Hill Sphere Research Articles

Eccentric mergers in AGN Discs: Influence of the supermassive black-hole on three-body interactions

Abstract There are indications that stellar-origin black holes (BHs) are efficiently paired up in binary black holes (BBHs) in Active Galactic Nuclei (AGN) disc environments, which can undergo interactions with single BHs in the disc. Such binary-single interactions can potentially lead to an exceptionally high fraction of gravitational-wave mergers with measurable eccentricity in LIGO/Virgo/KAGRA. We here take the next important step in this line of studies by performing post-Newtonian N-body simulations between migrating BBHs and single BHs set in an AGN disc-like configuration, with a consistent inclusion of the central supermassive black hole (SMBH) in the equations of motion. With this setup, we study how the fraction of eccentric mergers varies in terms of the initial size of the BBH semi-major axis relative to the Hill sphere, as well as how it depends on the angle between the BBH and the incoming single BH. We find that the fraction of eccentric mergers is still relatively large, even when the interactions are notably influenced by the gravitational field of the nearby SMBH. However, the fraction as a function of the BBH semi-major axis does not follow a smooth functional shape, but instead shows strongly varying features that originate from the underlying phase-space structure. The phase-space further reveals that many of the eccentric mergers are formed through prompt scatterings. Finally, we present the first analytical solution to how the presence of an SMBH in terms of its Hill sphere affects the probability for forming eccentric BBH mergers through chaotic three-body interactions.

Gas dynamics around a Jupiter-mass planet

Context. Giant planets grow and acquire their gas envelope during the disk phase. At the time of the discovery of giant planets in their host disk, it is important to understand the interplay between the host disk and the envelope and circum-planetary disk properties of the planet. Aims. Our aim is to investigate the dynamical and physical structure of the gas in the vicinity of a Jupiter-mass planet and study how protoplanetary disk properties, such as disk mass and viscosity, determine the planetary system as well as the accretion rate inside the planet’s Hill sphere. Methods. We ran global 3D simulations with the grid-based code fargOCA, using a fully radiative equation of state and a dust-to-gas ratio of 0.01. We built a consistent disk structure starting from vertical thermal equilibrium obtained by including stellar irradiation. We then let a gap open with a sequence of phases, whereby we deepened the potential and increased the resolution in the planet’s neighbourhood. We explored three models. The nominal one features a disk with surface density, ∑, corresponding to the minimum mass solar nebula at the planet’s location (5.2 au), characterised by an α viscosity value of 4 10−3 at the planet’s location. The second model has a surface density that is ten times smaller than the nominal one and the same viscosity. In the third model, we also reduced the viscosity value by a factor of 10. Results. During gap formation, giant planets accrete gas inside the Hill sphere from the local reservoir. Gas is heated by compression and cools according to opacity, density, and temperature values. This process determine the thermal energy budget inside the Hill sphere. In the analysis of our disks, we find that the gas flowing into the Hill sphere is approximately scaled as the product ∑ν, as expected from viscous transport. The accretion rate of the planetary system (envelope plus circum-planetary disk) is instead scaled as √Σv, with its efficiency depending on the thermal energy budget inside the Hill sphere. Conclusions. Previous studies have shown that pressure-supported or rotationally supported structures are formed around giant planets, depending on the equation of state (EoS) or on the opacity; namely, on the dust content within the Hill sphere. In the case of a fully radiative EoS and a constant dust to gas ratio of 0.01, we find that low-mass and low-viscosity circum-stellar disks favour the formation of a rotationally supported circum-planetary disk. Gas accretion leading to the doubling time of the planetary system of > 105 years has only been found in the case of a low-viscosity disk.

