Due and Scope
Due: [insert date]. Submit calculations, a short interpretation for each problem, and any spreadsheet or code used to check arithmetic.
Learning objectives: use Kepler scaling; identify resonances; infer interior properties from mass, radius, density, heat, and magnetic evidence; distinguish calculation from interpretation.
Instructions: Show equations, units, assumptions, and reasoning. Collaboration is allowed for discussion only. Revisions may be resubmitted with a change note.
Problems
1. Kepler Scaling and Resonance Check 25 points
Three small bodies orbit the Sun with semimajor axes 2.50 AU, 3.28 AU, and 5.20 AU. Use \(P^2=a^3\), with \(P\) in years and \(a\) in AU.
- Compute each orbital period.
- Compute the period ratios relative to the 5.20 AU object.
- Identify whether either inner body is close to a 2:1 or 3:1 mean-motion resonance with the outer body.
- Explain why resonance can either remove objects from some orbits or maintain tidal heating in others.
Deliverables: period table, resonance-ratio calculation, and 120-160 word interpretation.
2. Tidal Heating as an Energy Chain 25 points
Two moons have the same radius, density, and parent planet, but Moon A has eccentricity \(e=0.010\) and Moon B has \(e=0.004\). In a simplified model, tidal heating scales as \(e^2\) when all other properties are fixed.
- By what factor is Moon A heated more strongly than Moon B?
- If Moon B dissipates \(2.0\times10^{12}\) W, estimate Moon A's dissipation.
- Explain why a resonance is needed to keep eccentricity from damping away.
- Name one observation that would support active tidal heating and one ambiguity in that observation.
Deliverables: scaling calculation with units, a causal energy-chain diagram in words, and evidence/ambiguity pair.
3. Density, Differentiation, and Core Fraction 25 points
World X has \(M=0.055M_E\) and \(R=0.383R_E\). World Y has \(M=0.815M_E\) and \(R=0.949R_E\). Use \(\rho/\rho_E=(M/M_E)/(R/R_E)^3\).
- Compute \(\rho_X/\rho_E\) and \(\rho_Y/\rho_E\).
- Using \(\rho_E=5.51\,\mathrm{g\ cm^{-3}}\), compute both densities in g cm\(^{-3}\).
- Which world is more suggestive of a large metal core? Explain without claiming density alone proves the full interior.
- State two measurements, beyond mass and radius, that would help test differentiation.
Deliverables: density calculations, comparison, and evidence paragraph.
4. Interior Evidence Matrix 25 points
A spacecraft returns the following for a rocky planet: weak present magnetic field; large volcanic plains younger than nearby highlands; heat-flow estimate twice the lunar value but below Earth's; gravity anomalies over major basins; no global plate tectonics.
- Build a four-row evidence matrix using magnetic field, volcanism, heat flow, and gravity anomalies.
- For each row, state what interior property it constrains and one alternative explanation.
- Write a conclusion: is this world likely fully dead, weakly active, or Earth-like active? Defend your category.
Deliverables: evidence matrix and 180-220 word conclusion.
OpenStax Companion Practice
Recommended for additional practice: OpenStax Astronomy 2e, Chapter 3, Review Questions 1, 6, and 9; Thought Questions 23, 25, and 29; Figuring for Yourself 36-38; and Chapter 7, Review Questions 2 and 19, Thought Question 27, and Figuring for Yourself 31-33. Verify numbering against the adopted PDF before LMS release.