Worked Solutions
1. Mercury's Moment of Inertia and Core Size
Mercury: (0.4 - 0.346)/0.4 × 100 = 13.5%. Earth: 17.3%. Mercury's deviation below the uniform-sphere value is substantially larger than Earth's, implying an even more strongly core-concentrated interior relative to its total size than Earth's -- consistent with Mercury's core occupying an unusually large fraction (∼85%) of its radius, and with the giant-impact hypothesis that one or more early impacts stripped away a disproportionate share of Mercury's original silicate mantle, leaving its iron core relatively more dominant than on any other terrestrial planet.
| Criterion | Points |
|---|
| Correct percent-deviation calculation for Mercury | 6 |
| Correct comparison to Earth's deviation | 5 |
| Correct connection to the giant-impact hypothesis | 4 |
2. Radiogenic Heat Production 2 Gyr Ago
Fraction 2 Gyr ago relative to today = e⁺λt. For ⁴⁴K: e⁺(0.5545)(2) = 3.031, i.e., ⁴⁴K was about 3.03 times more abundant (and hence produced about 3.03 times more heat) 2 Gyr ago than today. For ²³⁶U: e⁺(0.1551)(2) = 1.364, only about 1.36 times more abundant. ⁴⁴K's much shorter half-life (1.25 Gyr versus 4.47 Gyr) means its heat contribution changed far more dramatically over this interval than ²³⁶U's, which decays much more slowly and so contributed comparably to both eras.
| Criterion | Points |
|---|
| Correct decay-law calculation for both isotopes | 8 |
| Correct identification and explanation of which isotope changed more | 7 |
3. Surface-to-Volume Scaling for Io versus the Moon
Because Io and the Moon are nearly the same size, the surface-to-volume scaling argument (heat loss/generation ∝ 1/R) predicts nearly identical cooling rates and hence nearly identical present-day geological activity for both bodies -- both should have long since radiated away their formation-era and radiogenic heat and become geologically quiescent. This prediction is correct for the Moon but dramatically wrong for Io, because Io receives an enormous additional, non-radiogenic heat source: tidal heating, sustained by its orbital resonance with Europa and Ganymede (Lecture 13), which has no analog for the Moon (whose orbit around Earth has already circularized and is not maintained in a forced-eccentricity resonance with another body).
| Criterion | Points |
|---|
| Correctly applies the surface-to-volume argument to predict similar activity | 7 |
| Correctly identifies tidal heating/resonance as the resolving mechanism | 8 |
4. Rayleigh Number Scaling
Since Ra ∝ d³, doubling d increases Ra by a factor of 2³ = 8. Starting from Ra ∼ 10⁶-10⁷ and multiplying by 8 gives Ra ∼ 8×10⁶-8×10⁷, still vastly above the critical value of ∼1000-2000 by four to five orders of magnitude. This change would not plausibly push Earth's mantle into a non-convecting regime; Earth's mantle convection is so strongly super-critical that even an order-of-magnitude change in a single parameter would not be sufficient to suppress convection.
| Criterion | Points |
|---|
| Correct cubic scaling calculation | 8 |
| Correct quantitative conclusion about remaining super-criticality | 7 |
5. Comparing Bulk Silicate Earth's Heat Budget to Observed Heat Flow
(47 - 20)/47 × 100 ≈ 57%. Just over half of Earth's total observed surface heat flow is not accounted for by present-day radiogenic heat production alone. Lecture 04 attributes this difference to secular cooling: Earth's core and mantle are still releasing heat left over from formation and differentiation (accretional and differentiation heat, discussed as one-time heat sources at the start of Lecture 04), in addition to their ongoing radiogenic heat production.
| Criterion | Points |
|---|
| Correct percent calculation | 7 |
| Correctly identifies secular cooling as the additional source | 8 |
Grading Notes
Award full marks for correct method, units, and a numeric result consistent with the arithmetic shown (allow reasonable rounding). Verify that the student re-derives the number for this problem's specific inputs rather than quoting a lecture example without adaptation. Watch specifically for unit-conversion errors (AU/m, days or years/seconds, bar/Pa) and for comparative-magnitude statements stated in the wrong direction (e.g., a tidal acceleration or greenhouse excess described as a multiple when it is actually a small fraction, or vice versa) -- these are the most common sources of silent error in this course.