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date: 25 November 2020

The Orbital Architecture of Exoplanetary Systemsfree

  • John C. B. PapaloizouJohn C. B. PapaloizouDepartment of Applied Mathematics and Theoretical Physics, University of Cambridge

Summary

The great diversity of extrasolar planetary systems has challenged our understanding of how planets form. During the formation process their orbits are modified while the protoplanetary disk is present. After its dispersal orbits may also be modified as a result of mutual gravitational interactions leading to their currently observed configurations in the longer term. A number of potentially significant phenomena have been identified. These include radial migration of solids in the protoplanetary disk, radial migration of protoplanetary cores produced by disk-planet interaction and how it can be halted by protoplanet traps, formation of resonant systems and subsystems, and gravitational interactions among planets or between a planet and an external stellar companion. These interactions may cause excitation of orbital inclinations and eccentricities which in the latter case may attain values close to unity. When the eccentricity approaches unity, tidal interaction with the central star could lead to orbital circularization and a close orbiting Hot Jupiter, providing a competitive process to direct migration through the disk or in-situ formation. Long-term dynamical instability may also account for the relatively small number of observed compact systems of super-Earths and Neptune class planets that have attained and subsequently maintained linked commensurabilities in the long term.

Introduction

The basic data for exoplanets and exoplanetary systems and relevant aspects of protoplanetary disks, the flattened rotating gaseous structures out of which the planets form are first reviewed. The article then moves on to consider the processes that cause solid material and protoplanetary cores to migrate radially through the protoplanetary disk and thus contribute to determining the original form that systems of protoplanets may take. Dynamical processes that can affect planetary systems after the protoplanetary disk has dispersed are also considered. These serve to excite orbital inclinations and eccentricities and reconfigure systems. The dynamical mechanisms proposed for explaining the close-in giant planets known as Hot Jupiters are reviewed. Multi-planet systems are considered reviewing the formation of resonant chains in general terms. Close-in systems of super-Earths some of which contain linked resonant chains are finally discussed, together with how their future evolution and long-term stability considerations are potentially able to reconcile formation theories with the observations.

Figure 1. The dependence of the planet mass, mp, on orbital period is shown in the left panel for all confirmed exoplanets. The right panel shows the radius mass relation. Note that radial velocity measurements determine mpsini, where i is the inclination of the orbit to the plane perpendicular to the line of sight, and this is used rather than mp. If i is distributed at random the mean value of sini~0.8.

Basic Data

The Relationship Between Planet Mass and Orbital Period

The relationship between planet mass orbital period is illustrated in the upper panel of Figure 1 for all exoplanets. Several features are apparent. The “Hot Jupiters” with masses clustered around 0.5 Jupiter masses (Mj) and orbital periods <10d. are represented. In the top left-hand corner of the plot, a notable clump in the period range 35d. and the mass range 100–300 earth masses (M) can be seen. Formation mechanisms for these are discussed in the “Origin of Hot Jupiters” section. The close-in Earths and super-Earths with masses <10M and periods between ~1d. and 100d. (Borucki et al., 2011) are represented in the bottom left. Neptune class planets with masses in the range 10100M are located in the central area of the plot. In the top right corner are giant planets with periods exceeding ~100d. There tends to be a gap between these and the more closely orbiting “Hot Jupiters.” Systems of these classes of planets are discussed in the “Multi-Planet Systems” section. It should be noted that detection of exoplanets becomes more difficult and eventually impossible as the orbital period increases and/or the mass is reduced. Hence the absence of planets in the lower right-hand region of the plot.

Mass–Radius Relation

The lower panel of Figure 1 shows the radius–mass relation for the relatively small number of exoplanets for which both these are known. There is no unique relation between mass and radius for Earth, super-Earth, and Neptune like planets, as they span a wide range of chemical compositions (e.g., Baraffe, Chabrier, Fortney, & Sotin, 2014). However, some probabilistic mass–radius, (mpRp), relations have been proposed. The mean power law relations proposed by Chen and Kipping (2017) are

mp/M=0.952(Rp/R)3.584forRp1.23Randmp/M=1.407(Rp/R)1.698forRp1.23R,(1)

The first of these applies to the Earths and super-Earths with mp<2M. However, there is a ±70% variation in mass for a given radius within a 95% confidence level.

The second, with 2M<mp<~100M, applies to more massive planets including Neptune class planets. In this case there is a greater range of masses for a given radius, being up to a factor of three within a 95% confidence level once the mean mass exceeds ~10M.

Finally, more general mean relations that do not assume power law fits have been proposed by Ning, Wolfgang, and Ghosh (2018). This should enable additional features in the mass–radius relation to be determined when more observations become available in the near future.

The planets with mp<2M have mean radii that makes their mean densities consistent with a rocky composition and may be considered to be bare rocky cores. Planets with mp>2M for which the second mean mass–radius relation applies are such that their expected mean density decreases as their mass increases. This implies the existence of a low density H/He gaseous envelope.

It has been argued that the bare rocky cores, which are very close to the central star (located in the lower left-hand region of the upper panel of Figure 1), have had their gaseous envelopes removed through stellar irradiation causing photoevaporation. Photoevaporation may also account for the lack of planets in the mass range 10100M with periods <3d. apparent in the upper panel of Figure 1 on account of their having lost their envelopes.

