1. Central pressure
M = 1.8084e+30 kg, R = 6.0108e+08 m. P_c = (3/8π)GM²/R⁴ = 1.996e+14 Pa.
| Criterion | Points |
|---|---|
| Correct unit conversions | 8 |
| Correct final pressure | 7 |
Worked Solutions
2. Comparison to the Sun
Alpha Centauri B's estimated central pressure (1.996e+14 Pa) is 1.483 times the Sun's (1.345e+14 Pa) -- lower, consistent with both its smaller mass (reducing M² in the numerator) and smaller radius (reducing R⁴ in the denominator, which would raise P_c) combining so that the mass effect dominates for these specific values.
| Criterion | Points |
|---|---|
| Correct computed ratio | 8 |
| Correct qualitative M vs. R⁴ scaling argument | 7 |
3. Electron-scattering opacity
κ_es = σ_T(1+X)/(2m_p) = 0.0338 m²/kg. This value depends only on composition (X) and fundamental constants, not on any star-specific density or temperature, so it is identical for every fully ionized star of the same hydrogen mass fraction.
| Criterion | Points |
|---|---|
| Correct arithmetic | 8 |
| Correct composition-only dependence explanation | 7 |
4. Radiation-to-gas pressure ratio
Because P_rad/P_gas ∝ T³/ρ and Alpha Centauri B has a lower core temperature than the Sun (and a comparable or somewhat higher core density, given its smaller radius at similar mass), the ratio is expected to be smaller than the Sun's already-negligible value -- radiation pressure is even less important in Alpha Centauri B's core than in the Sun's.
| Criterion | Points |
|---|---|
| Correct qualitative direction (smaller ratio) | 12 |
| Correct use of the T³/ρ scaling to justify the direction | 8 |
Grading Notes
Award full marks for correct method, units, and a numeric result consistent with the arithmetic shown (allow reasonable rounding). Verify that the student re-derives the number for this problem's specific inputs rather than quoting a lecture example without adaptation. Watch specifically for unit-conversion errors (km/AU/m, years/seconds, kg/M⊙) and for comparative-magnitude statements stated in the wrong direction (e.g., "X times greater" versus "a small fraction of") -- these are the most common sources of order-of-magnitude mistakes in this course.