arXiv subjects

R. Brasser

Publications and source records attributed to R. Brasser.

At least 19 recordsLinked to original sources

arXiv2026

The Enceladian crater production function

Interpreting Enceladus's past and present surface history and interior state remains challenging, owing to uncertain prescription of its impact bombardment history and limited interpretation of its crater statistics. Further progress in understanding its evolutionary history can be achieved with an improved crater chronology model and a thorough assessment of Enceladus's surface. Here we present the first step in the form of a comprehensive, global crater catalogue with geomorphology survey for Enceladus. From our dataset we build the crater production function (CPF), which is the underlying, unmodified crater size-frequency distribution of the satellite surface, assuming no subsequent modification. We obtained the CPF with a data-driven approach; therefore it makes no assumptions about the impactor source population or planet evolution models or the timing of impact. We fit the CPF with a high-order polynomial as is customary for the Moon and Mars, capturing the slope variations across different crater diameter ranges. The Enceladian CPF generally has a steeper cumulative slope than that of the Moon and Mars for small crater diameters D_cr < 10 km, as well as that of the size-frequency distribution of trans-Neptunian objects. This CPF serves as a critical observational input for an Enceladian crater chronology model, enabling the conversion of crater densities into absolute surface ages. Extending the crater cataloguing and CPF derivation of this study to other Saturnian satellites will determine whether the Enceladian CPF is unique; a shared CPF would indicate a common impactor population, providing observational constraints on the size-frequency distribution of small bodies in the outer Solar System.

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arXiv2025

Impact chronology of leftover planetesimals

After the formation of the Moon the terrestrial planets were pummelled by impacts from planetesimals left over from terrestrial planet formation. This work attempts to reproduce the impact rates set by modern crater chronologies using leftover planetesimals from three different dynamical models of terrestrial planet formation. I ran dynamical simulations for 1 billion years using leftover planetesimals from the Grand Tack, Depleted Disc and Implantation models of terrestrial planet formation with the GENGA N-body integrator. I fit the cumulative impacts on the Earth and Mars using a function that is a sum of exponentials with different weighing factors and e-folding times. Most fits require three or four terms. The fitted timescales cluster around t1=10 Myr, t2=35 Myr, t3=100 Myr and t4>200 Myr. I attribute them to dynamical losses of planetesimals through different mechanisms: high-eccentricity Earth crossers and the nu6 secular resonance, Earth crossers, Mars crossers, and objects leaking on to Mars crossing orbits from beyond Mars. I place a constraint on the initial population using the known Archean terrestrial spherule beds, and I conclude that the Archean impacts were mostly created by leftover planetesimals. The inferred mass in leftover planetesimals at the time of the Moon's formation was about 0.015 Earth masses. The third time constant is comparable to that of modern crater chronologies. As such, the crater chronologies are indicative of impacts by an ancient population of Mars crossers. The initial perihelion distribution of the leftovers is a major factor in setting the rate of decline: to reproduce the current crater chronologies the number of Earth crossers at the time of the Moon's formation had to be at most half of the Mars crossers. These results together place constraints on dynamical models of terrestrial planet formation.

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arXiv2025

The current cratering rate on the regular satellites of Jupiter, Saturn, and Uranus

We aim to compute the impact rates for objects with a diameter of 1 km onto the regular satellites of Jupiter, Saturn and Uranus using our latest dynamical simulations of the evolution of outer solar system coupled with the best estimates of the current population of objects beyond Neptune and their size-frequency distribution. We use the outcome of the last 3.5~Gyr of evolution of the outer solar system from our database of simulations and combine this with observational constraints of the population beyond Neptune to compute the flux of objects entering the Centaur region, with uncertainties. The initial conditions resemble the current population rather than a near-circular, near-planar disc usually assumed just before the onset of giant planet migration. We obtain a better estimate of the impact probability of a Centaur with the satellites from enacting simulations of planetesimals flying past the satellites on hyperbolic orbits, which agree with literature precedents. We find that our impact rate of objects greater than 1 km in diameter with Jupiter is 0.0012/yr, which is a factor of 3--6 lower than previous estimates of 0.0044/yr from Nesvorny et al. (2023) and 0.0075/yr from Zahnle et al. (2003). On the other hand our impact probabilities with the satellites scaled to the giant planets are consistent with these earlier literature estimates, as is the leakage rate of objects from beyond Neptune into the Centaur region. However, our absolute impact probabilities with the giant planets are lower. We attribute this to our choice of initial conditions.

