, making inverse Compton scattering less efficient, so it is also neglected.

2.3.2 Neutrino Production and Emission

Electron and muon neutrinos can be generated from the decay of secondary particles produced from the photohadronic interactions between protons and photons. As the neutrinos propagate from a source, the mass eigenstates of the neutrinos mix in a process called neutrino oscillations. The resulting neutrinos reach the Earth where they are detected by ground based neutrino telescopes, such as the IceCube neutrino detector. Detecting neutrino emission and ﬂares from blazars would be the smoking gun for lepto-hadronic modeling of blazar emission. Understanding the neutrino emission rates during quiescent and ﬂaring states can also help distinguish leptonic and lepto-hadronic models for blazar emission. Our code also takes into account the production rates of electron and muon neutrinos generated in muon and pion decays following photo hadronic interactions. The neutrino production rate depends on the number of charged pions that decay within a given time, Dπ(Eπ), which is given by the decay term in the pion Fokker-Planck equation:

nπ(γ, t) Dπ(Eπ) = (2.54) γtdecay With the pion decay rate, the neutrino production rate can be calculated as

Z ∞ dEπ Dπ(Eπ) Qν(Eν) = · (2.55) −1 Eν(1−rM) Eπ 1 − rM 2 2 where rM = mµ/mπ. The rate of muon decays is governed by the muon Fokker-Planck equation. The calculation of the spectrum of neutrinos generated by the decay of charged 68 muons is more diﬃcult than in the case of pion decay, since the system is a three body decay. We follow the procedure derived in [15] to ﬁnd the neutrino production rate for the three-body decay of muons:

Z ∞ dn Qν(Eν) = dEµDµ(Eµ) (2.56) Eν dEν

Using the dimensionless scalar variable m = Eν/Eµ, we can recast equation 2.56 into the form:

Z 1 Dµ(Eν/m) dn Qν(Eν) = dm · (2.57) 0 m dm where dn/dm represents the neutrino production rate in the laboratory frame in terms of the dimensionless variable m. Assuming that the neutrinos are traveling relativistically, we can cast the neutrino distribution function into the following form [15]:

dn = g (m) + g (m) (2.58) dm 0 1

The scalar functions g0(m) and g1(m) are listed in Table 2.1 and describe the laboratory-frame distributions of the neutrinos in the relativistic limit. Once we have computed the neutrino production rates within the emission region, we determine the expected ﬂuxes here on Earth and integrate over the IceCube sensitivity range in order to determine the expected number of detectable neutrinos.

Table 2.1: Laboratory-frame electron and muon neutrino distribution functions [15].

g0(m) g1(m) 5 − 2 4 3 1 − 2 8 3 νµ : 3 3m + 3 m 3 3m + 3 m 2 3 2 3 νe : 2 − 6m + 4m −2 + 12m − 18m + 8m 69

2.4 γγ Pair Production in Blazar Jets

High energy photons propagate through the emission region of the jet where they can interact with other photons and generate electron-positron pairs: γ + γ → e+ + e−. For example, photons with energies above 100 GeV can be strongly attenuated through pair production by interacting with photons of the extragalactic background light (EBL) that emit strongly in the IR and optical wavelengths. Measurements of the γ-ray attenuation at diﬀerent redshifts reveal clues on the total power generated by stars and AGN throughout the early history of the universe. As the γ-rays are absorbed and generate electron-positron pairs, they emit synchrotron and Compton scatter other photons to initiate other high energy γ-rays in an electromagnetic cascade from GeV to TeV energies. γγ pair production has integral importance in both leptonic and lepto-hadronic models at studying the highest energy γ rays at GeV-TeV scales. From quantum electrodynamics, the total cross section for γγ pair production is given by the relation [36]:

1 2 2 4 1 + βcm 2 σγγ(s) = πre (1 − βcm)((3 − βcm)ln( ) − 2βcm(2 − βcm)) (2.59) 2 1 − βcm √ √ −2 −1/2 −1 where βcm = (1 − γcm) = 1 − s , where s = γcm is the center-of-momentum frame Lorentz factor of the produced electron-positron pair. The strength of a given collision between two photons can be characterized by the invariant energy term, given by the relation [36]:

stot = 21(1 − µ) (2.60)

At the threshold for pair production, an electron-positron pair is generated with total

2 energy, 2mec . This implies that the invariant energy term is then stot = 4. Beyond the 70

2 threshold limit, the invariant term is then stot = 4γcm. The connection between the invariant energy and the Lorentz factor of the electron-positron pair in the rest frame is then:

1 s = γ2 = { (1 − µ)} (2.61) cm 2 1 For the γγ opacity in the comoving frame, the absorption probability per unit path length is given by the relation:

I Z ∞ dτγγ(1) = dΩ(1 − µ) dnph(, Ω)σγγ(s) (2.62) dx 0 Equation 2.62 represents the opacity for the combined photon field for both the leptonic and lepto-hadronic models. For the lepto-hadronic model, the synchrotron photon fields of the protons, pions, muons and electron-positron pairs originate from the emission region so they can be considered to be isotropic. For an isotropic photon field, we can rewrite equation 2.62 in the form:

Z 1 Z ∞ dτγγ(1) 1 = dµ(1 − µ) dnph()σγγ(s) (2.63) dx 2 −1 0 From Figure 2.13, we observe that the cross section for γγ absorption peaks at 2 as a

function of the interaction energy 12(1 − µ). A γ-ray of energy 1 will preferentially

interact with a photon of energy 2 = 2/2. Figure 2.14 illustrates the opacity for an

−4 isotropic power law photon ﬁeld, with spectral index α = 1.8 and cutoﬀs min = 10 and

4 max = 10 using equation 2.63. The opacity peaks at a photon energy of

4 pk ∼ 2/min ≈ 2.0 × 10 . Increasing the low energy cutoﬀ of the photon distribution causes the turnover in the γγ opacity to move to lower energies. As high energy photons interact with other photons in the emission region, electron/positron pairs are generated. The electrons and positrons serve as an additional injection term in the electron/positron Fokker-Planck equation. For the pair-production 71

Figure 2.13: Cross Section for γγ pair production as a function of the invariant interaction energy 12(1 − µ)[29].

γγ Opacity:

2 10

1 γγ 10 Opacity

0 10

-1 10

-2

) 10 ε ( γγ τ -3 10

-4 10

-5 10

-6 10

-7 10 -4 -3 -2 -1 0 1 2 3 4 5 6 10 10 10 10 10 10 10 10 10 10 10 ε Figure 2.14: Sample output of the γγ opacity from an isotropic power law distribution of 0 5 photons with normalization constant nph = 4.0 × 10 , spectral index (α = 1.8), cutoﬀs −4 4 16 min = 10 and max = 10 in an emission region of size R = 2.0 × 10 cm.

rate from γγ interactions, we use the approximation given by Aharonian et al. 1983. The injection spectrum is given by the relation [9]: 72

Z ∞ (1) Z ∞ (2) 3 nph (1) nph (2) n˙(γ) ≈ cσT d1 d2 w(γ, 1, 2) (2.64) 3 2 32 γ 1 1 4γ(1−γ) 2 1 2 where nph(1) and nph(2) represent the two photon ﬁelds that interact to produce the electron/positron pairs. The integration kernel, w(γ, 1, 2), is given by the relation:

2 2 4 4 4 γ( − γ) 2(212 − 1) 1 w γ, , 1 ln 2 1 − 1 − − 1 ( 1 2) = ( ) 8 1 2 + (1 ) 2 2 γ(1 − γ) 1 γ(1 − γ) 12 γ (1 − γ) (2.65) Equation 2.64 is able to reproduce the production rate of electron/positron pairs well with errors less than a few percent, see [25]. 73 3 Modeling the Broadband Emission of 3C 273, 3C 279 and

3C 454.3

3.1 General Considerations

With both the leptonic and lepto-hadronic models [37, 38], we model the broadband spectral energy distributions of three prominent blazars: 3C 273, 3C 279 and 3C 454.3. The input parameters from our SED fits will be used to study light curve patterns in the context of leptonic and lepto-hadronic models, see Chapter4. We begin by modeling 3C 273 with our one-zone leptonic model. We then use our lepto-hadronic model to produce the SED of the blazar 3C 279. Finally, we reproduce the broadband SED of the blazar 3C 454.3 in its quiescent state with both our leptonic and lepto-hadronic models. We show that both models can reproduce the broadband SED of 3C 454.3 in its quiescent state. This exemplifies the difficulty in determining the origin of the particle makeup that produces the emission from blazars. Our fits to the SEDs are in agreement with previous studies that have been performed on the aforementioned blazars, see [30, 41]. 74

3.2 Modeling the SED of 3C 273

3C 273 is a FSRQ located in the constellation Virgo. At a redshift of z = 0.158, it is one of the most luminous and closest quasars known, with an absolute optical magnitude of −26.7. 3C 273 is also the first quasar ever to be identified. 3C 273 is located at the heart of a giant elliptical galaxy, with an apparent size of 30 arcsec and an apparent magnitude of 12.9 in the V band. The quasar also exhibits a large scale jet, visible at different wavelengths, which measures about 23 arcsec across, or 200 kly (60 kpc) long, see Figure 3.1. The optical-UV spectrum of 3C 273 shows a prominent excess of emission which has been attributed to accretion disk emission [99, 104]. The detection of emission lines in the optical-UV spectrum, e.g. Ly − α, CIV, OVI, CIII, NIII and SVI is connected with the emission from the BLR [88]. Morphological studies done on 3C 273 using the Hubble Space Telescope (HST) show brightening of different knot structures exhibited by the jet, see [103]. We use 3C 273 as a target blazar to test our one-zone leptonic model. With the extensive observations done on 3C 273, we were able to obtain simultaneous, multi-wavelength data to model its broadband emission, [3]. We begin by performing a fit to the time-averaged SED of 3C 273 [data taken from3], based on an equilibrium solution obtained with our one-zone leptonic code with time-independent input parameters. The equilibrium model is fully determined through the following list of

input parameters: The magnetic ﬁeld B, the observed variability time scale, ∆tvar, the bulk

Lorentz factor Γ, the observing angle θobs, the low and high energy cutoﬀs γe,min and γe,max

and spectral index qe of the electron injection spectrum, the electron injection luminosity

Le,in j, the mass of the supermassive black hole MBH, the Eddington ratio of the accretion

disk ledd, the distance of the emission region from the central black hole, Raxis, the size of

the broad line region RBLR, the eﬀective blackbody temperature TBB of the BLR, and the

ratio between the acceleration and escape time scales, tacc/tesc. 75

Figure 3.1: Image of the FSRQ 3C 273 and its jet from Chandra [82].

Several of these parameters may be either directly measured or constrained through observations. Speciﬁcally, for 3C 273, we have the following observables [see 30, for

references to the observational data]: z = 0.158, β⊥,app = 13 (the apparent transverse velocity of individual jet components, normalized to the speed of light), ∆tvar ∼ 1 d,

47 −1 45 −1 Ldisk = 1.3 × 10 erg s , and LBLR = 9.1 × 10 erg s . The observed apparent superluminal speed implies a limit to the bulk Lorentz factor of Γ > 13. We choose the observing angle as θobs = 1/Γ so that δ = Γ. The size of the emission region is then constrained by R ≤ δc∆tvar/(1 + z). In the SED of 3C 273, the accretion disk component is directly visible as a prominent Big Blue Bump (BBB), peaking beyond 1015 Hz. The mass

9 of the supermassive black hole of 3C 273, MBH ≈ 1.0 × 10 M , is constrained through direct observations due to reverberation mapping [115]. The appearance of the BBB in the

−1 SED allows reliable estimates of the Eddington ratio, lEdd = Ldisk/LEdd ≈ 3.0 × 10 for the 76 accretion disk spectral component. The variability time scale allows estimates of the location of the emission region, Raxis ≈ 0.23 pc. With the observed bolometric luminosity of the BLR, we can determine its size through the relation

17 45 1/2 RBLR ≈ 10 (Ldisk/10 erg/s) cm [17]. This relation gives a size of the BLR of ≈ 1.1 × 1018 cm = 0.35 pc. The remaining set of input parameters are found through a ”fit-by-eye” procedure where a ballpark set is selected for the initial fit. These parameters are then adjusted until a satisfactory fit to the SED is obtained. The observed synchrotron peak frequency is related to the Doppler factor, magnetic field and peak electron energy through the relation

6 2 νc = 4.2 × 10 B(G) · (me/mi) · δ · γpk Hz. Assuming that SSC scattering is in the Thomson

2 regime, the peak location of the SSC power will then be located at νSSC = νsyn γpk. The variability time scale can give a reliable location of the emission region using the relation

2 17 Raxis = 2Γ c∆t/(1 + z) ≈ 5.0 × 10 cm. At this location, the emission zone lies within the broad line region. Using the location of the emission region, we compute the external Compton spectral components using the full Compton cross section for both the accretion disk and the BLR, see section 2.2.4. From this set of baseline parameters, we obtain satisfactory fits to the broadband SED of 3C 273, as shown in Figure 3.2. In accordance with leptonic model fits by other authors [30], the radio to optical emission is explained by electrons/positrons emitting synchrotron radiation in magnetic fields of around B ≈ 1.0 G. The radio emission from our model simulation is synchrotron-self-absorbed and therefore underpredicts the observed radio flux from 3C 273. This suggests that the observed radio emission likely originates in more extended regions of the jet, beyond the radiative zone considered in our model. The synchrotron radiation serves as a target photon field to be upscattered by the emitting electrons to produce SSC emission from soft X-rays to soft γ-rays. From the fits, parameters allowing for an observable minimum variability time scale of 7.2 × 104 s 77 Table 3.1: Parameter values used for the equilibrium fit to the SED of 3C 273.

