Radial Velocities

Second Authors:  Eric, Lauren


The Problem:

Use Kepler's Third Law and constancy of momentum to express the time variation of the line-of-sight velocity of the star in terms of the orbital period P , the mass of the star M and the mass of the planet m<<M.


The Solution:

Step 1: By definition, the frequency of the planet's orbit is 2π/P.


Step 2: The velocity is given by the product of this frequency and the distance from the center.


Step 3: Since the period for the star and the planet are the same, the velocity of the star can be expressed as shown.


Step 4: Constancy of momentum.


Step 5: Express velocity in terms of angular velocity and distance a.


Step 6: Express the distance of the star from the center of mass in terms of the desired quantities.  First, we begin with equation 5, and rearranging the equation to get a_star.  By definition, a_planet is given by a - a_star.  We ultimately get the expression shown with the help of Kepler's third law:

Step 7:  The amplitude of the star's line-of-sight velocity, K, is obtained by substituting our value of a from equation 6 into equation 3.


Step 8:  The line-of-sight velocity as a function of time is given by the amplitude K multiplied by cosine of the frequency/time product.


Evaluating our Results:
Using only our intuition, it is not difficult to see how the velocity K will scale directly with the mass of the planet and inversely with the mass of the star.  It is also simple to grasp that a larger period will mean a smaller velocity.  In our calculations, we have confirmed this.  We have shown that the K and m are directly proportional, while K scales inversely with M by a factor of 2/3 and with P by a factor of 1/3.