Solve Orbital Period for Two Identical Planets Around Star

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Homework Help Overview

The problem involves two identical planets orbiting a star in circular paths, positioned diametrically opposite each other. The task is to derive an expression for the orbital period in terms of the masses of the planets, the mass of the star, the radius of the orbits, and the gravitational constant.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss whether the gravitational interaction between the two planets is relevant to the orbital dynamics, with some suggesting that it may influence the acceleration of each planet.
  • One participant seeks resources for understanding similar problems, noting that their textbook only addresses single satellite systems.
  • There are attempts to set up equations involving gravitational forces and centripetal acceleration to relate the variables of the problem.

Discussion Status

The discussion is ongoing, with participants exploring different aspects of the problem and questioning the relevance of mutual gravitational effects. Some guidance has been offered regarding the setup of equations, but no consensus has been reached on the approach to take.

Contextual Notes

Participants note that the problem is complicated by the presence of two planets, which is not covered in their existing resources. There is an emphasis on the center of mass and its implications for the system's dynamics.

char808
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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?
 
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char808 said:

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



∑Fm=GMm/r^2 +Gmm/(2r)2

So can I say:?

mam= GMm/r2+Gmm/(2r)2 = mv2/r

and T=2∏r/v
 
Last edited:
char808 said:
Ok, do you have a resource on how to look at these problems? My book only covers 1 satellite around a planet.



∑Fm=GMm/r^2 +Gmm/(2r)2

So can I say:?

mam= 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.
 
char808 said:
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
 

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