Million-Body Problem: Star Clusters

For decades, the holy grail of star cluster simulations has been to perform a direct N-body calculation for the ten-billion year history of a globular star cluster, modeling the stars on a one-to-one basis. This implies solving the 1,000,000-body problem with staggering discrepancies in length and time scales. Neutronstars and black holes can form binary systems or ondergo unbound encounters with relevant time scales of milliseconds and length scales of kilometers. Given that globular clusters have dimensions of hundreds of light years and ages of more than ten billion years, the range in length scales spans more than fifteen orders of magnitude while the range in time scales exceeds twenty orders of magnitude. All this puts severe constraints on both hardware and software used in such simulations.

We are now in a position to carry out these types of realistic simulations. Progress may have seemed slow: we saw simulations with tens of stars in the sixties, hundreds in the seventies, thousands in the eighties, and finally tens of thousands in the nineties. The reason for this modest advance is that the computational cost of a full star cluster evolution scales proportional to the third power of the number of particles: two powers for the inter-particle interactions at each time step, and an additional power because the increase in particle number slows down the heat flow time by making two-body relaxation less effective.

Following the previous trend, we expect to see routine simulations of star clusters using several hundred of thousands of particles in the twothousandzeroes. The main challenge will now be to include enough physics to make these simulations sufficiently realistic to compare them in detail with the increasingly comprehensive observations of star clusters.

Dynamical Evolution

The first challenge is to model and interpret the rich set of phenomena already present in the purely gravitational N-body problem. Surprisingly, each increase of an order of magnitude in the number of stars has shown a qualitative chance in the properties of star cluster simulations, even though the underlying Newtonian dynamics is completely scale-free. There are various phenomena responsible for this, such as the occurrence of instabilities of a gravothermal nature, and the fact that binary star properties scale differently from single star properties. Here is a list of some of my papers that address problems in pure gravity:

Primordial Binaries

In the late eighties, based on various types of observations, it became clear that for most globular star clusters a significant fraction of their stars were part of primordial binary systems. Until then, almost all simulations of star cluster evolution had started with a collection of single stars, so it was time to go back to the drawing boards. Our first sketch of the implications for globular cluster evolution appeared in the paper:

We then carried out several 1000-body simulations containing significant fractions of primordial binaries:

Quite a bit later, we extended these calculations to include much larger number of particles, and also a much larger number of runs:

We published a brief review as:

while we also used more approximate methods, in:

Finally, we published a detailed survey review of observational and well as theoretical developments concerning primordial binaries in:

  • Binaries in Globular Clusters, by Hut, P., McMillan, S., Goodman, J., Mateo, M., Phinney, S., Pryor, T., Richer, H., Verbunt, F. & Weinberg, M., 1992, P.A.S.P. 104, 981-1034.

X-Ray Binaries

It was the discovery of an abundance of X-ray sources in globular clusters, in the early seventies, that changed our picture of those clusters. Rather than just being old and boring remnants of the early formation phases of our galactic environment, globular clusters were seen to be laboratories for rather unusual experiments in stellar evolution. It became clear that the high stellar densities in the centers of the clusters were responsible for the formation of X-ray sources, although many of the details long remained uncertain. Today, too, properties of globular cluster X-ray sources remain among the prime diagnostics for their evolutionary histories. Here are some papers we wrote on the subject:

Here is a summary of strong observational evidence that X-ray sources in globular clusters are indeed formed through dynamical encounters:

Following up on this paper, we showed that for cataclysmic variables, too, the majority has a dynamical origin:

  • Dynamical Formation of Close Binaries in Globular Clusters II: Cataclysmic Variables, by Pooley, D. & Hut, P.; 2006, Astrophys. J. Lett. xxx, xxx-xxx.

    Stellar Evolution: Ecology

    The most important progress in star cluster evolution in the nineties, from an astrophysical point of view, was the introduction of stellar evolution effects directly in the simulations. Here are some papers we wrote on these type of more realistic simulations:

    Evolution: a MODEST approach

    In 2002, we started a new MODEST initiative (the name stems from MOdeling DEnse STellar systems). Through a series of twice-yearly workshops, we bring together experts in in stellar evolution, stellar dynamics, and stellar hydrodynamics. Our goal is to develop a software framework to enable us combine in one simulation existing computer codes in these three areas. For a summary of the first two workshops, MODEST-1 and MODEST-2, see:

    • MODEST-1: Integrating Stellar Evolution and Stellar Dynamics, by Hut, P., Shara, M.M., Aarseth, S.J., Klessen, R.S., Lombardi, J.C., Makino, J., McMillan, S., Pols, O.R., Teuben, P.J., Webbink, R.F.; 2002, New Astronomy xxx, xxx-xxx (available in preprint form as astro-ph/0207318).
    • MODEST-2: A Summary, by Sills, A., Deiters, S., Eggleton, P., Freitag, M., Giersz, M., Heggie, D., Hurley, J., Hut, P., Ivanova, N., Klessen, R.S., Kroupa, P., Lombardi, J.C., McMillan, S., Portegies Zwart, S., Zinnecker, H., 2003, New Astronomy xxx, Lxx-Lxx (available in preprint form as astro-ph/0301478).

    A shorter review of the first three workshops, MODEST-1 through MODEST-3 can be found in the General Assembly proceedings of 2003:

    • MODEST: modeling stellar evolution and (hydro)dynamics, by Hut, P., 2003, Highlights of Astronomy 13, xx-xx (available in preprint form as astro-ph/0309395).

