Appendix/Ramblings/51BiPolytropeStability/NoAnalytic

Do Not Confine Search to Analytic Eigenvector

Overview

STEP01:
Develop an algorithm (for Excel) that numerically integrates the LAWEs from the center to the surface of a $(n_c,n_e)=(5,1)$ bipolytrope, for an arbitrary specification of the three parameters:   ${\mu }_e/{\mu }_c,{\xi }_i$, and ${\sigma }_c^2$.

• Enforce the proper interface matching condition(s) at the interface location, ${\xi }_i$.
• Note that in general, for an arbitrarily chosen set of the three parameter values, the resulting surface displacement function will not match the desired boundary condition.

STEP02:
Fix your chosen value of the parameter pair, $({\mu }_e/{\mu }_c,{\xi }_i)$, and vary ${\sigma }_c^2$ until the proper surface boundary condition is realized.

• In an accompanying discussion, we claim to have identified at what point along various ${\mu }_e/{\mu }_c$ sequences the fundamental mode of radial oscillation becomes unstable — that is, when ${\sigma }_c^2=0$. For a given choice of ${\mu }_e/{\mu }_c$, it would be wise to begin our eigenvector search at a value of ${\xi }_i<[{\xi }_i{]}_{\mathrm{F}\mathrm{M}}$, as specified in the following table:

  Marginally Unstable Fundamental Modes
  $\frac{{\mu }_e}{{\mu }_c}$	$[{\xi }_i{]}_{\mathrm{F}\mathrm{M}}$
  1	1.6686460157
  $\frac{1}{2}$	2.27925811317
  0.345	2.560146865247
  $\frac{1}{3}$	2.582007485476
  0.309	2.6274239687695
  $\frac{1}{4}$	2.7357711469398
  See orange-colored triangular markers in the associated Figure 4

• Keep steadily raising the value of the interface location until you find the 1^st overtone mode; our expectation is that, if this mode is unstable, the model will coincide with the turning point along the equilibrium sequence.

See Also