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> |
Derivation
We begin by defining the normalized total gas pressure as follows:
<math> p_\mathrm{total} \equiv \frac{1}{A_\mathrm{F}} \biggl[ P_\mathrm{gas} + P_\mathrm{deg} + P_\mathrm{rad} \biggr] . </math>
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),
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<math>~P_\mathrm{deg} = A_\mathrm{F} F(\chi) </math> |
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where: <math>F(\chi) \equiv \chi(2\chi^2 - 3)(\chi^2 + 1)^{1/2} + 3\sinh^{-1}\chi</math> |
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and: |
<math>\chi \equiv (\rho/B_\mathrm{F})^{1/3}</math> |
we see that,
<math> \frac{P_\mathrm{deg}}{A_\mathrm{F}} = F(\chi) . </math>
Normalized Ideal-Gas Pressure
Given the original expression for the pressure of an ideal gas,
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<math>~P_\mathrm{gas} = \frac{\Re}{\bar{\mu}} \rho T</math> |
along with the definitions of the physical constants, <math>~\Re</math>, <math>~A_\mathrm{F}</math>, and <math>~B_\mathrm{F}</math> provided in the accompanying Variables Appendix, we can write,
<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>
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,
<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>
where the atomic mass unit, <math>m_u \equiv 1/N_\mathrm{A}</math>.
Normalized Radiation Pressure
Given the original expression for the radiation pressure,
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<math>~P_\mathrm{rad} = \frac{1}{3} a_\mathrm{rad} T^4</math> |
along with the definitions of the physical constants, <math>~A_\mathrm{F}</math>, and <math>~a_\mathrm{rad}</math> provided in the accompanying Variables Appendix, we can write,
<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>
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© 2014 - 2021 by Joel E. Tohline |