2. The focusing phenomenon
The epicycle approximation allows us to study, in an analytical way, the evolution of an unbound system of stars under the influence of the galactic potential. Under such approximation, the equations of motion of stars can be expressed as:
in the coordinate system () centered at the current position of the Sun. points towards the galactic center (GC), is a linear coordinate measured along a circumference of radius R (galactocentric distance of the Sun) and positive in the sense of the galactic rotation (GR), and points towards the north galactic pole (NGP). is the epicyclic frequency , is the vertical frequency , t is the time (=0 at present) and is the angular velocity of the Galaxy at the current position of the Sun. and are integration constants related to the current position and velocity of a star (, , , , , ) as:
where the B and A are the Oort's constants. In Appendix A: we use Eqs. 1 to evaluate the evolution of space and velocity dispersions of a group of stars with low peculiar velocity with respect to their Regional Standard of Rest (RSR). Eqs. A2 show that dispersions in position and velocity components oscillate around constant values, except for the azimuthal coordinate , whose dispersion increases with t. After a few yr, this increase is dominated by the secular terms for typical position and velocity dispersions of young stellar groups, and it can be approximated by the linear relationship
If we consider a sample of stars with 0 pc and 0 km s-1, then all dispersions in position and velocity would oscillate around their mean values with an epicycle frequency . Stars would meet every (at the position of the Sun yr), a fact that has been referred to as "focusing phenomenon" (Yuan, 1977). However, when we apply Eq. 3 to a more realistic case where dispersions are close to those of the open clusters and associations, i.e. a few parsecs in and 1-2 km s-1 in , we obtain 1 kpc after a 5-10 yr - a period of time shorter than the age of some moving groups detected in the solar neighbourhood (e.g. Chen et al. 1997; Paper I). Thus, galactic differential rotation disrupts very efficiently unbound systems of stars. Nonetheless, if the distribution of stars in is supposed to be gaussian after the disruption of the stellar system, an important proportion of the original sample will be still concentrated in space after a long period of time; for instance, for = 1 kpc we can find still 24% of the initial stars in a region 600 pc long in after 5-10 yr.
© European Southern Observatory (ESO) 1999
Online publication: October 4, 1999