How to Constrain Your M Dwarf: measuring effective temperature, bolometric luminosity, mass, and radius

8.9 MB·English

by  Andrew W. Mann

ApJ accepted
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
HOW TO CONSTRAIN YOUR M DWARF:
MEASURING EFFECTIVE TEMPERATURE, BOLOMETRIC LUMINOSITY, MASS, AND RADIUS
Andrew W. Mann,1,2 Gregory A. Feiden,3 Eric Gaidos,4,5 Tabetha Boyajian 6, Kaspar von Braun 7
ApJ accepted
ABSTRACT
Precise and accurate parameters for late-type (late K and M) dwarf stars are important for char-
acterization of any orbiting planets, but such determinations have been hampered by these stars’
complex spectra and dissimilarity to the Sun. We exploit an empirically calibrated method to estimate
5 spectroscopic effective temperature (T ) and the Stefan–Boltzmann law to determine radii of 183
eff
1 nearby K7-M7 single stars with a precision of 2-5%. Our improved stellar parameters enable us
0
to develop model-independent relations between T or absolute magnitude and radius, as well as
2 eff
between color and T . The derived T –radius relation depends strongly on [Fe/H], as predicted
eff eff
n by theory. The relation between absolute K magnitude and radius can predict radii accurate to
S
u 3%. We derive bolometric corrections to the VR I grizJHK and Gaia passbands as a function of
C C S
J (cid:39)color, accurate to 1-3%. We confront the reliability of predictions from Dartmouth stellar evolution

2 models using a Markov Chain Monte Carlo to find the values of unobservable model parameters
(mass, age) that best reproduce the observed effective temperature and bolometric flux while satisfying

] constraints on distance and metallicity as Bayesian priors. With the inferred masses we derive a
R
semi-empirical mass–absolute magnitude relation with a scatter of 2% in mass. The best-agreement
S models over-predict stellar T s by an average of 2.2% and under-predict stellar radii by 4.6%, similar
eff
. to differences with values from low-mass eclipsing binaries. These differences are not correlated with
h
metallicity, mass, or indicators of activity, suggesting issues with the underlying model assumptions
p
e.g., opacities or convective mixing length.
-
o Subject headings: stars: fundamental parameters — stars: statistics — stars: abundances — stars:
r late-type — stars: low-mass – stars: planetary systems
t
s
a
[ 1. INTRODUCTION transit surveys such as TESS (Ricker 2014) and PLATO

(Rauer et al. 2014) are expected to detect hundreds of
3 Very-low-mass stars (0.1 < M < 0.6M ), i.e., late
(cid:12) (cid:63) (cid:12) rocky planets around nearby stars, including M dwarfs.
v K and M dwarf stars, have become prime targets in the
Bright host stars make transit spectroscopy to detect
5 search for exoplanets, particularly Earth-like planets that
3 orbitincircumstellar"habitablezones"wherethelevelof the planets’ atmospheres and search for biosignatures
6 stellar irradiation and a planet’s equilibrium temperature possiblewithobservatoriessuchastheJames Webb Space
1 permitstableliquidwater. Mdwarfsdominatethestellar Telescope.
0 population, comprising more than 70% of all stars in the Estimates of planet properties depend directly on prop-
. erties of the host stars as well as observables such as
1 Galaxy (Henry et al. 1994), and thus their planets weigh
transit depth or Doppler radial velocity amplitude, thus
0 heavily on the overall occurrence of planets in the Milky
it is critical that we have accurate stellar parameters to
5 Way. Because transit and Doppler signals increase with
match the precise measurements achievable with current
1 decreasing stellar radius (R ), stellar mass (M ), and
∗ ∗ and planned instrumentation. The radius of the host
: orbital period, Earth-size planets in the habitable zone
v star R is required to establish the radius of a transiting
are much easier to detect around M dwarfs. ∗
i planet. The stellar density ρ is required to determine
X Results based on observations by the NASA Kepler ∗
the probability that a planet with a given orbital period
mission suggest that M dwarfs are teeming with rocky
r is on a transiting orbit and infer the occurrence of such
a planets, with 1 planet per low-mass star with orbital
periods of less(cid:39)than 50 days (Cassan et al. 2012; Morton planets around a set of stars targeted by a transit survey.
The mass of a planet detected with a given orbital period
& Swift 2014; Gaidos et al. 2014). Future space-based
and radial velocity reflect amplitude scales with mass of
1Harlan J. Smith Fellow, University of Texas at Austin; the star as M∗2/3. The orbit-averaged irradiance on a
[email protected] planet is proportional to stellar luminosity L and thus
2Visiting Researcher, Institute for Astrophysical Research, ∗
BostonUniversity,USA the equilibrium temperature scales as L1∗/4. Among other
3DepartmentofPhysicsandAstronomy,UppsalaUniversity, applications,thisinformationisrequiredtodetermineifa
Box516,SE-75120,Uppsala,Sweden planetfallswithina“habitablezone” boundedbyrunaway
4DepartmentofGeologyandGeophysics,UniversityofHawai’i
greenhouse conditions and the maximum greenhouse ef-
atManoa,Honolulu,HI96822,USA
5VisitingScientist,MaxPlanckInstitutfürAstronomie,Hei- fectthataCO2 atmospherecanprovide(Kopparapuetal.
delberg,Germany 2013).
