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Consider a system of two masse $M$ and $m$ with $m<<M$ and the mass $m$ orbiting around $M$. Then $m$ describes a elliptic orbit with period $P$ and the third Kepler law states that: $$\frac{P^2}{a^3}=\frac{4\pi^2}{G(M+m)}\approx \frac{4\pi^2}{GM}$$ with $a$ the semi-major axis. It means that if I have a heavy planet with a satellite orbiting around it, I can know the mass of the planet by knowing only the period of the satellite? I would like to make sure that I understand well this law, that is why I am asking this question.

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    $\begingroup$ The left side doesn’t involve only the period. $\endgroup$ – G. Smith Jul 4 at 17:18
  • $\begingroup$ "... I can know the mass of the planet by knowing only the period of the satellite?" What makes you think that could be correct? Look carefully at your own equation. $\endgroup$ – Bill N Jul 4 at 17:37
  • $\begingroup$ You have to measure the semi-major axis, the period of the orbiting satellite, and the mass of the satellite to get enough information. Once this is done, the only unknown is M, which can be algebraically separated and solved for. $\endgroup$ – David White Jul 4 at 18:35
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Yes, you understand this law correctly. If you know the period $T$ of the orbit, and its major semi-axes $a$, you can estimate the mass $M$ of the planet. This is as well explained on wikipedia and many other sources.

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