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\begin{document}
\title{Frequency patterns as the key to unlock the secrets of pulsations in
\( \delta \) Scuti stars}
\author{Antonio García Hernández}
\affil{Affiliation not available}
\author{Susana Martín-Ruiz}
\affil{Affiliation not available}
\author{Awaiting Activation}
\affil{Affiliation not available}
\author{Juan Carlos Suárez}
\affil{Affiliation not available}
\author{Daniel Reese}
\affil{Affiliation not available}
\author{Awaiting Activation}
\affil{Affiliation not available}
\date{\today}
\maketitle
\selectlanguage{english}
\begin{abstract}
Delta Scuti (\( \delta \) Sct) stars are intermediate-mass
pulsators whose intrinsic oscillations have been studied for decades.
However, modelling their pulsations remains a real theoretical
challenge, thereby even hampering the precise determination of global
stellar parameters. Here we present a direct and precise measurement of
the mean density of \( \delta \) Sct stars. This measurement is
obtained from an observational relation that scales stellar mean density
with an oscillation frequency pattern analogous to the solar-like large
separation but in the low order regime. We also show that this relation
is independent of the rotation rate of the star, thus allowing us to
accurately determine the mass and the radius, constrain the evolutionary
stage, and estimate the rotational velocity. This places tight
constraints on stellar evolution theory and on the physical properties
of planets orbiting A-type stars.%
\end{abstract}%
Intermediate-mass stars with around 1.5 to 2.5 times the mass of the Sun, spectral types A and F, and effective temperatures between 7000 to 10000 K, display physical charateristics and develop internal processes during their lives that are the key to starting the next revolution in the theory of stellar interiors. Rotation, convection in the nucleus, chemical mixing and diffusion, angular momentum distribution and excitation of pulsations are some of these physical processes.
The classical pulsators known as $\delta$ Scuti (\ds) stars belong to this category and were considered during the last quarter of the XX century as the next Cepheids (the first "standard candels" to measure astronomical distances). They are the second most common and numerous group of pulsators in the Galaxy, right after pulsating white dwarfs \cite{1979PASP...91....5B}. In contrast with Cepheids, they are placed within the main-sequence part of the "classical instability strip" in the \hr\ (HR) diagram. For all these reasons, they have become the favourite objects with which to study those physical mechanisms which are not well understood yet.
The study of pulsations in solar-like stars triggered a revolution in the field of solar physics. In particular, the use of an obvious frequency pattern known as the large separation (\( \Delta \nu \)) has been used as the cornerstone to discover and analyse earth-size planets around solar analogues. This periodic pattern comes from the frequency differences between modes of consecutive radial orders ($n$) and with the same spherical degree (\ele), and is directly related to the mean density ($\rho$) of the star \cite{1980ApJS...43..469T}. This is only true when the radial order of the modes ($n$) is high compared to the spherical degree (\ele), a situation which is known as the asymptotic regime. Solar-like pulsations are within this range.
In contrast, interpreting the frequency spectra of \dss\ is not so straightforward. This is mainly due to the absence of any obvious pattern, since their pulsations are not in the asymptotic regime, and are strongly affected by rotation. Both the theory and the tools to extract the information from the raw data lack a proper parametrization of rotation. Thus, the results of these studies are less accurate and/or provide larger uncertainties.
Despite that, efforts have been made to properly account for rapid rotation in pulsation computations. The most recent theoretical works are able to produce the pulsation spectra of differentially rotating realistic models, including those based on the ROTORC code \cite{2009ApJ...693..677L}, 1D stellar models in which centrifugal distortion has been added \cite{2013arXiv1301.2496O}, models from the Self-Consistent Field (SCF) method \cite{2009A&A...506..189R}, and those from the ESTER code \cite{2014IAUS..301..169R}. Still, some approximations have to be taken into account, like a chemically homogeneous internal structure for the star (in the case of SCF models) or the lack of a self-consistent modelling of their evolution, as opposed to their slowly rotating 1D counterparts. This can lead to significant discrepancies between observed and computed frequencies, thereby hindering asteroseismic inferences. Furthermore, the numerical cost of calculating such spectra, and the time it takes to classify (by hand) the resultant modes, make it unfeasible to scan an entire parameter space for best-fitting models, as is commonly done in the 1D case.
