We present two new Lagrangian methods for hydrodynamics, in a systematic comparison with moving-mesh, smoothed particle hydrodynamics (SPH), and stationary (non-moving) grid methods. The new methods are designed to simultaneously capture advantages of both SPH and grid-based/adaptive mesh refinement (AMR) schemes. They are based on a kernel discretization of the volume coupled to a high-order matrix gradient estimator and a Riemann solver acting over the volume ‘overlap’. We implement and test a parallel, second-order version of the method with self-gravity and cosmological integration, in the code gizmo:1 this maintains exact mass, energy and momentum conservation; exhibits superior angular momentum conservation compared to all other methods we study; does not require ‘artificial diffusion’ terms; and allows the fluid elements to move with the flow, so resolution is automatically adaptive. We consider a large suite of test problems, and find that on all problems the new methods appear competitive with moving-mesh schemes, with some advantages (particularly in angular momentum conservation), at the cost of enhanced noise. The new methods have many advantages versus SPH: proper convergence, good capturing of fluid-mixing instabilities, dramatically reduced ‘particle noise’ and numerical viscosity, more accurate sub-sonic flow evolution, and sharp shock-capturing. Advantages versus non-moving meshes include: automatic adaptivity, dramatically reduced advection errors and numerical overmixing, velocity-independent errors, accurate coupling to gravity, good angular momentum conservation and elimination of ‘grid alignment’ effects. We can, for example, follow hundreds of orbits of gaseous discs, while AMR and SPH methods break down in a few orbits. However, fixed meshes minimize ‘grid noise’. These differences are important for a range of astrophysical problems.

A new class of accurate, mesh-free hydrodynamic simulation methods

As planets form and grow within gaseous protoplanetary disks, the mutual gravitational interaction between the disk and planet leads to the exchange of angular momentum and migration of the planet. We review current understanding of disk-planet interactions, focusing in particular on physical processes that determine the speed and direction of migration. We describe the evolution of low-mass planets embedded in protoplanetary disks and examine the influence of Lindblad and corotation torques as a function of the disk properties. The role of the disk in causing the evolution of eccentricities and inclinations is also discussed. We describe the rapid migration of intermediate-mass planets that may occur as a runaway process and examine the transition to gap formation and slower migration driven by the viscous evolution of the disk for massive planets. The roles and influence of disk self-gravity and magnetohydrodynamic turbulence are discussed in detail, as a function of the planet mass, as is the evolution...

/pdf/planet-disk-interaction-and-orbital-evolution-30h99z7v4g.pdf

Planet-Disk Interaction and Orbital Evolution

We study the torque on low-mass planets embedded in protoplanetary discs in the two-dimensional approximation, incorporating non-isothermal eects. 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 diusion act to keep the horseshoe drag unsaturated. We check this formula against numerical hydrodynamical simulations, using three independent numerical methods, and nd good agreement.

A torque formula for non-isothermal type I planetary migration – I. Unsaturated horseshoe drag

Context. Transition disks are believed to be the final stages of protoplanetary disks, during which a forming planetary system or photoevaporation processes open a gap in the inner disk, drastically changing the disk structure. From theoretical arguments it is expected that dust growth, fragmentation and radial drift are strongly influenced by gas disk structure, and pressure bumps in disks have been suggested as key features that may allow grains to converge and grow efficiently.Aims. We want to study how the presence of a large planet in a disk influences the growth and radial distribution of dust grains, and how observable properties are linked to the mass of the planet.Methods. We combined two-dimensional hydrodynamical disk simulations of disk-planet interactions with state-of-the-art coagulation/fragmentation models to simulate the evolution of dust in a disk, which has a gap created by a massive planet. We computed images at different wavelengths and illustrated our results using the example of the transition disk LkCa15.Results. The gap opened by a planet and the long-range interaction between the planet and the outer disk create a single large pressure bump outside the planetary orbit. Millimeter-sized particles form and accumulate at the pressure maximum and naturally produce ring-shaped sub-millimeter emission that is long-lived because radial drift no longer depletes the large grain population of the disk. For large planet masses around 9 M Jup , the pressure maximum and, therefore, the ring of millimeter particles is located at distances that can be more than twice the star-planet separation, creating a large spatial separation between the gas inner edge of the outer disk and the peak millimeter emission. Smaller grains do get closer to the gap and we predict how the surface brightness varies at different wavelengths.

