On the ellipsoidal figures of equilibrium of homogeneous masses

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The equilibrium and the stability of the Jeans spheroids

The equilibrium and the stability of homogeneous masses distorted by the tidal effects of a secondary (of mass M' at a distance R) are re-examined on the basis of the second-order virial equations. In agreement with known results, it is shown that, under circumstances when the figure of equilibrium is a prolate spheroid, there is a maximum value of π( = GM'/R3) which is compatible with equilibrium. The problem of the small oscillations of these Jeans spheroids is next considered. The characteristic frequencies of oscillation belonging to the second harmonics are determined both in case the mass is considered incompressible and in case it is considered compressible and subject to the gas laws governing adiabatic changes. In the former case, instability sets in when μ attains its maximum value; and in the latter case it sets in before that happens

On the occurrence of multiple frequencies and beats in the β Canis Majoris stars

An explanation is suggested for the occurrence of two nearly equal frequencies and associated beats in the light- and in the velocity-variations of the β Canis Majoris stars. It is shown that if the ratio of the specific heats γ is 1.6 and the star is rotating, any disturbance will excite two normal modes with nearly equal frequencies

Magnetoelliptic Instabilities

We consider the stability of a configuration consisting of a vertical magnetic field in a planar flow on elliptical streamlines in ideal hydromagnetics. In the absence of a magnetic field the elliptical flow is universally unstable (the ``elliptical instability''). We find this universal instability persists in the presence of magnetic fields of arbitrary strength, although the growthrate decreases somewhat. We also find further instabilities due to the presence of the magnetic field. One of these, a destabilization of Alfven waves, requires the magnetic parameter to exceed a certain critical value. A second, involving a mixing of hydrodynamic and magnetic modes, occurs for all magnetic-field strengths. These instabilities may be important in tidally distorted or otherwise elliptical disks. A disk of finite thickness is stable if the magnetic fieldstrength exceeds a critical value, similar to the fieldstrength which suppresses the magnetorotational instability.Comment: Accepted for publication in Astrophysical Journa

Shear-flow transition: the basin boundary

The structure of the basin of attraction of a stable equilibrium point is investigated for a dynamical system (W97) often used to model transition to turbulence in shear flows. The basin boundary contains not only an equilibrium point Xlb but also a periodic orbit P, and it is the latter that mediates the transition. Orbits starting near Xlb relaminarize. We offer evidence that this is due to the extreme narrowness of the region complementary to basin of attraction in that part of phase space near Xlb. This leads to a proposal for interpreting the 'edge of chaos' in terms of more familiar invariant sets.Comment: 11 pages; submitted for publication in Nonlinearit

On super-potentials in the theory of Newtonian gravitation

The character of the gravitational equilibrium of bodies in rotation and with prevalent magnetic fields depends on the tensor potential, and the associated tensors, and This paper is devoted to a consideration of these fundamental tensors It is shown, in particular, that the tensor potential can be expressed in the form where 58 is the gravitational potential as usually defined and X is a super-potential determined by the equation