OPTICAL–NEAR-INFRARED PHOTOMETRIC CALIBRATION OF M DWARF METALLICITY AND ITS APPLICATION

The SDSS is one of the most extensive surveys in astronomy. During its operations, it obtained multi-color images that covered more than a quarter of the sky. Multi-band photometry was collected using a 2.5 m wide-angle optical telescope at Apache Point Observatory, in New Mexico. The camera included thirty CCD chips each with 2048 × 2048 pixels (with a total of 120 Megapixels). These chips were organized in five columns of six chips per column. Each column had a different optical filter bandpass, designated u, g, r, i, and z (Fukugita et al. 1996), with average wavelengths of 355.1, 468.6, 616.5, 748.1, and 893.1 nm and with 95% completeness of point sources in typical seeing to magnitudes of 22.0, 22.2, 22.2, 21.3, and 20.5, respectively (Gunn et al. 1998). The SDSS imaging covers several thousand square degrees of sky, and over this region photometric calibrations achieved an accuracy of ≈0.01–0.03 mag in ugriz (Ivezić et al. 2004, 2007; Adelman-McCarthy et al. 2006; Tucker et al. 2006; Padmanabhan et al. 2008).

Our calibration uses both optical (the SDSS g) and NIR (the 2MASS JHK) broadband filters and we cross-matched the SDSS and 2MASS data for collecting our sample. The 2MASS was a survey of the whole sky in the three IR broadband filters J, H, and K, with average wavelengths of 1.2, 1.65, and 2.17 μm, respectively. The measurements were made from 1997 to 2001 using two highly automated 1.3 m telescopes, one at Mt. Hopkins, Arizona for the northern hemisphere observations and one at Cerro Tololo Inter-American Observatory, Chile for the southern hemisphere data. Each telescope had a three-channel camera, each containing a 256 × 256 array of infrared detectors to scan the sky simultaneously in the three filters. Point sources brighter than about 1 mJy, with signal-to-noise ratio greater than 10 in each band were characterized and compiled in a separate catalog, the 2MASS Point Source Catalog (2MASS–PSC, Cutri et al. 2003). These bright sources ($\leqslant$ 13 mag) generally have 1σ photometric uncertainty of <0.03 mag (Skrutskie et al. 2006).

We used the SQL Search tool in the SkyServer DR 9 to collect an M dwarf sample suitable for our study. The stars were taken from the SDSS DR 9 photometric catalog (identified by PhotoObj, Ahn et al. 2012) with the object class specified by Type = 6 (for stars). Each object in the SDSS DR 9 catalog has a unique SDSS identifier, called ObjID, which is also included in the 2MASS–PSC for objects in the regions that overlapped with the SDSS. We therefore selected those stars with the same ObjIDs in the two catalogs.

To minimize extinction, we chose stars with high Galactic latitude $b\geqslant 50{}^\circ$. It has long been shown that, on average, extinction decreases at higher latitudes, due to the low column densities found along these lines of sight (LOS; Larson & Whittet 2005; Zagury 2006).

We employed the dust maps of Schlegel et al. (1998, hereafter S98) to correct the stellar photometry for Galactic extinction. Currently, these maps provide the most comprehensive dust data of the Milky Way on a large scale, including two two-dimensional full-sky maps (one for the northern and one for the southern Galactic hemisphere) with total LOS dust column densities determined from far-infrared (100 and 240 μm) emission data. These maps have a resolution of 6.1 arcmin and are shown to predict reddening within 16% (S98). We also used the relative extinctions given in Table 6 of S98 to convert the extinction corrections in the V-band magnitude to SDSS–2MASS filter bands. While the maps mentioned above provide an efficient tool for estimating dust extinction, they refer to the total Galactic extinction along the LOS and so may overestimate the true extinction to nearby stars (Covey et al. 2007, hereafter C07, Jones et al. 2011, hereafter J11). Most of objects in the Galaxy lie behind only a portion of the Milky Way's total dust column and thus are likely attenuated and reddened by only a fraction of the total dust column.

