PROSPECTING IN LATE-TYPE DWARFS: A CALIBRATION OF INFRARED AND VISIBLE SPECTROSCOPIC METALLICITIES OF LATE K AND M DWARFS SPANNING 1.5 dex

Despite their intrinsic faintness, M dwarfs have become attractive targets for exoplanet searches (e.g., Charbonneau et al. 2009; Apps et al. 2010; Mann et al. 2011), as M dwarfs have smaller radii and lower masses than their solar-type counterparts, allowing for easier detection of low-mass exoplanets (Gaidos et al. 2007). For transiting M dwarf planets with high-precision observations, such as those done by Kepler (Borucki et al. 2010; Batalha et al. 2012), errors in planet parameters (primarily planet radius) are directly tied to errors in stellar parameters (e.g., Muirhead et al. 2012), which in turn depend on reliable measurements of stellar temperature and metallicity (Dotter et al. 2008; Demory et al. 2009; Allard et al. 2011). Accurate metallicities are also necessary to study any correlation between metallicity and planet frequency (e.g., Johnson et al. 2010; Mann et al. 2012). Further, because the stellar mass function peaks around mid-M, M dwarfs weigh heavily in any study of Galactic structure. The distribution of M dwarf metallicities can therefore be used to set limits on Milky Way formation models (Woolf & West 2012). However, unlike those of their solar-type counterparts, M-dwarf metallicities are difficult to determine, primarily due to the presence of complex molecular lines in their visible spectra, which result in line confusion and a lack of identifiable continuum, and do not always match with current M dwarf models (Mould 1976; Allard et al. 2011). The visible wavelengths usually used to derive metallicities for solar-type stars are dominated by TiO lines that increase in strength with decreasing effective temperature. These TiO lines obscure the continuum, making equivalent width measurements unreliable, and differ as a function of spectral type. Direct spectral synthesis has been tried on a small sample of stars (Bean et al. 2006; Önehag et al. 2012). However, such spectra is observationally expensive, and spectral synthesis is complicated by incomplete lines lists.

One common technique is to use wide binaries with a solar-type (late-F, G, or early-K) primary and a late-type (late K or M) dwarf companion. Since these stars presumably formed from the same molecular cloud, the metallicity of the M dwarf companion can be assumed to be the same as that of the FGK primary (Bonfils et al. 2005). From this one can create empirical calibrations of observable features to determine metallicities in M dwarfs. Techniques based on absolute photometry (Bonfils et al. 2005; Johnson & Apps 2009; Schlaufman & Laughlin 2010) require parallaxes, which are only available for a handful of the closest and brightest M dwarfs. Molecular indices at visible wavelengths (e.g., Woolf & Wallerstein 2006; Lépine et al. 2012) have proven useful for separating M dwarfs into luminosity/metallicity classes (i.e., dwarf, subdwarf, and extreme subdwarf), but saturate near solar metallicity and are less reliable for late-K and early-M dwarfs.

The use of spectral indices in the infrared has, however, been showing considerable promise. Rojas-Ayala et al. (2010, henceforth R10) showed that [Fe/H] can be inferred for M dwarfs using the Na i doublet and Ca i triplet in moderate resolution (R ≃ 2700) K-band spectra and assuming solar relative abundances. Rojas-Ayala et al. (2012, henceforth R12) expanded on this, showing that one can determine [M/H] for early to mid-M dwarfs as accurately as 0.1 dex using the same indices. Further, Terrien et al. (2012, henceforth T12) were able to demonstrate similar precision in the H band. However, calibrations for both H- and K-band metallicities were derived using a relatively small sample (18 and 22, respectively) of wide binaries. As a result, they were only verified for systems with near-solar metallicities (−0.4 ≲ [Fe/H] ≲ +0.3), and for a narrow range of spectral types (M0–M4).

In this paper we analyze visible and near-infrared (NIR) observations of 112 late K and M dwarfs with F, G, or early K star primaries. We provide determine metallicity estimates for 50 of the FGK stars. We also provide a full list of the 120 features we identify to be metal-sensitive features in visible and NIR spectra, with the expectation that these will be useful for future studies on M dwarf metallicities. In Section 2 we present our sample of wide-binary systems, followed by a description of our observations and reduction in Section 3. In Section 4 we explain how we determine basic properties for both the primary and companion stars. We discuss our search for metal-sensitive features in Section 5 and present the results from this analysis in Section 6, which includes our calibrations to determine metallicities at a range of wavelengths. We test existing techniques in Section 7. Last, we summarize the results in Section 8 and discuss possible drawbacks of our analysis as well as prospects for future studies. All wavelengths used in this work are stated as vacuum values.

We draw a sample of late K and M dwarfs with comoving FGK stars from a variety of literature sources, specifically Chanamé & Gould (2004), Gould & Chanamé (2004), Lépine & Bongiorno (2007), Schlaufman & Laughlin (2010), and Tokovinin & Lépine (2012). We also identify a number of new common-proper motion (CPM) pairs that contain an FGK star and a late K or M star using proper motions from (in order of preference) Hipparcos (van Leeuwen & Fantino 2005; van Leeuwen 2007), SUPERBLINK (Lépine & Shara 2005), or the PPMXL survey (Roeser et al. 2010). We identify new CPM pairs, as well as vet the literature sample, following the techniques of Lépine & Bongiorno (2007) and Dhital et al. (2010). Lépine & Bongiorno (2007) identify CPM pairs using the formula:

$$\begin{equation} \Delta X = [(\mu /0.15)^{-3.8}\Delta \theta \Delta \mu ]^{0.5}, \end{equation} \tag{ 1 }$$

where μ is the absolute proper motion of the primary star in arcsec yr⁻¹, Δθ is the separation between the two stars in arcseconds, and Δμ is the absolute difference in proper motion between the two stars in arcsec yr⁻¹. Lépine & Bongiorno (2007) consider two stars to be physically associated with each other if ΔX < 1, Δθ < 1500'', and Δμ < 100 mas yr⁻¹. We take a more conservative cut and require that our pairs have ΔX < 0.9, Δθ < 650'', and Δμ < 60 mas yr⁻¹ to cut down on the number of chance alignments. This technique takes advantage of the absolute proper motion, and not just the difference in proper motion, of the star, however, it does not factor in associated errors in proper motion. Thus we further force the constraint from Dhital et al. (2010):

$$\begin{equation} \left(\frac{\Delta \mu _\alpha }{\sigma _{\Delta \mu _{\alpha }}}\right)^2 + \left(\frac{\Delta \mu _\delta }{\sigma _{\Delta \mu _{\delta }}}\right)^2 \le 2, \end{equation} \tag{ 2 }$$

where Δμ_α and Δμ_δ are the differences in proper motion between the two components (right ascension and declination, respectively) and σ_Δμ is the error in the proper motion differences. Equation (2) is designed to ensure that the resulting sample has minimal contamination from chance alignment pairs, but at the cost of excluding a large number of true wide binaries. To increase our sample, we add pairs that do not satisfy Equation (2) but have published parallaxes for both the primary and companion star consistent to 1σ or the parallax uncertainty, which strongly suggests that the pair forms a physical system

We remove pairs with δ < −45° or δ > +68°, since these are outside the reach of telescopes in Hawaii, and pairs with Δθ < 3'' as these will be difficult to observe and may have inaccurate photometry. We further remove pairs with V_c − J_c < 2.5 (the subscript c denoting the companion and p the primary). This cut will remove almost all stars earlier than K5 (Lépine & Gaidos 2011) where metallicities can be measured using modified spectral synthesis techniques (Valenti & Piskunov 1996; Valenti & Fischer 2005). We cut out systems with V_c > 18, and K_c > 12, as the observation time required for these stars is highly prohibitive. For the same reason, we remove systems with V_p > 12 unless the primary star already has a published metallicity. The resulting sample contains 262 pairs with primaries with colors in the range 0.8 < V_p − J_p < 2.5 and/or published temperatures consistent with a late-F, G, or early-K dwarf (Fitzgerald 1970; Ducati et al. 2001). From here we prioritize our observations based on (1) V_c and K_c magnitudes (to minimize required telescope time), (2) V_c − J_c color (to ensure a range of spectral types), (3) availability of metallicities for the primary star in the literature or V_p magnitude if no literature metallicity is available (see Section 3.1), (4) availability of parallax information for the primary or secondary, which makes it more likely that these are true CPM pairs, and (5) metallicity of the primary (if available) to guarantee a range of metallicities for our analysis.

