© ESO, 2015

1. Introduction

Solar-like oscillation spectra are usually dominated by p-mode eigenfrequencies corresponding to waves trapped in the stellar interior with frequencies below a given cut-off frequency, ν_cut. However, the oscillation power spectrum of the resolved Sun shows a regular peak structure that extends well above ν_cut (e.g. Jefferies et al. 1988; Libbrecht 1988; Duvall et al. 1991). This signal is interpreted as travelling waves whose interferences produce a well-defined pattern corresponding to the so-called pseudo-mode spectrum (Kumar et al. 1990). The amount and quality of the space data provided by the Solar and Heliospheric Observatory (SoHO) satellite (Domingo et al. 1995) allowed us to measure these high-frequency peaks (HIPs), using Sun-as-a-star observations (García et al. 1998) from Global Oscillations at Low Frequency (GOLF, Gabriel et al. 1995) and from Variability of solar IRradiance and Gravity Oscillations (VIRGO, Fröhlich et al. 1995) instruments. The change in the frequency pattern between the acoustic and the pseudo-modes enabled the proper determination of the solar cut-off frequency (Jiménez 2006).

Theoretically, the cut-off frequency approximately scales as $\mathit{g}\sqrt{\mathit{\mu }\mathit{/}{\mathit{T}}_{\mathrm{eff}}}$, where T_eff is the effective temperature, g the gravity, and μ the mean molecular weight, with all values measured at the surface. Hence, the observed ν_cut can be used to constrain the fundamental stellar parameters, however, solar observations showed that ν_cut changes, for example, with the solar magnetic activity cycle (Jiménez et al. 2011). Therefore, the accuracy of the scaling relation given above needs to be discussed in a theoretical context and taking observational constraints into account.

For stars other than the Sun, the detection of HIPs is challenging because of the shorter length of available data sets and lower signal-to-noise ratio (S/N). Nevertheless, in this study we report on the analysis of the high-frequency part of the spectrum of six pulsating stars observed by Kepler (Borucki et al. 2010) for which we were able to characterize the HIP pattern and the cut-off frequency. The time series analysis and preparation of the spectra are detailed in Sect. 2. In Sect. 3 we describe how to estimate ν_cut and compare it with our theoretical expectations. In Sect. 4 we perform a detailed analysis of the HIPs and interpret the results as a function of the evolutionary stage of the stars. Finally, we provide our conclusions in Sect. 5.

2. Data analysis

In this study, we used ultra high-precision photometry obtained by NASA’s Kepler mission to study the high-frequency region of six stars with solar-like pulsations. The names of the stellar identifiers from the Kepler Input Catalogue (KIC, Brown et al. 2011) are given in the first column of Table 1. Short-cadence time series (sampling rate of 58.85s, Gilliland et al. 2010) up to quarter 17 have been corrected for instrumental perturbations and properly stitched together using the Kepler Asteroseismic Data Analysis and Calibration Software (KADACS, García et al. 2011).

For each star, the long time series (approximately four years) are divided into consecutive subseries and the average of all the power spectral density (AvPSD) is computed to reduce the high-frequency noise in the spectrum. As in the solar case (e.g. García et al. 1998; Jiménez et al. 2005, 2006), subseries of approximately four-days (4 × 1440 points, 3.92 days) were used because they are a good compromise between frequency resolution (which improves with longer subseries) and the increase of the S/N (which improves with the number of averaged spectra). For the same reasons as in the solar case, we use a boxcar function to smooth the AvPSD over 3 or 5 points, depending on the evolutionary state of the star and the S/N of the spectrum. From now on when we refer to the AvPSD, we mean the smoothed AvPSD. An example of this spectrum is given in the top panel of Fig. 1 for KIC 11244118. Because of the short length of the subseries, it is not necessary to interpolate the gaps in the data (see for more details García et al. 2014), so we prefer the simplest possible analysis. However, we have verified that the results remain the same when using series that were interpolated with an inpainting algorithm (Pires et al. 2015).

To avoid subseries of low quality and to increase the S/N in the AvPSD, we first remove all subseries with a duty cycle below 50%. Then, for each subseries of each star we compute the median of the flat noise at high frequencies above 2 to 5 mHz, depending on the frequency of maximum power of the star. From the statistical analysis of these medians, we reject those subseries in which the high-frequency noise is too high. We have verified that the selection of different high-frequency ranges does not affect the number of series retained. A detailed study of the rejected series show that there is an increase in the noise level when Kepler has lost the fine pointing and the spacecraft is in “coarse” pointing mode. This is particularly important in Q12 and Q16. A new revision of the KADACS software (Mathur, Bloemen, García, in prep.) systematically removes those data points from the final corrected time series. In Table 1 we summarize the number of four-day subseries computed for each star, the number of subseries finally retained for the calculation of the AvPSD, and the total number of short-cadence Kepler quarters available.

Figure 2 shows the place of the observed stars in a seismic HR diagram. The black lines are the evolution sequences computed using the Aarhus Stellar Evolution Code (ASTEC; Christensen-Dalsgaard 2008) in a range of masses from M = 0.9 M_⊙ to M = 1.6 M_⊙ in steps of M = 0.1 M_⊙ with a solar composition (Z_⊙ = 0.0246). The target stars cover evolutionary stages from late main-sequence stars to early red giants.

