User:Tohline/SR/Ptot QuarticSolution
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Determining Temperature from Density and Pressure
As has been derived elsewhere, the 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> |
To solve this algebraic equation for the normalized temperature <math>T/T_e</math>, given values of the normalized total pressure <math>p_\mathrm{total}</math> and the normalized density <math>\chi</math>, we first realize that the equation can be written in the form,
<math> a_4z^4 + a_1 z - a_0 = 0 , </math>
where,
<math> z \equiv \frac{T}{T_e} , </math>
and the coefficients,
<math>
a_4 \equiv \frac{8\pi^4}{15} ,
</math>
<math>
a_1 \equiv 8\biggl(\frac{\mu_e m_p}{\bar{\mu} m_u} \biggr) \chi^3 ,
</math>
<math>
a_0 \equiv \biggl[p_\mathrm{total} - F(\chi) \biggr] .
</math>
Mathematical Manipulation
Quartic Equation Solution
Following the Summary of Ferrari's method that is presented in Wikipedia's discussion of the Quartic Function to identify the roots of an arbitrary quartic equation, we can set the coefficients of the cubic and quadratic terms both to zero and deduce that the only root that will give physically relevant temperatures — for example, real and non-negative — is,
<math> 2z = \biggl[\frac{2a_1}{a_4 W}-W^2\biggr]^{1/2} - W , </math>
where,
<math>
\frac{1}{2}W^2 \equiv R^{-1/3}\biggl[R^{2/3} - \frac{a_0}{3a_4} \biggr] ,
</math>
<math>
R \equiv \biggl( \frac{a_1}{4a_4} \biggr)^2 \biggl[ 1 + (1 + \lambda^3)^{1/2} \biggr] ,
</math>
<math>
\lambda^3 \equiv \frac{2^8 a_0^3 a_4}{3^3 a_1^4} .
</math>
Defining,
<math> \phi \equiv \frac{2a_1}{a_4 W^3} ~~~~~\Rightarrow ~~~~~ W = 2\biggl(\frac{a_1}{4a_4 \phi} \biggr)^{1/3} , </math>
and realizing that, from one of the above expressions,
<math>
\frac{1}{2}W^2 = \biggl[ \frac{a_1^4}{2^8 a_4^4 R}\biggr]^{1/3}\biggl[\biggl( \frac{2^4 a_4^2 R}{a_1^2}\biggr)^{2/3} - \biggl( \frac{2^8 a_0^3 a_4}{3^3 a_1^4} \biggr)^{1/3} \biggr]
= \biggl[ \frac{a_1}{2^2 a_4}\biggr]^{2/3} \biggl[ 1 + (1 + \lambda^3)^{1/2} \biggr]^{-1/3}\biggl\{ \biggl[ 1 + (1 + \lambda^3)^{1/2} \biggr]^{2/3} - \lambda \biggr\} ,
</math>
we can rewrite the desired root of our quartic equation in the form,
<math> z = \biggl(\frac{a_1}{4a_4}\biggr)^{1/3} \phi^{-1/3}\biggl[ (\phi - 1)^{1/2} - 1 \biggr] , </math>
with,
<math> \phi = 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>
Relevant Cubic Formula
The relevant cubic equation is,
<math> y^3 +b_1 y +b_0 = 0 , </math>
where,
<math>
b_1 \equiv -4a_0 ~~~~~ \mathrm{and} ~~~~~ b_0 \equiv -a_1^2 .
