Editing Appendix/Ramblings/Radiation/RadHydro

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=Radiation-Hydrodynamics=

==Governing Equations==
===Hayes et al. (2006) &#8212; But Ignoring the Effects of Magnetic Fields===
First, referencing &sect;2 of [http://adsabs.harvard.edu/abs/2006ApJS..165..188H J. C. Hayes et al. (2006, ApJS, 165, 188 - 228)] &#8212; alternatively see &sect;2.1 of [http://adsabs.harvard.edu/abs/2012ApJS..199...35M D. C. Marcello &amp; J. E. Tohline (2012, ApJS, 199, id. 35, 29 pp)] &#8212; we see that the set of principal governing equations that is typically used in the astrophysics community to include the effects of radiation on self-gravitating fluid flows includes the,
<div align="center">
<span id="Poisson"><font color="#770000">'''Poisson Equation'''</font></span>

{{ Math/EQ_Poisson01 }}

[http://adsabs.harvard.edu/abs/2006ApJS..165..188H Hayes et al. (2006)], p. 190, Eq. (15)
</div>
the,
<div align="center">
<span id="Continuity"><font color="#770000">'''Continuity Equation'''</font></span>

{{ Math/EQ_Continuity01 }}
</div>

and &#8212; ignoring magnetic fields &#8212; a modified version of the, 

<div align="center">
<span id="ConservingMomentum:Lagrangian"><font color="#770000">'''Lagrangian Representation'''</font></span><br />
of the Euler Equation,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{d\vec{v}}{dt}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \frac{1}{\rho}\nabla P - \nabla \Phi + \frac{1}{\rho}\biggl(\frac{\chi}{c}\biggr) \vec{F} \, ,
</math>
  </td>
</tr>
</table>
</div>
plus the following pair of additional ''energy-conservation-based'' dynamical equations:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\rho \frac{d}{dt} \biggl( \frac{e}{\rho}\biggr) + P\nabla \cdot \vec{v} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
c\kappa_E E_\mathrm{rad} - 4\pi \kappa_p B_p \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\rho \frac{d}{dt} \biggl( \frac{E_\mathrm{rad}}{\rho}\biggr)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \biggl[ \nabla \cdot \vec{F} + \bold{P}_\mathrm{st}:\nabla{\vec{v}} + c\kappa_E E_\mathrm{rad} - 4\pi \kappa_p B_p \biggr]  \, ,
</math>
  </td>
</tr>
</table>
where, in this last expression, <math>\bold{P}_\mathrm{st}</math> is the radiation stress tensor.

===Various Realizations=== 
====First Law====

By combining the continuity equation with the

<div id="PGE:FirstLaw" align="center">
<font color="#770000">'''First Law of Thermodynamics'''</font>

{{ Math/EQ_FirstLaw01 }}
</div>

we can write,
<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>
\rho \frac{d\epsilon}{dt} - \frac{P}{\rho} \frac{d\rho}{dt}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\rho \frac{d\epsilon}{dt} + P\nabla\cdot \vec{v} \, .
</math>
  </td>
</tr>
</table>
Given that the specific internal energy <math>(\epsilon)</math> and the internal energy density <math>(e)</math> are related via the expression, <math>\epsilon = e/\rho</math>, we appreciate that the first of the above-identified ''energy-conservation-based'' dynamical equations is simply a restatement of the 1<sup>st</sup> Law of Thermodynamics in the context of a physical system whose fluid elements gain or lose entropy as a result of the (radiation-transport-related) source and sink terms,
<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>c\kappa_E E_\mathrm{rad} - 4\pi \kappa_p B_p \, .</math>
  </td>
</tr>
</table>

====Energy-Density of Radiation Field====
By combining the left-hand side of the second of the above-identified ''energy-conservation-based'' dynamical equations with the continuity equation, then replacing the Lagrangian (that is, the [https://en.wikipedia.org/wiki/Material_derivative ''material'']) time derivative by its Eulerian counterpart, the left-hand side can be rewritten as, 

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

<tr>
  <td align="right">
<math>\rho \frac{d}{dt} \biggl( \frac{E_\mathrm{rad}}{\rho}\biggr)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{dE_\mathrm{rad}}{dt} - \frac{E_\mathrm{rad}}{\rho}~\frac{d\rho}{dt}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{dE_\mathrm{rad}}{dt} + E_\mathrm{rad}\nabla\cdot \vec{v}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\partial E_\mathrm{rad}}{\partial t} + \vec{v}\cdot \nabla E_\mathrm{rad}+ E_\mathrm{rad}\nabla\cdot \vec{v}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\partial E_\mathrm{rad}}{\partial t} + \nabla\cdot (E_\mathrm{rad} \vec{v}) \, ,
</math>
  </td>
</tr>
</table>
which provides an alternate form of the expression, as found for example in equation (4) of [http://adsabs.harvard.edu/abs/2012ApJS..199...35M Marcello &amp; J. E. Tohline (2012)].

