10.2 Galactic dynamics and rotation

Galactic Structure and Rotation

The Milky Way doesn't rotate like a solid wheel. Different parts of the galaxy orbit at different angular velocities, and this differential rotation shapes everything from spiral arms to how chemical elements get distributed across the disk. Understanding the galaxy's rotation curve also leads to one of the biggest puzzles in modern astrophysics: the evidence for dark matter.

Differential Rotation in the Milky Way

Inner regions of the galaxy rotate faster (in angular velocity) than outer regions. This isn't surprising on its own since gravitationally bound systems generally behave this way, but the consequences are important:

• Spiral arm formation: Differential rotation shears gas clouds and star-forming regions, stretching them into the spiral patterns we observe. Density wave theory explains how these arms persist rather than winding up and disappearing.
• Chemical mixing: As different stellar populations orbit at different rates, they gradually mix chemical elements throughout the galactic disk.

The Oort constants A and B quantify differential rotation locally (near the Sun):

$A = \frac{1}{2}\left(\frac{V}{R} - \frac{dV}{dR}\right)$

$B = -\frac{1}{2}\left(\frac{V}{R} + \frac{dV}{dR}\right)$

Here, $V$ is the circular velocity and $R$ is the galactocentric distance. Current measured values are roughly $A \approx 15 \text{ km s}^{-1} \text{kpc}^{-1}$ and $B \approx -12 \text{ km s}^{-1} \text{kpc}^{-1}$.

A few useful relationships follow from these constants:

• The local angular velocity of the Sun: $\Omega_0 = A - B$
• The local shear rate (how quickly rotation speed changes with radius): captured by $A$
• The local vorticity of galactic rotation: captured by $B$

Rotation Curve of the Milky Way

A rotation curve plots circular orbital velocity against distance from the galactic center. Astronomers construct this curve using tracers like neutral hydrogen (HI 21 cm emission), CO molecular line emission, and masers in star-forming regions.

What we'd expect: If most of the galaxy's mass were concentrated in the visible bulge and disk, velocities at large radii should follow a Keplerian decline:

$V(R) \propto R^{-1/2}$

This is the same falloff you see in planetary orbits around the Sun, where almost all the mass sits at the center.

What we actually observe: The rotation curve stays roughly flat (around 220 km/s) out to large radii, well beyond where most visible matter is located. In some measurements, it even rises slightly.

This discrepancy has direct implications for the mass distribution:

• Visible mass (stars, gas, dust) is concentrated in the central bulge and disk.
• A flat rotation curve at radius $R$ implies enclosed mass growing as $M(R) \propto R$, meaning there's substantial mass at large radii that we can't see.
• This leads to the inference of an extended dark matter halo surrounding the visible galaxy.

Dark Matter and Galactic Dynamics

Dark Matter Evidence from Rotation

The flat rotation curve is the single most compelling piece of evidence for dark matter in the Milky Way. Here's the reasoning step by step:

1. Measure the rotation curve from gas and stellar tracers out to large galactocentric radii.
2. Calculate the mass required to produce the observed velocities using $V^2 = \frac{GM(R)}{R}$.
3. Compare that required mass to the mass you can actually see (stars, gas, dust).
4. Find that the required mass far exceeds the visible mass, especially at large radii.
5. Conclude that an additional, non-luminous mass component must be present.

Supporting this conclusion, the mass-to-light ratio increases with radius. In the inner galaxy, the ratio is modest and consistent with stellar populations. In the outer galaxy, it climbs steeply, meaning non-luminous matter increasingly dominates.

Several halo models attempt to describe the dark matter distribution:

• NFW (Navarro-Frenk-White) profile: Derived from cosmological N-body simulations, this predicts a density that scales as $\rho(r) \propto \frac{1}{(r/r_s)(1 + r/r_s)^2}$, where $r_s$ is a characteristic scale radius. It has a cuspy center and falls off as $r^{-3}$ at large radii.
• Isothermal sphere model: Assumes constant velocity dispersion throughout the halo, giving $\rho(r) \propto r^{-2}$. This naturally produces a flat rotation curve but is unphysical at very small and very large radii.

Modified Newtonian Dynamics (MOND) has been proposed as an alternative, modifying gravity at low accelerations rather than invoking unseen mass. MOND can reproduce some galaxy rotation curves, but it struggles with galaxy cluster observations and cosmological data (like the CMB power spectrum), where the dark matter framework succeeds.

Stellar Motions and Galactic Dynamics

Stellar kinematics give us a three-dimensional view of how the galaxy moves, going beyond the rotation curve alone.

• Proper motions measure angular displacement across the sky (in arcseconds per year).
• Radial velocities measure motion along the line of sight via Doppler shifts.
• Combining both yields full 3D velocity vectors for individual stars.

Velocity dispersions (the spread in stellar velocities within a population) carry important dynamical information. Older stellar populations and stars farther from the galactic plane tend to have higher velocity dispersions, reflecting their dynamical heating over time through encounters with molecular clouds and spiral arms.

The Jeans equations connect these velocity dispersions to the underlying gravitational potential. In simplified form, they relate the density distribution, velocity dispersion tensor, and gravitational field, allowing you to estimate the mass distribution even where you can't directly measure a clean rotation curve.

Two additional effects are worth knowing:

• Asymmetric drift: Stars with higher velocity dispersions tend to lag behind the local circular velocity. The greater the dispersion, the larger the lag. This means older, dynamically hotter populations orbit the galaxy more slowly on average than the local standard of rest.
• Vertical motions: Studying how stars oscillate above and below the galactic plane constrains the local matter density (including dark matter) in the disk. These measurements provide an independent check on dark matter estimates from the rotation curve.

The Gaia mission has transformed this entire field by providing precise parallaxes, proper motions, and radial velocities for over a billion stars. With Gaia data, astronomers can map the Milky Way's velocity field in unprecedented detail, revealing substructure like stellar streams, phase-space spirals from past satellite interactions, and refined constraints on the galaxy's total mass and dark matter distribution.