SSC/Structure/PolytropesEmbedded/n5 - Revision history

Joel2: /* Tabular Summary (n=5) */

2023-12-29T00:47:01Z

Tabular Summary (n=5)

← Older revision		Revision as of 20:47, 28 December 2023
Line 471:		Line 471:
	NOTE: None of the analytic expressions for the dimensionless radius, pressure, or mass presented in this table explicitly appear in the referenced articles by Horedt, by Whitworth, or by Stahler but, as is discussed fully above, they are straightforwardly derivable from the more general relations that appear in these papers. The final polynomial relating the dimensionless mass to the dimensionless radius ''does'' explicitly appear as equation (B13) in {{ Stahler83 }}.		NOTE: None of the analytic expressions for the dimensionless radius, pressure, or mass presented in this table explicitly appear in the referenced articles by Horedt, by Whitworth, or by Stahler but, as is discussed fully above, they are straightforwardly derivable from the more general relations that appear in these papers. The final polynomial relating the dimensionless mass to the dimensionless radius ''does'' explicitly appear as equation (B13) in {{ Stahler83 }}.

	Additional discussion of Stahler's analytic mass-radius relation is presented in an [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/~~Pt1~~#Plotting_Stahler.27s_Relation\|accompanying chapter]].		Additional discussion of Stahler's analytic mass-radius relation is presented in an [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#Plotting_Stahler.27s_Relation\|accompanying chapter]].
	</td>		</td>
	</tr>		</tr>

Joel2: /* Tabular Summary (n=5) */

2023-12-29T00:46:07Z

Tabular Summary (n=5)

← Older revision		Revision as of 20:46, 28 December 2023
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	<tr>		<tr>
	<td align="center" rowspan="1">		<td align="center" rowspan="1">
	~~[http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)]~~		{{ Horedt70 }}
	<br>for<br>		<br>for<br>
	fixed <math>~(M,K_n)</math>		fixed <math>~(M,K_n)</math>
Line 424:		Line 424:
	<tr>		<tr>
	<td align="center" rowspan="1">		<td align="center" rowspan="1">
	~~[http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)]~~		{{ Whitworth81 }}
	<br>for<br>		<br>for<br>
	fixed <math>~(M,K_n)</math>		fixed <math>~(M,K_n)</math>
Line 442:		Line 442:
	<tr>		<tr>
	<td align="center" rowspan="2">		<td align="center" rowspan="2">
	~~[http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)]~~		{{ Stahler83 }}
	<br>for<br>		<br>for<br>
	fixed <math>~(P_e,K_n)</math>		fixed <math>~(P_e,K_n)</math>
Line 469:		Line 469:
	<tr>		<tr>
	<td align="left" colspan="3">		<td align="left" colspan="3">
	NOTE: None of the analytic expressions for the dimensionless radius, pressure, or mass presented in this table explicitly appear in the referenced articles by Horedt, by Whitworth, or by Stahler but, as is discussed fully above, they are straightforwardly derivable from the more general relations that appear in these papers. The final polynomial relating the dimensionless mass to the dimensionless radius ''does'' explicitly appear as equation (B13) in ~~[http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)]~~.		NOTE: None of the analytic expressions for the dimensionless radius, pressure, or mass presented in this table explicitly appear in the referenced articles by Horedt, by Whitworth, or by Stahler but, as is discussed fully above, they are straightforwardly derivable from the more general relations that appear in these papers. The final polynomial relating the dimensionless mass to the dimensionless radius ''does'' explicitly appear as equation (B13) in {{ Stahler83 }}.

	Additional discussion of Stahler's analytic mass-radius relation is presented in an [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Plotting_Stahler.27s_Relation\|accompanying chapter]].		Additional discussion of Stahler's analytic mass-radius relation is presented in an [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt1#Plotting_Stahler.27s_Relation\|accompanying chapter]].
	</td>		</td>
	</tr>		</tr>

Joel2 at 00:02, 19 December 2023

2023-12-19T00:02:25Z

← Older revision		Revision as of 20:02, 18 December 2023
Line 1,170:		Line 1,170:
	</li>		</li>
	</ol>		</ol>

