Integrated Mass Loss of Evolved Stars in M4 using Asteroseismology

We established an initial M4 sample from the Gaia DR2 catalog (Gaia Collaboration et al., 2018; Lindegren et al., 2018) with Gaia magnitude $G_{\text{mag}}<15$ mag (to ensure targets are sufficiently bright to have detectable oscillations) and within the field of view of the M4 K2 superstamp (comprised of 16 target pixel files). Targets were classified into various evolutionary phases: RGB, RHB, blue HB, EAGB, and blue stragglers, based on their positions in the Gaia DR2 photometric CMD (Fig. 1a). We then assigned our own star identifications using the naming convention as follows: GC name (M4) + evolutionary phase (eg RGB, RHB or AGB) + star number (starting at 1).

Cluster membership was derived from Gaia DR2 proper motions (Fig. 1b), where stars were included in our sample if they were within a circle of radius $1.5~{}\mathrm{mas/yr}$ and centred at $(-12.5,-19)~{}\mathrm{mas/yr}$, in the proper motion right ascension and declination direction respectively. Our membership sample was cross checked with the Vasiliev & Baumgardt (2021) GC catalog, which calculated membership probabilities based on Gaia EDR3 parallaxes and proper motions. We found that all our stars, except M4AGB01 and M4RGB169, had a membership probability metric of $\geq 0.99$. The star M4AGB01 was contained in the Vasiliev & Baumgardt (2021) catalog, but they rejected this star because it had a much larger parallax than the M4 average. Upon inspection, M4AGB01 had a large Gaia EDR3 RUWE goodness-of-fit statistic of $4.03$, which could be indicative of unreliable astrometry or binarity. However, because the star M4AGB01 has been classified as a member using radial velocities and chemical abundances in MacLean et al. (2018, hereby MacLean18), we included it in our study. The star M4RGB169 was not contained within the Vasiliev & Baumgardt (2021) catalog. However, it was also included in the MacLean18 spectroscopic study and classified as a member. We also found three member stars not contained in the M4 superstamp, which had their own individual K2 target pixel files (EPIC203390179/M4AGB62, EPIC203362169/M4RGB225, & EPIC203394211/M4RGB408). These stars have also been included in our sample.

Only about half of M4 was observed in Campaign 2 of the K2 mission (Fig. 1c), with $3856$ 30-minute long cadences over the $\sim 80$ day observing period. Some cadences were missing data due to systematic events, such as thruster firing to roll the telescope. Each target pixel file consists of $50\times 50$ pixels. Due to the large pixel size, there is evidence of blending between stars, especially close to the centre of the cluster. We used a combination of automated aperture masks and custom individual masks to ensure there was no photometric contamination from neighbouring stars. The automated aperture masks were generated by including pixels that were within a threshold level of the brightest pixel over time (assumed to be the centre of the star). After inspection, these aperture masks were then individually adjusted to find the optimal mask for the star, based on the combined differential photometric precision metric (for further details see Christiansen et al., 2012).

We developed a pipeline using the Python package, LightKurve, (Lightkurve Collaboration et al., 2018) to detrend the time series in two stages; stage 1 used a pixel-level decorrelator (Deming et al., 2015) on the target pixel file itself and stage 2 used a self-flat fielding corrector (SFF) on the light curve (Vanderburg & Johnson, 2014). The pixel-level decorrelation method was developed for the EVEREST pipeline (Luger et al., 2016), and it removes systematic errors due to instrumental drifts from individual pixels. The SFF detrender then removed long-term trends to produce a finalised flattened light curve. For this detrender, we used a timescale of 9 days, which is significantly larger than the typical 1.5 days that is used in the literature (e.g. Stello et al., 2016; Wallace et al., 2019). In our tests, we found that at smaller timescales the low frequency end of the spectrum was attenuated. This will affect the background fitting to the spectrum, and also the measurement of the global seismic parameters (see Section 3), for evolved RGB and EAGB stars with low $\nu_{\text{max}}$ ($\nu_{\text{max}}\lesssim 5~{}\mu\mathrm{Hz}$). By increasing the timescale in the SFF detrender, we retained the power at lower frequencies, while not affecting the power for the rest of the spectrum. To illustrate this, Figure 2 shows the power spectra for two stars calculated with different timescales in the SFF detrender; M4RGB225 – an evolved RGB star with a $\nu_{\text{max}}\approx 3~{}\mu\mathrm{Hz}$ and M4RGB104 – a star with a $\nu_{\text{max}}\approx 35~{}\mu\mathrm{Hz}$ similar to the 6-hr roll noise inherent in the K2 photometry. For the star M4RGB225, a power excess can only be clearly detected when using the longer timescales. Because the SFF detrender was developed to remove systematics such as the 6-hr roll noise, we checked that any noise present was being efficiently removed at the longer timescale. Fig. 2b shows that varying the timescales for the star M4RGB104 does not alter the power excess of the solar-like oscillation.

