Locked differential rotation in core-helium burning red giants

Oscillation modes of a mixed character are able to probe the inner region of evolved low-mass stars and offer access to a range of information, in particular, the mean core rotation. Ensemble asteroseismology observations are then able to provide clear views on the transfer of angular momentum when stars evolve as red giants. Previous catalogs of core rotation rates in evolved low-mass stars have focussed on hydrogen-shell burning stars. Our aim is to complete the compilation of rotation measurements toward more evolved stages, with a detailed analysis of the mean core rotation in core-helium burning giants. The asymptotic expansion for dipole mixed modes allows us to fit oscillation spectra of red clump stars and derive their core rotation rates. We used a range of prior seismic analyses, complete with new data, to get statistically significant results. We measured the mean core rotation rates for more than 1500 red clump stars. We find that the evolution of the core rotation rate in core-helium-burning stars scales with the inverse square of the stellar radius, with a small dependence on mass. Assuming the conservation of the global angular momentum, a simple model allows us to infer that the mean core rotation and envelope rotation are necessarily coupled. The coupling mechanism ensures that the differential rotation in core-helium-burning red giants is locked.


Introduction
Solar-like oscillations in red giants have been extensively studied since their first unambiguous detection with the CoRoT mission (De Ridder et al. 2009).The NASA mission Kepler then provided longer time series and the yields have benefitted from enhanced methods for analysing the oscillation pattern and finer frequency resolution (e.g., Mosser et al. 2009;Bedding et al. 2010;Huber et al. 2010).Contrary to the Sun and low-mass main sequence (MS) stars, pressure waves excited in the upper stellar envelope of red giants can efficiently couple with gravity waves, whose propagation is restricted to the inner region delimited by the radiative core.Therefore, non-radial modes show a specific pattern of mixed modes, as theoretically predicted and depicted much before the large seismic surveys provided by space-borne missions (Shibahashi 1979;Unno et al. 1989).Taking advantage of these properties, dipole mixed modes have opened a new window in asteroseismology, due to their ability to probe the stellar core of red giants (Beck et al. 2011), unveil the evolutionary stage of evolved stars (Bedding et al. 2011), measure the mean core rotation (Beck et al. 2012), and/or detect strong magnetic fields in the red giant cores (Li et al. 2022).
In practice, the understanding of the mixed-mode pattern has proven to be very useful for a thorough analysis of solarlike oscillations in red giants.In the last ten years, the dipole mixed mode pattern of many red giants has been analyzed for a wealth of purposes, using the asymptotic expansion and theoretical developments (e.g., Goupil et al. 2013;Grosjean et al. 2014).Altogether, all the stars for which individual mixed modes were identified and fitted in previous work now form a rich, valuable data set of more than 600 stars on the red giant branch (RGB) and 700 in the red clump (RC).Gehan et al. (2018) already presented the analysis of the core rotation on the RGB.Here, we focus on core-helium burning stars, whose importance in terms of probing stellar densities, kinematics, and chemical abundances in the Galaxy has been illustrated by Girardi (2016) and references therein.
An extensive analysis of the core rotation of RC stars has not yet been made, with current information limited to the first catalog by Mosser et al. (2012), with about 310 red giant stars, including 225 located in the red clump.Such a data set is too small to enable a thorough study of the dependence on stellar evolution or on such parameters as the stellar mass.In this work, we use the consistent description of all seismic parameters provided by the asymptotic expansion (Mosser et al. 2018) to decipher the mixed-mode oscillation pattern of core-helium burning star to infer their mean core rotation.
In Section 2, we present the scientific context and the data used in this work.A specific seismic analysis for measuring rotation rates, based on stretched frequency spectra, is given in Section 3. Our results related to the mean core rotation in corehelium burning stars are discussed in Section 4. Section 5 is devoted to our conclusions.(2022).Typical uncertainties are below 0.1 µHz for ∆ν and below 4 s (1 s) for ∆Π 1 for, respectively, RC (RGB) stars.

Data
Rotational splittings in RC stars can be quite small (Mosser et al. 2012), with values comparable to the Kepler orbital cyclic frequency δν Kep ≃ 31.1 nHz, due to its revolution period (≃ 372.57days).This frequency can be spuriously introduced as an alias of genuine frequencies, when the photometric time series presents recurrent gaps introduced when one or more Kepler quarters are regularly missing.This occurs when a star falls on a damaged CCD (Handberg & Lund 2014).Therefore, it is necessary to take care to avoid stars with poor duty cycle.
The diversity of the origin of the stars previously analyzed prevents any firm information about potential bias selection when using them.For instance, Mosser et al. (2014) favored the analysis of stars with large masses in order to get a more detailed view on the secondary red clump.Therefore, in order to reduce the selection bias as much as possible, stars identified as RC stars were systematically analyzed.
The data set we obtained is shown in the ∆ν -∆Π 1 diagram, where the large separation, ∆ν, is representative of the envelope density and the period spacing, ∆Π 1 , of the core property (Fig. 1).This diagram allows us to infer the evolutionary stage of evolved low-mass stars.

Global fit
The global fit of the mixed-mode pattern relies first on the identification of radial modes (Mosser & Appourchaux 2009) in order to precisely locate pure pressure dipole modes that would still exist in the absence of coupling.This location must be precise enough to include the variation with frequency of the large separation due to second-order effects and to the modulation of pressure modes by acoustic glitches (Vrard et al. 2015).Then, the parameters of the gravity component of the mixed modes can be measured: the period spacing, ∆Π 1 , and the coupling factor, q, are automatically determined, whereas the gravity offset, ε g , is derived from the identification of a given mixed mode.The final step of the global fit comes from the identification of the rotational splittings (Mosser et al. 2018).

Rotational splittings
Both mean core and mean envelope rotation rates can be derived from dipole mixed modes (Goupil et al. 2013).Here, we used previous measurements to check that the mean envelope rotation is significantly smaller than the mean core rotation, aside from a few secondary clump stars (Deheuvels et al. 2015).Measuring the envelope rotation is however difficult and noisy.For sake of simplicity, in the following, we assume that the mean envelope rotation can be neglected in first approximation.
A convenient way to get rid of the complexity of the mixedmode frequency pattern consists of considering stretched spectra, constructed in such a way that mixed modes are expected to be regularly spaced.Here, contrary to the method used for inferring regular period spacings (Mosser et al. 2015), this approached is aimed at unveiling regular frequency spacings and thus making use of stretched frequencies, ν s , defined by: where ζ is the stretching function.Then, stretched rotational multiplets exhibit the same equidistance δν rot , which can be measured by the autocorrelation of the stretched spectrum.A typical identification is shown in Fig. 2, where the identification is complicated by the stochastic excitation of the peak; however, the value of the rotational splitting is secured by the many multiplets that are observed.For most stars in the red clump (and contrary to RGB stars), rotational splittings are small enough to avoid any confusion  with period spacings due to the overlap of different modes with different mixed-mode orders.However, two effects, combined with the stochastic excitation of modes, can complicate the identification of rotational doublets or triplets.First, for dim magnitudes or crowded fields, oscillation spectra show a poor signal.Second, low values of the inclination angle hamper the observation of |m| = 1 modes.A combination of the two effects complicates the identification further.
We could derive 1500 rotational splittings in RC stars, among which more than 850 were analysed in a systematic way, enlarging the previous catalog by a factor larger than six.We checked whether the rotational signatures close to the Kepler year alias δν Kep could be artifacts, which is not the case since many signatures near δν Kep are due to doublets separated by 2δν Kep , for stars seen nearly edge-on.We also specifically checked cases where missing Kepler observation quarters could introduce an alias signature.We could then infer the absence of spurious signatures due to the Kepler orbit, as shown by the histogram of the rotational splittings (Fig. 3), where the peak at low values is due to the most numerous low-mass RC stars, and is not affected by the Kepler orbital frequency.

Biases and completeness
The method we used for measuring the rotational splittings is efficient up to a value corresponding to twice the spacing between consecutive mixed modes, as tested for RGB stars.For RC stars, this corresponds to values much above the distribution shown by the histogram and proves the completeness of this distribution at large values of δν rot .The situation at the low end of the distribution of the rotational splittings is less clear, with rotational splittings down to the minimum measured value δν rot = 12 nHz, slightly above the frequency resolution of 7.8 nHz reached by the four-year time series.Lower splittings cannot be measured.Furthermore, low values of simple splittings in rotational triplets for low-inclinations stars are also missing, contrary to double splittings of rotational doublets of stars seen edge-on.These effects explain the decrease of the distribution of δν rot in the histogram for low splittings.However, the missing information is limited: the number of stars with a clear oscillation signal and not showing any rotational splitting is low, that is, below 15 %.This indicates that a vast majority of rotational splittings could be measured, so that the histogram in Fig. 3 is representative of the core rotation rates, but it fails at giving clear indication of the smallest core rotation rates.