Using spectra of more than 56,000 M dwarfs from the SDSS, J11 created a high-latitude, three-dimensional extinction map of the local Galaxy. In their technique, spectra from stars in the SDSS DR7 dwarf sample along low-extinction LOS were compared with other SDSS M dwarf spectra for deriving distances and accurate LOS extinction. The three-dimensional map is most appropriate for stars within around 500 pc from the Galactic plane (z $\leqslant \;500$ pc). As mentioned in J11, it can safely be assumed that stars with z-heights greater than 500 pc are essentially behind the entire dust column and the maps of S98 are more useful for such stars.

Due to partial overlaps between neighboring SDSS images, an object might be observed more than once. The best observation of an object is called PRIMARY detection and the other observations, if any, are assigned as SECONDARY detections. To avoid duplication, we chose only PRIMARY observations by setting the variable "mode" to 1. We also used point-spread function (PSF) magnitudes for SDSS data since these optimally measure the total fluxes for stars in this study.

By equating the flag CLEAN to 1, we selected those stars whose SDSS magnitudes has passed appropriate standards of clean photometry (i.e., the photometric flags: NOPROFILE, PEAKCENTER, NOTCHECKED, PSF_FLUX_INTERP, SATURATED, BAD_COUNTS_ERROR, DEBLEND_NOPEAK and INTERP_CENTER are not set). We also set the 2MASS read flag to "222," blend flag to "111," and contamination/confusion flag to "000" to select only those stars that have unsaturated, unblended, and uncontaminated photometry in the three IR bands (J, H, and K).

We selected stars that fall within typical color ranges for late-type K and M dwarfs, $r-i$ $\gtrsim \;0.47$ and $i-z$ $\gtrsim \;0.25$ (W11). There may be contamination by giants in any color-selected sample of M dwarfs, however, which needs to be addressed in statistical studies of the Galaxy. Bessell & Brett (1988, hereafter BB88) found in the JHK, there is a clear bifurcation between M giants and M dwarfs, beginning where TiO bands appear or the M dwarf sequence starts. By applying the NIR color ranges typical of M giants (BB88; C07), we found that around 1% of stars in our SDSS–2MASS sample are giants, consistent with other studies such as Covey et al. (2008) with less than 2% and W11 with around 0.5% giant contamination.

It should be mentioned that this approach for separating giants from dwarfs is not accurate for K-type stars (BB88), as they overlap in JHK color space. A better way to remove giants employs reduced proper motions of stars as described in M13b. By using a sample of stars with available proper motions, giants could be separated from dwarfs with higher accuracy.

The distance and z-height of our stars were estimated by means of a photometric parallax technique developed by Bochanski et al. (2010, hereafter B10). More specifically, the absolute magnitude in the r band, M_r, of low-mass stars can be obtained from a $(J-K)-$M_r relation (the first equation of Table 4 in B10), which is valid for the color range $0.50\lt (r-z)\lt 4.53$. We therefore selected only those stars which fall in this color range.

After meeting all the criteria above, as well as the color cuts applicable to our metallicity calibration outlined in Section 3, a sample of 1,330,179 M dwarfs was selected.

The relation between metallicity and distance z above the Galactic plane can be investigated by determining the metallicity distribution of stars as a function of z-height. We divided a subsample of our stars with z $\leqslant$ 1200 pc into 12 bins of equal Δz = 100 pc. The metallicity distribution associated with each bin is shown in Figures 6 and 7. There is a clear shift toward lower metallicities as z-height increases; the fraction of metal-rich stars slowly decreases and the distributions become more metal-poor with increasing height. The mean metallicity decreases (by around 0.19 dex) from 0 to 1200 pc, which is consistent with the age–metallicity–height relation. Studies of Galactic evolution have demonstrated that stars that formed at earlier times in the Galaxy's history generally have lower metallicities in general and are, on average, farther from the Galactic plane. Stellar systems are formed from interstellar gas and, at the end of their lifetimes, they may return a substantial fraction of their initial masses enriched with metals to the interstellar medium. As a result, succeeding generations of stars become more metal-rich than their predecessors, leading to an age–metallicity relation. The pioneering work on this was done by Twarog (1980a, 1980b), who derived the age–metallicity relation for the disk in the neighborhood of the Sun using a large sample of southern F dwarfs. They found that the mean metallicity of the Galactic disk increased by about a factor of five between 12 and 5 billion years ago and has changed only slightly since then. A recent evidence of this kind was offered by Casagrande et al. (2011, hereafter C11) who determined the metallicity distributions of three samples of solar-type stars with different ranges of age (Figure 16 of their paper). They showed that young stars (ages $\lt 1$ Gyr) have a quite narrow distribution around higher values of metallicity, whereas intermediate-age (between 1 and 5 Gyr) and old-age ($\gt 5$ Gyr) stars present broader distributions, significantly extending to lower metallicities.

