# Planetary orbital resonances

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The Kirkwood gaps in the asteroid belt are associated with orbital resonances with Jupiter yet planets seem to prefer resonant locations. Why don't asteroids accumulate near the "Kirkwood locations" rather than avoid them??

This is actually a very subtle question, much more so than the answers to the similar questions provided in the comments give it credit for. When I was in graduate school at Ohio State I routinely asked this question to visiting dynamicists and invariably got different answers.

The very basic answer is that if you have two sufficiently strong resonances sufficiently close together, then the resonance will be unstable. Otherwise, the resonance will be stable. But what determines "sufficiently strong" and "sufficiently close" is where things get very complicated quickly. A basic criterion is the Chirikov criterion. (The Scholarpedia article is somewhat more detailed.) However, the Chirikov criterion is not universally valid.

If you have overlapping resonances, then an object gets bounced back and forth between these two resonances chaotically. These different resonances perturb the orbit in different ways, and eventually they will perturb the orbit into an unstable orbit, thus leading to depletion of the resonance. If a resonance is "distant" from other resonances, then the resonance tends to keep objects locked in place, leading to an excess of objects in the resonance.

Most of the resonances in the asteroid belt are fairly close together, which leads to them being unstable. The Kirkwood Gaps are the most prominent manifestation of these instabilities. For example, the Alinda family of asteroids are in a 1:3 resonance with Jupiter, and are very close to a 4:1 resonance with the Earth. This leads to instability, and hence very few asteroids in this family. However, in the outer Solar System, the resonances are generally far apart, and so are mostly stable. The plutinos are one example of such a stable resonance, being in a 3:2 resonance with Neptune.

## Unique Planetary System With Rhythmic Orbital Resonance Revealed by Exoplanet Watcher Cheops

This is an artist’s impression of the TOI-178 planetary system, which was revealed by ESA’s exoplanet watcher Cheops. The system consists of six exoplanets, five of which are locked in a rare rhythmic dance as they orbit their central star. In this artist impression, the relative sizes of the planets are to scale, but not the distances and the size of the star. Credit: ESA

ESA’s exoplanet mission Cheops has revealed a unique planetary system consisting of six exoplanets, five of which are locked in a rare rhythmic dance as they orbit their central star. The sizes and masses of the planets, however, don’t follow such an orderly pattern. This finding challenges current theories of planet formation.

The discovery of increasing numbers of planetary systems, none like our own Solar System, continues to improve our understanding of how planets form and evolve. A striking example is the planetary system called TOI-178, some 200 light-years away in the constellation of Sculptor.

Astronomers already expected this star to host two or more exoplanets after observing it with NASA’s Transiting Exoplanet Survey Satellite (TESS). New, highly precise observations with Cheops, ESA’s Characterising Exoplanet Satellite that was launched in 2019, now show that TOI-178 harbors at least six planets and that this foreign solar system has a very unique layout. The team, led by Adrien Leleu of University of Geneva and the University of Bern in Switzerland, published their results today in Astronomy & Astrophysics.

One of the special characteristics of the TOI-178 system that the scientists were able to uncover with Cheops is that the planets – except the one closest to the star – follow a rhythmic dance as they move in their orbits. This phenomenon is called orbital resonance, and it means that there are patterns that repeat themselves as the planets go around the star, with some planets aligning every few orbits.

A similar resonance is observed in the orbits of three of Jupiter’s moons: Io, Europa, and Ganymede. For every orbit of Europa, Ganymede completes two orbits, and Io completes four (this is a 4:2:1 pattern).

In the TOI-178 system, the resonant motion is much more complex as it involves five planets, following an 18:9:6:4:3 pattern. While the second planet from the star (the first in the pattern) completes 18 orbits, the third planet from the star (second in the pattern) completes nine orbits, and so on.

This graphic shows a representation of the TOI-178 planetary system, which was revealed by ESA’s exoplanet watcher Cheops. In this graphic, the relative sizes of the planets are to scale, but not the distances and the size of the star. Credit: ESA/Cheops Mission Consortium/A. Leleu et al.

Initially, the scientists only found four of the planets in resonance, but by following the pattern the scientists calculated that there must be another planet in the system (the fourth following the pattern, the fifth planet from the star).

“We predicted its trajectory very precisely by assuming that it was in resonance with the other planets,” Adrien explains. An additional observation with Cheops confirmed that the missing planet indeed existed in the predicted orbit.

After they had uncovered the rare orbital arrangements, the scientists were curious to see whether the planet densities (size and mass) also follow an orderly pattern. To investigate this, Adrien and his team combined data from Cheops with observations taken with ground-based telescopes at the European Southern Observatory’s (ESO) Paranal Observatory in Chile.

