In two accompanying animation sequences (referred to somewhat nondescriptively as "Model A" and "Model B"), we have shown that rapidly rotating, self-gravitating, axisymmetric gas clouds that are unstable to a two-armed, spiral-mode instability ultimately evolve to a new nonaxisymmetric (triaxial or bar-like) configuration. The bar-like configuration appears to be spinning coherently (as though it were a solid object) about its shortest axis with a well-defined pattern speed. In order to demonstrate clearly that "Model A" and "Model B" have, in the end, both evolved to configurations that are, to a high degree, equilibrium and steady-state, we have extended both model evolutions through 20-30 additional dynamical times, in each case viewing the extended evolution from a frame of reference rotating with the overall figure (Cazes 1999; and Cazes & Tohline 1999). The following two movies, entitled "Steady-State Evolutions," show these two extended evolutions and demonstrate convincingly that the bar-like configurations are robust and dynamically stable.

Steady-State Evolutions
Model A	Model B
Quicktime (7,169K)	Quicktime (14.1M)

Although the configurations shown in these two animation sequences appear to be, overall, fairly static structures -- or, as viewed from an inertial frame, both appear to be spinning coherently as though they are solid objects -- in reality the configurations each possess strongly differential internal motions in addition to their overall coherent spinning motion. As viewed from the rotating reference frame, fluid streams supersonically along the length of the bar, then it passes through a mild standing shock front and slows to subsonic velocities in order to be able to "make the turn" and flow around the end of the bar to the other side (Cazes 1999; and Cazes & Tohline 1999). The slight trailing spiral "kink" that is visible in the density contour levels of both steady-state models identifies the location of the standing shock front. (Each model obviously possesses a pair of these shock fronts.) In the following animation sequences, we illustrate the differential internal motions late in the evolutions of these models. (See also Tohline, Cazes, & Cohl 1998.)

Equatorial Flow
Model A	Model B	Andalib's Model
Quicktime (5.75M)	Quicktime (5,976K)	Quicktime (5,754K)

In the first frame of each of the "Equatorial Flow" movies, approximately sixty test particles have been lined up along the major axis of the bar-like configuration. Thereafter the particles are followed as they move along equatorial-plane streamlines of the flow, as viewed in a frame of reference that is rotating with the overall pattern speed of the bar. The illustrated flow is entirely prograde and largely differential, but there is a small volume near the center of the configuration that is moving harmonically. Note the strong similarities between the equatorial flow that has developed in our three-dimensional, nonlinear dynamical simulations of Models A and B, and the flow that appears in Andalib's "prograde" model (also shown and discussed elsewhere) that Andalib (1998) has constructed using a self-consistent-field technique.

Meridional Flow Model A

Quicktime (3,375K)

In the first frame of the "Meridional Flow" movie, a vertical sheet of test particles has been aligned with the major axis of the Model A bar. The subsequent motion of these particles illustrates that there is relatively little vertical fluid motion and, although it varies in the radial and angular coordinate directions, the angular velocity is almost independent of Z. It may be possible, therefore, to understand this and simlar systems in terms of the properties of simpler, 2D nonaxisymmetric structures.

In virtually every respect, the configurations illustrated here appear to be a compressible analogs of the family of incompressible ellipsoids with internal motions that were discovered by Riemann over a century ago. (See Chandrasekhar 1969 for a thorough review of Riemann's incompressible ellipsoidal figures of equilibrium.) As we have argued in a paper presented at the "Numerical Astrophysics 1998" conference in Tokyo ( Tohline, Cazes, and Cohl 1998), proof of the existence of this type of dynamically stable nonaxisymmetric configuration with a compressible equation of state permits us to resurrect the "fission hypothesis of binary star formation." Such models are also important to studies of long-lived gaseous bars in galaxies, and to studies of compact stellar structures that may appear as continuous, rather than burst, sources of gravitational radiation.

• Andalib, S. (1998). Ph.D. Dissertation, Louisiana State University.
• Cazes, J. E. (1999). Ph.D. Dissertation, Louisiana State University.
• Cazes, J. E. and J. E. Tohline, (1999). The Astrophysical Journal, submitted. "Self-Gravitating Gaseous Bars. I. Compressible Analogs of Riemann Ellipsoids with Supersonic Internal Flows."
• Chandrasekhar, S. (1969), Ellipsoidal Figures of Equilibrium. Yale University Press, New Haven, CT, U.S.A.
• Tohline, J.E., J. E. Cazes, and H. S. Cohl (1999) in Numerical Astrophysics (Proceedings of the International Conference on Numerical Astrophysics 1998), eds. S.M.Miyama, K. Tomishak, and T. Hanawa, pp.155-158. " The Formation of Common-Envelope, Pre-Main-Sequence Binary Stars "

This work has been supported, in part, by the U.S. National Science Foundation through grant AST-9528424 and, in part, by grants of high-performance-computing time at the San Diego Supercomputer Center and through the PET program of the NAVOCEANO DoD Major Shared Resource Center in Stennis, MS.