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arXiv2024

Terrestrial planet formation during giant planet formation and giant planet migration I: The first 5 million years

Terrestrial planet formation (TPF) is a difficult problem that has vexed researchers for decades. Numerical models are only partially successful at reproducing the orbital architecture of the inner planets, but have generally not considered the effect of the growth of the giant planets. I dynamically model TPF as the gas giants Jupiter and Saturn are growing using GENGA. The evolution of the masses, radii and orbital elements of the gas giants are precomputed and read and interpolated within GENGA. The terrestrial planets are formed by planetesimal accretion from tens of thousands of self-gravitating planetesimals spread between 0.5 au and 8.5 au. The total mass of the inner planetesimal disc and outer disc are typically 2 and 3 Earth masses respectively, and the composition of the planetesimals changes from non-carbonaceous-like to carbonaceous-like at a prescribed distance, ranging from 2 au to 5 au. After 5 Myr of evolution approximately 10% to 25% of the mass of planetesimals in the Jupiter-Saturn region is implanted in the inner solar system, which is more than what cosmochemical models predict. The implantation initially sets up a composition gradient in the inner solar system, with the fraction of outer solar system material increasing with increasing distance to the Sun. The planetesimals that remain in the inner solar system have a mixed composition. The growth of the gas giants scatters planetesimals in their vicinity into the inner solar system, which changes the isotopic composition of the terrestrial planets. The planetesimal disc in the vicinity of the gas giants may not have been very massive, ~1 ME. The inner planetesimal disc may not have extended much farther than 2 au otherwise embryos do not grow fast enough to produce Mars analogues. This could mean that the region of the current asteroid belt never contained much mass to begin with.

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arXiv2023

Speeding up the GENGA N-body integrator on consumer-grade graphics cards

GPU computing is popular due to the calculation potential of a single card. The N-body integrator GENGA is built to for this, but it suffers a performance penalty on consumer-grade GPUs due to their truncated double precision (FP64) performance. We aim to speed up GENGA on consumer-grade cards by harvesting their high single-precision performance (FP32). We modified GENGA to be able to compute the long-distance forces between bodies in FP32 precision and tested this with 5 experiments. We ran simulations with similar initial conditions of 6600 planetesimals in both FP32 and FP64 precision. We also ran simulations that i) began with a mixture of planetesimals and planetary embryos, ii) planetesimal-driven giant planet migration, and iii) terrestrial planet formation with a gas disc. Second, we ran the same simulation beginning with 40 000 planetesimals using both FP32 and FP64 precision forces on a variety of consumer-grade and Tesla GPUs to measure the performance boost of FP32 computing. There are no statistical differences when running in FP32 or FP64 precision that can be attributed to the force prescription rather than stochastic effects. The uncertainties in energy are almost identical when using both precisions. However, the uncertainty in the angular momentum using FP32 rather than FP64 precision long-range forces is about two orders of magnitude greater, but still very low. Running the simulations in single precision on consumer-grade cards decreases running time by a factor of three and becomes within a factor of three of a Tesla A100 GPU. Additional tuning speeds up the simulation by a factor of two across all types of cards. The option to compute the long-range forces in single precision in GENGA when using consumer-grade GPUs dramatically improves performance at a little penalty to accuracy. There is an additional environmental benefit because it reduces energy usage.

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arXiv2023

Forming rocky exoplanets around K-dwarf stars

How multiple close-in super-Earths form around stars with masses lower than that of the Sun is still an open issue. Several recent modeling studies have focused on planet formation around M-dwarf stars, but so far no studies have focused specifically on K dwarfs, which are of particular interest in the search for extraterrestrial life. We aim to reproduce the currently known population of close-in super-Earths observed around K-dwarf stars and their system characteristics. We performed 48 high-resolution N-body simulations of planet formation via planetesimal accretion using the existing GENGA software running on GPUs. In the simulations we varied the initial disk mass and the solid and gas surface density profiles. Each simulation began with 12000 bodies with radii of between 200 and 2000 km around two different stars, with masses of 0.6 and 0.8 $M_{\odot}$. Most simulations ran for 20 Myr, with several simulations extended to 40 or 100 Myr. The mass distributions for the planets with masses between 2 and 12 $M_\oplus$ show a strong preference for planets with masses $M_p<6$ $M_\oplus$ and a lesser preference for planets with larger masses, whereas the mass distribution for the observed sample increases almost linearly. However, we managed to reproduce the main characteristics and architectures of the known planetary systems and produce mostly long-term angular-momentum-deficit-stable, nonresonant systems, but we require an initial disk mass of 15 $M_\oplus$ or higher and a gas surface density value at 1 AU of 1500 g cm$^{-2}$ or higher. Our simulations also produce many low-mass planets with $M<2$ $M_\oplus$, which are not yet found in the observed population, probably due to the observational biases. The final systems contain only a small number of planets, which could possibly accrete substantial amounts of gas, and these formed after the gas had mostly dissipated.

