<blockquote>
<p>In many articles they speak about 'mean motion resonance', <strong>what is the difference with a classic orbital resonance</strong>?</p>
</blockquote>
<p>I'm not really sure about a &quot;<em>classic</em> orbital resonance&quot; but at least in <a href="https://www.cambridge.org/core/books/solar-system-dynamics/108745217E4A18190CBA340ED5E477A2" rel="nofollow noreferrer">Murray &amp; Dermott: Solar System Dynamics</a> the concept of a <span class="math-container">$p+q:p$</span> orbit-orbit resonance / orbital resonance / coorbital resonance is introduced first by the integer relation arising from the periods of the bodies orbiting the pericentre of the system. This first idea is &quot;loose&quot; as it can be discussed. I would think that there's no difference between a mean motion resonance and an orbital resonance.</p>
<blockquote>
<p>If you have <strong>any recommandation of books/articles about this topic</strong>, I would be glad to hear it.</p>
</blockquote>
<p>The same applies to <a href="https://rads.stackoverflow.com/amzn/click/com/0691130272" rel="nofollow noreferrer" rel="nofollow noreferrer">Binney &amp; Tremaine: Galactic Dynamics</a>. These are good references to look at if interested in motion and mean motion resonances.</p>
<blockquote>
<p><strong>How can we define that two exoplanets are in resonance?</strong> Scientific articles are using the ratio of the orbital periods being related by an integer ratio, but in a book (exoplanet handbook-Michael Perryman) I saw &quot;mean motion resonance: definition is via the librating or oscillating behaviour of some linear combination of resonant arguments&quot;.</p>
</blockquote>
<p>When two planets are <em>near</em> to be <em>commensurable</em> (also known as a &quot;nominal resonance&quot;, even when this does not necessarily imply a capture in resonance) they <em>could</em> be captured in resonance if the migration process is slow enough. This can be seen from level curves of the Resonance Hamiltonian <span class="math-container">$K$</span></p>
<p><span class="math-container">$$K = -3 \Delta R + R^2 - 2 \sqrt{2R} \cos{r}$$</span>
corresponding to the &quot;<em>Second Fundamental Model of Resonance</em>&quot; (being the Simple Pendulum the First), that arises from simplifications when treating the 3-body problem. Here (<span class="math-container">$r$</span>,<span class="math-container">$R$</span>) are canonical variables and <span class="math-container">$\Delta$</span> is a dimensionless resonance distance that can be equivalently defined depending on the authors (e.g., see <a href="https://arxiv.org/pdf/2003.03388.pdf" rel="nofollow noreferrer">Chiang &amp; Choksi, 2020</a>, eqn. 1 or <a href="https://arxiv.org/pdf/1809.10042.pdf" rel="nofollow noreferrer">Terquem &amp; Papaloizou, 2019</a>, eqn. 44). A complete and enlightening aclaration to this respect is given at <a href="https://iopscience.iop.org/article/10.1088/0004-637X/770/1/24" rel="nofollow noreferrer">Petrovich, Malhotra, Tremaine: Planets Near Mean-Motion Resonances, 2013</a>. Paraphrasing a little:</p>
<blockquote>
<p>Generally, a planet is said to be &quot;in resonance&quot; if some dynamically significant critical argument librates - this is consistent with Murray &amp; Dermott SSD Chapt. 8, 2000 -  (...) This definition has several shortcomings (...): (1) not consistent (...) as planets can be resonant at <em>any</em> period ratio; (2) by canonical transformations one can change the critical argument from libration to circulation so the definition is coordinate-dependent; (3) interaction of nearby resonances arising from the degenerate frequencies in the Kepler problem can cause a critical argument to jump between libration and circulation at irregular intervals.</p>
</blockquote>
<p>(1) can be explained from the Hamiltonian K in the article above:</p>
<p><a href="https://i.sstatic.net/3JjyW.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/3JjyW.png" alt="Fig. 1 Petrovich, Malhotra, Tremaine. 2013" /></a></p>
<p>Here <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are canonical coordinates: momentum and coordinate, respectively, following <span class="math-container">$(x,y)= \sqrt{2R}(\cos{r}, \sin{r})$</span>. The interesting fact to note is that from <a href="https://en.wikipedia.org/wiki/Liouville%27s_theorem_(Hamiltonian)" rel="nofollow noreferrer">Liouville's Theorem</a> the phase-space volume (area in this case) must be conserved then there are &quot;adiabatic invariants&quot; in the problem: if you start a system @ some conditions, given values of <span class="math-container">$R,r$</span> for fixed <span class="math-container">$K$</span>, <span class="math-container">$\Delta = \Delta(t)$</span> will vary over time and this implies that a circulating solution could then start to librate, from where a capture could arise.</p>
<blockquote>
<p><strong>What is the difference between this or the ratio of periods to define the resonance?</strong></p>
</blockquote>
<p>Then <span class="math-container">$\Delta$</span> could keep changing over time (e.g., growing for a convergent migration) but a pair of planets could still be in a resonant motion, after passing by a <em>commensurability</em>. Altough I think this question requieres of someone who's more experienced on the subject, a thought about it is that: if one defines an orbital resonance from the period ratios only, there might be troubles justifying why it is a resonant behavior, e.g., if there's no capture (when the resonance is not stable).</p>
<p>By the way, these articles might also be useful:</p>
<ul>
<li><a href="https://adsabs.harvard.edu/full/1983CeMec..30..197H" rel="nofollow noreferrer">Henrard &amp; Lemaitre, 1983: A second fundamental model for Resonance</a></li>
<li><a href="https://www.sciencedirect.com/science/article/abs/pii/001910359090126T" rel="nofollow noreferrer">Malhotra &amp; Dermott, 1990: The role of secondary resonances in the orbital history of Miranda
</a></li>
<li>&quot;Winter O. C., Murray C. D., 1994a, Atlas of the Planar, Circular,Restricted Three-Body Problem. I. Internal Orbits, QMW Maths Notes, Murray Winter 1994&quot; but I couldn't find it anywhere.</li>
</ul>

