Editing SR/PressureCombinations (section)

===Dominant Contributions===

Let's examine which pressure contributions will dominate in various temperature-density regimes. 
Note, first, that {{ Template:Math/C_ProtonMass }}/{{ Template:Math/C_AtomicMassUnit }} &nbsp;<math>\approx 1</math> and, for fully ionized gases, the ratio {{ Template:Math/MP_ElectronMolecularWeight }}<math>/</math>{{ Template: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> {{ Template:Math/MP_ElectronMolecularWeight }}<math>/</math>{{ Template: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 [[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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