Editing SSC/Structure/Polytropes/Analytic (section)

====Primary E-Type Solution====
To derive the radial distribution of the Lane-Emden function <math>\Theta_H(r)</math> for an {{Math/MP_PolytropicIndex}} = 5 polytrope, we must solve,
<div align="center">
<math>\frac{1}{\xi^2} \frac{d}{d\xi}\biggl( \xi^2 \frac{d\Theta_H}{d\xi} \biggr) = - (\Theta_H)^5</math> ,
</div>
subject to the above-specified [[SSC/Structure/Polytropes#Boundary_Conditions|boundary conditions]].  Following [https://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden (1907)], [[Appendix/References#C67|[<font color="red">C67</font>] ]] (pp. 93-94) shows that by making the substitutions,
<div align="center">
<math>
\xi = \frac{1}{x} = e^{-t} \, ; ~~~~~\Theta_H = \biggl(\frac{x}{2}\biggr)^{1/2} z = \biggl(\frac{1}{2}e^t\biggr)^{1/2}z \, ,
</math>
</div>
the differential equation can be rewritten as,
<div align="center">
<math>
\frac{d^2 z}{dt^2} = \frac{1}{4}z (1 - z^4) \, .
</math>
</div>
This equation has the solution,
<div align="center">
<math>
z = \pm \biggl[ \frac{12 C e^{-2t}}{(1 + C e^{-2t})^2} \biggr]^{1/4} \, ,
</math>
</div>
that is,
<div align="center">
<math>
\Theta_H = \biggl[ \frac{3 C }{(1 + C \xi^2)^2} \biggr]^{1/4} \, .
</math>
</div>
where <math>C</math> is an integration constant.  Because <math>\Theta_H</math> must go to unity when <math>\xi = 0</math>, we see that <math>C=1/3</math>.  Hence,
<div align="center">
<math>
\Theta_H = \biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-1/2} \, .
</math>
</div>

From this Lane-Emden function solution, we obtain,
<div align="center">
<math>
\frac{\rho}{\rho_c} = \Theta_H^5 = \biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-5/2} \, ,
</math>
</div>
and,
<div align="center">
<math>
\frac{P}{P_c} = \biggr(\frac{\rho}{\rho_c}\biggr)^{6/5} = \biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-3} \, .
</math>
</div>
Notice that, for this polytropic structure, the density and pressure don't go to zero until <math>\xi \rightarrow \infty</math>.  Hence, <math>\xi_1 = \infty</math>. However, the radial scale length,
<div align="center">
<math>
a_5 = \biggr[ \frac{1}{4\pi G} \biggl( \frac{H_c}{\rho_c} \biggr) \biggr]^{1/2} = 
\biggr[ \frac{(n+1)K}{4\pi G} \rho_c^{(1/n - 1)} \biggr]^{1/2} =
\biggr[ \frac{3K}{2\pi G} \biggr]^{1/2}  \rho_c^{-2/5}  \, .
</math>
</div>
Hence, 
<div align="center">
<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
M_r(\xi)
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4\pi \rho_c a_5^3 \int_0^\xi \xi^2 \biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-5/2} d\xi  
</math>
  </td>  <td rowspan="3">
[[Image:WolframN5polytropeMass.jpg|border|240px|right]] 
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>  
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4\pi \biggr[ \frac{3K}{2\pi G} \biggr]^{3/2}  \rho_c^{-1/5} ~\biggl\{ \frac{\sqrt{3} \xi^3}{(3 + \xi^2)^{3/2}} \biggr\}
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>  
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2}  \rho_c^{-1/5} ~\{ \xi^3 ( 3 + \xi^2 )^{-3/2} \} \, .
</math>
  </td>
</tr>
</table>
</div>


<table border="1" align="center" width="80%"  cellpadding="5"><tr><td align="left" bgcolor="lightgreen">
<div align="center">'''n = 5 Polytrope'''</div>
Let's verify the expression for the pressure by integrating the hydrostatic-balance equation, 
<div align="center">
{{Math/EQ_SShydrostaticBalance01}} 
</div>
From our [[SSC/Structure/Polytropes|introductory discussion]] of the 
<div align="center">
<span id="LaneEmdenEquation"><font color="#770000">'''Lane-Emden Equation'''</font></span>
<br />
{{Math/EQ_SSLaneEmden01}}
</div>
we appreciate that, for a <math>n=5</math> polytrope,

