THE AGES OF EARLY-TYPE STARS: STRÖMGREN PHOTOMETRIC METHODS CALIBRATED, VALIDATED, TESTED, AND APPLIED TO HOSTS AND PROSPECTIVE HOSTS OF DIRECTLY IMAGED EXOPLANETS

Intermediate-mass stars ($1.5-3.0\ {{M}_{\odot }}$) have proven themselves attractive targets for planet search work. Hints of their importance first arose during initial data return from IRAS in the early 1980s, when several A-type stars (notably Vega but also β Pic and Fomalhaut) as well K-star Eps Eri—collectively known as "the fab four"—distinguished themselves by showing mid-infrared excess emission due to optically thin dust in Kuiper-Belt-like locations. Debris disks are signposts of planets, which dynamically stir small bodies resulting in dust production. Spitzer results in the late 2000s solidified the spectral type dependence of debris disk presence (e.g., Carpenter et al. 2006; Wyatt 2008) for stars of common age. For a random sample of field stars, however, the primary variable determining the likelihood of debris is stellar age (Kains et al. 2011).

The correlation in radial velocity studies of giant planet frequency with stellar mass (Fischer & Valenti 2005; Gaidos et al. 2013) is another line of evidence connecting planet formation efficiency to stellar mass. The claim is that while ∼14% of A stars have one or more $\gt 1{{M}_{{\rm Jupiter}}}$ companions at <5 AU, only ∼2% of M stars do (Johnson et al. 2010, see Lloyd 2013; Schlaufman & Winn 2013).

Consistently interpreted as indicators of hidden planets, debris disks finally had their long-awaited observational connection to planets with the watershed discovery of directly imaged planetary mass companions. These were—like the debris disks before them—found first around intermediate-mass A-type stars, rather than the solar-mass FGK-type stars that had been the subject of much observational work at high contrast during the 2000 s. HR 8799 (Marois et al. 2008, 2010) followed by Fomalhaut (Kalas et al. 2008) and β Pic (Lagrange et al. 2009, 2010) have had their planets and indeed one planetary system, digitally captured by ground-based and/or space-based high contrast imaging techniques. Of the known bona fide planetary mass ($\lt 10{{M}_{{\rm Jup}}}$) companions that have been directly imaged, six of the nine are located around the three A-type host stars mentioned above, with the others associated with lower mass stars including the even younger 5–10 Myr old star 1RXS 1609–2105 (Lafrenière et al. 2008; Ireland et al. 2011) and brown dwarf 2MASS 1207–3933 (Chauvin et al. 2004) and the probably older GJ 504 (Kuzuhara et al. 2013). Note that to date these directly imaged objects are all "super-giant planets" and not solar system giant planet analogs (e.g., Jupiter mass or below).

Based on the early results, the major direct imaging planet searches have attempted to optimize success by preferentially observing intermediate-mass, early-type stars. The highest masses are avoided due to the limits of contrast. Recent campaigns include those with all the major large aperture telescopes: Keck/NIRC2, VLT/NACO, Gemini/NICI, and Subaru/HiCAO. Current and near-future campaigns include Project 1640 (P1640; Hinkley et al. 2011) at Palomar Observatory, Gemini Planet Imager, operating on the Gemini South telescope, VLT/SPHERE, and Subaru/CHARIS. The next-generation TMT and E-ELT telescopes both feature high contrast instruments.

Mawet et al. (2012) compares instrumental contrast curves in their Figure 1. Despite the technological developments over the past decade, given the as-built contrast realities, only the largest, hottest, brightest, and therefore the youngest planets, i.e., those less than a few to a few hundred Myr in age, are still self-luminous enough to be amenable to direct imaging detection. Moving from the 3–10 ${{M}_{{\rm Jupiter}}}$ detections at several tens of AU that are possible today/soon, to detection of lower mass, more Earth-like planets located at smaller, more terrestrial zone, separations, will require pushing to higher contrast from future space-based platforms. The targets of future surveys, whether ground or space, are however not likely to be substantially different from the samples targeted in today's ground-based surveys.

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The most important parameter really is age, since the brightness of planets decreases so sharply with increasing age due to the rapid gravitational contraction and cooling (Burrows et al. 2004; Fortney et al. 2008). There is thus a premium on identifying the closest, youngest stars.

Unlike the other fundamental parameters of stellar mass (unambiguously determined from measurements of double-line eclipsing binaries and application of Kepler's laws) and stellar radius (unambiguously measured from interferometric measurements of angular diameters and parallax measurements of distances), there are no directly interpretable observations leading to stellar age.

Solar-type stars ($\sim 0.7-1.4\;{{M}_{\odot }}$, spectral types F6-K5) were the early targets of radial velocity planet searches and later debris disk searches that can imply the presence of planets. For these objects, although more work remains to be done, there are established activity-rotation-age diagnostics that are driven by the presence of convective outer layers and can serve as proxies for stellar age (e.g., Mamajek & Hillenbrand 2008).

For stars significantly different from our Sun, however, and in particular the intermediate-mass stars ($\sim 1.5-3.0\;{{M}_{\odot }}$, spectral types A0-F5 near the main sequence) of interest here, empirical age-dating techniques have not been sufficiently established or calibrated. Ages have been investigated recently for specific samples of several tens of stars using color–magnitude diagrams by Nielsen et al. (2013), Vigan et al. (2012), Moór et al. (2006), Su et al. (2006), Rhee et al. (2007), and Lowrance et al. (2000).

Perhaps the most robust ages for young BAF stars come from clusters and moving groups, which contain not only the early-type stars of interest, but also lower mass stars to which the techniques mentioned above can be applied. These groups are typically dated using a combination of stellar kinematics, lithium abundances, rotation-activity indicators, and placement along theoretical isochrones in a color–magnitude diagram. The statistics of these coeval stellar populations greatly reduce the uncertainty in derived ages. However, only four such groups exist within ∼60 pc of the Sun and the number of early-type members is small.

Field BAF stars having late-type companions at wide separation could have ages estimated using the methods valid for F6-K5 age dating. However, these systems are not only rare in the solar neighborhood, but considerable effort is required in establishing companionship (e.g., Stauffer & Hartmann (1995), Barrado et al. (1997), Song et al. (2000)). Attempts to derive fractional main sequence ages for A-stars based on the evolution of rotational velocities are ongoing (Zorec & Royer 2012), but this method is undeveloped and a bimodal distribution in $v\;{\rm sin} \;i$ for early type A-stars may inhibit its utility. Another method, asteroseismology that detects low-order oscillations in stellar interiors to determine the central density and hence age, is a heavily model-dependent method, observationally expensive, and best suited for older stars with denser cores.

The most general and quantitative way to age date A0-F5 field stars is through isochrone placement. As intermediate-mass stars evolve quickly along the H–R diagram, they are better suited for age dating via isochrone placement relative to their low-mass counterparts that remain nearly stationary on the main sequence for many Gyr (Soderblom 2010). Indeed, the mere presence of an early-type star on the main sequence suggests moderate youth, since the hydrogen burning phase is relatively short-lived. However, isochronal ages are obviously model-dependent and they do require precise placement of the stars on an H–R diagram implying a parallax. The major uncertainties arise from lack of information regarding metallicity (Nielsen et al. 2013), rotation (Collins & Smith 1985), and multiplicity (De Rosa et al. 2014).

Despite that many nearby BAF stars are well-studied, historically, there is no modern data set leading to a set of consistently derived stellar ages for this population of stars. Here we apply Strömgren photometric techniques, and by combining modern stellar atmospheres and modern stellar evolutionary codes, we develop the methods for robust age determination for stars more massive than the Sun. The technique uses specific filters, careful calibration, definition of photometric indices, correction for any reddening, interpolation from index plots of physical atmospheric parameters, correction for rotation, and finally Bayesian estimation of stellar ages from evolutionary models that predict the atmospheric parameters as a function of mass and age.

Specifically, our work uses high-precision archival $uvby\beta$ photometry and model atmospheres so as to determine the fundamental stellar atmospheric parameters ${{T}_{{\rm eff}}}$ and ${\rm log} g$. Placing stars accurately in an ${\rm log} {{T}_{{\rm eff}}}$ versus ${\rm log} g$ diagram leads to derivation of their ages and masses. We consider Bressan et al. (2012) evolutionary models that include pre-main sequence evolutionary times (2 Myr at 3 ${{M}_{\odot }}$ and 17 Myr at 1.5 ${{M}_{\odot }}$), which are a significant fraction of any intermediate mass star's absolute age, as well as Ekström et al. (2012) evolutionary models that self-consistently account for stellar rotation, which has non-negligible effects on the inferred stellar parameters of rapidly rotating early-type stars. Figure 1 shows model predictions for the evolution of both physical and observational parameters.

The primary sample to which our technique is applied in this work consists of 3499 BAF field stars within 100 pc and with $uvby\beta$ photometry available in the Hauck & Mermilliod (1998) catalog, hereafter HM98. The robustness of our method is tested at different stages with several control samples. To assess the uncertainties in our atmospheric parameters we consider (1) 69 T_eff standard stars from Boyajian et al. (2013) or Napiwotzki et al. (1993); (2) 39 double-lined eclipsing binaries with standard ${\rm log} g$ from Torres et al. (2010); and (3) 16 other stars from Napiwotzki et al. (1993), also for examining ${\rm log} g$. To examine isochrone systematics, stars in four open clusters are studied (31 members of IC 2602, 51 members of α Per, 47 members of the Pleiades, and 47 members of the Hyades). Some stars belonging to sample (1) above are also contained in the large primary sample of field stars.

The $uvby\beta$ photometric system is comprised of four intermediate-band filters (uvby) first advanced by Strömgren (1966) plus the Hβ narrow and wide filters developed by Crawford (1958); see Figure 2. Together, the two filter sets form a well-calibrated system that was specifically designed for studying earlier-type BAF stars, for which the hydrogen line strengths and continuum slopes in the Balmer region rapidly change with temperature and gravity.

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From the fluxes contained in the six passbands, five $uvby\beta$ indices are defined. The color indices, ($b-y$) and ($u-b$), and the β-index,

$$\begin{eqnarray}&&\beta ={\rm H}{{\beta }_{{\rm narrow}}}-{\rm H}{{\beta }_{{\rm wide}}},\end{eqnarray} \tag{ 1 }$$

are all sensitive to temperature and weakly dependent on surface gravity for late A- and F-type stars. The Balmer discontinuity index,

$$\begin{eqnarray}&&{{c}_{1}}=(u-v)-(v-b),\end{eqnarray} \tag{ 2 }$$

is sensitive to temperature for early type (OB) stars and surface gravity for intermediate (AF) spectral types. Finally, the metal line index,

$$\begin{eqnarray}&&{{m}_{1}}=(v-b)-(b-y),\end{eqnarray} \tag{ 3 }$$

is sensitive to the metallicity [M/H].

For each index, there is a corresponding intrinsic, dereddened index denoted by a naught subscript with, e.g., ${{c}_{0}},{{(b-y)}_{0}}$ and ${{(u-b)}_{0}}$, referring to the intrinsic, dereddened equivalents of the indices ${{c}_{1}},(b-y)$ and $(u-b)$, respectively. Furthermore, although reddening is expected to be negligible for the nearby sources of primary interest to us, automated classification schemes that divide a large sample of stars for analysis into groups corresponding to earlier than, around, and later than the Balmer maximum will sometimes rely on the reddening-independent indices defined by Crawford & Mandwewala (1976) for A type dwarfs:

$$\begin{eqnarray}&&[{{c}_{1}}]={{c}_{1}}-0.19(b-y)\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&[{{m}_{1}}]={{m}_{1}}+0.34(b-y)\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&[u-b]=[{{c}_{1}}]+2[{{m}_{1}}].\end{eqnarray} \tag{ 6 }$$

Finally, two additional indices useful for early A-type stars, a₀ and ${{r}^{*}}$, are defined as follows:

$$\begin{eqnarray}&&{{a}_{0}}=1.36{{(b-y)}_{0}}+0.36{{m}_{0}}+0.18{{c}_{0}}-0.2448\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&=\;{{(b-y)}_{0}}+0.18[{{(u-b)}_{0}}-1.36],\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{{r}^{*}}=0.35{{c}_{1}}-0.07(b-y)-(\beta -2.565).\end{eqnarray} \tag{ 9 }$$

Note that r^* is a reddening free parameter, and thus indifferent to the use of reddened or unreddened photometric indices.

Though the sample of nearby stars to which we apply the Strömgren methodology are assumed to be unextincted or only lightly extincted, interstellar reddening is significant for the more distant stars including those in the open clusters used in Section 6 to test the accuracy of the ages derived using our $uvby\beta$ methodology. In the cases where extinction is thought to be significant, corrections are performed using the UVBYBETA ³ and DEREDD ⁴ programs for IDL.

These IDL routines take as input $(b-y),{{m}_{1}},{{c}_{1}},\beta$, and a class value (between 1 and 8) that is used to roughly identify what region of the H–R diagram an individual star resides in. For our sample, stars belong to only four of the eight possible classes. These classes are summarized as follows: (1) B0-A0, III–V, $2.59\lt \beta \lt 2.88$, $-0.20\lt {{c}_{0}}\lt 1.00$, (5) A0-A3, III–V, $2.87\lt \beta \lt 2.93$, $-0.01\lt {{(b-y)}_{0}}\lt 0.06$, (6) A3-F0, III–V, $2.72\lt \beta \lt 2.88$, $0.05\lt {{(b-y)}_{0}}\lt 0.22$, and (7) F1-G2, III–V, $2.60\lt \beta \lt 2.72$, $0.22\lt {{(b-y)}_{0}}\lt 0.39$. The class values in this work were assigned to individual stars based on their known spectral types (provided in the XHIP catalog; Anderson & Francis 2011), and β values where needed. In some instances, A0–A3 stars assigned to class (5) with values of $\beta \lt 2.87$, the dereddening procedure was unable to proceed. For these cases, stars were either assigned to class (1) if they were spectral type A0–A1, or to class (6), if they were spectral type A2–A3.

Depending on the class of an individual star, the program then calculates the dereddened indices ${{(b-y)}_{0}},{{m}_{0}},{{c}_{0}}$, the color excess $E(b-y)$, $\delta {{m}_{0}}$, the absolute V magnitude, M_V, and the stellar radius and effective temperature. Notably, the β index is unaffected by reddening as it is the flux difference between two narrow band filters with essentially the same central wavelength. Thus, no corrections are performed on β and this index can be used robustly in coarse classification schemes.

To transform $E(b-y)$ to A_V, we use the extinction measurements of Schlegel et al. (1998) and to propagate the effects of reddening through to the various $uvby\beta$ indices we use the calibrations of Crawford & Mandwewala (1976):

$$\begin{eqnarray}&&E({{m}_{1}})=-0.33E(b-y)\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&E({{c}_{1}})=0.20E(b-y)\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&E(u-b)=1.54E(b-y).\end{eqnarray} \tag{ 12 }$$

From these relations, given the intrinsic color index ${{(b-y)}_{0}}$, the reddening free parameters ${{m}_{0}},{{c}_{0}},{{(u-b)}_{0}},$ and a₀ can be computed.

In Section 4.3 we quantify the effects of extinction and extinction uncertainty on the final atmospheric parameter estimation, ${{T}_{{\rm eff}}},{\rm log} g$.

From the four basic Strömgren indices—$b-y$ color, β, c₁, and m₁—accurate determinations of the stellar atmospheric parameters ${{T}_{{\rm eff}}},{\rm log} g$, and [M/H] are possible for B, A, and F stars. Necessary are either empirical (e.g., Crawford 1979; Lester et al. 1986; Olsen 1988; Smalley 1993; Smalley & Dworetsky 1995; Clem et al. 2004), or theoretical (e.g., Balona 1984; Moon & Dworetsky 1985; Napiwotzki et al. 1993; Balona 1994; Lejeune et al. 1999; Castelli & Kurucz 2004, 2006; Önehag et al. 2009) calibrations. Uncertainties of 0.10 dex in ${\rm log} g$ and 260 K in ${{T}_{{\rm eff}}}$ are claimed as achievable and we reassess these uncertainties ourselves Section 4.3.

