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HomeAll issuesVolume 519 (September 2010)A&A, 519 (2010) A105Full HTML
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A&A
Volume519, September 2010
Article NumberA105
Number of page(s)7
SectionPlanets and planetary systems
DOIhttps://doi.org/10.1051/0004-6361/201015016
Published online20 September 2010
A&A 519, A105 (2010)

A physically-motivated photometric calibration of M dwarf metallicity

K. C. Schlaufman - G. Laughlin

Astronomy and Astrophysics Department, University of California, Santa Cruz, CA 95064, USA

Received 18 May 2010 / Accepted 11 June 2010

Abstract
The location of M dwarfs in the$(V-K_{\rm s})-M_{K{\rm s}}$color-magnitude diagram (CMD) has been shown to correlate withmetallicity. We demonstrate that previous empirical photometriccalibrations of M dwarf metallicity exploiting this correlationsystematically underestimate or overestimate metallicity at theextremes of their range. We improve upon previous calibrations in threeways. First, we use both a volume-limited and kinematically-matchedsampleof F and G dwarfs from the Geneva-Copehnagen Survey (GCS) to inferthe mean metallicity of M dwarfs in the Solar Neighborhood.Second, we use theoretical models of M dwarf interiors andatmospheres to determine the effect of metallicity on M dwarfs in the$(V-K_{\rm s})-M_{K{\rm s}}$ CMD.Third, though we use the GCS to infer the mean metallicity ofM dwarfs in the Solar Neighborhood, our final calibration is basedpurely on high-resolution spectroscopy of FGK primaries withM dwarf companions as well as the trigonometric parallaxes andapparentV- and$K_{\rm s}$-bandmagnitudes of those M dwarf companions. As a result, ourphotometric calibration explains an order of magnitude more of thevariance in the calibration sample than previous photometriccalibrations. We use our calibration to non-parametrically quantify thesignificance of the observation that M dwarfs that host exoplanetsare preferentially in a region of the$(V-K_{\rm s})-M_{K{\rm s}}$ plane populated by metal-rich M dwarfs. We find that the probability p that planet-hosting M dwarfs are distributed across the$(V-K_{\rm s})-M_{K{\rm s}}$ CMD in the same way as field M dwarfs isp = 0.06 $\pm $0.008. Interestingly, the subsample of M dwarfs that hostNeptune and sub-Neptune mass planets may also be preferentially locatedin the region of the$(V-K_{\rm s})-M_{K{\rm s}}$ plane populated by high-metallicity M dwarfs. The probability of this occurrence by chance isp = 0.40 $\pm $0.02, and this observation hints that low-mass planets may be morelikely to be found around metal-rich M dwarfs. The confirmation ofthis hint would be in contrast to the result obtained forFGK stars, where it appears that metal-rich and metal-poor starshosts Neptune-mass planets with approximately equal probability.An increased rate of low-mass planet occurrence around metal-richM dwarfs would be a natural consequence of the core-accretionmodel of planet formation.

Key words:planets and satellites: formation - stars: abundances - stars: low-mass - stars: statistics

1 Introduction

The determination of metallicity for M dwarfs is a very difficult problem (e.g.Gustafsson 1989).Their cool atmospheres permit the existence of many molecules for whichmolecular opacities are currently poorly constrained.As a result, the estimation of the continuum level of aspectrum is challenging, rendering line-based metallicity indicatorsunreliable. The poorly constrained molecular opacity data currentlyavailable makes the determination of metallicity through spectralsynthesis difficult as well. For those reasons, alternativemethods must be employed to estimate M dwarf metallicities.

Table 1:  M dwarfs in binary systems with an FGK primary and those that host planets.

The main sequence lifetimes of M dwarfs are longer than theHubble time, so they have not yet departed much from the zero-agemain sequence. Consequently, M dwarfs might be expected to form atwo-parameter sequence in mass and metallicity, suggesting that atwo-color broad-band photometric calibration might constrain theirproperties. There have been several attempts to obtain the metallicityof M dwarfs using their photometric properties, including tworecent breakthroughs.Bonfils et al. (2005a) - bon05ahereafter - had the subtle insight to realize that M dwarfsin binary or multiple systems should have metallicities commensuratewith the easily-measured metallicity of an FGK primary in thesystem. In that way, bon05a identified a calibration sample ofM dwarfs with metallicities securely determined in one of twoways: (1) high-resolution spectroscopy of an FGK companionand (2) high-resolution spectroscopy of M dwarfs forwhich $T_{\rm eff}$and $\log{g}$couldbe fixed with photometric data. For the former,the metallicity is very likely the same as the metallicityinferred from high-resolution spectroscopy of its FGK companion.For the latter, spectral synthesis after fixing $T_{\rm eff}$and $\log{g}$with photometric data eliminates some degeneracy and produces a reasonable metallicityestimate. They noted that low-metallicity M dwarfs have blue$V-K_{\rm s}$ color at constant$K_{\rm s}$-band absolute magnitude $M_{K{\rm s}}$,and they fit a linear model to their calibration sample using $V-K_{\rm s}$and $M_{K{\rm s}}$to predict [Fe/H].Johnson & Apps (2009) -joh09 hereafter - addressed the relative lack of high-metallicityM dwarfs in thecalibration sample of bon05a and created an empirical model in whichthe distance of an M dwarf above the field M dwarf mainsequence (MS) in the$(V-K_{\rm s})-M_{K{\rm s}}$color-magnitude diagram (CMD) indicated its metallicity. The greatinsight of joh09 was that the mean metallicity of a population ofM dwarfs could be characterized by the easily-measured meanmetallicity of a similar population of FGK stars. Indeed, theyassumed that the field M dwarf MS was an isometallicity contourwith the same metallicity as a volume-limited sample of G andK stars and fit a linear model using the distance above the fieldM dwarf MS to predict [Fe/H].

