Monte-Carlo Applications for Partially Polarised Inverse External-Compton Scattering (MAPPIES) -- I. Description of the code and First Results
DDraft version February 2, 2021
Typeset using L A TEX twocolumn style in AASTeX63
Monte-Carlo Applications for Partially Polarised Inverse External-Compton Scattering (MAPPIES) -I. Description of the code and First Results
Lent´e Dreyer and Markus B¨ottcher Centre of Space Research, North-West University, Potchefstroom 2531, South Africa (Accepted November 10, 2020)
Submitted to ApJABSTRACTThe radiation mechanisms responsible for the multiwavelength emission from relativistic jet sourcesare poorly understood. The modelling of the spectral energy distributions (SEDs) and light curvesalone is not adequate to distinguish between existing models. Polarisation in the X -ray and γ -rayregime of these sources may provide new and unique information about the jet physics and radiationmechanisms. Several upcoming projects will be able to deliver polarimetric measurements of thebrightest X -ray sources, including active galactic nuclei (AGN) jets and γ -ray bursts (GRBs). Thisarticle describes the development of a new Monte-Carlo code – MAPPIES (Monte-Carlo Applicationsfor Partially Polarised Inverse External-Compton Scattering) – for polarisation-dependent Comptonscattering in relativistic jet sources. Generic results for Compton polarisation in the Thomson andKlein-Nishina regimes are presented. Keywords:
BL Lacertae objects: general – galaxies: active – galaxies: jets – gamma rays: galaxies– polarization – radiation mechanisms: non-thermal – relativistic processes – scattering – X -rays: galaxies INTRODUCTIONThe radiation from jetted astrophysical sources (e.g.active galactic nuclei (AGNs) and γ -ray bursts (GRBs))is characterised by their spectral energy distribution(SED) which can be modelled in many different ways,whilst being consistent with the spectral shape of theSED (e.g. Burrows et al. (2006); Walcher et al. (2011);B¨ottcher et al. (2013)). Additional constraints are there-fore required in order to distinguish between models.Relativistic jets are accompanied by the acceleration ofparticles up to very high energies, as well as the pro-duction of secondary, non-thermal radiation (B¨ottcheret al. 2012). Understanding the particle acceleration,radiation mechanisms, and the magnetic field structureof these jets is among the primary goals in the field ofhigh-energy astrophysics. Polarisation is fundamentallylinked to the internal geometry of astrophysical sources,and therefore carries important information about theastrophysical environment in terms of the how the mag-netic field is linked into the dynamics and accelerationof the energetic particles (see Trippe (2014) for a gen- Corresponding author: Lent´e [email protected] eral review). Polarisation of the emission from jet-likeastrophysical sources adds two essential parameters –the polarisation degree (PD) and the polarisation an-gle (PA) – to those already derived from spectra andvariability.Synchrotron polarisation of the radio/optical emissionfrom astrophysical jets has been a standard diagnos-tic to examine the magnetic field (e.g. Conway et al.(1993); Zhang et al. (2014); Gabuzda (2018)), since thepolarisation measurements combined with the spectraand variability of the emission reveal critical informa-tion about the magnetic field structure in the emissionregion. However, since the radio/optical polarisationmay come from regions that do not emit high-energyradiation, an important challenge that remains for high-energy astronomy is measuring the polarisation in theultraviolet (UV), X -ray, and γ -ray regimes in order toprobe the most active jet regions with powerful particleacceleration. High-energy polarisation measurementsmay provide unambiguous constraints on the geometryand structure of the astrophysical source, for example byconstraining the orientation of the accretion-disks withrespect to our line of sight (e.g. Schnittman & Krolik(2010, 2009); Laurent et al. (2011); Beheshtipour et al.(2017)). Compared to the orientation of the relativisticjet, multiwavelength polarisation, therefore, holds vital a r X i v : . [ a s t r o - ph . H E ] J a n Dreyer and B¨ottcher information on the extreme physical processes and mor-phology of the sources and their jets (see e.g. Krawczyn-ski et al. (2011); Zhang (2017); Krawczynski et al.(2019); Liodakis et al. (2020); B¨ottcher (2019); Raniet al. (2019) for reviews).The general formalism for calculating high-energy po-larisation has been well established. Synchrotron radia-tion of relativistic charged particles in ordered magneticfields is expected to be both linearly and circularly polar-ized (Westfold 1959; Rybicki & Lightman 1979). Comp-ton scattering off relativistic electrons will reduce thedegree of polarisation to about half of the seed photonfield’s polarisation (Bonometto et al. 1970). Since thereis no existing technology to measure high-energy circu-lar polarization, and the radiation is treated as a collec-tion of particles, rather than an electromagnetic wave,only linear polarisation is considered. The Klein-Nishinacross section is generally dependent on the polarisation.Polarised photons scatter preferentially in a directionperpendicular to their electrical field vector, and theelectric field vectors of the scattered photons tend toalign with the seed photons’ electric field vectors (Mattet al. 1996). Polarisation can therefore be induced whennon-relativistic electrons scatter an anisotropic photonfield, even if the seed photons are unpolarised. Comp-ton scattering off relativistic electrons, however, is notexpected to induce polarisation since the seed photonfield is approximately axisymmetric around the elec-tron momentum in the electron rest frame, making anyanisotropy of the photon field irrelevant. In a modelwhere thermal and a power-law tail of non-thermal elec-trons (in an emission region that moves along the jetwith a bulk Lorentz factor Γ jet ≥
10) scatter an externaloptical/UV radiation field, the hard X -ray/ γ -ray radia-tion will results from scattering off relativistic electronsand is thus expected to be unpolarised. The UV/soft X -ray radiation, on the other hand, results from scatteringoff non-relativistic thermal electrons and can thereforebe highly polarised (e.g. Schnittman & Krolik (2009)).A formalism for evaluating Compton polarisation inthe Thomson regime was developed by Bonometto et al.(1970), and applied to Synchrotron Self Compton (SSC)emission by Bonometto & Saggion (1973). Calculationsof Compton polarisation in the Thomson and Klein-Nishina regimes were provided by Sunyaev & Titarchuk(1984), following the Monte-Carlo approach. Krawczyn-ski (2012) and Beheshtipour et al. (2017) also followedthe Monte Carlo approach to calculate the Compton po-larisation in the Thomson and Klein-Nishina regimes,which included the contribution of non-thermal elec-trons in the emission region, and verified the analyti-cal results of Bonometto et al. (1970) in the Thomsonregime. This article describes the development of a newMonte-Carlo code – MAPPIES (Monte-Carlo Applica-tions for Partially Polarised Inverse External-ComptonScattering) – for polarisation dependent Compton scat-tering in jetted astrophysical sources. The potential of using high-energy polarisation as adiagnostic for different radiation mechanisms is brieflydiscussed in section 2. In section 3, the description ofthe MAPPIES code is presented, followed by the resultsfor Compton polarisation in the Thomson and KleinNishina regimes in section 4, and a summary in section5. SCIENTIFIC POTENTIAL OF HIGH-ENERGYPOLARISATIONHigh-energy polarimetry may play a crucial role inunderstanding the extreme physics of high-energy radi-ation, neutrino production, and particle acceleration injet-like astrophysical sources. Polarisation of X -ray/ γ -ray emission has remained largely unexplored, partlydue to the difficulty in the detection of high-energy po-larisation. However, advancements of new technologylead to several projects that may be able to deliver po-larimetric measurements of high-energy emission fromthe brightest X -ray sources, with estimates of a mini-mum detectable degree of polarisation (MDP) down to10% for moderately bright sources (McConnell & Ryan2004). Examples of recent, upcoming, and proposedmissions to measure the polarisation in the high-energyregime of the emission from jet-like astrophysical sourcesis listed in Table 1. It is thus timely to consider themodel predictions for different models of various astro-physical sources. In this section, a short overview ofthe model predictions for the radiation mechanisms ofhigh-energy emission from the AGN jets and GRBs isgiven. 2.1. High-energy emission from AGNs
Jet dominated AGNs, which are among the most pow-erful high-energy emitters in the Universe, harbor super-massive black holes (SMBHs) at their central engines. Inabout 10% of AGNs, mass accretion onto this SMBHis accompanied by the production of relativistic jets,whose bulk energy is converted into kinetic energy ofelectrons, multiwavelength radiation, and possibly par-ticle emission of ions and neutrinos. Blazars, in whichone of the jets is closely aligned to our line of sight (Urry& Padovani 1995), are the most numerous class of ex-tragalactic γ -ray sources detected (e.g. Aharonian et al.(2009); Ackermann et al. (2016)). The SEDs of blazarsare dominated by non-thermal radiation across the en-tire electromagnetic spectrum. The radio through opti-cal/UV emission is well explained by synchrotron emis-sion from relativistic electrons in the jet, which is con-sistent with moderate PDs (up to PD ∼ (3 − X -ray regime,and may thus be confirmed by X -ray polarisation (e.g.Krawczynski et al. (2011)).The origin of the high-energy ( X -ray/ γ -ray) compo-nent in the SEDs of blazars is still unclear with two APPIES I - Description and First Results Table 1.