Once equipped with $uvby\beta$ colors and indices and understanding the effects of extinction, arriving at the fundamental parameters ${{T}_{{\rm eff}}}$ and ${\rm log} g$ for program stars, proceeds by interpolation among theoretical color grids (generated by convolving filter sensitivity curves with model atmospheres) or explicit formulae (often polynomials) that can be derived empirically or using the theoretical color grids. In both cases, calibration to a sample of stars with atmospheric parameters that have been independently determined through fundamental physics is required (see, e.g., Figueras et al. (1991) for further description).

Numerous calibrations, both theoretical and empirical, of the $uvby\beta$ photometric system exist. For this work we use the Castelli & Kurucz (2006, 2004) color grids generated from solar metallicity (Z = 0.017, in this case) ATLAS9 model atmospheres using a microturbulent velocity parameter of $\xi {\mkern 1mu} =$ 0 km s⁻¹ and the new opacity distribution function (ODF). We do not use the alpha-enhanced color grids. The grids are readily available from F. Castelli⁵ or R. Kurucz.⁶

Prior to assigning atmospheric parameters to our program stars directly from the model grids, we first investigated the accuracy of the models on samples of BAF stars with fundamentally determined T_eff (through interferometric measurements of the angular diameter and estimations of the total integrated flux) and ${\rm log} g$ (from measurements of the masses and radii of double lined eclipsing binaries). We describe these validation procedures in Sections 4.1 and 4.2.

Atmospheric parameter determination occurs in three different observational Strömgren planes depending on the temperature regime (see Figure 3); this is in order to avoid the degeneracies that are present in all single observational planes when mapped onto the physical parameter space of ${\rm log} {{T}_{{\rm eff}}}$ and ${\rm log} g$.

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Building off of the original work of, e.g., Strömgren (1951, 1966), Moon & Dworetsky (1985), and later Napiwotzki et al. (1993), suggested assigning physical parameters in the following three regimes: for cool stars (${{T}_{{\rm eff}}}\leqslant 8500$ K), β or $(b-y)$ can be used as a temperature indicator and c₀ as a surface gravity indicator; for intermediate temperature stars (8500 K $\leqslant \;{{T}_{{\rm eff}}}{\mkern 1mu} \leqslant 11000$ K), the temperature indicator is a₀ and surface gravity indicator r^*; finally, for hot stars (${{T}_{{\rm eff}}}\gtrsim$ 11000 K), the c₀ or the $[u-b]$ indices can be used as a temperature indicator while β is a gravity indicator (note that the role of β is reversed for hot stars compared to its role for cool stars). We adopt here c₁ versus β for the hottest stars, a₀ versus r^* for the intermediate temperatures, and $(b-y)$ versus c₁ for the cooler stars.

Choosing the appropriate plane for parameter determination effectively means establishing a crude temperature sequence prior to fine parameter determination; in this, the β index is critical. Because the β index switches from being a temperature indicator to a gravity indicator in the temperature range of interest to us (spectral type B0-F5, luminosity class IV/V stars), atmospheric parameter determination proceeds depending on the temperature regime. For the T_eff and ${\rm log} g$ calibrations described below, temperature information existed for all of the calibration stars, though this is not the case for our program stars. In the general case we must rely on photometric classification to assign stars to the late, intermediate, and early groups, and then proceed to determine atmospheric parameters in the relevant $uvby\beta$ planes.

Moon (1985) provides a scheme, present in the UVBYBETA IDL routine, for roughly identifying the region of the H–R diagram in which a star resides. However, because our primary sample of field stars are assumed to be unextincted, and because the UVBYBETA program relies on user-inputted class values based on unverified spectral types from the literature, we opt for a classification scheme based solely on the $uvby\beta$ photometry.

Monguió et al. (2014), hereafter M14, designed a sophisticated classification scheme, based on the work of Strömgren (1966). The M14 scheme places stars into early (B0–A0), intermediate (A0–A3), and late (later than A3) groups based solely on β, the reddened color $(b-y)$, and the reddening-free parameters $[{{c}_{1}}],[{{m}_{1}}],{\rm and}\;[u-b]$. The M14 scheme improves upon the previous method of Figueras et al. (1991) by imposing two new conditions (see their Figure 2 for the complete scheme) intended to prevent the erroneous classification of some stars. For our sample of 3499 field stars (see Section 7), there are 699 stars lacking β photometry, all but three of which cannot be classified by the M14 scheme. For such cases, we rely on supplementary spectral type information and manually assign these unclassified stars to the late group. Using the M14 scheme, the final makeup of our field star sample is 85.9% late, 8.4% intermediate, and 5.7% early.

For all stars in this work, $uvby\beta$ photometry is acquired from the Hauck & Mermilliod (1998) compilation (hereafter HM98), unless otherwise noted. HM98 provides the most extensive compilation of $uvby\beta$ photometric measurements, taken from the literature and complete to the end of 1996 (the photometric system has seen less frequent usage/publication in more modern times). The HM98 compilation includes 105,873 individual photometric measurements for 63,313 different stars, culled from 533 distinct sources, and are presented both as individual measurements and weighted means of the literature values.

The HM98 catalog provides $(b-y),{{m}_{1}},{{c}_{1}},$ and β and the associated errors in each parameter if available. From these indices a₀ and r^* are computed according to Equations (7), (8), and (9). The ATLAS9 $uvby\beta$ grids provide a means of translating from ($b-y,{{m}_{1}},{{c}_{1}},\beta ,{{a}_{0}},{{r}^{*}}$) to a precise combination of (${{T}_{{\rm eff}}},{\rm log} g$). Interpolation within the model grids is performed on the appropriate grid: ($(b-y)$ versus c₁ for the late group, a₀ versus r^* for the intermediate group, and c₁ versus β for the early group).

The interpolation is linear and performed using the SciPy routine griddata. Importantly, the model ${\rm log} g$ values are first converted into linear space so that g is determined from the linear interpolation procedure before being brought back into log space. The model grids used in this work are spaced by 250 K in T_eff and 0.5 dex in ${\rm log} g$. To improve the precision of our method of atmospheric parameter determination in the future, it would be favorable to use model color grids that have been calculated at finer resolutions, particularly in ${\rm log} g$, directly from model atmospheres. However, the grid spacings stated above are fairly standardized among extant $uvby\beta$ grids.

Early-type stars are rapid rotators, with rotational velocities of $v{\rm sin} i\gtrsim 150$ km s⁻¹ being typical. For a rotating star, both surface gravity and effective temperature decrease from the poles to the equator, changing the mean gravity and temperature of a rapid rotator relative to a slower rotator (Sweet & Roy 1953). Vega, rotating with an inferred equatorial velocity of ${{v}_{{\rm eq}}}\sim 270$ km s⁻¹ at a nearly pole-on inclination, has measured pole-to-equator gradients in T_eff and ${\rm log} g$ that are ∼2400 K and ∼0.5 dex, respectively (Peterson et al. 2006). The apparent luminosity change due to rotation depends on the inclination: a pole-on ($i=0{}^\circ$) rapid rotator appears more luminous than a nonrotating star of the same mass, while an edge-on ($i=90{}^\circ$) rapid rotator appears less luminous than a nonrotating star of the same mass. Sweet & Roy (1953) found that a ${{(v{\rm sin} i)}^{2}}$ correction factor could describe the changes in luminosity, gravity, and temperature.

The net effect of stellar rotation on inferred age is to make a rapid rotator appear cooler, more luminous, and hence older when compared to a nonrotating star of the same mass (or more massive when compared to a nonrotating star of the same age). Optical colors can be affected since the spectral lines of early type stars are strong and broad. Kraft & Wrubel (1965) demonstrated specifically in the Strömgren system that the effects are predominantly in the gravity indicators (c₁, which then also affects the other gravity indicator r^*) and less so in the temperature indicators ($b-y$, which then affects a₀).

Figueras & Blasi (1998), hereafter FB98, used Monte-Carlo simulations to investigate the effect of rapid rotation on the measured $uvby\beta$ indices, derived atmospheric parameters, and hence isochronal ages of early-type stars. Those authors concluded that stellar rotation conspires to artificially enhance isochronal ages derived through $uvby\beta$ photometric methods by 30%–50% on average.

To mitigate the effect of stellar rotation on the parameters ${{T}_{{\rm eff}}}$ and ${\rm log} (g)$, FB98 presented the following corrective formulae for stars with ${{T}_{{\rm eff}}}\gt 11000$ K:

$$\begin{eqnarray}&&{\Delta}{{T}_{{\rm eff}}}=0.0167{{(v\;{\rm sin} \;i)}^{2}}+218,\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{\Delta}{\rm log} \;g=2.10\times {{10}^{-6}}{{(v\;{\rm sin} \;i)}^{2}}+0.034.\end{eqnarray} \tag{ 14 }$$

For stars with $8500\;{\rm K}\leqslant {{T}_{{\rm eff}}}\leqslant 11000\;{\rm K}$, the analogous formulae are:

$$\begin{eqnarray}&&{\Delta}{{T}_{{\rm eff}}}=0.0187{{(v\;{\rm sin} \;i)}^{2}}+150,\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{\Delta}{\rm log} \;g=2.92\times {{10}^{-6}}{{(v\;{\rm sin} \;i)}^{2}}+0.048.\end{eqnarray} \tag{ 16 }$$

In both cases, ${\Delta}{{T}_{{\rm eff}}}$ and ${\Delta}{\rm log} g$ are added to the T_eff and ${\rm log} g$ values derived from $uvby\beta$ photometry.

Notably, the rotational velocity correction is dependent on whether the star belongs to the early, intermediate, or late group. Specifically, FB98 define three regimes: ${{T}_{{\rm eff}}}\lt 8830$ K (no correction), 8830 K < T_eff < 9700 K (correction for intermediate A0-A3 stars), and T_eff > 9700 K (correction for stars earlier than A3).

Song et al. (2001), who performed a similar isochronal age analysis of A-type stars using $uvby\beta$ photometry, extended the FB98 rotation corrections to stars earlier and later than B7 and A4, respectively. In the present work, a more conservative approach is taken and the rotation correction is applied only to stars in the early or intermediate groups, as determined by the classification scheme discussed in Section 3.1. This decision was partly justified by the abundance of late-type stars that fall below the ZAMS in the open cluster tests (Section 6), for which the rotation correction would have a small (due to the lower rotational velocities of late-type stars) but exacerbating effect on these stars whose surface gravities are already thought to be overestimated.

We include these corrections and, as illustrated in Figure 4, emphasize that in their absence we would err on the side of over-estimating the age of a star, meaning conservatively overestimating rather than underestimating companion masses based on assumed ages. As an example, for a star with ${{T}_{{\rm eff}}}\;\approx$ 13,275 K and log g ≈ 4.1, assumed to be rotating edge-on at 300 km s⁻¹, neglecting to apply the rotation correction would result in an age of ∼100 Myr. Applying the rotation correction to this star results in an age of ∼10 Myr.

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Of note, the FB98 corrections were derived for atmospheric parameters determined using the synthetic $uvby\beta$ color grids of Moon & Dworetsky (1985). It is estimated that any differences in derived atmospheric parameters resulting from the use of color grids other than those of Moon & Dworetsky (1985) are less than the typical measurement errors in those parameters. In Section 4.3 we quantify the effects of rotation and rotation correction uncertainty on the final atmospheric parameter estimation, ${{T}_{{\rm eff}}},{\rm and}{\rm log} g$.

A fundamental determination of T_eff is possible through an interferometric measurement of the stellar angular diameter and an estimate of the total integrated flux. We gathered 69 stars (listed in Table 1) with fundamental T_eff measurements from the literature and determine photometric temperatures for these objects from interpolation of $uvby\beta$ photometry in ATLAS9 model grids.

Table 1.  Stars with Fundamental Determinations of T_eff through Interferometry

HD	Sp. Type	Tfund	Radius Ref.a	${{T}_{uvby\beta }}$	${\rm log} {{g}_{uvby\beta }}$	[Fe/H]	$v{\rm sin} i$	$(b-y)$	m1	c1	β
		(K)		(K)	(dex)	(dex)	(km s−1)	(mag)	(mag)	(mag)	(mag)
4614	F9V	5973 ± 8	3	5915	4.442	−0.28	1.8	0.372	0.185	0.275	2.588
5015	F8V	5965 ± 35	3	6057	3.699	0.04	8.6	0.349	0.174	0.423	2.613
5448	A5V	8070	18	8350	3.964	...	69.3	0.068	0.189	1.058	2.866
6210	F6Vb	6089 ± 35	1	5992	3.343	−0.01	40.9	0.356	0.183	0.475	2.615
9826	F8V	6102 ± 75	2,4	6084	3.786	0.08	8.7	0.346	0.176	0.415	2.629
16765	F7V	6356 ± 46	1	6330	4.408	−0.15	30.5	0.318	0.160	0.355	2.647
16895	F7V	6153 ± 25	3	6251	4.118	0.00	8.6	0.325	0.160	0.392	2.625
17081	B7V	12820	18	12979	3.749	0.24	23.3	−0.057	0.104	0.605	2.717
19994	F8.5 V	5916 ± 98	2	5971	3.529	0.17	7.2	0.361	0.185	0.422	2.631
22484	F9IV-V	5998 ± 39	3	5954	3.807	−0.09	3.7	0.367	0.173	0.376	2.615
30652	F6IV-V	6570 ± 131	3,6	6482	4.308	0.00	15.5	0.298	0.163	0.415	2.652
32630	B3V	17580	18	16536	4.068	...	98.2	−0.085	0.104	0.318	2.684
34816	B0.5IV	27580	18	28045	4.286	−0.06	29.5	−0.119	0.073	−0.061	2.602
35468	B2III	21230	18	21122	3.724	−0.07	53.8	−0.103	0.076	0.109	2.613
38899	B9IV	10790	18	11027	3.978	−0.16	25.9	−0.032	0.141	0.906	2.825
47105	AOIV	9240	18	9226	3.537	−0.28	13.3	0.007	0.149	1.186	2.865
48737	F5IV-V	6478 ± 21	3	6510	3.784	0.14	61.8	0.287	0.169	0.549	2.669
48915	A0mA1Va	9755 ± 47	7,8,9,10,11	9971	4.316	0.36	15.8	−0.005	0.162	0.980	2.907
49933	F2Vb	6635 ± 90	12	6714	4.378	−0.39	9.9	0.270	0.127	0.460	2.662
56537	A3Vb	7932 ± 62	3	8725	4.000	...	152	0.047	0.198	1.054	2.875
58946	F0Vb	6954 ± 216	3,18	7168	4.319	−0.25	52.3	0.215	0.155	0.615	2.713
61421	F5IV-V	6563 ± 24	11,13,14,15,18	6651	3.983	−0.02	4.7	0.272	0.167	0.532	2.671
63922	BOIII	29980	18	29973	4.252	0.16	40.7	−0.122	0.043	−0.092	2.590
69897	F6V	6130 ± 58	1	6339	4.290	−0.26	4.3	0.315	0.149	0.384	2.635
76644	A7IV	7840	18	8232	4.428	−0.03	142	0.104	0.216	0.856	2.843
80007	A2IV	9240	18	9139	3.240	...	126	0.004	0.140	1.273	2.836
81937	F0IVb	6651 ± 27	3	7102	3.840	0.17	146	0.211	0.180	0.752	2.733
82328	F5.5IV-V	6299 ± 61	3,18	6322	3.873	−0.16	7.1	0.314	0.153	0.463	2.646
90839	F8V	6203 ± 56	3	6145	4.330	−0.11	8.6	0.341	0.171	0.333	2.618
90994	B6V	14010	18	14282	4.219	...	84.5	−0.066	0.111	0.466	2.730
95418	A1IV	9181 ± 11	3,18	9695	3.899	−0.03	40.8	−0.006	0.158	1.088	2.880
97603	A5IV(n)	8086 ± 169	3,6,18	8423	4.000	−0.18	177	0.067	0.195	1.037	2.869
102647	A3Va	8625 ± 175	5,6,18	8775	4.188	0.07	118	0.043	0.211	0.973	2.899
102870	F8.5IV-V	6047 ± 7	3,18	6026	3.689	0.12	5.4	0.354	0.187	0.416	2.628
118098	A2Van	8097 ± 43	3	8518	4.163	−0.26	200	0.065	0.183	1.006	2.875
118716	B1III	25740	18	23262	3.886	...	113	−0.112	0.058	0.040	2.608
120136	F7IV-V	6620 ± 67	2	6293	3.933	0.24	14.8	0.318	0.177	0.439	2.656
122408	A3V	8420	18	8326	3.500	−0.27	168	0.062	0.164	1.177	2.843
126660	F7V	6202 ± 35	3,6,18	6171	3.881	−0.02	27.7	0.334	0.156	0.418	2.644
128167	F4VkF2mF1	6687 ± 252	3,18	6860	4.439	−0.32	9.3	0.254	0.134	0.480	2.679
130948	F9IV-V	5787 ± 57	1	5899	4.065	−0.05	6.3	0.374	0.191	0.321	2.625
136202	F8IV	5661 ± 87	1	6062	3.683	−0.04	4.9	0.348	0.170	0.427	2.620
141795	kA2hA5mA7V	7928 ± 88	3	8584	4.346	0.38	33.1	0.066	0.224	0.950	2.885
142860	F6V	6295 ± 74	3,6	6295	4.130	−0.17	9.9	0.319	0.150	0.401	2.633
144470	BlV	25710	18	25249	4.352	...	107	−0.112	0.043	−0.005	2.621
162003	F5IV-V	5928 ± 81	3	6469	3.916	−0.03	11.9	0.294	0.147	0.497	2.661
164259	F2V	6454 ± 113	3	6820	4.121	−0.03	66.4	0.253	0.153	0.560	2.690
168151	F5Vb	6221 ± 39	1	6600	4.203	−0.28	9.7	0.281	0.143	0.472	2.653
169022	B9.5III	9420	18	9354	3.117	...	196	0.016	0.102	1.176	2.778
172167	AOVa	9600	18	9507	3.977	−0.56	22.8	0.003	0.157	1.088	2.903
173667	F5.5IV-V	6333 ± 37	3,18	6308	3.777	−0.03	16.3	0.314	0.150	0.484	2.652
177724	A0IV-Vnn	9078 ± 86	3	9391	3.870	−0.52	316	0.013	0.146	1.080	2.875
181420	F2V	6283 ± 106	16	6607	4.187	−0.03	17.1	0.280	0.157	0.477	2.657
185395	F3 + V	6516 ± 203	3,4	6778	4.296	0.02	5.8	0.261	0.157	0.502	2.688
187637	F5V	6155 ± 85	16	6192	4.103	−0.09	5.4	0.333	0.151	0.380	2.631
190993	B3V	17400	18	16894	4.195	−0.14	140	−0.083	0.100	0.295	2.686
193432	B9.5 V	9950	18	10411	3.928	−0.15	23.4	−0.021	0.134	1.015	2.852
193924	B2IV	17590	18	17469	3.928	...	15.5	−0.092	0.087	0.271	2.662
196867	B9IV	10960	18	10837	3.861	−0.06	144	−0.019	0.125	0.889	2.796
209952	B7IV	13850	18	13238	3.913	...	215	−0.061	0.105	0.576	2.728
210027	F5V	6324 ± 139	6	6496	4.187	−0.13	8.6	0.294	0.161	0.446	2.664
210418	A2Vb	7872 ± 82	3	8596	3.966	−0.38	136	0.047	0.161	1.091	2.886
213558	A1Vb	9050 ± 157	3	9614	4.175	...	128	0.002	0.170	1.032	2.908
215648	F6V	6090 ± 22	3	6198	3.950	−0.26	7.7	0.331	0.147	0.407	2.626
216956	A4V	8564 ± 105	5,18	8857	4.198	0.20	85.1	0.037	0.206	0.990	2.906
218396	F0+(λ Boo)	7163 ± 84	17	7540	4.435	...	47.2	0.178	0.146	0.678	2.739
219623	F8V	6285 ± 94	1	6061	3.85	0.04	4.9	0.351	0.169	0.395	2.624
222368	F7V	6192 ± 26	3	6207	3.988	−0.14	6.1	0.330	0.163	0.399	2.625
222603	A7V	7734 ± 80	1	8167	4.318	...	62.8	0.105	0.203	0.891	2.826

