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model is needed, to explain at least the origin of the giant planets’ innermost satellites. Here, we consider a disk of solid material Formation of Regular Satellites around a planet, similar to Saturn’srings,where- in planetary tides prevent aggregation within the Roche radius rR (7) [supplementary text 1 (SM 1)]. from Ancient Massive Rings It is known that such a tidal disk will spread (8). Thus, the normalized disk lifetime can be de- in the . fined by tdisk = Mdisk/FTR,whereF is the flow through rR, TR is the at rR, 1 2,3 2 A. Crida * and S. Charnoz and Mdisk = pSrR is the disk’smass(S being the surface ). Using a prescription for the vis- When a planetary tidal disk—like Saturn’s rings—spreads beyond the Roche radius (inside cosity based on self-gravity and mutual colli- which planetary tides prevent aggregation), satellites form and migrate away. Here, we show sions (9), one finds that most regular satellites in the solar system probably formed in this way. According to our 2 (1) analytical model, when the spreading is slow, a retinue of satellites appear with increasing tdisk = 0.0425/D with distance to the Roche radius, in excellent agreement with Saturn’s, ’, and Neptune’s satellite systems. This suggests that Uranus and Neptune used to have massive rings that where D = Mdisk/Mp and Mp is the planet’s disappeared to give birth to most of their regular satellites. When the spreading is fast, only mass (SM 2.2.1). one large satellite forms, as was the case for Pluto and Earth. This conceptually bridges the gap As material migrates beyond rR, new between terrestrial and giant planet systems. form (10, 11). They are then repelled by the tidal disk through resonant angular momentum ex- atellites are generally thought to form con- possibly be lost) before the gas dissipates. These change and migrate outward as they grow. A currently with a giant planet, in a large models have their pros and cons, but none can satellite of mass M orbiting outside a tidal disk Scircumplanetary gaseous disk where there explain the surprising orbital architecture of Sat- experiences a positive gravitational torque (12) is inflow of solids. Two competing models exist urn’s, Uranus’, and Neptune’s satellite systems, 2 2 4 −2 −3 in the literature (1–4), in which solids aggregate where the smallest bodies accumulate at a dis- G =(32p /27)q SrR TR D (2) to form satellites that can migrate in the gas (and tance from the planet that is twice its radius (the Roche radius), and their masses increase with where q = M/Mp, D =(r – rR)/rR,andr is the F1 1Laboratoire Lagrange UMR 7293, Université de Nice Sophia– distance starting from this point (Fig. 1A). More- orbital radius. Thus, it migrates outward at a rate Antipolis, CNRS, Observatoire de la Côte d’Azur, BP4229, 2 over, in the frame of a circumplanetary gas disk, (SM 2.1) 06304 Nice Cedex 4, France. Laboratoire AIM, Université Paris Uranus’ satellites should orbit in the ecliptic plane, Diderot, CEA, IRFU-SAp, 91191 Gif-sur-Yvette Cedex France. dD dt 5 3 qDT −1D−3 3Institut Universitaire de France, 103 bd Saint Michel, 75005 and not the equatorial plane of the tilted planet / =(2/3 ) R (3) Paris, France. (5). Also, Uranus and Neptune might be too light *To whom correspondence should be addressed. E-mail: to have retained a massive enough gaseous disk (6). The migration rate increases with mass-ratio q [email protected]. These considerations suggest that an alternative and decreases with distance D. Based on the

Fig. 1. Distribution of the regular satellites of the giant planets. Saturn: 9 distance to rR, not to the center of the planets. (B) Satellite-to-planet mass ratio satellites from Pandora to . Uranus: (A) 18 from to ; (B) 14 q as a function of D =(r – rR)/rR.TheRocheradiusforeachplanetistaken satellites, from to Oberon (except Cupid and , out of scale). Neptune: consistently with the mean density of satellites, or with the orbit of the closest Naiad,Thalassa,Despina,Galatea,Larissa, and . Jupiter: Metis, Adrastea, one (SM 1). For Saturn, Uranus, and Neptune, rR = 140,000, 57,300, and Amalthea, These, , , , and . (A)Massasafunctionof 44,000 km, respectively. Short dashed curves: our model for the pyramidal regime 9/5 3.9 the orbital radius. The four systems do not extend all the way down to the Q(D) (Eq. S25, SM 6.3): for D < D2:1, q º D ;forD > D2:1, q º (D +1) , planetary radius (vertical line), but a pile-up of small satellites is observed at a where D2:1 = 0.59 is marked by the vertical dashed line. Jupiter’ssystem specific distance (the Roche limit, rR). The mass increases from zero with the does not fit well and is not shown (SM 7.3).

