Streaming Torque in Dust–Gas Coupled Protoplanetary Disks

Abstract We investigate the migration of low-mass protoplanets embedded in dust–gas coupled protoplanetary disks. Linear calculations are performed with respect to the Nakagawa–Sekiya–Hayashi (NSH) equilibrium within a shearing sheet. We find that the dusty quasi-drift mode dominates the dynamical behaviors in close proximity to the protoplanet. This mode exhibits an extremely short radial wavelength, characterized by a dispersion relation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mover> <mml:mi>ω</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:msub> <mml:mrow> <mml:mi>W</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>s</mml:mi> </mml:mrow> </mml:msub> <mml:mo>⋅</mml:mo> <mml:mrow> <mml:mi>k</mml:mi> </mml:mrow> </mml:math> . The emergence of this mode leads to a wake with a short radial length scale ahead of protoplanets, contributing to a positive torque, termed as “streaming torque” (ST). Furthermore, both Lindblad torque and corotation torque are affected by the NSH velocity. The total torque and planetary migration are contingent upon the coupling strength between dust and gas. In most scenarios, ST predominates, inducing outward migration for planets, thereby addressing the issue of rapid inward migration in their formation paradigm.