1. Introduction
As guest editor for the 1999 MarchApril issue of the new interdisciplinary (AIP & IEEE) magazine Computing in Science and Engineering (CiSE), the PI recently was afforded the opportunity to highlight the astrophysics community's accomplishments in the area of "Computational Cosmology" by assembling a set of expert theme articles on that topic. In that issue of CiSE, Bond et al. (1999) discussed progress that is being made with regard to measuring fluctuations in the cosmic microwave background (CMB); White and Springel (1999) reviewed how large Nbody simulations have been used to effectively simulate the natural development of CMBrelated fluctuations into nonlinearamplitude, dark matter structures (galaxies); Bryan (1999) described efforts by the GC^{3} collaboration to properly simulate the dynamics of gaseous (baryonic) material as it flows into the dark matter structures; and, on behalf of the Sloan Digital Sky Survey collaboration, Szalay (1999) reviewed efforts that are underway to document in digital form a "complete" and detailed description of the universe's present nonlinear structure at optical wavelengths. After reading such a collection of articles on modern cosmological investigations, one gathers the distinct impression that a fairly accurate and unified picture is emerging regarding the process by which galaxies form.
In contrast to this, the astrophysics community is still struggling to find a broad, general explanation of the process by which stars form from the gas that resides in galaxies. (This, of course, also presents a monumental barrier to cosmologists because we won't really understand how galaxies form until the gas dynamical simulations of, for example, the GC^{3} collaboration, actually produce stars.) It is generally accepted, of course, that gravity is responsible for gathering lowdensity clouds of gas in the interstellar medium into clumps that are dense enough to be called protostars, and it is generally believed that once gravity begins to control the process, it happens fairly rapidly  on the order of a dynamical time t_{d} ~ [ Gr_{0} ]^{1/2} as measured by the mean density r_{0} of the initial cloud state. (For interstellar molecular clouds with n ~ 10^{4}  10^{6} cm^{3}, for example, t_{d} ~ 10^{6}  10^{5} years.) But there is still a debate as to how the process of dynamical collapse usually gets started, a debate over what sets the typical mass scale for stars  a scale which must ultimately make sense in terms of observed properties of the stellar initial mass function (IMF)  and related debates over what processes are principally responsible for the breakup of single clouds into binary or small multiple star systems, or into larger groupings of stars that we call clusters.
In the context of the formation of individual, lowmass stars and their associated circumstellar disks Shu, Adams & Lizano (1987) offer a compelling model in which collapse proceeds from initially centrally condensed, magnetically supported molecular cloud cores. (For related models and ideas along these lines, see Mestel & Spitzer 1956; Nakano 1979; Lizano & Shu 1989; Tomisaka et al. 1989; Basu & Mouschovias 1994; Li & Shu 1997; and Galli et al. 1999. But for a recent cautionary discussion, see Nakano 1998.) In the model as outlined by Shu, Adams, & Lizano (1987), a star's mass is not primarily determined by the physical properties of a molecular cloud core but, rather, by processes associated with the onset of nuclear fusion, envelope convection, and winds. But based on the clustering and binary statistics of stars in the nearby TaurusAuriga region, Larson (1995) has found renewed support for the idea that it is the thermal Jeans mass in molecular cloud cores that sets the mass scale for young, lowmass stars. In this context it is interesting to note that, although it is widely acknowledged that magnetic fields play an important role in defining the observed properties of molecular cloud complexes, most research groups who have attempted over the past decade to simulate the process by which binary and small multiple star systems form have used conditions set by the thermal Jeans mass to define the onset of collapse (cf., Boss 1993, 1996, 1998; Boss & Myhill 1995; Burkert & Bodenheimer 1993, 1995; Matsumoto & Hanawa 1999; Monaghan 1994; Myhill & Kaula 1992; Nelson & Papaloizou 1993; Sigalotti & Klapp 1994, 1996, 1997; Truelove et al. 1997, 1998). Recently, however, Boss (1997, 1999) has tried to bring the two ideas together.
Given that star formation is ongoing in the solar neighborhood  that is, relatively speaking, it is a process that can be studied in our own backyard  it might seem strange to suggest that models of galaxy formation are more mature than are models of star formation. But the following few points illustrate why the star formation problem is in many respects the more challenging one:
With the advent of large digital infrared array detectors and the development of larger and more sophisticated millimeterwave radio telescope arrays, observational issues associated with this third point are being partially alleviated. In particular, millimeterwave observations are providing sufficient spatial resolution and signaltonoise to permit mapping of the structural and dynamical properties of starforming gas clouds at relatively high volume densities and with linear scales approaching the size of our own solar system (cf., Sargent & Welch 1993; André, WardThompson, & Motte 1996; Ohashi et al. 1997).
