Apparatus and Setup
This lab has two parts: (1) computing the transmitted intensity fraction through slabs of increasing optical depth (Lecture 03), and (2) computing the Boltzmann n=2/n=1 hydrogen excitation ratio across a range of real stellar temperatures spanning K to B spectral types (Lecture 04), reproducing the qualitative OBAFGKM Balmer-strength argument from ASTR 230 quantitatively.
Materials and Data
| Optical depth τ | Transmitted fraction e⁻τ |
|---|---|
| 0.1 | 0.9048 |
| 0.5 | 0.6065 |
| 1.0 | 0.3679 |
| 2.0 | 0.1353 |
| 3.0 | 0.0498 |
| 5.0 | 0.0067 |
| 7.0 | 0.0009 |
| T_eff (K) | Boltzmann n2/n1 ratio (E2−E1 = 10.2 eV, g2/g1 = 4) |
|---|---|
| 4000 | 5.697e-13 |
| 5750 | 4.628e-09 |
| 7500 | 5.632e-07 |
| 9900 | 2.580e-05 |
| 15000 | 1.501e-03 |
Procedure
- Using I(τ)/I(0) = e⁻τ, complete the transmitted-fraction table above for all seven listed optical depths, and plot transmitted fraction versus τ on both linear and semi-log axes; explain which axis choice makes the exponential relationship easiest to verify visually.
- Using n2/n1 = (g2/g1) exp[−(E2−E1)/kT], complete the Boltzmann-ratio table above for all five listed temperatures (spanning K-type Alpha Centauri B-like stars to B-type stars), showing the kT calculation explicitly at each temperature.
- Compute the ratio of the n2/n1 value at 9,900 K (A-type, near Vega/Sirius A) to the value at 5,750 K (G/K-type, near the Sun and Alpha Centauri A/B), and state whether this is consistent with the much stronger observed Balmer lines in A-type stars discussed in ASTR 230.
- Using only the Boltzmann equation (ignoring the Saha ionization turnover from Lecture 04), predict what the n2/n1 ratio would be at 25,000 K (a hot white dwarf's photospheric temperature, comparable to Sirius B's), and explain in one paragraph why this naive extrapolation overstates the expected Balmer-line strength at such a high temperature.
Analysis and Uncertainty
The Boltzmann-ratio calculation above ignores the Saha-equation ionization turnover entirely; state explicitly which physical process (not included in this lab's calculation) becomes important above roughly 10,000 K and would need to be added to obtain a fully correct line-strength prediction across the full OBAFGKM range.
Deliverables
- Completed transmitted-fraction table and both plots (linear and semi-log).
- Completed Boltzmann-ratio table with kT shown explicitly at each temperature.
- Ratio calculation (9,900 K vs. 5,750 K) with a written interpretation.
- Written paragraph on the naive high-temperature extrapolation and the missing Saha-equation physics.
Assessment Criteria
- Correct exponential arithmetic (not linear-approximation arithmetic) for all seven optical-depth values.
- Correct Boltzmann-ratio arithmetic with correctly converted units (eV to J) at all five temperatures.
- Correct qualitative connection to the OBAFGKM Balmer-strength sequence from ASTR 230.
- Correct identification that the Saha equation, not the Boltzmann equation alone, is needed for the high-temperature turnover.
References and Provenance
- OpenStax Astronomy 2e (local reference copy: references/openstax-astronomy-2e-extracted.txt), Chapter 5.5 and Chapter 17.3.
- Generated by
materials/ASTR310/src/generate_astr310_content.py.