Resonant Planetary Systems Research Articles

The role of disc torques in forming resonant planetary systems

Context. The most accurate method for modelling planetary migration and hence the formation of resonant systems is using hydrodynamical simulations. Usually, the force (torque) acting on a planet is calculated using the forces from the gas disc and the star, while the gas accelerations are computed using the pressure gradient, the star, and the planet’s gravity, ignoring its own gravity. For a non-migrating planet the neglect of the disc gravity results in a consistent torque calculation while for a migrating case it is inconsistent. Aims. We aim to study how much this inconsistent torque calculation can affect the final configuration of a two-planet system. We focus on low-mass planets because most of the multi-planetary systems, discovered by the Kepler survey, have masses around ten Earth masses. Methods. Performing hydrodynamical simulations of planet–disc interaction, we measured the torques on non-migrating and migrating planets for various disc masses as well as density and temperature slopes with and without considering the self-gravity of the disc. Using this data, we found a relation that quantifies the inconsistency, used this relation in an N-body code, and performed an extended parameter study modelling the migration of a planetary system with different planet mass ratios and disc surface densities, to investigate the impact of the torque inconsistency on the architecture of the planetary system. Results. Not considering disc self-gravity produces an artificially larger torque on the migrating planet that can result in tighter planetary systems. The deviation of this torque from the correct value is larger in discs with steeper surface density profiles. Conclusions. In hydrodynamical modelling of multi-planetary systems, it is crucial to account for the torque correction, otherwise the results favour more packed systems. We examine two simple correction methods existing in the literature and show that they properly correct this problem.

Inner rocky super-Earth formation: distinguishing the formation pathways in viscously heated and passive discs

Observations have revealed that super-Earths (planets up to 10 Earth masses) are the most abundant type of planets in the inner systems. Their formation is strongly linked to the structure of the protoplanetary disc, which determines growth and migration. In the pebble accretion scenario, planets grow to the pebble isolation mass, at which the planet carves a small gap in the gas disc halting the pebble flux and thus its growth. The pebble isolation mass scales with the disc’s aspect ratio, which directly depends on the heating source of the protoplanetary disc. I compare the growth of super-Earths in viscously heated discs, where viscous heating dissipates within the first million years, and discs purely heated by the central star with super-Earth observations from the Kepler mission. This allows two formation pathways of super-Earths to be distinguished in the inner systems within this framework. Planets growing within 1 Myr in the viscously heated inner disc reach pebble isolation masses that correspond directly to the inferred masses of the Kepler observations for systems that feature planets in resonance or not in resonance. However, to explain the period ratio distribution of Kepler planets – where most Kepler planet pairs are not in mean motion resonance configurations – a fraction of these resonant chains has to be broken. In case the planets are born early in a viscously heated disc, these resonant chains thus have to be broken without planetary mergers, for example through the magnetic rebound effect, and the final system architecture should feature low mutual inclinations. If super-Earths form either late or in purely passive discs, the pebble isolation mass is too small (around 2–3 Earth masses) to explain the Kepler observations, implying that planetary mergers have to play a significant role in determining the final system architecture. Resonant planetary systems thus have to experience mergers already during the gas disc phase, so the planets can get trapped in resonance after reaching 5–10 Earth masses. In case instabilities are dominating the system architecture, the systems should not be flat, but feature mutually inclined orbits. This implies that future observations of planetary systems with radial velocities (RV) and transits (for example through the Transiting Exoplanet Survey Satellite (TESS) and its follow up RV surveys) could distinguish between these two formation channels of super-Earth and thus constrain planet formation theories.

The role of dissipative evolution for three-planet, near-resonant extrasolar systems

Early dynamical evolution of close-in planetary systems is shaped by an intricate combination of planetary gravitational interactions, orbital migration, and dissipative effects. While the process of convergent orbital migration is expected to routinely yield resonant planetary systems, previous analyses have shown that the semi-major axes of initially resonant pairs of planets will gradually diverge under the influence of long-term energy damping, producing an overabundance of planetary period ratios in slight excess of exact commensurability. While this feature is clearly evident in the orbital distribution of close-in extrasolar planets, the existing theoretical picture is limited to the specific case of the planetary three-body problem. In this study, we generalise the framework of dissipative divergence of resonant orbits to multi-resonant chains, and apply our results to the current observational census of well-characterised three-planet systems. Focusing on the 2:1 and 3:2 commensurabilities, we identify three three-planet systems, whose current orbital architecture is consistent with an evolutionary history wherein convergent migration first locks the planets into a multi-resonant configuration and subsequent dissipation repels the orbits away from exact commensurability. Nevertheless, we find that the architecture of the overall sample of multi-planetary systems is incompatible with this simple scenario, suggesting that additional physical mechanisms must play a dominant role during the early stages of planetary systems’ dynamical evolution.

