View source for User:Tohline/SR/PressureCombinations

=Total Pressure=
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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">
<math>
~P_\mathrm{total} = P_\mathrm{gas} + P_\mathrm{deg} + P_\mathrm{rad} .
</math>
</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 [[User:Tohline/Appendix/Variables_templates|Variables Appendix]].  As derived below, this normalized total pressure can be written as,

<div align="center">
{{User:Tohline/Math/EQ_PressureTotal01}}
</div>

==Derivation==

We begin by defining the normalized total gas pressure as follows:
<div align="center">
<math>
p_\mathrm{total} \equiv \frac{1}{A_\mathrm{F}} \biggl[ P_\mathrm{gas} + P_\mathrm{deg} + P_\mathrm{rad} \biggr] .
</math>
</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.


===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),
<div align="center">
{{User:Tohline/Math/EQ_ZTFG01}}
</div>
we see that,
<div align="center">
<math>
\frac{P_\mathrm{deg}}{A_\mathrm{F}} = F(\chi) .
</math>
</div>


===Normalized Ideal-Gas Pressure===

Given the original expression for the pressure of an ideal gas,
<div align="center">
{{User:Tohline/Math/EQ_EOSideal0A}}
</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 [[User:Tohline/Appendix/Variables_templates|Variables Appendix]], we can write,
<div align="center">
<math>
\frac{P_\mathrm{gas}}{A_\mathrm{F}} = \frac{B_\mathrm{F}}{A_\mathrm{F}} \frac{\Re}{\bar{\mu}} \chi^3 T 
= \frac{\mu_e}{\bar{\mu}} \biggl[ \chi^3 T \biggr] \frac{8\pi m_p}{3} \biggl( \frac{m_e c}{h} \biggr)^3 \frac{3h^3}{\pi m_e^4 c^5} \biggl(k N_\mathrm{A} \biggr)
= \biggl(m_p N_\mathrm{A} \biggr)\frac{\mu_e}{\bar{\mu}} \biggl[8 \chi^3 T \biggr] \frac{k}{ m_e c^2}   .
</math>
</div>
Therefore, letting <math>T_e \equiv m_e c^2/k</math> represent the temperature associated with the rest-mass energy of the electron, the normalized ideal gas pressure is,
<div align="center">
<math>
\frac{P_\mathrm{gas}}{A_\mathrm{F}}  = \biggl(\frac{\mu_e m_p}{\bar{\mu} m_u} \biggr) \biggl[8 \chi^3 \frac{T}{T_e} \biggr] ,
</math>
</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>.


===Normalized Radiation Pressure===

Given the original expression for the radiation pressure,
<div align="center">
{{User:Tohline/Math/EQ_EOSradiation01}}
</div>
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">
<math>
\frac{P_\mathrm{rad}}{A_\mathrm{F}} = \biggl( \frac{T^4}{3} \biggr) \frac{a_\mathrm{rad}}{A_\mathrm{F}} 
= \biggl( \frac{T^4}{3} \biggr) \frac{8\pi^5}{15}\frac{k^4}{(hc)^3} \frac{3h^3}{\pi m_e^4 c^5} 
= \frac{8\pi^4}{15} \biggl( \frac{T}{T_e} \biggr)^4 .
</math>
</div>


==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, <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">
<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>

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 &#8212; namely, the normalized temperature, <math>~z</math> &#8212; 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">
<math>
z(\chi, p_\mathrm{total}) = \theta_\chi \phi^{-1/3}\biggl[ (\phi - 1)^{1/2} - 1 \biggr] ,
</math>
</div>

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" bgcolor="yellow">
<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" bgcolor="lightblue">
<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" bgcolor="pink">
<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.

===Dominant Contributions===

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}} &nbsp;<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 &#8212; 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">
<math>
\frac{P_\mathrm{rad}}{P_\mathrm{gas}} \approx \frac{\pi^4}{15} \biggl( \frac{z}{\chi} \biggr)^3
</math> .
</div>
This means that radiation pressure will dominate over ideal gas pressure in any regime where,
<div align="center">
<math>
T \gg T_e \biggl[\frac{15}{\pi^4} \biggl(\frac{\rho}{B_F} \biggr) \biggr]^{1/3}
</math> ,
</div>
that is, whenever,
<div align="center">
<math>
T_7 \gg 3.2 \biggl[\frac{\rho_1}{\mu_e} \biggr]^{1/3}
</math> ,
</div>
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 &#8212; see our [[User:Tohline/SR#Nonrelativistic_ZTF_Gas|separate discussion of the Zero-Temperature Fermi gas]] &#8212; in the limit <math>~\chi \ll 1</math>,
<div align="center">
<math>
F(\chi) \approx \frac{8}{5} \chi^5
</math> ;
</div>
and in the limit <math>\chi \gg 1</math>,
<div align="center">
<math>
F(\chi) \approx 2 \chi^4
</math> .
</div>
Hence, at low densities (<math>\chi \ll 1</math>), 
<div align="center">
<math>
\frac{P_\mathrm{gas}}{P_\mathrm{deg}} \approx \frac{5 z}{ \chi^{2}}
~~~~~ \mathrm{and} ~~~~~
\frac{P_\mathrm{rad}}{P_\mathrm{deg}} \approx \biggl(\frac{\pi^4}{3}\biggr) \frac{z^4}{ \chi^5} ;
</math>
</div>
and at high densities (<math>\chi \gg 1</math>),
<div align="center">
<math>
\frac{P_\mathrm{gas}}{P_\mathrm{deg}} \approx  \frac{4z}{\chi} 
~~~~~ \mathrm{and} ~~~~~
\frac{P_\mathrm{rad}}{P_\mathrm{deg}} \approx \frac{4 \pi^4}{15} \biggl( \frac{z}{\chi} \biggr)^4  .
</math>
</div>
  
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===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 z + C_r z^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>~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">
<math>\chi^3(p_\mathrm{total},z) =  \frac{1}{C_g z} \biggl[ p_\mathrm{total} - C_r z^4 \biggr] .</math>
</div>
-->


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