Dynamics of gravitationally interacting systems

Pasquale LondrilloINAF-Osservatorio Astronomico-Bologna (Italy)

Email pasquale.londrillo@bo.astro.it

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 equilibriaand/orRelaxation to statistical mechanics equilibria

The debate onfine-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 forcesF(x,t)=

2) The infinite-dimensional set of characteristic orbits issampled by a finite-N-set of particles moving under anЄ(N)- softened gravitational force Є(N) →0 N→infinity

3)Theorems are available to assure convergence (in some norm)of N-body representation to the limit Vlasov-Poisson solution

1- The set of N-representative particles aremoved in time using a simplectic integrator

2- the gravitational forces and potential are computed:

a)- either directely on particles, using thetwo-body softened Green-function

b)- or by solving the Poisson equation on agrid. Using a density distribution recovered

from particle positions by some interpolation (PIC)

Numerical procedures for N- body codes

Methods a) have clear (mathematical) advantages Now even

faster than b) => Multipole expansion techniques having O(N)computational complexity

Other numerical methods to solve thecollisionless 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)