M dwarfs are attractive targets around which to search forlow-mass planets because they have large reflex velocities and transitdepths even for low-mass and small-radius companions. Given that themetallicity of protoplanetary disks is a key parameter in models ofplanet formation (e.g.Laughlin et al. 2004;Ida & Lin 2004),the metallicity of M dwarfs that host planets will constrain theplanet formation process in low-mass protoplanetary disks. Indeed,it is well-established that metal-rich FGK stars are morelikely to host giant planets (e.g.Fischer & Valenti 2005;Santos et al. 2004),but there is also evidence to suggest that metal-rich FGK starsare not much more likely to host Neptune-mass planets than theirlow-metallicity counterparts (e.g.Sousa et al. 2008;Bouchy et al. 2009;Udry et al. 2006).Already, joh09 have used their model of M dwarf metallicity tosuggest that the M dwarfs that host planets are preferentiallymetal-rich. However, joh09 did not address whether the apparent lack ofa correlation between FGK host stellar metallicity and thepresenceof Neptune-mass planets extends to M dwarfs.

In this paper, we examine a calibration sample of M dwarfswith securely estimated metallicities and we show that the models ofbon05a and joh09 systematically underestimate or overestimatemetallicity at the extremes of the range of this calibration sample. Wedemonstrate that a volume-limited and kinematically-matched sample ofSun-like stars produces a better estimate of the mean M dwarfmetallicity in the Solar Neighborhood, and we use M dwarf modelsof different metallicities fromBaraffe et al. (1998) to improve on the technique described in joh09. The position of an M dwarf in the$(V-K_{\rm s})-M_{K{\rm s}}$ CMDremains an indicator of its metallicity, and we use that fact tonon-parametrically quantify the significance of the observation thatplanet-hosting M dwarfs are preferentially in a region of the$(V-K_{\rm s})-M_{K{\rm s}}$ planepopulated by metal-rich M dwarfs. Moreover, we identify for thefirst time a hint that the subsample of M dwarfs that host Neptuneand sub-Neptune mass planets may also be more likely to be in theregion of the$(V-K_{\rm s})-M_{K{\rm s}}$ CMDassociated with metal-rich M dwarfs. We describe our analysis inSect. 2 and summarize our findings in Sect. 3.

2 Analysis

2.1 Testing previous calibrations

We first collect from bon05a and joh09 a calibration sample ofM dwarfs in wide binary or multiple systems with anFGK primary. The metallicity of the FGK primary in the systemis straightforward to measure from a high-resolution spectrum, and ifthe M dwarf secondary and the FGK primary formed in the samemolecular core, then the expectation is that the two should havecommensurate metallicities. We collect 13 examples from bon05a,selectingonly those M dwarfs with preciseV-band magnitudes fromCCD photometry. We also collect six high-metallicity examples fromjoh09. We summarize this calibration sample of M dwarfs in thefirst 19 lines of Table 1.

We compute the metallicity predicted for this calibration sample fromboth the bon05a and joh09 relations and compare it to the observedvalues. Note that the bon05a relation was initially based on acalibration sample that included M dwarfs in binary or multiplesystems with FGK primaries (some withV-bandmagnitudes deduced from photographic plates) as well as low-metallicityM dwarfs with metallicity inferred from spectroscopy afterfixing $T_{\rm eff}$and $\log{g}$withphotometric data. Meanwhile, the joh09calibration sample included only the six metal-rich M dwarfs inbinary or multiple systems with FGK primaries listed inrows 14 through 19 in Table 1.For those reasons, we believe that the 19 M dwarfs withmetallicities inferred from high-resolution spectroscopy ofFGK primaries in Table 1is the largest and most reliable set of M dwarf metallicities fromwhich to verify previous calibrations. We apply both the bon05a andjoh09 relations to this sample and compute the residual between eachmodel and observation. We plot the distribution of residuals for bothmodels in Fig. 1, and we findthat the bon05a relation systematically underestimates M dwarfmetallicity and that the joh09 relation systematically overestimatesM dwarf metallicity.

