Kepler orbits for planets of similar masses

[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

 

[Filip Larsen]

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.

 

[vanhees71]

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 $M \gg m$.