Analysis of the acoustic cut-off frequency and HIPs in six Kepler stars with stochastically excited pulsations

Analysis of the acoustic cut-off frequency and HIPs in six Kepler stars with stochastically excited pulsations

1Introduction

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, . However, the oscillation power spectrum of the resolved Sun shows regular peak structure that extends well above [?]. This signal is interpreted as travelling waves whose interferences produce a well-defined pattern corresponding to the so-called pseudo-mode spectrum [?]. The amount and quality of the space data provided by the SoHO satellite [?] allowed us to measure these High-Frequency Peaks (HIPs) using Sun-as-a-star observations [?] from GOLF [?] and VIRGO [?] instruments. The change in the frequency pattern between the acoustic and the pseudo-modes enabled the proper determination of the solar cut-off frequency [?].

Theoretically, the cut-off frequency approximately scales as , where is the effective temperature, is the gravity, and the mean molecular weight, with all values measured at the surface. Hence, the observed can be used to constrain the fundamental stellar parameters. However, solar observations showed that changes, for example, with the solar magnetic activity cycle [?]. Therefore, the accuracy of the scaling relation given above needs to be discussed not only in a theoretical context but also taking into account observational constraints.

For stars other than the Sun, the detection of HIPs is challenging because of the shorter length of available datasets and the lower signal-to-noise ratio (SNR). Nevertheless, in this study we report on the analysis of the high-frequency part of the spectrum of six pulsating stars observed by Kepler [?] for which we were able to characterize the HIP pattern and the cut-off frequency. The time series analysis and the preparation of the spectra are detailed in Sect. 2. In Sect. 3 we describe how to estimate 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.

2Data Analysis

In this study ultra high-precision photometry obtained by NASA’s Kepler mission has been used to study the high-frequency region of six stars with solar-like pulsations. The stellar identifiers from the Kepler Input Catalogue names [?] are given in the first column of Table 1. Short-cadence time series [?] up to quarter 17 have been corrected for instrumental perturbations and properly stitched together using the Kepler Asteroseismic Data Analysis and Calibration Software [?].

For each star the long time series (4 years) are divided into consecutive subseries and the average of all the power spectral density (AvPSD) is computed in order to reduce the high-frequency noise in the spectrum. As in the solar case (e.g. [?], [?]) subseries of 4 days ( 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 SNR (which improves with the number of averaged spectra). For the same reasons (as also done in the solar case), we smooth – with a boxcar function – the AvPSD over 3 or 5 points depending on the evolutionary state of the star and the SNR of the spectra. 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 Figure 1 for KIC 11244118. Because of the short length of the subseries, it is not necessary to interpolate the gaps in the data [?], so we prefer the simplest possible analysis. However, we have verified that the results remain the same when using series that were interpolated with the inpainting algorithm [?].

Figure 1: Top panel: Smoothed averaged power spectral density of 4-day subseries of KIC 11244118. The red line is the severe smoothing used to normalize the spectrum (NAvPSD) shown in the lower panel (see text for details).
Figure 1: Top panel: Smoothed averaged power spectral density of 4-day subseries of KIC 11244118. The red line is the severe smoothing used to normalize the spectrum (NAvPSD) shown in the lower panel (see text for details).

To avoid subseries of low quality and to increase the SNR in the AvPSD we first remove all subseries with a duty cycle below . 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 preparation) systematically removes those data points from the final corrected time series. In Table 1 we summarize the number of 4-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.

Table 1: KIC name of each star, number of initial 4-day subseries, number of finally averaged subseries after removing those with a low SNR, and initial and final Kepler short-cadence quarters used in the analysis.
KIC Series Averaged series Qi - Qf
3424541 Q5-Q17
7799349 Q5-Q17
7940546 Q7-Q17
9812850 Q5-Q17
11244118 Q5-Q17
11717120 Q5-Q17

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 [?] in a range of masses from to in steps of with a solar composition (). The target stars cover evolutionary stages from late main sequence stars to early red giants.

Figure 2: Seismic HR diagram showing the position of the six stars analysed. The position of the Sun is indicated by the \odot symbol. The evolutionary tracks are computed using ASTEC .
Figure 2: Seismic HR diagram showing the position of the six stars analysed. The position of the Sun is indicated by the symbol. The evolutionary tracks are computed using ASTEC .

