User:Tohline/SR/PressureCombinations - Revision history

Tohline: /* Relationship Between State Variables */

2021-06-10T22:02:06Z

Relationship Between State Variables

← Older revision		Revision as of 22:02, 10 June 2021
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	<math>~\equiv</math>		<math>~\equiv</math>
	</td>		</td>
	<td align="left">		<td align="left" bgcolor="yellow">
	<math>~\biggl( \frac{3\cdot 5}{2^2 \pi^4} \biggr)^{1/3} C_g\chi \, ,</math>		<math>~\biggl( \frac{3\cdot 5}{2^2 \pi^4} \biggr)^{1/3} C_g\chi \, ,</math>
	</td>		</td>
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	<math>~\equiv</math>		<math>~\equiv</math>
	</td>		</td>
	<td align="left">		<td align="left" bgcolor="lightblue">
	<math>~ 2^{3/2} \biggl[ 1 + (1 + \lambda^3)^{1/2} \biggr]^{1/2}		<math>~ 2^{3/2} \biggl[ 1 + (1 + \lambda^3)^{1/2} \biggr]^{1/2}
	\biggl\{ \biggl[ 1 + (1 + \lambda^3)^{1/2} \biggr]^{2/3} - \lambda \biggr\}^{-3/2}\, ,</math>		\biggl\{ \biggl[ 1 + (1 + \lambda^3)^{1/2} \biggr]^{2/3} - \lambda \biggr\}^{-3/2}\, ,</math>
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	<math>~\equiv</math>		<math>~\equiv</math>
	</td>		</td>
	<td align="left">		<td align="left" bgcolor="pink">
	<math>~		<math>~
	\biggl(\frac{\pi^4}{2\cdot 3^4\cdot 5} \biggr)^{1/3} \biggl[\frac{p_\mathrm{total}-F(\chi)}{(C_g \chi)^{4}}\biggr] \, .		\biggl(\frac{\pi^4}{2\cdot 3^4\cdot 5} \biggr)^{1/3} \biggl[\frac{p_\mathrm{total}-F(\chi)}{(C_g \chi)^{4}}\biggr] \, .

Tohline: /* Relationship Between State Variables */

2016-11-18T20:51:55Z

Relationship Between State Variables

← Older revision		Revision as of 20:51, 18 November 2016
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	</div>		</div>

	If it is the two normalized state variables, <math>~\chi</math> and <math>~p_\mathrm{total}</math>, that are known, the third normalized state variable — namely, the normalized temperature, <math>~z</math> — also can be obtained analytically. But the governing expression is not as simple because it results from an inversion of the total pressure equation and, hence, the solution of a quartic equation. As is [[User:Tohline/SR/Ptot_QuarticSolution\|detailed in the accompanying discussion]], the desired solution is,		If it is the two normalized state variables, <math>~\chi</math> and <math>~p_\mathrm{total}</math>, that are known, the third normalized state variable — namely, the normalized temperature, <math>~z</math> — also can be obtained analytically. But the governing expression is not as simple because it results from an inversion of the total pressure equation and, hence, the solution of a quartic equation. As is [[User:Tohline/SR/Ptot_QuarticSolution#Determining_Temperature_from_Density_and_Pressure\|detailed in the accompanying discussion]], the desired solution is,

	<div align="center">		<div align="center">

Tohline: /* Relationship Between State Variables */ Clean up text a bit

2015-07-19T04:03:17Z

Relationship Between State Variables: Clean up text a bit

← Older revision		Revision as of 04:03, 19 July 2015
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	===Relationship Between State Variables===		===Relationship Between State Variables===

	If the two normalized state variables, <math>~\chi</math> and <math>~z</math>, are known, then the third normalized state variable can be obtained directly from the [[User:Tohline/SR/PressureCombinations#Total_Pressure\|above key expression for the total pressure]], that is,		If the two normalized state variables, <math>~\chi</math> and <math>~z</math>, are known, then the third normalized state variable, <math>~p_\mathrm{total}</math>, can be obtained directly from the [[User:Tohline/SR/PressureCombinations#Total_Pressure\|above key expression for the total pressure]], that is,

	<div align="center">		<div align="center">
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	</div>		</div>

	If it is the two normalized state variables, <math>~\chi</math> and <math>~p_\mathrm{total}</math>, that are known, the third normalized state variable — namely, the normalized temperature — also can be obtained analytically~~, but~~ the governing expression is not as simple because it results from the solution of a quartic equation. As is [[User:Tohline/SR/Ptot_QuarticSolution\|detailed in the accompanying discussion]], the ~~proper expression~~ is,		If it is the two normalized state variables, <math>~\chi</math> and <math>~p_\mathrm{total}</math>, that are known, the third normalized state variable — namely, the normalized temperature, <math>~z</math> — also can be obtained analytically. But the governing expression is not as simple because it results from an inversion of the total pressure equation and, hence, the solution of a quartic equation. As is [[User:Tohline/SR/Ptot_QuarticSolution\|detailed in the accompanying discussion]], the desired solution is,

