Difference between revisions of "User:Tohline/SR"

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(→‎Nonrelativistic ZTF Gas: Insert factor of mu_e from definition of B_F)
(→‎Supplemental Relations: Clean up lots of links throughout this chapter)
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=Supplemental Relations=
=Supplemental Relations=
{{LSU_HBook_header}}
{{LSU_HBook_header}}
Examining the set of [http://www.vistrails.org/index.php/User:Tohline/PGE 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.




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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 [http://www.vistrails.org/index.php/User:Tohline/PGE principal governing equations].  [Tassoul (1978) 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 [http://en.wikipedia.org/wiki/Barotropic 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 {{User:Tohline/Math/VAR_Pressure01}}({{User:Tohline/Math/VAR_Density01}}) 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]] .  [Tassoul (1978) 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.


<table width="80%" align="center" border=1 cellpadding=5>
<table width="80%" align="center" border=1 cellpadding=5>
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<br />
<br />
<font size="-1">
<font size="-1">
Reference (original): [http://adsabs.harvard.edu/abs/1935MNRAS..95..207C Chandraskehar, S. (1935)]
Reference (original): [http://adsabs.harvard.edu/abs/1935MNRAS..95..207C S. Chandraskehar (1935)]
</font>
</font>


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In the polytropic relation, the "polytropic index" {{User:Tohline/Math/MP_PolytropicIndex}} and the "polytropic constant" {{User:Tohline/Math/MP_PolytropicConstant}} are assumed to be independent of both {{User:Tohline/Math/VAR_PositionVector01}} and {{User:Tohline/Math/VAR_Time01}}.  In the zero-temperature Fermi gas relation, the two  constants {{User:Tohline/Math/C_FermiPressure}} and {{User:Tohline/Math/C_FermiDensity}} are expressible in terms of various fundamental physical constants, as detailed in the accompanying [http://www.vistrails.org/index.php/User:Tohline/Appendix/Variables_templates variables appendix].  This table also displays (<math>2^{nd}</math> row) the enthalpy as a function of mass density, <math>H(\rho)</math>, and (<math>3^{rd}</math> row) the inverted relation <math>\rho(H)</math> for both barotropic relations, where  
In the polytropic relation, the "polytropic index" {{User:Tohline/Math/MP_PolytropicIndex}} and the "polytropic constant" {{User:Tohline/Math/MP_PolytropicConstant}} are assumed to be independent of both {{User:Tohline/Math/VAR_PositionVector01}} and {{User:Tohline/Math/VAR_Time01}}.  In the zero-temperature Fermi gas relation, the two  constants {{User:Tohline/Math/C_FermiPressure}} and {{User:Tohline/Math/C_FermiDensity}} are expressible in terms of various fundamental physical constants, as detailed in the accompanying [[User:Tohline/Appendix/Variables_templates|variables appendix]].  This table also displays (2<sup>nd</sup> row) the enthalpy as a function of mass density, <math>~H(\rho)</math>, and (3<sup>rd</sup> row) the inverted relation <math>~\rho(H)</math> for both barotropic relations, where  
<div align="center">
<div align="center">
<math>H = \int\frac{dP}{\rho}</math> .
<math>H = \int\frac{dP}{\rho}</math> .
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===Steady-state Flow-Field Specification===
===Steady-state Flow-Field Specification===


A steady-state velocity flow-field must be specified for time-independent problems.  The specification can be as simple as stating that {{User:Tohline/Math/VAR_VelocityVector01}} = 0 everywhere in space, or that the system has uniform ("solid body") rotation.  Throughout the literature, efforts to generate equilibrium, axisymmetric configurations have adopted a variety of different [[User:Tohline/AxisymmetricConfigurations/SolutionStrategies#SRPtable|''simple rotation profiles'']].  In this H_Book, the flow-field specification will generally vary from chapter to chapter.
A steady-state velocity flow-field must be specified for time-independent problems.  The specification can be as simple as stating that <math>~\vec{v}=0</math> everywhere in space, or that the system has uniform ("solid body") rotation.  Throughout the literature, efforts to generate equilibrium, axisymmetric configurations have adopted a variety of different [[User:Tohline/AxisymmetricConfigurations/SolutionStrategies#SRPtable|''simple rotation profiles'']].  In this H_Book, the flow-field specification will generally vary from chapter to chapter.