3The acoustic cut-off frequency

3.1Observations

The observed stellar power spectrum has different patterns in the eigenmode and pseudo-mode regions. In particular, the mean frequency separation, , between consecutive peaks is different below and above the cut-off frequency. In the first case, it corresponds to half the mean large frequency spacing, while above it is the period of the interference pattern. In addition, a phase shift between both patterns appears in the transition region. The frequency where 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, , 3, and even, , 2, pairs of modes due to the small frequency resolution of Hz) from a few orders before up to the highest visible peak in the spectrum. To take proper account of the underlying background contribution (due to convective movements at different scales, faculae, and magnetic/rotation signal), we prefer to divide the AvPSD by the same spectrum heavily smoothed (see the red line in the top panel of Figure 1) instead of using a theoretical model [?]. Although it has been demonstrated that a two-component model usually fits properly the background of stars [?], the accuracy in the transition region between the eigenmode bump and the high-frequency region dominated by the HIPs is not properly described, which is the region we are interested in. 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 Figure 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 , and 3) the large amplitude dispersion of the peaks between 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 and the photon-noise dominated region into two regions. We arbitrarily choose a frequency where the amplitudes start to be very small and treat separately the power spectrum before and after this frequency. We denote by p-mode region the low-frequency zone and pseudo-modes region the other one. Several tests have been done 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 the SNR. We have 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 Figure 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. It is important to note that we are interested only in the frequency of the centroids of the peaks, and for that a Lorentzian profile is a good approximation. In Figs. Figure 3 and Figure 4 we show the results of the fits for KIC 3424541 in the p-mode and pseudo mode regions respectively. The figures for the other stars can be found in Appendix A.

Figure 3: Sections of the NAvPSD used to fit the eigenmode region by groups of six modes in KIC 3424541. The red line represents the final fit.
Figure 3: Sections of the NAvPSD used to fit the eigenmode region by groups of six modes in KIC 3424541. The red line represents the final fit.
Figure 4: Pseudo-mode region of the NAvPSD in KIC 3424541. The red line is the result of the fit.
Figure 4: Pseudo-mode region of the NAvPSD in KIC 3424541. The red line is the result of the fit.

After fitting the two parts of the spectra, we compute the frequency differences of consecutive peaks: . As an example, the resultant frequency differences for KIC 3424541 are plotted in Figure 5.

Figure 5: Consecutive frequency differences of KIC 3424541. The blue dashed and dotted lines are respectively the weighted mean value of the frequency differences in the p-mode (20.75\pm 0.20\,\muHz) and pseudo-mode regions (37.31\pm 1.58\,\muHz) . The red symbol represents the estimated cut-off frequency, \nu_{\mathrm{cut}}= 1210.67\pm 15.70 \,\muHz.
Figure 5: Consecutive frequency differences of KIC 3424541. The blue dashed and dotted lines are respectively the weighted mean value of the frequency differences in the p-mode (Hz) and pseudo-mode regions (Hz) . The red symbol represents the estimated cut-off frequency, Hz.

Two different regions are clearly visible in the frequency differences of KIC 3424541. Part of these differences – those at lower frequencies – are centred around Hz corresponding approximately to half of the large spacing, (see Table ?), owing to the alternation between odd and even modes. At a certain frequency, between 1200 and 1230 Hz, the frequency separations increase and remain roughly constant around Hz, although with a higher dispersion (see next section for further details). The observed acoustic cut-off frequency, , 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 Figure 5 for KIC 3424541. The results of this analysis for the six stars studied here are given in Table ?.

KIC number of each star, effective temperature from [?] with an error bar of 84 K, computed using the A2Z pipeline [?], , , , , , and computed as explained in the text.
KIC
Sun (GOLF) 5777 3097 116
Sun (VIRGO) 5777 3097 116
3424541
7799349
7940546
9812850
11244118
11717120

3.2Comparison between the observed and theoretical cut-off frequency

As noted in Sect. Section 1, the cut-off frequency at the stellar surface scales approximately as , 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 has large uncertainties [?] and a direct test is not possible. Alternatively, it has been demonstrated empirically that the frequency of maximum mode amplitude follows a similar scale relation [?]. As can be seen in Figure 6, the observational data show such a linear relation between and , although some stars deviate 2 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 can deviate by as much as from the scale relation for representative models of our stars. On the other hand, following [?], where theoretical values of based on stochastic mode excitation were computed, one can find departures from the scale relation as large as for a model of a star with and Hz.

Figure 6: Observed cut-off frequencies \nu_{\mathrm{cut}} versus maximum mode amplitude frequencies \nu_{\mathrm{max}}. Blue points with errors are values for the six stars considered in this paper and the Sun (the rightmost point). A straight line through the Sun has been drawn for guidance.
Figure 6: Observed cut-off frequencies versus maximum mode amplitude frequencies . Blue points with errors are values for the six stars considered in this paper and the Sun (the rightmost point). A straight line through the Sun has been drawn for guidance.