3. The acoustic cut-off frequency

3.1. Observations

The observed stellar power spectrum has different patterns in the eigenmode and pseudo-mode regions. In particular, the mean frequency separation, ⟨ Δν ⟩, between consecutive peaks is different below and above the cut-off frequency. In the first case, below the cut-off frequency corresponds to half the mean large frequency spacing, while above ν_cut it is the period of the interference pattern. In addition, a phase shift between both patterns appears in the transition region. The frequency in which the transition between the two regimes is observed corresponds to the cut-off frequency. Hence, we fit all the peaks in the spectrum (actually doublets of odd, ℓ = 1, 3, and even, ℓ = 0, 2, pairs of modes due to the small frequency resolution of 2.89 μHz) from a few orders before ν_max up to the highest visible peak in the spectrum. To account for the underlying background contribution properly, which is due to convective movements at different scales, faculae, and magnetic/rotation signal, we prefer to divide the AvPSD by the same spectrum after heavy smoothing (top panel of Fig. 1) instead of using a theoretical model (e.g. Mathur et al. 2011). Although it has been demonstrated that a two-component model usually properly fits the background of stars (for more details, see Kallinger et al. 2014), the accuracy in the transition region between the eigenmode bump and the high-frequency region dominated by the HIPs is not properly described. For the purposes of this study, we are primarily interested in the transition region. Therefore, we use a simpler description of the background based on the observed spectrum itself. The length of this smoothing varies according to the evolutionary stage of the star. An example of the normalized resultant AvPSD (NAvPSD) is given in the bottom panel of Fig. 1.

A global fit to all the visible peaks in the spectrum is not possible for the following reasons: 1) it requires a huge amount of computation time owing to the high number of free parameters in the model; 2) the background is not completely flat around ν_max; and 3) the large amplitude dispersion of the peaks between ν_max and the HIP pattern biases the results of the smallest peaks by overestimating their amplitudes. We therefore divide the spectrum into a few orders before ν_max and the photon-noise dominated region into two regions. We arbitrarily choose a frequency where the amplitudes start to be very small and separately treat the power spectrum before and after this frequency. We denote, with p-mode region, the low-frequency zone before the frequency mentioned above and the pseudo-modes region is the one after that frequency. Several tests have been performed in which the frequency separating these two regions were varied and in all cases the results remain the same.

In the p-mode region, we fit groups of several peaks at a time for which the underlying background can be considered flat. This fit depends on the evolutionary stage of the star and S/N. We checked that fitting a different number of peaks in each group does not change the final result. Appendix A shows the analysis of all the stars, and in the caption of each figure we explicitly mention the number of peaks fitted together. The pseudo-modes region can be fitted at once because 1) the background is flat (see the bottom panel of Fig. 1); 2) the range of the peak amplitudes is reduced; and 3) the number of free parameters is small. For both zones, p-mode and pseudo-mode, a Lorentzian profile is used to model the peaks using a maximum likelihood estimator. We are only interested in the frequency of the centroids of the peaks, and for that a Lorentzian profile is a good approximation. In Figs. 3 and 4 we show the results of the fits for KIC 3424541 in the p-mode and pseudo-mode regions. The figures for the other stars can be found in Appendix A.

After fitting the two parts of the spectra, we compute the frequency differences of consecutive peaks: ν_n − ν_{n − 1}. As an example, the resultant frequency differences for KIC 3424541 are plotted in Fig. 5.

Two different regions are clearly visible in the frequency differences of KIC 3424541. A portion of these differences, those at lower frequencies, are centred around 20.75 μHz, corresponding approximately to half of the large spacing, Δν_modes/ 2 (see Table 2) and owing to the alternation between odd and even modes. At a certain frequency, between 1200 and 1230 μHz, the frequency separations increase and remain roughly constant around 37.31 μHz, although with a higher dispersion; see the next section for further details. The observed acoustic cut-off frequency, ν_cut, lies in the transition region where the differences jump from the p-mode region to the pseudo-mode zone. We define this position as the mean frequency between the two closest points to this transition, represented by a red symbol in Fig. 5 for KIC 3424541. The results of this analysis for the six stars studied here are given in Table 2.

3.2. Comparison between the observed and theoretical cut-off frequency

As noted in Sect. 1, the cut-off frequency at the stellar surface scales approximately as ${\mathit{\nu }}_{\mathrm{cut}}\mathrm{\propto }\mathit{g}\sqrt{\mathit{\mu }\mathit{/}{\mathit{T}}_{\mathrm{eff}}}$, suggesting that we may compare the observational cut-off frequency with that computed from the spectroscopic parameters. However, for our set of stars, the spectroscopic log g has large uncertainties (see Bruntt et al. 2012) and a direct test is not possible. Alternatively, it has been demonstrated empirically that the frequency of maximum mode amplitude ν_max follows a similar scale relation (see Bedding & Kjeldsen 2003). As is seen in Fig. 6, the observational data show a linear relation between ν_max and ν_cut, although some stars deviate 2σ from it. In fact, from a theoretical point of view, one does not expect this linear relation to be accurate enough at the level of the observational errors. As shown below, values of ν_cut can deviate by as much as 10% from the scale relation ${\mathit{\nu }}_{\mathrm{cut}}\mathrm{\propto }\mathit{g}\sqrt{\mathit{\mu }\mathit{/}{\mathit{T}}_{\mathrm{eff}}}$ for representative models of our stars. On the other hand, following Chaplin et al. (2008), where theoretical values of ν_max based on stochastic mode excitation were computed, one can find departures from the scale relation as large as 25% for a model of a star with M = 1.3 M_⊙ and ν_max ~ 1500 μHz.

We can gain a deeper insight by comparing the observational and theoretical results in the ν_cut–Δν plane. For the theoretical calculations, we use a set of models computed with the “Code d’Évolution Stellaire Adaptatif et Modulaire” (CESAM) code (Morel & Lebreton 2008). These correspond to evolutionary tracks from the zero-age main sequence up to the red-giant phase, with masses between M = 0.9 M_⊙ and M = 2.0 M_⊙, and helium abundances between Y = 0.25 and Y = 0.28. Models with and without overshooting were considered. Other parameters were fixed to standard values; for example, the metallicity was fixed to Z = 0.02. Finally, the CEFF equation of state (Christensen-Dalsgaard & Däppen 1992) and a T(τ) relation derived from a solar atmosphere model were used.