Note.

^aInterferometric radii references: (1) Boyajian et al. (2013), (2) Baines et al. (2008), (3) Boyajian et al. (2012), (4) Ligi et al. (2012), (5) Di Folco et al. (2004), (6) van Belle & von Braun (2009), (7) Davis et al. (2011), (8) Hanbury Brown et al. (1974), (9) Davis & Tango (1986), (10) Kervella et al. (2003), (11) Mozurkewich et al. (2003), (12) Bigot et al. (2011), (13) Chiavassa et al. (2012), (14) Nordgren et al. (2001), (15) Kervella et al. (2004), (16) Huber et al. (2012), (17) Baines et al. (2012), (18) Napiwotzki et al. (1993). Note that (18) simply provides means of the T_eff values published by Code et al. (1976); Beeckmans (1977); Malagnini et al. (1986), all three of which used the radii of (9).

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Fundamental T_eff values were sourced from Boyajian et al. (2013), hereafter B13, and Napiwotzki et al. (1993), hereafter N93. Several stars have multiple interferometric measurements of the stellar radius, and hence multiple fundamental T_eff determinations. For these stars, identified as those objects with multiple radius references in Table 1, the mean T_eff and standard deviation were taken as the fundamental measurement and standard error. Among the 16 stars with multiple fundamental T_eff determinations by between 2 and 5 authors, there is a scatter of typically several percent (with 0.1%–4% range).

Additional characteristics of the T_eff "standard" stars are summarized as follows: spectral types B0–F9, luminosity classes III–V, 2 km s⁻¹ $\leqslant {\mkern 1mu} v{\rm sin} i\leqslant 316$ km s⁻¹, mean and median $v{\rm sin} i$ of 58 and 26 km s⁻¹, respectively, 2.6 pc $\leqslant \;d\;\leqslant 493$ pc, and a mean and median [Fe/H] of −0.08 and −0.06 dex, respectively. Line-of-sight rotational velocities were acquired from the Glebocki & Gnacinski (2005) compilation and [Fe/H] values were taken from SIMBAD. Variability and multiplicity were considered, and our sample is believed to be free of any possible contamination due to either of these effects.

From the HM98 compilation we retrieved $uvby\beta$ photometry for these "effective temperature standards." The effect of reddening was considered for the hotter, statistically more distant stars in the N93 sample. Comparing mean $uvby\beta$ photometry from HM98 with the dereddened photometry presented in N93 revealed that nearly all of these stars have negligible reddening ($E(b-y)\leqslant 0.001$ mag). The exceptions are HD 82328, HD 97603, HD 102870, and HD 126660 with color excesses of $E(b-y)=0.010,0.003,0.011,{\rm and}0.022$ mag, respectively. Inspection of Table 1 indicates that despite the use of the reddened HM98 photometry the T_eff determinations for three of these four stars are still of high accuracy. For HD 97603, there is a discrepancy of >300 K between the fundamental and photometric temperatures. However, the $uvby\beta$ T_eff using reddened photometry for this star is actually hotter than the fundamental T_eff. Notably, the author-to-author dispersion in multiple fundamental T_eff determinations for HD 97603 is also rather large. As such, the HM98 photometry was deemed suitable for all of the "effective temperature standards."

For the sake of completeness, different model color grids were investigated, including those of Fitzpatrick & Massa (2005) that were recently calibrated for early group stars, and those of Önehag et al. (2009) that were calibrated from MARCS model atmospheres for stars cooler than 7000 K. We found the grids that best matched the fundamental effective temperatures were the ATLAS9 grids of solar metallicity with no alpha-enhancement, microturbulent velocity of 0 km s⁻¹, and using the new ODF. The ATLAS9 grids with microturbulent velocity of 2 km s⁻¹ were also tested, but were found to worsen both the fractional T_eff error and scatter, though only nominally (by a few tenths of a percent).

For the early group stars, temperature determinations were attempted in both the ${{c}_{1}}-\beta$ and [$u-b$]-β planes. The c₁ index was found to be a far better temperature indicator in this regime, with the [$u-b$] index underestimating T_eff relative to the fundamental values >10% on average. Temperature determinations in the ${{c}_{1}}-\beta$ plane, however, were only ≈1.9% cooler than the fundamental values, regardless of whether c₁ or the dereddened index c₀ was used. This is not surprising as the ${{c}_{1}}-\beta$ plane is not particularly susceptible to reddening.

At intermediate temperatures, the ${{a}_{0}}-{{r}^{*}}$ plane is used. In this regime, the ATLAS9 grids were found to overestimate T_eff by ≈2.0% relative to the fundamental values.

Finally, for the late group stars, temperature determinations were attempted in the $(b-y)-{{c}_{1}}$ and $\beta -{{c}_{1}}$ planes. In this regime, $(b-y)$ was found to be a superior temperature indicator, improving the mean fractional error marginally and reducing the rms scatter by more than 1%. In this group, the model grids overpredict T_eff by ≈2.4% on average, regardless of whether the reddened or dereddened indices are used.

Figure 5 shows a comparison of the temperatures derived from the ATLAS9 $uvby\beta$ color grids and the fundamental effective temperatures given in B13 and N93. For the majority of stars the color grids can predict the effective temperature to within about 5%. A slight systematic trend is noted in Figure 5, such that the model color grids overpredict T_eff at low temperatures and underpredict T_eff at high temperatures. We attempt to correct for this systematic effect by applying T_eff offsets in three regimes according to the mean behavior of each group: late and intermediate group stars were shifted to cooler temperatures by 2.4% and 2.0%, respectively, and early group stars were shifted by 1.9% toward hotter temperatures. After offsets were applied, the remaining rms error in temperature determinations for these "standard" stars was 3.3%, 2.5%, and 3.5% for the late, intermediate, and early groups, respectively, or 3.1% overall.

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Taking the uncertainties or dispersions in the fundamental T_eff determinations as the standard error, there is typically a 5–6 σ discrepancy between the fundamental and photometric T_eff determinations. However, given the large author-to-author dispersion observed for stars with multiple fundamental T_eff determinations, it is likely that the formal errors on these measurements are underestimated. Notably, N93 does not publish errors for the fundamental T_eff values, which are literature means. However, those authors did find fractional errors in their photometric T_eff ranging from 2.5%–4% for BA stars.

In Section 6, we opted not to apply systematic offsets, instead assigning T_eff uncertainties in three regimes according to the average fractional uncertainties noted in each group. In our final T_eff determinations for our field star sample (Section 7) we attempted to correct for the slight temperature systematics and applied offsets, using the magnitude of the remaining rms error (for all groups considered collectively) as the dominant source of uncertainty in our T_eff measurement (see Section 4.3).

As demonstrated in Figure 6, rotational effects on our temperature determinations for the T_eff standards were investigated. Notably, the FB98 $v{\rm sin} i$ corrections appear to enhance the discrepancy between our temperature determinations and the fundamental temperatures for the late and intermediate groups, while moderately improving the accuracy for the early group. For the late group this is expected, as the correction formulae were originally derived for intermediate and early group stars. Notably, however, only two stars in the calibration sample exhibit projected rotational velocities $\gt 200$ km s⁻¹. We examine the utility of the $v{\rm sin} i$ correction further in Sections 4.2 and 6.

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The effect of metallicity on the determination of T_eff from the $uvby\beta$ grids is investigated in Figure 7 showing the ratio of the grid-determined temperature to the fundamental temperature as a function of [Fe/H]. The sample of temperature standards spans a large range in metallicity, yet there is no indication of any systematic effect with [Fe/H], justifying our choice to assume solar metallicity throughout this work (see further discussion of metallicity effects in the Appendix).

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The effect of reddening on our temperature determinations was considered but since the vast majority of sources with fundamental effective temperatures are nearby, no significant reddening was expected. Indeed, no indication of a systematic trend of the temperature residuals as a function of distance was noted.

In summary our findings that the ATLAS9 predicted T_eff values are ∼2% hotter than fundamental values for AF stars are consistent with the results of Bertone et al. (2004), who found 4%–8% shifts warmer in T_eff from fits of ATLAS9 models to spectrophotometry relative to T_eff values determined from the infrared flux method. We attempt systematic corrections with offsets of magnitude ∼2% according to group, and the remaining rms error between $uvby\beta$ temperatures and fundamental values is ∼3%.

To assess the surface gravities derived from the $uvby\beta$ grids, we compare to results on both double-lined eclipsing binary and spectroscopic samples.

4.2.1. Comparison with Double-lined Eclipsing Binaries

Torres et al. (2010) compiled an extensive catalog of 95 double-lined eclipsing binaries with fundamentally determined surface gravities for all 180 individual stars. Eclipsing binary systems allow for dynamical determinations of the component masses and geometrical determinations of the component radii. From the mass and radius of an individual component, the Newtonian surface gravity, $g=GM/{{R}^{2}}$, can be calculated.

From these systems, 39 of the primary components have $uvby\beta$ photometry available for determining surface gravities using our methodology. The spectral type range for these systems is O8-F2, with luminosity classes of IV and V. The mass ratio (primary/secondary) for these systems ranges from ≈1.00–1.79, and the orbital periods of the primaries range from ≈1.57–8.44 days. In the cases of low mass ratios, the primary and secondary components should have nearly identical fundamental parameters, assuming they are coeval. In the cases of high mass ratios, given that the individual components are presumably unresolved, we assume that the primary dominates the $uvby\beta$ photometry. For both cases (of low and high mass ratios), we assume that the photometry allows for accurate surface gravity determinations for the primary components and so we only consider the primaries from the Torres et al. (2010) sample.

It is important to note that the eclipsing binary systems used for the surface gravity calibration are more distant than the stars for which we can interferometrically determine angular diameters and effective temperatures for. Thus, for the surface gravity calibration it was necessary to compute the dereddened indices ${{(b-y)}_{0}},{{m}_{0}},{\rm and}\;{{c}_{0}}$ in order to obtain the highest accuracy possible for the intermediate-group stars, which rely on a₀ (an index using dereddened colors) as a temperature indicator. Notably, however, we found that the dereddened photometry actually worsened ${\rm log} g$ determinations for the early and late groups. Dereddened colors were computed using the IDL routine UVBYBETA.

The results of the ${\rm log} g$ calibration are presented in Table 2 and Figure 8. As described above, for the late group stars (${{T}_{{\rm eff}}}\lt 8500$ K), ${\rm log} g$ is determined in the $(b-y)-{{c}_{1}}$ plane. The mean and median of the ${\rm log} g$ residuals (in the sense of grid-fundamental) are −0.001 and −0.038 dex, respectively, and the rms error 0.145 dex. As in Section 4.1, we found that the $\beta -{{c}_{1}}$ plane produced less accurate atmospheric parameters, relative to fundamental determinations, for late group stars.

