Editing SR (section)

===Barotropic Structure===

For ''time-independent'' problems, a structural relationship between {{Template:Math/VAR_Pressure01}} and {{Template:Math/VAR_Density01}} is required to close the system of [[PGE#Principal_Governing_Equations|principal governing equations]].  [<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>] refers to this as a "geometrical" rather than a "structural" relationship; see the discussion associated with his Chapter 4, Eq. 14. Generally throughout this H_Book, we will assume that all time-independent configurations can be described as barotropic structures; that is, we will assume that {{Template:Math/VAR_Pressure01}} is only a function of {{Template:Math/VAR_Density01}} throughout such structures.  (The [[2DStructure/AxisymmetricInstabilities#Poincar.C3.A9-Wavre_Theorem|Poincar&eacute;-Wavre theorem]] provides additional support for, as well as additional implications of, this assumption.)  More specifically, we generally will adopt one of the two ''analytically'' prescribable <math>P(\rho)</math> relationships displayed in the first row of the following Table.

<table width="80%" align="center" border=1 cellpadding=5>
<tr>
<th colspan=2 align="center">
<font color="darkblue">Barotropic Relations</font>
</th>
</tr>
<tr>
<td align="center" width="30%"><font color="darkblue">Polytropic</font></td>
<td align="center"><font color="darkblue">Zero-temperature Fermi (degenerate electron) Gas</font></td>
</tr>
<tr>
<td align="center">
{{ Math/EQ_Polytrope01 }}
</td>
<td align="center">
{{ Math/EQ_ZTFG01 }}
<font size="-1">
Reference (original): {{ Chandrasekhar35full }}
</font>

</td>
</tr>
<tr>
<td align="center">
<!-- {{Math/EQ_Polytrope02}}-->
<math>H = (n+1)K_n \rho^{1/n} = \frac{(n+1)P}{\rho}</math>
</td>
<td align="center">
<math>
H = \frac{8A_\mathrm{F}}{B_\mathrm{F}} \biggl[(\chi^2 + 1)^{1/2} - 1  \biggr]
</math>
</td>
</tr>

<tr>
<td align="center">
{{Template:Math/EQ_Polytrope03}}
</td>
<td align="center">
<math>
\rho = B_\mathrm{F}  \biggl[\biggl(\frac{HB_\mathrm{F}}{8A_\mathrm{F}} + 1 \biggr)^{2} - 1  \biggr]^{3/2}
</math>
</td>
</tr>

</table>


In the polytropic relation, the "polytropic index" {{Template:Math/MP_PolytropicIndex}} and the "polytropic constant" {{Template:Math/MP_PolytropicConstant}} are assumed to be independent of both {{Template:Math/VAR_PositionVector01}} and {{Template:Math/VAR_Time01}}.  In the zero-temperature Fermi gas relation, the two  constants {{Template:Math/C_FermiPressure}} and {{Template:Math/C_FermiDensity}} are expressible in terms of various fundamental physical constants, as detailed in the accompanying [[Appendix/VariablesTemplates#Physical_Constants|variables appendix]].  This table also displays (2<sup>nd</sup> row) the enthalpy as a function of mass density, <math>H(\rho)</math>, and (3<sup>rd</sup> row) the inverted relation <math>\rho(H)</math> for both barotropic relations, where 
<div align="center">
<math>H = \int\frac{dP}{\rho}</math> .
</div>
In both cases, we have chosen an integration constant such that {{Template:Math/VAR_Enthalpy01}} is zero when {{Template:Math/VAR_Density01}} is zero.