			=Related Discussions=
			* [[SSC/Structure/BiPolytropes#BiPolytropes\|Constructing BiPolytropes]]
			* [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|Analytic description of BiPolytrope with <math>(n_c, n_e) = (5,1)</math>]]
			* [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Bonnor-Ebert spheres]]
			** [http://en.wikipedia.org/wiki/Bonnor-Ebert_mass Bonnor-Ebert Mass] according to Wikipedia
			** [http://www.astro.umd.edu/~cychen/MATLAB/ASTR320/matlabFrom320spring2011/Bonnor-EbertSphere/html/BonnorEbert.html A MATLAB script to determine the Bonnor-Ebert Mass coefficient] developed by [http://www.astro.umd.edu/people/cychen.html Che-Yu Chen] as a graduate student in the University of Maryland Department of Astronomy
			* [[SSC/Structure/LimitingMasses#Sch.C3.B6nberg-Chandrasekhar_Mass\|Schönberg-Chandrasekhar limiting mass]]
			* [[SSC/Structure/LimitingMasses#Relationship_Between_the_Bonnor-Ebert_and_Sch.C3.B6nberg-Chandrasekhar_Critical_Masses\|Relationship between Bonnor-Ebert and Schönberg-Chandrasekhar limiting masses]]

			* Wikipedia introduction to the [http://en.wikipedia.org/wiki/Lane-Emden_equation Lane-Emden equation]
			* Wikipedia introduction to [http://en.wikipedia.org/wiki/Polytrope Polytropes]

			{{ SGFfooter }}

Joel2 at 23:59, 18 December 2023

2023-12-18T23:59:53Z

@@ Line 1: / Line 1: @@
-==Polytropic Configurations with n = 5==
 Drawing from the [[SSC/Structure/Polytropes|earlier discussion of isolated polytropes]], we will reference various radial locations within a spherical {{Math/MP_PolytropicIndex}} = 5 polytrope by the dimensionless radius,
 <div align="center">

Joel2: Created page with " ==Polytropic Configurations with n = 5== Drawing from the earlier discussion of isolated polytropes, we will reference various radial locations within a spherical {{Math/MP_PolytropicIndex}} = 5 polytrope by the dimensionless radius,
$\xi \equiv \frac{r}{a_\mathrm{n=5}} ,$
where,
$a_{n=5} = \biggr[ \frac{(n+1)K}{4\pi G} \rho_c^{(1/n - 1)} \biggr]^{1/2}_{n=5} = \biggr[ \frac..."$

2023-12-18T23:41:47Z

Created page with " ==Polytropic Configurations with n = 5== Drawing from the earlier discussion of isolated polytropes, we will reference various radial locations within a spherical {{Math/MP_PolytropicIndex}} = 5 polytrope by the dimensionless radius, <div align="center"> <math> \xi \equiv \frac{r}{a_\mathrm{n=5}} , </math> </div> where, <div align="center"> <math> a_{n=5} = \biggr[ \frac{(n+1)K}{4\pi G} \rho_c^{(1/n - 1)} \biggr]^{1/2}_{n=5} = \biggr[ \frac..."

New page

==Polytropic Configurations with n = 5==
Drawing from the [[SSC/Structure/Polytropes|earlier discussion of isolated polytropes]], we will reference various radial locations within a spherical {{Math/MP_PolytropicIndex}} = 5 polytrope by the dimensionless radius,
<div align="center">
<math>
\xi \equiv \frac{r}{a_\mathrm{n=5}} ,
</math>
</div>
where,
<div align="center">
<math>
a_{n=5} = \biggr[ \frac{(n+1)K}{4\pi G} \rho_c^{(1/n - 1)} \biggr]^{1/2}_{n=5} =
\biggr[ \frac{3K}{2\pi G} \biggr]^{1/2} \rho_c^{-2/5} \, .
</math>
</div>

The solution to the Lane-Emden equation for <math>~n = 5</math> is,
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>
~\theta_5
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
\biggl(1+\frac{\xi^2}{3} \biggr)^{-1/2} \, ,
</math>
</td>
</tr>
</table>
</div>
hence,
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>
\frac{d\theta_5}{d\xi}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
- \frac{\xi}{3}\biggl(1+\frac{\xi^2}{3} \biggr)^{-3/2} \, .
</math>
</td>
</tr>
</table>
</div>

