Isochrone and chemical ages of stars in the old open cluster M67

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Isochrone and chemical ages of stars in the old

open cluster M67

Julia Ahlvind

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Supervisor: Andreas Korn

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_{& Bengt Gustafsson}

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Examiner: Bengt Edvardsson

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Advanced Physics - Project Course 15 ETCS

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_{Department of Physics and Astronomy – Uppsala University}

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The open cluster Messier 67 is known to have chemical composition, metallicity and age („ 4 Gyr) close to the Sun. Therefore, it is advantageous for stellar physical studies and of stellar evolution, in particular for solar like stars within the cluster. This work considers three such stars, the formerly studied solar twin M67-1194 and two more recently suggested solar twins M67-1787 & 2018. Most solar twins show a ratio of volatile to refractory elements that systematically depart from the Sun’s. Our targets do not follow this trend as closely. Their composition is closer to the Sun and they are, therefore, exquisite targets for studies of stellar evolution within the cluster. However, their solar likeness also provides studies regarding the origin and evolution of the Solar system. The stellar ages of the solar twins are established through a chemical clock [Y/Mg] and via stellar isochrones from BaSTI. The latter age assessment of the solar twins is supplemented with the analysis of two subgiant stars M67-1442 & 1844. We approach the isochrone-based method using spectroscopically, astrometrically and photometrically derived parameters. The different ages of the stars and methods thus estimate the age of the cluster itself. The chemical ages of the stars suggest a cluster age of 4.56 ˘0.44 Gyr and the isochrone-based estimates suggests a cluster age within the range 3.30-5.51 Gyr. Our results thus affirm and imply a near solar age of the cluster.

1 Introduction

gfdfdfgThe Sun is said to be a common, but controversially also a rather unique star. The characteristics of a star are sensitive to small changes in its stellar parameters that originate from some fundamental properties such as initial mass and initial chemical composition, Gustafsson (1998). Many stars are born from the same cloud, sharing these fundamental properties. This suggests that there are other stars similar to our Sun, thus, making it common. On the other hand, the Sun is, so far the only known star to host an orbiting planet which sustains life. This argues that the Sun provides a rather distinctive environment. Nevertheless, there are many stars within the Milky Way, which retain orbiting planets which also resemble the Sun in various ways. The so-called solar like stars are main sequence stars with similar spectral type as the Sun–commonly F, G or K type and therefore often referred to as FGK stars. The Sun is a spectral type G-star according to the Harvard classification scheme, Maury & Pickering (1897). Solar twins have an even higher degree of resemblance to the Sun. The main part of his project focus on this group of stars, the solar twins, one well-known candidate and two more recently proposed.

Stars with physical characteristics such as effective temperature (Teff), surface gravity (log g), chemical composition (like metallicity [Fe/H]), mass and age similar to the Sun are defined as solar twins according to Cayrel de Strobel et al. (1981) & Cayrel de Strobel & Bentolila (1989). The margins within which stars are labeled solar twins differ between authors. Baumann et al. (2010) defined solar twins as stars with iron abundance [Fe/H]=0.0 ˘0.1 and masses of M = (1.00 ˘0.04qM@ where the metallicity is defined as [Fe/H] = log10pNF e{NHq ´ log10pNF e{NHq_@. Nissen (2015) narrowed down a HARPS spectral sample of FGK stars to solar twins with the properties Teff “ Teff,@˘ 100 K, ˘0.15 dex in log g and ˘0.10 dex in [Fe/H]. Thus, a solar twin ought to show nearly identical spectrum to the Sun, accounting for margins of small errors

in various parameters such as effective temperature which is usually constrained within Teff˘ 50 K, Pasquini et al. (2008). Commonly used values for the Sun are T_eff@ “ 5777 K, logig_@ “ 4.44 cms´2, and rF e{Hs@=0.00, ¨Onehag et al. (2011). These values are likewise used within this work with a slight modification of Teff@ “ 5772 K, Ayres et al. (2006) & Heiter et al. (2015).

Studies of numerous surveys that are using multiple methods and definitions have resulted in a small sample of solar twin candidates, Mel´endez et al. (2009), Do Nascimento et al. (2009), Mahdi et al. (2016) and Datson et al. (2014). Within this work, we focus on three solar twins in the open cluster Messier 67 (M67) or NGC 2628. The solar twins mentioned are M67-1194, M67-1787 and M67-2018. The former solar twin M67-1194 is the first solar twin discovered within a cluster and has repeatedly been studied and used in the research of solar twins and of the cluster M67. The star was first proposed as a solar twin candidate by Pasquini et al. (2008) and has been further analysis with high resolution spectrum by ¨Onehag et al. (2011), Liu et al. (2016) and Liu et al. (2019).

With this work, we hope to invigorate earlier results regarding the stellar age of the cluster and the solar twins. The study is accompanied by the analysis of two subgiants also within the cluster, 1442 & M67-1844. They are used in the isochrone-based method thus helping to establish a common age of the cluster. Various methods result in different estimates with different uncertainties. Therefore, we expect that the combination of methods will result in a more reliable uncertainty of the cluster age. The two main stellar age estimating methods that will be used are the chemical clock based on the ratio of yttrium (Y) to magnesium (Mg) [Y/Mg] via an age-abundance relation presented by Nissen (2015). The abundances are acquired spectroscopically from equivalent width estimates from FLAMES/UVES spectra with the help of Fan Liu. Secondly we use fits of stellar isochrones based on spectroscopically assessed parameters such as Teff, log

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g as well as photometric and astrometric measurements from Gaia–apparent magnitudes mG in the G-band, parallax p–as well as mV from Yadav et al. (2008). The stellar isochrones considered are MESA Isochrones and Stellar Tracks (MIST), YONESI-YALE (Y2), PAdova & TRieste Stellar Evolution Code (PARSEC) and a Bag of Stellar Tracks and Isochrones (BaSTI), where the latter is the favoured one and what we base our results on.

1.1 Why look for solar twins?

fddssddsdfSolar twins provide both physical and technical benefits in astronomical research. A substantial collection of solar twins would improve our understanding of the commonness or uniqueness of the Sun, thus also broaden our view of the properties and formation of our solar system as well as extrasolar planetary systems. Furthermore, the only known stellar environment that supports life is the one we live in, the one around the Sun. Naturally, solar twins should represent good candidates for harbouring similar planetary systems like ours and thus possibilities for extraterrestrial life. The technical benefit of solar twins is that they are useful in calibrations of e.g. effective temperature scales, Gustafsson (1998). They are also advantageous for error estimations of e.g. cluster properties since their solar-like properties are known in great detail. This method is referred to as differential analysis. It is based on the assumption that systematical errors of the observed solar twin are very similar to those of the Sun. Therefore, they provide good calibrations opportunities and further error estimations for additional observations.

1.2 The cluster M67

Open clusters are essential in the study of stellar evolution. The cluster members are of similar age and chemical composition since they are thought to have been born out of the same molecular cloud. Therefore, we can observe stars of similar properties in various stages of their stellar evolution. This gives us clues to how a single star with distinct properties evolves. Moreover, the group of stars populates a small region in space relative to the distance from us, thus making the distance approximately equal to all members. As a result of these shared properties, open clusters prove beneficial for determining stellar properties and studies of stellar evolution.

