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Would it be possible to make artificial constructs (satellites) orbit each other? I looked at the gravitational laws, and nothing seems to suggest that this is not possible.

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    $\begingroup$ Your question doesn't make sense, yes two bodies can orbit each other, I'll give you an example The Earth and The Sun. $\endgroup$ – iharob Jul 4 '15 at 0:54
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It is certainly possible in principle.

That said, you should look up the Hill sphere and consider the amount of mass required to make the arrangement stable in the face of perturbation from the rest of the solar system.

If I wanted to plan this as a technology project in the near future I would probably propose the Earth-moon L4 or L5 point at the venue.

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Say, you want to these objects of equal masses and let them rotate on circular orbits (just to understand the scalse). You will thus have a force between them \begin{equation} F={Gm^2\over d^2}, \end{equation} where $d$ is a distance between them. Thus you'll get from $v^2/(d/2)=F$ \begin{equation} v^2=\frac{Gm^2}{2d}. \end{equation} So if you have, say, $m=50$ tons, $d=1$ km, they will orbit with \begin{alignat}{2} v=9.13 ~\text{cm/s (and with a period of}~P=96^{\rm h}=4^{\rm d}\text{)}. \end{alignat} Making them rotate elliptic orbits may change something, but not drastically.

So, theoretically, you can make this possible. The question is, how fast do you need them to rotate. For example, to make them rotate with at least $1$ m/s, you would require $5000$ tons.

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