Numerical Hydrodynamics of Mass-Transferring Binary Star Systems


Joel E. Tohline, Juhan Frank, & Patrick Motl
Department of Physics & Astronomy, Louisiana State University

This entire document, along with several illustrative movies is available on-line at

1. Summary of Proposed Research, Including Objectives and Goals

Over half of all stars in the sky are multiple star systems and most of these are binary stars. We recently have completed a review of the mechanisms that have been proposed to explain the origin of binary star systems (Tohline 2002). Approximately one half of the binary stars are close enough to one another for mass to be exchanged between the components at some point in their evolution (Trimble 1983). The components of these interacting binaries influence one another in a variety of ways including tidal forces, irradiation, and in the most extreme manifestation of the interaction, through mass transfer and mass exchange. As a result, the evolution and observational properties of the binary components and the binary itself are dramatically different from the behavior expected in the absence of mass transfer, even when this transfer proceeds at a leisurely pace.

Interacting binaries of one type or another, especially those in which the accretor is a compact object (Frank, King & Raine 2002), play central roles in many of the astrophysical sources of high energy photons, and are involved in the generation of gravitational and high-energy particle radiation. Most of the interacting binary types known observationally are presently undergoing long-term, stable mass-transfer on timescales dictated by secular angular momentum losses (e.g. gravitational radiation and magnetic braking) and/or nuclear and/or thermal evolution. Well-known examples of the above are cataclysmic variables (CVs) [Warner 1995, Beuermann 2002], low-mass X-ray binaries (LMXBs) [White et al. 1995; van Paradijs & McClintock 1995; Verbunt & van den Heuvel 1995], soft X-ray transients (SXTs) or X-ray novae [Chen, Shrader & Livio 1997], Algols (Batten 1989, Vesper Honeycutt & Hunt 2001; Zavala et al. 2002) and W Serpentis stars (Wilson 1989), contact binaries (W Ursae Majoris stars) [Rucinski 1989; Qian 2001], super-soft X-ray sources [Greiner 2000, Rappaport, Di Stefano & Smith 1994], and the ultra-luminous X-ray sources in nearby galaxies [King et al. 2001]. In these systems, the fraction of the donor star's mass that is transferred during one orbit is typically 10-12 to 10-9, many orders of magnitude less than what current numerical 3-D hydrocodes can resolve. All of the above systems, however, must have descended from binaries in which the accretor of today was initially the more massive component who evolved first off the main-sequence. The mass transfer in these progenitor systems was in many instances dynamically unstable, yielding mass-transfer rates many orders of magnitude above the currently observed rates and leading in some cases to a common envelope phase (see e.g. Warner 1995; Verbunt & van den Heuvel 1995; Nelson & Eggleton 2001) or a thermal mass-transfer phase (King & Schenker 2002).

In addition, there is a wide class of binary star systems not presently undergoing mass-transfer for which the astrophysical scenarios that have been proposed to explain their origin or ultimate fate involve dynamical or thermal mass-transfer in a close binary. Examples of such systems are millisecond pulsars (Bhattacharya 1995), some central stars of planetary nebulae (Iben & Livio 1993), double degenerate white dwarf binaries, perhaps leading to supernovae of type Ia through a merger (Iben & Tutukov 1984), or subdwarf sd0, sdB stars (Iben 1990), and double neutron star binaries, perhaps yielding g-ray bursts in a fireball when the neutron stars coalesce (Paczynski 1986, Mészáros & Rees 1992, Ruffert et al 1997, Mészáros 2001). The evolutionary scenarios that are drawn upon to explain the existence of some of these systems call for events in which 10% or more of the donor's mass can be transferred during a single orbit.

We are conducting a multi-year project that focuses on these relatively rapid phases of mass-transfer, investigating the conditions required for stable/unstable mass-transfer and following numerically their evolution in unstable cases. The dynamical mass-transfer that ensues in unstable binaries rapidly reaches nonlinear amplitudes and may lead to the formation of a massive disk-like 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 require large-scale numerical simulations to further our theoretical understanding of them. While the direct observation of these phases – with the possible exception of gravitational wave emission from mergers – remains unlikely in the near future, these investigations will help us understand the origin of these binaries and their mutual relationships. With our present numerical simulation tools, we are in a position to study both the tidal and the mass-transfer stability of close binaries in which the component stars obey a polytropic equation of state (EOS) but, in general, possess nonuniform entropy. These simple structural approximations enable us to treat, with some limitations, a very diverse set of binary systems containing main sequence (MS) and white dwarf (WD) stars. In addition, we are embarking on a collaborative project to extend our numerical algorithms to handle relativistic fluid flows and to incorporate adaptive-mesh-refinement (AMR) capabilities. In the future, this expanded toolset will permit us to spatially resolve accretion disk structures as they form around various types of compact stellar objects, and to accurately simulate mass-transfer in neutron star-neutron star (NS-NS), neutron star-black hole (NS-BH), and WD-BH systems – systems which are especially relevant to the LIGO and LISA experiments as sources of gravitational radiation.

In an effort to illustrate our early successes with this long-term, numerically intensive project, throughout this proposal we will reference a variety of short animation sequences (movies) that we have produced from the results of various extended simulations. If you're reading this proposal on-line, each movie can be accessed by clicking on the relevant "movie icon." The URL of each movie is also listed in the "Attachments" section at the end of the proposal text.

a. Linear Theory

We distinguish two types of dynamical instability that can occur in close binaries:   1) A dynamical instability which arises in close proximity of extended binary components with sufficiently stiff EOS due to the presence of tidal forces, even before contact and onset of mass-transfer; and  2) A dynamical instability which arises upon contact, even when total angular momentum and mass are conserved, as mass is transferred from one component to the other. We shall refer to these instabilities as the tidal and mass-transfer instabilities, respectively.

