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ABA A + (& 2 'AA A A &(8 +- .)AA%2A*)A+6?'AH+FF4F!+EED;F EED<FEED;FED?+)2&- Supplementary Material Formation of regular satellites from ancient massive rings in the Solar System. Aur´elien Crida & S´ebastien Charnoz 1 The Roche radius 2 Torque, migration, and disk of the giant planets evolution 2.1 Satellite-disk interaction All the calculations and the model described in this paper In the case where ∆ = (r rR)/rR 1, i.e., when a are independent of the expression of the Roche radius − ≪ rR. However, in order to make Figure 1b and to test our satellite is very close to the disk’s outer edge, the gravi- model against the actual satellites systems, we need to tational torque exerted by the disk onto the satellitesimal know the value of rR around every giant planet. is given at first order by (12) : 8 − The Roche radius around a planet of radius Rp and 2 7 2 3 Γ = q ΣrRΩR(r rR) (S2) density ρp, for material of density ρ, is given by (7) : 27 − 3 where ΩR = GMp/rR is the angular velocity at the Roche radius. In response to this torque, the orbital 1/3 ρp 1 dr radius r changes following Γ = M 2 GMp/r dt . Using rR = 2.456Rp (S1) 2 ρ TR = 2π/ΩR, Mdisk = πΣr , D = Mdisk/M , and r R p ≈ rR, this gives: dr 16 2 GMp −3 For Uranus, the average density(29) of the five largest M GMp/r = qΣrR ∆ −3 dt 27 rR regular satellites is 1627 kg.m , so that Eq.(S1) gives 5 1 d(r rR) 2 GMp − rR = 57309 km. This is between the orbits of the second 3 − = 3 qD 3/2 ∆ √rR r dt 3 innermost satellite of Uranus (Ophelia), and the third 2π rR one (Bianca). We adopt this value. 5 d∆ 2 −1 −3 = 3 q D TR ∆ (S3) For Jupiter, the average density(29) of the Galilean dt 3 −3 satellites is 2370 kg.m , leading to rR = 141 000 km. ∼ 2.2 Disk evolution In the case of Neptune, however, a similar calculation It is well-known that any viscous disk in Keplerian ro- gives a Roche radius that would be 66 000 km, out- tation spreads (e.g. (8) ). We note F the mass flow ∼ side the orbit of four out of the six main regular satel- through rR. Then, the life-time of the disk, normalized lites. We conclude that the density of these satellites to the orbital period at rR reads : must be ill determined. Thus, we take the Roche ra- Mdisk dius to be a little inside the orbit of Naiad (the inner- τdisk = . (S4) most one), at 44000 km. This corresponds to a density FTR ρ = 4240 kg.m−3 in Eq. (S1). As we shall see below, this dimensionless parameter is central in the system. In the case of Saturn, the Roche radius is known to be From a hypothesis on the viscosity of the tidal disk, F between 138000 and 141000 km from the planet center, and τdisk are estimated as a function of the disk to planet constrained by the presence of the rings. So, we take mass ratio D in the next subsection. However, in what rR = 140000 km. This corresponds to a density ρ = follows, all expressions will be given both as a function of −3 720 kg.m in Eq.(S1). Note that the density of Janus τdisk independently of any assumption on the viscosity, is estimated to be 630 kg.m−3. and as a function of D under the hypothesis below. 6 Supplementary Material Crida & Charnoz 2012, Science, 338, 1196- 2.2.1 Estimation of F and τdisk 3.1 End of the continuous regime The self-gravity and mutual collisions inside a disk cause This regime applies only when the satellite is close a strong outward angular momentum flux.