1. Core density
Core volume = (4/3)π(2.7 pc)³ = 82.4 pc³. Assuming half the total mass (200 M⊙) lies within it, density ≈ 2.43 M⊙/pc³.
| Criterion | Points |
|---|---|
| Correct volume arithmetic | 10 |
| Correct density arithmetic | 10 |
Worked Solutions
2. Half-mass density
Half-mass volume = (4/3)π(5.7 pc)³ = 775.7 pc³; density = 200 M⊙ / 775.7 pc³ = 0.258 M⊙/pc³, substantially lower than the core density, confirming the Hyades is centrally concentrated.
| Criterion | Points |
|---|---|
| Correct volume and density arithmetic | 8 |
| Correct core-vs-half-mass density comparison | 7 |
3. Metallicity comparison
Higher metallicity raises opacity (more bound-free/free-free absorbers), which raises the radiative temperature gradient (Lecture 03) needed to carry a given luminosity; for a fixed mass, this generally corresponds to a somewhat lower luminosity and hence a somewhat longer main-sequence lifetime than an otherwise identical solar-metallicity star.
| Criterion | Points |
|---|---|
| Correct opacity-metallicity connection | 8 |
| Correct qualitative lifetime-direction conclusion | 7 |
4. Turnoff-age discrepancy, revisited
The Lab 05 discrepancy is attributed to the fixed n=3.5 mass-luminosity exponent being an approximation, not to metallicity; while the Hyades' above-solar metallicity would be expected to modestly lengthen main-sequence lifetimes (Problem 3), this is a separate, second-order effect from the primary cause of the factor-of-two discrepancy, which is the exponent's inaccuracy at the Hyades' specific turnoff mass, not the cluster's metallicity.
| Criterion | Points |
|---|---|
| Correct identification that metallicity is a secondary, not primary, effect | 12 |
| Correct connection back to the fixed-exponent explanation from Lecture 07/09 | 8 |
Grading Notes
Award full marks for correct method, units, and a numeric result consistent with the arithmetic shown (allow reasonable rounding). Verify that the student re-derives the number for this problem's specific inputs rather than quoting a lecture example without adaptation. Watch specifically for unit-conversion errors (km/AU/m, years/seconds, kg/M⊙) and for comparative-magnitude statements stated in the wrong direction (e.g., "X times greater" versus "a small fraction of") -- these are the most common sources of order-of-magnitude mistakes in this course.