The resulting sample includes 112 stars for which we have obtained infrared spectra (Section 3.2), visible wavelength spectra (Section 3.3), and metallicities for the primary star from the literature or from our own analysis (Section 4.1). Our sample has −1.04 < [Fe/H] _p < 0.56, and companion star spectral types from K5.5 to M6. We list our sample in Table 1.

3.1. ESPaDOnS/CFHT

3.2. SpeX/IRTF

3.3. SNIFS/UH2.2m

3.4. Construction of a Combined M dwarf Spectrum

4.1. FGK Metallicities

We draw primary star metallicities from a variety of sources in the literature. We list the adopted metallicity and literature source for each binary in Table 1. In total, 33 primary star metallicities come from the SPOCS catalog (Valenti & Fischer 2005), 50 from observations taken as part of this project with CFHT/ESPaDOnS, and 29 from other literature sources. SPOCS consists of high-resolution echelle spectra of >1000 F-, G-, and K-type stars obtained with the Keck, Lick, or Anglo-Australian Telescope. Valenti & Fischer (2005) fit the observed spectrum to a synthetic spectrum using the software package SME (Spectroscopy Made Easy; Valenti & Piskunov 1996), which provides a set of observational parameters (T_eff, [Fe/H], [M/H], log g, etc.) for each star. We adopt an uncertainty of 0.03 dex for their derived [M/H] and [Fe/H] values.

To determine the stellar parameters of primaries observed with CFHT/ESPaDOnS we model each spectrum using the SME software (Valenti & Piskunov 1996), fitting the spectrum to a set of tuned lines from the SPOCS catalog (Valenti & Fischer 2005). We simultaneously solve for surface gravity, effective temperature, projected rotational velocity, and metallicity in addition to individual abundances of Fe, Na, Si, Ni, and Ti. Solar values are assumed for all of the initial models and after obtaining an initial fit, we then perturb T_eff by ±100 K and fit again. Our final model parameters are χ²-weighted averages of three runs. Corrections based on Vesta and stellar binary observations as detailed in Valenti & Fischer (2005) are then applied.

For stars with good parallax measurements (Hipparcos stars or their companions), we use Yonsei–Yale (Y²) isochrones (Demarque et al. 2004) to better constrain the surface gravity (Valenti et al. 2009). After we determine the stellar parameters as above, we use distance and B and V magnitudes to the derive bolometric luminosity and, combined with the SME T_eff, the stellar radius. Bolometric corrections are obtained by interpolating in the high temperature grid of VandenBerg & Clem (2003), and B, V magnitudes were drawn from Hipparcos (van Leeuwen & Fantino 2005). The SME determined ratio of Si to Fe is used as a proxy for alpha element enhancement. A best-fit evolutionary model is found by interpolating in the Y² grid which yielded a surface gravity for the star. This log g is compared to the value determined using SME and if the two did not match, a new set of SME models is found with the gravity fixed to the isochrone value. The process is repeated until the log g values agree to within 0.001 dex. Final stellar parameters (T_eff, log g, [M/H], etc.) for stars observed as part of our program are listed in Table 2.

