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Probing disk dynamics and dust evolution through shadows in protoplanetary disks: A case study of the HD 142527 disk

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read This paper shows that the temperature lag between a shadow cast by a misaligned inner disk and the outer disk's thermal response is a direct measure of the cooling timescale, and that in HD 142527 this timescale is 20–90 years and limits th

desk verdict Original method, but the cooling-time measurement sits at the resolution limit and is not beam-convolved, so the headline grain-size constraint is provisional. read the letter →

arxiv 2607.17648 v1 pith:O5UBYIGH submitted 2026-07-20 astro-ph.EP astro-ph.SR

Yuya Fukuhara , Ryuta Orihara , Satoshi Okuzumi , Takayuki Muto This is my paper · ORCID
classification astro-ph.EPastro-ph.SR PACS 97.10.Fy97.10.Gz
keywords protoplanetarydiskstransitiondiskshadowscoolingtimescaledustgrainsizeverticalshearinstabilityHD142527multi-wavelengthobservations
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper develops a way to measure the cooling timescale of a protoplanetary disk from the thermal wake left by a shadow cast by a misaligned inner disk. Applied to the transition disk HD 142527, the method uses near-infrared scattered light to fix the shadow boundaries and submillimeter continuum maps to trace the temperature response. The observed angular lag between the shadow edge and the temperature minimum implies a cooling time of a few percent of the orbital period — roughly 20–90 years at a radius of 170 au. Matching that measured timescale with an analytic cooling model constrains the maximum dust grain size to about 0.1–1 mm in the fiducial model, and shows that the vertical shear instability can sustain the turbulence inferred from the disk surface height. If the technique holds, shadows become general clocks for cooling and grain growth in transition disks.

What carries the argument

The mechanism is a two-sided comparison. Observationally, a geometric reconstruction of the outer-disk scattering surface identifies the shadow boundaries; the azimuthal lag between those boundaries and the brightness-temperature minimum is then fit with a first-order exponential relaxation law to extract the observed cooling timescale. Theoretically, an analytic model computes the local cooling timescale as the maximum of radiative diffusion, gas–dust collisional transfer, and emission timescales, evaluated at the dust thermal emission height where the vertical optical depth equals unity. Collisional transfer dominates in the outer disk. The intersection of the observed and calculated cooli

What would settle it

Re-observed the HD 142527 shadow at two epochs separated by a few years: if the shadow boundary moves by an angular amount comparable to the inferred lag (≳0.1 rad), the stationary-shadow assumption fails. Alternatively, a hydrodynamical simulation with a moving shadow would show a temperature minimum whose lag does not follow the exponential relaxation form, falsifying the interpretation.

Watch

Extended reading notes

Core claim

The central discovery is that the angular offset between a shadow boundary and the resulting temperature minimum on the outer disk is a direct read-off of the thermal relaxation timescale, and that this timescale translates into a dust grain size. For HD 142527, reconstructing the three-dimensional scattering surface from near-infrared polarized scattered light places the shadow boundary, while azimuthal brightness-temperature profiles at 170 au from submillimeter continuum show a downstream minimum. Fitting an exponential cooling law gives a dimensionless cooling time of 0.1–0.4, i.e., 20–90 years. An analytic model of the cooling timescale at the dust emission height, evaluated over dust s

Load-bearing premise

The interpretation assumes the shadow boundary is fixed and the temperature minimum's downstream offset is purely a thermal relaxation lag; if the inner-disk shadow moved between the epochs of the scattered-light and continuum observations, or if the finite synthesized beam (about 0.24 rad, comparable to the inferred 0.1–0.4 rad lag) shifts the apparent minimum, the retrieved cooling timescale is not physical.

