arXiv subjects

Sijme-Jan Paardekooper

Publications and source records attributed to Sijme-Jan Paardekooper.

At least 37 records · Page 2Linked to original sources

arXiv2021

Polydisperse Streaming Instability II. Methods for solving the linear stability problem

Occurring in protoplanetary discs composed of dust and gas, streaming instabilities are a favoured mechanism to drive the formation of planetesimals. The Polydispserse Streaming Instability is a generalisation of the Streaming Instability to a continuum of dust sizes. This second paper in the series provides a more in-depth derivation of the governing equations and presents novel numerical methods for solving the associated linear stability problem. In addition to the direct discretisation of the eigenproblem at second order introduced in the previous paper, a new technique based on numerically reducing the system of integral equations to a complex polynomial combined with root finding is found to yield accurate results at much lower computational cost. A related method for counting roots of the dispersion relation inside a contour without locating those roots is also demonstrated. Applications of these methods show they can reproduce and exceed the accuracy of previous results in the literature, and new benchmark results are provided. Implementations of the methods described are made available in an accompanying Python package psitools.

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arXiv2021

Polydisperse Streaming Instability III. Dust evolution encourages fast instability

Planet formation via core accretion requires the production of km-sized planetesimals from cosmic dust. This process must overcome barriers to simple collisional growth, for which the Streaming Instability (SI) is often invoked. Dust evolution is still required to create particles large enough to undergo vigorous instability. The SI has been studied primarily with single size dust, and the role of the full evolved dust distribution is largely unexplored. We survey the Polydispserse Streaming Instability (PSI) with physical parameters corresponding to plausible conditions in protoplanetary discs. We consider a full range of particle stopping times, generalized dust size distributions, and the effect of turbulence. We find that, while the PSI grows in many cases more slowly with a interstellar power-law dust distribution than with a single size, reasonable collisional dust evolution, producing an enhancement of the largest dust sizes, produces instability behaviour similar to the monodisperse case. Considering turbulent diffusion the trend is similar. We conclude that if fast linear growth of PSI is required for planet formation, then dust evolution producing a distribution with peak stopping times on the order of 0.1 orbits and an enhancement of the largest dust significantly above the single power-law distribution produced by a fragmentation cascade is sufficient, along with local enhancement of the dust to gas volume mass density ratio to order unity.

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arXiv2020

Polydisperse Streaming Instability I. Tightly coupled particles and the terminal velocity approximation

We introduce a polydisperse version of the streaming instability, where the dust component is treated as a continuum of sizes. We show that its behaviour is remarkably different from the monodisperse streaming instability. We focus on tightly coupled particles in the terminal velocity approximation and show that unstable modes that grow exponentially on a dynamical time scale exist. However, for dust to gas ratios much smaller than unity they are confined to radial wave numbers that are a factor $\sim 1/\overline{\rm St}$ larger than where the monodisperse streaming instability growth rates peak. Here $\overline{\rm St} \ll 1$ is a suitable average Stokes number for the dust size distribution. For dust to gas ratios larger than unity, polydisperse modes that grow on a dynamical time scale are found as well, similar as for the monodisperse streaming instability and at similarly large wave numbers. At smaller wave numbers, where the classical monodisperse streaming instability shows secular growth, no growing polydisperse modes are found under the terminal velocity approximation. Outside the region of validity for the terminal velocity approximation, we have found unstable epicyclic modes that grow on $\sim 10^4$ dynamical time scales.

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arXiv2020

Low-mass planet migration in three dimensional wind-driven inviscid discs: a negative corotation torque

We present simulations of low-mass planet--disc interactions in inviscid three-dimensional discs. We show that a wind-driven laminar accretion flow through the surface layers of the disc does not significantly modify the migration torque experienced by embedded planets. More importantly, we find that 3D effects lead to a dramatic change in the behaviour of the dynamical corotation torque compared to earlier 2D theory and simulations. Although it was previously shown that the dynamical corotation torque could act to slow and essentially stall the inward migration of a low-mass planet, our results in 3D show that the dynamical corotation torque has the complete opposite effect and speeds up inward migration. Our numerical experiments implicate buoyancy resonances as the cause. These have two effects: (i) they exert a direct torque on the planet, whose magnitude relative to the Lindblad torque is measured in our simulations to be small; (ii) they torque the gas librating on horseshoe orbits in the corotation region and drive evolution of its vortensity, leading to the negative dynamical corotation torque. This indicates that at low turbulent viscosity, the detailed vertical thermal structure of the protoplanetary disc plays an important role in determining the migration behaviour of embedded planets. If this result holds up under a more refined treatment of disc thermal evolution, then it has important implications for understanding the formation and early evolution of planetary systems.