====Thermodynamic Equilibrium====
In an optically thick environment that is in thermodynamic equilibrium at temperature, <math>T</math>, the energy-density of the radiation field is,

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

<tr>
  <td align="right">
<math>E_\mathrm{rad}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>a_\mathrm{rad}T^4 \, ,</math>
  </td>
</tr>
</table>
and each fluid element will radiate &#8212; and, hence lose some of its internal energy to the surrounding radiation field &#8212; at a rate that is governed by the integrated Planck function,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>B_p = \frac{\sigma}{\pi}T^4 </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{ca_\mathrm{rad}}{4\pi} T^4 \, ,</math>
  </td>
</tr>
</table>
where, <math>\sigma \equiv \tfrac{1}{4}c a_\mathrm{rad}</math>, is the Stefan-Boltzmann constant, and the ''radiation constant'' &#8212; which is included in an [[Appendix/VariablesTemplates|associated appendix]] among our list of key physical constants &#8212; is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
{{ Math/C_RadiationConstant }}
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{8\pi^5}{15}\frac{k^4}{(hc)^3} \, .</math>
  </td>
</tr>
</table>
Also under these conditions, it can be shown that &#8212; see, for example, discussion associated with equations (12) and (18) in [http://adsabs.harvard.edu/abs/2012ApJS..199...35M Marcello &amp; J. E. Tohline (2012)] &#8212;

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

<tr>
  <td align="right">
<math>\bold{P}_\mathrm{st} :\nabla{\vec{v}}</math>
  </td>
  <td align="center">
<math>\rightarrow</math>
  </td>
  <td align="left">
<math>\frac{E_\mathrm{rad}}{3} \nabla \cdot \vec{v} \, ,</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\vec{F}</math>
  </td>
  <td align="center">
<math>\rightarrow</math>
  </td>
  <td align="left">
<math>- \frac{1}{3}\biggl(\frac{c}{\chi}\biggr) \nabla E_\mathrm{rad} \, ,</math>
  </td>
</tr>
</table>
which implies,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl(\frac{\chi}{c}\biggr) \vec{F}</math>
  </td>
  <td align="center">
<math>\rightarrow</math>
  </td>
  <td align="left">
<math>-\nabla P_\mathrm{rad} \, ,</math>
  </td>
</tr>
</table>
where we have recognized that the radiation pressure,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>P_\mathrm{rad} = \frac{1}{3}E_\mathrm{rad}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{3}a_\mathrm{rad}T^4 \, .</math>
  </td>
</tr>
</table>

Hence, the modified Euler equation becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\rho ~ \frac{d\vec{v}}{dt}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \nabla (P+P_\mathrm{rad}) - \rho \nabla \Phi  \, ,
</math>
  </td>
</tr>
</table>

and the equation governing the time-dependent behavior of <math>E_\mathrm{rad}</math> becomes,

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

<tr>
  <td align="right">
<math>\frac{\partial E_\mathrm{rad}}{\partial t} + \nabla\cdot (E_\mathrm{rad} \vec{v}) + \frac{1}{3}E_\mathrm{rad} \nabla \cdot \vec{v} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \nabla \cdot \vec{F} - c\kappa_E E_\mathrm{rad} + 4\pi \kappa_p B_p   \, .
</math>
  </td>
</tr>
</table>

===Optically Thick Regime===
In the optically thick regime, the following conditions hold:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~c\kappa_E E_\mathrm{rad}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~4\pi \kappa_p B_p \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~E_\mathrm{rad}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~aT^4 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl(\frac{\chi}{c}\biggr) \vec{F}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~- \nabla \biggl(\frac{aT^4}{3} \biggr) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ \vec{\bold{P}}:\nabla{\vec{v}}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\frac{E_\mathrm{rad}}{3} \nabla \cdot \vec{v} \, .</math>
  </td>
</tr>
</table>

Start with,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~Tds_\mathrm{rad} = dQ</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
d\biggl(\frac{E_\mathrm{rad}}{\rho} \biggr) + P_\mathrm{rad~}d\biggl( \frac{1}{\rho} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\rho}~d E_\mathrm{rad} + E_\mathrm{rad~}d\biggl( \frac{1}{\rho} \biggr) + P_\mathrm{rad~}d\biggl( \frac{1}{\rho} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\rho}~d (aT^4 ) + \frac{4}{3} aT^4~d\biggl( \frac{1}{\rho} \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{4aT^3}{\rho}~dT + \frac{4}{3} aT^4~d\biggl( \frac{1}{\rho} \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{4aT}{3} \biggl[ \frac{3T^2}{\rho}~dT + T^3~d\biggl( \frac{1}{\rho} \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{4aT}{3} ~d\biggl( \frac{T^3}{\rho} \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ ds_\mathrm{rad}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
~d\biggl( \frac{4aT^3}{3\rho} \biggr) 
</math>
  </td>
</tr>
</table>

