Synchrotron self-absorption and the minimum energy of optically thick radio flares from stellar mass black holes

We consider the case of radio flares from black hole X-ray binaries in which the flare spectrum evolves from optically thick to optically thin, under the assumption that this is due to decreasing optical depth to synchrotron self-absorption. We are able to place upper and lower limits on the size of the emitting region associated with a radio flare, and determine the synchrotron source magnetic field and energy as a function of size. The energy has a clear minimum which occurs close to the condition that the magnetic field derived from synchrotron self-absorption equals that calculated from equipartition. This minimum energy estimate is independent of the rise time of the event, and so may be applied to any event for which the peak flux is measured and there is evidence for self-absorption. This is a much more accurate approach to minimum energy estimation than assuming expansion at close to the speed of light. We apply this method to four examples of optically thick radio flares and find that in each case either the filling factor of the synchrotron source is considerably less than unity, or the expansion speed is considerably less than the speed of light. The combination of unity filling factor and expansion speeds close to the speed of light is completely ruled out on energetic grounds for three of the four events we consider. The inferred slowed expansion is consistent with detailed modelling of such events, which has been recently reported in the literature. The minimum power requirements associated with the flares are found to be similar to 10(36) erg s(-1), which are easily accommodated in the context of stellar mass black hole accretion at near-Eddington levels, when these flares typically occur. However, the true jet power could still be orders of magnitude higher.