1. Mass-ratio splitting
M_A = total/(1+ratio) = 2.062 M⊙; M_B = total − M_A = 1.017 M⊙, consistent with the independently quoted masses.
| Criterion | Points |
|---|---|
| Correct splitting formula setup | 12 |
| Correct individual masses | 8 |
Worked Solutions
2. Europa’s Roche limit
ρ_Europa = 3014 kg/m³. Roche limit ≈ 66998 km. Europa's actual orbital distance (671100 km) is about 10.0 times the Roche limit, safely outside it.
| Criterion | Points |
|---|---|
| Correct density calculation | 8 |
| Correct Roche-limit calculation and comparison | 7 |
3. Europa’s tidal acceleration
Europa's surface gravity: g = GM_Europa/R_Europa² = 1.315 m/s². Δa(Europa) = 1.308e-03 m/s² = 0.10% of Europa's own surface gravity. Io's corresponding fraction (Lecture 10/Lab 05) is 0.34% of Io's own surface gravity. Io experiences the stronger relative tidal stress, because Io orbits much closer to Jupiter and tidal acceleration falls off as 1/d³, more steeply than the modest difference in the two moons' surface gravities can offset.
| Criterion | Points |
|---|---|
| Correct surface-gravity and tidal-acceleration calculation for Europa | 12 |
| Correct percentage comparison and correct 1/d³ explanation | 8 |
4. Physical interpretation
Because tidal acceleration falls steeply (as 1/d³) with orbital distance and Europa orbits substantially farther from Jupiter than Io, Europa's tidal flexing and resulting heating is weaker than Io's, consistent with Europa retaining an icy crust (implying much less surface heat flow than Io's continuous resurfacing by active volcanism) while still plausibly sustaining a subsurface ocean from a smaller but non-negligible tidal heat source.
| Criterion | Points |
|---|---|
| Correct qualitative reasoning tied to the computed ratio | 15 |
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.