Minimum Mass Solar Nebula Research Articles

Time evolution of snow regions and planet traps in an evolving protoplanetary disk

Aims. We track the time evolution of planet traps and snowlines in a viscously evolving protoplanetary disk using an opacity table that accounts for the composition of the dust material. Methods. We coupled a dynamical and thermodynamical disk model with a temperature-dependent opacity table (that accounts for the sublimation of the main dust components) to investigate the formation and evolution of snowlines and planet traps during the first million years of disk evolution. Results. Starting from a minimum mass solar nebula (MMSN), we find that the disk mid-plane temperature profile shows several plateaux (0.1-1 AU wide) at the different sublimation temperatures of the species that make up the dust. For water ice, the correspond- ing plateau can be larger than 1 AU, which means that this is a snow "region" rather than a snow "line". As a consequence, the surface density of solids in the snow region may increase gradually, not abruptly. Several planet traps and desert regions appear naturally as a result of abrupt local changes in the temperature and density profiles over the disk lifetime. These structures are mostly located at the edges of the temperature plateaux (surrounding the dust sublimation lines) and at the heat-transition barrier where the disk stellar heating and viscous heating are of the same magnitude (around 10 AU after 1 Myr). Conclusions. Several traps are identified: although a few appear to be transient, most of them slowly migrate along with the heat- transition barrier or the dust sublimation lines. These planet traps may temporarily favor the growth of planetary cores.

The JCMT Gould Belt Survey: low-mass protoplanetary discs from a SCUBA-2 census of NGC 1333

NGC1333 is a 1-2 Myr old cluster of stars in the Perseus molecular cloud. We used 850mu data from the Gould Belt Survey with SCUBA-2 on the JCMT to measure or place limits on disc masses for 82 Class II sources in this cluster. Eight disc-candidates were detected; one is estimated to have mass of about 9 Jupiter masses in dust plus gas, while the others host only 2-4 Jupiter masses of circumstellar material. None of these discs exceeds the threshold for the 'Minimum Mass Solar Nebula' (MMSN). This reinforces previous claims that only a small fraction of Class II sources at an age of 1-2 Myr has discs exceeding the MMSN threshold and thus can form a planetary system like our own. However, other regions with similarly low fractions of MMSN discs (IC348, UpSco, SigmaOri) are thought to be older than NGC1333. Compared with coeval regions, the exceptionally low fraction of massive discs in NGC1333 cannot easily be explained by the effects of UV radiation or stellar encounters. Our results indicate that additional environmental factors significantly affect disc evolution and the outcome of planet formation by core accretion.

