A new perspective on the irregular satellites of Saturn – I. Dynamical and collisional history (original) (raw)

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1Centre of Studies and Activities for Space ‘G. Colombo’, University of Padova, Via Venezia 15, 35131 Padova, Italy

2Physics Department, University of Padova, Via Marzolo 8, 35131 Padova, Italy

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2Physics Department, University of Padova, Via Marzolo 8, 35131 Padova, Italy

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3Laboratoire d'Astrophysique de Grenoble, UMR 5571 CNRS, Université J. Fourier, BP 53, 38041 Grenoble Cedex 9, France

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Published:

27 November 2008

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D. Turrini, F. Marzari, H. Beust, A new perspective on the irregular satellites of Saturn – I. Dynamical and collisional history, Monthly Notices of the Royal Astronomical Society, Volume 391, Issue 3, December 2008, Pages 1029–1051, https://doi.org/10.1111/j.1365-2966.2008.13909.x
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Abstract

The dynamical features of the irregular satellites of the giant planets argue against an in situ formation and are strongly suggestive of a capture origin. Since the last detailed investigations of their dynamics, the total number of satellites has doubled, increasing from 50 to 109, and almost tripled in the case of Saturn system. We have performed a new dynamical exploration of Saturn system to test whether the larger sample of bodies could improve our understanding of which dynamical features are primordial and which are the outcome of the secular evolution of the system. We have performed detailed _N_-body simulations using the best orbital data available and analysed the frequencies of motion to search for resonances and other possible perturbing effects. We took advantage of the hierarchical Jacobian symplectic algorithm to include in the dynamical model of the system also the gravitational effects of the two outermost massive satellites, Titan and Iapetus. Our results suggest that Saturn's irregular satellites have been significantly altered and shaped by the gravitational perturbations of Jupiter, Titan, Iapetus and the Sun and by the collisional sweeping effect of Phoebe. In particular, the effects on the dynamical evolution of the system of the two massive satellites appear to be non-negligible. Jupiter perturbs the satellites through its direct gravitational pull and, indirectly, via the effects of the Great Inequality, i.e. its near-resonance with Saturn. Finally, by using the hierarchical clustering method we found hints to the existence of collisional families and compared them with the available observational data.

1 INTRODUCTION

The outer Solar system is inhabited by different minor body populations: comets, Centaurs, trans-Neptunian objects (TNOs), Trojans and irregular satellites of the giant planets. Apart from comets, whose existence had long been known, the other populations have been discovered in the last century and extensively studied since then. Our comprehension of their dynamical and physical histories has greatly improved (see Jewitt 2008; Morbidelli 2008 for a general review on the subject) and recent models of the formation and evolution of the Solar system seem to succeed in explaining their evolution and overall structure (see Gomes et al. 2005; Morbidelli et al. 2005; Tsiganis et al. 2005 for details). There is still no clear consensus, however, on the origin of the irregular satellites. It is widely accepted that their orbital features are not compatible with an in situ formation, leading to the conclusion they must be captured bodies. At present, however, no single capture model is universally accepted (see Sheppard 2006 and Jewitt & Haghighipour 2007 for a general review). It is historically believed that the satellite capture occurred prior to the dissipation of the solar nebula (Pollack, Burns & Tauber 1979), since the gaseous drag is essential for the energy-loss process that leads to the capture of a body in a satellite orbit while crossing the sphere of influence of a giant planet.

A direct consequence and implicit assumption of this model is that no major removal of irregular satellites took place after the dissipation of the nebular gas for the actual satellite systems to be representative of the gas-captured ones. If instead this was the case, we are left with two possibilities.

The same issues apply also to the original formulation of the so-called Pull-Down scenario (Heppenheimer & Porco 1977), also based on the presence of the nebular gas but locating the capture events during the phase of rapid gas accretion and mass growth of the giant planets.

Recently, the plausibility of gas-based scenarios has been put on jeopardy. The comparative study performed by Jewitt & Sheppard (2005) pointed out that the giant planets possess similar abundances of irregular satellites, once their apparent magnitudes are scaled and corrected to match the same geocentric distance. Due to the different formation histories of gas and ice giants, the gas-based scenarios cannot supply a convincing explanation to this fact.

The Nice model (Gomes et al. 2005; Morbidelli et al. 2005; Tsiganis et al. 2005) instead undermines the physical relevance of the gas-based scenarios. Formulated to explain the present orbital structure of the outer Solar system, it postulates that the giant planets formed (or migrated due to the interaction with the nebular gas) in a more compact configuration than the current one. Successively, due to their mutual gravitational interactions, the giant planets evolved through a phase of dynamical rearrangement in which the ice giants and Saturn migrated outward while Jupiter moved slightly inward. The dynamical evolution of the giant planets was stabilized, during this phase, by the interaction with a disc of residual planetesimals. During the migration process, a fraction of the planetesimals can be captured as Trojans (Bottke et al. 2008) while the surviving outer planetesimal disc would slowly settle down as the present Kuiper Belt. The perturbations of the migrating planets on the planetesimals would have also caused the onset of the Late Heavy Bombardment (LHB) on the inner planets. The detailed description of this model and its implications are given in Gomes et al. (2005), Morbidelli et al. (2005), Tsiganis et al. (2005) and Morbidelli et al. (2007). A major issue with the original formulation of the Nice model was the ad hoc choice of the orbits of the giant planets after the dispersal of the nebular gas. Work is going on to solve the issue (Morbidelli et al. 2007) by assuming that migration of planets by interaction with the solar nebula could lead to a planetary resonant configuration subsequently destroyed by the interaction with the planetesimal disc.

The Nice model has two major consequences concerning the origin of the irregular satellite. Due to the large distance from their parent planets and their consequently looser gravitational bounds, no irregular satellites previously captured could have survived the phase of violent rearrangement taking place in the Solar system (Tsiganis et al. 2005). In addition, during the rearrangement phase planetesimals were crossing the region of the giant planets with an intensity orders of magnitude superior to the one we observe in the present Solar system (Tsiganis et al. 2005), favouring capture by different dynamical processes.

In other words, the Nice model does not rule out gas-based mechanisms for irregular satellites capture but it implies that they could not have survived till now due to the violent dynamical evolution of the planets. The only exception to the former statement could be represented by the Jovian system of irregular satellites, since Jupiter had a somehow quieter dynamical evolution and its satellites were more strongly tied to the planet due to its intense gravity. For the other giant planets, however, the chaotic orbital evolution created a favourable environment for the capture of new irregular satellites by three body processes (i.e. gravitational interactions during close encounters and collisions). These trapping processes do not significantly depend on the choice of the initial orbital configuration of the planets in the Nice model: they rely mainly on the presence of a residual disc of planetesimals which both stabilizes the dynamical evolution of the giant planets and supplies the bodies that can become irregular satellites.

In this paper we concentrate on the study of the dynamical features of Saturn's irregular satellites, stimulated by the unprecedented data gathered by the Cassini mission on Phoebe. The overall structure of the satellite system can in fact provide significant clues on its origin and capture mechanism. This dynamical exploration will be the first step to test the feasibility and implications of the collisional capture scenario which was originally proposed by Colombo & Franklin (1971), which will be the subject of a forthcoming paper. The work we will present is organized as follows.

2 COMPUTATION OF THE MEAN ORBITAL ELEMENTS

Our study of the dynamics of the irregular satellites starts from the computations of approximate proper orbital elements. Proper elements may be derived analytically from the non-linear theory developed by Milani & Knezevic (1994) or they can be approximated by the mean orbital elements, which can be numerically computed by averaging the osculating orbital elements over a reasonably long time interval which depends on the evolution of the system.

To compute the mean orbital elements of Saturn's irregular satellites we adopted the numerical approach and integrated over 108 yr the evolution of an _N_-body dynamical system composed of the Sun, the four giant planets and the satellites of Saturn in orbit around the planet. For the satellites, we adopted two different dynamical set-ups:

Model 1 basically reproduces the orbital scheme used by other authors in previous works (Carruba et al. 2002; Nesvorny et al. 2003; Cuk & Burns 2004). Model 2 includes the perturbing effects of the two outermost regular satellites of the Saturn system. We considered this alternative model to evaluate their contribution to the global stability and to the secular evolution of the irregular satellites.

In order to reproduce accurately the orbits of the satellites, the integration time-step has to be less than the shortest period associated with the frequencies of motion of the system: the common prescription for computational celestial mechanics is to use integration time-steps about 1/20 of the fastest orbital period, usually that of the innermost body. The time-steps we used in our simulations with Models 1 and 2 were, respectively,

The time-step used for Model 2 greatly limited the length of the simulations by imposing a heavier computational load. A time interval of 108 yr was a reasonable compromise between the need of accuracy in the computation of proper elements and CPU (i.e. computational time) requirements: this time interval has been used also by Nesvorny et al. (2003). We used it for both models in order to be able to compare their results.

The initial osculating orbital elements of the irregular satellites and of the major bodies have been derived, respectively, from two different ephemeris services:

The reference plane in all the simulations was the J2000 ecliptic and the initial elements of all bodies referred to the epoch 2005 January 30.

The numerical simulations have been performed with the public implementation of the HJS (hierarchical Jacobi symplectic) algorithm described in Beust (2003) and based on the swift code (Levison & Duncan 1994). The symplectic mapping scheme of HJS is applicable to any hierarchical system without any a priori restriction on the orbital structure. The public implementation does not include the effects of planetary oblateness or tidal forces which are not considered in our models.3 The HJS algorithm proved itself a valuable tool to study the dynamical evolution of satellite systems since its symplectic scheme supports multiple orbital centres. It easily allows us to include in the simulations both Titan and Iapetus which were not taken into account in the previous works on irregular satellites.

To test the reliability of HJS, we performed an additional run based on Model 2 using the RADAU algorithm (Everhart 1985) incorporated in the mercury 6.2 package by J. E. Chambers (Chambers 1999). Due to the high computational load of RADAU algorithm compared to symplectic mapping, we limited this simulation to the 26 irregular satellites known at the end of 2005 for Saturn, which means we excluded S/2004 S19 and the S/2006 satellites. Even with this set-up, RADAU's run required about 6 months of computational time: as a comparison, HJS's run based on full Model 2 took approximately a month.

The averaged (proper) orbital elements computed in Models 1 and 2 are given in Tables 1 and 2, together with the minimum and maximum values reached during the simulations. The same elements are visually displayed in Figs 1 and 2. At first sight, the mean elements obtained with Models 1 and 2 appear to be approximately the same. However, there are some relevant differences between the two sets: this is the case of Tarvos and S/2004 S18 and of Kiviuq and Ijiraq, whose radial ordering results reversed. The perturbations by Titan and Iapetus significantly affect the secular evolution of the satellite system. We will show in Section 3 that the changes are more profound that those shortly described here.

Table 1

Mean orbital elements for the 35 irregular satellites of Saturn computed with Model 1. For each orbital element, we report the minimum and maximum values attained during the simulations.