6DepartmentofAstronomy,YaleUniversity,NewHaven,CT Stellarparametersarealsousefulwhencomparingplan-
06511,USA
7Lowell Observatory, 1400 W. Mars Hill Rd., Flagstaff, AZ, etary systems in search of clues to the origins of their
USA diversity. The occurrence of giant planets around FGK
2 Mann et al.
dwarfs is a strong function of metallicity (Santos et al. measured to an accuracy of (cid:46)1% using well-calibrated,
2004; Fischer & Valenti 2005), and this correlation may high signal-to-noise (S/N) spectra. Metallicities of M
also hold for M dwarfs (Johnson et al. 2010; Mann et al. dwarfs have previously eluded precise determination be-
2013c; Gaidos & Mann 2014). Giant planet occurrence cause analysis of spectra visible wavelengths is compli-
also likely scales with M (Johnson et al. 2010; Gaidos cated by confusion of many overlapping lines and the
∗
et al. 2013), and the occurrence of small, close-in planets lack of a well-defined continuum. However, the isolated
appears to increase with decreasing M (Howard et al. lines in infrared spectra, combined with calibration by
∗
2012). observationsofcommonpropermotionpairsofsolar-type
Parameters of M dwarfs can be estimated by direct starsandMdwarfs,havebeenrecentlytoestimate[Fe/H]
comparison with model predictions (e.g., Casagrande toaprecisionasgoodas0.07dex(e.g., Nevesetal.2012;
et al. 2008; Gaidos 2013; Dressing & Charbonneau 2013). Terrien et al. 2012; Mann et al. 2013b; Newton et al.
Although models of M dwarf atmospheres have continued 2014). These estimates are model-independent to the
to advance (Allard et al. 2013; Rajpurohit et al. 2013), extent that the metallicities of the solar-type calibrators
there remain important molecular bands that are poorly are derived independently of any model assumptions but
describedorcompletelyomitted. Suchanapproachisalso will nonetheless be affected by any systematic errors in
ultimatelyunsatisfactoryastheinteriorsandatmospheres the latter (Mann et al. 2013a).
of these stars are dissimilar to the Sun and we wish to Precise parameter values for calibrator M dwarfs can
test the models, not trust them. be used to construct empirical relations between param-
Studies of low-mass eclipsing binary systems (LMEBs) eters that are, in principle, applicable to fainter, more
find that stellar models systematically under-predict stel- distant stars for which distances or angular diameters
lar radii and over-predict T s by up to 4% (e.g., Kraus are currently unobtainable (Mann et al. 2013b). They
eff
et al. 2011; Feiden & Chaboyer 2012a; Spada et al. 2013). can also be used to test models of M dwarf interiors and
The leading hypothesis is that magnetic phenomena atmospheres (e.g., Boyajian et al. 2015). However, only
(magneto-convection or spots) are responsible for the a small ( 30) number of M dwarfs are near and bright
observed discrepancies, motivated by theoretical predic- enough fo(cid:39)r observations by LBOI. This restricts the cali-
tions (e.g., Mullan & MacDonald 2001) and bolstered by bration to hotter values of T and sparsely samples the
eff
observational evidence (e.g., López-Morales 2007). Fur- relevantrangeof[Fe/H].Forexample,ithasbeendifficult
therinvestigationshaveleadtocontradictoryresults,with to evaluate whether R increases with [Fe/H] for a fixed
∗
some studies suggesting magnetic fields are adequate to T , a basic prediction of models.
eff
reconcile models and observations (e.g., MacDonald & If T and F can be determined without interferome-
eff bol
Mullan2012;Feiden&Chaboyer2013;Torresetal.2014), try, Equation 1 can be inverted to estimate θ, and with
whileothersquestionthevalidityofthemagnetichypoth- a trigonometric parallax, physical radius. Because pre-
esis(e.g.,Chabrieretal.2007;Feiden&Chaboyer2014a). cise spectroscopic temperatures, bolometric fluxes, and
It is clear that, if magnetic fields are to blame, single trigonometricparallaxescanbeobtainedforamuchlarger
inactive field stars are not expected to display significant number of stars than are reachable with existing LBOI
disagreements with model predictions. However, results instrumentation, this greatly expands the potential pool
from Boyajian et al. (2012b) indicate that single inactive of calibrators. A similar method called Multiple Optical-
fields stars may in fact show similar disagreements. Infrared Technique (MOITE) has been used to determine
Recentobservationaladvanceshaveledtomoreprecise, the radii of M dwarfs (Casagrande et al. 2008). However,
model-independent estimates of some M dwarf parame- that method is sensitive to the model temperatures and
ters. Long-baseline optical interferometry (LBOI) has photometric zero points.
beenusedtomeasuretheangulardiameters(θ),andwith In this work we present estimates of F , T and
bol eff
trigonometricparallaxes,physicaldiameters,ofsomevery [Fe/H] for 183 bright nearby M dwarfs with precise
nearby, bright M dwarfs with uncertainties of 1-5% (e.g., trigonometric parallaxes, which we use to determine R .
∗
Boyajian et al. 2012b; von Braun et al. 2014). The angu- We construct improved empirical relations between these
lar diameter plus the parallax yields the physical stellar parameters and confront the predictions of models of
radius. The angular diameter plus an estimate of the M dwarfs with these values. In Section 2 we define the
star’s bolometric flux (F ) yields a direct determination sample. We describe our spectroscopic observations in
bol
of T via the Stefan–Boltzmann law: Section 3 and how we turn spectra into determinations
eff
of F , T , [Fe/H], and R in Section 4. We compare
T =2341(cid:18)Fbol(cid:19)1/4, (1) our beosltimeafftes to values ava∗ilable from the literature in
eff θ2 Section 5. Using our derived M dwarf parameters, we de-
velop empirical relations between observables (i.e., color,
where F is in units of 10−8ergs−1cm−2. These esti- T , and M ) and fundamental parameters (R , L )
bol eff KS ∗ bol
mates are thus model-independent (with the exception inSection6,andprovideempiricalbolometriccorrections
of small limb darkening corrections to angular diameters) in Section 7. We compare predictions from the Dart-
and are applicable to single stars (close binaries being mouth stellar evolutionary model (Dotter et al. 2008) to
obvious in interferometric data). our values in Section 8, including exploring the impact of
Effective temperatures derived from fits to model grids changes in the model physics. We conclude in Section 9
can be compared to these bolometrically derived temper- with a summary of our work, a brief discussion of some
atures to calibrate or correct the former to within 60 K of the issues with our analysis, and potential applica-
(Mann et al. 2013b). Likewise, improvements in the ac- tionsofourmethodswiththeadventofextremelyprecise
curacy of photometric zero points and system passbands trigonometric parallaxes form the ESO Gaia astrometric
(Mann & von Braun 2015) allow bolometric fluxes to be mission.