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arXiv2023

Terrestrial planet formation from a ring

It has been long proposed that, if all the terrestrial planets form within a tiny ring of solid material at around 1 AU, the concentrated mass-distance distribution of the current system can be reproduced. Recent planetesimal formation models also support this idea. In this study, we revisit the ring model by performing a number of high-resolution N-body simulations for 10 Myr of a ring of self-interacting planetesimals, with various radial distributions of the gas disc. We found that even if all the planetesimals form at ~1 AU in a minimum mass solar nebula-like disc, the system tends to spread radially as accretion proceeds, resulting in a system of planetary embryos lacking mass-concentration at ~1 AU. Modifying the surface density of the gas disc into a concave shape with a peak at ~1 AU helps to maintain mass concentrated at ~1 AU and solve the radial dispersion problem. We further propose that such a disc should be short lived (<= 1 Myr) and with a shallower radial gradient in the innermost region (< 1 AU) than previously proposed to prevent a too-rapid growth of Earth. Future studies should extend to ~100 Myr the most promising simulations and address in a self-consistent manner the evolution of the asteroid belt and its role in the formation of the terrestrial planets.

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arXiv2022

Long-term tidal evolution of the TRAPPIST-1 system

The ultracool M-dwarf star TRAPPIST-1 is surrounded by seven planets configured in a resonant chain. Transit-timing variations have shown that the planets are caught in multiple three-body resonances and that their orbits are slightly eccentric, probably caused by resonant forcing. The current values of the eccentricities could be a remnant from their formation. Here we run numerical simulations using fictitious forces of trapping the fully-grown planets in resonances as they migrated in the gas disc, followed by numerical simulations detailing their tidal evolution. For a reduced disc scale height $h\sim 0.03$--0.05, the eccentricities of the planets upon capture in resonance are higher than their current values by factors of a few. We show that the current eccentricities and spacing of planets d to h are natural outcomes of coupled tidal evolution wherein the planets simultaneously damp their eccentricities and separate due to their resonant interaction. We further show that the planets evolve along a set of equilibrium curves in semimajor axis--eccentricity phase space that are defined by the resonances, and that conserve angular momentum. As such, the current 8:5--5:3--(3:2)$^2$--4:3--3:2 resonant configuration cannot be reproduced from a primordial (3:2)$^4$--4:3--3:2 resonant configuration from tidal dissipation in the planets alone. We use our simulations to constrain the long-term tidal parameters $k_2/Q$ for planets b to e, which are in the range $10^{-3}$ to $10^{-2}$, and show that these are mostly consistent with those obtained from interior modelling following reasonable assumptions.

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arXiv2022

Evidence of a primordial isotopic gradient in the inner region of the solar protoplanetary disc

Not only do the sampled terrestrial worlds (Earth, Mars, and asteroid 4 Vesta) differ in their mass-independent (nucleosynthetic) isotopic compositions of many elements (e.g. $\varepsilon^{48}$Ca, $\varepsilon^{50}$Ti, $\varepsilon^{54}$Cr, $\varepsilon^{92}$Mo), the magnitudes of some of these isotopic anomalies also appear to correlate with heliocentric distance. While the isotopic differences between the Earth and Mars may be readily accounted for by the accretion of mostly local materials in distinct regions of the protoplanetary disc, it is unclear whether this also applies to asteroid Vesta. Here we analysed the available data from our numerical simulation database to determine the formation location of Vesta in the framework of three planet-formation models: classical, Grand Tack, and Depleted Disc. We find that Vesta has a high probability of forming locally in the asteroid belt in models where material mixing in the inner disc is limited; this limited mixing is implied by the isotopic differences between the Earth and Mars. Based on our results, we propose several criteria to explain the apparent correlation between the different nucleosynthetic isotopic compositions of the Earth, Mars, and Vesta: (1) these planetary bodies accreted their building blocks in different regions of the disc, (2) the inner disc is characterised by an isotopic gradient, and (3) the isotopic gradient was preserved during the formation of these planetary bodies and was not diluted by material mixing in the disc (e.g. via giant planet migration).

