Nonthermal processes in hot accretion flows onto supermassive black holes: An inhomogeneous model
Eduardo M. Gutiérrez, Florencia L. Vieyro, Gustavo E. Romero
AAstronomy & Astrophysics manuscript no. aanda © ESO 2021February 25, 2021
Nonthermal processes in hot accretion flows onto supermassiveblack holes: An inhomogeneous model
E. M. Gutiérrez , F. L. Vieyro , , and G. E. Romero , Instituto Argentino de Radioastronomía (IAR, CONICET / CIC / UNLP), C.C.5, (1894) Villa Elisa, Buenos Aires, Argentina e-mail: [email protected] Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata, Paseo del Bosque s / n, 1900 La Plata, BuenosAires, ArgentinaSubmitted ABSTRACT
Context.
Many low-redshift active galactic nuclei harbor a supermassive black hole accreting matter at low or medium rates. Atsuch rates, the accretion flow usually consists of a cold optically thick disk, plus a hot, low density, collisionless corona. In the lattercomponent, charged particles can be accelerated to high energies by various mechanisms.
Aims.
We aim to investigate, in detail, nonthermal processes in hot accretion flows onto supermassive black holes, covering a widerange of accretion rates and luminosities.
Methods.
We developed a model consisting of a thin Shakura-Sunyaev disk plus an inner hot accretion flow or corona, modeledas a radiatively ine ffi cient accretion flow, where nonthermal processes take place. We solved the transport equations for relativisticparticles and estimated the spectral energy distributions resulting from nonthermal interactions between the various particle speciesand the fields in the source. Results.
We covered a variety of scenarios, from low accretion rates up to 10% of the Eddington limit, and identified the relevantcooling mechanisms in each case. The presence of hadrons in the hot flow is decisive for the spectral shape, giving rise to secondaryparticles and gamma-ray cascades. We applied our model to the source IC 4329A, confirming earlier results which showed evidenceof nonthermal particles in the corona.
Key words.
Relativistic processes – radiation mechanisms: nonthermal – black hole physics – accretion, accretion disks – galaxies:active
1. Introduction
The accretion of matter and magnetic fields onto compact ob-jects is the mechanism responsible for powering the most en-ergetic phenomena known in nature. In particular, active galac-tic nuclei (AGNs), which consist of a supermassive black holeaccreting material from the central region of a host galaxy, arethe sources that dominate the gamma-ray sky (Abdollahi et al.2020). Most gamma-ray emitting AGNs are blazars, whose lumi-nosity is beamed by the relativistic bulk motion of the plasma ina jet. Some extragalactic gamma-ray sources, however, are non-blazars: radio galaxies, usually associated with low-luminosityAGNs (LLAGNs), Narrow Line Seyfert 1 galaxies, which ac-crete close to the Eddington limit (Rieger 2017), and a fewSeyfert 2 galaxies (Wojaczy´nski et al. 2015; The Fermi-LAT col-laboration 2019). The latter usually present both an AGN and astarburst, and it is still unclear what is the relative contribution ofeach component to the total gamma-ray emission (Wojaczy´nskiet al. 2015).From a physical point of view, accretion flows are classi-fied into di ff erent regimes depending mainly on the accretionrate (Chen et al. 1995, see also Begelman 2014). At low accre-tion rates, the flow behaves as an optically thin radiatively in-e ffi cient accretion flow (RIAF). In many situations of interest,the RIAF may coexist with a standard Shakura-Sunyaev disk(SSD, Shakura & Sunyaev 1973; Novikov & Thorne 1973). Thisscenario leads to the so-called SSD + RIAF model (Bisnovatyi-Kogan & Blinnikov 1977; Narayan 1996; Dove et al. 1997), which has been applied to explain the various spectral states ofblack hole binaries (BHBs, Narayan 1996; Poutanen et al. 1997;Esin et al. 1997, 1998), the broadband spectrum of LLAGNs(Maraschi & Tavecchio 2003; Nemmen et al. 2014), and someSeyfert galaxies (Chiang & Blaes 2003; Yuan & Zdziarski 2004;Yuan & Narayan 2014 and references therein). This family ofmodels considers that an outer cold thin disk is truncated at aradius R tr > R ISCO , where R ISCO is the radius of the innermoststable circular orbit (ISCO). At the truncation radius, the diskevaporates into an inner hot advection-dominated accretion flowthat extends down to the black hole event horizon. Independentlyof the details of the physical mechanisms for this transition (see,e.g., Abramowicz et al. 1988), there must be a region of overlapbetween the two states. Moreover, an SSD extending down to theISCO is insu ffi cient to explain many features of the observed X-ray phenomenology in many luminous AGNs, and the presenceof a hot optically thin plasma above and below the disk is alsorequired (Bisnovatyi-Kogan & Blinnikov 1977; Poutanen 1998).Unlike SSDs, which are expected to be dense enough for theplasma to thermalize quickly, hot accretion flows (HAFs ) canbe weakly collisional or even collisionless and thus suitable forthe occurrence of particle acceleration and nonthermal processes Throughout the work, we use the terms HAF, RIAF, “corona”, and“advection-dominated accretion flow” to refer to the same physical sys-tem, namely the hot, inflated, optically thin component of the accretingstructure. See Yuan & Narayan (2014) for a complete review of theseflows. Article number, page 1 of 16 a r X i v : . [ a s t r o - ph . H E ] F e b & A proofs: manuscript no. aanda (Mahadevan & Quataert 1997). Although the gamma-ray emis-sion in AGNs is usually associated with the presence of a rel-ativistic jet, the detection of high-energy radiation from radio-quiet AGNs motivates the investigation of the contribution ofnonthermal processes in accretion flows to the overall energeticoutput of these sources. Moreover, the gamma-ray radiation froma few Seyfert 2 plus starburst galaxies not only shows variabilitybut also seems to lie well above both the known IR / γ and radio / γ correlation; this suggests that putative AGNs might be poweringmost of their gamma-ray output (Wojaczy´nski & Nied´zwiecki2017; Peng et al. 2019).There is also significant observational evidence that supportsthe idea that particles are accelerated in HAFs: The steady radioemission from Sgr A* is thought to be produced by a populationof nonthermal electrons within the HAF that feeds the centralblack hole (Yuan et al. 2003; Liu & Wu 2013). Additionally,the multiwavelength flaring activity of this source is likely re-lated to nonthermal activity in the flow (Yuan et al. 2003, 2004;Gutiérrez et al. 2020; Dexter et al. 2020). Recently, Inoue & Doi(2018) found evidence of nonthermal electron activity occurringin the corona of more luminous AGNs such as Seyfert I galaxies.The presence of nonthermal protons in HAFs is more di ffi cult totrace directly, but it can be indirectly inferred from the detec-tion of neutrinos or cosmic rays, as well as gamma-ray cascadeemission (Kimura et al. 2015, 2019a; Inoue et al. 2020). From atheoretical point of view, protons thermalize much slower thanelectrons, so it is expected that they retain a longer memory ofthe heating or acceleration process. Moreover, recent Particle-In-Cell (PIC) simulations show that protons are accelerated muchmore e ffi ciently than electrons in turbulent environments such asaccretion flows (Zhdankin et al. 2019).The study of the transport of nonthermal particles in coro-nae or HAFs around supermassive and stellar-mass black holesis a highly complex field. It involves numerous physical ingre-dients: the structure and dynamics of a disk, the thermal back-ground –including the thermal coupling of electrons and pro-tons and various mechanisms of cooling and heating of the gas–, and the possible presence of nonthermal particles with theircorresponding acceleration and transport. Several aspects of theproblem have been previously explored in the literature with dif-ferent approaches: Some studies include only leptonic contribu-tions (e.g., Coppi 1992; Vurm & Poutanen 2009; Veledina et al.2011; Bandyopadhyay et al. 2019), whereas other works includehadronic processes (Romero et al. 2010; Vieyro & Romero 2012;Rodríguez-Ramírez et al. 2019; Inoue et al. 2019; Kimura et al.2015, 2019a). Many previous models adopted a homogeneousspherical corona and rely on the one-zone approximation (e.g.,Vurm & Poutanen 2009; Romero et al. 2010; Vieyro & Romero2012; Kimura et al. 2015, 2019a; Inoue et al. 2019). On the otherhand, those models using actual hydrodynamic solutions for theRIAF do not usually solve the nonthermal transport equationsbut assume or fit the energy distribution for the relativistic parti-cles (e.g., Özel et al. 2000; Yuan et al. 2003; Wojaczy´nski et al.2015; Bandyopadhyay et al. 2019). Besides, di ff erent techniquesare also used to address the problem: Detailed numerical simu-lations allow treating in much more detail nonlinear phenomena,and are essential when one wants to tackle time evolution or mul-tidimensional phenomena (such as outflows) (e.g., Hilburn et al.2010; Yuan & Narayan 2014; Chael et al. 2017). Semi-analyticalprocedures, on the other hand, allow to study most of the relevantphysical processes that take place globally within the flow, withthe advantage that are more versatile to give a physical grasp ofthe situation (e.g., Vieyro & Romero 2012; Kimura et al. 2019a). In this work, we develop a new model of an HAF onto ablack hole with focus on the nonthermal processes. We use asemi-analytical treatment that combines several of the ingredi-ents mentioned above: a disk with both hot and cold compo-nents, thermal and nonthermal particles, and their interactionsand transport. In a first step, we solve the hydrodynamic equa-tions to obtain the accretion flow structure for various accretionregimes. Then, we calculate the electromagnetic emission of thethermal electrons in the flow. Once the thermal background isset, we inject a population of relativistic particles, both electronsand protons and estimate the outputs (photons and secondaryparticles) resulting from their interactions with the environment.To this end, we solve the transport equation for each particlespecies, and compute the spectral energy distribution (SED) pro-duced by all relevant processes, taking into account the radialdependence in the physical properties of the flow.The remainder of this article is organized as follows: In Sec-tion 2, we present in detail the accretion model. In Section 3, wediscuss the acceleration and transport of the relativistic particlesin the HAF and how we treat them. In Section 4 we describe thecalculation of the SED resulting from all the nonthermal pro-cesses discussed in the previous sections, for a set of specificmodels. In Section 5 we present the general results, and in Sec-tion 6 we apply the model to the Seyfert galaxy IC 4329A. InSection 7 we discuss various phenomena where nonthermal pro-cesses in HAFs might play important roles. Finally, we presenta summary and our conclusions in Section 8.
