On the orbital evolution and growth of protoplanets embedded in a gaseous disc

We present a new computation of the linear tidal interaction of a protoplanetary core with a thin gaseous disc in which it is fully embedded. For the first time a discussion of the orbital evolution of cores with eccentricity (e) significantly larger than the gas-disc scale height to radius ratio (H/r) is given. We find that the direction of orbital migration reverses for e>1.1H/r. This occurs as a result of the orbital crossing of resonances in the disc that do not overlap the orbit when the eccentricity is very small. Simple expressions giving approximate fits to the eccentricity damping rate and the orbital migration rate are presented. We go on to calculate the rate of increase of the mean eccentricity for a system of protoplanetary cores due to dynamical relaxation. By equating the eccentricity damping time-scale with the dynamical relaxation time-scale we deduce that an equilibrium between eccentricity damping and excitation through scattering is attained on a 10^3 to 10^4 yr time-scale, at 1au. The equilibrium thickness of the protoplanet distribution is such that it is generally well confined within the gas disc. By use of a three dimensional N-body code we simulate the evolution of a system of protoplanetary cores, incorporating our eccentricity damping and migration rates. Assuming that collisions lead to agglomeration, we find that the vertical confinement of the protoplanet distribution permits cores to build up from 0.1 to 1 earth mass in only ~10^4 yr, within 1au. The time-scale required to achieve this is comparable to the migration time-scale. We deduce that it is not possible to build up a massive enough core to form a gas giant planet before orbital migration ultimately results in the preferential delivery of all such bodies to the neighbourhood of the central star. [Abridged]Comment: Latex in MNRAS style, 13 pages with 6 figures, also available from http://www.maths.qmw.ac.uk/~jdl

Rings in the Planetesimal Disk of Beta Pic

The nearby main sequence star Beta Pictoris is surrounded by an edge-on disk of dust produced by the collisional erosion of larger planetesimals. Here we report the discovery of substructure within the northeast extension of the disk midplane that may represent an asymmetric ring system around Beta Pic. We present a dynamical model showing that a close stellar flyby with a quiescient disk of planetesimals can create such rings, along with previously unexplained disk asymmetries. Thus we infer that Beta Pic's planetesimal disk was highly disrupted by a stellar encounter in the last hundred thousand years.Comment: Accepted by ApJ Letters. LaTeX, 13 pages, 4 figures, full PostScript file available from http://www.maths.qmw.ac.uk/~jdl

Gas Giant Protoplanets Formed by Disk Instability in Binary Star Systems

We present a suite of three dimensional radiative gravitational hydrodynamics models suggesting that binary stars may be quite capable of forming planetary systems similar to our own. The new models with binary companions do not employ any explicit artificial viscosity, and also include the third (vertical) dimension in the hydrodynamic calculations, allowing for transient phases of convective cooling. The calculations of the evolution of initially marginally gravitationally stable disks show that the presence of a binary star companion may actually help to trigger the formation of dense clumps that could become giant planets. We also show that in models without binary companions, which begin their evolution as gravitationally stable disks, the disks evolve to form dense rings, which then break-up into self-gravitating clumps. These latter models suggest that the evolution of any self-gravitating disk with sufficient mass to form gas giant planets is likely to lead to a period of disk instability, even in the absence of a trigger such as a binary star companion.Comment: 52 pages, 28 figure

Close stellar encounters with planetesimal discs: The dynamics of asymmetry in the Beta Pictoris system

We numerically investigate the dynamics of how a close stellar fly-by encounter of a symmetrical circumstellar planetesimal disc can give rise to the many kinds of asymmetries and substructures attributed to the edge-on dusty disc of Beta Pic. In addition we present new optical coronagraphic observations of the outer parts of Beta Pic's disc, and report that the radial extent is significantly greater than was found in previous measurements. The northeasterly extension of the disc's midplane is now measured out to 1835au from the star; the southwesterly component is measured out to 1450au. Hence we use the length asymmetry induced in a distribution of simulation test particles as the principal diagnostic feature when modelling the disc response, in order to constrain fly-by parameters. In particular we favour a low inclination prograde and near-parabolic orbit perturber of mass approximately 0.5 Solar masses. These initial conditions suggest that the perturber could have been physically associated with Beta Pic prior to the encounter. Thus we also consider the possibility that the perturber could be bound to Beta Pic: a consideration also of general interest where dust discs are known to exist in binary star systems. In some of our models, we can relate groupings of perturbed particles to the large-scale structure of the Beta Pic disc. The groupings correspond to: high eccentricity and inclination particles that reach apocentre and maximum height in the southwest, moderately eccentric and low inclination particles that reach apocentre in the northeast, and relatively unperturbed particles inside approximately 200au radius.Comment: Accepted by MNRAS. 15 pages, 19 figures (mainly low resolution). High quality PostScript from http://www.maths.qmw.ac.uk/~jdl

