Particle dark matter in the Galactic halo may be bound to the solar system either by elastic scattering through weak interactions with nucleons in the Sun (weak scattering) or by gravitational interactions with the planets, mainly Jupiter (gravitational capture). In this thesis, I simulate weak scattering, gravitational capture, and the subsequent evolution of the bound orbits to determine the distribution of bound dark matter at the position of the Earth. Previous work on this subject suggested that the event rate in dark matter detection experiments due to bound particles could be of order the event rate of halo particles in direct detection experiments, and several orders of magnitude higher for neutrinos arising from dark matter annihilation in the Earth. I use direct integration of orbits in a simplified solar system consisting of the Sun and Jupiter. I follow bound orbits until the particles are either rescattered in the Sun onto orbits that no longer intersect the Earth, ejected from the solar system, or reach the lifetime of the solar system t_SS = 4.5 Gyr. Since many aspects of the particle orbits pose severe problems for traditional orbit integration methods, I develop a novel integration scheme for this problem, which has only small and oscillatory energy errors even for highly eccentric orbits over very long times. Using the bound dark matter distribution functions I generate from the simulations, I show that bound dark matter has a small effect on direct detection event rates, and that it will be almost impossible to detect neutrinos from dark matter annihilation in the Earth with the new generation of km³-scale neutrino telescopes. I also show how the distribution functions and resulting direct detection and neutrino telescope event rates can be scaled to other particle masses and elastic scattering cross sections.