Cygnus X-1 was the first X-ray source widely accepted to be a black hole candidate and remains among the most studied astronomical objects in its class. The detection of non-thermal radio, hard X-rays and gamma rays reveals the fact that this kind of objects are capable of accelerating particles up to very high energies. In order to explain the electromagnetic emission from Cygnus X-1 in the low-hard state we present a model of a black hole corona with both relativistic lepton and hadron content. We characterize the corona as a two-temperature hot plasma plus a mixed non-thermal population in which energetic particles interact with magnetic, photon and matter fields. Our calculations include the radiation emitted by secondary particles (pions, muons and electron/positron pairs). Finally, we take into account the effects of photon absorption. We compare the results obtained from our model with data of Cygnus X-1 obtained by the COMPTEL instrument.
The low-hard state of accreting black holes is characterized by the presence of a hot corona around the compact object. Figure 1 shows a scheme of the main components of the system. For this geometry, we assume a spherical corona with a radius R c and an accretion disk that penetrates the corona up to a radius R d < R c . We suppose that the corona is homogeneous and in steady state.
We assume that the luminosity of the corona is 1 % of the Eddington luminosity of the black hole. For a corona characterized by the parameters of Table 1, this yields L c = 1.3 × 10 37 erg s -1 . The corona is composed of a two-temperature plasma, with an electron temperature T e = 10 9 K and an ion temperature T i = 10 12 K (e.g. Ref. 1). This is a consequence of electron cooling being more efficient than proton.
In the low-hard state we can assume the system is in a steady state. We then consider equipartition of energy between the different components of the system and obtain the values of the relevant physical parameters.
The X-rays emission of the corona is characterized by a power law with an exponential cut-off at high energies, In accordance with what is observed in Cygnus X-1 (e.g. Ref.
2), we adopt α = 1.6 and E c = 150 keV. Table 1 shows the values of the parameters in our model. 0.1 (1) Typical value for Cygnus X-1 (Ref. 2).
(2) 35R G .
The injection function for relativistic protons and electrons is a power-law in the energy of the particles Q(E) = Q 0 E -Γ e -E/Emax , which is the consequence of diffusive particle acceleration by shock waves in the coronal plasma. Typical spectral indices are Γ ∼ 2.2 (Ref. 3). The normalization constant Q 0 can be obtained from the total power injected in relativistic protons and electrons, L rel = L p + L e . This power L rel was assumed to be a fraction of the luminosity of the corona, L rel = κL c , with κ = 10 -2 . The way in which energy is divided between hadrons and leptons is unknown, but different scenarios can be taken into account by setting L p = aL e . We consider models with an injection dominated by protons in which a = 100 and models with a = 1.
The injected particles will lose their energy by different radiative processes. There are three relevant processes of interaction of relativistic electrons with magnetic, matter and photon fields in this scenario: synchrotron radiation, inverse Compton scattering, and relativistic Bremsstrahlung. For protons there are also three relevant processes: synchrotron radiation, proton-proton inelastic collisions and photohadronic interactions. These interactions produce pions, which then decay producing muons. We also consider the energy loss of these particles; the processes for charged pions are the same as for protons and for muons the same as for electrons.
We consider two mechanisms of particle escape from the corona: advection and diffusion. In the case of advection, particles fall onto the compact object at a mean radial velocity v = 0.1c (Ref. 4). Therefore the advection rate is
In the case of diffusion, we consider that the corona is static and diffusion of relativistic particles occurs in the Bohm regime. The diffusion coefficient is D(E) = r g c/3, where r g = E/(eB) is the giroradius of the particle. The diffusion rate is
Figure 2 shows the cooling rates for different processes of energy loss, along with the acceleration and escape rates, for each type of particle considered. Under the physical conditions previously described, the main channel of energy loss for electrons is synchrotron radiation. For protons, both pp and pγ interactions are relevant. However, in the model with advection, most protons fall into the black hole before radiating their energy. For pions, the main channel of energy loss is πγ interaction, but an important fraction of pions decay before cooling. On the contrary, muons with energies above 10 11 eV cool mostly by inverse Compton scattering.
The maximum energy that a relativistic particle can attain depends on the acceleration mechanism and the different processes of energy loss. The acceleration rate t -1 acc for a particle of energy E in a magnetic field B is given by
where η ≤ 1 is a parameter that characterizes the efficiency of the acceleration. We have fixed η = 10 -2 , which describes an efficient acceleration. Fig. 2. Radiative losses in a corona characterized by the parameters of Table 1.
In order to obtain the spectral energy distributions produced by the different radiative processes, we solve the transport equation for each kind of particle. We calculate the SEDs of different processes following Ref. (5) as standard reference. We also calculate the radiation emitted by secondary pairs, which are injected mainly by photon-photon annihilations.
Figure 3 shows all contributions to the total luminosity for different advection regimes and for different values of the parameter a. Luminosities produced by hadrons and muons are higher in models with diffusion. This is because
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