On the filtering and processing of dust by planetesimals

Context. Circumstellar disks are known to contain a significant mass in dust ranging from micron to centimeter size. Meteorites are evidence that individual grains of those sizes were collected and assembled into planetesimals in the young solar system. Aims. We assess the efficiency of dust collection of a swarm of non-drifting planetesimals {\rev with radii ranging from 1 to $10^3$\,km and beyond. Methods. We calculate the collision probability of dust drifting in the disk due to gas drag by planetesimal accounting for several regimes depending on the size of the planetesimal, dust, and orbital distance: the geometric, Safronov, settling, and three-body regimes. We also include a hydrodynamical regime to account for the fact that small grains tend to be carried by the gas flow around planetesimals. Results. We provide expressions for the collision probability of dust by planetesimals and for the filtering efficiency by a swarm of planetesimals. For standard turbulence conditions (i.e., a turbulence parameter $\alpha=10^{-2}$), filtering is found to be inefficient, meaning that when crossing a minimum-mass solar nebula (MMSN) belt of planetesimals extending between 0.1 AU and 35 AU most dust particles are eventually accreted by the central star rather than colliding with planetesimals. However, if the disk is weakly turbulent ($\alpha=10^{-4}$) filtering becomes efficient in two regimes: (i) when planetesimals are all smaller than about 10 km in size, in which case collisions mostly take place in the geometric regime; and (ii) when planetary embryos larger than about 1000 km in size dominate the distribution, have a scale height smaller than one tenth of the gas scale height, and dust is of millimeter size or larger in which case most collisions take place in the settling regime. These two regimes have very different properties: we find that the local filtering efficiency $x_{filter,MMSN}$ scales with $r^{-7/4}$ (where $r$ is the orbital distance) in the geometric regime, but with $r^{-1/4}$ to $r^{1/4}$ in the settling regime. This implies that the filtering of dust by small planetesimals should occur close to the central star and with a short spread in orbital distances. On the other hand, the filtering by embryos in the settling regime is expected to be more gradual and determined by the extent of the disk of embryos. Dust particles much smaller than millimeter size tend only to be captured by the smallest planetesimals because they otherwise move on gas streamlines and their collisions take place in the hydrodynamical regime. Conclusions. Our results hint at an inside-out formation of planetesimals in the infant solar system because small planetesimals in the geometrical limit can filter dust much more efficiently close to the central star. However, even a fully-formed belt of planetesimals such as the MMSN only marginally captures inward-drifting dust and this seems to imply that dust in the protosolar disk has been filtered by planetesimals even smaller than 1 km (not included in this study) or that it has been assembled into planetesimals by other mechanisms (e.g., orderly growth, capture into vortexes). Further refinement of our work concerns, among other things: a quantitative description of the transition region between the hydro and settling regimes; an assessment of the role of disk turbulence for collisions, in particular in the hydro regime; and the coupling of our model to a planetesimal formation model.

OUTWARD MIGRATION OF JUPITER AND SATURN IN 3:2 OR 2:1 RESONANCE IN RADIATIVE DISKS: IMPLICATIONS FOR THE GRAND TACK AND NICE MODELS

Embedded in the gaseous protoplanetary disk, Jupiter and Saturn naturally become trapped in 3:2 resonance and migrate outward. This serves as the basis of the Grand Tack model. However, previous hydrodynamical simulations were restricted to isothermal disks, with moderate aspect ratio and viscosity. Here we simulate the orbital evolution of the gas giants in disks with viscous heating and radiative cooling. We find that Jupiter and Saturn migrate outward in 3:2 resonance in modest-mass ($M_{disk} \approx M_{MMSN}$, where MMSN is the "minimum-mass solar nebula") disks with viscous stress parameter $\alpha$ between $10^{-3}$ and $10^{-2} $. In disks with relatively low-mass ($M_{disk} \lesssim M_{MMSN}$) , Jupiter and Saturn get captured in 2:1 resonance and can even migrate outward in low-viscosity disks ($\alpha \le 10^{-4}$). Such disks have a very small aspect ratio ($h\sim 0.02-0.03$) that favors outward migration after capture in 2:1 resonance, as confirmed by isothermal runs which resulted in a similar outcome for $h \sim 0.02$ and $\alpha \le 10^{-4}$. We also performed N-body runs of the outer Solar System starting from the results of our hydrodynamical simulations and including 2-3 ice giants. After dispersal of the gaseous disk, a Nice model instability starting with Jupiter and Saturn in 2:1 resonance results in good Solar Systems analogs. We conclude that in a cold Solar Nebula, the 2:1 resonance between Jupiter and Saturn can lead to outward migration of the system, and this may represent an alternative scenario for the evolution of the Solar System.

Dynamical corotation torques on low-mass planets

We study torques on migrating low-mass planets in locally isothermal discs. Previous work on low-mass planets generally kept the planet on a fixed orbit, after which the torque on the planet was measured. In addition to these static torques, when the planet is allowed to migrate it experiences dynamical torques, which are proportional to the migration rate and whose sign depends on the background vortensity gradient. We show that in discs a few times more massive than the minimum-mass solar nebula, these dynamical torques can have a profound impact on planet migration. Inward migration can be slowed down significantly, and if static torques lead to outward migration, dynamical torques can take over, taking the planet beyond zero-torque lines set by saturation of the corotation torque in a runaway fashion. This means that the region in non-isothermal discs, where outward migration is possible, can be larger than what would be concluded from static torques alone.

