  Weakly Interacting Massive Particles (WIMPs), which are among the best
motivated dark matter (DM) candidates, could make up all or only a fraction of
the total DM budget. We consider a scenario in which WIMPs are a sub-dominant
DM component; such a scenario would affect both current direct and indirect
bounds on the WIMP-nucleon scattering cross section. In this paper we focus on
indirect searches for the neutrino flux produced by annihilation of
sub-dominant WIMPs captured by the Sun or the Earth via either spin-dependent
or spin-independent scattering. We derive the annihilation rate and the
expected neutrino flux at neutrino observatories. In our computation, we
include an updated chemical composition of the Earth with respect to the
previous literature, leading to an increase of the Earth's capture rate for
spin-dependent scattering by a factor three. Results are compared with current
bounds from Super-Kamiokande and IceCube. We discuss the scaling of bounds from
both direct and indirect detection methods with the WIMP abundance.