Satellite Mean a (106 km) Max a (106 km) Min a (106 km) Mean e Max e Min e Mean i (°) Max i (°) Min i (°)
Ijiraq 11.349 99 11.414 32 11.294 64 0.278 44 0.581 54 0.068 07 48.539 258 54.569 60 38.765 20
Kiviuq 11.363 45 11.429 28 11.309 60 0.226 97 0.597 59 0.000 07 48.926 008 55.127 70 38.226 50
Phoebe 12.932 74 13.015 01 12.865 42 0.163 20 0.189 57 0.138 17 175.049 98 177.9515 172.4591
Paaliaq 15.000 18 15.214 10 14.795 23 0.344 80 0.657 04 0.106 85 49.265 570 56.933 10 38.265 80
Skathi 15.571 64 15.767 62 15.393 62 0.274 97 0.386 04 0.175 90 151.940 91 157.0688 146.6004
Albiorix 16.328 61 16.695 12 16.036 89 0.476 53 0.646 54 0.322 14 37.163 084 45.953 80 28.120 50
S/2004 S11 16.965 89 17.383 27 16.635 28 0.477 77 0.657 18 0.315 52 38.055 491 47.025 70 28.697 70
S/2006 S8 17.532 87 17.906 87 17.248 63 0.491 02 0.609 71 0.376 23 158.947 55 165.8655 151.1970
Erriapo 17.538 85 18.026 54 17.143 92 0.470 78 0.647 29 0.310 49 37.244 669 46.096 40 28.226 80
Siarnaq 17.824 59 18.355 66 17.413 19 0.304 00 0.626 49 0.064 84 47.848 986 55.716 40 38.583 00
S/2004 S13 18.050 48 18.430 46 17.757 27 0.256 49 0.335 98 0.185 53 168.737 17 172.2910 164.9780
S/2006 S4 18.051 98 18.460 38 17.742 31 0.321 08 0.398 96 0.246 16 174.233 06 177.5732 170.9701
Tarvos 18.183 62 18.804 45 17.652 55 0.529 54 0.720 46 0.354 04 37.828 226 48.259 00 27.217 30
S/2004 S19 18.402 03 18.774 53 18.101 34 0.325 16 0.481 43 0.188 85 149.892 85 156.3719 143.0364
Mundilfari 18.554 62 18.954 05 18.221 02 0.218 32 0.300 69 0.149 59 167.024 86 170.6053 163.2256
S/2006 S6 18.633 91 19.013 89 18.325 74 0.217 71 0.296 09 0.145 97 162.737 32 166.6379 158.6233
S/2006 S1 18.828 39 19.163 49 18.535 18 0.124 79 0.196 73 0.064 75 156.273 35 160.0706 152.3567
Narvi 19.314 58 19.836 68 18.909 17 0.408 69 0.677 53 0.189 73 142.378 69 153.1620 132.3253
S/2004 S17 19.389 38 19.806 76 19.043 81 0.178 73 0.247 85 0.117 84 167.711 54 171.1124 164.2361
Suttungr 19.399 85 19.776 84 19.073 73 0.115 14 0.165 60 0.070 43 175.874 22 178.8399 173.3134
S/2004 S15 19.613 78 20.016 20 19.253 25 0.147 49 0.225 41 0.081 85 158.778 60 162.5818 154.7763
S/2004 S10 19.684 09 20.255 55 19.253 25 0.273 80 0.367 81 0.182 18 165.908 11 169.9454 161.5303
S/2004 S12 19.706 53 20.285 47 19.268 21 0.338 11 0.457 31 0.234 33 164.644 73 169.1534 159.5224
S/2004 S09 20.201 70 20.704 35 19.791 80 0.243 66 0.360 87 0.140 68 156.179 12 160.9251 150.9430
Thrymr 20.390 19 21.078 34 19.821 72 0.457 90 0.571 33 0.344 82 175.173 39 179.0080 171.3722
S/2004 S14 20.511 36 21.138 18 20.061 07 0.357 80 0.468 94 0.251 26 165.046 88 169.6972 159.7623
S/2004 S18 20.580 18 21.931 05 19.731 96 0.522 64 0.804 86 0.270 15 138.444 52 153.5911 125.2909
S/2004 S07 20.936 22 21.706 65 20.375 23 0.528 95 0.660 62 0.392 25 163.970 39 170.4156 156.1659
S/2006 S3 20.976 61 21.721 61 20.435 07 0.461 06 0.618 35 0.313 09 156.610 94 164.0096 148.0057
S/2006 S2 21.860 74 22.828 64 21.153 14 0.511 46 0.707 84 0.318 93 153.002 77 162.3733 142.1259
S/2006 S5 22.638 65 23.606 54 21.901 13 0.276 10 0.461 16 0.118 88 167.661 32 171.7822 162.8017
S/2004 S16 22.898 95 23.651 42 22.260 16 0.156 18 0.248 96 0.076 82 164.715 01 168.4223 160.7656
S/2006 S7 22.906 43 23.935 66 22.185 36 0.446 73 0.589 11 0.302 90 168.383 75 173.6354 161.7933
Ymir 23.029 10 24.025 42 22.305 04 0.337 42 0.457 24 0.221 75 173.058 57 176.9269 169.0612
S/2004 S08 24.210 92 25.237 16 23.397 11 0.218 86 0.328 16 0.117 58 170.169 30 173.7150 166.4152
Satellite Mean a (106 km) Max a (106 km) Min a (106 km) Mean e Max e Min e Mean i (°) Max i (°) Min i (°)
Ijiraq 11.349 99 11.414 32 11.294 64 0.278 44 0.581 54 0.068 07 48.539 258 54.569 60 38.765 20
Kiviuq 11.363 45 11.429 28 11.309 60 0.226 97 0.597 59 0.000 07 48.926 008 55.127 70 38.226 50
Phoebe 12.932 74 13.015 01 12.865 42 0.163 20 0.189 57 0.138 17 175.049 98 177.9515 172.4591
Paaliaq 15.000 18 15.214 10 14.795 23 0.344 80 0.657 04 0.106 85 49.265 570 56.933 10 38.265 80
Skathi 15.571 64 15.767 62 15.393 62 0.274 97 0.386 04 0.175 90 151.940 91 157.0688 146.6004
Albiorix 16.328 61 16.695 12 16.036 89 0.476 53 0.646 54 0.322 14 37.163 084 45.953 80 28.120 50
S/2004 S11 16.965 89 17.383 27 16.635 28 0.477 77 0.657 18 0.315 52 38.055 491 47.025 70 28.697 70
S/2006 S8 17.532 87 17.906 87 17.248 63 0.491 02 0.609 71 0.376 23 158.947 55 165.8655 151.1970
Erriapo 17.538 85 18.026 54 17.143 92 0.470 78 0.647 29 0.310 49 37.244 669 46.096 40 28.226 80
Siarnaq 17.824 59 18.355 66 17.413 19 0.304 00 0.626 49 0.064 84 47.848 986 55.716 40 38.583 00
S/2004 S13 18.050 48 18.430 46 17.757 27 0.256 49 0.335 98 0.185 53 168.737 17 172.2910 164.9780
S/2006 S4 18.051 98 18.460 38 17.742 31 0.321 08 0.398 96 0.246 16 174.233 06 177.5732 170.9701
Tarvos 18.183 62 18.804 45 17.652 55 0.529 54 0.720 46 0.354 04 37.828 226 48.259 00 27.217 30
S/2004 S19 18.402 03 18.774 53 18.101 34 0.325 16 0.481 43 0.188 85 149.892 85 156.3719 143.0364
Mundilfari 18.554 62 18.954 05 18.221 02 0.218 32 0.300 69 0.149 59 167.024 86 170.6053 163.2256
S/2006 S6 18.633 91 19.013 89 18.325 74 0.217 71 0.296 09 0.145 97 162.737 32 166.6379 158.6233
S/2006 S1 18.828 39 19.163 49 18.535 18 0.124 79 0.196 73 0.064 75 156.273 35 160.0706 152.3567
Narvi 19.314 58 19.836 68 18.909 17 0.408 69 0.677 53 0.189 73 142.378 69 153.1620 132.3253
S/2004 S17 19.389 38 19.806 76 19.043 81 0.178 73 0.247 85 0.117 84 167.711 54 171.1124 164.2361
Suttungr 19.399 85 19.776 84 19.073 73 0.115 14 0.165 60 0.070 43 175.874 22 178.8399 173.3134
S/2004 S15 19.613 78 20.016 20 19.253 25 0.147 49 0.225 41 0.081 85 158.778 60 162.5818 154.7763
S/2004 S10 19.684 09 20.255 55 19.253 25 0.273 80 0.367 81 0.182 18 165.908 11 169.9454 161.5303
S/2004 S12 19.706 53 20.285 47 19.268 21 0.338 11 0.457 31 0.234 33 164.644 73 169.1534 159.5224
S/2004 S09 20.201 70 20.704 35 19.791 80 0.243 66 0.360 87 0.140 68 156.179 12 160.9251 150.9430
Thrymr 20.390 19 21.078 34 19.821 72 0.457 90 0.571 33 0.344 82 175.173 39 179.0080 171.3722
S/2004 S14 20.511 36 21.138 18 20.061 07 0.357 80 0.468 94 0.251 26 165.046 88 169.6972 159.7623
S/2004 S18 20.580 18 21.931 05 19.731 96 0.522 64 0.804 86 0.270 15 138.444 52 153.5911 125.2909
S/2004 S07 20.936 22 21.706 65 20.375 23 0.528 95 0.660 62 0.392 25 163.970 39 170.4156 156.1659
S/2006 S3 20.976 61 21.721 61 20.435 07 0.461 06 0.618 35 0.313 09 156.610 94 164.0096 148.0057
S/2006 S2 21.860 74 22.828 64 21.153 14 0.511 46 0.707 84 0.318 93 153.002 77 162.3733 142.1259
S/2006 S5 22.638 65 23.606 54 21.901 13 0.276 10 0.461 16 0.118 88 167.661 32 171.7822 162.8017
S/2004 S16 22.898 95 23.651 42 22.260 16 0.156 18 0.248 96 0.076 82 164.715 01 168.4223 160.7656
S/2006 S7 22.906 43 23.935 66 22.185 36 0.446 73 0.589 11 0.302 90 168.383 75 173.6354 161.7933
Ymir 23.029 10 24.025 42 22.305 04 0.337 42 0.457 24 0.221 75 173.058 57 176.9269 169.0612
S/2004 S08 24.210 92 25.237 16 23.397 11 0.218 86 0.328 16 0.117 58 170.169 30 173.7150 166.4152

Table 1

Mean orbital elements for the 35 irregular satellites of Saturn computed with Model 1. For each orbital element, we report the minimum and maximum values attained during the simulations.

Satellite Mean a (106 km) Max a (106 km) Min a (106 km) Mean e Max e Min e Mean i (°) Max i (°) Min i (°)
Ijiraq 11.349 99 11.414 32 11.294 64 0.278 44 0.581 54 0.068 07 48.539 258 54.569 60 38.765 20
Kiviuq 11.363 45 11.429 28 11.309 60 0.226 97 0.597 59 0.000 07 48.926 008 55.127 70 38.226 50
Phoebe 12.932 74 13.015 01 12.865 42 0.163 20 0.189 57 0.138 17 175.049 98 177.9515 172.4591
Paaliaq 15.000 18 15.214 10 14.795 23 0.344 80 0.657 04 0.106 85 49.265 570 56.933 10 38.265 80
Skathi 15.571 64 15.767 62 15.393 62 0.274 97 0.386 04 0.175 90 151.940 91 157.0688 146.6004
Albiorix 16.328 61 16.695 12 16.036 89 0.476 53 0.646 54 0.322 14 37.163 084 45.953 80 28.120 50
S/2004 S11 16.965 89 17.383 27 16.635 28 0.477 77 0.657 18 0.315 52 38.055 491 47.025 70 28.697 70
S/2006 S8 17.532 87 17.906 87 17.248 63 0.491 02 0.609 71 0.376 23 158.947 55 165.8655 151.1970
Erriapo 17.538 85 18.026 54 17.143 92 0.470 78 0.647 29 0.310 49 37.244 669 46.096 40 28.226 80
Siarnaq 17.824 59 18.355 66 17.413 19 0.304 00 0.626 49 0.064 84 47.848 986 55.716 40 38.583 00
S/2004 S13 18.050 48 18.430 46 17.757 27 0.256 49 0.335 98 0.185 53 168.737 17 172.2910 164.9780
S/2006 S4 18.051 98 18.460 38 17.742 31 0.321 08 0.398 96 0.246 16 174.233 06 177.5732 170.9701
Tarvos 18.183 62 18.804 45 17.652 55 0.529 54 0.720 46 0.354 04 37.828 226 48.259 00 27.217 30
S/2004 S19 18.402 03 18.774 53 18.101 34 0.325 16 0.481 43 0.188 85 149.892 85 156.3719 143.0364
Mundilfari 18.554 62 18.954 05 18.221 02 0.218 32 0.300 69 0.149 59 167.024 86 170.6053 163.2256
S/2006 S6 18.633 91 19.013 89 18.325 74 0.217 71 0.296 09 0.145 97 162.737 32 166.6379 158.6233
S/2006 S1 18.828 39 19.163 49 18.535 18 0.124 79 0.196 73 0.064 75 156.273 35 160.0706 152.3567
Narvi 19.314 58 19.836 68 18.909 17 0.408 69 0.677 53 0.189 73 142.378 69 153.1620 132.3253
S/2004 S17 19.389 38 19.806 76 19.043 81 0.178 73 0.247 85 0.117 84 167.711 54 171.1124 164.2361
Suttungr 19.399 85 19.776 84 19.073 73 0.115 14 0.165 60 0.070 43 175.874 22 178.8399 173.3134
S/2004 S15 19.613 78 20.016 20 19.253 25 0.147 49 0.225 41 0.081 85 158.778 60 162.5818 154.7763
S/2004 S10 19.684 09 20.255 55 19.253 25 0.273 80 0.367 81 0.182 18 165.908 11 169.9454 161.5303
S/2004 S12 19.706 53 20.285 47 19.268 21 0.338 11 0.457 31 0.234 33 164.644 73 169.1534 159.5224
S/2004 S09 20.201 70 20.704 35 19.791 80 0.243 66 0.360 87 0.140 68 156.179 12 160.9251 150.9430
Thrymr 20.390 19 21.078 34 19.821 72 0.457 90 0.571 33 0.344 82 175.173 39 179.0080 171.3722
S/2004 S14 20.511 36 21.138 18 20.061 07 0.357 80 0.468 94 0.251 26 165.046 88 169.6972 159.7623
S/2004 S18 20.580 18 21.931 05 19.731 96 0.522 64 0.804 86 0.270 15 138.444 52 153.5911 125.2909
S/2004 S07 20.936 22 21.706 65 20.375 23 0.528 95 0.660 62 0.392 25 163.970 39 170.4156 156.1659
S/2006 S3 20.976 61 21.721 61 20.435 07 0.461 06 0.618 35 0.313 09 156.610 94 164.0096 148.0057
S/2006 S2 21.860 74 22.828 64 21.153 14 0.511 46 0.707 84 0.318 93 153.002 77 162.3733 142.1259
S/2006 S5 22.638 65 23.606 54 21.901 13 0.276 10 0.461 16 0.118 88 167.661 32 171.7822 162.8017
S/2004 S16 22.898 95 23.651 42 22.260 16 0.156 18 0.248 96 0.076 82 164.715 01 168.4223 160.7656
S/2006 S7 22.906 43 23.935 66 22.185 36 0.446 73 0.589 11 0.302 90 168.383 75 173.6354 161.7933
Ymir 23.029 10 24.025 42 22.305 04 0.337 42 0.457 24 0.221 75 173.058 57 176.9269 169.0612
S/2004 S08 24.210 92 25.237 16 23.397 11 0.218 86 0.328 16 0.117 58 170.169 30 173.7150 166.4152
Satellite Mean a (106 km) Max a (106 km) Min a (106 km) Mean e Max e Min e Mean i (°) Max i (°) Min i (°)
Ijiraq 11.349 99 11.414 32 11.294 64 0.278 44 0.581 54 0.068 07 48.539 258 54.569 60 38.765 20
Kiviuq 11.363 45 11.429 28 11.309 60 0.226 97 0.597 59 0.000 07 48.926 008 55.127 70 38.226 50
Phoebe 12.932 74 13.015 01 12.865 42 0.163 20 0.189 57 0.138 17 175.049 98 177.9515 172.4591
Paaliaq 15.000 18 15.214 10 14.795 23 0.344 80 0.657 04 0.106 85 49.265 570 56.933 10 38.265 80
Skathi 15.571 64 15.767 62 15.393 62 0.274 97 0.386 04 0.175 90 151.940 91 157.0688 146.6004
Albiorix 16.328 61 16.695 12 16.036 89 0.476 53 0.646 54 0.322 14 37.163 084 45.953 80 28.120 50
S/2004 S11 16.965 89 17.383 27 16.635 28 0.477 77 0.657 18 0.315 52 38.055 491 47.025 70 28.697 70
S/2006 S8 17.532 87 17.906 87 17.248 63 0.491 02 0.609 71 0.376 23 158.947 55 165.8655 151.1970
Erriapo 17.538 85 18.026 54 17.143 92 0.470 78 0.647 29 0.310 49 37.244 669 46.096 40 28.226 80
Siarnaq 17.824 59 18.355 66 17.413 19 0.304 00 0.626 49 0.064 84 47.848 986 55.716 40 38.583 00
S/2004 S13 18.050 48 18.430 46 17.757 27 0.256 49 0.335 98 0.185 53 168.737 17 172.2910 164.9780
S/2006 S4 18.051 98 18.460 38 17.742 31 0.321 08 0.398 96 0.246 16 174.233 06 177.5732 170.9701
Tarvos 18.183 62 18.804 45 17.652 55 0.529 54 0.720 46 0.354 04 37.828 226 48.259 00 27.217 30
S/2004 S19 18.402 03 18.774 53 18.101 34 0.325 16 0.481 43 0.188 85 149.892 85 156.3719 143.0364
Mundilfari 18.554 62 18.954 05 18.221 02 0.218 32 0.300 69 0.149 59 167.024 86 170.6053 163.2256
S/2006 S6 18.633 91 19.013 89 18.325 74 0.217 71 0.296 09 0.145 97 162.737 32 166.6379 158.6233
S/2006 S1 18.828 39 19.163 49 18.535 18 0.124 79 0.196 73 0.064 75 156.273 35 160.0706 152.3567
Narvi 19.314 58 19.836 68 18.909 17 0.408 69 0.677 53 0.189 73 142.378 69 153.1620 132.3253
S/2004 S17 19.389 38 19.806 76 19.043 81 0.178 73 0.247 85 0.117 84 167.711 54 171.1124 164.2361
Suttungr 19.399 85 19.776 84 19.073 73 0.115 14 0.165 60 0.070 43 175.874 22 178.8399 173.3134
S/2004 S15 19.613 78 20.016 20 19.253 25 0.147 49 0.225 41 0.081 85 158.778 60 162.5818 154.7763
S/2004 S10 19.684 09 20.255 55 19.253 25 0.273 80 0.367 81 0.182 18 165.908 11 169.9454 161.5303
S/2004 S12 19.706 53 20.285 47 19.268 21 0.338 11 0.457 31 0.234 33 164.644 73 169.1534 159.5224
S/2004 S09 20.201 70 20.704 35 19.791 80 0.243 66 0.360 87 0.140 68 156.179 12 160.9251 150.9430
Thrymr 20.390 19 21.078 34 19.821 72 0.457 90 0.571 33 0.344 82 175.173 39 179.0080 171.3722
S/2004 S14 20.511 36 21.138 18 20.061 07 0.357 80 0.468 94 0.251 26 165.046 88 169.6972 159.7623
S/2004 S18 20.580 18 21.931 05 19.731 96 0.522 64 0.804 86 0.270 15 138.444 52 153.5911 125.2909
S/2004 S07 20.936 22 21.706 65 20.375 23 0.528 95 0.660 62 0.392 25 163.970 39 170.4156 156.1659
S/2006 S3 20.976 61 21.721 61 20.435 07 0.461 06 0.618 35 0.313 09 156.610 94 164.0096 148.0057
S/2006 S2 21.860 74 22.828 64 21.153 14 0.511 46 0.707 84 0.318 93 153.002 77 162.3733 142.1259
S/2006 S5 22.638 65 23.606 54 21.901 13 0.276 10 0.461 16 0.118 88 167.661 32 171.7822 162.8017
S/2004 S16 22.898 95 23.651 42 22.260 16 0.156 18 0.248 96 0.076 82 164.715 01 168.4223 160.7656
S/2006 S7 22.906 43 23.935 66 22.185 36 0.446 73 0.589 11 0.302 90 168.383 75 173.6354 161.7933
Ymir 23.029 10 24.025 42 22.305 04 0.337 42 0.457 24 0.221 75 173.058 57 176.9269 169.0612
S/2004 S08 24.210 92 25.237 16 23.397 11 0.218 86 0.328 16 0.117 58 170.169 30 173.7150 166.4152

