Study of the Migration of Earth-Like Planets in Planetesimal Disks and the Formation of Debris Disks

In this paper, we further develop the model for the migration of planets introduced in Del Popolo et al. (2001). We first model the protoplanetary nebula as a time-dependent accretion disc and find self-similar solutions to the equations of the accretion disc that give to us explicit formulas for the spatial structure and the temporal evolution of the nebula. These equations are then used to obtain the migration rate of the planet in the planetesimal disc and to study how the migration rate depends on the disc mass, on its time evolution and on some values of the dimensionless viscosity parameter alpha. We find that planets that are embedded in planetesimal discs, having total mass of 10^{-4}-0.1 M_{\odot}, can migrate inward a large distance for low values of alpha (e.g., alpha \simeq 10^{-3}-10^{-2}) and/or large disc mass and can survive only if the inner disc is truncated or because of tidal interaction with the star. Orbits with larger $a$ are obtained for smaller value of the disc mass and/or for larger values of alpha. This model may explain several orbital features of the recently discovered giant planets orbiting nearby stars.

The existence of the Oort comet cloud, the Kuiper belt, and plausible inefficiencies in planetary core formation all suggest that there was once a residual planetesimal disk of mass ~10–100 M⊕ in the vicinity of the giant planets following their formation. Since removal of this disk requires an exchange of orbital energy and angular momentum with the planets, significant planetary migration can ensue. The planet migration phenomenon is examined numerically by evolving the orbits of the giant planets while they are embedded in a planetesimal disk having a mass of MD = 10–200 M⊕. We find that Saturn, Uranus, and Neptune evolve radially outward as they scatter the planetesimals, while Jupiter's orbit shrinks as it ejects mass. Higher mass disks result in more rapid and extensive planet migration. If orbital expansion and resonance trapping by Neptune are invoked to explain the eccentricities of Pluto and its cohort of Kuiper belt objects at Neptune's 3:2 mean motion resonance, then our simulations suggest that a disk mass of order MD ~ 50 M⊕ is required to expand Neptune's orbit by Δa ~ 7 AU, in order to pump up Plutino eccentricities to e ~ 0.3. Such planet migration implies that the solar system was more compact in the past, with the initial Jupiter-Neptune separation having been smaller by about 30%. We discuss the fate of the remnants of the primordial planetesimal disk. We point out that most of the planetesimal disk beyond Neptune's 2:1 resonance should reside in nearly circular, low-inclination orbits, unless there are (or were) additional, unseen, distant perturbers. The planetesimal disk is also the source of the Oort cloud of comets. Using the results of our simulations together with a simple treatment of Oort cloud dynamics, we estimate that ~12 M⊕ of disk material was initially deposited in the Oort cloud, of which ~4 M⊕ will persist over the age of the solar system. The majority of these comets originated from the Saturn-Neptune region of the solar nebula.

Planets orbiting a planetesimal circumstellar disc can migrate inward from their initial positions because of dynamical friction between planets and planetesimals. The migration rate depends on the disc mass and on its time evolution. Planets that are embedded in long-lived planetesimal discs, having total mass of $10^{-4}-0.01 M_{\odot}$, can migrate inward a large distance and can survive only if the inner disc is truncated or because of tidal interaction with the star. In this case the semi-major axis, a, of the planetary orbit is less than 0.1 AU. Orbits with larger $a$ are obtained for smaller value of the disc mass or for a rapid evolution (depletion) of the disc. This model may explain several of the orbital features of the giant planets that were discovered in last years orbiting nearby stars as well as the metallicity enhancement found in several stars associated with short-period planets.

In this paper, we further develop the model for the migration of planets introduced by Del Popolo, Gambera & Ercan and extended to time-dependent planetesimal accretion discs by Del Popolo & Eksi. More precisely, the assumption of Del Popolo & Eksi that the surface density in planetesimals is proportional to that of the gas was released. Indeed, the evolution of the radial distribution of solids is governed by many processes: gas-solid coupling, co- agulation, sedimentation, evaporation/condensation, so that the distribution of planetesimals emerging from a turbulent disc does not necessarily reflect that of the gas. In order to describe this evolution we use a method developed by Stepinski & Valageas, which, using a series of simplifying assumptions, is able to simultaneously follow the evolution of gas and solid particles for up to 10 7 yr. This model is based on the premise that the transformation of solids from dust to planetesimals occurs through hierarchical coagulation. Then, the distribution of planetesimals obtained after 10 7 yr is used to study the migration rate of a giant planet through the migration model introduced by Del Popolo, Gambera & Ercan. This allows us to investigate the dependence of the migration rate on the disc mass, on its time evolution and on the value of the dimensionless viscosity parameter α. We find that in the case of discs having a total mass of 10 −3 -10 −1 M� , and 10 −4 10 −3 and only in the case where α< 10 −3 do the planets move to a minimum value of orbital radius of �2 au. Moreover, the observed distribution of planets in the period range 0-20 d can be easily obtained from our model. Therefore, dynamical friction between planets and the plan- etesimal disc provides a good mechanism to explain the properties of observed extrasolar giant planets.

