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Appendix A: Cluster membership of M 34 Using extant kinematic measurements

There exist two extant kinematic studies of the M 34 cluster which can assist us further in establishing cluster membership for our gyrochronology sample. The first, by Jones & Prosser (1996), relies upon 2-d space motions in the form of proper motions, although their high-fidelity membership probabilities are magnitude limited to about . While JP96 proper motion results do extend to fainter magnitudes, their reliability reduces. That is to say, a fainter M 34 star with a  proper motion probability may indeed still be a bona fide member of the cluster, insomuch as Poisson-limited proper motion errors could lead to erroneously lower proper motion probabilities.

The second, by Jones et al. (1997), employing 1-d heliocentric radial velocities, can be used to establish cluster membership, or in the very least, establish cluster non-membership (or binarity). Unfortunately neither membership probabilities nor velocity errors were reported by Jones et al., however we can exploit their measurements in our membership assessment (see below). Interestingly, in the same manuscript, Jones et al. report Li I 6708 Å equivalent widths [EWs] for their target stars. Even though the detection of lithium in solar type stars is not a requisite for cluster membership, its absence is almost ubiquitous among the generally older, Galactic field stars. The very presence of a substantial lithium line in solar-type stellar spectra is in itself indicative of youth ( Myr; e.g., Soderblom et al. 1993b; James & Jeffries 1997; James et al. 2006).

For each of the M 34 stars in our sample, we have collated these additional membership criteria published by JP96 and Jones et al. (1997) in Table A.1. Its Lowell identifier is listed in Col. 1 in concert with its JP96 identifier in Col. 2. Each star's photometric and radial velocity membership flag, reproduced from Tables 2 and 3, are cited in Cols. 3 and 4 respectively, whereas a flag confirming the detection of lithium 6708 Å in its spectrum (from Jones et al. 1997) is listed in Col. 5. Corresponding Jones et al. radial velocities are detailed in Col. 6, as well as their associated membership probabilities in Col. 7 (see below and Fig. A.1). V-band magnitudes of each star are listed in Col. 8 and finally, cluster membership probabilities based on the proper motions reported in JP96 are reproduced in Col. 9.

In the first instance, we note that all stars in common between our gyrochronology sample and the Jones et al. sample have a significant lithium detection, indicative of stellar youth. These stars are therefore unlikely to be field star interlopers, which when combined with the photometric membership criterion, can safely be considered as bona fide cluster members. In the second, the radial velocity data from Jones et al. study do seem to cluster around -10 km s^-1, however, specific membership assignments are difficult to assess due to the associated scatter in the individual measurements. In order to understand whether the scatter in these radial velocities is due to a true dispersion in the data or is an admixture of single and binary member measurements, together with cluster non-members, we have constructed a histogram for all radial velocities published in the Jones et al. survey. This histogram, plotted in Fig. A.1, clearly shows a peak in the distribution representing the cluster's systemic heliocentric radial velocity. In order to investigate the dispersion about the central peak, we fit an unweighted Gaussian function to the velocity histogram, noting that the fit does not include background contamination due to binarity or cluster non-members, and assumes that the sample is complete. The Gaussian function is centred on -12.49 km s^-1 with a 1 width of 2.32 km s^-1, which we can exploit to determine a membership probability (reported in Col. 7 of Table A.1) for each individual star based on its radial velocity and these Gaussian fit parameters.

A.1 Results

Analysis of the additional membership data reported in Table A.1 yields few surprises. Of all the Lowell periodic variables identified as photometric and/or RV members in Table 2, only two of them, F3_0172 and F3_0215, have a zero probability of being proper motion members according to the JP96 survey. Interestingly however, their V-magnitudes lie right at the point at which the JP96 proper motions begin to become less reliable due to Poisson-error limits on their photographic plate measurements, which brings their validity into question. Furthermore, we have noted in Sect. 5.2 that the gyrochronology sample may indeed suffer from contamination due to one or two cluster non-members masquerading as bona fide members, but we argue that their contribution to the period variance, and hence gyro errors, is small. In fact, F3_0172 and F3_0215 contribute a combined period variance of 0.79 days², which is only a effect (see Table 5). In spite of their low contribution to the period variance, we remain in the uncomfortable position of choosing whether to include these photometric and kinematic members of the cluster in our gyrochronology sample, or to exclude them on the basis of their proper motion membership probabilities. In order to further assess their cluster membership status, we have recently obtained high resolution optical spectra of these two stars using the Canada France Hawai`i Telescope [CFHT], the analysis of which we discuss below (see Sect. A.2).

