  Among the different strategies aiming to detect WIMP dark matter (DM), a
neutrino signal coming from the Sun would be a smoking gun. This possibility
relies on the DM capture by the Sun driven by the local DM distribution
assumptions: the local mass density and the velocity distribution. In this
context, we revisit those astrophysical hypotheses (also relevant for direct
detection). We focus especially on the DM velocity distribution considering
different possibilities beyond the popular Maxwellian distribution. Namely,
some alternatives can be considered through analytical approaches and by
looking into cosmological simulations of spiral galaxies. Most of the fitting
formulas used to constrain the local velocity distribution function fail to
describe the peak and the high velocity tail of the velocity distribution
observed in simulations, the latter being improved when adding the local escape
velocity of DM into the benchmark fitting models. In addition we test the
predictions by the Eddington inversion method and also illustrate the
importance of the galactic dynamical history. We estimate the resulting
uncertainties on the DM capture rate by the Sun and conclude that different
velocity distributions will affect the capture rate of DM by the Sun up to a
$15-20\%$. On top of that, the calculation of the intrinsic variance of the
capture rate leads to poorly controlled uncertainties especially for high WIMP
masses ($>$30 GeV) raising concerns about the capture scenario.
