We use the formalism of Gould and Raffelt to compute the dimensionlessthermal conduction coefficients for scattering of dark matter particles withstandard model nucleons via cross-sections that depend on the relative velocityor momentum exchanged between particles. Motivated by models invoked toreconcile various recent results in direct detection, we explicitly compute theconduction coefficients $\alpha$ and $\kappa$ for cross-sections that go as$v_{\rm rel}^2$, $v_{\rm rel}^4$, $v_{\rm rel}^{-2}$, $q^2$, $q^4$ and$q^{-2}$, where $v_{\rm rel}$ is the relative DM-nucleus velocity and $q$ isthe momentum transferred in the collision. We find that a $v_{\rm rel}^{-2}$dependence can significantly enhance energy transport from the inner solar coreto the outer core. The same can true for any $q$-dependent coupling, if thedark matter mass lies within some specific range for each coupling. This effectcan complement direct searches for dark matter; combining these results withstate-of-the-art Solar simulations should greatly increase sensitivity tocertain DM models. It also seems possible that the so-called Solar AbundanceProblem could be resolved by enhanced energy transport in the solar core due tosuch velocity- or momentum-dependent scatterings.
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