Planetary Torques as the Viscosity of Protoplanetary Disks

By J. Goodman, R. Rafikov. 2000

223 citations recorded by the source corpus.

We revisit the idea that density wave wakes of planets drive accretion in protostellar disks. The effects of many small planets can be represented as a viscosity if the wakes damp locally but the viscosity is proportional to the damping length. Damping occurs mainly because of shocks even for Earth-mass planets. The excitation of the wake follows from standard linear theory including the torque cutoff. We use this as input to an approximate but quantitative nonlinear theory based on Burger's equation for the subsequent propagation and shock. Shock damping is indeed local, but weakly so. If all metals in a minimum-mass solar nebula are invested in planets of a few Earth masses each, dimensionless viscosities (α) of the order of -4 dex to -3 dex result. We compare this with observational constraints. Such small planets would have escaped detection in radial velocity surveys and could be ubiquitous. If so, then the similarity of the observed lifetime of T Tauri disks to the theoretical timescale for assembling a rocky planet may be fate rather than coincidence.

Subfields: Astronomy and Astrophysics, Instrumentation

View Paper: https://www.semanticscholar.org/paper/7a3a6577cf5f8f81cb2e0aa3d3904ad1d05ad359

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Citations in the corpus

The excitation of spiral density waves through turbulent fluctuations in accretion discs – II. Numerical simulations with MRI-driven turbulence (2008, background): “Shocks are expected even under conditions of weak excitation because as shearing waves propagate their radial wavelength shortens, resulting in the onset of non-linearity and shock formation (e.g. Goodman & Rafikov 2001).”
On the width and shape of gaps in protoplanetary disks (2005, methodology): “5 Gravity torque t_g Viscous torque t_v Pressure torque t_P Sum of the torques Viscous torque from (4)”
Three-dimensional Interaction between a Planet and an Isothermal Gaseous Disk. II. Eccentricity Waves and Bending Waves (2004, background): “Since Goodman & Rafikov (2001) showed that the waves are dissipated by a nonlinear effect after their excitation, the nonreflecting condition would be valid.”
Pebble-isolation mass: Scaling law and implications for the formation of super-Earths and gas giants (2018, methodology): “In contrast, the original formula from Lambrechts et al. (2014), which is at the base of the new, refined formula proposed here, is similar to the expression of the critical mass for the wake to shock in a disc (Goodman & Rafikov 2001; Rafikov 2002).”
Formation of the Galilean Satellites: Conditions of Accretion (2002, background): “The interaction of the satellites themselves with the disk provides a source of angular momentum transport (e.g., Goodman & Rafikov 2001).”
Multiple Disk Gaps and Rings Generated by a Single Super-Earth (2017, background): “Goodman & Rafikov (2001) showed that in an inviscid disk having a two-dimensional (2D) adiabatic index γ, waves launched to either side of the planet’s orbit travel a radial distance lsh ≈ 0.8 ( γ + 1 12/5 Mp Mthermal )−2/5 h (2) before shocking and dissipating (see also Rafikov 2002a).”
Massive Black Hole Binary Evolution (2004, background): “Some of the mechanisms that remove angular momentum of interstellar gas and thus channel it into the neighborhood of SBHs include the torquing of gas flow by the rapidly-fluctuating potential of merging galaxies [144] and by nested stellar bars [199], angular momentum transport by hydrodynamical tur”
OBSERVATIONAL SIGNATURES OF PLANETS IN PROTOPLANETARY DISKS: SPIRAL ARMS OBSERVED IN SCATTERED LIGHT IMAGING CAN BE INDUCED BY PLANETS (2015, background): “As predicted by Goodman & Rafikov (2001) and Rafikov (2002) and shown in simulations (Dong et al. 2011; Duffell & MacFadyen 2013; Zhu et al. 2013; Richert et al. 2015), density waves shock at a couple of scale heights away from the planet’s orbit when M M h rp 3( ) .”
Planet Migration and Gap Formation by Tidally Induced Shocks (2001, background): “…these mechanisms are probably ineffective in cold, weakly ionized, and optically thick systems such as protoplanetary disks (Hawley & Stone 1998; Goodman & Rafikov 2001, hereafter GR01); in this case tides raised by low-mass planets could propagate much farther than just a fraction of the disk…”
RINGED SUBSTRUCTURE AND A GAP AT 1 au IN THE NEAREST PROTOPLANETARY DISK (2016, background): “In principle, a single low-mass planet at ∼40 AU could potentially open a ‘double-gap’ feature if the disk is particularly inviscid (e.g., Goodman & Rafikov 2001; Dong et al. 2011; Duffell & MacFadyen 2012).”
The physics of star formation (2003, background): “…influence on the evolution of a disk by tidally extracting angular momentum from the inner part and transferring it to the outer part (Goldreich and Tremaine 1982; Lin and Papaloizou 1993; Goodman and Rafikov 2001), potentially driving disk evolution on a timescale of 106 years (Larson 1989).”
Dynamics of Protoplanetary Disks (2010, background): “Several authors have speculated that this apparent coincidence might, to the contrary, signal a causal relationship; perhaps the formation of planets itself catalyzes angular momentum transport and results in the viscous dispersal of the gas disk (Goodman & Rafikov 2001, Sari & Goldreich 2004).”
Dynamics and Origin of the 2:1 Orbital Resonances of the GJ 876 Planets (2001, unknown): “10−3 (Goodman & Rafikov 2001).”
HOW EMPTY ARE DISK GAPS OPENED BY GIANT PLANETS (2013, background): “Goodman and Rafikov (2001) calculated that nonlinear steepening of waves causes them to dissipate over lengthscales of∼3h (for q = 10−3 and h/r = 0.05; the damping length scales asq−0.4 and (h/r)2.2).”
The Disk Substructures at High Angular Resolution Project (DSHARP). VII. The Planet–Disk Interactions Interpretation (2018, background): “…those excited by low-mass planets, need to propagate in the radial direction for some distance to steepen into spiral shocks and induce gaps (Goodman & Rafikov 2001), (b) the horseshoe material has a slow relative motion with respect to the spiral shocks thus this material takes a long time…”
Enhanced Angular Momentum Transport in Accretion Disks (2003, background): “The question of whether an ensemble of planets embedded in a disk might serve as an effective viscosity has been examined recently by Goodman & Rafikov (2001).”
Eccentricity Evolution for Planets in Gaseous Disks (2002, background): “Apsidal waves are not traveling waves subject to the nonlinear steepening and shock dissipation that is the leading form of damping for density waves generated at ordinary Lindblad resonances (Goodman & Rafikov 2001).”