Faced with the tremendous dynamic range that separates the mean densities of molecular clouds from the mean densities of stars, and evidence that nearby star forming regions show structure at virtually all scales (Larson 1995), it would be foolish to believe that any single model that is based on a relatively simple geometry explains in full how stars form directly from conditions that are specified in molecular clouds. That's not to say that we haven't gained considerable insight into the process by which stars form by comparing observations to early spherically symmetric (e.g., Larson 1969) or more recent axisymmetric (e.g., Shu, Adams, & Lizano 1987) models of protostellar cloud evolution, but rather to emphasize that such models cannot possibly be painting for us the complete picture.
Taking advantage of the rapidly expanding computational resources that have emerged this decade, a significant number of research groups have attempted to model the star formation process in full threedimensional generality, as mentioned above. But the dynamic range of such models is still necessarily limited by their finite numerical resolution so, at best, these models are also only able to paint for us a portion of the picture. Generally speaking (see Matsumoto and Hanawa 1999 for a recent overview), published models of cloud fragmentation have focused on the early stages of collapse when it is generally understood that protostellar clouds cool very effectively and, hence, "direct fragmentation" based on the thermal Jeans criterion is not difficult to achieve. However, as we have been reminded most recently by Boss (1999), at densities above 10^{13}  10^{12} g cm^{3}, protostellar clouds become heated by compression and this increase in temperature serves to discourage further fragmentation.
Over the past 1520 years, the PI also has devoted a significant fraction of his research efforts toward modeling the dynamical evolution of protostellar clouds in full threedimensional generality with an eye toward understanding why stars preferentially form in pairs. This has been motivated by the observational investigations of, for example, Abt & Levy (1976), Abt (1983), Duquennoy & Mayor (1991) and Mathieu (1994) which quite convincingly indicate that, as Mathieu states, ''binary formation is the primary branch of the starformation process.'' The orbital periods of binary stars span an extremely wide range  from fractions of days to thousands of years  and they include systems with a wide variety of component mass ratios and a wide range of orbital eccentricities. Until we at least understand how binary systems with such a variety of physical properties are formed, we will be hard pressed to claim that we understand the star formation process.
Some of the PI's earliest simulations focused on the (lowest density) isothermal phase of collapse and demonstrated how direct fragmentation can produce binary stars with relatively long orbital periods (Tohline 1980; Bodenheimer, Tohline, & Black 1980). Subsequent work by numerous groups, as indicated above, have shown that it is possible to specify initial conditions in such a way that a binary system with practically any orbital period can be formed via direct fragmentation. But under realistic molecular cloud conditions, it seems most natural to expect that direct fragmentation will work effectively only at densities r_{0} < 10^{13}  10^{12} g cm^{3}  i.e., below the point when heating due to adiabatic compression begins to occur  and produce multiple systems whose orbital periods are of the same order as the dynamical time t_{d} that is associated with those densities  i.e., greater than a few hundred years. (See the related discussion connected with Table 1 in § C.3.c of this proposal.)
With this in mind, in the late '80s and early '90s the PI's simulation efforts turned toward modeling the (higher density) adiabatic phases of protostellar cloud evolution in an effort to understand how binary stars with relatively short orbital periods form. One advantage of focusing on models with mean densities r_{0} > 10^{12} g cm^{3} is that ionization fractions are expected to be extremely low, so magnetic fields are unlikely to significantly influence the cloud's evolution. Another is that the effective adiabatic exponent of the gas is large enough to permit the construction of configurations that are stable against further dynamical collapse. [In the protostellar models of Larson (1969) and Bodenheimer et al. (1990), for example, there is always a central core of material that is in hydrostatic balance. The highdensity cloud core continues to contract fairly rapidly as it cools (and/or adds additional mass through accretion), but its subsequent contraction occurs quasiadiabatically rather than dynamically.] If cloud cores of this type are to break into multiple pieces, however, it will have to occur as a result of an instability that arises spontaneously from an equilibrium state rather than via a Jeanstype instability that drives the "direct" fragmentation process.
In this context, progress has been made in following the nonlinear development of barlike (often spiralshaped), nonaxisymmetric instabilities in rapidly rotating, equilibrium gas clouds (Tohline, Durisen & McCollough 1985; Durisen et al. 1986; Williams & Tohline 1987, 1988). Also, building on ideas developed by Papaloizou & Pringle (1984), Goodman and Naryan (1988), and Adams, Ruden & Shu (1989), and tools developed by Hachisu (1986a,b), we have helped establish a better understanding of the behavior of gravitationally driven nonaxisymmetric instabilities that arise in relatively massive protostellar disks (Tohline & Hachisu 1990; Woodward, Tohline, & Hachisu 1994; Andalib, Tohline, & Christdoulou 1997). However, until very recently neither of these broad investigations into dynamical instabilities that arise in quasiadiabatically evolving systems has developed into a theory of binary star formation that is competitive with models of direct fragmentation. As a result, recent reviews have concluded that direct fragmentation during the collapse of molecular cloud cores provides the best explanation for the formation of most binary stars (Boss 1993a; Bodenheimer, Ruzmaikina, & Mathieu 1993; Boss 1999).