List of recent, upcoming, and future proposed missions to measure the polarisation of the high-energy emission fromjet-like astrophysical sources.
Polarimeter Energy [keV]
References Science objectives
Lightweight Asymmetry and Magnetism 0 .
25 She et al. (2015) Blazar jets, and thermalProbe (LAMP) Li et al. (2019) emission from pulsarsExperiment Demonstration of a Soft 0 . − . X -ray Polarimeter (REDSoX) Marshall et al. (2019) isolated pulsarsThe X -ray Polarization Probe (XPP) 0 . −
60 Jahoda et al. (2019) Black holes, neutron stars,Jahoda et al. (2019) magnetars, and AGNsEnhanced X -ray Timing and Polarimetry 2 . −
10 Zhang et al. (2016) Black holes, neutron stars,Mission (eXTP) Zhang et al. (2019b) and AGNsImaging X -ray Polarimetry Explorer (IXPE) 2 . − . . − . X -ray Polarimeter Experiment (POLIX) 5 . −
30 Paul et al. (2016) AGN jets, black holes, andaccretion powered pulsars.X Calibur and XL-Calibur 20 −
40 Kislat (2019) AGN jets, black holes, andAbarr & Krawczynski (2020) neutron stars. viable models, both consistent with the shape of theSED: The first leptonic model argues that the high-energy component is due to Compton scattering off thesame electrons that emit the radio to UV/ X -ray radia-tion. The seed photons can then either be synchrotronphotons or infrared (IR)/optical/UV photons externalto the jet (possibly from the accretion disk, the broadline region (BLR), or a dusty torus). In the second hadronic model, non-thermal protons radiate via protonsynchrotron emission and interact with low-energy pho-tons via photo-pair and photo-pion interactions leading,in most cases, to synchrotron-supported pair cascadesdeveloping in the emission region (for a general reviewon the features of these models, see B¨ottcher (2010)).High-energy polarisation provides an excellent diagnos-tic to distinguish between leptonic and hadronic emis-sion scenarios, since hadronic models intrinsically pre-dict higher degrees of X -ray and γ -ray polarisation thanleptonic models (Zhang & B¨ottcher 2013; Paliya et al.2018). The production of high-energy neutrinos provides ev-idence for hadronic interactions (e.g. Atoyan & Dermer(2001); Dermer & Menon (2010); Tavecchio & Ghisellini(2015)). If blazars accelerate enough high-energy pro-tons, the protons may interact with the local blazar ra-diation field and produce charged pions which decay andemit neutrinos. The recent IceCube-170922A neutrinoevent, which was reported to coincide with the blazarTXS 0506+056 in flaring state (IceCube Collaborationet al. 2018), indicates that hadronic processes may op-erate in a blazar jet. Many models have been put for-ward for an explanation of the corresponding SED ofTXS 0506+056 during the neutrino alert (e.g. Ansoldiet al. (2018); Keivani et al. (2018); Murase et al. (2018);Padovani et al. (2018); Cerruti et al. (2019); Reimeret al. (2019); Gao et al. (2019)), which can be categorisedinto two groups: The first is a leptonic scenario whereinverse-Compton dominates the high-energy emission,with a subdominant hadronic component which pro-duces the neutrinos as well as a considerable amountof X -rays through synchrotron emission from hadroni- Dreyer and B¨ottcher cally induced cascades. The second is a hadronic sce-nario where the X -ray emission consists of both protonsynchrotron and cascading synchrotron, while the γ -rayemission is dominated by proton synchrotron. X -raypolarisation can probe the secondary pair synchrotroncontribution complementary to the neutrino detection,while γ -ray polarisation can be used to distinguish be-tween the inverse-Compton and proton synchrotron sce-narios (Rani et al. 2019). For instance, Zhang et al.(2019a) found that the proton synchrotron (hadronic)scenarios generally predict higher PDs across the high-energy component than the inverse-Compton (leptonic)dominated scenarios.The SEDs of some blazars contain an excess in the UVand/or soft X -ray regime (e.g. Masnou et al. (1992);Grandi et al. (1997); Haardt et al. (1998); Pian et al.(1999); Raiteri et al. (2005); Palma et al. (2011); Acker-mann et al. (2012); Paliya et al. (2015); Pal et al. (2020))called the big blue bump (BBB). Various models for theemission responsible for the BBB have been proposed,which include: (1) Thermal emission from the accretiondisk that feeds the SMBH (e.g. Pian et al. (1999); Blaeset al. (2001); Paliya et al. (2015); Pal et al. (2020)),(2) a higher than Galactic dust-to-gas ratio towards thesource (e.g. Ravasio et al. (2003)), (3) a distinct syn-chrotron component from a different region in a multi-zone construction (e.g. Paltani et al. (1998); Raiteriet al. (2006); Ostorero et al. (2004); Roustazadeh &B¨ottcher (2012)), (4) the detection of a Klein-Nishinaeffect on the synchrotron spectrum (e.g. Ravasio et al.(2003); Moderski et al. (2005)), and (5) bulk Comptonemission (e.g. Sikora et al. (1994, 1997); B(cid:32)la˙zejowskiet al. (2000); Ackermann et al. (2012); Baring et al.(2017)). The polarisation of the UV/ X -ray emissionfrom blazars may yield significant insights in order todistinguish between different BBB emission scenarios.For instance, a BBB due to (unpolarised) thermal emis-sion from an accretion-disk predicts that the polarisa-tion will decrease with increasing frequency throughoutthe optical/UV regime, while Roustazadeh & B¨ottcher(2012) predicted that a BBB due to cascade synchrotronemission would result in PDs that show a weak depen-dence on the frequency over the optical/UV regime. Ifthe BBB arises from the bulk Compton feature, the ther-mal Comptonisation process should lead to significantpolarisation of the Compton emission from the UV/ X -ray excess in the SED (Baring et al. 2017).2.2. Gamma-ray burst prompt emission
GRBs are the strongest explosions in the Universe,separated into two phases: The initial burst of γ -rays(i.e. the prompt emission) that can last from a frac-tion of a second to hundreds of seconds, and a longerlasting (from days to weeks) afterglow emission. Thereare at least two classes of GRBs depending on the du-ration of the the prompt emission phase, believed to beassociated with the formation of two oppositely directed ultra-relativistic jets (McConnell et al. 2019): ShortGRBs ( ≤ > Band function (Band et al. 1993), which consistsof a broken power-law with a smooth break at a char-acteristic peak energy (i.e. the peak of the spectrumwhen plotted in terms of the energy output per decade).Many diverse models for the emission mechanism thatcan explain the Band like non-thermal prompt emis-sion spectra have been proposed (see e.g. Baring &Braby (2004); Pe’er (2015)), which include: (1) Opti-cally thin synchrotron radiation with either random orordered magnetic fields, (2) SSC emission, (3) Comptondrag models, and (4) photospheric models. The modelsshow that an integrated understanding of the geometryand physical processes close to the central engine mayonly be accomplished through high-energy polarimetry,since it depends on the emission processes involved thatproduce the prompt emission (see e.g. McConnell et al.(2019); Gill et al. (2020) for recent reviews).Emission due to the SSC process can be moderatelypolarised, with maximum PDs ∼
24% for a simplifiedmodel (Chang & Lin 2014). However, SSC has beendisfavored as a plausible emission mechanism by GRBenergetics (see e.g. Baring & Braby (2004); Piran et al.(2009)). The predicted linear polarisation for photo-spheric models is relatively low, although the polarisa-tion can be as high as PD ∼
40% depending on theline of he sight (Lundman et al. 2018). Models that ar-gue for synchrotron radiation and/or inverse-Comptonscattering between softer photons and relativistic elec-trons (i.e. Compton drag) predict high PDs (Lyutikovet al. 2003; Gill et al. 2020). Detection of highly po-larised prompt GRB emission would thus support bothsynchrotron and Compton drag models. However, Tomaet al. (2009) showed that a statistical study of a sampleof GRBs could then be used to differentiate between themodels that either invoke ordered or random magneticfields.