M67 is one of the most studied open clusters. As a result of its allegedly low dust obscuring and relative galactic nearness („ 860 pc, Yakut et al. (2009)) it is an ideal laboratory for studies of stellar evolution. Moreover, this benchmark cluster has shown to host many solar like stars and even a handful of solar twins like the ones used within this work. Despite the frequent studies of this cluster, properties such as age

and mass vary considerably between publications. One solar twin firs suggested by Pasquini et al. (2008) and later studied in more detail by ¨Onehag et al. (2011), is the solar twin M67-1194. Onehag et al. (2011)¨ presented a detailed line-by-line differential analysis of M67-1194 and the Sun by deducing fundamental parameters such as Teff, log g, microturbulence ξtand abundances [A/H]. This resulted in an estimated age (4.2 ˘ 1.6 Gyr) close to the solar age. An earlier study of the cluster M67 that included, among other stars, M67-1194 was presented by Yadav et al. (2008). They estimated the cluster age via stellar isochrones (Yonsei-Yale) and acquired an age within the range 3.5-4.8 Gyr. Furthermore, Barnes et al. (2016) used gyrochronology to evaluated the cluster’s age to 4.2 ˘ 0.7 Gyr by using rotation of FGK stars. They found rotation rates very similar to the solar rotation, thus concluding that the ages would be similar to the Sun. Finally, Richer et al. (1998) derived the cluster’s white dwarf cooling age and turnoff age to 4.3 Gyr and 4.0 Gyr respective.

1.3 The solar twins

The properties of M67 is not the only assets in studies of this cluster. It also provides a remote laboratory of which we can observe multiple stages of stellar evolution and draw parallels between the evolution of the solar system. The three solar twins considered in this analysis show unusual chemical properties closer to the Sun than what has been found for other solar twins. Mel´endez et al. (2009) established a trend for these kinds of stars regarding the abundances of volatile elements–elements with low condensation temperatures. The volatiles are seemingly more abundant in the Sun than in solar twins, while the opposite is true for the abundance of refractories–easily dust forming elements–which are under-abundant in the Sun (this trend is shown as a red dashed line in figure 7 in ¨Onehag et al. (2011)).

The recurring abundance difference between the Sun and solar twins have made some believe that the Sun is an uncommon star. Mel´endez et al. (2009) proposed an explanation for the Sun’s unusual abundance by analysing the influence of planetary formations around the star. Mel´endez et al. (2009) emphasised that the inner part of the solar system holds planets and meteorites that are rich in refractories against volatiles (Ciesla 2008) and that their abundance pattern is strikingly consistent with that of the solar pattern relative to the solar twins. This argues that the solar abundance pattern would look similar to that of solar twins if, e.g., the volatiles in the terrestrial planets were added to the Sun. Therefore, suggesting that the formation of these planets gave the Sun its unusual surface composition. However, these calculations are based on the present solar pattern and not on the formation stage or the early Sun. Thus, the calculations do not incorporate

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Figure 1: The FLAMES-UVES normalised (black) spectrum for the Sun (upper panel) M67-1194 (lower panel). Gaussian fit to the absorption line at 4900.11 ˚A for the Sun (green) and M67-1194 (blue) are also plotted. The figure also lists the estimated EWs for various absorption lines based on the Gaussian fits.

4.56 Gyrs evolution of the Sun which could include diffusion processes which alter the abundance pattern. This suggests that most solar twins which show this characteristic pattern, do not host terrestrial planets or as also proposed by Mel´endez et al. (2009), that the solar twins formed such planets but that the gas accretion occurred when the stellar convective zone was still deep. Alternatively, the solar twins formed terrestrial planets and lost their gas discs while the convective zone was still deep, in likeness to the Sun (Alexander et al. 2001) but in a later stage accreted the terrestrial planets and thereby enriching the twins in refractory elements. Yet an alternative explanation is the effect of gas giant formation. The core of our gas giants Jupiter, Saturn, Uranus and Neptune partly consist of heavy metals, which supposedly caused the accretion to start at an earlier stage. Thus removing refractory elements away from the Sun and to the outer region where they eventually became part of the gas giants’ cores. Consequently, stars with gas giants would show a closer chemical abundance pattern to the Sun, Booth & Owen (2020). However, observations of solar analogues and solar twins have given contrasting results, Mel´endez et al. (2009).

As mentioned before, our solar twins are unusual in this perspective. M67-1194 does not show this tendency relative to the Sun in chemical abundance. M67-1787 & M67-2018 show some trend, but less than what previous solar twins have shown. This fact makes these targets even more interesting as solar analogues. As shown by ¨Onehag et al. (2014) in figure 7 of their paper, M67-1194 has a very close abundance pattern to the Sun and does not follow the trend (red dashed line) found by Mel´endez et al. (2009). One gas giant has been found to orbit M67-1194, however, no planets have been discovered for the other solar twins. Thus questioning this relation between the chemical composition of the star and a giant planet companion. The chemical resemblance of the solar twins and the Sun provides good opportunities for a detailed study of the history and evolution of the Sun. It is also questioning former theories regarding the uniqueness of the Sun and opens up alternative theories of its formation and origin.

2 Observations and data process

2.1 Spectroscopic data

The spectroscopic data used within this work is based on spectra taken with the multi-object spectrograph FLAMES-UVES at ESO-VLT UT2. The observations covered, among other stars, one of our solar twins M67-1194. It was performed in the period between the end of January to the beginning of April in 2009 as a part of the 082.D-0726(A), P.I. Gustafsson project. The observation adapted a method such that the same number of photons were approximately attained for all observed targets, by adjusting the exposure time for fainter and brighter stars respectively. In total, 23 individual observations throughout 13 nights were done. The spectrograph (RED580) yielded a resolution of R “ λ{∆λ “ 47, 000p12_{fiber) in the wavelength} range 4800-6700 ˚A and the signal to noise ratio (S/N) per frame was 50 per unbinned pixel. For the Sun we use the FLAMES-UVES solar spectrum independently taken in 2005. The spectra of the remaining solar twins and subgiants were retrieved from the 094.D-0955(A), P.I. Gustafsson project in 2015. Note that the data quality of the later run is somewhat lower than for M67-1194.

Elemental chemical abundances for each star were derived through the equivalent width (EW) method on a line-by-line differential analysis of the solar twins M67-1194, 1787 & 2018 and the subgiants M67-1442 &1844 to the twilight Sun. For the first age estimating method using a chemical clock, the spectral absorption lines of interest were Mg I lines at 5711.090 ˚A & 6318.720 ˚A, and Y II lines at 4854.870 ˚A, 4883.685 ˚

A, 4900.110 ˚A, 5087.420 ˚A & 5200.410 ˚A. The EWs were measured using splot task in IRAF, and used by Fan Liu who derived the elemental chemical abundance

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through a 1D local thermal equilibrium (LTE) analysis with MOOG. Fig. (1) shows the normalised (black) FLAMES/UVES spectrum of the twilight Sun (upper panel) and M67-1194 (lower panel) around the Y-absorption line at 4900.11 ˚A. The figure also shows Gaussian fits of that spectral line (green for the Sun and blue for the M67-1194) which were used to estimate the EWs using a second method. These EWs were acquired simply by integrating the Gaussian curve using Pythons composite trapezoidal rule for the locally normalised spectrum in the limited range (shown in the figure) and subtracted by the estimated continuum («1). The EWs were calculated to identify any oddities and to crosscheck the formerly derived ones which were based on a more delicate approach with splot. The two methods were principally consistent, the later method using Gaussian fits showed for most lines somewhat higher values than the former. This is most probably due to the local normalisation and thus the continuum estimate which might differ between the two methods, see fig. (1). Since the differences in EWs were relatively small and consistent, they were not seen as a problem. The elemental abundances used were solely based on the splot-EW method performed by Fan Liu.

A further spectral comparison for the Sun and M67-1194 was made, by using more recently obtained spectra from the Keck telescope (2016), which were used in previous work by Liu et al. (2016). The EWs from each respective Gaussian fit are again comparable. The only notable difference is the line at 4854.87 ˚A, which is unsurprising since this line resides in the wing of the broad Hβ-line which the local continuum estimate might suffer from (see fig. (7)).

