# Saturns moons

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After receiving amazing help last time from Swansont, I really hope someone can help me again!

The question is the following:

Janus and Epimetheus are moons of Saturn. The difference in the radius of their circular orbit is 50 km. De radius of the inner moon is 2.51 times the radius of Saturn.

Calculate after how many orbits the inner moon passes the outer moon.

Below is my sketch of the question. Can someone tell me if I'm going about this the right way? Can I just call the radius of Saturn x? Or do I actually have to look up the radius of Saturn? Or am I approaching this completely wrong? Anyway, after some searching I think I need to use the proportion of the distance to the period. , with P in earthyears and R in AU (is that correct?).

I'm not really sure how to continue. Can anyone give me a push in the right direction or correct me somewhere?

Edited by Applejuice
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The distances used in problems like this are from centers, not surfaces. I think x will drop out of your equations.

If you have a circular orbit, the orbital speed related to the orbital radius; the force is gravitational and that's the centripetal force, so

mv2/r = GMm/r2 or v2 = GM/r

When you compare two orbits, M (Saturn's mass) should also drop out

You'll need to relate the speed to a period, knowing that each moon is traveling a circumference of a circle

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I've tried for a while now to understand how to solve the problem but I still can't figure it out; do you have an additional hint?

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Some questions to get you started:

• How long does it take for each of the moons to complete a full lap?
• Using this result, how many laps will each of the moons have completed after an arbitrary time t?
• What is the condition for the time when the inner moon passes the outer one?

That being said, I think you are already on to a good start from your mention of Kepler's third law.

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The formula you have is a good starting point. Now here's the trick, P does not have to measured in Earth years, nor does R have to be measured in AU. We only do that when we are comparing the orbits of the other planet's to that of the Earth.

In your case, you are comparing the orbits of two moons orbiting around Saturn. The thing to to remember, is that P^2/R^2 gives the same answer for both moons.

Another thing is that everything must be in the same units. In your example, you are told that the radius of the inner orbit is 2.51 that of Saturn and that the outer moon is 50 km further out. Saturn radii and km are different units. You'll have to convert one to the other to use the formula correctly, and for that you need to know how many km there are in one Saturn radius.

One further hint, after you find the relative periods of the moon's orbits, you will still have to find their synodic period to answer the question.

Edited by Janus

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