The highest mass Kepler red giants— I. Global asteroseismic parameters of 48 stars

$\nu_{\text{max}}$ itself is not a value that can be defined unambiguously. It measures the frequency of maximum power but, considering the stochastic nature of solar-like oscillations, the power of each oscillation mode may change over time as the modes are continually damped and re-excited. There are also several different ways to measure $\nu_{\text{max}}$. The most commonly used method is to fit both the background and a Gaussian for the power excess, the mean of which gives the $\nu_{\text{max}}$ estimate (see Hekker et al. 2011 and references therein). Another method, used by both Y18 and this work, is to again perform a background fit and subtraction, and then heavily smooth the remaining power excess. The peak of this smoothed power excess would then be the $\nu_{\text{max}}$ estimate (Kjeldsen et al., 2005; Huber et al., 2009). Generally, the difference between these two methods is very small, but for some stars they can differ considerably. This certainly impacts estimation of the mass.

We recalculate the $\nu_{\text{max}}$ for every star using the pySYD program (Chontos et al., 2022), which is a python adaptation of the SYD pipeline (Huber et al., 2009) used by Y18. pySYD’s default options are designed to fit the power spectra of cool dwarf stars instead of red giants, and we therefore used a few specific non-default options (see the documentation for explanation of these parameters): ind_width of 1, lower_bg of 1, smooth_ps of 0.5, lower_ex of 1, smooth_width of 5, and binning of 0.01. The majority of these parameters correspond to different smoothing parameters used for pySYD’s different fitting steps, as well as power spectrum boundaries. We highlight in particular that lower_bg of 1 indicates we are only fitting the background for $\nu$ > 1 $\mu$Hz, which is the cut-off frequency of our high-pass filter. pySYD chooses the number of background models dynamically, and chose 2 profiles for each of our stars. Note that granulation background is fit using a Harvey profile (Harvey, 1985) which is defined as a Lorentzian, however it is common for pipelines to fit super-Lorentzians or other similar functions to the background (see discussion in Mathur et al. 2011). Both this work and Y18 fit the background with profiles proportional to $1/(1+x^{2}+x^{4})$ where x is a linear function of frequency. pySYD outputs both a Gaussian and a smoothed estimate of the $\nu_{\text{max}}$, as discussed above. For our work we followed Y18 in adopting the smoothed estimate. pySYD calculates uncertainties by performing the fitting process a large number of times on perturbed power spectra and provides the standard deviations of the posterior distributions (Chontos et al., 2022). We used an iteration count of 200 for our stars.

We present the percentage difference between the $\nu_{\text{max}}$ measured in this work and those quoted in Y18 in Figure 1a. We see scatter in the measured $\nu_{\text{max}}$ consistent with the quoted uncertainties, and a small systematic offset introduced by our refitting process. The offset in $\nu_{\text{max}}$ is due to differences in background noise levels between the two works. The background noise in red giants is made up of frequency-dependent granulation and white noise. Both pySYD and SYD begin by fitting this background. We compared our background fits from pySYD and our power spectra to those generated for Y18 and found one key difference, illustrated by the example spectrum in Figure 2. The white noise component of the background fit is typically an order of magnitude larger in Y18 than in our work (see Figure 3). This is due to the difference in the data processing on the light curves used by each work. Y18 uses detrended Kepler SAP light curves (see Y18 for description of this process), and this work uses the high-pass filtered PDCSAP light curves. We find that the white noise estimates for this work correlate strongly with Kepler magnitude whereas the white noise estimates for Y18 correlate only weakly with Kepler magnitude. We additionally show the theoretical lower limit of white noise (the photon noise) for Kepler 30-minute data as shown in Jenkins et al. (2010), which neither Y18 nor this work reaches. We note that at Kepler magnitudes < 11, the white noise component of the fit is no longer dominated by photon noise, and therefore deviates from the Jenkins et al. (2010) relation. It is clear that the light curves used in this work have lower overall white noise measurements, which we prefer for this analysis. This difference in white noise levels is what causes the small systematic decrease in $\nu_{\text{max}}$ estimates. However, it is clear from Figure 1 that the reduction in white noise does not affect estimation of $\nu_{\text{max}}$ nor $\Delta\nu$ very strongly, and the seismic parameters in Y18 are still reasonable estimates.

