A few long gamma-ray bursts such as GRB 050421 show no afterglow emission
beyond the usual initial steep decay phase. It has been suggested that these
events correspond to "naked" bursts that occur in a very low density
environment. We reconsider this possibility in the context of various scenarios
for the origin of the afterglow. In the standard model where the afterglow
results from the forward shock as well as in the alternative model where the
afterglow comes from the reverse shock, we aim to obtain constraints on the
density of the environment, the microphysics parameters, or the Lorentz factor
of the ejecta, which are imposed by the absence of a detected afterglow. For
the two models we compute the afterglow evolution for different values of the
external density (uniform or wind medium) and various burst parameters. We then
compare our results to the Swift data of GRB 050421, which is the best example
of a long burst without afterglow. In the standard model we show that
consistency with the data imposes that the external density does not exceed
1E-5 cm-3 or that the microphysics parameters are very small with epsilon_e <~
1E-2 and epsilon_B <~ 1E-4. If the afterglow is caused by the reverse shock, we
find that its contribution can be strongly reduced if the central source has
mainly emitted fast-moving material (with less than 10 - 30 % of the kinetic
energy at Gamma<100 and was located in a dense environment. The two considered
scenarios therefore lead to opposite constraints on the circumburst medium. The
high-density environment, favored by the reverse shock model, better
corresponds to what is expected if the burst progenitor was a massive star.Comment: 6 pages, 3 figures, 1 table, to appear in A&
