Apparatus and Setup
This lab reproduces, with full unit-conversion detail, the Alpha Centauri AB total-mass determination from Lecture 08 and the shell-theorem/escape-velocity comparisons from Lecture 07, using the live-verified orbital elements and stellar parameters in ../data/alpha_centauri_orbit.csv and ../data/alpha_centauri_ab.csv.
Materials and Data
| Quantity | Value |
|---|---|
| Orbital period P | 79.762 yr |
| Semimajor axis a | 23.299 AU |
| Eccentricity e | 0.51947 |
| Alpha Centauri A (Rigil Kentaurus) measured mass | 1.0788 M⊙ |
| Alpha Centauri B (Toliman) measured mass | 0.9092 M⊙ |
Procedure
- Convert P to seconds and a to meters, then apply the generalized Kepler's third law, (m1+m2) = 4π²a³/(GP²), to obtain the total system mass in kilograms, then convert to solar masses.
- Compare your Kepler-derived total mass to the independently measured sum of the two stars' masses, and compute the percent agreement.
- Compute the surface escape velocity for Alpha Centauri A and for Alpha Centauri B using their measured masses and radii, and compare both to Earth's and the Sun's escape velocities (also compute these) and to Sirius B's escape velocity (a white dwarf, carried forward from ASTR230; data given in the lecture).
- Rank all five objects (Earth, Sun, Alpha Centauri A, Alpha Centauri B, Sirius B) by escape velocity, and explain in one paragraph why Sirius B's escape velocity is so much higher despite its unremarkable mass, tying the explanation explicitly to the 1/√R dependence in the escape-velocity formula.
Analysis and Uncertainty
The published semimajor axis (17.493 ± 0.0096 arcsec, converted using the measured parallax) and period (79.762 ± 0.019 yr) both carry small formal uncertainties; propagate the semimajor-axis uncertainty alone through Kepler's third law (noting the total mass scales as a³, so a fractional semimajor-axis uncertainty is tripled in the resulting fractional mass uncertainty) and compare the resulting mass uncertainty to the percent agreement found in step 2.
Deliverables
- Full unit-conversion chain and Kepler's-third-law total-mass calculation.
- Percent-agreement comparison to the independently measured mass sum.
- Escape-velocity table for all five objects with the full ranking and written explanation.
- Propagated semimajor-axis uncertainty and comparison to the observed agreement.
Assessment Criteria
- Correct unit conversions (years to seconds, AU to meters) before applying Kepler's third law -- this is the most common point of error in this lab.
- Correct percent-agreement arithmetic.
- Correct escape-velocity calculations and ranking for all five objects.
- Correct application of the tripled fractional-uncertainty scaling for the semimajor axis.
References and Provenance
- OpenStax Astronomy 2e (local reference copy: references/openstax-astronomy-2e-extracted.txt), Chapter 3.1-3.3.
- Alpha Centauri AB orbital elements and masses: live-verified this session; Sirius B: carried forward from ASTR230 (Bond et al. 2017, ApJ 840, 70), provenance already established there.