Exoplanet Detection: Radial Velocity and the Doppler Effect


Theory

When an exoplanet orbits a star, both the star and the exoplanet feel the same, but in opposite directions, force of gravity. The difference between the effect of gravity on each object is related to the mass difference of the objects. As a consequence while the less massive object (the exoplanet) moves quite a bit, the more massive object (the star) does not. In a reference frame where the star-exoplanet system has no velocity (the center of mass frame), the total momentum of the system must be zero. Therefore the ratio of the velocities is just vstar = - vexoplanet (mexoplanet/mstar).  Therefore by detecting the velocity of the star, one can deduce the existence of an exoplanet orbiting that star. 

One complication is that we only detect the motion of the star along our line of sight. Therefore it is the velocity along this line, or the radial velocity, that must be determined.  The magnitude of the star's radial velocity will be a maximum when the exoplanet is moving towards/away from Earth and it will be zero when the exoplanet is moving perpendicular to Earth. The the radial velocity, will therefore oscillate sinusoidally related to the position of the exoplanet. 

The radial velocity will also be affected by the relative orientation of the Earth and the star-exoplanet system.  If the motion of the exolanet is in the same plane as the star and Earth, the radial velocity will be a maximum.  If the motion of the exolanet is perpendicular to the plane of the star and Earth, the radial velocity will be zero.  The radial velocity detected will be vr = vr; max cos(θ), where θ is the inclination of the system relative to edge on.

Since the star is moving relative to Earth, its spectrum, the Fraunhofer lines, will be shifted in wavelength.  When the star is moving toward Earth its light will be blue shifted (longer wavelength) and when it is moving away from Earth its light will be red shifted (shorter wavelength).  Since the radial velocity of the star is at most a few hundred m/s, the Doppler shift will not manifest itself in a large shift in wavelength.  The simulation allows zooming in on the star's Fraunhofer lines so that these small shifts can be seen.

Mario Belloni (mabelloni@davidson.edu)