<h3>Bound states</h3>
<p>In retrospect the below answer is a bit long winded, so I will summarize the basic point: to get &quot;orbital&quot; motion, one simply needs that some object is trapped in the potential well of another, meaning it doesn't have enough kinetic energy to escape. If the potential well is rotationally symmetric then there will be conservation of angular momentum, and if the potential has <span class="math-container">$1/r$</span> dependence then the orbits will be <a href="https://en.wikipedia.org/wiki/Kepler_orbit" rel="nofollow noreferrer">Keplerian</a> (at least classically), but that is not necessarily important. All that matters is that you can have a hierarchy: some objects A, B, C, . . . with decreasing &quot;charges&quot; (e.g. gravitational mass, or electric charge) <span class="math-container">$m_A\gg m_B \gg m_C$</span>, . . . with separation distances <span class="math-container">$r_{AB}\gg r_{BC}\gg$</span> . . . so that B is trapped in the potential well of A, C is trapped in the potential well of B, etc. In the example below, I chose A=galactic core, B=a star, C=a planet, D=a moon (and a satellite can orbit the moon, and in principle even something very tiny could orbit the satellite, but it would be so weakly bound that tiny perturbations would give it enough energy to escape).</p>
<p>The quantum mechanical version of what I am describing is a <a href="https://en.wikipedia.org/wiki/Bound_state" rel="nofollow noreferrer">bound state</a>. Quarks are bound (by the strong force) into hadrons (protons and neutrons), which are bound (again by the strong force) into atomic nuclei, which form bound states with electrons (via the electromagnetic force) to make atoms, which can further form bound states (by chemical bonding, which is electromagnetic in origin) to make molecules, and so on until you get to sizes where electric charge is screened and Newtonian gravity takes over as described in the paragraph above and the original answer below.</p>
<h4>Original answer below</h4>
<p>There are many potential directions in which your question could be interpreted and answered. One, which comes from a physics perspective, is symmetry. In this case, rotational <a href="https://en.wikipedia.org/wiki/Symmetry_(physics)" rel="nofollow noreferrer">symmetry</a>. The laws of physics are rotationally invariant, and as a consequence of <a href="https://en.wikipedia.org/wiki/Noether%27s_theorem" rel="nofollow noreferrer">Noether's theorem</a>, <a href="https://en.wikipedia.org/wiki/Angular_momentum" rel="nofollow noreferrer">angular momentum</a> is a conserved quantity.</p>
<p>In orbital mechanics specifically, one generally is dealing with <a href="https://en.wikipedia.org/wiki/Central_force" rel="nofollow noreferrer">central potentials</a>, e.g. the potential energy of a massive object moving around the sun only depends on the distance from the sun (and decays is one over distance). Since this potential does not depend on the angle one makes with any particular axis through the sun, it is rotationally symmetric and thus angular momentum is conserved. (A note: although it is true that electrons do not &quot;orbit&quot; atomic nuclei, they still have conserved angular momentum!)</p>
<p>Now to the different scales portion of the question. At the scale of planetary motion, gravity is the only important force to consider, all other forces are essentially completely <a href="https://en.wikipedia.org/wiki/Electric-field_screening" rel="nofollow noreferrer">screened</a> at this scale (because the relevant objects, e.g. asteroids or planets, are charge-neutral). The motion of the objects in the solar system relative to each other can, to a good approximation, be approximated by only considering the gravitational force of the sun, as described in the previous paragraph. Inter-object forces (e.g. gravitational attraction between planets) can be treated <a href="https://en.wikipedia.org/wiki/Perturbation_theory" rel="nofollow noreferrer">perturbatively</a>.</p>