The spectroscopic stellar parameters log g and Teff were obtained by forcing excitation and ionisation balance of iron lines (Fe I and Fe II) based on the line-by-line differential analysis relative to the Sun, derived by Fan Liu. The effective temperatures were determined by requiring independence of excitation potential for the elemental abundances, assuming that the line belongs to the same element and ionisation stage. The Teff values were thus obtained by minimising the slope of the abundances, considering the excitation potentials, see table (2). The surface gravity was determined by forcing the abundance from lines of different ionisation stages to be in accord. The calculations resulted in log g values of: 4.42, 4.52, 4.51, 4.00 and 3.95 cms´2 _{for M67-1194, 1787, 2018, 1442} and 1844 respectively.

The log g values were later also derived using astrometric data and BaSTI evolutionary tracks. These calculations were done since the spectroscopically derived log g values were falling unexpectedly far from solar values. The spectroscopic

distance was derived using the input parameters such as Teff, log g and mass, mV, BCVand AV. mVis taken from Yadav et al. (2008), AV=0.10 and BCV from Alonso et al. (1999). For the effective temperatures, we used the aforementioned Teff. The input log g values are assumed close to the value of the solar twin M67-1194 which is kept at the formerly derived value. The masses are also taken to be close to the solar mass. If the spectroscopic distance diverges more than 1% from the astrometric distance, the surface gravity is altered until concordance. The new log g value is plotted against Teff together with theoretical evolutionary tracks to estimate the stellar mass. The new mass is thereafter used in a second iteration of deriving the log g. These steps are repeated until the agreement with the input stellar mass and from the evolutionary plot. The log g values and the stellar masses are listed in table (2) and seen in figs. (14 & 15). These later log g values derived from matching the spectroscopic and astrometric distances are the ones labelled chem 2 for the chemical analysis and used in the isochrone-based age estimation.

2.2 Gaia

The stellar isochrones were plotted with Teff and bolometric magnitude (Mbol). Parts of this data was retrieved from the Gaia catalogue, data release DR2 and the latest released EDR3 (early DR3) from December 2020. The European Space Agency space observatory Gaia is one of today’s greatest asset in stellar observations from space. Located in the Sun-Earth’s stable Lagrangian point L2, Gaia constantly scans the sky, thus charting the moving bodies. By doing so, it creates a three-dimensional map of remarkably „ 1.6 billion stars–equivalent to around one per cent of the stars in the Milky Way–with exquisite precision. The positions of targets brighter than magnitude 15 mag will have an accuracy of 24 microarcseconds, comparable to measuring the diameter of a human hair at a distance of 1000 km, and the distance accuracy of the nearest stars to 0.001% according to (ESA). This extraordinary accuracy will be possible at the end of the mission due to high accuracy of every single astrometric measurement and due to the repeated observations of each target that it will have made (it is expected to measure each target around 70 times) during its 5-year long mission.

Two bolometric magnitudes were derived using Gaia data. Firstly from the apparent magnitude (mG) (G for G-band in the wavelength range of 3000 to 11000 ˚

A) via the relation seen in eqn. (1). The distance D is equal to the inverse of the parallax (p), AG is the interstellar extinction and BCG the bolometric correction (BC), all for the G band. Both mGand p are provided in both data releases DR2 and EDR3, while AG only exists in DR2. Those AG are individually assessed for each star and are only available for

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Figure 2: Theoretical stellar isochrones from BaSTI along with our solar twins (black, orange, blue) and subgiants (turquoise & green) as well as the Sun in yellow. The stars correspond to Mbol calculations derived from luminosity (eqn. 2), the squares are Mbolcalculated from the distance formula (eqn. 1) and likewise for the triangles that neglects AG. The light gray errorbar connects to the stars, the darker gray to the triangles and the darkest to the squares. All data is retrieved from Gaia DR2.

the three solar twins, not for the subgiants. The reliability and accuracy of these interstellar reddening assessments will be discussed in a later section (4.1). Furthermore, the parallaxes were later proven to have a zero point problem. This is not taken into account for the following derivations of Mbol,but will be addressed in a later section. The BCs are not available in the Gaia archive and therefore, derived via a model based on MARCS library of fluxes (Gustafsson et al. 2008) presented by Casagrande & VandenBerg (2018). All above-mentioned parameters and uncertainties (σ) are recorded in table (3) and the results can be seen as triangles and squares in fig. (2) where the triangles assume AG “ 0.0. Note that the uncertainty of p and AG are given numerically in DR2 and EDR3 while as for mG we have derived it from Gaia’s fluxes. These derivations, as well as other uncertainties, can be seen in Appendix (A.2). The Mbol derivations via eqn. (1) uses DR2 data even if more recent and improved

EDR3 data is available. This is because the BCs from Casagrande & VandenBerg (2018) are not compatible with EDR3 which is using new transmission curves.

Mbol“ MG` BCG

MG“ mG´ 5log10pDq ` 5 ´ AG D “ 1

p

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It is worth mentioning that Gaia DR2 also provides values for Teff. Although, the uncertainties are considerable (ą 200 K) when compared to spectroscopically determined Teff (ď 20 K for solar twins and more generally ď 50-100 K). The effective temperatures in the Gaia archive are derived using the data processing system called Apsis-Priam. It estimates Teff from two colours, GBP ´ G and G ´ GRP. These colours exhibit a colour-temperature relation which can be used to derive Teff. The colours that come from the measured brightness exhibits extinctions which are indeed addressed. But even though synthetic colours would be preferable, the photometry is so far insufficient. Thus, an empirical approach is needed and, therefore, based on a sample of ą32 000 real stars of which literature values of Teffexist for. Since the sample is empirical, the real targets for Gaia exhibit non-zero extinctions which are estimated through literature values, The European Space Agency (ESA) (2020). The extinctions in Gaia measurements are thereafter derived based on the aforementioned literature values and effective temperature. However, there are large degeneracies between Teff and log g. Therefore, we rely on the spectroscopically estimated temperature values instead.

Mbol“ ´2.5log10p L L@

q ` Mbol,@ (2)

Secondly, the bolometric magnitudes were derived from the stellar luminosities (L). EDR3 does not contain luminosity measurements, however, these are available in DR2. The luminosity is originally derived from photometric measurements of the apparent magnitude mG. Therefore, the two Gaia-approaches are expected to be equal to some degree, Andrae et al. (2018). The method traces one step back in the very derivation of the parameter to attain Mbol via eqn. (2) where M_bol@was taken to be 4.74. The result is shown as stars in fig. (2)

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Figure 3: Theoretical stellar isochrones from BaSTI along with our solar twins (black, orange, blue) and subgiants (turquoise & green) as well as the Sun in yellow. The pentagons correspond to Mbol from the astrometric method and the hexagons from spectroscopy. The gray errorbar accords for the hexagons and the black for pentagons.

Further parameters retrieved from the Gaia archive were the colours BP and RP, positions as in celestial coordinates right ascension (ra) and declination (dec). Finally, the RUWE parameter from EDR3 was recovered. RUWE or Renormalised unit weight error, estimates how well a single-star model provides a good astrometric fit to the target. If RUWE is close to 1 the fit is reliable and the target is most likely a single star. However, if the value deviates from 1, e.g ą 1.4, it indicates that the source is not a single target but conceivably an unresolved multiple e.g. a binary. The parameter is calculated according to eqn. (3), where χ2 is the astrometrical chi-squared. It describes how well the model fits the data by the sum of the residuals over their standard uncertainty. nobs is the number of good observations, m is the number of parameters solved for and f is a renormalising function.

RU W E “ d χ2_{pn obs´ mq f pG, GBP ´ GRPq (3)

2.3 Astrometric method

The final methods of isochrone-based estimated stellar ages are based on astrometric data. Mbol,Gaia is the bolometric magnitude derived using mV from Yadav et al. (2008), assuming AV=0.10 and BCVfrom Alonso et al. (1999), similarly to Mbol derived from Gaia photometry as in eqn. (1). The method also requires parallaxes which are taken from Gaia EDR3 and corrected for the aforementioned zero point problem. The resulting Mbol,Gaia are listed in table (2) and seen in fig. (3).

