Editing SR (section)

==Time-Dependent Problems==
===Equation of State===

====Components of the Total Pressure====
For ''time-dependent'' problems we usually will supplement the set of [[PGE#Principal_Governing_Equations|principal governing equations]] by adopting a relationship between the state variables {{Template:Math/VAR_Pressure01}}, {{Template:Math/VAR_Density01}}, and {{Template:Math/VAR_Temperature01}} that is given by one of the expressions in the following Table, or by [[SR/PressureCombinations#Total_Pressure|some combination of these expressions]].  (For example, we could write <math>P_\mathrm{total} = P_\mathrm{gas} + P_\mathrm{deg} + P_\mathrm{rad}</math>.)  

<table width="95%" align="center" border=1 cellpadding=5>
<tr>
<th colspan=2 align="center">
<font color="darkblue">Analytic Equations of State</font>
</th>
<td align="center" rowspan="2" width="25%"><font color="darkblue">Radiation Pressure</font></td>
</tr>
<tr>
<td align="center" width="25%"><font color="darkblue">Ideal Gas</font></td>
<td align="center"><font color="darkblue">Degenerate Electron Gas</font></td>
</tr>
<tr>
<td align="center">
{{Template:Math/EQ_EOSideal0A}}
[[Image:OtherFormsButton.jpg|150px|link=SR/IdealGas#Ideal_Gas_Equation_of_State]]
</td>
<td align="center">
{{Template:Math/EQ_ZTFG01}}
</td>
<td align="center">
{{Template:Math/EQ_EOSradiation01}}
</td>
</tr>

<tr>
<td align="center" colspan=3><font color="darkblue">Normalized Total Pressure:</font></td>
</tr>
<tr>
<tr>
<td colspan=3 align="center">
{{Template:Math/EQ_PressureTotal01}}
[[Image:OriginButton.jpg|120px|link=SR/PressureCombinations#Total_Pressure]]
</td>
</tr>

</table>

In the so-called [[SR/IdealGas#Ideal_Gas_Equation_of_State|''ideal gas'' equation of state]], {{Template:Math/C_GasConstant}} is the gas constant and {{Template:Math/MP_MeanMolecularWeight}} is the mean molecular weight of the gas. In the equation that gives the electron degeneracy pressure, {{Template:Math/C_FermiPressure}} is the characteristic Fermi pressure and {{Template:Math/C_FermiDensity}} is the characteristic Fermi density.  And in the expression for the photon radiation pressure, {{Template:Math/C_RadiationConstant}} is the radiation constant.  The value of each of these identified physical constants can be found by simply scrolling the computer mouse over the symbol for the constant found in the text of this paragraph, and a definition of each constant can be found in the [[Appendix/VariablesTemplates|Variables Appendix]] of this H_Book.

All three of these equations are among the set of [[Appendix/EquationTemplates#Equations_of_State|key physical equations]] that provide the foundation for our discussion of the '''structure, stability, and dynamics of self-gravitating fluids.'''  A discussion of the physical principles that underpin each of these relations can be found in any of a number of different published texts &#8212; see, for example, the set of ''parallel references'' identified in the [[Appendix/EquationTemplates#Equations_of_State|Equations Appendix]] of this H_Book &#8212; or in the Wiki pages that can be accessed by clicking the linked "other forms" buttons in the above Table.  See also [<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>] &#8212; specifically the discussion associated with his Chapter 4, Eq. 13 &#8212; for a more general statement related to the proper specification of a supplemental, equation of state relationship.

====The Parameter, &beta;====
It should be pointed out that, in the astrophysics community, the dimensionless quantity <math>(1-\beta)</math> is sometimes used to denote the relative importance of radiation pressure in a gaseous configuration; specifically,
<div align="center">
<math>1-\beta = \frac{P_\mathrm{rad}}{P} \, .</math>
</div>
Hence, in the context of our present discussions, the parameter, <math>\beta</math>, itself is,
<div align="center">
<math>\beta = \frac{P_\mathrm{gas} + P_\mathrm{deg}}{P} \, .</math>
</div>
Examples include our discussion of [[SSC/Structure/BiPolytropes/Analytic1.53|bipolytropic configurations with]] <math>(n_c, n_e) = (\tfrac{3}{2}, 3)</math>, as introduced by {{ Milne30full }}; the study of [[Apps/SMS#Equation_of_State|rotating, supermassive stars]], especially as introduced by {{ BAC84full }}; and our reference to a [[SSC/Perturbations#Ledoux_and_Pekeris_.281941.29|derivation of the linear adiabatic wave equation]] by {{ LP41full }}.

