  Most rocky planets in the galaxy orbit a cool host star, and there is large
uncertainty among theoretical models whether these planets can retain an
atmosphere. The James Webb Space Telescope (JWST) might be able to settle this
question empirically, but most proposals for doing so require large
observational effort because they are based on spectroscopy. Here we show that
infrared photometry of secondary eclipses could quickly identify "candidate"
atmospheres, by searching for rocky planets with atmospheres thick enough that
atmospheric heat transport noticeably reduces their dayside thermal emission
compared to that of a bare rock. For a planet amenable to atmospheric
follow-up, we find that JWST should be able to confidently detect the heat
redistribution signal of an O(1) bar atmosphere with one to two eclipses. One
to two eclipses is generally much less than the effort needed to infer an
atmosphere via transmission or emission spectroscopy. Candidate atmospheres can
be further validated via follow-up spectroscopy or phase curves. In addition,
because this technique is fast it could enable a first atmospheric survey of
rocky exoplanets with JWST. We estimate that the TESS mission will find ~100
planets that are too hot to be habitable but that can be quickly probed via
eclipse photometry. Knowing whether hot, rocky planets around M dwarfs have
atmospheres is important not only for understanding the evolution of
uninhabitable worlds: if atmospheres are common on hot planets, then cooler,
potentially habitable planets around M dwarfs are also likely to have
atmospheres.
