Editing SR/PressureCombinations (section)

===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 [[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 [[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="white">
<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="white">
<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="white">
<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 [[SR/PressureCombinations#Total_Pressure|above key expression for the total pressure]] can be inverted to provide such a closed-form expression.