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restricted three-body problem, we assume that a newly formed moon before this new moon leaves This specific architecture is a testable obser- satellite accretes everything within 2 Hill radii, the continuous regime (SM 4.1). This corre- vational signature of this process. A compari- 1/3 rH, from its orbit (13, 14)[rH = r(q/3) ]. Thus, sponds to sonwithtoday’s giant planet systems of regular as a tidal disk spreads, a competition takes place moons reveals a very good match for Saturn, between accretion and migration. We assume D = Dd = ~3.1Dc (5) Uranus, and Neptune (Fig. 1B). Neptune’sinner that the satellites do not perturb each other’s moons (blue stars) match our model (blue dotted –3/2 3 orbit or the disk (SM 10), that D << 1, and that and q = qd ~20tdisk ~ 2200D as shown in line) very well, except for Despina, whose mass D and tdisk (thus F and S) are constant. We find SM 4. Equation 1 implies that this occurs after is underpredicted by a factor of 3. Uranus’ sat- from our analytical model (SM) that moon ac- ~100 orbits. Also, qd <0.1D provided D <6.7× ellites are somewhat more scattered, but the cretion proceeds in three steps, corresponding to 10−3. This is always the case around giant plan- system globally follows our model on the two three different regimes of accretion. ets (see below and SM 7), and it justifies the sides of r2:1 (SM 7.4), and the large number of When the disk starts spreading, a single moon assumption that D and tdisk are constant. After satellites outside rR appears to be a natural by- forms at the disk’s edge (moon 1). As long as D < Dd is reached, a third moon appears in the sys- product of the pyramidal regime; however, the 2rH/r, it will directly accrete the material flow- tem, and the discrete regime ends. four main bodies are not ranked by mass. Sat- ing through rR and grow linearly with time, while As moons of fixed mass (produced by the urn’s case is especially notable because the eight migrating outward. This is the “continuous re- discrete regime) appear successively at a given regular satellites from Pandora to Titan (consid- F2 gime” (Fig. 2A, SM 3). Integrating Eq. 3 with radius, they migrate outward with decreasing ering Janus plus Epimetheus as one object) have M = Ft, one finds that the condition D <2rH/r rate, and hence their mutual distance decreases; masses that closely follow our model (red dashed holds until eventually, they merge. Therefore, moons of dou- line), through four orders of magnitude in dis- ble mass are periodically formed, migrate out- tance and six in mass. Although unlikely, given 1/2 D = Dc =(3/tdisk) ward, merge again, and so on. Assuming that our present understanding of Saturn’s tides (15), – the satellites do not perturb each other’s orbit, this result suggests that even Titan might have q = q =~2t 3/2 c disk (4) mergers occur hierarchically—this is the “py- formed from the spreading of Saturn’s initially ramidal regime” (SM 6). The moons’ masses massive rings. Titan being about 50 times more Using Eq. 1, one finds Dc =8.4D, occurring after increase with distance, and an ordered orbital ar- massive than could be understood in the 2/3 ~10 orbits (SM 3.1). chitecture settles. In the region r < r2:1 =2 rR, frame of our model if the rings were initially −3 Then, a second moon forms at rR (moon 2). the migration is controlled by the disk’s torque very massive (D ~10 , see below), favoring the Moon 2 migrates away rapidly, approaches (Eq. 3); then the mass-distance relation fol- formation of one dominant moon, which mi- moon 1, and is caught by it. Another moon forms, lows M º D1.8, and the number density of grated fast outward, taking most of the mass. which is also eventually accreted by moon 1, moons is proportional to 1/D (SM 6.1). Con- Then, as the mass of the rings decreased, a stan- and so on. The growth of moon 1 thus proceeds sequently, just outside rR, an accumulation of dard pyramidal regime took place, giving birth at the same average rate as before, but step by small moons is expected, consistent with obser- to the other moons. However, Titan’s tidal age step through the accretion of moonlets: This is vations (Fig. 1A). Beyond r2:1, the migration is (the time needed to reach its orbit through Sat- the “discrete regime” (Fig. 2B, SM 4). When D > controlled by the planet’stidesandM º r3.9 urn’s tides) is