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In addition to the age–metallicity relation, there is also a relation between age and velocity dispersion. Observations have demonstrated that old stellar populations in the disk have larger velocity dispersions. Based on the sample of Twarog above, Carlberg et al. (1985) examined the age–velocity dispersion relation for stars in the solar neighborhood. They found an increase in velocity dispersion to a maximum at about 6 Gyr, and thereafter the trend remained roughly constant.⁷ The term "dynamical heating" is commonly applied to all processes that cause an increase in velocity dispersion with age. Different mechanisms are responsible for vertical dynamical heating (e.g., Nordström 2008). Whatever the mechanism(s), stars dissipate from the Galactic plane, increasing their vertical distance from the plane in the course of time. One therefore expects that older, more dynamically heated stars should be found, on average, at larger z-heights than the younger stars. As a consequence, stars farther from the Galactic plane should be, on average, more metal-poor than those closer to the plane.

Using a metal-sensitive ratio, (CaH2 + CaH3)/TiO5, from Lépine et al. (2003) as a proxy for M dwarf metallicity, West et al. (2008) demonstrated a decrease in metallicity as a function of z-height (up to 1000 pc), which implies that stars more distant from the Galactic plane have lower metallicities. In a simulation by West et al. (2006, 2008), a simple one-dimensional model of the thin disk was applied to investigate the vertical motions and positions of M dwarfs over the Galaxy's lifetime. The model also showed a decrease in activity fractions (as traced by Hα emission) as a function of z-height for all M dwarf spectral types, suggesting a decline in the magnetic activity of older M dwarfs. All results from this 1D dynamical simulation are consistent with observed activity and velocity trends.

Figure 8 indicates a comparison between the metallicity distribution (volume-corrected) from this work (solid line) and that from Woof & West (2012, hereafter WW12) who obtained the metallicity distribution of 4141 M dwarfs in the spectroscopic SDSS catalog (dashed line). As shown in the figure, there is a slight offset between the peaks of the two M dwarf distributions. It appears, however, that our sample includes a larger fraction of metal-poor stars than that of WW12. Moreover, the distribution of WW12 has a steeper slope toward lower metallicities than ours.

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Possible reasons for the discrepancy between these two distributions are as follows. First, the distributions were obtained by different methods. To estimate the metallicity of their sample, WW12 derived a linear relation (Equation (1) in their paper) between [Fe/H] and the metallicity-sensitive parameter ζ defined in L07. This relation was applicable to M dwarfs with 3500 K $\leqslant$ T $_{{\rm eff}}$ $\leqslant$ 4000 K and $-1.5\;{\rm dex}\;\leqslant$ [Fe/H] $\leqslant$ +0.05 dex. Dhital et al. (2012) recalibrated the definition of ζ and showed that this new ζ could be a significantly better indicator of metallicity for early M-type dwarfs (between M0 and M3). M13a tested the parameter ζ defined in L07 using their own sample and found that it is not only sensitive to metallicity (and temperature) but also to some other stellar characteristics such as activity and surface gravity. This could lead to an incorrect identification of some metal-poor stars as near-solar metallicity stars. They pointed out that although ζ correlates well with [Fe/H] for supersolar metallicities, it does not always diagnose metal-poor M dwarfs correctly and may identify such stars as more metal-rich than they are. Further, Lépine et al. (2013) remarked that the parameter ζ defined in L07 overestimated the metallicities of early-type M dwarfs whereas the parameter defined in Dhital et al. (2012) underestimated the metallicities of these stars, leading them to the redefinition of the index ζ. Due to the overestimation of the metallicity of metal-poor M dwarfs by the parameter ζ from L07, we could expect a bias toward metal-rich stars in the metallicity distribution of WW12, as seen in Figure 8.