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arXiv2021

Effects of pebble accretion on the growth and composition of planetesimals in the inner Solar System

Recent work has shown that aside from the classical view of collisions by increasingly massive planetesimals, the accretion of mm- to m-sized 'pebbles' can also reproduce the mass-orbit distribution of the terrestrial planets. Here, we perform N-body simulations to study the effects of pebble accretion onto growing planetesimals of different diameters located in the inner Solar System. The simulations are run to occur during the lifetime of the gas disc while also simultaneously taking Jupiter's growth into account. We find that pebble accretion can increase the mass in the solid disc by at least a few times its initial mass with reasonable assumptions that pebbles fragment to smaller-sized grains at the snow line and that gas-disc-induced orbital migration effects are in force. Such a large contribution in mass by pebbles would seem to imply that the isotopic composition of the inner Solar System should be similar to the pebble source (i.e. outer Solar System). This implication appears to violate the observed nucleosynthetic isotopic dichotomy of the sampled Solar System. Thus, pebble accretion played little or no role in terrestrial planet formation.

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arXiv2021

A new estimate for the age of highly-siderophile element retention in the lunar mantle from late accretion

Subsequent to the Moon's formation, late accretion to the terrestrial planets strongly modified the physical and chemical nature of silicate crusts and mantles. This alteration came in the form of melting through impacts, as well as the belated addition of volatiles and the highly siderophile elements (HSEs). Current debate centres on whether the lunar HSE record is representative of its whole late accretion history or alternatively that these were only retained in the mantle and crust after a particular time, and if so, when. Here we employ improved Monte Carlo impact simulations of late accretion onto the Moon and Mars and present an updated chronology based on new dynamical simulations of leftover planetesimals and the E-belt. We take into account the inefficient retention of colliding material. We compute the crater and basin densities on the Moon and Mars, the largest objects to strike these planets and the amount of material they accreted. Outputs are used to infer the mass in leftover planetesimals at a particular time period, which is then compared to the lunar HSE abundance. From this estimate we calculate a preferred lunar HSE retention age of ca. 4450 Ma which means that the modelled lunar mantle HSE abundances trace almost all of lunar late accretion. Based on our results, the surface ages of the lunar highlands are at least 4370 Ma. We find that the mass of leftover planetesimals with diameters Di<300 km at 4500 Ma that best fits the crater chronology is approximately 2x10^{-3} Earth mass (ME) while the mass of the E-belt was fixed at 4.5x10^{-4} ME. We also find that a leftover planetesimal mass in excess of 0.01 ME results in a lunar HSE retention age younger than major episodes of lunar differentiation and crust formation, which in turn violates geochemical constraints for the timing and intensity of late accretion to the Earth (Mojzsis et al., 2019).

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arXiv2021

Formation of Venus, Earth and Mars: Constrained by isotopes

We discuss the current state of knowledge of terrestrial planet formation from the aspects of different planet formation models and isotopic data from 182Hf-182W, U-Pb, lithophile-siderophile elements, 48Ca/44Ca isotope samples from planetary building blocks, 36Ar/38Ar, 20Ne/22Ne, 36Ar/22Ne isotope ratios in Venus' and Earth's atmospheres, the expected solar 3He abundance in Earth's deep mantle and Earth's D/H sea water ratios that shed light on the accretion time of the early protoplanets. Accretion scenarios that can explain the different isotope ratios, including a Moon-forming event after ca. 50 Myr, support the theory that the bulk of Earth's mass (>80%) most likely accreted within 10-30 Myr. From a combined analysis of the before mentioned isotopes, one finds that proto-Earth accreted 0.5-0.6 MEarth within the first ~4-5 Myr, the approximate lifetime of the protoplanetary disk. For Venus, the available atmospheric noble gas data are too uncertain for constraining the planet's accretion scenario accurately. However, from the available Ar and Ne isotope measurements, one finds that proto-Venus could have grown to 0.85-1.0 MVenus before the disk dissipated. Classical terrestrial planet formation models have struggled to grow large planetary embryos quickly from the tiniest materials within the typical lifetime of protoplanetary disks. Pebble accretion could solve this long-standing time scale controversy. Pebble accretion and streaming instabilities produce large planetesimals that grow into Mars-sized and larger planetary embryos during this early accretion phase. The later stage of accretion can be explained well with the Grand-Tack, annulus or depleted disk models. The relative roles of pebble accretion and planetesimal accretion/giant impacts are poorly understood and should be investigated with N-body simulations that include pebbles and multiple protoplanets.

