Generalized r-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 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 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.