# Kepler orbits for planets of similar masses

• I
• jaumzaum
In summary: So, if you want to calculate the period, distance and velocity of a two-body system in relation to an inertial frame, you first have to find the masses of the two bodies and use the original Kepler's law for the Sun plus the mass of the planet.
jaumzaum
When we use the third Kepler law to calculate the period, distance and velocity of the Earth, we consider that the Sun is fixed. We know this is not true, because the Sun is also attracted by the Earth. I was wondering, how could we use Kepler laws to calculate the period, distance and velocity of a 2-body-problem in relation to an inertial reference frame, if neithe of them has a mass much larger than the other.

Another doubt, in Third Kepler law seen below:
##\frac{T^2}{a^3}=\frac{4\pi^2}{G(M+m)}##
The "a" is calculated in relation to the other planet (the referential is the second planet) or in relation to an inertial frame?

To illustrate what I mean above, If we consider two bodies, of masses M and 3M, the maximal distance between them is x and the minimum is y. Haw can we calculate the period and the semi-major axes and eccentricities of both ellipses?

Thank you very much

jaumzaum said:
I was wondering, how could we use Kepler laws to calculate the period, distance and velocity of a 2-body-problem in relation to an inertial reference frame, if neithe of them has a mass much larger than the other.
You may want to look into the concept of reduced mass.

topsquark, PeroK and Ibix
jaumzaum said:
When we use the third Kepler law to calculate the period, distance and velocity of the Earth, we consider that the Sun is fixed. We know this is not true, because the Sun is also attracted by the Earth. I was wondering, how could we use Kepler laws to calculate the period, distance and velocity of a 2-body-problem in relation to an inertial reference frame, if neithe of them has a mass much larger than the other.

Another doubt, in Third Kepler law seen below:
##\frac{T^2}{a^3}=\frac{4\pi^2}{G(M+m)}##
The "a" is calculated in relation to the other planet (the referential is the second planet) or in relation to an inertial frame?

To illustrate what I mean above, If we consider two bodies, of masses M and 3M, the maximal distance between them is x and the minimum is y. Haw can we calculate the period and the semi-major axes and eccentricities of both ellipses?

Thank you very much
This is the equation of Kepler's 3rd Law under consideration of the finite mass of the Sun, ##M##. That's why the right-hand side also depends on the mass of the planet, ##m##, and thus it's not Kepler's original law, which states that ##T^2/a^3=\text{const}##, i.e., the same constant for all planets in our solar system. That's indeed a good approximation, because ##M \gg m##.

topsquark

## 1. How do Kepler orbits work for planets of similar masses?

Kepler's laws of planetary motion state that the orbit of a planet around a star is an ellipse with the star at one focus. This applies to planets of similar masses as well, meaning that both planets will orbit around their common center of mass.

## 2. What determines the shape of a Kepler orbit for planets of similar masses?

The shape of a Kepler orbit for planets of similar masses is determined by the eccentricity of the ellipse. The closer the eccentricity is to 0, the more circular the orbit will be, while an eccentricity closer to 1 indicates a more elongated, elliptical orbit.

## 3. Can planets of similar masses have different orbital periods?

Yes, planets of similar masses can have different orbital periods. According to Kepler's third law, the square of the orbital period is proportional to the cube of the semi-major axis of the orbit. Therefore, if the semi-major axes of two planets are different, their orbital periods will also be different.

## 4. How does the distance between planets affect their Kepler orbits?

The distance between planets does not directly affect their Kepler orbits, as the shape of the orbit is determined by the eccentricity. However, the distance between planets can affect the gravitational pull between them, which can influence the shape of their orbits over time.

## 5. Can planets of similar masses have stable Kepler orbits?

Yes, planets of similar masses can have stable Kepler orbits. As long as the planets are in a stable, elliptical orbit around their common center of mass, their orbits will remain stable. However, if there are other massive objects nearby, their gravitational pull can disrupt the stability of the orbits.

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