26 research outputs found
On the ellipsoidal figures of equilibrium of homogeneous masses
This article does not have an abstract
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