# Dark matter relative velocity

In Dark Matter Indirect Detection literature, it is usually said that the relative velocity of dark matter particles in the galactic halo is about $$v \sim 10^{-3}c$$ See for an example this lecture notes

I am wondering how one derives this result?

Any helps or comments would be appreciated.

The dark matter is expected to form (to first order) a virialized, non-rotating, smooth halo in which the visible matter of the Galaxy is embedded. In that respect it is quite similar to the other Galactic halo components (globular clusters etc.), that form a spherical distribution around the Galaxy with little net rotation. In this model the dark matter has a $$\sim$$ Maxwellian distribution, with an average velocity of zero and a velocity dispersion at the solar radius of around 200 km/s. The velocity distribution will be truncated at around 600 km/s by escape from the Galactic potential.
The solar system travels in an orbit with a speed of about 240 km/s ( $$8\times 10^{-4}c$$) with respect to a non-rotating Galactic frame. It therefore would encounter dark matter with an average speed of $$\simeq 270$$ km/s but with a minimum speed of zero and a tail out to around 800 km/s.
This first order model is complicated by possible substructure in the dark halo, tidal streams and the assembly history of the Milky way. At the Earth it would also be modulated by $$\pm 10$$% by the Earth's orbital motion and gravitational focusing by the Sun (see Freese et al. 2012 for much more detail).