Astron. Astrophys. 356, 1149-1156 (2000)

2. Free-free emission

Free-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 Gaunt factor . For radio observations of H II regions the low-frequency limit (LFL; GHz) of the Gaunt factor is sufficient, while at IR-wavelength the emissivity deviates considerably from the LFL and from the high energy approximation of Menzel & Pekeris (1935). The ff+ spectrum is given by Eq. 2. For K and Hz the Raleigh-Jeans approximation , and for Hz the Wien approximation apply. The optical depth in the R-J limit decreases with and according to

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 emission

For 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 g as the ratio of the quantum mechanical cross section to the semiclassical result of Kramers and Wentzel. In this context is the fine structure constant, the charge of the ion which is assumed to be hydrogenic, the charge, m the mass and v the velocity of the electron. Grant (1958) derived the Gaunt factor using Coulomb wave functions for initial and final states of free electrons in the form

with the complex function

expressed by hypergeometric functions F. This result was stated earlier by Menzel & Pekeris (1935) in a different way. The parameters and are related to the electron velocities by

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⁺, He, This Gaunt factor of Eq. 7 is used in the numerical treatment below.

2.2. Thermally averaged Gaunt factors

In 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 methods

The hypergeometric functions entering in Eq. 8 assume a large negative argument z with for most frequencies of interest with and the initial and final energy, respectively. To obtain an argument inside the unit circle in the complex plane we make use of a linear transformation of the hypergeometric function (e.g., Abramowitz & Stegun 1972)

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 y-axis at a midpoint

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.

Fig. 1. The Gaunt factor for free-free radiation according to Eqs. 10 and 7 calculated in the way described in Sect. 2.3. This is shown as the solid line and is the most accurate calculation. We choose K, i.e., a typical value for H II regions. The dotted line represents the classical approximation of Scheuer (1960) and Oster (1961), while the dashed curve is due to Eq. 23. The high-energy limit of Menzel & Pekeris (1935) is shown as dashed-dotted and the low frequency limit fit of Altenhoff et al. (1960) is dash-dotted.

Fig. 3. Frequency dependence of the free-free emissivity for different .

Table 1. Validity ranges for often used analytic approximations to the Gaunt factor for temperatures in the range of to K.

2.4. Low-frequency limit

Of 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 x we get the absolute value of in this limit

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
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