16014194
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. In this first paper we study the turbulent disc models prior to the introduction of a perturbing protoplanet. We study cylindrical disc models in which a central domain is in Keplerian rotation and unstable to the magnetorotational instability. Models of varying disc size and aspect ratio H/r are considered with magnetic fields having zero net flux. We relate the properties of the turbulent models to classical viscous disc theory. All models were found to attain a turbulent state in their Keplerian domains with volume averaged stress parameter α ∼ 5 × 10 −3 . At any particular time the vertically and azimuthally averaged value exhibited large fluctuations in radius. However, an additional time average over periods exceeding three orbital periods at the outer boundary of the Keplerian domain resulted in a more smoothly varying quantity with radial variations within a factor of 2 or so. The vertically and azimuthally averaged radial velocity showed much larger spatial and temporal fluctuations, requiring additional time averaging for at least seven to eight orbital periods at the outer boundary of the Keplerian domain 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 long enough for appropriate averaging to be carried out.
13904298
On disc driven inward migration of resonantly coupled planets with application to the system around GJ876
We consider two protoplanets gravitationally interacting with each other and a protoplanetary disc. The two planets orbit interior to a tidally maintained disc cavity while the disc interaction induces inward migration. When the migration is slow enough, the more rapidly migrating outer protoplanet approaches and becomes locked in a 2:1 commensurability with the inner one. This is maintained in subsequent evolution. We study this evolution using a simple analytic model, full hydrodynamic 2D simulations of the disc planet system and longer time N -body integrations incorporating simple prescriptions for the effects of the disc on the planet orbits. The eccentricities of the protoplanets are found to be determined by the migration rate and circularization rate induced in the outer planet orbit by the external disc. We apply our results to the recently discovered resonant planets around GJ876. Simulation shows that a disc with parameters expected for protoplanetary discs causes trapping in the 2:1 commensurability when the planets orbit in an inner cavity and that eccentricities in the observed range may be obtained.
15863571
On the growth and stability of Trojan planets
Aims. We investigate the stability of those low-mass Trojan planets that form in a protoplanetary disc and subsequently accrete gas to become gas giants. Methods. We calculate their evolution before, during, and after gas disc dispersal. A two-dimensional hydrodynamics code combined with an N-body solver is used to evolve the system of disc and planets. Gas disc dispersal is simulated in a simple manner by assuming global exponential decay of the disc mass, leading to the stalling of migration after semi-major axes have approximately halved from their initial values. We consider Trojan pairs with different initial masses and gas accretion rates and gas disc models with different masses and viscosities. An N-body code, adapted to model disc forces, is used to examine large-scale migration and the formation of very short period Trojan planets. Results. For each combination of planetary pair and disc model that we consider in our hydrodynamic simulations, each Trojan system remains stable before, during, and after disc dispersal. The long-term stability of these systems in the absence of gas is tested using N-body simulations, and all systems remain stable for those evolution times equal to 10 9 years. Eccentricities remain low (e < 0.02) in all cases. Increases in the amplitude of libration about the L4/L5 Lagrange points accompany the inward migration, and during very large-scale migration Trojan systems may be disrupted prior to the onset of disc dispersal. Conclusions. The stability of Trojan pairs during rapid type I migration, during the transition to type II migration with the accompanying gap formation in the gas disc, and during gas loss when the disc disperses, indicates that isolated Trojan planet systems are very stable. If a common mechanism exists for their formation, we suggest they may be readily observed in nature.
1737248
On the ionisation fraction in protoplanetary disks - I. Comparing different reaction networks
We calculate the ionisation fraction in protostellar disk models using a number of different chemical reaction networks, including gas-phase and gas-grain reaction schemes. The disk models we consider are conventional α -disks, which include viscous heating and radiative cooling. The primary source of ionisation is assumed to be X-ray irradiation from the central star. For most calculations we adopt a specific disk model (with accretion rate ${\dot M}=10^{-7}$ $M_{\odot}$ yr -1 and $\alpha=10_{}^{-2}$), and examine the predictions made by the chemical networks concerning the ionisation fraction, magnetic Reynolds number, and spatial extent of magnetically active regions. This is to aid comparison between the different chemical models. We consider a number of gas-phase chemical networks. The simplest is the five species model proposed by Oppenheimer & Dalgarno (1974). We construct more complex models by extracting species and reactions from the UMIST data base. In general we find that the simple models predict higher fractional ionisation levels and more extensive active zones than the more complex models. When heavy metal atoms are included the simple models predict that the disk is magnetically active throughout. The complex models predict that extensive regions of the disk remain magnetically uncoupled (“dead”) even when the fractional abundance of magnesium $x_{\rm Mg}=10^{-8}$. This is because of the large number of molecular ions that are formed, which continue to dominate the recombination with free electrons in the presence of magnesium. The addition of submicron sized grains with a concentration of $x_{\rm gr}=10^{-12}$ causes the size of the “dead zone” to increase dramatically for all kinetic models considered, as the grains are highly efficient at sweeping up the free electrons. We find that the simple and complex gas-grain reaction schemes agree on the size and structure of the resulting “dead zone”, as the grains play a dominant role in determining the ionisation fraction. We examine the effects of depleting the concentration of small grains as a crude means of modeling the growth of grains during planet formation. We find that a depletion factor of 10 -4 causes the gas-grain chemistry to converge to the gas-phase chemistry when heavy metals are absent. When magnesium is included a depletion factor of 10 -8 is required to reproduce the gas-phase ionisation fraction. This suggests that efficient grain growth and settling will be required in protoplanetary disks, before a substantial fraction of the disk mass in the planet forming zone between 1–10 AU becomes magnetically active and turbulent. Only after this has occurred can gas-phase chemical models be used to predict reliably the ionisation degree in protoplanetary disks.