==Time-Dependent Problems==
==Time-Dependent Problems==
===Equation of State===
===Equation of State===


For ''time-dependent'' problems we usually will supplement the set of [http://www.vistrails.org/index.php/User:Tohline/PGE principal governing equations] by adopting an equation of state &#8212; ''i.e.,'' a relationship between the state variables {{User:Tohline/Math/VAR_Pressure01}}, {{User:Tohline/Math/VAR_Density01}}, and {{User:Tohline/Math/VAR_Temperature01}} &#8212; that is given by one of the expressions in the following Table, or by [http://www.vistrails.org/index.php/User:Tohline/SR/PressureCombinations some combination of these expressions].  (For example, we could write <math>P_\mathrm{total} = P_\mathrm{gas} + P_\mathrm{deg} + P_\mathrm{rad}</math>.)   
For ''time-dependent'' problems we usually will supplement the set of [[User:Tohline/PGE#Principal_Governing_Equations|principal governing equations]] by adopting a relationship between the state variables {{User:Tohline/Math/VAR_Pressure01}}, {{User:Tohline/Math/VAR_Density01}}, and {{User:Tohline/Math/VAR_Temperature01}} that is given by one of the expressions in the following Table, or by [[User:Tohline/SR/PressureCombinations#Total_Pressure|some combination of these expressions]].  (For example, we could write <math>~P_\mathrm{total} = P_\mathrm{gas} + P_\mathrm{deg} + P_\mathrm{rad}</math>.)   


<table width="95%" align="center" border=1 cellpadding=5>
<table width="95%" align="center" border=1 cellpadding=5>
<tr>
<tr>
<th colspan=3 align="center">
<th colspan=2 align="center">
<font color="darkblue">Analytic Equations of State</font>
<font color="darkblue">Analytic Equations of State</font>
</th>
</th>
<td align="center" rowspan="2" width="25%"><font color="darkblue">Radiation Pressure</font></td>
</tr>
</tr>
<tr>
<tr>
<td align="center" width="25%"><font color="darkblue">Ideal Gas</font></td>
<td align="center" width="25%"><font color="darkblue">Ideal Gas</font></td>
<td align="center"><font color="darkblue">Degenerate Electron Gas</font></td>
<td align="center"><font color="darkblue">Degenerate Electron Gas</font></td>
<td align="center" width="25%"><font color="darkblue">Radiation</font></td>
</tr>
</tr>
<tr>
<tr>
<td align="center">
<td align="center">
{{User:Tohline/Math/EQ_EOSideal0A}}
{{User:Tohline/Math/EQ_EOSideal0A}}
[http://www.vistrails.org/index.php/User:Tohline/SR/IdealGas [[Image:LSU_OtherFormsButton.jpg|150px]] ]
[[Image:LSU_OtherFormsButton.jpg|150px|link=http://www.vistrails.org/index.php/User:Tohline/SR/IdealGas#Ideal_Gas_Equation_of_State]]
</td>
</td>
<td align="center">
<td align="center">
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<td colspan=3 align="center">
<td colspan=3 align="center">
{{User:Tohline/Math/EQ_PressureTotal01}}
{{User:Tohline/Math/EQ_PressureTotal01}}
[http://www.vistrails.org/index.php/User:Tohline/SR/PressureCombinations [[Image:LSU_OriginButton.jpg|150px]] ]
[[Image:LSU_OriginButton.jpg|120px|link=http://www.vistrails.org/index.php/User:Tohline/SR/PressureCombinations#Total_Pressure]]
</td>
</td>
</tr>
</tr>
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In the so-called ''ideal gas'' equation of state, {{User:Tohline/Math/C_GasConstant}} is the gas constant and {{User:Tohline/Math/MP_MeanMolecularWeight}} is the mean molecular weight of the gas. In the equation that gives the electron degeneracy pressure, {{User:Tohline/Math/C_FermiPressure}} is the characteristic Fermi pressure and {{User:Tohline/Math/C_FermiDensity}} is the characteristic Fermi density.  And in the expression for the photon radiation pressure, {{User:Tohline/Math/C_RadiationConstant}} is the radiation constant.  The value of each of these identified physical constants can be found by simply scrolling the computer mouse over the symbol for the constant found in the text of this paragraph, and a definition of each constant can be found in the [[User:Tohline/Appendix/Variables_templates|Variables Appendix]] of this H_Book.
In the so-called ''ideal gas'' equation of state, {{User:Tohline/Math/C_GasConstant}} is the gas constant and {{User:Tohline/Math/MP_MeanMolecularWeight}} is the mean molecular weight of the gas. In the equation that gives the electron degeneracy pressure, {{User:Tohline/Math/C_FermiPressure}} is the characteristic Fermi pressure and {{User:Tohline/Math/C_FermiDensity}} is the characteristic Fermi density.  And in the expression for the photon radiation pressure, {{User:Tohline/Math/C_RadiationConstant}} is the radiation constant.  The value of each of these identified physical constants can be found by simply scrolling the computer mouse over the symbol for the constant found in the text of this paragraph, and a definition of each constant can be found in the [[User:Tohline/Appendix/Variables_templates|Variables Appendix]] of this H_Book.


