# A dust spectrum and a weak SMA image of HD 48682

This experiment asks how much a broad-band spectrum and a marginally resolved image can tell us about one optically thin dust population. It preserves six measured photometry detections and a small restored SMA image. It fits a deliberately restricted family of synthetic annuli. It does not fit a planet or reproduce the original observing pipeline.

The public download is a flat directory. Python 3.9 or later and its standard library are sufficient; no account, private checkout, package install, network connection or Lab Engine is needed.

```sh
python3 test_tutorial.py .
python3 analyze_debris.py --input . --output rerun
python3 verify_outputs.py --expected expected --actual rerun
```

In the repository, the equivalent commands are:

```sh
python3 scripts/astronomy/ac16/tutorial/test_tutorial.py docs/astronomy/ac16/tutorial/input
python3 scripts/astronomy/ac16/tutorial/analyze_debris.py --input docs/astronomy/ac16/tutorial/input --output docs/astronomy/ac16/tutorial/output
```

The run evaluates 448 discrete families, solves one nonnegative dust-mass amplitude analytically for each, and writes eight small output files. Depending on the computer, it takes tens of seconds. Ten significant digits in saved floating results prevent irrelevant last-bit differences from changing the archive.

## What was measured

`HD48682_calibratedvis_cont_natural.fits` is a restored image of 56 Aur / HD 48682 from Luca Matrà's REASONS archive, Zenodo record 12583396, version v0.0, DOI [10.5281/zenodo.12583396](https://doi.org/10.5281/zenodo.12583396). The archive explicitly declares CC BY 4.0. We redistribute the original 74,880-byte image unchanged with that attribution; the supplied fit notes are unchanged too. Its SHA256 is `a3c410f53c4ba3bce3ca9166e3bfc293a8c108be8b0c611650a3865230fcad2e`.

Despite its filename, this is not a visibility table. The separate CASA Measurement Set in the original archive contains calibrated visibility data. This exercise reads the delivered restored map, masks its 4,799 NaN pixels, and keeps its coordinates and restoring beam. It does not recalibrate, CLEAN, primary-beam-correct or infer missing spatial frequencies. The header does not specify a primary-beam correction or supply its response map; our central-image comparison is conditional on the delivered response. The selected sky aperture has radius 15 arcsec.

The map's continuum reference frequency is **CRVAL3 = 229.4830981104 GHz**, or 1306.381431 micrometres. **RESTFRQ = 230.538 GHz** is retained as a separate header convention and is not substituted for the continuum frequency. The beam is 7.210853577 by 6.844987392 arcsec, with position angle 84.94906616 degrees east of north. The pixel increments are −0.6236127589 arcsec in east and +0.6236127589 arcsec in north. The negative east increment matters when projecting an offset or an inclined annulus.

`HD48682_published_cold_photometry.csv` transcribes six numerical detections from Table 2 of [Hengst et al. (2020), DOI10.1093/mnras/staa1972](https://doi.org/10.1093/mnras/staa1972). These are reported observations with their reported uncertainties, not synthetic measurements. We select that paper's PACS 70/100/160-micrometre reduction once; the alternate reduction of the same observations is not an extra dataset. SPIRE 250/350-micrometre entries cite Eiroa et al. (2013), and SCUBA-2 850-micrometre cites Holland et al. (2017). We transcribe numerical facts and attribution, not the publisher's table formatting or paper figures. The archive's license is not claimed to cover the paper. Unverified upper-limit conventions and adjacent correlated synthetic Spitzer bands are excluded.

The [9 October2020 erratum, DOI10.1093/mnras/staa2874](https://doi.org/10.1093/mnras/staa2874), expands and corrects the SMA observing/calibration description. The separate REASONS product has the beam recorded in its own header. The erratum describes how the earlier2020 SMA image was calibrated and combined; both product definitions remain explicit here.

