A TEST FOR RADIAL MIXING USING LOCAL STAR SAMPLES

The Geneva–Copenhagen survey of nearby stars (GCS; Nordström et al. 2004) used Hipparcos positions and proper motions, supplemented by spectroscopic observations, to construct a homogeneous sample of nearby mostly F- and G-dwarf stars. We employ their updated catalog (Holmberg et al. 2009) that uses their revised temperature calibration and the improved astrometry from the reanalysis of Hipparcos data by van Leeuwen (2007). This monumental effort has resulted in a large sample of stars in the solar neighborhood with distances and full space motions. Holmberg et al. (2009) do not supply individual uncertainties for each velocity, but assert that they are believed to be accurate to 1.5 km s⁻¹, with the greatest contribution coming from distance uncertainties.

The machine readable table of 16, 682 stars made available by Holmberg et al. (2009) includes sky positions, distances, and the (U, V, W) components of the star's motion relative to the Sun in Galactic coordinates.⁴ We discard those having no distance, no (U, V, W) velocities, or no estimate of T_eff. We do not use the disputed age estimates (e.g., Reid et al. 2007; Holmberg et al. 2007) in the present paper. We correct for solar motion relative to the local standard of rest (LSR) by adding (U, V, W)_☉ = (11.10, 12.24, 7.25) km s⁻¹ (Schönrich et al. 2010) to the tabulated velocities.

Nordström et al. (2004) note that they obtained [Fe/H] ≳ 0.4 for a small number of stars, though none is within 40 pc of the Sun. Furthermore, these high values are found only for stars with T_eff ≳ 6200 K. They argue that the coincidence of high T_eff and [Fe/H], together with other information, suggests that the extinction to these stars has been overestimated. We therefore discard a further 29 stars having [Fe/H] > 0.4.

In order to select nearby disk stars, we further restrict the sample to stars whose best estimate of the distance is within 500 pc, |U|, |V| < 80 km s⁻¹, and retain only those having an energy of vertical motion about the Galactic mid-plane, E_z = 0.5(z²ν² + W²) < 392 (km s⁻¹)², with the vertical frequency ν = 0.07 km s⁻¹ pc⁻¹ (Binney & Tremaine 2008), giving them a maximum vertical excursion of ±400 pc. Thus, we select only stars that have a high probability of being thin-disk stars, in the same spirit as Bensby et al. (2003), but not in exactly the same manner⁵ (see Section 4 for further discussion of selection effects). Our final sample contains 11, 877 stars that we use in this analysis.

The median distance of the selected stars is 75 pc and the sample within 40 pc of the Sun is believed to be near complete. We adopt R₀ = 8 kpc, the circular speed of the LSR to be V₀ = 220 km s⁻¹, and compute the Galactocentric angular momentum, L_z, using the in-plane distance of the star from the Sun and the peculiar velocities (U, V) corrected for the Sun's motion.

The red or solid contours in Figure 1 show the distribution of the selected GCS stars in the space of T_eff and [Fe/H]. The range of T_eff indicates that the sample extends outside the spectral types F and G, reflecting the selection criteria described by Nordström et al. (2004). These authors estimate uncertainties of ≲ 0.1 in [Fe/H] and 94 K in T_eff.

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Casagrande et al. (2011) present a recalibration of T_eff and [Fe/H] for the GCS stars that folds in the infrared flux from a star. They obtain slightly higher values for T_eff and [Fe/H] than those derived by Holmberg et al. (2009) and they too attempt to assign ages for the individual stars. The additional information used by Casagrande et al. (2011) should lead to improved values for the derived quantities, but they claim reliable results for fewer than half the GCS stars (their irfm sample) and for none of the cooler stars. We therefore continue to use the calibration for the whole GCS sample by Holmberg et al. (2009).

The RAVE survey (Steinmetz et al. 2006) plans to measure the heliocentric radial velocities and stellar parameters for about a million stars in the southern sky having apparent magnitudes in the range 9 < m_I < 12; the first 77,461 are available in the third data release (Siebert et al. 2011). The typical uncertainty in the radial velocity is <2 km s⁻¹, but the distance to most stars has to be judged photometrically and most proper motions are from ground-based data. Thus, three of the phase space coordinates for each star are of much lower quality than are those in the GCS, although this weakness will be compensated by a much larger sample size when the survey is complete.

