---
author:
- 'B. Mosser'
- 'K. Belkacem'
- 'M.J. Goupil'
- 'E.Michel'
- 'Y. Elsworth'
- 'C. Barban'
- 'T. Kallinger'
- 'S. Hekker'
- 'J. De Ridder'
- 'R. Samadi'
- 'F. Baudin'
- 'F.J.G. Pinheiro'
- 'M. Auvergne'
- 'A. Baglin'
- 'C. Catala'
bibliography:
- 'biblio\_tu.bib'
subtitle: an automated determination with CoRoT data
title: 'The universal red-giant oscillation pattern'
---

[The CoRoT and Kepler satellites have provided thousands of red-giant oscillation spectra. The analysis of these spectra requires efficient methods for identifying all eigenmode parameters.]{} [The assumption of new scaling laws allows us to construct a theoretical oscillation pattern. We then obtain a highly precise determination of the large separation by correlating the observed patterns with this reference. ]{}[We demonstrate that this pattern is universal and are able to unambiguously assign the eigenmode radial orders and angular degrees. This solves one of the current outstanding problems of asteroseismology hence allowing precise theoretical investigation of red-giant interiors.]{}

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Introduction\[introduction\]
============================

Red giants are evolved stars that have depleted the hydrogen in their cores and are no longer able to generate energy from core-hydrogen burning. The physical processes taking place in their interiors are currently rather poorly understood. Observations with the space-borne mission CoRoT have revealed the oscillation pattern [@2009Natur.459..398D] of many of these stars, which is a crucial step on the route to probing their internal structure. Before the advent of the CoRoT data, complex oscillation patterns were explained by short-lived modes of oscillation . The new era of the space-borne missions CoRoT and *Kepler* has dramatically increased the amount and quality of the available asteroseismic data of red giants . The analysis of the oscillation eigenmodes now allows seismic inferences to be drawn about the internal structure.

The identification of the angular degree and radial order of the eigenmodes represents a first and crucial step in an asteroseismic analysis. The values of the eigenfrequencies can be related to the order and degree of a mode with the commonly used asymptotic equation: $$\nu_{n,\ell} = \left[ n+{\ell\over 2} + \varepsilon \right] \, \Delta\nu - \dd_{0\ell}
\label{tassoul}$$ where $\nu_{n,\ell}$ is the eigenfrequency of a mode with radial order $n$ and angular degree $\ell$; $\Delta\nu$ is the mean value of the large separation ($\dnumoy \simeq \nu_{n+1,\ell} -
\nu_{n,\ell}$), and $\dd_{0\ell}$ is a second-order term, or small separation, dependent on the mode degree. This form, which is similar to the original expression developed for the Sun 30 years ago [@1980ApJS...43..469T], is useful for analysing the observations and performing the complete mode identification. It assumes $\dd_{00}$ equals 0. The parameter $\varepsilon$ comprises two parts: the offset due to the mode propagation in the upper-most layers of the star, and the second-order term of the asymptotic approximation which is sensitive to the gradient of sound speed in the stellar interior.

In the absence of accurate determinations of the individual mode frequencies, the global seismic parameters used in the asymptotic expression above are important indicators of the physical parameters of the star. The large separation gives a measure of the mean stellar density; the small separation $\dd_{0\ell}$ describes the stratification of the central regions. Unfortunately, the methods currently used to determine the global oscillation parameters suffer from various sources of uncertainty . First, the stochastic excitation of the modes gives rise to variability in the amplitudes, resulting in an apparently irregular comb structure; second, the finite mode lifetime blurs the estimates of the eigenfrequencies; third, estimates are affected by the stellar noise and granulation signal superimposed on the oscillations . In fact, simulations have shown that the impact of realization noise on the measurement of the large separation $\dnumoy$, can be much larger than the background noise for red giants [@2010arXiv1008.2959H].

In an analysis of a sub-sample of Kepler red giants, [@huber2010] have shown the regularity of the oscillation spectra of such stars. In this paper, we show that all red giants have a regular pattern, as modelled recently by [@2010ApJ...721L.182M]. We propose a method which allows us to tag all the modes with their appropriate radial order and angular degree, regardless of the presence of the perturbing effects described above.

