Turbulent Protoplanetary Disks Research Articles

Reduction of dust radial drift by turbulence in protoplanetary disks

Reduction of dust radial drift by turbulence in protoplanetary disks

Length and Velocity Scales in Protoplanetary Disk Turbulence

In the theory of protoplanetary disk turbulence, a widely adopted ansatz, or assumption, is that the turnover frequency of the largest turbulent eddy, Ω L , is the local Keplerian frequency Ω K . In terms of the standard dimensionless Shakura–Sunyaev α parameter that quantifies turbulent viscosity or diffusivity, this assumption leads to characteristic length and velocity scales given respectively by αH and αc , in which H and c are the local gas scale height and sound speed. However, this assumption is not applicable in cases when turbulence is forced numerically or driven by some natural processes such as vertical shear instability. Here, we explore the more general case where Ω L ≥ Ω K and show that, under these conditions, the characteristic length and velocity scales are respectively α/R′H and αR′c , where R′≡ΩL/ΩK is twice the Rossby number. It follows that α=α˜/R′ , where α˜c is the root-mean-square average of the turbulent velocities. Properly allowing for this effect naturally explains the reduced particle scale heights produced in shearing box simulations of particles in forced turbulence, and it may help with interpreting recent edge-on disk observations; more general implications for observations are also presented. For R′>1 , the effective particle Stokes numbers are increased, which has implications for particle collision dynamics and growth, as well as for planetesimal formation.

On corrugation mode radial wavelengths of the vertical shear instability

Abstract The vertical shear instability (VSI) is a promising mechanism to drive turbulence in protoplanetary disks. Numerical simulations in the literature demonstrate that the VSI non-linear saturation is predominated by the linear corrugation modes. These modes possess vertical wavelengths crucially longer than radial wavelengths. This paper aims to investigate the natural radial wavelength of corrugation modes upon VSI saturation, by a series of numerical simulations conducted in Athena++ at different grid resolutions, disk aspect ratios, and viscosity parameterized by ν. We find a sign of convergence emerges at 64 cells per gas scale height for fiducial simulations. Below it a continuous reduction of wavelengths with grid resolution is observed. Synthetic ALMA molecular line observations of $^{12}\rm CO(2-1)$ are performed to inspect the observability of the corrugation modes feature, which is significantly diminished with a resolution of 32 cells per scale height or above. Flared and viscous disks exhibiting longer saturation wavelengths may mitigate the observational difficulty.

Enhanced Size Uniformity for Near-resonant Planets

Super-Earths within the same close-in, compact planetary system tend to exhibit a striking degree of uniformity in their radius, mass, and orbital spacing, and this “peas-in-a-pod” phenomenon itself serves to provide one of the strongest constrains on planet formation at large. While it has been recently demonstrated from independent samples that such planetary uniformity occurs for both configurations near and distant from mean motion resonance, the question thus remains if the strength of this uniformity itself differs between near-resonant and nonresonant configurations such that the two modes may be astrophysically distinct in their evolution. We thus provide in this work a novel comparative size uniformity analysis for 48 near-resonant and 251 nonresonant multiplanet systems from the California Kepler Survey catalog, evaluating uniformity both across systems and between planetary pairs within the same system. We find that while multiplanet configurations exhibit strong peas-in-a-pod size uniformity regardless of their proximity to resonance, near-resonant configurations display enhanced intra-system size uniformity as compared to their analogous nonresonant counterparts at the level of both entire systems and subsystem planetary pairs and chains. These results are broadly consistent with a variety of formation paradigms for multiple-planet systems, such as convergent migration within a turbulent protoplanetary disk or planet–planet interactions incited by postnebular dynamical instabilities. Nevertheless, further investigation is necessary to ascertain whether the nonresonant and near-resonant planetary configurations respectively evolve via a singular process or mechanisms that are dynamically distinct.