Parameter Symbol Value

Magnetic f ield B 0.5 G Radius o f emission region R 2.79 × 1016 cm Constant multiple f or escape time scale η 18.0 Bulk Lorentz f actor Γ 14

−2 Observing angle θobs 7.14 × 10 rad

3 Minimum electron Lorentz f actor γe,min 1.0 × 10

4 Maximum electron Lorentz f actor γe,max 6.0 × 10

Electron in jection spectral index qe 3.6

42 −1 Electron in jection luminosity Le,inj 3.0 × 10 erg s

9 S upermassive blackhole mass MBH 1.0 × 10 M

−1 Eddington ratio lEdd 3.1 × 10

Blob location along the jet axis Raxis 0.23 pc

Radius o f external radiation f ield Rext 0.35 pc

4 Blackbody temperature radiation f ield TBB 5.0 × 10 K

Ratio between the acceleration and escape time scales tacc/tesc 0.5

provides better fits to both the synchrotron an SSC spectral components. Different variability time scales changes the size of the emission region affecting the SSC contribution, produce poor fits to the X-rays. We also find that an extensive contribution from inverse Compton scattering of the external radiation fields (accretion disk and BLR) is necessary to explain the low to high energy γ-ray emission in the quiescent state of 3C

46 273. From the input parameters, an accretion disk luminosity of Ld ≈ 9.5 × 10 erg/s is needed to provide satisfactory ﬁts to the SED of 3C 273, similar to the measured accretion 78

3C273

15 10

14 10

13 10

12 10 [Jy Hz] ν F ν 11 10

10 10

9 10 10 12 14 16 18 20 22 24 10 10 10 10 10 10 10 10 ν [Hz] Figure 3.2: The ﬁt between our one-zone leptonic model and the multi-wavelength data set (red data points) [3] for 3C 273. Green data points represent archival data. Black solid = total SED ﬁt; Red dashed = electron/positron synchrotron; green dashed = SSC; blue dashed = accretion disk; magenta dashed = EC (disk); indigo dashed = EC (BLR).

disk luminosity of 1.3 × 1047 erg/s. A full list of parameters which yield a satisfactory representation of the SED of 3C 273, is given in Table 3.1.

3.3 Modeling the SED of 3C 279

3C 279 (z = 0.538) was the first γ-ray blazar discovered using the Compton Gamma Ray Observatory and has been the target of several multi-frequency campaigns [e.g., 14, 46, 47, 61]. Many observational properties of 3C 279 have been well measured, including the accretion disk luminosity [47], the bolometric luminosity of the broad line region [116], the minimum variability time scale [27] and the apparent superluminal speed of relativistic jet components [48]. 3C 279 is one of only three FSRQs detected in VHE γ-rays by ground-based Cherenkov Telescope facilities. Specifically, 3C 279 was detected by the Major Atmospheric Gamma-Ray Imaging Cherenkov (MAGIC) Telescope during 79 an exceptional γ-ray flaring state in 2006 [11]. [28] have pointed out that this VHE detection, in combination with the rest of the SED and known variability properties of 3C 279, presents a severe challenge to single-zone leptonic jet models, and suggest a hadronic scenario as a viable alternative. For this reason, we choose this well-known blazar as a representative of γ-ray bright blazars in which hadronic processes might be important.

Figure 3.3: Time series of the superluminal motion exhibited by 3C 279 from the VLBA [85]. 80

Following our analysis on 3C 273, we perform a parameter study to provide a rough ﬁt to the average SED of 3C 279 [as presented in3], by running our time-dependent lepto-hadronic model code with time-independent input parameters and waiting for all particle and photon spectrum solutions to relax to an equilibrium. With the large number of input parameters in our model (see Table 3.2), we use independent constraints from observational data of 3C 279 to reduce as many unknown input parameters as possible for our model. For 3C 279, we have the following observational parameters [see 30, for references to the observational data]: z = 0.536, β⊥,app = 20.1, ∆tvar ∼ 2 d,

45 −1 44 −1 Ldisk = 2.0 × 10 erg s , and LBLR = 2.0 × 10 erg s . The superluminal motion speed sets a lower limit to the bulk Lorentz factor, Γ > 20.1. The observing angle is set by using the relation θobs = 1/Γ so that δ = Γ. From the variability time scale, we can constrain the

2 18 location of the emission region along the jet, Raxis = 2Γ c∆t/(1 + z) ≈ 10 cm. With the observed luminosity of the accretion disk, we can determine the characteristic size of the BLR [17]. The mass of the supermassive black hole in 3C 279 is constrained through the measured bolometric luminosity of the broad line region and is found to be

8 (4 − 8) × 10 Msol [115]. Within the parameter constraints listed above, we perform the same ”ﬁt by eye” procedure we did with 3C 273 to ﬁnd suitable values for the remaining parameters. In the context of most hadronic modeling, the X-ray to soft and intermediate γ-ray emission from FSRQs can be best explained by proton synchrotron radiation. Thus, the X-ray through HE γ-ray spectrum informs our choice of the proton injection luminosity, spectral index, and maximum proton energy. The VHE γ-ray emission detected by MAGIC [11] appears to constitute a separate radiation component beyond the Fermi-LAT γ-ray spectrum, and we here suggest that this component may be provided by muon and pion synchrotron radiation, which our code is uniquely able to handle in a time-dependent

10 fashion. By chosing Bγp,max & 5 × 10 G, our simulations will be in a parameter regime 81 in which muon and pion synchrotron is expected to make a signiﬁcant contribution to the γ-ray emission. A full list of parameters which yield a satisfactory representation of the SED of 3C 279, is given in Table 3.2.

Table 3.2: Parameter values used for the equilibrium ﬁt to the SED of 3C 279.

Parameter Symbol Value

Magnetic f ield B 150 G Radius o f emission region R 8.75 × 1015 cm Constant multiple f or escape time scale η 6.0 Bulk Lorentz f actor Γ 21

−2 Observing angle θobs 4.76 × 10 rad

Minimum proton Lorentz f actor γp,min 1.0

8 Maximum proton Lorentz f actor γp,max 4.5 × 10

Proton in jection spectral index qp 2.2

46 −1 Proton in jection luminosity Lp,inj 3.5 × 10 erg s

2 Minimum electron Lorentz f actor γe,min 5.1 × 10

4 Maximum electron Lorentz f actor γe,max 1.0 × 10

Electron in jection spectral index qe 3.2

41 −1 Electron in jection luminosity Le,inj 7.8 × 10 erg s

8 S upermassive blackhole mass MBH 6.0 × 10 M

−2 Eddington ratio lEdd 1.25 × 10

Blob location along the jet axis Raxis 0.324 pc

Radius o f external radiation f ield Rext 0.045 pc

4 Blackbody temperature radiation f ield TBB 5.0 × 10 K

Ratio between the acceleration and escape time scales tacc/tesc 16.0 82

With this set of baseline parameters, the broadband SED of 3C 279 can be reproduced quite well, as shown in Figure 3.4. The infrared to UV portion of the SED is fitted by synchrotron radiation from primary electrons/positrons. The X-ray to GeV γ-ray emission in our model SED is dominated by proton synchrotron radiation. The VHE γ-ray spectrum, as measured by MAGIC, is best explained by a combination of synchrotron radiation from the primary protons and secondary muons and pions generated via photo-pion production. We note that the proton synchrotron component slightly overshoots the Fermi data points. This is reasonable since the Fermi-LAT spectrum represents a long-term averaged high-state, while the MAGIC detection corresponds to an exceptional, short-term flaring event during which no GeV γ-ray observatory was operating, but one may expect that the HE γ-ray flux at that time was larger than the Fermi-LAT high-state flux presented in [3] and shown in Figure 3.4. The radio emission from our model simulation, again, underpredicts the observed radio flux from 3C 279. Strong emission from more extended regions of the jet beyond the one-zone likely explains this excess radio emission. In our simulation, the jet is — to within a factor of a few — in approximate equipartition between the powers carried in magnetic fields and in kinetic energy of particles: The power carried along the jet in the form of magnetic field (i.e., the Poynting Flux) is determined by the relation:

B2 L = πR2Γ2β c (3.1) B Γ 8π

48 −1 which, for our baseline ﬁt to 3C 279, yields LB = 2.8 × 10 erg s . The particle kinetic luminosities in the observer’s frame are calculated from the equilibrium particle distributions ni(γi) with the equation: 83

3C279

15 10

14 10

13 10 [Jy Hz] ν

F 12 ν 10

11 10

10 10 10 12 14 16 18 20 22 24 26 10 10 10 10 10 10 10 10 10 ν [Hz] Figure 3.4: The ﬁt between our one-zone lepto-hadronic model and the multi-wavelength data set (red data points) [3] for 3C 279. Green data points represent archival data. Black solid = total SED ﬁt; Red dashed = proton synchrotron; Green dashed = electron/positron synchrotron; blue dashed = accretion disk; magenta dashed = muon synchrotron; indigo dashed = pion synchrotron.

Z∞ 2 2 2 Li = πR Γ βΓc mic dγi ni(γi) γi (3.2) 1 where i denotes the particle species considered. From numerically integrating the solution to the Fokker-Planck equation for both the proton and electron/positron distributions when equilibrium is reached, we ﬁnd that the corresponding particle kinetic luminosities are

48 −1 43 −1 Lp = 9.7 × 10 erg s and Le = 3.5 × 10 erg s . With these values, the partition parameter between the combined particle kinetic luminosity and the power carried by the magnetic ﬁeld, B ≡ LB/Lkin, where Lkin = Le + Lp, is then B ≈ 0.29. Our value for Lp is similar to the values usually required by most previously published hadronic model interpretations of FSRQ SEDs [30]. However, previously published works usually require

−3 parameters far out of equipartition. For example, in [30], LB/Lp = 7.9 × 10 for their ﬁt 84 to the SED of 3C2 79, while our model produces a reasonable representation of the same SED with a jet near equipartition. This might be a consequence of the higher radiative eﬃciency in the parameter regime chosen here, with the inclusion of secondary muon and pion synchrotron radiation.

3.4 Modeling the SED of 3C 454.3

The ﬂat spectrum radio quasar 3C 454.3 (z = 0.859) is one of the most brightest γ-ray sources seen with Fermi and one of the most variable sources in the sky, with an

48 isotropic luminosity of Lγ ≥ 10 erg/s. 3C 454.3 has been the target of multi-wavelength campaigns that have determined many of its observational characteristics and monitored its variability across the entire electromagnetic spectrum [52, 107]. During the first year of Fermi operations, optical-infrared photometry of 3C 454.3 showed correlated variability between the UV and γ-ray bands [22]. High resolution radio core observations of 3C 454.3 have also been performed in order to understand superluminal features of the jet [51]. On November 2010, 3C 454.3 displayed a prominent flare that was detected with the Fermi telescope. The flare was followed up by observations in the optical, X-ray and mm bands. With the extensive multi-wavelength coverage of this blazar in its quiescent and flaring states, it serves as a suitable target for determining which model can best explain all the features of its emission. In this section, we apply our leptonic and lepto-hadronic models to the broadband spectral energy distribution of 3C 454.3. The fits from both models will then be used for our light curve analysis to determine which model best explains the Nov. flare, see section 4.5. With the observations carried out on 3C 454.3 [3], a set of input parameters for both models can be constrained. For 3C 454.3, we have the following observational parameters [see 30, for references to the observational data]. With the redshift (z = 0.857) the luminosity distance to the source can be determined, which sets the overall luminosity 85

Figure 3.5: All sky image of the galactic plane and the location of FSRQ 3C 454.3 from the Fermi Space Telescope [83].

scale. The variability time scale, ∆tvar ≈ 1 day, constrains the size of the emission region.

Superluminal motion of the jet of 3C 454.3, β⊥.app ≈ 15, places a lower limit on the bulk Lorentz factor Γ ≥ 15 of the emission region. The mass of the supermassive black hole,

9 47 −1 MBH ≈ 2.0 × 10 M [23, 41], and the accretion disk luminosity, Ldisk ≈ 1.7 × 10 erg s , allow the accretion disk spectral component to be estimated. The BLR luminosity,

45 −1 LBLR ≈ 2.5 × 10 erg s , allows the size of the BLR to be estimated [17]. We approximate the BLR spectrum as a thermal blackbody with a temperature of

4 TBLR ≈ 6.0 × 10 K so that it peaks in the UV. With the set constrained input parameters, we follow the same procedure as 3C 273 and 3C 279 and perform a ”fit-by-eye” method to determine the remaining parameters in both models. The spectral shape and normalization of the multiwavelength emission in different band passes allows the spectral shape and the normalization of the particle distributions to be determined. The magnetic field is also adjusted to provide reasonable fits to the peaks for the synchrotron, SSC and EC components for the one-zone leptonic model. For the lepto-hadronic model, we require that the proton distribution and the magnetic field are constrained in such a way so that muon and pion synchrotron emission 86 cannot be neglected in the SED fitting. We aim to set the input parameters in such a way that approximate equipartition between the particle energy and the energy density of the magnetic field is achieved for both models.

