Star cluster disruption by giant molecular clouds

We investigate encounters between giant molecular clouds (GMCs) and star clusters. We propose a single expression for the energy gain of a cluster due to an encounter with a GMC, valid for all encounter distances and GMC properties. This relation is verified with N-body simulations of cluster-GMC encounters, where the GMC is represented by a moving analytical potential. Excellent agreement is found between the simulations and the analytical work for fractional energy gains of Delta E/vertical bar E-0 vertical bar < 10, where vertical bar E-0 vertical bar is the initial total cluster energy. The fractional mass loss from the cluster scales with the fractional energy gain as (Delta M/M-0) = f(Delta E/vertical bar E-0 vertical bar), where f similar or equal to 0.25. This is because a fraction 1 - f of the injected energy goes to the velocities of escaping stars, that are higher than the escape velocity. We therefore suggest that the disruption time of clusters, t(dis), is best defined as the time needed to bring the cluster mass to zero, instead of the time needed to inject the initial cluster energy. We derive an expression for t(dis) based on the mass loss from the simulations, taking into account the effect of gravitational focusing by the GMC. Assuming spatially homogeneous distributions of clusters and GMCs with a relative velocity dispersion of sigma(cn), we find that clusters lose most of their mass in relatively close encounters with high relative velocities (similar to 2 sigma(cn)). The disruption time depends on the cluster mass (M-c) and half-mass radius (r(h)) as t(dis) = 2.0 S(M-c/10(4) M-circle dot)(3.75 pc/r(h))(3) Gyr, with S equivalent to 1 for the solar neighbourhood and S scales with the surface density of individual GMCs (Sigma(n)) and the global GMC density (rho(n)) as S proportional to (Sigma(n)rho(n))(-1). Combined with the observed relation between r(h) and M-c, that is, r(h) proportional to M-c(lambda), t(dis) depends on M-c as t(dis)proportional to M-c(gamma). The index gamma is then defined as gamma= 1 - 3 lambda. The observed shallow relation between cluster radius and mass (e.g. lambda similar or equal to 0.1), makes the value of the index gamma = 0.7 similar to that found from observations and from simulations of clusters dissolving in tidal fields (gamma similar or equal to 0.62). The constant of 2.0 Gyr, which is the disruption time of a 10(4) M circle dot cluster in the solar neighbourhood, is about a factor of 3.5 shorter than that found from earlier simulations of clusters dissolving under the combined effect of Galactic tidal field and stellar evolution. It is somewhat higher than the observationally determined value of 1.3 Gyr. It suggests, however, that the combined effect of tidal field and encounters with GMCs can explain the lack of old open clusters in the solar neighbourhood. GMC encounters can also explain the (very) short disruption time that was observed for star clusters in the central region of M51, since there rho(n) is an order of magnitude higher than that in the solar neighbourhood.