Why can't planets scoop up dark matter?

Question

The density of dark matter is way too low to affect solar system dynamics a measurable amount, see this question. High-speed particles will zip right through the planet (see this question), but what about the particles for which the hyperbolic excess velocity is small? Any two massive particles will attract each-other due to gravity and (at femtometer range) the the Higgs force. As the dark matter particle passes through the earth, these attractions will jostle the nuclei and dissipate energy through dynamical friction. Is this friction strong enough to trap enough low-velocity particles inside planets to feasibly measure the "extra" mass?

Answer 1

The gravitational force is too weak. For a proton with mass $m~\simeq~2\times 10^{-27}kg$ the ratio of the gravitational and electrostatic force between two of them at the same radius is $$ \frac{F_g}{F_e}~=~\frac{Gm^2}{ke^2}~=~1.7\times 10^{-37} $$ for $k~=~\frac{1}{4\pi\epsilon_0}$ $=~9\times 10^9N-m^2/coul^2$ and $G~=~6.67\times 10^{-11}N-m^2/kg^2$ and $e~=~1.6\times 10^{-19}coul$. This ratio for the electron is $2.3\times 10^{-43}$. This means there is very little of this friction due to gravitation, which is determined by the excitation of many additional states. This can apply to the weak interactions where the weak force is $F_w~\simeq~10^{-2}F_e$, so this ratio is only $4$ orders of magnitude larger than with the electrostatic force. So we can't expect gravitation to play a role of any significance with respect to elementary particles.

If we compute this ratio for a Planck mass $m_p~=~\sqrt{c\hbar/G}$ $=~2.2\times 10^{-8}kg$ so $$ \frac{F_g}{F_e}~=~\frac{Gm_p^2}{ke^2}~=~\frac{c}{ke^2\hbar}~=~\frac{1}{\alpha}~\simeq~137 $$ Here the ratio is rather large and gravity is more “muscular.” There are some ideas that dark matter might be black hole remnants that have a Planck mass. I do not think this hypothesis is likely to turn out. However, it might be reasonable to think black hole remnants might be captured by bulk material of a star or planet. This might also be one suggestion for why the black hole remnant idea is false.