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 / |E _ { 0 } | < 10 , where |E _ { 0 } | 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 / |E _ { 0 } | ) , where f \simeq 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 loose most of their mass in relatively close encounters with high relative velocities ( \sim 2 \mbox { $ \sigma _ { cn } $ } ) .
The disruption time depends on the cluster mass ( M _ { c } ) and half-mass radius ( r _ { h } ) as \mbox { $t _ { dis } $ } ~ { } = ~ { } 2.0 S \left ( \mbox { $M _ { c } $ } / 10 ^ { 4 } \mbox { $ { % M } _ { \odot } $ } \right ) \left ( 3.75 \mbox { pc } / \mbox { $r _ { h } $ } \right ) ^ { 3 } { % \mbox { Gyr } } , with S \equiv 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 \propto ( \mbox { $ \Sigma _ { n } $ } \mbox { $ \rho _ { n } $ } ) ^ { -1 } .
Combined with the observed relation between r _ { h } and M _ { c } , i.e . \mbox { $r _ { h } $ } \propto \mbox { $M _ { c } $ } ^ { \lambda } , t _ { dis } depends on M _ { c } as \mbox { $t _ { dis } $ } \propto \mbox { $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 \simeq 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 \simeq 0.62 ) .
The constant of 2.0 Gyr , which is the disruption time of a 10 ^ { 4 } \mbox { $ { M } _ { \odot } $ } cluster in the solar neighbourhood , is about a factor of 3.5 shorter than 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 in the solar neighbourhood .