irregular satellite capture during planetary resonance passage

Icarus 183 (2006) 362–372www.elsevier.com/locate/icarus

Irregular satellite capture during planetary resonance passage

Matija Cuk ∗, Brett J. Gladman

Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Rd., Vancouver, B.C. V6T 1Z1, Canada

Received 6 December 2005; revised 28 February 2006

Available online 27 April 2006

Abstract

The passage of Jupiter and Saturn through mutual 1:2 mean-motion resonance has recently been put forward as explanation for their relativelyhigh eccentricities [Tsiganis, K., Gomes, R., Morbidelli, A., Levison, H.F., 2005. Nature 435, 459–461] and the origin of Jupiter’s Trojans[Morbidelli, A., Levison, H.F., Tsiganis, K., Gomes, R., 2005. Nature 435, 462–465]. Additional constraints on this event based on other small-body populations would be highly desirable. Since some outer satellite orbits are known to be strongly affected by the near-resonance of Jupiterand Saturn (“the Great Inequality”; Cuk, M., Burns, J.A., 2004b. Astron. J. 128, 2518–2541), the irregular satellites are natural candidates forsuch a connection. In order to explore this scenario, we have integrated 9200 test particles around both Jupiter and Saturn while they went througha resonance-crossing event similar to that described by Tsiganis et al. [Tsiganis, K., Gomes, R., Morbidelli, A., Levison, H.F., 2005. Nature435, 459–461]. The test particles were positioned on a grid in semimajor axes and inclinations, while their initial pericenters were put at just0.01 AU from their parent planets. The goal of the experiment was to find out if short-lived bodies, spiraling into the planet due to gas drag (oralternatively on orbits crossing those of the regular satellites), could have their pericenters raised by the resonant perturbations. We found thatabout 3% of the particles had their pericenters raised above 0.03 AU (i.e. beyond Iapetus) at Saturn, but the same happened for only 0.1% of theparticles at Jupiter. The distribution of surviving particles at Saturn has strong similarities to that of the known irregular satellites. If saturnianirregular satellites had their origin during the 1:2 resonance crossing, they present an excellent probe into the early Solar System’s evolution. Wealso explore the applicability of this mechanism for Uranus, and find that only some of the uranian irregular satellites have orbits consistent withresonant pericenter lifting. In particular, the more distant and eccentric satellites like Sycorax could be stabilized by this process, while closer-inmoons with lower eccentricity orbits like Caliban probably did not evolve by this process alone.© 2006 Elsevier Inc. All rights reserved.

Keywords: Origin, Solar System; Celestial mechanics; Resonances; Satellites of Saturn; Satellites of Uranus

1. Introduction

The origin of irregular satellites is a controversial topicin planetary science, despite the rapid increase in constraintssince 1997. The number of known irregular satellites is nowan order of magnitude larger than it was in the pre-CCD era(Gladman et al., 1998; 2000, 2001; Sheppard and Jewitt, 2003;Holman et al., 2004; Kavelaars et al., 2004; Sheppard et al.,2005). Their orbits have been studied in great detail, mostlythrough numerical methods, and in many cases reveal intrigu-ing secular resonances with their parent planets’ orbits. Finally,last year the Cassini spacecraft has explored the longest-known

* Corresponding author. Fax: +1 604 822 2422.E-mail address: [email protected] (M. Cuk).

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irregular satellite, Phoebe, revealing a complex small worldthat was previously known only through its orbit and a low-resolution spectrum. There is an undisputed consensus that ir-regular satellites are captured bodies, originally formed on cir-cumsolar orbits (Burns, 1986).

Colombo and Franklin (1971) suggested that the two clustersof jovian irregulars recognized at the time (one correspondingto Himalia’s family and the other containing the four largest ret-rograde jovians) resulted from a collision between two asteroidswithin Jupiter’s Hill sphere. While ingenious, this hypothesis isunlikely to be correct for several reasons. For one, we now knowthat ejecta from catastrophic collisions tend to form a singledebris cloud, rather that two well-separated ones, as this hy-pothesis requires. Secondly, such a collision is a rather unlikelyevent (Heppenheimer and Porco, 1977), and the discoveries of

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Irregular satellite capture 363

many new irregular satellite families around all four giant plan-ets call for many separate collisions.

Recently, a similar mechanism where gravitational scatter-ing between the planet and a binary planetesimal producessatellite capture, has been proposed for the capture of Triton,with promising results (Agnor and Hamilton, 2004). A relatedmodel involving scattering between two planetesimals withina planet’s Hill sphere has also been advocated for the captureof irregular satellites (Jewitt and Sheppard, 2005). A perceivedadvantage of this scenario is its independence from the exactmodel of the planet’s formation (the structure of sub-nebula,mass-growth rate, etc.), which could account for the approxi-mately equal integrated mass of irregulars around each of thefour giant planets (Jewitt and Sheppard, 2005). However, theapplicability of this argument to neptunians is questionable, asthey were likely decimated through scattering by (and collisionswith) both Triton and Nereid (Cuk and Gladman, 2005).

The simple picture of interactions within a “sea of plan-etesimals” producing a few captured irregulars is unrealistic,as observed irregular satellites require large velocity “kicks”to move from unbound to their present orbits (unlike binaryKBOs which are more plausibly captured this way; Goldreichet al., 2002). This “kick” is typically greater than 1 km s−1,which would require a near-collision with Pluto-sized or largerobjects. While such three-way encounters likely happened,this scenario will generally produce very eccentric orbits, justlike those proposed for early Triton (Agnor and Hamilton,2004), while the large majority of irregulars have eccentric-ities smaller than 0.7. Therefore, in order to have a reason-able probability of producing more strongly bound irregularsone requires interaction with another full-fledged protoplanet.Such planet–planet scattering has been proposed by some re-cent models of early solar system evolution (Thommes, 2002;Tsiganis et al., 2005), so this variant of three-body-capture cer-tainly cannot be excluded, and is presently being explored (D.Nesvorný, 2005, private communication).

