Difference between revisions of "User:Tohline/SR/Ptot QuarticSolution"

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=Determining Temperature from Density and Pressure=
{{LSU_HBook_header}}
{{LSU_HBook_header}}
 
As has been [[User:Tohline/SR/PressureCombinations#Derivation|derived elsewhere]], the normalized total pressure can be written as,
=Determining Temperature from Density and Pressure=
As has been [http://www.vistrails.org/index.php/User:Tohline/SR/PressureCombinations#Derivation derived elsewhere], the normalized total pressure can be written as,


<div align="center">
<div align="center">
{{User:Tohline/Math/EQ_PressureTotal01}}
{{User:Tohline/Math/EQ_PressureTotal01}}
</div>
</div>
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,
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,
<div align="center">
<div align="center">
<math>
<math>~
z^4 + a_1 z + a_0 = 0 ,
a_4z^4 + a_1 z - a_0 = 0 \, ,
</math>
</math>
</div>
</div>
Line 16: Line 15:
<div align="center">
<div align="center">
<math>
<math>
z  \equiv \frac{T}{T_e} ,
~z  \equiv \frac{T}{T_e} \, ,
</math>
</div>
and the coefficients,
<div align="center">
<math>~
a_4 \equiv \frac{8\pi^4}{15} \, ,
</math><br/>
</math><br/>
<math>
<math>~
a_1 \equiv \frac{15}{\pi^4} \biggl(\frac{\mu_e m_p}{\bar{\mu} m_u} \biggr) \chi^3 ,
a_1 \equiv 8\biggl(\frac{\mu_e m_p}{\bar{\mu} m_u} \biggr) \chi^3 \, ,
</math><br/>
</math><br/>
<math>
<math>~
a_0 \equiv \frac{15}{8\pi^4}\biggl[p_\mathrm{total} - F(\chi)  \biggr] .
a_0 \equiv \biggl[p_\mathrm{total} - F(\chi)  \biggr] \, .
</math>
</math>
</div>
</div>
Line 30: Line 35:
===Quartic Equation Solution===
===Quartic Equation Solution===


Following the outline and notation used by [http://mathworld.wolfram.com/QuarticEquation.html mathworld.wolfram.com] to identify the roots of an arbitrary quartic equation, we can set <math>a_3 = a_2 = 0</math> and deduce that the only root that will give physically relevant temperatures &#8212; for example, real and non-negative &#8212; is,
Following the [http://en.wikipedia.org/wiki/Quartic_function#Summary_of_Ferrari.27s_method 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 &#8212; for example, real and non-negative &#8212; is,
<div align="center">
<div align="center">
<math>
<math>
z_3 = \frac{1}{2}\biggl[E-R\biggr] ,
2z = \biggl[\frac{2a_1}{a_4 W}-W^2\biggr]^{1/2} - W ,
</math>
</math>
</div>
</div>
Line 39: Line 44:
<div align="center">
<div align="center">
<math>
<math>
R \equiv y_1^{1/2} ,
\frac{1}{2}W^2 \equiv R^{-1/3}\biggl[R^{2/3} - \frac{a_0}{3a_4} \biggr] ,
</math><br/>
</math><br/>
<math>
<math>
E \equiv \biggl[ \frac{2a_1}{R} - R^2  \biggr]^{1/2} = y_1^{1/2}\biggl[ 2a_1 y_1^{-3/2} - 1  \biggr]^{1/2},
R \equiv \biggl( \frac{a_1}{4a_4} \biggr)^2 \biggl[ 1 + (1 + \lambda^3)^{1/2} \biggr] ,
</math>
</math><br />
</div>
and <math>y_1</math> is the <i>real</i> root of the following cubic equation:
<div align="center">
<math>
<math>
y^3 -4a_0 y -a_1^2 = 0 .
\lambda^3 \equiv \frac{2^8 a_0^3 a_4}{3^3 a_1^4} .
</math>
</math>
</div>
</div>
So, fully in terms of the real root of this cubic equation, the desired solution of our quartic equation is,
Defining,
<div align="center">
<div align="center">
<math>
<math>
z_3 = \frac{1}{2}y_1^{1/2} \biggl\{ \biggl[ 2a_1 y_1^{-3/2} - 1  \biggr]^{1/2}  -1 \biggr\} .
\phi \equiv \frac{2a_1}{a_4 W^3} ~~~~~\Rightarrow ~~~~~ W = 2\biggl(\frac{a_1}{4a_4 \phi} \biggr)^{1/3} ,
</math>
</math>
</div>
</div>
 