Additionally, we had three stars in our sample which we considered “anamolous peak” (Colman et al., 2017; Bedding et al., 2010) stars: KIC 6529078, KIC 3747623, and KIC 10384595. These stars exhibit a significantly large peak outside of the oscillation window, which may indicate the presence of a nearby close binary system or that the star is itself in a multiple system (see Section 3.1 for further discussion). Once the strong peak of these stars is removed, the $\nu_{\text{max}}$ must be recalculated to the proper location. For each of these stars, the change in $\nu_{\text{max}}$ from the original catalogue is large enough to remove the star from our final sample (see Figure 1).

There are a number of ways in which $\Delta\nu$ can be measured. The SYD pipeline and pySYD both use an auto-correlation of a window around the power-excess. This generally works well, but can be affected by the mixed mode characteristic of the $\ell$=1 and $\ell$=2 modes. Other authors choose instead to use a weighted least-squares fit to the radial modes, where the weights are chosen to decrease with distance from $\nu_{\text{max}}$ (e.g., White et al., 2011; Sharma et al., 2016; Stello & Sharma, 2022). These works tend to include work with models, where one would have an exact value for the frequency, whereas for observations determining the exact pulsation frequency is less straightforward. There are many other ways to define and calculate $\Delta\nu$, for example Kallinger et al. (2012) measures the separation between the three central radial orders, Mosser et al. (2011a) measures the average separation for all observed p-modes, Campante et al. (2017) uses the power spectrum of the power spectrum and, more recently, Dhanpal et al. (2022) uses machine learning techniques (this list is not exhaustive). Additionally, $\Delta\nu$ is not a constant as function of frequency. Indeed the definition of $\Delta\nu$ itself, also known as the asymptotic relation:

assumes that $\Delta\nu$ is strictly independent of $n$. The deviation of $\Delta\nu$ from the asymptotic relation can arise from abrupt changes, or glitches, in the interior stellar structure. It is known that higher mass red giants have stronger glitches than lower mass red giants, and red clump stars tend to also have glitches with stronger amplitudes (Vrard et al., 2015). When glitches are present, it is exceedingly difficult to define a precise $\Delta\nu$. However, when providing comparisons with other stars, $\Delta\nu$ is still a useful estimate, as long as each star is measured in the same way. Many of the stars in our sample have p-mode glitches (see Section 3.5 for further discussion), however they tend to have similar shapes, and $\Delta\nu$ is measured consistently across the sample.

For our sample we used pySYD to remeasure $\Delta\nu$ for all stars. There is very little systematic difference between the $\Delta\nu$ that we measure and the values quoted in Y18, which is insignificant compared to the roughly 1% scatter over the entire sample (see Figure 1b).

The temperatures quoted in Y18 were adopted from Mathur et al. (2017), which is a collection of stellar parameters from a variety of sources, explicitly citing estimates from spectroscopy for 14,813 stars, non-KIC photometry for 151,118 stars, and the KIC for 31,165 stars. While the method of collecting stellar parameters from various sources is good for studying a large sample of stars where the goal is coverage, for our smaller, more detailed sample we prefer to take all of our estimates from the same source if possible, to reduce systematic offsets.

We considered two large temperature estimation catalogues: APOGEE (Jönsson et al., 2020) and LAMOST (Cui et al., 2012). The APOGEE and LAMOST temperatures show within 2% agreement, with no systematic differences between the values estimated in each survey. It is not straightforward to place preference over either estimate when both are present in the same star. For this work, we adopt either the LAMOST or APOGEE temperatures if only one is available, otherwise we adopt the average of the two measurements. If neither of these temperatures are available, we adopt the temperature from Y18. Only 13 stars are not present in LAMOST nor APOGEE, 7 of which remain in our final sample. The $T_{\text{eff}}$ adoption source for every star is noted in our data set (see Table 1 notes). As we cannot provide homogenously measured $T_{\text{eff}}$ values with only these two surveys, the $T_{\text{eff}}$ remains a major source of potential uncertainty in our final mass estimates.

