B D Klee

J R Stellhorn

M Krbal

N Boudet

G Chahine

Nils Blanc

W C Pilgrim

T Wagner

S Hosokawa

M Krbal C

G A Chahine E

Distinguishing the local environments of

Keywords: Chalcogenide glasses, RMC, AXS, amorphous structure

come

Introduction

Chalcogenide glasses exhibit many interesting properties. Among them, infrared transparency and a high optical nonlinearity offer many potential applications in infrared optics and optoelectronics. [1][2][3][4] Since the high glass forming ability of many chalcogenide glasses is very good the composition can be varied in a wide range of element concentrations, which makes it possible to tune specific properties of these glasses. For example the incorporation of Ga in GeSe 2 was shown to increase the solubility of rare earth elements such as Er in the glass, which enables a huge range of accessible properties. [5,6] Since the physical properties are tightly linked to the structure (structural units) of the amorphous phase, the structure of chalcogenide glasses have been investigated for several decades. In the case of amorphous Ga-Ge-Se glasses it is not simple to differentiate between gallium and germanium, since the neutron scattering lengths b as well as the X-ray form factors f are very similar for both elements. In the last years multiple similar compositions of such Ga-Ge-Se glasses were investigated using Extended X-ray Absorption Fine Structure spectroscopy (EXAFS) [7,8], optical and Raman spectroscopy [9,10], X-ray photoelectron spectroscopy (XPS) [7], neutron scattering [8], and advanced nuclear resonance spectroscopy (NMR) techniques [11]. Each of these methods is either limited to nearest neighbor information, or lacks contrast to reliably differentiate between similar elements and will mostly yield averaged information. In contrast, Anomalous X-ray Scattering (AXS) combines the advantages of scattering methods, which is information about the nearest neighbor as well as intermediate range order, and element specific methods. [12][13][14][START_REF] Waseda | Novel Application of Anomalous X-ray Scattering for Structural Characterization of Disordered Materials[END_REF][START_REF] Stellhorn | [END_REF] Here, we demonstrate a combination of AXS and RMC as an effective tool to provide structural information about the Ga 14.3 Ge 21.4 Se 64.3 glass, which is a composition on the pseudobinary tie-line (Ga 2 Se 3 ) x (GeSe 2 ) 1-x where x=0.25.

Experimental

A total 5 g of pure elements were placed into a quartz ampoule, evacuated to 10 -4 Pa and sealed. Next, the ampoule was placed to a rocking furnace to 1050°C (rate 1° C/min) and then kept rocking for 12h at this temperature. In a further step the temperature was reduced to 850° C (2° C/min), then stop rocking for 1h and subsequently the ampoule was quenched in cold water for about 10s. The sample was immediately annealed at 350° C for 3h and then gradually cooled to room temperature. The resulting bulk was polished and the bulk composition was confirmed at multiple sample positions by Energy-dispersive X-ray spectroscopy (EDXS) at 30 kV electron energy. The scattering experiments were performed at the ESRF BM02 using a highly monochromatic primary beam in reflection geometry. Differential structure factors ∆S(q) were obtained using a 2D detection system coupled with an analyzer crystal. The beam energies were chosen to be 20 eV or 200 eV below the corresponding K-alpha absorption edge for the near and far edge measurements respectively. The total structure factor S(q) was measured at 17.008 keV and a scintillation point detector coupled with the same analyzer crystal and a slit system were used in this case. The weighting factors for these measurements at the structure factor maximum Q=2 Å -1 as well as for neutron scattering are shown in table 1. It can be seen that the information density of X-ray and neutron total scattering functions is small for correlations containing only Ga and Ge. The differential structure factors exhibit increased weighting factors for all correlations including the respective element. This way, different correlations can be distinguished more accurately by introducing a contrast between multiple scattering experiments. The high Energy structure factor, as well as the Se-far and all differential structure factors were fitted simultaneously using Reverse-Monte-Carlo modeling (RMC). [17] The sample density is assumed to be 4.417 g/cm 3 (taken from [10]). Our RMC simulations proceeded as follows: Firstly, a random configuration containing 20300 atoms was generated and equilibrated by simulating without including experimental datasets. This configuration was used as a starting point for further simulations. The minimum interatomic distance was chosen to be 2 Å from this point on. A model simulation was performed without excluding any bonds. In the next step, different bonds were forbidden by changing the proper minimum distance to 2.9 Å while changes in the quality of fit are monitored in the form of R w values, which are related to χ 2 from comparing the simulated structure factors with the experimental data (see [18] for more information on the exact definition of R w ). Low R w values correlate to a good agreement of experiment and simulation. In these simulations atoms were moved a total of 3⋅10 7 times with an acceptance rate of roughly 28%.

Results and discussion

The results of several simulations with different combinations of excluded bonds are summarized in table 2. The average R w value, showing the quality of the fit, represents the average of the R w values of all fitted experimental datasets. From each simulation dataset the generalized 8-N rule value is calculated following [19] and is expected to be 8. When excluding any Se bond the experimental data could not be reproduced, therefore no such simulation is discussed further. The following discussion will focus on so-called "wrong bonds" (Ga/Ge-Ga/Ge). The simulations presented here can be divided into two groups: Whenever Ga or Ge were not allowed to form bonds to any other Ga or Ge atom, which is the case for simulations 5 and 7, the R w value increased. In these simulations the general 8-N rule is less satisfied than in all others. Additionally, the averaged coordination numbers calculated from these simulations are significantly different compared to other simulations. These effects are weaker when constraining Ga (simulation 5), since the low Ga amount in the sample limits the information content of this element in the scattering data. Therefore the interpretation will focus on simulations 1 to 4 and 6. These simulations are very similar regarding their results. The simulations imply that the presence of wrong bonds for both Ga and Ge is supported by the presented data. Since excluding single bonds does not have a significant impact on the overall structure or the fit quality, Ga and Ge seem to play a similar role in the structure formation and are mostly interchangeable without change in the short range order. Partial coordination numbers obtained from the simulations are shown in table 3. We find that, on average, Ga and Ge form 1-1.5 wrong bonds while the Se-Se coordination number is 0.8±0.1. We decided to proceed with the simulation that combines both, lowest R w value and highest number of constraints, which is simulation 6 in this case. The partial coordination numbers from this simulation are comparable to those obtained by Pethes et. al. for a similar composition (Ga 10 Ge 20 Se 70 ), who excluded the same combination of bonds. [8] For simulation 6 the RMC fitted datasets and the resulting pair correlation functions are shown in figures 1 and 2 respectively. Bond lengths can be obtained by gauss fitting the first peak in the pair correlation functions and the second neighbor distances can be obtained in the same manner by fitting the second peak. The bond lengths and second neighbor distances obtained from all simulations are very similar, which indicates that they are well defined by the data. For simulations 1 and 6 they are shown exemplarily in table 4. The observed second neighbor distances support the assumption of corner sharing tetrahedral configurations for Ga and Ge, which can be deduced by simple trigonometric calculations. In all presented simulations the second neighbor peak in the Ga-Ga pair correlation function has the highest amplitude of all element pairs (see Fig. 2). This means there is a preference for Ga atoms to be located close to each other forming Ga-Se-Ga bonds. The incorporation of Ga into chalcogenide glasses was shown to increase the solubility of rare earth elements. [20,21] Clustering of Ga atoms in the glass structure may provide sites where rare earth atoms can be stabilized by excessive negative charges thus explaining the increased solubility. Similar findings are reported for Ga doped Sulphur-based glasses by Lee et. al. [22]

Conclusion

The structure of glassy (Ga 2 Se 3 ) 0.25 (GeSe 2 ) 0.75 was investigated by anomalous X-ray scattering combined with reverse Monte-Carlo modelling featuring increased contrast between the contained elements. It is shown that both gallium and germanium are 4-fold coordinated while selenium has a coordination number greater than 2. These findings are in accordance with previous publications on similar substances. [7][8][9][10][11]23] The local and intermediate range structure around Ga and Ge are almost identical and the presence of wrong bonds for both Ga and Ge is mandatory to reproduce the experimental data. There is no evidence that certain bonds are not present in the material. Furthermore a preference for Ga clustering on a second neighbor length scale is observed which hints at segregation on an intermediate range scale.

Fig. 1 .

 1 Fig. 1. Experimental structure factors and RMC fits for simulation 6.

Fig. 2 .

 2 Fig. 2. Partial pair correlation functions from simulation 6

Table 1 :

 1 Weighting factors for all experimental structure factors at Q=2 Å -1 and neutron scattering.

	energy	w Ga-Ga	w Ga-Ge	w Ga-Se	w Ge-Ge	w Ge-Se	w Se-Se
	17 keV	0.018	0.057	0.178	0.043	0.273	0.430
	Se-far	0.021	0.064	0.184	0.049	0.279	0.403
	∆Ga	0.096	0.177	0.555	0.020	0.086	0.067
	∆Ge	-0.011	0.117	-0.039	0.180	0.675	0.077
	∆Se	-0.001	-0.005	0.149	-0.004	0.220	0.641
	Neutron Scattering	0.017	0.058	0.170	0.049	0.286	0.419

Table 2 .

 2 Total coordination numbers (CN) acquired from simulations with different combinations of excluded bonds. Each forbidden bond is marked with X

	Sim No.	Ga-Ga	Ga-Ge	Ge-Ge	CN Ga	CN Ge	CN Se	average R w / %	8-N rule
	1				4.08	3.99	2.33	6.14	8.09
	2	X			3.97	3.99	2.34	6.12	8.12
	3		X		4.06	3.98	2.32	6.19	8.10
	4			X	4.13	3.86	2.34	6.18	8.14
	5	X	X		3.51	3.98	2.36	6.57	8.26
	6	X		X	4.03	3.86	2.35	6.16	8.17
	7		X	X	3.83	3.05	2.45	8.51	8.58

Table 3 .

 3 Partial coordination numbers obtained in different simulations

	Sim No.	1	2	3	4	6
	Ga-Ga	0.52	-	0.97	0.52	-
	Ga-Ge	0.88	1.05	-	1.22	1.31
	Ga-Se	2.68	2.91	3.09	2.39	2.71
	Ge-Ga	0.59	0.70	-	0.81	0.87
	Ge-Ge	0.83	0.83	1.25	-	-
	Ge-Se	2.57	2.46	2.73	3.04	2.99
	Se-Ga	0.60	0.65	0.69	0.53	0.60
	Se-Ge	0.86	0.82	0.91	1.01	1.00
	Se-Se	0.88	0.87	0.73	0.80	0.75

Table 3 .

 3 First and second neighbor distances in simulations 1 and 6.

	correlation	first neighbor distance / Å	second neighbor distance / Å
	Sim No.	1	6	1	6
	Ga-Ga	2.43	-	3.76	3.75
	Ga-Ge	2.41	2.40	3.64	3.67
	Ga-Se	2.41	2.42	3.93	3.94
	Ge-Ge	2.36	-	3.68	3.66
	Ge-Se	2.37	2.36	3.85	3.87
	Se-Se	2.37	2.38	3.89	3.89

Acknowledgements

We acknowledge the European Synchrotron Radiation Facility for provision of synchrotron radiation facilities and we would like to thank the staff for assistance in using beamline BM02 in the scope of the proposal HC-2534. Furthermore we acknowledge Projects No. LM2015082 and CZ.1.05/4.1.00/11.0251 Center of Materials and Nanotechnologies cofinanced by the European Fund of the Regional Development and the state budget of the Czech Republic.