Within our pipeline, we restricted the time series to 36 days (cadences 95717-97490) for stars with expected²²2The expected $\nu_{\text{max}}$ was determined by making a first pass of our data, and deriving a relation between Gaia DR2 $G_{\text{mag}}$ and $\nu_{\text{max}}$ ($\nu_{\text{max}}\approx 1.78\times 10^{-15}G_{\text{mag}}^{14.6}$) for stars showing clear oscillations. $\nu_{\text{max}}>20~{}\mu\mathrm{Hz}$. This step excluded noisy cadences, for example, the ‘unflagged pointing excursion’ at the start of the campaign where there is an extreme flux drop. We did not do this for the stars with predicted $\nu_{\text{max}}\leq 20~{}\mu\mathrm{Hz}$, because they require higher frequency resolution to properly resolve the oscillation excess power. We instead split the time series of these lower $\nu_{\text{max}}$ stars into two, corresponding to the change in the telescope’s roll direction, which occurred about halfway through the campaign. Each section was detrended individually and recombined at the end. Lomb-Scargle power spectral density periodograms (Lomb, 1976; Scargle, 1982) were then calculated for each star. To determine the efficacy of the data detrending, we used a white noise metric (calculated as the median power between $260-280~{}\mu\mathrm{Hz}$; Stello et al. 2015) and obtained a range of values between $\sim 50-1000~{}\mathrm{ppm}^{2}/\mu\mathrm{Hz}$. This is similar to what was achieved in the M16 study, but larger than K2 field stars of similar magnitudes (Stello et al., 2015). The larger noise levels of the M4 data could be because of the light contamination from neighbouring stars. This will have a direct impact on the global asteroseismology analysis and the final mass uncertainties.

Power spectra were manually searched for potential solar-like oscillations. Criteria based on the power spectra were introduced to determine which stars had clear solar-like oscillations or not. A star was removed from the initial membership sample if:

1. 1.

   no power excess was observed in the spectrum. For example, all blue HB and blue straggler stars were removed from the sample, due to the inability to detect solar-like oscillations in their power spectra. This included bright red giants ($G_{\text{mag}}<10$ mag) with expected $\nu_{\text{max}}\leq 2~{}\mu\mathrm{Hz}$. This $\nu_{\text{max}}$ is too low to be resolved properly with the short observing period of this K2 campaign.

2. 2.

   it was a horizontal branch star that was classified as a RR Lyrae by Stetson et al. (2014), or had an observed RR Lyrae-like spectrum.

3. 3.

   it had an ambiguous detection of a solar-like oscillation, typically due to a low signal-to-noise ratio (SNR) $\leq 5$. These targets were classified as tentative detections instead (see Table 1), and will not be used in our analysis. Note that the majority of the tentative detections are faint RGB stars. This is expected since the amplitude of the power excess decreases with increasing magnitude on the RGB, resulting in lower SNR.

After applying these criteria we retained a final sample of 75 stars with solar-like oscillations: 59 RGB, 11 RHB, and 5 EAGB. Our final sample is indicated by the coloured points in Figure 1.

M4 is located behind the Sco-Oph cloud complex and is affected by substantial reddening. Due to its spatially large size on the sky, it also suffers from significant differential reddening. Studies have suggested that the extinction of light from M4 is due to abnormal dust types (e.g. Ivans et al., 1999; Dixon & Longmore, 1993), hence the standard extinction ratio of $R_{V}=3.1$ is inappropriate for this cluster. Using optical and near-infrared photometry to investigate the dust type, Hendricks et al. (2012, hereby H12) determined an extinction ratio of $R_{\text{V}}=3.62\pm 0.07$, where $R_{V}=A_{V}/E(B-V)$ with $A_{V}$ being the total extinction in the visual band and $E(B-V)$ being the reddening in the $(B-V)$ colour.

