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Contact: F.X.Timmes
my one page vitae,
full vitae,
research statement, and
teaching statement.
The tool in coldwd.tbz generates models of stars in hydrostatic equilibrium with a cold electron Fermi gas equation of state: \begin{equation} \begin{split} x & = \left [ \dfrac{3}{8 \pi} \left ( \dfrac{h}{m_ec}\right )^3 N_A Y_e \rho \right ]^{1/3} \\ f(x) & = x (x^2 + 1)^{1/2}(2x^2 - 3) + 3\ln(x + (x^2 + 1)^{1/2}) \\ g(x) & = 8x^3 \left [ (x^2 + 1)^{1/2} -1) \right ] - f(x) \\ P_e & = \dfrac{\pi m_e^4 c^5}{3 h^3} \cdot f(x) \hskip 1.0in E_e = \dfrac{\pi m_e^4 c^5}{3 h^3} \cdot g(x) \end{split} \label{eq1} \tag{1} \end{equation} The derivatives of the pressure and energy with respct to the density are also returned by the equation of state module. A general relativistic Tolman-Oppenheimer-Volkoff (TOV) correction to the equation for hydrostatic equilibrium is avaliable as an option. A quote from Icko about generating white dwarf models comes to mind ...
The equations above suffer a loss of numerical precision for x ≪ 1 due to the subtraction of two nearly equal terms. These expansions are used instead \begin{equation} \begin{split} f(x) & = \frac{8}{5} x^5 - \frac{4}{7} x^7 + \frac{1}{3} x^9 - \frac{5}{22} x^{11} + \frac{35}{208} x^{13} - \frac{21}{160} x^{15} + \frac{231}{2176} x^{17} + \mathcal{O}(x^{19}) \\ g(x) & = \frac{12}{5} x^5 - \frac{3}{7} x^7 + \frac{1}{6} x^9 - \frac{15}{176} x^{11} + \frac{21}{416} x^{13} - \frac{21}{640} x^{15} + \frac{99}{4352} x^{17} + \mathcal{O}(x^{19}) \ . \end{split} \label{eq2} \tag{2} \end{equation}
The first plot below shows the central density vs mass relationship between a cold electron Fermi gas equation of state and a polytropic equation of state.
A cold electron Fermi gas at low central densities (x ≪ 1) approaches the well-known nonrelativistic form $P = 1.004 \times 10^{13} \ (Y_e \rho)^{5/3} \ {\rm erg} \ {\rm cm}^{-3}$, as can be seen by the leading order $x^5$ series expansion term for f(x) above. In this limit the electrons are well approximated by a n = 3/2, γ = 1 + 1 /n = 5/3 polytropic equation of state.
A cold electron Fermi gas at high central densities (x ≫ 1) approaches the relativistic form $P = 1.2435 \times 10^{15} \ (Y_e \rho)^{4/3} \ {\rm erg} \ {\rm cm}^{-3}$; expansions in this limit are in the source code for reference but are not used as they are not needed. In this limit the electrons are well approximated by a n = 3 γ = 1 + 1 /n = 4/3 polytropic equation of state – the celebrated Chandrasekhar limit.
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