  Photometry of short-period planetary systems allows astronomers to monitor
exoplanets, their host stars, and their mutual interactions. In addition to the
transits of a planet in front of its star and the eclipses of the planet by its
star, researchers have reported flux variations at the orbital frequency and
its harmonics: planetary reflection and/or emission and Doppler beaming of
starlight produce one peak per orbit, while ellipsoidal variations of a tidally
distorted star and/or planet produce two maxima per orbit. Researchers have
also reported significant photometric variability at three times the orbital
frequency, sometimes much greater than the predictions of tidal theory. The
reflected phase variations of a homogeneous planet contains power at even
orbital harmonics-important for studies of ellipsoidal variations-but cannot
contain odd orbital harmonics. We show that odd harmonics can, however, be
produced by an edge-on planet with a time-variable map, or an inclined planet
with a North-South (N-S) asymmetric map. Either of these scenarios entail
weather: short-period planets are expected to have zero obliquity and hence N-S
symmetric stellar forcing. North-South asymmetry in a giant planet would
therefore suggest stochastic localized features, such as weather. However, we
find that previous claims of large-amplitude odd modes in Kepler photometry are
artifacts of removing planetary transits rather than modeling them. The only
reliable claims of odd harmonics remain HAT-P-7b and Kepler-13Ab, for which the
third mode amplitude is 6-8% of the planetary flux. Although time-variable
albedo maps could in principle explain these odd harmonics, upper-limits on the
infrared variability of hot Jupiters make this hypothesis unlikely. We urge
theorists to study the effects of close-in planets on stellar atmospheres, as
this remains the only plausible hypothesis.