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Table 2.  Primary Components of Double-lined Eclipsing Binaries with Fundamental Determinations of ${\rm log} g$

Star	Sp. Type	Teff	Tuvby	${\rm log} {{g}_{{\rm EB}}}$	${\rm log} {{g}_{uvby}}$	$v{\rm sin} i$	[Fe/H]	$(b-y)$	m1	c1	β
		(K)	(K)	(dex)	(dex)	(km s−1)	(dex)	(mag)	(mag)	(mag)	(mag)
EM Car	O8V	34000 ± 2000	21987	3.855 ± 0.016	3.878	146.0	...	0.279	−0.042	0.083	2.617
V1034 Sco	O9V	33200 ± 900	28228	3.923 ± 0.008	3.969	159.0	−1.0	0.190	−0.024	−0.068	2.587
AH Cep	B0.5Vn	29900 ± 1000	24867	4.017 ± 0.009	4.115	154.0	...	0.290	−0.064	0.003	2.611
V578 Mon	B1V	30000 ± 740	25122	4.176 ± 0.015	4.200	107.0	...	0.206	−0.024	−0.003	2.613
V453 Cyg	B0.4IV	27800 ± 400	24496	3.725 ± 0.006	3.742	130.0	...	0.212	−0.004	−0.004	2.590
CW Cep	B0.5 V	28300 ± 1000	22707	4.050 ± 0.019	3.716	120.0	...	0.355	−0.077	0.050	2.601
V539 Ara	B3V	18100 ± 500	17537	3.924 ± 0.016	3.964	85.6	...	−0.033	0.089	0.268	2.665
CV Vel	B2.5 V	18100 ± 500	17424	3.999 ± 0.008	3.891	42.8	...	−0.057	0.083	0.273	2.659
AG Per	B3.4 V	18200 ± 800	15905	4.213 ± 0.020	4.311	92.6	−0.04	0.048	0.079	0.346	2.708
U Oph	B5V	16440 ± 250	15161	4.076 ± 0.004	3.954	350.0	...	0.081	0.050	0.404	2.695
V760 Sco	B4V	16900 ± 500	15318	4.176 ± 0.019	4.061	...	...	0.169	0.023	0.392	2.701
GG Lup	B7V	14750 ± 450	13735	4.298 ± 0.009	4.271	123.0	...	−0.049	0.115	0.514	2.747
ζ Phe	B6V	14400 ± 800	13348	4.121 ± 0.004	4.153	111.0	...	−0.039	0.118	0.559	2.747
${{\chi }^{2}}$ Hya	B8V	11750 ± 190	11382	3.710 ± 0.007	3.738	131.0	...	−0.020	0.110	0.841	2.769
V906 Sco	B9V	10400 ± 500	10592	3.656 ± 0.012	3.719	81.3	...	0.039	0.101	0.996	2.805
TZ Men	A0V	10400 ± 500	10679	4.224 ± 0.009	4.169	14.4	...	0.000	0.142	0.918	2.850
V1031 Ori	A6V	7850 ± 500	8184	3.559 ± 0.007	3.793	96.0	...	0.076	0.174	1.106	2.848
β Aur	A1m	9350 ± 200	9167	3.930 ± 0.005	3.894	33.2	−0.11	0.017	0.173	1.091	2.889
V364 Lac	A4m:	8250 ± 150	7901	3.766 ± 0.005	3.707	...	...	0.107	0.168	1.061	2.875
V624 Her	A3m	8150 ± 150	7902	3.832 ± 0.014	3.794	38.0	...	0.111	0.230	1.025	2.870
V1647 Sgr	A1V	9600 ± 300	9142	4.252 ± 0.008	4.087	...	...	0.040	0.174	1.020	2.899
VV Pyx	A1V	9500 ± 200	9560	4.087 ± 0.008	4.004	22.1	...	0.028	0.161	1.013	2.881
KW Hya	A5m	8000 ± 200	8053	4.078 ± 0.006	4.390	16.6	...	0.122	0.232	0.832	2.827
WW Aur	A5m	7960 ± 420	8401	4.161 ± 0.005	4.286	35.8	...	0.081	0.231	0.944	2.862
V392 Car	A2V	8850 ± 200	10263	4.296 ± 0.011	4.211	163.0	...	0.097	0.108	1.019	2.889
RS Cha	A8V	8050 ± 200	7833	4.046 ± 0.022	4.150	30.0	...	0.136	0.186	0.866	2.791
MY Cyg	F0m	7050 ± 200	7054	3.994 ± 0.019	3.882	...	...	0.219	0.226	0.709	2.756
EI Cep	F3V	6750 ± 100	6928	3.763 ± 0.014	3.904	16.2	0.27	0.234	0.199	0.658	2.712
FS Mon	F2V	6715 ± 100	6677	4.026 ± 0.005	3.992	40.0	0.07	0.266	0.148	0.594	2.688
PV Pup	A8V	6920 ± 300	7327	4.255 ± 0.009	4.386	66.4	...	0.200	0.169	0.636	2.722
HD 71636	F2V	6950 ± 140	6615	4.226 ± 0.014	4.104	13.5	0.15	0.278	0.157	0.496	...
RZ Cha	F5V	6450 ± 150	6326	3.905 ± 0.006	3.808	...	0.02	0.312	0.155	0.482	...
BW Aqr	F7V	6350 ± 100	6217	3.979 ± 0.018	3.877	...	...	0.328	0.165	0.432	2.650
V570 Per	F3V	6842 ± 50	6371	4.234 ± 0.019	3.998	44.9	0.06	0.308	0.165	0.441	...
CD Tau	F6V	6200 ± 50	6325	4.087 ± 0.007	3.973	18.9	0.19	0.314	0.178	0.436	...
V1143 Cyg	F5V	6450 ± 100	6492	4.322 ± 0.015	4.155	19.8	0.22	0.294	0.165	0.451	2.663
VZ Hya	F3V	6645 ± 150	6199	4.305 ± 0.003	4.182	...	−0.22	0.333	0.145	0.370	2.629
V505 Per	F5V	6510 ± 50	6569	4.323 ± 0.016	4.325	31.4	−0.03	0.287	0.142	0.435	2.654
HS Hya	F4V	6500 ± 50	6585	4.326 ± 0.005	4.471	23.3	0.14	0.287	0.160	0.397	2.648

Note. Spectral type, temperature, and fundamental ${\rm log} g$ information originate from Torres et al. (2010). The $uvby\beta {\rm log} g$ values are from this work. Projected rotational velocities are from Glebocki & Gnacinski (2005), [Fe/H] from Anderson & Francis (2012) and Ammons et al. (2006), and the $uvby\beta$ photometry are from HM98. The surface gravities in the column ${\rm log} {{g}_{uvby}}$ are derived together with T_uvby, and not for the T_eff values given by Torres et al. (2010).

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For the intermediate group stars (8500 K $\leqslant \;{{T}_{{\rm eff}}}\leqslant 11000$ K), ${\rm log} g$ is determined in the ${{a}_{0}}-{{r}^{*}}$ plane. The mean and median of the ${\rm log} g$ residuals are −0.060 and −0.069 dex, respectively, with rms error 0.091 dex. For the early group stars (${{T}_{{\rm eff}}}\gt 11000$ K), ${\rm log} g$ is determined in the ${{c}_{1}}-\beta$ plane. The mean and median of the ${\rm log} g$ residuals are −0.0215 dex and 0.024 dex, respectively, with rms error 0.113 dex. The $[u-b]-\beta$ plane was also investigated for early group stars, but was found to produce ${\rm log} g$ values of lower accuracy relative to the fundamental determinations.

When considered collectively, the mean and median of the ${\rm log} g$ residuals for all stars are −0.017 and −0.034, and the rms error 0.127 dex. The uncertainties in our surface gravities that arise from propagating the photometric errors through our atmospheric parameter determination routines are of the order ∼0.02 dex, significantly lower than the uncertainties demonstrated by the comparison to fundamental values of ${\rm log} g$.

As stated above, the main concern with using double-lined eclipsing binaries as surface gravity calibrators for our photometric technique is contamination from the unresolved secondary components. The ${\rm log} g$ residuals were examined as a function of both mass ratio and orbital period. While the amplitude of the scatter is marginally larger for low mass ratio or short period systems, in all cases our ${\rm log} g$ determinations are within 0.2 dex of the fundamental values ≈85% of the time.

To assess any potential systematic inaccuracies of the grids themselves, the surface gravity residuals were examined as a function of T_eff and the grid-determined ${\rm log} g$. Figures 9 show the ${\rm log} g$ residuals as a function of T_eff and ${\rm log} g$, respectively. No considerable systematic effects as a function of either effective temperature or ${\rm log} g$ were found in the $uvby\beta$ determinations of ${\rm log} g$.

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The effect of rotational velocity on our ${\rm log} g$ determinations was considered. As before, $v{\rm sin} i$ data for the surface gravity calibrators was collected from Glebocki & Gnacinski (2005). As seen in Figure 10, the majority of the ${\rm log} g$ calibrators are somewhat slowly rotating ($v{\rm sin} i\leqslant 150$ km s⁻¹). While the $v{\rm sin} i$ correction increases the accuracy of our ${\rm log} g$ determinations for the early group stars in most cases, the correction appears to worsen our determinations for the intermediate group, which appear systematically high to begin with.

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The potential systematic effect of metallicity on our ${\rm log} g$ determinations is considered in Figure 11, showing the surface gravity residuals as a function of [Fe/H]. Metallicity measurements were available for very few of these stars, and were primarily taken from Ammons et al. (2006) and Anderson & Francis (2012). Nevertheless, there does not appear to be a global systematic trend in the surface gravity residuals with metallicity. There is a larger scatter in ${\rm log} g$ determinations for the more metal-rich, late-type stars, however it is not clear that this effect is strictly due to metallicity.

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In summary, for the open cluster tests we assign ${\rm log} g$ uncertainties in three regimes: ±0.145 dex for stars belonging to the late group, ±0.091 dex for the intermediate group, and ±0.113 dex for the early group.

For our sample of nearby field stars we opt to assign a uniform systematic uncertainty of ±0.116 dex for all stars. We do not attempt to correct for any systematic effects by applying offsets in ${\rm log} g$, as we did with T_eff. As noted in discussion of the T_eff calibration, we do apply the $v{\rm sin} i$ correction to both intermediate and early group stars, as these corrections permit us to better reproduce open cluster ages (as presented in Section 6).

4.2.2. Comparison with Spectroscopic Measurements

The Balmer lines are a sensitive surface gravity indicator for stars hotter than ${{T}_{{\rm eff}}}\gtrsim$ 9000 K and can be used as a semi-fundamental surface gravity calibration for the early and intermediate group stars. The reason why surface gravities derived using this method are considered semi-fundamental and not fundamental is because the method still relies on model atmospheres for fitting the observed line profiles. Nevertheless, surface gravities determined through this method are considered of high fidelity and so we performed an additional consistency check, comparing our $uvby\beta$ values of ${\rm log} g$ to those with well-determined spectroscopic ${\rm log} g$ measurements.

N93 fit theoretical profiles of hydrogen Balmer lines from Kurucz (1979) to high resolution spectrograms of the Hβ and Hγ lines for a sample of 16 stars with $uvby\beta$ photometry. The sample of 16 stars was mostly drawn from the list of photometric β standards of Crawford (1966). We compared the ${\rm log} g$ values we determined through interpolation in the $uvby\beta$ color grids to the semi-fundamental spectroscopic values determined by N93. The results of this comparison are presented in Table 3.

Table 3.  Stars with Semi-fundamental Determinations of ${\rm log} g$ through Balmer-line Fitting

HR	Sp. Type	Teff	Tuvby	${\rm log} {{g}_{{\rm spec}}}$	${\rm log} {{g}_{uvby}}$	$(b-y)$	m1	c1	$[u-b]$	β
		(K)	(K)	(dex)	(dex)	(mag)	(mag)	(mag)	(mag)	(mag)
63	A2V	8970	9047	3.73	3.912	0.026	0.181	1.050	1.425	2.881
153	B2IV	20930	20635	3.78	3.872	−0.090	0.087	0.134	0.264	2.627
1641	B3V	16890	16528	4.07	4.044	−0.085	0.104	0.319	0.485	2.683
2421	AOIV	9180	9226	3.49	3.537	0.007	0.149	1.186	1.487	2.865
4119	B6V	14570	14116	4.18	4.176	−0.062	0.111	0.481	0.673	2.730
4554	AOVe	9360	9398	3.82	3.863	0.006	0.155	1.112	1.425	2.885
5191	B3V	17320	16797	4.28	4.292	−0.080	0.106	0.297	0.470	2.694
6588	B3IV	17480	17025	3.82	3.864	−0.065	0.079	0.292	0.418	2.661
7001	AOVa	9540	9508	4.01	3.977	0.003	0.157	1.088	1.403	2.903
7447	B5III	13520	13265	3.73	3.712	−0.016	0.088	0.575	0.743	2.707
7906	B9IV	10950	10838	3.85	3.861	−0.019	0.125	0.889	1.130	2.796
8585	A1V	9530	9615	4.11	4.175	0.002	0.170	1.032	1.373	2.908
8634	B8V	11330	11247	3.69	3.672	−0.035	0.113	0.868	1.077	2.768
8781	B9V	9810	9868	3.54	3.593	−0.011	0.128	1.129	1.380	2.838
8965	B8V	11850	11721	3.47	3.422	−0.031	0.100	0.784	0.969	2.725
8976	B9IVn	11310	11263	4.23	4.260	−0.035	0.131	0.831	1.076	2.833

Note. Spectral type, T_eff, and spectroscopic ${\rm log} g$ originate from N93. The $uvby\beta$ T_eff and ${\rm log} g$ values are from this work. Though N93 does not provide formal errors on the atmospheric parameters, those authors estimate uncertainties of ∼0.03 dex in their spectroscopically determined ${\rm log} g$. The fractional errors in their photometrically derived T_eff range from 2.5% for stars cooler than ≈11000 K to 4% for stars hotter than ≈20000 K. The photometry is from HM98.

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Though N93 provide dereddened photometry for the spectroscopic sample, we found using the raw HM98 photometry produced significantly better results (yielding an rms error that was three times lower). For the early group stars, the atmospheric parameters were determined in both the ${{c}_{0}}-\beta$ plane and the $[u-b]-\beta$ plane. In both cases, β is the gravity indicator, but we found that the ${\rm log} g$ values calculated when using c₀ as a temperature indicator for hot stars better matched the semi-fundamental spectroscopic ${\rm log} g$ values. This result is consistent with the result from the effective temperature calibration which suggests c₀ better predicted the effective temperatures of hot stars than $[u-b]$. As before, ${\rm log} g$ for intermediate group stars is determined in the ${{a}_{0}}-{{r}^{*}}$ plane.

We tested $uvby\beta$ color grids of different metallicity, alpha-enhancement, and microturbulent velocity and determined that the non-alpha-enhanced, solar metallicity grids with microturbulent velocity v_turb = 0 km s⁻¹ best reproduced the spectroscopic surface gravities for the sample of 16 early- and intermediate group stars measured by N93.

The ${\rm log} g$ residuals, in the sense of (spectroscopic—grid), as a function of the grid-calculated effective temperatures are plotted in Figure 9. There is no evidence for a significant systematic offset in the residuals as a function of either the $uvby\beta$-determined T_eff or ${\rm log} g$. For the early group, the mean and median surface gravity residuals are −0.007 and 0.004 dex, respectively, with rms 0.041 dex. For the intermediate group, the mean and median surface gravity residuals are −0.053 and −0.047 dex, respectively, with rms 0.081 dex. Considering both early- and intermediate-group stars collectively, the mean and median surface gravity residuals are −0.027 and −0.021 dex, and the rms 0.062 dex.

One issue that may cause statistically larger errors in the ${\rm log} g$ determinations compared to the T_eff determinations is the linear interpolation in a low resolution logarithmic space (the $uvby\beta$ colors are calculated at steps of 0.5 dex in ${\rm log} g$). In order to mitigate this effect one requires either more finely gridded models or an interpolation scheme that takes the logarithmic gridding into account.

Precise and accurate stellar ages are the ultimate goal of this work. The accuracy of our ages is determined by both the accuracy with which we can determine atmospheric parameters and any systematic uncertainties associated with the stellar evolutionary models and our assumptions in applying them. The precision, on the other hand, is determined almost entirely by the precision with which we determine atmospheric parameters and, because there are some practical limits to how well we may ever determine T_eff and ${\rm log} g$, the location of the star in the H–R diagram (e.g., stars closer to the main sequence will always have more imprecise ages using this method).

It is thus important to provide a detailed accounting of the uncertainties involved in our atmospheric parameter determinations, as the final uncertainties quoted in our ages will arise purely from the values of the ${{\sigma }_{{{T}_{{\rm eff}}}}}{\rm and}{{\sigma }_{{\rm log} g}}$ used in our ${{\chi }^{2}}$ calculations. Below we consider the contribution of the systematics already discussed, as well as the contributions from errors in interpolation, photometry, metallicity, extinction, rotational velocity, multiplicity, and spectral peculiarity.

Systematics: the dominant source of uncertainty in our atmospheric parameter determinations are the systematics quantified in Sections 4.1 and 4.2. All systematic effects inherent to the $uvby\beta$ method, and the particular model color grids chosen, which we will call ${{\sigma }_{{\rm sys}}}$, are embedded in the comparisons to the stars with fundamentally or semi-fundamentally determined parameters, summarized as approximately ∼3.1% in T_eff and ∼0.116 dex in ${\rm log} g$. We also found that for stars with available [Fe/H] measurements, the accuracy with which we can determine atmospheric parameters using $uvby\beta$ photometry does not vary systematically with metallicity, though we further address metallicity issues both below and in the Appendix.