Bare cores with a predominantly icy composition have been ruled out by the absence of objects of radius ~R with masses in the range 12M (see, for example, Jin & Mordasini, 2018; Lopez & Fortney, 2014). These would be expected to exist if planets containing such cores had migrated inward from beyond the ice line and lost their gaseous envelopes through photoevaporation.

Thus the data seem to support the view that most super-Earths and Neptune class planets with periods <~100d. were formed from silicates and iron together with a gaseous envelope interior to the ice line, with only a few with icy cores having possibly originated from beyond. Importantly this in turn indicates that the amount of radial migration of these planets induced through interacting with the protoplanetary disk, the flattened differentially rotating gaseous structure out of which they formed, could have been relatively small. However, the solids that make them up could have originated from larger radii in pebble form (e.g., Hansen & Murray, 2012).

The Relationship Between Orbital Eccentricity and Orbital Period

The relation between the orbital eccentricity, e, and orbital period for all exoplanets is shown in the lower panel of Figure 2. The same relation restricted to exoplanets with orbital periods less than ten days and masses exceeding 0.5MJ (Hot Jupiters) is shown in the upper panel. Apart from Hot Jupiters with periods <~3d., values of e up to 0.3 are common with values of, e, up to 0.9 occurring for periods exceeding ~100d. These values, which greatly exceed the putative protoplanetary disk aspect ratio, or ratio of semi-thickness to distance to the central star, h~0.05, are not expected to be produced as a result of interaction with the gaseous disk except possibly for masses exceeding around 3MJ (e.g., Kley & Dirksen, 2006; Papaloizou, Nelson, & Masset, 2001) and are thus suggestive of dynamical or secular interactions amongst planets or between planets and other distant objects as having caused their excitation.

The environment and processes that can influence the formation and/or change the orbital location of a protoplanet that operate during the stage when the gaseous protoplanetary disk is present are now documented. Then the dynamical processes involving gravitational interactions between protoplanets which could operate after the disk has dispersed will be discussed.

Figure 2. The relation between the orbital eccentricity and orbital period is shown in the right panel for all exoplanets. The same relation is shown in the left panel, although it restricted to exoplanets with orbital periods less than ten days and masses exceeding 0.5MJ (Hot Jupiters).

The Protoplanetary Disk

During the process of star formation, the angular momentum content of the collapsing cloud results in the formation of a flattened differentially rotating disk. During the early class I phase lasting around 105y the disk is massive with both accretion and strong outflows in the form of jets. This settles down to a longer-lived class II phase lasting ~107y where planetary accumulation is believed to occur (see, for example, Williams & Cieza, 2011). Figure 3 provides a sketch of the expected structure of a protoplanetary disk around one solar mass during the class II stage. The snow or ice line is at around 1.5–3 astronomical inits (au) (e.g., Lecar, Podolak, Sasselov, & Chiang, 2006) At this radius and beyond is an inert region or dead zone localized around the mid-plane (e.g., Terquem, 2008), shaded in blue, containing ice covered grains which can become the building bocks of planetesimals. The upper regions of the disk are penetrated by photons and cosmic rays which cause them to become ionized and support turbulent transport. The latter occurs throughout the disk at smaller radii <~0.2au. Turbulence is associated with kinetic energy dissipation that has to be provided from orbital energy. That in turn has to be provided by material in the inner disk regions moving closer to the central star while outer material moves outward, conserving angular momentum. In this way accretion on to the central star occurs. Accretion may also be driven by a disk wind (e.g., Bai, Goodman, & Yuan, 2016).

In the class I phase there is the possibility that giant planets form at larger disk radii directly through gravitational instability. During the later more long-lived class II phase grains accumulate to form pebbles, boulders, and planetesimals leading to the production of protoplanetary cores.

A rough and ready base model for a protoplanetary disk is the minimum mass solar nebula. This takes the mass within the planets of the solar system and smears it out to fill the spaces between them after making appropriate enhancements to restore solar composition. The surface density Σ at radius r is then r3/2, with a scaling that gives a mass of 2MJ within 5au. The disk temperature can be estimated as the temperature of a perfectly absorbing dust particle produced by radiation from the central star (here the sun) as

T=T*(R*2r)1/2,(2)

where T* is the solar effective temperature. The characteristic aspect ratio h~0.05.

Mobility and Accretion of Solid Particles and Radial Migration of Protoplanets

It is important to note that solids in the form of pebbles or boulders that have formed through adhesive collisions of interstellar grains migrate through the disk as a result of gas drag. The migration is in the direction of increasing pressure. It occurs because the pressure gradient causes the gas to rotate at a slightly different rate to solid objects that are unaffected by it. For meter size object in a minimum mass solar nebula with h~0.05, this can be on a time scale as short as =102y at 1au (see, for example, Papaloizou & Terquem, 2006). But note that such migration stalls near pressure maxima as does type-I migration, making such locations of interest for planet formation. Pebbles or boulders may accrete onto existing protoplanetary cores in a process known as pebble accretion or alternatively could accumulate to form planetesimals of size ~10km. The latter then accumulate further through collisions, eventually producing protoplanetary cores of several earth masses that can start to accrete gas and in turn form giant planets if enough is present. Details of these phenomena shall not be considered here but rather they are assumed to operate effectively. More detailed discussion may be found in Papaloizou and Terquem (2006), Johansen and Lambrechts (2017), and Paardekooper and Johansen (2018).