Properties of M Dwarfs 3
2. SAMPLESELECTION our derived parameters as a function of contrast ratio or
spectral type. We scaled all spectra in our sample so that
M dwarf stars were selected from the CONCH-SHELL
they are at the same distance (10pc, but the exact value
(Gaidos et al. 2014) or LG11 (Lépine & Gaidos 2011)
isarbitrary)thenaddedrandompairsofspectratogether,
catalogs. Wefirstselectedstarswithparallaxerrors<5%.
calculating F before and after addition (Figure 1). We
TrigonometricparallaxesweretakenfromHarringtonetal. bol
removedanytargetwithacompanionthatwouldincrease
(1993),Jaoetal.(2005),Henryetal.(2006),vanLeeuwen
F by more than 8%. This cut was selected because an
(2007), Gatewood (2008), Gatewood & Coban (2009), bol
8%increaseinF resultsina2%increaseinderivedR ,
Lépineetal.(2009),Riedeletal.(2010),Jaoetal.(2011), bol ∗
which is still less than our typical measurement errors.
and Dittmann et al. (2014). For targets with multiple
This binarity criterion removed 41 targets, or 18% of the
parallax measurements we adopted the distance from
sample.
the most precise measurement. This selection yielded
Using the binary period distribution from Raghavan
923 stars, about two thirds of which are visible to our
et al. (2010) and the overall binarity rate for M dwarfs
instruments/telescopes (Section 3). We prioritized our
from Fischer & Marcy (1992) we estimated that 17%
spectroscopic observations for targets based on: (i) error
of our sample contains binaries that are tight enou(cid:39)gh to
in stellar parallax, (ii) brightness, and (iii) estimated
be unresolved in our photometry/spectroscopy and not
spectral type or color (since late-type stars tend to be
resolved in SpeX/SNIFS finder images, but could have
removed by the first two criteria. Existing optical and/or
increased the derived F by (cid:38)8%. The actual binarity
near-infrared (NIR) spectra (Lépine et al. 2013; Mann bol
rate may differ in our sample because it is not volume-
etal.2013b;Gaidosetal.2014)wereusedwhereavailable.
limited while the binary fraction estimates are. However,
Ourprioritizationcoupledwithavailableobservingtime
the expected binary fraction is consistent with or lower
yielded 191 stars. To this we added an additional 36 M
dwarfs in wide ((cid:38)5(cid:48)(cid:48)) binaries from Mann et al. (2013a) than the fraction of targets that were removed because
theyareknownbinaries(18%),suggestingbinarycontam-
andMannetal.(2014). TheseMdwarfsdidnotmakethe
ination will not be a significant problem for our analysis.
original cut because they do not have trigonometric par-
Most of our targets are close, and have been studied ex-
allaxes or a sufficiently precise trigonometric parallaxes.
tensively for companions (e.g., Janson et al. 2012, 2014;
However, the their primaries have parallaxes, which we
Ansdell et al. 2015), and hence the agreement between
use to calculate the distance to the M dwarf companion.
the estimated and observed binarity is not surprising.
WealsoaddedthreeMdwarfswithLBOIθmeasurements
that were not in the LG11 or CONCH-SHELL catalogs.
We then removed three stars because of insufficient (<5
points) photometry to absolutely calibrate the spectrum 1.0
(see Section 3.3) or because there was poor agreement
F
between photometry and spectra, i.e., reduced χ2 > 10 Fbol0.8 ra
(χ2ν). e in 0.15 ction
s a
2.1. Binary Contamination ea 0.6 l C
Unresolved binaries are problematic for our analysis. ncr 0.10 ha
Unlike photometric variability, which changes the pho- al I 0.4 nge
tometry and any derived stellar properties randomly, an on in
ucrneraesseolFvebdol,aanndduhnecnocrerescytesdtemcoamtipcaalnlyionbiawsiltlhaelwraaydsiiionf- Fracti 0.2 0.05 Rad
the overall sample. It has been estimated that 40% iu
s
of M dwarfs in the solar neighborhood have at lea(cid:39)st one 0.0 0.00
companion within 1000 AU (Fischer & Marcy 1992; Lada 1 2 3 4
2006; Raghavan et al. 2010). The median distance to our Delta-K (mag)
targets is 10 pc, so the majority of these were sufficiently
separated (>5(cid:48)(cid:48) or >50 AU) such that the photometry Figure 1. The effect on the derived Fbol and R∗ from an unre-
solvedbinaryasafunctionoftheK-magnitudecontrastratio. The
and spectroscopy is unaffected. We flagged all binaries
dashed line indicates the maximum change in F (8%) from a
bol
that are resolved in the spectrographs finder cameras binary,abovewhichweremovethetargetfromoursample. The
(Section 3), which is capable of detecting companions scatter in the locus of points is larger than expected from mea-
with ∆K (cid:46)2 and as close as 1 2(cid:48)(cid:48) to the primary, de- surement errors because we are combining spectra of of a range
pending on conditions. Systems−with high contrast ratios ofmetallicities,andthereforehavevaryingFbol’sforagivenMK.
MoredetailscanbefoundinSection2.1.
