A tension between dust and gas radii: the role of substructures and external photoevaporation in protoplanetary disks

Abstract: Protoplanetary disk substructures are thought to play a crucial role in disk evolution and planet formation. Population studies of disks large-sample size surveys show that substructures, and their rapid formation, are needed to reproduce the observed spectral indices. Moreover, they enable the simultaneous reproduction of the observed spectral index and size-luminosity distributions. This study aims to investigate the necessity of substructures and predict their characteristics to reproduce gas-to-dust size ratios observed in the Lupus star-forming region. We performed a population synthesis study of gas and dust evolution in disks using a two-population model (two-pop-py) and the DustPy code. We considered the effects of viscous evolution, dust growth, fragmentation, transport, and external photoevaporation. The simulated population distributions were obtained by post-processing the resulting disk profiles of surface density, maximum grain size, and disk temperature. Although substructures help reduce the discrepancy between simulated and observed disk gas-to-dust size ratios; even when accounting for external photoevaporation, they do not fully resolve it. Only specific initial conditions in disks undergoing viscous evolution with external photoevaporation can reproduce the observations, highlighting a fine-tuning problem. While substructured disks reproduce dust size and spectral index, they tend to overestimate gas radii. The results ultimately highlight the main challenge of simultaneously reproducing gas and dust sizes. One possible explanation is that the outermost substructure is linked to the disk truncation radius, which determines the gas radius, or that substructures are frequent enough to always be near the gas outer radius.

• The paper demonstrates through population synthesis of 100,000 disks that smooth viscous disks overpredict observed gas-to-dust radius ratios, while planetary pressure traps shift typical values toward roughly 4 but do not fully match Lupus observations.
• The paper finds that external photoevaporation at 4 G₀ can produce observed ratios only for narrow combinations of stellar mass, disk size, mass, and trap location, and may erode traps near the truncation radius.
• The paper concludes that viscous evolution alone cannot generally reproduce gas radii, dust radii, spectral indices, and size–luminosity trends simultaneously, motivating tests of trap–truncation links and MHD-wind-driven disk evolution.

This paper presents a population synthesis and test-population study of protoplanetary disk evolution aimed at determining whether disk substructures and external photoevaporation can reconcile the observed gas-to-dust size ratios in the Lupus star-forming region with model predictions. Building on the population synthesis of Delussu et al. (2024), which showed that early and ubiquitous substructures are required to reproduce the observed spectral index and size–luminosity distributions, the authors extend the analysis to the joint evolution of gas and dust radii, testing the proposal by Toci et al. (2021) that unresolved pressure traps resolve the discrepancy between observed and simulated size ratios.

Motivation and observational context

Observations of the Lupus and Taurus regions show that most disks have gas-to-dust size ratios $R_{\mathrm{CO}}/R_{\mathrm{dust}}$ between 2 and 4, with only about 15% of sources exceeding 4. In contrast, models of viscously evolving, smooth disks with grain growth and pure radial drift predict ratios exceeding 5 within a short time. The tension arises because viscosity spreads the gas disk outward while radial drift shrinks the dust disk. Pressure maxima associated with substructures can trap dust and halt drift, thereby increasing $R_{\mathrm{dust}}$ and reducing the ratio; the question is whether this mechanism, possibly combined with external FUV photoevaporation that truncates the gas disk, suffices to match observations.

Methods

The study employs two numerical frameworks. The first is the two-population model (two-pop-py), used for population synthesis of $10^{5}$ disks evolved for 3 Myr, with initial conditions (disk mass, stellar mass, characteristic radius, $\alpha$, fragmentation velocity, and for substructured disks planet mass, position, and insertion time) drawn from prescribed PDFs. Substructures are modeled as planetary gaps via the Kanagawa et al. (2016) prescription, implemented as a local enhancement of $\alpha_{\mathrm{gas}}$. The second framework uses DustPy with an added external photoevaporation module, computing mass-loss rates by interpolation of the FRIEDv2 grid at a constant FUV flux of $4\,G_0$, the average field in Lupus, including dust entrainment in the wind. In DustPy, planetary gaps are imposed through torque-induced radial velocities rather than viscosity modifications, preserving the accretion flow across deep gaps. A comparison between the two codes shows good agreement in disk radii and size–luminosity distributions, with only modest differences in spectral indices near the gap, so the main conclusions are robust to the choice of model.

Simulated observables—dust effective radii ($R_{68\%}$, $R_{90\%}$), CO gas radii ($R_{\mathrm{CO},90\%}$ via the critical gas column density of Trapman et al. 2023, with a factor of 10 for carbon depletion), millimeter fluxes, and spectral indices between 0.89 and 3.1 mm—are computed by post-processing and compared against the Lupus samples of Sanchis et al. (2021), Tazzari et al. (2021), and Andrews et al. (2018), adopting the Ricci compact opacity, which was shown to be required for matching spectral indices.