All three of these equations of state are among the set of [[User:Tohline/Appendix/Equation_templates|key physical equations]] that provide the foundation for our discussion of the '''structure, stability, and dynamics of self-gravitating systems.'''  A discussion of the physical principles that underpin each of these relations can be found in any of a number of different published texts &#8212; see, for example, the set of ''parallel references'' identified in the [[User:Tohline/Appendix/Equation_templates|Equations Appendix]] of this H_Book &#8212; or in the Wiki pages that can be accessed by clicking the linked "other forms" buttons in the above Table.  See also Tassoul (1978) &#8212; specifically the discussion associated with his Chapter 4, Eq. 13 &#8212; for a more general statement related to the proper specification of a supplemental, equation of state relationship.
All three of these equations are among the set of [[User:Tohline/Appendix/Equation_templates#Equations_of_State|key physical equations]] that provide the foundation for our discussion of the '''structure, stability, and dynamics of self-gravitating systems.'''  A discussion of the physical principles that underpin each of these relations can be found in any of a number of different published texts &#8212; see, for example, the set of ''parallel references'' identified in the [[User:Tohline/Appendix/Equation_templates#Equations_of_State|Equations Appendix]] of this H_Book &#8212; or in the Wiki pages that can be accessed by clicking the linked "other forms" buttons in the above Table.  See also Tassoul (1978) &#8212; specifically the discussion associated with his Chapter 4, Eq. 13 &#8212; for a more general statement related to the proper specification of a supplemental, equation of state relationship.


===Initial Conditions===
===Initial Conditions===
For ''time-dependent'' problems, the [http://www.vistrails.org/index.php/User:Tohline/PGE 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 {{User:Tohline/Math/VAR_Density01}}( {{User:Tohline/Math/VAR_PositionVector01}}, {{User:Tohline/Math/VAR_Time01}} = 0), {{User:Tohline/Math/VAR_Pressure01}}( {{User:Tohline/Math/VAR_PositionVector01}}, {{User:Tohline/Math/VAR_Time01}} = 0), and {{User:Tohline/Math/VAR_VelocityVector01}}( {{User:Tohline/Math/VAR_PositionVector01}}, {{User:Tohline/Math/VAR_Time01}} = 0) 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.
 


{{LSU_HBook_footer}}
{{LSU_HBook_footer}}

Revision as of 23:04, 12 July 2015

Supplemental Relations

Whitworth's (1981) Isothermal Free-Energy Surface
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Examining the set of principal governing equations#Principal_Governing_Equations that describe adiabatic flows, we see that — apart from the independent variables <math>~t</math> and <math>~\vec{x}</math> — the equations involve the vector velocity <math>~\vec{v}</math>, and the four scalar variables, <math>~\Phi</math>, <math>~P</math>, <math>~\rho</math>, and <math>~\epsilon</math>. 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 <math>~T</math> and <math>~s</math> 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 <math>~P</math> and <math>~\rho</math>, and specify a steady-state flow-field. (B) For time-dependent problems, we will adopt an equation of state, and specify initial conditions.


Time-Independent Problems

Barotropic Structure

For time-independent problems, a structural relationship between <math>~P</math> and <math>~\rho</math> is required to close the system of principal governing equations . [Tassoul (1978) 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 <math>~P</math> is only a function of <math>~\rho</math> 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.

