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*Astron. Astrophys. 350, 434-446 (1999)*
## 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
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