The most accurate method for modelling planetary migration and hence the formation of resonant systems is using hydrodynamical simulations. Usually, the force (torque) acting on a planet is calculated using the forces from the gas disc and the star, while the gas accelerations are computed using the pressure gradient, the star, and the planet's gravity, ignoring its own gravity. For the non-migrating the neglect of the disc gravity results in a consistent torque calculation while for the migrating case it is inconsistent. We aim to study how much this inconsistent torque calculation can affect the final configuration of a two-planet system. Our focus will be on low-mass planets because most of the multi-planetary systems, discovered by the Kepler survey, have masses around 10 Earth masses. Performing hydrodynamical simulations of planet-disc interaction, we measure the torques on non-migrating and migrating planets for various disc masses as well as density and temperature slopes with and without considering the disc self-gravity. Using this data, we find a relation that quantifies the inconsistency, use it in an N-body code, and perform an extended parameter study modelling the migration of a planetary system with different planet mass ratios and disc surface densities, in order to investigate the impact of the torque inconsistency on the architecture of the planetary system. Not considering disc self-gravity produces an artificially larger torque on the migrating planet that can result in tighter planetary systems. The deviation of this torque from the correct value is larger in discs with steeper surface density profiles. In hydrodynamical modelling of multi-planetary systems, it is crucial to account for the torque correction, otherwise the results favour more packed systems.
(Abridged) We present the results of N-body simulations of planetary systems formation in radiatively-inefficient disc models, where positive corotation torques may counter the rapid inward migration of low mass planets driven by Lindblad torques. The aim of this work is to examine the nature of planetary systems that arise from oligarchic growth in such discs. We adapt the commonly-used Mercury-6 symplectic integrator by including simple prescriptions for planetary migration (types I and II), planetary atmospheres that enhance the probability of planetesimal accretion by protoplanets, gas acc…
We study the effects of diffusion on the non-linear corotation torque, or horseshoe drag, in the two-dimensional limit, focusing on low-mass planets for which the width of the horseshoe region is much smaller than the scale height of the disc. In the absence of diffusion, the non-linear corotation torque saturates, leaving only the Lindblad torque. Diffusion of heat and momentum can act to sustain the corotation torque. In the limit of very strong diffusion, the linear corotation torque is recovered. For the case of thermal diffusion, this limit corresponds to having a locally isothermal equat…
We study the torque on low-mass planets embedded in protoplanetary discs in the two-dimensional approximation, incorporating non-isothermal effects. We couple linear estimates of the Lindblad (or wave) torque to a simple, but non-linear, model of adiabatic corotation torques (or horseshoe drag), resulting in a simple formula that governs Type I migration in non-isothermal discs. This formula should apply in optically thick regions of the disc, where viscous and thermal diffusion act to keep the horseshoe drag unsaturated. We check this formula against numerical hydrodynamical simulations, usin…
We give an expression for the Lindblad torque acting on a low-mass planet embedded in a protoplanetary disk that is valid even at locations where the surface density or temperature profile cannot be approximated by a power law, such as an opacity transition. At such locations, the Lindblad torque is known to suffer strong deviation from its standard value, with potentially important implications for type I migration, but the full treatment of the tidal interaction is cumbersome and not well suited to models of planetary population synthesis. The expression that we propose retains the simplicit…
Planet migration is inherently a three-dimensional (3D) problem, because Earth-size planetary cores are deeply embedded in protoplanetary disks. Simulations of these 3D disks remain challenging due to the steep requirement in resolution. Using two different hydrodynamics code, FARGO3D and PEnGUIn, we simulate disk-planet interaction for a 1 to 5 Earth-mass planet embedded in an isentropic disk. We measure the torque on the planet and ensure that the measurements are converged both in resolution and between the two codes. We find that the torque is independent of the smoothing length of the pla…
