Aims. Trojans are defined as objects that share the orbit of a planet at the stable Lagrangian points L4 and L5. In the Solar System, these bodies show a broad size distribution ranging from micrometer (μm) to centimeter (cm) particles (Trojan dust) and up to kilometer (km) rocks (Trojan asteroids). It has also been theorized that earth-like Trojans may be formed in extra-solar systems. The Trojan formation mechanism is still under debate, especially theories involving the effects of dissipative forces from a viscous gaseous environment. Methods. We perform hydro-simulations to follow the evolution of a protoplanetary disk with an embedded 1–10 Jupiter-mass planet. On top of the gaseous disk, we set a distribution of μm–cm dust particles interacting with the gas. This allows us to follow dust dynamics as solids get trapped around the Lagrangian points of the planet. Results. We show that large vortices generated at the Lagrangian points are responsible for dust accumulation, where the leading Lagrangian point L4 traps a larger amount of submillimeter (submm) particles than the trailing L5, which traps mostly mm–cm particles. However, the total bulk mass, with typical values of ~Mmoon, is more significant in L5 than in L4, in contrast to what is observed in the current Solar System a few gigayears later. Furthermore, the migration of the planet does not seem to affect the reported asymmetry between L4 and L5. Conclusions. The main initial mass reservoir for Trojan dust lies in the same co-orbital path of the planet, while dust migrating from the outer region (due to drag) contributes very little to its final mass, imposing strong mass constraints for the in situ formation scenario of Trojan planets.
The substructures observed in protoplanetary disks may be the signposts of embedded planets carving gaps or creating vortices. The inferred masses of these planets often fall in the Jovian regime despite their low abundance compared to lower-mass planets, partly because previous works often assume that a single substructure (a gap or vortex) is caused by a single planet. In this work, we study the possible imprints of compact systems composed of Neptune-like planets (∼10–30 M ⊕) and show that long-standing vortices are a prevalent outcome when their interplanetary separation (Δa) falls below ∼8 times H p—the average disk’s scale height at the planet’s locations. In simulations where a single planet is unable to produce long-lived vortices, two-planet systems can preserve them for at least 5000 orbits in two regimes: (i) fully shared density gaps with elongated vortices around the stable Lagrange points L 4 and L 5 for the most compact planet pairs (Δa ≲ 4.6 H p), and (ii) partially shared gaps for more widely spaced planets (Δa ∼ 4.6–8 H p) forming vortices in a density ring between the planets through the Rossby wave instability. The latter case can produce vortices with a wide range of aspect ratios down to ∼3 and can occur for planets captured into the 3:2 (2:1) mean-motion resonances for disks’ aspects ratios of h ≳ 0.033 (h ≳ 0.057). We suggest that their long lifetimes are sustained by the interaction of spiral density waves launched by the neighboring planets. Overall, our results show that the distinguishing imprint of compact systems with Neptune-mass planets are long-lived vortices inside the density gaps, which in turn are shallower than single-planet gaps for a fixed gap width.
The Atacama Large Millimeter Array observations of the disk around HD 163296 have resolved a crescent-shape substructure at around 55 au, inside and off-center from a gap in the dust that extends from 38 to 62 au. In this work we propose that both the crescent and the dust rings are caused by a compact pair (period ratio ≃4:3) of sub-Saturn-mass planets inside the gap, with the crescent corresponding to dust trapped at the L 5 Lagrange point of the outer planet. This interpretation also reproduces well the gap in the gas recently measured from the CO observations, which is shallower than what is expected in a model where the gap is carved by a single planet. Building on previous works arguing for outer planets at ≈86 and ≈137 au, we provide a global model of the disk that best reproduces the data and shows that all four planets may fall into a long resonant chain, with the outer three planets in a 1:2:4 Laplace resonance. We show that this configuration is not only an expected outcome from disk–planet interaction in this system, but it can also help constrain the radial and angular position of the planet candidates using three-body resonances.
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