In theoretical computations, a standard upper boundary condition for eigenmodes is to impose in the uppermost layer the simple adiabatic evanescent solution for an isothermal atmosphere. In this case eigenmodes have frequencies below a cut-off frequency given by ω_cut = 2πν_cut = c/ 2H, where H is the density scale height. This is strictly speaking the cut-off frequency for radial oscillations, which is accurate for low-degree modes. To better fulfil the quasi-isothermal requirement, the boundary condition should be placed near the minimum temperature. For the Sun, this corresponds to an optical depth of about log τ = −4. Another advantage of using this sort of low optical depth is that at this position the oscillations are not too far from being adiabatic, at least compared to the photosphere. In our computations, we used this value as the uppermost point for all the models, but for some red giant stars we found that the maximum value of ω_cut(r) in the atmosphere can be located at higher optical depths; hence, we took that maximum value as representative of the observed ω_cut. From an interpolation to the solar mass and radius (with Y = 0.25 fixed), we obtained a value of ν_cut = 5125 μHz for the Sun from our set of models, which is in agreement with the observed value: ν_cut = 5106.41 ± 61.53μHz (Jiménez 2006). If model S (Christensen-Dalsgaard et al. 1996) is considered, the differences are a little larger, about 4%. It is important to remember that the solar ν_cut changes with the magnetic activity cycle (Jiménez et al. 2011); thus, some additional dispersion in the observed stellar values could be due to that effect.

For the large separation, the theoretical and observed values can be computed using the same approach. Specifically, we computed the large separation Δν_modes from a polynomial fit of the form (ν − ν₀)/(n − n₀) = ∑ _ia_iP_i(x), where n is the radial order and ν₀ is the radial frequency closest to ν_max with radial order n₀. For the models, ν₀ is estimated assuming a linear relation to the cut-off frequency. The frequencies considered are those of the ℓ = 0 modes with radial orders n = n₀ ± 4. P_i(x) is the Legendre polynomial of degree i, and x is the frequency normalized to the interval [−1,1 ]. We used third order polynomial. From the Tasoul equation we expect that a₀ ≡ Δν_modes ≃ Δν, whereas other terms mostly contain upper-layer information. We use only radial oscillations because, for evolved stars, the mixed character of some modes can introduce complications. For the Sun, we obtain Δν_modes = 134.9 μHz from the observations and Δν_modes = 136.0 μHz from model S.

Figure 7 shows ν_cut against the large separation Δν_modes for our set of models and the observed stars. At first glance, there is rough agreement between the observed and theoretical calculations, but for the group of stars with ν_cut ~ 2000 μHz, it seems hard to explain the dispersion in their Δν_modes values.

We can proceed by taking the values of T_eff into account. First, given the approximated scale relation for ν_cut, we compute the residuals ${\mathit{\delta }}_{\mathrm{c}}\mathrm{=}\mathrm{1}\mathrm{-}\frac{{\mathit{\nu }}_{\mathrm{cut}}^{\mathrm{0}}}{{\mathit{\nu }}_{\mathrm{cut}}}\hspace{0.17em}\frac{\mathit{g}\sqrt{\mathit{\mu }\mathit{/}{\mathit{T}}_{\mathrm{eff}}}}{{\mathrm{(}\mathit{g}{\sqrt{\mathit{\mu }\mathit{/}{\mathit{T}}_{\mathrm{eff}}}}^{\mathrm{)}}}_{\mathrm{0}}}\mathit{,}$(1)where ${\mathit{\nu }}_{\mathrm{cut}}^{\mathrm{0}}$ = 5079.5 μHz is a reference cut-off frequency, corresponding to a model in our data set close to the Sun (M = 1 M_⊙, R = 1.004 R_⊙, Y = 0.25, ${\mathit{T}}_{\mathrm{eff}}^{\mathrm{0}}$ = 5799.2 K). The T_eff values are shown in Fig. 8. As seen in the figure, the differences can be as large as | δ_c | ≈ 10%. In fact, the scale relation ${\mathit{\nu }}_{\mathrm{cut}}\mathrm{\propto }\mathit{g}\sqrt{\mathit{\mu }\mathit{/}{\mathit{T}}_{\mathrm{eff}}}$ is derived theoretically by approximating the density scale height, H, by the pressure scale height, H_p, which is strictly valid in an isothermal atmosphere, and further by taking the equation of state of a mono-atomic ideal gas. In Fig. 8, the blue points correspond to the relative differences δ_a obtained by replacing in Eq. (1) ω_cut = c/ 2H by ω_a = c/ 2H_p. Thus, the first condition introduces the highest departures from the scale relation, which was expected because in the range of T_eff considered, the gas is mainly mono-atomic at the surface.

Nevertheless, as seen in Fig. 8, δ_c is mainly a function of T_eff. Hence, the scale relation can be improved if we subtract a polynomial fit from δ_c. We only considered models with T_eff< 7000 K and ν_cut> 500μHz in that fit because this range includes all the stars used in the present study. For a better fit we also consider the dependence of δ_c on ν_cut. In particular, the red points in Fig. 8 are obtained by replacing ν_cut in Eq. (1) by $\begin{array}{ccc}{\mathit{\nu }}_{\mathrm{cut}}^{\mathrm{\ast }}& \mathrm{=}& {\mathit{\nu }}_{\mathrm{cut}}[\mathrm{1}\mathrm{-}\mathrm{(}\mathrm{0.0004}\mathrm{-}\mathrm{0.0057}\mathit{x}\mathrm{-}\mathrm{1.0684}{\mathit{x}}^{\mathrm{2}}\mathrm{+}\mathrm{0.0543}\mathit{y}\\ & & \end{array}$(2)where $\mathit{x}\mathrm{=}{\mathit{T}}_{\mathrm{eff}}\mathit{/}{\mathit{T}}_{\mathrm{eff}}^{\mathrm{0}}\mathrm{-}\mathrm{1}$ and $\mathit{y}\mathrm{=}{\mathit{\nu }}_{\mathrm{cut}}\mathit{/}{\mathit{\nu }}_{\mathrm{cut}}^{\mathrm{0}}\mathrm{-}\mathrm{1}$. The standard deviation for these residuals is 0.3%. Hence, the modified cut-off frequencies, which we calibrated on the Sun, deviate by 1σ = 0.3% from the scaling relation, at least for our set of models.