Pebble accretion in self-gravitating protostellar discs

Abstract Pebble accretion has become a popular component to core accretion models of planet formation, and is especially relevant to the formation of compact, resonant terrestrial planetary systems. Pebbles initially form in the inner protoplanetary disc, sweeping outwards in a radially expanding front, potentially forming planetesimals and planetary cores via migration and the streaming instability. This pebble front appears at early times, in what is typically assumed to be a low-mass disc. We argue this picture is in conflict with the reality of young circumstellar discs, which are massive and self-gravitating. We apply standard pebble accretion and streaming instability formulae to self-gravitating protostellar disc models. Fragments will open a gap in the pebble disc, but they will likely fail to open a gap in the gas, and continue rapid inward migration. If this does not strongly perturb the pebble disc, our results show that disc fragments will accrete pebbles efficiently. We find that in general the pebble-to-gas-density ratio fails to exceed 0.01, suggesting that the streaming instability will struggle to operate. It may be possible to activate the instability if 10 cm grains are available, and spiral structures can effectively concentrate them in regions of low gravito-turbulence. If this occurs, lunar mass cores might be assembled on time-scales of a few thousand years, but this is likely to be rare, and is far from proven. In any case, this work highlights the need for study of how self-gravitating protostellar discs define the distribution and properties of solid bodies, for future planet formation by core accretion.

Three planets around HD 27894

Aims. Our new program with HARPS aims to detect mean motion resonant planetary systems around stars which were previously reported to have a single bona fide planet, often based only on sparse radial velocity data. Methods. Archival and new HARPS radial velocities for the K2V star HD 27894 were combined and fitted with a three-planet self-consistent dynamical model. The best-fit orbit was tested for long-term stability. Results. We find clear evidence that HD 27894 is hosting at least three massive planets. In addition to the already known Jovian planet with a period $P_{\rm b}$ $\approx$ 18 days we discover a Saturn-mass planet with $P_{\rm c}$ $\approx$ 36 days, likely in a 2:1 mean motion resonance with the first planet, and a cold massive planet ($\approx$ 5.3 $M_{\mathrm{Jup}}$) with a period $P_{\rm d}$ $\approx$ 5170 days on a moderately eccentric orbit ($e_{\rm d}$ = 0.39). Conclusions. HD 27894 is hosting a massive, eccentric giant planet orbiting around a tightly packed inner pair of massive planets likely involved in an asymmetric 2:1 mean motion resonance. HD 27894 may be an important milestone for probing planetary formation and evolution scenarios.

On the formation and stability of resonant planetary systems

On the formation and stability of resonant planetary systems

The reversibility error method (REM): a new, dynamical fast indicator for planetary dynamics

We describe the Reversibility Error Method (REM) and its applications to planetary dynamics. REM is based on the time-reversibility analysis of the phase-space trajectories of conservative Hamiltonian systems. The round-off errors break the time reversibility and the displacement from the initial condition, occurring when we integrate it forward and backward for the same time interval, is related to the dynamical character of the trajectory. If the motion is chaotic, in the sense of non-zero maximal Characteristic Lyapunov Exponent (mLCE), then REM increases exponentially with time, as exp \lambda t, while when the motion is regular (quasi-periodic) then REM increases as a power law in time, as t^{\alpha}, where {\alpha} and {\lambda} are real coefficients. We compare the REM with a variant of mLCE, the Mean Exponential Growth factor of Nearby Orbits (MEGNO). The test set includes the restricted three body problem and five resonant planetary systems: HD 37124, Kepler-60, Kepler-36, Kepler-29 and Kepler-26. We found a very good agreement between the outcomes of these algorithms. Moreover, the numerical implementation of REM is astonishing simple, and is based on solid theoretical background. The REM requires only a symplectic and time-reversible (symmetric) integrator of the equations of motion. This method is also CPU efficient. It may be particularly useful for the dynamical analysis of multiple planetary systems in the KEPLER sample, characterized by low-eccentricity orbits and relatively weak mutual interactions. As an interesting side-result, we found a possible stable chaos occurrence in the Kepler-29 planetary system.

