How much does turbulence change the pebble isolation mass for planet formation
TL;DR: In this article, the effect of disk turbulence on the pebble isolation mass (PIM) was studied and its dependence on the gas turbulent viscosity, aspect ratio, and particles Stokes number was found.
Abstract: Context. When a planet becomes massive enough, it gradually carves a partial gap around its orbit in the protoplanetary disk. A pressure maximum can be formed outside the gap where solids that are loosely coupled to the gas, typically in the pebble size range, can be trapped. The minimum planet mass for building such a trap, which is called the pebble isolation mass (PIM), is important for two reasons: it marks the end of planetary growth by pebble accretion, and the trapped dust forms a ring that may be observed with millimetre observations.Aims. We study the effect of disk turbulence on the PIM and find its dependence on the gas turbulent viscosity, aspect ratio, and particles Stokes number.Methods. By means of 2D gas hydrodynamical simulations, we found the minimum planet mass to form a radial pressure maximum beyond the orbit of the planet, which is the necessary condition to trap pebbles. We then carried out 2D gas plus dust hydrodynamical simulations to examine how dust turbulent diffusion impacts particles trapping at the pressure maximum. We finally provide a semi-analytical calculation of the PIM based on comparing the radial drift velocity of solids and the root mean square turbulent velocity fluctuations around the pressure maximum.Results. From our results of gas simulations, we provide an expression for the PIM vs. disk aspect ratio and turbulent viscosity. Our gas plus dust simulations show that the effective PIM can be nearly an order of magnitude larger in high-viscosity disks because turbulence diffuse particles out of the pressure maximum. This is quantified by our semi-analytical calculation, which gives an explicit dependence of the PIM with Stokes number of particles.Conclusions. Disk turbulence can significantly alter the PIM, depending on the level of turbulence in regions of planet formation.
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Peking University1, University of Arizona2, Leiden University3, University of Leicester4, University of Cambridge5, University of Milan6, Vassar College7, Smith College8, University of Grenoble9, University of Victoria10, Chinese Academy of Sciences11, Max Planck Society12, Rice University13, University of Paris14, European Southern Observatory15, University of Chicago16, INAF17, Space Telescope Science Institute18, Ames Research Center19
TL;DR: In this article, the authors identify the most frequently revealed substructure in ALMA dust observations of protoplanetary disks, and measure their properties to investigate how they form, including axisymmetric rings and gaps.
Abstract: Rings are the most frequently revealed substructure in ALMA dust observations of protoplanetary disks, but their origin is still hotly debated. In this paper, we identify dust substructures in 12 disks and measure their properties to investigate how they form. This subsample of disks is selected from a high-resolution ($\sim0.12''$) ALMA 1.33 mm survey of 32 disks in the Taurus star-forming region, which was designed to cover a wide range of sub-mm brightness and to be unbiased to previously known substructures. While axisymmetric rings and gaps are common within our sample, spiral patterns and high contrast azimuthal asymmetries are not detected. Fits of disk models to the visibilities lead to estimates of the location and shape of gaps and rings, the flux in each disk component, and the size of the disk. The dust substructures occur across a wide range of stellar mass and disk brightness. Disks with multiple rings tend to be more massive and more extended. The correlation between gap locations and widths, the intensity contrast between rings and gaps, and the separations of rings and gaps could all be explained if most gaps are opened by low-mass planets (super-Earths and Neptunes) in the condition of low disk turbulence ($\alpha=10^{-4}$). The gap locations are not well correlated with the expected locations of CO and N$_2$ ice lines, so condensation fronts are unlikely to be a universal mechanism to create gaps and rings, though they may play a role in some cases.
395 citations
TL;DR: In this paper, the dependence of the pebble isolation mass on all relevant parameters of the protoplanetary disc was explored and a simple scaling law that captured the dependence on the local disc structure and the turbulent viscosity parameter α was derived.
Abstract: The growth of a planetary core by pebble accretion stops at the so-called pebble isolation mass, when the core generates a pressure bump that traps drifting pebbles outside its orbit. The value of the pebble isolation mass is crucial in determining the final planet mass. If the isolation mass is very low, gas accretion is protracted and the planet remains at a few Earth masses with a mainly solid composition. For higher values of the pebble isolation mass, the planet might be able to accrete gas from the protoplanetary disc and grow into a gas giant. Previous works have determined a scaling of the pebble isolation mass with cube of the disc aspect ratio. Here, we expand on previous measurements and explore the dependency of the pebble isolation mass on all relevant parameters of the protoplanetary disc. We use 3D hydrodynamical simulations to measure the pebble isolation mass and derive a simple scaling law that captures the dependence on the local disc structure and the turbulent viscosity parameter α. We find that small pebbles, coupled to the gas, with Stokes number τ f
< 0.005 can drift through the partial gap at pebble isolation mass. However, as the planetary mass increases, particles must be decreasingly smaller to penetrate the pressure bump. Turbulent diffusion of particles, however, can lead to an increase of the pebble isolation mass by a factor of two, depending on the strength of the background viscosity and on the pebble size. We finally explore the implications of the new scaling law of the pebble isolation mass on the formation of planetary systems by numerically integrating the growth and migration pathways of planets in evolving protoplanetary discs. Compared to models neglecting the dependence of the pebble isolation mass on the α-viscosity, our models including this effect result in higher core masses for giant planets. These higher core masses are more similar to the core masses of the giant planets in the solar system.
228 citations
Cites result from "How much does turbulence change the..."
...Adding the effects of particle diffusion in 2D disc simulations, Ataiee et al. (2018) found that diffusion can increase the pebble-isolation mass, in agreement with Pinilla et al. (2012) and our estimates....
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Peking University1, University of Arizona2, Leiden University3, University of Leicester4, University of Cambridge5, University of Milan6, Vassar College7, Smith College8, University of Grenoble9, University of Victoria10, Max Planck Society11, Chinese Academy of Sciences12, Rice University13, University of Paris14, European Southern Observatory15, University of Chicago16, INAF17, Space Telescope Science Institute18, Ames Research Center19
TL;DR: In this article, the authors identify the most frequently revealed substructure in ALMA dust observations of protoplanetary disks, and measure their properties to investigate how they form, including axisymmetric rings and gaps.
Abstract: Rings are the most frequently revealed substructure in ALMA dust observations of protoplanetary disks, but their origin is still hotly debated. In this paper, we identify dust substructures in 12 disks and measure their properties to investigate how they form. This subsample of disks is selected from a high-resolution ($\sim0.12''$) ALMA 1.33 mm survey of 32 disks in the Taurus star-forming region, which was designed to cover a wide range of sub-mm brightness and to be unbiased to previously known substructures. While axisymmetric rings and gaps are common within our sample, spiral patterns and high contrast azimuthal asymmetries are not detected. Fits of disk models to the visibilities lead to estimates of the location and shape of gaps and rings, the flux in each disk component, and the size of the disk. The dust substructures occur across a wide range of stellar mass and disk brightness. Disks with multiple rings tend to be more massive and more extended. The correlation between gap locations and widths, the intensity contrast between rings and gaps, and the separations of rings and gaps could all be explained if most gaps are opened by low-mass planets (super-Earths and Neptunes) in the condition of low disk turbulence ($\alpha=10^{-4}$). The gap locations are not well correlated with the expected locations of CO and N$_2$ ice lines, so condensation fronts are unlikely to be a universal mechanism to create gaps and rings, though they may play a role in some cases.
160 citations
Cites background from "How much does turbulence change the..."
...∝ ( H rp )a αb, (5) where rp is the distance of the planet to the star, α is the turbulence parameter, with power-law indices a ∈ [2, 3] and b ∈ [0, 1] (see derivations in Lin & Papaloizou 1993; Duffell & MacFadyen 2012, 2013; Ataiee et al. 2018)....
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TL;DR: In this article, it was shown that a difference of a factor of two in the pebble mass flux is enough to change the evolution from the terrestrial to the super-Earth growth mode.
Abstract: Super-Earths - planets with sizes between the Earth and Neptune - are found in tighter orbits than that of the Earth around more than one third of main sequence stars. It has been proposed that super-Earths are scaled-up terrestrial planets that also formed similarly, through mutual accretion of planetary embryos, but in discs much denser than the solar protoplanetary disc. We argue instead that terrestrial planets and super-Earths have two clearly distinct formation pathways that are regulated by the pebble reservoir of the disc. Through numerical integrations, which combine pebble accretion and N-body gravity between embryos, we show that a difference of a factor of two in the pebble mass flux is enough to change the evolution from the terrestrial to the super-Earth growth mode. If the pebble mass flux is small, then the initial embryos within the ice line grow slowly and do not migrate substantially, resulting in a widely spaced population of approximately Mars-mass embryos when the gas disc dissipates. Subsequently, without gas being present, the embryos become unstable due to mutual gravitational interactions and a small number of terrestrial planets are formed by mutual collisions. The final terrestrial planets are at most five Earth masses. Instead, if the pebble mass flux is high, then the initial embryos within the ice line rapidly become sufficiently massive to migrate through the gas disc. Embryos concentrate at the inner edge of the disc and growth accelerates through mutual merging. This leads to the formation of a system of closely spaced super-Earths in the five to twenty Earth-mass range, bounded by the pebble isolation mass. Generally, instabilities of these super-Earth systems after the disappearance of the gas disc trigger additional merging events and dislodge the system from resonant chains. Therefore, the key difference between the two growth modes is whether embryos grow fast enough to undergo significant migration. The terrestrial growth mode produces small rocky planets on wider orbits like those in the solar system whereas the super-Earth growth mode produces planets in short-period orbits inside 1 AU, with masses larger than the Earth that should be surrounded by a primordial H/He atmosphere, unless subsequently lost by stellar irradiation. The pebble flux - which controls the transition between the two growth modes - may be regulated by the initial reservoir of solids in the disc or the presence of more distant giant planets that can halt the radial flow of pebbles. (Less)
132 citations
TL;DR: In this article, the authors studied the growth and migration of an isolated planet in an evolving disc and showed that the growth of a giant planet requires a sufficiently high pebble flux to enable growth to outcompete migration.
