Reading one cube as maps, spectra and moments

Name the three coordinates first

A PPV cube has two sky coordinates and one recession-velocity coordinate. Fix velocity for a channel image. Fix a sky pixel for a spectrum. Take every velocity along an image row for a position–velocity cut.

The interactive record contains $128\times128\times96$ brightness samples. Its length unit is 0.1 parsec, the saved sky window is one parsec wide and velocity edges span $\pm4\,\mathrm{km\,s^{-1}}$. The observer remains on the positive-depth side. Rotating the display camera changes your view of the physical cloud without changing that observer’s data.

Colours in the physical cloud identify local velocities. Transfer through overlapping depths and beam mixing produces the measured quantity in the separate data row. A local velocity highlight is not the escaping intensity from a visible surface.

Every diagnostic and export reads the latest completely accepted record. While new settings compute, the previous record remains intact. Combining a new channel image with an old spectrum would create an inconsistency that no physical cloud produced.

Reading: Sandbox Physics independent molecular-line science producer and reader package · Mangum & Shirley, How to Calculate Molecular Column Density, arXiv 1501.01703v4 (2017)

Ask a distinct question of each moment

Let $T_i$ be the saved average brightness and $v_i$ the centre of channel $i$. The zeroth moment is line area. The first is brightness-weighted mean velocity. The second central moment measures velocity spread around that mean. The last two become fragile when their shared denominator approaches zero.

An equal double peak has its mean in the intervening dip. Mean velocity need not identify a parcel. Width can combine separate components, opacity broadening and a velocity gradient within the beam, so it does not by itself determine temperature or turbulent energy.

For a numerical example, take a transparent Gaussian with dispersion $250\,\mathrm{m\,s^{-1}}$ and channel width $83.333\,\mathrm{m\,s^{-1}}$. Replacing each bin’s brightness by a weight at its centre adds approximately $\Delta v^2/12=578.7\,\mathrm{m^2\,s^{-2}}$ to the variance in the smooth, fine-channel limit. The resulting discrete width is about $251.2\,\mathrm{m\,s^{-1}}$. This is a useful limit, not a universal correction for narrow or truncated lines.

Reading: Sandbox Physics independent molecular-line science producer and reader package

A beam brings neighbours into a spectrum

A finite beam, the telescope’s angular response, mixes nearby sky positions. When their peaks occur at different velocities, mixing can broaden the spectrum of one image pixel. The spatial averaging has broadened the line even though the local microscopic motion stayed fixed.

The model uses a nonnegative, pixel-integrated Gaussian kernel. It pads the sky with zeros, computes the full convolution and then crops the saved window. Total signal is conserved on the full convolution plane. Cropping can remove signal that the beam carried beyond the map.

A retained edge counterexample loses about 0.589 percent of integrated signal from the saved map. Renormalizing that crop to its input total would hide the loss and change its surface brightness. The interactive model keeps a tighter map-loss gate and retains the last valid record on rejection.

Refine sky and velocity sampling separately. Some original 64-pixel sky cases failed the independent reference; the 128-pixel version passed those registered comparisons. A quarter- or half-pixel translation tests whether a narrow feature happens to land at a pixel centre or between samples. Cold, thin, maximum-shear and these translated cases were added explicitly. An error theorem over every continuous control combination remains unproved; the evidence covers those particular calculations.

Reading: Sandbox Physics independent molecular-line science producer and reader package · Bendo, Avison & Freestone, Measuring the Rotation of TW Hydrae (19 June 2019)

A mask selects noise too

A mask selects which pixels and channels enter a calculation. Imagine a signal-free noise distribution symmetric about zero. Keep only samples above three standard deviations and every retained sample is positive. Adding them can manufacture a positive apparent line area.

The same threshold removes genuine weak wings, reducing signal area and width. The independent counterexamples retain both effects. The noiseless Lab instead uses a relative-brightness display mask to hide unstable moments in extremely weak pixels. That mask is not a significance test.

The real ALMA teaching cube has Hanning-related channel correlations. In the declared blank annulus and velocity window, the estimated adjacent-channel correlation is about 0.668. A two-channel sum then has roughly 1.29 times the noise standard deviation predicted by independence. This describes that empirical window, without certifying all channels or adaptive masks.

A fixed-window linear sum can propagate a covariance matrix. Selecting a mask from the same noisy data adds a selection effect. The real exercise therefore preserves signed fixed-window integrals alongside positive-threshold integrals.

Reading: Sandbox Physics independent molecular-line science producer and reader package · Bendo, Avison & Freestone, Measuring the Rotation of TW Hydrae (19 June 2019)