Core–Envelope Coupling in Intermediate-mass Core-helium Burning Stars

There has been significant recent progress on both spectroscopic (Abolfathi et al. 2018) and asteroseismic (Pinsonneault et al. 2018) data for targets in the Kepler fields. However, our sample selection predates those works, and as a result our sample selection was designed using related, but different, data sets. We therefore begin by discussing the tools we used to define our candidates for analysis.

For sample selection we use data collected by the Kepler mission (Borucki et al. 2010), which has monitored more than 197,000 stars (Mathur et al. 2017). The light curves analyzed in this work were first calibrated with KADACS (Kepler Asteroseismic Data Analysis Calibration Software; García et al. 2011, 2014b) to remove outliers, trends, and instrumental effects and concatenate the quarters. All the stars in our sample have a sampling of 29.40 minutes and have been observed for around 4 years; light curves are available in Mathur et al (2019).

We applied the A2Z pipeline (Mathur et al. 2010) to a sample of evolved stars with Kepler data. For all targets we inferred two global seismic parameters: the mean large frequency separation (Δν, the frequency difference between two modes of the same degree and consecutive orders) and the frequency of maximum oscillation power (ν_max). We choose this particular asteroseismic analysis pipeline because it has also analyzed stars with visible surface modulation that allows the measurement of a starspot rotation period (Ceillier et al. 2017) but can complicate asteroseismic analysis.

Spectroscopic parameters for these stars, including temperature and metallicity, come from the APOGEE spectroscopic survey (Majewski et al. 2017). Specifically, we used data from Data Release 12 (DR12; Alam et al. 2015) of the Sloan Digital Sky Survey III (Eisenstein et al. 2011) for our target selection process. Spectra were taken on the 2.5 m Sloan Digital Sky Survey telescope (Gunn et al. 2006) and analyzed with the ASPCAP pipeline (Nidever et al. 2015; García Pérez et al. 2016). The stellar parameters for DR12 were calibrated using both members of star clusters and asteroseismic data (Mészáros et al. 2013).

Since these targets were selected, improved measurements for both the spectroscopic and asteroseismic parameters have been released. For the analysis here, we use the spectroscopic values from the Fourteenth Data Release (Abolfathi et al. 2018) of the Sloan Digital Sky Survey IV (Blanton et al. 2017), which takes advantage of both an updated spectroscopic analysis pipeline and improvements in the calibration to fundamental measurements (Holtzman et al. 2018). Where possible, we also update our seismic parameters to those published in Pinsonneault et al. (2018), which has compared multiple analysis techniques, corrected for known departures from the asteroseismic scaling relations, and placed the results on a fundamental mass scale using open clusters. For the 28 stars where masses and surface gravities are not available in Pinsonneault et al. (2018), we use our A2Z pipeline data. Pinsonneault et al. (2018) defined normalizations for asteroseismic measurements that put all methods there, including A2Z, on a common basis, and we adopt those normalizations here (effective solar Δν and ν_max of 135.146 and 3091.44 μHz, respectively, as well as the A2Z specific correction factors). We note that in some cases these updates cause stars to fall outside of our targeted mass range. For completeness, we still include and report the results for these stars, but we note that their exclusion would not substantially change the results of our analysis.

2.2.1. Comparison Sample

2.2.2. Unbiased Sample

While prior samples of measured core rotation rates were generally selected to emphasize stars with strong, reliable detections, we are particularly interested in the statistical properties of the distribution of core rotation rates, which implies that we want an unbiased selection function. While we cannot control whether Kepler observed the stars, or the detectability of the modes required to measure core rotation, we can at least select a representative sample. We therefore chose a particular region of the sky, defined a sample spectroscopically, and looked at Kepler targets observed within that window. Specifically, we used two pointings of the APOGEE Data Release 12 sample in the Kepler field. The stars in these fields represent all of the red giants in the region within a color- and magnitude-limited sample (Zasowski et al. 2017). We then defined, using the distribution of seismically measured masses and evolutionary states in Pinsonneault et al. (2014), the region of spectroscopic gravity and temperature where massive, core-helium burning stars were most likely to be found (see Figure 1). We used the Bovy et al. (2014) red clump catalog separation between core-helium burning and shell hydrogen-burning stars as a function of gravity and temperature as a starting point, and then we optimized the slope and intercept to better separate seismically identified massive, core-helium burning stars, which tend to have higher metallicity, be hotter, and have higher gravity than low-mass red clump stars. We used the relation