We have downloaded the online table of the third data release from the RAVE Web site and selected a subset of stars for analysis.⁶ We estimate distances to these stars by fitting to the Yonsei–Yale isochrones (Demarque et al. 2004) using a method related to that described by Breddels et al. (2010, see also Burnett et al. 2011 for an alternative method). We adopt many of their selection criteria: we require the spectral signal-to-noise parameter S2N > 20 with a blank spectral warning flag field; the parameters [M/H], log(g), and T_eff to be determined; and the stars to have J and K_s magnitudes from Two Micron All Sky Survey (2MASS) with no warning flags about the identification of the star or the 2MASS photometry. Unlike those authors, however, we have kept stars with b < 25° on the grounds that extinction for the nearby stars that interest us will not be large enough in the near-IR to severely bias our distance estimates. As we are here interested only in nearby main-sequence stars that are members of the disk population, we also make preliminary cuts to eliminate stars with log(g) < 4, T_eff > 10⁴ K, and with |v_r| > 120 km s⁻¹. We have treated the tabulated [M/H] as equivalent to [Fe/H] since the corrections (Zwitter et al. 2008) are generally within the uncertainties. We also show in Section 2.3 that the [M/H] values for RAVE stars agree with the [Fe/H] values for the stars in common with the GCS.

We estimate the absolute J magnitude of each selected star by matching the estimated [Fe/H], log(g), T_eff, and J − K_s color to values in the isochrone tables for stars of all ages and all values of [α/Fe], rejecting a few more stars for which the best-match χ² > 6. We consider the closest match in the tables to the given input parameters to yield the best estimate of the absolute magnitude from which we estimate a photometric distance using the apparent J-band magnitude.⁷ We save the values of [Fe/H], log(g), T_eff, and J − K_s color of the closest matching model star in the isochrone table; Monte Carlo variation of the stellar parameters about this saved set of values suggests that distances have a fractional precision of 30%–50%, with some larger uncertainties.

We use the proper motions in equatorial coordinates tabulated in RAVE, mostly from Tycho-2 (Høg et al. 2000),⁸ which we then combine with the radial velocity and position to determine the heliocentric velocity in Galactic components (U, V, W) (Johnson & Soderblom 1987; Piatek et al. 2002). We estimate uncertainties in these velocities from 500 Monte Carlo re-selections of all the stellar parameters that affect the distance estimate, adopting σ(J) = 0.03 mag, σ(J − K_s) = 0.042 mag, σ(T_eff) = 300 K, σ(log  g) = 0.3 dex, and σ([Fe/H]) = 0.25 dex (Breddels et al. 2010), as well as the tabulated radial velocity and proper motion uncertainties.

We exclude stars more distant than 500 pc, correct the space velocities for solar motion, and apply the same restrictions as for the GCS sample to select only those with a high probability of being thin-disk stars.

Figure 2 shows histograms of distances and of uncertainties in L_z for the 9803 remaining stars. The median distance is 220 pc from the Sun and uncertainties in L_z are generally smaller than 5%.

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The blue or dotted contours in Figure 1 show the distribution of selected RAVE stars. The distribution in [Fe/H] values is slightly broader than for the GCS stars, reflecting in part at least the lower precision of the spectra. However, the rms scatter in [Fe/H] appears to be ∼0.2, suggesting that the uncertainty of 0.25 estimated by Siebert et al. (2011) is somewhat pessimistic for this sub-sample of main-sequence stars. These authors also suggest uncertainties of 300 K in their estimates of T_eff, and the broader range of T_eff than in the GCS sample is therefore real.

The overlap between the GCS and RAVE samples of stars is quite small but, of those we have selected, precisely 100 stars are positionally coincident. This small degree of overlap allows us to compare the spectral parameters, T_eff, [Fe/H], and radial velocity (v_r), derived separately by the two teams from their independent spectra.

We find that differences in the estimated v_r for the same stars average just 0.32 km s⁻¹ higher in RAVE than in GCS, with an rms scatter about this mean of only 2.3 km s⁻¹. Similarly, differences in T_eff have a mean of 203 K, again higher in RAVE than in GCS, with an rms scatter about the mean of 202 K. Finally, the mean difference in [Fe/H] is 0.057, lower in RAVE than in GCS, with an rms scatter of 0.18. We have looked in vain for systematic variations of these differences, which appear to be consistent with random scatter.

Thus, all three spectral parameters agree between the independent measures to an accuracy that is significantly better than expected from the quoted uncertainties, indicating that both teams have been very careful with their calibrations and conservative with their uncertainty estimates.