APPIES I - Description and First Results ∼ −
500 keV and detected 55 GRBs during 2016 and2017 (Xiong et al. 2017; Kole 2018). A time integratedanalysis for a number of GRBs observed by POLAR wasdone by Kole (2018), which showed low or unpolarisedprompt emission in the energy regime of ∼ − ∼
30% with an evolving PA. This indicatesthat low polarisation signals from the time-integratedanalysis could be a result of the summation of changingpolarisation signals for different epochs. The work ofKole et al. (2020) did not include energy resolved studieswhich have the potential to test predictions for differentmodels. Various components of the prompt emission canhave different distinct PDs, for example, Lundman et al.(2018) predicted significant polarisation in the energyregime of 10s of keV, but also predicted that the PDcan be lost at higher energies due to Comptonisation.Energy-dependent polarisation studies can therefore bea powerful diagnostic for different emission models. Thecode presented in this paper can be used for energy- andangle-dependent studies of Compton polarisation fromrelativistic jet sources. THE MAPPIES CODEIn this section, a newly developed Monte-Carlo code(MAPPIES) for polarisation-dependent Compton scat-tering of radiation fields in relativistic jets of e.g. GRBsand AGNs is presented. A flow diagram of the code isshown in Figure 1: An external radiation field, origi-nating in the laboratory frame (which can be e.g. therest frame of the AGN, or the rest frame of the GRBprogenitor), scatters off an arbitrary (thermal and non-thermal) electron distribution, assumed to be isotropicin the co-moving frame of the emission region that movesalong the jet with a relativistic speed (i.e. a bulk Lorentzfactor Γ jet (cid:29) single photon approach)through the computational domain. The Monte-Carlosingle photon approach is very flexible in terms of theComptonising medium, as well as the directional andenergy distributions of the seed photons and electrons. The input parameters (listed in Table 2) determine thecharacteristics of the emission region, the seed photondistribution (discussed in section 3.1), and the electronenergy distribution (discussed in section 3.2). The po-larisation signatures are calculated using the Stokes for-malism (Stokes 1851), and the polarisation-dependentCompton scattering of the seed photons is evaluated us-ing Monte-Carlo methods by Matt et al. (1996) (dis-cussed in section 3.3). A comparison to previously pub-lished results is given in Appendix A.In what follows, quantities in the laboratory frame andthe co-moving frame of the emission region are denotedwith subscripts lab and em , respectively. Quantities inthe electron rest frame and the observer’s frame are de-noted with subscripts e and obs , respectively, and thescattered photon quantities are denoted with a super-script sc . While following the single photon approach,an additional subscript i is used to label the quantitiesof the current, individual photon. All random numbersare denoted with ξ and are drawn with the MersenneTwister (Matsumoto & Nishimura 1998) between 0 and1, unless specified otherwise.3.1.
Seed photon fields
The seed photons are drawn in the laboratory framefrom either an isotropic, single-temperature black bodydistribution, or from a multi-temperature accretion-diskspectrum. In the first case, the polar angle Θ i,lab andazimuthal angle Φ i,lab is drawn from an isotropic distri-bution, so that Θ i,lab = cos − (2 ξ − i,lab = 2 πξ (1)with ξ and ξ are random numbers between 0 and 1.The seed photon energy (cid:15) i,lab is drawn from a blackbody distribution that corresponds to a single tempera-ture kT rad (defined as an input parameter), following theMonte-Carlo methods by Pozdnyakov et al. (1983). Thephoton is then boosted to the co-moving frame of theemission region with the bulk boost equations (B¨ottcheret al. 2012) (cid:15) i,em = Γ jet (cid:15) i,lab (1 − β jet cos Θ i,lab )cos Θ i,em = cos Θ i,lab − β jet − β jet cos Θ i,lab . (2)An illustration of how a single photon is drawn froman accretion-disk spectrum is given in Figure 2. Thecalculation requires the following input parameters: Themass of the black hole M BH , the inner-disk radius R inD ,the outer-disk radius R outD , the accretion-disk luminosity L D , and the height of the emission region h (i.e. thedistance between the central black hole and the emissionregion). The flux per unit radius is given by dF = (cid:0) πr cos Θ lab σ SB T rad ( r ) (cid:1) dr (3) Dreyer and B¨ottcher
Table 2.
The input parameters of the MAPPIES code for different seed photon and electron energy distributions.
Model description Parameter description
Emission region Lorentz factor of the jet, Γ jet
Redshift of the source, z Seed photon distributions (1) Isotropic, single-temperature Number of seed photons considered, N phot black body distribution. Temperature of the seed photons in eV, kT rad Number of seed photons considered, N phot (2) Multi-temperature Black hole mass in g, M BH accretion-disk Inner disk radius of the accretion-disk in cm, R inD spectrum. Outer disk radius of the accretion-disk in cm, R outD Distance between the central black hole and the emission region in cm, h Accretion-disk luminosity in erg · s − , L D Electron energy distributions (1) Purely thermal (Maxwell) distribution. Thermal temperature of electrons in eV, kT e (2) Hybrid (Maxwell + power-law) Fraction of non-thermal electrons, f nth distribution. Power-law index of the power-law distribution of non-thermal electrons, p Maximum Lorentz factor of non-thermal electrons, γ max (3) Input distribution File of electron distribution, n ( γ ) where kT rad ( r ) = k (cid:34) GM BH ˙ m πr σ SB (cid:32) − (cid:114) R inD r (cid:33)(cid:35) (4)is the radial temperature structure, andcos Θ lab = h √ r + h (5)is the cosine of the angle between the line of sight fromthe emission region to r and the normal to the disk.In Equation 4, ˙ m = ˙ M Edd ( L D /L Edd ) is the accretionrate of the disk, with ˙ M Edd = L Edd / (0 . c ) the Ed-dington accretion rate relative to the Eddington lumi-nosity L Edd = [(1 . × M BH ) / (10 M (cid:12) )] erg · s − , σ SB ∼ . × − erg · cm − · s − · K − is the Stefan-Boltzmann constant, G ∼ × − cm · g − · s − is thegravitational constant, c ∼ × cm · s − is the speedof light, and M (cid:12) ∼ × g is the solar mass.The cumulative probability to receive a photon froma radius r < r i is obtained from Equation 3 as P ( r < r i ) = 1 N (cid:90) r i R inD (cid:34) r √ r + h − (cid:112) R inD r √ r + h (cid:35) dr = N − ( I − (cid:113) R inD I ) (6)with the normalisation given by N = (cid:90) R outD R inD (cid:34) r √ r + h − (cid:112) R inD r √ r + h (cid:35) dr. (7) In Equation 6, I = (cid:90) r i R inD r √ r + h dr = (cid:34) − √ r + h h r (cid:35) r i R inD (8)and I = (cid:90) r i R inD r √ r + h dr (9)which is solved with the assumption that r (cid:28) h inmost relevant cases. The typical outer-disk radius R outD ∼ R G ≈ . × M cm, where M = M BH / (10 M (cid:12) ) and R G is the gravitational radius ofthe black hole. Therefore, r < h anywhere in the diskwhen M BH (cid:46) × M (cid:12) and/or h (cid:46) . The Taylorexpansion, 1 √ r + h = 1 z (cid:112) r/z ) = 1 z ∞ (cid:88) n =0 (cid:18) − n (cid:19) (cid:16) rz (cid:17) n (10)can then be used, so that, to first order I = ∞ (cid:88) n =0 (cid:18) − n (cid:19) z − (2 n +1) (cid:90) rR inD r n − dr (11) APPIES I - Description and First Results S T AR T MODEL PARAMETERS
Purely thermal (Maxwell) distribution. Hybrid (Maxwell/power-law) distribution. Input electron distribution.