Abstract: Giant planets migrate though the protoplanetary disc as they grow their solid core and attract their gaseous envelope. Previously, we have studied the growth and migration of an isolated planet in an evolving disc. Here, we generalise such models to include the mutual gravitational interaction between a high number of growing planetary bodies. We have investigated how the formation of planetary systems depends on the radial flux of pebbles through the protoplanetary disc and on the planet migration rate. Our N-body simulations confirm previous findings that Jupiter-like planets in orbits outside the water ice line originate from embryos starting out at 20-40 AU when using nominal type-I and type-II migration rates and a pebble flux of approximately 100-200 Earth masses per million years, enough to grow Jupiter within the lifetime of the solar nebula. The planetary embryos placed up to 30 AU migrate into the inner system (r P
< 1AU). There they form super-Earths or hot and warm gas giants, producing systems that are inconsistent with the configuration of the solar system, but consistent with some exoplanetary systems. We also explored slower migration rates which allow the formation of gas giants from embryos originating from the 5-10 AU region, which are stranded exterior to 1 AU at the end of the gas-disc phase. These giant planets can also form in discs with lower pebbles fluxes (50-100 Earth masses per Myr). We identify a pebble flux threshold below which migration dominates and moves the planetary core to the inner disc, where the pebble isolation mass is too low for the planet to accrete gas efficiently. In our model, giant planet growth requires a sufficiently high pebble flux to enable growth to out-compete migration. An even higher pebble flux produces systems with multiple gas giants. We show that planetary embryos starting interior to 5 AU do not grow into gas giants, even if migration is slow and the pebble flux is large. These embryos instead grow to just a few Earth masses, the mass regime of super-Earths. This stunted growth is caused by the low pebble isolation mass in the inner disc and is therefore independent of the pebble flux. Additionally, we show that the long-term evolution of our formed planetary systems can naturally produce systems with inner super-Earths and outer gas giants as well as systems of giant planets on very eccentric orbits.
127 citations
Cites background from "How much does turbulence change the..."
...…gap in the protoplanetary disc, where the planet generates an inversion in the radial pressure gradient of the disc exterior its orbit, halting the inward drift of pebbles (Paardekooper & Mellema 2006b; Morbidelli & Nesvorny 2012; Lambrechts et al. 2014; Bitsch et al. 2018a; Ataiee et al. 2018)....
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References
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2,001 citations
"How much does turbulence change the..." refers background in this paper
...…which is the case in our semi-analytic calculation and our 2D simulations, the stopping time is ts = πρss/2ΣΩ with ρs the particles internal density, s their physical radius, Σ and Ω the surface density and the angular velocity of the gas at the location of the particles (Weidenschilling 1977)....
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TL;DR: In this article, the authors measured the accretion rate onto seed masses ranging from a large planetesimal to a fully grown 10-Earth-mass core and test different particle sizes, concluding that pebble accretion can resolve the long-standing core accretion timescale conflict.
