arXiv subjects

Richard P. Nelson

Publications and source records attributed to Richard P. Nelson.

106 records · Page 6Linked to original sources

arXiv2005

On the accumulation of solid bodies in global turbulent protoplanetary disc models

We study the migration of solid bodies in turbulent protoplanetary accretion discs by means of global MHD simulations. The bodies range in size from 5 centimetres up to 1 metre, and so include objects whose migration is expected to be the most rapid due to gas drag interaction with the disc. As they drift inward through the disc, some of them are trapped in regions where gas pressure maxima are created by long lived anticyclonic vortices. This accumulation is very efficient, locally increasing the dust--to--gas ratio by a factor > 100 in some cases. We discuss the possible implications of this result for theories of planet formation.

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arXiv2005

On the Ionisation Fraction in Protoplanetary Disks II: The Effect of Turbulent Mixing on Gas--phase Chemistry

We calculate the ionisation fraction in protostellar disk models using two different gas-phase chemical networks, and examine the effect of turbulent mixing by modelling the diffusion of chemical species vertically through the disk. The aim is to determine in which regions of the disk gas can couple to a magnetic field and sustain MHD turbulence. We find that the effect of diffusion depends crucially on the elemental abundance of heavy metals (magnesium) included in the chemical model. In the absence of heavy metals, diffusion has essentially no effect on the ionisation structure of the disks, as the recombination time scale is much shorter than the turbulent diffusion time scale. When metals are included with an elemental abundance above a threshold value, the diffusion can dramatically reduce the size of the magnetically decoupled region, or even remove it altogther. For a complex chemistry the elemental abundance of magnesium required to remove the dead zone is 10(-10) - 10(-8). We also find that diffusion can modify the reaction pathways, giving rise to dominant species when diffusion is switched on that are minor species when diffusion is absent. This suggests that there may be chemical signatures of diffusive mixing that could be used to indirectly detect turbulent activity in protoplanetary disks. We find examples of models in which the dead zone in the outer disk region is rendered deeper when diffusion is switched on. Overall these results suggest that global MHD turbulence in protoplanetary disks may be self-sustaining under favourable circumstances, as turbulent mixing can help maintain the ionisation fraction above that necessary to ensure good coupling between the gas and magnetic field.

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arXiv2005

Numerical simulations of type I planetary migration in nonturbulent magnetized discs

Using 2D MHD numerical simulations performed with two different finite difference Eulerian codes, we analyze the effect that a toroidal magnetic field has on low mass planet migration in nonturbulent protoplanetary discs. The presence of the magnetic field modifies the waves that can propagate in the disc. In agreement with a recent linear analysis (Terquem 2003), we find that two magnetic resonances develop on both sides of the planet orbit, which contribute to a significant global torque. In order to measure the torque exerted by the disc on the planet, we perform simulations in which the latter is either fixed on a circular orbit or allowed to migrate. For a 5 earth mass planet, when the ratio βbetween the square of the sound speed and that of the Alfven speed at the location of the planet is equal to 2, we find inward migration when the magnetic field B_ϕ is uniform in the disc, reduced migration when B_ϕ decreases as r^{-1} and outward migration when B_ϕ decreases as r^{-2}. These results are in agreement with predictions from the linear analysis. Taken as a whole, our results confirm that even a subthermal stable field can stop inward migration of an earth--like planet.

