Period, epoch and prediction errors of ephemeris from continuous sets of timing measurements

TL;DR

This paper derives a simple formula for the period error in continuous timing measurements and discusses improved methods for predicting future events, applicable to space-based light curve analyses.

Contribution

It introduces a straightforward formula for period error estimation and proposes better practices for ephemeris epoch referencing in continuous time series.

Findings

Derived a formula for period error: sigma_P = sigma_T (12/(N^3 - N))^0.5

Verified the formula's accuracy through simulations

Recommended central epoch referencing for long, continuous time series

Abstract

Space missions such as Kepler and CoRoT have led to large numbers of eclipse or transit measurements in nearly continuous time series. This paper shows how to obtain the period error in such measurements from a basic linear least-squares fit, and how to correctly derive the timing error in the prediction of future transit or eclipse events. Assuming strict periodicity, a formula for the period error of such time series is derived: sigma_P = sigma_T (12/( N^3-N))^0.5, where sigma_P is the period error; sigma_T the timing error of a single measurement and N the number of measurements. Relative to the iterative method for period error estimation by Mighell & Plavchan (2013), this much simpler formula leads to smaller period errors, whose correctness has been verified through simulations. For the prediction of times of future periodic events, the usual linear ephemeris where epoch errors are quoted for the first time measurement, are prone to overestimation of the error of that prediction. This may be avoided by a correction for the duration of the time series. An alternative is the derivation of ephemerides whose reference epoch and epoch error are given for the centre of the time series. For long continuous or near-continuous time series whose acquisition is completed, such central epochs should be the preferred way for the quotation of linear ephemerides. While this work was motivated from the analysis of eclipse timing measures in space-based light curves, it should be applicable to any other problem with an uninterrupted sequence of discrete timings for which the determination of a zero point, of a constant period and of the associated errors is needed.