  We explore the momentum and velocity dependent elastic scattering between the
dark matter (DM) particles and the nuclei in detectors and the Sun. In terms of
the non-relativistic effective theory, we phenomenologically discuss ten kinds
of momentum and velocity dependent DM-nucleus interactions and recalculate the
corresponding upper limits on the spin-independent DM-nucleon scattering cross
section from the current direct detection experiments. The DM solar capture
rate is calculated for each interaction. Our numerical results show that the
momentum and velocity dependent cases can give larger solar capture rate than
the usual contact interaction case for almost the whole parameter space. On the
other hand, we deduce the Super-Kamiokande's constraints on the solar capture
rate for eight typical DM annihilation channels. In contrast to the usual
contact interaction, the Super-Kamiokande and IceCube experiments can give more
stringent limits on the DM-nucleon elastic scattering cross section than the
current direct detection experiments for several momentum and velocity
dependent DM-nucleus interactions. In addition, we investigate the mediator
mass's effect on the DM elastic scattering cross section and solar capture
rate.