## The conditional physical population

We fix distance to 16.65 pc, stellar luminosity to 1.752 solar luminosities and stellar temperature to 6086 K, using Table 1 of Hengst et al. Their stellar atmosphere fit differs from the blackbody approximation used here. Stellar radius is derived from the fixed luminosity and temperature through the Stefan–Boltzmann law; its far-infrared and millimetre contribution is added to every prediction. The stellar mass, 1.17 solar masses, is recorded for context but does not enter this static thermal fit.

The dust occupies three narrow annuli at 0.9,1.0,1.1 times a trial radius, with mass fractions0.25,0.5,0.25. This width is a specified quadrature family, not a measured width. Grains have density2700 kg per cubic metre, a maximum radius of1000 micrometres, and a prescribed number law proportional to radius to the power−3.5. Twenty-four logarithmic size bins preserve its exact integrated mass fractions; their geometric-mean radii approximate within-bin emissivity. The exponent is not a measured or emergent collision-cascade slope.

For grain radius \(s\), wavelength \(\lambda\) and a trial positive exponent \(p\), the teaching absorption law is

\[
Q_{\rm abs}(s,\lambda)=\min\{1,(2\pi s/\lambda)^p\}.
\]

It describes neither a particular mineral nor scattering. A finite log-wavelength Simpson integral from1 nm to1 m enforces radiative equilibrium separately for each actual grain radius and annulus:

\[
\int Q_{\rm abs}B_\lambda(T_s)\,d\lambda
=\frac{R_\star^2}{4r^2}\int Q_{\rm abs}B_\lambda(T_\star)\,d\lambda.
\]

The same grain counts, temperatures and absorption efficiencies give the spectrum and the image:

\[
F_{\nu,j}=\frac{N_j\pi s_j^2 Q_{\rm abs}(s_j,\lambda)B_\nu(T_j)}{D^2}.
\]

The factor of pi is the grain's projected area; isotropic luminosity and geometric dilution cancel the other solid-angle factor. The code uses SI internally and divides flux density by10 to the power−26 to obtain Jy. Each annulus' summed flux is distributed uniformly over64 physical sky points, then passed through the actual restoring beam. An inclination is a projection of those same rings; it does not independently change their optically thin total flux.

Heating of very small grains is treated as equilibrium, gas drag is absent, and the absorption law and size slope are fixed within each family. Filter passbands and their color corrections, calibration covariances, mineral mixtures, thickness, stellar parameter errors and interstellar/background-source models are omitted. Those choices matter before an astrophysical parameter claim could be made.

## A beam is an angular unit, not just a blur

A normalized discrete convolution has sum one and preserves a map expressed in Jy per pixel. It does not turn that map into Jy per beam. The required angular areas are

\[
\Omega_{\rm beam}=\frac{\pi\theta_{\rm maj}\theta_{\rm min}}{4\ln2},\qquad
\Omega_{\rm pixel}=|\Delta x\Delta y|,
\]

with all angles in radians. For this image there are143.8115495 pixels per beam. Integrating a restored Jy/beam map therefore requires multiplying its pixel sum by \(\Omega_{\rm pixel}/\Omega_{\rm beam}\).

`restored_image.py` uses an equivalent direct forward map: a point containing1 Jy is spread over a peak-one Gaussian, whose angular integral is the actual beam area, and therefore has peak1 Jy/beam. The Gaussian axes follow the actual beam position angle. A separate finite angular integral tests this normalization. A second test distributes the same thermal annulus fluxes into the sky image and recovers their total after integration.