Nonetheless, these works produced results that could be crucial to the analysis of the pulsation spectra of rapid rotators. First, they showed that even though rapid rotation strongly modifies the position of the frequencies, some patterns are still present \cite{2006A&A...455..607L, 2009A&A...500.1173L}, namely the large separation (\Dnu) and the rotational splitting, the frequency difference between different components of a given multiplet in a rotating star (although we do note that rapid rotation leads to a new type of rotational multiplet, as described in \cite{2012A&A...546A..11P}). Second, the difference between both patterns is that the \Dnu\ spacing leads to regular comb-like structures due its periodic behaviour whereas the rotational splittings leads to isolated groups of modes with this preferred spacing rather than a dirac comb. Still the question remains as to whether or not the pulsation modes of \ds\ stars allow a successful identification of the \Dnu\ spacing and, given their low radial orders, how this is related to the spacing predicted at a higher frequency domain (commonly named as asymptotic regime). In addition, it is crucial to determine up to what extent the \Dnu-$\rho$ relation is affected by rotation.
Recently, thanks to the high number of frequencies detected by space observations \cite{2009A&A...506...85P}, a periodic pattern was systematically identified in several \dss\ \cite{2009A&A...506...79G, 2013A&A...559A..63G}. These authors claimed to have found an analogous pattern to \Dnu\ in the low order regime. On the other hand, a recent theoretical study derived an equivalent \Dnu-$\rho$ relation in the low order regime of intermediate-mass main-sequence stars \cite{2014A&A...563A...7S}. In both cases, the use of non-rotating models limited their conclusions. However, recent work based on polytropic stellar models \cite{2008A&A...481..449R} and further calculations based on the more realistic SCF models have shown that this scaling relation continues to apply to within a few percent even at rapid rotation, provided one uses the true mean density of the star (which takes into account the centrifugal distortion). Furthermore, a very recent study using theoretical pulsation spectra of SCF models with realistic mode visibilities suggests that in favourable situations it may be possible to identify such patterns in observed stars (Reese et al. submitted).
Clearly, all these theoretical results needed an observational confirmation that was not available until the launch of space missions dedicated to precise photometry. Here we provide the observational confirmation of all these previous studies. In order to reach such a confirmation, we needed to avoid any dependency on stellar interior or pulsation modeling. This was achieved by using eclipsing binary systems with a \ds\ component. Eclipsing binaries allow us to determine the mass and the radius of both components of the system by means of geometrical considerations, Kepler's Third Law, and stellar atmosphere models, thereby significantly reducing the effects of rotation. Using these two quantities we were able to obtain a model-independent determination of the mean density of the pulsating component.
The second variable needed to derive a large separation-mean density relation is the large separation contained within the frequency spectrum of the \ds\ component. We followed the method described by \cite{2009A&A...506...79G, 2013A&A...559A..63G} to derive this value. As in the case analyzed by these authors, space observations were needed to assure a statistically significant number of frequencies. This does not necessarily need to be a huge number, but one which is sufficient to enable us to find a quasiperiodic behaviour. Consequently, the selected sample is formed by systems observed with the MOST \cite{Walker_2003}, \corot\ \cite{2006ESASP1306...33B} and \kepler\ \cite{2010Sci...327..977B} satellites.
We found a total of 7 systems fulfilling all the mentioned conditions: 6 eclipsing binaries and a binary for which the main \selectlanguage{greek}δ \selectlanguage{english}Sct component was resolved using optical interferometry. The relevant information used in this work, mainly the mass, the radius, the large separation, and the corresponding references are summarized in Table~\ref{tab:sample}. In order to compute the mean density, we used the Roche model approximation \cite{2009pfer.book.....M} assuming that the measured radii correspond to the equatorial ones. Such an approximation was validated through comparisons with more realistic SCF models (see Additional Material for details). Before deriving the large separation we treated the spectra in order to remove frequencies which could result from resonances or combinations, or were potentially associated with g-modes, as these might blur the periodicity. We also present in this work the frequency analysis of HD~172189 (see Additional Material for details).
%-table-%\selectlanguage{english}
\begin{table}
% \begin{footnotesize}
% \begin{center}
\label{tab:sample}
\begin{tabular}{cccccccc}
% \hline
\hline
System & \Dnu\ (\muhz) & M (M$_\odot$) & R (R$_\odot$) & $\rho$ $(\rho_{\odot})$ & \vsini\ & $i$ & $\Omega/\Omega_{\mathrm{K}}$ \\
%KIC 3858884 & KIC 10661783 & KIC 4544587 & ID 100866999 & HD 172189 & Rasalhague & HD 51844 & KIC 105906206 \\
\hline
KIC 3858884\footnote{\cite{2014A&A...563A..59M}} & $29\pm1$ & $1.86\pm0.04$ & $3.05\pm0.01$ & $0.0656\pm0.0021$ & $25.7\pm1.5$ & $88.176\pm0.002$ & 0.0754 \\
KIC 4544587\footnote{\cite{2013MNRAS.434..925H}} & $74\pm1$ & $1.61\pm0.06$ & $1.58\pm0.03$ & $0.547\pm0.044$ & $75.8\pm15$ & $87.9\pm3$ & 0.172 \\
KIC 10661783\footnote{\cite{2013A&A...557A..79L}} & $39\pm1$ & $2.100\pm0.028$ & $2.575\pm0.015$ & $0.1230\pm0.0038$ & $78\pm3$ & $82.39\pm0.23$ & 0.200 \\
% HD 51844\footnote{\cite{2014A&A...567A.124H}} & $20\pm1$ & $1.97\pm0.19$ & $3.52\pm0.17$ & $0.0452\pm0.0109$ & $41.73\pm1.7$ & $69.8\pm5$ & 0.136 \\
HD 172189\footnote{\cite{2009A&A...507..901C}} & $19\pm1$ & $1.78\pm0.24$ & $4.03\pm0.11$ & $0.0272\pm0.0059$ & $78\pm3$ & $73.2\pm0.6$ & 0.281 \\
ID 100866999\footnote{\cite{2013A&A...556A..87C}} & $56\pm1$ & $1.8\pm0.2$ & $1.9\pm0.2$ & $0.262\pm0.112$ & -- & $80\pm2$ & -- \\
ID 105906206\footnote{\cite{2014A&A...565A..55D}} & $20\pm1$ & $2.25\pm0.04$ & $4.24\pm0.02$ & $0.02952\pm0.00094$ & $47.8\pm0.5$ & $81.42\pm0.13$ & 0.152 \\
Rasalhague\footnote{\cite{2010ApJ...725.1192M} \& \cite{2011ApJ...726..104H}} & $38\pm1$ & $2.40^{+0.23}_{-0.37}$ & $\mathrm{R}_\mathrm{eq}=2.858\pm0.015$ & $0.112\pm0.019$ & $239\pm12$ & $87.5\pm0.6$ & 0.88 \\
& & & $\mathrm{R}_\mathrm{pol}=2.388\pm0.013$ & & & & \\
% \hline
\hline
\end{tabular}
\caption{{Characteristics of the systems taken from the literature {\bf INCLUIR EL MODO FUNDAMENTAL RADIAL}. The information corresponds to the pulsating component (which is not necessarily the primary). }}
\end{table}
%-table-%
The binarity also allowed us to estimate the rotation rate of the star. Assuming the rotation axis is perpendicular to the binary system plane, which is adequate in most cases, and using the observed values for \vsini\ and the radius, we computed $\Omega/\Omega_{K}$, $\Omega_{K}$ being the Keplerian break-up rotation rate (\textit{i.e.} the maximum velocity for which the star does not lose mass due to the centrifugal acceleration). This value allowed us to test the relation for any rotation rate. Indeed, our sample contains rotation rates which range from $0.0754$ to $0.88\,\Omega/\Omega_{K}$, this upper limit being reached by \starh\ \cite{2010ApJ...725.1192M}.
In Fig.~\ref{fig:dnu_rho}, we plotted all this information in a $\log$\Dnu\ versus $\log\bar{\rho}$ diagram and indicated the rotation rate through the colour of the various symbols. These symbols follow a clear linear trend. A weighted linear fit, taking uncertainties as weights, is given by the expression:
\begin{equation}
\bar{\rho}/\bar{\rho_{\odot}}=1.55^{+1.07}_{-0.68}(\Delta\nu/\Delta\nu_{\odot})^{2.035\pm0.095}.
\end{equation}
The normalization by the solar values (with $\Delta\nu_{\odot}$ = 134.8 \muhz, \cite{2008ApJ...683L.175K}) is useful because it allows us to compare this trend with the scaling relations from \cite{2014A&A...563A...7S}, and \cite{1980ApJS...43..469T} for solar-like pulsators, as depicted in Fig.~\ref{fig:dnu_rho}. To within the uncertainties, the slope of the linear trend fits the results found with non-rotating models, but this trend is offset by a multiplicative constant in comparison with the solar-like scaling relation. This was expected since the regime of \ds\ pulsations is different from the latter, but the agreement with the non-rotating case is the sign of the invariance of the relation with rotation as predicted by \cite{2008A&A...481..449R}. %Only a spurious point deviates the relation from the form by Suárez et al. (2014), \stard. But this system displays some differences with respect the other objects, like its peculiar metallicity or that it is not an eclipsing binary.%This is a special case, since it is not a strictly speaking eclipsing binary with eclipse events but a heart-beat binary showing brightening events. The uncertainties in the determination of the physical characteristics of the system are greater than in the pure eclipsing binary case. These facts make the point of \stard\ [suspect to be] less reliable than the others.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/Dnu-rho-weighted-Roche-7stars-Ras1/Dnu-rho-weighted-Roche-final}
\caption{{\label{fig:dnu_rho}
Large separation-mean density relation obtained for the sample of 7 binary systems of our sample. A linear fit to the points is also depicted, as well as the solar-like scaling relation \protect\cite{Tassoul_1980} and the theoretical scaling relation for non-rotating models of \ds\ stars \protect\cite{2014A&A...563A...7S}. Points are plotted with a gradient colour scale to account for the different rotation rates.%
}}
\end{center}
\end{figure}
Thus Fig.~\ref{fig:dnu_rho} confirms several hypotheses. The main one is that the large frequency separation calculated in the low order regime (relevant to \ds\ stars) is still related in a simple way to the mean density of the star, thereby enabling us to determine this quantity without additional information. The second consequence is that this relation does not depend on the rotation rate as clearly shown in the plot.
The last statement implies that non-rotating models might be useful to estimate, in some cases, other characteristics of the star, such as the evolutionary stage. Simple comparisons with non-rotating stellar evolution models, taking into account the observables \teff, \logg, \feh\ and \Dnu, provides such information \cite{Hareter_2014}. This is not a measure of the age but an indicator of the stage within the evolutionary sequence of the star.
Once the comb-like structure \Dnu\ is detected using the Fourier transform, we looked for other spacings present within the frequency spectra of our sample. Alternate techniques designed to discover other kinds of patterns may be used to that end. We computed histograms of frequency differences to look for a preferred spacing such as the rotational splitting. Once the orbital period is discarded, we found peaks in the lower part of the histograms that correspond to the rotational splitting for 5 out of our 7 stars (see Additional Material for details). We concluded that the combined use of the Fourier transform and histogram of frequency differences allows us to disentangle between both structures. Rotational splittings thus become an observable that can be studied in rapidly rotating stars through asteroseismology.\\
%Thus it opens the possibility of a direct determination of the equatorial velocity without the $\sin i$ dependence typical of spectroscopic observations. Pulsation calculations in rotating models suggest that the rotational splitting is present together with the large separation, but that it may not always be possible to disentangle both \cite{2010AN....331.1053L}. The relation presented in this work overcomes this limitation. Moreover, it might become a complementary and fundamental source of information for studies where rotational splittings have been detected \cite{2003Sci...300.1926A}. Rotation is then a physical phenomenon that can be studied in main-sequence rapidly-rotating stars through its interaction with pulsations. the mass {\bf DESCRIBIR CÓMO Y COMENTAR QUÉ CON LAS ESTRELLAS DE LA MUESTRA. ¿COINCIDE LA MASA CUANDO SE USA LA LUMINOSIDAD?¿CUÁL ES EL ERROR COMETIDO?} and the evolutionary stage. For the mass, just another measurement is needed, such as the paralax or the luminosity. Using the well established luminosity-mass (LM) relation for main-sequence stars \cite{Ibanoglu_2006}, one may compute the real value of the mass. The results for the main-sequence stars of our sample are really satisfactory, with a difference in the masses between the values in Table~\ref{tab:sample} and those determined using the LM relation of less than 10\%.
%because different patterns found in the frequency set can be compared with the information of the modelling and other complementary observations (for example, from spectroscopy) allowing such a distinction. , and the true rotational velocity, not the projected one. Thus the rotational splititng can be systematically identified within the frequency spectra of the object.
The determination of the mean density and the rotational splitting yields major clues for stars harbouring planetary systems. Using both quantities and the inclination angle of the system, the mass and the radius of a transiting planet can be determined \cite{2003ApJ...585.1038S}. Such a determination is feasible for main-sequence stars if a well determined luminosity exists. A mass-luminosity relation \cite{Ibanoglu_2006} will give the mass of the star. Using the derived luminosities of our sample, we tested such a relation. For the main-sequence stars, the agreement on the mass was found to be within 10\%. The Gaia mission \cite{2012Ap&SS.341...31D} will play an important role in obtaining better mass estimates.\par
Obtaining all of these characteristics in planets around A-F stars will complete our vision of the formation and evolution of planetary systems, poorly known for stars more massive and hotter than the Sun. The future space mission PLATO \cite{2014ExA....38..249R} will observe bright stars, in particular, many \ds\ pulsators. The observational confirmation presented in this work will be critical in accomplishing the mission goals of obtaining a precise characterisation of the full range of planetary systems.
%Finally, the determination of the mean density gives a major clue for stars harbouring planetary systems. Using the mean density of the star, the mean density of a transiting planet can be determined \cite{2003ApJ...585.1038S}. This allows us to establish the nature of the planet: gaseous or rocky. Additionally, the radius and the age are very important parameters when studying planetary systems because they give substantial information on the evolution history of the system. To distinguish all these characteristics in planets around A-F stars will complete our vision of the formation and evolution of planetary systems, poorly known for more massive and hotter than the Sun stars. The future space mission PLATO will study bright stars, so many \ds\ pulsators will be observed. The observational confirmation presented in this work will be critical in accomplishing the mission goals of obtaining a precise characterisation of the full range of planetary systems. The results shown here are a major step forward in that effort.
\section{Additional material}
\subsection{On the determination of rotational splittings}
We already demonstrated that a large separation pattern is feasible to be detected in the frequency spectra of \ds\ stars. Nonetheless, the rotational splitting was also predicted to be present, both in the slow rotating case and in the case of fast rotators \cite{2010AN....331.1053L}. We then tried to go further and searched for the rotational splitting. Due to the structure of such spacing, not a comb-like one, we used histograms of frequency differences to that end. This kind of histograms is sensitive to any recursive spacing, so identifying \Dnu\ is a previously required step.\par
We created the histograms using the 30 highest frequencies of the whole set for each star to be consistent with the Fourier transform analysis, for which we also selected that amount to compute it. In the cases where a less number of frequencies were detected (\starb, \starc, \starf), we used the whole set. The form of a histogram is sensitive to the size of the bin chosen to compute it. A bin size too large will result in hiding any periodic signal. On the other hand, a bin size too small might result in a flat histogram of just one difference per bin. Testing several values, we obtained a bin size of 1~\muhz\ (or 0.0864~\cd) as the best balance between both extremes.\par
Interestingly, in 5 cases the histogram shows a peak at lower frequency differences which is compatible with the rotational splitting (the splitting being computed with data of Table~\ref{tab:sample}). For the other 2 cases, we did not have information about the rotation of the star (\starf) or the number of frequencies detected were too low to obtain a clear histogram (\starc). The resulting histograms for the cases where we detected a peak are showed in Fig.XXX.\par
We also checked any possible coincidence of being the peak a harmonic of the orbital frequency. Just for one case, \starb, the rotational splitting was twice the orbital frequency, suggesting the stellar rotation being tidally locked. Indeed, this seems to be a system with strong interaction between its components \cite{2013MNRAS.434..925H}. Such peculiarity and the limited number of frequencies hinder the detection of the rotational splitting. Nevertheless, the peak around 1.814~\cd is about $2\Omega$. The double of the rotational splitting was predicted by the theory for some particular configurations of the visibility of the modes \cite{2010AN....331.1053L}.\par
The double of $\Omega$ can be found in the case of \stare, too. In addition, the histogram of this star shows a double peak of equal amplitude at low differences. Based on no additional information, one cannot differentiate the rotational splitting. The same happens for \starb, \starg\ and \starh, although in most of these cases a mean value is a good approximation. \starg\ represents the most tricky example.\par
%For the rest of the sample, a peak at lower values is clearly identifiable. Note that the peak at the lowest frequency differences is due in most cases to the presence of close frequencies (less that 1~\muhz).
It is worth to mention the cases of \stara\ and \stare, because they show the frequencies clustered around the large separation. \stara\ is a slow rotator, so such structure is expected due to mode trapping \cite{1990AcA....40...19D}. However, \stare\ cannot be defined as slow rotator. In this particular case, the reason for the clustering is the proportionality between \Dnu\ and $\Omega$, being $\Delta\nu/2\sim2\Omega$.
%Slow and high rotation seems to be easier. Mid-rotation is more complicated, also it might depends on the number of frequencies, orbital period and rotation synchronization, etc.
%The Fourier transform and the histogram of frequency differences become complementary tools to find and differentiate the periodic structures present within the frequency spectra.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/kic3858884.lis-hist30-marks/kic3858884-hist30-arrows}
\caption{{KIC3858884, $\Omega=0.1666 d^{-1}=1.929\mu Hz$%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/kic4544587.lis-hist16-marks/kic4544587-hist16-arrows}
\caption{{KIC4544587, $\Omega=0.9495 d^{-1}=11\mu Hz$%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{./static.png}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/hd172189.lis-hist30-marks/hd172189-Granado-hist30-arrows}
\caption{{HD172189, $\Omega=0.4 d^{-1}=4.63\mu Hz$%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/ID105906206-freq-clean.lis-hist30-marks/ID105906206-freq-clean.lis-hist30-arrows}
\caption{{ID105906206, $\Omega=0.2255 d^{-1}=2.61\mu Hz$%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Rasalhague-freq-clean.lis-hist30-marks/Rasalhague-freq-clean.lis-hist30-arrows}
\caption{{Rasalhague, $\Omega=1.803 d^{-1}=20.87\mu Hz$%
}}
\end{center}
\end{figure}
\subsection{The mean density of a rotating model}
In order to calculate the mean density of a rotating model, one needs to find its volume, as follows:
\begin{eqnarray}
V &=& \int_{\phi=0}^{2\pi} \int_{\theta=0}^{\pi} \int_{r=0}^{R(\theta)} r^2 \sin\theta \mathrm{d}\theta \mathrm{d}\phi \mathrm{d} r \nonumber \\
&=& \frac{2\pi}{3} \int_{\theta=0}^{\pi} \left[R(\theta)\right]^3 \sin\theta \mathrm{d}\theta,
\end{eqnarray}
where $V$ is the volume, and $R$ the radius, which depends on the co-latitude
$\theta$. One can also introduce a mean radius $\left< R \right>$ as follows:
\begin{equation}
V = \frac{4\pi}{3} \left< R \right>^3.
\end{equation}
This leads to the following formula:
\begin{equation}
\label{eq:mean_radius}
\left< R \right> = \sqrt[3]{\frac{1}{2} \int_{\theta=0}^{\pi} \left[R(\theta)\right]^3 \sin\theta \mathrm{d}\theta}.
\end{equation}
The mean density is then simply given by:
\begin{equation}
\left< \rho\right> = \frac{M}{\frac{4\pi}{3} \left< R \right>^3}.
\end{equation}
Ideally, one would want to calculate the mean densities of realistic stellar models. However, for the sake of simplicity, it is useful to consider the mean density of a Roche model. Indeed, the surface of such a model is determined through equipotentials of the following analytical function:
\begin{equation}
\Phi = -\frac{GM}{R(\theta)} - \frac{1}{2} \Omega^2 \sin^2\theta R^2(\theta).
\end{equation}
For a given mass and rotation rate, $R(\theta)$ can be determinednumerically with Newton's method, and then inserted into Eq.~(\ref{eq:mean_radius}) to yield the mean radius. Figure~\ref{fig:mean_radii} shows a comparison between the mean radii from Roche models and from a set of uniformly rotating SCF models with masses between 1.8 and 2.5 $M_{\odot}$ and rotation rates between $0.0$ and $0.8\,\Omega_{\mathrm{K}}$. The different curves are so close that they are difficult to distinguish, which implies that Roche models accurately reproduce the geometric shape of the more realistic SCF models. Hence, they provide a reliable estimate of the geometric properties of $\delta$ Scuti stars, provided their rotation rate is close to uniform.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/radii/radii}
\caption{{\label{fig:mean_radii}:
Mean radii of SCF models (solid blue curves) and Roche models (dotted red line) as a function of rotation rate.%
}}
\end{center}
\end{figure}
\subsection{Frequency extraction for HD\,172189}
HD\,172189 was discovered by \cite{2003ASPC..292...59M} and a preliminary study as detached binary system with a $\delta$~Scuti-type pulsating component is reported in \cite{2005A&A...440..711M}. HD\,171289 (ID\,8170) was selected as target of the asteroseismic core program of the \corot\ satellite mission and was observed during the Second Long Run in the centre direction (LRc02). The measurements was taken from April to September 2009 for 149.01 days with a time sampling of 32 s. Time series from N2-level \corot\ mission products were linearly interpolated in order to remove the spectral window introduced by wrong data due to the South Atlantic Anomaly. For the case of HD\,172189 this is not an efficient technique and, instead of the linear interpolation, we have used a new gap-filling method which is based on autoregressive moving average interpolation \cite{2014arXiv1410.0841P}. This method recover the signal as feasible as possible and in that way makes possible a better frequency determination. In addition, the curve was corrected for a decreasing trend. The final dataset consists of 402336 points extending over 26 orbital cycles.
The light curve has been analyzed using the SigSpec code \cite{Reegen_2007}. In order to avoid problems with the power close to zero frequency, the analysis has been performed in the range 0.05-100 \cd. The Rayleigh frequency resolution is (1/$\Delta$T)=0.00671 \cd\ and an over-sampling of 10 corresponding to a frequency spacing 0.00067~\cd. Usually, a S/N of 4.0 is the limit used to consider a frequency as significant \cite{1993A&A...271..482B};\cite{1997A&A...328..544K}. In SigSpec, the parameter used for the significance is the spectral significance (sig) with a default limit of 5.0. If we compare both values, a sig=5.0 is equivalent to S/N=3.8 or S/N=4 is equivalent to sig=5.46 \cite{Reegen_2007}. Recently, \cite{Balona_2014} has showed using simulations that it is not possible to extract reliable frequencies down to the expected noise level by means of successive pre-whitening. Hence, we introduced as limit a higher value of sig=20.0 in this analysis as input value in the SigSpec program achieving more than 500 frequencies. Analyzing the results, those frequencies with the sig <= 23 showed amplitudes of less than 8 ppm and S/N <= 4, so they have been excluded.