Population synthesis results with two-pop-py

Across the full parameter space, smooth disks systematically produce $R_{\mathrm{gas}(90\%)}/R_{\mathrm{dust}(90\%)}$ values larger than observed, extending the result of Toci et al. (2021) to the population level. Substructured disks, with one or two planets inserted at 0.1–0.4 Myr (and 0.5–0.8 Myr for the second planet), shift the population toward lower ratios, with the bulk lying near $R_{\mathrm{dust}}$0. The central finding, however, is that substructures narrow but do not close the gap with observations. This creates a tension with the earlier result that substructured disks can simultaneously match the spectral index and size–luminosity distributions: the same configurations that succeed for those diagnostics overpredict the gas radii. The implication is that reproducing all three observed distributions simultaneously requires more than the mere presence of pressure traps.

DustPy results with external photoevaporation

To test whether external photoevaporation supplies the missing ingredient, 27 test disks were evolved with DustPy under viscous evolution and an FUV field of $R_{\mathrm{dust}}$1. The outcome depends sensitively on stellar mass, characteristic radius, and substructure position:

• Disks around $R_{\mathrm{dust}}$2 stars evolve toward high ratios, since viscous spreading dominates and photoevaporation has only a modest effect on $R_{\mathrm{dust}}$3.
• Small disks ($R_{\mathrm{dust}}$4 au) around $R_{\mathrm{dust}}$5 stars also evolve toward high ratios because the photoevaporative truncation radius (~100 au) lies too far out to matter.
• Medium ($R_{\mathrm{dust}}$6 au) and large ($R_{\mathrm{dust}}$7 au) disks around $R_{\mathrm{dust}}$8 stars can evolve toward small ratios consistent with observations—but only under restrictive conditions. For $R_{\mathrm{dust}}$9 au and $10^{5}$0, a substructure at $10^{5}$1 reproduces the observed behavior in all three diagnostic spaces, while lighter disks, or massive disks with the trap at or beyond $10^{5}$2, produce excessive spectral indices and fluxes that are too low. For $10^{5}$3 au, traps placed farther out than $10^{5}$4 are eroded as they approach the truncation radius (~110 au), causing the disks to revert to smooth-disk behavior with discrepant spectral indices and fluxes; the $10^{5}$5 case, while acceptable in size and size–luminosity space, fails on the spectral index.

The authors state plainly that the narrow range of viable initial conditions constitutes a fine-tuning problem. External photoevaporation helps shrink gas radii but simultaneously prevents the large dust radii needed, and can even destroy pressure traps near the truncation radius.

Discussion

The paper frames the central result as a structural tension: substructures reproduce dust sizes and spectral indices but overpredict gas radii, whereas the smooth-disk configurations that reproduce gas radii underestimate dust radii. The physical picture is that viscosity inflates the gas disk while radial drift deflates the dust disk, and no combination of traps plus external photoevaporation within a viscous framework satisfies both constraints without fine-tuning. The authors propose two observational interpretations: either the outermost trap is physically linked to the disk truncation radius (which sets the gas radius), or substructures are so frequent that a trap always exists near the gas outer edge.

Alternative assumptions are examined and found insufficient. Reducing $10^{5}$6 to $10^{5}$7 suppresses spreading but yields fluxes that are too low and rings that are too small. Varying the initial surface density exponent $10^{5}$8 below unity initially lowers the gas-to-dust ratio, but with a radially constant $10^{5}$9 the disk relaxes back toward the $\alpha$0 configuration within a few viscous timescales; combining $\alpha$1 with a radially decreasing $\alpha$2 does not help either, because external photoevaporation dominates the shaping of the outer disk and erases the effect of the modified profile. This identifies external photoevaporation as the primary driver of the outer disk structure in these models, and explains the quantitative difference from Toci et al. (2021), whose smooth-disk models omitted it.

The most promising alternative left open is MHD-wind-driven evolution, which removes angular momentum without spreading the disk and could keep gas radii small while dust trapping operates; implementing a wind parametrization (e.g., Tabone et al. 2022) in the evolutionary codes remains untested.

Limitations and open questions

Several assumptions qualify the results. The FUV flux is held constant at $\alpha$3, neglecting dust extinction (which can shield disks for the first 0.5–1 Myr) and the time-variable flux from the relative motion of OBA stars, although the absence of early-type B and O stars in Lupus mitigates the latter. Substructures are static planet-induced gaps with no planetary migration; the authors argue Type II migration in the low-viscosity, strongly trapping regime is slow enough that disk radii are unaffected, but a quantitative treatment is deferred. Internal photoevaporation is not included and is expected to matter mainly at late stages by clearing residual gas. Stellar luminosity and temperature are held fixed, an approximation the authors show has limited impact for the stellar masses considered. Roughly 27% of the synthesized population has disk-to-star mass ratios $\alpha$4 and may be gravitationally unstable; removing this subset shifts mean continuum fluxes by 30–40% and radii by less than ~10%, leaving the population-level conclusions unchanged, but a self-consistent GI treatment remains outstanding.

Conclusions

The paper demonstrates, at the population level, that smooth disks cannot match the observed gas-to-dust size ratios, that substructures mitigate but do not resolve the discrepancy, and that adding external photoevaporation at the Lupus-average FUV field does not fully close it either—only narrowly tuned combinations of initial conditions succeed, while generically substructured viscous disks overpredict gas radii. The main unresolved issue is the simultaneous reproduction of gas and dust sizes within a viscous framework, pointing either to a link between the outermost pressure trap and the truncation radius or to a high frequency of traps, and motivating wind-driven disk evolution as the next testable scenario.