Electron energy distribution Bulk Lorentz factor.Redshift of the source.
Isotropic, single-temperature spectrum. Accretion disk spectrum.
Seed photon spectrumSTART MONTE CARLO SIMULATION FOR A SINGLE PHOTON
Draw the direction of propagation andthe energy of a photon.Calculate thescattered photon'spolarisation vector.
Boost the photon to the co-moving frame of the emission frame.
Draw a randompolarisation vector. Draw the Lorentz factor andthe momentum of a electron.
Transform the photon to the electron rest frame.
Calculate the Klein-Nishina cross-section.Calculate the scattering angles.Calculate the energy and direction ofthe scattered photon.
Simulate the Compton scatteringbetween the electron and the photon:
Calculate the polarisation degree due to scattering,Calculate the the contribution of thescattered photon to the Stokes vectors.Draw randompolarisation vector.Redshift the photon to the observer's frame.
Will the photon be scattered?Does the photon contribute to the polarized flux?
YES NOYESNO
Laboratory frame.Co-moving frame of the emission frame.Electron-rest frame.
Boost the photon to the laboratory frame.Transform the photon to the emission frame.
Figure 1.
A flow diagram for the Monte-Carlo simulation of the MAPPIES code. The input parameters define the characteristicsof the emission region (which moves along the jet with a bulk Lorentz factor Γ jet ), the seed photon distribution (drawn from anisotropic, single-temperature black body distribution, or a multi-temperature accretion-disk spectrum), and the electron energydistribution (drawn from a purely thermal distribution, a hybrid distribution, or an input electron spectrum). The Monte-Carlosimulation is shown for a single photon, and will continue for the number of seed photons considered. The photon is transformedbetween the laboratory frame (shown in the purple shaded area), the emission frame (shown in the blue shaded area), and theelectron rest frame (shown in the grey shaded area). After evaluating the full Klein-Nishina cross section in the electron restframe, the code continues with only the scattered photons.
For typical emission-region heights of h (cid:38) cm, r (cid:28) h everywhere in the disk, which gives I ( r (cid:28) h ) = (cid:20) − hr − √ rh (cid:21) r i R inD (12) The radius r i is drawn by calculating P ( r < r i ) for dif-ferent values of r i ∈ [ R inD , R outD ], until a given randomnumber ξ = P ( r < r i ). The temperature of the disk at r i , from which the photon energy is sampled, KT i,rad ,and polar angle of the seed photon, Θ i,lab , are subse-quently calculated with Equations 4 and 5, respectively. Dreyer and B¨ottcher
Figure 2.
An illustration of how a single photon (denotedwith a subscript i ) is drawn from an accretion-disk spectrum.The angle between the line of sight from the emission region r i and the normal to the disk is denoted with Θ i,lab , and h isthe distance between the central black hole and the emissionregion. The energy of the photon (cid:15) i,lab is drawn from a blackbody distribution that corresponds to KT i,rad (followingthe Monte-Carlo methods by Pozdnyakov et al. (1983)),and boosted to the co-moving frame of the emission re-gion with Equation 2.3.2. Electron energy distributions
The electrons are assumed to be isotropic in theco-moving frame of the emission region. An electronis randomly assigned to every photon as it is trans-ported through the computational domain, drawn from(1) a purely thermal (Maxwell) distribution, (2) a hy-brid (Maxwell + power-law) distribution, or (3) a user-defined input electron spectrum. In the first case, thethermal temperature kT e is given as an input param-eter, and a Lorentz factor γ i of an electron is drawnfrom a Maxwellian distribution, following the Monte-Carlo methods by Pozdnyakov et al. (1983). In thecase of thermal and a power-law tail of non-thermalelectrons three additional parameters are required: Thefraction of the electrons that are assumed to be non-thermal f nth , the power-law index of the non-thermalelectrons p (which are drawn from a power-law distribu-tion n pl ( γ ) = N pl γ − p , with N pl the normalisation con-stant), and the Lorentz factor that corresponds to thecut-off energy of the power-law tail γ max . The Lorentzfactor which corresponds to where the power-law tailbegins, γ min , is determined by iteration until f nth = n pl n th + n pl (13)is equal to the input parameter f nth defined, where n pl = N pl (cid:90) γ max γ min ˜ γ − p d ˜ γ (14)is the power-law tail and n th = N th (cid:90) γ min ˜ γ ˜ βe − ˜ γ Θ e d ˜ γ (15) is the number of thermal electrons, with ˜ β = (cid:112) − ˜ γ − ,Θ e = kT e / ( m e c ), and N th = 1 an arbitrary normalisa-tion constant. At the point of intersection, N pl γ − pmin = N th γ min β min e − γmin Θ e (16)During the Monte-Carlo simulation, a random number ξ ∈ [0 ,
1] is compared to f nth in order to draw γ i fromeither a power-law distribution or a Maxwellian distri-bution. If ξ ≤ f nth , γ i is drawn from a power-lawdistribution, γ i = (cid:104) ξ (cid:16) γ − pmax − γ − pmin (cid:17) + γ − pmin (cid:105) − p (17)where ξ ∈ [0 , ξ > f nth , γ i is drawn from aMaxwellian distribution, in which case, a new randomnumber ξ ∈ [0 ,
1] will be drawn if γ i > γ min , and theprocess will be repeated.In the case where the electron energy is drawn froma user-defined distribution, an input electron spectrum n ( γ ) alone is required. The cumulative distributionfunction (CDF), P ( γ < γ i ) = (cid:82) γ i n (˜ γ ) d ˜ γ (cid:82) γ max n (˜ γ ) d ˜ γ (18)is calculated from the input electron spectrum, and γ i is drawn such that ξ = P ( γ < γ i ).3.3. Polarisation-dependent Compton Scattering
The polarisation signatures of the photons are calcu-lated using the Stokes formalism (Stokes 1851). Thecontributions of the photon to the second ( Q i ) andthird ( U i ) Stokes parameters are calculated by follow-ing the Monte-Carlo methods by Matt et al. (1996),and summed over all the photons after the simulationis complete. Since the photons are individually trans-ported through the computational domain, every pho-ton is 100% polarised. However, the external radiationis assumed to be unpolarised. The polarisation vector (cid:126)P i,em (which points in the direction of the electric fieldvector) of every seed photon is thus randomly drawnperpendicular to the photon’s direction of propagation,which results in a zero net polarisation (see e.g. Tam-borra et al. (2018)).The probability of the photon to undergo Comptonscattering is determined by the (polarisation averaged)Compton cross section, σ KN σ T = 34 (cid:20) x e x e (cid:18) x e (1 + x e )1 + 2 x e − ln(1 + 2 x e ) (cid:19)(cid:21) + 34 (cid:34) x e ln(1 + 2 x e ) − x e (1 + 2 x e ) (cid:35) (19)most conveniently evaluated in the electron rest frame,with σ T ∼ . × − cm the Thomson cross sec-tion (B¨ottcher et al. 2012), and x e = (cid:15) e /m e c the di-mensionless seed photon energy. The seed photon is APPIES I - Description and First Results ξ ∈ [0 ,
