Just how "locked" are resonant-chains of exoplanets thought to be? (e.g. K2-138 and TOI-178)

Question

The Phys.org news item Discovery of new planet reveals distant solar system to rival our own outlines the recent announcement of results using AI to help search Kepler photometric (transit-method) data for exoplanets. Near the end is the paragraph:

Kepler-90i wasn't the only jewel this neural network sifted out. In the Kepler-80 system, they found a sixth planet. This one, the Earth-size Kepler-80g, and four of its neighboring planets form what is called a "resonant chain," where the planets are locked by their mutual gravity in a rhythmic orbital dance. The result is an extremely stable system, similar to the seven planets in the TRAPPIST-1 system, so precisely balanced that the length of Kepler-80g's year could be predicted with mathematics.

My question is:

about these "resonant-chains" of planets. If I understand correctly this would be a group of planets where due to mutual perturbative effects they are in orbits who's periods are in mutual rational number ratios, e.g. 3:2 or 7:9.

Is it known yet how long-lived (time domain) or narrow (frequency) domain these resonances are, or are likely to be? In other words, are they thought to be really "locked" into these fixed ratios for say tens or even hundreds of millions of orbits, or are the orbits just really close, but with occasional, irregular "slippage" events?

Since the span of data and its precision is limited, I would expect that one can not conclude from the data alone, and it's likely some modeling has been done on resonant-chains of planets to improve the understanding of the phenomenon, so I'm asking to understand what's known and believed, rather than a precise answer about a given system. Since the problem is mostly orbital mechanics it lends itself more easily to simulation than most problems.

Background:

Wikipedia's Orbital resonance addresses "locked" orbital resonances:

Under some circumstances, a resonant system can be stable and self-correcting, so that the bodies remain in resonance. Examples are the 1:2:4 resonance of Jupiter's moons Ganymede, Europa and Io, and the 2:3 resonance between Pluto and Neptune. (emphasis added)

Astronomy and Astrophysics Leleu et al. (2021) - : Six transiting planets and a chain of Laplace resonances in TOI-178 (and arXiv)

• Gizmodo: Enigmatic Star System Has 5 Planets Locked in Perfect Harmony

• Bad Astronomy A Six-Planet System Dances in Time to the Tune of Gravity

• YouTube: Artist’s animation of the TOI-178 orbits and resonances (sound on!)

• The BBC News article Citizen science bags five-planet haul announces the system K2-138, also described as a "resonant chain" of planets. However, see the actual paper in ArXiv and Ast. J. where the title calls it only a "near-resonant chain" instead; The K2-138 System: A Near-Resonant Chain of Five Sub-Neptune Planets Discovered by Citizen Scientists

Answer 1

Short Answer: MMR (Mean Motion Resonance) chains seem to be mostly unstable over the lifetime of a planetary system, since we suspect most planetary systems begin in some resonance lock, but we observe so few of them. They also seem to be somewhat unstable in Frequency since most of the chains we do observe have timing differences on the order of 10^-2 from perfect resonance. Pristine resonance chains (timing differences on the order of 10^-4) are even more rare, especially around mature stars.

Long Answer: During typical planetary formation in an accretion disk, the planets' orbits are suspected to move inward due to an exchange of momentum with gas and debris in the disk. Numerical models show that, as planets' orbits migrate, they pass through mutual resonances and are often caught together in MMR chains. Some modelers believe this is so ubiquitous they are surprised it isn't the majority case for stellar/planetary systems. If the MMR chains were stable in the long term, we would expect most of the stellar systems we observe to have them. Specifically, from Dai et al.

This process of resonant capture is considered to be so effective and robust that it is difficult to understand why only a few percent of Kepler multi-planet systems are near first-order MMR.

Instead, we observe very few stellar systems with MMRs, and we have no planetary MMRs in our own Solar System (since Pluto was demoted). Only a few percent have MMRs, and only about a percent of those are very close to perfect MMRs. If the stellar system creation modeling is correct, then most of the stellar system had planetary MMRs early in their formation, but no longer have them. Dai et al. have a nice diagram showing the early formation of these resonances and how most get broken up by the time they are mature (like our Solar System close to 5Gyr).

If The Grand Tack hypothesis is correct, our own Solar System had at least one MMR between Jupiter and Saturn, which no longer exists.

MMRs can be broken up by various mechanisms. Dai et al. list (as some of the mechanisms that break up MMR chains):

Planetesimal scattering (Chatterjee & Ford 2015), tidal dissipation (Lithwick & Wu 2012; Batygin & Morbidelli 2013a), secular chaos (Petrovich et al. 2018), and orbital instability (Pu & Wu 2015; Izidoro et al. 2017; Goldberg & Batygin 2022)

Some resonances are more easily broken in numerical simulations than others. Dai et al. performed Monte-Carlo orbital simulations of TOI-1136 with period ratios of 3:2, 2:1, 3:2, 7:5, and 3:2. In the majority of the simulations, the 7:5 resonance was the first to become disrupted. According to the authors, this is due to the second order resonance being weaker.

