  We simulate collisions involving red-giant stars in the centre of our galaxy.
Such encounters may explain the observed paucity of highly luminous red giants
within the central 0.2pc. The masses of the missing stars are likely to be in
the range 2 to 8 solar masses. Recent models of the galactic centre cluster's
density and velocity distributions are used to calculate two-body collision
rates. In particular we use stellar-evolution models to calculate the number of
collisions a star will have during different evolutionary phases. We find that
the number of two-body collisions per star is $\lo 1$ in the central 0.1 to 0.2
pc, depending strongly on the galactocentric radius. Using a 3D numerical
hydrodynamics code (SPH) we simulate encounters involving cluster stars of
various masses with 2 and 8 solar-mass red giants. The instantaneous mass loss
in such collisions is rarely enough to destroy either giant. A fraction of the
collisions do, however, lead to the formation of common envelope systems where
the impactor and giant's core are enshrouded by the envelope of the giant. Such
systems may evolve to expel the envelope, leaving a tight binary; the original
giant is destroyed. The fraction of collisions that produce common envelope
systems is sensitive to the local velocity dispersion and hence galactocentric
radius. Using our collision-rate calculations we compute the time-scales for a
giant star to suffer such a collision within the galactic centre. These
time-scales are >10^{9-10}years and so are longer than the lifetimes of stars
more-massive than 2 solar masses. Thus the observed paucity of luminous giants
is unlikely to be due to the formation of common envelope systems as a result
of two-body encounters involving giant stars.
