Radial Velocities Worksheet Problem 2

We can determine how much the sun is displaced by Jupiter's mass with just a few simple calculations. If we want the center of mass to be at the origin of the system, we have:

where J subscripts denote Jupiter, s subscripts denote the sun, M denotes mass, and a denotes distance from the center of mass. From the image, we see:

We want a_s, the distance from the center of the sun to the center of mass of the system. We can solve the center of mass equation for this distance and then substitute (a – a_p) for a_s.

In the last step, I've used the approximation M_J << M_s to simplify the expression. The mass of Jupiter is one-one-thousandth of the mass of the sun, and the distance a between the sun and Jupiter is 5 AU. We know that one AU is approximately 200 solar radii, so we can get a in terms of solar radii.

So we see that the sun is 1 solar radii away from the center of mass of the sun-Jupiter system. This means that the mass of Jupiter alone causes the sun to displace by a full solar radius!

Secondary authors: Daniel Lo, Monica He