THREE ANCIENT HALO SUBGIANTS: PRECISE PARALLAXES, COMPOSITIONS, AGES, AND IMPLICATIONS FOR GLOBULAR CLUSTERS^*,^†,^‡

Our astrometric measurements were made with the FGSs onboard the HST. They are a set of three interferometers that can sequentially measure precise positions of a target star and several surrounding astrometric reference stars with one FGS (FGS1r), while the other two (FGS2 and FGS3) are locked on guide stars in order to maintain telescope pointing. Our FGS observations of each halo subgiant consisted of two HST orbits each at five epochs, taken near the biannual dates of maximum parallax factor, for a total of 10 HST orbits per target (11 for HD 140283). The observations were made as follows: HD 84937 between 2003 November and 2005 November; HD 132475 between 2003 August and 2006 February; and HD 140283 between 2003 August and 2006 March, with an additional single orbit of observation in 2011 March.

During each HST orbit, we used FGS1r in "Position" mode to measure the relative positions of the subgiant and six or seven faint neighboring background reference stars. Each reference star was observed for about 30 s, three to four times per orbit, interspersed with five or more measurements of the subgiant. The data gathered over the course of an orbit were processed and calibrated as a single "plate." (The repeated exposures of the same stars are used to model and remove telescope pointing drift resulting from the changing thermal environment as the HST orbits the Earth.) Guide-star telemetry from the two guiding FGSs was used to remove high-frequency spacecraft pointing jitter from the FGS1r astrometric data. Because of refractive components in the FGS optical chain, color-dependent corrections to the astrometry were applied as functions of B − V color for each star (see Nelan & Bond 2013, their Section 3.3 for details). Finally, the FGS data were corrected for differential velocity aberration and geometric distortion ("optical field angle distortion") to produce the astrometric measurements from each HST orbit.

Data from the 10 or 11 orbits were combined using an updated version of the least-squares overlapping-plate GAUSSFIT program (Jeffreys et al. 1988), which solves for the plate constants (rotation, translation, and scale) along with the parallax and proper motion (PM) of each star. However, since the FGS measures only relative positions, the PMs and estimated parallaxes of the reference stars must be input as priors to the model in order to obtain an absolute parallax and PM of the subgiant. We estimated the input reference-star parallaxes based on ground-based photometry and spectroscopic observations (see next subsection). Their input PMs were obtained from the PPMXL (Roeser et al. 2010) or UCAC4 (Zacharias et al. 2013) catalogs; we chose the input catalog for each field that yielded the lower χ² value, which proved to be PPMXL for the HD 84937 field and UCAC4 for the other two. These reference-star data are provided to the model as observations with errors, not as fixed values. This allows GAUSSFIT freedom to adjust the values within their stated errors to find the overall best χ² solution. We assumed a 20% uncertainty in the distance of each reference star, corresponding to a ±0.4 mag uncertainty in their absolute magnitudes. (An overview of parallax measurements with the FGSs can be found in Nelan & Makidon 2002.)

The input ground-based PMs generally have large uncertainties, relative to the high precision of the HST measurements. This results in elevated residuals in all of the other quantities computed by the model. To mitigate this, we first executed the model with the ground-based measurements and errors, and then used the resulting FGS PMs and errors as input to the next iteration. We found that, while this does not appreciably change the PMs after the first iteration, the resulting χ² and residuals are generally reduced by a few tens of percent.

The V = 7.2 brightness of HD 140283 necessitated the use of the F5ND attenuator (required for targets brighter than V = 8) on the subgiant, while the fainter reference stars were observed using the F583W spectral element. This introduces a cross-filter wedge effect that shifts the position of HD 140283 relative to the field stars in a manner similar to parallax. Fortunately, this cross-filter shift (on the order of 8 mas) is well calibrated near the two locations in the FGS1r field of view where HD 140283 was observed over the course of the program. Nonetheless, there remains a ∼0.2 mas systematic uncertainty in the cross-filter calibration, which must be added in quadrature to the statistical errors computed by GAUSSFIT. This had not been included in the uncertainties reported for HD 140283 by Bond et al. (2013), but is incorporated now. For the other two targets, HD 84937 and HD 132475, the F5ND filter was not used for any of the observations.

We made distance estimates for the reference stars based on ground-based spectroscopy and photometry. For spectral classification, digital spectra in all three fields were obtained by us with the WIYN 3.5 m telescope and Hydra spectrograph at Kitt Peak National Observatory (KPNO) in 2004 and 2007, and by service observers on the 1.5 m SMARTS telescope⁸ and Ritchey–Chretien spectrograph at Cerro Tololo Interamerican Observatory (CTIO) in 2003 through 2005. The classifications were then determined through comparisons with a network of MK standards obtained with the same telescopes, assisted by equivalent-width measurements of lines sensitive to temperature and luminosity.

Photometry of the reference stars in the Johnson–Kron–Cousins BVI system was obtained with the SMARTS 1.3 m telescope at CTIO, using the ANDICAM CCD camera, and calibrated to the standard-star network of Landolt (1992). Each field was observed by Chilean service observers on four different photometric nights in 2003 through 2005. Additional photometry was obtained by H.E.B. with the CTIO 0.9 m telescope on one night in 2001 and with the KPNO 0.9 m telescope on two nights in 2007; and by D.H. with the KPNO 2.1 m telescope on single nights in 2003 and 2004.