Finally, all bar one of the Jones et al. (1997) stars are radial velocity members of the cluster, which correlates well with our own radial velocity data. The one Jones et al. star, F4_0136, that is formally a cluster non-member based on a Gaussian probability fit to their radial velocity data plotted in Fig. A.1, actually shows up as a single cluster star member in the long term synoptic velocity survey of Meibom et al. (in prep. - see also Sect. 2.1). Curiously, over the 9-epochs of observation covering 2.5-years, that Meibom et al. have for this star, its varies about their average radial velocity of -8.0 km s^-1 by only 0.68 km s^-1. Interestingly, a Gaussian fit to all radial velocities for the 70 single and binary M 34 stars for which Meibom et al. have obtained results yields a cluster systemic velocity of -7.59  1.02 km s^-1. Assuming the Jones et al. velocity is not in error, this star may be a binary member of the cluster, albeit with either a long-period orbit or a considerably eccentric one.

Table A.1:   Cluster membership assessments for Jones & Prosser (1996) stars in our period sample.

A.2 Cluster membership status for stars F3_0172 and F3_0215:

Two stars exhibiting variability in our differential photometric survey of the M 34 cluster, for which we derive periods, namely F3_0172 and F3_0215, present us with somewhat of a conundrum. While these stars have photometric and radial velocity properties consistent with cluster membership, with relatively short photometric periods indicative of youth (compared to the Galactic field), and result in gyrochronology ages appropriate for a 200 Myr group of stars, they possess proper motion vectors incongruent with the remainder of the cluster. In an attempt to establish or refute genuine cluster membership for these two objects, we have recently observed them at high resolution using the fibre-fed, bench-mounted ESPaDOns échelle spectrograph, located in a Coudé-like instrument chamber in the CFHT Observatory. The primary goal of these observations is to detect lithium at 6708 Å, and measure its equivalent width in these stars, thereby confirming their relative youth and increasing their probability of being bona fide cluster members.

	Figure A.1: Heliocentric radial velocities, in histogram form using 3 km s-1 bins, are plotted for M 34 stars using velocity data detailed in Jones et al. (1997). The red solid line depicts an unweighted Gaussian fit to the data, with a Gaussian centre of -12.49 km s-1 and a sigma of 2.32 km s-1.
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During the evening of 27 January 2010, high resolution ( ) spectra were acquired for stars F3_0172 and F3_0215 using the fibre-fed ESPaDOns spectrograph, in service with the 3.6-m Canada France Hawai`i Telescope located on top of Mauna Kea, Hawai`i, USA. The ESPaDOns spectrograph consists of a 79 lines mm^-1 échelle grating imaged onto a 2048  4608 EEV CCD detector, having 13.5 m square pixels, with photon input delivered by separate 100 m (1.6 arcsec) diameter sky and target fibres. This set-up yields a FWHM of cross correlated ThAr arc lines of 0.271 Å at 6700 Å, and a complete wavelength range of  Å. Using this set-up, F3_0172 and F3_0215 were observed for exposures totalling 2000 and 2140 s respectively, resulting in spectra with S/N  20 at 6700 Å.

For each of the targets, we exploit our CFHT ESPaDOns spectra in order to measure their heliocentric radial velocity and their Li I 6708 Å EW. The removal of CCD instrumental effects, as well as the extraction of wavelength-calibrated spectra, have been achieved by two independent data reduction methodologies. In the first, we performed the bias-subtraction, flat-fielding, spectral order tracing, optimal extraction and wavelength calibration using standard IRAF procedures. In the second, we used the direct output from Libre-ESpRIT (Donati et al. 1997), the dedicated pipeline software for the ESPaDOns spectrograph. A cross-match of radial velocities and EWs for each target from the two reduction methods reassuringly yields consistent results to within  km s^-1 and 3 mÅ respectively.

