How do Astronomers Measure the Mass of a Planet using Doppler Spectroscopy?

The planet and the star both rotate around their center of mass. That's how star "wobbling" occurs. When the star in our line-of-sight moves away from us, photons reaching us are red-shifted and when the star moves towards us, its photons are blue-shifted - this is the so-called Doppler effect. The star mass $M$ is extracted from its spectrum, luminosity and other parameters. Then over the years, astronomers measure the radial velocity of star, which is extrapolated from the Doppler effect.

In the graph, the highest radial velocity of the star is marked as $K$. From the same observations, the wobbling period $P$ is also detected. This wobbling period is the time duration between adjacent maximum blue-shifts (or between maximum red-shifts). In essence, astronomers compile such a star radial velocity graph:

Given this data, one can calculate exoplanet mass (given that I haven't made any formulas substitution error):

$$ m = frac{M^{2/3}K}{sin(i)} left(frac{P}{2pi G}right)^{1/3} $$

where $i$ is inclination - angle between reference plane and orbital plane.

EDIT

The equation above is suitable only if the exoplanet orbit is circular. This is rarely the case. According to this article, ones needs to take into account orbit eccentricity when it is elliptic. We need to make yet another assumption, that $ frac{m}{M} ll 1$, for example $ m_{earth}/M_{sun} approx 3.004×10^{-6} $. ("$ll$" means A LOT smaller than). Assuming that, planet mass can be reduced to:

$$ m = frac{M^{2/3}Ksqrt{1-e^2}}{sin(i)} left(frac{P}{2pi G}right)^{1/3} $$

where $e$ is orbital eccentricity