3.4.1 Leptonic Model Fits

Figure 3.6 shows the SED of the blazar 3C 454.3 and our leptonic model fit. The input parameters used for the leptonic fit are given in Table 3.3. We find that the leptonic fitting is satisfactory for the quiescent state of 3C 454.3. The IR/optical/UV emission is well explained by synchrotron emission from the electrons/positrons. Typical values of B ≈ 1 G are needed to produce satisfactory fits for our leptonic model, consistent with previous findings for magnetic field strength for one-zone leptonic models [30, 41]. The synchrotron emission is opaque at longer wavelengths due to synchrotron self absorption. The observed radio emission is due to more extended regions that have lower densities and hence lower SSA opacity. We also find from the fits that the soft to intermediate X-rays are best explained by synchrotron self Compton emission, in accordance with leptonic modeling. While there are no observational features in the SED indicative of a thermal component from the disk, we include the spectral component of the accretion disk for completeness. We find that for a radiation field indicative of a BLR, the radiation field needs to be of size Rext ≈ 0.42 pc in order to provide satisfactory fits to the SED. The size of the emission region is consistent with other one-zone leptonic model fits for 3C 454.3

[30, 41]. An emission site location of Raxis ≈ 0.12 pc was also needed to provide good ﬁts to the HE γ-ray band. This location places the emission zone within the broad line region. Photons in excess of 100 GeV preferentially interact and produce electron/positron pairs with photons from the BLR, causing the emission beyond 100 GeV to become opaque to γγ absorption, see section 2.4. Using the location of the emission region, we compute the 87

3C454.3

15 10

14 10

13 10

12 [Jy Hz]

ν 10 F ν

11 10

10 10

9 10 10 12 14 16 18 20 22 24 26 10 10 10 10 10 10 10 10 10 ν [Hz] Figure 3.6: The ﬁt between our one-zone leptonic model and the multi-wavelength data set [3] for 3C 454.3. Black solid = total SED ﬁt; Red dashed = electron/positron synchrotron; green dashed = SSC; blue dashed = accretion disk; magenta dashed = EC (disk); indigo dashed = EC (BLR).

external Compton spectral components using the full Compton cross section for both the accretion disk and the BLR. We find that an extensive contribution from inverse Compton scattering of the external radiation fields (accretion disk and BLR) is necessary to explain the low to high energy γ-ray emission in the quiescent state of 3C 454.3. Below a spectral break centered at 2 GeV, the Fermi emission is explained by external Compton scattering from the accretion disk, while above the break it is almost entirely from the Compton scattering of the BLR. The combination of the two external Compton components is able to reproduce the spectral break observed at 2 GeV, consistent with previous studies on Leptonic modeling of 3C 454.3 [30, 41]. From the input parameters, we find that the

46 accretion disk luminosity of Ld ≈ 10 erg/s, and the BLR luminosity of

44 LBLR ≈ 10 erg/s are needed to provide satisfactory ﬁts to the SED of 3C 454.3. However, these values are in accordance with previous parameter studies made on 3C 88

454.3 using leptonic models [41, 52]. Relatively long escape time scales are also needed,

7 tesc ≈ 10 s, to ensure that electrons have enough time to cool to produce the low energy tail needed for the EC disk component at hard X-rays/soft γ-rays.

Table 3.3: Parameter values used for the leptonic and lepto-hadronic equilibrium ﬁt to the SED of 3C 454.3. Parameter Symbol Leptonic Value Hadronic Value

Magnetic f ield B 1.5 G 125 G Radius o f emission region R 2.51 × 1016 cm 2.51 × 1016 cm Multiple f or escape time scale η 15.0 15.0 Bulk Lorentz f actor Γ 15 15

−2 −2 Observing angle θobs 6.66 × 10 rad 6.66 × 10 rad

Minimum proton Lorentz f actor γp,min - 1.0

8 Maximum proton Lorentz f actor γp,max - 4.85 × 10

Proton in jection spectral index qp - 2.25

46 −1 Proton in jection luminosity Lp,inj - 3.75 × 10 erg s

2 1 Minimum electron Lorentz f actor γe,min 9.0 × 10 5.0 × 10

4 3 Maximum electron Lorentz f actor γe,max 6.0 × 10 2.5 × 10

Electron in jection spectral index qe 2.9 2.9

43 −1 42 −1 Electron in jection luminosity Le,inj 2.45 × 10 erg s 3.64 × 10 erg s

9 9 S upermassive blackhole mass MBH 2.0 × 10 M 2.0 × 10 M

−2 −2 Eddington ratio lEdd 3.5 × 10 3.5 × 10

Blob location along the jet axis Raxis 0.12 pc 0.12 pc

Radius o f external radiation f ield Rext 0.42 pc 0.42 pc

Ratio o f acc and esc time scales tacc/tesc 0.1 4.0 89

With the magnetic ﬁeld strength and the solution to the electron Fokker-Planck equation, we can compute the equipartition value between the magnetic and particle

−1 energy densities. We find an equipartition value of Be = LB/Le ≈ 10 . This result is also consistent with previous findings on the power from the magnetic and the particle kinetic luminosities [41]. This implies that the jet is particle dominated. The jet being particle dominated at these locations within the jet is troublesome for models arguing that electromagnetic energy is extracted from the rotation of a Kerr black hole in a Blandford-Znajek mechanism [19], through which the first few fractions of a parsec down the jet would be Poynting flux dominated rather than particle dominated.

3.4.2 Hadronic Model Fits

Figure 3.7 shows the SED fitting for our lepto-hadronic model used in this study. Table 3.3 gives the corresponding set of input parameters used for the fitting. We also find reasonable fits for the lepto-hadronic model to the quiescent state of 3C 454.3. The IR/optical/UV emission is again explained by synchrotron emission of electrons/positrons.

The electron injection requires low electron injection energies, γe,min ≈ 100, and soft spectral indices, qe ≈ 2.9, to provide satisfactory ﬁts. The soft X-ray to intermediate/hard γ-ray emission is explained by the synchrotron radiation from protons. The spectral ﬁts

17 required maximum energies for the protons of around Ep ∼ 10 eV to reproduce the γ-ray spectra. The curvature of the proton synchrotron spectrum is able to reproduce the spectral break in the γ-rays observed at ∼ 2 GeV. A large magnetic field strength of B ∼ 100 G is used in order to produce the necessary proton synchrotron radiation to provide adequate fits to the X-ray to γ-ray broadband emission. The protons then interact with the primary electron and proton synchrotron radiation and, through photohadronic interactions, produce pions and muons. With the maximum proton Lorentz factor and magnetic field strength used, the muon 90

3C454.3

15 10

14 10

13 10 [Jy Hz] ν

F 12

ν 10

11 10

10 10 10 12 14 16 18 20 22 24 26 10 10 10 10 10 10 10 10 10 ν [Hz] Figure 3.7: The ﬁt between our one-zone lepto-hadronic model and the multi-wavelength data set [3] for 3C 454.3. Black Solid = Total SED ﬁt; Red dashed = proton synchrotron; Green dashed = electron/positron synchrotron; blue dashed = accretion disk; magneta dashed = muon synchrotron; indigo dashed = pion synchrotron.

synchrotron cooling timescale becomes smaller than the decay time scale in the blob frame. As a result, muon synchrotron radiation becomes non-negligible and produces high energy γ-rays, centered at ∼ 20 GeV. The combination of proton and muon synchrotron is able to reproduce the shoulder necessary to ﬁt the high energy data point centered at 20 GeV, see Figure 3.7. The charged pions produce their own synchrotron spectral component, but the emission is negligible in comparison to the muon synchrotron emission. The muons subsequently decay and produce electron/positron pairs that generate their own synchrotron spectral component. However, the synchrotron emission from the electron/positron pairs generated from charged muon decay is also negligible compared to the proton synchrotron. The broadband ﬁts require a kinetic power in relativistic protons around

48 −1 Lp ≈ 7.46 × 10 erg s and a power carried by the magnetic ﬁeld around 91

48 −1 LB ≈ 8.37 × 10 erg s . This produces an equipartition parameter for the broadband fits of Bp = LB/Lp ≈ 1.12. This suggests that the jet is slightly magnetically dominated and that magnetic energy is efficiently converted into particle kinetic energy, favoring a Poynting flux dominated scenario necessary to power the jet from the black hole. 92 4 Light Curve Modeling of Blazar Emission

The results of this chapter have been published in the Journal of High Energy Astrophysics and the Astrophysical Journal: Diltz, C., & Bottcher,¨ M., 2014, Journal of High Energy Astrophysics, 1, 63D: Time dependent leptonic modeling of Fermi II processes in the jets of ﬂat spectrum radio quasars. Diltz, C., Bottcher,¨ M., & Fossati, G., 2015, The Astrophysical Journal, 802, 133D: Time Dependent Hadronic Modeling of Flat Spectrum Radio Quasars Diltz, C., & Bottcher,¨ M., submitted: Leptonic and Lepto-Hadronic Modeling of the Nov. 2010 ﬂare from 3C 454.3

4.1 General Considerations

One zone leptonic and lepto-hadronic models produce different spectral components to explain the SEDs of blazar emission. Different input parameters for both models are able to successfully reproduce the SED of a blazar. The success of both models to reproduce the broadband SED of a blazar makes determining which model (leptonic/lepto-hadronic) best explains the radiation difficult. However, blazars undergo flaring episodes, in which the different spectral components change and evolve. Applying perturbations to any one of the input parameters of both models causes the spectral components that produce the SED to evolve on different timescales. How these spectral components change and the rate at which they change can serve as a diagnostic tool to distinguish which model can best explain both the quiescent and flaring states exhibited by blazars. In this section, starting from the parameter sets acquired from the SED fits to the blazars 3C 273 and 3C 279, I apply perturbations to a subset of the input parameters in order to study the light curve behavior for both the one-zone leptonic and lepto-hadronic models, respectively, in order to find distinguishing characteristics between the two 93 models. Following our analysis between the leptonic and lepto-hadronic models, we then apply perturbations to the input parameters of both model SED fits of 3C 454.3 in order to explain a prominent flare that it exhibited in Nov. 2010. We begin by allowing both the leptonic and lepto-hadronic models to reach equilibrium for all three blazars, see Chapter3. We then modify the time step to ∆t ≈ 1.0 × 106 s in the comoving frame, corresponding to ∼ (5 − 10) × 104 s in the observer’s frame for all three blazars. The choice of the time step allows us to resolve light curve patterns on time scales characteristic for cooling effects of the relativistic protons and the acceleration time scales for the individual particle distributions. However, we are unable to model the shorter-term variability, potentially caused by the radiative cooling of high energy electron-positron pairs generated from the decay of charged mesons, since their cooling time scales are significantly shorter than the size of the time step selected for these simulations. We do therefore not explore predicted variability patterns on such short time scales, as this would increase the required computational time significantly. Note, again, that the implicit Crank-Nicholson scheme implemented for the solution of the Fokker-Planck equations guarantees that a stable solution for the electrons/positrons, muons, and pions is obtained even if the time step is longer than the radiative cooling time scale. The choice of the subset of input parameters selected for the light curve analysis is based on the expected physics during the production of strong shock. During the development of a shock, the upstream material crosses the shock front and increases in density, while decreasing in bulk speed. As particles cross the shock front, they reflect off of magnetic inhomogeneities on both sides of the shock front leading to first-order Fermi acceleration. A power law distribution of the non-thermal particles results and this represents the injected particle distribution. As a shock proceeds, the magnetic field in the acceleration region can be amplified due to a build up of cosmic ray pressure or it can be 94 reduced due to magnetic reconnection events taking place. During magnetic reconnection, magnetic energy is converted into particle kinetic energy, causing the background magnetic field to decrease. Increased stochastic acceleration can also cause flaring events, see [38]. Starting from a shock, current driven or Kelvin-Helmholtz instabilities cause turbulence to develop in regions downstream. The turbulence cascades to smaller scales, increasing the density of magnetic inhomogeneities with which particles can have gyro-resonant interactions. This causes a decrease in the acceleration time scale due to gyro-resonant interactions of the particles with magnetohydrodynamic turbulence. Thus, four parameters are chosen to modify in both models for our light curve analysis: the particle injection luminosity, the particle spectral indices, the background magnetic field and the stochastic acceleration timescale. We modify the perturbations for the input parameters in the form of a Gaussian function in time:

2 2 −(t−t0) /2σ Lin j,i(t) = Lin j,0,i + KL,i · e (4.1)

2 2 −(t−t0) /2σ qi(t) = qi,0 + Kq,i · e (4.2)

2 2 −(t−t0) /2σ B(t) = B0 + KB · e (4.3)

tacc,0 tacc(t) = 2 2 (4.4) −(t−t0) /2σ 1 + Ktacc · e where i represents the particle species injected, the constant K represents the amplitude of

the perturbations. The terms Lin j,0,i, qi,0, B0 and tacc,0 represent the particle injection luminosity, spectral index, background magnetic ﬁeld and stochastic acceleration timescale during quiescence, respectively. The variable σ parametrizes the duration of the

perturbation in the comoving frame and t0 is the time where the perturbation peaks in our 95 simulation. The value of σ is related to the size scale and the shock speed. For an emission region of size R and a shock moving with a speed of vsh, the input parameters will change at a rate of σ ∼ tsh ≈ R/vsh. For our light curve analysis, we assume a shock speed of vsh ∼ 0.1c. This value is in accordance for shock speed values obtained from numerical simulations of strong shocks in plasmas [102]. The diﬀerent sizes of the emission region with the same shock speed produce diﬀerent values of σ for each blazar. The value used for σ determines the rise time for a light curve in a given bandpass. If the cooling time scale in a given bandpass is larger than σ, then the light curve will decay on a time scale of order of the cooling time scale. If, however, the cooling timescale is smaller, then the decay time scale of the light curve will be of order σ.