With a similar approach, we compute the deviation of Δν_modes from its expected scale relation, namely, ${\mathit{\delta }}_{\mathrm{\Delta }}\mathrm{=}\mathrm{1}\mathrm{-}\frac{\mathrm{(}\mathrm{\Delta }{\mathit{\nu }}_{\mathrm{modes}}{\mathrm{)}}^{\mathrm{0}}}{\mathrm{\Delta }{\mathit{\nu }}_{\mathrm{modes}}}\sqrt{\frac{\mathit{M}\mathit{/}{\mathit{M}}_{\mathrm{0}}}{\mathrm{(}\mathit{R}\mathit{/}{\mathit{R}}_{\mathrm{0}}{\mathrm{)}}^{\mathrm{3}}}}\mathit{,}$(3)where (Δν_modes)⁰ is the value for our reference model, the same as the value used in ν_cut. This quantity is shown in Fig. 9 for our set of models.

From Fig. 9 we can determine that the differences between the large separation computed from p-modes and the scaling relation can be as large as 5% for solar-like pulsators with T_eff< 7000 K. However, as shown in this figure, the differences δ_Δ depend mainly on T_eff. This was previously noted by White et al. (2011) and allows for a correction to Δν_modes to obtain a better estimate of the mean density. Here we use a fit similar to that considered for ν_cut. As in the previous case, we have considered only those models with T_eff< 7000 K and Δν_modes> 10 μHz, which include all the stars used in the present work. The red points in Fig. 9 correspond to the residuals obtained after replacing Δν_modes in Eq. (3) by $\begin{array}{ccc}\mathrm{\Delta }{\mathit{\nu }}_{\mathrm{modes}}^{\mathrm{\ast }}& & \mathrm{=}\mathrm{\Delta }{\mathit{\nu }}_{\mathrm{modes}}[\mathrm{1}\mathrm{-}\mathrm{(}\mathrm{0.0023}\mathrm{-}\mathrm{0.1123}\mathit{x}\mathrm{-}\mathrm{0.7309}{\mathit{x}}^{\mathrm{2}}\\ & & \end{array}$(4)where $\mathit{x}\mathrm{=}{\mathit{T}}_{\mathrm{eff}}\mathit{/}{\mathit{T}}_{\mathrm{eff}}^{\mathrm{0}}\mathrm{-}\mathrm{1}$ and z = Δν_modes/ (Δν_modes)⁰−1 with (Δν_modes)⁰ = 135.4μHz. The standard deviation for these residuals is 0.6%. Given the differences between the observed and theoretical values for the Sun, we calibrated Eq. (4) to the observed solar value.

Figure 10 shows ${\mathit{\nu }}_{\mathrm{cut}}^{\mathrm{\ast }}$ against $\mathrm{\Delta }{\mathit{\nu }}_{\mathrm{modes}}^{\mathrm{\ast }}$ for our set of models (black points) and the observed stars including the Sun (blue points). Whereas the scattering in the theoretical values are substantially reduced, the observational values are not. Hence, if the error estimates are correct, we must conclude that the observational cut-off frequencies do not completely agree with the theoretical estimates. Moreover, the red points in Fig. 10 are proportional to the observed values of the frequency of maximum amplitude scaled to the Sun, (ν_cut/ν_max)_⊙ν_max. For our limited set of stars, ν_max follows the scaling relation better than ${\mathit{\nu }}_{\mathrm{cut}}^{\mathrm{\ast }}$. This is a little surprising because, as noted before, theoretically, we would expect the opposite, even more so when the effective ${\mathit{\nu }}_{\mathrm{cut}}^{\mathrm{\ast }}$ is used.

This result can be understood in terms of the inferred surface gravity since we can use either ν_max or ν_cut to estimate log g. The results are summarized in Table 3 and compared to the spectroscopic log g_sp. To estimate the errors in log g_νmax we only considered the observational errors in ν_max and T_eff, while for log g_νcut we have included the 1σ value of 0.3% found in the scaling relation plus an error in μ_s derived from assuming an unknown composition with standard stellar values. In particular, we have taken ranges of Y = [ 0.24,0.28 ] and Z = [ 0.01,0.03 ] for the helium abundance and the metallicity, respectively, and thus estimated the error in the mean molecular weight μ_s by some 5%. As expected, the values obtained from log g_νmax are quite similar to those reported by Bruntt et al. (2012) because they are based on the same global parameters with very similar values. The values derived from ν_max and ν_cut are also much closer to each other than those derived spectroscopically, which, as mentioned before, have large observational uncertainties.

4. The HIP region

4.1. Observations

Using Sun-as-a-star observations from the GOLF instrument on board SoHO, García et al. (1998) uncovered the existence of a sinusoidal pattern of peaks above ν_cut, as the result of the interference between two components of a travel wave generated on the front side of the Sun with a frequency ν ≥ ν_cut, where the inward component returns to the visible side after a partial reflection on the far side of the Sun (see Fig. 3 in García et al. 1998). Therefore, the frequency spacing of ~70 μHz found in the Sun corresponded to the time delay between the direct emitted wave and that coming from the back of the Sun, corresponding to waves behaving like low-degree modes, i.e. a delay of four times the acoustic radius of the Sun (~3600 s). This value is roughly half of the large frequency spacing of the star, which we call Δν₁. García et al. (1998) also speculated that, above a given frequency, a second pattern should become visible with a double frequency spacing (close to the large frequency separation), Δν₂ ~ 140 μHz. This pattern (for a theoretical description, see Kumar 1993), which is usually visible in imaged instruments (Duvall et al. 1991), corresponds to the interference between outward emitted waves and the inward components, which arrive at the visible side of the Sun after the refraction at the inner turning point (non-radial waves).

