Lecture 10: The Milky Way: Structure, Rotation, and the Case for Dark Matter

ASTR 230 Stars, Galaxies, and the Universe

Context and Why This Matters

The Milky Way’s rotation curve was one of the first and remains one of the most direct pieces of evidence for dark matter: if all of the Galaxy’s mass were in the visible stars and gas, rotation speeds should decline with radius outside the visible disk, the way planets slow down at greater distances from the Sun. They do not.

Learning Goals

  1. Describe the Milky Way’s major structural components (disk, bulge, halo).
  2. Explain how a galaxy’s rotation curve is measured and what a "flat" rotation curve implies about its mass distribution.
  3. Estimate the enclosed mass of the Milky Way at different radii from rotation-curve data.

Opening Phenomenon

Stars and gas clouds far outside the Sun’s orbital radius (measured via 21-cm radio observations of neutral hydrogen, unaffected by dust) are observed orbiting the Galactic center at roughly the same speed (about 220 km/s) as material near the Sun’s orbit, rather than slowing down as visible starlight and gas density drop off.

Terms Developed in Context

These terms form the reasoning chain from observation to model, calculation, and interpretation. Each entry expands the corresponding term-explorer panel and should be read alongside the worked example.

galactic disk

If nearly all mass were interior to the visible disk (as for the solar system, where nearly all mass is in the Sun), V(R) should decline as 1/√R beyond the visible mass, the way planetary orbital speeds decline with distance from the Sun.

Use and relationship: Stars and gas clouds far outside the Sun’s orbital radius (measured via 21-cm radio observations of neutral hydrogen, unaffected by dust) are observed orbiting the Galactic center at roughly the same speed (about 220 km/s) as material near the Sun’s orbit, rather than slowing down as visible starlight and gas density drop off. The lecture's quantitative connection is \( M(R) = \dfrac{V(R)^{2} R}{G} \), which shows how this term participates in the model rather than functioning as an isolated label.

Boundary or confusion: Assuming "dark matter" means "dark, ordinary matter we just haven’t seen yet" (like cold gas or dim stars). Multiple independent lines of evidence (rotation curves, gravitational lensing, the cosmic microwave background, large-scale structure) point to a non-baryonic form of matter that does not emit, absorb, or scatter light at all, distinguishing it from merely faint ordinary matter.

galactic bulge

Stars and gas clouds far outside the Sun’s orbital radius (measured via 21-cm radio observations of neutral hydrogen, unaffected by dust) are observed orbiting the Galactic center at roughly the same speed (about 220 km/s) as material near the Sun’s orbit, rather than slowing down as visible starlight and gas density drop off.

Use and relationship: The Sun orbits the Galactic center at radius R₀ ≈ 8.2 kpc with orbital speed V₀ ≈ 220 km/s. The lecture's quantitative connection is \( M(R) = \dfrac{V(R)^{2} R}{G} \), which shows how this term participates in the model rather than functioning as an isolated label.

Boundary or confusion: Assuming "dark matter" means "dark, ordinary matter we just haven’t seen yet" (like cold gas or dim stars). Multiple independent lines of evidence (rotation curves, gravitational lensing, the cosmic microwave background, large-scale structure) point to a non-baryonic form of matter that does not emit, absorb, or scatter light at all, distinguishing it from merely faint ordinary matter.

galactic halo

The Galaxy’s visible stellar and gas mass is estimated at only about 6.0e+10 M☉ — far less than the 3.07e+11 M☉ implied by the rotation curve at 25 kpc, requiring roughly 5.1 times more mass than is visible, attributed to a dark matter halo.

Use and relationship: Stars and gas clouds far outside the Sun’s orbital radius (measured via 21-cm radio observations of neutral hydrogen, unaffected by dust) are observed orbiting the Galactic center at roughly the same speed (about 220 km/s) as material near the Sun’s orbit, rather than slowing down as visible starlight and gas density drop off. The lecture's quantitative connection is \( M(R) = \dfrac{V(R)^{2} R}{G} \), which shows how this term participates in the model rather than functioning as an isolated label.

Boundary or confusion: Assuming "dark matter" means "dark, ordinary matter we just haven’t seen yet" (like cold gas or dim stars). Multiple independent lines of evidence (rotation curves, gravitational lensing, the cosmic microwave background, large-scale structure) point to a non-baryonic form of matter that does not emit, absorb, or scatter light at all, distinguishing it from merely faint ordinary matter.

rotation curve

Explain how a galaxy’s rotation curve is measured and what a "flat" rotation curve implies about its mass distribution.

