Improved secular stability limits for differentially rotating polytropes and degenerate dwarfs

An alternative to the tensor virial equation (TVE) method applied to rotating polytropes and degenerate dwarfs is proposed. The trial eigenfunction satisfies the Kelvin circulation theorem and yields a sufficient condition for the secular instability of barlike modes when used in a Lagrangian variational formulation. Using the sufficient condition, differentially rotating polytropes and degenerate dwarfs are secularly unstable for T/absolute value of W is approximately 0.14, where T is the rotational kinetic energy, for a wide range of compressibilities and rotation laws. The presented stability limit is 1-7% greater than the TVE result. The difference increases with the degree of the central concentration of mass and the angular momentum of the equilibrium model.