2.2 A physically-motivated empirical model of M dwarf metallicity

As discussed in Sect. 2.1 and Fig. 1, the models of bon05a and joh09 have non-negligible residuals when applied to the calibration sample in Table 1. Still, there is a correlation between the metallicity of an M dwarf and itsdistance in the$(V-K_{\rm s})-M_{K{\rm s}}$ planefrom the field M dwarf MS. We attempted to improve the joh09 modelby reassessing both the zero point of the model and the direction fromthe M dwarf MS in the$(V-K_{\rm s})-M_{K{\rm s}}$ plane best correlated with metallicity.

Recall that joh09 set the mean metallicity of the Solar NeighborhoodM dwarf sample equal to the mean metallicity of a volume-limitedsample of G0-K2 stars (4.0 <MV < 6.5) from the SPOCS catalog ofValenti & Fischer (2005).The SPOCS sample of joh09 was based on a catalog of stars selectedto have absorption lines deep enough to enable high-precision radialvelocity detection of exoplanets. As a result, theSPOCS sample is biased against metal-poor stars and thereforepotentially unsuitable for the determination of the average metallicityin the Solar Neighborhood (as noted inValenti & Fischer 2005). Alternatively, the Geneva-Copenhagen Survey (GCS -Holmberg et al. 2009,2007;Nordström et al. 2004)of Solar Neighborhood F and G dwarfs is magnitude-complete,kinematically-unbiased, and free of the line depth bias inherent in theSPOCS catalog. Thoughthe GCS metallicity estimates are based on Strömgren$uvby\beta$ photometryand not high-resolution spectroscopy, the precision of theGCS metallicities are sufficient when combined with the reducedbias of the sample to provide a better estimate of the mean SolarNeighborhoodmetallicity than the SPOCS sample.

\begin{figure}\par\includegraphics[width=9cm,clip]{15016fg1.eps}\end{figure}Figure 1:

Optimally-smoothed residual distributions forBonfils et al. (2005a) - bon05a hereafter - andJohnson & Apps (2009) -joh09 hereafter. In both cases the vertical dashed line indicates themean of the distribution. The mean value of the bon05a residualsis 0.12 with standard deviation 0.16, while the mean value ofthe joh09 residuals is -0.12 with standard deviation 0.12.Note that the bon05a distribution has a heavy tail at large positivevalues (indicating systematically low [Fe/H] estimates) and thejoh09 distribution has a heavy tail at large negative values(indicating systematically high [Fe/H] estimates).

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In addition, theUVW kinematics of a volume-limited sampleof M dwarfs is not necessarily equivalent to the kinematics of avolume-limited sample of FGK dwarfs. Since the mean of a sample issensitive to outliers, and because Sun-like stars with outlierkinematics are also likely to be outliers in metallicity,a kinematic-match is important to determine the mean metallicityof Solar Neighborhood M dwarfs in this way. To address thispoint, we use the M dwarfUVW distribution described byHawley et al. (1996) to create a volume-limited and kinematically-matched sample of F and G dwarfs fromHolmberg et al. (2009) from which we infer the average metallicity of the Solar NeighborhoodM dwarf population. In Fig. 2 we superimpose theUVW velocity-space distribution of local M dwarfs derived byHawley et al. (1996) on top of theUVWvelocity-space distribution of F and G stars from the GCS withparallax-based distance estimates that place them within 20 pc ofthe Sun. We bootstrap resample from the subset of GCS stars within20 pc and with kinematics consistent with the M dwarfvelocity ellipsoid as defined inHawley et al. (1996). We ensure that 68% of the GCS stars in each bootstrap sample haveUVW velocities that place them within the one-sigma contour ofHawley et al. (1996)and that the rest of each bootstrap sample lies within the two-sigmacontour. In the end, we find that a volume-limited andkinematically-matched sample of F and G dwarfs from the GCS surveyhas a mean metallicity of [Fe/H] = -0.14 $\pm $0.06. We obtain a similar result with a sample of GCS starsvolume-limited in the same way as the volume-limited SPOCS sampleof joh09, for which we find a mean Solar Neighborhood metallicity of[Fe/H] = -0.15 $\pm $0.02. In this case, the superior statistics of the largervolume-limited sample is enough to formally achieve a higher precisionthan the volume-limited and kinematically-matched sample, though thevolume-limited sample is subject to a greater degree of possiblesystematic error. For that reason, we regard the volume-limited andkinematically-matched result as likely more reliable.

\begin{figure}\par\includegraphics[width=9cm,clip]{15016fg2.eps}\end{figure}Figure 2:

Velocity ellipsoids inferred for a volume-limited sample of early M dwarfs fromHawley et al. (1996) superimposed on theUVW velocity distribution of a volume-limited sample ($d< 20~{\rm pc}$)of Sun-like stars from the Geneva-Copenhagen Survey (gray points -Holmberg et al. 2009).The light curve denotes the one-sigma region while the heavy curve denotes the two-sigma region. Bootstrap resampling of theHolmberg et al. (2009)sample with the constraint that 68% of each bootstrap sample lieswithin the one-sigma contour and that the other 32% lies within thetwo-sigmacontour produces a volume-limited and kinematically-matched populationmetallicity of [Fe/H] = -0.14 $\pm $ 0.06.