The Herschel view of circumstellar discs: a multiwavelength study of Chamaeleon-I

We present the results of a multi-wavelength study of circumstellar discs around 44 young stellar objects in the 3 Myr old nearby Chamaeleon I star-forming region. In particular, we explore the far-infrared/submm regime using Herschel fluxes. We show that Herschel fluxes at 160-500$\,\mu$m can be used to derive robust estimates of the disc mass. The median disc mass is 0.005$M_{\odot}$ for a sample of 28 Class IIs and 0.006$M_{\odot}$ for 6 transition disks (TDs). The fraction of objects in Chamaeleon-I with at least the `minimum mass solar nebula' is 2-7%. This is consistent with previously published results for Taurus, IC348, $\rho$ Oph. Diagrams of spectral slopes show the effect of specific evolutionary processes in circumstellar discs. Class II objects show a wide scatter that can be explained by dust settling. We identify a continuous trend from Class II to TDs. Including Herschel fluxes in this type of analysis highlights the diversity of TDs. We find that TDs are not significantly different to Class II discs in terms of far-infrared luminosity, disc mass or degree of dust settling. This indicates that inner dust clearing occurs independently from other evolutionary processes in the discs.

CONSTRAINING THE DISK MASSES OF THE CLASS I BINARY PROTOSTAR GV Tau

We present new spatially resolved 1.3 mm imaging with CARMA of the GV Tau system. GV Tau is a Class I binary protostar system in the Taurus Molecular Cloud, the components of which are separated by 12. Each protostar is surrounded by a protoplanetary disk, and the pair may be surrounded by a circumbinary envelope. We analyze the data using detailed radiative transfer modeling of the system. We create synthetic protostar model spectra, images, and visibilities and compare them with CARMA 1.3 mm visibilities, a Hubble Space Telescope near-infrared scattered light image, and broadband spectral energy distributions from the literature to study the disk masses and geometries of the GV Tau disks. We show that the protoplanetary disks around GV Tau fall near the lower end of estimates of the Minimum Mass Solar Nebula, and may have just enough mass to form giant planets. When added to the sample of Class I protostars from Eisner, we confirm that Class I protostars are on average more massive than their Class II counterparts. This suggests that substantial dust grain processing occurs between the Class I and Class II stages, and may help to explain why the Class II protostars do not appear to have, on average, enough mass in their disks to form giant planets.

THE FORMATION OF URANUS AND NEPTUNE: CHALLENGES AND IMPLICATIONS FOR INTERMEDIATE-MASS EXOPLANETS

In this paper we investigate the formation of Uranus and Neptune, according to the core-nucleated accretion model, considering formation locations ranging from 12 to 30 AU from the Sun, and with various disk solid-surface densities and core accretion rates. It is shown that in order to form Uranus-like and Neptune-like planets in terms of final mass and solid-to-gas ratio, very specific conditions are required. We also show that when recently proposed high solid accretion rates are assumed, along with solid surface densities about 10 times those in the minimum-mass solar nebula, the challenge in forming Uranus and Neptune at large radial distances is no longer the formation timescale, but is rather finding agreement with the final mass and composition of these planets. In fact, these conditions are more likely to lead to gas-giant planets. Scattering of planetesimals by the forming planetary core is found to be an important effect at the larger distances. Our study emphasizes how (even slightly) different conditions in the protoplanetary disk and the birth environment of the planetary embryos can lead to the formation of very different planets in terms of final masses and compositions (solid-to-gas ratios), which naturally explains the large diversity of intermediate-mass exoplanets.