The migration of planets on nearly circular, non-inclined orbits in protoplanetary discs is entirely described by the disc's torque. This torque is a complex function of the disc parameters, and essentially amounts to the sum of two components: the Lindblad torque and the corotation torque. Known torque formulae do not reproduce accurately the torque actually experienced in numerical simulations by low- and intermediate- mass planets in radiative discs. One of the main reasons for this inaccuracy is that these formulae have been worked out in two-dimensional analyses. Here we revisit the torqu…
The present work is a critical revision of the hypothesis of the planetary tidal influence on solar activity published by Abreu et al. (Astron. Astrophys. 548, A88, 2012; called A12 here). A12 describes a hypothesis that planets can have an impact on the solar tachocline and therefore on solar activity. We checked the procedure and results of A12, namely the algorithm of planetary tidal torque calculation and the wavelet coherence between torque and heliospheric modulation potential. We found that the claimed peaks in long-period range of the torque spectrum are artefacts caused by the calcula…
The University of Tokyo（东京大学）；National Astronomical Observatory of Japan（日本国立天文台）；University of Tokyo（东京大学）；Université Côte d’Azur（蔚蓝海岸大学）；Observatoire de la Côte d’Azur（蔚蓝海岸天文台）；CNRS（法国国家科学研究中心）；Centre national d’études spatiales (CNES)（法国国家空间研究中心）；Charles University（查理大学）；University of Michigan（密歇根大学）；Johns Hopkins University Applied Physics Lab（约翰斯·霍普金斯大学应用物理实验室）
We provide torque formulae for low mass planets undergoing type I migration in gaseous disks. These torque formulae put special emphasis on the horseshoe drag, which is prone to saturation: the asymptotic value reached by the horseshoe drag depends on a balance between coorbital dynamics (which tends to cancel out or saturate the torque) and diffusive processes (which tend to restore the unperturbed disk profiles, thereby desaturating the torque). We entertain here the question of this asymptotic value, and we derive torque formulae which give the total torque as a function of the disk's visco…
We explore the arguments presented in the past linking changes in the angular momentum $modulus$, $|{\bf L}|$, of the Sun's barycentric orbit, with the discharges of Po River in Europe and Paraná River in South America, looking for any evidence regarding to a possible underlying physical mechanism. We clarify the planetary effect on solar torque presenting new analyses and results; we also improve prior results on Paraná River's cycles finding significant spectral lines around 6.5 yr, 7.6 yr, 8.7 yr, and 10.4 yr. We show that the truly important dynamical parameter in this issue is the {\it ve…
Fast inward migration of planetary cores is a common problem in the current planet formation paradigm. Even though dust is ubiquitous in protoplanetary disks, its dynamical role in the migration history of planetary embryos has not been assessed. In this Letter, we show that the scattered pebble-flow induced by a low-mass planetary embryo leads to an asymmetric dust-density distribution that is able to exert a net torque. By analyzing a large suite of multifluid hydrodynamical simulations addressing the interaction between the disk and a low-mass planet on a fixed circular orbit, and neglectin…
We present the results of high resolution 2D simulations of low mass planets on fixed eccentric orbits embedded in protoplanetary discs. The aim of this study is to determine how the strength of the sustained, non-linear corotation torque experienced by embedded planets varies as a function of orbital eccentricity, disc parameters, and planetary mass. In agreement with previous work we find that the corotation torque diminishes as orbital eccentricity, $e$, increases. Analysis of the time-averaged streamlines in the disc demonstrates that the width of the horseshoe region narrows as the eccent…
(Abridged) The following describes a model of a broad planetary ring whose sharp edge is confined by a satellite's m^th Lindblad resonance (LR). This model uses a streamline formalism to calculate the ring's internal forces, namely, ring gravity, pressure, viscosity, as well as a hypothetical drag force. The model calculates the streamlines' forced orbit elements and surface density throughout the perturbed ring. The model is then applied to the outer edge of Saturn's B ring, which is maintained by an m=2 inner LR with the satellite Mimas. Ring models are used to illustrate how a ring's pertur…
Quantifying the Impact of the Dust Torque on the Migration of Low-mass Planets II: The Role of Pebble Accretion in Planet Growth within a Global Planet Formation Model
