Context. Giant planets are expected to form with near-zero obliquities. It has recently been shown that the fast migration of Titan could be responsible for the current 26.7°-tilt of Saturn’s spin axis. Aims. We aim to quantify the level of generality of this result by measuring the range of parameters allowing for this scenario to happen. Since Titan continues to migrate today, we also aim to determine the obliquity that Saturn will reach in the future. Methods. For a large variety of migration rates for Titan, we numerically propagated the orientation of Saturn’s spin axis both backwards and forwards in time. We explored a broad range of initial conditions after the late planetary migration, including both small and large obliquity values. Results. In the adiabatic regime, the likelihood of reproducing Saturn’s current spin-axis orientation is maximised for primordial obliquities between about 2° and 7°. For a slightly faster migration than expected from radio-science experiments, non-adiabatic effects even allow for exactly null primordial obliquities. Starting from such small tilts, Saturn’s spin axis can evolve up to its current state provided that: (i) the semi-major axis of Titan changed by more than 5% of its current value since the late planetary migration, and (ii) its migration rate does not exceed ten times the nominal measured rate. In comparison, observational data suggest that the increase in Titan’s semi-major axis exceeded 50% over 4 Gyr, and error bars imply that the current migration rate is unlikely to be larger than 1.5 times its nominal value. Conclusions. If Titan did migrate substantially before today, tilting Saturn from a small obliquity is not only possible, but it is the most likely scenario. Saturn’s obliquity is still expected to be increasing today and could exceed 65° in the future. Maximising the likelihood would also put strict constraints on Saturn’s polar moment of inertia. However, the possibility remains that Saturn’s primordial obliquity was already large, for instance as a result of a massive collision. The unambiguous distinction between these two scenarios would be given by a precise measure of Saturn’s polar moment of inertia.
Aims. We aim to determine whether Jupiter’s obliquity is bound to remain exceptionally small in the Solar System, or if it could grow in the future and reach values comparable to those of the other giant planets. Methods. The spin-axis of Jupiter is subject to the gravitational torques from its regular satellites and from the Sun. These torques evolve over time due to the long-term variations of its orbit and to the migration of its satellites. With numerical simulations, we explore the future evolution of Jupiter’s spin axis for different values of its moment of inertia and for different migration rates of its satellites. Analytical formulas show the location and properties of all relevant resonances. Results. Because of the migration of the Galilean satellites, Jupiter’s obliquity is currently increasing, as it adiabatically follows the drift of a secular spin-orbit resonance with the nodal precession mode of Uranus. Using the current estimates of the migration rate of the satellites, the obliquity of Jupiter can reach values ranging from 6° to 37° after 5 Gyr from now, according to the precise value of its polar moment of inertia. A faster migration for the satellites would produce a larger increase in obliquity, as long as the drift remains adiabatic. Conclusions. Despite its peculiarly small current value, the obliquity of Jupiter is no different from other obliquities in the Solar System: It is equally sensitive to secular spin-orbit resonances and it will probably reach comparable values in the future.
Context. Dynamically linking a meteor shower with its parent body can be challenging. This is in part due to the limit of today's tools (such as D-criteria) and in part due to the complex dynamics of meteoroid streams.Aims. We choose a method to study chaos in meteoroid streams and apply it to the Geminid meteoroid stream.Methods. We decide to draw chaos maps. We show that the Orthogonal Fast Lyapunov Indicator is well-suited to our problem, amongst the chaos indicator we studied. The maps are drawn for three size bins, ranging from 10 −1 to 10 −4 m.Results. We show the influence of mean-motion resonances with the Earth and with Venus, which tend to trap the largest particles. The chaos maps present 3 distinct regimes in eccentricity, reflecting close encounters with the planets. We also study the effect of non-gravitational forces. We determine a first approximation of the particle size r lim needed to counterbalance the resonances with the diffusion due to the non-gravitational forces. We find that, for the Geminids, r lim lies in the range [3; 8] × 10 −4 m. However, r lim depends on the orbital phase space.
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