To estimate the line of sight extinction to individual stars, we used the H12 differential dust map for M4. This map can be used to find stars that deviate from the mean extinction of $E(B-V)=0.37$ mag with a star-to-star scatter of $\sigma=\pm 0.02$ mag. The map contains the centre of the cluster, but doesn’t cover our outskirt stars ($\sim$15% of our sample was positioned outside of the H12 map). For the stars that were within the dust map, we calculated an average extinction and star-to-star scatter of $E(B-V)=0.36$ mag ($\sigma=\pm 0.02$ mag), slightly lower than the reported average extinction by H12. This is because of the bias in the position of our sample above the cluster centre. The stars not located in the H12 map were assumed to have our average extinction value.

2MASS $JHK$ magnitudes (Skrutskie et al., 2006) and $BV$ photometry (Momany et al., 2002; Mochejska et al., 2002) were collected for each star and used to calculate the photometric temperatures. Campbell et al. (2017) showed that calculating the temperature from the $(V-K)$ colour was the most reliable (and had the smallest formal uncertainties) for evolved stars. The $E(B-V)$ values were converted to $E(V-K)$ using Equation 8 in Fitzpatrick & Massa (2007). Using the dereddened colours, $(V-K)_{0}=(V-K)-E(V-K)$, temperatures were found using the colour-$T_{\text{eff}}$ relation for giant stars from González Hernández & Bonifacio (2009), with a metallicity of [Fe/H] $=-1.1$ (Harris, 1996; Marino et al., 2008; MacLean et al., 2018). The largest source of uncertainty in the photometric $T_{\text{eff}}$ were the reddening corrections.

Ideally, we need an extinction-independent method of calculating $T_{\text{eff}}$, such as spectroscopy. The MacLean18 spectroscopic sample included an overlap of 39 RGB and 3 EAGB stars. An offset of $81~{}\mathrm{K}$ (Fig. 6) was found between the spectroscopic and photometric $T_{\text{eff}}$, with the spectroscopic temperatures being typically hotter. Since photometric temperatures are dependent on reddening corrections, and often show systematic differences between various colour-$T_{\text{eff}}$ relations (e.g. Campbell et al. 2017), we offset the photometric temperatures by $+81~{}\mathrm{K}$ for stars without a spectroscopic temperature estimate. An uncertainty of $108~{}\mathrm{K}$ was adopted for the scaled temperatures, derived from the addition in quadrature of the average spectroscopic uncertainty and the scatter of the difference between the two temperature methods. Our final $T_{\text{eff}}$ values are provided in Table LABEL:tab:final_results.

Luminosities were calculated using the bolometric correction ($BC$) for giant stars (Eq. 18 from Alonso et al., 1999) and the dereddened visual band magnitude $V_{0}$ using:

For the bolometric magnitude of the Sun, the value of $M_{\text{bol},\odot}=4.74\pm 0.13$ (Cox, 2000; Torres, 2010; Mamajek et al., 2015a) was adopted. For the true distance modulus $(m-M)_{0}$ for M4, we used the average literature value calculated in Baumgardt & Vasiliev (2021) of $11.34\pm 0.02$. Luminosities can be found in Table LABEL:tab:final_results.

We calculated the seismic stellar radius for our sample using:

where $\nu_{\mathrm{max,\odot}}=3090\pm 30~{}\mu\mathrm{Hz}$, $\Delta\nu_{\odot}=135.1\pm 0.1~{}\mu\mathrm{Hz}$ (Huber et al., 2011, scaled to the SYD pipeline) and $T_{\text{eff},\odot}=5772\pm 0.8~{}\mathrm{K}$ (Mamajek et al., 2015b). These seismic values were compared to independent radius estimates calculated using the Stefan-Boltzmann law (Fig. 7). For the ratio of the radius estimates, we found a median residual of $2\%$ with a $1\sigma$ scatter of $\pm 20\%$. Seven stars in our sample did not agree within the 2$\sigma$ uncertainties, which could indicate errors in the measurements of the seismic parameters, such as $\Delta\nu$.

