Worked Solutions
1. Pressure balance at a different phase pair
P/k(WIM)=1,200 cm^-3 K, P/k(HIM)=1,750 cm^-3 K, a ratio of 1.46 -- notably farther from equilibrium than the CNM/WNM pair's ratio of 1.00, consistent with the WIM/HIM not being expected to be in simple two-phase thermal-pressure balance the way CNM/WNM are (they are governed by different heating sources, ionizing radiation and supernova energy input respectively, not the same Field-instability mechanism).
| Criterion | Points |
|---|
| Correct P/k arithmetic for both phases | 6 |
| Correct ratio comparison to CNM/WNM | 5 |
| Correct qualitative interpretation | 4 |
2. H II region pressure vs its surroundings
P/k(H II)=1,000,000 cm^-3 K versus P/k(WNM)=2,400 cm^-3 K, a ratio of 416.7 -- a large mismatch, but not a problem for the framework: H II regions are actively, freshly ionized and pressurized by a nearby hot star's ionizing flux (Unit 6) and are expected to expand and eventually dissipate rather than sit in long-term pressure equilibrium with the surrounding diffuse ISM.
| Criterion | Points |
|---|
| Correct P/k arithmetic | 6 |
| Correct ratio computed | 5 |
| Correct physical reasoning about transience | 4 |
3. A hypothetical seventh phase
T = P/(k n) = 2,400 cm^-3 K / 3.0 cm^-3 = 800 K -- closest in temperature to the H II region phase (T=10,000 K) among the six real phases, though at a much lower density than real H II regions.
| Criterion | Points |
|---|
| Correct algebraic solve for T | 6 |
| Correct identification of closest real phase | 5 |
| Clear units | 4 |
4. Filling factor sanity check
Sum of published filling factors = 0.9910, close to but not exactly 1 -- the small deviation is expected given that these six phases' filling factors come from different studies using somewhat different definitions and survey methods, not from a single self-consistent partition of the ISM's volume.
| Criterion | Points |
|---|
| Correct summation | 5 |
| Correct, non-overclaiming interpretation of the near-1 sum | 5 |
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 (cm^-3 vs m^-3, pc vs m, K km/s integrals), for comparative-magnitude statements given in the wrong direction, and for level-1/level-2/level-3 data-provenance conflation (e.g., presenting the illustrative synthetic HI spectrum shape as an exact measurement of a specific real sightline) -- these are the most common sources of mistakes in this course.