Regular and chaotic orbits in the dynamics of exoplanets

Many of exoplanetary systems consist of more than one planet and the study of planetary orbits with respect to their long-term stability is very interesting. Furthermore, many exoplanets seem to be locked in a mean-motion resonance (MMR), which offers a phase protection mechanism, so that, even highly eccentric planets can avoid close encounters. However, the present estimation of their initial conditions, which may change significantly after obtaining additional observational data in the future, locate most of the systems in chaotic regions and consequently, they are destabilized. Hence, dynamical analysis is imperative for the derivation of proper planetary orbital elements. We utilize the model of spatial general three body problem, in order to simulate such resonant systems through the computation of families periodic orbits. In this way, we can figure out regions in phase space, where the planets in resonances should be ideally hosted in favour of long-term stability and therefore, survival. In this review, we summarize our methodology and showcase the fact that stable resonant planetary systems evolve being exactly centered at stable periodic orbits. We apply this process to co-orbital motion and systems HD 82943, HD 73526, HD 128311, HD 60532, HD 45364 and HD 108874.

Ias15: a fast, adaptive, high-order integrator for gravitational dynamics, accurate to machine precision over a billion orbits

We present IAS15, a 15th-order integrator to simulate gravitational dynamics. The integrator is based on a Gau\ss-Radau quadrature and can handle conservative as well as non-conservative forces. We develop a step-size control that can automatically choose an optimal timestep. The algorithm can handle close encounters and high-eccentricity orbits. The systematic errors are kept well below machine precision and long-term orbit integrations over $10^9$ orbits show that IAS15 is optimal in the sense that it follows Brouwer's law, i.e. the energy error behaves like a random walk. Our tests show that IAS15 is superior to a mixed-variable symplectic integrator (MVS) and other popular integrators, including high-order ones, in both speed and accuracy. In fact, IAS15 preserves the symplecticity of Hamiltonian systems better than the commonly-used nominally symplectic integrators to which we compared it. We provide an open-source implementation of IAS15. The package comes with several easy-to-extend examples involving resonant planetary systems, Kozai-Lidov cycles, close encounters, radiation pressure, quadrupole moment, and generic damping functions that can, among other things, be used to simulate planet-disc interactions. Other non-conservative forces can be added easily.

Dissipation in planar resonant planetary systems

Close-in planetary systems detected by the Kepler mission present an excess of periods ratio that are just slightly larger than some low order resonant values. This feature occurs naturally when resonant couples undergo dissipation that damps the eccentricities. However, the resonant angles appear to librate at the end of the migration process, which is often believed to be an evidence that the systems remain in resonance. Here we provide an analytical model for the dissipation in resonant planetary systems valid for low eccentricities. We confirm that dissipation accounts for an excess of pairs that lie just aside from the nominal periods ratios, as observed by the Kepler mission. In addition, by a global analysis of the phase space of the problem, we demonstrate that these final pairs are non-resonant. Indeed, the separatrices that exist in the resonant systems disappear with the dissipation, and remains only a circulation of the orbits around a single elliptical fixed point. Furthermore, the apparent libration of the resonant angles can be explained using the classical secular averaging method. We show that this artifact is only due to the severe damping of the amplitudes of the eigenmodes in the secular motion.

Formation of the resonant system HD 60532

Among multi-planet planetary systems there are a large fraction of resonant systems. Studying the dynamics and formation of these systems can provide valuable informations on processes taking place in protoplanetary disks where the planets are thought have been formed. The recently discovered resonant system HD 60532 is the only confirmed case, in which the central star hosts a pair of giant planets in 3:1 mean motion resonance. We intend to provide a physical scenario for the formation of HD 60532, which is consistent with the orbital solutions derived from the radial velocity measurements. Observations indicate that the system is in an antisymmetric configuration, while previous theoretical investigations indicate an asymmetric equilibrium state. The paper aims at answering this discrepancy as well. We performed two-dimensional hydrodynamical simulations of thin disks with an embedded pair of massive planets. Additionally, migration and resonant capture are studied by gravitational N-body simulations that apply properly parametrized non-conservative forces. Our simulations suggest that the capture into the 3:1 mean motion resonance takes place only for higher planetary masses, thus favouring orbital solutions having relatively smaller inclination i=20 degrees. The system formed by numerical simulations qualitatively show the same behaviour as HD 60532. We also find that the presence of an inner disk (between the inner planet and the star) plays a very important role in determining the final configurations of resonant planetary systems. Its damping effect on the inner planet's eccentricity is responsible for the observed antisymmetric state of HD 60532.

