View source for User:Tohline/SSC/Structure/Polytropes

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=Polytropic Spheres=
{{LSU_HBook_header}}
[[Image:LSU_Structure_still.gif|74px|left]]
Here we will supplement the [[User:Tohline/SphericallySymmetricConfigurations/PGE|simplified set of principal governing equations]] with a polytropic equation of state, as defined in our overview of [[User:Tohline/SR#Time-Independent|Problems supplemental relations for time-independent problems]].  Specifically, we will assume that {{User:Tohline/Math/VAR_Density01}} is related to {{User:Tohline/Math/VAR_Enthalpy01}} through the relation, 


<div align="center">
{{User:Tohline/Math/EQ_Polytrope03}}
</div>

It will be useful to note as well that, for any polytropic gas, the three key state variables are always related to one another through the simple expression,
<div align="center"><font size="+1">
<math> ~(n+1) P = H\rho</math> .
</font></div>


==Governing Relations==

===Lane-Emden Equation===

Adopting [[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#Technique_2|solution technique #2]], we need to solve the following second-order ODE relating the two unknown functions, {{User:Tohline/Math/VAR_Density01}} and {{User:Tohline/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 {{User:Tohline/Math/VAR_Enthalpy01}} and {{User:Tohline/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 {{User:Tohline/Math/VAR_Enthalpy01}} with <math>~H_c \Theta_H</math> on the left-hand-side of the governing differential equation and we replace {{User:Tohline/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 />
{{User:Tohline/Math/EQ_SSLaneEmden01}}
</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, {{User:Tohline/Math/MP_PolytropicIndex}}.


<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 [[User:Tohline/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 [[User:Tohline/SphericallySymmetricConfigurations/Virial#Structural_Form_Factors|accompanying discussion of structural form factors]] &#8212; see, also, [[User:Tohline/SSC/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 [[User:Tohline/SSC/FreeEnergy/PowerPoint#Case_M_Equilibrium_Conditions|accompanying summary supporting a PowerPoint presentation]].

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

===Boundary Conditions===

Given that it is a <math>2^\mathrm{nd}</math>-order ODE, a solution of the Lane-Emden equation will require specification of two boundary conditions.  Based on our definition of the variable <math>~\Theta_H</math>, one obvious boundary condition is to demand that <math>~\Theta_H = 1</math> at the center <math>~(\xi=0)</math> of the configuration.  In astrophysically interesting structures, we also expect the first derivative of many physical variables to go smoothly to zero at the center of the configuration &#8212; see, for example, the radial behavior that was derived for {{User:Tohline/Math/VAR_Pressure01}}, {{User:Tohline/Math/VAR_Enthalpy01}}, and {{User:Tohline/Math/VAR_NewtonianPotential01}} in a [[User:Tohline/SSC/Structure/UniformDensity#Summary|uniform-density sphere]].  Hence, we will seek solutions to the Lane-Emden equation where <math>~d\Theta_H /d\xi = 0</math> at <math>~\xi=0</math> as well.

==Known Analytic Solutions==
While the Lane-Emden equation has been studied for over 100 years, to date, analytic solutions to the equation (subject to the above specified boundary conditions) have been found only for three values of the polytropic index, {{User:Tohline/Math/MP_PolytropicIndex}}.  We will review these three solutions here.

===n = 0 Polytrope===
When the polytropic index, {{User:Tohline/Math/MP_PolytropicIndex}}, is set equal to zero, the right-hand-side of the Lane-Emden equation becomes a constant <math>~(-1)</math>, so the equation can be straightforwardly integrated, twice, to obtain the desired solution for <math>~\Theta_H(\xi)</math>.  Specifically, the first integration along with enforcement of the boundary condition on <math>~d\Theta_H/d\xi</math> at the center gives,

<div align="center">
<math>
\xi^2 \frac{d\Theta_H}{d\xi} = - \frac{1}{3}\xi^3 .
</math>
</div>

Then the second integration along with enforcement of the boundary condition on <math>~\Theta_H</math> at the center gives,

<div align="center">
<math>
~\Theta_H = 1 - \frac{1}{6}\xi^2 .
</math>
</div>
This function varies smoothly from unity at <math>~\xi = 0</math> (as required by one of the boundary conditions) to zero at <math>~\xi = \xi_1 = \sqrt{6}</math> (by tradition, the subscript "1" is used to indicate that it is the "first" zero of the Lane-Emden function), then becomes negative for values of <math>~\xi > \xi_1</math>.  

The astrophysically interesting surface of this spherical configuration is identified with the first zero of the function, that is, where the dimensionless enthalpy first goes to zero.  In other words, the dimensionless radius <math>~\xi_1</math> should correspond with the dimensional radius of the configuration, <math>~R</math>.  From the definition of <math>~\xi</math>, we therefore conclude that,

<div align="center">
<math>~
a_{n=0} = \frac{R}{\xi_1} = \frac{R}{\sqrt{6}} ,
</math>
</div>
and

<div align="center">
<math>
\xi = \sqrt{6} \biggl(\frac{r}{R} \biggr) ,
</math>
</div>
Hence, the Lane-Emden function solution can also be written as,

<div align="center">
<math>
\Theta_H = \frac{H}{H_c} = 1 - \biggl(\frac{r}{R}\biggr)^2 .
</math>
</div>
Since,
<div align="center">
<math>
a_{n=0}^2 = \frac{1}{4\pi G} \biggl(\frac{H_c}{\rho_c}\biggr) = \frac{R^2}{6} , 
</math>
</div>

we also conclude that,
<div align="center">
<math>~
H_c = \frac{2\pi G}{3} \rho_c R^2 .
</math>
</div>
This, combined with the Lane-Emden function solution, tells us that the run of enthalpy through the configuration is,
<div align="center">
<math>~
H(r) = \frac{2\pi G}{3} \rho_c R^2 \biggl[ 1 - \biggl(\frac{r}{R}\biggr)^2 \biggr].
</math>
</div>

Now, it is always true for polytropic structures &#8212; see, for example, expressions at the top of this page of discussion &#8212; that {{User:Tohline/Math/VAR_Density01}} can be related to {{User:Tohline/Math/VAR_Enthalpy01}} through the expression,
<div align="center">
<math>~
\biggl( \frac{\rho}{\rho_c} \biggr) = \biggl( \frac{H}{H_c} \biggr)^n = \Theta_H^n .
</math>
</div>
Hence, for the specific case of an {{User:Tohline/Math/MP_PolytropicIndex}} = 0 polytrope, we deduce that
<div align="center">
<math>~
\frac{\rho}{\rho_c} = 1 .
</math>
</div>
This means that an {{User:Tohline/Math/MP_PolytropicIndex}} = 0 polytropic sphere is also a uniform-density sphere.  It should come as no surprise to discover, therefore, that the functional behavior of {{User:Tohline/Math/VAR_Enthalpy01}}<math>(r)~</math> we have derived for the {{User:Tohline/Math/MP_PolytropicIndex}} = 0 polytrope is identical to the {{User:Tohline/Math/VAR_Enthalpy01}}<math>(r)~</math> function that we have [[User:Tohline/SSC/Structure/UniformDensity|derived elsewhere for uniform-density spheres]].  All of the other [[User:Tohline/SSC/Structure/UniformDensity#Summary|summarized properties of uniform-density spheres]] can therefore also be assigned as properties of {{User:Tohline/Math/MP_PolytropicIndex}} = 0 polytropes.

===n = 1 Polytrope===
====Primary E-Type Solution====
When the polytropic index, {{User:Tohline/Math/MP_PolytropicIndex}}, is set equal to unity, the Lane-Emden equation takes the form of an inhomogeneous, <math>2^\mathrm{nd}</math>-order ODE that is linear in the unknown function, <math>~\Theta_H</math>.  Specifically, to derive the radial distribution of the Lane-Emden function <math>~\Theta_H(r)</math> for an {{User:Tohline/Math/MP_PolytropicIndex}} = 1 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</math> ,
</div>
subject to the above-specified [[User:Tohline/SSC/Structure/Polytropes#Boundary_Conditions|boundary conditions]].  If we multiply this equation through by <math>~\xi^2</math> and move all the terms to the left-hand-side, we see that the governing ODE takes the form,
<div align="center">
<math>~\xi^2 \frac{d^2\Theta_H}{d\xi^2} + 2\xi \frac{d\Theta_H}{d\xi}  + \xi^2 \Theta_H = 0 \, ,</math>
</div>
which is a relatively familiar <math>2^\mathrm{nd}</math>-order ODE (the [http://mathworld.wolfram.com/SphericalBesselFunction.html ''spherical Bessel differential equation'']) whose general solution involves a linear combination of the [http://en.wikipedia.org/wiki/Bessel_function order zero spherical Bessel functions] of the first and second kind, respectively,
<div align="center">
<math>
~j_0(\xi) = \frac{\sin\xi}{\xi} ,
</math>
</div>
and,
<div align="center">
<math>
~y_0(\xi) = - \frac{\cos\xi}{\xi} .
</math>
</div>
Given the boundary conditions that have been imposed on our astrophysical problem, we can rule out any contribution from the <math>~y_0</math> function.  The desired solution is,
<div align="center">
<math>
\Theta_H(\xi) = j_0(\xi) = \frac{\sin\xi}{\xi} .
</math>
</div>
This function is also referred to as the (unnormalized) [http://en.wikipedia.org/wiki/Sinc_function sinc function].

<table border="1" align="center" cellpadding="8" width="70%">
<tr>
  <th align="center" bgcolor="yellow">
LaTeX mathematical expressions cut-and-pasted directly from
<br />
NIST's ''Digital Library of Mathematical Functions''
  </th>
</tr>
<tr>
  <td align="left">
As an additional point of reference, note that according to [http://dlmf.nist.gov/10.47 &sect;10.47 of NIST's ''Digital Library of Mathematical Functions''], a Spherical Bessel Function is the solution to the 2<sup>nd</sup>-order ODE,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
z^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+2z\frac{\mathrm{d}w}{\mathrm{d}z}+\left(z^{2}-m(m+1)\right)w
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0.</math>
  </td>
</tr>
</table>

This is our governing ODE if we set the parameter, <math>~m\rightarrow 0</math>, in which case, according to [http://dlmf.nist.gov/10.49 &sect;10.49 of NIST's ''Digital Library of Mathematical Functions''], the solutions are,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\mathsf{j}_{0}\left(z\right)
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\sin z}{z},</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\mathsf{y}_{0}\left(z\right)
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\frac{\cos z}{z}.</math>
  </td>
</tr>
</table>

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


Because, by definition, <math>~H/H_c = \Theta_H</math>, and for an {{User:Tohline/Math/MP_PolytropicIndex}} = 1 polytrope <math>~\rho/\rho_c = H/H_c</math>, we can immediately conclude from this Lane-Emden function solution that,
<div align="center">
<math>
~\frac{\rho(\xi)}{\rho_c} = \frac{H(\xi)}{H_c} = \frac{\sin\xi}{\xi} .
</math>
</div>
Furthermore, because the relation ({{User:Tohline/Math/MP_PolytropicIndex}} + 1){{User:Tohline/Math/VAR_Pressure01}} = {{User:Tohline/Math/VAR_Enthalpy01}}{{User:Tohline/Math/VAR_Density01}} holds for all polytropic gases, we conclude that the pressure distribution inside an {{User:Tohline/Math/MP_PolytropicIndex}} = 1 polytrope is,
<div align="center">
<math>
~\frac{P(\xi)}{P_c} = \biggl( \frac{\sin\xi}{\xi} \biggr)^2 .
</math>
</div>
The functions {{User:Tohline/Math/VAR_Pressure01}}<math>(\xi)~</math>, {{User:Tohline/Math/VAR_Enthalpy01}}<math>(\xi)~</math>, and {{User:Tohline/Math/VAR_Density01}}<math>(\xi)~</math> all first drop to zero when <math>~\xi = \pi</math>.  Hence, for an {{User:Tohline/Math/MP_PolytropicIndex}} = 1 polytrope, <math>~\xi_1 = \pi</math> and, in terms of the configuration's radius, <math>~R</math>, the polytropic scale length is,
<div align="center">
<math>
~a_{n=1} = \frac{R}{\xi_1} = \frac{R}{\pi} .
</math>
</div>
So, throughout the configuration, we can relate <math>~\xi</math> to the dimensional spherical coordinate <math>~r</math> through the relation,
<div align="center">
<math>
~\xi = \pi \biggl(\frac{r}{R}\biggr) ;
</math>
</div>
and, from the general definition of <math>~a_n</math>, the central value of {{User:Tohline/Math/VAR_Enthalpy01}} can be expressed in terms of <math>~R</math> and <math>~\rho_c</math> via the relation,
<div align="center">
<math>
~H_c = \frac{4G}{\pi}\rho_c R^2 .
</math>
</div>
Again because the relation ({{User:Tohline/Math/MP_PolytropicIndex}} + 1){{User:Tohline/Math/VAR_Pressure01}} = {{User:Tohline/Math/VAR_Enthalpy01}}{{User:Tohline/Math/VAR_Density01}} must hold everywhere inside a polytrope, this means that the central pressure is given by the expression,
<div align="center">
<math>
~P_c = \frac{2G}{\pi}\rho_c^2 R^2 .
</math>
</div>

