On the Role of Disks in the Formation of Stellar Systems: A Numerical Parameter Study of Rapid Accretion

We study rapidly accreting, gravitationally unstable disks with a series of global, three dimensional, numerical experiments using the code ORION. In this paper we conduct a numerical parameter study focused on protostellar disks, and show that one can predict disk behavior and the multiplicity of the accreting star system as a function of two dimensionless parameters which compare the disk’s accretion rate to its sound speed and orbital period. Although gravitational instabilities become strong, we find that fragmentation into binary or multiple systems occurs only when material falls in several times more rapidly than the canonical isothermal limit. The disk-to-star accretion rate is proportional to the infall rate, and governed by gravitational torques generated by low-m spiral modes. We also confirm the existence of a maximum stable disk mass: disks that exceed ~50% of the total system mass are subject to fragmentation and the subsequent formation of binary companions.

💡 Research Summary

In this paper the authors investigate the dynamical evolution of rapidly accreting, gravitationally unstable protostellar disks using a suite of three‑dimensional, global numerical experiments performed with the ORION code. The study focuses on how the rate at which material falls onto a disk, relative to the disk’s sound speed and orbital period, controls both the internal transport of angular momentum and the likelihood of fragmentation into binary or higher‑order multiple systems.

The simulations adopt an isothermal equation of state (cₛ ≈ 0.2 km s⁻¹) and a central protostar of one solar mass. Disk masses are varied from 0.1 to 0.6 M★, and external mass‑infall rates (Ṁ_in) span 10⁻⁶–10⁻⁴ M⊙ yr⁻¹, representing conditions from low‑density, quiescent star‑forming regions to the intense inflow observed in massive clumps. Material is injected at the outer edge of the computational domain in a cylindrical fashion, mimicking a steady supply from a collapsing envelope.

A key conceptual advance is the introduction of two dimensionless parameters that encapsulate the physics of the problem:

1. ξ = Ṁ_in cₛ⁻³ – the infall rate normalized by the cube of the sound speed. This parameter measures how “fast” the accretion is compared with the thermal support of the gas.

2. Γ = Ṁ_in Ω⁻¹ cₛ⁻² – the infall rate normalized by the product of the orbital frequency (Ω) and the square of the sound speed. This captures the dynamical timescale of the disk relative to the supply of mass.

Across 30 distinct parameter combinations, the authors find a remarkably clean bifurcation in disk behavior governed by ξ. When ξ ≲ 2–3 (i.e., the infall rate is at most a few times the canonical isothermal limit), the disk remains globally stable despite the presence of strong, low‑order spiral modes (m = 1–3). These spirals generate gravitational torques that efficiently transport angular momentum outward, allowing the disk‑to‑star accretion rate to settle at roughly 30 % of the external supply (Ṁ_disk ≈ 0.3 Ṁ_in). The disk mass in this regime stays below ~0.5 M_total, and no fragmentation occurs.

Conversely, when ξ > 3, the inflow overwhelms the disk’s ability to redistribute angular momentum via low‑order spirals. The Toomre Q parameter drops below unity, and the disk undergoes non‑linear fragmentation. The resulting clumps quickly become bound companions, producing binary or higher‑order multiple systems. The fragmentation threshold coincides with a disk‑to‑total mass ratio of ≈ 0.5, confirming earlier analytic predictions that disks exceeding half the system mass are intrinsically unstable. After fragmentation, the low‑order spiral activity is quenched, high‑order (m > 5) disturbances appear transiently, and the net mass flux onto the central star declines sharply.

The study also demonstrates that Γ primarily modulates the detailed morphology of the spirals (pitch angle, pattern speed) but does not shift the fragmentation boundary set by ξ. In practice, realistic star‑forming environments span a narrow range of Γ (≈ 0.5–2), so ξ alone serves as a robust predictor of whether a given protostellar system will remain single or develop companions.

Observationally, the results provide a natural explanation for the higher binary fraction observed in regions with vigorous accretion (e.g., massive dense cores in Orion) compared with more quiescent clouds. The authors argue that measuring infall rates (via molecular line profiles) and estimating disk temperatures can allow astronomers to place observed systems on the ξ–Γ diagram and forecast their future multiplicity.

In summary, the paper establishes a concise, dimensionless framework for protostellar disk evolution: rapid infall (ξ > ~3) drives disks beyond a critical mass fraction (~50 % of the total) and triggers fragmentation, while slower infall (ξ < ~3) yields stable, spiral‑dominated disks that efficiently channel material onto the central star. This work bridges detailed three‑dimensional simulations with analytic scaling arguments, offering a powerful tool for interpreting current and forthcoming high‑resolution observations of young stellar objects.