Heppenheimer and Porco (1977) proposed that irregularsatellites of Jupiter were captured during a phase of rapid in-crease in the planet’s mass through gas accretion. This mech-anism has been explored more recently by Vieira Neto et al.(2004), who used numerical simulation to show that rapid andsubstantial mass increase by Jupiter (>10%) is needed for theretrograde jovians to evolve from initially unbound to their cur-rent stable orbits. Still, there are at least two major problemswith this scenario. First, gas that is being accreted onto theplanet would first have to cross the satellite’s orbit, leading tofriction between the two. In general, an amount of gas muchmore modest than that explored by Vieira Neto et al. (2004) canlead to substantial evolution through aerodynamic drag (cf. Cukand Burns, 2004a). Before self-consistent simulations includingboth gas drag and mass increase are published, we will have toassume that satellites captured by mass increase would likelyrapidly spiral into the planet due to the dense gas environmentsurrounding them. Secondly, gas accretion onto Uranus seemsinsufficient to capture many of its irregular satellites, as most ofthem have rather strongly bound orbits.

Capture by gas drag was first proposed by Pollack et al.(1979). They suggested that friction within a circumplanetaryenvelope might have slowed down passing asteroids enough forthem to become permanent satellites. The basic deficiency ofgas-drag capture—the lack of a protection mechanism againstcomplete orbital collapse—is resolved by having the enveloperapidly fall onto the planet once the capture is complete. Cukand Burns (2004a) have explored this scenario numerically andfound that a reasonably massive circumjovian disk is capableof capturing Himalia (arguably the most tightly-bound irregu-lar satellite). However, they still needed the envelope to eithershrink or completely collapse on 104-year timescale in order tohave Himalia survive on its present orbit. Other irregular satel-lites of Jupiter, being smaller and mostly retrograde, would havean even shorter lifetime in this nebula, requiring some othermechanism for stopping their collapse into Jupiter.

It is still unclear if there is one uniform mechanism for ir-regular satellite capture, or if the outer satellites of each ofthe four giant planets had very different histories. This can beexplored by comparing the overall architecture of the four ir-regular systems. Unfortunately, the data available are equivocal,pointing to both the common characteristics and important dif-ferences among the irregular satellite systems. Cuk and Burns(2004b), after mapping the location of secular resonances withthe Sun within each planet’s Hill sphere, noted that there issome clustering of satellite families around the apsidal align-ment resonance in which the longitude of the Moon’s pericenteris locked in either alignment or anti-alignment with that of itsparent planet (see Cuk and Burns, 2004b, for detailed analy-sis). This is the same resonance that has already been foundto affect Pasiphae (Whipple and Shelus, 1993), Sinope (Sahaand Tremaine, 1993) and Siarnaq (Cuk and Burns, 2004b). Thisclustering is strongest among retrograde jovians and progradesaturnians, possibly suggesting that there is more than one waysecular resonances can affect irregulars. Finally, some of thelargest irregulars, like Himalia and Phoebe, are also the mostdistant from resonance. This is interesting, as Cuk and Burns(2004a) argued, on the basis of their large size and low e, thatthey are the best candidates for simple gas-drag capture (seeSection 4).

One peculiarity of orbits within Saturn’s Hill sphere is thestrong “Great Inequality” resonance associated with the 2:5near-commensurability of Jupiter and Saturn. This semisecularresonance affects prograde moons with the apsidal precessionperiod of about 1850 years, which is twice the period of theGreat Inequality. Test particles satisfying this condition sufferstrong perturbations to their eccentricities and are generally un-stable on 106-year timescales. Since the orbital geometry ofthe early Solar System was likely very different from the onetoday (Hahn and Malhotra, 1999), it is conceivable that otherresonances of the similar type might may have affected the pri-mordial irregular satellite systems of some or all of the giantplanets. Such resonances would generally vary from planet toplanet, and may plausibly leave a signature in the orbital dis-tribution of the irregulars, making the resonance crossings agood candidate for a mechanism sculpting the observed irregu-lar satellite systems. Numerical experiments set up to test this

364 M. Cuk, B.J. Gladman / Icarus 183 (2006) 362–372

hypothesis are described in Sections 2 and 3, we discuss thelarger picture in Section 4, and our conclusions are summarizedin Section 5.

2. The 1:2 resonance of Jupiter and Saturn

While divergent resonance crossing has been suggested be-fore as a possible source of eccentricity in extrasolar planetarysystems (Chiang et al., 2002), only very recently has this mech-anism been applied to our own Solar System. Tsiganis et al.(2005) show that the eccentricities of Jupiter and Saturn can beelegantly explained assuming their formation on near-circularorbits more closely packed than a mutual 1:2 resonance (semi-major axis ratio of 1.59) and subsequent divergent migrationdue to interaction with a planetesimal disk. Jupiter’s and Sat-urn’s eccentricities and inclinations are “pumped up” to highervalues and are subsequently damped to their current values bydynamical friction. In order to maintain the planets’ eccentric-ities, the authors invoke either scatterings between Saturn andthe two “ice giants” (Uranus and Neptune), or possibly a moredynamically excited disk. For our purposes here, how the ec-centricity was maintained is not relevant, but the question ofpossible close encounters of Saturn with other planets is, asthey would affect their irregular satellites. In particular, if wecan demonstrate that present irregulars were captured duringthe Jupiter–Saturn resonance crossing, then strong constraintscan be put on subsequent close encounters involving Saturn.Constraints from irregular satellites of the ice giants are lessuseful, as the resonance crossings involving them could havehappened after the planet–planet scattering epoch (cf. Fig. 1 inTsiganis et al., 2005).