	<div align="center">		<div align="center">
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	It also would be desirable to have an analytic expression for the function, <math>~\chi(z, p_\mathrm{total})</math>, in order to be able to immediately determine the normalized density from any specified values of the normalized temperature and normalized pressure. However, it does not appear that the [[User:Tohline/SR/PressureCombinations#Total_Pressure\|above key expression for the total pressure]] can be inverted to provide such a closed-form expression.		It also would be desirable to have an analytic expression for the function, <math>~\chi(z, p_\mathrm{total})</math>, in order to be able to immediately determine the normalized density from any specified values of the normalized temperature and normalized pressure. However, it does not appear that the [[User:Tohline/SR/PressureCombinations#Total_Pressure\|above key expression for the total pressure]] can be inverted to provide such a closed-form expression.


	===Dominant Contributions===		===Dominant Contributions===

Tohline: Improved order in which topics are "discussed" including presentation of general solution to quartic equation

2015-07-18T17:37:27Z

Improved order in which topics are "discussed" including presentation of general solution to quartic equation

← Older revision		Revision as of 17:37, 18 July 2015
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	{{~~LSU_WorkInProgress~~}}		==Discussion==
			For simplicity of presentation, in what follows we will use
			<div align="center">
			<math>
			z \equiv \frac{T}{T_e} \, ,
			</math>
			</div>
			to represent a normalized temperature, in addition to using <math>~\chi</math> to represent (the cube root of) the normalized mass density, and <math>~p_\mathrm{total}</math> to represent the normalized total pressure.


			===Relationship Between State Variables===

			If the two normalized state variables, <math>~\chi</math> and <math>~z</math>, are known, then the third normalized state variable can be obtained directly from the [[User:Tohline/SR/PressureCombinations#Total_Pressure\|above key expression for the total pressure]], that is,

			<div align="center">
			<math>p_\mathrm{total}(\chi, z) = 8(C_g \chi)^3 z + F(\chi) + \biggl(\frac{8\pi^4}{15}\biggr) z^4 \, ,</math>
			</div>
			where,
			<div align="center">
			<math>C_g \equiv \biggl(\frac{\mu_e m_p}{\bar\mu m_u}\biggr)^{1/3} \, .</math>
			</div>

	~~NOTE: Still need to provide derivation of ZTF pressure in limit of very large densities. Also, in~~ the ~~subsection where degeneracy pressure is ignored~~, ~~it would be good to introduce "standard" definition of~~ <math>~\~~beta~~</math>.		If it is the two normalized state variables, <math>~\chi</math> and <math>~p_\mathrm{total}</math>, that are known, the third normalized state variable — namely, the normalized temperature — also can be obtained analytically, but the governing expression is not as simple because it results from the solution of a quartic equation. As is [[User:Tohline/SR/Ptot_QuarticSolution\|detailed in the accompanying discussion]], the proper expression is,

	~~==Discussion==~~
	~~For simplicity of presentation, in what follows we will use~~
	<div align="center">		<div align="center">
	<math>		<math>
	\~~tau~~ \~~equiv~~ \~~frac~~{T}{~~T_e~~}		z(\chi, p_\mathrm{total}) = \theta_\chi \phi^{-1/3}\biggl[ (\phi - 1)^{1/2} - 1 \biggr] ,
	</math>		</math>
	</div>		</div>
	~~to represent a normalized temperature~~, ~~in addition to using~~ <math>~\chi</math> ~~to represent~~ (~~the cube root of~~) the ~~normalized mass density and~~ <math>~p_\mathrm{total}</math> to ~~represent~~ the normalized total pressure.
			where,

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

			<tr>
			<td align="right">
			<math>~\theta_\chi</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~\biggl( \frac{3\cdot 5}{2^2 \pi^4} \biggr)^{1/3} C_g\chi \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\phi</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~ 2^{3/2} \biggl[ 1 + (1 + \lambda^3)^{1/2} \biggr]^{1/2}
			\biggl\{ \biggl[ 1 + (1 + \lambda^3)^{1/2} \biggr]^{2/3} - \lambda \biggr\}^{-3/2}\, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\lambda</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~
			\biggl(\frac{\pi^4}{2\cdot 3^4\cdot 5} \biggr)^{1/3} \biggl[\frac{p_\mathrm{total}-F(\chi)}{(C_g \chi)^{4}}\biggr] \, .
			</math>
			</td>
			</tr>
			</table>
			</div>

			It also would be desirable to have an analytic expression for the function, <math>~\chi(z, p_\mathrm{total})</math>, in order to be able to immediately determine the normalized density from any specified values of the normalized temperature and normalized pressure. However, it does not appear that the [[User:Tohline/SR/PressureCombinations#Total_Pressure\|above key expression for the total pressure]] can be inverted to provide such a closed-form expression.