estimated at about 10 billion years Dc +2rH/r, moon 1 is too far away to accrete a (SM 6.2). (11); thus, Titan’s fit with our model may be a coincidence. Jupiter’ssystemisn’t compatible with the py- Fig. 2. Sketches of the ramidal regime (SM 7.3), suggesting a differ- three accretion regimes, ent formation mechanism, even if the Laplace where the accretion re- resonance may have erased the initial configu- gions are defined as T2rH ration. The system of the Galilean moons is sat- around a moon’slocation. isfactorily explained by the circumplanetary disk The tidal disk is in gray models (1–4). These models may also explain with F denoting the mass the formation of Saturn’s moons Titan and flow at the edge. The first but not those of the other satellites. Indeed, the moon to form is in blue, conditions inside Saturn’s circumplanetary disc the second in red, and the were very different. Jupiter opened up a gap in third in green. (A)Con- the protoplanetary nebula early in the history of tinuous regime: only one the solar system, whereas Saturn, which formed moon is present, directly later and is less massive, hardly did (16). ’ fed by the disk smass Our model also predicts the conditions under flow. (B) Discrete regime: which one moon is formed rather than numer- When the first moon has ous ones: Surprisingly, the mass and distance r > r = r (D +1),anew c R c of moon 1 at the end of the discrete regime depend moonlet forms (in red). on only one parameter, t (or D). Figure 3 F3 The first moon (blue) con- disk shows the different regimes encountered dur- tinues to grow by accret- D t ing these moonlets (red), ing the recession of a moon in the ( , disk) space t which are fed by the disk (SM 8). When disk is large, a moon that forms in the discrete regime reaches a low mass and and have masses ≤ Mc.(C) Pyramidal regime: When small D, as shown by Eq. 5. Thus, the pyramidal the first moon has r – 2r > regime dominates, where many moons coexist H ’ rc, several moons can form and migrate away. For each of the solar system s andgrowuptoM ≥ Mc. planets, the mass of the putative tidal disk is Moons with r – 2rH > rc chosen to be about 1.5 times the mass of its cur- grow through merging events between moons of similar masses. rent satellite system (SM 7) (Fig. 3). For all the

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Fig. 3. Zones of the three grinding of the rings through meteoritic bom- regimes in the D – tdisk bardment (22) could all be invoked, but their role space obtained through is still not well understood. However, our model numerical integration of is compatible with existing ring-formation sce- the exact disk’storque narios, and thus the most common mechanism (SM 8). The right verti- of satellite formation is likely to be the spread- calaxisdisplaysthecorre- ing of a tidal disk surrounding a planet, terrestrial D t sponding to disk through or giant, besides other processes (1–4). The struc- Eq.1.Whenadiskspreads ture of the satellite system then depends only on and forms satellites, a sat- the disk’s lifetime. ellite follows a horizontal line (D increases and t disk References and Notes is constant), from left to 1. R. Canup, W. R. Ward, Astron. J. 124, 3404 (2002). right. First, the satellite 2. R. M. Canup, W. R. Ward, Nature 441, 834 (2006). appears in the continu- 3. I. Mosqueira, P. R. Estrada, Icarus 163, 198 (2003). ous regime (black region) 4. I. Mosqueira, P. R. Estrada, Icarus 163, 232 (2003). where it is fed directly 5. G. Boué, J. Laskar, Astrophys. J. Lett. 712, L44 (2010). from the disk, while mi- 6. B. A. Ayliffe, M. B. Bate, Mon. Not. R. Astron. Soc. 397, 657 (2009). grating away. Then it en- 7. Roche, E. Mémoire de la section des sciences, Académie des ters the discrete regime sciences et des lettres de Montpellier 1, 243-262 (1849). (red region) where it grows by regularly accreting new moons appearing at the disk’s edge, in the black 8. J. Salmon, S. Charnoz, A. Crida, A. Brahic, Icarus 209, region. Finally, it leaves the discrete regime, and many satellites may form and accrete all together— 771 (2010). this is the pyramidal regime (white region). The boundaries between the regions follow exactly our 9. H. Daisaka, H. Tanaka, S. Ida, Icarus 154, 296 (2001). D 10. S. Charnoz, J. Salmon, A. Crida, Nature 465, 752 (2010). analytical expressions for small (Eqs. 4 and 5). Refined equations for the boundaries are provided in 11. S. Charnoz et al., Icarus 216, 535 (2011). SM 9. The horizontal lines show the path that may have been followed around the solar system’s 12. D. N. C. Lin, J. Papaloizou, Mon. Not. R. Astron. Soc. 186, planets. The corresponding values of D were computed in SM 7 and for Jupiter, Saturn, Uranus, and 799 (1979). − − − − Neptune are 1.6 × 10 4,1.3×105,1.7×104, and 7.3 × 10 7,respectively. 