On detecting the large separation in the autocorrelation of stellar oscillation times series

The observations carried out by the space missions CoRoT and Kepler provide a large set of asteroseismic data. Their analysis requires an efficient procedure first to determine if the star is reliably showing solar-like oscillations, second to measure the so-called large separation, third to estimate the asteroseismic information that can be retrieved from the Fourier spectrum. We develop in this paper a procedure, based on the autocorrelation of the seismic Fourier spectrum. We have searched for criteria able to predict the output that one can expect from the analysis by autocorrelation of a seismic time series. First, the autocorrelation is properly scaled for taking into account the contribution of white noise. Then, we use the null hypothesis H0 test to assess the reliability of the autocorrelation analysis. Calculations based on solar and CoRoT times series are performed in order to quantify the performance as a function of the amplitude of the autocorrelation signal. We propose an automated determination of the large separation, whose reliability is quantified by the H0 test. We apply this method to analyze a large set of red giants observed by CoRoT. We estimate the expected performance for photometric time series of the Kepler mission. Finally, we demonstrate that the method makes it possible to distinguish l=0 from l=1 modes. The envelope autocorrelation function has proven to be very powerful for the determination of the large separation in noisy asteroseismic data, since it enables us to quantify the precision of the performance of different measurements: mean large separation, variation of the large separation with frequency, small separation and degree identification.Comment: A&A, in pres

On the detection of Lorentzian profiles in a power spectrum: A Bayesian approach using ignorance priors

Aims. Deriving accurate frequencies, amplitudes, and mode lifetimes from stochastically driven pulsation is challenging, more so, if one demands that realistic error estimates be given for all model fitting parameters. As has been shown by other authors, the traditional method of fitting Lorentzian profiles to the power spectrum of time-resolved photometric or spectroscopic data via the Maximum Likelihood Estimation (MLE) procedure delivers good approximations for these quantities. We, however, show that a conservative Bayesian approach allows one to treat the detection of modes with minimal assumptions (i.e., about the existence and identity of the modes). Methods. We derive a conservative Bayesian treatment for the probability of Lorentzian profiles being present in a power spectrum and describe an efficient implementation that evaluates the probability density distribution of parameters by using a Markov-Chain Monte Carlo (MCMC) technique. Results. Potentially superior to "best-fit" procedure like MLE, which only provides formal uncertainties, our method samples and approximates the actual probability distributions for all parameters involved. Moreover, it avoids shortcomings that make the MLE treatment susceptible to the built-in assumptions of a model that is fitted to the data. This is especially relevant when analyzing solar-type pulsation in stars other than the Sun where the observations are of lower quality and can be over-interpreted. As an example, we apply our technique to CoRoT observations of the solar-type pulsator HD 49933.Comment: 12 pages, 11 figures, accepted for publication in Astronomy and Astrophysic

Mode width fitting with a simple bayesian approach. Application to CoRoT targets HD 181420 and HD 49933

We investigate the asteroseismology of two solar-like targets as observed with the CoRoT satellite, with particular attention paid to the mode fitting. HD 181420 and HD 49933 are typical CoRoT solar-like targets (156 and 60-day runs). The low signal-to-noise ratio (SNR) of about 3-10 prevents us from unambiguously identifying the individual oscillation modes. In particular, convergence problems appear at the edges of the oscillation spectrum. HD 181420 and HD 49933 are typical CoRoT solar-like targets (156 and 60-day runs). The low signal-to-noise ratio (SNR) of about 3-10 prevents us from unambiguously identifying the individual oscillation modes. In particular, convergence problems appear at the edges of the oscillation spectrum. We apply a Bayesian approach to the analysis of these data. We compare the global fitting of the power spectra of this time series, obtained by the classical maximum likelihood (MLE) and the maximum a posteriori (MAP) estimators. We examine the impact of the choice of the priors upon the fitted parameters. We also propose to reduce the number of free parameters in the fitting, by replacing the individual estimate of mode height associated with each overtone by a continuous function of frequency (Gaussian profile). The MAP appears as a powerful tool to constrain the global fits, but it must be used carefully and only with reliable priors. The mode width of the stars increases with the frequency over all the oscillation spectrum.Comment: 10 pages, 9 figures, 2 table

The art of fitting p-mode spectra: Part II. Leakage and noise covariance matrices

In Part I we have developed a theory for fitting p-mode Fourier spectra assuming that these spectra have a multi-normal distribution. We showed, using Monte-Carlo simulations, how one can obtain p-mode parameters using 'Maximum Likelihood Estimators'. In this article, hereafter Part II, we show how to use the theory developed in Part I for fitting real data. We introduce 4 new diagnostics in helioseismology: the $(m,\nu)$ echelle diagramme, the cross echelle diagramme, the inter echelle diagramme, and the ratio cross spectrum. These diagnostics are extremely powerful to visualize and understand the covariance matrices of the Fourier spectra, and also to find bugs in the data analysis code. These diagrammes can also be used to derive quantitative information on the mode leakage and noise covariance matrices. Numerous examples using the LOI/SOHO and GONG data are given.Comment: 17 pages with tex and ps files, submitted to A&A, [email protected]

The art of fitting p-mode spectra: Part I. Maximum Likelihood Estimation

In this article we present our state of the art of fitting helioseismic p-mode spectra. We give a step by step recipe for fitting the spectra: statistics of the spectra both for spatially unresolved and resolved data, the use of Maximum Likelihood estimates, the statistics of the p-mode parameters, the use of Monte-Carlo simulation and the significance of fitted parameters. The recipe is applied to synthetic low-resolution data, similar to those of the LOI, using Monte-Carlo simulations. For such spatially resolved data, the statistics of the Fourier spectrum is assumed to be a multi-normal distribution; the statistics of the power spectrum is \emph{not} a $\chi^{2}$ with 2 degrees of freedom. Results for $l=1$ shows that all parameters describing the p modes can be obtained without bias and with minimum variance provided that the leakage matrix is known. Systematic errors due to an imperfect knowledge of the leakage matrix are derived for all the p-mode parameters.Comment: 13 pages, ps file gzipped. Submitted to A&

On deriving p-mode parameters for inclined solar-like stars

Thanks to their high quality, new and upcoming asteroseismic observations - with CoRoT, Kepler, and from the ground... - can benefit from the experience gained with helioseismology. We focus in this paper on solar-like oscillations, for which the inclination of the rotation axis is unknown. We present a theoretical study of the errors of p-mode parameters determined by means of a maximum-likelihood estimator, and we also analyze correlations and biases. We have used different, complementary approaches: we have performed either semi-analytical computation of the Hessian matrix, fitting of single mean profiles, or Monte Carlo simulations. We give first analytical approximations for the errors of frequency, inclination and rotational splitting. The determination of the inclination is very challenging for the common case of slow rotators (like the Sun), making difficult the determination of a reliable rotational splitting. Moreover, due to the numerous correlations, biases - more or less significant - can appear in the determination of various parameters in the case of bad inclination fittings, especially when a locking at 90 degrees occurs. This issue concerning inclination locking is also discussed. Nevertheless, the central frequency and some derived parameters such as the total power of the mode are free of such biases.Comment: 9 pages, 6 figures, to appear in A&