====Nonrelativistic ZTF Gas====

At sufficiently low densities, specifically, when

<div align="center">
<math>~\chi \ll 1 \, ,</math>
</div>

the zero-temperature Fermi (ZTF) equation of state describes the pressure-density behavior of a nonrelativistic (NR) degenerate electron gas.  We can determine the expression for <math>~P_\mathrm{deg}</math> in this limit by writing the function, <math>~F(\chi)</math>, in terms of two relevant series expansions &#8212; for the inverse hyperbolic sine function, see for example, the [http://en.wikipedia.org/wiki/Inverse_hyperbolic_function#Series_expansions Wikipedia presentation] &#8212; then keeping only the highest order terms.  
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>F(\chi)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- 3\chi \biggl(1 - \frac{2}{3}\chi^2 \biggr) \biggl( 1 + \frac{1}{2}\chi^2 - \frac{1}{2^3}\chi^4 +\frac{1}{2^4}\chi^6 - \frac{5}{2^7} \chi^8+ \cdots  \biggr)
+3 \displaystyle\sum_{m=0}^{\infty} \biggl[ \frac{(-1)^m (2m)!}{2^{2m}(m!)^2} \cdot \frac{\chi^{2m+1}}{(2m + 1)}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- 3\chi \biggl( 1 + \frac{1}{2}\chi^2 - \frac{1}{2^3}\chi^4 +\frac{1}{2^4}\chi^6 - \frac{5}{2^7}\chi^8 + \cdots  \biggr) +
2\chi^3 \biggl( 1 + \frac{1}{2}\chi^2 - \frac{1}{2^3}\chi^4 +\frac{1}{2^4}\chi^6 - \frac{5}{2^7}\chi^8 + \cdots  \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+3\biggl\{\chi - \frac{1}{6}\chi^3 + \frac{3}{2^3 \cdot 5} \chi^5 - \frac{5}{2^4\cdot 7} \chi^7 + \cdots \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left"><math>-3\chi + 3\chi + \chi^3\biggl(-\frac{3}{2} + 2 - \frac{1}{2}  \biggr) + \chi^5 \biggl(\frac{3}{8} +1 +\frac{3^2}{2^3\cdot 5}  \biggr)
+\chi^7 \biggl( -\frac{3}{2^4} -\frac{1}{2^2} - \frac{3\cdot 5}{2^4\cdot 7} \biggr) + \cdots </math> 
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left"><math>\chi^5 \biggl(\frac{15 + 40 + 9}{2^3\cdot 5}  \biggr)
-\chi^7 \biggl( \frac{21 + 28 + 15}{2^4\cdot 7} \biggr) + \cancelto{0}{\cdots} </math> 
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left"><math>\chi^5 \biggl(\frac{2^3}{5}  \biggr)
-\chi^7 \biggl( \frac{2^2}{7} \biggr) \, .</math> 
  </td>
</tr>
</table>
</div>

This agrees with, for example, the asymptotic form presented as equation (24) in &sect;X.1 (p. 361) of [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>].  Keeping only the leading term leads to the expression,

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

<tr>
  <td align="right">
<math>P_\mathrm{deg}\biggr|_\mathrm{NR}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{2^3}{5} A_F \biggl( \frac{\rho}{B_F}\biggr)^{5/3}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{2^3}{5} \biggl( \frac{\pi m_e^4 c^5}{3h^3} \biggr) \biggl[ \frac{8\pi m_p}{3} \biggl( \frac{m_e c}{h} \biggr)^3 \mu_e \biggr]^{-5/3} \rho^{5/3}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\mu_e^{-5/3} \biggl[ \frac{2^9 \cdot 3^5 \pi^3}{2^{15}\cdot 3^3\cdot  5^3 \pi^5} \biggr]^{1/3} \biggl( \frac{m_e^4 c^5}{h^3} \biggr) 
\biggl( \frac{h}{m_e c} \biggr)^5 \biggl( \frac{\rho}{m_p}\biggr)^{5/3}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{2^2 \cdot 5}\biggl( \frac{3}{\pi} \biggr)^{2/3} \biggl( \frac{h^2}{m_e} \biggr) \biggl( \frac{\rho}{m_p \mu_e}\biggr)^{5/3} \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], &sect;X.1, Eq. (27)<br />
[<b>[[Appendix/References#Clayton68|<font color="red">Clayton68</font>]]</b>], Eq. (2-32)<br />
[<b>[[Appendix/References#H87|<font color="red">H87</font>]]</b>], Eq. (11.42)<br />
[<b>[[Appendix/References#KW94|<font color="red">KW94</font>]]</b>], Eq. (15.23)<br />
[<b>[[Appendix/References#P00|<font color="red">P00</font>]]</b>], upper expression in Eq. (5.163)<br />
[<b>[[Appendix/References#Choudhuri10|<font color="red">Choudhuri10</font>]]</b>], Eqs. (5.9)-(5.10)
  </td>
</tr>
</table>
<!--
This matches equation (27) in &sect;X.1 (p. 362) of [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>]; see also, equation (2-32) of [<b>[[Appendix/References#Clayton68|<font color="red">Clayton68</font>]]</b>], equation (11.42) of [<b>[[Appendix/References#H87|<font color="red">H87</font>]]</b>], equation (15.23) of [<b>[[Appendix/References#KW94|<font color="red">KW94</font>]]</b>], the upper expression labeled as equation (5.163) in [<b>[[Appendix/References#P00|<font color="red">P00</font>]]</b>], and equations (5.9)-(5.10) of [<b>[[Appendix/References#Choudhuri10|<font color="red">Choudhuri10</font>]]</b>].
-->

