arXiv subjects

S. -J. Paardekooper

Publications and source records attributed to S. -J. Paardekooper.

At least 19 recordsLinked to original sources

arXiv2026

A multi-fluid approach for polydisperse pebble accretion: From particles to fluids, establishing the multifluid framework

Pebble accretion offers an efficient pathway to form planets, driven by a constant supply of inward drifting mass and an accretion efficiency enhanced by gas drag. While most studies assume a single pebble size (monodisperse), real discs contain a range of sizes (polydisperse) that drift, interact, and accrete at different rates. We aim to model polydisperse pebble accretion with a fluid approach, validating the method and exploring how gas disc evolution, solid-to-gas back-reaction, and a polydisperse size distribution affect growth. We used FARGO3D, modified to allow pebble accretion, to run 2D hydrodynamic simulations in a global disc with multiple pebble species representing an underlying continuous pebble size distribution. With our multi-fluid approach, we find values for pebble accretion efficiency consistent with earlier studies for a static gas disc. This confirms that our approach gives an accurate representation of pebble accretion. Evolving the gas disc, we find lower efficiencies compared to an unperturbed gas disc for high Stokes numbers ($\gtrsim 0.3$) and higher efficiencies for smaller Stokes numbers ($\lesssim0.3$). This effect increases for higher planet masses. The accretion rate is mostly dominated by the highest Stokes numbers in our parameter study ($\mathrm{St}\in[10^{-2},10^0]$). The ratio we find between the polydisperse and monodisperse pebble accretion rates is higher than previous estimations. We constructed a multi-fluid model framework capable of accurately simulating polydisperse pebble accretion consistent with previous studies. This framework offers advantages for simulating higher planet masses and for modelling multiple pebble species coupled to the gas. We find that the protoplanet's perturbation of the gas-disc lowers the accretion rate when assuming an MRN-distribution of solids.

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arXiv2018

Investigating the possibility of reversing giant planet migration via gap edge illumination

A massive planet in a protoplanetary disc will open a gap in the disc material which acts as a transition between Type I and Type II planetary migration. Type II migration is slower than Type I migration, however it is still desirable to slow down Type II migration to allow gas giant planets with semi-major axis in the range 5 to 10AU to exist, similarly to our Solar system. We investigate a method of slowing down and reversing Type II migration by heating the outer gap edge due to incident radiation from the central star. Using an approximate vertically averaged heating method we find that Type II migration can be slowed or in extreme cases reversed if we assume near maximum allowed irradiation from the central star. Therefore, we believe this is a very interesting phenomenon that should be investigated in greater detail using three dimensional hydrodynamic and radiative transfer simulations.

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arXiv2016

Modelling circumbinary protoplanetary disks II. Gas disk feedback on planetesimal dynamical and collisional evolution in the circumbinary systems Kepler-16 and 34

Aims. We investigate the feasibility of planetesimal growth in circumbinary protoplanetary disks around the observed systems Kepler- 16 and Kepler-34 under the gravitational influence of a precessing eccentric gas disk. Methods. We embed the results of our previous hydrodynamical simulations of protoplanetary disks around binaries into an N-body code to perform 3D, high-resolution, inter-particle gravity-enabled simulations of planetesimal growth and dynamics that include the gravitational force imparted by the gas. Results. Including the full, precessing asymmetric gas disk generates high eccentricity orbits for planetesimals orbiting at the edge of the circumbinary cavity, where the gas surface density and eccentricity have their largest values. The gas disk is able to efficiently align planetesimal pericenters in some regions leading to phased, non-interacting orbits. Outside of these areas eccentric planetesimal orbits become misaligned and overlap leading to crossing orbits and high relative velocities during planetesimal collisions. This can lead to an increase in the number of erosive collisions that far outweighs the number of collisions that result in growth. Gravitational focusing from the static axisymmetric gas disk is weak and does not significantly alter collision outcomes from the gas free case. Conclusions. Due to asymmetries in the gas disk, planetesimals are strongly perturbed onto highly eccentric orbits. Where planetesimals orbits are not well aligned, orbit crossings lead to an increase in the number of erosive collisions. This makes it difficult for sustained planetesimal accretion to occur at the location of Kepler-16b and Kepler-34b and we therefore rule out in-situ growth. This adds further support to our initial suggestions that most circumbinary planets should form further out in the disk and migrate inwards.