Second, there is an important difference between our sample and that of WW12. As shown in Section 4, our sample is taken mostly from the Galactic thin disk, and a large fraction of its stars are in the solar neighborhood with $z\lesssim 500\;{\rm pc}$. On the other hand, a significant portion of stars in the sample of WW12 lies outside the thin disk and is not representative of the local neighborhood: the WW12's sample is expected to have more low-metallicity stars than ours, but that is not what is seen in Figure 7.

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It should be noted that the distribution of WW12 in Figure 8 is volume-corrected due to the metallicity-dependent volume coverage. It should be mentioned that this is different from the volume correction (owing to only volume effects) that we made above. WW12 argued that since low-metallicity main sequence stars (subdwarfs) are less luminous than higher-metallicity stars of the same temperature or spectral type, higher metallicity stars should be overrepresented in magnitude-limited samples. In other words, metal-poor M dwarfs must be, on average, closer than the metal-rich ones in such samples. Using a sample of stars from Woolf et al. (2009) for which parallax data were available, WW12 determined the luminosity variation with metallicity for M dwarfs in the temperature range 3500 K $\leqslant$ T_eff $\leqslant$ 4000 K. Based on this variation, they calculated volume-correction factors for different values of [Fe/H] (Table 1 in their paper) and implemented a volume correction for their metallicity distribution. Accordingly, to have a more reliable metallicity distribution, we need to determine the metallicity-dependent correction factors appropriate to our magnitude-limited sample. Nevertheless, we would not expect these corrections to change the location of the peak and the slope of the low-metallicity tail significantly (Figure 2 of WW12). Therefore, the uncorrected metallicity distribution can give us, to some extent, important information about the local Galactic disk.

Figure 9 compares our M dwarf metallicity distributions with those of G and K dwarfs in the SEGUE survey from Schlesinger et al. (2012, hereafter S12) for three different Galactic height ranges. As noted in S12, the G and K dwarf metallicity distributions have more metal-poor stars as height increases. It can be seen that the M dwarf distribution exhibits the same trend, which implies a similar star formation history as G and K dwarfs. There is also a discrepancy between the distributions of different spectral types; the cooler spectral type, the more metal rich the distribution includes.

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JA09 argued that the M and FGK dwarfs in the local Galactic thin disk should have the same metallicity distributions and that there should not be a systematic offset between M- and FGK dwarf metallicity distributions. They related the offset between the metallicity distributions of M dwarfs and FGK dwarfs in some published papers to the reliability of the methods by which the metallicity of M dwarfs were determined. Accordingly, they adopted the mean M dwarf metallicity of their sample to be the same as the mean metallicity of a volume-limited sample of FGK dwarfs ($\sim -0.05$ dex) from VF05. SL10 improved upon JA09 and remarked that for a fair comparison between samples of M dwarfs and FGK dwarfs in the local Galactic disk, the samples must not only be volume corrected but also have equivalent kinematics. By selecting two volume-completed, kinematically matched samples of FG and M dwarfs, SL10 derived a mean metallicity of around −0.14 dex for M dwarfs in the solar neighborhood. It should be mentioned that we compared our distribution to those of previous works without any consideration of kinematics of the samples under study. To make proper comparisons between the metallicity distributions of dwarf stars with different spectral types, volume-corrected samples with equivalent kinematics have to be considered.

On the other hand, the offset between FGK- and M dwarf distributions might be explained in other ways. For example, similar to S12, one may expect that since M dwarfs have longer lifetimes than G and K dwarfs, there is a possibility that more metal-rich G and K dwarfs have evolved off the MS, causing a metallicity bias in G- and K dwarf distributions against high-metallicity values which has not happened for cooler M dwarfs. Furthermore, if more lower-mass dwarfs are born than higher-mass stars as the metallicity of interstellar gas increases, there should be fewer metal-poor M dwarfs than G and K dwarfs.

