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Joel2: /* See Also */
2024-01-15T22:30:29Z
<p><span class="autocomment">See Also</span></p>
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<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ul></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> <li>[https://ui.adsabs.harvard.edu/abs/1944ApJ...100..343H/abstract M. Hall Harrison (1944, ApJ, 100, 343 - 346)], ''The Generalized Cowling Model'' &#8212; Bibliographic Code: 1944ApJ...100..343H<br />(3<sup>rd</sup> paragraph on p. 343): <font color="darkgreen">"We shall consider a composite model made up of a central core described by the Lane-Emden function of index <math>n=\tfrac{3}{2}</math> and a point-source envelope."</font></li></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> <li>[https://ui.adsabs.harvard.edu/abs/1946ApJ...103..193H/abstract M. Hall Harrison (1946, ApJ, 103, 193 - 206)], ''Stellar Models with Partially Degenerate Isothermal Cores and Point-Source Envelopes'' &#8212; Bibliographic Code: 1946ApJ...103..193H</li></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> <li>[https://ui.adsabs.harvard.edu/abs/1947ApJ...105..322H/abstract M. Hall Harrison (1947, ApJ, 105, 322 - 326)], ''Stellar Models with Isothermal Cores and Point-Source Envelopes'' &#8212; Bibliographic Code: 1947ApJ...105..322H</li></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> <li>[https://ui.adsabs.harvard.edu/abs/1960PASJ...12...21S/abstract K. Suda &amp; Z. Hitotuyanagi (1960, PASJapan, 12, 21 - 27)], ''Stellar Models with Partially Degenerate Isothermal Cores''</li></ins></div></td></tr>
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Joel2
https://selfgravitatingfluids.education/JETohline/index.php?title=SSC/Structure/BiPolytropes/Analytic1.53/Pt2&diff=1062&oldid=prev
Joel2: Created page with "=BiPolytrope with (n<sub>c</sub>, n<sub>e</sub>) = (3/2, 3)= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" bgcolor="lightblue" width="33%"><br />Part I: Milne's (1930) EOS </td> <td align="center" bgcolor="lightblue" width="33%"><br />Part II: Point-Source Model </td> <td align="center" bgcolor="lightblue"><..."
2024-01-15T22:29:11Z
<p>Created page with "=BiPolytrope with (n<sub>c</sub>, n<sub>e</sub>) = (3/2, 3)= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" bgcolor="lightblue" width="33%"><br /><a href="/JETohline/index.php/SSC/Structure/BiPolytropes/Analytic1.53" title="SSC/Structure/BiPolytropes/Analytic1.53">Part I: Milne's (1930) EOS</a> </td> <td align="center" bgcolor="lightblue" width="33%"><br /><a href="/JETohline/index.php/SSC/Structure/BiPolytropes/Analytic1.53/Pt2" title="SSC/Structure/BiPolytropes/Analytic1.53/Pt2">Part II: Point-Source Model</a> </td> <td align="center" bgcolor="lightblue"><..."</p>
<p><b>New page</b></p><div>=BiPolytrope with (n<sub>c</sub>, n<sub>e</sub>) = (3/2, 3)=<br />
<br />
<table border="1" align="center" width="100%" colspan="8"><br />
<tr><br />
<td align="center" bgcolor="lightblue" width="33%"><br />[[SSC/Structure/BiPolytropes/Analytic1.53|Part I:&nbsp; Milne's (1930) EOS]]<br />
&nbsp;<br />
</td><br />
<td align="center" bgcolor="lightblue" width="33%"><br />[[SSC/Structure/BiPolytropes/Analytic1.53/Pt2|Part II:&nbsp; Point-Source Model]]<br />
&nbsp;<br />
</td><br />
<td align="center" bgcolor="lightblue"><br />[[SSC/Structure/BiPolytropes/Analytic1.53/Pt3|Part III:&nbsp; Our Derivation]]<br />
&nbsp;<br />
</td><br />
</tr><br />
</table><br />
<br />
<br />
==The Point-Source Model==<br />
<br />
According to Chapter IX.3 (p. 332) of [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], in the so-called "point-source" model, <font color="darkgreen">"&hellip; it is assumed that the entire source of energy is liberated at the center of the star; analytically, the assumption is that <math>L_r =~\mathrm{constant}~= L</math>."</font><br />
<br />
===Handling Radiation Transport===<br />
<br />
Here we begin with the [[PGE/FirstLawOfThermodynamics#Example_B|familiar expression for the radiation flux]], <br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><br />
<math>\vec{F}_\mathrm{rad}</math><br />
</td><br />
<td align="center"><br />
<math>=</math><br />
</td><br />
<td align="left"><br />
<math>- \frac{c}{3\rho\kappa_R} \nabla (a_\mathrm{rad}T^4) </math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>-\chi_\mathrm{rad} \nabla T \, ,</math><br />