Gradual drift away from MMRs is probably the reason why we see many exoplanet chains with mildly degraded MMRs (10^-2 MMR deviations). In one of the papers linked in the original question, The K2-138 System: a near-resonant chain of five sub-Neptune planets discovered by citizen scientists, Christiansen et al. have ratios for the orbital periods of planets b through f as:

1.513, 1.518, 1.528 and 1.544 for the b-c, c-d, d-e, and e-f pairs respectively, just outside the 3:2 resonance

So K2-138 would sit in the bottom middle part of the above diagram, since a perfect 3:2 resonance is 1.5, so they are on the order of 10^-2 away from perfect resonance. Of course K2-138 is only estimated to be about 2.3 Gyr old, according to wikipedia, so we might expect additional departure from perfect resonance as the system ages. Christiansen et al. paraphrase another paper as to the suspected migration from perfect resonance in this system:

They offer several possible explanations for this, including gravitational scattering slightly out of resonance by the additional bodies in the system, or tidal dissipation preferentially acting to drag the inner planets inward from the resonance

Tidal dissipation acting to pull the inner planets inward I guess would imply the planets are in retrograde spin direction. If they were in prograde spin, tidal dissipation would act to push the inner planets outward. Then the period ratios would be less than 1.5, rather than greater.

Summary: Resonant chains of exoplanets naturally form in some young stellar systems as their planetary orbits gradually migrate due to angular momentum exchange with the original accretion disk. Most of these resonant chains are unstable on the order of billions of years, so they are very rare around mature stars. Of those that are near resonance, most of them vary by on the order of 10^-2 from perfect resonance. Very few are within 10^-4 of perfect resonance.

Answer 2

Is it known yet how long-lived (time domain) or narrow (frequency) domain these resonances are, or are likely to be? In other words, are they thought to be really "locked" into these fixed ratios for say tens or even hundreds of millions of orbits, or are the orbits just really close, but with occasional, irregular "slippage" events?

TL;DR

Yes! Well .... sorta.

This incredibly powerful and comprehensive article on Resonant Chains of Exoplanets: Libration Centers for Three-body Angles answers every possible question you have about these resonances. It studies three-body angles as a diagnostic of resonant chains through tidally damped $N$-body integrations.

Long answer

Definitions

Now, what is a three body angle:

In resonant systems, three-body resonances can engage and link the planets' dynamics together. Such resonances are characterized by the general three-body angle, which is found through a linear combination of the planets' mean longitudes. These angles are easily detected in transit data because they strongly rely on the observed transit phase (with a weak dependence on eccentricity) and are a powerful diagnostic of system architecture.

A general three-planet resonant chain obeys $\tfrac{{P}_{2}}{{P}_{1}}\approx \tfrac{j+1}{j}$ and $\tfrac{{P}_{3}}{{P}_{2}}\approx \tfrac{k+1}{k}$, where $j$ and $k$ are integers and $P_i$ are the periods of subsequent planets $(i = 1, 2, 3)$; we refer to such chains by $(j + 1: j, k + 1: k)$. For a given chain, the system can be characterized by the two critical angles:

$$\phi_{12}=(j+1)\lambda_2-j\lambda_1-\varpi_2$$

$$\phi_{23}=(k+1)\lambda_3-k\lambda_2-\varpi_2$$

Simulations

From our numerical integrations, we find 180° is the preferred libration center for nearly all three-body angles. The notable exceptions are the (3:2, 4:3), (4:3, 3:2), and (5:4, 4:3) configurations, where the three-body angle admits equilibria off 180°; we discuss these three configurations in detail in Section 3.3.

To illustrate this behavior, in Figure 1 we present the evolution of the three-body angle equilibria for $S_1$ (see Table 1 for model parameters). We also report the libration center and amplitude for each equilibrium in Table 2, again for $S_1$. The reported values are taken from a 1 kyr window beginning when the inner period ratio has spread 0.5% from the assigned commensurability. With the exception of the (5:4, 4:3) chain, we find the three-body angle does not evolve considerably with additional spreading. _{Evolution of the three-body angles in ${{ \mathcal S }}_{1}$, where τa = 108 days and K = 103; the three-body angles are dependent on the adopted migration timescales (see Section 3.2). For each resonant configuration, the 30 models are grouped based on their three-body angle. The mean libration center of each group is represented as a line, the mean libration amplitude is shown as a shaded region, and the colors correspond to the different groups. The vertical dashed black line demarcates the end of the migration regime and the start of the tidal-dissipation forces. Configurations without any libration of the three-body angle are shaded gray.}