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Sandbox Physics
Debris disks from grain collisions to thermal evidence

AC16 · L3

Turn one grain record into a thermal image and spectrum

Sandbox Physics ·

The image and spectrum should be two sums over the same grains. A different picture drawn for each wavelength would break the link we want to test.

An optical law connects dynamics to temperature

For a controlled calculation, specify an absorption efficiency that approaches one at wavelengths short compared with grain circumference and declines as a power law at long wavelengths. This is a useful teaching boundary, not a mineral identification. The default exponent is one. Stellar heating uses the same function as thermal cooling, and the absorption-only momentum model uses its stellar average to compute beta.

Balance absorbed stellar power against emitted grain power. The star is a blackbody with radius derived consistently from luminosity and temperature. Its radiance is geometrically diluted at the actual grain position. After canceling the grain area, the equilibrium equation relates two Planck integrals. Small inefficient emitters can be hotter than blackbodies at the same distance.

Qabs(s,λ)=min⁡ ⁣[1,(2πsλ)p],L⋆=4πR⋆2σT⋆4Q_{\rm abs}(s,\lambda)=\min\!\left[1,\left(\frac{2\pi s}{\lambda}\right)^p\right],\qquad L_\star=4\pi R_\star^2\sigma T_\star^4
All optical predictions in this model inherit the chosen exponent and the omission of scattering.
(R⋆2r)2∫0∞QabsBλ(T⋆) dλ=∫0∞QabsBλ(Ts) dλ\left(\frac{R_\star}{2r}\right)^2\int_0^\infty Q_{\rm abs}B_\lambda(T_\star)\,d\lambda=\int_0^\infty Q_{\rm abs}B_\lambda(T_s)\,d\lambda
The factor of four comes from projected absorption area versus full emitting surface area.

Reading:

Check the integral before trusting the lookup

The model evaluates Planck integrals on logarithmic wavelength from one nanometre to one metre. A logarithmic integration variable needs its wavelength Jacobian. Omitting it changes the physical integral, even if a smooth temperature curve still appears. An independent high-resolution quadrature recovers the blackbody integral and the analytic blackbody equilibrium temperature.

The browser interpolates equilibrium temperature on a bounded radial table. Trajectories and populations continue to come from the actual integration. Held-out radii are compared with a fourfold finer quadrature. There is no dynamical surrogate behind the planet drag. Distances outside the 0.2 to 160 astronomical-unit table are outside the declared model domain.

As a saved analytic example, a blackbody at 20 astronomical units from a solar-luminosity star has an equilibrium temperature near 62.2 kelvin. Moving it to 80 astronomical units halves that temperature. Small-grain temperatures must instead solve the efficiency-weighted equation; changing the radius while keeping the blackbody temperature would erase part of the wavelength-dependent morphology.

∫0∞Bλ(T) dλ=σT4π,TBB=(L⋆16πσr2)1/4\int_0^\infty B_\lambda(T)\,d\lambda=\frac{\sigma T^4}{\pi},\qquad T_{\rm BB}=\left(\frac{L_\star}{16\pi\sigma r^2}\right)^{1/4}
The first identity independently checks the finite quadrature; the second follows only when efficiency is unity.

Reading:

From one parcel to spectral flux

For an optically thin parcel, every grain emits independently. Its spectral luminosity is emitting area times pi times radiance; dividing by the sphere around the observer leaves projected grain area times radiance divided by distance squared. Multiplying by physical grain number gives the parcel flux. Sum these terms for the complete synthetic spectral distribution.

The plotted ordinate is frequency-density flux in janskys, even when wavelength labels the horizontal axis. Bolometric integration requires frequency measure, or the correct Jacobian when switching to wavelength. Simply adding values along a logarithmic wavelength axis does not yield luminosity. The same caution applies when comparing a broad instrumental band with a monochromatic calculation; the tutorial explicitly uses published monochromatic flux conventions.

fν,i=Niπsi2Qabs(si,ν)Bν(Ti)D2,Fν=∑ifν,if_{\nu,i}=\frac{N_i\pi s_i^2Q_{\rm abs}(s_i,\nu)B_\nu(T_i)}{D^2},\qquad F_\nu=\sum_i f_{\nu,i}
One jansky equals ten to the minus twenty-six watts per square metre per hertz.
Fλ=Fνcλ2,Fbol=∫Fν dν=∫Fλ dλF_\lambda=F_\nu\frac{c}{\lambda^2},\qquad F_{\rm bol}=\int F_\nu\,d\nu=\int F_\lambda\,d\lambda
Frequency and wavelength spectral densities are different quantities.

Reading:

Project the actual positions, then convolve

Choose east left and north up on the sky. Tilt the physical grain coordinates by the stated inclination, place each parcel flux into a pixel, and keep any flux outside the field as a separate image-edge loss. No ideal annulus is substituted for the active record. The visible stage selects grains above 1.5 micrometres for legibility, but the thermal sums retain all resolved bins.

A discrete beam kernel with sum one preserves the sum of pixel flux if the image is large enough. Finite edges leak flux; recording that leakage makes the closure test meaningful. A wider field and different pixel size should recover the same integrated source flux, within the reported loss and discretization error. Beam smoothing changes the appearance and spatial information, not the physical grain temperatures.

