The Influence of the Fractal Dimension on Dust Evolution in Protoplanetary Disks

Abstract: Context: During the first stages of dust coagulation in protoplanetary disks, the dust aggregates are expected to have a high degree of porosity. Most models of dust growth, however, do not take this into account. The reason for this is the technical complexity of this problem. Furthermore, the coagulation/fragmentation kernel for colliding porous or fractal dust aggregates is not well understood. Aims: We wish to explore the effect of aggregate porosity on the evolution of the dust population in protoplanetary disks, with an emphasis on the fragmentation and the bouncing barrier. Methods: We use the DustPy code, and implement porosity as a prescribed function of particle mass with the fractal dimension as a free parameter. In this way, we parameterize the ill-constrained physics of colliding porous/fractal aggregates, and we can explore the effect of different porosity prescriptions. We take into account the effect of porosity on the dust dynamics, while neglecting its effect on the collision outcomes. Results: We find that larger particle masses are reached for lower fractal dimensions. The maximum Stokes numbers that are reached do not depend on the fractal dimension in the case of fragmentation-limited growth and decrease with decreasing fractal dimension in the case of bouncing-limited growth. Furthermore, particle growth is slower for smaller fractal dimensions in our models. Conclusions: The dust evolution is strongly influenced by the fractal dimension. Although larger masses are reached for smaller fractal dimensions, the particles are still much smaller than planetesimals. Under the assumption that the bouncing/fragmentation velocity does not depend on the fractal dimension or filling factor, fractal growth is not beneficial for the streaming instability to occur in the case of fragmentation-limited growth and even disadvantageous in the case of bouncing-limited growth.

• The paper models dust evolution for fractal dimensions D = 2, 2.5, and 3 over 1 million years, showing that lower D shifts fragmentation and bouncing barriers to larger masses.
• The simulations find that fractal aggregates grow more slowly in the adopted models because reduced relative velocities and settling outweigh their larger collision cross sections, though this result is model-dependent.
• Lower porosity does not improve streaming-instability conditions under the assumed collision physics: fragmentation-limited particles reach similar maximum Stokes numbers, while bouncing-limited fractal particles reach smaller ones.

Motivation and approach

Dust coagulation in protoplanetary disks is the first step of planet formation, but the growth of dust aggregates is limited by several barriers: the drift, fragmentation, bouncing, and electrostatic barriers. Since the earliest growth phase produces highly porous, fractal aggregates, porosity should in principle be included in dust evolution models. Doing so self-consistently is computationally expensive: in Smoluchowski-type population models, treating the volume filling factor $\phi$ as an independent variable adds a dimension to the problem, while Monte Carlo approaches sacrifice dynamic range.

Schöll, Dullemond, and Dominik adopt a deliberately simplified strategy: they prescribe the filling factor as a function of mass, $\phi(m)$, parameterized by a free fractal dimension $D$ (with $m \propto a^D$), and use the DustPy code (a 1D radial advection-diffusion plus Smoluchowski solver) to evolve the dust distribution for $10^6$ years. This trades predictivity and falsifiability for generality: because the ill-constrained collision physics of porous aggregates is not modeled explicitly, the results remain valid across a family of porosity prescriptions rather than tied to one specific microphysical model. Crucially, they assume that collision outcomes (sticking, bouncing, fragmentation thresholds) do not depend on filling factor — an assumption whose consequences are discussed below.

The models span $D = 3$ (compact), $D = 2.5$, and $D = 2$ (highly fractal), with a minimum filling factor $\phi_\mathrm{min} = 10^{-5}$ beyond which growth proceeds non-fractally, $\alpha = 10^{-4}$ (with variations to $\phi(m)$0 and $\phi(m)$1), monomer size $\phi(m)$2 cm, fragmentation velocity $\phi(m)$3, and rolling force $\phi(m)$4 dyn. Two cross-section prescriptions are compared: simple spherical cross sections ("SC" models) and semi-analytic fractal cross sections from Tazaki et al. ("AC" models). A notable technical contribution is the treatment of area dispersion: since two aggregates of equal mass can have slightly different cross sections, Stokes numbers within a mass bin follow a narrow log-normal distribution with width parameter $\phi(m)$5. The authors derive modified relative-velocity expressions averaged over this dispersion; without it, the special case $\phi(m)$6 (where St is mass-independent under simple cross sections) yields qualitatively wrong distributions that are also hypersensitive to $\phi(m)$7.

Main results

Barrier locations shift with fractal dimension. Lowering $\phi(m)$8 reduces the Stokes number at fixed mass ($\phi(m)$9 in the Epstein regime for SC models), so both the fragmentation and bouncing barriers move to larger masses. Models with smaller $D$0 therefore reach larger aggregate masses, with or without bouncing. The analytical estimate for the bouncing barrier overestimates its mass for $D$1: fractal particles can be very massive while still in the $D$2 regime, where relative velocities are small but the mass-dependent bouncing velocity $D$3 has already been exceeded. The authors also derive a "lower bouncing barrier" describing the lower edge of the mass distribution, which agrees reasonably well with the simulations.