CC BY-SA 4.0
2022-02-01T01:37:26.253Z
2022-02-01T01:37:26.253Z
nuwe
<p>I am studying for a project the dynamics of multi-planetary system in resonance. I have two questions about this domain.</p>

<ul>
<li><p>In many articles they speak about 'mean motion resonance', what is the difference with a classic orbital resonance?</p></li>
<li><p>How can we define that two exoplanets are in resonance? Scientific articles are using the ratio of the orbital periods being related by an integer ratio, but in a book (exoplanet handbook-Michael Perryman) I saw "mean motion resonance: definition is via the librating or oscillating behaviour of
some linear combination of resonant arguments". What is the difference between this or the ratio of periods to define the resonance?</p></li>
</ul>

<p>Also, if you have any recommandation of books/articles about this topic, I would be glad to hear it.</p>

<p>Thank you!</p>

CC BY-SA 4.0
2020-03-27T21:02:49.990Z
different but related: [Just how “locked” are resonant-chains of exoplanets thought to be?](https://astronomy.stackexchange.com/q/23980/7982)
uhoh
CC BY-SA 4.0
2022-02-23T16:13:06.397Z
take a look at https://sites.google.com/view/mmresonances/
regards,
tabare
Tabare
<astrophysics><exoplanet><celestial-mechanics><orbital-resonance>
Explanation about the resonance, mean motion resonance and libration
Adrien Chimay
# Explanation about the resonance, mean motion resonance and libration

(Asked by: username_0 on 2020-03-27)

I am studying for a project the dynamics of multi-planetary system in resonance. I have two questions about this domain.

* In many articles they speak about 'mean motion resonance', what is the difference with a classic orbital resonance?
* How can we define that two exoplanets are in resonance? Scientific articles are using the ratio of the orbital periods being related by an integer ratio, but in a book (exoplanet handbook-Michael Perryman) I saw "mean motion resonance: definition is via the librating or oscillating behaviour of
  some linear combination of resonant arguments". What is the difference between this or the ratio of periods to define the resonance?

Also, if you have any recommandation of books/articles about this topic, I would be glad to hear it.

Thank you!

**Tags:** astrophysics, exoplanet, celestial-mechanics, orbital-resonance

> **Comments:**
> **username_1**: different but related: [Just how “locked” are resonant-chains of exoplanets thought to be?](https://astronomy.stackexchange.com/q/23980/7982)  
> **username_2**: take a look at https://sites.google.com/view/mmresonances/ regards, tabare  

---

## 1 answers

### Answer by username_3 on 2022-02-01. Score: 2

> In many articles they speak about 'mean motion resonance', **what is the difference with a classic orbital resonance**?

