^{*}

Edited by: Joyce Ann Guzik, Los Alamos National Laboratory (DOE), United States

Reviewed by: Jason Jackiewicz, New Mexico State University, United States; Ariane Schad, University of Freiburg, Germany

This article was submitted to Stellar and Solar Physics, a section of the journal Frontiers in Astronomy and Space Sciences

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Since the advent of CoRoT, and NASA ^{4} targets. With the recent launch of NASA TESS space mission, we have confirmed our entrance to the era of all-sky observations of oscillating stars. TESS is currently releasing good quality datasets that already allow for the characterization and identification of individual oscillation modes even from single 27-days shots on some stars. When ESA PLATO will become operative by the next decade, we will face the observation of several more hundred thousands stars where identifying individual oscillation modes will be possible. However, estimating the individual frequency, amplitude, and lifetime of the oscillation modes is not an easy task. This is because solar-like oscillations and especially their evolved version, the red giant branch (RGB) oscillations, can vary significantly from one star to another depending on its specific stage of the evolution, mass, effective temperature, metallicity, as well as on its level of rotation and magnetism. In this perspective I will present a novel, fast, and powerful way to derive individual oscillation mode frequencies by building on previous results obtained with D

Despite both CoRoT (Baglin et al.,

First works focusing on extracting and exploiting oscillation mode frequencies, amplitudes, and linewidths were already carried out (see e.g., Benomar et al.,

My view on the problem is that we require a simple approach that can be at the same time powerful, flexible to be adapted to different conditions, and fast in performing a peak bagging analysis for each star. This approach should also be accessible by the global asteroseismic community. For this purpose, I present a novel methodology, based on the public code D

An efficient and robust approach to the fitting of individual oscillation modes exploits the Bayesian inference based on the nested sampling Monte Carlo algorithm (Skilling,

With the development of D^{1}

The first approach, uni-modal and high-dimensional (hereafter Approach 1), requires priors set up for each of the individual mode properties of frequency centroid, amplitude, and linewidth. The resulting sampling of the parameter space will exhibit a single global maximum of the likelihood distribution. Approach 1 was also used extensively to perform peak bagging analysis in about 90 red giant stars (C15; Corsaro et al., ^{2.4} power law^{2}

The second approach, originally proposed by C14, is multi-modal and low-dimensional (hereafter Approach 2). The initial recipe accounted for a model with three Lorentzian profiles trying to reproduce the ℓ = 2, 0, 1 triplet typical of main sequence solar-like pulsators. This was done through the fitting of the small frequency spacings between adjacent ℓ = 2, 0 modes and adjacent ℓ = 0, 1 modes and of a frequency centroid ν_{1} of the dipolar mode used to locate the triplet of peaks in the spectrum. With this peak bagging model, the resulting sampling exhibits a series of clustered regions in the parameter space, each one corresponding to a local maximum (island) of the likelihood (see Figure 13 of Corsaro and De Ridder,

In the following sections, I describe how to revise the original Approach 2 presented by C14 to improve significantly its speed, simplicity in use, and also its reliability in producing accurate and precise estimates of individual oscillation frequencies.

The choice of the model to fit the so-called power spectral density (PSD), namely the power spectrum normalized by the frequency resolution of the dataset, requires that the output sampling from D

which I refer to as the _{0} its frequency centroid. The model is superimposed on a fixed background level that can be fitted in a previous step (see e.g., Corsaro et al., _{0} are considered as free parameters since Γ is fixed to a value that is related to the actual ν_{max} of the star. By fixing the FWHM of the peak we can stabilize the fit and obtain a better resolving power of the actual oscillation peak structures that are present in the dataset. The choice of Γ proves not to be particularly critical, but the general idea behind is that it should be set to a value equal to or below the minimum linewidth of an oscillation peak among those of a given star. I refer to section 3 for more discussion about the choice of Γ for the application presented in this work. With this islands model, we therefore account for only two free parameters, clearly expecting a considerable gain in speed when performing the fit. In addition, because of the adoption of just one Lorentzian profile and the only need for a rough assumption about the linewidth and amplitude of the observed peak structures, the sampling from D

Once the islands model is constructed, one can perform the actual fit to the PSD by providing as uniform prior distribution for the frequency centroid ν_{0} an input range that spans the actual region of the PSD that we intend to analyze, and a uniform prior range for the amplitude that resembles the average level of the amplitude of the peaks in the spectrum. This is the way we practically make the islands model a peak bagging model producing a multi-modal likelihood distribution. In the test performed, an input range for ν_{0} comparable to the large frequency separation Δν offers an optimal choice in terms of resulting frequency resolution of the sampling for detecting all the oscillation peaks reported by C15.

Before performing the fit, it is essential to provide some considerations on the parameters that configure the nested sampling algorithm of D_{live} = 500 live points is adequate for Approach 1 to converge to an accurate solution even if a large number of dimensions is involved (up to _{live} = 2, 000 and performed the fit until the termination condition of δ_{final} = 0.01 was reached (see also Keeton, _{final} constitutes a severe problem. The result is that not only the fit takes a long time to be performed in its entirety (many nested iterations are required), but that it is also very likely that the computation will fail before reaching the end because the sampling has to take place from all or most of the local maxima at each attempt to draw a new sampling point. Clearly, the more we approach to the threshold, the more difficult it will be to find a new point that satisfies the new higher likelihood constraint.

