Basic physical principles of the
Hénon
Monte Carlo method
The MC schemes rely on the following assumptions concerning the stellar
cluster under consideration:
- Statistical treatment.
The system can be treated from the point of view of (local) statistical
properties, like velocity distribution or mass function. More
technically, we assume that the notion of 1-particle distribution
function is everywhere (and at all times) significant, although
distribution functions are nevere explicitly used (unless in the
Shapiro code). This means that, in principle, we shouldn't try to use
the MC code to stuy systems in which a small number of particles
(<10) play a key role for the evolution of the whole cluster. This
is sometimes too tempting, though! Note that, although this is not
always aknowledged, this limitation is shared by direct N-body
simulations because of the extreme sensitivity of individual
trajectories to initial conditions (and to details of the numerical
integration).
- Dynamical equilibrium.
The cluster is always in dynamical equilibrium. If it were only for
dynamical processes (occuring on the time scale of the crossing time),
the cluster would preserve its structure for ever. We are here
interested in the long-term evolution of the system which is driven by
"collisional" effects. Of central importance is relaxation, i.e. the
effect of the fluctuating "graininess" of the gravitational force.
- Spherical symmetry. The
system is assumed to have perfect spherical symmetry. Note that this
does not imply the velocity distribution to be isotropic but to depend
only on the radial and tangential component of the velocity vector
(i.e. there is axial symmetry in velocity space). From Jeans theory and
assumptions 1 and 2, we know that (if all stars are identical) the
system can me described by a distribution function in phase-space which
depends on position and velocity only through the energy E and modulus
of angular momentum J.
- Diffusive relaxation. The
relaxation is treated as a diffusive process, with the classical
Chandrasekhar theory. The long-term effects on orbits of departures of
the gravitational forces from a smooth stationary potential are assumed
to be that of a large number of uncorrelated small angle hyperbolic
2-body encounters.
This list of simplifying assumptions is identical to the ones used in
almost all so-called direct Fokker-Planck (FP) methods. In particular
the
treatment of relaxation is done is the Fokker-Planck
approximation:
- An individual 2-body encounter is considered to affect the
velocity instantaneously, without change of position.
- Only 2-body encounters resulting in very small angle scatterings
are considered. With the assumption that they are uncorrelated, this
results in relaxation to produce diffusion in energy space. For a test
particle of mass M1
travelling in field of particles of mass M2 with number density n during δt on an
(initially) straight trajectory, this produces a deflection of the
direction of the velocity vector with the following statistical
properties:
The higher moments are negelected,
another aspect of the Fokker-Planck approach. ln Λ is the Coulomb
logarithm which measures how many order of magnitudes in impact
parameter contribute to relaxation. For a self-gravitating cluster, Λ
is proportional to the number of stars.
- As every logarithmic intervall of impact parameter contributes
equally, relaxation is dominated by "relatively close" encounters so
when integrateing the effects of all field particles on the trajectory
of the test particle, one uses the local properties (density, velocity
and mass distributions) of the field.
The basic difference between the MC code and direct FP methods is that
the cluster is represented as a set of particles in the MC approach
while the FP methods deal with distribution function(s). The advantage
of the particle approach is that aditional physical effects, that,
unlike diffusive relaxation, do not produce quasi-continuous change of
particle properties can be included with relative ease and
realism. Most importanty, let's mention the role of binaries
(included in the Giersz and MIT/NU codes), physical collisions between
stars (treated in Freitag code), interactions between the stars and a
central BH (Freitag), role of the rare large angle scatterings
(Freitag, unpublished)... A continuous stellar mass spectrum is handled
without difficulty by the MC codes, while the mass function has to be
discretized into 5-25 components in FP simulation (with the computing
time increasing with the number of components). On the other hand,
unlike FP models, MC simulations suffer from statistical noise despite
the use of 1-10 million particles. Also, FP codes for rotating,
axi-symmetrical clusters have been developed (Goodman thesis, Einsel
& Spurzem 99, Kim
et al 2002, see also this link),
while all MC codes heavily rely on spherical symmetry.
For more information about the dynamics of stellar clusters, look at
the references on the literature MODEST
page.
