arXiv subjects

Alexander J. Mustill

Publications and source records attributed to Alexander J. Mustill.

At least 19 recordsLinked to original sources

arXiv2026

A homogeneous three-dimensional view of Molecular Cloud kinematics out to 2.5 kpc. Using Young Stellar Objects and Open Clusters as complementary tracers

Understanding the large-scale dynamics of molecular clouds (MCs) is crucial for constraining the processes that govern star formation and the structure and evolution of the Galaxy. While gas tracers have traditionally been used to map MC kinematics, stellar tracers such as young stellar objects (YSOs) and open clusters (OCs) provide a complementary approach that enables direct comparisons between the stellar and gaseous components. We aim to validate OCs as complementary tracers by testing whether they retain the same bulk kinematic imprint as YSOs, and to reconstruct the three-dimensional (3D) motions of the main MC complexes within 2.5 kpc of the Sun using YSOs and young OCs as tracers. Using Gaia DR3 astrometry together with complementary spectroscopic surveys for radial velocities, we compiled a unified sample of 24,732 stellar tracers. We applied robust clustering in proper motion space to identify co-moving YSOs and derived cloud-averaged motions via Monte Carlo sampling. These were compared with the kinematics of OCs younger than 30 Myr. Finally, we performed orbital integrations in a realistic Galactic potential to trace the past evolution of the clouds and quantify their expansion and rotation. We derive homogeneous 3D kinematics for 15 MC complexes within 2.5 kpc. YSOs and OCs exhibit strongly consistent kinematics, with a median spatial velocity offset of $\simeq 2$ km s$^{-1}$, confirming that both populations trace the bulk motion of their parent clouds. The resulting cloud kinematics show a median peculiar velocity of $\simeq 8.7$ km s$^{-1}$ with respect to Galactic rotation. We trace back the Solar System's voyage through the Orion cloud and the common origin of Lupus, Ophiuchus, and Corona Australis in Sco-Cen. Internally, we detect significant expansion in Orion and Ophiuchus ($5σ$) and coherent rotation in at least seven complexes.

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arXiv2026

The evolution of exocomets and their source populations

We review the current state of knowledge of the long-term evolution of the small bodies that give rise to comets and exocomets, as well as their reservoirs. The active cometary phase is only transitory, and bodies that become comets pass from a source population, such as the Kuiper Belt, Oort Cloud or their extra-solar analogues, through the active cometary phase, to eventual dormancy or destruction. We discuss dynamical delivery channels that can move comets from their source reservoirs to orbits with small periapsides, and the depletion of these reservoirs by dynamical and collisional means. We also discuss the physical evolution of cometary nuclei, especially in light of recent advances from missions to Solar System comets such as Rosetta's visit to 67P. We then describe our current knowledge of interstellar objects, which can originate from the same source regions as exocomets but be amenable to detailed study when they enter the Solar System. We include a summary of stellar winds emanating from different types of stars, which become increasingly strong once stars leave the Main Sequence. This is followed by a description of how small bodies are affected by stellar evolution, and the range of comet-like phenomena observed towards white dwarf stars. Overall, while we have an increasingly good picture of the physical and dynamical evolution of Solar System comets, a number of large gaps remain in our knowledge of the physics of exocomets, related to our inability to directly probe these bodies and many of the planets that might be affecting their orbits.

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arXiv2025

Starshades as technosignatures in direct imaging phase curves: Application to the Habitable Worlds Observatory targets

A star's luminosity increases as it evolves along the Main Sequence (MS), which inevitably results in a higher surface temperature for planets in orbit around the star. Technologically advanced civilizations may tackle this issue by installing artificial structures -- starshades -- which can reduce the radiation received by the planet. Starshades, if they exist, are potentially detectable with current or near-future technology. We have simulated phase curve signatures in direct imaging of hypothetical starshades in systems targeted by the upcoming Habitable Worlds Observatory (HWO), which will be tasked with searching for Earth-like exoplanets orbiting nearby stars. The starshade is assumed to be a circular, reflecting surface placed at the inner Lagrange point between the star and the planet. Our results show that the phase curve of a starshade has a distinct shape compared to that of a typical planet. The phase curve signature lies above the expected $1σ=10^{-11}$ single-visit precision in contrast ratio of the telescope for 70.8% of the target stars for the expected inner working angle (IWA) of around 60 mas. If the IWA can be reduced to 45 mas, the percentage of stars above the $1σ$ limit increases to 96.7%. With a sufficiently small IWA, HWO should be able to detect anomalies in light curves caused by starshades or similar highly-reflective surfaces -- which could serve as key indicators for technologically advanced civilizations.

