ASTEROSEISMIC CLASSIFICATION OF STELLAR POPULATIONS AMONG 13,000 RED GIANTS OBSERVED BY KEPLER

We have used simple aperture photometry (SAP) data for our analysis (Jenkins et al. 2010), downloaded from the MAST database.⁹ Jumps in each time series (e.g., across quarterly gaps) were corrected by fitting and removing a linear regression to five-day light curve segments before and after each gap. The remaining slow instrumental trends were then removed by subtracting a smoothed version of the light curve that was calculated by applying a boxcar filter with a width of 10 days. The high-pass-filtered light curves were then used to calculate a frequency spectrum. This was analyzed with an automated pipeline (Huber et al. 2009) to detect the presence of solar-like oscillations. The results from the pipeline were verified by visual inspection of its graphical output of each star, and a few percent of the stars were discarded in this process because the pipeline results were deemed incorrect. Oscillations were detected in 13,412 red giant stars. We found a similar number (13,182) when using the method described in Mosser & Appourchaux (2009). Hence, for ∼2000 stars, predominantly low and high log  g giants, we "missed" the oscillations because of the high-pass filter and the limits set by the Nyquist frequency of the data, respectively. For each star, the analysis provided a frequency spectrum corrected for the stellar granulation signal, as well as measurement of ν_max and Δν (Figure 1(a)).

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Using an approach similar to Bedding et al. (2011), we smoothed the frequency spectra and measured the frequency of each peak reaching above three times the noise level within ±5Δν of ν_max. A smoothing width of about 0.1–0.2 μHz provided a good compromise, enabling the detection of closely spaced narrow peaks from long-lived modes while keeping the number of spurious detections low around broad peaks arising from single short-lived modes (Figure 1(b–d)).

We associated the frequencies, ν, with radial or quadrupole (l = 0 and 2) modes if − 0.3 < ν/Δν mod 1 < + 0.1, where was derived using the Δν– relation fitted to red giant branch stars by Corsaro et al. (2012) based on the formulation by Mosser et al. (2011b; Figure 1(b–d), shaded regions). The remaining frequencies were associated with dipole modes (l = 1), assuming negligible contamination from l = 3 modes (panel (e)). To ensure robust results, we discarded stars with fewer than five detected dipole modes overall. There were 13,031 stars that passed this criterion.

For each star, we calculated the pairwise period spacings, ΔP, between adjacent dipole modes, hence providing at least four spacings per star and usually many more. We plot these versus Δν in Figure 2(a).

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In this figure we show the ΔP regime that is populated by red giant branch (first ascent) stars (below about 100 s; see Bedding et al. 2011). We find a remarkably clear upper envelope of the ΔP distribution in the range 6 < Δν/μHz <16 at values of roughly ΔP =70–90 s, which we interpret as a signature of the "true" g-mode period spacing of dipole modes.

To confirm this, we show the asymptotic g-mode period spacing for dipole modes, ΔP$_g=\sqrt{2}\pi ^2(\int N/r dr)^{-1}$, where N is the buoyancy frequency integrated over the radius, along a stellar model track representative for the typical stellar masses in our sample of stars (see Section 3). This is the period spacing of the g-modes if they did not couple with the p-modes. The alignment between ΔP_g from the model and the upper envelope of the observed ΔP is striking. The sharp edge of this upper envelope, and the fact that all tracks of representative masses fall almost on top of each other in this region (Section 3; see also Bedding et al. 2011; White et al. 2011; Mosser et al. 2012a), is strong evidence that many of the detected dipole modes are essentially separated by the true g-mode spacing, which is now directly measurable for a significant fraction of stars. In order words, these modes resemble almost pure g-modes (see example in Figure 1(b)). Except for some special cases, we have previously only been able to infer the true g-mode spacing when making certain assumptions about the coupling between p- and g-modes (Stello 2012; Mosser et al. 2012a).

In addition, the overdensity of data points just below the model track (Figure 2(a)) gives us further confidence that we are essentially detecting the true g-mode spacing directly in a large number of stars. To illustrate this point, we show in Figure 3 the period spacings of a sequence of l = 1 modes from a representative 1.5 M_☉ model. The most common period spacings in the model are those close to the true g-mode period spacing, arising from the least bumped modes between the "dips" in ΔP where the slope of the ΔP(ν) "curve" is close to zero. The same effect could explain the slightly higher number of modes spaced at about 50 s (as the slope of the ΔP(ν) curve also becomes zero at the bottom of each dip). These two extremes are marked as Max(ΔP) and Min(ΔP) in Figure 2(b).

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Stars at the bottom of the red giant branch with high values of Δν, toward the subgiants, typically show stronger coupling between p- and g-modes, and hence wider dips in the period spacing, compared to what is shown in Figure 3 (Dupret et al. 2009). In addition, their larger Δν and ΔP makes their frequency spectra less dense, implying fewer modes spaced just below the true g-mode spacing. This seems to be supported by the absence of a clear overdensity of points near the model track toward lower luminosity (higher Δν) in Figure 2(a).

Another clear feature in this diagram is a large number of points with ΔP ∼10–30 s. Many arise from rotationally split modes and are discussed in Section 3.2.

