  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 ~ 1e36 erg/s, which are easily accomodated 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.
