Worked Solutions
1. Phobos and Mars's Roche Limit
Phobos's density: ρ = 1.0659e+16 / [(4/3)π(1.127e+04)³] ≈ 1876 kg/m³. Mars's mean density (from Lecture 01's table) is 3934 kg/m³. Fluid Roche limit: d = 2.44 R_Mars (ρ_Mars/ρ_Phobos)¹⁄³ = 10775 km. Phobos's current orbital radius (9376 km) is larger than this computed Roche limit but is decreasing over time due to tidal dissipation inside Mars, implying that Phobos will eventually cross inside the Roche limit and be tidally disrupted, likely within the next tens of millions of years.
| Criterion | Points |
|---|
| Correct Phobos density and Mars mean density calculations | 6 |
| Correct fluid Roche-limit calculation | 6 |
| Correct qualitative conclusion about Phobos's eventual fate | 3 |
2. Saturn versus Jupiter: Why Does Density Differ Despite Similar Composition?
Jupiter's substantially larger mass produces much stronger self-gravity throughout its interior, compressing its hydrogen/helium envelope (and any core material) to higher density at any given pressure level than Saturn's weaker self-gravity can achieve, exactly as Lecture 03 showed that Earth's stronger self-compression (relative to Mercury) raises its bulk density above what its material composition alone would suggest. This self-compression effect, not any difference in bulk hydrogen/helium/heavy-element composition, is sufficient on its own to explain why Jupiter is denser than Saturn despite both planets sharing a broadly similar protosolar-like composition.
| Criterion | Points |
|---|
| Correctly identifies self-compression, not composition difference, as the explanation | 9 |
| Correctly connects this to the analogous terrestrial-planet argument from Lecture 03 | 6 |
3. Neptune's Ring Arcs
A shepherd moon in resonance with a ring can exert a gravitational torque that varies periodically as the moon and a given ring particle repeatedly return to the same relative orbital geometry; this resonant torque can trap ring material at specific longitudes relative to the moon's orbit (librating around stable resonance points) rather than allowing it to spread freely around the entire ring circumference, producing discrete, confined arcs of enhanced ring-particle density at those resonance points instead of a smooth, uniform ring.
| Criterion | Points |
|---|
| Correctly explains resonant trapping/libration as the confinement mechanism | 15 |
4. Jupiter's Zonal Wind Depth and Gravity Harmonics
As depth increases inside Jupiter, pressure and temperature both rise, and hydrogen's electrical conductivity increases correspondingly (eventually reaching the fully metallic state described in Lecture 11). Once the surrounding fluid becomes sufficiently electrically conductive, magnetic forces (generated by Jupiter's own dynamo-produced field) become strong enough to resist and damp out organized large-scale zonal flow, effectively braking the winds at the depth where conductivity becomes significant -- roughly the depth Juno's gravity data indicate, ∼3000 km.
| Criterion | Points |
|---|
| Correctly connects increasing conductivity with depth to magnetic braking of the winds | 15 |
5. Why Ice Giants Are Not Simply Small Gas Giants
The gas-giant/ice-giant distinction is not simply a matter of Uranus and Neptune being smaller versions of Jupiter and Saturn; if Uranus and Neptune had captured hydrogen/helium gas as efficiently as Jupiter and Saturn did, they would be expected to show a similarly dominant (∼90%+) hydrogen/helium mass fraction regardless of their smaller total mass. Instead, their measured ∼10-20% hydrogen/helium fraction reflects a genuine difference in formation history: their cores are thought to have formed more slowly (in a lower-density, more slowly orbiting part of the disk, per Lecture 02's core-accretion argument) and so captured comparatively little nebular gas before the gas disk dispersed, leaving them dominated by ices and rock rather than hydrogen/helium -- a compositional outcome, not merely a size difference.
| Criterion | Points |
|---|
| Correctly explains why size alone would not predict the observed composition difference | 8 |
| Correctly connects the composition difference to core-accretion formation timing | 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, days or years/seconds, bar/Pa) and for comparative-magnitude statements stated in the wrong direction (e.g., a tidal acceleration or greenhouse excess described as a multiple when it is actually a small fraction, or vice versa) -- these are the most common sources of silent error in this course.