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arXiv2016

The evolution of a supermassive retrograde binary embedded in an accretion disk

In this note we discuss the main results of a study of a massive binary with unequal mass ratio, q, embedded in an accretion disk, with its orbital rotation being opposed to that of the disk. When the mass ratio is sufficiently large, a gap opens in the disk, but the mechanism of gap formation is very different from the prograde case. Inward migration occurs on a timescale of t_ev ~ M_p/(dot M), where M_p is the mass of the less massive component (the perturber), and dot M is the accretion rate. When q<< 1, the accretion takes place mostly onto the more massive component, with the accretion rate onto the perturber being smaller than, or of order of, q^(1/3)M. However, this rate increases when supermassive binary black holes are considered and gravitational wave emission is important. We estimate a typical duration of time for which the accretion onto the perturber and gravitational waves could be detected.

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arXiv2015

Modelling circumbinary protoplanetary disks: I. Fluid simulations of the Kepler-16 and 34 systems

The Kepler mission's discovery of a number of circumbinary planets orbiting close (a_p < 1.1 au) to the stellar binary raises questions as to how these planets could have formed given the intense gravitational perturbations the dual stars impart on the disk. The gas component of circumbinary protoplanetary disks is perturbed in a similar manner to the solid, planetesimal dominated counterpart, although the mechanism by which disk eccentricity originates differs. This is the first work of a series that aims to investigate the conditions for planet formation in circumbinary protoplanetary disks. We present a number of hydrodynamical simulations that explore the response of gas disks around two observed binary systems: Kepler-16 and Kepler-34. We probe the importance of disk viscosity, aspect-ratio, inner boundary condition, initial surface density gradient, and self-gravity on the dynamical evolution of the disk, as well as its quasi steady-state profile. We find there is a strong influence of binary type on the mean disk eccentricity, e_d, leading to e_d = 0.02 - 0.08 for Kepler-16 and e_d = 0.10 - 0.15 in Kepler-34. The value of alpha-viscosity has little influence on the disk, but we find a strong increase in mean disk eccentricity with increasing aspect-ratio due to wave propagation effects. The choice of inner boundary condition only has a small effect on the surface density and eccentricity of the disk. Our primary finding is that including disk self-gravity has little impact on the evolution or final state of the disk for disks with masses less than 12.5 times that of the minimum-mass solar nebula. This finding contrasts with the results of self-gravity relevance in circumprimary disks, where its inclusion is found to be an important factor in describing the disk evolution.

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arXiv2014

The evolution of a binary in a retrograde circular orbit embedded in an accretion disk

Supermassive black hole binaries may form as a consequence of galaxy mergers. Both prograde and retrograde orbits have been proposed. We study a binary of a small mass ratio, q, in a retrograde orbit immersed in and interacting with a gaseous accretion disk in order to estimate time scales for inward migration leading to coalescence and the accretion rate to the secondary component. We employ both semi-analytic methods and two dimensional numerical simulations, focusing on the case where the binary mass ratio is small but large enough to significantly perturb the disk. We develop the theory of type I migration for this case and determine conditions for gap formation finding that then inward migration occurs on a time scale equal to the time required for one half of the secondary mass to be accreted through the unperturbed disk, with accretion onto the secondary playing only a minor role. The semi-analytic and fully numerical approaches are in good agreement, the former being applicable over long time scales. Inward migration induced by interaction with the disk alleviates the final parsec problem. Accretion onto the secondary does not significantly affect the orbital evolution, but may have observational consequences for high accretion efficiency. The binary may then appear as two sources of radiation rotating around each other. This study should be extended to consider orbits with significant eccentricity and the effects of gravitational radiation at small length scales. Note too that torques acting between a circumbinary disk and a retrograde binary orbit may cause the mutual inclination to increase on a timescale that can be similar to, or smaller than that for orbital evolution, depending on detailed parameters. This is also an aspect for future study (abridged).

