# How many 'recursive' gravitational orbits are possible?

I was thinking about how the moon orbits around Earth, which orbits around the Sun, which orbits around the center of the Milky Way.

I think of these kinds of orbits as recursive. For each body $B_n$, we can imagine adding another smaller body $B_{n+1}$ which orbits around $B_n$ such that the center of mass of the system remains within the radius of the larger body and the system as a whole is stable.

Is there a limit to how many times this can be repeated? Is there an ideal ratio of masses for these bodies? Assuming $B_0$ is the supermassive black hole at the center of the Milky Way, how small can we get before quantum effects take over?

• I am not aware that nature stacks these orbits more than about half a dozen deep, if we start with large galactic structures, rotation of globular clusters around the centers of galaxies, binary stars within a globular cluster (that's rare, I believe), planets around an individual star of the binary, moon around a planet and some debris around the moon... that's probably it. There are no orbits in QM, so the buck stops where gravity becomes too weak to bind objects against outside forces like radiation pressure. – CuriousOne Aug 14 '16 at 23:14
• What ideas do you have about the answers to your questions? – sammy gerbil Aug 14 '16 at 23:20
• @sammygerbil Other than what we can already observe, I don't really know. I wasn't able to find any resources about this online. I don't have a particularly strong background in physics to do it myself. Why the downvote? – Curtis Bechtel Aug 14 '16 at 23:30
• I down-voted because I think your question "does not show any research effort... and is not useful." How have you tried to apply your knowledge of physics to this problem? What resources have you found which might be useful in solving it? Why do you think this question would be useful to other users? – sammy gerbil Aug 15 '16 at 0:02
• Good question, but it's not quite right to imagine the Sun in orbit about the centre of the Galaxy. The central black hole doesn't dominate gravitationally the background stars in a galaxy except very close to the core. To good approximation the stars move within a smooth potential in non-Keplerian orbits. – AGML Aug 15 '16 at 3:31

Star systems.

There's already a word for this called Hierarchy, described "here". The largest recognized is 4, but that may be unstable. The largest stable one known is 3 hierarchies and when star systems form, they can have 7, 8 or 9 stars, and perhaps more than 4 hierarchies, but those systems tend to be unstable. I think your question only makes sense with stable orbits, so 3 might be the limit just looking at stars, just maybe 4.

Galaxies

Galaxies can orbit other galaxies. 2 Dwarf Galaxies orbit the Milky way.

Does our local group of galaxies orbit a larger group? Does any local group orbit another larger one? There's no evidence that this is the case. Question on that here, and there's limitations on how far galaxy and local group orbits can be extended due to gravity dropping off by the square of the distance and space being so empty and space expanding. I see no logical reason why a galaxy like our Milky way couldn't orbit (instead of crash into) Andromeda, but that's just theoretical. When you get into local groups orbiting something, and not being just an N-body system, but an orbit, that gets more convoluted, and you'd be talking about orbits tens if not hundreds of millions of years in period. As a hunch, I'm going to limit the heircarchy of galaxy orbits to 3. Dwarf galaxy around a spiral galaxy, around a larger galaxy or clump of galaxies. - A limit of 3 is kind of a guess on my part. I also think that's a kind of ugly part of the question. Yes, Galaxies can orbit other galaxies, but it's on such a different time scale.

Stars obviously orbit within a galaxy, but orbits are jostled by other stars, sometimes they even get kicked out or their orbits can measurably change. It's more like orbiting within, not orbiting around.

Planets and Moons.

There's a problem we need to address first and that's the type of star system the planets orbit.

If the 2 stars orbit close to each other, it's very hard to get a planet around one of those stars. (It's possible to have a planet around both), but you only get to count the next orbit once if the planet orbits around both stars.

See diagram

So, we need "the other kind", not "Circumbinary".

And for "the other kind" to have planets with moons, you need a fair bit of distance between the stars, so heirchy 3 or 4 star systems, we'd need to get a closer look at the outer stars in that system to see if there's room for planets and moons. A close binary, there wouldn't be.

So, there's a natural limitation to this process. When orbiting objects get too close, there's limitations to how many layers of orbits you can go and maintain stability.

