User:Tohline/SR - Revision history

Tohline: /* Barotropic Structure */

2019-08-09T15:26:53Z

Barotropic Structure

← Older revision		Revision as of 15:26, 9 August 2019
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	<td align="center">		<td align="center">
	{{User:Tohline/Math/EQ_ZTFG01}}		{{User:Tohline/Math/EQ_ZTFG01}}
	~~<br />~~
	<font size="-1">		<font size="-1">
	Reference (original): [http://adsabs.harvard.edu/abs/1935MNRAS..95..207C S. Chandraskehar (1935)]		Reference (original): [http://adsabs.harvard.edu/abs/1935MNRAS..95..207C S. Chandraskehar (1935)]

Tohline: /* Barotropic Structure */

2019-06-26T01:02:41Z

Barotropic Structure

← Older revision		Revision as of 01:02, 26 June 2019
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	===Barotropic Structure===		===Barotropic Structure===

	For ''time-independent'' problems, a structural relationship between {{User:Tohline/Math/VAR_Pressure01}} and {{User:Tohline/Math/VAR_Density01}} is required to close the system of [[User:Tohline/PGE#Principal_Governing_Equations\|principal governing equations]]. [<b>[[User:Tohline/Appendix/References#T78\|<font color="red">T78</font>]]</b>] refers to this as a "geometrical" rather than a "structural" relationship; see the discussion associated with his Chapter 4, Eq. 14. Generally throughout this H_Book, we will assume that all time-independent configurations can be described as barotropic structures; that is, we will assume that {{User:Tohline/Math/VAR_Pressure01}} is only a function of {{User:Tohline/Math/VAR_Density01}} throughout such structures. More specifically, we generally will adopt one of the two ''analytically'' prescribable <math>~P(\rho)</math> relationships displayed in the first row of the following Table.		For ''time-independent'' problems, a structural relationship between {{User:Tohline/Math/VAR_Pressure01}} and {{User:Tohline/Math/VAR_Density01}} is required to close the system of [[User:Tohline/PGE#Principal_Governing_Equations\|principal governing equations]]. [<b>[[User:Tohline/Appendix/References#T78\|<font color="red">T78</font>]]</b>] refers to this as a "geometrical" rather than a "structural" relationship; see the discussion associated with his Chapter 4, Eq. 14. Generally throughout this H_Book, we will assume that all time-independent configurations can be described as barotropic structures; that is, we will assume that {{User:Tohline/Math/VAR_Pressure01}} is only a function of {{User:Tohline/Math/VAR_Density01}} throughout such structures. (The [[User:Tohline/2DStructure/AxisymmetricInstabilities#Poincar.C3.A9-Wavre_Theorem\|Poincaré-Wavre theorem]] provides additional support for, as well as additional implications of, this assumption.) More specifically, we generally will adopt one of the two ''analytically'' prescribable <math>~P(\rho)</math> relationships displayed in the first row of the following Table.

	<table width="80%" align="center" border=1 cellpadding=5>		<table width="80%" align="center" border=1 cellpadding=5>

Tohline: /* Supplemental Relations */

2019-01-29T18:47:02Z

Supplemental Relations

← Older revision		Revision as of 18:47, 29 January 2019
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	__FORCETOC__		__FORCETOC__
	=Supplemental Relations=		=Supplemental Relations=

			<table border="0" cellpadding="3" align="center" width="60%">
			<tr><td align="left">
			<font color="darkgreen">
			"The equilibrium configuration of a stellar system depends crucially on the equation of state. Similarly, the time evolution of the perturbation will depend on the relationship that is assumed between the perturbed values of pressure, temperature, and density. The simplest assumption regarding these perturbations will be to involve an adiabatic relationship between perturbed pressure and density."</font>
			</td></tr>
			<tr><td align="right">
			— Drawn from §7, p. 172 of [<b>[[User:Tohline/Appendix/References#P00\|<font color="red">P00</font>]]</b>]
			</td></tr></table>

