Solve Orbital Period for Two Identical Planets Around Star

[char808]

Homework Statement

Two identical planets (equal masses, m) move in identical circular orbits around a star (mass M) diametrically opposed to each other (opposite sides of the planet). Find an expression in terms of m, r, M and G for the orbital period T.

Homework Equations

T^2=(4pi^2/GM)r^3

F=Gm1m2/r^2

The Attempt at a Solution

I haven't really gotten to far on this because I can't decide if the planets are affecting each other. It would seem that they are because all mass exerts a gravitation force on other mass. But they are not orbiting around each other...So the force between the two is irrelevant to the problem?

 

[Dick]

Homework Statement

Two identical planets (equal masses, m) move in identical circular orbits around a star (mass M) diametrically opposed to each other (opposite sides of the planet). Find an expression in terms of m, r, M and G for the orbital period T.

Homework Equations

T^2=(4pi^2/GM)r^3

F=Gm1m2/r^2

The Attempt at a Solution

I haven't really gotten to far on this because I can't decide if the planets are affecting each other. It would seem that they are because all mass exerts a gravitation force on other mass. But they are not orbiting around each other...So the force between the two is irrelevant to the problem?

The force between the two is very relevant. The acceleration of each of the two planets is determined by the combined forces of the star and the other planet.

 

[char808]

Ok, do you have a resource on how to look at these problems? My book only covers 1 satellite around a planet.



∑F_m=GMm/r^2 +Gmm/(2r)²

So can I say:?

ma_m= GMm/r²+Gmm/(2r)² = mv²/r

and T=2∏r/v

 

[Dick]

Ok, do you have a resource on how to look at these problems? My book only covers 1 satellite around a planet.



∑F_m=GMm/r^2 +Gmm/(2r)²

So can I say:?

ma_m= GMm/r²+Gmm/(2r)² = mv²/r

and T=2∏r/v

Good! Now all you have to do is solve for v and put that into your formula for T.

 

[Andrew Mason]

Ok, do you have a resource on how to look at these problems? My book only covers 1 satellite around a planet.

It is easier to do if you have two identical planets on opposite sides of the star like this.

The centre of rotation is always the center of mass of the system. In a one-planet system, this depends on the relative masses. In this case, you know that the centre of rotation is the centre of star regardless of the value of m.

AM