B Mosser

C Gehan

K Belkacem

R Samadi

E Michel

M-J Goupil

Period spacings in red giants IV. Toward a complete description of the mixed-mode pattern

Keywords: oscillations -Stars, interiors -Stars, evolution

Context. Oscillation modes with a mixed character, as observed in evolved low-mass stars, are highly sensitive to the physical properties of the innermost regions. Measuring their properties is therefore extremely important to probe the core, but requires some care, due to the complexity of the mixed-mode pattern. Aims. The aim of this work is to provide a consistent description of the mixed-mode pattern of low-mass stars, based on the asymptotic expansion. We also study the variation of the gravity offset εg with stellar evolution. Methods. We revisit previous works about mixed modes in red giants and empirically test how period spacings, rotational splittings, mixed-mode widths, and heights can be estimated in a consistent view, based on the properties of the mode inertia ratios. Results. From the asymptotic fit of the mixed-mode pattern of a large set of red giants at various evolutionary stages, we derive unbiased and precise asymptotic parameters. As the asymptotic expansion of gravity modes is verified with a precision close to the frequency resolution for stars on the red giant branch (10 -4 in relative values), we can derive accurate values of the asymptotic parameters. We decipher the complex pattern in a rapidly rotating star, and explain how asymmetrical splittings can be inferred. We also revisit the stellar inclinations in two open clusters, NGC 6819 and NGC 6791: our results show that the stellar inclinations in these clusters do not have privileged orientation in the sky. The variation of the asymptotic gravity offset with stellar evolution is investigated in detail. We also derive generic properties that explain under which conditions mixed modes can be observed.

Introduction

Probing the cores of stars is difficult since, generally, stellar information arises from their photosphere. Fortunately, asteroseismology of evolved stars reveals stellar interiors in a unique and powerful way: gravity waves that propagate throughout the core couple with pressure waves and construct mixed modes that can be observed (Beck et al. 2011;Bedding et al. 2011;Benomar et al. 2014). The measurement of the global seismic properties of these mixed modes then carries unique information on the core structure (e.g., Montalbán et al. 2013;Lagarde et al. 2016;Bossini et al. 2015Bossini et al. , 2017)). Observations with the space missions CoRoT and Kepler have provided the measurement of the asymptotic period spacings (Mosser et al. 2012b;Vrard et al. 2016), of the differential-rotation profile in red giants (Beck et al. 2012;Deheuvels et al. 2014Deheuvels et al. , 2015)), and of the core rotation for about 300 stars analyzed by Mosser et al. (2012c).

Most of the previous studies are based on the measurement and analysis of global seismic parameters, such as the asymptotic large separation ∆ν and the asymptotic period spacings ∆Π 1 (e.g., Miglio et al. 2017). It is now time to access the properties of individual frequencies in red giants. Up to now, most of the studies (e.g., Baudin et al. 2012;Di Mauro et al. 2016) were limited to stars on the red giant branch (RGB). Two main reasons explain this restriction: first, the oscillation spectra benefit from a better relative frequency resolution for this evolutionary stage; second, the oscillation spectra remain simple, with rotational splittings smaller than period spacings. When stars evolve, these features become intricate, so that confusion is possible. For the most evolved stars, mixed modes are no longer observable (e.g., Baudin et al. 2012;Mosser et al. 2013;Stello et al. 2014).

The understanding of any complicated mixed-mode oscillation pattern must be based on an unambiguous identification of the modes. Up to now, the most efficient method has relied on the use of the asymptotic expansion, completed by a clear description of the influence of rotation (Mosser et al. 2015). New insights on rotation were provided by the analysis depicted in Gehan et al. (2016), who have developed a methodology to measure rotational splittings in an automated way; Gehan et al. (2017) and Gehan et al. (2018) showed how rapid rotation can be addressed efficiently. This efficiency derives from the use of stretched oscillation spectra.

In this work, we first examine in Section 2 how the different frequency spacings in the asymptotic mixed-mode expansion can be expressed as a function of the mode inertia. New expressions are proposed for the mixed-mode spacings and rotational splittings. Case studies are examined in Section 3 to test and validate these expressions. In Section 4, we take advantage of the precision of the fits to derive accurate asymptotic period spacings and gravity offsets; for the first time, we can exhibit the global evolution of these gravity offsets as a function of stellar evolution. New insights on the rotational splittings are proposed in Section 5; in particular, we show how the asymptotic expansion can be used to provide priors based upon physical assumptions for any fitting code used later in the analysis. Finally, we assess the conditions for observing mixed modes, based on global asymptotic parameters only (Section 6). Section 7 is devoted to our conclusions.

Mixed-mode parameters

Following the work of Shibahashi (1979) and [START_REF] Unno | Nonradial oscillations of stars[END_REF], we derived asymptotic expansions of mixed modes for different seismic parameters: eigenfrequencies (Mosser et al. 2012b), period spacings (Christensen-Dalsgaard 2012), rotational splittings (Goupil et al. 2013;Deheuvels et al. 2015), and mode widths and mode heights (Grosjean et al. 2014;Belkacem et al. 2015b,a;Mosser et al. 2017a).

Here, we intend to revisit all these parameters that depict the mixed-mode spectrum in order to provide a more precise and unified view.

Asymptotic expansion

The asymptotic expansion of mixed modes is an implicit relation between the phases θ p and θ g of the pressureand gravity-wave contributions to the mixed modes, respectively. It reads tan θ p = q tan θ g ,

where q is the coupling factor (Mosser et al. 2017b). The phases are related to the large separation ∆ν and the period spacing ∆Π 1 . The most convenient expressions of the phase refer respectively to the pure1 p and g mode spectra

θ g = π 1 ∆Π 1 1 ν - 1 ν g , (2)
θ p = π ν -ν p ∆ν p , (3)
where ν p and ν g are the asymptotic frequencies of pure pressure and gravity modes, respectively, and ∆ν p is the frequency difference between the consecutive pure pressure radial modes with radial orders n p and n p +1. In this work, we consider that the radial modes and pure dipole pressure modes obey the universal red giant oscillation pattern (Mosser et al. 2011b) and that the dipole gravity modes follow the asymptotic comb-like pattern

1 ν g = (-n g + ε g ) ∆Π 1 , (4)
where ∆Π 1 is the period spacing and ε g is the gravity offset. Mosser et al. (2015) derived that the variation of the oscillation period P with the mixed radial order n writes dP dn = ζ ∆Π 1 .

(5)

A convenient way to write the parameter ζ is (Hekker & Christensen-Dalsgaard 2017)

ζ(ν) = 1 + q N 1 q 2 cos 2 θ p + sin 2 θ p -1 , (6)
where N = ∆ν/(ν 2 ∆Π 1 ) is the density of gravity modes compared to pressure modes, in other words the number of mixed modes in a ∆ν-wide interval. Compared to the original form presented in Mosser et al. (2015), the rapidly varying phase θ g has been replaced by a function of θ p that varies in a smooth way.

As demonstrated by Goupil et al. (2013) and used by subsequent work (Benomar et al. 2014;Deheuvels et al. 2015), the function ζ is connected to the inertia of mixed modes. Introducing the contributions of the envelope and of the core,

ζ = I core I env + I core , (7)
and assuming that the envelope contribution of a mixed mode is similar to the inertia of the closest radial mode (I env ≡ I np,0 ), we find that the inertia of the dipole mode with mixed radial order n varies as

I n,1 = I np,0 1 -ζ . ( 8
)
For the sake of simplicity, we use hereafter the abridged notation I n for the inertia of the dipole mixed modes and I 0 for the closest radial modes, and follow the same convention for the mode heights and widths.

Seismic parameters

With ζ, we now intend to express the different seismic parameters.

Period spacing

Following Christensen-Dalsgaard (2012) and Mosser et al. (2015), period spacings can be expressed as

∆P = P n -P n+1 = ζ ∆Π 1 . ( 9
)
This expression is however ambiguous, since ζ may vary significantly between the periods P n+1 and P n (> P n+1 ). Therefore, we prefer to consider the expression resulting from the integration of Eq. ( 5)

∆P = P n -P n+1 = ∆Π 1 n+1 n ζ(ν) dn = ∆Π 1 ζ n , ( 10
)
where we consider that the mixed-mode radial order n is a continuous variable defined by dn = dτ /∆Π is derived from the universal red giant oscillation pattern (Mosser et al. 2011b), whereas n g is given by

n g = - 1 ν ∆Π 1 - 1 4
on the RGB, ( 12)

n g = - 1 ν ∆Π 1 + 1 4 in the red clump, ( 13
)
where the correcting terms ±1/4 that depend on the evolutionary stage are justified in Section 4.4. They differ by 1/2, as depicted by the asymptotic relation (e.g., Tassoul 1980;Benomar et al. 2013). In red giants, the high density N of mixed modes implies that |n g | n p , so that the mixed-mode orders are negative.

From the definition of the stretched period, Eq. ( 10) reduces to

∆P = νn+1 νn dν ν 2 . ( 14
)
This evident relation justifies the relevance of Eq. ( 10) instead of Eq. ( 9): using ζ n is necessarily more accurate than using ζ for computing period spacings.

Rotational splitting

As introduced by Goupil et al. ( 2013), the function ζ is used to express the mixed-mode rotational splitting as a function of the mean rotational splittings related to pure gravity or pure pressure modes:

δν rot = ζ δν rot,g + (1 -ζ) δν rot,p . (15)
As shown by subsequent works (e.g., Deheuvels et al. 2014;Di Mauro et al. 2016;Triana et al. 2017), it is difficult to derive from the observed rotational splittings more than these two mean quantities. Again, we have to solve the ambiguity of the meaning of ζ in Eq. ( 15), since we can either consider the value2 ζ(ν n,0 ), in the framework of the perturbation of the non-rotating frequency ν n,0 , or ζ(ν n,m ), considering that the inertia to be considered corresponds to the actual frequency ν n,m . By analogy with the equation dealing with the period spacing, we propose to rewrite the rotational splitting δν rot = ν n,mν n,0 , in the limit case where the mean envelope rotation is negligible compared to the mean core rotation, as

δν rot = δν rot,core νn,m νn,0 ζ dm = δν rot,core ζ m , (16)
where δν rot,core ≡ δν rot,g . As for the radial order n in Eq. ( 10), we consider the azimuthal order m as a continuous variable varying from 0 to ±1. So, we have introduced two mean values of ζ,

ζ n = n+1 n ζ dn = νn+1,m νn,m dν ∆Π 1 ν 2 , ( 17
)
ζ m = ±1 0 ζ dm = νn,±1 νn,0 dν ∆Π 1 ν 2 , ( 18
)
to account for the period spacings and rotational splittings. The relevance of ζ n is already proven by Eqs. (10) and (11), whereas the relevance of ζ m has yet to be demonstrated. If we succeed, we will also have understood the relevance of the use of stretched periods for analyzing the mixed modes (Eq. 11).

Mixed-mode width, height, and amplitude

The work performed by the gas during one oscillation cycle is the same for all modes, associated with surface damping, when the radiative damping in the Brunt-Väisälä cavity is considered as negligible. Hence, Benomar et al. (2014) have estimated that the mode width of the mixed modes writes

Γ n = Γ 0 I 0 I n = Γ 0 (1 -ζ). (19)
From this relation, we verify that mixed modes have smaller mode widths than radial modes. However, we recall that a family of stars behave differently, when mixed modes are depressed because of an extra damping in the radiative inner region (Mosser et al. 2012a;García et al. 2014;Mosser et al. 2017a).

From Belkacem et al. (2015a) we also derive that the amplitude of a resolved dipole mixed mode is

A 2 n = A 2 0 (1 -ζ), (20)
when the geometrical factor that conducts to a visibility of about 1.54 for red giant dipole modes (Mosser et al. 2012a(Mosser et al. , 2017a) ) is omitted. Such amplitudes correspond to similar heights for radial and dipole modes since A 2 = πΓH/2. When, for non-resolved mixed modes, the width Γ n is less than the frequency resolution δf res , a dilution factor must be considered (Dupret et al. 2009). It expresses

H n = π 2 Γ n δf res H 0 , (21)
when radial modes are resolved, which is the common case.

Synthetic mixed-mode pattern

The previous ingredients can be used to depict an oscillation pattern. Figure 1 shows the synthetic spectrum of a typical star on the low RGB, based on Eq. ( 16) for the rotational splittings, on Eq. ( 19) for the mode widths, and on Eq. ( 21) for the mode heights of unresolved modes. This spectrum resembles the description derived by Grosjean et al. (2014) from non-adiabatic computations, with a timedependent treatment of convection which provides the lifetimes of radial and non-radial mixed modes.

