The Lane-Emden equation is a second order differential equation which is used to model a spherically symmetric gas cloud, eg a stellar interior. Although it is a considerably simplified model (eg, no rotation), it still provides a good starting point. Traditionally the LE equation is written in terms of a linear independent variable, say x, which represents the relative radius of the gas cloud. The variable x goes from 0 (center of cloud) to 1 (outer boundary of cloud). With the assumption of a particular type of model of the gas cloud’s thermodynamics — called a polytropic model — one ends up with the LE equation. The model is characterized by a parameter n, called the polytropic index (n is a non-negative real number, not just an integer) which determines a function f (which depends upon the choice of polytropic index n), which in turn determines the density profile of the cloud, and thereby its other profiles, eg its mass profile.
It turns out, however, that the function f is an even function, and its power series expansion, for instance, involves only even powers of x. For instance, f(x) equals 1 – (1/6)*x^2 + (n/120)*x^4 + … and so on, with terms for x^6, x^8 etc. So one asks, whether it might be useful to write the LE equation (a second order differential equation) in terms not of independent variable x, but in terms of independent variable s = x^2. The variable s is the relative surface area of a spherical shell of relative radius x. At x=1 (outer boundary of the gas cloud), the value of s=1, ie relative area of that outer shell, is also 1.
There is a possibility for getting some physical insight from such a rewrite of the LE equation. The physics which is being modeled involves transport of energy and force between nested shells, and such transport may be conceptually more meaningful if it considered as a function of relative area rather than relative radius.
I have prepared a short pdf (2 pages) which describes such a rewrite of the Lane-Emden equation. Nothing new there; I’m sure such a rewrite has been done by others, as the LE equation has been a subject of study for over a century, and there is a considerable literature. Still, it may be of interest to others who are working their way into the details of the Lane-Emden equation and the stellar interior models for which it is a starting place. I give a few references, including to Chandrasekhar’s classic book on Stellar Interiors, which is still a better explanation than some of the more recent books, as it illustrates some motivations. And also a reference to a very interesting little paper by Klaus Rohe, who used Python to calculate rational expressions for the first 15 coefficients of a power series representation of the LE equation (ie, up to the x^28 term), as expressions in the polytropic index n. His paper is at Arxiv 1409.2008 if you want to go there directly.
Best wishes,
Ken Roberts
07-Mar-2015
Link to pdf file mentioned above, with my rewrite of Lane-Emden equation using relative area…
https://lasi2.files.wordpress.com/2015/03/lane-emden-rewrite.pdf
Logic, Astronomy, Science, and Ideas Too 