The parameters mentioned in the former section were as previously discussed used to derive the log g. The surface gravity is also plotted with the stellar isochrones in order to estimate the stellar ages. This is seen in fig. (4). With the derived log g and stellar mass, we also obtain the ”spectroscopic magnitude” Mbol,sp via eqn. (4). However, the astrometric parameters have been used in the derivation of log g which are used within the derivation of Mbol,sp. Therefore, we do not expect the result from Mbol,sp and Mbol,Gaia to differ much, „ 0.02 mag. The relation (eqn. 4) is derived from eqns. (17, 16, & 2) and is seen in appendix (A.2). The method of Mbol,spwill be referred to as the spectroscopic method in the following discussions.

Mbol“ ´2.5 ´ 4log10 `_T eff 5780 ˘ ` log10 ´ M MSun ¯ ´ log10 ´ g gSun ¯¯ ` 4.74 (4)

2.4 Stellar isochrones & Evolutionary

tracks

A great part of this work is based on fits of stellar isochrones and stellar tracks. Four such isochrones and stellar tracks were considered in the analysis.

These were PARSEC (Bressan et al. (2012), Chen et al. (2015), Tang et al. (2014), Marigo et al. (2017) and Chen et al. (2019)), MIST (Dotter (2016), Choi et al. (2016), Paxton et al. (2011), Paxton et al. (2013), Paxton et al. (2015) and Paxton et al. (2018)), Y2 (Spada et al. 2013) and finally BaSTI (Pietrinferni et al. (2004), Pietrinferni et al. (2006), Cordier et al. (2007) and Percival et al. (2009)). The respective set of isochrones were adapted to a solar-normalised scale. Regardless, additional corrections were required in order to match solar values. The isochrones were manually shifted along the evolutionary path to match the solar parameters and the solar age with the theoretical track. This was done by cross-matching the position of the Sun of the two axes, e.g.

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Figure 4: Theoretical stellar isochrones from BaSTI along with our solar twins (black, orange, blue) and subgiants (turquoise & green) as well as the Sun in yellow.

Teff “ 5772 K and log g=4.44 cms´2 and that this point coincides with the point of solar age 4.56 Gyr on the theoretical tracks. If not, we manually moved the isochrones along the y and x-axes in the individual stellar track direction, i.e. along the evolutionary path an individual star would move on the isochrone, until the desired result was achieved.

The PARSEC isochrones are derived based on a Z (metallicity) relation concerning the initial Sun and not the present which is mildly affected by diffusion. However, despite the supposedly solar normalised parameters, alike Z=0.014711, the Sun appears to have an offset to the isochrones near the age of the Sun for both log g-Teff and Mbol´ Teff. The MIST isochrones were chosen to match present solar values including a correction for atomic diffusion over its lifetime 4.56 Gyr, [Fe/H]=+0.06. These isochrones showed a smaller shift than PARSEC, but where nonetheless manually shifted. Y2 isochrones had fixed groups of isochrones were the one stated to be solar-normalised with X=0.70952 (hydrogen abundance), Z=0.01631

and mixing length α=1.875 was used. The isochrones of bolometric magnitude appear more deviant than surface gravity plots which were not shifted afterwards. The only package of isochrones that did not require any corrections was BaSTI. They are solar scaled with heavy element mixing [α/Fe]=0.0, z=0.017210, [Fe/H]=+0.06 and Y=0.2695. Furthermore, these tracks also consider overshooting and Diffusion. Overshooting or convective boundary mixing assumes that the stars have chemical mixing beyond the formal convective boundary due to coactions of physical processes in stellar evolution (Hidalgo et al. 2018). This phenomenon has been proposed to have an important effect for the cluster M67, VandenBerg et al. (2007). Diffusion is incorporated into the models for hydrogen, helium and metals where diffusion alters the surface abundances by carrying heavy elements closer to the stellar core and bringing the lighter hydrogen (H) up. The latter is assuredly crucial for the solar normalisation, the effects of diffusion and overshooting can be seen in figs. (5 & 6). Diffusion is seemingly important for both solar twins on the main sequence (MS) and the subgiants on the subgiant branch (SGB) while overshooting is less crucial to consider for the solar twins, and more so for the subgiants. Since BaSTI were the only tracks without additional corrections, we used these isochrones in our stellar age estimations to obtain as trustworthy results as possible without additional uncertainties. However, all isochrones with Teff-Mbol or log g are available in figs. (18 & 19).

The estimated stellar masses were based on evolutionary tracks from the same four sources PARSEC, MIST, Y2 and BaSTI as the stellar isochrones. Similarly to the isochrones, several evolutionary tracks also needed further adjustments. Therefore, we shifted the tracks to match the solar values of Teff “ 5772 K and log g=4.44 cms´2assuming an age of 4.56 Gyr and mass 1M@. Evolutionary tracks in log g-Teff and Mbol were, as mentioned before both adjusted, however, the masses listed (table (2)) were established on the basis of log g. Once more we base our estimations on the BaSTI evolutionary tracks (see fig. (14) & 15) which did not require additional correction to represent the Sun. The final masses were as mentioned above, iteratively derived along with log g and fixed Teff. Stellar evolutionary tracks for all sources can be seen in figs. (16 & 17).

3 Analysis

3.1 Chemical analysis

Nissen (2015) presented a tight empirical relationship between the elements magnesium, yttrium and stellar age. The abundances of metals in a star, like [Mg/Fe] increases with stellar age in contrast to yttrium [Y/Fe] which decreases, similar to what has previously been proven true for barium, Edvardsson et al. (1993).

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Figure 5: The figure shows a limited region of the MS on the BaSTI isochrones and their dependance on overshoot and diffusion. Upper: The effects of including overshooting (solid) and neglecting it (dashed). Lower: The effects of including diffusion (solid) and neglecting it (dashed). The stars represent our solar twins (black, orange, blue) and subgiants (turquoise & green) as well as the Sun in yellow, with bolometric magnitude derived from luminosity retrieved from Gaia DR2, see eqn. (2)

The abundance ratio between Mg and Y, therefore, becomes a sensitive indicator of stellar age. The linear relation (Nissen 2015) connects the ratio of [Y/Mg] with stellar age based on a maximum likelihood fit to abundance data and previous fits of Yonsei-Yale isochrones. The precise correlation used in our calculations can be seen in eqn. (5). The trend is constructed based on solar twins and may, therefore, not be applicable with the same accuracy for the subgiant stars. The abundances might change for the more massive stars due to their higher degree of diffusion processes. However, we test the model on our two subgiant stars to see whether or not it can produce any results in the proximity as for the solar twins.

rY {M gs “ 0.175p˘0.011q ´ 0.0404p˘0.0019qAgerGyrs (5)

The abundance of each element was derived by differential elemental abundances based on the

Figure 6: The figure is equal to fig. (5) but focused on the SGB.

