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The aim of the Center is to solve the important and long-standing problem of astrophysical thermonuclear flashes. Thermonuclear flashes are relevant to a diverse array of phenomena in astrophysics, and include phenomena such as x-ray bursts (which are due to combined hydrogen-plus-helium and pure helium flashes in the accreted envelopes of neutron stars), classical novae (which are due to hydrogen flashes in the accreted envelopes of white dwarfs), and Type I supernovae (which are thought to be due to a carbon flash in the core of a white dwarf in the case of Type Ia supernovae.

These phenomena are fascinating, in and of themselves. They are also important for the light they shed on other fundamental questions in astrophysics: x-ray bursts for what they tell us about the masses and radii of neutron stars; classical novae for the contribution they make to the abundances of intermediate-mass elements in the galaxy, and for what they say about how the masses of white dwarfs change with time in close binary systems; and Type I supernovae for the birth rate and the masses of neutron stars, and for the contribution they make to the abundances of intermediate mass and heavy elements in the galaxy. Type I supernovae are also important for their crucial role as ``standard candles'' in determining the Hubble constant, and therefore whether or not the universe is closed, flat, or open.

While seemingly diverse phenomena, x-ray bursts, classical novae, and Type I supernovae all involve a close binary system in which matter from a companion star accretes onto the surface of a compact star (neutron star or white dwarf). And of particular interest to our Center, all have in common the ignition of a nuclear fuel under degenerate conditions, followed by the propagation of thermonuclear burning via a convective or turbulent flame front (or deflagration wave), or via a strong shock front (or detonation wave).

X-ray bursts are due to combined hydrogen-helium or pure helium flashes in a shell at the bottom of a thin layer (~ 10 meters) of hydrogen-rich or pure helium material that has accreted onto the surface of a neutron star (Tamm 1995, Lewin et al. 1993, Taam et al. 1993). This phenomenon is somewhat simpler than the other phenomena we consider, in the sense that the nuclear energy released per gram of accreted matter is a factor ~ 20-100 less than the gravitational binding energy of the same gram of matter. Consequently, the flash is not quenched by expansion of the envelope; rather, the helium and heavier elements in the accreted envelope are incinerated to iron-peak nuclei.

Novae are due to hydrogen flashes in the shell at the bottom of a thin (~ 108 cm) layer of hydrogen-rich material that has accreted onto the surface of a dwarf (Truran 1982, Shara 1989, Livio 1994). But in contrast to the case of x-ray bursts, the nuclear energy released per gram of accreted matter is a factor of ~ 100 more than the gravitational binding energy of the same gram of matter. As a result, the flash leads to an enormous expansion of the envelope of the white dwarf; the envelope engulfs the companion star, forming a common envelope binary. At the same time, the work done against gravity in the expansion of the envelope cools the hydrogen burning shell and quenches the flash. Steady hydrogen burning then ensues.

Type I supernovae are thought to be due to carbon flashes that ignite in the core of a white dwarf whose mass has grown by accretion (in the case of Type Ia supernovae; Woosley and Weaver 86, Nomoto et al. 1994, Niemeyer 1995a, Niemeyer et al. 95b}. Neither deflagration nor detonation alone can account for both the abundances of intermediate-mass nuclei, and the abundances of iron-peak nuclei as well as the large expansion velocity of the ejecta that are produced in a Type Ia supernova. Consequently, current Type Ia supernova models invoke a transition from a deflagration wave to a detonation wave. One possibility is that the intial deflagration wave becomes a detonation wave as it travels outward in the star (Niemeyer 1995a, Niemeyer 1995b, Niemeyer 1997). Another possibility is that the inital deflagration wave fails when it reaches the outer part of the star, leading to recollapse of the white dwarf and nearly complete mixing of the nuclear fuel, followed by detonation (Blinnikov and Kohkhlov 1987, Boisseau et al. 1996, Khokhlov 1995, Khokhlov et al. 1997). In either case, much of the white dwarf is incinerated to iron-peak nuclei, and the white dwarf is blown apart.

In all three of these phenomena, it is highly likely that ignition of the nuclear fuel occurs at a single point, due to the long time scale over which the nuclear fuel is accreted and the extremely short time scale on which the thermonuclear runaway occurs, and that the burning propagates around the surface of the compact star via a detonation wave in the case of x-ray bursts, a deflagration wave in the case of novae, and a deflagration wave, a detonation wave, or a combination of both in the case of Type I supernovae.

