3. Reaction rate formalism
Below we describe a formalism, which further allows us to calculate the annihilation rate, energy losses and energy dispersion rate due to Coulomb scattering, bremsstrahlung, and Comptonization.
where is the cross section of a reaction, and are correspondingly the differential number density and velocity of particles of type i in the laboratory system (LS), is the relative velocity of the particles, the factor corrects for double counting if the interacting particles are identical.
We consider energetic particles which interact with particles of a thermal gas. Let masses of both types of particles be equal (). For isotropic distributions, Eq. (8) can be reduced to the triple integral over particle momenta, , and the relative angle, ,
is the relative Lorentz factor of two colliding particles (invariant). Putting the relativistic Maxwell-Boltzmann distribution for the electron gas (pay attention to the normalization) together with the monoenergetic distribution for the beamed particles,
into Eq. (9) yields
where is the j -order modified Bessel function.
Using Eq. (10) to eliminate in favor of and changing variables from to one can find
where . After integrating over , the reaction rate can be exhibited in the form (Dermer 1985)
Another form of the reaction rate for interacting isotropic distributions of particles (Eqs. [ 11], [ 12]) was found useful for some purposes (Dermer 1984)
where is the Lorentz factor of the center-of-mass system (CMS), and .
If we are interested in energy losses suffered by the energetic particles in an isotropic gas, it is necessary to weight the cross section in Eq. (14) or (16) by the average LS energy change per collision . The concrete form for depends on the studied process. Hereafter we will consider the reaction rate and energy losses per one positron in the unit volume (), while will denote the electron number density.
© European Southern Observatory (ESO) 1997
Online publication: May 5, 1998