Two sine waves are fitted above the cut-off frequency to obtain a global estimate of the frequency of the interference patterns in the HIP region of our sample of stars. The amplitudes and frequencies of both sine waves are summarized in Table 2. We also reanalysed the GOLF and VIRGO data, following the same procedure using at a higher frequency range of between 7 and 8 mHz, compared to the original analyses performed by García et al. (1998) and Jiménez (2006). With this approach, we have been able to obtain the second periodicity at ~140 μHz (see Table 2).

We tried fitting various functions and opted for the simplest one. Indeed, the amplitude of the interference patterns decreases with frequency in a manner that is close to an exponential decrease. The additional parameters required give more unstable fits with a heavy dependence on the guess parameters. We preferred this approach because we obtained the same qualitative results leading to the same classification of stars. In future studies, we will look for a better function for the fit, i.e. one more suited to the observations, in a larger set of stars. We are already working in this direction but this investigation is beyond of the scope of this paper.

According to the fitted amplitudes, we can classify the stars into two groups: those in which the two amplitudes of the sine waves, A₁ and A₂, are similar (KIC 7940546, KIC 9812850, KIC 11244118), and those in which A₂ is much larger than A₁ (KIC 3424541, KIC 7799349, KIC 11717120). An example of each group of stars is given in Fig. 11 for stars KIC 7940546 and KIC 11717120. The error bars of the fitted amplitudes are large because the actual amplitudes of the interference patterns decrease in a quasi-exponential way that has not been taken into account here. We favoured the fits with constant amplitudes to avoid adding more unknown parameters to the fit.

The stars of the first group have larger Δν_modes than the other three stars, which implies a correlation with the evolutionary state of the stars. Starting with the Sun on the main sequence, the HIP pattern is dominated by interference waves partially reflected at the back of the Sun (for the solar case A₁ is much bigger than A₂). When the stars evolve, the HIP pattern due to the interference of direct with refracted waves in the stellar interior is more and more visible and becomes dominant for evolved RGB stars with Δν_modes below ~40 μHz.

4.2. HIPs and stellar evolution

To reproduce the periodic signals expected from the pseudo-mode spectrum theoretically, we follow the interpretation given by Kumar (1993) and García et al. (1998) and assume that waves are excited isotropically at a point very close to the photosphere. The observed spectrum of the pseudo-modes is then interpreted as an interference pattern between outgoing and ingoing components, possibly including successive surface reflections. In what follows, we summarize the basic concepts and apply them to stars at different evolutionary stages.

We start with a gravito–acoustic ray theory approximation with the following dispersion relation (see e.g. Gough 1986, 1993): ${\mathit{k}}^{\mathrm{2}}\mathrm{=}{\mathit{k}}_{\mathrm{r}}^{\mathrm{2}}\mathrm{+}{\mathit{k}}_{\mathrm{h}}^{\mathrm{2}}\mathrm{=}\frac{\mathrm{1}}{{\mathit{c}}^{\mathrm{2}}}\mathrm{(}{\mathit{\omega }}^{\mathrm{2}}\mathrm{-}{{\mathit{\omega }}_{\mathit{c}}^{\mathrm{2}}}^{\mathrm{)}}\mathrm{+}\frac{{\mathit{N}}^{\mathrm{2}}}{{\mathit{\omega }}^{\mathrm{2}}}{\mathit{k}}_{\mathrm{h}}^{\mathrm{2}}\mathit{,}$(5)where k_r is the radial component of the wave number, ${\mathit{k}}_{\mathrm{h}}^{\mathrm{2}}\mathrm{=}{\mathit{L}}^{\mathrm{2}}\mathit{/}{\mathit{r}}^{\mathrm{2}}$, L = l(l + 1) the horizontal component, N is the buoyancy frequency, ω_c is a generalized cut-off frequency given by${\mathit{\omega }}_{\mathit{c}}^{\mathrm{2}}\mathrm{=}\frac{{\mathit{c}}^{\mathrm{2}}}{\mathrm{4}{\mathit{H}}^{\mathrm{2}}}\left(\mathrm{1}\mathrm{-}\mathrm{2}\frac{\mathrm{d}\mathit{H}}{\mathrm{d}\mathit{r}}\right)\mathit{,}$(6)and H is the density scale height. In the ray approximation, the turning points are given by k_r = 0, whereas the ray path is determined by the group velocity. For the spherically symmetric case, the rays are contained in a plane with a path given by dθ/dr = v_θ/ (rv_r) . Here, r and θ are the usual polar coordinates and the group velocity v_g = (v_r,v_θ) = (∂ω/∂k_r,∂ω/∂k_h) has the following components: ${{v}}_{\mathrm{g}}\mathrm{=}\left(\hspace{0.17em}\frac{{\mathit{k}}_{\mathit{r}}{\mathit{\omega }}^{\mathrm{3}}{\mathit{c}}^{\mathrm{2}}}{{\mathit{\omega }}^{\mathrm{4}}\mathrm{-}{\mathit{k}}_{\mathrm{h}}^{\mathrm{2}}{\mathit{c}}^{\mathrm{2}}{\mathit{N}}^{\mathrm{2}}}\mathit{,}{\mathit{k}}_{\mathrm{h}}\hspace{0.17em}\mathit{\omega }\hspace{0.17em}{\mathit{c}}^{\mathrm{2}}\left[\frac{{\mathit{\omega }}^{\mathrm{2}}\mathrm{-}{\mathit{N}}^{\mathrm{2}}}{{\mathit{\omega }}^{\mathrm{4}}\mathrm{-}{\mathit{k}}_{\mathrm{h}}^{\mathrm{2}}{\mathit{c}}^{\mathrm{2}}{\mathit{N}}^{\mathrm{2}}}\right]\hspace{0.17em}\right)\mathrm{·}$(7)According to the ray theory, the general solution of the wave equation can be expressed as a superposition of rays of the form ψ_ω(r,t) = A_ω(r)exp [ i(ωt ± ^∫_sk ds) ], where the two signs corresponds to outgoing and ingoing waves and the integral is computed over the ray path s, including additional reflections where appropriate. For every reflection, a constant phase shift must be introduced. In particular, for a ray travelling from an inner turning point r₁ to an outer turning point r₂, this integral can be expressed as $\frac{\mathit{\omega }}{\mathrm{2}\mathrm{\Delta }{\mathit{\nu }}_{\mathit{\omega ,\ell }}}\mathrm{\equiv }{\mathrm{\int }}_{\mathit{s}}\mathit{k}\hspace{0.17em}\mathrm{d}\mathit{s}\mathrm{=}{\mathrm{\int }}_{{\mathit{r}}_{\mathrm{1}}}^{{\mathit{r}}_{\mathrm{2}}}\mathit{k}\frac{{\mathit{v}}_{\mathrm{g}}}{{\mathit{v}}_{\mathit{r}}}\hspace{0.17em}\mathrm{d}\mathit{r}\mathrm{-}\mathit{\pi }\mathit{/}\mathrm{4.}$(8)For low-degree acoustic waves $\mathit{\omega }\mathit{/}\mathrm{\left(}\mathrm{2}\mathrm{\Delta }{\mathit{\nu }}_{\mathit{\omega ,\ell }}\mathrm{\right)}\mathrm{\simeq }\mathit{\omega }{{}^{\mathrm{\int }}}_{\mathrm{0}}^{\mathit{R}}\mathrm{(}\mathrm{1}\mathit{/}\mathit{c}\mathrm{)}\mathrm{d}\mathit{r}\mathrm{-}\mathrm{(}\mathrm{1}\mathrm{+}\mathit{\ell }\mathrm{)}\mathit{\pi }\mathit{/}\mathrm{2}\hspace{0.17em}$, that is, Δν_ω,ℓ ≈ Δν.