<font color="darkblue">
===Review===
</font>

Again, from the [[SSC/Structure/Polytropes|earlier discussion]], we can describe the properties of an isolated, spherical {{Math/MP_PolytropicIndex}} = 5 polytrope as follows:
* <font color="red">Mass</font>:
: In terms of the central density, <math>\rho_c</math>, and {{Math/MP_PolytropicConstant}}, 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>
: and, expressed as a function of <math>M</math>, the mass that lies interior to the dimensionless radius <math>\xi</math> is,
<div align="center">
<math>
\frac{M_\xi}{M} = \xi^3 (3 + \xi^2)^{-3/2} \, .
</math>
</div>
: Hence,
<div align="center">
<math>
M_\xi = \biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2} \rho_c^{-1/5} \biggl[ \xi^3 (3 + \xi^2)^{-3/2} \biggr] \, .
</math>
</div>

* <font color="red">Pressure</font>:
: The central pressure of the configuration is,
<div align="center">
<math>
P_c = \biggr[ \frac{\pi M^2 G^3}{2\cdot 3^4} \biggr]^{1/3} \rho_c^{4/3} = \biggr[ \frac{\pi G^3}{2\cdot 3^4}
\biggr( \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr) \rho_c^{-2/5}\biggr]^{1/3} \rho_c^{4/3} = K\rho_c^{6/5}
</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_\xi= P_c \biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-3}</math> .
</div>
: Hence,
<div align="center">
<math>
P_\xi = K \rho_c^{6/5} \biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-3} =
3^3K \rho_c^{6/5} [ 3 + \xi^2 ]^{-3}
</math> .
</div>

===Extension to Bounded Sphere===
Eliminating <math>\rho_c</math> between the last expression for <math>M_\xi</math> and the last expression for <math>P_\xi</math> gives,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<td align="right"><math>~P_\xi</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
3^3K [ 3 + \xi^2 ]^{-3} \biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{3} M_\xi^{-6} \biggl[ \xi^3 (3 + \xi^2)^{-3/2} \biggr]^6
</math>
</td>
</tr>

<tr>
<td align="right"> </td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
\biggl( \frac{2^3\cdot 3^{15} K^{10}}{\pi^3 M_\xi^{6} G^9} \biggr) \frac{\xi^{18}}{(3 + \xi^2)^{12}} \, .
</math>
</td>
</tr>
</table>
</div>
Now, if we rip off an outer layer of the star down to some dimensionless radius <math>\xi_e < \infty</math>, the interior of the configuration that remains — containing mass <math>M_{\xi_e}</math> — should remain in equilibrium if we impose the appropriate amount of externally applied pressure <math>P_e = P_{\xi_e} </math> at that radius. (This will work only for spherically symmetric configurations, as the gravitation acceleration at any location only depends on the mass contained inside that radius.) If we rescale our solution such that the mass enclosed within <math>\xi_e</math> is the original total mass <math>M</math>, then the pressure that must be imposed by the external medium in which the configuration is embedded is,
<div align="center">
<math>P_e= \biggr( \frac{2^3\cdot 3^{15} K^{10}}{\pi^3 M^{6} G^9} \biggr) \frac{\xi_e^{18}}{(3 + \xi_e^2)^{12}} </math> .
</div>
The associated equilibrium radius of this pressure-confined configuration is,
<div align="center">
<math>
R_\mathrm{eq} = \xi_e a_\mathrm{n=5} = \biggl[ \frac{3K}{2\pi G} \biggr]^{1/2} \rho_c^{-2/5} \xi_e =
\biggl[ \frac{\pi M^4 G^5}{2^3 \cdot 3^7 K^5} \biggr]^{1/2} \frac{(3+\xi_e^2)^3}{\xi_e^5} \, .
</math>
</div>