These stars have been previously published in Y18 and thus have previously estimated asteroseismic masses. However, those asteroseismic masses depend on derived global oscillation parameters as follows in Eqn 2. (Ulrich, 1986; Christensen-Dalsgaard, 1988; Brown et al., 1991; Kjeldsen & Bedding, 1995; Stello et al., 2008; Kallinger et al., 2010):

Thus, after re-calculating our estimates for $\nu_{\text{max}}$, $\Delta\nu$, and $T_{\text{eff}}$, we now re-estimate the masses for each of our stars. For the solar values, we use $\nu_{\rm max,\odot}$ = 3090 $\pm$ 30 $\mu$Hz, $\Delta\nu_{\odot}$ = 135.1 $\pm$ 0.1 $\mu$Hz, and T_eff,⊙ = 5772 K (Huber et al., 2011a; Prša et al., 2016). For the two correction factors, $f_{\Delta\nu}$ and $f_{\nu_{\rm max}}$, we used asfgrid (Sharma et al., 2016; Stello & Sharma, 2022) to estimate $f_{\Delta\nu}$ (assuming our stars are core He-burning; see Section 3.2 for justification of this assumption) and assume $f_{\nu_{\rm max}}$ to be one, following the convention used by current works in red giant asteroseismology (such as Yu et al. 2018; Zinn et al. 2019; Li et al. 2023). asfgrid requires input of $\nu_{\text{max}}$, $T_{\text{eff}}$, and log(g), the latter two of which we adopt from APOGEE and LAMOST (see Section 2.3). We note that the stellar model grid used in Sharma et al. (2016) does not go beyond 4 $\text{M}_{\odot}$, however this upper mass limit was extended to 5.5 $\text{M}_{\odot}$ in Stello & Sharma (2022). There are also more recent works such as Li et al. (2023) which calculate $f_{\Delta\nu}$ by taking into account the surface correction necessary in evolutionary models. These models span an even smaller range of masses, from 0.8 to 1.8 $\text{M}_{\odot}$, and we therefore cannot use the Li et al. (2023) models for our purposes. However, we note that the $f_{\Delta\nu}$ values found when including the surface correction terms tends to decrease $f_{\Delta\nu}$ by $\sim$1–2%, which would decrease the mass estimated by roughly $\sim$4–8%.

Once we have validated the mass estimates for our stars, we again impose a cut in mass to 3.0 $\text{M}_{\odot}$, as the re-estimation process has decreased the mass estimates for the vast majority of stars. We additionally remove any stars with $\nu_{\text{max}}$ < 20 $\mu$Hz as those lower than this threshold have $\Delta\nu$ values that are difficult to measure due to the small number of excited modes. This narrows our final sample to 48 stars. The sample selection can be seen in Figure 4. While the re-estimation of $\nu_{\text{max}}$, $\Delta\nu$, and $T_{\text{eff}}$ each effect the overall distribution of the masses (see the preceding subsections), it is clear that the systematic decrease in stellar mass from the Y18 catalogue is predominantly due to a systematic decrease in $\nu_{\text{max}}$. Section 2.1 gives a full discussion of the origin of this offset. Our quoted uncertainties for the mass include the uncertainties on $\nu_{\text{max}}$, $\Delta\nu$, and $T_{\text{eff}}$, but we do not include systematic uncertainties such as the potential breakdown of the scaling relation at high mass. Future studies will reduce this uncertainty and refine the mass estimates.

High-mass red giants are an interesting laboratory for studying binary systems. Binary fraction is known to increase with stellar mass, with some current estimates finding that up to 50 to 70 % of young A- and late B-type stars (the progenitors of the high-mass red giants) are in binaries (see review by Lee et al. 2020). It is also known that binarity, especially close binarity, weakens oscillation amplitudes in red giants (Gaulme et al., 2014; Schonhut-Stasik et al., 2020). A review of binary orbital period as a function of primary mass and mass ratio can be found in Moe & Di Stefano (2017). We therefore check if our sample has any indications of binarity, and additionally check for the likelihood of contamination.

Yu+18 and the catalogues from which those data were collated removed any obvious binary signals (i.e. eclipsing systems), but there may still be signals hidden within. An excellent example of these is the class of Kepler stars with “anomalous peaks" which may be due to nearby (chance-aligned) or associated close binaries (in hierarchical triples) (Colman et al., 2017). As discussed in Section 2.1, our original set of 114 contained three such stars: KIC 6529078 and KIC 3747623, which were previously known, and KIC 10384595, which was not previously published. These signals are understood to come either from chance alignments with other systems, or from physically associated systems such as hierarchical triples. As mentioned in Section 2.1, these stars with anomalous peaks are not present in the final sample.

