All radial velocity detections hinge on one thing: that the star is orbiting some center of mass in a system. Isaac Newton theorized that it is not planets which go around a star, nor stars which go around planets, but rather, both going around a shared center of mass. Thus, the star’s “wobble” about the C.O.M. is indicative of a planetary companion.
The ideal conditions to observe this would be when our line of sight is “edge-on” of the planet’s orbit, since the Doppler shift of light as the star moves depends only on the velocity of the star along our line of sight. In other words, the optimal orientation would be an inclination angle of 90 degrees.
If we assume the orbit of the star is perfectly circular and edge on, simple Newtonian physics yields the amount of velocity we expect to see:
To generalize this for all inclination angles, we add a term to account for the projection of the star’s Doppler-shifted velocity along our line of sight (and we can also approximate that the star's mass is much greater than the planet's) to get:
From the radial velocity detection technique, we can determine the planet’s minimum mass. Combined with the transit technique, in which we discover the radius of the planet, we can derive the planet’s average density, and from that can come surface gravity and existence of atmosphere.
*Side note:* Planet masses are usually given a minimum bound by the term “Msin(i).” This is because often, from using the radial velocity technique, we don’t know the inclination angle (i) of the planet’s orbit.