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

On the tilting of protostellar disks by resonant tidal effects

We consider the dynamics of a protostellar disk surrounding a star in a circular-orbit binary system. Our aim is to determine whether, if the disk is initially tilted with respect to the plane of the binary orbit, the inclination of the system will increase or decrease with time. The problem is formulated in the binary frame in which the tidal potential of the companion star is static. We consider a steady, flat disk that is aligned with the binary plane and investigate its linear stability with respect to tilting or warping perturbations. The dynamics is controlled by the competing effects of the m=0 and m=2 azimuthal Fourier components of the tidal potential. In the presence of dissipation, the m=0 component causes alignment of the system, while the m=2 component has the opposite tendency. We find that disks that are sufficiently large, in particular those that extend to their tidal truncation radii, are generally stable and will therefore tend to alignment with the binary plane on a time-scale comparable to that found in previous studies. However, the effect of the m=2 component is enhanced in the vicinity of resonances where the outer radius of the disk is such that the natural frequency of a global bending mode of the disk is equal to twice the binary orbital frequency. Under such circumstances, the disk can be unstable to tilting and acquire a warped shape, even in the absence of dissipation. The outer radius corresponding to the primary resonance is always smaller than the tidal truncation radius. For disks smaller than the primary resonance, the m=2 component may be able to cause a very slow growth of inclination through the effect of a near resonance that occurs close to the disk center. We discuss these results in the light of recent observations of protostellar disks in binary systems.Comment: 21 pages, 7 figures, to be published in the Astrophysical Journa

Evolutionary outcomes for pairs of planets undergoing orbital migration and circularization: second order resonances and observed period ratios in Kepler's planetary systems

In order to study the origin of the architectures of low mass planetary systems, we perform numerical surveys of the evolution of pairs of coplanar planets in the mass range (1-4)\ \rmn{M}_{\oplus}. These evolve for up to 2\times10^7 \rmn{yr} under a range of orbital migration torques and circularization rates assumed to arise through interaction with a protoplanetary disc. Near the inner disc boundary, significant variations of viscosity, interaction with density waves or with the stellar magnetic field could occur and halt migration, but allow ircularization to continue. This was modelled by modifying the migration and circularization rates. Runs terminated without an extended period of circularization in the absence of migration torques gave rise to either a collision, or a system close to a resonance. These were mostly first order with a few $\%$ terminating in second order resonances. Both planetary eccentricities were small $< 0.1$ and all resonant angles liberated. This type of survey produced only a limited range of period ratios and cannot reproduce Kepler observations. When circularization alone operates in the final stages, divergent migration occurs causing period ratios to increase. Depending on its strength the whole period ratio range between $1$ and $2$ can be obtained. A few systems close to second order commensurabilities also occur. In contrast to when arising through convergent migration, resonant trapping does not occur and resonant angles circulate. Thus the behaviour of the resonant angles may indicate the form of migration that led to near resonance.Comment: 15 pages, 12 figures, 2014, MNRAS, 449, 304

Interchange instability in an accretion disc with a poloidal magnetic field

We investigate the stability to nonaxisymmetric perturbations of an accretion disc in which a poloidal magnetic field provides part of the radial support against gravity. Interchange instability due to radial gradients in the magnetic field are strongly stabilized by the shear flow in the disc. For smooth field distributions this instability is restricted to discs in which the magnetic energy is comparable to the gravitational energy. An incompressible model for the instability akin to the Boussinesq approximation for convection is given which predicts the behaviour of the instability accurately. Global axisymmetric disturbances are also considered and found to be stable for a certain class of models. The results indicate that accretion discs may be able to support poloidal fields which are strong enough to suppress other forms of magnetic instability. These strong and stable field distributions are likely to be well suited for the magnetic acceleration of jets and winds.Comment: uuencoded gzip'ed postscript, 9 page