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arXiv2005

On the Orbital Evolution of Low Mass Protoplanets in Turbulent, Magnetised Disks

(Abridged).We present the results of MHD simulations of low mass protoplanets interacting with turbulent disks. We calculate the orbital evolution of `planetesimals' and protoplanets with masses in the range 0 < m_p < 30 M_Earth. Planetesimals and protoplanets undergo stochastic migration due to interaction with turbulent density fluctuations. Over run times of ~ 150 planet orbits, stochastic migration dominates over type I migration for many models. Fourier analysis of the torques experienced by planets indicates that the torque fluctuations contain components with significant power whose time scales of variation are similar to the simulation run times. These low frequency fluctuations partly explain the dominance of stochastic torques, and may provide a powerful means of counteracting the type I migration of some planets in turbulent disks. Turbulence is a source of eccentricity driving. Planetesimals attained eccentricities in the range 0.02 < e < 0.14, m_p=1 M_Earth planets attained eccentricities 0.02 < e < 0.08, and m_p=10 M_Earth protoplanets reached 0.02 < e < 0.03. This is in basic agreement with a model in which turbulence drives e-growth, and interaction with disk material at coorbital Lindblad resonances causes e-damping. These results are significant for planet formation. Stochastic migration may prevent some planet cores migrating into their star via type I before becoming gas giants. The growth of planetary cores may be enhanced by preventing isolation. Eccentricity excitation by turbulence, however, may reduce growth rates of planetary cores during the runaway and oligarchic growth stages, and cause collisions between planetesimals to become destructive.

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arXiv2005

Oligarchic and giant impact growth of terrestrial planets in the presence of gas giant planet migration

We present the results of N--body simulations which examine the effect that gas giant planet migration has on the formation of terrestrial planets. The models incorporate a 0.5 Jupiter mass planet undergoing type II migration through an inner protoplanet--planetesimal disk, with gas drag included. Each model is initiated with the inner disk being at successively increased levels of maturity, so that it is undergoing either oligarchic or giant impact style growth as the gas giant migrates. In all cases, a large fraction of the disk mass survives the passage of the giant, either by accreting into massive terrestrial planets shepherded inward of the giant, or by being scattered into external orbits. Shepherding is favored in younger disks where there is strong dynamical friction from planetesimals and gas drag is more influential, whereas scattering dominates in more mature disks where dissipation is weaker. In each scenario, sufficient mass is scattered outward to provide for the eventual accretion of a set of terrestrial planets in external orbits, including within the system's habitable zone. An interesting result is the generation of massive, short period, terrestrial planets from compacted material pushed ahead of the giant. These planets are reminiscent of the short period Neptune mass planets discovered recently, suggesting that such `hot Neptunes' could form locally as a by-product of giant planet migration.

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arXiv2004

Models of Accreting Gas Giant Protoplanets in Protostellar Disks

(Abridged) We consider models of gas giant planets forming in protoplanetary disks consisting of solid cores with gaseous envelopes in contact with their critical Hill spheres while accreting gas from the surrounding disk.We suppose the luminosity derives from gas accretion alone.We label such models as type A and follow their evolution which may occur on a time scale similar to the protostellar disk lifetime until rapid gas accretion. We consider another set of models, we label type B, with a free surface, powered by gravitational contraction, while accreting through a disk.We find these models rapidly attain a radius <~ 2x10^(10)cm without subsequent expansion.We speculate that giant planet formation is initially described by models of type A, until at the onset of rapid gas accretion, there is a transition to models of type B. Protoplanet migration in standard models tends to be most effective near this transition where it also changes from type I to type II.If a mechanism prevents type I migration of low mass protoplanets, a rapid inward migration might occur near the transitional mass regime. Such protoplanets would end up in the inner disk regions undergoing type II migration and further accretion potentially becoming sub Jovian close orbiting planets. Noting that dustier more massive cores spend longer at a larger transitional mass where faster migration is expected, these may be more prone to end in close orbiters.We find the luminosity of the protoplanets during the later stages is dominated by the circumplanetary disk and protoplanet disk boundary layer.For one Jupiter mass the luminosity range is 10^-(1.5-4) L_sun$ depending on the evolutionary stage and external conditions.

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arXiv2004

The interaction of planets with a disc with MHD turbulence III: Flow morphology and conditions for gap formation in local and global simulations