By testing the radii, we can also deduce whether systematics could arise within the asteroseismic scaling relations for certain stars. Zinn et al. (in press) did a comparison study between the seismic radius and radius computed from Gaia parallaxes. They reported that the two types of radius estimates for $R<50~{}\mathrm{R}\textsubscript{$\odot$}$³³3This is an upward change from the $30~{}\mathrm{R}\textsubscript{$\odot$}$ limit in Zinn et al. (2019). agreed within $2\%\pm 2\%$(sys), and stars with a radius $R>50~{}\mathrm{R}\textsubscript{$\odot$}$ may have systematics in the measurement of the asteroseismic parameters. All of our stars are below this radius limit. The seismic radii estimates are provided in Table LABEL:tab:final_results.

Using our estimates for the parameters $\nu_{\text{max}}$, $\Delta\nu$, luminosity, and $T_{\text{eff}}$ (Table LABEL:tab:final_results), masses of individual stars were determined using the seismic mass equations 1-4.

As mentioned in Sec. 1, there is some evidence that mass loss begins at around the RGB bump luminosity (Bharat Kumar et al., 2015; Mullan & MacDonald, 2003, 2019a). Since we are interested in measuring the total integrated mass loss between the initial/main-sequence turnoff mass and the HB, including stars above the bump may bias our result. To avoid this problem, we separate the RGB sample into the lower RGB (LRGB; at the luminosity bump and below) and the upper RGB (URGB; above the luminosity bump), where the magnitude of the bump is $V=13.6$ mag ($G\sim 13$ mag; Song et al. 2018).

It is known that there are deviations from the seismic scaling relation $\Delta\nu\propto\overline{\rho}^{1/2}$, which can introduce systematics into the calculated mass (White et al., 2011; Miglio et al., 2012, 2013). These departures are likely due to differing stellar structures, and it is common to apply corrections to $\Delta\nu$ obtained from models. We implemented $\Delta\nu$ corrections derived from asfgrid (Sharma et al., 2016). We note that 44% of our stellar masses were below the lower mass limit of this grid of models, which will result in an under-estimation of the $\Delta\nu$ corrections for these stars. Also, the asfgrid did not have specific $\Delta\nu$ corrections for EAGB stars. We instead treated them as RGB stars to determine their corrections, because they have similar structures.

We determined average masses for the LRGB, URGB, RHB, and EAGB evolutionary phases using both corrected and uncorrected $\Delta\nu$ mass estimates separately (Table 3 and Figure 8). Stars with marginal detections (see Sec. 3 and Table LABEL:tab:final_results) have been included in these average calculations. We note that without the MD stars our RHB and EAGB samples would be smaller.

Uncertainties for the average masses for each mass relation were taken as the standard error on the mean. It can be see from Table 3 that the mean masses calculated with Eq. 3 generally have the smallest uncertainties. The Eq. 3 average mass also closely matches the expected masses for the LRGB and RHB from detailed M4 models (MacLean18; McDonald & Zijlstra 2015; also see Sec. 6.1). In addition, M16 found a final RGB mass of $0.84~{}\mathrm{M}\textsubscript{$\odot$}$, in agreement with our Eq. 3 mean mass of $0.83\pm 0.01~{}\mathrm{M}\textsubscript{$\odot$}$. This suggests that Eq. 3 may be the most accurate as well as being the most precise.

We found that after $\Delta\nu$ corrections, the masses from Eqs. 1, 2 and 4 still had larger uncertainties. In Section 3 we highlighted that our $\Delta\nu$ measurements are very unreliable due to the poor SNR of the M4 photometric data. Also, as mentioned above, the $\Delta\nu$ corrections were under-estimated due to asfgrid not covering our mass range for a large fraction of our stars. From Figure 8 it is apparent that the $\Delta\nu$ corrections are not sufficient to improve the mass estimates from the $\Delta\nu$-based relations. For these reasons, we only consider our Eq. 3 mass estimates as reliable, because they are independent of $\Delta\nu$. Hence, we will only use Eq. 3 mass estimates for the following analysis and discussion.

Figure 9 shows the individual masses (Eq. 3) for our entire sample. Six stars (indicated by star symbols) were identified as obvious mass outliers (approximately $2\sigma$ from the average mass in each evolutionary phase): two over-massive RGB stars, one over-massive EAGB star, and three under-massive RHB stars. We discuss the possible causes for the outliers in Sections 5.3 and 6.4. The outliers were removed from the mean mass calculation, to avoid the averages being biased. Our final mean mass estimates were $0.826\pm 0.008~{}\mathrm{M}\textsubscript{$\odot$}$ (LRGB), $0.658\pm 0.006~{}\mathrm{M}\textsubscript{$\odot$}$ (RHB), and $0.54\pm 0.01~{}\mathrm{M}\textsubscript{$\odot$}$ (EAGB). The final mean mass estimates are provided in Table 3 (denoted by the ‘without outliers’ subscript) and shown in Figure 8. We identified that the EAGB mean mass is significantly lower than predicted model estimates (Fig. 8c), which will be discussed further in Section 6.3.

