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Continue Search for Marginally Unstable (5,1) Bipolytropes

This Ramblings Appendix chapter — see also, various trials — provides some detailed trial derivations in support of the accompanying, thorough discussion of this topic.


Whitworth's (1981) Isothermal Free-Energy Surface
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Key Differential Equation

In an accompanying discussion, we derived the so-called,

Linear Adiabatic Wave (or Radial Pulsation) Equation

~
\frac{d^2x}{dr_0^2} + \biggl[\frac{4}{r_0} - \biggl(\frac{g_0 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{dr_0} 
+ \biggl(\frac{\rho_0}{\gamma_\mathrm{g} P_0} \biggr)\biggl[\omega^2 + (4 - 3\gamma_\mathrm{g})\frac{g_0}{r_0} \biggr]  x = 0

whose solution gives eigenfunctions that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations. After adopting an appropriate set of variable normalizations — as detailed here — this becomes,

~0

~=

~
\frac{d^2x}{dr*^2} + \biggl\{ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}\frac{1}{r^*} \frac{dx}{dr*} 
+ \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr\}  x \, ,

where, ~\alpha_g \equiv (3 - 4/\gamma_g). Alternatively — see, for example, our introductory discussion — for polytropic configurations we may write,

~0

~=

~\frac{d^2x}{d\xi^2} + \biggl[4 - (n+1) \biggl(- \frac{d\ln \theta}{d\ln \xi} \biggr)\biggr] \frac{1}{\xi} \frac{dx}{d\xi} + 
\biggl\{ \frac{(n+1)}{\theta} \biggl[ \frac{\sigma_c^2}{6\gamma_g}\biggr]  
- 
(n+1) \biggl(- \frac{d\ln \theta}{d\ln \xi} \biggr) \frac{\alpha_g}{\xi^2 } \biggr\}  x \, .

Applied to the Core

As we have already summarized in an accompanying discussion, throughout the core we have,

~r^*

~=

~\biggl( \frac{3}{2\pi} \biggr)^{1/2} \xi  \, ;

     

~\frac{\rho^*}{P^*}

~=

~\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{1 / 2}  \, ;

     

~\frac{M_r^*}{r^*}

~=

~
2 \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} 
\, .

So the relevant core LAWE becomes,

~0

~=

~
\biggl( \frac{2\pi}{3} \biggr) \frac{d^2x}{d\xi^2} 
+ \biggl( \frac{2\pi}{3} \biggr) \biggl\{ 4 -  \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{1 / 2} \biggl[ 2 \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3 / 2} \biggr]\biggr\}\frac{1}{\xi} \frac{dx}{d\xi} 
+  \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{1 / 2}\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\biggl( \frac{2\pi}{3} \biggr)\frac{\alpha_\mathrm{g} }{\xi^2} \biggl[ 2 \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3 / 2} \biggr] \biggr\}  x

~\Rightarrow ~~~ \biggl( \frac{3}{4\pi} \biggr) \cdot 0

~=

~
\frac{1}{2}\cdot \frac{d^2x}{d\xi^2} 
+ \biggl[ 2 -  \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \biggr] \frac{1}{\xi} \frac{dx}{d\xi} 
+  \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{1 / 2}\biggl[ \frac{\sigma_c^2}{2\gamma_\mathrm{g}}  ~-~\alpha_\mathrm{g} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3 / 2}  \biggr]  x \, .

Now, following our separate discussion of an analytic solution to this LAWE, we try,

~x_P\biggr|_\mathrm{core}

~\equiv

~1 - \frac{\xi^2}{15}

~\Rightarrow~~~\frac{dx_P}{d\xi}\biggr|_\mathrm{core}

~\equiv

~- \frac{2\xi}{15}

~\Rightarrow~~~\frac{d\ln x_P}{d\ln \xi}\biggr|_\mathrm{core}

~\equiv

~- \frac{2\xi^2}{15} \biggl[ \frac{(15 - \xi^2)}{15} \biggr]^{-1} = - \frac{2\xi^2}{(15 - \xi^2)}  \, .

