  The next generation of dark matter (DM) direct detection experiments and
neutrino telescopes will probe large swaths of dark matter parameter space. In
order to interpret the signals in these experiments, it is necessary to have
good models of both the halo DM streaming through the solar system and the
population of DM bound to the solar system. In this paper, the first in a
series of three on DM in the solar system, we present simulations of orbits of
DM bound to the solar system by solar capture in a toy solar system consisting
of only the Sun and Jupiter, assuming that DM consists of a single species of
weakly interacting massive particle (WIMP). We describe how the size of the
bound WIMP population depends on the WIMP mass, spin-independent cross section,
and spin-dependent cross section. Using a standard description of the Galactic
DM halo, we find that the maximum enhancement to the direct detection event
rate, consistent with current experimental constraints on the WIMP-nucleon
cross section, is < 1% relative to the event rate from halo WIMPs, while the
event rate from neutrinos from WIMP annihilation in the center of the Earth is
unlikely to meet the threshold of next-generation, km^3-sized (IceCube, KM3NeT)
neutrino telescopes.