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	<div align="center">		<div align="center">
	<math>		<math>
	\frac{P_\mathrm{rad}}{P_\mathrm{gas}} \approx \frac{\pi^4}{15} \biggl( \frac{~~\tau~~}{\chi} \biggr)^3		\frac{P_\mathrm{rad}}{P_\mathrm{gas}} \approx \frac{\pi^4}{15} \biggl( \frac{z}{\chi} \biggr)^3
	</math> .		</math> .
	</div>		</div>
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	<div align="center">		<div align="center">
	<math>		<math>
	\frac{P_\mathrm{gas}}{P_\mathrm{deg}} \approx 5 ~~\tau~~ \chi^{-2}		\frac{P_\mathrm{gas}}{P_\mathrm{deg}} \approx \frac{5 z}{ \chi^{2}}
	~~~~~ \mathrm{and} ~~~~~		~~~~~ \mathrm{and} ~~~~~
	\frac{P_\mathrm{rad}}{P_\mathrm{deg}} \approx \biggl(\frac{\pi^4}{3}\biggr) \~~tau~~^4 \chi^{-5} ;		\frac{P_\mathrm{rad}}{P_\mathrm{deg}} \approx \biggl(\frac{\pi^4}{3}\biggr) \frac{z^4}{ \chi^5} ;
	</math>		</math>
	</div>		</div>
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	<div align="center">		<div align="center">
	<math>		<math>
	\frac{P_\mathrm{gas}}{P_\mathrm{deg}} \approx ~~4 \biggl(~~ \frac{~~\tau~~}{\chi} ~~\biggr)~~		\frac{P_\mathrm{gas}}{P_\mathrm{deg}} \approx \frac{4z}{\chi}
	~~~~~ \mathrm{and} ~~~~~		~~~~~ \mathrm{and} ~~~~~
	\frac{P_\mathrm{rad}}{P_\mathrm{deg}} \approx \frac{4 \pi^4}{15} \biggl( \frac{~~\tau~~}{\chi} \biggr)^4 .		\frac{P_\mathrm{rad}}{P_\mathrm{deg}} \approx \frac{4 \pi^4}{15} \biggl( \frac{z}{\chi} \biggr)^4 .
	</math>		</math>
	</div>		</div>

			<!-- OMIT NEXT SHORT SECTION
	===Just Ideal-Gas and Radiation===		===Just Ideal-Gas and Radiation===

	In certain density-temperature regimes, contributions from the electron degeneracy pressure can be ignored and, to a good approximation, the normalized total pressure will take the form,		In certain density-temperature regimes, contributions from the electron degeneracy pressure can be ignored and, to a good approximation, the normalized total pressure will take the form,
	<div align="center">		<div align="center">
	<math>~p_\mathrm{total} = C_g \chi^3 ~~\tau~~ + C_r ~~\tau~~^4 ,</math>		<math>~p_\mathrm{total} = C_g \chi^3 z + C_r z^4 ,</math>
	</div>		</div>
	where the coefficients,		where the coefficients,
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	</math>		</math>
	</div>		</div>
	Given any values for the pair of state variables, <math>~\chi</math> and <math>~~~\tau~~</math>, the third state variable can be calculated analytically from this specified function, <math>~p_\mathrm{total}(\chi,~~\tau~~)</math>. It is easy to see as well that, given any values for the pair of state variables, <math>~p_\mathrm{total}</math> and <math>~~~\tau~~</math>, the third state variable can be calculated analytically from the function,		Given any values for the pair of state variables, <math>~\chi</math> and <math>~z</math>, the third state variable can be calculated analytically from this specified function, <math>~p_\mathrm{total}(\chi,z)</math>. It is easy to see as well that, given any values for the pair of state variables, <math>~p_\mathrm{total}</math> and <math>~z</math>, the third state variable can be calculated analytically from the function,
	<div align="center">		<div align="center">
	<math>\chi^3(p_\mathrm{total},~~\tau~~) = \frac{1}{C_g ~~\tau~~} \biggl[ p_\mathrm{total} - C_r ~~\tau~~^4 \biggr] .</math>		<math>\chi^3(p_\mathrm{total},z) = \frac{1}{C_g z} \biggl[ p_\mathrm{total} - C_r z^4 \biggr] .</math>
	</div>		</div>
	~~The third desirable relationship between the three state variables, <math~~>~\tau(p_\mathrm{total},\chi^3)</math>, can also be expressed analytically, but the resulting expression is not as simple as the first two because it results from the solution of a quartic equation. As is [[User:Tohline/SR/Ptot_QuarticSolution\|detailed in the accompanying discussion]], the proper expression is,		-->


	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Just Ideal-Gas and Radiation */ Clean up a few paragraphs