But while the planets in the TOI-178 system orbit their star in a very orderly manner, their densities do not follow any particular pattern. One of the exoplanets, a dense, terrestrial planet like Earth is right next to a similar-sized but very fluffy planet ­­– like a mini-Jupiter, and next to that is one very similar to Neptune.

“This is not what we expected, and is the first time that we observe such a setup in a planetary system,” says Adrien. “In the few systems we know where the planets orbit in this resonant rhythm, the densities of the planets gradually decrease as we move away from the star, and it is also what we expect from theory.”

Catastrophic events such as giant impacts could normally explain large variations in planet densities, but the TOI-178 system would not be so neatly in harmony if that had been the case.

“The orbits in this system are very well ordered, which tells us that this system has evolved quite gently since its birth,” explains co-author Yann Alibert from the University of Bern.

Revealing the complex architecture of the TOI-178 system, which challenges current theories of planet formation, was made possible thanks to almost 12 days of observations with Cheops (11 days of continuous observations, plus two shorter observations).

“Solving this exciting puzzle required quite some effort to plan, in particular to schedule the 11-day continuous observation needed in order to catch the signatures of the different planets,” says ESA Cheops project scientist Kate Isaak. “This study highlights very nicely the follow-up potential of Cheops – not only to better characterize known planets, but to hunt down and confirm new ones.”

Adrien and his team want to continue to use Cheops to study the TOI system in even more detail.

“We might find more planets that could be in the habitable zone – where liquid water might be present on the surface of a planet – which begins outside of the orbits of the planets that we discovered to date,” says Adrien. “We also want to find out what happened to the innermost planet that is not in resonance with the others. We suspect that it broke out of resonance due to tidal forces.”

Astronomers will use Cheops to observe hundreds of known exoplanets orbiting bright stars.

“Cheops will not only deepen our understanding of the formation of exoplanets, but also that of our own planet and the Solar System,” adds Kate.

Reference: “Six transiting planets and a chain of Laplace resonances in TOI-178” by A. Leleu, Y. Alibert, N. C. Hara, M. J. Hooton, T. G. Wilson, P. Robutel, J.-B. Delisle, J. Laskar, S. Hoyer, C. Lovis, E. M. Bryant, E. Ducrot, J. Cabrera, J. Acton, V. Adibekyan, R. Allart, C. Allende Prieto, R. Alonso, D. Alves, D. R. Anderson et al., 25 January 2021, Astronomy & Astrophysics.
DOI: 10.1051/0004-6361/202039767

Cheops is an ESA mission developed in partnership with Switzerland, with a dedicated consortium led by the University of Bern, and with important contributions from Austria, Belgium, France, Germany, Hungary, Italy, Portugal, Spain, Sweden and the UK.

ESA is the Cheops mission architect, responsible for procurement and testing of the satellite, the launch and early operations phase, and in-orbit commissioning, as well as the Guest Observers’ Programme through which scientists world-wide can apply to observe with Cheops. The consortium of 11 ESA Member States led by Switzerland provided essential elements of the mission. The prime contractor for the design and construction of the spacecraft is Airbus Defence and Space in Madrid, Spain.

The Cheops mission consortium runs the Mission Operations Centre located at INTA, in Torrejón de Ardoz near Madrid, Spain, and the Science Operations Centre, located at the University of Geneva, Switzerland.

## "Planet Nine" update: Possible resonances beyond the Kuiper belt?

When Konstantin Batygin and Mike Brown announced the possible existence of a distant planet, my first question was "what do the dynamicists think?" Several were quoted in the media surrounding the announcement, but yesterday leading dynamicist Renu Malhotra (with coauthors Kat Volk and Xianyu Wang) posted to ArXiv the first formal response I've seen. In brief, Malhotra and coauthors are on board with the idea of a possible outer planet, and found that it may have shaped the orbits of extremely distant Kuiper belt objects in another way beyond the several ways that Batygin and Brown proposed.

Before I continue with the story, I want to mention that you can hear from Batygin and Brown directly tonight through a Planetary Radio Live webcast! I'll be onstage as well.

Possible orbit of a perturbing "ninth planet" The six most distant known objects in the solar system with orbits exclusively beyond Neptune (magenta), including Sedna (dark magenta), all mysteriously line up in a single direction. Also, when viewed in three dimensions, they tilt nearly identically away from the plane of the solar system. Another population of Kuiper belt objects (cyan) are forced into orbits that are perpendicular to the plane of the solar system and clustered in orientation. Batygin and Brown show that a planet with 10 times the mass of the earth in a distant eccentric orbit (orange) anti-aligned with the magenta orbits and perpendicular to the cyan orbits is required to maintain this configuration. Image: Caltech/R. Hurt (IPAC) [Diagram was created using WorldWide Telescope.]

