  We present a method that employs the secondary eclipse light curves of
transiting extrasolar planets to probe the spatial variation of their thermal
emission. This technique permits an observer to resolve the surface of the
planet without the need to spatially resolve its central star. We evaluate the
feasibility of this technique for the HD 209458 system [..]. We consider two
representations of the planetary thermal emission; a simple model parameterized
by a sinusoidal dependence on longitude and latitude, as well as the results of
a three-dimensional dynamical simulation of the planetary atmosphere previously
published by Cooper & Showman. We find that observations of the secondary
eclipse light curve are most sensitive to a longitudinal offset in the
geometric and photometric centroids of the hemisphere of the planet visible
near opposition. To quantify this signal, we define a new parameter, the
``uniform time offset,'' which measures the time lag between the observed
secondary eclipse and that predicted by a planet with a uniform surface flux
distribution. We compare the predicted amplitude of this parameter for HD
209458 with the precision with which it could be measured with IRAC. We find
that IRAC observations at 3.6um a single secondary eclipse should permit
sufficient precision to confirm or reject the Cooper & Showman model of the
surface flux distribution for this planet. We quantify the signal-to-noise
ratio for this offset in the remaining IRAC bands (4.5um, 5.8um, and 8.0um),
and find that a modest improvement in photometric precision (as might be
realized through observations of several eclipse events) should permit a
similarly robust detection.
