Contents

Chandrasekhar's white dwarf equation

Inastrophysics, 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

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

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[edit]

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

${\displaystyle \mu _{e}}$ 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

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

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

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

gives

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

The mass interior to a specified point can then be calculated

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[edit]

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

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[edit]

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

to obtain at leading order, the following equation

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[edit]

When the central density becomes large, i.e., ${\displaystyle y_{o}\rightarrow \infty }$ or equivalently ${\displaystyle C\rightarrow 0}$, the governing equation reduces to

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

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

The Chandrasekhar limit follows from this limit.

See also[edit]

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

References[edit]