2. Accretion flow model
The model is constructed so that it is general enough to be ap-plied to a wide variety of accretion flows onto black holes, fromthose in LLAGNs like Sgr A*, which are usually modeled aspure RIAFs (Narayan et al. 1998; Yuan et al. 2003), to thosepowering moderately luminous Seyfert galaxies. In the latterclass of objects, an SSD disk may penetrate down to the ISCO(Wojaczy´nski et al. 2015; Inoue et al. 2019). A general picturewith the basic geometry of the model is shown in Fig. 1. It con-sists of an SSD extending from a radius R out , SSD ∼ − R S downto the truncation radius R tr , and an HAF extending from a ra-dius R out ∼ − R S ≥ R tr down to the event horizon. Here, R S = GM BH / c denotes the Schwarzschild radius, M BH is theblack hole mass, G is the gravitational constant, and c is thespeed of light in vacuum. In the region R tr < R < R out the twostates coexist, and the HAF plays the role of a corona above andbelow the thin disk. The total accretion rate is divided into theSSD and the corona:˙ M ( R ) = ˙ M d ( R ) + ˙ M c ( R ) , (1)where ˙ M ( R ) is the total accretion rate at radius R , ˙ M c ( R ) is theaccretion rate through the corona, and ˙ M d ( R ) is the accretion ratethrough the SSD. We write˙ M ( R ) = ˙ M out [1 − w ( R )] , (2)˙ M d ( R ) = ˙ M out f ( R ) , (3)˙ M c ( R ) = ˙ M out g ( R ) , (4)so that f ( R ) + g ( R ) = − w ( R ), where w ( R ) accounts for the totalmass loss rate via winds integrated from R out to R . Here, ˙ M out isthe accretion rate at the outer boundary of the system, and f ( R )makes the transition between the two components smooth. It iswell-known that HAFs present magneto-centrifugal winds thatdecrease the amount of matter that actually reaches the black Article number, page 2 of 16utiérrez, Vieyro & Romero: Nonthermal processes around supermassive black holes
Fig. 1.
Cartoon representing the accretion system. A jet might be presentbut it is not considered in our model. hole (Narayan & Yi 1995; Stone et al. 1999; Yuan et al. 2012).This decrease is usually parameterized in a phenomenologicalway as (Blandford & Begelman 1999)˙ M c ( R ) = ˙ M out (cid:32) RR out (cid:33) s , (5)with 0 < s < g ( R ) in such a way that when f ( R ) =
0, that is to say, when theSSD has completely evaporated, ˙ M c ∝ R s . Expressions for f ( R ), g ( R ) and w ( R ) are given in Appendix A. We modeled the hot, inflated, optically thin inner component ofthe accretion flow as an RIAF. Since this is the site where weaim to explore nonthermal phenomena, we treat with detail boththe hydrodynamics and the radiative outputs of this component.
We obtain the hydrodynamical structure of the RIAF (mass den-sity ρ , magnetic field B , proton and electron temperatures T p , T e ,etc) by solving the height-integrated, steady-state hydrodynamicequations (Abramowicz et al. 1988; Yuan et al. 2003) via theshooting method with appropriate boundary conditions (Yuanet al. 2000). The remaining parameters to determine the structureof the flow are the viscosity α -parameter, the plasma β -parameter(gas pressure to magnetic pressure ratio), and the fraction ofenergy released by turbulence that directly heats electrons, δ .Early studies on ADAFs considered δ to be small ( (cid:46) − ), butmore recent numerical (Quataert & Gruzinov 1999; Sharma et al.2007) and observational works (Yu et al. 2011) suggest that itmay be as high as ∼ .
5. Unless explicitly stated, we fix theseparameters to standard values: α = . β = δ = . s = . When R out ∼ R S , the outer proton temperature is T out , p = . T vir ,and the electron temperature is T out , e = . T vir ; here, T vir = . × ( R S / R ) K is the virial temperature. The Mach number at R out is M s ≡ v / c s = . the black hole event horizon at the speed of light, the hydrody-namical solution must be transonic. The calculation necessaryto obtain such a transonic solution involves adjusting iterativelyan eigenvalue (the specific angular momentum accreted bythe black hole) in such a way that the sonic radius is crossedsmoothly (see Yuan et al. 2000 for details).Although we consider that the plasma has both thermal andnonthermal components, we assume that the thermal gas domi-nates energetically. Then, the kinetic energy density of protons(p) and electrons (e) is simply (Chandrasekhar 1939) u q ≈ u thq = a ( θ q ) n q m q c θ q , (6)where a ( θ q ) = θ q (cid:34) K (1 /θ q ) + K (1 /θ q )4 K (1 /θ q ) − (cid:35) . (7)Here, K n are the modified Bessel functions of n th order, θ q = k B T q / m q c , m q and n q are the mass and the number density ofparticles of the species q = p , e; k B is the Boltzmann constant. Radiatively ine ffi cient accretion flows are optically thin andhence both the cooling function and the shape of the emittedspectrum depend strongly on the details of the radiative pro-cesses that take place in the flow. Electrons reach relativis-tic temperatures ( θ e (cid:38)
1) and cool via synchrotron radiation,Bremssthralung radiation, and inverse Compton up-scattering oflow-energy photons. Whereas the first two processes are local,inverse Compton is not: Photons may su ff er multiple scatteringsin di ff erent regions of the flow before escaping. Nevertheless,to estimate the cooling function of the flow, a local treatmentis usually adopted giving fairly accurate results (Dermer et al.1991; Esin et al. 1996; Manmoto et al. 1997). We use this ap-proximation when solving the hydrodynamical structure, thoughwe do take into account nonlocal scatterings in the calculation ofthe spectrum (see Sect. 4).For thermal protons, the three cooling mechanisms men-tioned above are completely negligible, but since they reachmuch higher temperatures than electrons (almost virial, ∼ Kin the inner regions), those at the tail of the Maxwellian dis-tribution have su ffi cient energy to produce neutral pions (andthus gamma rays) via proton-proton (pp) collisions (Mahade-van et al. 1997; Oka & Manmoto 2003).We solve the hydrodynamic equations and obtain the tem-peratures T e , p ( R ), magnetic field B ( R ), mass density ρ ( R ), radialvelocity v ( R ), and height H ( R ) as a function of the distance fromthe event horizon. To calculate the thermal SED emitted by theflow, we divide its volume into N logarithmically-spaced cylin-drical shells, and solve the radiative transfer taking into accountthe coupling (through Comptonization) among them. When anSSD is present in the model, we consider its emission as a seedfor Compton scattering as well. In Section 4, we describe in de-tail the radiation transfer calculation. We calculate the emission from the SSD using the standard mul-ticolor blackbody method (e.g., Frank et al. 2002). We take intoaccount the coupling between the SSD and the RIAF by means Neutral pions decay with a mean lifetime of 8 . × − s into twogamma rays π → γ . Article number, page 3 of 16 & A proofs: manuscript no. aanda of two mechanisms: heating of the SSD by absorption of inci-dent radiation from the RIAF and inverse Compton scattering ofSSD photons by hot electrons in the RIAF (Narayan et al. 1997).For simplicity, we ignore X-ray reflection in the SSD; it shouldbe taking into account, however, to investigate in detail the X-rayphenomenology of the AGN.