Type-I Migration

This affects protoplanets with masses up to those characteristic of super-Earths embedded in a protoplanetary disk that make a relatively small perturbation on it. An example of a five Earth mass planet embedded in a non-turbulent protoplanetary disk with a structure similar to that of the minimum mass solar nebula is illustrated in Figure 4. Wakes directed downstream interior and exterior to the protoplanet that are excited by its gravitational interaction with the disk are visible. A dominant torque is produced by the gravitational attraction of the more pronounced outer one. This acts to drag the protoplanet backward, thus causing it to slow down and migrate inwards. For the model disk considered, the time scale can be ~105y at 5 au, well within the typical disk life time. In addition, in this regime interaction with the disk causes the orbit to circularize on a time scale shorter than that to migrate by a factor h2~1/400. Thus, this time scale is expected to be ~250y and circularization to be very effective (see, for example, Baruteau et al., 2014, for more discussion).

The Effect of Coorbital Material

For disks with surface density profiles as that illustrated in Figure 4, the effect of coorbital material turns out not to be significant. However, this is not the case for other profiles. In particular, when there is a rapid decline of the surface density going inward, as occurs at the inner edge of the dead zone illustrated in Figure 3 or at the boundary of an inner cavity near the central star. At such location coorbital material may produce a torque that acts to reverse the inward migration produced by the outer wake. This can be thought of as being produced by material passing through the coorbital region, transferring angular momentum to the protoplanet as it goes, the dynamics of the maintenance of the profile does not allow for this material to return. Such locations where type-I migration may be stalled are known as protoplanet traps (e.g., Masset et al., 2006). Clearly these may play an important role in planet formation. Coorbital torques can also play an important role in restricting radial migration depending on the thermal properties of the disk (e.g., Paardekooper & Mellema, 2006) and local heating arising from protoplanetary accretion (Ben’tez-Llambay, Masset, Koenigsberger, & Szulágyi,2015).

Figure 3. Putative protoplanetary disk model with ice grains existing at larger radii, in practice exceeding the snow or ice line located at 2–3 astronomical units (au). These tend to accumulate in the mid-plane region known as a dead zone. The upper regions of the disk remain active on account of exposure to stellar radiation and cosmic rays.

Type-II Migration

Type-II migration applies to protoplanets that are big enough to make a gap in the disk. Typically their mass should exceed that of Saturn (for further discussion, see Papaloizou & Terquem, 2006). Nonlinear effects must operate, and these tend to adjust the surface density profile in a way that results in the wake-driven migration being suppressed. In this case radial migration ends up being on the same time scale as the general local evolution of the disk, i.e. the time for dissipative processes to cause the interior disk mass to be removed. This is typically 2×105y in standard models incorporating a minimum mass solar nebula with h=0.05 at 5au (e.g., Nelson, Papaloizou, Masset, & Kley,2000). However, it increases with orbital period and if processes act to restrict turbulent transport or disperse the disk during the late stages of its life, such as photo-evaporation driven by the stellar UV flux (e.g., Alexander & Pascucci, 2012). A Jupiter mass orbiting in a prominent gap in a turbulent disk is illustrated in Figure 4.

Figure 4. (Left-hand panel) A five Earth mass planet orbits in the type-I migration regime that is simultaneously embedded in a non-turbulent protoplanetary disk with a surface density profile similar to that of the minimum mass solar nebula with aspect ratio h=0.05. The disc planet interaction produces prominent wakes interior and exterior to the planet. The outer one is the stronger so that the net torque exerted on the protoplanet drags it backward, causing it to migrate inward. (Right-hand panel) A five Jupiter mass orbiting in a prominent gap in a turbulent disk. Note that in these plots the direction of orbital rotation is anticlockwise.

Dynamical Interactions in Planetary Systems

After the protoplanetary disk has dispersed, protoplanetary orbits can evolve as a result of the gravitational forces acting between the protoplanets. In a rough and ready manner, the evolution of many planet systems resulting from their mutual gravitational interactions can be characterized as either requiring consideration of short orbital time scales, possibly for potentially long time periods, or secular evolution associated with much longer time scales. In the latter case, behavior on the orbital time scale is averaged out.

Dynamical Relaxation

Phenomena such as close encounters of pairs of planets leading to scattering involve significant changes on orbital time scales. Interactions associated with orbital resonances can also lead to particularly strong interactions that can have important consequences for significant repositioning of the planets on relatively short time scales.

Figure 5. Dependence of the obliquity, or the angle between the orbital plane and the equatorial plane of the central rotating star, on the effective temperature of the star.