However, as we explain in §C.2 of this proposal, improvements in modeling and simulation tools over the past few years have made it possible for the PI's group to examine the fission hypothesis of binary star formation in a more realistic manner than has heretofore been possible. We have done so in the context of the hypothesis as recently formulated by Lebovitz (1987) for quasistatically contracting, inviscid protostellar gas clouds. Our results offer the most convincing evidence, to date, that binary stars can form during the quasiadiabatic phase of a protostellar cloud's evolution by a process of fission, rather than via a Jeanstype fragmentation process. Because fission should work best at relatively high gas densities when protostellar gas clouds do not cool effectively under compression, it offers a mechanism for forming binary (or small multiple) star systems that complements the mechanism of direct fragmentation. Over the next few years we propose to extend our study of the structural and stability properties of quasiadiabatically evolving protostellar gas clouds with an emphasis on the fission problem.
Many classes of interesting astronomical phenomena are known for which the proposed astrophysical scenarios involve dynamical or thermal masstransfer in a close binary star system, sometimes leading to a common envelope phase and/or the complete destruction of the binary system through merger of its two components. Wellknown examples of such systems are cataclysmic variables (CVs) [Warner 1995], lowmass Xray binaries (LMXBs) [White et al. 1995; van Paradijs & McClintock 1995; Verbunt & van den Heuvel 1995], millisecond pulsars (Bhattacharya 1995), soft Xray transients (SXTs) or Xray novae [Chen, Shrader & Livio 1997], Algols (Batten 1989) and W Serpentis stars (Wilson 1989), contact binaries (W Ursae Majoris stars) [Rucinski 1985], some central stars of planetary nebulae (Iben & Livio 1993), double degenerate white dwarf (WD) binaries, perhaps leading to supernovae (SNe) of type Ia through a merger (Iben & Tutukov 1984), or subdwarf sdO, sdB stars (Iben 1990), and double neutron star (NS) or NSblack hole binaries, perhaps yielding gammaray bursts in a fireball when the system coalesces (Mészáros & Rees 1992; Ruffert et al. 1997; Kluzniak & Lee 1998).
In many cases, the presentday systems are known to be undergoing longterm, stable masstransfer driven by angular momentum losses or nuclear evolution. Although such longterm, secular phases of evolution may be easier to study observationally, they are too slow to be amenable to present numerical simulations. But when a system becomes dynamically unstable toward masstransfer, the transfer rate can rapidly reach nonlinear amplitudes and may lead to the formation of a massive disklike object, maybe a common envelope phase (Iben & Livio 1993), some mass loss from the system, and in some cases, end in a merger. These phenomena are truly three dimensional in character and largescale numerical simulations are required to further our theoretical understanding of them. Realizing that the numerical tools that we have constructed to examine fission and gravitational fragmentation processes in protostellar clouds and protostellar disks may in some cases be used effectively to simulate such phenomena, we have begun to examine tidal and masstransfer instabilities in short period, compact binary star systems. As is documented in §G (Current and Pending Support) of this proposal, we recently have received partial support through NASA's Astrophysics Theory Program to pursue this investigation; the research is being conducted in collaboration with J. Frank [LSU] who is lead PI on the NASA project.
Although, in the context of this NSF proposal, the study of masstransfer in evolved binary systems will not represent the principal focus of our research efforts over the next few years, we have decided to include a brief discussion of the subject here for several reasons. First, as is explained more fully in § C.2.e, below, our investigation into the relative stability against merger of close binary star systems began during the period of time that was covered by our most recent NSF award. Second, by demonstrating that our simulation tools can be used very effectively to follow through many orbits the motion of dynamically stable, close binary star systems, we have gained considerable additional confidence in the physical correctness of our simulations that address binary star formation processes. Finally, as we have grown to appreciate, dynamical processes that prove to be important in the context of evolved, close binary star systems often prove to be important in the binary formation problem as well. Hence, our plans to expand our studies of tidal and masstransferring instabilities in evolved stars over the next few years will both overlap and complement the research activities being proposed herein.