and realizing that, from one of the above expressions,  
===Relevant Cubic Formula===
The relevant cubic equation is,
<div align="center">
<div align="center">
<math>
<math>
y^3 +b_1 y +b_0 = 0 ,
\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]
</math>
= \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\} ,
</div>
where,
<div align="center">
<math>
b_1 \equiv -4a_0 ~~~~~ \mathrm{and} ~~~~~ b_0 \equiv -a_1^2 .
</math><br/>
</math><br/>
</div>
</div>
According to [http://mathworld.wolfram.com/CubicFormula.html mathworld.wolfram.com], the roots of a cubic equation having this form include a real root given by the expression,
we can rewrite the desired root of our quartic equation in the form,
<div align="center">
<div align="center">
<math>
<math>
y_1 = \mathcal{S} + \mathcal{T} ,
z = \biggl(\frac{a_1}{4a_4}\biggr)^{1/3} \phi^{-1/3}\biggl[ (\phi - 1)^{1/2} - 1 \biggr] ,
</math>
</math>
</div>
</div>
where,
with,
<div align="center">
<div align="center">
<math>
<math>
\mathcal{D}  \equiv \biggl( \frac{b_1}{3} \biggr)^2 + \biggl(\frac{b_0}{2}\biggr)^2
\phi = 2^{3/2} \biggl[ 1 + (1 + \lambda^3)^{1/2} \biggr]^{1/2}
= \biggl( \frac{4a_0}{3} \biggr)^2 + \biggl(\frac{a_1^2}{2}\biggr)^2
\biggl\{ \biggl[ 1 + (1 + \lambda^3)^{1/2} \biggr]^{2/3} - \lambda \biggr\}^{-3/2} .
= \biggl(\frac{a_1^2}{2}\biggr)^2 \biggl[ 1 + \biggl( \frac{8a_0}{3a_1^2} \biggr)^2 \biggr] ,
</math><br />
<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><br/>
<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>
</math>
</div>
</div>


===Summary===
===Key Dimensionless Parameters===
Hence, defining,
From the above solution, there appear to be two key dimensionless parameters that can be formed from a strategic combination of the coefficients of the original quartic equation. They are,
<div align="center">
<div align="center">
<math>
<math>~
\mathcal{K} \equiv \biggl( \frac{8a_0}{3a_1^2} \biggr)^2 ,
\lambda^3 \equiv \frac{2^8 a_0^3 a_4}{3^3 a_1^4} , ~~~~~\mathrm{and}~~~~~
\theta^3 \equiv \frac{a_1}{2^2 a_4} .
</math>
</math>
</div>
</div>
and
[Note, as well, that the product <math>~\lambda\theta = (4a_0)/(3a_1)</math>.]
The desired solution of our quartic equation is a product of <math>~\theta</math> and an expression that is only a function of <math>~\lambda</math>.  Specifically, we can write,
<div align="center">
<div align="center">
<math>
<math>~
f_Q(\mathcal{K}) \equiv y_1 \biggl[ \frac{2}{a_1^2} \biggr]^{1/3} =  
\frac{z}{\theta} = \mathcal{K}(\lambda) ,
\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>
</math>
</div>
</div>
we can write the desired solution of the quartic equation as,
where, in terms of the above-defined function <math>~\phi(\lambda)</math>,
<div align="center">
<div align="center">
<math>
<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\} .
\mathcal{K}(\phi(\lambda)) \equiv \phi^{-1/3}\biggl[ (\phi - 1)^{1/2} - 1 \biggr] .
</math>
</math>
</div>
</div>
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,
 
 
===Limiting Regimes===
We can immediately see that this solution makes sense in the present context.  In order for the temperature &#8212; that is, <math>~z</math> &#8212; to be real and nonnegative, the function <math>~\phi(\lambda)</math> must be greater than or equal to <math>~2</math>.  This limiting value occurs when the dimensionless parameter, <math>~\lambda = 0</math>.  The constraint <math>~\lambda \ge 0</math> is satisfied as long as the three coefficients of the quartic equation are real and nonnegative, which is certainly true for our specific problem.
 