From here on, we shall express the black hole mass in units ofsolar masses: M BH = m M (cid:12) , and accretion rates in units of theEddington accretion rate, ˙ M = ˙ m ˙ M Edd , where ˙ M Edd = L Edd /η c , L Edd = . × m erg s − is the Eddington luminosity, and η = . ffi ciency . We also express radii in unitsof the Schwarzschild radius: R = r R S . Of the many parametersof the model, the accretion rate is the most important one. Hence,we fix the mass of the central black hole to m = , and the HAFouter radius to r out = ff erent scenarios with increasing accretionrate: ˙ m out = − , − , − , − . In the latter three scenar-ios, we assume that a thin disk penetrates down to r tr = . We also make the transition softer as the disk penetratesmore into the RIAFs, thus we take b = , , .
5, respectively (seeAppendix A). This is reasonable since a high value of b impliesan abrupt transition between the two accretion states at the trun-cation radius, and hence a too small corona when the SSD pen-etrates down to the innermost regions. Figure 2 shows the SEDresulting from thermal processes in each of these four scenarios.At low accretion rates, three peaks can be clearly distinguishedin the spectrum, and they correspond to synchrotron emission,a first inverse Compton scattering of these synchrotron photons,and Bremsstrahlung radiation, respectively. As the accretion rateincreases, the SSD penetrates deeper into the HAF and its black-body emission starts to compete with the synchrotron emissionfrom the HAF, and second and higher-order inverse Comptonscatterings overtake the Bremsstrahlung X-ray radiation, hard-ening the spectrum. Also, the Comptonization of the SSD pho-tons increases. In the most luminous scenario, the emission iscompletely dominated by the radiation from an SSD penetratingdown to the ISCO, and the power-law tail is caused by the super-position of many inverse Compton scatterings of the blackbodyphotons from the SSD by the hot electrons in the corona. Thespectral index of the spectrum in this latter scenario is deter-mined mainly by the b parameter (see Figure 11 for comparisonwith this case).
3. Nonthermal particles
Many numerical studies show that particles can be acceleratedin hot accretion flows (see, e.g., Li & Miller 1997; Lynn et al.2014). This is expected since an RIAF is a collisionless plasmawith strong and turbulent magnetic fields. In such an environ-ment, several mechanisms can accelerate particles to relativis-tic energies. The most plausible ones are magnetic reconnec-tion (Hoshino & Lyubarsky 2012; de Gouveia Dal Pino et al.2010), stochastic acceleration by turbulence (Dermer et al. 1996; The actual radiative e ffi ciency is in general much lower than 0 . Accretion flows with very low accretion rate ( ˙ m out (cid:46) − ) are usu-ally modeled as pure RIAFs. Although an optically thick componentmight be present at outer radii, its e ff ect would be negligible. The spectral index of the power-law is consistent with the analyticalexpression α = − log τ T / log A , where τ T is the Thomson optical depthand A = + θ e + θ (Rybicki & Lightman 1979) [ ] [] = . , == , == , == Fig. 2.
Thermal emission from the accretion flow around a supermas-sive black hole of mass M BH = M (cid:12) for four di ff erent models. Inthe three flows with highest accretion rate, we show in thin dotted linesthe direct emission from the thin disk, in dot-dashed lines with trian-gle markers the Comptonization of the thin-disk photons, and in dashedlines the Comptonization of the local Synchrotron and Bremsstrahlungphotons from the RIAF. Zhdankin et al. 2019), and di ff usive acceleration mediated byshocks (Drury 1983; Blandford & Eichler 1987). Regardless of the acceleration mechanism, we assume that afraction of the particles is pushed out from the thermal distri-bution and get accelerated to high energies. We assume that afterbeing accelerated, the particle population can be described by apower-law injection function of the form Q ( γ, r ) = Q ( r ) γ − p exp (cid:2) − γ/γ cut ( r ) (cid:3) , (8)characteristic of di ff usive acceleration mechanisms. We shallconsider two values for the spectral index: p =
2, where thepower is distributed equally along the whole energy range, and p = .
2, where most of the power is in the highest energy par-ticles. The cuto ff Lorentz factor at each radius is estimated bythe balance between the acceleration timescale and the cool-ing / escape timescale (see Section 3.2). The transport equationthat governs the evolution of the population of nonthermal parti-cles is ∂ N ∂ t + ∇ · ( v N − D r ∇ N ) = ∂∂γ (cid:16) | ˙ γ rad | N (cid:17) − Nt esc + Q , (9)where N ( γ, r ) d γ is the number of particles per unit volumewhose Lorentz factor lies in the range ( γ, γ + d γ ), D r is the radialdi ff usion coe ffi cient, ˙ γ rad is the radiative cooling rate, and t esc isthe escape timescale . The spatial derivatives on the left-hand-side in Eq. 9 are responsible for the radial transport: The firstterm represents advection toward the black hole, and the secondterm represents radial di ff usion. In cylindrical coordinates, as-suming axisymmetry, homogeneity in the vertical direction, and Only in regions where the magnetic energy density is much smallerthan the gas energy density. We assume escape in the vertical direction.Article number, page 4 of 16utiérrez, Vieyro & Romero: Nonthermal processes around supermassive black holes steady state, we can re-write Eq. 9 in the form ∂∂ r (cid:34) A ( γ, r ) ∂ ˜ N ∂ r + B ( γ, r ) ˜ N (cid:35) − ∂∂γ (cid:16) | ˙ γ rad | ˜ N (cid:17) + ˜ Nt esc = ˜ Q , (10)where ˜ N ( γ, r ) : = rN ( γ, r ), A ( γ, r ) : = − D r ( γ, r ), and B ( γ, r ) : = −A / r + v r ( r ).Equation 10 is a two-dimensional advection-di ff usion equa-tion in the ( γ, r )-space. Nevertheless, since energy and radialtransport have in general very di ff erent natural timescales, Eq. 10can be simplified when the transport in one dimension dominatesover the other. For electrons, radiative cooling is much fasterthan radial transport at almost all energies. Only electrons with γ <
