Why is the concept of center of mass relevant for the radial velocity (RV) method of detecting exoplanets? What is radial velocity? Why is that the only part of a star’s velocity that the Doppler effect allows us to measure? If a certain star’s Doppler shift at a given instance is zero, can we infer that the star must be not moving relative to us at that moment? Explain. What exactly do astronomers measure about a star’s spectrum over time in order to come up with its radial velocity curve?

In astronomy and astrophysics, the observational study of the movements and the motions of the stars is known as stellar kinematics. Essentially, the main goal of stellar kinematics is to measure the stellar velocities of our galaxy (the Milky Way) and the closest galaxies to it, and the internal kinematics of the stars from the more distant galaxies. The motion of a star relative to the observing point is defined as its space velocity. Space velocity is divided into two types: radial and tangential velocity.
The radial velocity is defined as the relative movement or the kinematics of a star toward or away from the Earth. It is, basically, the distance from the Earth to the star, which is usually and often accurately measured with a high-resolution spectrum. Thus, it is called spectroscopic radial velocity.
The radial velocity is the motion of the star along our line of sight. If the star is moving away from the observer, or if the distance between the star and the observer is increasing, then we have a positive radial velocity. This shifts the spectral lines to the red end of the spectrum, which makes the shift a red shift; if the star is moving toward the observer, or the distance between the star and the observer is decreasing, then we have a negative radial velocity. This moves the spectral lines toward the blue end of the spectrum, which makes the shift a blue shift.
This entire shift or effect is defined as the Doppler Effect and is most commonly used to measure the radial velocity and the speed of the stars and/or other astronomical objects as they move toward or away from the Earth.
If these kinds of shifts are regular and fixed, then the star is moving back and forth toward and away from the Earth, usually in a circular or elliptic motion. These motions mean that there is a body orbiting around the star, and if it has a low enough mass, then it can be called a planet. Doppler spectroscopy, also known as the radial velocity method, is also the most reliable and effective method for locating extra-solar planets.
The Doppler Effect and the star’s radial velocity are graphically shown with the radial velocity curve and are mathematically calculated with the following equation:
vrad/c = (lambdashift – lambdarest ) / lambdarest
In which lambdashift is the shifted or the observed value, lambdarest is the rested or the un-shifted value, and c is the speed of light. If a certain star’s Doppler shift or its radial velocity at a given instance is zero, then we measure the tangential velocity.
The majority of stars move at an angle to the observer’s line of sight. The part of the star’s total velocity that is perpendicular to the line of sight is called tangential velocity. Essentially, this is the motion of the star that is perpendicular to the direction to the Sun. If it weren’t for their tangential velocities, the planets of our solar system, for instance, would’ve crashed into the Sun. If an object’s radial velocity is zero, then its orbit must be circular. Measuring the tangential or proper velocity of a star is a challenging process which can take years or decades to complete, as the star moves across our line of sight. Thus, the radial velocity is more often calculated.