</td><br />
</tr><br />
<tr><br />
<td align="center" colspan="3"><br />
[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Vol. I, &sect;2, p. 17, Eq. (2.17)<br />
</td><br />
<td align="left" colspan="2">and &nbsp; &nbsp;[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;3.4, p. 57, Eq. (67)<br />
</td><br />
</tr><br />
</table><br />
</div><br />
where [<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>] refers to <br />
<div align="center"><br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><br />
<math>\chi_\mathrm{rad}</math><br />
</td><br />
<td align="center"><br />
<math>\equiv</math><br />
</td><br />
<td align="left"><br />
<math><br />
\frac{4c a_\mathrm{rad} T^3}{3\kappa \rho} \, ,<br />
</math><br />
</td><br />
</tr><br />
</table><br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;3.4, p. 57, Eq. (68)<br />
</div><br />
as the coefficient of ''radiative'' conductivity. When modeling spherically symmetric configurations, the radiation flux has only a radial component, that is, <math>\vec{F}_\mathrm{rad} = \hat{e}_r(F_r)</math>. And, as pointed out in the context of Eq. (170) on p. 214 of [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>] <font color="darkgreen">&hellip; the quantity <math>L_r \equiv 4\pi r^2 F_r</math>, which is the net amount of energy crossing a spherical surface of radius <math>r</math>, is generally introduced instead of <math>F_r</math>.</font> We therefore have,<br />
<div align="center"><br />
<table border="0" cellpadding="5"><br />
<br />
<tr><br />
<td align="right"><math>\frac{L_r}{4\pi r^2}</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math><br />
- \frac{c}{3\rho\kappa_R} \frac{d}{dr} (a_\mathrm{rad}T^4)<br />
</math></td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><br />
<math>\Rightarrow ~~~\frac{dT}{dr}<br />
</math><br />
</td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math>- \frac{3}{4ca_\mathrm{rad}} \frac{\rho\kappa_R}{T^3} \frac{L_r}{4\pi r^2}</math></td><br />
</tr><br />
</table><br />
[<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter V, Eq. (171)<br /><br />
[<b>[[Appendix/References#Clayton68|<font color="red">Clayton68</font>]]</b>], &sect;6, Eq. (6-4a)<br /><br />
[<b>[[Appendix/References#KW94|<font color="red">KW94</font>]]</b>], &sect;9.1, Eq. (9.6)<br /><br />
[<b>[[Appendix/References#HK94|<font color="red">HK94</font>]]</b>], &sect;7.1, Eq. (7.8)<br /><br />
[<b>[[Appendix/References#BLRY07|<font color="red">BLRY07</font>]]</b>], &sect;5.2, Eq. (5.15)<br />
</div><br />
<br />
<table border="1" align="center" cellpadding="8" width="60%"><tr><td align="left"><br />
Dimensional Analysis:<br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><math>\frac{\kappa_R L}{ca_\mathrm{rad}}</math></td><br />
<td align="center"><math>\sim</math></td><br />
<td align="left"><math><br />
\frac{rT^4}{\rho} \sim m^{-1}\ell^4 ( {^\circ}K)^4 \, .<br />
</math></td><br />
</tr><br />
</table><br />
NOTE: This is consistent with the opacity, <math>\kappa_R \sim (\ell^2 m^{-1})</math>.<br />
</td></tr></table><br />
<br />
===Harrison's Approach===<br />
Following {{ Harrison46full }}, we seek to solve this last expression in concert with solutions to a pair of additional key governing relations for spherically symmetric equilibrium configurations, namely,<br />
<table border="0" align="center" cellpadding="5"><br />
<tr><br />
<td align="center">{{Math/EQ_SShydrostaticBalance01}}</td><br />
<td align="center">; &nbsp; &nbsp; &nbsp;</td><br />
<td align="center">{{Math/EQ_SSmassConservation01}}</td><br />
</tr><br />
<br />
<tr><br />
<td align="center" colspan="3"><br />
{{ Harrison46 }}, p. 196, Eq. (20)<br />
</td><br />
</tr><br />
</table><br />
<br />
while adopting (see [[SR/IdealGas#Consequential_Ideal_Gas_Relations|related discussion]])<br />
<br />
<div align="center"><br />
<span id="IdealGas:FormA"><font color="#770000">'''Form A'''</font></span><br /><br />
of the Ideal Gas Equation of State,<br />
<br />
{{ Template:Math/EQ_EOSideal0A }}<br />
</div><br />
and adopting [https://en.wikipedia.org/wiki/Kramers'_opacity_law Kramers' opacity law], that is,<br />
<table border="0" align="center" cellpadding="5"><br />
<br />
<tr><br />