The possibility of close encounters induced by their mutual gravitational attraction in planetary systems with members initially in circular orbits as may be expected for them after protoplanetary disk dispersal is considered. An important parameter is the separation, Δ, between a pair of planets measured in Hill radii. For a pair of planets of masses, m1 and m2, orbiting at radii, a1 and a2, the Hill radius defined as

RH=((m1+m2)3M*)1/3(a1+a2)2.(3)

Here M* is the central stellar mass. Thus, for a central solar mass and two Jovian mass planets, RH=0.09[(a1+a2)/2]. For such a system of two planets, Gladman (1993) found that in order to avoid instability due to close encounters leading to a scattering, Δ should exceed 23. For the case, this requires that the initial separation exceed 0.3[(a1+a2)/2]. For additional aspects on the problem of stability for two planets orbiting a central star, see Barnes and Greenberg (2006), Deck, Payne, and Holman (2013), and Ford and Rasio (2008).

If an ensemble of Jovian mass planets is set up with such a characteristic separation between neighboring planets, strong and chaotic interactions are expected as it undergoes dynamical relaxation. Notably, orbits with high eccentricity may be produced.

Figure 6. (Right panel) histogram of the period ratio distribution obtained for all neighboring pairs within confirmed planetary systems–their period ratio was defined so as to make it exceed unity. Period ratios less than three are indicated. The blue vertical lines indicate the locations of the 5:4, 4:3, 3;2 and 2:1 resonances. The bins between adjacent first order resonances are seen to be of equal width in period ratio, but with these widths increasing by a factor of 5 as the period ratio increases from 1 to 2 Beyond the latter value the width is proportional to the period ratio. This enables resolution of the structure near resonances with some smoothing at larger period ratios. (Left panel) Same distribution but with the restriction that each planet making up a pair has a mass exceeding 0.5Mj. In this case period ratios up to 3.5 are considered.

Linked Chains of N Lower Mass Protoplanets

Extending the discussion to the super-Earth mass regime, a system of N protoplanets with mass ratio, mp/M*=105, in circular orbits with uniform separation Δ measured in Hill radii is considered. Chambers, Wetherill, and Boss (1996) find that such systems, composed of three protoplanets, become unstable leading to close encounters, which may lead to physical collisions or expulsion of protoplanets, on a time scale, measured in orbital periods at the innermost mass, that increases with Δ according to

log(t)~16.5log(Δ)-6.(4)

For a system with each consecutive pair close to a 4:3 commensurability, and innermost period of four days, this gives a time ~4×109y which is comparable to the age of the solar system. Of course this time scale decreases with increasing protoplanet mass and increasing numbers of them (see Chambers et al., 1996)

Secular Evolution

As the separations in a multi-planet system increase the dynamical evolution slows down and short time scale phenomena can be averaged out. A protoplanet then acts as if its mass, mi, is averaged around its orbit. One then has a system of interacting orbits rather than individual protoplanets. Considering first the case when the orbits are coplanar, it is clear that no evolution occurs when the orbits are exactly circular as the system is axisymmetric with forces remaining central. In order for evolution to occur orbits must be eccentric so that variations in orientation allow them to exert torques on each other as they precess. These torques change the orbital angular momentum but not the orbital energy. Accordingly individual semi-major axes, ai, are conserved while associated eccentricities, ei, vary. The total scaled angular momentum, G is conserved with

G=i=1Nmiai(1ei2).(5)

Given the conservation of ai, a conserved quantity known as the angular momentum deficit may be constructed

C=i=1Nmiai(1(1ei2)).(6)

This is an important quantity that limits the amount of variation that can occur for the system. In general the system acts like a set of coupled oscillators. When the masses are small or the system is well separated and orbital eccentricities are small they collectively participate in normal modes of oscillation that are of small amplitude and linear. When the masses increase and/or separations reduce, interactions strengthen, becoming nonlinear and chaotic, leading to significant changes if allowed by the angular momentum deficit.

The discussion can be extended to allow the system to be non-coplanar. Then additional degrees of freedom arise through having non-zero orbital inclinations with respect to some reference plane. In this more general case, the conserved angular momentum deficit becomes

C=i=1Nmiai(1(1ei2)cosIi),(7)

where Ii is the inclination of the orbit of planet i.

For the solar system, most of the angular momentum deficit is contributed by Jupiter, here labeled as planet 1. In order for the eccentricity of Mercury, subscripted as 2, to be increased to a value close to unity while that of Jupiter, subscripted as 1, remains non-zero, with nothing else changing and orbital inclinations being assumed to remain close to zero, conservation of the angular momentum deficit requires that C~m1a1(1(1e12>m2a2(1e22) w1with current values. Thus nonlinear interactions are not prevented from making large changes of this type by the constancy of the angular momentum deficit. Such changes are found to have a non-negligible probability of occurring over long time scales (Laskar & Gastineau, 2009).

This discussion considered interactions in strictly or almost coplanar systems; however, dynamical and secular interactions may also be responsible for generating high orbital eccentricities in systems for which initial orbital planes start significantly misaligned. The simplest example of this, in the secular case, is the Kozai-Lidov effect which occurs when an external binary has a circular orbit that is significantly misaligned with the initial supposed circular orbit of the protoplanet. In that case, the protoplanet orbit will oscillate between its initial orbit and an orbit that is aligned with that of the binary and may have very high eccentricity (see, for example, Lin, Papaloizou, Terquem, Bryden, & Ida,2000). The latter is determined by using the conservation of its angular momentum component parallel to that of the binary. From this it follows that in order to generate very high eccentricities the binary orbital plane has to be almost perpendicular to the initial orbital plane of the protoplanet, which may be unlikely in practice.

