For an idealized infinite DM mass (compared to the other energy scales), the velocity of the outgoing Xe nucleus is twice the DM velocity.

Its energy in natural units is 1/2 mXe v^2 = 2 mXe vDM^2

So: 250 keV = 2*131 GeV * vDM^2

If we approximate 250 ~ 262 this results in

v=0.001 (units of c)

so

v=300 km/s
as the candidate Dark Matter velocity for the scattering DM particle

Our velocity through the galaxy is about 230 km/s, so this is quite fast, but in a plausible range

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