Extraction of seismic indices and stellar granulation parameters for CoRoT and Kepler red giants using the MLEUP method . Main results and perspectives

In the framework of the SPACEInn project, a Stellar Seismic Indices (SSI http://ssi.lesia.obspm. fr/) database was developed in order to provide the scientific community with oscillations and granulation signatures for a large set of red-giant type stars. For this purpose, we have developed the method MLEUP able to extract simultaneously the seismic indices (the equidistance ∆ν, the frequency νmax and the height Henv of the maximum oscillation power) and granulation parameters (the e-folding time τeff and the variance of the integrated brightness fluctuations σ2). This method has been tested in terms of precision and accuracy, using Monte Carlo simulations. Then we applied it to all stars observed by CoRoT and all long-cadence Kepler lightcurves. In total, we yield seismic indices and granulation parameters for about 5,000 stars for CoRoT and more than 13,000 for Kepler. In this paper, we focus on the main results for both seismic indices ∆ν and νmax as well as for the stellar parameters (mass, radius and luminosity) seismically inferred. Then, in the perspective of Gaia, we discuss about the possibility to derive other seismic quantities like e.g. a seismic effective temperature.


Introduction
We have developed a new automatic method, called MLEUP, taking advantage of the MLE (Maximum Likelihood Estimate) algorithm combined with the parametrized representation of the red giants pulsation spectrum following the UP (Universal Pattern, cf.[1]) in order to measure simultaneously the oscillations and the granulation signatures.The full description of this method is in [2].We describe briefly in Sect. 2 the mainlines of our method.In Sect.3, we present the main results of the analysis of the CoRoT and Kepler datasets: PUBLI ! the mean large separation ∆ν, the peak frequency ν max as well as the "seismic" masses and radii.In Sect.4, we discuss about the prospect, such as the possibility to derive the "seismic" effective temperature using the Gaia data.Finally, we conclude in Sect. 5.

Mainlines of the Method MLEUP
The model used to fit the power density spectrum is based on two Lorentzian-like functions for the granulation and activity component, a red giant parametric oscillations pattern based on the Universal Pattern [1] for the oscillations and a constant for the white noise (see Fig. 1).The oscillations are characterised by three seismic indices: The mean large separation, ∆ν, corresponding to the mean frequency spacing between two consecutive p-modes with same angular degree; H env which is the maximum height e-mail: raphael.peralta@obspm.fr of the oscillation envelope and ν max , the corresponding frequency.Concerning the granulation, it is characterised by two parameters: the effective timescale τ eff (or the efolding time), which measures the temporal coherence of the granulation in the time domain and the characteristic amplitude σ2 which corresponds to the variance brightness fluctuation of the granulation.The method has been tested in order to characterize its bias and dispersion using Monte Carlo simulations.These simulations revealed that MLEUP presents low dispersions, especially for ∆ν and ν max , and that the internal errors are reasonably representative of the real dispersions 3 Mean large separation ∆ν and peak frequency ν max -"seismic" mass and radius We analysed all CoRoT data1 with a duration of observation larger than 50 days and all long-cadence (dt = 29.42min) Kepler data 2 .We yield 4783 CoRoT stars and 13,689 Kepler stars for which we get the seismic indices, and 1520 CoRoT stars and 12,879 Kepler stars for which we get also the granulation parameters.The results of the analysis of the large sets of Kepler and CoRoT stars are presented and discussed in [2].Here, we focus on the results of both seismic indices ∆ν and ν max , and the stellar parameters inferred by them.We plotted in figure 2 ∆ν as a function of ν max from the 4783 CoRoT and 13,689 Kepler stars for which we extracted the seismic indices.We can see that the dispersion for CoRoT and Kepler is small, as well as the internal error (cf.Tab. 1).We deduced the two following scaling relations by adjusting the CoRoT and Kepler datasets: The previous seismic indices are very valuable to characterised large sets of stars.To illustrate this, we considered 13,408 of the Kepler selected dataset for which we have the effective temperature from [5].Combining with ν max and ∆ν, one can estimate the mass, radius and luminosity [e.g.6] seismically inferred with a high precision (cf.Tab. 1).The figure 3 shows the distribution of our results in the Hertzsprung-Russell (H-R) diagram where the red giant branch (RGB) and the red clump are well recognizable.

Perspective -The "seismic" effective temperature T eff
The arrival of Gaia data opens new perspectives.As an illustration, we introduce the "seismic" effective temperature.Indeed, if we combine the apparent magnitude m, the parallax p and the interstellar extinction A m with both seismic indices ν max and ∆ν, it is possible to derive the T eff via the following equation (cf.Sect. 5 for detailed derivation): with A m the interstellar extinction (mag), m the apparent magnitude (mag), p the parallax (arcsec), ν max the peak frequency (µHz) and ∆ν, the large separation (µHz).The constant T eff, corresponds to the solar values and the constants ν ref and ∆ν ref , to the reference values defined by [7].
The corresponding error is given by: The MLEUP method provides the seismic indices ν max and ∆ν with a high precision.The relative errors obtained with the selected Kepler dataset are 0.55% and 0.04% respectively (cf.Tab. 1).Regarding m, p and A m , they could be provided by Gaia.The precision that Gaia will soon provide should be of the order δp ∼ 10 µas for parallaxes, δm ∼ 3 mmag for apparent magnitudes and δA m 0.1 mag for extinctions 3 .Thereby, the relative incertitude on T eff would be about 2%, mainly dominated by the interstellar extinction.For a star with T eff = 5000 K, we have δT eff ∼ 100 K, which is very competitive with spectroscopic determination.
Alternatively, we could also imagine to derive the seismic interstellar extinction given the spectroscopic T eff .

Conclusion
The MLEUP method allowed us to extract simultaneously with a high precision the seismic indices and the granulation parameters from a large number of Kepler and CoRoT stars.We yield 4783 CoRoT stars and 13,689 Kepler stars for which we extracted the seismic indices, and 1520 CoRoT stars and 12,879 Kepler stars for which we get both the seismic indices and the Concerning the luminosity L, it can be expressed as a function of the stellar absolute magnitude M (mag): with the constant M corresponding to the solar values.
And given that M is dependant of the apparent magnitude m (mag), the distance d (pc) and the extinction A m (mag), as: One can finally deduce the seismic effective temperature as a function of A m , m, d, ν max and ∆ν, as: Furthermore, the stellar parallax p being small, one can approximate the distance d by 1/p.Thus, the seismic T eff can also be expressed with the parallax p (arcsec): .
The associated relative error of this seismic T eff is then: Note that the bolometric correction have to be taken into account in the calculation of the seismic effective temperature.

Figure 1 :
Figure 1: Results of the final adjustment of the background and oscillations of MLEUP (black line).In grey, the raw PSD of KIC 2850913.The dash-dot green line and the dashed blue one correspond respectively to the activity and granulation component.The dotted magenta line represents the white noise component and the solid red one is the Universal Pattern.

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∆ν = (0.282 ± 0.001) ν 0.7506±0.0010max (Kepler) The Kepler scaling relation is consistent with the [3]'s scaling relation (∆ν = (0.274 ± 0.002) ν 0.751±0.002max ), determined by an average of scaling relations obtained by various different methods with the Kepler data, as well as the theoretical one [β = 0.75, e.g.4].Between our CoRoT and Kepler scaling relations, we note slopes significantly different.This difference is not yet understood.It could come from the difference in the stellar populations considered.