<td align="right"><math>\kappa_R ~~ \rightarrow ~~ \kappa_\mathrm{Kramers}</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math>\kappa_0 \rho T^{-7 / 2} \, .</math></td><br />
</tr><br />
</table><br />
<br />
<table border="1" align="center" cellpadding="8" width="60%"><tr><td align="left"><br />
Dimensional Analysis:<br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><math>\biggl[ \frac{\kappa_0 L}{ca_\mathrm{rad}} \biggr]</math></td><br />
<td align="center"><math>\sim</math></td><br />
<td align="left"><math><br />
\frac{rT^{15/2}}{\rho^2} \sim m^{-2}\ell^7 ( {^\circ}K)^{15/2}<br />
</math></td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><math>\frac{\mathfrak{R}}{\mu}</math></td><br />
<td align="center"><math>\sim</math></td><br />
<td align="left"><math><br />
t^{-2}\ell^2 ( {^\circ}K)^{-1}<br />
</math></td><br />
</tr><br />
</table><br />
<br />
----<br />
<br />
We note as well that the leading coefficient in Kramers' opacity is,<br />
<div align="center"><br />
<table border="0" cellpadding="5"><br />
<br />
<tr><br />
<td align="right"><math>\kappa_0</math></td><br />
<td align="center"><math>\approx</math></td><br />
<td align="left"><math><br />
5 \times 10^{22} ~\mathrm{cm}^5~\mathrm{g}^{-2} ({^\circ}K)^{7 / 2} \, .<br />
</math></td><br />
</tr><br />
</table><br />
[<b>[[Appendix/References#Clayton68|<font color="red">Clayton68</font>]]</b>], &sect;3.3, Eq. (3-170)<br /><br />
[<b>[[Appendix/References#KW94|<font color="red">KW94</font>]]</b>], &sect;17.2, Eq. (17.5)<br /><br />
[<b>[[Appendix/References#HK94|<font color="red">HK94</font>]]</b>], &sect;4.4.2, Eq. (4.35)<br />
</div><br />
<br />
</td></tr></table><br />
<br />
===Does a Polytropic Relation Work?===<br />
<br />
Let's examine whether a point-source model can be represented by a polytropic relation.<br />
<br />
====Adopting Temperature (instead of enthalpy)====<br />
<table border="0" align="center" cellpadding="5"><br />
<br />
<tr><br />
<td align="right"><math>P</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math>K\rho^{1 + 1/n} ~~~~\Rightarrow~~~~ \rho = \biggl(\frac{P}{K}\biggr)^{n/(n+1)} \, ;</math><br />
</tr><br />
<br />
<tr><br />
<td align="right"><math>\biggl(\frac{\mathfrak{R}}{\mu}\biggr)\rho T</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math>K\rho^{1 + 1/n} ~~~~\Rightarrow~~~~ T = \biggl(\frac{K}{\mathfrak{R}/\mu}\biggr)\rho^{1/n} \, ;</math><br />
</tr><br />
<br />
<tr><br />
<td align="right"><math>T^{n}</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math>\biggl(\frac{K}{\mathfrak{R}/\mu}\biggr)^{n}\rho <br />
~~~~\Rightarrow~~~~ <br />
T^n = \biggl(\frac{K}{\mathfrak{R}/\mu}\biggr)^{n}\biggl(\frac{P}{K}\biggr)^{n/(n+1)} <br />
~~~~\Rightarrow~~~~ <br />
T^{(n+1)} = \biggl(\frac{K}{\mathfrak{R}/\mu}\biggr)^{(n+1)}\biggl(\frac{P}{K}\biggr) <br />
\, .</math><br />
</tr><br />
</table><br />
Hydrostatic balance is governed by the single 2<sup>nd</sup> order ODE,<br />
<br />
<table border="0" align="center" cellpadding="5"><br />
<tr><br />
<td align="right"><math>\frac{1}{r^2} \frac{d}{dr} \biggl[ \frac{r^2}{\rho} \frac{dP}{dr} \biggr]</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math>- 4\pi G\rho \, .</math><br />
</tr><br />
</table><br />
Normally in order to arrive at the Lane-Emden equation, <math>P</math> is converted to <math>\rho</math>; here, let's convert both <math>P</math> and <math>\rho</math> to <math>T</math>. First, on the RHS we have,<br />
<br />
<table border="0" align="center" cellpadding="5"><br />
<tr><br />
<td align="right"><math>\rho</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math>\biggl(\frac{\mathfrak{R}/\mu}{K}\biggr)^n T^n \, ;</math><br />
</tr><br />
</table><br />
and second, on the LHS we have,<br />
<br />
<table border="0" align="center" cellpadding="5"><br />
<br />
<tr><br />
<td align="right"><math>\frac{1}{\rho} \frac{dP}{dr}</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><br />
<math><br />
\biggl(\frac{K}{\mathfrak{R}/\mu}\biggr)^n T^{-n} \cdot \frac{d}{dr}\biggl[<br />
K^{-n}(\mathfrak{R}/\mu)^{(n+1)} T^{(n+1)}<br />
\biggr]<br />
</math><br />
</tr><br />
<br />
<tr><br />
<td align="right">&nbsp;</td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><br />
<math><br />
\biggl(\frac{\mathfrak{R}}{\mu}\biggr) T^{-n} \cdot \frac{d}{dr}\biggl[T^{(n+1)}<br />
\biggr]<br />
</math><br />
</tr><br />
<br />
<tr><br />
<td align="right">&nbsp;</td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><br />