2015-07-13T02:47:13Z

Just Ideal-Gas and Radiation: Clean up a few paragraphs

← Older revision		Revision as of 02:47, 13 July 2015
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	In certain density-temperature regimes, contributions from the electron degeneracy pressure can be ignored and, to a good approximation, the normalized total pressure will take the form,		In certain density-temperature regimes, contributions from the electron degeneracy pressure can be ignored and, to a good approximation, the normalized total pressure will take the form,
	<div align="center">		<div align="center">
	<math>p_\mathrm{total} = C_g \chi^3 \tau + C_r \tau^4 ,</math>		<math>~p_\mathrm{total} = C_g \chi^3 \tau + C_r \tau^4 ,</math>
	</div>		</div>
	where the coefficients,		where the coefficients,
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	</math>		</math>
	</div>		</div>
	Given any values for the pair of state variables, <math>\chi</math> and <math>\tau</math>, the third state variable can be calculated analytically from this specified function, <math>p_\mathrm{total}(\chi,\tau)</math>. It is easy to see as well that, given any values for the pair of state variables, <math>p_\mathrm{total}</math> and <math>\tau</math>, the third state variable can be calculated analytically from the function,		Given any values for the pair of state variables, <math>~\chi</math> and <math>~\tau</math>, the third state variable can be calculated analytically from this specified function, <math>~p_\mathrm{total}(\chi,\tau)</math>. It is easy to see as well that, given any values for the pair of state variables, <math>~p_\mathrm{total}</math> and <math>~\tau</math>, the third state variable can be calculated analytically from the function,
	<div align="center">		<div align="center">
	<math>\chi^3(p_\mathrm{total},\tau) = \frac{1}{C_g \tau} \biggl[ p_\mathrm{total} - C_r \tau^4 \biggr] .</math>		<math>\chi^3(p_\mathrm{total},\tau) = \frac{1}{C_g \tau} \biggl[ p_\mathrm{total} - C_r \tau^4 \biggr] .</math>
	</div>		</div>
	The third desirable relationship between the three state variables, <math>\tau(p_\mathrm{total},\chi^3)</math>, can also be expressed analytically, but the resulting expression is not as simple as the first two because it results from the solution of a quartic equation. As is [~~http://www.vistrails.org/index.php/~~User:Tohline/SR/Ptot_QuarticSolution detailed in the accompanying discussion], the proper expression is,		The third desirable relationship between the three state variables, <math>~\tau(p_\mathrm{total},\chi^3)</math>, can also be expressed analytically, but the resulting expression is not as simple as the first two because it results from the solution of a quartic equation. As is [[User:Tohline/SR/Ptot_QuarticSolution\|detailed in the accompanying discussion]], the proper expression is,

	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: Begin cleaning up discussion of "combined pressures"

2015-07-12T23:57:28Z

Begin cleaning up discussion of "combined pressures"

← Older revision		Revision as of 23:57, 12 July 2015
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	=Total Pressure=		=Total Pressure=
	{{LSU_HBook_header}}		{{LSU_HBook_header}}
	In our overview of [~~http://www.vistrails.org/index.php/~~User:Tohline/SR#Time-Dependent_Problems equations of state], we identified analytic expressions for the pressure of an ideal gas, <math>~P_\mathrm{gas}</math>, electron degeneracy pressure, <math>~P_\mathrm{deg}</math>, and radiation pressure, <math>~P_\mathrm{rad}</math>. Rather than considering these ~~equations of state~~ one at a time, in general we should consider the contributions to the pressure that are made by all three ~~of these equations of state~~ simultaneously. That is, we should examine the total pressure,		In our overview of [[User:Tohline/SR#Time-Dependent_Problems\|equations of state]], we identified analytic expressions for the pressure of an ideal gas, <math>~P_\mathrm{gas}</math>, electron degeneracy pressure, <math>~P_\mathrm{deg}</math>, and radiation pressure, <math>~P_\mathrm{rad}</math>. Rather than considering these relations one at a time, in general we should consider the contributions to the pressure that are made by all three simultaneously. That is, we should examine the total pressure,

	<div align="center">		<div align="center">
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	</div>		</div>

	In order to assess which of these three contributions will dominate <math>~P_\mathrm{total}</math> in different density and temperature regimes, it is instructive to normalize <math>~P_\mathrm{total}</math> to the characteristic Fermi pressure, {{User:Tohline/Math/C_FermiPressure}}, as defined in the accompanying [~~http://www.vistrails.org/index.php/~~User:Tohline/Appendix/Variables_templates Variables Appendix]. As derived below, this normalized total pressure can be written as,		In order to assess which of these three contributions will dominate <math>~P_\mathrm{total}</math> in different density and temperature regimes, it is instructive to normalize <math>~P_\mathrm{total}</math> to the characteristic Fermi pressure, {{User:Tohline/Math/C_FermiPressure}}, as defined in the accompanying [[User:Tohline/Appendix/Variables_templates\|Variables Appendix]]. As derived below, this normalized total pressure can be written as,

	<div align="center">		<div align="center">
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	</div>		</div>

	To derive the expression for <math>p_\mathrm{total}</math> shown in the opening paragraph above, we begin by normalizing each component pressure independently.		To derive the expression for <math>~p_\mathrm{total}</math> shown in the opening paragraph above, we begin by normalizing each component pressure independently.


