Open AccessSeries solutions for polytropes and the isothermal sphere
Author(s) -
Hunter C.
Publication year - 2001
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.1365-8711.2001.04914.x
Subject(s) - polytropic process , physics , series (stratigraphy) , singularity , radius , power series , euler's formula , isothermal process , polytrope , gravitational singularity , classical mechanics , limit (mathematics) , mathematical analysis , mathematics , thermodynamics , quantum mechanics , paleontology , biology , computer security , computer science
The Lane–Emden equation for polytropic index and its limit of the isothermal sphere equation are singular at some negative value of the radius squared. This singularity prevents the real power series solutions about the centre from converging all the way to the outer surface when . However, a simple Euler transformation gives series that do converge all the way to the outer radius. These Euler‐transformed series converge significantly faster than the series in the contained mass derived by Roxburgh & Stockman, which are limited to finite radii whenever by a complex conjugate pair of singularities. We construct some compact analytical approximations to the isothermal sphere, and give one for which the density profile is accurate to 0.001 per cent out to the limit of stability against gravothermal collapse.