Difference between revisions of "User:Tohline/Appendix/Ramblings/RadiationHydro"

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{{LSU_HBook_header}}
{{LSU_HBook_header}}


==Principal Governing Equations==
==Governing Equations==
===Ignoring the Effects of Magnetic Fields===
===Hayes et al. (2006) — But Ignoring the Effects of Magnetic Fields===
First, referencing §2 of [http://adsabs.harvard.edu/abs/2006ApJS..165..188H J. C. Hayes et al. (2006, ApJS, 165, 188 - 228)] — alternatively see §2.1 of [http://adsabs.harvard.edu/abs/2012ApJS..199...35M D. C. Marcello & J. E. Tohline (2012, ApJS, 199, id. 35, 29 pp)] — 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,
First, referencing §2 of [http://adsabs.harvard.edu/abs/2006ApJS..165..188H J. C. Hayes et al. (2006, ApJS, 165, 188 - 228)] — alternatively see §2.1 of [http://adsabs.harvard.edu/abs/2012ApJS..199...35M D. C. Marcello & J. E. Tohline (2012, ApJS, 199, id. 35, 29 pp)] — 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">
<div align="center">
Line 50: Line 50:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\frac{d}{dt} \biggl( \frac{e}{\rho}\biggr)</math>
<math>~\rho \frac{d}{dt} \biggl( \frac{e}{\rho}\biggr) + P\nabla \cdot \vec{v} </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 57: Line 57:
   <td align="left">
   <td align="left">
<math>~
<math>~
- \frac{P}{\rho}\nabla \cdot \vec{v} + \frac{1}{\rho} \biggl[c\kappa_E E_\mathrm{rad} - 4\pi \kappa_p B_p\biggr] \, ,
c\kappa_E E_\mathrm{rad} - 4\pi \kappa_p B_p \, ,
</math>
</math>
   </td>
   </td>
Line 64: Line 64:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\frac{d}{dt} \biggl( \frac{E_\mathrm{rad}}{\rho}\biggr)</math>
<math>~\rho \frac{d}{dt} \biggl( \frac{E_\mathrm{rad}}{\rho}\biggr)</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 71: Line 71:
   <td align="left">
   <td align="left">
<math>~
<math>~
- \frac{1}{\rho} \biggl[ \nabla \cdot \vec{F} + \vec{\bold{P}}:\nabla{\vec{v}} + c\kappa_E E_\mathrm{rad} - 4\pi \kappa_p B_p \biggr] \, .
- \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>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</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>
{{User:Tohline/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 [[User:Tohline/Appendix/Variables_templates|associated appendix]] among our list of key physical constants &#8212; is,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
{{ User:Tohline/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===
===Optically Thick Regime===
Line 130: Line 399:
</table>
</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>
{{ User:Tohline/Math/EQ_SShydrostaticBalance01 }}
<br />
<font color="#770000">'''Mass Conservation'''</font>
{{ User:Tohline/Math/EQ_SSmassConservation01 }}
<br />
<font color="#770000">'''Energy Conservation'''</font>
{{ User:Tohline/Math/EQ_SSenergyConservation01 }}
<br />
<font color="#770000">'''Radiation Transport'''</font>
{{ User:Tohline/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=
=Related Discussions=

Latest revision as of 17:45, 5 July 2021


Radiation-Hydrodynamics

Whitworth's (1981) Isothermal Free-Energy Surface
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Governing Equations

Hayes et al. (2006) — But Ignoring the Effects of Magnetic Fields

First, referencing §2 of J. C. Hayes et al. (2006, ApJS, 165, 188 - 228) — alternatively see §2.1 of D. C. Marcello & J. E. Tohline (2012, ApJS, 199, id. 35, 29 pp) — 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,

Poisson Equation

LSU Key.png

<math>\nabla^2 \Phi = 4\pi G \rho</math>

Hayes et al. (2006), p. 190, Eq. (15)

the,

Continuity Equation

LSU Key.png

<math>\frac{d\rho}{dt} + \rho \nabla \cdot \vec{v} = 0</math>

and — ignoring magnetic fields — a modified version of the,

Lagrangian Representation
of the Euler Equation,

<math>~\frac{d\vec{v}}{dt}</math>

<math>~=</math>

<math>~ - \frac{1}{\rho}\nabla P - \nabla \Phi + \frac{1}{\rho}\biggl(\frac{\chi}{c}\biggr) \vec{F} \, , </math>

plus the following pair of additional energy-conservation-based dynamical equations:

<math>~\rho \frac{d}{dt} \biggl( \frac{e}{\rho}\biggr) + P\nabla \cdot \vec{v} </math>

<math>~=</math>

<math>~ c\kappa_E E_\mathrm{rad} - 4\pi \kappa_p B_p \, , </math>

<math>~\rho \frac{d}{dt} \biggl( \frac{E_\mathrm{rad}}{\rho}\biggr)</math>

<math>~=</math>

<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>

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

First Law of Thermodynamics

LSU Key.png

<math>T \frac{ds}{dt} = \frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr)</math>

we can write,

<math>~\rho T\frac{ds}{dt}</math>

<math>~=</math>

<math>~ \rho \frac{d\epsilon}{dt} - \frac{P}{\rho} \frac{d\rho}{dt} </math>

 