The strong magnetic fields (B ~ 106 - 109 G) of white dwarfs and the superstrong magnetic fields (B ~ 109 - 1012 G) of neutron stars are capable of funneling the flow of accreting matter onto the magnetic polar caps of the star. The accreted matter, which constitutes the nuclear fuel, may or may not then spread over the surface of the star before ignition occurs, depending on the strength of the magnetic field and the depth in the envelope at which ignition occurs. In the case of magnetic neutron stars, but probably not magnetic white dwarfs, the three-dimensional character of the problem is compounded by the effects of the magnetic field on opacities (Lamb et al. 1996). In the case of both neutron stars and white dwarfs, it is compounded by the inhibition of fuel transport and mixing, due to the magnetic field.

Thus, the thermonuclear flash problem is inherently three-dimensional even in the absence of a magnetic field. Fully three-dimensional calculations are only now starting to be done; with the exception of a few recent studies (see, e.g., Fryxell and Woosley 1982a, Fryxell and Woosley 1982b, Shankar at al. 1992, Shankar and Arnett 1994, Livne 1993, Glasner et al. 1995, Glasner et al. 1997, Boisseau et al. 1996}), most existing calculations have been one-dimensional. The reason is that astrophysical thermonuclear flashes involve a variety of distinct and complex nonlinear physical processes which have widely disparate length and time scales. Because a number of the physical processes involved are inherently three-dimensional, even the results of the two-dimensional models are highly suspect.

To make matters worse, observational information exists only on the results of the flashes, rather than on the details of the flashes themselves. Thus, the astrophysical thermonuclear flash problem has many of the same features that the DOE Science-based Stockpile Stewardship program faces: The physics is highly varied and complex; the range of spatial and temporal scales that must be treated is huge; and the ultimate physical system being modeled is not directly accessible to experiment, or even to observation. The daunting computational challenges are also similar, and validation closely coupled to the computations is essential in order to know whether the ultimate results are at all correct.

A sequence of important physics, computational, and computer science questions must be answered in order to develop a complete theory of astrophysical thermonuclear flashes. These are discussed below, starting first with the astrophysical and physics challenges.

How does surface accretion occur? -- The accretion of material onto the surfaces of white dwarfs and neutron stars is an enormously complex problem, whose details depend very much on the specifics of the binary system in which it occurs. We do not know exactly where the accreted material is deposited, and what its physical state is. Is the accreted material (ever) deposited uniformly over the surface, or does it preferentially gather at the magnetic poles? That is, what is the role of the magnetosphere in modulating the accretion process? If accretion occurs inhomogeneously, does the accreted matter diffuse or otherwise re-distribute over the surface of the compact star? The answers to these questions can only be established by carrying out two and three-dimensional compressible magnetohydrodynamic simulations.

How does mixing operate at the interface of the accreted layer and the star? How does `dredge-up' work? -- Once the accreted matter has been deposited onto the white dwarf, it is crucial to determine whether or not there is substantial mixing between the accreted envelope and the core of the star. This question emerges both during the accretion phase, and at the time when the accreted envelope begins to convect; in the latter case, convection undershoot may dredge up material from the white dwarf and mix it into the accreted hydrogen envelope (Truran 1982, Shara 1989). Mixing is critical because without the addition of the intermediate-mass elements from the white dwarf itself, the hydrogen burning will be too slow to produce a nova. In addition, it would not be possible to produce the overabundances of intermediate-mass nuclei that are observed in the ejected envelope, which forms the nova nebula. Reliable calculations of this mixing do not exist.

In the case of a neutron star, essentially no mixing of this sort is expected to occur, both because even a weak magnetic field is sufficient to cause the accreting matter to flow radially onto the star, and because the convection zone is strongly inhibited from penetrating into the core of the star by the very large change in mean molecular weight in the burning shell.

How does the nuclear fuel ignite? -- In each case, ignition is preceded by a very slow buildup of accreted material on the surface of the compact star. During this process, the density and temperature at the bottom of the accreted nuclear fuel -- and consequently the burning rate -- gradually increase. Eventually, sufficient heat is generated that the material above the ignition site becomes convectively unstable (Truran 1982, Taam et al. 1985, Shankar et al. 1992, Shankar and Arnett 1994). The convection has two important consequences: It mixes fresh fuel into the site of the runaway, keeping the burning rate high for a longer period of time, and transports heat outward, thus enlarging the radial extent of the runaway. This mixing is fundamentally not understood.

How does the transition from a point runaway to a propagating burning front occur? -- In x-ray bursts and novae, the burning front is a subsonic deflagration wave (Bildsten 95, Glasner et al. 1995, Glasner et al. 1997}. However, if the burning rate is sufficiently large, it is possible to produce a detonation wave. This is thought to occur in Type Ia supernova (Boisseau et al. 1996, Khokhlov 1995, Khokhlov et al. 1997, Niemeyer 1995, Niemeyer and Hillebrandt 1995, Niemeyer and Woosley 1997}. How these differences in behavior come about is not understood.

