Lecture arc
Rotation curves made the mass problem quantitative: outer disk speeds do not fall as a luminous point-mass model predicts. Start by asking students what the telescope actually measures. Then introduce circular speed and mass; 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
Rotation curve
The circular speed \(v_c(R)\) inferred as a function of Galactocentric radius. Its outer shape is the central dynamical evidence for mass beyond the luminous disk. Under an idealized spherical interpretation, \(M(\lt R)=Rv_c^2/G\). In this lecture, use this term while reasoning about circular speed and mass, and distinguish the measured quantity from the model-dependent quantity in \(M(\lt R)=Rv_c^2/G\). A rotation curve is inferred from tracers and geometry; it is not a direct plot of all stellar orbital speeds. This boundary is important when comparing tracers or carrying the result into the worked example.
Circular speed
The speed required for a circular orbit in a specified gravitational potential at a given radius. It links the measured kinematics to the radial gravitational force. It satisfies \(v_c^2=R\,\partial\Phi/\partial R\) in an axisymmetric mid-plane model. In this lecture, use this term while reasoning about circular speed and mass, and distinguish the measured quantity from the model-dependent quantity in \(M(\lt R)=Rv_c^2/G\). It differs from the mean azimuthal speed of a population with random motion. This boundary is important when comparing tracers or carrying the result into the worked example.
Enclosed mass
The mass inside radius \(R\) that produces the inferred gravitational acceleration under a stated geometry. It converts a flat outer rotation curve into a growing dynamical mass estimate. For spherical symmetry, \(M(\lt R)=Rv_c^2/G\), so constant \(v_c\) implies \(M\propto R\). In this lecture, use this term while reasoning about circular speed and mass, and distinguish the measured quantity from the model-dependent quantity in \(M(\lt R)=Rv_c^2/G\). The simple formula is not exact for a flattened disk and does not separate baryons from dark matter. This boundary is important when comparing tracers or carrying the result into the worked example.
Baryonic mass
Mass in ordinary matter, chiefly stars and gas, that emits, absorbs, or otherwise interacts electromagnetically. Its predicted gravitational contribution can be compared with the observed rotation curve. Stellar mass-to-light ratios and gas maps build the baryonic rotation model. In this lecture, use this term while reasoning about circular speed and mass, and distinguish the measured quantity from the model-dependent quantity in \(M(\lt R)=Rv_c^2/G\). Baryonic mass is not identical to visible light because faint stars and cold gas also contribute. This boundary is important when comparing tracers or carrying the result into the worked example.
Dark matter
A non-luminous mass component inferred from gravity whose distribution extends beyond the stellar disk. It provides the standard explanation for flat outer rotation curves and multiple independent dynamical probes. A halo contribution is added in quadrature to bulge, disk, and gas contributions to \(v_c^2\). In this lecture, use this term while reasoning about circular speed and mass, and distinguish the measured quantity from the model-dependent quantity in \(M(\lt R)=Rv_c^2/G\). A mass discrepancy establishes missing gravity under the model, not a direct image or unique particle identity. This boundary is important when comparing tracers or carrying the result into the worked example.