User:Tohline/Appendix/Ramblings/PPToriPt2

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Stability Analyses of PP Tori (Part 2)

Whitworth's (1981) Isothermal Free-Energy Surface
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This is a direct extension of our Part 1 discussion. Here we continue our effort to check the validity of the Blaes85 eigenvector. The relevant reference is:

Start From Scratch

Basic Equations from Blaes85

  Blaes85

Eq. No.

<math>~(\beta\eta)^2</math>

<math>~=</math>

<math>~x^2(1+xb) \, ;</math>

(2.6)

<math>~b</math>

<math>~\equiv</math>

<math>~3\cos\theta - \cos^3\theta \, ;</math>

(2.6)

<math>~f</math>

<math>~=</math>

<math>~1-\eta^2 \, .</math>

(2.5)

  Blaes85

Eq. No.

<math>~LHS \equiv \hat{L}W</math>

<math>~=</math>

<math> ~fx^2 \cdot \frac{\partial^2 W}{\partial x^2} + f \cdot \frac{\partial^2 W}{\partial \theta^2} + \biggl[ \frac{fx(1-2x\cos\theta)}{(1-x\cos\theta)} + nx^2\cdot \frac{\partial f}{\partial x}\biggr]\frac{\partial W}{\partial x} </math>

 

 

 

<math> + \biggl[ \frac{fx\sin\theta}{(1-x\cos\theta)} + n\cdot \frac{\partial f}{\partial \theta}\biggr]\frac{\partial W}{\partial \theta} + \biggl[ \frac{2nx^2m^2}{\beta^2(1-x\cos\theta)^4} - \frac{m^2 x^2 f}{(1-x\cos\theta)^2} \biggr]W </math>

(4.2)

<math>~RHS</math>

<math>~=</math>

<math> ~-\frac{2nm^2}{\beta^2} \cdot (\beta\eta)^2 \biggl[ M \biggl(\frac{\nu}{m}\biggr)^2 + \frac{N}{m} \biggl(\frac{\nu}{m}\biggr)\biggr] W </math>

(4.1)

 

<math>~=</math>

<math> ~-\frac{2nm^2}{\beta^2} \biggl[ x^2 \biggl(\frac{\nu}{m}\biggr)^2 + \frac{2x^2}{(1-x\cos\theta)^2} \biggl(\frac{\nu}{m}\biggr)\biggr] W </math>

(4.2)

<math>~\frac{W}{A_{00}}</math>

<math>~=</math>

<math> ~1 + \beta^2 m^2 \biggl\{ 2\eta^2\cos^2\theta - \frac{3\eta^2}{4(n+1)} - \frac{(4n+1)}{4(n+1)^2} ~\pm~i~\biggl[ \frac{2^3\cdot 3}{(n+1)}\biggr]^{1/2} \eta\cos\theta \biggr\} </math>

(4.13)

<math>~\frac{\nu}{m}</math>

<math>~=</math>

<math> ~-1 ~\pm ~ i~\biggl[ \frac{3}{2(n+1)} \biggr]^{1/2} \beta </math>

(4.14)

Our Manipulation of These Equations

<math>~\Lambda \equiv \frac{2^2(n+1)^2}{m^2}\biggl[\frac{W}{A_{00}}-1\biggr]</math>

<math>~=</math>

<math>~\beta^2 \biggl\{ 2^3(n+1)^2 \eta^2\cos^2\theta - 3\eta^2(n+1)^2 - (4n+1) ~\pm~i~[ 2^7\cdot 3(n+1)^3 ]^{1/2} \eta\cos\theta \biggr\} </math>

 

<math>~=</math>

<math>~- (4n+1)\beta^2 + (\beta\eta)^2 (n+1)^2[ 2^3 \cos^2\theta - 3] ~\pm~i~\beta [ 2^7\cdot 3(n+1)^3 ]^{1/2} (\beta\eta) \cos\theta \, ; </math>

<math>~\Rightarrow~~~~\frac{W}{A_{00}} </math>

<math>~=</math>

<math>~1+ \biggl[ \frac{m}{2(n+1)} \biggr]^2 \Lambda </math>


<math>~\frac{LHS}{A_{00}} </math>

<math>~=</math>

<math>~\biggl[ \frac{m}{2(n+1)} \biggr]^2 f ~\biggl[ x^2 \cdot \frac{\partial^2 \Lambda}{\partial x^2} + \frac{\partial^2 \Lambda}{\partial \theta^2}\biggr] + \biggl[ \frac{m}{2(n+1)} \biggr]^2\biggl[ \frac{fx(1-2x\cos\theta)}{(1-x\cos\theta)} + nx^2\cdot \frac{\partial f}{\partial x}\biggr]\frac{\partial \Lambda}{\partial x} </math>

 

 

<math> + \biggl[ \frac{m}{2(n+1)} \biggr]^2\biggl[ \frac{fx\sin\theta}{(1-x\cos\theta)} + n\cdot \frac{\partial f}{\partial \theta}\biggr]\frac{\partial \Lambda}{\partial \theta} + \biggl[ \frac{2nx^2m^2}{\beta^2(1-x\cos\theta)^4} - \frac{m^2 x^2 f}{(1-x\cos\theta)^2} \biggr]\biggl\{1+ \biggl[ \frac{m}{2(n+1)} \biggr]^2 \Lambda\biggr\} </math>

 