1] is drawn todetermine whether the seed photon will be scattered. If ξ > ( σ i,KN /σ T ) (where σ i,KN is the full Klein-Nishinacross section for the current photon), the photon willcontinue in the same direction without scattering, oth-erwise the Compton scattering event will be simulated.The geometry of the Compton effect for a single pho-ton in the electron rest frame is illustrated in Figure 3.The energy of the scattered photon is given by (cid:15) sci,e = (cid:15) i,e − x i,e cos Θ sci,e (20)The probability of a photon to have a scattering angle ofΘ sci,e (the angle between the seed and scattered photon)is given by P (Θ sci,e < Θ sce ) = x i,e (cid:2) + µ sce (1 − µ sce ) (cid:3) + (cid:2) µ sce ) (cid:3) + 2 x i,e + x i,e ln(1 + 2 x i,e ) − x i,e (cid:26) ln [1 + x i,e (1 − µ sce )] − ln(1 + 2 x i,e ) (cid:27) + 2 x i,e + x i,e ln(1 + 2 x i,e ) (21)where µ sce = cos Θ sce (Matt et al. 1996). The scat-tering angle Θ sci,e is thus calculated by drawing a ran-dom number ξ and finding the value of Θ sce for which P (Θ sci,e < Θ sce ) = ξ .The azimuthal distribution of the photons is depen-dent on the seed photon polarisation. When the seedphotons are unpolarised, the angle between (cid:126)P sci,e and theplane of scattering, Φ sci,e , can be assumed to be isotrop-ically distributed in the electron rest frame (Matt et al.1996). However, while following a single photon ap-proach, every individual photon is fully polarised. Theazimuthal angle Φ sci,e is thus drawn by calculating theprobability of a fully polarised photon to have an angleΦ sci,e (Matt et al. 1996), P (Φ sci,e < Φ sce ) = 12 π Φ sce − sin Θ sce sin Φ sce cos Φ scex i,e x sci,e + x sci,e x sci,e − sin Θ sce (22)and finding the value of Φ sce for which P (Φ sci,e < Φ sce ) = ξ , where ξ is a newly drawn random number.The PD due to scattering is then calculated for a fullypolarised photon asPD sci,e = 2 − sin Θ sci,e cos Φ sci,ex i,e x sci,e + x sci,e x i,e − Θ sci,e cos Φ sci,e (23)in the electron rest frame (Matt et al. 1996), whichdetermines whether the photon will be polarised after Figure 3.
Illustration of the geometry of the Compton ef-fect for a single photon (denoted with a subscript i ) in theelectron rest frame. The seed photon, moving in a direc-tion (cid:126)D i,e , with an energy (cid:15) i,e , and a polarisation vector (cid:126)P i,e (which points in the direction of the electric field vector) isshown in purple. The scattered photon is shown in blue witha direction (cid:126)D sci,e , an energy (cid:15) sci,e , and a polarisation vector (cid:126)P sci,e .The polar scattering angle Θ sci,e is the angle between the seedand scattered photon, while the azimuth scattering angle Φ sci,e is the angle between (cid:126)P i,e and the plane of scattering. scattering. A random number ξ ∈ [0 ,
1] is drawn andcompared to PD sci,e in order to calculate the polarisationvector (cid:126)P sci,e of the scattered photon. If ξ < PD sci,e , thephoton will contribute to the partially polarised Comp-ton emission with a polarisation vector of (cid:126)P sci,e = ( (cid:126)P i,e × (cid:126)D sci,e ) × (cid:126)D sci,e | (cid:126)P sci,e | (24)where (cid:126)D sci,e is the scattered photon’s direction of propa-gation (Angel 1969). Otherwise, (cid:126)P sci,e is randomly drawnperpendicular to (cid:126)D sci,e , and will contribute to the unpo-larised fraction of the Compton emission (Matt et al.1996; Tamborra et al. 2018).The scattered photon is transformed back into the co-moving frame of the emission region with the Lorentzmatrix, where the photon’s contribution to the second( Q sci ) and third ( U sci ) Stokes parameters is calculated.The photon is boosted into the laboratory frame withthe bulk boost equations (B¨ottcher et al. 2012), (cid:15) sci,lab = Γ jet (cid:15) sci,em (cid:0) β jet cos Θ sci,em (cid:1) cos Θ sci,lab = cos Θ sci,em + β jet β jet cos Θ sci,em (25)and shifted to the observer’s frame where (cid:15) sci,obs = (cid:15) sci,lab / (1 + z ), with z the redshift of the source.The Compton polarisation signatures are calculatedafter the simulation is complete by summing the contri-0 Dreyer and B¨ottcher butions of the scattered photons to the Stokes parame-ters, so thatQ = N scphot (cid:88) i =0 Q sci and U = N scphot (cid:88) i =0 U sci (26)where N scphot is the number of the scattered photons inthe specified direction. The polarisation signatures ofthe Compton emission are determined asPD = (cid:112) Q + U N scphot and PA = 12 tan − UQ (27)and binned in viewing angles, Θ sci,lab , and energy, (cid:15) sci,obs .This allows us to identify the viewing angle and photonenergy range at which the maximum PD occurs, thusoffering the best opportunities to measure Compton po-larisation. COMPTON POLARISATION IN THE THOMSONAND KLEIN-NISHINA REGIMESThe MAPPIES code presented in this paper can beused to simulate the polarisation due to Compton scat-tering of different seed photon fields and electrons withdifferent energy distributions. In this section, genericresults for the Compton polarisation in the Thomsonand Klein-Nishina regimes are presented. Only resultsfor the isotropic black body target photons are shown inthis paper, while results for an accretion-disk spectrumwill be presented in a companion paper (Dreyer andB¨ottcher 2020, in preparation) for applications to spe-cific AGN. The results are shown for the combination offree parameters listed in Table 3. The seed photons aredrawn in the laboratory frame from an isotropic, single-temperature black body distribution (shown in the toppanel of Figure 4) with kT rad = 0 . kT rad = 50keV, and kT rad = 500 keV. The electrons (shown in thebottom panel of Figure 4) are assumed to be isotropic inthe co-moving frame of the emission region (that movesalong the jet with a bulk Lorentz factor of Γ jet = 10)with thermal temperatures of kT e = 50 keV, kT e = 500keV, and kT e = 5000 keV. The electron energy is drawnfrom either a purely thermal distribution (shown withsolid lines) or a hybrid (Maxwell + power-law) distribu-tion (shown with dashed lines). In all the figures dis-cussed, results for soft X -rays ( kT rad = 0 . X -rays ( kT rad = 50keV) are shown in blue, and results for γ -rays ( kT rad =500 keV) are shown in grey.4.1. The Compton spectra
The external Compton spectra resulting from the dif-ferent combinations of target radiation fields and elec-tron distributions mentioned above, are shown in Figure5. Compton scattering in the Thomson regime ( x e (cid:28) Table 3.
The input parameters for the generic results ofCompton polarisation in the Thomson and Klein-Nishinaregimes.
Input Parameter
ValueLorentz factor of the jet, Γ jet Temperature of the seed photons, kT rad . kT e
50 keV500 keV5000 keVFraction of non-thermal electrons, f nth . p . γ max . × energy exchange between the seed photon and electronbecomes substantial, along with a reduction of the crosssection, in the Klein-Nishina regime ( x e (cid:29) (cid:15) sclab ∼ γ Γ jet (cid:15) lab (where γ is the averaged Lorentz factor of the elec-trons) in the laboratory frame. Compton scattering off apower-law distribution of non-thermal electrons resultsin a power-law distribution of scattered photons (shownwith dashed lines). The photons that are scattered inthe Klein-Nishina regime have cut-off energies in the lab-oratory frame that correspond to the reduction of thecross section in the electron rest frame. For electronswith thermal energies of kT e = 50 keV, all the elec-trons are non-relativistic. Soft X -rays and hard X -raysare thus scattered in the Thomson regime with ener-gies ∼ γ Γ jet higher than the seed photon energies. Formildly-relativistic electrons, the peak of the electron dis-tribution is at γ ∼
2. Hard X -rays and γ -rays are thusscattered in the Klein-Nishina regime with very similarhigh-energy spectra (shown in blue and grey in Figure5). 4.2. The polarisation degree in the Thomson andKlein-Nishina regimes
The Compton cross section is generally dependent onpolarisation. The most familiar form of the polarisation-dependent differential cross section is given by dσ KN d Ω sce = 14 r e (cid:18) x e x sce (cid:19) (cid:20) x e x sce + x sce x e − θ (cid:21) (28) APPIES I - Description and First Results Figure 4.