10 in the innermost regions can be advected without beingcompletely cooled (see Section 3.2), and we can neglect the firstterm and solve the equation under the one-zone approximation ateach cylindrical shell. The opposite is true for protons, meaningthat they carry their energy throughout the flow via advectionand di ff usion. Radiative cooling, however, cannot be neglectedat the highest energies. To tackle this e ff ect, we approximate thesecond term in the left-hand side of Eq. 10 as ≈ − ˜ N / t rad , where t rad : = | ˙ γ rad | /γ is the cooling timescale. We solve Eq. 10 via fi-nite di ff erences with the Chang & Cooper (1970) discretizationas outlined in Park & Petrosian (1996).We normalize the injection function by assuming that a frac-tion ε NT of the total accretion power goes to nonthermal parti-cles. This power is divided into protons and electrons with theprescription L e = ε e L NT , and L p = ε p L NT = (1 − ε e ) L NT . Inaddition, we must give a functional form for the radial depen-dence of the injected nonthermal power. We conservatively as-sume Q ( r ) ∝ B ( r ) u th ( r ). Thus, L e , p = ε NT ε e , p ˙ M out c = q (cid:90) dV B ( r ) u th ( r ) (cid:90) γ max γ min d γ γ − p + , (11)where q is a normalization constant. Finally, after solving thetransport equation we check that p CR9 (cid:46) . p gas for all radii,assuring that our initial assumption that nonthermal particles donot a ff ect the flow structure is fulfilled. The timescale for the acceleration of a particle of mass m andcharge e to an energy γ mc depends on the acceleration mech-anism. We enclose the uncertainties into the acceleration e ffi -ciency parameter η acc <
1, and write the acceleration timescaleas (e.g., Aharonian et al. 2002) t acc ( γ ) ∼ η − r L c = η − γ mceB , (12)where r L is the relativistic Larmor radius of the particle.High-energy particles in the RIAF radiate and lose energy byseveral processes. Electrons cool mainly via synchrotron emis-sion and inverse Compton up-scattering of low-energy photons.The latter include those emitted by the thermal particles in theRIAF and the SSD as well as the synchrotron photons emitted bythemselves (Synchrotron Self-Compton, SSC). The synchrotroncooling time for a charged particle of mass m and Lorentz factor γ moving in a medium with magnetic field B is t syn ( γ ) = mc σ T U B (cid:32) mm e (cid:33) γ − , (13) The cosmic ray pressure at the position r is p CR ( r ) : = m p c (cid:82) d γ γ N ( r , γ ). where U B = B / π is the magnetic energy density. Let n ph ( (cid:15) ) bethe isotropically-averaged spectral density of low-energy pho-tons (see Sect. 4), then the inverse Compton cooling timescale is(Moderski et al. 2005) t IC ( γ ) = mc σ T U ph (cid:32) mm e (cid:33) γ − (cid:34) U ph (cid:90) d (cid:15) (cid:15) n ph ( (cid:15) ) f KN (˜ b ) (cid:35) − , (14)where U ph is the photon energy density, and f KN (˜ b ) = g (˜ b ) / ˜ b ,where g (˜ b ) = (cid:32)
12 ˜ b + + b (cid:33) ln(1 + ˜ b ) − (cid:32) b + b + b + (cid:33) (15) × (cid:16) + ˜ b (cid:17) − + ( − ˜ b ) . (16)Here, ˜ b = γ(cid:15)/ ( mc ).In addition to the mechanisms mentioned above, relativisticprotons lose energy through inelastic pp and p γ interactions. Thecooling timescale due to pp collisions is t pp ( γ ) = n p σ pp c κ pp , (17)where σ pp is the total cross-section and κ pp ∼ .
17 is the inelas-ticity of the process. An accurate parametrization for the totalcross-section is given by (Kelner et al. 2006) σ pp (cid:39) (cid:16) . + . L + . L (cid:17) − (cid:32) E π, thr E p (cid:33) mb , (18)for E p > E π, thr , where E p = γ p m p c is the proton energy, E π, thr = .
22 GeV is the threshold energy for pion production,and L = log (cid:16) E p / (cid:17) . Photohadronic inelastic collisions coolprotons via two channels: photomeson production (p γ ) and pho-topair production—the so-called Bethe-Heitler (BH) channel.The cooling timescale for the first process is given by t − γ = c γ p (cid:90) ∞ ¯ ε thr , p γ d ¯ εσ p γ ( ¯ ε ) κ p γ ( ¯ ε ) ¯ ε (cid:90) ∞ ¯ ε/ (2 γ p ) d (cid:15) U ph ( (cid:15) ) (cid:15) , (19)where ¯ ε thr , p γ =
145 MeV is the threshold energy, σ p γ is thecross-section, and κ p γ is the inelasticity of the process. Usefulparametrizations for these functions are given in Atoyan & Der-mer (2003). For the Bethe-Heitler cooling channel, an equivalentexpression is obtained by replacing the cross-section, inelastic-ity, and threshold energy in Eq. 19 for their correspondent values σ BH , κ BH (see, e.g., Begelman et al. 1990), and ¯ ε thr , BH = m e c .Both species of particles can escape from the system by twoprocesses: energy-dependent di ff usion in the vertical directionor standard RIAF winds. The escape timescale via winds is in-dependent of the particle energy and can be parameterized as t − = d ˙ Mdr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) wind v r ˙ M ( r ) ≈ s | v r | r . (20)The di ff usion timescale is strongly dependent on the modeladopted for the turbulence spectrum: P ( k ) ∝ k − q , where k isthe wavenumber. We assume a value of q = /
3, namely aKolmogorov spectrum. Thus, the spatial di ff usion coe ffi cient forisotropically turbulent magnetic fields is D r ≈ c ζ r L ( k min r L ) − q , (21) Article number, page 5 of 16 & A proofs: manuscript no. aanda [] == == [] == == [] == == [] == == Fig. 3.
Timescales for nonthermal electrons in the various scenarios.Di ff erent rows show di ff erent scenarios (accretion rate increases down-ward), and each column shows di ff erent regions in the RIAF; left col-umn: inner regions ( ∼ R S ), right column: outer regions ( ∼ R S ).Plain solid lines show acceleration timescales, where a proper flag indi-cates the value of the acceleration e ffi ciency η acc ; dotted lines shows ac-celeration timescales for SDA (see Sect. 3.1). Dashed lines show the dif-fusion timescale, dash-dotted lines show the advection timescale. Solidlines with markers show cooling timescales for the relevant processes;with triangles: synchrotron, with stars: inverse Compton. where k min ∼ H − is the minimum wave number of the turbu-lence spectrum, and the di ff usive escape timescale is (Stawarz &Petrosian 2008) t di ff ≈ H D r (cid:39) Hc ζ (cid:18) r L H (cid:19) q − γ q − , (22)where ζ = π (cid:82) P ( k ) dk / B is the strength ratio of turbulencefields against the background ordered component. Finally, thetotal escape timescale is t esc = (cid:16) t − + t − ff (cid:17) − . (23)Figure 3 shows acceleration, cooling, and escape timescalesas a function of energy for the nonthermal electrons in the vari-ous scenarios considered. Each row shows a di ff erent set of pa-rameters (accretion rate increasing downward), and the columnscorrespond to di ff erent regions in the RIAF; inner regions: ∼ R S and outer regions: ∼ R S . The acceleration timescale is shownfor two values of the acceleration e ffi ciency: η acc = − , − , [] == == [] == == [] == == [] == == Fig. 4.