The Origin of Hot Jupiters: The Role of Disk Migration, Dynamical and Secular Interactions and In-Situ Formation

The Hot Jupiters have masses >0.3MJ and orbital periods <10d. The first exoplanet to be discovered, 51 Pegasi, with an orbital period of 4.5d was in this category (Mayor & Queloz, 1995). Subsequently it has been found that about one in ten giant planet systems contain a Hot Jupiter (Mayor et al., 2011). The question immediately arises as to how they could be so close to their central stars given that conditions for the formation of their cores were thought to be optimal beyond the ice line at 23au. Two processes have been proposed for producing orbital migration from beyond the ice line to present close-in orbits and while it is likely that both have operated at some level, there is no consensus as to which is the dominant process. However, the data plotted in Figure 1 indicate a separation between giant planets with orbital periods >~100d that may have formed beyond an ice line with some later modest migration and the Hot Jupiters that underwent a more extreme form.

An alternative to radial migration is to consider that these planets were formed in situ. However, in that case the cores would have to be formed from silicates and/or iron without the help of ices, and it is unclear how the constituents can be brought to and accreted at small disk radii. Protoplanetary cores of adequate mass are not expected to be produced, unless the local surface density is enhanced by a factor of 100 as compared to standard models. Accretion of pebbles brought in from the outer disk may help, but this is hindered by gap formation for core masses that are too small (see Dawson & Johnson, 2018).

Type-II migration of the protoplanet driven by the disk planet interaction (or a more extreme type-III form proposed by Masset & Papaloizou, 2003) is a possibility. However, a stopping process needs to be invoked such as an inner edge to the disk produced by a stellar magnetic field. A challenging aspect for the disk migration proposal is that the inclination of the orbital plane can be significantly misaligned with the equatorial plane of the star. This feature shows up through the Rossiter–Mclaughlin effect which refers to the changing form of spectral lines as the planet transits across the star. A plot of the distribution of the obliquity as a function of the effective temperature of the central star is given in Figure 5. It will be seen that about one-third of the systems show significant obliquity exceeding ~23° in magnitude. This would not be expected for a planet formed within and migrating through the protoplanetary disk. For a time it was thought that this ruled out that picture in favor of one involving strong dynamical interactions, scattering or the Kozai–Lidov effect (e.g., Winn, Fabrycky, Albrecht, & Johnson,2010, and the “Secular Evolution” section). However, it was subsequently realized that the disk itself may be torqued out of alignment with the stellar equatorial plane by external objects (e.g., Batygin, 2012; Xiang-Gruess & Papaloizou, 2014) or that it could be sourced with material whose angular momentum direction varied with time (Bate, Lodato, & Pringle, 2010). These processes may account for a primordial origin of misalignment.

Interactions in multi-planet systems may readily produce orbits with high eccentricity that were initially nearly circular. The idea is that an orbit starting as circular, and beyond the ice line, may attain a high eccentricity, close to unity, such that a close approach to the central star takes place at perihelion. Tidal interaction with the central star then takes place, which circularizes the orbit at fixed angular momentum. If the pericenter distance of the highly eccentric, e~1, orbit is Rp, conservation of angular momentum requires the final semi-major axis, ap, to be 2Rp (this is because the pericenter velocity is a factor of 2 larger than the local circular value). This completed process has been termed “high eccentricity migration.”

For Hot Jupiters, the observations require Rp~0.02au. A significant quantity is the Hill radius, defined here for a single protoplanet of mass, mp, as

RH=23(mp3M*)Rp.(8)

This should exceed the protoplanet radius in order to avoid tidal disruption. For a Jupiter mass protoplanet and a central solar mass, it is found that this implies that Rp should exceed 106km, with the corresponding, ap, giving a minimum orbital period of, 1d. This is marginally violated by a few objects though there may be a possibility of accounting for this through additional tidal evolution occurring after the initial circularization. Note that the data plotted in the upper panel of Figure 2 indicate significantly smaller eccentricities for orbital periods <~3d, with most values being consistent with zero, given the error limits, thus being consistent with the operation of orbital circularization.

Since multi-planet systems provide an environment where strong interactions occurring after disk dispersal can lead to the development of orbits with high eccentricity, Papaloizou and Terquem (2001) and Adams and Laughlin (2003) have shown that dynamical relaxation in crowded systems of five to ten giant planets occupying the radial range 530au produces such orbits. Less crowded systems can operate through secular dynamics, but note that they need some eccentricity and/or inclination excitation in order to attain non-zero angular momentum deficit in order for the process to work as well as explain the observed eccentricities as indicated in Figure 2. This may require dynamical relaxation though more limited than described. As an example of a system producing an orbit of high eccentricity through secular interactions. Wu and Lithwick (2011) consider three giant planets of masses 0.5,1 and 1.5 Jupiter masses at 1,5, and 16 astronomical units with modest eccentricities and inclinations (though somewhat larger than would be expected to be produced in a protoplanetary disk, and thus requiring a prior dynamical relaxation). They find that a Hot Jupiter is produced on a 3×108y time scale.

