(i) Backward integration — Reconstructing the globular cluster orbit in the last 5 Gyr.
First, for each Galactic globular cluster in our list, we determine their current positions and velocities. We then integrate the orbit of a single point mass, representing the cluster barycenter, backwards in time for 5 Gyr, and in this way we retrieve its position and velocity at that time in the chosen Galactic potential.
(ii) Forward integration — Test-particle streams from the past to the present day.
Once the positions and velocities of the barycenter of each cluster, 5 Gyr ago, are determined, we build the corresponding N-body system, with N=100000 particles. The phase-space coordinates of these particles are generated following a Plummer distribution, with total mass and half-mass radius as described here. The barycenter of this N-body cluster is then assigned initial positions and velocities in the Galactic model, as those retrieved at step (i) and the cluster is then integrated forward in time, until the present day. Particles in this N-body system are modeled as test-particles, that is they experience the gravitational field exerted by the globular cluster itself (i.e. Plummer sphere) and by the Galaxy (see here), but not generate any gravitational field themselves. This allows us to maintain a computational time which scales as O(N) and not as O(N2), as it would be the case for a direct N-body self-consistent computation. The Plummer sphere, representing the cluster potential, moves through the Galaxy along the orbit retrived at step (i), travelled this time in the opposite direction, from 5 Gyr ago to the present day.