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arXiv2014

Convergence of simulations of self-gravitating accretion discs II: Sensitivity to the implementation of radiative cooling and artificial viscosity

Recently it has been suggested that the fragmentation boundary in Smoothed Particle Hydrodynamic (SPH) and FARGO simulations of self-gravitating accretion discs with beta-cooling do not converge as resolution is increased. Furthermore, this recent work suggests that by carefully optimising the artificial viscosity parameters in these codes it can be shown that fragmentation may occur for much longer cooling times than earlier work suggests. If correct, this result is intriguing as it suggests that gas giant planets could form, via direct gravitational collapse, reasonably close to their parent stars. This result is, however, slightly surprising and there have been a number of recent studies suggesting that the result is likely an indication of a numerical problem with the simulations. One suggestion, in particular, is that the SPH results are influenced by the manner in which the cooling is implemented. We extend this work here and show that if the cooling is implemented in a manner that removes a known numerical artefact in the shock regions, the fragmentation boundary converges to a value consistent with earlier work and that fragmentation is unlikely for the long cooling times suggested by this recent work. We also investigate the optimisation of the artificial viscosity parameters and show that the values that appear optimal are likely introducing numerical problems in both the SPH and FARGO simulations. We therefore conclude that earlier predictions for the cooling times required for fragmentation are likely correct and that, as suggested by this earlier work, fragmentation cannot occur in the inner parts (r < 50 au) of typical protostellar discs.

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arXiv2010

Vortex migration in protoplanetary disks

We consider the radial migration of vortices in two-dimensional isothermal gaseous disks. We find that a vortex core, orbiting at the local gas velocity, induces velocity perturbations that propagate away from the vortex as density waves. The resulting spiral wave pattern is reminiscent of an embedded planet. There are two main causes for asymmetries in these wakes: geometrical effects tend to favor the outer wave, while a radial vortensity gradient leads to an asymmetric vortex core, which favors the wave at the side that has the lowest density. In the case of asymmetric waves, which we always find except for a disk of constant pressure, there is a net exchange of angular momentum between the vortex and the surrounding disk, which leads to orbital migration of the vortex. Numerical hydrodynamical simulations show that this migration can be very rapid, on a time scale of a few thousand orbits, for vortices with a size comparable to the scale height of the disk. We discuss the possible effects of vortex migration on planet formation scenarios.

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arXiv2010

A torque formula for non-isothermal Type I planetary migration - II. Effects of diffusion

We study the effects of diffusion on the non-linear corotation torque, or horseshoe drag, in the two-dimensional limit, focusing on low-mass planets for which the width of the horseshoe region is much smaller than the scale height of the disc. In the absence of diffusion, the non-linear corotation torque saturates, leaving only the Lindblad torque. Diffusion of heat and momentum can act to sustain the corotation torque. In the limit of very strong diffusion, the linear corotation torque is recovered. For the case of thermal diffusion, this limit corresponds to having a locally isothermal equation of state. We present some simple models that are able to capture the dependence of the torque on diffusive processes to within 20% of the numerical simulations.

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arXiv2010

Orbital migration of low-mass planets in evolutionary radiative models: Avoiding catastrophic infall

Outward migration of low-mass planets has recently been shown to be a possibility in non-barotropic disks. We examine the consequences of this result in evolutionary models of protoplanetary disks. Planet migration occurs towards equilibrium radii with zero torque. These radii themselves migrate inwards because of viscous accretion and photoevaporation. We show that as the surface density and temperature fall, the planet orbital migration and disk depletion timescales eventually become comparable, with the precise timing depending on the mass of the planet. When this occurs, the planet decouples from the equilibrium radius. At this time, however, the gas surface density is already too low to drive substantial further migration. A higher mass planet, of 10 Earth masses, can open a gap during the late evolution of the disk, and stops migrating. Low mass planets, with 1 or 0.1 Earth masses, released beyond 1 AU in our models, avoid migrating into the star. Our results provide support for the reduced migration rates adopted in recent planet population synthesis models.

