Is there a limit to the "degrees of orbits"?

During the Apollo space mission (11-17) there was a star called The Sun orbiting the center of a galaxy. That star had the planet Earth orbiting it and that planet had The Moon orbiting it and that moon had a space capsule orbiting it. Is there a mathematical or physical limit to how many things can orbit each other before everything collapses or at least no more “sub-orbits” can be achieved?

Electron orbiting the nucleus?

Just because I like to be fussy I must say that an electron does not orbit a nucleus in the way a planet orbits a star. An electrons “trajectory” around a nucleus has nothing to do with gravity.

No, there is no theoretical limit on the number of levels of things orbiting other things. Practically it may be hard to arrange the objects so that the number of levels is much higher than the example you suggest.

Gravity gets weak at short distances, so tiny orbits must be much slower.

A lost wrench could orbit the space shuttle, but the shuttle is partly in the ionosphere, so the “wind” would mess things up. The same goes for solar wind and small rocks which orbit asteroids: they get blown off like a comet’s tail. (If not, then the solar system would be choked with dust, where the bigger dust is orbiting the smaller dust.)

Even if you’re out between galaxies, at some point the photon pressure of light from distant stars or the electric forces from a few surface ions on an object will knock away anything in a sub-sub-sub orbit.

Surely it gets stronger as the distances shorten?

Automagic, are you limiting the question to orbits that are maintained through gravity?

At the large scale, there is a size limit imposed by the finite speed of light and age of the universe. Distances greater than this size pose the problem that each of the objects could not have started to exist yet, from the point of view of the other object, so they can’t pull on one another.

First of all, it only gets weak relative to other forces. Second, it depends on what you mean by “slower”: If you’re orbiting close to the surface of your parent object, and it’s spherical, the time it takes for one orbit depends only on the density of the parent object. A low orbit around the Earth takes about an hour and a half, and so does a low orbit around a small asteroid. A low orbit around Jupiter or the Sun would take longer, because they’re less dense than the Earth.

To get erntirely technical, no one thing actually orbits another – they both orbit the common center of gravity of the two-mass system they comprise. However, when one is vastly larger/more massive than the other, as in artificial Earth satellites or Mars and its moons, the common center of gravity is displaced a small fraction of a centimeter from the center of the larger object.

For kicks, let’s set up a hypothetical system. A type B “blue giant” star many times the mass of the sun has a type M red dwarf companion star a fraction of the sun’s mass, in a quite distant orbit. For all practical purposes, the red dwarf orbits the blue giant. Now, suppose the red dwarf has a superjovian planet, seven times the mass of Jupiter (2225 Earth masses) – not massive enough to begin fusion but big. It in turn is being orbited by a Mars-size body – technically a satellite, not a planet, but only by the definition of whether the primary is a star or planet. And this pseudo-Mars has small moons of its own. In each case the predominant gravitational effect on the smaller body is that of its larger primary – the dictionary definition of “to orbit”. And that one we took to five natural layers. And the B/M star system may be part of a star cluster with internal orbital cohesion, which in turn is orbiting the center of the Galaxy. Further, let’s put this in a Magellanic-cloud type of galaxy which is itself orbiting a larger galaxy, as the two Magellanic clouds do the Milky Way and M32 does the Andromeda Galaxy. We now have an eight-layer sequence. Technically each pair is in an orbit around its common center of gravity – but for all practical purposes that center is very close to the center of the primary taken by itself.

And, of course, the Milky Way and M31 orbit around each other, too, though there you’d be hard-pressed to say that one is the primary (M31 is a bit bigger than us, but only a bit). And the Milky Way-M31 pair is also gravitationally bound to (and therefore orbiting) the Virgo cluster.