Apparatus and Setup
This capstone lab combines the tidal-acceleration framework from Lecture 13 with the real, live-verified TRAPPIST-1 system data introduced in Lecture 14, synthesizing the semester's tidal and radiative-balance tools on a single exoplanet system.
Materials and Data
| Quantity | Value |
|---|---|
| TRAPPIST-1 stellar mass | 0.0898 M⊙ |
| TRAPPIST-1e semimajor axis | 0.0293 AU |
| TRAPPIST-1e orbital period | 6.1 days |
| Io tidal acceleration fraction (Lecture 13, for comparison) | 0.34% of Io's own surface gravity |
Procedure
- Using the tidal-acceleration formula from Lecture 13, and approximating TRAPPIST-1e as Earth-sized and Earth-massed (a stated simplification, since TRAPPIST-1e's exact mass carries larger uncertainty than its radius), compute the tidal acceleration TRAPPIST-1e would experience from its host star, and express it as a fraction of Earth's own surface gravity (never as a multiple).
- Compare this fraction to Io's tidal-acceleration fraction from Lecture 13 (recomputed there as 0.34% of Io's own gravity) and state which body experiences the larger relative tidal stress.
- Because TRAPPIST-1e's orbital eccentricity is measured to be very low (consistent with strong tidal circularization, though some residual forced eccentricity from the system's resonant chain is expected), explain in one paragraph why a small tidal acceleration combined with even a small non-zero eccentricity could still produce meaningful sustained tidal heating, referencing Lecture 13's resonance-sustained-eccentricity argument.
- Recompute TRAPPIST-1e's zero-albedo equilibrium temperature (identical formula to Lecture 14's worked example) and compare it to Earth's zero-albedo equilibrium temperature computed in Lecture 08.
- Synthesize your results in a short written summary (150-250 words) addressing whether TRAPPIST-1e is a plausible candidate for a temperate, geologically active world, explicitly citing which of this semester's tools (equilibrium temperature, tidal heating, atmospheric escape, magnetic shielding) most strongly support or undercut that conclusion, and stating clearly which parts of your answer are well-constrained versus speculative.
Analysis and Uncertainty
Approximating TRAPPIST-1e as Earth-sized/massed for the tidal-acceleration calculation is a genuine simplification (its true mass is less precisely measured than its radius); students must state this explicitly rather than presenting the resulting tidal-heating estimate as equally well-constrained as the equilibrium-temperature calculation, which uses only directly measured quantities (semimajor axis and stellar luminosity).
Deliverables
- Computed tidal-acceleration fraction for TRAPPIST-1e, correctly labeled as a fraction of Earth's own gravity.
- A comparison to Io's tidal fraction with a correctly stated conclusion about relative tidal stress.
- A written paragraph on resonance-sustained eccentricity and its implications for TRAPPIST-1e.
- Recomputed equilibrium temperature for TRAPPIST-1e compared to Earth's.
- A synthesis paragraph explicitly distinguishing well-constrained from speculative conclusions.
Assessment Criteria
- Correct tidal-acceleration calculation, correctly expressed as a fraction (not a multiple) of surface gravity.
- Correct, properly directioned comparison to Io.
- Correct equilibrium-temperature recomputation.
- Synthesis paragraph explicitly and correctly separates well-constrained results from speculative extrapolations, per this course's standard of honest uncertainty characterization.
References and Provenance
- OpenStax Astronomy 2e (local reference copy: references/openstax-astronomy-2e-extracted.txt), Chapter 13.
- TRAPPIST-1 system parameters: live-verified this session via direct web fetch (Wikipedia "TRAPPIST-1," citing Agol et al. 2021, PSJ 2, 1); see
materials/ASTR320/reference-log.md. - Generated by
materials/ASTR320/src/generate_astr320_labs_psets.py.