In the paper, Malhotra points out that because the orbits of the extremely distant Kuiper belt objects like Sedna, 2010 GB174, 2004 VN112, 2012 VP113, and 2013 GP136 are so eccentric, then they are likely to have had close encounters with the putative planet. Close encounters with a massive planet change orbits of smaller worlds. It's relatively easy for small worlds, so tenuously connected to the Sun, to get ejected from the solar system entirely. If there is an undiscovered distant planet affecting their orbits, the fact that the little worlds still remain in our solar system means either that they've had relatively few encounters, or else they're protected from close encounters with the planet by being in resonances. This is how Pluto is still a member of our solar system even though its orbit crosses Neptune's: because Pluto orbits the Sun twice for every three times Neptune does, Pluto and Neptune are never actually close to each other, so Neptune doesn't get a chance to eject Pluto.

Armed with this hypothesis, Malhotra, Volk, and Wang investigated whether the worlds we know about could be in resonances with the one that Batygin and Brown suggested. In short, they can. It's complicated because we have short observational arcs on these distant, slow-moving worlds, so the analysis has to include a detailed understanding of the uncertainties on the worlds' orbits. The analysis suggests that Sedna's orbital period is in a 3:2 resonance with the putative planet 2010 GB174 in a 5:2 2994 VN112 in a 3:1 resonance 2004 VP113 in 4:1 and 2013 GP136 in 9:1.

If all this is true (and I should note here that the paper has not been peer-reviewed yet), Malhotra et al.'s work constrains the mass and location of the possible planet in different ways than Batygin and Brown's does. In this new paper, in order to keep the smaller worlds corralled into resonant orbits, the possible planet has to have a mass of at least 10 times that of Earth. The orbital plane can be one of two: either inclined at 18 degrees or 48 degrees. In the low-inclination case, the orbit eccentricity would be less than 0.18 in the high-inclination case, it could be much larger. There are many places along the possible orbits that the putative planet could not be, or else it would have close encounters with the discovered worlds.

Is this proof for a ninth planet? No. From their conclusion:

Our analysis supports the distant planet hypothesis, but should not be considered definitive proof of its existence. The orbital period ratios have significant uncertainties, so the near-coincidences with N/1 and N/2 ratios may simply be by chance for a small number of bodies. The long orbital timescales in this region of the outer solar system may allow formally unstable orbits to persist for very long times, possibly even to the age of the solar system, depending on the planet mass if so, this would weaken the argument for a resonant planet orbit. It would be pertinent to examine this question quantitatively in future work.

More work is always needed -- but this work is cool because it suggests new constraints on where to look for the possible undiscovered world.

## Contents

### Gas disk Edit

Protoplanetary gas disks around young stars are observed to have lifetimes of a few million years. If planets with masses of around an Earth mass or greater form while the gas is still present, the planets can exchange angular momentum with the surrounding gas in the protoplanetary disk so that their orbits change gradually. Although the sense of migration is typically inwards in locally isothermal disks, outward migration may occur in disks that possess entropy gradients.

### Planetesimal disk Edit

During the late phase of planetary system formation, massive protoplanets and planetesimals gravitationally interact in a chaotic manner causing many planetesimals to be thrown into new orbits. This results in angular-momentum exchange between the planets and the planetesimals, and leads to migration (either inward or outward). Outward migration of Neptune is believed to be responsible for the resonant capture of Pluto and other Plutinos into the 3:2 resonance with Neptune.

There are many different mechanisms by which planets' orbits can migrate, which are described below as disk migration (Type I migration, Type II migration, or Type III migration), tidal migration, planetesimal-driven migration, gravitational scattering, and Kozai cycles and tidal friction. This list of types is not exhaustive or definitive: Depending on what is most convenient for any one type of study, different researchers will distinguish mechanisms in somewhat different ways.

Classification of any one mechanism is mainly based on the circumstances in the disk that enable the mechanism to efficiently transfer energy and / or angular momentum to and from planetary orbits. As the loss or relocation of material in the disk changes the circumstances, one migration mechanism will give way to another mechanism, or perhaps none. If there is no follow-on mechanism, migration (largely) stops and the stellar system becomes (mostly) stable.