In order to test the effects of the 1:2 Jupiter–Saturn res-onance crossing on their irregulars we captured the essentialphysics by setting up a numerical integration containing station-ary Jupiter, smoothly migrating Saturn and 9200 test particlesorbiting Saturn (the experiment was later repeated with test par-ticles orbiting Jupiter). We used our own numerical integratorbased on the algorithm of Wisdom and Holman (1991), whichassumes planetocentric orbits (perturbed by the Sun) for par-ticles within the planet’s Hill sphere. The test particles wereplaced on highly eccentric orbits with pericenters of 0.01 AU.

Our initial conditions were designed to mimic a steady-statepopulation of planetesimals captured onto very eccentric orbitsby gas drag within the protosatellite nebula. The planetesimalsrepresented by these initial conditions would eventually spiralinto the planet unless some other process can alter their orbitsfirst. However, we assume that new objects were continuouslycaptured to take their place, without changing the average char-acteristics of the population, justifying our use of non-evolvingsatellites. In Section 4 we will discuss the possible effects ofsubstitution of a static population for the steady-state one.

The angular variables were chosen to be ω = 90◦ and 2� −2λSun = 0 for prograde and 2� − 2λSun = 180◦ for retrogradeparticles, so that the initial conditions would represent the maxi-mum eccentricity (and minimum pericenter) during both Kozaiand evection cycles (Cuk and Burns, 2004b). The semimajoraxes and inclinations were distributed on a grid 0.05 < a �

(a)

(b)

Fig. 1. (a) Distribution of particles with final q > 0.03 AU (green symbols)in semimajor axis–inclination space (retrograde test particles with a < 0.1 AUmarked with red symbols). Real saturnian irregulars with well-determined or-bits are marked with large blue asterisks, while the recent discoveries with nopublished mean elements are plotted as smaller violet asterisks. (b) Same as(a), but in semimajor axis–eccentricity space. Retrograde satellites generallyhave lower eccentricities than our surviving test particles (note that the violetasterisks stand for poorly known osculating elements of very recent discover-ies which, once their orbits are better known, will likely cluster with some ofthe old satellites). Note that Phoebe (a = 0.087 AU) does not correspond to anytest particles.

0.2 AU (100 values) and 0 � i � 50◦ (prograde, 51 values)and 140◦ � i � 180◦ (retrograde particles; 41 values). As theseinitial conditions place particles at the point in the Kozai cy-cle when the eccentricities are largest but the inclinations aresmallest, the average inclinations of our particles will be shiftedtoward 90◦ (relative to their initial values), which makes ourhigh-i “hole” in initial conditions less dramatic that it appears.We excluded those initial conditions simply because there areno irregulars with such inclinations, as such orbits are highlyunstable due to solar perturbations (Gladman et al., 2001;Carruba et al., 2002).

The protosatellite nebula invoked here is about the size ofthe regular satellite systems of Jupiter and Saturn. We have ourtest particles “captured” with pericenters at 0.01 AU, close to


where major satellites (like Ganymede and Titan) are today, rea-soning that the disk is thickest at that point. Such a disk shouldsurvive for at least 106 years, in order to allow for an undifferen-tiated Callisto (Canup and Ward, 2002; Mosqueira and Estrada,2003). At the end of the simulation, particles were consideredstable if their pericenters stayed outside 0.02 AU from Jupiter(1.6 Callisto distances) or 0.03 AU from Saturn (1.25 Iapetusdistances) during the last 20,000 years of the simulation. Wepoint out that the disk used here is much smaller and longerlived than one Cuk and Burns (2004a) require to capture Hi-malia; the outer part they discuss should have already thinned atthis stage. Their conclusion that the disk beyond about 0.05 AUhad a lifetime shorter than 105 years is in no way inconsistentwith a long lived (>106 years) compact disk closer to the planet(r < 0.03 AU).

Jupiter and Saturn were initially put on orbits with e = 0.01just interior to the 1:2 resonance (by reducing Saturn’s a), sothat the crossing happened during the first 20,000 years of thesimulations. We opted for this setup to avoid losing rapidly un-stable particles before the resonance passage. We also assumedthat the low pre-crossing eccentricities of Jupiter and Saturnmade strong secular interactions unlikely, which was confirmedpost-facto. The simulation ran for about 7.6×105 years, the last2 × 104 years of which we used to obtain final averaged orbitalelements. We then discarded all the particles with pericentersreaching below 0.03 and 0.02 AU for saturnians and jovians,respectively. This way, we were left with 325 surviving saturni-ans and just 37 jovians (8 of which had q > 0.03 AU).

Fig. 1 gives the a–i and a–e distributions of the survivingsaturnians, compared to its known irregular satellites. It is clearfrom the plots that there is a qualitative match between thetwo populations, especially in a and i. The prograde bodies inboth populations are concentrated at relatively high inclinations(35◦–55◦), while the retrograde bodies span a greater range ofinclinations, but with a concentration towards 180◦. While itappears that most saturnian irregulars can be generated by thisprocess, the model has trouble producing the lowest-e orbits(Fig. 1b), especially retrograde ones, with Phoebe being theworst outlier. This is clearly visible in Fig. 1b, where retrogradetest particles with a < 0.1 AU (red symbols) have eccentricitieswell above that of Phoebe. We think that the lack of gas dragin our model is the cause of this discrepancy, and that a moresophisticated model may be able to address the issue. There isno such discrepancy for prograde bodies, which is consistentwith them being less affected by gas drag. Finally, the lack ofgas drag in the simulation is likely to result in an overestimateof the survival rate for the retrograde particles with smaller a,which would be among shortest lived, explaining why there areno retrograde saturnians (apart from anomalous Phoebe) witha < 0.1 AU.