Plugging this trial function into the relevant LAWE gives,

LAWE

~=

~
\frac{1}{2} \biggl( -\frac{2}{3\cdot 5}\biggr)
+ \biggl( -\frac{2}{3\cdot 5}\biggr)\biggl[ 2 -  \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \biggr]
+  \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{1 / 2}\biggl[ \frac{\sigma_c^2}{2\gamma_\mathrm{g}}  ~-~\alpha_\mathrm{g} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3 / 2}  \biggr]  \biggl[1 - \frac{\xi^2}{15}\biggr]

 

~=

~
- \frac{1}{3}
+ \biggl( \frac{2}{3\cdot 5}\biggr)\biggl[ \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1} \biggr]
+  \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{1 / 2}\biggl[ \frac{\sigma_c^2}{2\gamma_\mathrm{g}}  ~-~\alpha_\mathrm{g} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3 / 2}  \biggr]  \biggl[1 - \frac{\xi^2}{15}\biggr]

Now, if we set ~\sigma_c^2 = 0 and ~\gamma_g = \gamma_c = \tfrac{6}{5} ~~\Rightarrow ~~ \alpha_g = -1/3, we find that the terms on the RHS sum to zero. It therefore appears that we have identified a dimensionless displacement function that satisfies the core LAWE.

Applied to the Envelope

And as we have also summarized in the same accompanying discussion, throughout the envelope we have,

~r^*

~=

~\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2}\eta  \, ;

   

~\frac{\rho^*}{P^*}

~=

~
\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \phi(\eta)^{-1}
\, ;

   

~\frac{M_r^*}{r^*}

~=

~
2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta_i \biggl(-\eta \frac{d\phi}{d\eta} \biggr)
\, .

So the relevant envelope LAWE becomes,

~\mathrm{LAWE}

~=

~
\frac{d^2x}{dr*^2} + \biggl\{ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{r^*}\biggr\}\frac{1}{r^*} \frac{dx}{dr*} 
+ \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr\}  x

 

~=

~
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2} \biggr]^{-2}\frac{d^2x}{d\eta^2} 
+ \biggl\{ 4 - \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \phi^{-1}\biggr] \biggl[ 2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta_i  \biggl(-\eta \frac{d\phi}{d\eta} \biggr) \biggr]
\biggr\}\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2} \biggr]^{-2}\frac{1}{\eta} \frac{dx}{d\eta}

 

 

~
+ \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \phi^{-1}\biggr]\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\frac{\alpha_\mathrm{g}}{\eta^2} 
\biggl[ 2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta_i  \biggl(-\eta \frac{d\phi}{d\eta} \biggr) \biggr] \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2} \biggr]^{-2}\biggr\}  x

~\Rightarrow ~~~ \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{2} \theta^{4}_i (2\pi) \biggr]^{-1} \cdot~ \mathrm{LAWE}

~=

~
\frac{d^2x}{d\eta^2} 
+ \biggl\{ 4 - \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \phi^{-1}\biggr] \biggl[ 2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta_i  \biggl(-\eta \frac{d\phi}{d\eta} \biggr) \biggr]
\biggr\} \frac{1}{\eta} \frac{dx}{d\eta}

 

 

~
+ \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{-1}_i \phi^{-1}\biggr]\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{2} \theta^{4}_i (2\pi) \biggr]^{-1}  
~-~\frac{\alpha_\mathrm{g}}{\eta^2} 
\biggl[ 2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta_i \biggl(-\eta \frac{d\phi}{d\eta} \biggr) \biggr]  \biggr\}  x

 

~=

~
\frac{d^2x}{d\eta^2} 
+ \biggl\{ 4 - \biggl[ 2  \biggl(-\frac{d\ln \phi}{d\ln \eta} \biggr) \biggr]
\biggr\} \frac{1}{\eta} \frac{dx}{d\eta} 
+ \biggl\{ \frac{\sigma_c^2}{3\gamma_\mathrm{g}}  \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-5}_i \phi^{-1}\biggr]
~-~\frac{\alpha_\mathrm{g}}{\eta^2} \biggl[ 2 \biggl(- \frac{d\ln \phi}{d\ln \eta} \biggr) \biggr]  \biggr\}  x

 

~=

~
\frac{d^2x}{d\eta^2} 
+ \biggl\{ 4 - 2Q_\eta
\biggr\} \frac{1}{\eta} \frac{dx}{d\eta} 
+ \biggl\{ \frac{\sigma_c^2}{3\gamma_\mathrm{g}}  \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-5}_i \phi^{-1}\biggr]
~-~(2Q_\eta)\frac{\alpha_\mathrm{g}}{\eta^2}  \biggr\}  x

where,

~\phi(\eta)

~=

~\frac{A\sin(\eta - B)}{\eta}

    and    

~Q_\eta

~\equiv

~- \frac{d\ln \phi}{d\ln\eta} = \biggl[1 - \eta\cot(\eta-B) \biggr] \, .