====Overlap with Whitworth's Presentation====
The curve labeled <math>~n=5</math> in the top two panels of Figure 1 shows how <math>R_\mathrm{eq}</math> varies with the applied external pressure <math>P_e</math>; as shown, the curve has two segments — configurations that are stable (blue diamonds) and configurations that are unstable (red squares). Following the lead of [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth] (1981, MNRAS, 195, 967) — for clarification, read the [[SSC/Structure/PolytropesASIDE1|accompanying ASIDE]] — these two quantities have been respectively normalized (or, "referenced") to,
<div align="center">
<math>
R_\mathrm{rf}\biggr|_\mathrm{n=5} \equiv \frac{2^6}{3^3} \biggl( \frac{\pi}{5^5} \biggr)^{1/2} \biggl[ \frac{G^5 M^4}{K^5} \biggr]^{1/2} ~~~\Rightarrow ~~~ \frac{R_\mathrm{eq}}{R_\mathrm{rf}} = \biggl( \frac{5^5}{2^{15}\cdot 3} \biggr)^{1/2} \frac{(3+\xi_e^2)^3}{\xi_e^5} \, ;
</math>
</div>
and,
<div align="center">
<math>
P_\mathrm{rf}\biggr|_\mathrm{n=5} \equiv \frac{3^{12} 5^9}{2^{26} \pi^3} \biggl( \frac{K^{10}}{G^9 M^6} \biggr) ~~~\Rightarrow ~~~ \frac{P_e}{P_\mathrm{rf}} = \biggl( \frac{2^{29}\cdot 3^{3} }{5^9} \biggr) \frac{\xi_e^{18}}{(3 + \xi_e^2)^{12}} \, .
</math>
</div>
We see that this <math>~n=5</math> model sequence bends back on itself. That is to say, for this polytropic index there is an externally applied pressure above which no equilibrium configuration exists. This limiting pressure arises along the curve where,
<div align="center">
<math>\frac{dP_e}{dR_\mathrm{eq}} = \biggl( \frac{dP_e}{d\xi_e} \biggr) \biggl( \frac{dR_\mathrm{eq}}{d\xi_e} \biggr)^{-1} = 0 \, .</math>
</div>
Evaluation of this expression shows that the limiting pressure occurs precisely at <math>\xi_e = 3</math>, that is,
<div align="center">
<math>
\biggl( \frac{P_e}{P_\mathrm{rf}} \biggr)_\mathrm{max} = \biggl( \frac{2^{29}\cdot 3^{3} }{5^9} \biggr) \frac{3^{18}}{12^{12}} = \frac{2^5 \cdot 3^9}{5^9} \, ,
</math>
</div>
and the radius of this limiting configuration is,
<div align="center">
<math>
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr) = \biggl( \frac{5^5}{2^{15}\cdot 3} \biggr)^{1/2} \frac{12^3}{3^5} = \biggl( \frac{5^5}{2^3 \cdot 3^5} \biggr)^{1/2} \, .
</math>
</div>
On the log-log plot displayed in the top-right panel of Figure 1, the location of this special point is <math>[ \log(P_e/P_\mathrm{rf}) , \log(R_\mathrm{eq}/R_\mathrm{rf}) ] \approx [ -0.49149, +0.10308 ] \, .</math>

We note as well that a conversion from Whitworth's normalizations to the normalizations adopted by Horedt produce the following coordinates for the limiting model configuration:
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>
~p_a|_\mathrm{max}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
~\frac{3^{12}}{2^{24}} \, ,
</math>
</td>
</tr>
</table>
</div>
and, at this bounding pressure, the model has an equilibrium radius,
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>
~r_a
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
\frac{2^6}{3^3} \, .
</math>
</td>
</tr>
</table>
</div>

====Overlap with Stahler's Presentation====
We can invert the above expression for <math>~P_e(K,M)</math> to obtain the following expression for <math>~M(K,P_e)</math>:
<div align="center">
<math>M^{6}= \biggr( \frac{2^3\cdot 3^{15} K^{10}}{\pi^3 P_e G^9} \biggr) \frac{\xi_e^{18}}{(3 + \xi_e^2)^{12}} </math> .
</div>

