Apparatus and Setup
This lab applies Lecture 03's radiative-transfer solution and Lecture 04's column-density formula to a representative two-component (CNM+WNM) 21 cm brightness-temperature spectrum, and compares the result to two real, published all-sky-survey HI column densities.
Materials and Data
| Component | T_spin (K) | Peak T_B (K) | FWHM (km/s) |
|---|---|---|---|
| CNM | 100 | 25 | 3.0 |
| WNM | 8000 | 4 | 25 |
T_spin values and characteristic linewidths are real, standard published CNM/WNM values (level 2, Kalberla & Kerp 2009 review); the specific peak-T_B synthetic Gaussian profile shape is an illustrative representative spectrum (level 3), not a claimed measurement of one specific real sightline (see ../reference-log.md). Real comparison sightlines: 3C 273 (N(HI)=1.79e+20 cm^-2, HI4PI Collaboration 2016) and Cygnus X-1 (N(H)=6.0e+21 cm^-2, X-ray absorption studies).
Procedure
- Using the Boltzmann population-ratio formula, compute n1/n0 for both the CNM and WNM spin temperatures and confirm both are close to, but measurably below, the high-temperature limit of 3.0.
- Using the Gaussian-integral column-density formula, compute N(HI) separately for the CNM and WNM components and sum them for the total.
- Compare your total N(HI) to the real, published 3C 273 sightline value and to the real, published Cygnus X-1 sightline value; state which real sightline your representative spectrum's total is closer to in order of magnitude, and why this is only a loose comparison (your spectrum is illustrative, not a fit to either specific sightline).
- Recompute N(HI) for the CNM component using the full (not optically thin) radiative-transfer solution at a specified optical depth of tau=0.05, and confirm the optically thin approximation is accurate to better than 1% at this value.
Analysis and Uncertainty
Discuss what would happen to your total N(HI) estimate if the CNM component were in fact optically thick (tau approaching or exceeding 1) rather than the optically thin tau=0.05 used above, and why narrow, high-peak-brightness CNM features are the ones most likely to violate the optically thin assumption in real spectra.
Deliverables
- Both components' population ratios, column densities, and the summed total.
- A comparison table against the two real published sightline column densities.
- The optically thin vs. full radiative-transfer cross-check at tau=0.05.
Assessment Criteria
- Correct Boltzmann population-ratio arithmetic for both spin temperatures.
- Correct Gaussian-integral column-density arithmetic with the 1.8224e18 coefficient applied correctly.
- Correctly disclosed level-1/level-2/level-3 distinctions when comparing the illustrative spectrum to real sightline values.
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.
- 3C 273 and Cygnus X-1 sightline column densities: standard published, level 2; see
materials/ASTR360/reference-log.md. - Generated by
materials/ASTR360/src/generate_astr360_content.py.