Searching for Additional Eigenvectors of Zero-Zero Bipolytropes
This chapter is an extension of two accompanying discussions: The original discovery and detailed derivation; and the more readable, summary.
In our accompanying summary, we demonstrated how analytically specified eigenvectors can be constructed for the mode labeled, <math>~(\ell, j) = (2,1)</math>. This was done by specifying <math>~\gamma_e</math>, then solving a quartic equation for <math>~q</math>. Shortly after completing this summary chapter, we noticed that an alternate approach may be to specify <math>~q</math>, then solve for <math>~\gamma_e</math>; and this path may be simpler because it may only involve solution of a quadratic equation. If this proves to be the case, then it may also be possible to analytically construct eigenvectors of additional modes. Let's see.
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