Constraining turbulence in protoplanetary discs using the gap contrast: an application to the DSHARP sample

ABSTRACT Constraining the strength of gas turbulence in protoplanetary discs is an open problem that has relevant implications for the physics of gas accretion and planet formation. In this work, we gauge the amount of turbulence in six of the discs observed in the DSHARP programme by indirectly measuring the vertical distribution of their dust component. We employ the differences in the gap contrasts observed along the major and the minor axes due to projection effects, and build a radiative transfer model to reproduce these features for different values of the dust scale heights. We find that (a) the scale heights that yield a better agreement with data are generally low ($\lesssim 4\, \mathrm{AU}$ at a radial distance of $100\, \mathrm{AU}$), and in almost all cases we are only able to place upper limits on their exact values; these conclusions imply (assuming an average Stokes number of ≈10−2) low turbulence levels of αSS ≲ 10−3 − 10−4; (b) for the 9 other systems we considered out of the DSHARP sample, our method yields no significant constraints on the disc vertical structure; we conclude that this is because these discs have either a low inclination or gaps that are not deep enough. Based on our analysis we provide an empirical criterion to assess whether a given disc is suitable to measure the vertical scale height.

Impact of local pressure enhancements on dust concentration in turbulent protoplanetary discs

The standard core accretion model for planetesimal formation in protoplanetary discs (PPDs) is subject to a number of challenges. One is related to the vertical settling of dust to the disc mid-plane against turbulent stirring. This is particularly relevant in the presence of the vertical shear instability (VSI), a purely hydrodynamic instability applicable to the outer parts of PPDs, which drives moderate turbulence characterized by large-scale vertical motions. We investigate the evolution of dust and gas in the vicinity of local pressure enhancements (pressure bumps) in a PPD with turbulence sustained by the VSI. Our goal is to determine the morphology of dust concentrations and if dust can concentrate sufficiently to reach conditions that can trigger the streaming instability (SI). We performed a suite of global 2D axisymmetric and 3D simulations of dust and gas for a range of values for Σd∕Σg (ratio of dust-to-gas surface mass densities or metallicity), particle Stokes numbers, τ, and pressure bump amplitude, A. Dust feedback onto the gas is included. For the first time, we use global 3D simulations to demonstrate the collection of dust in long-lived vortices induced by the VSI. These vortices, which undergo a slow radial inward drift, are the dusty analogs of large long-lived vortices found in previous dust-free simulations of the VSI. Without a pressure bump and for solar metallicity Z ≈ 0.01 and Stokes numbers τ ~ 10−2, we find that such vortices can reach dust-to-gas density ratios slightly below unity in the discs’ mid-plane, while for Z ≳ 0.05, long-lived vortices are largely absent. In the presence of a pressure bump, for Z ≈ 0.01 and τ ~ 10−2, a dusty vortex forms that reaches dust-to-gas ratios of a few times unity, such that the SI is expected to develop, before it eventually shears out into a turbulent dust ring. At intermediate metallicities, Z ~ 0.03, this occurs for τ ~ 5 × 10−3, but with a weaker and more short-lived vortex, while for larger τ, only a turbulent dust ring forms. For Z ≳ 0.03, we find that the dust ring becomes increasingly axisymmetric for increasing τ and dust-to-gas ratios reach order unity for τ ≳ 5 × 10−3. Furthermore, the vertical mass flow profile of the disc is strongly affected by dust for Z ≳ 0.03, such that gas is transported inward near the mid-plane and outward at larger heights, which is the reversed situation compared to simulations with zero or small amounts of dust. We find viscous α-values to drop moderately as ~10−3–10−4 for metallicities increasing as Z = 0–0.05. Our results suggest that the VSI can play an active role in the formation of planetesimals through the formation of vortices for plausible values of metallicity and particle size. Also, it may provide a natural explanation for the presence or absence of asymmetries of observed dust rings in PPDs, depending on the background metallicity.

Impact of Magnetorotational Instability on Grain Growth in Protoplanetary Disks. II. Increased Grain Collisional Velocities

Abstract Turbulence is the dominant source of collisional velocities for grains with a wide range of sizes in protoplanetary disks. So far, only Kolmogorov turbulence has been considered for calculating grain collisional velocities, despite the evidence that turbulence in protoplanetary disks may be non-Kolmogorov. In this work, we present calculations of grain collisional velocities for arbitrary turbulence models characterized by power-law spectra and determined by three dimensionless parameters: the slope of the kinetic energy spectrum, the slope of the autocorrelation time, and the Reynolds number. The implications of our results are illustrated by numerical simulations of the grain size evolution for different turbulence models. We find that for the modeled cases of the Iroshnikov–Kraichnan turbulence and the turbulence induced by the magnetorotational instabilities, collisional velocities of small grains are much larger than those for the standard Kolmogorov turbulence. This leads to faster grain coagulation in the outer regions of protoplanetary disks, resulting in rapid increase of dust opacity in millimeter wavelength and possibly promoting planet formation in very young disks.

