Synthesize the full semester’s toolkit -- equilibrium temperature, tidal locking, detection-method bias, and comparative architecture -- using real TRAPPIST-1 and Kepler-186f data.
Kepler-186f orbits a red dwarf star with luminosity 0.041 L⊙ at a semimajor axis of 0.432 AU (orbital period 129.9 days), and has a measured radius of 1.17 Earth radii. Using the identical equilibrium-temperature formula from Lecture 08/14, compute Kepler-186f's zero-albedo equilibrium temperature and compare it to TRAPPIST-1e's (250.8 K, from Lecture 14) and Earth's (278.3 K, from Lecture 08).
Using the equilibrium-temperature table from Lecture 14, TRAPPIST-1b's zero-albedo equilibrium temperature is 400.3 K and TRAPPIST-1h's is 172.6 K. Compute the ratio T_eq(b)/T_eq(h) and compare it to the ratio of their semimajor axes (a_h/a_b), verifying that the ratio of equilibrium temperatures follows the expected inverse-square-root scaling with orbital distance (T_eq ∝ a⁻¹⁄²).
Tidal-locking timescales decrease steeply with decreasing orbital distance and increasing host-star mass (roughly as a⁶ for fixed stellar/planetary parameters, from standard tidal-despinning theory, not derived in this course but stated here for use). Given that all seven TRAPPIST-1 planets orbit within 0.062 AU (compared to Mercury's 0.39 AU orbit around the much more massive Sun), explain in a short paragraph why essentially all TRAPPIST-1 planets are expected to be tidally locked today, even though Mercury itself is not (Mercury is in a 3:2 spin-orbit resonance rather than a strict 1:1 lock).
Explain, in a short paragraph, why the radial-velocity method's greater sensitivity to massive, close-in planets caused early exoplanet surveys to discover a disproportionate number of 'hot Jupiters,' and why this early sample is not representative of the exoplanet population as a whole, referencing Lecture 14's discussion of detection bias.
In 200-300 words, synthesize this course's comparative-planetology themes (formation, interior structure, atmospheres, magnetism, tides) to address the open question posed in Lecture 01: how representative is our solar system's architecture of planetary systems generally? Your answer must cite at least three specific pieces of evidence developed across the semester (e.g., the frost-line argument, the exoplanet mass-radius degeneracy, TRAPPIST-1's compact resonant architecture) and must distinguish which parts of your answer are well-supported by current data versus still genuinely open questions in the research literature.
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, Chapter 13.
materials/ASTR320/src/generate_astr320_content.py where referenced above.