We can gain a deeper insight by comparing the observational and theoretical results in the plane. For the theoretical calculations we use a set of models computed with the CESAM code [?]. These correspond to evolutionary tracks from the zero-age main sequence up to the red-giant phase, with masses between and , and helium abundances between and . Models with and without overshooting were considered. Other parameters were fixed to standard values, such as the metallicity to . Finally, the CEFF equation of state [?] and a 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 turn to have frequencies below a cut-off frequency given by , where 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 . Another advantage of using such a 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 have used this value for all the models as the uppermost point, but for some red-giant stars we found that the maximum value of in the atmosphere can be located at higher optical depths; hence, we have taken that maximum value as representative of the observed . From an interpolation to the solar mass and radius (with fixed) we obtain from our set of models a value of = 5125 Hz for the Sun, which is in agreement with the observed one: Hz [?]. If model S [?] is considered, the differences are a little larger, about . It is important to remember that the solar changes with the magnetic activity cycle [?]; 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 in the same way. Specifically, we have computed the large separation from a polynomial fit of the form , where is the radial order and is the radial frequency closest to with radial order . For the models, is estimated by assuming a linear relation to the cut-off frequency. The frequencies considered are those of the modes with radial orders . is the Legendre polynomial of degree , and is the frequency normalized to the interval . A third order polynomial has been used. From the Tasoul equation we expect that , whereas other terms will 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 Hz from the observations and Hz from model S.

Figure 7 shows against the large separation 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 2000 Hz it seems hard to explain the dispersion in their values.

Figure 7: \nu_{\mathrm{cut}} versus \Delta \nu_{\mathrm{modes}} for our set of theoretical models (grey dots). Blue points correspond to the observed stars, including the Sun (upper-right point).
Figure 7: versus for our set of theoretical models (grey dots). Blue points correspond to the observed stars, including the Sun (upper-right point).

We can proceed forward by taking into account the values of . First, given the approximated scale relation for , we compute the residuals:

where = 5079.5 Hz is a reference cut-off frequency, corresponding to a model in our dataset close to the Sun (, , , = 5799.2 K). The values are shown in Figure 8. As seen in the figure, the differences can be as large as . In fact, the scale relation is derived theoretically by approximating the density scale height, , by the pressure scale height, , which is strictly valid in an isothermal atmosphere, and further by taking the equation of state of a mono-atomic ideal gas. In Figure 8, the blue points correspond to the relative differences obtained by replacing in Equation 1 by . Thus, the first condition introduces the highest departures from the scale relation, which was expected because in the range of considered, the gas is mainly mono-atomic at the surface.

Nevertheless, as seen in Figure 8, is mainly a function of . Hence, the scale relation can be improved if we subtract a polynomial fit from . Only models with K and Hz have been considered 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 on . In particular, the red points in Figure 8 are obtained by replacing in Equation 1 by:

where and . The standard deviation for these residuals is . Hence, the modified cut-off frequencies, that we calibrated on the Sun deviate by from the scaling relation, at least for our set of models.

Figure 8: Relative differences \delta_{\rm c} as a function of T_{\mathrm{eff}} (black dots). Blue points are the same relative differences but considering the isothermal cut-off frequency \omega_{\rm a} instead of \omega_{\rm c}. Red points are the residuals after correcting by the polynomial fit indicated in the text. The fit is limited to models with T_{\mathrm{eff}}<7000 K and \nu_{\mathrm{cut}}>500 \muHz.
Figure 8: Relative differences as a function of (black dots). Blue points are the same relative differences but considering the isothermal cut-off frequency instead of . Red points are the residuals after correcting by the polynomial fit indicated in the text. The fit is limited to models with K and Hz.
Table 2: Values of derived from spectroscopy, and for each star in our sample. The mean density is in g/cm and the large separation in Hz. See text for details.
KIC
3424541
7799349
7940546
9812850
11244118
11717120

In a similar way, we compute the deviation of from its expected scale relation, namely:

where is the value for our reference model, the same as the one used in . This quantity is shown in Figure 9 for our set of models.

Figure 9: Relative differences \delta_{\Delta} versus T_{\mathrm{eff}} for the set of models indicated in the text (black dots). The red points are the residuals after the polynomial fit. This fit is limited to models with T_{\mathrm{eff}}<7000\,K and \Delta\nu>10\muHz.
Figure 9: Relative differences versus for the set of models indicated in the text (black dots). The red points are the residuals after the polynomial fit. This fit is limited to models with K and Hz.