Aims. We investigate the process of Neptune trojan capture and permanence in resonance up to the present time based on a planetary instability migration model.Methods. We do a numerical simulation of the migration of the giant planets in a planetesimal disk. Several planetesimals became trapped in coorbital resonance with Neptune, but no trojan survived to the end of the integration at 4.5 Gy. We increased the statistics by running synthetic integrations with cloned particles from the original integration and keeping the same migration rates of the planets.Results. For the synthetic integrations, Neptune trojans survived to the end of the simulations. The total mass that corresponds to these surviving trojans is about 1.6 × 10-4 Earth mass and the distributions of eccentricities, inclinations, and libration amplitudes are respectively 0.007−0.173, 4.9°−32.9°, and 6.9°−64.3°. In a specific run where Neptune to Uranus mean motion ratio reached 1.963 and decreased to its present value (1.961), many more trojans escaped the coorbital resonance with Neptune and in the end there was an equivalent mass of 5 × 10-5 Earth mass of Neptune trojans.Conclusions. The simulations yielded Neptune trojans that match the orbital distribution of real Neptune trojans quite well. Since planetary migration in an instability model shows the possibility that in the past Neptune was a little farther from the Sun than it is today, it is reasonable to consider this possibility to explain the relatively low mass of Neptune trojans.

The HR 8799 system hosts four giant planets between a warm and cold debris disk and has an extended dusty tail beyond. It serves as an ideal laboratory for studying planetary formation and evolution. The debris disks have been observed at various wavelengths, and the planetary properties are well constrained. Nonetheless, there are still open questions regarding the role of the planets in shaping the debris disks. We investigated the system evolution with the aim of understanding how planetary migration shaped its architecture in terms of planets and the disk. We performed N-body simulations to model the HR 8799 system. We examined the orbital evolution of the known four super-Jupiter planets through the course of simple, imposed migration in a gaseous disk as they perturb an external massless planetesimal disk. We also explored the impact of introducing a fifth planet on the dynamical and morphological aspects of the disk. The planets migrate outward as a result of their imposed interactions with the gaseous disk while maintaining their resonant configuration. This outward migration excites the planetesimal disk and produces a transient scattered population. While a four-planet system partially reproduces the observed cavity between the star and the cold debris disk, the inclusion of a fifth low-mass planet appears to be crucial for better reproducing key morphological aspects of the cold debris disk. This model provides a novel explanation for the architecture of HR 8799. Outward planetary migration, combined with mean-motion resonant interactions and a fifth low-mass planet, can effectively replicate the observed planetary architecture and the characteristics of the cold debris disk. Our findings underscore the potential important role of planetary migration in shaping debris disks.

Planetary migration in a planetesimal disk: why did Neptune stop at 30 AU?

Several hundred stars older than 10 million years have been observed to have infrared excesses. These observations are explained by dust grains formed by the collisional fragmentation of hidden planetesimals. Such dusty planetesimal discs are known as debris discs. In a dynamically cold planetesimal disc, collisional coagulation of planetesimals produces planetary embryos which then stir the surrounding leftover planetesimals. Thus, the collisional fragmentation of planetesimals that results from planet formation forms a debris disc. We aim to determine the properties of the underlying planetesimals in debris discs by numerically modelling the coagulation and fragmentation of planetesimal populations. The brightness and temporal evolution of debris discs depend on the radial distribution of planetesimal discs, the location of their inner and outer edges, their total mass, and the size of planetesimals in the disc. We find that a radially narrow planetesimal disc is most likely to result in a debris disc that can explain the trend of observed infrared excesses of debris discs around G-type stars, for which planet formation occurs only before 100 million years. Early debris disc formation is induced by planet formation, while the later evolution is explained by the collisional decay of leftover planetesimals around planets that have already formed. Planetesimal discs with underlying planetesimals of radii $\sim 100\,$km at $\approx 30$ AU most readily explain the Spitzer Space Telescope 24 and 70$ \mu$m fluxes from debris discs around G-type stars.