We present the results of both global cylindrical disc simulations and local shearing box simulations of protoplanets interacting with a disc undergoing MHD turbulence with zero net flux magnetic fields. We investigate the nature of the disc response and conditions for gap formation. This issue is an important one for determining the type and nature of the migration of the protoplanet, with the presence of a deep gap being believed to enable slower migration. For both types of simulation we find a common pattern of behaviour for which the main parameter determining the nature of the response is $M_p R^3/(M_* H^3)$, with $M_p$, $M_*$, $R$, and $H$ being the protoplanet mass, the central mass, the orbital radius and the disc semi-thickness respectively. We find that as this parameter is increased towards 0.1, the presence of the protoplanet is first indicated by the appearance of the well known trailing wake which, although it may appear erratic on account of the turbulence, appears to be well defined. Once the above parameter exceeds a number around unity a gap starts to develop inside which the magnetic energy density tends to be concentrated in the high density wakes. This gap formation condition can be understood from simple dimensional considerations of the conditions for nonlinearity, and the balance of angular momentum transport due to Maxwell and Reynolds' stresses with that due to tidal torques. An important result is that the basic flow morphology in the vicinity of the protoplanet is very similar in both the local and global simulations. This indicates that local shearing box simulations, which are computationally less demanding, capture much of the physics of disc-planet interaction. Thus they may provide a useful tool for studying the local interaction between forming protoplanets and turbulent, protostellar discs.

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arXiv2003

Disc-planet, planet-planet, and star-planet interactions during planet formation

In this article we present results from three on-going projects related to the formation of protoplanets in protostellar discs. We present the results of simulations that model the interaction between embedded protoplanets and disc models undergoing MHD turbulence. We review the similarities and differences that arise when the disc is turbulent as opposed to laminar (but viscous), and present the first results of simulations that examine the tidal interaction between low mass protoplanets and turbulent discs. We describe the results of simulations of Jovian mass protoplanets forming in circumbinary discs, and discuss the range of possible outcomes that arise in hydrodynamic simulations. Finally, we report on some preliminary simulations of three protoplanets of Jovian mass that form approximately coevally within a protostellar disc. We describe the conditions under which such a system can form a stable three planet resonance.

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arXiv2003

The interaction of planets with a disc with MHD turbulence IV: Migration rates of embedded protoplanets

(Abridged) We present global disc and local shearing box simulations of planets interacting with a MHD turbulent disc. We examine the torque exerted by the disc on the embedded planets as a function of planet mass, and thus make a first study of orbital migration of planets due to interaction with turbulent discs. Global simulations were performed for a disc with H/R=0.07 and planet masses M_p=3,10,30 Earth masses, and 3 Jupiter masses. Shearing box runs were performed for values of (M_p/M_*)/(H/R)^3=0.1,0.3,1.0 and 2.0, M_* being the central mass. These allow embedded and gap forming planets to be examined. In all cases the instantaneous torque exerted on a planet showed strong fluctuations. In the embedded cases it oscillated between negative and positive values, and migration occurs as a random walk, unlike the usual type I migration. Running time averages for embedded planets over 20-25 orbital periods show that large fluctuations occur on longer time scales, preventing convergence of the average torque to well defined values, or even to a well defined sign. Fluctuations become relatively smaller for larger masses, giving better convergence, due to the planet's perturbation of the disc becoming larger than the turbulence in its vicinity. Eventually gap formation occurs, with a transition to type II migration. The existence of significant fluctuations occurring in turbulent discs on long time scales is important for lower mass embedded protoplanets. If significant fluctuations occur on the longest disc evolutionary time scales, convergence of torque running averages for practical purposes will not occur, and the migration behaviour of low mass protoplanets considered as an ensemble would be very different from predictions of type I theory for laminar discs.

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arXiv2002

The interaction of a giant planet with a disc with MHD turbulence I: The initial turbulent disc models

This is the first of a series of papers aimed at developing and interpreting simulations of protoplanets interacting with turbulent accretion discs. Here we study the disc models prior to the introduction of a protoplanet.We study models in which a Keplerian domain is unstable to the magnetorotational instability (MRI). Various models with B-fields having zero net flux are considered.We relate the properties of the models to classical viscous disc theory.All models attain a turbulent state with volume averaged stress parameter alpha ~ 0.005. At any particular time the vertically and azimuthally averaged value exhibited large fluctuations in radius. Time averaging over periods exceeding 3 orbital periods at the outer boundary of the disc resulted in a smoother quantity with radial variations within a factor of two or so. The vertically and azimuthally averaged radial velocity showed much larger spatial and temporal fluctuations, requiring additional time averaging for 7-8 orbital periods at the outer boundary to limit them. Comparison with the value derived from the averaged stress using viscous disc theory yielded schematic agreement for feasible averaging times but with some indication that the effects of residual fluctuations remained. The behaviour described above must be borne in mind when considering laminar disc simulations with anomalous Navier--Stokes viscosity. This is because the operation of a viscosity as in classical viscous disc theory with anomalous viscosity coefficient cannot apply to a turbulent disc undergoing rapid changes due to external perturbation. The classical theory can only be used to describe the time averaged behaviour of the parts of the disc that are in a statistically steady condition for long enough for appropriate averaging to be carried out.

