Extended Data Fig. 4: Various models for the collisional restitution coefficient.
From: A dense ring of the trans-Neptunian object Quaoar outside its Roche limit
Left: Model 1: Frost-covered ice spheres at temperature T = 210 K, \({\epsilon }({v}_{{\rm{n}}})={({v}_{{\rm{n}}}/{v}_{{\rm{c}}})}^{-0.234}\), with vc = 0.0077 cm s−1 (ref. 25); Model 2: Frost-covered ice at T = 123 K, \({\epsilon }({v}_{{\rm{n}}})=0.48{{v}_{{\rm{n}}}}^{-0.20}\) (ref. 8); Model 3: as Model 1 but with vc = 0.077 cm s−1 (ten times the value of Model 1); Model 4: particles of radius R = 20 cm with compacted frost at T = 123 K, \({\epsilon }({v}_{{\rm{n}}})=0.90\,\exp (-0.22{v}_{{\rm{n}}})+0.01{{v}_{{\rm{n}}}}^{-0.6}\) (ref. 8). Right: Theoretical relation for the dependence of critical coefficient of restitution ϵcr on optical depth, required for the balance between dissipation and the viscous gain of energy due to local viscosity 43. In case of velocity-dependent elasticity, the system adjusts its impact velocities (via velocity dispersion) so that the effective mean ϵ corresponds to the ϵcr(τ). In case of constant ϵ < ϵcr the system flattens to a near-monolayer state with a minimum c ≈ 2 to 3Rn ≈ 0.01 cm s−1R/1 m (where n is the mean motion) determined by the non-local viscous gain associated with the finite size of the particles. If the constant ϵ > ϵcr, no thermal balance is possible and the system disperses via exponentially increasing c.
