Worked Solutions
1. Kepler-186f: A Second Real Exoplanet Comparison
T_eq = [L/(16πσa²)]¹⁄⁴ = [(1.569e+25) / (16π×5.670374e-8×(6.463e+10)²)]¹⁄⁴ = 190.6 K. This is comparable to TRAPPIST-1e's 250.8 K and to Earth's 278.3 K, placing Kepler-186f, like TRAPPIST-1e, within the temperature range commonly associated with the habitable zone, though (as with TRAPPIST-1e) this equilibrium-temperature calculation alone cannot confirm habitability without atmospheric and albedo information.
| Criterion | Points |
|---|
| Correct equilibrium-temperature calculation with correct unit conversions | 12 |
| Correct, appropriately hedged comparison to TRAPPIST-1e and Earth | 8 |
2. TRAPPIST-1b and 1h: The Hottest and Coldest Planets in the System
T_eq(b)/T_eq(h) = 400.3/172.6 = 2.320. √(a_h/a_b) = √(0.0619/0.0115) = 2.320. These two ratios agree closely (both ≈ 2.32), confirming the expected T_eq ∝ a⁻¹⁄² scaling directly from real TRAPPIST-1 data.
| Criterion | Points |
|---|
| Correct temperature-ratio calculation | 8 |
| Correct semimajor-axis-ratio square-root calculation and correct verification of the scaling law | 12 |
3. Tidal Locking Timescale Reasoning
Because tidal-despinning timescale falls off steeply with decreasing orbital distance (roughly as a⁶), even a modest decrease in orbital distance produces an enormous decrease in locking timescale. All seven TRAPPIST-1 planets orbit at less than one-sixth of Mercury's orbital distance from the Sun, and TRAPPIST-1 itself, though much less massive than the Sun, is old (∼7.6 Gyr), giving even a slower-acting tidal torque ample time to act; the combination of extremely small orbital distance (steeply reducing the timescale) and a old system age is expected to be more than sufficient to have tidally locked essentially every TRAPPIST-1 planet, whereas Mercury's larger orbital distance from a more massive star left it in a more weakly torqued regime where it settled into a 3:2 rather than a strict 1:1 resonance.
| Criterion | Points |
|---|
| Correctly invokes the steep (a^6-type) distance scaling of tidal locking timescale | 8 |
| Correctly contrasts with Mercury's different regime rather than treating all close orbits identically | 7 |
4. Detection-Method Bias in the Exoplanet Census
Radial-velocity amplitude scales with planet mass and inversely with orbital distance and stellar mass, so the method most easily detects massive planets orbiting very close to their star (the 'hot Jupiter' population); lower-mass planets, or planets on wider orbits, produce a smaller reflex stellar wobble that is harder to distinguish from instrumental and stellar-activity noise. Because early radial-velocity surveys were the dominant discovery method before large-scale transit surveys (e.g., Kepler, TESS) came online, the earliest confirmed exoplanets were disproportionately hot Jupiters, giving an initially skewed impression of typical exoplanet system architecture that later, less biased transit-survey statistics have substantially corrected.
| Criterion | Points |
|---|
| Correctly explains the radial-velocity sensitivity bias | 8 |
| Correctly explains how this biased early demographic conclusions | 7 |
5. Capstone Synthesis: Our Solar System in Context
[Model answer, 200-300 words; graded for content and correct hedging rather than for an exact match] A strong answer should note: (1) the frost-line argument (Lecture 02) shows our solar system's terrestrial/giant-planet split is a specific, well-understood consequence of where the frost line fell in our particular protoplanetary disk, not a universal necessity -- other disks with different temperature profiles or disk masses could plausibly produce very different architectures; (2) the exoplanet mass-radius degeneracy (Lecture 14) shows that our confidence in solar-system-like interior compositions elsewhere is genuinely limited by current data, an honestly open question; (3) TRAPPIST-1's compact, resonant, tidally locked architecture (Lecture 14) has no solar-system analog at all, and such compact systems appear statistically common in the broader exoplanet census, suggesting our solar system's wider-orbit, non-resonant architecture may not be the 'default' outcome of planet formation. A strong answer explicitly separates well-supported claims (the existence of common compact systems, the basic frost-line physics) from open questions (whether Earth-like habitability is common, whether TRAPPIST-1 planets can retain atmospheres) rather than presenting a falsely confident overall conclusion.
| Criterion | Points |
|---|
| Cites at least three specific pieces of evidence from across the semester | 12 |
| Correctly and explicitly distinguishes well-supported claims from open questions | 12 |
| Coherent, well-organized synthesis within the word limit | 6 |
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.