Given the radial distribution of {{User:Tohline/Math/VAR_Density01}}, we can determine the functional behavior of the integrated mass.  Specifically, 
<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>
~\int_0^r 4\pi r^2 \rho~ dr  
</math>
  </td>  <td rowspan="3">
[[Image:WolframN1polytropeMass.jpg|border|240px|right]] 
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>  
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~4\pi \rho_c \biggl(\frac{R}{\pi}\biggr)^3 \int_0^\xi \xi\sin\xi ~d\xi
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>  
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{4}{\pi^2} \rho_c R^3 [ \sin\xi - \xi\cos\xi  ]  \, .
</math>
  </td>
</tr>
</table>
</div>
Because <math>~\xi = \pi</math> at the surface of this spherical configuration &#8212; in which case the term inside the square brackets is <math>~\pi</math> &#8212; we conclude as well that the total mass of the configuration is,
<div align="center">
<math>
~M = \frac{4}{\pi}\rho_c R^3 .
</math>
</div>

====Summary====

From the above derivations, we can describe the properties of a spherical {{User:Tohline/Math/MP_PolytropicIndex}} = 1 polytrope as follows:
* <font color="red">Mass</font>:  
: Given the density, <math>\rho_c</math>, and the radius, <math>R</math>, of the configuration, the total mass is,
<div align="center">
<math>M = \frac{4}{\pi} \rho_c R^3 </math> ;
</div>

: and, expressed as a function of <math>M</math>, the mass that lies interior to radius <math>r</math> is,
<div align="center">
<math>\frac{M_r}{M} = \frac{1}{\pi} \biggl[ \sin\biggl(\frac{\pi r}{R} \biggr) - \biggl(\frac{\pi r}{R} \biggr)\cos\biggl(\frac{\pi r}{R} \biggr) \biggr]</math> .
</div>

* <font color="red">Pressure</font>: 
: Given values for the pair of model parameters <math>( \rho_c , R )</math>, or <math>( M , R )</math>, or <math>( \rho_c , M )</math>, the central pressure of the configuration is,
<div align="center">
<math>P_c = \frac{2 G}{\pi} \rho_c^2 R^2 = \frac{\pi G}{8}\biggl( \frac{M^2}{R^4} \biggr) = \biggl[ \frac{1}{2\pi} G^3 \rho_c^4 M^2 \biggr]^{1/3}</math> ; 
</div>

: and, expressed in terms of the central pressure <math>P_c</math>, the variation with radius of the pressure is,
<div align="center">
<math>P(r)= P_c \biggl[\frac{R}{\pi r} \sin\biggl(\frac{\pi r}{R}\biggr)  \biggr]^2</math> .
</div>

* <font color="red">Enthalpy</font>: 
: Throughout the configuration, the enthalpy is given by the relation,
<div align="center">
<math>H(r) = \frac{2 P(r)}{ \rho(r)} = \frac{GM}{R} \biggl[\frac{R}{\pi r} \sin\biggl(\frac{\pi r}{R}\biggr)  \biggr]</math> .
</div>

* <font color="red">Gravitational potential</font>: 
: Throughout the configuration &#8212; that is, for all <math>r \leq R</math> &#8212; the gravitational potential is given by the relation,
<div align="center">
<math>\Phi_\mathrm{surf} - \Phi(r) = H(r) = \frac{GM}{R} \biggl[\frac{R}{\pi r} \sin\biggl(\frac{\pi r}{R}\biggr)  \biggr] </math> .
</div>
: Outside of this spherical configuration&#8212; that is, for all <math>r \geq R</math> &#8212;  the potential should behave like a point mass potential, that is,
<div align="center">
<math>\Phi(r) = - \frac{GM}{r} </math> .
</div>
: Matching these two expressions at the surface of the configuration, that is, setting <math>\Phi_\mathrm{surf} = - GM/R</math>, we have what is generally considered the properly normalized prescription for the gravitational potential inside a spherically symmetric, {{User:Tohline/Math/MP_PolytropicIndex}} = 1 polytropic configuration:
<div align="center">
<math>\Phi(r) = - \frac{G M}{R} \biggl\{ 1 + \biggl[\frac{R}{\pi r} \sin\biggl(\frac{\pi r}{R}\biggr)  \biggr] \biggr\} </math> .
</div>

* <font color="red">Mass-Radius relationship</font>:
: We see that, for a given value of <math>\rho_c</math>, the relationship between the configuration's total mass and radius is,
<div align="center">
<math>M \propto R^3  ~~~~~\mathrm{or}~~~~~R \propto M^{1/3} </math> .
</div>

* <font color="red">Central- to Mean-Density Ratio</font>:
: The ratio of the configuration's central density to its mean density is,
<div align="center">
<math>\frac{\rho_c}{\bar{\rho}} = \biggl(\frac{\pi M}{4 R^3}  \biggr)\biggl(\frac{3 M}{4 \pi R^3}  \biggr) = \frac{\pi^2}{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/n1.xlsx --- worksheet = Sheet1]]Figure 1: &nbsp; Mass vs. Radius <br />for n = 1 polytrope</th>
</tr>
<tr>
  <td align="center">[[File:N100SequenceA.png|300px|n = 1 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 [[User:Tohline/SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|accompanying discussion of pressure-truncated polytropic configurations]]. In the case of an <math>~n=1</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}}\biggr|_{n=1}
</math>
  </td>
  <td align="center">
<math>~\equiv~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{1}{4\pi} \biggr)^{1/2} \xi   \, ;
</math>
  </td>
</tr>

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

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

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

As Figure 1 illustrates, this normalized mass increases monotonically with radius.  Given that the surface of the configuration is associated with the parameter value, <math>~\xi = \pi</math>, we recognize that, at the surface,
<math>~R/R_\mathrm{SWS} = \sqrt{\pi/4} \approx 0.8862269</math> and <math>~M/M_\mathrm{SWS}</math> formally climbs to infinity.


====Published n = 1 Tabulations====

<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="7">Published Tabulations of n = 1 Polytropic Structure (Primary E-Type Solution)</th>
</tr>
<tr>
  <td align="center" colspan="3">
Copied from p. 75 of [https://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden (1907)]
  </td>
  <td align="center" width="10%">&nbsp;</td>
  <td align="center" colspan="3">
Copied from p. 73 of [http://adsabs.harvard.edu/abs/2004ASSL..306.....H Horedt (2004)]
  </td>
</tr>
<tr>
  <td align="center"><math>~\mathfrak{r}_1</math></td>
  <td align="center"><math>~u_1</math></td>
  <td align="center"><math>~- \frac{du_1}{d\mathfrak{r}_1}</math></td>
  <td align="center">&nbsp;</td>
  <td align="center"><math>~\xi</math></td>
  <td align="center"><math>~\Theta_H</math></td>
  <td align="center"><math>~- \frac{d\Theta_H}{d\xi}</math></td>
</tr>
<tr>
  <td align="right"><math>~0</math></td>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~0</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~0</math></td>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~0</math></td>
</tr>
<tr>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~\tfrac{1}{10}</math></td>
  <td align="center"><math>~9.983342\mathrm{E-01}</math></td>
  <td align="center"><math>~3.330001\mathrm{E-02}</math></td>
</tr>
<tr>
  <td align="right"><math>~\tfrac{1}{4}</math></td>
  <td align="right"><math>~0.98960</math></td>
  <td align="right"><math>~0.08280</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~\tfrac{1}{2}</math></td>
  <td align="right"><math>~0.95882</math></td>
  <td align="right"><math>~0.16250</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~\tfrac{1}{2}</math></td>
  <td align="center"><math>~9.588511\mathrm{E-01}</math></td>
  <td align="center"><math>~1.625370\mathrm{E-01}</math></td>
</tr>
<tr>
  <td align="right"><math>~\tfrac{3}{4}</math></td>
  <td align="right"><math>~0.90886</math></td>
  <td align="right"><math>~0.23623</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~1</math></td>
  <td align="right"><math>~0.84148</math></td>
  <td align="right"><math>~0.30117</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~1</math></td>
  <td align="center"><math>~8.414710\mathrm{E-01}</math></td>
  <td align="center"><math>~3.011687\mathrm{E-01}</math></td>
</tr>
<tr>
  <td align="right"><math>~1 \tfrac{1}{4}</math></td>
  <td align="right"><math>~0.75918</math></td>
  <td align="right"><math>~0.35511</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~1\tfrac{1}{2}</math></td>
  <td align="right"><math>~0.66500</math></td>
  <td align="right"><math>~0.39622</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~2</math></td>
  <td align="right"><math>~0.45464</math></td>
  <td align="right"><math>~0.43541</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>2</math></td>
  <td align="center"><math>~4.546487\mathrm{E-01}</math></td>
  <td align="center"><math>~4.353978\mathrm{E-01}</math></td>
</tr>
<tr>
  <td align="right"><math>~2\tfrac{1}{2}</math></td>
  <td align="right"><math>~0.23938</math></td>
  <td align="right"><math>~0.41621</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~3</math></td>
  <td align="right"><math>~0.04703</math></td>
  <td align="right"><math>~0.34569</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~3</math></td>
  <td align="center"><math>~4.704000\mathrm{E-02}</math></td>
  <td align="center"><math>~3.3456775\mathrm{E-01}</math></td>
</tr>
<tr>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~3.140</math></td>
  <td align="center"><math>~5.072143\mathrm{E-04}</math></td>
  <td align="center"><math>~3.186325\mathrm{E-01}</math></td>
</tr>
<tr>
  <td align="right"><math>~\pi</math></td>
  <td align="right"><math>~0</math></td>
  <td align="right"><math>~0.31831</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~\pi</math></td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~3.183099\mathrm{E-01}</math></td>
</tr>
<tr>
  <td align="right"><math>~3\tfrac{1}{4}</math></td>
  <td align="right"><math>~-0.03330</math></td>
  <td align="right"><math>~0.29564</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
</table>

===n = 5 Polytrope===
====Primary E-Type Solution====
To derive the radial distribution of the Lane-Emden function <math>\Theta_H(r)</math> for an {{User:Tohline/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 [[User:Tohline/SSC/Structure/Polytropes#Boundary_Conditions|boundary conditions]].  Following [https://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden (1907)], [[User:Tohline/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>
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 [[User:Tohline/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 [[User:Tohline/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>

