I Kepler orbits for planets of similar masses

AI Thread Summary
The discussion revolves around applying Kepler's laws to a two-body problem where neither mass is significantly larger than the other. It questions how to calculate the period, distance, and velocity of such a system in an inertial reference frame. The third Kepler law is highlighted, noting that the semi-major axis "a" should be considered in relation to the combined masses of the two bodies. The concept of reduced mass is suggested as a potential solution for these calculations. Overall, the thread emphasizes the need to adapt Kepler's laws for more complex gravitational interactions.
jaumzaum
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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
 
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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.
 
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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##.
 
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