We compared our mean mass estimates to the Eq. 3 mean mass results of the M16 study. Their RGB mean mass of $0.84\pm 0.01~{}\mathrm{M}\textsubscript{$\odot$}$ is consistent with ours within the 2$\sigma$ uncertainties. However, the M16 RHB mass of $0.61\pm 0.08~{}\mathrm{M}\textsubscript{$\odot$}$ is outside the 2$\sigma$ uncertainties for our RHB mean mass. This could be because the RHB mass estimate from M16 was determined using only a single star, compared to our sample of 11 stars. In Sec. 6.1, we compare our masses to other studies that used independent mass determination methods.

Total integrated mass loss on the RGB was determined by taking the difference between the average mass of the LRGB and RHB phases of evolution, denoted by $\Delta\overline{M}$. The standard error on the means for the LRGB and RHB were added in quadrature to determine the associated uncertainties for the mass differences. These mass difference uncertainties were of the same order of magnitude as the Miglio et al. (2012) open cluster mass loss study, which used a different error propagation method. We found an integrated mass loss on the RGB of $\Delta\overline{M}=0.17\pm 0.01~{}\mathrm{M}\textsubscript{$\odot$}$. This value is similar to the expected total integrated RGB mass loss $\sim 0.2~{}\mathrm{M}\textsubscript{$\odot$}$ for GCs (e.g. McDonald et al. 2011b; Salaris et al. 2016; also see Sec. 6.1).

Upon inspection of the 18 URGB stars, we saw evidence of a decline in masses (Table 3 and Fig. 9), which supports the theory that mass loss starts becoming significant at the RGB bump. We also saw that a substantial amount of matter is lost straight after the bump. This is in contradiction to the Reimers’ and Schröder & Cuntz schemes, where the mass loss rates slowly increase on the RGB until the majority of the mass is lost at the RGB tip. We further explore this observation in detail in an upcoming paper.

In Fig. 9 the mass outlier stars are displayed as large star symbols. Here we consider if systematic uncertainties can explain the deviant masses. In Sec. 6.4 we discuss the possible interpretations if these masses are correct.

Our results for the average LRGB mass, $0.826\pm 0.008~{}\mathrm{M}\textsubscript{$\odot$}$, and average RHB mass, $0.658\pm 0.006~{}\mathrm{M}\textsubscript{$\odot$}$ (Table 3), can be used as constraints for mass loss in stellar models. Here we make the assumption that the initial masses of the LRGB and RHB stars are equal, because the mass difference ($\sim 10^{-3}~{}\mathrm{M}\textsubscript{$\odot$}$) is smaller than our mass uncertainties (see Sec. 1). In particular, the masses can be used to calibrate mass-loss scaling parameters, such as $\eta_{\text{R}}$ in the most commonly used Reimers (1975) formula. To explore this we calculated a series of models using the MESA stellar code (Paxton et al. 2011; Paxton et al. 2019; version 12778). The models all have identical initial parameters and input physics except that $\eta_{\text{R}}$ is varied. The initial mass of the models was taken as our measured average mass for the M4 LRGB stars, and the initial metallicity as $Z=0.0015$, based on the measured [Fe/H]$=-1.15$ (MacLean et al. 2018) plus an alpha-enhancement. The initial helium abundance was taken as Y $=0.245$ to match that for the first subpopulation of M4, which populates the red end of the HB (Marino et al. 2011; MacLean et al. 2018). We used the standard MESA nuclear network (‘basic.net’), standard equation of state (see Paxton et al. 2019 for details), and mixing length parameter $\alpha=2.0$. Convective boundary locations were based on the Schwarzschild criterion, extended with exponential overshoot (Herwig et al. 1997; $f_{os}=0.016$) at the bottom of the RGB envelope and at the top of the convective core during core helium burning.

