# Kepler's Law of planetary motion

1. Jan 13, 2015

### Tulatalu

1. The problem statement, all variables and given/known data
Two stars of masses M and m, separated by a distance d, revolve in circular orbits around their center of mass.Show that each star has a period given by
T^2= (4π^2)(d^3)/ G(M+m)

2. Relevant equations

3. The attempt at a solution
I
know the Kepler's Laws can be expressed as T^2= (4π^2)(d^3)/ G(M) but i don't know how it is applied when 2 planets interact with each other (both in circular motion)

2. Jan 13, 2015

### BvU

It's a given that the orbits are circular. Simple kinematics, therefore.

3. Jan 13, 2015

### Tulatalu

Can you explain a bit further please, how does it relate to the Kepler's law. I thought their must be something to do with the gravitational force

4. Jan 13, 2015

### Staff: Mentor

Kepler's 3rd Law is a special case of Newton's more general formula relating the period to the size of the orbit (the semi-major axis for elliptical orbits, the orbit radius for circular orbits). Kepler's Law makes the unstated assumption that the mass M of the Sun is much, much greater than that of the planet m in orbit so that (M + m) ≈ M. Newton's laws can be applied without making such an assumption.

In this problem you are to show that the individual periods of the orbits of the objects about their common center of mass are as given. You should be able to determine the radius of each orbit by locating the center of mass. Hint: use the angular motion form for centripetal acceleration (involving ω) since there's a simple relationship between ω and period.

5. Jan 13, 2015

### Tulatalu

I still don't knowhow to apply Kepler's Law to this problem. I can mathematically prove the Kepler's Law in case of satellite moving around earth but with 2 planet in circular orbit I have no idea. Can you please explain a bit further and use the equation so that it would be easier for me to follow.

6. Jan 13, 2015

### DEvens

You don't need Kepler's law to derive the period of he orbits. You are given that the orbits are circular, the masses, and the radius. You should be able to work out the rest fairly easily. What speed must the stars be moving at for the circular orbits to be stable? And once you have the speed, what is the period?

7. Jan 13, 2015

### Staff: Mentor

You use the same methodology: Pick one of the planets. Determine the radius of the orbit. It's moving in a circle so equate the gravitational acceleration (due to the other planet, which takes the role of the "Sun" in this case) to the centripetal acceleration.

8. Jan 13, 2015

### Tulatalu

Thanks everybody. It turn out to be not as complicated as I think it is :D

9. Jan 13, 2015

### Tulatalu

Coukd I ask one more question : how about the period of 3 identical planets moving around another planet and they are positioned one third of a revolution apart from each other? Do we have consider the force between the three planets or just simply the force between them and the planet in the centre?

10. Jan 13, 2015

### Staff: Mentor

Um, there would be no planet at the center if they are spaced equally around the orbit. Like the two-planet scenario, the center of the orbits is empty.

But yes, you must take into account all the forces acting and resolve them (vectors!) into the net force that provides the centripetal force for each planet.

Note that many configurations of multiple objects in mutual orbit are not stable over long periods of time. If you're interested in the topic, I might suggest starting with a search on "Klemperer rosette" :)

11. Jan 13, 2015

### Tulatalu

Now it's too much for me :D but thanks anyway.