4.2 Light Curve Modeling of Leptonic Models (3C 273)

For one-zone leptonic models, one particle population (electrons/positrons) is responsible for the broadband emission. Inverse Compton scattering of the synchrotron or external photon fields produces the broadband emission from low energy X-rays to high energy γ-rays. Changing any one of the input parameters will have profound effects on the electron distribution. As a result, the spectral components will evolve on similar but different timescales. For our light curve analysis of 3C 273, we use the parameter set from our steady-state fit as our baseline model. Once the model has reached equilibrium, we apply perturbations to the magnetic field, acceleration time scale, and electron injection luminosity in order to investigate the effects of these perturbations in different bands. From the SED fits, the size of the emission region is R ≈ 1016 cm. For a shock speed of vsh ∼ 0.1c, this produces a perturbation time scale for our simulation of

6 σ ∼ R/vsh ≈ 3.0 × 10 s. As the code runs in time, ﬂuxes will be computed in the following bands: radio (73 MHz − 50 GHz), optical (R-band), X-ray (0.1 keV − 10 keV), and HE γ-rays (20 MeV − 300 GeV). 96

We start our light curve analysis by changing the magnetic ﬁeld in the form given by equation 4.3. We change the magnetic ﬁeld from its quiescent state of B = 0.5 G to

B(t = t0) = 2.5 G. An increase in the magnetic field produces an increase in the magnetic energy density. The higher magnetic energy density results in increased synchrotron emission, producing a flare in the optical R band. The higher magnetic energy density also causes an increase in the synchrotron self absorption coefficient, see equation 2.2.1. Higher SSA absorption of synchrotron photons results in a decrease in the emission of the radio bandpass. Raising the value of the magnetic field also produces stronger cooling for the electrons, see Figure 4.1. The stronger cooling produces a pile up of electrons at lower energies. The pile up produces an initial increase in the flux in X-rays in the (0.1 keV − 10 keV) band. However, as the B field continues to rise, the pile up produces a deficit of high energy electrons that upscatter the different photon fields in the emission region to higher energies. For the SSC component, the lower energy electrons still upscatter the increased synchrotron photon field. But the deficit of high energy electrons, due to increased cooling, causes the spectral component to decrease slightly, resulting in a decrease in X-ray emission. Similar to SSC, the deficit of high energy electrons causes reduced upscattering of the external photon fields to larger energies, producing a drop in HE γ-rays, see Figure 4.2. Following the magnetic field perturbation, we change the injection luminosity in the form given by equation 4.1. We change the electron injection luminosity from its

42 42 quiescent value of Lin j = 3.0 × 10 erg/s to Lin j(t = t0) = 9.0 × 10 erg/s. A temporary increase in the injection luminosity causes flares in all the bands across the SED. The synchrotron emission coefficient increases due to the injection of more electrons into the magnetic field, producing a flare in the optical R band. However the accretion disk contributes significantly to the flux in the optical R band, thus reducing the amplitude of the flux increase due to synchrotron radiation. Higher electron injection causes an 97

3C 273 EED: B Field Perturbation:

6 10

5 EED: Quiescent Fit 10 EED: Peak of B Field Perturbation

4 10

3 10 ]

-3 2 10 [cm 2 γ ) 1 γ

( 10 e n

0 10

-1 10

-2 10

-3 10 -1 0 1 2 3 4 5 6 7 10 10 10 10 10 10 10 10 10 γ

Figure 4.1: Electron energy distribution in the quiescent state and at the peak of the magnetic ﬁeld perturbation in the leptonic model scenario.

3C 273: B Field Perturbation:

Radio X-ray R Band HE γ-ray

1 0 F/F

0.1 5e+06 5.5e+06 6e+06 6.5e+06 7e+06 7.5e+06 8e+06 t - t0 [s] Figure 4.2: Normalized light curves for the magnetic field perturbation in the leptonic model scenario. Black dashed = Radio light curve; Red dashed = X-ray light curve; Green dashed = R Band light curve; Blue dashed = HE γ-ray light curve. 98 increased density of low energy electrons responsible for SSA, causing a weak flare in the radio bandpass followed by a slight drop, see Figure 4.3. The higher density of electrons also causes increased scattering of the external photon fields from the accretion disk and the BLR, generating a flare in the HE γ-rays. The increase in the synchrotron and EC components are the same since the emission coefficients are directly proportional to the number of electrons. However, SSC emission depends on both the electron distribution and the synchrotron photon field. The higher density synchrotron photon field is then upscattered by the higher density electrons to produce a large flare for the SSC component, see Figure 4.3.

3C 273: Linj Perturbation:

Radio X-ray R Band HE γ-rays 0 F/F

5e+06 5.5e+06 6e+06 6.5e+06 7e+06 7.5e+06 8e+06 8.5e+06 9e+06 t - t0 [s] Figure 4.3: Normalized light curves for the electron injection luminosity perturbation in the leptonic model scenario. Black dashed = Radio light curve; Red dashed = X-ray light curve; Green dashed = R Band light curve; Blue dashed = HE γ-ray light curve.

After the perturbation of the electron injection luminosity, we change the acceleration time scale as given by equation 4.4. The acceleration time scale is decreased from its 99 quiescent value by a factor of 30.0. A reduction in the stochastic acceleration timescale leads to more efficient acceleration of electrons/positrons to higher energies, see Figure 4.4. As the electron/positron distribution is shifted to higher energies, all the spectral components are shifter to higher frequencies. As low energy electrons are shifted to higher energies, the effective number density of electrons for SSA decreases. The decreased SSA absorption causes a prominent flare in the radio bandpass. Higher energy electrons and positrons interact with the magnetic field in the emission region, increasing synchrotron emission which generates a flare in the optical R band. As the electrons are being accelerated to higher energies, see Figure 4.4, they interact with the increased synchrotron photon field to produce higher energy SSC photons. The shift of the SSC component to higher frequencies produces an initial increase in the flux in the X-ray band, see Figure 4.5. However, as the SSC component continues to move to higher frequencies, the integrated X-ray flux then drops. As the acceleration time scale perturbation subsides, the SSC component shifts back to lower frequencies, increasing the flux until it reaches its original quiescent state. The higher energy electrons also interact with the external radiation fields boosted into the emission region, upscattering the photons to higher energies. This results in a prominent flare in the HE γ-ray band. In summary, changing the magnetic field, electron injection luminosity and the stochastic acceleration time scale produces several unique features exhibited by one-zone leptonic models: 1.) B Field Perturbation: Higher magnetic fields produce stronger SSA absorption and electron cooling. A higher SSA opacity results in a reduced flux of radio emission. Stronger electron cooling results in reduced upscattering of synchrotron (SSC) and external photon fields (EC). This produces a reduced flux in the X-ray and HE γ-ray bands. A higher magnetic field also produce increased synchrotron emission in the optically thin regime, resulting in a flare in the optical R band. 100

3C 273 EED: tacc Perturbation:

6 10

5 EED: Quiescent Fit 10 EED: Peak of tacc Perturbation

4 10

3 10 ]

-3 2 10 [cm 2 γ ) 1 γ

( 10 e n

0 10

-1 10

-2 10

-3 10 -1 0 1 2 3 4 5 6 7 10 10 10 10 10 10 10 10 10 γ

Figure 4.4: Electron energy distribution in the quiescent state and at the peak of the stochastic acceleration timescale perturbation in the leptonic model scenario.

2.) Lin j Perturbation: Higher injection luminosities produces stronger SSA absorption, linear variations between the synchrotron and EC components and a quadratic amplitude in the SSC component. A higher SSA opacity again results in a reduced flux of radio emission. Increasing the electron injection luminosity produces correlated optical and HE γ-ray flares. The quadratic amplitude relation in the SSC component from the increased electron injection results in prominent flaring in the X-ray band.

3.) tacc Perturbation: Decreasing the stochastic acceleration time scale produces weaker SSA absorption and a shifting of the spectral components to higher frequencies. Increased acceleration reduces the number of low energy electrons responsible for SSA absorption. The reduced SSA opacity results in a prominent ﬂare in the radio band. Higher energy electrons produced increased synchrotron and EC emission, resulting in ﬂares in the optical R and HE γ-ray bands. Higher energy electrons also produced higher 101

3C 273: tacc Perturbation:

Radio X-ray R Band HE γ-ray 0 F/F

5.4e+06 6e+06 6.6e+06 7.2e+06 7.8e+06 8.4e+06 9e+06 9.6e+06 t - t0 [s] Figure 4.5: Normalized light curves for the stochastic acceleration timescale perturbation in the leptonic model scenario. Black dashed = Radio light curve; Red dashed = X-ray light curve; Green dashed = R Band light curve; Blue dashed = HE γ-ray light curve.

energy SSC photons. The shifting of the SSC component to higher frequencies results in a decreased ﬂux in the X-ray band.

4.3 Light Curve Modeling of Lepto-Hadronic Models (3C 279)

In the context of one-zone lepto-hadronic models, proton synchrotron can explain the broadband emission from blazars from X-rays to high energy γ-rays. Large magnetic field of around ∼ 100 G are used to reproduce satisfactory fits to the SED of blazars, see Figure 3.3. Changing the magnetic field or the stochastic acceleration timescale, for example, will cause the spectral components to evolve in different ways and on different timescales when compared to the one-zone leptonic model. Similar to our light curve analysis to 3C 273, we use the parameter set from our steady-state fit to the SED of 3C 279 as our baseline model. Once the model has reached equilibrium, we apply perturbations to the 102

magnetic ﬁeld, acceleration time scale, proton spectral index and proton injection luminosity in order to investigate the eﬀects of these perturbations on the resulting multi-wavelength light curves in the context of one-zone lepto-hadronic models. For 3C

16 279, the size of the emission region is R ≈ 10 cm. For a shock speed of vsh ∼ 0.1c, this

6 produces a perturbation time scale for our simulation of σ ∼ R/vsh ≈ 2.0 × 10 s. As the simulations are running, we integrate the fluxes in the following bands: optical (R-band), X-ray (0.1 keV − 10 keV), HE γ-rays (20 MeV − 300 GeV), VHE γ-rays (30 GeV − 100 TeV) and IceCube (300 TeV − 1 EeV) for the expected electron neutrino flux from 3C 279 detected on Earth. We begin by changing the magnetic field in the form given by equation 4.3.A temporary increase in the magnetic field leads to a marked increase in the proton synchrotron (primarily HE γ-rays) and electron/positron synchrotron (IR – optical) spectral components. The corresponding increase of the synchrotron photon energy density leads to larger pion and muons production rates. The resulting pions and muons are also subjected to the increased magnetic field, thus strongly increasing the contribution of muon and pion synchrotron to the SED, see Figure 4.6. This increase in synchrotron emission from secondary particles leads to a distinct VHE γ-ray flare. The pions and muons cool very rapidly compared to the protons from the increased magnetic field, see Equation 2.10, resulting in a delay between the VHE and HE γ-ray bands. The temporary shift of the muon and pion synchrotron components to lower energies results in a drop of the flux in the VHE γ-ray band back to its equilbirum level. The protons responsible for the synchrotron emission in the HE γ-ray bandpass cool and return to their quiescent levels. The protons responsible for the synchrotron emission in the X-ray bandpass cool due to adiabatic losses and return to their original quiescent levels. The secondary electron/positrons generated from pair production also produce synchrotron emission in the optical band and cool along with the protons to their original levels before the flare. 103

The strong synchrotron cooling reduces the number of high energy protons responsible for pion and muon production, leading to a modest flare of neutrinos. The increased neutrino flux from the source that is approximately coincident with the secondary proton synchrotron flare in the X-rays.

3C 279: B Field Perturbation:

R Band X-ray HE γ--ray VHE γ-ray ν e Band 0 F/F

0 10

5.4e+06 5.7e+06 6e+06 6.3e+06 6.6e+06 6.9e+06 7.2e+06 7.5e+06 t - t0 [s] Figure 4.6: Normalized light curves for the magnetic ﬁeld perturbation in the lepto- hadronic scenario. Black dashed = R Band light curve; Red dashed = X-ray light curve; Green dashed = HE γ-ray light curve; Blue dashed = VHE γ-ray light curve; Magenta dashed = Electron neutrino light curve.

A perturbation in the proton injection luminosity causes the proton synchrotron emission to increase, producing a prominent HE γ-ray flare. This increase in both the number of protons and proton-synchrotron photons leads to strongly enhanced pion (and subsequently, muon) production rates. The muons and pions emit synchrotron radiation, producing a prominent VHE γ-ray flare. The pions and muons again cool more rapidly than the protons, see Equation 2.10, resulting in a delay between the VHE and HE γ-ray bands. As in the case of the magnetic-field perturbation, the protons responsible for the 104

synchrotron emission in the HE γ-ray band cool via synchrotron losses to their original levels. Protons responsible for the synchrotron emission in the X-ray band cool via adiabatic losses. The additional secondary electrons/positrons generated from pair production then produce a delayed synchrotron flare in the optical R band. The increase in the proton injection luminosity leads to a larger number of high energy protons responsible for secondary particle and neutrino production. The enhanced pion and muon decay rates again lead to a prominent flare of neutrinos that is concurrent with the secondary proton synchrotron flare in the X-rays.

3C 279: Linj Perturbation:

R Band X-ray HE γ-ray VHE γ-ray 1 ν 10 e Band 0 F/F

0 10

5.4e+06 5.7e+06 6e+06 6.3e+06 6.6e+06 6.9e+06 7.2e+06 7.5e+06 t - t0 [s] Figure 4.7: Normalized light curves for the proton injection luminosity perturbation in the lepto-hadronic scenario. Black dashed = R Band light curve; Red dashed = X-ray light curve; Green dashed = HE γ-ray light curve; Blue dashed = VHE γ-ray light curve; Magenta dashed = Electron neutrino light curve.

