ASTR 472 Lecture 13 Notes: Galactic H I: Rotation, Geometry, and Mapping

Learning objectives

Use the flat-rotation inner-Galaxy relation to infer Galactocentric radius from longitude and radial velocity.
Distinguish measured LSR velocity from inferred distance and radius.
Explain near-far ambiguity and non-circular motions.

Prerequisites in use: ASTR 340; PHYS 341 electromagnetism is recommended for receiver and radiation physics.

Terms Developed in Context

These terms connect the measured radio product to the physical model and delimit what the observation can support.

Local standard of rest

A conventional velocity frame that removes an adopted Solar motion before Galactic interpretation. In Galactic H I: Rotation, Geometry, and Mapping, it helps resolve the question At fixed longitude, how does a positive non-circular velocity perturbation propagate into inferred R in this model? Give the derivative sign.

Use in the reasoning chain: The term links the observable or diagnostic The 21-cm line is observable through much of the Galactic disk, including dusty sightlines to the model \(R=\frac{R_0}{1+v_{\rm LSR}/(\Theta_0\sin l)}\). Compare it directly with Tangent point rather than treating either quantity in isolation.

Scope: The flat, axisymmetric rotation curve is an idealization; streaming velocities of order tens of km/s can shift inferred radii and distances substantially. The term therefore supports an inference only when its measurement convention, calibration stage, and stated approximation are retained.

Tangent point

The minimum Galactocentric radius sampled along an inner-Galaxy line of sight. In Galactic H I: Rotation, Geometry, and Mapping, it helps resolve the question At fixed longitude, how does a positive non-circular velocity perturbation propagate into inferred R in this model? Give the derivative sign.

Use in the reasoning chain: The term links the observable or diagnostic The 21-cm line is observable through much of the Galactic disk, including dusty sightlines to the model \(R=\frac{R_0}{1+v_{\rm LSR}/(\Theta_0\sin l)}\). Compare it directly with Terminal velocity rather than treating either quantity in isolation.

Terminal velocity

The extremal line-of-sight velocity associated with gas near the tangent point under circular rotation. In Galactic H I: Rotation, Geometry, and Mapping, it helps resolve the question At fixed longitude, how does a positive non-circular velocity perturbation propagate into inferred R in this model? Give the derivative sign.

Use in the reasoning chain: The term links the observable or diagnostic The 21-cm line is observable through much of the Galactic disk, including dusty sightlines to the model \(R=\frac{R_0}{1+v_{\rm LSR}/(\Theta_0\sin l)}\). Compare it directly with Kinematic distance rather than treating either quantity in isolation.

Kinematic distance

A distance inferred by matching line-of-sight velocity to an adopted Galactic rotation model. In Galactic H I: Rotation, Geometry, and Mapping, it helps resolve the question At fixed longitude, how does a positive non-circular velocity perturbation propagate into inferred R in this model? Give the derivative sign.

Use in the reasoning chain: The term links the observable or diagnostic The 21-cm line is observable through much of the Galactic disk, including dusty sightlines to the model \(R=\frac{R_0}{1+v_{\rm LSR}/(\Theta_0\sin l)}\). Compare it directly with Rotation curve rather than treating either quantity in isolation.

Rotation curve

The circular orbital speed of Galactic material as a function of Galactocentric radius. In Galactic H I: Rotation, Geometry, and Mapping, it helps resolve the question At fixed longitude, how does a positive non-circular velocity perturbation propagate into inferred R in this model? Give the derivative sign.

Use in the reasoning chain: The term links the observable or diagnostic The 21-cm line is observable through much of the Galactic disk, including dusty sightlines to the model \(R=\frac{R_0}{1+v_{\rm LSR}/(\Theta_0\sin l)}\). Compare it directly with Local standard of rest rather than treating either quantity in isolation.

Conceptual entry point

A 21-cm velocity cube turns a line-of-sight Doppler field into a model-dependent map of Galactic gas and kinematics. Begin with the receiver product: a spectrum, time series, image, or set of complex visibilities. The object we ultimately discuss is several inference steps removed from that product.

Radio astronomy's history makes this distinction concrete. Karl Jansky's 1930s rotating antenna detected persistent celestial interference while he investigated radio communication noise; Grote Reber then built a purpose-designed reflector and mapped the radio sky. Instrumental curiosity became a new window on the Galaxy.

The 21-cm line is observable through much of the Galactic disk, including dusty sightlines.

Evidence and measurement layers

The 21-cm line is observable through much of the Galactic disk, including dusty sightlines.
For a flat circular rotation curve, v_LSR=(Theta(R)R0/R-Theta0)sin(l) on the inner-Galaxy sightline.
Streaming motions, spiral arms, and the solar motion perturb the simple axisymmetric conversion.

A calibrated brightness scale is not equivalent to a flux measurement, and a catalog parameter is not equivalent to a direct observable. State the processing stage whenever comparing results.

Derivation and assumptions

Set Theta(R)=Theta0 and solve v_LSR=Theta0(R0/R-1)sin(l) for R=R0/[1+v_LSR/(Theta0 sin(l))]. This gives Galactocentric radius for a chosen longitude under circular rotation. Heliocentric distance requires a second geometric step and has a near/far ambiguity inside the solar circle.

\(R=\frac{R_0}{1+v_{\rm LSR}/(\Theta_0\sin l)}\)

The approximation is useful because it compresses a full electromagnetic/instrument problem into a scale relation. It is not a substitute for the response model: explicitly retain the beam, bandpass, convention, sampling, or plasma assumptions relevant to this lecture.

Worked example

Inputs: R0=8.2 kpc, Theta0=236 km/s, l=30 deg, v_LSR=100 km/s.

Programmatically evaluated result: R_GC = 4.439 kpc.

The flat, axisymmetric rotation curve is an idealization; streaming velocities of order tens of km/s can shift inferred radii and distances substantially. Recompute independently from the equation before using the result to compare facilities or source models.

Observational interpretation

For a flat circular rotation curve, v_LSR=(Theta(R)R0/R-Theta0)sin(l) on the inner-Galaxy sightline. A defensible interpretation records the competing hypotheses, checks whether the instrument response can mimic the feature, and identifies a measurement that would separate the alternatives. The next layer of evidence should be chosen for discriminating power, not simply because it is available.

Common conceptual traps

Confusing total flux with flux per beam, or a brightness temperature with a thermodynamic temperature.
Assuming a narrow thermal-noise estimate includes flux-scale or selection uncertainty.
Treating spatial resolution, field of view, and largest recoverable scale as interchangeable.
Using a Doppler or calibration convention without recording its reference frame.

Lecture-specific limitation: The flat, axisymmetric rotation curve is an idealization; streaming velocities of order tens of km/s can shift inferred radii and distances substantially.

Study and transfer questions

At fixed longitude, how does a positive non-circular velocity perturbation propagate into inferred R in this model? Give the derivative sign.
Change the dominant input by 10%. Predict the direction of the output change before computing it.
Name one raw diagnostic and one independent data product that could falsify the preferred interpretation.

Reading

OpenStax Astronomy 2e §20.2 introduces interstellar gas and 21-cm observations; HI4PI (2016) provides an all-sky spectral data product for modern mapping.

See materials/ASTR472/reference-log.md for sources, verification dates, exact OpenStax anchors, and human spot-check caveats.