Case studies

In this Section, we use RGB stars showing clear oscillation spectra as case studies, in order to test the description of the mixed-mode spacings, widths, heights, and rotational splittings, which were previously introduced. The first steps consist in identifying their oscillation spectra and in fitting as many dipole mixed modes as possible. One of the two stars considered here, KIC 6144777 was already investigated in many previous articles (e.g., Corsaro et al. 2015;García Saravia Ortiz de Montellano et al. 2018). The other one, KIC 3955033, was less studied since it shows a complicated mixed-mode spectrum; it belongs to the list of red giants with period spacings automatically computed by Vrard et al. (2016). We used data downloaded from the KASOC site 3 , processed using the Kepler pipeline developed by Jenkins et al. (2010), and corrected from outliers, occasional jumps, and drifts (see García et al. 2011, for details).

Identification of the mixed modes

The location of the mixed modes primarily relies on the firm identification of the pure pressure-mode spectrum.

The determination of the large separation ∆ν, first derived from the envelope autocorrelation function (Mosser & Appourchaux 2009), is based on the universal red giant oscillation pattern. This method provides the efficient identification of the radial modes and helps to locate the frequency ranges where mixed modes cannot be mistaken for radial or quadrupole modes. For = 1 modes, the secondorder asymptotic expansion writes

ν p = n p + ε p + 1 2 + d 01 + α 2 [n p -n max ] 2 ∆ν, ( 22
)
where ε p is the acoustic offset, n max = ν max /∆ν -ε p , and α = 0.076/n max . The parameter d 01 is function of the large separation, under the form A + B log ∆ν (where ∆ν is expressed in µHz), with A = 0.0553 and B = -0.036, as determined from the large-scale analysis along the RGB conducted by Mosser et al. (2014). The accurate determination of d 01 is crucial for the determination of the pure dipole pressure modes, hence for the determination of the minima of the function ζ. In that respect, the small modulation of the radial-mode pattern induced by the soundspeed glitches (Miglio et al. 2010;Vrard et al. 2015) must be considered also. Therefore, we fit the actual position of the radial modes first, then use them to refine the pure pressure dipole-mode frequencies according to

ν p = ν np,0 + ν np+1,0 /2 + d 01 ν np+1,0 -ν np,0 . (23)
The background parameters, derived as in Mosser et al. (2012a), are used to correct the granulation contribution in the frequency range around ν max . Hence, mixed modes can be automatically identified in frequency ranges that have no radial and quadrupole modes when their heights 3 http://kasoc.phys.au.dk are significantly above the background. The automatic selection of the modes relies on a statistical test: the heightto-background ratio of the modes must be higher than a threshold level R p in order to reject the null hypothesis to a low probability p. According to [START_REF] Appourchaux | The CoRoT Mission Pre-Launch Status -Stellar Seismology and Planet Finding[END_REF], the relation between R p and p depends for long-lived modes on the observation duration T obs and on the width ∆ ν of the frequency range where a mode is expected. This relation expresses

R p ln T obs ∆ ν p , (24)
when expressed in noise unit. This situation applies here, since the precise identification of the mixed-mode pattern is based on gravity-dominated mixed modes. With 4-year observations and the search of a couple of modes in a frequency range ∆ ν = ∆ν/N , the threshold is typically 10 for a secure probability rejection at the 10 -2 level. In practice, mixed modes with a height-to-background ratio higher than 10 are used to initiate the fit. A lower threshold is enough for the final agreement, when the synthetic mixedmode pattern based on secure modes can be used to search for long-lived mixed modes in narrow frequency ranges. We benefit from the fact that the asymptotic fit is precise and enables to search for thin modes in a frequency range ∆ ν narrower than 0.1 µHz. Therefore, a threshold of 7 is enough for rejecting the null hypothesis at the 1 %-level for these modes whose detection benefits from the information gained by larger peaks. The thin mode widths (Eq. 19) are of great use to map the observed spectrum: a thin gravitydominated mixed mode must be found in the close vicinity, less than 4 times the mode width, of its expected position.

For unresolved peaks, this condition is relaxed to 4 times the frequency resolution. The global seismic parameters of the gravity component are then derived from the methods described in Vrard et al. (2016) and Mosser et al. (2017b), with a least-square fit between the observed and asymptotic patterns. At this stage, global seismic parameters are measured and mixed modes are identified, so that it is possible to measure their individual properties.

Individual fitting procedure

When fitting individually mixed modes, we aim at testing the validity of the asymptotic expression, but not at reaching the ultimate precision, which is the role of a dedicated fit of individual modes (e.g., Gaulme et al. 2009). Therefore, in order to simplify the fit, we supposed (and checked a posteriori) that all multiplets can be fitted independently. This is not the case in all red giant spectra, but it is verified for most stars on the early RGB or in the red clump.

From the asymptotic fit, we identify in the backgroundcorrected spectrum the power excess associated to each mode. Then, we determine the central frequency of the peak as the barycenter of the power excess. The height H and full width at half maximum Γ are simultaneously derived from the Lorentzian fit of the mode. We use Eqs. ( 19) and (21) as priors. Modes are fitted individually when the mode density is low, or simultaneously when the Lorentzians used as priors overlap.

The fitted spectrum and the seismic parameters of KIC 6144777, used as a first study case, are given in Fig. 2 and Fig. 2. Fit of the oscillation pattern of the RGB star KIC 6144777, showing the pressure radial orders np from 9 to 12. The power spectrum density has been divided by the fit of the background. Radial and quadrupole modes are highlighted in red and green. The expected locations of dipole mixed modes are labelled with their mixed radial orders. When detected, mixed modes are highlighted in dark blue (m = -1), light blue (m = 0), or purple (m = 1). = 3 modes, which are also mixed, are located near the abscissa 0.22; extra peaks in the range [-0.2, -0.05] are mixed quadrupole modes. The gray dashed lines indicate the two thresholds used in this work, corresponding to height-to-background ratios of 7 and 10. in Table A.1. We note the large agreement between the observed and asymptotic peaks. As in other stars showing a seismic signal with a high signal-to-noise ratio (S/R), outliers with a height-to-background value R higher than 7 are present. Their detection does not invalidate the method presented above: they correspond either to = 2 or 3 mixed modes, possibly also to = 4 modes, or to aliases (since the duty cycle is about 93 %), or even to noise since the detection of 1 noisy peak with R ≥ 8 is expected in a 30-µHz frequency range after 4 years of observation, assuming that the noise statistic is a χ 2 with two degrees of freedom.

The quality of the fit is shown by the small residuals between the observed frequencies and the asymptotic fits (Fig. 3); we note that these residuals are comparable to the uncertainties, derived from Libbrecht (1992) or slightly larger when the quality of the fit may be affected by the high mode density. These residuals are of about the frequency resolution. When pressure-dominated mixed modes are excluded, the standard deviation of the asymptotic fit is 11 nHz. This value represents 1.3 times the frequency resolution δf res , or ∆ν/1000, or a relative precision at ν max of about 10 -4 . The quality of the fits is based on a small number of parameters: the radial mode frequencies, the mean location d 01 of the expected pure pressure dipole modes, and four asymptotic parameters: the period spacing ∆Π 1 , the coupling factor q, gravitational offset ε g , and the mean core rotational splitting δν rot . Residuals reach maximum values near the pressure-dominated mixed modes: there, deviations of about ∆ν/200 are observed, to be compared to the amplitudes of pressure glitches of about ∆ν/40 (Vrard et al. 2015). We suspect that these residuals are mostly due to the variation of the parameter d 01 with frequency.

Relationships with ζ

With the identification of the mixed-mode pattern, we aim to verify the relevance of the use of ζ n for the period spacings, to test the relevance of ζ m for the rotational splittings, and further test the predictions for the mode widths and heights.

Period spacings

Period spacings were fitted with different functions of ζ, according either to the integrated value ζ n (Eq. 10) or to the arithmetical mean ζ = (ζ n + ζ n+1 )/2. The resulting plots are shown in Fig. 4. When ζ n is not used, one remarks that the ∆P (ζ) relation shows a modulation that results from the concavity of ζ. When ζ is close to unity for gravity-dominated mixed modes, no modulation is seen; in the range [0.7, 0.9], where the function is convex, the period spacings are larger than predicted; below 0.7, where the function is concave, the period spacings are smaller than expected. The relation between ∆P and ζ n does not show such a modulation. Furthermore, the fit with ζ n is nearly linear, with residuals two times smaller. From this comparison, we confirm that the use of ζ n is preferable for fitting the period spacings.

Rotational splittings

We performed similar test for the rotational splittings. We a priori excluded a dependence on ζ(ν n,0 ), since we clearly observe asymmetrical splittings (see below, Section 5.1) that cannot be reproduced with ζ(ν n,0 ). In fact, the rotation rate of KIC 6144777 is not important enough to observe any difference between the variations with either ζ or ζ m . We therefore performed the fit of the star KIC 3955033 (Fig. A.3), which shows a much more rapid rotation (Fig. 5). From the comparison of δν rot (ζ) and δν rot ( ζ m ), we derive that this latter expression is more convenient since it provides a χ 2 ten times smaller than when using ζ, associated with a much more precise estimate of the core rotation: δν rot,core = 765 ± 10 nHz with ζ m , versus δν rot,core = 730 ± 50 nHz with ζ. From this test, we conclude positively about the relevance of the use of ζ m for the rotational splittings.

Widths, amplitudes, and heights

As expected from Eq. ( 19), the mixed-mode width shows large variations: pressure-dominated mixed modes have widths comparable to those of the radial modes, contrary to gravity-dominated modes that are much thinner (Fig. 6, top panel). Figure 6 also shows the validity of Eq. ( 19), with the mixed-mode width proportional to (1 -ζ), except for low values where the observations resolution hampers the measurement of very thin widths. The precision of the fit is limited by the stochastic excitation, especially for longlived peaks: the presence or absence of signal in a single frequency bin can modify the width in large proportion. This limit added to the limitation in frequency resolution does not allow us to test if small additional radiative damping affects the gravity-dominated mixed modes (Dupret et al. 2009;Grosjean et al. 2014).

As shown by Mosser et al. (2015), Eq. ( 20) has a strong theoretical justification, since it expresses the conservation of energy: the sum of all the energy distributed in the mixed modes corresponds to the energy expected in the single pure pressure mode that should exist in absence of any coupling. So, our result is in line with the findings of Mosser et al. (2012a), who measured that, except for depressed modes, the observed total visibility of dipole modes matches the theoretical expectations.

Due to the stochastic nature of the excitation, the mode heights show a large spread (Fig. 7). Dips in the distributions occur when modes are not resolved. It is however clear in Fig. 7 bottom that the dipole mode heights follow the radial distribution according to the trend of Eq. ( 21). We note that all mixed modes associated with a given pressure radial order show a systematic behavior. For instance, all mixed modes of KIC 6144777 in the frequency range [137, 143 µHz] associated with the pressure mode n p = 11 show lower amplitudes than expected from the Gaussian fit of the power excess. Such a behavior recalls us that the excitation of a mixed mode is due to its acoustic component.

Validation

From these two case studies and from other examples shown in Appendix, we can conclude that the asymptotic fits are relevant at all evolutionary stages, when the signal-tonoise ratio is high enough. So, the equations developed in Section 2 allow us to depict the mixed-mode spectrum with a very high accuracy, when the integrated values ζ n and ζ m are considered for the period spacings and the rotational splittings, respectively. Up to now, only red clump stars showing buoyancy glitches cannot be fitted with a single set of parameters. max used as a proxy for the luminosity. The color codes the stellar mass. Stars on the RGB are plotted with + symbols, red clump stars with . The dotted and dashed lines indicate the limit of the visibility of mixed modes for RGB and clump stars, respectively, as defined by Eq. ( 31). Evolution tracks on the RGB, computed with MESA for solar metallicity (Gehan et al. 2018), are shown for the stellar masses 1.0, 1.3, 1.6, and 1.9 M . The error box indicates the typical 1-σ uncertainties on T eff and νmax.