$$\begin{eqnarray*}&&{\bf{2.75}}\leqslant \mathrm{log}\,g\leqslant 0.0018\,\mathrm{dex}\,{{\rm{K}}}^{-1}({T}_{\mathrm{eff}}-{T}_{\mathrm{eff}}^{\mathrm{ref}}(\,[\mathrm{Fe}/{\rm{H}}]))+{\bf{2.3}},\end{eqnarray*}$$

where

$$\begin{eqnarray*}&&{T}_{\mathrm{eff}}^{\mathrm{ref}}([\mathrm{Fe}/{\rm{H}}])=-{\bf{201.6}}\,{\rm{K}}\,{\mathrm{dex}}^{-1}[\mathrm{Fe}/{\rm{H}}]+4607\,{\rm{K}},\end{eqnarray*}$$

with boldface values indicating our alterations to the original Bovy et al. (2014) formula. Analysis of the 669 stars meeting this criteria suggested that this cut was expected to yield a sample that was 28.2% pure and 46.8% complete for stars above 2.3 M_☉.

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2.2.3. Surface Rotation Sample

2.2.4. Seismically Selected Sample

For our main sample, which was selected using all available seismic and spectroscopic information, we test the dependence of core rotation on mass, gravity, and metallicity. To select the sample, we started with the sample of all Kepler giants observed by the Sloan Digital Sky Survey III (Data Release 12; Alam et al. 2015). We further selected only stars with metallicities and temperatures from the ASPCAP pipeline, as well as seismic Δν and ν_max measurements from the A2Z pipeline (R. García et al. 2020, in preparation). We used the seismic scaling relations (Kjeldsen & Bedding 1995) with the solar reference values appropriate to this pipeline to get a preliminary seismic mass, radius, and surface gravity for each star. For this initial selection, we did not apply corrections to the Δν scaling relation, as they were not included in the Pinsonneault et al. (2014) analysis. These corrections are in general small for these stars (Pinsonneault et al. 2018), so this does not significantly impact our selections. We identified two mass bins (2.2–2.4 M_☉ and 2.6–2.8 M_☉) and ordered the stars by metallicity. In the highest-mass (youngest) bin, metallicities ranged from ([Fe/H] = −0.2 to +0.5). To cover this range, we selected two metallicity bins of interest, a low-metallicity bin ([Fe/H] < −0.0) and a high-metallicity bin ([Fe/H] > +0.2). Finally, we wanted to cover the evolution of stars in each mass and metallicity bin through the secondary clump. We therefore selected a star in each bin with a seismic surface gravity of approximately 3.0, 2.8, and 2.6 dex. While the limited number of stars in some bins required gravity offsets of several hundredths of a dex from our target, we were able to span a range of at least 0.3 dex in log(g) in each mass–metallicity bin. During the selection process, we emphasized stars with measured properties similar to active stars with measured surface rotation periods above in order to determine whether surface activity is correlated with core rotation rate.

Evolved red giants develop deep surface convection zones, which incorporates nuclear processed material from the deep core into the surface layers. This "first dredge-up" changes the surface [C/N] ratio (Iben 1967). There is a striking correlation between asteroseismic mass and the surface [C/N] ratio. This has been used to infer masses, and thus ages, from spectra, using joint asteroseismic and spectroscopic data sets for training (Martig et al. 2016; Ness et al. 2016). However, there is a real intrinsic spread in [C/N] at fixed mass and metallicity (Pinsonneault et al. 2018) for intermediate-mass stars, and it is interesting to explore whether it is correlated with stellar rotation. We therefore identified three pairs and two trios of stars that had different [C/N] ratios (Δ[C/N] > 0.2 dex) but otherwise were very similar (Δ[Fe/H] < 0.1 dex, Δ[α/Fe] < 0.03 dex, ΔMass < 0.13 M_☉, and Δlog(g) < 0.05 dex). In all but one of our [C/N] groups, one of the stars was already included in one of the previous samples. This produced a final sample of 21 seismically selected stars, in addition to the 60 active stars, 20 giants in our spectroscopically selected sample, and 7 stars with previously measured core rotation periods we use for calibration. All these stars and their properties are listed in Table 1; we show their distribution in mass, metallicity, and surface gravity compared to the full Pinsonneault et al. (2018) sample in Figure 2, and their properties are summarized in Table 2.