In principle the Kozai–Lidov mechanism could attain a similar result through the action of a binary companion in a highly inclined orbit. However, there is not enough companions of the required type for this to be viable except in a small number of cases (see Dawson & Johnson, 2018, for further discussion).

The high eccentricity migration mechanisms cannot account for all Hot Jupiters for a number of reasons. The problem of providing the initial conditions required to produce them at the necessary rate has been alluded to. Indeed a study of six Hot Jupiters by Becker, Vanderburg, Adams, Khain, & Bryan (2017) found that orbits of outer companions are most likely well aligned and so have insufficient mutual inclination to drive high eccentricity migration. In addition, on account of the large eccentricities produced, the system needs to be cleared of other objects to avoid the process being disrupted (e.g., Wu & Murray, 2003). Most Hot Jupiters are found not to have companions in support of this. However, there are exceptions such as WASP-47, and recently Hot Jupiters have been discovered orbiting TTauri stars (Donati et al., 2017; Yu et al., 2017), which have formed within a 1067y. time scale and most likely while the gas disk was present.

Table 1. Properties of the HD 155358, 24 Sextantis, HD 6805 and HD 60532 Systems

Central Mass

Inner Planet Mass

Outer Planet Mass

Inner Semi-Major Axis

Outer Semi-Major Axis

M*

m1

m2

a1

a2

HD 155358

0.92

0.85

0.82

0.64

1.02

Sextantis

1.54

1.99

0.86

1.33

2.08

HD 6805

1.7

2.5

3.3

1.27

1.93

HD 60532

1.44

3.15

7.46

0.77

1.58

Note. These contain giant planets in or near a commensurability The first three are possibly in 2:1 resonance, while the fourth is in a 3:1 resonance. The first column identifies the system, and the second column gives the mass of the central star in solar masses. The third and fourth columns give the masses of the planets in Jupiter masses and the fifth and sixth columns give their semi-major axes in au.System.

Multi-Planet Systems

Systems of planets, their architecture and interactions, and how these relate to their origin are now considered, by first discussing the distribution of period ratios between neighboring members of a system and the occurrence of near commensurabilities.

The Period Ratio Distribution in Exoplanetary Systems

The period ratio distribution for adjacent pairs of planets in exoplanetary systems is illustrated in Figure 6. The upper panel is for all systems and considers period ratios <3. The lower panel restricts consideration to pairs with masses exceeding 0.5MJ where they are known and period ratios <3.5.

Both panels indicate an excess of pairs of planets at locations that are close to first-order commensurabilities but with period ratios that predominantly exceed the exactly commensurable value. In the upper panel 411 systems are listed with 75 pairs of neighboring planets with period ratios differing from exact resonance by <1.5% for the 5:4, 4:3, and 3:2 resonances and <2% for the 2:1 resonance. Most of these are closest to the 3:2 resonance. In the lower panel where planets with masses exceeding 0.5MJ are considered 42 systems are listed with 15 pairs of neighboring planets similarly close to resonance. But in this case most of these are closest to the 2:1 resonance.

Table 2. Details of the Planetary Systems GJ876 and HR 8799 which Contain Giant Planets that Participate in a Three-Body Laplace Resonance

GJ876

Mass

Semi-Major Axis Au

Orbital Period Days

Eccentricity

d

6.83 M

0.02080665

1.937780

0.207

c

0.7142 MJ

0.129590

30.0081

0.25591

b

2.2756MJ

0.208317

61.1166

0.0324

e

14.6 M

0.3343

124.26

0.055

HR 8799

b

5 MJ

68

1.67×105

-

c

7 MJ

42.8

83255

-

d

7 MJ

27

41628

-

e

7 MJ

17

20816

-

Note. For GJ876, it is the outer three planets that participate in the commensurability, while for HR8799 it is the inner three. The first column identifies the system and component, the second column gives the masses of the planets in Jupiter masses with the fifth and sixth columns respectively giving their semi-major axes in au and their eccentricities when known.

Convergent Migration and the Formation of Resonant Systems

A protoplanet’s radial migration rate depends on its mass and the local disk properties This dependence may cause a pair to converge or diverge. In the former case convergence continues until a resonance (or period commensurability) occurs. At that point, provided that convergence is slow enough, the planets couple resonantly and migrate inward while retaining their relative orbital configuration. Additional protoplanets may join later through a further convergent migration episode in a similar manner. In this way a resonant chain may be produced.

Assuming the system then evolves, maintaining its resonant structure, it is inferred that period ratios and hence semi-major axis ratios remain fixed. If the eccentricities are small enough to be neglected, the total angular momentum of the system of N planets, i1,2,N, is

J=m1GM*a1i=1Nmim1aia1(9)

and the total orbital energy is

E=m1GM*2a1i=1Nmiaim1a1,(10)

where G is the gravitational constant. Now as the system maintains the same relative orbital configuration as it evolves, the quantities occurring after the summation signs do not change. Accordingly the result is obtained:

dEdJ=dE/dtdJ/dt=n1i=1Nmia1m1aii=1Nmim1aia1, (11)

where ni=GM*/aj3 is the mean motion of planet i.