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arXiv2020

No evidence for interstellar planetesimals trapped in the Solar System

In two recent papers published in MNRAS, Namouni and Morais (2018, 2020) claimed evidence for the interstellar origin of some small Solar System bodies, including i) objects in retrograde co-orbital motion with the giant planets, and ii) the highly-inclined Centaurs. Here, we discuss the flaws of those papers that invalidate the authors' conclusions. Numerical simulations backwards in time are not representative of the past evolution of real bodies. Instead, these simulations are only useful as a means to quantify the short dynamical lifetime of the considered bodies and the fast decay of their population. In light of this fast decay, if the observed bodies were the survivors of populations of objects captured from interstellar space in the early Solar System, these populations should have been implausibly large (e.g. about 10 times the current main asteroid belt population for the retrograde coorbital of Jupiter). More likely, the observed objects are just transient members of a population that is maintained in quasi-steady state by a continuous flux of objects from some parent reservoir in the distant Solar System. We identify in the Halley type comets and the Oort cloud the most likely sources of retrograde coorbitals and highly-inclined Centaurs.

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arXiv2019

Impact bombardment chronology of the terrestrial planets from 4.5 Ga to 3.5 Ga

Subsequent to the Moon's formation, late accretion to the terrestrial planets modified their silicate crusts and mantles. We combine dynamical N-body and Monte Carlo simulations to determine impact probabilities, impact velocities, and expected mass augmentation onto the terrestrial planets from three sources: planetesimals left over from primary accretion, asteroids from the E-belt, and comets arriving from the outer Solar System. We present estimates of the amount of cometary material striking the terrestrial planets in an early episode of planetesimal-driven giant planet migration (Mojzsis et al., 2019). The Moon and Mars suffer proportionally higher cometary accretion than Venus and Earth. The background mass addition from small leftover planetesimals to Earth and Mars is far less than independent estimates based on their mantle abundances of highly-siderophile elements. This supports the theory that both planets were struck by single large bodies that delivered most of their mass augmentation since primary accretion. We present fits to the impact chronologies valid from 4500 Ma to ca. 3700 Ma. The lunar timeline thus obtained is at odds with both the calibrated Neukum (Neukum et al., 2001) and Werner (Werner et al., 2014; Werner, 2019) chronologies. For Mars, the match with its calibrated Werner chronology is no better; by increasing the mass of the E-belt by a factor of four the dynamical lunar and martian chronologies are in line with that of Werner (2019) and match constraints from the current population of Hungaria asteroids. Yet, neither of our dynamical timelines fit well with that of Neukum. The dynamical lunar and martian chronologies are also different from each other. Consequently, the usual extrapolation of such chronologies from one planetary body to the other is technically inappropriate.

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arXiv2019

The tidal parameters of TRAPPIST-1 b and c

The TRAPPIST-1 planetary system consists of seven planets within 0.05 au of each other, five of which are in a multi-resonant chain. {These resonances suggest the system formed via planet migration; subsequent tidal evolution has damped away most of the initial eccentricities. We used dynamical N-body simulations to estimate how long it takes for the multi-resonant configuration that arises during planet formation to break. From there we use secular theory to pose limits on the tidal parameters of planets b and c. We calibrate our results against multi-layered interior models constructed to fit the masses and radii of the planets, from which the tidal parameters are computed independently.} The dynamical simulations show that the planets typically go unstable 30 Myr after their formation. {Assuming synchronous rotation throughout} we compute $\frac{k_2}{Q} \gtrsim 2\times 10^{-4}$ for planet b and $\frac{k_2}{Q} \gtrsim 10^{-3}$ for planet c. Interior models yield $(0.075-0.37) \times 10^{-4}$ for TRAPPIST-1 b and $(0.4-2)\times 10^{-4}$ for TRAPPIST-1 c. The agreement between the {dynamical and interior} models is not too strong, but is still useful to constrain the dynamical history of the system. We suggest that this two-pronged approach could be of further use in other multi-resonant systems if the planet's orbital and interior parameters are sufficiently well known.