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arXiv2019

Constraining the masses of planets in protoplanetary discs from the presence or absence of vortices -- Comparison with ALMA observations

A massive planet in a protoplanetary disc will open a gap in the disc material. A steep gap edge can be hydrodynamically unstable, which results in the formation of vortices that can act as tracers for the presence of planets in observational results. However, in a viscous disc, the potential formation of these vortices is dependent on the timescale over which the massive planet accretes mass and with a sufficiently long timescale it is possible for no vortices to form. Hence, there is a connection between the presence of vortices and the growth timescale of the planet and it may therefore be possible to exclude a planetary interpretation of observed structure from the absence of vortices. We have investigated the effect of the planet growth timescale on vortex formation for a range of planet masses and viscosities and have found an approximate relation between the planet mass, viscosity and planet growth timescale for which vortices are not formed within the disc. We then interpret these results in the light of recent observations. We have also found that planets do not need to be close to a Jupiter mass to form vortices in the disc if these discs have low viscosity, as these can be caused by planets as small as a few Neptune masses.

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arXiv2019

Dynamics of Dusty Vortices I: Extensions and limitations of the terminal velocity approximation

Motivated by the stability of dust laden vortices, in this paper we study the terminal velocity approximation equations for a gas coupled to a pressureless dust fluid and present a numerical solver for the equations embedded in the FARGO3D hydrodynamics code. We show that for protoplanetary discs it is possible to use the baricenter velocity in the viscous stress tensor, making it trivial to simulate viscous dusty protoplanetary discs with this model. We also show that the terminal velocity model breaks down around shocks, becoming incompatible with the two fluid model it is derived from. Finally we produce a set of test cases for numerical schemes and demonstrate the performance of our code on these tests. Our implementation embedded in FARGO3D using an unconditionally stable explicit integrator is fast, and exhibits the desired second order spatial convergence for smooth problems.

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arXiv2019

Multiplanet systems in inviscid discs can avoid forming resonant chains

Convergent migration involving multiple planets embedded in a viscous protoplanetary disc is expected to produce a chain of planets in mean motion resonances, but the multiplanet systems observed by the Kepler spacecraft are generally not in resonance. We demonstrate that under equivalent conditions, where in a viscous disc convergent migration will form a long-term stable system of planets in a chain of mean motion resonances, migration in an inviscid disc often produces a system which is highly dynamically unstable. In particular, if planets are massive enough to significantly perturb the disc surface density and drive vortex formation, the smooth capture of planets into mean motion resonances is disrupted. As planets pile up in close orbits, not protected by resonances, close encounters increase the probability of planet-planet collisions, even while the gas disc is still present. While inviscid discs often produce unstable non-resonant systems, stable, closely packed, non-resonant systems can also be formed. Thus, when examining the expectation for planet migration to produce planetary systems in mean motion resonances, the effective turbulent viscosity of the protoplanetary disc is a key parameter.