2. Results from Prior NSF Support

More than three years have passed since the starting date of award AST9528424 from the MPS division of the NSF, and four years have passed since the PI last submitted a full proposal requesting renewed NSF support. Over this period of time, support from the NSF has been acknowledged in eighteen separate publications (7 refereed journal articles, 6 articles in conference proceedings, and 5 doctoral dissertations, as itemized by the "Jnn" reference notation in §D of this proposal). As itemized by the "Mnn" reference notation in §D, eighteen Quicktime movies also have been produced in an effort to illustrate more completely our various numerical simulation results. Our research activities have focused on the following five separate, but overlaping areas:
Over the past four years, five graduate students have completed their doctoral dissertation research under the PI's direction [student names and dissertation titles are cited in §D as references J2, J11, J12, J15, J16]. The PI also has supervised one postdoctoral research associate (D.M. Christodoulou; a position funded primarily through stateallocated funds to the LSU Department of Physics and Astronomy) and the research projects of three undergraduate students, one of whom (D. Sherfesee) was awarded a 1998 NSF Graduate Research Fellowship and is presently enrolled in Berkeley's graduate astronomy program.
Over the past four years, the PI also has been heavily involved in a significant science outreach project in the Parish of East Baton Rouge, Louisiana. He was lead investigator on a project that secured $115,000 in funds from the Louisiana State Board of Regents and $26,000 from the LSU Foundation to purchase and install a modern, 20" optical telescope in a new public park observatory building that was constructed at a cost of approximately $300,000 by the Park and Recreation Commission of the Parish of East Baton Rouge (BREC). Since the opening of "Highland Road Park Observatory" in November, 1997, in close collaboration with the Baton Rouge Astronomical Society (BRAS), LSU and BREC have hosted an open house and public viewing session every Friday evening; have hosted numerous monthly K12 field trips; have run summer astronomy camps and teacher training sessions; and have provided an exceptional optical instrument for use by local amateur astronomers. For example, BRAS members have discovered over 30 new asteroids using the telescope and its CCD camera. (See URL http:// www.phys.lsu.edu/ observatory and [J10] for more details.) This science outreach project was initiated and completed without direct funding support from federal agencies, but is mentioned here as an indication of the scope of the PI's activities outside of normal research and universitylevel instruction.
In connection with the research area itemized as i, above, two particularly significant results have emerged. First, Andalib [J11] has developed a modified selfconsistentfield technique for constructing equilibrium models of rapidly rotating, selfgravitating, gas dynamical configurations with compressible equations of state, nontrivial internal motions, and a variety of different nonaxisymmetric structures. To date, the technique has been applied only toward the construction of twodimensional systems (i.e., either infinitesimally thin disks or systems having infinite vertical extent) with nonaxisymmetric structures, but we understand in principle how to extend the technique to fully threedimensional systems (see further discussion in §C.3.b, below). Andalib has used the technique, for example, to construct infinitesimally thin disks with steadystate elliptical, dumbbell, and "binary" shapes, as viewed from a frame that is rotating with a specific system pattern frequency [J11, J13]. Invariably, as viewed from that same rotating frame, the fluid exhibits nontrivial internal flows [M3, M4, M5]  somewhat analogous to the classical incompressible Riemann Stype ellipsoids. The Quicktime movie [M5] associated with Fig. 1 illustrates the differential internal flow in three of Andalib's steadystate models of centrally condensed, elliptical disks (one with flow that is entirely retrograde with respect to the figure motion; one that is entirely prograde; and one with two offaxis vortices separating a region of prograde flow from one of regrograde flow), and illustrates the flow in one of his models of an equalmass, commonenvelope binary. Although Andalib's models are only twodimensional, his work represents a significant breakthrough in the sense that, historically, it has not been possible to construct equilibrium models of rapidly rotating nonaxisymmetric systems with nontrivial internal flows and equations of state that are sufficiently compressible to be of astrophysical interest in the context of protostellar clouds, gasrich galaxies, and most normal stars. When it is not possible to construct individual equilibrium models of such systems, of course, it is impossible to diagnose the relative stability of such sytems or to say anything quantitatively meaningful about related evolutionary sequences.
Figure 1 Four Andalib Models [M5] 
In connection with the research area itemized as v, above, we are continuously working (with the manpower at our disposal) to improve our numerical simulation tools in order to be able to more accurately and efficiently simulate realistic astrophysical fluid flows. To date, our efforts have been focused on the accurate, threedimensional representation and analysis of rapidly rotating, selfgravitating dynamical flows, while incorporating relatively simple equations of state and virtually ignoring the effects of magnetic fields, radiation transport, and relativistic fields. This approach has been reasonable because it is the competing effects of rotation and Newtonian selfgravity that are expected to dominate the dynamics of the systems in which we have been most interested, and it is still computationally impractical to treat all of these physical processes simultaneously in a realistic manner. Athough there are certainly improvements that remain to be made in our techniques and tools  some of which are discussed in §C.3, below  indications are that our present tools and techniques are competitive. Through the peer review process that has been established to evaluate proposals for highperformancecomputing time on hardware maintained by NSF's national computing alliances, over the past two years we have been allocated and have utilized over 100,000 service units on Cray T3E platforms. We recently have submitted a proposal to NRAC requesting an additional 65,000 SUs for the coming year in support of our ongoing research activities.