The seed photon spectra (top panel) and electron energy distributions (bottom panel). The photons (top panel) aredrawn in the laboratory frame from an isotropic, single-temperature black body distributions with kT rad = 0 . kT rad = 50 keV (shown in blue), and kT rad = 500 keV (shown in grey). The electrons (bottom panel) are isotropic inthe co-moving frame of the emission region with thermal temperatures of kT e = 50 keV, kT e = 500 keV, and kT e = 5000 keVthat increase with the shade of grey in the bottom panel. The electron energy is drawn from either a purely thermal distribution(shown with solid lines) or a hybrid (Maxwell + power-law) distribution (shown with dashed lines) for the combination of freeparameters listed in Table 3. in the electron rest frame, where θ is the angle betweenthe polarisation vector of the seed photon (cid:126)e e and thepolarisation vector of the scattered photon (cid:126)e sce (Mattet al. 1996). From Equation 28, dσ KN d Ω sce ∝ x e x sce + x sce x e − (cid:126)e e · (cid:126)e sce ) . (29)Since the polarisation term in Equation 29 dominatesfor x e (cid:28)
1, photons that are scattered in the Thomsonregime are expected to be polarised. The polarisationterm becomes negligible for x e (cid:29)
1, and polarisation isthus not expected to be induced for Compton scatteringin the Klein-Nishina regime.The polarisation signatures are shown as a functionof the scattered photon energy (cid:15) sclab in Figure 6, forviewing angles of Θ sclab ∼ Γ − jet rad. The PD decreaseswith increasing photon energies and further decreasesfor higher electron temperatures, because the polarisa-tion arises only for photons that are scattered in theThomson regime. The maximum PD for Compton emis- sion in the Thomson regime occurs where the thermal,non-relativistic electrons scatter the seed photons (i.e.to an energy of Γ jet kT rad ). For electron temperaturesof kT e = 5 × keV, all the electrons are highly rela-tivistic with γ (cid:38)
10, and no Compton polarisation is in-duced, irrespective of whether the photons are scatteredin the Thomson or Klein-Nishina regime. There aremore photons produced at higher energies for scatter-ing off a power-law distribution of non-thermal electrons(shown with dashed lines) than in the case of scatter-ing off purely thermal electrons (shown with solid lines),but those photons are unpolarised since the non-thermalelectrons are relativistic.4.3.
The polarisation signatures in the Thomsonregime
The polarisation signatures due to scattering off non-relativistic and mildly-relativistic electrons are given asa function of the scattered photon viewing angle Θ sclab in Figure 7 (averaged over all photon energies). Since2
Dreyer and B¨ottcher
Figure 5.
The external Compton spectrum for soft X -rays ( kT rad = 0 . X -rays ( kT rad = 50 keV;shown in blue), and γ -rays ( kT rad = 500 keV; shown in grey). The results are shown for scattering off non-relativistic ( kT e = 50keV), mildly-relativistic ( kT e = 500 keV), and relativistic ( kT e = 5000 keV) electrons with thermal temperatures that increaseswith the shade of color in each panel. The electrons are drawn from either a purely thermal distribution (shown with solidlines) or a hybrid (Maxwell + power-law) distribution (shown with dashed lines) for the combination of free parameters listedin Table 3. the orientation of the polarisation does not change sig-nificantly for different electron energy distributions, thepolarisation signatures are similar for scattering off ther-mal electrons (shown with solid lines) and a power-law distribution of non-thermal electrons (shown withdashed lines). The PA is shown as a function of (cid:15) sclab andΘ sclab in the bottom panels of Figures 6 and 7, respec-tively. In both cases, the PA for Compton emission that are polarised assumes a constant value of PA = π/ sclab (Figure 7) shows at whichangles the maximum polarisation occurs. For scatter-ing to happen in the Thomson regime, the electronshave to move in the same direction as the seed photons(backwards in the jet). The scattered photons move APPIES I - Description and First Results Figure 6.
The polarisation signatures as a function of the scattered photon energy for soft X -rays ( kT rad = 0 . X -rays ( kT rad = 50 keV; shown in blue), and γ -rays ( kT rad = 500 keV; shown in grey). The results are shown forscattering off non-relativistic ( kT e = 50 keV), mildly-relativistic ( kT e = 500 keV), and relativistic ( kT e = 5000 keV) electronswith thermal temperatures that increases with the shade of color in each panel. The electrons are drawn from either a purelythermal distribution (shown with solid lines) or a hybrid (Maxwell + power-law) distribution (shown with dashed lines) for thecombination of free parameters listed in Table 3. perpendicular to their incoming direction in the elec-tron rest frame which appears at an angle of ∼ γ − radwith respect to the backward direction. For photons that are scattered in the Thomson regime, the maxi-mum PD occurs at a right angle of Θ sce ∼ π/ Dreyer and B¨ottcher
Figure 7.
The polarisation signatures as a function of the scattered photon viewing angle for soft X -rays ( kT rad = 0 . X -rays ( kT rad = 50 keV; shown in blue). The results are shown for scattering off non-relativistic( kT e = 50 keV) and mildly-relativistic ( kT e = 500 keV) electrons with thermal temperatures that increases with the shadeof purple and blue. The electrons are drawn from either a purely thermal distribution (shown with solid lines) or a hybrid(Maxwell + power-law) distribution (shown with dashed lines) for the combination of free parameters listed in Table 3. Thegrey line indicates the viewing angle of Θ sclab = Γ − jet rad, where Γ jet = 10. in the emission-region rest frame for scattering off non-relativistic ( kT e = 50 keV; when there are hardly anyrelativistic motions) electrons. Due to relativistic boost-ing, photons at Θ scem ∼ π/ sclab ∼ Γ − jet rad in thelaboratory frame. The maximum PDs for scatteringoff non-relativistic electrons occur thus at angles ofΘ sclab ∼ Γ − jet rad, indicated with a grey dashed line inFigure 7.The peak of the electron distribution for mildly-relativistic ( kT e = 500 keV) electrons is around γ ∼ X -rays with kT rad = 0 . ∼ (0 . jet = 5 . x em =Γ jet . kT rad m e c ∼ .
03. Similarly, hard X -rays with kT rad = 50 keV are boosted to ∼
500 keV into theemission frame, with the black body spectrum peakingat x em ∼ .
