Given a known parent object, we determine the epoch and velocity distribution of an orbital break-up given unassociated observations of the debris cloud positions at multiple times. Using a tentative guess of the break-up time, we solve Lambert’s problem to determine an initial velocity distribution for each debris cloud observation. Agreement of proposed initial velocity distributions depends on the assumed break-up time. We use maximal mean discrepancy in a reproducing kernel Hilbert space to quantify the difference between tentative initial velocity distributions, and optimize this metric to determine the time of break-up.