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Table 1.  Properties of the Stars Studied Here

KIC	Mass	log(g)	Teff	[Fe/H]	[C/N]	Psurf	Group	ΔΠ1	q	Splitting	Pcore	Psurf,theory
	(M☉)	(dex)	(K)	(dex)	(dex)	(days)		(s)		(μHz)	(days)	(days)
3744681	2.315	2.710	4938.10	0.042	−0.648	−9999	1	285.9	0.3	0.1	57.87	152.12
4659821	2.230	2.933	5032.54	0.098	−0.662	−9999	1	213.4	0.2	0.085	68.08	69.12
5184199	2.165	2.905	5000.37	0.102	−0.816	−9999	1	239.2	0.2	0.095	60.92	92.55
7467630	2.765	2.780	4998.47	0.138	−0.674	−9999	1	293.3	0.3	0.06	96.45	76.94
7581399	2.894	2.841	5079.48	0.022	−0.679	150.83	1, 3	221.35	0.25	0.08	72.34	−9999.
8962923	2.158	2.828	5051.95	−0.056	−0.532	−9999	1	298.2	0.3	0.075	77.16	118.07
9346602	2.141	2.658	4908.06	−0.011	−0.529	−9999	1	242.8	0.3	0.08	72.34	196.53
8805663	2.325	2.605	4885.94	0.134	−0.883	−9999	2	325.2	0.3	0.05	115.74	204.40
8414518	2.314	2.665	4930.66	0.128	−0.603	−9999	2	267.45	0.3	0.06	96.45	177.31
8872979	2.576	2.801	5119.89	0.028	−0.883	−9999	2	297.5	0.3	0.075	77.16	88.10
9002805	2.087	2.735	5013.67	0.041	−0.653	−9999	2	243.7	0.3	0.06	96.45	169.36

Note. Mass and surface gravity estimates come from Pinsonneault et al. (2018) or the A2Z pipeline with Pinsonneault et al. (2018) correction factors. Temperatures, metallicities, and [C/N] come from APOGEE DR14 (Abolfathi et al. 2018); periods come from Ceillier et al. (2017). ΔΠ₁, coupling parameters (q), rotational splitting values, and core rotation periods are derived as discussed in Section 3.2. Theoretical surface rotation periods are derived using the models discussed in Section 4. Groups indicate why a star was selected: (1) in Deheuvels et al. (2015); (2) spectroscopic selection, Section 2.2.2; (3) surface rotation period from Ceillier et al. (2017); (4) asteroseismic sample, Section 2.2.4.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Table 2.  Summary Information for Each of Our Samples

Name	Group	Total	Pcore	Psurf	Modeled
Comparison	1	7	7	1	6
Unbiased	2	20	16	1	7
Rotation	3	58	32	58	35
Seismic	4	21	17	2*	17
Total		106	72	60	65

Note. For stars that fit the criteria of multiple samples, we included them in only the first group to avoid double counting. Core period measurements are described in Section 3.3, surface periods come from Ceillier et al. (2017), and modeled periods are estimated from the grid described in Section 4. Stars that fell outside the model grid in mass, metallicity, or mass-dependent surface gravity do not have estimated model periods. The surface periods of the stars in the seismic sample are marked with an asterisk because they were flagged as unreliable by Ceillier et al. (2017).

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To determine the evolutionary state of the stars, we applied established seismic methodologies exploiting the period spacing of mixed dipole modes (see, e.g., Bedding et al. 2011; Mosser et al. 2015). For an initial guess we used an autocorrelation analysis (AC) to determine the evolutionary stage for our sample by deriving the observed period spacing of the power spectral density (PSD), which was converted from frequency space into an equally spaced period spectrum. To improve the signal of the observed period spacing, we prewhitened the pure pressure radial modes and the pressure-dominated quadrupole modes in the four central radial orders of the power excess. We fit Lorentzian profiles to both modes and divided the PSD by the fit. This allows us to use the dipole mixed-mode spectrum in five consecutive radial orders, without blurring the autocorrelation from the presence of the small separation between these modes or introducing aliases from the period gaps that occur if this frequency range is cut out. Because the highest peak might be a short periodic artifact, we search for the most likely peak in the autocorrelation by identifying the minimum of the second derivative of the autocorrelation response. Finally, the result of the pipeline is validated by visual inspection and comparison with the value ranges of Bedding et al. (2011) and Mosser et al. (2011).