For the reasons just presented, I have implemented another termination condition in D^{3}_{live} = 2, 000 and a high threshold stopping point, set to _{nest} = 6, 000, produces a very stable and reliable sampling of the structures observed in the PSD for the application presented in this work. All the other configuring parameters of D

_{0} (orange dots) as a function of the nested iteration, for the red giant KIC 12008916. Each nested iteration corresponds to one sampling point. The sampling presented is built as follows: (1) D_{nest} = 6, 000 iterations (hence 6,000 sampling points) using 2,000 live points; (2) the run is completed by adding up the set of 2,000 live points that is left at the end of the nested sampling, thus resulting in a total of 8,000 effective nested iterations (or equivalently sampling points); (3) the sampling from the first 1,500 nested iterations was removed to improve the detection of the peak structures in the PSD. This results in a final set of 6,500 nested iterations, hence of sampling points used for the analysis.

By using the sampling cloud obtained from D_{0}, we can proceed with the construction of a corresponding counts histogram. I adopt a resolution per bin in ν_{0} given by 1/100 of the length of the parameter range, where each bin in frequency will contain the number of sampling points falling in the given bin. This proves to be adequate to detect at least 11 different peaks within a frequency range of length Δν and to get rid of spurious structures arising from the noise. As a result, when the sampling matches an actual oscillation peak in the PSD, a corresponding peak pops up in the counts histogram. The subsequent step is to consider a detection threshold in counts in order to pick up only those peaks of the histogram that exceed the threshold. A value for the threshold of 3% of the maximum number of counts found in the histogram appear to provide an optimal condition for the test presented in this work in order to detect all the oscillation peaks reported by C15. Once the threshold is defined, one can rely on a relatively simple hill climbing algorithm to progressively gather the local maxima present in the histogram by starting from the left edge of the frequency range. The result for this application is shown in the right panel of

The example shown in

Comparison between the oscillation frequencies extracted from D_{0}) and those published by C15 (ν_{Corsaro15}). The values are reported in percentage of deviation with respect to the literature values. The uncertainty error bars obtained with the new approach are also overlaid for each oscillation frequency, after they were rescaled by ν_{Corsaro15}. The horizontal lines at (ν_{0} − ν_{Corsaro15})/ν_{Corsaro15} = 0% denote the perfect matching. The panels from left to right depict the result obtained by considering four different methods to extract the frequencies from the counts histogram:

It is important to consider that the newly presented Approach 2 has the advantage that can be applied to the complex pattern of the RGB oscillations, as demonstrated in this work, contrary to the original Approach 2 that was limited to main sequence stars only. A direct comparison between the new and old Approach 2 is therefore not possible for this dataset, but one can in general expect to achieve precisions on the frequency estimates that are of the same order of magnitude given that both approaches are based on the sampling of a multi-modal posterior distribution and involve the use of a low number of free parameters.

Lastly, I note that the extraction of the oscillation frequencies using the new Approach 2 takes about 2 s on a single 2.6 GHz CPU core for the PSD chunk presented in

The methodology presented in section 2, building on a previous result by C14, was able not only to find all the actual oscillation frequency peaks in a range of 25 μHz containing 22 different oscillation modes of KIC 12008916—both pressure and mixed dipolar modes and some of which extremely narrow in width and small in height as compared to the background level—but also to deliver frequencies that are as accurate as 0.01% for the entire set analyzed, and precise to a level of 0.1 %. This implies that these frequencies could be used for modeling purposes, although precisions do not reach the level of those extracted from a peak bagging analysis using Approach 1, which ranges from 10^{−2} to 10^{−3} % for the range of frequencies used in this work, as obtained by C15. This new Approach 2 has two important advantages: (1) it is extremely convenient in terms of computational speed, yielding a factor of about 100 improvement with respect to Approach 1; (2) it does not require any assumption on the location of the frequency peaks, but only a simple average estimate of the oscillation amplitude in a chunk and an estimate of a minimum linewidth for each star, making it an adequate approach also for the complex oscillation patterns found in evolved stars. For the test presented here, I have adopted a FWHM of three times the frequency resolution of the dataset, because this was shown by C15 to be the typical FWHM of a fully unresolved oscillation peak, which is the narrowest observed. Therefore, Approach 2 is automated because it does not require any human supervision, with the only requirement of using the global oscillation properties of ν_{max} and Δν to set up the model and priors, proving that it is perfectly suitable for the analysis of a large sample of targets.

Approach 2 can be easily coupled to Approach 1 in an automated sequence where Approach 2 is used to rapidly define the number of peaks, locate their frequency position and build relatively narrow prior ranges that are then feed in Approach 1 to perform a full peak bagging analysis. As a result, also the peak significance test performed with Approach 1 can be fully automated. This is possible because, thanks to the high accuracy of the ν_{0} values extracted with Approach 2, the frequency priors can be set up as the boundaries given by the extracted uncertainties on the frequency centroids from Approach 2, thus eliminating the problem of locating the peaks before performing the actual fit. Future work will aim at investigating the performances of Approach 2, as well as assessing its calibration, in the light of different ν_{max} values, level of background photon noise, actual linewidth of the peaks (e.g., in the presence of blending as for F-type stars), and of different frequency resolutions typical of the new observations from K2 and TESS.

EC has developed the new features implemented in D

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

I thank Dr. Joris De Ridder for helpful discussions.

^{1}

^{2}See the D

^{3}The nested iteration number reported in