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arXiv2010

Planetesimal collisions in binary systems

We study the collisional evolution of km-sized planetesimals in tight binary star systems to investigate whether accretion towards protoplanets can proceed despite the strong gravitational perturbations from the secondary star. The orbits of planetesimals are numerically integrated in two dimensions under the influence of the two stars and gas drag. The masses and orbits of the planetesimals are allowed to evolve due to collisions with other planetesimals and accretion of collisional debris. In addition, the mass in debris can evolve due to planetesimal-planetesimal collisions and the creation of new planetesimals. We show that it is possible in principle for km-sized planetesimals to grow by two orders of magnitude in size if the efficiency of planetesimal formation is relatively low. We discuss the limitations of our two-dimensional approach.

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arXiv2009

Disc-planet interactions in sub-keplerian discs

One class of protoplanetary disc models, the X-wind model, predicts strongly subkeplerian orbital gas velocities, a configuration that can be sustained by magnetic tension. We investigate disc-planet interactions in these subkeplerian discs, focusing on orbital migration for low-mass planets and gap formation for high-mass planets. We use linear calculations and nonlinear hydrodynamical simulations to measure the torque and look at gap formation. In both cases, the subkeplerian nature of the disc is treated as a fixed external constraint. We show that, depending on the degree to which the disc is subkeplerian, the torque on low-mass planets varies between the usual Type I torque and the one-sided outer Lindblad torque, which is also negative but an order of magnitude stronger. In strongly subkeplerian discs, corotation effects can be ignored, making migration fast and inward. Gap formation near the planet's orbit is more difficult in such discs, since there are no resonances close to the planet accommodating angular momentum transport. In stead, the location of the gap is shifted inwards with respect to the planet, leaving the planet on the outside of a surface density depression. Depending on the degree to which a protoplanetary disc is subkeplerian, disc-planet interactions can be very different from the usual Keplerian picture, making these discs in general more hazardous for young planets.

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arXiv2009

On corotation torques, horseshoe drag and the possibility of sustained stalled or outward protoplanetary migration

We study the torque on low mass protoplanets on fixed circular orbits, embedded in a protoplanetary disc in the isothermal limit. For low mass protoplanets and large viscosity the corotation torque behaves as expected from linear theory. However, when the viscosity becomes small enough to enable horseshoe turns to occur, the linear corotation torque exists only temporarily after insertion of a planet into the disc, being replaced by the horseshoe drag first discussed by Ward. This happens after a time that is equal to the horseshoe libration period reduced by a factor amounting to about twice the disc aspect ratio. This torque scales with the radial gradient of specific vorticity, as does the linear torque, but we find it to be many times larger. If the viscosity is large enough for viscous diffusion across the coorbital region to occur within a libration period, we find that the horseshoe drag may be sustained. If not, the corotation torque saturates leaving only the linear Lindblad torques. As the magnitude of the non linear coorbital torque (horseshoe drag) is always found to be larger than the linear torque, we find that the sign of the total torque may change even for for mildly positive surface density gradients. In combination with a kinematic viscosity large enough to keep the torque from saturating, strong sustained deviations from linear theory and outward or stalled migration may occur in such cases (abridged).

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arXiv2009

On the width and shape of the corotation region for low-mass planets

We study the coorbital flow for embedded, low mass planets. We provide a simple semi-analytic model for the corotation region, which is subsequently compared to high resolution numerical simulations. The model is used to derive an expression for the half-width of the horseshoe region, x_s, which in the limit of zero softening is given by x_s/r_p = 1.68(q/h)^(1/2), where q is the planet to central star mass ratio, h is the disc aspect ratio and r_p the orbital radius. This is in very good agreement with the same quantity measured from simulations. This result is used to show that horseshoe drag is about an order of magnitude larger than the linear corotation torque in the zero softening limit. Thus the horseshoe drag, the sign of which depends on the gradient of specific vorticity, is important for estimates of the total torque acting on the planet. We further show that phenomena, such as the Lindblad wakes, with a radial separation from corotation of ~ a pressure scale height H can affect x_s, even though for low-mass planets x_s << H. The effect is to distort streamlines and to reduce x_s through the action of a back pressure. This effect is reduced for smaller gravitational softening parameters and planets of higher mass, for which x_s becomes comparable to H.

