On the gravitational instability in thin gaseous Kepler disks

Abstract

The idea that the TitiusrBode law reflects an unstable mode of a self-gravitational 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 (6r/r 2 0.5) the well-known short-wave approximation is not appropriate for definite assertions. 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 (6r/r = 0.1) to be a serious challenge for a better, i.e. non-local theory.Die Idee, daD die Titius-Bodesche Reihe die instabile Mode einer gravitativen Instabilitat in einer dunnen Gasscheibe darstellt, erfordert eine sorgfaltige Behandlung der Poisson-Gleichung. Die haufig benutzte Kurzwellen-Approximation ist wegen der Abstandsverhaltnisse im Planetensystem ( F r / r 2: 0.5) fur giiltige Aussagen ungeeignet. Die von uns verwendete Reihenentwicklung in der Beziehung zwischen radialer und vertikaler Wellenzahl erlaubt auch die Behandlung ma0ig-langer sowie nicht-axialsymmetrisher Storungen. Die vorwiegend numerische Diskussion der Dispersionsrelation IaBt das Vorhandensein einer zudtzlichen instabilen Mode erkennen, deren Wellenlange die der bisher allein bekannten um etwa das Hundertfache ubersteigt. Obwohl selbst damit die geometrischen Verhaltnisse des solaren Planetensystems noch nicht wiedergegeben werden, halten wir die Existenz instabiler Mittelwellen (Fr/r 2: 0.1) doch fur eine ernste Herausforderung zur Konstruktion besserer, also nicht-lokaler Ansitze.

Nonlocal density wave theory for gravitational instability of protoplanetary disks without sharp boundaries

“…For protoplanetary disks Rüdiger & Tschäpe (1987) tried to overcome the shortcomings of the short-wave approximation by employing a relation between potential disturbances and density disturbances given by Bertin and Mark (1979). In contrast to the earlier treatments (cf.…”