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arXiv2025

The Origins & Reservoirs of Exocomets

Small bodies exist in distinct populations within their planetary systems. These reservoir populations hold a range of compositions, which to first order are dependent on formation location relative to their star. We provide a general overview of the nature of the reservoirs that source exocomets, from the influence of the stellar environment through planetesimal formation to comparisons with Solar System populations. Once transitioned from a young protoplanetary disc to a debris disc, a star can expect to be rained with exocomets. While exocomets are predominantly detected to date at A-type stars, planetesimals plausibly exist across a range of stellar masses, based on exoplanet abundance, debris disc occurrence and white dwarf infall.

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arXiv2025

TOI-1438: A rare system with two short-period sub-Neptunes and a tentative long-period Jupiter-like planet orbiting a K0V star

We present the detection and characterisation of the TOI-1438 multi-planet system discovered by TESS. We collected a series of follow-up observations including high-spectral resolution observations with HARPS-N over a period of five years. Our modelling shows that the K0V star hosts two transiting sub-Neptunes with Rb = 3.04 +/- 0.19 RE, Rc = 2.75 +/- 0.14 RE, Mb = 9.4 +/- 1.8 ME, and Mc = 10.6 +/- 2.1 ME. The orbital periods of planets b and c are 5.1 and 9.4 days, respectively, corresponding to instellations of 145 +/- 10 and 65 +/- 4 FE. The bulk densities are 1.8 +/- 0.5 and 2.9 +/- 0.7 g cm-3, respectively, suggesting a volatile-rich interior composition. We computed a set of planet interior structure models. Planet b presents a high-metallicity envelope that can accommodate up to 2.5 % in H/He in mass, while planet c cannot have more than 0.2 % as H/He in mass. For any composition of the core considered (Fe-rock or ice-rock), both planets would require a volatile-rich envelope. In addition to the two planets, the radial velocity (RV) data clearly reveal a third signal, likely coming from a non-transiting planet, with an orbital period of 7.6 +1.6 -2.4 years and a radial velocity semi-amplitude of 35+3-5 m s-1. Our best fit model finds a minimum mass of 2.1 +/- 0.3 MJ and an eccentricity of 0.25+0.08-0.11. However, several RV activity indicators also show strong signals at similar periods, suggesting this signal might (partly) originate from stellar activity. More data over a longer period of time are needed to conclusively determine the nature of this signal. If it is confirmed as a triple-planet system, TOI-1438 would be one of the few detected systems to date characterised by an architecture with two small, short-period planets and one massive, long-period planet, where the inner and outer systems are separated by an orbital period ratio of the order of a few hundred.

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arXiv2024

TOI-1408: Discovery and Photodynamical Modeling of a Small Inner Companion to a Hot Jupiter Revealed by TTVs

We report the discovery and characterization of a small planet, TOI-1408 c, on a 2.2-day orbit located interior to a previously known hot Jupiter, TOI-1408 b ($P=4.42$ d, $M=1.86\pm0.02\,M_\mathrm{Jup}$, $R=2.4\pm0.5\,R_\mathrm{Jup}$) that exhibits grazing transits. The two planets are near 2:1 period commensurability, resulting in significant transit timing variations (TTVs) for both planets and transit duration variations (TDVs) for the inner planet. The TTV amplitude for TOI-1408 c is 15% of the planet's orbital period, marking the largest TTV amplitude relative to the orbital period measured to date. Photodynamical modeling of ground-based radial velocity (RV) observations and transit light curves obtained with the Transiting Exoplanet Survey Satellite (TESS) and ground-based facilities leads to an inner planet radius of $2.22\pm0.06\,R_\oplus$ and mass of $7.6\pm0.2\,M_\oplus$ that locates the planet into the Sub-Neptune regime. The proximity to the 2:1 period commensurability leads to the libration of the resonant argument of the inner planet. The RV measurements support the existence of a third body with an orbital period of several thousand days. This discovery places the system among the rare systems featuring a hot Jupiter accompanied by an inner low-mass planet.