i) Tidal Instability

Tidal Instability
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The classical tidal instability, first studied by Darwin (1887), deals with the secular evolution away from synchronous rotation due to dissipation. More recent treatments of this secular instability include: an analysis of tidal stability of coplanar binaries (spin axes perpendicular to orbital plane) by Counselman (1973), who shows that in unstable systems evolution away from synchronism is accompanied by a secular decrease or increase in the orbital separation; and a more general treatment by Hut (1980) allowing for arbitrary orientations of spin and orbital angular momenta. In a series of papers Lai, Rasio & Shapiro (1993a; 1993b; 1994a; 1994b) studied the equilibrium and stability of binaries with polytropic EOS, making use of analytic techniques and a variational energy method. They were able to extend their work to unequal masses and allow for internal motions. They showed that in addition to the classical secular instability, a new dynamical instability exists not far from the point of secular instability, which leads to rapid binary inspiral under certain conditions. They showed that the instability occurred before contact was reached for stiff EOS. The nonlinear development of one such tidal instability is illustrated by Movie-1, drawn from the work of New and Tohline (1997). In this simulation, the initial configuration is a pair of detached, equal-mass, n = 0.5 polytropic stars.

ii) Mass-Transfer Instability

In binaries with unequal components, in general one of the stars will reach contact with its Roche lobe first (the donor) and mass-transfer to its companion star (the accretor) will commence either because the donor star is evolving or because the binary separation is reduced by systemic angular momentum losses. What happens next, i.e. the stability of binaries to mass-transfer, depends crucially on the reaction of the donor star to mass-loss, and on the changes in Roche geometry that ensue from the mass-transfer.   Important factors determining the evolution of the Roche geometry are: 1) whether all the mass transferred by the donor is accreted (conservative mass-transfer); and   2) if some of the mass transferred is lost from the system, what are the consequential angular momentum losses (Webbink 1985; King & Kolb 1995). If upon contact and onset of mass-transfer, the radius of the donor and the radius of the Roche lobe vary in such a way that contact becomes deeper, mass-transfer will proceed at an ever increasing rate and the situation is unstable. If, however, the contact becomes shallower, the donor will detach and mass-transfer will cease. In the latter case, if some physical effect other than mass-transfer insists on driving the system back into contact, stable mass-transfer may be possible for as long as the driving is present. (As mentioned above, in CVs and LMXBs, systemic angular momentum losses drive stable secular mass-transfer from a less massive donor onto a compact object.) In a series of simulations that we have completed over the past few years, we have focused our investigation on the effect that the initial system mass ratio q Mdonor/Maccretor has on the stability of such mass-tranfer events.

b. Nonlinear Treatment of Stable Systems

Before outlining what evolutionary scenarios are likely to develop as we follow the nonlinear evolution of systems that are found to be unstable toward either a tidal or mass-transfer instability, it is important to point out that throughout the execution of this project we also are making a concerted effort to demonstrate the degree to which our nonlinear simulation techniques are able to accurately represent the equilibrium properties of close binary systems that are dynamically stable.

q = 1 Contact Binary
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Stable, q = 0.84 Binary
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To illustrate our early successes with this aspect of our investigation, we point to two movies: Movie-2 is drawn from New & Tohline (1997); Movie-3 is drawn from one simulation discussed by Motl, Tohline & Frank (2002). Movie-2 shows that, for a sufficiently soft equation of state – in this case, an n = 3/2 polytrope – an equal-mass binary system can be stable against the tidal instability even when the two stars are so close to one another that they are in contact and sharing a common envelope. Movie-2 follows the evolution of one such system through 3.5 orbits; although the system is being viewed with its orbital plane face-on, orbital motion is not apparent because the evolution is being shown in a rotating reference frame. Movie-3 demonstrates how well we are able to follow through more than 5 orbits the evolution of an unequal-mass (q = 0.84) system in which the less massive component (on the right in the Movie-3 icon) very nearly fills its Roche lobe. The movie displays equatorial-plane mass densities on a logarithmic color scale, as viewed from a frame of reference rotating with the binary system's initial orbital frequency.

c. Nonlinear Development of Unstable Systems

In those cases where the mass-transfer is initially unstable, we now are able to follow the evolution through ~ 20 orbits to investigate how initial conditions (e.g., mass ratio, entropy gradients, or polytropic index) influence the nonlinear growth of the mass-transfer and the final outcome. Depending on the assumed conditions, some binaries may become stable after a certain mass excess is transferred, some may exchange mass, and still others may evolve through a common envelope phase. This schematic division is adopted for simplicity since there may well exist cases where a combination of the above processes operate sequentially.

i) Dynamically Unstable Mass Transfer

When the donor is more massive than the accretor (q > 1) and has a deep convective envelope, the mass-transfer almost certainly will be dynamically unstable. This is because, as mass is transferred from the more massive star to the less massive one, the orbital separation should actually decrease with time while the radius of the donor is increasing. It is generally thought that once the rate of transfer is high enough, the accretor is swamped and driven out of thermal equilibrium, and the matter transferred forms a common envelope. Most calculations of the subsequent evolution have dealt with a relatively compact star (the accretor) orbiting or plunging into the convective (polytropic) atmosphere of a giant star (Rasio & Livio 1996; Terman, Taam & Hernquist 1994; 1995 and references therein). Some calculations by other groups have specifically focused on the ejection of the envelope (Taam et al. 1997; Sandquist et al. 1998).

q = 1.32
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Movie-4 shows results from our first investigation of an n = 3/2 polytropic system in which both stars have comparable radii, but the more massive star is the donor. Specifically, the illustrated system has an initial mass ratio q = 1.32 and, at the beginning of the simulation, the donor is just marginally filling its Roche lobe. Eventually, however, it develops a well-defined mass-transfer stream (as illustrated in the Movie-4 icon) and the mass-transfer rate steadily increases thereafter. It is interesting to note that, in this case, the center of mass of the system (and, hence, the orbital axis for the system) is positioned to the right of the L1 Lagrange point and is actually inside the donor envelope. The evolution is shown in a rotating frame of reference that is rotating with the initial angular velocity of the system and, as Movie-4 illustrates, we have been able to follow this system evolution through 12 full orbits. Eventually the objects merge into a single, rotationally flattened object.