Table 1 gives the designations, coordinates, photometry, and spectral types of the reference stars in Columns 1 through 7. The uncertainties in the magnitudes and colors, based on the internal agreement, are typically about ±0.004 and ±0.005 mag, respectively. To estimate the reddening of the reference stars (assumed to be the same for all stars in each field, since their distances place them well beyond the dust of the Galactic disk), we compared the observed B − V color of each star with the intrinsic (B − V)₀ color corresponding to its spectral type, and calculated the average E(B − V) for each field. The intrinsic colors were taken from literature compilations⁹ assembled by E. Mamajek. These averages agreed very well with E(B − V) values from the extinction maps of Schlafly & Finkbeiner (2011, hereafter SF11), as implemented at the NASA/IPAC Web site.¹⁰ Our adopted values of E(B − V) were 0.037, 0.100, and 0.130 mag for the background fields of HD 84937, HD 132475, and HD 140283, respectively.

For the distance estimations, we used a purely empirical approach, which is based on determining the luminosity class from the spectroscopic data, and then using the dereddened photometry to estimate the absolute magnitude, M_V. For subgiants and giants, we found the absolute magnitudes by interpolation in a fiducial sequence [M_V versus (V − I)₀] for the old open cluster M 67 (Sandquist 2004), which we took to be representative of the faint population at high galactic latitudes (b = +455, +319, and +336, in turn, for HD 84937, HD 132475, and HD 140283).

For the reference stars classified as dwarfs, we followed the algorithm described in Bond et al. (2013), which in brief is based on calibrations of the visual absolute magnitude against B − V and V − I colors through polynomial fits to a sample of 791 single main-sequence stars with accurate BVI photometry and Hipparcos or USNO parallaxes of 40 mas or higher (provided online by I. N. Reid¹¹). This calibration includes a correction for metallicity, estimated from each star's position in the B − V versus V − I diagram. We tested our algorithm by applying it to 136 nearby stars with accurate parallaxes and a wide range of metallicities listed by Casagrande et al. (2010). We reproduced their known absolute magnitudes with an rms scatter of only 0.28 mag. At the distances of the reference stars, ranging mostly from about 500 to 1700 pc, this scatter corresponds to parallax errors of 0.08 to 0.24 mas.

The final two columns in Table 1 give the input estimated parallaxes and associated errors for the reference stars, and the final adjusted parallax that was output by the GAUSSFIT routine. In most cases, the adjustments were quite small.

Table 2 presents our final astrometric results for the three halo subgiants. Columns 2 through 4 show the Hipparcos PM components and parallaxes, and Columns 5 through 7 show our FGS results. Note that we have reanalyzed the FGS data for HD 140283 that were presented by Bond et al. (2013), using the identical procedures employed for the other two targets in the present paper. Our results for this star are thus very slightly different from those published in 2013. The error bar for HD 140283 also now includes the uncertainty due to use of the neutral-density filter, as discussed above in Section 2.1.

The FGS PMs agree very well with those found by Hipparcos. However, there are slight offsets in the PM zero-points for the FGS measurements compared to the absolute PMs of Hipparcos, because the FGS reference frames are based on only about half a dozen background stars chosen from the PPMXL or UCAC4 catalogs. These catalogs have errors in the absolute PMs per star of several mas yr⁻¹ (e.g., Zacharias et al. 2013).

Our parallaxes likewise agree extremely well with Hipparcos for two of our targets, but our statistical uncertainties are about 2.5–4 times smaller. For HD 84937, our FGS parallax measurement is smaller than that found by Hipparcos, by about 2σ in units of the Hipparcos uncertainty. We note that offsets of a similar amount and sign between Hipparcos parallaxes and precise FGS and/or radio-interferometric parallaxes have sometimes been found by other investigators, e.g., for the Pleiades cluster (Soderblom et al. 2005; Melis et al. 2013).

The effective temperatures given in Table 4 are based on the calibration of the infrared flux method (IRFM) by Casagrande et al. (2010) and taken from Meléndez et al. (2010), except in the case of HD 140283. For this star, we detect weak interstellar Na i D lines (EW = 23.0 and 11.5 mÅ), which implies E(B − V) = 0.004 according to the calibration of Alves-Brito et al. (2010). This increases the IRFM temperature of HD 140283 from T_eff = 5777 K, as reported by Meléndez et al., who assumed E(B − V) = 0.0, to T_eff = 5797 K. The statistical errors associated with the IRFM temperatures are on the order of ±60 K according to Casagrande et al. (2010), but in addition, there could be systematic errors of the particular T_eff scale that is adopted. For instance, using a different implementation of the IRFM based on 2MASS photometry, González Hernández & Bonifacio (2009) derived T_eff values for HD 19445, HD 84937, and HD 140283 that are an average of 80 K cooler than those by Casagrande et al. In the case of HD 132475, they find a higher temperature (5815 K), but this is probably because they assume a reddening of E(B − V) = 0.03 instead of the value that we favor, E(B − V) = 0.007, which is based on the strength of interstellar Na i D lines.

As a check of the T_eff scale, we have compared the IRFM-based temperatures with those derived from the wings of the Balmer lines. This spectroscopic determination has the advantage of being independent of interstellar reddening. For the best studied star, HD 140283, Gehren et al. (2004) obtained T_eff = 5773 K from Hα and Hβ, while Asplund et al. (2006) found T_eff = 5753 K from Hα and Nissen et al. (2007) determined T_eff = 5849 K from Hβ. The average of these values, 5791 K, agrees well with our IRFM temperature of 5797 K, but as discussed by Nissen et al., there may be systematic errors in the Balmer-line temperatures related to the temperature structures of model atmospheres, non-LTE effects, and line-broadening theory. For 10 halo stars having T_eff from both Hα and Hβ, the Hβ temperatures are systematically higher than those from Hα by ΔT_eff = 64 ± 28 K. This suggests that the spectroscopic T_eff scale is uncertain by approximately ±70 K. Furthermore, for a sample of 30 metal-poor halo stars, Casagrande et al. (2010) compared their IRFM temperatures with those derived from the hydrogen lines by Bergemann (2008) and Nissen et al. (2007). As seen in their Figure 12, there is good agreement between the two scales at 5600 < T_eff < 6000 K, whereas the IRFM temperatures tend to be higher by 50 to 100 K at T_eff > 6100 K. In summary, we conclude that there is no evidence from the presently available spectroscopic temperatures of halo stars that the Casagrande et al. (2010) IRFM scale is too low, but it may be 50–100 K too high for metal-poor halo stars in the turnoff region.

Surface gravities are derived from the standard relation

$$\begin{eqnarray} &&\log \frac{g}{g_{\odot }} = \log \frac{\cal {M}}{\cal {M}_{\odot }} + 4 \log \frac{T_{\rm eff}}{T_{\rm eff,\odot }} + 0.4 (M_{\rm bol} - M_{{\rm bol},\odot }),\quad \end{eqnarray} \tag{ 1 }$$

where $\cal {M}$ is the mass of the star and M_bol is the absolute bolometric magnitude (see Section 5). From the errors, which are estimated to be ${\pm }0.03 \,\cal {M}_{\odot }$ in mass, ±60 K in T_eff, and ±0.03 in the bolometric correction, as well as the error of M_V arising from the parallax uncertainty, we obtain σ(log g) = ± 0.04 dex for HD 19445 and σ(log g) = ± 0.03 dex for the other stars.

The abundances of oxygen and iron were determined from the lines that are listed in Table 3. The oscillator strengths (log  gf) of the O i triplet lines are from the theoretical work by Hibbert et al. (1991), while those adopted for the Fe ii lines are from Meléndez & Barbuy (2009), who determined accurate gf values based on lifetime measurements of the atomic energy levels and calculations of relative line ratios within a given multiplet. The results from individual lines clearly agree very well. In the case of oxygen, the rms scatter of A(O) ¹² is on the order of ±0.04 dex and the scatter of A(Fe) is ±0.06 dex.

Non-LTE corrections for the oxygen triplet lines are adopted from the calculations of Fabbian et al. (2009a), who considered a model atom with 54 energy levels and adopted electron collision cross sections from Barklem (2007). Inelastic collisions with hydrogen atoms were described by the classical Drawin formula (Drawin 1968), scaled by a factor S_H. The calculations were performed for both S_H = 0 and 1, which enables us to interpolate the non-LTE corrections to S_H = 0.85, i.e., the value determined by Pereira et al. (2009) from a study of the solar center-to-limb variation of the O i triplet lines. This leads to mean non-LTE corrections for the three O i lines ranging from −0.11 dex in the case of HD 19445 to −0.19 dex in HD 84937. By comparison, the non-LTE correction for the Sun is −0.20 dex.

The high excitation O i triplet lines are formed deep in the stellar atmospheres, where the effects of atmospheric inhomogeneities on the temperature structure are small. According to inhomogeneous models of metal-poor halo stars, the 3D–1D, MARCS correction of oxygen abundances derived from the O i triplet is negligible (Asplund 2005, his Figure 8).

The Fe abundance determined from Fe ii lines is practically unaffected by departures from LTE (Mashonkina et al. 2011; Lind et al. 2012). There is, however, a small 3D–1D,MARCS correction of approximately +0.05 dex in metal-poor stars (Nissen et al. 2002; Asplund 2005).

The abundance determinations are summarized in Table 4. The statistical errors arising from uncertainties in the EWs and those of the T_eff and log  g values are on the order of ±0.07 dex for A(O) and ±0.04 dex in the case of A(Fe). Systematic errors due to uncertainties in the gf values, the non-LTE effects, and the 3D–1D corrections are larger: we estimate total uncertainties of ±0.15 dex for A(O) and ±0.10 dex for A(Fe). The table also lists [Fe/H] and [O/Fe] values corresponding to the solar abundances reported by Asplund et al. (2009, who give A(O)_☉ = 8.69 and A(Fe)_☉ = 7.50). As noted, [O/Fe] in HD 84937 is about 0.15 dex lower than in the other three stars, but given that the error of the differential values of [O/Fe] is ±0.08 dex (mostly due to the temperature uncertainties) this could be an accidental 2σ deviation. (Alternatively, the lower abundance may be due to gravitational settling, which is expected to have bigger effects on the surface abundances of turnoff stars than those on the MS or SGB; see Richard et al. 2002.)

