Formation of multiple dust rings and gaps in protoplanetary discs by a single migrating planet. A parameter study in locally isothermal discs

This paper investigates whether a single migrating planet can produce the multiple concentric dust rings and gaps observed in protoplanetary discs by ALMA, particularly at large radii (out to 150 au). The authors use 2D hydrodynamical simulations (FargoCPT) of locally isothermal discs with a single migrating planet, varying the disc aspect ratio (h0 = 0.06–0.09), viscosity parameter (α = 10−3, 10−4, 10−5), and planet mass (30–400 M⊕). The planet is initially placed at 50 au and allowed to migrate freely.

Key results:

1. Migration regimes: The simulations reveal two distinct migration regimes depending on disc parameters:

• Intermittent type-III migration (for h0 = 0.06–0.07, and for planets ≳100 M⊕): The planet alternates between slow migration episodes and rapid type-III runaway episodes, producing a staircase-shaped migration track. This occurs when the planet mass exceeds roughly twice the local thermal mass: the first runaway of the intermittent type-III migration typically occurs when the planet mass exceeds twice the local thermal mass (see Eq. 1): Mp ≈ 2Mth(rp).

• Smooth or vortex feedback regime (for h0 = 0.08–0.09, and for lower-mass planets): The planet opens a partial gap, migration slows, and the planet eventually stalls in a deep gap.

2. Three mechanisms for ring formation:

• Rings at the stalled planet's gap edges: The planet eventually stalls in a deep gap, with pressure maxima at both gap edges that can trap dust. These stalls typically occur at r ≲ 50 au, with the outermost stall observed at 35 au (for a 70 M⊕ planet in h0 = 0.07, α = 10−4).

• Secondary rings from secondary Lindblad spirals: At low viscosity (α ≤ 10−4), secondary and tertiary gaps open inside the stalled planet's orbit (r ≲ 20 au), created by shocks from secondary Lindblad spirals.

• Remnant rings from type-III migration: During intermittent migration, partial gaps built up during slow migration episodes are left behind when the planet undergoes rapid runaway migration. These remnants can extend to large radii: "our simulation with a 320 M⊕ planet starting at rp0 = 200 au illustrates that planets massive enough to enter the intermittent type-III regime at large radii can leave remnant rings far out in the disc – up to Rring ≈ 160 au in this case."

3. Remnant lifetimes and vortex smear-out:

• At α = 10−3, remnants diffuse within a few kyr and dissipate entirely within ≲100 kyr.

• At α ≤ 10−4, remnants persist for at least 300–500 kyr, a significant fraction of the disc lifetime.

• At α ≤ 10−4, the pressure maxima start as large-scale vortices that gradually smear out into axisymmetric rings. The smear-out timescale increases with lower α and with distance from the star. At α = 10−4, ring-like structures outlast their progenitor vortices by 1.5 to 5 times the smear-out timescale.

4. Constraints on planet and disc properties: The authors conclude that "the planetary origin hypothesis of multi-ringed structures out to r ∼ 150 au is viable for planets of approximately one Jupiter mass or above in discs with low viscosity, α ≲ 10−4, that are sufficiently massive to enable type-III migration."

5. Caveats: The study omits explicit dust evolution, disc self-gravity (though the Toomre parameter Q > 1, they note that Q < r/H in most of the disc, suggesting self-gravity should be included for accurate vortex modelling), and uses a locally isothermal equation of state, which may overestimate gap-opening capabilities. The authors recommend future work with self-gravity, radiative cooling, and explicit dust components.

Improvement 1: Physics-Informed Constraint Enforcement for Planet–Disc Interaction Models

The AI system can be enhanced to automatically enforce the identified migration regime criteria—e.g., triggering intermittent type-III migration only when planet mass exceeds twice the local thermal mass (Mp ≈ 2Mth(rp)) and disc aspect ratio h0 ≤ 0.07—thereby preventing unphysical parameter combinations in forward simulations. The improved system can generate synthetic ALMA ring/gap morphologies that match observed multi-ring structures out to 150 au, while rejecting configurations that would produce smooth migration or rapid gap-crossing without remnant rings, as seen in the paper’s regime boundaries.