Method\[method\]
================

The method to mitigate the effects of realization noise uses Eq. \[tassoul\] in a dimensionless form: $${\nu_{n,\ell} \over \dnumoy} = n+{\ell\over 2}  + \varepsilon (\dnumoy) - d_{0\ell} (\dnumoy) .
\label{tassoul_m}$$ In a departure from the previous practice, we have assumed that $\varepsilon$ obeys a scaling law $\varepsilon = A + B \log
\dnumoy$, as derived from the observation of thousands of CoRoT targets and as observed by [@huber2010]. This is justified by the observation that scaling laws apparently govern *all* global asteroseismic parameters and is equivalent to assume that the underlying physics of $\varepsilon$ varies with the global stellar parameters. As the mixed nature of dipole modes ($\ell=1$) is more pronounced, we did not include them in the template, but only doublets corresponding to the eigenmodes with even degrees ($\nu_{n-1,2}$ and $\nu_{n,0}$), with equal amplitudes. As a first guess, we set the small separation $d_{02}$ at $- 0.14$ and then allowed it to vary with the value of $\dnumoy$ according to the same relationship as given for $\varepsilon$. For constructing the peaks of the template, we have also used the scaling laws of the Gaussian excess power derived by . We have assumed that the mode lifetime varies as $\dnumoy^{-1}$ and have used mode widths equal to about a few percent of $\dnumoy$. Finally, we stress that no background model is needed.

![CoRoT red giant power spectra stacked into an image after sorting on the large separation. One line corresponds to one star. [**a)**]{} The first sorting is based on the large separation before any correction. [**b)**]{} The blurred aspect disappears once the correction has been performed and reveals a clear comb-structure common to all red giants. The vertical lines at 23.2$\mu$Hz are the signature of the low-Earth orbit . \[deter\]](TU_avant.ps "fig:"){width="8.2cm"} ![CoRoT red giant power spectra stacked into an image after sorting on the large separation. One line corresponds to one star. [**a)**]{} The first sorting is based on the large separation before any correction. [**b)**]{} The blurred aspect disappears once the correction has been performed and reveals a clear comb-structure common to all red giants. The vertical lines at 23.2$\mu$Hz are the signature of the low-Earth orbit . \[deter\]](TU_apres.ps "fig:"){width="8.2cm"}

![The spectra presented in Fig. \[deter\]b have been rearranged in order to have the dimensionless frequency on the abscissa and the mean large separation on the ordinate. They show that the red-giant oscillation pattern is universal. The hyperbolic branch is here the signature of the low-Earth orbit. \[universal\]](TU_univer.ps){width="8.7cm"}

The measurement of the large separation is performed in two steps. First, an initial-guess value $\dnug$ of the large separation is computed by an automated pipeline . This is used to form the initial synthetic template to correlate with the real spectrum. The best correlation between the observed and synthetic spectra provides then the corrected value of the large separation. The template was iteratively adjusted by varying its parameters to maximize the correlation.

In Fig. \[deter\], we show the results obtained with all high signal-to-noise CoRoT data . In both cases the graphs show the spectra arranged in strips with the colour representing the strength of the signal, as was done by @2010PASP..122..131G. The spectra are sorted by increasing large separation with the smallest large separation at the top of the plots: in the upper plot we use the output from a conventional pipeline and in the lower one we use the corrected value. The remarkably regular structure within the oscillation spectra in the lower plot reveals the signature of comb-like structure of the asymptotic relationship in Eq. (\[tassoul\_m\]) already reported . Further, it validates the scaling law in $\varepsilon$ included in the reference template. The global agreement of all high signal-to-noise spectra of bright targets with the synthetic pattern (Fig. \[universal\]) shows that these oscillation patterns are homologous and that the red-giant oscillation pattern is universal.

We further found that the template is significantly improved if it takes account of the linear dependence of the large separation in frequency, expressed by the degree-dependent gradient $\alpha_\ell
= ({\mathrm{d}}\log \dnumoy / {\mathrm{d}}n)_\ell$: $${\nu_{n,\ell} \over \dnumoy} = n+{\ell\over 2}  + \varepsilon
(\dnumoy) - d_{0\ell} (\dnumoy) + {\alpha_\ell (\dnumoy) \over2}
\left( n - {\numax \over \dnumoy} \right)^2 \label{tassoul_mod}$$ with $\numax$ the frequency of maximum oscillation amplitude. The corrected values of $\dnumoy$ are derived from this template. The values of the 12 free parameters that account for the variations in frequency of the parameter $\varepsilon$, of the small separations $d_{0\ell}$ and of the gradients $\alpha_\ell$ as derived from the fits to more than 6000 eigenmodes (Fig. \[identi\]) are given in Table \[fits\]. The value of $\varepsilon$, defined modulo 1, is fixed thanks to the extrapolation to the Solar case. As noted by and [@huber2010], the small separation $d_{01}$ is negative. All these fits are consistent with Kepler results.

