ASTR 310 Problem Set 03: Hydrostatic Equilibrium for Alpha Centauri A and the Virial Mass of 47 Tucanae

Problem set for Unit 3.

Problems

1. Central pressure of Alpha Centauri A 15 points

Using P_c ≈ (3/8π)GM²/R⁴, estimate the central pressure of Alpha Centauri A (M = 1.0788 M⊙, R = 1.223 R⊙), showing all unit conversions.

2. Comparison to the Sun 15 points

Compare your Problem 1 answer to the Sun's estimated central pressure from Lecture 05/Lab 03 (1.345e+14 Pa), and explain, using the P_c ∝ M²/R⁴ scaling, why Alpha Centauri A's slightly larger mass and radius than the Sun's produce the sign and rough size of the difference you find.

3. Virial mass of 47 Tucanae 20 points

The globular cluster 47 Tucanae has a published core velocity dispersion of 11.5 km/s and a half-light radius of 3.8 pc. Using M ≈ 5σ²R_h/G, estimate its virial mass in solar masses.

4. Agreement check 15 points

The commonly cited dynamical mass of 47 Tucanae is about 6.4e+05 M⊙. Compute the percent agreement with your Problem 3 answer, and state whether this level of agreement is consistent with the same order-of-magnitude uncertainty seen for M15 in Lab 03.

Due and Scope

Submit a complete derivation with equations, units, labeled quantities, and a short narrative interpretation for each problem. Show every step; a correct final number without a visible derivation receives partial credit at most, and quoting a lecture number without adapting it to this problem's specific inputs receives no credit for that step.

OpenStax Companion Reading

OpenStax Astronomy 2e (local reference copy: references/openstax-astronomy-2e-extracted.txt), Chapter 15.4-15.5.

References and Data Sources