The SDSS (York et al. 2000) and Segue2 (Yanny et al. 2009) surveys are complete, but sample fainter stars than RAVE (the magnitude range for Segue2 was 14.0 < g < 20.3) that are therefore generally more distant. Since the Sloan spectral parameter pipeline (Lee et al. 2008) does not attempt to fit stars with T_eff ≲ 4500 K, almost all main-sequence stars with estimated parameters in SDSS DR8 are more distant than 500 pc.

Fortunately, West et al. (2011) provided a catalog of 70,841 M-dwarfs from SDSS DR7. Their table provides a photometric estimate of the distance to each star, as well as a spectroscopic radial velocity, and proper motion from USNO-B/SDSS catalog (Munn et al. 2004, 2008). They suggest that distance uncertainties are typically about 20% and uncertainties in radial velocity are 7–10 km s⁻¹. None of these stars are in RAVE (because they are in different parts of the sky) and all are much cooler than the GCS stars.

As West et al. (2011) recommend, we selected stars with the "goodPM" and "goodPhot" flags set to "true," and the "WDM" flag set to "false" to eliminate possible binaries with a white dwarf companion, which reduces the sample to 39,151 stars. We further excluded stars having no radial velocity as well as those with no distance estimate or for which the estimated distance exceeded 500 pc. As for the GCS and RAVE stars, we correct the space velocities for solar motion and apply the same restrictions to select only those with a high probability of being thin-disk stars, leaving us with a final sample of 11, 022 stars.

We estimated uncertainties in the velocity components, (U, V, W), by combining the 20% distance uncertainty, a 10 km s⁻¹ uncertainty in the radial velocity, and the proper motion uncertainties. The distribution of distances and fractional L_z uncertainties is shown in Figure 3; many of these intrinsically faint stars lie within 200 pc and, again, angular momentum uncertainties are typically ≲ 5%.

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As estimates of T_eff or [Fe/H] are not easily derived from low-resolution spectra of such cool stars, West et al. (2011) do not attempt to provide these quantities in their catalog. Instead, they provide spectral sub-class, which can be related to T_eff, and a titanium oxide index, ζ_TiO, which is believed to be an indicator of metallicity.

Figures 4 and 5 show the metallicity versus angular momentum of GCS and RAVE stars each separately binned into various temperature ranges. The bins are chosen so as to have equal numbers of stars, and the average T_eff of the stars in each panel is shown. The density of points in each panel is indicated by the contours.

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If the rotation curve of the Milky Way were flat at a constant circular speed of 220 km s⁻¹, then stars with L_z = 1210 km s⁻¹ kpc would have home radii of 5.5 kpc, while the home radii of those with L_z = 2200 km s⁻¹ kpc would be 10 kpc. (Note that these numerical values will simply scale with our choices of R₀ and V₀.) Thus, although the stars in our sample are passing close to the Sun right now, many are visiting from quite far afield.

A number of trends can be seen: the spread in L_z is greater for the cooler stars, reflecting the fact that the higher velocity dispersions of older stars allow stars having a wider range of L_z to visit the solar neighborhood. The rms scatter in [Fe/H] is in the range 0.19 ≲ σ ≲ 0.23 for the RAVE stars in each panel, which is less than the nominal uncertainty, as noted in Section 2.2. The spread in [Fe/H] for the GCS stars is generally smaller, rising from 0.14 in the second panel to 0.21 for the coolest stars. The spread for the hottest GCS stars (0.16) is anomalously high, due to incomplete removal of stars that were overcorrected for reddening, as noted in Section 2.1. While the large spread in [Fe/H] is partly due to uncertainties and partly real, each panel has a weak declining trend of metallicity with increasing L_z that reflects the metallicity gradient in the Galaxy. We have added the open circles, which show the variation of the median [Fe/H] in bins of L_z in each panel, simply to make this last trend more clearly visible.

We adopt Theil's non-parametric method (Hollander & Wolfe 1999) to quantify the metallicity gradient in each panel, which we plot as filled symbols in Figure 6, with the dashed error bars showing the 95% confidence limits. This method returns the median slope between every pair of points in the sample with the confidence limits being determined by the range of the pairwise values. We have converted the gradient with angular momentum to the more usual dex per kpc by multiplying by our adopted circular speed V₀ = 220 km s⁻¹. Our abundance gradient estimates therefore assume only a locally flat rotation curve and a distance scale set by adopting R₀ = 8 kpc.

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While these estimates of the metallicity gradient are derived from stars that happen to be close to the Sun at the present time, the sample includes stars from a broad range of home radii, as noted above. Stars whose ages greatly exceed the radial epicycle period, ∼170 Myr at the solar radius, should be uniformly distributed in radial phase around their epicycles. Thus, both the spread in metallicities of the stars in our sample and the gradient we measure are characteristic of the stars at their home radii, and our estimates of the metallicity gradient are based on a radial range of several kiloparsec.