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arXiv2024

The evolution and delivery of rocky extra-solar materials to white dwarfs

Understanding stellar evolution and its effect on planetary systems is crucial for correctly interpreting the chemical constraints of exo-planetary material that can be given to us by white dwarfs. This article will describe how asteroids, moons, and comets, as well as boulders, pebbles and dust, evolve into eventual targets for chemical spectroscopy, and how planets and companion stars play a vital role in reshaping system architectures for this purpose.

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arXiv2024

The formation of transiting circumplanetary debris discs from the disruption of satellite systems during planet-planet scattering

Several stars show deep transits consistent with discs of roughly 1 Solar radius seen at moderate inclinations, likely surrounding planets on eccentric orbits. We show that this configuration arises naturally as a result of planet-planet scattering when the planets possess satellite systems. Planet-planet scattering explains the orbital eccentricities of the discs' host bodies, while the close encounters during scattering lead to the exchange of satellites between planets and/or their destabilisation. This leads to collisions between satellites and their tidal disruption close to the planet. Both of these events lead to large quantities of debris being produced, which in time will settle into a disc such as those observed. The mass of debris required is comparable to a Ceres-sized satellite. Through N-body simulations of planets with clones of the Galilean satellite system undergoing scattering, we show that 90 percent of planets undergoing scattering will possess debris from satellite destruction. Extrapolating to smaller numbers of satellites suggests that tens of percent of such planets should still possess circumplanetary debris discs. The debris trails arising from these events are often tilted at tens of degrees to the planetary orbit, consistent with the inclinations of the observed discs. Disruption of satellite systems during scattering thus simultaneously explains the existence of debris, the tilt of the discs, and the eccentricity of the planets they orbit.

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arXiv2023

Company for the ultra-high density, ultra-short period sub-Earth GJ 367 b: discovery of two additional low-mass planets at 11.5 and 34 days

GJ 367 is a bright (V $\approx$ 10.2) M1 V star that has been recently found to host a transiting ultra-short period sub-Earth on a 7.7 hr orbit. With the aim of improving the planetary mass and radius and unveiling the inner architecture of the system, we performed an intensive radial velocity follow-up campaign with the HARPS spectrograph -- collecting 371 high-precision measurements over a baseline of nearly 3 years -- and combined our Doppler measurements with new TESS observations from sectors 35 and 36. We found that GJ 367 b has a mass of $M_\mathrm{b}$ = 0.633 $\pm$ 0.050 M$_{\oplus}$ and a radius of $R_\mathrm{b}$ = 0.699 $\pm$ 0.024 R$_{\oplus}$, corresponding to precisions of 8% and 3.4%, respectively. This implies a planetary bulk density of $ρ_\mathrm{b}$ = 10.2 $\pm$ 1.3 g cm$^{-3}$, i.e., 85% higher than Earth's density. We revealed the presence of two additional non transiting low-mass companions with orbital periods of $\sim$11.5 and 34 days and minimum masses of $M_\mathrm{c}\sin{i_\mathrm{c}}$ = 4.13 $\pm$ 0.36 M$_{\oplus}$ and $M_\mathrm{d}\sin{i_\mathrm{d}}$ = 6.03 $\pm$ 0.49 M$_{\oplus}$, respectively, which lie close to the 3:1 mean motion commensurability. GJ 367 b joins the small class of high-density planets, namely the class of super-Mercuries, being the densest ultra-short period small planet known to date. Thanks to our precise mass and radius estimates, we explored the potential internal composition and structure of GJ 367 b, and found that it is expected to have an iron core with a mass fraction of 0.91$^{+0.07}_{-0.23}$. How this iron core is formed and how such a high density is reached is still not clear, and we discuss the possible pathways of formation of such a small ultra-dense planet.