Asymptotic period spacings and gravity offsets

In this section, we show how previous findings can be used to derive accurate period spacings. We also explore the variation of the gravity offsets ε g with stellar evolution. These studies rely on the determination of the pure-gravity mode pattern.

Observations

Our analysis was conducted over 372 red giants at various evolutionary stages, mainly from Mosser et al. (2014) and Vrard et al. (2016), with stars also considered in Beck et al. (2012), Kallinger et al. (2012), Deheuvels et al. (2014), and Corsaro et al. (2015). Data were obtained as for the two stars considered in Section 3. When available, effective temperatures are from APOGEE spectra (Albareti et al. 2017). Selection criteria are mainly based upon the noise level, with Kepler magnitudes brighter than 12 on the low RGB or 14 for more evolved stars. Following the method exposed in Section 3.1, we need data with a S/R high enough to allow the identification of gravity-dominated mixed modes. When such modes are too few, measurements are impossible. This condition induces a selection bias, specifically addressed in Section 6.

The 372 stars that were analyzed are shown in a seismic diagram (Fig. 8). We considered stars from the low RGB (Fig. ). In fact, except for red-clump stars with large buoyancy glitches (Cunha et al. 2015;Mosser et al. 2015), the asymptotic expansion provides a relevant fit. We could then obtain precise measurements of the asymptotic gravity parameters in Eq. ( 4) and of their uncertainties for a large number of stars. We must report one exception: KIC 3216736 is the only RGB star of our sample where we found buoyancy glitches and could not provide a relevant fit of the spectrum, but only an échelle diagram based on stretched periods (Fig. 9). Since we have tested more than 160 stars on the RGB, with a systematic approach, we can conclude that the most common case on the RGB is the absence of buoyancy glitches, as expected theoretically (Cunha et al. 2015).

Characterizing the sample we studied in terms of bias is difficult. Apart from the RGB stars that were already studied in detail in previous works, we have mostly treated the stars with increasing KIC numbers. This systematic method implies that we did not introduce any further bias compared to the Kepler sample of red giants. Considering a high enough S/R, which is almost equivalent to select bright stars in the red giant domain, is not supposed to introduce biases either. Contrary to many previous studies, we are not limited to stars showing rotational splittings smaller than the confusion limit (δν rot ≤ ν 2 max ∆Π 1 ). However, the presence of a strong cutoff (Section 6) limits the sample, when gravity-dominated mixed modes disappear. Red-clump stars with strong buoyancy glitches are absent in our data set since the fitting process requires then to account for the extra-modulation, which can be quite large (about ∆Π 1 /10). When mixed modes are depressed, the low height-to-background ratio of the mixed modes allows the measurement of ∆Π 1 (Mosser et al. 2017a) but is not enough for fitting the pattern. Both cases deserve specific care beyond the scope of this work.

Pure gravity modes

The identification of the mixed modes depicted in Section 3.1 allows us to retrieve the periods of the pure gravity modes and to infer global asymptotic parameters of the gravity components. We compute these periods from the mixed-mode frequencies ν, using Eqs. ( 1) and ( 2),

1 ν g = 1 ν - ∆Π 1 π atan tan θ p q . ( 25
)
Close to each radial mode, when θ p varies from values less than but close to π/2 to values higher than but close to -π/2, the atan correcting term introduces an offset of -∆Π 1 , which in fact allows to relate the (N + 1) mixed modes in the ∆ν-wide interval to N only gravity modes. In order to use all mixed modes, including the |m| = 1 components, we corrected first the rotational splittings, using Eq. ( 16) in order to obtain ν values that are corrected from the rotational splitting.

From the periods of the gravity modes 1/ν g , we could then derive the asymptotic parameters ∆Π 1 and ε g , assuming the first-order asymptotic expression for pure gravity modes (Eq. 4). In practice, a first estimate of ∆Π 1 derived from the formalism of Mosser et al. (2015) and Vrard et al. (2016) is used in Eq. ( 25), then iterated with a least-square fit of the linear variation of the gravity modes (Eq. 4).

Asymptotic period spacings

Up to now, measurements of ∆Π 1 considering that ε g is a free parameter have been obtained for a few stars only (Buysschaert et al. 2016;Hekker et al. 2018, for 3 and 22 observed stars, respectively). The offset ε g being arbitrarily fixed, Mosser et al. (2012b) and Mosser et al. (2014) reported a very high precision for the period spacings, of typically 0.1 s for stars on the RGB and 0.3 s in the red clump. However, owing to the choice of ε g = 0, their period spacings were slightly affected by a bias of about a fraction of ν max ∆Π 2

1 . The values reported by Vrard et al. (2016), free of any hypothesis on ε g , are not biased but show uncertainties typically five to fifteen times higher than the new values. Their comparison with our data confirms the absence of systematic offsets (Fig. 10). So, the new method ensures accuracy, in the sense that the measurements of ∆Π 1 are now free of any hypothesis on ε g and prove that the asymptotic expansion for gravity modes (Eq. 4) is relevant. The relative accuracy we obtained for the period Fig. 12. Left: variation of εg with ∆ν, with the same style as Fig. 11. The horizontal dark gray domain corresponds to the expected range predicted for RGB stars by Takata (2016a), whereas the dot-dashed line shows the value εg,as = 1/4 derived from the asymptotic expansion (Provost & Berthomieu 1986). Uncertainties on εg are indicated by vertical lines; uncertainties on ∆ν are smaller than the symbol size. Right: histograms of the distributions of εg on the RGB (blue curve) and in the red clump (red curve). The dot-dashed line and the gray domain have the same meaning as indicated above.

spacings, assuming Eq. ( 4), is shown in Fig. 11. The median relative accuracies on the RGB and in the red clump are similar, of about 7 10 -4 . They translate, respectively, into 0.06 s on the RGB and 0.22 s in the red clump; Hekker et al. ( 2018) reach a similar precision.

Gravity offsets ε g

We could measure ε g for a large set of stars. We however have to face the indetermination of ε g modulo 1: we simply assume that ε g is in the range [-0.5, 0.5]. The ε g values computed for the set of stars presented in the paper is given in Table 1 and plotted in Fig. 12, where an histogram is also given. The complete table is given online only. Uncertainties on ε g are small, related to the uncertainties in ∆Π 1 by

δε g = δ∆Π 1 ν max ∆Π 2 1 . ( 26
)
This relation comes from the derivative of Eq. ( 4). As a result, the median uncertainties are of about 0.08 on the RGB and 0.06 in the red clump. We noticed that the median value of ε g on the RGB is in fact close to 1/4, which is the expected asymptotic value in absence of stratification below the convection zone (Provost & Berthomieu 1986), derived from the contribution /2 -ε as with = 1 and ε as = 1/4. Hence, we inferred that the degeneracy on the determination of ε g is removed.

We then noted a slight decrease in ε g when stars evolve on the RGB, with an accumulation of values close to 0 for redclump stars. Hekker et al. (2018) reported values of ε g in the range [-0.2, 0.5] for 21 stars on the RGB, but did not identify the accumulation of values in the range [0.20, 0.35] predicted by Takata (2016a) for stars on the low RGB. Our measurements fully confirm this prediction. From a check of their data set, we interpret the differences in ε g as resulting from less precise gravity spacings when large rotational splittings apparently modify the period spacings. As made clear by the recent theoretical developments of the asymptotic expansion (Takata 2016b,a), the accurate measurement of the leading-order term ∆Π 1 is necessary to provide reliable estimates of ε g .

We can study the variation of ε g along stellar evolution. On the RGB, the asymptotic expansion predicts ε g = 1/4 -ϑ (Provost & Berthomieu 1986), where ϑ is a measure of the stratification just below the convection zone. From this dependence, we can infer that the term ϑ is certainly very small for most stars on the low RGB. Higher values are suspected for evolved RGB stars, but with too few stars to firmly conclude, whereas lower values are seen for evolved RGB. We checked that the change of regime of ε g is not associated with the luminosity bump since it occurs for more evolved stars than our sample (Khan et al. 2018). In the red clump, the ϑ correction seems important, on the order of 0.3, with a larger spread than observed on the RGB.

An extended study of ε g can now be performed to use this parameter as a probe of the stratification occurring in the radiative region. This study is however beyond the scope of this work.

Rotation

The fits based on the function ζ also allow us to analyze rotational splittings in detail.

Splitting asymmetry

Recently, asymmetries in the rotational splittings were reported by Deheuvels et al. (2017), as the signature of the combined effects of rotation and mode mixing. Using both perturbative and non-perturbative approaches, they computed near-degeneracy effects and could fit the data. In fact, the asymptotic development of mixed modes also describes the combined effects of rotation and mode mixing, so that the rotational splittings based on ζ m (Eqs. 16 and 18) are not symmetric. Inversely, the symmetrical rotational splitting based on ζ (Eq. 15) does not reproduce the observed asymmetry. Hence, observing asymmetrical triplets is a way to prove the relevance of the use of ζ m instead of ζ.

Observing the asymmetry is challenging but possible for stars with a rapid rotation rate. As explained by Gehan et al. (2017Gehan et al. ( , 2018)), rapid rotation means δν rot ≥ ∆ν/N for seismology. This rotation is however very slow in terms of interior structure, so that the formalism developed by Goupil et al. ( 2013) and Deheuvels et al. (2014), summarized by Eq. ( 16), remains relevant. It simplifies the study, as shown by Ouazzani et al. (2013) who treated the case where rotational splittings can be as large as ∆ν. We fitted the mixed-mode spectrum of KIC 3955033 with both the symmetrical and asymmetrical splitting. At high radial order n p , it is hard to distinguish them. At low orders, when the rotational splittings exceed the mixed-mode spacings, the symmetrical splittings fail whereas the asymmetrical one provides a consistent solution along the whole spectrum. The radial order n p = 8 is shown in Fig. 13, the whole spectrum is shown in

Surface rotation

For stars on the low RGB, surface rotation can be inferred from the rotational splittings (Eq. 15). The measurement is however difficult, since it results from an extrapolation at ζ = 0, when values are mostly obtained above ζ = 0.6 only (Fig. 5). The highest level of precision, hence the use of ζ m instead of ζ, is required for deriving a correct estimate of the surface rotation. The case of KIC 3955033 is illustrative, with a negative surface rotation when using ζ; the use of ζ m provides a null value (5 ± 20 nHz). This case also confirms the general situation shown by previous works (Goupil et al. 2013;Di Mauro et al. 2016;Triana et al. 2017): deriving surface rotation can be achieved for the low RGB only.

Stellar inclination

From its ability to fit the gravity-dominated modes that carry useful information, the asymptotic fit can be used to derive the stellar inclination i. The amplitude of the m = 0 component of the dipole multiplet is proportional to sin 2 i whereas the sum of the amplitudes of the m = ±1 mode is proportional to cos 2 i. From Eq. ( 20), a correction factor of 1/(1 -ζ) should be applied on the amplitudes: its differential effect is however much below the precision one can get on i.

We tested our results on a set of stars for which the inclinations measured with other methods have been obtained. We checked that our results are relevant, with a precision limited by the uncertainties on the amplitude measurements. In order to avoid bias, we consider only peaks with a height-to-background ratio larger than 8. Nevertheless, we noted that the stochastic excitation of the modes induces a small bias for large inclinations. Equator-on inclinations, near 90 • , cannot be retrieved precisely, with measurements reduced toward the range 70-80 • . As a consequence, they are rare in our analysis. However, many stars show inclinations that, according to the uncertainties, are compatible with equator-on measurement, so that the bias does not affect the following analysis.

