Gravitational Thermodynamics

A thermodynamic treatment of self-gravitating systems is fraught with peril: from a formal point of view, it cannot even be defined, because there is no thermodynamic limit. The technical reason that we cannot scale the system to an infinitely large size stems from the fact that gravitation is a long-range force, which implies that the gravitational potential energy grows faster than linear with the mass of the system. While intensive thermodynamic quantities like density and temperature stay constant under scaling, and extensive quantities such as energy and particle number grow linearly with mass, potential energy turns out to be a superextensive quantity, scaling like the five-thirds power of the mass. What happens when we scale a self-gravitating system in size while keeping the temperature and density constant is that it starts to fall apart into separate clumps as soon as the systems exceeds a critical mass, called the Jeans mass in astrophysics.

In practice, however, the deviations from a true thermodynamic equilibrium are often not that large, and N-body simulations show a relatively smooth behavior of the approximate thermodynamic parameters as a function of time. This has been the reason that various approximation methods, such as Fokker-Planck treatments and conducting gas sphere models, have worked so well. Here I list a few areas in gravitational thermodynamics where I have made contributions.

If you sprinkle point masses into a limited area in space, and weight for the initial transient effects to settle down, you will find that some of the point objects spontaneously form binaries (double stars, in case the points model stars). For wide binaries, the distribution of binaries is proportional to the amount of phase space volume available. It is a tedious job to compute this volume, since one has to integrate over all six Kepler orbital elements. A much easier way to derive this distribution is to use the correspondence principle between classical and quantum mechanics. Starting with the distribution of energy levels of the hydrogen atom, and taking the classical limit of extremely high-n levels, we rediscover the thermal distribution of double star orbital parameters, as I illustrated 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 tighter binaries, the Boltzmann factor has to be added, at least formally. In practice, the population of hard binaries is stationary but not static, and the ever-hardening binaries never succeed in filling up the Boltzmann distribution. See my discussion of gravitational thermal equilibrium.

I provided a unified treatment for various time scales in self-gravitating systems, from local encounter times and global crossing times to relaxation times and binary formation times, in the paper:

• Fundamental Timescales in Star Cluster Evolution, by Hut, P., 1989, in Dynamics of Dense Stellar Systems. ed. David Merritt (Cambridge University Press), pp. 229-236.

A numerical microscopic study of two-body relaxation was published in our paper:

• Relaxation in small N-Body Systems,

  McMillan, S., Casertano, S. & Hut, P., 1988, The Few Body Problem I.A.U. Colloq. 96, Turku, Finland, ed. M.J. Valtonen, (Dordrecht: Kluwer), pp. 313-317.

For a comparison between two-body relaxation and exponential divergence, see our paper:

• Orbital Divergence and Relaxation in the Gravitational N-Body Problem, by Hut, P. and Heggie, D.C., 2002, in the proceedings of the 84th Statistical Mechanics Conference (to celebrate the 65th birthdays of David Ruelle and Yasha Sinai), J. Stat. Phys xxx, xxx-xxx (available in preprint form as astro-ph/0111015).

If we enclose a self-gravitating system in a box with a sufficiently high temperature, the system as a whole will have a positive heat capacity and behave like a normal thermodynamic system. However, when the temperature (velocity dispersion) drops too far, or equivalently, if the central density becomes too high, the system will develop a negative heat capacity. At this point the central regions will start shrinking in an accelerated `collapse' on a thermal conduction time scale. This phenomenon has been dubbed `gravothermal collapse' by Donald Lynden-Bell. We have illustrated the onset of this phenomenon by presenting a linear stability analysis of the isothermal sphere in the paper:

• On Core Collapse, by Makino, J. & Hut, P., 1991, Astrophys. J. 383, 181-191.

Gravothermal collapse is not only the product of a theoretical exercise. By studying the distribution of core parameters for the globular clusters in our galaxy, we have used statistical arguments to argue that core collapse is currently going on for many of these clusters. See our paper:

• Is there Life after Core Collapse in Globular Clusters? by Cohn, H. & Hut, P., 1984, Astrophys. J. Lett.277, L45-L48.

We have presented the results of a detailed numerical investigation of the self-similar contraction phase leading up to core collapse in:

• Parameters of core-collapse, by Baumgardt, H., Heggie, D.C., Hut, P. & Makino, J. 2003, Mon. Not. R. astr. Soc. 341, 247-250 (available in preprint form as astro-ph/0301166).

An analysis based solely on two-body relaxation would predict that gravothermal collapse proceeds until the central density in a star cluster becomes infinitely high. In physics, whenever a theory predicts the occurrence of singularities, it has been a sign that other physical effects, which have been overlooked, will kick in before actual infinities are reached. The case of gravitational thermodynamics is no different. In a system of point masses, binaries will form just before core collapse. When they shrink, their negative binding energy will increase which means that they supply positive energy to their surroundings. They thus for a heating mechanism that can power post-collapse expansion. The first quantitative estimate of this effect based on detailed numerical scattering experiments was published in

• Binaries as a Heat Source in Stellar Dynamics: Release of Binding Energy, by Hut, P., 1983, Astrophys. J. Lett. 272, L29-L33.

During the re-expansion phase, new instabilities form when the number of particles is sufficiently large, greater than about 10,000. In this case, the central regions of the expanding core in a star cluster have the tendency to re-collapse and again re-expand, a phenomenon known as gravothermal core oscillations. We have studied these in detail in the following papers:

• Large Scale Calculations of Core Oscillations in Globular Clusters, by Cohn, H., Wise, M., Yoon, T., Statler, T., Ostriker, J. & Hut, P. 1986, in The Use of Supercomputers in Stellar Dynamics, eds. P. Hut and S. McMillan (Springer), pp. 206-211.
• Long-Term Evolution of Cores of Globular Clusters after Core Collapse, by Inagaki, S. & Hut, P., 1988, in The Few Body Problem I.A.U. Colloq. 96, Turku, Finland ed. M.J. Valtonen, (Dordrecht: Kluwer), pp. 319-324.
• Gravothermal Oscillations after Core Collapse in Globular Cluster Evolution, by Cohn, H., Hut P. & Wise, M., 1989, The Astrophys. J. 342, 814-822.
• Realistic models for evolving globular cluster - II. Post core collapse with a mass spectrum, by Murphy, B.W., Haldan, N & Hut, P., 1990, Mon. Not. R. astr. Soc. 245, 335-349.
• Core Oscillations in Globular Cluster Evolution: Recent Results, by Cohn, H.N., Lugger, P.M., Grabhorn, R.P., Breeden, J.L., Packard, N.H., Murphy, B.W. & Hut, P., 1991, in The Formation and Evolution of Star Clusters, A.S.P. Conference Series, ed. K. Janes, Vol. 13, 381-384.
• The Onset of Gravothermal Oscillations in Globular Cluster Evolution, by Breeden, J.L., Cohn, H.N. & Hut, P., 1994, Astrophys. J. 421, 195-205.

For a general introduction and review of physical as well as astrophysical aspects of the thermodynamics of self-gravitating systems, see:

• Gravitational Thermodynamics, by Hut, P., 1997, Complexity, 3, No. 1, pp. 38-45 (available in preprint form as astro-ph/9704286).