Table 2

Mean orbital elements for the 35 irregular satellites of Saturn computed with Model 2. For each orbital element, we report the minimum and maximum values attained during the simulations.

Satellite Mean a (106 km) Max a (106 km) Min a (106 km) Mean e Max e Min e Mean i (°) Max i (°) Min i (°)
Kiviuq 11.345 50 11.429 28 11.279 68 0.261 36 0.585 66 0.000 07 48.144 251 54.300 30 38.177 60
Ijiraq 11.351 49 11.429 28 11.279 68 0.300 76 0.568 49 0.100 84 48.085 587 54.212 80 39.007 40
Phoebe 12.946 20 13.029 97 12.880 38 0.161 90 0.188 62 0.136 91 175.045 66 177.9671 172.4479
Paaliaq 14.946 32 15.199 14 14.705 47 0.346 83 0.657 90 0.107 13 49.179 928 56.891 50 38.039 60
Skathi 15.597 07 15.782 58 15.438 50 0.274 99 0.386 29 0.177 05 151.778 71 157.0443 146.1022
Albiorix 16.253 81 16.680 16 15.872 33 0.474 72 0.645 81 0.319 28 37.148 303 45.807 70 28.101 80
S/2004 S11 16.901 57 17.413 19 16.440 81 0.476 28 0.660 72 0.306 82 38.199 374 47.394 60 28.685 10
Erriapo 17.511 93 18.071 42 17.084 08 0.467 27 0.645 20 0.306 02 37.237 629 45.982 80 28.289 60
S/2006 S8 17.592 71 18.026 54 17.263 59 0.489 69 0.610 15 0.372 43 158.907 49 165.8853 151.1704
Siarnaq 17.800 65 18.310 78 17.383 27 0.303 01 0.619 90 0.062 62 47.806 963 55.699 50 38.759 80
S/2006 S4 18.002 61 18.385 58 17.727 35 0.323 98 0.397 49 0.252 21 174.256 62 177.5717 171.0669
Tarvos 18.037 02 18.864 29 17.398 23 0.524 01 0.715 99 0.346 81 37.896 709 48.348 80 27.335 50
S/2004 S13 18.059 45 18.430 46 17.742 31 0.255 07 0.332 15 0.185 84 168.795 86 172.3424 165.1410
S/2004 S19 18.391 56 18.759 57 18.086 38 0.322 58 0.481 27 0.186 05 149.835 05 156.3264 143.1073
Mundilfari 18.608 48 18.969 01 18.310 78 0.212 35 0.281 77 0.148 79 167.079 64 170.6035 163.4678
S/2006 S6 18.654 85 19.013 89 18.355 66 0.216 90 0.295 38 0.147 41 162.704 89 166.6243 158.5058
S/2006 S1 18.837 36 19.163 49 18.535 18 0.124 57 0.196 78 0.064 25 156.265 51 160.0878 152.3155
Narvi 19.251 75 19.731 96 18.864 29 0.420 33 0.711 40 0.196 85 141.706 17 152.9912 130.2114
Suttungr 19.366 94 19.746 92 19.028 85 0.115 63 0.165 92 0.070 91 175.879 27 178.8359 173.3126
S/2004 S17 19.377 41 19.791 80 19.013 89 0.179 44 0.248 28 0.119 06 167.738 42 171.1409 164.2874
S/2004 S15 19.609 29 20.016 20 19.253 25 0.150 33 0.229 44 0.083 77 158.759 16 162.5991 154.7317
S/2004 S12 19.699 05 20.270 51 19.253 25 0.334 79 0.448 65 0.229 25 164.651 64 169.1637 159.6312
S/2004 S10 19.787 31 20.285 47 19.402 84 0.265 22 0.357 15 0.182 70 165.949 75 169.9036 161.7321
S/2004 S09 20.191 22 20.704 35 19.776 84 0.246 17 0.364 85 0.144 08 156.238 85 161.0478 150.9973
S/2004 S18 20.201 70 20.868 90 19.702 04 0.501 76 0.781 85 0.250 99 138.170 34 152.6666 125.7233
Thrymr 20.337 83 21.093 30 19.806 76 0.476 65 0.591 83 0.357 65 175.134 48 179.0279 171.3395
S/2004 S14 20.508 37 21.123 22 20.061 07 0.356 62 0.467 76 0.250 58 165.054 83 169.7195 159.8302
S/2004 S07 20.934 73 21.706 65 20.375 23 0.528 53 0.661 39 0.393 30 163.941 12 170.4116 156.0144
S/2006 S3 21.084 32 21.931 05 20.479 95 0.463 78 0.632 40 0.307 06 156.660 42 164.1352 147.7268
S/2006 S2 21.871 21 22.828 64 21.183 06 0.492 23 0.687 29 0.310 75 153.045 66 161.9840 142.2631
S/2006 S5 22.649 12 23.531 75 21.946 01 0.231 84 0.387 93 0.101 94 167.750 79 171.5942 163.5061
S/2006 S7 22.907 92 23.950 62 22.155 44 0.455 52 0.592 55 0.315 50 168.843 16 173.7958 162.9456
S/2004 S16 22.909 42 23.666 38 22.275 12 0.156 19 0.248 86 0.076 16 164.674 91 168.4193 160.6690
Ymir 23.033 58 24.025 42 22.305 04 0.347 88 0.475 84 0.223 87 173.646 63 177.9365 169.0400
S/2004 S08 24.204 94 25.237 16 23.412 07 0.218 45 0.328 04 0.117 97 170.094 07 173.6581 166.2385
Satellite Mean a (106 km) Max a (106 km) Min a (106 km) Mean e Max e Min e Mean i (°) Max i (°) Min i (°)
Kiviuq 11.345 50 11.429 28 11.279 68 0.261 36 0.585 66 0.000 07 48.144 251 54.300 30 38.177 60
Ijiraq 11.351 49 11.429 28 11.279 68 0.300 76 0.568 49 0.100 84 48.085 587 54.212 80 39.007 40
Phoebe 12.946 20 13.029 97 12.880 38 0.161 90 0.188 62 0.136 91 175.045 66 177.9671 172.4479
Paaliaq 14.946 32 15.199 14 14.705 47 0.346 83 0.657 90 0.107 13 49.179 928 56.891 50 38.039 60
Skathi 15.597 07 15.782 58 15.438 50 0.274 99 0.386 29 0.177 05 151.778 71 157.0443 146.1022
Albiorix 16.253 81 16.680 16 15.872 33 0.474 72 0.645 81 0.319 28 37.148 303 45.807 70 28.101 80
S/2004 S11 16.901 57 17.413 19 16.440 81 0.476 28 0.660 72 0.306 82 38.199 374 47.394 60 28.685 10
Erriapo 17.511 93 18.071 42 17.084 08 0.467 27 0.645 20 0.306 02 37.237 629 45.982 80 28.289 60
S/2006 S8 17.592 71 18.026 54 17.263 59 0.489 69 0.610 15 0.372 43 158.907 49 165.8853 151.1704
Siarnaq 17.800 65 18.310 78 17.383 27 0.303 01 0.619 90 0.062 62 47.806 963 55.699 50 38.759 80
S/2006 S4 18.002 61 18.385 58 17.727 35 0.323 98 0.397 49 0.252 21 174.256 62 177.5717 171.0669
Tarvos 18.037 02 18.864 29 17.398 23 0.524 01 0.715 99 0.346 81 37.896 709 48.348 80 27.335 50
S/2004 S13 18.059 45 18.430 46 17.742 31 0.255 07 0.332 15 0.185 84 168.795 86 172.3424 165.1410
S/2004 S19 18.391 56 18.759 57 18.086 38 0.322 58 0.481 27 0.186 05 149.835 05 156.3264 143.1073
Mundilfari 18.608 48 18.969 01 18.310 78 0.212 35 0.281 77 0.148 79 167.079 64 170.6035 163.4678
S/2006 S6 18.654 85 19.013 89 18.355 66 0.216 90 0.295 38 0.147 41 162.704 89 166.6243 158.5058
S/2006 S1 18.837 36 19.163 49 18.535 18 0.124 57 0.196 78 0.064 25 156.265 51 160.0878 152.3155
Narvi 19.251 75 19.731 96 18.864 29 0.420 33 0.711 40 0.196 85 141.706 17 152.9912 130.2114
Suttungr 19.366 94 19.746 92 19.028 85 0.115 63 0.165 92 0.070 91 175.879 27 178.8359 173.3126
S/2004 S17 19.377 41 19.791 80 19.013 89 0.179 44 0.248 28 0.119 06 167.738 42 171.1409 164.2874
S/2004 S15 19.609 29 20.016 20 19.253 25 0.150 33 0.229 44 0.083 77 158.759 16 162.5991 154.7317
S/2004 S12 19.699 05 20.270 51 19.253 25 0.334 79 0.448 65 0.229 25 164.651 64 169.1637 159.6312
S/2004 S10 19.787 31 20.285 47 19.402 84 0.265 22 0.357 15 0.182 70 165.949 75 169.9036 161.7321
S/2004 S09 20.191 22 20.704 35 19.776 84 0.246 17 0.364 85 0.144 08 156.238 85 161.0478 150.9973
S/2004 S18 20.201 70 20.868 90 19.702 04 0.501 76 0.781 85 0.250 99 138.170 34 152.6666 125.7233
Thrymr 20.337 83 21.093 30 19.806 76 0.476 65 0.591 83 0.357 65 175.134 48 179.0279 171.3395
S/2004 S14 20.508 37 21.123 22 20.061 07 0.356 62 0.467 76 0.250 58 165.054 83 169.7195 159.8302
S/2004 S07 20.934 73 21.706 65 20.375 23 0.528 53 0.661 39 0.393 30 163.941 12 170.4116 156.0144
S/2006 S3 21.084 32 21.931 05 20.479 95 0.463 78 0.632 40 0.307 06 156.660 42 164.1352 147.7268
S/2006 S2 21.871 21 22.828 64 21.183 06 0.492 23 0.687 29 0.310 75 153.045 66 161.9840 142.2631
S/2006 S5 22.649 12 23.531 75 21.946 01 0.231 84 0.387 93 0.101 94 167.750 79 171.5942 163.5061
S/2006 S7 22.907 92 23.950 62 22.155 44 0.455 52 0.592 55 0.315 50 168.843 16 173.7958 162.9456
S/2004 S16 22.909 42 23.666 38 22.275 12 0.156 19 0.248 86 0.076 16 164.674 91 168.4193 160.6690
Ymir 23.033 58 24.025 42 22.305 04 0.347 88 0.475 84 0.223 87 173.646 63 177.9365 169.0400
S/2004 S08 24.204 94 25.237 16 23.412 07 0.218 45 0.328 04 0.117 97 170.094 07 173.6581 166.2385

Table 2

Mean orbital elements for the 35 irregular satellites of Saturn computed with Model 2. For each orbital element, we report the minimum and maximum values attained during the simulations.

Satellite Mean a (106 km) Max a (106 km) Min a (106 km) Mean e Max e Min e Mean i (°) Max i (°) Min i (°)
Kiviuq 11.345 50 11.429 28 11.279 68 0.261 36 0.585 66 0.000 07 48.144 251 54.300 30 38.177 60
Ijiraq 11.351 49 11.429 28 11.279 68 0.300 76 0.568 49 0.100 84 48.085 587 54.212 80 39.007 40
Phoebe 12.946 20 13.029 97 12.880 38 0.161 90 0.188 62 0.136 91 175.045 66 177.9671 172.4479
Paaliaq 14.946 32 15.199 14 14.705 47 0.346 83 0.657 90 0.107 13 49.179 928 56.891 50 38.039 60
Skathi 15.597 07 15.782 58 15.438 50 0.274 99 0.386 29 0.177 05 151.778 71 157.0443 146.1022
Albiorix 16.253 81 16.680 16 15.872 33 0.474 72 0.645 81 0.319 28 37.148 303 45.807 70 28.101 80
S/2004 S11 16.901 57 17.413 19 16.440 81 0.476 28 0.660 72 0.306 82 38.199 374 47.394 60 28.685 10
Erriapo 17.511 93 18.071 42 17.084 08 0.467 27 0.645 20 0.306 02 37.237 629 45.982 80 28.289 60
S/2006 S8 17.592 71 18.026 54 17.263 59 0.489 69 0.610 15 0.372 43 158.907 49 165.8853 151.1704
Siarnaq 17.800 65 18.310 78 17.383 27 0.303 01 0.619 90 0.062 62 47.806 963 55.699 50 38.759 80
S/2006 S4 18.002 61 18.385 58 17.727 35 0.323 98 0.397 49 0.252 21 174.256 62 177.5717 171.0669
Tarvos 18.037 02 18.864 29 17.398 23 0.524 01 0.715 99 0.346 81 37.896 709 48.348 80 27.335 50
S/2004 S13 18.059 45 18.430 46 17.742 31 0.255 07 0.332 15 0.185 84 168.795 86 172.3424 165.1410
S/2004 S19 18.391 56 18.759 57 18.086 38 0.322 58 0.481 27 0.186 05 149.835 05 156.3264 143.1073
Mundilfari 18.608 48 18.969 01 18.310 78 0.212 35 0.281 77 0.148 79 167.079 64 170.6035 163.4678
S/2006 S6 18.654 85 19.013 89 18.355 66 0.216 90 0.295 38 0.147 41 162.704 89 166.6243 158.5058
S/2006 S1 18.837 36 19.163 49 18.535 18 0.124 57 0.196 78 0.064 25 156.265 51 160.0878 152.3155
Narvi 19.251 75 19.731 96 18.864 29 0.420 33 0.711 40 0.196 85 141.706 17 152.9912 130.2114
Suttungr 19.366 94 19.746 92 19.028 85 0.115 63 0.165 92 0.070 91 175.879 27 178.8359 173.3126
S/2004 S17 19.377 41 19.791 80 19.013 89 0.179 44 0.248 28 0.119 06 167.738 42 171.1409 164.2874
S/2004 S15 19.609 29 20.016 20 19.253 25 0.150 33 0.229 44 0.083 77 158.759 16 162.5991 154.7317
S/2004 S12 19.699 05 20.270 51 19.253 25 0.334 79 0.448 65 0.229 25 164.651 64 169.1637 159.6312
S/2004 S10 19.787 31 20.285 47 19.402 84 0.265 22 0.357 15 0.182 70 165.949 75 169.9036 161.7321
S/2004 S09 20.191 22 20.704 35 19.776 84 0.246 17 0.364 85 0.144 08 156.238 85 161.0478 150.9973
S/2004 S18 20.201 70 20.868 90 19.702 04 0.501 76 0.781 85 0.250 99 138.170 34 152.6666 125.7233
Thrymr 20.337 83 21.093 30 19.806 76 0.476 65 0.591 83 0.357 65 175.134 48 179.0279 171.3395
S/2004 S14 20.508 37 21.123 22 20.061 07 0.356 62 0.467 76 0.250 58 165.054 83 169.7195 159.8302
S/2004 S07 20.934 73 21.706 65 20.375 23 0.528 53 0.661 39 0.393 30 163.941 12 170.4116 156.0144
S/2006 S3 21.084 32 21.931 05 20.479 95 0.463 78 0.632 40 0.307 06 156.660 42 164.1352 147.7268
S/2006 S2 21.871 21 22.828 64 21.183 06 0.492 23 0.687 29 0.310 75 153.045 66 161.9840 142.2631
S/2006 S5 22.649 12 23.531 75 21.946 01 0.231 84 0.387 93 0.101 94 167.750 79 171.5942 163.5061
S/2006 S7 22.907 92 23.950 62 22.155 44 0.455 52 0.592 55 0.315 50 168.843 16 173.7958 162.9456
S/2004 S16 22.909 42 23.666 38 22.275 12 0.156 19 0.248 86 0.076 16 164.674 91 168.4193 160.6690
Ymir 23.033 58 24.025 42 22.305 04 0.347 88 0.475 84 0.223 87 173.646 63 177.9365 169.0400
S/2004 S08 24.204 94 25.237 16 23.412 07 0.218 45 0.328 04 0.117 97 170.094 07 173.6581 166.2385
Satellite Mean a (106 km) Max a (106 km) Min a (106 km) Mean e Max e Min e Mean i (°) Max i (°) Min i (°)
Kiviuq 11.345 50 11.429 28 11.279 68 0.261 36 0.585 66 0.000 07 48.144 251 54.300 30 38.177 60
Ijiraq 11.351 49 11.429 28 11.279 68 0.300 76 0.568 49 0.100 84 48.085 587 54.212 80 39.007 40
Phoebe 12.946 20 13.029 97 12.880 38 0.161 90 0.188 62 0.136 91 175.045 66 177.9671 172.4479
Paaliaq 14.946 32 15.199 14 14.705 47 0.346 83 0.657 90 0.107 13 49.179 928 56.891 50 38.039 60
Skathi 15.597 07 15.782 58 15.438 50 0.274 99 0.386 29 0.177 05 151.778 71 157.0443 146.1022
Albiorix 16.253 81 16.680 16 15.872 33 0.474 72 0.645 81 0.319 28 37.148 303 45.807 70 28.101 80
S/2004 S11 16.901 57 17.413 19 16.440 81 0.476 28 0.660 72 0.306 82 38.199 374 47.394 60 28.685 10
Erriapo 17.511 93 18.071 42 17.084 08 0.467 27 0.645 20 0.306 02 37.237 629 45.982 80 28.289 60
S/2006 S8 17.592 71 18.026 54 17.263 59 0.489 69 0.610 15 0.372 43 158.907 49 165.8853 151.1704
Siarnaq 17.800 65 18.310 78 17.383 27 0.303 01 0.619 90 0.062 62 47.806 963 55.699 50 38.759 80
S/2006 S4 18.002 61 18.385 58 17.727 35 0.323 98 0.397 49 0.252 21 174.256 62 177.5717 171.0669
Tarvos 18.037 02 18.864 29 17.398 23 0.524 01 0.715 99 0.346 81 37.896 709 48.348 80 27.335 50
S/2004 S13 18.059 45 18.430 46 17.742 31 0.255 07 0.332 15 0.185 84 168.795 86 172.3424 165.1410
S/2004 S19 18.391 56 18.759 57 18.086 38 0.322 58 0.481 27 0.186 05 149.835 05 156.3264 143.1073
Mundilfari 18.608 48 18.969 01 18.310 78 0.212 35 0.281 77 0.148 79 167.079 64 170.6035 163.4678
S/2006 S6 18.654 85 19.013 89 18.355 66 0.216 90 0.295 38 0.147 41 162.704 89 166.6243 158.5058
S/2006 S1 18.837 36 19.163 49 18.535 18 0.124 57 0.196 78 0.064 25 156.265 51 160.0878 152.3155
Narvi 19.251 75 19.731 96 18.864 29 0.420 33 0.711 40 0.196 85 141.706 17 152.9912 130.2114
Suttungr 19.366 94 19.746 92 19.028 85 0.115 63 0.165 92 0.070 91 175.879 27 178.8359 173.3126
S/2004 S17 19.377 41 19.791 80 19.013 89 0.179 44 0.248 28 0.119 06 167.738 42 171.1409 164.2874
S/2004 S15 19.609 29 20.016 20 19.253 25 0.150 33 0.229 44 0.083 77 158.759 16 162.5991 154.7317
S/2004 S12 19.699 05 20.270 51 19.253 25 0.334 79 0.448 65 0.229 25 164.651 64 169.1637 159.6312
S/2004 S10 19.787 31 20.285 47 19.402 84 0.265 22 0.357 15 0.182 70 165.949 75 169.9036 161.7321
S/2004 S09 20.191 22 20.704 35 19.776 84 0.246 17 0.364 85 0.144 08 156.238 85 161.0478 150.9973
S/2004 S18 20.201 70 20.868 90 19.702 04 0.501 76 0.781 85 0.250 99 138.170 34 152.6666 125.7233
Thrymr 20.337 83 21.093 30 19.806 76 0.476 65 0.591 83 0.357 65 175.134 48 179.0279 171.3395
S/2004 S14 20.508 37 21.123 22 20.061 07 0.356 62 0.467 76 0.250 58 165.054 83 169.7195 159.8302
S/2004 S07 20.934 73 21.706 65 20.375 23 0.528 53 0.661 39 0.393 30 163.941 12 170.4116 156.0144
S/2006 S3 21.084 32 21.931 05 20.479 95 0.463 78 0.632 40 0.307 06 156.660 42 164.1352 147.7268
S/2006 S2 21.871 21 22.828 64 21.183 06 0.492 23 0.687 29 0.310 75 153.045 66 161.9840 142.2631
S/2006 S5 22.649 12 23.531 75 21.946 01 0.231 84 0.387 93 0.101 94 167.750 79 171.5942 163.5061
S/2006 S7 22.907 92 23.950 62 22.155 44 0.455 52 0.592 55 0.315 50 168.843 16 173.7958 162.9456
S/2004 S16 22.909 42 23.666 38 22.275 12 0.156 19 0.248 86 0.076 16 164.674 91 168.4193 160.6690
Ymir 23.033 58 24.025 42 22.305 04 0.347 88 0.475 84 0.223 87 173.646 63 177.9365 169.0400
S/2004 S08 24.204 94 25.237 16 23.412 07 0.218 45 0.328 04 0.117 97 170.094 07 173.6581 166.2385

Comparison in the a–i plane between the mean orbital elements of Saturn's irregular satellites computed with Model 2 using HJS and RADAU algorithms (upper plot), Model 1 and Model 2 using HJS algorithm (middle plot) and Model 1 with standard and strict double precision (lower plot). The vertical and horizontal bars show the variation ranges of the elements in the simulations. Distances are expressed in 106 km while angles are expressed in degrees. In the simulation with RADAU algorithm we considered only the 26 irregular satellites known till 2005.

Figure 1

Comparison in the a_–_i plane between the mean orbital elements of Saturn's irregular satellites computed with Model 2 using HJS and RADAU algorithms (upper plot), Model 1 and Model 2 using HJS algorithm (middle plot) and Model 1 with standard and strict double precision (lower plot). The vertical and horizontal bars show the variation ranges of the elements in the simulations. Distances are expressed in 106 km while angles are expressed in degrees. In the simulation with RADAU algorithm we considered only the 26 irregular satellites known till 2005.

Comparison in the a–e plane between the mean orbital elements of Saturn's irregular satellites computed with Model 2 using HJS and RADAU algorithms (upper plot), Model 1 and Model 2 using HJS algorithm (middle plot) and Model 1 with standard and strict double precision (lower plot). The vertical and horizontal bars show the variation ranges of the elements in the simulations. Distances are expressed in 106 km while angles are expressed in degrees. In the simulation with RADAU algorithm we considered only the 26 irregular satellites known till 2005.

Figure 2

Comparison in the a_–_e plane between the mean orbital elements of Saturn's irregular satellites computed with Model 2 using HJS and RADAU algorithms (upper plot), Model 1 and Model 2 using HJS algorithm (middle plot) and Model 1 with standard and strict double precision (lower plot). The vertical and horizontal bars show the variation ranges of the elements in the simulations. Distances are expressed in 106 km while angles are expressed in degrees. In the simulation with RADAU algorithm we considered only the 26 irregular satellites known till 2005.

The differences in the mean elements computed with RADAU and HJS on the same model (Model 2) are shown in the top panels of Figs 1 and 2 and are significantly less important: they appear to be due to the presence of chaos in the system. Finally, in Fig. 3 we confronted the mean elements computed from Model 2 with the average orbital elements available on the JPL Solar System Dynamics website.4 In this case the differences between the two data sets are more marked, probably depending on the different integration-averaging time used to compute them.

Comparison between the mean orbital elements of Saturn's irregular satellites computed with Model 2 using the HJS algorithm and those from the JPL Solar System Dynamics website. The comparison is shown in the a–i (upper graph) and a–e (lower graph) planes. The vertical and horizontal bars show the variation ranges of the elements in our simulation. Distances are expressed in 106 km while angles are expressed in degrees.

Figure 3

Comparison between the mean orbital elements of Saturn's irregular satellites computed with Model 2 using the HJS algorithm and those from the JPL Solar System Dynamics website. The comparison is shown in the a_–_i (upper graph) and a_–_e (lower graph) planes. The vertical and horizontal bars show the variation ranges of the elements in our simulation. Distances are expressed in 106 km while angles are expressed in degrees.

In the following discussions, when not stated differently, we will always be implicitly referring to the mean elements computed with Model 2 since it better represents the real dynamical system. We would like to stress that the validity of our mean elements is proved over 108 yr and only through the analysis of the satellites' secular evolution we can be able to assess if they can be meaningful on longer time-scales.

As a final remark, the two major gaps in the radial distribution of Saturn's irregular satellites, the first centred at Phoebe and extending from 11.22 × 106 to 14.96 × 106 km and the second located between 20.94 × 106 and 22.44 × 106 km, will be discussed, respectively, in Sections 4 and 3.

3 EVALUATION OF THE DYNAMICAL EVOLUTION

The mean orbital elements given in the previous section give a global view of the dynamical evolution of the satellite system. However, to understand better the features of the mean elements and of the differences observed between the different models we need to have a better insight on the individual secular evolution of the satellites. We start our analysis by looking for hints of chaotic or resonant behaviour. We adopt a variant of the mean actions criterion (see Morbidelli 2002 and references therein) introduced by Cuk & Burns (2004) to study the dynamics of irregular satellites. This modified criterion is based on the computation of the η parameter, defined as

formula

1

where

formula

2

and

formula

3

The η parameter is a measure of the temporal uniformity of the secular evolution of the eccentricity. As shown in equations (1), (2) and (3), its value is estimated by summing the differences between the mean values 〈e_〉_j of eccentricity computed on a fixed number of time intervals (10 in equations 1 and 2) and the mean value 〈_e_〉 on the whole time-span covered by the simulations. If the motion of the considered body is regular or quasi-periodic, the mean value of eccentricity should change little in time and the η parameter should approach zero. On the contrary, if the motion is resonant or chaotic, the mean value of eccentricity would depend on the time interval over which it is computed and, therefore, η would assume increasing values. Obviously, the reliability of this method depends on the relationship between the number and extension of the time intervals considered and the frequencies of motion: if the time-scale of the variations is shorter than the length of each time interval, the averaging process can mask the irregular behaviour of the motion and output an η value lower than the one really describing the satellite dynamics. On the contrary, a satellite having a regular behaviour but whose orbital elements oscillate with a period longer than the considered time intervals may spuriously appear chaotic. There is no a priori prescription concerning the number of time intervals to be used: after a set of numerical experiments, we opted for the value (10 time intervals) suggested in the original work by Cuk & Burns (2004).

In the following we extend the idea of Cuk & Burns (2004) by applying a similar analysis also to inclination and semimajor axis, thus introducing η_a_ and η_i_ as

formula

4

and

formula

5

with corresponding definitions of the quantities involved.

We present the η values in Fig. 4, where we compare the output of Model 1, Model 2 and of an alternative version of Model 1 where a different precision in the floating point calculations have been used. We will shortly explain the reason for this duplication.

Values of the η parameters for the irregular satellites of Saturn. From top to bottom the plots show the η values for eccentricity, inclination and semimajor axis, respectively. The results concern Models 1 and 2 and the additional simulation based on Model 1 where strict double precision (64 bit) was adopted in the computations instead of the standard extended precision (80 bit). The change in the numerical precision has major effects on various satellites which, as a consequence, evolve on chaotic orbits.

