J.E. TOHLINE, J.E. CAZES, AND H.S. COHL

*Louisiana State University*

*Department of Physics & Astronomy,
202 Nicholson Hall,
Baton Rouge, LA 70803-4001 U.S.A.*

NOTE: Belorussian translation

**1. Introduction**

Recent observational investigations of the frequency of occurrence
of pre-main-sequence binary stars have reinforced earlier
suspicions that ''binary formation
is the primary branch of the star-formation process''
(Mathieu 1994). As
Bodenheimer *et al.* (1993) have reviewed,
a number of different theories have been proposed to
explain the preponderance of binary stars. Klein
*et al.* (1998) show how the direct
fragmentation of protostellar gas clouds may occur in early phases of
collapse (at cloud densities
n = 10^{3} - 10^{10} cm^{-3}).
But at higher densities, clouds are unable to cool
efficiently upon contraction. Consequently, direct fragmentation
becomes problematical. Because higher mean densities are
associated with systems having shorter dynamical times, one is
led to consider mechanisms other than direct cloud fragmentation
for forming binary systems with orbital periods less than a few hundred
years. Here we investigate whether such binaries can form
by spontaneous fission of rapidly rotating protostars.

**2. The Classical Fission Hypothesis**

As Chandrasekhar (1969) has reviewed (see also Durisen & Tohline 1985), if protostellar objects are assumed to be self-gravitating, incompressible fluids with uniform vorticity, one can show analytically that their allowed equilibrium configurations are defined by spheroids or ellipsoids. Classically, models describing the slow contraction of rotating protostellar gas clouds have been formulated around such analytically prescribable equilibrium configurations. For example, a large, slowly rotating gas cloud with a relatively small ratio of rotational to gravitational potential energy T/|W| will resemble a Maclaurin spheroid. As it contracts conserving angular momentum and mass, its evolution will proceed along the Maclaurin sequence through progressively flatter configurations of higher T/|W|. At a sufficiently high T/|W|, one finds that the axisymmetric configuration is no longer the lowest energy state available. Instead, there is an ellipsoidal configuration to which the gas cloud will prefer to evolve. Furthermore, if one follows evolution along a more and more distorted ellipsoidal sequence (such as the Jacobi sequence or any one of the Riemann sequences), one finds that eventually other configurations with even higher order surface distortions become energetically favorable. For example (see Fig. 3 of Durisen & Tohline 1985), there is a ''dumbbell-binary sequence'' that branches smoothly off of the Jacobi ellipsoid sequence. One might imagine, therefore, that binary stars form from the slow contraction of a rapidly rotating gas cloud along the Maclaurin, then Jacobi (or Riemann), then dumbbell-binary sequences. In reality, the picture is not this clear. Most significantly, detailed work on ellipsoidal figures of equilibrium has only been completed for incompressible fluid systems. It is not at all clear to what extent the results carry over to more realistic structures having compressible equations of state.

**3. Problems Promoting the Fission Hypothesis **

While examining the structure of rotating
gas clouds that form the compressible analogues of Maclaurin spheroids,
Ostriker & Bodenheimer (1968) showed that
models with reasonable degrees of compressibility
must incorporate a significant degree of differential rotation
if they are to possess reasonably high values of T/|W| and, therefore,
be physically interesting in the context of the fission hypothesis.
Employing 3D numerical hydrodynamics techniques,
Durisen *et al.* (1986),
Williams & Tohline (1988), and
Houser *et al.* (1994) have examined the
relative stability of rapidly rotating, compressible gas clouds that are initially
in axisymmetric equilibrium but which reside just
past a critical bifurcation point along the axisymmetric sequence
according to
Ostriker & Bodenheimer (1968). Invariably these simulations
have shown that models with T/|W| > 0.27 are dynamically unstable
toward the growth of a nonaxisymmetric deformation
but, unlike their uniformly rotating, incompressible counterparts, the
eigenmode to which these structures appear to be unstable has a
spiral character.

Movie1 |

Quicktime (5,907K) |

**4. Fission Hypothesis Revived**

Interestingly, the instability illustrated by
Movie1
produces a final steady-state object (hereafter referred to as the
''final bar'')
that is dynamically stable, has a T/|W| = 0.25, and has a decidedly
nonaxisymmetric structure. In many respects this final bar appears
to be a compressible analog of a Riemann ellipsoid but, as
**Movie2**
illustrates, the configuration possesses nontrivial internal motions.
In the first frame of Movie2,
108 test particles have
been lined up along the major axis of the final bar.
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 final 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.
In **Movie3**
we illustrate the 3D flow-field of the final bar.
In the first frame of Movie3, a
vertical *sheet* of test particles
has been alligned with the major axis of the final bar. The subsequent
motion of these particles illustrates that
there is relatively little vertical fluid motion and,
although it varies with R and q, the angular velocity
w(**x**) is almost independent of z.
It may be possible, therefore, to understand this
and similar systems in terms of the properties of simpler, 2D
nonaxisymmetric structures.

Andalib (1998) recently has developed a self-consistent-field technique that can be used to

Movie4 |

Quicktime (6,927K) |

The similarity between the flow illustrated in Movie2 and the flow in Andalib's model P (Movie4) is striking. Apparently Andalib's model provides a good 2D analog of the 3D ''final bar'' that formed as a result of our fully hydrodynamic simulation of the two-armed, spiral mode instability (Movie1). Furthermore, Andalib's work demonstrates that model P is just one among a series of compressible models with nontrivial internal flows that defines a smooth elliptical-dumbbell-binary sequence. We suspect, therefore, that the final bar sits on an analogous (3D) sequence and that, if it is cooled slowly, it will evolve along the sequence to a common-envelope binary configuration such as the one illustrated by model D in Movie4. Additional support for this conjecture comes from New & Tohline (1997) who have demonstrated that stable, equal-mass common-envelope binaries can be constructed for fully 3D fluid systems with a sufficiently compressible equation of state. In summary, it seems clear that a wide variety of rapidly rotating, nonaxisymmetric systems can be constructed with compressible equations of state. This work gives us renewed confidence that fission offers a viable route to binary star formation. Future investigations designed to model the slow cooling and contraction of initially nonaxisymmetric configurations like the final bar described above should demonstrate whether or not this scenario is correct.

**5. Acknowledgments**

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.

**6. References**

Durisen, R.H., Gingold, R.A., Tohline, J.E. and Boss, A.P. (1986),
*Ap.J.*, **305 **, p. 281

Houser, J.L., Centrella, J.M. and Smith, S.C. (1994),
*Phys. Rev. Lett.*, **72**, p. 1314

Klein, R.I, McKee, C.F. and Fisher, R. (1998), *These proceedings*

Mathieu, R.D. (1994), *Ann. Rev. Astr. Ap.*, **32**, p. 465

New, K.B.C. and Tohline, J.E. (1997), *Ap.J.*, **490**, p. 311

Ostriker, J.P. and Bodenheimer, P. (1968), *Ap.J.*, **151**, p. 1089

Williams, H.A. and Tohline, J.E. (1988), *Ap.J.*, **334**, p. 449