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Sandbox Physics
Molecular line cubes: space, velocity and hidden depth

Molecular line cubes · L3

Replay real ALMA velocity slices of the TW Hya disk

Sandbox Physics ·

We now work with a real CO image cube. Read its frequency coordinates and per-beam units first, then derive connected plots from one saved observation. The exercise teaches measurement operations without fitting a prescribed cloud to a real disk.

Identify the disk and the processing starting point

TW Hya is a protoplanetary disk around a young star. The image traces the ALMA Science Verification observation of 22 April 2011, project ADS/JAO.ALMA#2011.0.00001.SV. The official guide by Bendo, Avison and Freestone is dated 19 June 2019.

The named teaching member, TWHydra_CO3_2line.image.fits, is already imaged; its history includes continuum subtraction. We start there, without rerunning visibility calibration or CLEAN. We use the disk to study measurement methods. A molecular-cloud observation would be needed to test cloud-specific claims.

Unpack the standalone science ZIP into a new directory. Its optional reader obtains only two fixed files: the 4,763,520-byte teaching FITS and the 444,204-byte LAMDA table, checking complete SHA256 identities. The ZIP does not redistribute either file or observation-derived images. Public scientific access alone does not establish redistribution rights for a separately served teaching derivative.

python3 scripts/astronomy/ac36/optional_reader.py --source-dir /absolute/private/sources --download
python3 -W error::RuntimeWarning scripts/astronomy/ac36/replay.py --native-source-dir /absolute/private/sources --full-mesh --output /absolute/new/path/alma-review

Reading: Bendo, Avison & Freestone, Measuring the Rotation of TW Hydrae (19 June 2019) · ALMA Science Verification: TW Hya Band 7 CO(3–2), ADS/JAO.ALMA#2011.0.00001.SV · Sandbox Physics independent molecular-line science producer and reader package

Recover sky and velocity from the header

FITS axis order reverses the displayed array order. This image has 100 by 100 sky samples and 118 frequencies; the Python array is indexed channel, row, column. The strict reader follows all 513 header cards to the actual data offset of 43,200 bytes.

The spectral axis is FREQ, the frame is LSRK and VELREF is 257. Recover frequency with one-based FITS pixels, then use the native rest frequency 345.795990 GHz345.795990\,\mathrm{GHz} for radio velocity. Decreasing frequency gives increasing positive recession velocity.

Samples are approximately 0.120 km s−10.120\,\mathrm{km\,s^{-1}} apart. The original effective resolution is about 0.2 km s−10.2\,\mathrm{km\,s^{-1}}, with Hanning and regridding correlations. Sampling and resolution are distinct. The synthetic cloud’s rectangular channels do not reproduce this exact instrumental response.

Sky conversion uses the native SIN projection and reference position. The producer verifies forward and inverse coordinates before labeling an angular map. Relabeling raw row and column indices would bypass that projection check.

ν(p)=CRVAL3+(p−CRPIX3)CDELT3,v=c(1−ννrest)\nu(p)=\mathrm{CRVAL3}+(p-\mathrm{CRPIX3})\mathrm{CDELT3},\qquad v=c\left(1-\frac{\nu}{\nu_{\rm rest}}\right)
Pixels here are one-based. The radio convention defines the conversion; LSRK identifies its velocity reference frame.

Reading: Bendo, Avison & Freestone, Measuring the Rotation of TW Hydrae (19 June 2019) · ALMA Science Verification: TW Hya Band 7 CO(3–2), ADS/JAO.ALMA#2011.0.00001.SV · Sandbox Physics independent molecular-line science producer and reader package

Distinguish per-beam brightness from aperture flux

The image unit is Jy/beam. Its Gaussian restoring beam is about 1.68065×1.53193 arcsec1.68065\times1.53193\,\mathrm{arcsec} at position angle 22.30 degrees. Converting per-beam flux to intensity uses the beam’s solid angle. A chosen aperture defines a separate angular sum.

The computed beam solid angle is 6.85696×10−11 sr6.85696\times10^{-11}\,\mathrm{sr}. Divide a value in Jy/beam by this angle and apply the Jy-to-SI factor. Converting intensity to RJ-equivalent kelvin then uses the actual frequency of each channel.

The saved analysis contains both the mean Jy/beam spectrum inside radius 0.85 arcsecond and integrated Jy inside radius six arcseconds. The latter sums pixels with pixel-solid-angle over beam-solid-angle weights. They have different units and answer different angular questions.

Ωb=πθmajθmin4ln⁡2,Iν=Sν,beam 10−26Ωb,TRJ=c2Iν2kν2\Omega_b=\frac{\pi\theta_{\rm maj}\theta_{\rm min}}{4\ln2},\quad I_\nu=\frac{S_{\nu,\rm beam}\,10^{-26}}{\Omega_b},\quad T_{\rm RJ}=\frac{c^2I_\nu}{2k\nu^2}
Convert beam axes to radians. Aperture flux requires an angular sum in addition to this intensity conversion.

