Dance of a star: Part I - (Problem 1-3)

by Mee Wong-u-railertkun, David Vartanyan, John Pharo

We present the solution to problems from the worksheet "Planet Host Stars Wobble But They Don't Fall Down" (which could be found here.) The outline of this worksheet is as following. First, we are going to see how a planet affects the movement of a star. From that knowledge, we can observe this "weird" characteristic of a star to find exoplanets. In this post, we present solution to question one to three.

We are provided with useful approximation, as following.

(1) In general, we always say that planets orbit stars. Technically, it is not quite true. Actually, both planets and stars orbit around a vacant point, their mutual center of mass. In Kepler's Third Law, the variable "a" is really the mean semi major axis: a = a_p + a_*. Here, we are going to see how do a_p and a_* depend on the masses of the star and planet, M_* and m^p.

Linear momentum is equal around the center of mass of the system, so we could say that,

Substitute the relationship between velocity and frequencies, we get

Now, what is the relationship between period of the star and the planet?

If their period are not the same, the situation would be as following,

As you can see, a line connecting a star and a planet doesn't go through the center of mass. That is impossible. Thus, we could say that period of a star and a planet must be equal. That is,

Hence, we get the relationship of,

(2) Now, we want to know how much is the Sun displaced from the Solar System's center of mass as a result of Jupiter's orbit. So, we can use the relationship we just discovered.

which could be converted as 1.075 solar radius. Thus, the center of mass of the solar system is approximately on the Sun's radius.

(3) From relationship in part 1, we can write a_* in term of mean semi-axis.

From Kepler's Third law, we reduce by assuming that m<<M.

Use the same assumption on the first equation in part (3) and substitute the Kepler's Third law,

From the definition of the frequency of planet's orbit,

Thus, we can find the velocity of the star as the frequency of planet's orbit times its semi-axis which we already found,

It evolves with time as the orbit of star, i.e. it has the same period as the star.

Thus, if we observed this movement of the star, we can know the mass of the planet orbiting a star.

Acknowledgement

Two figures are created by me using Microsoft Powerpoint. All equations are made by online Latex editor. We thank Jackie for pointing out that period of the star and the planet are the same.