Trans-Neptunian objects (TNOs) are remnants of a collisionally and dynamically evolved planetesimal disk in the outer solar system. This complex structure, known as the trans-Neptunian belt (or Edgeworth-Kuiper belt), can reveal important clues about disk properties, planet formation, and other evolutionary processes. In contrast to the predictions of accretion theory, TNOs exhibit surprisingly large eccentricities, e, and inclinations, i, which can be grouped into distinct dynamical classes. Several models have addressed the origin and orbital evolution of TNOs, but none have reproduced detailed observations, e.g., all dynamical classes and peculiar objects, or provided insightful predictions. Based on extensive simulations of planetesimal disks with the presence of the four giant planets and massive planetesimals, we propose that the orbital history of an outer planet with tenths of Earth's mass can explain the trans-Neptunian belt orbital structure. This massive body was likely scattered by one of the giant planets, which then stirred the primordial planetesimal disk to the levels observed at 40-50 AU and truncated it at about 48 AU before planet migration. The outer planet later acquired an inclined stable orbit (>100 AU; 20-40 deg) because of a resonant interaction with Neptune (an r:1 or r:2 resonance possibly coupled with the Kozai mechanism), guaranteeing the stability of the trans-Neptunian belt. Our model consistently reproduces the main features of each dynamical class with unprecedented detail; it also satisfies other constraints such as the current small total mass of the trans-Neptunian belt and Neptune's current orbit at 30.1 AU. We also provide observationally testable predictions.

Planetary migration is a crucial stage in the early Solar System, explaining many observational phenomena and providing constraints on details related to the Solar System’s origins. This paper aims to investigate the acceleration during planetary migration in detail using numerical simulations, delving deeper into the early Solar System’s preserved information. We confirm that planetary migration is a positive feedback process: the faster the migration, the more efficient the consumption of planetesimals; once the migration slows down, Neptune clears the surrounding space, making further migration more difficult to sustain. Quantitatively, a tenfold increase in the migration rate corresponds to a reduction of approximately 30% in the mass of planetesimals consumed to increase per unit of angular momentum of Neptune. We also find that Neptune’s final position is correlated with the initial surface density of planetesimals at that location, suggesting that the disk density at 30 au was approximately 0.009 M⊕/au2 in the early Solar System. Furthermore, we identify two mechanisms that can accelerate planetary migration. The first is mean motion resonance between Uranus and Neptune: migration acceleration will be triggered whenever these two giant planets cross their major mean motion resonance. The second mechanism is the ring structure within the planetesimal disk, as the higher planetesimal density in this region can provide the material support necessary for migration acceleration. Our research indicates that Neptune in the current Solar System occupies a relatively delicate position. In case Neptune crossed the 1:2 resonance with Uranus, it could have migrated to a much more distant location. Our results demonstrate that giant planet instability is fundamentally required; otherwise, reconstructing the migration histories of Uranus and Neptune would yield physically implausible orbital configurations. Furthermore, even without introducing the giant planet instability, under the influence of the positive feedback mechanism, the evolution of the Solar System to its current configuration might still be a stochastic outcome rather than an inevitable consequence.

Context. Prevailing N-body planet formation models typically start with lunar-mass embryos and show a general trend of rapid migration of massive planetary cores to the inner Solar System in the absence of a migration trap. This setup cannot capture the evolution from a planetesimal to embryo, which is crucial to the final architecture of the system. Aims. We aim to model planet formation with planet migration starting with planetesimals of ~10−6−10−4 M⊕ and reproduce the giant planets of the Solar System. Methods. We simulated a population of 1000-5000 planetesimals in a smooth protoplanetary disc, which was evolved under the effects of their mutual gravity, pebble accretion, gas accretion, and planet migration, employing the parallelized N-body code SyMBAp. Results. We find that the dynamical interactions among growing planetesimals are vigorous and can halt pebble accretion for excited bodies. While a set of results without planet migration produces one to two gas giants and one to two ice giants beyond 6 au, massive planetary cores readily move to the inner Solar System once planet migration is in effect. Conclusions. Dynamical heating is important in a planetesimal disc and the reduced pebble encounter time should be considered in similar models. Planet migration remains a challenge to form cold giant planets in a smooth protoplanetary disc, which suggests an alternative mechanism is required to stop them at wide orbits.