	===Normalized Degenerate Electron Pressure===		===Normalized Degenerate Electron Pressure===

	This normalization is trivial. Given the original expression for the pressure due to a degenerate electron gas (or a zero-temperature Fermi gas),		This normalization is trivial. Given the original expression for the pressure due to a degenerate electron gas (or a zero-temperature Fermi gas),
	<div align="center">		<div align="center">
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	</math>		</math>
	</div>		</div>


	===Normalized Ideal-Gas Pressure===		===Normalized Ideal-Gas Pressure===

	Given the original expression for the pressure of an ideal gas,		Given the original expression for the pressure of an ideal gas,
	<div align="center">		<div align="center">
	{{User:Tohline/Math/EQ_EOSideal0A}}		{{User:Tohline/Math/EQ_EOSideal0A}}
	</div>		</div>
	along with the definitions of the physical constants, {{User:Tohline/Math/C_GasConstant}}, {{User:Tohline/Math/C_FermiPressure}}, and {{User:Tohline/Math/C_FermiDensity}} provided in the accompanying [~~http://www.vistrails.org/index.php/~~User:Tohline/Appendix/Variables_templates Variables Appendix], we can write,		along with the definitions of the physical constants, {{User:Tohline/Math/C_GasConstant}}, {{User:Tohline/Math/C_FermiPressure}}, and {{User:Tohline/Math/C_FermiDensity}} provided in the accompanying [[User:Tohline/Appendix/Variables_templates\|Variables Appendix]], we can write,
	<div align="center">		<div align="center">
	<math>		<math>
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	</math>		</math>
	</div>		</div>
	where, by definition, the [http://en.wikipedia.org/wiki/Atomic_mass_unit atomic mass unit] is, <math>m_u \equiv (1/N_\mathrm{A})~\mathrm{g} = 0.992776 m_p</math>, that is, <math>m_p/m_u = 1.007276</math>.		where, by definition, the [http://en.wikipedia.org/wiki/Atomic_mass_unit atomic mass unit] is, <math>m_u \equiv (1/N_\mathrm{A})~\mathrm{g} = 0.992776 m_p</math>, that is, <math>~m_p/m_u = 1.007276</math>.


	===Normalized Radiation Pressure===		===Normalized Radiation Pressure===

	Given the original expression for the radiation pressure,		Given the original expression for the radiation pressure,
	<div align="center">		<div align="center">
	{{User:Tohline/Math/EQ_EOSradiation01}}		{{User:Tohline/Math/EQ_EOSradiation01}}
	</div>		</div>
	along with the definitions of the physical constants, {{User:Tohline/Math/C_FermiPressure}}, and {{User:Tohline/Math/C_RadiationConstant}} provided in the accompanying [~~http://www.vistrails.org/index.php/~~User:Tohline/Appendix/Variables_templates Variables Appendix], we can write,		along with the definitions of the physical constants, {{User:Tohline/Math/C_FermiPressure}}, and {{User:Tohline/Math/C_RadiationConstant}} provided in the accompanying [[User:Tohline/Appendix/Variables_templates\|Variables Appendix]], we can write,
	<div align="center">		<div align="center">
	<math>		<math>
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	</div>		</div>


			{{LSU_WorkInProgress}}

			NOTE: Still need to provide derivation of ZTF pressure in limit of very large densities. Also, in the subsection where degeneracy pressure is ignored, it would be good to introduce "standard" definition of <math>~\beta</math>.

	==Discussion==		==Discussion==
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	</math>		</math>
	</div>		</div>
	to represent a normalized temperature, in addition to using <math>\chi</math> to represent (the cube root of) the normalized mass density and <math>p_\mathrm{total}</math> to represent the normalized total pressure.		to represent a normalized temperature, in addition to using <math>~\chi</math> to represent (the cube root of) the normalized mass density and <math>~p_\mathrm{total}</math> to represent the normalized total pressure.