<p>Similarly at other scales, e.g. a single planet gravitationally interacting with its moons can be approximated to a good approximation as a collection of two-body problems. Each object orbits in the near-spherically-symmetric potential well of the nearest object which has a mass orders of magnitude larger than itself, and the gravitational attraction of smaller nearby objects act only perturbatively relative to the dominant nearby attractive body.</p>
<p>Let me try to be slightly more precise about that. Let's start at a large scale: the solar system orbits around the galactic core. For simplicity let's model the galactic core as a point mass.<span class="math-container">$^\dagger$</span> The solar system is about <span class="math-container">$10^{20}$</span> meters from the galactic core, and has a radius of about <span class="math-container">$10^{10}$</span> meters. The size of the solar system relative to the size of the galaxy is about the same as the size of an atom relative to the size of your body, so we can treat it as a point mass. Treating this as a 2-body problem, the gravitational potential that the solar system orbits in might look something like this:</p>
<p><a href="https://i.sstatic.net/vIuDs.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/vIuDs.png" alt="enter image description here" /></a></p>
<p>where <span class="math-container">$a_{\mathrm{galactic}}$</span> is a lengthscale on the order <span class="math-container">$10^{20}$</span> meters. But of course, the solar system itself is made of a bunch of small massive objects, the largest of the which is the sun, and so itself has a gravitational potential. If I <em>greatly</em> exaggerate that, and throw the Earth in (I must stress this is <em>not</em> to scale <em>at all</em>, only for illustrative purposes), the potential looks something like this:</p>
<p><a href="https://i.sstatic.net/of8Nl.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/of8Nl.png" alt="enter image description here" /></a></p>
<p>We see that while the solar system moves in the potential well of the galactic core, the Earth also moves in the potential well of the sun. Of course, the Earth itself also has a potential, so we could further break that down (again, only illustrative, and throwing the moon in too) as</p>
<p><a href="https://i.sstatic.net/YcPg1.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/YcPg1.png" alt="enter image description here" /></a></p>
<p>So the lesson is this: each object moves in the local potential well of the nearest massive gravitational source. Each object then creates a smaller local potential well around itself that smaller objects can orbit in, and so on. In fact, if we were to correct the above plot to accurately indicate the distances and masses involved, we would see that at each of these scales the <span class="math-container">$1/r$</span> potential of the next larger object is <em>almost completely flat</em>. That is to say, for example <span class="math-container">$1/(a_{\mathrm{galactic}}+a_{\mathrm{solar\,system}}) \approx 1/(a_{\mathrm{galactic}})$</span> to one part in <span class="math-container">$10^{10}$</span>. This means that <em>every object in the solar system feels essentially the same gravitational force from the galactic core</em>, regardless of where in the solar system they are, because the galactic core is, practically, infinitely far away, and its gravitational force points uniformly in the same direction at each point within the solar system. Similarly the sun is very far away from the Earth and moon, relative to their separation distance. All that is to say that <a href="https://en.wikipedia.org/wiki/Tidal_force" rel="nofollow noreferrer">tidal forces</a> are negligible, and these systems can be treated to a very good approximation as two-body orbital pairs, with their center of mass subject to an external uniform force.</p>
<p><span class="math-container">${}^\dagger$</span> <em>this approximation is not for realistic modelling, just to make a point, see the comments.</em></p>

