Globular clusters offer a powerful way to test the properties of stellar populations and the late stages of lowmass stellar evolution. In this paper we study oscillating giant stars and overtone RR Lyraetype pulsators in the nearest globular cluster M4 with the help of highprecision continuous light curves collected by the Kepler space telescope in the K2 mission. We determine the frequency composition of five RRc stars and model their physical parameters with a grid of linear pulsation models. We are able for the first time to compare seismic masses of RR Lyrae stars directly to the masses of the very similar red horizontal branch stars in the same stellar population independently determined from asteroseismic scaling relations. We find a close match with an average seismic mass of 0.651pm0.028Modot for RR Lyrae stars and 0.657pm0.034Modot for red horizontalbranch stars. While the validity of our RR Lyrae masses still relies on the similarity of neighboring horizontal branch subgroups this result strongly indicates that RRc stars may indeed exhibit highdegree l  8 and 9 nonradial modes and modeling these modes can provide realistic mass estimates. We also determine the He content of the cluster to be Y  0.266pm 0.008 and compare the seismic masses for our sample of RR Lyrae to theoretical mass relations and highlight the limitations of these relations.
['Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Astrophysics of Galaxies']
Matching seismic masses for RR Lyraetype and oscillating red horizontalbranch stars in M4
{'Matching seismic masses for RR Lyrae-type and oscillating red horizontal-branch stars in M4': 'László Molnár 1 , 2 , Henryka Netzel 3 , Madeline Howell 4 , 5 , Csilla Kalup 1 , 2 , and Meridith Joyce 1 , 6  \n- 1 Konkoly Observatory, HUN-REN Research Centre for Astronomy and Earth Sciences, MTA Centre of Excellence, Konkoly-Thege Miklós út 15-17, H-1121, Budapest, Hungary\n- e-mail: molnar.laszlo@csfk.org; lmolnar@konkoly.hu\n- 2 ELTE Eötvös Loránd University, Institute of Physics and Astronomy, 1117, Pázmány Péter sétány 1 / A, Budapest, Hungary\n- 3 Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland\n- 4 School of Physics and Astronomy, Monash University, Clayton, VIC 3800, Australia\n- 5 ARC Centre of Excellence for Astrophysics in Three Dimensions (ASTRO-3D), Australia\n- 6 University of Wyoming, 1000 E University Ave, Laramie, WY USA', 'ABSTRACT': 'Globular clusters o ff er a powerful way to test the properties of stellar populations and the late stages of low-mass stellar evolution. In this paper we study oscillating giant stars and overtone RR Lyrae-type pulsators in the nearest globular cluster, M4, with the help of high-precision, continuous light curves collected by the Kepler space telescope in the K2 mission. We determine the frequency composition of five RRc stars and model their physical parameters with a grid of linear pulsation models. We are able, for the first time, to compare seismic masses of RR Lyrae stars directly to the masses of the very similar red horizontal branch stars in the same stellar population, independently determined from asteroseismic scaling relations. We find a close match, with an average seismic mass of 0 . 651 ± 0 . 028 M ⊙ for RR Lyrae stars and 0 . 657 ± 0 . 034 M ⊙ for red horizontal-branch stars. While the validity of our RR Lyrae masses still relies on the similarity of neighboring horizontal branch subgroups, this result strongly indicates that RRc stars may indeed exhibit high-degree, l = 8 and 9 non-radial modes, and modeling these modes can provide realistic mass estimates. We also determine the He content of the cluster to be Y = 0 . 266 ± 0 . 008, and compare the seismic masses for our sample of RR Lyrae to theoretical mass relations and highlight the limitations of these relations.  \nKey words. globular star clusters - asteroseismology - RR Lyrae variable stars', '1. Introduction': "Masses of RR Lyrae stars are notoriously di ffi cult to determine. This is a problem, because the masses of Horizontal Branch (HB) stars, and in particular the masses of RR Lyrae stars, located at the intersection of the HB and the instability strip of classical pulsating stars, are crucial pieces of information for stellar evolution theories and models (Catelan 2009). Masses would provide information not only on the mass distribution along the HB, but also on the amount of mass loss during the red giant phase of stellar evolution and on the accuracy of mass estimates based on pulsational envelope models, as well. Thus, many avenues of research have been already explored to determine the masses of RR Lyrae stars.  \nBinarity, for example, would be a straightforward way to infer dynamical masses. However, not only do we not know any eclipsing binaries with an RR Lyrae member, it is very di ffi cult to find systems that contain an RR Lyrae star at all (Wade et al. 1999; Kennedy et al. 2014; Hajdu et al. 2015). Since bona fide RR Lyrae stars are considered to be very old, their companions have either evolved into white dwarfs or other stellar remnants already, or must be smaller stars, most likely still evolving along the lower main sequence as K-M dwarfs. As such, the contribution of RR Lyr companions to the total light output of the system is expected to be minimal, precluding the detection via spectral energy distributions or as double-lined spectroscopic binaries.  \nThe dearth of eclipsing RR Lyrae stars can also be explained by the evolutionary pathways of these pulsators. These stars moved past the red giant phase without any significant interactions with a companion that could have directed the primary away from evolving onto the HB and towards binary evolution scenarios. Companions thus must be distant to avoid any mass transfer, and a distant companion orbiting a now shrunken horizontal-branch star would make the probability of observing eclipses from the system very low. The only eclipsing binary candidate system turned out to be an impostor, a Binary Evolution Pulsator (Pietrzy'nski et al. 2012; Karczmarek et al. 2017). In this case, mass transfer directed a much lighter primary to rapidly cross the instability strip and to experience RR Lyraelike pulsations. We note that binarity and a limited amount of mass transfer has been proposed to explain metal-rich RR Lyrae stars, but those systems are expected to have wide orbits, too (Bobrick et al. 2024).  \nMore indirect methods have shown increased promise, but all have their own caveats. For example, radial velocity searches are hindered by the expected long orbital periods and small signals relative to the much larger pulsation velocities. A recent work proposed new binary candidates, but it also failed to confirm the expected periastron passage of the oldest binary candidate, TU UMa (Wade et al. 1999; Barnes et al. 2021).  \nAn alternative way to detect binaries and estimate masses is the light-time e ff ect, but this method has been impeded by the  \nprevalence of the Tseraskaya-Blazhko e ff ect 1 , the quasi-periodic modulation of the pulsation amplitude and phase (Blažko 1907; Shapley 1916; Kurtz 2022). Nevertheless, several binary candidates have been identified in the OGLE survey (Hajdu et al. 2015, 2021). And while Hajdu et al. (2021) did not address the masses of the primaries directly, these authors found a trimodal distribution in the mass functions of the secondaries.  \nAs we have shown, dynamical methods to determine masses of RR Lyrae stars su ff er from various theoretical and / or practical shortcomings. This limits us to other, more indirect, modeldependent methods. Fundamental physical parameters such as luminosity, T e ff and log g , along with elemental abundances, can be fitted with evolutionary tracks and atmosphere models. A relation to calculate masses from other physical parameters (period, luminosity, T e ff ) was first proposed by van Albada & Baker (1971) and by multiple authors since. However, they come with strong uncertainties in various model parameters that are very loosely constrained by observations, if at all. These include the He content of the models, the approximations chosen for internal processes like convection and overshoot, or various aspects of mass loss along the evolutionary pathway (see, e.g., Marconi et al. 2018; Anders & Pedersen 2023; Joyce & Tayar 2023; Joyce et al. 2024). Nemec et al. (2011) used such relations to calculate masses for the RR Lyrae stars in the Kepler field, for example, but without providing uncertainties for them.  \nOne way to limit uncertainties in stellar parameters is to study RR Lyrae stars in globular clusters. There are currently close to three thousand RR Lyrae stars known in 115 Galactic globular clusters (Cruz Reyes et al. 2024). For these stars, parameters like distance and metallicity can be determined accurately for the members, and using that information, the van Albada & Baker (1971) relation was employed in many cases to estimate masses. M3, for example, was studied by multiple authors, and Valcarce & Catelan (2008) determined the mass distribution for the entire HB of the cluster based on evolutionary models, and found the distribution to be potentially bimodal. However, they discuss how that finding can be complicated by limitations present in the stellar tracks. A di ff erent approach was taken by Kumar et al. (2024), who fitted the light curves of the M3 RR Lyrae variables directly, based on a grid of non-linear pulsation models. Direct light curve fitting is a promising technique (see also, Bellinger et al. 2020, and Das et al., in prep), but current one-dimensional non-linear pulsation models still cannot reproduce light curves very accurately, due to limitations in handling convection, for example (Kovács et al. 2023).  \nUncertainties in evolutionary and non-linear pulsation models can also be mitigated if we turn towards asteroseismology, and model the frequencies of the oscillatory modes observed in the star (see, e.g., Kjeldsen & Bedding 1995; Aerts 2021). Incorporating asteroseismic constraints has shown great potential recently for stars outside the instability strip. However, classical pulsators like RR Lyrae stars face obstacles in this aspect, too: in these stars, most often only a single mode can be observed, severely limiting our ability to constrain physical parameters from frequencies alone. Masses for a frew doublemode RR Lyrae stars have been calculated from non-linear nonadiabatic hydrodynamical simulations, (Molnár et al. 2015b), but, as mentioned before, the accuracy of non-linear models have been called into question (Smolec & Moskalik 2008).  \nMore recently, a new method based on the discovery of lowamplitude signals in overtone RR Lyrae stars has been proposed  \n(Netzel et al. 2015b,c). Theoretical predictions by Dziembowski (2016) suggested that these modes may be high-order, ℓ = 8, 9 non-radial modes, which can be calculated from linear pulsation models. Unlike modeling radial modes in the non-linear regime, fitting radial and non-radial modes together using a linear model grid is a technique very similar to the asteroseismology of red giant stars. This method was developed by Netzel &Smolec (2022), and was further explored, incorporating other observables such as T e ff , L and [Fe / H] information into the fit in Netzel et al. (2023).  \nBut, as we have shown, we still lack a way to test the validity of these new RR Lyrae masses through direct methods. We therefore searched for alternatives. RR Lyrae stars make up only a small portion of the He-core-burning regime, and stars exist blue- and redward of them on the HB. Indeed, the evolution of HB stars and their positions on the HB are largely defined by the mass of their H-rich envelopes, as well as the mass loss they experience before reaching the HB. At temperatures below the red edge of the instability strip, we find the red horizontal branch (rHB) stars. These stars are very similar to the RR Lyrae pulsators, except for hosting a slightly more extended envelope. Since their envelopes are also convective, these stars show solarlike oscillations instead of pulsations, and these solar-like oscillations can be analysed in a way very similar to that of red giant and red clump stars (Matteuzzi et al. 2023). If rHB and RR Lyrae stars can be observed and modeled within the same population of stars, such as in globular clusters, they can potentially o ff er a way to compare asteroseismic masses determined by independent methods.  \nThis opportunity came with the Kepler space telescope. Kepler observed multiple globular clusters during the K2 mission, including the cluster closest to the Sun, Messier 4 (M4). This cluster was bright enough that seismic data was obtainable not only for the bright red giants, but fainter stars, such as the rHB stars in it, as well. Howell et al. (2022) calculated seismic masses for several stars and compared the averages of various evolutionary phases to estimate the integrated mass loss between stages in the cluster. M4 also contains several RR Lyrae stars, including overtone ones that have been observed by Kepler . If non-radial modes can be detected in the overtone stars, their seismic masses could be compared directly to the seismic masses of rHB stars, validating the results.  \nIn Section 2, we process the Kepler data of five RRc stars observed in M4 and determine their physical parameters. In Section 3, we analyse the photometry and the frequency content of the RRc stars and fit them with seismic models. In Section 4, we compare the RRc physical parameters to the (updated) parameters of the red giant stars of the cluster, as well as to theoretical mass relations. Finally, we summarize our findings in Section 5.", '2. Data and methods': 'The Kepler space telescope observed M4 during Campaign 2 of the K2 mission in late 2014 for 80 days nearly continuously. The cluster was at the edge of module 12 of the detector; hence M4 was only partially covered by the campaign.', '2.1. K2 photometry': 'A large portion of the cluster was covered with an extended custom pixel aperture. We identified three RRc stars within this aperture: M4 V6, V42 and V61. To extract their light curves, we followed the same di ff erential-image method based on the  \nFITSH photometry package of Pál (2012) that we used on M80 (Molnár et al. 2023b), and on other crowded, faint, and / or moving targets (see, e.g., Molnár et al. 2015a, 2018; Kalup et al. 2021). The output for these stars is flux variations relative to the subtracted master frame. We shifted the median variations to the Gaia DR3 G-band average brightnesses (Gaia Collaboration et al. 2021; Riello et al. 2021).  \nSince M4 has a large angular diameter, it extends beyond the K2 custom aperture. We located two additional RRc stars farther away from the center (V43 and V76), on the outskirts of the cluster: these were recorded with individual pixel apertures. For these two stars, we used the simple aperture photometry (SAP) light curves provided by the mission.  \nRaw K2 photometric data contains systematic signals due to the limited pointing capabilities experienced by the telescope during the mission. It was found in earlier works focusing on RRLyrae stars (Plachy et al. 2019; Bódi et al. 2022; Molnár et al. 2023b) that the best method to separate the instrumental signals from the pulsations is the K2 Systematics Correction (K2SC) method developed by Aigrain et al. (2016). We applied the same corrections here to remove fast systematics. Slow trends were then removed with the algorithm developed by Bódi et al. (2022), fitting a high-order polynomial to the light curve that is optimized for best overlap of the pulsation cycles via phase dispersion minimization. The resulting light curves are shown in Fig. 1. Photometric data is available in Appendix B.  \nOne more target, M4 V75, which was identified as a possible RRc star by Yao et al. (1988), is located within the K2 custom aperture as well. This star, however, shows no periodic variation, confirming previous non-detections by Stetson et al. (2014) and Safonova et al. (2016).', '2.2. Observational constraints': 'A detailed asteroseismic analysis requires further constraints on the physical parameters (such as L , T e ff , log g or [Fe / H]) of the stars. Howell et al. (2022) used scaling relations for the red giants in the cluster, which relies on luminosities and e ff ective temperatures. These quantities can be calculated from photometry, but they require accurate corrections for interstellar extinction. Here we used the reddening map and E(B-V) zero point determined specifically for M4 by Hendricks et al. (2012).  \nIn contrast, our approach for the RRc stars is akin to peakbagging, where we collect and fit individual oscillation peaks (Appourchaux 2003). However, since we are limited to very few modes, it is still important to constrain our modeling space with classical observables such as allowed luminosity, T e ff and metallicity ranges.  \nWe collected the average brightness values of the RRc stars both in Gaia DR3 passbands, and in Johnson passbands from the observations of Stetson et al. (2014) 2 . The Johnson data set is displayed in Fig. 2, with the RRc stars and the stars studied by Howell et al. (2022) highlighted. However, since input parameters for previous RRc model fits were based on Gaia brightnesses, we decided to use those here, as well.  \nFirst, we calculated the interstellar extinction in the Gaia passbands with the seismolab software package, which uses the mwdust code and the Bayestar 3D dust map for that purpose (Bódi et al. 2022; Bovy et al. 2016; Green et al. 2019). We then used the distance modulus provided by Baumgardt & Vasiliev  \nFig. 1. Corrected K2 light curves of the five RRc stars that were targeted by the mission.  \n<!-- image -->  \n(2021), µ = 11 . 337 ± 0 . 018 mag, to convert the extinctioncorrected values into absolute magnitudes.  \nWe used M G and ( BP -RP )0 to derive luminosity and effective temperature. The e ff ective temperature T e ff was calculated using the relation from eq. 1 from Mucciarelli & Bellazzini (2020), which requires metallicity, [Fe / H], and colour. We employed the relation derived for giant stars and colour ( BP -RP )0. We set the metallicity to [Fe / H] ≈ -1 . 1 ± 0 . 07 (Harris 1996; Marino et al. 2008; MacLean et al. 2018). The luminosity was calculated using the formula:  \nL / L ⊙ = 10 ( -0 . 4 ( M bol -( M ⊙ bol )) , (1)  \nwhere M bol is a bolometric brightness of a star, and M ⊙ bol is a bolometric brightness of the Sun, set to 4.74 mag. We converted M G to M bol using bolometric correction calculated with the Gaiadr3 Bcg tool by Creevey et al. (2023). The input parameters to calculate bolometric correction are gravity (log g ), metallicity, and e ff ective temperature. We assumed log g = 3.  \nMetal, Z , and helium, Y , contents were calculated based on [Fe / H]. First, we transformed [Fe / H] to [M / H] using formula by Salaris et al. (1993):  \nFig. 2. Color-magnitude diagram of M4 in Johnson passbands using photometry from Momany et al. (2002) and Mochejska et al. (2002). Stars targeted by our study and by Howell et al. (2022) are highlighted with colored points. The Gaia membership sample is from Vasiliev & Baumgardt (2021).  \n<!-- image -->  \n[M / H] ≈ [Fe / H] + log GLYPH<16> 0 . 638 · 10 [ α/ Fe] + 0 . 362 GLYPH<17> , (2)  \nwhere [ α / Fe] is enhancement by α elements. Then, we transformed [M / H] to Z , using the relation:  \n[M / H] = log Z Z ⊙ -log X X ⊙ , (3)  \nwhere Z ⊙ and X ⊙ are solar values, which we set to Z ⊙ = 0 . 0134 and X ⊙ = 0 . 7381 (Asplund et al. 2009).  \nAssuming that hydrogen content, X , can vary from 0.70 to 0.76, and α -element enhancement can vary from 0 to 0.4, we calculated the modeling range for Z to be 0.00086 - 0.0032.', '3. Photometric results on the RRc stars': 'We used the Period04 software to determine the frequency composition of each star (Lenz & Breger 2005). The detection limit was set to S / N > 4.0, relative to the nearby average noise level in the frequency spectrum. First, we subtracted the pulsation frequency and its harmonics, then searched for any extra signals above or below the pulsation frequency. Folded light curves and frequency spectra are displayed in Fig. 3.', '3.1. Extra modes': 'According to Dziembowski (2016), the signals around period ratios P / P O1 ≈ 0 . 63 and 0.61, relative to the first overtone, correspond to (the first harmonic of) l = 8 and l = 9 modes, respectively. Here we label these as f 63 and f 61 frequencies, although this family of modes is also called f 61 or fX modes collectively. The actual mode frequencies are the subharmonics of these signals at f 61 / 2 and f 63 / 2, but those are usually harder to detect due to geometric cancellation e ff ects.  \nWe clearly identified the modes in four stars: V6, V42, V43 and V76. We also see subtle di ff erences between them. In V6, the subharmonic at the true pulsation frequency is the strongest, which is unusual among RRc stars as we expect the harmonic to be stronger. In V42 and V76, we detect multiple peaks and the corresponding harmonics and subharmonics. In V43, the peak appears to be incoherent and can be fitted with three close-by frequencies. Here we observe a very wide structure at the subharmonic that extends below the expected range of the f 61 modes. However, these frequencies are not low enough to be either the fundamental mode or an f 68 mode.  \nWhile f 61 modes are prevalent in the RRc population of M4, we did not detect the low-frequency f 68 mode clearly in either of the targets. The latter is a separate group of extra modes, discovered recently in RRc stars, which appear at lower frequencies ( f / f O1 ≈ 0 . 686), and whose origin has not been identified yet (Netzel et al. 2015a; Benk"o & Kovács 2023). The lack of detection is in agreement with the mode abundance results of Netzel et al. (2024). They found that while the f 61 modes are most frequent around [Fe / H] ≈ -1 . 0, the frequency of the f 68 mode drops drastically above -1.3, and the metallicity of M4 is higher than that limit.  \nIn contrast to the results above, V61 turned out to be the rather di ff erent from the four other stars. Here we only detect significant extra frequency peaks at the subharmonic range, but not at the expected f 61 range. The two peaks appear at period ratios 0.7335 and 0.7877. The former value may potentially correspond to the fundamental mode, with a period of 0.36168 d: however, that would put the star to [Fe / H] values higher than that of M4 in the Petersen diagram (Szabó et al. 2004; Chen et al. 2023). We investigate the modulation properties of the star in Appendix A.', '3.2. Petersen diagram': 'We present the identified f 61 signals on a Petersen diagram in Fig. 4, showing the period ratios against the longer period (here fixed to the first overtone). We plot the M4 results in red against the points presented by Molnár et al. (2023b). We find that the points line up well with other samples that have moderate metallicities, such as the bulge and NGC 6362. In this regime most frequencies clearly fall onto the main ridges at period ratios P 1 / PX ≈ 0 . 632 and 0.613, corresponding to the ℓ = 8 and 9 modes. We also detect signals in the middle ridge, which Dziembowski (2016) interprets as combination frequencies. Further signals appear below, at P 1 / PX ≈ 0 . 605 which might also be combination peaks. We did not detect any ℓ = 10 peaks which should be at P 1 / PX ≈ 0 . 6 or below, according to Dziembowski (2016).', '3.3. Comparison with other photometric results': 'The K2 observations were analysed before by Wallace et al. (2019a) who searched the cluster for new variables. They de-  \nFig. 3. Left: Fourier spectra of the five RRc stars. Here we removed the main pulsation frequency and its harmonics belonging to the first overtone to reveal the low-amplitude extra modes. Modes are labeled with red, and combination frequencies with black. Right: light curves folded with the first overtone period.  \n<!-- image -->  \ntected higher scatter in V61 than in other RR Lyrae stars, but did not recognize it as modulated. They also detected two stars they classified as millimagnitude RR Lyrae stars (Wallace et al. 2019b). Both of those two stars show only a single periodicity.  \nGround-based multicolor photometry was collected for the RRLyrae members (among others) by Stetson et al. (2014). They did not recognize V61 as a modulated star either, although upon reanalysis of their photometry, a side peak is visible. The only other star common with our targets that has a useful amount of time series photometry is V6. There a frequency signal is marginally detectable at the position of the subharmonic ( f 63 / 2). This shows that seismic modeling of RRc stars requires extensive, high-precision photometry to detect the low-amplitude extra modes.', '3.4. RRc pulsation models': 'For each star, we calculated pulsation models to match the observed first-overtone period and period ratio(s). To calculate the theoretical models, we used the envelope pulsation code of  \nDziembowski (1977). The input physical parameters required by the code are mass, luminosity, e ff ective temperature, hydrogen and metal abundances. Additionally, this code requires an approximate value of dimensionless frequency for the non-radial mode. We followed the approach of Netzel & Smolec (2022) for individual calculations. Namely, we used the estimates that relate non-radial mode frequencies from the linear fits to sequences in the Petersen diagram (see Equations 4 and 5 in Netzel & Smolec 2022), which were then converted to dimensionless frequencies, σ , using the formula from Eq. 6 in Netzel & Smolec (2022). Furthermore, we set the range of possible starting values of nonradial mode frequencies to σ ± 0 . 1. Consequently, for each model we were able to get the non-radial mode with the highest driving rate, i.e., it is the most strongly trapped in the envelope.  \nIn order to find the best matches between periods and period ratios in theoretical models and in observed values for each star, we employed genetic algorithms 3 , which used the pulsation code with di ff erent input physical parameters. The ranges of physical  \nTable 1. Input parameter ranges, pulsation periods and period ratios for each RRc star that we used for modeling. PR1 and PR2 are the mode period ratios relative to the first overtone. Mass ranges are left deliberately wide to accommodate realistic values over the grid.  \nFig. 4. Petersen diagram of the f 61-type modes. Here we plot signals detected in M4 (in red) over existing literature data. Frequency peaks fall onto the main ridges discovered in the OGLE data. Positions of the f 61 and f 63 ridges are labelled respectively.  \n<!-- image -->  \nparameters were set based on observational constraints (see Table 1). In the case of mass, we set a wide range of 0 . 5 -0 . 9 M ⊙ to accommodate any realistic mass result. For each star, we executed 100 separate runs. For each run we set the maximum number of iterations to 150, population size of 100, mutation probability of 0.1, elitist ratio to 0.01, crossover probability to 0.5, and parents portion to 0.3. We note that we performed calculations with di ff erent parameters for genetic algorithms beforehand, including the numbers of populations and iterations, to ensure that the results are robust. The values and errors of physical parameters were derived as the means and standard deviations of the results from each of the hundred individual runs.  \nCalculated ranges of physical parameters used to constrain the models for each star are summarized in Table 1.', '4. New and revised physical parameters': 'With the RRc physical parameters obtained, we proceeded to compare them to parameters of other RR Lyrae stars, as well as to the average properties of M4. For further comparison, we  \nFig. 5. Positions of the seismically determined T e ff and log g values for the five RRc stars in M4, against the spectroscopically observed RRc stars in the T e ff -log g plane, as collected by Molnár et al. (2023b).  \n<!-- image -->  \nrecalculated the masses for the oscillating rHB stars originally studied in Howell et al. (2022), too.', '4.1. RR Lyrae masses and other physical parameters': 'As listed in Table 2, the masses of four out of the five RRc stars are within 0.06 M ⊙ , between 0.631 and 0.683 M ⊙ with an average uncertainty of ± 0 . 022 M ⊙ . The only outlier is the modulated star, V61, at 0.751 M ⊙ , but for that star our mode identification was more uncertain. However, mass outliers exist among the other evolutionary stages, such as among the rHB stars in the cluster as well. These masses align well with the mass-period distribution of RRc stars published by Netzel et al. (2023).  \nWe compared the calculated T e ff and log g values to the distribution of RR Lyrae stars presented by Molnár et al. (2023b). As Fig. 5 shows, the stars belong to the cooler RRc stars, lying between 6500-7000 K, with log g values being very close to 3.0 for all five stars. For chemical composition, we find average bulk abundances of X = 0 . 734 ± 0 . 0075 and Z = 0 . 0019 ± 0 . 0002. These values indicate a He abundance of Y = 0 . 266 ± 0 . 008 for M4.  \nThe comparison of the mass ratio of heavy elements, Z , to the observed [Fe / H] index is not straightforward, as it is influenced by the abundance di ff erences of individual elements (see,  \nTable 2. Modeled physical parameters of the five RRc stars based on seismic modeling.  \ne.g., Hinkel et al. 2022). Nevertheless, we can give an estimate for the correctness of our seismic Z values. Following the relations described in Section 2.2, in reverse order, and using an α -enhancement of [ α / Fe] = 0 . 39 ± 0 . 05 (Marino et al. 2008), we get an average seismic [Fe / H] of -1 . 13 ± 0 . 