From Figure 9 we can determine that the differences between the large separation computed from p-modes and the scaling relation can be as large as for solar-like pulsators with K. However, as shown in this figure, the differences depend mainly on . This was previously noted by [?] and allows for a correction to in order to get better a estimate of the mean density. Here we will use a fit similar to that considered for . As in the previous case, we have considered only those models with K and Hz, which include all the stars used in the present work. The red points in Figure 9 correspond to the residuals obtained after replacing in Equation 3 by:

where and with Hz. The standard deviation for these residuals is . Given the differences between the observed and theoretical values for the Sun, we have calibrated Equation 4 to the observed solar value.

Figure 10: Effective \nu_{\mathrm{cut}}^* versus the effective \Delta \nu_{\mathrm{modes}}^* for our set of models (grey dots) and for the observed stars (blue dots). Red points are observed values of the frequencies of maximum amplitude properly scaled, (\nu_{\mathrm{cut}}/\nu_{\mathrm{max}})_{\odot}\nu_{\mathrm{max}}.
Figure 10: Effective versus the effective for our set of models (grey dots) and for the observed stars (blue dots). Red points are observed values of the frequencies of maximum amplitude properly scaled, .

Figure 10 shows against 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 ones 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 Figure 10 are proportional to the observed values of the frequency of maximum amplitude scaled to the Sun, . For our limited set of stars, follows the scaling relation better than . This is a little surprising because, as noted before, theoretically, we would expect the opposite, even more so when the effective is used.

This result can be understood in terms of the inferred surface gravity since we can use either or to estimate . The results are summarized in Table 2 and compared to the spectroscopic . To estimate the errors in we have considered only the observational errors in and , while for we have included the 1- value of found in the scaling relation plus an error in derived from assuming an unknown composition with standard stellar values. In particular, we have taken ranges of and for the helium abundance and the metallicity respectively and thus estimated the error in the mean molecular weight by some . As expected, the values obtained from are quite similar to those reported by [?] because they are based on the same global parameters with very similar values. The values derived from and are also much closer to each other than those derived spectroscopically, which, as mentioned before, have large observational uncertainties.

4The HIP region

4.1Observations

Using Sun-as-a-star observations from the GOLF instrument on board SoHO, [?] uncovered the existence of a sinusoidal pattern of peaks above as the result of the interference between two components of a travel wave generated on the front side of the Sun with a frequency , where the inward component returns to the visible side after a partial reflection on the far side of the Sun [?]. Therefore, the frequency spacing of 70 Hz 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 ( 3600s). This value is roughly half of the large frequency spacing of the star and we will call it . [?] also speculated that, above a given frequency, a second pattern should become visible with a double frequency spacing (close to the large frequency separation), 140 Hz. This pattern [?] – usually visible in imaged instruments [?] – corresponds to the interference between outward emitted waves and the inward components that arrive at the visible side of the Sun after the refraction at the inner turning point (non-radial waves).

In order to get a global estimate of the frequency of the interference patterns in the HIP region of our sample of stars, two sine waves are fitted above the cut-off frequency. The amplitudes and the frequencies of both sine waves are summarized in Table ?. We have also re-analysed the GOLF and VIRGO data following the same procedure but at a higher frequency range, between 7 and 8 mHz, compared to the original analyses performed by [?] and [?]. In this way, we have been able to obtain the second periodicity at 140 Hz (see Table ?).

We tried fitting various functions and opted for the simplest one. Indeed, the amplitude of the interference patterns decreases with frequency in a way close to an exponential decrease. The additional parameters required give more unstable fits with a heavy dependence on the guess parameters. Because we obtained the same qualitative results leading to the same classification of stars, we preferred this approach. 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 where the two amplitudes of the sine waves, and , are similar (KIC 7940546, KIC 9812850, KIC 11244118), and those where is much larger than (KIC 3424541, KIC 7799349, KIC 11717120). An example of each group of stars is given in Figure 11 for stars KIC 7940546 and KIC 11717120. It is important to notice that 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. In this pioneering work, we have favoured the fits with constant amplitudes to avoid adding more unknown parameters to the fit.

Figure 11: HIPs region of KIC 11717120 (top), where the HIP pattern is dominated by the sine wave (red line) owing to the interference between direct and refracted waves, and of KIC 7940546 (bottom), where the two HIP patterns can be determined. The blue and green line in the bottom panel correspond to the two fitted sine waves with frequencies \Delta\nu_1 and \Delta\nu_2 respectively. The green and blue lines have been shifted by 0.05 for clarity. The red line is the actual fit.
HIPs region of KIC 11717120 (top), where the HIP pattern is dominated by the sine wave (red line) owing to the interference between direct and refracted waves, and of KIC 7940546 (bottom), where the two HIP patterns can be determined. The blue and green line in the bottom panel correspond to the two fitted sine waves with frequencies \Delta\nu_1 and \Delta\nu_2 respectively. The green and blue lines have been shifted by 0.05 for clarity. The red line is the actual fit.
Figure 11: HIPs region of KIC 11717120 (top), where the HIP pattern is dominated by the sine wave (red line) owing to the interference between direct and refracted waves, and of KIC 7940546 (bottom), where the two HIP patterns can be determined. The blue and green line in the bottom panel correspond to the two fitted sine waves with frequencies and respectively. The green and blue lines have been shifted by 0.05 for clarity. The red line is the actual fit.