A Thermodynamic Criterion for the Formation of Circumplanetary Disks

The formation of circumplanetary disks is central to our understanding of giant planet formation, influencing their growth rate during the post-runaway phase and observability while embedded in protoplanetary disks. We use three-dimensional global multifluid radiation hydrodynamics simulations with the FARGO3D code to define the thermodynamic conditions that enable circumplanetary disk formation around Jovian planets on wide orbits. Our simulations include stellar irradiation, viscous heating, static mesh refinement, and active calculation of opacity based on multifluid dust dynamics. We find a necessary condition for the formation of circumplanetary disks in terms of a mean cooling time: When the cooling time is at least 1 order of magnitude shorter than the orbital timescale, the specific angular momentum of the gas is nearly Keplerian at scales of one-third of the Hill radius. We show that the inclusion of multifluid dust dynamics favors rotational support because dust settling produces an anisotropic opacity distribution that favors rapid cooling. In all our models with radiation hydrodynamics, specific angular momentum decreases as time evolves, in agreement with the formation of an inner isentropic envelope due to compressional heating. The isentropic envelope can extend up to one-third of the Hill radius and shows negligible rotational support. Thus, our results imply that young gas giant planets may host spherical isentropic envelopes, rather than circumplanetary disks.

Cosymplectic Geometry, Reductions, and Energy-Momentum Methods with Applications

Classical energy-momentum methods study the existence and stability properties of solutions of t-dependent Hamilton equations on symplectic manifolds whose evolution is given by their Hamiltonian Lie symmetries. The points of such solutions are called relative equilibrium points. This work devises a new cosymplectic energy-momentum method providing a new and more general framework to study t-dependent Hamilton equations. In fact, cosymplectic geometry allows for using more types of distinguished Lie symmetries (given by Hamiltonian, gradient, or evolution vector fields), relative equilibrium points, and reduction methods, than symplectic techniques. To make our work more self-contained and to fill some gaps in the literature, a review of the cosymplectic formalism and the cosymplectic Marsden–Weinstein reduction is included. Known and new types of relative equilibrium points are characterised and studied. Our methods remove technical conditions used in previous energy-momentum methods, like the Ad∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Ad}^*$$\\end{document}-equivariance of momentum maps. Eigenfunctions of t-dependent Schrödinger equations are interpreted in terms of relative equilibrium points in cosymplectic manifolds. A new cosymplectic-to-symplectic reduction is developed and a new associated type of relative equilibrium points, the so-called gradient relative equilibrium points, are introduced and applied to study the Lagrange points and Hill spheres of a restricted circular three-body system by means of a not Hamiltonian Lie symmetry of the system.

Delivery of DART Impact Ejecta to Mars and Earth: Opportunity for Meteor Observations

NASA’s Double Asteroid Redirection Test (DART) and ESA’s Hera missions offer a unique opportunity to investigate the delivery of impact ejecta to other celestial bodies. We performed ejecta dynamical simulations using 3 million particles categorized into three size populations (10 cm, 0.5 cm, and 30 μm) and constrained by early postimpact LICIACube observations. The main simulation explored ejecta velocities ranging from 1 to 1000 m s−1, while a secondary simulation focused on faster ejecta with velocities from 1 to 2 km s−1. We identified DART ejecta orbits compatible with the delivery of meteor-producing particles to Mars and Earth. Our results indicate the possibility of ejecta reaching the Mars Hill sphere in 13 yr for launch velocities around 450 m s−1, which is within the observed range. Some ejecta particles launched at 770 m s−1 could reach Mars's vicinity in 7 yr. Faster ejecta resulted in a higher flux delivery toward Mars and particles impacting the Earth Hill sphere above 1.5 km s−1. The delivery process is slightly sensitive to the initial observed cone range and driven by synodic periods. The launch locations for material delivery to Mars were predominantly north of the DART impact site, while they displayed a southwestern tendency for the Earth–Moon system. Larger particles exhibit a marginally greater likelihood of reaching Mars, while smaller particles favor delivery to Earth–Moon, although this effect is insignificant. To support observational campaigns for DART-created meteors, we provide comprehensive information on the encounter characteristics (orbital elements and radiants) and quantify the orbital decoherence degree of the released meteoroids.