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The mean metallicity of our volume-limited and kinematically-matchedGCS sample suggests that the field M dwarf MS defined byjoh09 is an isometallicity contour with [Fe/H] $\approx $-0.14. Note that if the isometallicity contour corresponded to [Fe/H] = -0.05 as in joh09, five stars from Table 1with [Fe/H] < -0.05 would be to the right of the isometallicitycontour indicating [Fe/H] > -0.05. Alternatively, if weassume that the isometallicity contour corresponds to [Fe/H] $\approx $-0.14, then only two of the 19 stars are on the wrong side of contour.

We now determine which direction in the$(V-K_{\rm s})-M_{K{\rm s}}$ plane an isochrone moves as a function of metallicity. In Fig. 3 we plot the M dwarfs with securely determined metallicity from Table 1, along with the M dwarf MS from joh09 andtwo different isochrones fromBaraffe et al. (1998). We use the transformation ofCarpenter (2001) to transform the $K_{\rm CIT}$given inBaraffe et al. (1998) into $K_{\rm s}$.The left-most isochrone corresponds to a population with [Fe/H] = -0.5 andY = 0.25 while the right-most isochrone corresponds to a population with [Fe/H] = 0 andY = 0.275. Both isochrones use mixing-length parameterl = 1 for a 5 Gyr population (there is no detectable evolution in$(V-K_{\rm s})-M_{K{\rm s}}$ CMD after 3 Gyr). The horizontal lines connect points of constant mass. With all otherparameters constant, metallicity should best correlate with horizontal shifts in the$(V-K_{\rm s})-M_{K{\rm s}}$ plane.For that reason, we compute the distance from the M dwarf MSin the horizontal direction for each M dwarf with securemetallicity from Table 1. We then fit a linear model using this distance as a predictor with [Fe/H] as the response. We find that

 $\displaystyle \mbox{[Fe/H]} = 0.79 \Delta\left(V-K_{\rm s}\right) - 0.17$ 
 $\displaystyle \Delta\left(V-K_{\rm s}\right) \equiv \left(V-K_{\rm s}\right)_{\rm obs}-\left(V-K_{\rm s}\right)_{\rm iso}$(1)

is the optimal model. In this case,$M_{K{\rm s}}$as function of $V-K_{\rm s}$is given by the fifth-order polynomial with coefficients in increasingorder (-9.58933, 17.3952, -8.88365, 2.22598, 0.258854, 0.0113399) fromjoh09. To aid in the calculation of$\left(V-K_{\rm s}\right)_{\rm iso}$,we give the same curve with $V-K_{\rm s}$as a function $M_{K{\rm s}}$:it is a fifth-order polynomial with coefficients in increasingorder (51.1413, -39.3756, 12.2862, -1.83916, 0.134266, -0.00382023).

\begin{figure}\par\includegraphics[width=9cm,clip]{15016fg3.eps}\end{figure}Figure 3:

Position of M dwarfs with secure metallicities from Table 1 (blue points) in the$(V-K_{\rm s})-M_{K{\rm s}}$ CMD in relation to the field M dwarf MS from joh09 (black line) and the theoretical isochrones ofBaraffe et al. (1998) (blue lines). The color of the isochrone line gives its metallicity: [Fe/H] = -0.5 andY = 0.25 on the left and [Fe/H] = 0 andY = 0.275 on the right. Both isochrones assume mixing-length parameterl = 1 for a 5 Gyr population, as there is no detectable evolution in the$(V-K_{\rm s})-M_{K{\rm s}}$CMDafter about 3 Gyr. The horizontal lines connect points of constantmass. The models indicate that differences in metallicity should bestcorrelate with horizontal shifts in the$(V-K_{\rm s})-M_{K{\rm s}}$ plane.

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\begin{figure}\par\includegraphics[width=8.5cm,clip]{15016fg4a.eps}\hspace*{5mm}\includegraphics[width=8.5cm,clip]{15016fg4b.eps}\end{figure}Figure 4:

Left: position of M dwarfs known to host Jupiter-mass planets(dark blue triangles) and Neptune-mass (or below) planets (bluesquares) in relation to a control sample of field M dwarfs (graypoints) and the field M dwarf MS from joh09 (black line). Again,like the high-metallicity M dwarfs, the M dwarfs that hostplanets are concentrated to the right of the field M dwarf MS. Right: distribution of cumulative sample distances from the field M dwarf MS of joh09, which we assume to be a [Fe/H] $\approx $ -0.17 isometallicity contour in the$(V-K_{\rm s})-M_{K{\rm s}}$ CMD.Points to the right of the field M dwarf MS add their distance tothe sum, while points left of MS subtract their distance from the sum.We generate each distribution with a Monte Carlo simulation.First, we randomly select from the field M dwarf sample a numberof stars equal to the number of M dwarfs known to host planets ofa certain type. We then compute the cumulative horizontal distance ofthat random subsample from the field M dwarf MS. We repeat thisprocess 1000 times to generate the distribution of samplecumulative horizontal distances from the field M dwarf MS given nocorrelation between the presence of an exoplanet and the location ofits host in the$(V-K_{\rm s})-M_{K{\rm s}}$ CMD.In all cases, we confirm the findings of joh09 that theM dwarfs that host exoplanets are preferentially to the right ofthe field M dwarf MS. In particular, we find that theprobability p that there is no correlation between the location of an M dwarf in the$(V-K_{\rm s})-M_{K{\rm s}}$ CMD and its status as an exoplanet host isp = 0.06 $\pm $0.008. For the subsample that hosts Jupiter-mass planets, we find that the probability isp = 0.04 $\pm $0.005. More interestingly, we find that the probability that there is no correlation between the location in the$(V-K_{\rm s})-M_{K{\rm s}}$CMD and an M dwarf's status as the host of a Neptune-mass (or below) exoplanet isp = 0.40 $\pm $0.02. If M dwarfs to the right of the field M dwarf MS are metal-rich as suggested by theBaraffe et al. (1998)models and argued by joh09, then this observation may be evidence foran increased incidence of low-mass planets around metal-rich low-massstars, a trend which is not observed in FGK stars.

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We use two of the model selection criteria given inHocking (1976) to evaluate all three models. First, we compute the residual mean square (rms), defined as

$\displaystyle %\mbox{rms}_{p} = \frac{\mbox{SSE}_{p}}{n-p}$ (2)

wheren is the number of data points,p is the number of predictors in the model, and$\mbox{SSE}_{p}$ is the residual sum of squares for ap-term model. In general, models withsmaller values of $\mbox{rms}_{p}$are best-suited to prediction. For our model, we find that$\mbox{rms}_{p} = 0.02$;for the model of joh09 the value is$\mbox{rms}_{p} = 0.04$while for the model of bon05a the value is$\mbox{rms}_{p} = 0.05$.Next, we compute the adjusted square of the multiple correlation coefficient Rap2, which is widely used to judge the fit of a model. A value ofRap2 = 1 indicates that a model explains all of the variance in a sample, whileRap2 = 0 indicates that the model explains none of the variance.Rap2 is defined as

 $\displaystyle R_{ap}^{2} = 1 - \left(n-1\right) \frac{\mbox{rms}_{p}}{\mbox{SST}}$ 
 $\displaystyle \mbox{SST} \equiv \sum{(y_i-\bar{y})^2}$(3)

whereyi and$\bar{y}$are the sample and its mean, respectively. For our model, we find thatRap2 = 0.49; for the model of joh09 the value isRap2 = 0.059 while for the model of bon05a the value isRap2 < 0.05. We note that ourmodel explains almost an order of magnitude more of the variance in thecalibration sample than either model presented in bon05a or joh09.

Differing$T_{\rm eff}$scales have been well-noted as a source of metallicity discrepancies in metallicity studies of the Solar Neighborhood (e.g.Holmberg et al. 2007). The differing$T_{\rm eff}$ calibrationsbetween the GCS and other surveys will not affect our results,as our calibration (including the metallicity of the M dwarfMS) is based on the horizontal distance in the$(V-K_{\rm s})-M_{K{\rm s}}$ CMDfrom the mean M dwarf MS of joh09 of M dwarfs withmetallicities known from high-resolution spectroscopy ofFGK primaries. We only used the GCS Strömgren-based metallicitiesto establish the fact that the mean Solar Neighborhood metallicity iscloser to [Fe/H] $\approx $-0.15 than it is to [Fe/H] $\approx $-0.05 as argued by joh09.For that reason, our calibration is based purely onhigh-resolution spectroscopy of FGK primaries as well as thetrigonometric parallaxes and apparentV- and$K_{\rm s}$-band magnitudes of their M dwarf companions.

The M dwarfs in binary systems with FGK primaries that we use to fixour calibration are not a volume-limited or kinematically-matchedsample. The volume-limit and kinematic-match were only necessary toverify the fact that the mean metallicity of the M dwarfpopulation in the SolarNeighborhood is a well-defined quantity. That verification is anecessary precondition that must be established before any joh09 stylecalibration using distance from the field M dwarf MS in the$(V-K_{\rm s})-M_{K{\rm s}}$CMD can even be considered. Once the points along the field M dwarf MS in the$(V-K_{\rm s})-M_{K{\rm s}}$ CMD are fixed to the mean metallicity of the Solar Neighborhood, the metallicity of an M dwarf with given$V-K_{\rm s}$ color and absolute magnitude $M_{K{\rm s}}$along the curve is specified regardless of its position or velocity.Indeed, when we build our calibration using only the metallicities ofM dwarfs in binaries with FGK primaries, theirV- and$K_{\rm s}$-bandmagnitudes, and trigonometric parallaxes, we find that the meanmetallicity of the Solar Neighborhood M dwarf MS based on thecalibration sample ([Fe/H] = -0.17 $\pm $ 0.07)is statistically indistinguishable from the mean metallicity inferredfrom the volume-limited and kinematically-matched sample([Fe/H] = -0.14 $\pm $ 0.06).