The average value of the correction from $\dnug$ to $\dnumoy$ is of the order of 2.5%. For the largest giants, with the smallest values of $\dnumoy$ and the smallest ratio $\numax / \dnumoy$, the correction can be as high as 6%. The absolute difference $|\dnug-\dnumoy|$ can be 10 times the estimate of the stellar noise contribution. At low frequency, with an observing run not much longer than the mode lifetimes [@baudin2010], the realization noise dominates the background noise and the mean accuracy of the determination of $\dnumoy$ is uniform, at about 0.015$\mu$Hz. We are aware that any bias in the $\varepsilon
(\dnumoy)$ input relation will induce a bias in the results. We therefore took care to insure the iterative process to be unbiased. Analysis of synthetic data indicates that the precision gained with this method is of order 10 times better than that obtained with conventional methods [@2010arXiv1008.2959H].

  --- --------------- ------------------------------ -------------------- -----------------
                       gradient of $\Delta\nu_\ell$                       
                                 $A_\ell$                  $B_\ell$         $\alpha_\ell$
  0   $\varepsilon$         $ 0.634 \pm 0.008$        $ 0.546 \pm 0.008$   $0.008\pm0.001$
  1   $d_{01}$              $-0.056 \pm 0.012$        $-0.002 \pm 0.010$   $0.003\pm0.002$
  2   $d_{02}$              $ 0.131 \pm 0.008$        $-0.033 \pm 0.009$   $0.005\pm0.001$
  3   $d_{03}$              $ 0.280 \pm 0.012$               $0$           $0.005\pm0.002$
  --- --------------- ------------------------------ -------------------- -----------------

  : Fits of the ridges, for $\dnumoy$ expressed in $\mu$Hz[]{data-label="fits"}

![Complete identification of the ridges, automatically derived from the eigenfrequencies extracted with a height-to-background ratio greater than 3. Each colour correspond to a different mode degree (radial modes in red, dipole modes in dark blue, $\ell=2$ modes in green, $\ell=3$ modes in light blue). The solid grey lines superimposed on the ridges indicate for each radial order the fits of $\varepsilon$ (with indication of the radial order $n$). The fits of $d_{01}$, $d_{02}$ and $d_{03}$ are superimposed on the respective ridges (respectively dash-dot, dot and dash lines for $\ell = $1, 2 and 3). The dark dashed lines, derived from the scaling law dealing with the oscillation excess power, delineate the region where the modes have noticeable amplitudes . \[identi\]](etap71.ps){width="8.8cm"}

Discussion\[results\]
=====================

This new method based on a simple hypothesis and an automated procedure removes any ambiguity on the identification of the modes (Fig. \[identi\]), despite the complexity induced by mixed modes. Mode identification is derived by looking at the closest ridge. In particular, we provide a straightforward determination of the mode radial orders, which were previously unknown. Radial eigenfrequencies are located at: $$\label{identin}
   \nu_{n,0} = \left[ n + \varepsilon (\dnumoy) \right] \; \dnumoy$$ Ridges were already shown in previous works. While [@2009Natur.459..398D] and looked at single stars separately, [@2010ApJ...713L.176B] and [@huber2010] used manual fine-tuning of the large separation to align the radial modes of a large sample of stars. However, the radial modes were identified in only one third of the spectra by [@huber2010], but they also showed the ridges with varying $\varepsilon$ in the folded and collapsed power spectrum.

In most regions of the oscillation spectra we observe the presence of both radial and non-radial modes. Realization noise causes the height of the individual modes to show considerable variability, but on average, the ratio between the dipole and radial mode height is approximately independent of $\numax$. Although, at very low $\numax$ there is some reduction in the strength of the dipole mode. We also make clear that the larger spread of the ridges corresponding to dipole modes (Fig. \[identi\]) is due to the presence of many mixed modes, as already noticed . The universal pattern makes it easier to identify them opening up the possibility of exploring the conditions in the inner layers of the red giants.

Despite their low amplitudes and the resulting poor signal, $\ell=3$ modes have been detected in Kepler data on red giants [@2010ApJ...713L.176B; @huber2010]. Our results represent the first such detection in CoRoT data. Their identification gives access to the fine structure of the oscillation spectra, as modes of different angular degree probe different depths within the star. Their detection and complete characterization will first be derived from the universal pattern, then the small differences to this pattern will be exploited to characterize in detail a given object [@2010arXiv1009.1024M].

