A second-order ordinary differential equation that applies to polytropic profiles in density, defined as those which have the form

where P is pressure, K is a constant, is the density, and

with n is an integer (Chandrasekhar 1960).

The Lane-Emden equation is also applicable to magnetohydrodynamic fluids under the action of force-free magnetic fields.This implies that mass configurations obtained for neutral fluids through the Lane-Emden equation also exist for conducting fluids in force-free magnetic fields.Such fields are believed to exist in several astrophysical situations (Krishan 1999).

Poisson's equation for gravity states

where is the gravitational potential,G is the gravitational constant, and is the mass density. For spherical symmetry, in spherical coordinates is

Now, plug (1) into the hydrostatic law

where g is the gravitational acceleration, giving


Plugging (6) into (4),

This is actually a form of the Lane-Emden equation





Subject to the boundary conditions


These boundary conditions establish the allowed values of R and M. By appropriate change of variables


equation (10) can be transformed to the Lane-Emden equation


and the boundary conditions become


The cases n = 0 and 1 can be solved analytically; the others must be obtained numerically. The mass of a spherical body of radius R is given by the integral


so the central density is

giving



The central pressure is then given by


and the moment of inertia by



so


From the ideal gas law,

where is the mean molecular mass, and k is Boltzmann's constant. Therefore, the central temperature is


Finally, the gravitational potential energy is




For n = 0( ), the Lane-Emden equation is







The boundary condition then gives and so

so is parabolic. The first zero is found by solving






For n = 1 ()in space, the differential equation becomes


which is the spherical Bessel differential equation

with k = 1 and n = 0, so the solution is

Applying the boundary condition gives

In space,


]
The solution is

The boundary condition (11) requires B = 0. Applying (12) to (30) with B = 0,

for n = 1, 2, .... It is physically impossible for to equal 0 anywhere but at the planet's boundary. Therefore, it must be true that the first place where (30) is 0 is also the boundary. This implies that (31) must equal

which implies the surprising result that R is independent of M!

Therefore, the solution (30) is constrained by boundary conditions to

A can be expressed in terms of M using the condition

Let


The integral (35) then becomes











Let

then

To find , solve (33) for K


To find I,