8. Calculations and analysis
The rates obtained in the paper were integrated over the Maxwellian distribution in order to compare with well-known results for the thermal plasma. The annihilation rate was tested with annihilation rate of an plasma (Ramaty & Mészáros 1981), bremsstrahlung energy losses were compared with -, ee -, and ep -bremsstrahlung luminosities of thermal plasmas (Haug 1985c). Two more tests on Coulomb energy losses and bremsstrahlung were carried out with calculations by Dermer & Liang (1989). An excellent agreement was found. Compton energy loss Eq. (38)-(39) coincides with the Thomson limit as . Besides, we have found that the formulas obtained can be also successfully applied for the calculation of the bremsstrahlung luminosity and annihilation rate of the thermal plasma by replacing the positron Lorentz factor with the average one over the Maxwellian distribution .
The relevant energy loss rates () and annihilation rate per one positron are shown in Fig. 1 and 2. All values are provided dimensionless, in units , the Coulomb logarithm was taken a constant . Moller and Bhabha energy losses show negligible difference and dominate over the others except Compton scattering, which is quite effective and can prevail at large Lorentz factors of positrons (electrons). Low energy particles gain energy in Coulomb scattering with thermal electrons that appears as the sign change of . Energy losses due to bremsstrahlung are negligible in comparison with others. Annihilation rate is small in comparison with the relaxation rate, so that most of positrons annihilate after their distribution approaches the steady-state one.
The energy losses due to Compton scattering (Fig. 2) have been calculated in the Thomson limit Eq. (40) and in the Klein-Nishina regime for a Planckian spectrum , and the -function approximation. The energy loss rates due to the Comptonization on Planck's photons are shown for two photon temperatures, the -function approximation of Planck's distribution with gives similar results. For the clear comparison with Fig. 1 the photon number density have been taken equal to that of the plasma electrons easily generalizing for an arbitrary by trivial vertical shift of the curves. For the coherence, in all calculations the energy density of photons was taken equal to that of Planck's distribution . Shown also is the dispersion coefficient calculated in the Thomson limit Eq. (41). The radiation can provide some heating for the cold particles similar to that in the Coulomb scattering. Very low-energy particles gain energy due to Comptonization that appears as a sign change of the energy losses (see the inset in Fig. 2). Clearly, the effect results from using the Klein-Nishina cross section.
At small positron Lorentz factors, the Coulomb energy losses dominate the losses due to Comptonization over the variety of photon temperatures and densities (cf. Fig. 1 and 2). Qualitatively it means that high photon density leads to the cooling of plasma preferentially through high-energy particles. Herewith, the Coulomb scattering mixes particles so that the plasma remains nearly Maxwellian. Therefore the energy losses due to Comptonization would be only important for the high-energy tail of the particle distribution, which becomes narrower. The precise shape of the distribution would be driven by the balance of income and outcome energy fluxes.
Positrons could be injected into the hydrogen plasma volume by an external source or produced in the bulk of the plasma. In the latter case the form of the source function is governed by the nature of the processes involved. Electron-positron pair production in ep -collisions becomes possible when the electron interacting with a stationary proton has the Lorentz factor exceeding 3, for ee -collisions one should exceed 7 when one interacting particle is at rest. If the pair is to be produced in two-photon collisions, the photon energies, , and the relative angle, , must satisfy the condition . Low plasma temperature is consistent with a small positron fraction in the plasma since the positrons could be produced by the relatively small number of head on collisions of energetic photons and/or electrons from the tail of Maxwellian distribution.
If the particle production is not balanced by annihilation it could lead to escape of -plasma, since the gravitation near a compact object can't prevent pairs from escaping. Two independent mechanisms, at least, diffusion and the radiation pressure result in escaping of particles from the plasma volume. We, therefore, explore these factors separately. If particles escape due to the radiation pressure, it is natural to suppose that the escape probability is a weak function of the particle Lorentz factor, we thus put it a constant. In the case the escape is of diffusive origin, the diffusion coefficient is a function of particle speed . We thus consider two functional forms for the escape probability , and which simulates the case when both mechanisms operate simultaneously.
Calculations of the distorted function have been made (Fig. 3) for the source function in the form of monoenergetic distribution , power-law , and Gaussian . The escape rate was taken energy-independent , 10, and 100 in units , which is negligible, medium and very high in comparison with the time scale of the Coulomb energy losses (cf. Fig. 1). It demonstrates an effect of blowing away of (unbound) electron-positron pairs by radiation pressure.
The behavior of the solution depending on the injection function and escape rate is quite clear from the figure. One can show that the right side of the Eq. (4) and (5) is negligible at , the solution is therefore Maxwellian-like. Beginning from some point, the term becomes non-negligible that leads to some increasing of the derivative and deviation of the solution from Maxwellian. Thus, a bump is forming. At some Lorentz factor the last term in the right side of Eqs. (4) and (5) is switching on, which leads to some decreasing of the derivative or could even change it to a negative value. At large Lorentz factors the right side of the equations again approaches zero (see Eq. [ 6]). Generally, if the energy of injected particles essentially exceeds the average one of plasma particles it leads to an extended tail, while the correct normalization of the whole solution thus requires some deficit at low energies.
Typical spectra of photons from annihilation of these positrons with Maxwellian electrons are shown in Fig. 4 for electron temperatures , and 0.1. It is seen that as plasma temperature grows the annihilation line widens, its height decreases and distortions of its shape become relatively more intensive.
Another case is shown in Fig. 5. The distorted functions were calculated for electron temperatures , 0.3, and 0.5 while the escape probability in all cases was taken the same (in units ). The actual values of the escape rate in these cases could be inferred from the value of the integral , which is equal to , , and , correspondingly. Particle injection was taken monoenergetic with energy equal to the average energy of plasma electrons. In all cases, the escape leads to some deficit of energetic particles in the tail of distribution, while the particle injection appears as a bump. Although the distributions of positrons differ from Maxwellians, their annihilation with thermal electrons does not lead to large distortions of the annihilation line form. This latter is very similar to the line from annihilation of two Maxwellian distributions.
Although only few cases have been discussed, the performed calculations have shown that the functional dependence of the escape rate is not very important. In all three cases , , and we obtained similar results for the same injection function, the difference appears only at very low temperatures . It is quite clear, since increases from 0 to in a narrow region remaining further a constant. The particle distribution actually depends on the value , energy of the injected particles and their distribution (cf. Figs. 3 and 5). In absence of the particle injection, the escape of particles operates as an additional mechanism for the plasma cooling.
© European Southern Observatory (ESO) 1997
Online publication: May 5, 1998