(∆K 2) which were more likely to be missed have a
neglig(cid:29)ible effect on the F (see below).
bol
To identify tighter binaries we crosschecked our sample The final sample contains 183 stars with spectral types
withtheFourthCatalogofInterferometricMeasurements ranging from K7 to M7 (median spectral type of M3),
of Binary Stars (Hartkopf et al. 2001), the Washington [Fe/H] from -0.60 to +0.53 (median [Fe/H] of -0.03), and
Double Star Catalog (Mason et al. 2001), and SIMBAD. distances from 1.8pc (Barnard’s Star) to 38pc (median
Each of these catalogs reports heterogeneous information, distance of 11pc).
e.g., contrast ratios are not always given in the same
filter and sometimes just spectral types are given rather 3. OBSERVATIONSANDREDUCTION
than flux differences. To deal with this heterogeneity,
3.1. Optical Spectra from SNIFS and STIS
we use our sample to estimate the impact of binarity on
4 Mann et al.
Optical spectra of all stars were obtained with the Su- are fit variables that also depend on wavelength (λ). We
perNova Integral Field Spectrograph (SNIFS, Aldering compared this model derived from all photometric nights
etal.2002;Lantzetal.2004)ontheUniversityofHawai’i to a correction derived using just standards taken in a
2.2m telescope on Mauna Kea. SNIFS provides simulta- single night. We found that the differences between these
neous coverage from 3200–9700Å by splitting the beam corrections is only 1% in the red and (cid:46)2% in the blue,
with a dichroic mirror onto blue (3200-5200Å) and red excluding regions (cid:39)of significant telluric contamination.
(5100-9700Å) spectrograph channels. Although the spec- For a more accurate flux calibration we took the median
tralresolutionofSNIFSisonlyR 1000, theinstrument derived correction from all standard stars observed in a
provides excellent spectrophotom(cid:39)etric precision (Mann single night (without an airmass fit), which we used to
et al. 2011, 2013b). This is in part a consequence of the derive an additional wavelength dependent adjustment
stability of the atmosphere over Mauna Kea (Hill et al. weappliedtoκ(inadditiontoEquation2). Thefinalcor-
1994; Buton et al. 2013) and because the instrument is rectionwasthenappliedtoallstarsfromthatnight. This
an integral field spectrograph and does not suffer from assumesthatthatwhiletheatmospherictransparency(a)
wavelength dependent slit losses that can be difficult to varies between nights, the effect of airmass (b, c4) does
accurately remove. not.
Spectra were obtained between 2010 and 2014 under Five of our stars are included in the Next Genera-
varying conditions, although most nights were photo- tion Spectral Library (NGSL, Gregg et al. 2006; Heap &
metric or nearly photometric (few to zero cloud cover). Lindler 2007), which consists of spectra obtained with
Integration times varied from 2 to 600 s, yielding S/N the Space Telescope Imaging Spectrograph (STIS) on the
> 100 per resolution element in the red channel for all Hubble Space Telescope (HST) using the G230LB, G430L,
targets, while avoiding the non-linear regime of the detec- and G750L gratings. NGSL spectra cover 2000-10000Å
tor. For exposure times < 5s, multiple exposures were withspectrophotometricprecisionof 0.5%(Bohlinetal.
takenandstackedafterreductiontosuppressscintillation 2001; Bohlin & Gilliland 2004a,b). (cid:39)We used the NGSL
noise. instead of the SNIFS data for these five stars.
SNIFS data reduction was broken into two parts: ini- Totestthequalityofourspectrophotometriccalibration
tial reduction from the SuperNova Factory (SNF) team wecomparedourSNIFSdatatotheNGSLspectraforthe
pipeline, which was performed concurrently with all ob- fiveoverlappingtargets. Wefoundthatwecanreproduce
servations,andamoreprecisefluxcalibrationandtelluric theHSTspectrawithascatter(1σ)of1.1%intheredand
correction, which we applied to the SNF pipeline reduced 1.5% in the blue, with median offsets of 0.1% and 0.3%
data. Detailed information on the SNF pipeline can be respectively. We added this estimate of the systematic
found in Bacon et al. (2001) and Aldering et al. (2006). error in quadrature to the formal errors in each spectrum
To summarize, the SNF pipeline performed dark, bias, for our error analysis of F (Section 4).
bol
and flat-field corrections, and cleaned the data of bad
3.2. Near-infrared (NIR) Spectra from SpeX
pixels and cosmic rays. It then fit and extracted each of
spectra from the 15 15 "spaxels" in the red and blue Between 2012 and 2014 we obtained NIR spectra of all
channels and convert×ed these into spectral-spatial image stars using the SpeX spectrograph (Rayner et al. 2003)
cubes. The pipeline applied a wavelength calibration attached to the NASA Infrared Telescope Facility (IRTF)
to the cubes based on arc lamp exposures taken at the on Mauna Kea. As with the SNIFS observations, data
sametelescopepointingandtimeasthesciencedata,and were taken under mixed conditions, but most nights were
a rough flux calibration based on an archive correction. photometricornear-photometric. SpeXobservationswere
Lastly, the pipeline extracted one-dimensional spectra taken in the short cross-dispersed (SXD) mode using the
from each of the cubes using an analytic model of the 0.3 15(cid:48)(cid:48) slit, yielding simultaneous coverage from 0.8 to
point spread function. 2.4µ×m withasmallgapat 1.8µm,andataresolutionof
Our part of the reduction is described in some detail R 2000. Each target was(cid:39)placed at two positions along
in Lépine et al. (2013) and Gaidos et al. (2014). We the(cid:39)slit (A and B) and observed in an ABBA pattern
restate most of the process here as there has been some in order to subsequently subtract the sky background.
improvements in our data reduction since the publication For each star we took 6-10 exposures following this pat-
of those papers. Over several years of observing with tern, which, when stacked, provided a S/N per resolution
SNIFS we collected more than 400 observations of 42 element of > 100, and typically > 150 in the H- and
spectrophotometric standards8. We first identified all K-bands. In2014JulytheSpeXchipwasupgraded(now
standard star observations taken on photometric nights called uSpeX), which resulted in better wavelength cover-
based on extinction estimates from the CFHT skyprobe9. age(0.7-2.5µm), includingcoverageoftheregionbetween
We divided the "true" spectrum of each standard star, as the H and K bands that SpeX lacked. In total, 18 of 183
taken from the literature (Oke 1990; Hamuy et al. 1994; stars were observed with uSpeX. Observation procedures
Bessell 1999; Bohlin et al. 2001), by the corresponding and reduction were nearly identical for the two detectors.