ii) Mild Transfer and Return to Stability

q = 0.5
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If the mass ratio is not extreme and the polytropic indexes are similar, it should be possible for the binary to transfer some mass and regain stability. In systems that have a mass ratio q < 1, for example, the binary separation should increase with time as material flows from the donor to the accretor. But, based on one-dimensional linear stability theory, if q is not too far from unity and the donor star's radius expands as it loses mass (as is the case for n = 3/2 polytropes) the Roche lobe should stay in contact with the donor – at least for a while – and thereby promote a continuous mild transfer event. Once q drops to a sufficiently small value, however, the Roche lobe should separate from the surface of the donor and the mass transfer event should be terminated. If the donor has a polytropic index n = 3/2, linear theory predicts that mass transfer should stop when q 2/3. In an effort to test this hypothesis, we have followed the evolution of one system in which both stars are n = 3/2 polytropes and the initial system mass ratio q = 0.5 (that is, q is initially somewhat smaller than the predicted critical value of 2/3). Movie-5 follows the evolution of this system through 20 orbits! What we have discovered, contrary to the prediction of (the over-simplified) linear theory, is that mass continues to be transferred from the donor to the accretor at a steady rate and the orbit continues to slowly shrink throughout the evolution. As is discussed more fully, below, additional simulations are required in order to ascertain the proper critical value of the initial mass ratio. With the PACI resources we are requesting this year, we expect to solve this problem.

iii) Mergers

The coalescence of WD-WD and NS-NS binaries is important to examine since this process may produce a number of astrophysically interesting objects and events. Double white dwarf binary mergers have been suggested as precursors to some Type Ia Supernovae (Iben & Tutukov 1984; Iben 1988, 1991; Iben & Webbink 1989; Branch et al.  1995) and to long g-ray bursts (Katz & Canel 1996). WD-WD mergers may also lead to the formation of massive single white dwarfs or neutron stars (Colgate & Petschek 1982; Saio & Nomoto 1985; Iben & Tutukov 1986; Kawai, Saio, & Nomoto 1987; Chen & Leonard 1993); to the formation of subdwarf stars; or to the formation of hydrogen deficient, highly luminous stars (Iben 1990 and references therein; Webbink 1984). NS-NS mergers may result in bursts of g-rays and neutrinos, in the production of r-process elements, and in the formation of black holes (BH) (Eichler et al. 1989; Meyer 1989; Narayan, Paczynski, & Piran 1992; Rasio & Shapiro 1992; Davies, Benz, Piran, & Thielemann 1994; Katz & Canel 1996; Lipunov et al. 1995; Ruffert, Janka, & Schäfer 1996; New & Tohline 1997; Swesty, Wang & Calder 2000; Rosswog, Davies, Thielemann & Piran 2000). Merging compact binaries are also expected to be relatively strong sources of gravitational radiation (Abramovici et al. 1992; Cutler et al. 1993; Thorne 1995).

Most of the numerical simulations of mergers that have been carried out in recent years have been focused on the NS-NS or NS-BH problem because of their direct relevance to LIGO (and its sister instruments, like TAMA and VIRGO); and most, although not all (cf. Rasio & Shapiro 1994; Zhuge et al. 1996; Rosswog et al. 2000), have focused on equal-mass (q = 1) systems. Even when relativistic effects are ignored in such simulations, the stars usually merge on a dynamical timescale because one star (or both simultaneously, in the equal-mass case) encounters a tidal instability before either star fills its Roche lobe (Rasio & Shapiro 1994; New & Tohline 1997). This occurs, in turn, because the EOS of a NS is relatively stiff. As Rasio & Shapiro (1994) have summarized, the existence of a Roche limit and any associated mass-transfer instability has little importance for most NS star binaries because this "dynamical instability always arises before the Roche limit." When mass transfer begins, the system is already in "a state of dynamical coalescence," so the mass-transfer rates can be much larger than 10% per orbit. One such evolution, taken from New & Tohline (1997), was highlighted above as Movie-1.

Although we are interested in continuing to pursue this type of merger problem, the focus of our present research is on systems in which the secondary star has a relatively soft EOS (e.g., n = 3/2 polytrope) and the star fills its Roche lobe before the system encounters a tidal instability. As we have discovered – see, for example, Movie-4, above – these mass-transfer events can also lead to a merger of the two stars.

2. Computational Methodology/Algorithms

In carrying out a number of additional binary mass-transfer simulations this year, as outlined in §2.d, below, we will follow the same procedures as have lead to the various simulation results discussed above and illustrated through Movies 1-5. For each simulation, we will initially utilize a 3D self-consistent field (SCF) technique to generate accurate equilibrium structures of synchronously rotating binaries in circular orbits. Each rotationally flattened and tidally distorted binary model will then serve as input into (initial conditions for) a dynamical simulation. Each dynamical simulation will be performed using a 3D, finite-difference hydrodynamic (FDH) technique that follows in a self-consistent fashion the motion of every fluid element in both stars. Our present SCF and FDH codes, as well as forerunners of these codes, have been effectively used to study gravitationally driven instabilities in protostellar gas clouds (Durisen et al. 1986; Woodward, Tohline & Hachisu 1994; Cazes & Tohline 2000); rapidly rotating, compact stellar objects (New, Centrella & Tohline 2000; Lindblom, Tohline & Vallisneri 2001, 2002; Ou & Tohline 2004); and close binary systems (New & Tohline 1997; Motl, Tohline, & Frank 2002). A detailed description of the present structure of both codes and their capabilities is given in Motl, Tohline, & Frank (2002). Here we briefly summarize the principal elements of both.

a. Construction of Equilibrium Models

It is important that we have the capability to construct accurate equilibrium structures as initial input into each dynamical simulation so that the simulations are not dominated by dynamical transients that depend heavily on initial conditions. We use the iterative SCF technique described by Hachisu (1986a,b) in its most general form in order to construct unequal-mass polytropic binaries with the internal structure of both stellar components fully resolved. Models are constructed on a uniformly zoned, cylindrical mesh that conforms exactly to the mesh that is used in the FDH code (see below) so that no interpolation is needed when the converged initial state is introduced into the dynamical code. To date, we have used this SCF technique to generate a wide variety of synchronous, polytropic binaries with mass ratios 0.1 < q < 1.5 in which both stars have uniform (but different) entropy distributions (homentropic). We can readily build detached, semi-detached or, in the limit of identical components, contact systems. These models have been constructed on computational grids ranging in size from ~ 1283 to ~ 2563 as necessary to conform to the chosen resolution of the FDH code. As measured by the global virial error, these models typically converge to better than one part in 105. As a further test, we have compared parameters from our equilibrium binaries with those calculated from the point mass Roche model by Mochnacki (1984). As expected, our results differ from Mochnacki's because we use a self-consistent gravitational potential as opposed to the point mass potential as is appropriate in the Roche model. Our results converge to Mochnacki's in cases where the mass ratio becomes small and/or the separation increases.