While there have been discrepancies between the metallicity distributions of different spectral types from various studies and some effort has been made to explain them, we can mention the work of C11 whose metallicity distribution of solar type stars (obtained by a photometric method) has a peak around $[{\rm Fe}/{\rm H}]\simeq 0,$ close to those of the M dwarf distributions shown in Figure 8. This is in agreement with the statement of JA09 who pointed out that there should not be any offset of different metallicity distributions. To reach an accurate conclusion, more careful investigations are needed.

The simplest model of GCE, the SCBM, is based on three assumptions.

• 1.  
  The system is isolated and there is no mass inflow or outflow,
• 2.  
  There are two types of stars; those that live forever and those that die right after their birth, indicating instantaneous recycling,
• 3.  
  The system initially starts off with entirely metal-free gas and eventually ends up full of stars.

Studies of the chemical evolution of local G dwarfs (e.g., van den Bergh 1962; Pagel & Patchett 1975; Wyse & Gilmore 1995) have shown the existence of the "G dwarf problem," which indicates that the SCBM predicts many more low metallicity G dwarfs than are observed. Similarly, the "K dwarf problem," a paucity of metal-poor K dwarfs relative to the prediction of the SCBM, has also been observed (e.g., Casuso & Beckman 2004, hereafter CB04). WW12 were the first to recognize the "M dwarf problem" (as shown in Figure 2 of their paper); the number of low metallicity M dwarfs is insufficient to match the SCBM, as with G and K dwarfs.

To test the SCBM, we compare our metallicity distribution with that predicted by the model. It can be shown (e.g., Pagel 2009, hereafter P09; Mo et al. 2010) that the mass in stars (M_S) with metallicities between Z and Z + dZ is given by

$$\begin{eqnarray}&&d{{M}_{{\rm S}}}\propto {\rm exp} (-\frac{Z}{p})dZ\end{eqnarray} \tag{ 2 }$$

where p is the metal yield, defined as the mass of newly synthesized and ejected metals per unit mass permanently locked up in the stars. Considering [Fe/H] ∼ [M/H] $\sim {\rm log} \left( \frac{Z}{{{Z}_{}}} \right)$, the mass of stars with metal abundances between [Fe/H] and [Fe/H]+d[Fe/H] can be estimated as

$$\begin{eqnarray}&&d{{M}_{{\rm S}}}\propto Z{\rm exp} (-\frac{Z}{p})d[{\rm Fe}/{\rm H}],\end{eqnarray} \tag{ 3 }$$

which can be rewritten as

$$\begin{eqnarray}&&\frac{d{{M}_{{\rm S}}}}{d{\rm log} ({\rm Z})}\propto \ Z\;{\rm exp} (-\frac{Z}{p})\end{eqnarray} \tag{ 4 }$$

The proportionality (4) is the metallicity distribution function predicted by the SCBM. The histogram in Figure 10 depicts our volume-corrected metallicity distribution. The black curve in this figure shows the normalized distribution from the SCBM with p = 0.025, which has a maximum at the same metallicity ([Fe/H] = +0.1 dex) as our histogram.

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Obviously, the SCBM cannot reproduce the low metallicity tail of the distribution, suggesting the M dwarf problem. It is clear that one or more of the assumptions in the SCBM, such as instantaneous recycling, the system's isolation without gas flow or the assumption of a zero initial metallicity are unrealistic. The G and K dwarf problems have motivated a generation of more complicated GCE models. We will examine a few such models and test them using our M dwarf metallicity distribution in the following subsections.