Considering a wave excited very close to the surface with an outgoing component of amplitude A_ωℓ and an ingoing component that emerges at the surface with an amplitude ${\mathit{A}}_{\mathit{\omega \ell }}^{\mathrm{\prime }}$ after its first internal traversal, and with an amplitude of ${\mathit{A}}_{\mathit{\omega ,\ell }}^{\mathrm{\prime }}\mathit{R}\mathrm{\left(}\mathit{\omega }\mathrm{\right)}$ after a subsequent partial surface reflection on the back side of the star, the contribution to the amplitude spectrum can be expressed as $\sum _{\mathit{\omega ,\ell }}\left({\mathit{V}}_{\mathit{\omega ,\ell }}{\mathit{A}}_{\mathit{\omega ,\ell }}\mathrm{+}{\mathit{A}}_{\mathit{\omega ,\ell }}^{\mathrm{\prime }}{\mathrm{e}}^{\mathrm{i}\mathit{\delta }}[{\mathit{V}}_{\mathit{\omega ,\ell }}^{\mathrm{\prime }}{\mathrm{e}}^{\mathrm{-}\mathrm{i}\mathit{\omega }\mathit{/}\mathrm{\Delta }{\mathit{\nu }}_{\mathit{\omega ,\ell }}}\mathrm{+}{\mathit{V}}_{\mathit{\omega ,\ell }}^{\mathrm{\prime \prime }}\mathit{R}\mathrm{\left(}\mathit{\omega }\mathrm{\right)}{\mathrm{e}}^{\mathrm{-}\mathrm{2}\mathrm{i}\mathit{\omega }\mathit{/}\mathrm{\Delta }{\mathit{\nu }}_{\mathit{\omega ,\ell }}}]\right)\mathit{,}$(9)where V_ω,ℓ, ${\mathit{V}}_{\mathit{\omega ,\ell }}^{\mathrm{\prime }}$ and ${\mathit{V}}_{\mathit{\omega ,\ell }}^{\mathrm{\prime \prime }}$ are visibility factors and δ is a phase constant that depends on the type of radiation emitted. The reflection coefficient R is expected to be a monotonically decreasing function of frequency (for further details see Kumar 1993). We also assume that, for stochastically excited waves, the amplitudes and the visibility factors are a smooth function of frequency. Hence, the amplitude spectrum is modulated with the periodic functions cos(ω/ Δν_ω,ℓ) and cos(2ω/ Δν_ω,ℓ). Although in principle different values of Δν_ω,ℓ can be expected in different frequency ranges and for different degrees ℓ, their differences are too weak, and in the present study we fitted the data to just two periodic components, Δν₂ = Δν_ω,ℓ and Δν₁ = Δν_ω,ℓ/ 2. In addition, in some stars one of the signals could be masked by low-visibility factors or low reflection coefficients.

To illustrate the problem, we considered a 1.1 M_⊙ evolution sequence and plotted rays for two typical frequencies in the pseudo-mode range and three evolutionary stages in Fig. 12. Top panels are for frequencies ω = 1.1ω_a while bottom panels are for ω = 1.5ω_a. On the other hand, the star evolves from left (zero age main sequence, ZAMS) to right (red giant branch, RGB). For clarity, surface reflections have been omitted.

For the model close to the ZAMS (leftmost panels), the inward rays emerge at the surface making angles between 160° and 180° with the outward component. In this case, the outward and inward components of a given wave can only be simultaneously visible if they lie too close to the limb. Hence, their signal corresponding to Δν₂ ≃ Δν is highly attenuated in the power spectrum. In contrast, after one surface reflection the angle between the outward and inward components lies in the range 0° to 35°. This interference would give the Δν₁ ≃ Δν/ 2 separation and, hence, if it had the same intrinsic amplitude as the former, the Δν₁ separation would be easier to observe. In this case, however, the inward rays are only partially reflected on the far side (e.g. Kumar 1993) and hence the attenuation factor, corresponding to R(ω) in Eq. (9), is high. Since the actual observations show that in the Sun the dominant separation in the HIP pattern is Δν₁ (García et al. 1998), we can use this case as a qualitative reference between the two competing factors. For model S, we obtain Δν₁ = 72 μHz with our fit, for which a frequency range ~[5200, 6600] μHz was used. This result is in agreement with the observational value of 70.46 ± 2 μHz found by García et al. (1998). As mentioned in section 4.1, we reanalysed the solar data (GOLF and VIRGO Sun photometers) and obtained Δν₁ ~ 70 μHz (see Table 2), in agreement with previous values given in the literature. At higher frequencies we refitted the data with two sinusoidal components and found Δν₂ ~ 140 μHz (Table 2). This value is in agreement with the theoretical prediction already described by Garcia et al. (1998) if we assume that the photosphere is the only source of partial wave reflection. However, in this case the amplitude is much smaller because only waves close to the limb contribute to this interference pattern.

