<p>As the universe undergoes expansion, small perturbations grow denser, leading some to eventually collapse to form high-density structures.  Simultaneously, these perturbations acquire angular momentum from torques acting on them.  The torques arise from the tidal fields generated by neighbors acting on the quadrupole moment of the perturbation's mass.  Statistically, this process results in an average spin parameter of 0.035 across all scales of bound structures, with a wide dispersion.</p>
<p>The spin parameter (<span class="math-container">$\lambda$</span>) is defined as <span class="math-container">$$\lambda = \frac{J}{\sqrt{2} M_{vir} V_{vir} R_{vir}},$$</span> where <span class="math-container">$J$</span> is the angular momentum, and <span class="math-container">$M_{vir}, V_{vir},$</span> and <span class="math-container">$R_{vir}$</span> are the mass, circular velocity, and radius at the virial radius.</p>
<p>Starting with this amount of spin, collapsing objects spin up to higher rotational velocities, and expanding objects spin down as they adhere to the conservation of angular momentum.</p>
<p>To counter expansion, a critical density must be surpassed, a condition met only below supercluster scales. Consequently, at and above supercluster scales, three dimensional collapse does not occur and so rotation velocities become insignificant.</p>
<p>Superclusters, possessing lower densities (closer to the mean density of the universe because they are so large), may collapse predominantly in one dimension while continuing to expand in the other two dimensions.</p>
<p>This observed variation in densities with structure size can be attributed to the initial conditions of the early universe, specifically the primordial power spectrum.   Dissipation of energy also plays a role in the final extents of the baryonic matter.  And, the story is a bit different for planetary systems and stellar clusters which form from clouds of gas in galaxies after collapse.  They would have a different distribution of spin parameter values.</p>

CC BY-SA 4.0
2024-03-11T04:42:54.863Z
Thank you for the answer! Tell me, how was the formula for the spin parameter obtained?
ayr
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2024-03-11T16:36:07.330Z
That is its definition.  It is a unitless parameterization of the amount of rotation an object has,  so it works equally well on all scales.
eshaya
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2024-03-12T06:15:07.483Z
Great! And I asked this question because, as follows from your answer, a fairly extensive mathematical basis should be used to study the stable states of such dynamic systems (@JamesK mentioned them in his answer). It’s just still unclear why physically everything is this way.
ayr
CC BY-SA 4.0
2024-02-06T20:00:02.820Z
2024-02-07T03:38:06.387Z
2024-02-07T03:38:06.387Z
eshaya
<p>The two-body problem in Newtonian mechanics is soluble and has a simple solution of the two bodies rotating around each other.</p>
<p>If there are more than two bodies then generally the situation is unstable, unless either 1) there are pairs of bodies that are close enough to act like two bodies or 2) one of the bodies is massive enough that the others will orbit around it, as if there were a collection of two body problems, or 3) a hierarchical combination of 1) and 2).</p>
<p>So the reason that most objects are rotating around each other is that no other configuration is stable in anything but the shortest term. The motion of galaxies in clusters isn't always like this, because &quot;in the short term&quot; for galaxies might be billions of years.</p>

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2024-02-04T17:50:58.423Z
I needed time.  Thanks for the answer!  This is what I thought about now... What will we see if we look at the entire universe from the outside?  Will there be the same large structures rotating around a certain center of mass, or is everything going differently at this scale?
ayr
CC BY-SA 4.0
2024-02-04T19:16:35.717Z
There is no outside.  There isn't much rotation above the level of galaxies, actually.
James K
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2024-02-05T06:07:54.947Z
If on such a scale the amount of rotational motion is small, but the law of conservation of angular momentum is still satisfied, then what could be seen on a scale even larger than the scale on which the galactic filaments are located? And is it possible to assess the degree of self-similarity of cosmic structures on different scales and evaluate the influence of the parameters of this dynamic system on the boundary of the transition to expansion?
ayr
CC BY-SA 4.0
2024-01-29T19:55:27.417Z
2024-01-29T19:55:27.417Z
James K
<p>I decided to return to the <a href="https://astronomy.stackexchange.com/questions/55195/hierarchy-of-gravitational-interactions-of-astronomical-objects-from-single-to">question</a> first written here after some time, since a short and succinct phrase had formed in my head that described the phenomenon that I was trying to understand.</p>
<p>What are the generally accepted explanations or hypotheses for why the universe is a structure of objects rotating around a center of mass, which in turn is a structure of objects rotating around a center of mass, which in turn is a structure of objects rotating around a center of mass... etc? From superclusters of galaxies to star systems.</p>
<p>I don't think I phrased the question well enough last time, but the <a href="https://astronomy.stackexchange.com/a/55213/48182">answer</a> mentions a lot of interesting effects that weren't originally intended. So I decided to write a shorter and more focused question.</p>

