Apparatus and Setup
This lab evaluates the dimensionless radiative temperature gradient ∇_rad (Lecture 03) at two illustrative solar-envelope sample points and applies the Schwarzschild criterion (Lecture 04) to classify each as radiative or convective, reproducing (with schematic, order-of-magnitude opacity coefficients rather than full OPAL/OP tables) the Sun's real radiative-core/convective-envelope structure.
Materials and Data
| Sample point | T (K) | P (Pa) | ρ (kg/m³) | κ (m²/kg) |
|---|---|---|---|---|
| Deep envelope (r ≈ 0.7 R⊙) | 2.0e+06 | 6.0e+13 | electron-scattering (Lecture 02) | 0.0338 |
| Near-surface | 5.0e+04 | 2.0e+10 | 1.0e-03 | 153.84 (schematic Kramers) |
Procedure
- Using ∇_rad = 3κLP/(16πacGMT⁴), compute ∇_rad at the deep-envelope sample point, showing every substituted value and its units.
- Repeat the calculation at the near-surface sample point.
- Compare each computed ∇_rad to the ideal monatomic-gas adiabatic gradient ∇_ad = 2/5, and classify each sample point as radiatively stable or convectively unstable using the Schwarzschild criterion.
- The Sun's real convection-zone base is helioseismically measured at r ≈ 0.713 R⊙. State whether your two classifications (steps 1-3) are qualitatively consistent with this real boundary, and explain in one paragraph which specific input (opacity, temperature, or pressure) is most responsible for the difference between your two computed ∇_rad values.
Analysis and Uncertainty
Both sample points use schematic, order-of-magnitude opacity coefficients (electron scattering exactly, Kramers-law approximately) rather than precision OPAL/OP tabulated opacities; state explicitly why this is an acceptable simplification for classifying stability (a qualitative radiative-versus-convective question) but would not be acceptable for computing the Sun's precise internal temperature profile.
Deliverables
- Both ∇_rad calculations with full substituted values and units.
- Explicit Schwarzschild-criterion classification for each point.
- Written paragraph connecting the results to the Sun's real convection-zone boundary and identifying the dominant driver of the difference between the two points.
Assessment Criteria
- Correct substitution and arithmetic in both ∇_rad calculations.
- Correct Schwarzschild-criterion classification for each point.
- Correct identification of opacity (not temperature or pressure alone) as the dominant driver of the difference between the two computed gradients.
References and Provenance
- OpenStax Astronomy 2e (local reference copy: references/openstax-astronomy-2e-extracted.txt), Chapter 16.2.
- Envelope temperature, pressure, and density sample values: standard published order-of-magnitude solar-model values, not independently re-verified this session.
- Generated by
materials/ASTR330/src/generate_astr330_content.py.