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,^[1] in his study of gravitational potential of a completely degenerate white dwarf stars. The equation reads as^[2]

{\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 $C=0.01-0.8$ .

References

↑ Chandrasekhar, Subrahmanyan, and Subrahmanyan Chandrasekhar. An introduction to the study of stellar structure. Vol. 2. Chapter 11 Courier Corporation, 1958.
↑ Davis, Harold Thayer. Introduction to nonlinear differential and integral equations. Courier Corporation, 1962.

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[1] Chandrasekhar, Subrahmanyan, and Subrahmanyan Chandrasekhar. An introduction to the study of stellar structure. Vol. 2. Chapter 11 Courier Corporation, 1958.

[2] Davis, Harold Thayer. Introduction to nonlinear differential and integral equations. Courier Corporation, 1962.

Chandrasekhar's white dwarf equation

Derivation

Approximate solution

See also

References