Accreting intermediate-mass black holes ( IMBHs ) have been proposed as an explanation for ultraluminous X-ray sources ( ULXs ) .
Runaway core collapse inside a massive cluster is a possible mechanism for IMBH formation .
But if so , why are ULXs only rarely found associated with a cluster ?
We use a simple analytical approximation to show that rapid core collapse can occur in two physical regimes .
For cluster masses \sim 10 ^ { 6 } M _ { \odot } , an IMBH may be formed if the collapse timescale is \buildrel { \lower 3.0 pt \hbox { $ < $ } } \over { \lower 2.0 pt \hbox { $ \sim$ } } 3 Myr , as already well known ; the parent cluster is expected to survive .
For cluster masses \sim 10 ^ { 5 } M _ { \odot } , an IMBH may result from a core collapse on even shorter timescales ( \approx 0.5 Myr ) , when the protocluster is still embedded in optically thick gas .
Most clusters in this latter regime are disrupted “ explosively ” as soon as the gas is ionized by the OB stars .
We speculate that this process may produce isolated ULXs with masses up to a few 10 ^ { 2 } M _ { \odot } , surrounded by a loose OB association , and perhaps by a nebula , remnant of the expanding gas from the disrupted protocluster .