Wave propagation in protoplanetary disks: Formation of twin planets by ‘disk–brown dwarf’ collisions?

In this paper we derive a special linear non-vortical wave propagation solution in the shearing sheet, a model of a compressible two-dimensional fluid system with constant density, constant shear and constant Coriolis force, but without self-gravity. The linear analysis of the shearing sheet leads to a single differential equation for the azimuthal velocity perturbation. A detailed derivation of a special solution with a prescribed azimuthal wavenumber k is presented. More general wave solutions, eventually excited by large local ‘impacts’, can be derived by superimposing all k-modes. The special wave functions so obtained describe the formation of two independent spiral wave arms originating out of a ring-shaped structure. The motivation for this investigation lies in the fact that similar wave propagators can be excited by the transit of a solid or ‘clumpy’ object through a protoplanetary disk. We speculate that a disk–brown dwarf collision can produce in the disk a pair of two spiral density wave fragments triggering the rapid accretion of two giant planets by a gravitational shear instability simultaneously (Hypothesis of a mechanism for the production of giant planets in pairs).