Why do the particles’ eccentricities get affected at all bythe planets’ resonance passage? By examining the histories ofa number of particles, we concluded that different secular ef-fects affect prograde and retrograde particles. Fig. 2a plots theeccentricity evolution of a prograde particle. The singular fea-ture is a sharp drop in e around 2 × 105 years. Note that thisevent happens well after the resonance crossing (which itself

(a)

(b)

Fig. 2. (a) Evolution of eccentricity for one of the prograde test particles inFig. 1. The Jupiter–Saturn resonance crossing happened some 20,000 years intothe simulation, and is not coincident with the sharp drop in eccentricity shownhere. (b) Evolution of the secular resonance argument Ψ = � − �S for thesame particle. The short period of Ψ -libration is clearly associated with thesharp drop in eccentricity seen in (a).

is at about 2 × 104 years). Inspecting other orbital elements ofthe particle, we found that it went through a short episode of li-bration of the secular argument Ψ = � − �Saturn. This is thesame argument involved in the current secular resonance of thesaturnian moon Siarnaq (Cuk and Burns, 2004b). Today, the in-teraction with the secular resonance is mild, as Saturn’s ownorbit evolves slowly and regularly. However, as Saturn’s eccen-tricity at that epoch varied greatly on the timescale of the 1:2inequality (∼103 years), the resonance was much more intenseand chaotic, leading to dramatic consequences. We found thatall prograde survivors had their orbits stabilized through thisprocess, typically crossing the resonance more than once. It isimportant to understand that this interaction is basically chaoticand that the eccentricity can both increase and decrease duringthis resonance, destroying some particles and stabilizing others.

Exploring the nature of the retrograde particles’ evolutionwas more challenging. Among these particles, the most dra-matic drop in eccentricity was typically coincident with theplanetary 1:2 resonance crossing. However, there were no


(a)

(b)

Fig. 3. (a) Evolution of eccentricity of a saturnian retrograde test particlein one of the additional simulations (in which the output interval was de-creased by a factor of ten). Jupiter–Saturn resonance crossing happened some1.7 × 105 years into this simulation, and coincides with the first sharp drop inthe particle’s eccentricity. (b) Evolution of the planetary near-resonant inverseperiod 1

2π(2nS − nJ ) (solid line) and the inverse period of the apsidal preces-

sion 12π

� for the same particle. It is clear that biggest jumps of eccentricity inpanel (a) happen when the two frequencies shown in (b) overlap.

slowly-moving secular angles in this case; both � and ω werevarying too fast for their precession periods to be recoverablefrom the simulations’ output, the interval of which was about150 years. To clarify the issue, we re-integrated all the initialconditions with i > 160◦, using one-tenth the output interval.Since we were now mostly interested in the events coincidingwith the resonance crossing itself, we set up our initial con-ditions so that the crossing happened closer to the middle ofthe simulation (�1.7 × 105 years). Fig. 3a shows the eccentric-ity history in one of the new simulations. Since the drop in e

is not caused by any purely secular mechanism, we decided totest commensurabilities between the circulation frequency ofthe planetary resonant argument φ21 = 2λS − λJ and the parti-cle’s longitude of pericenter. Fig. 3b plots how the frequenciesof the resonant argument φ21 and the particle’s � change overtime. Here, ˙φ21 was calculated using the semimajor axes ofJupiter and Saturn from the simulation’s output, while � was

Fig. 4. Orbital period vs minimum inclination plot for surviving test particles(small pluses) and saturnian irregulars with well determined orbits (asterisks).The area between vertical lines corresponds to a < 0.06 AU. The solid curvesmark the boundary between the apsidal precession periods longer (center) andshorter (sides) than 1000 years (calculated using the secular model of Cuk andBurns, 2004b, assuming emax = 0.6 and Saturn’s current orbit). The two curveslabeled “Sec” and “K” mark the location of secular and the lower edge of theKozai resonance, respectively. The present location of the Great Inequality res-onance (labeled “G.I.”) will be strongly destabilized, and particles around it willbe removed once the planets approach their present configuration. Likewise, thevery high-i Kozai resonators are likely unstable on the age of the Solar System(Carruba et al., 2002).

taken directly from the change of � in the output. The twocurves intersect at the times of greatest variability in e (Fig. 3a),indicating that the argument φ21 − � = 2λS − λJ − � is re-sponsible for the variability. This is similar to the present GreatInequality resonance that affects saturnians whose orbits satisfythe condition 5λS − 2λJ − 2� − �S (Cuk and Burns, 2004b).

How do these resonant interactions affect the orbital distri-bution of the surviving particles? Fig. 4 superposes contoursof constant � precession (calculated using the secular modelof Cuk and Burns, 2004b) with the distribution of survivingtest particles and extant saturnians in orbital period–minimuminclination space. In this figure we used the convention that ret-rograde bodies have negative periods but i < 90◦ (after Cuk andBurns, 2004b) in order for the lines of constant precession toappear smoother. “Minimum inclination” used here is the onesatellite has when ω = 90◦ and, unlike the mean inclination, isuniquely defined. All the precession contours are calculated forsatellites which have e = 0.6 when ω = 90◦, and assume thepresent orbital period for Saturn. Vertical dotted lines just in-dicate the periods for which a = 0.06 AU, between which noreal satellites are known and no test particles survived (all 736particles with 0.05 < a < 0.0635 AU retained low pericenters).