Editorial extensions

If this is right

  • If the method is correct, shadows in transition disks provide a general, directly observable measure of local cooling timescales, needing only scattered-light geometry and continuum temperature maps.
  • For HD 142527, the inferred sub-millimeter maximum grain sizes imply dust growth is limited by fragmentation or bouncing at low collision velocities (roughly 1–2.4 m/s for compact grains).
  • The short cooling timescale satisfies the vertical shear instability criterion, so the vertical shear instability is a viable source of the turbulence that shapes the dust distribution and the disk surface height.
  • The approach transfers to other transition disks with misaligned inner disks, allowing systematic comparisons of cooling across disks.
  • The single cooling-timescale constraint leaves the maximum grain size and dust surface density degenerate; combining with multi-wavelength continuum data would break that degeneracy, as the paper notes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension is to treat the shadow as a moving boundary: multi-epoch scattered-light images would reveal precession of the inner disk, and the full time-dependent thermal response could separate kinematic shadow motion from thermal relaxation, tightening the inferred cooling time.
  • The inferred lag (0.1–0.4 rad) is comparable to the synthesized beam's azimuthal extent (about 0.24 rad); observations at higher angular resolution, or deconvolution of the azimuthal temperature profile, would test whether the retrieved cooling timescale is resolution-limited.
  • Because the analytic cooling time is dominated by gas–dust collisional heat transfer, the same shadow-clock observable could be used as an independent probe of the dust-to-gas ratio, complementary to continuum optical depth estimates.
  • Surveying transition disks of different ages with shadows could reveal whether the sub-millimeter size cap seen in HD 142527 is universal or evolves as dust coagulation, fragmentation, and bouncing balance change.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes a method to constrain dust grain sizes in protoplanetary disks by measuring the local cooling timescale from the azimuthal thermal relaxation of the outer disk behind an inner-disk shadow. The method is applied to HD 142527: the shadow geometry is reconstructed from a 2017 VLT/SPHERE H-band polarized image using the Orihara & Momose (2025) framework, and the azimuthal brightness-temperature profile at R=170 au from 2015 ALMA Band 9 data is fitted to a first-order exponential relaxation model (Eq. 26). The inferred dimensionless cooling timescale is t_cool,est Ω_K = 0.1–0.4 (20–90 yr). An analytic model for the cooling timescale at the ALMA Band 9 emission height is then used to map this constraint onto dust maximum grain size, yielding a_max ≈ 150–900 μm at Σ_dust = 10 g cm⁻² in the fiducial model. The paper also checks whether the vertical shear instability (VSI) can sustain the assumed vertical diffusion α_z = 2×10⁻³, finding a consistent parameter region for the fiducial and several alternative parameter choices.

Significance. If the central measurement is robust, the paper introduces a genuinely new observational probe: shadows as 'cooling clocks' in transition disks. The inferred sub-mm maximum grain size in HD 142527's outer disk and the consistency with a VSI-active layer would connect disk thermodynamics, turbulence, and dust evolution in a falsifiable way. The analytic model is transparent and the parameter study is unusually thorough (gas surface density, size distribution, porosity, opacity models, VSI condition). The observational estimate of t_cool,est is a forward-model comparison, not a fit that returns the model inputs, so the inversion is not circular in the strong sense. However, the load-bearing measurement is made at the resolution limit and without beam convolution, and the shadow is taken from an epoch two years away from the temperature map; these issues must be addressed before the claimed grain-size and VSI conclusions become convincing.