One of the solutions proposed for the G dwarf problem is to assume an initial metallicity of ${{Z}_{0}}\ne 0$ (Samland & Hensler 1996). If, for some reason, there is a finite initial abundance of order [Fe/H]$_{0}\approx -1$ dex, a good fit to the distribution is obtained at the expense of rejecting some stars with low metallicity. In this case, all metallicities (Z) in Equation (4) are replaced by $Z-{{Z}_{0}}$ yielding:

$$\begin{eqnarray}&&\frac{d{{M}_{{\rm S}}}}{d{\rm log} (Z)}\ \propto \ (Z-{{Z}_{0}}){\rm exp} (-\frac{(Z-{{Z}_{0}})}{p}).\end{eqnarray} \tag{ 5 }$$

The normalized distributions calculated by the SCBM with two selected initial metal abundances of ${{Z}_{0}}$ = 0.002 (the red curve) and 0.003 (the blue curve) are shown in Figure 10. We set the yields for the red and blue curves to p = 0.023 and 0.022, respectively, to have a maximum at [Fe/H] = +0.1 (the same as our distribution). As can be seen, the distributions of the SCBM with a pre-enrichment give a better fit to our measured metallicity distribution than that without any initial abundances (the black curve).

There have been several hypotheses to justify such a pre-enrichment. For example, this might arise from prior star formation activity in the halo (Ostriker & Thuan 1975) or the bulge. It was argued (Prantzos 2007), however, that while the Galactic halo reached a maximum metallicity of Z ∼ 0.002 (∼0.1 ${{Z}_{}}$), its mean metallicity is Z ∼ 0.0006 (∼0.03 ${{Z}_{}}$) and its total mass (2 × 10⁹ ${{M}_{}}$) is 20 times less than that of the disk (4.5 × 10¹⁰ ${{M}_{}}$). Köppen & Arimoto (1990) suggested a model in which the bulge evolves with a large yield (p ≈ 0.034) and ends star formation by ejecting 1/10 of its mass as highly enriched gas with an abundance of 0.05 (∼2.5 ${{Z}_{}}$) in a terminal wind, which is captured by the proto-disk. Assuming that the proto-disk has a mass equal to that of the bulge, this mass transfer results in an initial disk abundance of 0.005 (∼0.25 ${{Z}_{}}$) after mixing. Such a model is consistent with the metallicity distribution functions of both the bulge and disk (for more discussions, see P09).

On the other hand, the p values derived from these models above are larger than the estimated value in the solar neighborhood: ${{p}_{{\rm SolN}}}$ ∼ 0.7 ${{Z}_{}}$ = 0.014 (P09). We therefore need to establish models that not only match the metallicity distributions of the solar neighborhood but also give yields more consistent with observational data.

In more realistic models, the disk does not evolve as a closed box and gas inflow or outflow (or both) is present. Inflow models have been attractive because of their capacity to reduce the number of low-metallicity stars born at early times, providing an elegant solution of the G dwarf problem, especially in view of the fact that gas accretion is expected to be a common phenomenon in the universe (e.g., Prantzos 2007). There is indirect observational evidence for gas infall into the Galactic disk. The star-formation rate in the disk is about a few solar masses per year, while the total gas mass therein is $\sim 5\;\times \;{{10}^{9}}\;{{M}_{}}$. Thus, if there were no accretion of new material, our disk would run out of gas in a fraction of the Hubble time (Mo et al. 2010).

The inflowing gas could originate from the gaseous halo, the bulge, the outer parts of the disk, or the accretion of small satellite galaxies. The disk itself may be considered as having formed completely from such inflowing gas or as having an initial mass (usually small compared with its final mass) with or without metals. Still more physically realistic models assume gas inflow with variable rates, and those with declining rates appear especially to provide in better agreement with recent observations (P09). In this section, two inflow models will be considered and the resulting metallicity distributions will be compared with that from this study.

6.3.1. The Exponential Inflow Model (EIM)

Some studies have shown that an exponentially decreasing infall rate with a long characteristic time scale ∼7 Gyr can provide a reasonably good fit to the data. For example, CB04 demonstrated that such a model seems to reproduced the observed metallicity distributions of G and K dwarfs in the disk.