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arXiv2023

Making hot Jupiters in stellar clusters II: efficient formation in binary systems

Observations suggested that the occurrence rate of hot Jupiters (HJs) in open clusters is largely consistent with the field ($\sim1\%$) but in the binary-rich cluster M67, the rate is $\sim5\%$. How does the cluster environment boost HJ formation via the high-eccentricity tidal migration initiated by the extreme-amplitude von Zeipel-Lidov-Kozai (XZKL) mechanism forced by a companion star? Our analytical treatment shows that the cluster's collective gravitational potential alters the companion's orbit slowly, which may render the star-planet-companion configuration XZKL-favourable, a phenomenon only possible for very wide binaries. We have also performed direct Gyr $N$-body simulations of the star cluster evolution and XZKL of planets' orbit around member stars. We find that an initially-single star may acquire a companion star via stellar scattering and the companion may enable XZKL in the planets' orbit. Planets around an initially-binary star may also be XZKL-activated by the companion. In both scenarios, the companion's orbit has likely been significantly changed by star scattering and the cluster potential before XZKL occurs in the planets' orbits. Across different cluster models, 0.8\%-3\% of the planets orbiting initially-single stars have experienced XZKL while the fraction is 2\%-26\% for initially-binary stars. Notably, the ejection fraction is similar to or appreciably smaller than XZKL. Around a star that is binary at 1 Gyr, 13\%-32\% of its planets have undergone XZKL, and combined with single stars, the overall XZKL fraction is 3\%-21\%, most affected by the cluster binarity. If 10\% of the stars in M67 host a giant planet, our model predicts an HJ occurrence rate of $\sim1\%$. We suggest that HJ surveys target old, high-binarity, not-too-dense open clusters and prioritise wide binaries to maximise HJ yield.

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arXiv2022

Making hot Jupiters in stellar clusters: the importance of binary exchange

It has been suggested that the occurrence rate of hot Jupiters (HJs) in open clusters might reach several per cent, significantly higher than that of the field ($\sim$ a per cent). In a stellar cluster, when a planetary system scatters with a stellar binary, it may acquire a companion star which may excite large amplitude von Zeipel-Lidov-Kozai oscillations in the planet's orbital eccentricity, triggering high-eccentricity migration and the formation of an HJ. We quantify the efficiency of this mechanism by modelling the evolution of a gas giant around a solar mass star under the influence of successive scatterings with binary and single stars. We show that the chance that a planet $\in(1,10)$ au becomes an HJ in a Gyr in a cluster of stellar density $n_*=50$ pc$^{-3}$ and binary fraction $f_\mathrm{bin}=0.5$ is about 2\% and an additional 4\% are forced by the companion star into collision with or tidal disruption by the central host. An empirical fit shows that the total percentage of those outcomes asymptotically reaches an upper limit determined solely by $f_\mathrm{bin}$ (e.g., $10\%$ at $f_\mathrm{bin}=0.3$ and 18\% at $f_\mathrm{bin}=1$) on a timescale inversely proportional to $n_*$ ($\sim$ Gyr for $n_*\sim100$ pc$^{-3}$). The ratio of collisions to tidal disruptions is roughly a few, and depends on the tidal model. Therefore, if the giant planet occurrence rate is 10~\%, our mechanism implies an HJ occurrence rate of a few times 0.1~\% in a Gyr and can thus explain a substantial fraction of the observed rate.

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arXiv2022

A low-eccentricity migration pathway for a 13-h-period Earth analogue in a four-planet system

It is commonly accepted that exoplanets with orbital periods shorter than 1 day, also known as ultra-short period (USP) planets, formed further out within their natal protoplanetary disk, before migrating to their current-day orbits via dynamical interactions. One of the most accepted theories suggests a violent scenario involving high-eccentricity migration followed by tidal circularization. Here, we present the discovery of a four planet system orbiting the bright (V=10.5) K6 dwarf star TOI-500. The innermost planet is a transiting, Earth-sized USP planet with an orbital period of $\sim$ 13 hours, a mass of 1.42 $\pm$ 0.18 M$_{\oplus}$, a radius of $1.166^{0.061}_{-0.058}$ R$_{\oplus}$, and a mean density of 4.89$^{+1.03}_{-0.88}$ gcm$^{-3}$. Via Doppler spectroscopy, we discovered that the system hosts three outer planets on nearly circular orbits with periods of 6.6, 26.2, and 61.3d and minimum masses of 5.03 $\pm$ 0.41 M$_{\oplus}$, 33.12 $\pm$ 0.88 M$_{\oplus}$ and 15.05$^{+1.12}_{-1.11}$ M$_{\oplus}$, respectively. The presence of both a USP planet and a low-mass object on a 6.6-day orbit indicates that the architecture of this system can be explained via a scenario in which the planets started on low-eccentricity orbits, then moved inwards through a quasi-static secular migration. Our numerical simulations show that this migration channel can bring TOI-500 b to its current location in 2 Gyrs, starting from an initial orbit of 0.02au. TOI-500 is the first four planet system known to host a USP Earth analog whose current architecture can be explained via a non-violent migration scenario.