More than 75% of the red-giant candidates with brighter magnitude than $m{_{\mathrm{R}}}=13$ observed with CoRoT show solar-like oscillations. In the remainder, we observe a large proportion of classical pulsators or of giants with a so large radius that the oscillations occur at a too low frequency for a positive detection. In a very limited number of cases at very low frequency, the possible confusion between radial and dipole modes is not clearly solved. This confusion increases toward dimmer targets with lower quality time series. Among the positive detection of bright stars, we did not observe any outliers when performing the correlation with the universal pattern. For this procedure to be effective, we require that the modes have a significant height-to-background ratio. Hence for all high signal-to-noise targets we are able to derive corrected values for the large frequency spacing.

It is recognized that the majority of the red giants in the CoRoT field of view are in their post-flash helium-burning phase . In terms of stellar evolution, the demonstration of the universal regular pattern of red giants proves that these red giants have similar and homologous interior structures. On the other hand, despite the agreement of the fit in $\varepsilon$ with the Solar value, we have verified that the method does not work with subgiants or main-sequence stars . We explain this by the wide range of evolutionary phases pertaining outside the red clump, which certainly will cause a complex dependence of $\varepsilon$ with more parameters than just the large separation. This disagreement reinforces the homogenous properties of red-giant stars. The clearly observed pattern confirms that, on average, the linewidths of the modes are significantly smaller than the separation of the even mode pairs and hence makes short mode lifetimes unlikely [@2004SoPh..220..207S]. From the quasi-uniform width of the ridges (Fig. \[identi\]), we can see that the lifetimes of the modes increase significantly with decreasing large separation, contrary to [@huber2010]. It also suggests that very complex oscillation spectra previously observed may be an artefact of noise.

![Variation in stellar mass and radius as a function of the large separation $\dnumoy$. The vertical line corresponds to the error box in large separation and radius. \[masserayon\]](figure4_M_R.ps){width="8.8cm"}

Finally, the better determination of the large separation $\dnumoy$ helps us to enhance the accuracy of the estimates of the stellar mass and radius as done in , with typical error-bars of 12 and 5% respectively, instead of 20 and 8% (Fig. \[masserayon\]). This accuracy, achieved without the need of stellar modeling [@kallinger2010], constitutes an important progress compared to the current photometric determination and demonstrates the power of asteroseismology.

Conclusion
==========

We have shown with CoRoT observations that the red-giant oscillation spectrum is very regular and can be described by its underlying universal pattern. This was modelled in parallel by [@2010ApJ...721L.182M]. As a consequence, the precise measures of the large separation and the scaling relation of the parameter $\varepsilon$ allow us to provide an unambiguous detection of the radial orders and angular degrees of the modes. Since the method is able to mitigate the realization noise, we consider it to give the most precise determination of the large separation available.

It remains important to interpret the physical meaning of the scaling law for the term $\varepsilon$ in the Tassoul expression. We will have to disentangle the contributions of the surface and of the inner region. This will require an investigation of the term $\varepsilon$ in the context of the second-order corrections of the Tassoul development. Very long-duration observations with *Kepler* will help for this task.

Despite the uniform aspect of the oscillation spectra, many differences invisible in the global approach are revealed by a detailed analysis of each individual spectrum, as, for example, the modulation with frequency of the large separation . Study of this variation will give access to the most accurate analysis of the red-giant interior structure. This summarizes the power of asteroseismology: first, the regular pattern provides the identification of the individual modes; second, the difference to this regular pattern unveils the detailed interior structure. We are confident that the shifts to the regular pattern will be explained by mass, age and/or metallicity effects.

Similar analysis can be performed for oscillations in subgiants and solar-like stars . Due to the large variety of evolutionary stages among those stars, we expect the Tassoul parameter $\varepsilon$ to depend on more than the large separation.

This work was supported by the Centre National d’Études Spatiales (CNES). It is based on observations with CoRoT. The research has made use of the Exo-Dat database, operated at LAM-OAMP, Marseille, France, on behalf of the CoRoT/Exoplanet program. KB acknowledges financial support through a postdoctoral fellowship from the Subside fédéral pour la recherche 2010, University of Liège. FJGP acknowledges FCT’s grant SFRH/BPD/37491/2007. YE and SH acknowledge financial support from the UK Science and Technology Facilities Council (STFC). The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement n$^\circ$227224 (PROSPERITY), as well as from the Research Council of K.U.Leuven grant agreement GOA/2008/04.