Same timescales as Fig. 3 but for protons. The cooling timescalesshown are for synchrotron (triangles), pp (squares), and photohadroniclosses (circles, includes p γ and BH). and for the case of stochastic di ff usive acceleration (SDA, seeSect. 7.1). Synchrotron radiation and inverse Compton scatter-ing are the dominant cooling processes for electrons and deter-mine the maximum energy they can achieve. Inverse Comptonbecomes dominant at higher accretion rates, mainly because ofthe addition of copious amounts of seed photons emitted by theSSD penetrating the inner regions.Figure 4 shows the same timescales as Fig. 3, but for rela-tivistic protons. At low accretion rates, the losses are completelydominated by escape processes, mainly by di ff usion. At higheraccretion rates, the magnetic field intensifies and advection com-petes with di ff usion. Also, photohadronic losses start to be rele-vant and dominate at the highest energies as the RIAF luminosityand the photon density increase. The hadronic processes described above lead to the produc-tion of secondary mesons and leptons. These particles will inturn emit radiation and su ff er further interactions and decays(Reynoso & Romero 2009). Inelastic pp and p γ collisions notonly create neutral pions but also charged pions. The main chan- Article number, page 6 of 16utiérrez, Vieyro & Romero: Nonthermal processes around supermassive black holes nels for these processes arep + p → p + p + ζ π + ζ ( π + + π − ) , (24)p + p → p + n + π + + ζ π + ζ ( π + + π − ) , (25)andp + γ → p + ζ π + ζ ( π + + π − ) , (26)p + γ → n + π + + ζ π + ζ ( π + + π − ) , (27)where ζ and ζ are the multiplicities. Charged pions have a meanlifetime of τ π ± (cid:39) . × − s in their proper frame, and primarilydecay into a muon and a muonic neutrino. The muon in turndecays with a mean lifetime of τ µ ± (cid:39) . × − s into an electron(or positron), an electronic neutrino, and a muon neutrino: π ± → µ ± + ν µ (¯ ν µ ) , (28) µ ± → e ± + ¯ ν µ ( ν µ ) + ν e (¯ ν e ) . (29)Collisions between protons and photons also inject secondarypairs directly via the Bethe-Heitler (BH) process:p + γ → p + e + + e − . (30)The last channel for the injection of secondary pairs is photon-photon annihilation. We include the presence of these particlesand their interaction in our study. Useful approximations for theinjection functions of pions and muons can be found in Atoyan &Dermer (2003); Kelner et al. (2006); Lipari et al. (2007), whereasphotopair production is studied in Aharonian et al. (1983). Secondary pairs are created with very high energies and theycool locally. Similarly, charged pions and muons decay or coolbefore being significantly transported in the radial direction. Forsimplicity, we track the evolution of secondary particles via theone-zone approximation at each radius in the RIAF. The evolu-tion of pairs and photons is coupled through photo-annihilationabsorption and must be solved iteratively. The coupled kineticequations at the position r for secondary pairs, charged pions,muons, and photons, respectively, are the following: N e ± ( γ e ± ) (cid:104) t − + t − ff (cid:105) = Q p γ → e ± ( γ e ± ) + Q µ ± → e ± ( γ e ± ) + Q γγ → e ± ( γ e ± ) . (31) N π ± ( γ π ) (cid:104) t − + t − ff + t − (cid:105) = Q p γ → π ± ( γ π ) + Q pp → π ± ( γ π ) . (32) N µ ± ( γ µ ) (cid:104) t − + t − ff + t − (cid:105) = Q π ± → µ ± ( γ µ ) . (33) N γ ( (cid:15) γ ) (cid:104) t − + t − γγ → e ± (cid:105) = Q γ ( (cid:15) γ ) , (34)where N q ( γ q ) denotes the steady state particle energy distributionfor the q-species (in units of erg − cm − ), and t dec = γ q τ q is itsmean lifetime in the laboratory frame. In Eq. 34, the term Q γ ( E γ )includes the injection of photons by all radiative processes fromevery particle species.
4. Spectral energy distributions
Once the steady-state particle distributions are obtained, wecalculate the radiative spectrum emitted by the primary andsecondary particles. For thermal synchrotron, we use theparametrization of the emissivity given in Mahadevan et al.(1996). For electron-ion Bremsstrahlung we use the expressiongiven in Stepney & Guilbert (1983), with a small correction byNarayan & Yi (1995), whereas we follow Svensson (1982) forelectron-electron Bremsstrahlung; see also Yarza et al. (2020)for a discussion on the di ff erent approximations used in the liter-ature. To estimate the emission from the nonthermal populations,we use expressions for the emissivities per particle and integrateover the energy distribution. For the synchrotron emissivity weuse the exact formula given in Blumenthal & Gould (1970); wealso take into account synchrotron-self absorption. For inverseCompton scattering, we assume an isotropic background pho-ton field and use the formula given by Moderski et al. (2005)(Eq. 16); for the neutral pion decay emissivity via hadronic in-teractions, we use the formalism of the delta-approximation asoutlined in Kelner et al. (2006) for pp collisions and the methoddescribed in Kelner & Aharonian (2008) for p γ collisions.We follow Manmoto et al. (1997), and solve the radiativetransfer in the vertical direction under the two-stream approxi-mation (Rybicki & Lightman 1979). The flux arising from eachsurface of the disk, without accounting for Comptonization, is F ν = π √ S ν (cid:104) − exp (cid:16) − √ τ ∗ ν (cid:17)(cid:105) , (35)where S ν = j ν /κ ν is the source function, τ ∗ ν = ( π / / κ ν H is thehalf optical depth in the vertical direction, and κ ν and j ν are theabsorption and emission coe ffi cients. For the thermal spectrum j ν = j sync ν + j bremss ν , κ ν = κ th ν = j ν / B ν , and thus S ν = B ν . The N cylindrical shells are centered at r j and have boundaries at l j − and l j + . Hence, the luminosity arising from the j -shell is L ν, j = × π ( l j + − l j − ) × F ν, j , (36)where the factor 2 comes from the two faces of the disk. In orderto calculate the global coupling between di ff erent cells throughComptonization, we follow a similar approach to Narayan et al.(1997), namely we calculate scattering probability matrices cou-pling the di ff erent shells in the RIAF between themselves andwith the cold disk, and we find iteratively the Comptonized lu-minosity as L k C , out ( ν ) = (cid:90) d ν (cid:48) (cid:18) νν (cid:48) (cid:19) L k C , in ( ν (cid:48) ) P ( ν, ν (cid:48) , T e ) , (37)where P ( ν ; ν (cid:48) , T e ) is the probability for a photon of frequency ν (cid:48) to be scattered to a frequency ν by and electron in a relativisticMaxwellian distribution of temperature T e (Coppi & Blandford1990), and L k C , in is the luminosity emitted by all the shells inthe RIAF and by the cold disk that reach the k -shell and getsscattered . We iterate until convergence (see Narayan et al. 1997for more details).For the nonthermal emission we restrict to local interactionsand use Eqs. 35 and 36 with the appropriate absorption and emis-sion coe ffi cients, namely κ ν = κ th ν + κ SSA ν at low frequencies, and κ γγν = (cid:82) d (cid:15) (cid:48) n ph ( (cid:15) (cid:48) ) σ γγ ( (cid:15) γ , (cid:15) (cid:48) ) in the high-energy band accounting We emphasize that to account for multiple Comptonization, this lu-minosity is updated at each iteration and in general includes previousorders of the Comptonization. Article number, page 7 of 16 & A proofs: manuscript no. aanda for internal photon-photon annihilation, where σ γγ is the cross-section for the process (Gould & Schréder 1967). We include inthe photon density n ph both thermal and synchrotron nonthermalcontributions. For the thermal photons, the most copious ones,we take into account nonlocal e ff ects, and calculate the photondensity in the shell centered at r j as n ph ( (cid:15) ) = L NL j ( (cid:15) ) (cid:15) × t cell V j , (38)where L NL j ( (cid:15) ) is the spectral luminosity emitted by all the shellsin the RIAF that reach the shell j (itself included). It is calculatedvia probability matrices that couple di ff erent shells in a similarway to how is done for Comptonization, and it includes redshifte ff ects. The timescale t cell is the average time a photon lives inshell j , and we estimate it as ≈ ( H / c ) × (1 + τ es ) for photonsemitted in the shell j and ≈ ( l j + − l j ) / c × (1 + τ es ) for thosecoming from any other shell. Here, τ es = n e σ T H is the opticaldepth for Thomson scattering. The nonthermal synchrotron pho-tons are added to n ph ( (cid:15) ) at each shell and are calculated by Eq.38 though considering only local emission. Finally, given the lu-minosity emitted by all the radiative processes at each shell, wecalculate the total spectral luminosity measured by a distant ob-server as L ν o = (cid:88) j L ν e , j (cid:104) + z ( r j ) (cid:105) , (39)where ν e = [1 + z ( r )] ν o , and z ( r ) = [(1 − r − ) (1 − β v )] − / − r .
5. Results
We have chosen the value of the power injected into protons suchthat for the models with p = p CR (cid:46) . p CR turns out to be (cid:28) .
2. This is explainedbecause of two facts: Radial di ff usion occurs mainly outward,and the prescription we choose for the injected power at di ff er-ent radii is fairly conservative. It is plausible that acceleration ismuch more e ffi cient in the inner regions than in the outer ones.The models with p = . p CR becausemore particles at higher energies imply more e ffi cient coolingand di ff usive escape. The power injected in electrons is less con-strained; we choose it in such a way that synchrotron emissiondoes not heavily overtake the background thermal spectrum.As discussed in Secs. 2.3 and 3.1, we have chosen four ac-cretion regimes, and two spectral slopes for particle injection. Inturn, for the harder spectrum, we consider two values for particleacceleration e ffi ciency: η acc = − as in the p = ffi cient scenario with η acc = − . Table 1 summarizes themain parameters of the di ff erent models. Here, ε NT and ε e , p arechosen as described above. We show representative particle distributions of Models B1 andB3 in Figs. 5 and 6, respectively. Protons approximately main-tain the injection spectral index, though little changes by the ra-dial transport are seen in the outer layers. Electrons are cooled bysynchrotron / inverse Compton mechanism. In the outer regions, alittle hardening is seen at high energies, where inverse Comptonscattering enters into the Klein-Nishina regime. Table 1.