Table A.2:   CFHT spectroscopic data products for M 34 candidate members F3_0172 and F3_0215.

Heliocentric radial velocities were derived, relative to the well-exposed standard star HD 32963, in the spectral region of the Mg triplet lines (  Å). Cross correlation of HD 32963 with other radial velocity standard stars observed during the CFHT programme shows that the zero-point in placing our velocities onto the standard system is 0.2 km s^-1. The relatively low S/N of our target spectra results in radial velocity precision errors of 0.5 km s^-1.

In order to measure equivalent widths for the Li I 6708 Å line in our targets, each spectrum between  Å, was normalized using continuum fitting after spectra extraction. Each EW was calculated using both the direct integration and the Gaussian fitting methods, whose values were within a few percent of each other. In the lithium region, Li I EWs include contributions from the small Fe I+CN line at 6707.44 Å, leading to measured Li I EWs which are representative of a slightly (10-20 mÅ) over-estimated photospheric Li presence. Soderblom et al. (1993b) report that this Fe line blend has an EW = [20 (B-V)₀ -3] mÅ, determined through an empirical relationship for main sequence, solar-type stars. For each target star, we removed the Fe I line contribution before transforming Li I EWs into abundances, N(Li) - on a scale where log N(H) = 12, using the effective temperature-colour (B-V) relation from Soderblom et al. (1993c), and the curves of growth presented in Soderblom et al. (1993b). Data products, radial velocity and lithium measurements, from the CFHT spectroscopic observations of F3_0172 and F3_0215 are presented in Table A.2.

CFHT radial velocities of F3_0172 and F3_0215 are consistent with cluster membership of M 34 irrespective of whether we compare their individual values to the Jones et al. (1997) sample or to the Meibom et al. (in prep.) one. For the Jones et al. sample, their kinematic membership probabilities are non-zero although they are quite low at and  (for F3_0172 and F3_0215 respectively). Their membership probabilities are far more convincing when compared to the Meibom et al. sample (with respective values of  and >99%).

Figure A.2: Logarithmic lithium abundances N(Li) = 12 + log(Li/H) versus effective temperature are plotted for stars in the M 34 cluster (data taken from Jones et al. 1997). Red filled squares represent abundances for F3_0172 and F3_0215 (JP 49 and JP 41), determined using data reported in Table A.2, with error parallelograms based on temperature errors of 100 K and equivalent width uncertainties.

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The lithium content of both F3_0172 and F3_0215 is substantial indicating that these stars are relatively young compared to the general Galactic field, whose solar-type stellar content is typically old and has had sufficient time to have proton-burned considerable fractions of its natal lithium. In order for these stars to be judged as likely members of the M 34 cluster, not only must they contain lithium in their atmospheres, but it must quantitatively fit into the mass-dependent lithium abundance distribution for the cluster. In Fig. A.2, we plot the lithium abundances for F3_0172 and F3_0215 that we have determined from our CFHT spectra in concert with the extant mass-dependent lithium distribution for M 34 stars (data taken from Jones et al. 1997, who employed identical (B-V)_o-Temperature, Fe I line correction and lithium curves of growth as we have used). It is clear that both F3_0172 and F3_0215 have lithium abundances consistent with the mass-dependent lithium distribution of the M 34 cluster, and must be considered lithium abundance members.

JP96 proper motion vectors for F3_0172 and F3_0215 indicate cluster non-membership. However their V-magnitudes render JP96 2-d kinematic membership probabilities questionable, because at or about this magnitude, JP96 proper motion accuracy and precisions begin to have strong dependencies on the Poissonian errors of their centroiding measurements. In consideration that these two stars are both photometric and kinematic members of the cluster, have photometric periods consistent with the remainder of the cluster's distribution, and have measured lithium abundances which lie right along the trend of the mass-dependent lithium distribution for the cluster, they are probable M 34 cluster members and we retain them in our gyrochronology sample.