Core rotation rates along stellar evolution
In order to study the core rotation along stellar evolution and its dependence on stellar parameters, we considered the rotational splittings as a function of the large separation ∆ν (Fig. 4).We use the seismic scaling relations to derive proxies of the stellar mass, with the additional information of effective temperatures (Majewski et al. 2017).
Core rotations of RC stars are clearly distinct from the values measured for stars on the RGB (Fig. 4).Contrary to the flat distribution on the RGB shown by Gehan et al. (2018), the fit for red-clump stars shows a clear dependence on the large separation, δν rot ≃ (5.21 ± 0.13) ∆ν 1.322±0.039, (2) for ∆ν in µHz and δν rot in nHz.
In order to emphasize any mass dependence, we investigated the variation of median values of the rotational splittings in different bins of mass, as a function of ∆ν (Fig. 5).All curves are close to each other and close to the mean power law in ∆ν, expressed by Eq. ( 2).The variations of the parameters (exponent, factor) with stellar masses are not significant.Seismic parameters and rotational splittings are listed in Table 1.The full table is available online.

Discussion
In this section, ensemble asteroseismology is considered to derive information about the transfer of angular momentum from the measurements of the core rotation rate.Then, the comparison of our measurement with the surface rotation rates of MS stars is used to provide a consistent picture of the mean stellar rotation along stellar evolution.

Rotation and stellar evolution
The power law (Eq.2) used to fit the mean core rotation rate is compatible with an evolution as δν rot ∝ ∆ν 4/3 .According to seismic scaling relations (∆ν ∝ M 1/2 R −3/2 , e.g.Belkacem et al.Fig. 5. Mean evolution of the core rotational splittings of RC stars as a function of the stellar mass, with bins of 0.2 M ⊙ (full lines, with the same color code as in Fig. 4).The black dotted line shows a scaling as ∆ν 4/3 .Extrapolations of the mean rotation from the MS rotation rates are shown for similar mass bins (dashed lines).The blue dot-dashed line, for [0.7, 0.9]M ⊙ mass bin, a 0.2-M ⊙ mass loss into account.
2011; Miglio et al. 2012), this corresponds to the core rotation period scaling as where M and R are, respectively, the stellar mass and radius; and T 0 ≃ 1.62 days.From our modelling, we found that there is no scaling relation between T core and the core radius (Appendix A).Thus, the observations indicate that the mean core rotation period evolves as R 2 .A similar strong correlation was found in secondary-clump stars (Fig. 6 of Tayar et al. 2019).

Locking of the core and envelope rotation rates
In fact, such a relationship between the rotation and the stellar radius also exists for the mean envelope rotation, as derived from a simple analytical three-zone model composed of a rigid, dense helium core, an intermediate radiative layer with a rotation profile allowed to vary smoothly, and a rigid convective envelope.This analytical view is presented in Appendix B. For such a model, the total angular momentum writes: where M c = 4πρ c r 3 HBS /3 is the core mass below the heliumburning shell of radius, r HBS , and α is a structural parameter that accounts for the non-uniform mass distribution inside the helium core.This parameter α is expected not to vary significantly during the red clump evolution.Therefore, supposing the conservation of J and the predominant sensitivity of J to the radius variations yields With both core and envelope rotation rates evolving together as R −2 , we conclude that differential rotation is locked in corehelium burning red giants: The variation of the mean core rotation period as R 2 indicates that a mechanism for transferring angular momentum from the core to surface is able to couple the stellar radius and the core rotation.Such a transfer mechanism exists on the RGB, but is not efficient enough for spinning down their core rotation while the envelope radius drastically increases (Gehan et al. 2018).Evolution in the RC is much slower than on the RGB, by a factor of 80 for a 1.0-M ⊙ star or a factor of 15 for 1.8-M ⊙ star, according to the modelling.So, the measurement of the core rotation of RC stars enables us to constrain the efficiency of the mechanism(s) transferring angular momentum over a longer evolutionary phase: for RGB stars, the mechanism must ensure a stable core rotation period; for RC stars, it must ensure a rotation rate locked to the stellar radius, assuming the conservation of the global angular momentum.

