A&A 423, 793-795 (2004)
DOI: 10.1051/0004-6361:20040377
Research Note
E. Haug
Institut für Astronomie und Astrophysik, Universität Tübingen, 72076 Tübingen, Auf der Morgenstelle 10, Germany
Received 3 March 2004 / Accepted 25 May 2004
Abstract
The average energy loss due to bremsstrahlung emission of electrons
passing through a plasma is calculated. In particular, the contribution of
electron-electron bremsstrahlung is specified. The bremsstrahlung losses
are compared with the energy loss by collisions with ambient electrons.
Above electron energies of 510 MeV the energy loss due to bremsstrahlung
predominates.
Key words: radiation mechanisms: nonthermal
Electrons passing through a plasma lose energy by collisions with ambient particles and by emission of bremsstrahlung. Whereas in most cases the energy loss of electrons is almost entirely due to collisions, the emission of bremsstrahlung photons with energies of the order of magnitude of the electron kinetic energy will be predominant at very high electron energies, as was already pointed out by Heitler (1954). The emitted radiation is composed of electron-nucleus (e-n) and of electron-electron (e-e) bremsstrahlung. The calculation of the energy loss by e-n bremsstrahlung causes no difficulties since astrophysical plasmas are composed mainly of hydrogen and helium with minor admixtures of heavier elements. Therefore the relativistic cross section derived by Bethe & Heitler (1934) in Born approximation is sufficient. In addition there exist accurate approximations to this cross section (Haug 1997). On the other hand, the evaluation of the e-e energy loss is complicated by the lengthy formulae for the cross section (Haug 1998). It has been performed in a rough approximation by substituting the factor Z(Z+1) for the factor Z^{2} in the e-n bremsstrahlung cross section, where Z is the atomic number of the target nucleus. Moreover, the extreme relativistic approximation of the bremsstrahlung cross section is used (Blumenthal & Gould 1970). The relative error of this procedure is rather high, 28% at electron energy E=100 keV, and still 10% at E=3.5 MeV. The aim of the present note is to provide simple formulae which lead to accurate values of the energy loss due to bremsstrahlung. The plasma is assumed to be sufficiently thin and hot so that the weak shielding and complete ionization approximations can be applied.
The average energy loss per unit time by bremsstrahlung emission of an
electron of kinetic energy
in a plasma is given by
For e-e bremsstrahlung the integral over k occurring in Eq. (1) has
to be calculated numerically. Here the upper integration limit is
.
Using the differential cross section including
the Coulomb correction factor (Haug 1975) the cross section
Assuming a completely ionized plasma the electron number density is
given by
.
Together, the energy loss due to emission of e-n and e-e bremsstrahlung takes the form
Figure 1: Energy loss rate of electrons with kinetic energy E due to bremsstrahlung according to Eq. (9) (H) and according to the approximation of Blumenthal and Gould (BG), compared with the energy loss rate by e-e collisions (coll. loss) in a completely ionized plasma with hydrogen number density cm^{-3} and ratio . | |
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It is interesting to compare the radiative energy loss with the loss
by collisions which dominate at lower energies. Considering a completely
ionized plasma, where no ionization losses occur, the latter is given by
losses in electron-electron collisions,
Figure 1 shows in a doubly logarithmic scale the brems-strahlung energy loss rate according to Eq. (9) and the approximation of Blumenthal & Gould (1970), compared with the energy loss rate by electron-electron collisions. As was to be expected, the approximation is poor at low electron energies E. But even in the MeV energy range there are considerable differences between the accurate calculation (Eq. (9)) and the approximation. Only at high energies E>80 MeV the two curves converge. At nonrelativistic and mildly relativistic energies the loss rate by collisions exceeds the bremsstrahlung loss rate by several orders of magnitude. The bremsstrahlung loss rate dominates only at E>512 MeV. Here the approximation of Blumenthal & Gould (1970) is accurate.