A major feature in Fig. 4 is that both satellites and test parti-cles tend to have � precession periods on the order of or longerthan 1000 years (continuous lines). Both resonances describedabove require satellites to have relatively slow precession. Theregion of phase space available to retrograde particles is larger,as the evection perturbation tends to slow down precession oftheir � , while it speeds up precession of prograde orbits. Thetwo fine-dashed lines on top are those of stationary pericenter


(i.e. � = 0; bottom) and the lower boundary of Kozai librations(top). Particles above the latter line have librating ω’s while allthose below have circulating ω’s. The � precession of Kozairesonators is generally fast (as � = Ω), therefore they likelymigrated from the slow-precession region once the orbit wassufficiently stabilized (low-e bodies can cross the circulation–libration boundary much more easily than high-e ones). In thecase of the secular resonance, it is currently restricted to theimmediate neighborhood of the � = 0 line, as �S = g6 is veryslow. However, at the epoch of the resonance crossing secu-lar frequencies wandered throughout the regions between theboundaries of fast precession and Kozai libration. Similarly, theresonant argument φ21 swept much of the equivalent region onthe retrograde side, stabilizing the large number of particles dis-tributed in that region. Therefore, there is a clear relationshipbetween the resonant stabilization mechanisms and the orbitaldistribution of the surviving particles. Also, long after the epochcovered our simulation, the sweeping by the Great Inequalityresonance destabilized at least some of the particle orbits ap-pearing in Fig. 1 (see Section 4).

While the resonant crossing produced dramatic results forSaturn’s irregulars, it has little effect on our initial conditionsaround Jupiter. Fig. 5 shows the a–i and a–e distributions ofthe few stabilized particles and the known irregular satellites.There is basically no correspondence between the two groups.A number of close-in retrograde survivors (not correspondingto any real moons) are likely just an artifact of our model, justlike their counterparts at Saturn (see discussion of Fig. 1b in thetext). The only jovian irregulars that could plausibly be capturedthrough this process are isolated prograde Themisto and Carpo(S/2003 J20), as some high-inclination prograde test particlesdid survive the simulation. However, the resonance-crossing isunable to account for the two main jovian irregular groups: Hi-malia’s directly orbiting family and the retrograde complex, andthis will be discussed in Section 4.

We have a relatively simple physical explanation of whythere is such a strong effect on saturnians but not on jovians.First, it is clearly the indirect perturbations from the other planetthat are affecting the satellite, rather than direct ones. To putit differently, the other planet is causing significant perturba-tions in the apparent (planetocentric) motion of the Sun, themain perturber of the irregular satellites’ orbits. In the case ofthe saturnian moons, Jupiter’s perturbations were in the samephase from one Saturn’s orbit to the next, as Jupiter was com-pleting about two full orbits for each one of Saturn. On theother hand, Saturn completed only half of its orbit for each or-bit of Jupiter, putting its indirect perturbation on the joviansin opposite phases for alternate jovian orbits. This way, theperturbations could not simply add up as those on the saturni-ans, making capture through the φ21 resonant-argument weak.However, the secular interactions between the two planets stillaffected Jupiter’s eccentricity on 103-years timescales, makinga capture through secular resonance still viable. The differencein the planets’ masses and the smaller “slow-precession” phase-space region at Jupiter (due to its faster mean motion) makeresonant capture even less likely, accounting for the meager sur-viving population in Fig. 5.

(a)

(b)

Fig. 5. Same as Fig. 1, but for Jupiter. Surviving test particles are plotted as solidsquares (if q > 0.03 AU) and small pluses (if 0.03 AU > q > 0.02 AU), whilethe real irregulars satellites are shown as asterisks (panels (a) and (b) show theira–i and a–e distributions, respectively). Themisto and Carpo (S/2003 J20) areindicated by “T” and “C,” respectively. There is very little correspondence be-tween the two sets, and only two isolated high-i prograde satellites appear to bepotential candidates for capture by the resonance crossing. Note that Thermis-tor’s pericenter is in fact below 0.03 AU.

In general, whenever two planets are migrating away fromeach other across their mutual 1:2 (or 1:3, 1:4, etc.) reso-nance, satellites of the outer planet will always be affected morestrongly. While the strength of perturbations on satellites doesdepend on the masses of the planets involved, only the moonsof the outer planet can feel coherent indirect perturbations ofthe inner body, and their Sun-induced precession will typicallybe slower, making them more vulnerable to long-period near-resonant perturbations. In particular, if Uranus or Neptune everencountered a mean-motion resonance with Saturn of type 1:n,their irregulars’ orbits could have been affected by strong reso-nant perturbations.

3. Application to Uranus

In the Tsiganis et al. (2005) model, the “ice giants” Uranusand Neptune had much more complex orbital histories than


Jupiter and Saturn. The model predicts that the migration ex-panded the ice giants’ orbits by many AU, and that they alsosuffered gravitational scattering off each other, and possiblySaturn. The importance of planet–planet scattering will be dis-cussed in the next section, while here we address the possibil-ity of a relatively slow resonance crossing between Saturn andUranus. We decided to model only the crossing of the 1:2 reso-nance of these planets, as it is likely to produce more dramaticresults than 2:5 or other resonances. We also do not considerNeptune here, as its irregular satellite system was likely re-arranged during Triton’s capture and orbital evolution (Holmanet al., 2004; Cuk and Gladman, 2005), making the present or-bital distribution differ significantly from the primordial one.This mechanism could also in principle work for Neptune, ei-ther through higher-order (1:3, 1:4) resonances with Saturn orcloser resonances with Uranus (e.g., 2:3, as seen in Tsiganis etal., 2005, Fig. 1).

We replicated our previous experiment by placing Saturnand Uranus just inside their mutual 1:2 resonance and evolv-ing Uranus outwards. Jupiter and Saturn were on their presentorbits, while Uranus was given a pre-crossing eccentricity of0.01. Once again, a grid of 9200 particles was used (identical tothe one around Saturn) and the results are presented in Fig. 6.Overall, the distribution of surviving particles is analogous tothat around Saturn. The prograde stabilized particles tend tohave 30◦ < i < 50◦, while the retrograde ones cover a greaterrange of inclinations. This is unremarkable, as this resonance-crossing physically differs little from that of Jupiter and Saturn(except for the lower mass of the inner perturber). Comparisonwith extant uranians shows some strong similarities betweenthe two populations, although less complete than in the case ofsaturnians. The wide “Sycorax cluster” (comprising large a andlarge e retrograde bodies) and lone prograde uranian Margaret(S/2000 U3), can be produced this way while the closer-in retro-grade moons like Caliban cannot. The efficiency of the processwas found to be lower (137 bodies with q > 0.02 AU, of which27 had q > 0.03 AU), probably due to Saturn being a less pow-erful perturber than Jupiter. The condition of q > 0.02 AU isprobably somewhat artificial for Uranus, the outermost regularof which is Oberon at only a = 4×10−3 AU. Our goal when de-vising this experiment was to see if the resonance crossing canhave an effect on uranians, using the same set-up as for Saturn.Only a study with closer-in initial conditions tailored specifi-cally to Uranus will be capable of assessing the efficiency ofthe process.

