Open AccessPower series solutions of the polytrope equations
Author(s) -
Roxburgh Ian W.,
Stockman Lynne M.
Publication year - 1999
Publication title -
monthly notices of the royal astronomical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.058
H-Index - 383
eISSN - 1365-2966
pISSN - 0035-8711
DOI - 10.1046/j.1365-8711.1999.02219.x
Subject(s) - polytropic process , polytrope , series (stratigraphy) , physics , power series , alternating series , star (game theory) , convergence (economics) , taylor series , series expansion , mathematical analysis , astrophysics , mathematics , mathematical physics , quantum mechanics , paleontology , economic growth , economics , biology
We derive recurrence relations for the coefficients a k in the power series expansion θ(ξ)=∑ a k ξ 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 θ, ξ in powers of m=q 2/3 , where q=‐ξ 2 dθ/dξ is the polytropic mass. These series appear to converge everywhere within the star for all n ≤ 5. Finally we show that θ(ξ) can be satisfactorily approximated (∼ 1 per cent) by (1‐cξ 2 )/(1+eξ 2 ) m , and give the values of the constants determined by a Padé approximation to the series, and by a two‐parameter fit to the numerical solutions.