Transcript Dynamics of gravitationally interacting systems

Dynamics of gravitationally
interacting systems
Pasquale Londrillo
INAF-Osservatorio Astronomico-Bologna (Italy)
Email [email protected]
Trying to look at possible intersections with beam-dynamics
==
seemingly none
Fenomenology
• Dark-Matter dominated Large Scale
structures => collisionless regime
• The Galaxies => collisionless regime
• The Globular Clusters => collisionality
important
Elliptical Galaxies
Violent relaxation (Lynden-Bell (1967)
Relaxation to dynamical large-scale equilibria
and/or
Relaxation to statistical mechanics equilibria
The debate on
fine-graining and coarse-graining
The Numerical approach:
The Particle method
Well posed Mathematical set-up:
1) the Vlasov-Poisson equation for f(x,v,t) is replaced
by a lagrangian fluid ot orbits [x(t),v(t)] preserving f
f(x,v,t)=f(x(t),v(t)] and moving under the collective forces
F(x,t)=
2) The infinite-dimensional set of characteristic orbits is
sampled by a finite-N-set of particles moving under an
Є(N)- softened gravitational force
Є(N) →0 N→infinity
3)Theorems are available to assure convergence (in some nor
of N-body representation to the limit Vlasov-Poisson solution
Numerical procedures for N- body codes
1- The set of N-representative particles are
moved in time using a simplectic integrator
2- the gravitational forces and potential are computed:
a)- either directely on particles, using the
two-body softened Green-function
b)- or by solving the Poisson equation on a
grid. Using a density distribution recovered
from particle positions by some interpolation (PIC)
Methods a) have clear (mathematical) advantages Now even
faster than b) => Multipole expansion techniques having O(N)
computational complexity
Other numerical methods to solve the
collisionless Vlasov-Poisson equation
Methods to solve f(x,v,t) directely on a six-dimensional eulerian grid
Proposed mainly for Plasma-Physics computations (where
Electromagnetic fields can be represented on a grid in a natural way)