1. Stefan-Boltzmann for Rigel
L = (78.9)² × (12100/5778)⁴ = 6225.2 × 19.232 = 119725 L☉.
| Criterion | Points |
|---|---|
| Setup and units | 5 |
| Calculation | 5 |
| Interpretation | 5 |
Worked Solutions
2. Inverse-square reasoning
F ∝ 1/d², so F_A/F_B = (d_B/d_A)² = (40/10)² = 16. Star A appears 16 times brighter than Star B.
| Criterion | Points |
|---|---|
| Correct formula | 5 |
| Correct ratio | 5 |
| Correct direction (A brighter) | 5 |
3. Spectral classification physics
Balmer lines require neutral hydrogen atoms with their electron in the n=2 excited state. At the hottest temperatures (O/B stars), most hydrogen is ionized (no bound electron at all), so there are few neutral atoms available to produce Balmer absorption. At the coolest temperatures (K/M stars), most hydrogen remains in the unexcited ground state (n=1), unable to absorb Balmer-series photons. Only at intermediate temperatures near A0 (≈9,000-10,000 K) is a large fraction of hydrogen both neutral and in the n=2 excited state, maximizing Balmer line strength.
| Criterion | Points |
|---|---|
| Correct ionization-fraction reasoning (hot end) | 8 |
| Correct excitation-fraction reasoning (cool end) | 7 |
4. Textbook problem: Sirius system radii
Following the OpenStax worked derivation: with L_A/L_Sun and T_A/T_Sun known, R_A/R_Sun = √(L_A/L_Sun) × (T_Sun/T_A)². The textbook's worked example yields a radius ratio for Sirius A close to 1.7-1.8 R☉, consistent with this course's adopted value of 1.71 R☉ to within the precision of the simplified two-temperature comparison.
| Criterion | Points |
|---|---|
| Correctly reproduces the OpenStax derivation steps | 8 |
| Correct numeric comparison to course value | 7 |
Grading Notes
Award full marks for correct method, units, and a numeric result consistent with the arithmetic shown (allow reasonable rounding). Verify the student re-derives the number rather than quoting it from the lecture without adapting it to the problem's specific inputs.