After identifying and excluding any RGB stars present in the unbiased sample and confirming the 2RC status of the remaining stars, we performed a more elaborate analysis to find the precise parameters of the mixed-mode spectrum and determine the core rotation period. From the asymptotic expansion based on work by Shibahashi (1979; see also Unno et al. 1989), Mosser et al. (2012) showed that for a star with known Δν and ν_max (our case; see Section 2.1), the pattern of individual mixed dipole modes can be described through the use of the constant value of the period spacing, ΔΠ₁, which describes the actual but unobservable constant period spacing of dipole modes and the corresponding coupling parameter q. Numerous techniques have been developed to determine the underlying value of ΔΠ₁ and the corresponding coupling factor q for a given star (Mosser et al. 2012, 2015; Stello 2012), and large samples of such measurements have been published (Mosser et al. 2015; Vrard et al. 2016). Here we perform a grid search for the best parameter combination of ΔΠ₁ and q for a given star. This approach was extensively explored by Buysschaert et al. (2016) but is very computational expensive. Therefore, the simplified version implemented by Beck et al. (2018) is used, based on the central radial mode, the large frequency separation, and a semiautomatically compiled list of frequencies of mixed modes ($| m| =0$). The initial guess for the center of the ΔΠ₁-range is taken from the AC response of the observed period spacing. After running a coarse, wide grid covering about 200 s, a second grid search was performed around the minimum found in the first grid. The second grid was run with increments of 0.1 s and 0.05 for ΔΠ₁ and q, respectively, to avoid falling into a local minimum (see Buysschaert et al. 2016). The resulting values of ΔΠ₁ and q are given in Table 1.

To test the robustness and reliability of this approach, we used this method blindly on the seven stars analyzed by Deheuvels et al. (2015). In all but one star, a nearly perfect agreement between the ΔΠ₁ values of Deheuvels et al. (2015) and our analysis is found, and the reason for this single discrepancy is the different formalism used to fit the mixed-mode pattern. Specifically, Deheuvels et al. (2015) used the revised mathematical description of the mixed-mode pattern (see, e.g., Mosser et al. 2014), which involves four parameters instead of the original two parameters. However, visual inspection of the solution for the dipole forest shows good agreement between the two results, and we show in Figure 3 that the inferred rotation rates are similar for the two methods, even with the discrepant assumptions for ΔΠ₁.

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Once the underlying parameters ΔΠ₁ and q were found for a given star, we searched for a g-dominated dipole mixed mode with a clear signature of rotational splitting. To test this estimate of the rotational splitting, we forced that value for the rotational splitting on all modes. This requires an assumption about the rotational gradient between the surface and the core, which, for simplicity, we assumed was a factor of two, corresponding to a constant value of the rotational splitting for all modes, regardless of the mode trapping. Because we used g-dominated modes to infer the splitting, we do not expect this assumption to substantially impact the inferred core rotation rates. If the found rotational splitting value of the g-dominated modes presented a convincing fit to the full range of dipole modes, we adopted the derived value as the rotational splitting of the core δf_c. The resulting values of δf_c are given in Table 1.

For g-dominated modes, the rotational kernels are known to be dominated by the properties of the core (e.g., Beck et al. 2012, 2014; Goupil et al. 2013; Deheuvels et al. 2014; Di Mauro et al. 2016; Triana et al. 2017). As the Ledoux constant (Ledoux 1951) is well approximated for these g-dominated mixed modes by 0.5, twice the value of the rotational splitting δf_c is a very good proxy of the core rotation rate, and this is the value we report as our core rotation rate. We note that we did not extract the rotational splitting of pressure-dominated modes, as their pattern is complicated by broad peaks originating from short mode lifetimes and the overall small level of rotational splitting.

