The Milky Way Galaxy

The Sun sits inside a barred spiral galaxy of some $1 0^{11}$ stars, embedded in a disk of gas and dust that obscures the view toward the center. Mapping our own Galaxy is a problem in reconstructing three-dimensional structure from a vantage point buried in the midplane. The tools are star counts, the kinematics of tracer populations, and the 21-cm line that penetrates the dust. This lesson assembles the structural components, derives the differential-rotation formalism and the Oort constants that turn local stellar velocities into a rotation curve, treats spiral arms as density waves rather than fixed features, and uses the resolved orbits of stars around the compact radio source Sgr A $^{*}$ to weigh the central black hole.

Structural components

The Galaxy separates into distinct populations distinguished by geometry, age, metallicity, and kinematics:

Thin disk. Scale height $\sim 300 pc$ , scale length $\sim 2.6 kpc$ , containing the gas, dust, and young metal-rich (Population I) stars on nearly circular orbits. The Sun belongs to it, at $R_{0} \approx 8.2 kpc$ from the center.
Thick disk. Scale height $\sim 1 kpc$ , older and more metal-poor than the thin disk, with larger vertical velocity dispersion.
Bulge and bar. A boxy, bar-shaped concentration a few kpc across, mixing old and intermediate-age stars and reaching high metallicity.
Stellar halo. A roughly spheroidal, metal-poor, pressure-supported distribution of ancient (Population II) stars extending beyond $30 kpc$ , containing the $\sim 150$ globular clusters on plunging, randomly oriented orbits.
Dark halo. An extended, nearly spherical mass distribution inferred from the rotation curve and the motions of satellites, dominating the mass beyond the optical disk (developed in the dark-matter lesson).

The surface-brightness profile of the disk falls exponentially with radius, $I (R) = I_{0} e^{- R / h_{R}}$ , and the vertical profile of an isothermal self-gravitating sheet follows $ρ (z) \propto sech^{2} (z /2 z_{0})$ , flattening to $e^{- ∣ z ∣/ z_{0}}$ at large height.

Edge-on and face-on schematic of the Galactic components: the thin gas- and-young-star disk, the boxy bar and bulge, and the spheroidal old halo with its globular clusters and the Sun at eight kpc from the center.

Star counts realize an old program of Galactic astronomy. The number of stars per unit solid angle brighter than apparent magnitude $m$ constrains the density profile along the line of sight, provided the extinction and the luminosity function are known. Interstellar dust reddens and dims distant stars, so counts must be corrected for an extinction $A_{λ}$ that grows with path length; ignoring it made the early star-count models place the Sun near the center. Only when the reddening was measured did the globular-cluster system reveal the true center, $8 kpc$ away toward Sagittarius.

Differential rotation and the Oort constants

The disk does not rotate as a rigid body. Each star follows a nearly circular orbit with a circular speed $Θ (R)$ set by the enclosed mass, so the angular speed $Ω (R) = Θ (R) / R$ varies with galactocentric radius. Interior stars overtake the Sun; exterior stars lag. This differential rotation imprints a characteristic pattern on the velocities of nearby stars.

Place the Sun at radius $R_{0}$ with circular speed $Θ_{0}$ and angular speed $Ω_{0} = Θ_{0} / R_{0}$ . A star at radius $R$ and Galactic longitude $ℓ$ , seen at heliocentric distance $d$ , has a line-of-sight (radial) velocity and a tangential velocity, both measured relative to the local standard of rest,

v_{r} = R_{0} (Ω - Ω_{0}) sin ℓ, v_{t} = R_{0} (Ω - Ω_{0}) cos ℓ - Ω d .

The local standard of rest (LSR) is the velocity of a fictitious point on a perfectly circular orbit at $R_{0}$ ; the Sun's own peculiar velocity relative to the LSR (about $13 km s^{- 1}$ ) is removed before comparison. For stars near the Sun, $∣ R - R_{0} ∣ ≪ R_{0}$ , expand the angular speed to first order,

Ω (R) \approx Ω_{0} + (\frac{d Ω}{d R})_{R_{0}} (R - R_{0}),

and use the geometry $R - R_{0} \approx - d cos ℓ$ for small $d$ . Substituting into $v_{r}$ and $v_{t}$ and applying the double-angle identities produces the Oort formulae,

v_{r} \approx A d sin 2 ℓ, v_{t} \approx d (A cos 2 ℓ + B),

with the two Oort constants

A = - \frac{1}{2} (\frac{d Θ}{d R} - \frac{Θ}{R})_{R_{0}} = - \frac{1}{2} R_{0} (\frac{d Ω}{d R})_{R_{0}}, B = - \frac{1}{2} (\frac{d Θ}{d R} + \frac{Θ}{R})_{R_{0}} .