	{{LSU_HBook_header}}		{{LSU_HBook_header}}
	Examining the set of [[User:Tohline/PGE#Principal_Governing_Equations\|principal governing equations]] that describe ''adiabatic'' flows, we see that — apart from the independent variables {{User:Tohline/Math/VAR_Time01}} and {{User:Tohline/Math/VAR_PositionVector01}} — the equations involve the vector velocity {{User:Tohline/Math/VAR_VelocityVector01}}, and the four scalar variables, {{User:Tohline/Math/VAR_NewtonianPotential01}}, {{User:Tohline/Math/VAR_Pressure01}}, {{User:Tohline/Math/VAR_Density01}}, and {{User:Tohline/Math/VAR_SpecificInternalEnergy01}}. Because the variables outnumber the equations by one, a supplemental relationship between the physical variables must be specified in order to close the set of equations. (If nonadiabatic flows are considered, additional supplemental relations must be specified because the scalar variables {{User:Tohline/Math/VAR_Temperature01}} and {{User:Tohline/Math/VAR_SpecificEntropy01}} enter the discussion as well.) Also, in order to complete the unique specification of a particular physical problem, either a steady-state flow field or initial conditions must be specified, depending on whether one is studying a time-independent (structure) or time-dependent (stability or dynamics) problem, respectively. Throughout this H_Book, the following strategy will be adopted in order to complete the physical specification of each examined system: (A) For time-independent problems, we will adopt a structural relationship between {{User:Tohline/Math/VAR_Pressure01}} and {{User:Tohline/Math/VAR_Density01}}, and specify a steady-state flow-field. (B) For time-dependent problems, we will adopt an equation of state, and specify initial conditions.		Examining the set of [[User:Tohline/PGE#Principal_Governing_Equations\|principal governing equations]] that describe ''adiabatic'' flows, we see that — apart from the independent variables {{User:Tohline/Math/VAR_Time01}} and {{User:Tohline/Math/VAR_PositionVector01}} — the equations involve the vector velocity {{User:Tohline/Math/VAR_VelocityVector01}}, and the four scalar variables, {{User:Tohline/Math/VAR_NewtonianPotential01}}, {{User:Tohline/Math/VAR_Pressure01}}, {{User:Tohline/Math/VAR_Density01}}, and {{User:Tohline/Math/VAR_SpecificInternalEnergy01}}. Because the variables outnumber the equations by one, a supplemental relationship between the physical variables must be specified in order to close the set of equations. (If nonadiabatic flows are considered, additional supplemental relations must be specified because the scalar variables {{User:Tohline/Math/VAR_Temperature01}} and {{User:Tohline/Math/VAR_SpecificEntropy01}} enter the discussion as well.) Also, in order to complete the unique specification of a particular physical problem, either a steady-state flow field or initial conditions must be specified, depending on whether one is studying a time-independent (structure) or time-dependent (stability or dynamics) problem, respectively. Throughout this H_Book, the following strategy will be adopted in order to complete the physical specification of each examined system: (A) For time-independent problems, we will adopt a structural relationship between {{User:Tohline/Math/VAR_Pressure01}} and {{User:Tohline/Math/VAR_Density01}}, and specify a steady-state flow-field. (B) For time-dependent problems, we will adopt an equation of state, and specify initial conditions.

Tohline: /* Time-Dependent Problems */

2018-10-23T17:03:21Z

Time-Dependent Problems

← Older revision		Revision as of 17:03, 23 October 2018
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	</table>		</table>
	</div>		</div>

			===Entropy Tracer===
			We begin with the basic equation of state,
			<div align="center">
			<math>~P = (\gamma_g - 1)\epsilon\rho \, ,</math>
			</div>
			and the 1<sup>st</sup> Law of Thermodynamics,
			<div align="center">
			{{ User:Tohline/Math/EQ_FirstLaw01 }}
			</div>
			Adopting the concept of an ''entropy tracer'',
			<div align="center">
			<math>~\tau \equiv (\epsilon\rho)^{1/\gamma_g} \, ,</math>
			</div>
			the 1<sup>st</sup> becomes,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{\tau}{c_p} \frac{ds}{dt}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{\partial\tau}{\partial t} + \nabla\cdot (\tau \vec{v}) \, .</math>
			</td>
			</tr>
			</table>
			See equation (33) of Motl et al. (2002)


	===Initial Conditions===		===Initial Conditions===

Tohline: /* See Also */

2017-07-05T20:36:38Z

Tohline at 00:17, 5 July 2017

2017-07-05T00:17:54Z

← Older revision		Revision as of 00:17, 5 July 2017
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	===Initial Conditions===		===Initial Conditions===
	For ''time-dependent'' problems, the [[User:Tohline/PGE#Principal_Governing_Equations\|principal governing equations]] must be supplemented further through the specification of initial conditions. Frequently throughout this H_Book, we will select as initial conditions a specification of <math>~\rho(\vec{x}, t=0)</math>, <math>~P(\vec{x}, t=0)</math>, and <math>~\vec{v}(\vec{x}, t=0)</math> that, as a group themselves, define a static or steady-state equilibrium '''structure'''. Perturbation or computational fluid dynamic (CFD) techniques can be used to test the '''stability''' or nonlinear '''dynamical behavior''' of such structures.		For ''time-dependent'' problems, the [[User:Tohline/PGE#Principal_Governing_Equations\|principal governing equations]] must be supplemented further through the specification of initial conditions. Frequently throughout this H_Book, we will select as initial conditions a specification of <math>~\rho(\vec{x}, t=0)</math>, <math>~P(\vec{x}, t=0)</math>, and <math>~\vec{v}(\vec{x}, t=0)</math> that, as a group themselves, define a static or steady-state equilibrium '''structure'''. Perturbation or computational fluid dynamic (CFD) techniques can be used to test the '''stability''' or nonlinear '''dynamical behavior''' of such structures.

