The Structure, Stability, and Dynamics
of Self-Gravitating Systems

Fall Semester, 1997
Project #1

Joseph Hodges

Numerical Models of Polytropic Spheres With 0 < n <

Assignment (by J.E. Tohline):

Present a clear discussion of the equilibrium structure of nonrotating polytropes having polytropic indices in the range 0 < n < . The emphasis should be on models for which there is no known analytical solution to the Lane-Emden equation. The presentation should be in an HTML format, similar to the presentation that already has been made for "Spherical, n = 1 Polytropes."

To complete this project, it will be necessary to write a numerical algorithm that will effectively solve the Lane-Emden equation for arbitrary values of the polytropic index, n. Rather than looking through the literature for a numerical technique that will accurately integrate a nonlinear, second-order, ordinary differential equation and applying it to the Lane-Emden equation directly, however, I recommend that you adopt the Self-Consistent Field Technique. Before sitting down to write an SCF algorithm based on my outline of the technique, I recommend that you try to implement the technique analytically to derive the solution for Maclaurin Spheroids, as described in the accompanying homework problem.

As a check of your results, you should at least compare your numerically derived model structures for n = 1 and n = 5 polytropes to the analytically known results. You may wish to present a summary table of model properties like the one I've presented in the accompanying discussion of polytrope properties.

Home Page | Preface | Context | Applications | Appendices | Search Index