====Published n = 5 Tabulations====

<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="7">Published Tabulations of n = 5 Polytropic Structure (Primary E-Type Solution)</th>
</tr>
<tr>
  <td align="center" colspan="3">
Copied from p. 76 of [https://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden (1907)]
  </td>
  <td align="center" width="10%">&nbsp;</td>
  <td align="center" colspan="3">
Copied from p. 75 of [http://adsabs.harvard.edu/abs/2004ASSL..306.....H Horedt (2004)]
  </td>
</tr>
<tr>
  <td align="center"><math>~\mathfrak{r}_1</math></td>
  <td align="center"><math>~u_1</math></td>
  <td align="center"><math>~- \frac{du_1}{d\mathfrak{r}_1}</math></td>
  <td align="center">&nbsp;</td>
  <td align="center"><math>~\xi</math></td>
  <td align="center"><math>~\Theta_H</math></td>
  <td align="center"><math>~- \frac{d\Theta_H}{d\xi}</math></td>
</tr>
<tr>
  <td align="right"><math>~0</math></td>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~0</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~0</math></td>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~0</math></td>
</tr>
<tr>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~\tfrac{1}{10}</math></td>
  <td align="center"><math>~9.983375\mathrm{E-01}</math></td>
  <td align="center"><math>~3.316736\mathrm{E-02}</math></td>
</tr>
<tr>
  <td align="right"><math>~\tfrac{1}{4}</math></td>
  <td align="right"><math>~0.98974</math></td>
  <td align="right"><math>~0.08079</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~\tfrac{2}{4}</math></td>
  <td align="right"><math>~0.96078</math></td>
  <td align="right"><math>~0.14781</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~\tfrac{1}{2}</math></td>
  <td align="center"><math>~9.607689\mathrm{E-01}</math></td>
  <td align="center"><math>~1.478106\mathrm{E-01}</math></td>
</tr>
<tr>
  <td align="right"><math>~\tfrac{3}{4}</math></td>
  <td align="right"><math>~0.91768</math></td>
  <td align="right"><math>~0.19320</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~1</math></td>
  <td align="right"><math>~0.86602</math></td>
  <td align="right"><math>~0.21650</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~1</math></td>
  <td align="center"><math>~8.660254\mathrm{E-01}</math></td>
  <td align="center"><math>~2.165064\mathrm{E-01}</math></td>
</tr>
<tr>
  <td align="right"><math>~\tfrac{3}{2}</math></td>
  <td align="right"><math>~0.75593</math></td>
  <td align="right"><math>~0.21598</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~2</math></td>
  <td align="right"><math>~0.65465</math></td>
  <td align="right"><math>~0.18704</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~\tfrac{5}{2}</math></td>
  <td align="right"><math>~0.56950</math></td>
  <td align="right"><math>~0.15392</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~3</math></td>
  <td align="right"><math>~0.50000</math></td>
  <td align="right"><math>~0.12500</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~\tfrac{7}{2}</math></td>
  <td align="right"><math>~0.44353</math></td>
  <td align="right"><math>~0.10180</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~4</math></td>
  <td align="right"><math>~0.39736</math></td>
  <td align="right"><math>~0.08365</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~5</math></td>
  <td align="right"><math>~0.32733</math></td>
  <td align="right"><math>~0.05845</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~5</math></td>
  <td align="center"><math>~3.273268\mathrm{E-01}</math></td>
  <td align="center"><math>~5.845122\mathrm{E-02}</math></td>
</tr>
<tr>
  <td align="right"><math>~6</math></td>
  <td align="right"><math>~0.27735</math></td>
  <td align="right"><math>~0.04267</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~7</math></td>
  <td align="right"><math>~0.24020</math></td>
  <td align="right"><math>~0.03233</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~8</math></td>
  <td align="right"><math>~0.21160</math></td>
  <td align="right"><math>~0.02527</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~10</math></td>
  <td align="right"><math>~0.17066</math></td>
  <td align="right"><math>~0.01657</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~10</math></td>
  <td align="center"><math>~1.706640\mathrm{E-01}</math></td>
  <td align="center"><math>~1.656932\mathrm{E-02}</math></td>
</tr>
<tr>
  <td align="right"><math>~12</math></td>
  <td align="right"><math>~0.14286</math></td>
  <td align="right"><math>~0.01166</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~16</math></td>
  <td align="right"><math>~0.10763</math></td>
  <td align="right"><math>~0.00665</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~20</math></td>
  <td align="right"><math>~0.08628</math></td>
  <td align="right"><math>~0.00428</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~30</math></td>
  <td align="right"><math>~0.05764</math></td>
  <td align="right"><math>~0.00192</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~50</math></td>
  <td align="center"><math>~3.462025\mathrm{E-02}</math></td>
  <td align="center"><math>~6.915751\mathrm{E-04}</math></td>
</tr>
<tr>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~100</math></td>
  <td align="center"><math>~1.731791\mathrm{E-02}</math></td>
  <td align="center"><math>~1.731272\mathrm{E-04}</math></td>
</tr>
<tr>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~500</math></td>
  <td align="center"><math>~3.464081\mathrm{E-03}</math></td>
  <td align="center"><math>~6.928079\mathrm{E-06}</math></td>
</tr>
<tr>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~1000</math></td>
  <td align="center"><math>~1.732048\mathrm{E-03}</math></td>
  <td align="center"><math>~1.732043\mathrm{E-06}</math></td>
</tr>
<tr>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~\infty</math></td>
  <td align="center"><math>~0.000000\mathrm{E+00}</math></td>
  <td align="center"><math>~0.000000\mathrm{E+00}</math></td>
</tr>
</table>

====Srivastava's F-Type Solution====
=====Demonstration of Function's Validity=====

In a short paper, [http://adsabs.harvard.edu/abs/1962ApJ...136..680S S. Srivastava (1968, ApJ, 136, 680)] presents another, analytically prescribable solution to the Lane-Emden equation of index <math>~n = 5</math> that we will call upon in our [[User:Tohline/SSC/Structure/BiPolytropes/Analytic1_5#BiPolytrope_with_nc_.3D_1_and_ne_.3D_5|discussion of one category of bipolytropic configurations]].  Rather than repeat Srivastava's derivation here, we will simply specify his functional solution then demonstrate that it satisfies the Lane-Emden equation.  Srivastiva's Lane-Emden function is (see his equations 12 &amp; 13),

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\theta_{5F}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\sin[\ln(A\xi)^{1/2})]}{\xi^{1/2}\{3-2\sin^2[\ln(A\xi)^{1/2}]\}^{1/2}} \, ,</math>
  </td>
</tr>

</table>
</div>
where, <math>~A</math> is an arbitrary (positive) constant.  Adopting the shorthand notation,
<div align="center">
<math>\Delta \equiv \ln(A\xi)^{1/2}\, ,</math>
</div>
and, recognizing that,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\frac{d}{d\ln(A\xi)}\biggl[ \ln(A\xi)^{1/2} \biggr] = \frac{1}{2}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; <math>~\Rightarrow~</math>&nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\frac{d\Delta}{d\xi} = \frac{1}{2\xi} \, ,
</math>
  </td>
</tr>

</table>
</div>
the first derivative of Srivastava's Lane-Emden function is,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{d\theta_{5F}}{d\xi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\cos\Delta}{2\xi^{3/2}(3-2\sin^2\Delta)^{1/2}} 
- \frac{\sin\Delta}{2\xi^{3/2}(3-2\sin^2\Delta)^{1/2}} 
+ \frac{\sin^2\Delta \cos\Delta }{\xi^{3/2}(3-2\sin^2\Delta)^{3/2}} 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3\cos\Delta-3\sin\Delta + 2\sin^3\Delta }{2\xi^{3/2}(3-2\sin^2\Delta)^{3/2}}  \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, the left-hand-side of the,
<div align="center">
<span id="LaneEmdenEquation"><font color="#770000">'''Lane-Emden Equation'''</font></span>
<br />
{{User:Tohline/Math/EQ_SSLaneEmden01}}
</div>
is,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
LHS
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\xi^2}\frac{d}{d\xi}\biggl[
\frac{\xi^{1/2}(3\cos\Delta-3\sin\Delta + 2\sin^3\Delta )}{2(3-2\sin^2\Delta)^{3/2}} 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\xi^2}\biggl[
\frac{(3\cos\Delta-3\sin\Delta + 2\sin^3\Delta )}{4\xi^{1/2}(3-2\sin^2\Delta)^{3/2}} 
+ \frac{(-3\sin\Delta-3\cos\Delta + 6\sin^2\Delta \cos\Delta )}{4\xi^{1/2}(3-2\sin^2\Delta)^{3/2}} 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^{-2}\xi^{-5/2}(3-2\sin^2\Delta)^{-5/2}
[ (3-2\sin^2\Delta) (3\cos\Delta-3\sin\Delta + 2\sin^3\Delta ) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+ (3-2\sin^2\Delta) (-3\sin\Delta-3\cos\Delta + 6\sin^2\Delta \cos\Delta ) 
+ 6(3\cos\Delta-3\sin\Delta + 2\sin^3\Delta )\sin\Delta \cos\Delta ]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^{-2}\xi^{-5/2}(3-2\sin^2\Delta)^{-5/2}
[ (3-2\sin^2\Delta) (-6\sin\Delta + 2\sin^3\Delta + 6\sin^2\cos\Delta) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ + 6(3\cos\Delta-3\sin\Delta + 2\sin^3\Delta )\sin\Delta \cos\Delta ]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^{-2}\xi^{-5/2}(3-2\sin^2\Delta)^{-5/2}
[ -18\sin\Delta + 6\sin^3\Delta + 18\sin^2\cos\Delta + 12\sin^3\Delta - 4\sin^5\Delta -12\sin^4\cos\Delta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ + 18\sin\Delta \cos^2\Delta-18\sin^2\Delta \cos\Delta + 12\sin^4\Delta \cos\Delta ]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^{-2}\xi^{-5/2}(3-2\sin^2\Delta)^{-5/2}[ -4\sin^5\Delta  ]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \theta_{5F}^5 \, .
</math>
  </td>
</tr>
</table>
</div>
This demonstrates that Srivastava's function satisfies the Lane-Emden equation of index <math>~n=5</math>.

=====Function Properties=====
The function, <math>~\theta_{5F}</math>, looks like a damped oscillator with the following specific properties:
* As <math>~\xi</math> increases from zero, the function oscillates with an ever increasing period; the function goes through zero when <math>~\Delta = \pm \pi m</math> (m is an integer), that is, when <math>~(A\xi) = e^{\pm 2\pi m}</math>.
* The amplitude of the oscillation drops approximately as <math>~\xi^{-1/2}</math>.
* In an astrophysical context, the function can be used as a physically realistic representation of a spherical shell inside of a self-gravitating configuration only over the interval of a single oscillation for which <math>~\theta_{5F}</math> is  positive (ensuring that the mass density is everywhere positive) and, at the same time,  <math>~d\theta_{5F}/d\xi</math> is negative (ensuring that the density and pressure are a decreasing function of the radial coordinate).  In the following example, the astrophysically relevant segment of the function is identified with the parameter interval, <math>~\xi_\mathrm{crit} \le (A\xi) \le e^{2\pi}</math>.

=====Example Interval=====
As an example, let's set <math>~A=1</math> and examine the oscillation interval between <math>~m=0</math> and <math>~m=1</math>, that is, over the range, <math>~0 \le \Delta \le \pi</math> which corresponds to the parameter interval <math>~\xi = [1, e^{2\pi}]</math>.  The denominator of <math>~\theta_{5F}</math> is positive for all values of <math>~\xi</math> and, over this specified interval, the numerator of <math>~\theta_{5F}</math> is also always positive.  The blue curve in the following figure presents a plot of <math>~\theta_{5F}(x)</math> and the green curve presents a plot of the first derivative (the slope) of the function <math>~d\theta_{5F}(x)/d\xi</math> over the desired interval, where <math>~x \equiv \xi/e^{2\pi}</math>; note that the horizontal axis is shown in logarithmic units.