The O and Fe abundances agree fairly well with results from recent studies. In a large survey of 825 stars in the local disk and halo, Ramírez et al. (2013) derived oxygen abundances from the O i triplet lines using their own non-LTE calculations. The four stars discussed in this paper are included in their survey. The mean differences (this paper—Ramírez) and rms deviations are: Δ T_eff = 30 ± 35 K, Δ log g = −0.02 ± 0.06, Δ [Fe/H] = −0.09 ± 0.05, Δ [O/H] = −0.05 ± 0.13, and Δ [O/Fe] = +0.04 ± 0.12. The rms deviations correspond well to the estimated errors in the two studies. There seems, however, to be a significant systematic difference in the [Fe/H] values in the sense that the Ramírez et al. metallicity scale is about 0.10 dex higher than ours. On the other hand, our metal abundances agree almost exactly (Δ[Fe/H] = +0.01 ± 0.02) with those of Casagrande et al. (2010) and Meléndez et al. (2010), who adopted mean values of [Fe/H] from a number of recent high-resolution studies.

The oxygen abundance may also be determined from the forbidden [O i] line at 6300 Å, which is, however, very weak in spectra of metal-poor F and G dwarf stars. This line could not be detected in HD 19445 and HD 84937, but in our spectrum for HD 132475, we measured an equivalent width of EW = 1.7 mÅ (after a small correction for the contribution of the Ni i blend (Allende Prieto et al. 2001), and in the case of HD 140283, Nissen et al. (2002) obtained EW = 0.5 mÅ. Non-LTE corrections are negligible, but there is a significant 3D–1D correction on the order of −0.2 dex (see Nissen et al. 2002), because the [O i] line is formed in the upper layers of the atmosphere, where the temperature is lower in 3D models than in 1D models. Taking this into account, we derive A(O) = 7.60 for HD 132475 and A(O) = 6.95 for HD 140283. In the case of HD 132475, the oxygen abundance from the [O i] line is ∼0.2 dex lower than derived from the triplet, whereas the two O abundances happen to agree for HD 140283. As the oxygen abundances derived from the [O i] line have statistical errors on the order of ±0.2 dex due to the weakness of this line, we consider the results from the O i triplet to be more reliable.

The [O i] λ6300 line is of sufficient strength in metal-poor K giants to allow a precise determination of their oxygen abundances. In an extensive study of such stars, Cayrel et al. (2004) found a plateau of [O/Fe] ≃ 0.7 at [Fe/H] <−2 based on a 1D model atmosphere analyses. Applying 3D–1D corrections of −0.1 dex (Collet et al. 2007) for giants, the plateau value decreases to [O/Fe] ≃ 0.6, in good agreement with the average ratio, [O/Fe] =0.57 that has been derived in this paper for the four dwarfs and subgiants.

The abundances of carbon and nitrogen relative to that of iron are not far from the solar ratios. From equivalent widths of C i lines near 9100 Å, we get [C/Fe] =−0.01 for HD 84937 and [C/Fe] =0.08 for HD 140283 if non-LTE corrections from Fabbian et al. (2009b) corresponding to S_H = 1 are adopted. If S_H = 0, these abundance ratios are decreased by 0.08 dex. In the case of HD 132475, Nissen et al. (2014) derive [C/Fe] = 0.15 from the weak C i lines at 5250 and 5380 Å. We have no data for HD 19445, but Takeda & Takada-Hidai (2013) obtained [C/Fe] =+0.03 from a set of infrared C i lines at 1.068–1.069 μm. Within the estimated errors, typically ±0.15 dex, all of these values are compatible with a solar C/Fe ratio. Concerning nitrogen, we note that a model-atmosphere synthesis by Israelian et al. (2004) of the NH band at 3360 Å in UVES spectra yields [N/Fe] =0.35 for HD 19445, <0.0 for HD 84937, and 0.05 for HD 140283. These values are based on 1D model atmospheres. According to Asplund (2005), large negative 3D corrections should be applied, because the NH band is formed high in the atmospheres, which decreases [N/Fe] by 0.4 to 0.5 dex. For HD 132475, we have not been able to find any determinations of the nitrogen abundance based on high-resolution spectra, but intermediate-resolution observations of the NH band yield [N/Fe] ∼−0.5 (Carbon et al. 1987). Hence, although the N abundances are uncertain, it is clear that none of the four stars considered here belong to the rare class of N-rich halo stars (Bessell & Norris 1982), and that nitrogen gives a negligible contribution to the total abundance of the CNO elements.

The abundances of the α-capture elements, Mg, Si, and Ca, have been determined from one Mg i line (λ5711.1), two Si i lines (λ6155.1 and λ6237.3), and eight Ca i lines in the wavelength range 6100–6440 Å. The sources of the relevant gf values are Chang & Tang (1990) for Mg, Shi et al. (2009) for Si, and Smith & Raggett (1981) for Ca. Non-LTE corrections of about +0.15 dex for the Mg i line were adopted from Zhao & Gehren (2000), whereas the corrections for the Si and Ca lines are vanishingly small (Shi et al. 2009; Mashonkina et al. 2007). In addition, we include small 3D–1D corrections from Asplund (2005). Based on all of this, we derive the [α/Fe] values given in Table 4 (adopting solar abundances from Asplund et al. 2009). As shown in this table, the [α/Fe] values of HD 19445, HD 84937, and HD 132475 are close to 0.40 dex, while that for HD 140283 (0.26 dex) is lower. This may be a statistical fluctuation because the error of [α/Fe] is on the order of ±0.1 dex. Alternatively, HD 140283 could belong to the population of "low-α" halo stars discovered by Nissen & Schuster (2010), although it would then be puzzling why HD 140283 does not have a lower value of [O/Fe] than the other stars. We note in this connection that the possible low [α/Fe] has no influence on the abundance determinations: if a MARCS model with [α/Fe] =+0.26 is applied instead of the canonical value of +0.40 dex, there is no significant change of the derived abundances.

