Editing SSC/Structure/IsothermalSphere (section)

=Our Numerical Integration=
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|- 
! style="height: 125px; width: 125px; background-color:white;" |
<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures|<b>via<br />Direct<br />Numerical<br />Integration</b>]]</font>
|}
The [[#keyExpression|above governing relation]] &#8212; see especially [[#Chandrasekhar|Chandrasekhar's notation]] &#8212; may be rewritten as (see also, for example, &sect;19.8, eq. 19.35 of [<b>[[User:Tohline/Appendix/References#KW94|<font color="red">KW94</font>]]</b>]),
<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>
where we appreciate that,
<div align="center">
<math>~w \equiv \ln\biggl(\frac{\rho}{\rho_c}\biggr) \, .</math>
</div>

We'll adopt the following finite-difference approximations for the first and second derivatives on a grid of radial spacing, <math>~\Delta_r</math>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~w_i'</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\frac{w_+ - w_-}{2\Delta_r}</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~w_i''</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\frac{w_+ - 2w_i +w_-}{\Delta_r^2} \, .</math>
  </td>
</tr>
</table>
</div>
Our finite-difference approximation of the governing equation is, then,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r_i \biggl[ \frac{w_+ - 2w_i +w_-}{\Delta_r^2} \biggr] + 2\biggl[  \frac{w_+ - w_-}{2\Delta_r} \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_i e^{-w_i} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ r_i [ w_+ - 2w_i +w_- ] + \Delta_r [ w_+ - w_- ]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Delta_r^2 r_i e^{-w_i} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~w_+ 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\Delta_r^2 r_i e^{-w_i} +  2r_i w_i  + w_- (\Delta_r - r_i)}{( \Delta_r + r_i)} \, .</math>
  </td>
</tr>
</table>
</div>

Now, for the first two steps away from the center &#8212; where, <math>~w_i = w_0 = 0</math> and <math>~r_i = r_0 = 0</math> &#8212;  we will use the following [[Appendix/Ramblings/PowerSeriesExpressions#IsothermalLaneEmden|power-series expansion]] (see, for example, eq. 377 from &sect;22 in Chapter IV of [<b>[[User:Tohline/Appendix/References#C67|<font color="red">C67</font>]]</b>]) to determine the value of <math>~w_i</math>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~w_1 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\Delta_r^2}{6} - \frac{\Delta_r^4}{120} + \frac{\Delta_r^6}{1890} \, ,</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~w_2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(2\Delta_r)^2}{6} - \frac{(2\Delta_r)^4}{120} + \frac{(2\Delta_r)^6}{1890} \, .</math>
  </td>
</tr>
</table>
</div>