====Adiabatic Exponent====

Consider, first, the evolution of a system that is composed entirely of an ideal gas; that is,  <math>P_\mathrm{rad} = P_\mathrm{deg} = 0</math> and <math>\beta = 1</math>.  It is widely appreciated that if the entropy of such a system remains constant during a phase of expansion/contraction, then the variation of pressure with density can be properly described by the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{d\ln P}{d\ln \rho}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\gamma_\mathrm{g} \, ,</math>
  </td>
</tr>
</table>
</div>
where the value of the adiabatic exponent, <math>\gamma_\mathrm{g}</math>, is given by the ratio of specific heats of the (ideal) gas.  Now, according to the ideal-gas equation of state, changes in the three state variables must always be related to one another via the differential expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>dP\biggr|_\mathrm{gas}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\Re}{\bar{\mu}} \biggl[ \rho dT + T d\rho \biggr]_\mathrm{gas}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ d\ln P</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>d\ln \rho +  d\ln T \, .
</math>
  </td>
</tr>
</table>
</div>

We therefore also deduce that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{d\ln P}{d\ln \rho}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>1 +  \frac{d\ln T}{d\ln\rho} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{d\ln T}{d\ln\rho} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\gamma_\mathrm{g}-1  \, ;
</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{d\ln P}{d\ln T}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{d\ln \rho}{d\ln T} +1 = \biggl( \frac{1}{\gamma_\mathrm{g} -1}\biggr) + 1
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\gamma_\mathrm{g}}{\gamma_\mathrm{g}-1 } \, .
</math>
  </td>
</tr>
</table>
</div>

Following the lead of [[Appendix/References#C67|[<b><font color="red">C67</font></b>]]], the astrophysics community has found that, when radiation pressure is included in the mix &#8212; that is, when we consider situations in which,
<div align="center">
<math>P = P_\mathrm{gas} + P_\mathrm{rad}</math>
</div>
&#8212; it can be useful to characterize the adiabatic compression/expansion of fluid elements in terms of three ''separate'' adiabatic exponents, <math>\Gamma_1, \Gamma_2, \Gamma_3</math>, that are defined via similar differential expressions, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{d\ln P}{d\ln \rho}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\Gamma_1 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{d\ln P}{d\ln T}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\Gamma_2}{\Gamma_2-1} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{d\ln T}{d\ln \rho}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\Gamma_3-1 \, .</math>
  </td>
</tr>
<tr>
<td align="center" colspan="3">
[[Appendix/References#C67|[<b><font color="red">C67</font></b>]]], Chapter II, &sect;12, Eqs. (123 - 125)<br />
[<b>[[Appendix/References#Clayton68|<font color="red">Clayton68</font>]]</b>], &sect;2-2, Eqs. (2-121)<br />
[[Appendix/References#KW94|[<b><font color="red">KW94</font></b>]]], &sect;13.2, Eqs. (13.27 - 13.29)<br />
[[Appendix/References#HK94|[<b><font color="red">HK94</font></b>]]], &sect;3.7.1, Eqs. (3.87 - 3.98)<br />
[[Appendix/References#P00|[<b><font color="red">P00</font></b>]]], Vol. I, &sect;5.6, Eqs. (5.85)
</td>
</tr>
</table>
</div>
In this case, though, each of the three adiabatic exponents is a function of <math>\beta</math> as well as <math>\gamma_\mathrm{g}</math>; specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\Gamma_1</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\beta + \frac{(4-3\beta)^2 (\gamma_\mathrm{g}-1)}{\beta + 12(\gamma_\mathrm{g}-1)(1-\beta)}
\, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Gamma_2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 + \frac{(4-3\beta)(\gamma_\mathrm{g} - 1)}{\beta^2 + 3(\gamma_\mathrm{g} - 1)(1-\beta)(4+\beta)}
\, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Gamma_3</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 + \frac{(4-3\beta)(\gamma_\mathrm{g}-1)}{\beta+12(\gamma_\mathrm{g} - 1)(1-\beta)}
\, .</math>
  </td>
</tr>
<tr>
<td align="center" colspan="3">[[Appendix/References#C67|[<b><font color="red">C67</font></b>]]], Chapter II, &sect;12</td>
</tr>
</table>
</div>

===Entropy Tracer===
We begin with the basic equation of state,
<div align="center">
<math>P = (\gamma_g - 1)\epsilon\rho \, ,</math>
</div>
and the 1<sup>st</sup> Law of Thermodynamics,
<div align="center">
{{ Template:Math/EQ_FirstLaw01 }}
</div>
Adopting the concept of an [[PGE/FirstLawOfThermodynamics#Entropy_Tracer|''entropy tracer'']],
<div align="center">
<math>\tau \equiv (\epsilon\rho)^{1/\gamma_g} \, ,</math>
</div>
the 1<sup>st</sup> becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\tau}{c_p} \frac{ds}{dt}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\partial\tau}{\partial t} + \nabla\cdot (\tau \vec{v}) \, .</math>
  </td>
</tr>
</table>
See &sect;4.1, Eq. (33) of {{ MTF2002 }} or &sect;2.2, Eq. (31) of {{ MT2012 }}.

===Initial Conditions===
For ''time-dependent'' problems, the [[PGE#Principal_Governing_Equations|principal governing equations]] must be supplemented further through the specification of initial conditions.  Frequently throughout this H_Book, we will select as initial conditions a specification of <math>~\rho(\vec{x}, t=0)</math>, <math>~P(\vec{x}, t=0)</math>, and <math>~\vec{v}(\vec{x}, t=0)</math> that, as a group themselves, define a static or steady-state equilibrium '''structure'''.  Perturbation or computational fluid dynamic (CFD) techniques can be used to test the '''stability''' or nonlinear '''dynamical behavior''' of such structures.