Lab 05: Jeans Mass, Jeans Length, and Free-Fall Time

Apparatus and Setup

This lab computes Jeans mass, Jeans length, and free-fall time for the OMC-1 dense core and Barnard 68 side by side, and evaluates each against its own observed (or, for OMC-1, representative) mass.

Materials and Data

Standard published values, level 2 (Bergin & Tafalla 2007 for the representative dense-core values; Alves, Lada & Lada 2001 for Barnard 68's real measured mass, radius, and density).

Procedure

1. Compute the Jeans length and Jeans mass for both cores using Lecture 09's formulas.
2. Compute the free-fall time for both cores using Lecture 10's formula.
3. For Barnard 68, compare the computed Jeans mass to its real observed mass (2.1 Msun) and state whether it is above, below, or comparable to the instability threshold.
4. Using the observed star-formation efficiency per free-fall time (epsilon_ff~1-2%, Lecture 10), estimate an effective star-formation timescale for the OMC-1 core, and compare it to the core's own free-fall time.

Analysis and Uncertainty

Discuss explicitly why the simple Jeans criterion's idealized assumptions (uniform density, no turbulence, no magnetic support, no rotation) mean Barnard 68's near-critical Jeans-mass result should be read as "consistent with marginal instability," not as a precise, exact prediction of imminent collapse.

Deliverables

• Jeans length, Jeans mass, and free-fall time for both cores with full derivations.
• The Barnard 68 instability-threshold comparison.
• The OMC-1 effective star-formation timescale estimate.

Assessment Criteria

• Correct Jeans length/mass arithmetic with correct SI-to-cm^-3 density conversion.
• Correct free-fall time arithmetic.
• Correctly hedged (not overclaimed) Barnard 68 instability conclusion.

References and Provenance

• 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.
• OMC-1 dense-core and Barnard 68 parameters: standard published, level 2; see materials/ASTR360/reference-log.md.
• Generated by materials/ASTR360/src/generate_astr360_content.py.