\section{Introduction}
\label{sec:intro}

Stellar physics is undergoing a revolution thanks to the wealth of asteroseismic data that have been made available by space missions such as CoRoT \citep{Baglin2006} and \textit{Kepler} \citep{Gilliland2010}.
With the seismic analyses of these stars providing the frequencies of the stellar eigenmodes, asteroseismology is rapidly becoming a tool for understanding stellar physics.  
 
Solar-type stars have been 
observed over periods exceeding
six months using CoRoT and \textit{Kepler} providing many lists of mode frequencies required for seismic analysis \citep[See][and references therein]{Appourchaux2012a}.  Additional invaluable information about the evolution of stars is provided by the study of the internal structure of red giants  \citep{Bedding2011a, Beck2011, Beck2012, Mosser2012,Mosser2012a} and of sub giants  \citep{Deheuvels2012, Benomar2013}.  The large asteroseismic database of \textit{Kepler} allowed us to estimate the properties of an ensemble of solar-type stars that is large enough to perform statistical studies \citep{Chaplin2013a}.  

Solar-like oscillations are stochastically excited and damped by the convection.  Thus measurements of mode linewidths and mode heights provide information about how the stellar modes are excited and damped.  The processes involved are related to the generation of the acoustic noise and the dissipation of energy at the surface of the star \citep[See][]{GH99,Samadi2011}.  


For solar-like stars, a scaling relation for mode linewidth related to the stellar effective temperature has been proposed by \citet{Chaplin2009} using ground-based observations, \citet{Baudin2011} using CoRoT data and Appourchaux et al. (2012a) using \textit{Kepler} data. Those relations are based upon the linewidth measured at the frequency of maximum mode height and have been found by \citet{Belkacem2012} to be in qualitatively good agreement with the theoretical predictions. The relation was extended to lower effective temperature for red giants \citep{Corsaro2012}.


These previous scaling studies do not provide the frequency dependence of the linewidth.  Using a simple modelling approach, \citet{Gough1980} suggested that solar linewidths might have a local decrease, or dip, at the frequency of maximum power.  With more accurate and detailed modelling, \citet{Balmforth1992} showed that there is indeed such a linewidth dip for the Sun.  \citet{GH99} found that stellar mode linewidths show either a dip or {\it plateau} close to the maximum of mode height.  The plateau is located at the frequency of the maximum of the mode height as shown by \citet{Belkacem2011}, which is also
related to the Mach number (${\cal M}_a$), the ratio of convective velocity to the sound speed.  This dip was first observed but not acknowledged in the solar p-mode linewidths by \citet{Libbrecht1988}, while a small dip or plateau was indeed observed by \citet{Chaplin1997}.    The dip is caused by a resonance between the thermal adjustment time of the superadiabatic boundary layer and the mode frequency \citep{Balmforth1992}.  Since the thermal adjustment time is proportional to the acoustic cut-off frequency $\nu_{c}$, the two frequencies follow a scaling relation as shown by \citet{Belkacem2011}.   \citet{Claus1997} observed a very pronounced dip during solar minimum, hypothesizing that the depth of the dip may be modulated by solar activity,
as confirmed later by \citet{Komm2000}.  These variations of the solar mode damping with solar activity were thought to be related to the change of the solar granule properties with the increasing magnetic field \citep{Houdek2001, Muller2007}, but these changes were not confirmed using space-based data \citep{Muller2011}.  The variations are likely to be affected by the change in the global magnetic field during a solar cycle.  Very recently, \citet{Benomar2013} studied the frequency dependence of mode linewidth of 4 sub-giant stars having mixed modes.  They found that the linewidth of $l=0$ modes showed a clear dip at the location of the maximum of mode power.


 It was shown by \citet{Appourchaux2012} that different fits of the same data could provide significantly different results for stellar linewidths.  Understanding the source of systematic errors  will result in a better understanding of how physics operate in stars.  \citet{Appourchaux2012} provided some insight on the various sources of systematic errors  related to stellar background estimation and the mode height ratio.   \citet{Chaplin2008a} showed that biased linewidths are also obtained when measuring mode linewidth of the order of 1 to 7 times larger than the frequency resolution.   Apart from these two papers, the understanding of the origin of systematic errors on  mode linewidth and height has not been widely studied.