The abundances described in the previous subsections refer to the present compositions of the stellar atmospheres. As discussed by Bond et al. (2013), the observed surface abundances should be adjusted for the effects of diffusive processes in order to yield the bulk composition of the stellar interior. (If gravitational settling is not treated, the age of HD 140283 would exceed the age of the universe as determined from Planck and WMAP observations by ${\mathrel {{\raise0.6 ex {>}{\sim }}}}1.5$ Gyr.) However, it is clear from the studies by Lind et al. (2008); Nordlander et al. (2012, and references therein), and Gruyters et al. (2013) that diffusive stellar models are not able to reproduce the observed variations of the metal abundances of stars lying between the TO and the base of the giant branch in GC CMDs unless additional mixing (below envelope convection zones) is invoked to limit the efficiency of diffusion in their surface layers (also see Richard et al. 2002). In the case of NGC 6397, which has [Fe/H] ≈−2.0 (e.g., Carretta et al. 2009), Nordlander et al. concluded that the difference between the initial and observed metal abundances in the case of cluster subgiants with T_eff ≈ 5800 K (similar to that of HD 140283) is 0.1–0.15 dex, depending on how this extra mixing is treated. Similar, or slightly smaller, differences were obtained by Gruyters et al. in their investigation of NGC 6752 ([Fe/H] ≈−1.55).

Based on these findings, we decided to correct the [Fe/H] value that we determined for HD 132475 by +0.10 dex and those of the other three stars by +0.15 dex. (As mentioned in Section 3.2, a larger adjustment could well be more appropriate for HD 84937.) If the same corrections are applied to the absolute abundances of all of the metals, the inferred [m/Fe] values remain unaffected. However, since the ages of very metal-poor subgiants, at their observed absolute magnitudes, depend primarily on their absolute oxygen abundances (see below), the adoption of increased [O/H] values by 0.1–0.15 dex does have the effect of reducing the ages derived for them by 0.3–0.5 Gyr, which is not negligible. That is, the ages of HD 132475 and HD 140283 that are derived in this study would have been higher by up to 0.5 Gyr if the aforementioned adjustments to the observed abundances had not been made.

Figure 5 plots the same 14.0 Gyr isochrone that appeared in Figure 1, but on the [(V − I)₀, M_V]-plane. On this CMD, the inferred age of HD 140283 is about 0.30 Gyr younger than our best estimate from the (log T_eff, M_V)-diagram, though virtually the same age that was derived in Section 5 is implied by both the B − V and V − K_S colors of HD 140283 (not shown). As discussed by Casagrande & VandenBerg (2014), whose color–T_eff relations are used throughout this investigation, uncertainties in the zero-points of the transformations at the level of 0.01–0.015 mag are easily possible (for any color index). Alternatively, it may be the observed V − I color of HD 140283 that is anomalously too blue by a small amount (for whatever reason). This is a moot point. What is troubling is that the fit of a fully consistent ZAHB to the cluster counterpart in M 92 yields an apparent distance modulus, (m − M)_V = 14.75, that is 0.22 mag higher than the value, which is obtained by matching the cluster CMD to HD 140283 (see the previous figure). If the shorter modulus is adopted, not only would the cluster HB stars be fainter than our ZAHB models (which, as already mentioned, do a good job of satisfying empirical RR Lyrae luminosity constraints; see VandenBerg et al. 2013), but the cluster main sequence (MS) would be much fainter, at a given color, than the MS portion of the isochrone for [Fe/H] =−2.23 that has been plotted in Figure 5. This is problematic because the separations between isochrones for low metallicities are predicted to be quite small at $M_V \mathrel {{\raise0.6 ex {>}\lower.5 ex{\sim }}}5$ on the V − I, V plane (see VandenBerg et al. 2010, their Figure 11).

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These difficulties are almost certainly telling us that M 92 and HD 140283 cannot have the same age—the field halo subgiant must be older—and perhaps even that they are chemically dissimilar. To examine the implications of possible age and chemical abundance differences, we have opted to fit isochrones and ZAHB loci to the F606W, F814W photometry that was obtained by Sarajedini et al. (2007) using the Advanced Camera for Surveys on the HST. (The same data were recently used in the determination of the ages of 55 of the GCs in their sample by VandenBerg et al. 2013.) The main reason for this choice is that a deeper and tighter CMD for M 92 can be derived from this archive¹⁶ than from any other publicly available source. In addition, since we intend to carry out similar fits of stellar models to the CMD of M 5, it is an important advantage of using the Sarajedini et al. observations that they represent a very homogeneous data set.