Table 3. Planetary Systems with >3 Planets and Examples of Pairs of Close-In Planets That Are Close to a First-Order Commensurability

Number of Planets

Innermost Period (Days)

Successive Close Commensurabilities

Kepler-60

3

7.132

5:4, 4:3

Kepler-80

6 (Outer 5)

3.0722

3:2, 3:2, 4:3, 3:2

Kepler-223

4

7.3845

4:3,3:2,4:3

TRAPPIST-1

7

1.5109

3:2, 3:2, 3:2, 3:2, 4:3, 3:2

K2-138

6 (Inner 5)

2.353

3:2,3:2,3:2.3:2

Kepler-27

2

15.33

2:1

Kepler-59

2

11.87

3:2

Kepler-181

2

7.3845

4:3

Kepler-11

6 (Innermost pair)

10.30

5:4

Kepler-50

2

7.81

6:5

Note. The first five rows indicate planetary systems with >3 planets that either form a resonant chain in entirety or contain such a subsystem as indicated in brackets in the second column. In the TRAPPIST-1 system, the first two listed period ratios have significantly increased above 1.5, but they still satisfy the condition for linkage by a Laplace resonance. Rows six to ten list examples of pairs of close-in planets that are close to a first-order commensurability. Apart from the ninth entry, the systems contain only two planets. The innermost period listed in the third column is for the resonant subsystem considered.

As planet 1 is the outermost, ai/a1<1, for i1 equation (11) implies that dE/dJ>n1, or equivalently dJ/dE<1/n. However, if the system is driven by an external torque that removes angular momentum from planet 1, while maintaining its orbit circular, it would produce dJ/dE=1/n1. This means that the external torque removes too much angular momentum to allow the required orbital energy to be removed from the system while retaining circular orbits as it contracts. The situation can only be rectified if inner orbits start to become eccentric, allowing them to contain less angular momentum for a given semi-major axis, hence making it possible for the resonant configuration to be maintained.

Accordingly if the external torque causes the system to contract on a time scale TM, a characteristic eccentricity is typically expected to start increasing according to

de2dt=1/Tm.(12)

However, this can be balanced by orbital circularization, which is expected to occur through interaction with the disk, leading to

de2dt=1/Tm2e2/TC,(13)

where TC is the circularization time scale. Thus an equilibrium is possible such that e2~Tc/(2Tm). As Tc/Tm~h2 was expected, the implied eccentricities are small and of order the disk aspect ratio, h.

Note that as the period ratio approaches the commensurable value, convergent migration tends to be halted with this exceeding that in agreement with the tendency displayed in Figure 6. However, this situation could change under further evolution after the disk disperses.

Pairs of Giant Planets Near Commensurabilities

How disk driven migration is expected to lead to commensurabilities has been discussed above. Their occurrence or lack thereof may thus give information about the nature of disk migration. Although not all neighboring pairs of planets in multi-planet systems are close to commensurability, incidence reaches at around the 8%level, which doubles if only neighboring pairs close enough to be within the 3:1 commensurability are considered. The data set used respectively consisted of 431, 143, 51,18, 5, 4, 1, and 2 systems comprised of 2, 3, 4, 5, 6, 7, 8, and 9 planets.

In Table 1 the properties of four pairs of giant planets with semi-major axes in the au range are tabulated. Three are near to 2:1 resonance and the third to 3:1 resonance. The formation of these though convergent disk migration in the manner described in the “Convergent Migration and the Formation of Resonant Systems” section was studied by André and Papaloizou (2016) who employed 3D numerical hydrodynamic simulations. They found that their configurations could be understood if the planets formed at sufficient distances and at a late stage in the disk lifetime so that they achieved present locations at the time of disk dispersal.

The fact that only a relatively small number of pairs, in any mass range, is currently close to commensurability indicates either that migration was restricted, or that such pairs became either unstable, by for example processes listed in the “Linked Chains of N Lower Mass Protoplanets” section, or removed from resonance by other means. It seems feasible that all of these processes could contribute. In the case of the super-Earths the rocky plus H/He composition is consistent with formation interior to the ice line with potential limited migration (see also Terquem & Papaloizou, 2019).

Three-Body Laplace Resonances and Linked Resonant Chains

How resonant chains could be formed through convergent migration was discussed. In such a chain, neighboring planets are close to a commensurability. A special situation occurs when three-body resonances are set up. Suppose planets 1 and 2 are near a p:p+1 commensurability and planets 2 and 3 are near a q:q+1 commensurability. A Laplace resonance occurs when

(p+1)n1pn2=(q+1)n2qn3or(p+1)n1(p+q+1)n2+qn3=0,(14)

with this relation being found to be satisfied to much higher precision than the near two-body resonances (p+1)n1~pn2 and (q+1)n1~qn3. Note that, in systems with more than three planets, it is possible that all successive triples are linked in this way. Thus another interior planet 4 could be added, which together with planets 2 and 3 satisfies the Laplace relation

(q+1)n2qn3=(l+1)n3ln4or(q+1)n2(l+q+1)n3+ln4=0,(15)

with the approximate two-body commensurability (l+1)n3~ln4. Such configurations tend to be attained after a chain has formed and migrated for some time as a unit or if the migration stops but dissipative processes causing orbital circularization continue (e.g., Papaloizou, 2015). An aspect of conditions (14) and (15) is that they are invariant under rotation, which means that any angular rotation rate, ω, may be added to the individual mean motions without changing these relations. In addition a special rotation rate can be added that transforms the near two-body commensurabilities to become exact. This in turn means that in a particular rotating frame the linked planetary configuration looks periodic indicating that this particular solution can provide a long-term stable attractor.