05 for the five RRc stars, which agrees with the spectroscopic [Fe / H] value of the cluster.', '4.2. Seismic masses for red giants and the average rHB mass': "Pressure modes in solar-like oscillators - such as red giants are characterized by two global seismic parameters; the large frequency separation, ∆ ν , and the frequency of the maximum acoustic power, ν max. These quantities are correlated to stellar properties, which are used to derive seismic mass scaling relations (Ulrich 1986; Brown et al. 1991; Kjeldsen & Bedding 1995). For time-series photometry with short baselines (e.g. the K2 mission), it has been shown that accurate asteroseismic masses can be measured independently of a ∆ ν estimate (e.g. Howell et al. 2022, 2024). This method uses the following scaling relation:  \nM M ⊙ ! ≃ ν max ν max , ⊙ ! L L ⊙ ! T e ff T e ff , ⊙ ! -7 / 2 . (4)  \nWeuse the red giant sample and their corresponding K2 light curves from Howell et al. (2022) to measure seismic masses using this relation above. They detected solar-like oscillations in 75 red giants across three phases of evolution (RGB, rHB and early AGB), and used the pySYD pipeline (Chontos et al. 2022) to measure the asteroseismic parameters. We adopt the Howell et al. (2022) estimates for T e ff and luminosity, although not their ν max determinations. Here we report updated measurements of ν max and the corresponding uncertainty for each star using a new asteroseismic pipeline, pyMON (Howell et al., in prep).  \nThe pyMON pipeline is an adaptation of pySYD (see Chontos et al. 2022 for more details), where the main di ff erence is that ∆ ν is not measured. In both pipelines, the power spectrum is smoothed using an estimate for ∆ ν . pyMON substitutes this measurement with an estimate from a calibrated ∆ ν -ν max scaling relation (Stello et al. 2009; Howell et al. 2022). This is useful for low signal-to-noise data (e.g. our M4 photometry), because it is di ffi cult to measure an accurate ∆ ν . A comparison between the pySYD and pyMON methods shows that the measurement of ν max remains consistent, however there is a reduction by a factor of ∼ 2 in the ν max uncertainty, due to the reduced uncertainty in ∆ ν . Consequently, this results in a decrease in the seismic mass uncertainties by a factor of ∼ 1 . 5.  \nWe provide our individual seismic masses in the bottom panel of Figure 6. Note that we have removed the stars identified as mass outliers in Howell et al. (2022), and also limited the RGB sample to below the luminosity bump (hereby known as  \nlower RGB or LRGB). This ensured there was no bias to lower masses due to any mass loss that might have occurred. In the top panel of Figure 6, we represent the distribution of our masses as kernel density estimation (KDE) functions for each evolutionary phase. A KDE is calculated by modelling each data point as a Gaussian, adopting the mass as the mean and the uncertainty as the standard deviation. The final KDE is the summation of the Gaussian components, where the area has been normalised to a value of one.  \nTo estimate a new average mass for the HB sample, we determine the peak value of the corresponding KDE. This represents the mode mass for this sample, and was measured to be 0 . 655 ± 0 . 007 M ⊙ The uncertainty is calculated as the standard error on the mean. Our average HB mass is consistent with the value found previously in Howell et al. (2022). This was similarly found for the RGB and early AGB samples, where we calculated average masses of 0 . 80 ± 0 . 009 M ⊙ and 0 . 55 ± 0 . 01 M ⊙ respectively 4 . Individual and average masses relative to the calculated luminosities are plotted in Fig. 7.  \nThe integrated mass loss rates between evolutionary phases are also consistent with Howell et al. (2022). The mass loss from the lower RGB to the HB decreases slightly to ∆ M RGB -HB = 0 . 145 ± 0 . 011, whereas we find a mass loss from ∆ M HB -AGB = 0 . 105 ± 0 . 013 from the HB to the early AGB.  \nWe plot the distributions of the masses along the HRD in Fig. 8. We also include a MIST 5 (MESA Isochrones and Stellar Tracks; Choi et al. 2016) isochrone for comparison. This isochrone has an age of 12.2 Gyr, an average of various literature sources for M4 (Cabrera-Ziri & Conroy 2022). Since MIST does not o ff er α -enhanced isochrones, we use a value of [Fe / H] = -0.9 to emulate the higher [ α / Fe] abundances, based on the scaling described by Joyce et al. (2023). The mass loss prescriptions used by MIST for low-mass stars are η R = 0 . 1 and η B = 0 . 2 for the RGB and AGB phases, following the (Reimers 1975) and Bloecker (1995) schemes, respectively. Colors represent masses: it is clear that the conservative mass loss prescription used in MIST's pre-computed isochrone database does not capture the true amount of mass loss in the cluster. This is most evident in the later evolutionary stages: while the isochrone has masses around 0.82-0.83 M ⊙ in the RGB, this only drops to 0.79 by the time of the HB and early AGB, which is significantly higher than the observed values. A consequence of the overly conservative mass loss prescription is that the isochrone barely reaches the red edge of the RHB region and therefore cannot reproduce the structure of the HB at all. We note that mass loss parameters cannot be adjusted in MIST through the web interface, either.  \nFig. 6. a) The mass distributions (calculated as KDEs) for the M4 sample of red giants and RR Lyraes. We have separated the KDEs into evolutionary phases. For the HB sample, we have included a KDE consisting of just the rHB sample (blue), and a KDE which combines both the rHB and RR Lyrae samples (cyan). We note that the area of each KDE is normalised to one, and as such the heights of the distributions do not correspond to sample sizes. b) Individual masses for the red giant and RR Lyrae samples plotted against Gaia magnitude.  \n<!-- image -->  \nTable 3. Average masses of each evolutionary group.  \nOur results indicate that for old, low-mass stellar populations, such as globular clusters, pre-packaged MIST isochrones (and any other isochrone set with similarly conservative mass loss prescriptions) can only reliably reproduce the masses of stars up to the lower RGB, before reaching the red giant branch bump. This highlights the importance of cluster seismology and realistic mass loss estimates, because these studies can provide the first observational constraints on the required level of mass loss through the later stages of low-mass stellar evolution.", '4.3. Comparison of the rHB and RRc masses': 'As we discussed in Section 1, estimating the masses of RR Lyrae stars in a model-independent way is an extremely di ffi cult task. While evolutionary models can be used to constrain HB masses in globular clusters (see, e.g., Gratton et al. 2010; Howell et al.  \n2024; Kumar et al. 2024), and masses for RRd stars can be estimated from non-linear models (Molnár et al. 2015a), these approaches still need independent verification. Here we present, for the first time, simultaneous mass estimates based on two independent seismic approaches for stars along the HB in a globular cluster. And although these seismic models and scaling relations could still contain their own model uncertainties, they can now be verified against each other.  \nFor M4, we found a combined HB mass of 0 . 655 ± 0 . 007 M ⊙ . If we split this sample into their constituent RR Lyrae and rHB parts, we find average masses of M rHB = 0 . 657 ± 0 . 034 M ⊙ and M RRc = 0 . 651 ± 0 . 028 M ⊙ , respectively. A small non-zero di ff erence is in agreement with the evolutionary predictions that stellar (envelope) masses along the HB decrease towards the blue (Salaris & Cassisi 2006). Therefore, we can conclude that our two seismic mass estimates, the scaling relations for rHB stars, and the linear RRc model grid of Netzel & Smolec (2022), are in agreement. Thus, we may conclude that fitting the f 61 modes o ff ers a reliable way to estimate RR Lyrae star masses, even if it is only available for overtone stars.', '4.4. Comparison with mass relations': 'Relations to calculate masses for RR Lyrae, or more broadly for HB stars from observational constraints, have been given by multiple authors, derived from pulsation and evolutionary  \nFig. 7. The distribution of masses with increasing luminosity among the evolutionary groups. Lines indicate the average masses and uncertainties.  \n<!-- image -->  \nmodels. Here we compare seven prescriptions with our seismic masses. The first widely adopted relation on dependence of the pulsation period on the physical parameters ( M, L, T e ff ) was published by van Albada & Baker (1971). This was later refined and updated by multiple authors, also incorporating a metallicity term (Bono et al. 1997; Jurcsik 1998; Marconi et al. 2003; Di Criscienzo et al. 2004; Marconi et al. 2015). A more general relation between mass, color and metallicity was given for HB stars by Gratton et al. (2010).  \nThe accuracy of these prescriptions, however, strongly depend on the modeling choices and uncertainties intrinsic to parameter choices, which are not often explored. Furthermore, many prescriptions are based on T e ff and L values, which have to be converted from observations and have many uncertainties and potential parameter degeneracies themselves. Converting apparent brightness to theoretical luminosity, for example, involves uncertainties in the distance of the object, in the amount of interstellar absorption between the observer and the object, and even in the calculation of the bolometric correction. Similarly, converting observed colors to T e ff involves uncertainties coming from interstellar absorption and the choice of extinction law parameter, as well as from the accuracy of the calibration of the colorT e ff conversion scales themselves.  \nWith these limitations in mind, we calculated the RR Lyrae masses using these relations and compared them to our seismic results in two ways. First we used the classical constraints, L  \nFig. 8. The seismic HRD of M4. Here we show stars in the log L -T e ff plane for which masses are available. Colors indicate stellar mass. The gap in the HB between RRc and rHB stars is where RRab stars reside, for which no mass estimate is available. The solid line is a MIST isochrone with an age of 12.2 Gyr and an [Fe / H] index of -0.9.  \n<!-- image -->  \neff  \nand T e ff values, computed from the Gaia photometry, which also served as initial values for our seismic model fits. These results are plotted in the left panel of Fig. 9. As the figure shows, the values are generally in the right range, between 0.55 and 0.8 M ⊙ , however, di ff erences from the seismic values can reach 0.1 M ⊙ , and uncertainty ranges can be even wider than that, ranging from ± 0 . 03 to ± 0 . 12 M ⊙ .  \nWe then recalculated the masses using the L and T e ff values obtained from the seismic model fits listed in Table 2, and plotted them in the right panel of Fig. 9. In contrast with the results above, these correlate very well with our seismic masses. The matches are not perfect, however, as clear systematic di ff erences are visible. Most relations give results very similar to the original one published by van Albada & Baker (1971), underestimating the masses by an average di ff erence of -0.020 to -0.028 M ⊙ , relative to the seismic masses. The only relation that overestimates the masses is that of Marconi et al. (2015), with an average difference of + 0.032 M ⊙ . These results highlight the true di ff erence between various models and their parameter choices, rather than the uncertainties coming from the observations. The main source of observational uncertainty left in these fits is in the frequencies of the modes we fit.  \nGratton et al. (2010) determined a di ff erent mass relation based on the B -V (or V -I ) color indices and [Fe / H] indices  \n<!