1. Extrapolated luminosity
L = 10^3.5 = 3162 L⊙.
| Criterion | Points |
|---|---|
| Correct exponent application | 12 |
| Correct final value | 8 |
Worked Solutions
2. Extrapolated lifetime
t_MS = 10 Gyr × 10/3162 = 0.0316 Gyr = 31.6 Myr.
| Criterion | Points |
|---|---|
| Correct lifetime arithmetic | 8 |
| Correct Gyr-to-Myr conversion | 7 |
3. Comparison to Sirius A
The 10 M⊙ star's lifetime (31.6 Myr) is much shorter than Sirius A's (812.2 Myr), consistent with t_MS ∝ M^-2.5: a 4.85× mass increase produces a (4.85)^-2.5 = 0.019× lifetime factor, a substantial reduction from a comparatively modest mass increase.
| Criterion | Points |
|---|---|
| Correct quantitative M^-2.5 scaling calculation | 8 |
| Correct comparison to Sirius A | 7 |
4. Limits of the extrapolation
The n=3.5 exponent is calibrated primarily from stars near solar mass; at 10 M☉, radiation pressure (Lecture 02) becomes progressively more important relative to gas pressure, which changes the star's effective equation of state and hence the true mass-luminosity exponent, so this problem set's answer should be treated as order-of-magnitude only, not a precision prediction.
| Criterion | Points |
|---|---|
| Correct acknowledgment of order-of-magnitude-only validity | 12 |
| Correct identification of radiation pressure as the missing physics | 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.