<math>~=</math>

<math>~ \rho \frac{d\epsilon}{dt} + P\nabla\cdot \vec{v} \, . </math>

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 1st 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,

<math>~\rho T \frac{ds}{dt}</math>

<math>~=</math>

<math>~c\kappa_E E_\mathrm{rad} - 4\pi \kappa_p B_p \, .</math>

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 material) time derivative by its Eulerian counterpart, the left-hand side can be rewritten as,

<math>~\rho \frac{d}{dt} \biggl( \frac{E_\mathrm{rad}}{\rho}\biggr)</math>

<math>~=</math>

<math>~ \frac{dE_\mathrm{rad}}{dt} - \frac{E_\mathrm{rad}}{\rho}~\frac{d\rho}{dt} </math>

 

<math>~=</math>

<math>~ \frac{dE_\mathrm{rad}}{dt} + E_\mathrm{rad}\nabla\cdot \vec{v} </math>

 

<math>~=</math>

<math>~ \frac{\partial E_\mathrm{rad}}{\partial t} + \vec{v}\cdot \nabla E_\mathrm{rad}+ E_\mathrm{rad}\nabla\cdot \vec{v} </math>

 

<math>~=</math>

<math>~ \frac{\partial E_\mathrm{rad}}{\partial t} + \nabla\cdot (E_\mathrm{rad} \vec{v}) \, , </math>

which provides an alternate form of the expression, as found for example in equation (4) of Marcello & 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,

<math>~E_\mathrm{rad}</math>

<math>~=</math>

<math>~a_\mathrm{rad}T^4 \, ,</math>

and each fluid element will radiate — and, hence lose some of its internal energy to the surrounding radiation field — at a rate that is governed by the integrated Planck function,

<math>~B_p = \frac{\sigma}{\pi}T^4 </math>

<math>~=</math>

<math>~\frac{ca_\mathrm{rad}}{4\pi} T^4 \, ,</math>

where, <math>~\sigma \equiv \tfrac{1}{4}c a_\mathrm{rad}</math>, is the Stefan-Boltzmann constant, and the radiation constant — which is included in an associated appendix among our list of key physical constants — is,

<math>~a_\mathrm{rad}</math>

<math>~\equiv</math>

<math>~\frac{8\pi^5}{15}\frac{k^4}{(hc)^3} \, .</math>

Also under these conditions, it can be shown that — see, for example, discussion associated with equations (12) and (18) in Marcello & J. E. Tohline (2012)

<math>~ \bold{P}_\mathrm{st} :\nabla{\vec{v}}</math>

<math>~\rightarrow</math>

<math>~\frac{E_\mathrm{rad}}{3} \nabla \cdot \vec{v} \, ,</math>

and,

<math>~\vec{F}</math>

<math>~\rightarrow</math>

<math>~- \frac{1}{3}\biggl(\frac{c}{\chi}\biggr) \nabla E_\mathrm{rad} \, ,</math>

which implies,

<math>~\biggl(\frac{\chi}{c}\biggr) \vec{F}</math>

<math>~\rightarrow</math>

<math>~-\nabla P_\mathrm{rad} \, ,</math>

where we have recognized that the radiation pressure,

<math>~P_\mathrm{rad} = \frac{1}{3}E_\mathrm{rad}</math>

<math>~=</math>

<math>~\frac{1}{3}a_\mathrm{rad}T^4 \, .</math>

Hence, the modified Euler equation becomes,

<math>~\rho ~ \frac{d\vec{v}}{dt}</math>

<math>~=</math>

<math>~ - \nabla (P+P_\mathrm{rad}) - \rho \nabla \Phi \, , </math>

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

<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>

<math>~=</math>

<math>~ - \nabla \cdot \vec{F} - c\kappa_E E_\mathrm{rad} + 4\pi \kappa_p B_p \, . </math>


Optically Thick Regime

In the optically thick regime, the following conditions hold:

<math>~c\kappa_E E_\mathrm{rad}</math>

<math>~\rightarrow</math>

<math>~4\pi \kappa_p B_p \, ,</math>

<math>~E_\mathrm{rad}</math>

<math>~\rightarrow</math>

<math>~aT^4 \, ,</math>

<math>~\biggl(\frac{\chi}{c}\biggr) \vec{F}</math>

<math>~\rightarrow</math>

<math>~- \nabla \biggl(\frac{aT^4}{3} \biggr) \, ,</math>

<math>~ \vec{\bold{P}}:\nabla{\vec{v}}</math>

<math>~\rightarrow</math>

<math>~\frac{E_\mathrm{rad}}{3} \nabla \cdot \vec{v} \, .</math>

Start with,

<math>~Tds_\mathrm{rad} = dQ</math>

<math>~=</math>

<math>~ d\biggl(\frac{E_\mathrm{rad}}{\rho} \biggr) + P_\mathrm{rad~}d\biggl( \frac{1}{\rho} \biggr) </math>