The lack of close-in retrograde satellites in our results is in-teresting, but may once again be a consequence of gas-drag notbeing present in our simulations. Caliban and the other close-inirregulars have relatively low eccentricities (e < 0.25) which dopoint to the action of gas drag. It is possible that a more com-plete model could produce the “Caliban” group from the “Syco-rax” group if gas drag is active during the resonance crossing.Note that the “Sycorax” group is likely not a true collisionalcluster, but a collection of bodies stabilized by the same reso-nance. This might explain why Grav et al. (2004) found theircolors to be heterogeneous, but we keep in mind that the visible

(a)

(b)

Fig. 6. Same as Fig. 1, but for Uranus (following the crossing of 1:2 reso-nance with Saturn). (a) Semimajor axis–inclination distribution of survivingtest particles (small pluses; those with q > 0.03 AU are marked as filledsquares) and real uranian irregulars (asterisks). (b) Same as (a), just in semima-jor axis–eccentricity space. The outer retrograde satellites and the lone progrademoon (Margaret) appear to be successfully reproduced by the simulation, un-like inner retrograde irregulars.

colors of Sycorax are not consistent through the literature (e.g.,Rettig et al., 2001).

There is a widespread perception that Uranus and Neptune,having much less gas than Jupiter and Saturn, are unable to cap-ture satellites with similar efficiency (cf. the review of Jewittand Sheppard, 2005). This is even seen as a proof that gas draghad no role in the capture of irregulars, as they appear to bepresent around all giant planets in equal numbers. However,this argument does not survive closer scrutiny. Let us assumea “toy model” with a uniform circumplanetary disk extend-ing out to a sharp edge, and interacting with a constant influxof planetesimals. At any time, one expects to have a steady-state population of captured planetesimals which are decayinginto the planet. How does the number of objects bigger thansay, 10 km, in this equilibrium population depend on the disk’sthickness? Obviously, if the disk is too thin to capture themat all, they should not be present. But for disk surface densi-ties high enough to allow capture of 10-km bodies, increasing


the disk mass will not only allow the capture of larger bodies,but also accelerate the decay of smaller ones. If the originalsize distribution was steeper than n(r) ∼ r−1 (as the timescalefor gas-drag evolution is proportional to radius), the increasein number of captured (large) bodies is more than offset by themore rapid decay of the smaller ones. Therefore, a planet witha denser uniform disk could actually have fewer temporary ir-regulars than the one with less gas. Of course, disks were likelynot uniform and capture might have been concentrated on itsouter edge, but the purpose of this toy model is to show that therelationship between the gas density and number of survivingirregular satellites can be counterintuitive.

4. Discussion

Fig. 1b showed that the resonance-crossing mechanismalone, while successful in reproducing most saturnian irreg-ulars, cannot result in an object with orbit similar to that ofPhoebe, which simply has an eccentricity too low to be pro-duced by this method. Since its pericenter is at 0.07 AU, Phoebecould not have interacted with the compact protosatellite diskenvisioned in our model after the end of resonant interactions(as friction in prograde gas cannot raise a retrograde body’spericenter). In Section 2 we argued that the incorporation ofgas drag into the simulation could help bring down the aver-age eccentricity of the surviving test particles, and that gas draglikely eliminated the close-in retrograde particles (small plusesin Fig. 1b) before they had time to interact with the resonance.Due to its large size and high pericenter, Phoebe may be im-mune to this decay, and could have survived much longer inthis environment.

Therefore, it is likely that Phoebe’s orbit underwent signifi-cant evolution before the resonance crossing, at an epoch whenthe circumplanetary disk was much more dense. One cannotsay much more without directly simulating Phoebe’s captureand evolution, which is beyond the scope of this work. Our pre-liminary simulations of the effects of the resonance crossingon Phoebe’s present orbit show changes in eccentricity smallerthan 0.1, but these results are not definite as we used only tenclones. While it is still possible that resonance crossing had sig-nificant effects on Phoebe (if already present), it is more likelythat Phoebe has already been captured and evolved by the timeof the resonance crossing. The latter scenario would make itsimilar to Himalia, which according to Cuk and Burns (2004a)could have been captured by gas drag alone. Note that one of thereasons Himalia was selected as the object of that study was itsdynamical reversibility. It was the only jovian satellite which,when evolved backward in time in the presence of gas drag, didnot encounter any secular resonances that could compromisethe whole method of time-reversed evolution. So it is no sur-prise that Himalia’s (and possibly Phoebe’s) origins likely havelittle or nothing to do with the resonance crossing discussedhere.

The failure of the resonance-crossing model to account forthe capture of a significant number of jovian irregulars is some-what surprising. When we first realized that presently feeblesecular resonances would become violent during the Jupiter–

Saturn 1:2 resonance crossing, we expected resonant jovianmoons like Pasiphae and Sinope to be logical candidates forthis effect. However, Fig. 5 shows that there are few, if any,retrograde jovians captured by this process. While Section 2explained why interactions with the resonant argument 2λS −λJ − � are unlikely for jovians, there is no a priori reason whysecular resonance itself (i.e. argument � − �J ) should not beable to stabilize some of the test particles.