We measured inclinations of red giants in the open clusters NGC 6819 observed by Kepler (e.g., [START_REF] Basu | [END_REF]Stello et al. 2011;Miglio et al. 2012). We selected the stars that exhibit mixed modes and could fit 20 mixed-mode patterns with the asymptotic expansion. In one case, the asymptotic fit is impossible, due to a low S/R. In two other cases, different possible solutions exist, based either on different period spacings, or on different rotational splittings, but without any ambiguity for the inclination measurement: when two peaks dominate per period spacing, the inclination is necessarily high, whereas it is low when one single peak only is present. We completed this list with other NGC 6819 members listed in Handberg et al. (2017) and could fit two additional stars, which incidentally show a large inclination. Results for the inclinations and rotational splittings are given in Table 2. As shown in Fig. 14, the distribution of the stellar inclinations mimics the sin i relation expected for random inclinations, except near 90 • , due to bias mentioned above. A similar test performed on the open cluster NGC 6791 reaches the same conclusion. Low stellar inclinations in NGC 6819 and 6791 were measured by Corsaro et al. (2017), using a Bayesian analysis, from which aligned spins were inferred. Our measurements however contradict their claim, as shown in Fig. 15 for NGC 6819. In fact, our measurements compared to theirs agree for high inclinations, whereas they mostly disagree for low inclinations. Comparison with the asymptotic fits shows that their Bayesian rotational splittings are most often overestimated and that the related inclinations are most often underestimated.

In fact, the asymptotic fit can be used as a prior for the Bayesian fit. It indicates that the rotational splitting is derived from the thin gravity-dominated mixed modes, with 2017) are plotted on the x-axis, while inclinations derived from the asymptotic analysis are on the y-axis; 1-σ uncertainties are indicated by vertical and horizontal error bars. The red symbol shows the case where no asymptotic fit could be found, and the green ones to cases without Bayesian fit. The dashed line corresponds to the 1:1 relation.

narrow widths (Eq. 19) and average rotational splittings slightly inferior to the mean core rotation δν rot (Eq. 16). Mixed modes at low pressure radial orders, with frequencies much below ν max , are especially informative, since previous work has shown that their radial mode widths, hence their mixed-mode widths according to Eq. ( 19), are the thinnest possible (Fig. 5 of [START_REF] Unno | Nonradial oscillations of stars[END_REF]. Figure 1 of the supplementary material of Corsaro et al. (2017) provides an explanation of the discrepant Bayesian values. Their fit of the star KIC 5112373 in NGC 6819 provides nearly uniform large mode widths, relevant for the pressuredominated mixed modes but much too high for gravitydominated modes, in contradiction with the physical variation indicated by Eq. ( 19). As a consequence, their fit assumes that all the power is concentrated in the m = 0 mode; the resulting stellar inclination is 20 ± 8 • . We show the asymptotic solution of KIC 5112373 in Fig. 16, with thin gravity dominated mixed modes and the clear identification of triplets. Since m = ±1 modes are observed all along the spectrum, our solution for the inclination is larger, about 47 ± 18 • .

We provide another example with the star KIC 2437976, a NGC 6791 member. As shown in Fig. 17, rotational splittings are explained in a consistent way with thin unresolved gravity-dominated mixed modes and a rotation rate rapid enough to ensure that close modes do not belong to the same multiplets. All peaks can be explained by the m = ±1 modes. In practice, m = 0 modes are absent, so that this star has necessarily an inclination close to 90 • , whereas Corsaro et al. (2017) measured i 0 • . We conclude that some of the low inclinations reported in Corsaro et al. (2017) are incompatible with the analysis presented here. It seems that the difference is due to a too low range of the linewidth priors in the Bayesian analysis, which favors a solution with a low inclination angle and a high splitting. As a result, stellar spins in old open clusters are neither aligned nor quasi parallel to the line of sight. Our study emphasizes a major role for the asymptotic analysis: pro- Fig. 17. Fit of the mixed modes corresponding to np = 9 in KIC 2437976 (NGC 6791 member). The color codes the azimuthal order: m = +1 in purple, m = -1 in blue. The location of m = 0 modes is indicated in light blue, but none shows a large height for this star seen equator-on. The gray dashed lines indicate the two thresholds used in this work, corresponding to height-tobackground ratios of 7 and 10. Many peaks above the threshold value 5.5 that rejects the null hypothesis at the 5 %-level follow the mixed-mode pattern.

viding relevant estimates of all features of the mixed-mode pattern, including mode widths.

Observability of the mixed modes

All the information derived from mixed modes relies on their observability. The properties of the function ζ can be used to assess under which conditions mixed modes can be actually observed. To achieve this, we investigate first the domain where pressure-dominated mixed modes are observed, then the condition for observing gravity-dominated mixed modes.

Pressure-dominated mixed modes

We can define the frequency range where mixed modes are pressure-dominated (pm) from the full width at half minimum of the ζ function. So, these modes cover a range, expressed in terms of the pressure phase θ p (Eq. 6), verifying

δθ p | pm = 2q 1 + 1 N q (27)
under the assumption that q is small, which is verified for all stars except at the transition between subgiants and red giants (Mosser et al. 2017b). When expressed in frequency and compared to the large separation, this condition corresponds to a frequency range surrounding each pure pressure modes with a width δν pm defined by

δν pm ∆ν = 2q π 1 + 1 N q . ( 28
)
The variations in q and N explain the narrowing of the region with pressure-dominated mixed modes when stars evolve on the RGB. An example is shown in the Appendix (Fig. A.4). The expression of δν pm also shows that redclump stars, with larger q show pressure-dominated mixed modes in a broader region than RGB stars.

Visible gravity-dominated mixed modes

The non-dilution of the mode height expressed by Eq. 21 can be used to define a criterion of visibility of the gravitydominated (gm) mixed modes. So, they are clearly visible when they show heights similar to those of the pressure modes (H n = H 0 ), hence when Γ 0 (1 -ζ) ≥ 2δf res /π (Eq. 19). This condition translates into

(1 -q 2 ) sin 2 θ p ≤ q N π 2 Γ 0 δf res -1 -q 2 . ( 29
)
Except at the transition from subgiants to red giants, where mixed modes are unambiguously visible (Benomar et al. 2013;Deheuvels et al. 2014), the terms q 2 are negligible, so that modes are clearly visible if

| sin θ p | gm ≤ q N π 2 Γ 0 δf res -1 . (30)
This condition for observing gravity-dominated mixed modes has many consequences: -It can be fulfilled only if the definition of the right term is ensured, which requires a frequency resolution low enough compared to the radial mode width. With Γ 0 in the range [100, 150 nHz], the observation must last 50-75 days at least. In fact, mixed modes were observable with CoRoT runs lasting about 150 days (Mosser et al. 2011a), but are hardly observable with K2 80-day time series (Stello et al. 2017).

-When stars evolve on the RGB, the decrease in q and increase in N contribute to the narrowing of observable modes. Mixed modes are more easily visible in the red clump, owing to larger q values. This criterion is implicitly used by Elsworth et al. (2017) for their determination of the evolutionary state of red-giant stars.

-All mixed modes are clearly visible when the condition expressed by Eq. ( 30) is always met, that is when N ≤ q(πΓ 0 /2δf res -1). This condition is met for subgiants, on the lower RGB, and for secondary-clump stars (Mosser et al. 2014).

-No mixed mode can be observed when the condition is so drastic that only pressure-dominated mixed modes can be observed. The combination of the conditions expressed by Eq. ( 27) and Eq. ( 30) yields the limit of visibility of gravity-dominated mixed modes, expressed by a condition on the mixed-mode density

N ≤ 1 4q π 2 Γ 0 δf res -5 . (31)
In the conditions of observation of Kepler, with typical parameters defined as in Mosser et al. (2017a), this limit corresponds to a mode density N of about 25, for RGB and clump stars, over which no gravity-dominated mixed modes can be identified. This theoretical estimate is observed in practice, with a few exceptions with larger N (Fig. 8). On the RGB, observation of mixed modes with Kepler is limited to ∆ν ≥ 6 µHz, whereas the limit is around 3 µHz for clump stars. As a consequence, visible mixed modes in an oscillation spectrum with ∆ν in the range [3, 6 µHz] most often indicate a red-clump star. Incidentally, the location of the RGB bump was recently identified by Khan et al. (2018) in the range [5, 6 µHz], depending on the stellar mass and metallicity. This means that sounding the bump with mixed modes will be very difficult, if not impossible.

Conclusion

The asymptotic analysis allows us to depict the whole properties of the mixed-mode spectrum in a consistent way. Period spacings, rotational splittings, mode widths, and mode heights, all depend on the mode inertia so that all are related to the parameter ζ. We could derive interesting properties:

-The asymptotic fit of the mixed modes proves to be precise and unbiased. Its precision for the RGB stars is so high that the asymptotic expansion of gravity modes can be validated when buoyancy glitches are absent. This ensures the delivery of accurate asymptotic parameters ∆Π 1 , q, and ε g . We found only one RGB star with such buoyancy glitches; on the contrary, buoyancy glitches are often present in red-clump stars.

-The period spacings and rotational splittings are better estimated with integrated values of the function ζ. The use of these mean values ζ n and ζ m is useful for evolved RGB stars and is mandatory for stars with intricate splittings and spacings. Using the stretched period (Mosser et al. 2015) is in fact equivalent.

-The gravity asymptotic parameters ∆Π 1 and ε g can now be accurately determined, with typical accuracy of respectively 0.06 s and 0.1 on the RGB, and 0.22 s and 0.08 in the red clump. This opens the way to a fruitful dialogue with theoretical developments (Takata 2006(Takata , 2016b,a) ,a) and modeling (e.g., Bossini et al. 2015;Cunha et al. 2015).

-We have made clear that observing mixed modes in evolved red giants requires an observation duration longer than 100 days. However, gravity-dominated mixed modes are no longer observable when the stars are more evolved than ∆ν 6 µHz on the RGB, or ∆ν 3 µHz in the red clump. These thresholds are indicative values: the natural spread of the seismic parameters with respect to their mean values explain slight differences.

-We have demonstrated the non-alignment of the rotation axis of the stars belonging to the old open clusters NGC 6791 and NGC 6819. These results contradict previous findings by Corsaro et al. (2017) and illustrate how useful the asymptotic fit will be in the future when used to define priors to any Bayesian or other type of fit of mixed modes. Owing to the small radial orders, small shifts are seen between observed and asymptotic spectra. Same style as Fig. 2, but = 3 modes appear near the abscissa 0.28 A.4. Fit of the oscillation pattern of the evolved RGB star KIC 2443903, near the limit of capability of identification, with a large crowding due to the high mode density. The second rotation crossing, where all m components apparently coincide, occurs at n = -189 (with an abscissa 0.1 and np = 9); the third crossing, where |m| = 1 components apparently coincide with m = 0 inbetween, occurs at n = -233 (with an abscissa 0.25 and np = 7). Same style as Fig. 2. Note that the modes with large heights at an abscissa 0.2 are = 3 modes.

Fig. 1 .

 1 Fig. 1. Description of the radial order closest to νmax of the oscillation power spectrum of a typical RGB star. Frequencies, widths, and heights are estimated according to the function ζ. Quadupole and radial modes are plotted in gray, dipole mixed modes in dark blue (m = -1), light blue (m = 0), or purple (m = 1), respectively.

Fig. 3 .

 3 Fig. 3. Relative residuals, multiplied by 1000, between the observed and asymptotic mixed-mode frequencies in KIC 6144777. The color of the symbols indicates the azimuthal order: dark blue squares for m = -1, light blue diamonds for m = 0, or purple triangles for m = 1; 1-σ uncertainties are also shown. The dashed line corresponds to a perfect fit. The dotted lines show the frequency resolution plus an extra-modulation ∆ν(1 -ζ)/100, which is empirically used to define the quality of the fit.

Fig. 4 .

 4 Fig. 4. Period spacings of the RGB star KIC 6144777. Top: plot as a function of the arithmetical mean value (ζ(νn)+ζ(νn+1))/2. Bottom: plot as a function of the mean value ζ n. The colors code the azimuthal orders, as in Fig. 2; the dashed line indicates the 1:1 relation; 1-σ uncertainties on both the spacings and the mean values of ζ are indicated by vertical and horizontal error bars.

Fig. 5 .

 5 Fig. 5. Mean rotation splittings of the RGB star KIC 3955033. Top: plot as a function of ζ. Bottom: plot as a function of the mean value ζ m. The colors code the azimuthal orders; the dashed line indicates the 1:1 relation; 1-σ uncertainties on both the splittings and the mean values of ζ are indicated by vertical and horizontal error bars.

Fig. 6 .