If, following Stahler's lead, we normalize this expression by <math>~M_\mathrm{SWS}</math> (evaluated for <math>~n=5</math>) and we normalize the above expression for <math>~R_\mathrm{eq}</math> by <math>~R_\mathrm{SWS}</math> (evaluated for <math>~n=5</math>), we obtain,
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
\frac{M}{M_\mathrm{SWS}}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
\biggr( \frac{2^3\cdot 3^{15} K^{10}}{\pi^3 P_e G^9} \biggr)^{1/6} \frac{\xi_e^{3}}{(3 + \xi_e^2)^{2}}
\biggl[ \biggl( \frac{2\cdot 3}{5G} \biggr)^{3/2} K^{5/3} P_\mathrm{ex}^{-1/6} \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
\biggr( \frac{3^{2} \cdot 5^3 }{4\pi } \biggr)^{1/2} \frac{\xi_e^{3}}{(3 + \xi_e^2)^{2}} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\frac{R_\mathrm{eq}}{R_\mathrm{SWS}}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\pi M^4 G^5}{2^3 \cdot 3^7 K^5} \biggr]^{1/2} \frac{(3+\xi_e^2)^3}{\xi_e^5}
\biggl[ \biggl( \frac{2\cdot 3}{5G} \biggr)^{1/2} K^{5/6} P_\mathrm{ex}^{-1/3}
\biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
\biggl[ \biggr( \frac{2^3\cdot 3^{15} K^{10}}{\pi^3 P_e G^9} \biggr)^{1/3} \frac{\xi_e^{6}}{(3 + \xi_e^2)^{4}} \biggr]
\biggl[ \frac{\pi G^5}{2^3 \cdot 3^7 K^5} \biggr]^{1/2} \frac{(3+\xi_e^2)^3}{\xi_e^5}
\biggl( \frac{5G}{2\cdot 3} \biggr)^{1/2} \biggl[ K^{-5/6} P_\mathrm{ex}^{1/3} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
\biggr( \frac{3^{2} \cdot 5}{2^2 \pi } \biggr)^{1/2} \frac{\xi_e}{(3 + \xi_e^2)} \, .
</math>
</td>
</tr>
</table>
</div>
This set of parametric relations that relate the mass of the truncated configuration to its radius via the parameter, <math>~\xi_e</math>, has been recorded to the immediate right of Stahler's name in our [[SSC/Structure/PolytropesEmbedded#n5Summary|<math>~n=5</math> summary table]], below.

Stahler points out (see his equation B13) that, for this particular pressure-bounded polytropic sequence, <math>~\xi_e</math> can be eliminated between the expressions to obtain the following direct algebraic relationship between <math>~M</math> and <math>~R_\mathrm{eq}</math>:
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>
\biggl( \frac{M}{M_\mathrm{SWS}} \biggr)^2 - 5 \biggl( \frac{M}{M_\mathrm{SWS}} \biggr)\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)
+ \frac{20\pi}{3} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
~0 \, .
</math>
</td>
</tr>
</table>
</div>
Viewed as a quadratic equation in the mass, the roots of this expression give,
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>
\frac{M}{M_\mathrm{SWS}}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
\frac{5}{2} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr) \biggl\{ 1 \pm \biggl[ 1 - \frac{16\pi}{15}
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^2 \biggr]^{1/2} \biggr\} \, .
</math>
</td>
</tr>
</table>
</div>
[<font color="red">CORRECTION: Changed factor inside square root from <math>~16\pi/3</math> to <math>~16\pi/15</math> on 24 December 2014.</font>] We have used this expression to generate the complete <math>~n=5</math> sequence shown here in the top panel of Figure 2 — the solid green segment of the curve shows the negative root and the solid red segment of the curve was generated using the positive root.

ASIDE: In his Appendix B, [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)] claims that the quadratic equation relating <math>~M</math> directly to <math>~R_\mathrm{eq}</math> (his equation B13) can be obtained by analytically integrating the first-order ordinary differential equation presented as his equation B10. I don't think that this is possible without knowing ahead of time how <math>~M</math> relates to <math>~R_\mathrm{eq}</math> through the above-derived parametric relations in <math>~\xi_e</math>.

[<font color="red">29 September 2014</font> by J. E. Tohline] Now that (I think) I've finished deriving the properly defined [[SSC/Virial/Polytropes#Nonrotating_Adiabatic_Configuration_Embedded_in_an_External_Medium|virial equilibrium condition for embedded polytropes]] and have reconciled that equilibrium expression with Horedt's corresponding specification of the equilibrium radius and surface-pressure, it's time to [[SSC/Structure/StahlerMassRadius|revisit the concern]] that was expressed in this "ASIDE" regarding the mass-radius relationship for embedded, <math>~n=5</math> polytropes presented by Stahler.

