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We consider the effects of inverse Compton scattering of synchrotron photonsfrom relativistic electrons in GRB afterglows. We compute the spectrum of theinverse Compton emission and find that it can dominate the total cooling rateof the afterglow for several months or even years after the initial explosion.We demonstrate that the presence of strong inverse Compton cooling can bededuced from the effect it has on the time-evolution of the cooling break inthe synchrotron spectral component, and therefore on the optical and X-rayafterglow lightcurves. We then show how the physical interpretation of theobserved characteristics of the synchrotron spectrum must be modified to takeinto consideration this extra source of cooling, and give a revisedprescription for computing physical parameters characterizing the expandingshock wave from the observed quantities. We find that for a given set ofobservables (synchrotron break frequencies and fluxes) there is either noconsistent physical interpretation or two of them. Finally we discuss theprospects of directly detecting the inverse Compton emission with Chandra. Weargue that such a detection is possible for GRBs exploding in a reasonablydense (n>1 cm^-3) medium.Comment: 21 pages, ApJ submitte

On the Synchrotron Self‐Compton Emission from Relativistic Shocks and Its Implications for Gamma‐Ray Burst Afterglows