Interpolation Precision: to estimate the errors in atmospheric parameters due to the numerical precision of the interpolation procedures employed here, we generated 1000 random points in each of the three relevant $uvby\beta$ planes. For each point, we obtained ten independent ${{T}_{{\rm eff}}},{\rm log} g$ determinations to test the repeatability of the interpolation routine. The scatter in independent determinations of the atmospheric parameters were found to be $\lt {{10}^{-10}}$ K, dex, and thus numerical errors are assumed zero.

Photometric Errors: considering the most basic element of our approach, there are uncertainties due to the propagation of photometric errors through our atmospheric parameter determination pipeline. As discussed in Section 7, the photometric errors are generally small (∼0.005 mag in a given index). Translating the model grid points in the rectangular regions defined by the magnitude of the mean photometric error in a given index, and then interpolating to find the associated atmospheric parameters of the perturbed point, we take the maximum and minimum values for T_eff and ${\rm log} g$ to calculate the error due to photometric measurement error.

To simplify the propagation of photometric errors for individual stars, we performed simulations with randomly generated data to ascertain the mean uncertainty in T_eff, ${\rm log} g$ that results from typical errors in each of the $uvby\beta$ indices.

We begin with the HM98 photometry and associated measurement errors for our sample (3499 stars within 100 pc, B0-F5, luminosity classes IV–V). Since the HM98 compilation does not provide a₀ or r^*, as these quantities are calculated from the four fundamental indices, we calculate the uncertainties in these parameters using the crude approximation that none of the $uvby\beta$ indices are correlated. Under this assumption, the uncertainties associated with a₀ and r^* are as follows:

$$\begin{eqnarray}&&{{\sigma }_{{{a}_{0}}}}=\sqrt{{{1.36}^{2}}\sigma _{b-y}^{2}+{{0.36}^{2}}\sigma _{{{m}_{1}}}^{2}+{{0.18}^{2}}\sigma _{{{c}_{1}}}^{2}}\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{{\sigma }_{{{r}^{*}}}}=\sqrt{{{0.07}^{2}}\sigma _{b-y}^{2}+{{0.35}^{2}}\sigma _{{{c}_{1}}}^{2}+\sigma _{\beta }^{2}}.\end{eqnarray} \tag{ 18 }$$

A model for the empirical probability distribution function (hereafter PDF) for the error in a given $uvby\beta$ index is created through a normalized histogram with 25 bins. From this empirical PDF, one can randomly draw values for the error in a given index. For each $uvby\beta$ plane, 1,000 random points in the appropriate range of parameter space were generated with photometric errors drawn as described above. The eight (T_eff, ${\rm log} g$) values corresponding to the corners and midpoints of the "standard error rectangle" centered on the original random data point are then evaluated. The maximally discrepant (T_eff, ${\rm log} g$) values are saved and the overall distributions of ${\Delta}{{T}_{{\rm eff}}}/{{T}_{{\rm eff}}}$ and ${\Delta}{\rm log} g$ are then analyzed to assess the mean uncertainties in the atmospheric parameters derived in a given $uvby\beta$ plane due to the propagation of typical photometric errors.

For the late group, points were generated in the range of $(b-y)-{{c}_{1}}$ parameter space bounded by 6500 K $\leqslant \;{{T}_{{\rm eff}}}\leqslant 9000$ K and $3.0\leqslant {\rm log} g\leqslant 5.0$. In this group, typical photometric uncertainties of $\langle {{\sigma }_{{\rm b}-{\rm y}}}\rangle$ = 0.003 mag and $\langle {{\sigma }_{{{{\rm c}}_{1}}}}\rangle$ = 0.005 mag lead to average uncertainties of 0.6 % in T_eff and 0.055 dex in ${\rm log} g$. For the intermediate group, points were generated in the range of ${{a}_{0}}-{{r}^{*}}$ parameter space bounded by 8500 K $\leqslant \;{{T}_{{\rm eff}}}\leqslant 11000$ K and $3.0\leqslant {\rm log} g\leqslant 5.0$. In this group, typical photometric uncertainties of $\langle {{\sigma }_{{{{\rm a}}_{0}}}}\rangle$ = 0.005 mag and $\langle \sigma _{{\rm r}}^{*}\rangle$ = 0.005 mag lead to average uncertainties of 0.8 % in T_eff and 0.046 dex in ${\rm log} g$. For the early group, points were generated in the range of ${{c}_{1}}-\beta$ parameter space bounded by 10000 K $\;\leqslant \;{{T}_{{\rm eff}}}\leqslant$ 30000 K and $3.0\leqslant {\rm log} g\leqslant 5.0$. In this group, typical photometric uncertainties of $\langle {{\sigma }_{{{{\rm c}}_{1}}}}\rangle$ = 0.005 mag and $\langle {{\sigma }_{\beta }}\rangle$ = 0.004 mag lead to average uncertainties of 1.1 % in T_eff and 0.078 dex in ${\rm log} g$. Across all three groups, the mean uncertainty due to photometric errors is $\approx 0.9\%$ in T_eff and $\approx 0.060$ dex in ${\rm log} g$.

Metallicity Effects: for simplicity and homogeneity, our method assumes solar composition throughout. However, our sample can more accurately be represented as a Gaussian centered at −0.109 dex with $\sigma \approx 0.201$ dex. Metallicity is a small, but non-negligible, effect and allowing [M/H] to change by ±0.5 dex can lead to differences in the assumed T_eff of ∼1–2% for late, intermediate, and some early group stars, or differences of up to 6% for stars hotter than ∼17000 K (of which there are few in our sample). In ${\rm log} g$, shifts of ±0.5 dex in [M/H] can lead to differences of ∼0.1 dex in the assumed ${\rm log} g$ for late or early group stars, or ∼0.05 dex in the narrow region occupied by intermediate group stars.

Here, we estimate the uncertainty the metallicity approximation introduces to the fundamental stellar parameters derived in this work. We begin with the actual $uvby\beta$ data for our sample, and [Fe/H] measurements from the XHIP catalog (Anderson & Francis 2012), which exist for approximately 68% of our sample. Those authors collected photometric and spectroscopic metallicity determinations of Hipparcos stars from a large number of sources, calibrated the values to the high-resolution catalog of Wu et al. (2011) in an attempt to homogenize the various databases, and published weighted means for each star. The calibration process is described in detail in Section 5 of Anderson & Francis (2012).

For each of the stars with available [Fe/H] in our field star sample, we derive ${{T}_{{\rm eff}}},{\rm log} g$ in the appropriate $uvby\beta$ plane for the eight cases of [M/H] = −2.5, −2.0, −1.5, −1.0, −0.5, 0.0, 0.2, and 0.5. Then, given the measured [Fe/H], and making the approximation that [M/H] = [Fe/H], we perform a linear interpolation to find the most accurate values of ${{T}_{{\rm eff}}}{\rm and}{\rm log} g$ given the color grids available. We also store the atmospheric parameters a given star would be assigned assuming [M/H] = 0.0. Figure 12 shows the histograms of ${{T}_{{\rm eff}}}/{{T}_{{\rm eff},[{\rm M}/{\rm H}]=0}}$ and ${\rm log} g-{\rm log} {{g}_{[{\rm M}/{\rm H}]=0}}$. We take the standard deviations in these distributions to reflect the typical error introduced by the solar metallicity approximation. For T_eff, there is a 0.8% uncertainty introduced by the true dispersion of metallicities in our sample, and for ${\rm log} g$, the uncertainty is 0.06 dex. These uncertainties in the atmospheric parameters are naturally propagated into uncertainties in the age and mass of a star through the likelihood calculations outlined in Section 5.2.1.

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Reddening Effects: for the program stars studied here, interstellar reddening is assumed negligible. Performing the reddening corrections (described in Section 2.2) on our presumably unreddened sample of stars within 100 pc, we find for the ∼80% of stars for which dereddening proved possible, that the distribution of A_V values in our sample is approximately Gaussian with a mean and standard deviation of $\mu =0.007,\sigma =0.125$ mag, respectively (see Figure 19). Of course, negative A_V values are unphysical, but applying the reddening corrections to our $uvby\beta$ photometry and deriving the atmospheric parameters for each star in both the corrected and uncorrected cases gives us an estimate of the uncertainties in those parameters due to our assumption of negligible reddening out to 100 pc. The resulting distributions of ${{T}_{{\rm eff},0}}/{{T}_{{\rm eff}}}$ and ${\rm log} {{g}_{0}}-{\rm log} g$, where the naught subscripts indicate the dereddened values, are sharply peaked at 1 and 0, respectively. The FWHM of these distributions indicate an uncertainty of $\lt 0.2\%$ in T_eff and ∼0.004 dex in ${\rm log} g$. For the general case of sources at larger distances that may suffer more significant reddening, the systematic effects of under-correcting for extinction are illustrated in Figure 13.

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Uncertainties in Projected Rotational Velocities: the Glebocki & Gnacinski (2005) compilation contains mean $v{\rm sin} i$ measurements, as well as individual measurements from multiple authors. Of the 3499 stars in our sub-sample of the HM98 catalog, 2547 stars have $v{\rm sin} i$ values based on 4893 individual $v{\rm sin} i$ measurements, 1849 of which have an accompanying measurement error. Of these measurements, 646 are for intermediate or early groups, for which rotation corrections are performed in our method. The mean fractional error in $v{\rm sin} i$ for this subset of measurements is ∼13%. Caclulating the atmospheric parameters for these stars, then performing the FB98 $v{\rm sin} i$ corrections using v_rot and ${{v}_{{\rm rot}}}\pm {{\sigma }_{{{v}_{{\rm rot}}}}}$ allows us to estimate the magnitude of the uncertainty in ${{T}_{{\rm eff}}},{\rm log} g$ due to the uncertainties in $v{\rm sin} i$ measurements. The resulting rms errors in ${{T}_{{\rm eff}}},{\rm log} g$ are 0.7% and 0.01 dex, respectively. When $v{\rm sin} i$ measurements are not available, an average value based on the spectral type can be assumed, resulting in a somewhat larger error. The systematic effects of under-correcting for rotation are illustrated in Figure 4.

Influence of Multiplicity: in a large study such as this one, a high fraction of stars are binaries or higher multiples. Slightly more than 30% of our sample stars are known as members of multiple systems. We choose not to treat these stars differently, given the unknown multiplicity status of much of the sample, and caution our readers to use due care regarding this issue.

Influence of Spectral peculiarities: finally, early-type stars possess several peculiar subclasses (e.g., Ap, Bp, Am, etc. stars) for which anomalous behavior has been reported in the $uvby\beta$ system with respect to their "normal-type" counterparts. Some of these peculiarities have been linked to rotation, which we do account for. We note that peculiar subclasses constitute ∼4% of our sample and these stars could suffer unquantified errors in the determination of fundamental parameters when employing a broad methodology based on calibrations derived from mostly normal-type stars (see Tables 1 and 2 for a complete accounting of the spectral types used for calibrations). As these subclasses were included in the atmospheric parameter validation stage (Section 3), and satisfactory accuracies were still obtained, we chose not to adjust our approach for these stars and estimate the uncertainties introduced by their inclusion is negligible.

Final Assessment: our final atmospheric parameter uncertainties are dominated by the systematic effects quantified in Sections 4.1 and 4.2, with the additional effects outlined above contributing very little to the total uncertainty. The largest additional contributor comes from the photometric error. Adding in quadrature the sources ${{\sigma }_{{\rm sys}}},{{\sigma }_{{\rm num}}},{{\sigma }_{{\rm phot}}},{{\sigma }_{[{\rm Fe}/{\rm H}]}},{{\sigma }_{v{\rm sin} i}}$, and ${{\sigma }_{{{{\rm A}}_{{\rm V}}}}}$ results in final error estimates of 3.4% in T_eff and 0.14 dex in ${\rm log} g$.

The use of $uvby\beta$ photometry to determine fundamental stellar parameters is estimated in previous literature to lead to uncertainties of just 2.5% in ${{T}_{{\rm eff}}}$ and 0.1 dex in ${\rm log} g$ (Asiain et al. 1997), with our assessment of the errors somewhat higher.

The uncertainties that we derive in our Strömgren method work can be compared with those given by other methods. The Geneva photometry system ($U,B1,B2,V1,G$ filters), like the Strömgren system, has been used to derive ${{T}_{{\rm eff}}},{\rm log} g$, and [M/H] values based on atmospheric grids (Kobi & North 1990; Kunzli et al. 1997), with Kunzli et al. (1997) finding 150–250 K (few percent) errors in ${\rm log} {{T}_{{\rm eff}}}$ and 0.1–0.15 dex errors in log g, comparable to our values. From stellar model atmosphere fitting to high dispersion spectra, errors of 1%–5% in T_eff and 0.05–0.15 dex (typically 0.1 dex) in ${\rm log} g$ are quoted for early-type stars (e.g., Nieva & Simón-Díaz 2011), though systematic effects in log g on the order of an additional 0.1 dex may be present. Wu et al. (2011) tabulate the dispersions in atmospheric parameters among many different studies, finding author-to-author values that differ for OBA stars by 300–5000 K in ${{T}_{{\rm eff}}}$ (3%–12%) and 0.2–0.6 dex in ${\rm log} g$ (cm s⁻²), and for FGK stars 40–100 K in ${{T}_{{\rm eff}}}$ and 0.1–0.3 dex in ${\rm log} g$ (cm s⁻²).

Once ${{T}_{{\rm eff}}}$ and ${\rm log} g$ have been established, ages are determined through a Bayesian grid search of the fundamental parameter space encompassed by the evolutionary models. In this section we discuss the selection of evolution models, the Bayesian approach, numerical methods, and resulting age/mass uncertainties.

Two sets of isochrones are considered in this work. The model families are compared in Figure 14. The PARSEC solar-metallicity isochrones of Bressan et al. (2012), hereafter B12, take into account in a self-consistent manner the pre-main-sequence phase of evolution. The PARSEC models are the most recent iteration of the Padova evolutionary models, with significant revisions to the major input physics such as the equation of state, opacities, nuclear reaction rates and networks, and the inclusion of microscopic diffusion. The models are also based on the new reference solar composition, Z = 0.01524 from Caffau et al. (2011), but can be generated for a wide range of metallicities. The B12 models cover the mass range $0.1-12\;{{M}_{\odot }}$.

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PARSEC isochrones are attractive because early-type dwarfs have relatively rapid evolution with the pre-main-sequence evolution constituting a significant fraction of their lifetimes, i.e., ${{\tau }_{{\rm PMS}}}/{{\tau }_{{\rm MS}}}$ is larger compared to stars of later types. For stars with effective temperatures in the range 6500—25000 K (approximately spectral types B0–F5), the B12 models predict pre-main sequence lifetimes ranging from ∼0.2–40 Myr, main-sequence lifetimes from ∼14 Myr—2.2 Gyr, and the ratio ${{\tau }_{{\rm PMS}}}/{{\tau }_{{\rm MS}}}\sim 1.6-2.4\%$. A star of given initial mass thus can be followed consistently through the pre-MS, MS, and post-MS evolutionary stages. As a consequence, most points in ${{T}_{{\rm eff}}}-{\rm log} g$ space will have both pre-ZAMS and post-ZAMS ages as possible solutions. Figure 1 illustrates the evolution of atmospheric and corresponding photometric properties according to the PARSEC models.

The solar-metallicity isochrones of Ekström et al. (2012), hereafter E12, also use updated opacities and nuclear reaction rates, and are the first to take into account the effects of rotation on global stellar properties at intermediate masses. They are available for both non-rotating stars and stars that commence their lives on the ZAMS with a rotational velocity of 40 $\%$ their critical rotational velocity (${{v}_{{\rm rot},{\rm i}}}/{{v}_{{\rm crit}}}$ = 0.4); however, the Ekström et al. (2012) models do not take the pre-main sequence phase into account. The E12 models currently exist only for solar metallicity (Z = 0.014 is used), but cover a wider range of masses ($0.8-120\;{{M}_{\odot }}$).