$A$ measures the local shear (the departure from rigid rotation) and $B$ the local vorticity. Their combinations recover the rotation directly: $Ω_{0} = A - B$ and $(d Θ/ d R)_{R_{0}} = - (A + B)$ . Modern astrometry gives $A \approx 15.3 km s^{- 1} kp c^{- 1}$ and $B \approx - 11.9 km s^{- 1} kp c^{- 1}$ , so $Θ_{0} = (A - B) R_{0} \approx 27.2 \times 8.2 \approx 223 km s^{- 1}$ .¹

The double sinusoid of radial velocity with Galactic longitude produced by differential rotation: nearby stars show a sine of twice the longitude with amplitude set by the Oort A constant.

At the tangent point along a line of sight interior to the Sun, the circular orbit is tangent to the sight line and $v_{r}$ reaches its maximum for that longitude. Measuring that maximum with the 21-cm line, which traces neutral hydrogen through the dust, yields $Θ (R)$ at $R = R_{0} sin ℓ$ and builds the inner rotation curve directly.

The rotation curve and the mass discrepancy

For a spherically symmetric mass distribution, a circular orbit balances gravity against the centripetal requirement,

\frac{Θ ^{2} ( R )}{R} = \frac{G M ( < R )}{R ^{2}} ⟹ Θ (R) = \frac{G M ( < R )}{R} .

If the mass were concentrated in the visible bulge and disk, then beyond the light $M (< R)$ would be nearly constant and the speed would fall as $Θ \propto R^{- 1/2}$ — the Keplerian decline. The observed curve does the opposite: it rises through the inner Galaxy and stays flat, $Θ \approx 220 km s^{- 1}$ , far beyond the optical edge. A flat curve requires

M (< R) = \frac{Θ ^{2} R}{G} \propto R,

mass growing linearly with radius even where there is essentially no light. The enclosed mass at $20 kpc$ reaches $\sim 2 \times 1 0^{11} M_{⊙}$ , several times the stellar mass. This discrepancy is the local, dynamical case for a dark halo, whose density must fall as $ρ \propto R^{- 2}$ to give constant $Θ$ .

The observed rotation curve stays flat well beyond the luminous disk, whereas the mass of the visible stars alone would give a Keplerian decline; the gap is the dynamical signature of a dark halo.

Spiral structure as density waves

Spiral arms trace the young, luminous stars and the H II regions of the disk, yet they cannot be fixed material features. If an arm were a persistent collection of the same stars, differential rotation would shear it: an arm at the solar circle completes an orbit in $\sim 2.4 \times 1 0^{8} yr$ , while material a few kpc inward orbits noticeably faster, so any material arm winds itself into a tightly coiled spiral within a few Galactic rotations. Grand-design spirals persist far longer than that. This is the winding problem.

The resolution is that the arms are a density wave: a spiral-shaped pattern in the gravitational potential that rotates rigidly at a single pattern speed $Ω_{p}$ , distinct from the orbital speeds of the stars and gas. Stars and gas move through the pattern, slowing and bunching up where the potential is deepest, the way cars bunch at a slow-moving traffic constriction. The arm is a standing enhancement in density, not a fixed set of stars. Because the compression triggers cloud collapse and star formation, the arms glow with short-lived massive stars even though individual stars drift in and out.

The pattern speed sets three resonances where the stellar epicyclic motion beats against the wave. A star perturbed from a circular orbit oscillates radially at the epicyclic frequency

κ^{2} (R) = R \frac{d Ω ^{2}}{d R} + 4 Ω^{2} = - 4 B (A - B),

evaluated locally through the Oort constants. Resonances occur where

Ω_{p} = Ω (R) (corotation), Ω_{p} = Ω (R) \pm \frac{κ ( R )}{m} (Lindblad),

with $m$ the number of arms. The spiral pattern is maintained between the inner and outer Lindblad resonances; at corotation the stars and the pattern move together. For a flat rotation curve $κ = 2 Ω$ , so the epicyclic and orbital periods are locked in a fixed ratio throughout the disk.

Density-wave spiral arms: material orbits at the local angular speed while the spiral pattern turns rigidly at a slower pattern speed, so stars overtake and pile up in the arms where the potential is deepest.