Abstract: The observed lifetimes of gaseous protoplanetary discs place strong constraints on gas and ice giant formation in the core accretion scenario. The approximately 10-Earth-mass solid core responsible for the attraction of the gaseous envelope has to form before gas dissipation in the protoplanetary disc is completed within 1–10 million years. Building up the core by collisions between km-sized planetesimals fails to meet this timescale constraint, especially at wide stellar separations. Nonetheless, gas-giant planets are detected by direct imaging at wide orbital distances. In this paper, we numerically study the growth of cores by the accretion of cm-sized pebbles loosely coupled to the gas. We measure the accretion rate onto seed masses ranging from a large planetesimal to a fully grown 10-Earth-mass core and test different particle sizes. The numerical results are in good agreement with our analytic expressions, indicating the existence of two accretion regimes, one set by the azimuthal and radial particle drift for the lower seed masses and the other, for higher masses, by the velocity at the edge of the Hill sphere. In the former, the optimally accreted particle size increases with core mass, while in the latter the optimal size is centimeters, independent of core mass. We discuss the implications for rapid core growth of gas-giant and ice-giant cores. We conclude that pebble accretion can resolve the long-standing core accretion timescale conflict. This requires a near-unity dust-to-gas ratio in the midplane, particle growth to mm and cm and the formation of massive planetesimals or low radial pressure support. The core growth timescale is shortened by a factor 30–1000 at 5 AU and by a factor 100–10 000 at 50 AU, compared to the gravitationally focused accretion of, respectively, low-scale-height planetesimal fragments or standard km-sized planetesimals.
769 citations
"How much does turbulence change the..." refers background in this paper
...Pebbles – solids in the mm-cm size range – are important ingredients for planet formation in protoplanetary disks as they can efficiently speed up the formation of planetary cores (e.g. Ormel & Klahr 2010; Johansen & Lacerda 2010; Lambrechts & Johansen 2012)....
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TL;DR: In this paper, the authors consider the fraction of the gravity torque that is evacuated by pressure supported waves and show that the flux of angular momentum carried by the waves corresponds to a pressure torque.
648 citations
"How much does turbulence change the..." refers background in this paper
...The minimum planet-to-star mass ratio (qgap) that satisfies the gap-opening criterion of Crida et al. (2006), 3 4 h (q/3)1/3 + 50αh2 q = 1 (3) is given by Eq....
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...This certainly differs from the gap-opening criterion formulated in Crida et al. (2006), which assumes planets carve a gap with a gas density drop of about 90%....
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...We note that the trapping location increases with decreasing the alpha turbulent viscosity, which is due to planets carving wider gaps in less viscous disks (e.g. Crida et al. 2006)....
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...However, the formation of even a partial gap in the disk gas around the orbit of the planet is sensitive to Article number, page 1 of 10 the turbulent viscosity of the disk, as shown for instance in the gap-opening criterion formulated in Crida et al. (2006)....
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TL;DR: In this article, the Schmidt number (ratio of gas to particle diusivity) is shown to rise quadratically, not linearly, with stopping time, and the particle layer becomes thinner with the strength of turbulent diusion held xed.
621 citations
"How much does turbulence change the..." refers background or methods in this paper
...Because turbulent diffusion depends both on the turbulent viscosity of the disk and the size of the solids via their stopping time (Youdin & Lithwick 2007), the PIM should therefore also depend on solids size....
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...(37) of Youdin & Lithwick (2007) to calculatie the radial profile of vturb, and their Eqs....
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...For the dust’s turbulent diffusion Dd, we use Dd = ν(1 + 4St 2)/(1 + St2)2 from (Youdin & Lithwick 2007)....
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...…particles stopping time ts (the time particles need to adjust their motion to the gas because of drag forces), and the eddy turnover time teddy, that is the correlation timescale of turbulent fluctuations, which is typically of the order of the orbital timescale (see e.g. Youdin & Lithwick 2007)....
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TL;DR: In this paper, the authors present an efficient and simple modification of the standard transport algorithm used in explicit eulerian fixed polar grid codes, aimed at getting rid of the average azimuthal velocity when applying the Courant condition.
Abstract: We present an efficient and simple modification of the standard transport algorithm used in explicit eulerian fixed polar grid codes, aimed at getting rid of the average azimuthal velocity when applying the Courant condition. This results in a much larger timestep than the usual procedure, and it is particularly well-suited to the description of a Keplerian disk where one is traditionally limited by the very demanding Courant condition on the fast orbital motion at the inner boundary. In this modified algorithm, the timestep is limited by the perturbed velocity and by the shear arising from the differential rotation. FARGO stands for “Fast Advection in Rotating Gaseous Objects”. The speed-up resulting from the use of the FARGO algorithm is problem dependent. In the example presented here, which shows the evolution of a Jupiter sized protoplanet embedded in a minimum mass protoplanetary nebula, the FARGO algorithm is about an order of magnitude faster than a traditional transport scheme, with a much smaller numerical diffusivity.
610 citations
"How much does turbulence change the..." refers background in this paper
...We carried out our gas and gas plus dust hydrodynamical simulations with the code Dusty FARGO-ADSG....
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...It is an extended version of the public code FARGO-ADSG (Masset 2000; Baruteau & Masset 2008b,a), which includes dust (Baruteau & Zhu 2016)....
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