Main parameters adopted in the models.
Model ˙ m out r tr b p η acc ε NT [%] ε e A1 10 − − − − . − A2 10 − − − . − . − A3 10 − − − . − . − B1 10 −
100 2 2 10 − − B2 10 −
100 2 1 . − − B3 10 −
100 2 1 . − − C1 10 −
30 1 2 10 − − C2 10 −
30 1 1 . − − C3 10 −
30 1 1 . − − D1 0 . . − . . × − D2 0 . . . − . . × − D3 0 . . . − . . × − ()[] [] . ()[] [] Fig. 5.
Steady particle energy distributions for Model B1 (see Table 1).Di ff erent colored lines show di ff erent regions in the RIAF, which areindicated in the colorbar. Upper panel: protons. Lower panel: electrons. ˙ m out = − Models A correspond to the scenario with the lowest accretionrate ( ˙ m out = − ), where the flow is modeled as a pure RIAF atall radii; the calculated SEDs are shown in Fig. 7. The left panelshows Model A1, where nonthermal synchrotron emission is rel-atively strong, and a low electron power is enough to produce abump in the radio band. If more power is injected into electrons,synchrotron emission will easily overcome the thermal luminos-ity. In the gamma-ray band, pp emission is also comparable tothe broadband luminosity, but p γ emission is completely negligi-ble due to the very low photon density. This low density impliesthat only the highest energy photons are absorbed. Electron-positron pairs are created mainly via muon decay and less byphoto-annihilation, and their synchrotron emission is dominantin the megaelectronvolt band. Article number, page 8 of 16utiérrez, Vieyro & Romero: Nonthermal processes around supermassive black holes ()[] [] . ()[] [] Fig. 6.
Steady particle energy distributions for Model B3 (see Table 1).Di ff erent colored lines show di ff erent regions in the RIAF, which areindicated in the colorbar. Upper panel: protons. Lower panel: electrons. The right panel shows Models A2 and A3. A harder spec-trum decreases the contribution of synchrotron emission to theradio band. Again, pp emission dominates in the very-high-energy band, but now the contribution of pairs is dominant ina broader range. The little bump in the pp spectrum at ∼ ff ects if the black hole rotates and we are seeing thedisk at the proper inclination. ˙ m out = − , truncation radius r tr = Models B correspond to the scenario with an accretion rate of˙ m out = − . Here, we have considered an outer thin disk trun-cated at r tr = ∼ γ emission becomes comparable to pp emission in Model B2,and dominates in Model B3. This emission peaks at the PeV, andit is absorbed and reprocessed by the secondary pairs. The syn-chrotron emission of these secondaries dominates in the gamma-ray band and peaks at ∼
10 GeV. ˙ m out = − , truncation radius r tr = m out = − . Here, we have considered an outer thin disk trun-cated at r tr =
30; the calculated SEDs are shown in Fig. 9.The left panel shows in detail the contributions from the var-ious processes in Model C1. Now, radio synchrotron emissionis almost completely self-absorbed, and the gamma rays abovethe GeV are absorbed. The high-energy emission that escapes issynchrotron from secondary pairs.The right panel shows the SEDs for Models C2 and C3. Now,p γ emission is higher than pp emission. The absorbed emissionis reprocessed by the secondary pairs, whose synchrotron emis-sion now peaks at ∼ ˙ m out = . , truncation radius r tr = ∼ r ISCO
Models D correspond to the scenario with an accretion rate of˙ m out = .
1. Here, we have considered an outer thin disk pene-trating down the ISCO: r tr =
3; the calculated SEDs are shownin Fig. 10. The left panel shows in detail the contributions fromthe various processes in Model D1. The physics of this flowis dominated by the emission from the SSD. The temperatureof the corona is lower than in the other models because thereare many more seed photons for inverse Compton cooling. Ra-dio synchrotron emission is self-absorbed at lower energies and,again, produces a bump at 10 GHz. The spectral index of theX-ray coronal emission depends on the transition parameter (seeSect. 6) and, in this case, it is Γ ≈ − .
5. The high-energy emis-sion above the GeV is absorbed, and the reprocessed emissionhas a contribution from both synchrotron and inverse Comptonpair emission. Inverse Compton from primary electrons now be-comes more intense than synchrotron, but it is subdominant dueto the low value of direct power going into electrons we havechosen.The right panel shows the SEDs for Models D2 and D3. Now,p γ emission is very strong, and the reprocessed radiation by sec-ondary pairs produces a bump at energies above the megaelec-tronvolt range.
6. An application: The corona in the Seyfert galaxyIC 4329A
IC 4329A is a bright X-ray Seyfert 1.2 galaxy (Véron-Cetty &Véron 2006) at z = . .
61 Mpc, assuming the cosmologicalparameters H =
70 km s − Mpc − , Ω Λ = .
7, and Ω m = . ∼ . × M (cid:12) (Markowitz 2009; de La Calle Pérez et al. 2010). The host galaxyof the AGN is an edge-on spiral galaxy; the inclination of thedisk of the host galaxy with respect to the axes of the AGN isthought to be the result of the interaction between IC 4329A andthe companion galaxy IC 4329 at ∼ F X ∝ E − Γ , where [ F X ] = photons / keV / cm / s. The class Seyfert 1.2 is used to describe objects with relativelyweaker narrow H β components, intermediate between Seyfert 1.0 and1.5 (Véron-Cetty & Véron 2006). Article number, page 9 of 16 & A proofs: manuscript no. aanda [ ] [] ( ) ±± [ ] [] ( ) ±± == Fig. 7.
Accretion rate: ˙ m out = − . Left panel: Model A1. Right panel: Models A2 is shown in detail with the individual contributions and theabsorbed total emission (solid dark line), and for Model A3 only the absorbed total emission (dashed dark line) is shown. [ ] [] ( ) ±± [ ] [] ( ) ±± == Fig. 8.
Accretion rate: ˙ m out = − . Left panel: Model B1. Right panel: Models B2 is shown in detail with the individual contributions and theabsorbed total emission (solid dark line), and for Model B3 only the absorbed total emission (dashed dark line) is shown. [ ] [] ( ) ±± [ ] [] ( ) ±± == Fig. 9.
Accretion rate: ˙ m out = − . Left panel: Model C1. Right panel: Models C2 is shown in detail with the individual contributions and theabsorbed total emission (solid dark line), and for Model C3 only the absorbed total emission (dashed dark line) is shown. the hot plasma of the RIAF or "corona". Although modest vari-ability is observed, the power-law index in the Swift-BAT band(14 −
195 keV) is estimated as
Γ = . + . − . , and the total in-tegrated flux is F − = (263 . + . − . ) × − erg s − cm − ,which at a distance of 69 .
61 Mpc corresponds to a luminosity of L − (cid:39) . erg s − (Oh et al. 2018).A moderated broadened Fe K α line has been reported by sev-eral authors, possibly indicating that the cold disk is truncated(Done et al. 2000). Nevertheless, as discussed in Mantovani et al.2014, the nature of the emission line in this source is still underdebate, and the high bolometric luminosity of the source seemsto favor a radiatively e ffi cient flow down to low radii.Inoue & Doi (2018) have shown that observations in the mil-limeter band of two Seyferts, one of them IC 4329A, are wellexplained assuming nonthermal synchrotron emission in a hotcorona. Additionally, inverse Compton emission by these elec-trons would contribute to the cosmic MeV background emission(Inoue et al. 2019). Figure 11 shows the SED predicted by our model. The parameters chosen are shown in Table 2. To modelthis source, Inoue et al. (2019) assumed an homogeneous coronaof radius 40 R S , where the required magnetic field is of ∼
10 G.Our model predicts higher magnetic fields that increase towardthe inner regions. We can explain the radio features as producedby synchrotron emission from nonthermal electrons in the outerlayers of the corona. Moreover, since we considered the emissionfrom the secondary particles, the contribution at high energies isdi ff erent; in our model it is dominated by synchrotron radiationfrom secondary pairs, whereas in Inoue et al. (2019) it is dom-inated by direct inverse Compton from primaries. Inoue & Doi(2018) already suggested that the corona in this system is likelyan advection-heated hot accretion flow. This approach naturallyexplains the magnetic fields required for both e ffi cient particleacceleration and synchrotron emission, and it is consistent withrecent numerical results (Kimura et al. 2019b). The inhomoge-neous nature of the flow in our model also allows di ff erent re-gions of the flow with di ff erent magnetic fields and nonthermalpower to produce the various features in the spectrum. Moreover, Article number, page 10 of 16utiérrez, Vieyro & Romero: Nonthermal processes around supermassive black holes [ ] [] ( ) ±± [ ] [] ( ) ±± == Fig. 10.