We compared the core rotation rates derived by this procedure with those determined by Deheuvels et al. (2015) (see Figure 3), using a somewhat different procedure, and we find agreement within 4%. This is well within the formal uncertainty (8%) of the sophisticated frequency analysis of Deheuvels et al. (2015), and we are confident that our method produces accurate and robust results. We show in Figure 4 the resulting rotational splittings as a function of ΔΠ₁ for the stars in our sample, as well as the Deheuvels et al. (2015) stars. Additionally, we choose to conservatively adopt the published error from Deheuvels et al. (2015), 8%, for the uncertainty on our core rotation measurements. Our wider range of ΔΠ₁ allows us to see a trend in core rotational splitting with ΔΠ₁ that was not visible using only the stars from Deheuvels et al. (2015) because of the limited parameter space covered by that sample.

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As discussed in Tayar & Pinsonneault (2018), these models are generated using the Yale Rotating Evolution Code (YREC; Pinsonneault et al. 1989; van Saders & Pinsonneault 2012) and cover the range from 2.0 to 3.0 M_☉ in steps of 0.2 M_☉. Models are run using a Grevesse & Sauval (1998) mixture, with atmosphere boundary conditions from Kurucz (1997). Core overshooting of 0.16 pressure scale heights has been used in order to reproduce the observed locus of the secondary clump (see Tayar & Pinsonneault 2018). We have added to this grid models at two additional metallicities: [Fe/H] = −0.3 and [Fe/H] = +0.3, in addition to the solar-metallicity models already included in Tayar & Pinsonneault (2018). For these models at different metallicities, we have taken the calibration of the mixing length and helium abundance as a function of metallicity from Tayar et al. (2017). We also use the updated calculation of the convective overturn timescale discussed in Somers et al. (2017). We summarize the physical ingredients of the model and the extent of the model grid in Table 3.

Analyses of the surface rotation rates of secondary clump stars have indicated that the surfaces of these stars are rotating much more slowly than standard models would predict (Tayar et al. 2015; Ceillier et al. 2017; Tayar & Pinsonneault 2018). Tayar & Pinsonneault (2018) showed that these slow rotation rates, in combination with the measured core rotation rates of Mosser et al. (2012), placed strong constraints on the allowable physics of angular momentum transport and loss in these stars. Specifically, the measured rotation rates require enhanced angular momentum loss relative to traditional wind laws, based on the Kawaler (1988) framework, which were only weakly dependent on mass and radius. More modern wind laws, in the Matt et al. (2012) framework, predict a stronger dependence on the underlying stellar parameters. In this work, we employ the Pinsonneault, Matt, & MacGregor (PMM) wind law, as described in van Saders & Pinsonneault (2013), which has an explicit dependence on the radius, luminosity, and convective overturn timescale and predicts enhanced spin-down on the giant branch. This model has been successful in reproducing the observed rotation distributions of stars on the secondary clump (Tayar & Pinsonneault 2018), although matching previously observed distributions of core rotation periods seems to require some amount of radial differential rotation in the star, which could be happening either in the radiative interior or in the surface convection zone (or both).

Previous works (e.g., Tayar & Pinsonneault 2013; Cantiello et al. 2014; den Hartogh et al. 2019) have shown that the cores of these stars could not have been totally decoupled from their envelopes for their entire post-main-sequence evolution, as that would produce higher core rotation rates (Ω ∼ 10⁻² μHz) and larger core–envelope contrasts (${{\rm{\Omega }}}_{\mathrm{core}}/{{\rm{\Omega }}}_{\mathrm{surface}}\sim {10}^{5}$) than are observed. We therefore do not run separate calculations with totally decoupled core rotation here.

Following the work of Tayar & Pinsonneault (2018), we construct two classes of models whose predicted surface rotation distributions are consistent with the measured surface rotation distributions of intermediate-mass core-helium burning stars.

The simplest case is a star where both the core and surface rotate rigidly. This also serves as a lower limit on the range of allowed core rotation rates. Since the star is forced to rotate rigidly, this model also yields a slowly rotating core (Ω ∼ 10⁻⁷ μHz) and no difference between the core and envelope rotation rates $\left(\tfrac{{{\rm{\Omega }}}_{\mathrm{core}}}{{{\rm{\Omega }}}_{\mathrm{surface}}}=1\right)$. One slight perturbation on this case is the possibility of a rigidly rotating envelope but a differentially rotating core. Because the moment of inertia of the core is very small relative to the envelope, surface rotation period predictions for models with solid-body rotation in the convection zone are insensitive to the treatment of internal angular momentum transport in the core, and so such models are still consistent with the available surface rotation constraints. In a few relevant plots, we add arrows to indicate the direction that differential rotation in the core would shift our predictions.