Table 2. Parameters of Primary Stars observed at CFHT

Name	Teff	±	log g	±	[Fe/H]	±	[M/H]	±	[Na/H]	±	[Ti/H]	±	[Si/H]	±	[Ni/H]	±	χ2red	Run Typea
HIP 1224	5141	44	4.53	0.06	+0.07	0.03	+0.03	0.03	+0.12	0.03	+0.03	0.05	+0.02	0.02	+0.08	0.03	2.9	ITER
HIP 5110	4648	44	4.61	0.06	−0.09	0.03	−0.13	0.03	−0.25	0.03	−0.18	0.05	−0.15	0.02	−0.11	0.03	6.0	ITER
HIP 5286	4676	53	4.60	0.06	+0.25	0.03	+0.21	0.05	+0.30	0.09	+0.14	0.05	+0.09	0.06	+0.26	0.03	14.7	ITER
HIP 6431	4858	44	4.57	0.06	+0.10	0.03	+0.04	0.03	+0.10	0.03	+0.00	0.05	+0.06	0.02	+0.11	0.03	6.3	ITER
HIP 6456	5222	52	4.44	0.06	+0.45	0.03	+0.36	0.06	+0.62	0.05	+0.27	0.08	+0.43	0.02	+0.53	0.03	6.8	ITER
HIP 11572	5093	44	4.51	0.06	−0.08	0.03	+0.00	0.03	+0.05	0.03	+0.03	0.05	+0.05	0.02	−0.02	0.03	3.6	ITER
HIP 15126	5285	44	4.64	0.06	−0.92	0.03	−0.66	0.03	−0.84	0.03	−0.56	0.05	−0.51	0.02	−0.86	0.03	2.2	ITER
HIP 16467	5539	44	4.33	0.06	−0.01	0.03	−0.01	0.03	−0.09	0.03	−0.04	0.05	+0.02	0.02	−0.04	0.03	2.1	ITER
HIP 16563	5788	44	4.52	0.06	+0.20	0.03	+0.16	0.06	+0.10	0.03	+0.21	0.05	+0.04	0.02	+0.05	0.04	4.4	ITER
HIP 31127	5158	44	3.83	0.06	−0.54	0.03	−0.45	0.03	−0.56	0.03	−0.50	0.05	−0.34	0.02	−0.56	0.03	3.1	ITER
HIP 31597	5428	44	4.44	0.06	+0.09	0.03	+0.08	0.03	+0.14	0.03	+0.08	0.05	+0.10	0.02	+0.12	0.03	2.0	ITER
HIP 32423	4817	44	4.64	0.06	−0.26	0.03	−0.21	0.04	−0.32	0.04	−0.27	0.07	−0.24	0.03	−0.27	0.03	5.6	ITER
HIP 35449	6156	44	4.36	0.06	+0.21	0.03	+0.18	0.03	+0.17	0.03	+0.21	0.05	+0.19	0.02	+0.18	0.03	1.9	ITER
HIP 40298	5618	44	4.51	0.06	−0.07	0.03	−0.09	0.03	−0.09	0.03	−0.04	0.05	−0.08	0.02	−0.14	0.03	1.7	ITER
NLTT 12373	5838	93	4.62	0.09	−0.08	0.05	−0.00	0.08	−0.08	0.05	+0.05	0.08	−0.04	0.03	−0.12	0.06	2.2	VESTA
HIP 45863	5172	44	4.51	0.06	−0.12	0.03	−0.13	0.03	−0.07	0.03	−0.10	0.05	−0.09	0.02	−0.12	0.03	4.3	ITER
NLTT 23002	5259	44	4.58	0.06	−0.15	0.03	−0.11	0.03	−0.12	0.03	−0.11	0.05	−0.14	0.02	−0.16	0.03	3.1	VESTA
HIP 50802	4472	44	4.70	0.06	−0.01	0.03	−0.06	0.03	−0.25	0.04	−0.11	0.05	−0.21	0.06	−0.14	0.03	13.1	ITER
HIP 54155	5547	50	4.54	0.06	+0.16	0.03	+0.11	0.07	+0.03	0.03	+0.14	0.06	+0.05	0.02	+0.05	0.03	2.9	ITER
HIP 55486	5371	44	4.51	0.06	+0.46	0.03	+0.42	0.03	+0.58	0.07	+0.37	0.05	+0.39	0.02	+0.48	0.03	9.6	ITER
HIP 56729	5421	44	4.47	0.06	−0.09	0.03	−0.04	0.03	−0.10	0.03	−0.02	0.05	−0.04	0.02	−0.11	0.03	2.7	ITER
HIP 56930	5025	44	4.61	0.06	−0.12	0.03	−0.17	0.05	−0.20	0.03	−0.15	0.05	−0.14	0.02	−0.18	0.03	7.0	ITER
HIP 59080	5423	44	4.41	0.06	−0.16	0.03	−0.13	0.03	−0.12	0.03	−0.16	0.05	−0.11	0.02	−0.16	0.03	4.9	ITER
HIP 61081	5298	44	4.59	0.06	−0.54	0.03	−0.42	0.03	−0.49	0.03	−0.33	0.05	−0.26	0.02	−0.48	0.03	2.5	ITER
HIP 61189	4655	44	4.93	0.06	+0.11	0.03	+0.05	0.03	−0.01	0.03	+0.02	0.05	+0.08	0.02	+0.09	0.03	16.6	VESTA
HIP 61589	5700	44	4.49	0.06	−0.05	0.03	−0.08	0.05	−0.08	0.03	−0.05	0.05	−0.05	0.02	−0.06	0.03	2.3	ITER
HIP 64345	5495	44	4.40	0.06	−0.57	0.03	−0.39	0.03	−0.49	0.03	−0.33	0.05	−0.25	0.02	−0.52	0.03	1.8	ITER
HIP 64797	5041	44	4.60	0.06	−0.12	0.03	−0.13	0.03	−0.16	0.03	−0.11	0.05	−0.16	0.02	−0.17	0.03	15.1	ITER
HIP 65636	4619	44	4.64	0.06	+0.13	0.03	+0.01	0.03	−0.11	0.03	−0.08	0.05	+0.00	0.02	+0.02	0.03	13.4	ITER
HIP 65963	5478	44	4.52	0.06	−0.14	0.03	−0.08	0.03	−0.19	0.03	−0.08	0.05	−0.09	0.02	−0.14	0.03	1.7	ITER
HIP 68799	5492	44	4.42	0.06	−0.03	0.03	−0.06	0.03	+0.00	0.03	−0.05	0.05	−0.02	0.02	−0.04	0.03	2.7	ITER
HIP 70100	4886	44	4.57	0.06	+0.15	0.03	+0.13	0.03	+0.26	0.03	+0.09	0.05	+0.12	0.02	+0.16	0.03	9.9	ITER
HIP 70426	4801	44	4.58	0.06	+0.09	0.03	+0.03	0.04	+0.13	0.04	−0.00	0.05	+0.06	0.02	+0.10	0.03	12.8	ITER
HIP 74396	5124	44	4.60	0.06	−0.09	0.03	−0.07	0.03	−0.08	0.03	−0.06	0.05	−0.12	0.02	−0.13	0.03	3.5	ITER
HIP 74734	5822	44	4.34	0.06	−0.32	0.03	−0.27	0.03	−0.37	0.03	−0.20	0.05	−0.23	0.02	−0.39	0.03	2.1	ITER
HIP 75069	5196	44	4.61	0.06	−0.38	0.03	−0.34	0.03	−0.36	0.03	−0.33	0.05	−0.33	0.02	−0.40	0.03	2.8	ITER
HIP 76668	4636	44	4.65	0.06	−0.06	0.03	−0.14	0.03	−0.22	0.06	−0.17	0.05	−0.10	0.02	−0.12	0.03	13.7	ITER
HIP 78969	4899	44	4.58	0.06	+0.19	0.03	+0.12	0.03	+0.23	0.06	+0.08	0.05	+0.13	0.02	+0.20	0.03	9.0	ITER
HIP 79629	5600	50	4.48	0.06	−0.25	0.04	−0.25	0.06	−0.28	0.03	−0.25	0.05	−0.20	0.02	−0.31	0.03	2.0	ITER
HIP 84616	4826	44	4.62	0.06	−0.12	0.03	−0.09	0.03	−0.09	0.03	−0.09	0.05	−0.10	0.02	−0.14	0.03	6.4	ITER
PM J1742+1645	5492	44	4.50	0.06	−0.09	0.03	+0.01	0.03	−0.06	0.03	−0.00	0.05	−0.11	0.02	−0.15	0.03	4.2	VESTA
HIP 87082	5795	44	4.40	0.06	−0.05	0.03	−0.03	0.05	−0.17	0.03	+0.02	0.05	−0.02	0.02	−0.09	0.03	2.7	ITER
HIP 88188	5299	53	4.54	0.06	+0.05	0.04	−0.04	0.07	−0.06	0.03	−0.05	0.13	+0.02	0.03	+0.04	0.03	3.2	ITER
HIP 88365	5371	44	4.59	0.06	−0.66	0.03	−0.34	0.03	−0.52	0.03	−0.33	0.05	−0.33	0.02	−0.57	0.03	3.7	ITER
HIP 90246	4494	44	4.69	0.06	−0.01	0.03	−0.08	0.03	−0.26	0.05	−0.08	0.05	−0.08	0.05	−0.09	0.03	11.3	ITER
HIP 104097	4428	44	4.72	0.06	−0.38	0.03	−0.27	0.03	−0.51	0.03	−0.15	0.05	−0.37	0.04	−0.40	0.03	15.1	ITER
PM J2206+4322E	5652	44	4.62	0.06	+0.30	0.03	+0.15	0.03	+0.37	0.03	+0.22	0.06	+0.29	0.02	+0.33	0.03	6.2	VESTA
HIP 114156	4314	66	4.72	0.06	−0.04	0.08	−0.06	0.05	−0.37	0.15	−0.08	0.05	−0.24	0.07	−0.15	0.03	15.5	ITER
HIP 117960	5252	44	4.54	0.06	+0.12	0.03	+0.06	0.03	+0.14	0.03	+0.04	0.05	+0.12	0.02	+0.11	0.03	4.0	ITER
HIP 118282	5178	44	4.64	0.06	−0.67	0.03	−0.47	0.03	−0.54	0.03	−0.36	0.05	−0.35	0.03	−0.57	0.03	2.0	ITER

Notes. ^aITER: parameters determined using Hipparcos parallaxes and Y² isochrones. VESTA: parameters determined using classical SME fitting (no parallax information included) with a correction using Vesta as described in Valenti & Fischer (2005).

Download table as:  ASCIITypeset image

Our analysis of the ESPaDOnS spectra is designed to keep our metallicities consistent with those from the SPOCS catalog (both are based on SME analysis and use the same set of spectral lines). As an extra check on consistency we have obtained CFHT spectra for three stars in the SPOCS sample. The derived stellar parameters from these three spectra are consistent (within errors) with those listed in the SPOCS catalog, confirming that there is no systematic offset between metallicities from SPOCS and CFHT.

Metallicities from other literature sources are not necessarily determined in the same way as our spectral analysis, and thus may have small systematic inconsistencies. We correct for this by checking for overlap between the SPOCS samples and any given literature source. For us to use a metallicity derived from any other literature source we require (1) at least 30 stars in both samples that can be used as a control sample to check for differences, (2) metallicities for our primary stars from the literature source fall within the range of metallicities of the control sample, (3) the mean difference between the SPOCS metallicities and the literature metallicities in the control sample Δ[Fe/H]_control ⩽0.07 dex, and (4) the resulting scatter in the control sample σ_control ⩽0.08 dex. These limits are designed to keep uncertainties in the primary star metallicities well below the precision already obtained for determining M dwarf metallicities (e.g., R12 and T12). We adopt σ_control as the uncertainty in [Fe/H] for a given literature source. As an example, we show metallicities from both Ramírez et al. (2007) and SPOCS in Figure 1. We list all sources of metallicities, the adopted systematic offset (which we apply for all calculations in this paper) for that source, and the adopted uncertainty in Table 3.

Zoom In Zoom Out Reset image size

Table 3. Corrections Applied to Primary Star Metallicities

Source	Control Stars	Δ[Fe/H]	σcontrol	No. of Stars Useda
VF05	...	0.0	0.03	33
CFHT	...	0.0	0.03b	50
C01	294	0.02	0.07	1
M04	178	0.04	0.08	1
LH05	174	0.02	0.06	1
T05	127	0.01	0.05	1
B06	33	0.07	0.07	1
Ra07	112	0.08	0.05	4
Ro07	127	0.00	0.07	5
F08	165	0.03	0.06	2
C11, c	614	0.00	0.08	10
S11	50	0.03	0.04	1
N12, d	125	0.00	0.05	2

Notes. See Table 1 for the abbreviations on references. ^aNumber of stars from listed source used in our final wide binary sample. ^bTypical uncertainty. Errors for individual stars listed in Table 2. ^cMetallicities determined from Strömgren photometry. ^dBased on a control sample from Santos et al. (2004) and Sousa et al. (2011), on which abundances in N12 are anchored.

Download table as:  ASCIITypeset image

4.2. Late-K/M Dwarf Spectral Types

We determine M dwarf spectral types using indices at both visible and NIR wavelengths. We use the empirical spectral type–band index relations from Lépine et al. (2012), which have been calibrated to work on the SNIFS/UH2.2m. Lépine et al. (2012) determined spectral types accurate to ≃ 0.2 subtypes based on empirical relations between spectral type and the strengths of TiO and CaH bands (Reid et al. 1995). It has been shown that CaH is sensitive to spectral type, and that TiO is sensitive to both spectral type and metallicity (Woolf & Wallerstein 2006). As a result, for stars with [Fe/H] <−0.5 (the metallicity of the primary star) we base our visible wavelength spectral types solely on relations using CaH bands.

R12 showed that one can determine temperatures and spectral types using a modified version of the H₂O-K (H₂O-K2, Covey et al. 2010) index. R12 calibrated their spectral types based on K-band spectra of stars from the Research Consortium on Nearby Stars Measuring (RECONS; Henry et al. 1994). Their calibration is accurate to ≃ 0.6 subtypes.

Figure 2 compares the spectral types derived from visible wavelength indices versus those derived using the H₂O-K2 index. Although there is good agreement between the two techniques for the later-type stars in our sample, for stars earlier than M1 (as determined by TiO and CaH bands), spectral types determined from H₂O-K2 are systematically later than those from visible wavelength indices. The H₂O features become quite weak in the spectra of late-K and early-M stars and R12 caution using it on stars with T_eff > 4000. Further, the spectral-type calibration from R12 does not include any K stars, and is therefore unreliable for the warmest stars in our sample. As a result, we choose to use spectral types determined from our visible wavelength spectra for the entire sample.

Zoom In Zoom Out Reset image size

To determine which features in the companion dwarf spectra best correlate with metallicity we perform a systematic analysis of our sample of spectra and metallicities. Our analysis proceeds as follows:

• 1.  
  A center wavelength is selected, starting at the blue end of the spectrum (≃ 0.33 μm) and incrementally increasing by 0.00015 μm (1.5 Å) after all other steps are complete. This process is repeated until the center is at the red end of the spectrum (≃ 2.4 μm) and excludes the gap in all SpeX spectra at 1.85 μm.
• 2.  
  For each feature center, we select a feature width starting at 0.002 μm (20 Å), and then increased incrementally by 0.00015 μm (1.5 Å) after completing all following steps. We use 20 Å as a minimum, as features smaller than this have considerable Poisson noise (making their measurement difficult). We use an upper limit of 0.01 μm (100 Å) for the feature width, as regions of the spectra larger than this likely contain multiple features that should be treated separately.
• 3.  
  The equivalent width is calculated for each feature using the approximation:

  $$\begin{equation} {\rm EW}_\lambda \simeq \sum _{i=0}^{n-1}\left[1 - \frac{F(\lambda _i)}{F_c(\lambda _i)} \right]\Delta \lambda _i, \end{equation} \tag{ 3 }$$

  where λ_i is the wavelength at pixel i, F is the flux at λ_i, F_c is the pseudo-continuum at λ_i, and the sum is computed over all n pixels within a given feature. We compensate for the low resolution by interpolating the spectrum near the edge to a much higher resolution (R > 10, 000). We experiment with different techniques to calculate the pseudo-continuum (see below). The list of continuum regions used is listed in Table 4, many of which are taken from T12.
• 4.  
  A temperature-sensitive parameter τ, is calculated from each spectrum. τ is defined based on the center wavelength of the randomly selected feature from step (2). Specifically, τ is set to be the K-band H₂O (H₂O-K2) index as defined by R12 if the feature is centered in the K band, the H-band H₂O (H₂O-H) index as defined by T12 if the feature is in the H band, or the Color1 index from Hawley et al. (2002) if the feature is drawn from visible wavelengths. If the feature falls within the J band, we use a new H₂O-J index defined as:

  $$\begin{equation} \mathrm{H}_2\mathrm{O} \mathit {-} \mathrm{J} = \frac{\langle \mathcal {F}(1.210\hbox{--}1.230) \rangle /\langle \mathcal {F}(1.313\hbox{--}1.333) \rangle }{\langle \mathcal {F}(1.313\hbox{--}1.313) \rangle /\langle \mathcal {F}(1.311\hbox{--}1.351) \rangle }, \end{equation} \tag{ 4 }$$

  where $\langle \mathcal {F}(a-b) \rangle$ indicates the median flux level in a wavelength range between a and b (in μm). H₂O-J is defined to select regions relatively clear of atomic or molecular features and to correlate well with the H₂O-K and H₂O-H indices.
• 5.  
  Using least-squares, the best fit is found for the equation:

  $$\begin{equation} \mathrm{[Fe/H]}_{i} = A + B\times {\rm EW}_{i} + C\times \tau _{i}, \end{equation} \tag{ 5 }$$

  where [Fe/H]_i is the metallicity of ith primary star (assumed to be the metallicity of the companion M dwarf), EW_i is the calculated equivalent width of the selected feature in the ith late-K or M dwarf companion spectrum, and A, B, and C are fitting parameters. The quality of the fit is measured by the adjusted square of the multiple correlation coefficient (R²_ap), which is defined as:

  $$\begin{eqnarray} &&R_{{\rm ap}}^2 = 1 - \frac{(n - 1)\sum (y_{i,{\rm model}}- {y_i})^2}{(n-p)\sum (y_i - \bar{y})^2},\\ &&\nonumber \end{eqnarray} \tag{ 6 }$$

  where p is the number of changeable parameters (i.e., A, B, and C), n is the number of data points in the fit, y_i is the metallicity of the ith primary star, y_{i, model} is the metallicity of the ith star predicted by the fit, and $\bar{y}$ is the average of y_i. A R²_ap closer to 1 implies that the model accurately explains the variance of the sample, while R²_ap = 0 implies that it can explain none. For Equation (5), p = 3, y_i = [Fe/H]_i, and y_{i, model} = A + B × EW_i + C × τ_i. Note that for n ≫ p, R²_ap ≃ R².
• 6.  
  To assess the significance of the assigned R²_ap value, [Fe/H] (or [M/H]) values are randomly reassigned among the stars, and step 5 is repeated 1000 times (re-randomizing the metallicities each time). The resulting distribution of the 1000 R²_ap values gives the level above which the R²_ap value (determined from non-random metallicities) can be considered significant. We consider the given feature center to be a bona-fide metal sensitive feature if R²_ap is higher than the 99.9% highest R²_ap value from the randomly assigned metallicities (henceforth R²_rand).

For each increment in feature center and width, we record the resulting R²_ap and R²_rand. We show the resulting distribution of R² and R²_rand values as a function of the feature's central wavelength in Figure 3.

Zoom In Zoom Out Reset image size

We repeat our analysis using various methods of fitting for the pseudo-continuum. Better estimates of the continuum should result in more accurate line measurements, and therefore higher R²_ap values for the same features. In one experiment we tried to fit the global spectrum with a high order (>10) polynomial. We tested fitting each band (visible, JHK) with third- through sixth-order polynomials, as well as with third- through sixth-order Legendre polynomials. Interestingly, we found we had the best (highest R²_ap with respect to R²_rand) results when fitting the pseudo-continuum using a linear fit of the continuum regions immediately blueward and redward of the selected feature. We use this fitting procedure for all calibrations derived in Section 6.

We run additional experiments to test the influence of spectral type on determination of metallicities: we repeat our analysis on just the early-type stars in our sample (K5.5–M2.0) and again with just the late-type stars (M2.0–M6). The split roughly corresponds to our median spectral type (≃M2). It is possible to parse our data into smaller spectral-type ranges, although this will proportionately shrink each sample. Thus this would make it difficult to identify metal sensitive features that have not been previously discovered.

Last, we rerun our analysis using [M/H] rather than [Fe/H] for the metallicity of the primary star. R12 were able to derive better fits between K-band atomic lines and [M/H] than for [Fe/H], perhaps due to variations in [α/Fe] creating discrepancies between the actual measured quantities (Na and Ca in the case of R12) and Fe abundances. However, many of the literature sources we draw from do not provide [M/H] values for the primary stars (just [Fe/H]). As a result, there are 7 fewer binaries when using [M/H] and 15 primary stars have a different source for their [M/H] values than their [Fe/H] values (see Table 1 for a full list of [Fe/H] and [M/H] sources used).