 6 Fig. 6. Mode widths as a function of the cyclic frequency (top) or as a function of (1 -ζ)Γ0 (bottom), for the RGB star KIC 6144777. Radial modes are plotted with square symbols and dipole mixed modes with +; 1-σ uncertainties on Γ are indicated by vertical error bars. The value 2δfres/π (Eq. 21) plotted as a dotted line is proportional to the 4-year long frequency resolution. In the bottom plot, radial modes have been considered too, assuming they have ζ = 0 as pure pressure modes. The dashed line indicates the 1:1 relation.

Fig. 7 .

 7 Fig. 7. Mode heights as a function of the cyclic frequency (top) or as a function of the radial mode height, modified when the modes are not resolved, (bottom), for the RGB star KIC 6144777. Radial modes are plotted with square symbols and dipole mixed modes with +. The Fourier spectrum is plotted in red (green) for emphasizing the radial (quadrupole) modes. The dot-dashed line provides the expected heights of dipole modes, under the assumption that the power excess mimics a Gaussian relation (dotted line). The dashed line indicates the 1:1 relation.

Fig. 8 .

 8 Fig. 8. Seismic diagram of the 372 red giants studied in this work, with ν -1max used as a proxy for the luminosity. The color codes the stellar mass. Stars on the RGB are plotted with + symbols, red clump stars with . The dotted and dashed lines indicate the limit of the visibility of mixed modes for RGB and clump stars, respectively, as defined by Eq. (31). Evolution tracks on the RGB, computed with MESA for solar metallicity(Gehan et al. 2018), are shown for the stellar masses 1.0, 1.3, 1.6, and 1.9 M . The error box indicates the typical 1-σ uncertainties on T eff and νmax.

  A.1) to more evolved RGB stars (Fig. A.2). The spectrum of the evolved RGB star KIC 2443903 (Fig. A.4) corresponds to a case near the limit of visibility of gravitydominated mixed modes, with a mode density N 22.4 close to the limit value above which the detection is impossible (Section 6). The fitting process for red clump stars can be achieved only when the amplitude of the buoyancy glitches remains limited (Fig. A.5); the same limitation appears in the secondary red clump (Fig. A.6

Fig. 9 .

 9 Fig. 9. Stretched-period échelle diagram of KIC 3216736, the only RGB star in our sample showing buoyancy glitches. The spectrum is simple, since only m = 0 dipole mixed modes are present, but shows a large-period modulation instead of the expected vertical alignment. Modes plotted in light blue are pressure dominated; extra peaks that do not follow the global trend are either = 3 modes or = 2 mixed modes. Red figures indicate the radial orders of the radial modes. For clarity, the power spectrum density is also plotted twice, top to tail.

Fig. 10 .

 10 Fig.10. Comparison of the asymptotic period spacings with previous values. Light blue triangles show the bias in period spacings computed under the assumption εg = 0(Mosser et al. 2014), whereas dark blue diamonds are free of this hypothesis(Vrard et al. 2016).

Fig. 11 .

 11 Fig. 11. Relative precision of the asymptotic period spacings. Same style as in Fig. 8.

Fig. 13 .

 13 Fig. 13. Splitting asymmetry at low frequency in KIC 3955033. Each dipole mixed mode of the spectrum is labelled with its radial and azimuthal orders. The rotational splittings of the radial orders from -142 to -140, plotted with diamonds, do no match the function ζ. Only the multiplet with n = -141 is complete: the m = +1 splitting is much larger than the m = -1 splitting; the colored regions indicate the ranges over which the function ζ is integrated for the components of the multiplet n = -141. The dashed lines indicate height-to-background values of 7 and 10.

  Fig. A.3.

Fig. 14 .

 14 Fig. 14. Histogram of the inclinations measured in NGC 6819. The dashed line indicates the sin i distribution.

Fig. 15 .

 15 Fig. 15. Comparison of the inclinations of the spin axis of the stars in NGC 6819. Inclinations measured by Corsaro et al. (2017) are plotted on the x-axis, while inclinations derived from the asymptotic analysis are on the y-axis; 1-σ uncertainties are indicated by vertical and horizontal error bars. The red symbol shows the case where no asymptotic fit could be found, and the green ones to cases without Bayesian fit. The dashed line corresponds to the 1:1 relation.

Fig. 16 .

 16 Fig. 16. Fit of the mixed modes corresponding to np = 7 in KIC 5112373 (NGC 6819 member). The color codes the azimuthal order: m = +1 in purple, m = -1 in blue. The gray dashed lines indicate the two thresholds used in this work, corresponding to height-to-background ratios of 7 and 10. Contrary to the analysis conducted by Corsaro et al. (2017), modes with m = ±1 are clearly identified.

Fig

  Fig. A.1. Fit of the oscillation pattern of the low RGB star KIC 10272858, at the limit of validity of the asymptotic pattern.Owing to the small radial orders, small shifts are seen between observed and asymptotic spectra. Same style as Fig.2, but = 3 modes appear near the abscissa 0.28

Fig. A. 2 .

 2 Fig. A.2. Fit of the oscillation pattern of the RGB star KIC 11353313. Same style as Fig. 2.

Fig

  Fig. A.3. Fit of the oscillation pattern of the RGB star KIC 3955033. The overlap of mixed modes with different mixed-mode orders is the signature of the rapid core rotation. The second rotation crossing, where all components of the multiplets overlap (Gehan et al. 2017), occurs at the mixed-order n = -122. Same style as Fig. 2.

Fig.

  Fig. A.4. Fit of the oscillation pattern of the evolved RGB star KIC 2443903, near the limit of capability of identification, with a large crowding due to the high mode density. The second rotation crossing, where all m components apparently coincide, occurs at n = -189 (with an abscissa 0.1 and np = 9); the third crossing, where |m| = 1 components apparently coincide with m = 0 inbetween, occurs at n = -233 (with an abscissa 0.25 and np = 7). Same style as Fig.2. Note that the modes with large heights at an abscissa 0.2 are = 3 modes.

Fig

  Fig. A.5. Fit of the oscillation pattern of the red-clump star KIC 1723700. Buoyancy glitches explain the small shifts between observed and asymptotic spectra but do no hamper the mode identification. Same style as Fig. 2.

Fig. A. 6 .

 6 Fig. A.6. Fit of the oscillation pattern of the secondary-clump star KIC 1725190. Buoyancy glitches explain the small shifts between observed and asymptotic spectra but do no hamper the mode identification. Same style as Fig. 2.

Fig

  Fig. B.1. Fit of the oscillation pattern of the RGB star KIC 5024476, member of the open cluster NGC 6819.The dim magnitude of the cluster stars explains the low S/R. However, unambiguous doublets are identified all along the mixed-mode spectrum; m = 0 modes are mostly absent and |m| = 1 modes dominate the mixed-mode spectrum, so that a nearly pole-on inclination is not possible. Same style as Fig.2.

Fig. B. 2 .

 2 Fig. B.2. Fit of the oscillation pattern of the RGB star KIC 2437976, member of the open cluster NGC 6791.The dim magnitude of the cluster stars explains the low S/R. The identification at radial order 9, supported by the radial orders 8 and 10, is unambiguously conclusive: m = 0 modes are mostly absent and |m| = 1 modes dominate the mixed-mode spectrum, so that a pole-on inclination is not possible. Same style as Fig.2.

Table 1 .

 1 Period spacings and gravity offsets The list of the full data set with 372 red giants showing an uncertainty in εg less than 0.15 is available on line as a CDS/VizieR document.

	KIC	νmax	∆ν	∆Π1	q	εg	δνrot
		(µHz)	(µHz)	(s)			(nHz)
	1576469	90.60 ± 0.98	7.41 ± 0.04 284.80 ± 0.64 0.23 ± 0.03 -0.102 ± 0.096	67 ± 6
	1723700	39.42 ± 0.57	4.48 ± 0.04 323.40 ± 0.17 0.24 ± 0.04	0.066 ± 0.043	57 ± 5
	2437976	89.37 ± 1.10	8.22 ± 0.05	74.70 ± 1.00 0.10 ± 0.02 -0.006 ± 0.095 320 ± 30
	2443903	66.76 ± 0.90	7.01 ± 0.04	71.10 ± 0.02 0.12 ± 0.02 -0.184 ± 0.078	360 ± 4
	3955033 106.10 ± 1.24	9.23 ± 0.05	74.65 ± 0.06 0.13 ± 0.02	0.207 ± 0.115 765 ± 10
	5024476	68.66 ± 0.75	5.73 ± 0.04 299.60 ± 1.00 0.27 ± 0.03 -0.199 ± 0.102	63 ± 6
	5112373	43.82 ± 0.59	4.63 ± 0.04 240.30 ± 0.14 0.19 ± 0.02 -0.246 ± 0.058	37 ± 3
	6144777 128.23 ± 1.50 11.03 ± 0.05	79.05 ± 0.04 0.13 ± 0.02	0.210 ± 0.055	244 ± 5
	10272858 341.45 ± 6.16 22.71 ± 0.14	96.90 ± 0.30 0.19 ± 0.02	0.338 ± 0.098 660 ± 20
	11353313 127.29 ± 1.46 10.75 ± 0.05	76.95 ± 0.06 0.14 ± 0.02	0.290 ± 0.088	465 ± 7

Table 2 .

 2 Asymptotic and rotational parameters in NGC 6819

	KIC ID	∆ν	∆Π1	q	δνrot	i
		(µHz)	(s)		(nHz)	( • )
	4937056	4.76 291.0 0.21	90 60±15
	4937257 a	4.13 292.1 0.19	27 72±13
	4937770 b	7.82 161.0 0.18	×	×
	4937775 a	7.33 226.3 0.21	110 75±15
	5023953	4.74 293.9 0.24	50 51±28
	5024327	4.72 269.5 0.20	55 56±13
	5024404	4.78 242.6 0.25	110 80±10
	5024414	6.47 283.0 0.30	90 45±20
	5024476	5.73 299.5 0.24	56 71±11
	5024582	4.76 323.5 0.22	70 55±18
	5111718	10.59	88.4 0.12	410 69±21
	5111949	4.81 319.0 0.28	35 66±15
	5112072	10.08	91.9 0.15	350 72±12
	5112361 c	6.19	99.0 0.12	350 70±20
	5112373	4.63 240.2 0.19	37 47±18
	5112387	4.70 267.2 0.28	84 25±17
	5112401	4.03 311.0 0.26	50 54±13
	5112467	4.75 285.2 0.25	90 61±12
	5112491	4.68 324.3 0.30	150 31±16
	5112730	4.56 320.0 0.25	45 56±18
	5112938	4.73 320.0 0.30	65 45±11
	5112950	4.35 319.5 0.38	38 61±18
	5112974	4.32 309.6 0.24	60 50±12
	5113441 c	11.75	89.0 0.13	730 18±18
	5200152	4.73 327.2 0.28	50 70±15

a : KIC 4937257 and KIC 4937775 are absent in

Corsaro et al. (2017)

. b : × symbols indicate the absence of any reliable asymptotic fit for KIC 4937770. c : Different solutions in ∆Π1 are possible for KIC 5112361, which all provide a high inclination. Different solutions in δνrot are possible for KIC 5113441, which all provide a low inclination. Typical uncertainties for those stars with low S/R spectra are 0.7 % in ∆ν and ∆Π1, 12 % in q, and 8 % in δνrot.

Table A .

