Jeans Instability and Collapse Timescales
M_J(8 K) = 1.36 Msun, lower than the real-temperature result (1.90 Msun at 10 K) -- colder gas has a smaller Jeans mass (M_J ∝ T^{3/2}), so Barnard 68 would be *more* prone to instability at this lower, hypothetical temperature, since less mass would be required to exceed the (now smaller) critical threshold.
| Criterion | Points |
|---|---|
| Correct Jeans-mass recomputation at 8 K | 8 |
| Correct qualitative direction (more, not less, unstable) | 7 |
t_ff(1.0e6 cm^-3)=33,718 yr, a factor of 3.16 shorter than the OMC-1 core's free-fall time (106,624 yr) -- consistent with the t_ff ∝ rho^(-1/2) scaling, since a factor-of-10 density increase should shorten t_ff by a factor of sqrt(10)=3.16.
| Criterion | Points |
|---|---|
| Correct free-fall-time recomputation | 8 |
| Correct confirmation of the sqrt(10) scaling factor | 7 |
Jeans length = 0.0948 pc = 19,558 AU -- roughly 652 times Neptune's orbital radius, i.e. a Jeans-unstable region at this core's density and temperature spans a region many times larger than the entire solar system.
| Criterion | Points |
|---|---|
| Correct pc-to-AU conversion | 5 |
| Correct, physically intuitive comparison | 5 |
At epsilon_ff=1%: t_SF≈10,662,437 yr. At epsilon_ff=2%: t_SF≈5,331,218 yr -- a factor of exactly 2 difference between the two timescale estimates, directly reflecting the inverse relationship between assumed efficiency and effective star-formation timescale.
| Criterion | Points |
|---|---|
| Correct arithmetic for both efficiency assumptions | 8 |
| Correct factor-of-2 relationship identified | 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.