Oscillatory migration of accreting protoplanets driven by a 3D distortion of the gas flow

Context. The dynamics of a low-mass protoplanet accreting solids is influenced by the heating torque, which was found to suppress inward migration in protoplanetary disks with constant opacities. Aims. We investigate the differences of the heating torque between disks with constant and temperature-dependent opacities. Methods. Interactions of a super-Earth-sized protoplanet with the gas disk are explored using 3D radiation hydrodynamic simulations. Results. Accretion heating of the protoplanet creates a hot underdense region in the surrounding gas, leading to misalignment of the local density and pressure gradients. As a result, the 3D gas flow is perturbed and some of the streamlines form a retrograde spiral rising above the protoplanet. In the constant-opacity disk, the perturbed flow reaches a steady state and the underdense gas responsible for the heating torque remains distributed in accordance with previous studies. If the opacity is non-uniform, however, the differences in the disk structure can lead to more vigorous streamline distortion and eventually to a flow instability. The underdense gas develops a one-sided asymmetry which circulates around the protoplanet in a retrograde fashion. The heating torque thus strongly oscillates in time and does not on average counteract inward migration. Conclusions. The torque variations make the radial drift of the protoplanet oscillatory, consisting of short intervals of alternating rapid inward and outward migration. We speculate that transitions between the positive and oscillatory heating torque may occur in specific disk regions susceptible to vertical convection, resulting in the convergent migration of multiple planetary embryos.Comment: Accepted for publication in A&A, 19 pages, 18 figure

Influence of grain growth on the thermal structure of protoplanetary discs

The thermal structure of a protoplanetary disc is regulated by the opacity that dust grains provide. However, previous works have often considered simplified prescriptions for the dust opacity in hydrodynamical disc simulations, e.g. by considering only a single particle size. In the present work we perform 2D hydrodynamical simulations of protoplanetary discs where the opacity is self-consistently calculated for the dust population, taking into account the particle size, composition and abundance. We first compare simulations using single grain sizes to two different multi-grain size distributions at different levels of turbulence strengths, parameterized through the $\alpha$-viscosity, and different gas surface densities. Assuming a single dust size leads to inaccurate calculations of the thermal structure of discs, because the grain size dominating the opacity increases with orbital radius. Overall the two grain size distributions, one limited by fragmentation only and the other determined from a more complete fragmentation-coagulation equilibrium, give similar results for the thermal structure. We find that both grain size distributions give less steep opacity gradients that result in less steep aspect ratio gradients, in comparison to discs with only micrometer sized dust. Moreover, in the discs with a grain size distribution, the innermost outward migration region is removed and planets embedded is such discs experience lower migration rates. We also investigate the dependency of the water iceline position on the alpha-viscosity, the initial gas surface density at 1 AU and the dust-to-gas ratio and find $r_{ice} \propto \alpha^{0.61} \Sigma_{g,0}^{0.8} f_{DG}^{0.37}$ independently of the distribution used. The inclusion of the feedback loop between grain growth, opacities and disc thermodynamics allows for more self-consistent simulations of accretion discs and planet formation.Comment: Accepted by A&A, 27 pages, 19 figure

The growth of planets by pebble accretion in evolving protoplanetary discs

The formation of planets depends on the underlying protoplanetary disc structure, which influences both the accretion and migration rates of embedded planets. The disc itself evolves on time-scales of several Myr during which both temperature and density profiles change as matter accretes onto the central star. Here we use a detailed model of an evolving disc to determine the growth of planets by pebble accretion and their migration through the disc. Cores that reach their pebble isolation mass accrete gas to finally form giant planets with extensive gas envelopes, while planets that do not reach pebble isolation mass are stranded as ice giants and ice planets containing only minor amounts of gas in their envelopes. Unlike earlier population synthesis models, our model works without any artificial reductions in migration speed and for protoplanetary discs with gas and dust column densities similar to those inferred from observations. We find that in our nominal disc model the emergence of planetary embryos preferably occurs after approximately 2 Myr in order to not exclusively form gas giants, but also ice giants and smaller planets. The high pebble accretion rates ensure that critical core masses for gas accretion can be reached at all orbital distances. Gas giant planets nevertheless experience significant reduction in semi-major axes by migration. Considering instead planetesimal accretion for planetary growth, we show that formation time-scales are too long to compete with the migration time-scales and the dissipation time of the protoplanetary disc. Altogether, we find that pebble accretion overcomes many of the challenges in the formation of ice and gas giants in evolving protoplanetary discs.Comment: Accepted by A&A, now with language editin