Thermal atmospheric tides can torque telluric planets away from spin-orbit synchronous rotation, as observed in the case of Venus. They thus participate to determine the possible climates and general circulations of the atmospheres of these planets. In this work, we write the equations governing the dynamics of thermal tides in a local vertically-stratified section of a rotating planetary atmosphere by taking into account the effects of the complete Coriolis acceleration on tidal waves. This allows us to derive analytically the tidal torque and the tidally dissipated energy, which we use to di…
We evaluate the thermal torques exerted on low-mass planets embedded in gaseous protoplanetary discs with thermal diffusion, by means of high-resolution three-dimensional hydrodynamics simulations. We confirm that thermal torques essentially depend on the offset between the planet and its corotation, and find a good agreement with analytic estimates when this offset is small compared to the size of the thermal disturbance. For larger offsets that may be attained in discs with a large pressure gradient or a small thermal diffusivity, thermal torques tend toward an asymptotic value broadly compa…
With hundreds of exoplanets detected, it is necessary to revisit giant planets accretion models to explain their mass distribution. In particular, formation of sub-jovian planets remains unclear, given the short timescale for the runaway accretion of massive atmospheres. However, gas needs to pass through a circum-planetary disc. If the latter has a low viscosity (as expected if planets form in "dead zones"), it might act as a bottleneck for gas accretion. We investigate what the minimum accretion rate is for a planet under the limit assumption that the circum-planetary disc is totally invisci…
We investigate the effects of viscosity on disk-planet interaction and discuss how type I migration of planets is modified. We have performed a linear calculation using shearing-sheet approximation and obtained the detailed, high resolution density structure around the planet embedded in a viscous disk with a wide range of viscous coefficients. We use a time-dependent formalism that is useful in investigating the effects of various physical processes on disk-planet interaction. We find that the density structure in the vicinity of the planet is modified and the main contribution to the torque …
Massive cores of the giant planets are thought to have formed in a gas disk by accretion of pebble-size particles whose accretional cross-section is enhanced by aerodynamic gas drag [1][2]. A commonly held view is that the terrestrial planet system formed later (30-200 Myr after the dispersal of the gas disk) by giant collisions of tens of roughly Mars-size protoplanets [3]. Here we propose, instead, that the terrestrial planets formed earlier by gas-driven convergent migration of protoplanets toward $\sim\!1\,{\rm au}$ (related ref. [4] invoked a different process to concentrate planetesimals…
This paper presents a generalized treatment of Type I planetary migration in the presence of stochastic perturbations. In many planet-forming disks, the Type I migration mechanism, driven by asymmetric torques, acts on a short time scale and compromises planet formation. If the disk also supports MHD instabilities, however, the corresponding turbulent fluctuations produce additional stochastic torques that modify the steady inward migration scenario. This work studies the migration of planetary cores in the presence of stochastic fluctuations using complementary methods, including a Fokker-Pla…
We present N-body simulations of planetary system formation in thermally-evolving, viscous disc models. The simulations incorporate type I migration (including corotation torques and their saturation), gap formation, type II migration, gas accretion onto planetary cores, and gas disc dispersal through photoevaporation. The aim is to examine whether or not the oligarchic growth scenario, when combined with self-consistent disc models and up-to-date prescriptions for disc-driven migration, can produce planetary systems similar to those that have been observed. The results correlate with the init…
This paper uses an inductive method to investigate the factors responsible for variations in planetary-rotation periods. I began by showing the presence of a correlation between the masses of planets and their rotation periods. Then I tested the impact of planetary radius, acceleration, velocity, and torque on rotation periods. I found that velocity, acceleration, and radius are the most important factors in explaining rotation periods. The effect of mass may be rather on influencing the size of the radii of planets. That is, the larger the mass of a planet, the larger its radius. Moreover, ma…