 A 1. Oscillation pattern of the RGB star KIC 6144777 Table A.1. continued.

	np np	n n	m m	ζ ζ	νas νas (µHz) (µHz)	ν ν (µHz) (µHz)	x x	Γas Γas (µHz) (µHz)	Γ Γ (µHz) (µHz)	H H (ppm 2 µHz -1 ) (ppm 2 µHz -1 )	R R
	11	Radial modes -80 -1 0.8644 139.037 139.015 ± 0.007 0.403	0.024	0.012 ± 0.004	2323 ± 388	36.5
	8 11 9 11 10 11 11 11 12 11 13 11 11 8 -112 -80 -80 -79 -1 0.4672 140.063 140.116 ± 0.009 101.916 101.916 ± 0.014 0 0.8183 139.247 139.230 ± 0.006 112.612 112.612 ± 0.011 1 0.7594 139.431 139.438 ± 0.007 123.726 123.622 ± 0.006 134.574 134.534 ± 0.008 -0.003 0.040 0.422 0.010 0.441 0.008 0.503 -79 0 0.4628 140.175 140.185 ± 0.012 0.509 145.842 145.585 ± 0.017 -0.002 -79 1 0.4939 140.295 140.352 ± 0.009 0.524 156.846 156.846 ± 0.042 0.019 -78 -1 0.8100 140.998 141.019 ± 0.006 0.585 -78 0 0.8550 141.194 141.202 ± 0.006 0.601 Dipole mixed modes 1 0.9522 105.827 105.836 ± 0.007 0.396 11 -78 1 0.8894 141.409 141.419 ± 0.009 0.621 8 -111 1 0.8506 106.633 106.648 ± 0.007 0.469 11 -77 -1 0.9561 142.436 142.432 ± 0.006 0.713 8 -110 -1 0.7311 106.957 106.985 ± 0.014 0.500 11 -77 0 0.9617 142.668 142.663 ± 0.006 0.734 8 -110 0 0.6467 107.134 107.137 ± 0.010 0.514 11 -77 1 0.9667 142.902 142.895 ± 0.009 0.755 8 -110 1 0.5989 107.278 107.322 ± 0.008 0.530 12 -74 -1 0.9778 147.312 147.297 ± 0.007 0.154 8 -109 -1 0.6428 107.551 107.604 ± 0.009 0.556 12 -74 0 0.9761 147.549 147.551 ± 0.007 0.177 8 -109 0 0.7161 107.706 107.730 ± 0.010 0.567 12 -74 1 0.9739 147.785 147.778 ± 0.006 0.197 8 -109 1 0.7983 107.899 107.909 ± 0.007 0.584 12 -73 -1 0.9511 148.996 148.980 ± 0.009 0.306 8 -108 -1 0.8911 108.264 108.263 ± 0.010 0.616 12 -73 0 0.9422 149.226 149.215 ± 0.007 0.327 8 -108 0 0.9206 108.482 108.484 ± 0.006 0.636 12 -73 1 0.9306 149.452 149.441 ± 0.006 0.348 8 -108 1 0.9406 108.710 108.700 ± 0.011 0.655 12 -72 -1 0.7456 150.556 150.584 ± 0.008 0.452 8 -107 -1 0.9617 109.140 109.138 ± 0.007 0.695 12 -72 0 0.6672 150.737 150.764 ± 0.011 0.468 8 -107 0 0.9683 109.373 109.375 ± 0.008 0.716 12 -72 1 0.5911 150.880 150.951 ± 0.011 0.485 8 -106 0 0.9822 110.304 110.274 ± 0.008 0.798 12 -71 -1 0.4722 151.509 151.598 ± 0.010 0.543 8 -104 0 0.9889 112.235 112.235 ± 0.007 -0.024 12 -71 0 0.5300 151.624 151.720 ± 0.009 0.554 9 -103 -1 0.9889 112.989 113.018 ± 0.012 0.047 12 -71 1 0.6122 151.772 151.861 ± 0.012 0.567 9 -103 0 0.9889 113.228 113.226 ± 0.006 0.065 12 -70 1 0.9339 153.268 153.289 ± 0.007 0.697 9 -102 -1 0.9872 113.999 113.996 ± 0.007 0.135 12 -69 0 0.9717 154.828 154.849 ± 0.007 -0.162 9 -102 1 0.9856 114.477 114.478 ± 0.007 0.179 12 -69 1 0.9733 155.064 155.083 ± 0.007 -0.141	0.033 0.043 0.096 0.097 0.091 0.034 0.026 0.009 0.020 0.027 0.008 0.048 0.007 0.064 0.006 0.072 0.004 0.064 0.004 0.051 0.005 0.036 0.009 0.020 0.010 0.014 0.013 0.011 0.046 0.007 0.060 0.006 0.074 0.003 0.095 0.002 0.085 0.002 0.070 0.002 0.012 0.002 0.005 0.003 0.005	0.120 ± 0.023 0.020 ± 0.005 0.106 ± 0.016 0.029 ± 0.006 0.054 ± 0.007 0.078 ± 0.015 0.087 ± 0.012 0.128 ± 0.031 0.244 ± 0.057 0.071 ± 0.012 0.258 ± 0.074 0.013 ± 0.004 0.009 ± 0.004 0.007 ± 0.004 0.027 ± 0.007 0.011 ± 0.004 0.007 ± 0.004 0.068 ± 0.017 0.006 ± 0.004 0.042 ± 0.009 0.009 ± 0.004 0.040 ± 0.007 0.005 ± 0.004 0.055 ± 0.011 0.006 ± 0.004 0.036 ± 0.008 0.006 ± 0.004 0.018 ± 0.005 0.010 ± 0.004 0.016 ± 0.005 0.009 ± 0.004 0.007 ± 0.004 0.011 ± 0.004 0.014 ± 0.005 0.041 ± 0.008 0.006 ± 0.004 0.051 ± 0.011 0.005 ± 0.004 0.060 ± 0.012 0.006 ± 0.004 0.067 ± 0.012 0.005 ± 0.004 0.026 ± 0.006 0.006 ± 0.004 0.077 ± 0.018 0.005 ± 0.004 0.007 ± 0.004 0.006 ± 0.004 0.005 ± 0.004 0.005 ± 0.004 0.006 ± 0.004	1390 ± 316 2879 ± 427 3314 ± 497 942 ± 147 23358 ± 3059 2479 ± 552 10451 ± 1305 1534 ± 491 1491 ± 441 1831 ± 330 317 ± 121 3675 ± 536 3583 ± 541 1185 ± 281 1200 ± 332 2124 ± 440 1831 ± 358 454 ± 149 1912 ± 278 658 ± 145 434 ± 133 1497 ± 250 528 ± 146 1272 ± 292 557 ± 102 630 ± 157 665 ± 120 1646 ± 323 747 ± 193 770 ± 254 756 ± 160 1554 ± 335 726 ± 122 505 ± 186 545 ± 112 1303 ± 313 298 ± 79 783 ± 323 444 ± 99 803 ± 175 650 ± 126 959 ± 335 1788 ± 329 739 ± 269 380 ± 113 1736 ± 480 572 ± 100 789 ± 174 587 ± 160 919 ± 297 546 ± 97	50.3 24.1 18.2 85.2 102.8 338.5 74.4 195.7 49.4 26.6 61.6 10.7 58.3 44.1 10.1 30.4 26.0 33.3 14.5 7.3 13.9 9.4 30.6 11.1 17.8 12.0 11.2 18.1 18.0 13.9 12.2 14.3 14.0 18.2 8.8 19.8 11.9 23.6 7.2 27.4 8.4 66.9 9.3 20.2 8.1 12.3 17.3 11.7 8.2 9.4 11.8
	9	-101	0 0.9800 115.262 115.254 ± 0.008	0.249	0.004	0.005 ± 0.004	701 ± 221	7.3
	9 -100 -1 0.9672 116.058 116.058 ± 0.006	0.322	0.006	0.006 ± 0.004	1721 ± 254	19.0
	9 -100	0 0.9600 116.292 116.285 ± 0.007	0.343	0.007	0.006 ± 0.004	1150 ± 273	12.1
	9 -100	1 0.9506 116.523 116.525 ± 0.006	0.364	0.009	0.006 ± 0.004	2571 ± 532	27.2
	9	-99 -1 0.9056 117.072 117.076 ± 0.009	0.414	0.017	0.022 ± 0.006	459 ± 134	12.7
	9	-99	0 0.8694 117.291 117.318 ± 0.025	0.436	0.023	0.040 ± 0.013	77 ± 34	7.2
	9	-99	1 0.8189 117.489 117.520 ± 0.006	0.455	0.033	0.011 ± 0.004	4320 ± 641	54.2
	9	-98 -1 0.6256 117.926 117.973 ± 0.007	0.496	0.067	0.043 ± 0.007	2105 ± 327	32.6
	9	-98	0 0.5622 118.077 118.138 ± 0.009	0.511	0.079	0.033 ± 0.007	5534 ± 1222	91.7
	9	-98	1 0.5461 118.211 118.282 ± 0.009	0.524	0.082	0.034 ± 0.007	3206 ± 652	69.8
	9	-97 -1 0.6989 118.597 118.638 ± 0.008	0.556	0.054	0.034 ± 0.007	3294 ± 723	142.5
	9	-97	0 0.7744 118.766 118.815 ± 0.007	0.572	0.041	0.013 ± 0.004	3831 ± 566	66.2
	9	-97	1 0.8394 118.969 118.994 ± 0.006	0.588	0.029	0.010 ± 0.004	10109 ± 1478	154.7
	9	-96 -1 0.9267 119.532 119.539 ± 0.008	0.638	0.013	0.017 ± 0.005	1339 ± 307	23.5
	9	-96	0 0.9428 119.758 119.758 ± 0.006	0.658	0.010	0.008 ± 0.004	4038 ± 588	46.3
	9	-96	1 0.9539 119.989 119.991 ± 0.007	0.679	0.008	0.008 ± 0.004	2271 ± 444	25.7
	9	-95 -1 0.9717 120.623 120.617 ± 0.006	0.735	0.005	0.006 ± 0.004	3853 ± 586	44.1
	9	-95	1 0.9778 121.096 121.088 ± 0.006	0.778	0.004	0.006 ± 0.004	6667 ± 971	77.7
	9	-94 -1 0.9828 121.763 121.749 ± 0.006 -0.162	0.003	0.006 ± 0.004	5234 ± 769	61.2
	9	-93 -1 0.9867 122.933 122.925 ± 0.010 -0.055	0.002	0.007 ± 0.004	2021 ± 647	27.2
	10	-92	0 0.9861 124.367 124.355 ± 0.006	0.074	0.002	0.006 ± 0.004	1007 ± 169	13.4
	10	-91 -1 0.9833 125.344 125.329 ± 0.007	0.162	0.003	0.005 ± 0.004	2714 ± 740	33.8
	10	-91	0 0.9817 125.582 125.568 ± 0.006	0.184	0.003	0.005 ± 0.004	1574 ± 381	19.7
	10	-91	1 0.9800 125.819 125.796 ± 0.006	0.205	0.004	0.006 ± 0.004	5347 ± 831	67.2
	10	-90 -1 0.9706 126.575 126.561 ± 0.007	0.274	0.005	0.006 ± 0.004	1987 ± 407	25.3
	10	-90	0 0.9661 126.810 126.799 ± 0.008	0.296	0.006	0.009 ± 0.004	2797 ± 714	35.8
	10	-90	1 0.9594 127.042 127.032 ± 0.006	0.317	0.007	0.006 ± 0.004	5283 ± 1186	67.9
	10	-89 -1 0.9167 127.792 127.789 ± 0.006	0.385	0.015	0.010 ± 0.004	2454 ± 424	31.9
	10	-89	0 0.8900 128.014 128.018 ± 0.007	0.406	0.020	0.007 ± 0.004	1441 ± 297	18.8
	10	-89	1 0.8528 128.220 128.231 ± 0.007	0.425	0.026	0.009 ± 0.004	6599 ± 1274	92.5
	10	-88 -1 0.6144 128.822 128.875 ± 0.008	0.484	0.069	0.021 ± 0.005	6832 ± 1038	132.3
	10	-88	0 0.5406 128.970 129.032 ± 0.008	0.498	0.083	0.043 ± 0.007	3925 ± 617	91.8
	10	-88	1 0.5044 129.095 129.168 ± 0.009	0.510	0.089	0.034 ± 0.007	5624 ± 1181	130.5
	10	-87 -1 0.6672 129.569 129.608 ± 0.009	0.550	0.060	0.049 ± 0.010	3066 ± 755	157.3
	10	-87	0 0.7439 129.731 129.772 ± 0.007	0.565	0.046	0.014 ± 0.004	12352 ± 1855	257.9
	10	-87	1 0.8133 129.927 129.955 ± 0.006	0.582	0.034	0.011 ± 0.004	16357 ± 2408	226.5
	10	-86 -1 0.9294 130.677 130.670 ± 0.006	0.647	0.013	0.007 ± 0.004	4084 ± 855	55.9
	10	-86	0 0.9433 130.903 130.896 ± 0.006	0.667	0.010	0.007 ± 0.004	7019 ± 1024	96.4
	10	-86	1 0.9533 131.134 131.122 ± 0.006	0.688	0.008	0.006 ± 0.004	10575 ± 2196	145.8
	10	-85 -1 0.9728 131.986 131.962 ± 0.009	0.764	0.005	0.009 ± 0.004	1419 ± 431	19.8
	10	-85	0 0.9756 132.221 132.196 ± 0.009	0.785	0.004	0.006 ± 0.004	530 ± 151	7.4
	10	-84	0 0.9833 133.590 133.573 ± 0.006 -0.090	0.003	0.006 ± 0.004	3228 ± 566	46.4
	11	-83	0 0.9844 134.993 134.973 ± 0.006	0.037	0.003	0.006 ± 0.004	4520 ± 719	66.5
	11	-82 -1 0.9806 136.185 136.161 ± 0.006	0.144	0.004	0.006 ± 0.004	1267 ± 184	20.0
	11	-81 -1 0.9639 137.632 137.609 ± 0.006	0.275	0.007	0.006 ± 0.004	1148 ± 170	18.7
	11	-81	0 0.9578 137.866 137.837 ± 0.007	0.296	0.008	0.006 ± 0.004	900 ± 151	13.9
	11	-81	1 0.9494 138.096 138.075 ± 0.006	0.318	0.009	0.011 ± 0.004	2426 ± 366	39.1