<div align="center" id="Fig3">
<table border="2" cellpadding="10">
<tr>
  <th align="center">
Figure 3: &nbsp; Segment of <math>~\theta_{5F}</math> Function <p></p> as derived by [http://adsabs.harvard.edu/abs/1962ApJ...136..680S S. Srivastava (1968, ApJ, 136, 680)]
  </th>
<tr>
  <td>
[[File:PlotTheta5F.png|450px|center|Srivastava's Lane-Emden function for n = 5]]
  </td>
</tr>
</table>
</div>

At both ends of the chosen parameter interval &#8212; that is, at <math>~\Delta = 0</math> and at <math>~\Delta = \pi</math> &#8212; the function <math>~\theta_{5F} = 0</math> and, correspondingly as depicted in the figure, the blue curve touches the horizontal axis.  At the beginning of the interval (<math>~\Delta =0</math>), the slope of the function and, correspondingly, the green curve, has the (positive) value,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{d\theta_{5F}}{d\xi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3\cos(0)-3\sin(0) + 2\sin^3(0) }{2\xi^{3/2}[3-2\sin^2(0)]^{3/2}}  
= \frac{3}{2(3^{3/2})} = (2^2 \cdot 3)^{-1/2} \approx 0.28868 
\, .
</math>
  </td>
</tr>
</table>
</div>
At the end of the interval (<math>~\Delta=\pi</math>), the slope of the function as well as the green curve, has the (negative) value,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{d\theta_{5F}}{d\xi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3\cos(\pi)-3\sin(\pi) + 2\sin^3(\pi) }{2\xi^{3/2}[3-2\sin^2(\pi)]^{3/2}}  
= \frac{-3}{2e^{3\pi}(3^{3/2})} = -e^{-3\pi} (2^2 \cdot 3)^{-1/2} \approx  -2.3296 \times 10^{-5} 
\, .
</math>
  </td>
</tr>
</table>
</div>

Over this interval, <math>~\theta_{5F}</math> reaches its maximum when the slope of the function is zero, that is, at the value of <math>~\Delta</math> where,

<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>~3\cos\Delta -3\sin\Delta +2(1-\cos^2\Delta)\sin\Delta</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3\cos\Delta -\sin\Delta -2\cos^2\Delta \sin\Delta</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3\cot\Delta -2\cos^2\Delta \, .</math>
  </td>
</tr>
</table>
</div>
Rewriting both of these trigonometric functions in terms of the tangent function and adopting the shorthand notation,
<div align="center">
<math>~y \equiv \tan\Delta \, ,</math>
</div>
this condition becomes,

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

<tr>
  <td align="right">
<math>~1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3}{y} -\frac{2}{1+y^2} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ y(y^2 + 1)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3(y^2+1) - 2y </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ y^3 - 3y^2 + 3y - 3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</div>

<div align="center" id="CubicRoot">
<table border="1" cellpadding="8" width="80%">
<tr>
  <td align="left">
<font color="red">'''ASIDE:'''</font>  As is well known and documented &#8212; see, for example [http://mathworld.wolfram.com/CubicFormula.html Wolfram MathWorld] or [http://en.wikipedia.org/wiki/Cubic_function Wikipedia's discussion] of the topic &#8212; the roots of any cubic equation can be determined analytically.  In order to evaluate the root(s) of our particular cubic equation, we have drawn from the utilitarian [http://www.math.vanderbilt.edu/~schectex/courses/cubic/ online summary provided by Eric Schechter at Vanderbilt University].  For a cubic equation of the general form,
<div align="center">
<math>~ay^3 + by^2 + cy + d = 0 \, ,</math>
</div> 
a real root is given by the expression,
<div align="center">
<math>~
y = p + \{q + [q^2 + (r-p^2)^3]^{1/2}\}^{1/3} + \{q - [q^2 + (r-p^2)^3]^{1/2}\}^{1/3}
\, ,</math>
</div> 
where,
<div align="center">
<math>~p \equiv -\frac{b}{3a} \, ,</math> &nbsp;&nbsp;&nbsp;&nbsp;
<math>~q \equiv \biggl[p^3 + \frac{bc-3ad}{6a^2} \biggr] \, ,</math>
&nbsp;&nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp; 
<math>~r=\frac{c}{3a} \, .</math>
</div>
In our particular case,
<div align="center">
<math>~a =1\, ,</math> &nbsp;&nbsp;&nbsp;&nbsp;
<math>~b =-3\, ,</math> &nbsp;&nbsp;&nbsp;&nbsp;
<math>~c = +3 \, ,</math>
&nbsp;&nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp; 
<math>~d = - 3 \, .</math>
</div>
[[File:WolframAlphaCubicSolver.png|thumbnail|right|100px|WolframAlpha]]Hence, interestingly enough,
<div align="center">
<math>~p = q = r = + 1 \, ,</math>
</div>
which implies that the real root is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 + \{2\}^{1/3} + \{0\}^{1/3} \, .</math>
  </td>
</tr>
</table>
</div>
(There is also a pair of imaginary roots, but they are irrelevant in the context of our overarching astrophysical discussion.)

Just for fun, we have also used WolframAlpha's online "cubic equation solver" widget to find the root(s) of our specific cubic equation.  Clicking on the thumbnail image provided here, on the right, displays the key result that was returned by this WolframAlpha widget.
  </td>
</tr>
</table>
</div>


The single, real root of this cubic equation is,
<div align="center">
<math>~y = 1 + 2^{1/3} \, ,</math>
</div>
which corresponds to,
<div align="center">
<math>~\Delta = \tan^{-1}(1 + 2^{1/3}) \, .</math>
</div>

[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 17 April 2015:  As far as I have been able to determine, this analytic prescription of xi_crit has not previously been derived, although, as is made clear in what follows, Murphy (1983) has assessed its value numerically to six significant digits.]]Hence, over this example interval, the maximum of Srivastava's <math>~\theta_{5F}</math> function &#8212; and, hence also, the location at which the function's slope transitions from positive to negative values (denoted by the vertical red line in the above figure) &#8212; occurs at,
<div align="center">
<math>\xi_\mathrm{crit} \equiv e^{2\tan^{-1}(1+2^{1/3})} = 10.05836783\, .</math>
</div>
The corresponding value of the function at this critical radial location is,
<div align="center">
<math>\theta_{5F}|_\mathrm{max} \equiv \theta_{5F}(\xi_\mathrm{crit}) = (1+2^{1/3})[3 + (1+2^{1/3})^2]^{-1/2} e^{-\tan^{-1}(1+2^{1/3})} = 0.250260848 \, .</math>
</div>
This agrees precisely with the determination made by [http://adsabs.harvard.edu/abs/1983PASAu...5..175M J. O. Murphy (1983)] &#8212; see the excerpts from his paper displayed in the following boxed-in image &#8212; that the portion of the <math>~\theta_{5F}</math> function that falls in the interval <math>~1 \le (A\xi) < \xi_\mathrm{crit}</math> (the segment of the blue curve that lies to the left of the vertical red line in the above figure) is unphysical because the slope of the function is positive throughout that interval.  
<!--
<table border="1" cellpadding="10" align="center">
<tr>
  <th align="center">Excerpts (edited) from 
[http://adsabs.harvard.edu/abs/1983PASAu...5..175M J. O. Murphy (1983, Proc. Astr. Soc. Australia, 5, 175)]
  </th>
<tr>
  <td>
[[File:Murphy1983Extremum02.png|600px|center|Murphy's (1983) examination of Srivastava's function]]
[[Image:AAAwaiting01.png|600px|center|Murphy's (1983) examination of Srivastava's function]]
  </td>
</tr>
</table>
-->

<div align="center">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center">
Equation and text extracted<sup>&dagger;</sup> from p. 177 of [http://adsabs.harvard.edu/abs/1983PASAu...5..175M J. O. Murphy (1983)]<p></p>
"''Composite and Analytical Solutions of the Lane-Emden Equation with Polytropic Indices N = 1 and N = 5''"<p></p>
Proc. Astronomical Soc. Australia, vol. 5, pp. 175-179 &copy; Astronomical Society of Australia
</td></tr>
<tr>
  <td align="center">
<!-- [[File:Murphy1983Extremum02.png|700px|center|Murphy's (1983) examination of Srivastava's function]] -->
<!-- [[Image:AAAwaiting01.png|400px|center|Norman &amp; Wilson (1978)]] -->
<table border="0" align="center" cellpadding="8" width="70%">
<tr>
  <td align="center">
on the interval <math>~[1, e^{2\pi} ]</math> &nbsp; &nbsp; &hellip; &nbsp; &nbsp; <math>~d\theta_{5F}/d\xi > 0</math> in the range [1, 10.0583]

<p></p>
----
<p></p>

<math>~\theta_{5F}(\zeta)_\mathrm{MAX} = 0.2503 \sqrt{A}</math> &nbsp; &nbsp; at &nbsp; &nbsp;  <math>~\zeta = 10.0583/A</math>
  </td>
</tr>
</table>
  </td>
</tr>
<tr><td align="left"><sup>&dagger;</sup>Equations and text displayed here,  with presentation order &amp; layout modified from the original publication.</td></tr>
</table>
</div>

On the other hand, the segment that falls in the interval, <math>~\xi_\mathrm{crit} \le (A\xi) \le e^{2\pi}</math>, whose function values lie in the range, <math>~\theta_{5F}|_\mathrm{max} \ge (A^{-1/2} \theta_{5F}) \ge 0</math> &#8212; that is, the segment of the blue curve that lies to the right of the vertical red line in the above figure &#8212; can be used to describe the <math>~n=5</math> "envelope" of a bipolytropic configuration because the function value is positive while it's first derivative is negative.

====Other (All) Solutions====
In a very clearly written article titled, ''All Solutions of the n = 5 Lane-Emden Equation'', [http://adsabs.harvard.edu/abs/2012JMP....53f2503M Patryk Mach (2012, J. Math. Phys., 53, 062503)] has pointed out that there are other families of solutions to the Lane-Emden equation of index, <math>~n=5</math>, in addition to the two solutions that have just been detailed, which he includes as his equations (3) and (5):  

<!--
<div align="center">
<table border="2" cellpadding="10">
<tr>
  <th align="center">
Extracted (with minor editing) from [http://adsabs.harvard.edu/abs/2012JMP....53f2503M Mach (2012)]
  </th>
<tr>
  <td>
[[File:Mach2012Eqs.png|300px|center|Mach (2012)]]
[[Image:AAAwaiting01.png|300px|center|Mach (2012)]]
  </td>
</tr>
</table>
</div>
-->
<div align="center">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center">
Equations extracted<sup>&dagger;</sup> from pp. 062503-1 &amp; -2 of [http://adsabs.harvard.edu/abs/2012JMP....53f2503M Mach (2012)]<p></p>
"''All Solutions of the n = 5 Lane-Emden Equation''"<p></p>
Journal of Mathematical Physics, vol. 53, pp. 062503-062503-6 &copy; American Institute of Physics
</td></tr>
<tr>
  <td align="center">
<!-- [[File:Mach2012Eqs.png|400px|center|Mach (2012)]] -->
<!-- [[Image:AAAwaiting01.png|400px|center|Norman &amp; Wilson (1978)]] -->
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\theta(\xi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pm \frac{1}{\sqrt{1 + \xi^2/3}}</math>
  </td>
  <td align="right">
&nbsp; &nbsp; &nbsp; <math>~(3)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\theta(\xi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pm \frac{\sin(\ln \sqrt{\xi})}{\sqrt{3\xi - 2\xi\sin^2(\ln\sqrt{\xi}) }}</math>
  </td>
  <td align="right">
&nbsp; &nbsp; &nbsp; <math>~(5)</math>
  </td>
</tr>
</table>
  </td>
</tr>
<tr><td align="left"><sup>&dagger;</sup>Equations displayed here, with layout modified from the original publication.</td></tr>
</table>
</div>


For completeness, Mach mentions a well-known solution that works for all indexes, <math>~n > 3</math>, which we have discussed separately in the context of  [[User:Tohline/SSC/Structure/PowerLawDensity#Power-Law_Density_Distributions|power-law density distributions]], namely,
<div align="center">
<math>\theta^n(\xi) = 
\frac{\rho}{\rho_c} = \biggl[ \frac{2(n-3)}{(n-1)^2} \biggr]^{n/(n-1)} \xi^{- 2n/(n-1)} \, .
</math>
</div>

In addition, Mach identifies the rarely referenced work of [http://adsabs.harvard.edu/abs/2000JMP....41.7029G H. Goenner &amp; P. Havas (2000, J. Math. Phys., 41, 7029)], which presents a family of solutions that is expressed in terms of the Weierstrass elliptic function; and he derives a new family of solutions &#8212; see equation (10) in ''his'' &sect;2.1 &#8212; that can be expressed entirely in terms of Jacobi elliptic functions.  Mach's new solutions, in particular, are oscillatory (like Srivastava's solution) but have no zeros, so in isolation they are not likely to be useful for astrophysical models.  But, as Mach suggests, they "can be used in composite stellar models on the same footing as Srivastava's solution" &#8212; see our [[User:Tohline/SSC/Structure/BiPolytropes/Analytic1_5#Step_6:__Envelope_Solution|accompanying description of a composite model using Srivastava's solution]].

==Numerical Solutions==
Here we explain how an Excel workbook can be used to numerically solve the Lane-Emden equation, evaluating the Lane-Emden function across a one-dimensional, discrete grid.  

