Editing SSC/Structure/PolytropesEmbedded (section)

===Kimura's Presentation===

At the same time Whitworth's work was being published, [http://adsabs.harvard.edu/abs/1981PASJ...33..273K Kimura (1981a)] also published a derivation of the equations that define the equilibrium properties of embedded, pressure-truncated polytropic configurations.  (Note that an [http://adsabs.harvard.edu/abs/1981PASJ...33..749K erratum] has been published correcting typographical errors that appear in a few equations of the original paper.)  When compared with, for example, [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|Horedt's published work]] &#8212; which Kimura references &#8212; Kimura's set of structural equations are a bit more difficult to digest because they include (a) an equation-of-state index that is different from the traditional ''polytropic index'' &#8212; specifically (see his equation 6),
<div align="center">
<math>~\sigma \equiv (n+1)^{-1} \, </math>
</div>
&#8212; which was Kimura's effort to more gracefully accommodate discussions of isothermal <math>~(n=\infty)</math> configurations; and (b) an additional integer index, <math>~m</math>, so that a single set of equations can be used to specify the structure of planar <math>~(m = 1)</math> and cylindrical <math>~(m=2)</math> as well as spherical <math>~(m=3)</math> configurations.  In the present context, we will fix the value to <math>~m = 3</math>.  Kimura also chose to express his structural solutions in terms of a dimensionless radius, <math>~\zeta</math>, instead of the traditional variable, <math>~\xi</math> &#8212; note that the two are related via the expression,
<div align="center">
<math>~\zeta = (n+1)^{1/2} \xi \, ;</math>
</div>
and in terms of a dimensionless gravitational potential, <math>~\phi</math>, instead of the traditional dimensionless enthalpy variable, <math>~\theta_n</math> &#8212; note that the two are related via the expression (see Kimura's equation 12),
<div align="center">
<math>~\phi = \sigma^{-1}(1 - \theta_n) \, .</math>
</div>
Given this relationship, we note as well that,
<div align="center">
<math>~\phi^' \equiv \frac{d\phi}{d\zeta} = -\frac{d\theta_n}{d\xi} \cdot \biggl[ \sigma^{-1} \frac{d\xi}{d\zeta} \biggr] = 
-\frac{d\theta_n}{d\xi} (n+1)^{1/2} \, .</math>
</div>

The set of equilibrium equations derived by [http://adsabs.harvard.edu/abs/1981PASJ...33..273K Kimura (1981a)] in what he identifies as "Paper I" &#8212; see especially his equations number (16) and (23) &#8212; are summarized most succinctly in Table 1 of  his "Paper II" ([http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura 1981b]).  The equations he presents for "radial distance," "pressure," and "fractional mass within <math>~\tilde{\zeta}</math>" are, respectively,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{Kimura}}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\tilde\zeta = (n+1)^{1/2} \tilde\xi \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\frac{P_\mathrm{e}}{P_\mathrm{Kimura}}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\tilde\theta_n^{n+1}\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\frac{M}{M_\mathrm{Kimura}}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\tilde\zeta^2 {\tilde\phi}^' = (n+1)^{3/2} \biggl[ - \xi^2 \frac{d\theta_n}{d\xi} \biggr]_{\tilde\xi} \, ,
</math>
  </td>
</tr>
</table>
</div>
where, expressed in terms of the central pressure, <math>~p_*</math>, and the polytropic constant, <math>~K_n, ~[</math>note that, in Kimura's paper, <math>~H = K_n^{n/(n+1)}]</math>, the relevant normalization parameters are,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~R_\mathrm{Kimura}
</math>
  </td>
  <td align="center">
<math>~\equiv~</math>
  </td>
  <td align="left">
<math>
(4\pi G)^{-1/2} H p_*^{\sigma - 1/2} = 
(4\pi G)^{-1/2} K_n^{n/(n+1)} p_*^{(1-n)/[2(n+1)]} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~P_\mathrm{Kimura}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
p_* \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~M_\mathrm{Kimura}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
(4\pi G)^{-3/2} (4\pi) H^2 p_*^{2\sigma - 1/2} =
(4\pi G^3)^{-1/2} K_n^{2n/(n+1)} p_*^{(3-n)/[2(n+1)]} \, .
</math>
  </td>
</tr>
</table>
</div>