			=See Also=
			For derivations and additional discussions of the triplet of adiabatic exponents, <math>~\Gamma_1, \Gamma_2, \Gamma_3</math>, see:
			* [[User:Tohline/Appendix/References#C67\|[<b><font color="red">C67</font></b>]]] — especially pp. 56-58; note that <math>~d\ln V = - d\ln \rho</math>;
			* [http://adsabs.harvard.edu/abs/1968psen.book.....C Donald D. Clayton (1968)]; especially p. 118; note that <math>~d\ln V = - d\ln \rho</math>;
			* [[User:Tohline/Appendix/References#KW94\|[<b><font color="red">KW94</font></b>]]] — especially p. 106;
			* [http://adsabs.harvard.edu/abs/1994sipp.book.....H Carl J. Hansen & Steven D. Kawaler (1994)] — especially pp. 136-138.


	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Adiabatic Exponent */

2017-07-04T23:59:54Z

Adiabatic Exponent

← Older revision		Revision as of 23:59, 4 July 2017
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	</div>		</div>

	Following the lead of [[User:Tohline/Appendix/References#C67\|[<b><font color="red">C67</font></b>]]], the astrophysics community has found that, when radiation pressure is included in the mix, it can be useful to characterize the adiabatic compression/expansion of fluid elements in terms of three ''separate'' adiabatic exponents, <math>~\Gamma_1, \Gamma_2, \Gamma_3</math>, that are defined via similar differential expressions, namely,		Following the lead of [[User:Tohline/Appendix/References#C67\|[<b><font color="red">C67</font></b>]]], the astrophysics community has found that, when radiation pressure is included in the mix — that is, when we consider situations in which,
			<div align="center">
			<math>~P = P_\mathrm{gas} + P_\mathrm{rad}</math>
			</div>
			— it can be useful to characterize the adiabatic compression/expansion of fluid elements in terms of three ''separate'' adiabatic exponents, <math>~\Gamma_1, \Gamma_2, \Gamma_3</math>, that are defined via similar differential expressions, namely,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<td align="left">		<td align="left">
	<math>~\Gamma_3-1 \, .</math>		<math>~\Gamma_3-1 \, .</math>
			</td>
			</tr>
			</table>
			</div>
			In this case, though, each of the three adiabatic exponents is a function of <math>~\beta</math> as well as <math>~\gamma_\mathrm{g}</math>; specifically,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\Gamma_1</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\beta + \frac{(4-3\beta)^2 (\gamma_\mathrm{g}-1)}{\beta + 12(\gamma_\mathrm{g}-1)(1-\beta)}
			\, ;</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Gamma_2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			1 + \frac{(4-3\beta)(\gamma_\mathrm{g} - 1)}{\beta^2 + 3(\gamma_\mathrm{g} - 1)(1-\beta)(4+\beta)}
			\, ;</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Gamma_3</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			1 + \frac{(4-3\beta)(\gamma_\mathrm{g}-1)}{\beta+12(\gamma_\mathrm{g} - 1)(1-\beta)}
			\, .</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Adiabatic Exponent */

2017-07-04T23:49:07Z

Adiabatic Exponent

← Older revision		Revision as of 23:49, 4 July 2017
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	</div>		</div>

	Following the lead of [[User:Tohline/Appendix/References#C67\|[<b><font color="red">C67</font></b>]]], the astrophysics community has found that, when radiation pressure is included in the mix, it can be useful to characterize the adiabatic compression/expansion of fluid elements ~~— if not entire fluid configurations —~~ in terms of three ''separate'' adiabatic exponents, <math>~\Gamma_1, \Gamma_2, \Gamma_3</math>, that are defined via similar differential expressions, namely,		Following the lead of [[User:Tohline/Appendix/References#C67\|[<b><font color="red">C67</font></b>]]], the astrophysics community has found that, when radiation pressure is included in the mix, it can be useful to characterize the adiabatic compression/expansion of fluid elements in terms of three ''separate'' adiabatic exponents, <math>~\Gamma_1, \Gamma_2, \Gamma_3</math>, that are defined via similar differential expressions, namely,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</table>		</table>
	</div>		</div>