Barotropic Relations

Polytropic Zero-temperature Fermi (degenerate electron) Gas

<math>~P = K_\mathrm{n} \rho^{1+1/n}</math>

LSU Key.png

<math>~P_\mathrm{deg} = A_\mathrm{F} F(\chi) </math>

where:  <math>F(\chi) \equiv \chi(2\chi^2 - 3)(\chi^2 + 1)^{1/2} + 3\sinh^{-1}\chi</math>

and:   

<math>\chi \equiv (\rho/B_\mathrm{F})^{1/3}</math>


Reference (original): S. Chandraskehar (1935)

<math>~H = (n+1)K_\mathrm{n} \rho^{1/n}</math>

<math> H = \frac{8A_\mathrm{F}}{B_\mathrm{F}} \biggl[(\chi^2 + 1)^{1/2} - 1 \biggr] </math>

<math>~\rho = \biggl[ \frac{H}{(n+1)K_\mathrm{n}} \biggr]^n </math>

<math> \rho = B_\mathrm{F} \biggl[\biggl(\frac{HB_\mathrm{F}}{8A_\mathrm{F}} + 1 \biggr)^{2} - 1 \biggr]^{3/2} </math>


In the polytropic relation, the "polytropic index" <math>~n</math> and the "polytropic constant" <math>~K_\mathrm{n}</math> are assumed to be independent of both <math>~\vec{x}</math> and <math>~t</math>. In the zero-temperature Fermi gas relation, the two constants <math>~A_\mathrm{F}</math> and <math>~B_\mathrm{F}</math> are expressible in terms of various fundamental physical constants, as detailed in the accompanying variables appendix. This table also displays (2nd row) the enthalpy as a function of mass density, <math>~H(\rho)</math>, and (3rd row) the inverted relation <math>~\rho(H)</math> for both barotropic relations, where

<math>H = \int\frac{dP}{\rho}</math> .

In both cases, we have chosen an integration constant such that <math>~H</math> is zero when <math>~\rho</math> is zero.

Nonrelativistic ZTF Gas

At sufficiently low densities, specifically, when

<math>~\chi \ll 1 \, ,</math>

the zero-temperature Fermi (ZTF) equation of state describes the pressure-density behavior of a nonrelativistic (NR) degenerate electron gas. We can determine the expression for <math>~P_\mathrm{deg}</math> in this limit by writing the function, <math>~F(\chi)</math>, in terms of two relevant series expansions then keeping only the highest order terms.

<math>~F(\chi)</math>

<math>~=</math>

<math>~- 3\chi \biggl(1 - \frac{2}{3}\chi^2 \biggr) \biggl( 1 + \frac{1}{2}\chi^2 - \frac{1}{2^3}\chi^4 +\frac{1}{2^4}\chi^6 - \frac{5}{2^7} \chi^8+ \cdots \biggr) +3 \displaystyle\sum_{m=0}^{\infty} \biggl[ \frac{(-1)^m (2m)!}{2^{2m}(m!)^2} \cdot \frac{\chi^{2m+1}}{(2m + 1)}\biggr] </math>

 

<math>~=</math>

<math>~- 3\chi \biggl( 1 + \frac{1}{2}\chi^2 - \frac{1}{2^3}\chi^4 +\frac{1}{2^4}\chi^6 - \frac{5}{2^7}\chi^8 + \cdots \biggr) + 2\chi^3 \biggl( 1 + \frac{1}{2}\chi^2 - \frac{1}{2^3}\chi^4 +\frac{1}{2^4}\chi^6 - \frac{5}{2^7}\chi^8 + \cdots \biggr) </math>

 

 

<math>~ +3\biggl\{\chi - \frac{1}{6}\chi^3 + \frac{3}{2^3 \cdot 5} \chi^5 - \frac{5}{2^4\cdot 7} \chi^7 + \cdots \biggr\} </math>

 

<math>~=</math>

<math>~-3\chi + 3\chi + \chi^3\biggl(-\frac{3}{2} + 2 - \frac{1}{2} \biggr) + \chi^5 \biggl(\frac{3}{8} +1 +\frac{3^2}{2^3\cdot 5} \biggr)

+\chi^7 \biggl( -\frac{3}{2^4} -\frac{1}{2^2} - \frac{3\cdot 5}{2^4\cdot 7} \biggr) + \cdots </math>

 

<math>~=</math>

<math>~\chi^5 \biggl(\frac{15 + 40 + 9}{2^3\cdot 5} \biggr)

-\chi^7 \biggl( \frac{21 + 28 + 15}{2^4\cdot 7} \biggr) + \cancelto{0}{\cdots} </math>

 

<math>~\approx</math>

<math>~\chi^5 \biggl(\frac{2^3}{5} \biggr)

-\chi^7 \biggl( \frac{2^2}{7} \biggr) \, .</math>

Note that the series expansion for the inverse hyperbolic sine has been obtained from Wikipedia's presentation. Keeping only the leading term leads to the expression,