    Another short review appeared in the General Assembly proceedings of 2006:

    • Modeling Dense Stellar Systems: Background, by Hut, P., 2006, Highlights of Astronomy 14, xx-xx (available in preprint form as astro-ph/0610223).

    In the same proceedings, we published a more general review:

    • Neutron Stars and Black Holes in Star Clusters, by Rasio, F.A., Baumgardt, H., Corongiu, A., D'Antona, F., Fabbiano, G., Fregeau, J.M., Gebhardt, K., Heinke, C.O., Hut, P., Ivanova, N., Maccarone, T.J., Ransom, S.M., Webb, N.A. 2007, Highlights of Astronomy 14, xx-xx (available in preprint form as astro-ph/0611615).

    Here is a review that advocates the notion of creating MODEST centers:

    • Dense Stellar Systems as Laboratories for Fundamental Physics, by Hut, P.; 2006, in A Life With Stars eds. L. Kaper, M. van der Klis and R. Wijers [Amsterdam: Elsevier] (available in preprint form as astro-ph/0601232).

    And here is a review that focuses on software issues and the question of how theorists can collaborate in creating robust computer codes:

    • Virtual Laboratories, by Hut, P.; 2007, Prog. Theor. Phys. xxx, xxx-xxx. (available in preprint form as astro-ph/0610222).

    Yet another review forms a general introduction to the study of dense stellar systems:

    • Modeling Dense Stellar Systems, by Hut, P., Mineshige, S., Heggie, D.C. & Makino, J. 2007, Prog. Theor. Phys. Suppl. xxx, xxx-xxx. (available in preprint form as

    Ultimate Fate

    While it is interesting and important to model the histories of individual globular clusters, we are also interested in the evolution of the globular cluster system as a whole. While our galaxy currently has only a little over a hundred globulars left, there may well have been far more originally, with many of them having been destroyed by a variety of mechanisms. We have addressed this issue in the following papers:

    Galactic Nuclei and Intermediate Mass Black Holes

    Interesting as the study of globular clusters can be for its own sake, many of the lessons learned there can be applied to the study of galactic nuclei, even more exciting systems often harboring massive black holes of millions or sometimes even billions of solar masses. In addition, evidence has been growing for the possibility that some star clusters may contain an intermediate-mass black hole in their center. Here are some of our contributions:

    For a short popular summary of the last paper, see a New Scientist article. A second paper in which we discuss the possible presence of an Intermediate Mass Black Hole (IMBH), following up on the previous one, is:

    See also:

    • Black Holes in Massive Star Clusters, by McMillan, S., Baumgardt, H., Portegies Zwart S., Hut, P. & Makino, J.; 2005, in Formation and Evolution of Massive Young Star Clusters, eds. H.J.G.L.M. Lamers, A. Nota & L.J. Smith, pp. xx-xx (available in preprint form as astro-ph/0411166).

    In the following paper, we have simulated the inner 100pc of the Milky-Way Galaxy to study the formation and evolution of the population of star clusters and intermediate mass black holes. We predict that region within about 10 parsec of the central supermassive black hole is populated by about 50 Intermediate-mass black holes of some 1000 solar masses.

    • The ecology of star clusters and intermediate mass black holes in the Galactic bulge, by Portegies Zwart S., Baumgardt, H., McMillan, S., Makino, J. & Hut, P.; 2006, Astrophys. J. 641, 319-326 (available in preprint form as astro-ph/0511397).

    The following articles investigates a claim that the globular clusters M15 and G1 may harbor a central black hole:

    An important question concerns the observational characteristics of a globular cluster harboring an intermediate-mass black hole. We address this question in:

    An extension of this work, in which we study the presence of primordial binaries as well as an intermediate-mass black hole can be found in

    • Primordial Binaries and Intermediate Mass Black Holes in Globular Clusters, by Trenti, M., Ardi, E., Mineshige, S. & Hut, P.; 2006, Mon. Not. R. astr. Soc. 374, 857-866 (available in preprint form as astro-ph/0610342; in an earlier version as astro-ph/0508517).

    This work was followed up in another paper, in which we derived an explicit expression for the size of the core radius as a function of the black hole mass:

    • The Core Radius of a Star Cluster Containing a Massive Black Hole, by Heggie, D.C., Hut, P., Mineshige, S., Makino, J. & Baumgardt, H. 2007, PASJ Letters xxx, xxx-xxx (available in preprint form as astro-ph/0611950).


    Here are some early reviews of globular cluster dynamics:

    • Dynamical Evolution of Globular Clusters, by Elson, R., Hut, P. & Inagaki, S., 1987, Ann. Rev. Astron. Astrophys. 25, 565-601.
    • New Directions in Globular Cluster Modeling, by Hut, P., 1992, in X-ray Binaries and Recycled Pulsars, eds. E.P.J. van den Heuvel & S.A.Rappaport (Dordrecht: Kluwer Acad. Publ.), pp. 317-348.
    • Star Clusters, Globular, Gravothermal Instability, by Hut, P., 1992, in The Astronomy and Astrophysics Encyclopedia, ed. S.P. Maran (New York: Van Nostrand Reinhold), pp. 675-677.

    A much more detailed discussion can be found in our new book:

    Although some of the technical parts are on the level of a graduate text, the bulk of the book is written so as to be accessible to interested undergraduates, as well as a wide range of mathematicians, physicists, computer scientists and others who are interested to see an overview of fun aspects of gravitational thermodynamics. Here are previews of

    Jun Makino and I are currently working on a series of books on simulations of star clusters, titled

    Michele Trenti and I have written a review article for Scholarpedia, a more scholarly version of Wikipedia: