Power series solutions of the polytrope equations

Details Power series solutions of the polytrope equations Power series solutions of the polytrope equations

Mar 1999

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1999mnras.303..466r&link_type=abstract

Monthly Notices of the Royal Astronomical Society, Volume 303, Issue 3, pp. 466-470.

Astronomy and Astrophysics

Astronomy

7

Methods: Analytical, Stars: Interiors

Scientific paper

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 nn_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.

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