CC BY-SA 4.0
2021-12-07T01:04:30.353Z
The shell theorem only works for spherically-symmetric objects. The galaxy is not spherically symmetric.
notovny
CC BY-SA 4.0
2021-12-07T01:50:09.900Z
@notovny true but the majority of the matter in the interior of the galaxy is dark matter which is mostly spherically symmetric, and the shell theorem [also applies for a disk](https://en.wikipedia.org/wiki/Shell_theorem#Converses_and_generalizations). The point source approximation is still quite reasonable.
Kai
CC BY-SA 4.0
2021-12-07T08:51:13.923Z
Actually, dark matter haloes can exhibit quite a bit of ellipticity. https://arxiv.org/abs/1004.4214
Daddy Kropotkin
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2021-12-07T11:20:14.740Z
While maybe a reasonable assumption for dark matter haloes, this answer edges on misrepresenting a very important fact of the galactic mass distribution: The galaxy *cannot* be treated as a point mass for the purposes of stellar dynamics. Furthermore the galactic center only carries a negligible amount of the baryonic mass (which in itself is negligible compared to the dark matter halo). If you treat the entire galactic mass as a point mass inside the stellar radius, you will get the wrong answer concerning the orbits of the stars in the milky way.
AtmosphericPrisonEscape
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2021-12-07T18:17:05.513Z
@AtmosphericPrisonEscape I completely agree. I'm not trying to make claims about galactic dynamics, I just wanted to use it to make the point about why at different scales you get orbits. In the end the specific shape of the potential is not important if we just want to know whether or not things orbit. Conservation of angular momentum and being in a potential well is sufficient. If I have time later I'll clarify
Kai
CC BY-SA 4.0
2021-12-07T22:40:31.663Z
@Kai Yes, a point well-taken. That an orbit is bound by the min/max values of the potential is also a result of classical mechanics (i.e. the second part of your comment), but it does not guarantee the regularity of the orbit. Particularly, it will lead to orbit crossings, hence scattering, hence orbit migration between stars inside the galaxy, without even keeping the galactic potential constant. So I guess my main point is cautioning that non-simple potentials lead to all sorts of complications that cannot be encapsulated easily.
AtmosphericPrisonEscape
CC BY-SA 4.0
2021-12-08T14:45:38.113Z
@AtmosphericPrisonEscape "Orbit crossing" and "scattering" are not an issue for stars in galaxies, except in very dense subsystems (e.g. globular clusters) or on timescales much longer than the current age of the universe.
Peter Erwin
CC BY-SA 4.0
2021-12-08T14:56:54.600Z
@notovny There are ellipsoidal generalizations of the shell theorem, so it does apply to (some) flattened, axisymmetric systems. (Kai's Wikipedia link is not relevant, though.)
Peter Erwin
CC BY-SA 4.0
2021-12-08T16:48:28.907Z
I have updated the answer
Kai
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2021-12-08T17:11:32.647Z
@PeterErwin: That's oversimplified considerations from star density and crossing times calculations. The galactic community refers to 'blurring' and 'churning' as mechanisms migrating stars in the galaxy. Those are effects of the collective potentials on individual stars. No matter what you call it, the galaxy doesn't have an $1/r$ potential.
AtmosphericPrisonEscape
CC BY-SA 4.0
2021-12-08T19:20:59.480Z
@AtmosphericPrisonEscape I'm quite familiar with those effects; as you point out, they are the collective effects of large-scale mass distributions (bars, spiral arms), not 2-body "scattering" interactions. Orbit-crossing is relevant for *gas* flows, but not for stars.
Peter Erwin
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2021-12-07T00:50:41.270Z
2021-12-08T16:52:00.610Z
2021-12-08T16:52:00.610Z
Kai
<p>The concept you seem to be looking for is <a href="https://en.wikipedia.org/wiki/Scale_invariance" rel="noreferrer">scale invariance</a>.<br />
This is a property of certain simple systems, such as a particle under the action of the gravitational one or the electrostatic force.</p>
<p>If the system is simple enough, one can prove that it is scale-invariant, such as the planetary orbital configurations that you mentioned. Note that 'scale invariance' is often used as a comparing qualifier, such as 'two orbits are scale-invariant' w.r.t. each other.</p>
<p>The (approximate) invariance of planetary orbits around the sun can be understood as an expression of Kepler's 3rd law. More precisely, rescaling the spatial axis (e.g. 'zooming out') and the temporal axis (e.g. 'slowing down'), one finds that the slow-down factor f and the zoom-out-factor a, have to be related by a constant, <span class="math-container">$f^2/a^3=\rm const.$</span> in order to fulfill the equations of motion.</p>
<p>This is a branch of classical mechanics, i.e. the first or second-semester theoretical physics course that you'll usually take in a physics degree. If you want some more gory detail you can try some <a href="http://www.pas.rochester.edu/%7Erajeev/phy411/LectureNotesf11.pdf" rel="noreferrer">lecture notes I googled</a> or take up one of the theroetical lectures teaching books in your language.</p>
<p>As last bit, note that one example you have mentioned, the galactic motion of the sun, is not invariant w.r.t to the planetary motions. The galactic motion of stars happens in a complex gravitational potential of mutual interactions, and cannot be simply rescaled.</p>