EW-technique. The absorption lines available for this analysis were Mg I λ “ 5711.090 ˚A & 6318.720 ˚A, Y II λ “4883.690, 4854.870, 4900.110, 5087.420 & 5200.410 ˚A. Numerous calculations (five different cases) using this chemical clock were tested for various spectral lines. All five cases consider a different set of absorption lines, some of which were done in order to compare the results to the previous study of M67-1194 by Önehag et al. (2011) by using their published individual line abundances. Results from all calculations based on Fan Liu’s abundances and Önehag et al. (2011) are seen in tables (4 & 5). The first analysis included all available spectral lines except λ “4883.690 ˚A of which the abundance was later assessed. The resulting ages via chem 1 (chem 1 is the chemical age results from spectroscopically derived log g without astrometric match) were lower than anticipated (AgeM 671194=3.66 ˘0.84 Gyr, Mean=AgeM 67=3.64 ˘0.43 Gyr). The results were also lower compared to the results using all available lines in the work of Önehag et al. (2011) (4.25 ˘0.74 Gyr). The results correspond to Case 1 in tables (4 & 5). In a second analysis we excluded the line at 5200.42 ˚A, which resulted in slightly higher ages of the stars and hence, the cluster itself. This set of wavelengths where chosen such that a more accurate comparison for M67-1194 could be made between our results and Önehag et al. (2011) since the included

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Figure 7: The figure shows a limited section of the FLAMES-UVES spectrum of the Sun (green) and M67-1194 (black) centred around the Hβ line at 4854.870 ˚A. The vertically dahsed lines mark three of the absorption lines used within this work.

lines are the common lines measured in both works. These resulted in somewhat closer chem 1 ages for both set of abundances, AgeM 671194=3.70 ˘0.87 Gyr, AgeM 67=3.66 ˘0.44 Gyr and with the lines from

¨

Onehag et al. (2011) 3.88 ˘0.64. These are listed as Case 2 in tables (4 & 5).

In a third variant of the analysis (Case 3 in tables (4 & 5)) we excluded yet another Y line at 4854.870 ˚

A. The line resides in the wing of the broad Hβ-line as seen in fig. (7). This could lead to misplacement of the continuum in this part of the spectrum and must thus be treated with caution. Therefore, we argue that this line might cause biased abundances and thereby alter the chemical age, thus this should be neglected in further derivations. The chem 1 age of M67-1194 is then AgeM 671194=3.94 ˘0.85 Gyr and the mean AgeM 67=3.69 ˘0.44 Gyr. The equivalent age using ¨Onehag et al. (2011) data is 4.46 ˘0.60 Gyr. The significantly increased age for the ¨Onehag et al. (2011) data could be an indication of a misplaced continuum around the line as previously discussed. For the following case, we have incorporated abundances of the Y-line at 4883.49 ˚A which was also included in

¨

Onehag et al. (2011). The results are somewhat lower for M67-1194, chem 1 AgeM 671194=3.81 ˘0.86 Gyr and marginally higher for the average AgeM 67=3.71 ˘0.45 Gyr, while ¨Onehag et al. (2011) gives 4.37 ˘0.65 Gyr. The results are presented in Case 4 in tables (4 & 5).

In the final variant, we incorporate all spectral lines excluding the Y-line near the Hβ-line (4854.870 ˚

A). This final case (Case 5a in table (4)) is the one we argue for to be the preferable case with as many spectral lines as possible, but neglecting the uncertain ones. The chem 1 age of M67-1194 is again somewhat higher AgeM 671194=3.89 ˘0.79 Gyr but the mean lower, AgeM 67=3.61 ˘0.43 Gyr. All the above-mentioned results from the chemical clock are based on the spectroscopic derived log g (referred to as chem 1 ages), that is to say not the log g values seen in table (2). These spectroscopically derived surface gravities are somewhat questionable since they deviate significantly (2σ) from each other. The near-perfect solar twin M67-1194 show a near solar surface gravity while the remaining solar twins deviate significantly from that. Therefore, the final case (Case 5b in table (4)) was recalculated using new abundances based on the newly derived log g using astrometric data (ages referred to as chem 2 ages). Furthermore, in the final case, there is a further correction of the Mg line at 6318 ˚A which was affected by telluric pollution for the twilight solar spectrum, this is true for both sets of log g values. With the new log g values we indeed get ages closer to anticipated, chem 2 AgeM 671194=3.89 ˘0.79 Gyr and for a mean age of the solar twins and the cluster, AgeM 67=4.56 ˘0.44 Gyr. The final average results agree well with the age of the Sun, more so than many previous aforementioned studies. There is a discrepancy between the chemical ages of the stars

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(see table (1)) where M67-2018 indicates an age above 5 Gyr. Since this is the star with a deviating log g, from both chem 1 and chem 2, the star probably drives the mean age up. The average cluster age using only M67-1194 and M67-1787 is 4.31 ˘0.55 Gyr.

3.2 Isochrones

The numerous ways of deriving bolometric magnitudes have already been touched upon in previous sections. With the four different approaches using Mbol and log g, we estimated the stellar ages via isochrones. We have also discussed that three out of four sets of isochrones required additional alteration to be solar normalised, therefore, the analysis presented below is based on the one set of isochrones that did not need further corrections, namely BaSTI.

First and foremost, Mbol was derived via eqn. (2). As previously mentioned the luminosity measurements originate from photometrically measured magnitudes by Gaia. The resulting magnitudes are listed in table (3) as Mbol,lum and can be seen as stars in fig. (2). The solar twins suggest an age of „ 4.50 Gyr apart from M67-2018 which suggest „ 6.50 Gyr. The subgiants signify a slightly lower age „ 4.20 Gyr. The uncertainties (shown as the lightest grey bars in fig. (2)) cover a considerable age span for the solar twins „ 2.5 Gyr or more. The uncertainties are smaller for the subgiants „ 0.4 Gyr due to the faster evolution of the subgiants which makes the isochrones more sensitive to age at the SGB. Consequently, we can not establish stellar ages within adequate margins. However, the average age and thus the estimated cluster age based on all stars except the eccentric M67-2018 is « 4.31 ˘1.08 Gyr. Note that the age and error estimation for each star is by eye read from the figures.

Another method, closely related to the previous, is deriving Mbol via apparent magnitude in the G-pass band mG, from Gaia DR2. The results are shown in fig. (2) where both the triangles and squares represent Mbol for our five stars derived from mGand p from Gaia DR2 together with BC from Casagrande & VandenBerg (2018). The triangles assume AG “ 0.0 while the squares assume the individually assessed AG also from DR2. The resulting Mbol with AG “ 0.0 presented as triangles, are listed as (DR2) Mbol in table (3). The darker grey and black error bars correspond to the uncertainty of Mbol for DR2s AG and AG=0.0 respectively.

Clearly, the latter case estimates exceedingly high stellar ages that are not near satisfactory. All three solar twins are estimated to be well above 6.00 Gyr, near „ 9.00 Gyr. The subgiants show instead lower ages than expected, however, not nearly as excessive as the solar twins. Nevertheless, even when assuming zero reddening AG “ 0.0 the outcome is surprisingly high.

AGis not provided for the subgiants which are why the squares and triangles are equal in the upper panel of fig. (2). One would expect the extinction to be similar for all stars within the cluster and thus, one could accept an extinction for the subgiants, equal to the average of the solar twins. The subgiants are younger than anticipated and the solar twins older, around „ 6.00 Gyr except for M67-2018 which consistently indicates a much higher age. The exceedingly low Mbol (squares) does not suffice to estimate an average age, nor is it justified to assume zero extinction for the cluster, Taylor (2007). Therefore, we leave the average age estimation for these two but will come back to the reasoning around them in later sections. It is curious how these results, that are supposedly equal to the method mentioned above, deviate by „ 0.1 mag for all three solar twins, leading to an age-shift of „ 1.50 Gyr. In later sections, we will analyse this discrepancy further, by considering the zero point problem of Gaia parallax and the interstellar reddening.

The final two isochrone-based methods are the Mbol,spec and Mbol,Gaia. As previously mentioned, both Mbol were derived on the basis of mV, AV and p. The parallaxes originate from Gaia DR3 with an additional correction. In the previous section, we mentioned a zero-point problem for the Gaia parallaxes, this is true for both DR2 and EDR3. The parallaxes used in the derivation of these Mbol values accommodates for this offset by increasing the parallaxes by δp=0.04 for the solar twins and 0.03 for the subgiants. A comprehensive discussion of this offset is presented in a later section (4.2). The Mbol,spec values are derived using the log g values. Thus this method takes both mass estimation, spectroscopic Teff and astrometrically based log g into account. Respective results are seen in fig. (3) where the hexagons with lighter gray error bars correspond to Mbol,spec and the pentagons with black error bars to Mbol,Gaia. The spectroscopic Mbol has greater uncertainty („ 0.1) than Mbol,Gaia which has 0.06. Furthermore, the spectroscopic results consistently show moderately lower Mbol thus, somewhat higher ages. The average age via Mbol,spec appears to be „4.46 ˘1.52 Gyr, and 4.30 ˘0.9 Gyr via Mbol,Gaia. Persistently as for previous isochron-based age assessment, all stars except the solar twin M67-2018 are included in the average, and the ages and errors are by eye estimated.

