Period Law Mismatch Three-Row Scan - Observational Astrophysics: Distance, Flux, Transits, and Spectra Step by Step

A period-law-mismatch scan varies the candidate period across three rows, reading off how far each T^2/r^3 value strays from the source constant, so a mismatch stands out as a nonzero residual. The arithmetic is exact only for the stated periods, radii, source values, and rows in the ideal circular two-body model. Real observations add cadence, bandpass, stellar variability, limb darkening, detector calibration, background/noise, finite precision, and measurement uncertainty.

Three exact rows hold the ratio fixed while the law changes. 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 same observed ratio is judged by the cited law

The observed ratio stays fixed; only the cited law constant changes, so the acceptance bit flips only for the matching row.

\begin{array}{c}K=\frac{1}{4}\\{T^{2}\over R^{3}}={10\over20}=\frac{1}{2}\\\text{accept}=0\end{array}

The same observed ratio is judged by the cited law

The observed ratio stays fixed; only the cited law constant changes, so the acceptance bit flips only for the matching row.

\begin{array}{c}K=\frac{1}{2}\\{T^{2}\over R^{3}}={10\over20}=\frac{1}{2}\\\text{accept}=1\end{array}

The same observed ratio is judged by the cited law

The observed ratio stays fixed; only the cited law constant changes, so the acceptance bit flips only for the matching row.

\begin{array}{c}K=\frac{2}{3}\\{T^{2}\over R^{3}}={10\over20}=\frac{1}{2}\\\text{accept}=0\end{array}