1. Central pressure 15 points
Using P_c ≈ (3/8π)GM²/R⁴, compute Alpha Centauri B's estimated central pressure (M = 0.9092 M⊙, R = 0.864 R⊙), showing all unit conversions.
Problems
2. Comparison to the Sun 15 points
Compare your Problem 1 answer to the Sun's estimated central pressure (1.345e+14 Pa, Lab 01), and explain, using the P_c ∝ M²/R⁴ scaling, why Alpha Centauri B's smaller mass and radius than the Sun's produce the sign and rough size of the difference you find.
3. Electron-scattering opacity 15 points
Compute the electron-scattering opacity κ_es = σ_T(1+X)/(2m_p) for a hydrogen mass fraction X = 0.70, and state why this value does not depend on which star it is being applied to.
4. Radiation-to-gas pressure ratio, order of magnitude 20 points
Alpha Centauri B's core temperature is expected to be somewhat lower than the Sun's (a cooler, lower-mass K1V star). Without computing an exact value, explain qualitatively, using the P_rad ∝ T⁴/ρ versus P_gas ∝ T/ρ scaling from Lecture 02, whether you expect Alpha Centauri B's radiation-to-gas pressure ratio to be larger or smaller than the Sun's.
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 reference copy: references/openstax-astronomy-2e-extracted.txt), Chapter 16.1-16.2.
References and Data Sources
- Corresponding lecture slides and notes for this unit.
- OpenStax Astronomy 2e (local reference copy: references/openstax-astronomy-2e-extracted.txt).
- Course datasets under
materials/ASTR330/data/where referenced above.