Molecular Cloud Mass Estimators
Since M ∝ n at fixed volume, doubling density exactly doubles the direct mass: M=48,227 Msun, a factor of 2.00 above OMC-1's direct mass (24,114 Msun) -- confirming the predicted factor-of-2 scaling.
| Criterion | Points |
|---|---|
| Correct algebraic factor-of-2 derivation | 7 |
| Correct numeric mass | 8 |
L_CO(5.0e3 Msun) = 1561.2 K km/s pc^2, versus L_CO(Taurus real mass, 15,000 Msun) = 4683.5 K km/s pc^2 -- the 5.0e3 Msun hypothetical cloud's implied CO luminosity is smaller than Taurus's real value, consistent with its smaller assumed mass.
| Criterion | Points |
|---|---|
| Correct inverse X_CO conversion | 8 |
| Correct comparison direction | 7 |
M_vir(sigma doubled)=62,762 Msun, a factor of 4.00 above the original Taurus virial mass (15,690 Msun) -- matching the predicted M_vir ∝ sigma^2 scaling (doubling sigma should quadruple M_vir).
| Criterion | Points |
|---|---|
| Correct virial-mass recomputation | 7 |
| Correct confirmation of the sigma^2 scaling | 8 |
Taurus: M_CO/M_vir=0.96. OMC-1: M_CO/M_vir=1.98. Both ratios are within roughly a factor of 2 of unity, but OMC-1's ratio is closer to 1, suggesting it is somewhat closer to virial balance, while Taurus's lower ratio is more consistent with a cloud that is comparatively less centrally concentrated/bound relative to its turbulent support -- stated as a comparative, not absolute, conclusion given the simplified uniform-sphere virial model's known limitations (Lecture 08).
| Criterion | Points |
|---|---|
| Correct ratio computation for both clouds | 8 |
| Correctly hedged, comparative (not absolute) conclusion | 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 (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.