Luminosity enhancement factor for thermal Comptonization and the electron energy balance

Abstract

Comptonization of soft photons by hot thermal electrons is an important cooling mechanism governing the electron energy balance in astrophysical plasmas. We treat thermal Comptonization in terms of the luminosity enhancement factor η, which gives the average change in energy of a photon between injection into and escape from a scattering medium. The region in which the photon scatters is characterized by the dimensionless electron temperature θ = kT/m<SUB>e</SUB>c<SUP>2</SUP>, Thomson depth τ<SUB>T</SUB>, and geometry. We derive an approximate expression for η given by <P />η = <P />where P = 1 - exp (-τ×), A = 1 + 4θ + 16θ<SUP>2</SUP>, and x = hv/m<SUB>e</SUB>c<SUP>2</SUP> is the photon energy at injection. Equation (1), with τ<SUB>T</SUB> replaced by an effective scattering depth T<SUB>eff</SUB>(θ, τ<SUB>T</SUB>), is used to fit Monte Carlo results for η at relativistic temperatures in three geometries: a point source of photons located at the center of a sphere; a uniform source of photons distributed throughout the volume of a sphere; and a uniform source of photons distributed throughout the volume of a disk. Equation (1) also provides a good approximation to the nonrelativistic luminosity enhancement factor in the diffusion regime when - ln P/ln A is identified with the nonrelativistic photon spectral index α derived by Sunyaev and Titarchuk. Integration of equation (1) over the bremsstrahlung source term provides a generalization of the result derived by Kompaneets for the enhancement of energy-loss rate of thermal electrons due to Comptonization of internally produced bremsstrahlung photons. Expressions for the electron energy-loss rate due to Comptonization of thermal cyclosynchrotron photons and nonthermal soft photons are also derived.