</math>
According to mathworld.wolfram.com, the roots of a cubic equation having this form include a real root given by the expression,
<math> y_1 = \mathcal{S} + \mathcal{T} , </math>
where,
<math>
\mathcal{D} \equiv \biggl( \frac{b_1}{3} \biggr)^2 + \biggl(\frac{b_0}{2}\biggr)^2
= \biggl( \frac{4a_0}{3} \biggr)^2 + \biggl(\frac{a_1^2}{2}\biggr)^2
= \biggl(\frac{a_1^2}{2}\biggr)^2 \biggl[ 1 + \biggl( \frac{8a_0}{3a_1^2} \biggr)^2 \biggr] ,
</math>
<math>
\mathcal{S} \equiv \biggl[ -\frac{b_0}{2} + \mathcal{D}^{1/2} \biggr]^{1/3}
= \biggl[ \frac{a_1^2}{2} + \mathcal{D}^{1/2} \biggr]^{1/3}
= \biggl[ \frac{a_1^2}{2} \biggr]^{1/3} \biggl\{1 + \biggl[ 1 + \biggl( \frac{8a_0}{3a_1^2} \biggr)^2 \biggr]^{1/2} \biggr\}^{1/3} ,
</math>
<math>
\mathcal{T} \equiv \biggl[ -\frac{b_0}{2} - \mathcal{D}^{1/2} \biggr]^{1/3}
= \biggl[ \frac{a_1^2}{2} - \mathcal{D}^{1/2} \biggr]^{1/3}
= \biggl[ \frac{a_1^2}{2} \biggr]^{1/3} \biggl\{1 - \biggl[ 1 + \biggl( \frac{8a_0}{3a_1^2} \biggr)^2 \biggr]^{1/2} \biggr\}^{1/3} .
</math>
Summary
Hence, defining,
<math> \mathcal{K} \equiv \biggl( \frac{8a_0}{3a_1^2} \biggr)^2 , </math>
and
<math> f_Q(\mathcal{K}) \equiv y_1 \biggl[ \frac{2}{a_1^2} \biggr]^{1/3} = \biggl[1 + \biggl( 1 + \mathcal{K} \biggr)^{1/2} \biggr]^{1/3} + \biggl[1 - \biggl( 1 + \mathcal{K} \biggr)^{1/2} \biggr]^{1/3} , </math>
we can write the desired solution of the quartic equation as,
<math> z_3 = 2^{-7/6} a_1^{1/3} [f_Q(\mathcal{K})]^{1/2} \biggl\{ \biggl[ 8^{1/2} [f_Q(\mathcal{K})]^{-3/2} - 1 \biggr]^{1/2} -1 \biggr\} . </math>
Note that, in order for the solution, <math>z_3</math>, to be real and non-negative, the function <math>f_Q(\mathcal{K})</math> must be limited to a range of values given by the expression,
<math>
8^{1/2} [f_Q(\mathcal{K})]^{-3/2} \ge 2
</math>
<math>
\Rightarrow ~~ f_Q(\mathcal{K}) \le 2^{1/3} .
</math>
This, in turn, implies that the dimensionless parameter, <math>\mathcal{K}</math>, must be limited to values,
<math> \mathcal{K} \ge 0 . </math>
Our solution takes on a somewhat cleaner form if we define a function,
<math> g_Q(\mathcal{K}) \equiv 2^{1/2} f_Q^{-3/2} = 2^{1/2} \biggl\{ \biggl[1 + \biggl( 1 + \mathcal{K} \biggr)^{1/2} \biggr]^{1/3} + \biggl[1 - \biggl( 1 + \mathcal{K} \biggr)^{1/2} \biggr]^{1/3} \biggr\}^{-3/2} , </math>
which is limited to values,
<math> g_Q(\mathcal{K}) \ge 1 . </math>
Then we can write,
<math> z_3 a_1^{-1/3} = \frac{1}{2}\biggl[\frac{1}{g_Q(\mathcal{K})}\biggr]^{1/3} \biggl\{ \biggl[ 2g_Q(\mathcal{K}) - 1 \biggr]^{1/2} -1 \biggr\} . </math>
Physical Implications
Clearly, a key dimensionless physical parameter for this problem is,
<math> \mathcal{K} \equiv \biggl( \frac{8a_0}{3a_1^2} \biggr)^2 = \biggl\{ \frac{1}{5} \biggl[ \frac{\pi^2}{3} \biggl(\frac{\bar{\mu} m_u}{\mu_e m_p} \biggr) \biggr]^2 \biggl[ \frac{p_\mathrm{total} - F(\chi)}{\chi^6} \biggr] \biggr\}^2 . </math>
And, since <math>z_3 \propto T</math> and <math>a_1 \propto \rho</math>, the above solution tells us that the product <math>T \rho^{-1/3}</math> can be expressed as a function of this single parameter, <math>\mathcal{K}</math>.
Related Wikepedia Links
Quartic Function
mathworld.wolfram.com
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© 2014 - 2021 by Joel E. Tohline |