Reading: Mangum & Shirley, How to Calculate Molecular Column Density, arXiv 1501.01703v4 (2017) · Bendo, Avison & Freestone, Measuring the Rotation of TW Hydrae (19 June 2019) · Sandbox Physics independent molecular-line science producer and reader package

Expose mask choices and correlated channels

Begin with a fixed line window from −0.5 to 6.5 km s−1-0.5\text{ to }6.5\,\mathrm{km\,s^{-1}} and the six-arcsecond source aperture. Integrate signed brightness. Next compare masks retaining only positive values above two and three empirical noise dispersions. Predict both weak-wing loss and positive-noise selection before opening the plots.

The noise sample comes from a nine-to-thirteen-arcsecond annulus, conditionally treated as source-free. The calculation estimates channel covariance and checks it against independent scalar dots and the variance of a weighted projection. Its adjacent-channel correlation makes the independent-sample square-root rule inappropriate.

The native-audit output saves one NPZ containing processed brightness, three masks, aperture spectra and covariance. Native-figure reads that record to draw channels, PV, moments and rebinning comparisons. The plotting selection uses zero-based indices 52, 58 and 64; the figure labels the corresponding one-based channels 53, 59 and 65. Input negative noise has not been clipped away.

Open the channel montage beside the saved PV cut. Follow a red-side feature through neighbouring channels, then locate the same sky row in PV. This connects an image patch to its spectral position before any dynamical interpretation.

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

State the conclusion at the supported scale

The disk’s red–blue pattern reveals ordered line-of-sight motion. A stellar-mass fit would additionally need inclination, distance, beam and spectral response, disk structure and an uncertainty model. Historical guide distances and inclinations are not upgraded here to current precision constants.

The optional native species calculation sums the 41 supplied ground-vibrational rotational levels in LTE. It does not solve collisional excitation or certify the omitted physical CO partition tail. Brightness temperature, excitation and kinetic temperature retain their different meanings.

A successful exercise leaves traceable plots and diagnoses: a named velocity frame, explicit angular weights, visible mask choices and retained channel correlation. These outputs establish the measurement operations on which a later physical fit would depend.

Reading: Bendo, Avison & Freestone, Measuring the Rotation of TW Hydrae (19 June 2019) · ALMA Science Verification: TW Hya Band 7 CO(3–2), ADS/JAO.ALMA#2011.0.00001.SV · Leiden Atomic and Molecular Database, fixed CO data file

Exercises and explanations

  1. What is lost if the 0.12-kilometre-per-second spacing is treated as independent resolution elements?

    Read the explanation

    The effective response and off-diagonal channel covariance are omitted. Additional samples do not automatically provide the same number of independent measurements; check rebinned noise against the saved covariance or direct blank-region statistics.

  2. Why is mean Jy/beam inside radius 0.85 arcsecond not the disk’s integrated flux in Jy?

    Read the explanation

    It is a local average, without the pixel-area over beam-area sum, and it uses a different aperture. Specify angular domain and weights before comparing fluxes.

Try the prediction

Compare measurement operations

  1. Link a sightline, channel and spectrum in the Lab, then perform the corresponding operation in the local real outputs. Compare coordinates and operations while retaining the disk/cloud distinction.
  2. Compare signed and positive-threshold integrals locally. Identify weak-wing loss and noise-selection bias separately.
  3. Compare the measured noise of an adjacent-channel average with the covariance prediction.

Standalone scientific materials

Download the scientific ZIP · Read the setup and calculation scope

The standalone ZIP recomputes independent synthetic physics. The optional real exercise obtains two exact files locally, then generates actual observed-data plots. Original observations, native spectroscopy and observation-derived plots remain local.

Sources and calculation scope

  1. Mangum & Shirley, How to Calculate Molecular Column Density, arXiv 1501.01703v4 (2017)

    Fixed v4: transfer and brightness equations 13–32, partition section 7, approximations 11–14 and appendices A–C. The printed Gaussian-width power in A4/A5 is checked against the normalized profile; no blanket non-LTE solution is inherited.

  2. Bendo, Avison & Freestone, Measuring the Rotation of TW Hydrae (19 June 2019)

    Complete eight-page official teaching guide and named original image member. The local tutorial uses native header frequencies, beam area and saved processing history; historical distance/inclination are not promoted to current precision.

  3. ALMA Science Verification: TW Hya Band 7 CO(3–2), ADS/JAO.ALMA#2011.0.00001.SV

    Original 2011-04-22 disk observation and Hanning/effective-resolution provenance. Public SV access/acknowledgment is distinct from precise redistribution terms for separately served teaching derivatives.

  4. Leiden Atomic and Molecular Database, fixed CO data file

    Exact native table identity is checked in the optional local reader. Its 41 ground-vibrational levels and spectroscopy are separate from the own two-level Lab material; collisional rates are not solved or redistributed here.

  5. Sandbox Physics independent molecular-line science producer and reader package

    Own synthetic input, independent frequency-space/continuous-ray references and retained resolution failures. The native reader obtains exact source files locally; no observed image, native table or app/Node Engine is bundled.

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