Three-Body Problem: Encounters

The gravitational three-body problem has been called the oldest unsolved problem in mathematical physics. Unlike the two-body problem, there is no closed analytical solution and we have to use numerical orbit integrations to determine the evolution of a typical three-body system. Most of those are unstable, and decay either into three separate stars moving away to infinity, or into a binary star and a single star. There are some stable configurations, most of which have been known for centuries through the work of some of the most famous mathematical physicists of the eighteenth and nineteenth centuries. Interestingly, and completed unexpected, a few years ago a new category of stable three-body orbits has been discovered in which the three stars chase each other in a figure-eight orbit. Unfortunately, it is unlikely that there is even one such system in our galaxy, given the low formation rate; see the article A new outcome of binary-binary scattering, by Heggie, DC. 2000, MNRAS, 318, L61-63.

The sun can shine for billions of years because nuclear reactions deep in its interior generate the energy that is lost through the sun's radiation at its surface. On a completely different scale but in an analogous way, stars are lost from the `surface' of a star cluster by `evaporation', and there is a similar need to replenish the energy in the central regions. In fact, the mechanism is remarkably similar in both cases: the sun burns hydrogen through slow nuclear fusion into helium, while star clusters `burn' single stars through a kind of gravitational fusion into binary stars.

In the eighties, it was impossible to model the evolution of a star cluster with more than a few thousand stars through direct N-body calculations. In those days, the best one could do was to approximate the cluster by various means, such as Fokker-Planck techniques or conducting gas sphere models. However, such approximations were only accurate for two-body relaxation effects, and did not include three-body scattering effects. In order to model these, the very energy sources of cluster evolution, it is useful to have estimates for cross sections and reaction rates such gravitational scattering processes. Recetnly, advances in computer speed, as well as the development of special-purpose computers have made it possible to start simulating the full history of a modest-size globular star cluster. However, we are still interested in using scattering rates to analyze the outcomes of those large simulations, in order to gain an understanding of the underlying microphysics.

Here is a list of the papers I have written with my co-authors, describing our research in gravitational scattering. For a brief discussion of the techniques used, see my web page on intelligent tools.

• Binary - Single Star Scattering. I. Numerical Experiments for Equal Masses, by Hut, P. & Bahcall, J.N., 1983, Astrophys. J. 268, 319-341.
• Binary - Single Star Scattering. II. Analytic Approximations for High Velocity, by Hut, P., 1983, Astrophys. J. 268, 342-355.
• Binary - Single Star Scattering. III. Numerical Experiments for Equal Mass Hard Binaries, by Hut, P., 1993, Astrophys. J. 403, 256-270.
• Binary - Single Star Scattering. IV. Analytic Approximations and Fitting Formulae, by Heggie, D.C. & Hut, P., 1993, Astrophys. J. Suppl. 85, 347-409.
• Binary - Single Star Scattering. V. Steady State Binary Distribution in a Homogeneous Static Background of Single Stars, by Goodman, J. & Hut, P., 1993, Astrophys. J. 403, 271-277.
• Binary - Single Star Scattering. VI. Automatic Determination of Interaction Cross Sections, by McMillan, S. & Hut, P., 1996, Astrophys. J. 467, 348-358.
• Binary - Single Star Scattering. VII. Hard Binary Exchange Cross Sections for Arbitrary Mass Ratios: Numerical Results and Semi-Analytic Fits, by Heggie, D.C., Hut, P. & McMillan, S., 1996, Astrophys. J. 467, 359-369.
• Binaries as a Heat Source in Stellar Dynamics: Release of Binding Energy, by Hut, P., 1983, Astrophys. J. Lett. 272, L29-L33.
• Hard Binary - Single Star Scattering Cross Sections for Equal Masses, by Hut, P., 1984, Astrophys. J. Suppl. 55, 301-317.
• The Topology of Three-Body Scattering, by Hut, P., 1983, Astron. J. 88, 1549-1559.
• The Three-Body Problem in Stellar Dynamics, by Hut, P., 1984, in The Big Bang and Georges Lemaitre International Symposium G. Lemaitre, ed. A. Berger (Dordrecht: Reidel), pp. 239-256.
• Rates of Collisions and Exchange Reactions in Globular Clusters, by Hut, P., 1995, in Millisecond Pulsars: a Decade of Surprise, eds. A.C. Fruchter, M. Tavani and D.C. Backer, ASP Conference Series 72, pp. 46-56.
• Dynamics and Binary (Trans)Formation in Globular Clusters, by Hut, P., 1996, in Compact Stars in Binaries, I.A.U. Symp. 165, eds. J. van Paradijs, E. Kuulkers, and E.P.J. van den Heuvel (Dordrecht: Kluwer), pp. 377-388.
• Exchange Cross Sections For Hard Binaries, by Heggie, D.C., Hut, P. & McMillan, S., 1996 in Dynamical Evolution of Star Clusters, I.A.U. Symp. 174, eds. P. Hut and J. Makino (Dordrecht: Kluwer), pp. 371-372.
• Star Cluster Ecology II: Binary Evolution with Single-Star Encounters, by Portegies Zwart, S.F., Hut, P., McMillan, S.L.W. & Verbunt, F., 1997 Astron. & Astrophys. 328, 143-157 (available in preprint form as astro-ph/9706090).

Most stars in our galaxy are part of either a binary system or a more complex multiple system of stars, such as a triple or quadruple or even larger collection of stars on stable hierarchical orbits. It is therefore natural to think about the possible presence of triple stars when encountering unsolved problems in stellar dynamics. Here are some papers I have written on that topic.

• Precession and System Parameters in Early-Type Binary Models for SS 433, by Hut, P. & van den Heuvel, E.P.J., 1981, Astron. Astrophys. 94, 327-332.
• A Phenomenological Triple Star Scenario for SS433, by Fabian, A.C., Eggleton, P.P., Hut, P. & Pringle, J.E., 1986, Astrophys. J. 305, 333-335.
• Binary-Binary Interactions and the Formation of the PSR B1620-26 Triple System in M4, by Rasio, F.R., McMillan, S. & Hut, P., 1995, Astrophys. J. Lett. 438, L33-L36.
• Predictions for Triple Stars and Pulsar Triples in Star Clusters, by Trenti, M., Ransom, S., Hut, P. & Heggie, D.C., 2007, Mon. Not. R. astr. Soc.xxx, xxx-xxx.

In thermal equilibrium, wide binaries are formed and destroyed constantly. To derive the equilibrium distribution of such double stars, the simplest approach is to use the correspondence principle. Starting from the known distribution of energy levels in the hydrogen atom, I derived the equilibrium distribution of binary stars in the paper Binary Formation and Interaction with Field Stars, by Hut, P., 1985, in Dynamics of Star Clusters, I.A.U. Symp. 113, eds. J. Goodman and P. Hut (Dordrecht: Reidel), pp. 231-249.

For the time-dependent evolution of this distribution function, see the paper Binary - Single Star Scattering. V. Steady State Binary Distribution in a Homogeneous Static Background of Single Stars, by Goodman, J. & Hut, P., 1993, Astrophys. J. 403, 271-277.

In many cases, even slight deviations in the way we set up a three-body scattering experiment will lead to a complete different outcome. The extreme sensitivity to initial conditions resembles the occurrence of mathematical chaos. We explored this in our paper Round-off Sensitivity in the N-Body Problem, by Dejonghe, H. & Hut, P., 1986, in The Use of Supercomputers in Stellar Dynamics, eds. P. Hut and S. McMillan (Springer), pp. 212-218.