AN H-BAND SPECTROSCOPIC METALLICITY CALIBRATION FOR M DWARFS

The first step in measuring the EWs was to define a standard width and location for each feature. The width had to be sufficient to contain the feature at all strengths but not so wide as to contain nearby features. Our fixed feature widths, W, were

$$\begin{eqnarray*} W_{{{\rm Na,}K\hbox{-}{\rm band}}} & = & 71 \,\mathrm{\AA}, \\ W_{{{\rm Ca,}K\hbox{-}{\rm band}}} & = & 74 \,\mathrm{\AA}, \\ W_{{{\rm Ca}(1),H\hbox{-}{\rm band}}} & = & 25 \,\mathrm{\AA}, \\ W_{{{\rm Ca}(2),H\hbox{-}{\rm band}}} & = & 33 \,\mathrm{\AA}, \\ W_{{{\rm K,}H\hbox{-}{\rm band}}} & = & 24 \,\mathrm{\AA}. \\ \end{eqnarray*}$$

As discussed in Section 3.2, the feature widths were chosen to produce the best agreement between our analytic and Monte Carlo error estimates, signifying that the EW measurements were robust to small changes in the spectrum. These windows were positioned on centers C:

$$\begin{eqnarray*} C_{{{\rm Na},K\hbox{-}{\rm band}}} & = & 2.2074 \,\mu \mathrm{m}, \\ C_{{{\rm Ca},K\hbox{-}{\rm band}}} & = & 2.2638 \,\mu \mathrm{m}, \\ C_{{{\rm Ca}(1),H\hbox{-}{\rm band}}} & = & 1.6159 \,\mu \mathrm{m}, \\ C_{{{\rm Ca}(2),H\hbox{-}{\rm band}}} & = & 1.6203 \,\mu \mathrm{m}, \\ C_{{{\rm K},H\hbox{-}{\rm band}}} & = & 1.5171 \,\mu \mathrm{m}. \\ \end{eqnarray*}$$

Calculation of the EWs also requires an estimate of the pseudo-continuum for normalization. To obtain this, we selected regions in each order that bracketed our features and contained no strong features themselves. The continuum regions are listed in Figure 1. These regions were fit by a fourth-order Legendre polynomial using the IDL svdfit routine, and the fits were visually confirmed.

We then accounted for radial velocity (RV), opting to use the strong features provided by the entire K-band order and not the comparatively noisy H band. As a template, we used the high-S/N spectrum of the M1.5V star HD36395 from the IRTF spectral library (Rayner et al. 2009). A fit to the cross-correlation peak yielded a velocity offset, which was applied to all the feature and continuum regions for each target.

EW measurement for each feature then proceeded in the standard fashion:

$$\begin{equation} \mathrm{EW}_{\lambda } = \sum _{i} \left[ 1 - \frac{I(\lambda _{i})}{I_{c}(\lambda _{i})} \right]\,\Delta \lambda _{i}, \end{equation} \tag{ 1 }$$

where the feature spans pixels i, λ_i is the wavelength at pixel i, I is the intensity, and I_c is the pseudo-continuum intensity. We included in this sum fractional "edge" pixels which fell inside the feature windows, weighted by the proportion of the pixel contained in the window. For the H-band Ca i feature, we combined the two components:

$$\begin{equation} \mathrm{EW}_{{{\rm Ca},H\hbox{-}{\rm band}}}=\mathrm{EW}_{{{\rm Ca}(1),H\hbox{-}{\rm band}}}+\mathrm{EW}_{{{\rm Ca}(2),H\hbox{-}{\rm band}}}. \end{equation} \tag{ 2 }$$

As suggested by RA10, we accounted for temperature effects by including in our model H₂O indices in both the K and H bands, as the strength of H₂O absorption in these bands tracks spectral type (Covey et al. 2010). These indices are defined as