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arXiv2002

The interaction of a giant planet with a disc with MHD turbulence II: The interaction of the planet with the disc

We present a global MHD simulation of a turbulent accretion disc interacting with a protoplanet of 5 Jupiter masses. The disc model had H/r=0.1,and a value of the Shakura & Sunyaev alpha ~ 0.005. The protoplanet opened a gap in the disc, with the interaction leading to inward migration on the expected time scale. Spiral waves were launched by the protoplanet and although they were diffused and dissipated through interaction with the turbulence, they produced an outward angular momentum flow which compensated for a reduced flux associated with the turbulence, so maintaining the gap. When compared with laminar disc models with the same estimated alpha, the gap was found to be deeper and wider indicating that the turbulent disc behaved as if it possessed a smaller alpha. This may arise for two reasons. First, the turbulence does not provide a source of constantly acting friction in the near vicinity of the planet that leads to steady mass flow into the gap region. Instead the turbulence is characterised by large fluctuations in the radial velocity, and time averaging over significant time scales is required to recover the underlying mass flow through the disc. Near the planet the disc material experiences high amplitude perturbations on time scales that are short relative to the time scale required for averaging. The disc response is thus likely to be altered relative to a Navier--Stokes model. Second, the simulation indicates that an ordered magnetic connection between the inner and outer disc can occur enabling angular momentum to flow out across the gap, helping to maintain it independently of the protoplanet's tide. This type of effect may assist gap formation for smaller mass protoplanets which otherwise would not be able to maintain them.

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arXiv2002

Possible Commensurabilities Among Pairs of Extrasolar Planets

We investigate the possible commensurabilities to be expected when two protoplanets in the Jovian mass range, gravitationally interacting with each other and an external protoplanetary disc, are driven by disc induced orbital migration of the outer protoplanet into a commensurability which is then maintained in subsequent evolution. We find that for a variety of protoplanet masses and typical protoplanetary disc properties, as well as the setting up of 2:1 commensurabilities of the type recently observed in GJ876, 3:1 commensurabilities are often formed, in addition to 4:1, 5:1, and 5:2 commensurabilities which occur less frequently. The higher order commensurabilites are favoured when either one of the planets is massive, or the inner planet begins with a significant orbital eccentricity. Detection of such commensurabilities could yield important information relating to the operation of protoplanet disc interactions during and shortly post formation.

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arXiv2000

Hydrodynamic Simulations of the Bardeen-Petterson Effect

We present SPH simulations of accretion discs in orbit about rotating compact objects such as black holes and neutron stars, and study the structure of warped discs produced by the Bardeen-Petterson effect. We calculate the transition radius out to which the disc specific angular momentum vector is aligned with that of the black hole. We focus on the parameter regime where the warp dynamics are controlled by bending wave propagation, but also consider models in which warps are subject to diffusion rather than wave transport, and are able to consider the fully nonlinear regime. Because of hydrodynamic or pressure effects, for the parameter range investigated, the transition radius is always found to be much smaller than that obtained by Bardeen & Petterson (1975). For discs with midplane Mach numbers of about 10, the transition occurs between 10 - 16 gravitational radii, whereas for a Mach number of about 30 it occurs at around 30 gravitational radii. A thicker disc with a Mach number of 5 is found to produce no discernible warped structure. The rate of black hole - disc alignment is found to be consistent with the ideas of Ress (1978), with the alignment torque behaving as if it arises from the accreted material transferring its misaligned component of angular momentum at the larger transition radius of Bardeen & Petterson (1975). The inclusion of Einstein precession in the calculations modified both the warped disc structure and, consistent with linear analysis, produced an increased alignment rate by up to a factor of 4 because of the effect that a non Keplerian potential has on the propagation of warps.

