Editing SSC/Structure/Polytropes (section)

===Lane-Emden Equation===

Adopting [[SSCpt2/SolutionStrategies#Technique_2|solution technique #2]], we need to solve the following second-order ODE relating the two unknown functions, {{Math/VAR_Density01}} and {{Math/VAR_Enthalpy01}}:

<div align="center">
<math>\frac{1}{r^2} \frac{d}{dr}\biggl( r^2 \frac{dH}{dr} \biggr) =-  4\pi G \rho</math> .
</div>

It is customary to replace {{Math/VAR_Enthalpy01}} and {{Math/VAR_Density01}} in this equation by a dimensionless polytropic enthalpy, <math>~\Theta_H</math>, such that,
<div align="center">
<math>
~\Theta_H \equiv \frac{H}{H_c} = \biggl( \frac{\rho}{\rho_c} \biggr)^{1/n} ,
</math>
</div>
where the mathematical relationship between <math>~H/H_c</math> and <math>~\rho/\rho_c</math> comes from the adopted barotropic (polytropic) relation identified above.  To accomplish this, we replace {{Math/VAR_Enthalpy01}} with <math>~H_c \Theta_H</math> on the left-hand-side of the governing differential equation and we replace {{Math/VAR_Density01}} with <math>~\rho_c \Theta_H^n</math> on the right-hand-side, then gather the constant coefficients together on the left.  The resulting ODE is,

<div align="center">
<math>\biggl[ \frac{1}{4\pi G}~ \biggl( \frac{H_c}{\rho_c} \biggr) \biggr] \frac{1}{r^2} \frac{d}{dr}\biggl( r^2 \frac{d\Theta_H}{dr} \biggr) = - \Theta_H^n</math> .
</div>

The term inside the square brackets on the left-hand-side has dimensions of length-squared, so it is also customary to define a dimensionless radius,
<div align="center">
<math>
\xi \equiv \frac{r}{a_\mathrm{n}} ,
</math>
</div>
where,
<div align="center">
<math>~
a_\mathrm{n} \equiv \biggl[\frac{1}{4\pi G}~ \biggl( \frac{H_c}{\rho_c} \biggr)\biggr]^{1/2} 
= \biggl[\frac{(n+1)K_n}{4\pi G} \cdot \rho_c^{(1-n)/n} \biggr]^{1/2} \, ,
</math>
</div>
in which case our governing ODE becomes what is referred to in the astronomical literature as the,
<div align="center">
<span id="LaneEmdenEquation"><font color="#770000">'''Lane-Emden Equation'''</font></span>
<br />
{{Math/EQ_SSLaneEmden01}}

&sect;IV.2 of [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], p. 88, Eq. (11)
<br />
</div>
Our task is to solve this ODE to determine the behavior of the function <math>~\Theta_H(\xi)</math> &#8212; and, from it in turn, determine the radial distribution of various dimensional physical variables &#8212; for various values of the polytropic index, {{Math/MP_PolytropicIndex}}.  <span id="IntegralMass">In particular, from time to time we will find it useful to realize that the mass interior to <math>r</math> is given by the expression, </span>

<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>M_r</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\int_0^r 4\pi \rho r^2 dr = 4\pi \rho_c a_n^3 \int_0^\xi \Theta_H^n \xi^2 d\xi</math></td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>- 4\pi \rho_c a_n^3 \int_0^\xi \biggl[\frac{1}{\xi^2} \frac{d}{d\xi}\biggl(\xi^2 \frac{d\Theta_H}{d\xi}\biggr) \biggr]  \xi^2d\xi</math></td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>- 4\pi \rho_c a_n^3  \biggl(\xi^2 \frac{d\Theta_H}{d\xi}\biggr) \, .</math></td>
</tr>

<tr>
  <td align="center" colspan="3">
&sect;IV.5.b of [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], p. 97, Eq. (67)

  </td>
</tr>
</table>

<div align="center" id="MassRadiusRelation">
<table border="1" cellpadding="8" align="center" width="90%">
<tr><td align="left">
<font size="+1" color="red"><b>ASIDE:</b></font>  In an [[SSC/FreeEnergy/PolytropesEmbedded#Case_M_Free-Energy_Surface|accompanying discussion of pressure-truncated polytropes]], we adopt the following length normalization:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~R_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[\biggl(\frac{G}{K_n}\biggr)^n M_\mathrm{tot}^{n-1}  \biggr]^{1/(n-3)} \, .</math>
  </td>
</tr>
</table>
</div>
Let's see how the traditional Lane-Emden length scale, <math>~a_n</math>, relates.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a_n^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{(n+1)K_n}{4\pi G}\biggr] \biggl[ \frac{\rho_c}{\bar\rho}\biggr]^{(1-n)/n} \biggl[ \frac{3M_\mathrm{tot}}{4\pi (a_n \xi_1)^3} \biggr]^{(1-n)/n}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{(n+1)K_n}{3 G} \cdot M_\mathrm{tot}^{(1-n)/n}\biggr] \mathfrak{f}_M^{(n-1)/n} \xi_1^{3(n-1)/n} \biggl(\frac{3}{4\pi}\biggr)^{1/n} a_n^{3(n-1)/n}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ a_n^{(n-3)/n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{3}{(n+1)} \biggl( \frac{G}{K_n}  \biggr) M_\mathrm{tot}^{(n-1)/n}\biggr]  \biggl[\mathfrak{f}_M \xi_1^3\biggr]^{(1-n)/n} \biggl(\frac{4\pi}{3}\biggr)^{1/n} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{a_n}{R_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{3}{(n+1)} \biggr]^{n/(n-3)}  \biggl[\mathfrak{f}_M \xi_1^3\biggr]^{(1-n)/(n-3)} \biggl(\frac{4\pi}{3}\biggr)^{1/(n-3)} \, . 
</math>
  </td>
</tr>
</table>
</div>
where, we have made use of the relation drawn from our [[SSCpt1/Virial#Structural_Form_Factors|accompanying discussion of structural form factors]] &#8212; see, also, [[SSCpt1/Virial/FormFactors#PTtable|here]] &#8212; 
<div align="center">
<math>~\mathfrak{f}_M = \biggl[ - \frac{3\theta^'}{\xi} \biggr]_{\xi_1} \, ,</math>
</div> 

denotes the equilibrium ratio of the mean-to-central density. We conclude, therefore, that, in terms of <math>~R_\mathrm{norm}</math>, the equilibrium radius of an isolated polytrope is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[\frac{R_\mathrm{eq}}{R_\mathrm{norm}}\biggr]^{(n-3)} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{a_n \xi_1}{R_\mathrm{norm}}\biggr]^{(n-3)} =
\biggl[ \frac{3}{(n+1)} \biggr]^{n}  \biggl[\mathfrak{f}_M \xi_1^3\biggr]^{(1-n)} \biggl(\frac{4\pi}{3}\biggr) \xi_1^{(n-3)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{4\pi}{(n+1)^n}  \biggl[(-\theta^') \xi^2\biggr]_{\xi_1}^{(1-n)}  \xi_1^{(n-3)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~\frac{R_\mathrm{eq}}{R_\mathrm{norm}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)} \biggl[(-\theta^') \xi^2\biggr]_{\xi_1}^{(1-n)/(n-3)} \xi_1 \, .
</math>
  </td>
</tr>
</table>
</div>
This matches the expression presented in an [[SSC/FreeEnergy/Powerpoint#Case_M_Equilibrium_Conditions|accompanying summary supporting a PowerPoint presentation]].

</td></tr>
</table>
</div>