ASTR 410 Lecture 04: Stellar Motions and the Galactic Phase Space
Galactic Astronomy ยท study notes
Learning objectives
Interpret proper motions, radial velocities, and velocity ellipsoids.
Explain the role of 6D phase space and asymmetric drift.
Audit an inference from observable to physical conclusion.
Lecture arc
The disk is not a rigid carousel: random motions encode heating, birth conditions, and resonances. Start by asking students what the telescope actually measures. Then introduce 6D phase space and asymmetric drift; delay the equation until the geometry and assumptions are visible. Use the visual as an argument: students should identify what is data, what is a model, and what is inferred.
The lecture should return to the one-dimensional worked calculation and ask whether the answer is stable under a plausible change in distance, tracer selection, or model family.
Terms Developed in Context
Phase space
The six-dimensional specification of three positions and three velocities for each star. Galactic dynamics predicts how stellar distributions evolve in this space rather than on a sky map alone. Gaia astrometry plus radial velocities approximates 6D phase-space coordinates after coordinate transformations. In this lecture, use this term while reasoning about 6D phase space and asymmetric drift, and distinguish the measured quantity from the model-dependent quantity in \(v_t=4.74047\,\mu d\), \quad \sigma_R>\sigma_\phi>\sigma_z\). A catalogue with six reported numbers still has correlated errors and selection effects. This boundary is important when comparing tracers or carrying the result into the worked example.
Velocity ellipsoid
A covariance description of the distribution of stellar velocities along chosen axes. Its axis lengths and orientation quantify random motions, disk heating, and population differences. Disk populations often satisfy \(\sigma_R>\sigma_\phi>\sigma_z\), linking dispersions to epicyclic dynamics. In this lecture, use this term while reasoning about 6D phase space and asymmetric drift, and distinguish the measured quantity from the model-dependent quantity in \(v_t=4.74047\,\mu d\), \quad \sigma_R>\sigma_\phi>\sigma_z\). It describes a distribution, not a solid object or a single star's orbit. This boundary is important when comparing tracers or carrying the result into the worked example.
Asymmetric drift
The lag of a pressure-supported stellar population's mean rotation behind the local circular speed. It explains why hotter populations rotate more slowly even in the same gravitational potential. The lag is related through the Jeans equations to radial dispersion and density gradients. In this lecture, use this term while reasoning about 6D phase space and asymmetric drift, and distinguish the measured quantity from the model-dependent quantity in \(v_t=4.74047\,\mu d\), \quad \sigma_R>\sigma_\phi>\sigma_z\). It is not evidence that every lagging star belongs to the halo. This boundary is important when comparing tracers or carrying the result into the worked example.
Solar motion
The Sun's velocity relative to a chosen local standard of rest or Galactic reference frame. Observed stellar velocities must be corrected for the observer's own motion before Galactic patterns are inferred. It enters transformations from heliocentric radial velocity and proper motion to Galactocentric components. In this lecture, use this term while reasoning about 6D phase space and asymmetric drift, and distinguish the measured quantity from the model-dependent quantity in \(v_t=4.74047\,\mu d\), \quad \sigma_R>\sigma_\phi>\sigma_z\). Different reference-frame conventions produce different quoted solar-motion values. This boundary is important when comparing tracers or carrying the result into the worked example.
Tangential velocity
The linear component of motion perpendicular to the line of sight. It is the physical speed inferred from angular proper motion and distance. The conversion \(v_t=4.74047\mu d\) couples astrometric and distance uncertainties. In this lecture, use this term while reasoning about 6D phase space and asymmetric drift, and distinguish the measured quantity from the model-dependent quantity in \(v_t=4.74047\,\mu d\), \quad \sigma_R>\sigma_\phi>\sigma_z\). Tangential speed alone is not the total space speed. This boundary is important when comparing tracers or carrying the result into the worked example.
Define the symbols and units aloud. The relation is useful because it turns an observable into a dynamical or population constraint, but it is conditional on the stated approximation. In discussion, distinguish a parameter that is directly measured from one that is inferred after adopting a model.
Velocity Vector: a lecture-specific schematic or model plot. Axes and annotations identify the inference being made; it is not presented as a raw survey image.
Worked example
For mu=1.8 mas yr^-1 and d=2.4 kpc, v_t = 20.479 km s^-1.
The conversion assumes the proper motion is calibrated and the distance is the adopted estimate. Have students estimate the order of magnitude before revealing the computed value. Require units and one sentence of physical interpretation. This example is generated by the course source and can be regenerated if a shared constant changes.
Misconceptions and evidence
A large proper motion can mean proximity rather than extreme space velocity. A map or fitted curve is not a direct inventory. Ask students to list a calibration, selection, or projection effect that could mimic the displayed trend. Then connect the answer to the paired data activity.
Study guidance
Students should be able to reproduce the calculation, explain the approximation, and compare the result with at least one independent tracer. Suggested practice: OpenStax Astronomy 2e Chapter 25 review and exercise material after checking the adopted PDF; exact local index entries are recorded in the reference log rather than guessed.