major comments (3)
  1. [Section 4.1.2, Eq. (26), Fig. 4(b)] The synthesized beam subtends ~0.24 rad in azimuth at R=170 au, while the inferred t_cool Ω_K = 0.1–0.4 corresponds to an e-folding lag of 0.1–0.4 rad. A physically instantaneous temperature drop at the shadow boundary, smoothed by a Gaussian beam of this size, produces an apparent exponential relaxation with a scale comparable to the beam width. The paper acknowledges that the pre-boundary decrease is 'likely affected by convolution' but does not quantify the bias. Since the model fit is performed in the image plane and the model is not convolved with the beam, the reported t_cool,est may be dominated by beam smearing rather than physical cooling. The authors should forward-model the synthesized beam (e.g., convolve the model brightness distribution, or fit in the visibility domain) and show that the inferred t_cool range survives this treatment.
  2. [Sections 3.1, 3.2, 4.1.2] The shadow boundary is derived from the 2017 SPHERE epoch, while the azimuthal temperature profile is from the 2015 ALMA program (project 2015.1.00614.S). The central observable is the azimuthal offset between the shadow boundary and the temperature minimum; any radial or azimuthal motion of the inner-disk shadow between 2015 and 2017 enters directly as an apparent thermal lag. Section 6.2 cites detections of moving shadows in other disks but gives no bound on the shadow angular velocity for HD 142527. The authors should either justify that the shadow is static on these timescales (e.g., from multi-epoch images or polarimetric data) or explicitly fold the unknown shadow motion into the uncertainty of t_cool,est. Without this, the measured offset cannot be unambiguously interpreted as a thermal relaxation timescale.
  3. [Section 4.1.2, Eq. (26), Section 6.2] The interpretation of Eq. (26) assumes a single, stationary shadow boundary, a constant target temperature behind the shadow, and first-order relaxation without azimuthal transport or dynamical feedback. The paper itself discusses in Section 6.2 that shadow-induced spirals, warps, or non-axisymmetric dust traps could reshape the temperature pattern on timescales comparable to or shorter than the cooling time. These effects are not included in the fiducial fitting model or in the uncertainty budget. At minimum, the authors should test whether the inferred t_cool is stable when the fit is restricted to a narrower azimuthal window around the shadow boundary or when the model includes a smooth transition region for the illumination profile. Without such tests, the inferred t_cool = 0.1–0.4 may reflect a combination of thermal lag and azimuthal structure rather than cooling alone.
minor comments (5)
  1. [Section 2.2.1 / Eq. (27)] The conversion from I_ν to T_b uses full Planck inversion; it may be worth stating explicitly that this is a brightness temperature and that the optically-thick assumption is revisited in Section 4.2.1. Currently the assumption is introduced in the calculation procedure but not prominently flagged as a modeling assumption.
  2. [Section 4.2.1 / Figure 8] The abstract states 'maximum grain size consistent with the observations is approximately 0.1–1 mm', but this is the fiducial-model result at Σ_dust = 10 g cm⁻². The paper is careful to show the strong degeneracy with Σ_dust (20–200 μm at 1 g cm⁻², 20–50 μm at 0.1 g cm⁻²) and with other parameters (e.g., single-size model allows 20–30 μm). The abstract and conclusions should be reworded to emphasize the fiducial-model conditionality, or framed as 'sub-millimeter to millimeter' only after quoting the full parameter dependence.
  3. [Figure 4(b) / Section 4.1.2] The horizontal scale bar indicating one synthesized beam is helpful, but the figure could also mark the 68% credible interval of t_cool Ω_K directly on the axis to make the resolution-limit comparison more transparent. Also clarify why the orange fitting points exclude the data immediately at the shadow boundary.
  4. [Section 5.1, Eqs. (29)–(31)] The VSI criterion uses |q_temp| = 0.5, but the paper does not justify this value for HD 142527's outer disk. Since Section 5's conclusion that VSI operates depends on this choice, a brief justification or a sensitivity check for |q_temp| would strengthen the claim.
  5. [General / notation] The text sometimes writes t_cool Ω_K and sometimes t_coolΩ_K without a space; please normalize the notation. Also, the symbol t_cool,cal is introduced without an explicit definition in the calculation procedure (step 8); define it at first use.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: t_cool,est and t_cool,cal are independent quantities compared as a forward model.

full rationale

The derivation chain is self-contained rather than circular. The observed cooling timescale t_cool,est is obtained by fitting Eq. (26) to the ALMA azimuthal brightness-temperature profile, with the shadow boundary fixed by a separate SPHERE scattered-light reconstruction; no parameter of the analytic dust model enters that fit. The analytic t_cool,cal is computed from dust opacities, vertical distribution, and emission height (Eqs. 1–25) and compared with t_cool,est only at the final step, so the 150–900 μm grain-size range is the output of a forward-model comparison, not an input. Self-citations (Orihara & Momose 2025 for shadow reconstruction; Fukuhara & Okuzumi 2024 and Fukuhara et al. 2025 for the cooling model and VSI criterion) are prior published methods, not uniqueness theorems, and the central grain-size constraint does not reduce to them. The concerns raised in the paper itself—beam convolution in §4.1.2, moving shadows and steady-state/dynamical effects in §6.2—are observational/systematic-error limitations, not circular reductions of the kind required to raise the circularity score.

