## 2. Free-free emissionFree-free emission, or Bremsstrahlung, is emitted as electrons are
scattered by ions. The emissivity can be derived in a semiclassical
way due to Kramers and Wentzel (Kramers 1923), which deviates from the
exact quantum mechanical treatment. Gaunt (1930) quantified the ratio
of the exact to the semiclassical value by introducing a corresponding
factor which today usually is named after him the with the emission measure. is the Gaunt factor for free-free and free-bound transitions averaged over a Maxwellian velocity distribution which in its rigorous form is rather difficult to handle. There exist a number of approximations for different regimes in -space, see for instance the work of Brussaard & van de Hulst (1962) and Karzas & Latter (1962). ## 2.1. Rigorous treatment of free-free emissionFor K the free-free contribution to the emission in H II regions requires a detailed treatment. The cross section for the emission of one photon is with the Gaunt factor expressed by hypergeometric functions The indices refer to the initial (i) and final (f) state of the
electron. The derivation of Eq. 7 assumes a point-like scattering
object. Accordingly it is strictly correct only for fully ionized
atoms, like H ## 2.2. Thermally averaged Gaunt factorsIn many cases the electrons can be assumed to obey a thermal distribution of temperature and the free-free emission stated in Eq. 13 of a dilute thermal plasma is obtained from the Maxwellian average of the Gaunt factor (see Appendix A) by with the dimensionless kinetic energy of the electrons after scattering scaled to the mean electron energy The spectral emissivity depends on frequency through the Gaunt factor times the exponential which follows since only electrons with initial energies in excess of contribute to the emissivity (see also Appendix A). The free-free emissivity is given by Inserting Eqs. 6 and 10 results in The corresponding optical depth for thermal electrons is In the Rayleigh-Jeans limit we obtain optical depth which is the free-free contribution () in Eq. 5. ## 2.3. Numerical methodsThe hypergeometric functions entering in Eq. 8 assume a large
negative argument which is especially useful for a numerical treatment. The complex -function defined in Eq. 8 is evaluated with the hypergeometric series for which has a good numerical convergence if This is sufficient for very high frequencies but does not dominate
the thermal average. If we can take
as the starting argument for the
hypergeometric series and continue to solve the hypergeometric
differential equation by the Bulirsch-Stör method. If the value
of exceeds 0.5 we choose the
transformation given in Eq. 17, because we find that the stepsize
of the integration scheme does not underflow in that case. For very
large the approximation of
Eq. 23 can be employed preventing numerical inaccurate results.
This scheme provides the argument for the integration in Eq. 10.
The integration uses the extended midpoint rule and a split of the
All Gaunt factors shown in the attached figures are calculated in this way if not stated otherwise. In Fig. 1 we compare our results for a temperature of K with approximations for the LFL and the high energy limit. For different temperatures we always find a frequency interval where both approximations are in error by more than 10 %. The method presented here can be used for an accurate calculation of free-free Gaunt factors for a wide range of temperatures and frequencies as shown in Fig. 3. In Table 1 we summarize the validity ranges of often used analytical approximations.
## 2.4. Low-frequency limitOf special interest for radio observations is the low-frequency limit, where the frequency of the emitted photon is much smaller than the final energy . So we can introduce a variable and explore the Gaunt factor in the limit . This corresponds to and allows the use of the hypergeometric series for the evaluation
of Eq. 17. For large the series
converges rapidly and can be used for numerical purposes, if
is not too large. Expanding
to first order in which was obtained by Oster (1963) as an exact result. We rather derive this result as the limiting case for . In the LFL, which corresponds to the real part of the Digamma-Function, , is approximately and we obtain the classical value For H II regions and Hz the LFL of Eq. 24 can be used in the thermal average. The corresponding approximation derived by Oster (1961, 1970) is widely used. For the radio regime there exists an even simpler approximation by Altenhoff et al. (1960) which, combined with Eq. 5, yields with a correction factor (Mezger & Henderson 1967) which accounts for the difference between this approximation and the one by Oster (1961). We have computed and show in Fig. 1 the averaged Gaunt factor together with its approximations by Oster and Altenhoff et al., respectively. © European Southern Observatory (ESO) 2000 Online publication: April 17, 2000 |