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Perturbative solution to the Lane-Emden equation: An eigenvalue approach | | Kenny L. S. Yip
; T. K. Chan
; P. T. Leung
; | | Date: |
22 Nov 2016 | | Abstract: | Under suitable scaling, the structure of self-gravitating polytropes is
described by the standard Lane-Emden equation (LEE), which is characterised by
the polytropic index $n$. Here we use the known exact solutions of the LEE at
$n=0$ and $1$ to solve the equation perturbatively. We first introduce a scaled
LEE (SLEE) where polytropes with different polytropic indices all share a
common scaled radius. The SLEE is then solved perturbatively as an eigenvalue
problem. Analytical approximants of the polytrope function, the radius and the
mass of polytropes as a function of $n$ are derived. The approximant of the
polytrope function is well-defined and uniformly accurate from the origin down
to the surface of a polytrope. The percentage errors of the radius and the mass
are bounded by $8.1 imes 10^{-7}$ per cent and $8.5 imes 10^{-5}$ per cent,
respectively, for $nin[0,1]$. Even for $nin[1,5)$, both percentage errors are
still less than $2$ per cent. | | Source: | arXiv, 1611.7202 | | Services: | Forum | Review | PDF | Favorites | |
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