### Disk migration Edit

Disk migration arises from the gravitational force exerted by a sufficiently massive body embedded in a disk on the surrounding disk's gas, which perturbs its density distribution. By the reaction principle of classical mechanics, the gas exerts an equal and opposite gravitational force on the body, which can also be expressed as a torque. This torque alters the angular momentum of the planet's orbit, resulting in a variation of the semi-major axis and other orbital elements. An increase over time of the semi-major axis leads to outward migration, i.e., away from the star, whereas the opposite behavior leads to inward migration.

Three sub-types of disk migration are distinguished as Types I, II, and III. The numbering is not intended to suggest a sequence or stages.

#### Type I migration Edit

Small planets undergo Type I disk migration driven by torques arising from Lindblad and co-rotation resonances. Lindblad resonances excite spiral density waves in the surrounding gas, both interior and exterior of the planet's orbit. In most cases, the outer spiral wave exerts a greater torque than does the inner wave, causing the planet to lose angular momentum, and hence migrate toward the star. The migration rate due to these torques is proportional to the mass of the planet and to the local gas density, and results in a migration timescale that tends to be short relative to the million-year lifetime of the gaseous disk. [1] Additional co-rotation torques are also exerted by gas orbiting with a period similar to that of the planet. In a reference frame attached to the planet, this gas follows horseshoe orbits, reversing direction when it approaches the planet from ahead or from behind. The gas reversing course ahead of the planet originates from a larger semi-major axis and may be cooler and denser than the gas reversing course behind the planet. This may result in a region of excess density ahead of the planet and of lesser density behind the planet, causing the planet to gain angular momentum. [2] [3]

The planet mass for which migration can be approximated to Type I depends on the local gas pressure scale height and, to a lesser extent, the kinematic viscosity of the gas. [1] [4] In warm and viscous disks, Type I migration may apply to larger mass planets. In locally isothermal disks and far from steep density and temperature gradients, co-rotation torques are generally overpowered by the Lindblad torques. [5] [4] Regions of outward migration may exist for some planetary mass ranges and disk conditions in both local isothermal and non-isothermal disks. [4] [6] The locations of these regions may vary during the evolution of the disk, and in the local-isothermal case are restricted to regions with large density and/or temperature radial gradients over several pressure scale-heights. Type I migration in a local isothermal disk was shown to be compatible with the formation and long-term evolution of some of the observed Kepler planets. [7] The rapid accretion of solid material by the planet may also produce a "heating torque" that causes the planet to gain angular momentum. [8]

#### Type II migration Edit

A planet massive enough to open a gap in a gaseous disk undergoes a regime referred to as Type II disk migration. When the mass of a perturbing planet is large enough, the tidal torque it exerts on the gas transfers angular momentum to the gas exterior of the planet's orbit, and does the opposite interior to the planet, thereby repelling gas from around the orbit. In a Type I regime, viscous torques can efficiently counter this effect by resupplying gas and smoothing out sharp density gradients. But when the torques become strong enough to overcome the viscous torques in the vicinity of the planet's orbit, a lower density annular gap is created. The depth of this gap depends on the temperature and viscosity of the gas and on the planet mass. In the simple scenario in which no gas crosses the gap, the migration of the planet follows the viscous evolution of the disk's gas. In the inner disk, the planet spirals inward on the viscous timescale, following the accretion of gas onto the star. In this case, the migration rate is typically slower than would be the migration of the planet in the Type I regime. In the outer disk, however, migration can be outward if the disk is viscously expanding. A Jupiter-mass planet in a typical protoplanetary disk is expected to undergo migration at approximately the Type II rate, with the transition from Type I to Type II occurring at roughly the mass of Saturn, as a partial gap is opened. [9] [10]

Type II migration is one explanation for the formation of hot Jupiters. [11] In more realistic situations, unless extreme thermal and viscosity conditions occur in a disk, there is an ongoing flux of gas through the gap. [12] As a consequence of this mass flux, torques acting on a planet can be susceptible to local disk properties, akin to torques at work during Type I migration. Therefore, in viscous disks, Type II migration can be typically described as a modified form of Type I migration, in a unified formalism. [10] [4] The transition between Type I and Type II migration is generally smooth, but deviations from a smooth transition have also been found. [9] [13] In some situations, when planets induce eccentric perturbation in the surrounding disk's gas, Type II migration may slow down, stall, or reverse. [14]

From a physical viewpoint, Type I and Type II migration are driven by the same type of torques (at Lindblad and co-rotation resonances). In fact, they can be interpreted and modeled as a single regime of migration, that of Type I appropriately modified by the perturbed gas surface density of the disk. [10] [4]

