Apparatus and Setup
This lab computes TRAPPIST-1's full chain of adjacent orbital-period ratios from the system's real, live-verified periods and evaluates each against the nearest simple integer resonance (Lecture 05), then computes bulk densities for all seven planets to evaluate the interior mass-radius composition degeneracy (Lecture 06).
Materials and Data
| Adjacent pair | Period ratio |
|---|---|
| b-c | 1.6026 |
| c-d | 1.6736 |
| d-e | 1.5062 |
| e-f | 1.5098 |
| f-g | 1.3409 |
| g-h | 1.5198 |
Procedure
- For each adjacent period ratio in the table, identify the nearest simple integer ratio (e.g., 3:2, 4:3, 5:3, 8:5) and compute the percent deviation from that exact ratio.
- State whether the full six-ratio chain is consistent with a genuine resonant chain (small, consistent percent deviations throughout) or a coincidental near-resonance (large or inconsistent deviations).
- Using this course's real TRAPPIST-1 eccentricity dataset (essentially circular orbits) and Lecture 05's tidal-circularization argument, explain why circularization is expected to be efficient for all seven planets despite their being in mean-motion resonance.
- Compute the bulk density of each of the seven TRAPPIST-1 planets and rank them from most to least likely to carry a substantial water/ice layer, per Lecture 06's composition-degeneracy reasoning.
Analysis and Uncertainty
State explicitly that identifying a near-integer period ratio is necessary but not sufficient evidence of a true dynamical resonance (Lecture 05's pitfall); a full confirmation requires transit-timing-variation or stability-simulation evidence this lab does not itself compute, only cites from the published literature.
Deliverables
- Completed period-ratio table with nearest-integer-ratio identification and percent deviations.
- Density ranking table for all seven TRAPPIST-1 planets with composition-degeneracy discussion.
Assessment Criteria
- Correct period-ratio arithmetic and reasonable nearest-integer-ratio identification for all six adjacent pairs.
- Correct density calculations and ranking for all seven planets.
- Explicit acknowledgment that period-ratio proximity alone does not prove dynamical resonance.
References and Provenance
- 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.
- TRAPPIST-1 orbital periods and mass/radius values: reused from ASTR320's live verification (Wikipedia, 2026-09-24, citing Agol et al. 2021 and Ducrot et al. 2020); see
materials/ASTR350/reference-log.md.