Assumptions & free parameters 10 free parameters · 7 assumptions · 0 invented entities

The central inversion (cooling time → grain size) carries ~9 hand-set or fitted parameters beyond the constrained (a_max, Σ_dust) pair: Σ_gas, α_z, p, a_min, f_dust, opacity model, q_temp, μ', and the VSI layer threshold. The paper is honest about most of these (Tables 1-2) and tests sensitivity via model variants; α_z is the least-tested, being fixed at 2e-3 for the headline constraint and only checked for VSI consistency downstream. The shadow-stationarity premise and the optically-thick interpretation are domain assumptions that the paper partially self-validates. No new physical entities are introduced; the VSI-active layer and global VSI condition are diagnostic criteria, not entities.

free parameters (10)
  • Σ_gas (gas surface density) = 10 g cm^-2 (fiducial); 1 g cm^-2 (Low-Gas)
    Sets the Stokes number and dust settling via Eqs. (10)-(13). Varied by hand over two values; not measured.
  • α_z (vertical diffusion coefficient) = 2e-3 (fixed)
    Assumed for the headline constraint (§4.2), chosen to match the scattering surface height and Tazaki et al. 2021; checked against VSI only in §5, never varied while deriving a_max.
  • p (size-distribution slope) = -2.5 (fiducial); -3.5 (Steep)
    Power-law index in Eq. (1); shifts the allowed a_max by an order of magnitude (Fig. 10).
  • a_min (minimum grain size) = 1 μm (fiducial); 0.1 μm (variant)
    Minor effect for p > -3; large effect when combined with p = -3.5.
  • f_dust (filling factor) = 1.0 (fiducial); 0.3, 0.1 (Mid/Low-Fill)
    Dust porosity rescales internal density and opacities; with f_dust = 0.1 the allowed a_max rises to ~1 cm.
  • Opacity model = Ricci (fiducial); DIANA; DSHARP
    Changes the a_max constraint from 150-900 μm to 200-1000 μm / 250-1500 μm (appendix 6).
  • |q_temp| (temperature gradient for VSI) = 0.5
    Input to the VSI critical cooling time, Eq. (30); hand-chosen typical value.
  • μ' (scattering-surface grazing angle) = 0.1
    Sets the near-IR scattering surface in Eq. (23); justified by the fitted surface aspect ratio z_sca/R ≈ 0.1.
  • ΔL_VSI > 2 H_gas threshold = 2 gas scale heights
    Global VSI condition (Eq. 31) calibrated in the authors' own simulations (Fukuhara & Okuzumi 2024; Fukuhara et al. 2025).
  • a_max and Σ_dust (constrained pair) = a_max = 150-900 μm at Σ_dust = 10 g cm^-2 (fiducial)
    Fit outputs: matched so that t_cool,cal = t_cool,est. Degenerate (one observable, two parameters); allowed a_max spans 20 μm to >1 cm across the Table 1 models.
assumptions (7)
  • domain assumption First-order thermal relaxation with a single cooling timescale and constant T_ini, T_tar (Eq. 26)
    The entire observational t_cool,est rests on this exponential model; no nonlocal transport, temperature-dependent t_cool, or shadow motion is included.
  • domain assumption Band 9 emission is optically thick, so brightness temperature equals local dust temperature (Eq. 27)
    Used in §2.2.1 step 2 and Fig. 4b; partly self-validated by the model's τ_all > 1 (§4.2.1, Fig. 17), but the observational channel ignores the scattering term the paper's own Eq. (4) includes.
  • domain assumption Vertical dust distribution from settling-diffusion balance (Eqs. 10-13) in the Epstein regime
    Standard result (Takeuchi & Lin 2002; Dubrulle et al. 1995); assumes all grain radii are small relative to the gas mean free path.
  • domain assumption Cooling timescale is the maximum of diffusion, collision, and emission timescales (Eq. 24)
    From Malygin et al. 2017 and Fukuhara & Okuzumi 2024; maps (a_max, Σ_dust) to t_cool,cal; the collisional term dominates at z_emi.
  • ad hoc to paper VSI active where t_cool < t_crit (Eqs. 29-30) and globally when ΔL_VSI > 2 H_gas (Eq. 31)
    The 2 H_gas threshold is calibrated in the authors' own simulations; |q_temp| = 0.5 is assumed.
  • domain assumption The shadow-casting inner disk is static, axisymmetric, centered on the star, with h_r = 0.16
    Needed to apply the 2017 SPHERE shadow boundary to the 2015 ALMA temperature profile; §6.2 concedes that moving shadows have been detected in other disks (Mullin et al. 2026).
  • standard math Keplerian rotation maps azimuthal angle to time (φ_R = Ω_K t)
    Needed for Eq. (26); ignores shadow-induced spirals, warps, and vortices that §6.2 discusses as possible contaminants.