#### Type III disk migration Edit

Type III disk migration applies to fairly extreme disk / planet cases and is characterized by extremely short migration timescales. [15] [16] [10] Although sometimes referred to as "runaway migration", the migration rate does not necessarily increase over time. [15] [16] Type III migration is driven by the co-orbital torques from gas trapped in the planet's libration regions and from an initial, relatively fast, planetary radial motion. The planet's radial motion displaces gas in its co-orbital region, creating a density asymmetry between the gas on the leading and the trailing side of the planet. [10] [1] Type III migration applies to disks that are relatively massive and to planets that can only open partial gaps in the gas disk. [1] [10] [15] Previous interpretations linked Type III migration to gas streaming across the orbit of the planet in the opposite direction as the planet's radial motion, creating a positive feedback loop. [15] Fast outward migration may also occur temporarily, delivering giant planets to distant orbits, if later Type II migration is ineffective at driving the planets back. [17]

### Gravitational scattering Edit

Another possible mechanism that may move planets over large orbital radii is gravitational scattering by larger planets or, in a protoplantetary disk, gravitational scattering by over-densities in the fluid of the disk. [18] In the case of the Solar System, Uranus and Neptune may have been gravitationally scattered onto larger orbits by close encounters with Jupiter and/or Saturn. [19] [20] Systems of exoplanets can undergo similar dynamical instabilities following the dissipation of the gas disk that alter their orbits and in some cases result in planets being ejected or colliding with the star.

Planets scattered gravitationally can end on highly eccentric orbits with perihelia close to the star, enabling their orbits to be altered by the tides they raise on the star. The eccentricities and inclinations of these planets are also excited during these encounters, providing one possible explanation for the observed eccentricity distribution of the closely orbiting exoplanets. [21] The resulting systems are often near the limits of stability. [22] As in the Nice model, systems of exoplanets with an outer disk of planetesimals can also undergo dynamical instabilities following resonance crossings during planetesimal-driven migration. The eccentricites and inclinations of the planets on distant orbits can be damped by dynamical friction with the planetesimals with the final values depending on the relative masses of the disk and the planets that had gravitational encounters. [23]

### Tidal migration Edit

Tides between the star and planet modify the semi-major axis and orbital eccentricity of the planet. If the planet is orbiting very near to its star, the tide of the planet raises a bulge on the star. If the star's rotational period is longer than the planet's orbital period the location of the bulge lags behind a line between the planet and the center of the star creating a torque between the planet and the star. As a result, the planet loses angular momentum and its semi-major axis decreases with time.

If the planet is in an eccentric orbit the strength of the tide is stronger when it is near perihelion. The planet is slowed the most when near perihelion, causing its aphelion to decrease faster than its perihelion, reducing its eccentricity. Unlike disk migration – which lasts a few million years until the gas dissipates – tidal migration continues for billions of years. Tidal evolution of close-in planets produces semi-major axes typically half as large as they were at the time that the gas nebula cleared. [24]

### Kozai cycles and tidal friction Edit

A planetary orbit that is inclined relative to the plane of a binary star can shrink due to a combination of Kozai cycles and tidal friction. Interactions with the more distant star cause the planets orbit to undergo an exchange of eccentricity and inclination due to the Kozai mechanism. This process can increase the planet's eccentricity and lower its perihelion enough to create strong tides between the planet on the star increases. When it is near the star the planet loses angular momentum causing its orbit to shrink.

The planet's eccentricity and inclination cycle repeatedly, slowing the evolution of the planets semi-major axis. [25] If the planet's orbit shrinks enough to remove it from the influence of the distant star the Kozai cycles end. Its orbit will then shrink more rapidly as it is tidally circularized. The orbit of the planet can also become retrograde due to this process. Kozai cycles can also occur in a system with two planets that have differing inclinations due to gravitational scattering between planets and can result in planets with retrograde orbits. [26] [27]

### Planetesimal-driven migration Edit

The orbit of a planet can change due to gravitational encounters with a large number of planetesimals. Planetesimal-driven migration is the result of the accumulation of the transfers of angular momentum during encounters between the planetesimals and a planet. For individual encounters the amount of angular momentum exchanged and the direction of the change in the planet's orbit depends on the geometry of the encounter. For a large number of encounters the direction of the planet's migration depends on the average angular momentum of the planetesimals relative to the planet. If it is higher, for example a disk outside the planet's orbit, the planet migrates outward, if it is lower the planet migrates inward. The migration of a planet beginning with a similar angular momentum as the disk depends on potential sinks and sources of the planetesimals. [28]

