On the gravitational instability in thin gaseous Kepler disks

The idea that the Titius-Bode law reflects an unstable mode of a self-gravitationl instability in very thin Keplerian disks makes a careful discussion of the Poisson equation especially necessary. Due to the planetary distances in the solar system (δr/r ⋍ 0·5) the well-known short-wave approximation is not appropriate for definite asertions. We will here use a simple series expansion of the relation between the radial and vertical wave numbers of the disturbances which is additionally valid for medium-scale and non-zonal modes. The numerical solution of the dispersion relation reveals an extra unstable branch for wave-lengths of rings and spirals two orders of magnitudes larger than those already known. Though we are not yet able to consider modes long enough for application to the planetary system, we feel the existence of the medium-wave instability (δr/r ⋍ 0·1) to be a serious challenge for a better, i.e. non-local theory.