On the dynamics of extrasolar planetary systems under dissipation: Migration of planets

We study the dynamics of planetary systems with two planets moving in the same plane, when frictional forces act on the two planets, in addition to the gravitational forces. The model of the general three-body problem is used. Different laws of friction are considered. The topology of the phase space is essential in understanding the evolution of the system. The topology is determined by the families of stable and unstable periodic orbits, both symmetric and non symmetric. It is along the stable families, or close to them, that the planets migrate when dissipative forces act. At the critical points where the stability along the family changes, there is a bifurcation of a new family of stable periodic orbits and the migration process changes route and follows the new stable family up to large eccentricities or to a chaotic region. We consider both resonant and non resonant planetary systems. The 2/1, 3/1 and 3/2 resonances are studied. The migration to larger or smaller eccentricities depends on the particular law of friction. Also, in some cases the semimajor axes increase and in other cases they are stabilized. For particular laws of friction and for special values of the parameters of the frictional forces, it is possible to have partially stationary solutions, where the eccentricities and the semimajor axes are fixed.

Influence of an inner disc on the orbital evolution of massive planets migrating in resonance

The formation of resonant pairs of planets in exoplanetary systems involves planetary migration in the protoplanetary disc. After a resonant capture, the subsequent migration in this configuration leads to a large increase of planetary eccentricities if no damping mechanism is applied. This has led to the conclusion that the migration of resonant planetary systems cannot occur over large radial distances and has to be terminated sufficiently rapidly through disc dissipation. In this study, we investigate whether the presence of an inner disc might supply an eccentricity damping of the inner planet, and if this effect could explain the observed eccentricities in some systems. To investigate the influence of an inner disc, we first compute hydrodynamic simulations of giant planets orbiting with a given eccentricity around an inner gas disc, and measure the effect of the latter on the planetary orbital parameters. We then perform detailed long term calculations of the GJ 876 system. We also run N-body simulations with artificial forces on the planets mimicking the effects of the inner and outer discs. We find that the influence of the inner disc can not be neglected, and that it might be responsible for the observed eccentricities. In particular, we reproduce quite well the orbital parameters of a few systems engaged in 2:1 mean motion resonances: GJ 876, HD 73526, HD 82943 and HD 128311. Finally, we derive analytically the effect that the inner disc should have on the inner planet to reach a specific orbital configuration with a given damping effect of the outer disc on the outer planet.

On periodic orbits and resonance in extrasolar planetary systems

We study the evolution of an extrasolar planetary system with two planets, for planar motion, starting from an exact resonant periodic motion and increasing the deviation from the equilibrium solution. We keep the semimajor axes and the eccentricities of the two planets fixed and we change the initial conditions by rotating the orbit of the outer planet by Δω. In this way the resonance is preserved, but we deviate from the exact periodicity and there is a transition from order to chaos as the deviation increases. There are three different routes to chaos, as far as the evolution of (ω2 − ω1) is concerned: (a) Libration → rotation → chaos, with intermittent transition from libration to rotation in between, (b) libration → chaos and (c) libration → intermittent interchange between libration and rotation → chaos. This indicates that resonant planetary systems where the angle (ω2 − ω1) librates or rotates are not different, but are closely connected to the exact periodic motion.

Stability Limits in Resonant Planetary Systems

The relationship between the boundaries for Hill and Lagrange stability in orbital element space is modified in the case of resonantly interacting planets. Hill stability requires the ordering of the planets to remain constant while Lagrange stability also requires all planets to remain bound to the central star. The Hill stability boundary is defined analytically, but no equations exist to define the Lagrange boundary, so we perform numerical experiments to estimate the location of this boundary. To explore the effect of resonances, we consider orbital element space near the conditions in the HD 82943 and 55 Cnc systems. Previous studies have shown that, for non-resonant systems, the two stability boundaries are nearly coincident. However the Hill stability formula are not applicable to resonant systems, and our investigation shows how the two boundaries diverge in the presence of a mean-motion resonance, while confirming that the Hill and Lagrange boundaries are similar otherwise. In resonance the region of stability is larger than the domain defined by the analytic formula for Hill stability. We find that nearly all known resonant interactions currently lie in this extra stable region, i.e. where the orbits would be unstable according to the non-resonant Hill stability formula. This result bears on the dynamical packing of planetary systems, showing how quantifying planetary systems' dynamical interactions (such as proximity to the Hill-stability boundary) provides new constraints on planet formation models.

Planetary migration and extrasolar planets in the 2/1 mean-motion resonance

In this paper, we present a new set of corotational solutions for the 2/1 commensurability, including previously known solutions and new results. Comparisons with observed exoplanets show that current orbital fits of three proposed resonant planetary systems are consistent with apsidal corotations. We also discuss the possible relationship between the current orbital elements fits of known exoplanets in the 2/1 mean-motion resonance and the expected orbital configuration due to migration. We find that, as long as the orbital decay was sufficiently slow to be approximated by an adiabatic process, all captured planets should be in apsidal corotations. In other words, they should show a simultaneous libration of both the resonant angle and the difference in longitudes of pericenter.