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arXiv2021

Hot Jupiters, cold kinematics: High phase space densities of host stars reflect an age bias

Context. The birth environments of planetary systems are thought to influence planet formation and orbital evolution, through external photoevaporation and stellar flybys. Recent work has claimed observational support for this, in the form of a correlation between the properties of planetary systems and the local Galactic phase space density of the host star. In particular, Hot Jupiters are found overwhelmingly around stars in regions of high phase space density, which may reflect a formation environment with high stellar density. Aims. We instead investigate whether the high phase space density may have a galactic kinematic origin: Hot Jupiter hosts may be biased towards being young and therefore kinematically cold, because tidal inspiral leads to the destruction of the planets on Gyr timescales, and the velocity dispersion of stars in the Galaxy increases on similar timescales. Methods. We use 6D positions and kinematics from Gaia for the Hot Jupiter hosts and their neighbours, and construct distributions of the phase space density. We investigate correlations between the stars' local phase space density and peculiar velocity. Results. We find a strong anticorrelation between the phase space density and the host star's peculiar velocity with respect to the Local Standard of Rest. Therefore, most stars in "high-density" regions are kinematically cold, which may be caused by the aforementioned bias towards detecting Hot Jupiters around young stars before the planets' tidal destruction. Conclusions. We do not find evidence in the data for Hot Jupiter hosts preferentially being in phase space overdensities compared to other stars of similar kinematics, nor therefore for their originating in birth environments of high stellar density.

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arXiv2021

Dynamical orbital evolution scenarios of the wide-orbit eccentric planet HR 5183b

The recently-discovered giant exoplanet HR5183b exists on a wide, highly-eccentric orbit ($a=18$\,au, $e=0.84$). Its host star possesses a common proper-motion companion which is likely on a bound orbit. In this paper, we explore scenarios for the excitation of the eccentricity of the planet in binary systems such as this, considering planet-planet scattering, Lidov-Kozai cycles from the binary acting on a single-planet system, or Lidov-Kozai cycles acting on a two-planet system that also undergoes scattering. Planet-planet scattering, in the absence of a binary companion, has a $2.8-7.2\%$ probability of pumping eccentricities to the observed values in our simulations, depending on the relative masses of the two planets. Lidov-Kozai cycles from the binary acting on an initially circular orbit can excite eccentricities to the observed value, but require very specific orbital configurations for the binary and overall there is a low probability of catching the orbit at the high observed high eccentricity ($0.6\%$). The best case is provided by planet-planet scattering in the presence of a binary companion: here, the scattering provides the surviving planet with an initial eccentricity boost that is subsequently further increased by Kozai cycles from the binary. We find a success rate of $14.5\%$ for currently observing $e\ge0.84$ in this set-up. The single-planet plus binary and two-planet plus binary cases are potentially distinguishable if the mutual inclination of the binary and the planet can be measured, as the latter permits a broader range of mutual inclinations. The combination of scattering and Lidov-Kozai forcing may also be at work in other wide-orbit eccentric giant planets, which have a high rate of stellar binary companions.

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arXiv2021

Metal pollution of the solar white dwarf by solar system small bodies

White dwarfs (WDs) often show metal lines in their spectra, indicating accretion of asteroidal material. Our Sun is to become a WD in several Gyr. Here, we examine how the solar WD accretes from the three major small body populations: the main belt asteroids (MBAs), Jovian trojan asteroids (JTAs), and trans-Neptunian objects (TNOs). Owing to the solar mass loss during the giant branch, 40\% of the JTAs are lost but the vast majority of MBAs and TNOs survive. During the WD phase, objects from all three populations are sporadically scattered onto the WD, implying ongoing accretion. For young cooling ages $\lesssim 100$ Myr, accretion of MBAs predominates; our predicted accretion rate $\sim10^6$ g/s falls short of observations by two orders of magnitude. On Gyr timescales, thanks to the consumption of the TNOs that kicks in $\gtrsim 100$ Myr, the rate oscillates around $10^6-10^7$ g/s until several Gyr and drops to $\sim10^5$ g/s at 10 Gyr. Our solar WD accretion rate from 1 Gyr and beyond agrees well with those of the extrasolar WDs. We show that for the solar WD, the accretion source region evolves in an inside-out pattern. Moreover, in a realistic small body population with individual sizes covering a wide range as WD pollutants, the accretion is dictated by the largest objects. As a consequence, the accretion rate is lower by an order of magnitude than that from a population of bodies of a uniform size and the same total mass and shows greater scatter.

