# Find the mass of a planet

1. Feb 10, 2010

### vivic

1. The problem statement, all variables and given/known data
A planet orbits around the Sun with an orbital period of 14.5 years. It is observed it has a moon. At opposition, the moon has an almost circular orbit, and a radius of 7 arcminutes as seen from the telescope. Its orbital period around the planet is 3 days. What is the mass of the planet?

2. Relevant equations
Newton's ver Kepler's 3rd law: p^2 = [4pi^2/(G*M1+M2)]*a^3
G=6.67x10^-11 m^3/kgs^2

3. The attempt at a solution
I used the small angle formula to get "a";
at oppositon, a=(7")(1deg/60")(1.5x10^8km)(2pi/360deg)= 3.05x10^5km
p=72hrs*(3600sec/hr)
(Mplanet+Mmoon)= [4pi^2*a^3]/[G*p^2]

Am I doing this correctly? How do I find the mass of the moon since it doesn't state the relative mass is small enough to be negligible. Where does the orbital period of the planet come into play?

2. Feb 10, 2010

### Chewy0087

Check your Keplers 3rd law, the orbit of a planet around another is independent of its own mass. You may have some difficultly accepting then why we don't see the Sun orbiting the earth once every 100 years or so but remember all of the other planets etc, it's being pulled from many angles resulting in little effect. Whereas in this case it's safe to assume that the only force on the moon is that of the planet.

(Hint : derive it yourself by considering the equations )

$$\frac{mv^2}{r} = \frac{GMm}{r^2}$$
$$v = \frac{2\pi r}{T}$$