We consider a feature to be metal-sensitive if a given feature's center and width has an R²_ap value are above its corresponding R²_rand value. This criterion ensures that no features are identified simply by coincidence.

Once features are identified, we then attempt to derive a calibration for a given wavelength range (e.g., J band) by solving the equation (by least squares):

$$\begin{equation} \mathrm{[Fe/H]}_i = A + \sum _{j=1}^{N}(B_j \times {\rm EW}_{i,j}) + C \times \tau _i, \end{equation} \tag{ 7 }$$

where [Fe/H]_i is the metallicity of the ith primary star, N is the total number of features of interest among M_λ features identified as metal-sensitive in a given wavelength range, τ_i is the temperature sensitive parameter selected based on the wavelength regime (see above), EW_{i, j} is the equivalent width of the jth feature measured for the ith star, and A, C, and the B_j's are fitting parameters. We find the best fit for N = 1,2,3,...<M_λ until the increase in R²_ap is negligible (ΔR²_ap < 0.03) or it is clear from visual inspection of the data that the adding of further variables is over fitting the data. This limit is usually hit at N = 3–4. Although we use [Fe/H] in Equations (5) and (7), we also perform the same procedure using [M/H].

The first thing our analysis gives us is a catalog of metal-sensitive features, which we list in order of λ_c in Table 5. In total we find 120 features that are statistically significant predictors of metallicity, although only 20 of these are used in our final calibrations. We identify a number of previously known metal-sensitive features, as well as many of new ones. One of the most metal-sensitive features is the Na i doublet in the K band (2.208 μm), already identified by R10. Our analysis identifies the Ca i (1.616 μm and 1.621 μm) and K i (1.5176 μm) lines shown to be metal-sensitive by T12. In fact, our analysis locks on to very similar wavelength centers and widths as those found by T12 for both H-band and K-band features. Since our analysis covers all wavelengths and is completely blind (e.g., they have no a-priori line lists or knowledge of feature size) this suggests that our purely empirical analysis is identifying metal-sensitive atomic and molecular lines.

Because our only restriction was the size of the feature (⩽100 Å), our technique can easily identify areas of the spectrum corresponding to several, even unrelated lines. Some of these regions may be associated with doublets/triplets from the same atomic species, with broad molecular bands, or with sets of lines that are blended at our resolution. Some features in Table 5 may not correspond to any one specific element or molecule, but simply to a region of the spectrum that undergoes overall changes as a function of the metallicity of the star.

We show the distribution of R²_ap values as a function of feature width and center for two example wavelength regions in Figure 4. Interestingly, features in the H- and K-band features yield similar R²_ap for a wide range of feature widths. However, the opposite is seen in the visible end of the spectrum, where features perform better near the minimum feature width (20 Å). This is most likely due to crowding at visible wavelengths.

Zoom In Zoom Out Reset image size

We find far better metal-sensitive features by doing a separate analysis for earlier-type dwarfs (K5.5–M2) and for later-type dwarfs (M2–M6). Specifically, the best fit we achieve when fitting for [Fe/H] using all stars in our sample yields R²_ap = 0.54, whereas when we split up the sample by spectral type we achieve R²_ap = 0.84 for K5.5–M2.0 and R²_ap = 0.68 for M2.0–M6.0. This is not unexpected, many of the most metal-sensitive features for K5.5–M2 dwarfs become blended with molecular bands (which grow as a function of spectral type) at the resolution of SNIFS. Further, features blueward of 0.5 μm tend to have very low S/N for stars later than M2, and are not to be very useful. This also suggests that better results could be achieved on later-type M dwarfs with modest improvements in resolution, to better distinguish the lines.

We find the best empirical fits to Equation (7) for each wavelength regime (visible, J, H, and K) and metallically metric ([Fe/H] and [M/H]). They are:

$$\begin{eqnarray} \mathrm{[Fe/H]_{V,e}} &=& 0.53F_{07} + 0.26F_{01} - 0.16F_{02} \nonumber\\ &&- 0.784\mathrm{(Color1)} - 0.34 \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} \mathrm{[M/H]_{V,e}} &=& 0.38F_{07} + 0.21F_{01} + 0.29F_{08} \nonumber\\ &&- 0.504\mathrm{(Color1)} - 0.79 \end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray} \mathrm{[Fe/H]_{V,l}} &=& {-}0.20F_{05} + 0.48F_{08} + 0.24F_{07} \nonumber\\ && + 0.14F_{03} - 0.204\mathrm{(Color1)} - 0.32 \end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray} \mathrm{[M/H]_{V,l}} &=& {-}0.065F_{05} - 0.071F_{04} - 0.30F_{06} \nonumber\\ && + 0.719\mathrm{(Color1)} - 0.24 \end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray} \mathrm{[Fe/H]_J} &=& 0.29F_{10} + 0.21F_{09} + 0.26F_{12} \nonumber\\ && - 0.26F_{13} - 0.190\mathrm{(\mathrm{H}_2\mathrm{O} \mathit {-} \mathrm{J})} - 1.03 \end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray} \mathrm{[M/H]_J} &=& 0.32F_{10} + 0.46F_{11} + 0.076F_{09} \nonumber\\ && + 1.213\mathrm{(\mathrm{H}_2\mathrm{O} \mathit {-} \mathrm{J})} - 1.97 \end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray} \mathrm{[Fe/H]_H} &=& 0.40F_{17} + 0.51F_{14} - 0.28F_{18} \nonumber\\ && - 1.460\mathrm{(\mathrm{H}_2\mathrm{O} \mathit {-} \mathrm{H})} + 0.71 \end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray} \mathrm{[M/H]_H} &=& 0.38F_{17} + 0.40F_{16} + 0.41F_{15} \nonumber\\ && + 0.194\mathrm{(\mathrm{H}_2\mathrm{O} \mathit {-} \mathrm{H})} - 0.76 \end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray} \mathrm{[Fe/H]_K} &=& 0.19F_{19} + 0.069F_{22} + 0.083F_{20} \nonumber\\ && + 0.218\mathrm{(\mathrm{H}_2\mathrm{O} \mathit {-} \mathrm{K})} - 1.55 \end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray} \mathrm{[M/H]_K} &=& 0.12F_{19} + 0.086F_{22} + 0.13F_{21} \nonumber\\ && + 0.245\mathrm{(\mathrm{H}_2\mathrm{O} \mathit {-} \mathrm{K})} - 1.18 \end{eqnarray} \tag{ 17 }$$

where F# refer to features listed in Table 5, the subscripts refer to the wavelength bands where the calibration is useful (V referring to visible wavelengths). An additional subscript is added (e or l) for calibrations in visible wavelengths to denote which formula is valid for early (K5.5–M2) and late (M2–M6) dwarfs. We show the primary star metallicity as a function of the derived metallicity for the companion dwarf for each of these 10 calibrations in Figure 5 and list reduced χ², R²_ap, root mean square error (RMSE), in Table 6. The RMSE indicates how useful a model is at prediction (lower numbers indicate the fit is a better predictor) and is defined as:

$$\begin{equation} {\rm RMSE} = \sqrt{\sum _{i=0}^n\frac{(y_{i,{\rm model}}- {y_i})^2}{(n-p)}}. \end{equation} \tag{ 18 }$$

Zoom In Zoom Out Reset image size

Lower R²_ap and higher RMSE values may in part be due to differences in S/N as a function of wavelength. We estimate measurement noise sources by adding synthetic noise to each spectrum consistent with the observed S/N of that spectrum, then recalculating the metallicity of the M dwarf using the appropriate equation above. The standard deviation in the metallicity estimate from 1000 different additions of noise pattern is assumed to be the measurement error. This error is what we use for our calculation of the reduced χ² for each fit. Thus the reduced χ² values probe how much of the noise comes from measurement (errors)A reduced χ² close to 1 would suggest that most or all of the error from the fit is due to measurement (e.g., Poisson) noise.