The stars of the first group have larger 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 is much bigger than ). 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 below 40 Hz.

4.2HIPs and stellar evolution

In order to reproduce theoretically the periodic signals expected from the pseudo-mode spectrum we will follow the interpretation given by [?] and [?] 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.

Let us start by using a gravito–acoustic ray theory approximation with the following dispersion relation [?]:

where is the radial component of the wave number, , the horizontal component, is the buoyancy frequency, is a generalized cut-off frequency given by

and is the density scale height. In the ray approximation the turning points are given by , 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 . Here, and are the usual polar coordinates and the group velocity has the following components:

According to the ray theory, the general solution of the wave equation can be expressed as a superposition of rays of the form , where the two signs corresponds to outgoing and ingoing waves and the integral is computed over the ray path , 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 to an outer turning point , this integral can be expressed as

For low-degree acoustic waves , that is, .

Considering a wave excited very close to the surface with an outgoing component of amplitude and an ingoing component that emerges at the surface with an amplitude after its first internal traversal, and with an amplitude of after a subsequent partial surface reflection on the back side of the star, the contribution to the amplitude spectrum can be expressed as:

where , and are visibility factors, and is a phase constant that depends on the type of radiation emitted. The reflection coefficient is expected to be a monotonically decreasing function of frequency [?]. We also assume that, for stochastically excited waves, the amplitudes and the visibility factors will be a smooth function of frequency. Hence, the amplitude spectrum will be modulated with the periodic functions and . Although in principle different values of can be expected in different frequency ranges and for different degrees , their differences are too weak, and in the present study we fitted the data to just two periodic components, and . 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 have considered a evolution sequence and plotted in Figure 12 rays for two typical frequencies in the pseudo-mode range and three evolutionary stages. Top panels are for frequencies while bottom panels are for . On the other hand the star evolves from left (ZAMS) to right (RGB). For clarity, surface reflections have been omitted.

Figure 12:  Ray path for waves with angular degrees \ell=1 (red) and \ell=2 (green) for a 1.1M_{\odot} model evolution sequence. For clarity, surface reflections have been omitted. The upper part shows rays with a frequency a 10% greater than the acoustic cut-off frequency \omega_{\rm a} while the bottom panels corresponds to rays with a frequency 50% greater than \omega_{\rm a}. The large separation indicated is the integral (2\int_0^R 1/c)^{-1} and can be used to identified the models in Fig. . The leftmost panels correspond to a model near the ZAMS, the middle panels are for a model near the TAMS, and the rightmost panels are for a model in the RGB. The dotted circles indicate the base of the convection zone.
Figure 12: Ray path for waves with angular degrees (red) and (green) for a model evolution sequence. For clarity, surface reflections have been omitted. The upper part shows rays with a frequency a 10% greater than the acoustic cut-off frequency while the bottom panels corresponds to rays with a frequency 50% greater than . The large separation indicated is the integral and can be used to identified the models in Fig. . The leftmost panels correspond to a model near the ZAMS, the middle panels are for a model near the TAMS, and the rightmost panels are for a model in the RGB. The dotted circles indicate the base of the convection zone.

For the model close to the ZAMS (leftmost panels) the inward rays emerge at the surface making angles between and with the outward component. In this case, the outward and inward components of a given wave can be simultaneously visible only if they lie too close to the limb. Hence, their signal corresponding to will be highly attenuated in the power spectrum. On the other hand, after one surface reflection the angle between the outward and inward components lies in the range to . This interference would give the separation and, hence, had it the same intrinsic amplitude as the former, the separation would be easier to observe. But in this case the inward rays are only partially reflected on the far side (e.g. [?]) and hence the attenuation factor, corresponding to in Equation 6, is high. Since the actual observations show that in the Sun the dominant separation in the HIP pattern is [?] we can use this case as a qualitative reference between the two competing factors. For model S, we obtain = 72 Hz with our fit, for which a frequency range [5200,6600] Hz was used. This result is in agreement with the observational value of 70.46 2 Hz found by [?]. As mentioned in section 4.1, we re-analysed the solar data (GOLF and VIRGO/SPM) and obtained 70 Hz (see Table ?), in agreement with previous values given in the literature. At higher frequencies we re-fitted the data with two sinusoidal components and found 140Hz (Table ?). 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 pseudo-modes. This is clearly apparent in Figure 12. Hence, the interference pattern becomes easier to observe (higher observed 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 Figure 12 (upper and bottom right panels) the waves have a gravity character close to the centre that bends the ray into a loop.