	If we examine the evolution of a system that is entirely composed of an ideal gas — that is, <math>~P_\mathrm{rad} = P_\mathrm{deg} = 0</math> — then from the ideal-gas equation of state we can write,
	~~<div align="center">~~
	~~<table border="0" cellpadding="5" align="center">~~

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~dP\biggr\|_\mathrm{gas}</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~\frac{\Re}{\bar{\mu}} \biggl[ \rho dT + T d\rho \biggr]_\mathrm{gas}~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~\Rightarrow ~~~ d\ln P</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~d\ln \rho + d\ln T \, .~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~
	~~</div>~~
	~~But if the first of the ''adiabatic exponent'' expressions also properly describes how the pressure of a fluid element varies with its density during an evolution, then~~

	===Initial Conditions===		===Initial Conditions===

Tohline: /* Adiabatic Exponent */

2017-07-04T23:42:05Z

Adiabatic Exponent

← Older revision		Revision as of 23:42, 4 July 2017
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	====Adiabatic Exponent====		====Adiabatic Exponent====

	Consider, first, the evolution of a system that is composed entirely of an ideal gas; that is, <math>~P_\mathrm{rad} = P_\mathrm{deg} = 0</math>. It is widely appreciated that if the ~~enthalpy~~ of such a system remains constant during a phase of expansion/contraction, then the variation of pressure with density can be properly described by the expression,		Consider, first, the evolution of a system that is composed entirely of an ideal gas; that is, <math>~P_\mathrm{rad} = P_\mathrm{deg} = 0</math> and <math>~\beta = 1</math>. It is widely appreciated that if the entropy of such a system remains constant during a phase of expansion/contraction, then the variation of pressure with density can be properly described by the expression,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~d\ln \rho + d\ln T		<math>~d\ln \rho + d\ln T \, .
			</math>
			</td>
			</tr>
			</table>
			</div>

			We therefore also deduce that,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{d\ln P}{d\ln \rho}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~1 + \frac{d\ln T}{d\ln\rho}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ \frac{d\ln T}{d\ln\rho} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\gamma_\mathrm{g}-1 \, ;
			</math>
			</td>
			</tr>
			</table>
			</div>
			and,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{d\ln P}{d\ln T}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{d\ln \rho}{d\ln T} +1 = \biggl( \frac{1}{\gamma_\mathrm{g} -1}\biggr) + 1
	</math>		</math>
	</td>		</td>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	~~<math>~\Rightarrow ~~~ \frac{d\ln P}{d\ln \rho}</math>~~
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~~~1 +~~ \frac{d\~~ln T~~}{d\ln\~~rho~~} \, .		<math>~\frac{\gamma_\mathrm{g}}{\gamma_\mathrm{g}-1 } \, .
	</math>		</math>
	</td>		</td>
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	</table>		</table>
	</div>		</div>
	~~But if the first of the ''adiabatic exponent'' expressions also properly describes how the pressure of a fluid element varies with its density during an evolution, then~~

	Following [[User:Tohline/Appendix/References#C67\|[<b><font color="red">C67</font></b>]]], the astrophysics community has found it useful to characterize the adiabatic compression/expansion of fluid elements — if not entire fluid configurations — in terms of ~~one or more~~ ''adiabatic exponents'', <math>~\Gamma_1, \Gamma_2, \Gamma_3</math>, defined ~~such that~~,		Following the lead of [[User:Tohline/Appendix/References#C67\|[<b><font color="red">C67</font></b>]]], the astrophysics community has found that, when radiation pressure is included in the mix, it can be useful to characterize the adiabatic compression/expansion of fluid elements — if not entire fluid configurations — in terms of three ''separate'' adiabatic exponents, <math>~\Gamma_1, \Gamma_2, \Gamma_3</math>, that are defined via similar differential expressions, namely,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~~~\frac{1}{~~\Gamma_3-1} \, .</math>		<math>~\Gamma_3-1 \, .</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Adiabatic Exponent */

2017-07-04T23:09:49Z

Adiabatic Exponent

← Older revision		Revision as of 23:09, 4 July 2017
Line 393:		Line 393:
	====Adiabatic Exponent====		====Adiabatic Exponent====