The dependence on τ varies significantly between equations. This is most likely due to different features capturing some of the temperature-dependence and/or that the Color1 and H₂O indices are not accurately modeling temperature dependencies across the full sample. It is also interesting that some coefficients of features are negative. This may be due to a combination of factors, including changes in [Fe/H] versus [α/Fe] (e.g., Ca is an α element) or complex relations between T_eff and [Fe/H], that vary for each feature. Whatever the case, since these fits are purely empirical, we should be cautious not to overinterpret the physical meaning of any particular coefficient or feature.

In addition to defining our own metallicity calibrations, we can use our sample to test existing metallicity estimators, as well as improve the existing calibrations. Like before, we use R²_ap, and RMSE as our standard metrics to asses the quality of a calibration. We summarize our refits in Table 7.

7.1. ζ_TiO/CaH

Much effort has gone into determine M dwarf metallicities using visible wavelength spectra. Most of this has been focused on the ζ_TiO/CaH (henceforth ζ) parameter (e.g., Lépine et al. 2007; Woolf et al. 2009; Dhital et al. 2012). However, the setup of our analysis means that we would not be able to identify ζ at all, because ζ is based on spectroscopic indices (not equivalent widths). Band indices (e.g., TiO5, CaH3, etc.) are calculated from the ratio of the flux in region a to the flux in region b using the approximation:

$$\begin{equation} \mathrm{Index} \simeq \frac{[\sum _{i=a}F(\lambda _i)]/[w_a]}{[\sum _{i=b}F(\lambda _i)]/[w_b]}, \end{equation} \tag{ 19 }$$

where w_a and w_b are the widths of region a and b in angstroms, respectively. The sums are computed over all pixels i in region a and b, respectively. Our analysis only makes use of equivalent widths (see Equation (3)). Further, we do not allow high-order terms, while ζ generally requires third- or fourth-order polynomials of the CaH index (e.g., Lépine et al. 2012). However, this does not prevent us from using our data to test the performance of ζ.

We calculate the CaH2, CaH3, and TiO5 indices following the definitions from Reid et al. (1995). We compute corrected indices (CaH2_c, CaH3_c, and TiO5_c) using the formula from Lépine et al. (2012), which include corrections for the SNIFS instrument. We use these to compute ζ following the formula as defined by Lépine et al. (2007):

$$\begin{equation} \zeta = \frac{1-\mathrm{TiO5}}{1-\mathrm{[TiO5]}_{Z_\odot }}, \end{equation} \tag{ 20 }$$

where ${\rm [TiO5]}_{Z_\odot }$ is a function of CaH = CaH2 + CaH3. We use the formula for [TiO5]$_{Z_\odot }$ from Lépine et al. (2012).

$$\begin{eqnarray} \nonumber \mathrm{[TiO5]}_{Z_\odot } &=& -0.050 - 1.118(\mathrm{CaH_{c}}) \\ &&+ 0.670(\mathrm{CaH_{c}})^2 - 0164 (\mathrm{CaH_{c}})^3. \end{eqnarray} \tag{ 21 }$$

We plot the primary star metallicities as a function of the derived ζ values as filled points in Figure 6. ζ shows a weak trend with metallicity in both [Fe/H] and [M/H]. From this we derive the following relationships:

$$\begin{eqnarray} \mathrm{[Fe/H]} &=& 0.98\zeta - 1.04 \end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray} \mathrm{[M/H]} &=& 0.68\zeta - 0.74. \end{eqnarray} \tag{ 23 }$$

Like before, we randomly reassign the metallicities to different CPM pairs, and attempt to compute an R²_rand value. We find that both Equations (22) and (23) give R²_ap <R²_rand. Further, the ζ parameter only correctly identifies one companion as a subdwarf (LHS 1812/PM I06032+1921S). Although this is the most metal-poor star in our sample, there are 12 other stars in our sample with [Fe/H] <−0.5 but ζ values consistent with solar metallicity (ζ > 0.825). According to Woolf et al. (2009), M dwarfs with [Fe/H] < −0.34 should have ζ < 0.825, and be labeled as sdM, suggesting a problem with these stars. If we remove these pairs, we derive the following relations:

$$\begin{eqnarray} \mathrm{[Fe/H]} &=& 1.26\zeta - 1.25 \end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray} \mathrm{[M/H]} &=& 0.88\zeta - 0.89. \end{eqnarray} \tag{ 25 }$$

These formulae are highly significant; they yield R²_ap values of 0.58 and 0.52, and RMSE values of 0.22 and 0.20, respectively. This suggests that ζ may be useful at predicting metallicities for [Fe/H] >+0.05 (the limit of the Woolf et al. (2009) calibration), provided the high-zeta, low-metallicity stars can be explained.

Zoom In Zoom Out Reset image size

Woolf et al. (2009, using R ≃ 3000 spectra) derive a relation between metallicity and the ζ parameter using a mix of wide binaries (for which the primary star metallicity is known) and single (K/M) stars with high-resolution spectra, analyzed using the MOOG software (Sneden 1973) with NEXTGEN models (Hauschildt et al. 1999). Although many of their wide binaries are also in our binary sample (including LHS 1812), there is insufficient overlap between the stars in Woolf & Wallerstein (2005, 2006), and Woolf et al. (2009) and those from SPOCS or our CFHT samples to detect any systematic offsets between the two samples. This is further complicated by fact that Woolf et al. (2009) have very few subdwarf binaries in their sample (most of their subdwarfs are single stars). To test weather these low-metallicity, high-ζ dwarfs are anomalous, we observed an additional set of stars from Woolf et al. (2009).

In total we observed 22 stars used in the Woolf et al. (2009) calibration with SNIFS. We specifically select stars to cover a wide range of metallicities to get a wide range of ζ values. We list the 22 stars in Table 8 and show them in Figure 6 as unfilled points. These 22 points form a clear metallicity sequence, showing that the revised ζ can reproduce the results of Woolf et al. (2009) and that ζ can be measured using modest resolution spectra.

Table 8. Stars Observed from Woolf et al. (2009)

Name	[M/H]a	[Fe/H]a	σ[Fe/H]a	ζb
LHS 38	−0.43	0.05	−0.40	0.94
HIP 1386	0.16	0.10	0.15	0.98
HIP 59514	−0.05	0.08	−0.03	1.06
HIP 89490	−0.53	0.08	−0.44	0.85
HIP 98906	−0.62	0.10	−0.52	0.56
HIP 105932	−0.37	0.05	−0.30	0.87
HD 18143B	0.19	0.11	0.18	1.03
HD 88230	−0.03	0.18	−0.05	1.00
HD 95735	−0.42	0.07	−0.40	1.10
GJ 129	−1.66	0.05	−1.33	0.09
GJ 1177B	−0.09	0.10	−0.06	0.91
GJ 3212	−0.08	0.05	−0.06	0.70
LHS364	−1.41	0.00	−1.15	−0.00
GJ 9722	−0.83	0.04	−0.70	0.58
LHS 174	−1.11	0.05	−0.95	0.59
LHS 182	−2.15	0.03	−1.88	−0.23
LHS 491	−0.93	0.08	−0.78	0.45
LHS 3084	−0.73	0.05	−0.64	0.78
LHS156	−1.00	0.00	−0.85	0.61
LHS161	−1.30	0.00	−1.06	0.30
LHS318	−1.26	0.00	−1.03	0.25
LSPMJ2205+5353	−1.29	0.00	−1.06	0.17

Notes. ^aDetermined from Woolf & Wallerstein (2005), Woolf & Wallerstein (2006), or Woolf et al. (2009). ^bDetermined from SNIFS spectra.