In addition to polytropic binaries, we are able to construct self-consistent, Newtonian models of compact binaries by choosing an appropriate equation of state such as the zero temperature WD equation of state (Hachisu 1986) or any of several realistic NS equations of state as described by New & Tohline (1997). Binary models using homentropic equations of state are, however, inadequate for examining the stability of mass transfer in main sequence stars without deep convective envelopes (stars with masses greater than but on the order of one solar mass). By allowing the specific entropy to be a function of the density, we are able to construct equilibrium configurations that are both convectively stable and that reproduce the desired mass-radius relationship. In the context of this present proposal, however, we will confine our study to homentropic binaries with n = 3/2 polytropic equations of state.

b. Dynamical Simulations

Our primary dynamical simulation tool is an mpi (message passing interface) based algorithm designed to perform multidimensional, gravitational CFD (computational fluid dynamic) simulations efficiently on parallel computing platforms. As has been described in detail by Motl et al. (2002), the evolution of a system is governed through a finite-difference representation of the multidimensional equations that describe the dynamics of inviscid, Newtonian self-gravitating, compressible gas flows. More specifically, the employed finite-difference hydrodynamics (FDH) algorithm is based on the second-order accurate, van Leer monotonic interpolation scheme described by Stone and Norman (1992), but extended to three dimensions on a uniform, cylindrical coordinate mesh. Most of our simulations, to date (e.g., those associated with Movies 4-7), have been performed on a computational lattice containing 16298256 grid zones in the R, Z, and q directions, respectively, with no assumed geometric symmetries. This has been sufficient to resolve both binary components, along with the mass-transfer stream. In order to properly resolve both stars in a system with a mass ratio q < 0.2 – and thereby test the dynamical stability criterion alluded to above – we will find it necessary to adopt a lattice containing significantly more grid zones because in such a low q system the volume of the Roche-lobe surrounding the less-massive (donor) star is significantly smaller than the Roche-lobe of the more massive, accretor. (As discussed further, below, a lattice having dimensions 194130512 appears to be sufficient.) The fluid equations are advanced in time via an explicit integration scheme so an upper limit to the size of each integration time step is set by the familiar Courant condition. At each time step, the Poisson equation is solved implicitly using a combined Fourier transformation and iterative ADI scheme (Cohl, Sun & Tohline 1997) in order to determine, in a self-consistent fashion, how fluid elements are to be accelerated in response to the fluid's own Newtonian self-gravity. Boundary values for the Poisson equation are determined via a compact cylindrical Green's function technique (Cohl & Tohline 1999). By performing our simulations in a frame of reference that is rotating with an angular velocity W0 equal to the initial angular velocity of the binary orbit, we are able to minimize the effects of numerical viscosity (see New & Tohline 1997; Swesty, Wang and Calder 2000), which invariably plays a role in any FDH algorithm.

c. Visualization Tools

We have constructed a heterogeneous computing environment through which we are able to routinely create volume-rendered images and 3D animation sequences of our evolving fluid configurations. Relatively small amounts of data are regularly passed across the network from the parallel machine that is performing the FDH simulation to a machine that is more suited to visualization tasks; images that will become part of an extended animation sequence are created from this data; then the 3D data sets are immediately destroyed in order to minimize the impact on disk storage. We are utilizing the scriptable attributes of Maya (formerly Alias-Wavefront) software on SGI systems in order to perform these critical visualization tasks. (The 3D "Movie" sequences referred to throughout this proposal have been produced in this fashion. This visualization tool and the heterogeneous computing environment in which it functions will provide a critical component of the simulations being proposed here.

d. Proposed Simulations and Code Development
i) Simulations

This is a computationally intensive, multi-year research project that will provide a more complete understanding of tidal and mass-transfer instabilities in close binary systems. With the allocation of PACI resources that we are requesting this year, we expect to ascertain whether any n = 3/2 polytropic binary system that is initially unstable toward mass-transfer can return to stability before the system merges. As explained above, linear theory predicts that the critical mass ratio is qcrit = 2/3. However, simulations that we have completed that have started with a mass-ratio near the value 2/3 have not regained stability; instead, they have each led to progressively higher mass transfer rates and, ultimately, to merger of the binary system. (These simulations each resembled the much higher mass-ratio evolution referenced above as Movie-4.) It appears as though these relatively high-q simulations do not return to stability for two reasons: First, because the accretion stream directly impacts the accretor, angular momentum carried by the stream material is robbed from the orbit and deposited onto the accretor. Hence, the assumption that orbital angular momentum is conserved during the mass-transfer event, which is built into the linear theory prediction, is invalidated. Second, the mass-transfer rates grow to such high (nonlinear) levels during the evolution of these dynamically unstable systems that the transfer is difficult to shut off even when q drops to a level below which the binary separation starts to increase.

q = 0.41
Quicktime (60 MB)
We have attempted to modify the standard linear stability analysis to account for the fact that angular momentum is being drained from the orbit in these types of evolutions. Our re-analysis suggests that qcrit is more likely between 0.2 and 0.4 for n = 3/2 polytropic stars. With this revised prediction in hand, this past year we have followed through more than 17 orbits the evolution of a sysem with an initial q = 0.409; the result is illustrated here by Movie-6. Very early in this model evolution, the binary system began to separate but the corresponding rate of expansion of the radius of the donor was sufficiently fast that, overall, the mass-transfer rate steadily increased. At 17 orbits (model configuration at the end of Movie-6), however, we can see that the rate of mass transfer is beginning to level off, so there is some expectation that the model is approaching the critical mass ratio. In order to locate qcrit, we will need to follow this q = 0.409 evolution through at least an additional 20 orbits. It will be valuable to follow the q = 0.5 model (described earlier in connection with Movie-5) further in time as well, to the point where that system's mass ratio drops below q = 0.409 in order to determine whether qcrit depends sensitively on the initial system mass ratio. Finally, we must study the evolution of a model that has a much lower initial mass ratio (q < qcrit, say, q = 0.2) in order to test whether such a system is stable at all times. We propose to use NCSA's "tungsten" to carry out these three simulations.

ii) Code Development

Through NSF's medium-ITR program, we have been awarded funds to significantly expand the capabilities of our existing numerical tools. A primary goal is to be able to couple our fluid-flow simulations to the general-relativistic (GR) Einstein equations to permit us to accurately study accretion in NS-NS, NS-BH, and WD-BH systems. We have made a commitment to incorporate adaptive mesh refinement (AMR) techniques that execute efficiently in parallel computing environments and the ability to transition to general relativistic settings by implementing the general relativistic hydrodynamic equations. We will build upon our own experience, described above, of accurately simulating accretion flows in nonrelativistic, Newtonian gravitational fields, and will draw upon the expertise of our relativity colleagues (who are Co-PIs on the NSF/ITR grant) in the implementation of AMR and General Relativity codes.

