Extend this unit's scale-height and Jeans-escape reasoning.
Using this unit's reading, explain quantitatively (using the scale-height formula) why a planet with the same temperature but twice HD 209458 b's surface gravity would have a transmission-spectroscopy signal only half as large, all else equal.
Compute the Jeans escape parameter for Earth-mass hydrogen atoms at Earth's exobase (assume T≈1000 K, use Earth's real mass and radius from materials/ASTR350/data/), and compare it to HD 209458 b's value (10.93). State which planet is in more danger of hydrodynamic hydrogen escape and why this is consistent with Earth having retained only trace atmospheric hydrogen today.
If HD 209458 b's real mass-loss rate had instead been measured as 9.0e+08 kg/s (three times the real value), recompute the naive mass-loss timescale and state whether this would change the conclusion about the planet's long-term survival at the currently observed rate.
Connect this unit's escape-parameter reasoning explicitly to Lecture 04's radius-valley discussion: explain why a lower-mass sub-Neptune at HD 209458 b's orbital distance would have a smaller Jeans parameter (all else equal) and therefore be more vulnerable to complete envelope loss than HD 209458 b itself.
Submit a complete derivation with equations, units, labeled quantities, and a short narrative interpretation for each problem. Show every step; a correct final number without a visible derivation receives partial credit at most, and quoting a lecture number without adapting it to this problem's specific inputs receives no credit for that step.
OpenStax Astronomy 2e (local extracted reference copy: references/openstax-astronomy-2e-extracted.txt) introduces exoplanet detection methods in the "Exoplanets" section of Chapter 7 (Other Worlds: An Introduction to the Solar System) and astrobiology topics (habitable zone, biosignatures, the Drake equation) in Chapter 30 (Life in the Universe); these citations follow the ASTR320/330 precedent of using native OpenStax chapter numbering, extrapolated by analogy rather than independently cross-checked against embedded figure/table numbering this session -- flagged for human spot-check in reference-log.md. Nearly every quantitative derivation in this course (the radial-velocity semi-amplitude equation, transit depth/duration, the mass-radius interior degeneracy, the radiative-equilibrium habitable-zone boundary, and the Drake equation as a structured estimation framework) goes well beyond OpenStax Astronomy 2e’s introductory, descriptive treatment; each lecture states explicitly where this course extends beyond the assigned reading.
materials/ASTR350/data/ where referenced above.