Old pre-main-sequence stars

Young stars show evidence of accretion discs which evolve quickly and disperse with an e-folding time of $\sim$ 3Myr. This is in striking contrast with recent observations that suggest evidence for numerous $>30$ Myr old stars with an accretion disc in large star-forming complexes. We consider whether these observations of apparently old accretors could be explained by invoking Bondi-Hoyle accretion to rebuild a new disc around these stars during passage through a clumpy molecular cloud. We combine a simple Monte Carlo model to explore the capture of mass by such systems with a viscous evolution model to infer the levels of accretion that would be observed. We find that a significant fraction of stars may capture enough material via the Bondi-Hoyle mechanism to rebuild a disc of mass $\gtrsim$ 1 minimum-mass solar nebula, and $\lesssim 10\%$ accrete at observable levels at any given time. A significant fraction of the observed old accretors may be explained with our proposed mechanism. Such accretion may provide a chance for a second epoch of planet formation, and have unpredictable consequences for planetary evolution.

THE SPITZER INFRARED SPECTROGRAPH DEBRIS DISK CATALOG. I. CONTINUUM ANALYSIS OF UNRESOLVED TARGETS

During the Spitzer Space Telescope cryogenic mission, Guaranteed Time Observers, Legacy Teams, and General Observers obtained Infrared Spectrograph (IRS) observations of hundreds of debris disk candidates. We calibrated the spectra of 571 candidates, including 64 new IRAS and Multiband Imaging Photometer for Spitzer (MIPS) debris disks candidates, modeled their stellar photospheres, and produced a catalog of excess spectra for unresolved debris disks. For 499 targets with IRS excess but without strong spectral features (and a subset of 420 targets with additional MIPS 70 μm observations), we modeled the IRS (and MIPS data) assuming that the dust thermal emission was well-described using either a one- or two-temperature blackbody model. We calculated the probability for each model and computed the average probability to select among models. We found that the spectral energy distributions for the majority of objects (~66%) were better described using a two-temperature model with warm (T gr ~ 100-500 K) and cold (T gr ~ 50-150 K) dust populations analogous to zodiacal and Kuiper Belt dust, suggesting that planetary systems are common in debris disks and zodiacal dust is common around host stars with ages up to ~1 Gyr. We found that younger stars generally have disks with larger fractional infrared luminosities and higher grain temperatures and that higher-mass stars have disks with higher grain temperatures. We show that the increasing distance of dust around debris disks is inconsistent with self-stirred disk models, expected if these systems possess planets at 30-150 AU. Finally, we illustrate how observations of debris disks may be used to constrain the radial dependence of material in the minimum mass solar nebula.

ACCRETION OF JUPITER-MASS PLANETS IN THE LIMIT OF VANISHING VISCOSITY

In the core-accretion model the nominal runaway gas-accretion phase brings most planets to multiple Jupiter masses. However, known giant planets are predominantly Jupiter-mass bodies. Obtaining longer timescales for gas accretion may require using realistic equations of states, or accounting for the dynamics of the circumplanetary disk (CPD) in low-viscosity regime, or both. Here we explore the second way using global, three-dimensional isothermal hydrodynamical simulations with 8 levels of nested grids around the planet. In our simulations the vertical inflow from the circumstellar disk (CSD) to the CPD determines the shape of the CPD and its accretion rate. Even without prescribed viscosity Jupiter's mass-doubling time is $\sim 10^4$ years, assuming the planet at 5.2 AU and a Minimum Mass Solar Nebula. However, we show that this high accretion rate is due to resolution-dependent numerical viscosity. Furthermore, we consider the scenario of a layered CSD, viscous only in its surface layer, and an inviscid CPD. We identify two planet-accretion mechanisms that are independent of the viscosity in the CPD: (i) the polar inflow -- defined as a part of the vertical inflow with a centrifugal radius smaller than 2 Jupiter-radii and (ii) the torque exerted by the star on the CPD. In the limit of zero effective viscosity, these two mechanisms would produce an accretion rate 40 times smaller than in the simulation.

