WS 7, Problem 1: Kepler's Third Law

For a planet of mass m orbiting a star of mass M$_{*}$, st a distance a, start with the Virial Theorem and derive Kepler's Third Law of motion.  Assume that m<<M$_{*}$.  Remember that since m is so small, the semimajor axis, which is formally 
a=a$_{p}$+a$_{*}$, reduces to a=a$_{p}$.


We can say that a=a$_{p}$ because the planet is so much smaller than the star that it can essentially be treated as a point mass by comparison to the star.  The distance to the planet's center is too small compared to the distance to the star's center to significantly change a.

We started by drawing this picture:

We were told to start with the Virial Theorem, which defines the relationship between the kinetic and potential energies of a system bound by potential forces.  

$K = -\frac{1}{2}U$

We substituted the equations for the kinetic and potential energies and ended up with this equation:

$\frac{1}{2}mv^2=\frac{1}{2}\frac{GM_* m}{a}$

We knew from previous worksheets that Kepler's Third Law related orbital period to the semimajor axis of the orbit.  We just couldn't remember exactly what the equation was (which was good, because that was the equation we were trying to find, and knowing it beforehand would have ruined all the fun).  Since orbital period measured in time, we decided to solve the above equation for seconds.

There was a brief moment of panic when we saw that there aren't any seconds in the equation, but that moment quickly passed when we remembered that velocity is distance per time.  We quickly realized that, in an orbit, the distance traveled in the velocity is circumference of orbit per orbital period.  That led us to this equation:

  $\left ( \frac{2\pi a}{P} \right )^2= \frac{GM_*}{a}$

where P is the orbital period and we assume a circular orbit when solving for its circumference.  Simplifying the equation gives us these equations:

$\frac{4\pi ^2a^2}{P^2}=\frac{GM_*}{a}$

$P^2=\frac{4\pi^2a^3}{GM_*}$

We compared our answer to the equation provided to us on an old worksheet and it was correct!

I've said "we" a lot, and by that, I mean Dennis Lee, Delfina Martinex-Pandiani, and myself.