Power series solutions of the polytrope equations
Abstract
We derive recurrence relations for the coefficients a_k in the power series expansion theta(xi)=∑ a_kxi^2k of the solution of the Lane-Emden equation, and examine the convergence of these series. For values of the polytropic index n<n_1~1.9 the series appear to converge everywhere inside the star. For n>n_1 the series converge in the inner part of the star but then diverge. We also derive the series expansions for theta, xi in powers of m=q^2/3, where q=-xi^2dtheta/dxi is the polytropic mass. These series appear to converge everywhere within the star for all n <= 5. Finally we show that theta(xi) can be satisfactorily approximated (~ 1 per cent) by (1-cxi^2)/(1+exi^2)^m, and give the values of the constants determined by a Pade approximation to the series, and by a two-parameter fit to the numerical solutions.
- Publication:
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Monthly Notices of the Royal Astronomical Society
- Pub Date:
- March 1999
- DOI:
- 10.1046/j.1365-8711.1999.02219.x
- Bibcode:
- 1999MNRAS.303..466R
- Keywords:
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- METHODS: ANALYTICAL;
- STARS: INTERIORS