Accretion rate: ˙ m out = .
1. Left panel: Model D1. Right panel: Models D2 is shown in detail with the individual contributions and theabsorbed total emission (solid dark line), and for Model D3 only the absorbed total emission (dashed dark line) is shown.
Table 2.
Parameters of our RIAF + thin disk model for the Seyfer galaxy IC 4329A. Parameter [units] Value m black hole mass [ × ] 1 . m out outer accretion rate 0 . r out outer radius 200 r tr truncation radius 4 α viscosity parameter 0 . β plasma parameter 10 δ fraction of energy heating electrons 0 . s wind parameter 0 . b transition parameter 0 . ε NT fraction of the accretion power going to nonthermal particles [%] 5 ε e fraction of nonthermal power into electrons 10 − η acc acceleration e ffi ciency 10 − p spectral index of injection 2 [ ] [] ( ) ±± [ ] [] [ ] [] ( )- ( ) - ( )( )( ) - Fig. 11.
Seyfert galaxy IC 4329A. The radio data are from Inoue & Doi (2018), and the X-ray data are from Oh et al. (2018). We also show thesensitivity of future MeV missions: COSI-X (300 days, Tomsick et al. 2019), e-ASTROGAM (1 year), GRAMS (35 days and 1 year, Aramakiet al. 2020), and AMEGO (1 year, McEnery et al. 2019. For reference, we also included the sensitivity of three gamma-ray instruments: MAGIC(operating; above 100 GeV), CTA (forthcoming; above ∼
30 GeV), and
Fermi (operating; ∼ . −
100 GeV). Article number, page 11 of 16 & A proofs: manuscript no. aanda the secondary particle (hadronic in nature) origin for the pre-sumed high-energy tail in the megaelectronvolt range is favoredby recent particle-in-cell simulations (Zhdankin et al. 2019).
7. Discussion
Particle acceleration is not self-consistently treated in our model,but it is included via the injection functions of primary electronsand protons. The most plausible acceleration mechanisms in hotcollisionless RIAFs or coronae are magnetic reconnection andstochastic acceleration by magnetic turbulence. A turbulent mag-netized flow naturally gives rise to fast magnetic reconnection,since turbulence induces magnetic fluxes of opposite polarity toencounter each other at high velocities ( ∼ V A ). Under these con-ditions, magnetic energy is transferred to the particles in the formof thermal, bulk and kinetic energy of individual particles. Thelatter involves particle acceleration (see Hoshino 2013 for a re-view). Fast reconnection leads to e ffi cient particle accelerationat a rate t − ∝ γ − a , with 0 . < a < .
6, and power-law indices N ( γ ) ∝ γ − , − (del Valle et al. 2016; Ball et al. 2018; Werneret al. 2018).Magnetic reconnection also serves as a mechanism to pushparticles out from the thermal bath facilitating further stochasticacceleration via collisions between the particles and scatteringcenters produced by the turbulence. SDA produces hard spectrathat deviates from a simple power-law (Park & Petrosian 1996;Becker et al. 2006; Kimura et al. 2015), and the accelerationtimescale di ff ers from Eq. 12. This timescale can be estimatedas t SDA = ¯ p D ¯ p , (40)where ¯ p is the momentum of the particle and D ¯ p is the di ff usioncoe ffi cient in the momentum space; according to the quasi-lineartheory (Dermer et al. 1996), it is given by D ¯ p (cid:39) ( mc ) ( ck min ) (cid:18) v A c (cid:19) ζ ( r L k min ) j − γ , (41)where v A = B / (cid:112) πρ is the Alfvén speed. Figures 3 and 4 in-clude the SDA timescale, for which we have taken ζ = . j = / ff ective to accel-erate electrons since they cool too fast. Protons cool much lesse ffi cient and are able to reach high energies, though lower thanin our models.Another acceleration process that has been considered inthe literature is di ff usive shock acceleration (Drury 1983; Inoueet al. 2019). Nevertheless, this process requires that the plasma iscompressible, and hence not highly magnetized ( ρ v (cid:29) B / π ,see, e.g., Romero et al. 2018). Since we are dealing with mag-netized plasmas, the two quantities above are comparable andstrong shocks are not expected to occur. Despite most of the high-energy electromagnetic emission is in-ternally absorbed, neutrinos produced by photomeson interac-tions (see Eqs. 28 and 29) escapes almost freely. The study ofneutrino production in accretion flows is of particular interestgiven that an excess of 2 . σ over the neutrino background wasfound in the ten-year survey data of IceCube, coincident with thedirection of a nearby type-2 Seyfert galaxy (Aartsen et al. 2020). [ ] [] ( , = °)( , = °) Fig. 12.
Total neutrino flux predicted for the Seyfert galaxy IC 4329A.Sensitivity curves for IceCube and IceCube-Gen2 are shown (van San-ten & IceCube-Gen2 Collaboration 2017; Aartsen et al. 2019)
Inoue et al. (2020) studied the production of neutrinos in thecorona of this source (see also Inoue et al. 2019), and found thatit can explain the excess, within a certain range of parameters intheir model.A rough analysis of our results indicates that we should ex-pect significant neutrino emission only for sources with moder-ate / high accretion rates, since, as discussed in Sect. 4, for lowaccretion rate p γ is irrelevant and the luminosities achieved bypp interactions are low. We show in Figure 12 the expected neu-trino flux for IC 4329A. Our hadronic-dominated model predictsa high neutrino emission that could be detected in the future byIceCube-Gen2. The total contribution of the population of HAFsin the Universe to the neutrino background will be investigatedin a future work. The soft-gamma ray extragalactic background ( ∼ ff erent sources, in-cluding SN Ia (Ruiz-Lapuente et al. 2016), Fermi blazars peak-ing in the MeV band (Giommi & Padovani 2015, although mostcontribution from blazars is above 10 MeV), and also radio-quietAGN if the accretion flow contains nonthermal particles (Steckeret al. 1999; Inoue et al. 2007). Inoue et al. (2019) studied thecontribution to the MeV background by hot accretion flows, andobtained that the measured fluxes can be explained as inverseCompton emission by nonthermal electrons in Seyfert coronae.The main di ff erence with respect to our work is that they ob-tain primary electron fluxes to be dominant, hence cascades arenot relevant. In our model, synchrotron emission from secondarypairs usually dominates over direct inverse Compton in the MeVband. In some extreme scenarios, proton and muon synchrotronemission can also contribute significantly in the megalectronvoltrange (see, e.g., Romero & Gutiérrez 2020). The hadronic con-tent is, then, an important component to be taken into accountwhen studying the contribution of hot accretion flows to the MeVbackground. Radiatively ine ffi cient accretion flows are usually associatedwith the launching and collimation of relativistic jets in AGNs Article number, page 12 of 16utiérrez, Vieyro & Romero: Nonthermal processes around supermassive black holes and microquasars. Jets launched by the Blandford-Znajek (BZ)process (Blandford & Znajek 1977) requires the accumulationof magnetic flux in the innermost regions close to the black holeergosphere (Tchekhovskoy et al. 2011), which is favored by theadvective nature of HAFs. On the other hand, both the high H / R ratio of the flow and the ubiquitous presence of wind in these sys-tems help to collimate the jet at its first stage (Yuan & Narayan2014). Since BZ jets are launched as purely Poynting fluxes, animportant problem to deal with is how this electromagnetic out-flow can be loaded with mass at the base of the jet. The presenceof matter is inferred by very-high-resolution observations of thenearby AGN M87, which show that radiation associated withthe jet is being produced at distances down to 5 Schwarzschildradii from the central supermassive black hole (Hada et al. 2013;Event Horizon Telescope Collaboration et al. 2019).The nonthermal processes in accretion flows onto black holethat we discussed may play a non-negligible role in loadingjets with charged particles. The interaction of relativistic pro-tons with matter (pp) and radiation (p γ ) produce neutrons via thechannels given by Eqs. 25 and 27. These neutrons freely escapefrom the corona and decay into relativistic protons and electronsat long distances; a