Planetesimal-driven migration of terrestrial planet embryos

We develop a model for planetesimal-driven migration (PDM) in the context of rocky planetary embryos in the terrestrial planet region during the runaway and oligarchic growth phases of inner planet formation. We develop this model by first showing that there are five necessary and sufficient criteria that must be simultaneously satisfied in order for a rocky inner Solar System embryo to migrate via PDM. To investigate which embryos within a given disk satisfy the five criteria, we have developed a Monte Carlo planetesimal merger code that simulates the growth of embryos from a planetesimal disk with nebular gas. The results of our Monte Carlo planetesimal merger code suggest that, for typical values of the minimum mass solar nebula for the inner Solar System, an average of 0.2 embryos capable of PDM emerge over the lifetime of the disk. Many disks in our simulations produce no migration candidates, but some produced as many as 3. The number of embryos that experience PDM in a disk increases with increasing disk mass and decreasing il planetesimal mass, although we were not able to simulate disks where the average initial planetesimal size was smaller than 50km. For disks 4× more massive than the standard minimum mass solar nebula, we estimate that an average of 1.5 embryos capable of PDM emerge, with some producing as many as 7.

Effect of lift force on the aerodynamics of dust grains in the protoplanetary disk

We newly introduce lift force into the aerodynamics of dust grains in the protoplanetary disk. Although many authors have so far investigated the effects of the drag force, gravitational force and electric force on the dust grains, the lift force has never been considered as a force exerted on the dust grains in the gas disk. If the grains are spinning and moving in the fluid, then the lift force is exerted on them. We show in this paper that the dust grains can be continuously spinning due to the frequent collisions so that the lift force continues to be exerted on them, which is valid in a certain parameter space where the grain size is larger than ~ 1 m and where the distance from the central star is larger than 1 AU for the minimum mass solar nebula. In addition, we estimate the effects of the force on the grain motion and obtain the result that the mean relative velocity between the grains due to the lift force is comparable to the gas velocity in the Kepler rotational frame when the Stokes number and lift-drag ratio are both ~ 1. This estimation is performed under the assumptions of the steady state and the isotropic spin angular momentum. We also estimate the mean relative velocity when the grains keep spinning and conclude that the lift force marginally affects the mean relative velocity in the minimum mass solar nebula. If there is a grain-concentrated part in the disk, the relative velocity due to the lift force may dominate there because of high collision rate.

INSIDE-OUT PLANET FORMATION

The compact multi-transiting planet systems discovered by Kepler challenge planet formation theories. Formation in situ from disks with radial mass surface density, $\Sigma$, profiles similar to the minimum mass solar nebula (MMSN) but boosted in normalization by factors $\gtrsim 10$ has been suggested. We propose that a more natural way to create these planets in the inner disk is formation sequentially from the inside-out via creation of successive gravitationally unstable rings fed from a continuous stream of small (~cm--m size) "pebbles", drifting inwards via gas drag. Pebbles collect at the pressure maximum associated with the transition from a magneto-rotational instability (MRI)-inactive ("dead zone") region to an inner MRI-active zone. A pebble ring builds up until it either becomes gravitationally unstable to form an $\sim 1\ M_\oplus$ planet directly or induces gradual planet formation via core accretion. The planet may undergo Type I migration into the active region, allowing a new pebble ring and planet to form behind it. Alternatively if migration is inefficient, the planet may continue to accrete from the disk until it becomes massive enough to isolate itself from the accretion flow. A variety of densities may result depending on the relative importance of residual gas accretion as the planet approaches its isolation mass. The process can repeat with a new pebble ring gathering at the new pressure maximum associated with the retreating dead zone boundary. Our simple analytical model for this scenario of inside-out planet formation yields planetary masses, relative mass scalings with orbital radius, and minimum orbital separations consistent with those seen by Kepler. It provides an explanation of how massive planets can form with tightly-packed and well-aligned system architectures, starting from typical protoplanetary disk properties.