Figure 4

Values of the η parameters for the irregular satellites of Saturn. From top to bottom the plots show the η values for eccentricity, inclination and semimajor axis, respectively. The results concern Models 1 and 2 and the additional simulation based on Model 1 where strict double precision (64 bit) was adopted in the computations instead of the standard extended precision (80 bit). The change in the numerical precision has major effects on various satellites which, as a consequence, evolve on chaotic orbits.

The coefficient showing the largest variations is ηe (see Fig. 4, top panel) and the extent of the dynamical variations for each satellite can be roughly evaluated by the square root of ηe. These variations are on average around 5–10 per cent with peak values of about 30 per cent. The corresponding variations in semimajor axis and inclination, as derived from the η_a_ and η_i_, are on average more limited ranging from 1 to 2 per cent. Since the values of ηe are not significantly correlated with those of η_a_ and η_i_, our guess is that the different coefficients are indicators of different dynamical effects.

We used a Model 1 characterized by a different numerical precision in the calculations to reveal the presence of chaos in the system. The standard set-up on ×86 machines for double precision computing is the so-called extended precision, where the floating point numbers are stored in double precision variables (64 bit) but a higher precision (80 bit) is employed during the computations. This is also the set-up we originally used in Model 1, which in the graphs is labelled ‘Model 1 (80 bit)’. We ran an additional simulation forcing the computer to use strict double precision, thus employing 64 bits also during the computations. The results of this simulation are labelled as ‘Model 1 (64 bit)’. Hereafter, unless differently stated, when we refer generically to Model 1 we intend the 80-bit case. From a numerical point of view, the change in computing precision should in principle affect only the last few digits of each floating point number, since the values are stored in 64-bit variables in both cases. Regular motions should be slightly affected by the change while chaotic motions should show divergent trajectories. By comparing the different values of the η parameters for ‘Model 1 (64 bit)’ and ‘Model 1 (80 bit)’ we find some satellites with drastically different values, a clear indication of chaotic motion.

After this preliminary global study of the satellite system, we have performed a more detailed analysis of selected objects whose behaviours have been already investigated in previous papers. Carruba et al. (2002), Nesvorny et al. (2003) and Cuk & Burns (2004) reported four cases of resonant motion between the irregular satellites of Saturn:

The authors computed the orbital evolution of these satellites with a modified version of the swift_N_-body code, where the planets were integrated in the heliocentric reference frame and the irregular satellites in planetocentric frame. This dynamical structure is similar to that of our Model 1. We extracted from our simulations the data concerning the same objects studied in Carruba et al. (2002), Nesvorny et al. (2003) and Cuk & Burns (2004) and we analysed their behaviour. The motion of Ijiraq as from our Models 1 and 2 integrated with HJS (see Fig. 5, first three panels from top left-hand side in counterclockwise direction) appears to evolve in a stable Kozai regime with the Sun. The longitude of pericentre librates around 90° with an amplitude of ±30°. The simulations based on Model 2 and Model 1 (80-bit precision) show a regular secular behaviour of the orbital elements while the simulation based on Model 1 (64-bit precision) shows periods in which the range of variation of the eccentricity shrinks by a few per cent in correspondence to a similar behaviour of the longitude of pericentre (see Fig. 6 for details). The output obtained from Model 2 with RADAU algorithm (Fig. 7) is instead significantly different. Ijiraq is in a stable Kozai regime for the first 6 × 107 yr, then it experiences a change in the secular behaviour of both eccentricity and inclination and, finally, the longitude of pericentre ϖ circulates for about 1.3 × 107 yr. From then on, ϖ alternates phases of circulation and libration around both 90° and 270°.

Secular evolution of Ijiraq's longitude of pericentre in (from top left-hand side, clockwise direction) Model 2 computed with HJS algorithm, Model 2 computed with RADAU algorithm, Model 1, Model 1 with strict (64-bit) double precision. Angles are expressed in degrees and time in years. In all cases computed with HJS code, Ijiraq's longitude of pericentre is into a stable librational regime, indicating that the satellite is locked into a Kozai resonance with the Sun. In the case computed with RADAU, the satellite is initially trapped into the same resonance but it breaks the resonant regime during the last 2 × 107 yr of the simulation. Escape from the resonance is not permanent, since the satellite is temporarily captured at least other two times before the end of the simulation. Ijiraq's behaviour is probably due to the perturbations of Titan and Iapetus.

Figure 5

Secular evolution of Ijiraq's longitude of pericentre in (from top left-hand side, clockwise direction) Model 2 computed with HJS algorithm, Model 2 computed with RADAU algorithm, Model 1, Model 1 with strict (64-bit) double precision. Angles are expressed in degrees and time in years. In all cases computed with HJS code, Ijiraq's longitude of pericentre is into a stable librational regime, indicating that the satellite is locked into a Kozai resonance with the Sun. In the case computed with RADAU, the satellite is initially trapped into the same resonance but it breaks the resonant regime during the last 2 × 107 yr of the simulation. Escape from the resonance is not permanent, since the satellite is temporarily captured at least other two times before the end of the simulation. Ijiraq's behaviour is probably due to the perturbations of Titan and Iapetus.

Evolution of Ijiraq's longitude of pericentre and eccentricity in Model 1 simulated with HJS algorithm using strict (64-bit) double precision. The longitude of pericentre and the eccentricity have limited non-periodic changes which may be ascribed to the perturbations of Jupiter since Titan and Iapetus are not included in the model.

Figure 6

Evolution of Ijiraq's longitude of pericentre and eccentricity in Model 1 simulated with HJS algorithm using strict (64-bit) double precision. The longitude of pericentre and the eccentricity have limited non-periodic changes which may be ascribed to the perturbations of Jupiter since Titan and Iapetus are not included in the model.

Evolution of Ijiraq's longitude of pericentre, eccentricity and inclination (from top to bottom) in Model 2 integrated with RADAU. The dynamical evolution of the satellite has a sudden change around 6 × 107 yr leading to a progressive increase in the eccentricity oscillations. Also the libration amplitude slightly grows until the longitude of pericentre starts to circulate at 8 × 107 yr. Since Fig. 6 proved that the effects of the giant planets are quite limited after the onset of the Kozai resonance with the Sun, Titan and Iapetus are responsible for the perturbed evolution of the satellite.

Figure 7

Evolution of Ijiraq's longitude of pericentre, eccentricity and inclination (from top to bottom) in Model 2 integrated with RADAU. The dynamical evolution of the satellite has a sudden change around 6 × 107 yr leading to a progressive increase in the eccentricity oscillations. Also the libration amplitude slightly grows until the longitude of pericentre starts to circulate at 8 × 107 yr. Since Fig. 6 proved that the effects of the giant planets are quite limited after the onset of the Kozai resonance with the Sun, Titan and Iapetus are responsible for the perturbed evolution of the satellite.

Kiviuq shows an even more complex behaviour (see Fig. 8). All the simulations (Models 1 and 2 and both HJS and RADAU algorithms) predict transitions between periods of circulation and libration of ϖ. The centre of libration changes depending on the numerical algorithm: the secular evolution computed with RADAU algorithm show a prevalence of librational phases around 270°. In the case of HJS algorithm the dominant libration mode is around 90°. While the libration cycles were all characterized by an amplitude of about ±30°, their durations vary from case to case. In Model 2 computed with HJS ϖ mostly circulates with only a few short-lived librational periods. In Model 1 (80 bit) the satellite enters the Kozai regime with the Sun only after 8 × 107 yr, while Model 1 (64 bit) shows a behaviour more similar to that of Model 2, even if with longer lived librational phases. Model 2 computed with RADAU shows a behaviour similar to that of Model 1. As for Ijiraq, such different dynamical evolutions confirm the chaotic behaviour of the satellite orbit.

Secular evolution of Kiviuq's longitude of pericentre. From top left-hand side in clockwise direction we plot the outcome of Model 2 computed with HJS algorithm, Model 2 computed with RADAU algorithm, Model 1, Model 1 with strict (64-bit) double precision. Angles are expressed in degrees and time in years. Transitions between circulation and libration appear in both the dynamical models and independently from the numerical codes. Librations in Model 2 – HJS occur between 6 × 107 and 7 × 107 yr and are extremely short-lived (the angle mainly circulates). The librations are also evenly distributed around 90° and 270°. The same model computed with RADAU shows long-lived librational periods centred at 270°. Also in Model 1 the librations are long lived but they concentrated around 90°. In Model 1 (bottom right-hand panel) at the end of the simulation the satellite enters a libration phase which lasts about 2 × 107 yr. The differences between the four cases argue for the chaotic nature of Kiviuq's dynamical evolution. Titan and Iapetus intervene altering the dynamical history of the satellite.

Figure 8

Secular evolution of Kiviuq's longitude of pericentre. From top left-hand side in clockwise direction we plot the outcome of Model 2 computed with HJS algorithm, Model 2 computed with RADAU algorithm, Model 1, Model 1 with strict (64-bit) double precision. Angles are expressed in degrees and time in years. Transitions between circulation and libration appear in both the dynamical models and independently from the numerical codes. Librations in Model 2 – HJS occur between 6 × 107 and 7 × 107 yr and are extremely short-lived (the angle mainly circulates). The librations are also evenly distributed around 90° and 270°. The same model computed with RADAU shows long-lived librational periods centred at 270°. Also in Model 1 the librations are long lived but they concentrated around 90°. In Model 1 (bottom right-hand panel) at the end of the simulation the satellite enters a libration phase which lasts about 2 × 107 yr. The differences between the four cases argue for the chaotic nature of Kiviuq's dynamical evolution. Titan and Iapetus intervene altering the dynamical history of the satellite.

Ijiraq and Kiviuq are examples of a different balancing between the perturbations acting on the trajectories of these satellites. Ijiraq appears dominated by the Kozai resonance caused by the Sun's influence while Titan, Iapetus and the other planets play only a minor role. On the opposite side, Kiviuq is significantly perturbed by Titan, Iapetus and the other planets which prevent its settling into a stable Kozai regime with the Sun and cause its chaotic evolution.

For Siarnaq and Paaliaq our updated dynamical model do not confirm the resonant motions found in previous publications. In presence of Titan and Iapetus (Model 2) both the satellites show evidence of a limited but systematic variation of their semimajor axes. They monotonically migrate inwards by ∼2 per cent during the time-span covered by our simulations (see Fig. 9). Paaliaq has a regular evolution of the eccentricity and inclination (Fig. 9, left-hand column), while Siarnaq (Fig. 9, right-hand column) during its radial migration presents phases of chaotic variations of both eccentricity and inclination (see Figs 10 and 11 for details). We cannot rule out major changes in semimajor axis (possibly periodic?) on longer time-scales. Paaliaq, during its orbital evolution, might enter more dynamically perturbed regions (see Fig. 26 for further details).

Comparison between the evolution of semimajor axis, eccentricity and inclination (from top to bottom) of Paaliaq (left-hand column) and Siarnaq (right-hand column) in Model 2 integrated with HJS. Both satellites suffer a slight inward displacement, limited to about ∼1–2 per cent, yet Paaliaq's eccentricity and inclination show a regular behaviour whereas Siarnaq's ones have a coupled variation at about 1 × 107 and 5 × 107 yr. This behaviour is reproduced in both models (see Figs 10 and 11 for further details on Siarnaq's evolution).

Figure 9

Comparison between the evolution of semimajor axis, eccentricity and inclination (from top to bottom) of Paaliaq (left-hand column) and Siarnaq (right-hand column) in Model 2 integrated with HJS. Both satellites suffer a slight inward displacement, limited to about ∼1–2 per cent, yet Paaliaq's eccentricity and inclination show a regular behaviour whereas Siarnaq's ones have a coupled variation at about 1 × 107 and 5 × 107 yr. This behaviour is reproduced in both models (see Figs 10 and 11 for further details on Siarnaq's evolution).

Secular evolution of the eccentricity of Siarnaq. From top left-hand side, clockwise direction, the plots show the outcome of Model 2 – HJS algorithm, Model 2 – RADAU algorithm, Model 1 – HJS with standard precision and Model 1 – HJS with strict (64-bit) double precision. Siarnaq's eccentricity clearly shows evidence of chaotic evolution in Model 2 and Model 1 (64 bit). In Model 1 (80-bit) Siarnaq's eccentricity appears quasi-stationary at all time-scales.

Figure 10

Secular evolution of the eccentricity of Siarnaq. From top left-hand side, clockwise direction, the plots show the outcome of Model 2 – HJS algorithm, Model 2 – RADAU algorithm, Model 1 – HJS with standard precision and Model 1 – HJS with strict (64-bit) double precision. Siarnaq's eccentricity clearly shows evidence of chaotic evolution in Model 2 and Model 1 (64 bit). In Model 1 (80-bit) Siarnaq's eccentricity appears quasi-stationary at all time-scales.

Secular evolution of the inclination of Siarnaq. From top left-hand side, clockwise direction, the plots show the outcome of Model 2 – HJS algorithm, Model 2 – RADAU algorithm, Model 1 – HJS with standard precision and Model 1 – HJS with strict (64-bit) double precision. While Paaliaq radial displacement is coupled to a regular evolution of the other orbital elements (see Fig. 9), Siarnaq's eccentricity (see Fig. 10) and inclination show clearly the presence of chaotic features in Model 2 and Model 1 (64 bit). In Model 1 (80-bit) Siarnaq's inclination appears quasi-stationary at all time-scales.

Figure 11

Secular evolution of the inclination of Siarnaq. From top left-hand side, clockwise direction, the plots show the outcome of Model 2 – HJS algorithm, Model 2 – RADAU algorithm, Model 1 – HJS with standard precision and Model 1 – HJS with strict (64-bit) double precision. While Paaliaq radial displacement is coupled to a regular evolution of the other orbital elements (see Fig. 9), Siarnaq's eccentricity (see Fig. 10) and inclination show clearly the presence of chaotic features in Model 2 and Model 1 (64 bit). In Model 1 (80-bit) Siarnaq's inclination appears quasi-stationary at all time-scales.

Comparison between the ηe parameters of prograde (top panels) and retrograde (bottom panels) test particles moving close to the Phoebe gap (i.e. the region extending from 10.47 × 106 to 14.96 × 106 km from Saturn) in Model 1 (left-hand column) and Model 2 (right-hand column). By comparing the plots pairwise, we note that the differences increase with larger inclinations of the test particles. Those orbits with inclination in the range 30°–60° for the prograde case and 120°–150° for the retrograde one are the most influenced by the presence of Titan and Iapetus. This holds particularly true in the inner part of this region (i.e. up to Phoebe's orbital distance). Prograde test particles are less affected by the presence of Titan and Iapetus than retrograde ones. In Model 1, the retrograde test particles have smaller values of ηe than the prograde ones. In Model 2, the retrograde test particles show a marked increase in their ηe values for high inclination orbits (i.e. ∼120°). Such orbits should likely be subject to a strongly chaotic evolution and could be short lived on a time-scale longer than the one (106 yr) we considered. A noteworthy feature present in all plots is the peak at ∼13.015 × 106 km (the orbital distance of Phoebe) for both prograde and retrograde test particles. A second peak, peculiar to retrograde orbits, is located at ∼14.21 × 106 km. These peaks appear only for inclination of 30° (150° for retrograde orbits). These values bring the orbits of the test particles nearer to the equatorial plane of Saturn (once corrected for the giant planet's axial tilt) and thus nearer to the orbital plane of Titan and, to a minor extent, Iapetus. The peak at ∼13.015 × 106 km is ∼3 times lower for retrograde particles than for prograde ones in Model 1. In Model 2 the behaviour is opposite and the ηe value is increased by the same factor of ∼3. Phoebe's evolution has been protected from the effects of these perturbations by its inclination value, which put the satellite into a dynamically safe region.

