Next: 7. Kilobyte Needles Up: No Title Previous: 5. The Software

6. The Physics Barrier: Recipes for Stellar Evolution

What will happen when a W Uma contact binary encounters another double star, say a red giant in orbit around a black hole (`Bambi meets Godzilla'), after the four stars begin a chaotic four-body dance? It is not inconceivable, in the dense environment of a post-collapse cluster core, that an unrelated triple system will saunter by, say in the form of a white dwarf closely orbiting a neutron star, having just captured a main-sequence star in a wide elliptic orbit.

Farfetched, such a scenario, of a tightly-interacting seven-body system? Not really. With stars in a globular cluster core, during a post-collapse period of years, and a core crossing time of years, occasional traffic jams like the one sketched above are bound to happen.

A glance at the number of free parameters involved in such a complex encounter will dispel any thought of an approach based on some type of table lookup. Tabulating (or giving functional fits to) equal-mass three-body encounters in the point-mass limit is certainly doable (Heggie &Hut 1993), but in the much more complex cases such as the one sketched above such an approach simply won't work. To avoid our computer code giving up in despair, we are aiming at constructing a hierarchical set of recipes, which hopefully will handle encounters of the type sketched above.

But, one cannot help wondering, do we really need to model a globular cluster on this level of detail? The answer is simple: we do. The reason is that there is no simpler simulation worth doing, because of the large gap between an equal-mass point-particle simulation of cluster evolution (unrealistic, but at least consistent), and a more realistic simulation. This is an important point, not generally appreciated. Let me be more specific.


As soon as we introduce a mass spectrum in a star cluster simulation, we will see that the heavier stars start sinking toward the center, on the dynamical friction time scale, shorter than the two-body relaxation time by a factor proportional to the mass of individual heavy stars. The reason is that relaxation tends toward equipartition of energy, which implies that heavier stars will move more slowly and therefore gather at the bottom of the cluster potential well.

If stars would live forever, there would be a large overconcentration of heavy stars in the core of a star cluster. However, in reality there is a important counter-effect: heavy stars burn up much faster than lighter ones. They may or may not leave degenerate remnants, that may or may not be heavier than the average stellar mass in the cluster (a quantity that also decreases in time). Clearly, it would be grossly unrealistic to introduce a mass spectrum without removing most of the mass of the heaviest stars on the time scale of their evolution off the main sequence and past the giant branches.

Another reason for introducing finite life times for stars comes from abandoning the very restrictive point mass model for stars. As soon as we do that, giving our stars a finite radius will give rise to stellar collisions. The heavier stars produced in the collision of two turn-off stars will burn up in one or two billion years. Again, we have to take this into account to be consistent, especially since the merger products themselves are prime candidates for further merging collisions.

The need to let many stars shed most of their mass, together with the fact that most of the energy in a globular cluster is locked up in binaries, poses a formidable consistency problem. Since binaries play a central role in cluster dynamics, consistency requires that we follow their complex stellar evolution, which involves mass overflow (which can be stable or unstable, and take place on dynamical or thermal or nuclear time scales) and the possibility of a phase of common-envelope evolution. On top of all that, we will have to find simple recipes for the hydrodynamic effects occurring in three-body and four-body reactions, and in occasional reactions, as indicated above.

To sum up: there does not seem to be a half-way stopping point, at which we can expect to carry out consistent cluster evolution simulations. Either we study the interesting but unrealistic mathematical-physics problem of an equal-mass point particle model, or we opt for the realistic model with some set of stellar-evolution bells and whistles. The only question is: what is the simplest set that is still consistent?

Getting Started

Another way of posing the question is: how to mimic some form of stellar evolution that is utterly simple but not totally silly. We would be more than happy with a simple toy-model for starters, something which makes errors of factors-of-a-few in many places, without being altogether ridiculous (no order-of-magnitude errors). From there on, we can then make further progress through a series of closely placed stepping stones, by further improving each of the many ingredients in the recipes hinted at above.

However, to build a not-altogether-silly toy model is far from trivial. The main problem is that a simulation with 100,000 stars will give rise to so many different types of interactions that human intervention will become impractical. The code will have to contain some rudimentary knowledge about each foreseeable (and probably as yet unforeseen!) encounter.

Once a minimum treatment of stellar evolution effects is included in future simulations, we have to face the question of the extent to which the initial rudimentary modeling can be improved. Here the prospects may well be limited, due to fundamental uncertainties posed by such processes as common-envelope evolution. Until detailed three-dimensional hydrodynamical modeling (including a full radiative treatment!) becomes available for such cases, there seems to be little reason to extend our treatment much beyond the simplest type of consistent implementation.

The scope for stellar evolution modeling thus seems well-determined by limitations to the allowed complexity on either end of the scale. I am confident that in a couple years time we will succeed in an implementation of this rather well-determined set of recipes. By then, the full wealth of observations of X-ray binaries, millisecond pulsars and blue stragglers can be brought in, and compared with our simulations. This will finally allow us to obtain a coherent picture of the so-far-elusive quantitative aspects of the structure and evolution of globular clusters.

Next: 7. Kilobyte Needles Up: No Title Previous: 5. The Software

Thu Feb 24 00:52:57 EST 1994