For a single planet system, planetesimals can only be lost (a sink) due to their ejection, which would cause the planet to migrate inward. In multiple planet systems the other planets can act as sinks or sources. Planetesimals can be removed from the planet's influence after encountering an adjacent planet or transferred to that planet's influence. These interactions cause the planet's orbits to diverge as the outer planet tends to remove planetesimals with larger momentum from the inner planet influence or add planetesimals with lower angular momentum, and vice versa. The planet's resonances, where the eccentricities of planetesimals are pumped up until they intersect with the planet, also act as a source. Finally, the planet's migration acts as both a sink and a source of new planetesimals creating a positive feedback that tends to continue its migration in the original direction. [28]

Planetesimal-driven migration can be damped if planetesimals are lost to various sinks faster than new ones are encountered due to its sources. It may be sustained if the new planetesimals enter its influence faster than they are lost. If sustained migration is due to its migration only, it is called runaway migration. If it is due to the loss of planetesimals to another planets influence, it is called forced migration [28] For a single planet orbiting in a planetesial disk the shorter timescales of the encounters with planetesimals with shorter period orbits results in more frequent encounters with the planetesimals with less angular momentum and the inward migration of the planet. [29] Planetesimal-driven migration in a gas disk, however, can be outward for a particular range of planetesimal sizes because of the removal of shorter period planetesimals due to gas drag. [30]

The migration of planets can lead to planets being captured in resonances and chains of resonances if their orbits converge. The orbits of the planets can converge if the migration of the inner planet is halted at the inner edge of the gas disk, resulting in a systems of tightly orbiting inner planets [31] or if migration is halted in a convergence zone where the torques driving Type I migration cancel, for example near the ice line, in a chain of more distant planets. [32]

Gravitational encounters can also lead to the capture of planets with sizable eccentricities in resonances. [33] In the Grand tack hypothesis the migration of Jupiter is halted and reversed when it captured Saturn in an outer resonance. [34] The halting of Jupiter's and Saturn's migration and the capture of Uranus and Neptune in further resonances may have prevented the formation of a compact system of super-earths similar to many of those found by Kepler. [35] The outward migration of planets can also result in the capture of planetesimals in resonance with the outer planet for example the resonant trans-Neptunian objects in the Kuiper belt. [36]

Although planetary migration is expected to lead to systems with chains of resonant planets most exoplanets are not in resonances. The resonance chains can be disrupted by gravitational instabilities once the gas disk dissipates. [37] Interactions with leftover planetesimals can break resonances of low mass planets leaving them in orbits slightly outside the resonance. [38] Tidal interactions with the star, turbulence in the disk, and interactions with the wake of another planet could also disrupt resonances. [39] Resonance capture might be avoided for planets smaller than Neptune with eccentric orbits. [40]

The migration of the outer planets is a scenario proposed to explain some of the orbital properties of the bodies in the Solar System's outermost regions. [41] Beyond Neptune, the Solar System continues into the Kuiper belt, the scattered disc, and the Oort cloud, three sparse populations of small icy bodies thought to be the points of origin for most observed comets. At their distance from the Sun, accretion was too slow to allow planets to form before the solar nebula dispersed, because the initial disc lacked enough mass density to consolidate into a planet. The Kuiper belt lies between 30 and 55 AU from the Sun, while the farther scattered disc extends to over 100 AU, [41] and the distant Oort cloud begins at about 50,000 AU. [42]

According to this scenario the Kuiper belt was originally much denser and closer to the Sun: it contained millions of planetesimals, and had an outer edge at approximately 30 AU, the present distance of Neptune. After the formation of the Solar System, the orbits of all the giant planets continued to change slowly, influenced by their interaction with the large number of remaining planetesimals. After 500–600 million years (about 4 billion years ago) Jupiter and Saturn divergently crossed the 2:1 orbital resonance, in which Saturn orbited the Sun once for every two Jupiter orbits. [41] This resonance crossing increased the eccentricities of Jupiter and Saturn and destabilized the orbits of Uranus and Neptune. Encounters between the planets followed causing Neptune to surge past Uranus and plough into the dense planetesimal belt. The planets scattered the majority of the small icy bodies inwards, while moving outwards themselves. These planetesimals then scattered off the next planet they encountered in a similar manner, moving the planets' orbits outwards while they moved inwards. [43] This process continued until the planetesimals interacted with Jupiter, whose immense gravity sent them into highly elliptical orbits or even ejected them outright from the Solar System. This caused Jupiter to move slightly inward. This scattering scenario explains the trans-Neptunian populations' present low mass. In contrast to the outer planets, the inner planets are not believed to have migrated significantly over the age of the Solar System, because their orbits have remained stable following the period of giant impacts. [44]

