Visual Binary: Total Mass from Kepler III
Known separation a in AU. Time the period at several a and recover M₁+M₂ from a fit of T² versus a³ (solar units).
Goal
Determine M = M₁+M₂ from T² = a³/M (AU, years, solar masses). The slope of T² versus a³ is 1/M.
Equipment
- Visual binary (unknown masses)
- Separation a (AU)
- Stopwatch
Experiment
Theory
In AU–year–solar-mass units, Kepler III for the relative orbit is T² = a³ / (M₁+M₂). The bench hides T, both masses and any ready-made T(a) curve. a is known. A stopwatch reports T after you record. The on-screen orbit is cosmetic and must not be timed.
Procedure
- You only change the known separation a (AU). Masses and the period are hidden.
- Record. A stopwatch logs T in years with small timing noise. There is no live T or M.
- The notebook computes T² and a³. Repeat for at least 6 separations from about 1.2 AU to 4.0 AU.
- Fit T² versus a³; M = 1 / slope. Compare with the reference.
Conclusion
The fitted total mass agrees with the hidden binary. Main uncertainties: stopwatch noise and treating the orbit as circular Keplerian.