Impact of dust diffusion on the rim shape of protoplanetary disks

Context. Multiple mechanisms are known to give rise to turbulence in protoplanetary disks, which facilitates the accretion onto the central star. Small dust particles that are well coupled to the gas undergo diffusion due to this turbulent motion. Aims. This paper investigates the influence of turbulence-induced dust diffusion on the equilibrium of protoplanetary disks. Methods. The model accounts for dust sublimation, radiative transfer with the flux-limited diffusion approximation, and dust diffusion. It predicts the density and temperature profiles as well as the dust-to-gas ratio of the disk. Results. It is shown that dust diffusion can have a large impact: Assuming the dust survives for 104 s or longer before it can be evaporated, this leads the dust diffusion to widen the inner disk considerably. This effect is generated through a feedback mechanism as the diffusion length is much shorter than the disk width. With increasing dust diffusion, the inclination of the inner rim toward the stellar radiation becomes steeper until it is almost vertical. The temperature range of evaporation and condensation, which is linked to the dust composition, has no influence on this effect. Conclusions. For realistic parameters, dust diffusion cannot be neglected when determining the equilibrium of the disk. Stronger turbulence inside the disk induces more dust diffusion. Therefore, the dust density grows more gradually over a greater distance and less radiation reaches the disk surface. The new equilibrium shape of the disk is more inclined toward the star. This effect is universal and independent of the specific dust composition.

Effects of the Compressibility of Turbulence on the Dust Coagulation Process in Protoplanetary Disks

Abstract Planetesimals are believed to be formed by the coagulation of dust grains in the protoplanetary disk turbulence. However, the bouncing and fragmentation barriers have not been completely solved, particularly for silicate dust. To circumvent these barriers, the turbulent clustering of dust particles must be properly treated. According to the minimum-mass solar nebula (MMSN) model, the Mach number of the turbulence ranges from M rms ≃ 0.01–0.32, and thus the turbulence is often regarded as essentially incompressible. However, it has not been quantitatively investigated whether the incompressible limit is adequate for protoplanetary disk simulations. We therefore compare in this study the motions of inertial particles in direct numerical simulations (DNSs) of the Navier–Stokes equation between weakly compressible turbulence and incompressible turbulence. In the DNSs of compressible turbulence, we use an external force to set the total dissipation and the dilatational-to-solenoidal dissipation ratio. The DNSs reveal that despite the small Mach number M rms( ≲ 0.3), the compressible turbulence field notably differs from the incompressible field in terms of the density fluctuations, pressure fluctuations, and shocklet generation, depending on the ratio of the dilatational forcing. However, we quantitatively confirmed that these effects on the particle collision statistics are weak and that the motion of inertial particles in weakly compressible turbulence is dominated by the solenoidal velocity components. Therefore we can conclude that the incompressible assumption is appropriate for an investigation of the dust coagulation process in protoplanetary disk turbulence, as assumed in the MMSN model.

Turbulence Regulates the Rate of Planetesimal Formation via Gravitational Collapse

Abstract We study how the interaction between the streaming instability (SI) and intrinsic gas-phase turbulence affects planetesimal formation via gravitational collapse in protoplanetary disks. Turbulence impedes the formation of particle clumps by acting as an effective turbulent diffusivity, but it can also promote planetesimal formation by concentrating solids, for example, in zonal flows. We quantify the effect of turbulent diffusivity using numerical simulations of the SI in small local domains, forced with velocity perturbations that establish approximately Kolmogorov-like turbulence. We find that planetesimal formation is suppressed by turbulence once velocity fluctuations exceed ≃ (10−3.5–10−3) . Turbulence whose strength is just below the threshold reduces the rate at which solids are bound into clumps. Our results suggest that the well-established turbulent thickening of the midplane solid layer is the primary mechanism by which turbulence influences planetesimal formation and that planetesimal formation requires a midplane solid-to-gas ratio ϵ ≳ 0.5. We also quantify the initial planetesimal mass function using a new clump-tracking method to determine each planetesimal mass shortly after collapse. For models in which planetesimals form, we show that the mass function is well described by a broken power law, whose parameters are robust to the inclusion and strength of imposed turbulence. Turbulence in protoplanetary disks is likely to substantially exceed the threshold for planetesimal formation at radii where temperatures T ≳ 103 K lead to thermal ionization. Planetesimal formation may therefore be unviable in the inner disk out to 2–3 times the dust sublimation radius.