spectrafromtheSNFpipeline. Wethenfoundthebest-fit SpeXanduSpeXspectrawereextractedusingtheSpeX-
correction as a function of airmass: Tool package (Cushing et al. 2004), which performed flat-
field correction, wavelength calibration, sky subtraction,
κ(λ)=a(λ)+b(λ)z+c(λ)z2, (2)
andextractionoftheone-dimensionalspectrum. Multiple
where κ is the true atmospheric (including telluric) and exposures were combined using the IDL routine xcombx-
instrumental correction, z is the airmass, and a, b, and c pec. Tocorrectfortelluriclines,weobservedanA0V-type
star within 1 hr and 0.1 airmass of the target observation
8 Fulllistathttp://snfactory.lbl.gov/snf/snf-specstars.html (usually much closer in time and airmass). A telluric
9 http://www.cfht.hawaii.edu/Instruments/Skyprobe/ correction spectrum was constructed from each A0V star
Properties of M Dwarfs 5
and applied to the relevant spectrum using the xtellcor the formula:
package(Vaccaetal.2003). Separateorderswerestacked (cid:82)
using the xcombspec tool, which also shifts the flux level f (λ)S (λ)dλ
λ x
in each orders to match. These corrections are generally fspec,x = (cid:82) , (3)
1% or less per order. S (λ)dλ
x
Rayner et al. (2009) noted that using the 0.3(cid:48)(cid:48) slit can
leadtoerrorsintheslopeofthespectrumby1-3%dueto where fλ(λ) is the spectrum (radiative flux density) as
changes in seeing, guiding, and differential atmospheric a function of wavelength (λ) and Sx(λ) is the system
refraction between target and standard star observation. throughput the filter x (e.g., U, B, V) in energy units.
We reduced the impact of this problem by observing at As with the zero points, filter profiles were taken from
the parallactic angle and minimizing the time between Cohenetal.(2003),Jarrettetal.(2011),andMann&von
target and standard star observations. We compared the Braun(2015). Wecalculatedtheratioofthesyntheticflux
slopes of unstacked observations of the same star and tothatderivedfromthephotometryforeachbandaswell
found that slope errors are 1 2%, which is backed up as corresponding uncertainties accounting for errors in
by a comparison JHK magn−itudes from the literature the spectrum, photometry, and photometric zero points.
(Mann & von Braun 2015). This source of error was Excluding the five stars with STIS spectra, each of
included in all analyzes. our stars had a spectrum composed of three components
from the SNIFS blue channel, the SNIFS red channel,
3.3. Absolute Flux Calibration of Spectra andSpeX.Theblueandredchannelswererelativelyflux-
SpeX spectra have a gap at 1.8µm between the H calibrated independently from our reduction explained in
and K bands, although there is(cid:39)no gap for uSpeX data. Section 3.1. However, repeat measurements and compari-
Also, several regions in the NIR are strongly affected by son with spectra from other instruments suggested that
telluric absorption. Although our observations of A0V therelativefluxcalibrationbetweenthesetwochannelsis
starsenabledustocorrectfortelluricabsorption,theS/N only accurate to 3%, while the calibration is accurate to
in some regions was too low for accurate reconstruction 1% within a channel. The overlapping region covers only
of the spectra. We replaced these regions with the best- 50Å andthisspectralregionoftheinstrumenthaslittle
fit atmospheric model from the BT-SETTL grid (Allard (cid:39)throughput and hence low S/N. For the SpeX to SNIFS
et al. 2011, 2013). We did the same for 2.4–10µm. Our transition there is significantly more overlap (> 600Å),
method for finding the best model spectrum is explained but this region is complicated by telluric H O absorption,
2
inSection4. Asatest,weexaminedthespectraofthe12 low throughput for SpeX, and small but significant shape
targets with S/N >500 in the NIR, and found that using errors ((cid:46) 1% in the optical and 2% in the NIR) above
a model to replace the regions of telluric contamination the Poisson and measurement errors (typically 1%).
changes the derived F by 0.15 0.26%, and hence adds WeusedtheoverlappingregionbetweenSNIFSan(cid:28)dSpeX
bol
negligible error. ± (excluding 50Å on each end) data to calculate an offset
Blueward of our cutoff at 0.32µm (0.2µm for targets and error, but the result is dominated by errors in the
with STIS spectra) we assumed the spectrum follows shape of each spectral component.
Wein’s approximation and redward of 10µm we assumed The absolute fluxes of the three components of each
it follows the Rayleigh-Jeans law. We fit these functions spectrum were adjusted to minimize the difference (in
using the data at 0.3–0.4µm and 2.0–10µm, respectively. standard deviations) between SNIFS and SpeX overlap-
Flux in each of these regions represented 1% of the ping regions, the difference between photometric and
total flux from a typical star, and compariso(cid:28)n to photom- spectroscopic fluxes (as described above), and the in-
etry from the Wide-field Infrared Survey Explorer (WISE, dependent flux calibrations of the SNIFS red and blue
Wright et al. 2010) and UV spectra from STIS (Heap & channels. This renormalization process served to both
Lindler 2007) indicated that these approximations are combine the three components and absolutely flux cali-
negligible compared to other measurement errors. brate each complete spectrum. For the targets with STIS
Foreachstarweretrieved(whereavailable)JHK pho- datathereisnoseparatebluechannel,sotheanalysishad
S
tometry from the Two-Micron All-Sky Survey (2MASS, one fewer free parameter, but otherwise the method was
Skrutskie et al. 2006), BV photometry from the AAVSO the same. We show combined and absolutely calibrated
All-SkyPhotometricSurvey(APASS,Hendenetal.2012), spectra of four representative stars (Gl 699, Gl 876, Gl
H from Hipparcos (van Leeuwen 2007), V and B 880, and Gl 411) in Figure 2. These four are highlighted
P T T
from Tycho-2 (Høg et al. 2000), W1 and W2 from WISE because they span a wide range of parameter space in
(Wrightetal.2010),andallphotometryfromtheGeneral terms of stellar mass and metallicity; two are below the
Catalog of Photometric Data (Mermilliod et al. 1997). fullyconvectiveboundary(Gl699andGl876),twoabove
We discarded APASS data for stars brighter than V =10 (Gl411andGl880), twometal-poor(Gl699andGl411),
where images were potentially saturated. Much of the and two metal-rich (Gl 876 and Gl 880).
