Read dust rings and clumping thresholds without skipping a step

Start with the actual brightness profile

Open the official DSHARP release for Elias24. Its public ASCII product has 300 radial samples and eight columns. The original README describes radius, intensity per synthesized beam, approximate pointwise errors and two brightness-temperature conversions. The current file contains the radius-column correction made in 2019 and propagated to the release pages in 2020.

The measured profile below uses every saved intensity sample, including negative noisy outer values. In a declared 45–65 au window the minimum lies at 57.5 au; in a 65–95 au window the maximum lies at 76.5 au. Their brightness ratio is 43.3 for these selected bins. The numerical ratio is a property of the measured profile and search windows, not a planet mass or gas-pressure contrast.

Bins spaced by one au do not become independent observations: the synthesized beam spans several au. Read the plotted error bars as the published pointwise uncertainties. A confidence interval for a fitted ring width would need spatial covariance and calibration treatment.

Observed DSHARP Elias24 continuum intensity and published pointwise errors, all300source rows. Radius is in au; the vertical axis uses a fixed asinh stretch to retain the inner peak, outer ring and negative noise. Samples are correlated by the synthesized beam. This is brightness, separate from the synthetic density fields.

Reading: ALMA DSHARP official data release and revised radial profiles · DSHARP radial-profile original README · Huang et al.2018, DSHARP II Annular Substructures, v1 · Andrews et al.2018, DSHARP I Motivation, Sample, Calibration and Overview

Check the beam before converting intensity

The frozen FITS primary header gives frequency $231.943014843\,\mathrm{GHz}$ and Gaussian beam widths $0.0367535\,\mathrm{arcsec}$ by $0.0342672\,\mathrm{arcsec}$ . These are measured imaging metadata. The public package contains the exact header blocks, not the large image or its pixels.

Convert the beam widths to radians and use $\Omega_{\rm beam}=\pi\theta_{\rm maj}\theta_{\rm min}/(4\ln2)$ . Divide Jy/beam by beam solid angle and multiply by $10^{-26}$ to obtain spectral radiance in SI. The public brightness replay reproduces the archive’s Rayleigh–Jeans and full-Planck columns within $0.000854\,\mathrm K$ and $0.000738\,\mathrm K$ , respectively; the small residual includes rounded source columns.

The Planck inversion is defined only for positive intensity. There are 31 nonpositive intensity samples in this file. They remain in the intensity plot; their brightness temperature is stored as null. Temperature and opacity still have to enter a physical interpretation of intensity, so an emission ring alone cannot identify its formation mechanism.

T_{\rm B}=\frac{h\nu/k}{\ln[1+2h\nu^3/(c^2I_\nu)]}

This is the temperature of a blackbody with the same radiance. Relating it to the material temperature requires the emitting optical depth.

Reading: DSHARP radial-profile original README · ALMA DSHARP official data release and revised radial profiles · Huang et al.2018, DSHARP II Annular Substructures, v1

Read a clumping map with the right density ratio

Now open Li & Youdin 2021 version 2. Section 2 sets up vertically stratified simulations. Figure 1 maps strong particle clumping against Stokes number and surface metallicity $Z=\Sigma_{\rm d}/\Sigma_{\rm g}$ . This is not the local volume ratio $\epsilon=\rho_{\rm d}/\rho_{\rm g}$ used in the uniform linear calculation.

Inspect how the paper measures the dust layer thickness before it translates a surface abundance into a midplane loading. Keep pressure support and stirring with the plotted thresholds. Reading a threshold from one slice of parameter space and applying it to a different disk silently changes the calculation.

The map concerns strong clumping in the specified simulations. A criterion for gravitationally bound bodies additionally needs gravity, tidal conditions and a particle-collapse calculation. A concentration threshold and a prediction of planet formation are different steps.

Reading: Li & Youdin 2021, Thresholds for Particle Clumping, v2

Compare the paper’s linear maps with its nonlinear runs

In the same paper, Section 3.5 and Figure 7 impose wavelength and resolution restrictions on unstratified linear modes. Check which waves fit within the measured dust layer and which the numerical grid can resolve. A formal fastest root at an arbitrarily short wavelength may be unavailable in the actual simulation.

Figure 8 compares the relevant linear rates with stratified clumping outcomes. Read it as a test of the tempting idea that faster infinitesimal growth always gives stronger clumping. The paper finds a poor correlation. It explains three changes between the questions: linear to nonlinear evolution, unstratified to stratified gas, and growth of a mode to concentration of particles.

This is a useful place to compare your tutorial. Its periodic unstratified grid tests the implementation of one linear mode. The paper’s clumping map answers a later question about a settling and mixing layer. They can inform each other while retaining their different states and observables.

Reading: Li & Youdin 2021, Thresholds for Particle Clumping, v2 · Youdin & Goodman 2005, Streaming Instabilities, arXiv v1

Ask what a diffusion coefficient represents

Open Umurhan, Estrada & Cuzzi 2020 version 4, Sections 3.1–3.3 and Appendix A. Locate the gas viscosity, particle diffusion and particle-pressure terms separately. The background may also contain viscous forcing. These terms enter more than a single subtraction from the final growth rate.