Figure 26

Comparison between the ηe parameters of prograde (top panels) and retrograde (bottom panels) test particles moving close to the Phoebe gap (i.e. the region extending from 10.47 × 106 to 14.96 × 106 km from Saturn) in Model 1 (left-hand column) and Model 2 (right-hand column). By comparing the plots pairwise, we note that the differences increase with larger inclinations of the test particles. Those orbits with inclination in the range 30°–60° for the prograde case and 120°–150° for the retrograde one are the most influenced by the presence of Titan and Iapetus. This holds particularly true in the inner part of this region (i.e. up to Phoebe's orbital distance). Prograde test particles are less affected by the presence of Titan and Iapetus than retrograde ones. In Model 1, the retrograde test particles have smaller values of ηe than the prograde ones. In Model 2, the retrograde test particles show a marked increase in their ηe values for high inclination orbits (i.e. ∼120°). Such orbits should likely be subject to a strongly chaotic evolution and could be short lived on a time-scale longer than the one (106 yr) we considered. A noteworthy feature present in all plots is the peak at ∼13.015 × 106 km (the orbital distance of Phoebe) for both prograde and retrograde test particles. A second peak, peculiar to retrograde orbits, is located at ∼14.21 × 106 km. These peaks appear only for inclination of 30° (150° for retrograde orbits). These values bring the orbits of the test particles nearer to the equatorial plane of Saturn (once corrected for the giant planet's axial tilt) and thus nearer to the orbital plane of Titan and, to a minor extent, Iapetus. The peak at ∼13.015 × 106 km is ∼3 times lower for retrograde particles than for prograde ones in Model 1. In Model 2 the behaviour is opposite and the ηe value is increased by the same factor of ∼3. Phoebe's evolution has been protected from the effects of these perturbations by its inclination value, which put the satellite into a dynamically safe region.

The comparison of our results with previously published papers suggest that the three-body approximation adopted in previous analytical works was not accurate enough to be used as a reference model for the dynamical evolution of Saturn's irregular satellites. An additional feature arguing against a simplified three-body approximation is the following one. When we illustrate the secular evolution of the satellites in the e_–_i plane, in several cases we obtained a thick arc (see Fig. 12 for details) having opposite orientation for prograde and retrograde satellites. It indicates a global anticorrelation of the two orbital elements (see Fig. 13 for details) and it appears more frequently among the satellites integrated with Model 2. This anticorrelation of eccentricity and inclination is a characteristic feature of the Kozai regime. The arc-like feature is in fact manifest in the two previously discussed cases: Kiviuq and Ijiraq. By inspecting the dynamical histories of all the other satellites, we found the following with the same feature: Paaliaq, Skathi, Albiorix, Erriapo, Siarnaq, Tarvos, Narvi, Bebhionn (S/2004 S11), Bestla (S/2004 S18), Hyrrokkin (S/2004 S19). Three of these satellites have an inclination close to the critical one leading to Kozai cycles. Farbauti (S/2004 S9), Kari (S/2006 S2), S/2006 S3 and Surtur (S/2006 S7) show a more dispersed arc-like feature suggesting that external gravitational perturbations by the massive satellites and planets interfere with the Kozai regime due to solar gravitational effects. This hypothesis is confirmed by the fact that in a few cases the arc-like feature is present only in Model 2 where Titan and Iapetus are included.

Anticorrelation of inclination and eccentricity (see also Fig. 13) in the dynamical evolution of (clockwise from top left-hand side) Ijiraq, Paaliaq, Bestla (S/2004 S18) and Narvi. These features, observed in both Models 1 and 2, might imply that these satellites were prevented from entering the Kozai cycle with the Sun by the combined gravitational perturbations of the other giant planets. It is noteworthy that the only satellites in a full Kozai regime are the two innermost ones, for which the external perturbations are less effective.

Figure 12

Anticorrelation of inclination and eccentricity (see also Fig. 13) in the dynamical evolution of (clockwise from top left-hand side) Ijiraq, Paaliaq, Bestla (S/2004 S18) and Narvi. These features, observed in both Models 1 and 2, might imply that these satellites were prevented from entering the Kozai cycle with the Sun by the combined gravitational perturbations of the other giant planets. It is noteworthy that the only satellites in a full Kozai regime are the two innermost ones, for which the external perturbations are less effective.

Phase displacement of inclination and eccentricity in the dynamical evolution of (clockwise from top left-hand side) Ijiraq, Paaliaq, Hyrrokkin (S/2004 S19) and Narvi. Both eccentricity and inclination have been rescaled and shifted to show variations of the same order of magnitude. The plots referring to prograde satellites show the relation between eccentricity and inclination, those referring to retrograde satellites show the relation between eccentricity and the inclination supplemental angle (i.e. 180°−i). Hyrrokkin's evolution show alternating phases of correlation and anticorrelation between eccentricity and inclination: anticorrelation phases last longer than correlation phases. The evolutions of Hyrrokkin's inclination and eccentricity are moderately coupled (see Fig. 27 for details) and this coupling interferes with solar perturbations causing the Kozai regime.

Figure 13

Phase displacement of inclination and eccentricity in the dynamical evolution of (clockwise from top left-hand side) Ijiraq, Paaliaq, Hyrrokkin (S/2004 S19) and Narvi. Both eccentricity and inclination have been rescaled and shifted to show variations of the same order of magnitude. The plots referring to prograde satellites show the relation between eccentricity and inclination, those referring to retrograde satellites show the relation between eccentricity and the inclination supplemental angle (i.e. 180°−i). Hyrrokkin's evolution show alternating phases of correlation and anticorrelation between eccentricity and inclination: anticorrelation phases last longer than correlation phases. The evolutions of Hyrrokkin's inclination and eccentricity are moderately coupled (see Fig. 27 for details) and this coupling interferes with solar perturbations causing the Kozai regime.

Additional differences in the chaotic evolution of some satellites can be ascribed to the presence of Titan and Iapetus. According to our simulations 23 satellites show a chaotic behaviour. Aside from the previously mentioned Kiviuq, Ijiraq, Paaliaq and Siarnaq, the list is completed by Albiorix, Erriapo, Tarvos, Narvi, Thrymr, Ymir, Mundilfari, Skathi, S/2004 S10, Bebhionn (S/2004 S11), S/2004 S12, S/2004 S13, S/2004 S18, S/2006 S2, S/2006 S3, S/2004 S4, S/2006 S5, S/2006 S7 and S/2006 S8 (see Fig. 14). Some of them are chaotic only in one of the two dynamical models (e.g. Figs 15–17) while others in both the models (see Fig. 18 as an example). The most appealing interpretation is that Titan and Iapetus perturb the dynamical evolution of the irregular satellites either by stabilizing otherwise chaotic orbits (Fig. 16) or causing chaos (Fig. 17). Another interesting feature is that in most cases the chaotic features appeared in the secular evolution of the semimajor axis with eccentricity and inclination exhibiting a regular, quasi-periodic evolution. As a consequence, the present structure of the outer Saturnian system could be not representative of the primordial one. We will further explore this issue in Section 5.

Chaotic and resonant irregular satellites of Saturn's system. The square symbols identify the irregular satellites showing resonant or chaotic features in their dynamical evolution. The data are presented in the a–e (left-hand plot) and a–i (right-hand plot) planes.

Figure 14

Chaotic and resonant irregular satellites of Saturn's system. The square symbols identify the irregular satellites showing resonant or chaotic features in their dynamical evolution. The data are presented in the a_–_e (left-hand plot) and a_–_i (right-hand plot) planes.

Secular evolution of the inclination of S/2006 S7 in Model 2 (left-hand panel) and Model 1 computed with strict (64-bit) double precision (right-hand panel). While the first case show a regular behaviour, the last one presents an evident chaotic jump in inclination. This jump may be related to the entry of the satellite into a chaotic region. This event does not occur in the Model 1 computed with standard precision, probably due to the different numerical set-up.

Figure 15

Secular evolution of the inclination of S/2006 S7 in Model 2 (left-hand panel) and Model 1 computed with strict (64-bit) double precision (right-hand panel). While the first case show a regular behaviour, the last one presents an evident chaotic jump in inclination. This jump may be related to the entry of the satellite into a chaotic region. This event does not occur in the Model 1 computed with standard precision, probably due to the different numerical set-up.

Secular evolution of the semimajor axis of Mundilfari. From top left-hand side, clockwise direction, the plots show the outcome of Model 2 – HJS algorithm, Model 2 – RADAU algorithm, Model 1 – HJS with standard precision and Model 1 – HJS with strict (64-bit) double precision. The simulations based on Model 2 show a regular behaviour on the whole time-span, independently of the employed algorithm. Both cases based on Model 1 show instead an irregular evolution diagnostic of chaotic behaviour. Titan and Iapetus act like orbital stabilizers for Mundilfari.

Figure 16

Secular evolution of the semimajor axis of Mundilfari. From top left-hand side, clockwise direction, the plots show the outcome of Model 2 – HJS algorithm, Model 2 – RADAU algorithm, Model 1 – HJS with standard precision and Model 1 – HJS with strict (64-bit) double precision. The simulations based on Model 2 show a regular behaviour on the whole time-span, independently of the employed algorithm. Both cases based on Model 1 show instead an irregular evolution diagnostic of chaotic behaviour. Titan and Iapetus act like orbital stabilizers for Mundilfari.

Secular evolution of the semimajor axis of Erriapo. From top left-hand side, clockwise direction, the plots show the outcome of Model 2 – HJS algorithm, Model 2 – RADAU algorithm, Model 1 – HJS with standard precision and Model 1 – HJS with strict (64-bit) double precision. Contrary to the case showed in Fig. 16, Titan and Iapetus cause an irregular evolution of Erriapo. In Model 1 the satellite has a regular behaviour.

Figure 17

Secular evolution of the semimajor axis of Erriapo. From top left-hand side, clockwise direction, the plots show the outcome of Model 2 – HJS algorithm, Model 2 – RADAU algorithm, Model 1 – HJS with standard precision and Model 1 – HJS with strict (64-bit) double precision. Contrary to the case showed in Fig. 16, Titan and Iapetus cause an irregular evolution of Erriapo. In Model 1 the satellite has a regular behaviour.

Secular evolution of the eccentricity of S/2006 S5 in Model 2 computed with HJS (left-hand panel) and Model 1 computed using standard precision (right-hand panel). The chaotic nature of the satellite orbital motion can be easily inferred by the secular behaviour of its eccentricity. Similar conclusions can be drawn from the inspection of semimajor axis and inclination.

Figure 18

Secular evolution of the eccentricity of S/2006 S5 in Model 2 computed with HJS (left-hand panel) and Model 1 computed using standard precision (right-hand panel). The chaotic nature of the satellite orbital motion can be easily inferred by the secular behaviour of its eccentricity. Similar conclusions can be drawn from the inspection of semimajor axis and inclination.

The presence of chaos in the dynamical evolution of the irregular satellites appears to be the major driver of the differences between the outcome of Model 1 (80 and 64 bit) and Model 2 (HJS and mercury RADAU algorithms). The chaotic nature of the trajectories is probably also at the origin of the differences (e.g. the alternation between resonant and non-resonant phases in Kiviuq's evolution) we noted between our simulations based on Model 1 and those published in the literature by Carruba et al. (2002), Nesvorny et al. (2003) and Cuk & Burns (2004), which were based on a similar dynamical scheme. The interplay between the inclusion of Titan's and Iapetus' gravitational perturbations and the presence of chaos can finally be invoked to explain the differences between the results obtained with Models 1 and 2.

To explore in more detail the effects of Jupiter, Titan and Iapetus on the dynamics of irregular satellites we performed two additional sets of simulations where we sampled with a large number of test particles the phase space populated by the satellites. We distributed 100 test particles in between 10.47 × 106 and 25.43 × 106 km on initially circular orbits and integrated their trajectories for 106 yr. 10 simulations have been performed each with a different value of the initial inclination of the particles. We considered _i_= 0°, 15°, 30°, 45°, 60° for prograde orbits, and _i_= 120°, 135°, 150°, 165°, 180° for retrograde orbits. To compute the orbital evolution we used the HJS _N_-body code and the dynamical structure of Models 1 and 2. The initial conditions for the massive perturbing bodies (Sun, giant planets and major satellites) and the time-steps were the same used in our previous simulations.

To analyse the output data we looked at the mean elements and the η parameters. In Figs 19 and 20 we show some examples of the outcome where the influence of Titan and Iapetus is manifest while comparing the outcome of Model 1 versus Model 2. For particles in the inclination range [30°–60°] and [120°–150°] the values of the ηe are significantly different. In addition, we verified the leading role of Jupiter in perturbing the system. We computed the test particle orbits switching off the gravitational attraction of the planet. The dynamically perturbed regions shown Fig. 21 disappear or are reduced (Titan and Iapetus are still effective) when Jupiter's pull is switched off.

Evolution of the η parameters with semimajor axis for prograde test particles in Model 1 (left-hand column) and Model 2 (right-hand column). From top to bottom, the plots show the values of ηe, ηi, ηa, respectively. Values of η near 0 indicate quasi periodic motion, higher values indicate increasingly chaotic or resonant behaviour. Distances are expressed in 106 km.

Figure 19

Evolution of the η parameters with semimajor axis for prograde test particles in Model 1 (left-hand column) and Model 2 (right-hand column). From top to bottom, the plots show the values of ηe, η_i_, η_a_, respectively. Values of η near 0 indicate quasi periodic motion, higher values indicate increasingly chaotic or resonant behaviour. Distances are expressed in 106 km.

Evolution of the η parameters with semimajor axis for retrograde test particles in Model 1 (left-hand column) and Model 2 (right-hand column). From top to bottom, the plots show the values of ηe, ηi, ηa, respectively. Values of η near 0 indicate quasi periodic motion, higher values indicate increasingly chaotic or resonant behaviour. Distances are expressed in 106 km.

Figure 20

Evolution of the η parameters with semimajor axis for retrograde test particles in Model 1 (left-hand column) and Model 2 (right-hand column). From top to bottom, the plots show the values of ηe, η_i_, η_a_, respectively. Values of η near 0 indicate quasi periodic motion, higher values indicate increasingly chaotic or resonant behaviour. Distances are expressed in 106 km.

