Chandrasekhar's white dwarf equation

In astrophysics, Chandrasekhar's white dwarf equation is an initial value ordinary differential equation introduced by the Indian American astrophysicist Subrahmanyan Chandrasekhar, in his study of gravitational potential of a completely degenerate white dwarf stars. The equation reads as

${\frac {1}{\eta ^{2}}}{\frac {d}{d\eta }}\left(\eta ^{2}{\frac {d\varphi }{d\eta }}\right)+(\varphi ^{2}-C)^{3/2}=0$

with initial conditions

$\varphi (0)=1,\quad \varphi '(0)=0$

where $\varphi$ measures the density of white dwarf, $\eta$ is the non-dimensional radial distance from the center and $C$ is a constant which is related to the density of the white dwarf at the center. When $C=0$, this equation reduces to Lane–Emden equation with polytropic index $3$. In the Lane-Emden equation, the density at the centre can be scaled out of the equation, but for white-dwarfs, the central density is directly tied to the equation.

Derivation

From quantum statistics of completely degenerate electron gas(all the lowest quantum states are occupied), the pressure and the density of a white dwarf is given by

$P=Af(x),\quad \rho =Bx^{3}$

where

${\begin{aligned}&A=6.01\times 10^{22},\ B=9.82\times 10^{5}\mu _{e},\\&f(x)=x(2x^{2}-3)(x^{2}+1)^{1/2}+3\sinh ^{-1}x,\end{aligned}}$

where $\mu _{e}$ is the mean molecular weight of the gas. 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

${\frac {1}{r^{2}}}{\frac {d}{dr}}\left(r^{2}{\frac {d{\sqrt {x^{2}+1}}}{dr}}\right)=-{\frac {\pi GB^{2}}{2A}}x^{3}$

and letting $y^{2}=x^{2}+1$, we have

${\frac {1}{r^{2}}}{\frac {d}{dr}}\left(r^{2}{\frac {dy}{dr}}\right)=-{\frac {\pi GB^{2}}{2A}}(y^{2}-1)^{3/2}$

If we denote the density at the origin as $\rho _{o}=Bx_{o}^{3}=B(y_{o}^{2}-1)^{3/2}$, then a non-dimensional scale

$r=\left({\frac {2A}{\pi GB^{2}}}\right)^{1/2}{\frac {\eta }{y_{o}}},\quad y=y_{o}\varphi$

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 $C=1/y_{o}^{2}$. In other words, once the above equation is solved the density is given by

$\rho =By_{o}^{3}\left(\varphi ^{2}-{\frac {1}{y_{o}^{2}}}\right)^{3/2}$

Approximate solution

In the neighborhood of the origin, $\eta \ll 1$, Chandrasekhar provided an asymptotic expansion as

${\begin{aligned}\varphi ={}&1-{\frac {q^{3}}{6}}\eta ^{2}+{\frac {q^{4}}{40}}\eta ^{4}-{\frac {q^{5}(5q^{2}+14)}{7!}}\eta ^{6}\\[6pt]&{}+{\frac {q^{6}(339q^{2}+280)}{3\times 9!}}\eta ^{8}-{\frac {q^{7}(1425q^{4}+11346q^{2}+4256)}{5\times 11!}}\eta ^{10}+\cdots \end{aligned}}$

where $q^{2}=C-1$. He also provided numerical solutions for the range $C=0.01-0.8$.

See also

• Chandrasekhar equation
• Lane–Emden equation