What is the speed of the burning front? -- Detonation and deflagration waves are distinctly different in their computational difficulty. Computing the speed of a detonation wave is relatively easy, and in most cases, the structure of the front can be ignored and treated as a simple discontinuity. (The exception arises if -- in some cases -- the front develops a cellular structure (Williams 1985), which may have an effect on the speed of the front and especially on the resulting nucleosynthesis; if this effect is important, extremely high spatial resolution in the front is required.). In contrast, simulating deflagration waves is always much trickier. Unlike the case of detonation, determining the speed of a laminar deflagration wave always requires resolving the structure of the front. For novae, this is not particularly difficult, since the width of the burning front is comparable to the width of the accreted envelope (Fryxell and Woosley 1982). However, for Type I supernovae, the width of the laminar deflagration front can be more than ten orders of magnitude smaller than the radius of the star (Khokhlov 1995, Khokhlov et al. 1997). In principle, the speed of the front can be determined in this case from high-resolution one-dimensional simulations (Timmes and Woosley 1992, Timmes and Woosley 1994), and the result can then be treated as a parameter in the three-dimensional simulations. How well this can be done has not been established.

How does mixing occur at deflagration fronts? -- The actual physics at deflagration fronts is very likely not well-described by laminar deflagration, making the problem of front tracking much harder. That is, due to buoyancy-driven instabilities in the burning region and to Kelvin-Helmholtz instabilities along the burning front, the propagation speed may be dominated by small-scale turbulent mixing and energy transport (Khokhlov 1995, Khokhlov et al. 1997). These processes must be studied using high-resolution multidimensional calculations. Furthermore, especially in the supernova problem, the flame front is Rayleigh-Taylor unstable, which results in a large-scale wrinkling of the flame front. This dramatically increases the surface area of the front and therefore the burning rate. Thus, the evolution of the nuclear burning is highly dependent on physical processes which are extremely difficult to treat correctly.

What are the effects of magnetic fields on the nuclear burning? -- One complication in thermonuclear flashes on magnetic compact stars is the unequal depth of nuclear fuel over the stellar surface, due to the funnelling of accreted matter onto the magnetic polar caps. Ignition is likley to occur at one of the polar caps, due to the larger depth of accreted matter there, followed by propagation of the burning front around the surface of the star. This deflagration wave may rapidly die out, due to the shallower depth of the accreted matter away from the magnetic polar cap, or may reach the opposite magnetic polar cap, igniting the accreted matter that is pooled there. Another complication is determining how the magnetic field alters the velocity of the burning front. If the field is sufficiently strong, it can prevent the development of convective motions and turbulence, significantly slowing the propagation of the front. Even in cases where the field seems too weak to have much effect, the convective eddies could wind up the field until its strength becomes significant; that is, even initially weak fields may ultimately have major effects on local transport processes (Cattaneo et al. 1994).

Coupling between complex physics and computational strategy. -- The complexity and variety of the physical effects in stellar thermonuclear burning necessitates an intimate connection between the specifics of the physics (e.g., where the burning takes place) and the choice of methodology used to solve the problem. Take burning fronts as a prototype: While the deflagration wave which forms in a nova is quite broad, and can be resolved on a grid using no special techniques (although the calculation can be made much more efficient by using adaptive mesh refinement), this is not the case for Type Ia supernovae: In this latter case, there is no hope of resolving the structure of the front in a simulation of the entire star. The only reasonable approach is to do a high-resolution simulation of a small section of the burning front, in order to obtain its speed, then use the result as a parameter in the full model, combined with a front tracking method. Another potential complication in Type Ia supernovae is that in some models, the burning front starts as a deflagration wave but switches to a detonation wave in the outer layers of the star (Khokhlov 1995, Khokhlov et al. 1997, Niemeyer 95, Niemeyer and Hillebrandt 1995, Niemeyer and Woosley 1997). In this case, one must carry out a detailed study of the transition to detonation.

Similarly, simple estimates of the burning front velocity for novae vary by orders of magnitude, depending on the assumptions that are made about the effects of turbulence. However, the speed of the deflagration front is likely to be highly subsonic; indeed, the propagation of the front is so slow that the use of the explicit schemes that can be successfully used to model (for example) detonation waves is not sensible. It is necessary, instead, to use an implicit hydrodynamic scheme to follow the propagation of the burning front a significant distance around the star. The issue then arises of how such a subsonic front can propagate around the star before the expansion of the envelope quenches it; this problem requires one to model the entire surface layer of the star, rather than just a piece of it.


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