Stokes numbers behave differently under the two barriers. In fragmentation-limited growth, the maximum Stokes number is independent of $D$4, because $D$5 depends only on $D$6, $D$7, and $D$8 — larger masses are reached simply because more mass is needed to reach the same St. In bouncing-limited growth, by contrast, maximum and mean Stokes numbers decrease with decreasing $D$9: although lower-$m \propto a^D$0 particles reach higher masses, their larger masses depress $m \propto a^D$1, so bouncing sets in at smaller St. For $m \propto a^D$2, the bouncing barrier is reached entirely within the fractal regime regardless of $m \propto a^D$3, since St is constant with mass and only $m \propto a^D$4 decreases.

Growth is slower for smaller $m \propto a^D$5 — apparently contradicting prior work. The paper finds that fractal particles grow more slowly than compact ones during the sticking phase. The authors trace this to a near-cancellation between competing effects: smaller $m \propto a^D$6 increases the collision cross section (raising the growth rate) but simultaneously reduces relative velocities and vertical settling (lowering it). The velocity/settling effect wins slightly in their setup, but they explicitly caution that the outcome is sensitive to model specifics — different treatments could reverse the sign, which may explain why earlier studies (e.g., Okuzumi et al., Estrada et al., Michoulier et al.) reported faster fractal growth.

Cross-section prescription matters for masses, not for fragmentation-limited St. Using the advanced Tazaki cross sections changes mean masses by factors of ~20 ($m \propto a^D$7) to ~100 ($m \propto a^D$8) in the outer disk, but leaves the fragmentation-limited maximum Stokes number unchanged. Because the AC prescription includes monomer overlap effects, $m \propto a^D$9 becomes physically allowed, and such highly fractal AC models converge toward the SC $10^6$0 case, which represents the maximal possible aerodynamical cross section.

Implications for planetesimal formation via streaming instability

The streaming instability requires Stokes numbers of order $10^6$1 or above and enhanced dust-to-gas ratios. Under the stated assumption that threshold velocities are independent of filling factor and fractal dimension:

• In the fragmentation-limited case, fractal growth provides no benefit: the maximum St reached is identical for all $10^6$2, so porosity does not help aggregates cross the streaming-instability threshold any sooner.
• In the bouncing-limited case, fractal growth is actively disadvantageous: for $10^6$3, the maximum St remains well below $10^6$4 throughout the disk, whereas $10^6$5 and $10^6$6 models reach the threshold in the outer regions.

This is a strong and somewhat counterintuitive claim: making aggregates more porous does not bring them closer to planetesimal formation conditions, despite yielding larger physical masses. Even the most porous models produce particles far short of planetesimal sizes, so an additional concentration mechanism remains necessary.

Limitations and open questions

The central caveat, acknowledged plainly by the authors, is the assumption that collision outcomes are filling-factor-independent. Laboratory and numerical studies conflict: some simulations find no bouncing below filling factors of ~0.3–0.4, while laboratory experiments observe bouncing at much lower filling factors (though still far above the $10^6$7 explored here); Oshiro et al. find the bouncing velocity increases as filling factor decreases. If porous aggregates resist bouncing more strongly, the disadvantage of low $10^6$8 for the streaming instability could be reduced or reversed. Similarly, Wada et al. find the fragmentation velocity drops by roughly a factor of two from $10^6$9 to $D = 3$0, which would erode the mass advantage of fractal growth. Resolving these dependencies is the key open question the paper leaves unanswered.

Further simplifications include: a spatially constant $D = 3$1 (in reality radially dependent), neglect of compaction by gas ram pressure and self-gravity, omission of dust traps that could locally boost the dust-to-gas ratio, and a bouncing-barrier estimate that cannot be made accurate because collisions between unequal masses dominate. The finding that fractal growth is slower rests on a fine balance between cross-section and velocity effects and is explicitly flagged as potentially model-dependent.

Conclusion

By prescribing porosity through a free fractal dimension rather than modeling it dynamically, this work isolates how fractality shapes dust evolution in global disk models. Lower fractal dimensions shift growth barriers to larger masses but do not raise — and under bouncing, actually lower — the maximum Stokes numbers attained. Consequently, if threshold velocities are insensitive to porosity, fractal growth neither helps nor hinders reaching streaming-instability conditions when fragmentation-limited, and hinders them when bouncing-limited. The results underscore that whether porosity benefits planetesimal formation hinges on poorly constrained microphysics: the dependence of bouncing and fragmentation thresholds on filling factor and fractal dimension is the decisive unknown.