Linear, isentropic oscillations of the compressible MacLaurin spheroids

Smeyers' procedure (1986) for the determination of linear, isentropic oscillations of the incompressible MacLaurin spheroids is extended to the compressible MacLaurin spheroids. It is shown that the solutions can be constructed by a direct integration of a finite set of differential equations written in spherical coordinates. Oblate spheroidal coordinates are used with regard to the boundary conditions that must be satisfied at the surface of the MacLaurin spheroid. For compressible MacLaurin spheroids with eccentricities e varying from zero to unity, the modes are determined that stem from the fundamental radial mode and the second-harmonic Kelvin modes in the non-rotating equilibrium sphere with uniform mass density. The modes obtained agree with the modes determined earlier by Chandrasekhar and Lebovitz (1962a, 1962b) by means of the second-order tensor virial equations. Next, four axisymmetric modes are determined that stem from the first radial overtone, the second-harmonic p(1)- and g(1)(-)-mode, and the fourth-harmonic Kelvin mode in the non-rotating equilibrium sphere with uniform mass density. The g(1)(-)-mode becomes dynamically stable at the eccentricity e = 0.7724 and again dynamically unstable at e = 0.9952.