Extrapolation from the main sequence
Since the core rotation is tightly coupled with the surface rotation, we also investigated the possible relationship between the core rotation in RC stars and the envelope rotation in MS stars (Chaplin et al. 2014;García et al. 2014), in the limit case of solid rotation.The significant mass redistribution inside the star along stellar evolution was taken into account with MESA simulations, which we used to infer the variations of the ratio β used for expressing moments of inertia as a function of the stellar fundamental parameters (Appendix C): As solid rotation is assumed, the envelope rotation Ω MS is representative of the mean rotation.From the conservation of the total angular momentum, and assuming negligible mass loss, we can extrapolate the mean rotation Ω during the clump phase from where the radius, R MS , and the rotation rate, Ω MS , are derived from the catalogs of MS rotation, while the coefficients β MS and β RC are derived from MESA.For stars above 1.1 M ⊙ , the mean rotation rates extrapolated from the MS are much slower than the core rotation measured in the red clump (Fig. 5).However, this is not the case in the range of [0.7 − 0.9] M ⊙ , where extrapolated mean rotation rates are much significantly larger.In order to explain this apparently paradoxical situation for the lowest mass, two solutions are possible, which both involve mass loss: a strong mass loss may evacuate a large amount of angular momentum; the too high inferred rotation may come from an overestimated mass redistribution due to a too low mass in the MS.Assuming a 0.2 M ⊙ mass loss in this second scenario, the change in the coefficient β RC , which expresses the mass redistribution in Eq. ( 8), moves the extrapolated value (Fig. 5) toward a trend that agrees with other evolutions.These simple arguments point toward a mass loss of at least 0.2 M ⊙ for the less massive stars of our data set.Extensive modelling to confirm this result is beyond the scope of this work.

Conclusion
We measured rotational splittings for 1500 core-helium burning stars observed by Kepler.The evolution of their mean core rotation rates scales approximately as ∆ν 4/3 , which corresponds to an evolution in R −2 dominated by the expansion of the stellar envelope.This indicates that mechanisms for transferring angular momentum from the core to the envelope are efficient in the core-helium burning phase.
Assuming the global conservation of angular momentum, a simple analytical model allows us to infer that the envelope rotation rate also evolves as R −2 in core-helium burning stars.Therefore, we infer that the rotation rates in the core and in the envelope jointly evolve as R −2 .Despite this efficient spinning down of the core associated with the expansion of the envelope, stars in the red clump do not rotate rigidly, but with a core rotation and an envelope rotation that are tightly coupled.The extrapolation of the surface rotation of MS stars provides similar information.For stars with a mass below 0.9 M ⊙ , a mass loss of about 0.2 M ⊙ at the tip of the RGB appears necessary to reconcile the extrapolated values and compare them with values in the helium-burning phase.

Fig. 1 .
Fig. 1. ∆ν -∆Π 1 diagram of the core-helium burning stars used in this work.The seismic estimate of the stellar mass is color coded.The low branch corresponds to stars ascending the RGB.Most of the outliers below the global trend of the RGB were identified as stars with a mass transfer by Deheuvels et al. (2022).Typical uncertainties are below 0.1 µHz for ∆ν and below 4 s (1 s) for ∆Π 1 for, respectively, RC (RGB) stars.

Fig. 2 .
Fig.2.Power density spectrum of rotational ℓ = 1 multiplets in a typical red clump star (KIC 4459025).Each multiplet is centered on its mean frequency, indicated on the left side; the y-axis shows the different multiplets, with normalized heights, ranked in frequency.The thickness of the lines is scaled by the mean height of the multiplets.The color codes the mean parameter ζ of the multiplet: gravity-dominated (pressure-dominated) mixed modes appear in purple (blue), respectively.The grey areas indicate the expected width of the components of the multiplets(Mosser et al. 2018).

Fig. 3 .
Fig. 3. Histograms of δν rot in red-clump stars (blue curve for all stars; grey curve for those systematically analyzed).The vertical dashed line at 31.1 nHz shows the Kepler orbital frequency.Typical uncertainties are provided by the horizontal bars, for small and large splittings.

Fig. 4 .
Fig. 4. Rotational splittings δν rot as a function of the large separation ∆ν.Large symbols represent red-clump stars, whereas small symbols are stars on the RGB.The color codes the stellar mass.Typical uncertainties for clump stars vary from 12 nHz at low ∆ν to 25 nHz at large ∆ν.