Using Ly α transits to constrain models of atmospheric escape

ABSTRACT Ly $\alpha$ transits provide an opportunity to test models of atmospheric escape directly. However, translating observations into constraints on the properties of the escaping atmosphere is challenging. The major reason for this is that the observable parts of the outflow often comes from material outside the planet’s Hill sphere, where the interaction between the planetary outflow and circumstellar environment is important. As a result, 3D models are required to match observations. Whilst 3D hydrodynamic simulations are able to match observational features qualitatively, they are too computationally expensive to perform a statistical retrieval of properties of the outflow. Here, we develop a model that determines the trajectory, ionization state, and 3D geometry of the outflow as a function of its properties and system parameters. We then couple this model to a ray tracing routine in order to produce synthetic transits. We demonstrate the validity of this approach, reproducing the trajectory of the outflows seen in 3D simulations. We illustrate the use of this model by performing a retrieval on the transit spectrum of GJ 436 b. The bound on planetary outflow velocity and mass-loss rates are consistent with a photoevaporative wind.

Irregular fixation – II. The orbits of irregular satellites

ABSTRACT Irregular satellites (ISs) are believed to have been captured during the Solar system’s dynamical history and provide clues for the Solar system’s formation and evolution. ISs occupy a large fraction of the Hill sphere of their host planet and their orbits are highly perturbed by the Sun. We use a novel formalism developed in Paper I to characterize their orbits in terms of an effective secular Hamiltonian (the Brown Hamiltonian) that accounts for their large orbital separations. We find that prograde satellites generally follow the Brown Hamiltonian, while retrograde satellites (which extend further) deviate more significantly. Nevertheless, the phase portrait is much better described by the Brown Hamiltonian for all satellites. We construct a semi-analytical criterion that predicts the librating orbit based on the effective energy due to the Brown Hamiltonian. We also check our results with highly accurate N-body integrations of satellite orbits, where initial conditions are loaded directly from the updated ephemeris from the NASA Horizons data base. Although the retrograde librating orbits occupy more area in the parameter space, the vast majority of librating ISs are prograde. Using our method, we find 13 librating satellites, 8 of them previously known to librate, and the rest shown to librate for the first time. Further observations of existing and new satellites could shed more light on the dynamical history of the Solar system and satellite formation and test our results.

Disc novae: thermodynamics of gas-assisted binary black hole formation in AGN discs

ABSTRACT We investigate the thermodynamics of close encounters between stellar mass black holes (BHs) in the gaseous discs of active galactic nuclei (AGNs), during which binary black holes (BBHs) may form. We consider a suite of 2D viscous hydrodynamical simulations within a shearing box prescription using the Eulerian grid code athena++. We study formation scenarios where the fluid is either an isothermal gas or an adiabatic mixture of gas and radiation in local thermal equilibrium. We include the effects of viscous and shock heating, as well as optically thick cooling. We co-evolve the embedded BHs with the gas, keeping track of the energetic dissipation and torquing of the BBH by gas and inertial forces. We find that compared to the isothermal case, the minidiscs formed around each BH are significantly hotter and more diffuse, though BBH formation is still efficient. We observe massive blast waves arising from collisions between the radiative minidiscs during both the initial close encounter and subsequent periapsis periods for successfully bound BBHs. These ‘disc novae’ have a profound effect, depleting the BBH Hill sphere of gas and injecting energy into the surrounding medium. In analysing the thermal emission from these events, we observe periodic peaks in local luminosity associated with close encounters/periapses, with emission peaking in the optical/near-infrared (IR). In the AGN outskirts, these outbursts can reach 4 per cent of the AGN luminosity in the IR band, with flares rising over 0.5–1 yr. Collisions in different disc regions, or when treated in 3D with magnetism, may produce more prominent flares.

Concurrent Accretion and Migration of Giant Planets in Their Natal Disks with Consistent Accretion Torque