Collisional Cascades in Planetesimal Disks II. Embedded Planets

We use a multiannulus planetesimal accretion code to investigate the growth of icy planets in the outer regions of a planetesimal disk. In a quiescent minimum mass solar nebula, icy planets grow to sizes of 1000--3000 km on a timescale t = 15-20 Myr (a/30 AU)^3 where a is the distance from the central star. Planets form faster in more massive nebulae. Newly-formed planets stir up leftover planetesimals along their orbits and produce a collisional cascade where icy planetesimals are slowly ground to dust. The dusty debris of planet formation has physical characteristics similar to those observed in beta Pic, HR 4796A, and other debris disks. We derive dust masses for small particles, 1 mm and smaller, and large particles, 1 mm and larger, as a function of the initial conditions in the planetesimal disk. The dust luminosities derived from these masses are similar to those observed in Vega, HR 4796A, and other debris disks. The calculations produce bright rings and dark gaps. Bright rings occur where 1000 km and larger planets have recently formed. Dark gaps are regions where planets have cleared out dust or shadows where planets have yet to form.Comment: to be published in the Astronomical Journal, January 2004; 7 pages of text; 17 figures at http://cfa-www.harvard.edu/~kenyon/pf/emb-planet-figures.pdf; 2 animations at http://cfa-www.harvard.edu/~kenyon/pf/emb-planet-movies.htm

Angular Momentum Transfer in Star-Discs Encounters: The Case of Low-Mass Discs

A prerequisite for the formation of stars and planetary systems is that angular momentum is transported in some way from the inner regions of the accretion disc. Tidal effects may play an important part in this angular momentum transport. Here the angular momentum transfer in an star-disc encounter is investigated numerically for a variety of encounter parameters in the case of low mass discs. Although good agreement is found with analytical results for the entire disc, the loss {\it inside} the disc can be up to an order of magnitude higher than previously assumed. The differences in angular momentum transport by secondaries on a hyperbolic, parabolic and elliptical path are shown, and it is found that a succession of distant encounters might be equally, if not more, successful in removing angular momentum than single close encounter.Comment: 11pages, 8 figures, 1 tabl

Precessing warped discs in close binary systems

We describe some recent nonlinear three dimensional hydrodynamic simulations of accretion discs in binary systems where the orbit is circular and not necessarily coplanar with the disc midplane. The calculations are relevant to a number of observed astrophysical phenomena, including the precession of jets associated with young stars, the high spectral index of some T Tauri stars, and the light curves of X-ray binaries such as Hercules X-1 which suggest the presence of precessing accretion discs

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.Comment: 14 pages, 9 figures, to appear in MNRA

The tidally induced warping, precession and truncation of accretion discs in binary systems: three-dimensional simulations

We present the results of non linear, hydrodynamic simulations, in three dimensions, of the tidal perturbation of accretion discs in binary systems where the orbit is circular and not necessarily coplanar with the disc mid-plane. The accretion discs are assumed to be geometrically thin, and of low mass relative to the stellar mass so that they are governed by thermal pressure and viscosity, but not self-gravity. The parameters that we consider in our models are the ratio of the orbital distance to the disc radius, D/R, the binary mass ratio, the initial inclination angle between the orbit and disc planes and the Mach number in the outer parts of the unperturbed disc. For binary mass ratios of around unity and D/R in the range 3 to 4, we find that the global evolution of the discs is governed primarily by the value of the Mach number. For relatively low Mach numbers (i.e. 10 to 20) we find that the discs develop a mildly warped structure, are tidally truncated, and undergo a near rigid body precession at a rate which is in close agreement with analytical arguments. For higher Mach numbers (approx. 30), the evolution is towards a considerably more warped structure, but the disc none the less maintains itself as a long-lived, coherent entity. Additionally, it is found that the inclination angle between the disc and orbital angular momentum vectors evolves on a longer timescale which is probably the viscous evolution timescale of the disc. The calculations are relevant to a number of observed astrophysical phenomena, including the precession of jets associated with young stars, the high spectral index of some T-Tauri stars, and the light curves of X-ray binaries such as Hercules X-1 which suggest the presence of precessing accretion discs.Comment: Accepted by MNRAS, received in original form 23 January 1996. LaTeX, 22 pages. 15 figures available from [email protected], or by anonymous ftp (see http://www.maths.qmw.ac.uk/~jdl/preprint.html for instructions