| Abstract: | In this paper we investigate both the global and the local hydrodynamics of axisymmetric accretion disks around young stellar objects under the simultaneous action of viscosity, self-gravity and pressure forces. For simplicity, we take for the global model a polytropic equation of state, make the infinitely thin disk approximation and characterize the surface density and temperature profiles in the disk as power laws in the radial distance r from the protostar. We solve the problem of the general density profile of a Keplerian disk showing that self-gravity could not be an important factor for the fast formation of the rocky cores of giant gaseous planets in our solar system. Under the hypothesis that the unperturbed rotation curve of the disk is nearly Keplerian throughout the radial extent, we can estimate with our polytropic model a lower limit for the resulting masses Md(r) of stable disks up to 100 AU. These masses are in the range of the so-called minimum mass solar nebular (d/Ms ≈ 0.01–0.02).By adopting a simplified viscosity model, where the height-integrated turbulent dynamical viscosity ν is a function of the surface density σ like η ∝ σΓ, we derive in the local shearing sheet model linearized evolution equations for small density perturbations describing both a diffusion process and the propagation of acoustic density waves. We solve a special initial value problem and calculate the appropriate Green's function. The analytical solutions so obtained describe in the case Γ < 0 the successive formation of quasi-stationary ring-shaped density structures in a disk with a definite mode of maximum instability, whereas in the case Γ > Γc the density wave equation describes the propagation of an “overstable” ring-shaped acoustic density wavelet to the outer ranges of the accretion disk. Whereas the group velocity of the wave packet is subsonic, the phase velocities of individual wave crests in the wave packet are supersonic. The mode of maximum instability, the growth rate and the number of growing waves in the wavelet are controlled by Γ and α. Our present knowledge concerning turbulent viscosity in protoplanetary disks is not sufficient to decide whether or not the case Γ > Γc is realized.The suggested structuring processes in the linear theory should initiate in the non-linear regime the formation of narrow ring-shaped density shock waves moving through the protoplanetary disk. These non-linear waves could produce extremely spatially and temporally heterogeneous temperature regions in the disk. We speculate that ring-shaped density waves, excited by inner boundary conditions and which have dominated the disk's evolution at early times, are responsible both for the fast growth of dust to planetesimals and at least for the rapid accretion of the rocky cores of giant gaseous planets in the protoplanetary accretion disk (shock wave trigger hypothesis). We derive provisional scaling rules for planetary systems regarding the spacing of orbits as a function of the mass ratio of the protoplanetary disk to the protostar. However, further analytical work and linear as well as nonlinear numerical simulations of density waves excited by inner boundary conditions are needed to consolidate the results and speculations of our linear wave mechanics in the future. |