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arXiv2021

Accretion of tidally disrupted asteroids onto white dwarfs: direct accretion versus disk processing

Atmospheric heavy elements have been observed in more than a quarter of white dwarfs (WDs) at different cooling ages, indicating ongoing accretion of asteroidal material, whilst only a few per cent of the WDs possess a dust disk, and all these WDs are accreting metals. Here, assuming that a rubble-pile asteroid is scattered inside a WD's Roche lobe by a planet, we study its tidal disruption and the long-term evolution of the resulting fragments. We find that after a few pericentric passages, the asteroid is shredded into its constituent particles, forming a flat, thin ring. On a timescale of Myr, tens of per cent of the particles are scattered onto the WD, and are therefore directly accreted without first passing through a circularised close-in disk. Fragment mutual collisions are most effective for coplanar fragments, and are thus only important in $10^3-10^4$ yr before the orbital coplanarity is broken by the planet. We show that for a rubble pile asteroid with a size frequency distribution of the component particles following that of the near earth objects, it has to be roughly at least 10 km in radius such that enough fragments are generated and $\ge10\%$ of its mass is lost to mutual collisions. At relative velocities of tens of km/s, such collisions grind down the tidal fragments into smaller and smaller dust grains. The WD radiation forces may shrink those grains' orbits, forming a dust disk. Tidal disruption of a monolithic asteroid creates large km-size fragments, and only parent bodies $\ge100$ km are able to generate enough fragments for mutual collisions to be significant. Hence, those large asteroids experience a disk phase before being accreted.

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arXiv2021

The entry geometry and velocity of planetary debris into the Roche sphere of a white dwarf

Our knowledge of white dwarf planetary systems predominately arises from the region within a few Solar radii of the white dwarfs, where minor planets break up, form rings and discs, and accrete onto the star. The entry location, angle and speed into this Roche sphere has rarely been explored but crucially determines the initial geometry of the debris, accretion rates onto the photosphere, and ultimately the composition of the minor planet. Here we evolve a total of over 10^5 asteroids with single-planet N-body simulations across the giant branch and white dwarf stellar evolution phases to quantify the geometry of asteroid injection into the white dwarf Roche sphere as a function of planetary mass and eccentricity. We find that lower planetary masses increase the extent of anisotropic injection and decrease the probability of head-on (normal to the Roche sphere) encounters. Our results suggest that one can use dynamical activity within the Roche sphere to make inferences about the hidden architectures of these planetary systems.

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arXiv2021

How the formation of Neptune shapes the Kuiper belt

Inward migration of giant planets is predicted by hydrodynamical simulations during the gas phase of the protoplanetary disc. The phenomenon is also invoked to explain resonant and near-resonant exoplanetary system structures. The early inward migration may also have affected our Solar System and sculpted its different minor planet reservoirs. In this study we explore how the early inward migration of the giant planets shapes the Kuiper Belt. We test different scenarios with only Neptune and Uranus and with all the four giant planets, including also some models with the subsequent outward planetesimal-driven migration of Neptune after the gas dispersal. We find objects populating mean motion resonances even when Neptune and Uranus do not migrate at all or only migrate inwards. When the planets are fixed, planetesimals stick only temporarily to the mean motion resonances, while inwards migration yields a new channel to populate the resonances without invoking convergent migration. In these cases, however, it is hard to populate mean motion resonances that do not cross the planetesimal disc (such as 2:1 and 5:2) and there is a lack of resonant KBOs that cross Neptune's orbit. These Neptune crossers are an unambiguous signature of the outward migration of Neptune. The starting position and the growth rate of Neptune matters for the contamination of the classical Kuiper belt region from neighbouring regions. The eccentricity and inclination space of the hot classicals and the scattered disc region become much more populated when all the giant planets are included. The 5:2 resonance with Neptune becomes increasingly populated with deeper inward migrations of Neptune. The overall inclination distribution, however, is still narrower than from observations, as is generally the case for Kuiper belt population models.

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