The stochastic nature of migration of disc instability protoplanets in three-dimensional hydrodynamical and MHD simulations of fragmenting discs

We present a detailed analysis of the nature of migration of protoplanetary clumps formed via disc instability in self-consistent 3D hydrodynamical (HD) and magneto-hydrodynamical (MHD) simulations of self-gravitating discs. Motivated by the complex structure of protoplanetary clumps we do not introduce sink particles. We find that the orbital evolution of the clumps has a stochastic character but also exhibits recurrent properties over many orbits. Clump migration is governed by two sources of gravitational torques: a torque originating from a region about twice the Hill sphere around each clump’s orbit, and the torque resulting from clump-clump interactions. Compared to non-magnetized companion runs, the latter are more frequent in MHD simulations, which give rise to more numerous clumps starting off at smaller masses, often below a Neptune mass. Clump-clump interactions can lead to temporary strong accelerations of migration in both directions, but integrated over time provide a lesser impact than disc-driven torques. They can also lead to clump mergers but do not cause ejections; a difference to previous works which adopted sink particles. The local “Hill torque” is responsible for the fast migration, inward or outward. Estimating the characteristic timescales of conventional migration in our regime, we find that the disc-driven migration timescales are in agreement with Type III migration. However, the dominant local torque is rapidly fluctuating, which reflects the turbulent nature of the flow. The resulting stochastic migration pattern is markedly different from Type III runaway migration and appears to be a distinctive feature of orbital dynamics in a fragmenting disc.

💡 Research Summary

This paper presents a comprehensive investigation of the orbital migration of protoplanetary clumps that form via gravitational instability (GI) in self‑gravitating protoplanetary discs. Using fully three‑dimensional simulations performed with the GIZMO code, the authors explore both pure hydrodynamical (HD) and magnetohydrodynamical (MHD) regimes, deliberately avoiding the use of sink particles so that the internal structure of each clump is resolved by thousands to hundreds of thousands of SPH particles.

Simulation Setup
The initial disc extends from 5 to 25 AU, follows Σ∝r⁻¹ and T∝r⁻¹⁄², and has a total mass of ≈0.1 M⊙, representing an early, massive stage where GI is expected. A weak toroidal magnetic field is seeded; in the MHD runs it is amplified by the so‑called GI dynamo. Cooling is implemented via a β‑prescription: the disc is first evolved with a slow cooling (β = 8) to develop a quasi‑steady spiral pattern, then a rapid cooling (β = 3) is applied to trigger fragmentation, and finally the cooling is set back to β = 2π to avoid runaway collapse. Particle masses are 2 × 10⁻⁶ M_Jup for MHD and 2.4 × 10⁻⁵ M_Jup for HD, giving a spatial resolution of 0.1–0.9 AU for the clumps. Four independent realizations are run for each physics case, providing statistical robustness.

Clump Identification and Torque Decomposition
Clumps are identified as gravitationally bound particle groups; their centre of mass is tracked throughout the simulation. At each snapshot the gravitational acceleration a_g exerted on a clump by all disc particles (excluding the clump’s own particles) is computed. The specific torque is then μ = r × a_g, where r is the position vector from the central star. Assuming near‑circular orbits, the torque directly translates into a radial drift rate. The total torque is split into two physically distinct components:

1. Hill Torque – the contribution from material located within roughly twice the Hill radius of the clump’s orbit. This torque reflects the local disc response (spiral wakes, co‑orbital density asymmetries) and is the dominant driver of migration.
2. Clump‑Clump Interaction Torque – the torque arising from direct gravitational encounters between neighbouring clumps. These events can produce short‑lived, large accelerations but are intermittent.

Key Results

Mass Spectrum and Number of Clumps – In the MHD runs the magnetic pressure and tension suppress large‑scale spiral amplitudes while fostering small‑scale turbulence. Consequently, fragmentation yields many more low‑mass clumps, many below Neptune’s mass, whereas the HD runs produce fewer, more massive fragments (tens of Jupiter masses).

Torque Behaviour – The Hill torque exhibits rapid fluctuations in both magnitude and sign, driven by the turbulent, magnetically‑mediated flow. Its time‑averaged magnitude corresponds to migration timescales typical of Type III (runaway) migration (a few tens of orbital periods), but the stochastic sign changes prevent a monotonic inward drift. The clump‑clump interaction torque is highly episodic; in MHD runs encounters are more frequent because of the higher clump number density, yet when averaged over many orbits its contribution to net radial migration is modest compared with the Hill torque.

Migration Pattern – Because the dominant torque is stochastic, clumps undergo a “random‑walk” migration: periods of rapid inward drift alternate with outward excursions, often repeating over many orbits. This behaviour is markedly different from the classic Type III runaway, which is characterized by a sustained, unidirectional torque. The stochastic migration is especially pronounced in the MHD simulations, where magnetic turbulence amplifies torque variability.

Clump‑Clump Outcomes – Close encounters frequently lead to mergers, especially when a low‑mass clump is captured by a more massive neighbour. No ejections of clumps from the disc are observed, in contrast to earlier studies that employed sink particles. The absence of ejections is attributed to the resolved internal structure of the clumps, which allows tidal forces and shear to dissipate encounter energy without unbinding the bodies.

Comparison with Previous Work – Earlier migration studies of GI discs often inserted a single sink particle into a marginally unstable disc, thereby missing the collective dynamics of multiple fragments. By resolving the clumps, this work demonstrates that the “Hill torque” dominates migration, while clump‑clump torques, though dramatic in individual events, do not set the long‑term migration rate. The stochastic nature of the torque, driven by gravito‑magneto‑turbulence, constitutes a new migration regime that the authors term “stochastic Type III”.

Implications and Future Directions
The findings suggest that planets formed by disc instability may not rapidly plunge into their host star nor be flung outward as free‑floating objects; instead, they can linger in a turbulent, stochastic migration phase that can retain them at intermediate radii for many orbital periods. This could help explain the observed diversity of wide‑orbit giant planets and the presence of massive companions at tens of AU.

Future work should aim to (i) extend the integration time to cover thousands of orbits, possibly using adaptive time‑stepping or sub‑cycling techniques; (ii) incorporate more realistic radiative transfer and dust physics to assess how cooling variations affect torque statistics; and (iii) explore a broader parameter space of magnetic field geometry and strength, as well as disc mass, to map out where stochastic migration dominates over classical Type I/II/III regimes.

In summary, the paper provides the first self‑consistent, sink‑free, three‑dimensional MHD study of fragment migration in a GI‑active disc, revealing that stochastic, Hill‑torque‑driven migration is the hallmark of such environments, fundamentally different from the deterministic runaway migration traditionally associated with Type III.