Our primary simulation tool is a threedimensional, finitedifference hydrodynamics code patterned after ZEUS2D (Stone & Norman 1992). Our specific treatment of advection and source terms in the principal dynamical equations has been detailed by New [J2] and Cazes [J16]; our technique for efficiently solving the global Poisson equation has been detailed by Cohl [J6, J15, J17]; and the heterogeneous computing environment that we have developed to permit the routine visualization of complex, timedependent fluid flows (see, for example, the Quicktime movie sequences [M1  M18]) has been described by Cazes et al. [J14]. The mpi (message passing interface) version of our code  recently designed and implemented by LSU graduate student Patrick Motl  can now routinely simulate selfgravitating flows of the type discussed throughout this proposal on Eulerian grids containing 16 MegaCells. (Given the continuing upward trend in hardware, networking, and parallel compiler technologies, we comfortably expect this capability to increase by a factor of 4  8 over the next few years.)
Of all the improvements that we have made in our simulation tools over the past few years, the one that is most likely to make a longterm impact on the astrophysical community as a whole is the one reported in Cohl & Tohline [J17; see also J15]. As the abstract of that paper summarizes, we have discovered that an exact expression for the Green's function in cylindrical coordinates (v, f, z) is,
 x  x' ^{1}  =  [ p^{2} v v' ]^{1/2}  S  e^{im(ff') } Q_{m1/2}( c ), 
Q_{m1/2} is the halfinteger degree Legendre function of the second kind, and the summation over the index "m" is from minus infinity to infinity. This expression is significantly more compact and easier to evaluate numerically than the more familiar cylindrical Green's function expression which involves infinite integrals over products of Bessel functions and exponentials. It also contains far fewer terms in its series expansion  and is therefore more amenable to accurate evaluation  than does the familiar expression for x  x'^{1} that is given in terms of spherical harmonics. This compact cylindrical Green's function (CCGF) expression is wellsuited for the solution of potential problems in a wide variety of astrophysical contexts because it adapts readily to extremely flattened (or extremely elongated), isolated mass distributions. We now use it exclusively in our algorithm that calculates the value of the gravitational potential on the boundary of our computational grid; these values are then used as boundary conditions for our solution of the global Poisson equation in all of our dynamical simulations.
The idea that the quasiadiabatic contraction of a rotating protostellar gas cloud may lead in a very natural way to the formation of binary stars through a process of "fission" was suggested over 100 years ago. (See, for example, the reviews by Lyttleton 1953; Chandrasekhar 1969; Durisen & Tohline 1985; and Lebovitz 1987.)
Figure 2 Rotating Fluid Drop  

Results from a 1992 space shuttle (STS50) Drop Dynamics Experiment. 
Initially as the model cooled, it became even more elongated and its overall spin pattern frequency increased  in concert with a shrinking principal moment of inertia  but, it remained centrally condensed. After the polytropic constant dropped to 55% of its initial value, however, the overall configuration began to oscillate between two welldefined states: a highly elongated, centrally condensed bar, and an equalmass, "commonenvelope" binary. This phase of the slow cooling evolution can be most fully appreciated by viewing several of our Quicktime movie sequences [M13, M14, M15], but the two states between which the oscillation occurred are also illustrated here in Fig. 4 (see the last page of §C of this proposal). In the righthand column of Fig. 4, the binary state is most clearly delineated by the nested 3D isodensity surfaces (middle frame) and by the surface plot (bottom frame) of the effective potential in the equatorial plane, as viewed from a frame rotating with the overall pattern frequency of the configuration  i.e., the orbital frequency of the binary. The effective potential (density profile) in the equatorial plane of the system is also illustrated by the solid (dashed) nested contour lines in the top frame of Fig. 4. (The solid circle identifies corotation.)
Notice the similarities between the effective potential shown on the righthandside of Fig. 4 and the familiar effective potential one derives in the binary Roche problem: two offaxis potential minima; two maxima associated with the "L4 and L5" Lagrange points; two saddle points (at which envelope material is paritally spilling out of the system) associated with the "L2 and L3" Lagrange points; and, finally, an "L1" saddle point separating the two potential minima. As the velocity vectors in the top frame of Fig. 4 also illustrate, when the model is in its binary state, a portion of the flow continues to stream along the full length of the "barlike" configuration, but another portion of the flow is isolated around the separate binary components, effectively giving the components a net spin. These three physical features  offaxis density maxima, close correspondence with the Roche potential, and fluid flow that is isolated around each offaxis component  give us considerable confidence that the cloud configuration shown on the right in Fig. 4 is indeed a fully selfconsistent (commonenvelope) binary state.