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arXiv2019

Migrating super-Earths in low-viscosity discs: unveiling the roles of feedback, vortices, and laminar accretion flows

We present the highest resolution study to date of super-Earths migrating in inviscid and low-viscosity discs, motivated by the connection to laminar, wind-driven models of protoplanetary discs. Our models unveil the critical role of vortices in determining the migration behaviour for partial gap-opening planets. Vortices form in pressure maxima at gap edges, and prevent the disc-feedback stopping of migration for intermediate planets in low-viscosity and inviscid discs, contrary to the concept of the `inertial limit' or `disc feedback' halting predicted from analytical models. Vortices may also form in the corotation region, and can dramatically modify migration behaviour through direct gravitational interaction with the planet. These features become apparent at high resolution, and for all but the highest viscosities there exist significant difficulties in obtaining numerically converged results. The migration of partial gap-opening planets, however, clearly becomes chaotic for sufficiently low viscosities. At moderate viscosity, a smooth disc-feedback regime is found in which migration can slow substantially, and the migration time-scale observed corresponds to migration being driven by diffusive relaxation of the gap edges. At high viscosity classical Type I migration is recovered. For Jupiter-analogue planets in inviscid discs, a wide, deep gap is formed. Transient Type II migration occurs over radial length-scales corresponding to the gap width, beyond which migration can stall. Finally, we examine the particle trapping driven by structures left in inviscid discs by a migrating planet, and find that particle traps in the form of multiple rings and vortices can persist long after the planet has passed. In this case, the observation of particle traps by submillimetre interferometers such as ALMA cannot be used to infer the current presence of an adjacent planet.

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arXiv2018

Local numerical simulations of warped discs

We study the hydrodynamical stability of the laminar flows associated with warped astrophysical discs using numerical simulations of warped shearing boxes. We recover linear growth rates reported previously due to a parametric resonance of inertial waves, and show that the nonlinear saturated state can significantly reduce the laminar flows, meaning that the warp would evolve on much longer time scales than would be concluded from the internal torques due to these laminar flows. Towards larger warp amplitudes, we find first of all a reversal of angular momentum flux, indicating that the mass distribution would evolve in an anti-diffusive manner, and second that the linear growth rates disappear, possibly because of the very strong shear in the laminar flows in this regime. For discs with small enough viscosity, a nonlinear state can still be found when linear growth rates are absent by introducing a large enough perturbation, either by starting from a nonlinear state obtained at smaller warp amplitude, or by starting from a state with no laminar flows.

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arXiv2018

Low mass planet migration in Hall-affected disks

Recent developments in non-ideal magnetohydrodynamic simulations of protoplanetary disks suggest that instead of being traditional turbulent (viscous) accretion disks, they have a largely laminar flow with accretion driven by large-scale wind torques. These disks are possibly threaded by Hall-effect generated large-scale horizontal magnetic fields. We have examined the dynamics of the corotation region of a low mass planet embedded in such a disk and the evolution of the associated migration torque. These disks lack strong turbulence and associated turbulent diffusion, and the presence of a magnetic field and radial gas flow presents a situation outside the applicability of previous corotation torque theory. We summarize the analytical analysis of the corotation torque, give details on the numerical methods used, and in particular the relative merits of different numerical schemes for the inviscid problem.

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arXiv2018

Low mass planet migration in magnetically torqued dead zones - II. Flow-locked and runaway migration, and a torque prescription

We examine the migration of low mass planets in laminar protoplanetary discs, threaded by large scale magnetic fields in the dead zone that drive radial gas flows. As shown in Paper I, a dynamical corotation torque arises due to the flow-induced asymmetric distortion of the corotation region and the evolving vortensity contrast between the librating horseshoe material and background disc flow. Using simulations of laminar torqued discs containing migrating planets, we demonstrate the existence of the four distinct migration regimes predicted in Paper I. In two regimes, the migration is approximately locked to the inward or outward radial gas flow, and in the other regimes the planet undergoes outward runaway migration that eventually settles to fast steady migration. In addition, we demonstrate torque and migration reversals induced by midplane magnetic stresses, with a bifurcation dependent on the disc surface density. We develop a model for fast migration, and show why the outward runaway saturates to a steady speed, and examine phenomenologically its termination due to changing local disc conditions. We also develop an analytical model for the corotation torque at late times that includes viscosity, for application to discs that sustain modest turbulence. Finally, we use the simulation results to develop torque prescriptions for inclusion in population synthesis models of planet formation.