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arXiv1999

The Migration and Growth of Protoplanets in Protostellar Discs

We investigate the gravitational interaction of a Jovian mass protoplanet with a gaseous disc with aspect ratio and kinematic viscosity expected for the protoplanetary disc from which it formed. Different disc surface density distributions have been investigated. We focus on the tidal interaction with the disc with the consequent gap formation and orbital migration of the protoplanet. Nonlinear hydrodynamic simulations are employed using three independent numerical codes. A principal result is that the direction of the orbital migration is always inwards and such that the protoplanet reaches the central star in a near circular orbit after a characteristic viscous time scale of approximately 10,000 initial orbital periods. This was found to be independent of whether the protoplanet was allowed to accrete mass or not. Inward migration is helped through the disappearance of the inner disc, and therefore the positive torque it would exert, because of accretion onto the central star.Our results indicate that a realistic upper limit for the masses of closely orbiting giant planets is approximately 5 Jupiter masses, because of the reduced accretion rates obtained for planets of increasing mass. Assuming some process such as termination of the inner disc through a magnetospheric cavity stops the migration, the range of masses estimated for a number of close orbiting giant planets (Marcy, Cochran, & Mayor 1999; Marcy & Butler 1998) as well as their inward orbital migration can be accounted for by consideration of disc--protoplanet interactions during the late stages of giant planet formation. Maximally accreting protoplanets reached about four Jovian masses on reaching the neighbourhood of the central star.

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arXiv1999

Hydrodynamic Simulations of Propagating Warps and Bending Waves In Accretion Discs

We present the results of a study of propagating warp or bending waves in accretion discs. Three dimensional hydrodynamic simulations were performed using SPH, and the results of these are compared with calculations based on the linear theory of warped discs. We consider primarily the physical regime in which the dimensionless viscosity parameter `alpha' < H/r, the disc aspect ratio, so that bending waves are expected to propagate. We also present calculations in which `alpha' > H/r, where the warps are expected to behave diffusively. Small amplitude perturbations are studied in both Keplerian and slightly non Keplerian discs, and we find that the SPH results can be reasonably well fitted by those of the linear theory. The main results of these calculations are: (1) the warp in Keplerian discs when `alpha' < H/r propagates with little dispersion and damps at a rate expected from estimates of the code viscosity, (2) warps evolve diffusively when `alpha' > H/r, (3) the non Keplerian discs exhibit a substantially more dispersive behaviour of the warps. Initially imposed higher amplitude nonlinear warping disturbances were studied in Keplerian discs. The results indicate that nonlinear warps can lead to the formation of shocks, and that the evolution of the warp becomes less wave-like and more diffusive in character. This work is relevant to the study of the warped accretion discs that may occur around Kerr black holes or in misaligned binary systems. The results indicate that SPH can accurately model the hydrodynamics of warped discs, even when using rather modest numbers of particles.

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arXiv1994

Variable Smoothing Lengths and Energy Conservation in Smoothed Particle Hydrodynamics

We present a new formulation of the equations of motion used in smoothed particle hydrodynamics (SPH). The spatial resolution in SPH is determined by the smoothing length, $h$, and it has become common practice for each particle to be given its own adaptive smoothing length, $h_i$. Consequently, the dynamic range that may be spatially resolved is greatly increased, but additional ($\nabla h$) terms, which account for the variability of the smoothing lengths, should appear in the equations of motion in order to satisy conservation requirements. Previous implementations of SPH have neglected these additional terms, whereas we have included them. This is achieved by defining a functional form for the $h_i$s, that depends on inter-particle distances only, and then deriving the equations of motion using a Hamiltonian formalism. A number of test calculations are presented. We find that including the $\nabla h$ terms has no detrimental effect on the ability of SPH to model known problems with reasonable accuracy. For problems where the energy conservation was rather poor, including the $\nabla h$ terms results in a dramatic improvement. In particluar non-conservation of energy during a collision between two polytropes can occur at the level of 10% when the $\nabla h$ terms are neglected. When the $\nabla h$ terms are included, then this error reduces to 0.8%.

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