We attempt to quantify the probability that binarity or contamination are occurring through various metrics, which we record in our extended data table (available online). The first of these is through visual inspection of the Kepler TPFs and the pipeline apertures. Using the lightkurve (Lightkurve Collaboration et al., 2018a) package function interact_sky, one can view the Gaia DR2 (Gaia Collaboration et al., 2018) positions overlain on the Kepler Target Pixel File (TPF). We used this to view whether there were any known sources nearby the target star, specifically within the Kepler aperture. For each star, we added a flag in Table 1 to indicate whether the star has (i) no nearby potential contaminating sources (denoted with 0), (ii) nearby sources that were more than 3 mag fainter than the target star, and thus not expected to be strongly contaminating the signal (denoted with 2), or (iii) a nearby star of similar brightness (denoted with 1). For the final high-mass sample, these three scenarios represented 44%, 46%, and 10% of the sample, respectively. Note that we were quite strict with the definition of “potentially contaminated" as meaning that there was a star within the TPF of a brightness within 3 mag of the target. Interestingly, none of the three anomalous peak stars have potentially contaminating nearby stars that have been catalogued by Gaia DR2. The package we use for this study (the lightkurve function interact_sky) does not allow us to compare with the Gaia DR3 positions, but we do not expect the update from DR2 to DR3 to shift targets enough to affect this analysis.

The Kepler database quotes a crowding metric (CROWDSAP), which is defined as the ratio of the target flux to the total flux within the aperture. We find only two stars in the final sample with values < 0.95 (three in the large sample), only one of which has a potentially contaminating source. The other star has no nearby sources resolved by Gaia DR2. We include this value in Table 1, for completeness. We also include Gaia’s renormalized unit weight error (RUWE) parameter, which may indicate binarity for values higher than 1.4 (Fabricius et al., 2021). The large sample has 15 stars with RUWE > 1.4, whereas the final sample has seven. The final sample has no star with RUWE > 3.5, however the large sample has three, with the maximum in the large sample being KIC 5380617 which has RUWE = 43.331. Of the three anomalous peak stars, only KIC 3747623 has a RUWE value above the threshold, with RUWE = 11.013. We emphasize that a low RUWE value does not exclude the possibility of binarity, nor is a large RUWE value exclusive to binary systems.

We additionally find eleven stars in our large sample that have been suggested as potential spectroscopic binaries in three different articles. KIC 9655167 and KIC 9655101 were reported as single-lined spectroscopic binaries by Molenda-Żakowicz et al. (2014) in their study of the Kepler open cluster NGC 6811. These two stars are also listed as NGC 6811 cluster members by Stello et al. (2011). Note that this is a young cluster with high-mass stars currently in the red giant/clump phase, consistent with the mass estimates of these stars. We find KIC 9836930 in the reference sample (Sample R) of Jorissen et al. (2020) as a potential single-lined spectroscopic binary. Finally, stars with KIC IDs 8365782, 10094550, 6866251, 11297585, 5080332, 11413158, 5534910, and 5707338 appear in the Gaia DR3 non-single star catalogue (Gaia Collaboration, 2022; Gaia Collaboration et al., 2023) as potential single-lined spectroscopic binaries, although only one of these (KIC 11413158) has a strong significance estimation.

To summarize, we cannot rule out the fact that some of our stars may either be in multiple systems or otherwise contaminated in some way. With the combination of the aforementioned metrics, we have an understanding of which stars are more likely to be contaminated or in binaries. That being said, we do not currently believe any of our stars are having strong interactions with a companion source, nor a nearby star contaminating the oscillation signals, as we do not see those signatures in the Kepler pixel data nor the oscillation spectra. In fact, we do not see any clear indications that any of the stars in our sample are in binaries.

Near the red giant branch (RGB) region of the HRD, there is significant overlap between stars of different evolutionary stages, specifically the RGB itself, the red clump (often called the horizontal branch in globular clusters), and the asymptotic giant branch (AGB). These three represent the evolutionary phases corresponding to shell hydrogen burning, core helium burning (CHeB), and shell hydrogen and helium burning¹¹1Note that there are a few differing definitions of the beginning of the AGB. For this work we define the beginning of the AGB to be at core He exhaustion and the onset of shell He burning., respectively. Additionally, at masses greater than $\sim$ 2 $\text{M}_{\odot}$, stars do not undergo explosive ignition of core-He burning (“the Helium Flash"), and therefore settle onto a region called the “secondary clump", which lies separate from the lower-mass red clump (Girardi, 1999). All of our stars have masses greater than 2 $\text{M}_{\odot}$, so if they are in the core He-burning phase, they are going to lie in the secondary clump. In the following text, we will refer to these three phases as RGB, CHeB, and AGB, and use the term “red giants" as a general term that encompasses these three phases. For our sample, we explore the potential evolutionary phases for each star.