COAGULATION CALCULATIONS OF ICY PLANET FORMATION AROUND 0.1-0.5M☉STARS: SUPER-EARTHS FROM LARGE PLANETESTIMALS

We investigate formation mechanisms for icy super-Earth mass planets orbiting at 2-20 AU around 0.1-0.5 solar mass stars. A large ensemble of coagulation calculations demonstrates a new formation channel: disks composed of large planetesimals with radii of 30-300 km form super-Earths on time scales of roughly 1 Gyr. In other gas-poor disks, a collisional cascade grinds planetesimals to dust before the largest planets reach super-Earth masses. Once icy Earth-mass planets form, they migrate through the leftover swarm of planetesimals at rates of 0.01-1 AU per Myr. On time scales of 10 Myr to 1 Gyr, many of these planets migrate through the disk of leftover planetesimals from semimajor axes of 5-10 AU to 1-2 AU. A few per cent of super-Earths might migrate to semimajor axes of 0.1-0.2 AU. When the disk has an initial mass comparable with the minimum mass solar nebula scaled to the mass of the central star, the predicted frequency of super-Earths matches the observed frequency.

Trace element geochemistry of CR chondrite metal

AbstractWe report trace element analyses by laser ablation inductively coupled plasma mass spectrometry (LA‐ICP‐MS) of metal grains from nine different CR chondrites, distinguishing grains from chondrule interior (“interior grains”), chondrule surficial shells (“margin grains”), and the matrix (“isolated grains”). Save for a few anomalous grains, Ni‐normalized trace element patterns are similar for all three petrographic settings, with largely unfractionated refractory siderophile elements and depleted volatile Au, Cu, Ag, S. All three types of grains are interpreted to derive from a common precursor approximated by the least‐melted, fine‐grained objects in CR chondrites. This also excludes recondensation of metal vapor as the origin of the bulk of margin grains. The metal precursors were presumably formed by incomplete condensation, with evidence for high‐temperature isolation of refractory platinum‐group‐element (PGE)‐rich condensates before mixing with lower temperature PGE‐depleted condensates. The rounded shape of the Ni‐rich, interior grains shows that they were molten and that they equilibrated with silicates upon slow cooling (1–100 K h−1), largely by oxidation/evaporation of Fe, hence their high Pd content, for example. We propose that Ni‐poorer, amoeboid margin grains, often included in the pyroxene‐rich periphery common to type I chondrules, result from less intense processing of a rim accreted onto the chondrule subsequent to the melting event recorded by the interior grains. This means either that there were two separate heating events, which formed olivine/interior grains and pyroxene/margin grains, respectively, between which dust was accreted around the chondrule, or that there was a single high‐temperature event, of which the chondrule margin records a late “quenching phase,” in which case dust accreted onto chondrules while they were molten. In the latter case, high dust concentrations in the chondrule‐forming region (at least three orders of magnitude above minimum mass solar nebula models) are indicated.

TURBULENT DISKS ARE NEVER STABLE: FRAGMENTATION AND TURBULENCE-PROMOTED PLANET FORMATION

A fundamental assumption in our understanding of disks is that when the Toomre Q>>1, the disk is stable against fragmentation into self-gravitating objects (and so cannot form planets via direct collapse). But if disks are turbulent, this neglects a spectrum of stochastic density fluctuations that can produce rare, high-density mass concentrations. Here, we use a recently-developed analytic framework to predict the statistics of these fluctuations, i.e. the rate of fragmentation and mass spectrum of fragments formed in a turbulent Keplerian disk. Turbulent disks are never completely stable: we calculate the (always finite) probability of forming self-gravitating structures via stochastic turbulent density fluctuations in such disks. Modest sub-sonic turbulence above Mach number ~0.1 can produce a few stochastic fragmentation or 'direct collapse' events over ~Myr timescales, even if Q>>1 and cooling is slow (t_cool>>t_orbit). In trans-sonic turbulence this extends to Q~100. We derive the true Q-criterion needed to suppress such events, which scales exponentially with Mach number. We specify to turbulence driven by MRI, convection, or spiral waves, and derive equivalent criteria in terms of Q and the cooling time. Cooling times >~50*t_dyn may be required to completely suppress fragmentation. These gravoturbulent events produce mass spectra peaked near ~M_disk*(Q*M_disk/M_star)^2 (rocky-to-giant planet masses, increasing with distance from the star). We apply this to protoplanetary disk models and show that even minimum mass solar nebulae could experience stochastic collapse events, provided a source of turbulence.