===Techniques===
====HSCF Technique====
On the first spreadsheet within the workbook, we establish the following columns of number:
<ul>
  <li>'''Column A:''' &nbsp; Labeled <math>~r_i/R</math> (for ''i'' between 1 and N), that represents a discrete radial grid of spacing, <math>~~\Delta = (N-1)^{-1}</math>; each row gives the radial coordinate location of the ''i''<sup>th</sup> zone, starting from <math>~r_1/R = 0</math> and ending at <math>~r_N/R = 1</math>.</li>
  <li>'''Column B:''' &nbsp; Labeled <math>~\mathrm{rhf}_i</math> (for ''i'' between 1 and N-1); each row gives the radial coordinate of the mid-point of a grid zone.</li>
<div align="center">
<math>~\mathrm{rhf}_i \equiv \frac{1}{2}\biggr[\frac{r_i}{R} + \frac{r_{i+1}}{R}  \biggr] \, .</math>
</div>
  <li>'''Column C:''' &nbsp; Labeled <math>~\rho_i</math> (for ''i'' between 1 and N-1); each row provides an initial ''guess'' for the mass-density of the grid zone.  Usually it is sufficient to guess, <math>~\rho_i = 1</math> throughout.  For an <math>~n=0</math> polytrope, this proves also to be the correct ''final'' density profile.</li>
  <li>'''Column D:''' &nbsp; Labeled <math>~M_i</math> (for ''i'' between 1 and N); the ''i''<sup>th</sup> row gives the integrated mass enclosed interior to the radial grid coordinate, <math>~r_i/R</math>.  Specifically, <math>~M_1 = 0</math>, and thereafter, beginning with zone, <math>~i = 2</math>,</li>
<div align="center">
<math>~M_i = M_{i-1} + \frac{4\pi \rho_{i-1}}{3}\biggl[ \biggl(\frac{r_{i}}{R} \biggr)^3 - \biggl(\frac{r_{i-1}}{R} \biggr)^3  \biggr] \, .</math>
</div>
  <li>Note that, <math>~M_\mathrm{tot} = M_N \, .</math></li>
  <li>'''Column E:''' &nbsp; Labeled <math>~g_i</math> (for ''i'' between 2 and N); each row tabulates the inwardly directed gravitational acceleration that is felt at the outer edge of each grid zone.  Specifically,</li>
<div align="center">
<math>g_i = \frac{GM_i}{ (r_i/R)^2} \, .</math>
</div>
  <li>'''Column F:''' &nbsp; Labeled <math>~\Phi_i</math> (for ''i'' between 1 and N); each row gives the value of the gravitational potential at the mid-point of a grid zone.  Here, we start by specifying the ''value'' of the potential just (specifically, half a radial grid-zone) outside the surface of the configuration, where it should be, <math>~\Phi_N = -GM_\mathrm{tot}/(1+\Delta/2)</math>.  Then, working from the surface, inward and, given that, <math>~g = d\Phi/dr</math>, we use the corresponding finite-difference representation of the radial derivative and set,</li>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\Phi_i - \Phi_{i-1}}{\Delta} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~g_i </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \Phi_{i-1} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \Phi_i - g_i \Delta \, .</math>
  </td>
</tr>
</table>
</div>
<li>Note that the value of the gravitational potential ''at'' the surface is not <math>~\Phi_N</math> but, rather, must be <math>~\Phi_\mathrm{surf} = -GM_\mathrm{tot}/R</math>.</li>
<li>Furthermore, note that a lop-sided Taylor-series expansion about the center of the configuration provides the following good approximation to the gravitational potential ''at'' the center: &nbsp; <math>~\Phi_c \approx (9\Phi_1 - \Phi_2)/8</math>.</li>
<li>Note as well that all of these numerically determined values of the gravitational potential can be checked against the [[User:Tohline/SSC/Structure/UniformDensity#UniformSpherePotential|known analytic expression]] for the radial profile of the potential in a uniform-density sphere.</li>
<li>'''Column G:''' &nbsp; Labeled <math>~H_i</math> (for ''i'' between 1 and N-1); each row provides the value of the fluid enthalpy at the center of a grid cell.  Adopting the convention that the enthalpy is zero at the surface of the configuration, and given that [[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#Technique_3|the enthalpy and the gravitational potential must sum to zero]] throughout the configuration, we have,</li>
<div align="center">
<math>H_i = \Phi_\mathrm{surf} - \Phi_i \, .</math>
</div>
<li>At the center of the configuration, we have, <math>~H_c = \Phi_\mathrm{surf} - \Phi_c</math>. </li>
<li>'''Column H:''' &nbsp; Labeled <math>~H_\mathrm{norm}</math> (for ''i'' between 1 and N-1); each row provides the value of the fluid enthalpy, renormalized to the central value, specifically,</li>
<div align="center">
<math>~[H_\mathrm{norm}]_i = \frac{H_i}{H_c} \, .</math>
</div>
</ul>


The second spreadsheet within the workbook should be initially created by generating a copy of the first spreadsheet.  Then:
<ul>
  <li>'''Column C:''' &nbsp; Labeled <math>~\rho_i</math> (for ''i'' between 1 and N-1); generate a new, improved ''guess'' for the normalized mass-density at each grid zone based on the corresponding value of the normalized enthalpy from the previous spreadsheet/iteration. Specifically, given that the [[User:Tohline/SR#Barotropic_Structure|relationship between the density and enthalpy in a polytrope]] of index, <math>~n</math>, is, <math>~\rho \propto H^n</math>, we should set,</li>
<div align="center">
<math>~\biggl\{ \frac{\rho_i}{\rho_c} \biggr\}_\mathrm{sheet2}= \biggr\{[H_\mathrm{norm}]_i^n \biggr\}_\mathrm{sheet1} \, .</math>
</div>
</ul>


====Straight Numerical Integration====
The [[#LaneEmdenEquation|above governing relation]] may be rewritten as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ 
\xi \theta^{''} + 2 \theta^' 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \xi \theta^n \, .</math>
  </td>
</tr>
</table>
</div>

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

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

<tr>
  <td align="right">
<math>~\theta_i''</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\frac{\theta_+ - 2\theta_i +\theta_-}{\Delta_\xi^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>~\xi_i \biggl[ \frac{\theta_+ - 2\theta_i +\theta_-}{\Delta_\xi^2} \biggr] + 2\biggl[  \frac{\theta_+ - \theta_-}{2\Delta_\xi}\biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \xi_i \theta_i^n </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \xi_i [ \theta_+ - 2\theta_i +\theta_-] + \Delta_\xi [  \theta_+ - \theta_- ]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \Delta_\xi^2\xi_i \theta_i^n </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 
\theta_+
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2\xi_i \theta_i  + \theta_-(\Delta_\xi - \xi_i)  - \Delta_\xi^2\xi_i \theta_i^n }{\Delta_\xi + \xi_i} \, .</math>
  </td>
</tr>
</table>
</div>

Now, for the first two steps away from the center &#8212; where, <math>~\theta_i = \theta_0 = 1</math> and <math>~\xi_i = \xi_0 = 0</math> &#8212;  we will use the following [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#PolytropicLaneEmden|power-series expansion]] (see, for example, eq. 62 from &sect;5 in Chapter IV of [<b>[[User:Tohline/Appendix/References#C67|<font color="red">C67</font>]]</b>]) to determine the value of <math>~\theta_i</math>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\theta_1 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>1 - \frac{\Delta_\xi^2}{6} + \frac{n \Delta_\xi^4}{120}  - \frac{n}{378} \biggl( \frac{n}{5} - \frac{1}{8} \biggr) \Delta_\xi^6 \, ,</math> 
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\theta_2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>1 - \frac{(2\Delta_\xi)^2}{6} + \frac{n (2\Delta_\xi)^4}{120}  - \frac{n}{378} \biggl( \frac{n}{5} - \frac{1}{8} \biggr) (2\Delta_\xi)^6 \, .</math> 
  </td>
</tr>
</table>
</div>

===Results===
====Tabulated Global Properties====

Here, drawing from tables that have been previously published by other authors, we record numerically determined properties of polytropic models having a fairly wide range of polytropic indexes.  First, we draw from Table 4 (p.96) of [[User:Tohline/Appendix/References#C67|[<b><font color="red">C67</font></b>] ]].  To convert from his tabulated variables to our desired [[User:Tohline/SSC/Virial/FormFactors#PTtable|3 structural form-factors]], our normalized equilibrium radius (see [[#Lane-Emden_Equation|above ASIDE]]), and the "virial" (drawn from a [[User:Tohline/SSC/FreeEnergy/PowerPoint#Pressure-Truncated_Polytropes|more general overview]]), note that for ''isolated'' polytropes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{f}_\mathrm{M}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\rho_c}{\bar\rho}\biggr)^{-1}  = \biggl[ - \frac{3\theta^'}{\xi}\biggr]_{\xi_1} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~(5-n)\mathfrak{f}_\mathrm{W}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~5 \mathfrak{f}_\mathrm{M}^2 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~(5-n)\mathfrak{f}_\mathrm{A}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\biggl( \frac{4\pi}{3}\biggr) W_n\biggr]^{-1}  = 3(n+1)(-\theta^')^2_{\xi_1} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~x_\mathrm{eq} \equiv \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)} \xi_1 (- \xi^2 \theta^')^{(1-n)/(n-3)}_{\xi_1} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
Virial
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(5-n)\biggl[ \frac{b}{n} \cdot x_\mathrm{eq}^{(n-3)/n} - \frac{a}{3} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl(\frac{3}{4\pi} \biggr)^{1/n} \frac{(5-n)\mathfrak{f}_A}{\mathfrak{f}_M^{(n+1)/n}}  \cdot x_\mathrm{eq}^{(n-3)/n} - 1 \, .</math>
  </td>
</tr>
</table>
</div>


<div align="center" id="Chandrasekhar">
<table border="1" cellpadding="5" align="center">
<tr><th align="center" colspan="1">From Table 4 of [[User:Tohline/Appendix/References#C67|[<font color="red">C67</font>] ]]</th></tr>
<tr>
  <th align="center">Copied Directly from Table (1<sup>st</sup> 5 columns) &nbsp; &nbsp; &hellip; &nbsp; &nbsp; Implied Values of 3 Structural Form Factors, <math>x_\mathrm{eq}</math>, and Virial (last 5 columns)</th>
</tr>
<tr>
  <td align="left" colspan="1">
<pre>
n           xi_1            "mass"       rho_c/rho_avg       W_n             f_M             (5-n)f_W          (5-n)f_A        x_eq        Virial

0          2.4494           4.8988         1              0.119366       1                 5                 2.00000         0.620335        ---
0.5        2.7528           3.7871         1.8361         0.26227        0.544632645       1.483123592       0.910254        0.831089     -0.19009
1          3.14159          3.14159        3.28987        0.392699       0.303963378       0.461968677       0.607927        1.253313      3.0E-06
1.5        3.65375          2.71406        5.99071        0.77014        0.166925122       0.139319982       0.309857        2.357285      2.4E-06
2          4.35287          2.41105       11.40254        1.63818        0.087699758       0.038456238       0.145730        7.517481      1.4E-06
2.5        5.35528          2.18720       23.40646        3.90906        0.042723248       0.00912638        0.061072      186.3666        1.6E-08
3          6.89685          2.01824       54.1825        11.05066        0.018456144       0.001703146       0.0216035          ---          ---
3.25       8.01894          1.94980       88.153         20.365          0.011343913       0.000643422        0.0117227      3.3265E-06    2.9E-06
3.5        9.53581          1.89056      152.884         40.9098         0.006540907       0.000213917       0.00583558      0.00166854    2.2E-06
4         14.97155          1.79723      622.408        247.558          0.001606663       1.29068E-05       0.00096435      0.051854      6.1E-06
4.5       31.83646          1.73780     6189.47        4922.125          0.000161565       1.30516E-07       4.8502E-05      0.284868     -5.9E-05
4.9      169.47             1.73205        9.348E+05      3.693E+06      1.06975E-06       5.7218E-12        6.4645D-08      2.129056      1.5E-04
</pre>
  </td>
</tr>
<tr>
  <td align="left">
The column labeled "mass" contains the tabulated quantity, <math>(-\xi^2 \theta^')_{\xi_1}</math>.
  </td>
</tr>
</table>
</div>




