Appendix/Ramblings/51BiPolytropeStability/NoAnalytic: Difference between revisions
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In an [[SSC/Stability/BiPolytropes#Fundamental_Modes|accompanying discussion]], we claim to have identified at what point along various <math>\mu_e/\mu_c</math> sequences the fundamental mode of radial oscillation becomes unstable — that is, when <math>\sigma_c^2 = 0</math>. For a given choice of <math>\mu_e/\mu_c</math>, it would be wise to begin our eigenvector search at a value of <math>\xi_i</math> that is smaller than specified in the following table: | In an [[SSC/Stability/BiPolytropes#Fundamental_Modes|accompanying discussion]], we claim to have identified at what point along various <math>\mu_e/\mu_c</math> sequences the fundamental mode of radial oscillation becomes unstable — that is, when <math>\sigma_c^2 = 0</math>. For a given choice of <math>\mu_e/\mu_c</math>, it would be wise to begin our eigenvector search at a value of <math>\xi_i</math> that is smaller than specified in the following table: | ||
<table border="1" cellpadding="10" align="center"> | <table border="1" cellpadding="10" align="center" width="40%"> | ||
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<td align="center"><math>\frac{\mu_e}{\mu_c}</math></td> | <td align="center"><math>\frac{\mu_e}{\mu_c}</math></td> | ||
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<td align="center"><math>\tfrac{1}{4}</math></td> | <td align="center"><math>\tfrac{1}{4}</math></td> | ||
<td align="right">2.7357711469398</td> | <td align="right">2.7357711469398</td> | ||
</tr> | |||
<tr> | |||
<td align="left" colspan="2">See orange-colored triangular markers in the associated [[SSC/Stability/BiPolytropes#Figure4|Figure 4]]</td> | |||
</tr> | </tr> | ||
</table> | </table> | ||
Revision as of 19:37, 19 July 2022
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 bipolytrope, for an arbitrary specification of the three parameters: , and .
- Enforce the proper interface matching condition(s) at the interface location, .
- 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, , and vary until the proper surface boundary condition is realized.
-
In an accompanying discussion, we claim to have identified at what point along various sequences the fundamental mode of radial oscillation becomes unstable — that is, when . For a given choice of , it would be wise to begin our eigenvector search at a value of that is smaller than specified in the following table:
1 1.6686460157 2.27925811317 0.345 2.560146865247 2.582007485476 0.309 2.6274239687695 2.7357711469398 See orange-colored triangular markers in the associated Figure 4
See Also
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Appendices: | VisTrailsEquations | VisTrailsVariables | References | Ramblings | VisTrailsImages | myphys.lsu | ADS | |