We can first test the evolutionary phase using the value of $\epsilon$, or the position of the radial mode in units of $\Delta\nu$ (sometimes called the phase offset), as defined by the asymptotic relation (Eqn 1). We measure this by calculating the location of the maximum value of a collapsed échelle (where the values for each radial order of a typical échelle diagram are summed, see Figure 7). The uncertainty associated with $\epsilon$ is strongly tied to the uncertainty in $\Delta\nu$ because changing $\Delta\nu$ directly affects the measurement of $\epsilon$. We therefore adopt an uncertainty on $\epsilon$ equivalent to the fractional error on $\Delta\nu$ multiplied by the approximate radial order near $\nu_{\text{max}}$, calculated as $\sigma_{\Delta\nu}(\nu_{\rm max}/\Delta\nu)$ for each star. The value of $\epsilon$ differs for CHeB and RGB stars due to the change in the acoustic depth of the partial Helium ionization zone (Bedding et al., 2011; Kallinger et al., 2012; Christensen-Dalsgaard et al., 2014). In Figure 8 we show the relation between $\Delta\nu$ and $\epsilon$ for our sample compared to the central $\epsilon_{c}$ values from Kallinger (2019) as measured using the method in Kallinger et al. (2012). The dashed line is a fitted relation from Corsaro et al. (2012) based on RGB stars in Kepler clusters. The shaded region denotes $\epsilon$ $\pm$ 0.1 from this relation, as was done in Stello et al. (2016a) to isolate the red giant branch stars. The majority of our stars lie clearly offset from this relation, indicating that they are likely in the CHeB phase (Bedding et al., 2011). This agrees as well with the position of the secondary clump stars in Kallinger et al. (2012), as seen in the comparison sample. We however note that the works of Kallinger et al. (2012) and Christensen-Dalsgaard et al. (2014) use the central 3 radial modes to derive the central phase shift ($\epsilon_{c}$), which is more robust to acoustic glitches, whereas using the collapsed èchelle to measure $\epsilon$ as we have done may fold in extra errors from these acoustic glitches.

We can also test the evolutionary phase using the dipole modes of these stars. Due to their mixed mode nature, the dipole oscillation modes of red giants have the important capability to probe the near core region. Successive dipole modes have a period spacing ($\Delta\Pi$) that allows us to discern whether or not the star is undergoing core burning (Bedding et al., 2011; Mosser et al., 2011b).

For stars that have clearly identifiable dipole modes (see Section 3.4) and do not have weakly coupled dipole modes (i.e. the dipole ridge shows mixed mode spacing), we can measure the period spacing. One can measure the period spacing by calculating the mean pairwise difference in periods of dipole modes, by aligning a period échelle, or using the stretched period échelle, which attempts to ‘decouple’ the p-modes from the mixed-mode dipole ridge and view the g-mode contribution to them, as described in Vrard et al. (2016); Ong & Gehan (2023). The mean period spacings (often $\Delta$P_obs) measured in Bedding et al. (2011) and Mosser et al. (2011b) are significantly smaller than the period spacings measured from the other two methods ($\Delta\Pi$) due to the mixed modes nearest to the pure p-mode having a different period spacing. We measure the period spacing using the stretched period échelle and adopt a conservative uncertainty of 10 seconds on these estimates.

We compare the $\Delta\Pi$ values measured for our final sample to Vrard et al. (2016) in Figure 9. For the 20 stars which overlap in these two samples, our estimates are consistent within uncertainties (within 2% agreement). It is clear from Figure 9 that our stars lie in the region occupied by typical CHeB stars. However, one must be careful when comparing the higher-mass stars of this sample to the lower-mass stars studied by Vrard et al. (2016) or, more specifically, when comparing the secondary clump stars to the red clump stars. As shown in Fig.4 of Stello et al. (2013) and reproduced in Figure 9, the predicted $\Delta\Pi$ values of high-mass models pass twice through this region of the diagram, once during the RGB phase and again in the CHeB phase. For low-mass stars, $\Delta\Pi$ is very useful for distinguishing these two evolutionary phases²²2We note that the 1.2 $\text{M}_{\odot}$ track’s red clump $\Delta\Pi$ estimates do not replicate the majority of the observed low-mass $\Delta\Pi$ values. This is due to the implementation of overshooting in models, as known and discussed in Constantino et al. (2015); Bossini et al. (2015), and does not persist in the high-mass tracks.. In the high-mass stars, $\Delta\Pi$ is less useful as the RGB and CHeB tracks are closer together. However, if the mass of the star is unambiguously known, then $\Delta\Pi$ can still be used to distinguish these two phases, because the RGB and CHeB for each track do not overlap. $\Delta\Pi$ serves as a probe of the near core region, which for low-mass stars is very different during the RGB and CHeB phases, because the stars have degenerate (or near-degenerate) cores on the former and not during the latter. High-mass stars, on the other hand, do not have degenerate cores at any point during their evolution, and therefore the near-core regions of RGB and CHeB are much more similar (Montalbán et al., 2013).