I'm not really sure about a "*classic* orbital resonance" but at least in [Murray & Dermott: Solar System Dynamics](https://www.cambridge.org/core/books/solar-system-dynamics/108745217E4A18190CBA340ED5E477A2) the concept of a $p+q:p$ orbit-orbit resonance / orbital resonance / coorbital resonance is introduced first by the integer relation arising from the periods of the bodies orbiting the pericentre of the system. This first idea is "loose" as it can be discussed. I would think that there's no difference between a mean motion resonance and an orbital resonance.

> If you have **any recommandation of books/articles about this topic**, I would be glad to hear it.

The same applies to [Binney & Tremaine: Galactic Dynamics](https://rads.stackoverflow.com/amzn/click/com/0691130272). These are good references to look at if interested in motion and mean motion resonances.

> **How can we define that two exoplanets are in resonance?** Scientific articles are using the ratio of the orbital periods being related by an integer ratio, but in a book (exoplanet handbook-Michael Perryman) I saw "mean motion resonance: definition is via the librating or oscillating behaviour of some linear combination of resonant arguments".

When two planets are *near* to be *commensurable* (also known as a "nominal resonance", even when this does not necessarily imply a capture in resonance) they *could* be captured in resonance if the migration process is slow enough. This can be seen from level curves of the Resonance Hamiltonian $K$

$$K = -3 \Delta R + R^2 - 2 \sqrt{2R} \cos{r}$$
corresponding to the "*Second Fundamental Model of Resonance*" (being the Simple Pendulum the First), that arises from simplifications when treating the 3-body problem. Here ($r$,$R$) are canonical variables and $\Delta$ is a dimensionless resonance distance that can be equivalently defined depending on the authors (e.g., see [Chiang & Choksi, 2020](https://arxiv.org/pdf/2003.03388.pdf), eqn. 1 or [Terquem & Papaloizou, 2019](https://arxiv.org/pdf/1809.10042.pdf), eqn. 44). A complete and enlightening aclaration to this respect is given at [Petrovich, Malhotra, Tremaine: Planets Near Mean-Motion Resonances, 2013](https://iopscience.iop.org/article/10.1088/0004-637X/770/1/24). Paraphrasing a little:

> Generally, a planet is said to be "in resonance" if some dynamically significant critical argument librates - this is consistent with Murray & Dermott SSD Chapt. 8, 2000 - (...) This definition has several shortcomings (...): (1) not consistent (...) as planets can be resonant at *any* period ratio; (2) by canonical transformations one can change the critical argument from libration to circulation so the definition is coordinate-dependent; (3) interaction of nearby resonances arising from the degenerate frequencies in the Kepler problem can cause a critical argument to jump between libration and circulation at irregular intervals.

(1) can be explained from the Hamiltonian K in the article above:

[![Fig. 1 Petrovich, Malhotra, Tremaine. 2013](https://i.sstatic.net/3JjyW.png)](https://i.sstatic.net/3JjyW.png)

Here $x$ and $y$ are canonical coordinates: momentum and coordinate, respectively, following $(x,y)= \sqrt{2R}(\cos{r}, \sin{r})$. The interesting fact to note is that from [Liouville's Theorem](https://en.wikipedia.org/wiki/Liouville%27s_theorem_(Hamiltonian)) the phase-space volume (area in this case) must be conserved then there are "adiabatic invariants" in the problem: if you start a system @ some conditions, given values of $R,r$ for fixed $K$, $\Delta = \Delta(t)$ will vary over time and this implies that a circulating solution could then start to librate, from where a capture could arise.

> **What is the difference between this or the ratio of periods to define the resonance?**

Then $\Delta$ could keep changing over time (e.g., growing for a convergent migration) but a pair of planets could still be in a resonant motion, after passing by a *commensurability*. Altough I think this question requieres of someone who's more experienced on the subject, a thought about it is that: if one defines an orbital resonance from the period ratios only, there might be troubles justifying why it is a resonant behavior, e.g., if there's no capture (when the resonance is not stable).

By the way, these articles might also be useful:

* [Henrard & Lemaitre, 1983: A second fundamental model for Resonance](https://adsabs.harvard.edu/full/1983CeMec..30..197H)
* [Malhotra & Dermott, 1990: The role of secondary resonances in the orbital history of Miranda](https://www.sciencedirect.com/science/article/abs/pii/001910359090126T)
* "Winter O. C., Murray C. D., 1994a, Atlas of the Planar, Circular,Restricted Three-Body Problem. I. Internal Orbits, QMW Maths Notes, Murray Winter 1994" but I couldn't find it anywhere.


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