Exoplanet Demographics
ρ=M/(4/3 π R^3). TRAPPIST-1h: M=0.331 Mearth, R=0.755 Rearth, ρ=4240 kg/m^3. TRAPPIST-1b: ρ=4034 kg/m^3 -- b is denser than h.
| Criterion | Points |
|---|---|
| Correct derivation from sphere volume | 6 |
| Correct numeric density for both planets | 6 |
| Correct comparison | 3 |
The transition occurs roughly in the several-to-tens-of-Earth-mass range (consistent with this course's sample: TRAPPIST-1 planets, all under 1.2 Mearth-scale, are rocky-density; Jupiter/HD 209458 b, hundreds of Earth masses, are gas-giant-density); it is not sharp because increasing envelope mass fraction is a continuum, not a step function, and electron-degeneracy-pressure support becomes progressively more important rather than switching on suddenly.
| Criterion | Points |
|---|---|
| Reasonable transition mass range identified from the dataset | 7 |
| Correct physical explanation of gradualness | 8 |
A close-in planet sits in a much higher stellar X-ray/UV flux environment, giving its exobase a higher temperature and thus a smaller Jeans escape parameter (lambda ∝ 1/T) for the same planet mass/radius, pushing it toward the hydrodynamic-escape regime and faster envelope loss than an identical planet farther out with a cooler exobase and larger lambda.
| Criterion | Points |
|---|---|
| Correct linkage between irradiation, exobase temperature, and lambda | 8 |
| Correct qualitative escape-rate conclusion | 7 |
ρ = 8.0 Mearth × 5.972e+24 kg / (4/3 π (2.2×6.371e+06 m)^3) = 4142 kg/m^3 -- well below Earth's 5514 kg/m^3 but well above HD 209458 b's 360 kg/m^3, consistent with a transitional, volatile-rich sub-Neptune-class classification.
| Criterion | Points |
|---|---|
| Correct density calculation | 8 |
| Correct, dataset-grounded classification | 7 |
Transit surveys require a near-edge-on inclination and are biased toward close-in orbits (higher transit probability, Lecture 02); this systematically under-represents wide-separation and low-inclination (more face-on) systems, meaning a transit-dominated sample cannot be used to claim wide-orbit planets are rare without first correcting for this geometric selection effect.
| Criterion | Points |
|---|---|
| Correct identification of the inclination/separation bias | 8 |
| Correct statement of what is under-represented | 7 |
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.