  Dark matter (DM) self-interactions affect the gravitational capture of DM in
the Sun and Earth differently as a simple consequence of the differing
kinematics of collisions within the two potential wells: the dominant effect of
self-interactions in the Sun is to provide an additional channel for capture,
while the dominant effect in the Earth is to eject previously captured DM. We
point out that this simple observation can be used to deduce the existence of
DM self-interactions by comparing the annihilation rates of DM gravitationally
bound within the Sun and Earth. We compute the Sun and Earth annihilation
fluxes for DM with spin-independent nuclear cross-sections and thermal
annihilation cross-sections and demonstrate that, for cross-sections allowed by
direct detection, self-interactions can easily suppress the expected Earth flux
by multiple orders of magnitude. This suppression is potentially significant
even for self-interaction cross-sections orders of magnitude below the Bullet
Cluster bounds, making this solar system comparison a leading test of dark
matter self-interactions. Additionally, we consider thermalization of the
captured DM population with the nuclei of the capturing body in some detail,
accounting for both nuclear and self-interactions, and point out some
consequential and broadly applicable considerations.