Location of the irregular satellites in the a–e (upper plot) and a–i (lower plot) planes respect to the unstable regions identified by the simulations computing the evolution of regularly sampled test particles. The unstable regions, perturbed by Jupiter, are marked by dashed and dotted lines. Dotted lines indicate regions which are excited for all values of inclination. Dashed lines are related to those regions which get perturbed for high absolute inclination values (i.e. i∼ 150°).

Figure 21

Location of the irregular satellites in the a_–_e (upper plot) and a_–_i (lower plot) planes respect to the unstable regions identified by the simulations computing the evolution of regularly sampled test particles. The unstable regions, perturbed by Jupiter, are marked by dashed and dotted lines. Dotted lines indicate regions which are excited for all values of inclination. Dashed lines are related to those regions which get perturbed for high absolute inclination values (i.e. _i_∼ 150°).

By inspecting the mean orbital elements of irregular satellites in Fig. 21 one may note that those orbiting close to the perturbed regions have a ‘stratification’ in eccentricity typical of a resonant population. It is possible that the dynamical features we observe today are not primordial but a consequence of a secular interaction with Jupiter. One of these features is the gap between about 20.94 × 106 and 22.44 × 106 km in the radial distribution of the mean elements of the irregular satellites. This gap was particularly evident till 2006 June, since no known satellite populated that region. With the recent announcement of nine additional members of Saturn's irregular satellites, two objects are now known to cross this region: S/2006 S2 and S/2006 S3. Both these bodies have their mean semimajor axis falling near to perturbed regions. According to our simulations, the semimajor axis of both satellites show irregular jumps (see Fig. 22) typical of chaotic behaviours. S/2006 S2 has phases of regular evolution alternating with periods of chaos lasting from a few ×104 up to about 105 yr (see Fig. 22, left-hand plot). In the case of S/2006 S3, we observed a few large jumps in semimajor axis over the time-span covered by our simulations (see Fig. 22, right-hand plot). The evolutions of both the eccentricity and the inclination of the satellites appear more regular, with long period modulations and beats.

Secular evolution of the semimajor axis of S/2006 S2 Kari (detail, left-hand graph) and S/2006 S3 (detail, right-hand graph) in Model 2. While on different time-scales and with different intensities, both the satellites have a chaotic evolution possibly due to Jupiter's perturbations through the Great Inequality with Saturn.

Figure 22

Secular evolution of the semimajor axis of S/2006 S2 Kari (detail, left-hand graph) and S/2006 S3 (detail, right-hand graph) in Model 2. While on different time-scales and with different intensities, both the satellites have a chaotic evolution possibly due to Jupiter's perturbations through the Great Inequality with Saturn.

We applied frequency analysis in order to identify the source of the perturbations leading to the chaotic evolution of the two satellites. By analysing the h and k non-singular variables defined with respect to the planet, we found among the various frequencies two with period of 900 and 1800 yr, respectively. These values are close to the Great Inequality period (∼883 yr) of the near-resonance between Jupiter and Saturn. This is possibly one source of the chaotic behaviour of the satellites. In addition, inspection of the upper plot in Fig. 20 shows that the η_a_ parameters of the test particles populating this radial region increase when Titan and Iapetus are included. By comparing the frequencies of motion of the two irregular satellites with those of Titan and Iapetus we find an additional commensurability. The frequencies 7.1 × 10−4 and 7.4 × 10−4yr−1 that are present in the power spectrum of S/2006 S2 and S/2006 S6, respectively, are about twice the frequency 3.4 × 10−4yr−1 in the power spectrum of Titan. The cumulative effects of the Great Inequality and of Titan and Iapetus lead to the large values of the η parameters in Figs 4 and 20. The chaotic evolution of the two satellites does not lead to destabilization in the time-span of our integration (108 yr), however longer simulations are needed to test the long-term stability. It is possible that the irregular behaviour takes the two satellites into other chaotic regions ultimately leading to their expulsion from the system. It is also possible that, during their chaotic wandering, they cross the paths of more massive satellites and are collisionally removed. This could have been the fate of possible other satellites which originally populated the region encompassed between 20.94 × 106 and 22.44 × 106 km.

4 EVALUATION OF THE COLLISIONAL EVOLUTION

To understand the present orbital structure of Saturn's satellite system and how it evolved from the primordial one we have to investigate the collisional evolution within the system. Impacts between satellites, in fact, may have removed smaller bodies and fragmented the larger ones. Minor bodies in heliocentric orbits like comets and Centaurs may have also contributed to the system shaping by colliding with the satellites as addressed by Nesvorny, Beaugé & Dones (2004). At present, however, such events are not frequent because of the reduced flux of minor bodies and the small sizes of irregular satellites (Zahnle et al. 2003; Nesvorny et al. 2004).

The last detailed evaluation of the collisional probabilities between the irregular satellites around the giant planets was the one performed by Nesvorny et al. (2003), which showed that the probabilities of collisions between pairs of satellites were rather low and practically unimportant. The computed average collisional lifetimes were tens to hundreds of times longer than Solar system's lifetime. The only notable exceptions were those pairs involving one of the big irregular satellites (e.g. Himalia, Phoebe, etc.) in the systems. In the Saturn system, Phoebe is between one and two orders of magnitudes more active than any other satellite. The authors computed a cumulative number of collisions between 6 and 7 in 4.5 × 109 yr (Nesvorny et al. 2003).

However, at the time of the publication of the work by Nesvorny et al. (2003) only 13 of the 35 currently known irregular satellites of Saturn had been discovered. We extend here their analysis taking advantage of the larger number of known bodies and of the improved orbital data. Using the mean orbital elements computed with Model 2 we have estimated the collisional probabilities using the approach described by Kessler (1981). Since in the scenario described by the Nice model the LHB represents a lower limit for the capture epoch of the irregular satellites (Gomes et al. 2005; Tsiganis et al. 2005), we considered a time interval for the collisional evolution of 3.5 × 109 yr, making the conservative assumption that the LHB took place after about 109 yr since the beginning of Solar system formation. Since the collisional probability depends linearly on time, our results can be immediately extended to longer time-scales.

The results of our computations (see Fig. 23) confirm that the only pairs of satellites with high probability of collisions involve Phoebe. The satellite pairs with the highest number of collisions over the considered time-span are Phoebe–Kiviuq (0.7126), Phoebe–Ijiraq (0.7099) and Phoebe–Thrymr (0.6436). The remaining pairs involving Phoebe have a number of collisions between 0.1 and 0.35 (see Fig. 23, line/column 3), with the highest values associated with the prograde satellites Paaliaq, Siarnaq, Tarvos, Albiorix, Erriapo and Bebhionn (S/2004 S11). All the other satellite pairs, due also to their small radii, have negligible (<10−2) collisional probabilities. The predicted total number of collisions in Saturn's irregular satellite system, obtained by summing over all the possible pairs, is of about 12 collisions over the considered time-span. Half of these impacts involve Phoebe. This is probably at the cause of the gap centred at Phoebe and radially extending from about 11.22 × 106 to about 14.96 × 106 km from Saturn (i.e. between Ijiraq's and Paaliaq's mean orbits) for both prograde and retrograde satellites. To further confirm this hypothesis we evaluated the impact probability for a cloud of test particles populating this region.

Colour map of the collisional probabilities between the 35 presently known irregular satellites of Saturn. For each pair of satellites the probability of collision is evaluated in terms of the number of impacts during the system lifetime (3.5 × 109 yr). The colour coding used to illustrate the impact probabilities ranges from yellow (highest values) to purple (lowest values). Black is used when the orbits do not intersect. The only pairs of satellites for which the impact probability is not negligible always include Phoebe as it can be argued by the fact that only column/line 3 (Phoebe) is characterized by bright colours.

Figure 23

Colour map of the collisional probabilities between the 35 presently known irregular satellites of Saturn. For each pair of satellites the probability of collision is evaluated in terms of the number of impacts during the system lifetime (3.5 × 109 yr). The colour coding used to illustrate the impact probabilities ranges from yellow (highest values) to purple (lowest values). Black is used when the orbits do not intersect. The only pairs of satellites for which the impact probability is not negligible always include Phoebe as it can be argued by the fact that only column/line 3 (Phoebe) is characterized by bright colours.

We filled with test bodies a volume in the phase space defined in the following way:

The sampling step size were δ_a_= 4.488 × 104km, δ_e_= 0.1 and δ_i_= 2°, for a total of 2860 prograde orbits and 2860 retrograde ones. A radius of 3 km has been adopted for the test particles to compute the cross-sections. The results are presented in the colour maps of Fig. 24 for the prograde cases and 25 for the retrograde ones. Our results show that lowest collision probabilities (of the order of 0.1 collisions) are related to orbits with high values of eccentricity and inclination (e > 0.5 and i > 25° or i < 155°) as illustrated in the top right-hand quadrants of each plot of Fig. 24 and the bottom right-hand quadrants of each plot of Fig. 25. Prograde low-inclination orbits with 0° < _i_≤ 10° and e < 0.5 have more than four collisions over the given time-span (see the bottom left-hand quadrants of each plot of Fig. 24). Retrograde orbits with 174° < i < 180° have at least one collision for any value of eccentricity (see the top part of each plot of Fig. 25). The number of collisions with Phoebe is generally higher for the prograde test particles (one to five collisions) compared to the retrograde orbits (0.5–1). These results support the existence of a Phoebe gap caused by collisional erosion. It cannot be due to dynamical clearing mechanisms since, according to an additional set of simulations, we show that the region around the Phoebe gap is stable. 100 test particles are uniformly distributed across the Phoebe gap with the following criteria:

Colour maps of the collisional probabilities between Phoebe and a cloud of prograde, 3 km radius test particles filling the region of Phoebe gap at six values of semimajor axis: 10.47 × 106, 11.37 × 106, 12.27 × 106, 13.16 × 106, 14.06 × 106, 14.96 × 106 km. The probabilities of collision are evaluated in terms of the number of impacts during the system lifetime (3.5 × 109 yr) and are represented through a colour code going from yellow (highest values) to purple (lowest values) with black representing the case in which the orbits do not intersect. Note that due to the high values of collisional probabilities reached in the prograde cases (up to 30 collisions in the time-span considered) we had to limit the colour range to a maximum of 10 collisions to maintain the scale readability.

Figure 24

Colour maps of the collisional probabilities between Phoebe and a cloud of prograde, 3 km radius test particles filling the region of Phoebe gap at six values of semimajor axis: 10.47 × 106, 11.37 × 106, 12.27 × 106, 13.16 × 106, 14.06 × 106, 14.96 × 106 km. The probabilities of collision are evaluated in terms of the number of impacts during the system lifetime (3.5 × 109 yr) and are represented through a colour code going from yellow (highest values) to purple (lowest values) with black representing the case in which the orbits do not intersect. Note that due to the high values of collisional probabilities reached in the prograde cases (up to 30 collisions in the time-span considered) we had to limit the colour range to a maximum of 10 collisions to maintain the scale readability.

Colour maps of the collisional probabilities between Phoebe and a cloud of retrograde, 3 km radius test particles filling the region of Phoebe gap at six values of semimajor axis: 10.47 × 106, 11.37 × 106, 12.27 × 106, 13.16 × 106, 14.06 × 106, 14.96 × 106 km. The probabilities of collision are evaluated in terms of the number of impacts during the system lifetime (3.5 × 109 yr) and are represented through a colour code going from yellow (highest values) to purple (lowest values) with black representing the case in which the orbits do not intersect.

Figure 25

Colour maps of the collisional probabilities between Phoebe and a cloud of retrograde, 3 km radius test particles filling the region of Phoebe gap at six values of semimajor axis: 10.47 × 106, 11.37 × 106, 12.27 × 106, 13.16 × 106, 14.06 × 106, 14.96 × 106 km. The probabilities of collision are evaluated in terms of the number of impacts during the system lifetime (3.5 × 109 yr) and are represented through a colour code going from yellow (highest values) to purple (lowest values) with black representing the case in which the orbits do not intersect.

The orbits have been integrated for 106 yr and for each of them a value of η has been computed (see Fig. 26). The η values indicate that the perturbations of Titan and Iapetus are negligible and that test particles positioned in regions of high collision probability (i.e. for _i_≤ 10° and _i_≥ 174°) are not dynamically unstable.

The collisional origin of the Phoebe's gap is also confirmed by observational data showing that the irregular satellites moving closer to Phoebe are those located in regions of the phase space where the impact probability with Phoebe is lower. In addition, the images of Phoebe taken by ISS onboard the Cassini spacecraft revealed a strongly cratered surface, with a continuous crater size distribution ranging from about 50 m, a lower value imposed by the resolution of ISS images, to an upper limit of about 100 km comparable to the dimension of the satellite. A highly cratered surface was predicted by Nesvorny et al. (2003), who also suggested that the vast majority of Phoebe's craters should be due to either impacts with other irregular satellites (mainly prograde ones) or the result of a past intense flux of bodies crossing Saturn's orbit. Comets give a negligible contribution having a frequency of collision with Phoebe of about 1 impact every 109 yr (Zahnle et al. 2003). Our results confirm those of Nesvorny et al. (2003) showing that indeed Phoebe had a major role in shaping the structure of Saturn's irregular satellites. The existence of a primordial now extinct population of small irregular satellites or collisional shards between 11.22 × 106 and 14.96 × 106 km from Saturn could explain in a natural way the abundance of craters on Phoebe surface.

Phoebe's sweeping effect appears to have another major consequence, related to the existence of Phoebe gap: it argues against the hypothesis of a Phoebe family. The existence of Phoebe's family has been a controversial subject since its proposition by Gladman et al. (2001). Its existence was guessed on the close values of inclination of Phoebe and other retrograde satellites. The dynamical inconsistency of this criterion has been pointed out by Nesvorny et al. (2003), who showed that the velocity dispersion required to relate the retrograde satellites to Phoebe would be too high to be accounted as realistic in the context of the actual knowledge of fragmentation and disruption processes. Our results suggest also that if a break-up event involved Phoebe, the fragments would have been ejected within the Phoebe gap for realistic ejection velocities. As a consequence, they would have been removed by impacting on Phoebe.

5 EXISTENCE OF COLLISIONAL FAMILIES

The identification of possible dynamical families between the irregular satellites of the giant planets had been a common task to all the studies performed on the subject. The existence of collisional families could in principle be explained by invoking the effects of impacts between pairs of satellites and between satellites and bodies on heliocentric orbits. The impact rate between satellite pairs is low even on time-scales of the order of the Solar system age, with the sole exception of impacts amongst the most massive irregular satellites. The gravitational interactions between the giant planets and the planetesimals in the early Solar system may have pushed some of them into planet-crossing orbits. This process is still active at the present time and Centaurs may cross the Hill's sphere of the planets. However, Zahnle et al. (2003) showed that the present flux of bodies, combined with the small size of irregular satellites, is unable to supply an adequate impact rate. If collisions between planetesimals and satellites are responsible for the formation of families, these events should date back sometime between the formation of the giant planets and the LHB.