Dust delivery and entrainment in photoevaporative winds

ABSTRACT We model the gas and dust dynamics in a turbulent protoplanetary disc undergoing extreme-UV photoevaporation in order to better characterize the dust properties in thermal winds (e.g. size distribution, flux rate, trajectories). Our semi-analytic approach allows us to rapidly calculate these dust properties without resorting to expensive hydrodynamic simulations. We find that photoevaporation creates a vertical gas flow within the disc that assists turbulence in supplying dust to the ionization front. We examine both the delivery of dust to the ionization front and its subsequent entrainment in the overlying wind. We derive a simple analytic criterion for the maximum grain size that can be entrained and show that this is in good agreement with the results of previous simulations where photoevaporation is driven by a range of radiation types. We show that, in contrast to the case for magnetically driven winds, we do not expect large-scale dust transport within the disc to be effected by photoevaporation. We also show that the maximum size of grains that can be entrained in the wind (smax) is around an order of magnitude larger than the maximum size of grains that can be delivered to the front by advection alone ($s_{\mathrm{crit}}\lesssim 1 \,\, \mu {\mathrm{m}}$ for Herbig Ae/Be stars and $\lesssim 0.01 \,\, \mu {\mathrm{m}}$ for T Tauri stars). We further investigate how larger grains, up to a limiting size slimit, can be delivered to the front by turbulent diffusion alone. In all cases, we find smax ≳ slimit so that we expect that any dust that is delivered to the front can be entrained in the wind and that most entrained dust follows trajectories close to that of the gas.

Streaming Instability in Turbulent Protoplanetary Disks

Abstract The streaming instability for solid particles in protoplanetary disks is reexamined assuming the familiar alpha (α) model for isotropic turbulence. Turbulence always reduces the growth rates of the streaming instability relative to values calculated for globally laminar disks. While for small values of the turbulence parameter, α < 10−5, the wavelengths of the fastest growing disturbances are small fractions of the local gas vertical scale height H, we find that for moderate values of the turbulence parameter, i.e., α ∼ 10−5–10−3, the length scales of maximally growing disturbances shift toward larger scales, approaching H. At these moderate turbulent intensities and for local particle to gas mass density ratios ϵ < 0.5, the vertical scales of the most unstable modes begin to exceed the corresponding radial scales so that the instability appears in the form of vertically oriented sheets extending well beyond the particle scale height. We find that for hydrodynamical turbulent disk models reported in the literature, with α = 4 × 10−5–5 × 10−4, together with state-of-the-art global evolution models of particle growth, the streaming instability is predicted to be viable within a narrow triangular patch of α–τ s parameter space centered on Stokes numbers, τ s ∼ 0.01 and α ∼ 4 × 10−5, and further, exhibits growth rates on the order of several hundreds to thousands of orbit times for disks with 1% (Z = 0.01) cosmic solids abundance or metallicity. Our results are consistent with, and place in context, published numerical studies of streaming instabilities.

The resonant drag instability of dust streaming in turbulent protoplanetary disc

ABSTRACT Damping of the previously discovered resonant drag instability (RDI) of dust streaming in the protoplanetary disc is studied using the local approach to dynamics of gas–dust perturbations in the limit of the small dust fraction. Turbulence in a disc is represented by the effective viscosity and diffusivity in equations of motion for gas and dust, respectively. In the standard case of the Schmidt number (ratio of the effective viscosity to diffusivity) Sc = 1, the reduced description of RDI in terms of the inertial wave (IW) and the streaming dust wave (SDW) falling in resonance with each other reveals that damping solution differs from the inviscid solution simply by adding the characteristic damping frequency to its growth rate. RDI is fully suppressed at the threshold viscosity, which is estimated analytically, first, for radial drift, next, for vertical settling of dust, and at last, in the case of settling combined with a radial drift of the dust. In the last case, RDI survives up to the highest threshold viscosity, with a greater excess for smaller solids. Once Sc ≠ 1, a new instability specific for dissipative perturbations on the dust settling background emerges. This instability of the quasi-resonant nature is referred to as settling viscous instability (SVI). The mode akin to SDW (IW) becomes growing in a region of long waves provided that Sc > 1 (Sc < 1). SVI leads to an additional increase in the threshold viscosity.