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arXiv2008

On disc protoplanet interactions in a non-barotropic disc with thermal diffusion

We study the disc planet interactions of low-mass protoplanets embedded in a circumstellar disc. We extend the standard theory of planet migration from the usual locally isothermal assumption to include non-barotropic effects, focusing on the validity of linear theory. We compared solutions of the linear equations with results from non-linear hydrodynamic simulations, where in both cases we adopted a background entropy gradient and solved the energy equation. We show that the migration behavior of embedded planets depends critically on the background radial entropy gradient in the disc. The presence of such a gradient not only changes the corotation torque on the planet, but also always guarantees a departure from linear behavior, which gives a singular density response at corotation, in the absence of thermal or viscous diffusion. A negative entropy gradient tends to give rise to positive, non-linear corotation torques apparently produced as material executes horseshoe turns at approximately constant entropy. These torques have no counterpart in linear theory, but can be strong enough to push the planet outwards until saturation starts to occur after a horseshoe libration period. Increased thermal diffusion acts to reduce these non-linear torques, but, at the same time, it can help to prevent their saturation. In combination with a small kinematic viscosity that is able to maintain a smooth density profile the positive torque could be sustained.

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arXiv2008

Planetesimal and gas dynamics in binaries

Observations of extrasolar planets reveal that planets can be found in close binary systems, where the semi-major axis of the binary orbit is less than 20 AU. The existence of these planets challenges planet formation theory, because the strong gravitational perturbations due to the companion increase encounter velocities between planetesimals and make it difficult for them to grow through accreting collisions. We study planetesimal encounter velocities in binary systems, where the planetesimals are embedded in a circumprimary gas disc that is allowed to evolve under influence of the gravitational perturbations of the companion star. We find that the encounter velocities between planetesimals of different size strongly depend on the gas disc eccentricity. In all cases studied, inclusion of the full gas dynamics increases the encounter velocity compared to the case of a static, circular gas disc. Full numerical parameter exploration is still impossible, but we derive analytical formulae to estimate encounter velocities between bodies of different sizes given the gas disc eccentricity. The gas dynamical evolution of a protoplanetary disc in a binary system tends to make planetesimal accretion even more difficult than in a static, axisymmetric gas disc.

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arXiv2007

Growing and moving low-mass planets in non-isothermal disks

We study the interaction of a low-mass planet with a protoplanetary disk with a realistic treatment of the energy balance by doing radiation-hydrodynamical simulations. We look at accretion and migration rates and compare them to isothermal studies. We used a three-dimensional version of the hydrodynamical method RODEO, together with radiative transport in the flux-limited diffusion approach. The accretion rate, as well as the torque on the planet, depend critically on the ability of the disk to cool efficiently. For densities appropriate to 5 AU in the solar nebula, the accretion rate drops by more than an order of magnitude compared to isothermal models, while at the same time the torque on the planet is positive, indicating outward migration. It is necessary to lower the density by a factor of 2 to recover inward migration and more than 2 orders of magnitude to recover the usual Type I migration. The torque appears to be proportional to the radial entropy gradient in the unperturbed disk. These findings are critical for the survival of protoplanets, and they should ultimately find their way into population synthesis models.

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arXiv2006

Dust accretion onto high-mass planets

We study the accretion of dust particles of various sizes onto embedded massive gas giant planets, where we take into account the structure of the gas disk due to the presence of the planet. The accretion rate of solids is important for the structure of giant planets: it determines the growth rate of the solid core that may be present as well as their final enrichment in solids. We use the RODEO hydrodynamics solver to solve the flow equations for the gas, together with a particle approach for the dust. The solver for the particles' equations of motion is implicit with respect to the drag force, which allows us to treat the whole dust size spectrum. We find that dust accretion is limited to the smallest particle sizes. The largest particles get trapped in outer mean-motion resonances with the planet, while particles of intermediate size are pushed away from the orbit of the planet by the density structure in the gas disk. Only particles smaller than approximately s_max =10 micron may accrete on a planet with the mass of Jupiter. For a ten times less massive planet s_max=100 micron. The strongly reduced accretion of dust makes it very hard to enrich a newly formed giant planet in solids.

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