<p>Based merely on an article titled &quot;<a href="https://authors.library.caltech.edu/27789/" rel="nofollow noreferrer">Onset of secular chaos in planetary systems: period doubling and strange attractors</a>, it seems that this term indicates the behavior of planetary objects when there are no collisions or other effects that collapse the system.   'Period doubling' itself is a well-known phenomenon in basic chaos, aka strange attractor, math and can make for fun reading.</p>

CC BY-SA 4.0
2022-04-26T11:51:09.010Z
2022-04-26T11:51:09.010Z
Carl Witthoft
<p>Even <a href="https://www.nature.com/articles/261459a0" rel="nofollow noreferrer">seemingly simple systems can result in chaotic behavior</a>. Animal populations can be modeled by a very simple equation:
<span class="math-container">$$x_{t+1} = \lambda\,x_t\,(1-x_t)$$</span>
This is called the logistic map. Here <span class="math-container">$x$</span> is the population relative to the maximum population at some point in time, the subscripts <span class="math-container">$t$</span> and <span class="math-container">$t+1$</span> refer to the populations at time step <span class="math-container">$t$</span> the next time step (<span class="math-container">$t+1)$</span>, and <span class="math-container">$\lambda$</span> is a number between 0 and 4.</p>
<p>The value <span class="math-container">$x=0$</span> is an obvious solution. For small values of <span class="math-container">$\lambda$</span>, this is the only ultimate solution: Populations collapse. Another steady state solution is <span class="math-container">$x=1-1/\lambda$</span>, which yields a non-negative solution if <span class="math-container">$\lambda&gt;1$</span>. All initial values of <span class="math-container">$x$</span> other than zero or one eventually settle to this value if <span class="math-container">$\lambda$</span> is between one and three. Two other steady state solutions arise when <span class="math-container">$\lambda&gt;3$</span>. Populations alternate between a high value and a low value. Oak trees appear to have found this set of solutions in their wars with squirrels. Some species of oaks have evolved to alternate between very small yields of acorns one year and very heavy yields the next. Even more bifurcations appear as <span class="math-container">$\lambda$</span> increases beyond three. At a certain point it becomes chaos. All from a seemingly simple equation.</p>
<p>Next, secular equations. Modeling the orbits of planets about a star is not easy. One way to do this is to simplify the multi-body interactions. The result is a set of secular equations. In this context, &quot;secular&quot; essentially means &quot;averaged out&quot;. For example, the effects of Jupiter on Mercury's orbit can be secularized by treating Jupiter as a ring of mass about the Sun at Jupiter's orbital distance from the Sun as opposed to a point mass.</p>
<p>The result is a set of fairly nice and fairly simple time-dependent set of orbital elements for each planet. The question then is can these seemingly simple secular equations result in chaotic behavior? The answer is apparently yes. That is &quot;secular chaos&quot;.</p>

CC BY-SA 4.0
2022-04-26T21:41:17.390Z
Is this similar to Kozai mechanism? https://en.m.wikipedia.org/wiki/Kozai_mechanism In that article it seems to mention a 3-body system which causes changes in eccentricity though it does not mention chaotic behaviour.
WDUK
CC BY-SA 4.0
2022-04-27T12:05:32.523Z
@WDUK The Kozai mechanism is but one of many mechanisms that make the multi-body problem challenging to solve. Resonances are even more challenging. That even seemingly simple systems can result in chaotic behavior -- that is more challenging yet. One of the problems with predicting the stability of Mercury's orbit over the long term is the potential for resonances between the orbits of Jupiter and Mercury.
David Hammen
CC BY-SA 4.0
2022-04-27T21:25:49.930Z
So is Kozai mechanism a type of chaos aswell ? Or is it a more stable system that happens to cause changes to orbits?
WDUK
CC BY-SA 4.0
2022-04-26T14:10:17.880Z
2022-04-26T16:32:56.427Z
2022-04-26T16:32:56.427Z
David Hammen
<p>I have been reading about planet migration, and have seen secular chaos being mentioned a few times. There is no Wikipedia article on it so I am struggling to find a summary explanation of what secular chaos is. I can only find academic papers on it which go beyond my comprehension.</p>
<p>All I know so far is that secular chaos can cause high eccentricity to occur leading to tidal migration of planets. And is a theory proposed for hot Jupiters.</p>
<p>Can any one paint a picture of what secular chaos is and how it causes eccentric orbits to form ?</p>
<p>One article i read : <a href="https://www.pnas.org/doi/10.1073/pnas.1308261110" rel="nofollow noreferrer">https://www.pnas.org/doi/10.1073/pnas.1308261110</a></p>
<blockquote>
<p>Secular chaos causes planets’ eccentricities to randomly wander. When
one of the planets attains high enough e that it suffers collision,
ejection, or tidal capture, the removal of that planet can then lead
to a more stable system, with a longer chaotic diffusion time.</p>
</blockquote>
<p>It's a bit lacking in explaining secular chaos for the average person.  It dives into the theory showing the math which is then too difficult to understand or visualize what happens (at least for me)...</p>
<p>Hoping some one can simplify and explain it.</p>

CC BY-SA 4.0
2022-04-26T03:45:50.430Z
Someone could probably give a better, more complete answer but it appears based off of this paper that secular refers to orbit-averaged chaos: https://ui.adsabs.harvard.edu/abs/2011ApJ...739...31L/abstract
Justin T
CC BY-SA 4.0
2022-04-26T06:10:59.113Z
That looks like its similar to the paper i linked, although not any more simplified for the average person to understand
WDUK
CC BY-SA 4.0
2022-04-26T11:47:04.040Z
A quick search finds a bunch of papers I'll link here in another note, but first I have to share this gem of absurdity:  "Secular Chaos or Christian Truth—The Educational Options" yes, a real publication.
Carl Witthoft
CC BY-SA 4.0
2022-04-27T11:59:29.437Z
@CarlWitthoft The word "secular" has multiple meanings. But yes, that is a gem.
David Hammen
<orbital-migration>
What is secular chaos?
WDUK
# What is secular chaos?