Finally, we can approach this problem of deciding the evolutionary phase with a probabilistic approach by considering the relative lifetimes of these evolutionary phases. We use the set of evolutionary tracks provided by MIST (Dotter, 2016; Choi et al., 2016) to estimate the relative amount of time each model spends in the RGB and the CHeB phases with $\Delta\nu$ between 2 and 8 $\mu$Hz. To do this we calculate the time spent in the RGB and CHeB phases (as indicated by MIST’s phase column) in this $\Delta\nu$ range, and use the percentage of the CHeB phase lifetime compared to the total RGB + CHeB lifetime as a proxy for the probability that a star found here is in the CHeB phase. Note that MIST reports an estimate of $\Delta\nu$ that it calculates as the inverse of the sound travel time across the diameter of the star, as in Christensen-Dalsgaard (1988); Kjeldsen & Bedding (1995). From this exercise, we find that solar-metallicity models with masses of 3–4 $\text{M}_{\odot}$ spend 97–98% of their time in the CHeB phase. For completeness, we also checked models with [Fe/H] $\pm$ 0.25 and found the same result. For comparison, a solar-metallicity 1-$\text{M}_{\odot}$ star only spends $\sim$ 45 % of its time as a red giant in the CHeB phase. One can see confirmation of this in Figure 9, where the low-mass track has a much higher density of points in the RGB phase than do the high-mass tracks. Therefore, since the stars in our sample have masses greater than 3 $\text{M}_{\odot}$, it is significantly more likely for them to be CHeB secondary clump stars than RGB stars. We would only expect to approximately one star in the RGB phase for the entire sample of 114 stars. Therefore, our conclusion that all stars are in the CHeB phase of evolution, and belong to the secondary clump, is reasonable.

In Figure 10 we compare three oscillation envelope parameters from our sample with the set of red clump stars studied by Y18. Figure 10(a) shows the mean amplitude per mode for each star, defined as

This equation includes a sinc function correction to account for the attenuation of signals approaching the Nyquist frequency due to the non-zero integration time (Huber et al., 2010; Murphy, 2012; Chaplin et al., 2014). Here, H_env is the height of the oscillation excess (A_smooth in the pySYD output) and $c$ is the effective number of modes per order, adopted as 3.04 (Kjeldsen et al., 2008). This adopted value for $c$ is useful for comparison to the other red giant stars, but we note that many of our stars have suppressed dipole modes, and thus have fewer effective modes per order. Thus, the mean mode amplitude calculated using this method is lower for stars with suppressed dipole modes than for ‘normal’ stars.

The trends of our mean mode amplitude estimates with $\nu_{\text{max}}$ and mass are consistent with those found in Y18, where the highest mass stars have much lower mode amplitudes than intermediate mass stars. The mean mode amplitudes estimated in this work are on average 20% larger than those reported in Y18, independent of stellar $\nu_{\text{max}}$. This is again due to the reduction in white noise in this work from adopting the PDCSAP light curves (see Section 2.1). We illustrate this using our example power density spectrum in Figure 2. Here we see that the reduction in the white noise component of the power density does not reduce the power in the oscillation modes, and therefore this work must estimate a larger H_env, and by extension mean mode amplitude, to properly model the background and oscillations. Our mean mode amplitudes are still consistent with the trend reported in Y18, and our high mass stars have lower mean mode amplitudes than lower mass CHeB stars, as expected (Mosser et al., 2012).

Figure 10(b) shows the granulation power at $\nu_{\text{max}}$, which excludes the white noise and is also corrected for attenuation. These two are known to be tightly correlated, as the convective properties (and thus the granulation background) are well correlated with the acoustic cutoff frequency and therefore $\nu_{\text{max}}$ (Kjeldsen & Bedding, 2011; Mathur et al., 2011; Kallinger et al., 2014). Our results are again consistent with Y18.

