Apparatus and Setup
This lab has two parts using real data: (1) applying the distance modulus and Stefan-Boltzmann cross-check to two contrasting stars from the Lab 01 sample, and (2) estimating the Sun's nuclear fuel supply and main-sequence lifetime from its measured luminosity, mass, and the proton-proton chain mass defect (Lecture 04).
Materials and Data
| Quantity | Value |
|---|---|
| Sirius A: V = -1.46, parallax = 379.21 mas, T_eff = 9940 K, R = 1.71 R☉ | |
| Alpha Centauri A: V = -0.01, parallax = 747.10 mas, T_eff = 5790 K, R = 1.22 R☉ | |
| Solar mass | 1.989e+30 kg |
| Solar luminosity (measured) | 3.828e+26 W |
| Energy released per proton-proton chain reaction | 26.73 MeV = 4.282e-12 J |
| Fraction of the Sun's mass available as core hydrogen fuel (typical estimate) | 10% |
Procedure
- For Sirius A and Alpha Centauri A, compute distance, distance modulus, and absolute magnitude (as in Lab 01), then compute luminosity two independent ways: from M_V (L = 10^((4.83 − M_V)/2.5) L☉) and from Stefan-Boltzmann (L = R²(T/5778)⁴ L☉, using the given radius and T_eff).
- Compute the percent difference between the two luminosity estimates for each star. State which star's two estimates agree more closely, and explain why, referencing the "V magnitude ≈ bolometric magnitude" simplifying assumption from Lecture 03.
- Using the proton-proton chain mass defect above, compute how many reactions per second are needed to produce the Sun's measured luminosity (3.828e+26 W).
- Each reaction consumes 4 hydrogen-1 nuclei (mass 1.007825 u = 1.673 × 10⁻²⁵ kg each). Compute the mass of hydrogen consumed per second, then per year.
- Assuming 10% of the Sun's mass (1.989e+30 kg) is available as core fuel, compute the Sun's total main-sequence lifetime implied by this consumption rate, and compare it to the mass-luminosity-relation estimate from Lecture 06 (10 Gyr for a 1 M☉ star).
Analysis and Uncertainty
The two luminosity estimation methods (magnitude-based and Stefan-Boltzmann-based) rely on different simplifying assumptions (bolometric correction neglect vs. adopted radius/temperature precision); state which source of uncertainty you judge to be larger for each of your two sample stars and why.
Deliverables
- Completed two-method luminosity comparison table for Sirius A and Alpha Centauri A, with percent differences.
- Fuel-consumption-rate calculation (reactions/second, kg H/year).
- Implied main-sequence lifetime from fuel-supply reasoning, compared numerically to the Lecture 06 mass-luminosity-relation estimate.
- A short paragraph explaining any discrepancy between the two lifetime estimates.
Assessment Criteria
- Correct two-method luminosity calculation and percent difference for both stars.
- Correct identification of which star's estimates agree more closely, with a physically grounded explanation.
- Correct reaction-rate and fuel-consumption arithmetic with units shown at each step.
- Lifetime comparison correctly references the Lecture 06 mass-luminosity relation result.
References and Provenance
- OpenStax Astronomy 2e (local reference copy: references/openstax-astronomy-2e-extracted.txt), Chapter 19.1-19.2 and Chapter 16.1-16.2.
- Star data:
materials/ASTR230/data/nearby_bright_stars.csv.