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Pith. "Pith review of Probing disk dynamics and dust evolution through shadows in protoplanetary disks: A case study of the HD 142527 disk." pith.science (2026). https://pith.science/paper/O5UBYIGH

@misc{pith2026260717648,
  author       = {Pith},
  title        = {Pith review of: Probing disk dynamics and dust evolution through shadows in protoplanetary disks: A case study of the HD 142527 disk},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O5UBYIGH}},
  note         = {Machine review of arXiv:2607.17648}
}
read the original abstract

Planet formation begins with dust growth and planetesimal formation within protoplanetary disks surrounding young stars. To understand these processes, it is essential to estimate dust grain sizes from disk observations. In this study, we develop a new method to constrain grain size based on the estimation of cooling timescales. Our approach applies to transitional disks that possess an inclined inner disk casting shadows on the outer disk, whose temperature variations serve as a tracer of dust properties. By constructing a three-dimensional model of the disk surface using near-infrared scattering light images and comparing it with submillimeter dust continuum maps, we estimate the spatial offset between the irradiated and shadowed regions to derive the cooling timescale. We then build an analytic model that calculates the cooling timescale at the dust thermal emission height with an assumed turbulent diffusion intensity to infer the dust surface density and dust grain size. Applying this method to the protoplanetary disk around HD~142527, we find that the disk's northern shadowed region cools on a timescale of a few percent of the orbital period and that the maximum grain size consistent with the observations is approximately 0.1-1 mm. We also find that the conditions required for the vertical shear instability, which needs a short cooling timescale, are satisfied, allowing turbulence with an intensity consistent with near-infrared observations. This study demonstrates that estimating cooling timescales is an effective tool for constraining dust grain size. Our approach can be generally applied to other transition disks with inner-disk-induced shadows.

Figures

Figures reproduced from arXiv: 2607.17648 by the authors.