## Title: Orbital resonances and planetary accretion in the early solar system evolution

The solar system, in its early evolution, is thought to have consisted of an accretion disk around a growing central protostar. The accretion disk from which the planets ultimately formed can play a significant role in the processes of planetary and solar formation. As well as leading, by thermalization of orbital motions in the disk, to bipolar flows in the T Tauri stage of stellar evolution, the disk can influence the course of planetary accumulation. By virtue of its essentially solar composition, Jupiter was formed before the accretion disk was removed. This first-formed planet then gravitationally imposed a harmonic strucnture on the planetesimal swarm through its commensurability resonances. Accelerated growth of planetesimals in orbital resonance with Jupiter resulted in runaway growth producing planetary embryos. These embryos accelerated growth at their own resonances in a process that propagated inward and outward forming a resonant configuration of embryos. When the accretion disk is eventually dispersed, the radial force law changes so that this resonant structure of preplanetary zones is transformed into the present non-resonant structure. During this process, the strong resonances of Jupiter swept through the asteroid zone. Motion of commensurability resonances leads to a new celestial mechanical effect where eccentricities aremore » permanently increased and semimajor axes are permanently decreased by significant amounts. The eccentricity excitation, producing collision velocities resulting in catastrophic fragmentation, can explain the lack of a planet in that region. The semimajor axis reduction can account for the clearing of the Kirkwood gaps. « less

## Chaotic orbits

The French astronomer Michel Hénon and the American astronomer Carl Heiles discovered that when a system exhibiting periodic motion, such as a pendulum, is perturbed by an external force that is also periodic, some initial conditions lead to motions where the state of the system becomes essentially unpredictable (within some range of system states) at some time in the future, whereas initial conditions within some other set produce quasiperiodic or predictable behaviour. The unpredictable behaviour is called chaotic, and initial conditions that produce it are said to lie in a chaotic zone. If the chaotic zone is bounded, in the sense that only limited ranges of initial values of the variables describing the motion lead to chaotic behaviour, the uncertainty in the state of the system in the future is limited by the extent of the chaotic zone that is, values of the variables in the distant future are completely uncertain only within those ranges of values within the chaotic zone. This complete uncertainty within the zone means the system will eventually come arbitrarily close to any set of values of the variables within the zone if given sufficient time. Chaotic orbits were first realized in the asteroid belt.

A periodic term in the expansion of the disturbing function for a typical asteroid orbit becomes more important in influencing the motion of the asteroid if the frequency with which it changes sign is very small and its coefficient is relatively large. For asteroids orbiting near a mean motion commensurability with Jupiter, there are generally several terms in the disturbing function with large coefficients and small frequencies that are close but not identical. These “resonant” terms often dominate the perturbations of the asteroid motion so much that all the higher-frequency terms can be neglected in determining a first approximation to the perturbed motion. This neglect is equivalent to averaging the higher-frequency terms to zero the low-frequency terms change only slightly during the averaging. If one of the frequencies vanishes on the average, the periodic term becomes nearly constant, or secular, and the asteroid is locked into an exact orbital resonance near the particular mean motion commensurability. The mean motions are not exactly commensurate in such a resonance, however, since the motion of the asteroid orbital node or perihelion is always involved (except for the 1:1 Trojan resonances).

For example, for the 3:1 commensurability, the angle θ = λA - 3λJ + ϖA is the argument of one of the important periodic terms whose variation can vanish (zero frequency). Here λ = Ω + ω + l is the mean longitude, the subscripts A and J refer to the asteroid and Jupiter, respectively, and ϖ = Ω + ω is the longitude of perihelion (see Figure 2 ). Within resonance, the angle θ librates, or oscillates, around a constant value as would a pendulum around its equilibrium position at the bottom of its swing. The larger the amplitude of the equivalent pendulum, the larger its velocity at the bottom of its swing. If the velocity of the pendulum at the bottom of its swing, or, equivalently, the maximum rate of change of the angle θ, is sufficiently high, the pendulum will swing over the top of its support and be in a state of rotation instead of libration. The maximum value of the rate of change of θ for which θ remains an angle of libration (periodically reversing its variation) instead of one of rotation (increasing or decreasing monotonically) is defined as the half-width of the resonance.

