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 the gravitational potential of completely degenerate white dwarf stars. The equation reads as^[2]

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

with initial conditions

$${\displaystyle \varphi (0)=1,\quad \varphi '(0)=0}$$

where ${\displaystyle \varphi }$ measures the density of white dwarf, ${\displaystyle \eta }$ is the non-dimensional radial distance from the center and ${\displaystyle C}$ is a constant which is related to the density of the white dwarf at the center. The boundary ${\displaystyle \eta =\eta _{\infty }}$ of the equation is defined by the condition

$${\displaystyle \varphi (\eta _{\infty })={\sqrt {C}}.}$$

such that the range of ${\displaystyle \varphi }$ becomes ${\displaystyle {\sqrt {C}}\leq \varphi \leq 1}$. This condition is equivalent to saying that the density vanishes at ${\displaystyle \eta =\eta _{\infty }}$.

Derivation

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 ${\displaystyle p_{0}}$standardized as ${\displaystyle x=p_{0}/mc}$, with pressure ${\displaystyle P=Af(x)}$ and density ${\displaystyle \rho =Bx^{3}}$, where

$${\displaystyle {\begin{aligned}&A={\frac {\pi m_{e}^{4}c^{5}}{3h^{3}}}=6.02\times 10^{21}{\text{ Pa}},\\&B={\frac {8\pi }{3}}m_{p}\mu _{e}\left({\frac {m_{e}c}{h}}\right)^{3}=9.82\times 10^{8}\mu _{e}{\text{ kg/m}}^{3},\\&f(x)=x(2x^{2}-3)(x^{2}+1)^{1/2}+3\sinh ^{-1}x,\end{aligned}}}$$

${\displaystyle \mu _{e}}$ is the mean molecular weight of the gas, and ${\displaystyle h}$ is the height of a small cube of gas with only two possible states.

When this is substituted into the hydrostatic equilibrium equation

$${\displaystyle {\frac {1}{r^{2}}}{\frac {d}{dr}}\left({\frac {r^{2}}{\rho }}{\frac {dP}{dr}}\right)=-4\pi G\rho }$$

where ${\displaystyle G}$ is the gravitational constant and ${\displaystyle r}$ is the radial distance, we get

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

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

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

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

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

gives

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

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

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

The mass interior to a specified point can then be calculated

$${\displaystyle M(\eta )=-{\frac {4\pi }{B^{2}}}\left({\frac {2A}{\pi G}}\right)^{3/2}\eta ^{2}{\frac {d\varphi }{d\eta }}.}$$

The radius-mass relation of the white dwarf is usually plotted in the plane ${\displaystyle \eta _{\infty }}$-${\displaystyle M(\eta _{\infty })}$.

Solution near the origin

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

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

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

Equation for small central densities

When the central density ${\displaystyle \rho _{o}=Bx_{o}^{3}=B(y_{o}^{2}-1)^{3/2}}$ is small, the equation can be reduced to a Lane–Emden equation by introducing

$${\displaystyle \xi ={\sqrt {2}}\eta ,\qquad \theta =\varphi ^{2}-C=\varphi ^{2}-1+x_{o}^{2}+O(x_{o}^{4})}$$

to obtain at leading order, the following equation

$${\displaystyle {\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}\left(\xi ^{2}{\frac {d\theta }{d\xi }}\right)=-\theta ^{3/2}}$$

subjected to the conditions ${\displaystyle \theta (0)=x_{o}^{2}}$ and ${\displaystyle \theta '(0)=0}$. Note that although the equation reduces to the Lane–Emden equation with polytropic index ${\displaystyle 3/2}$, 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., ${\displaystyle y_{o}\rightarrow \infty }$ or equivalently ${\displaystyle C\rightarrow 0}$, the governing equation reduces to

$${\displaystyle {\frac {1}{\eta ^{2}}}{\frac {d}{d\eta }}\left(\eta ^{2}{\frac {d\varphi }{d\eta }}\right)=-\varphi ^{3}}$$

subjected to the conditions ${\displaystyle \varphi (0)=1}$ and ${\displaystyle \varphi '(0)=0}$. This is exactly the Lane–Emden equation with polytropic index ${\displaystyle 3}$. Note that in this limit of large densities, the radius

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

tends to zero. The mass of the white dwarf however tends to a finite limit

$${\displaystyle M\rightarrow -{\frac {4\pi }{B^{2}}}\left({\frac {2A}{\pi G}}\right)^{3/2}\left(\eta ^{2}{\frac {d\varphi }{d\eta }}\right)_{\eta =\eta _{\infty }}=5.75\mu _{e}^{-2}M_{\odot }.}$$

The Chandrasekhar limit follows from this limit.

See also

• Emden–Chandrasekhar equation
• Tolman–Oppenheimer–Volkoff equation

References