Editing SSC/Perturbations (section)

==Assembling the Key Relations==

===Governing Equations===
After combining the Euler equation with the Poisson equation in essentially the manner outlined by the "structural solution strategy" we have called [[SSCpt2/SolutionStrategies#Solution_Strategies|Technique 1]], the relevant set of time-dependent governing equations is:

<div align="center">
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />

<math>\frac{d\rho}{dt} = - \rho \biggl[\frac{1}{r^2}\frac{d(r^2 v_r)}{dr}  \biggr] 
= -\rho \biggl[ \frac{dv_r}{dr} + \frac{2v_r}{r} \biggr] </math><br />


<span id="PGE:Euler"><font color="#770000">'''Euler + Poisson Equations'''</font></span><br />

<math>\frac{dv_r}{dt} = - \frac{1}{\rho}\frac{dP}{dr} - \frac{GM_r}{r^2} </math><br />



<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />

{{ Math/EQ_FirstLaw02 }}

</div>
where, 

<div align="center">
<math>v_r \equiv \frac{dr}{dt}</math> ,
</div>

and, as before, the mass enclosed inside radius <math>r</math> is,

<div align="center">
<math>M_r \equiv \int_0^r dm_r = \int_0^r 4\pi r^2 \rho dr</math> .
</div>

===Consistent Lagrangian Formulation===

The (Lagrangian) time derivatives in these equations define how a given physical parameter &#8212; for example, {{Math/VAR_Density01}}, <math>v_r</math>, or {{Math/VAR_SpecificInternalEnergy01}} &#8212; should vary with time in a (Lagrangian) fluid element that is not fixed in space but, rather, is moving along with the flow.  However, the radial derivatives describe the spatial variation of various physical parameters as measured at fixed locations in space; that is, as written, the radial derivatives do not track conditions as viewed by a (Lagrangian) fluid element that is moving along with the flow because the position <math>r</math> of each (Lagrangian) fluid element is itself changing with time.  A proper Lagrangian representation of the spatial derivatives can be formulated in the case of one-dimensional, spherically symmetric flows by using <math>M_r</math> (or, equivalently, <math>m_r</math>) instead of <math>r</math> as the independent variable. Making the substitution,
<div align="center">
<math>\frac{d}{dr} = \frac{dM_r}{dr}\frac{d}{dM_r} = 4\pi \rho r^2 \frac{d}{dM_r} \, ,</math> 
</div>
in the first two equations above gives, respectively,

<div align="center">
<math>\frac{d\rho}{dt} = - 4\pi \rho^2 r^2  \frac{dv_r}{dM_r} - \frac{2\rho v_r}{r}  \, ,</math> 
</div>

and,

<div align="center">
<math>\frac{dv_r}{dt} = - 4\pi r^2 \frac{dP}{dM_r} - \frac{GM_r}{r^2} \, .</math>
</div>

===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)
</div>
As a result, the adiabatic form of the <math>1^\mathrm{st}</math> law of thermodynamics can be written as,
<div align="center">
<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>

===Summary Set of Nonlinear Governing Relations===
In summary, the following three one-dimensional ODEs define the physical relationship between the three dependent variables {{Math/VAR_Density01}}, {{Math/VAR_Pressure01}}, and <math>~r</math>, each of which should be expressible as a function of the two independent (Lagrangian) variables, {{Math/VAR_Time01}} and <math>~M_r</math>:

<div align="center">
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />
<math>\frac{d\rho}{dt} = - 4\pi \rho^2 r^2  \frac{d}{dM_r}\biggl(\frac{dr}{dt}\biggr) - \frac{2\rho}{r} \biggl(\frac{dr}{dt}\biggr)  </math><br /> ,

<span id="PGE:Euler"><font color="#770000">'''Euler + Poisson Equations'''</font></span><br />
<math>\frac{d^2 r}{dt^2} = - 4\pi r^2 \frac{dP}{dM_r} - \frac{GM_r}{r^2} </math><br />


<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />
<math>
\rho \frac{dP}{dt} - \gamma_\mathrm{g} P \frac{d\rho}{dt} = 0 .
</math>

</div>