13. R. Karjalainen, Icarus 189, 523 (2007). 14. A. Crida, J. C. B. Papaloizou, H. Rein, S. Charnoz, J. Salmon, Astron. J. 140, 944 (2010). −4 6 giant planets, D <2×10 ,sothattdisk >10 , These results strongly suggest that, like the 15. V. Lainey et al., Astrophys. J. 752, 14 (2012). making the pyramidal regime the final out- Moon and , most regular satellites of 16. T. Sasaki, G. R. Stewart, S. Ida, Astrophys. J. 714, 1052 (2010). come, which explains the presence and the dis- Saturn, Uranus, and Neptune formed from the 17. See SM 9 for a refinement of Eq. 4 at large D. tribution of their numerous innermost regular spreading of a tidal disk. It may be that only the 18. R. M. Canup, E. Asphaug, Nature 412, 708 (2001). satellites. giant planets’ most massive regular satellites 19. E. Kokubo, S. Ida, J. Makino, Icarus 148, 419 (2000). 20. R. M. Canup, Science 307, 546 (2005). Conversely, when tdisk is short, the contin- (the Galilean moons, Titan, and Iapetus) formed ’ 1–4 21. M. Jutzi, E. Asphaug, Nature 476, 69 (2011). uous and discrete regimes are more efficient, directly from the planet s subnebula ( ). Many 22. A. Harris, in R. Greenberg, A. Brahic, Eds., Planetary Rings and thus a massive satellite is built and mi- models have been proposed for the formation of (Univ. of Arizona Press, Tucson, 1984), pp. 641–659. grates to large distances (17). This applies to the giant planets’ massive rings. Rings could be 23. S. Charnoz, A. Morbidelli, L. Dones, J. Salmon, Icarus Pluto and Earth, which have only one moon. remnants of disrupted satellites [either by an 199, 413 (2009). N 22 23 24 24. R. M. Canup, Nature 468, 943 (2010). Our model agrees well with -body numerical impact ( , )orbytides( )], an explanation 25. L. Dones, Icarus 92, 194 (1991). simulations of the formation of Charon and the that is favored for Saturn, or remnants from 26. A. Morbidelli, K. Tsiganis, K. Batygin, A. Crida, R. Gomes, Moon (10, 18–20), but neglects thermodynam- tidally disrupted comets (23, 25), which is more Icarus 219, 737 (2012). ical effects. This limitation is investigated in SM likely for Uranus and Neptune (23). Uranus’ 27. L. W. Esposito, J. E. Colwell, Nature 339, 605 (1989). 7.1. In this case, where a single moon forms as and Neptune’s large inclinations indicate that 28. D. P. Rubincam, Icarus 184, 532 (2006). D the disk spreads, varies strongly with time, giant impacts capable of generating massive Acknowledgments: This work was partially funded by the making Eqs. 4 and 5 inaccurate. The system can rings were common on the ice giants during the French Programme National de Planétologie. S. Charnoz was still be solved provided F is constant, which is formation of the solar system (26). However, the supported by a “Labex UnivEarths” grant of Université likely in a disk at thermodynamical equilibrium. many uncertainties in the initial conditions of Paris Diderot and by CEA, IRFU, SAP. We thank A. Morbidelli, ’ D. Nesvorny, and N. Murdoch for comments and help in The Earth s Moon probably finished its accre- giant planet formation make it hard to conclude the writing of this manuscript. tion in the discrete regime, allowing the possi- how massive rings formed. The way Uranus’ and bility of a short-lived and low-mass companion Neptune’s massive rings disappeared is also an Supplementary Materials moon. As the companion approached the proto- open question. Viscous spreading by itself would www.sciencemag.org/cgi/content/full/VOL/ISSUE/PAGE/DC1 Supplementary Text Moon, it may have been trapped in a horseshoe not be efficient enough to make the rings disap- Figs. S1 and S2 orbit, and later impacted the proto-Moon, as was pear because it becomes increasingly inefficient References (29–39) recently suggested to explain the Moon’s high- when the disk loses mass (8). Atmospheric gas 22 June 2012; accepted 12 October 2012 lands (21). drag (22, 27), the Yarkvosky effect (28), or slow 10.1126/science.1226477

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