The E12 models are attractive because they explictly account for rotation, though at a fixed percentage of breakup velocity. All output of stellar evolutionary models (e.g., lifetimes, evolution scenarios, and nucleosynthesis) are affected by axial stellar rotation which for massive stars enhances the MS lifetime by about 30% and may increase isochronal age estimates by about 25% (Meynet & Maeder 2000). In terms of atmospheres, for A-type stars, stellar rotation increases the strength of the Balmer discontinuity relative to a non-rotating star with the same color index (Maeder & Peytremann 1970). In the E12 models, the convective overshoot parameter was selected to reproduce the observed main sequence width at intermediate masses, which is important for our aim of distinguishing the ages of many field stars clustered on the main sequence with relatively large uncertainties in their surface gravities. Figure 14 shows, however, that there is close agreement between the B12 and the rotating E12 models. Thus, there is not a significant difference between the two models in regards to the predicted width of the MS band.

It should be noted that the $uvby\beta$ grids of Castelli & Kurucz (2006, 2004) were generated assuming a solar metallicity value of Z = 0.017. As discussed elsewhere, metallicity effects are not the dominant uncertainty in our methods and we are thus not concerned about the very small metallicity differences between the two model isochrone sets nor the third metallicity assumption in the model atmospheres.

In matching data to evolutionary model grids, a general issue is that nearly any given point in an H–R diagram (or equivalently in T_eff–${\rm log} g$ space), can be reproduced by multiple combinations of stellar age and mass. Bayesian inference can be used to determine the relative likelihoods of these combinations, incorporating prior knowledge about the distributions of the stellar parameters being estimated.

A simplistic method for determining the theoretical age and mass for a star on the Hertzsprung–Russell (H–R) diagram is interpolation between isochrones or evolutionary models. Some problems with this approach, as pointed out by Takeda et al. (2007) and Pont & Eyer (2004), are that interpolation between isochrones neither accounts for the nonlinear mapping of time onto the H–R diagram nor the non-uniform distribution of stellar masses observed in the galaxy. As a consequence, straightforward interpolation between isochrones results in an age distribution for field stars that is biased toward older ages compared to the distribution predicted by stellar evolutionary theory.

Bayesian inference of stellar age and mass aims to eliminate such a bias by accounting for observationally and/or theoretically motivated distribution functions for the physical parameters of interest. As an example, for a given point with error bars on the H–R diagram, a lower stellar mass should be considered more likely due to the initial mass function (IMF). Likewise, due to the longer main-sequence timescales for lower mass stars, a star that is observed to have evolved off the main sequence should have a probability distribution in mass that is skewed toward higher masses, i.e., because higher mass stars spend a more significant fraction of their entire lifetime in the post-MS stage.

5.2.1. Bayes Formalism

Bayesian estimation of the physical parameters can proceed from comparison of the data with a selection of models. Bayes' Theorem states

$$\begin{eqnarray}&&P({\rm model}|{\rm data})\propto P({\rm data}|{\rm model})\times P({\rm model}).\end{eqnarray} \tag{ 19 }$$

The probability of a model given a set of data is proportional to the product of the probability of the data given the model and the probability of the model itself. In the language of Bayesian statistics, this is expressed as

$$\begin{eqnarray}&&{\rm posterior}\propto {\rm likelihood}\times {\rm prior}.\end{eqnarray} \tag{ 20 }$$

Our model is the set of stellar parameters, age (τ) and mass (M_*), and our data are the measured effective temperature, T_eff, and surface gravity, ${\rm log} g$, for a given star. At any given combination of age and mass, the predicted T_eff and ${\rm log} g$ are provided by stellar evolutionary models. The ${{\chi }^{2}}$ statistic for an individual model can be computed as follows:

$$\begin{eqnarray}&&{{\chi }^{2}}(\tau ,{{M}_{*}})=\sum \frac{{{(O-E)}^{2}}}{{{\sigma }^{2}}}\end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}&&=\;\frac{{{[{{({{T}_{{\rm eff}}})}_{O}}-{{({{T}_{{\rm eff}}})}_{E}}]}^{2}}}{\sigma _{{{T}_{{\rm eff}}}}^{2}}+\frac{{{[{{({\rm log} g)}_{O}}-{{({\rm log} g)}_{E}}]}^{2}}}{\sigma _{{\rm log} g}^{2}},\end{eqnarray} \tag{ 22 }$$

where the subscripts O and E refer to the observed and expected (or model) quantities, respectively, and σ is the measurement error in the relevant quantity.

Assuming Gaussian statistics, the relative likelihood of a specific combination of (${{T}_{{\rm eff}}},{\rm log} g$) is

$$\begin{eqnarray}&&P({\rm data}|{\rm model})=P({{T}_{{\rm eff},\;{\rm obs}}},{\rm log} {{g}_{{\rm obs}}}|\tau ,{{M}_{*}})\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&\propto {\mkern 1mu} {\rm exp} \left[ -\frac{1}{2}{{\chi }^{2}}(\tau ,{{M}_{*}}) \right].\end{eqnarray} \tag{ 24 }$$

Finally, the joint posterior probability distribution for a model with age τ and mass M_*, is given by

$$\begin{eqnarray}&&P({\rm model}|{\rm data})=P(\tau ,{{M}_{*}}|{{T}_{{\rm eff},\;{\rm obs}}},{\rm log} {{g}_{{\rm obs}}})\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&\propto {\mkern 1mu} {\rm exp} \left[ -\frac{1}{2}{{\chi }^{2}}(\tau ,{{M}_{*}}) \right]P(\tau )P({{M}_{*}}),\end{eqnarray} \tag{ 26 }$$

where $P(\tau )$ and $P({{M}_{*}})$ are the prior probability distributions in age and mass, respectively. The prior probabilities of age and mass are assumed to be independent such that $P(\tau ,{{M}_{*}})=\;P(\tau )P({{M}_{*}})$.

5.2.2. Age and Mass Prior Probability Distribution Functions

Standard practice in the Bayesian estimation of stellar ages is to assume an age prior that is uniform in linear age (e.g., Pont & Eyer 2004; Jørgensen & Lindegren 2005; Takeda et al. 2007; Nielsen et al. 2013). There are two main justifications for choosing a uniform age prior: 1) it is the least restrictive choice of prior and 2) at this stage the assumption is consistent with observations that suggest a fairly constant star formation rate in solar neighborhood over the past 2 Gyr (Cignoni et al. 2006).

Since the evolutionary models are logarithmically gridded in age, the relative probability of age bin i is given by the bin width in linear age divided by the total range in linear age:

$$\begin{eqnarray}&&P({\rm log} ({{\tau }_{i}})\leqslant {\rm log} (\tau )\lt {\rm log} ({{\tau }_{i+1}}))=\frac{{{\tau }_{i+1}}-{{\tau }_{i}}}{{{\tau }_{n}}-{{\tau }_{0}}},\end{eqnarray} \tag{ 27 }$$

where ${{\tau }_{n}}$ and ${{\tau }_{0}}$ are the largest and smallest allowed ages, respectively. This weighting scheme gives a uniform probability distribution in linear age.

As noted by Takeda et al. (2007), it is important to understand that assuming a flat prior in linear age corresponds to a highly non-uniform prior in the measured quantities of ${\rm log} {{T}_{{\rm eff}}}$ and ${\rm log} g$. This is due to the non-linear mapping between these measurable quantities and the physical quantities of mass and age in evolutionary models. Indeed, the ability of the Bayesian approach to implicitly account for this effect is considered one of its main strengths.

As is standard in the Bayesian estimation of stellar masses, an IMF is assumed for the prior probability distribution of all possible stellar masses. Several authors point out that Bayesian estimates of physical parameters are relatively insensitive to the mass prior (i.e., the precise form of the IMF assumed), especially in the case of parameter determination over a small or moderate range in mass space. For this work considering BAF stars, the power law IMF of Salpeter (1955) is assumed for the mass prior, so that the relative probability of mass bin i is given by the following expression:

$$\begin{eqnarray}&&P({{M}_{i}}\leqslant M\lt {{M}_{i+1}})\propto M_{i}^{-2.35}.\end{eqnarray} \tag{ 28 }$$

5.2.3. Numerical Methods

5.2.4. Age and Mass Uncertainties

Membership probabilities, $uvby\beta$ photometry, and projected rotational velocities are obtained for members of these open clusters via the WEBDA open cluster database.⁷ For the Pleiades, membership information was augmented and cross-referenced with Stauffer et al. (2007). Both individual $uvby\beta$ measurements and calculations of the mean and scatter from the literature measurements are available from WEBDA in each of the photometric indices. As the methodology requires accurate classification of the stars according to regions of the H–R diagram, we inspected the spectral types and β indices and considered only spectral types B0–F5 and luminosity classes III–V for our open cluster tests.

In contrast to the field stars studied in the next section, the open clusters studied here are distant enough for interstellar reddening to significantly affect the derived stellar parameters. The photometry is thus dereddened as described in Section 2.2. Figure 15 shows the histograms of the visual extinction A_V for each cluster, with the impact of extinction on the atmospheric parameter determination illustrated above in Figure 13.

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In many cases, individual cluster stars have multiple measurements of $v{\rm sin} i$ in the WEBDA database and we select the measurement from whichever reference is the most inclusive of early-type members. In very few cases does a cluster member have no rotational velocity measurement present in the database; for these stars we assume the mean $v{\rm sin} i$ according to the ${{T}_{{\rm eff}}}-v{\rm sin} i$ relation presented in Appendix B of Gray (2005).

Atmospheric parameters are determined for each cluster member, as described in Section 3. Adopting our knowledge from the comparison to fundamental and semi-fundamental atmospheric parameters (Sections 4.1 & 4.2), a uniform 1.6 $\%$ shift toward cooler T_eff was applied to all temperatures derived from the model color grids to account for systematic effects in those grids. The FB98 $v{\rm sin} i$ corrections were then applied to the atmospheric parameters. The $v{\rm sin} i$ corrections prove to be a crucial step in achieving accurate ages for the open clusters (particularly for the Pleiades).

The results of applying our procedures to open cluster samples appear in Figure 16. While the exact cause(s) of the remaining scatter observed in the empirical isochrones for each cluster is not known, possible contributors may be systematic or astrophysical in nature, or due to incorrect membership information. Multiplicity, variability, and spectral peculiarities were among the causes investigated for this scatter, but the exclusion of objects on the basis of these criteria did not improve age estimation for any individual cluster. The number of stars falling below the theoretical ZAMS, particulary for stars with ${\rm log} {{T}_{{\rm eff}}}\lesssim 3.9$, is possibly systematic and may be due to an incomplete treatment of convection by the ATLAS9 models. This source of uncertainty is discussed in further detail in Section 8.2.

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For each cluster, we publish the individual stars considered, along with relevant parameters, in Tables 8, 9, 10, and 11. In each table, the spectral types and $v{\rm sin} i$ measurements are from WEBDA, while the dereddened $uvby\beta$ photometry and atmospheric parameters are from this work.

6.2.1. Ages from Bayesian Inference

Once atmospheric parameters have been determined, age determination proceeds as outlined in Section 5. For each individual cluster member, the ${{\chi }^{2}}$, likelihood, and posterior probability distribution are calculated for each point on a grid ranging from log(age yr⁻¹) = 6.5–10, with masses restricted to $1\leqslant \;M/{{M}_{\odot }}\leqslant 10$. The resolution of the grid is 0.0175 dex in log(age yr⁻¹) and 0.045 ${{M}_{\odot }}$ in mass. The 1D marginalized posterior PDFs for each individual cluster member are normalized and then summed to obtain an overall posterior PDF in age for the cluster as a whole. This composite posterior PDF is also normalized prior to the determination of statistical measures (mean, median, confidence intervals). Additionally, the posterior PDFs in log(age) for each member are multiplied to obtain the total probability in each log(age) bin that all members have a single age. While the summed PDF better depicts the behavior of individual stars or groups of stars, the multiplied PDF is best for assigning a single age to the cluster and evaluating any potential systematics of the isochrones themselves.

As shown in Figure 17, the summed age PDFs for each cluster generally follow the same behavior: (1) the peaks are largely determined by the early group (B-type) stars that have well-defined ages due to their unambiguous locations in the ${{T}_{{\rm eff}}}-{\rm log} g$ diagram; (2) examining the age posteriors for individual stars, the intermediate group stars tend to overpredict the cluster age relative to the early group stars, and the same is true for the late group stars with respect to the intermediate group stars, resulting in a large tail at older ages for each of the summed PDFs due to the relatively numerous and broad PDFs of the later group stars. For IC 2602 and the Pleiades, the multiplied PDFs have median ages and uncertainties that are in close agreement with the literature ages. Notably, the results of the open cluster tests favor an age for the Hyades that is older (∼800 Myr) than the accepted value, though not quite as old as the recent estimate of 950 ± 100 Myr from Brandt & Huang (2015). The Bayesian age analysis also favors an age for α Per that is younger (∼70 Myr) than the accepted value based on lithium depletion, but older than the canonical 50 Myr from the Upper Main Sequence Turnoff Mermilliod (1981). In the Appendix, we perform the same analysis for the open clusters on p(τ) rather than p(${\rm log} \tau$), yielding similar results.

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The results of the open cluster test are presented in Table 4. It is noted that all statistical measures of the marginalized age PDFs quoted hereafter are from PDFs normalized in log(age), as opposed to converting to linear age and then normalizing. This choice was made due to the facts that (1) the isochrones are provided in uniform logarithmic age bins, and (2) the marginalized PDFs of individual stars are more symmetric (and thus better characterized by traditional statistical measures) in log(age) than in linear age. Notably, the median age is equivalent regardless of whether one chooses to analyze prob(${\rm log} \tau$) or prob(τ). This issue is discussed further in an Appendix. In general, there is very close agreement in the Bayesian method ages between B12 and rotating E12 models. For IC 2602 and the Pleiades, our analysis yields median cluster ages (as determined from the multiplied PDFs) that are within 1σ of accepted values, regardless of the evolutionary models considered. The Bayesian analysis performed with the PARSEC models favor an age for α Persei that is ∼20% younger than the currently accepted value, or ∼20% older for the Hyades.

Table 4.  Open Cluster Ages

			Summed PDF	Summed PDF	Multiplied PDF	Multiplied PDF
Cluster	Lit. Age	Models	Median	68% C.I.	Median	68% C.I.	$\chi _{{\rm min} }^{2}$
	(Myr)		(Myr)	(Myr)	(Myr)	(Myr)	(Myr)
IC 2602	46$_{-5}^{+6}$	Ekström et al. (2012)	80	32–344	42	41–46	39
		Bressan et al. (2012)	79	27–284	46	44–50	37
α Persei	90$_{-10}^{+10}$	Ekström et al. (2012)	234	83–1618	71	68–74	50
		Bressan et al. (2012)	226	74–1500	70	69–74	48
Pleiades	125$_{-8}^{+8}$	Ekström et al. (2012)	277	81–899	128	126–130	126
		Bressan et al. (2012)	271	85–948	123	121–126	115
Hyades	625$_{-50}^{+50}$	Ekström et al. (2012)	872	518–1940	827	812–837	631
		Bressan et al. (2012)	844	487–1804	764	747–780	501

Note. Literature ages (column 2) come from the sources referenced in Section 6. For each set of evolutionary models, the median and 68% confidence interval are computed for both the summed PDF (columns 4,5) and multiplied PDF (columns 6,7). The final column indicates the best-fit isochrone found through ${{\chi }^{2}}$-minimization of all cluster members in ${\rm log} ({{T}_{{\rm eff}}})-{\rm log} g$ space. Note, the Hyades analysis includes the blue straggler HD 27962 and the spectroscopic binary HD 27268. Excluding these outliers results in a median and 68% confidence interval of 871 Myr [517–1839 Myr] of the summed PDF or 832 Myr [812–871 Myr] of the multiplied PDF, using the B12 models.

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6.2.2. Ages from Isochrone Fitting

As a final test of the two sets of evolutionary models, we used ${{\chi }^{2}}$-minimization to find the best-fitting isochrone for each cluster. By fitting all members of a cluster simultaneously, we are able to assign a single age to all stars, test the accuracy of the isochrones for stellar ensembles, and test the ability of our $uvby\beta$ method to reproduce the shapes of coeval stellar populations in ${{T}_{{\rm eff}}}-{\rm log} g$ space. For this exercise, we did not interpolate between isochrones, choosing instead to use the default spacing for each set of models (0.1 and 0.0125 dex in log(age/yr) for the E12 and B12 models, respectively). For the best results, we consider only the sections of the isochrones with ${\rm log} g$ between 3.5 and 5.0 dex. The results of this exercise are shown in Figure 18. The best-fitting E12 isochrone (including rotation) is consistent with accepted ages to within 1% for the Pleiades and Hyades, ∼15% for IC 2602, and ∼44% for α-Per. For the B12 models, the best-fit isochrones are consistent with accepted ages to ∼8% for the Pleiades, ∼20% for the Hyades and IC 2602, and ∼47% for α-Per. The B12 models produce systematically younger ages than the E12 models, by a fractional amount that increases with absolute age.

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As detailed above, the open cluster tests revealed that our method is able to distinguish between ensembles of differing ages, from tens to hundreds of Myr, at least in a statistical sense. For individual stars, large uncertainties may remain, particularly for the later types, owing almost entirely to the difficulty in determining both precise and accurate surface gravities. The open cluster tests also demonstrate the importance of a $v{\rm sin} i$ correction for early (B0–A0) and intermediate (A0–A3) group stars in determining accurate stellar parameters. While the $v{\rm sin} i$ correction was not applied to the late group (A3–F5 in this case) stars, it is likely that stars in this group experience non-negligible gravity darkening. The typically unknown inclination angle, i, also contributes significant uncertainties in derived stellar parameters and hence ages.

From our newly derived set of ages and masses of solar-neighborhood B0-F5 stars, we can determine an empirical mass–age relation. Using the mean ages and masses for all stars in our sample, we performed a linear least squares fit using the NumPy polyfit routine, yielding the following relation, valid for stars $1.04\lt M/{{M}_{\odot }}\lt 9.6$:

$$\begin{eqnarray}&&{\rm log} ({\rm age}\;{\rm y}{{{\rm r}}^{-1}})=9.532-2.929\;{\rm log} \left( \frac{M}{{{M}_{\odot }}} \right).\end{eqnarray} \tag{ 29 }$$

The rms error between the data and this relation is a fairly constant 0.225 dex as a function of stellar mass.

We can also derive empirical spectral-type-age and spectral-type-mass relations for the solar neighborhood, using the mean masses derived from our 1D marginalized posterior PDFs in age, and spectral type information from XHIP. These relations are plotted in Figure 23, and summarized in Table 7.

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Table 7.  Empirical Spectral-type Relations for Main Sequence B0–F5 Stars

Sp. Type	$\langle \tau \rangle$	${{\sigma }_{\tau }}$	$\langle M\rangle$	${{\sigma }_{M}}$	No. of Stars
	(Myr)	(Myr)	(${{M}_{\odot }}$)	(${{M}_{\odot }}$)
B0	19	...	4.75	...	1
B1	6	...	9.59	...	1
B2	15	13	6.96	0.81	2
B3	41	16	5.22	0.50	3
B4	26	12	4.94	0.59	4
B5	44	16	3.94	0.49	5
B6	84	51	3.69	0.23	4
B7	140	209	3.23	0.60	13
B8	99	43	3.28	0.38	18
B9	154	86	2.88	0.88	67
A0	285	437	2.47	0.40	120
A1	313	217	2.23	0.29	132
A2	373	320	2.11	0.30	144
A3	462	412	2.07	0.96	100
A4	540	333	1.84	0.19	37
A5	514	350	1.86	0.81	81
A6	628	265	1.85	0.49	46
A7	574	262	1.78	0.30	79
A8	642	272	1.64	0.11	62
A9	800	339	1.62	0.21	102
F0	994	544	1.52	0.19	324
F1	948	352	1.51	0.13	68
F2	1280	526	1.42	0.19	441
F3	1687	633	1.34	0.23	605
F4	1856	600	1.30	0.12	129
F5	2326	697	1.27	0.18	905

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Table 8.  IC 2602 Members Dereddened $uvby\beta$ Photometry and Atmospheric Parameters

Star	Sp. Type	${{(b-y)}_{0}}$	m0	c0	β	Teff	${\rm log} g$	$v{\rm sin} i$
		(mag)	(mag)	(mag)	(mag)	(K)	(dex)	(km s−1)
HD 91711	B8 V	−0.062	0.146	0.457	2.745	14687 ± 235	4.467 ± 0.113	153
HD 91839	A1 V	0.025	0.178	1.033	2.904	9509 ± 152	4.188 ± 0.091	146
HD 91896	B7 III	−0.081	0.093	0.346	2.660	16427 ± 263	3.782 ± 0.113	155
HD 91906	A0 V	0.016	0.177	1.005	2.889	9799 ± 157	4.146 ± 0.113	149
HD 92275	B8 III/IV	−0.056	0.125	0.562	2.709	13775 ± 220	3.852 ± 0.113	153
HD 92467	B95III	−0.026	0.168	0.833	2.851	11178 ± 179	4.423 ± 0.113	110
HD 92478	A0 V	0.010	0.183	0.978	2.925	9586 ± 153	4.431 ± 0.091	60
HD 92535	A5 V n	0.104	0.194	0.884	2.838	8057 ± 129	4.344 ± 0.145	140
HD 92536	B8 V	−0.043	0.131	0.705	2.795	13183 ± 211	4.423 ± 0.113	250
HD 92568	A M	0.209	0.237	0.625	2.748	7113 ± 114	4.341 ± 0.145	126
HD 92664	B8 III P	−0.083	0.118	0.386	2.702	15434 ± 247	4.145 ± 0.113	65
HD 92715	B9 V nn	−0.027	0.136	0.882	2.836	12430 ± 199	4.362 ± 0.113	290
HD 92783	B85V nn	−0.033	0.124	0.835	2.804	12278 ± 196	4.130 ± 0.113	230
HD 92837	A0 IV nn	−0.007	0.160	0.953	2.873	10957 ± 175	4.322 ± 0.113	220
HD 92896	A3 IV	0.114	0.193	0.838	2.831	8010 ± 128	4.425 ± 0.145	139
HD 92938	B3 V n	−0.075	0.105	0.384	2.690	15677 ± 251	4.015 ± 0.113	120
HD 92966	B95V nn	−0.019	0.158	0.930	2.878	11372 ± 182	4.445 ± 0.113	225
HD 92989	A05Va	0.008	0.180	0.982	2.925	9979 ± 160	4.480 ± 0.091	148
HD 93098	A1 V s	0.017	0.180	0.993	2.915	9688 ± 155	4.385 ± 0.091	135
HD 93194	B3 V nn	−0.078	0.105	0.357	2.668	17455 ± 279	4.015 ± 0.113	310
HD 93424	A3 Va	0.060	0.197	0.950	2.890	8852 ± 142	4.247 ± 0.113	95
HD 93517	A1 V	0.052	0.196	0.976	2.919	9613 ± 154	4.510 ± 0.091	220
HD 93540	B6 V nn	−0.065	0.116	0.476	2.722	15753 ± 252	4.308 ± 0.113	305
HD 93549	B6 V	−0.066	0.123	0.454	2.729	15579 ± 249	4.422 ± 0.113	265
HD 93607	B25V n	−0.084	0.102	0.292	2.675	17407 ± 279	4.098 ± 0.113	160
HD 93648	A0 V n	0.041	0.188	1.025	2.890	9672 ± 155	4.157 ± 0.091	215
HD 93714	B2 IV-V n	−0.092	0.100	0.201	2.647	18927 ± 303	3.979 ± 0.113	40
HD 93738	A0 V nn	−0.027	0.158	0.842	2.817	12970 ± 208	4.336 ± 0.113	315
HD 93874	A3 IV	0.071	0.203	0.947	2.896	8831 ± 141	4.367 ± 0.091	142
HD 94066	B5 V n	−0.068	0.117	0.439	2.680	15096 ± 242	3.792 ± 0.113	154
HD 94174	A0 V	0.046	0.193	0.946	2.907	9305 ± 149	4.391 ± 0.113	149

A machine-readable version of the table is available.

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Table 9.  α Persei Members Dereddened $uvby\beta$ Photometry and Atmospheric Parameters

Star	Sp. Type	${{(b-y)}_{0}}$	m0	c0	β	Teff	${\rm log} g$	$v{\rm sin} i$
		(mag)	(mag)	(mag)	(mag)	(K)	(dex)	(km s−1)
BD+49 868	F5 V	0.261	0.165	0.459	2.683	6693 ± 107	4.455 ± 0.145	20
HD 19767	F0 V N	0.176	0.178	0.756	2.765	7368 ± 118	4.174 ± 0.145	140
HD 19805	A0 Va	−0.000	0.161	0.931	2.887	10073 ± 161	4.344 ± 0.113	20
HD 19893	B9 V	−0.031	0.131	0.850	2.807	12614 ± 202	4.176 ± 0.113	280
HD 19954	A9 IV	0.150	0.200	0.794	2.792	7632 ± 122	4.297 ± 0.145	85
HD 20135	A0 P	−0.011	0.186	0.970	2.848	10051 ± 161	3.998 ± 0.113	35
BD+49 889	F5 V	0.292	0.156	0.418	2.656	6430 ± 103	4.352 ± 0.145	65
BD+49 896	F4 V	0.261	0.168	0.472	2.686	6686 ± 107	4.410 ± 0.145	30
HD 20365	B3 V	−0.079	0.103	0.346	2.681	16367 ± 262	4.025 ± 0.113	145
HD 20391	A1 Va n	0.026	0.179	1.006	2.901	10415 ± 167	4.386 ± 0.091	260
HD 20487	A0 V N	−0.016	0.151	0.976	2.856	11659 ± 187	4.198 ± 0.113	280
BD+47 808	F1 IV N	0.183	0.179	0.759	2.763	7281 ± 116	4.062 ± 0.145	180
BD+48 892	F3 IV-V	0.246	0.167	0.524	2.696	6800 ± 109	4.359 ± 0.145	20
BD+48 894	F0 IV	0.174	0.202	0.734	2.770	7416 ± 119	4.284 ± 0.145	75
HD 20809	B5 V	−0.074	0.109	0.395	2.696	15934 ± 255	4.097 ± 0.113	200
HD 20842	A0 Va	−0.005	0.157	0.950	2.886	10258 ± 164	4.325 ± 0.113	85
HD 20863	B9 V	−0.034	0.134	0.810	2.813	12154 ± 194	4.267 ± 0.113	200
BD+49 914	F5 V	0.281	0.170	0.431	2.664	6520 ± 104	4.395 ± 0.145	120
HD 20919	A8 V	0.168	0.191	0.757	2.775	7463 ± 119	4.259 ± 0.145	50
BD+49 918	F1 V N	0.186	0.183	0.770	2.755	7235 ± 116	3.977 ± 0.145	175
HD 20931	A1 Va	0.018	0.174	0.979	2.911	9588 ± 153	4.342 ± 0.113	85
BD+47 816	F4 V	0.271	0.155	0.452	2.672	6600 ± 106	4.399 ± 0.145	28
HD 20961	B95V	−0.019	0.163	0.920	2.875	10537 ± 169	4.344 ± 0.113	25
BD+46 745	F4 V	0.274	0.169	0.462	2.674	6566 ± 105	4.332 ± 0.145	160
HD 20969	A8 V	0.186	0.192	0.715	2.758	7291 ± 117	4.239 ± 0.145	20
HD 20986	A3 V N	0.046	0.190	1.004	2.896	9584 ± 153	4.243 ± 0.091	210
HD 21005	A5 V N	0.074	0.189	0.987	2.862	8266 ± 132	4.197 ± 0.145	250
HD 21091	B95IV nn	−0.019	0.152	0.938	2.856	12477 ± 200	4.416 ± 0.113	340
HD 21092	A5 V	0.054	0.218	0.938	2.893	8775 ± 140	4.311 ± 0.091	75
TYC 3320–1715-1	F4 V	0.281	0.153	0.469	2.663	6495 ± 104	4.220 ± 0.145	110
HD 21152	B9 V	−0.018	0.158	0.943	2.868	11306 ± 181	4.353 ± 0.113	225
HD 232793	F5 V	0.311	0.172	0.377	2.645	6274 ± 100	4.362 ± 0.145	93
HD 21181	B85V N	−0.038	0.122	0.784	2.766	13726 ± 220	4.119 ± 0.113	345
HD 21239	A3 V N	0.045	0.190	0.997	2.910	9182 ± 147	4.320 ± 0.091	145
HD 21278	B5 V	−0.073	0.111	0.398	2.705	15274 ± 244	4.152 ± 0.113	75
HD 21302	A1 V N	0.022	0.177	0.989	2.888	10269 ± 164	4.301 ± 0.091	230
BD+48 923	F4 V	0.270	0.153	0.464	2.673	6603 ± 106	4.362 ± 0.145	20
HD 21345	A5 V N	0.051	0.208	0.969	2.893	9435 ± 151	4.324 ± 0.091	200
HD 21398	B9 V	−0.030	0.145	0.825	2.837	11615 ± 186	4.372 ± 0.113	135
HD 21428	B3 V	−0.077	0.105	0.363	2.686	16421 ± 263	4.076 ± 0.113	200
HD 21481	A0 V N	−0.013	0.164	0.993	2.858	11187 ± 179	4.141 ± 0.113	250
HD 21527	A7 IV	0.093	0.231	0.855	2.856	8231 ± 132	4.486 ± 0.145	80
HD 21551	B8 V	−0.048	0.118	0.673	2.746	14869 ± 238	4.220 ± 0.113	380
HD 21553	A6 V N	0.072	0.206	0.921	2.872	8381 ± 134	4.414 ± 0.145	150
HD 21619	A6 V	0.052	0.221	0.935	2.894	8843 ± 141	4.329 ± 0.091	90
BD+49 957	F3 V	0.258	0.168	0.500	2.687	6699 ± 107	4.334 ± 0.145	56
HD 21641	B85V	−0.042	0.131	0.721	2.747	12914 ± 207	3.929 ± 0.113	215
BD+49 958	F1 V	0.198	0.188	0.732	2.739	7137 ± 114	3.989 ± 0.145	155
HD 21672	B8 V	−0.050	0.119	0.649	2.747	13473 ± 216	4.071 ± 0.113	225
BD+48 944	A5 V	0.063	0.220	0.931	2.886	8799 ± 141	4.305 ± 0.091	120
HD 21931	B9 V	−0.029	0.147	0.835	2.829	11998 ± 192	4.343 ± 0.113	205

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Table 10.  Pleiades Members Dereddened $uvby\beta$ Photometry and Atmospheric Parameters

