Generalizedr-modes of the Maclaurin spheroids

Analytical solutions are presented for a class of generalized $r$-modes of rigidly rotating uniform density stars—the Maclaurin spheroids—with arbitrary values of the angular velocity. Our analysis is based on the work of Bryan; however, we derive the solutions using slightly different coordinates that give purely real representations of the $r$-modes. The class of generalized $r$-modes is much larger than the previously studied “classical” $r$-modes. In particular, for each $l$ and $m$ we find $l-m$ (or $l-1$ for the $m=0\mathrm{}$ case) distinct $r$-modes. Many of these previously unstudied $r$-modes (about 30% of those examined) are subject to a secular instability DRIVEN by gravitational radiation. The eigenfunctions of the “classical” $r$-modes, the $l=m+1\mathrm{}$ case here, are found to have particularly simple analytical representations. These $r$-modes provide an interesting mathematical example of solutions to a hyperbolic eigenvalue problem.