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Wednesday, November 1, 2023

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 the gravitational potential of completely degenerate white dwarf stars. The equation reads as

with initial conditions

where measures the density of white dwarf, is the non-dimensional radial distance from the center and is a constant which is related to the density of the white dwarf at the center. The boundary of the equation is defined by the condition

such that the range of becomes . This condition is equivalent to saying that the density vanishes at .

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 standardized as ,

with pressure

and density

where

is the mean molecular weight of the gas, and is the height of a small cube of gas with only two possible states.

When this is substituted into the hydrostatic equilibrium equation

where is the gravitational constant and is the radial distance, we get

and letting , we have

If we denote the density at the origin as , then a non-dimensional scale

gives

where . 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 -.

Solution near the origin

In the neighborhood of the origin, , Chandrasekhar provided an asymptotic expansion as

where . He also provided numerical solutions for the range .

Equation for small central densities

When the central density 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 and . Note that although the equation reduces to the Lane-Emden equation with polytropic index , 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., or equivalently , the governing equation reduces to

subjected to the conditions and . This is exactly the Lane-Emden equation with polytropic index . 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.

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