In order to compare Kimura's equilibrium expressions  for <math>~R_\mathrm{eq}</math> and <math>~P_e</math> with the corresponding expressions presented by [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|Horedt]] and by [[SSC/Structure/PolytropesEmbedded#Whitworth.27s_Presentation|Whitworth]], we need to replace <math>~p_*</math> by <math>~M</math> in both expressions.  Inverting Kimura's expression for <math>~M</math>, we have,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~p_*^{(3-n)/[2(n+1)]}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
M (n+1)^{-3/2}( - \tilde\xi^2 \tilde\theta^' )^{-1} (4\pi G^3)^{1/2} K_n^{-2n/(n+1)} \, .
</math>
  </td>
</tr>
</table>
</div>
Hence,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~P_\mathrm{Kimura}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
[ M (n+1)^{-3/2}( - \tilde\xi^2 \tilde\theta^' )^{-1} (4\pi G^3)^{1/2} K_n^{-2n/(n+1)} ]^{2(n+1)/(3-n)}
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
[ M^{-2} (n+1)^{3}( - \tilde\xi^2 \tilde\theta^' )^{2} (4\pi G^3)^{-1} K_n^{4n/(n+1)} ]^{(n+1)/(n-3)} 
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~P_\mathrm{Horedt}
[ ( - \tilde\xi^2 \tilde\theta^' )^{2} ]^{(n+1)/(n-3)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~~ P_e</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~P_\mathrm{Horedt} ~\tilde\theta^{n+1}
( - \tilde\xi^2 \tilde\theta^' )^{2(n+1)/(n-3)} 
\, ,
</math>
  </td>
</tr>
</table>
</div>
which matches Horedt's expression for <math>~P_e</math>.  Also after replacement we obtain,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~R_\mathrm{Kimura}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
(4\pi G)^{-1/2} K^{n/(n+1)} [ M (n+1)^{-3/2}( - \tilde\xi^2 \tilde\theta^' )^{-1} (4\pi G^3)^{1/2} K_n^{-2n/(n+1)} ]^{(1-n)/(3-n)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
M^{(n-1)/(n-3)} ( - \tilde\xi^2 \tilde\theta^' )^{(1-n)/(n-3)} (n+1)^{3(1-n)/2(n-3)} 
(4\pi)^{[(1-n)-(3-n)]/[2(3-n)]} G^{[3(1-n)- (3-n)]/[2(3-n)]} [ K_n^{n(3-n)-2n(1-n)} ]^{1/[(n+1)(3-n)]}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
M^{(n-1)/(n-3)} ( - \tilde\xi^2 \tilde\theta^' )^{(1-n)/(n-3)} (n+1)^{3(1-n)/2(n-3)} 
(4\pi)^{1/(n-3)} G^{n/(n-3)} K_n^{-n/(n-3)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~~ R_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>\tilde\xi( - \tilde\xi^2 \tilde\theta^' )^{(1-n)/(n-3)} 
(n+1)^{[3(1-n)+(n-3)]/2(n-3)} 
\biggl[ 4\pi \biggl( \frac{G}{K_n} \biggr)^{n} M^{(n-1)}  \biggr]^{1/(n-3)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math> 
R_\mathrm{Horedt}~ \tilde\xi( - \tilde\xi^2 \tilde\theta^' )^{(1-n)/(n-3)} \, ,
</math>
  </td>
</tr>
</table>
</div>
which exactly matches Horedt's expression for <math>~R_\mathrm{eq}</math>.