Apparatus and Setup
Part 1 reproduces the mass-ratio splitting of the Alpha Centauri AB total mass (Lecture 09) into individual stellar masses. Part 2 computes the Roche limit for the Earth-Moon system and Io's differential tidal acceleration from Jupiter (Lecture 10), using the standard published Io/Jupiter values in ../data/io_jupiter_tidal.csv.
Materials and Data
| Quantity | Value |
|---|---|
| Alpha Cen AB Kepler total mass (Lab 04) | 1.9877 M⊙ |
| Mass ratio M_B/M_A (independently measured) | 0.8428 |
| Earth mass, radius, mean density | 5.972e+24 kg, 6.371e+06 m, 5513 kg/m³ |
| Moon mass, radius, mean density | 7.342e+22 kg, 1.737e+06 m, 3342 kg/m³ |
| Jupiter mass, mean density | 1.898e+27 kg, 1326 kg/m³ |
| Io mass, radius, semimajor axis, eccentricity | 8.932e+22 kg, 1.822e+06 m, 4.217e+08 m, 0.0041 |
Procedure
- Using the total mass from Lab 04 and the independently measured mass ratio M_B/M_A above, split the total mass into individual masses for Alpha Centauri A and B, and compare your result to the independently quoted individual masses (M_A = 1.0788 M⊙, M_B = 0.9092 M⊙).
- Compute the rigid-body Roche limit for the Earth-Moon system (using each body's mean density) and compare it to the Moon's actual orbital distance (384400 km); state whether the Moon is inside or outside its own Roche limit and by what factor.
- Compute the differential tidal acceleration across Io due to Jupiter, Δa = 2GM_Jupiter R_Io/a_Io³, and express the result as a percentage of Io's own surface gravity (g = GM_Io/R_Io², not Earth's), since Io's own gravity is the physically relevant comparison for whether the tidal stress is dynamically significant.
- Given that Io's orbit is forced to remain slightly eccentric (e = 0.0041) by an orbital resonance with Europa and Ganymede, explain in one paragraph why a perfectly circular orbit at the same semimajor axis would produce far less tidal heating, even though the differential tidal acceleration computed in step 3 would be identical at every point in a circular orbit.
Analysis and Uncertainty
The mass-ratio splitting in step 1 uses an independently measured mass ratio that was itself derived using stellar-structure/asteroseismic modeling consistent with the quoted individual masses; state explicitly why this makes the "agreement" in step 1 a consistency check on the arithmetic rather than a fully independent confirmation, in contrast to the genuinely independent agreement demonstrated in Lecture 08 between the Kepler-derived and measured total mass.
Deliverables
- Individual-mass splitting calculation and comparison to the quoted individual masses.
- Earth-Moon Roche-limit calculation and comparison to the Moon's actual orbital distance.
- Io tidal-acceleration calculation expressed as a percentage of Io's own surface gravity.
- Written explanation of why orbital eccentricity, not just tidal acceleration magnitude, drives tidal heating.
Assessment Criteria
- Correct mass-ratio splitting arithmetic.
- Correct Roche-limit calculation with correctly computed mean densities.
- Correct tidal-acceleration calculation with correct unit conversions (km to m).
- Written explanation correctly distinguishes tidal-acceleration magnitude from tidal-flexing (heating), which requires a changing tidal configuration over an orbit.
References and Provenance
- OpenStax Astronomy 2e (local reference copy: references/openstax-astronomy-2e-extracted.txt), Chapter 18.4-18.5 and Chapter 3.6.
- Io/Jupiter parameters: standard published values, not independently re-verified this session; flagged in
materials/ASTR310/reference-log.md.