Following the injection luminosity perturbation, we decrease the stochastic acceleration time scale in the form of equation 4.4. With an increase in the stochastic acceleration efficiency, all particle distributions (protons, electrons/positrons, pions and 105 muons) are shifted to larger energies at the same rate. As protons are accelerated to higher energies, see Figure 4.8, the proton synchrotron emission increases, leading to a prominent HE γ-ray flare. The ultrarelativistic protons interact with the enhanced synchrotron radiation field, thus increasing the pion and muon production rates. As the pions and muons emit synchrotron radiation, they cool very rapidly and shift to lower frequencies, resulting in a drop in flux for the VHE γ-ray bandpass back to equilibrium, see Figure 4.9. The protons continue to emit synchrotron radiation, but cool at a slower rate, leading to a delayed HE γ-ray flare. The protons that produce synchrotron emission in the X-ray band cool via adiabatic losses and return to equilibrium. The combination of proton synchrotron and muon synchrotron in the FERMI band leads to a slightly delayed flare between the X-rays and the HE γ-rays. The synchrotron emission from the secondary electrons/positrons generated from the increased proton synchrotron photon field reduces the emission in the optical R band, returning to equilibrium. The acceleration of protons to higher energies, produces increased amounts secondary particles, leading to a prominent flare of neutrinos. Finally, after the acceleration time scale perturbation, we modify the proton spectral index in the form of equation 4.2. Due to the harder proton spectrum, see Figure 4.10, the primary proton synchrotron emission increases in the HE γ-ray band, producing a prominent flare. As in the case of the other perturbations discussed above, this also leads to increased pion and muon production rates, producing a prominent flare in the VHE γ-ray band. The increased number of secondary particles generated from photohadronic interactions cool very rapidly, shifting the synchrotron spectral component to lower frequencies. This shift to lower frequencies produces a drop in the VHE γ-ray flux back to its quiescent levels. The protons generating the synchrotron emission in the X-ray band cool via adiabatic losses, producing a delayed flare in that band. The combination of synchrotron emission from protons and muons causes the flux in the HE γ-ray band to be 106

3C 279 PED: tacc Perturbation:

6 10

PED: Quiescent Fit

PED: Peak of tacc Perturbation

5 10 ] -3

[cm 4 2

γ 10 ) γ ( p n

3 10

2 10 0 2 4 6 8 10 12 10 10 10 10 10 10 10 γ

Figure 4.8: Proton energy distribution in the quiescent state and at the peak of the stochastic acceleration timescale perturbation in the lepto-hadronic scenario.

slightly delayed with respect to the X-ray band. The secondary electrons/positrons produced from the increased proton synchrotron photon field produce a delayed flare in the optical R before the perturbation subsides and radiative equilibrium is re-established. The increased number of high energy protons from the harder spectra, also produces a prominent flare of neutrinos. In summary, changing the magnetic field, proton injection luminosity, stochastic acceleration time scale produces several unique features exhibited by one-zone lepto-hadronic models: 1.) B Field Perturbation: Higher magnetic fields produce stronger cooling for all particle distributions, but at different rates. Increasing the magnetic field produces increased synchrotron emission from all particle distributions, resulting flares in the optical R, X-ray, HE and VHE γ-ray bands. However, the stronger cooling for the muons and pions results in an initial VHE γ-ray flare, followed by a HE γ-ray flare. As the 107

3C 279: tacc Perturbation:

2 10

R Band X-ray HE γ-ray VHE γ-ray ν e Band

1 10 0 F/F

0 10

5.4e+06 5.7e+06 6e+06 6.3e+06 6.6e+06 6.9e+06 7.2e+06 7.5e+06 t - t0 [s] Figure 4.9: Normalized light curves for the stochastic acceleration time scale perturbation in the lepto-hadronic scenario. Black dashed = R Band light curve; Red dashed = X-ray light curve; Green dashed = HE γ-ray light curve; Blue dashed = VHE γ-ray light curve; Magenta dashed = Electron neutrino light curve.

secondary particles decay to produce electron/positron pairs, X-ray and optical ﬂares follow afterwards. A perturbation leads to a modest ﬂare for neutrinos. Stronger synchrotron cooling reduces the number of high energy protons responsible for photo-hadronic interactions that generate neutrinos.

2.) Lin j Perturbation: Higher injection luminosities produces correlated flares in all bands. The different cooling timescales for the particle distributions that make up the broadband emission reproduce the same order of delayed flares between the different bands as the magnetic field perturbation. Increased proton injection luminosity leads to more secondary particle production and decay, leading to a prominent flare of neutrinos.

3.) tacc Perturbation: Decreasing the stochastic acceleration time scale produces eﬃcient acceleration for all the particle distributions at the same rate. As the protons are accelerated to higher energies and generate pions and muons, the secondary particles cool 108

3C 279 PED: qp Perturbation:

6 10

PED: Quiescent Fit

PED: Peak of qp Perturbation

5 10 ] -3

[cm 4 2

γ 10 ) γ ( p n

3 10

2 10 0 2 4 6 8 10 12 10 10 10 10 10 10 10 γ

Figure 4.10: Proton energy distribution in the quiescent state and at the peak of the proton spectral index perturbation in the lepto-hadronic scenario.

very rapidly and shift to lower frequencies, producing a rapid flare in the VHE γ-ray band. The combination of synchrotron emission from protons and muons causes the HE γ-ray band to be slightly delayed with respect to the X-ray band. The HE γ-ray flux subsides and returns to equilibrium values. The electron/positron pairs generated from pair production then radiate in the optical R band an return to equilibrium. Increased proton acceleration leads to increased secondary particle production and decay, producing a flare in the neutrinos.

4.) qp Perturbation: Changing the proton spectral index produces a substantial increase in high energy protons. As with the previous perturbations, the high energy protons enhance the muon and pion production rates and synchrotron emission, leading to a VHE γ-ray ﬂare. The pions and muons are subjected to strong synchrotron cooling, returning to their equilibrium values. The protons responsible for the emission in the X-ray band cool due to adiabatic losses. The combination of proton and muon 109

3C 279: qp Perturbation:

2 10

R Band X-ray HE γ-ray VHE γ-ray ν e Band 1 10 0 F/F

0 10

5.4e+06 5.7e+06 6e+06 6.3e+06 6.6e+06 6.9e+06 7.2e+06 7.5e+06 t - t0 [s] Figure 4.11: Normalized light curves for the proton spectral index perturbation in the lept- hadronic scenario. Black dashed = R Band light curve; Red dashed = X-ray light curve; Green dashed = HE γ-ray light curve; Blue dashed = VHE γ-ray light curve; Magenta dashed = Electron neutrino light curve.

synchrotron emission leads to a delayed flare in the HE γ-ray band. Electron/positron pairs responsible for the synchrotron flux in the optical R band then cool and return to equilibrium. As with the previous two perturbations, an increased number of high energy protons produces a prominent flare of neutrinos.

4.4 Discrete Correlation Analysis

The light curves for the one zone leptonic and lepto-hadronic models possess correlated/anti-correlated behavior with time lags between different bands. In this section, we calculate the discrete correlation function (DCF) [39] between the light curves in the various bands for both models. The discrete correlation function gives more precise values of the time lags between different bands which we can compare with different cooling rates from the models. A discrete correlation function compares two time series as a 110

function of the time lag of one series relative to the other. Correlated time series have a DCF with a positive magnitude, uncorrelated time series have a magnitude of zero and anti-correlated time series have negative values of the DCF. In order to quantify the preferred values of the strength of the correlation (the maximum amplitude of the DCF) and inter-band time lag, a Gaussian ﬁt to the DCFs was performed that minimized the chi-square between the data set and a ﬁtting function of the form:

2 2 −(τ−τpk) /2σ DCF(τ) = F1 · e (4.5)

where τ represents the time lag between two sets of data. For this purpose, in order to be able to evaluate a χ2 value, we arbitrarily assumed a relative flux error of 1 % for each simulated light curve point when calculating the DCFs and their errors. For the one-zone leptonic model, we consider the time lag between the optical R band and the X-ray/HE γ-rays. For the lepto-hadronic model, we focus on the X-ray through γ-ray portion of the spectrum, and thus, on the DCFs between X-rays, HE γ-rays, and VHE γ-rays. This is largely motivated to compare the different cooling timescales in the different bands for both models used. The best fit parameters for the various flaring scenarios are listed in Table 3. The plots for the discrete correlation functions between the various bands can be found in Section A.2. The time lags between any two bands can be estimated from the different cooling rates of the particle populations that emit the light in the given bands. For example, in our one-zone leptonic model, we changed the electron injection luminosity from its quiescent

42 42 value of Lin j = 3.0 × 10 erg/s to Lin j(t = t0) = 9.0 × 10 erg/s. Electron synchrotron emission is responsible for the emission seen in the optical R band while SSC emission is responsible for the emission in the X-ray band. The synchrotron cooling timescale for a given electron with Lorentz factor γ can be found from equation 2.10: 111 Table 4.1: Best-fit parameters of the Gaussian fit to the discrete correlation functions between selected bands of light curves for the leptonic fit of 3C 273. Negative values for the normalization indicate an anti-correlation between bands. Negative values for the peak time indicate a lag between the first and second bands.

Bands Scenario F1 σ[s] τpk[s] R-X B −0.37 (3.86 ± 1.82) × 105 (3.09 ± 1.58) × 105 R-HE B −0.41 (3.30 ± 1.45) × 105 (3.78 ± 1.11) × 104

5 5 R-X Le,inj 0.54 (2.28 ± 0.98) × 10 (1.42 ± 0.83) × 10

5 4 R-HE Le,inj 0.53 (1.96 ± 0.62) × 10 (2.22 ± 1.86) × 10

5 5 R-X tacc −0.42 (3.32 ± 0.94) × 10 (2.86 ± 0.79) × 10

5 4 R-HE tacc 0.57 (1.60 ± 0.39) × 10 (0.58 ± 4.25) × 10

Table 4.2: Best-fit parameters of the Gaussian fit to the discrete correlation functions between selected bands of light curves for the lepto-hadronic fit of 3C 279. Negative values for the normalization indicate an anti-correlation between bands. Negative values for the peak time indicate a lag between the first and second bands.

Bands Scenario F1 σ[s] τpk[s] X-HE B 1.01 (2.29 ± 0.39) × 105 (−3.13 ± 2.75) × 104 HE-VHE B 1.00 (2.40 ± 0.35) × 105 (−1.53 ± 2.96) × 104

5 4 X-HE Lp,inj 1.00 (2.44 ± 0.41) × 10 (−0.88 ± 2.96) × 10

5 4 HE-VHE Lp,inj 1.00 (2.29 ± 0.34) × 10 (−3.70 ± 2.78) × 10

5 4 X-HE tacc 0.98 (1.94 ± 0.24) × 10 (0.36 ± 2.24) × 10

5 5 HE-VHE tacc 1.01 (1.55 ± 0.19) × 10 (−4.91 ± 2.11) × 10

5 4 X-HE qp 0.99 (2.09 ± 0.29) × 10 (1.68 ± 2.56) × 10

5 4 HE-VHE qp 0.99 (1.83 ± 0.24) × 10 (−3.66 ± 2.36) × 10

γ 6πm c2 t e cool = = 2 (4.6) |γ˙ syn| cσT B γ 112

The SSC cooling rate can be found from the energy density of the synchrotron photon ﬁeld:

4 u γ − cσ ˙SSC = T 2 (4.7) 3 mec

where u = uB + usyn = uB(1 + k) represents the sum of the magnetic and synchrotron energy densities in the emission region in terms of the Compton parameter k. The Compton parameter, k, can be found from the ratio of the energy densities between the magnetic ﬁeld and the synchrotron radiation ﬁeld that is Compton scattered

Lsyn/LC = uB/usyn. Rewriting the SSC cooling rate in terms of the Compton parameter, the SSC cooling timescale is then given by the relation:

6πm c2 t e cool = 2 (4.8) cσT B (1 + k)γ

obs 6 2 From the synchrotron frequency, νsyn = 4.2 × 10 B(G)δγ Hz, we ﬁnd that electrons with Lorentz factors of γ ≈ 3.8 × 103 are responsible for the synchrotron radiation in the R

obs obs 2 band. From the SSC frequency, νSSC = νsynγ , we ﬁnd that electrons with Lorentz factors of γ ≈ 4.2 × 102 are responsible for the SSC emission in the X-ray band. At the peak of the electron injection luminosity perturbation, the observed ratio between the SSC and

synchrotron peaks is LC/Lsyn = k ≈ 2.63. The time delay in the comoving frame between the optical R and the X-ray bands is then:

6πm c2 1 1 tCM e − ≈ . × 6 s delay = 2 ( ) 1 20 10 (4.9) cσT B γSSC(1 + k) γsyn

obs CM 4 The observed time delay is then tdelay ≈ tdelay/δ ≈ 8.5 × 10 s, which is within the error limits of the time lag between the X-ray and optical R bands from our discrete correlation analysis, see Table 4.1. 113

For the one-zone lepto-hadronic model, synchrotron emission from different particle populations produces the broadband emission. The different synchrotron cooling rate for the particle distributions generates the time lags between the different bands. For example, we change the magnetic field from a quiescent strength of B = 150 G to a peak strength of

B(t = t0) = 300 G. At the peak of the perturbation, protons with Lorentz factors of γ ≈ 1.4 × 108 are responsible for the synchrotron emission around ν ≈ 3.0 × 1023 Hz in the Fermi band. Similarly, muons with Lorentz factors of γ ≈ 7.4 × 108 produce synchrotron emission around ν ≈ 7.5 × 1025 Hz in the VHE γ-ray band. The time delay in the comoving frame is then found by the diﬀerence in the synchrotron cooling timescales:

6πm c2 1 m 1 m tCM e p 3 − µ 3 ≈ . × 5 s delay = 2 ( ( ) ( ) ) 3 77 10 (4.10) cσT B γp me γµ me

obs CM 4 The observed time delay between the two bands is then tdelay ≈ tdelay/δ ≈ 1.77 × 10 s, which is consistent with the time lags between the two bands found through the discrete correlation analysis, see Table 4.2.