2.3 The metallicity of M dwarfs that host planets

We plot the location of M dwarfs that host planets in the left-hand panel of Fig. 4. The M dwarf models ofBaraffe et al. (1998) suggest that horizontal distance in the$(V-K_{\rm s})-M_{K{\rm s}}$ CMDbest correlates with metallicity. To non-parametrically determinethe degree to which planet-hosting M dwarfs are preferentiallyfound to the right of the M dwarf MS, we need to quantify thelikelihood that the cumulative horizontal distance from theisometallicity contour of a randomly selected sample of fieldM dwarfs can be as large as that observed in the sample ofM dwarfs that host planets simply by chance.

To address this issue, we create a control sample of field M dwarfs selected from the Hipparcos (van Leeuwen 2007) and Yale Parallax Catalogs (van Altena et al. 1995). We include in the control sample those M dwarfs from the Hipparcos catalog that have parallaxes$\pi > 100$ mas precise to better than 5% and those M dwarfs from the Yale catalog that have parallaxes$\pi > 100$ mas. We useV-band photometry from each catalog and we obtain$K_{\rm s}$ photometry for both samples from the 2MASS database (Skrutskie et al. 2006). For a sample of size n we can compute the statistic $\Sigma$:

$\displaystyle %\Sigma = \sum_{i=1}^{n} \left(V-K_{\rm s}\right)_i-\left(V-K_{\rm s}\right)_{\rm iso}.$ (4)

To characterize the likelihood that an observed value of $\Sigma$for a subsample with size mof M dwarfs that host planets can be produced by chance, we use aMonte Carlo simulation. We randomly select a sample of size m from the 127 M dwarfs in the control sample, compute $\Sigma$for that sample, save the result, and repeat the calculation1000 times. In that way, we can determine the distributionof $\Sigma$expected under the null hypothesis that M dwarfs that host planets are distributed in the$(V-K_{\rm s})-M_{K{\rm s}}$ CMDin the same way as field M dwarfs. We consider three sub-samples:(1) all planets hosts; (2) hosts of Jupiter-mass planets; and(3) hosts of Neptune-mass (and below) planets. We find that incase (1)$\Sigma = 3.43$indicating only ap = 0.06 $\pm $0.008 probability that the cumulative distance of the sample fromthe isometallicity contour occurred by chance. In case (2) we find$\Sigma = 2.39 \Rightarrow p = 0.04$ $\pm $0.005 and in case (3) we find$\Sigma = 1.04 \Rightarrow p = 0.40$ $\pm $0.02. We summarize this calculation in the right-hand panel of Fig. 4.

2.4 Discussion

The apparent position of planet-hosting M dwarfs in the region of the$(V-K_{\rm s})-M_{K{\rm s}}$ CMDassociated with known high-metallicity M dwarfs tentativelysuggests that metal-rich M dwarfs are more likely to hostJupiter-mass and possibly Neptune-mass planets as well.If this correlation is confirmed in the future, it can beunderstood as a natural consequence of the core-accretion model ofplanet formation (e.g.Laughlin et al. 2004;Ida & Lin 2004).Indeed, a more metal-rich protoplanetary disk will almostcertainly have a higher surface density of solids, and that increasedsurface density enables the rapid formation of the several Earth-masscores necessary to accrete gas from the protoplanetary disk before thegaseous disk is dissipated. Moreover, it would be especiallyinteresting if the tentatively suggested correlation extends to thehosts of Neptune-mass planets, as current evidence seems to suggestthat the probability that an FGK star hosts a Neptune-mass planetis not a strongfunction of metallicity (e.g.Sousa et al. 2008;Bouchy et al. 2009;Udry et al. 2006).