fraction of them will decay within the jet fun-nel. This mechanism was proposed as a means to load Poynting-dominated outflows with baryons (Toma & Takahara 2012; Vilaet al. 2014). Moreover, neutrons may collide with photons creat-ing pions through the process n + γ → p + π − . Pion decay willquickly inject pairs in the funnel (Romero & Gutiérrez 2020).Pairs can also be created via the annihilation of photons emit-ted in the corona; this process can be separated into two cate-gories: a) MeV-MeV collisions, which will take place even whennonthermal processes are not significant (thermal MeV photons),and b) collisions between a high-energy (nonthermal) gammaray and a soft photon from the accretion disk. The latter mecha-nism is highly dependent on the details of the nonthermal mech-anisms occurring in the corona (see Romero & Gutiérrez 2020for numerical estimates of the particle loading through these var-ious mechanisms). The semi-analytical approach we have taken has pros andcons when compared with detailed numerical simulations. Aswe mentioned, simulations are unavoidable to study nonlin-ear physics and multidimensional phenomena (like outflows) orcomplex time variability (Yuan & Narayan 2014). Nevertheless,in many situations they require focusing on only a small part ofthe system (as it is the case in Particle-in-Cell simulations) orupon a few of all the relevant physical processes that take place:Nonthermal processes are usually neglected in magnetohydrody-namic simulations of accretion flows. Semi-analytical models,on the other hand, rely on their versatility. They allow to treatglobally the flow, including several physical processes at di ff er-ent time or spatial scales, and the interpretation of the results ismore direct.Several previous semi-analytical approaches to investigatenonthermal processes in RIAFs were presented in the litera-ture. Kimura et al. (2015) developed a model considering theRIAF as a homogeneous spherical hot plasma where the phys-ical fields are obtained either from the self-similar solution(Narayan & Yi 1994) or from numerical simulations (Kimuraet al. 2019a). They propose stochastic acceleration of protonsas a means to produce nonthermal processes that could leadto multi-messenger outputs. Recently, Inoue et al. (2019) de-veloped a more observationally-motivated corona model based on the detection of nonthermal coronal activity in two nearbySeyfert galaxies; they also considered a homogeneous spher-ical flow. Our model presents improvements with respect tothese works: We combine the detailed treatment of nonther-mal processes with actual hydrodynamic solutions of hot ac-cretion flows. The latter provides a radial dependence of thephysical fields in the flow, which in turn permits to calculatea more accurate thermal background. In addition, we includedself-consistently the presence of a cold thin disk coexisting withthe hot flow. Over this background, we introduced the presenceof a population of nonthermal particles and study both the spatialand energy transport, the production of secondary particles, andcalculate all the relevant outputs.
8. Conclusions
We have developed a detailed model to study nonthermal pro-cesses occurring in hot accretion flows. The model is flexibleenough to be applied to a broad range of accretion rates andluminosities. It consists of a hot accretion flow, modeled as anRIAF, plus a cold thin disk. For various sets of parameters, weinvestigated the most relevant nonthermal processes that occur,including the particle transport and the nonthermal radiative out-put.In models with accretion rate ˙ m > − , the radiation aboveGeV is highly attenuated due to self-absorption, hence all emis-sion in this domain is expected to originate in jets. At lower ac-cretion rates (Models A) most of the radiation can escape. On theother hand, the contribution from both thermal and nonthermalRIAF emission to the MeV / sub-GeV band might be dominant inthe local Universe.We also applied our model to the source IC 4329A, confirm-ing results from previous works, in which millimetric observa-tions of this source can be well reproduced by synchrotron emis-sion in a hot corona. In our model, the corona is heated by themagnetorotational instability, and the millimetric excess comesfrom further distances to the hole than those considered by theprevious authors (Inoue & Doi 2018).Our model presents some improvements to previous modelsof nonthermal processes in HAFs: • By solving the actual hydrodynamics equations for ahot accretion flow, we obtain a description of the flowthat takes into account the radial dependence of the fluidproperties, and where the values of these properties ariseself-consistently from the solution itself. • We include the presence of a thin disk that coexists andinteracts with the hot accretion flow, as it is expected tooccur in many AGNs. • The thermal radiative emission is calculated with greatdetail, taking into account nonlocal e ff ects, and it is consis-tent with the fluid properties. This, together with the twoprevious improvements, set up a more realistic backgroundover which nonthermal processes are calculated. • The nonthermal transport includes the spatial advection anddi ff usion of particles when these processes are important,namely for protons. Besides, all relevant secondary pro-cesses are calculated.The transition from thermal to nonthermal emission fromRIAFs would take place in the MeV band of the SED. Hence- Article number, page 13 of 16 & A proofs: manuscript no. aanda forth, it is fundamental to cover this energy range to better un-derstand particle acceleration in these systems. Besides, studieson gamma-ray polarization can be useful to disentangle the ori-gin of the radiation (coronae vs jets, or starbursts when present),as it was used in the case of BHBs (Laurent et al. 2011). Af-ter the successful COMPTEL instrument, on board the ComptonGamma-Ray Observatory (1991-2000), we lack a detector in theMeV band. Future MeV gamma-ray missions, such as GRAMS(Aramaki et al. 2020), and in particular, those with polarimetricfacilities, as AMEGO (0 . −
10 GeV McEnery et al. 2019),and COSI-X (0 . − Acknowledgements.
We thank the referee, D. Khangulyan, for useful commentsthat helped to improve the manuscript. We thank Enrico Peretti for discussionsabout the transport of relativistic particles. This work was supported by the Ar-gentine agency CONICET (PIP 2014-00338), the National Agency for Scientificand Technological Promotion (PICT 2017-0898 and PICT 2017-2865). G.E.R.acknowledges the support by the Spanish Ministerio de Ciencia e Innovación(MICINN) under grant PID2019-105510GB-C31 and through the “Center of Ex-cellence María de Maeztu 2020-2023” award to the ICCUB (CEX2019-000918-M).
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Article number, page 15 of 16 & A proofs: manuscript no. aanda
Appendix A: Accretion rate parametrization
The phenomenological function that parametrize the smoothtransition between the SSD and the RIAF is f ( R ) = R ≤ R tr1 − ( R tr / R ) b − ( R tr / R out ) b if R tr < R ≤ R out R out < R . . (A.1)The normalized mass loss rate via winds w ( R ) is w ( R ) = ˙ M − (cid:90) RR out dR (cid:48) d ˙ MdR (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) winds = − s (cid:90) RR out dR (cid:48) g ( R (cid:48) ) R (cid:48) , (A.2)where we have taken d ˙ MdR = s ˙ M c ( R ) R . (A.3)Di ff erentiating the relation f ( R ) + g ( R ) = − w ( R ), we obtain afirst-order linear ordinary di ff erential equation for g ( R ): f (cid:48) ( R ) + g (cid:48) ( R ) = s g ( R ) R , (A.4)whose solution is g ( R ) = ( R / R out ) s − f ( R ) + s (cid:90) R out R dR (cid:48) (cid:18) RR (cid:48) (cid:19) s f ( R (cid:48) ) R (cid:48) . (A.5)Inserting Eq. A.1 into Eq. A.5, an analytical solution for g ( R ) iseasily obtained. This solution satisfy that g ( R ) = R > R out and g ( R ) ∝ ( R / R tr ) s for R < R tr .Figure A shows the accretion rates as a function of the radiusfor s = . α = R tr =
30, and R out = . [ ] / Thin diskRIAF/coronaTotal
Fig. A.1.
Accretion rates involved in the model for R tr = R out = , s = .
2, and α ==