<math>~=</math>

<math>~\biggl[ \frac{m}{2(n+1)} \biggr]^2 f \biggl\{ ~\biggl[ x^2 \cdot \frac{\partial^2 \Lambda}{\partial x^2} + \frac{\partial^2 \Lambda}{\partial \theta^2}\biggr] + \biggl[ \frac{x(1-2x\cos\theta)}{(1-x\cos\theta)} \biggr]\frac{\partial \Lambda}{\partial x} + \biggl[ \frac{x\sin\theta}{(1-x\cos\theta)} \biggr]\frac{\partial \Lambda}{\partial \theta} - \biggl[ \frac{m^2 x^2 }{(1-x\cos\theta)^2} \biggr] \biggl[ \frac{2^2(n+1)^2}{m^2} + \Lambda\biggr]\biggr\} </math>

 

 

<math> + n\biggl[ \frac{m}{2(n+1)} \biggr]^2 \biggl\{ x^2\cdot \frac{\partial f}{\partial x}\cdot \frac{\partial \Lambda}{\partial x} ~+~ \frac{\partial f}{\partial \theta}\cdot \frac{\partial \Lambda}{\partial \theta} ~+~ \biggl[ \frac{2x^2m^2}{\beta^2(1-x\cos\theta)^4} \biggr]\biggl[ \frac{2^2(n+1)^2}{m^2} + \Lambda\biggr]\biggr\} </math>

 

<math>~=</math>

<math>~\frac{x^2 f}{(1-x\cos\theta)^2} \biggl[ \frac{m}{2(n+1)} \biggr]^2 \biggl\{ ~(1-x\cos\theta)^2\biggl[ \frac{\partial^2 \Lambda}{\partial x^2} + \frac{1}{x^2}\cdot \frac{\partial^2 \Lambda}{\partial \theta^2}\biggr] + \frac{(1-x\cos\theta)}{x} \biggl[ (1-2x\cos\theta) \frac{\partial \Lambda}{\partial x} + \sin\theta\cdot \frac{\partial \Lambda}{\partial \theta} \biggr] - [ 2^2(n+1)^2 + m^2\Lambda ]\biggr\} </math>

 

 

<math> + ~\frac{x^2 n}{\beta^2(1-x\cos\theta)^4} \biggl[ \frac{m}{2(n+1)} \biggr]^2 \biggl\{\beta^2 (1-x\cos\theta)^4\biggl[ \frac{\partial f}{\partial x}\cdot \frac{\partial \Lambda}{\partial x} ~+~ \frac{1}{x^2}\cdot \frac{\partial f}{\partial \theta}\cdot \frac{\partial \Lambda}{\partial \theta} \biggr] ~+~ [ 2^3(n+1)^2 + 2m^2\Lambda ]\biggr\} \, . </math>

Also,

<math>~\frac{RHS}{A_{00}}</math>

<math>~=</math>

<math> ~-\frac{2n x^2}{\beta^2(1-x\cos\theta)^2} \biggl[ \frac{m}{2(n+1)} \biggr]^2 \biggl[ (1-x\cos\theta)^2\biggl(\frac{\nu}{m}\biggr)^2 + 2\biggl(\frac{\nu}{m}\biggr)\biggr] [ 2^2(n+1)^2 + m^2\Lambda ] </math>

 

<math>~=</math>

<math> ~-\frac{x^2n}{\beta^2(1-x\cos\theta)^4} \biggl[ \frac{m}{2(n+1)} \biggr]^2 \biggl[ (1-x\cos\theta)^4\biggl(\frac{\nu}{m}\biggr)^2 + 2(1-x\cos\theta)^2\biggl(\frac{\nu}{m}\biggr)\biggr] [ 2^3(n+1)^2 + 2m^2\Lambda ] \, . </math>

Putting the two together implies,

<math>~0</math>

<math>~=</math>

<math>~\frac{1}{x^2}\biggl[\frac{LHS}{A_{00}} - \frac{RHS}{A_{00}}\biggr]\biggl[ \frac{2(n+1)}{m} \biggr]^2 (1-x\cos\theta)^4</math>

 

<math>~=</math>

<math>~f (1-x\cos\theta)^2 \biggl\{ ~(1-x\cos\theta)^2\biggl[ \frac{\partial^2 \Lambda}{\partial x^2} + \frac{1}{x^2}\cdot \frac{\partial^2 \Lambda}{\partial \theta^2}\biggr] + \frac{(1-x\cos\theta)}{x} \biggl[ (1-2x\cos\theta) \frac{\partial \Lambda}{\partial x} + \sin\theta\cdot \frac{\partial \Lambda}{\partial \theta} \biggr] - [ 2^2(n+1)^2 + m^2\Lambda ]\biggr\} </math>

 

 

<math> + ~\frac{n}{\beta^2} \biggl\{\beta^2 (1-x\cos\theta)^4\biggl[ \frac{\partial f}{\partial x}\cdot \frac{\partial \Lambda}{\partial x} ~+~ \frac{1}{x^2}\cdot \frac{\partial f}{\partial \theta}\cdot \frac{\partial \Lambda}{\partial \theta} \biggr] ~+~ \biggl[ (1-x\cos\theta)^4\biggl(\frac{\nu}{m}\biggr)^2 + 2(1-x\cos\theta)^2\biggl(\frac{\nu}{m}\biggr)+ 1 \biggr] [ 2^3(n+1)^2 + 2m^2\Lambda ] \biggr\} \, . </math>

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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