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arXiv2018

Impact bombardment on the regular satellites of Jupiter and Uranus during an episode of giant planet migration

The intensity and effects of early impact bombardment on the major satellites of the giant planets during an episode of giant planet migration is still poorly known. We use a combination of dynamical N-body and Monte Carlo simulations to determine impact probabilities, impact velocities, and expected masses that collide with these satellites to determine the chronology of impacts during the migration. Volatile loss through bombardment is typically 20% for Miranda, a few percents for the larger Uranian satellites and negligible for the Galilean satellites. Due to its small size and the high impact velocity there is a >99% chance that Miranda suffered a catastrophic impact that shattered the satellite. Subsequent re-accretion from a circum-Uranian ring could account for its peculiar surface morphology and low density. The probability to destroy Ariel and Umbriel is 15% and 1% for Titania and Oberon. Approximately 90% of the mass in planetesimals that passes through the Jovian and Uranian satellite systems (about $4 {\rm \ M_{\oplus}}$ and $2 {\rm \ M_{\oplus}}$ respectively) does so in about 15 Myr. This extremely rapid and intense bombardment causes repeated local crustal melting on all satellites. The combination of these effects results in an entirely different impact chronology than that of the inner solar system. We conclude that the simple extrapolation of the lunar chronology to the outer solar system satellites is not correct. The tail end (after 25 Myr) of the chronology function has an e-folding time of 100 Myr at Jupiter, but follows a cumulative Weibull distribution at Uranus, making direct comparisons between the gas and ice giant planets difficult. Based on our results the surfaces of the Uranian satellites, Callisto, and possibly Ganymede, are all about the same age, and are roughly 150 Myr younger than the timing of the dynamical instability.

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arXiv2018

Plausible home stars of the interstellar object 'Oumuamua found in Gaia DR2

The first detected interstellar object 'Oumuamua that passed within 0.25au of the Sun on 2017 September 9 was presumably ejected from a stellar system. We use its newly determined non-Keplerian trajectory together with the reconstructed Galactic orbits of 7 million stars from Gaia DR2 to identify past close encounters. Such an "encounter" could reveal the home system from which 'Oumuamua was ejected. The closest encounter, at 0.60pc (0.53-0.67pc, 90% confidence interval), was with the M2.5 dwarf HIP 3757 at a relative velocity of 24.7km/s, 1Myr ago. A more distant encounter (1.6pc) but with a lower encounter (ejection) velocity of 10.7km/s was with the G5 dwarf HD 292249, 3.8Myr ago. Two more stars have encounter distances and velocities intermediate to these. The encounter parameters are similar across six different non-gravitational trajectories for 'Oumuamua. Ejection of 'Oumuamua by scattering from a giant planet in one of the systems is plausible, but requires a rather unlikely configuration to achieve the high velocities found. A binary star system is more likely to produce the observed velocities. None of the four home candidates have published exoplanets or are known to be binaries. Given that the 7 million stars in Gaia DR2 with 6D phase space information is just a small fraction of all stars for which we can eventually reconstruct orbits, it is a priori unlikely that our current search would find 'Oumuamua's home star system. As 'Oumuamua is expected to pass within 1pc of about 20 stars and brown dwarfs every Myr, the plausibility of a home system depends also on an appropriate (low) encounter velocity.

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arXiv2018

Trapping low-mass planets at the inner edge of the protostellar disc

The formation of multiple close-in low-mass exoplanets is still a mystery. The challenge is to build a system wherein the outermost planet is beyond 0.2 AU from the star. Here we investigate how the prescription for type I planet migration affects the ability to trap multiple planets in a resonant chain near the inner edge of the protostellar disc. A sharp edge modelled as a hyperbolic tangent function coupled with supersonic corrections to the classical type I migration torques results in the innermost planets being pushed inside the cavity through resonant interaction with farther planets because migration is starward at slightly supersonic eccentricities. Planets below a few Earth masses are generally trapped in a resonant chain with the outermost planet near the disc edge, but long-term stability is not guaranteed. For more massive planets the migration is so fast that the eccentricity of the innermost resonant pair is excited to highly supersonic levels due to decreased damping on the innermost planet as it is pushed inside the cavity; collisions frequently occur and the system consists one or two intermediate-mass planets residing closer to the star than the disc's inner edge. We found a neat pileup of resonant planets outside the disc edge only if the corotation torque does not rapidly diminish at high eccentricity. We call for detailed studies on planet migration near the disc's inner edge, which is still uncertain, and for an improved understanding of eccentricity damping and disc torques in the supersonic regime.

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