In Figure 10(c) we show the full-width at half-maximum (FWHM) of the power envelope, which is measured by fitting a Gaussian distribution to the power excess. The tight trend with $\nu_{\text{max}}$ is also visible in Figure 6. Our stars have quite wide power envelopes compared to typical CHeB stars, and especially compared to the RGB stars, as previously found for high-mass stars (Mosser et al., 2012; Yu et al., 2018). However, this effect is inflated due to the reduction in white noise once again allowing for wider envelopes, just as it allowed for larger average mode amplitudes. The error bars on the FWHM show that the few stars with anomalously large FWHM values also have relatively large uncertainties. The power excess width should increase with $\nu_{\text{max}}$, as is shown by the comparison data from Y18 in this figure, however that does not necessarily imply that more orders are being excited. If we divide the FWHM by $\Delta\nu$ we can get a lower estimate of the number of excited orders, shown in Figure 11. We emphasize that this estimate is not the same nor as robust as simply counting the number of visible radial modes in the power spectrum. Our sample has on average $\sim$6 strongly excited orders using this metric, with many stars exciting at least 8 radial orders. We therefore have an excellent opportunity to study acoustic glitches with this sample (see Section 3.5).

In most red giants the radial, dipole, and quadrupole modes are easily identified. The ratios of the amplitudes for the three sets of modes, also known as their visibilities, are interesting probes of the stellar interior. One typically considers the ratios of the amplitudes of dipole modes and the quadrupole modes to those of the radial modes. Typical Kepler RGB stars have dipole mode visibilities of about $\sim$1.4 and quadrupole mode visibilities of $\sim$0.7. Some stars show strong suppression of their dipole modes, which translates to low dipole mode visibilities and has been observed to be a strong function of mass in RGB stars (Mosser et al., 2012; Kallinger et al., 2014; Fuller et al., 2015; Cantiello et al., 2016; Stello et al., 2016b; Stello et al., 2016a). Mosser et al. (2017) studies both RGB and CHeB visibilities, but does not quote any strong differences between the two populations.

We measure the dipole mode visibilities of our stars by following the framework set by Mosser et al. (2012) and Stello et al. (2016a); Stello et al. (2016b). However, these two groups measured the visibilities slightly differently. Mosser et al. (2012) measured a weighted average mode amplitude for each star by dividing a Gaussian weight, amplifying the contribution from the modes further from $\nu_{\text{max}}$. Stello et al. (2016a); Stello et al. (2016b) instead used a simple, non-weighted integration of the mode amplitudes before calculating the visibility ratios. Additionally, Stello et al. integrated over 4 orders, whereas Mosser et al. integrated over 5 orders. While these two differences aren’t expected to produce a systematic offset, they will nevertheless have measured slightly different values. We choose to follow the Stello et al. formulation, and we will be comparing to their values in our analysis. We additionally calculated the visibilities by integrating over 5 orders rather than 4 orders, and find agreement between the two cases within uncertainties. However, one critical difference from the Stello et al. formulation is regarding the identification of modes. For our work, we begin by measuring the frequencies of the radial modes for each star (‘peak-bagging’), whereas Stello et al. (2016a) identified modes via autocorrelation with template functions at a range of $\Delta\nu$ values. Additionally, we calculate error bars for our visibility estimates by calculating the mode visibilities for each available quarter of Kepler data and taking the standard error in the mean (standard deviation divided by the square root of the number of observed quarters). Our results are shown in Fig 12, plotted over the estimates from Stello et al. (2016a) for comparison.

As we do not have any overlap with the sample of Stello et al. sample, which included only RGB stars, we tested our methodology on 10 typical stars from the Stello et al. sample covering a range of $\nu_{\text{max}}$. We found good agreement at large $\nu_{\text{max}}$ but for $\nu_{\text{max}}$ in the range 50–100 $\mu$Hz, our estimates are systematically larger by about 10%, albeit consistent within uncertainties. Whether this systematic offset is due to the Kepler data processing, the visibility calculation, or the crowdedness of the modes at low $\nu_{\text{max}}$ is unclear.

