Fitting formulae for the equation of state of a perfect, semirelativistic Boltzmann gas

Abstract

Concise formulae are given which accurately yield the pressure, density, specific heats, and adiabatic indices of a perfect Boltzmann gas. The formulae are valid uniformly from the nonrelativistic, through the semi-relativistic, to the ultrarelativistic regimes. Subject headings: equation of state-relativity It sometimes happens that functions useful for astrophysical application, while they are conceptually simple, are for one reason or another difficult to compute numerically. In such cases, it proves useful to have approximate fitting formulae available for practical use. The present author has previously provided such formulae for the Lane-Emden functions (Service 1977) and for the isothermal function and its derivatives (Appendix in Flannery and Krook 1978 1). Present cosmological interest in exotic, stable, particles has fueled a resurgent interest in the thermodynamics and statistical mechanics of a gas of massive particles which cools from ultrarelativistic, through the semirelativistic regime (where kinetic energy and rest-mass energy are comparable) to a nonrelativistic, cold state. As we will now see, existing analytic formulae for the equation of state of such a gas are not very tractable numerically. Formulae for a relativistic Boltzmann gas have been given by, among others, Jiittner (1911), Chandrasekhar (1939, Chap. 10), Israel (1963), Stewart (1971, pp. 70-74), and Lightman et al (1975, § § 5.32-5.35). The following summary derives principally from the last three references. Hereafter, we adopt units with the speed of light c and Boltzmann's constant k equal to 1, so masses and temperatures are given in energy units. We are given a gas of particles each of rest mass m in a relativistic Boltzmann distribution at temperature T. Among the intrinsic thermodynamical quantities associated with the gas are its total proper density of mass and energy p, its pressure P, its entropy per particle s, and its proper particle number density n. The parameter that varies as we pass from the nonrelativistic to the relativistic regimes is T/m. The classical gas law P = nT (1) can be shown to hold exactly in all regimes. Other thermody-namic relations require the evaluation of the relativistic enthalpy rj, defined in terms of modified Bessel functions by K 3 (m/T) K 2 (m/T) (2) The parameter rj varies from 1 at T-» 0 to oo at T-> oo. In terms of rj one can derive P T p + P mrj 5 (3) (4) (5) (6) It is familiar that F varies from 5/3 to 4/3 as T varies from 0 to oo. For the speed of sound ü s , one has v 2 s TP p + P' (7) Apart from the problem of evaluating the modified Bessel functions, it turns out that equation (4) above is extremely delicate numerically, especially in the transition from the non-relativistic to semirelativistic regimes. For example, to derive the simple, nonrelativistic formula for the speed of sound, lim vl T->0 5T 3 m (8) involves expansion of the Bessel functions to third order in their asymptotic series before noncancelling terms are obtained. The first relativistic correction to equation (8) would require even higher expansions. Matters are worse, not better, on the computer, where the loss of significance in equation (4) from subtraction of nearly equal terms is greater than the accuracy of most routines for calculating the Bessel functions. Taking some care, however, one can design appropriate numerical routines for the desired calculations and then apply Chebyschev fitting methods to summarize the results as concise fitting formulae. Useful algorithms for these tasks are given in Press et al. (1986, Chaps. 5-6). We now give the results of such an effort. One family of fitting formulas is based on the ansatz 1 There is a typographical error in this reference: in eq. (Al) the value of a 3 should be-2.2999683(-4). 60 T/m 0.36 + T/m ' (9)