Period Square over Radius Cubed Matches - Observational Astrophysics: Distance, Flux, Transits, and Spectra Step by Step

Kepler's third law ties an orbital period to its radius through T^2 / r^3 = constant, so two orbits sharing a central mass share that same ratio as an observational fingerprint. The arithmetic is exact only for stated periods, radii, and a common central mass under the circular two-body approximation. Real observations add inclination, eccentricity, stellar mass uncertainty, cadence and noise, calibration, and source-traceability limits.

Two normalized period/radius ledgers cite the same K source. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The first candidate cites the normalized period law

The period-law source is 1/2. The first candidate has 4 over 8.

K=\tfrac{1}{2},\quad \frac{T^{2}}{a^{3}}=\frac{4}{8}=\tfrac{1}{2}

A second exact candidate matches the same source

The second candidate has 32 over 64, which also reduces to 1/2.

\frac{T^{2}}{a^{3}}=\frac{32}{64}=\tfrac{1}{2}