-- image -->  \nFig. 9. Comparison of the RRc masses with results from various mass relations based on physical parameters. On the left: masses calculated from classical observational constraints obtained for L and T e ff . Right: masses calculated from the L and T e ff outputs produced by our seismic results listed in Table 2. Seismic masses are shifted by small amounts in both plots to make individual error bars visible. The dashed line is the identity line in both plots.  \n<!-- image -->  \nFig. 10. Comparison of the seismic HB masses, both for the RRc and rHB stars, with the mass relation published by Gratton et al. (2010). Empty symbols indicate marginal detections. Large grey symbols indicate the averages for both groups. The dashed line is the identity line.  \n<!-- image -->  \nof HB stars. We calculated the masses of both the RRc and the rHB stars using our dereddened B -V colors. The results are displayed in Fig. 10. This relation results in a larger mass di ff erence (0.037 M ⊙ ) between the two groups than what we found from the seismic data. We suspect that this di ff erence is coming from the fact that the relation is based on zero-age HB stellar models, whereas the stars in M4 are not necessarily on the ZAHB, but rather at evolutionary stages dictated by their common ages and thus common masses.', '5. Conclusions': "Determining the masses of RR Lyrae variables is a notoriously di ffi cult task. While relations based on physical parameters exist, and pulsational masses can, in principle, provide masses for  \ndouble-mode stars, these are strongly model-dependent methods that could not be verified by independent mass estimates. Asteroseismology o ff ers a new way to determine stellar masses (Aerts 2021). While seismic masses still depend on stellar envelope models and on the validity of scaling relations, they can be tested against other observational techniques, e.g., dynamical masses in eclipsing binaries.  \nGlobular clusters, such as M4, provide an opportunity to probe the late stages of low-mass stellar evolution. Building on the unique capabilities of the Kepler space telescope, we calculated new, more accurate masses for the oscillating giant stars using asteroseismic scaling relations and ν max values determined with the new pyMON code (Howell et al., in prep.). We then analysed the overtone RR Lyrae stars in the cluster and determined their masses using peak-bagging and fitting the frequencies with the seismic model grid of Netzel & Smolec (2022). With these, we were able to relate RR Lyrae seismic masses to seismic masses of red HB stars, and found that they match each other closely. Furthermore, we estimate a He content of Y = 0 . 266 ± 0 . 008 for the cluster from our models.  \nWhile our result is not yet a direct measurement of RR Lyrae masses, as it relies on the assumption that RR Lyrae and rHB stars in the cluster have similar masses, we were able to test model-dependent seismic (or pulsational) masses against independent mass estimates for the first time. We find that the average RRc and rHB masses agree within the errors, with the RRc stars having slightly lower masses, as expected from the stucture of the HB. We compared the modeled physical parameters of the RRc stars to various relations between M, L, T e ff and the pulsation periods, and found that while the generally agree, systematical di ff erences do exist. We also show, however, that the usability of these relations is rather limited, as L and T e ff values have to be determined very accurately, otherwise the calculated masses will be very uncertain.  \nThis study not only tests RR Lyrae masses for the first time, but it also lends further support to the hypothesis of the f 61 modes being high-order ℓ modes, as proposed by Dziembowski (2016). Therefore, it is now possible to estimate masses for field RRc and RRd stars as well, where enough modes can be detected from high-precision photometry (Netzel & Smolec 2022; Netzel et al. 2023). Furthermore, RRc stars now o ff er a way to measure masses on the HB in globular clusters, even where rHB stars are too faint for asteroseismology. RR Lyrae stars thus open up a new way to study mass loss in more clusters, either with Kepler , or with future instruments, such as the Roman Space Telescope  \n(Howell et al. 2024; Molnár et al. 2023a) or the HAYDN telescope project, which is designed specifically to observe dense stellar fields (Miglio et al. 2021).  \nAcknowledgements. This research was supported by the 'SeismoLab' KKP137523 Élvonal grant of the Hungarian Research, Development and Innovation O ffi ce (NKFIH). Cs.K. was supported by the ÚNKP-23-3, New National Excellence Program of the Ministry of Culture and Innovation from the source of the National Research, Development and Innovation Fund. This research has received support from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant Agreement No. 947660). 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We calculated the Fourier-spectra of both the amplitude and phase variations. The former is also shown in Fig. B.1; the spectrum of the phase variation looks very similar. Both show to separate frequency peak, from which we calculated two modulation periods: Pm 1 = 11 . 39 ± 0 . 05 d and Pm 2 = 19 . 63 ± 0 . 25 d.', 'Appendix B: Data tables': 'In Table B.1, we present the reduced K2 light curves for the five RRc stars analysed in this work. The full table will be available online. In Table B.4, we list the updated seismic and physical parameters for the RGB, AGB and rHB stars. In Tables B.2 and B.3 we list the mass estimates we show in Figures 9 and 10.  \nTable B.1. K2SC- and trend-corrected photometric data for the five RRc stars. The full table is available in machine-readable format online.Table B.2. Mass values for RR Lyrae and rHB stars based on the mass relation of Gratton et al. (2010), compared to our seismic masses.  \nTable B.3. Mass values calculated from the various mass relations, using either the classical observational constraints or the physical parameters from our seismic fits as constraints.Table B.4. The updated global seismic properties and calculated physical param-  \neters for the giants in M4.  \nTable B.4. continued.'}