Integrating then gives us,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~s_\mathrm{rad}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
~\frac{4aT^3}{3\rho} + \mathrm{const.} 
</math>
  </td>
</tr>
</table>

[http://adsabs.harvard.edu/abs/1968psen.book.....C D. D. Clayton (1968)], Eq. (2-136)<br />
[<b>[[User:Tohline/Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Vol. I, &sect;9, immediately following Eq. (9.22)
</div>

This also means that,

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

<tr>
  <td align="right">
<math>~\rho \frac{d}{dt} \biggl( \frac{E_\mathrm{rad}}{\rho}\biggr) + \frac{E_\mathrm{rad}}{3} \nabla\cdot\vec{v}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{dE_\mathrm{rad}}{dt} - \frac{E_\mathrm{rad}}{\rho} \frac{d\rho}{dt}
+ \frac{E_\mathrm{rad}}{3} \nabla\cdot\vec{v}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{dE_\mathrm{rad}}{dt} + \frac{4E_\mathrm{rad}}{3} \nabla\cdot\vec{v}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4E_\mathrm{rad}}{3} 
\biggl[ \frac{3}{4} \cdot \frac{d\ln E_\mathrm{rad}}{dt} + \nabla\cdot\vec{v} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4E_\mathrm{rad}}{3} 
\biggl[ \frac{d\ln (E_\mathrm{rad})^{3/4}}{dt} + \nabla\cdot\vec{v} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4E_\mathrm{rad}}{3} 
\biggl[ \frac{d\ln T^3}{dt} - \frac{d\ln\rho}{dt} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4E_\mathrm{rad}}{3} 
\biggl[ \frac{d\ln (T^3/\rho)}{dt}  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4aT^4}{3} \biggl( \frac{\rho}{T^3}\biggr)
\biggl[ \frac{d(T^3/\rho)}{dt}  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\rho T\biggl[ \frac{ds_\mathrm{rad}}{dt}  \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Hence, the equation governing the time-dependent behavior of <math>~E_\mathrm{rad}</math> becomes an expression detailing the time-dependent behavior of the specific entropy, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho T~\frac{ds_\mathrm{rad}}{dt}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \nabla \cdot \vec{F} - c\kappa_E E_\mathrm{rad} + 4\pi \kappa_p B_p   \, .
</math>
  </td>
</tr>
</table>

[<b>[[User:Tohline/Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], &sect;9, Eq. (9.22)
</div>

=Traditional Stellar Structure Equations=

<div align="center">
<font color="#770000">'''Hydrostatic Balance'''</font>
{{ Math/EQ_SShydrostaticBalance01 }}
<br />
<font color="#770000">'''Mass Conservation'''</font>
{{ Math/EQ_SSmassConservation01 }}
<br />
<font color="#770000">'''Energy Conservation'''</font>
{{ Math/EQ_SSenergyConservation01 }}
<br />
<font color="#770000">'''Radiation Transport'''</font>
{{ Math/EQ_SSradiationTransport01 }}
<br />

[http://adsabs.harvard.edu/abs/1958ses..book.....S M. Schwarzschild (1958)], Chapter III, &sect;12, Eqs. (12.1), (12.2), (12.3), (12.4)<br />
[http://adsabs.harvard.edu/abs/1968psen.book.....C D. D. Clayton (1968)], Chapter 6, Eqs. (6-1), (6-2), (6-3a), (6-4a)<br />
[<b>[[User:Tohline/Appendix/References#HK94|<font color="red">HK94</font>]]</b>], Eqs. (1.5), (1.1), (1.54), (1.57)<br />
[<b>[[User:Tohline/Appendix/References#KW94|<font color="red">KW94</font>]]</b>], Eqs. (1.2), (2.4), (4.22), (5.11)<br />
[http://adsabs.harvard.edu/abs/1998asa..book.....R W. K. Rose (1998)], Eqs. (2.27), (2.28), (2.xx), (2.80)<br />
[<b>[[User:Tohline/Appendix/References#P00|<font color="red">P00</font>]]</b>], Vol. II, Eqs. (2.1), (2.2), (2.18), (2.8)<br />
[http://adsabs.harvard.edu/abs/2010asph.book.....C A. R. Choudhuri (2010)], Chapter 3, Eqs. (3.2), (3.1), (3.15), (3.16)<br />
[http://adsabs.harvard.edu/abs/2016asnu.book.....M D. Maoz (2016)], &sect;3.5, Eqs. (3.56), (3.57), (3.59), (3.58)


</div>

=Related Discussions=
* Euler equation viewed from a [[User:Tohline/PGE/RotatingFrame|rotating frame of reference]].
* An [[User:Tohline/PGE/ConservingMomentum#Euler_Equation|earlier draft of this "Euler equation" presentation]].

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