Suppose the ZAHB-based distance is the correct one and that M 92 and HD 140283 are chemically indistinguishable. As shown in panel (a) of Figure 6, a 12.20 Gyr isochrone for the derived initial abundances of HD 140283 provides quite a good match to the MS and TO of M 92, aside from being slightly too red (by only ≈0.01 mag). To make this age determination, isochrones for different ages were adjusted in the horizontal direction by whatever amount was needed in order to match the turnoff color: the one that provided the best fit to the cluster stars from ∼1 mag below the TO to ∼0.6 mag above it was taken to be the best estimate of the turnoff age. A thorough discussion and justification of this procedure is provided by VandenBerg et al. (2013), who obtained a higher age by about 0.5 Gyr because the models that they employed assumed slightly lower values of [Fe/H] and [O/Fe] and their ZAHB models yielded a smaller distance modulus by 0.03 mag (which, by itself, accounts for about a 0.3 Gyr age difference). If M 92 has the same [Fe/H] as HD 140283, but a somewhat lower oxygen abundance (say, [O/Fe] =+0.50 instead of +0.64), the same distance modulus is obtained, but the predicted age increases to ≈12.6 Gyr (see panel (b)). This isochrone also follows along the red edge of the cluster CMD. To obtain an age near 13.0 Gyr, the models would need to assume [O/Fe] $\mathrel {{\raise0.6 ex {<}\lower.5 ex{\sim }}}0.3$.

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Note that the ZAHB for higher [O/Fe] extends to redder colors. In these and all similar plots presented in this paper, the mass of the reddest ZAHB model is identical to the mass at the tip of the RGB that is predicted by the isochrone which is fitted to the turnoff photometry, without taking any mass loss whatsoever into account. The intrinsic color of the reddest ZAHB star in M 92 appears to be ≈0.20. At this color, the ZAHB models shown in panels (a) and (b) have masses of 0.730 and 0.748 ${\cal M_{\odot }}$, respectively, whereas the corresponding RGB tip (and TO) masses are 0.790 and 0.797 ${\cal M_{\odot }}$. Thus, relatively little mass loss appears to be necessary to explain the reddest ZAHB stars in M 92, though ${>} 0.2 {\cal M_{\odot }}$ must be lost to explain the bluest stars (assuming that mass loss is the primary cause of the dispersion in color along the HB). The decreased importance of mass loss, compared with the predictions of older models, is one of the consequences of the reduced rate of the ¹⁴N(p, γ)¹⁵O reaction (Marta et al. 2008). As shown by Pietrinferni et al. (2010), this revision causes a ZAHB model for a fixed mass and chemical abundances to be shifted to significantly higher temperatures—an effect that is especially noticeable at low metal abundances.

The small offset between the predicted and observed MS and TO colors that are apparent in panels (a) and (b) can be eliminated if models for either a slightly higher helium abundance (Y = 0.260 instead of 0.250) or a somewhat lower metallicity ([Fe/H] =−2.55 rather than −2.23) are overlaid onto the observed CMD (see panels (c) and (d)). To be sure, T_eff, color, and other uncertainties are large enough that this is no more than suggestive. Still, the improved agreement leads one to wonder if VandenBerg et al. (2013) generally found it necessary to shift isochrone (but not ZAHB) colors to the blue by typically ∼0.02 in their large survey of GC ages because they adopted helium abundances that were too low, or more likely, [Fe/H] values that were too high. Although Y = 0.250 is within current uncertainties of recent determinations of the primordial abundance (0.2485 ± 0.0016; Komatsu et al. 2011), it is possible that a somewhat larger value is more appropriate for Population II stars—perhaps especially those found in GCs, since some processing of the gas through the AGB stars of an earlier stellar generation (see, e.g., Gratton et al. 2012) or via an initial supermassive star (Denissenkov & Hartwick 2014) is suggested by the chemical abundance variations found in them.

Insofar as the possibility that M 92 has [Fe/H] $\mathrel {{\raise0.6 ex {<}\lower.5 ex{\sim }}}-2.5$ is concerned, we note that Roederer & Sneden (2011) have obtained [Fe/H] =−2.70 ± 0.03 from a spectroscopic analysis of 19 of its red giants. Similar investigations by Preston et al. (2006) and Sobeck et al. (2011) have yielded [Fe/H] $\mathrel {{\raise0.6 ex {<}\lower.5 ex{\sim }}}-2.6$ for M 15, which has generally been found to have nearly the same iron abundance as M 92 (see, e.g., Zinn & West 1984; Carretta & Gratton 1997; Kraft & Ivans 2003). According to Roederer & Sneden (also see Sobeck et al.) the 0.3 dex larger [Fe/H] values that some of the same investigators had obtained previously for both clusters (e.g., Sneden et al. 2000a, 2000b) are due to differences in the atomic data, the adopted model atmosphere grids, and the treatment of Rayleigh scattering. Additional support for a reduced metallicity is provided in Figure 7, which compares the relative locations of the M 92 RGB and the very metal-deficient field giant HD 122563 (π = 4.22 ± 0.55 mas from Hipparcos; van Leeuwen 2007) on the [(B − V)₀, M_V]-diagram. HD 122563 has measured [Fe/H] values that range from ∼ − 2.8 to ∼ − 2.6 (e.g., Cayrel et al. 2004; Ramírez et al. 2010; Mashonkina et al. 2011) and it seems to have normal α-element and oxygen abundances for its metallicity ([α/Fe] ≈0.45, Kirby & Cohen 2012; [O/Fe] ≳  0.7, Barbuy et al. 2003); consequently, it should lie on the blue side of the cluster giant branch if M 92 has, e.g., [Fe/H] =−2.35 (Carretta et al. 2009).