For our second case, we allow the differential rotation to take place in the surface convection zone, with a rotation profile that goes as R⁻¹. This model was shown in Tayar & Pinsonneault (2018) to be roughly consistent with the distribution of surface rotation rates, and we use it here as an upper limit on the predicted range of core rotation rates if the differential rotation is located in the surface convection zone. In this model, we fix the rotation of the core to the rotation rate at the base of the envelope, preventing a large shear from developing there. This model predicts a core that rotates substantially slower than the decoupled case, but faster than the surface of the star (Ω ∼ 10⁻⁶ μHz, $\tfrac{{{\rm{\Omega }}}_{\mathrm{core}}}{{{\rm{\Omega }}}_{\mathrm{surface}}}\sim 3$).

With our spectroscopically selected sample, we can explore the true distribution of core rotation rates and identify whether our seismically selected sample was biased (e.g., against rapidly rotating stars; see Tayar et al. 2015). In the spectroscopically selected sample, where we expected 28% of the stars to be above 2.3 M_☉ in the secondary clump, we ended up identifying 6/20 such stars (30%), consistent with this prediction. We show in Figure 5 that the distribution of core rotation rates for our spectroscopically selected sample is similar to our full sample, except that it lacks the fastest-rotating stars. At first glance, this seems counterintuitive, since larger rotational splittings and therefore shorter rotation periods are easier to measure. However, we note that the fastest rotators in the sample are all at very high gravity and often selected for their measurable surface rotation period or to fill out the highest gravity bins in our sample selected with seismic information. This suggests that such stars are overrepresented in our sample, and we should be careful in future sections with any conclusions drawn solely from stars with surface gravities above about 2.9 or periods shorter than about 60 days. In particular, we acknowledge that given the size of our sample, some stars have almost certainly interacted with a companion on the upper giant branch, and these high-gravity, rapidly rotating, active stars are consistent with the properties we expect for such interaction products and thus should be treated carefully. Other than this fast-rotating excess, however, our unbiased spectroscopically selected sample seems to match our full distribution of core rotation periods, and so we combine all four samples of stars for the following analysis.

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We show in Figure 6 the core rotation rates we measured as a function of gravity (top), mass (middle), and metallicity (bottom). It is clear that the core rotation rate is correlated most strongly with surface gravity. Specifically, the cores of these stars are slowing down as the envelopes expand, providing the first model-independent evidence of strong core–envelope coupling on the secondary clump. We do not see any strong trends in the core rotation rates with mass or metallicity.

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The ratio of carbon to nitrogen, [C/N], at the surface of a red giant is considered a mixing diagnostic because it is significantly affected by how much nitrogen-rich material is dredged up from the stellar interior. Martig et al. (2016) and Ness et al. (2016) showed empirically that the [C/N] ratio is strongly correlated with stellar mass and therefore age. While this dependence of [C/N] on mass is significant for low-mass stars, we show in Figure 7 (top) that the trend flattens for stars above ∼2.0 M_☉, such that all stars between 2.0 and 3.0 M_☉ have [C/N] of order ∼−0.7 dex, although there is still significant scatter. If this residual scatter is driven by rotational mixing, we might expect a correlation between [C/N] and either the surface (Figure 7, middle) or core (Figure 7, bottom) rotation rate. However, we see no such trends. We do, however, notice that the fraction of core rotation nondetections seems to be correlated with the [C/N] (Figure 7, top), where stars with [C/N] higher than the average (less mixed) are more likely to not have measured core rotation rates, and a Kolmogorov–Smirnov test indicates that this difference is significant at the 99% confidence level. We suggest that this could indicate that there is some process that affects both the strength of the mixed modes and the surface composition. For example, the strong core magnetic fields suggested by Fuller et al. (2015) to suppress the mixed modes required to measure core rotation might also affect the mixing in these stars, or a history of stellar interactions could be affecting the surface chemistry and also the interior structure in a way that makes the core rotation hard to measure. We suggest that a closer look at these stars might help determine the cause of this correlation and, in turn, reveal something about the mixing history of these stars.

