ASTR 230 Problem Set 05: Black Holes and the Milky Way Rotation Curve/Dark Matter

Problem set for Unit 5.

Problems

1. Schwarzschild radius of a stellar-mass black hole 15 points

Compute the Schwarzschild radius of a 5 M☉ black hole (a plausible mass for many observed stellar-mass black holes). Show your unit conversions.

2. Earth's Schwarzschild radius 15 points

Compute the Schwarzschild radius for Earth's mass (5.97 × 10²⁴ kg) and compare it to Earth's actual radius (6,371 km). State by what factor Earth would need to be compressed to become a black hole.

3. Enclosed mass from the rotation curve 15 points

Using M(R) = V²R/G, compute the Milky Way's enclosed mass at R = 14 kpc, given a circular velocity of 224 km/s at that radius (from this unit's rotation-curve data set). Show your unit conversions.

4. Textbook problem: dark matter fraction 15 points

OpenStax Astronomy 2e (local reference copy: references/openstax-astronomy-2e-extracted.txt), Chapter 25, Review Questions (evidence for dark matter from rotation curves): using your Problem 3 result and a visible-mass estimate of 6.0e+10 M☉, compute the implied dark-to-visible mass ratio at R = 14 kpc.

Due and Scope

Submit a clear solution with equations, units, labeled quantities, and a short narrative interpretation for each problem. Show every step; a correct final number without visible work receives partial credit at most.

OpenStax Companion Reading

OpenStax Astronomy 2e (local reference copy: references/openstax-astronomy-2e-extracted.txt), Chapter 24.1-24.3 (black holes) and Chapter 25.1-25.4 (the Milky Way, dark matter).

References and Data Sources