Lab 01: Verifying the Blackbody Radiation Law for the Sun and Alpha Centauri A/B

Apparatus and Setup

This is a data-analysis lab using real, precisely measured stellar parameters for the Sun and for Alpha Centauri A and B (live-verified this session via a direct fetch of the Alpha Centauri system's measured orbital and stellar parameters, citing Akeson et al. 2021, AJ 162, 14; see ../reference-log.md for full provenance). No telescope time is required; the lab reproduces, from first principles, the Stefan-Boltzmann and Wien's-law cross-checks introduced in Lecture 02.

Materials and Data

Procedure

1. For each of the three stars, compute the Wien's-law peak wavelength λ_max = b/T_eff using b = 2.8971e-03 m·K, and identify which part of the electromagnetic spectrum each peak falls in.
2. For each star, compute the Stefan-Boltzmann luminosity L = 4πR²σT⁴ from its measured radius and temperature (convert R to meters and T to kelvin first), then convert the result to solar luminosities.
3. Compare each star's Stefan-Boltzmann luminosity to its independently measured (asteroseismic/interferometric) luminosity from the data table, and compute the percent agreement for each star.
4. Rank the three stars by T_eff, by radius, and by Stefan-Boltzmann luminosity, and explain in one paragraph why Alpha Centauri B is both cooler and smaller than the Sun yet not simply proportionally less luminous (i.e., discuss how L depends on both R and T, not R or T alone).
5. Propagate a plausible ±1% uncertainty in each star's measured radius through the Stefan-Boltzmann formula (noting L ∝ R², so a fractional radius uncertainty is doubled in the resulting fractional luminosity uncertainty) and state whether this uncertainty alone could explain the percent agreement found in step 3.

Analysis and Uncertainty

Because L ∝ R²T⁴, the fractional uncertainty in a Stefan-Boltzmann luminosity estimate is approximately 2×(fractional radius uncertainty) + 4×(fractional temperature uncertainty) added in quadrature; the T⁴ dependence means that even a small temperature uncertainty can dominate the total luminosity uncertainty budget.

Deliverables

• Completed Wien's-law and Stefan-Boltzmann tables for all three stars, with full unit conversions shown.
• Percent-agreement calculation between the Stefan-Boltzmann and measured luminosity for each star.
• Written comparison paragraph addressing the R²T⁴ dependence.
• Uncertainty-propagation estimate and a one-paragraph judgment of whether measurement uncertainty alone explains the observed agreement.

Assessment Criteria

• Correct unit conversions (R⊙ to m, T_eff already in K) before applying the Stefan-Boltzmann formula.
• Correct percent-agreement arithmetic for all three stars.
• Written comparison correctly explains the R²T⁴ scaling rather than attributing luminosity differences to radius or temperature alone.
• Uncertainty propagation correctly applies the doubled/quadrupled fractional-uncertainty scaling for R and T respectively.

References and Provenance

• OpenStax Astronomy 2e (local reference copy: references/openstax-astronomy-2e-extracted.txt), Chapter 5.4 and Chapter 18.1.
• Alpha Centauri A/B parameters: live-verified this session (Wikipedia "Alpha Centauri," citing Akeson et al. 2021, AJ 162, 14); see materials/ASTR310/reference-log.md.
• Generated by materials/ASTR310/src/generate_astr310_content.py.