ASTR 360 Problem Set 06: H II Region Diagnostics and the Stromgren Sphere

Extend this unit's ionization-balance and diagnostic tools to a second real H II region and sensitivity checks.

Problems

1. The Rosette Nebula 15 points

Using the Rosette Nebula's real published parameters (Q_H=6.0e+49 photons/s from the NGC 2244 cluster, n_e=15 cm^-3), compute its Stromgren radius and compare it to its real observed radius (10 pc). State whether the Rosette Nebula's model-observation discrepancy is larger or smaller (in percentage terms) than the Orion Nebula's from the lab, and suggest one physical reason low-density, evolved H II regions like the Rosette might show a different-sized discrepancy than compact, young ones like Orion.

2. Ionizing photon rate needed for a target radius 15 points

Using the Orion Nebula's real n_e, compute what value of Q_H would be required to produce a Stromgren radius exactly equal to the nebula's observed radius (0.6 pc), and compare this required Q_H to theta1 Ori C's real published value. State what this comparison implies about whether Orion is more consistent with an ionization-bounded or a density-bounded picture.

3. Recombination-line ratio sanity check 10 points

Using the theoretical case B Balmer decrement value (Halpha/Hbeta=2.86 at Te=10^4 K, given in the lecture), state what an observed ratio of 4.0 in a specific region of a nebula would imply about that region's internal dust extinction, and explain in one sentence why this diagnostic is independent of the Stromgren-radius calculation in this unit's other problems.

4. Electron density sensitivity 15 points

Recompute the Orion Nebula's Stromgren radius assuming n_e were instead 1200 cm^-3 (double the real published value, with Q_H unchanged), and state the exact factor by which the Stromgren radius changes, connecting your answer to the R_s ∝ n_e^(-2/3) scaling derived in the lecture.

Due and Scope

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 Companion Reading

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.

References and Data Sources