/pdf/ring-shaped-dust-accumulation-in-transition-disks-1i8fvrl1er.pdf

Ring shaped dust accumulation in transition disks

We use two-dimensional hydrodynamic simulations of viscous disks to examine whether dynamically interacting multiple giant planets can explain the large gaps (spanning over one order of magnitude in radius) inferred for the transitional and pre-transitional disks around T Tauri stars. In the absence of inner disk dust depletion, we find that it requires three to four giant planets to open up large enough gaps to be consistent with inferences from spectral energy distributions, because the gap width is limited by the tendency of the planets to be driven together into 2:1 resonances. With very strong tidal torques and/or rapid planetary accretion, fewer planets can also generate a large cavity interior to the locally formed gap(s) by preventing outer disk material from moving in. In these cases, however, the reduction of surface density produces a corresponding reduction in the inner disk accretion rate onto the star; this makes it difficult to explain the observed accretion rates of the pre-transitional/transitional disks. We find that even with four planets in disks, additional substantial dust depletion is required to explain observed disk gaps/holes. Substantial dust settling and growth, with consequent significant reductions in optical depths, is inferred for typical T Tauri disks in any case, and an earlier history of dust growth is consistent with the hypothesis that pre-transitional/transitional disks are explained by the presence of giant planets. We conclude that the depths and widths of gaps and disk accretion rates in pre-transitional/transitional disks cannot be reproduced by a planet-induced gap opening scenario alone. Significant dust depletion is also required within the gaps/holes. Order-of-magnitude estimates suggest that the mass of small dust particles (1 μm) relative to the gas must be depleted to 10 −5 to 10 −2 of the interstellar medium value, implying a very efficient mechanism of small dust removal or dust growth.

/pdf/transitional-and-pre-transitional-disks-gap-opening-by-4v7oh8ryff.pdf

Transitional and pre-transitional disks: gap opening by multiple planets?

We perform numerical simulations of a disc-planet system using various grid-based and smoothed particle hydrodynamics (SPH) codes. The tests are run for a simple setup where Jupiter and Neptune mass planets on a circular orbit open a gap in a protoplanetary disc during a few hundred orbital periods. We compare the surface density contours, potential vorticity and smoothed radial profiles at several times. The disc mass and gravitational torque time evolution are analysed with high temporal resolution. There is overall consistency between the codes. The density profiles agree within about 5 per cent for the Eulerian simulations. The SPH results predict the correct shape of the gap although have less resolution in the low-density regions and weaker planetary wakes. The disc masses after 200 orbital periods agree within 10 per cent. The spread is larger in the tidal torques acting on the planet which agree within a factor of 2 at the end of the simulation. In the Neptune case, the dispersion in the torques is greater than for Jupiter, possibly owing to the contribution from the not completely cleared region close to the planet.

/pdf/a-comparative-study-of-disc-planet-interaction-1y9fphq7ij.pdf

A comparative study of disc–planet interaction

We perform numerical simulations of a disc-planet system using various grid-based and smoothed particle hydrodynamics (SPH) codes. The tests are run for a simple setup where Jupiter and Neptune mass planets on a circular orbit open a gap in a protoplanetary disc during a few hundred orbital periods. We compare the surface density contours, potential vorticity and smoothed radial profiles at several times. The disc mass and gravitational torque time evolution are analyzed with high temporal resolution. There is overall consistency between the codes. The density profiles agree within about 5% for the Eulerian simulations while the SPH results predict the correct shape of the gap although have less resolution in the low density regions and weaker planetary wakes. The disc masses after 200 orbital periods agree within 10%. The spread is larger in the tidal torques acting on the planet which agree within a factor 2 at the end of the simulation. In the Neptune case the dispersion in the torques is greater than for Jupiter, possibly owing to the contribution from the not completely cleared region close to the planet.

E. J. Delgado-Donate

Papers

A comparative study of disc–planet interaction

A comparative study of disc-planet interaction