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...

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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.

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Ring shaped dust accumulation in transition disks

In planetary systems with two or more giant planets, dynamical instabilities can lead to collisions or ejections through strong planet-planet scattering. Previous studies for initial conditions with two equal-mass planets revealed two discrepancies between the results of simulations and the observed orbits of exoplanets: potentially frequent collisions between giant planets and a narrow distribution of final eccentricities. We show that simulations with two unequal-mass planets starting on nearly circular orbits predict fewer collisions and a broader range of final eccentricities. Thus, the two-planet scattering model can reproduce the observed eccentricities with a plausible distribution of planet mass ratios. The model also predicts a maximum eccentricity of 0.8, independent of the distribution of planet mass ratios, provided that both planets are initially placed on nearly circular orbits. This compares favorably with observations and will be tested by future planet discoveries. Moreover, the combination of planet-planet scattering and tidal circularization may explain the existence of some giant planets with very short period orbits. Orbital migration due to planet scattering could play an important role in explaining the distribution of orbital periods found by radial velocity surveys. We also reexamine and discuss various possible correlations between eccentricities and other properties of exoplanets. We find that radial velocity observations are consistent with planet eccentricities being correlated with the ratio of the escape velocity from the planet's surface relative to the escape velocity from the host star at the planet's location. We demonstrate that the observed distribution of planet masses, periods, and eccentricities can provide constraints for models of planet formation and evolution.

https://iopscience.iop.org/article/10.1086/590926/pdf

Origins of Eccentric Extrasolar Planets: Testing the Planet-Planet Scattering Model

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.

A comparative study of disc-planet interaction

We study the evolution and emission of circumbinary disks around close classical T Tauri binary systems. High resolution numerical hydrodynamical simulations are employed to model a system consisting of a central eccentric binary star within an irradiated accretion disk. A detailed energy balance including viscous heating, radiative cooling and irradiation from the central star is applied to calculate accurately the emitted spectral energy distribution. Numerical simulations using two different methods, the previously developed Dual-Grid technique with a finite difference discretization, and the Smoothed Particle Hydrodynamics method are employed to compare the hydrodynamical features and strengthen our conclusions. Physical parameters of the setup are chosen to model the close systems of DQ Tau and AK Sco. Using the self-consistent models, we are able to fit the observed spectral energy distributions by constraining parameters such as disk mass, density profile and radial extension for those systems. We find that the incorporation of irradiation effects is necessary to obtain correct disk temperatures.

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Evolution of irradiated Circumbinary Disks

We have performed Smoothed Particle Hydrodynamics (SPH) simulations to study the time evolution of one and two protoplanets embedded in a protoplanetary accretion disc. We investigate accretion and migration rates of a single protoplanet depending on several parameters of the protoplanetary disc, mainly viscosity and scale height. Additionally, we consider the influence of a second protoplanet in a long time simulation and examine the migration of the two planets in the disc, especially the growth of eccentricity and chaotic behaviour. One aim of this work is to establish the feasibility of SPH for such calculations considering that usually only grid-based methods are adopted. To resolve shocks and to prevent particle penetration, we introduce a new approach for an artificial viscosity, which consists of an additional artificial bulk viscosity term in the SPH-representation of the Navier-Stokes equation. This allows for an accurate treatment of the physical kinematic viscosity to describe the shear, without the use of an artificial shear viscosity.

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C. Schaefer

Papers

A comparative study of disc-planet interaction

Evolution of irradiated Circumbinary Disks

Simulations of planet-disc interactions using Smoothed Particle Hydrodynamics

Simulations of planet-disc interactions with SPH.