Apply moment-of-inertia interpretation, the Rayleigh-number criterion, and the radiogenic decay law from Lectures 03-04 to Mercury, Io, and Earth's overall heat budget.
MESSENGER measurements give Mercury's moment of inertia factor as C/MR² = 0.346. Compute its percent deviation below the uniform-sphere value of 0.4, and compare this deviation to Earth's (17.3% computed from C/MR² = 0.3307). State what this comparison implies about Mercury's core, in light of Lecture 03's giant-impact hypothesis for Mercury's anomalously large core.
Using the decay constants λ = ln2/t₁⁄² for ⁴⁴K (t₁⁄² = 1.25 Gyr) and ²³⁶U (t₁⁄² = 4.47 Gyr), compute what fraction of each isotope's present-day abundance existed 2 Gyr ago (i.e., the ratio of abundance at t = -2 Gyr relative to today, noting this ratio is e⁺λt for t = 2 Gyr since we are looking backward in time). Explain which isotope's heat contribution changed more dramatically over this interval and why.
Io (radius 1821.6 km) and the Moon (radius 1737.4 km) are nearly the same size, yet Io is the most volcanically active body in the solar system while the Moon is essentially geologically dead. Using the surface-to-volume cooling argument from Lecture 04, explain why this argument alone would predict similar (low) geological activity for both bodies, and then explain what additional physical mechanism (introduced in Lecture 04 as a preview and developed fully in Lecture 13) resolves this apparent contradiction.
The Rayleigh number is Ra = (ρgαΔTd³)/(κμ). If a planet's mantle thickness d were doubled while all other quantities (ρ, g, α, ΔT, κ, μ) stayed fixed, by what factor would its Rayleigh number change? Given that Earth's mantle Rayleigh number is estimated at 10⁶-10⁷, vastly above the critical value of ∼1000-2000, would this change plausibly push a convecting mantle below the critical threshold into a non-convecting regime? Justify your answer quantitatively.
Earth's present-day radiogenic heat production is estimated at roughly 20 TW, while its total observed surface heat flow is roughly 47 TW. Compute the percent of Earth's total heat flow that is NOT accounted for by present-day radiogenic heat production, and explain, using the concept introduced in Lecture 04, what additional heat source is thought to make up this difference.
Submit a complete derivation with equations, units, labeled quantities, and a short narrative interpretation for each problem. Show every step; a correct final number without a visible derivation receives partial credit at most, and quoting a lecture number without adapting it to this problem's specific inputs receives no credit for that step.
OpenStax Astronomy 2e, Section 9.1-9.2.
materials/ASTR320/src/generate_astr320_content.py where referenced above.