We found a model value of $\eta_{\text{R}}=0.39$ closely matched our measured integrated RGB mass loss, giving an RHB mass of 0.657 M_⊙, compared to our measured average RHB mass of 0.658 M_⊙. The mass loss rate along the HB itself is unknown, however a commonly used assumption is to extend the Reimers mass loss formalism to the HB. At the lower luminosity of the HB (compared to the upper RGB), there is expected to be little mass loss. For our $\eta_{\text{R}}=0.39$ model we find a total mass loss of $0.01~{}\mathrm{M}\textsubscript{$\odot$}$ using this assumption. This is smaller than our mass uncertainties on individual HB stars, and of a similar magnitude to the $1\sigma$ scatter of $0.02~{}\mathrm{M}\textsubscript{$\odot$}$ among our HB masses.

To illustrate the sensitivity of modelled HB mass to $\eta_{\text{R}}$, we show the RHB masses for three values of $\eta_{\text{R}}=0.30,0.40,0.50$ in Figure 8b. Taking Eq. 3 as the most reliable (as discussed earlier) it can be seen that $\eta_{\text{R}}=0.30$ gives an HB mass that is too high (0.70 M_⊙) compared to our observed average mass of 0.658 $\mathrm{M}\textsubscript{$\odot$}$, and $\eta_{\text{R}}=0.50$ gives mass that is to low (0.60 M_⊙). However, the observed average RHB mass is reasonably well matched by $\eta_{\text{R}}=0.40$, which has an RHB mass of 0.65 M_⊙ As mentioned above, our $\eta_{\text{R}}=0.39$ model gives an almost exact match, however given the inherent uncertainties in stellar models we believe it unwise to place too much credence at this level of precision.

Previous studies have estimated $\eta_{\text{R}}$ for reproducing the observed characteristics of M4 RHB stars using independent methods. By comparing a grid of models with observed CMDs, McDonald & Zijlstra (2015) found $\eta_{\text{R}}=0.40^{+0.05}_{-0.06}$. This matches with our optimal value of 0.39. MacLean et al. (2018) used a similar method and found $\eta_{\text{R}}=0.44$, consistent (within uncertainties) with the value of McDonald & Zijlstra (2015), and slightly higher than our optimal value.

Although the $\eta_{\text{R}}$ values all agree, this could be a coincidence, since the value is degenerate with different initial masses and HB masses. However this doesn’t appear to be the case with M4, since previous studies also agree on the initial mass and HB mass: McDonald & Zijlstra (2015) found an initial mass of $0.84^{+0.02}_{-0.02}$, consistent with our LRGB mass. Likewise, MacLean et al. (2018) found an initial mass of 0.827 M_⊙, almost identical to our value of 0.826 M_⊙. For their average HB mass McDonald & Zijlstra (2015) report $0.66\pm 0.01~{}\mathrm{M}\textsubscript{$\odot$}$, also in agreement with our value of $0.658\pm 0.006~{}\mathrm{M}\textsubscript{$\odot$}$. MacLean et al. (2018) did not report a HB mass.

Origlia et al. (2014) determined an empirical mass-loss relation for Pop. II giants based on mid-IR photometry of RGB stars. Their Equation 4, which is dependent only on [Fe/H], enables us to calculate the total expected RGB mass loss for M4, giving $0.15\pm 0.03~{}\mathrm{M}\textsubscript{$\odot$}$. This matches with our integrated mass-loss measurement of $\Delta\overline{M}=0.17\pm 0.01~{}\mathrm{M}\textsubscript{$\odot$}$.

The remarkable agreement of initial mass, HB mass, $\eta_{\text{R}}$, and total RGB mass loss between these four independent studies, and three very different methods – CMDs+models, IR observations, and asteroseismology – is very reassuring. It confirms that the traditional methods are accurate. Asteroseismology does however give higher precision – even with the low-quality K2 data used in the current study.

Turning to the next phase of stellar evolution, in Fig. 8c we include the model EAGB masses for the same set of model $\eta_{\text{R}}$ values. In this case, none of the models can reproduce the average EAGB mass. This surprising mismatch could have a number of different explanations, which we discuss in Sec. 6.3.

It is well established that GCs contain multiple populations defined by their light elemental abundances (e.g. C, N, O, Na, O, and He; Sneden, 1999; Gratton et al., 2012). Among these abundance variations, helium is a key element for stellar evolution: a star that has a higher He abundance will evolve faster (MacLean et al., 2018; Jang et al., 2019). Because GC stellar populations are essentially co-eval, this means that the multiple populations will have different current masses – the more ‘massive’ He-poor stars and the lower-mass He-rich stars.