Migration commonly occurs during the epoch of planet formation. For emerging gas giant planets, it proceeds concurrently with their growth through the accretion of gas from their natal protoplanetary disks. A similar migration process should also be applied to the stellar-mass black holes embedded in active galactic nucleus disks. In this work, we perform high-resolution 3D and 2D numerical hydrodynamical simulations to study the migration dynamics for accreting embedded objects over the disk viscous timescales in a self-consistent manner. We find that an accreting planet embedded in a predominantly viscous disk has a tendency to migrate outward, in contrast to the inward orbital decay of nonaccreting planets. 3D and 2D simulations find the consistent outward migration results for the accreting planets. Under this circumstance, the accreting planet’s outward migration is mainly due to the asymmetric spiral arms feeding from the global disk into the Hill radius. This is analogous to the unsaturated corotation torque although the imbalance is due to material accretion within the libration timescale rather than diffusion onto the inner disk. In a disk with a relatively small viscosity, the accreting planets clear deep gaps near their orbits. The tendency of inward migration is recovered, albeit with suppressed rates. By performing a parameter survey with a range of disks’ viscosity, we find that the transition from outward to inward migration occurs with the effective viscous efficiency factor α ∼ 0.003 for Jupiter-mass planets.

How planets grow by pebble accretion

The characterization of super-Earth- to Neptune-sized exoplanets relies heavily on our understanding of their formation and evolution. In this study, we link a model of planet formation by pebble accretion to the planets’ long-term observational properties by calculating the interior evolution, starting from the dissipation of the protoplanetary disk. We investigate the evolution of the interior structure in 5–20 M⊕ planets, accounting for silicate redistribution caused by convective mixing, rainout (condensation and settling), and mass loss. Specifically, we have followed the fate of the hot silicate vapor that remained in the planet’s envelope after planet formation as the planet cools. We find that disk dissipation is followed by a rapid contraction of the envelope from the Hill or Bondi radius to about one-tenth of that size within 10 Myr. Subsequent cooling leads to substantial growth of the planetary core through silicate rainout accompanied by inflated radii, in comparison to the standard models of planets that formed with core-envelope structure. We examined the dependence of rainout on the planet’s envelope mass, on the distance from its host star, on its silicate mass, and on the atmospheric opacity. We find that the population of planets that formed with polluted envelopes can be roughly divided into three groups based on the mass of their gas envelopes: bare rocky cores that have shed their envelopes, super-Earth planets with a core-envelope structure, and Neptune-like planets with diluted cores that undergo gradual rainout. For polluted planets that formed with envelope masses below 0.4 M⊕, we anticipate that the inflation of the planet’s radius caused by rainout will enhance the mass loss by a factor of 2–8 compared to planets with unpolluted envelopes. Our model bridges the gap between the predicted composition gradients in massive planets and the core-envelope structure in smaller planets.

The β Pictoris b Hill sphere transit campaign

The β Pictoris system is the closest known stellar system with directly detected gas giant planets, an edge-on circumstellar disc, and evidence of falling sublimating bodies and transiting exocomets. The inner planet, β Pictoris c, has also been indirectly detected with radial velocity (RV) measurements. The star is a known δ Scuti pulsator, and the long-term stability of these pulsations opens up the possibility of indirectly detecting the gas giant planets through time delays of the pulsations due to a varying light travel time. We search for phase shifts in the δ Scuti pulsations consistent with the known planets β Pictoris b and c and carry out an analysis of the stellar pulsations of β Pictoris over a multi-year timescale. We used photometric data collected by the BRITE-Constellation, bRing, ASTEP, and TESS to derive a list of the strongest and most significant δ Scuti pulsations. We carried out an analysis with the open-source python package maelstrom to study the stability of the pulsation modes of β Pictoris in order to determine the long-term trends in the observed pulsations. We did not detect the expected signal for β Pictoris b or β Pictoris c. The expected time delay is 6 s for β Pictoris c and 24 s for β Pictoris b. With simulations, we determined that the photometric noise in all the combined data sets cannot reach the sensitivity needed to detect the expected timing drifts. An analysis of the pulsational modes of β Pictoris using maelstrom showed that the modes themselves drift on the timescale of a year, fundamentally limiting our ability to detect exoplanets around β Pictoris via pulsation timing.