===Tabular Summary (n=5) ===
<span id="n5Summary">
<div align="center">
<table border="1" cellpadding="8" width="95%">
<tr>
<th align="center" colspan="3">
Table 2:  Properties of <math>~n=5</math> Polytropes Embedded in an External Medium of Pressure <math>~P_e</math>
<br>
(and, accordingly, truncated at radius <math>~\xi_e</math>)
</th>
</tr>
<tr>
<td align="center" colspan="3">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>
~\theta_5 = \biggl( 1 + \frac{\xi_e^2}{3} \biggr)^{-1/2}
</math>
</td>

<td align="center">
        and        
</td>

<td align="right">
<math>
~\frac{d\theta_5}{d\xi} \biggr|_{\xi_e} = - \frac{\xi_e}{3} \biggl( 1 + \frac{\xi_e^2}{3} \biggr)^{-3/2}
</math>
</td>
</tr>
</table>
</td>
</tr>

<tr>
<td align="center" rowspan="1">
[http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)]
<br>for<br>
fixed <math>~(M,K_n)</math>
</td>
<td align="center">
<math>
~r_a = \frac{R_\mathrm{eq}}{R_\mathrm{Horedt}} = \biggl\{ 3 \biggl[ \frac{(\xi_e^2/3)^5}{(1+\xi_e^2/3)^{6}} \biggr] \biggr\}^{-1/2}
</math>
</td>
<td align="center">
<math>
~p_a = \frac{P_e}{P_\mathrm{Horedt}} = 3^3 \biggl[ \frac{(\xi_e^2/3)^3}{(1+\xi_e^2/3)^{4}} \biggr]^3
</math>
</td>
</tr>

<tr>
<td align="center" rowspan="1">
[http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)]
<br>for<br>
fixed <math>~(M,K_n)</math>
</td>
<td align="center">
<math>
\frac{R_\mathrm{eq}}{R_\mathrm{rf}} = \biggl\{ \frac{2^{15}}{5^5} \biggl[ \frac{(\xi_e^2/3)^5}{(1+\xi_e^2/3)^{6}} \biggr] \biggr\}^{-1/2}
</math>
</td>
<td align="center">
<math>
\frac{P_e}{P_\mathrm{rf}} = \frac{2^{29}}{5^9} \biggl[ \frac{(\xi_e^2/3)^3}{(1+\xi_e^2/3)^{4}} \biggr]^3
</math>
</td>
</tr>

<tr>
<td align="center" rowspan="2">
[http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)]
<br>for<br>
fixed <math>~(P_e,K_n)</math>
</td>
<td align="center">
<math>
\frac{R_\mathrm{eq}}{R_\mathrm{SWS}} = \biggl\{ \frac{3\cdot 5}{2^2 \pi} \biggl[ \frac{\xi_e^2/3}{(1+\xi_e^2/3)^{2}} \biggr] \biggr\}^{1/2}
</math>
</td>
<td align="center">
<math>
\frac{M}{M_\mathrm{SWS}} = \biggl[ \biggl( \frac{3 \cdot 5^3}{2^2\pi} \biggr) \frac{(\xi_e^2/3)^3}{(1+\xi_e^2/3)^{4}} \biggr]^{1/2}
</math>
</td>
</tr>

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

<tr>
<td align="left" colspan="3">
NOTE: None of the analytic expressions for the dimensionless radius, pressure, or mass presented in this table explicitly appear in the referenced articles by Horedt, by Whitworth, or by Stahler but, as is discussed fully above, they are straightforwardly derivable from the more general relations that appear in these papers. The final polynomial relating the dimensionless mass to the dimensionless radius ''does'' explicitly appear as equation (B13) in [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)].