The two populations in M4 have been observed and modelled by multiple studies (e.g. Lardo et al., 2017; Marino et al., 2017a, 2019; MacLean et al., 2018; Tailo et al., 2019; Dondoglio et al., 2022). In MacLean18’s two population models, the first sub-population (SP1; lower light metal abundances and more massive) and the second sub-population (SP2; higher light metal abundances and less massive) had an initial mass difference of $0.04~{}\mathrm{M}\textsubscript{$\odot$}$. This mass difference is of a similar magnitude to our individual mass uncertainties (Table LABEL:tab:final_results), and hence could be too small to detect in the RGB sample.

To test this, we plotted the mass distributions for the LRGB, RHB, and EAGB (without outliers) in kernel density estimation (KDE) histograms (Fig. 12a). We used the random uncertainties of the masses for the Gaussian widths. We find a broad mass distribution on the LRGB, larger than expected for a single population. In Fig. 12a we over-plot the MacLean18 initial masses for their sub-population models. Although our mass distribution is not clearly bi-modal, the MacLean18 SP1 and SP2 initial masses fall at either end of the broad peak. This suggests that this distribution could be due to the two sub-populations, but is smoothed by our (relatively) large uncertainties.

Our RHB mass distribution is consistent with the presence of only one sub-population. This concurs with spectroscopic studies that generally show that the RHB samples in GCs only contain first sub-population stars, and the blue HBs only contain second population (or higher) stars. For M4, Marino et al. (2011) showed that the RHB is populated only by Na-poor (SP1) stars. In Fig. 12 we also show the observed SP1 and predicted SP2 HB sub-population masses, where SP1 experiences our observed integrated RGB mass loss of $0.17\pm 0.01~{}\mathrm{M}\textsubscript{$\odot$}$ (Sec. 5.2), and SP2 has an adopted mass loss that is $0.027~{}\mathrm{M}\textsubscript{$\odot$}$ larger than SP1 (as calculated by Tailo et al. 2019). As can be seen, there is no peak in our distributions near the predicted HB SP2 mass, as expected.

Interestingly, we do not detect a bi-modality in our EAGB mass distribution. This is in agreement with the observations by MacLean18, who report that M4 lacks SP2 stars at this evolutionary phase (although there is some disagreement in the literature on this, see Marino et al. 2017b).

To extend our investigations into the multiple populations of M4, we require chemical information for sub-population membership. Despite the limitations (low SNR of the K2 data, photometric $T_{\text{eff}}$, and small sample size), our study hints at what could be accomplished with the synergy between asteroseismology and spectroscopy in studying the multiple populations of globular clusters.

From our 5 EAGB stars, we measured a mean mass of $0.54\pm 0.01~{}\mathrm{M}\textsubscript{$\odot$}$, significantly lower than the modelled EAGB mass of $0.64~{}\mathrm{M}\textsubscript{$\odot$}$ for $\eta_{R}=0.40$ (Fig. 8). For our measured RHB mean mass of $0.658\pm 0.006~{}\mathrm{M}\textsubscript{$\odot$}$, the lower mass of the EAGB suggests there is a significant mass loss between the helium burning stage and the EAGB of $\sim 0.12~{}\mathrm{M}\textsubscript{$\odot$}$. Mass loss on the HB in low mass stars has been modelled, but normally in the context of explaining HB morphology, such as the extreme HB (stars hotter than the blue HB; e.g. Yong et al. 2000; Vink & Cassisi 2002). There are no detailed studies that consider the mass loss of cooler RHB stars, and it is typically assumed there is either very little ($\sim 0.01~{}\mathrm{M}\textsubscript{$\odot$}$, see Sec. 6.1), or no mass loss during this phase.

Interestingly, our EAGB mean mass estimate is consistent with the measured white dwarf mass for GCs of $0.5-0.55~{}\mathrm{M}\textsubscript{$\odot$}$ (Kalirai et al., 2009). For low mass stars, the AGB phase of evolution can be separated into two parts; the EAGB and the thermal-pulsing AGB stage, where intermittent helium shell flashes occur. After the thermal-pulsing AGB stage, a star then evolves to become a white dwarf. If our low EAGB masses are correct, it would suggest that these stars do not reach the thermal-pulsing AGB stage, and become white dwarfs earlier in their evolution. These objects are similar to the theorised AGB-manqué stars (Dorman 1995), which are low-mass stars that avoid the AGB phase of evolution. This theory has also been explored by McDonald et al. (2011b), who estimated the integrated RGB and AGB mass losses in the GC $\omega$ Centauri with a combination of empirical calculations and HB models.

