An alternative method of finding bifurcation points in pseudo‐barotropes

Abstract

AbstractStarting from the equations of hydrodynamics, which include both collisional and collisionless ideal fluids, the special case of pseudo‐arotropes with anisotropic pressure distribution along tangential or vertical direction (in respect to a “vertical” rotation axis) is considered. We find that vertical motions are equivalent to an imaginary rotation, and call “pseudo‐otation” the combination of rotation and real or imaginary peculiar circular motions. By use of a necessary condition for equilibrium, i.e. the coincidence of the boundary with an isopotential (gravitational + rotational centrifugal + real peculiar or imaginary peculiar centrifugal) surface, a new method of finding bifurcation points from axisymmetric to triaxial pseudo‐arotropic sequences – discussed in an earlier paper – is dealt with for the systems under discussion. A simple application to the special case where the isopycnic surfaces are spheroidal discloses that, on one hand, steadily pseudo‐otating, heterogeneous spheroids, cannot be equilibrium configurations and, on the other hand, the same results obtained by use of the virial technique hold when the current method can work. Then the coincidence of the boundary with an isopotential surface and the validity of the virial equations of the second order make two independent (necessary) conditions for equilibrium and both must be satisfied. Under the further assumptions of steady rotation and homeoidally or focaloidally striated distribution of matter, the configurations for which a bifurcation point must necessarily occurr, are determined. In the former alternative a connection is also established, between local (on the boundary) and global anisotropy of pressure distribution; in particular, it is found that local isotropy (on the boundary) involves global isotropy and vice versa. In both cases, the right amount of of anisotropy turns out to yield violation of a treshold for stability, ϵrot = −Erot/Epot ⩽ 0.14, conjectured by Ostriker and Peebles (1973). This result gives additional support to a conclusion established in an earlier paper: it seems more germane to speak about Ostriker‐eebles conjecture for stability in connection with visible bodies of galaxies, instead of Ostriker‐eebles criterion for stability in connection with self‐ravitating fluids.