In order to reproduce the observed power spectrum schematically we used Equation 6, and considered ray traces for a continuum spectrum of waves with frequencies between the acoustic cut-off frequency at the surface and about a times that value. This frequency interval approximately spans the observed pseudo-mode frequency range. Since we are interesting only in reproducing the periodicities of the pseudo-mode spectra, constant amplitudes were considered, and visibility factors either constant or proportional to , where , is the angular distance between the outward and inward waves. We determine the periodic signal in the spectrum by 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 for models in a , evolution sequence with the code and physics indicated in section ?. 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 and 2 are averaged separately. Since no surface reflection is considered, these waves form the pattern.

Looking at the behaviour of the pseudo-modes in Figure 13 (red points) we may conclude that for stars that evolve to a below some given threshold between Hz and Hz (corresponding to evolved main sequence stars as the one shown in the middle panel of Figure 12), the pattern should become visible. Indeed, a smaller angular distance means more disc-centred interferences favouring a higher . The critical slightly changes with mass, being lower for higher masses. In any case all the stars in Table ?, except the Sun, are evolved to a point where the pattern with could be easily observed, as is in fact the case.

Figure 13: Angular distance between inward and outward waves against \Delta \nu for a 1.1M_{\odot} evolution sequence. Frequencies between the acoustic cut-off frequency at the surface and about 50\% greater than \omega_{\rm a} are considered. For every model in the sequence, frequency-averaged values of the angular distance and 1\sigma dispersions are calculated for \ell=1 (red points) and \ell=2 (green points) waves separately. An angle of 180^{\circ} means no inner reflection.
Figure 13: Angular distance between inward and outward waves against for a evolution sequence. Frequencies between the acoustic cut-off frequency at the surface and about greater than are considered. For every model in the sequence, frequency-averaged values of the angular distance and dispersions are calculated for (red points) and (green points) waves separately. An angle of means no inner reflection.

Regarding the pseudo-modes, it can be seen in Figure 13 that the angle distance always remains above some and hence this degree contributes only to the period in the pseudo-mode spectrum. The higher dispersion in the angular distance for models with a large separation between and Hz (evolved main sequence stars and subgigants), clearly visible in Figure 13, deserves a comment. According to the dispersion relation used in this study, for models evolved near the TAMS the characteristic frequencies of the 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 at the bottom middle panel of Figure 12 (green line). Here, the ray trace close to the centre is not of the acoustic type and of the emerging ray is consequently scattered. Although 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 dataset the interference pattern coming from waves reflected back at the surface with period is not observed. Because, as noted before, for evolved stars the signal from the pseudo-modes do not suffer any attenuation from surface reflections, it is possible that this explains why for stars evolved to the RGB the signal becomes completely masked. However, it is interesting to note the another, possible 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, thence with a period . 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 , as for the one dimensional problem.1

If the equation is used with , being the point where , it happens that, for red giants 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 signal and hence the period would hardly be observed.

Let us now discuss the relation between the large separation obtained from the eigenmodes, and those corresponding to the pseudo-modes, and . First, we have verified that for waves with the phase travel time is always close to the asymptotic acoustic value, . Hence, when these acoustic waves contribute to the HIPs after a surface reflection they give . In addition, most of the and, in some stars, travelling waves have similar acoustic characteristics, which also contributed to the same interference period. This is in agreement with Table ?, where both frequency separations match within the errors.

The case of is different. The grey points in Fig. ? are the periods of the interference pattern computed according to Equation 6 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 values while blue points are when observed, including the Sun. The black line corresponds to . From Fig. ? we can see that for Hz there are models with . 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 (Hz) has a large separation – too high for this delay to appear. We obtain a mass of either by applying the scaling relations to and by doing an isochrone fit. The upper right corner of Figure 12 is representative of this star. Hence we expect to observe the signal from waves reflected in the interior but our computations do not shown any significant delay in the phase of the compared to the radial oscillations. Further work is need here; in particular, the asymptotic theory that we have used could be inadequate for such details.

For the stars with the lower two of them have and one star (KIC 11717120) has . 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, . The observed period will be a weighted average, but for a better comparison with the observations proper amplitudes need to be computed.

