## Hyperbolic Transitions or Free Free Opacity

A free electron of velocity v moving in the Coulomb field of an ion of charge Zj e can absorb (ff absorption) or emit (Bremsstrahlung) a photon of frequency v (Fig.

8.1). In the hydrogenic approximation, the absorption coefficient for ff absorption for an ion and a free electron is by volume unity

where gf is the Gaunt factor 1) for the process. This absorption is continuous, without jumps and has no frequency limit. This coefficient must first be integrated over the Maxwell-Boltzmann distribution (C.62) expressed in term of velocities

Aj muJo where dne(v) is the concentration of free electrons with velocities between v and v + dv. Then one must sum up over the various elements j,

j Jo Ajmu

In (8.26), one has a dependence on 1/(v v3). In thermal equilibrium v scales like T1/2. As noted in Sect. 8.3, a dependence on v-3 leads to a scaling with T-3, making thus an overall dependence on T-3-5, when account is given to the velocity. In addition the process depends on the electron density, which according to (7.42) yields a term q (1 + X) and the abundances of hydrogen and helium lead to a factor (X + Y). The mean Rosseland opacity (3.22) for the free-free absorption only is also roughly given by a Kramers law [523], see also [610],

Kff ~ K0,ff q T-3-5 with K0ff ~ 3-7 x 1022 (X + Y)(1 + X), (8.29)

where Kff is in cm2 g-1. The T and q dependences are the same as for bound-free opacity, thus ff transitions produce the same slope of k as a function of T (cf. Fig.

8.2). The coefficient Kf is smaller than K0 bf (8.24) for standard composition, so that bf transitions dominate over ff effects. However, the free-free opacity is essentially independent of metallicity Z and thus it dominates over bound-free opacity in low-Z stars.

In case of partial degeneracy, the distribution dne to be considered is the Fermi-Dirac rather than the Maxwell-Boltzmann distribution. Degeneracy limits the ff transitions, because the final stages of low energies are occupied. When electron concentration is high, screening effects (Sect. 9.4) play a role. Thus, the complete theory contains more effects, including also a Gaunt factor and a guillotine effect as mentioned for bf opacities.

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