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If V − K_S colors are plotted instead of B − V, the resultant comparison (not shown) looks qualitatively nearly identical. (Photometry for HD 122563 has been taken from the studies by Cayrel et al. 2004 and Creevey et al. 2012.) The main advantage of examining BV observations is that B − V colors are considerably more sensitive than V − K_S to metallicity differences at low [Fe/H] values. Whereas the 12 Gyr isochrones for [Fe/H] =−2.8, −2.5, and −2.2 that have been plotted are clearly separated from one another, they are nearly coincident on the [(V − K_S)₀, M_V] diagram. Why HD 122563 is offset by such a large amount from M 92 giants, at the same color or at the same luminosity, is not known, but it may be that its measured parallax, which has a fairly large uncertainty, is too high. Consistency with M 92 (if the latter has [Fe/H] $\mathrel {{\raise0.6 ex {<}\lower.5 ex{\sim }}}-2.6$) would be obtained if the correct parallax of HD 122563 were π ≈ 3.2 mas, which is just within the 2σ error bar of the Hipparcos value.

Based on the available evidence, we are inclined to adopt [Fe/H] =−2.5 ± 0.15 for M 92. The error bar encompasses earlier spectroscopic determinations near −2.4 by, e.g., Kraft & Ivans (2003) and Carretta et al. (2009), and recent values $\mathrel {{\raise0.6 ex {<}\lower.5 ex{\sim }}}-2.6$ that have been either derived for M 92 (Roederer & Sneden 2011) or implied by studies of M 15 (Preston et al. 2006; Sobeck et al. 2011) and HD 122563 (as discussed above). If M 92 has [Fe/H] <−2.55, it would need to have [O/Fe] >0.64 in order for our ZAHB models to provide a satisfactory match to the red end of the cluster HB. As shown in panel (d) of Figure 6, the ZAHB model for a mass that is equal to the TO mass of the best-fit isochrone is predicted to have (m_F606W − m_F814W)₀ = 0.19, which is just barely compatible with the cluster HB (assuming that the few HB stars with redder colors are evolved stars since they tend to be brighter than a ZAHB; see the other panels). The adoption of a lower metallicity would imply an even bluer ZAHB unless a higher absolute oxygen abundance is assumed: higher [O/H] values imply younger ages at a given TO luminosity. Our analysis suggests that M 92 is younger than 13 Gyr, and it may even be younger than 12 Gyr if its stars have a very high oxygen abundance. Pending further advances in our understanding, an age near 12.5 Gyr is our current best estimate. Thus the subgiants in M 92 appear to be different from the field subgiant, HD 140283, in having both a younger age and (probably) a lower [Fe/H] value.

Before briefly considering the RR Lyrae constraint on cluster distances, the distance and age of M 5 will be derived from fits of isochrones and ZAHBs to its CMD.

As in the case of M 92, one could produce many different fits of stellar models to the CMD of M 5 given current uncertainties in [Fe/H], [O/Fe], and Y, as well as those associated with the cluster distance and reddening. We have chosen to present just two of them. The left-hand panel of Figure 8 shows that a ZAHB for essentially the derived abundances of HD 132475 yields an apparent modulus of 14.43 mag, if E(B − V) = 0.032 (from SF11), which implies a TO age of ≈11.5 Gyr. The ZAHB-based true distance modulus is thus ≈0.10 mag larger than that inferred from HD 132475 (see Figure 4)—a difference that is slightly larger than the 2σ error bar on the M_V of the field subgiant. Indeed, this explains why the ages that have been determined for M 5 and HD 132475 (see Figure 3) differ by about 1 Gyr. (VandenBerg et al. 2013 also obtained an age of 11.5 Gyr for M 5 using models that assumed the solar abundances given by Grevesse & Sauval 1998 as the base mixture, with suitably enhanced α-element abundances, and then scaled to [Fe/H] =−1.33 (Carretta et al. 2009). Although they found a smaller modulus by about 0.05 mag, compared with our estimate, the effects on the age of adopting slightly larger values of [Fe/H] and [O/H] in their study compensates almost exactly for this difference.)

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The main difficulty with the fit to the observations in the left-hand panel is that the predicted colors for the MS and TO are too red (though by only 0.01–0.015 mag, which is well within the color uncertainties). As suggested in the previous subsection, this discrepancy may be telling us that the adopted [Fe/H] value is too high. The right-hand panel shows that the best-fit isochrone for [Fe/H] =−1.55 reproduces the MS and TO photometry of M 5 very well without requiring any horizontal offset. (This is obviously not a compelling argument for a reduced [Fe/H] value, but neither can one rule out this possibility given current uncertainties in metal abundance determinations.) Not unexpectedly, the same age, to within the small fitting uncertainties, is obtained because both the ZAHB and the TO luminosity at a given age become brighter as the [Fe/H] value is reduced (though not at identical rates).