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arXiv2017

Low mass planet migration in magnetically torqued dead zones - I. Static migration torque

Motivated by models suggesting that the inner planet forming regions of protoplanetary discs are predominantly lacking in viscosity-inducing turbulence, and are possibly threaded by Hall-effect generated large-scale horizontal magnetic fields, we examine the dynamics of the corotation region of a low-mass planet in such an environment. The corotation torque in an inviscid, isothermal, dead zone ought to saturate, with the libration region becoming both symmetrical and of a uniform vortensity, leading to fast inward migration driven by the Lindblad torques alone. However, in such a low viscosity situation, the material on librating streamlines essentially preserves its vortensity. If there is relative radial motion between the disc gas and the planet, the librating streamlines will no longer be symmetrical. Hence, if the gas is torqued by a large scale magnetic field so that it undergoes a net inflow or outflow past the planet, driving evolution of the vortensity and inducing asymmetry of the corotation region, the corotation torque can grow, leading to a positive torque. In this paper we treat this effect by applying a symmetry argument to the previously studied case of a migrating planet in an inviscid disc. Our results show that the corotation torque due to a laminar Hall-induced magnetic field in a dead zone behaves quite differently from that studied previously for a viscous disc. Furthermore, the magnetic field induced corotation torque and the dynamical corotation torque in a low viscosity disc can be regarded as one unified effect.

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arXiv2017

The role of gap edge instabilities in setting the depth of planet gaps in protoplanetary discs

It is known that an embedded massive planet will open a gap in a protoplanetary disc via angular momentum exchange with the disc material. The resulting surface density profile of the disc is investigated for one dimensional and two dimensional disc models and, in agreement with previous work, it is found that one dimensional gaps are significantly deeper than their two dimensional counterparts for the same initial conditions. We find, by applying one dimensional torque density distributions to two dimensional discs containing no planet, that the excitement of the Rossby wave instability and the formation of Rossby vortices play a critical role in setting the equilibrium depth of the gap. Being a two dimensional instability, this is absent from one dimensional simulations and does not limit the equilibrium gap depth there. We find similar gap depths between two dimensional gaps formed by torque density distributions, in which the Rossby wave instability is present, and two dimensional planet gaps, in which no Rossby wave instability is present. This can be understood if the planet gap is maintained at marginal stability, even when there is no obvious Rossby wave instability present. Further investigation shows the final equilibrium gap depth is very sensitive to the form of the applied torque density distribution, and using improved one dimensional approximations from three dimensional simulations can go even further to reducing the discrepancy between one and two dimensional models, especially for lower mass planets. This behaviour is found to be consistent across discs with varying parameters.

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arXiv2017

Multidimensional upwind hydrodynamics on unstructured meshes using Graphics Processing Units I. Two-dimensional uniform meshes

We present a new method for numerical hydrodynamics which uses a multidimensional generalisation of the Roe solver and operates on an unstructured triangular mesh. The main advantage over traditional methods based on Riemann solvers, which commonly use one-dimensional flux estimates as building blocks for a multidimensional integration, is its inherently multidimensional nature, and as a consequence its ability to recognise multidimensional stationary states that are not hydrostatic. A second novelty is the focus on Graphics Processing Units (GPUs). By tailoring the algorithms specifically to GPUs we are able to get speedups of 100-250 compared to a desktop machine. We compare the multidimensional upwind scheme to a traditional, dimensionally split implementation of the Roe solver on several test problems, and we find that the new method significantly outperforms the Roe solver in almost all cases. This comes with increased computational costs per time step, which makes the new method approximately a factor of 2 slower than a dimensionally split scheme acting on a structured grid.

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arXiv2017

Exploring plausible formation scenarios for the planet candidate orbiting Proxima Centauri

