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Molecular line cubes: space, velocity and hidden depth

Molecular line cubes · L1

Why a velocity channel cannot select one depth

Sandbox Physics ·

A channel image groups gas by its speed along our line of sight. Material near the front of a cloud and material far behind it can enter the same channel if their motions toward us are alike.

One frequency can contain two places

Imagine a sightline through a cloud. One parcel is nearby and another lies much farther along it. Both contain the same molecule and both recede at the same speed. Light from the same molecular transition reaches us at the same shifted frequency.

If the line is sufficiently transparent, light from the rear parcel passes through the front one. Their contributions add in one image pixel. Move the rear parcel farther back while preserving the stated angular extent, temperature, velocity and total molecular column, and the channel image can remain almost unchanged. The image has combined locations that were separate in space.

Only the component of motion toward or away from us supplies this first-order Doppler shift. A parcel can move rapidly across the sky without acquiring a large recession speed. A spectrum and a picture describe different parts of its motion.

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

Molecules supply a recognizable frequency

A rotating molecule has particular allowed energy levels. A downward transition emits light at a characteristic frequency. Many molecules produce a narrow peak at that frequency, called a spectral line. Several useful rotational lines of CO lie at millimetre or submillimetre wavelengths, where radio telescopes can receive them.

Motion shifts the peak. Approaching gas is received at a slightly higher frequency; receding gas arrives at a lower one. A receiver divides the frequency range into small intervals. Each interval gives a channel image of the sky.

There is a width on both sides of this measurement. Molecules have a distribution of thermal velocities, and the receiver channel has a finite frequency width. A channel therefore admits a band of speeds with different weights. Narrower channels may reveal finer velocity structure, until physical line broadening becomes the limiting width.

Reading: Mangum & Shirley, How to Calculate Molecular Column Density, arXiv 1501.01703v4 (2017) · Leiden Atomic and Molecular Database, fixed CO data file

Rotation moves the bright region between channels

Tilt a cloud’s rotation axis away from the line of sight. One side moves toward us and the other recedes. The approaching side contributes to the higher-frequency channels; the receding side enters the lower-frequency ones. Scan through the line and the bright region moves across the cloud.

The receiver is selecting different parts of one snapshot. No searchlight has crossed the gas. A smooth distribution of molecules can make distinct patches in a narrow channel because different locations have different velocities.

Another useful view takes a line across the image and stacks the full spectrum at each position. This produces a position–velocity diagram. One axis is sky position and the other is velocity. Rotation can make a tilted bright band. The vertical coordinate of that diagram still describes motion.

Uniform density with a parabolic velocity-versus-depth relation produces enhanced low-velocity signal after Gaussian broadening.
An independent synthetic example. Density is constant in the upper panel, but velocity changes with depth. The lower panel combines equal-velocity contributions and applies the declared Gaussian width. More physical depths contribute near the slow end of the velocity distribution.

Reading: Lazarian & Pogosyan, Velocity Modification of HI Power Spectrum, astro-ph/9901241v2 (2000) · Sandbox Physics independent molecular-line science producer and reader package

Try sorting three parcels

Suppose one parcel approaches at six hundred metres per second. Two others recede at two hundred metres per second, one behind the other. A channel near the latter recession speed mainly selects those two parcels. Move to the approaching speed and the first parcel becomes bright.

This separation works cleanly when the physical line widths are small compared with the gap between the two velocity groups. Broad thermal or microscopic velocity distributions can make the peaks overlap. Temperature and absorption can also change how strongly different layers contribute.

For a real cloud, begin by following a patch through adjacent channels. Then inspect its full spectrum and ask whether the patch survives when the channels are integrated. These connected views help turn an attractive image into a more specific question about gas motion.

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

Exercises and explanations

  1. Two transparent parcels on one sightline have equal temperatures and velocities. Must the channel image change when their separation in true depth changes?

    Read the explanation

    No. Under the stated fixed angular extent and molecular column, the same channel selects both parcels and their light can add. Their separation has supplied no new frequency label. Additional geometric or physical information is needed to distinguish the depths.

  2. A bright patch moves as you scan from the blue side to the red side of a saved line. Has the cloud travelled across the picture during the scan?

    Read the explanation

    The frequency selection has moved. Different velocities already belonged to different gas locations in the saved snapshot, so successive locations become bright. Actual motion with time requires a dynamical calculation or observations at multiple times.

Try the prediction

Follow a rotating cloud through its line

  1. Keep the physical axis fixed. Predict which channels select approaching and receding gas, then scan and compare the velocity colours with the channel image.
  2. Choose Look along the axis and scan again. Explain how rotation can remain present while the spatial gradient of line-of-sight speed becomes small.
  3. Select a sightline in the image and read its spectrum and position–velocity row. Explain each axis in a complete sentence.

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. Lazarian & Pogosyan, Velocity Modification of HI Power Spectrum, astro-ph/9901241v2 (2000)

    Fixed 11 January 2000 version. Main PPV mapping/channel-window derivation, figures 1–3 / table 1, finite-cloud appendix E and stated optical-depth/statistical limits. The later generated 2018 draft date is not the scientific version.

  3. 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.

  4. 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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