User:Tohline/Appendix/Ramblings/RadiationHydro - Revision history

Tohline: /* Optically Thick Regime */

2021-07-05T17:45:45Z

Optically Thick Regime

← Older revision		Revision as of 17:45, 5 July 2021
Line 521:		Line 521:

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

Tohline: /* Traditional Stellar Structure Equations */

2018-10-28T19:05:42Z

Traditional Stellar Structure Equations

← Older revision		Revision as of 19:05, 28 October 2018
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	[http://adsabs.harvard.edu/abs/1958ses..book.....S M. Schwarzschild (1958)], Chapter III, §12, Eqs. (12.1), (12.2), (12.3), (12.4)<br />		[http://adsabs.harvard.edu/abs/1958ses..book.....S M. Schwarzschild (1958)], Chapter III, §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 />		[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 />		[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/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)], §3.5, Eqs. (3.56), (3.57), (3.59), (3.58)~~<br />~~		[http://adsabs.harvard.edu/abs/2016asnu.book.....M D. Maoz (2016)], §3.5, Eqs. (3.56), (3.57), (3.59), (3.58)
	~~[<b>[[User:Tohline/Appendix/References#Shu92\|<font color="red">Shu92</font>]]</b>], §9, immediately following Eq. (9.22)~~

	</div>		</div>

Tohline: /* Traditional Stellar Structure Equations */

2018-10-28T04:18:34Z

Traditional Stellar Structure Equations

← Older revision		Revision as of 04:18, 28 October 2018
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	<div align="center">		<div align="center">
			<font color="#770000">'''Hydrostatic Balance'''</font>
			{{ User:Tohline/Math/EQ_SShydrostaticBalance01 }}
			<br />
	<font color="#770000">'''Mass Conservation'''</font>		<font color="#770000">'''Mass Conservation'''</font>
	{{ User:Tohline/Math/EQ_SSmassConservation01 }}		{{ User:Tohline/Math/EQ_SSmassConservation01 }}
	<br />		<br />
	<font color="#770000">'''~~Hydrostatic Balance~~'''</font>		<font color="#770000">'''Energy Conservation'''</font>
	{{ User:Tohline/Math/~~EQ_SShydrostaticBalance01~~ }}		{{ User:Tohline/Math/EQ_SSenergyConservation01 }}
	<br />		<br />
	<font color="#770000">'''Radiation Transport'''</font>		<font color="#770000">'''Radiation Transport'''</font>
	{{ User:Tohline/Math/EQ_SSradiationTransport01 }}		{{ User:Tohline/Math/EQ_SSradiationTransport01 }}
	<br />		<br />
	<font color="~~#770000~~">~~'''Energy Conservation'''~~</font>
	~~{{ User:Tohline/Math/EQ_SSenergyConservation01 }}~~		[http://adsabs.harvard.edu/abs/1958ses..book.....S M. Schwarzschild (1958)], Chapter III, §12, Eqs. (12.1), (12.2), (12.3), (12.4)<br />
	<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 />
			[http://adsabs.harvard.edu/abs/1998asa..book.....R W. K. Rose (1998)], Eqs. (2.27), (2.28), (2.xx), (2.80)<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)], §3.5, Eqs. (3.56), (3.57), (3.59), (3.58)<br />
			[<b>[[User:Tohline/Appendix/References#Shu92\|<font color="red">Shu92</font>]]</b>], §9, immediately following Eq. (9.22)

	</div>		</div>

Tohline at 03:44, 28 October 2018

2018-10-28T03:44:54Z

← Older revision		Revision as of 03:44, 28 October 2018
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	[<b>[[User:Tohline/Appendix/References#Shu92\|<font color="red">Shu92</font>]]</b>], §9, Eq. (9.22)		[<b>[[User:Tohline/Appendix/References#Shu92\|<font color="red">Shu92</font>]]</b>], §9, Eq. (9.22)
			</div>

			=Traditional Stellar Structure Equations=

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

	</div>		</div>

Tohline: /* Governing Equations */

2018-10-27T17:12:19Z

Governing Equations

← Older revision		Revision as of 17:12, 27 October 2018
Line 642:		Line 642:
	</tr>		</tr>
	</table>		</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>], §9, Eq. (9.22)
	</div>		</div>

Tohline: /* Optically Thick Regime */

2018-10-27T17:03:16Z

Optically Thick Regime

← Older revision		Revision as of 17:03, 27 October 2018
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\~~rho~~ \~~biggl[ \frac~~{d}{dt} ~~\biggl(~~ \frac{E_\mathrm{rad}}{\rho}~~\biggr) +~~ \frac{1}{3} ~~\biggl(~~\frac{E_\mathrm{rad}}{~~\rho~~} ~~\biggr)~~ \nabla\cdot\vec{v} ~~\biggr]~~		\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>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{E_\mathrm{rad}}{3} \biggl[ \frac{d\ln(E_\mathrm{rad}~~/\rho~~)^3}{dt} + \nabla\cdot\vec{v} \biggr]		\frac{dE_\mathrm{rad}}{dt} + \frac{4E_\mathrm{rad}}{3} \nabla\cdot\vec{v}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</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">

			</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">

			</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">

			</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">

			</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">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\rho T\biggl[ \frac{ds_\mathrm{rad}}{dt} \biggr] \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Governing Equations */

2018-10-27T05:17:22Z

Governing Equations

← Older revision		Revision as of 05:17, 27 October 2018
Line 522:		Line 522:
	[http://adsabs.harvard.edu/abs/1968psen.book.....C D. D. Clayton (1968)], Eq. (2-136)<br />		[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>], §9, immediately following Eq. (9.22)		[<b>[[User:Tohline/Appendix/References#Shu92\|<font color="red">Shu92</font>]]</b>], §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>~
			\rho \biggl[ \frac{d}{dt} \biggl( \frac{E_\mathrm{rad}}{\rho}\biggr) + \frac{1}{3} \biggl(\frac{E_\mathrm{rad}}{\rho} \biggr) \nabla\cdot\vec{v} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

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

Tohline: /* Governing Equations */

2018-10-27T04:40:12Z

Governing Equations

← Older revision		Revision as of 04:40, 27 October 2018
Line 398:		Line 398:
	</tr>		</tr>
	</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">

			</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">

			</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">

			</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">

			</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">

			</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>], §9, immediately following Eq. (9.22)
			</div>

	=Related Discussions=		=Related Discussions=

Tohline: /* Governing Equations */

2018-10-22T20:22:43Z

Governing Equations

← Older revision		Revision as of 20:22, 22 October 2018
Line 78:		Line 78:
	where, in this last expression, <math>~\bold{P}_\mathrm{st}</math> is the radiation stress tensor.		where, in this last expression, <math>~\bold{P}_\mathrm{st}</math> is the radiation stress tensor.

	===Various ~~Manipulations~~===		===Various Realizations===
	====First Law====		====First Law====

Line 246:		Line 246:
	</tr>		</tr>
	</table>		</table>
			Also under these conditions, it can be shown that — see, for example, discussion associated with equations (12) and (18) in [http://adsabs.harvard.edu/abs/2012ApJS..199...35M Marcello & J. E. Tohline (2012)] —

			<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===

Tohline: /* Thermodynamic Equilibrium */

2018-10-22T19:45:23Z

Thermodynamic Equilibrium

← Older revision		Revision as of 19:45, 22 October 2018
Line 200:		Line 200:

	====Thermodynamic Equilibrium====		====Thermodynamic Equilibrium====
	In thermodynamic equilibrium at temperature, <math>~T</math>, the energy-density of the radiation field is,		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">		<table border="0" cellpadding="5" align="center">