Appendix B: A period vs. equatorial velocity analysis

A comparison of photometric period and projected equatorial rotation rate for M 34 stars is plotted in Fig. B.1. For the most part, photometric periods are determined from the Lowell campaign described in this manuscript, except for a few cases where Irwin et al. periods are employed where Lowell ones do not exist. Spectroscopic velocities for stars in common to the Lowell campaign and Irwin et al. survey are obtained from Jones et al. (1997). Three loci are also shown in the figure, representing equality between period and data for G0, K0 and M0 dwarfs, where is assumed for each.

Figure B.1: Projected equatorial velocities () are plotted for those M 34 stars having corresponding photometric periods derived during the course of the Lowell campaign.  data are taken from Jones et al. (1997). For those cases where there is no Lowell campaign period available, similar data from the Irwin et al. (2006) sample are presented. One exception is shown for the star F3_0258, linked by a straight solid line, whose Lowell campaign and Irwin et al. periods are seriously discrepant (11.0  1.0 and 6.655 days respectively). Three loci are depicted representing equal period-vs- relationships (assuming ) for G0, K0 and M0 dwarfs (solid, dashed and dotted lines respectively). In order of descending mass, stellar radii of  = 1.08, 0.81 and 0.61 were employed (Gray 1992).

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There are three notable characteristics to the plot. First, the clustering of data-points for stars with periods  days and 10-14 km s^-1 lie to the right of the G0 dwarf locus. Given that the M 34 stars having photometric periods are late-F to early M-dwarfs, these data-points are most likely indicative of those stars whose  values are . Second, there are four stars with periods <3 days and 10 < < 25 km s^-1, whose period, data place them considerably below even the M0 dwarf locus in the diagram. Assuming that these stars are bona fide cluster members, whose period and values are correctly determined, these stars must be high inclination systems ( ; i.e., becoming more pole-on), whose true equatorial velocities are considerably higher.

Finally, there is one M 34 star (F3) whose period determinations, from our Lowell data and by the Irwin et al. study, are seriously discrepant (see also Fig. 2). The periodicity for this star was detected in three filter/exposure time observations during our Lowell campaign. It is interesting to note that the Irwin et al. period is almost half that of our Lowell one, and their lower value could be due to phase aliasing, power leakage in the power spectrum as a consequence of their shorter observing window or multiple spot groups on the surface on the star during its observation. In any case, either both period determinations are incorrect, or either value from the Lowell campaign or the Irwin et al. study is in error.

If we assume that one of the periods for this star is correct, we can make some predictive statements as to its equatorial velocity. With an intrinsic B-V colour of 0.95, its spectral type on the main sequence would be K2 or K3, placing it close to the central locus of the three plotted in Fig. B.1 (assuming the star is inclined  to the line-of-sight). This scenario is consistent with its Irwin et al. period of 6.655 days. Conversely, if the Lowell period of 11.0 days is correct for this object, and it is a single member of the cluster lying on the main sequence, its inclination angle must be higher than ( ), whose appearance is more face-on to the line-of-sight. Hopefully, more extensive photometric monitoring of this star will reveal its true nature.

Appendix C: Lowell light-curves for photometrically variable stars in the field of M 34

	Figure C.1: Period-phased photometric lightcurves for M 34 variables, derived from short-V observations.
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	Figure C.1: continued.
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	Figure C.2: Period-phased photometric lightcurves for M 34 variables, derived from short-I observations.
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	Figure C.2: continued.
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	Figure C.3: Period-phased photometric lightcurves for M 34 variables, derived from long-V observations.
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	Figure C.3: continued.
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	Figure C.3: continued.
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	Figure C.4: Period-phased photometric lightcurves for M 34 variables, derived from long-I observations.
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	Figure C.4: continued.
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	Figure C.4: continued.
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