The apparent flattening of the metallicity gradient with decreasing T_eff is suggested by the RAVE stars (left panel of Figure 6) and is more convincing in the better-quality data of the GCS sample (right panel). As this is our main result, we use Monte Carlo simulations to check whether the uncertainties in the slopes are small enough to justify our claim of a trend. We generate 100 new samples, by randomly resampling stars, adding normally distributed values to T_eff, [Fe/H], and L_z to simulate measurement errors. The size of the errors is determined by the appropriate uncertainties from each survey as described above or, for the L_z values of the RAVE stars, by propagating our own estimates of the distance, proper motion, and radial velocity uncertainties. We divide each sample into equal numbers of stars in successive bins of T_eff and compute the slopes. The means and standard errors of these re-estimates are also shown by the crosses and solid error bars in Figure 6. Note that the solid error bars from our Monte Carlo simulations indicate the 1σ dispersion, whereas the dashed error bars from the Theil algorithm show the 95% confidence interval; the two independent estimates are therefore consistent, and the flattening of the slope with decreasing T_eff seems to be confirmed. Note also that the somewhat shallower trend in the Monte Carlo re-estimates from the RAVE data is expected, since resampling the data in this manner inserts additional uncertainties over and above those already present in the data, which inevitably weakens a trend. This effect is more marked in the RAVE sample because uncertainties in their data are larger.

It also occurred to us that the change of slope could perhaps be a systematic effect due to the above-noted increasing spread in L_z with decreasing T_eff—i.e., the greater spread in L_z could simply make the fitted slope appear shallower, while in fact the data may be consistent with a constant slope. We tested for this possibility by creating some pseudo star samples that adopt a linear relation between [Fe/H] and L_z that is the same for all the stars in either the RAVE or the GCS sample, irrespective of their temperature, which is the null hypothesis that we argue Figure 6 disproves. For each star in the sample, we use the measured L_z to assign a new [Fe/H] from a Gaussian distribution about the mean from the adopted trend with a dispersion equal to that about the linear regression line for the entire sample. We then bin this pseudosample by T_eff as before and test whether the cooler stars, which have the larger spread in L_z, have a shallower apparent slope. With 500 realizations for each of the GCS and RAVE samples, we confirmed that we recover the original input slope in every bin. The absence of any systematic trend with T_eff in this test increases our confidence that the trend we find in the data in Figure 6 is real.

Furthermore, both samples also hint at a break in the trend with temperature around T_eff ≲ 6 000 K, although these points alone are also consistent with a uniform trend. In order to find extra evidence for this possible break of slope, we attempt to include the M-dwarf stars into our analysis.

Figure 7 shows the distribution of L_z and ζ_TiO for the M-dwarf stars separated into two groups by spectral class. This metallicity indicator again has a noticeable gradient with L_z.

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As for the hotter stars, we have used Theil's algorithm to estimate the gradient dζ_TiO/dL_z, dividing the M-dwarf sample into four groups by stellar sub-class, and plot the results in green or as squares in Figure 8. (The abscissae were determined from the relation T_eff = 3700–150S K, where S is the spectral sub-class, fitted by eye to the compilation by Reylé et al. (2011).) The M-dwarfs have a constant metallicity gradient that appears independent of T_eff.

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Because they are measured from a different quantity, the values of dζ_TiO/dL_z cannot be compared directly with those of d[Fe/H]/dL_z from the hotter stars, shown in red or crosses and blue or circles. We tried using the empirical relation between ζ_TiO and [Fe/H] suggested by Woolf et al. (2009), but many [Fe/H] values required an extrapolation beyond the range calibrated by their study, and the resulting mean [Fe/H] is unreasonably high. Thus, we lack a direct basis to relate the green points in Figure 8 to the others.

However, we note that if ζ_TiO is proportional to [Fe/H], the gradient d[Fe/H]/dL_z will scale with dζ_TiO/dL_z, and we would merely need to shift the green points vertically by some fixed factor to bring them onto the same scale. Since an anti-correlation between ζ_TiO and [Fe/H] seems unlikely, the green points probably could not be shifted above d[Fe/H]/dL_z = 0. Thus, the data from the M-dwarf stars do seem to confirm that the decreasing trend of metallicity gradient with T_eff among the hotter stars really does break to no trend around T_eff ≲ 6000 K, i.e., for stars whose main-sequence lifetimes are ∼10¹⁰ yr.