<div align="center" id="Horedt2004">
<table border="1" cellpadding="5" align="center">
<tr><th align="center" colspan="1">From Table 2.5.2 (p. 77) of [http://adsabs.harvard.edu/abs/2004ASSL..306.....H Horedt (2004)]
 &nbsp; &mdash; &nbsp; "Polytropic Spheres (N = 3)"</th></tr>
<tr>
  <th align="center">Copied Directly from Table (1<sup>st</sup> 3 columns) &nbsp; &nbsp; &hellip; &nbsp; &nbsp; Implied Values (last 7 columns)</th>
</tr>
<tr>
  <td align="left" colspan="1">
<pre>
n	    xi_1	    theta'	 "mass"	       rho_c/rho_avg	    W_n	           f_M	          (5-n)f_A	   x_eq	        Virial
0	2.44948974	-8.164966E-01	4.898980	1.000000	1.193662E-01	1.000000E+00	2.000000E+00	6.2035049E-01	
0.5	2.75269805	-4.999971E-01	3.788651	1.835143	2.122091E-01	5.449168E-01	1.124987E+00	8.3099030E-01	0.0E+00
1	3.14159265	-3.183099E-01	3.141593	3.289868	3.926990E-01	3.039636E-01	6.079272E-01	1.2533141E+00	0.0E+00
1.5	3.65375374	-2.033013E-01	2.714055	5.990704	7.701402E-01	1.669253E-01	3.099856E-01	2.3572860E+00	0.0E+00
2	4.35287460	-1.272487E-01	2.411047	1.140254E+01	1.638182E+00	8.769977E-02	1.457301E-01	7.5164793E+00	0.0E+00
2.5	5.35527546	-7.626491E-02	2.187199	2.340646E+01	3.909062E+00	4.272324E-02	6.107153E-02	1.8636634E+02	0.0E+00
3	6.89684862	-4.242976E-02	2.018236	5.418248E+01	1.105068E+01	1.845615E-02	2.160341E-02		
3.5	9.53580534	-2.079098E-02	1.890557	1.528837E+02	4.090983E+01	6.540920E-03	5.835575E-03	1.6685566E-03	0.0E+00
4	1.49715463E+01	-8.018079E-03	1.797230	6.224079E+02	2.475594E+02	1.606664E-03	9.643439E-04	5.1854394E-02	0.0E+00
4.5	3.18364632E+01	-1.714549E-03	1.737799	6.189473E+03	4.921842E+03	1.615646E-04	4.850469E-05	2.8486849E-01	0.0E+00
4.99	1.75818915E+03	-5.598955E-07	1.730765	1.046736E+09	4.237887E+10	9.553503E-10	5.633289E-12	2.3460204E+01	2.2E-15</pre>
  </td>
</tr>
<tr>
  <td align="left">
The column labeled "mass" contains the tabulated quantity, <math>(-\xi^2 \theta^')_{\xi_1}</math>.
  </td>
</tr>
</table>
</div>

====Plotted Structural Profiles====
Using the just-described numerical techniques, we have solved the polytropic Lane-Emden equation on a 200-zone, uniform grid for a variety of values of the polytropic index.  In each case we have recorded how the dimensionless enthalpy, <math>~\theta_n(\xi)</math>, and its first radial derivative, <math>~\theta_n^'(\xi) \equiv d\theta_n/d\xi</math>, vary with <math>~\xi</math>, from the center of the polytropic configuration to its surface.  For the record, these tabulated results reside in the following DropBox files:

* <b>n = 2.5:</b> (10 SCF iterations) &nbsp; WorkFolder/Wiki edits/HSCF/n25.xlsx
* <b>n = 3:</b> (19 SCF iterations) &nbsp; WorkFolder/Wiki edits/HSCF/n300.xlsx
* <b>n = 3.005:</b> (15 SCF iterations) &nbsp; WorkFolder/Wiki edits/HSCF/n3005.xlsx
* <b>n = 3.05:</b> (15 SCF iterations) &nbsp; WorkFolder/Wiki edits/HSCF/n305.xlsx
* <b>n = 3.5:</b> (18 SCF iterations) &nbsp; WorkFolder/Wiki edits/HSCF/n25.xlsx
* <b>n = 6:</b> (direct integration) &nbsp; WorkFolder/Wiki edits/EmbeddedPolytropes/N6.xlsx

For each of these models, as indicated (n = 2.5, 3, 3.005, 3.05, 3.5, 6), [[#Fig4|Figure 4]] illustrates 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,
<div align="center">
<table border="0" cellpadding="3">

<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{n^3}{4\pi} \biggr)^{1/2} \theta_n^{(n-3)/2} \xi^2 
\biggl| \frac{d\theta_n}{d\xi} \biggr|  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\frac{R}{R_\mathrm{SWS}}
</math>
  </td>
  <td align="center">
<math>~\equiv~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{n}{4\pi} \biggr)^{1/2} \xi \theta_n^{(n-1)/2}  \, ,
</math>
  </td>
</tr>
</table>
</div>

has been drawn from an [[User:Tohline/SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|accompanying discussion of pressure-truncated polytropic configurations]].  In four of the Figure 4 panels, we have compared the profile of our numerically determined polytropic function (curve defined by <math>~\sim 200</math> small, black circular markers) to results (7 - 9 larger, blue circular markers) taken from Table 2.5.1 of [http://adsabs.harvard.edu/abs/2004ASSL..306.....H Horedt (2004)] &#8212; see, specifically the segment of his table on pp. 74 - 75 that applies to polytropic spheres &#8212; in an effort to demonstrate that our numerically determined solutions are accurate.   

<div align="center" id="Fig4">
<table border="1" align="center" cellpadding="8">
<tr>
  <th align="center" colspan="3">
Figure 4: &nbsp; Numerically Determined Solutions to the Polytropic Lane-Emden Equation
  </th>
</tr>
<tr>
  <td align="center">[[File:n25SequenceA.png|250px|n = 2.5 equilibrium sequence]]</td>
  <td align="center">[[File:n30SequenceA.png|250px|n = 3 equilibrium sequence]]</td>
  <td align="center">[[File:n3005SequenceA.png|250px|n = 3.005 equilibrium sequence]]</td>
</tr>
<tr>
  <td align="center" colspan="1" align="center">[[File:DataFileButton02.png|75px|file = Dropbox/WorkFolder/Wiki edits/HSCF/n25.xlsx --- worksheet = Horedt_n25]] <font size="+2">&#x21b2;</font></td>
  <td align="center" colspan="1" align="center">[[File:DataFileButton02.png|75px|file = Dropbox/WorkFolder/Wiki edits/LinearPerturbation/n300.xlsx --- worksheet = Horedt_n300]] <font size="+2">&#x21b2;</font></td>
  <td align="center" colspan="1" align="center">[[File:DataFileButton02.png|75px|file = Dropbox/WorkFolder/Wiki edits/HSCF/n3005.xlsx --- worksheet = Horedt_n3005]] <font size="+2">&#x21b2;</font></td>
</tr>
<tr>
  <td align="center">[[File:n305SequenceA.png|250px|n = 3.05 equilibrium sequence]]</td>
  <td align="center">[[File:n35SequenceA.png|250px|n = 3.5 equilibrium sequence]]</td>
  <td align="center">[[File:n600SequenceA.png|250px|n = 6 equilibrium sequence]]</td>
</tr>
<tr>
  <td align="center" colspan="1" align="center">[[File:DataFileButton02.png|75px|file = Dropbox/WorkFolder/Wiki edits/HSCF/n305.xlsx --- worksheet = Horedt_n305]] <font size="+2">&#x21b2;</font></td>
  <td align="center" colspan="1" align="center">[[File:DataFileButton02.png|75px|file = Dropbox/WorkFolder/Wiki edits/HSCF/n35.xlsx --- worksheet = Horedt_n35]] <font size="+2">&#x21b2;</font></td>
  <td align="center" colspan="1" align="center">[[File:DataFileButton02.png|75px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/N6.xlsx --- worksheet = PolytropeN6 (2)]] <font size="+2">&#x21b2;</font></td>
</tr>
<tr>
  <td align="left" colspan="3">
Data examples from Table 2.5.1 (pp. 74 - 75) of [http://adsabs.harvard.edu/abs/2004ASSL..306.....H Horedt (2004)]:
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="center"><math>~n</math></td>
  <td align="center"><math>~\xi</math></td>
  <td align="center"><math>~\theta_n</math></td>
  <td align="center"><math>~\frac{d\theta_n}{d\xi}</math></td>
  <td align="center"><math>~\frac{R}{R_\mathrm{SWS}}</math></td>
  <td align="center"><math>~\frac{M}{M_\mathrm{SWS}}</math></td>
</tr>
<tr>
  <td align="center">2.5</td>
  <td align="right">4.00000</td>
  <td align="right">0.1376807</td>
  <td align="left">- 0.1340534</td>
  <td align="right">0.4032551</td>
  <td align="right">3.926310</td>
</tr>
<tr>
  <td align="center">3.0</td>
  <td align="right">5.00000</td>
  <td align="right">0.1108198</td>
  <td align="left">- 0.08012604</td>
  <td align="right">0.2707342</td>
  <td align="right">2.936234</td>
</tr>
<tr>
  <td align="center">3.5</td>
  <td align="right">5.00000</td>
  <td align="right">0.1786843</td>
  <td align="left">- 0.07362030</td>
  <td align="right">0.3065541</td>
  <td align="right">2.210326</td>
</tr>
<tr>
  <td align="center">6.0</td>
  <td align="right">5.00000</td>
  <td align="right">0.3973243</td>
  <td align="left">- 0.05113662</td>
  <td align="right">0.3437981</td>
  <td align="right">1.327430</td>
</tr>
</table>
  </td>
</tr>
</table>
</div>

=Related Discussions=

* [http://en.wikipedia.org/wiki/Lane-Emden_equation Lane-Emden equation]

* [http://en.wikipedia.org/wiki/Polytrope Polytrope]

* [http://adsabs.harvard.edu/abs/1977ApJ...211..908S A. T. Service (1977, ApJ 211, 908)] &#8212; ''Concise Approximation Formulae for the Lane-Emden Functions''

=Lane's 1870 Article=

In what follows, text presented in a dark green font (and/or, as indicated, each equation) has been copied ''verbatim'' from a scientifically impactful 1870 article by [https://en.wikipedia.org/wiki/Jonathan_Homer_Lane J. Homer Lane].  Note that we have broken the article into separate '''segments''' in order to facilitate the insertion of interspersed comments.

==SEGMENT I==

<table border="1" align="center" cellpadding="10" width="90%">
<tr align="left"><tr><td align="left">
[https://archive.org/details/mobot31753002152772/page/56 J. H. Lane (1870)], '''The American Journal of Science and Arts''', Vol. 50, pp. 57 - 74: ''On the Theoretical Temperature of the Sun under the Hypothesis of a Gaseous Mass Maintaining Its Volume by Its Internal Heat and Depending on the Laws of Gases Known to Terrestrial Experiment''

<div align="center">'''SEGMENT I''' (beginning on p. 57)</div>

</td></tr>
<tr><td align="left">
<div align="center">
[Read (by J. Homer Lane at age 49) before the National Academy of Sciences at the session of April 13-16, 1869.]
</div>

<font color="darkgreen">
Many years have passed since the suggestion was thrown out by Helmholtz, and afterwards by others, that the present volume of the sun is maintained by his internal heat, and may become less in time.  Upon this hypothesis it was proposed to account for the renewal of the heat radiated from the sun, by means of the mechanical power of the sun's mass descending toward his center.  Calculations made by Prof. Pierce, and I believe by others, have shown that this provides a supply of heat far greater than it is possible to attribute to the meteoric theory of Prof. Wm. Thomson, which, I understand, has been abandoned by Prof. Thomson himself as not reconcilable with astronomical facts.  Some years ago the question occurred to me in connection with this theory of Helmholtz whether the entire mass of the sun might not be a mixture of transparent gases, and whether Herschel's clouds might not arise from the precipitation of some of these gases, say carbon, near the surface, with their revaporization when fallen or carried into the hotter subjacent layers of atmosphere beneath; the circulation necessary for the play of this Espian theory being of course maintained by the constant disturbance of equilibrium due to the loss of heat by radiation from the precipitated clouds.  Prof. Espy's theory of storms I first became acquainted with more than twenty years ago from lectures delivered by himself, and, original as I suppose it to be, and well supported as it is in the phenomena of terrestrial meteorology, I have long thought that Prof. Espy's labors deserve a more general recognition than they have received abroad.  It is not surprising, therefore, in a time when the constitution of the sun was exciting so much discussion, that the above suggestions should have occurred to myself before I became aware of the very similar, and in the main identical, views of Prof. Faye, put forth in the [https://en.wikipedia.org/wiki/Comptes_rendus_de_l'Académie_des_Sciences Comptes Rendus].  I sought to determine how far such a supposed constitution of the sun could be made to connect with the laws of the gases as known to us in terrestrial experiments at common temperatures.  Some calculations based upon conjectures of the highest temperature and least density thought supposable at the sun's photosphere led me to the conclusion that it was extremely difficult if not impossible, to make out the connection in a credible manner.  Nevertheless, I mentioned my ideas to Prof. Henry, Secretary of the Smithsonian Institution, when he immediately referred me to a number of the [https://en.wikipedia.org/wiki/Comptes_rendus_de_l'Académie_des_Sciences Comptes Rendus], then recently received, containing Faye's exposition of his theory.  Of course nothing is further from my purpose than to make any kind of claim to any thing in that publication.  After becoming acquainted with his labors I still regarded the theory as seriously lacking, in its physical or mechanical aspect, the direct support of confirmatory observations, and even as being subject to grave difficulty in that direction.  In this attitude I allowed the subject to rest until my friend Dr. Craig, in charge of the Chemical Laboratory of the Surgeon General's office, without any knowledge of Faye's memoir, or of my own suggestions previously made to Prof. Henry and another scientific friend, fell upon the same ideas of the sun's constitution, availing himself, precisely as I had done, of Espy's theory of storms.  Dr. Craig's ideas were communicated to a company of scientific gentlemen early last spring, and soon after, Prof. Newcomb, of the U. S. Naval Observatory, entered into a general survey of the nebular hypothesis.  These communications of Dr. Craig and Prof. Newcomb led me to enter into a renewed examination of the mechanical embarrassment under which I had believed the theory to labor.  Not any longer relying on my first rough estimate based on assumed high temperatures at the photosphere, the question was now inverted.  Assuming the gaseous constitution, and assuming the laws expressed in Poisson's formul&aelig;, known to govern the constitution of gases at common temperatures and densities, what shall we find to be the temperatures and densities corresponding to the observed volume of the sun supposing it were composed of some known gas such as hydrogen, or supposing it to be composed of such a mixture of gases as would be represented by common air.  Pure hydrogen will, of course, give us the lowest temperature of all known substances, under the general hypothesis.

