  We place the first constraints on binary planets and exomoons from Doppler
monitoring of directly imaged exoplanets. We model radial velocity observations
of HR 8799 b, c, and d from Ruffio et al. (2021) and determine upper limits on
the $m\sin{i}$ of short-period binary planets and satellites. At 95%
confidence, we rule out companions orbiting the three planets more massive than
$m\sin{i} = 2 M_J$ with orbital periods shorter than 5 days. We achieve our
tightest constraints on moons orbiting HR 8799 c, where with 95% confidence we
rule out out edge-on Jupiter-mass companions in periods shorter than 5 days and
edge-on half-Jupiter-mass moons in periods shorter than 1 day. These radial
velocity observations come from spectra with resolution 20 times lower than
typical radial velocity instruments and were taken using a spectrograph that
was designed before the first directly imaged exoplanet was discovered. Similar
datasets from new and upcoming instruments will probe significantly lower
exomoon masses.
