View source for User:Tohline/SR

__FORCETOC__
=Supplemental Relations=
{{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 &#8212; apart from the independent variables {{User:Tohline/Math/VAR_Time01}} and {{User:Tohline/Math/VAR_PositionVector01}} &#8212; 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.


==Time-Independent Problems==
===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.

<table width="80%" align="center" border=1 cellpadding=5>
<tr>
<th colspan=2 align="center">
<font color="darkblue">Barotropic Relations</font>
</th>
</tr>
<tr>
<td align="center" width="30%"><font color="darkblue">Polytropic</font></td>
<td align="center"><font color="darkblue">Zero-temperature Fermi (degenerate electron) Gas</font></td>
</tr>
<tr>
<td align="center">
{{User:Tohline/Math/EQ_Polytrope01}}
</td>
<td align="center">
{{User:Tohline/Math/EQ_ZTFG01}}
<br />
<font size="-1">
Reference (original): [http://adsabs.harvard.edu/abs/1935MNRAS..95..207C Chandraskehar, S. (1935)]
</font>

</td>
</tr>
<tr>
<td align="center">
{{User:Tohline/Math/EQ_Polytrope02}}
</td>
<td align="center">
<math>
H = \frac{8A_\mathrm{F}}{B_\mathrm{F}} \biggl[(\chi^2 + 1)^{1/2} - 1  \biggr]
</math>
</td>
</tr>

<tr>
<td align="center">
{{User:Tohline/Math/EQ_Polytrope03}}
</td>
<td align="center">
<math>
\rho = B_\mathrm{F}  \biggl[\biggl(\frac{HB_\mathrm{F}}{8A_\mathrm{F}} + 1 \biggr)^{2} - 1  \biggr]^{3/2}
</math>
</td>
</tr>

</table>


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 
<div align="center">
<math>H = \int\frac{dP}{\rho}</math> .
</div>
In both cases, we have chosen an integration constant such that {{User:Tohline/Math/VAR_Enthalpy01}} is zero when {{User:Tohline/Math/VAR_Density01}} is zero.

====Nonrelativistic ZTF Gas====

At sufficiently low densities, specifically, when

<div align="center">
<math>~\chi \ll 1 \, ,</math>
</div>

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

<tr>
  <td align="right">
<math>~F(\chi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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}+ \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>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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}+ \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}+ \cdots  \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+3 \displaystyle\sum_{m=0}^{\infty} \biggl[ \frac{(-1)^m (2m)!}{2^{2m}(m!)^2} \cdot \frac{\chi^{2m+1}}{(2m + 1)}\biggr]
+3\biggl\{\chi - \frac{1}{6}\chi^3 +   \biggr\}
</math>
  </td>
</tr>
</table>
</div>
where, the series expansion for the inverse hyperbolic sine has been obtained from [http://en.wikipedia.org/wiki/Inverse_hyperbolic_function#Series_expansions Wikipedia's presentation]

===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.

==Time-Dependent Problems==
===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>.)  

<table width="95%" align="center" border=1 cellpadding=5>
<tr>
<th colspan=3 align="center">
<font color="darkblue">Analytic Equations of State</font>
</th>
</tr>
<tr>
<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" width="25%"><font color="darkblue">Radiation</font></td>
</tr>
<tr>
<td align="center">
{{User:Tohline/Math/EQ_EOSideal0A}}
[http://www.vistrails.org/index.php/User:Tohline/SR/IdealGas [[Image:LSU_OtherFormsButton.jpg|150px]] ]
</td>
<td align="center">
{{User:Tohline/Math/EQ_ZTFG01}}
</td>
<td align="center">
{{User:Tohline/Math/EQ_EOSradiation01}}
</td>
</tr>

<tr>
<td align="center" colspan=3><font color="darkblue">Normalized Total Pressure:</font></td>
</tr>
<tr>
<tr>
<td colspan=3 align="center">
{{User:Tohline/Math/EQ_PressureTotal01}}
[http://www.vistrails.org/index.php/User:Tohline/SR/PressureCombinations [[Image:LSU_OriginButton.jpg|150px]] ]
</td>
</tr>

</table>

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.

===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.

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