User:Tohline/SR/PressureCombinations
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Total Pressure
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In our overview of equations of state, we identified analytic expressions for the pressure of an ideal gas, , electron degeneracy pressure, , and radiation pressure, . 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,
In order to assess which of these three contributions will dominate in different density and temperature regimes, it is instructive to normalize to the characteristic Fermi pressure, , as defined in the accompanying Variables Appendix. As derived below, this normalized total pressure can be written as,
Derivation
We begin by defining the normalized total gas pressure as follows:
To derive the expression for 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 zerotemperature Fermi gas),
we see that,
Normalized IdealGas Pressure
Given the original expression for the pressure of an ideal gas,
along with the definitions of the physical constants, , , and provided in the accompanying Variables Appendix, we can write,
Therefore, letting represent the temperature associated with the restmass energy of the electron, the normalized ideal gas pressure is,
where, by definition, the atomic mass unit is, , that is, .
Normalized Radiation Pressure
Given the original expression for the radiation pressure,
along with the definitions of the physical constants, , and provided in the accompanying Variables Appendix, we can write,
Discussion
For simplicity of presentation, in what follows we will use
to represent a normalized temperature, in addition to using to represent (the cube root of) the normalized mass density, and to represent the normalized total pressure.
Relationship Between State Variables
If the two normalized state variables, and , are known, then the third normalized state variable, , can be obtained directly from the above key expression for the total pressure, that is,
where,
If it is the two normalized state variables, and , that are known, the third normalized state variable — namely, the normalized temperature, — 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 detailed in the accompanying discussion, the desired solution is,
where,









It also would be desirable to have an analytic expression for the function, , 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 above key expression for the total pressure can be inverted to provide such a closedform expression.
Dominant Contributions
Let's examine which pressure contributions will dominate in various temperaturedensity regimes. Note, first, that / and, for fully ionized gases, the ratio is of order unity — more precisely, the ratio of these two molecular weights falls within the narrow range . Hence, we can assume that the numerical coefficient of the first term in our expression for is approximately , so the ratio of radiation pressure to gas pressure is,
.
This means that radiation pressure will dominate over ideal gas pressure in any regime where,
,
that is, whenever,
,
where is the temperature expressed in units of and is the matter density expressed in units of .
Second, note that the function can be written in a simpler form when examining regions of either very low or very high matter densities. Specifically — see our separate discussion of the ZeroTemperature Fermi gas — in the limit ,
;
and in the limit ,
.
Hence, at low densities (),
and at high densities (),
© 2014  2019 by Joel E. Tohline 