Re-examination of the transportation abilities of the 5:2 MMR with Jupiter

Resonances in the main asteroid belt play a significant role in the dynamical evolution of small bodies. They are capable of driving objects into the near-Earth object (NEO) region as well. This work re-examines the transportation abilities of the 5:2 mean motion resonance (MMR) with Jupiter. We focus on a greater portion of the resonance than the previous study that used a similar method. We are also interested in an elimination course along $q 0.26\,$au that was discovered previously. Moreover, we search for the orbits of potentially hazardous asteroids and for orbits that correspond to recent L chondrite meteorites. Firstly, short-term fast Lyapunov indicator maps of the 5:2 MMR were computed in order to distinguish between stable and unstable orbits. Then over $10\,000$ unstable particles were selected and integrated for a longer period of time, up to $10\,$Myr, to reveal the transportation abilities of the resonance. During our simulation, $99.45<!PCT!>$ of test particles became NEOs, $9.43<!PCT!>$ reached the orbit with a semi-major axis, $a<1\,$au, and over $27<!PCT!>$ of particles migrated to low perihelion distances, $q<0.005\,$au. In addition, $92.8<!PCT!>$ of the particles entered the Hill sphere of the Earth and over $97<!PCT!>$ reached an orbit at which we would classify them as potentially hazardous if they were sufficiently large. However, our simulation did not confirm ejections along $q 0.26\,$au. Our results suggest that there is some kind of discrepancy between using the MERCURIUS integrator (REBOUND package) and the ORBIT9 integrator (OrbFit package). This subject is worth additional examination.

Formation and structure of circumplanetary disks and envelopes during the late stages of giant planet formation

Giant planets are expected to form within circumstellar disks, which shape their formation history and the local environment. Here, we consider the formation and structure of circumplanetary disks that arise during the late stages of giant planet formation. During this phase, when most of the final mass is accumulated, incoming material enters the Hill sphere and falls toward the planet. In the absence of torques, the falling parcels of gas conserve their specific angular momentum and collect into a circumplanetary disk. Generalizing previous work, we consider a range of possible geometries for the flow entering the sphere of influence of the planet. Specifically, we consider five geometric patterns for the inward flow, ranging from concentration toward the rotational poles of the system to isotropic flow to concentration along the equatorial plane. For each case, we derive analytic descriptions for the density field of the infall region, the disk surface density in the absence of viscosity, and steady-state solutions for viscous disks. These results, in turn, specify the luminosity contributions of the planet, the circumplanetary disk, and the envelope. These power sources, in conjunction with the surrounding material, collectively determine the observational appearance of the forming planet. We conclude with an approximate determination of these radiative signatures.

Global N-body simulation of gap edge structures created by perturbations from a small satellite embedded in Saturn’s rings

Small satellites embedded in Saturn’s ring – Daphnis and Pan – open a clear gap. Observations by the Voyager and Cassini spacecrafts have revealed various striking features of the gap structure, such as the density waves, sharp edge, and vertical wall structure. In order to explain these features in a single simulation, we perform a high-resolution (N∼106−107) global full N-body simulation of gap formation by an embedded satellite considering gravitational interactions and inelastic collisions among all ring particles and the satellite, while these features have been mostly investigated separately with different theoretical approaches: the streamline models, 1D diffusion models, and local N-body simulation. As a first attempt of a series of papers, we here focus on the gap formation by separating satellite migration with fixing the satellite orbit in a Keplerian circular orbit. We reveal how the striking gap features – the density waves, sharp edge, and vertical wall structure – are simultaneously formed by an interplay of the satellite-ring and ring particle–particle interactions. In particular, we propose a new mechanism to quantitatively explain the creation of the vertical wall structure at the gap edge. Inelastic collisions between ring particles damp their eccentricity excited by the satellite’s perturbations to enhance the surface density at the gap edge, making its sharp edges more pronounced. We find the eccentricity damping process inevitably raises the vertical wall structures the most effectively in the second epicycle waves. Particle–particle collisions generally convert their lateral epicyclic motion into vertical motion. Because the excited epicyclic motion is the greatest near the ring edge and the epicycle motions are aligned in the first waves, the conversion is the most efficient in the gap edge of the second waves and the wall height is scaled by the satellite Hill radius, which are consistent with the observations.

