Chandrasekhar's white dwarf equation

From the quantum statistics of a completely degenerate electron gas (all the lowest quantum states are occupied), the pressure and the density of a white dwarf are calculated in terms of the maximum electron momentum p_0standardized as x = p_0 / mc, with pressure P = A f(x) and density \rho = B x^3, where \begin{align} & A = \frac{\pi m_\text{e}^4 c^5}{3h^3} = 6.02\times 10^{21} \text{ Pa}, \\ & B = \frac{8\pi}{3}m_p\mu_\text{e}\left(\frac{m_\text{e} c}{h}\right)^3 = 9.82\times 10^8 \mu_\text{e} \text{ kg/m}^3, \\ & f(x) = x(2x^2-3)(x^2+1)^{1/2} + 3 \sinh^{-1} x, \end{align} \mu_\text{e} is the mean molecular weight of the gas, and h is the Planck constant. When this is substituted into the hydrostatic equilibrium equation \frac 1 {r^2} \frac{d}{dr}\left(\frac{r^2}{\rho}\frac{dP}{dr}\right)=-4\pi G \rho where G is the gravitational constant and r is the radial distance, we get \frac{1}{r^2} \frac{d}{dr}\left(r^2\frac{d\sqrt{x^2+1}}{dr}\right)=-\frac{\pi G B^2}{2A}x^3 and letting y^2 = x^2 + 1, we have \frac{1}{r^2} \frac{d}{dr}\left(r^2\frac{dy}{dr}\right)=-\frac{\pi G B^2}{2A}(y^2-1)^{3/2} If we denote the density at the origin as \rho_\text{o} = B x_\text{o}^3 = B (y_\text{o}^2-1)^{3/2} , then a non-dimensional scale r = \left(\frac{2A}{\pi G B^2}\right)^{1/2} \frac{\eta}{y_\text{o}}, \quad y = y_\text{o} \varphi gives \frac{1}{\eta^2} \frac{d}{d\eta}\left(\eta^2 \frac{d\varphi}{d\eta}\right) + (\varphi^2 - C)^{3/2} = 0 where C=1/y_\text{o}^2. In other words, once the above equation is solved the density is given by \rho = B y_\text{o}^3 \left(\varphi^2 - \frac 1 {y_\text{o}^2}\right)^{3/2}. The mass interior to a specified point can then be calculated M(\eta) = -\frac{4\pi}{B^2}\left(\frac{2A}{\pi G}\right)^{3/2}\eta^2\frac{d\varphi}{d\eta}. The radius–mass relation of the white dwarf is usually plotted in the plane \eta_\infty–M(\eta_\infty). == Solution near the origin ==

In the neighborhood of the origin, \eta\ll 1, Chandrasekhar provided an asymptotic expansion as \begin{align} \varphi = {} & 1 - \frac{q^3} 6 \eta^2 + \frac{q^4}{40} \eta^4 - \frac{q^5(5q^2+14)}{7!} \eta^6 \\[6pt] & {} + \frac{q^6(339q^2 + 280)}{3\times 9!}\eta^8 - \frac{q^7(1425q^4 + 11346q^2 + 4256)}{5\times 11!}\eta^{10} + \cdots \end{align} where q^2 = C-1. He also provided numerical solutions for the range C = 0.01 - 0.8 . == Equation for small central densities ==

When the central density \rho_\text{o} = B x_\text{o}^3 = B (y_\text{o}^2-1)^{3/2} is small, the equation can be reduced to a Lane–Emden equation by introducing \xi = \sqrt{2}\eta, \qquad \theta = \varphi^2-C= \varphi^2-1+x_\text{o}^2 + O(x_\text{o}^4) to obtain at leading order, the following equation \frac{1}{\xi^2}\frac{d}{d\xi}\left(\xi^2\frac{d\theta}{d\xi}\right) = - \theta^{3/2} subjected to the conditions \theta(0)=x_\text{o}^2 and \theta'(0)=0. Note that although the equation reduces to the Lane–Emden equation with polytropic index 3/2, the initial condition is not that of the Lane–Emden equation. == Limiting mass for large central densities ==

When the central density becomes large, i.e., y_\text{o}\rightarrow \infty or equivalently C\rightarrow 0, the governing equation reduces to \frac{1}{\eta^2}\frac{d}{d\eta}\left(\eta^2\frac{d\varphi}{d\eta}\right) = - \varphi^{3} subjected to the conditions \varphi(0)=1 and \varphi'(0)=0. This is exactly the Lane–Emden equation with polytropic index 3. Note that in this limit of large densities, the radius r = \left(\frac{2A}{\pi G B^2}\right)^{1/2} \frac{\eta}{y_\text{o}} tends to zero. The mass of the white dwarf however tends to a finite limit M\rightarrow - \frac{4\pi}{B^2}\left(\frac{2A}{\pi G}\right)^{3/2}\left(\eta^2 \frac{d\varphi}{d\eta}\right)_{\eta=\eta_\infty}=5.75 \mu_\text{e}^{-2}M_\odot. The Chandrasekhar limit follows from this limit. == See also ==