A moving cloud through a spectral line

Molecular Line Cubes and Velocity Channels

Why does the same cloud look different in different velocity channels? Each channel collects gas with a similar speed along the line of sight. Rotation sends different parts of the cloud into different channels. Gas at widely separated depths can share that speed, so a channel image groups motion rather than selecting one physical depth.

Drag a rotating cloud and its axis, then scan its line. Follow the same gas into a channel image, a spectrum and a position–velocity cut.

Probe in this channel $T_{\rm ch}=4.56\,\mathrm K$

Probe spectral peak $T_{\rm pk}=11.3\,\mathrm K$

Probe integrated line $W=12.4\,\mathrm{K\,km\,s^{-1}}$

Probe velocity centroid $v_{\rm c}=0.0163\,\mathrm{km\,s^{-1}}$

A cloud at one velocity

Velocity channel

0.625

v_{\rm ch}/(\mathrm{km\,s^{-1}})

Rotation speed at the long radius

1.00

v_{\rm rot}/(\mathrm{km\,s^{-1}})

Axis angle to the observer

60^\circ

i/{}^\circ

Central species column

17.0

\log_{10}[N_0/(\mathrm{cm^{-2}})]

Drag the cloud or its axis; drag empty space to orbit the camera. Bright points emphasise relative local channel-selection probability, not observed intensity.

Calculating the complete default line cube.

Positions, material and model

Cloud centre across the image

0\,\mathrm{pc}

Cloud centre up the image

0\,\mathrm{pc}

Own two-level teaching line. Material temperature and microscopic dispersion change only with presets.

This is a static isothermal rotating ellipsoid with LTE excitation, a Planck source and no inward flow. The physical observer is fixed along the positive depth axis.

Cloud points sample prescribed density and velocity. The local Gaussian channel weight is divided by that point’s largest weight among the selectable equal-width channels; faint context preserves the full cloud. This display ratio is not escaping intensity. Arrow direction is physical velocity, with compressed lengths. The data image alone includes transfer and the declared beam.

Finite registered and compound-extreme samples passed independent finer references. They do not supply one error theorem for every continuous setting.

The relative-brightness display mask excludes faint model wings from centroid colours. It is not a calibrated detection threshold.

Recompute the separate real ALMA disk exercise

Model assumptions

This static prescribed ellipsoid uses an exactly two-level teaching transition at 345 GHz, local LTE, an isothermal Planck source and rigid rotation. It is separate from real CO and the fixed ALMA TW Hya disk exercise. Thermal and preset microscopic dispersion, positive finite-channel averaging and a normalized pixel-integrated Gaussian beam produce one 128-by-128-by-96 cube. The fixed physical observer lies on the positive depth axis; display-camera rotation leaves that observer unchanged. Depth in the physical scene is model geometry; the cube’s third coordinate is recession velocity. Cloud-centre and axis changes recompute a full state. Unsupported or failed candidates retain the last accepted state; finite spectral-tail, beam-flux and finite-map-loss limits are checked. Fixed and withheld cases and compound cold/thin/shear/phase endpoints were compared to independent finer references; this finite evidence is not a uniform error theorem over every continuous setting. Temperature and microdispersion change only through tested presets. There is no dynamical collapse, solved turbulence, non-LTE excitation, dust continuum, scattering, interferometric imaging or inferred true cloud mass/depth. Points sample the prescribed physical field. Arrow directions follow local velocity and lengths are schematically scaled; data images below read only the same accepted cube. The relative-brightness display mask is a model-viewing choice, not a calibrated detection rule.

Start with the essentials

Focus question: Which part of a rotating cloud contributes to the selected channel?
One-sentence intuition: A channel groups line-of-sight velocities. The physical locations that share that velocity can lie at different depths.

Core mathematical model

Radio velocity convention

v=c(1-\nu/\nu_0)

The reference frequency belongs to the declared transition. Positive velocity means recession; the physical observer stays fixed while the display camera moves.

A line through one isothermal ray

\Delta T(v)=[J_{\nu_0}(T)-J_{\nu_0}(T_{\rm bg})](1-e^{-\tau(v)})

The same gas supplies the opacity and the Planck source. Increasing column first brightens the line, then saturates its peak and extends its wings.

Integrated line and velocity centroid

W=\sum_i T_i\Delta v_i,\qquad v_{\rm c}=\frac{\sum_i v_iT_i\Delta v_i}{W}

Both sums read the saved finite-channel cube. The centroid is a brightness-weighted velocity, with a declared display mask, rather than the gas speed at one known depth.

Common difficulties

Velocity is not a depth ruler

Typical misconceptionMoving through the channels moves through successive physical layers.

Better mental modelThe same speed can occur at several depths. The three-dimensional gas view shows the prescribed geometry; the separate image combines every point along each sightline.

Brightness and molecule number

Typical misconceptionTen times as many molecules always gives a tenfold peak.

Better mental modelOpacity reduces how much material deeper in the cloud changes the escaping line. The saturated preset keeps the peak near the source brightness while the wings grow.

Run the experiment

01
Scan a turning cloud
Start with A turning cloud and move the velocity-channel control across the line. Compare the bright patch with the gas velocity colours.
What to observe: Receding and approaching gas enter different channels; the position–velocity cut follows the same cube.
02
Change the physical axis
Drag the purple axis tip, then choose Look along the axis. Orbit the display camera separately.
What to observe: Changing the physical axis changes projected velocities. Moving only the camera changes the physical scene view while the measured cube stays fixed.
03
Move the probe on actual data
Drag in the channel image to select a sightline. Follow its spectrum and the row used for the position–velocity cut.
What to observe: The spectrum, moments and export use that same accepted generation. The relative-brightness mask is a viewing choice, not a noise significance.
04
Compare thin and saturated lines
Choose A nearly transparent line, then More molecules, same peak. Compare spectral peak and integrated brightness.
What to observe: The relation between brightness and column changes with opacity; a peak alone does not provide a cloud mass.

Molecular Line Cubes and Velocity Channels

One cube, three ways to read it

Watch one cloud move between velocity channels

Core mathematical model

Radio velocity convention

A line through one isothermal ray

Integrated line and velocity centroid

Common difficulties

Velocity is not a depth ruler

Brightness and molecule number

Run the experiment

Scan a turning cloud

Change the physical axis

Move the probe on actual data

Compare thin and saturated lines

Related experiments and articles

Molecular line cubes and velocity channels