The Habitable Zone
d_inner=sqrt(0.5/1.1)=0.674 AU, d_outer=sqrt(0.5/0.32)=1.250 AU -- both closer in than the Sun's 0.953-1.768 AU by a factor of sqrt(0.5)=0.707, i.e. about 1.41× closer.
| Criterion | Points |
|---|---|
| Correct HZ boundary calculation | 8 |
| Correct scaling-factor statement | 7 |
Using L∝M^3.5: 51 Pegasi (M=1.11 Msun) gives L≈1.44 Lsun, HZ≈1.14-2.12 AU; HD 209458 (M=1.15 Msun) gives L≈1.63 Lsun, HZ≈1.22-2.26 AU. Both 51 Pegasi b (a=0.052 AU) and HD 209458 b (a=0.0471 AU) orbit far interior to their star's habitable zone -- neither is anywhere near habitable-zone distance, consistent with both being hot, gas-giant worlds.
| Criterion | Points |
|---|---|
| Correct mass-luminosity scaling application | 8 |
| Correct HZ comparison and conclusion for both planets | 7 |
S(Venus)/S(Earth) = 1.91 -- above the inner-edge threshold (1.1 S_Earth), meaning Venus lies inside (interior to) the classical habitable zone's inner edge, consistent with its real runaway-greenhouse outcome.
| Criterion | Points |
|---|---|
| Correct insolation ratio calculation | 8 |
| Correct classification relative to the inner edge | 7 |
The classical HZ answers only "is stellar insolation compatible with surface liquid water assuming an Earth-like atmosphere"; Europa's ocean is sustained by tidal heating, an entirely different energy source unrelated to stellar flux, so its existence outside the classical HZ does not contradict the HZ concept -- it simply falls outside the specific physical question the classical HZ was designed to answer, exactly as Lecture 10 states explicitly.
| Criterion | Points |
|---|---|
| Correct statement that classical HZ is insolation-specific | 8 |
| Correct, non-contradictory framing of the Europa counter-case | 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.