An Upper Limit on Turbulent Viscosity in the Circumjovian Nebula

Abstract

Circumplanetary disks are open, dynamic systems sustained by a meridional circulation that draws gas and dust from the parent nebula along nearly polar streamlines, and ultimately returns some of this material through a viscously driven, in-plane outflow. In such a disk, the water ice line acts as a site of solid-mass accumulation: inward-drifting icy solids sublimate interior to the ice line, the resulting vapor is advected outward by the gas, and recondensation beyond the ice line resets the material to a small Stokes number. The operation of this drift-mediated loop requires the fragmentation-limited Stokes number of the mass-dominant icy aggregates to exceed the equilibrium Stokes number at which the radial drift of solids reverses. Because the former scales inversely with the Shakura-Sunyaev viscosity parameter $\alpha$ while the latter is directly proportional to it, this requirement yields a compact upper bound on the vigor of turbulence in the satellite-forming region. Adopting an actively heated, steady-state circumjovian disk truncated at the tidal radius, we find a marginal bound of $\alpha\lesssim 10^{-3}$ for an icy-particle fragmentation threshold of $v_f\simeq1\,{\rm m\,s^{-1}}$ - appropriate for cold ice with sticking properties similar to silicate dust. This bound scales linearly with $v_f$, reaching $\alpha\lesssim5\times10^{-3}$ for more adhesive, warm ice ($v_f\simeq5\,{\rm m\,s^{-1}}$). The bound carries no explicit dependence on the disk's mass flux, and is set entirely by the local thermal state, pressure gradient, and geometry of the circumjovian nebula.