Finally, we assessed the isochrone-based stellar ages via log g, thus skipping the intermediate step of deriving the bolometric magnitude. It is evident in fig. (4) that M67-2018 shows a lower surface gravity than the two other solar twins and thereby a higher age. The remaining four stars point to a cluster age of „4.51 ˘1.36 Gyr. This age agrees, unsurprisingly well with the other spectroscopic method using Mbol,spec, but also with the solar age.

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Table 1: Tabulated results of the stellar ages using stellar isochrones from BaSTI and the chemical clock [Y/Mg]. The isochron results can be visualised in fig. ( 2, 4 & 3). The average in the rightmost column does not consider the values that are seemingly overestimated and listed in the table with one or two ą and is by eye estimated from isochrone plots, furthermore, the mean of the chemical age is solely based on the solar twins.

Star M67-1194 M67-1787 M67-2018 M67-1442 M67-1844 Average

[Gyr] [Gyr] [Gyr] [Gyr] [Gyr] [Gyr]

log g 5.55 4.50 (ą6.00) 3.90 4.10 4.51 ˘1.36 (AG=0.0) Mbol,lum 4.25 4.70 (ą6.00) 4.20 4.10 4.31 ˘1.08 (AG‰0.0) (DR2) Mbol (ąą6.00) (ąą6.00) (ąą6.00) 3.80 3.80 3.80 Mbol,spec 5.50 4.40 (ąą6.00) 3.80 4.15 4.46 ˘1.52 Mbol,Gaia 5.30 3.90 (ąą6.0) 3.80 4.20 4.30 ˘0.9 Chemical 3.89˘0.79 4.73˘0.76 5.07˘0.74 (5.79˘0.69) (5.15˘0.70) 4.56˘0.44

For three out of four isochrone-based methods we recognise trustworthy values for the stellar ages and the cluster. The results of each method agree rather well with each other. We also recognise that the discrepancy for each star is extensive and that the stars themselves can deviate between the estimates, especially between the solar twins and the subgiants. However, it is truly interesting that they all reckon a near solar age of the cluster, all ě 4.30 Gyr. All estimated ages based on the isochrones and the final chemical age are tabled in table (1). We argue for that the main result is the chemical age (chem 2 ) due to the relative small uncertainty. The most accurate isochrone-based results are evidently those derived from Mbol,Gaia. This method uses few parameters which minimise the error. However, it does include a corrected for the zero point problem in Gaia EDR3 and assumes an extinction which is not definite for M67. Furthermore, this isochrone-based method shows the smallest uncertainty which makes it the most accurate one.

3.3 Colour magnitude diagram

Two out of three solar twins have shown strong resemblance to the Sun in numerous ways, bolometric magnitude, effective temperature and seemingly in age while the solar twin M67-2018 shows a similar Teff but somewhat lower log g and Mbol. A further perspective that can be analysed is the colours of the stars by plotting a colour-magnitude diagram for the stars over the members of M67. This is shown in fig. (8). The figure shows the main sequence (MS), turnoff point and the subgiant (subG) branch of M67 together with our three solar twins (black, orange, blue) and two subgiants (turquoise & green). The data for our five stars and the cluster members were retrieved from Gaia DR2, where the members were selected based on some criteria: Firstly, the region covered on the celestial sphere is a circle with radius 0.5˝_{centred at the cluster} centre coordinates, ra=132.833˝ _{and dec=11.8167}˝_.

Stars within the distance 1.088masă p ă 1.156mas (equating to approximately „892 ˘ 3% pc) and with magnitude 12 ă mG ă 17 were included. In the left panel of fig. (8) the mG is plotted against the colours BP-RP (Blue passband - Red passband) and in the right, Mbol derived from luminosity according to eqn. (2). In both cases, we see that all three solar twins reside near each other and closely on the MS while the subgiants are relatively close and fit well with the SGB, in particular for the left panel, mG. The BP-RP colour of the Sun in the rightmost panel is taken from Casagrande & VandenBerg (2018) and shows an offset to the MS and the solar twins. The offset from the solar twin M67-1194 suggesting an extinction of E(B-R)« 0.05.

4 Discussion

While the numerous results of the isochrone-based stellar ages struggle to agree on definite stellar ages within a reasonable uncertainty, a final age of the cluster M67 was to some extent settled. At the same time, the chemical clocks seem to demonstrate a more narrow result. Our most trusted chemical based result (chem 2 age from case 5b) showed an average age of 4.56 ˘0.44 Gyr with a relatively small uncertainty. Worth noting, for all various chemical approaches with chem 1 and chem 2 results, the star M67-1194 seems to give the most consistent age. In other words, the estimated age of M67-1194 from all six derivations of different absorption line combinations (Case1-5a & 5b) seems to be relatively stable. This star does have better data quality than the other stars which might suggest a better model-fit for all spectral lines. Other factors that might affect the different elemental abundances could be the assumed log g and Teff which could affect the model fit to the stellar spectrum. The various processes lead to many surprising results and more enigmas have surfaced. However, all methods seem to suggest a cluster age above 4.3 Gyr, near the solar age which raises many interesting questions.

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Figure 8: Colour magnitude diagram of M67 where the gray dots are cluster members and the coloured dots (black, orange, blue) represent our solar twins and (turquoise & green) the subgiants. The colour of the Sun (yellow) in the right most figure is according to Casagrande & VandenBerg (2018). All data is retrieved from Gaia DR2.

4.1 Interstellar extinction

While the bolometric magnitudes derived via spectroscopy, astrometry and photometrically acquired luminosities, agree rather well for the bolometric magnitudes and thus stellar ages, the one method using photometrically measured magnitudes, estimated via eqn. (1) does not. The squares in fig. (2) represent Mbol with the individually assessed AG in DR2. The derived ages are not adequate, which rises wariness of the calculation of Mbol. As previously mentioned, AG is individually estimated for each star and they have prodigious uncertainties, almost of the same extent as the parameter itself. It is also argued that these estimates are rather poor and sometimes set to zero (Andrae et al. 2018) which makes AG estimates hard to rely on. Nevertheless, even when omitting this parameter (AG=0.0) the age estimates are surprisingly high, see the triangles in fig. (2). It is worth mentioning that Gaia does not have the most common reference stars in its system, namely the Sun or Vega. This leads to the fact that the scale is normalised to solar twins that already possess some reddening. This could lead to some offset and errors when assuming a differential analysis with the Sun. Leaving out the lastly mentioned possible impact, all of our calculations that were using stellar reddening of that order, based on different types of data and bolometric corrections,

show this discrepancy. Since AG is the reddening measured in Gaias G-band it is problematic to compare other estimates of the extinction from previous work since the closest estimate to AG are extinction in the visual range, V-band. There seems to exist a near linear relation between AG and AV, however, it does not appear explicit (Andrae et al. 2018). A single value for the interstellar reddening for M67, as for AV has not yet been established. Previously published paper accepts various values, however, most are found around „0.1, e.g AV “ 0.13, Taylor (2007). From our results it is not unlikely to believe that the extinction is seemingly overestimated, however, it is improbable and against former research, to suggest zero extinction which seems favourable for more reasonable stellar ages. It is noteworthy that our results assuming AG “ 0.0 lie closer to the outcome derived from luminosity. This is somewhat expected since AG values are allegedly not included in the derivation of luminosities in Gaia DR2 according to Andrae et al. (2018).

