Worked Solutions
1. Why Mercury Has (Essentially) No Scale Height to Compute
Mercury's low mass and correspondingly low escape velocity (4.25 km/s), combined with its very high dayside temperature (∼700 K, giving even heavy atmospheric molecules substantial thermal speeds), together give even relatively heavy gas species a Jeans escape parameter λ small enough for rapid, geologically fast atmospheric loss (the same λ = v_esc²/v_mp² reasoning applied quantitatively to Earth and Mars in Lecture 09). Any substantial primordial atmosphere Mercury may have had would therefore have been lost long ago, leaving only an extremely tenuous, transient exosphere (supplied by solar-wind sputtering and micrometeorite vaporization) far too thin to be usefully described by a hydrostatic scale-height formula.
| Criterion | Points |
|---|
| Correctly connects low escape velocity and high temperature to a small Jeans parameter | 8 |
| Correctly concludes the scale-height formula is not meaningfully applicable | 7 |
2. Mars's Small Greenhouse Excess
92 bar / 6 mbar = 92000/6 ≈ 15,300, i.e., Venus's surface pressure (and hence total atmospheric mass and infrared opacity) is roughly four orders of magnitude greater than Mars's, despite both atmospheres being >95% CO₂ by composition. This confirms that the greenhouse effect's magnitude depends on total atmospheric mass and column opacity to outgoing infrared radiation, not merely on which gas is present: Mars's atmosphere, though chemically similar to Venus's, is simply too thin to trap much outgoing thermal radiation, giving it only a small greenhouse excess despite its CO₂-dominated composition.
| Criterion | Points |
|---|
| Correct pressure-ratio calculation | 6 |
| Correctly explains that mass/opacity, not composition alone, controls greenhouse magnitude | 9 |
3. Earth's Scale Height with a Different Surface Temperature
H(250 K) = H(288 K) × (250/288) = 8.72 × (250/288) = 7.57 km, a decrease of 13.2% relative to the standard 288 K value, since H ∝ T linearly at fixed g and m.
| Criterion | Points |
|---|
| Correctly applies the linear T-scaling of the scale-height formula | 8 |
| Correct percent-change calculation | 7 |
4. Bond Albedo and Cloud Cover
T_eq(A=0.088) = [(1-0.088)/(1-0.76)]¹⁄⁴ × T_eq(A=0.76) = [3.800]¹⁄⁴ × 229.1 K = 319.8 K. This hypothetical value is higher than Venus's real computed equilibrium temperature because a lower albedo means Venus would absorb a larger fraction of the sunlight it intercepts, requiring a higher equilibrium temperature to radiate that larger absorbed power back to space.
| Criterion | Points |
|---|
| Correct proportional recomputation using the (1-A)^(1/4) scaling | 9 |
| Correct one-sentence physical explanation | 6 |
5. The Runaway Greenhouse and D/H Ratio
Because hydrogen (light) escapes far more readily than deuterium (heavier by roughly a factor of two in mass) via Jeans escape, any water vapor that is photodissociated high in Venus's atmosphere loses its hydrogen preferentially, progressively enriching the remaining hydrogen inventory in deuterium over time. The presently observed, strongly elevated D/H ratio is therefore best explained as the cumulative fossil signature of a much larger original water/hydrogen inventory having been lost over billions of years, rather than as evidence that Venus's water was unusually deuterium-rich from the start (which would require an ad hoc, unexplained difference in Venus's initial volatile delivery relative to Earth's, for which there is no independent supporting evidence).
| Criterion | Points |
|---|
| Correctly explains the differential-escape mechanism | 8 |
| Correctly argues why this is stronger evidence than an alternative "born deuterium-rich" hypothesis | 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.