Expert: Philip A. Stahl - 12/30/2008

Question
Can you tell me what are polytropic gas spheres and how they factor into stellar evolution? Thanks!

Answer
Hello,

Polytropic gas spheres are basically mathematical entities used for modelling of actual stars. As usual, some basic assumptions are made (often in terms of temperatures, pressures, potential energies etc.) and these are then used to develop one or more "polytropic" models to test to see if they can work for a given star. Or, more likely, be employed as a guide to model a star.

A primary objective is to develop a basis for a self-gravitating sphere. In the most desirable of cases, one works to attain a simple relationship between the pressure P, and density (rho) of a form:

P  =  K (rho)^ (1 + 1/n)

where K and n are constants, and n is known as "the polytropic index" and K the "polytropic constant".

The polytropic index n can be defined:

n = 1/ (y - 1)

where 'y' is the ratio of specific heats.

In a non-relativistic limit, for example, one will have y = 5/3 and

n = 1 / (5/3 - 1)  =  1/ (2/3)  =  3/2

in which case,

P =  K (rho) ^1 + 1/(3/2)  =  K (rho) ^5/3

so that we see also:  K (rho)^ (1 + 1/n) = K (rho) ^y

in the relativistic limit, meanwhile, y = 4/3 so:

n = 1 / (4/3 - 1)  =  3  and

P =  K (rho) ^4/3


By way of general consideratios of the behavior of polytropic gas spheres, the relation:

V  =   3/ (n - 5)   GM^2/ R

has been established. Note what happens for n = 5:

V  = 3/ (5 - 5)   GM^2/ R  so V  ->  oo


so a polytropic index of n = 5 essentially blows up for the potential energy. (Mass infinitely concentrated at the center of the configuration)


(Note: In actual stellar structure modelling, one uses plots of dimensionless variables, generally called U,V to assess where singularities occur, and what polytropic indices work best. One finds that some solutions are "singular" - one valued, others exhibit no solutions, and others "blow up". The general rubric here is that for the solution to have physical meaning one needs: U, V > 0)

The polytropic gas spheres of general interest are those of indices: n = 3, n = 3/2 and n = oo.

The aim of most explorations of polytropic gas spheres is to arrive at reasonable solutions of what is called "the Lane-Emden equation" after J.H. Lane and R. Emden. This equation lays the basis for the development of polytropic stellar models - limiting solutions to those which are finite at the model star's center (e.g. z = 0 or r = 0).

Unfortunately, this is far outside the purview of this topic- question, but if you have any further questions on polytropic gas spheres feel free to ask!