The 21 cm Line and HI Column Density
n1/n0 = 3 exp(-0.0682/500) = 2.99959 -- much closer to the high-temperature limit of 3.0 than the CNM's ratio (2.99796), because 500 K is already large compared to T*=68.2 mK.
| Criterion | Points |
|---|---|
| Correct formula application | 7 |
| Correct qualitative comparison | 8 |
N(HI) = 1.8224e18 x 12.0 x 6.0 x sqrt(pi/(4 ln2)) = 1.397e+20 cm^-2 -- intermediate in order of magnitude between this course's CNM (1.45e+20 cm^-2) and WNM (1.94e+20 cm^-2) components, consistent with its intermediate peak temperature and linewidth.
| Criterion | Points |
|---|---|
| Correct Gaussian-integral formula application | 8 |
| Correct order-of-magnitude comparison | 7 |
Ratio = 33.5 -- Cygnus X-1 lies much closer to the Galactic plane (b=3.07 deg) than 3C 273 (b=64.36 deg), and HI is strongly concentrated toward the Galactic plane, so a low-latitude sightline like Cygnus X-1's passes through far more atomic hydrogen than a high-latitude sightline like 3C 273's.
| Criterion | Points |
|---|---|
| Correct ratio computed | 7 |
| Correct qualitative Galactic-latitude reasoning | 8 |
Full solution: T_B = 100(1-e^-0.5) = 39.347 K. Optically thin approximation: T_B ≈ 100 x 0.5 = 50.0 K. Percent error = 27.1%, with the optically thin approximation over-estimating the true brightness temperature at this larger, no-longer-safely-thin optical depth.
| Criterion | Points |
|---|---|
| Correct full radiative-transfer calculation | 7 |
| Correct percent-error direction (over-, not under-, estimate) | 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 (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.