7. Therefore, photons that are scatteredby mildly-relativistic electrons have angles of Θ scem =( π − γ − ) rad in the emission-region rest frame. Rela-tivistic aberration into the laboratory frame causes themaximum PD to occur at angles that are larger thanthose in the case of scattering off non-relativistic elec-trons. SUMMARY AND CONCLUSIONThe MAPPIES code presented in this paper is capa-ble of predicting the Compton polarisation in differentjet-like astrophysical sources for different photon ener- gies and electron temperatures (see Appendix A for acomparison to previously published results). The effectsof Compton scattering depend on the temperature ofthe seed photons, as well as the Lorentz factors and en-ergy distribution of the electrons. The photons scatterto higher energies (with a factor of Γ jet γ ) in the Thom-son regime, and have cut-off energies that correspond tothe reduction of the cross section in the Klein-Nishinaregime. The PD of the scattered photons depends onthe effects of Compton scattering due to the polarisationdependence of the Klein-Nishina cross section, given byEquation 28. The PD decreases with the increase ofphoton energies and higher electron temperatures. Theenergy regimes with non-negligible PDs shift to higherenergies and become smaller for higher seed photon tem-peratures, while narrowing further for higher electrontemperatures. Polarisation is therefore expected to beinduced for photons that are scattered in the Thomsonregime, and no polarisation is expected to be induced forphotons that are scattered in the Klein-Nishina regime.For electron temperatures of kT e = 5000 keV, essentiallyall the electrons are highly relativistic and no Comp-ton polarisation is induced irrespective of whether thephotons are scattered in the Thomson or Klein Nishinaregime.The maximum PD for scattering in the Thomsonregime occurs at viewing angles of Θ sclab ∼ Γ − jet rad(shown in Figure 7). The maximum PD occurs atlarger angles for scattering off mildly-relativistic elec- APPIES I - Description and First Results γ (cid:38)
2. The PA forthe fraction of the scattered photons that are polarisedassumes a constant value of PA ∼ π/ X -ray excess in the SED. This will reinforce futureprospects of using measurements of polarisation signa-tures to distinguish between different radiation mecha-nism models for the sources of interest.ACKNOWLEDGMENTSWe thank the anonymous referee for giving us an ex-peditious, helpful, and constructive report. The work ofM.B. is supported through the South African ResearchChair Initiative of the National Research Foundation and the Department of Science and Innovation of SouthAfrica, under SARChI Chair grant No. 64789.APPENDIX A. COMPARISON TO PREVIOUSLY PUBLISHEDRESULTSThe Monte-Carlo code developed by Krawczynski(2012) can be used to numerically compute the polari-sation due to Compton scattering in the Thomson andKlein-Nishina regimes. The numerical results were com-pared to analytical results of Bonometto et al. (1970)that were based on quantum mechanical scattering cal-culations in the Thomson regime. The numerical formu-lation was subsequently used to study the polarisation ofCompton radiation emitted in the Klein-Nishina regime.In this Appendix, the results for Compton polarisationin the Thomson and Klein-Nishina regimes from the sim-ulations of the MAPPIES code are compared to the nu-merical results of Krawczynski (2012). The simulations of Krawczynski (2012) were in goodagreement with the analytical calculations of Bonomettoet al. (1970) for Compton polarisation in the Thom-son regime. An important implication of the calcula-tions of Bonometto et al. (1970) is that the PD van-ishes for unpolarised photons scattered by electronswith Lorentz factors γ (cid:38)
10. Krawczynski (2012)tested this prediction by simulating the Compton scat-tering of an isotropic distribution of ∼ . ∼ .
26% due to scattering off mono-energetic elec-trons with Lorentz factors of γ = 10 , assumed to beisotropic in the co-moving frame of the emission region(which moves along the jet with a bulk Lorentz factorof Γ jet = 5). The MAPPIES code is used to simulateComptonisation with the same initial conditions to thoseof Krawczynski (2012). The Stokes vectors correspondto a net PD ∼ . ∼ . kT rad = 3 . lab , Φ lab ) = (1 . ,
0) rad.In all figures discussed, the results from the MAPPIEScode are shown in the left panels, while the numericalresults of Krawczynski (2012) are shown in the rightpanels. The intensity and PD are given as a function ofthe scattered photon energy in Figures 8 and 9, respec-tively. The scattered photon energy is shown in unitsof the maximum energy allowed kinematically y = x sclab x maxlab where x maxlab = 4 γx lab γx lab (1)with x lab = (cid:15) lab / ( m e c ) and x sclab = (cid:15) sclab / ( m e c ) the di-mensionless energy of the seed and scattered photons,respectively. The intensity of the scattered photonsshifts to higher energies for larger Lorentz factors, andpeak towards x maxlab deeper in the Klein-Nishina regime.The PD decreases for larger Lorentz factors, and isstrongly suppressed for γ (cid:29)
10. In Figure 10, the netpolarisation is shown as a function of the Lorentz factorsof the electrons, where the function PD = 0 . / (1 + x e )from Bonometto et al. (1970) is indicated with a red6 Dreyer and B¨ottcher line. The net polarisation decreases approximately withthe inverse of the seed photon energy in the electronrest frame, consistent with the analytical prediction of PD = 0 . x e ) from Bonometto et al. (1970) in theThomson regime. The results from the MAPPIES codeare therefore overall consistent with those of Krawczyn-ski (2012).REFERENCES Abarr, Q., & Krawczynski, H. 2020, ApJ, 889, 111,doi: 10.3847/1538-4357/ab5fdfAckermann, M., Ajello, M., Ballet, J., et al. 2012, ApJ, 751,159, doi: 10.1088/0004-637X/751/2/159Ackermann, M., Anantua, R., Asano, K., et al. 2016, ApJL,824, L20, doi: 10.3847/2041-8205/824/2/L20Aharonian, F., Akhperjanian, A. G., Anton, G., et al. 2009,A&A, 499, 273, doi: 10.1051/0004-6361/200811564Angel, J. R. P. 1969, ApJ, 158, 219, doi: 10.1086/150185Ansoldi, S., Antonelli, L. A., Arcaro, C., et al. 2018, ApJL,863, L10, doi: 10.3847/2041-8213/aad083Atoyan, A., & Dermer, C. D. 2001, PhRvL, 87, 221102,doi: 10.1103/PhysRevLett.87.221102Band, D., Matteson, J., Ford, L., et al. 1993, ApJ, 413, 281,doi: 10.1086/172995Baring, M. G., B¨ottcher, M., & Summerlin, E. J. 2017,MNRAS, 464, 4875, doi: 10.1093/mnras/stw2344Baring, M. G., & Braby, M. L. 2004, ApJ, 613, 460,doi: 10.1086/422867Beheshtipour, B., Krawczynski, H., & Malzac, J. 2017,ApJ, 850, 14, doi: 10.3847/1538-4357/aa906aBlaes, O., Hubeny, I., Agol, E., & Krolik, J. H. 2001, ApJ,563, 560, doi: 10.1086/324045B(cid:32)la˙zejowski, M., Sikora, M., Moderski, R., & Madejski,G. M. 2000, ApJ, 545, 107, doi: 10.1086/317791Bonometto, S., Cazzola, P., & Saggion, A. 1970, A&A, 7,292Bonometto, S., & Saggion, A. 1973, A&A, 23, 9B¨ottcher, M. 2010, arXiv e-prints, arXiv:1006.5048.https://arxiv.org/abs/1006.5048—. 2019, Galaxies, 7, 20, doi: 10.3390/galaxies7010020B¨ottcher, M., & Dermer, C. D. 1998, ApJL, 499, L131,doi: 10.1086/311366B¨ottcher, M., Harris, D. E., & Krawczynski, H. 2012,Relativistic Jets from Active Galactic Nuclei (WileyOnline Library)B¨ottcher, M., Reimer, A., Sweeney, K., & Prakash, A.2013, ApJ, 768, 54, doi: 10.1088/0004-637X/768/1/54Burrows, D. N., Grupe, D., Capalbi, M., et al. 2006, ApJ,653, 468, doi: 10.1086/508740Cerruti, M., Zech, A., Boisson, C., et al. 2019, MNRAS,483, L12, doi: 10.1093/mnrasl/sly210Chang, Z., & Lin, H.-N. 2014, ApJ, 795, 36,doi: 10.1088/0004-637X/795/1/36 Conway, R. G., Garrington, S. T., Perley, R. A., & Biretta,J. A. 1993, A&A, 267, 347Dermer, C. D., & Menon, G. 2010, in AmericanAstronomical Society Meeting Abstracts, Vol. 215,American Astronomical Society Meeting Abstracts
APPIES I - Description and First Results Figure 8.
The intensity of the Compton emission as a function of the scattered photon energy in units of the maximumkinematically allowed energy y = x sclab /x maxlab . The seed photons are uni-directional and mono-energetic in the laboratory framewith x lab = (cid:15) lab / ( m e c ) = 0 . γ = 10 , , , , , Figure 9.