Use and relationship: The Galaxy’s visible stellar and gas mass is estimated at only about 6.0e+10 M☉ — far less than the 3.07e+11 M☉ implied by the rotation curve at 25 kpc, requiring roughly 5.1 times more mass than is visible, attributed to a dark matter halo. The lecture's quantitative connection is \( M(R) = \dfrac{V(R)^{2} R}{G} \), which shows how this term participates in the model rather than functioning as an isolated label.

Boundary or confusion: Assuming "dark matter" means "dark, ordinary matter we just haven’t seen yet" (like cold gas or dim stars). Multiple independent lines of evidence (rotation curves, gravitational lensing, the cosmic microwave background, large-scale structure) point to a non-baryonic form of matter that does not emit, absorb, or scatter light at all, distinguishing it from merely faint ordinary matter.

dark matter halo

The Galaxy’s visible stellar and gas mass is estimated at only about 6.0e+10 M☉ — far less than the 3.07e+11 M☉ implied by the rotation curve at 25 kpc, requiring roughly 5.1 times more mass than is visible, attributed to a dark matter halo.

Use and relationship: A flat rotation curve, combined with Newtonian gravity, implies a mass distribution that keeps growing well beyond the visible disk — the central piece of evidence for the Milky Way’s dark matter halo. The lecture's quantitative connection is \( M(R) = \dfrac{V(R)^{2} R}{G} \), which shows how this term participates in the model rather than functioning as an isolated label.

Boundary or confusion: Assuming "dark matter" means "dark, ordinary matter we just haven’t seen yet" (like cold gas or dim stars). Multiple independent lines of evidence (rotation curves, gravitational lensing, the cosmic microwave background, large-scale structure) point to a non-baryonic form of matter that does not emit, absorb, or scatter light at all, distinguishing it from merely faint ordinary matter.

21-cm line

If nearly all mass were interior to the visible disk (as for the solar system, where nearly all mass is in the Sun), V(R) should decline as 1/√R beyond the visible mass, the way planetary orbital speeds decline with distance from the Sun.

Use and relationship: Because V(R) instead stays roughly flat out to large R, M(R) must keep growing linearly with R, implying an extended, largely invisible ("dark") mass distribution well beyond the visible disk. The lecture's quantitative connection is \( M(R) = \dfrac{V(R)^{2} R}{G} \), which shows how this term participates in the model rather than functioning as an isolated label.

Boundary or confusion: Assuming "dark matter" means "dark, ordinary matter we just haven’t seen yet" (like cold gas or dim stars). Multiple independent lines of evidence (rotation curves, gravitational lensing, the cosmic microwave background, large-scale structure) point to a non-baryonic form of matter that does not emit, absorb, or scatter light at all, distinguishing it from merely faint ordinary matter.

Evidence

Model

Working Equation

\[ M(R) = \dfrac{V(R)^{2} R}{G} \]

Worked Example

  1. At the Sun’s radius, R₀ = 8.2 kpc = 2.530e+20 m, V₀ = 220 km/s = 2.200e+05 m/s.
  2. M(R₀) = (2.200e+05)² × 2.530e+20 / 6.674e-11 = 1.835e+41 kg = 9.226e+10 M☉.
  3. At R = 25 kpc, V ≈ 230 km/s (still nearly flat): M(25 kpc) = 3.074e+11 M☉, roughly 3.33 times the enclosed mass at the Sun’s radius, even though the radius only increased by a factor of 3.05.
  4. The Galaxy’s visible stellar and gas mass is estimated at only about 6.0e+10 M☉ — far less than the 3.07e+11 M☉ implied by the rotation curve at 25 kpc, requiring roughly 5.1 times more mass than is visible, attributed to a dark matter halo.

Common Misconception

Assuming "dark matter" means "dark, ordinary matter we just haven’t seen yet" (like cold gas or dim stars). Multiple independent lines of evidence (rotation curves, gravitational lensing, the cosmic microwave background, large-scale structure) point to a non-baryonic form of matter that does not emit, absorb, or scatter light at all, distinguishing it from merely faint ordinary matter.

Active-Learning Guidance

Using M(R) = V²R/G, predict what a rotation curve would look like (rising, flat, or falling with R) if the Galaxy’s mass were entirely enclosed within the Sun’s orbital radius, and compare to the observed flat curve.

Lab Connection

Lab 05 computes enclosed mass at several radii from a representative rotation-curve data set and compares the result to the Galaxy’s visible mass.

Synthesis Questions

References