This paper aims at providing the frequency dependence of mode linewidth $\Gamma$, mode height $H$ and mode amplitude $A$ ($A=\sqrt{\pi H \Gamma /2}$) for 23 \textit{Kepler} main-sequence observed for nearly 2 years by \textit{Kepler}, as well as an understanding of the source of systematic errors  affecting these parameters.  The paper also aims at providing the dependence of the linewidth dip as a function of effective temperature.

Section 2 describes how the time series and power spectra were obtained.  Section 3 describes the peak fitting procedure.  Section 4 details the sources of systematic errors on the mode linewidth and mode heights provided by the fitters.  Section 5 provides a procedure for correcting the systematic errors with respect to a reference fit.    We then discuss the detection of the dip as a function of effective temperature and the implications for stellar physics. The paper includes two examples of mode linewidth and mode height and an example of systematic error correction, while tables of the parameters of the 23 stars and correction for 22 stars are available online.  


 
   





\section{Time series and power spectra}
\textit{Kepler} observations are obtained in two different operating modes: long cadence (LC) and short cadence (SC) \citep{Gilliland2010,Jenkins2010}.  This work is based on SC data. For the brightest stars (down to \textit{Kepler} magnitude $Kp \approx12$), SC observations can be obtained for a limited number of stars  (up to 512 at any given time)  with a faster sampling cadence of 58.84876~s (Nyquist frequency of $\sim$ 8.5~mHz), which permits a more precise exoplanet transit timing and improves the performance of asteroseismology.  \textit{Kepler} observations are divided into three-month-long {\it quarters} (Q).  A subset of 23 stars from the 61 stars analyzed by \citet{Appourchaux2012a} has been used in the present analysis.  The subset of stars, observed during quarters Q5 to Q12 (March  22, 2010 to March 22, 2012), were chosen because they have oscillation modes spanning more than 10 radial orders. Therefore, the longest length of data gives a frequency resolution of about 16~nHz.  The stars used in this study are listed in Table~\ref{table1}.

To maximise the signal-to-noise ratio for asteroseismology, the time series were corrected for outliers, occasional jumps, and drifts \citep[see][]{RAG2011}, and the mean level for each quarter was normalised.  Finally, the resulting light curves were high-pass filtered using a triangular smoothing with full-width-at-half-maximum of one day, to minimise the effects of the long-period instrumental drifts.   The amount of data missing from the time series ranges from 3\% to 7\%, depending on the star.  All of the power spectra were produced by one of the co-authors using the Lomb-Scargle periodogram \citep{Scargle82}, properly calibrated to comply with
Parseval's theorem \citep[see][]{Appourchaux2011}.  All power spectra are single sided.








 



   

\section{Mode parameter extraction}
\subsection{Power spectrum model}


The mode-parameter extraction was performed by eight teams of fitters whose leaders are listed in Table~\ref{methods}.  The power spectra were modelled over a frequency range typically covering 10 to 20 large separations ($\Delta\nu$) between successive radial orders.  The stellar background was modelled using a multi-component Harvey model \citep{Harvey85}, each component with up to three parameters,  and a white noise component.  The Harvey model is a modified Lorentzian profile with a different exponent.  The number of components for the stellar background is given in Table~\ref{methods}.  The stellar background was fitted prior to the extraction of the mode parameters and then held at a fixed value. For each radial order, the model parameters were mode frequencies (one for each degree, $l=0,1,2$), a single mode height (with assumed ratios between degrees given in Table~\ref{methods2}), and a single mode linewidth for all degrees; a total of 5 parameters per order.  Since we have no stars with mixed modes, we choose to set a common mode linewidth and mode height to reduce the number of fitted parameters.  Other choices could be implemented depending on hypotheses.  The relative heights $h_{(l,m)}$ (where $m$ is the azimuthal order) of the rotationally split components of the modes depend on the stellar inclination angle, as given by \citet{Gizon2003}.  For 
each star, the rotational splitting and stellar inclination angle were chosen to be common for all of the 
modes; it adds 2 additional free parameters.
The mode profile was assumed to be Lorentzian.  In total, the number of free parameters for 15 orders was at least $5 \times 15+2=77$.  The potential misidentification of the pair $l=0-2$ for the $l=1-3$ pair as in by \citet{Appourchaux2008} was avoided by having the fitters using the same correct identification.

The model described above was used to fit the parameters of the 23 stars using maximum likelihood estimators (MLE) and with a Bayesian approach.  For the MLE, formal uncertainties in each parameter were derived from the inverse of the Hessian matrix \citep[for more details on MLE, significance, and formal errors, see][]{Appourchaux2011}.  For the Bayesian approach, the uncertainties (or credible intervals) were derived from the marginal posterior distribution of each parameter \citep[for more details on credible intervals, see][]{Benomar2009,Handberg2011}. 




Tables~\ref{methods} and~\ref{methods2} provides a summary of the model used by each fitter team.


\subsection{Guess parameter and fitting procedures}
The procedure for the initial guess of the parameters is described in \citet{Appourchaux2012a}, in which the steps of the fitting procedure are also described. These steps are repeated here for completeness:
\begin{enumerate}
\item We fit the power spectrum as the sum of a stellar background made up of a combination of modified Lorentzian profiles (one or two) and white noise, as well as a Gaussian oscillation mode envelope with three parameters (the frequency of the maximum mode power, the maximum power, and the width of the mode power).  Some fitters chose to exclude the p-mode region and fit the stellar background alone.
\item We fit the power spectrum with $n$ orders using the mode profile model described above, with no splitting and the stellar background fixed as determined in step 1
\item We follow step 2 but leave the rotational splitting and the stellar inclination angle as free parameters, and then apply a likelihood ratio test to assess the significance of the fitted splitting and inclination angle.  For the Bayesian fit, step 2 {\ is by-passed then step 3 is directly} applied; {\ this is because} credible intervals are always provided for any parameter, hence the significance test is only applied by MLE fitters.
\end{enumerate}
The steps above were sometimes varied slightly depending on the assumptions that were made.  For instance, the mode height ratio could instead be defined as a free parameter to study the impact of its variations on the derivation of the mode linewidth and mode height.   


\section{Sources of mode-linewidth and mode-height systematic errors and their correction}
When comparing the results that we obtained, it was clear that there are large differences in the mode linewidth and mode height measured by the different fitters.  Figure~\ref{fig_3733735} provides a typical example of large discrepancies between the fitters.
The different sources of bias leading to the systematic errors observed are listed in Appendix A.  The understanding of the different sources of bias provides a way to correct the systematic errors with respect to a {\it reference} fit.  In short, the main sources of systematic error in the measurement of the mode linewidth and mode height are: 1) splitting bias and 2) stellar background bias.  There are additional sources of systematic errors such as a different mode-height ratio and a different mean frequency which have been solved by assuming that the fitters use the same mode height ratio and mean frequency (see Appendix A for a full description of the various contributors).  

In order to check that all of the sources of systematic errors encountered when comparing the different results can be understood and compensated for, we have implemented a correction scheme based upon the one-fit correction of \citet{TT2005b}.  The procedure that they devised is based on fitting a spectrum model with no realization noise (i.e. the limit spectrum) with a different spectrum model.  The assumption behind that procedure is that the systematic errors are the same with the one-fit approach compared to a full Monte-Carlo simulation of the systematic errors resulting from using two different spectrum models.  The correction scheme was applied for comparing with the results of the reference fit, which were provided by Appourchaux.  The reference fit is simply used as a basis for understanding the source of the systematic errors.  The reference fit value is in no way the {\it best} set but simply a data set that can provide a basis for understanding the impact of using a different fitting model due to different theoretical or observational constraints.  The correction scheme is detailed in Appendix A. 




\begin{figure*}[htbp]
\centering
\includegraphics[width=10.75 cm,angle=90]{bias_comparison_3733735_new_nc.ps}
\caption{Mode linewidth, mode height and mode amplitude (Top) and relative values of these parameters with respect to the reference fit (Bottom) as a function of mode frequency for various fitters for KIC 3733735.  The grey band indicates the range of systematic error around the reference fit values of $\pm$ 15\% for mode linewidth and mode height, of $\pm$ 5\% for mode amplitude. The 1-$\sigma$ error bars are those of Appourchaux.}
\label{fig_3733735}
\end{figure*}


\begin{figure*}[htbp]
\centering
\includegraphics[width=10.75 cm,angle=90]{bias_comparison_3733735_new.ps}
\caption{Corrected mode linewidth, mode height and mode amplitude (Top) and relative values of these parameters with respect to the reference fit (Bottom) as a function of mode frequency for KIC 3733735. The grey band indicates the range of systematic error around the reference fit values of $\pm$ 15\% for mode linewidth and mode height, of $\pm$ 5\% for mode amplitude.  The error bars are those of the reference fit.}
\label{bias_comparison_3733735_new}
\end{figure*}









The correction has been implemented for all fitters who provided the information on the stellar background, the splitting and inclination angle and compared to a reference fit.  Figure~\ref{bias_comparison_3733735_new} shows the correction results for the largest discrepancies already shown in Fig.~\ref{fig_3733735}.  Figures B.1 to B.22 of the online material provide details of how the correction operates for all other stars on the mode linewidth and mode height.  For 17 out of 23 stars, the correction scheme provides mode linewidths, mode heights and mode amplitudes agreeing with the reference fit within $\pm$15\% for the first two parameters, and within $\pm$5\% for the latter.  As we showed above, the systematic error on the mode amplitude is bound to be smaller than for the mode linewidth or mode height alone.  The correction is really effective especially when there are large discrepancies (Antia, R\'egulo) with respect to the reference fit for which the different stellar background is the major source, with in addition a different splitting in some cases.  The remaining discrepancies (not larger than $\pm$10\%) mainly occur for the amplitude, especially when the mode height ratio used in the correction is far from the one used by the fitter.  This is the case for KIC1435467, KIC3733735, KIC3424541 and KIC12258514 for some fitters (Benomar, Davies, R\'egulo) for which even the modified ratios of 1.0 / 2.0 / 1.0 / 0.0 were underestimating their fitted mode height ratio.  Similar discrepancies (not larger than +30\%) occur for mode linewidth when using a far-from-nominal mode height ratio for KIC2837475 and KIC7103006 (R\'egulo).  In the latter case, the nominal mode height ratios of 1.0 / 1.5 / 0.5 / 0.05 overestimate the fitted mode height ratio.  For Antia's results, there are also large discrepancies for 13 stars that are traced back to a mode height ratio varying largely from mode to mode and to the use of flat stellar background.  In that latter case, neither a varying mode height ratio nor a flat stellar background are realistic assumptions regarding the fitting model.

Given the success encountered with the {\it simple} correction scheme, we chose not to make the correction model more detailed.  Therefore, we assume that the reference fit values given in this paper are corrected for known systematic errors.  





\section{Results and discussion}
Tables 4 to 26 provide the mode linewidth, mode height and mode amplitude provided by the reference fit for all 23 stars of this study and the associated error bars.  Figure~\ref{modeh} shows the dependence of the mode linewidth for the 23 stars of this study together with the $l=0$ mode linewidths of 4 sub-giant stars of \citet{Benomar2013} and that of the Sun measured by the Luminosity Oscillations Imager (LOI) of VIRGO \citep{CFJR95,TABA97}.  We computed the mean power spectra of 16 years of full-disk time series of the LOI; the data were then analysed in a similar manner as for the 23 stars but using a dual modified Harvey profile for the background.   For the data of \citet{Benomar2013}, we divided the published amplitude by $\sqrt{2}$ because the values given in their paper  were incorrectly quoted as $A=\sqrt{\pi \Gamma H}$ (for a double-sided power spectrum), instead of $A=\sqrt{\pi \Gamma H/2}$ (for a single-sided power spectrum as here); the mode height is unaffected by the correction since the former fit was performed on a single-sided spectrum.  The figures are centred with respect to the frequency of the maximum of mode height.  Figure~\ref{modeh} shows clearly that the linewidth dip disappears as the effective temperature increases.  Figure~\ref{modeh1} shows also that the maximum mode height and amplitude decrease with the effective temperature, as already shown by \citet{HK1995}.  It is also interesting to note that the location of maximum mode height also corresponds to the location of the linewidth dip as anticipated by \citet{Belkacem2011}, a location which does not coincide with the frequency of maximum amplitude, which is about half a large separation higher than the frequency of maximum mode height.


\begin{figure*}[htbp]
\centering
\includegraphics[width=9 cm,angle=90]{linewidth_com_sun.ps}
\caption{Mode linewidth as a function of the mode order for the 23 stars with their colour identified by their effective temperature, for the $l=0$ mode linewidths of 4 sub-giant stars of \citet{Benomar2013} (dashed lines) and for the solar LOI data (solid black line).  For the 23 stars of this study the relative 1-$\sigma$ error bars range from 5\% to 10\%, while those of \citet{Benomar2013} range from 10\% to 20\%.  The 10\% and 20\% error bars are shown as an example.  The longer LOI data set has relative error bars of about 2\%.} 
\label{modeh}
\end{figure*}

\begin{figure*}[htbp]
\centering
\hbox{
\hspace{+1.2 cm}
\includegraphics[width=5.5 cm,angle=90]{modeh_com_sun.ps}
\hspace{+1.2 cm}
\includegraphics[width=5.5 cm,angle=90]{amplitude_com_sun.ps}
}
\caption{Mode height and amplitude as a function of the mode order for the 23 stars with their color identified by their effective temperature (as in Fig. \ref{modeh}), for the 4 sub-giant stars of \citet{Benomar2013} (dashed lines) and for the solar LOI data (solid black line).  The relative 1-$\sigma$ error bars for mode height are similar to those of the mode linewidth, while they are typically 3 times smaller for mode amplitude.}
\label{modeh1}
\end{figure*}




In order to understand the dependence of the linewidth dip on effective temperature, we modelled the frequency dependence of the mode linewidth $\Gamma$ as a combination of the power law dependence and the dip modelled as a Lorentzian profile in $\ln \nu$:
\begin{equation}
\ln(\Gamma)=(\alpha \ln(\nu/\nu_{\max})+\ln \Gamma_{\alpha})+\left(\frac{\ln \Delta \Gamma_{\rm dip}}{1+\left(\frac{2\ln(\nu/\nu_{\rm dip})}{\ln (W_{\rm dip}/\nu_{\rm max})}\right)^2}\right)
\label{Eq_fit}
\end{equation}
where $\nu$ is the mode frequency, $\nu_{\rm max}$ is the frequency of maximum mode height, $\alpha$ is the power law index, $\Gamma_{\alpha}$ is the factor of the power law, $\Delta \Gamma_{\rm dip}$ is the depth of the dip, $W_{\rm dip}$ is the width of the dip and $\nu_{\rm dip}$ is the frequency of the dip.  Figure~\ref{fit} shows an example of a fitted linewidth using Eq.~(\ref{Eq_fit}), clearly showing a dip.  Figure~\ref{dip} shows the result of the fit of the linewidth for all the stars of Fig. \ref{modeh} and their values are given in Table~\ref{table_last}.  The left-hand panels of Fig.~\ref{dip} show the fit of the power-law only, for all 28 stars.   We checked whether the additional parameters of the Lorentzian profile of Eq.~(\ref{Eq_fit}) were significant by examining the decrease of the reduced $\chi^2$ with respect to a $\chi^2$ law with 3 degrees of freedom; the probability cut was $\frac{1}{23}$ ensuring that on average only one fit would be due to noise.  In total there were 11 out of the 23 stars for which the 5-parameter fit was significant.  In order to check the gaussianity of the probability distribution of the parameters, we used Monte-Carlo simulations of the fit for returning the median value and its associated credible intervals.  The non-gaussianity of the distribution occurs only for two stars but the Monte-Carlo scheme is applied to all stars for coherence.  The right-hand panels of Fig.~\ref{dip} show the Lorentzian fit of the dip for 15 stars for which the dip was significant.

Fig.~\ref{dip1} shows the result of the fit as a function of the frequency of maximum power, $\nu_{\rm max}$, which is approximately proportional to $g/\sqrt{T_{\rm eff}}$  \citep{Brown1991, Belkacem2011}, where $g$ is the surface gravity and $T_{\rm eff}$ is the effective temperature .  

\citet{Chaplin1997} found that the solar mode linewidth follows a power law of 7$\pm 1.5$, at frequencies below 1800 $\mu$Hz in agreement with the theoretical result of \citet{Balmforth1992} and \citet{PG1994a}.  \citet{Komm2000} found a power law index of about 3.3 for mode frequency excluding the linewidth dip extending from 2400 $\mu$Hz to 3750 $\mu$Hz for the Sun.  In our case, the solar power law index is closer to that of \citet{Komm2000} because we made a global fit over a larger frequency range than that of the fit performed by \citet{Chaplin1997}.  

The width of the solar dip as measured by \citet{Komm2000} is about 1350 $\mu$Hz wide (full width) which is about twice as large as the full width at half maximum that we found for the Sun.  The right-hand panels of Fig.~\ref{dip} clearly shows that the amplitude and the width of the dip decrease with the effective temperature.  On the left top panel of Fig.~\ref{dip}, the dip also manifests itself as a deviation with respect to the linewidth measured at the maximum mode height (the orange line); the deviation becomes very small at high effective temperature.  

It was shown by \citet{Komm2000} that the solar linewidth dip is reduced for increasing solar activity.  The reduction was suggested to be due to the fact that radiative processes occurring in the upper superadiabatic boundary layer of the
convection zone that are locally destabilizing, would be in turn less unstable because of an increasing magnetic field \citep{Komm2000}.  But for other stars what is the source of the vanishing linewidth dip?  Using data from a high-resolution spectrograph, \citet{Karoff2013} measured the flux of the chromospheric lines Ca II H\&K of 11 stars in our study having a level of activity similar to that of the Sun.  All of these stars fall in the category IV (low activity stars) of \citet{Vaughan1980}.  KIC3733735 and KIC8379927 show a very high level of activity compared to the others of \citet{Karoff2013}.  KIC3733735 has the highest effective temperature of our sample, and shows a dip that is at the limit of detection with very large error bars (see Fig.~\ref{bias_comparison_3733735_new} and Fig.~\ref{dip}).  On the other hand KIC8379927 shows a measurable dip that might imply that this star is at minimum activity (see Fig.~\ref{8379927}).  Therefore the reason for the absence of a dip in some active stars could be due to the stars being at their maximum activity.  This can only be confirmed by measuring the activity of these stars over longer period of time.

\citet{Balmforth1992} showed that the linewidth dip would disappear with an increasing mixing length parameter ($\alpha$).  Numerical simulations performed by \citet{Ludwig1999}, \citet{Freytag1999} and \citet{Trampedach2011} showed that the mixing length parameter decreases with effective temperature, and therefore a smaller mixing length parameter at high effective temperature would increase the depth of  the linewidth dip contrary to what is observed in Fig.~\ref{dip}.  Results based on stellar modelling by \citet{Pinheiro2013} show an opposite dependence to that of the numerical simulations, e.g. the mixing length parameter increases with the effective temperature, being thus consistent with the findings of Fig.~\ref{dip}.  

\citet{GH96} showed that the linewidth dip becomes more pronounced with decreasing surface density if the mixing-length parameter and anisotropy of the turbulent velocity field are kept constant in the model computations.  For the stars studied in this paper, the surface density variations are dominated by changes in the effective temperature, i.e. the surface density decreases with increasing effective temperature provided the surface gravity stays approximately constant.  For that case the linewidth dip would increase with increasing surface temperature, according to \citet{GH96}, which is the opposite to what we found in Fig.~\ref{dip}.  However, the properties of the linewidth dip also depend crucially on the anisotropy of the turbulent velocity field which calibration may lead to better agreements between model computations and observations.

These contradictory findings may have important impact for the modelling of convection and turbulence in stars.
\begin{figure}[htbp]
\centering
\includegraphics[width=6.0 cm,angle=90]{fit_6116048.ps}
\caption{Mode linewidth as a function of frequency for KIC 6116048 (solid line) with its 1-$\sigma$ error bars compared with the fit following Eq.~(\ref{Eq_fit}) (Orange line)}
\label{fit}
\end{figure}




\section{Conclusions}
We have analysed the oscillation power spectra of 23 main-sequence stars for which we obtained the mode linewidths, mode heights and mode amplitude parameters.  The parameters were obtained by a team of 8 independent fitters.  We found large systematic errors between the parameters that could be traced to the way that the stellar background of the power spectrum was modelled; and to the fitted values of the rotational splitting and the stellar inclination angle.  Other sources of systematic errors related to the mean frequency definition and to the mode height ratio were also studied.  Finally using  a correction scheme derived from the one-fit approach of \citet{TT2005b}, we could explain all sources of systematic errors, which could be reduced to less than $\pm$15\% for mode linewidth and mode height, and to less than $\pm$5\% for amplitude, when compared to a reference fit value.  A different stellar background will give rise to frequency-dependent systematic errors that might affect the comparison with theoretical mode linewidth and mode height, therefore affecting the understanding of the physical nature of these parameters.  All other sources of relative systematic errors are independent of frequency.

Using the 23 stars of this study, 4 additional sub-giant stars of \citet{Benomar2013} and solar data, we also derived that the amplitude of the linewidth dip close to the maximum of frequency decreases with effective temperature.  The dependence of the dip with effective temperature is linked to the behaviour of convection in the stellar atmosphere, implying that either the mixing length or the level of activity may increase with effective temperature.








\begin{figure*}[htbp]
\centering
\hbox{
\includegraphics[width=7 cm,angle=90]{linewidth_fit_1.ps}
\includegraphics[width=7 cm,angle=90]{linewidth_fit_2.ps}
}
\caption{Parameters of Eq.~(\ref{Eq_fit}) as a function of the effective temperature for all 28 stars, for the power law dependence (Left) and for the Lorentzian fit (Right).  The median value together with the credible intervals at 33\% and 66\% were derived from a Monte-Carlo simulation of the fit.  
The orange line shows the temperature dependence of the linewidth at the frequency of maximum mode height as derived by \citet{Appourchaux2012}.  The open diamond is the result of the fit for the solar data of the LOI.  The open triangles are the results of the fit for the sub-giant stars of \citet{Benomar2013}.  The solid lines show a linear fit of the parameters with respect to the effective temperature.  The Lorentzian parameters for stars for which the Lorentzian fit is not significant are not plotted.}
\label{dip}
\end{figure*}



\begin{figure*}[htbp]
\centering
\hbox{
\includegraphics[width=7 cm,angle=90]{linewidth_fit_1yes.ps}
\includegraphics[width=7 cm,angle=90]{linewidth_fit_2yes.ps}
}
\caption{Parameters of Eq.~(\ref{Eq_fit}) as a function of the frequency of maximum mode height for all 28 stars, for the power law dependence (Left) and for the Lorentzian fit (Right).  The median value together with the credible intervals of 33\% and 66\% were derived from a Monte-Carlo simulation of the fit.  The open diamond is the result of the fit for the solar LOI data.  The open triangles are the result of the fit for the sub-giant stars of \citet{Benomar2013}.   The error bars are derived from a Monte-Carlo simulation using credible intervals of 33\% and 66\%.  The solid lines show a linear fit of the parameters with respect to the frequency.  The Lorentzian parameters for stars for which the Lorentzian fit is not significant are not plotted.}
\label{dip1}
\end{figure*}









\begin{acknowledgements}
We are grateful to the referee for their comments which considerably improved the structure and readability to the paper.  The authors wish to thank the entire \textit{Kepler} team, without whom
these results would not have been possible. Funding for this Discovery mission is provided by NASA's
Science Mission Directorate.  We also thank all funding
councils and agencies that have supported the activities
of KASC Working Group~1, as well as the International Space Science
Institute (ISSI).  TA gratefully acknowledges the financial support of the Centre National d'Etudes Spatiales (CNES) under a PLATO grant.  TLC, WJC and GRD acknowledge financial support from the UK Science and Technology Facilities Council (STFC).  Funding for the Stellar Astrophysics Centre is provided by The Danish National Research Foundation. The research is supported by the ASTERISK project (ASTERoseismic Investigations with SONG and \textit{Kepler}) funded by the European Research Council (Grant agreement no.: 267864).  RAG and GRD has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 269194.  RH acknowledges computing support from the National Solar Observatory.  SOHO is a mission of international collaboration between ESA and NASA.  We thank J\'er\^ome Ballot for useful discussions.
\end{acknowledgements}

\bibliographystyle{aa}
