<math>~P_\mathrm{deg}\biggr|_\mathrm{NR}</math>

<math>~=</math>

<math>~\frac{2^3}{5} A_F \biggl( \frac{\rho}{B_F}\biggr)^{5/3}</math>

 

<math>~=</math>

<math>~\frac{2^3}{5} \biggl( \frac{\pi m_e^4 c^5}{3h^3} \biggr) \biggl[ \frac{8\pi m_p}{3} \biggl( \frac{m_e c}{h} \biggr)^3 \mu_e \biggr]^{-5/3} \rho^{5/3}</math>

 

<math>~=</math>

<math>~\mu_e^{-5/3} \biggl[ \frac{2^9 \cdot 3^5 \pi^3}{2^{15}\cdot 3^3\cdot 5^3 \pi^5} \biggr]^{1/3} \biggl( \frac{m_e^4 c^5}{h^3} \biggr) \biggl( \frac{h}{m_e c} \biggr)^5 \biggl( \frac{\rho}{m_p}\biggr)^{5/3}</math>

 

<math>~=</math>

<math>~\frac{1}{2^2 \cdot 5}\biggl( \frac{3}{\pi} \biggr)^{2/3} \biggl( \frac{h^2}{m_e} \biggr) \biggl( \frac{\rho}{m_p}\biggr)^{5/3} \mu_e^{-5/3} \, .</math>

Steady-state Flow-Field Specification

A steady-state velocity flow-field must be specified for time-independent problems. The specification can be as simple as stating that <math>~\vec{v}=0</math> everywhere in space, or that the system has uniform ("solid body") rotation. Throughout the literature, efforts to generate equilibrium, axisymmetric configurations have adopted a variety of different simple rotation profiles. In this H_Book, the flow-field specification will generally vary from chapter to chapter.

Time-Dependent Problems

Equation of State

For time-dependent problems we usually will supplement the set of principal governing equations by adopting a relationship between the state variables <math>~P</math>, <math>~\rho</math>, and <math>~T</math> that is given by one of the expressions in the following Table, or by some combination of these expressions. (For example, we could write <math>~P_\mathrm{total} = P_\mathrm{gas} + P_\mathrm{deg} + P_\mathrm{rad}</math>.)

Analytic Equations of State

Radiation Pressure
Ideal Gas Degenerate Electron Gas

LSU Key.png

<math>~P_\mathrm{gas} = \frac{\Re}{\bar{\mu}} \rho T</math>

LSU OtherFormsButton.jpg

LSU Key.png

<math>~P_\mathrm{deg} = A_\mathrm{F} F(\chi) </math>

where:  <math>F(\chi) \equiv \chi(2\chi^2 - 3)(\chi^2 + 1)^{1/2} + 3\sinh^{-1}\chi</math>

and:   

<math>\chi \equiv (\rho/B_\mathrm{F})^{1/3}</math>

LSU Key.png

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

Normalized Total Pressure:

LSU Key.png

<math>~p_\mathrm{total} = \biggl(\frac{\mu_e m_p}{\bar{\mu} m_u} \biggr) 8 \chi^3 \frac{T}{T_e} + F(\chi) + \frac{8\pi^4}{15} \biggl( \frac{T}{T_e} \biggr)^4</math>

LSU OriginButton.jpg

In the so-called ideal gas equation of state, <math>~\Re</math> is the gas constant and <math>~\bar{\mu}</math> is the mean molecular weight of the gas. In the equation that gives the electron degeneracy pressure, <math>~A_\mathrm{F}</math> is the characteristic Fermi pressure and <math>~B_\mathrm{F}</math> is the characteristic Fermi density. And in the expression for the photon radiation pressure, <math>~a_\mathrm{rad}</math> is the radiation constant. The value of each of these identified physical constants can be found by simply scrolling the computer mouse over the symbol for the constant found in the text of this paragraph, and a definition of each constant can be found in the Variables Appendix of this H_Book.

All three of these equations are among the set of key physical equations that provide the foundation for our discussion of the structure, stability, and dynamics of self-gravitating systems. A discussion of the physical principles that underpin each of these relations can be found in any of a number of different published texts — see, for example, the set of parallel references identified in the Equations Appendix of this H_Book — or in the Wiki pages that can be accessed by clicking the linked "other forms" buttons in the above Table. See also Tassoul (1978) — specifically the discussion associated with his Chapter 4, Eq. 13 — for a more general statement related to the proper specification of a supplemental, equation of state relationship.

Initial Conditions

For time-dependent problems, the 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.


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