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

By M. Zechmeister, M. Kürster. 2009

1,159 citations recorded by the source corpus.

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 (χ 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.

Subfields: Astronomy and Astrophysics, Computational Mechanics, Instrumentation

View Paper: https://www.semanticscholar.org/paper/1a57908211b7d48cfd50c68abfaaf4452163a4e0

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In Astronomy and Astrophysics

1. D. Charbonneau
2. J. Southworth
3. J. Johnson
4. Timothy Morton
5. Jon Fortney
6. J. Winn
7. G. Marcy
8. M. Perryman

In Computational Mechanics

1. J. Vanderplas
2. T. Mazeh
3. A. McQuillan
4. Suzanne Aigrain
5. C. Lovis
6. D. Queloz
7. A. Reiners
8. Željko Ivezic´

In Instrumentation

1. D. Charbonneau
2. J. Southworth
3. Timothy Morton
4. J. Winn
5. Suzanne Aigrain
6. J. Johnson
7. G. Marcy
8. T. Mazeh

Citations in the corpus

• The CARMENES search for exoplanets around M dwarfs (2018, methodology): “2, we show the generalized Lomb-Scargle periodograms (GLS, Zechmeister & Kürster 2009) of different data products obtained with SERVAL and our CCF analysis.”
• Planets and Stellar Activity: Hide and Seek in the CoRoT-7 system (2013, methodology): “We produced a generalised Lomb-Scargle periodogram (Zechmeister & Kürster 2009) of the 2012 RV data, shown in Figure 3.”
• The HARPS search for southern extra-solar planets - XXVIII. Up to seven planets orbiting HD 10180: probing the architecture of low-mass planetary systems (2010, methodology): “We use the generalized Lomb-Scargle (GLS) periodogram as described in Zechmeister & Kürster (2009).”
• Measuring the rotation period distribution of field M dwarfs with Kepler (2013, methodology): “Standard approaches to period detection in light curves are based on Fourier decomposition or, for irregularly sampled data, least-squares fitting of sinusoidal models and variants thereof (Scargle 1982; Zechmeister & Kürster 2009).”
• A detached stellar-mass black hole candidate in the globular cluster NGC 3201 (2018, methodology): “The analysis of the radial velocity variation is done using the generalized Lomb-Scargle (GLS) periodogram (Zechmeister & Kürster 2009), the likelihood function approach of Baluev (2008), and a final fit of a Keplerian orbit.”
• Chromospheric activity catalogue of 4454 cool stars (2018, methodology): “This makes the period determination more robust and less prone to aliasing (for a detailed analytic solution to the generalised Lomb-Scargle technique, we refer to Zechmeister & Kürster 2009).”
• THE KEPLER-10 PLANETARY SYSTEM REVISITED BY HARPS-N: A HOT ROCKY WORLD AND A SOLID NEPTUNE-MASS PLANET (2014, methodology): “In this section, we do not utilize this filtering and instead search for significant signals in the GLS (Zechmeister & Kürster 2009) of the raw RVs.”
• The Kepler characterization of the variability among A- and F-type stars I. General overview (2011, methodology): “…comparison, subsamples of the stars were analysed using different analysis methods, such as SigSpec (Reegen, 2007, 2011), Period04 (Lenz & Breger 2005), the generalized Lomb-Scargle periodogram (Zechmeister & Kürster 2009), and the non-interactive code,freqfind (Leroy & Gutiérrez-Soto, in prep.).”
• The NASA Exoplanet Archive: Data and Tools for Exoplanet Research (2013, methodology): “The service supports three algorithms: Lomb-Scargle (Scargle 1982; Horne & Baliunas 1986; Zechmeister & Kurster 2009), box-fitting least squares (BLS; Kovács et al. 2002) and Plavchan (Plavchan et al. 2008; Stellingwerf 1978).”
• The HARPS search for southern extra-solar planets - XXXI. The M-dwarf sample (2011, methodology): “We follow the formalism developed in Zechmeister & Kürster (2009), and choose a power normalization where 1.0 means that a sinusoidal fit is perfect (χ2 = 0) and where 0.0 means that a sinusoidal fit does not improve χ2 over a constant model.”
• ROTATION PERIODS OF 34,030 KEPLER MAIN-SEQUENCE STARS: THE FULL AUTOCORRELATION SAMPLE (2014, methodology): “They applied the Generalized Lomb-Scargle periodogram (Zechmeister & Kürster 2009) to LCs binned to 2 hr sampling, in order to reduce computation time, and accepted periods in the range 12hrs–45days.”
• RADIAL VELOCITY PLANETS DE-ALIASED: A NEW, SHORT PERIOD FOR SUPER-EARTH 55 Cnc e (2010, background): “See Cumming et al. (1999) and Zechmeister & Kürster (2009). the same even sampling.”
• Random forest automated supervised classification of Hipparcos periodic variable stars (2011, methodology): “…search methods such as Deeming (1975), Lomb-Scargle (Lomb 1976; Scargle 1982), harmonic least squares analysis of generalized Lomb-Scargle methods (Zechmeister & Kürster 2009), String Length methods (Lafler & 5 The problem of spurious and aliased periods (or frequencies) typically showing up at…”
• The planet search programme at the ESO CES and HARPS - IV. The search for Jupiter analogues around solar-like stars (2012, methodology): “Analogous to Cumming et al. 2008 and Zechmeister & Kürster (2009), we calculated the probabilities 8 Prob(p > pbest) = (1− pbest) N−5 2 and (11) Prob(pKep > pKep,best) = (1+ N−72 pKep,best)(1− pKep,best) N−7 2 , (12) respectively.”
• Pulsar timing analysis in the presence of correlated noise (2011, methodology): “The Cholesky method essentially extends the L-S or Z-K methods by fitting a sine-cosine pair at each frequency, but first whitening both the data and the model (the sine-cosine pair) with the Cholesky transformation.”
• Planetary detection limits taking into account stellar noise - I. Observational strategies to reduce stellar oscillation and granulation effects (2010, methodology): “Since the periodogram technique is equivalent to a weighted sine wave fitting (Scargle 1982; Zechmeister & Kürster 2009), to pass from the spectrum to RVs, we have to calculate for the desired epochsti: RV(ti) = ∑ ν √ VPSD(ν)(sin(2πνti + phase(ν))), (6) where the frequencyν goes from 1/T to the…”