Dynamical instability in multi-orbiter systems with gas friction

ABSTRACT Closely packed multiplanet systems are known to experience dynamical instability if the spacings between the planets are too small. Such instability can be tempered by the frictional forces acting on the planets from gaseous discs. A similar situation applies to stellar-mass black holes embedded in active galactic nuclei discs around supermassive black holes. We use N-body integrations to evaluate how the frictional damping of orbital eccentricity affects the growth of dynamical instability for a wide range of K (the difference in the planetary semimajor axes in units of the mutual Hill radius) and (unequal) planet masses. We find that, in general, the stable region (large K) and unstable region (small K) are separated by a “grey zone”, where the (in)stability is not guaranteed. We report the numerical values of the critical spacing for stability Kcrit and the “grey zone” range in different systems, and provide fitting formulae for arbitrary frictional forcing strength. We show that the stability of a system depends on the damping time-scale τ relative to the zero-friction instability growth time-scale tinst: two-planet systems are stable if tinst ≳ τ; three-planet systems require tinst ≳ 10τ−100τ. When K is sufficiently small, tinst can be less than the synodic period between the planets, which makes frictional stabilization unlikely to occur. As K increases, tinst tends to grow exponentially, but can also fluctuate by a few orders of magnitude. We also devise a linear map to analyse the dynamical instability of the “planet + test mass” system, and find qualitative agreement with N-body simulations.

Geysers' Dust Dynamics Inside the Hill Sphere of Enceladus

AbstractIn this work, we study the trajectories of the dust particles ejected in the geysers of the moon Enceladus, inside its Hill sphere, where its gravitational influence dominates. Our goal is to determine the properties of the particles that escape from the moon or of those that eventually impact on its surface. The analysis considers the gravity of Enceladus, the Lorentz force due to Saturn's magnetic field and the gas drag from the geysers. The local magnetic field around Enceladus is derived from an interpolation of Cassini data from six flybys. In general, we obtain that grain sizes and launching speeds define whether grains escape, though the launching angle may also play an important role in some cases.

Chaotic Capture of a Retrograde Moon by Venus and the Reversal of Its Spin

Planets are surrounded by fractal surfaces (traditionally called Hill spheres), separating the inner zones of long-term stable orbital motion of their satellites from the outer space where the gravitational pull from the Sun takes over. Through this surface, external minor bodies in trajectories loosely co-orbital to a planet can be stochastically captured by the planet without any assistance from external perturbative forces, and can become moons chaotically orbiting the planet for extended periods of time. Using state-of-the-art orbital integrators, we simulate such capture events for Venus, resulting in long-term attachment phases by reversing the forward integration of a moon initially attached to the planet and escaping it after an extended period of time. Chaotic capture of a retrograde moon from a prograde heliocentric orbit appears to be more probable because the Hill sphere is almost four times larger in area for a retrograde orbit than for a prograde orbit. Simulated capture trajectories include cases with attachment phases up to 860,000 years for prograde moons and up to 370,000 years for retrograde moons. Although the probability of a long-term chaotic capture from a single encounter is generally low, the high density of co-orbital bodies in the primordial protoplanetary disk makes this outcome possible, if not probable. The early Venus was surrounded by a dusty gaseous disk of its own, which, coupled with the tidal dissipation of the kinetic energy in the moon and the planet, could shrink the initial orbit and stabilize the captured body within the Hill surface. The tidal torque from the moon, for which we use the historical name Neith, gradually brakes the prograde rotation of Venus, and then reverses it, while the orbit continues to decay. Neith eventually reaches the Roche radius and disintegrates, probably depositing most of its material on Venus’ surface. Our calculations show that surface density values of about 0.06 kg m−2 for the debris disk may be sufficient to stabilize the initial chaotic orbit of Neith and to bring it down within several radii of Venus, where tidal dissipation becomes more efficient.

Using the Scorpion as an Interplanetary Transportation System

The Scorpion is a nuclear powered transportation system that can extend the human infrastructure beyond low Earth orbit. Past work has shown its capability in high Earth orbits out to the Hill sphere and Lunar surface operations. However, the Scorpion was also intended to conduct interplanetary missions. The functionality required for interplanetary missions is the same, the only major differences are the mission length and the mission velocities. This means the Scorpion will need to carry additional supplies and also requires a boost to leave low Earth orbit. The boost can be provided in one of three ways; using another Scorpion as a boost stage, using a specialist cargo derivative of the Scorpion (called Taurus) or departing from a Spaceport in the Lunar environment used to support the infrastructure on the Moon surface. It is shown the Scorpion can reach Venus, Mars and many asteroids, but has problems reaching the asteroid main belt and gas giants beyond.