Separating gas-giant and ice-giant planets by halting pebble accretion

In the Solar System giant planets come in two flavours: 'gas giants' (Jupiter and Saturn) with massive gas envelopes and 'ice giants' (Uranus and Neptune) with much thinner envelopes around their cores. It is poorly understood how these two classes of planets formed. High solid accretion rates, necessary to form the cores of giant planets within the life-time of protoplanetary discs, heat the envelope and prevent rapid gas contraction onto the core, unless accretion is halted. We find that, in fact, accretion of pebbles (~ cm-sized particles) is self-limiting: when a core becomes massive enough it carves a gap in the pebble disc. This halt in pebble accretion subsequently triggers the rapid collapse of the super-critical gas envelope. As opposed to gas giants, ice giants do not reach this threshold mass and can only bind low-mass envelopes that are highly enriched by water vapour from sublimated icy pebbles. This offers an explanation for the compositional difference between gas giants and ice giants in the Solar System. Furthermore, as opposed to planetesimal-driven accretion scenarios, our model allows core formation and envelope attraction within disc life-times, provided that solids in protoplanetary discs are predominantly in pebbles. Our results imply that the outer regions of planetary systems, where the mass required to halt pebble accretion is large, are dominated by ice giants and that gas-giant exoplanets in wide orbits are enriched by more than 50 Earth masses of solids.Comment: Accepted for publication in Astronomy and Astrophysic

The structure of protoplanetary discs around evolving young stars

The formation of planets with gaseous envelopes takes place in protoplanetary accretion discs on time-scales of several millions of years. Small dust particles stick to each other to form pebbles, pebbles concentrate in the turbulent flow to form planetesimals and planetary embryos and grow to planets, which undergo substantial radial migration. All these processes are influenced by the underlying structure of the protoplanetary disc, specifically the profiles of temperature, gas scale height and density. The commonly used disc structure of the Minimum Mass Solar Nebular (MMSN) is a simple power law in all these quantities. However, protoplanetary disc models with both viscous and stellar heating show several bumps and dips in temperature, scale height and density caused by transitions in opacity, which are missing in the MMSN model. These play an important role in the formation of planets, as they can act as sweet spots for the formation of planetesimals via the streaming instability and affect the direction and magnitude of type-I-migration. We present 2D simulations of accretion discs that feature radiative cooling, viscous and stellar heating, and are linked to the observed evolutionary stages of protoplanetary discs and their host stars. These models allow us to identify preferred planetesimal and planet formation regions in the protoplanetary disc as a function of the disc's metallicity, accretion rate and lifetime. We derive simple fitting formulae that feature all structural characteristics of protoplanetary discs during the evolution of several Myr. These fits are straightforward to apply for modelling any growth stage of planets where detailed knowledge of the underlying disc structure is required.Comment: Accepted by A&A, v3 corrected small typo in the fitting formula

Dust clearing by radial drift in evolving protoplanetary discs

Recent surveys have revealed that protoplanetary discs typically have dust masses that appear to be insufficient to account for the high occurrence rate of exoplanet systems. We demonstrate that this observed dust depletion is consistent with the radial drift of pebbles. Using a Monte Carlo method we simulate the evolution of a cluster of protoplanetary discs, using a 1D numerical method to viscously evolve each gas disc together with the radial drift of dust particles that have grown to 100 $\mu$m in size. For a 2 Myr old cluster of stars, we find a slightly sub-linear scaling between the gas disc mass and the gas accretion rate ($M_\mathrm{g}\propto\dot{M}^{0.9}$). However, for the dust mass we find that evolved dust discs have a much weaker scaling with the gas accretion rate, with the precise scaling depending on the age at which the cluster is sampled and the intrinsic age spread of the discs in the cluster. Ultimately, we find that the dust mass present in protoplanetary disc is on the order of 10-100 Earth masses in 1-3 Myr old star-forming regions, a factor of 10 to 100 depleted from the original dust budget. As the dust drains from the outer disc, pebbles pile up in the inner disc and locally increase the dust-to-gas ratio by a factor of up to 4 above the initial value. In these high dust-to-gas ratio regions we find conditions that are favourable for planetesimal formation via the streaming instability and subsequent growth by pebble accretion. We also find the following scaling relations with stellar mass within a 1-2 Myr old cluster: a slightly super-linear scaling between the gas accretion rate and stellar mass ($\dot{M}\propto M_\star^{1.4}$), a slightly super-linear scaling between the gas disc mass and the stellar mass ($M_\mathrm{g}\propto M_\star^{1.4}$) and a super-linear relation between the dust disc mass and stellar mass ($M_\mathrm{d}\propto M_\star^{1.4-4.1}$).Comment: 18 pages, 18 figures, accepted for publication in A&

Disc population synthesis: decrease of the solid mass reservoir through pebble drift