The Galactic center and Sgr A $^{*}$

The dynamical center of the Galaxy coincides with a compact, non-thermal radio source, Sgr A $^{*}$ . Near-infrared astrometry over two decades resolves individual stars, the S-stars, on Keplerian orbits around an invisible focus. The star S2 completes an orbit in $P \approx 16.0 yr$ with a semimajor axis $a \approx 0.12 3^{''} \approx 1020 AU$ at the Galactic-center distance of $8.2 kpc$ , and its orbit is a clean ellipse with Sgr A $^{*}$ at one focus.

Kepler's third law converts the orbit into a mass:

M_{∙} = \frac{4 π ^{2} a ^{3}}{G P ^{2}} .

The measured $4.3 \times 1 0^{6} M_{⊙}$ within $120 AU$ implies a mean density above $1 0^{15} M_{⊙} p c^{- 3}$ , orders of magnitude beyond any stable stellar system. The Schwarzschild radius of such a black hole, $r_{s} = 2 G M_{∙} / c^{2} \approx 0.08 AU$ , subtends about $10 μ as$ , resolved as the central shadow by very-long-baseline interferometry. The physics of the horizon and the accretion flow is developed in the black-hole lesson; here the orbits alone fix the mass.

Resolved elliptical orbits of the innermost stars around Sgr A-star, each with the black hole at a common focus; applying Kepler's third law to the S2 orbit yields a central mass of about four million solar masses.

Stellar populations and chemical enrichment

The components differ in more than geometry. Population I stars of the thin disk are metal-rich ( $Z \sim Z_{⊙}$ ) and young; Population II stars of the halo and old bulge are metal-poor (down to $Z \sim 1 0^{- 3} Z_{⊙}$ ) and ancient. The trend records the chemical enrichment of the Galaxy: each stellar generation returns metals synthesized in its cores and supernovae to the interstellar medium, so later generations form from more enriched gas.

A one-zone model captures the trend. Let $Z$ be the gas metallicity, $y$ the yield of metals per unit mass locked into long-lived stars, and treat the system as closed with no inflow or outflow. Metal conservation gives $d (Z M_{g}) = y d M_{*} - Z d M_{*}$ , which, with $d M_{g} = - d M_{*}$ for gas turning into stars, integrates to the closed-box (Simple) model relation

Z (t) = y ln (\frac{M _{g} ( 0 )}{M _{g} ( t )}) = - y ln f_{g},

where $f_{g} = M_{g} / M_{tot}$ is the gas fraction. Metallicity rises as gas is consumed, without an adjustable normalization once the yield is fixed. The model predicts far more metal-poor stars than the disk actually contains — the G-dwarf problem — indicating that the disk was not a closed box but was fed by continuing infall of low-metallicity gas, which dilutes the metals and suppresses the metal-poor tail. The abundance ratios add a clock: the ratio of $α$ -elements (O, Mg, Si, produced promptly in core-collapse supernovae) to iron (produced with a delay in Type Ia supernovae) distinguishes rapid early enrichment from prolonged star formation, and the halo's high $[α / Fe]$ marks its rapid, ancient formation.

Summary

The Milky Way is a barred spiral of a thin metal-rich disk, a thick disk, a boxy bar-bulge, and an old metal-poor halo of stars and $\sim 150$ globular clusters, mapped through star counts corrected for dust extinction. The disk rotates differentially: the Oort constants $A$ (shear) and $B$ (vorticity) encode the local rotation through $Ω_{0} = A - B$ and give $Θ_{0} \approx 220 km s^{- 1}$ at $R_{0} \approx 8.2 kpc$ , with $A + B \approx 0$ signaling a flat rotation curve. The flat curve requires $M (< R) \propto R$ far beyond the light, the dynamical case for a dark halo with $ρ \propto R^{- 2}$ . Spiral arms are density waves turning at a rigid pattern speed, resolving the winding problem, with resonances set by the epicyclic frequency $κ$ . Resolved S-star orbits around Sgr A $^{*}$ apply Kepler's third law to weigh a $4.3 \times 1 0^{6} M_{⊙}$ central black hole confined within $120 AU$ . The population and $[α / Fe]$ gradients record the Galaxy's chemical enrichment, with the G-dwarf problem pointing to continued gas infall onto the disk.

Carroll & Ostlie, §24.3 — Galactic rotation, the Oort constants, and the local standard of rest; the differential-rotation formulae and measured values of $A$ and $B$ . ↩