1. Radiative gradient
∇_rad = 3κLP/(16πacGMT⁴) = 4.1045e-03, using κ = 0.0338 m²/kg, L = 9.723e+27 W, P = 1.0 × 10¹² Pa, M = 3.898e+30 kg, T = 3.0 × 10⁶ K.
| Criterion | Points |
|---|---|
| Correct substitution and unit handling | 12 |
| Correct final value | 8 |
Worked Solutions
2. Schwarzschild classification
∇_rad (4.1045e-03) is far greater than ∇_ad (0.4), so this illustrative envelope point is convectively unstable in this model.
| Criterion | Points |
|---|---|
| Correct comparison and classification | 8 |
| Correct numeric justification | 7 |
3. Comparison to the Sun
Since ∇_rad ∝ L directly, and Sirius A is about 25 times more luminous than the Sun, a proportionally larger ∇_rad is expected for otherwise similar opacity, pressure, mass, and temperature conditions -- consistent with the much larger value found in Problem 1 relative to the Sun's 0.0982.
| Criterion | Points |
|---|---|
| Correct use of the direct L proportionality | 8 |
| Correct qualitative comparison | 7 |
4. Physical interpretation
This problem set's single illustrative envelope point, using electron-scattering opacity only, finds convective instability there, which is not the same as real A-type stars' radiative envelopes; real A-type stellar envelopes have lower opacity and different pressure/temperature structure than this simplified illustrative point assumes. Correctly classifying Sirius A's actual envelope and core would require a full, depth-resolved stellar model with a realistic opacity table, not a single schematic sample point.
| Criterion | Points |
|---|---|
| Correct acknowledgment that the illustrative point does not match the real structure | 12 |
| Correct identification of a full depth-resolved model as the missing ingredient | 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.