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2024-01-29T19:01:09.070Z
I think you mean clusters and not superclusters -- the latter are not gravitationally bound and are just expanding like the universe.
Sten
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2024-01-30T05:24:31.567Z
@Sten What is the reason for the transition to a different structure at this level?
ayr
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2024-02-03T04:46:09.853Z
@eshaya By the way, why, what is the reason for this change in behavior?
ayr
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2024-02-04T17:45:09.713Z
@eshaya my question, if formulated very specifically, would be this: why at a certain scale the rotational motion of some objects around others disappears
ayr
CC BY-SA 4.0
2024-02-05T01:19:46.080Z
Objects that collapse rotate faster and faster as they collapse.  Objects that expand rotate less and less.  It is the conservation of angular momentum, mass X radius X tangential motion is a constant.  There is a critical density required to overcome expansion and that density is reached only below supercluster scales, as I explained.
eshaya
CC BY-SA 4.0
2024-02-05T06:02:23.790Z
@eshaya Is it possible to predict/assess the degree of fractality...or rather, the degree of self-similarity of the subsystems under consideration on the appropriate scale and assess the influence of the parameters of this large dynamic system on the boundary of transition to expansion? And is the galactic filament the ultimate structure, or will there be even larger filaments at the next scale? In general, I liked your comments and I think they should be formulated as an answer and appreciated.
ayr
CC BY-SA 4.0
2024-02-05T15:17:44.433Z
It sounds like you have another question to ask.
eshaya
CC BY-SA 4.0
2024-02-05T16:06:21.207Z
@eshaya not exactly, but it seems to me that it can be related to the original question
ayr
<astrophysics><gravity><cosmology><orbital-mechanics><filaments>
Fractal structure of the universe
ayr
# Fractal structure of the universe

(Asked by: username_0 on 2024-01-29)

I decided to return to the [question](https://astronomy.stackexchange.com/questions/55195/hierarchy-of-gravitational-interactions-of-astronomical-objects-from-single-to) first written here after some time, since a short and succinct phrase had formed in my head that described the phenomenon that I was trying to understand.

What are the generally accepted explanations or hypotheses for why the universe is a structure of objects rotating around a center of mass, which in turn is a structure of objects rotating around a center of mass, which in turn is a structure of objects rotating around a center of mass... etc? From superclusters of galaxies to star systems.

I don't think I phrased the question well enough last time, but the [answer](https://astronomy.stackexchange.com/a/55213/48182) mentions a lot of interesting effects that weren't originally intended. So I decided to write a shorter and more focused question.

**Tags:** astrophysics, gravity, cosmology, orbital-mechanics, filaments
> **Comments:**
> **username_1**: I think you mean clusters and not superclusters -- the latter are not gravitationally bound and are just expanding like the universe.  
> **username_0**: @Sten What is the reason for the transition to a different structure at this level?  
> **username_0**: @eshaya By the way, why, what is the reason for this change in behavior?  
> **username_0**: @eshaya my question, if formulated very specifically, would be this: why at a certain scale the rotational motion of some objects around others disappears  
> **username_2**: Objects that collapse rotate faster and faster as they collapse.  Objects that expand rotate less and less.  It is the conservation of angular momentum, mass X radius X tangential motion is a constant.  There is a critical density required to overcome expansion and that density is reached only below supercluster scales, as I explained.  
> **username_0**: @eshaya Is it possible to predict/assess the degree of fractality...or rather, the degree of self-similarity of the subsystems under consideration on the appropriate scale and assess the influence of the parameters of this large dynamic system on the boundary of transition to expansion? And is the galactic filament the ultimate structure, or will there be even larger filaments at the next scale? In general, I liked your comments and I think they should be formulated as an answer and appreciated.  
> **username_2**: It sounds like you have another question to ask.  
> **username_0**: @eshaya not exactly, but it seems to me that it can be related to the original question  