4.5 Modeling the Exceptional Nov. 2010 Flare of 3C 454.3

3C 454.3 has exhibited many outbursts in the pre Fermi era. In 2005, 3C 454.3 displayed a prominent optical outburst, reaching its historical maximum with R = 12.0, [109]. Follow up observations at X-ray wavelengths with INTEGRAL [89], Swift [43] and observations made with AGILE [106] as well as a flare in the radio band with about a one year delay [110]. After the launch of Fermi, in June 2008, 3C 454.3 has exhibited many large bursts in γ-rays and across the whole electromagnetic spectrum. 3C 454.3 displayed prominent outbursts in late 2009, April 2010 and late 2010 [7, 24, 92, 113]. On October 2010, observers noticed that 3C 454.3 was undergoing a pronounced flaring near IR wavelengths [31]. The flaring was followed up in optical, UV, X-ray and γ-ray band passes [107]. In the following month, 3C 454.3 obtained its highest flux in γ-rays, peaking 114

−5 −2 −1 around November 19-20 with a flux of Fγ(E > 100 MeV) = (6.8 ± 1.0) × 10 ph cm s [6, 95, 101]. During this period, 3C 454.3 was extensively monitored in the radio, IR, optical and X-ray bands. Observations before the main γ-ray flare showed an increase in the form of a plateau, days before the main γ-ray flare occurred [112]. However, cross correlation analysis of the flares between the Fermi γ-ray band and the 160 µm, 1.3 mm bands from Herschel shows no delays between them [113]. With the extensive coverage of this flaring event, careful analysis can be done to understand the conditions of the emission site of the flare and the physical mechanism through which the flare was generated. In this section, I use my one-zone leptonic and lepto-hadronic models to reproduce the SED of 3C 454.3 during the peak of the Nov 2010 flare. Once a satisfactory SED fit can be obtained with either model, I simulate flaring scenarios to generate light curves in the optical R, Swift XRT and Fermi γ-ray bands to reproduce the fluxes during the peak of the flare. No extensive simultaneous SED data was collected during the periods of enhanced activity that occurred both before and after the main flare on November 19-20. As a result, broadband SED fits to 3C 454.3 cannot be performed during these enhanced periods. Without extensive SED coverage, attempting to simulate the periods of enhanced activity in the optical R, Swift XRT and Fermi γ-ray bands introduces additional free parameters for both models, making it difficult to ascertain which model can best produce the periods of enhanced activity. As a result, the flare is modeled from the quiescent state to the peak state and the periods of enhanced activity are neglected.

4.5.1 Leptonic Light Curve Modeling of 3C 454.3

From the four input parameters selected to modify, a suitable combination is desired to reproduce both the SED of 3C 454.3 and the light curves in the optical R, Swift XRT and Fermi bandpass during the Nov 2010 ﬂare for our one-zone leptonic model, see Figures 3.6. The combination of perturbations has to reproduce the similar levels of ﬂux 115

increase in the optical and HE γ-ray bands and the much weaker ﬂux increase in the X-ray band of the SED, see 3.6. We begin by decreasing the acceleration time scale from its background value to tacc(t = t0) = tacc,0/35. This decrease is motivated to ensure that the acceleration timescale in the comoving frame drops below the cooling timescale in the same frame for electrons producing synchrotron emission in the optical bandpass,

5 tcool ≈ 10 s. The decrease in the acceleration timescale causes the electrons to be accelerated to higher energies. As a result, the synchrotron emission, SSC and external Compton components are shifted to higher energies, and the SSC-dominated X-ray ﬂux drops. Higher energy electrons also produce increased external Compton emission, see [37]. The decreased acceleration timescale also changes the electron energy distribution, causing the synchrotron emission to display a pronounced curvature in the spectrum. Shifting the electrons to higher energies also causes the spectral components to become narrower since the low energy tail of the electron energy distribution is disrupted by the acceleration. The optical and high energy γ-ray emission increases while the X-ray emission subsequently decreases. To oﬀset the decrease in X-ray emission from altering the acceleration time scale, we

43 −1 increase the electron injection luminosity from Lin j,0,e = 2.5 × 10 erg s to

44 −1 Lin j,e(t = t0) = 8.25 × 10 erg s . This subsequently causes all spectral components to increase in flux. The effect is most pronounced in the X-rays since the SSC flux scales as

2 ne. Increasing the electron injection luminosity also causes the non-thermal and thermal particle densities to increase. This, in turn, causes the Alfven velocity of the plasma to decrease, resulting in decreased diffusion due to stochastic acceleration. The result is that both the electron synchrotron and SSC spectral components reach their peak levels for the SED, but the EC emission from the accretion disk and BLR still underpredicts the flux levels in the γ-ray bandpass. Increasing the electron injection causes the flux levels in the optical R and X-ray to overshoot 116

The background magnetic ﬁeld is then decreased from B0 = 1.5 G to

B(t = t0) = 0.5 G. Weakening the magnetic field inhibits synchrotron cooling for the highest energy electrons and causes them to be accelerated to even higher energies. A magnetic field decrease can represent a magnetic reconnection event where the magnetic energy is converted into particle kinetic energy. With the lower B field, the synchrotron emission decreases as a result. This, in turn, causes the SSC emission to decrease as well. The higher energy electrons push the external Compton scattering of the accretion disk to higher fluxes and energies. Decreasing the magnetic field any further causes the model to over-predict the Fermi γ-ray flux, while at the same time under predicting the flux in the optical R and Swift XRT band passes. From the fits, we find that changing the spectral index of the electron injection is unnecessary. Since the electrons are already heavily accelerated, changing the electron spectral index will only increase the number of high energy electrons rather than the flux levels in the HE γ-ray band. With the chosen combination of variations of the magnetic field, acceleration time scale and electron injection luminosity, see Table 4.3, we obtain flux levels that are in rough agreement with the SED of 3C 454.3. However, the spectral fits in the X-ray and HE γ-rays are inconsistent with the observations of the SED during the peak of the flare. While the decreased acceleration timescale is able to shift the SSC and EC components to higher energies, the decreased number of low energy electrons depletes the low energy SSC and EC photons, generating poor spectral fits to the X-ray and HE γ-ray bands during the peak of the flare. A high acceleration efficiency is necessary to upscatter photons from external sources to the flux levels of the Nov 2010 flare. Reducing the acceleration efficiency would improve the fits to the X-ray spectra, but it then under predicts the flux levels in the HE γ-ray band during the peak of the flare. A reduced magnetic field is also necessary since weaker synchrotron cooling produces even more energetic electrons to upscatter the photons in the emission region. Increasing the magnetic field produces 117

3C454.3

15 10

14 10

13 10

12 [Jy Hz]

ν 10 F ν

11 10

10 10

9 10 10 12 14 16 18 20 22 24 26 10 10 10 10 10 10 10 10 10 ν [Hz] Figure 4.12: Broadband fit to the SED of 3C 454.3 using our leptonic model during the quiescent and flaring states. The quiescent-state data points included in the fit are plotted in red [3]; additional, archival data are plotted in grey. Broadband data during the Nov 2010 flare are plotted in cyan [113]. The model curves are: Black solid = total spectrum; Red dashed = synchrotron emission from electrons/positrons; Green dashed = synchrotron self Compton emission; Blue dashed = thermal emission from accretion disk; Magneta dashed = EC emission from accretion disk; Indigo dashed = EC emission from BLR.

stronger cooling and less high energy electrons, inhibiting Compton scattering processes. The reduced synchrotron emission in the IR/optical/UV is then offset by a larger electron injection luminosity. The strength of the perturbations used in the SED fits for our one-zone leptonic model is given in Table 4.3. The resultant fits to the SED during the peak of the flare is given in Figure 4.12. With these considerations, the one-zone leptonic model can not reproduce the SED fits during the peak of the Nov 2010 flare for 3C 454.3. This result, motivates us to perform a second light curve analysis on 3C 454.3 using our one-zone lepto-hadronic model. 118 Table 4.3: 3C 454.3 model light curve fit parameters for the leptonic and lepto-hadronic models. The negative value for the perturbation of the particle spectral index indicates spectral hardening. Conversely, a positive value indicates a spectral softening.

−1 Scenario KL [erg s ] Kq KB [G] Ktacc Electron (Leptonic) 8.0 × 1044 − −1.0 34.0 Proton (Lepto-Hadronic) − −0.3 −50.0 3.0 Electron (Lepto-Hadronic) 4.5 × 1043 − −50.0 3.0

4.5.2 Lepto-Hadronic Light Curve Modeling of 3C 454.3

For a one-zone lepto-hadronic model, different particle distributions produce the non-thermal emission in different band passes. Changing any one of the four originally selected input parameters produces different effects to the individual particle distributions in comparison to the leptonic model. Following the same procedure as our one-zone leptonic model, we decrease the stochastic acceleration time scale to tacc(t = t0) = tacc,0/4. Since the magnetic field is much higher and the stochastic acceleration diffusion coefficient is larger, a modest change to the acceleration timescale is needed to accelerate protons to larger energies in comparison to the leptonic model. Decreasing the acceleration time scale causes low energy protons to pile up at higher energies, producing flares in the X-ray and HE γ-ray bands. The higher energy protons interact with the increased proton synchrotron photon field and produce more energetic pions and muons, which then decay to produce high energy electrons/positrons. The increased amount of synchrotron radiation from both the muons and pions produces a major VHE γ-ray flare beyond 20 GeV. Unfortunately, no observations were carried out with HESS or MAGIC that could have potentially detected flaring activity in the VHE γ-ray band. Altering the stochastic acceleration timescale produces no effect on the electron distribution responsible for the optical emission. This is due to the large magnetic field present in the 119 emission region and the extremely short electron cooling timescales that dominate the evolution of the particle distribution in the Fokker-Planck equation. This signature is unique to lepto-hadronic models in that an X-ray and γ-ray flare can present itself while leaving the optical emission unaffected by simply changing the stochastic acceleration timescale. In order to offset the unchanged level emission from electrons, we increase the

42 −1 electron injection luminosity from Lin j,0,e = 3.64 × 10 erg s to

43 −1 Lin j,e(t = t0) = 4.9 × 10 erg s . This has the effect of increasing the electron synchrotron radiation to produce a strong flare in the IR/optical/UV bands. As with the leptonic model light curve analysis of 3C 454.3, we find that it is unnecessary to change the electron spectral index during the peak of the flare. A harder or softer spectral index will worsen the spectral fits during the peak of the flare on the SED. However, changing the proton spectral index is indeed necessary to improve the fits for the harder spectral index from the X-rays to high energy γ-rays, see Figure 4.13. Using a common spectral component between the X-rays and HE γ-rays and producing a harder index, from qp = 2.2 to qp(t = t0) = 1.9 produces a strong flare in the HE γ-ray band with a moderate flare in the X-ray band. The choice of changing the proton spectral index is able to vastly improve the fits of the SED during the flare in comparison to the one-zone leptonic model. For a developing shock that is responsible for a flare, an increased compression ratio can lead to a harder spectral index, which in turn, reproduces the harder synchrotron spectra seen in the peak SED of the flare. The proton spectral index and acceleration time scale are adjusted to better improve the quality of the fits to the SED during the peak of the flare. With the change in the acceleration timescale, the protons move to higher energies, but produced poor quality fits to the location of the observed frequency of the peak of the flare in the γ-rays. As a result, we decrease the background magnetic field from

B = 125 G to B(t = t0) = 75 G. A reduced magnetic field can indicate magnetic 120 reconnection events taking place in the acceleration zone in the jet producing the flare. Lowering the background magnetic field, leads to reduced synchrotron cooling, producing even more energetic electrons and protons. The combination of a reduced magnetic field and increased particle acceleration causes the proton synchrotron spectra to peak at

pk 23 νobs ≈ 1.5 × 10 Hz, consistent with the peak flux levels seen in the SED fit. Lowering the magnetic field also has the added effect of lowering the proton and electron synchrotron flux. The acceleration time scale and electron injection luminosity are adjusted to offset the reduction in the magnetic field until a satisfactory fit is obtained for the flux levels in the broadband SED during the peak of the flare.

3C454.3

15 10

14 10

13 10 [Jy Hz] ν

F 12 ν 10

11 10

10 10

10 12 14 16 18 20 22 24 26 10 10 10 10 10 10 10 10 10 ν [Hz] Figure 4.13: Broadband ﬁt to the SED of 3C 454.3 using our lepto-hadronic model during the quiescent and ﬂaring states. The model curves are: black solid = total spectrum; red dashed = proton synchrotron emission; green dashed = synchrotron emission from electrons/positron; blue dashed = thermal emission from accretion disk; magenta dashed = pion synchrotron; violet dashed = pion synchrotron.