4.2 Zero point problem

With the excessive results of Mbol via eqn. (1) when assuming AG “ 0.0 as analysed in the above section, we derive much too high stellar ages. Thus, there is

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Figure 9: Theoretical stellar isochrones from BaSTI along with our solar twins (black, orange, blue) and subgiants (turquoise & green) as well as the Sun in yellow. The stars correspond to Mbolcalculations derived from luminosity (eqn. 2), the triangles and squares are Mbolcalculated from the distance formula (eqn. 1) assuming AG “ 0.0 & AG “ 0.1 respective, they are also corrected for with δp “ 0.029. The lightest gray error bars connects to the stars, the slightly darker gray to the triangles and the darkest gray to the squares.

reason to believe that there are further problems within the data. It has been recognised that Gaia DR2 parallaxes have a zero point problem, Sch¨onrich et al. (2019). According to Sch¨onrich et al. (2019), the parallaxes in Gaia DR2 need to be increased since they suffer from a mean offset of δp=-0.054 mas for bright stars and a general mean of δp=-0.029 mas according to Lindegren et al. (2018). If this offset (δp=-0.029) is accounted for, Mbol increases and the ages decrease. This can be is seen in fig. (9) where the stars are the unchanged Mbol,lum for reference, the triangles and squares are derived via eqn. (1) with p “ p ` δp and the extinction AG “ 0.0 and 0.1 respectively. Note that the δpoffset found is negative but the shift added to the parallaxes are positive to increase the parallaxes. Markedly, Mbol with AG “ 0.0 are closer to the other results, but the results more argued for, with AG“ 0.1, are still giving to high ages.

A zero-point problem is also noted in Gaia EDR3. This is considered for in the parallaxes that are used within calculations of Mbol,spec and Mbol,Gaia. Even though these parallaxes of EDR3 are more accurately estimated (the uncertainty is decreased by „ 0.01 mas from DR2 uncertainties) and seem to have been corrected for some of this offset since the parallaxes of our stars have increased by „0.01-0.04, they still possess a zero point problem, Lindegren et al. (2020). In the publication of Lindegren et al. (2020) it is discussed how a correction of the parallaxes given by Gaia EDR3 can be applied based on five or six-parameter astrometric, source magnitude, colour and celestial position solution. The correction was implemented through the Python code provided with the publication, with Gaia EDR3 parameter such as mG, ecliptic latitude, pseudocolour, the effective wavenumber of the source used in the astrometric solution and which parameters that have been solved for. We found that the zero point offsets were δp “-0.044, -0.044, -0.044, -0.026 and -0.026 mas for the stars M67-1194, 1787, 2018, 1442 and 1844 respectively. These offsets were taken into account in the derivation of the spectroscopic and astrometric bolometric magnitudes Mbol,spec and Mbol,Gaia, with a general offset-increment taken to be δp “ 0.04 mas and δp “ 0.03 mas the solar twins and the subgiants accordingly. Including this correction yields consistent solutions via the stellar isochrones to former results as well as accepts an interstellar reddening of AV “ 0.10. Without adjustments for δp and AV “ 0.10, the resulting ages are seemingly higher than expected, but with the correction, we can maintain a reasonable extinction and acquire results closer to anticipated ones.

Based on the discussion above, we can go back to the method via eqn. (1). Given that the offset in EDR3 is 0.04 mas for our solar twins and the parallaxes have been increased by 0.013, 0.045 and 0.034 mas respectively from DR2, we can assume that the offset for these stars in DR2 is supposedly greater than 0.029 mas. Since the bolometric corrections derived from Casagrande & VandenBerg (2018) are not compatible with mG from EDR3 we can not derive the bolometric magnitudes with full EDR3 data. However, we reviewed the derivations of (DR2) Mbol using mG from DR2, the parallax from EDR3 with δp “ 0.04 mas for the solar twins and 0.03 mas for the subgiants as suggested in the previous section. Furthermore, we also assumed AG “ 0.1. Fig. (10) shows the result, where the solar twin M67-1787 is closer to Mbol,lum. M67-1194 still gives a too high age but considerably closer to previous estimates. Likewise for M67-2018, the age is too high but closer to the results from other methods. For the subgiant M67-1442, Mbol has not

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Figure 10: Theoretical stellar isochrones from BaSTI along with our solar twins (black, orange, blue) and subgiants (turquoise & green) as well as the Sun in yellow. The stars correspond to Mbol calculations derived from luminosity (eqn. 2), the squares are Mbol calculated from the distance formula (eqn. 1) with mG from DR2 and p from EDR3, assuming AG“ 0.1 and corrected for with δp“ 0.04 &0.03 for the solar twins and subgiants respective. The lightest gray error bars connects to the stars and the darker gray to the squares.

changed much while for M67-1884 Mbol it is now almost identical to Mbol,lum. Through an average estimation of the cluster age based on the four stars M67-1194, 1787, 1442 and 1884, we get a cluster age of « 4.7 ˘1.8 Gyr. Again the age is close to the solar age, but with significant uncertainty. Since there still seem to be an offset, we do not rely on these resulting ages.

4.3 M67-2018

M67-2018 consistently demonstrates in all calculations, a significant age difference to what is expected for a solar twin within the cluster M67. Markedly so for all isochron-based ages which showed an age well beyond

6.00 Gyr. Given the consistency in our age estimation of this star, the chemical clock indicates an age of « 5.07 Gyr and all isochrones ąą6.00 Gyr. This raises predicaments to whether or not to accept this star as a solar twin within the cluster. According to the definitions of a solar twin stated in the very beginning of this report, this star has Teff “ 5749 K, well within the margins Teff “ Teff@ ˘50 K, log g=4.38 cms´2 included in the range of solar surface gravity ˘0.1 dex, iron abundance [Fe/H]=+0.03 also within the margins ([Fe/H]=0.0 ˘0.1) and finally mass, M=0.98M@ in accordance with M=(1.00 ˘0.1)M@. Thus, this star is a prodigious solar twin, but with a much higher stellar age than anticipated. The temperatures suggested by Gaia DR2 have, as previously mentioned, much higher uncertainties: For M67-2018 Teff=5849`457_´388 K which would suggest an age of « 5.75 Gyr via log g and ăă 3.50 Gyr for Mbol,lum plots. Meanwhile, Gaia DR2 temperatures also suggest 5807 K for M67-1194, which is not a big difference to the spectroscopically achieved Teff, „ 27 K difference. But for M67-1787 DR2 gives Teff “ 5756 K which is 95 K cooler than the spectroscopic Teff. With this temperature, M67-1787 would give an age of « 5.70 Gyr via log g and ąą 6.00 Gyr in Mbol,lum-Teff space. Assuming such big errors in the spectroscopic effective temperature based on Gaia comparisons is, therefore, not justified. However, there is still a possibility that the effective temperature of M67-2018 is underestimated in the spectroscopic assessment. This would be a simple solution to the problem.

Is there a possibility that the star is an unresolved binary? The measured magnitude e.g. mV of an unresolved binary is generally overestimated since the fluxes from both stars are acquired in one measurement. In that case, Mbol results in a higher stellar age when plotted with isochrones. Furthermore, Zackrisson et al. (2018) investigated a potential Dyson sphere(a hypothetical artificial megastructure that encompasses the star and utilises the stellar flux output) target with data from Gaia DR1 and found that the explanation to the stars unusual appearance was in fact because of the star being an unresolved binary. Spectroscopic distances showed that the Gaia parallax was severely overestimated due to the binary properties. If a similar circumstance has effected our star, we would expect a smaller parallax and a greater distance, however, this leads to a controversy to the latter argument. This would in that case decrease (DR2) Mbol and thus make the star even older. Furthermore, if the magnitude is changed significantly due to a binary or superposition of two or more stars, one would expect a change in the spectral continuum and with spectral lines. The spectroscopic analysis does not suggest this.