Improvement 2: Time-Dependent Ring Persistence and Viscosity-Aware Prediction

The AI can be trained to predict ring lifetimes and morphological evolution (e.g., vortex-to-ring smear-out timescales) as a function of disc viscosity α and radial distance, using the paper’s quantitative results (e.g., remnants persist >300–500 kyr for α ≤ 10−4, but dissipate <100 kyr for α = 10−3). The improved system can then forecast whether observed rings at a given radius are likely transient or long-lived, enabling better discrimination between planetary-origin and alternative (e.g., dead-zone or MHD-wind) explanations in real ALMA observations.

Improvement 3: Multi-Scale Gap/Spiral Feature Detection and Classification

The AI can be upgraded to identify and classify the three distinct ring-formation mechanisms from simulated or observed surface brightness maps: (a) stalled-planet gap-edge rings (r ≲ 50 au), (b) secondary Lindblad spiral-induced inner rings (r ≲ 20 au, low viscosity), and (c) remnant rings from type-III runaway migration (extending to 160 au). The improved system can automatically annotate which mechanism dominates for a given disc–planet parameter set, aiding in inverse modeling of exoplanet masses and disc viscosities from ring spacings and radial extents.

Improvement 4: Self-Gravity and Cooling-Aware Correction Module

Given the paper’s caveat that Q < r/H in most of the disc (indicating self-gravity matters) and that locally isothermal EOS overestimates gap opening, the AI can incorporate a correction layer that adjusts predicted ring depths and migration rates when self-gravity or radiative cooling is present. The improved system can flag simulations where self-gravity would suppress vortex formation or alter remnant ring stability, and can recommend whether to include explicit dust evolution for accurate dust-trap efficiency predictions.

Improvement 5: Observational Feasibility Ranking for Exoplanet Detection

The AI can be enhanced to output a ranked list of observable ring configurations most likely to harbor a single migrating planet, based on the paper’s constraints (planet mass ≳1 Jupiter mass, α ≲ 10−4, disc mass sufficient for type-III migration). The improved system can prioritize ALMA targets with rings at large radii (50–160 au) and low-viscosity signatures (sharp, persistent gaps), and can estimate the minimum planet mass required to produce each observed ring pattern, directly aiding in survey design and follow-up observations.

ALMA observations show that large protoplanetary discs usually contain multiple concentric dust rings separated by dark gaps. A natural explanation is dust-trapping at the edges of gaps opened by newly formed planets. However, planets typically migrate inward on timescales shorter than disc lifetimes, seemingly at odds with rings at large radii. We aim to investigate the conditions under which migrating planets can form long-lived, multi-ringed structures out to r about 150 au to constrain the parameter space for the planetary origin hypothesis of rings. Using the FargoCPT hydrodynamics code, we ran two-dimensional, locally isothermal disc models with a single migrating planet, varying disc aspect ratio, viscosity (alpha), and planetary mass. In all models, the planet eventually stalls in a deep gap. At alpha at most 10-4, secondary spirals launched by the planet can open additional gaps at smaller radii. When planets exceed twice the local thermal mass before stalling, they enter a regime of alternating slow and type-III rapid migration, leaving partial gaps outside their orbit. The gap edges consistently feature pressure maxima that trap dust. These begin as large vortices at alpha at most 10-4, but gradually smear out into rings before dissipating. The type-III remnant rings dissipate quickly at alpha=10-3, but persist for at least 300-500 kyr at alpha at most 10-4. Both smear-out and dissipation timescales increase with lower alpha. Our results show that migrating planets can reproduce observed multi-ringed structures in discs with alpha 10-4 through their stall (r 50 au), secondary gap-opening (r 20 au), and type-III migration remnants (extending to r about 150 au for Jupiter-mass planets in sufficiently massive discs). Longer simulations will be required to compare the statistics of ring-to-vortex occurrence to observations.