Perhaps the main problem faced by most general relativistic simulations in three dimensions is the lack of computational resources. Such evolutions require a tremendous amount of storage as well as a huge number of floating point operations. As an example, consider a model to solve the Einstein equations with roughly 100 field variables. To model a black hole of mass M with boundaries at a distance about 100M away with a resolution of some small fraction of M, say M/10, requires roughly 1,000 points per side. The total storage required is then 100 fields × 10003 (dimensions)× 8 Bytes ~ 750 GB. Such an estimate is quite rough and assumes this is enough for a scheme to be capable of evolving the system in a stable way. Perhaps a much larger number of fields will be necessary; and a numerically stable scheme may ultimately require a much improved resolution. Furthermore, the numerical problem involves a large disparity in scales requiring improved resolution. For example, one must simultaneously resolve the nonlinear distortions around a black hole (small fractions of M) as well as the outgoing gravitational waves ~ 10's of M.

These huge computational demands outstrip the ability of current conventional supercomputers, and, even if they did not, such demands will likely increase at a rate faster than that of computer hardware. Add to such demands the fact that once we have good codes, we have a huge parameter space to explore and the turnaround for tests/experiments becomes a huge burden. Thankfully, two computational methods provide some hope. Adaptive Mesh Refinement: AMR is an algorithm by which improved resolution (over the initial grid) is added where and when needed (e.g., Berger & Oliger 1984). The dynamical creation and destruction of fine sub-grids limits the computational work so that it scales with the dynamics one models. Additionally, by breaking the computational problem into pieces – each of which can then be solved separately on different computational nodes – one takes advantage of massively parallel architectures. Achieving efficient distributed AMR requires minimizing the communication between nodes and keeping them busy instead of waiting for information (so-called load balancing). These are nontrivial tasks that we will address as part of our ITR goals. We will draw from the experience that has been gained by other groups in the development of efficient AMR algorithms for a variety of problems – for example, the codes ENZO, CHOMBO, PARAMESH, and CARPET (a thorn of CACTUS) all utilize AMR and we have extensive experience utilizing ENZO for cosmological simulations.

At the same time, we will work on modifying the Newtonian code to incorporate the General Relativistic Hydrodynamic equations. In the first stage this will be done assuming a fixed background spacetime metric (corresponding to a single black hole). This will serve as a preliminary project with our ultimate, multi-year goal being to model accretion flows in fully dynamical spacetime settings. A significant amount of computer time is required to develop and test this implementation. Once this stage is passed we will also incorporate distributed AMR technique into the relativistic FDH code. This segment of our proposed project should be considered developmental work. Nevertheless, a considerable amount of computer time will be required to test the scalability, performance and efficiency of the distributed AMR techniques in a realistic setting (i.e. when dealing with the full equations rather than a toy model). Additionally, the development of general relativistic hydrodynamical codes requires substantial computer time for development and testing.

The development work described in the preceding few paragraphs is being done as we look toward future years of research when we will begin to focus more of our simulation efforts on relativistic systems. At the present time, we are not requesting PACI resources to directly support these development efforts as we anticipate that LSU's large Linux Cluster (SuperMike) will support the development work adequately. Nevertheless, we felt that it was important to include a brief outline of this work in the context of our request for an allocation of PACI resources because the efforts are tightly linked.

3. Preliminary Results and Progress to Date

a. Code Development and Testing on PACI (and LSU) Machines

Over the past eight years, we have been awarded a total of approximately 1,200,000 SUs at the SDSC, UT-Austin, and most recently NCSA to study dynamically evolving astrophysical systems in connection with our NSF- and NASA-sponsored research into the birth and destruction of binary star systems. Our current request for PACI resources will allow us to build directly upon the successful simulation work that has grown out of this earlier, multi-year support. It is important to note, as well, that over the past several years the State of Louisiana has provided generous support to LSU in the broad field of information technology, and through this support LSU has built a 2.2 TeraFlop Linux SuperCluster (called "SuperMike") initially based on Intel's 1.8GHz Xeon DP processors and the Myrinet 2000 interconnect. (See §5, below, for a more complete description of SuperMike.) As we shall show in what follows, in 2002 this Linux SuperCluster generally performed as well as Blue Horizon (the IBM SP3) at the SDSC and, because it was dedicated to the support of LSU reseachers, a larger number of processors were routinely available to us on SuperMike than we could secure on Blue Horizon. So, in 2002 SuperMike became the system of choice for our development and simulation work, and our experience with this system has led us to select NCSA's "tungsten" as our preferred PACI hardware system.

i) Parallel CFD Algorithm

Our initial simulations on the Cray T3E at the SDSC were carried out with a version of our code that had been developed on other parallel platforms and written in high-performance-fortran (HPF). Using the available Cray HPF compiler (PGHPF, developed by the Portland Group), we found that relatively few unexpected code modifications were required in order to successfully compile and execute our code on the SDSC system. We discovered, however, that the PGHPF compiler was producing executable code that performed computational instructions on each T3E processor with extraordinarily low efficiency. In 1998, we successfully rewrote our entire FDH algorithm, incorporating explicit message passing instructions via mpi. We found that this new, mpi-based code executed four to five times faster than the old HPF version of the code! We also found that the code scaled very well as the number of processors was increased from 4 to 128.

Shortly thereafter, we migrated our mpi-based FDH code to the SP3 at SDSC. Table 1 illustrates the level of performance and degree of scalability of the code on the SDSC SP3, as it was configured in 2002. The first column lists the number of processors used in each test; and the numbers at the top of every other column specify each test's grid size as well as the total number of grid cells [in millions of cells]. The numbers in the body of the table specify the amount of time that is required to take one integration timestep in a typical simulation; the values quoted are in units of nanoseconds per grid cell. (The wall-clock time required to take one timestep is obtained by multiplying the tabulated number by the number of grid cells; for example, running on 128 SP3 processors, a simulation with a grid resolution of 2582×256 requires 166 ns/cell × 17M cells = 2.8 seconds per timestep.) Listed in this manner, if our code scaled perfectly linearly with problem size, the numbers across a given row would all be the same; and if our code parallelized perfectly, the numbers down each column would drop by exactly a factor of two each time the number of processors is doubled. In both directions, the scaling gets closer to linear as we take fuller advantage of the machine's available memory, that is, as we perform simulations with larger and larger grid sizes.