	===Dominant Contributions===		===Dominant Contributions===

	Let's examine which pressure contributions will dominate in various temperature-density regimes.		Let's examine which pressure contributions will dominate in various temperature-density regimes.
	Note, first, that {{User:Tohline/Math/C_ProtonMass}}/{{User:Tohline/Math/C_AtomicMassUnit}}  <math>\approx</math> 1 and, for fully ionized gases, the ratio {{User:Tohline/Math/MP_ElectronMolecularWeight}}/{{User:Tohline/Math/MP_MeanMolecularWeight}} is of order unity — more precisely, the ratio of these two molecular weights falls within the narrow range <math>1 < </math> {{User:Tohline/Math/MP_ElectronMolecularWeight}}/{{User:Tohline/Math/MP_MeanMolecularWeight}} <math>\le 2</math>. Hence, we can assume that the numerical coefficient of the first term in our expression for <math>p_\mathrm{total}</math> is approximately 8, so the ratio of radiation pressure to gas pressure is,		Note, first, that {{User:Tohline/Math/C_ProtonMass}}/{{User:Tohline/Math/C_AtomicMassUnit}}  <math>~\approx 1</math> and, for fully ionized gases, the ratio {{User:Tohline/Math/MP_ElectronMolecularWeight}}<math>~/</math>{{User:Tohline/Math/MP_MeanMolecularWeight}} is of order unity — more precisely, the ratio of these two molecular weights falls within the narrow range <math>~1 < </math> {{User:Tohline/Math/MP_ElectronMolecularWeight}}<math>~/</math>{{User:Tohline/Math/MP_MeanMolecularWeight}} <math>\le 2</math>. Hence, we can assume that the numerical coefficient of the first term in our expression for <math>~p_\mathrm{total}</math> is approximately <math>~8</math>, so the ratio of radiation pressure to gas pressure is,
	<div align="center">		<div align="center">
	<math>		<math>
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	</math> ,		</math> ,
	</div>		</div>
	where <math>T_7</math> is the temperature expressed in units of <math>10^7</math> K and <math>\rho_1</math> is the matter density expressed in units of <math>\mathrm{g~cm}^{-3}</math>.		where <math>~T_7</math> is the temperature expressed in units of <math>~10^7~K</math> and <math>~\rho_1</math> is the matter density expressed in units of <math>~\mathrm{g~cm}^{-3}</math>.


	Second, note that the function <math>F(\chi)</math> can be written in a simpler form when examining regions of either very low or very high matter densities. Specifically (see~~, for example, various discussions in the "parallel references" cited in conjunction with this equation of state in the accompanying~~ [~~http://www.vistrails.org/index.php/~~User:Tohline/~~Appendix/Equation_templates~~#~~EOS Appendix~~ of ~~Key Equations~~]), in the limit <math>\chi \ll 1</math>,		Second, note that the function <math>~F(\chi)</math> can be written in a simpler form when examining regions of either very low or very high matter densities. Specifically — see our [[User:Tohline/SR#Nonrelativistic_ZTF_Gas\|separate discussion of the Zero-Temperature Fermi gas]] — in the limit <math>~\chi \ll 1</math>,
	<div align="center">		<div align="center">
	<math>		<math>

Tohline: /* Total Pressure */

2015-05-29T19:42:17Z

Total Pressure

← Older revision		Revision as of 19:42, 29 May 2015
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	=Total Pressure=		=Total Pressure=
	{{LSU_HBook_header}}		{{LSU_HBook_header}}
	In our overview of [http://www.vistrails.org/index.php/User:Tohline/SR#Time-Dependent_Problems equations of state], we identified analytic expressions for the pressure of an ideal gas, <math>P_\mathrm{gas}</math>, electron degeneracy pressure, <math>P_\mathrm{deg}</math>, and radiation pressure, <math>P_\mathrm{rad}</math>. Rather than considering these equations of state one at a time, in general we should consider the contributions to the pressure that are made by all three of these equations of state simultaneously. That is, we should examine the total pressure,		In our overview of [http://www.vistrails.org/index.php/User:Tohline/SR#Time-Dependent_Problems equations of state], we identified analytic expressions for the pressure of an ideal gas, <math>~P_\mathrm{gas}</math>, electron degeneracy pressure, <math>~P_\mathrm{deg}</math>, and radiation pressure, <math>~P_\mathrm{rad}</math>. Rather than considering these equations of state one at a time, in general we should consider the contributions to the pressure that are made by all three of these equations of state simultaneously. That is, we should examine the total pressure,

	<div align="center">		<div align="center">
	<math>		<math>
	P_\mathrm{total} = P_\mathrm{gas} + P_\mathrm{deg} + P_\mathrm{rad} .		~P_\mathrm{total} = P_\mathrm{gas} + P_\mathrm{deg} + P_\mathrm{rad} .
	</math>		</math>
	</div>		</div>

	In order to assess which of these three contributions will dominate <math>P_\mathrm{total}</math> in different density and temperature regimes, it is instructive to normalize <math>P_\mathrm{total}</math> to the characteristic Fermi pressure, {{User:Tohline/Math/C_FermiPressure}}, as defined in the accompanying [http://www.vistrails.org/index.php/User:Tohline/Appendix/Variables_templates Variables Appendix]. As derived below, this normalized total pressure can be written as,		In order to assess which of these three contributions will dominate <math>~P_\mathrm{total}</math> in different density and temperature regimes, it is instructive to normalize <math>~P_\mathrm{total}</math> to the characteristic Fermi pressure, {{User:Tohline/Math/C_FermiPressure}}, as defined in the accompanying [http://www.vistrails.org/index.php/User:Tohline/Appendix/Variables_templates Variables Appendix]. As derived below, this normalized total pressure can be written as,

