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]
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
.
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
[edit]
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
[edit]
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
[edit]
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.