Although systems with three-body resonances are relatively few, they can occur over a range of radial scales and planet masses. Table 2 gives the parameters for the systems GJ 876 and HR 8799 each of which contain three planets linked by a Laplace resonance with associated 2:1 commensurabilities. In the case of GJ 876 the orbital periods are 30d., 61d. and 124d., and for HR 8799 they are 83255d., 41628d., and 20816d. In the latter case the protoplanetary disk would have to have been about four times more massive than the minimum mass solar nebula over the radial range of the planets.

If it is argued that the planets in this type of system formed in situ, dissipative processes arising from the protoplanetary disk, in particular orbital circularization would be required to attain the attractor. In these circumstances orbital migration is generically expected, and while this may be limited, it is hard to this could be completely suppressed in all cases.

Systems of Close-in Super-Earths

Table 3 lists properties of close-in planetary systems with linked chains satisfying the three-body Laplace relation. In addition some systems containing just one commensurable pair are also listed. As for the cases with more massive planets the existence of such systems indicates the action of dissipative processes such as orbital migration and circularization. However, the extent of any migration is not well constrained. The data indicates that extensive migration from beyond the ice line is unlikely. However, if the late stages of the assembly of the systems involved convergent migration, the question arises as to why there are not many more such systems observed. As indicated in the “Effect of Coorbital Material” section and proposed by many authors (e.g., Brasser, Matsumura, Muto, & Ida, 2018; Cossou, Raymond, Hersant, & Pierens,2014; Terquem & Papaloizou, 2007) such migration episodes may end with the system stalling at a protoplanet trap of the type that may be provided by the inner disk boundary. Its subsequent evolution after migration has halted then needs to be considered.

Longer-Term Evolution of Resonant Chains

If such a system is linked with a series of Laplace resonances, as is Kepler 223, or like Kepler 60, a system of three planets with a single Laplace resonance, the presence of residual dissipative effects can maintain resonant angles in a state of liberation and the system close to a stable periodic solution as seen in an appropriately rotating frame (e.g., Papaloizou, 2015). Such dissipation could arise through tides on the planets induced by the central star which act to circularize their orbits in competition with resonant gravitational interactions which tend to make them eccentric. It could also arise through interaction with the disk if that is present.

During this evolution the planetary system spreads reducing its orbital energy to account for the dissipation while conserving its total angular momentum. This can be seen in a simple way from equation (13) which expresses the scaling of an equilibrium eccentricity with the circularization and migration rates. If the system is considered to be gradually increasing Tm while Tc is fixed as migration becomes halted, the eccentricity is decreasing. This means that the system has to be separating away from resonance because proximity to resonance increases the strength of interaction and accordingly eccentricity (e.g., Papaloizou, 2011). This form of evolution is generic, applies to systems with arbitrary numbers of planets, and indicates why there should be a preference for period ratios to exceed values corresponding to precise commensurability. However, as the evolution is driven by having a finite eccentricity which is decreasing away from resonance, period ratios never depart greatly from commensurability.

But note that other processes can be involved. Chatterjee and Ford (2015) propose that significant numbers of planetesimals may remain in the systems, eventually causing disruption of resonances and increasing period ratios. Similarly, Baruteau and Papaloizou (2013) find that the wakes produced by the planets (see Figure 4), while the disk is still present, can perturb their neighbors and also cause them to separate. For systems made up of just two planets, it is unclear that the proposed processes can disrupt resonances to an extent that is consistent with the observations, indicating that the degree of convergent migration during and post formation has been modest (Terquem & Papaloizou, 2019). However, chains of more than two planets may undergo dynamical instability and be disrupted on long time scales. To what extent this occurs depends on the number of planets in the chain, their masses, and the disk properties on formation. Izidoro et al. (2017) have found that while it is possible that most systems have gone through a resonant-chain stage, robust mechanisms are needed to account for why up to 95% of chains produced by migrating accumulating cores have to be eventually disrupted in order to be consistent with observations. Although some mechanisms for this listed have been identified, whether the observations can be accounted for is uncertain.

Summary

Exoplanet systems exhibit a diversity of spacings, and their properties reflect their origin and evolution. Formation in situ, radial migration either caused by interaction with the gaseous protoplanetary disk or by gravitational interactions among planets followed by tidal interaction with the central star and long-term instability are all likely to have been involved. Significantly, the extended gaseous atmospheres of close-in super-Earths and the existence of regular systems linked by a chain of commensurabilities point to the presence of the protoplanetary disk during formation. However, it seems clear that a single scenario involving a specific combination of these processes cannot be responsible for determining the outcome in all cases of a particular class of system. Because in some aspects they can act as alternatives, it is difficult to evaluate the level of importance of each of them. This must be determined by making more theoretical studies, enabling specific predictions of observable properties associated with each mechanism, and then making comparisons with the more extensive observational data we can look forward to obtaining in the future.

References