The coexistence of the streaming instability and the vertical shear instability in protoplanetary disks

The streaming instability is a leading candidate mechanism to explain the formation of planetesimals. However, the role of this instability in the driving of turbulence in protoplanetary disks, given its fundamental nature as a linear hydrodynamical instability, has so far not been investigated in detail. We study the turbulence that is induced by the streaming instability as well as its interaction with the vertical shear instability. For this purpose, we employ the FLASH Code to conduct two-dimensional axisymmetric global disk simulations spanning radii from 1 to 100 au, including the mutual drag between gas and dust as well as the radial and vertical stellar gravity. If the streaming instability and the vertical shear instability start their growth at the same time, we find the turbulence in the dust midplane layer to be primarily driven by the streaming instability. The streaming instability gives rise to vertical gas motions with a Mach number of up to ~10−2. The dust scale height is set in a self-regulatory manner to about 1% of the gas scale height. In contrast, if the vertical shear instability is allowed to saturate before the dust is introduced into our simulations, then it continues to be the main source of the turbulence in the dust layer. The vertical shear instability induces turbulence with a Mach number of ~10−1 and thus impedes dust sedimentation. Nonetheless, we find the vertical shear instability and the streaming instability in combination to lead to radial dust concentration in long-lived accumulations that are significantly denser than those formed by the streaming instability alone. Therefore, the vertical shear instability may promote planetesimal formation by creating weak overdensities that act as seeds for the streaming instability.

Global Simulations of the Vertical Shear Instability with Nonideal Magnetohydrodynamic Effects

Abstract The mechanisms of angular momentum transport and the level of turbulence in protoplanetary disks (PPDs) are crucial for understanding many aspects of planet formation. In recent years, it has been realized that the magneto-rotational instability tends to be suppressed in PPDs due to nonideal magnetohydrodynamic (MHD) effects, and the disk is primarily laminar with accretion driven by magnetized disk winds. In parallel, several hydrodynamic mechanisms have been identified that likely also generate vigorous turbulence and drive disk accretion. In this work, we study the interplay between MHD winds in PPDs with the vertical shear instability (VSI), one of the most promising hydrodynamic mechanisms, through 2D global nonideal MHD simulations with ambipolar diffusion (AD) and ohmic resistivity. For typical disk parameters, MHD winds can coexist with the VSI with accretion primarily wind-driven accompanied by vigorous VSI turbulence. The properties of the VSI remain similar to the unmagnetized case. The wind and overall field configuration are not strongly affected by the VSI turbulence, showing a modest level of variability and corrugation of the midplane current sheet. Weak AD strength or the enhanced coupling between gas and magnetic fields weakens the VSI. The VSI is also weakened with increasing magnetization, and characteristic VSI corrugation modes transition to low-amplitude breathing mode oscillations with strong magnetic fields.

Effect of nucleation on icy pebble growth in protoplanetary discs

Solid particles in protoplanetary discs can grow by direct vapour deposition outside of ice lines. The presence of microscopic silicate particles may nevertheless hinder growth into large pebbles, since the available vapour is deposited predominantly on the small grains that dominate the total surface area. Experiments on heterogeneous ice nucleation, performed to understand ice clouds in the Martian atmosphere, show that the formation of a new ice layer on a silicate surface requires a substantially higher water vapour pressure than the deposition of water vapour on an existing ice surface. In this paper, we investigate how the difference in partial vapour pressure needed for deposition of vapour on water ice versus heterogeneous ice nucleation on silicate grains influences particle growth close to the water ice line. We developed and tested a dynamical 1D deposition and sublimation model, where we include radial drift, sedimentation, and diffusion in a turbulent protoplanetary disc. We find that vapour is deposited predominantly on already ice-covered particles, since the vapour pressure exterior of the ice line is too low for heterogeneous nucleation on bare silicate grains. Icy particles can thus grow to centimetre-sized pebbles in a narrow region around the ice line, whereas silicate particles stay dust-sized and diffuse out over the disc. The inhibition of heterogeneous ice nucleation results in a preferential region for growth into planetesimals close to the ice line where we find large icy pebbles. The suppression of heterogeneous ice nucleation on silicate grains may also be the mechanism behind some of the observed dark rings around ice lines in protoplanetary discs, as the presence of large ice pebbles outside ice lines leads to a decrease in the opacity there.