Another term with nearly zero frequency when the asteroid is near the 3:1 commensurability has the argument θ′ = λA - λJ + 2ϖJ. The substitution of the longitude of Jupiter’s perihelion for that of the asteroid means that the rates of change of θ and θ′ will be slightly different. As the resonances are not separated much in frequency, there may exist values of the mean motion of the asteroid where both θ and θ′ would be angles of libration if either resonance existed in the absence of the other. The resonances are said to overlap in this case, and the attempt by the system to librate simultaneously about both resonances for some initial conditions leads to chaotic orbital behaviour. The important characteristic of the chaotic zone for asteroid motion near a mean motion commensurability with Jupiter is that it includes a region where the asteroid’s orbital eccentricity is large. During the variation of the elements over the entire chaotic zone as time increases, large eccentricities must occasionally be reached. For asteroids near the 3:1 commensurability with Jupiter, the orbit then crosses that of Mars, whose gravitational interaction in a close encounter can remove the asteroid from the 3:1 zone.

By numerically integrating many orbits whose initial conditions spanned the 3:1 Kirkwood gap region in the asteroid belt, Jack Wisdom, an American dynamicist who developed a powerful means of analyzing chaotic motions, found that the chaotic zone around this gap precisely matched the physical extent of the gap. There are no observable asteroids with orbits within the chaotic zone, but there are many just outside extremes of the zone. Other Kirkwood gaps can be similarly accounted for. The realization that orbits governed by Newton’s laws of motion and gravitation could have chaotic properties and that such properties could solve a long-standing problem in the celestial mechanics of the solar system is a major breakthrough in the subject.

## This study

This study aims at reinvestigating the mean-motion resonances in the systems of Jupiter and Saturn in the light of a quantity, kcrit, which has been introduced in the context of exoplanetary systems by Goldreich & Schlichting (2014). This quantity is to be compared with a constant of the system, in the absence of dissipation, and the comparison will tell us whether an inner circulation zone appears or not. In that sense, this study gives an alternative formulation of the results given by the Second Fundamental Model of the Resonance. The conclusion is that the resonances should be classified into two groups. The first group contains Mimas-Tethys and Titan-Hyperion, which have large libration amplitudes, and for which the inner circulation zone exists (here presented as overstability). The other group contains the resonances with a small amplitude of libration, i.e. not only Enceladus-Dione, but also Io-Europa and Europa-Ganymede, seen as independent resonances.

## The Faintest Dwarf Galaxies

Joshua D. Simon
Vol. 57, 2019

### Abstract

The lowest luminosity ( L) Milky Way satellite galaxies represent the extreme lower limit of the galaxy luminosity function. These ultra-faint dwarfs are the oldest, most dark matter–dominated, most metal-poor, and least chemically evolved stellar systems . Read More

### Supplemental Materials

Figure 1: Census of Milky Way satellite galaxies as a function of time. The objects shown here include all spectroscopically confirmed dwarf galaxies as well as those suspected to be dwarfs based on l.

Figure 2: Distribution of Milky Way satellites in absolute magnitude () and half-light radius. Confirmed dwarf galaxies are displayed as dark blue filled circles, and objects suspected to be dwarf gal.

Figure 3: Line-of-sight velocity dispersions of ultra-faint Milky Way satellites as a function of absolute magnitude. Measurements and uncertainties are shown as blue points with error bars, and 90% c.

Figure 4: (a) Dynamical masses of ultra-faint Milky Way satellites as a function of luminosity. (b) Mass-to-light ratios within the half-light radius for ultra-faint Milky Way satellites as a function.

Figure 5: Mean stellar metallicities of Milky Way satellites as a function of absolute magnitude. Confirmed dwarf galaxies are displayed as dark blue filled circles, and objects suspected to be dwarf .

Figure 6: Metallicity distribution function of stars in ultra-faint dwarfs. References for the metallicities shown here are listed in Supplemental Table 1. We note that these data are quite heterogene.

Figure 7: Chemical abundance patterns of stars in UFDs. Shown here are (a) [C/Fe], (b) [Mg/Fe], and (c) [Ba/Fe] ratios as functions of metallicity, respectively. UFD stars are plotted as colored diamo.

Figure 8: Detectability of faint stellar systems as functions of distance, absolute magnitude, and survey depth. The red curve shows the brightness of the 20th brightest star in an object as a functi.

Figure 9: (a) Color–magnitude diagram of Segue 1 (photometry from Muñoz et al. 2018). The shaded blue and pink magnitude regions indicate the approximate depth that can be reached with existing medium.

## Planets orbital resonance

Planets and moons in our solar system have orbital resonance. Even planets and asteroids have forms of orbital resonance.

The most famous are Jupiter and its moons – Ganymede, Europa and Io. You also have Neptune and Pluto, Titan and Hyperion, Dione and Enceladus.

### Stars and binary stars orbital resonance

Stars and especially binary stars have an orbital resonance, they are in harmony or a form of frequency with each other.

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