Temperature of thermal X-ray and γ-ray sources

A framework is developed for the accurate determination of the temperature of a source, assuming that it is optically thin and emits a bremsstrahlung spectrum. The temperature can be measured from the shape of the spectral distribution, and an explicit relation is derived for the temperature in terms of this slope and a sum of correction terms to the limiting form of the bremsstrahlung formula. The corrections can be evaluted easily in terms of simple, though accurate, asymptotic formulae and are due to (i) correction to an asymptotic form of a modified Bessel function, (ii) a lowest-order correction to the Born approximation to the bremsstrahlung cross-section, (iii) relativistic corrections to the plasma electron energy distribution and to the nonrelativistic limit to the electron-ion bremstrahlung cross section, (iv) the contribution from electron-electron bremsstrahlung. For the case where there is a distribution of temperatures in the plasma, an additional correction is derived in terms of the rms variation in the plasma temperature. Some corresponding results for highly relativistic plasmas are given. The intermediate domain (kTapprox.m/sub e/c2) is also discussed, and a rough general formula is suggested