<math><br />
(n+1)\biggl(\frac{\mathfrak{R}}{\mu}\biggr) \frac{dT}{dr}<br />
</math><br />
</tr><br />
<br />
<tr><br />
<td align="right">&nbsp;</td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><br />
<math><br />
(n+1)\biggl(\frac{\mathfrak{R}}{\mu}\biggr) \biggl\{<br />
- \frac{3}{4ca_\mathrm{rad}} \frac{\rho\kappa_R}{T^3} \frac{L}{4\pi r^2}<br />
\biggr\}<br />
</math><br />
</tr><br />
<br />
<tr><br />
<td align="right">&nbsp;</td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><br />
<math><br />
- (n+1)\biggl(\frac{\mathfrak{R}}{\mu}\biggr) \biggl[<br />
\frac{3L}{16\pi ca_\mathrm{rad}} <br />
\biggr] \biggl[ \frac{\rho}{r^2T^3} \biggr]\kappa_0 \rho T^{-7 / 2} <br />
</math><br />
</tr><br />
<br />
<tr><br />
<td align="right"><math>\Rightarrow ~~~ \frac{r^2}{\rho} \frac{dP}{dr}</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><br />
<math><br />
- (n+1)\biggl(\frac{\mathfrak{R}}{\mu}\biggr) \biggl[<br />
\frac{3\kappa_0 L}{16\pi ca_\mathrm{rad}} <br />
\biggr] \rho^2 T^{-13 / 2} <br />
</math><br />
</tr><br />
<br />
<tr><br />
<td align="right">&nbsp;</td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><br />
<math><br />
- (n+1)K^{-2n}\biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{2n+1} \biggl[<br />
\frac{3\kappa_0 L}{16\pi ca_\mathrm{rad}} <br />
\biggr] T^{(4n-13 )/ 2} \, .<br />
</math><br />
</tr><br />
</table><br />
So, the hydrostatic-balance condition becomes,<br />
<br />
<table border="0" align="center" cellpadding="5"><br />
<br />
<tr><br />
<td align="right"><br />
<math><br />
- (n+1)K^{-2n}\biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{2n+1} \biggl[<br />
\frac{3\kappa_0 L}{16\pi ca_\mathrm{rad}} <br />
\biggr] \cdot \frac{1}{r^2}\frac{dT^{(4n-13 )/ 2}}{dr} <br />
</math><br />
</td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math>- 4\pi G\biggl(\frac{\mathfrak{R}/\mu}{K}\biggr)^n T^n </math></td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><br />
<math>\Rightarrow ~~~<br />
\frac{(n+1)}{4\pi G K^{2n}}\biggl(\frac{\mathfrak{R}/\mu}{K}\biggr)^{-n}\biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{2n+1} \biggl[<br />
\frac{3\kappa_0 L}{16\pi ca_\mathrm{rad}} <br />
\biggr] \cdot \frac{1}{r^2}\frac{dT^{(4n-13 )/ 2}}{dr} <br />
</math><br />
</td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math> T^n </math></td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><br />
<math>\Rightarrow ~~~<br />
\frac{\mathcal{A}}{r^2}\frac{dT^{(4n-13 )/ 2}}{dr} <br />
</math><br />
</td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math> T^n </math></td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><math>\Rightarrow ~~~ {\mathcal{A}}^{-1}r^2 dr </math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><br />
<math><br />
T^{-n} \cdot dT^{(4n-13 )/ 2} \, ,<br />
</math><br />
</td><br />
</tr><br />
</table><br />
where,<br />
<table border="0" align="center" cellpadding="5"><br />
<br />
<tr><br />
<td align="right"><math>\mathcal{A}</math></td><br />
<td align="center"><math>\equiv</math></td><br />
<td align="left"><br />
<math><br />
\frac{(n+1)}{4\pi G K^n}\biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{n+1} \biggl[<br />
\frac{3\kappa_0 L}{16\pi ca_\mathrm{rad}} <br />
\biggr] \, . <br />
</math><br />
</td><br />
</tr><br />
</table><br />
<br />
<br />
<table border="1" align="center" cellpadding="8" width="60%"><tr><td align="left"><br />
Dimensional Analysis:<br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><math>\biggl[ \frac{\kappa_0 L}{ca_\mathrm{rad}} \biggr]</math></td><br />
<td align="center"><math>\sim</math></td><br />
<td align="left"><math><br />
m^{-2}\ell^7 ( {^\circ}K)^{15/2}<br />
</math></td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><math>\frac{\mathfrak{R}}{\mu}</math></td><br />
<td align="center"><math>\sim</math></td><br />
<td align="left"><math><br />
t^{-2}\ell^2 ( {^\circ}K)^{-1}<br />
</math></td><br />
</tr><br />
<br />
<tr><br />
<td align="right">polytropic <math>K</math></td><br />
<td align="center"><math>\sim</math></td><br />
<td align="left"><math><br />
t^{-2} \ell^{2 + 3/n} m^{-1 / n}<br />
</math></td><br />
</tr><br />
</table><br />
Hence,<br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><math>\mathcal{A}</math></td><br />
<td align="center"><math>\sim</math></td><br />
