Editing SSC/Perturbations (section)

===Supplemental Relations===
As has been discussed [[SR#Time-Dependent_Problems|elsewhere]], in any analysis of time-dependent flows, the principal governing equations must be supplemented by adopting an equation of state for the gas and by specifying initial conditions.  Here, initial conditions will be given by the structural properties &#8212; for example, {{Math/VAR_Density01}}<math>(M_r)~</math> and {{Math/VAR_Pressure01}}<math>(M_r)~</math> &#8212; of one of our derived, spherically symmetric equilibrium structures &#8212; for example, a [[SSC/Structure/UniformDensity#Summary|uniform-density sphere]] or an [[SSC/Structure/Polytropes#Summary|<math>~n = 1</math> polytrope]].  We will adopt what has been referred to in an [[SR/IdealGas#IdealGasFormB|accompanying discussion]] as 
<div align="center">
<span id="FormB"><font color="#770000">'''Form B'''</font></span><br />
of the Ideal Gas Equation of State

{{ Template:Math/EQ_EOSideal02 }}

[<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter II, Eq. (5)<br />
[<b>[[Appendix/References#HK94|<font color="red">HK94</font>]]</b>], &sect;1.3.1, Eq. (1.22)<br />
[<b>[[Appendix/References#BLRY07|<font color="red">BLRY07</font>]]</b>], &sect;6.1.1, Eq. (6.4)
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As a result, the adiabatic form of the <math>1^\mathrm{st}</math> law of thermodynamics can be written as,
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<math>
\rho \frac{dP}{dt} - \gamma_\mathrm{g} P \frac{d\rho}{dt} = 0 .
</math>
</div>

<table border="1" align="center" cellpadding="8" width="80%"><tr><td align="left">
<font color="red"><b>ASIDE:</b>&nbsp;</font> When we introduced the [[PGE/FirstLawOfThermodynamics#Incorporation_Into_the_First_Law|concept of the ''entropy tracer'' in the context of our introductory discussion of the first law of thermodynamics]], we showed that a useful expression for the time-rate-of-change of the specific entropy, <math>s</math>, is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\rho T ~\frac{ds}{dt} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{P}{(\gamma_g - 1)} ~\frac{d}{dt}\biggl[
\ln\biggl( \frac{P}{\rho^{\gamma_g}}\biggr)
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~\frac{ds}{dt} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d}{dt}\biggl[
\frac{\mathfrak{R}/\bar\mu}{(\gamma_g - 1)} \cdot
\ln\biggl( \frac{P}{\rho^{\gamma_g}}\biggr)
\biggr]\, .
</math>
  </td>
</tr>
</table>

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