Table 1
FDH Code Timings on the SDSC SP3 in 2002
(nanoseconds per cell per integration timestep)
[ 1.1 M cells ]
[ 2.2 M cells ]
[ 4.3 M cells ]
[ 8.6 M cells ]
[ 17.0 M cells ]
[ 34.0 M cells ]
32 550 516 570 581 -- --
64 392 315 308 312 306 --
128 -- 216 192 169 166 187
256 -- -- 150 124 101 100

In the summer of 2001, we participated in a short workshop held at NCSA that introduced users to an early Linux SuperCluster (512 nodes of 1.0GHz Intel Xeon DP processors). Within a matter of two days, our FDH code was successfully ported to this cluster and obtained reasonably good run times. We were therefore in a position to immediately take advantage of SuperMike, when it was installed at LSU in August of 2002. Table 2 shows the performance that we have achieved on SuperMike for problems of comparable size to the ones we had tested earlier on Blue Horizon at SDSC. The timings and the degree of scaling depicted in Table 2 are amazingly similar to the timings reported in Table 1, especially for simulations requiring large grids. For example, on 128 processors, both Blue Horizon and SuperMike require 170 - 180 ns/cell; and this number drops to approximately 100 ns/cell when twice as many processors are invoked.

Table 2
FDH Code Timings on LSU's Linux Cluster, SuperMike, in 2003
(nanoseconds per cell per integration timestep)
[ 1.1 M cells ]
[ 1.6 M cells ]
[ 3.3 M cells ]
[ 6.7 M cells ]
[ 10.7 M cells ]
[ 21.4 M cells ]
32 -- -- -- 609 -- --
64 378 349 -- 341 302 326
128 -- 208 -- 208 179 182
256 -- -- -- -- 104 106

In April, 2004, we received an allocation of 400,000 SUs on NCSA's Linux SuperCluster, "tungsten." However, tungsten did not become fully available for production runs until the Fall. Since that time, we have made significant use of its resources, including one very successful week-long dedicated run on 256 processors in early January, 2005. Table 3 contains the timing information that we have gathered this month (January, 2005) on tungsten. A comparison with data provided in Table 2 shows that, generally speaking, tungsten provides us with a 5-10% speedup over SuperMike. (The 133 ns/cell/timestep reported in the last row of Table 3 has been calculated from the dedicated production run that contained 12.9 M cells. It is a bit slower than what we would obtain in a standard test run because the production run included a significant amount of data I/O.)

Table 3
FDH Code Timings on NCSA's tungsten, in 2005
(nanoseconds per cell per integration timestep)
[ 1.1 M cells ]
[ 2.2 M cells ]
[ 4.3 M cells ]
[ 8.6 M cells ]
[ 17.0 M cells ]
[ 34.0 M cells ]
32 -- 401 -- 571 496 --
64 -- 335 -- 352 297 --
128 -- 190 -- 158 160 --
256 -- -- -- 133 --

(ii) A Significantly Improved Poisson Solver

It has been clear to us for quite some time that one of the most computationally demanding components of our FDH code is the algorithm used to calculate boundary values for the gravitational potential that must be fed into the Poisson solver in order to be able to solve for the Newtonian gravitational potential throughout the volume of the fluid. Historically, we have used an algorithm based on a multipole expansion in terms of spherical harmonics (Tohline 1980; see also Stone & Norman 1992), but spherical harmonics are not a natural basis set when simulations are being conducted on a cylindrical coordinate mesh. In the late '90s, we discovered that the traditional Green's function expansion in cylindrical coordinates which involves an infinite integral over products of Bessel Functions can be written in a much more compact form (Cohl & Tohline 1999) that is extremely well-suited for implementation on cylindrical coordinate meshes. A parallel implementation of this "compact cylindrical Green's function" technique is now part of our FDH code. It will be used in all of our future Newtonian simulations because it is both faster and more accurate than the one based on a multipole expansion in terms of spherical harmonics. In addition, its implementation has permitted us to bring the top (and bottom) cylindrical grid boundary down closer to the rotationally flattened mass distribution and reduce the computational lattice size in the vertical direction.

(iii) Establishing a Heterogeneous Computing Environment

As we have described above in §2.c, we have constructed a heterogeneous computing environment through which we are able to routinely create volume-rendered images and 3D animation sequences of our evolving fluid configurations. Movies 4-7, pointed to above, are products of this environment. Initially we developed this heterogeneous computing environment at the SDSC, routinely exchanging data between the Cray T3E and the SGI machines in SDSC's VizLab. We consider the successful establishment of this environment – which smoothly couples high-end, SGI visualization tools with our simulation efforts on PACI (and now, LSU) machines – to have been a significant accomplishment.

b. Simulations to Date using PACI Resources

Our first two years of research utilizing NPACI facilities were focused on the problem of binary star formation. As detailed in the accompanying attachment entitled, "Publications Resulting from NPACI Research," the results of this work have been summarized in a number of different publications, including one feature article in the April-June, 1999 issue of NPACI's EnVision Magazine. In the spring of 2000, while on sabbatical at Caltech, the PI collaborated with K. Thorne and L. Lindblom to model the nonlinear growth of an unstable "r-mode" in isolated, rotating neutron stars, and to simulate the tidal disruption of a neutron star that is in orbit about a more massive, Kerr black hole. Through this collaboration, the PI gained access to the HP Exemplar at CACR, a partner in the NPACI alliance. Our initial application of the PI's simulation tools (as described here in §2) to the "r-mode" problem proved to be very successful (Lindblom, Tohline, & Vallisneri 2001, 2002). The results of this work (and images produced with our visualization tools) were featured in the April-June 2001 issue of NPACI & SDSC's EnVision Magazine. For his dissertation work, LSU graduate student Shangli Ou has extended this work to examine the nonlinear development of the secular "bar-mode" instability in rotating neutron stars (Ou 2004; Ou & Tohline 2004).