---

## 2 answers

### Answer by username_2 on 2024-02-06. Score: 2 | Accepted answer

As the universe undergoes expansion, small perturbations grow denser, leading some to eventually collapse to form high-density structures. Simultaneously, these perturbations acquire angular momentum from torques acting on them. The torques arise from the tidal fields generated by neighbors acting on the quadrupole moment of the perturbation's mass. Statistically, this process results in an average spin parameter of 0.035 across all scales of bound structures, with a wide dispersion.

The spin parameter ($\lambda$) is defined as $$\lambda = \frac{J}{\sqrt{2} M\_{vir} V\_{vir} R\_{vir}},$$ where $J$ is the angular momentum, and $M\_{vir}, V\_{vir},$ and $R\_{vir}$ are the mass, circular velocity, and radius at the virial radius.

Starting with this amount of spin, collapsing objects spin up to higher rotational velocities, and expanding objects spin down as they adhere to the conservation of angular momentum.

To counter expansion, a critical density must be surpassed, a condition met only below supercluster scales. Consequently, at and above supercluster scales, three dimensional collapse does not occur and so rotation velocities become insignificant.

Superclusters, possessing lower densities (closer to the mean density of the universe because they are so large), may collapse predominantly in one dimension while continuing to expand in the other two dimensions.

This observed variation in densities with structure size can be attributed to the initial conditions of the early universe, specifically the primordial power spectrum. Dissipation of energy also plays a role in the final extents of the baryonic matter. And, the story is a bit different for planetary systems and stellar clusters which form from clouds of gas in galaxies after collapse. They would have a different distribution of spin parameter values.

> **Comments:**
> **username_0**: Thank you for the answer! Tell me, how was the formula for the spin parameter obtained?  
> **username_2**: That is its definition.  It is a unitless parameterization of the amount of rotation an object has,  so it works equally well on all scales.  
> **username_0**: Great! And I asked this question because, as follows from your answer, a fairly extensive mathematical basis should be used to study the stable states of such dynamic systems (@JamesK mentioned them in his answer). It’s just still unclear why physically everything is this way.  

---

### Answer by username_3 on 2024-01-29. Score: 4

The two-body problem in Newtonian mechanics is soluble and has a simple solution of the two bodies rotating around each other.

If there are more than two bodies then generally the situation is unstable, unless either 1) there are pairs of bodies that are close enough to act like two bodies or 2) one of the bodies is massive enough that the others will orbit around it, as if there were a collection of two body problems, or 3) a hierarchical combination of 1) and 2).

So the reason that most objects are rotating around each other is that no other configuration is stable in anything but the shortest term. The motion of galaxies in clusters isn't always like this, because "in the short term" for galaxies might be billions of years.

> **Comments:**
> **username_0**: I needed time.  Thanks for the answer!  This is what I thought about now... What will we see if we look at the entire universe from the outside?  Will there be the same large structures rotating around a certain center of mass, or is everything going differently at this scale?  
> **username_3**: There is no outside.  There isn't much rotation above the level of galaxies, actually.  
> **username_0**: If on such a scale the amount of rotational motion is small, but the law of conservation of angular momentum is still satisfied, then what could be seen on a scale even larger than the scale on which the galactic filaments are located? And is it possible to assess the degree of self-similarity of cosmic structures on different scales and evaluate the influence of the parameters of this dynamic system on the boundary of the transition to expansion?  

---

