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[tex]T=2\pi\sqrt{\frac{a^3}{G(M_1+M_2)}}[/tex]

I do not understand where does this M

_{1}+M

_{2}can possibly come from. My thinking is as follows: This formula must also be valid for circular orbits, so for simplicity I am considering a as the radius, instead of semi-major axes. Let M

_{1}=M be our sun, and M

_{2}=m is the orbiting planet. Equating the accelerations found from Newton's law of universal gravitation and from circular motion:

[tex]m\left(\frac{2\pi}{T}\right)^2r=G\frac{Mm}{r^2}[/tex]

Therefore,

[tex]T=2\pi\sqrt{\frac{a^3}{GM}}[/tex]

Where does m come into the picture, doesn't it just cancel? For M>>m, the given formula simplifies into what I found alright, but why, in the general case, it is true? The first thing that came into my mind was that I ignored general relativity by using Newton's law of gravity, but I think the effect of general relativity must be far smaller than the difference M+m makes. Another possibility that occurred to me is that in my derivation I assumed M stationary, which could be wrong since the planet also pulls the sun, but I do not know how to put that into the picture. Is that the reason of that M+m, or am I missing something else?