One possible conclusion is that resonance crossing had noconnection to the capture of the retrograde jovians. In thatcase an alternative mechanism is needed to explain their ori-gin. Three-body interaction during a retrograde passage ofa large planetesimal by Jupiter is one promising mechanism(D. Nesvorný, 2005, personal communication). This mecha-nism can explain why the retrograde satellites cluster in a nar-row range of semimajor axes. Secular (Whipple and Shelus,1993) and mean-motion resonances (Saha and Tremaine, 1993)observed today would have been established during the epochof planetary migration, without any evolution by the satellitesthemselves (Nesvorný et al., 2003).

We mention a different hypothesis: retrograde jovians werelikely stabilized during the resonance crossing but, unlike sat-urnians, they were not recruited from a steady state swarmof highly-eccentric ephemeral gas-drag captures. Since all ret-rograde jovians have large semimajor axes (a > 0.13 AU),it is plausible that they might have been stabilized directlyfrom classic temporary captures, rather than the already-boundsteady-state bodies we postulated. For example, Nesvorný et al.(2003) show that Pasiphae and its family, while stable for atleast 108 years, are relatively close to the limit of orbital sta-bility. Capture of temporary satellites is much more difficult tomodel than stabilization of eccentric bodies (Section 2), as thetransient moons cannot be easily modeled by a stationary distri-bution. To test this hypothesis more realistic direct simulationsare required, probably similar to those Morbidelli et al. (2005)used to model the capture of Jupiter Trojans.

As mentioned above, scattering between planets and largeplanetesimals, or even planets themselves, is sometimes viewedas a potential mechanism for satellite capture. Such scatterings(following the Jupiter–Saturn resonance crossing) are actuallyrequired by the Tsiganis et al. (2005) model to reproduce thecurrent planetary orbits. Their model allows both for histo-ries including only close encounters between the “ice giants,”and those incorporating scatterings of these planets off Saturn.Clearly, as the Hill spheres of two planets overlap, the orbitsof irregular satellites should suffer significant “kicks.” The factthat we can produce saturnian and outer uranian irregulars dur-ing the resonance crossing seems to contradict the scenario ofrepeated planet–planet scatterings stripping planets of irregu-lar moons (Tsiganis et al., 2005). Aside from such catastrophicevents, encounters with large planetesimals could lead to some“random walk” in satellite elements, and spread out their dis-tributions in a, e and i, possibly helping broaden the range ofparticle eccentricities in Fig. 1b. However, only direct simula-tions can put quantitative limits on either of these processes, aswell as to explore the potential of gravitational scattering forsatellite capture.


Another mechanism that likely affected Saturn’s irregularsatellite orbits long after the 1:2 resonance passage is sweep-ing by the Great Inequality. This near-resonance is associatedwith the arguments of type 5λS − 2λJ , and is currently knownto have an influence on the dynamics in the saturnian irregularsatellite region. Cuk and Burns (2004b) show that the progradeparticles for which the condition 5nS − 2nJ − 2� − �S = 0is satisfied (implying P� = 1850 years; n stands for mean mo-tion) are unstable on short timescales. The long-dashed line inFig. 4 plots the location of this resonance in period–inclinationspace. The Great Inequality instability clearly cuts the popu-lation of prograde satellites in half. Naturally, no real satellitesare located on this resonance, but they cluster at inclinations be-low and above the resonant one. The natural conclusion is thatthe primordial population might have been continuous, but wasbroken into two inclination groups by “sweeping” of the GreatInequality resonance, which destabilized the moons on inter-mediate orbits. In this view, neither of the inclination groupsneeds to be a collisional family, which resolves the problemof their large velocity dispersion (Nesvorný et al., 2003). Notethat this resonance affects prograde bodies because the planetsare presently situated interior to the 2:5 resonance. In contrast,the 1:2 resonance affected retrograde objects because it oper-ated during the immediate post-crossing epoch. It is possiblethat the members of the low-i prograde Gallic cluster (Albiorix,Erriapo, Tarvos) felt the sweeping of the Great Inequality reso-nance as the planets were approaching their present configura-tion. However, since the inequality had to have a period of only350 years at that epoch (as opposed to about 900 years today),it is unlikely that its effects were particularly strong.

There is some tentative evidence that the Great Inequal-ity may also affect retrograde satellites. Thrymr (formerlyS/2000 S7; corresponds to the asterisk with retrograde periodof 3 years, on the left solid curve in Fig. 4) has an apsidal pre-cession period of about 880 years, which is very close to that ofthe Great Inequality. This seems to affect at least its mean mo-tion, which undergoes periodic perturbation, apparently causedby Jupiter, with the period of about 1850 years (R. Jacobson,2005, personal communication). As the exact resonant argu-ment relevant in this case is still unknown, we cannot say muchabout this resonance (or even if it really exists). We just notethat, if the Great Inequality could capture retrograde moons ofSaturn into an apsidal resonance, this would provide us with yetanother way of modifying their eccentricities.

Apart from the dynamical considerations, the resonance-crossing capture model is additionally constrained by the com-position of the irregular satellites of Saturn. If they were cap-tured during the Jupiter–Saturn 1:2 resonance crossing, wewould expect them to come from the same source as the otherpopulation that has been putatively traced back to that event:the Jupiter Trojans. Morbidelli et al. (2005) show how Trojanscan be captured from the large swarm of bodies scattering offJupiter and Saturn; these bodies in turn fuel the gas giants’ mi-gration. These are the same bodies that are being captured bygas drag and then stabilized by resonance crossing in our model.Therefore, similar compositions of Jupiter Trojans and Saturn’sirregulars would serve as a constraining check for our model,

while the lack of spectral affinity between these groups wouldbe a strong argument against it.

