Worked Solutions
1. Jeans Escape of Molecular Oxygen: Earth versus Mars
O₂ thermal speed at 1000 K: v_mp = √(2kT/m) = √(2 × 1.381e-23 × 1000 / (5.314e-26)) ≈ 721 m/s; at Earth, λ_O2 = (11200/721)² ≈ 241. At Mars (T=300 K), v_mp ≈ 395 m/s; λ_O2 = (5030/395)² ≈ 162. Both λ_O2 values are far larger than the corresponding λ_H2 values from Lecture 09 (∼15 for Earth, ∼3 for Mars), confirming that molecular oxygen is retained far more effectively than hydrogen on both planets, because oxygen's much larger mass gives it both a lower thermal speed and (through λ ∝ m) a much larger Jeans parameter at the same temperature and escape velocity.
| Criterion | Points |
|---|
| Correct thermal-speed and Jeans-parameter calculations for both planets | 9 |
| Correct comparison to Lecture 09's hydrogen values with correct physical explanation | 6 |
2. Mars's Ion Pickup Loss and MAVEN
A present-day measured loss rate gives only a snapshot of current conditions; extrapolating it across 4.5 Gyr would implicitly assume the loss rate has been constant throughout Mars's history, which is almost certainly false, since both the solar wind's intensity (much stronger from the young, more active Sun) and Mars's own atmosphere (likely much thicker early on) have changed dramatically over time. Making a reliable historical extrapolation would require independent modeling of the young Sun's solar-wind and XUV output (informed by observations of young solar-type stars) combined with a model of how Mars's atmospheric thickness itself evolved over time, not simply multiplying today's measured rate by 4.5 Gyr.
| Criterion | Points |
|---|
| Correctly identifies that a present-day rate cannot simply be extrapolated linearly | 8 |
| Correctly identifies what additional modeling would be required | 7 |
3. Comparing Uranus’s and Neptune’s Magnetic Fields
Ratio of magnetic pressures (B²) = (1.42e-05)² / (2.28e-05)² = 0.39. Since magnetopause standoff distance scales as B²/6 (the sixth root of B², from the dipole field's r⁻³ falloff and the pressure balance derived in Lecture 10), a magnetic-pressure ratio of 0.39 implies Neptune's standoff distance (in its own planetary radii) should be only modestly larger than Uranus's (roughly 0.85 times as large), since the sixth-root dependence strongly compresses even a substantial pressure-ratio difference into a much smaller distance-ratio difference.
| Criterion | Points |
|---|
| Correct magnetic-pressure ratio calculation | 7 |
| Correctly applies the sixth-root scaling to standoff distance rather than a linear or squared scaling | 8 |
4. Venus's Induced Magnetosphere
Lecture 10 established that a self-sustaining dynamo requires a convecting, electrically conducting fluid organized by planetary rotation into helical flow; Venus's induced magnetosphere, by contrast, arises passively from the solar wind interacting with Venus's ionosphere and does not require any internal dynamo process at all. An induced field is generally much weaker and more variable than an intrinsic dipole field, and it does not extend nearly as far from the planet, meaning it stands off the solar wind much closer to Venus's ionosphere/atmosphere than Earth's intrinsic magnetosphere does relative to Earth's atmosphere, providing correspondingly less protective standoff distance against direct atmospheric erosion.
| Criterion | Points |
|---|
| Correctly distinguishes induced from intrinsic (dynamo-generated) fields | 8 |
| Correctly explains why an induced field provides less standoff protection | 7 |
5. Escape Velocity and the Retention of Helium
Helium thermal speed at 1000 K: v_mp = √(2 × 1.381e-23 × 1000 / (4 × 1.66054e-27)) ≈ 2029 m/s. λ_He = (11200/2029)² ≈ 30.4, about twice the hydrogen value (∼15) from Lecture 09, since helium is roughly twice as massive as H₂. Earth should retain helium somewhat more effectively than hydrogen via Jeans escape alone. However, Earth's atmosphere contains only trace helium today primarily because Earth's helium is continuously but only slowly replenished by radioactive decay (alpha decay of uranium and thorium, Lecture 04) and, unlike nitrogen or oxygen, has no significant chemical sink that would otherwise retain it at the surface; the modest but non-negligible Jeans and non-thermal escape over geological time is therefore sufficient to keep helium's atmospheric abundance low despite its relatively large λ, since its supply rate is also very low.
| Criterion | Points |
|---|
| Correct thermal-speed and Jeans-parameter calculation for helium | 8 |
| Correctly explains the low supply-rate resolution rather than treating Jeans escape as the full explanation | 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.