User:Tohline/Appendix/Ramblings/PowerSeriesExpressions
Approximate Power-Series Expressions
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Polytropic Lane-Emden Function
We seek a power-series expression for the polytropic, Lane-Emden function, <math>~\Theta_\mathrm{H}(\xi)</math> — expanded about the coordinate center, <math>~\xi = 0</math> — that approximately satisfies the Lane-Emden equation,
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<math>~\frac{1}{\xi^2} \frac{d}{d\xi}\biggl( \xi^2 \frac{d\Theta_H}{d\xi} \biggr) = - \Theta_H^n</math> |
A general power-series should be of the form,
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<math>~\Theta_H</math> |
<math>~=</math> |
<math>~ \theta_0 + a\xi + b\xi^2 + c\xi^3 + d\xi^4 + e\xi^5 + f\xi^6 + \cdots </math> |
Isothermal Lane-Emden Function
Displacement Function for Polytropic LAWE
Displacement Function for Isothermal LAWE
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