Astron. Astrophys. 346, 713-720 (1999)

3. The comptonization

We follow the method of Comptonization described by Górecki & Wilczewski (1984).

3.1. Basic concepts

The differential cross section for Compton scattering is given by the following formula (Akhiezer & Berestetski 1965):

where , , and are respectively the energy and the direction of the photon before and after the scattering, is the velocity of the electron, is the Lorentz factor and is the classical electron radius. The symbol X denotes the invariant part of the cross section:

where

are the energies of the incoming photon and of the scattered photon, respectively, expressed in units of in the reference frame of the electron. The energies and are related by the Compton formula

We use the total Compton cross section which is given by (Berestetski et al. 1972)

The basic concept of this method is to follow the photon trajectory from the moment of emission until the photon leaves the flow. The probability that a photon leaves the flow without scattering is

where the integral is taken along the photon trajectory from the point of the last (i-th) scattering to the boundary of the flow , is the electron density, is the electron velocity distribution, and

is a mean cross section averaged over the electron velocity distribution. .

The probability enables us to find the statistical weights of the number of photons leaving the flow without scattering (and thus contributing to the emerging spectrum) and the photons which remain in the flow and undergo the next (-th) scattering. These weights are given by and , respectively, where , 1, 2, 3, is the index denoting the succeeding scatterings. We assume . We follow the trajectory of the photon until w becomes less than a certain minimal value . Since ADAFs are optically thin (in our model the Thomson optical depth is about 0.1 in equatorial directions) the mean number of scatterings is 4-5 for .

3.2. Generating the random variables

The random variables are generated from the probability distributions using Monte Carlo methods: the inversion of the cumulative distribution function or von Neumann's rejection technique. Multi-dimensional distributions are modeled using the conditional probability distributions.

(i) At first we generate the position vector at which the photon is initially emitted. According to our approximation this position is uniformly distributed in space within each of the annuli jk. The probability that a photon is emitted from a region of the given number jk is:

where are given in Eq. 28. In Eq. 28 we take into account the optical depth due to the absorption, so the number of the low frequency photons we use in the simulation is the expected number of the photons which have a chance of escape. The absorption does not have to be considered on the further photon trajectory. (After a scattering with relativistic electrons a photon gains so much energy that the possibility of its absorption can be neglected. The probability of absorption on the original trajectory is included in Eq. 28.)

(ii) As the position of input photon is determined we can generate the initial energy of the photon using a probability distribution specified by the photon spectrum of synchrotron or bremsstrahlung emission

where is the photon spectrum approximated by formulae of Pacholczyk (1970) and Mahadevan et al. (1996) for synchrotron emission or Svensson (1982) for bremsstrahlung emission. The photon spectrum is determined from the energy spectrum by dividing the last one by .

(iii) We assume that the emission of input photons is isotropic. Hence the direction of the photon in a comoving Cartesian coordinate frame = , , is generated from uniform distributions in the ranges and .

In this way we determine the set of parameters {, , , } describing the initial point of the trajectory of the photons beam. We calculate the next points of the trajectory, i.e. {, , , } (, 1, 2, 3, ) until . The way of computing the weights is described in Sect. 3.1. Below we present the way of computing the position, energy and direction of a photon after following scatterings.

(iv) The position is found on the photon trajectory at the proper distance l from the starting point , from the probability distribution:

where

(v) The two remaining parameters and of the point of the photon trajectory are obtained by simulating the scattering of the photon of energy and direction by an electron with velocity . To describe the probability distribution of this scattering we use the differential cross section (1):

We model the multi-dimensional probability distribution (13) as a product of the probability distribution of and the conditional probability distribution of

where

and

We generate the velocity from Eq. 42 and then the direction from Eq. 43. Having , the energy of the scattered photon can be obtained from the Compton formula (32). The detailed description of the method of modeling the probability distributions and can be found in Górecki & Wilczewski (1984).

© European Southern Observatory (ESO) 1999

Online publication: May 21, 1999
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