Additional discussion of Stahler's analytic mass-radius relation is presented in an [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Plotting_Stahler.27s_Relation|accompanying chapter]].
</td>
</tr>
</table>
</div>
</span>

===Equilibrium Sequences===

====Example Sequences====

<table border="1" align="center" cellpadding="5"><tr><td align="left">
Pulling from, and setting n = 5 in, our [[#Chieze's_Presentation|above discussion of Chieze's presentation]], we find the following …

<table border="0" cellpadding="8" align="center">

<tr>
<td align="right">
<math>\frac{P_e}{P_\mathrm{Ch}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
{\theta}^{n+1} = \biggl(1 + \frac{\xi^2}{3}\biggr)^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{R_\mathrm{eq}}{R_\mathrm{Ch}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{3}{2\pi} \biggr]^{1 / 2} \xi</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{M_\mathrm{tot}}{M_\mathrm{Ch}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{1 / 2}(- {\xi}^2 {\theta}^')
=
\biggl[ \frac{2^3\cdot 3^3}{2^2\pi} \biggr]^{1 / 2}\biggl\{ \frac{\xi^3}{3} \biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2} \biggr\}
=
\biggl[ \frac{2\cdot 3}{\pi} \biggr]^{1 / 2}\biggl\{ \xi^3 \biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2} \biggr\}
</math>
</td>
</tr>
</table>

where,

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

<tr>
<td align="right">
<math>P_\mathrm{Ch}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>K\rho_c^{(n+1)/n} = K\rho_c^{6/5}</math>
</td>
</tr>

<tr>
<td align="right">
<math>R_\mathrm{Ch}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl[\biggl(\frac{K}{G}\biggr) \rho_c^{1/n-1}\biggr]^{1 / 2}
=
\biggl[\biggl(\frac{K}{G}\biggr) \rho_c^{-4/5}\biggr]^{1 / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{Ch}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl[\biggl(\frac{K}{G}\biggr)^3 \rho_c^{-2/5}\biggr]^{1 / 2} \, .</math>
</td>
</tr>
</table>

Hence,

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

<tr>
<td align="right">
<math>P_e</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3}K\rho_c^{6/5}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R_\mathrm{eq}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{3}{2\pi} \biggr]^{1 / 2}
\biggl(\frac{K}{G}\biggr)^{1 / 2} \rho_c^{-2/5} \xi
</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_\mathrm{tot}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[ \frac{2\cdot 3}{\pi} \biggr]^{1 / 2}
\biggl(\frac{K}{G}\biggr)^{3/2} \rho_c^{-1/5}
\xi^3\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}
</math>
</td>
</tr>
</table>

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

<ul><li>
External Pressure vs. Volume (fixed mass); displayed in panel "a" of [[#Fig3|Figure 3]]:
</li></ul>

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

<tr>
<td align="right">
<math>P_e \biggl[G^9 K^{-10} M^6 \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{2\cdot 3}{\pi}\biggr)^3 \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl(\frac{4\pi R^3}{3} \biggr) \biggl[\biggl(\frac{K}{G}\biggr)^{15/2} M^{-6} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{\pi}{2\cdot 3}\biggr)^{5/2} \xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9}
</math>
</td>
</tr>
</table>

<ul><li>
Mass vs. Radius (fixed external pressure); displayed in panel "b" of [[#Fig3|Figure 3]]:
</li></ul>

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

<tr>
<td align="right">
<math>M \biggl[G^{3/2} K^{-5/3} P_e^{1/6} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>R \biggl[ G^{1 / 2} K^{-5/6} P_e^{1/3} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \xi \biggl(1 + \frac{\xi^2}{3} \biggr)^{-1}
</math>
</td>
</tr>
</table>

<ul><li>
Mass vs. Central Density (fixed external pressure); displayed in panel "c" of [[#Fig3|Figure 3]]:
</li></ul>

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

<tr>
<td align="right">
<math>M \biggl[G^{3/2} K^{-5/3} P_e^{1/6} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}
</math>
</td>
</tr>

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

<ul><li>
Mass vs. Central Density (fixed radius); displayed in panel "d" of [[#Fig3|Figure 3]]:
</li></ul>

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

<tr>
<td align="right">
<math>R_\mathrm{eq}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{3}{2\pi} \biggr]^{1 / 2}
\biggl(\frac{K}{G}\biggr)^{1 / 2} \rho_c^{-2/5} \xi
</math>
</td>
</tr>