$$\begin{eqnarray*} \mathrm{H}_{2}\mathrm{O\hbox{-}}K&=&\frac{ \langle \mathcal {F}(2.180 \hbox{--} 2.200) \rangle / \langle \mathcal {F}(2.270\hbox{--}2.290) \rangle }{ \langle \mathcal {F}(2.270\hbox{--}2.290) \rangle / \langle \mathcal {F}(2.360\hbox{--}2.380) \rangle }, \\ \mathrm{H}_{2}\mathrm{O\hbox{-}}H&=&\frac{ \langle \mathcal {F}(1.595\hbox{--}1.615) \rangle / \langle \mathcal {F}(1.680\hbox{--}1.700) \rangle }{ \langle \mathcal {F}(1.680 \hbox{--} 1.700) \rangle / \langle \mathcal {F}(1.760 \hbox{--} 1.780) \rangle }, \end{eqnarray*}$$

where $\langle \mathcal {F}(a\hbox{--}b) \rangle$ is the mean flux in the range bounded by a and b in μm. A calibration of H₂O-K based on the stars in the IRTF Spectral Library (Rayner et al. 2009) provides the spectral-type estimates listed in Table 1.

We used two independent methods to estimate uncertainties in our EW measurements. The first was an analytic formalism described in Sembach & Savage (1992). Briefly, this method accounts both for errors in intensity and errors in the continuum fit, using the measured uncertainty from the extraction and the covariance matrix from the least-squares fit to the pseudo-continuum. This yielded average relative errors for our EWs of 1%–3%.

Since this method did not account for errors induced by our RV correction, we also computed Monte Carlo errors. We performed 100 trials for each target, jittering each pixel by a draw from a normal distribution defined by its extracted error (∼0.3%–0.5%), and repeated the entire EW analysis on each iteration. The spread in the recovered EW values included errors in RV shifts, intensity, and pseudo-continuum fitting. In Table 1, we report the more conservative Monte Carlo uncertainties. We selected our feature definitions by minimizing (through trial-and-error) the discrepancies between our analytic and Monte Carlo errors. For all except two metallicity calibrators, the Monte Carlo and analytic error estimates were consistent, their uncertainties being dominated by intensity and continuum-fitting uncertainties rather than RV uncertainties.

We note that some of our observations were performed in conditions with highly variable cloud cover and water column. We diagnosed this issue by reducing individual frames of a selection of targets which had high S/N in each frame. In poor conditions, we found that the EW of the weak H-band K i feature could vary between frames by as much as 20%–30% (∼0.2 Å). We suspect this is due to strong nearby H₂O features (as shown in Figure 1). Simulating strong H₂O features (Clough et al. 2005) in varying water column and calculating the resultant EWs showed that a realistic variation of 1 mm of water column yielded ΔEW ≈ 0.2 Å, enough to account for the observed fluctuations.

We performed two least-squares linear regressions on our calibration targets, with metallicity as the response: one with the K-band features and the H₂O-K index, and one with the H-band features and the H₂O-H index. For the K band, we found the best-fit equation

$$\begin{eqnarray*} [\mathrm{Fe}/\mathrm{H}]_{{K\hbox{-}{\rm band}}}&=&0.132(\mathrm{EW}_{\mathrm{Na}}) + 0.083(\mathrm{EW}_{\mathrm{Ca}})\\ && - 0.403(\mathrm{H}_{2}\mathrm{O\hbox{-}}K) - 0.616, \\ \sigma ([\mathrm{Fe}/\mathrm{H}]_{{K\hbox{-}{\rm band}}}) &=& 0.12. \\ \end{eqnarray*}$$

For the H band, we found

$$\begin{eqnarray*} [\mathrm{Fe}/\mathrm{H}]_{{H\hbox{-}{\rm band}}} &=& 0.340(\mathrm{EW}_{\mathrm{Ca}}) + 0.407(\mathrm{EW}_{\mathrm{K}})\\ && + 0.436(\mathrm{H}_{2}\mathrm{O\hbox{-}}H)-1.485, \\ \sigma ([\mathrm{Fe}/\mathrm{H}]_{{H\hbox{-}{\rm band}}}) &=& 0.12, \\ \end{eqnarray*}$$

where σ is the dispersion of the residuals, and an average of the K- and H-band metallicity estimates also yields a residual dispersion of σ([Fe/H]_avg) = 0.12. Residuals to these fits are shown in Figure 2. The observed fluctuations of the weak K i feature suggest that the H-band metallicity estimate is more susceptible to atmospheric variability. For example, a variation of 0.2 Å in the H₂O feature would yield Δ[Fe/H] ≈ 0.08 dex.