WISE photometry was saturated or had measurement The median reduced χ2 (χ2) value for our fits was 1.5,
ν
quality flags and hence was discarded. Photometric mag- and the highest was 7.0 (for Gl 725B). Many of the χ2
ν
nitudes were converted to fluxes using the zero points values >1 may be due stellar variability, which was not
from Cohen et al. (2003) for 2MASS, Jarrett et al. (2011) accounted for in our error analysis. Conversely, the low
for WISE, and Mann & von Braun (2015) for all other χ2 ofthesamplesuggeststhateitherthenumberofhighly
ν
photometry. The number of photometric points per star variablestarsinthesampleisrelativelylow,orthaterrors
varied from 5 to 179, with a median of 22. are somewhat overestimated.
We calculated fluxes from the spectrum (synthetic pho-
tometry)correspondingtotheavailablephotometryusing 4. DERIVATIONOFSTELLARPARAMETERS
6 Mann et al.
3.0
Data Data
−1Å) 0.8 BT−SEPThT o L t o M m o e d t re y l −1Å) 2.5 BT−SEPThT o L t o M m o e d t re y l
−1s Synthetic Photometry −1s Synthetic Photometry
−2m 0.6 Gl 411 −2m 2.0 Gl 699
c c
g g 1.5
er er
−110 0.4 −120 1.0
1 1
ux ( 0.2 ux ( 0.5
Fl Fl
)σ 3 )σ 3
al ( al (
u 0 u 0
d d
si si
e −3 e −3
R R
1 2 3 4 1 2 3 4
Wavelength (µm) Wavelength (µm)
3.0
Data 1.5 Data
−1Å) 2.5 BT−SEPThT o L t o M m o e d t re y l −1Å) BT−SEPThT o L t o M m o e d t re y l
−1s Synthetic Photometry −1s Synthetic Photometry
−2m 2.0 Gl 880 −2m 1.0 Gl 876
c c
g 1.5 g
er er
−120 1.0 −120 0.5
1 1
x ( x (
u 0.5 u
Fl Fl
)σ 3 )σ 3
al ( al (
u 0 u 0
d d
si si
e −3 e −3
R R
1 2 3 4 1 2 3 4
Wavelength (µm) Wavelength (µm)
Figure 2. Combinedandabsolutelyflux-calibratedspectraoffourrepresentativestarsinoursample. Thespectraareshowninblack,
withregionsreplacedbymodelsingray. Photometryisshowninred,withthehorizontal‘errorbars’indicatingthewidthofthefilter,and
verticalerrorsrepresentingcombinedmeasurementandzeropointerrors. Bluepointsindicatethecorrespondingsyntheticfluxescalculated
using Equation 3. Residuals are plotted in the bottom subpanels in units of standard deviations. More details on observations can be
foundinSection3andabsolutefluxcalibrationinSection3.3. Thesestarsareselectedbecausetheyroughlyspantheparameterspacein
metallicityandmassofourwholesample.
Using the absolutely flux-calibrated spectra (Sec- (2003). We then applied the empirical relations between
tion 3.3) and trigonometric parallaxes from the litera- these indices and stellar spectral type provided by Lépine
ture (Section 2) we calculated F , T , [Fe/H], R , M , etal.(2013). Theassignedspectraltypewastheweighted
bol eff ∗ ∗
synthetic photometry (Equation 3), and spectral types (by the measurement error of the index) mean of spectral
for all targets in our sample. For these calculations we types from each empirical relation (six in total, but VO1
ignoredinterstellarextinction,asallstarsarewithin40pc andVO2wereonlyusedifthestarislaterthanM3). This
and Aumer & Binney (2009) showed that reddening is method gives consistent spectral types to those assigned
0 for stars within 70pc due to the Local Bubble. Al- by-eyeandhasanaccuracyof 0.3subtypes(Lépineetal.
(cid:39)lowed reddening to float had negligible effects on the F 2013). The use of indices inst±ead of template matching
bol
determinations, reinforcing this assumption. We also as- allowedustoassignfractionalsubtypes, whichisjustified
sumedeverystarissingle,oraresolvablebinary,i.e.,each given the uncertainties.
spectrum and photometric measurement is from just one
object (also see Section 2.1). Details of our calculations 4.2. Bolometric Flux
follow. Values and formal errors for all stellar parameters The bolometric flux (F ) was calculated by integrat-
bol
are reported in Table 5, with the synthetic photometry ing over the radiative flux density (the spectrum). Errors
given in Table 6. on F were calculated from Monte Carlo randomization
bol
of our flux calibration. Specifically, we randomly varied
4.1. Spectral Type
the photometry, spectra, and photometry zero points ac-
Spectraltypesweredeterminedfromouropticalspectra cording to their estimated errors. We then repeated the
following the procedure from Lépine et al. (2013). We process described in Section 3.3 10,000 times, recalcu-
calculated CaH2, CaH3, TiO5, TiO6, VO1, and VO2 lating F each time. We used the standard deviation
bol
band indices for each of the optical spectra following the of these values as the error on F . While many of our
bol
definitions given in Reid et al. (1995) and Lépine et al. starshaveS/N 100and 20photometricpoints,other
(cid:29) (cid:29)
Properties of M Dwarfs 7
sources of error (e.g., flux calibration, photometric zero Hinkel et al. 2014). We conservatively adopt the larger
points) are typically 0.5-2% and usually dominate the value as the calibration error, which, because the S/N
error budget. As a result no star has a F error <0.5%, of our spectra is generally very high, dominate the total
bol
and typically the error is (cid:38)1%. error in [Fe/H].
