  Assuming an adiabatic evolution of a Gamma-Ray Burst (GRB) remnant
interacting with an external medium, we calculate the injection, cooling, and
absorption break frequencies, and the afterglow flux for plausible orderings of
the break and observing frequencies. The analytical calculations are restricted
to a relativistic remnant and, in the case of collimated ejecta, to the phase
where there is an insignificant lateral expansion. Results are given for both a
homogeneous external medium and for a wind ejected by the GRB progenitor.
  We compare the afterglow emission at different observing frequencies, for
each type of external medium. It is found that observations at sub-millimeter
frequencies during the first day provide the best way of discriminating between
the two models. By taking into account the effect of inverse Compton
scatterings on the electron cooling, a new possible time-dependence of the
cooling break is identified. The signature of the up-scattering losses could be
seen in the optical synchrotron emission from a GRB remnant interacting with a
pre-ejected wind, as a temporary mild flattening of the afterglow decay. The
up-scattered radiation itself should be detected in the soft X-ray emission
from GRB remnants running into denser external media, starting few hours after
the main event.
