ASTR 330 Problem Set 05: Hyades Cluster Structure and Dynamical Timescales

OpenStax Astronomy 2e (local reference copy: references/openstax-astronomy-2e-extracted.txt), Chapter 21.1.

Problems

1. Core density 20 points

Using the Hyades' core radius (2.7 pc) and assuming, as an order-of-magnitude estimate, that about half the cluster's total mass (400 M⊙) lies within the core, compute the core's mean mass density in M⊙/pc³.

2. Half-mass density 15 points

Using the half-mass radius (5.7 pc), by definition enclosing exactly half the cluster's total mass, compute the mean density within the half-mass radius in M⊙/pc³, and compare it to your Problem 1 core-density estimate.

3. Metallicity comparison 15 points

The Hyades cluster's overall metallicity is measured at +0.14 dex (live-verified this session), meaning its stars are more metal-rich than the Sun (metallicity 0 dex by definition). Using the fact that higher metallicity increases opacity (Lecture 02), explain qualitatively whether you would expect Hyades stars of a given mass to have systematically longer or shorter main-sequence lifetimes than otherwise-identical solar-metallicity stars.

4. Turnoff-age discrepancy, revisited 20 points

Lab 05 found the fixed n=3.5 mass-luminosity-lifetime scaling overestimates the Hyades' true age by a factor of about 2.0. Using your Problem 3 answer, state whether the Hyades' above-solar metallicity would be expected to make this specific discrepancy larger or smaller, or whether it is unrelated to the discrepancy's likely cause (the fixed mass-luminosity exponent, per Lecture 07/09).

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 21.1.

References and Data Sources