Looking at the limiting functional behavior of our solution, we see that when <math>~0 \le \lambda \ll 1</math>,
<div align="center">
<div align="center">
<math>
<math>~
8^{1/2} [f_Q(\mathcal{K})]^{-3/2} \ge 2
\phi \approx 2 + \frac{3}{2^{2/3}}\lambda
</math><br />
</math><br />
<math>
 
\Rightarrow ~~ f_Q(\mathcal{K}) \le 2^{1/3} .
<math>~
\Rightarrow ~~~~~ z \approx \frac{3\lambda}{2^2} \biggl( \frac{a_1}{2^2a_4} \biggr)^{1/3} = \frac{a_0}{a_1} .
</math>
</math>
</div>
</div>
This, in turn, implies that the dimensionless parameter, <math>\mathcal{K}</math>, must be limited to values,
 
 
We see, as well, that when <math>~\lambda \gg 1</math>,
<div align="center">
<div align="center">
<math>
<math>~
\mathcal{K} \ge 0 .
\phi \approx \biggl(\frac{3\lambda}{2^2} \biggr)^{3/2}
</math><br />
 
<math>~
\Rightarrow ~~~~~ z \approx \biggl(\frac{3\lambda}{2^2} \biggr)^{1/4} \biggl( \frac{a_1}{2^2a_4} \biggr)^{1/3} = \biggl( \frac{a_0}{a_4} \biggr)^{1/4} .
</math>
</math>
</div>
</div>
Our solution takes on a somewhat cleaner form if we define a function,
<div align="center">
<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>
</div>
which is limited to values,
<div align="center">
<math>
g_Q(\mathcal{K}) \ge 1 .
</math>
</div>
Then we can write,
<div align="center">
<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>
</div>


==Physical Implications==
==Physical Implications==
Clearly, a key dimensionless physical parameter for this problem is,
Clearly, a key dimensionless physical parameter for this problem is,
<div align="center">
<div align="center">
<math>
<math>~
\mathcal{K} \equiv \biggl( \frac{8a_0}{3a_1^2} \biggr)^2 =  
\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{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
\biggl[ \frac{p_\mathrm{total} - F(\chi)}{\chi^6} \biggr] \biggr\}^2 .
</math>
</math>
</div>
</div>
And, because <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=
* [http://en.wikipedia.org/wiki/Quartic_function Quartic Function]
* [http://mathworld.wolfram.com/QuarticEquation.html mathworld.wolfram.com]


{{LSU_HBook_footer}}
{{LSU_HBook_footer}}

Latest revision as of 03:00, 13 July 2015

Determining Temperature from Density and Pressure

Whitworth's (1981) Isothermal Free-Energy Surface
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As has been derived elsewhere, the normalized total pressure can be written as,

LSU Key.png

<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>

Key Dimensionless Parameters

From the above solution, there appear to be two key dimensionless parameters that can be formed from a strategic combination of the coefficients of the original quartic equation. They are,

<math>~ \lambda^3 \equiv \frac{2^8 a_0^3 a_4}{3^3 a_1^4} , ~~~~~\mathrm{and}~~~~~ \theta^3 \equiv \frac{a_1}{2^2 a_4} . </math>

[Note, as well, that the product <math>~\lambda\theta = (4a_0)/(3a_1)</math>.] The desired solution of our quartic equation is a product of <math>~\theta</math> and an expression that is only a function of <math>~\lambda</math>. Specifically, we can write,

<math>~ \frac{z}{\theta} = \mathcal{K}(\lambda) , </math>

where, in terms of the above-defined function <math>~\phi(\lambda)</math>,

<math>~ \mathcal{K}(\phi(\lambda)) \equiv \phi^{-1/3}\biggl[ (\phi - 1)^{1/2} - 1 \biggr] . </math>


Limiting Regimes

We can immediately see that this solution makes sense in the present context. In order for the temperature — that is, <math>~z</math> — to be real and nonnegative, the function <math>~\phi(\lambda)</math> must be greater than or equal to <math>~2</math>. This limiting value occurs when the dimensionless parameter, <math>~\lambda = 0</math>. The constraint <math>~\lambda \ge 0</math> is satisfied as long as the three coefficients of the quartic equation are real and nonnegative, which is certainly true for our specific problem.

Looking at the limiting functional behavior of our solution, we see that when <math>~0 \le \lambda \ll 1</math>,

<math>~ \phi \approx 2 + \frac{3}{2^{2/3}}\lambda </math>

<math>~ \Rightarrow ~~~~~ z \approx \frac{3\lambda}{2^2} \biggl( \frac{a_1}{2^2a_4} \biggr)^{1/3} = \frac{a_0}{a_1} . </math>


We see, as well, that when <math>~\lambda \gg 1</math>,

<math>~ \phi \approx \biggl(\frac{3\lambda}{2^2} \biggr)^{3/2} </math>

<math>~ \Rightarrow ~~~~~ z \approx \biggl(\frac{3\lambda}{2^2} \biggr)^{1/4} \biggl( \frac{a_1}{2^2a_4} \biggr)^{1/3} = \biggl( \frac{a_0}{a_4} \biggr)^{1/4} . </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, because <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

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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