<td align="left"><math><br />
G^{-1} K^{-n} \biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{n+1} \biggl[ m^{-2}\ell^7 ( {^\circ}K)^{15/2} \biggr]<br />
</math></td><br />
</tr><br />
<br />
<tr><br />
<td align="right">&nbsp;</td><br />
<td align="center"><math>\sim</math></td><br />
<td align="left"><math><br />
\biggl[\ell^{-3} m t^{2}\biggr] \biggl[ t^{-2} \ell^{2 + 3/n} m^{-1 / n} \biggr]^{-n} <br />
\biggl[ t^{-2}\ell^2 ( {^\circ}K)^{-1}\biggr]^{n+1} \biggl[ m^{-2}\ell^7 ( {^\circ}K)^{15/2} \biggr]<br />
</math></td><br />
</tr><br />
<br />
<tr><br />
<td align="right">&nbsp;</td><br />
<td align="center"><math>\sim</math></td><br />
<td align="left"><math><br />
\biggl[\ell^{-3} \biggr] \biggl[ \ell^{2 + 3/n} \biggr]^{-n} <br />
\biggl[ \ell^2 \biggr]^{n+1} \biggl[ \ell^7\biggr] ( {^\circ}K)^{13/2-n} <br />
\sim<br />
\ell^3( {^\circ}K)^{13/2-n} <br />
\, .<br />
</math></td><br />
</tr><br />
</table><br />
<br />
Now, the [[SSC/Structure/Polytropes#Lane-Emden_Equation|characteristic length scale for polytropic configurations]] is given by the expression,<br />
<br />
<div align="center"><br />
<math>~<br />
a_\mathrm{n}^2 \equiv \biggl[\frac{(n+1)K}{4\pi G} \cdot \rho_c^{(1-n)/n} \biggr] \, .<br />
</math><br />
</div><br />
If we divide <math>\mathcal{A}</math> by <math>a_n^3</math>, the resulting expression should give us the characteristic temperature of the envelope. Specifically, we find that,<br />
<br />
<table border="0" align="center" cellpadding="5"><br />
<br />
<tr><br />
<td align="right"><math>\frac{\mathcal{A}}{a_n^3}</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><br />
<math><br />
\frac{(n+1)}{4\pi G K^n}\biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{n+1} \biggl[<br />
\frac{3\kappa_0 L}{16\pi ca_\mathrm{rad}} <br />
\biggr] <br />
\times<br />
\biggl[\frac{(n+1)K_n}{4\pi G} \cdot \rho_c^{(1-n)/n} \biggr]^{-3 / 2} <br />
</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right">&nbsp;</td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><br />
<math><br />
(n+1)^{- 1 / 2}<br />
\rho_c^{3(n-1)/2n} <br />
\biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{n+1} \biggl[<br />
\frac{3\kappa_0 L}{16\pi ca_\mathrm{rad}} <br />
\biggr] (4\pi G)^{ 1 / 2} K^{-n - 3 / 2}</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right">&nbsp;</td><br />
<td align="center"><math>\sim</math></td><br />
<td align="left"><br />
<math><br />
\biggl[m \ell^{-3}\biggr]^{3(n-1)/2n} <br />
\biggl[ t^{-2}\ell^2 ( {^\circ}K)^{-1}\biggr]^{n+1} \biggl[<br />
m^{-2}\ell^7 ( {^\circ}K)^{15/2} <br />
\biggr] \biggl[ \ell^3 m^{-1} t^{-2} \biggr]^{ 1 / 2} <br />
\biggl[ t^{-2} \ell^{2 + 3/n} m^{-1 / n} \biggr]^{-n - 3 / 2}</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right">&nbsp;</td><br />
<td align="center"><math>\sim</math></td><br />
<td align="left"><br />
<math><br />
t^{-2n - 2 - 1 + 2n + 3}<br />
m^{ 3(n-1)/2n - 2 - 1 / 2 + 1 + 3/2n}<br />
\ell^{ -9(n-1)/2n + 2n + 2 + 7 + 3 / 2}<br />
\biggl[ \ell^{(2n + 3)/n} \biggr]^{-(2n + 3) / 2}<br />
( {^\circ}K)^{13/2 - n } <br />
</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right">&nbsp;</td><br />
<td align="center"><math>\sim</math></td><br />
<td align="left"><br />
<math><br />
\ell^{ (9 + 4n^2 + 12n)/2n }<br />
\ell^{-(2n + 3)^2 / 2n}<br />
( {^\circ}K)^{13/2 - n } <br />
</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right">&nbsp;</td><br />
<td align="center"><math>\sim</math></td><br />
<td align="left"><br />
<math><br />
( {^\circ}K)^{(13-2n)/2 } \, .<br />
</math><br />
</td><br />
</tr><br />
</table><br />
<br />
<br />
</td></tr></table><br />
<br />
Quite generally we can write,<br />
<table border="0" align="center" cellpadding="5"><br />
<br />
<tr><br />
<td align="right"><math>\frac{1}{\alpha}~\frac{dT^\alpha}{dr}</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><br />
<math><br />
T^{\alpha - 1} \frac{dT}{dr} \, .<br />
</math><br />
</td><br />
</tr><br />
</table><br />
Rewriting the hydrostatic-balance condition, we find that,<br />
<table border="0" align="center" cellpadding="5"><br />
<br />
<tr><br />
<td align="right"><math>{\mathcal{A}}^{-1}r^2 </math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><br />
<math><br />
T^{-n} \cdot \frac{d}{dr}\biggl[T^{(4n-13 )/ 2}\biggr]<br />
=<br />
T^{-n} \biggl(\frac{4n-13}{2} \biggr) T^{(4n-15 )/ 2} \cdot \frac{dT}{dr}<br />
=<br />
\biggl(\frac{4n-13}{2} \biggr) T^{(2n-15 )/ 2} \cdot \frac{dT}{dr} \, .<br />
</math><br />
</td><br />
</tr><br />
</table><br />
Associating the exponents,<br />
<table border="0" align="center" cellpadding="5"><br />
<br />
<tr><br />
<td align="right"><math>\alpha - 1</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><br />
<math><br />
\frac{2n-15}{2} ~~~\Rightarrow ~~~ \alpha = \frac{2n-13}{2} \, , <br />
</math><br />
</td><br />
</tr><br />
</table><br />
we can write,<br />
<table border="0" align="center" cellpadding="5"><br />
<br />
<tr><br />
<td align="right"><math>{\mathcal{A}}^{-1}r^2 </math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><br />
<math><br />
\biggl(\frac{4n-13}{2} \biggr) \biggl(\frac{2}{2n-13} \biggr)\frac{d}{dr} \biggl[ T^{(2n-13)/2 } \biggr]<br />
</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><math>\Rightarrow ~~~ 3d(r^3)</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><br />
<math><br />
\mathcal{A} \biggl(\frac{4n-13}{2n-13} \biggr) d\biggl[ T^{(2n-13)/2 } \biggr]<br />
</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><math>\Rightarrow ~~~ 0</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><br />
<math><br />
d\biggl[ \mathcal{A} \biggl(\frac{13-4n}{13 - 2n} \biggr) T^{(2n-13)/2 } - 3r^3\biggr]<br />
</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><math>\Rightarrow ~~~</math> constant</td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><br />
<math><br />
\frac{\mathcal{A}}{a_n^3} \biggl(\frac{13-4n}{13 - 2n} \biggr) T^{(2n-13)/2 } - 3\biggl( \frac{r}{a_n}\biggr) ^3 \, .<br />
</math><br />
</td><br />
</tr><br />
</table><br />
<br />
====Adopting Enthalpy (instead of Temperature)====<br />
<br />
For [[SR#Barotropic_Structure|polytropic configurations]] the enthalpy, <math>H</math>, can easily be adopted in place of temperature via the relation,<br />
<table border="0" align="center" cellpadding="8"><br />
<br />
<tr><br />
<td align="right"><math>\biggl(\frac{\mathfrak{R}}{\mu}\biggr) T</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math>\frac{H}{(n+1)} \, .</math></td><br />
</tr><br />
</table><br />
Hence,<br />
<br />
<table border="0" align="center" cellpadding="5"><br />
<br />
<tr><br />
<td align="right"><math>P</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math>K\rho^{1 + 1/n}\, ;</math></td><br />
<td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td><br />
<td align="right"><math>H</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math>K\rho^{1/n}\, ;</math></td><br />
<td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td><br />
<td align="right"><math>H^{n+1}</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math>K^n P\, .</math></td><br />
</table><br />
And the radiation-transport equation can be rewritten in the form,<br />
<table border="0" cellpadding="5" align="center"><br />
<br />
<tr><br />
<td align="right"><math>\frac{L_r}{4\pi r^2}</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math><br />
- \frac{ca_\mathrm{rad}}{3\rho\kappa_R (n+1)^4} <br />
\biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{-4} <br />
\frac{dH^4}{dr}<br />
</math></td><br />
</tr><br />
<br />
<tr><br />
<td align="right">&nbsp;</td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math><br />
- \frac{ca_\mathrm{rad}}{3\kappa_0 (n+1)^4} <br />
\biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{-4} \biggl[\rho^{-2} T^{7 / 2}\biggr]<br />
\frac{dH^4}{dr}<br />
</math></td><br />
</tr><br />
<br />
<tr><br />
<td align="right">&nbsp;</td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math><br />
- \frac{ca_\mathrm{rad}}{3\kappa_0 (n+1)^4} <br />
\biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{-4} <br />
\biggl\{\biggl(\frac{K}{H}\biggr)^{2n} \biggl[ \biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{-1} \frac{H}{(n+1)}\biggr]^{7 / 2}\biggr\}<br />
4 H^3 \frac{dH}{dr}<br />
</math></td><br />
</tr><br />
<br />
<tr><br />
<td align="right">&nbsp;</td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math><br />
- \frac{4ca_\mathrm{rad} K^{2n}}{3\kappa_0 (n+1)^{15 / 2}} <br />
\biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{-15 / 2} <br />
H^{(13 - 4n) / 2}<br />
\frac{dH}{dr}<br />
</math></td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><math>\Rightarrow ~~~ r^2 \frac{dH}{dr} </math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math><br />
- \biggl[ \frac{3\kappa_0 L (n+1)^{15 / 2}}{16\pi ca_\mathrm{rad} K^{2n}} <br />
\biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{15 / 2} \biggr]<br />
H^{(4n - 13 ) / 2} \ .<br />
</math></td><br />
</tr><br />
</table><br />
<br />
In terms of the enthalpy, the hydrostatic-balance expression becomes,<br />
<br />
<table border="0" align="center" cellpadding="5"><br />
<tr><br />
<td align="right"><math>- 4\pi G\biggl[\frac{H^n}{K^n}\biggr]</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math>\frac{1}{r^2} \frac{d}{dr} \biggl\{r^2\biggl[\frac{K^n}{H^n}\biggr] <br />
\frac{d}{dr} \biggl[K^{-n} H^{n+1} \biggr]<br />
\biggr\}</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><math>\Rightarrow ~~~ - 4\pi GH^n</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math>\frac{K^n}{r^2} \frac{d}{dr} \biggl\{<br />
\biggl[\frac{r^2}{H^n}\biggr] <br />
\frac{d}{dr} \biggl[H^{n+1} \biggr]<br />
\biggr\}</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right">&nbsp;</td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math>\frac{(n+1)K^n}{r^2} \frac{d}{dr} \biggl\{<br />
r^2 \frac{dH}{dr} <br />
\biggr\} \, .</math><br />
</td><br />
</tr><br />
</table><br />
Combining these two equations gives,<br />
<br />
<table border="0" align="center" cellpadding="5"><br />
<br />
<tr><br />
<td align="right"><math>r^2 H^n</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math><br />
\mathcal{B} \cdot \frac{d}{dr} \biggl\{ H^{(4n - 13 ) / 2} \biggr\} <br />
</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right">&nbsp;</td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math><br />
\frac{(4n - 13)\mathcal{B}}{2} \cdot H^{(4n - 15 ) / 2} \frac{dH}{dr} <br />
</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><math>\Rightarrow ~~~ \biggl[\frac{2}{(4n - 13)\mathcal{B}}\biggr] r^2</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math><br />
H^{(2n - 15 ) / 2} \frac{dH}{dr} <br />
</math><br />
</td><br />
</tr><br />
</table><br />
where,<br />
<br />
<table border="0" align="center" cellpadding="5"><br />
<br />
<tr><br />
<td align="right"><math>\mathcal{B}</math></td><br />
<td align="center"><math>\equiv</math></td><br />
<td align="left"><math> <br />
\biggl[ \frac{3\kappa_0 L (n+1)^{17 / 2}}{64\pi^2 G ca_\mathrm{rad} K^{n}} <br />
\biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{15 / 2} \biggr] \, .<br />
</math><br />
</td><br />
</tr><br />
</table><br />
As above, quite generally we can write,<br />
<table border="0" align="center" cellpadding="5"><br />
<br />
<tr><br />
<td align="right"><math>\frac{1}{\alpha}~\frac{dH^\alpha}{dr}</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><br />
<math><br />
H^{\alpha - 1} \frac{dH}{dr} \, .<br />
</math><br />
</td><br />
</tr><br />
</table><br />
So, associating the exponents, we appreciate that,<br />
<table border="0" align="center" cellpadding="5"><br />
<br />
<tr><br />
<td align="right"><math>\alpha - 1</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><br />
<math><br />
\frac{2n-15}{2} ~~~\Rightarrow ~~~ \alpha = \frac{2n-13}{2} \, . <br />
</math><br />
</td><br />
</tr><br />
</table><br />
Hence, we have,<br />
<table border="0" align="center" cellpadding="5"><br />
<br />
<tr><br />
<td align="right"><math>\biggl[\frac{2}{(4n - 13)\mathcal{B}}\biggr] r^2</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math><br />
\frac{2}{(2n - 13)}\frac{d}{dr}\biggl[ H^{(2n - 13 ) / 2} \biggr] <br />
</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><math>\Rightarrow ~~~ r^2</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math><br />
\frac{(4n - 13)\mathcal{B}}{(2n - 13)}\frac{d}{dr}\biggl[ H^{(2n - 13 ) / 2} \biggr] <br />
</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><math>\Rightarrow ~~~ d\biggl[\frac{r^3}{a_n^3}\biggr]</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math><br />
\frac{3(4n - 13)\mathcal{B}}{(2n - 13)a_n^3}\cdot d\biggl[ H^{(2n - 13 ) / 2} \biggr] <br />
</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><math>\Rightarrow ~~~ d\biggl[\xi^3\biggr]</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math><br />
\biggl[\frac{3(4n - 13)}{(2n - 13)}\biggr] \biggl[\frac{\mathcal{B}}{a_n^3} \cdot H_\mathrm{norm}^{(2n-13)/2}\biggr]<br />
d\biggl[ \frac{H}{H_\mathrm{norm}} \biggr]^{(2n - 13 ) / 2} \, ;<br />
