I can't seem to find this answer anywhere online. I am trying to work through a solid state physics textbook for research and this popped into my head. I don't have any mathematical background on quantum mechanics and other more advanced physics topics.

Obviously, atoms can't be described using similar terms to a star system for the most part. Knowing that, I was wondering if having an atom with two or more orbiting nuclei would be a possibility. From my understanding, nuclei can really only be stable up to certain sizes which is why you get atomic decaying. Would there be a set of subatomic forces that might allow for this sort of equilibrium to occur?

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    $\begingroup$ A nucleus has a positive charge. Once you break it into two smaller parts, assuming both qualify as a "nucleus", then the repulsive forces become significant. If you keep these parts close enough that attractive forces dominate, you don't really have a "binary" system. $\endgroup$ – Floris Jul 31 '17 at 18:00
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    $\begingroup$ An "atom" with a binary nucleus, which I interpret as two of them, is a molecule. A hydrogen molecule $H_2$ is a sort of binary atom. $\endgroup$ – Lawrence B. Crowell Jul 31 '17 at 18:23

In star systems you have a lot of options for exotic systems like this (one famous example). That flexibility comes about because star systems are large compared with the de Broglie wavelengths of any of their components, so that stellar orbits are not quantized. With atoms and their electrons and nuclei you are further constrained.

As a commenter observes, if you have two nuclei and enough electrons to make things neutral, and remove thermal energy until particles are forced to bind to each other, what you get are molecules. For example, in the dihydrogen ($\mathrm H_2$) molecule, the two protons are separated by roughly an ångstrom, which is very different from the $10^{-5}\,\mathrm Å$ that separates nucleons within a nucleus. The $\mathrm H_2$ molecule has two electrons, which orbit both nuclei. The low-energy excitations of the hydrogen molecule are rotations, which have energy $E_J = \frac{\rm 15\,meV}{2} J(J+1)$ for orbital quantum number $J=0,1,2,\ldots$; these states correspond to different amounts of angular momentum as the two nuclei orbit each other.

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    $\begingroup$ And there is also a bound system with two protons (two hydrogen nuclei) and just one single electron. The electron "orbits" both nuclei and hold them together. Of course the total charge is nonzero, so we have a molecular ion. This system is called the dihydrogen cation (Wikipedia) or similar names, symbol $\mathrm{H}_2^+$. $\endgroup$ – Jeppe Stig Nielsen Aug 28 '17 at 8:43

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