====ZTF Gas in Relativistic Limit====

At sufficiently high densities, specifically, when

<div align="center">
<math>\chi \gg 1 \, ,</math>
</div>

the zero-temperature Fermi (ZTF) equation of state describes the pressure-density behavior of a degenerate electron gas in the (special) relativistic limit (RL).  We can determine the expression for <math>P_\mathrm{deg}</math> in this limit by writing the function, <math>F(\chi)</math>, in terms of two relevant series expansions &#8212; for the inverse hyperbolic sine function, see for example, [http://dlmf.nist.gov/4.38 NIST's ''Digital Library of Mathematical Functions''] &#8212; then keeping only the highest order terms.  
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>F(\chi)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(2\chi^4 - 3\chi^2 \biggr) \biggl( 1 + \frac{1}{2\chi^2} - \frac{1}{2^3\chi^4} +\frac{1}{2^4\chi^6} - \frac{5}{2^7\chi^8} + \cdots  \biggr)
+3 \biggl[ \ln(2\chi) + \frac{1}{2^2 \chi^2} - \frac{3}{2^5\chi^4} + \frac{5}{2^5\cdot 3\chi^6} - \cdots \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( 2\chi^4 + \chi^2 - \frac{1}{2^2} +\frac{1}{2^3\chi^2} - \frac{5}{2^6\chi^4} + \cdots  \biggr)
- \biggl( 3\chi^2 + \frac{3}{2} - \frac{3}{2^3\chi^2} +\frac{3}{2^4\chi^4} + \cdots  \biggr)
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2\chi^4 - 2 \chi^2 + 3\ln(2\chi) - \frac{7}{4}  +\frac{5}{4\chi^2} - \frac{35}{2^6\chi^4} + \cdots  
</math>
  </td>
</tr>
</table>
</div>

This agrees with, for example, the asymptotic form presented as equation (25) in &sect;X.1 (p. 361) of [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>].  Keeping only the leading term leads to the expression,

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

<tr>
  <td align="right">
<math>P_\mathrm{deg}\biggr|_\mathrm{RL}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2 A_F \biggl( \frac{\rho}{B_F}\biggr)^{4/3}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2 \biggl[ \frac{\pi m_e^4 c^5}{3h^3} \biggr] \biggl[ \frac{8\pi m_p \mu_e}{3} \biggl( \frac{m_e c}{h}\biggr)^3\biggr]^{-4/3} \rho^{4/3}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{2^3}\biggl(\frac{3}{\pi}\biggr)^{1/3} (hc) \biggl(\frac{\rho}{m_p \mu_e}\biggr)^{4/3} \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], &sect;X.1, Eq. (28)<br />
<!--  [<b>[[Appendix/References#Clayton68|<font color="red">Clayton68</font>]]</b>], Eq. (2-32)<br /> -->
[<b>[[Appendix/References#H87|<font color="red">H87</font>]]</b>], Eq. (11.43)<br />
[<b>[[Appendix/References#KW94|<font color="red">KW94</font>]]</b>], Eq. (15.26)<br />
[<b>[[Appendix/References#P00|<font color="red">P00</font>]]</b>], lower expression in Eq. (5.163)<br />
[<b>[[Appendix/References#Choudhuri10|<font color="red">Choudhuri10</font>]]</b>], Eqs. (5.11)-(5.12)
  </td>
</tr>
</table>
<!--
This matches equation (28) in &sect;X.1 (p. 362) of [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>]; see also, equation (11.43) of [<b>[[Appendix/References#H87|<font color="red">H87</font>]]</b>], equation (15.26) of [<b>[[Appendix/References#KW94|<font color="red">KW94</font>]]</b>], the lower expression labeled as equation (5.163) in [<b>[[Appendix/References#P00|<font color="red">P00</font>]]</b>], and equations (5.11)-(5.12) of  [<b>[[Appendix/References#Choudhuri10|<font color="red">Choudhuri10</font>]]</b>].
-->