Difference between revisions of "User:Tohline/Appendix/Ramblings/PowerSeriesExpressions"

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==Isothermal Lane-Emden Function==
==Isothermal Lane-Emden Function==
We seek a power-series expression for the isothermal, Lane-Emden function, <math>~w(r)</math> &#8212; expanded about the coordinate center, <math>~r = 0</math> &#8212; that approximately satisfies the isothermal Lane-Emden equation,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
  <td align="right">
<math>~\frac{d^2w}{dr^2} +\frac{2}{r} \frac{d w}{dr}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~e^{-w} \, . </math>
  </td>
</tr>
</table>
</div>


A general power-series should be of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~w</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
w_0 + ar + br^2 + cr^3 + dr^4 + er^5 + fr^6 + gr^7 + hr^8 +\cdots
</math>
  </td>
</tr>
</table>
</div>
Result:
<div align="center" id="PolytropicLaneEmden">
<table border="1" width="80%" cellpadding="8" align="center"><tr><td align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~w(r)
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{r^2}{6} - \frac{r^4}{120} + \frac{r^6}{1890} + \cdots \, .</math>
  </td>
</tr>
</table>
</td></tr></table>
</div>
See also:
* Equation (377) from &sect;22 in Chapter IV of [[User:Tohline/Appendix/References#C67|C67]]).


==Displacement Function for Polytropic LAWE==
==Displacement Function for Polytropic LAWE==

Revision as of 22:26, 25 February 2017

Approximate Power-Series Expressions

Whitworth's (1981) Isothermal Free-Energy Surface
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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,

LSU Key.png

<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,

<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>


Result:

<math>~\theta</math>

<math>~=</math>

<math>~ 1 - \frac{\xi^2}{6} + \frac{n}{120} \xi^4 - \frac{n}{378} \biggl( \frac{n}{5} - \frac{1}{8} \biggr) \xi^6 + \biggl[ \frac{n(122n^2 -183n + 70)}{3265920} \biggr] \xi^8 + \cdots </math>

Isothermal Lane-Emden Function

We seek a power-series expression for the isothermal, Lane-Emden function, <math>~w(r)</math> — expanded about the coordinate center, <math>~r = 0</math> — that approximately satisfies the isothermal Lane-Emden equation,

<math>~\frac{d^2w}{dr^2} +\frac{2}{r} \frac{d w}{dr} </math>

<math>~=</math>

<math>~e^{-w} \, . </math>

A general power-series should be of the form,

<math>~w</math>

<math>~=</math>

<math>~ w_0 + ar + br^2 + cr^3 + dr^4 + er^5 + fr^6 + gr^7 + hr^8 +\cdots </math>


Result:

<math>~w(r) </math>

<math>~=</math>

<math>~\frac{r^2}{6} - \frac{r^4}{120} + \frac{r^6}{1890} + \cdots \, .</math>


See also:

  • Equation (377) from §22 in Chapter IV of C67).

Displacement Function for Polytropic LAWE

Displacement Function for Isothermal LAWE

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation