Keplerian periodogram for Doppler exoplanets detection: optimized computation and analytic significance thresholds

TL;DR

This paper introduces an optimized computational method and an analytic significance threshold for the Keplerian periodogram, enhancing the detection of high-eccentricity exoplanets in Doppler surveys.

Contribution

It presents a new adaptive algorithm for computing the Keplerian periodogram and derives a fast analytic approximation for false alarm probability levels.

Findings

Keplerian periodogram outperforms Lomb-Scargle for high-eccentricity orbits.

New algorithm reduces computational demands and improves reliability.

Analytic false alarm probability approximation is accurate and efficient.

Abstract

We consider the so-called Keplerian periodogram, in which the putative detectable signal is modelled by a highly non-linear Keplerian radial velocity function, appearing in Doppler exoplanetary surveys. We demonstrate that for planets on high-eccentricity orbits the Keplerian periodogram is far more efficient than the classic Lomb-Scargle periodogram and even the multiharmonic periodograms, in which the periodic signal is approximated by a truncated Fourier series. We provide new numerical algorithm for computation of the Keplerian periodogram. This algorithm adaptively increases the parameteric resolution where necessary, in order to uniformly cover all local optima of the Keplerian fit. Thanks to this improvement, the algorithm provides more smooth and reliable results with minimized computing demands. We also derive a fast analytic approximation to the false alarm probability levels of the Keplerian periodogram. This approximation has the form $(P z^{3/2} + Q z) W \exp(-z)$, where $z$ is the observed periodogram maximum, $W$ is proportional to the settled frequency range, and the coefficients $P$ and $Q$ depend on the maximum eccentricity to scan.