Apparatus and Setup
You will use a representative Milky Way circular-velocity (rotation) curve, ../data/milky_way_rotation_curve.csv, consistent with published 21-cm and stellar-tracer rotation curves (e.g. Eilers et al. 2019, ApJ 871, 120; see ../reference-log.md for exact provenance and spot-check status of these specific data points), to compute the Milky Way's enclosed mass at several radii and compare it to the visible (stellar + gas) mass.
Materials and Data
| Radius (kpc) | Circular velocity (km/s) | Enclosed mass (M☉) |
|---|---|---|
| 2.0 | 200 | |
| 4.0 | 220 | |
| 6.0 | 225 | |
| 8.2 | 220 | |
| 10.0 | 222 | |
| 14.0 | 224 | |
| 18.0 | 226 | |
| 22.0 | 228 | |
| 25.0 | 230 |
Solar radius R₀ = 8.2 kpc, solar circular velocity V₀ = 220 km/s. Visible (stellar + gas) mass estimate for comparison: 6.0e+10 M☉.
Procedure
- Using M(R) = V(R)²R/G, compute the enclosed mass at R = 2 kpc, R = R₀ = 8.2 kpc, and R = 25 kpc. Show unit conversions (kpc to meters, km/s to m/s) explicitly.
- Plot enclosed mass versus radius for all nine data points in the table. Describe the shape of the curve (does mass grow linearly with radius, faster, or slower, over the range where velocity is roughly flat?).
- If all of the Galaxy's mass were interior to R = 8 kpc (visible disk), predict qualitatively what the rotation curve should do beyond 8 kpc, and explain why the observed flat curve is inconsistent with that prediction.
- Compute the ratio of enclosed mass at 25 kpc to the visible mass estimate (6.0e+10 M☉), and state what fraction of the Galaxy's total mass within 25 kpc must be non-luminous (dark matter) under this comparison.
- A hypothetical galaxy has a flat rotation curve at 300 km/s out to 40 kpc. Compute its enclosed mass at 40 kpc and compare it (as a ratio) to the Milky Way's enclosed mass at 25 kpc.
Analysis and Uncertainty
The dominant uncertainty in this method is not the velocity measurement itself (21-cm line centroids are measured very precisely) but the assumption of circular, planar orbits and a well-determined distance to the Galactic center; state qualitatively how an error in R₀ (e.g. if the true solar radius were 8.5 kpc rather than 8.2 kpc) would propagate into the enclosed-mass estimate at the solar radius.
Deliverables
- Enclosed-mass calculations at R = 2, 8.2, and 25 kpc with unit conversions shown.
- Completed enclosed-mass-vs-radius table and plot for all nine data points.
- Dark-matter mass-fraction calculation at 25 kpc.
- Hypothetical-galaxy comparison calculation.
Assessment Criteria
- Correct unit conversions and enclosed-mass arithmetic at all three specified radii.
- Correct qualitative description of the mass-vs-radius trend and its implication for a flat rotation curve.
- Correct dark-matter mass-fraction calculation and physically sound interpretation.
- Correct hypothetical-galaxy mass ratio.
References and Provenance
- OpenStax Astronomy 2e (local reference copy: references/openstax-astronomy-2e-extracted.txt), Chapter 25.1-25.4.
- Rotation curve:
materials/ASTR230/data/milky_way_rotation_curve.csv; representative of published results, logged with spot-check status inmaterials/ASTR230/reference-log.md.