Tidal dissipation in satellites prevents Hill sphere escape

ABSTRACT The transit method is a promising means of detecting exomoons, but few candidates have been identified. For planets close to their stars, the dynamical interaction between a satellite’s orbit and the star must be important in their evolution. Satellites beyond synchronous orbit spiral out due to the tide raised on their planet, and it has been assumed that they would likely escape the Hill sphere. Here we follow the evolution with a three-body code that accounts for tidal dissipation within both the planet and the satellite. We show that tidal dissipation in satellites often keeps them bound to their planet, making exomoons more observable than previously thought. The probability of escape depends on the ratio of tidal quality factors of the planet and satellite; when this ratio exceeds 0.5, escape is usually avoided. Instead, the satellite moves to an equilibrium in which the spin angular momentum of the planet is not transferred into the orbit of the satellite, but is transferred into the orbit of the planet itself. While the planet continues spinning faster than the satellite orbits, the satellite maintains a semi-major axis of approximately 0.41 Hill radii. These states are accompanied with modest satellite eccentricity near 0.1 and are found to be stable over long time-scales.

Motion of Planetesimals in the Hill Sphere of the Star Proxima Centauri

The motion of planetesimals initially located in the feeding zone of the planet Proxima Centauri c, at distances of 500 AU from the star to the star’s Hill sphere radius of 1200 AU was considered. In the analyzed non-gaseous model, the primary ejection of planetesimals from most of the feeding zone of an almost formed planet c to distances greater than 500 AU from the star occurred during the first 10 million years. Only for planetesimals originally located at the edges of the planet’s feeding zone, the fraction of planetesimals that first reached 500 AU over the time greater than 10 million years was more than half. Some planetesimals could reach the outer part of the star’s Hill sphere over hundreds of millions of years. Approximately 90% of the planetesimals that first reached 500 AU from Proxima Centauri first reached 1200 AU from the star in less than 1 million years, given the current mass of the planet c. No more than 2% of planetesimals with aphelion orbital distances between 500 and 1200 AU followed such orbits for more than 10 million years (but less than a few tens of millions of years). With a planet mass equal to half the mass of the planet c, approximately 70–80% of planetesimals increased their maximum distances from the star from 500 to 1200 AU in less than 1 million years. For planetesimals that first reached 500 AU from the star under the current mass of the planet c, the fraction of planetesimals with orbital eccentricities greater than 1 was 0.05 and 0.1 for the initial eccentricities of their orbits eo = 0.02 and eo = 0.15, respectively. Among the planetesimals that first reached 1200 AU from the star, this fraction was approximately 0.3 for both eo values. The minimum eccentricity values for planetesimals that have reached 500 and 1200 AU from the star were 0.992 and 0.995, respectively. In the considered model, the disk of planetesimals in the outer part of the star’s Hill sphere was rather flat. Inclinations i of the orbits for more than 80% of the planetesimals that first reached 500 or 1200 AU from the star did not exceed 10°. With the current mass of the planet c, the percentage of such planetesimals with i 20° did not exceed 1% in all calculation variants. The results may be of interest for understanding the motion of bodies in other exoplanetary systems, especially those with a single dominant planet. They can be used to provide the initial data for models of the evolution of the disk of bodies in the outer part of Proxima Centauri’s Hill sphere, which take into account gravitational interactions and collisions between bodies, as well as the influence of other stars. The strongly inclined orbits of bodies in the outer part of Proxima Centauri’s Hill sphere can primarily result from bodies that entered the Hill sphere from outside. The radius of Proxima Centauri’s Hill sphere is an order of magnitude smaller than the radius of the outer boundary of the Hills cloud in the Solar System and two orders of magnitude smaller than the radius of the Sun’s Hill sphere. Therefore, it is difficult to expect the existence of a similarly massive cloud around this star as the Oort cloud around the Sun.