Figure 1
Figure 1. Schematic overview of our methodology for estimating the cooling timescale and constraining the dust grain size by combining multi-wavelength obser￾vations with an analytic model. High-resolution dust thermal emission observations at submillimeter wavelengths provide the emission intensity and azimuthal temperature profile. From near-infrared scattered-light observations, we determine the scattering surface and shad… view at source ↗
Figure 2
Figure 2. Schematic illustration of the observational method used to estimate the cooling timescale from the azimuthal temperature variation across a disk shadow. (a) Geometry of the disk at a representative radius R. The shadowed sector is shown in black and is bounded by the shadow edges indicated in red. The azimuthal angular distance ϕR is measured from the upstream shadow boundary along the direction of disk rotation. (b… view at source ↗
Figure 3
Figure 3. Best-fit scattering-surface model for the HD 142527 disk. (a) H-band Qϕ image overlaid with the best-fit model curves. The blue curve indicates the horizon of the outer disk, the red curves show the reconstructed shadow boundaries, and the green curve shows the apparent outer edge of the outer disk. The blue cross and red star mark the positions of the outer-disk center and the central star, respectively. (b) Three-… view at source ↗
Figures from the paper (17 more)
Figure 4
Figure 4. Figure 4: Comparison between the reconstructed shadow geometry and the azimuthal temperature variation in the HD 142527 outer disk. (a) ALMA Band 9 brightness-temperature image overlaid with the surface grid of the best-fit scattering-surface model. The red curves indicate the s…
Figure 5
Figure 5. Figure 5: Calculated vertical optical depth, τν(z) at ALMA Band 9 [upper panel; equation (22)], and cooling timescale, tcool [lower panel; equation (24)], as functions of height z for amax = 100 µm and Σdust = 10 g cm−2 in the fiducial model. In the upper panel, the horizontal a…
Figure 8
Figure 8. Figure 8 [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 7
Figure 7. Figure 7: Calculated cooling timescale at the ALMA Band 9 emission height, tcool,cal, as a function of amax for Σdust = 10 g cm−2 in the fiducial model. The hor￾izontal dash-dotted lines mark the upper (tcoolΩK = 0.4) and lower (0.1) limits of the observationally estimated range…
Figure 9
Figure 9. Figure 9: Same as figure 8, but for the Low-Gas model (Σgas = 1.0 g cm−2 , see table 1 for other parameter choices). grains [equations (10)–(13)], which suppresses their vertical dif￾fusion. Consequently, dust grains become more concentrated near the midplane, causing the ALMA B…
Figure 10
Figure 10. Figure 10: Same as figure 8, but for the Single-Size, Small-amin, Steep-SizeDis, and Steep-SizeDis-Small-amin models from left to right (see table 1 for the parameter choice of each model). the dust density [equation (13)]. Both effects change the emission height at ALMA Band 9 …
Figure 12
Figure 12. Figure 12: Thickness of the VSI-active region ∆LVSI as a function of amax for different values of Σdust in the fiducial model (see table 1 for the param￾eter choice). The horizontal dash-dotted line marks the critical threshold ∆LVSI = 2Hgas. The shaded area of ∆LVSI > 2Hgas hig…
Figure 11
Figure 11. Figure 11: Same as figure 8, but for the Mid-Fill model (fdust = 0.3; upper panel) and the Low-Fill model (fdust = 0.1; lower panel). For VSI-driven turbulence to maintain a strong vertical diffusion of αz = 2 × 10−3 in a self-consistent manner, the VSI-active layer must be thic…
Figure 13
Figure 13. Figure 13: Area diagram in the amax–Σdust plane for the models presented in table 1, excluding the varying opacity models. The vertically and horizontally hatched areas indicate the parameter spaces satisfying the global VSI condition [∆LVSI > 2Hgas; equation (31)] and the cooli…
Figure 14
Figure 14. Figure 14: Images and gradient maps used for the shadow-geometry reconstruction. (a) Inverse-hyperbolic-sine-stretched H-band Qϕ image. (b) Inverse-hyperbolic￾sine-stretched H-band Uϕ image. (c) Inverse-hyperbolic-sine-stretched S/N map, defined as sinh−1 (Qϕ/Uϕ,rms), overlaid w…
Figure 15
Figure 15. Figure 15: Posterior distributions of the disk model parameters listed in [PITH_FULL_IMAGE:figures/full_fig_p022_15.png]
Figure 16
Figure 16. Figure 16: Dust absorption opacity (left panel), effective dust scattering opacity (middle panel), and asymmetry parameter (right panel) for different opacity models and porosity models as a function of the dust grain size with the frequency of ν = 697 GHz. 10 3 10 2 10 1 10 0 a…
Figure 17
Figure 17. Figure 17: Calculated disk’s vertical extinction optical thickness at ALMA Band 9 (left panel), disk temperature (middle panel), and gas scale height (right panel) as a function of the maximum dust grain size for different values of Σdust in the fiducial model. The horizontal do…
Figure 18
Figure 18. Figure 18: Upper row: Same as the left of figure 17, but for the different models of the size distribution (left panel), porosity (middle panel), and opacity (right panel), with Σdust = 10 g cm−2 . The horizontal dotted lines show τall = µ. Lower row: Same as the middle panel of…
Figure 19
Figure 19. Figure 19: Same as figure 6, but for the different models of the Low-Gas, size distribution, porosity, and opacity from left to right, with Σdust = 10 g cm−2 except for the Low-Gas model. both models, tcool,cal increases slightly compared to that for the fiducial model, as shown…
Figure 20
Figure 20. Figure 20: Same as figure 8, but for the DIANA model (upper panel) and DSHARP model (lower panel). 0.1 < tcool,calΩK < 0.4 and αz = 2 × 10−3 with Σdust = 10 g cm−2 becomes 200 µm ≲ amax ≲ 600 µm and 250 µm ≲ amax ≲ 730 µm for the DIANA and DSHARP models, respectively. For both m…
Figure 21
Figure 21. Figure 21: Same as figure 13, but for the DIANA model (upper panel) and DSHARP model (lower panel) [PITH_FULL_IMAGE:figures/full_fig_p025_21.png]

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.