Periods, phases and O−C diagrams

For a periodic signal with period $P$ and a reference epoch $T_0$, the phase of an observation at time $t$ is the fractional part

and the cycle number is the integer part, $n = \lfloor(t-T_0)/P\rfloor$. Plotting the data against $\varphi$ instead of $t$ (phase folding) stacks all cycles on top of one another. A correct period gives a clean curve; a slightly wrong one smears it out.

The 2019 triple-star problem asks for exactly this with $P_1 = 0.9056768$ d and $P_2 = 45.4711$ d. With so many cycles elapsed, keep every digit of the period.

• Count cycles over a long baseline. If $N$ cycles separate two identical features, $P = \Delta t/N$, and the error in $P$ is the timing error divided by $N$. Long baselines win.
• Use zero crossings or extrema of a sinusoid: the time between two upward crossings of the mean level is one period. Extrema are flat, so their times are less precise than crossings.
• Watch for aliases. With one observation per night, periods of $P$ and $1/(1/P \pm 1\,{\rm d}^{-1})$ can fit the same data. Check that the folded curve is smooth.

Predict the time of each event with a trial ephemeris, $T_{\rm calc} = T_0 + nP$, and plot the difference $O-C = T_{\rm obs} - T_{\rm calc}$ against the cycle number $n$:

• a horizontal line means the ephemeris is right,
• a sloped line means the period is off: slope $= \delta P$, intercept $= \delta T_0$,
• a parabola means the period is changing ($\dot P\neq0$),
• a sinusoid means a light-travel-time effect from an orbiting third body (the ETV term in 2019).

In 2016 D1 the O−C points rise by 0.220 d over 529 cycles. The slope, $4.3\times10^{-4}$ d/cycle, corrects the period from 1.12570 to 1.12613 d: about 2000 times more precise than the first estimate.

A star in a circular orbit has

where $\gamma$ is the systemic velocity and $K$ the semi-amplitude. To fit by eye:

• $\gamma$ is the level halfway between the maxima and minima,
• $K = (v_{\max} - v_{\min})/2$,
• $P$ from successive crossings of $\gamma$ in the same direction.

An eccentric orbit gives a skewed, non-sinusoidal curve. Then the time spent above and below $\gamma$ differs.

The 2018 TRES data cover about one and a half cycles: $P\approx83$ d and $K\approx45$ km/s.

• Eclipsing binaries: the four contacts (I, II, III, IV) give the sizes of the stars relative to the orbit. For a central eclipse, the time from I to II (ingress) measures the smaller star's diameter, I to III the larger star's, and I to IV the sum of the two. The ratio of eclipse depths gives the surface brightness (temperature) ratio.
• Cepheids: a fast rise and slow decline (a sawtooth), with the star bluest and hottest near maximum light, while the radial velocity curve is roughly a mirror image of the light curve.
• Novae and supernovae: characterised by the peak and the decline times $t_2$, $t_3$ (time to fade by 2 or 3 mag) or $\Delta m_{15}$. Faster decliners are intrinsically fainter for Type Ia supernovae and brighter for novae.