## 3. Reaction rate formalismBelow 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. The relativistic reaction rate where is the cross section of a reaction,
and are correspondingly
the differential number density and velocity of particles of type
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, , where is the number density of particles of
type 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 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 |