1. Unit conversion
P = 50.13 yr × 3.1558e+07 s/yr = 1.5820e+09 s. a = 19.78 AU × 1.4960e+11 m/AU = 2.9591e+12 m.
| Criterion | Points |
|---|---|
| Correct period conversion | 8 |
| Correct semimajor-axis conversion | 7 |
Worked Solutions
2. Total mass
(m1+m2) = 4π²a³/(GP²) = 3.079 M⊙.
| Criterion | Points |
|---|---|
| Correct formula setup with converted units | 12 |
| Correct final total mass | 8 |
3. Comparison
Sum of individual masses = 2.063 + 1.018 = 3.081 M⊙. Percent agreement with the Kepler total (3.079 M⊙) = 0.1%.
| Criterion | Points |
|---|---|
| Correct sum of individual masses | 8 |
| Correct percent-agreement arithmetic | 7 |
4. Escape velocity contrast
v_esc(Sirius A) = 678.3 km/s; v_esc(Sirius B) = 6926 km/s, a ratio of 10. Sirius B has less than half of Sirius A's mass yet a radius over 200 times smaller, and because v_esc ∝ 1/√R at fixed mass (and Sirius B's mass is not even fixed, only modestly smaller), its escape velocity is far higher — exactly the 1/√R sensitivity emphasized in Lecture 07.
| Criterion | Points |
|---|---|
| Correct escape-velocity arithmetic for both stars | 8 |
| Correct physical explanation referencing 1/√R scaling | 7 |
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 (AU/m, years/seconds, km/Mpc, Msun/kg) -- these are the most common source of order-of-magnitude mistakes in this course.