With the variations of the magnetic ﬁeld, acceleration time scale, electron injection luminosity and proton spectral index, see Table 4.3, we obtain ﬁts to the SED that are in 121

good agreement for the Nov 2010 flare, see Figure 4.13. A shock scenario that produces a harder proton spectral index, reproduces the spectral fits to the SED of 3C 454.3 in the X-rays and γ-rays during the peak of the Nov 2010 flare. Increasing the electron injection leads to a prominent flare in the IR/optical/UV bands of the SED that is in good agreement with the data. Decreasing both the magnetic field and acceleration time scale produces more energetic protons that reproduce the peak proton synchrotron spectra in the HE γ-rays. With these improved SED fits, we can then integrate the fluxes in the Swift R, XRT and Fermi γ-ray bands to model the light curves of the flare. The light curves in all three bands rise on a time scale of order of the shock timescale in the observer’s frame

5 trise,obs ∼ σ/δ ≈ 1.6 × 10 s ≈ 2 days. As the perturbations subsides, the particles cool in the Fermi γ-ray bandpass with a synchrotron cooling timescale in the observer’s frame of

5 tcool,obs ≈ 3.5 × 10 s ≈ 4 days, consistent with the decay timescale for the light curve in the Fermi bandpass. In our model, we considered both radiative and adiabatic losses. At the highest energies, protons lose energy due to synchrotron losses, but at lower energies, adiabatic losses dominate. As the acceleration timescale drops below the adiabatic loss timescale in the comoving frame, the lower energy particles are accelerated, producing a ﬂare in the X-ray band. Once the perturbation subsides, the particles then cool on a time

5 scale of order of the adiabatic loss timescale, tad,obs ≈ 2.5 × 10 s ≈ 3 days, in rough agreement with the decay timescale of the light curve in the Swift XRT bandpass during the ﬂare. The electrons producing IR/optical/UV radiation cool via synchrotron losses at

2 an extremely fast rate, tcool,obs ≈ (1 − 5) × 10 s. As a result, the optical R band decays on a timescale similar to the shock timescale for the electron injection luminosity,

5 tdecay,obs = σ/δ ≈ 1.6 × 10 s ≈ 2 days. The ﬁts to the light curves in the Fermi, Swift XRT and R band passes are shown in Figures 4.14, 4.15, and 4.16. With these considerations, I conclude that the one-zone lepto-hadronic model can best explain the SED and peak 122

fluxes in the optical R, Swift XRT and Fermi γ-ray bands from the quiescent state to the flare state for the Nov 2010 flare of 3C 454.3.

FERMI Flare (Nov 2010):

FERMI Band 0.0001 Model Fit (Lepto-Hadronic) ] -1 s -2 F [ph cm 1e-05

55512 55514 55516 55518 55520 55522 55524 55526 MJD Figure 4.14: Light curve ﬁts between the data [113], and our lepto-hadronic model in the Fermi bandpass (20 − 300) GeV. 123

Swift XRT Flare (Nov 2010):

XRT Band Model Fit (Lepto-Hadronic) ] -1 s -2

1e-10 F[erg cm

55510 55512 55514 55516 55518 55520 55522 55524 55526 55528 MJD Figure 4.15: Light curve ﬁt between data and our lepto-hadronic model in the Swift XRT bandpass (0.2 − 10) keV.

R Band Flare (Nov 2010):

2 10

R Band Lepto-Hadronic Model

1 10 F [mJy]

0 10 55512 55514 55516 55518 55520 55522 55524 55526 55528 MJD Figure 4.16: Light curve ﬁt between the data and the lepto-hadronic model in the R band. 124 5 Discussion

One zone leptonic and lepto-hadronic models have had great success in reproducing the broadband SEDs of blazars. Both models use different sets of input parameters to produce nearly identical fits to the multi wavelength spectrum, making it difficult to determine which model can best explain all attributes of blazar emission. In this work, I have developed time dependent one-zone leptonic and lepto-hadronic models in order to investigate the differences between the models under different flaring scenarios. With my one zone leptonic and lepto-hadronic models, I was able to reproduce the broadband SEDs of the blazars 3C 273, 3C 279 and 3C 454.3 using input parameters that are similar to previous one-zone models that have been done [30, 41]. My lepto-hadronic model incorporates the radiation emitted from secondary particles that are created from photohadronic interactions between protons and photons in the emission region. This approach differs from conventional hadronic models that neglect the radiation from pions and muons and their contribution to the SED of blazars [30, 53]. This allows me to extend the parameter range of higher magnetic fields and proton Lorentz factors that are expected for one-zone lepto-hadronic models. The inclusion of synchrotron radiation from secondary muons and pions in my lepto-hadronic code can reproduce the high energy shoulders in the SED beyond 20 GeV in the fits of 3C 279 and 3C 454.3, see Figures 3.4 and 3.7. My time dependent one-zone lepto hadronic model also considers the expected neutrino flux from blazars based on the neutrino production rates from the decay of secondary particles created from photohadronic interactions. The fits to the SEDs of 3C 273, 3C 279 and 3C 454.3 serve as a starting point to investigate differences between the one-zone leptonic and lepto-hadronic models under a set of flaring scenarios in my light curve analysis. I chose four input parameters to modify to simulate flaring events: (a) the magnetic field, (b) the electron and proton injection luminosities, (c) the acceleration time scale, and (d) the particle injection spectral index. 125

The choice of input parameters selected for the light curve analysis is based on the expected physics during the production of a strong shock. The selected input parameters have been modeled in the form of a Gaussian function in time, see equations 4.1-4.4. From my analysis, I found that perturbations indicative of magnetic field amplification produce profound differences in the X-ray and HE γ-ray bands that can distinguish leptonic and lepto-hadronic models. The difference arises in the spectral components responsible for the emission in the X-ray and high energy γ-ray bands for both models. In the leptonic scenario, low energy SSC photons produce the emission in the X-ray band while EC photons produce the emission in the HE γ-ray band. In the hadronic scenario, proton synchrotron produces the emission from X-rays to high energy γ-rays. In the leptonic scenario, increasing the magnetic field depletes the number of high energy electrons necessary for Compton scattering of radiation fields in the emission region. This will result in a drop in the SSC and EC components and a drop of flux in the X-ray and HE γ-ray bands. Conversely, for the lepto-hadronic model, an increase in the magnetic field will produce increased proton synchrotron emission, generating flares in the X-ray and high energy γ-ray bands. In my light curve analysis, I also found distinct differences from decreasing the acceleration timescale between both models. In the leptonic scenario, increasing the acceleration efficiency shifts the electrons to higher energies. This results in the spectral components that produce the SED to shift to larger frequencies. I found that the SSC component shifting to larger frequencies results in a drop in the flux in the X-ray band. Conversely, in the lepto-hadronic scenario, decreasing the acceleration timescale causes more energetic protons to produce more synchrotron emission, resulting in an increase in flux in the X-ray band. These two perturbations cause unique signatures that can distinguish the models apart from one another. The light curve analysis also revealed interesting signatures for the neutrino flux. As a result of increasing the magnetic field, the 126

stronger synchrotron cooling depletes the number of high energy protons. For this reason, secondary particles created from photohadronic interactions decay to produce a modest flare in comparison to other perturbations investigated. This contrasting behavior between the magnetic field perturbation and the other three investigated could represent a signature of strong proton cooling taking place during a flare. These light curve investigations motivated a comprehensive study on the blazar 3C 454.3. Both my leptonic and lepto-hadronic models were able to reproduce the broadband SED, exemplifying the difficulty in distinguishing which model could best explain the emission. However, 3C 454.3 exhibited a prominent γ-ray flare on Nov 2010 that was also observed in the optical R and X-ray. No extensive simultaneous SED data was collected during the enhanced state seen in the Fermi γ-ray band both before and after the main flare on November 19-20. As a result, the flare is modeled from the quiescent state to the peak state and the enhanced states are neglected. With the extensive SED coverage of the main flare, 3C 454.3 served as a suitable target for my light curve analysis. From the light curve analysis, I found that while the leptonic model was able to produce flux levels in the optical R, Swift XRT and Fermi bands that were consistent with the Nov. 2010 flare, it was unable to reproduce the broadband SED for 3C 454.3 during the peak of the flare. The poor fits to the X-ray spectra were largely due to the increased acceleration efficiency used for the fits. Decreasing the acceleration efficiency would have resulted in a drop in the EC component, hindering the fits to the Fermi band. These results make it unlikely that a leptonic scenario can best explain the features of this flare. However, the lepto-hadronic model was able to reproduce both the broadband fits and the flux levels in the same bands. The improved spectral fit was due to a harder proton spectral index during the course of the flare. The harder spectral index provided satisfactory fits in the optical R, Swift XRT and Fermi bands at the peak of the flare. These results suggest that a one-zone lepto-hadronic model can explain the features exhibited by the blazar 3C 454.3 from its 127

quiescent state to the peak of the flare that occurred on November 19-20. This case study shows that while both leptonic and lepto-hadronic models can reproduce the SED during quiescence, flaring events can reveal which model is best suited to explain all the features of a given blazar. Time dependent leptonic and lepto-hadronic models have been instrumental in explaining many features of blazar emission. Recent multi-wavelength observations of FSRQs have shown correlations between different wavelength bands that can be attributed to the flaring scenarios discussed here. Multi-wavelength observations of the FSRQ 3C 454.3 from August-December 2008 have shown pronounced flaring activity in the IR, UV, X-ray and γ-ray bands with correlations for all bands except the X-rays [22]. These correlations are consistent with a model in which a change in the injection luminosity of higher energy electrons takes place and interacts with external photons, causing the flaring observed in the γ-rays [22]. The much longer cooling time of the low-energy electrons responsible for the X-ray emission leads to much delayed variability, on much longer time scales compared to the optical and γ-ray bands, which might be washed out by super-imposed longer-term variability. Correlated multi-wavelength campaigns have also been done on the FSRQ 3C 273 that reveal a correlation between the IR and X-ray bands, with time lags on the order of a few hours [67]. This is consistent with the results presented here and supports the notion that the X-ray emission is dominated by synchrotron self Compton radiation [67]. Bonnoli et al. (2011) used a one-zone leptonic model to explain a very prominent flare for 3C 454.3 during the first week of December 2009. This flare exhibited the behavior that the γ-ray emission doubled relative to the optical emission. This observation conflicted with the EC mechanism in which the synchrotron and EC emission should rise in direct proportion to each other since the emission coefficients are directly proportional to the electron spectrum. These authors were able to reproduce the large γ-ray flare by changing the particle injection luminosity 128 by a factor of 10 and increasing the bulk Lorentz factor from 15 to 20. The behavior of flux doubling between the optical and high energy γ-rays was attributed to decreasing the magnetic field as the overall jet luminosity got stronger. A decrease in the magnetic field and increase in particle injection luminosity have also been invoked in the modeling of flaring events for 3C 279 in the Fermi γ-ray bandpass [13]. 129 6 Outlook

One-zone time dependent modeling of blazar jets using leptonic and lepto-hadronic models gives a wealth of information on the acceleration mechanisms, radiative properties and the evolution of the particle distributions that produce the broadband emission. Perturbations to any one of the input parameters can affect the spectral components from both models that produce the SEDs of blazars. The evolution of the spectral components from the same perturbation applied can produce different patterns in the same wavelength band, distinguishing which model best explains a given flare. However, not any single perturbation is enough to sufficiently explain both the SED and the flux in different bands for any given flare. A combination of perturbations is needed to explain all the features exhibited by a blazar during a given flare. A self consistent approach that incorporates the development of a shock in a plasma and considers how the shock energy is distributed between the magnetic field and MHD turbulence can put constraints on how strong the perturbations for the magnetic field and stochastic acceleration efficiency need to be to model a given flare of a blazar. Particle-in-cell simulations have shown magnetic field generation in relativistic collisionless shocks [86, 100]. MHD simulations of inhomogeneous media have also shown that magnetic field amplification depends on the direction of the preshock magnetic field and that the time scale for magnetic field growth depends on the strength of the shock [72]. Magnetic fields perpendicular to the shock flow will have more shock energy partitioned to the magnetic field since the initial magnetic field is already compressed by the shock before any amplification from turbulence can occur. MHD shock simulations with different magnetic field geometries can help understand how the magnetic field and acceleration time scales are correlated and how they evolve. MHD shock simulations can also help reveal how particle spectral indices evolve and how they are correlated with shock strength and magnetic field amplification. These correlations would eliminate the number of free parameters to simulate a shock, 130 placing greater constraints on the dynamics of the shock that produce a blazar flare. Coupling MHD simulations with one-zone leptonic and lepto-hadronic models can reveal new and interesting clues on how to distinguish which model is best suited to explain flaring events for blazars. Fermi II acceleration represents a unique ingredient in time dependent modeling of particles in jet environments. The standard approach to modeling stochastic acceleration of non-thermal particles is done by solving the transport equation that incorporates acceleration, radiative losses, injection and escape. A wide variety of different solutions exist for different astrophysical settings [96]. Dermer et al. 1996 considered the stochastic acceleration of particles in black hole magnetospheres and showed that the diffusion

p p−2 coefficient and escape time scaled as D(γ) ∝ γ and tesc ∝ γ with respect to particle energy, where p represents the spectral index of the MHD turbulence [35]. In our analysis, we investigated Fermi II acceleration between particles and MHD waves with a turbulent spectral index of p = 2, indicative of hard sphere scattering. This produced a diffusion coefficient that scaled as D(γ) ∝ γ2 and an escape time scale that was independent of particle energy. However, particle distributions in the emission region can interact with magnetohydrodynamic waves with different turbulent spectral indices, such as a Kolomogorov p = 5/3, or a Kraichnan p = 3/2 spectrum. These different spectral indices produce energy dependent escape time scales that can affect how much radiation particles emit before they escape the emission region [16]. Modeling the diffusion coefficient with Kraichnan or Kolomogorov MHD spectral indices can also produce different energy dependent acceleration time scales [36]. In this study, we have focused on the stochastic acceleration of particles by Alfven waves. Fermi II acceleration by Alfven waves has been used in a number of time dependent leptonic models investigate variability features of blazars, such as 1 ES 1011+496 [114] and Mrk 421 [12, 13]. Fermi II acceleration has also been invoked to explain flaring properties of the blazar PKS 0208-517 in optical and 131