In this work, we apply kinematic and vertical motion cuts to eliminate most thick disk and halo stars, which may well also eliminate some old thin-disk stars that should perhaps be included in our samples. However, any bias in our analysis created by excluding older stars because of their higher peculiar velocities will cause lower temperature bins, which are the only bins to include stars of the old disk, to have unrepresentatively few older disk stars.

Radial mixing implies that these possibly missing stars would be members of the most well-mixed population with the lowest radial metallicity gradient, and adding significant numbers of them to the lower T_eff bins would only increase the trend we observe. In fact, we find that the trend of metallicity gradient with T_eff is little affected by limiting the vertical excursions of selected stars to either 300 pc or 500 pc, suggesting that this cut is not biasing our result. Further restricting the vertical oscillations to <300 pc, on the other hand, does exclude a disproportionate fraction of older stars and the trend with T_eff is weakened, as our argument predicts. Naturally also, the number of selected stars decreases with a more severe restriction increasing the statistical uncertainty in our estimate of the metallicity gradient.

We do not use any measure of chemical composition to select stars, and our sample may therefore include some stars with enhanced [α/Fe]. Whether such abundances best characterize thick disk stars is the subject of current debate (Ivezić et al. 2008; Lee et al. 2011), while the existence of a thick disk that is distinct from the thin has been questioned (Bovy et al. 2012). Our study is unaffected by these questions, because we are interested only in how the metallicity gradient changes with the mean age of the stars and, provided we have excluded halo stars that are little affected by spiral activity, we do not need to consider whether each star may or may not belong to a particular population. Note that Solway et al. (2012) have shown that mixing is only gradually weakened by increased random motion and vertical thickness.

A separate issue is that all the stars in this study are close to the Sun at the present time, and we therefore cannot include stars with home radii far from the Sun but with radial excursions that are too small to bring them into the solar neighborhood. However, provided the stars we do see have abundances that are representative of those of their T_eff and home radius, the omitted stars will not affect our estimate of the slope. Since we are not selecting on metallicity, there should be no bias in our slope estimates.

Casagrande et al. (2011) re-analyzed the GCS using a revised calibration of T_eff and [Fe/H] and new estimates of the individual stellar ages. We have tried to compare our results with theirs, which is not straightforward because our bins of T_eff each contain stars having a broad range of ages. We have used isochrone tables (Demarque et al. 2004) to estimate the main-sequence lifetimes of the individual stars in the GCS sample, using the T_eff and [Fe/H] of each, and so find the average main-sequence lifetime of the stars in our separate T_eff bins. The mean maximum age of stars defined in this way increases from about 2 Gyr for our hottest bin to 10 Gyr for the coolest. Using these mean maximum ages to replot the trend of metallicity gradient in our Figure 8, we find reasonable agreement with the gradient values as a cumulative function of age shown in Figure 18(d) of Casagrande et al. (2011).

Our estimated gradient is in the range of other measurements from stars. For example, Coşkunoğlu et al. (2011) derived a metallicity gradient of −0.051 dex kpc⁻¹ for F-dwarfs, and they also find a shallower gradient for the slightly older G-dwarfs. While their sample clearly overlaps with ours, they adopted different selection criteria and employed a different technique to determine the home radii, and so it is encouraging that they reached similar conclusions. Using disk Cepheids, which are young objects, Luck et al. (2011) derived [Fe/H] = −0.055R + 0.475, where R is the Galactocentric radius, again in tolerable agreement with our gradient estimate for the hotter stars.

Chen et al. (2003) estimate a radial gradient of [Fe/H] of −0.063 ± 0.008 dex kpc⁻¹ from 118 star clusters. Both they and Friel et al. (2002) report that, when the clusters are divided by age, the slope appears steeper for the older sample, which is the opposite of our finding here. Note that the radial distribution of star clusters in these papers extends outward from the solar radius, whereas our sample of stars overlaps the solar radius and extends inward to ∼5 kpc—thus the different variations with age could indicate the absence of mixing in the outer disk. However, other systematic differences may complicate this interpretation; e.g., the outer disk open clusters in the sample assembled by Chen et al. (2003) are farther from the mid-plane than are those near the solar radius.

Maciel et al. (2006) also claim a steeper abundance gradient for older PNe, although their estimated gradients for the different elements show a lot of scatter and ages are less reliable than for star clusters. It should be noted that Stanghellini & Haywood (2010) reached the opposite conclusion finding, for a sample of PNe concentrated over the radial range 3 kpc < R < 12 kpc, a shallower abundance gradient for the older PNe, which is in better agreement with our results.