	~~If we examine~~ the evolution of a system that is entirely ~~composed~~ of an ideal gas ~~&#8212~~; that is, <math>~P_\mathrm{rad} = P_\mathrm{deg} = 0</math> ~~—~~ then ~~from~~ the ~~ideal-gas equation~~ of ~~state we~~ can ~~write~~,		Consider, first, the evolution of a system that is composed entirely of an ideal gas; that is, <math>~P_\mathrm{rad} = P_\mathrm{deg} = 0</math>. It is widely appreciated that if the enthalpy of such a system remains constant during a phase of expansion/contraction, then the variation of pressure with density can be properly described by the expression,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~dP\~~biggr\|_~~\~~mathrm~~{~~gas~~}</math>		<math>~\frac{d\ln P}{d\ln \rho}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\~~frac{~~\~~Re}~~{~~\bar{\mu}~~} \~~biggl[ \rho dT + T d\rho \biggr]_\mathrm{gas}~~		<math>~\gamma_\mathrm{g} \, ,</math>
	</math>
	</td>		</td>
	</tr>		</tr>
			</table>
			</div>
			where the value of the adiabatic exponent, <math>~\gamma_\mathrm{g}</math>, is given by the ratio of specific heats of the (ideal) gas. Now, according to the ideal-gas equation of state, changes in the three state variables must always be related to one another via the differential expression,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\~~Rightarrow ~~~ d~~\~~ln P~~</math>		<math>~dP\biggr\|_\mathrm{gas}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~d\ln \rho + d\~~ln T~~		<math>~\frac{\Re}{\bar{\mu}} \biggl[ \rho dT + T d\rho \biggr]_\mathrm{gas}
	</math>		</math>
	</td>		</td>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Rightarrow ~~~ ~~\frac{~~d\ln P~~}{d\ln \rho}~~</math>		<math>~\Rightarrow ~~~ d\ln P</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~1 + ~~\frac{~~d\ln T~~}{d\ln\rho} \, .~~		<math>~d\ln \rho + d\ln T
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	~~</table>~~
	~~</div>~~
	Now, it is widely appreciated that if the enthalpy of a fluid element remains constant during a phase of expansion/contraction — that is, if the evolution is ''adiabatic'' — then the variation of pressure with density can be properly described by the expression,
	~~<div align="center">~~
	~~<table border="0" cellpadding="5" align="center">~~

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{d\ln P}{d\ln \rho}</math>		<math>~\Rightarrow ~~~ \frac{d\ln P}{d\ln \rho}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\~~gamma_~~\~~mathrm~~{g} \, ,</math>		<math>~1 + \frac{d\ln T}{d\ln\rho} \, .
			</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	</div>		</div>
	~~where the value of the adiabatic exponent, <math>~\gamma_\mathrm{g}</math>, is given by the ratio of specific heats of the gas.~~ But if the first of the ''adiabatic exponent'' expressions also properly describes how the pressure of a fluid element varies with its density during an evolution, then		But if the first of the ''adiabatic exponent'' expressions also properly describes how the pressure of a fluid element varies with its density during an evolution, then

	Following [[User:Tohline/Appendix/References#C67\|[<b><font color="red">C67</font></b>]]], the astrophysics community has found it useful to characterize the adiabatic compression/expansion of fluid elements — if not entire fluid configurations — in terms of one or more ''adiabatic exponents'', <math>~\Gamma_1, \Gamma_2, \Gamma_3</math>, defined such that,		Following [[User:Tohline/Appendix/References#C67\|[<b><font color="red">C67</font></b>]]], the astrophysics community has found it useful to characterize the adiabatic compression/expansion of fluid elements — if not entire fluid configurations — in terms of one or more ''adiabatic exponents'', <math>~\Gamma_1, \Gamma_2, \Gamma_3</math>, defined such that,

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	* [http://adsabs.harvard.edu/abs/1968psen.book.....C Donald D. Clayton (1968)]; especially p. 118; note that <math>~d\ln V = - d\ln \rho</math>;		* [http://adsabs.harvard.edu/abs/1968psen.book.....C Donald D. Clayton (1968)]; especially p. 118; note that <math>~d\ln V = - d\ln \rho</math>;
	* [[User:Tohline/Appendix/References#KW94\|[<b><font color="red">KW94</font></b>]]] — especially p. 106;		* [[User:Tohline/Appendix/References#KW94\|[<b><font color="red">KW94</font></b>]]] — especially p. 106;
	* [http://adsabs.harvard.edu/abs/1994sipp.book.....H Carl J. Hansen & Steven D. Kawaler (1994)] — especially pp. 136-138.		* [http://adsabs.harvard.edu/abs/1994sipp.book.....H Carl J. Hansen & Steven D. Kawaler (1994)] — especially pp. 136-138;
			* Our own brief presentation of [[User:Tohline/Apps/SMS#Expression_for_Adiabatic_Exponent\|several equivalent expressions for <math>~\Gamma_1</math>]].


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