γ-ray bands [32]. However, Alfven waves represent only a branch of the cold plasma dispersion relation of transverse electromagnetic waves propagating in a plasma. Whistler waves represent another branch that charged particles can interact with and be accelerated to high energies. Stochastic acceleration by whistler waves has been considered in a number of different works to model ion and electron acceleration in solar flares [66, 70, 71] and ion transport in the accretion disks around super massive black holes [35]. However, time dependent simulations that consider the evolution of particle spectra from the interaction of whistler waves have not been extensively investigated. Time dependent models of blazars with different spectral indices of MHD turbulence that also take into account both Alfven and whistler wave acceleration can potentially produce unique features that could help distinguish leptonic and lepto-hadronic models. In this study, I explored the nature of time dependent modeling of blazars in the setting of a single emission region. This setup does not consider light travel time effects from multiple zones that interfere to produce the integrated emission seen by an observer of a blazar. Time dependent multi-zone leptonic models have been invoked to explain blazar emission and features during flaring episodes. Marscher (2014) developed a turbulent extreme multi-zone leptonic model to explain flaring features exhibited by 3C 454.3 [113]. The model consisted of a Mach disk interacting with a conical shock and compressing the pre shock magnetic field. The compressed magnetic field produced synchrotron emission. The synchrotron radiation and emission from a dusty torus were upscattered to produce high energy γ-rays. This model was used to explain the broadband emission and fluxes in the optical R and Fermi bands during the Nov 2010 flare of 3C 454.3. However, the model had difficulty in reproducing the spectral shape and flux in the X-ray band. Chen at al. (2013) developed a time dependent multi-zone leptonic model to look at the flaring patterns of the blazar PKS 0208-512 observed with the SMARTS, Fermi and Swift telescopes [32]. PKS 0208-512 was observed to exhibit correlated flaring 132 activity in the optical and γ-ray bands with one case where there was a lone optical flare and no γ-ray flare. They concluded that an EC scenario with a magnetic field amplification or stochastic acceleration increase could explain the flaring behavior. Recently, multi-zone leptonic models have also been used to investigate magnetic field topology and evolution from the polarization swings exhibited by blazars during flaring events [118, 119]. Synchrotron emission is known to be polarized, with polarization degrees of around a few tens of percent [94]. The polarized synchrotron emission can be upscattered, producing SSC emission that retains a fraction of the original polarization [69, 117]. EC emission produces even less polarization at high energy γ-rays [56, 69]. Polarization degrees in the X-ray to high energy γ-rays are much higher for hadronic models where the bulk of the emission is caused by proton synchrotron radiation [117]. A huge advance in blazar physics would consist of developing a multi-zone lepto-hadroic model that produces broadband SEDs, light curves and polarization signatures during the flare of a given blazar. The predictions on polarization swings and behavior for both multi-zone leptonic and lepto-hadronic models can be compared with the next generation of high energy polarimeters, such as X-Calibur and ASTRO-PH to resolve the long standing question on which mechanism, leptonic or hadronic is best suited to explain blazar emission [45, 73]. 133 References

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Correlation Analysis

A.1 Numerically Solving the Fokker-Planck Equation

We solve the one-dimensional Fokker-Planck equation using the Crank-Nichelson (CN) method. This approach has the advantage of being unconditionally stable and immune to spurious oscillations appearing in the approximate solutions for a given diﬀerential equation. We consider a one dimensional partial diﬀerential equation (PDE):

∂u ∂u ∂2u = F(u, x, t, , ) (A.1) ∂t ∂x ∂x2

i where u is a function of x and t. By letting u(i∆x, j∆t) = u j, the CN method ﬁnds the solution of the PDE through the equation:

i+1 i 2 u j − u j ∂u ∂ u = Fi+1(u, x, t, , ) (A.2) ∆t j ∂x ∂x2 This method is implicit in time, obtaining the ”next” value of u. Using this method, we obtain a set of algebraic equations. If the set of algebraic equations is linear, a tridiagonal matrix algorithm can be used to obtain a numerically fast solution. We rewrite the one dimensional FP equation into the follow form:

∂ni(γ, t) ∂ 2 ∂ni(γ, t) ∂ ni(γ, t) ni(γ, t) = [Kγ ] − (˙γ · ni(γ, t)) + Qi(γ, t) − − (A.3) ∂t ∂γ ∂γ ∂γ tesc γtdecay where the diffusion coefficient, 1/(a + 2)tacc, has been replaced with the constant K. Carrying out the differentiation for the second order term, we have:

∂n (γ, t) ∂n (γ, t) ∂2n (γ, t) ∂ n (γ, t) n (γ, t) i Kγ i Kγ2 i − γ·n γ, t Q γ, t − i − i = 2 + 2 (˙ i( ))+ i( ) (A.4) ∂t ∂γ ∂γ ∂γ tesc γtdecay 141

Using Equation A.2, we can discretize the one dimensional Fokker-Planck equation, into the following relation:

i+1 i 2 i+1 i i+1 i+1 n − n 2Kγ Kγ (˙γ j+1n − γ˙ jn j) n n j j j ni+1 −ni+1 j ni+1 − ni+1 n j+1 − j+1 Qi − j − j = ( j+1 j )+ 2 ( j+1 2 j + j−1) + j ∆t ∆γ ∆γ ∆γ tesc γ jtdecay (A.5) where ∆γ = γ j+1 − γ j. We then rewrite Equation A.5 into a set of linear equations by placing the terms forward in time on the left side of the equation:

i+1 i+1 i+1 i i a jn j+1 + b jn j + c jn j−1 = n j + Q j∆t (A.6)

where the coeﬃcients, a j, b j and c j are given by the equations:

2 Kγ j ∆t a = − (A.7) j ∆γ2

∆t ∆t γ˙ ∆t 2Kγ ∆t 2Kγ2∆t b − j j j j = 1 + + + + 2 (A.8) tesc γ jtdecay ∆γ ∆γ ∆γ

2 γ˙ j ∆t 2Kγ j∆t Kγ j ∆t c = +1 − − (A.9) j ∆γ ∆γ ∆γ2 We solve the set of linear equations using a tridiagonal matrix algorithm that uses back substitution to determine the approximate solution for the next time step. The back substitution is done by modifying the coeﬃcients in the following form:

 c  j ; j = 1, 0  b j c j =  (A.10)  c j  − 0 ; j = 2, 3,..., n. b j a jc j−1 and the coeﬃcient:

  d j  b ; j = 1, d0  j j =  d −a d0 (A.11)  j j j−1  − 0 ; j = 2, 3,..., n. b j a jc j−1 142

The solution to the Fokker-Planck equation is then found through the recursive routine:

i+1 0 nn = dn (A.12)

i+1 0 0 i+1 n j = d j − c jn j+1 ; j = n - 1, n - 2,...,1 (A.13)

A.2 Discrete Correlation Functions of the Light Curves for the Leptonic and Lepto-Hadronic Models

DCF: R - X-ray: B Field: 3C 273

0.5

τ DCFR-X( ) Gaussian Fit ) τ ( 0 R-X DCF

-0.5

-6e+05 -4e+05 -2e+05 0 2e+05 4e+05 6e+05 8e+05 τ [s] Figure A.1: Discrete correlation function between the optical R and X-ray bands for the magnetic ﬁeld perturbation in the leptonic model scenario. The red curve represents the Gaussian ﬁt to the DCF using equation 4.5. 143

DCF: R - HE γ-ray: B Field: 3C 273

τ DCFR-HE γ( ) Gaussian Fit

0 ) τ ( γ R-HE DCF

-0.5

-8e+05 -6e+05 -4e+05 -2e+05 0 2e+05 4e+05 6e+05 8e+05 τ [s] Figure A.2: Discrete correlation function between the optical R and HE γ-ray bands for the magnetic ﬁeld perturbation in the leptonic model scenario. The red curve represents the Gaussian ﬁt to the DCF using equation 4.5.

DCF: R - X-ray: Linj: 3C 273

τ DCFR-X( ) Gaussian Fit 0.5 ) τ ( R-X DCF 0

-0.5 -8e+05 -6e+05 -4e+05 -2e+05 0 2e+05 4e+05 6e+05 8e+05 τ [s] Figure A.3: Discrete correlation function between the optical R and X-ray bands for the electron injection luminosity perturbation in the leptonic model scenario. The red curve represents the Gaussian ﬁt to the DCF using equation 4.5. 144 γ DCF: R - HE -ray: Linj: 3C 273

τ DCFR-HE γ( ) Gaussian Fit 0.5 ) τ ( γ R-HE

DCF 0

-0.5 -8e+05 -6e+05 -4e+05 -2e+05 0 2e+05 4e+05 6e+05 8e+05 τ [s] Figure A.4: Discrete correlation function between the optical R and HE γ-ray bands for the electron injection luminosity perturbation in the leptonic model scenario. The red curve represents the Gaussian ﬁt to the DCF using equation 4.5.

DCF: R - X-ray: tacc: 3C 273

0.4

τ DCFR-X( ) Gaussian Fit 0.2 ) τ

( 0 R-X DCF

-0.2

-0.4

-8e+05 -6e+05 -4e+05 -2e+05 0 2e+05 4e+05 6e+05 8e+05 τ [s] Figure A.5: Discrete correlation function between the optical R and X-ray bands for the stochastic acceleration timescale perturbation in the leptoic model scenario. The red curve represents the Gaussian ﬁt to the DCF using equation 4.5. 145 γ DCF: R - HE -ray: tacc: 3C 273

τ DCFR-HE γ( )

0.5 ) τ ( γ R-HE

DCF 0

-0.5 -8e+05 -6e+05 -4e+05 -2e+05 0 2e+05 4e+05 6e+05 8e+05 τ [s] Figure A.6: Discrete correlation function between the optical R and HE γ-ray bands for the stochastic acceleration timescale perturbation in the leptonic model scenario. The red curve represents the Gaussian ﬁt to the DCF using equation 4.5.

DCF: X - HE γ-ray: B Field: 3C 279

DCF (τ) 1 X-HE γ Gaussian Fit )

τ 0.5 ( γ X-HE DCF

-0.5 -8e+05 -6e+05 -4e+05 -2e+05 0 2e+05 4e+05 6e+05 8e+05 τ [s] Figure A.7: Discrete correlation function between the X-ray and HE γ-ray bands for the magnetic ﬁeld perturbation in the lepto-hadronic model scenario. The red curve represents the Gaussian ﬁt to the DCF using equation 4.5. 146

DCF: HE - VHE γ-ray: B Field: 3C 279

DCF (τ) 1 HE-VHE γ Gaussian Fit ) τ (

γ 0.5 HE -VHE DCF

-0.5 -8e+05 -6e+05 -4e+05 -2e+05 0 2e+05 4e+05 6e+05 8e+05 τ [s] Figure A.8: Discrete correlation function between the HE and VHE γ-ray bands for the magnetic ﬁeld perturbation in the lepto-hadronic model scenario. The red curve represents the Gaussian ﬁt to the DCF using equation 4.5.

γ DCF: X - HE -ray: Linj: 3C 279

DCF (τ) 1 X-HE γ Gaussian Fit )

τ 0.5 ( γ X-HE DCF

-0.5 -8e+05 -6e+05 -4e+05 -2e+05 0 2e+05 4e+05 6e+05 8e+05 τ [s] Figure A.9: Discrete correlation function between the X-ray and HE γ-ray bands for the proton injection luminosity perturbation in the lepto-hadronic model scenario. The red curve represents the Gaussian ﬁt to the DCF using equation 4.5. 147 γ DCF: HE - VHE -ray: Linj: 3C 279

DCF (τ) 1 HE-VHE γ Gaussian Fit ) τ (

γ 0.5 HE -VHE DCF

-0.5 -8e+05 -6e+05 -4e+05 -2e+05 0 2e+05 4e+05 6e+05 8e+05 τ [s] Figure A.10: Discrete correlation function between the HE and VHE γ-ray bands for the proton injection luminosity perturbation in the lept-hadronic model scenario. The red curve represents the Gaussian ﬁt to the DCF using equation 4.5.

γ DCF: X - HE -ray: tacc: 3C 279

DCF (τ) 1 X-HE γ Gaussian Fit )

τ 0.5 ( γ X-HE DCF

-0.5 -8e+05 -6e+05 -4e+05 -2e+05 0 2e+05 4e+05 6e+05 8e+05 τ [s] Figure A.11: Discrete correlation function between the X-ray and HE γ-ray bands for the stochastic acceleration timescale perturbation in the lepto-hadronic model scenario. The red curve represents the Gaussian ﬁt to the DCF using equation 4.5. 148 γ DCF: HE - VHE -ray: tacc: 3C 279

DCF (τ) 1 HE-VHE γ Gaussian Fit ) τ (

γ 0.5 HE -VHE DCF

-0.5 -8e+05 -6e+05 -4e+05 -2e+05 0 2e+05 4e+05 6e+05 8e+05 τ [s] Figure A.12: Discrete correlation function between the HE and VHE γ-ray bands for the stochastic acceleration timescale perturbation in the lepto-hadronic model scenario. The red curve represents the Gaussian ﬁt to the DCF using equation 4.5.

γ DCF: X - HE -ray: qp: 3C 279

DCF (τ) 1 X-HE γ Gaussian Fit )

τ 0.5 ( γ X-HE DCF

-0.5 -8e+05 -6e+05 -4e+05 -2e+05 0 2e+05 4e+05 6e+05 8e+05 τ [s] Figure A.13: Discrete correlation function between the X-ray and HE γ-ray bands for the proton spectral index perturbation in the lepto-hadronic model scenario. The red curve represents the Gaussian ﬁt to the DCF using equation 4.5. 149 γ DCF: HE - VHE -ray: qp: 3C 279

DCF (τ) 1 HE-VHE γ Gaussian Fit ) τ (

γ 0.5 HE -VHE DCF

-0.5 -8e+05 -6e+05 -4e+05 -2e+05 0 2e+05 4e+05 6e+05 8e+05 τ [s] Figure A.14: Discrete correlation function between the HE and VHE γ-ray bands for the proton spectral index perturbation in the lepto-hadronic model scenario. The red curve represents the Gaussian ﬁt to the DCF using equation 4.5. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

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