Concurrent Accretion and Migration of Giant Planets in their Natal Disks with Consistent Accretion Torque (II): Parameter Survey and Condition for Outward Migration
Population synthesis studies into planet formation have suggested that distributions consistent with observations can only be reproduced if the actual Type I migration timescale is at least an order of magnitude longer than that deduced from linear theories. Although past studies considered the effect of the Type I migration of protoplanetary embryos, in most cases they used a conventional formula based on static torques in isothermal disks, and employed a reduction factor to account for uncertainty in the mechanism details. However, in addition to static torques, a migrating planet experience…
Gas giants are thought to form by runaway accretion: an instability driven by the self-gravity of growing atmospheres that causes accretion rates to rise super-linearly with planet mass. Why runaway should stop at a Jupiter or any other mass is unknown. We consider the proposal that final masses are controlled by circumstellar disc gaps (cavities) opened by planetary gravitational torques. We develop a fully time-dependent theory of gap formation and couple it self-consistently to planetary growth rates. When gaps first open, planetary torques overwhelm viscous torques, and gas depletes as if …
Planetary migration is the process by which a forming planet undergoes a drift of its semi-major axis caused by the tidal interaction with its parent protoplanetary disc. One of the key quantities to assess the migration of embedded planets is the tidal torque between the disc and planet, which has two components: the Lindblad torque and the corotation torque. We review the latest results on both torque components for planets on circular orbits, with a special emphasis on the various processes that give rise to additional, large components of the corotation torque, and those contributing to th…
During its 5 year mission, the Kepler spacecraft has uncovered a diverse population of planetary systems with orbital configurations ranging from single-transiting planets to systems of multiple planets co-transiting the parent star. By comparing the relative occurrences of multiple to single-transiting systems, recent analyses have revealed a significant over-abundance of singles. Dubbed the "Kepler Dichotomy," this feature has been interpreted as evidence for two separate populations of planetary systems: one where all orbits are confined to a single plane, and a second where the constituent…
Detailed observational characterization of transiting exoplanet systems has revealed that the spin-axes of massive (M > ~1.2 solar masses) stars often exhibit substantial misalignments with respect to the orbits of the planets they host. Conversely, lower-mass stars tend to only have limited obliquities. A similar trend has recently emerged within the observational dataset of young stars' magnetic field strengths: massive T-Tauri stars tend to have dipole fields that are ~10 times weaker than their less-massive counterparts. Here we show that the associated dependence of magnetic star-disk tor…
We present radiation hydrodynamic simulations in which binary planets form by close encounters in a system of several super-Earth embryos. The embryos are embedded in a protoplanetary disk consisting of gas and pebbles and evolve in a region where the disk structure supports convergent migration due to Type I torques. As the embryos accrete pebbles, they become heated and thus affected by the thermal torque (Benítez-Llambay et al. 2015) and the hot-trail effect (Chrenko et al. 2017) which excites orbital eccentricities. Motivated by findings of Eklund & Masset (2017), we assume the hot-trail e…
In order to study the origin of the architectures of low mass planetary systems, we perform numerical surveys of the evolution of pairs of coplanar planets in the mass range $(1-4)\ \rmn{M}_{\oplus}.$ These evolve for up to $2\times10^7 \rmn{yr}$ under a range of orbital migration torques and circularization rates assumed to arise through interaction with a protoplanetary disc. Near the inner disc boundary, significant variations of viscosity, interaction with density waves or with the stellar magnetic field could occur and halt migration, but allow ircularization to continue. This was modelle…
The structure of planetary systems around their host stars depends on their initial formation conditions. Massive planets will likely be formed as a consequence of rapid migration of planetesimals and low mass cores into specific trapping sites in protoplanetary discs. We present analytical modeling of inhomogeneities in protoplanetary discs around a variety of young stars, - from Herbig Ae/Be to classical T Tauri and down to M stars, - and show how they give rise to planet traps. The positions of these traps define the initial orbital distribution of multiple protoplanets. We investigate both…