	<div align="center">		<div align="center">

Tohline: Move title image

2012-11-06T17:47:03Z

Move title image

← Older revision		Revision as of 17:47, 6 November 2012
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			=Total Pressure=
	{{LSU_HBook_header}}		{{LSU_HBook_header}}

	~~=Total Pressure=~~

	In our overview of [http://www.vistrails.org/index.php/User:Tohline/SR#Time-Dependent_Problems equations of state], we identified analytic expressions for the pressure of an ideal gas, <math>P_\mathrm{gas}</math>, electron degeneracy pressure, <math>P_\mathrm{deg}</math>, and radiation pressure, <math>P_\mathrm{rad}</math>. Rather than considering these equations of state one at a time, in general we should consider the contributions to the pressure that are made by all three of these equations of state simultaneously. That is, we should examine the total pressure,		In our overview of [http://www.vistrails.org/index.php/User:Tohline/SR#Time-Dependent_Problems equations of state], we identified analytic expressions for the pressure of an ideal gas, <math>P_\mathrm{gas}</math>, electron degeneracy pressure, <math>P_\mathrm{deg}</math>, and radiation pressure, <math>P_\mathrm{rad}</math>. Rather than considering these equations of state one at a time, in general we should consider the contributions to the pressure that are made by all three of these equations of state simultaneously. That is, we should examine the total pressure,

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	</div>		</div>
	The third desirable relationship between the three state variables, <math>\tau(p_\mathrm{total},\chi^3)</math>, can also be expressed analytically, but the resulting expression is not as simple as the first two because it results from the solution of a quartic equation. As is [http://www.vistrails.org/index.php/User:Tohline/SR/Ptot_QuarticSolution detailed in the accompanying discussion], the proper expression is,		The third desirable relationship between the three state variables, <math>\tau(p_\mathrm{total},\chi^3)</math>, can also be expressed analytically, but the resulting expression is not as simple as the first two because it results from the solution of a quartic equation. As is [http://www.vistrails.org/index.php/User:Tohline/SR/Ptot_QuarticSolution detailed in the accompanying discussion], the proper expression is,


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Tohline: /* Just Ideal-Gas and Radiation */ Insert link to quartic-solution discussion

2010-03-02T18:11:10Z

Just Ideal-Gas and Radiation: Insert link to quartic-solution discussion

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	<math>\chi^3(p_\mathrm{total},\tau) = \frac{1}{C_g \tau} \biggl[ p_\mathrm{total} - C_r \tau^4 \biggr] .</math>		<math>\chi^3(p_\mathrm{total},\tau) = \frac{1}{C_g \tau} \biggl[ p_\mathrm{total} - C_r \tau^4 \biggr] .</math>
	</div>		</div>
	The third desirable relationship between the three state variables, <math>\tau(p_\mathrm{total},\chi^3)</math>, can also be expressed analytically, but the resulting expression is not as simple as the first two because it results from the solution of a quartic equation. ~~The~~ proper expression is,		The third desirable relationship between the three state variables, <math>\tau(p_\mathrm{total},\chi^3)</math>, can also be expressed analytically, but the resulting expression is not as simple as the first two because it results from the solution of a quartic equation. As is [http://www.vistrails.org/index.php/User:Tohline/SR/Ptot_QuarticSolution detailed in the accompanying discussion], the proper expression is,


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Tohline: /* Discussion */ Present quartic-equation solution for temperature

2010-02-28T23:10:52Z

Discussion: Present quartic-equation solution for temperature

← Older revision		Revision as of 23:10, 28 February 2010
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	==Discussion==		==Discussion==
			For simplicity of presentation, in what follows we will use
			<div align="center">
			<math>
			\tau \equiv \frac{T}{T_e}
			</math>
			</div>
			to represent a normalized temperature, in addition to using <math>\chi</math> to represent (the cube root of) the normalized mass density and <math>p_\mathrm{total}</math> to represent the normalized total pressure.