Download table as:  ASCIITypeset image

Assuming the metallicities from Woolf et al. (2009) are reliable and consistent with our own, we use the combined set of our binaries and the 22 additional late-type dwarfs from Woolf et al. (2009) to derive the following relations:

$$\begin{eqnarray} \mathrm{[Fe/H]} &=& 1.55\zeta - 1.62 \end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray} \mathrm{[M/H]} &=& 1.29\zeta - 1.35 \end{eqnarray} \tag{ 27 }$$

which we show in Figure 6 as dashed lines. The resulting fits yield R²_ap values of 0.58 and 0.61, respectively, both well above the R²_rand (0.17 and 0.14, respectively). The RMSE values are 0.28 and 0.23, although it is notably higher for dwarfs with ζ > 0.825 and lower for dwarfs with ζ < 0.825. If we remove the 11 stars with [Fe/H] <−0.5 but ζ > 0.825, ζ follows a clear trend over the full range of metallicities covered. Interestingly, these 11 dwarfs appear to follow a completely different sequence in [Fe/H] (or [M/H]) versus ζ, and are well separated from their single-star, metal-poor counterparts, suggesting that they are unique in some way. However, further inspection of these 11 pairs does not reveal anything that could explain their discrepancy: none exhibit H-α significant emission (likely inactive), they cover a wide range of spectral types (K7–M5), and they have metallicities from 5 different sources (including from SPOCS and our own CFHT spectra). We revisit the issue of these stars in Section 8.

7.2. J − K versus Metallicities

Johnson et al. (2012) find a relation between the J − K and V − K colors and the metallicity of M dwarfs, based in part on relations noted by Leggett (1992) and Lépine & Shara (2005). They find a best-fit relation of:

$$\begin{equation} [{\rm Fe/H}] = -0.050 + 3.520\Delta (J-K), \end{equation} \tag{ 28 }$$

where Δ(J − K) is defined as:

$$\begin{equation*} \Delta (J-K) = \left\lbrace \begin{array}{@{} ll}(J-K) - 0.835 & : V-K < 5.5\\ \ms (J-K) - \sum _{i=0} a_i (V-K)^i & : V-K \ge 5.5 \end{array} \right. \end{equation*}$$

and {a} = {1.637, −0.2910, 0.02557}. Johnson et al. (2012) note that this metallicity relation is only valid for stars with −0.1 < Δ(J − K) < 0.1 and V − K > 3.8, but that this technique yields metallicities accurate to ±0.15 dex. When we apply these two restrictions to our sample, we have 118 M dwarfs with known metallicities for their primary stars. This includes stars without SNIFS/IRTF spectra that were therefore not included earlier analyses. We find a higher RMSE of 0.20 dex, and an R²_ap of 0.20. One possible issue is the quality of V magnitudes in our sample, which come from a variety of sources. However, when we remove stars with V − K ⩾ 5.5 unless they have more reliable V magnitudes from Tycho (Høg et al. 2000), the quality of the fit does not change in any significant way (for stars with V − K < 5.5, Δ(J − K) is independent of V so these are not removed).

We attempt to improve on the calibration and derive a relation for [M/H] and find:

$$\begin{eqnarray} \mathrm{[Fe/H]} &=& -0.11 + 3.14\Delta (J-K) \end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray} \mathrm{[M/H]} &=& -0.09 + 2.14\Delta (J-K) \end{eqnarray} \tag{ 30 }$$

which results in a slightly improved RMSE = 0.19 and 0.16, and improved R²_ap = 0.30 and 0.25 for [Fe/H] and [M/H], respectively. These R²_ap values are significantly larger than the R²_rand values (0.08 and 0.09). The major difference between our fit and that of Johnson et al. (2012) is that they fix the constant term to −0.05 in order to keep [Fe/H] = −0.05 at Δ(J − K) = 0, consistent with a volume limited sample of stars from Johnson & Apps (2009), whereas we make no such restrictions.

7.3. K-band Metallicities

R10 have shown that one could derive M dwarf metallicities from K-band spectra using the Na i and Ca i lines (at 2.21 μm and 2.26 μm). T12 refine the calibration of R12 using SpeX data, and find that metallicities derived this way are accurate to ±0.12 dex. However, both R12 and T12 use relatively few wide binary pairs (18 and 22, respectively), and there is significant overlap in their two samples. Our sample has overlap with theirs, but is large enough to serve as a robust check on their calibrations. Because the work of T12 was optimized for SpeX, we perform our test on their calibration. We follow their method as closely as possible (including altering our continuum fitting procedure to match theirs).

We find that following the calibration of T12 yields RMSE = 0.16 and R²_ap = 0.69. We improve this calibration, and find a best fit of the form:

$$\begin{eqnarray} \nonumber \mathrm{[Fe/H]} &=& 0.19\times {\rm EW}_{{\rm Na}} + 0.074 \times {\rm EW}_{{\rm Ca}} \\ && + 2.13\times (\mathrm{H}_2\mathrm{O} \mathit {-} \mathrm{K}) - 3.18. \end{eqnarray} \tag{ 31 }$$

This new form yields metallicities accurate to RMSE = 0.14 and R²_ap = 0.76. The new calibration noticeably improves the fit for stars with −0.3 <[Fe/H]<+0.0; however, both calibrations do a relatively poor job fitting the most metal-poor stars ([Fe/H] <−0.5) in the sample. Adding square terms improves the fit only negligibly (ΔR²_ap < 0.02) and does not significantly improve the results for the most metal-poor stars. This, combined with our results from Section 6, suggests that fitting metal-poor stars requires a different set of lines, rather than simply higher order terms. Improvements may also be possible by deriving a separate calibration for [Fe/H] <−0.5; however, our sample has only 12 stars in this range, which is insufficient to derive a reliable calibration.

We also find a calibration for determining [M/H] of the form:

$$\begin{eqnarray} \nonumber [{\rm M/H}] &=& 0.16\times {\rm EW}_{{\rm Na}} + 0.039 \times {\rm EW}_{{\rm Ca}}\\ && + 2.29\times (\mathrm{H}_2\mathrm{O} \mathit {-} \mathrm{K}) - 3.04 \end{eqnarray} \tag{ 32 }$$

which gives RMSE = 0.13 and R²_ap = 0.75.

7.4. H-band Metallicities

In addition to refining the calibration of R10, T12 derive metallicities from H-band spectra. The technique relies on the Ca and K lines in the H band and a separate H₂O band defined for the H band (H₂O-H). As we did with K-band metallicities in Section 7.3, we use our sample to test the quality of the T12 technique. As before, we follow their prescription, including using the same continuum regions to fit the continuum to a fourth-order Legendre polynomial.

We find the T12 calibration gives RMSE = 0.16, and R²_ap = 0.71. As before, we improve this calibration, and find a best fit of the form:

$$\begin{eqnarray} \nonumber \mathrm{[Fe/H]} &=& 0.55\times {\rm EW}_{K} + 0.32 \times {\rm EW}_{{\rm Ca}}\\ && + 1.1\times (\mathrm{H}_2\mathrm{O} \mathit {-} \mathrm{H}) - 2.09. \end{eqnarray} \tag{ 33 }$$

The new calibration gives an almost negligible improvement over T12; yielding RMSE = 0.14, and R²_ap = 0.74. As with the K-band metallicities, adding square terms improves the fit negligibly (increase in R²_ap < 0.01), again suggesting that more lines are needed to fit the metal-poor stars.

Fitting these features to [M/H] we find a best fit of the form:

$$\begin{eqnarray} \nonumber \mathrm{[M/H]} &=& 0.41\times {\rm EW}_{K} + 0.24\times {\rm EW}_{{\rm Ca}}\\ && + 1.04 \times (\mathrm{H}_2\mathrm{O} \mathit {-} \mathrm{H}) - 1.77, \end{eqnarray} \tag{ 34 }$$

which gives RMSE = 0.12 and R²_ap = 0.71.