The possible existence of dynamical families in the Saturn satellite system has been explored by using different approaches. Photometric comparisons has been exploited by Grav et al. (2003), Grav & Holman (2004), Buratti, Hicks & Davies (2005) and were limited to a few bright objects. Dynamical methods have been used by Grav et al. (2003), Nesvorny et al. (2003), Grav & Holman (2004) but on the limited sample (about one-third of the presently known population) of irregular satellites available at that time. These methods aimed to identify those satellites which could have originated from a common parent body following one or more break-up events. The identification was based on the evaluation through Gauss equations of the dispersion in orbital element space due to the collisional ejection velocities. In this paper we apply the hierarchical clustering method (hereafter HCM) described in Zappalá et al. (1990, 1994) to the irregular satellites of Saturn. HCM is a cluster-detection algorithm which looks for groupings within a population of minor bodies with small nearest neighbour distances in orbital element space. These distances are translated into differences in orbital velocities via Gauss equations and the membership to a cluster or family is defined by giving a limiting velocity difference (cut-off). Nesvorny et al. (2003) adopted a cut-off velocity value of 100 m s−1 according to hydrocode models (Benz & Aspaugh 1999). Here we prefer to relax this value to 200 m s−1 considering the possible range of variability of the mean orbital elements because of dynamical effects. The results we obtained are summarized in Table 3 and interpreted as in the following.

Table 3

Dynamical clustering of Saturn's irregular satellites. The term cluster used in the table refers to the merging of previously reported families.

Family name Family members Dispersion (m s−1)
Prograde satellites
Kiviuq Kiviuq, Ijiraq 102
Albiorix Albiorix, Erriapo, Tarvos, S/2004 S11 129
Siarnaq Paaliaq, Siarnaq 314
Siarnaq + Albiorix Albiorix &Siarnaq families 443
Prograde All prograde satellites 532
Retrograde satellites
S/2004 S15 S/2004 S15, S/2006 S1 114
Mundilfari Mundilfari, S/2004 S13, S/2004 S17 116
S/2006 S2 S/2006 S2, S/2006 S3 132
S/2004 S10 S/2004 S10, S/2004 S12, S/2004 S14 144
S/2004 S8 S/2004 S8, S/2006 S5, S/2004 S16 168
Narvi Narvi, S/2004 S18 200
Ymir Ymir, S/2006 S2 family, 259
S/2004 S7, S/2006 S7
Cluster A Mundilfari family, S/2006 S6, 150
S/2004 S10 family
Cluster B Cluster A, S/2004 S15, S/2006 S1 202
Cluster C Cluster B, S/2004 S8 family, 240
S/2004 S9, S/2006 S4
Cluster D Cluster C, Ymir family 267
Retrograde − Phoebe All retrograde satellites except Phoebe 315
Retrograde All retrograde satellites 658
Family name Family members Dispersion (m s−1)
Prograde satellites
Kiviuq Kiviuq, Ijiraq 102
Albiorix Albiorix, Erriapo, Tarvos, S/2004 S11 129
Siarnaq Paaliaq, Siarnaq 314
Siarnaq + Albiorix Albiorix &Siarnaq families 443
Prograde All prograde satellites 532
Retrograde satellites
S/2004 S15 S/2004 S15, S/2006 S1 114
Mundilfari Mundilfari, S/2004 S13, S/2004 S17 116
S/2006 S2 S/2006 S2, S/2006 S3 132
S/2004 S10 S/2004 S10, S/2004 S12, S/2004 S14 144
S/2004 S8 S/2004 S8, S/2006 S5, S/2004 S16 168
Narvi Narvi, S/2004 S18 200
Ymir Ymir, S/2006 S2 family, 259
S/2004 S7, S/2006 S7
Cluster A Mundilfari family, S/2006 S6, 150
S/2004 S10 family
Cluster B Cluster A, S/2004 S15, S/2006 S1 202
Cluster C Cluster B, S/2004 S8 family, 240
S/2004 S9, S/2006 S4
Cluster D Cluster C, Ymir family 267
Retrograde − Phoebe All retrograde satellites except Phoebe 315
Retrograde All retrograde satellites 658

Table 3

Dynamical clustering of Saturn's irregular satellites. The term cluster used in the table refers to the merging of previously reported families.

Family name Family members Dispersion (m s−1)
Prograde satellites
Kiviuq Kiviuq, Ijiraq 102
Albiorix Albiorix, Erriapo, Tarvos, S/2004 S11 129
Siarnaq Paaliaq, Siarnaq 314
Siarnaq + Albiorix Albiorix &Siarnaq families 443
Prograde All prograde satellites 532
Retrograde satellites
S/2004 S15 S/2004 S15, S/2006 S1 114
Mundilfari Mundilfari, S/2004 S13, S/2004 S17 116
S/2006 S2 S/2006 S2, S/2006 S3 132
S/2004 S10 S/2004 S10, S/2004 S12, S/2004 S14 144
S/2004 S8 S/2004 S8, S/2006 S5, S/2004 S16 168
Narvi Narvi, S/2004 S18 200
Ymir Ymir, S/2006 S2 family, 259
S/2004 S7, S/2006 S7
Cluster A Mundilfari family, S/2006 S6, 150
S/2004 S10 family
Cluster B Cluster A, S/2004 S15, S/2006 S1 202
Cluster C Cluster B, S/2004 S8 family, 240
S/2004 S9, S/2006 S4
Cluster D Cluster C, Ymir family 267
Retrograde − Phoebe All retrograde satellites except Phoebe 315
Retrograde All retrograde satellites 658
Family name Family members Dispersion (m s−1)
Prograde satellites
Kiviuq Kiviuq, Ijiraq 102
Albiorix Albiorix, Erriapo, Tarvos, S/2004 S11 129
Siarnaq Paaliaq, Siarnaq 314
Siarnaq + Albiorix Albiorix &Siarnaq families 443
Prograde All prograde satellites 532
Retrograde satellites
S/2004 S15 S/2004 S15, S/2006 S1 114
Mundilfari Mundilfari, S/2004 S13, S/2004 S17 116
S/2006 S2 S/2006 S2, S/2006 S3 132
S/2004 S10 S/2004 S10, S/2004 S12, S/2004 S14 144
S/2004 S8 S/2004 S8, S/2006 S5, S/2004 S16 168
Narvi Narvi, S/2004 S18 200
Ymir Ymir, S/2006 S2 family, 259
S/2004 S7, S/2006 S7
Cluster A Mundilfari family, S/2006 S6, 150
S/2004 S10 family
Cluster B Cluster A, S/2004 S15, S/2006 S1 202
Cluster C Cluster B, S/2004 S8 family, 240
S/2004 S9, S/2006 S4
Cluster D Cluster C, Ymir family 267
Retrograde − Phoebe All retrograde satellites except Phoebe 315
Retrograde All retrograde satellites 658

By inspecting our data, we conclude that, as already argued by Nesvorny et al. (2003), the velocity dispersion of prograde and retrograde satellites (about 500 m s−1 for progrades and more than 600 m s−1 for retrogrades) makes extremely implausible that each of the two groups originated from a single parent body. In addition, the classification in dynamical groups based on the values of the orbital inclination originally proposed by Gladman et al. (2001) and reported by other authors (see Sheppard 2006 and references within) is probably misleading. Prograde satellites like Kiviuq, Ijiraq, Siarnaq and Paaliaq do share the same inclination but, as a group, they have a velocity dispersion of over 450 m s−1, hardly deriving from the break-up of a single parent body. The enlarged population of retrograde satellites we have analysed show that the clustering around a single inclination (Gladman et al. 2001) and their association to Phoebe (Gladman et al. 2001) is not an indication of a common origin. The required velocity dispersion is in fact about 650 m s−1.

We found two potential dynamical families between the prograde satellites: the couple Kiviuq–Ijiraq and what we term as Albiorix family, composed of Albiorix, Erriapo, Tarvos and S/2004 S11. The analysis of the retrograde satellites is more complex. There are three possible groups each composed of three satellites and two others by two satellites, all characterized by acceptable values of the velocity dispersion (between 100 and 170 m s−1). A sixth possible group satisfying our acceptance criterion is composed of Narvi and S/2004 S18, but its interpretation is quite tricky. This group shows a high velocity dispersion, at the upper limit of our range, but the dynamical evolution of both satellites is uncertain on a time-scale of 109 yr. The orbits of both bodies have the most extreme values of inclination among all retrograde satellites. Our numerical experiments with test particles showed that for such bodies the eccentricity is strongly coupled to the inclination (see Fig. 27). In our simulations, initially circular orbits became highly eccentric in less than 106 yr. It is possible that Narvi and S/2004 S18 had similar orbits in the past which later diverged due to the inclination-eccentricity link.

Mean eccentricities of the retrograde test particles computed with Model 2 for different values of inclination. The test particles were on initially circular orbits. Most of the irregular satellites on retrograde orbits lie in the range of inclination where the increase in eccentricity is limited on a 106 yr time-scale. The Narvi family is a different case. Narvi spends a significant fraction of its dynamical evolution in the excited region while S/2004 S18 is inside it most of the time. As it can be argued from the plot, the subsequent divergent evolutions could be the source of the high velocity dispersion within the group, assuming that the two satellites originated from a common parent body.

Figure 27

Mean eccentricities of the retrograde test particles computed with Model 2 for different values of inclination. The test particles were on initially circular orbits. Most of the irregular satellites on retrograde orbits lie in the range of inclination where the increase in eccentricity is limited on a 106 yr time-scale. The Narvi family is a different case. Narvi spends a significant fraction of its dynamical evolution in the excited region while S/2004 S18 is inside it most of the time. As it can be argued from the plot, the subsequent divergent evolutions could be the source of the high velocity dispersion within the group, assuming that the two satellites originated from a common parent body.

Some of our candidate families merge at higher values of the velocity dispersion forming bigger groups we called clusters. The most relevant one is that termed as cluster A in Table 3. It is made of two three-body families and an individual satellite and it is defined at a velocity cut-off of 150 m s−1. At a velocity cut-off of 202 m s−1 cluster A merges with the two-body family related to S/2004 S15 forming cluster B. Confirming these dynamical groups by comparing their colour indices is a difficult task because of the limited amount of data available in the literature. The only spectrophotometric data concerning Saturn's retrograde satellites are those of Phoebe and Ymir, which, according to our analysis, are separated by a velocity dispersion of more than 600 m s−1. Phoebe appears to have colours not compatible with any other irregular satellite of the system, supporting our claim that Phoebe is not related to the rest of Saturn's present population of irregular satellites.

The situation looks more favourable for prograde families: the colours of three members of the possible Albiorix family are available, with two sets of data for Albiorix itself. The colour data are reported in Table 4 with the corresponding 1σ errors. These data seem to be compatible with the hypothesis of a common origin of the group at a 3σ level.

Table 4

Colour indices of three candidate members of the Albiorix family: for each colour index the 1σ error range is reported (Grav et al. 2003).

Satellite B_−_V V_−_R V_−_I
Tarvos 0.77 ± 0.12 0.57 ± 0.09 0.88 ± 0.11
Albiorix 0.89 ± 0.07 0.50 ± 0.05 0.91 ± 0.05
0.98 ± 0.07 0.47 ± 0.04 0.92 ± 0.04
Erriapo 0.83 ± 0.09 0.49 ± 0.06 0.61 ± 0.12
Satellite B_−_V V_−_R V_−_I
Tarvos 0.77 ± 0.12 0.57 ± 0.09 0.88 ± 0.11
Albiorix 0.89 ± 0.07 0.50 ± 0.05 0.91 ± 0.05
0.98 ± 0.07 0.47 ± 0.04 0.92 ± 0.04
Erriapo 0.83 ± 0.09 0.49 ± 0.06 0.61 ± 0.12

Table 4

Colour indices of three candidate members of the Albiorix family: for each colour index the 1σ error range is reported (Grav et al. 2003).

Satellite B_−_V V_−_R V_−_I
Tarvos 0.77 ± 0.12 0.57 ± 0.09 0.88 ± 0.11
Albiorix 0.89 ± 0.07 0.50 ± 0.05 0.91 ± 0.05
0.98 ± 0.07 0.47 ± 0.04 0.92 ± 0.04
Erriapo 0.83 ± 0.09 0.49 ± 0.06 0.61 ± 0.12
Satellite B_−_V V_−_R V_−_I
Tarvos 0.77 ± 0.12 0.57 ± 0.09 0.88 ± 0.11
Albiorix 0.89 ± 0.07 0.50 ± 0.05 0.91 ± 0.05
0.98 ± 0.07 0.47 ± 0.04 0.92 ± 0.04
Erriapo 0.83 ± 0.09 0.49 ± 0.06 0.61 ± 0.12

6 CONCLUSIONS

The aim of this paper was to investigate the dynamical and collisional nature of the Saturn system of irregular satellites. We analysed the secular dynamical evolution of the satellites on a time-span of 108 yr, computing their mean orbital elements. We found evidences of resonant and chaotic behaviours in the motion of about two-thirds of the satellites. We also explored the dynamical features of the phase space close to them and verified the influence of Titan and Iapetus as well as of the Great Inequality in shaping the satellite system.

In this paper we have also verified that the present orbital structure is long-lived against collisions but we found indications that in the past a more intense collisional activity could have taken place, mainly due to Phoebe's sweeping effect. By considering the impact rates of the present population, we deduce that the original population could have been at least ∼30 per cent more abundant. We also suggest that the absence of prograde and low inclination (∼170°–180°) retrograde irregular satellites in the region encompassed between 10.47 × 106 and 14.96 × 106 km is a by-product of the sweeping effect of Phoebe. It is less clear if the absence of retrograde satellites with lower inclinations (i.e. high-velocity ejecta from Phoebe or other captured bodies) in the same radial region could be due to the same reason or if it is a primordial structure related to the capture mechanism.

We also found evidences of dynamical groupings among prograde and retrograde satellites, even if their interpretation in terms of families is not straightforward due to the effects of chaotic and resonant evolution. By applying the HCM algorithm and assuming a velocity cut-off of about 200 m s−1, we retrieved two candidate families between the prograde and six between the retrograde satellites. Some of these families merge in bigger clusters at higher but possibly still acceptable velocity cut-offs. This might be an indication that the system suffered intense post-break-up collisional evolution which could have dispersed the original families.

3

Preliminary tests on a time-span of several 106 yr, performed with a new library (on development) which allows the treatment of such effects in HJS algorithm, show no major discrepancies in the computed mean orbital elements and no qualitative differences on the chaos measures we will describe in Section 3, thus supporting the analysis and the results reported in the present paper.

DT wishes to thank David Nesvorny for fruitful discussions and help in studying the collisional evolution of the irregular satellites in Saturn system. All the authors wish to thank the anonymous referee and A. R. Dobrovolskis for help and suggestions to improve this paper.

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