Issue 
A&A
Volume 537, January 2012



Article Number  A9  
Number of page(s)  8  
Section  Stellar structure and evolution  
DOI  https://doi.org/10.1051/00046361/201117158  
Published online  20 December 2011 
Solarlike oscillations in the G9.5 subgiant β Aquilae^{⋆}
^{1} Department of Physics and Astronomy, Astrophysics Section, University of Catania, via S. Sofia 78, 95123 Catania, Italy
email: eco@oact.inaf.it
^{2} INAF – Astrophysical Observatory of Catania, via S. Sofia 78, 95123 Catania, Italy
^{3} Department of Physics and Astronomy, Aarhus University, 8000 Aarhus C, Denmark
^{4} INAF – Astronomical Observatory of Capodimonte, Salita Moiariello 16, 80131 Napoli, Italy
Received: 29 April 2011
Accepted: 4 October 2011
Context. An interesting asteroseismic target is the G9.5 IV solarlike star β Aql. This is an ideal target for asteroseismic investigations, because precise astrometric measurements are available from Hipparcos that greatly help in constraining the theoretical interpretation of the results. The star was observed during six nights in August 2009 by means of the highresolution échelle spectrograph SARG operating with the TNG 3.58 m Italian telescope (Telescopio Nazionale Galileo) on the Canary Islands, exploiting the iodine cell technique.
Aims. We present the result and the detailed analysis of highprecision radial velocity measurements, where the possibility of detecting individual pmode frequencies for the first time and deriving their corresponding asymptotic values will be discussed.
Methods. The Fourier analysis technique based on radial velocity time series and the fitting of asymptotic relation to the power spectrum were the main tools used for the detection of the asteroseismic parameters of the star.
Results. The timeseries analysis carried out from ~800 collected spectra shows the typical pmode frequency pattern with a maximum centered at 416 μHz. In the frequency range 300–600 μHz we identified for the first time six high signaltonoise ratio (S/N ≳ 3.5) modes with ℓ = 0,2 and 11 < n < 16 and three possible candidates for mixed modes (ℓ = 1), although the pmode identification for this type of star appears to be quite difficult owing to a substantial presence of avoided crossings. The large frequency separation and the surface term from the set of identified modes by means of the asymptotic relation were derived for the first time. Their values are Δν = 29.56 ± 0.10 μHz and ϵ = 1.29 ± 0.04, consistent with expectations. The most likely value for the small separation is δν_{02} = 2.55 ± 0.71 μHz.
Key words: stars: individual: βAquilae / stars: latetype / stars: oscillations / techniques: radial velocities
© ESO, 2012
1. Introduction
The search for solarlike oscillations (see ChristensenDalsgaard 2004; Elsworth & Thompson 2004, for a summary) in mainsequence and subgiant stars showed a tremendous growth in the last decade (for reviews see e.g. Bedding & Kjeldsen 2003, 2006), especially by means of the photometric spacebased missions CoRoT (Baglin et al. 2006; Michel et al. 2008) and Kepler (Borucki et al. 2010; Koch et al. 2010). The latter in particular is presently providing an enormous amount of highquality asteroseismic data (e.g. see Gilliland et al. 2010, for more details on the KAI – Kepler Asteroseismic Investigation). Photometric studies of a large number of solartype stars are fundamental for statistical investigations of intrinsic stellar properties and for testing theories of stellar evolution (e.g. Chaplin et al. 2011). However, as is known from the theory of solarlike oscillations, highprecision Doppler shift measurements are more effective for detecting p modes of higher angular degrees. At the present time the échelle spectrometers such as CORALIE, HARPS, UCLES, UVES, SOPHIE and SARG, attaining highprecision radial velocity (RV) measurements (Marcy & Butler 1992), offer a way to detect solarlike oscillations in bright asteroseismic targets (e.g. see Bedding & Kjeldsen 2008, for a review on groundbased campaigns). The spectroscopic approach to the detection of solarlike oscillations will also be used in the groundbased SONG project (Grundahl et al. 2009) in the near future.
In this paper we report the detection of excess of power and perform a detailed analysis based on the radial velocity measurements for the evolved subgiant star β Aql (HR 7602, HD 188512, HIP 98036). This star has (UBV photometric measurements from Oja 1986), spectral type G9.5 IV (Gray et al. 2006), distance 13.70 ± 0.04 pc derived from the Hipparcos parallax π = 73.00 ± 0.20 mas (van Leeuwen 2007), T_{eff} = 5160 ± 100 K (Luck & Heiter 2005), log g = 3.79 ± 0.06 (Valenti & Fischer 2005) and [Fe/H] = −0.17 ± 0.07 (Fuhrmann 2004). The radius R = 3.05 ± 0.13 R_{⊙} was measured by means of Long Baseline Interferometry (Nordgren et al. 1999) that is listed in the CHARM2 catalogue for high angular resolution measurements (Richichi et al. 2005). An excess of power in the power spectrum (PS) of β Aql data acquired with HARPS was already found by Kjeldsen et al. (2008), who estimated a mean large separation Δν = 30 μHz from stellar parameters. In Sect. 2 we describe the observations, the data reduction and the RV extraction developed for this star. The Fourier analysis and the mode characterization that led us to derive the asteroseismic parameters are presented in Sect. 3. In Sect. 4 we discuss the scaled mass, mode amplitude, and frequency of maximum power of the star, where a comparison with expectations and a global list of stellar parameters also derived by means of the SEEK package (Quirion et al. 2010).
2. Observation and data reduction
The data were acquired with observations carried out during six nights (2009 August 5–11) by means of the highresolution crossdispersed échelle spectrograph SARG (Gratton et al. 2001; Claudi et al. 2009) mounted on the 3.58 m Italian telescope TNG at the La Palma observing site. SARG operates in both singleobject and longslit (up to 26′′) observing modes and covers a spectral wavelength range from 370 nm up to about 1000 nm, with a resolution ranging from R = 29 000 up to R = 164 000. Our spectra were obtained at R = 164 000 in the wavelength range 462–792 nm. The calibration iodine cell works only in the blue part of the spectrum (500–620 nm), which was used for measuring Doppler shifts. During the observing period we collected spectra with a signaltonoise ratio (S/N) varying from 150 to 300, a typical exposure of ~150 s and a sampling time of ~190 s. A total of 828 spectra were collected with the following distribution over the six nights: 91, 133, 110, 146, 163, 185. The spectra were then reduced and calibrated in wavelength with a ThAr lamp, using standard tasks of the IRAF package facilities. No flatfielding was applied to the spectra because of a degradation of the S/N level.
2.1. Radial velocity measurements
The RV measurements were determined by means of the iSONG code, developed for the SONG project (Grundahl et al. 2009). iSONG models the instrumental profile (PSF), stellar, and iodine cell spectra to measure Doppler shifts. The observed spectrum was fitted with a reconstructed one by using a convolution between the oversampled stellar template, the very highresolution iodine cell spectrum, and the measured spectrograph instrumental profile. Essential to this process are the template spectra of β Aql taken with the iodine cell removed from the beam, and the iodine cell itself superimposed on a rapidly rotating Btype star, the same for all the measurements. The Btype star spectra allow us to construct the instrumental profile, while the β Aql spectra acquired without the iodine cell create the original stellar template when deconvolved with the PSF (see Butler et al. 1996, for a detailed explanation of the method). The velocities were corrected to the solar system barycenter (Hrudkov et al. 2006) and no other corrections, such as decorrelation or filtering by removing polynomial fits to the timeseries, were applied.
Fig. 1 Upper panel: cumulative histograms of r_{i}/σ_{i} for SARG data. The diamonds show the observed data, and the solid curve shows the result expected for a Gaussiandistributed noise. Lower panel: ratio f of the observed to the expected histograms, indicating the fraction of “good” data points. An excess of outliers is evident for r_{i}/σ_{i} ≳ 1.5. 
Fig. 2 Radial velocity measurements of β Aql for the entire time of observation, obtained with the iSONG code from the SARG spectra (lower panel) and their corresponding noisescaled and outlierscorrected uncertainties σ_{i} (upper panel). 
