Instructor Notes

Accept small rounding differences. Require explicit units for periods, power, and density. Do not accept resonance labels without calculated ratios.

Problem 1. Kepler Scaling and Resonance Check

Worked solution: \(P=a^{3/2}\). For 2.50 AU, \(P=2.50^{3/2}=3.95\) yr. For 3.28 AU, \(P=3.28^{3/2}=5.94\) yr. For 5.20 AU, \(P=5.20^{3/2}=11.9\) yr. Ratios with the outer body are \(11.9/3.95=3.01\) and \(11.9/5.94=2.00\), so 2.50 AU is near a 3:1 resonance and 3.28 AU is near a 2:1 resonance. Resonances repeatedly apply gravitational tugs at the same orbital phase; depending on geometry and damping, they can pump eccentricity, clear gaps, or maintain flexing.

Criterion	Points	Notes
Periods	9	Years required.
Ratios and resonance labels	8	Must compare periods, not distances.
Physical interpretation	6	Credit destabilizing and maintaining cases.
Communication	2	Clear table and assumptions.

Common errors: using \(P=a^3\), labeling resonance from semimajor-axis ratios, or treating resonance as always stable.

Problem 2. Tidal Heating as an Energy Chain

Worked solution: Heating ratio \(=(0.010/0.004)^2=(2.5)^2=6.25\). If Moon B dissipates \(2.0\times10^{12}\) W, Moon A dissipates \(6.25\times2.0\times10^{12}=1.25\times10^{13}\) W. Eccentricity creates changing tidal distortion; flexing dissipates mechanical energy as heat. Without resonance or another forcing source, tidal dissipation damps eccentricity and reduces the heating. Supporting observations include volcanic plumes, young resurfacing, high heat flow, or induced magnetic/ocean evidence. Ambiguities include radiogenic heat, impact resurfacing, observational cadence, or surface composition effects.

Criterion	Points	Notes
\(e^2\) scaling	7	Factor 6.25 expected.
Power estimate	5	Units in W.
Resonance/damping reasoning	7	Must explain persistence.
Evidence and ambiguity	6	Specific observation required.

Common errors: using a linear eccentricity ratio, omitting units, or saying resonance creates heat directly rather than maintaining eccentricity.

Problem 3. Density, Differentiation, and Core Fraction

Worked solution: For X, \(\rho_X/\rho_E=0.055/(0.383)^3=0.055/0.0562=0.979\), so \(\rho_X=5.39\,\mathrm{g\ cm^{-3}}\). For Y, \(\rho_Y/\rho_E=0.815/(0.949)^3=0.815/0.854=0.954\), so \(\rho_Y=5.26\,\mathrm{g\ cm^{-3}}\). X is slightly denser relative to Earth and, given its small size and lower compression, is more suggestive of a large metal fraction. This remains an inference: thermal state, porosity, compression, and composition can produce similar mean densities. Useful additional measurements include moment of inertia, magnetic field, gravity harmonics, seismic data, heat flow, and surface/meteorite composition.

Criterion	Points	Notes
Two ratio calculations	8	Radius must be cubed.
Densities in g cm\(^{-3}\)	5	Use \(\rho_E=5.51\).
Core-fraction reasoning	7	Reward caveats about compression.
Additional measurements	5	Must be interior-relevant.

Common errors: concluding exact core size from mean density or forgetting small bodies are less compressed.

Problem 4. Interior Evidence Matrix

Worked solution: Magnetic field constrains dynamo state and conducting fluid motion; an alternative is remanent crustal magnetism or measurement geometry. Volcanic plains constrain recent melt production and resurfacing; alternatives include impact melt or misdated surfaces. Heat flow constrains internal energy loss; alternatives include local crustal enrichment or measurement uncertainty. Gravity anomalies constrain buried basins, crustal thickness, and mass distribution; alternatives include uncompensated topography or density contrasts unrelated to present activity. The best category is weakly active: younger plains and heat flow above lunar values argue against fully dead, while no plate tectonics and weak field argue against Earth-like activity.

Criterion	Points	Notes
Four-row matrix	8	Each row must connect evidence to interior property.
Alternative explanations	6	Specific alternatives, not vague uncertainty.
Conclusion category	7	Weakly active is expected but other defended answers possible.
Evidence weighting	4	Must balance all data.

Common errors: treating weak magnetic field as proof of no interior heat, or ignoring relative ages of volcanic plains.

OpenStax Companion Practice Notes

Chapter 3 entries reinforce Kepler scaling and resonance-adjacent orbital reasoning. Chapter 7 entries reinforce differentiation, density, and age evidence. Assign these as practice or adapt selected parts; verify against the adopted PDF before release.