## How the ranking is calculated

For each fixed radius/size/opacity/geometry choice, call its dust prediction per reference mass \(d_k\), the fixed stellar contribution \(s_k\), the datum \(y_k\) and the trial nonnegative mass amplitude \(A\). We minimize

\[
S(A)=\sum_k w_k\,[y_k-s_k-A d_k]^2,
\quad
A=\max\left(0,\frac{\sum_k w_kd_k(y_k-s_k)}{\sum_k w_kd_k^2}\right).
\]

Differentiating the quadratic with respect to amplitude gives the ratio; the positivity constraint selects the zero boundary when its numerator is negative. The six photometry weights are reciprocal reported variances. The image contributes every second finite pixel within15 arcsec. Its noise scale is1.4826 times the median absolute deviation in the22–35-arcsec annulus after subtracting that annulus' median. Each sampled map point receives weight proportional to its sampled angular area divided by beam area, as well as reciprocal noise variance.

This approximate beam-area weight prevents oversampled pixels from being counted as independent measurements. It does **not** supply a CLEAN covariance matrix, calibrated chi-squared distribution, posterior or confidence interval. The image and SED scores are saved separately. No significance threshold or credible contour is derived from their sum.

The stellar point source is placed at the FITS reference pixel. We do not infer a propagated stellar position from this image, and a fitted dust offset is not converted into an eccentricity. The grid deliberately stays small: radii25,40,60,80,100,120,160 au; minimum sizes0.2,1,5,15 micrometres; absorption exponents1,2; inclinations0,60 degrees; position angles0,110 degrees; and either a centered ring or an east/north offset of−1/+3 arcsec. For a face-on ring the position angle is redundant, so the source quadrature uses one canonical orientation and reports its duplicate grid entries explicitly. Grid alternatives and offsets are teaching assumptions, not priors or astrometric measurements. A minimum at a grid boundary calls for an expanded radius or optical grid before assessing that parameter.

## Read the saved result before changing the model

The selected grid family has radius40 au, minimum grain radius5 micrometres and absorption exponent2. Its conditional spectrum score is5.0561 and image score9.2627. A very different25-au family with minimum size15 micrometres has combined score14.3248, almost the same as14.3188 for the first family. A60-au family with0.2-micrometre grains and exponent1 has combined score16.1689. These examples expose a radius/size/emissivity degeneracy within this limited model. They are not a calibrated probability set.

The derived masses differ too: about9.17,6.33 and7.09 times10 to the power21 kg for those three examples. Their values depend on the imposed size range, density and opacity. They are not total planetesimal-belt masses. The first family's continuum dust flux is about1.47 mJy. The source archive's own weak-geometry warning is preserved in `HD48682_fitnotes.txt`; our small grid does not supersede its visibility fit.

The deliberate wavelength-independent blackbody failure case searches a continuous0.5–160-au radius interval and positive normalization using the SED alone. Sixty-four golden-section steps locate the minimum; the radius is rounded to0.001 au before the final evaluation. Its best saved radius is about6.715 au and SED score28.73, appreciably worse than the grain-law alternatives. Calling the blackbody temperature a measured geometric radius would hide the emissivity dependence that this exercise is meant to expose.

`model-sed.csv` gives every measured flux, reported error, stellar term, dust term and residual. `conditional-profile.csv` keeps the best geometry for every radius/optical family, including poor fits. `geometry-profile.csv` retains all448 rows. `aperture-audit.csv` shows how finite aperture choice affects integrated map flux and gives an approximate beam-area noise scale. `model-image.csv` preserves all441 sampled sky points, model terms and residuals in Jy/beam, plus the explicit conversion to Jy per original pixel. `population.csv` saves the72 physical size/annulus populations with their grain counts and temperatures; these are the exact common inputs to both predicted observables. `fits-header.json` preserves the interpreted input header. `summary.json` records units, assumptions, numerical checks and the alternatives above.

The checks recover blackbody radiative balance, compare small-grain cooling with an independent dimensionless-frequency integral, integrate the elliptical beam numerically, close image/SED flux for one population, and refine the grain and thermal quadratures. These checks establish the defined numerical calculation. Grain optical measurements, stronger spatial constraints and dynamical alternatives would be needed to turn such a conditional fit into a physical system interpretation.