Period Ratio Acceptance Cross-Scan - Observational Astrophysics: Distance, Flux, Transits, and Spectra Step by Step

A period-ratio-acceptance cross-scan tests candidate period ratios against the Kepler T^2/r^3 invariant across rows, accepting those that close the ratio and rejecting the rest. The verdict is exact only for the stated periods, radii, and rows in the ideal circular two-body model. Real observations add calibration, finite cadence/bandpass, noise, stellar variability, line-identification uncertainty, detector response, and measurement uncertainty.

Three exact rows show accepted and rejected period ratios. 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 period ratio must match the law source

The checked source divides period square eight by radius cubed sixteen, giving one half, so the acceptance bit is one.

\frac{T^{2}}{R^{3}}=\frac{8}{16}=\tfrac{1}{2},\quad \text{accept}=1

Same ratio accepts; nearby wrong ratio rejects

Two rows preserve the one-half law. The last row changes period square alone, so the ratio no longer matches.

\begin{array}{c|c|c|c}T^{2}&R^{3}&T^{2}/R^{3}&\text{accept}\\4&8&\frac{1}{2}&1\\8&16&\frac{1}{2}&1\\9&16&\frac{9}{16}&0\\\end{array}