TURBULENCE IN THE OUTER REGIONS OF PROTOPLANETARY DISKS. II. STRONG ACCRETION DRIVEN BY A VERTICAL MAGNETIC FIELD

We carry out a series of local, vertically stratified shearing box simulations of protoplanetary disks that include ambipolar diffusion and a net vertical magnetic field. The ambipolar diffusion profiles we employ correspond to 30AU and 100AU in a minimum mass solar nebula (MMSN) disk model, which consists of a far-UV-ionized surface layer and low-ionization disk interior. These simulations serve as a follow up to Simon et al. (2013), in which we found that without a net vertical field, the turbulent stresses that result from the magnetorotational instability (MRI) are too weak to account for observed accretion rates. The simulations in this work show a very strong dependence of the accretion stresses on the strength of the background vertical field; as the field strength increases, the stress amplitude increases. For gas to magnetic pressure ratios of 1e4 and 1e5, we find accretion rates between 1e-8 and 1e-7 solar masses per year. These accretion rates agree with observational constraints, suggesting a vertical magnetic field strength between 60 and 200 microgauss at 30AU and 10 and 30 microgauss at 100AU in a MMSN disk. Furthermore, the stress has a non-negligible component due to a magnetic wind. For sufficiently strong vertical field strengths, MRI turbulence is quenched, and the flow becomes largely laminar, with accretion proceeding through large scale correlations in the radial and toroidal field components as well as through the magnetic wind. In all simulations, the presence of a low ionization region near the disk mid-plane, which we call the ambipolar damping zone, results in reduced stresses there.

The formation of systems with closely spaced low-mass planets and the application to Kepler-36

The Kepler-36 system consists of two planets that are spaced unusually close together, near the 7:6 mean motion resonance. While it is known that mean motion resonances can easily form by convergent migration, Kepler-36 is an extreme case due to the close spacing and the relatively high planet masses of 4 and 8 times that of the Earth. In this paper, we investigate whether such a system can be obtained by interactions with the protoplanetary disc. These discs are thought to be turbulent and exhibit density fluctuations which might originate from the magneto-rotational instability. We adopt a realistic description for stochastic forces due to these density fluctuations and perform both long term hydrodynamical and N-body simulations. Our results show that planets in the Kepler-36 mass range can be naturally assembled into a closely spaced planetary system for a wide range of migration parameters in a turbulent disc similar to the minimum mass solar nebula. The final orbits of our formation scenarios tend to be Lagrange stable, even though large parts of the parameter space are chaotic and unstable.

PROTOSTELLAR DISK EVOLUTION OVER MILLION-YEAR TIMESCALES WITH A PRESCRIPTION FOR MAGNETIZED TURBULENCE

Magnetorotational instability (MRI) is the most promising mechanism behind accretion in low-mass protostellar disks. Here we present the first analysis of the global structure and evolution of non-ideal MRI-driven T-Tauri disks on million-year timescales. We accomplish this in a 1+1D simulation by calculating magnetic diffusivities and utilizing turbulence activity criteria to determine thermal structure and accretion rate without resorting to a 3-D magnetohydrodynamical (MHD) simulation. Our major findings are as follows. First, even for modest surface densities of just a few times the minimum-mass solar nebula, the dead zone encompasses the giant planet-forming region, preserving any compositional gradients. Second, the surface density of the active layer is nearly constant in time at roughly 10 g/cm2, which we use to derive a simple prescription for viscous heating in MRI-active disks for those who wish to avoid detailed MHD computations. Furthermore, unlike a standard disk with constant-alpha viscosity, the disk midplane does not cool off over time, though the surface cools as the star evolves along the Hayashi track. The ice line is firmly in the terrestrial planet-forming region throughout disk evolution and can move either inward or outward with time, depending on whether pileups form near the star. Finally, steady-state mass transport is a poor description of flow through an MRI-active disk. We caution that MRI activity is sensitive to many parameters, including stellar X-ray flux, grain size, gas/small grain mass ratio and magnetic field strength, and we have not performed an exhaustive parameter study here.