CC BY-SA 4.0
2021-12-06T13:32:29.927Z
2021-12-06T16:42:58.673Z
2021-12-06T16:42:58.673Z
AtmosphericPrisonEscape
<p>I'm certainly not the first to notice this, but there is a commonality between the structures of objects in space at different levels of magnification.  For instance...</p>
<p>The moon orbits earth...</p>
<p>The earth orbits the sun...</p>
<p>Our solar system orbits the center of the Milky Way...</p>
<p>Etc. Etc. ad infinitum</p>
<p>There seems to be a repetition of this arrangement where there is one object (or a cluster of objects) at a center, and then others orbit around it.  The repetition repeats on the micro scale as well (cells, atoms) but that's beyond the scope of astronomy SE.</p>
<p>I want to learn more, but don't know where to start.  Is there a name for this phenomenon?</p>

CC BY-SA 4.0
2021-12-06T15:57:18.923Z
Cells do not orbit, nor do atoms
Daddy Kropotkin
CC BY-SA 4.0
2021-12-06T16:43:51.687Z
@DaddyKropotkin If you want to leave a minor comment, leave a minor comment, don't edit other peoples' answers.
AtmosphericPrisonEscape
CC BY-SA 4.0
2021-12-06T16:59:33.170Z
@AtmosphericPrisonEscape People edit my answers for minor things, didn't know it was such a touchy subject, sorry. I always considered it in the vein of teamwork.
Daddy Kropotkin
CC BY-SA 4.0
2021-12-06T17:21:40.357Z
@Daddy Kropotkin true, cells do not orbit.  Atoms do not either, although electrons are often imagined as orbiting, right?  I suppose I'm referring more to the structure of a nucleus of some kind and other related objects away from it, not the orbital motion per se.  Feel free to edit that part for clarity if you like
nuggethead
CC BY-SA 4.0
2021-12-07T02:04:28.683Z
@nuggethead electrons orbiting a nucleus is a model that only kind of approximates what's actually happening. It's not really comparable to astronomical orbits
Drake P
CC BY-SA 4.0
2021-12-07T10:49:35.697Z
@Drake P yes it's a model.  I'm not a scientist at all, just interested in what the phenomen where things look alike at different scales is called.
nuggethead
CC BY-SA 4.0
2021-12-09T16:55:47.627Z
What you're describing sounds like fractals to me, but I don't know enough about them to write an answer. @atmosphericprisonescape 's "scale invariance" is accurate too.
Beer Me
<orbit>
Repetition of orbital structure on many scales - what is it called?
nuggethead
# Repetition of orbital structure on many scales - what is it called?

(Asked by: username_0 on 2021-12-06)

I'm certainly not the first to notice this, but there is a commonality between the structures of objects in space at different levels of magnification. For instance...

The moon orbits earth...

The earth orbits the sun...

Our solar system orbits the center of the Milky Way...

Etc. Etc. ad infinitum

There seems to be a repetition of this arrangement where there is one object (or a cluster of objects) at a center, and then others orbit around it. The repetition repeats on the micro scale as well (cells, atoms) but that's beyond the scope of astronomy SE.

I want to learn more, but don't know where to start. Is there a name for this phenomenon?

**Tags:** orbit

> **Comments:**
> **username_1**: Cells do not orbit, nor do atoms  
> **username_2**: @username_1 If you want to leave a minor comment, leave a minor comment, don't edit other peoples' answers.  
> **username_1**: @username_2 People edit my answers for minor things, didn't know it was such a touchy subject, sorry. I always considered it in the vein of teamwork.  
> **username_0**: @username_1 true, cells do not orbit.  Atoms do not either, although electrons are often imagined as orbiting, right?  I suppose I'm referring more to the structure of a nucleus of some kind and other related objects away from it, not the orbital motion per se.  Feel free to edit that part for clarity if you like  
> **username_3**: @username_0 electrons orbiting a nucleus is a model that only kind of approximates what's actually happening. It's not really comparable to astronomical orbits  
> **username_0**: @username_3 yes it's a model.  I'm not a scientist at all, just interested in what the phenomen where things look alike at different scales is called.  
> **username_4**: What you're describing sounds like fractals to me, but I don't know enough about them to write an answer. @username_2 's "scale invariance" is accurate too.  

---

## 2 answers

### Answer by username_2 on 2021-12-06. Score: 15 | Accepted answer

The concept you seem to be looking for is [scale invariance](https://en.wikipedia.org/wiki/Scale_invariance).  
This is a property of certain simple systems, such as a particle under the action of the gravitational one or the electrostatic force.

If the system is simple enough, one can prove that it is scale-invariant, such as the planetary orbital configurations that you mentioned. Note that 'scale invariance' is often used as a comparing qualifier, such as 'two orbits are scale-invariant' w.r.t. each other.

The (approximate) invariance of planetary orbits around the sun can be understood as an expression of Kepler's 3rd law. More precisely, rescaling the spatial axis (e.g. 'zooming out') and the temporal axis (e.g. 'slowing down'), one finds that the slow-down factor f and the zoom-out-factor a, have to be related by a constant, $f^2/a^3=\rm const.$ in order to fulfill the equations of motion.

This is a branch of classical mechanics, i.e. the first or second-semester theoretical physics course that you'll usually take in a physics degree. If you want some more gory detail you can try some [lecture notes I googled](http://www.pas.rochester.edu/%7Erajeev/phy411/LectureNotesf11.pdf) or take up one of the theroetical lectures teaching books in your language.

As last bit, note that one example you have mentioned, the galactic motion of the sun, is not invariant w.r.t to the planetary motions. The galactic motion of stars happens in a complex gravitational potential of mutual interactions, and cannot be simply rescaled.