As the stars evolve, the angle between the outward ray and the first surface appearance of the inward refracted ray becomes smaller for the ℓ = 1 pseudo-modes. This is clearly apparent in Fig. 12. Hence, the Δν₂ ≃ Δν interference pattern becomes easier to observe (higher observed A₂ amplitudes). This phenomenon is due to the transition between acoustic rays smoothly bending through the stellar interior, and waves where the density scale height in the stellar inner structure becomes close to their wavelength and the ray behaviour at the inner turning point is closer to a two-layer reflection. Finally, in the most evolved model in Fig. 12 (upper and bottom right panels) the ℓ = 1 waves have a gravity character close to the centre that bends the ray into a loop.

We used Eq. (9) to reproduce the observed power spectrum schematically, and considered ray traces for a continuum spectrum of waves with frequencies between the acoustic cut-off frequency at the surface and about 1.5 times that value. This frequency interval spans approximately the observed pseudo-mode frequency range. Since we are only interested in reproducing the periodicities of the pseudo-mode spectra, constant amplitudes were considered as well as visibility factors either constant or proportional to cosΘ/2, where Θ, is the angular distance between the outward and inward waves. We determine the periodic signal in the spectrum with a non-linear fit to the above equation.

Figure 13 shows the angular distance between the inward and outward components before any surface reflection against the large separation Δν for models in a 1.1 M_⊙, Y = 0.28 evolution sequence with the code and physics indicated in Sect. 3.2. Here, mean values of the angular distance are computed by averaging those of the ray traces in the frequency range indicated above. Pseudo-modes with ℓ = 1 and 2 are averaged separately. Since no surface reflection is considered, these waves form the Δν₂ pattern.

Looking at the behaviour of the ℓ = 1 pseudo-modes in Fig. 13 (red points), we may conclude that for M = 1.1 M_⊙ stars that evolve to a Δν below some given threshold between 90 μHz and 60 μHz (corresponding to evolved main-sequence stars as that shown in the middle panel of Fig. 12), the Δν₂ pattern should become visible. Indeed, a smaller angular distance means more disc-centred interferences favouring a higher A₂. The critical Δν slightly changes with mass, which is lower for higher masses. In any case, all the stars in Table 2, except the Sun, are evolved to a point where the pattern with Δν₂ could be easily observed, as is in fact the case.

Regarding the ℓ = 2 pseudo-modes, it can be seen in Fig. 13 that the angle distance always remains above some 150° and hence this degree only contributes to the Δν₁ period in the pseudo-mode spectrum. The higher dispersion in the angular distance for models with a large separation between Δν = 60 and 90 μHz (evolved main-sequence stars and subgiants), clearly visible in Fig. 13, deserves a comment. According to the dispersion relation we used, for models evolved near the Terminal Age Main Sequence (TAMS) the characteristic frequencies of the ℓ = 2 waves become complex in a small radius interval above the core, thus increasing the resonant cavity of the pseudo-modes within a limited frequency range. An example of the ray path of these kinds of waves is shown in the middle of the bottom panel of Fig. 12 (green line). Here, the ray trace close to the centre is not of the acoustic type and the emerging ray is consequently scattered. Althought these results may be questionable in terms of the validity of the approximation used here, they do not have any observable consequences.

For the three most evolved stars in our data set the interference pattern coming from waves reflected back at the surface with period Δν₁ is not observed. The fact that for evolved stars the signal from the ℓ = 1 pseudo-modes do not suffer any attenuation from surface reflections may explain why for stars evolved to the RGB the Δν₁ signal becomes completely masked. However, it is interesting to note the another, possibly superimposed, cause for this circumstance. In principle, one might expect that radial waves propagates all the way from the surface to the centre and hence contributes to the signal only once reflected back, hence, with a period Δν₁. However there is a point in the evolution where the cut-off frequency in the core rises above the typical pseudo-mode frequencies, in which case the ray theory introduces an inner reflection. Although a plano-parallel approximation is questionable in terms of the wavelength-radius relation when we are too close to the centre, this approximation allows us to estimate the transmission coefficient T, as in the one-dimensional problem¹. If the equation T = 1/(1 + e^2K) is used with $\mathit{K}\mathrm{=}\mathrm{2}{{}^{\mathrm{\int }}}_{\mathrm{0}}^{{\mathit{r}}_{\mathrm{1}}}\sqrt{\mathrm{(}{\mathit{\omega }}_{\mathit{c}}^{\mathrm{2}}\mathrm{-}{\mathit{\omega }}^{\mathrm{2}}\mathrm{)}\mathit{/}{\mathit{c}}^{\mathrm{2}}}$, r₁ being the point where k_r = 0, it happens that, for red giant stars, radial waves in the observed frequency range of the pseudo-mode spectrum are mostly reflected at the core edge. Thus, at this stage, radial pseudo-modes also contribute to the Δν₂ signal and, hence, the Δν₁ period would hardly be observed.

We now discuss the relation between the large separation obtained from the eigenmodes, Δν_modes, and those corresponding to the pseudo-modes, Δν₁ and Δν₂. First, we verified that for waves with ℓ = 0 the phase travel time is always close to the asymptotic acoustic value, ${}^{\mathrm{\int }}\mathit{k}\mathrm{d}\mathit{l}\mathrm{\simeq }{{}^{\mathrm{\int }}}_{\mathrm{0}}^{\mathit{R}}\mathrm{1}\mathit{/}\mathit{c}\hspace{0.17em}\mathrm{d}\mathit{r}$. Hence, when these acoustic waves contribute to the HIPs after a surface reflection they give Δν₁ ≃ Δν_modes/ 2. In addition, most of the ℓ = 2 and, in some stars, ℓ = 1 travelling waves have similar acoustic characteristics, which also contributed to the same interference period. This is in agreement with Table 2, where both frequency separations match within the errors.