Normally, a binary system with pulsating components is modeled in order to subtract the light curve solution and after carrying out a period analysis of the the residual data. In our case, the first step has not been performed to avoid adding non-real frequencies to the data if the binary fitting is not correct. Therefore, the frequency corresponding to the orbital period of the binary system, 5.701986~\cd, besides their numerous harmonics have been removed from our frequency list. As in \cite{2013A&A...559A..63G}, also we investigated harmonics and combinations between the main peaks using a range of $\pm0.005$~\cd\ as well as those peaks related with the satellite orbital frequency $\mathrm{f_s}=13.972$~\cd.
The frequencies used in this study are listed in Table XX together with the most relevant parameters: amplitude, phase, sig, S/N, residuals and errorbars. The signal-to-noise ratio corresponding to each peak is calculated using the residual file provided by SigSpec. Each S/N value was calculated on a box of width 5~\cd\ centered in the corresponding peak, as usual for this parameter \cite{1993A&A...271..482B}. The values of rms for frequency "{\it i}" correspond to the sigma of the residuals before extracting the peak "{\it i}". Errors for the frequencies, amplitudes and phases are estimated using \cite{1999DSSN...13...28M} approximation. Frequencies are in \cd, amplitudes in ppm and phases in radians.
{\bf FROM HERE, OUT OF THE PAPER}\par
But the consequences of this result extend further. The fundamental radial mode is also directly related to the mean density of the star, $\Pi_{0}\simeq\left(G\bar{\rho}\right)^{-1/2}$ \cite{Shapley_1914}, where $\Pi_{0}$ is the period of the fundamental radial mode, G is the gravitational constant and $\rho$ the mean density of the star. Thus it is straightforward to compute the value of this frequency just from the large separation\selectlanguage{ngerman} {\bf [CALCULAR EL MODO FUNDAMENTAL RADIAL PARA LA MUESTRA Y COMPARAR CON EL LÍMITE DE MODOS G EN AQUELLOS CASOS DADOS POR LOS AUTORES]}. The position of the fundamental radial mode then gives an estimate of the limit between the two pulsating regimes, namely the $p$ mode regime, for which the restoring force is pressure, and the $g$ mode region, for which the restoring force is buoyancy. It is true that avoided crossings due to evolved stages in the evolution as well as rotation (REF), may modify this limit but the value of the fundamental radial mode will always remain directly related to the mean density of the star. Being able to distinguish both regimes allows us to study different depths of the internal stellar structure, since the $g$ modes propagate deeper and the $p$ modes concentrate mainly near the surface. Hence, hybrid \ds-\gdor\ stars, which show both kinds of pulsations (see, e.g., \cite{2010ApJ...713L.192G}), can be used to obtain critical information on the whole interior of the star. {\bf COMENTAR CASO DE \cite{Van_Reeth_2015} }
The \Dnu-$\rho$ relation also has an impact on the period-luminosity-colour (PLC) relation too. If the large separation is used to compute the latter relation, it would solve various discrepancies found in the observations of \dss. Those discrepancies were mainly due to the fact that an identification of the modes (namely, the identification of the fundamental radial mode) is not feasible for these pulsators. Some studies during the last decade showed that a well constrained period-luminosity relation could indeed be established between the fundamental radial mode and the luminosity of \ds\ stars in the main sequence \cite{2002ApJ...576..963T}. Recently, based on non-rotating models, \cite{2013MNRAS.429.1466B} derived different PLC relations taking into account several radial modes. Using our small sample of stars, we fitted an analogous relation using \Dnu\ instead of a certain pulsating mode:
\begin{equation}
\log{\Delta\nu}= (-9.5\pm6.7) + (2.5\pm1.8)\log{\mathrm{T_{eff}}} - (0.82\pm0.32)\log{\mathrm{L/L_{\odot}}}.
\end{equation}
This relation allows us to estimate the distance to the object.
\selectlanguage{english}
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