</math><br />
</td><br />
</tr><br />
</table><br />
so, if we adopt the definition,<br />
<table border="0" align="center" cellpadding="5"><br />
<br />
<tr><br />
<td align="right"><math>H_\mathrm{norm}</math></td><br />
<td align="center"><math>\equiv</math></td><br />
<td align="left"><br />
<math><br />
\biggl[\frac{(2n - 13)}{3(4n - 13)} \cdot \frac{a_n^3}{\mathcal{B}} \biggr]^{2/(2n-13)} \, ,<br />
</math><br />
</td><br />
</tr><br />
</table><br />
the relation becomes,<br />
<table border="0" align="center" cellpadding="5"><br />
<br />
<tr><br />
<td align="right"><math>d\biggl[\xi^3\biggr]</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math><br />
d\biggl[ \frac{H}{H_\mathrm{norm}} \biggr]^{(2n - 13 ) / 2} \, .<br />
</math><br />
</td><br />
</tr><br />
</table><br />
<br />
===Power-Law Density Distribution===<br />
<br />
In an [[SSC/Structure/PowerLawDensity#Power-Law_Density_Distributions|accompanying discussion]], we have demonstrated that power-law density distributions can provide analytic solutions of the Lane-Emden equation, although the associated boundary conditions do not naturally conform to the boundary conditions that are suitable to astrophysical configurations. We have just shown that the point-source envelope configuration appears to admit a power-law temperature (alternatively, enthalpy) solution. Via the polytropic relation, <math>H = K\rho^{1 / n}</math>, we can convert to the density-radius relation,<br />
<table border="0" align="center" cellpadding="5"><br />
<br />
<tr><br />
<td align="right"><math>d\biggl[\xi^3\biggr]</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math><br />
\biggl[ \frac{K\rho_c^{1/n}}{H_\mathrm{norm}} \biggr]^{(2n - 13 ) / 2} d\biggl[ \frac{\rho}{\rho_c} \biggr]^{(2n - 13 ) / 2n}<br />
</math><br />
</td><br />
</tr><br />
</table><br />
which, upon integration gives,<br />
<table border="0" align="center" cellpadding="5"><br />
<br />
<tr><br />
<td align="right">constant</td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math><br />
\biggl[ \frac{\rho}{\rho_c} \biggr]^{(2n - 13 ) / 2n} - \xi^3 \, ,<br />
</math><br />
</td><br />
</tr><br />
</table><br />
if we adopt the definition,<br />
<table border="0" align="center" cellpadding="5"><br />
<br />
<tr><br />
<td align="right"><math>\rho_c</math></td><br />
<td align="center"><math>\equiv</math></td><br />
<td align="left"><math><br />
\biggl(\frac{H_\mathrm{norm}}{K}\biggr)^n \, .<br />
</math><br />
</td><br />
</tr><br />
</table><br />
<br />
Setting the integration constant to zero, our result gives,<br />
<table border="0" align="center" cellpadding="5"><br />
<br />
<tr><br />
<td align="right"><math>\frac{\rho}{\rho_c}</math></td><br />
<td align="center"><math>\propto</math></td><br />
<td align="left"><math><br />
\xi^{6n/(2n-13)} \, .<br />
</math><br />
</td><br />
</tr><br />
</table><br />
In astrophysically relevant configurations, the exponent on <math>\xi</math> must be negative, which means that we are confined to models for which <math>n < \tfrac{7}{2}</math>.<br />
<br />
Now, from our [[SSC/Structure/PowerLawDensity#Derivation|associated discussion of power-law density distributions]] in polytropes, we discovered that hydrostatic balance can be established at all radial positions within a spherically symmetric configuration for power-law density distributions of the form,<br />
<div align="center"><br />
<math><br />
\frac{\rho}{\rho_c} \propto \xi^{- 2n/(n-1)}<br />
</math><br />
</div><br />
This matches our just-derived point-source model if,<br />
<table border="0" align="center" cellpadding="5"><br />
<br />
<tr><br />
<td align="right"><math>6n/(2n-13)</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math>- 2n/(n-1)</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><math>\Rightarrow ~~~ 6(n-1)</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math>2(13 - 2n)</math><br />
</td><br />
</tr><br />
<br />
<tr><br />
<td align="right"><math>\Rightarrow ~~~ n</math></td><br />
<td align="center"><math>=</math></td><br />
<td align="left"><math>\frac{16}{5} \, ,</math><br />
</td><br />
</tr><br />
</table><br />
which ''is'' less than <math>\tfrac{7}{2}</math>, so it is an astrophysically viable result.<br />
<br />
=See Also=<br />
<br />
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Joel2