<div align="center">
<math>
\rho = \rho_c \Theta_H^5 = \rho_c \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \, ,
</math>
</div>
and,
<div align="center">
<math>
r = \biggl( \frac{3K_5}{2\pi G}\biggr)^{1 / 2}\rho_c^{-2/5} \xi
</math>.
</div>
Combining these expressions with our  above-derived expression for <math>M_r</math>, namely,
<table align="center" border="0" cellpadding="5">

<tr>
  <td align="right">
<math>
~M_r(\xi)
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggr( \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr)^{1/2}  \rho_c^{-1/5} ~\{ \xi^3 ( 3 + \xi^2 )^{-3/2} \} \, ,
</math>
  </td>
</tr>
</table>

the RHS of the hydrostatic-balance relation can be written as,
<table align="center" cellpadding="8">

<tr>
  <td align="right"><math>\mathrm{RHS} = - G \biggl[M_r\biggr]~\biggl[\rho\biggr]~\biggl[ r \biggr]^{-2}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- G \biggl[\biggr( \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr)^{1/2}  \rho_c^{-1/5} ~\{ \xi^3 ( 3 + \xi^2 )^{-3/2} \}\biggr]
~\biggl[ \rho_c \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \biggr]
~\biggl[ \biggl( \frac{3K_5}{2\pi G}\biggr)^{1 / 2}\rho_c^{-2/5} \xi \biggr]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- G ~3^{5/2}\biggr( \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr)^{1/2}  \rho_c^{-1/5} \rho_c \biggl[ \biggl( \frac{2\pi G}{3K_5}\biggr)\rho_c^{4/5} \biggr]
\biggl[~ \xi ( 3 + \xi^2 )^{-4} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- ~\biggr( 2^3\cdot 3^7\pi \cdot G K  \biggr)^{1/2}  \rho_c^{8/5} 
\biggl[~ \xi ( 3 + \xi^2 )^{-4} \biggr]
</math>
  </td>
</tr>

</table>

Now, let's integrate the hydrostatic-balance equation:

<table align="center" cellpadding="8">
<tr>
  <td align="right"><math>\int_{P_c}^{P}dP</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- ~\biggr( 2^3\cdot 3^7\pi \cdot G K  \biggr)^{1/2}  \rho_c^{8/5} \biggl( \frac{3K_5}{2\pi G}\biggr)^{1 / 2}\rho_c^{-2/5}
\int_0^\xi \biggl[~ \xi ( 3 + \xi^2 )^{-4} \biggr] d\xi
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ P - P_c</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2\cdot 3^4 \cdot K   \rho_c^{6/5} 
\biggl\{
\frac{1}{ 6(3+\xi^2)^3 }
\biggr\}_0^\xi
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ P </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
P_c + 
3^3 \cdot K   \rho_c^{6/5} 
\biggl[
\frac{1}{ (3+\xi^2)^3 }
-\frac{1}{ (3)^3 }
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
P_c + 
K   \rho_c^{6/5} 
\biggl[(1+\xi^2/3)^{-3} -1 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
P_c \biggl(1+\frac{\xi^2}{3} \biggr)^{-3} \, ,
</math>
  </td>
</tr>

</table>
where, <math>P_c = K\rho_c^{6/5}</math>.
</td></tr></table>
The function of <math>\xi</math> inside the curly brackets of this last expression goes to unity as <math>\xi \rightarrow \infty</math>, so the integrated mass is finite even though the configuration extends to infinity.  Specifically, the total mass is,
<div align="center">
<math>M = \biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2}  \rho_c^{-1/5}  \, .</math>
</div>

We can invert this formula to obtain an expression for <math>K</math> in terms of <math>M</math> and <math>\rho_c</math>, namely,
<div align="center">
<math>
K = \biggr[ \frac{\pi M^2 G^3}{2\cdot 3^4} \biggr]^{1/3}  \rho_c^{2/15}  \, .
</math>
</div>
This, in turn, means that the central pressure,
<div align="center">
<math>
P_c = K\rho_c^{6/5} = \biggr[ \frac{\pi M^2 G^3}{2\cdot 3^4} \biggr]^{1/3}  \rho_c^{4/3}  \, ,
</math>
</div>
and,
<div align="center">
<math>
H_c = \frac{6P_c}{\rho_c} = \biggr[ \frac{2^2 \pi M^2 G^3}{3} \biggr]^{1/3}  \rho_c^{1/3}  \, .
</math>
</div>