However, we can make use of an additional constraint to shed some light on this issue; namely, subdwarfs in the solar neighborhood with [Fe/H] values similar to that of M 5 (and HD 132475) that also have well determined parallaxes from Hipparcos. In Figure 9, all of the subdwarfs that we have been able to identify with M_V > 5.0, σ_π/π < 0.10 (van Leeuwen 2007), and [Fe/H] values within ±0.15 dex of our determination for HD 132475 have been superimposed onto the CMD of M 5. For the latter, we have adopted E(B − V) = 0.032 (from SF11) and two different values of the apparent distance modulus. The higher value, (m − M)_V = 14.44 (left-hand panel), implies the same true distance modulus as in the left-hand panel of the previous figure when the differences in A_V and A_F606W (see, e.g., McCall 2004) are taken into account. The lower value, (m − M)_V = 14.34 (right-hand panel) is obtained when the CMD of M 5 is fitted to HD 132475 (recall Figure 4). The [Fe/H] values, the photometry, and the adopted reddenings of the subdwarfs have all been taken from the study by Casagrande et al. (2010), while the M 5 observations are the same as those employed in the analysis of the MARCS color–T_eff relations by VandenBerg et al. (2010).

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It turns out that the mean [Fe/H] value of the six subdwarfs that satisfied our criteria is identical to the value that we have derived for HD 132475 (i.e., −1.51). (Due to diffusive effects, the initial [Fe/H] values are expected to be larger than the observed values by ∼0.1 dex; recall the discussion in Section 3.4.) Small blueward or redward color offsets (<0.01 mag) were applied to the subdwarf colors, as necessary, to compensate for the effects of differences between the observed and mean metallicities. These corrections were based on the differences in the predicted colors, at the subdwarf M_V values, of isochrones for the observed range in [Fe/H]. Lutz–Kelker corrections have not been applied to the absolute magnitudes of the subdwarfs: they are, in any case, quite small (0.03–0.05 mag for the two brightest stars, <0.02 mag for the others).

Effectively, Figure 9 is telling us that the apparent distance modulus of M 5, as derived from fits of the cluster main-sequence to the sample of subdwarf calibrators that we have considered is approximately (m − M)_V = 14.40. (Although not shown, the adoption of a value as low as 14.30 or as high as 14.50 causes obvious discrepancies between the subdwarfs and the M 5 CMD.) While a common age for M 5 and HD 132475 is within the uncertainties of the subdwarf-based distance (see the right-hand panel), distance moduli based on RR Lyrae stars tend to favor values of (m − M)_V ∼ 14.50 (e.g., Coppola et al. 2011). For this reason, and because the modulus based on our ZAHB models is within the uncertainties of the values derived from the subdwarf and RR Lyrae standard candles (see the next section), we consider (m − M)_V = 14.45 to be our "best estimate" (rounded to the nearest 0.05 mag) of the M 5 modulus. In this case, M 5 is predicted to be ≈1 Gyr younger than HD 132475.

At the present time, the best empirical determination of the slope of the RR Lyrae M_V versus [Fe/H] relation is Δ M_V/Δ [Fe/H] =0.214 ± 0.047 by Clementini et al. (2003) from their analysis of >100 variables in the Large Magellanic Cloud (LMC). If the true distance modulus of the LMC is 18.50, which is within the 2% uncertainty of the distance derived from eight long-period eclipsing binary members by Pietryński et al. (2013), the mean V magnitude of the RR Lyraes observed by Clementini et al. (〈V₀〉 =19.064 ± 0.064 at [Fe/H] =−1.50) implies that their mean absolute magnitude is M_V = 0.564 ± 0.064 mag at the reference metallicity. These results are plotted in Figure 10 as the open circle and the dashed line. Dotted lines have been included in this figure to illustrate the effect of adopting a steeper or shallower slope by the derived 1σ uncertainty of this quantity. The findings of Benedict et al. (2011), who obtained M_V = 0.45 ± 0.05 from FGS parallaxes of a small sample of field RR Lyraes, and of Kollmeier et al. (2013), who derived M_V = 0.59 ± 0.10 using the statistical parallax technique, are also shown.

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According to M. Catelan (2014, private communication), the mean apparent magnitudes of the RR Lyrae variables in M 5 and M 92 are 15.062 and 15.081, respectively. Judging from the results presented in the two previous subsections, M 5 has (m − M)_V = 14.45 ± 0.07 and M 92 has (m − M)_V = 14.77 ± 0.07. Moreover, [Fe/H] =−1.40 ± 0.12 and −2.50 ± 0.15 appear to be reasonable estimates of the iron abundances of M 5 and M 92, in turn, based on the available (direct and indirect) evidence.¹⁷ Needless to say, it is encouraging to find rather good consistency of these determinations with the empirical results obtained by Clementini et al. (2003) and Kollmeier et al. (2013), to well within their 1σ uncertainties, and with those reported by Benedict et al. (2011), to within 1.5σ (see Figure 10). Furthermore, when ZAHB-based distance moduli are adopted, our isochrones, which have been shown to provide good fits to the local subdwarf calibrators (VandenBerg et al. 2013), are able to reproduce the observed main sequences of M 5 and M 92 to within ∼0.01 mag in color (or ∼0.05 mag in magnitude).

As the uncertainties associated with both the RR Lyrae and subdwarf standard candles are still rather large, it is easily possible that the distance moduli derived here are too large or too small by several hundredths of a magnitude. However, something would have to be seriously wrong with our understanding of the HB stars in M 5 and (especially) M 92 if the short distance moduli implied by HD 132475 and HD 140283, respectively, are correct. Our finding that these two field halo subgiants are significantly older than well studied GCs of similar chemical compositions seems compelling.