jupiter's mass? need help.. please?

Question

in 1610, Galileo discovered four of the 16 moons in Jupiter. the largest of which is Ganemede. this Juvian moon revolves around the planet on 7.16 days in a nearly circular orbit whose radius is about 1.07 x 10^6 km, using this data, find the mass of Jupiter? can u please explain why your answer is that way. thanks

Answer 1

Let's assume the orbit of the moon is a perfect circle and all the mass of Jupiter is centred in its midpoint.

According to Newton's laws an object keeps moving in a straight line unless there is a force working on it.
The moon is circling around Jupiter so there must be a force holding bending its path.
This force is gravity. This magnitude of this force depends on the mass of both Jupiter and the moon.

F_g = (G * M_jup * M_moon) / r^2
(G is a known constant)

Because the moon is moving on a circle we also now that the centripetal force ([1]) is:

F_g = M_moon * v^2 / r

So we can combine these two equations and get (after dividing away M_moon and r):

G * M_jup / r = v^2

M_jup = r * v^2 / G

Now all we have to do is find v (the speed of the moon)

It takes 7.16 days = 618 624 seconds to move on a circle of radius 1.07 x 10^6 km.
The length of the path is 2*pi*r = 6 723 x 10^6 m.

So v = distance/time = 10 868 m/s.

Filling all this in (and knowing G = 6.673x10^-11 m^3 s^2 / kg [2]):

M_jupiter = 1.89 × 10^27 kg

Answer 2

1.9 10¨27 kg