To separate the stars into distinct groups, we need to assign a single representative value of ΔP to each star. The median of all period spacings per star provided an efficient and robust measure for this purpose. The absolute values of ΔP from this method is most likely somewhat different than those found using previously published methods (Bedding et al. 2011; Mosser et al. 2011a, 2012a), but for our purpose that is not of concern. We show the result of the median period spacing in Figure 4(a), which for clarity only includes modes that reach above six times the noise level (30% of all stars). The masses indicated by color and shown as a histogram in the inset are derived from the scaling relation

$$\begin{equation} M/{M}_\odot \simeq (\nu _\mathrm{max}/\nu _{\mathrm{max},\odot })^3(\Delta \nu /\Delta \nu _\odot)^{-4}(T_\mathrm{eff}/{T}_{\mathrm{eff},\odot })^{3/2}\,, \end{equation} \tag{ 2 }$$

which shows a relative scatter below 10% for samples of equal-mass stars (Miglio et al. 2012). This relation is derived from the scaling relations for Δν and ν_max (Kjeldsen & Bedding 1995). For T_eff, we used the values from Pinsonneault et al. (2012). We found qualitative agreement for this plot when comparing it with values derived using the method described in Mosser et al. (2011a).

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To compare qualitatively with our observations, we plot ΔP_g for a grid of stellar models in Figure 4(b) using the MESA 1M_pre_ms_to_wd test suite case (Paxton et al. 2011, 2013). In this diagram the models evolve from right to left along the red giant branch. The hook at ΔP ≃ 200 s along each track is the bottom of the red giant branch. When helium ignites at the tip of the red giant branch with Δν values close to zero (outside the plotted range), the tracks quickly reappear on the helium-burning ZAMS (open diamonds; known as the zero-age horizontal branch for low-metallicity stars). Although the tracks were computed through this very rapid phase, for clarity we do not show this. About 2–3 stars per 100 red clump stars are expected to be in this transition phase (Bildsten et al. 2012). Again, to simplify the plot, the helium-core-burning phase is only shown from the helium-burning ZAMS until the core helium mass fraction has reached 0.14, except for the 1.0 M_☉ track (dark blue), which shows the evolution all the way to the early phases of the asymptotic giant branch (dashed). As core convection due to helium-burning ends, ΔP drops from 190 s to 100 s in less than 1 Myr. At the point where helium is completely exhausted in the center of the star, which happens at ΔP ≃ 80 s for the 1.0 M_☉ track, the period spacing has been reduced back to something comparable to that of the red giant branch stars. We note that the change in ΔP as a function of mass along the helium-burning ZAMS follows quite tightly that of the helium core mass, which is lowest for M ∼ 2.4 M_☉ without overshooting (Cassisi & Salaris 2013; Montalban et al. 2013).

From the speed at which models move across this diagram (Figure 4(b)), seen by the density of filled dots, we expect most observed stars to occupy the region of 50 < ΔP <90 (red giant branch stars with M ≲ 1.8 M_☉), 200 < ΔP <250 (red clump stars with M ≲ 1.8 M_☉), and 150 < ΔP <200 (secondary clump stars with M ≳ 1.8 M_☉). Although the median of our observed ΔP (panel (a)) is not exactly the same quantity as ΔP_g (panel (b)), the qualitative agreement between the models and the data is surprisingly good, and demonstrates that we can clearly separate the different stellar populations. How much the observed median ΔP deviate from ΔP_g depends on the fraction of individual period spacings that we observe within, relative to outside, the dips (Figure 3), which again depends on both the strength of the coupling between the p- and g-modes and the number of dipole modes per radial p-mode order that we detect (see, e.g., Mosser et al. 2012a).

One significant difference between panels (a) and (b) is the observed low-mass (blue and green) stars in the region bracketed by ΔP ∼80–250 s and Δν ∼5–7 μHz, which is unexpected when compared to the models. These stars lie in the region where Bildsten et al. (2012) predicted the helium flashing stars to be while they establish stable core burning. However, spot checks of the frequency spectra show that these stars are mainly red giant branch stars whose measured median ΔP are not representative of the actual spacing between adjacent modes caused by our limited frequency resolution. Hence, the median ΔP is artificially increased. Toward lower values of Δν, red giant branch stars show smaller period spacings, making it harder to resolve the individual mixed modes, which is somewhat exacerbated because we smooth the frequency spectra before we extract the frequencies. This also explains the lack of red giant branch stars with Δν ≲ 5 μHz.

In Figure 4(c) we finally look at how metallicity affects the position in the ΔP–Δν diagram, in order to judge how a bias in metallicity could be introduced when selecting samples of stars from this diagram. It is quite clear that within the range of Z shown (corresponding roughly to −0.2 < [Fe/H] < 0.2), the models do not show any significant shift in ΔP_g, and only a slight shift in Δν, due to a change in mean stellar density.

In Figure 5 we show the distribution of the observed median ΔP. Like Figure 2, it clearly shows a large fraction of stars with period spacings well below ΔP_g for red giant branch stars. Visual inspection of the frequency spectra shows that these are red giant branch stars with rotationally split dipole modes, as reported in a few stars by Beck et al. (2012; see also the ensemble analysis by Mosser et al. 2012b). Figures 1(c) and (d) show some typical examples of such stars. Most of the pairwise period spacings for these stars are due to the rotational splitting of the dipole modes, which is why they show up with low values of the median ΔP. As illustrated in Figures 1(c) and (d), the dipole modes split into two or three peaks depending on their angle of inclination. We see two peaks when the inclination is high (close to 90°), three peaks are seen when the inclination is intermediate (∼40°–60°), and just one peak is seen when the star is viewed pole-on (Gizon & Solanki 2003). This sample of rotating red giant branch stars provides an interesting selection for further studies into transport of angular momentum (Mosser et al. 2012b), age–rotation relations, and the connection between surface and internal rotation including their angles of inclination.

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