Grey points are periods of the pseudo-mode interference pattern for evolution sequences with masses between 0.9M_{\odot} and 2M_{\odot}. For the sake of clarity only waves with no surface reflections are shown. Red points correspond to the observed \Delta\nu_2 values and blue points correspond to 2\Delta\nu_1 when observed. The black line corresponds to \Delta \nu_{\mathrm{modes}}=\Delta\nu_2
Grey points are periods of the pseudo-mode interference pattern for evolution sequences with masses between and . For the sake of clarity only waves with no surface reflections are shown. Red points correspond to the observed values and blue points correspond to when observed. The black line corresponds to

.

5Conclusions

In this study the acoustic cut-off frequency and the characteristics of the HIPs have been measured in six star with solar-like pulsations observed by the Kepler mission. A comparison with the observed shows a linear trend with all stars lying in 2-. As a result, the values of derived from agree to within the same accuracy with those derived from but are substantially different in some cases from the values derived from spectroscopic fits (see Table 2). When comparing the stars with the models in the plane we found a departure in from the expected values. While using theoretical information it is possible to calculate a measurable function that scales as with no more than deviation (for our set of models representative of our star sample and the Sun), we find that the observational does not follow this relation so accurately. Rather surprisingly, we found that the frequency of maximum power, , follows the linear relation within errors. Hence, for the evolved stars considered in the present study it must be concluded that gives an even better estimate of than 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 the one 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 due to 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 , 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, the large separation derived from the pseudo-modes being closer to ). However, the period 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 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 such a phenomenon is expected with the same order of magnitude.

The pseudo-mode spectrum reveals information not only from the surface properties of the stars through but also from the interior, at least in the cases where the period 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 done.