We present our sample of 112 late K and M dwarfs in wide binary systems, which we use to locate the most metal-sensitive features and recalibrate existing methods to determine late K and M dwarf metallicities. We combine published metallicities of 62 of the primary stars with 50 from our own CFHT spectra. We use moderate-resolution visible and NIR spectra of the late K and M dwarfs to identify the largest possible set of metal-dependent spectral features in late K to mid-M dwarfs for each of the JHK and visible wavelength bands. We utilize the metallicities of the primaries to calibrate these metal-dependent features and obtain optimal relationships to estimate metallicity in M dwarfs. Our sample covers a wide range of spectral types (from K5 to M6) and 1.5 dex in metallicity. This enables us to search for dependencies on spectral type, which was previously impossible with the relatively small samples used. We draw five important conclusions from our analysis:

• 1.  
  It is possible to determine accurate (RMSE < 0.1 dex) metallicities for late-K to mid-M dwarfs using modest resolution spectra (1000 < R < 2000) from a variety of different wavelengths. Although features in the K band perform best, metallicities can be estimated from spectra of any of the four wavelength regions.
• 2.  
  Determining reliable metallicities at visible wavelengths requires different calibrations depending on the spectral type of the star. The results are most accurate for K5.5–M2 dwarfs, most likely because the atomic lines we use are less contaminated by molecular lines and the pseudo-continuum is easier to estimate for these dwarfs. It is not known if our calibrations are applicable for stars later than M6. This will be the subject of a future investigation on metallicities for late M and brown dwarfs.
• 3.  
  Existing methods to determine metallicities using H- and K-band spectra (e.g., those from T12) work well for stars of near solar-metallicity, but have difficulties with most metal-poor stars in our sample ([Fe/H] <−0.5), even after applying our re-calibrations. Instead, determining metallicities for these stars requires the use of additional lines/features to improve the fit. This is most likely due to differences in [α/Fe], which are not being accurately captured by the K, Ca, and Na lines (Na and K are not α elements) used by R10 and T12.
• 4.  
  We are approaching the limits of what is possible with moderate resolution spectra. There is a diminishing return on adding additional lines to a given fit after three to four features, even if there are many more metal-sensitive features present in a given wavelength range. Thus going to higher S/N or adding more wide binaries of near solar-metallically is unlikely to improve the calibration. However, improvements could probably be made by including later spectral types (later than M5), getting more [M/H] values, a larger number of more metal-poor stars ([Fe/H] <−1.0), or obtaining spectra with higher resolution.
• 5.  
  Although the ζ parameter, commonly used to place stars into metallicity classes, correctly identifies metal-poor stars used in Woolf et al. (2009), classifications based on ζ incorrectly identify 12 of the 13 K/M companions with [Fe/H] <−0.5 as near solar-metallicity. This suggests that the ζ parameter is sensitive to stellar characteristics other than temperature and metallicity (e.g., activity, gravity, etc.), and may incorrectly identify some metal-poor stars as having near-solar abundances.

Our calibrations may be useful for both existing and future catalogs of M dwarf spectra. In particular, our calibration for visible wavelength spectra can be used on existing catalogs of local M dwarfs such as Lépine et al. (2012) to better probe the metallicity distribution of the local neighborhood especially since this sample is mostly early M dwarfs, where our calibration performs best. Sloan Digital Sky Survey also has ≃ 70, 000 visible wavelength spectra of M dwarfs (West et al. 2011; Bochanski et al. 2011) with similar resolution to our own, which could be used in conjunction with our calibrations to map out the metallicity distribution of the sample. Work has already been done in this area to confirm the existence of an "M dwarf problem" (Woolf & West 2012), but this depends on less metallicities derived from ζ parameter.

Although our fits have better RMSE values for [M/H] than they do for [Fe/H], this does not necessarily mean those fits are superior. In fact, the R²_ap values for fits to [M/H] are all inferior to those calibrations done on [Fe/H]. The reason is that the distribution of values for [Fe/H] is not the same as it is for [M/H]. Some of the most metal-poor stars do not have [M/H] values, and those that do generally have higher [M/H] values due to large differences in α abundance (as determined for the primary star).

We confirm the claim of Johnson et al. (2012), that one can get approximate M dwarf metallicities using J − K versus V − K colors. However, the technique has a limited range of metallicities (−0.4 <[Fe/H] <+0.2) and is only accurate to ≃ 0.2 dex. Thus this technique is probably best used in special applications, such as biasing a planet-search toward metal-rich M dwarfs.

One possible explanation for the poor performance of ζ on our sample compared to that of Woolf et al. (2009) is the presence of unresolved binaries. It is likely that most wide binaries form as higher order systems (Kouwenhoven et al. 2010). Thus many of our wide pairs may include unresolved M+M dwarf pairs. There is evidence of radius inflation in low-mass eclipsing binary systems (López-Morales 2007; Irwin et al. 2011; Kraus et al. 2011) and may also be cooler than their single star counterparts (Boyajian et al. 2012). Further, atmospheric models indicate that the TiO5 and CaH2/CaH3 indices, on which ζ is based, are sensitive to temperature and gravity (Jao et al. 2008; Allard et al. 2011). However, none of our most metal-poor companions show H-α emission, whereas radius inflation in tight binaries is usually associated with high chromospheric activity (López-Morales 2007; Kraus et al. 2011; Stassun et al. 2012). Additional metallicities of M dwarf with known multiplicity (e.g., low-mass eclipsing binaries and spectroscopic binaries) are needed to confirm if this is the source of the discrepancy.

Another complication is the possibility of having false binaries (chance alignments) in our sample. We can estimate the number of interlopers by cross referencing our sample with that of Tokovinin & Lépine (2012). Tokovinin & Lépine (2012) calculate the probability that stars with commiserate proper motions are actually physically associated with each other (P_phys). Although Tokovinin & Lépine (2012) caution that their probabilities are purely based on models (and therefore only approximate), the numbers can be used to give a rough estimate of contamination from chance alignments. By summing up P_phys values for all of our binaries included in the Tokovinin & Lépine (2012) sample we find that >90% of our binaries are physically associated with each other. However, some of the pairs with low P_phys values have parallax information for both the primary and companion that are consistent with each other. If we assume pairs with consistent parallaxes have P_phys = 1 and repeat our calculation, we find that 94% of our binaries are physically associated with each other. Although our metal-poor stars ([Fe/H] <−0.5) tend to be more distant, and therefore only 3 of the 13 are listed in Tokovinin & Lépine (2012), two of them have P_phys > 93% (the other has P_phys = 72%). Three more of our [Fe/H] <−0.5 stars have parallaxes that are consistent with the primary to 1σ, indicating that even our metal-poor stars are almost all physical pairs.

We do not claim to have identified every single metal-sensitive feature at the resolution of our spectra, however, the nature of our analysis means that it is unlikely that we missed any of the most useful ones. We perform a rough test on our recovery rate by introducing artificial metal-sensitive lines of various usefulness and repeating our analysis. Specifically, we select a sample of the most metal-sensitive features (those used in Equations (8)– (17)) and insert them elsewhere in the spectrum of the stars. When moving features from NIR to visible wavelengths we convolve the lines with a Gaussian profile to reduce the resolution of the feature (features moved into the NIR are not changed). We then repeat our analysis as described in Section 5. We find that metal-sensitive features are sometimes not identified when they are placed on very strong telluric features, blueward of 0.4 μm (where the S/N is very low), or when they overlap with other strong features (e.g., the Mg Ib line) that make clean measurements difficult. We also note that features identified as metal-sensitive in the NIR appear less metal-sensitive (although they are still identified) when placed in visible wavelengths; this is likely due to the lower resolution and/or difficulties measuring features that are convolved with strong molecular lines in the visible. In spite of these exceptions, we still recover >88% of the lines on average, and >93% when we exclude telluric regions and low S/N regions of the spectrum. This indicates that our analysis is quite robust, and that expanding on our findings will require observations later-type stars (past M5), more metal-poor stars, or higher resolution visible spectra.

This work was supported by NSF grant AST-0908419 (Astronomy & Astrophysics) and NNX11AC33G (Origins of Solar Systems) to E.G., as well as NSF grants AST 06-07757 and AST 09-08419 to S.L. We thank Andrew West for his useful suggestions this paper.

SNIFS on the UH 2.2 m telescope is part of the Nearby Supernova Factory project, a scientific collaboration among the Centre de Recherche Astronomique de Lyon, Institut de Physique NuclŽaire de Lyon, Laboratoire de Physique NuclŽaire et des Hautes Energies, Lawrence Berkeley National Laboratory, Yale University, University of Bonn, Max Planck Institute for Astrophysics, Tsinghua Center for Astrophysics, and the Centre de Physique des Particules de Marseille.

Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/IRFU, at the Canada–France–Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Science de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at Terapix available at the Canadian Astronomy Data Centre as part of the Canada–France–Hawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS.

Based on observations at the Infrared Telescope Facility, which is operated by the University of Hawaii under Cooperative Agreement No. NNX-08AE38A with the National Aeronautics and Space Administration, Science Mission Directorate, Planetary Astronomy Program.

Facilities: IRTF - Infrared Telescope Facility, CFHT - Canada-France-Hawaii Telescope, UH:2.2m - University of Hawaii 2.2 meter Telescope