The Compton polarisation as a function of the scattered photon energy in units of the maximum kinematicallyallowed energy y = x sclab /x maxlab . The seed photons are uni-directional and mono-energetic in the laboratory frame with x lab = (cid:15) lab / ( m e c ) = 0 . γ = 10 , , , , , Figure 10.
The net PD of the Compton emission as a function of Lorentz factors of the electrons. The seed photons are uni-directional and mono-energetic in the laboratory frame with x lab = (cid:15) lab / ( m e c ) = 0 . γ = 10 , , , , , , . × , . × . Results from Krawczynski (2012) are shown in the right panel, and theresults from the MAPPIES code are shown in the left panel. The function PD = 0 . / (1 + x e ) from Bonometto et al. (1970) isindicated with a red line.Laurent, P., Rodriguez, J., Wilms, J., et al. 2011, Science,332, 438, doi: 10.1126/science.1200848 Li, W., Xing, Y., Yu, Y., et al. 2019, Journal ofAstronomical Telescopes, Instruments, and Systems, 5,019003, doi: 10.1117/1.JATIS.5.1.019003 Dreyer and B¨ottcher
Liodakis, I., Peirson, L., & Romani, R. 2020, in AmericanAstronomical Society Meeting Abstracts, AmericanAstronomical Society Meeting Abstracts, 305.09Lundman, C., Vurm, I., & Beloborodov, A. M. 2018, ApJ,856, 145, doi: 10.3847/1538-4357/aab3e8Lyutikov, M., Pariev, V. I., & Blandford, R. D. 2003, ApJ,597, 998, doi: 10.1086/378497Marshall, H. L., Garner, A., Heine, S. N., et al. 2019, inUV, X-Ray, and Gamma-Ray Space Instrumentation forAstronomy XXI, Vol. 11118, International Society forOptics and Photonics, 111180AMarshall, H. L., Schulz, N. S., Trowbridge Heine, S. N.,et al. 2017, in Society of Photo-Optical InstrumentationEngineers (SPIE) Conference Series, Vol. 10397,Proc. SPIE, 103970K, doi: 10.1117/12.2274107Masnou, J. L., Wilkes, B. J., Elvis, M., McDowell, J. C., &Arnaud, K. A. 1992, A&A, 253, 35Matsumoto, M., & Nishimura, T. 1998, ACM Trans. Model.Comput. Simul., 8, 3–30, doi: 10.1145/272991.272995Matt, G., Feroci, M., Rapisarda, M., & Costa, E. 1996,Radiation Physics and Chemistry, 48, 403,doi: 10.1016/0969-806X(95)00472-AMcConnell, M., Ajello, M., Baring, M., et al. 2019, BAAS,51, 100McConnell, M. L., & Ryan, J. M. 2004, NewAR, 48, 215,doi: 10.1016/j.newar.2003.11.029Moderski, R., Sikora, M., Coppi, P. S., & Aharonian, F.2005, MNRAS, 363, 954,doi: 10.1111/j.1365-2966.2005.09494.xMurase, K., Oikonomou, F., & Petropoulou, M. 2018, ApJ,865, 124, doi: 10.3847/1538-4357/aada00Ostorero, L., Villata, M., & Raiteri, C. M. 2004, A&A, 419,913, doi: 10.1051/0004-6361:20035813Padovani, P., Giommi, P., Resconi, E., et al. 2018,MNRAS, 480, 192, doi: 10.1093/mnras/sty1852Pal, M., Kushwaha, P., Dewangan, G. C., & Pawar, P. K.2020, ApJ, 890, 47, doi: 10.3847/1538-4357/ab65eePaliya, V. S., Sahayanathan, S., & Stalin, C. S. 2015, ApJ,803, 15, doi: 10.1088/0004-637X/803/1/15Paliya, V. S., Zhang, H., B¨ottcher, M., et al. 2018, ApJ,863, 98, doi: 10.3847/1538-4357/aad1f0Palma, N. I., B¨ottcher, M., de la Calle, I., et al. 2011, ApJ,735, 60, doi: 10.1088/0004-637X/735/1/60Paltani, S., Courvoisier, T. J. L., & Walter, R. 1998, A&A,340, 47. https://arxiv.org/abs/astro-ph/9809113Paul, B., Gopala Krishna, M. R., & Puthiya Veetil, R. 2016,in 41st COSPAR Scientific Assembly, Vol. 41, E1.15–8–16Pe’er, A. 2015, Advances in Astronomy, 2015, 907321,doi: 10.1155/2015/907321 Pian, E., Urry, C. M., Maraschi, L., et al. 1999, ApJ, 521,112, doi: 10.1086/307548Piran, T., & Granot, J. 2001, in Gamma-ray Bursts in theAfterglow Era, ed. E. Costa, F. Frontera, & J. Hjorth,300, doi: 10.1007/10853853 80Piran, T., Sari, R., & Zou, Y.-C. 2009, MNRAS, 393, 1107,doi: 10.1111/j.1365-2966.2008.14198.xPozdnyakov, L. A., Sobol, I. M., & Syunyaev, R. A. 1983,Astrophys. Space Phys. Res., 2, 189Racusin, J. L., Oates, S. R., Schady, P., et al. 2011, ApJ,738, 138, doi: 10.1088/0004-637X/738/2/138Raiteri, C. M., Villata, M., Kadler, M., et al. 2006, A&A,452, 845, doi: 10.1051/0004-6361:20054409Raiteri, C. M., Villata, M., Ibrahimov, M. A., et al. 2005,A&A, 438, 39, doi: 10.1051/0004-6361:20042567Rani, B., Zhang, H., Hunter, S. D., et al. 2019, BAAS, 51,348. https://arxiv.org/abs/1903.04607Ravasio, M., Tagliaferri, G., Ghisellini, G., et al. 2003,A&A, 408, 479, doi: 10.1051/0004-6361:20031015Reimer, A., B¨ottcher, M., & Buson, S. 2019, ApJ, 881, 46,doi: 10.3847/1538-4357/ab2bffRoustazadeh, P., & B¨ottcher, M. 2012, ApJ, 750, 26,doi: 10.1088/0004-637X/750/1/26Rybicki, G. B., & Lightman, A. P. 1979, Radiativeprocesses in astrophysics (John Wiley & Sons)Sari, R. 1997, ApJL, 489, L37, doi: 10.1086/310957Schnittman, J. D., & Krolik, J. H. 2009, ApJ, 701, 1175,doi: 10.1088/0004-637X/701/2/1175—. 2010, ApJ, 712, 908, doi: 10.1088/0004-637X/712/2/908Sgr`o, C., & IXPE Team. 2019, Nuclear Instruments andMethods in Physics Research A, 936, 212,doi: 10.1016/j.nima.2018.10.111She, R., Feng, H., Muleri, F., et al. 2015, in Society ofPhoto-Optical Instrumentation Engineers (SPIE)Conference Series, Vol. 9601, Proc. SPIE, 96010I,doi: 10.1117/12.2186133Sikora, M., Begelman, M. C., & Rees, M. J. 1994, ApJ, 421,153, doi: 10.1086/173633Sikora, M., Madejski, G., Moderski, R., & Poutanen, J.1997, ApJ, 484, 108, doi: 10.1086/304305Stokes, G. G. 1851, Trans. Cambridge Philos., 9, 399Sunyaev, R. A., & Titarchuk, L. G. 1984, in High EnergyAstrophysics and Cosmology, 245Tamborra, F., Matt, G., Bianchi, S., & Dovˇciak, M. 2018,A&A, 619, A105, doi: 10.1051/0004-6361/201732023Tavecchio, F., & Ghisellini, G. 2015, MNRAS, 451, 1502,doi: 10.1093/mnras/stv1023Toma, K., Sakamoto, T., Zhang, B., et al. 2009, ApJ, 698,1042, doi: 10.1088/0004-637X/698/2/1042
APPIES I - Description and First Results19