Exoplanets and Wobbly Stars

Exoplanets are planets found outside our solar system, and remain the focus of a rapidly growing field of astronomy. Detecting them, however, can be difficult, since they don't radiate light like stars, and are much smaller. In this post I'll be introducing the physics of one of the methods we use to detect exoplanets. The method utilizes the fact that exoplanets exert a gravitational force on their host star, resulting in a "wobbling" of the star as it orbits ever so slightly around the two bodies' center of mass.

Let's start from the basics. Consider a star and a planet, orbiting each other around their mutual center of mass:

The equation for the center of mass is:

In this case, we have set the center of mass to be at x=0. Substituting in our variables, we have:

Simplifying, we have the following relationship:

Thus, the distance from the star/planet to the center of mass is inversely proportional to its mass.

What can we do with this? We can use Kepler's Third Law to find out how much the Sun wobbles. Kepler's Third Law:

where a is the mean semimajor axis, the sum of a_p and a_*, and M is the sum of the mass of the star and the mass of the planet. If we assume the mass of the star to be much greater than the mass of the planet, we can use Kepler's Third law for the planet's orbit and solve for either a or T (since the mass of the star is so large, a is effectively equal to the planet's orbital radius). For example, if we consider the effect of Jupiter on

the sun:

Solving for a_* (using the fact that the mass of Jupiter is about 1000th of the Sun's mass), we have:

Thus, the displacement of the Sun due to Jupiter's gravitational field is roughly a solar radius. That's an incredibly small distance to measure! That's also why we measure velocity, instead of displacement, when searching for exoplanets.