We present a model in which planetesimal disks are built from the combination of planetesimal formation and accretion of radially drifting pebbles onto existing planetesimals. In this model, the rate of accretion of pebbles onto planetesimals quickly outpaces the rate of direct planetesimal formation in the inner disk. This allows for the formation of a high mass inner disk without the need for enhanced planetesimal formation or a massive protoplanetary disk. Our proposed mechanism for planetesimal disk growth does not require any special conditions to operate. Consequently, we expect that high mass planetesimal disks form naturally in nearly all systems. The extent of this growth is controlled by the total mass in pebbles that drifts through the inner disk. Anything that reduces the rate or duration of pebble delivery will correspondingly reduce the final mass of the planetesimal disk. Therefore, we expect that low mass stars (with less massive protoplanetary disks), low metallicity stars and stars with giant planets should all grow less massive planetesimal disks. The evolution of planetesimal disks into planetary systems remains a mystery. However, we argue that late stage planet formation models should begin with a massive disk. This reinforces the idea that massive and compact planetary systems could form in situ but does not exclude the possibility that significant migration occurs post-planet formation.

HR 8799 is a four planet system that also hosts a debris disk. By numerically integrating both planets and a planetesimal disk, we find interactions between an exterior planetesimal disk and the planets can influence the lifetime of the system. We first consider resonant planetary configurations that remained stable for at least 7 Myrs sans debris disk. An exterior debris disk with only 1 per cent the mass of the outermost planet (approximately a Neptune mass) was sufficiently large enough to pull the system out of resonance after 2 to 6 Myrs. Secondly, we consider configurations which are unstable in less than a few hundred thousand years. We find that these can be stabilized by a debris disk with a mass of more than 10 per cent that of the outermost planet. Our two sets of simulations suggest that estimates of the long term stability of a planetary system should take into account the role of the debris disk.

Context. The observed clumpy structures in debris disks are commonly interpreted as particles trapped in mean-motion resonances with an unseen exo-planet. Populating the resonances requires a migrating process of either the particles (spiraling i nward due to drag forces) or the planet (moving outward). Because the drag time-scale in resolved debris disks is generally long compared to the collisional time-scale, the planet migration scenario mig ht be more likely, but this model has so far only been investigated for planets on circular orbits. Aims. We present a thorough study of the impact of a migrating planet on a planetesimal disk, by exploring a broad range of masses and eccentricities for the planet. We discuss the sensitivi ty of the structures generated in debris disks to the basic pl anet parameters. Methods. We perform many N-body numerical simulations, using the symplectic integrator SWIFT, taking into account the gravitational influence of the star and the planet on massless test pa rticles. A constant migration rate is assumed for the planet . Results. The effect of planetary migration on the trapping of particles in mean motion resonances is found to be very sensitive to the initial eccentricity of the planet and of the planetesim als. A planetary eccentricity as low as 0.05 is enough to smear out all the resonant structures, except for the most massive planets. The planetesimals also initially have to be on orbits with a mean eccentricity of less than than 0.1 in order to keep the resonant clumps visible. Conclusions. This numerical work extends previous analytical studies and provides a collection of disk images that may help in interpreting the observations of structures in debris disk s. Overall, it shows that stringent conditions must be fulfil led to obtain observable resonant structures in debris disks. Theoretical models of the origin of planetary migration will therefore have to explain how planetary systems remain in a suitable configuration to repr oduce the observed structures.

The newly formed giant planets may have migrated and crossed a number of mutual mean motion resonances (MMRs) when smaller objects (embryos) were accreting to form the terrestrial planets. We investigated the effects of the planetesimal-driven migration of Jupiter and Saturn, and the influence of their mutual 1:2 MMR crossing on terrestrial planet formation for the first time, by performing N-body simulations. These simulations considered distinct timescales of MMR crossing and planet migration. In total, 68 high-resolution simulation runs using 2000 disk planetesimals were performed, which was a significant improvement on previously published results. Even when the effects of the 1:2 MMR crossing and planet migration were included in the system, Venus and Earth analogs (considering both orbits and masses) successfully formed in several runs. In addition, we found that the orbits of planetesimals beyond a ~1.5-2 AU were dynamically depleted by the strengthened sweeping secular resonances associated with Jupiter's and Saturn's more eccentric orbits (relative to present-day) during planet migration. However, this depletion did not prevent the formation of massive Mars analogs (planets with more than 1.5 times Mars' mass). Although late MMR crossings (at t > 30 Myr) could remove such planets, Mars-like small mass planets survived on overly excited orbits (high e and/or i), or were completely lost in these systems. We conclude that the orbital migration and crossing of the mutual 1:2 MMR of Jupiter and Saturn are unlikely to provide suitable orbital conditions for the formation of solar system terrestrial planets. This suggests that to explain Mars' small mass and the absence of other planets between Mars and Jupiter, the outer asteroid belt must have suffered a severe depletion due to interactions with Jupiter/Saturn, or by an alternative mechanism (e.g., rogue super-Earths).