The other possibility is that our measured EAGB masses are incorrect. The low masses could be a result of systematic discrepancies in the scaling relations for EAGB stars. Some studies have measured the solar-like oscillations of post-red clump stars that are moving towards the EAGB (e.g. Kallinger et al., 2012; Mosser et al., 2014). The stellar luminosity and effective temperature of AGB stars are similar to the RGB, making it difficult to distinguish between these two phases in CMDs. Consequently, EAGB stars are often grouped with RGB stars in asteroseismic studies (e.g. Elsworth et al., 2017; Pinsonneault et al., 2018), and can be seen as over-densities in seismic Hertzsprung-Russel diagrams. Recently, Dreau et al. (2021) detected differences in the seismic signatures of the helium second-ionisation zones between RGB and EAGB stars, and were therefore able to separate the evolutionary phases of a sample of field stars. However, because these stars are across a wide distribution of masses and metallicities, the scaling relations cannot be robustly tested. The tightness of the CMDs of GCs makes it easier to distinguish between the RGB and EAGB evolutionary phases with photometry. Our study presents the first detections of solar-like oscillations in GC EAGB stars. Further work is needed to study how the thinner convective envelopes of EAGB stars would affect the seismic mass scale.

We analysed the reddening dependency on the photometry and mass estimates in Section 5.3 for the six mass outliers, finding that we require spectroscopic $T_{\text{eff}}$ in order to determine the effect of reddening on the masses. If we assume that the measured masses are correct, then these outlier masses could instead be explained in the context of non-standard evolutionary events.

Blue stragglers (BS) are massive stars located above the main sequence turn-off and are thought to be the result of either stellar mergers of two main sequence stars (Hills & Day, 1976) or mass-transfer events in binary systems (McCrea, 1964). First observed in the GC M3 (Sandage, 1953), BSs have now been found to be common objects in all GCs, numbering between 40-400 in each cluster (Piotto et al., 2002; Davies et al., 2004). These stars have been modelled to follow the evolution of standard $\sim 1.2-1.6~{}\mathrm{M}\textsubscript{$\odot$}$ stars, and follow an evolutionary track that is slightly bluer than the $\sim 0.8~{}\mathrm{M}\textsubscript{$\odot$}$ RGB in a CMD (Tian et al., 2006; Sills et al., 2009). However, evolved BSs are practically indistinguishable from ‘normal’ GC evolved stars in photometry and will only explicitly stand out in mass. Attempts to identify evolved BSs have used asteroseismology to identify candidates in open clusters (Brogaard et al., 2012; Corsaro et al., 2012; Handberg et al., 2017; Brogaard et al., 2021). Due to the slightly hotter temperatures and larger masses of M4RGB36 and M4RGB217, they could be ‘hidden’ evolved BSs in our RGB sample. M4AGB58 could also be an evolved BSs in the post core He burning phase.

The estimated mass of a BS star on the main sequence is $\sim 1.2-1.6~{}\mathrm{M}\textsubscript{$\odot$}$. For our over-massive RGB stars with masses of $\sim 1~{}\mathrm{M}\textsubscript{$\odot$}$, this would imply a mass loss of $0.2-0.6~{}\mathrm{M}\textsubscript{$\odot$}$ before the end of the RGB. The mass of $0.75~{}\mathrm{M}\textsubscript{$\odot$}$ for our M4AGB58 star would then suggest that a further $~{}0.25~{}\mathrm{M}\textsubscript{$\odot$}$ would be lost before the post core He burning phase of BSs. This is a large amount of mass to lose in the short time between the two phases, and suggests that these outliers are not massive enough to be evolved BSs. However, the evolution of BSs is not well understood, and there could be other unknown channels of mass loss.

The under-massive RHB stars have masses of $\sim 0.53~{}\mathrm{M}\textsubscript{$\odot$}$. This is consistent with GC white dwarf masses (Kalirai et al., 2009). Further, an HB star with such a low mass would be found on the extreme end of the blue HB (Dorman, 1995), not on the RHB. Thus we conclude that there must be inaccuracies in one or more of their stellar or seismic parameters for the under-massive stars, and we cannot speculate further without more data.