Until now, the Lane-Emden equation has been solved by simply entering a value for the polytropic index and solving the differential equation. With an n=° polytrope, however, things become more difficult; a way must be found around the infinity to obtain a solution. The equilibrium structure of a spherically symmetric, n=° polytrope can be found by first working with the equation of hydrostatic equilibrium,
If we assume an ideal gas, we can say that
where MH is the mass of hydrogen and m is the mean molecular weight. Upon substitution into the equation for hydrostatic equilibrium and with some rearranging, one finds
and substitute these variables into the rearranged equation of hydrostatic equilibrium, manipulating the equation such that the final result is
This form is quite reminiscent of the Lane-Emden equation, with the boundary conditions
Numerical integration yields an interesting solution; the density never goes to zero for any value of x; the sphere extends to infinity. This polytrope, while not suited for modelling stars, has been used to describe stellar distributions in globular clusters and galaxy distributions in galaxy clusters. Another application of the isothermal gas sphere is called the Bonnor-Ebert sphere, which has been used to describe nebulae. A Bonnor-Ebert sphere results from enveloping an isothermal gas sphere in a hot medium which applies an external pressure on the sphere. This pressure causes the sphere to collapse. The derivation of the equations for Bonnor-Ebert spheres are similar to the arguments used above.
First, begin with a sphere in equilibrium with radius R, a definite central density, pressure p, volume V, and number density N, all of which are functions of the radius r. If there are slight, spherically symmetric pressure and volume fluctuations throughout the sphere, one finds (with N and T held constant)
where ¶ indicates the partial derivative. After a considerable amount of algebra, one can obtain
where p0 and V0 are the outside pressure, the fixed number of particles in the sphere, and the volume of the sphere, respectively. The quantity A represents
while the quantity B represents
where N0 is the number of molecules in the sphere. Now, if one recalls the substitutions made previously for r and r, that is,
one can rewrite A as
and quantity B can be rewritten as
In these forms, values for these expressions can easily be calculated. For small x (or r), both A and B are positive and ¶p/¶V is negative. If r is large, they oscillate in sign, with A reaching zero before B.
What happens if the system is "thumped"? One can find out by letting v be the smallest volume for which A goes to zero; then the partial derivative of pressure with respect to volume is less than zero, that is,
and for a small fluctuation dV in the volume of a sphere with mean volume v,
This means that if there is a fluctuation resulting in a slight decrease in the volume of the sphere, a decrease in the internal pressure will follow, which will in turn lead to a further reduction in volume. Therefore, an isothermal gas sphere with a volume greater than v will be unstable; small fluctuations will cause it to collapse.
More information on polytropes and isothermal gas spheres can be obtained from:
Bowers, R. and Deeming T., Astrophysics I: Stars. Boston: Jones & Bartlett Publishers, 1984.
For a more in-depth discussion of Bonnor-Ebert spheres, the reader is referred to the original (English) literature:
Bonnor, W.B., MNRAS, 116, 351, 1956.