Jupiter Trojans are predominantly of spectral type D, witha minority of P-type bodies and a few more exotic types(Bendjoya et al., 2004). Spectral class D is widely seen as themost primitive asteroid type found within 10 AU of the Sun,which is consistent with Trojans having originating in the outerSolar System, as envisaged by Morbidelli et al. (2005). Sat-urnian irregulars other than Phoebe were classified by Grav etal. (2003) as “light red,” which corresponds to D and P classesin the more traditional terminology. Therefore a common ori-gin of Trojans and Saturn’s irregulars is not contradicted bytheir colors. Once again, Phoebe appears atypical, and exceptthat it comes from the outer Solar System, not much more canbe said about its source (Johnson and Lunine, 2005). Phoebe’sneutral spectrum is another reason to think that it was capturedat a different epoch, probably by a non-resonant mechanism. Itis intriguing that the isolated prograde jovian Themisto also hasa “light red” spectrum, allowing for the possibility that it wascaptured during the same event as the saturnians.

While the model presented in the present work appears tobe consistent with the resonance crossing scenario of Tsiganiset al. (2005) and Morbidelli et al. (2005), there is an appar-ent inconsistency with the timing proposed in the related pa-per of Gomes et al. (2005). In order to stabilize saturnian ir-regular satellites by sweeping secular resonances, we requirethe prior presence of a substantial steady-state population ofbodies captured by gas drag onto eccentric orbits and decay-ing towards the planet. The gas in our model is concentratedin the protosatellite disks with sizes comparable to the mod-ern regular satellite systems. These disks are thought to havelasted for at least 106 years (Canup and Ward, 2002), whichis about the minimum timescale on which planets can migrateinto the 1:2 resonances (Tsiganis et al., 2005, supplementarymaterial). However, Gomes et al. (2005) propose that the morecompact pre-resonance crossing Solar System lasted for about7 × 108 years, after which the passage finally occurred and,among other things, caused the “Late Heavy Bombardment”(LHB) of the Moon. It is highly unlikely that protosatellite diskswere still present at this time, making it hard for the planetsto permanently capture any planetesimals using aerodynamicdrag.

We personally favor the scenario that the Jupiter–Saturn 1:2resonance crossing happened only a few million years after theformation of the planets themselves, and that the cause of LHBmust be sought elsewhere. However, scenarios consistent withGomes et al. (2005) timing can be imagined. The only departurefrom our favored model would be the mechanism of the initialplanetesimal capture. This modified scenario would still predictphysical similarity between Trojans and saturnian irregulars, asboth groups are captured from the same small-body popula-tion dominant during the great resonance crossing, regardlessof its absolute timing. If a Myr-lifetime gas disk had initiallycaptured a population of objects with pericenters in the regularsatellite system, we believe it implausible that this populationcould survive the long interval between gas dissipation and theLHB. If instead the pre-resonance-crossing “irregulars” were


bodies captured by either binary-exchange interactions (Agnorand Hamilton, 2004) or scattering of planetesimals off majorregular satellites (K. Tsiganis, 2006, personal communication),one could imagine the production of a low-q population justbefore the LHB. However, it is still unclear if the purely grav-itational mechanisms can produce a large enough populationwith orbital elements suitable for resonant pericenter lifting.Needless to say, testing of this alternative hypothesis requires aseparate dedicated study which would need to explicitly includeboth regular satellites and possible binary nature of planetesi-mals, which were ignored here.

5. Conclusions

We find that the orbital distribution of irregular satellites ofSaturn supports the idea that Jupiter and Saturn crossed their1:2 mean-motion resonance in the past (Tsiganis et al., 2005).Using numerical simulation, we show that a steady-state cap-tured population of saturnian satellites on very eccentric orbitscan be saved from collapse due to gas drag by resonant interac-tion associated with the Jupiter–Saturn 1:2 resonance crossing.These resonant interactions can decrease the eccentricity andlift the pericenter of an irregular’s orbit above the edge of aplanet’s protosatellite gas disk. In particular, prograde orbitsare found to be affected by the terms with argument � − �S ,while the retrograde ones feel perturbations associated with ar-gument 2λS − λJ − � . The orbital distribution created by ourmodel is very similar to that of the real irregular satellites ofSaturn. It may also naturally explain the inclination groups ascollections of moons captured via the same resonant argument.In some cases, the original groups were further partitioned bysubsequent destabilizing resonances.

For Uranus, the same model shows that its outer group of ir-regular satellites could have been stabilized during a putativeresonance crossing with Saturn. However, our model was un-able to account for the great majority of jovian irregulars, as theeffects of the resonant crossing are not symmetric for Jupiterand Saturn. While the Himalia group could have been capturedby gas drag alone (Cuk and Burns, 2004a), we still expect somevariant of the present mechanism may be able to capture theretrograde jovians, given their close association with secularresonances.

The colors of saturnian irregular satellites (Grav et al., 2003)and jovian Trojans (Bendjoya et al., 2004) are consistent witha common origin during the Jupiter–Saturn resonance crossing(Morbidelli et al., 2005). However, our model postulates thatthe protosatellite disks still be present at the time of the reso-nance crossing, in contradiction with the timing proposed byGomes et al. (2005). Our model assumes that the great reso-nance crossing happened in the first few million years of theSolar System’s history, and is completely unrelated to the “LateHeavy Bombardment.” Nevertheless, alternative scenarios con-sistent with the results of Gomes et al. (2005) that do not requiregas drag are also possible and need to be explored. In any case,if the capture of irregular satellites can be dated by tying it tospecific events during planetary migration, their present orbits

hold great potential for constraining the early history of the So-lar System.

Acknowledgments

The numerical experiments described here were performedon the Leverrier computing cluster at Department of Physicsand Astronomy, University of British Columbia. We gratefullyacknowledge hardware and salary support from the CanadianCFI and NSERC agencies. We also thank Menios Tsiganis andan anonymous referee for their insightful reviews.

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