<table border="1" cellpadding="8" align="right">
<tr>
  <th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/NewN5.xlsx --- worksheet = AnalyticMR]]Figure 2: &nbsp; Mass vs. Radius <br />for n = 5 polytrope</th>
</tr>
<tr>
  <td align="center">[[File:N500SequenceB.png|300px|n = 5 mass vs. radius diagram]]</td>
</tr>
</table>
For the purposes of comparing the internal structure of configurations having different polytropic indexes &#8212; see, for example [[#Fig4|Figure 4, below]] &#8212; we have found it useful in each case to graphically illustrate how the normalized mass, <math>~M/M_\mathrm{SWS}</math>, varies with the normalized radius, <math>~R/R_\mathrm{SWS}</math>, where the definition of these two functions is drawn from an [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|accompanying discussion of pressure-truncated polytropic configurations]]. In the case of an <math>~n=5</math> polytrope, both functions are expressible analytically; specifically, we have, 
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~\frac{R}{R_\mathrm{SWS}}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{5}{4\pi} \biggr)^{1/2} \biggl[ \xi \theta^{2}  \biggr]_{n=5}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{5}{4\pi} \biggr)^{1/2} \xi \biggl[1 + \frac{\xi^2}{3}  \biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{5}{4\pi} \biggr)^{1/2} \biggl[\frac{3\xi}{3 + \xi^2}  \biggr] \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\frac{M}{M_\mathrm{SWS}}
</math>
  </td>
  <td align="center">
<math>~\equiv~</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{5^3}{4\pi} \biggr)^{1/2} \biggl[\theta \xi^2 
\biggl| \frac{d\theta}{d\xi} \biggr| ~\biggr]_{n=5}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\equiv~</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{5^3}{4\pi} \biggr)^{1/2} \biggl\{\xi^2 \biggl[1 + \frac{\xi^2}{3}\biggr]^{-1 / 2} 
\frac{\xi}{3} \biggl[1 + \frac{\xi^2}{3}\biggr]^{-3 / 2}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\equiv~</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{5^3}{4\pi} \biggr)^{1/2} \frac{3\xi^3}{(3 + \xi^2)^2}\, .
</math>
  </td>
</tr>
</table>
</div>
As [[SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|Stahler has pointed out]], for an <math>~n = 5</math> polytrope, this mass-radius relation can also be precisely couched in the form of a quadratic equation, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{M}{M_\mathrm{SWS}} \biggr)^2 - 5 \biggl( \frac{M}{M_\mathrm{SWS}} \biggr)\biggl( \frac{R}{R_\mathrm{SWS}} \biggr)
+ \frac{2^2 \cdot 5 \pi}{3} \biggl( \frac{R}{R_\mathrm{SWS}} \biggr)^4
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{M}{M_\mathrm{SWS}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{5}{2} \biggl( \frac{R}{R_\mathrm{SWS}} \biggr) \biggl[ 1 \pm \sqrt{1-  \frac{2^4 \pi}{3\cdot 5}\biggl( \frac{R}{R_\mathrm{SWS}} \biggr)^2} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

As Figure 2 illustrates, this mass-radius relationship exhibits two turning points: &nbsp; 
The maximum radius occurs at coordinate location,
<div align="center">
<math>~\biggl[ \frac{R}{R_\mathrm{SWS}}, \frac{M}{M_\mathrm{SWS}}\biggr]_\mathrm{R\_turn} = \biggl[ \biggl( \frac{3\cdot 5}{2^4 \pi} \biggr)^{1 / 2}, \biggl( \frac{3\cdot 5^3}{2^6 \pi} \biggr)^{1 / 2} \biggr] 
\approx \biggl[ 0.5462742,  1.3656855\biggr] \, ;</math>
</div>
and the maximum mass occurs at coordinate location,
<div align="center">
<math>~\biggl[ \frac{R}{R_\mathrm{SWS}}, \frac{M}{M_\mathrm{SWS}}\biggr]_\mathrm{M\_turn} = \biggl[ \biggl( \frac{3^2\cdot 5}{2^6 \pi} \biggr)^{1 / 2}, \biggl( \frac{3^4\cdot 5^3}{2^{10} \pi} \biggr)^{1 / 2} \biggr] 
\approx \biggl[ 0.4730873,  1.7740776\biggr] \, .</math>
</div>