Over the past several years, we have turned our primary attention to problems associated with mass-transfer in binary star systems during the late stages of stellar evolution, as described in this proposal. Using NPACI facilities both at SDSC and Austin, TX, we invested a considerable amount of effort fine-tuning the capabilities of our SCF and FDH algorithms so that they would be able to perform the required modeling tasks with a high degree of quantitative accuracy. Motl et al. (2002) describe in detail how well our code presently performs on a variety of benchmark simulations. One of these simulations – a stable, detached binary system with mass ratio q = 0.84 – was evolved through over five orbital periods on a grid having 130 × 98 × 256 zones in R, Z, q respectively. This evolution is illustrated by Movie-3, above, which shows how the equatorial-plane density distribution of the system changes with time: Only extraordinarily small changes are observed in this stable system. Even after five orbits, the centers-of-mass of the two stars and of the system as a whole essentially have not moved from their initial positions. And although a careful examination reveals very low-amplitude, epicyclic motion, quantitative measurements show that the orbits of the stars are circular to the level of two parts in 104. This provides strong confirmation that our SCF code produces excellent initial states and that the hydrodynamical equations are being integrated forward in time in a physically realistic manner.

For the past two years, we have been in a position to study mass-transfer instabilities in unequal-mass binaries with a degree of quantitative accuracy that heretofore has not been possible. We not only have improved on the simulation capabilities of New & Tohline (1997), but as Motl et al. (2002) detail, the epicyclic amplitude of our binary orbits are an order of magnitude smaller and angular momentum conservation is an order of magnitude better than in the equal-mass binary simulations of Swesty, Wang, and Calder (2000). As Motl et al. (2002) predicted, and as our most recent simulations have demonstrated, with our current simulation tools we are able to follow mass-transfer events in which the fraction of the donor's mass that is transferred is greater than or on the order of a few × 10-3 per orbit for at least 20 orbits! We are now in a position to accurately determine mass-transfer instability boundaries of the type discussed elsewhere in this project proposal.

q = 0.5 stream
Quicktime (8.5 MB)
Highlights of this past year's effort include the two extended evolutions described briefly, earlier in this proposal (see the discussions associated with Movie-5 and Movie-6). The model with an initial mass ratio q = 0.4 has evolved through more than 17 orbits (requiring over 1.6 million integration time steps) and the model with q = 0.5 has evolved through more than 20 orbits (approximately 0.9 million time steps); both have reached a phase of "steady-state" mass-transfer. Results of this type have not previously been obtained by any other group in the field: We've been able to follow both evolutions a factor of 3-4 longer in time than any previously published simulations of this type; the systems are not merging; and the simulations are behaving sufficiently well that we can probably continue to follow each system through as many as 100 orbits (if sufficient computing time is made available). This represents an unprecedented level of numerical accuracy and stability in the modeling of such complex, astrophysical fluid flows. As a result, we have discovered that both accretion streams are able to excite nonaxisymmetric structure in the accretor. For example, a standing wave develops along the equatorial belt of the accreting star in both simulations that produces a noticeable "bump" on each star that slightly leads the location of the mass-stream. Movie-7, which is a magnified view of the q = 0.5 model stream late in the evolution, shows this feature clearly.

In the q = 0.5 evolution, the binary separation decreased for a while but after 20 orbits it appears as though the separation is beginning to increase. In contrast to this, in the q = 0.4 evolution, the binary separation steadily increases throughout the evolution but the mass-transfer rate has continued to increase as well. Neither model has reached a phase of evolution where the mass-transfer rate is decreasing, so neither model has yet identified the mass-ratio below which stability against mass-transfer occurs. (Recall that linear theory predicted stability at any q < 2/3.) This past year we also have begun to follow an evolution that has been the most computationally demanding, to date. It is a binary system that starts with a very low mass-ratio, q = 0.2, and is almost certain to be in the "stable" mass-transfer regime. Due to the inherent properties of such a system, however, each time step is significantly shorter than in the q = 0.4 or q = 0.5 model evolutions. (See further discussion, below.) Even after a week-long, dedicated run on 256 tungsten processors, this model has evolved through only 2.1 orbits. A major focus of our efforts this coming year will be to push this evolution to a point where its stability against mass-transfer has been confirmed; this will require evolving the system through approximately 8 additional orbits. In addition, we propose to follow both the q = 0.4 and q = 0.5 models far enough in time so that their mass ratios drop below the critical stability limit.

4. Justification of Resources Requested

i) Required Platforms

To complete the tasks outlined for this project, we are requesting an allocation of 466,000 SUs on "tungsten" (the Xeon Linux SuperCluster) at NCSA (see Table 3 for details). We feel that Tungsten is the best platform on which to conduct production runs this year because we have demonstrated good performance of our simulation code on this machine, and its basic architecture, operating system, and compilers are very similar to the LSU Linux SuperCluster – "SuperMike" – so maintainance of our code will be minimized as we move back and forth between these machines. Tungsten and SuperMike both are Intel Xeon DP processor-based systems with Myrinet 2000 interconnects. It will be an advantage for us to use Tungsten (instead of SuperMike) to carry out our most demanding model simulations primarily for two reasons: (1) Tungsten has slightly faster Intel processors than SuperMike, and tungsten's recent hardware upgrade ensures robustness of all arithmetic operations. (2) Because Tungsten has many more hardware nodes than does SuperMike, we will be much more likely to secure 256-512 processors for dedicated runs on Tungsten and therefore complete our most demanding simulations in a reasonable number of days.

ii) Code Requirements and Optimization Issues

Since we have discussed code optimization issues fairly thoroughly in earlier sections of this proposal, we will not comment further on them here. From the information provided above – in §3.a.ii and Table 3, for example – we can estimate how much computing time will be required to make significant progress on this project. On 128 processors, both our q = 0.4 and q = 0.5 models require approximately 160 ns/cell/timestep and both models have been run with a grid resolution of 16298256 [4.1 M grid cells]. Hence, each integration timestep requires approximately 0.65 seconds. Since, at this resolution, a single binary orbit for model q = 0.4 (q = 0.5) requires approximately 100,000 (45,000) integration timesteps, extending the q = 0.4 (q = 0.5) evolution through an additional 20 orbits will require approximately 360 (160) wall-clock hours, that is, approximately 46,000 (20,000) SUs.