Our stars do not appear anomalous from the overall sample of red giants tested by these works. However, unlike the high-$\nu_{\text{max}}$ portion of the Stello et al. sample, we do not see a clear distinction between our suppressed and not suppressed stars. We instead see a smooth distribution across visibility space, similar to the highest mass bin in Figure 2 of Stello et al. (2016b), and seen at the low-$\nu_{\text{max}}$ end of their sample, where the suppressed theoretical model turns upwards, and therefore we do not adopt a threshold value upon which we categorise our stars as either “suppressed" or “not suppressed". We do confirm the trend that the highest mass stars in our sample tend to have weaker dipole mode visibilities. In fact we only measure dipole mode visibilities greater than 1.1 at masses below 3.5 $\text{M}_{\odot}$. Additionally, we note that there appears to be a weak correlation between $\nu_{\text{max}}$ and the dipole mode visibilities. Recall that we do not see any trends with the mass of the star and its $\nu_{\text{max}}$. We believe the trend between $\nu_{\text{max}}$ and dipole visibility is coincidental, especially considering that model evolutionary tracks for these high masses show that the Zero-age CHeB main sequence lies at very different values of $\nu_{\text{max}}$ for each mass, and it is therefore likely that these stars are at different points in their CHeB lifetimes. We also point out that there is a weak trend in dipole mode visibilities and $\epsilon$, visible to a careful eye in Figure 7. This can be interpreted as the same trend that is seen in $\nu_{\text{max}}$, as $\nu_{\text{max}}$ is known to vary with $\Delta\nu$, and we have shown that $\Delta\nu$ also correlates with $\epsilon$ in Figure 8. Thus we are skeptical that this trend is more than a small number statistical effect.

For quadrupole modes, suppression is a much more subtle effect. As shown in Stello et al. (2016a), the quadrupole mode suppression is much more clear at large values of $\nu_{\text{max}}$. This is also clear in the lower panel of Figure 12 where the estimates from Stello et al. (2016a) are plotted. At low $\nu_{\text{max}}$ there is very little bimodality in the quadrupole mode visibility. Therefore it is not necessarily surprising that we see a very localized distribution of the $\ell$=2 visibilities for our sample. However, we do generally see that the highest-mass stars in our sample tend to have quadrupole visibilities on the low end of our distribution.

Of the non-radial modes, the dipole modes are more strongly coupled to the core (Dupret et al., 2009), and therefore probe the interior more strongly, and may be more affected by core characteristics, which are the main difference between the RGB and the CHeB stars. As discussed in Section 3.2, our sample is expected to be dominated by CHeB stars, whose visibilities have been discussed theoretically by Cantiello et al. (2016) and observed by Mosser et al. (2017). The leading theory is that the suppression of these non-radial modes in RGB stars is due to fossil magnetic fields left from the convective cores of the main sequence phase (Fuller et al., 2015; Stello et al., 2016b; Stello et al., 2016a; Cantiello et al., 2016). This “magnetic greenhouse effect" (Fuller et al., 2015) arises as the mode energy that leaks into the g-mode cavity is dissipated due to the core magnetic field. Cantiello et al. (2016) describes three possible scenarios for suppression in CHeB stars, relating to whether or not the fossil magnetic field is destroyed in the helium flashes of intermediate-mass stars. Our stars will not exhibit helium flashes before the CHeB phase, and thus the core magnetic field should not be affected. Additionally, our sample of high-mass CHeB stars should have convective cores during the current phase, so perhaps they are regenerating a magnetic dynamo in their core, driving further suppression. However, there are other recent works such as Mosser et al. (2017) which dispute the magnetic greenhouse effect, showing that suppression does not disrupt mixed modes as one would expect from the magnetic greenhouse. The effects of magnetic fields on the g-mode cavities in the cores of red giants have begun to be explored by works such as Loi & Papaloizou (2018), and our high-mass sample provides an interesting lens through which to study the source of dipole mode suppression.

Deviations of $\Delta\nu$ from the asymptotic relation are often called acoustic “glitches", and are indicative of abrupt structural changes within the envelope of the star, such as the edges of partial ionization zones (especially the Helium partial ionization zone) and the bottom of the convection zone (Vrard et al., 2015). Perhaps owing to their wide excitation envelopes and thus high number of excited orders (see Section 3.3 and Figure 10), all stars in our sample show radial mode glitches. Additionally, many of our stars share a similar glitch morphology, which may point to shared structure within the secondary clump. In this work we do not measure properties (amplitude, phase, or period) of radial mode glitches. However, we note that we can see the glitch effect in the broadening of the collapsed radial mode ridge in Figure 7, for example. This also adds an extra uncertainty onto the measured epsilon value (see Figure 8). We are also sensitive to glitches in our visibility calculation, but we account for these by peak-bagging the radial modes, rather than assuming they are spaced exactly $\Delta\nu$ from each other. We intend to further study these glitches in later work.