Due to slight modifications in our procedure for com- For17ofthewidewidebinariesamongourcurrentsam-
bining optical and NIR spectra, updates to our SNIFS ple we adopted the metallicity reported for the primary.
reduction pipeline, and changes in the photometry zero These metallicities are more precise (errors typically 0.03-
points and filter profiles (Mann & von Braun 2015), our 0.05 dex) than those derived from our NIR spectra for
F values are slightly different from those presented in the M dwarf.
bol
Mann et al. (2013b) for the interferometry stars. How- Metallicities reported here differ slightly from those in
ever, these differences are almost all <1σ. Further, our Gaidos et al. (2014). Gaidos et al. (2014) based their
F values are in good agreement with those determined [Fe/H] values on the optical metallicity calibration of
bol
using the (similar) procedure outlined in van Belle et al. Mann et al. (2013a), while ours are based on the NIR
(2008) and von Braun et al. (2011) provided the same calibration. The optical calibration gives [Fe/H] accurate
photometric zero points are used. to 0.10–0.13 dex, depending on spectral type, while the
NIR calibration is good to 0.08 dex. For the stars with
4.3. Effective Temperature [Fe/H] measurements in both samples the standard devi-
ation of the differences is 0.10 dex with a mean offset of
T was calculated by comparing our optical spectra
witheffthe CFIST suite10 of the BT-SETTL version of 0.017 0.007, consistent with the expected errors.
±
the PHOENIX atmosphere models (Allard et al. 2013).
4.5. Synthetic Photometry
More details of this procedure are given in Mann et al.
We synthesized VR I JHK photometry using the
(2013b) and Gaidos et al. (2014), although we modified C C S
absolutely flux-calibrated spectra using Equation 3 and
theirmethodsslightly. UnlikeGaidosetal.(2014)wedid
procedure from Section 3.3. We generated SDSS griz
not fit and remove a polynomial trend in wavelength to
magnitudes using the zero point and filter profiles from
correct for slit losses, as the flux calibration of SNIFS is
theSloanDigitalSkySurvey11 (Fukugitaetal.1996). We
better than many of the instruments used in the Gaidos
alsocalculatedGaiaG,G ,andG magnitudes(Jordi
et al. (2014) study. Like Gaidos et al. (2014) we used BP RP
et al. 2010). The Gaia system profiles were provided by
linearcombinationsofthethreebestmodelstointerpolate
C. Bailer-Jones (2015, personal communication), and the
between grid points, while Mann et al. (2013b) used only
zero point was estimated using a calibrated STIS spec-
two. Mann et al. (2013b) used [Fe/H] to restrict which
trum of Vega (Bohlin 2007) and assuming G =0.03.
models are allowed in the fit, while this work and Gaidos Vega
Readers are cautioned that the system throughput we
et al. (2014) added [Fe/H] as a term in the equation
for χ2. These differences resulted in almost negligible used for Gaia passbands may prove to be inaccurate once
the mission does its own on-sky calibrations. Hence our
changes to the overall sample T (median change of 8 K)
eff derived magnitudes could require corrections for small
between the three papers, although several stars changed
color terms or zero point errors when real Gaia magni-
by >100K.
tudes become available.
Errors in T were calculated as the quadrature sum of
eff Generatingsyntheticphotometryensuredthatwehada
the calibration error found by Mann et al. (2013b) and
homogeneoussetofallrelevantmagnitudesforeverystar.
the scatter in T from the model comparison. Generally
eff This also provided us with SDSS griz magnitudes, which
thefirsttermdominates, resultingintypicalerrorsinT
eff are not available for nearly all the stars in our sample.
of 60K.
(cid:39) Further, in many cases our synthetic photometry was
superiortotheliteraturephotometryintermsofprecision
4.4. Metallicity
andaccuracy. Whilealmostallstarshave2MASSJHK
S
Metallicities were calculated from the NIR spectra us- magnitudes, many of the objects are brighter than the
ing the empirical relations from Mann et al. (2013a) for linearity/saturation limit, and hence the 2MASS JHK
S
K7-M4.5dwarfsandfromMannetal.(2014)forM4.5-M7 magnitudes are unreliable.
dwarfs. Mann et al. (2013a, 2014) provide relations be-
tween the equivalent widths of atomic features (e.g., Na, 4.6. Mass
Ca) in NIR spectra and the metallicity of the star, cali-
Masses for each target were derived using the empirical
brated using wide binaries with an FGK primary and an
mass-luminosity (M –mass) relation in Delfosse et al.
K
M dwarf companion. Errors on [Fe/H] were calculated by
(2000), which was based on mass measurements of M
adding(inquadrature)measurementerrorsinthespectra
dwarf eclipsing and astrometric binaries. We converted
andscatterintheempiricalcalibration. Comparisonwith
oursyntheticK magnitudestoCITK magnitudesusing
S
other methods of measuring M dwarf metallicities (e.g.,
the corrections in Carpenter (2001), i.e., a shift of
Terrien et al. 2012; Neves et al. 2013; Newton et al. 2014) 0.02mag. Based on the scatter between the relation an(cid:39)d
suggests that our internal errors are only 0.04-0.06 dex
the binaries in Delfosse et al. (2000) we assumed 10%
(Gaidos & Mann 2014), which is significantly lower than
errors on our masses derived this way.
theerrorestimatedbyMannetal.(2013a)andMannetal.