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Figure 11: The figure shows position of cluster members of M67 along with our solar twins (black, orange, blue) and subgiants (turquoise & green). The top figure shows ra against dec, the middle shows distance from us to the stars vs dec and finally, at the bottom the distance vs ra.

Two further checks to the statement of M67-2018 being a binary is the RUWE parameter and the radial velocity of the star. RUWE, as described in an early section, tells whether the single-star model has a good fit (RUWE=1.0) or not (e.g. RUWEą1.4). For M67-2018, a good fit is indeed the case, RUWE=1.04. Therefore, it is not likely that the star is in fact a binary. Furthermore, the radial velocity of M67-2018 is -31.57 kms´1_{. It is the star with the lowest velocity} when compared to the four other stars. Furthermore, Yadav et al. (2008) estimated the mean radial velocity of the cluster to v=-33.673 ˘0.208 meaning that M67-2018 is indeed the star that deviates the most from the mean. However, this is a small difference which one can not argue for that the star is not a cluster member nor a binary star.

4.4 Membership probability

Fig. (11) shows the position of the cluster members of M67 in three dimensions, right ascension, declination and distance (parallax). The data was retrieved from Gaia DR3, where the supposed cluster members were selected by limiting circular region of radius 0.5˝ _{centred at ra=132.833}˝_{, dec=11.8167}˝_–measured to be the centre of M67. Further restrains on the stellar parameters such as magnitude and parallaxes 10 ă mG ă 20, 1.0555 ă p ă 1.1666 equating to « 902 ˘ 5% pc were made. As seen in fig. (11) all solar twins seem to lie relatively close to each other and the centre of the cluster in all three dimensions. M67-2018 is the furthest out from the centre in dec & ra and lies at nearly the same distance from us as M67-1787. The subgiant M67-1442 is seemingly located very close to the centre of M67 in all three dimensions whereas M67-1844 is centred in the upper panel for ra, dec. However, looking at the third dimension along the line of sight, it appears that this star lies at the outer part of the cluster. This subgiant resides around 70 pc closer to us than the other considered stars. With the selection criteria of the cluster members, there are most likely stars that do not belong to the actual cluster and since the distance to the cluster centre is not accurately determined, it is with bigger difficulty to conclude whether or not this star is an unusual cluster member or if it is not a member at all. But it is evident that the subgiant lies significantly further away from the other four stars. At the same time, the subgiant shows similarities to other stars within M67 such as in age and radial velocity vrad(see Yadav et al. (2008) tabulated values in table (2)) where the latter is expected to be similar for all cluster members.

There is a further indication that M67-1844 is somewhat of an outcast. Yadav et al. (2008) have listed the Proper-motion membership probability (Prob. in table (2)) which is defined as the ratio between the distribution of cluster stars divided by the sum of the distribution of cluster members plus the distribution of field stars. The cluster stars are a local sample

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Figure 12: Left: The proper motions of the cluster members of M67 as well as the solar twins (black, orange & blue) and subgiants (turquoise & green). Right: The colour magnitude diagram of M67 stars and the three solar twins and two subgiants. The data is taken from Gaia EDR3.

selected to represent, as closely as possible, stars with properties of the target stars such as magnitudes and proper motion dispersion. All solar twins show a high probability of being a cluster member, 99, 98 and 98 respective. Likewise, M67-1442 shows a great probability (100), contrarily M67-1844 shows a lower probability in comparison (91). The proper motions from Gaia DR3 are seen in fig. (12) and a wider field of view in fig. (13). The figure does not suggest that the subgiant M67-1844 is far from the others, thus it is a small probability that this solar twin is an outcast and not a member.

4.5 Origin of the Sun

It has previously been suggested ( ¨Onehag et al. 2011) that the Sun might have formed within a cluster sharing many of its properties, such as metallicity and age. A good candidate is the open cluster M67. Our results suggest an even closer age to the Sun than previously suggested (Barnes et al. (2016) &

¨

Onehag et al. (2011))–chemical age of 4.56 ˘0.44–thus strengthening this hypothesis. The cluster resides on a distance from the galactic centre such that its orbit encloses the sun within its apocentre and pericentre. The solar orbit is currently at a distance z=26 ˘3 pc above the galactic plane (Majaess et al. 2009). Furthermore, it has never exceeded z=80 pc according to Innanen et al. (1978), while M67 lies near its vertical apex at z=0.41 kpc according to a dynamical study of Carraro & Chiosi (1994) where they projected the orbit

of the cluster from its birth, Davenport & Sandquist (2010). Thus, for the Sun to originate from M67, it must have parted from the cluster into an orbit of the plane within the Galactic disc. According to Carraro & Chiosi (1994), M67 has made 17 passes through the Galactic disc. This interaction region causes stars to leave the cluster, but the co-planar orbit of the Sun can not be produced through such an event. For such an orbit, it is more likely that the Sun left the cluster before M67 diverted into its present orbit, Gustafsson et al. (2016). As we know clusters lose much of their mass in an early stage due to supernovae and stellar winds. When the cluster has lost a great amount of its mass, its members are less bound and can escape more easily. The dynamical requirements for the Sun to escape M67 is therefore plausible. Furthermore, our results also suggest a cluster age close to the solar age, thus strengthen the hypothesis of the Sun possibly born within the cluster.

5 Summary

Within this work, we have presented analyses of isochrone-based and chemical ages of stars within the frequently studied open cluster M67. The stars being the solar twins M67-1194, M67-1787 and M67-2018 and the subgiant stars M67-1442 and M67-1884. The stellar ages were estimated via two main methods: With BaSTI stellar isochrones and the chemical clock [Y/Mg] proposed by Nissen (2015). The chemical

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assessment applied to the solar twins resulted in stellar ages comparable to the Sun. The average age of all chem 2 solar twin ages suggests an age of 4.56 ˘0.44 Gyr, in numerical change agreement with the solar age. The average age when excluding the star with somewhat deviating log g (M67-2018) the chem 2 age of the cluster results in 4.31 ˘0.55 Gyr. Three methods using spectroscopically determined stellar parameters (log g & Teff) astrometric, p (EDR3) and photometry from Yadav et al. (2008), all provides similar ages for all five stars. However, we note the possibility of the overestimated age of M67-2018. The cluster age based on the well-fitted stars was 4.51, 4.46 and 4.18 Gyr respectively. Photometric data from Gaia DR2 showed somewhat conflicting results to former methods. Mbol,lum seemed to give adequate results while (DR2) Mbol did not. With alterations of these calculations, replacing p from DR2 to EDR3 and correcting for the zero point problem (Lindegren et al. 2020) we constricted the results to that of other methods, however, there is still peculiar results. The zero point problem is successfully corrected for in the spectroscopic and astrometric derivations of Mbol and result in an estimated cluster age of 4.46 ˘1.52 Gyr and 4.30 ˘0.9 Gyr respectively. In conclusion, all results suggest a cluster age near that of the Sun, all ě 4.30 Gyr. That is indeed promising for studies using solar twins, investigations of M67 and finally for the hypothesis that the Sun may originate from this very cluster. With these results we hope to learn more about the stellar evolution of the solar twins and due to the near solar age, learn more about the evolution of the solar system.

Acknowledgments

I want to express my deepest gratitude to my two supervisors Andreas Korn and Bengt Gustafsson for an interesting and fun project, for many educative meetings and discussions and all the support with the project. A big thanks to Fan Liu who provided data and insightful comments on multiple occasions and on short notice. Thanks also to Bengt Edvardsson for the Fortran support and informative comments.

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Isochrone and chemical ages of stars in the old open cluster M67

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