Moons have a very hard time having moons. In our solar system there's no known cases of Moons having Moons. There are cases of asteroids orbiting asteroids, but no known cases of asteroids orbiting moons, even though Moons are quite a bit larger on average. Part of this may be that we don't have telescopes good enough to see, but there's also reasons why it's difficult. Three or more body systems tend towards instability and tidal forces would tend to disrupt such systems. Also, the sphere of influence gets quite small for small objects orbiting significantly more massive ones. In addition to small sphere's of influence, perturbations grow with more than one gravitational object in the vicinity. Earth's moon, for example, is perturbed by the sun and it's not gravitationally smooth, so orbits around our Moon are quite unstable long term they are impossible.

For a moon to have a moon, it helps if the planet is quite distant from it's sun and it helps if the Moon is quite distant from it's planet to avoid tidal forces pulling the two objects together. Planet X, for example, could have a moon that has a moon, cause it's so far from the sun, it's almost like a mini solar system. Pluto is quite small but gravitationally it's like a binary system of it's own. Distance helps, but even Pluto-Charon have only 1 hierarchy of moons.

So, there are practical limits of how many orbits around orbits you could build and have them be reasonably long-term stable because you need a significant amount of space and, for the most part, a couple of orders of magnitude between the mass of each object.

If I was to guess, if we say 3 for galaxy orbits, 1 for the system around a galaxy, 3 for stars, and then planet, Moon and moon around the moon, that would be 10 layers of orbits and That would probably be quite rare, as a 3 star hierarchy with enough space to have a planet, moon, moon is a bit of a stretch. That would be my guess as an outside limit.

Now, in practice, if you wanted to build your own orbits in the middle of nowhere, you could go higher than that, but given the nature of galaxies, I think 10 orders of orbit is about as far as it's likely to go and that might even be too high. Might be 9 or maybe even 8.

Since no exo-moon has ever been observed (Exo-Ring systems have), and I'm not sure if any planets have ever been observed on 3 or more heirarchy star systems, I think the observed limit to layers of orbits is quite a bit less than my estimated limit.

The other answers have nicely addressed the scale of increasing size. This answer provides detail on the scale of decreasing size. There are no known cases of moons having moons in the solar system. Two key things get in the way of moons having moonlets having even smaller moonlets (etc.).

One issue is the ever-decreasing volume surrounding an object about which orbits are stable. One way to express this volume is via the Hill sphere. which provides a rough estimate of the volume surrounding an object about which orbits are stable. As scale decreases, the Hill sphere will eventually become smaller than the moon itself.

Another issue is that as small objects tend to be less round than larger ones. Mimas at about 400 km in diameter is the smallest object rounded by self-gravitation in the solar system. Smaller objects look more like a lumpy potato than a sphere. This results in a non-spherical gravitational field. Even objects as large as our own Moon have a significantly non-spherical gravitational field. This means that orbits close to our Moon will be unstable. For example, read about the case of bizarre lunar orbits suffered by the PFS-1 and PFS-2 subsatellites released by Apollo 15 and 16.

There are multiple star systems. We can start with that as an ideal case. Algol is a triple star system with two bodies in a tight orbit and the other star further out. Even more interesting is the recently discovered quadruple star system HD98800 with a pair of close orbiting stars in a larger orbit. These are relatively stable systems. The small perturbation of the dipole oscillation of the pairs is below a criterion for chaos in the KAM theory. You will have to look up KAM (Kolmogoroff-Arnold-Moser) theory of punctured tori in phase space. It is a long and complicated theorem.

Can this in principle continue, in a Newtonian framework? I would be tempted to say yes, but this is working some sort of renormalization scaling scheme in KAM theory. In other words, could the two stars in each of these pairs in a quadruple system be composed of two stars. This would result in the sequence $4~\rightarrow~8~\rightarrow~16~\dots$. The phase space doubles in dimension as do the tori of motion. So this appears to put some sort of renormalization group flow condition, similar to problems with turbulent eddies that have sub-eddies, and so forth in hydrodynamics. This would be a very difficult problem to work on, and would be easily worth a Ph'D thesis topic.

Does nature continue onwards indefinitely? This is instead of pure mathematics. The largest gravitational bound systems are galaxy superclusters. The Milky Way is in the Virgo supercluster. Further down are galaxy clusters, the local group being what the Milky Way is in. Then galaxies, gravitational clumps in galaxies (globular clusters, gas clouds etc), multiple star systems and planetary systems, down to planets and moons, and there are are asteroids that have smaller asteroids as moonlets. Things can't get bigger because dark energy and exponential expansion will tear them apart. Things like filaments and domain walls are being stretched by cosmic expansion. Things can't get smaller than asteroids because gravitation is too weak.