ζ is derived from the best asymptotic fit; νas are the asymptotic frequencies, whereas ν correspond to the observed values; x = ν/∆ν -(np -εp) is the reduced frequency; Γas are the asymptotic mode widths, whereas Γ correspond to the observed values; H are the observed heights, and R is the height-to-background ratio.

Table A .

 A 2. Oscillation pattern of the RGB star KIC 3955033 Radial modes and mixed modes identified in KIC 3955033 with a height-to-background ratio R larger than 7. Same caption as TableA.1

	np	n	m	ζ	νas	ν	x	Γas	Γ	H	R
					(µHz)	(µHz)	(µHz)	(µHz)	(ppm 2 µHz -1 )
						Radial modes
	8				84.745	84.745 ± 0.077	0.033		0.198 ± 0.089	1015 ± 656	16.2
	9				93.958	93.762 ± 0.012	0.010		0.094 ± 0.017	3985 ± 872	27.3
	10				102.911 103.010 ± 0.013 0.012		0.138 ± 0.027	7641 ± 1745	30.2
	11				112.215 112.143 ± 0.009 0.002		0.067 ± 0.012	18117 ± 3832	114.7
	12				121.869 121.474 ± 0.016 0.013		0.122 ± 0.025	2495 ± 637	19.6
						Dipole mixed modes
	8	-144	1 0.8744	88.882	88.894 ± 0.006	0.483	0.019	0.007 ± 0.004	5748 ± 1146	15.9
	8	-141 -1 0.8051	89.078	89.075 ± 0.009	0.503	0.029	0.020 ± 0.005	2011 ± 512	7.1
	8	-142	0 0.7444	89.214	89.205 ± 0.008	0.517	0.038	0.032 ± 0.006	2964 ± 565	10.9
	8	-143	1 0.6908	89.338	89.338 ± 0.010	0.531	0.046	0.041 ± 0.009	2226 ± 536	15.9
	8	-140 -1 0.6640	89.507	89.496 ± 0.007	0.548	0.050	0.035 ± 0.006	6623 ± 1063	21.4
	8	-141	0 0.6955	89.623	89.602 ± 0.011	0.560	0.046	0.036 ± 0.008	1250 ± 297	10.1
	8	-142	1 0.7483	89.743	89.725 ± 0.012	0.573	0.038	0.043 ± 0.010	1315 ± 389	10.1
	8	-139 -1 0.8432	89.969	89.956 ± 0.009	0.598	0.023	0.024 ± 0.006	2118 ± 541	9.8
	8	-141	1 0.9111	90.251	90.238 ± 0.009	0.629	0.013	0.006 ± 0.004	2235 ± 601	7.4
	9	-127 -1 0.9047	97.755	97.779 ± 0.011	0.446	0.014	0.023 ± 0.007	1041 ± 358	9.9
	9	-126 -1 0.6934	98.346	98.392 ± 0.007	0.512	0.046	0.014 ± 0.004	36038 ± 5534	121.7
	9	-127	0 0.6800	98.375	98.395 ± 0.007	0.512	0.048	0.024 ± 0.005	21840 ± 3803	121.7
	9	-128	1 0.6615	98.413	98.395 ± 0.007	0.512	0.051	0.027 ± 0.005	19513 ± 3160	121.7
	9	-125 -1 0.6969	98.825	98.818 ± 0.007	0.558	0.045	0.014 ± 0.004	10668 ± 1614	47.6
	9	-126	0 0.7049	98.843	98.879 ± 0.007	0.565	0.044	0.028 ± 0.006	6104 ± 971	27.3
	9	-127	1 0.7220	98.874	98.883 ± 0.007	0.565	0.042	0.031 ± 0.006	5427 ± 941	27.3
	9	-124 -1 0.9050	99.430	99.435 ± 0.006	0.625	0.014	0.006 ± 0.004	9343 ± 1572	29.3
	9	-126	1 0.9124	99.474	99.470 ± 0.006	0.629	0.013	0.012 ± 0.004	7355 ± 1118	23.3
	9	-125	1 0.9647 100.170 100.166 ± 0.007 0.704	0.005	0.006 ± 0.004	2664 ± 543	9.3
	10 -120	1 0.9883 103.940 103.940 ± 0.007 0.113	0.002	0.006 ± 0.004	3119 ± 574	11.1
	10 -118 -1 0.9880 104.023 104.025 ± 0.010 0.122	0.002	0.007 ± 0.004	2677 ± 836	10.2
	10 -118	0 0.9848 104.778 104.780 ± 0.011 0.204	0.002	0.007 ± 0.004	2318 ± 827	9.3
	10 -117 -1 0.9844 104.838 104.842 ± 0.009 0.211	0.002	0.006 ± 0.004	4417 ± 1446	14.7
	10 -116 -1 0.9748 105.657 105.666 ± 0.006 0.300	0.004	0.005 ± 0.004	4718 ± 1166	15.8
	10 -115 -1 0.9438 106.481 106.502 ± 0.022 0.391	0.008	0.019 ± 0.011	1349 ± 1022	16.4
	10 -116	1 0.8515 107.096 107.070 ± 0.008 0.452	0.022	0.008 ± 0.004	1890 ± 368	10.1
	10 -115	0 0.8294 107.167 107.194 ± 0.011 0.466	0.026	0.041 ± 0.011	1090 ± 354	9.5
	10 -114 -1 0.8023 107.238 107.259 ± 0.006 0.473	0.030	0.016 ± 0.004	5430 ± 799	21.0
	10 -115	1 0.5811 107.710 107.706 ± 0.007 0.521	0.063	0.019 ± 0.005	22251 ± 3274	99.4
	10 -114	0 0.5753 107.757 107.776 ± 0.011 0.529	0.064	0.082 ± 0.021	3573 ± 1204	70.9
	10 -113 -1 0.5782 107.819 107.818 ± 0.008 0.533	0.063	0.042 ± 0.009	6466 ± 1487	71.0
	10 -114	1 0.7802 108.260 108.253 ± 0.008 0.580	0.033	0.018 ± 0.005	7228 ± 1753	32.4
	10 -113	0 0.8149 108.344 108.337 ± 0.009 0.589	0.028	0.027 ± 0.007	3135 ± 809	35.8
	10 -112 -1 0.8468 108.441 108.429 ± 0.009 0.599	0.023	0.029 ± 0.007	3843 ± 1006	25.7
	10 -113	1 0.9383 109.023 109.019 ± 0.006 0.663	0.009	0.008 ± 0.004	11460 ± 1670	41.2
	10 -112	1 0.9728 109.876 109.870 ± 0.006 0.756	0.004	0.006 ± 0.004	6439 ± 938	23.4
	11 -104 -1 0.9239 115.807 115.818 ± 0.006 0.400	0.011	0.007 ± 0.004	3908 ± 613	15.8
	11 -105	1 0.8432 116.274 116.279 ± 0.006 0.450	0.023	0.009 ± 0.004	6992 ± 1054	26.2
	11 -104	0 0.7745 116.468 116.470 ± 0.009 0.471	0.034	0.033 ± 0.008	1247 ± 360	18.8
	11 -103 -1 0.6842 116.648 116.670 ± 0.007 0.492	0.047	0.028 ± 0.006	2247 ± 418	28.7
	11 -104	1 0.5371 116.961 116.976 ± 0.010 0.525	0.069	0.054 ± 0.011	2255 ± 530	17.6
	11 -103	0 0.5477 117.092 117.077 ± 0.015 0.536	0.068	0.082 ± 0.021	1275 ± 415	19.0
	11 -102 -1 0.6181 117.248 117.211 ± 0.010 0.551	0.057	0.028 ± 0.007	1479 ± 387	11.9
	11 -103	1 0.8015 117.612 117.600 ± 0.007 0.593	0.030	0.017 ± 0.005	2633 ± 464	14.5
	11 -101 -1 0.9071 118.072 118.063 ± 0.017 0.643	0.014	0.031 ± 0.013	479 ± 289	10.9

Pure p (or g) modes are hypothetical modes that could be formed in the pressure (or gravity) cavity without any coupling with the other cavity.

Since we consider dipole modes only, we use a simplified notation νn,m instead of ν n, ,m .

Acknowledgements. We thank the entire Kepler team, whose efforts made these results possible. BM warmly thanks Yvonne Elsworth, James Kuszlewicz and Masao Takata for their comments on the draft submitted for internal review on the website of the Kepler Asteroseismic Science Operations Center. We acknowledge financial support from the Programme National de Physique Stellaire (CNRS/INSU). BM acknowledges the support of the International Space Institute (ISSI) for the program AsteroSTEP (Asteroseismology of STEllar Populations)

Appendix A: Seismic parameters

We used KIC 6144777 as a case study (Fig. 2). Table A.1 provides the fit of its radial dipole mixed modes. Our results are in agreement with those published by Corsaro et al. (2015) and derive a similar number of modes (about 100), but also show differences: -The determination of the frequencies in Corsaro et al. (2015) can be as precise as 0.3 nHz. This precision of about δf res /30 was corrected into about δf res /10 in their corrigendum (Corsaro et al. 2018), which remains surprisingly good; the frequencies we obtain are given with a precision that is at best about half the frequency resolution ( 4 nHz).

-Their mode widths are quite different and, most often, larger than ours; -Heights also differ, which can come from a different treatment of the time series. A large agreement is also met with the results obtained by García Saravia Ortiz de Montellano et al. (2018) with a peak detection algorithm that works in a fully blind manner, if we relax their uncertainties that can be as low as δf res /20.

The potential of the comparison between methods based on different principles is very high: coupling the physics of the asymptotic expansion and the power of a pure numerical approach is the next step for delivering duly identified mixed modes.

The échelle diagrams of the stars mentioned in the main text are also presented: -KIC 10272858 lies on the low part of the RGB (Fig

Appendix B: Stars in open clusters

All stars studied by Corsaro et al. (2017) were investigated. The fitting process is challenging, due to the dim magnitudes of such dim stars in open clusters. However, the combination of all pressure radial orders near ν max provides in most cases an unambiguous fit, and at least a few mixed-mode radial orders provide clear splittings.

- These stars are representative of the whole data set treated by Corsaro et al. (2017): the inability of the fitting process to identify thin short-lived mixed modes translates into the identification of a single broad m = 0 peak. In such cases, stellar inclinations derived from the Bayesian fits are necessarily underestimated and biased toward low values.