Orbital Architecture and Interiors
Direct: P_h/P_b = 12.4305. Product of six adjacent ratios: 12.4305. They agree (telescoping product of ratios) because each intermediate period cancels: (P_c/P_b)(P_d/P_c)...(P_h/P_g) = P_h/P_b algebraically, regardless of the specific intermediate values.
| Criterion | Points |
|---|---|
| Correct direct ratio | 5 |
| Correct product-of-ratios computation matching the direct value | 6 |
| Correct telescoping explanation | 4 |
t_circ(e)/t_circ(b) ≈ (a_e/a_b)^(13/2) = (1.7)^6.5 ≈ 31 -- TRAPPIST-1e's circularization timescale is roughly 31× longer than b's, but both are still almost certainly far shorter than the system's multi-Gyr age given how close-in even TRAPPIST-1e is, consistent with both planets' observed essentially-zero eccentricities.
| Criterion | Points |
|---|---|
| Correct application of the 13/2 power law | 8 |
| Correct qualitative conclusion that both are circularized | 7 |
TRAPPIST-1d has ρ=3346 kg/m^3 versus TRAPPIST-1g's 4398 kg/m^3; d is markedly less dense (a larger fractional departure from Earth's 5514 kg/m^3), consistent with d carrying a proportionally larger ice/water mass fraction than g.
| Criterion | Points |
|---|---|
| Correct density values cited | 6 |
| Correct comparative conclusion with quantitative support | 9 |
Mass and radius alone cannot distinguish a smaller iron core from an added ice/water layer, both of which lower bulk density from a pure-silicate baseline; breaking the degeneracy would require an independent atmospheric composition measurement (detecting or ruling out a substantial water-vapor/volatile atmosphere, Lecture 07) or, in a favorable case, a measured tidal Love number probing internal structure directly.
| Criterion | Points |
|---|---|
| Correct statement of the degeneracy | 7 |
| Correct identification of at least one degeneracy-breaking measurement | 8 |
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/seconds, Jupiter/Earth mass-radius units), for comparative-magnitude statements given in the wrong direction, and for level-1/level-2/level-3 data-provenance conflation (e.g., presenting an illustrative Miller-Urey bar height as a precise measured yield) -- these are the most common sources of mistakes in this course.