The migration of planets plays an important role in the early planet-formation process. An important problem has been that standard migration theories predict very rapid inward migration, which poses problems for population synthesis models. However, it has been shown recently that low-mass planets (20-30 Earth Masses) that are still embedded in the protoplanetary disc can migrate outwards under certain conditions. Simulations have been performed mostly for planets at given radii for a particular disc model. Here, we plan to extend previous work and consider different masses of the disc to qua…
The Kepler satellite has discovered a number of transiting planets around close binary stars. These circumbinary systems have highly aligned planetary and binary orbits. In this paper, we explore how the mutual inclination between the planetary and binary orbits may reflect the physical conditions of the assembly of protoplanetary disks and the interaction between protostellar binaries and circumbinary disks. Given the turbulent nature of star-forming molecular clouds, it is possible that the gas falling onto the outer region of a circumbinary disk and the central protostellar binary have diff…
Context: Several recent studies have found that planet migration in adiabatic discs differs significantly from migration in isothermal discs. Depending on the thermodynamic conditions, i.e., the effectiveness of radiative cooling, and the radial surface density profile, planets migrate inward or outward. Clearly, this will influence the semimajor axis - mass distribution of planets as predicted by population synthesis simulations. Aims: Our goal is to study the global effects of radiative cooling, viscous torque desaturation and gap opening as well as stellar irradiation on the tidal migration…
When they first appear in the HR diagram, young stars rotate at a mere 10\% of their break-up velocity. They must have lost most of the angular momentum initially contained in the parental cloud, the so-called angular momentum problem. We investigate here a new mechanism by which large amounts of angular momentum might be shed from young stellar systems, thus yielding slowly rotating young stars. Assuming that planets promptly form in circumstellar disks and rapidly migrate close to the central star, we investigate how the tidal and magnetic interactions between the protostar, its close-in pla…
A key process in planet formation is the exchange of angular momentum between a growing planet and the protoplanetary disc, which makes the planet migrate through the disc. Several works show that in general low-mass and intermediate-mass planets migrate towards the central star, unless corotation torques become dominant. Recently, a new kind of torque, called the thermal torque, was proposed as a new source that can generate outward migration of low-mass planets. While the Lindblad and corotation torques depend mostly on the properties of the protoplanetary disc and on the planet mass, the th…
We present simulations of low-mass planet--disc interactions in inviscid three-dimensional discs. We show that a wind-driven laminar accretion flow through the surface layers of the disc does not significantly modify the migration torque experienced by embedded planets. More importantly, we find that 3D effects lead to a dramatic change in the behaviour of the dynamical corotation torque compared to earlier 2D theory and simulations. Although it was previously shown that the dynamical corotation torque could act to slow and essentially stall the inward migration of a low-mass planet, our resul…
Planetary systems are born in the disks of gas, dust and rocky fragments that surround newly formed stars. Solid content assembles into ever-larger rocky fragments that eventually become planetary embryos. These then continue their growth by accreting leftover material in the disc. Concurrently, tidal effects in the disc cause a radial drift in the embryo orbits, a process known as migration. Fast inward migration is predicted by theory for embryos smaller than three to five Earth masses. With only inward migration, these embryos can only rarely become giant planets located at Earth's distance…
Recent studies on the planet-dominated regime of Type II migration showed that, contrary to the conventional wisdom, massive planets can migrate outwards. Using `fixed-planet' simulations these studies found a correlation between the sign of the torques acting on the planet and the parameter $K'$ (which describes the depth of the gap carved by the planet in the disc). We perform `live-planet' simulations exploring a range of $K'$ and disc mass values to test and extend these results. The excitation of planet eccentricity in live-planet simulations breaks the direct dependence of migration rate…