(Asked by: username_0 on 2022-04-26)

I have been reading about planet migration, and have seen secular chaos being mentioned a few times. There is no Wikipedia article on it so I am struggling to find a summary explanation of what secular chaos is. I can only find academic papers on it which go beyond my comprehension.

All I know so far is that secular chaos can cause high eccentricity to occur leading to tidal migration of planets. And is a theory proposed for hot Jupiters.

Can any one paint a picture of what secular chaos is and how it causes eccentric orbits to form ?

One article i read : <https://www.pnas.org/doi/10.1073/pnas.1308261110>

> Secular chaos causes planets’ eccentricities to randomly wander. When
> one of the planets attains high enough e that it suffers collision,
> ejection, or tidal capture, the removal of that planet can then lead
> to a more stable system, with a longer chaotic diffusion time.

It's a bit lacking in explaining secular chaos for the average person. It dives into the theory showing the math which is then too difficult to understand or visualize what happens (at least for me)...

Hoping some one can simplify and explain it.

**Tags:** orbital-migration
> **Comments:**
> **username_1**: Someone could probably give a better, more complete answer but it appears based off of this paper that secular refers to orbit-averaged chaos: https://ui.adsabs.harvard.edu/abs/2011ApJ...739...31L/abstract  
> **username_0**: That looks like its similar to the paper i linked, although not any more simplified for the average person to understand  
> **username_2**: A quick search finds a bunch of papers I'll link here in another note, but first I have to share this gem of absurdity:  "Secular Chaos or Christian Truth—The Educational Options" yes, a real publication.  
> **username_3**: @CarlWitthoft The word "secular" has multiple meanings. But yes, that is a gem.  

---

## 2 answers

### Answer by username_3 on 2022-04-26. Score: 5 | Accepted answer

Even [seemingly simple systems can result in chaotic behavior](https://www.nature.com/articles/261459a0). Animal populations can be modeled by a very simple equation:
$$x\_{t+1} = \lambda\,x\_t\,(1-x\_t)$$
This is called the logistic map. Here $x$ is the population relative to the maximum population at some point in time, the subscripts $t$ and $t+1$ refer to the populations at time step $t$ the next time step ($t+1)$, and $\lambda$ is a number between 0 and 4.

The value $x=0$ is an obvious solution. For small values of $\lambda$, this is the only ultimate solution: Populations collapse. Another steady state solution is $x=1-1/\lambda$, which yields a non-negative solution if $\lambda>1$. All initial values of $x$ other than zero or one eventually settle to this value if $\lambda$ is between one and three. Two other steady state solutions arise when $\lambda>3$. Populations alternate between a high value and a low value. Oak trees appear to have found this set of solutions in their wars with squirrels. Some species of oaks have evolved to alternate between very small yields of acorns one year and very heavy yields the next. Even more bifurcations appear as $\lambda$ increases beyond three. At a certain point it becomes chaos. All from a seemingly simple equation.

Next, secular equations. Modeling the orbits of planets about a star is not easy. One way to do this is to simplify the multi-body interactions. The result is a set of secular equations. In this context, "secular" essentially means "averaged out". For example, the effects of Jupiter on Mercury's orbit can be secularized by treating Jupiter as a ring of mass about the Sun at Jupiter's orbital distance from the Sun as opposed to a point mass.

The result is a set of fairly nice and fairly simple time-dependent set of orbital elements for each planet. The question then is can these seemingly simple secular equations result in chaotic behavior? The answer is apparently yes. That is "secular chaos".

> **Comments:**
> **username_0**: Is this similar to Kozai mechanism? https://en.m.wikipedia.org/wiki/Kozai_mechanism In that article it seems to mention a 3-body system which causes changes in eccentricity though it does not mention chaotic behaviour.  
> **username_3**: @WDUK The Kozai mechanism is but one of many mechanisms that make the multi-body problem challenging to solve. Resonances are even more challenging. That even seemingly simple systems can result in chaotic behavior -- that is more challenging yet. One of the problems with predicting the stability of Mercury's orbit over the long term is the potential for resonances between the orbits of Jupiter and Mercury.  
> **username_0**: So is Kozai mechanism a type of chaos aswell ? Or is it a more stable system that happens to cause changes to orbits?  

---

### Answer by username_2 on 2022-04-26. Score: -1

Based merely on an article titled "[Onset of secular chaos in planetary systems: period doubling and strange attractors](https://authors.library.caltech.edu/27789/), it seems that this term indicates the behavior of planetary objects when there are no collisions or other effects that collapse the system. 'Period doubling' itself is a well-known phenomenon in basic chaos, aka strange attractor, math and can make for fun reading.


---