Demonstration that a binary configuration does become available to a quasiadiabatically contracting protostellar gas cloud as envisioned by proponents of the fission hypothesis of binary star formation over the past 100 years (see especially the modified scenario outlined recently by Lebovitz 1987) is the most important scientific result to have emerged from the PI's NSFsponsored research efforts over the past few years. This result forms a principal component of Cazes' Ph.D. dissertation [J16] (degree awarded August, 1999), and is presently being written up for publication.
In connection with the research area itemized as iv, above, we have constructed equilibrium sequences of synchronously rotating, equalmass binaries in circular orbit with a single parameter  the binary separation  varying along each sequence. Sequences have been constructed with various polytropic as well as realistic white dwarf and neutron star equations of state. Then, using our Newtonian, gravitational hydrodynamics code, we have examined the dynamical stability of individual models along these equilibrium sequences [J2, J7].
Our simulations indicate that no
Figure 3 Binary MassTransfer [M17] 
Very recently, graduate student P. Motl has begun to conduct a similar investigation involving unequalmass polytropic binary star systems. This investigation is both computationally more challenging and broader in physical complexity than the study of equalmass systems. For example, for a given polytropic index and component separation, there is an infinite range of component mass ratios from which to choose; and even when the system is dynamically stable against a global tidal instability, it may be unstable toward an entirely different type of masstransferring instability. One of our Quicktime movie sequences [M16] illustrates the degree to which we have been able to follow a detached system through more than four full orbits. Two other movie sequences [M17, M18] (one frame of which is shown here as Fig. 3) illustrate the early phase of a masstransfer instability in a system with a mass ratio q = 0.88.
3. Proposed Research
Cazes' [J16] demonstration (see the discussion in §C.2.d, above) that a quasistatically contracting gas cloud can spontaneously evolve to a binary state breaths new life into the fission hypothesis of binary star formation, as most recently outlined by Lebovitz (1987), and thereby represents a milestone in modern star formation research. As explained earlier, because it can operate in high density cloud cores that heat up under compression, fission offers an alternate but complementary mechanism for the formation of binary stars to the Jeanstype process of direct fragmentation which works most effectively in relatively lowdensity clouds. Since the binary state arises spontaneously from an otherwise dynamically stable, equilibrium configuration, this mechanism for forming binary stars brings with it the hope (expectation) that the massspectrum of binary systems with orbital periods less than a few hundred years can be explained by processes unrelated to the spectrum of initial fluctuations in interstellar clouds.
However, with the tools in hand and the time available, Cazes was only able to study one slow cooling evolution, and that particular evolution did not actually end with the cloud in a binary state. After watching the system oscillate back and forth several times between the two states depicted here in Fig. 4, Cazes stopped cooling the model in order to determine which configuration represented the absolute minimum energy state at that particular point in the evolution. As the Quicktime movie [M13] associated with Fig. 4 illustrates, in response to this query the system settled back into the barlike configuration. But we are confident that, had the cooling been continued, and had Cazes been able to maintain adequate spatial resolution to accurately follow the cloud's continued radial contraction, the system would have eventually settled into the binary equilibrium state. Clearly this speculation needs to be verified if the fission hypothesis is to be fully resurrected. Other examples, and examples employing more realistic cooling processes, need to be given as well.
We should point out that the classical idea that fission can only produce binary systems with equalmass components is almost certain to be dispelled by carrying to completion this type of simulation. As we appreciate from models of masstransferring binary systems, when the components do not share a common envelope, the equalmass configuration is not preferred. Hence, although the fission process may initially create two equalmass offaxis condensations, we suspect that as those condensations individually contract (via cooling processes) and begin to separate from one another, an equalmass configuration also is unlikely to be preferred.
This is one principal avenue of research that we propose to follow over the next few years. Employing the more efficient mpibased hydrodynamics algorithm, mentioned earlier, that has recently been developed in the PI's group and an adaptive grid technique like the one proposed by Berger & Colella (1989) and implemented by Truelove et al. (1997,1998) and Bryan (1999), we propose to follow Cazes' cooling evolution with higher sustained grid resolution, further in time to document how and when the system settles permanently into the binary state. As a second example, we propose to slowly cool either the second of Cazes' CARE models (his Model B), or a new 3D CARE constructed via a selfconsistentfield technique (see § C.3.c, below). Third, in order to determine whether or not the process of fission is sensitive to the rate at which the cloud contracts, we propose to repeat at least one of these simulations but introduce a rate of cooling that is two or three times slower that the rate employed by Cazes. Finally, as discussed more fully below, we propose to incorporate a radiation transport algorithm into our primary hydrodynamic simulation code in an effort to model spatially differential cooling processes more realistically.
