Extend this unit's resonance-chain and density-inference reasoning.
Using the period-ratio table from this unit's lab, compute the overall ratio P_h/P_b (TRAPPIST-1's outermost to innermost known planet) two ways: (a) directly from the two periods, and (b) as the product of the six adjacent-pair ratios. Confirm the two methods agree, and explain why they must.
Using the steep a-dependence of the tidal-circularization timescale from Lecture 05 (t_circ ∝ a^(13/2)), and given that TRAPPIST-1e (a=0.0293 AU) has an orbit only about 1.7 times larger than TRAPPIST-1b (a=0.0115 AU), estimate the ratio of their tidal-circularization timescales, and state whether both should be expected to be circularized within the system's age.
Using your computed densities for TRAPPIST-1d and TRAPPIST-1g (this problem set's reading), state which is more likely to carry a larger ice/water mass fraction and justify your answer quantitatively using the density comparison to Earth.
Explain, using Lecture 06's degeneracy argument, what additional measurement (beyond mass and radius) would be needed to determine whether TRAPPIST-1e's slightly-lower-than-Earth density reflects a smaller iron core, a modest ice/water layer, or some combination of both.
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.