{title2=$\phi_i$}

<Phase folding of a light curve> maps timestamps onto a trial cycle of duration $P>0$. Given a reference time $t_0$, define the phase in cycles by
$$
\phi_i=\operatorname{frac}\left(\frac{t_i-t_0}{P}\right),\qquad 0\le\phi_i<1,
$$
where $\operatorname{frac}(x)=x-\lfloor x\rfloor$ is the <fractional part>, using the <floor function> also for negative $x$. Plot brightness against $\phi_i$ rather than $t_i$; measurements separated by an integer multiple of $P$ occupy the same phase. For a <periodic function> $F(t+P)=F(t)$, this reveals the shared shape of many cycles. Folding changes coordinates; it neither averages samples nor chooses the correct $P$ by itself. Welsh et al.'s folded Kepler plot separates the <exoplanet transit> and orbital brightness modulation. https://www.ucolick.org/~jfortney/papers/Welsh10.pdf[Welsh et al. (2010), figure 1]{ref}.

The second figure uses a completely synthetic trapezoidal dip: $P=3$ days, $t_0=1$ day, and the same $1\%$ depth and four-hour duration as the first figure. Different colours identify four successive cycles. Correct folding superimposes the dips; a wrong trial <orbital period> spreads them out. If $P'=P+\epsilon$, the centre of the $n$th repeated event shifts by approximately $-n\epsilon/P$ cycles when folded on $P'$, for $|\epsilon|\ll P$. With $P'=3.03$ days, the fourth plotted event is displaced by $-3(0.03)/3.03\simeq-0.0297$ cycles. <Transit-timing variations> can also smear a constant-period fold, so retain the original timestamps and inspect individual events.

\Image[/light-curve-phase-folding-magnitudes.png]
{title=Toy phase folding: repeated synthetic dips align at the correct period; a flux dip becomes a positive magnitude change}

One may combine nearby phases with a <weighted mean>, but the uncertainty reduction requires assumptions. For $N$ independent measurements with common <standard deviation> $\sigma$ and effectively the same expected brightness within the bin, their <arithmetic mean> has <standard deviation> $\sigma/\sqrt N$. Thus sixteen independent $0.0006$-precision samples give $0.00015$ precision on that phase-bin mean. Shared calibration errors, changing <starspots>, and bins wide enough to mix ingress with out-of-transit brightness violate these assumptions. A folded plot can conceal changes between cycles. <Gaussian process> models can represent correlated variability and interpolate irregular samples, but their predictions and uncertainties depend on their assumed <covariance function>.

The bottom panels show the same model as relative brightness $f=F/F_0$ and as <magnitude difference> in one fixed <photometric passband> with a fixed <magnitude zero point>:
$$
\Delta m(t)=-2.5\log_{10}f(t),\qquad f(t)>0.
$$
A dip to $f=0.99$ gives $\Delta m\simeq+0.010912$ magnitudes: fainter means a larger <apparent magnitude>. The figure places larger <magnitude differences> lower on the page so the two dip shapes are easy to compare; reversing that plotting convention does not change the formula. For $f=1-\delta$ and small $\delta$, expansion of the <logarithm> gives $\Delta m\simeq(2.5/\ln10)\delta\simeq1.085736\delta$. The exact transform is nonlinear: transforming an exposure's mean brightness is generally different from averaging its instantaneous <apparent magnitudes>. The ordinary <logarithm> cannot be applied to a nonpositive background-subtracted measurement; keep such measurements and their uncertainties in brightness units for an appropriate statistical model.


 Phase folding of a light curve (source code)