The effects of Hall electromotive forces (HEMFs) on the linear stability of protostellar disks are examined. Earlier work on this topic focused on axial field and perturbation wavenumbers. Here we treat the problem more generally. Both axisymmetric and nonaxisymmetric cases are investigated. Though seldom explicitly included in calculations, HEMFs appear to be important whenever Ohmic dissipation is. They allow for the appearance of electron whistler waves, and since these have right-handed polarization, a helicity factor is also introduced into the stability problem. This factor is the product of the components of the angular velocity and magnetic field along the perturbation wavenumber, and it is destabilizing when negative. Unless the field and angular velocity are exactly aligned, it is always possible to find destabilizing wavenumbers. HEMFs can destabilize any differential rotation law, even those with angular velocity increasing outward. Regardless of the sign of the angular velocity gradient, the maximum growth rate is always given in magnitude by the local Oort A value of the disk, as in the standard magnetorotational instability. The role of Hall EMFs may prove crucial to understanding how turbulence is maintained in the ``low state'' of eruptive disk systems.Comment: 34 pages, 10 figures, AAS LaTEx, v.4.0. Submitted to Ap
The existence of extrasolar planets with short orbital periods suggests that planetary migration induced by tidal interaction with the protoplanetary disk is important. Cores and terrestrial planets may undergo migration as they form. In this paper we investigate the evolution of a population of cores with initial masses in the range 0.1Y1 M È embedded in a disk. Mutual interactions lead to orbit crossing and mergers, so that the cores grow during their evolution. Interaction with the disk leads to orbital migration, which results in the cores capturing each other in mean motion resonances. As the cores migrate inside the disk inner edge, scatterings and mergers of planets on unstable orbits, together with orbital circularization, causes strict commensurability to be lost. Near commensurability however is usually maintained. All the simulations end with a population typically between two and five planets, with masses depending on the initial mass. These results indicate that if hot super-Earths or Neptunes form by mergers of inwardly migrating cores, then such planets are most likely not isolated. We would expect to always find at least one, more likely a few, companions on close and often near-commensurable orbits. To test this hypothesis, it would be of interest to look for planets of a few to about 10 M È in systems where hot super-Earths or Neptunes have already been found. Subject headingg s: planetary systems: formation -planetary systems: protoplanetary disks
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