User:Tohline/SR/PressureCombinations
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Total Pressure
In our overview of equations of state that are used to supplement our set of principal governing equations when studying time-dependent problems, 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,
<math> P_\mathrm{total} = P_\mathrm{gas} + P_\mathrm{deg} + P_\mathrm{rad} . </math>
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, <math>~A_\mathrm{F}</math>, as defined in the accompanying Variables Appendix. As derived below, this normalized total pressure can be written as,
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<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> |
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