You are here:
Home >
Activities >
Astrophysics >
ThreeBody Problem
ThreeBody Problem: Encounters
The gravitational threebody problem has been called the oldest
unsolved problem in mathematical physics. Unlike the twobody
problem, there is no closed analytical solution and we have to use
numerical orbit integrations to determine the evolution of a typical
threebody 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 threebody orbits has been discovered in which the
three stars chase each other in a
figureeight
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 binarybinary scattering,
by Heggie, DC. 2000, MNRAS, 318, L6163.
Scattering Experiments
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 Nbody
calculations. In those days, the best one could do was to approximate
the cluster by various means, such as FokkerPlanck techniques or
conducting gas sphere models. However, such approximations were only
accurate for twobody relaxation effects, and did not include
threebody 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
specialpurpose computers
have made it possible to start simulating the full history of a
modestsize 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 coauthors,
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, 319341.

Binary  Single Star Scattering. II. Analytic Approximations for High Velocity, by
Hut, P., 1983, Astrophys. J. 268, 342355.

Binary  Single Star Scattering. III.
Numerical Experiments for Equal Mass Hard Binaries, by
Hut, P., 1993, Astrophys. J. 403, 256270.

Binary  Single Star Scattering. IV.
Analytic Approximations and Fitting Formulae, by
Heggie, D.C. & Hut, P., 1993,
Astrophys. J. Suppl. 85, 347409.

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, 271277.

Binary  Single Star Scattering. VI.
Automatic Determination of Interaction Cross Sections, by
McMillan, S. & Hut, P., 1996,
Astrophys. J. 467, 348358.

Binary  Single Star Scattering. VII.
Hard Binary Exchange Cross Sections for Arbitrary Mass Ratios:
Numerical Results and SemiAnalytic Fits, by
Heggie, D.C., Hut, P. & McMillan, S., 1996,
Astrophys. J. 467, 359369.

Binaries as a Heat Source in Stellar Dynamics: Release of Binding Energy, by
Hut, P., 1983, Astrophys. J. Lett. 272, L29L33.

Hard Binary  Single Star Scattering Cross Sections for Equal Masses, by
Hut, P., 1984, Astrophys. J. Suppl. 55, 301317.

The Topology of ThreeBody Scattering, by
Hut, P., 1983,
Astron. J. 88, 15491559.

The ThreeBody 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. 239256.

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. 4656.

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. 377388.

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. 371372.

Star Cluster Ecology II: Binary Evolution with SingleStar Encounters, by
Portegies Zwart, S.F., Hut, P., McMillan, S.L.W. & Verbunt, F., 1997
Astron. & Astrophys. 328, 143157 (available in
preprint form as
astroph/9706090).
Triple Stars
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 EarlyType Binary Models for SS 433, by
Hut, P. & van den Heuvel, E.P.J., 1981, Astron.
Astrophys. 94, 327332.

A Phenomenological Triple Star Scenario for SS433, by
Fabian, A.C., Eggleton, P.P., Hut, P. & Pringle, J.E., 1986,
Astrophys. J. 305, 333335.

BinaryBinary Interactions and the Formation of the PSR B162026 Triple System in M4, by
Rasio, F.R., McMillan, S. & Hut, P., 1995,
Astrophys. J. Lett. 438, L33L36.

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, xxxxxx.
Equilibrium Distribution
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. 231249.
For the timedependent 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, 271277.
Chaos
In many cases, even slight deviations in the way we set up a
threebody 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
Roundoff Sensitivity in the NBody Problem, by
Dejonghe, H. & Hut, P., 1986, in The Use of Supercomputers in Stellar
Dynamics, eds. P. Hut and S. McMillan (Springer), pp. 212218.
Back to astrophysics.
Back to activities
or to table of contents.
Back to Piet Hut's home page.