Dust settling against hydrodynamic turbulence in protoplanetary discs

Enhancing the local dust-to-gas ratio in protoplanetary discs is a necessary first step to planetesimal formation. In laminar discs, dust settling is an efficient mechanism to raise the dust-to-gas ratio at the disc midplane. However, turbulence, if present, can stir and lift dust particles, which ultimately hinders planetesimal formation. In this work, we study dust settling in protoplanetary discs with hydrodynamic turbulence sustained by the vertical shear instability. We perform axisymmetric numerical simulations to investigate the effect of turbulence, particle size, and solid abundance or metallicity on dust settling. We highlight the positive role of drag forces exerted onto the gas by the dust for settling to overcome the vertical shear instability. In typical disc models we find particles with a Stokes number $\sim 10^{-3}$ can sediment to $\lesssim 10\%$ of the gas scale-height, provided that $\Sigma_\mathrm{d}/\Sigma_\mathrm{g}\gtrsim 0.02$-$0.05$, where $\Sigma_\mathrm{d,g}$ are the surface densities in dust and gas, respectively. This coincides with the metallicity condition for small particles to undergo clumping via the streaming instability. Super-solar metallicities, at least locally, are thus required for a self-consistent picture of planetesimal formation. Our results also imply that dust rings observed in protoplanetary discs should have smaller scale-heights than dust gaps, provided that the metallicity contrast between rings and gaps exceed the corresponding contrast in gas density.

The role of cooling dust particles in gravitational turbulent protoplanetary disks

The role of cooling dust particles in gravitational turbulent protoplanetary disks

Dust Segregation in Hall-dominated Turbulent Protoplanetary Disks

Abstract Imaging of the dust continuum emitted from disks around nearby protostars reveals diverse substructure. In recent years, theoretical efforts have been intensified to investigate how far the intrinsic dynamics of protoplanetary disks (PPDs) can lead to such features. Turbulence in the realm of non-ideal magnetohydrodynamics (MHD) is one candidate for explaining the generation of zonal flows which can lead to local dust enhancements. Adopting a radially varying cylindrical disk model, and considering combinations of vertical and azimuthal initial net flux, we perform 3D non-ideal MHD simulations aimed at studying self-organization induced by the Hall effect in turbulent PPDs. To this end, new modules have been incorporated into the Nirvana-iii and FARGO3D MHD codes. We moreover include dust grains, treated in the fluid approximation, in order to study their evolution subject to the emerging zonal flows. In the regime of a dominant Hall effect, we robustly obtain large-scale organized concentrations in the vertical magnetic field that remain stable for hundreds of orbits. For disks with vertical initial net flux alone, we confirm the presence of zonal flows and vortices that introduce regions of super-Keplerian gas flow. Including a moderately strong net-azimuthal magnetic flux can significantly alter the dynamics, partially preventing the self-organization of zonal flows. For plasma beta-parameters smaller than 50, large-scale, near-axisymmetric structures develop in the vertical magnetic flux. In all cases, we demonstrate that the emerging features are capable of accumulating dust grains for a range of Stokes numbers.

Dynamics and Formation of the Near-resonant K2-24 System: Insights from Transit-timing Variations and Radial Velocities

Abstract While planets between the size of Uranus and Saturn are absent within the solar system, the star K2-24 hosts two such planets, K2-24b and c, with radii equal to 5.4 and 7.5 , respectively. The two planets have orbital periods of 20.9 days and 42.4 days, residing only 1% outside the nominal 2:1 mean-motion resonance. In this work, we present results from a coordinated observing campaign to measure planet masses and eccentricities that combines radial velocity measurements from Keck/HIRES and transit-timing measurements from K2 and Spitzer. K2-24b and c have low, but nonzero, eccentricities of . The low observed eccentricities provide clues to the formation and dynamical evolution of K2-24b and K2-24c, suggesting that they could be the result of stochastic gravitational interactions with a turbulent protoplanetary disk, among other mechanisms. K2-24b and c are and , respectively; K2-24c is 20% less massive than K2-24b, despite being 40% larger. Their large sizes and low masses imply large envelope fractions, which we estimate at % and %. In particular, K2-24c’s large envelope presents an intriguing challenge to the standard model of core-nucleated accretion that predicts the onset of runaway accretion when ≈ 50%.