We present a comprehensive theoretical study on the spin evolution of a planet under the combined effects of tidal dissipation and gravitational perturbation from an external companion. Such a "spin + companion" system (called Colombo's top) appears in many [exo]planetary contexts. The competition between the tidal torque (which drives spin-orbit alignment and synchronization) and the gravitational torque from the companion (which drives orbital precession of the planet) gives rise to two possible spin equilibria ("Tidal Cassini Equilibria", tCE) that are stable and attracting: the "simple" tC…
We carry out local three dimensional (3D) hydrodynamic simulations of planet-disk interaction in stratified disks with varied thermodynamic properties. We find that whenever the Brunt-Vaisala frequency (N) in the disk is nonzero, the planet exerts a strong torque on the disk in the vicinity of the planet, with a reduction in the traditional "torque cutoff". In particular, this is true for adiabatic perturbations in disks with isothermal density structure, as should be typical for centrally irradiated protoplanetary disks. We identify this torque with buoyancy waves, which are excited (when N i…
Beyond solar metallicity: How enhanced solid content in disks re-shape low-mass planet torques
HUN-REN CSFK Konkoly Observatory（匈牙利科学院研究网络天体物理研究所孔科利天文台）；ELTE Eötvös Loránd University, Institute of Physics and Astronomy, Department of Astronomy（厄特沃什·罗兰大学物理与天文学研究所天文学系）
As planets grow the exchange of angular momentum with the gaseous component of the protoplanetary disc produces a net torque resulting in a variation of the semi-major axis of the planet. For low-mass planets not able to open a gap in the gaseous disc this regime is known as type I migration. Pioneer works studied this mechanism in isothermal discs finding fast inward type I migration rates that were unable to reproduce the observed properties of extrasolar planets. In the last years, several improvements have been made in order to extend the study of type I migration rates to non-isothermal d…
The torques exerted by a locally isothermal disk on an embedded planet lead to rapid inward migration. Recent work has shown that modeling the thermodynamics without the assumption of local isothermality reveals regions where the net torque on an embedded planet is positive, leading to outward migration of the planet. When a region with negative torque lies directly exterior to this, planets in the inner region migrate outwards and planets in the outer region migrate inwards, converging where the torque is zero. We incorporate the torques from an evolving non-isothermal disk into an N-body sim…
The strength and direction of migration of low mass embedded planets depends on the disk's thermodynamic state, where the internal dissipation is balanced by radiative transport, and the migration can be directed outwards, a process which extends the lifetime of growing embryos. Very important parameters determining the structure of disks, and hence the direction of migration, are the viscosity and the adiabatic index. In this paper we investigate the influence of different viscosity prescriptions (alpha-type and constant) and adiabatic indices on disk structures and how this affects the migra…
Semidiurnal atmospheric thermal tides are important for terrestrial exoplanets in the habitable zone of their host stars. With solid tides, they torque these planets, thus contributing to determine their rotation states as well as their climate. Given the complex dynamics of thermal tides, analytical models are essential to understand its dependence on the structure and rotation of planetary atmospheres and the tidal frequency. In this context, the state of the art model proposed in the 60's by Lindzen and Chapman explains well the properties of thermal tides in the asymptotic regime of Earth-…
Context. The origin of giant planets at moderate separations $\simeq$$1$$-$$10$ au is still not fully understood because numerical studies of Type II migration in protoplanetary disks often predict a decay of the semi-major axis that is too fast. According to recent 2D simulations, inward migration of a gap-opening planet can be slowed down or even reversed if the outer gap edge becomes heated by irradiation from the central star, and puffed up. Aims. Here we study how stellar irradiation reduces the disk-driven torque and affects migration in more realistic 3D disks. Methods. Using 3D hydrody…
