## Hénon Monte Carlo method for stellar cluster dynamics

On these pages, we describe biefly a class of numerical schemes used to follow the long term evolution of spherically symmetric stellar clusters. We concentrate on the algorithm invented by Michel Hénon in the early seventies because it is the inspiration of all codes presently in use. However, it should be noted that, in the past, two other, quite different, Monte Carlo (MC) techniques were developed, by Spitzer et al. and by Shapiro et al (this link).

After the pioneering work of Hénon, his method was adopted and improved in a variety of directions by Stoldolkiewicz and Giersz in Warsaw, Joshi, Rasio, Fregeau and Gürkan at the MIT and Northwestern University and Freitag and Benz at Geneva Observatory and the ARI in Heidelberg. There is presently three such codes in use across the world that have been described in published papers: the "Giersz" MC code, the "MIT/NU" code and the "Freitag" one. Although all share basic concepts they also have important differences. Consequently, we start by a presentation of the common ideas and structures and then present the features specific to each of them. Follow this link for references.

In short, the Monte Carlo method borrows both from the N-body simulations and from methods that treat the stellar cluster as a contimuum. Like in N-body simulations, the cluster is treated as a set of reprsentative particles (but they are spherical shells of matter, not point-masses and their orbital motion is not explicitelly followed). Like in Fokker-Planck methods, dynamical equilibrium in a smooth (and spherical) potential is assumed and the secular evolution of the system (due to relaxation, collisions, stellar evolution, binaries and/or interaction with a central BH) is investigated. The treatment of relaxation is based on the same assumptions that underly the Fokker-Planck equation.

Compared to Fokker-Planck and gaseous methods, the MC scheme, being particle-based, allows easier and more realistic account of a variety of physical processes. It is also slower and more noisy. Compared to the direct N-body methods, it is much faster but more approximate. On a standard single-processor computer, one can simulate the evolution of a cluster over a few relaxation times using 0.5 to a few million particles in a few CPU-days or weeks. The required CPU time scales with the number of particles N like N*ln(N) (for a given number of relaxation times), while the scaling is Nk with k of order 2-3 for direct N-body codes.

Physical principles underlying the Monte Carlo approach.

Numerical implementation.

My MC code, nammed ME(SSY)**2 (Pau's suggestion), is publicly available HERE

Other information on the web.

Evolution of the central region of a spherical cluster of stars, as computed with the Monte Carlo algorithm. For these figures, realised for the autumn 2003 issue of inSiDE newsletter, the angular position of each star on the spherical shell dealt with by the Monte Carlo code (i.e. a "particle") has been picked up randomly for the purpose of visualisation. All the stars within a slice containing the centre are depicted. For clarity, their radii are highly magnified. The white circles represent spheres containing 1, 3 and 10% of the total cluster mass. Note how the massive, large stars concentrate to the centre, through "mass segregation".  See  Gürkan, Freitag & Rasio 04

Maintained by Marc Freitag; comments and contributions welcome. Last Update: 2004-06-01