Fig. 3 Detail of the fifth night of observation of radial velocity measurements obtained with the iSONG code from the SARG spectra (filled black circles). The solid blue line represents a 3.30 min wide smoothing to enhance the pmode oscillations pattern. 
iSONG provides an estimate of the uncertainty in the velocity measurements, σ_{i}; these values were derived from the scatter of the velocities measured from many (≃650), small (≃1.8 Å) segments (chunks) of the échelle spectrum. To include the accuracy of the measurements in a weighted Fourier analysis of the data, we firstly verified that these σ_{i} reflected the noise properties of the RV measurements and their Fourier transform, following a slightly modified Butler et al. (2004) approach. First we considered that the variance deduced from the noise level σ_{amp} in the amplitude spectrum has to satisfy the relationship (1)according to Kjeldsen & Frandsen (1992) and Kjeldsen & Bedding (1995). This procedure yields new RV uncertainties scaled down by a factor of 1.51. Then we performed the following three steps. (i) The highfrequency noise in the power spectrum (PS), well beyond the stellar signal (>1000 μHz), reflects the properties of the noise in the RV data, and because we expected that the oscillation signal is the dominant cause of variations in the velocity time series, we need to remove it to analyze the noise. This we made iteratively by finding the strongest peak in the PS of the velocity timeseries and subtracting the corresponding sinusoid from the timeseries data (see Frandsen et al. 1995, Sect. 4.3). This procedure, namely the prewhitening, was carried out for the strongest peaks in the oscillation spectrum in the frequency range 0–1.5 mHz, until the spectral leakage into high frequencies from the remaining power was negligible (see also Leccia et al. 2007; Bonanno et al. 2008). This left us with a time series of residual velocities, r_{i}, that reflects the noise properties of the measurements. (ii) We then analyzed the ratio r_{i}/σ_{i}, expected to be Gaussiandistributed, so that the outliers correspond to the data points that exceed the given distribution. The cumulative histogram of the residuals is shown as a solid red curve in the upper panel of Fig. 1, indicating the fraction of “good” data points. An excess of outliers is evident for r_{i}/σ_{i} ≳ 1.5. In this step we used the theoretical Gaussian function for the cumulative histogram, given by the expression (2)where (3)is the error function, N is the total number of data points, , and x_{i} = r_{i}/σ_{i}. The zero point fixed to x_{0} = 0.89 allows us to adjust the fit for a noiseoptimized distribution, which means that the chosen distribution minimizes the noise level in the weighted PS (see Bedding et al. 2007, for more details on different kinds of weight optimizations). The reason for this optimization relies on the possibility to improve the mode identification by increasing the S/N of the frequency peaks. (iii) The lower panel of Fig. 1 shows the ratio f of the values of the observed points to the corresponding ones of the cumulative distribution function, i.e. the fraction of data points that could be considered as “good” observations, namely those that are close to unity. The quantities were adopted as weights in the computation of the weighted PS.
Fig. 4 Power spectrum of the weighted radial velocity measurements of β Aql extracted with the iSONG code from the SARG spectra. An excess of power is clearly visible, with a maximum centered at 422 μHz. The inset shows the normalized power spectrum of the window function for a sinewave signal of amplitude 1 m s^{1}, sampled in the same way as the observations. 
The timeseries of the whole data set is presented in Fig. 2 (lower panel) with the corresponding noisescaled and outlierscorrected uncertainties σ_{i} (upper panel). Figure 3 shows the details of the oscillation observations during the fifth night, overlaid with a solid blue curve representing a smoothing of 3.30 min for enhancing the pmode oscillations pattern. The final data point number of the full observation was reduced with respect to the number of observed spectra owing to a consistent improvement of the oscillation envelope, so that a total of 818 data points was taken into account for computing the timeseries analysis.
3. Timeseries analysis
The amplitude spectrum of the velocity time series was calculated as a weighted leastsquares fit of sinusoids (Frandsen et al. 1995; Arentoft et al. 1998; Bedding et al. 2004; Kjeldsen et al. 2005) with a weight assigned to each point according to its uncertainty estimate obtained from the radial velocity measurement, as explained in Sect. 2.1. The result is shown in Fig. 4, where a clear excess of power around 420 μHz is visible, with the typical pattern for pmode oscillations in a G9.5 subgiant star. This feature is apparent in the power spectra of individual nights, and its frequency agrees with theoretical expectations, as we will discuss in Sect. 4. To determine the S/N of the peaks in the PS, we measured the noise level σ_{amp} in the amplitude spectrum in the range 1200–1500 μHz, far from the excess of power. By means of the new weights introduced above, it was reduced from 14.4 cm s^{1} to the final value of 12.8 cm s^{1}, which corresponds to a noise level in the PS of 0.02 m^{2} s^{2}. Since this is based on 818 measurements, we can deduce the velocity precision on the corresponding timescales using the relation , as derived by Kjeldsen & Bedding (1995), which gives a scatter per measurement of 2.1 m s^{1}.