16377798
On the Formation and Dynamical Evolution of Planets in Binaries
Among the extrasolar planetary systems about 30 are located in a stellar binary orbiting one of the stars, preferably the more massive primary. The dynamical influence of the second companion alters firstly the orbital elements of the forming protoplanet directly and secondly the structure of the disk from which the planet formed which in turn will modify the planet's evolution. We present detailed analysis of these effects and present new hydrodynamical simulations of the evolution of protoplanets embedded in circumstellar disks in the presence of a companion star, and compare our results to the system $\gamma$ Cep. To analyse the early formation of planetary embryos, we follow the evolution of a swarm of planetesimals embedded in a circumstellar disk. Finally, we study the evolution of planets embedded in circumbinary disks.
2170770
Models of accreting gas giant protoplanets in protostellar disks
We present evolutionary models of gas giant planets forming in protoplanetary disks. We first consider protoplanet models that consist of solid cores surrounded by hydrostatically supported gaseous envelopes that are in contact with the boundaries of their Hill spheres, and accrete gas from the surrounding disk. We neglect planetesimal accretion, and suppose that the luminosity arises from gas accretion alone. This generally occurs on a long time scale which may be comparable to the protostellar disk lifetime. We classify these models as being of type A, and follow their quasi static evolution until the point of rapid gas accretion is reached. We consider a second class of protoplanet models that have not hitherto been considered. These models have a free surface, their energy supply is determined by gravitational contraction, and mass accretion from the protostellar disk that is assumed to pass through a circumplanetary disk. An evolutionary sequence is obtained by specifying the accretion rate that the protostellar disk is able to supply. We refer to these models as being of type B. An important result is that these protoplanet models contract quickly to a radius ∼2 × 10 10 cm and are able to accrete gas from the disk at any reasonable rate that may be supplied without any consequent expansion (e.g. a Jupiter mass in ∼few ×10 3 years, or more slowly if so constrained by the disk model). We speculate that the early stages of gas giant planet formation proceed along evolutionary paths described by models of type A, but at the onset of rapid gas accretion the protoplanet contracts interior to its Hill sphere, making a transition to an evolutionary path described by models of type B, receiving gas through a circumplanetary disk that forms within its Hill sphere, which is in turn fed by the surrounding protostellar disk. We consider planet models with solid core masses of 5 and 15 M⊕, and consider evolutionary sequences assuming different amounts of dust opacity in the gaseous envelope. The initial protoplanet mass doubling time scale is very approximately inversely proportional to the magnitude of this opacity. Protoplanets with 5 M⊕ cores, and standard dust opacity require ∼3 × 10 8 years to grow to a Jupiter mass, longer than reasonable disk life-times. A model with 1% of standard dust opacity requires ∼3 × 10 6 years. Rapid gas accretion in both these cases ensues once the planet mass exceeds � 18 M⊕, with substantial time spent in that mass range. Protoplanets with 15 M⊕ cores grow to a Jupiter mass in ∼3 × 10 6 years if standard dust opacity is assumed, and in ∼10 5 years if 1% of standard dust opacity is adopted. In these cases, the planet spends substantial time with mass between 30−40 M⊕ before making the transition to rapid gas accretion. We emphasize that these growth times apply to the gas accretion phase and not to the prior core formation phase. According to the usual theory of protoplanet migration, although there is some dependence on disk parameters, migration in standard model disks is most effective in the mass range where the transition from type A to type B occurs. This is also the transitional regime between type I and type II migration. If a mechanism prevents the type I migration of low mass protoplanets, they could then undergo a rapid inward migration at around the transitional mass regime. Such protoplanets would end up in the inner regions of the disk undergoing type II migration and further accretion potentially becoming sub Jovian close orbiting planets. Noting that more dusty and higher mass cores spend more time at a larger transitional mass that in general favours more rapid migration, such planets are more likely to become close orbiters. We find that the luminosity of the forming protoplanets during the later stages of gas accretion is dominated by the circumplane- tary disk and protoplanet-disk boundary layer. For final accretion times for one Jupiter mass in the range 10 5−6 y, the luminosities are in the range ∼10 −(3−4) Land the characteristic temperatures are in the range 1000−2000 K. However, the luminosity may reach ∼10 −1.5 Lfor shorter time periods at the faster rates of accretion that could be delivered by the protoplanetary disk.
119470781
Some Applications of the ADM Formalism
The ADM Formalism is discussed in the context of 2+1--dimensional gravity, uniting two areas of relativity theory in which Stanley Deser has been particularly active. For spacetimes with topology R x T^2 the partially reduced and fully reduced ADM formalism are related and quantized, and the role of "large diffeomorphisms" (the modular group) in the quantum theory is illustrated.
119509119
The MINOS Magnets
The MINOS long baseline neutrino oscillation experiment will consists of two magnetized tracking calorimeters one on the Fermilab site and one in Minnesota. This paper summarizes the design, magnetic performance, and status of the MINOS magnets.