AFigures of the other stars

Figure 14: Fitted (red line) NAvPSD of KIC 7799349 by groups of nine peaks in the low-frequency range (p modes).
Figure 14: Fitted (red line) NAvPSD of KIC 7799349 by groups of nine peaks in the low-frequency range (p modes).
Figure 15: Fitted (red line) NAvPSD of KIC 7799349 in the high-frequency range (pseudo modes).
Figure 15: Fitted (red line) NAvPSD of KIC 7799349 in the high-frequency range (pseudo modes).
Figure 16: Consecutive frequency differences (KIC 7799349); that is, the separations between the fitted peaks for NAvPSD. Two levels appear, one around \Delta\nu/2 corresponding to p-modes (blue dashed line with a weighted mean of 16.64 \pm 0.09 \muHz) and other around \Delta\nu corresponding to pseudo-modes (blue dotted line with a weighted mean of 35.23 \pm 1.52 \muHz). The red symbol is the estimate of the acoustic cut-off frequency (895.47 \pm 15.38 \muHz).
Figure 16: Consecutive frequency differences (KIC 7799349); that is, the separations between the fitted peaks for NAvPSD. Two levels appear, one around corresponding to p-modes (blue dashed line with a weighted mean of 16.64 0.09 Hz) and other around corresponding to pseudo-modes (blue dotted line with a weighted mean of 35.23 1.52 Hz). The red symbol is the estimate of the acoustic cut-off frequency (895.47 15.38 Hz).
Figure 17: Fitted (red line) NavPSD of KIC 7940546 by groups of five peaks in the low-frequency range (p modes).
Figure 17: Fitted (red line) NavPSD of KIC 7940546 by groups of five peaks in the low-frequency range (p modes).
Figure 18: Fitted (red line) NAvPSD of KIC 7940546 in the high-frequency range (pseudo-modes).
Figure 18: Fitted (red line) NAvPSD of KIC 7940546 in the high-frequency range (pseudo-modes).
Figure 19: Consecutive frequency differences (KIC 7940546), that is the separations between the fitted peaks for NAvPSD. Two levels appear, one around \Delta\nu/2 corresponding to p-modes (blue dashed line with a weighted mean of 29.50 \pm 0.07 \muHz) and other around \Delta\nu corresponding to pseudo-modes (blue dotted line with a weighted mean of 45.71 \pm 2.12 \muHz). The red symbol is the estimate of the acoustic cut-off frequency (1940.59\pm 20.71 \muHz).
Figure 19: Consecutive frequency differences (KIC 7940546), that is the separations between the fitted peaks for NAvPSD. Two levels appear, one around corresponding to p-modes (blue dashed line with a weighted mean of 29.50 0.07 Hz) and other around corresponding to pseudo-modes (blue dotted line with a weighted mean of 45.71 2.12 Hz). The red symbol is the estimate of the acoustic cut-off frequency (1940.59 20.71 Hz).
Figure 20: Fitted (red line) NAvPSD of KIC 9812850 by groups of four peaks in the low-frequency range (p modes).
Figure 20: Fitted (red line) NAvPSD of KIC 9812850 by groups of four peaks in the low-frequency range (p modes).
Figure 21: Fitted (red line) NAvPSD of KIC 9812850 in the high-frequency range (pseudo-modes).
Figure 21: Fitted (red line) NAvPSD of KIC 9812850 in the high-frequency range (pseudo-modes).
Figure 22: Consecutive frequency differences (KIC 9812850); that is, the separations between the fitted peaks for NAvPSD. Two levels appear, one around \Delta\nu/2 corresponding to p-modes (blue dashed line with a weighted mean of 32.53 \pm 0.12 \muHz) and other around \Delta\nu corresponding to pseudo-modes (blue dotted line with a weighted mean of 58.13 \pm 2.39 \muHz). The red symbol is the estimate of the acoustic cut-off frequency (1898.08 \pm 24.29 \muHz).
Figure 22: Consecutive frequency differences (KIC 9812850); that is, the separations between the fitted peaks for NAvPSD. Two levels appear, one around corresponding to p-modes (blue dashed line with a weighted mean of 32.53 0.12 Hz) and other around corresponding to pseudo-modes (blue dotted line with a weighted mean of 58.13 2.39 Hz). The red symbol is the estimate of the acoustic cut-off frequency (1898.08 24.29 Hz).
Figure 23: Fitted (red line) NAvPSD of KIC 11244118 by groups of five peaks in the low-frequency range (p modes).
Figure 23: Fitted (red line) NAvPSD of KIC 11244118 by groups of five peaks in the low-frequency range (p modes).
Figure 24: Fitted (red line) NAvPSD of KIC 11244118 in the high-frequency range (pseudo modes)
Figure 24: Fitted (red line) NAvPSD of KIC 11244118 in the high-frequency range (pseudo modes)
Figure 25: Consecutive frequency differences (KIC 11244118); that is, the separations between the fitted peaks for NAvPSD. Two levels appear, one around \Delta\nu/2 corresponding to p-modes (blue dashed line with a weighted mean of 35.54 \pm 0.08 \muHz) and other around \Delta\nu corresponding to pseudo-modes (blue dotted line with a weighted mean of 61.02 \pm 2.19 \muHz). The red symbol is the estimation of the acoustic cut-off frequency (1968.69 \pm 21.48 \muHz).
Figure 25: Consecutive frequency differences (KIC 11244118); that is, the separations between the fitted peaks for NAvPSD. Two levels appear, one around corresponding to p-modes (blue dashed line with a weighted mean of 35.54 0.08 Hz) and other around corresponding to pseudo-modes (blue dotted line with a weighted mean of 61.02 2.19 Hz). The red symbol is the estimation of the acoustic cut-off frequency (1968.69 21.48 Hz).
Figure 26: Fitted (red line) NAvPSD of KIC 11717120 by groups of sixteen peaks in the low-frequency range (p-modes).
Figure 26: Fitted (red line) NAvPSD of KIC 11717120 by groups of sixteen peaks in the low-frequency range (p-modes).
Figure 27: Fitted (red line) NAvPSD of KIC 11717120 in the high-frequency range (pseudo-modes).
Figure 27: Fitted (red line) NAvPSD of KIC 11717120 in the high-frequency range (pseudo-modes).
Figure 28: Consecutive frequency differences (KIC 11717120); that is, the separations between the fitted peaks for NAvPSD. Two levels appear, one around \Delta\nu/2 corresponding to p-modes (blue dashed line with a weighted mean of 16.71 \pm 0.09 \muHz) and other around \Delta\nu corresponding to pseudo-modes (blue dotted line with a weighted mean of 29.85 \pm 0.66 \muHz). The red symbol is the estimation of the acoustic cut-off frequency (958.38 \pm 11.56 \muHz).
Figure 28: Consecutive frequency differences (KIC 11717120); that is, the separations between the fitted peaks for NAvPSD. Two levels appear, one around corresponding to p-modes (blue dashed line with a weighted mean of 16.71 0.09 Hz) and other around corresponding to pseudo-modes (blue dotted line with a weighted mean of 29.85 0.66 Hz). The red symbol is the estimation of the acoustic cut-off frequency (958.38 11.56 Hz).

Footnotes

  1. For radial oscillations the full adiabatic equations are of second order and hence a WKB analysis can be done without the approximations assumed in the dispersion relation Eq. (Equation 5). The qualitative results given in this paragraph rely solely on the assumption of a one-dimensional problem.
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