---

### Answer by username_5 on 2021-12-07. Score: 1

### Bound states

In retrospect the below answer is a bit long winded, so I will summarize the basic point: to get "orbital" motion, one simply needs that some object is trapped in the potential well of another, meaning it doesn't have enough kinetic energy to escape. If the potential well is rotationally symmetric then there will be conservation of angular momentum, and if the potential has $1/r$ dependence then the orbits will be [Keplerian](https://en.wikipedia.org/wiki/Kepler_orbit) (at least classically), but that is not necessarily important. All that matters is that you can have a hierarchy: some objects A, B, C, . . . with decreasing "charges" (e.g. gravitational mass, or electric charge) $m\_A\gg m\_B \gg m\_C$, . . . with separation distances $r\_{AB}\gg r\_{BC}\gg$ . . . so that B is trapped in the potential well of A, C is trapped in the potential well of B, etc. In the example below, I chose A=galactic core, B=a star, C=a planet, D=a moon (and a satellite can orbit the moon, and in principle even something very tiny could orbit the satellite, but it would be so weakly bound that tiny perturbations would give it enough energy to escape).

The quantum mechanical version of what I am describing is a [bound state](https://en.wikipedia.org/wiki/Bound_state). Quarks are bound (by the strong force) into hadrons (protons and neutrons), which are bound (again by the strong force) into atomic nuclei, which form bound states with electrons (via the electromagnetic force) to make atoms, which can further form bound states (by chemical bonding, which is electromagnetic in origin) to make molecules, and so on until you get to sizes where electric charge is screened and Newtonian gravity takes over as described in the paragraph above and the original answer below.

#### Original answer below

There are many potential directions in which your question could be interpreted and answered. One, which comes from a physics perspective, is symmetry. In this case, rotational [symmetry](https://en.wikipedia.org/wiki/Symmetry_(physics)). The laws of physics are rotationally invariant, and as a consequence of [Noether's theorem](https://en.wikipedia.org/wiki/Noether%27s_theorem), [angular momentum](https://en.wikipedia.org/wiki/Angular_momentum) is a conserved quantity.

In orbital mechanics specifically, one generally is dealing with [central potentials](https://en.wikipedia.org/wiki/Central_force), e.g. the potential energy of a massive object moving around the sun only depends on the distance from the sun (and decays is one over distance). Since this potential does not depend on the angle one makes with any particular axis through the sun, it is rotationally symmetric and thus angular momentum is conserved. (A note: although it is true that electrons do not "orbit" atomic nuclei, they still have conserved angular momentum!)

Now to the different scales portion of the question. At the scale of planetary motion, gravity is the only important force to consider, all other forces are essentially completely [screened](https://en.wikipedia.org/wiki/Electric-field_screening) at this scale (because the relevant objects, e.g. asteroids or planets, are charge-neutral). The motion of the objects in the solar system relative to each other can, to a good approximation, be approximated by only considering the gravitational force of the sun, as described in the previous paragraph. Inter-object forces (e.g. gravitational attraction between planets) can be treated [perturbatively](https://en.wikipedia.org/wiki/Perturbation_theory).

Similarly at other scales, e.g. a single planet gravitationally interacting with its moons can be approximated to a good approximation as a collection of two-body problems. Each object orbits in the near-spherically-symmetric potential well of the nearest object which has a mass orders of magnitude larger than itself, and the gravitational attraction of smaller nearby objects act only perturbatively relative to the dominant nearby attractive body.

Let me try to be slightly more precise about that. Let's start at a large scale: the solar system orbits around the galactic core. For simplicity let's model the galactic core as a point mass.$^\dagger$ The solar system is about $10^{20}$ meters from the galactic core, and has a radius of about $10^{10}$ meters. The size of the solar system relative to the size of the galaxy is about the same as the size of an atom relative to the size of your body, so we can treat it as a point mass. Treating this as a 2-body problem, the gravitational potential that the solar system orbits in might look something like this:

[![enter image description here](https://i.sstatic.net/vIuDs.png)](https://i.sstatic.net/vIuDs.png)

where $a\_{\mathrm{galactic}}$ is a lengthscale on the order $10^{20}$ meters. But of course, the solar system itself is made of a bunch of small massive objects, the largest of the which is the sun, and so itself has a gravitational potential. If I *greatly* exaggerate that, and throw the Earth in (I must stress this is *not* to scale *at all*, only for illustrative purposes), the potential looks something like this:

[![enter image description here](https://i.sstatic.net/of8Nl.png)](https://i.sstatic.net/of8Nl.png)

We see that while the solar system moves in the potential well of the galactic core, the Earth also moves in the potential well of the sun. Of course, the Earth itself also has a potential, so we could further break that down (again, only illustrative, and throwing the moon in too) as

[![enter image description here](https://i.sstatic.net/YcPg1.png)](https://i.sstatic.net/YcPg1.png)

So the lesson is this: each object moves in the local potential well of the nearest massive gravitational source. Each object then creates a smaller local potential well around itself that smaller objects can orbit in, and so on. In fact, if we were to correct the above plot to accurately indicate the distances and masses involved, we would see that at each of these scales the $1/r$ potential of the next larger object is *almost completely flat*. That is to say, for example $1/(a\_{\mathrm{galactic}}+a\_{\mathrm{solar\,system}}) \approx 1/(a\_{\mathrm{galactic}})$ to one part in $10^{10}$. This means that *every object in the solar system feels essentially the same gravitational force from the galactic core*, regardless of where in the solar system they are, because the galactic core is, practically, infinitely far away, and its gravitational force points uniformly in the same direction at each point within the solar system. Similarly the sun is very far away from the Earth and moon, relative to their separation distance. All that is to say that [tidal forces](https://en.wikipedia.org/wiki/Tidal_force) are negligible, and these systems can be treated to a very good approximation as two-body orbital pairs, with their center of mass subject to an external uniform force.

${}^\dagger$ *this approximation is not for realistic modelling, just to make a point, see the comments.*

> **Comments:**
> **username_6**: The shell theorem only works for spherically-symmetric objects. The galaxy is not spherically symmetric.  
> **username_5**: @username_6 true but the majority of the matter in the interior of the galaxy is dark matter which is mostly spherically symmetric, and the shell theorem [also applies for a disk](https://en.wikipedia.org/wiki/Shell_theorem#Converses_and_generalizations). The point source approximation is still quite reasonable.  
> **username_1**: Actually, dark matter haloes can exhibit quite a bit of ellipticity. https://arxiv.org/abs/1004.4214  
> **username_2**: While maybe a reasonable assumption for dark matter haloes, this answer edges on misrepresenting a very important fact of the galactic mass distribution: The galaxy *cannot* be treated as a point mass for the purposes of stellar dynamics. Furthermore the galactic center only carries a negligible amount of the baryonic mass (which in itself is negligible compared to the dark matter halo). If you treat the entire galactic mass as a point mass inside the stellar radius, you will get the wrong answer concerning the orbits of the stars in the milky way.  
> **username_5**: @username_2 I completely agree. I'm not trying to make claims about galactic dynamics, I just wanted to use it to make the point about why at different scales you get orbits. In the end the specific shape of the potential is not important if we just want to know whether or not things orbit. Conservation of angular momentum and being in a potential well is sufficient. If I have time later I'll clarify  
> **username_2**: @username_5 Yes, a point well-taken. That an orbit is bound by the min/max values of the potential is also a result of classical mechanics (i.e. the second part of your comment), but it does not guarantee the regularity of the orbit. Particularly, it will lead to orbit crossings, hence scattering, hence orbit migration between stars inside the galaxy, without even keeping the galactic potential constant. So I guess my main point is cautioning that non-simple potentials lead to all sorts of complications that cannot be encapsulated easily.  
> **username_7**: @username_2 "Orbit crossing" and "scattering" are not an issue for stars in galaxies, except in very dense subsystems (e.g. globular clusters) or on timescales much longer than the current age of the universe.  
> **username_7**: @username_6 There are ellipsoidal generalizations of the shell theorem, so it does apply to (some) flattened, axisymmetric systems. (Kai's Wikipedia link is not relevant, though.)  
> **username_5**: I have updated the answer  
> **username_2**: @username_7: That's oversimplified considerations from star density and crossing times calculations. The galactic community refers to 'blurring' and 'churning' as mechanisms migrating stars in the galaxy. Those are effects of the collective potentials on individual stars. No matter what you call it, the galaxy doesn't have an $1/r$ potential.  
> **username_7**: @username_2 I'm quite familiar with those effects; as you point out, they are the collective effects of large-scale mass distributions (bars, spiral arms), not 2-body "scattering" interactions. Orbit-crossing is relevant for *gas* flows, but not for stars.  

---