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

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

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

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

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

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

<div align="center" id="Fig3">
<table border="1" align="center" cellpadding="8" width="1050px">
<tr>
<td align="center" colspan="6">
<b>Figure 3:</b>   Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres<br />(viewed from several different astrophysical perspectives)
</td>
</tr>
<tr>
<td align="center"><font color="black" size="+2">●</font></td><td align="center"><math>~\xi_e</math></td>
<td align="center" width="300px"><sup>&dagger;</sup>External Pressure vs. Volume<br /><font size="-1">(Fixed Mass)</font></td>
<td align="center" width="300px">Mass vs. Radius<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px"><sup>&Dagger;</sup>Mass vs. Central Density<br /><font size="-1">(Fixed External Pressure)</font></td>
<td align="center" width="300px">Mass vs. Central Density<br /><font size="-1">(Fixed Radius)</font></td>
</tr>
<tr>
<td align="center" colspan="1"><font color="yellow" size="+2">●</font></td> <td align="center" colspan="1">√3</td>
<td align="center" colspan="1" rowspan="4">(a)<br />
[[File:N5Sequence01B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(b)<br />
[[File:N5Sequence02B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(c)<br />
[[File:N5Sequence03B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
<td align="center" colspan="1" rowspan="4">(d)<br />
[[File:N5Sequence04B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="darkgreen" size="+2">●</font></td> <td align="center" colspan="1">3</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="purple" size="+2">●</font></td> <td align="center" colspan="1">√15</td>
</tr>
<tr>
<td align="center" colspan="1"><font color="red" size="+2">●</font></td> <td align="center" colspan="1">9.01</td>
</tr>
<tr>
<td align="center" colspan="2"> </td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^3 \biggl[ \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12} \biggr]_\tilde\xi</math><br /> vs. <br />
<math>\biggl(\frac{\pi}{2\cdot 3}\biggr)^{5/2} \biggl[ \xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2}\biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs.
<br /> <math>\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \biggl[ \xi \biggl(1 + \frac{\xi^2}{3} \biggr)^{-1} \biggr]_\tilde\xi</math></td>
<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>
</td>
<td align="center" colspan="1"><math>\biggl[ \frac{2^3\cdot 3}{\pi} \biggr]^{1 / 4} \biggl[ \xi^{5/2}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \frac{3}{2\pi} \biggr]^{5 / 4} \tilde\xi^{5 / 2}</math>
</td>
</tr>
</table>
</div>

====Example Extrema====

<ol>
<li>
External Pressure vs. Volume (fixed mass):
<ol type="a">
<li>
Maximum <math>P_e</math> (green circular marker) …

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

<tr>
<td align="right">
<math>0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{d}{d\xi} \biggl\{ \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12} \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
18 \xi^{17} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12}
-
12 \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-13} \frac{2\xi}{3}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\xi^{17} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-13}\biggl[
18 \biggl(1 + \frac{\xi^2}{3} \biggr)
-
8\xi^2
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\xi^{17} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-13}\biggl[
18 - 2\xi^2
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \xi^2_\mathrm{green}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
9 \, .
</math>
</td>
</tr>
</table>
</li>
<li>
Minimum Volume (purple circular marker) …

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

<tr>
<td align="right">
<math>0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{d}{d\xi}\biggl\{ \xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9} \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- 15 \xi^{-16} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9}
+
9\xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{8} \frac{2\xi}{3}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\xi^{-16}\biggl(1 + \frac{\xi^2}{3} \biggr)^{8} \biggl[
- 15 \biggl(1 + \frac{\xi^2}{3} \biggr)
+
6\xi^2
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\xi^{-16}\biggl(1 + \frac{\xi^2}{3} \biggr)^{8} \biggl[
\xi^2 - 15
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \xi^2_\mathrm{purple}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
15 \, .
</math>
</td>
</tr>
</table>
</li>
</ol>
</li>

<li>
Mass vs. Central Density (fixed radius):
<ol type="a">
<li>
Maximum mass (??? circular marker) …

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

<tr>
<td align="right">
<math>0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{d}{d\xi} \biggl\{\xi^{5/2}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2} \biggr\}
</math>
</td>
</tr>

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

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

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math> \frac{\xi^{3/2}}{6}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-5 / 2} \biggl[
15 - \xi^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \xi^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
15 = \xi^2_\mathrm{purple} \, .
</math>
</td>
</tr>
</table>
</li>
</ol>
</li>
</ol>