If the tentatively suggested correlation between the presence ofplanets and the metallicity of their host M dwarfs is eventuallyconfirmed, it might indicate a lower-limit on the amount of solidmaterial necessary to form planets. To see why, recall that themass of a protoplanetary diskscales roughly as$M_{\rm disk} \propto M_{\ast}$and that the fraction of solid material in a disk $f_{\rm solid}$scales roughly as $Z_{\ast}$where $Z_{\ast}$is the metal content of the host star. The total amount of solid material in a protoplanetary disk will then scale like$M_{\rm solid} \propto f_{\rm solid} M_{\rm disk} \propto Z_{\ast} M_{\ast}$.Minimum-mass Solar Nebula models (MMSN -Hayashi 1981)and observations of T Tauri disks in star-forming regionssuggest that protoplanetary disks around young Solar-type stars areabout 1% the mass of their host stars, albeit with significantscatter (e.g.Hartmann et al. 1998). Combined with the fact that the metal content of the Sun is$Z_{\odot} = 0.0176$by mass, the total solid mass in the MMSN was about$M_{\rm solid} \approx 60~M_{\oplus}$.This is a lower-limit, as more careful calculations suggest that the protoplanetary disk around the Sun had$M_{\rm solid} \sim 100~M_{\oplus}$(e.g.Lissauer 1993). In either case, this is a factor of a few to ten greater than the$10~M_{\oplus}$of material necessary to form the core of a gas or ice giant planet inthe core-accretion model of planet formation. In the case of aSolar-metallicity mid-M dwarf with$M_{\ast} = 0.3~M_{\odot}$,the total amount of solid material in the disk is 70% less, about$M_{\rm solid} \approx 20~M_{\oplus}$.This is factor of order unity to a few times the mass necessary to formthe core of a gas or ice giant. Since planet formation likely does notlock-up the entire solid component of a protoplanetary disk in planets,reducing the total mass of solids in the disk - either by reducingthe metallicity or mass - will also reduce the chances of forminga$10~M_{\oplus}$ core (and therefore a gas or ice giant) before the parent protoplanetary disk is dissipated.

The confirmation of the hint of a correlation between the presence oflow-mass planets and M dwarf metallicity could be evidence of thisthreshold solid mass necessary to form Neptune-mass planets. AroundFGK stars, the same threshold solid mass suggests that acorrelation between the presence of low-mass planets and host starmetallicity might occur at one-third Solar metallicity,or [Fe/H] = -0.5. This is just below the typicalmetallicity of stars observed at high radial velocity precision withHARPS (Sousa et al. 2008).This expected correlation might be verified as larger samples oflow-metallicity stars are surveyed at high radial velocity precision orby transit surveys of nearby low-metallicity open clusters(e.g. NGC 752 or IC 4756).

The hint of a correlation between the presence of Neptune-mass(and below) planets and M dwarf metallicity tentativelysuggests that searches for low-mass planets around M dwarfs likethe MEarth Project (Nutzman & Charbonneau 2008) could improve their yield by shading their target list toward M dwarfs that have red$V-K_{\rm s}$ colors at constant$K_{\rm s}$-band absolute magnitude $M_{K{\rm s}}$.Note that since the absolute magnitude $M_{K{\rm s}}$of an M dwarf depends only on the logarithm of its often poorly-known distance, while$V-K_{\rm s}$depends linearly on its often poorly-knownV-band magnitude, the collection of high-quality CCD-basedV-bandmagnitudes for M dwarfs in the Solar Neighborhood could be thefirst step towards maximizing the yield of planets aroundM dwarfs.

3 Conclusion

We showed that previous empirical photometric calibrations ofM dwarf metallicity systematically underestimate or overestimatemetallicity at the extremes of their range. We derived aphysically-motivated model that explains an order of magnitude more ofthe variance in the calibration sample than either theBonfils et al. (2005a) orJohnson & Apps (2009) models. We used the correlation underlying our model to non-parametrically show that the probability p that there is no relationship between position of an M dwarf in the$(V-K_{\rm s})-M_{K{\rm s}}$ CMD and the presenceor absence of planets isp = 0.06 $\pm $0.008. For the subsample of M dwarfs that host Jupiter-mass planets, the probability that there is no correlation isp = 0.04 $\pm $0.005. Meanwhile, for the subsample of M dwarfs that hostNeptune-mass (or below) planets, we find that the probability thatthere is no correlation isp = 0.40 $\pm $0.02. Since the models ofBaraffe et al. (1998) suggest that the position of an M dwarf in the$(V-K_{\rm s})-M_{K{\rm s}}$ CMDis a qualitative indicator of metallicity, this observation tentativelysuggests that metal-rich M dwarfs are more likely to host planetsand hints that the correlation may extend to low-mass planetsas well. If this correlation is confirmed in the future,it will be in contrast to planetary systems around FGK stars,in which there appears to be only a weak connection betweenmetallicityand the presence of Neptune-mass planets.

Acknowledgements
We thank Connie Rockosi for usefulcomments and conversation and the anonymous referee for many insightfulsuggestions that improved this paper significantly. This research hasmade use of NASA's Astrophysics Data System Bibliographic Services, theExoplanet Orbit Database and the Exoplanet Data Explorer atexoplanets.org, and the NASA/IPAC Infrared Science Archive, which isoperated by the Jet Propulsion Laboratory, California Institute ofTechnology, under contract with the NationalAeronautics and Space Administration. This material is based upon worksupported under a National Science Foundation Graduate ResearchFellowship.

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All Tables

Table 1:   M dwarfs in binary systems with an FGK primary and those that host planets.