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For comparison with other empirical methods, we present the residual mean squares (RMS_p) and the adjusted squared multiple correlation coefficients R²_a for our models and others (from RA10 and Schlaufman & Laughlin 2010). Our H- and K-band models provide equivalent fits: both have RMS_p = 0.02 and they have R²_a = 0.74 and R²_a = 0.73, respectively. Our results are an improvement over those presented by Rojas-Ayala et al. (2011; RMS_p = 0.02, R²_a = 0.67), RA10 (RMS_p = 0.02, R²_a = 0.63), JA09 (RMS_p = 0.04, R²_a = 0.059), SL10 (RMS = 0.02, R²_a = 0.49), and Neves et al. (2012; RMS_p = 0.03, R²_a = 0.43). Each quantity is reported for the calibration set with which it was created.

As shown in Figure 2, our calibration is well constrained for approximately −0.25  <  [Fe/H]  <  0.3. It is poorly constrained for late (>M5), low-metallicity M dwarfs. We note that increasing the order of our model (e.g., to quadratic) does improve our fit, but this is mostly due to the single point with [Fe/H]_SPOCS = −0.69 for which our calibrations overestimate the metallicity compared to the primary star (as does RA10).

The main disadvantage of the H-band method is that the H-band features are much weaker than those in the K band, and therefore higher S/N spectra and non-variable conditions are required in order to accurately measure the feature strengths. However, the H band offers three important advantages.

• 1.  
  The H-band calibration allows an independent metallicity estimate from the K-band calibration, in a spectral region less plagued by thermal noise. The information content of the H band may be useful to other groups which are pursuing similar programs to ours.
• 2.  
  The H-band relation is superior to previous models in terms of R²_a. That this relation performs as well as it does despite its sensitivity to atmospheric effects is quite promising. Moreover, higher resolution observations can mitigate much of the atmospheric effects with detailed telluric modeling.
• 3.  
  The H-band model locates features that are strongly correlated with metallicity. These features and our calibration can be used as a starting point for M dwarf metallicity studies with high-resolution surveys like APOGEE. A recent analysis by O11 demonstrates that high-resolution spectroscopic modeling of early M dwarfs is feasible in the J band, raising the possibility that the H band might also be amenable to such techniques.

We have also applied our calibration to several planet hosts, listed in Table 2. Our measurement of GJ 317 represents the first empirically calibrated spectroscopic metallicity estimate of this object. Our estimates are [Fe/H]_K = 0.26 and [Fe/H]_H = 0.31 dex, supporting the findings of Anglada-Escudé et al. (2012) and O11 that this star is metal-rich.

We compare our results to other spectroscopic studies in Table 2. Generally, our results are consistent within errors with those of both RA10 and O11. For the planets we have in common with RA10, our K-band estimate differences have mean = −0.11 and σ = 0.08 dex, and the H band yields differences with mean = −0.14 and σ = 0.12 dex. For planets we have in common with O11, our K-band estimate differences have mean = 0.04 and σ = 0.09 dex, and the H-band differences have mean = −0.01 and σ = 0.10 dex. The most discrepant points, the H-band estimates for GJ 849 and GJ 876, have high S/N which allowed us to probe frame-by-frame variability. The individual frames yielded highly variable (0.1–0.2 Å) EW measurements of the H-band K i feature in both objects.

Since we have obtained metallicity estimates for five M dwarfs with giant planets (of seven known) and four M dwarf planet hosts (of 12 known) without known giant planets, we can statistically asses whether M dwarfs with giant planets are metal-rich. We elect to use the nonparametric Mann–Whitney–Wilcoxon rank-sum test (Mann & Whitney 1947) on the averages of our K- and H-band metallicity estimates. This test determines whether one set of observations has preferentially higher values than another. For the samples of non-giant planet hosts (n = 4, median = −0.03) and giant planet hosts (n = 5, median = 0.27), we find that the distributions differed significantly (P < 0.01). The P-value is the probability of obtaining a rank-sum at least as extreme as the one obtained under the null hypothesis that neither group has preferentially higher metallicities, and our alternative hypothesis was that the giant planet hosts were more metal-rich. Thus, for the set of M dwarf planet hosts for which we have consistent metallicity estimates, we find that giant planet hosts are indeed preferentially metal-rich compared to M dwarfs with less massive planets.