If we set ~\sigma_c^2 = 0 and ~\gamma_g = \gamma_e = 2 ~~\Rightarrow ~~ \alpha_g = +1, the envelope LAWE simplifies to the form,

~ \biggl(\frac{r^*}{\eta}\biggr)^2 \cdot~ \mathrm{LAWE}

~=

~
\frac{d^2x}{d\eta^2} 
+ \biggl\{ 4 - 2Q_\eta
\biggr\} \frac{1}{\eta} \frac{dx}{d\eta} 
- \biggl\{ \frac{2Q_\eta}{\eta^2}  \biggr\}  x \, .


In yet another Ramblings Appendix derivation we have explored a trial dimensionless displacement for the envelope of the form,

~x_P\biggr|_\mathrm{env}

~= \frac{3c_0}{\eta^2} \cdot Q_\eta \, .

In this case,

~\frac{1}{3c_0}\cdot \frac{dx_P}{d\eta}

~=

~\frac{1}{\eta^2} \frac{dQ_\eta}{d\eta} - \frac{2Q_\eta}{\eta^3}

 

~=

~
\frac{1}{\eta^2}\biggl[\eta -\cot(\eta - B)
+\eta\cot^2(\eta - B) 
\biggr] 
- \frac{2Q_\eta}{\eta^3}

 

~=

~
\frac{1}{\eta^3 \sin^2(\eta-B)} \biggl[\eta^2 + \eta\sin(\eta-B)\cos(\eta-B)  -2\sin^2(\eta-B)\biggr]

 

~=

~
\frac{1}{\eta\sin^2(\eta - B)} + \frac{\cot(\eta-B)}{\eta^2} - \frac{2}{\eta^3}
\, ;

~\frac{1}{3c_0}\cdot \frac{d^2x_P}{d\eta^2}

~=

~
\frac{d}{d\eta}\biggl[\frac{1}{\eta\sin^2(\eta - B)} + \frac{\cot(\eta-B)}{\eta^2} - \frac{2}{\eta^3} \biggr]

 

~=

~
\frac{6}{\eta^4} - \frac{2\cot(\eta-B)}{\eta^3} - \frac{2}{\eta^2\sin^2(\eta-B)} - \frac{2\cos(\eta-B)}{\eta \sin^3(\eta-B)}
\, ,

and it can be shown that the simplified envelope LAWE is perfectly satisfied. Notice that, with this adopted segment of the eigenfunction for the envelope, we have,

~\frac{d\ln x_P}{d\ln\eta}\biggr|_\mathrm{env} = \frac{\eta^3}{3c_0 Q_\eta}\cdot \frac{dx_P}{d\eta}

~=

~
\frac{\eta^3}{Q_\eta} \biggl\{ \frac{1}{\eta^2}\biggl[\eta -\cot(\eta - B)
+\eta\cot^2(\eta - B) 
\biggr] 
- \frac{2Q_\eta}{\eta^3} \biggr\}

 

~=

~
\frac{1}{Q_\eta} \biggl[\eta^2 - \eta \cot(\eta - B)
+\eta^2 \cot^2(\eta - B) 
\biggr] 
- 2

 

~=

~
\frac{[\eta^2 - 2 + \eta \cot(\eta - B)+\eta^2 \cot^2(\eta - B) ] }{[1 - \eta\cot(\eta-B)]}

 

~=

~
\frac{[\eta^2  - 2\sin^2(\eta-B) + \eta \sin(\eta-B) \cos(\eta - B) ] }{[\sin^2(\eta-B) - \eta \sin(\eta-B) \cos(\eta-B)]}  \, .

Interface Matching

According to our accompanying discussion of the interface matching condition — as we presently understand it — the proper eigenfunction will exhibit a discontinuity in the slope of the dimensionless displacement function such that,

~\frac{d\ln x_\mathrm{env}}{d\ln \eta} \biggr|_{\eta=\eta_i}

~=

~3\biggl(\frac{\gamma_c}{\gamma_e}  -1\biggr) + \frac{\gamma_c}{\gamma_e} \biggl( \frac{d\ln x_\mathrm{core}}{d\ln \xi} \biggr)_{\xi=\xi_i}

 

~=

~\frac{3}{5}\biggl[ \biggl( \frac{d\ln x_\mathrm{core}}{d\ln \xi} \biggr)_{\xi=\xi_i} -2 \biggr] \, .

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

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