Pj=∑i↦jfν,i,Iν,j=PjΩpix,Pjconv=∑kKj−kPk,∑kKk=1P_j=\sum_{i\mapsto j}f_{\nu,i},\quad I_{\nu,j}=\frac{P_j}{\Omega_{\rm pix}},\quad P_j^{\rm conv}=\sum_kK_{j-k}P_k,\quad\sum_kK_k=1
Pixel flux has units Jy/pixel; dividing by pixel solid angle gives Jy per steradian. The normalized convolution retains the pixel-flux convention.

Reading:

Worked unit conversion for the actual SMA image

The real HD 48682 image has an approximately 7.21 by 6.85 arcsecond restored Gaussian beam and 0.624 arcsecond square pixels. A beam covers about 144 pixel solid angles. An image value in janskys per beam is therefore not a pixel flux. Summing such values without conversion overestimates integrated flux by roughly that factor.

The tutorial reads the actual FITS header, including negative right-ascension increment, beam position angle and continuum reference frequency. It masks nonfinite edge pixels. The continuum axis reference frequency is not silently replaced with the nearby rest-frequency keyword. After converting the map to pixel flux, an aperture sum has a clear unit and a model can be convolved with the actual elliptical beam.

CLEAN-restored pixels have correlated noise, and some extended emission may be filtered by missing interferometer spacings. Beam-area weighting can make a useful conditional diagnostic, but it does not turn every pixel into an independent Gaussian measurement. This exercise reports conditional comparisons, not calibrated confidence intervals on a planet or a unique belt geometry.

Ωb=πθmajθmin4ln⁡2,Pj[Jy/pixel]=Ij[Jy/beam]ΩpixΩb\Omega_b=\frac{\pi\theta_{\rm maj}\theta_{\rm min}}{4\ln2},\qquad P_j[\mathrm{Jy/pixel}]=I_j[\mathrm{Jy/beam}]\frac{\Omega_{\rm pix}}{\Omega_b}
Angular dimensions must use radians in solid-angle calculations. A sum-one kernel alone does not perform this conversion.

Reading: Matrà · REASONS archive 12583396, version v0.0, CC BY 4.0 · Hengst et al. 2020 · HD 48682, Tables 1–2

A joint fit has to survive a joint check

Changing minimum grain size, optical exponent and belt radius can compensate for one another in a spectral fit. A spatial image adds information, but a weak and broad beam may leave substantial alternatives. Use the same physical population in both predictions, include the stellar contribution where relevant, and inspect residuals before comparing scores.

The public exercise saves a deliberately restricted blackbody alternative as well as more flexible grain models. A bad restricted fit is evidence against that conditional model. A successful flexible fit still inherits its adopted star, grain law, size limits, geometry and noise treatment. The next page turns these statements into a flat-folder calculation with original inputs and expected outputs.

Reading:

Saved calculation, readable without the Lab

Solar star, planet mass 0.003 solar masses, circular 13 AU orbit, initial normal optical depth 0.0003. Eight grain bins, 384 initial weighted parcels, RK4 at 128 steps per planet orbit, finite-cell catastrophic collisions and a 128-per-bin compression limit. The representative record is eight planet periods old.

QuantitySaved resultInterpretation
Elapsed time374.42 yr374.42\,\mathrm{yr}A real finite integration window
Weighted collision events590Statistical packet events, not individual grain count
Maximum normal optical depth0.0019440.001944Finite-cell column estimate
Largest raw kinetic compression jump3.16%3.16\%Before explicit velocity-moment correction
Conservative probe angle range4.6919 rad4.6919\,\mathrm{rad}Separate 150-period record, not the whole collisional cloud

The stochastic four-period refinement changes 850 µm flux by about 0.7–4.0% and 70 µm flux by 8.7–32%. Centroid and low-order density moments change enough that individual cloud clumps are not a converged planet inference. The independent conservative probe only establishes bounded angle behavior for its specified initial state and saved 150-period window.

Inspect actual refinement records · Inspect the all-parent finite stress counterexample

Test one-record radiative closure

  1. Pause a preset and change observing wavelength while retaining its physical grain record.
  2. Compare integrated map flux with the marked spectral point, then examine image-edge loss.

Exercises with explanations

  1. A sum-one convolution leaves 100 janskys summed across a model image whose units are Jy/pixel. Is the new sum automatically 100 janskys per beam?

    Read the explanation

    No. The total is still 100 janskys and the individual values still follow the pixel-flux convention. Converting each value to Jy/beam needs multiplication by beam solid angle divided by pixel solid angle.

  2. Why does a small temperature error matter more on the short-wavelength side of a cold dust spectrum?

    Read the explanation

    On the Wien side, Planck radiance depends exponentially on inverse temperature. A modest fractional temperature change can produce a much larger fractional flux change. Validate flux errors in the relevant bands as well as temperature errors.

Sources and calculation scope

  1. Matrà · REASONS archive 12583396, version v0.0, CC BY 4.0

    Original unchanged HD 48682 restored SMA FITS and fit notes; image units and actual beam retained. The filename does not make it a visibility table. Source-paper context: arXiv 2501.09058v1, 15 January 2025, A&A 693 A151.

  2. Hengst et al. 2020 · HD 48682, Tables 1–2

    Published stellar values and six numerical cold-photometry detections; one PACS reduction only. Unverified upper-limit conventions and correlated synthetic spectral bins are excluded.

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