The Earth's velocity for direct detection experiments

The Earth's velocity relative to the Sun in galactic coordinates is required in the rate calculation for direct detection experiments. We provide a rigorous derivation of this quantity to first order in the eccentricity of the Earth's orbit. We also discuss the effect of the precession of the equinoxes, which has hitherto received little explicit discussion. Comparing with other expressions in the literature, we confirm that the expression of Lee, Lisanti and Safdi is correct, while the expression of Lewin and Smith, the de facto standard expression, contains an error. For calculations of the absolute event rate, the leading order expression is sufficient while for modulation searches, an expression with the eccentricity is required for accurate predictions of the modulation phase.