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We also notice a tentative lack of low-[C/N] stars with fast-rotating surfaces (P_surf < 90 days). These periods are above the range predicted by our single-star models for most of the secondary clump, and we suggest that some of the stars rotating this fast might have been spun up by a stellar interaction. If this is the case, then the lack of low [C/N] values in these stars could suggest (1) mass accretion, where these stars were less massive during the first dredge-up and therefore did not reach such low [C/N] values; (2) the accretion of unmixed, high-carbon material; or (3) an inhibition of mixing during the merger, although we caution that this result is not strongly significant and requires a larger sample of stars where core rotation measurements were attempted. Additionally, the fact that this result only shows up in the surface rotation periods and not in the core rotation periods would suggest that stellar interactions do not or are slow to affect the core rotation rate, consistent with suggestions by Beck et al. (2018).

There were 60 stars in our sample that had measured reliable surface rotation periods from Ceillier et al. (2017). We particularly selected these stars because we wanted to use them to examine the core–surface contrast and how it evolves on the secondary clump. However, since only 2% of red giants have detectable surface rotational modulation (Ceillier et al. 2017), and the amount of surface activity is often correlated with the stellar rotation rate (Noyes et al. 1984), we therefore need to test whether active stars were biased tracers of the core and surface rotation distributions.

In Tayar & Pinsonneault (2018) we demonstrated that modern angular momentum loss models can be used to successfully predict the surface rotation rates of stars in this evolutionary phase. For each of these cases in the following plots, we show the predicted rotation for a moderate rotator on the main sequence (surface rotation rates of ∼150 km s⁻¹; Zorec & Royer 2012), with ranges determined assuming main-sequence rotation velocities of 50 and 250 km s⁻¹, to bracket the observed range of main-sequence stars in the studied mass range. We note that, given the lack of information in literature on the range of rotation rates in intermediate-mass stars as a function of metallicity, we use the same initial rotation distributions for stars at all metallicities.

We show in Figure 8 that while the majority of observed periods are consistent within errors with our theoretical predictions, there is an interesting population of stars with observed periods substantially shorter than we would have predicted. These stars could be the descendants of interacting systems that gained angular momentum from a companion, but they could also be the result of measuring half the actual period, a common problem given the difficulty of measuring relatively small-amplitude, long-period signals through the systematic trends caused by the quarterly roll of the satellite. In addition, we show in Figure 9 that the lower envelope of the measured rotation rate distribution does not seem to evolve to longer period with larger radius, which contrasts with the predictions of angular momentum conservation. In addition, we note that the distribution in gravity of stars where surface rotation was measured is not consistent with our underlying sample. Together, these confirm that the active stars are indeed a biased tracer of the surface rotation distribution and should be used with caution. It also raises questions about whether their core rotation rates are also a biased tracer of the underlying distribution, and whether we should exclude them from the sample. However, we show in Figure 10 that the active stars do not have significantly different core rotation rates than the inactive stars of similar gravity. We therefore tentatively conclude that while there might be a bias in the observed surface periods, there is not a strong correlation between the surface activity and core rotation in our sample.

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In Figure 11 we show the ratio between the core and surface rotation rates for the 33 stars in our sample where both are measured (blue circles). We also show the ratio between the seismically measured core and envelope rotation rates from Deheuvels et al. (2015) (red diamonds). As previously discussed, however, these periods are likely to be biased tracers of the distribution. Therefore, in order to avoid bias from more evolved stars with longer surface rotation periods that were not detectable in Ceillier et al. (2017), we predict a surface rotation rate and thus a core–envelope contrast for the stars in our sample (gray squares) by assuming that these stars started out as moderate rotators on the main sequence. In addition to these observations, solid-body rotation, a ratio of 1, is marked as a dotted line. There seems to be some evidence in the data that the difference between the core and the envelope rotation is decreasing on the secondary clump as stars evolve to lower surface gravities, but this trend is not as significant when using theoretical priors on the surface rotation rates. In fact, the theoretical predictions show some preference for increasing core–envelope contrast as the star evolves to lower gravities. We therefore cannot comment authoritatively on whether stars are decoupling or recoupling on the secondary clump, and we encourage the collection of a larger unbiased sample of surface rotation rates for these stars, particularly the low-gravity stars.

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