Extend this unit's Jeans-mass and free-fall-time tools to new temperatures, densities, and efficiency assumptions.
Recompute Barnard 68's Jeans mass assuming its temperature were instead 8 K (colder than the real published 10 K, with density unchanged). State whether this makes Barnard 68 more or less prone to gravitational instability, and connect your answer to the Jeans mass's explicit T^(3/2) dependence.
Compute the free-fall time for a hypothetical dense core with n(H2)=1.0e6 cm^-3 (ten times the OMC-1 core's density), and state the exact factor by which it differs from the OMC-1 core's free-fall time, connecting your answer to the t_ff ∝ rho^(-1/2) scaling.
Convert the OMC-1 dense core's Jeans length (computed in the lab, in pc) into astronomical units (AU), and compare it to the solar system's scale (e.g., Neptune's orbital radius, about 30 AU) to give a physically intuitive sense of the Jeans length's size.
Using the OMC-1 core's free-fall time from the lab, compute the effective star-formation timescale for two different assumed efficiencies, epsilon_ff=1% and epsilon_ff=2%, and state the factor by which the two timescale estimates differ from each other.
Submit a complete derivation with equations, units, labeled quantities, and a short narrative interpretation for each problem. Show every step; a correct final number without a visible derivation receives partial credit at most, and quoting a lecture number without adapting it to this problem's specific inputs receives no credit for that step.
OpenStax Astronomy 2e (local extracted reference copy: references/openstax-astronomy-2e-extracted.txt) covers the interstellar medium in Chapter 20 ("Between the Stars: Gas and Dust in Space"): 20.1 The Interstellar Medium, 20.2 Interstellar Gas (including the 21 cm line and H II regions), 20.3 Cosmic Dust (extinction and reddening), 20.4 Cosmic Rays, 20.5 The Life Cycle of Cosmic Material, and 20.6 Interstellar Matter around the Sun. Chapter 20’s section numbering was cross-checked directly against its own embedded figure numbering this session (Figures 20.1 through 20.20 all appear under their matching section headers with no offset), so this citation is verified against the local extracted text rather than extrapolated by analogy. OpenStax’s treatment throughout Chapter 20 is descriptive and qualitative; nearly every quantitative derivation in this course (the 21 cm hyperfine radiative-transfer solution and HI column-density formula, the Jeans mass/length instability criterion, the free-fall collapse time, the Stromgren-sphere ionization-balance derivation, the Rankine-Hugoniot shock-jump conditions, and the cosmic-ray power-law spectrum) goes well beyond OpenStax Astronomy 2e’s introductory level; each lecture states explicitly where this course extends beyond the assigned reading.
materials/ASTR360/data/ where referenced above.