The question was resolved, and the results were communicated in graphical and numerical form in May or June last to two or three scientific friends, but their publication has been delayed by an unavoidable absence of several months from home.

Premising that the unit of density shall correspond to a unit of mass in the cube of the unit of length, the unit of force to the force of terrestrial gravity in the unit of mass, and the unit of pressure or elasticity in the gas to the unit of force on a surface equal to the square of the unit of length:
</font>
<table border="0" cellpadding="5" align="center" width="60%">
<tr>
  <td align="center" colspan="3">'''Lane's Notation'''<br />
----
  </td>
  <td align="center">Modern Notation<br />
----
  </td>
</tr>

<tr>
  <td align="right" width="20%">
Let &nbsp;<math>~r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
the distance of an element of the sun's mass from the sun's center,
  </td>
  <td align="center"><math>~r</math></td>
</tr>

<tr>
  <td align="right">
<math>~t</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
the temperature of the element,
  </td>
  <td align="center">{{ User:Tohline/Math/VAR_Temperature01 }}</td>
</tr>

<tr>
  <td align="right">
<math>~\sigma t</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
its atmospheric [https://en.wikipedia.org/wiki/Subtangent subtangent], referred to the force of gravity at the earth's surface, or height of the column of homogeneous gas, whose terrestrial gravitating force would equal its elasticity,
  </td>
  <td align="center"><math>~\frac{R_\oplus^2}{GM_\oplus}\biggl( \frac{\Re}{\bar\mu} \biggr) T</math></td>
</tr>

<tr>
  <td align="right">
<math>~\rho</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
its density, or mass of its unit volume
  </td>
  <td align="center"><math>~\rho</math></td>
</tr>

<tr>
  <td align="right">
<font size="+1">?</font>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
force of terrestrial gravity in its unit volume,
  </td>
  <td align="center"><font size="+1">?</font></td>
</tr>

<tr>
  <td align="right">
<math>~\rho\sigma t</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
its elasticity, or elastic force per unit surface,
  </td>
  <td align="center"><math>\biggl( \frac{R_\oplus^2}{GM_\oplus} \biggr)</math> {{ User:Tohline/Math/VAR_Pressure01 }}</td>
</tr>

<tr>
  <td align="right">
<math>~R</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
the earth's radius,
  </td>
  <td align="center"><math>~R_\oplus</math></td>
</tr>

<tr>
  <td align="right">
<math>~M</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
the earth's mass,
  </td>
  <td align="center"><math>~M_\oplus</math></td>
</tr>

<tr>
  <td align="right">
<math>~m</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
the mass of the part of the sun's body contained in the concentric sphere whose radius is <math>~r</math>,
  </td>
  <td align="center"><math>~M_r</math></td>
</tr>

<tr>
  <td align="right">
<math>~\frac{M~r^2}{m~R^2} ~\sigma t</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
the [https://en.wikipedia.org/wiki/Subtangent subtangent] of the gas under its actual gravitating force in the sun.
  </td>
  <td align="center"><math>~\frac{r^2}{GM_r}\biggl( \frac{\Re}{\bar\mu} \biggr) T</math></td>
</tr>
</table>

<font color="darkgreen">
The condition of equilibrium between the gravitating force of a thin horizontal layer of gas whose thickness is <math>~dr</math>, and the difference of elastic force between its lower and upper surfaces is expressed by the equation,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="3">'''Lane's Notation'''<br />
----
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; </td>
  <td align="center" colspan="3">Modern Notation<br />
----
  </td>
</tr>

<tr>
  <td align="right">
<math>~d(\rho \sigma t)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{m~R^2}{M~r^2} ~\rho dr \, .</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\biggl( \frac{R_\oplus^2}{GM_\oplus} \biggr)dP</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl( \frac{R_\oplus^2}{GM_\oplus} \biggr)\frac{GM_r}{r^2} ~\rho dr \, .</math>
  </td>
</tr>
</table>
</font>

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

We see that, drawing upon his background in meteorology, Lane is suggesting that the radial dependence of various internal properties of the Sun is governed by the ''key'' integro-differential equation that is now commonly referred to as a statement of,
<div align="center">
<font color="maroon">'''Hydrostatic Balance'''</font>

{{ User:Tohline/Math/EQ_SShydrostaticBalance01 }}
</div>
where,
<div align="center">
{{ User:Tohline/Math/EQ_SSmassConservation01 }}
</div>

Our derivation of this mathematical statement of hydrostatic balance has been tagged, ''[[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#Technique_1|Technique 1]]'', in our accompanying discussion of various [[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#Solution_Strategies|solution strategies]].

==SEGMENT II==

<table border="1" align="center" cellpadding="10" width="90%">
<tr align="left"><tr><td align="left">
[https://archive.org/details/mobot31753002152772/page/56 J. H. Lane (1870)], '''The American Journal of Science and Arts''', Vol. 50, pp. 57 - 74: ''On the Theoretical Temperature of the Sun under the Hypothesis of a Gaseous Mass Maintaining Its Volume by Its Internal Heat and Depending on the Laws of Gases Known to Terrestrial Experiment''

<div align="center">'''SEGMENT II''' (beginning on p. 59)</div>

</td></tr>
<tr><td align="left">

<font color="darkgreen">
Under the hypothesis that the [http://www.thermopedia.com/content/598/ law of Mariotte] and the law of Poisson prevail throughout the whole mass, and that this mass is in convective equilibrium, we have
<table border="0" cellpadding="5" align="center" width="50%">
<tr>
  <td align="center" colspan="4">'''Lane's Notation'''<br />
----
  </td>
  <td align="center">Modern Notation<br />
----
  </td>
</tr>

<tr>
  <td align="left">&nbsp;</td>
  <td align="right" width="20%">
<math>~\sigma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
a constant,
  </td>
  <td align="center"><math>~\biggl(\frac{\Re}{\bar\mu}\biggr)</math>&nbsp; is a constant</td>
</tr>

<tr>
  <td align="left">(Eq. 1)</td>
  <td align="right" width="20%">
<math>~t</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~t_1 \rho^{k-1} \, ,</math>
  </td>
  <td align="center"><math>~T = T_0 \biggl( \frac{\rho}{\rho_0}\biggr)^{\gamma_g-1} \, ,</math></td>
</tr>
</table>
<math>~t_1</math> representing the value of <math>~t</math> in the part of the mass where the density is a unit.
</font>
<p></p>
----
<p></p>
In this last expression, the parameter <math>~k</math> that appears in the exponent <font color="darkgreen">represents the [[User:Tohline/SR/IdealGas#gamma_g|ratio of specific heat]] of a gas under constant pressure to its specific heat under constant volume.</font>  Instead of <math>~k</math>, it is more common in modern notation to use, <math>~\gamma_g\equiv c_P/c_V</math>.
<p></p>
----
<p></p>
<font color="darkgreen">
The theoretical difficulties which, if the supply of solar heat is to be kept up by the potential due to the mutual approach of the parts of the sun's mass consequent on the loss of heat by radiation, come in when we suppose a material departure from these laws of [http://www.thermopedia.com/content/598/ Mariotte] and of Poisson at the extreme temperatures and pressures in the sun's body, or how far such difficulties intervene, will be considered further on.

By means of the constant value of <math>~\sigma</math>, and the value of <math>~t</math> given in (1), the above differential equation is transformed into
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="3">'''Lane's Notation'''<br />
----
  </td>
</tr>

<tr>
  <td align="right">
<math>~\sigma t_1 d(\rho^k)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{m~R^2}{M~r^2} ~\rho dr </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ k \sigma t_1 \rho^{k-2} d\rho</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{m~R^2}{M~r^2} ~dr </math>
  </td>
</tr>
</table>
the integral of which gives

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="4">'''Lane's Notation'''<br />
----
  </td>
</tr>

<tr>
  <td align="right">
<math>~- \frac{R^2}{M} ~\int_0^r\frac{m dr}{r^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~k \sigma t_1 \int_{\rho_0}^\rho \rho^{k-2} d\rho</math>
  </td>
  <td align="right">&nbsp; &nbsp; &nbsp; </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{k \sigma t_1}{k-1}\biggr] \biggl[\rho^{k-1} -  \rho_0^{k-1} \biggr]</math>
  </td>
  <td align="right">&nbsp; &nbsp; &nbsp;</td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \biggl[1 - \biggl( \frac{\rho}{\rho_0}\biggr)^{k-1} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~+ \biggl[ \frac{k-1}{k \sigma t_1}\biggr] \frac{R^2}{M \rho_0^{k-1}} ~\int_0^r\frac{m dr}{r^2} \, ,</math>
  </td>
  <td align="right">&nbsp; &nbsp; &nbsp; (Eq. 2)</td>
</tr>
</table>
in which <math>~\rho_0</math> is the value of <math>~\rho</math> at the sun's center.

We have also,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~m </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi \int_0^r \rho r^2 dr = 4\pi \rho_0 \int_0^r \biggl(\frac{\rho}{\rho_0}\biggr)r^2 dr \, .</math>
  </td>
  <td align="right">&nbsp; &nbsp; &nbsp; (Eq. 3)</td>
</tr>

</table>

If now we put

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~r </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{k\sigma M t_1}{4(k-1)R^2 \pi \rho_0^{2-k}} \biggr]^{1 / 2} x\, ,</math>
  </td>
  <td align="right">&nbsp; &nbsp; &nbsp; (Eq. 4)</td>
</tr>

</table>

we shall have

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~m </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi \rho_0\biggl[ \frac{k\sigma M t_1}{4(k-1)R^2 \pi \rho_0^{2-k}} \biggr]^{3 / 2} \mu \, ,</math>
  </td>
  <td align="right">&nbsp; &nbsp; &nbsp; (Eq. 5)</td>
</tr>

</table>

in which

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\mu </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\int_0^x \biggl(\frac{\rho}{\rho_0} \biggr) x^2 dx \, ,</math>
  </td>
  <td align="right">&nbsp; &nbsp; &nbsp; (Eq. 6)</td>
</tr>

</table>

and equation (2) becomes

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="4">'''Lane's Notation'''<br />
----
  </td>
</tr>

<tr>
  <td align="right">
<math>~1 - \biggl( \frac{\rho}{\rho_0}\biggr)^{k-1} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\int_0^\xi \frac{\mu dx}{x^2} \, .</math>
  </td>
  <td align="right">&nbsp; &nbsp; &nbsp; (Eq. 7)</td>
</tr>

</table>
</font>

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


Drawing upon ''[[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#Technique_2|Technique 2]]'' from our discussion of various solution strategies, in a [[User:Tohline/SSC/Structure/Polytropes#Lane-Emden_Equation|separate chapter]] we have derived what is now commonly referred to as the,
<div align="center">
<font color="maroon">'''Polytropic Lane-Emden Equation'''</font>

