  Axions with masses of order keV can be produced in great abundance within the
Solar core. The majority of Sun-produced axions escape to infinity, but a small
fraction of the flux is produced with speeds below the escape velocity. Over
time, this process populates a basin of slow-moving axions trapped on bound
orbits. These axions can decay to two photons, yielding an observable
signature. We place the first limits on this solar basin of axions using recent
quiescent solar observations made by the NuSTAR X-ray telescope. We compare
three different methodologies for setting constraints, and obtain world-leading
limits for axions with masses between 5 and 30 keV, in some cases improving on
stellar cooling bounds by more than an order of magnitude in coupling.
