Dynamical modeling

MPE   OPINAS    Dynamics Group

The main goal of dynamical modeling is to recover the distribution of a galaxy's stars in phase space from observational data, such as the surface brightness distribution and the mean line-of-sight velocity field, and to constrain the distribution of mass and the structure of stellar orbits. These are interesting because the mass distribution gives some information on the distribution of dark matter and the mass of the central black hole, whereas the orbit structure is likely to have preserved some record of the formation of the galaxy since the phase space density is conserved.

Evolution of collisionless systems

The evolution of a stellar system, where the stars can be considered as point masses, is determined by the mutual gravitational forces of the stars. The collision rate of a star in a galaxy with equal mass stars and a constant total mass is inversely proportional to the total number of stars. This can be understood as follows: If the total number of stars is doubled, then also the number of encounters is doubled, but the strength of a single scattering event is reduced by a factor of four, since the mass of each star is halved and the gravitational force is proportional to the square of the mass (cf. Hockney & Eastwood, 1988). One can show that for galaxies, which typically have N ~ 1011 stars, encounters are unimportant. Hence, galaxies can be considered as collisionless systems (e.g. Binney & Tremaine, 1987) and the motions of the stars are governed by the smooth gravitational potential, generated by the entire system. Such collisionless systems can be completely described by the collisonless Boltzmann equation.

The collisionless Boltzmann equation

A collisionless stellar system can be described by its phase-space distribution function (DF) f(x,v,t), which gives the density of stars in the six-dimensional phase space of x,v and is positive everywhere. For example, the surface brightness μ and the mean line-of-sight velocity vlos are then given by integrals:

Eqn 1

where the line-of-sight is assumed along the z-direction. The evolution of the DF under the influence of the total gravitational potential Φ, is determined by the collisionless Boltzmann equation (CBE) (cf. Binney & Tremaine, 1987):

Eqn 2

which follows from the conservation of stars in phase-space. The CBE states that the phase-space flow is incompressible. The total gravitational potential Φ(x,t) is generated by the combined stellar mass and dark matter distributions and is given by

Eqn 3

where the stellar potential Φ is related to the DF via Poisson's equation

Eqn 4

with the volume density

Eqn 5

The Poisson equation together with the CBE are the fundamental equations in stellar dynamics. Once a solution of these equations has been found, all the interesting information about the system can be extracted as illustrated in equations (1).

Jeans theorem

Often one is interested in solutions of the CBE that describe stars moving in a gravitational potential which is constant in time, such that the DF is also constant. For such a steady-state solution (∂ f/∂ t=0) of the CBE Jeans theorem states that the DF depends on the phase-space coordinates only through the integrals of motion for the stellar orbits in the gravitational potential. On the other hand, any non-negative function of the integrals of motion is a steady-state solution of the CBE. For a steady-state system, the Jeans theorem implies, that the phase-space density is constant along individual orbits. This is the basis of many methods for constructing equilibrium models of galaxies, such as the Schwarzschild orbit superposition technique (Schwarzschild, 1979), DF-based methods or particle based methods, such as the made-to-measure algorithm (Syer & Tremaine, 1996, De Lorenzi et al., 2007).