For simplicity, some inflow models are based on a specific star formation rate given by

$$\begin{eqnarray}&&\frac{d{{M}_{{\rm S}}}}{dt}=w{{M}_{{\rm G}}}(t)\end{eqnarray} \tag{ 6 }$$

where t is time, ${{M}_{{\rm G}}}$(t) is the gas mass as a function of time and w is a constant. By defining a time-like variable u by

$$\begin{eqnarray}&&u=wt,\end{eqnarray} \tag{ 7 }$$

Sommer-Larsen (1991) constructed a dynamical model with the assumption

$$\begin{eqnarray}&&F(u)=A\;w\;{\rm exp} (-u)\end{eqnarray} \tag{ 8 }$$

in which F is the accretion rate of material from outside the system and A is a constant. If the initial mass and metallicity of the system are assumed to be zero and if the inflow gas has no metal content (${{Z}_{{\rm F}}}$ = 0) then, as u becomes very large, the expression for Z and $d{{M}_{{\rm S}}}$/dlog(Z) can simply be written as (P09):

$$\begin{eqnarray}&&Z(u)=\left( \frac{pu}{2} \right)\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&\frac{d{{M}_{{\rm S}}}}{d{\rm log} ({\rm Z})}\ \propto \ {{\left( \frac{Z}{p} \right)}^{2}}\ {\rm exp} (-2Z/p).\end{eqnarray} \tag{ 10 }$$

The expression on the right side of the proportionality (10) is essentially the square of that corresponding to the SCBM. The normalized distribution from the EIM for p = 0.025, which peaks at [Fe/H] = +0.1, is shown in Figure 11 (the red curve); it presents the metallicity distributions from this work (the histogram) and from the SCBM (the black curve) as well. Evidently, the EIM predicts fewer metal-poor M dwarfs and a steeper slope in the low metallicity tail than the SCBM, and is therefore in better agreement with the data. There is, however, an inconsistency between the value of p from this model and ${{P}_{{\rm SolN}}}$, which is somewhat problematic. It should be remarked here that if our distribution had a peak at a lower metallicity, say, [Fe/H] $\approx -0.1$ dex, the model would result in a more reasonable yield. Accordingly, the accuracy of the metallicity distribution is critical for a comparison of GCE models.

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6.3.2. Clayton's Models

Clayton (1985, 1987, 1988) has derived a series of models in which the inflow rate is parameterized by

$$\begin{eqnarray}&&F(t)=\frac{{\rm k}}{t+{{t}_{0}}}{{M}_{{\rm G}}}(t)\end{eqnarray} \tag{ 11 }$$

where k is an integer and ${{t}_{0}}$ (or ${{u}_{0}}=w{{t}_{0}}$) is arbitrary. By considering Equations (6) and (7) and assuming ${{Z}_{0}}={{Z}_{F}}=0$, expressions for Z and $d{{M}_{{\rm S}}}/d{\rm log} ({\rm Z})$ in terms of u can be derived (Equations (8.38) and (8.39) in P09). The red curve in Figure 12 shows a distribution of this kind with k = 7 and p = 0.017, having a peak at [Fe/H] = +0.1 dex. The P value in this case is in better agreement with ${{P}_{{\rm SolN}}}$. Although the model fits the low-metallicity tail of our distribution (the histogram) well compared with the SCBM (the black curve), it underestimates the number of supersolar-metallicity stars, and a modification of this model for the high-metallicity tail is needed.

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There have been other models such as merger models (Nagashima & Okamoto 2006) that resolve the G and K dwarf problems and can be tested using M dwarf metallicity distributions in future. However, larger, more accurate data sets are required in order to discriminate among these models and find the more realistic one.

The metallicity distribution of long-lived stars (i.e., G, K, and M dwarfs) as well as the age–metallicity relation traced by the metallicity of these dwarfs are the most important observational constraints for GCE. These low-mass stars belong to stellar populations of different ages and their metallicity distribution provides a fairly complete record of the evolutionary history of the Galaxy. However, CB04 pointed out that G dwarfs are sufficiently massive that they have begun to evolve away from the MS, and consequently, K dwarfs would make a cleaner sample for the local metal abundance distribution. For the same reason, we can expect that M dwarfs would provide an even better sample to represent the local metallicity distribution. M dwarfs are numerous, making up around half the total stellar mass of the Galaxy (Reid & Gizis 1997), and their metallicity distribution therefore offers a robust tool for studying the chemical evolution of the Milky Way. However, more precise metallicity calibrations are essential.