Surveys of star-forming regions reveal that the dust mass of protoplanetary discs decreases by several orders of magnitude on a timescale of a few million years. This decrease in the mass budget of solids is likely due to the gas-drag-induced radial drift of mm-sized solids, called pebbles. However, quantifying the evolution of this dust component in young stellar clusters is difficult due to the inherent large spread in stellar masses and formation times. Therefore, we aim to model the collective evolution of a cluster to investigate the effectiveness of radial drift in clearing the discs of mm-sized particles. We use a protoplanetary disc model that numerically solves for disc formation, and the viscous evolution and photoevaporative clearing of the gas component, while also including the drift of particles limited in size by fragmentation. We find that discs are born with dust masses between 50 Earth masses and 1000 Earth masses, for stars with, respectively, masses between 0.1 solar masses and 1 solar masses. The majority of this initial dust reservoir is typically lost through drift before photoevaporation opens a gap in the gas disc for models both with and without strong X-ray-driven mass loss rates. We conclude that the decrease in time of the mass locked in fragmentation-limited pebbles is consistent with the evolution of dust masses and ages inferred from nearby star-forming regions when assuming viscous evolution rates corresponding to mean gas disc lifetimes between 3 Myr and 8 Myr.Comment: 16 pages, 11 figures, accepted for publication in A&A. Addressed additional language comment

Pebble-driven Planet Formation around Very Low-mass Stars and Brown Dwarfs

We conduct a pebble-driven planet population synthesis study to investigate the formation of planets around very low-mass stars and brown dwarfs, in the (sub)stellar mass range between $0.01 \ M_{\odot}$ and $0.1 \ M_{\odot}$. Based on the extrapolation of numerical simulations of planetesimal formation by the streaming instability, we obtain the characteristic mass of the planetesimals and the initial masses of the protoplanets (largest bodies from the planetesimal size distributions), in either the early self-gravitating phase or the later non-self-gravitating phase of the protoplanetary disk evolution. We find that the initial protoplanets form with masses that increase with host mass, orbital distance and decrease with disk age. Around late M-dwarfs of $0.1 \ M_{\odot}$, these protoplanets can grow up to Earth-mass planets by pebble accretion. However, around brown dwarfs of $0.01 \ M_{\odot}$, planets do not grow larger than Mars mass when the initial protoplanets are born early in self-gravitating disks, and their growth stalls at around $0.01$ Earth-mass when they are born late in non-self-gravitating disks. Around these low mass stars and brown dwarfs, we find no channel for gas giant planet formation because the solid cores remain too small. When the initial protoplanets form only at the water-ice line, the final planets typically have ${\gtrsim} 15\%$ water mass fraction. Alternatively, when the initial protoplanets form log-uniformly distributed over the entire protoplanetary disk, the final planets are either very water-rich (water mass fraction ${\gtrsim}15\%$) or entirely rocky (water mass fraction ${\lesssim}5\%$).Comment: 12 pages, 8 figures, accepted in A&

Formation of pebbles in (gravito-)viscous protoplanetary disks with various turbulent strengths

Aims. Dust plays a crucial role in the evolution of protoplanetary disks. We study the dynamics and growth of initially sub-$\mu m$ dust particles in self-gravitating young protoplanetary disks with various strengths of turbulent viscosity. We aim to understand the physical conditions that determine the formation and spatial distribution of pebbles when both disk self-gravity and turbulent viscosity can be concurrently at work. Methods. We perform the thin-disk hydrodynamics simulations of self-gravitating protoplanetary disks over an initial time period of 0.5 Myr using the FEOSAD code. Turbulent viscosity is parameterized in terms of the spatially and temporally constant $\alpha$-parameter, while the effects of gravitational instability on dust growth is accounted for by calculating the effective parameter $\alpha_{\rm GI}$. We consider the evolution of dust component including momentum exchange with gas, dust self-gravity, and also a simplified model of dust growth. Results. We find that the level of turbulent viscosity strongly affects the spatial distribution and total mass of pebbles in the disk. The $\alpha=10^{-2}$ model is viscosity-dominated, pebbles are completely absent, and dust-to-gas mass ratio deviates from the reference 1:100 value no more than by 30\% throughout the disk extent. On the contrary, the $\alpha=10^{-3}$ model and, especially, the $\alpha=10^{-4}$ model are dominated by gravitational instability. The effective parameter $\alpha+\alpha_{\rm GI}$ is now a strongly varying function of radial distance. As a consequence, a bottle neck effect develops in the innermost disk regions, which makes gas and dust accumulate in a ring-like structure. Abridged.Comment: Accepted for publication in Astronomy and Astrophysic