The virial method and the classical ellipsoids

Abstract

Chandra completed his book Hydrodynamic and Hydromagnetic Stability (HHS) in 1960 and, as was his custom, turned his attention to a new area of research. He began to study general relativity at this time, and it appeared that this would be his exclusive new direction. However, as the result of a pair of accidents, Chandra in fact devoted much of the period from 1960 through 1968 to the virial method and to an analysis of the figures of the classical ellipsoids and their stability. This subject and the general theory of relativity competed for his attention during these years. It was only after completion of his book Ellipsoidal Figures of Equilibrium (EFE) (Chandrasekhar 1969) in 1968 that he felt able to devote himself primarily to the subject of relativity, which then was the principal occupation of the remainder of his research career. His enthusiasm for the development of the classical ellipsoids waxed and waned during this period, and he wrote that parts of it were performed ‘under protest,’ his sense of responsibility to the subject taking precedence over his inclination to enter more fully into the study of relativity. The virial theorem, in scalar form, has a history in astronomy (cf. Ambartsumyan 1958). In the theory of stellar pulsations, it was employed by Ledoux (Ledoux 1945) to obtain an approximate expression for the lowest mode of radial pulsation and the effect on that mode of a slow rotation. Tensor forms of the virial theorem had been employed by Rayleigh (Rayleigh 1903) and by Parker (Parker 1957) in special contexts, and Chandra had long had in the back of his mind the notion that one could use this form of the theorem to obtain useful, approximate information about figures seriously distorted from the spherical by rotation or magnetic fields. He had included the basic equations in his preparation of HHS with this in mind. I was at this time one of his research students and it was therefore natural that he suggest to me, as part of my dissertation, the development of this method. The application he proposed was the problem of the oscillations and stability of the Maclaurin spheroids. This would represent a test case: the frequencies of the Maclaurin spheroids were known, and the virial equations, along with a linear ansatz for the Lagrangian displacement as in Ledoux's problem, would lead to approximate frequencies which could then be compared with the exact values. The ansatz would be needed because the virial method is a moment method which would require some kind of approximate closure procedure, such as that provided by the ansatz. What neither of us