I study the process of dark matter capture by the Sun, under the assumption of a Weakly Interacting Massive Particle (WIMP), in the framework of non-relativistic effective field theory. Hypothetically, WIMPs from the galactic halo can scatter against atomic nuclei in the solar interior, settle to thermal equilibrium with the solar core and annihilate to produce an observable flux of neutrinos. In particular, I examine the thermalization process using Monte-Carlo integration of WIMP trajectories. I consider WIMPs in a mass range of 10-1000 GeV and WIMP-nucleon interaction operators with different dependence on spin and transferred momentum. I find that the density profiles of captured WIMPs are in accordance with a thermal profile described by the Sun's gravitational potential and core temperature. Depending on the operator that governs the interaction, the majority of the thermalization time is spent in either the solar interior or exterior. If normalizing the WIMP-nuclei interaction strength to a specific capture rate, I find that the thermalization time differs at most by 3 orders of magnitude between operators. In most cases of interest, the thermalization time is many orders of magnitude shorter than the age of the solar system.