Because they are very broadly applicable to studies of the structure and evolution of protostellar clouds, compact stellar objects, and gasrich galaxies, we also propose to pursue further the development of techniques for constructing compressible analogs of Riemann ellipsoids (CAREs) with a wide range of physical characteristics. We expect to build directly on the selfconsistentfield technique that has been successfully applied by Andalib to 2D systems, extending his technique to fully threedimensional configurations. In order to illustrate how this might be done, we must first outline certain features of Andalib's 2D technique.
In a frame of reference that is rotating with a constant angular velocity kW_{f}, Euler's equation governing the motion of a barotropic gas may be written in the form (cf., LyndenBell & Katz 1981; Papaloizou & Savonije 1991),
where H is the gas enthalpy, F is the gravitational potential, v is the gas velocity as measured in the rotating frame, z º Ñ ´ v is the fluid vorticity, and v is cylindrical radius. For a steadystate system in which the gas is confined to an infinitesimally thin disk, the lefthandside of Euler's equation may be replaced by,
where, making use of the continuity equation and following the lead of Papaloizou & Savonije (1991), we have related the product of the surface mass density S and velocity to a pseudostream function Y through the expression,
Hence, in such a steadystate system it must be possible to express the vortensity [ ( z + 2 W_{f} ) / S ] (the inverse of what LyndenBell & Katz 1981 refer to as the "load") as a function only of Y (Papaloizou & Savonije 1991) and then move the term that is customarily on the lefthandside of Euler's equation to the righthandside and group it with all the other terms under the gradient operator. As a reasonable first pass at the problem, Andalib [J11] assumed that the vortensity was either uniform in space or, at worst, only a linear function of Y, i.e.,
in which case Euler's equation can be transformed into the following relatively simple algebraic expression defining the spatial relationship among the various key physical variables:
Following the procedure laid out in traditional selfconsistentfield techniques (Ostriker & Mark 1968; Hachisu 1986a,b), therefore, in Andalib's technique it is this algebraic expression that must be satisfied simultaneously with the selected barotropic equation of state and with the Poisson equation relating the gravitational potential to the surface density in order to construct a steadystate, equilibrium configuration.
But in the case of nonaxisymmetric flows, an additional mathematical expression must be used to constrain the functional form of Y in a manner that properly satisfies the continuity equation. Based on the definition of vorticity and the relationship given above between the momentum density Sv and Y, Andalib [J11] realized that it is the following selfadjoint PDE that governs the spatial behavior of the pseudostream function in the case where the vortensity is, at worst, a linear function of Y:
Andalib has devised an iterative algorithm that successfully solves this secondorder PDE in concert with the other three traditional SCF equations for a variety of different nonaxisymmetric surface boundary conditions, a variety of different degrees of gas compressibility (polytropic indices 0.1 < n < 1.3 ), and a variety of different values of the constants C_{0} and C_{1}. It is with this tool that he has been able to build the various 2D, nonaxisymmetric equilibrium models described above.
In extending the technique to nonaxisymmetric structures with finite vertical extent, we expect to be able to break the problem down into a series of vertical slices perpendicular to the figure's overall spin axis, in a manner similar to that used by Eriguchi & Hachisu (1985). We should be able to solve for the structure of each 2D slice using Andalib's technique and adopt slicetoslice variations in the various "constants" of the flow such that the slices blend together vertically in a physically reasonable manner. In order to ascertain what is physically reasonable, we will take certain cues from the properties of the two CAREs that have been constructed by Cazes using dynamical simulation tools, as well as from the somewhat simpler "Dedekindlike" configurations that have been examined recently by Uryu and Eriguchi (1999). For example, both Cazes [J16, J18] and Uryu & Eriguchi (1999) have noted that, locally throughout the fluid, the vorticity vector should be slightly missaligned with respect to the spin axis of the figure. Also, as we move from 2D to 3D structures, it will be important to relate the pseudostream function to the load, as defined by LyndenBell & Katz (1981), rather than to the vortensity which has been defined earlier only in terms of a surface density.
Finally we note that, although Andalib successfully introduced gas compressibility into his equilibrium models using a polytropic prescription for the equation of state, he was unable to converge to models with polytropic indices n > 1.3. Based on Cazes' results, we suspect that it will be necessary to introduce a "violinshaped mach surface" and standing shock fronts into model configurations that have relatively large degrees of compressibility. This will be a nontrivial, but we think not insurmountable, task. For example, in Chapter XII of Landau & Lifshitz's (1959) classical text on Fluid Mechanics, a discussion is presented on how to use the method of characteristics to determine the location and strength of weak shocks in curved flows that resemble in many respects the flows observed in Cazes' CAREs. As long as the shocks are weak, treating the flow as homentropic will introduce relatively little error in the determination of the primary flow variables because the entropy jump across a weak shock goes as the third power of the (small) pressure jump. This approximation should simplify the model construction task, then afterward an estimate of the timescale for secular evolution of the converged model could be acquired by accounting for the actual (small) amount by which entropy is created by each passage of the fluid through one of the standing shocks.