However, particular care has to be taken with the noise evaluation within the region of solarlike oscillations. Indeed, by evaluating the noise in the amplitude spectrum in the intervals 100 − 300 μHz and 600–800 μHz, just below and above the excess of power, we obtained the two noise levels cm s^{1} and cm s^{1}, which appear to be quite different. We then adopted a noise decaying accordingly to a linear trend law, ranging between the two values within the region 200–700 μHz.
3.1. Search for a comblike pattern
The mode frequencies for lowdegree, high radial order pmode oscillations in Sunlike stars are reasonably well approximated by the asymptotic relation (Tassoul 1980) (4)where n and ℓ are integers that define the radial order and angular degree of the mode, respectively, Δν = ⟨ ν_{n,ℓ} − ν_{n − 1,ℓ} ⟩ is the socalled mean large frequency separation and reflects the average stellar density, δν_{02} = ⟨ ν_{n,0} − ν_{n − 1,2} ⟩ is the small frequency separation, a quantity sensitive to the sound speed gradient near the core, and ϵ is a quantity on the order of unity sensitive to the stratification of the surface layers. On attempting to find the peaks in our power spectrum that match the asymptotic relation, we were severely hampered by the singlesite window function, whose power is visible in normalized units in the inset of Fig. 4, giving an effective observation time of ~1.80 days. As is well known, daily gaps in a time series produce aliases in the power spectrum at spacings ± 11.57 μHz and multiples, which are difficult to disentangle from the genuine peaks. Various methods to search for regular series of peaks have been discussed in the literature, such as autocorrelation, comb response and histograms of frequencies. To find a starting value for the Δν investigation we used the combresponse function method, where a combresponse function CR(Δν) is calculated for all sensible values of Δν (see Kjeldsen et al. 1995, for details), representing a generalization of the PS of a PS and consequently allowing us to search for any regularity in the spectral pattern. In particular we used the generalized combresponse function discussed in Bonanno et al. (2008)(5)so that a peak in the CR at a particular value of Δν indicates the presence of a regular series of peaks in the PS, centered at ν_{max} and having a spacing of Δν/2. It differs from a correlation function because the product of individuals terms is considered rather than the sum. For actual calculations we used N = 2 but we checked that our result was stable for N > 2 as well.
To reduce the uncertainties caused by noise, only peaks above 300 μHz and with amplitude > 0.5 m s^{1} in the amplitude spectrum, corresponding to a S/N ≳ 3.5, were used to compute the CR. We determined the local maxima of the response function CR(Δν) in the range 8 ≤ Δν ≤ 50 μHz as first step. In this case the result was seriously affected by two very strong peaks at 11.57 μHz and 23.14 μHz, corresponding to once and twice the value of the daily gap, respectively, which consistently reduced the strength of the peak corresponding to Δν. We then decided to restrict the search range to 26 ≤ Δν ≤ 50 μHz as second step to completely exclude the daily alias peaks. The resulting cumulative combresponse function for this range, obtained by summing the contributions of all the response functions, had the most prominent peak centered at 28.90 ± 0.45 μHz as shown in Fig. 5, where the uncertainty was computed by considering the FWHM of the Gaussian used to fit the peak, and represented by the blue dotdashed curve. The peak corresponding to three times the daily spacing is also visible in the right side of the plot, centered at 34.71 μHz.
Fig. 5 Cumulative combresponse obtained as the sum of the individual combresponses for each central frequency ν_{0} with amplitude >0.5 m s^{1} (S/N ≳ 3.5) in the amplitude spectrum. The maximum peak is centered at Δν = 28.90 ± 0.45 μHz where the uncertainty is evaluated as the FWHM of the Gaussian used to fit the peak (blue dotdashed curve). The second marked peak on the right, centered at 34.71 μHz, represents three times the daily spacing. 
3.2. Oscillation frequencies
The combresponse function provided a guess of the large separation, which was then used as a starting point to investigate the most reliable value that could represent the observed data. The investigation involved a parallel checking and tradeoff between two different methods, i.e. the folded PS and the échelle diagram. Firstly we computed the folded PS, namely the PS collapsed at the large separation value, for different values of Δν subsequent to the guess number. The final result for the folding was computed for Δν = 29.56 μHz and is shown in Fig. 6, where the peaks corresponding to ℓ = 0 and ℓ = 2 ridges are marked by a dashed and a dotted line, respectively. The daily sidelobes for ±11.57 μHz are also clearly visible and are marked with the same linestyles. The overlaid ridges represent the result of a leastsquares fit to the asymptotic relation (4). It is noticeable how the ℓ = 1 ridge, marked by a dotdashed line, does not correspond to any peak in the folded PS. This could be caused by avoided crossings, although it could also be explained by a wrong identification of the modes, as we will discuss in more detail in Sect. 4. We note that the ℓ = 2 ridge appears to be slightly shifted with respect to the position of the maximum of the relative peak, a result that can be explained by the limit of our frequency resolution.
Fig. 6 Folded PS in normalized units in the case of Δν = 29.56 μHz. The overlaid ridges represent the result for ℓ = 2,0,1 mode degrees as derived by a leastsquares fit to the asymptotic relation, which are marked as dotted, dashed and dotdashed lines, respectively. The daily sidelobes for ± 11.57 in the case of ℓ = 0,2 are also clearly visible and marked with the same linestyle for each mode degree. 
Fig. 7 Échelle diagram overlaid on the amplitude spectrum with a colored background scale. The filled symbols (white and orange) represent the identified modes for ℓ = 0 (circles), ℓ = 1 (triangles) and ℓ = 2 (squares). The orange symbols are the frequencies shifted for the daily gap of ± 11.57 μHz while the open symbols correspond to the original unshifted values. The ridges derived from the fit to the asymptotic relation (4) are also marked. 
The second method adopted for the investigation of the large separation is represented by the well known échelle diagram, an essential tool for frequency identification in asteroseismic data. The échelle diagram was computed for a list of 9 high S/N ( ≳ 3.5) frequencies directly obtained from the PS by means of the CLEAN algorithm, although a further consideration regarding this aspect is required to explain the way the final list was attained. In fact, two different ways of CLEANing the PS were adopted for this work. In the first case the algorithm was applied to the weighted PS for a sufficiently large number of frequencies ( ≃ 50 to find all the peaks with S/N > 3) in the range 0.01 − 700 μHz, then the resulting values were restricted to the interval 200–700 μHz, obtaining a list of 20 frequencies. For the second case the PS region below 200 μHz was at first completely prewhitened. Then 20 frequencies were identified on the new resulting weighted PS in the range 200–700 μHz. Comparing both lists of frequencies, the most unstable ones, i.e. those that were not present in both lists, were rejected and only the first nine frequencies were considered. These frequencies, showing an amplitude >0.5 m s^{1} (or S/N ≳ 3.5), were almost the same in both cases, however, the first frequency list was selected because the chance of the identification of one more ℓ = 2 mode was possible for this case. In Fig. 7 we show the final result for the échelle diagram superimposed on a colored scale background representing the amplitude spectrum, where the plot was computed in the same way as in Mathur et al. (2011). The filled symbols (white and orange) represent the 10 identified modes for ℓ = 0 (circles), ℓ = 1 (triangles) and ℓ = 2 (squares); the orange symbols are the corrected frequencies, i.e. those shifted for the daily gap of ± 11.57 μHz from their original values (the corresponding open symbols) and reported in Table 1. The aliasing considerably affects the amplitude spectrum with the presence of several fictitious peaks, which appear as strong spots in the diagram without frequency symbols overlaid. In particular, the two daily sidelobes arising from the strongest mode (ℓ = 0) are clearly visible as two red spots on the righthand part of the diagram. The uncertainties on frequencies are listed in Table 1 and were evaluated by using the analytical relation provided by Montgomery & O’Donoghue (1999)(6)where N = 818 is the total number of data points, T = 5.21 is the total duration of the run in days, ⟨ σ_{v} ⟩ = 2.72 m s^{1} is the average uncertainty on radial velocity each data point and A is the amplitude per mode as derived in Sect. 3.3. This relation holds exactly for coherent oscillations, hence we remark that the estimated uncertainties represent only a lower limit to the real uncertainty value because these oscillations are not fully coherent. An upper limit to these uncertainties can be fixed to the formal frequency resolution, given as the reciprocal of the total duration of the run, which is 2.2 μHz for this data set. The complete identification of the p modes is reported in Table 1 together with their S/N, where only three frequencies out of nine were shifted by the daily alias. The ℓ = 1 modes reported without any radial order number are potential candidates for mixed modes.
Mode identification for β Aql, in the frequency range 300–600 μHz.
By means of a linear weighted leastsquares fit to the asymptotic relation of the ℓ = 0 frequencies, the final value of Δν = 29.56 ± 0.10 μHz was obtained, together with the constant ϵ = 1.29 ± 0.04. The most likely value for the small separation was derived by using the definition from the asymptotic relation with the frequency pairs (ν_{12,0},ν_{11,2}), (ν_{13,0},ν_{12,2}) and (ν_{16,0},ν_{15,2}), where ν_{15,2} = 508.47 μHz was computed directly from ν_{16,2} = 538.03 μHz by adopting our value of the large separation. This led to δν_{02} = 2.55 ± 0.71 μHz, as reported in Table 2, but because it is comparable to the frequency resolution, its uncertainty is relatively high (Fig. 7). As a consequence, the reliability of this result requires additional investigations and a longer data set.
Fig. 8 Smoothed amplitude spectrum showing the amplitude per radial mode computed in the range 200–700 μHz. The maximum amplitude A_{max} = 76 ± 13 cm s^{1} occurs at ν_{max} = 416 μHz. The positions of the identified ℓ = 0 frequencies as derived from asymptotic relationship (4) are also marked. 
3.3. Mode amplitudes
The evaluation of mode amplitudes is interesting for the discussion on the L/M scaling by extrapolating from the Sun, as we will discuss in Sect. 4. As is known from the theory of solarlike oscillations, the amplitudes of individual modes are affected by the stochastic nature of the excitation and damping (e.g. see Kjeldsen & Bedding 1995). To measure the oscillation amplitude per mode in a way that is independent of these effects, we followed the approach explained in Kjeldsen et al. (2005, 2008). This involves the following steps: (i) heavy smoothing of the weighted PS by convolving it with a Gaussian whose FWHM is fixed to 4Δν (which is the value considered as a standard that allows comparisons since the amount of smoothing affects the exact height of the smoothed amplitude spectrum), which is large enough to produce a single hump of power that is insensitive to the discrete nature of the oscillation spectrum; (ii) conversion of the smoothed PS to power density spectrum (PSD) by multiplying by the effective length of the observing run (that is given by the reciprocal of the area under the spectral window in power, namely 6.40 μHz for this data set); (iii) subtraction of the background noise, which we computed as a linear trend in the interval 200–700 μHz, ranging from 26.2 cm s^{1} to 15.6 cm s^{1}; (iv) multiplication of the result by Δν/c where c = 4.09 (which represents the effective number of modes that fall in each segment of length Δν as evaluated in the case of radial velocities) and taking the square root to convert to amplitude per oscillations mode. The result is shown in Fig. 8 for the range 200–700 μHz, where A_{max} = 76 ± 13 cm s^{1} centered at ν_{max} = 416 μHz, which is assumed to be the frequency of maximum power and agrees with the result of Kjeldsen et al. (2008). The uncertainty on the amplitude is evaluated by means of the analytical estimation relation (7)(Montgomery & O’Donoghue 1999), where, as for the frequencies case, N = 818 is the total number of data points and ⟨ σ_{v} ⟩ is once more the average uncertainty on each data point. Again, the uncertainty in amplitude represents a lower limit to the real value. The amplitude distribution is evaluated for the radial modes only, but the calculation of the amplitude in the case of ℓ = 1 and ℓ = 2 modes is straightforward, namely it can be obtained by multiplying the result for a factor of 1.35 and 1.02 respectively, representing the relative strength given by the spatial response function (see Kjeldsen et al. 2008, for more details). The result derived in this work is not far from the value obtained by Kjeldsen et al. (2008).
4. Discussion
A complete discussion on the evolutionary state of this star goes beyond the scope of this work. Nevertheless, the identified modes provided a reliable estimate for the mean large separation, as described in Sect. 3.2, which can be used to derive the scaled mass for this star according to the fact that Δν scales approximately with the square root of the mean density of the star (Cox & Smith 1981). From the scaling relation extrapolating from the solar case (e.g. see Bedding et al. 2007) (8)where the radius is provided by Nordgren et al. (1999) and Δν_{⊙} = 134.9 μHz, we obtained a mass of M = 1.36 ± 0.17 M_{⊙}, which agrees very well with the value found by Fuhrmann (2004). By considering a luminosity of L = 5.63 ± 0.16 L_{⊙}, as derived by means of visual magnitude (Oja 1986), bolometric correction (Flower 1996), Sun bolometric magnitude (Cox & Pilachowski 2000) and Hipparcos parallax (van Leeuwen 2007), and the scaled value for the mass, we were able to compute the value for the amplitude by using the relation provided by Kjeldsen & Bedding (1995): (9)which gives A_{osc} = 97 ± 6 cm s^{1}, which fairly agrees with the value obtained in Sect. 3.3. Moreover, the expected frequency of maximum power can be evaluated from scaling the acoustic cutoff frequency from the solar case. We computed this frequency by considering the relationship (10)(Kjeldsen & Bedding 1995), whose result gives ν_{max} = 472 ± 72 μHz, consistent with the value obtained on the smoothed amplitude spectrum. Concerning the values for the mean large separation and the frequency of maximum power it is outstanding that they fit the power law relation of Stello et al. (2009) quite well, which represents a considerable validation to the reliability of our results. Moreover, an independent measure for the large separation by using Eq. (8), with the radius from Nordgren et al. (1999) and the mass from Fuhrmann (2004) is also compatible with the value presented in Sect. 3.2.
Lastly, an important note regards the dipole frequencies reported in this work, which are expected to deviate strongly from their asymptotic values, especially for an evolved subgiant star like β Aql. Indeed, the presumed ℓ = 1 modes identified here are possibly bumped because of avoided crossings. Although the lack of a clear ℓ = 1 ridge in Fig. 6 could also be explained by a wrong identification of the modes, we are confident that the large separation is correct. The reason of our belief is that different tools for its investigation, such as the combresponse function, the folded PS, the échelle diagram and the fit to the asymptotic relation, show consistent results. In addition, its compatibility with the independent estimation and the agreement with the Δν − ν_{max} power law relation mentioned in the above paragraph ensure a robust derivation of the result presented. Nonetheless, we are talking about a very difficult star, such as ν Ind (Carrier et al. 2007), because in the region of the HR diagram to which the star belongs, avoided crossings considerably hamper the pmode identification (see Bedding 2011, for a summary on groundbased observations across the HR diagram). Therefore, a theoretical confirmation is required before adopting the ℓ = 1 frequencies reported in Table 1 as real frequencies of oscillations for mixed modes, and more observations by a multisite groundbased project such as SONG are required to firmly solve the mode identification.
Global list of stellar parameters for β Aql.
4.1. Stellar parameters
The SEEK package (Quirion et al. 2010) is developed for the analysis of asteroseismic data from the Kepler mission and is able to estimate stellar parameters in a form that is statistically welldefined. It is based on a large grid of stellar models computed with the Aarhus Stellar Evolution Code (ASTEC), which allow us to derive additional stellar parameters giving as input astrophysical quantities such as T_{eff} (Luck & Heiter 2005), log g (Valenti & Fischer 2005) and [Fe/H] (Fuhrmann 2004), and the asteroseismic values derived in this work, i.e. the large and small separation and the frequency of maximum power. The output list of parameters for β Aql is shown in Table 3, where the 1σ error bars on the SEEK values are computed using the Bayesian evaluation of the posterior distributions. The mass, radius and luminosity computed by SEEK also agree with the values presented in this work.
5. Conclusions
Our observations of β Aql show an evident excess of power in the PS region centered at 415 μHz, clearly very well separated from the lowfrequency power, and with a position and amplitude that agree with expectations. Although consistently hampered by the singlesite window, the comb analysis and the échelle diagram show clear evidence for regularity in the peaks at the spacing expected from the asymptotic theory. The complete identification of six high S/N modes for ℓ = 0,2 led to a wellconstrained mean large separation of Δν = 29.56 ± 0.10 μHz, compatible with a scaled value from the Sun and the value obtained by the power law relation (Stello et al. 2009), and to a most likely value for the small separation of δν_{02} = 2.55 ± 0.71 μHz, whose reliability has yet to be confirmed. The ℓ = 1 modes found are presumably mixed modes but a theoretical confirmation is needed before adopting these values as real modes of oscillations for this star. Moreover, our results provide a valuable proof that oscillations in an evolved subgiant solarlike star show amplitudes that scale as L/M by extrapolating from the Sun.
This campaign of observations attained with SARG led to highprecision RV measurements by means of the iSONG code, which was used for the first time in this work. The timeseries analysis of the given data set was able to provide for the first time global asteroseismic parameters and individual p modes together with the evidence for mixed modes. Moreover, this result will be extremely important to develop theoretical models for this star. Multisite observation campaign with the SONG project is highly desirable in a near future. That would then allow us to explore the solarlike oscillations for this target in a detailed way by providing a large number of identified modes. The asteroseismic and astrophysical parameters of this star will then be constrained properly, yielding a deeper understanding of solarlike oscillations in the very difficult region of the HR diagram to which β Aql belongs.
Acknowledgments
We thank the Italian Foundation Galileo for the chance to acquire spectra at the TNG Italian Telescope in La Palma; the Department of Physics and Astronomy of the Aarhus University for hosting part of this work; Antonio Frasca for suggestions in the calculation of the stellar luminosity; Christoffer Karoff for the evaluation of additional stellar parameters with the SEEK package; the Sydney Institute for Astronomy (SIfA), School of Physics of the University of Sydney, for hosting part of this work; Tim Bedding and Dennis Stello for important discussions and advice in improving the paper.
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All Tables
All Figures
Fig. 1 Upper panel: cumulative histograms of r_{i}/σ_{i} for SARG data. The diamonds show the observed data, and the solid curve shows the result expected for a Gaussiandistributed noise. Lower panel: ratio f of the observed to the expected histograms, indicating the fraction of “good” data points. An excess of outliers is evident for r_{i}/σ_{i} ≳ 1.5. 

In the text 
Fig. 2 Radial velocity measurements of β Aql for the entire time of observation, obtained with the iSONG code from the SARG spectra (lower panel) and their corresponding noisescaled and outlierscorrected uncertainties σ_{i} (upper panel). 

In the text 
Fig. 3 Detail of the fifth night of observation of radial velocity measurements obtained with the iSONG code from the SARG spectra (filled black circles). The solid blue line represents a 3.30 min wide smoothing to enhance the pmode oscillations pattern. 

In the text 
Fig. 4 Power spectrum of the weighted radial velocity measurements of β Aql extracted with the iSONG code from the SARG spectra. An excess of power is clearly visible, with a maximum centered at 422 μHz. The inset shows the normalized power spectrum of the window function for a sinewave signal of amplitude 1 m s^{1}, sampled in the same way as the observations. 

In the text 
Fig. 5 Cumulative combresponse obtained as the sum of the individual combresponses for each central frequency ν_{0} with amplitude >0.5 m s^{1} (S/N ≳ 3.5) in the amplitude spectrum. The maximum peak is centered at Δν = 28.90 ± 0.45 μHz where the uncertainty is evaluated as the FWHM of the Gaussian used to fit the peak (blue dotdashed curve). The second marked peak on the right, centered at 34.71 μHz, represents three times the daily spacing. 

In the text 
Fig. 6 Folded PS in normalized units in the case of Δν = 29.56 μHz. The overlaid ridges represent the result for ℓ = 2,0,1 mode degrees as derived by a leastsquares fit to the asymptotic relation, which are marked as dotted, dashed and dotdashed lines, respectively. The daily sidelobes for ± 11.57 in the case of ℓ = 0,2 are also clearly visible and marked with the same linestyle for each mode degree. 

In the text 
Fig. 7 Échelle diagram overlaid on the amplitude spectrum with a colored background scale. The filled symbols (white and orange) represent the identified modes for ℓ = 0 (circles), ℓ = 1 (triangles) and ℓ = 2 (squares). The orange symbols are the frequencies shifted for the daily gap of ± 11.57 μHz while the open symbols correspond to the original unshifted values. The ridges derived from the fit to the asymptotic relation (4) are also marked. 

In the text 
Fig. 8 Smoothed amplitude spectrum showing the amplitude per radial mode computed in the range 200–700 μHz. The maximum amplitude A_{max} = 76 ± 13 cm s^{1} occurs at ν_{max} = 416 μHz. The positions of the identified ℓ = 0 frequencies as derived from asymptotic relationship (4) are also marked. 

In the text 
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