The case of Δν₂ is different. The grey points in Fig. 14 are the periods of the interference pattern computed according to Eq. (9) for travelling waves and no surface reflection. A wide range of masses and evolutionary stages from main sequence to the base of the red-giant branch were included. The red points are the observed Δν₂ values while blue points are 2Δν₁ when observed, including the Sun. The black line corresponds to Δν_modes = Δν₂. From Fig. 14 we can see that, for Δν_modes = 60–80 μHz, there are models with Δν₂< Δν_modes. In fact, they correspond to evolved stars up to the end of the main sequence or the subgiant phase, as it is the case for our sample of stars. Although the order of magnitude of the phase delay found for our models is similar to the observed one for this type of star, it is also apparent from the figure that KIC 1124411 (Δν_modes ~ 70 μHz) has a large separation, which is too high for this delay to appear. We obtain a mass of ~1 M_⊙ either by applying the scaling relations to ν_max and Δν_modes via an isochrone fit. The upper right corner of Fig. 12 is representative of this star. Hence we expect to observe the signal Δν₂ from waves reflected in the interior but our computations do not show any significant delay in the phase of the ℓ = 1 compared to the radial oscillations. Further work is need here; in particular, the asymptotic theory that we have used could be inadequate for these kinds of details.

For stars with the lower Δν_modes, two of them have Δν₂ ≃ Δν_modes and one star (KIC 11717120) has Δν₂< Δν_modes. With our isochrones fitting, we find that the latter is at the base of the RGB as well as KIC 7799349, thus we have two different results for stars with very similar parameters. With our simple simulation for models at this evolutionary stage we have both signals since, as noted above, radial oscillations are partially reflected at the core edge and, for these stars, Δν₂ ≃ Δν_modes. The observed period is a weighted average, but for a better comparison with the observations proper amplitudes need to be computed.

5. Conclusions

We measured the acoustic cut-off frequency and characteristics of the HIPs in six stars with solar-like pulsations observed by the Kepler mission. A comparison with the observed ν_max shows a linear trend with all stars lying in ~2σ. As a result, the values of log g derived from ν_cut agree to within the same accuracy with those derived from ν_max, but are substantially different in some cases from the values derived from spectroscopic fits (see Table 3). When comparing the stars with the models in the ν_cut–Δν plane, we found a departure in ν_cut from the expected values. It is possible to calculate, with theoretical information, a measurable function ${\mathit{\nu }}_{\mathrm{cut}}^{\mathrm{\ast }}\mathrm{=}\mathit{f}\mathrm{\left(}{\mathit{T}}_{\mathrm{eff}}\mathit{,}{\mathit{\nu }}_{\mathrm{cut}}\mathrm{\right)}$ that scales as $\mathit{g}\mathit{/}\sqrt{{\mathit{T}}_{\mathrm{eff}}}$ with no more than 0.3% deviation (for our set of models representative of our star sample and the Sun), however, we find that the observational ${\mathit{\nu }}_{\mathrm{cut}}^{\mathrm{\ast }}$ does not follow this relation so accurately. Rather surprisingly, we found that the frequency of maximum power, ν_max, follows the linear relation within errors. Hence, for the evolved stars considered in the present study it must be concluded that ν_max gives an even better estimate of log g than ν_cut with the current uncertainties.

Given the observed characteristics of the HIPs, our set of six stars can be divided into two groups. In three stars: KIC 3424541, KIC 7799349, and KIC 11717120, the only visible pattern is that due to the interference with inner refracted waves in the visible disc of the star and not close to the limb. In the other three stars (KIC 7940546, KIC 9812850, and KIC 11244118), the pattern is because partial reflection of the inward waves at the back of the star is also detected. This different behaviour is related to the wide spacing of the star, which reveals a dependence on the evolutionary stage.

When present, the period Δν₁, which corresponds to the interference of outward waves with their inward counterpart once reflected on the far side, always agrees with half the large separation, although, like the Sun, their values are a little higher. This result is also found theoretically, and the large separation derived from the pseudo-modes is closer to ^∫dr/c. However, the period Δν₂ corresponding to waves refracted in the inner part of the star can be substantially different from the large frequency separation. We interpret this by claiming the presence of waves of a mixed nature. For these stars, the phase ^∫kdl is smaller compared to the radial acoustic case. Although a detailed analysis is beyond the scope of the present study, simple ray theory calculations reveal that this kind of a phenomenon is expected with the same order of magnitude.

The pseudo-mode spectrum reveals information from the surface properties of the stars through ν_cut and from the interior, at least in the cases where the period Δν₂ is lower than the large separation derived from the eigenmodes. We are aware that the interference periods derived observationally are power-weighted averages and for a proper comparison with theoretical expectation some work along these lines should be addressed. This might be accomplished by computing transmission coefficients so that more realistic theoretical simulations of the interference phenomenon can be performed.

Online material

Appendix A: Figures of the other stars

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Acknowledgments

The authors of this paper thank Dr. J. Ballot, Dr. G. R. Davies, and P. L. Pallé for useful comments and discussions, as well as the entire Kepler team, without whom these results would not be possible. Funding for this Discovery mission is provided by NASA Science Mission Directorate. This research was supported in part by the Spanish National Research Plan under project AYA2010-17803. This research was supported in part by the National Science Foundation under Grant No. NSF PHY05-51164. R.A.G. has received funding from the European Community Seventh Framework Program (FP7/2007-2013) under grant agreement No. 269194 (IRSES/ASK), from the ANR (Agence Nationale de la Recherche, France) program IDEE (No. ANR-12-BS05-0008) “Interaction Des Étoiles et des Exoplanètes”, and from the CNES. S.M. acknowledges the support of the NASA grant NNX12AE17G.

References

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