Observational data from arrays of millimeterwave radio telescopes now provide sufficient spatial resolution and signaltonoise to permit mapping of the structure and dynamical properties of starforming gas clouds with linear scales approaching the size of our own solar system (cf., Sargent & Welch 1993; Ohashi et al. 1997). Hence, there may be an opportunity to directly compare the properties of our models of quasiadiabatically contracting protostellar cloud cores  both as CAREs and as commonenvelope binaries  with the observed properties of starforming clouds having comparable scales. To illustrate this point, Table 1 (drawn from reference [J16]) scales one of Cazes' dimensionless 3D, barlike CAREs to a cloud of one solar mass having five different sizes ranging between 1 and 100 AU. These represent a possible protostellar cloud core configuration at different stages of its contraction (mean densities between 10^{13} and 10^{7} g cm^{3}). Although Cazes' model employs a simple n = 3/2 polytropic equation of state, when scaled to the sizes given in Table 1 it predicts mean cloud temperatures and values of the total angular momentum that are quite realistic.
Table 1  

P_{spin}  R_{major}  r_{mean}  n_{H2}  T  J 
[years]  [AU]  [gm cm^{3}]  [cm^{3}]  [°K]  [g cm^{2} s^{1}] 
           
1  1.0  1.5 x 10^{7}  4.5 x 10^{16}  4860  2.5(52) 
10  4.6  1.5 x 10^{9}  4.5 x 10^{14}  1050  5.5(52) 
100  21  1.5 x 10^{11}  4.5 x 10^{12}  224  1.2(53) 
250  39  2.4 x 10^{12}  7.3 x 10^{11}  122  1.6(53) 
1000  99  1.5 x 10^{13}  4.5 x 10^{10}  49  2.5(53) 
During the first year of this project, we propose to incorporate radiative transfer techniques and realistic equations of state into our modeling algorithm along the lines of those developed and implemented by Boss (1984), Bodenheimer et al. (1990), Boss & Myhill (1992), and Sigalotti (1998). In addition to helping us simulate the slow contraction of our gas clouds more realistically, as mentioned above, such techniques will permit us to produce spatially resolved surface brightness and velocity maps from our dynamical models for comparison with radiofrequency observations of star forming regions.
Although directed primarily toward a better understanding of how binary star systems are formed, as has been discussed in § C.1.b of this proposal most of the PI's recent modeling efforts have been of a sufficiently general nature that they also can provide insight into the structural and stability properties of other selfgravitating systems, such as galaxies or highly evolved stars with or without accompanying accretion disks. In particular, if our CARE models are scaled to the size of compact stars, as longlived nonaxisymmetric configurations they could be a source of continuouswave gravitational radiation. We also propose, therefore, to determine (in a postNewtonian approximation) what the observational signature of such a continuouswave source of gravitational radiation would be. Building on the techniques outlined by Finn & Evans (1990) and Rasio & Shapiro (1992), New [J2] has already incorporated into the PI's basic hydrodynamic algorithm the tools that are necessary to compute this signature. What remains to be determined is the most realistic size scale to which our CARE model should be scaled and whether it would be expedient to incorporate a more realistic equation of state into the model.
In connection with these plans to compare our steadystate and evolving cloud models to observations of molecular cloud cores and compact stellar objects, the PI has arranged to spend his upcoming sabbatical leave (Spring, 2000) at Caltech, working in collaboration with both Anneila Sargent and Kip Thorne. As Director of the Owens Valley Radio Observatory (OVRO), Sargent leads a strong radio astronomy group whose efforts are largely focused on studies of star forming regions in our Galaxy. By spending an extended period of time interacting with this radio astronomy group, the PI hopes to gain a much better appreciation of the variety and quality of data that is being collected in connection with ongoing star formation processes in the solar neighborhood, and can effectively begin to establish a relationship between his models and the observations. Under Kip Thorne's direction, Caltech is widely recognized as the home base for one of the world's leading theory groups whose efforts are directed largely toward modeling sources that should be detectable by the laser interferometer gravitationalwave observatory (LIGO). By participating in the activities of this group on a regular basis while on leave, the PI hopes to achieve the second observationallyrelated objective as outlined above. No NSF funds are being requested in support of the PI's sabbatical leave at Caltech during the Spring of 2000. It is mentioned here, however, because the PI's planned research activities during this leave period will significantly complement the research activities for which NSF support is being requested.