The generalised Lomb-Scargle periodogram. A new formalism for the floating-mean and Keplerian periodograms

TL;DR

This paper introduces a generalized Lomb-Scargle periodogram that improves frequency detection accuracy and aliasing resistance for uneven data, and extends it to Keplerian orbits for exoplanet searches.

Contribution

It provides an analytic solution for a full sine wave fit with offsets and weights, unifying several existing methods and enabling Keplerian orbit detection in radial velocity data.

Findings

Enhanced frequency accuracy over traditional Lomb-Scargle

Reduced susceptibility to aliasing

Effective detection of eccentric exoplanet orbits

Abstract

The Lomb-Scargle periodogram is a common tool in the frequency analysis of unequally spaced data equivalent to least-squares fitting of sine waves. We give an analytic solution for the generalisation to a full sine wave fit, including an offset and weights ($\chi^{2}$ fitting). Compared to the Lomb-Scargle periodogram, the generalisation is superior as it provides more accurate frequencies, is less susceptible to aliasing, and gives a much better determination of the spectral intensity. Only a few modifications are required for the computation and the computational effort is similar. Our approach brings together several related methods that can be found in the literature, viz. the date-compensated discrete Fourier transform, the floating-mean periodogram, and the "spectral significance" estimator used in the SigSpec program, for which we point out some equivalences. Furthermore, we present an algorithm that implements this generalisation for the evaluation of the Keplerian periodogram that searches for the period of the best-fitting Keplerian orbit to radial velocity data. The systematic and non-random algorithm is capable of detecting eccentric orbits, which is demonstrated by two examples and can be a useful tool in searches for the orbital periods of exoplanets.