Follow how their turbulent dilution model links stirring to the local volume loading in Section 5. Changing turbulence can alter both the perturbation damping and the background dust layer. This is why changing one diffusion coefficient while fixing every other property is a controlled teaching comparison with a narrower purpose.

Li & Youdin’s Section 3.5.3 offers a complementary warning: a simple diffusion-damped linear-rate estimate fails to predict their nonlinear stratified clumping. Retain the physical closure, fitted or measured diffusion, geometry and the available modes when comparing different threshold claims.

Reading: Umurhan, Estrada & Cuzzi 2020, Streaming Instability in Turbulent Disks, v4 · Li & Youdin 2021, Thresholds for Particle Clumping, v2

Write a next question that can be measured

For the observed Elias24 ring, a useful next question is whether gas kinematics support a pressure maximum at the same location. For a linear calculation, ask whether the growth is resolved and survives the specified mixing model. For a dense simulated clump, ask whether self-gravity overcomes tides and dispersion. Each question suggests a different measurement or calculation.

Huang and colleagues’ DSHARP II Section 5 discusses several possible origins of annular substructures. Keep the real brightness profile beside those alternatives. The synthetic radial trap makes gas–dust transport tangible, and the linear tutorial tests a collective mechanism; neither supplies a unique explanation for this observed ring.

Reading: Huang et al.2018, DSHARP II Annular Substructures, v1 · Li & Youdin 2021, Thresholds for Particle Clumping, v2 · Youdin & Goodman 2005, Streaming Instabilities, arXiv v1

Exercises and explanations

Why does a high local dust-to-gas ratio not specify a unique surface-metallicity threshold?
Read the explanation
The local ratio depends on how vertically concentrated dust is relative to gas. Stirring, settling and layer geometry connect volume density to surface density. Those can differ between models even at the same surface metallicity.
Your linear root is positive only at wavelengths shorter than a grid can resolve. What can that simulation test?
Read the explanation
It can test the modes represented by its domain and stencil. An unresolved continuum root is not evidence that the numerical calculation follows that mode. Refine the grid or choose a resolved mode, then measure the actual growth.

Try the prediction

Keep the three stages beside each other

Prepare a radial trap and identify the synthetic gas-pressure and dust-density quantities.
Choose a linear wave and identify the perturbation amplitude and its validity stop.
Compare those with the measured DSHARP intensity figure. Name the additional physics required to move from one stage to another.

Save these scientific files together and install the pinned NumPy/SciPy versions in requirements.txt. They run without a private checkout or interactive Lab engine.

Sources and calculation scope

Youdin & Goodman 2005, Streaming Instabilities, arXiv v1
10 September 2004 version 1; ApJ 620,459–469(2005). Full text read, Section 2 equations 1–7/12–17/24–31, Figures 1–3 and Section 4 mode geometry; Section 5 equation 39 and its restricted terminal-velocity domain; Section 6 distinguishes arbitrary linear amplitude from subsequent collapse. The PDF itself prints a later Draft Modified July 5, 2018 header; the archive version is explicitly frozen.
Li & Youdin 2021, Thresholds for Particle Clumping, v2
14 October 2021 version 2; DOI 10.3847/1538-4357/ac0e9f. Full-text Sections 2,3.1–3.5 and 4: stratified simulation setup, surface metallicity versus volume loading, threshold map Figure 1, linear-growth maps Figure 7 and clumping comparison Figure 8. Equation 21 diffusion diagnostic fails to predict all nonlinear stratified clumping, explicitly cautioning against universal linear thresholds.
Umurhan, Estrada & Cuzzi 2020, Streaming Instability in Turbulent Disks, v4
25 May 2020 version 4, PDF header 26 May. Sections 3.1–3.3, equations 4–18 and Appendix A: explicit viscosity, dust diffusion/pressure and equilibrium forcing; Section 5 interprets growth with a turbulent dilution model. The Lab’s equal damping of all linear fields is a much simpler declared teaching closure, not a reproduction of this alpha-disk model.
ALMA DSHARP official data release and revised radial profiles
Project2016.1.00484.L, release snapshot28October2026. Exact Elias24 ASCII profile last-modified5August2019, all300rows/eight columns; original README updated5February2020; exact8640-byte FITS primary header bounded-range fetched, no image pixels. Hashes, units, source terms and observing/header metadata are retained in the public package.
DSHARP radial-profile original README
All eight column definitions, Jy/beam intensity/errors, Rayleigh–Jeans and full-Planck temperature conversions, negative-intensity NaNs and the August2019 radius correction propagated in February2020. Pointwise uncertainties are not an independent-bin covariance matrix.
Huang et al.2018, DSHARP II Annular Substructures, v1
10December2018 version 1; ApJL869,L42. Full-text Section 3.1 radial-profile construction and Section 5 interpretations; annular emission can have multiple origins. The real Elias24 brightness profile is read separately from the synthetic gas/dust calculations. DSHARP I, ApJL869,L41, is also cited by the official release for calibration/beam provenance.
Andrews et al.2018, DSHARP I Motivation, Sample, Calibration and Overview
Published ApJL869,L41; the official public release requests this calibration/survey citation together with DSHARP II. Survey abstract, official release description and original Elias24 FITS header were consulted. We reuse the released profile and header, not its CASA calibration pipeline or visibilities.