Secular ring instability in the protoplanetary accretion disk

Abstract

The aim of the present paper is to show that an accretion disk, the material of which satisfies the Navier-Stokes equations of a compressible fluid, is secularly unstable to axisymmetric density disturbances even if Toomre's parameter Q is greater than one. This instability process, which can also be interpreted as negative diffusion, leads immediately to the formation of rings in the accretion disk as an intermediate step in the formation of planets, at least for the outer gaseous ones. We believe that the same process is also responsible for the ringlet-structure of planetary rings. Only a dynamical instability of axisymmetric density wave perturbations, represented by Toomre's condition Q < 1, does not exist in a viscous self-gravitating disk. On the base of a local theory we calculate the characteristic wave number and the corresponding time scale of the mode of maximum secular instability. In the global theory, we formulate a complex linear integral equation describing the diffusion instability in the accretion disk. The radial distances of the rings to the protosun obey in the outer range of the disk an exponential law provided that the quantity vo (the vertically integrated viscosity) is nearly constant throughout the disk. Our treatment gives arguments in favour for the result, that the spacing ratio 4 = r,,+I/r,, between two consecutive protoplanetary orbits is approximately given by the formula CJ = exp(2r+r"), where p is close to the ratio of the accreted ring-masses in the disk and the mass of the central body. We believe that the formation process is shorter for the outer gaseous than for the inner rocky planets, because the global secular ring instability is evanescent in the inner regions of the disk.

A gas-kinetic stability analysis of self-gravitating and collisional particulate disks with application to Saturn’s rings

“…Notice also that the critical value of Q ≈ 2 predicted in our analysis is close to Salo (1992Salo ( , 1995, Willerding (1992), Osterbart and Willerding (1995), Sterzik et al (1995), andGriv et al (1999b) numerical results. 15 Summarizing, collective motion connected with the Jeans-unstable mode is excited in the plane of a disk of mutually gravitating particles when the random velocity dispersion is not sufficiently large, c r < (2Ω/κ)c T , in other words, if the effective Toomre's Q-value is Q < 2Ω/κ ∼ 2.…”

“…In fact, it should be made clear right from the start that the suggestion of hyperfine < ∼ 100 m structure in Saturn's rings due to the combined effects of gravity and interparticle collisions is not an entirely new idea. Salo (1992Salo ( , 1995, Willerding (1992), Richardson (1994), Osterbart and Willerding (1995), and Sterzik et al (1995) have already predicted such a structure in Saturn's rings by N -body computer simulations.…”

On the Gravitational Stability of Gravito-Turbulent Accretion Disks

“…Viscosity-driven gravitational instability Viscous disks can also develop GI even when Q > 1. This is because, as demonstrated below, viscosity removes rotational support against self-gravity for longwavelength disturbances (Lynden- Bell & Pringle 1974;Willerding 1992;Gammie 1996). A similar effect occurs in dusty fluids where the required frictional forces are provided by dust-gas drag (Goodman & Pindor 2000;Ward 2000;Takahashi & Inutsuka 2014).…”

On the stability of Saturn's rings: a quasi-linear kinetic theory

“…They claimed this instability to be analogous to the well‐known viscous mechanism that converts Maclaurin spheroids to Jacobi ellipsoids. Morozov, Torgashin & Fridman (1985), Willerding (1992), Griv et al (2000a) improved Lynden‐Bell and Pringle's calculation by taking into account the effect of non‐uniform rotation in a two‐dimensional self‐gravitating system 2 . It seems that N ‐body simulations have indicated that the densest parts of the B ring with optical depth τ > 1 can exhibit spontaneous viscous overstability (Salo, Schmidt & Spahn 2001).…”