astrophysics - Direct Dark Matter Detection: relative velocity between WIMPs & Nuclei

In direct dark matter detection it is said that the relative velocity between the WIMPs, which form a DM halo, and the target nuclei on earth is of order $100 \frac{\text{km}}{\text{s}}$. How does one derive this result?

Answer

This is an order of magnitude characterization of the velocity dispersion (typical velocities) in the milky way. The sun for example is moving at about 200 km/s in the milky way, which should be comparable to local Dark Matter particles.

A common equation for the velocity dispersion is simply the virial velocity:

$$\sigma^2 \sim \frac{GM}{r}$$

Where $\sigma$ is the velocity dispersion, and $r$ and $M$ are the characteristic mass and radius of the system. If one uses a mass of $M \sim 10^{11} \, M_\odot$, and a radius $r \sim 10^4 \, \mathrm{pc}$, the velocity is indeed: $\sigma \approx 200 \, \mathrm{km/s}$.

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Why is there a relative velocity between the sun and dark matter?

Because Dark matter (DM) only interacts gravitationally, it's much harder to dissipate its energy and angular momentum so it doesn't collapse into disks, or settle into ordered rotation. Dark matter is on more chaotic, random orbits throughout galactic halos. So overall, the dark matter doesn't tend to be moving in the same direction as the sun.

Why should there be annual variation in a direct detection experiment?

Even though DM is on chaotic orbits, the earth will still be moving faster through the background of DM particles at one time of the year (when it is moving in the same direction as the sun) as during another (when it is moving in the opposite direction). You can imagine the DM as a fairly continuous fluid of particles that the earth (and sun) are passing through. While each particles is on a random(ish) orbit, overall the density of DM particles in a given region will stay roughly the same. Over the course of the year, the earth's velocity through that background changes. There can, however, be variations to the exact type and magnitude of seasonal variation depending on different models for the milky way and dark matter halo.