Solve Orbital Period for Two Identical Planets Around Star

In summary, the orbital period T for two identical planets moving in identical circular orbits around a star can be expressed as T=2π√(r^3/(GM+m)) where r is the distance between the planets and m is their mass. The force between the two planets is relevant and can be determined using the equation ∑Fm=GMm/r^2 +Gmm/(2r)2. The centre of rotation is always the centre of mass of the system, in this case, the centre of the star.
  • #1
char808
27
0

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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  • #2
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.
 
  • #3
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:
  • #4
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.
 
  • #5
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
 

FAQ: Solve Orbital Period for Two Identical Planets Around Star

1. How do you calculate the orbital period for two identical planets around a star?

To calculate the orbital period for two identical planets around a star, you can use Kepler's third law, which states that the square of a planet's orbital period is proportional to the cube of its semi-major axis. This means that if you know the semi-major axis of the planets' orbits, you can calculate the orbital period using the formula: T^2 = (4π^2/GM) * a^3, where T is the orbital period, G is the gravitational constant, M is the mass of the star, and a is the semi-major axis.

2. Can the orbital period of two identical planets around a star vary?

Yes, the orbital period of two identical planets around a star can vary depending on their distance from the star. The closer the planets are to the star, the shorter their orbital period will be due to the stronger gravitational force pulling them towards the star. Similarly, the farther the planets are from the star, the longer their orbital period will be.

3. How does the mass of the star affect the orbital period of two identical planets?

The mass of the star does not affect the orbital period of two identical planets, as long as the distance between the planets and the star remains the same. This is because the mass of the star only affects the strength of the gravitational force, but the distance between the planets and the star determines their orbital period according to Kepler's third law.

4. What other factors can affect the orbital period of two identical planets around a star?

Besides the distance between the planets and the star, other factors that can affect the orbital period include the eccentricity of the planets' orbits, the gravitational pull of nearby planets or moons, and any external forces acting on the planets such as tidal forces.

5. Can the orbital period of two identical planets be the same as another set of planets orbiting the same star?

Yes, it is possible for the orbital period of two identical planets to be the same as another set of planets orbiting the same star. This can occur if the planets have the same semi-major axis and are in a similar orbital configuration. However, it is unlikely for two sets of planets to have identical orbital periods due to the many variables that can affect their orbits.

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