			===Dominant Contributions===
	Let's examine which pressure contributions will dominate in various temperature-density regimes.		Let's examine which pressure contributions will dominate in various temperature-density regimes.
			Note, first, that {{User:Tohline/Math/C_ProtonMass}}/{{User:Tohline/Math/C_AtomicMassUnit}}  <math>\approx</math> 1 and, for fully ionized gases, the ratio {{User:Tohline/Math/MP_ElectronMolecularWeight}}/{{User:Tohline/Math/MP_MeanMolecularWeight}} is of order unity — more precisely, the ratio of these two molecular weights falls within the narrow range <math>1 < </math> {{User:Tohline/Math/MP_ElectronMolecularWeight}}/{{User:Tohline/Math/MP_MeanMolecularWeight}} <math>\le 2</math>. Hence, we can assume that the numerical coefficient of the first term in our expression for <math>p_\mathrm{total}</math> is approximately 8, so the ratio of radiation pressure to gas pressure is,
	~~===Simplifications===~~
	Note, first, that {{User:Tohline/Math/C_ProtonMass}}/{{User:Tohline/Math/C_AtomicMassUnit}}  <math>\approx</math> 1 and, for fully ionized gases, the ratio {{User:Tohline/Math/MP_ElectronMolecularWeight}}/{{User:Tohline/Math/MP_MeanMolecularWeight}} is of order unity — more precisely, the ratio of these two molecular weights falls within the narrow range <math>1 < </math> {{User:Tohline/Math/MP_ElectronMolecularWeight}}/{{User:Tohline/Math/MP_MeanMolecularWeight}} <math>\le 2</math>. ~~For simplicity, then~~, we can assume that the numerical coefficient of the first term in our expression for <math>p_\mathrm{total}</math> is approximately 8, so the ratio of radiation pressure to gas pressure is,
	<div align="center">		<div align="center">
	<math>		<math>
	\frac{P_\mathrm{rad}}{P_\mathrm{gas}} \approx \frac{\pi^4}{15} \biggl( \frac{T}{~~T_e~~} \biggr)^3 ~~\frac{B_F}{\rho}~~		\frac{P_\mathrm{rad}}{P_\mathrm{gas}} \approx \frac{\pi^4}{15} \biggl( \frac{\tau}{\chi} \biggr)^3
	</math> .		</math> .
	</div>		</div>
	~~Hence,~~ radiation pressure will dominate over ideal gas pressure in any regime where,		This means that radiation pressure will dominate over ideal gas pressure in any regime where,
	<div align="center">		<div align="center">
	<math>		<math>
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	<math>		<math>
	\frac{P_\mathrm{gas}}{P_\mathrm{deg}} \approx 5 \~~frac{T}{T_e} \biggl( \frac{B_F}{\rho}~~ \~~biggr)~~^{2/3}		\frac{P_\mathrm{gas}}{P_\mathrm{deg}} \approx 5 \tau \chi^{-2}
	~~~~~ \mathrm{and} ~~~~~		~~~~~ \mathrm{and} ~~~~~
	\frac{P_\mathrm{rad}}{P_\mathrm{deg}} \approx \frac{\pi^4}{3~~} \biggl( \frac{T}{T_e~~} \biggr)^4 \~~biggl( \frac{B_F}{\rho} \biggr)~~^{5/3} ;		\frac{P_\mathrm{rad}}{P_\mathrm{deg}} \approx \biggl(\frac{\pi^4}{3}\biggr) \tau^4 \chi^{-5} ;
	</math>		</math>
	</div>		</div>
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	<div align="center">		<div align="center">
	<math>		<math>
	\frac{P_\mathrm{gas}}{P_\mathrm{deg}} \approx 4 ~~\frac{T}{T_e}~~ \biggl( \frac{~~B_F~~}{\~~rho~~} \biggr)~~^{1/3}~~		\frac{P_\mathrm{gas}}{P_\mathrm{deg}} \approx 4 \biggl( \frac{\tau}{\chi} \biggr)
	~~~~~ \mathrm{and} ~~~~~		~~~~~ \mathrm{and} ~~~~~
	\frac{P_\mathrm{rad}}{P_\mathrm{deg}} \approx \frac{4 \pi^4}{15} \biggl( \frac{~~T}{T_e} \biggr)^4 \biggl(~~ \~~frac{B_F~~}{\~~rho~~} \biggr)^{4~~/3}~~ .		\frac{P_\mathrm{rad}}{P_\mathrm{deg}} \approx \frac{4 \pi^4}{15} \biggl( \frac{\tau}{\chi} \biggr)^4 .
	</math>		</math>
	</div>		</div>

			===Just Ideal-Gas and Radiation===

			In certain density-temperature regimes, contributions from the electron degeneracy pressure can be ignored and, to a good approximation, the normalized total pressure will take the form,
			<div align="center">
			<math>p_\mathrm{total} = C_g \chi^3 \tau + C_r \tau^4 ,</math>
			</div>
			where the coefficients,
			<div align="center">
			<math>
			C_g \equiv 8\biggl(\frac{\mu_e m_p}{\bar{\mu} m_u} \biggr) ~~~~~ \mathrm{and} ~~~~~ C_r \equiv \frac{8\pi^4}{15} .
			</math>
			</div>
			Given any values for the pair of state variables, <math>\chi</math> and <math>\tau</math>, the third state variable can be calculated analytically from this specified function, <math>p_\mathrm{total}(\chi,\tau)</math>. It is easy to see as well that, given any values for the pair of state variables, <math>p_\mathrm{total}</math> and <math>\tau</math>, the third state variable can be calculated analytically from the function,
			<div align="center">
			<math>\chi^3(p_\mathrm{total},\tau) = \frac{1}{C_g \tau} \biggl[ p_\mathrm{total} - C_r \tau^4 \biggr] .</math>
			</div>
			The third desirable relationship between the three state variables, <math>\tau(p_\mathrm{total},\chi^3)</math>, can also be expressed analytically, but the resulting expression is not as simple as the first two because it results from the solution of a quartic equation. The proper expression is,
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