We present a study of four different formation scenarios that may be able to explain the origin of the recently announced planet (`Proxima b') orbiting Proxima Centauri. The aim is to examine how the formation scenarios differ in their predictions for the multiplicity of the Proxima system, the water/volatile content of Proxima b and its orbital eccentricity, so that these can be tested by future observations. A scenario of in situ formation via giant impacts from a locally enhanced disc of planetary embryos/planetesimals predicts that Proxima b will be in a multiplanet system with a measurably finite eccentricity. Assuming that the local solid enhancement needed to form a Proxima b analogue arises because of the inwards drift of solids in the form of small planetesimals/boulders, this scenario also results in Proxima b analogues being only moderately endowed with water/volatiles. A scenario in which multiple embryos form, migrate and mutually collide within a gas disc results in Proxima b being a member of a multiple system, possibly displaying mean motion resonances, but where the constituent members are Ocean planets due to accretion occurring mainly outside of the snowline. A scenario in which a single accreting embryo forms outside the snowline, and migrates inwards while accreting planetesimals/pebbles results in Proxima b being an isolated Ocean planet on a circular orbit. A scenario in which Proxima b formed via pebble accretion interior to the snowline produces a dry planet on a circular orbit. Future observations that characterise the physical and orbital properties of Proxima b, and any additional planets in the system, will provide valuable insights into the formation history of this neighbouring planetary system.

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arXiv2016

A terrestrial planet candidate in a temperate orbit around Proxima Centauri

At a distance of 1.295 parsecs, the red-dwarf Proxima Centauri ($α$ Centauri C, GL 551, HIP 70890, or simply Proxima) is the Sun's closest stellar neighbor and one of the best studied low-mass stars. It has an effective temperature of only $\sim$ 3050 K, a luminosity of $\sim$0.1 per cent solar, a measured radius of 0.14 R$_\odot$ and a mass of about 12 per cent the mass of the Sun. Although Proxima is considered a moderately active star, its rotation period is $\sim$ 83 days, and its quiescent activity levels and X-ray luminosity are comparable to the Sun's. New observations reveal the presence of a small planet orbiting Proxima with a minimum mass of 1.3~Earth masses and an orbital period of $\sim$11.2 days. Its orbital semi-major axis is $\sim0.05$ AU, with an equilibrium temperature in the range where water could be liquid on its surface.

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arXiv2016

Planet Formation Imager (PFI): science vision and key requirements

The Planet Formation Imager (PFI) project aims to provide a strong scientific vision for ground-based optical astronomy beyond the upcoming generation of Extremely Large Telescopes. We make the case that a breakthrough in angular resolution imaging capabilities is required in order to unravel the processes involved in planet formation. PFI will be optimised to provide a complete census of the protoplanet population at all stellocentric radii and over the age range from 0.1 to about 100 Myr. Within this age period, planetary systems undergo dramatic changes and the final architecture of planetary systems is determined. Our goal is to study the planetary birth on the natural spatial scale where the material is assembled, which is the "Hill Sphere" of the forming planet, and to characterise the protoplanetary cores by measuring their masses and physical properties. Our science working group has investigated the observational characteristics of these young protoplanets as well as the migration mechanisms that might alter the system architecture. We simulated the imprints that the planets leave in the disk and study how PFI could revolutionise areas ranging from exoplanet to extragalactic science. In this contribution we outline the key science drivers of PFI and discuss the requirements that will guide the technology choices, the site selection, and potential science/technology tradeoffs.

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arXiv2015

Spiral arms in scattered light images of protoplanetary discs: Are they the signposts of planets?

One of the striking discoveries of protoplanetary disc research in recent years are the spiral arms seen in several transitional discs in polarised scattered light. An interesting interpretation of the observed spiral features is that they are density waves launched by one or more embedded (proto-)planets in the disc. In this paper we investigate whether planets can be held responsible for the excitation mechanism of the observed spirals. We use locally isothermal hydrodynamic simulations as well as analytic formulae to model the spiral waves launched by planets. Then H-band scattered light images are calculated using a 3D continuum radiative transfer code to study the effect of surface density and pressure scale height perturbation on the detectability of the spirals. We find that a relative change of about 3.5 in the surface density is required for the spirals to be detected with current telescopes in the near-infrared for sources at the distance of typical star-forming regions (140pc). This value is a factor of eight higher than what is seen in hydrodynamic simulations. We also find that a relative change of only 0.2 in pressure scale height is sufficient to create detectable signatures under the same conditions. Therefore, we suggest that the spiral arms observed to date in protoplanetary discs are the results of changes in the vertical structure of the disc (e.g. pressure scale height perturbation) instead of surface density perturbations.

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