  The role of Compton cooling in the standard model for the afterglows of
gamma-ray bursts is considered. When electrons cool by scattering off their own
synchrotron photons, three cooling regimes are identified in which the observed
synchrotron radiation exhibits qualitatively different characteristics.
Depending on the values of the source parameters, an afterglow may evolve
through one, two or all three of these regimes. Since synchrotron radiation can
be regarded as Compton scattering of the virtual photons due to the magnetic
field, in one of these regimes the instantaneous synchrotron spectrum has
properties identical to those when Compton cooling is negligible. During this
phase in the evolution, the synchrotron radiation falls mainly in the near
infrared/optical/UV spectral range. It is argued that the observed afterglows
of GRB 980923 and GRB 971214 are outside this degenerate phase. The derived
values of the source parameters show Compton cooling to have been important in
both of these afterglows.
