Atmospheres and Escape
H=kT/(mu g); doubling g halves H, and the transit-signal amplitude formula delta(N)-delta(0) ∝ H, so doubling g halves the predicted signal amplitude, all else (temperature, N, Rp, Rstar) held fixed.
| Criterion | Points |
|---|---|
| Correct H ∝ 1/g relation stated | 7 |
| Correct propagation to the transit-signal formula | 8 |
lambda(Earth, hydrogen, T=1000 K) = GMm/(kTR) = 7.5 -- far larger than HD 209458 b's 10.93, meaning Earth's hydrogen is much more tightly gravitationally bound relative to its thermal energy, so Earth is in the negligible-escape (classical Jeans, not hydrodynamic) regime for hydrogen, consistent with the fact that Earth would have far more atmospheric hydrogen today if it escaped as readily as on HD 209458 b.
| Criterion | Points |
|---|---|
| Correct Jeans-parameter calculation for Earth | 8 |
| Correct comparison and physically consistent conclusion | 7 |
Timescale scales inversely with mass-loss rate: 136.7 Gyr / 3 = 45.6 Gyr -- still comparable to or longer than a typical few-Gyr system age, so the qualitative conclusion (not in danger of complete evaporation at this constant rate) would likely still hold, though with reduced margin.
| Criterion | Points |
|---|---|
| Correct inverse scaling | 8 |
| Correct, appropriately hedged conclusion | 7 |
lambda=GMm/(kTR) decreases for smaller planet mass M at fixed T (exobase temperature set mainly by irradiation, not planet mass) and R (which also shrinks less than proportionally with M for a similar-density planet); a lower-mass sub-Neptune therefore has a smaller lambda than HD 209458 b at the same orbital distance, placing it more deeply in the hydrodynamic-escape regime and making full envelope loss (crossing into the radius valley's "stripped core" population) far more plausible than for a hot-Jupiter-mass planet like HD 209458 b.
| Criterion | Points |
|---|---|
| Correct mass-dependence argument for lambda | 8 |
| Correct explicit connection to the radius valley | 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.