 

<math>~=</math>

<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>

 

<math>~=</math>

<math>~ \frac{1}{\rho}~d (aT^4 ) + \frac{4}{3} aT^4~d\biggl( \frac{1}{\rho} \biggr) </math>

 

<math>~=</math>

<math>~ \frac{4aT^3}{\rho}~dT + \frac{4}{3} aT^4~d\biggl( \frac{1}{\rho} \biggr) </math>

 

<math>~=</math>

<math>~ \frac{4aT}{3} \biggl[ \frac{3T^2}{\rho}~dT + T^3~d\biggl( \frac{1}{\rho} \biggr) \biggr] </math>

 

<math>~=</math>

<math>~ \frac{4aT}{3} ~d\biggl( \frac{T^3}{\rho} \biggr) </math>

<math>~\Rightarrow ~~~ ds_\mathrm{rad}</math>

<math>~=</math>

<math>~ ~d\biggl( \frac{4aT^3}{3\rho} \biggr) </math>

Integrating then gives us,

<math>~s_\mathrm{rad}</math>

<math>~=</math>

<math>~ ~\frac{4aT^3}{3\rho} + \mathrm{const.} </math>

D. D. Clayton (1968), Eq. (2-136)
[Shu92], Vol. I, §9, immediately following Eq. (9.22)

This also means that,

<math>~\rho \frac{d}{dt} \biggl( \frac{E_\mathrm{rad}}{\rho}\biggr) + \frac{E_\mathrm{rad}}{3} \nabla\cdot\vec{v}</math>

<math>~=</math>

<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>

 

<math>~=</math>

<math>~ \frac{dE_\mathrm{rad}}{dt} + \frac{4E_\mathrm{rad}}{3} \nabla\cdot\vec{v} </math>

 

<math>~=</math>

<math>~\frac{4E_\mathrm{rad}}{3} \biggl[ \frac{3}{4} \cdot \frac{d\ln E_\mathrm{rad}}{dt} + \nabla\cdot\vec{v} \biggr] </math>

 

<math>~=</math>

<math>~\frac{4E_\mathrm{rad}}{3} \biggl[ \frac{d\ln (E_\mathrm{rad})^{3/4}}{dt} + \nabla\cdot\vec{v} \biggr] </math>

 

<math>~=</math>

<math>~\frac{4E_\mathrm{rad}}{3} \biggl[ \frac{d\ln T^3}{dt} - \frac{d\ln\rho}{dt} \biggr] </math>

 

<math>~=</math>

<math>~\frac{4E_\mathrm{rad}}{3} \biggl[ \frac{d\ln (T^3/\rho)}{dt} \biggr] </math>

 

<math>~=</math>

<math>~\frac{4aT^4}{3} \biggl( \frac{\rho}{T^3}\biggr) \biggl[ \frac{d(T^3/\rho)}{dt} \biggr] </math>

 

<math>~=</math>

<math>~ \rho T\biggl[ \frac{ds_\mathrm{rad}}{dt} \biggr] \, . </math>

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,

<math>~\rho T~\frac{ds_\mathrm{rad}}{dt} </math>

<math>~=</math>

<math>~ - \nabla \cdot \vec{F} - c\kappa_E E_\mathrm{rad} + 4\pi \kappa_p B_p \, . </math>

[Shu92], §9, Eq. (9.22)

Traditional Stellar Structure Equations

Hydrostatic Balance

LSU Key.png

<math>~\frac{dP}{dr} = - \frac{GM_r \rho}{r^2}</math>


Mass Conservation

LSU Key.png

<math>~\frac{dM_r}{dr} = 4\pi r^2 \rho</math>


Energy Conservation

LSU Key.png

<math>~\frac{dL_r}{dr} = 4\pi r^2 \rho \epsilon_\mathrm{nuc}</math>


Radiation Transport

LSU Key.png

<math>~\frac{dT}{dr} = - \frac{ 3 }{ 4a_\mathrm{rad} c} \biggl(\frac{ \kappa \rho }{ T^3 }\biggr) \frac{ L_r }{ 4\pi r^2 }</math>


M. Schwarzschild (1958), Chapter III, §12, Eqs. (12.1), (12.2), (12.3), (12.4)
D. D. Clayton (1968), Chapter 6, Eqs. (6-1), (6-2), (6-3a), (6-4a)
[HK94], Eqs. (1.5), (1.1), (1.54), (1.57)
[KW94], Eqs. (1.2), (2.4), (4.22), (5.11)
W. K. Rose (1998), Eqs. (2.27), (2.28), (2.xx), (2.80)
[P00], Vol. II, Eqs. (2.1), (2.2), (2.18), (2.8)
A. R. Choudhuri (2010), Chapter 3, Eqs. (3.2), (3.1), (3.15), (3.16)
D. Maoz (2016), §3.5, Eqs. (3.56), (3.57), (3.59), (3.58)


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Whitworth's (1981) Isothermal Free-Energy Surface

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Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation