Blackbody Radiation: Wien’s Law and Stefan-Boltzmann for Sirius A
λ_max = b/T = 2.8971e-03/9940 = 291 nm, in the ultraviolet/blue-visible part of the spectrum, consistent with Sirius A's observed blue-white color.
| Criterion | Points |
|---|---|
| Correct formula and unit handling | 8 |
| Correct spectral-region identification | 7 |
R = 1.711 × 6.957e+08 m = 1.190e+09 m. L = 4πR²σT⁴ = 9.856e+27 W = 25.75 L⊙.
| Criterion | Points |
|---|---|
| Correct unit conversion R⊙ to m | 8 |
| Correct final luminosity in solar units | 7 |
Percent agreement = |25.75 − 25.4| / 25.4 × 100% = 1.4%. A residual difference this size is well within plausible measurement uncertainty in Sirius A's published radius and temperature.
| Criterion | Points |
|---|---|
| Correct percent-agreement arithmetic | 8 |
| Valid physical reason for residual disagreement | 7 |
L_Sirius/L_AlphaCenA ≈ (R ratio)² × (T ratio)⁴ = (1.399)² × (1.729)⁴ = 17.5, close to the 17× luminosity ratio quoted, showing the T⁴ dependence dominates over the near-unity radius ratio.
| Criterion | Points |
|---|---|
| Correct computed ratio using R²T⁴ scaling | 12 |
| Correct identification that T⁴ dominates | 8 |
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 (AU/m, years/seconds, km/Mpc, Msun/kg) -- these are the most common source of order-of-magnitude mistakes in this course.