Problems
1. Why Mercury Has (Essentially) No Scale Height to Compute 15 points
The scale-height formula H = kT/(mg) requires a bound atmosphere in hydrostatic equilibrium. Mercury's escape velocity is 4.25 km/s and its dayside surface temperature reaches ∼700 K. Using the Jeans-escape reasoning previewed in Lecture 07 and developed fully in Lecture 09, explain in a short paragraph why Mercury cannot sustain a substantial bound atmosphere, making a meaningful scale-height calculation for a thick Mercury atmosphere physically inapplicable.
2. Mars's Small Greenhouse Excess 15 points
Using the equilibrium-temperature table values from Lecture 08, Mars's greenhouse excess is ΔT_gh = +0 K, compared to Venus's +508 K, despite both planets having CO₂-dominated atmospheres. Compute the ratio of Venus's to Mars's surface pressure (92 bar versus 6 mbar) and use it to explain, in a short paragraph, why atmospheric mass/opacity, not merely the presence of CO₂, controls the size of the greenhouse excess.
3. Earth's Scale Height with a Different Surface Temperature 15 points
Recompute Earth's atmospheric scale height (using N₂ as the dominant species) for a hypothetical mean surface temperature of 250 K (an ice-age-like scenario) instead of the standard 288 K, holding gravity fixed. Compare your result to the standard 8.72 km value from Lecture 07 and state the percent change.
4. Bond Albedo and Cloud Cover 15 points
Venus has the highest Bond albedo (0.76) of any planet, due to its global sulfuric-acid cloud deck, while Mercury has the lowest (0.088), due to its dark, airless regolith surface. Using the equilibrium-temperature formula, compute what Venus's zero-greenhouse equilibrium temperature would be if its albedo were instead equal to Mercury's (0.088), holding its semimajor axis fixed, and explain in one sentence why this hypothetical value is higher than Venus's real computed equilibrium temperature.
5. The Runaway Greenhouse and D/H Ratio 15 points
Venus's atmospheric deuterium-to-hydrogen ratio is roughly 100-150 times Earth's. Using the differential-escape reasoning previewed in Lecture 08 and developed in Lecture 09 (lighter isotopes escape preferentially), explain in a short paragraph why this elevated D/H ratio is considered strong indirect evidence that Venus once had a much larger water inventory than it has today, rather than simply having formed with unusually deuterium-rich water.