{{ User:Tohline/Math/EQ_SSLaneEmden01 }}
</div>
where, given that <math>~n = (\gamma_\mathrm{g}-1)^{-1}</math> and <math>~K_n = P_c \rho_c^{-(n+1)/n}</math>, we have,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="3">'''Modern Notation'''<br />
----
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; </td>
  <td align="center" colspan="3">'''Lane's Notation'''<br />
----
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Theta_H</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\rho}{\rho_c} \biggr)^{1/n}</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\Theta_H</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\rho}{\rho_0} \biggr)^{k-1}</math>
  </td>
</tr>
</table>

and the dimensionless radial coordinate, <math>~\xi</math>, is defined via the relation,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="3">'''Modern Notation'''<br />
----
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; </td>
  <td align="center" colspan="3">'''Lane's Notation'''<br />
----
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{\gamma_\mathrm{g}}{4\pi G(\gamma_\mathrm{g}-1)} \cdot \frac{P_c}{\rho_c^2} \biggr]^{1/2} \xi</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{k}{4\pi G(k-1)} \biggl(\frac{GM_\oplus}{R_\oplus^2}\biggr)\frac{\sigma t_c}{\rho_c} \biggr]^{1/2} \xi</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
&nbsp;
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{k}{4\pi (k-1)} \biggl(\frac{M_\oplus}{R_\oplus^2}\biggr)\frac{\sigma t_1 \rho_c^{k-1}}{\rho_c} \biggr]^{1/2} \xi</math>
  </td>
</tr>
</table>
Comparing this last expression with Lane's equation (4), it is therefore clear that Lane's dimensionless radial function, <math>~x</math>, is identical to the now more commonly used dimensionless radial function, <math>~\xi</math>.  Note, as well, that Integrating the ''Polytropic Lane-Emden Equation'' once gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\xi^2 \frac{d\Theta_H}{d\xi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \int_0^\xi \xi^2 \Theta_H^n d\xi</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \xi^2 \frac{d}{d\xi}\biggl( \frac{\rho}{\rho_0}  \biggr)^{k-1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \int_0^\xi \xi^2 \biggl( \frac{\rho}{\rho_0}  \biggr) d\xi</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \mu_\mathrm{Lane} \, ,</math>
  </td>
</tr>
</table>
where the function, <math>~\mu_\mathrm{Lane}</math>, is as defined by Lane's equation (6); and integrating a second time gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\int_{\rho_0}^\rho d\biggl( \frac{\rho}{\rho_0}  \biggr)^{k-1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \int_0^\xi \frac{\mu_\mathrm{Lane}}{\xi^2} d\xi</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 1 - \biggl( \frac{\rho}{\rho_0}  \biggr)^{k-1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~+ \int_0^\xi \frac{\mu_\mathrm{Lane}}{\xi^2} d\xi \, ,</math>
  </td>
</tr>
</table>
which is identically Lane's equation (7).  It is therefore clear that Lane's equation (7) provides a mathematical model of the same stellar-interiors problem as &#8212; and is simply an ''integrated'' form of &#8212; the 2<sup>nd</sup>-order ODE that now customarily carries his name (along with Emden's).

==SEGMENT III==

<table border="0" align="center" cellpadding="10" width="100%">
<tr>
  <th align="center"><math>~n = \tfrac{3}{2}</math></th>
</tr>
<tr><td align="left">
<table border="1" align="center" cellpadding="10">
<tr><td align="left">
<pre>
Emden (1907) Tabelle 4 [n = 3/2]					
1.5	<-- n				
					
xi	theta	(-theta_prime)	theta^(n+1)	"""mass"""	rho_c/rho_avg
0.00	1.00000	0.00000	1.0000	0.00000	1.0000
0.25	0.98966	0.08268	0.9744	0.00517	1.0079
0.50	0.95911	0.16057	0.9009	0.04014	1.0380
0.75	0.91008	0.22988	0.7901	0.12931	1.0875
1.00	0.84516	0.28727	0.6567	0.28727	1.1603
1.25	0.76761	0.33061	0.5162	0.51658	1.2603
1.50	0.68132	0.35752	0.3832	0.80442	1.3985
1.75	0.58994	0.37168	0.2673	1.1383	1.5695
2.00	0.49670	0.37209	0.1739	1.4884	1.7917
2.25	0.40477	0.36119	0.1042	1.8285	2.0765
2.50	0.31678	0.34120	0.0565	2.1325	2.4424
2.75	0.23468	0.34750	0.0267	2.6280	2.6379
2.8085	0.21617	0.30788	0.0217	2.4285	3.0407
3.00	0.15972	0.28442	0.0102	2.5598	3.5159
3.25	0.09258	0.25261	0.0026	2.6682	4.2886
3.50	0.03335	0.22147	0.0002	2.7130	5.2678
3.625	0.00659	0.20680	0.0000	2.7175	5.8430
3.64	0.00350	0.20511	0.0000	2.7176	5.9155
3.6571	0.00000	0.20316	0.0000	2.7171	6.0004
</pre>
</td></tr>
</table>
<table border="1" align="center" cellpadding="10">
<tr><td align="left">
<pre>
Horedt  2004 pp. 73-74  [n = 3/2]					
1.5	<-- n				
					
					
					
					
					
					
xi	theta	(-theta_prime)	theta^(n+1)	"""mass"""	rho_c/rho_avg
0.00	1.00000	0.00000	1.0000	0.00000	1.0000
0.10	0.99833	0.03328	0.9958	0.00033	1.0015
0.50	0.95910	0.16054	0.9009	0.04014	1.0381
1.00	0.84517	0.28726	0.6567	0.2873	1.1604
3.00	0.15886	0.28425	0.0101	2.5583	3.5180
3.60	0.01109	0.20939	0.0000	2.7137	5.7309
3.650	0.00076	0.20372	0.0000	2.7141	5.9723
3.6538	0.00000	0.20330	0.0000	2.7141	5.9907</pre>
</td></tr>
</table>

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

<table border="0" align="center" cellpadding="10" width="100%">
<tr>
  <th align="center"><math>~n = \tfrac{5}{2}</math></th>
</tr>
<tr><td align="left">
<table border="1" align="center" cellpadding="10">
<tr><td align="left">
<pre>
Emden (1907) Tabelle 6 [n = 5/2]						
2.5	<-- n					
						
xi	theta	(-theta_prime)	theta^(n+1)	"""mass"""	rho_c/rho_avg	
0.00	1.00000	0.00000	1.0000		0.00000	1.0000	
0.25	0.98971	0.08226	0.9644		0.00514	1.0130	
0.50	0.95961	0.15676	0.8656		0.03919	1.0632	
0.75	0.91242	0.21798	0.7256		0.12261	1.1469	
1.00	0.85196	0.26282	0.5708		0.26282	1.2683	
1.25	0.78246	0.29036	0.4238		0.45369	1.4350	
1.50	0.70809	0.30213	0.2988		0.67979	1.6549	
1.75	0.63246	0.29532	0.2012		0.9044	1.9753	
2.00	0.55961	0.28614	0.1311		1.1446	2.3299	
2.35	0.46331	0.26290	0.0677		1.4516	2.9793	
2.50	0.42473	0.25080	0.0499		1.5675	3.3227	
3.00	0.31000	0.20793	0.0166		1.8714	4.8093	
3.5000	0.21752	0.16783	0.0048		2.0559	6.9515	
4.00	0.14300	0.13445	0.0011		2.1512	9.9169	
4.50	0.08263	0.10813	0.0001622	2.1896	13.8722	<-- Emden's value is 10.539
5.00	0.03384	0.08796	7.129E-06	2.1990	18.9480	
5.400	0.00128	0.07545	7.503E-11	2.2001	23.8569	
5.4172	0.00000	0.07500	0.0000		2.2010	24.0764	</pre>
</td></tr>
</table>
<table border="1" align="center" cellpadding="10">
<tr><td align="left">
<pre>
NNN
</pre>
</td></tr></table>


===Published n = 3/2 and n = 5/2 Tabulations===

<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="7">Published Tabulations of Numerically Constructed n = 3/2 Polytrope</th>
</tr>
<tr>
  <td align="center" colspan="3">
Copied from p. 79 of [https://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden (1907)]
  </td>
  <td align="center" width="10%">&nbsp;</td>
  <td align="center" colspan="3">
Copied from p. 73 of [http://adsabs.harvard.edu/abs/2004ASSL..306.....H Horedt (2004)]
  </td>
</tr>
<tr>
  <td align="center"><math>~\mathfrak{r}_1 \leftrightarrow \xi</math></td>
  <td align="center"><math>~u_1 \leftrightarrow \Theta_\mathrm{H}</math></td>
  <td align="center"><math>~- \frac{du_1}{d\mathfrak{r}_1} \leftrightarrow - \frac{d\Theta_\mathrm{H}}{d\xi} </math></td>
  <td align="center">&nbsp;</td>
  <td align="center"><math>~\xi</math></td>
  <td align="center"><math>~\theta \leftrightarrow \Theta_H</math></td>
  <td align="center"><math>~- \theta^' \leftrightarrow - \frac{d\Theta_H}{d\xi}</math></td>
</tr>
<tr>
  <td align="right"><math>~0</math></td>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~0</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~0</math></td>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~0</math></td>
</tr>
<tr>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~0.10</math></td>
  <td align="center"><math>~9.983346\mathrm{E-01}</math></td>
  <td align="center"><math>~3.328337\mathrm{E-02}</math></td>
</tr>
<tr>
  <td align="right"><math>~0.25</math></td>
  <td align="right"><math>~0.98966</math></td>
  <td align="right"><math>~0.08268</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~0.50</math></td>
  <td align="right"><math>~0.95911</math></td>
  <td align="right"><math>~0.16057</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~0.50</math></td>
  <td align="center"><math>~9.591039\mathrm{E-01}</math></td>
  <td align="center"><math>~1.605449\mathrm{E-01}</math></td>
</tr>
<tr>
  <td align="right"><math>~0.75</math></td>
  <td align="right"><math>~0.91008</math></td>
  <td align="right"><math>~0.22988</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~1</math></td>
  <td align="right"><math>~0.84516</math></td>
  <td align="right"><math>~0.28727</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~1.00</math></td>
  <td align="center"><math>~8.451698\mathrm{E-01}</math></td>
  <td align="center"><math>~2.872555\mathrm{E-01}</math></td>
</tr>
<tr>
  <td align="right"><math>~1.25</math></td>
  <td align="right"><math>~0.76761</math></td>
  <td align="right"><math>~0.33061</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~1.50</math></td>
  <td align="right"><math>~0.68132</math></td>
  <td align="right"><math>~0.35752</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~2</math></td>
  <td align="right"><math>~0.49670</math></td>
  <td align="right"><math>~0.37209</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~2.25</math></td>
  <td align="right"><math>~0.40477</math></td>
  <td align="right"><math>~0.36119</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~2.50</math></td>
  <td align="right"><math>~0.31678</math></td>
  <td align="right"><math>~0.34120</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~2.75</math></td>
  <td align="right"><math>~0.23468</math></td>
  <td align="right"><math>~0.31475</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~2.8085</math></td>
  <td align="right"><math>~0.21617</math></td>
  <td align="right"><math>~0.30788</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~3</math></td>
  <td align="right"><math>~0.15972</math></td>
  <td align="right"><math>~0.28442</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~3.00</math></td>
  <td align="center"><math>~1.588576\mathrm{E-01}</math></td>
  <td align="center"><math>~2.842527\mathrm{E-01}</math></td>
</tr>
<tr>
  <td align="right"><math>~3.25</math></td>
  <td align="right"><math>~0.09258</math></td>
  <td align="right"><math>~0.25261</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~3.50</math></td>
  <td align="right"><math>~0.03335</math></td>
  <td align="right"><math>~0.22147</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~3.60</math></td>
  <td align="center"><math>~1.109099\mathrm{E-02}</math></td>
  <td align="center"><math>~2.093927\mathrm{E-01}</math></td>
</tr>
<tr>
  <td align="right"><math>~3.625</math></td>
  <td align="right"><math>~0.00659</math></td>
  <td align="right"><math>~0.20680</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~3.64</math></td>
  <td align="right"><math>~0.00350</math></td>
  <td align="right"><math>~0.20511</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~3.65</math></td>
  <td align="center"><math>~7.639242\mathrm{E-04}</math></td>
  <td align="center"><math>~2.037196\mathrm{E-01}</math></td>
</tr>
<tr>
  <td align="right"><math>~3.6571</math></td>
  <td align="right"><math>~0</math></td>
  <td align="right"><math>~0.20316</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~3.65375374</math></td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~2.033013\mathrm{E-01}</math></td>
</tr>
</table>


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