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tokens: ( 16 ), the BR emissivity in the classical limit then reads where p min is the minimal electron momentum that is required to produce a photon of energy ω = h ν/ m e c 2 . This is determined by ω = γ min − 1, which yields . In the last step of equation ( 17 ), we used a non-relativistic Maxwell–Boltzmann distribution function, with dimensionless electron temperature θ e = kT e / m e c 2 to carry out the integral. The above expressions explicitly assume p 1 , ω ≪ 1. Even without quantum corrections, to generalize the Kramers approximation to higher temperatures/energies, we shall use the relativistic Maxwell–Boltzmann distribution function where K 2 ( x ) is the modified Bessel function of second kind. 9 We also keep the full relativistic expression for and furthermore realize that at low frequencies the overall BR cross-section scales as towards higher electron energies (see the discussion about Bethe–Heitler limit). Thus, after multiplying the Kramers approximation, equation (2), by and carrying out the thermal average with the above modification, we have the relativistically improved Kramers approximation for the BR emissivity: This shows that in the classical treatment the improved asymptotic scales as for low temperatures. The origin of this correction is not quantum-mechanical but simply due to special relativistic effects. This modification absorbs the leading order corrections towards the BH limit, as we discuss next. We also mention that in equation (19) the temperature-dependent factor, is directly related to the differences in the normalization of the non-relativistic and relativistic Maxwell–Boltzmann distributions. At high temperatures, the corrections can become sizable, giving at and at , and thus should be taken into account for accurate calculations. 4.2 Definition of thermally averaged Gaunt factors The main quantity that enters the BR emission term in the photon Boltzmann equation, as well as the electron temperature evolution equation, is the thermally averaged Gaunt factor. It can be simply obtained by comparing the total plasma emissivity with the emissivity in the Kramers limit and is usually computed as In the last step, we explicitly assumed that the electrons follow a non-degenerate, relativistic Maxwell–Boltzmann distribution. 10 In the non-relativistic limit (γ 1 ≃ 1), equation ( 21 ) is a very good choice. However, for p 1 ≳ 1, the Gaunt factor scales at low frequencies (see Section 2.3 ). It is thus useful to multiply the Kramers cross-section by , which then yields a slightly modified definition for the BR Gaunt factor: This redefinition reduces the dynamic range of the Gaunt factor and is thus very useful for compressing the data in tabulations. The final BR emission term then takes the form where is given by equation ( 19b ) and by equation ( 20 ). Both definitions, of course, give exactly the same answer for the overall BR emission term. Nevertheless, in applications, is beneficial since it does not scale as