The particular, low "q" binary mass-transfer simulation that we need to pursue, however, demands somewhat higher spatial resolution in R and Z, and twice the azimuthal resolution (specifically, we have found that 194130512 = 12.9 M grid cells will suffice) and approximately 400,000 time steps per orbit. (More timesteps are required because the Courant-limited timestep is smaller.) Utilizing 256 processors on tungsten, this evolution has required 133 ns/cell/step, that is, 1.7 s/timestep, so each additional orbit will require approximately 190 hours (8 days), or about 50,000 SUs. To evolve this system through 8 additional orbits will therefore require 400,000 SUs. (In order to reduce the number of weeks that are required to complete this model evolution, we may run the model on 512 processors, in which case 400,000 service units may only allow us to complete 6 or 7 additional orbits.)

Our largest proposed simulation, at a resolution of 194×130×512 will require approximately 12 GB of RAM. Hence, when spread across 256 or 512 processors, there will be no problem fitting the code into Tungsten's available memory.

iii) Proposed Production Runs

Table 4 summarizes the SU requirements for the three key production simulations that we have just described.

Table 4: SU Requirements
Proposed Simulations on NCSA's Xeon Linux Supercluster (Tungsten) Nmodels Norbits SU
A. Extending the q = 0.4 and q = 0.5 Model Evolutions
  Grid resolution of 162×98×256 [4.1 M cells] on 128-procs:
Each orbit requires 100,000 (45,000) time steps @ 160 ns/cell/timestep ⇒
2300 (1000) SU per orbit.
2 20 66,000
B. Searching for Critical Initial Mass-Ratio: q = 0.2
  Grid resolution of 194×130×512 [12.9 M cells] on 256-procs:
Each orbit requires 400,000 time steps @ 133 ns/cell/timestep ⇒
50,000 SU per orbit.
1 8 400,000
Total requirement.

v) Special Requirements

Our primary simulation tools (the SCF code and the FDH code) have been developed entirely by our group at LSU and require no special packages other than standard mathematics utilities and message-passing libraries that are available on NCSA's Linux clusters.

5. Local Computing Environment

In the astrophysics group at LSU, we have access to two, dual-processor SGI Octanes on which we have installed Maya (TM), the Alias-Wavefront software that is being used to routinely render 3D images and generate movie sequences from our dynamical simulations, as explained in §2.c. Within the group, we also have acquired a half a dozen Linux PC platforms on which most of our day-to-day work is done and through which we gain access to all remote computing facilities.

In 2001, the Louisiana legislature funded a new, statewide information technology initiative through which LSU received $7M in new, recurring funding. For two years, the PI (Tohline) served as interim director of the Center for Applied Information Technology and Learning (LSU CAPITAL; now called "CCT") that has been established at LSU through this initiative. A portion of the first year's funding was used to purchase a 1024-processor, Linux SuperCluster, nicknamed "SuperMike": 1.8 GHz Intel P4 Xeon processors; Myrinet 2000 network; 1 GByte RAM per processor (see for more details.) When this hardware system was installed in August, 2002, it was clocked at 2.2 TeraFlops on standard high-performance linpack benchmarks and it was ranked by as the 11th fastest supercomputer in the world. As described elsewhere in this proposal, this system has proven to be a superb workhorse for most of our development efforts and some of our simulation efforts over the past couple of years. We will continue to have access to this system over the coming year and expect to perform (lower resolution) simulations as well as conduct a lot of code development on SuperMike that will complement the project described herein.

6. Other Supercomputer Support

Our group does not presently have access to any other supercomputing resources, beyond the PACI and LSU hardware architectures mentioned elsewhere in this proposal.

7. Project Team Qualifications

a. Background of the PI

A major component of the PI's research work over the past twenty-five years has involved the large-scale numerical simulation of multidimensional gas flows in astrophysical systems utilizing a variety of different high- performance- computing platforms. In the early 1980s, the PI held a postdoctoral research position in group T-6 at Los Alamos National Laboratories at which time he broadened his exposure to state-of-the-art CFD techniques and learned to develop efficient algorithms for vector supercomputers. From the mid-1980s through the early 1990s, Tohline was PI on a large account at the Cornell Theory Center during which time he and his students experimented with the parallel implementation of CFD algorithms on the IBM 3090 and various accompanying FPS Array Processors. When an 8K-node MasPar MP-1 system arrived at LSU in 1991, the PI abandoned the CTC facilities because his group was having considerable success utilizing the MasPar system for its largest simulation efforts. As has been detailed elsewhere in this proposal, in recent years the PI and his students have gained experience developing gravitational CFD algorithms for execution on a variety of parallel platforms including MasPar's MP-2, the CM-5, the Cray T3E, the IBM SP2 & SP3, and large Linux Clusters. The group also has developed and implemented a heterogeneous computing environment (at LSU and at the SDSC) that tightly couples its parallel simulation efforts to sophisticated visualization tools.

Co-PI J. Frank has broad experience in studies of the astrophysics of interacting binary systems, particularly when such interactions involve fluid flow and accretion events. He is, for example, lead author of a widely referenced technical CUP publication entitled, "Accretion Power in Astrophysics." His expertise within this project helps ensure that the results of the proposed complex multi-dimensional and nonlinear simulations are physically correct and quantitatively meaningful when placed in the context of the extensive literature on this subject.

Co-PI Motl is presently a postdoc in the LSU Astrophysics group. He developed and tested the earliest mpi-based version of our primary CFD simulation code while he was a graduate student in our group at LSU. Subsequently, while holding postdoctoral positions at Missouri and CASA (Colorado), he worked extensively with the ENZO code (developed principally by G. Bryan in M. Norman's group) to simulate cosmological fluid flows. He brings to the group broad experience in large-scale fluid-flow simulations that include AMR techniques.

b. Others Working on the Project

Mario D'Souza, Xiaomeng Peng, Ilsoon Park, and Wes Even are graduate students enrolled in the Ph.D. physics program at LSU. D'Souza, in his 6th year of graduate work, will be heavily involved in the simulation and code development work outlined in this proposal as his dissertation research focuses on mass-transferring binary systems. The three other students have just begun to utilize our simulation tools, but expect to become experienced users by the end of this coming year.

A1: Bibliography
A2: Publications Resulting from PACI Research
B1: Curriculum Vitae for J. E. Tohline
B2: Curriculum Vitae for J. Frank
B3: Curriculum Vitae for P. Motl

Movie URLs