All Figures

  \begin{figure}\par\includegraphics[width=9cm,clip]{15016fg1.eps}\end{figure}Figure 1:

Optimally-smoothed residual distributions forBonfils et al. (2005a) - bon05a hereafter - andJohnson & Apps (2009) -joh09 hereafter. In both cases the vertical dashed line indicates themean of the distribution. The mean value of the bon05a residualsis 0.12 with standard deviation 0.16, while the mean value ofthe joh09 residuals is -0.12 with standard deviation 0.12.Note that the bon05a distribution has a heavy tail at large positivevalues (indicating systematically low [Fe/H] estimates) and thejoh09 distribution has a heavy tail at large negative values(indicating systematically high [Fe/H] estimates).

Open with DEXTER
In the text

  \begin{figure}\par\includegraphics[width=9cm,clip]{15016fg2.eps}\end{figure}Figure 2:

Velocity ellipsoids inferred for a volume-limited sample of early M dwarfs fromHawley et al. (1996) superimposed on theUVW velocity distribution of a volume-limited sample ($d< 20~{\rm pc}$)of Sun-like stars from the Geneva-Copenhagen Survey (gray points -Holmberg et al. 2009).The light curve denotes the one-sigma region while the heavy curve denotes the two-sigma region. Bootstrap resampling of theHolmberg et al. (2009)sample with the constraint that 68% of each bootstrap sample lieswithin the one-sigma contour and that the other 32% lies within thetwo-sigmacontour produces a volume-limited and kinematically-matched populationmetallicity of [Fe/H] = -0.14 $\pm $ 0.06.

Open with DEXTER
In the text

  \begin{figure}\par\includegraphics[width=9cm,clip]{15016fg3.eps}\end{figure}Figure 3:

Position of M dwarfs with secure metallicities from Table 1 (blue points) in the$(V-K_{\rm s})-M_{K{\rm s}}$ CMD in relation to the field M dwarf MS from joh09 (black line) and the theoretical isochrones ofBaraffe et al. (1998) (blue lines). The color of the isochrone line gives its metallicity: [Fe/H] = -0.5 andY = 0.25 on the left and [Fe/H] = 0 andY = 0.275 on the right. Both isochrones assume mixing-length parameterl = 1 for a 5 Gyr population, as there is no detectable evolution in the$(V-K_{\rm s})-M_{K{\rm s}}$CMDafter about 3 Gyr. The horizontal lines connect points of constantmass. The models indicate that differences in metallicity should bestcorrelate with horizontal shifts in the$(V-K_{\rm s})-M_{K{\rm s}}$ plane.

Open with DEXTER
In the text

  \begin{figure}\par\includegraphics[width=8.5cm,clip]{15016fg4a.eps}\hspace*{5mm}\includegraphics[width=8.5cm,clip]{15016fg4b.eps}\end{figure}Figure 4:

Left: position of M dwarfs known to host Jupiter-mass planets(dark blue triangles) and Neptune-mass (or below) planets (bluesquares) in relation to a control sample of field M dwarfs (graypoints) and the field M dwarf MS from joh09 (black line). Again,like the high-metallicity M dwarfs, the M dwarfs that hostplanets are concentrated to the right of the field M dwarf MS. Right: distribution of cumulative sample distances from the field M dwarf MS of joh09, which we assume to be a [Fe/H] $\approx $ -0.17 isometallicity contour in the$(V-K_{\rm s})-M_{K{\rm s}}$ CMD.Points to the right of the field M dwarf MS add their distance tothe sum, while points left of MS subtract their distance from the sum.We generate each distribution with a Monte Carlo simulation.First, we randomly select from the field M dwarf sample a numberof stars equal to the number of M dwarfs known to host planets ofa certain type. We then compute the cumulative horizontal distance ofthat random subsample from the field M dwarf MS. We repeat thisprocess 1000 times to generate the distribution of samplecumulative horizontal distances from the field M dwarf MS given nocorrelation between the presence of an exoplanet and the location ofits host in the$(V-K_{\rm s})-M_{K{\rm s}}$ CMD.In all cases, we confirm the findings of joh09 that theM dwarfs that host exoplanets are preferentially to the right ofthe field M dwarf MS. In particular, we find that theprobability p that there is no correlation between the location of an M dwarf in the$(V-K_{\rm s})-M_{K{\rm s}}$ CMD and its status as an exoplanet host isp = 0.06 $\pm $0.008. For the subsample that hosts Jupiter-mass planets, we find that the probability isp = 0.04 $\pm $0.005. More interestingly, we find that the probability that there is no correlation between the location in the$(V-K_{\rm s})-M_{K{\rm s}}$CMD and an M dwarf's status as the host of a Neptune-mass (or below) exoplanet isp = 0.40 $\pm $0.02. If M dwarfs to the right of the field M dwarf MS are metal-rich as suggested by theBaraffe et al. (1998)models and argued by joh09, then this observation may be evidence foran increased incidence of low-mass planets around metal-rich low-massstars, a trend which is not observed in FGK stars.

Open with DEXTER
In the text

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