The equilibrum and the stability of the Riemann ellipsoids. I

Abstract

Riemann's problem is concerned with the ellipsoidal figures of equilibrium of homogeneous masses rotating uniformly with an angular velocity Ω and with internal motions having a uniform vorticity ζΩ in the rotating frame. In this paper, the equilibrium and the stability of these Riemann ellipsoids are considered in the special case the axes of rotation and vorticity coincide with a principal axis of the ellipsoid. It is shown that for the case considered (1) the equilibrium figures can be delineated into sequences-the Riemann sequences-along which the ratio ƒ= ζΩ/Ω is a constant; (2) an ellipsoid which is a figure of equilibrium for some given ƒ is also a figure of equilibrium for ƒ† = (a₁² + a₂²)²/a₁²a₂², where a_l and a₂ are the semi-axes of the ellipsoid in the equatorial plane; (3) the two states of internal motion, corresponding to ƒ and ƒ†, lead to configurations which are adjoint in the sense of a theorem due to Dedekind; (4) the first member of a Riemann sequence is a Maclaurin spheroid which is stable in the absence of any dissipative mechanism; (5) from each point of the stable part of the Maclaurin sequence two Riemann sequences bifurcate; (6) there exist two self-adjoint sequences along which ƒ = ƒ† = ± (a₁² + a₂²)/a_la₂ and that limit the domain ofthe Riemann sequences in the (a₂/a_l, a₃/a₁) = plane; and (7) the bifurcation of the Jacobian and the Dedekind sequences from, what is usually called, the point of bifurcation is a special case of a much more general phenomenon. The stability of the Riemann ellipsoids with respect to modes of oscillation belonging to the second and the third harmonics is also investigated. With respect to modes of oscillation belonging to the second harmonics it is shown that (1) the Riemann ellipsoids allow a non-trivial neutral mode of oscillation; (2) the characteristic frequencies of oscillation of an ellipsoid and its adjoint are the same; (3) the Riemann ellipsoids with ƒ ≥-2 are stable with respect to these modes; and (4) instability by one of these modes arises along the sequences for ƒ <-2. With respect to modes of oscillation belonging to the third harmonics, it is shown that along all Riemann sequences instability first arises by a mode which deforms the ellipsoid into a pear-shaped configuration. The points at which instability sets in along the different Riemann sequences and the loci, which separate the regions of stability from the regions of instability in the domain of the Riemann ellipsoids considered, are also determined.