HD	Sp. Type	${{(b-y)}_{0}}$	m0	c0	β	Teff	${\rm log} g$	$v{\rm sin} i$
		(mag)	(mag)	(mag)	(mag)	(K)	(dex)	(km s−1)
HD 23157	A9 V	0.168	0.190	0.725	2.778	7463 ± 121	4.369 ± 0.145	100
HD 23156	A7 V	0.111	0.215	0.815	2.837	8046 ± 130	4.498 ± 0.145	70
HD 23247	F3 V	0.237	0.174	0.527	2.704	6863 ± 111	4.424 ± 0.145	40
HD 23246	A8 V	0.170	0.184	0.758	2.773	7409 ± 120	4.234 ± 0.145	200
HD 23288	B7 V	−0.051	0.120	0.636	2.747	13953 ± 226	4.151 ± 0.113	280
HD 23302	B6 III	−0.054	0.098	0.638	2.690	13308 ± 216	3.478 ± 0.113	205
HD 23289	F3 V	0.244	0.164	0.521	2.699	6796 ± 110	4.387 ± 0.145	40
HD 23326	F4 V	0.250	0.164	0.514	2.691	6741 ± 109	4.358 ± 0.145	40
HD 23324	B8 V	−0.052	0.116	0.634	2.747	13748 ± 223	4.126 ± 0.113	255
HD 23338	B6 IV	−0.061	0.104	0.553	2.702	13696 ± 222	3.772 ± 0.113	130
HD 23351	F3 V	0.249	0.176	0.507	2.695	6755 ± 109	4.391 ± 0.145	80
HD 23361	A25Va n	0.069	0.201	0.959	2.872	8356 ± 135	4.309 ± 0.145	235
HD 23375	A9 V	0.180	0.187	0.710	2.765	7336 ± 119	4.318 ± 0.145	75
HD 23410	A0 Va	0.004	0.164	0.975	2.899	10442 ± 169	4.382 ± 0.113	200
HD 23409	A3 V	0.070	0.202	0.980	2.892	8903 ± 144	4.270 ± 0.091	170
HD 23432	B8 V	−0.039	0.127	0.758	2.793	12695 ± 206	4.250 ± 0.113	235
HD 23441	B9 V N	−0.029	0.135	0.858	2.822	11817 ± 191	4.209 ± 0.113	200
HD 23479	A9 V	0.188	0.166	0.716	2.755	7239 ± 117	4.212 ± 0.145	150
HD 23489	A2 V	0.033	0.183	1.012	2.907	9170 ± 149	4.239 ± 0.091	110
HD 23512	A2 V	0.057	0.196	1.035	2.909	8852 ± 143	4.214 ± 0.091	145
HD 23511	F5 V	0.279	0.174	0.412	2.674	6521 ± 106	4.477 ± 0.145	28
HD 23514	F5 V	0.285	0.179	0.443	2.668	6450 ± 104	4.307 ± 0.145	40
HD 23513	F5 V	0.278	0.170	0.423	2.673	6528 ± 106	4.447 ± 0.145	30
HD 23568	B95Va n	−0.024	0.139	0.914	2.847	11731 ± 190	4.301 ± 0.113	240
HD 23567	F0 V	0.159	0.196	0.735	2.788	7560 ± 122	4.407 ± 0.145	50
HD 23585	F0 V	0.168	0.185	0.713	2.780	7472 ± 121	4.405 ± 0.145	100
HD 23608	F5 V	0.278	0.177	0.482	2.673	6492 ± 105	4.185 ± 0.145	110
HD 23607	F0 V	0.108	0.203	0.814	2.841	8085 ± 131	4.534 ± 0.145	12
HD 23629	A0 V	−0.001	0.163	0.986	2.899	10340 ± 168	4.342 ± 0.113	170
HD 23632	A0 Va	0.006	0.167	1.009	2.899	10461 ± 169	4.312 ± 0.113	225
HD 23628	A4 V	0.090	0.189	0.904	2.853	8163 ± 132	4.381 ± 0.145	215
HD 23643	A35V	0.079	0.194	0.943	2.862	8258 ± 134	4.301 ± 0.145	185
HD 23733	A9 V	0.207	0.177	0.672	2.736	7066 ± 114	4.174 ± 0.145	180
HD 23732	F5 V	0.258	0.172	0.460	2.688	6695 ± 108	4.473 ± 0.145	50
HD 23753	B8 V N	−0.046	0.113	0.712	2.736	13096 ± 212	3.859 ± 0.113	240
HD 23791	A9 V+	0.139	0.214	0.758	2.811	7776 ± 126	4.480 ± 0.145	85
HD 23850	B8 III	−0.048	0.102	0.701	2.695	13446 ± 218	3.483 ± 0.113	280
HD 23863	A8 V	0.116	0.201	0.857	2.826	7926 ± 128	4.354 ± 0.145	160
HD 23872	A1 Va n	0.032	0.182	1.013	2.894	10028 ± 162	4.247 ± 0.091	240
HD 23873	B95Va	−0.023	0.143	0.907	2.852	10897 ± 177	4.255 ± 0.113	90
HD 23886	A4 V	0.068	0.214	0.915	2.880	8974 ± 145	4.343 ± 0.091	165
HD 23912	F3 V	0.274	0.154	0.481	2.671	6531 ± 106	4.242 ± 0.145	130
HD 23924	A7 V	0.100	0.223	0.852	2.852	8121 ± 132	4.460 ± 0.145	100
HD 23923	B85V N	−0.033	0.124	0.839	2.794	12911 ± 209	4.159 ± 0.113	310
HD 23948	A1 Va	0.033	0.191	0.984	2.905	9237 ± 150	4.307 ± 0.091	120
HD 24076	A2 V	0.008	0.168	0.923	2.867	10196 ± 165	4.298 ± 0.091	155
HD 24132	F2 V	0.245	0.149	0.597	2.692	6744 ± 109	4.182 ± 0.145	230

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Table 11.  Hyades Members Dereddened $uvby\beta$ Photometry and Atmospheric Parameters

HD	Sp. Type	${{(b-y)}_{0}}$	m0	c0	β	Teff	${\rm log} g$	$v{\rm sin} i$
		(mag)	(mag)	(mag)	(mag)	(K)	(dex)	(km s−1)
HD 26015	F3 V	0.252	0.174	0.537	2.693	6732 ± 109	4.244 ± 0.145	25
HD 26462	F1 IV-V	0.230	0.165	0.596	2.710	6916 ± 112	4.291 ± 0.145	30
HD 26737	F5 V	0.274	0.168	0.477	2.674	6558 ± 106	4.263 ± 0.145	60
HD 26911	F3 V	0.258	0.176	0.525	2.690	6682 ± 108	4.228 ± 0.145	30
HD 27176	A7 m	0.172	0.187	0.785	2.767	7380 ± 120	4.087 ± 0.145	125
HD 27397	F0 IV	0.171	0.194	0.770	2.766	7410 ± 120	4.173 ± 0.145	100
HD 27429	F2 VN	0.240	0.171	0.588	2.693	6828 ± 111	4.270 ± 0.145	150
HD 27459	F0 IV	0.129	0.204	0.871	2.812	7782 ± 126	4.198 ± 0.145	35
HD 27524	F5 V	0.285	0.161	0.461	2.656	6461 ± 105	4.213 ± 0.145	110
HD 27561	F4 V	0.270	0.162	0.482	2.677	6594 ± 107	4.284 ± 0.145	30
HD 27628	A2 M	0.133	0.225	0.707	2.756	7944 ± 129	4.743 ± 0.145	30
HD 27819	A7 IV	0.080	0.209	0.982	2.857	8203 ± 133	4.170 ± 0.145	35
HD 27901	F4 V N	0.238	0.178	0.597	2.704	6837 ± 111	4.233 ± 0.145	110
HD 27934	A5 IV-V	0.064	0.201	1.053	2.867	8506 ± 138	3.884 ± 0.091	90
HD 27946	A7 V	0.149	0.192	0.840	2.783	7584 ± 123	4.112 ± 0.145	210
HD 27962	A3 V	0.020	0.193	1.046	2.889	9123 ± 148	4.004 ± 0.091	30
HD 28024	A9 IV-N	0.165	0.175	0.947	2.753	7279 ± 118	3.503 ± 0.145	215
HD 28226	A M	0.164	0.213	0.771	2.775	7493 ± 121	4.248 ± 0.145	130
HD 28294	F0 IV	0.198	0.173	0.694	2.745	7174 ± 116	4.194 ± 0.145	135
HD 28319	A7 III	0.097	0.198	1.011	2.831	7945 ± 129	3.930 ± 0.145	130
HD 28355	A7 m	0.112	0.226	0.908	2.832	7930 ± 128	4.207 ± 0.145	140
HD 28485	F0 V+N	0.200	0.192	0.717	2.740	7129 ± 115	4.035 ± 0.145	150
HD 28527	A5 m	0.085	0.218	0.964	2.856	8180 ± 133	4.194 ± 0.145	100
HD 28546	A7 m	0.142	0.234	0.796	2.809	7726 ± 125	4.354 ± 0.145	30
HD 28556	F0 IV	0.147	0.202	0.814	2.795	7645 ± 124	4.244 ± 0.145	140
HD 28568	F5 V	0.274	0.168	0.466	2.676	6564 ± 106	4.315 ± 0.145	55
HD 28677	F2 V	0.214	0.176	0.654	2.725	7032 ± 114	4.161 ± 0.145	100
HD 28911	F5 V	0.283	0.163	0.459	2.663	6481 ± 105	4.249 ± 0.145	40
HD 28910	A9 V	0.144	0.200	0.830	2.796	7659 ± 124	4.213 ± 0.145	95
HD 29169	F2 V	0.236	0.183	0.567	2.708	6880 ± 111	4.321 ± 0.145	80
HD 29225	F5 V	0.276	0.171	0.461	2.675	6547 ± 106	4.316 ± 0.145	45
HD 29375	F0 IV-V	0.187	0.187	0.740	2.754	7257 ± 118	4.106 ± 0.145	155
HD 29388	A5 IV-V	0.062	0.199	1.047	2.870	8645 ± 140	3.927 ± 0.091	115
HD 29499	A M	0.140	0.231	0.826	2.810	7713 ± 125	4.266 ± 0.145	70
HD 29488	A5 IV-V	0.080	0.196	1.017	2.852	8127 ± 132	4.025 ± 0.145	160
HD 30034	A9 IV-	0.150	0.195	0.813	2.791	7610 ± 123	4.218 ± 0.145	75
HD 30210	A5 m	0.091	0.252	0.955	2.845	8126 ± 132	4.181 ± 0.145	30
HD 30780	A9 V+	0.122	0.207	0.900	2.813	7823 ± 127	4.141 ± 0.145	155
HD 31845	F5 V	0.294	0.165	0.439	2.658	6396 ± 104	4.229 ± 0.145	25
HD 32301	A7 IV	0.079	0.202	1.034	2.847	8116 ± 131	3.975 ± 0.145	115
HD 33254	A7 m	0.132	0.251	0.835	2.824	7797 ± 126	4.306 ± 0.145	30
HD 33204	A7 m	0.149	0.245	0.803	2.796	7634 ± 124	4.270 ± 0.145	30
HD 25202	F4 V	0.206	0.172	0.695	2.724	7082 ± 115	4.064 ± 0.145	160
HD 28052	F0 IV-V N	0.153	0.183	0.934	2.767	7431 ± 120	3.733 ± 0.145	170
HD 18404	F5 IV	0.269	0.169	0.481	2.680	6605 ± 107	4.299 ± 0.145	0
HD 25570	F4 V	0.249	0.147	0.557	2.688	6752 ± 109	4.183 ± 0.145	34
HD 40932	A2 M	0.079	0.205	0.978	2.853	8224 ± 133	4.191 ± 0.145	18

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Table 12.  Open Cluster Ages: Analysis in Linear Age

			Summed PDF	Summed PDF	Multiplied PDF	Multiplied PDF
Cluster	Lit. Age	Models	Median	68 $\%$ C.I.	Median	68 $\%$ C.I.
	(Myr)		(Myr)	(Myr)	(Myr)	(Myr)
IC 2602	46$_{-5}^{+6}$	Ekström et al. (2012)	22	3–39	41	41–42
		Bressan et al. (2012)	24	3–40	40	37–43
α Persei	90$_{-10}^{+10}$	Ekström et al. (2012)	41	3–68	63	61–68
		Bressan et al. (2012)	45	3–71	62	58–66
Pleiades	125$_{-8}^{+8}$	Ekström et al. (2012)	61	3–113	125	122–131
		Bressan et al. (2012)	77	3–117	112	107–120
Hyades	625$_{-50}^{+50}$	Ekström et al. (2012)	118	3–403	677	671–690
		Bressan et al. (2012)	288	17–593	738	719–765

Note. Literature ages (column 2) come from the sources referenced in Section 6. For each set of evolutionary models, the median and 68% confidence interval are computed for both the summed PDF (columns 4 and 5) and multiplied PDF (columns 6 and 7). Note, the Hyades analysis includes the blue straggler HD 27962 and the spectroscopic binary HD 27268. Excluding these outliers results in a median and 68% confidence interval of 322 Myr [17–650 Myr] of the summed PDF or 784 Myr [749–802 Myr] of the multiplied PDF, using the B12 models.

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In this section we place our work on nearby open cluster stars and approximately 3500 nearby field stars in the context of previous age estimation methods for BAF stars.

Song et al. (2001) utilized a method quite similar to ours, employing $uvby\beta$ data from the catalogs of Hauck & Mermilliod (1980), Olsen (1983), Olsen & Perry (1984), the color grids of Moon & Dworetsky (1985) including a temperature-dependent gravity modification suggested by Napiwotzki et al. (1993), and isochrones from Schaller et al. (1992), to determine the ages of 26 Vega-like stars.

For A-type stars, Vican (2012) determined ages for Herschel DEBRIS survey stars by means of isochrone placement in log(${{T}_{{\rm eff}}}$)–log(g) space using Li & Han (2008) and Pinsonneault et al. (2004) isochrones, and atmospheric parameters from the literature. Rieke et al. (2005) published age estimates for 266 B- & A-type main sequence stars using cluster/moving group membership, isochrone placement in the H–R diagram, and literature ages (mostly coming from earlier application of $uvby\beta$ photometric methods).

Among later type F dwarfs, previous age estimates come primarily from the Geneva-Copenhagen Survey (Casagrande et al. 2011), but their reliability is caveated by the substantially different values published in various iterations of the catalog (Nordström et al. 2004; Holmberg et al. 2007, 2009; Casagrande et al. 2011) and the inherent difficulty of isochrone dating these later type dwarfs.

More recently, Nielsen et al. (2013) applied a Bayesian inference approach to the age determination of 70 B- and A-type field stars via M_V versus $B-V$ color–magnitude diagram isochrone placement, assuming a constant star formation rate in the solar neighborhood and a Salpeter IMF. De Rosa et al. (2014) estimated the ages of 316 A-type stars through placement in a M_K versus $V-K$ color–magnitude diagram. Both of these broad-band photometric studies used the theoretical isochrones of Siess et al. (2000).

Considering the above sources of ages, the standard deviation among them suggests scatter among authors of only 15% for some stars up to 145%, with a typical value of 40%. The full range (as opposed to the dispersion) of published ages is 3%–300%, with a peak around the 80% level. The value of the age estimates presented here resides in the large sample of early-type stars and the uniform methodology applied to them.

In Figure 21 it may be noted that many stars, particularly those with ${\rm log} {{T}_{{\rm eff}}}\leqslant 3.9$, are located well below the model isochrones. Using rotation-corrected atmospheric parameters, ∼540 stars, or ∼15% of the sample, fall below the theoretical ZAMS.

Prior studies also faced a similarly large fraction of stars falling below the main sequence. Song et al. (2001) arbitrarily assigned an age of 50 Myr to any star lying below the 100 Myr isochrone used in that work. Tetzlaff et al. (2011) arbitrarily shifted stars toward the ZAMS and treated them as ZAMS stars.

Several possibilities might be invoked to explain the large population of stars below the ${\rm log} g-{\rm log} {{T}_{{\rm eff}}}$ isochrones: (1) failure of evolutionary models to predict the true width of the MS, (2) spread of metallicities, with the metal-poor MS residing beneath the solar-metallicity MS, (3) overaggressive correction for rotational velocity effects, or (4) systematics involved in surface gravity or luminosity determinations. Of these explanations, we consider (4) the most likely, with (3) also contributing somewhat. Valenti & Fischer (2005) found a 0.4 dex spread in ${\rm log} g$ among their main sequence FGK stars along with a 0.1 dex shift downward relative to the expected zero metallicity main sequence.

The Bayesian age estimates for stars below the MS are likely to be unrealistically old, so we compared the ages for these stars with interpolated ages. Using the field star atmospheric parameters and Bressan et al. (2012) models, we performed a 2D linear interpolation with the SciPy routine griddata. Stars below the main sequence could be easily identified by selecting objects with ${\rm log} {{({\rm age}\;{\rm y}{{{\rm r}}^{-1}})}_{{\rm Bayes}}}-{\rm log} {{({\rm age}\;{\rm y}{{{\rm r}}^{-1}})}_{{\rm interp}}}\gt 1.0$. Notably, for these stars below the MS, the linear interpolation produces more realistic ages than the Bayesian method. A comparison of the Bayesian and interpolated ages for all stars is presented in Figure 24. Of note, there is closer agreement between the Bayesian and interpolation methods in regards to estimating masses.

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Figure 24 further serves to illustrate the difference between the Bayesian ages and the interpolated ages, which scatters over an order magnitude from a 1:1 relationship. A number of stars that fall below the MS and have independently constrained ages are examined in detail in n Section 8.3. These stars have interpolated ages that are more in line with prior studies, and in light of this, we publish the interpolated ages in addition to the Bayesian ages in the final electronic table.

In this section we discuss stars of particular interest given that they have either spatially resolved debris disks, detected possibly planetary mass companions, or both. As a final test of the Bressan et al. (2012) evolutionary models and our Bayesian age and mass estimation method, we performed our analysis on these stars, including the Sun.

8.3.1. Sun