(2014). This might be due to overlap in the calibration 4.7. Angular Diameter and Physical Radius
sample of the aforementioned studies, or underestimated
errors in the FGK star metallicities (Torres et al. 2012;
11 We followed the procedure outlined at
http://classic.sdss.org/dr7/algorithms/fluxcal.html, so these
10 http://phoenix.ens-lyon.fr/Grids/BT-Settl/CIFIST2011/ aretrueSDSSmagnitudes.
8 Mann et al.
Angular diameters were calculated from T and F
eff bol
using Equation 1. Radii were then determined from θ 4200
+0.4
and the trigonometric parallax for each star. Errors on 4000
R∗ and θ were estimated by Monte Carlo simulation. K) 3800
We randomly perturbed the Teff and Fbol according to BI ( 3600 +0.2
estimated errors (see above), and the parallax according O
L 3400
taollerrerloervsarnetppoartreadmientetrhsefloitreeraatcuhres.taWr.eWtheenrerpeecaatlceudlattheids T eff3200 +0.0Fe/H]
[
process 10,000 times. We adopted the standard deviation 3000
-0.2
of these 10,000 iterations as the error in each parameter.
2800
Therederivedvaluesshowsignificantcorrelationsbetween 200
al 100
parameters (e.g., Teff and R∗), which are accounted for du 0 -0.4
by the MC error estimate. esi -100
R
Errors on R are typically 3 4%, although they -200
∗
tendtobelargerwithdecreasin(cid:39)gR−. OurF valuesare 2800 3000 3200 3400 3600 3800 4000 4200
∗ bol T This Work (K)
generally very precise (1–2%) as are our constraints on eff
T ( 2%). However, because θ F 1/2 and θ T −2,
eff bol eff
errors(cid:39)in θ (and hence R ) are pr∝imarily driven b∝y errors
∗ 1.6
in T . Trigonometric parallax errors are typically 2%, +0.4
eff
and hence also noticeably contribute to errors in R . 1.4
∗ s)
ma 1.2 +0.2
5. COMPARISOPNAWRAITMHEPTREERVVIOAULUSLEYSPUBLISHED BI ( 1.0
O
5.1. Interferometric Teff and θ θ L 0.8 +0.0Fe/H]
[
Twenty-nine stars in our sample have interferometric 0.6
radiiinLaneetal.(2001),Ségransanetal.(2003),Berger ual 0.4 -0.2
d
vetonalB.r(a2u00n6e)t,aKl.er(v2e0l1la1)e,tvaoln. B(2r0a0u8n),eDtaelm.(o2r0y1e2t),aBl.o(y2a0j0ia9n), Resi 00..0150 -0.4
et al. (2012b), von Braun et al. (2014), or Boyajian et al. nal -00..0005
o -0.10
(2015, in preparation). We derived radii for these stars in cti
thesamemanneraswasdoneforeveryotherstar,i.e.,we Fra 0.4 0.6 0θ.8 This W1.0ork (m1a.2s) 1.4 1.6
took SNIFS and SpeX spectra of these stars, absolutely
flux-calibrated the spectra, measured the temperature by
fitting to atmospheric models, and then derived θ and Figure 3. ComparisonofTeff(left)andangulardiameter(θ,right)
determined through our methods (Section 4) compared to those
R . We also recalculated interferometric T for these
∗ eff determined from interferometry (LBOI). Bottom panels in both
starsusingtheLBOIθ measurementsandourF values
bol plotsshowtheresiduals. Allpointsarecolor-codedbymetallicity.
usingEquation1(althoughourconclusionsdonotchange DetailsareinSection5.1.
significantly if we use the T and F values from the
eff bol
original reference). dius and the LBOI T . However, F would need to be
eff bol
Agreement between the T and θ determinations is ex- increased by (cid:38) 20% to reach 1σ agreement between
cellent(Figure3). ThemeaneffdifferenceinT is20 11K the two determinations (althou≤gh increased by just 4%
eff
and the χ2 is 0.90. Similarly, the mean difference±in θ is increase to achieve agreement at 3σ). Variability cannot
ν
1.4 0.7% with a χ2 of 1.01. Our method for measuring explain the difference; the scatter in V-band measure-
T ±was tuned (choνice of atmospheric model and spectral ments for Gl 725B is (cid:46) 0.02% and the LBOI visibility
eff
regionsutilized)tomatchLBOImeasurements(seeMann curve shows no sign of multiplicity.
et al. 2013b, for more details), so consistency is not sur-
5.2. M -R from Low-mass Eclipsing Binaries
prising. However, Mann et al. (2013b) had only 18 LBOI ∗ ∗
stars in this T range, while the expanded sample used We compared our derived masses and radii to those
eff
here includes 29 LBOI stars, so this comparison is still from low-mass eclipsing binaries (LMEBs). Since unre-
demonstrative. Thenear-unityχ2 valuessuggestthatour solved binaries hamper our F and T estimates, and
ν bol eff
assigned errors are reasonable. Importantly, there is no most LMEBs do not have trigonometric parallaxes (or
statically significant correlation between the R residuals sufficiently precise parallaxes) we did not make a direct
∗
and[Fe/H],orT ,suggestingthatourR determinations comparison as we did with the interferometry targets.
eff ∗
are clean of systematic errors. Instead we could only see if the trends in the data are
The largest discrepancy between our values and those in agreement. We compare with a collection of detached,
from LBOI is for the star Gl 725B. Our T was 3.6σ double-line eclipsing binaries with mass and radius er-
eff
warmer, and our radius was 3.4σ smaller than LBOI es- rors less than 5% in Figure 4. LMEB parameters are
timates. Assuming the interferometric parameters are taken from Torres & Ribas (2002), Ribas (2003). López-
correct, Gl 725B has a significantly larger radius than Morales & Ribas (2005), Lopez-Morales et al. (2006),
stars of similar T . Photometry is also in relatively poor López-Morales & Shaw (2007), Morales et al. (2009b),
eff
agreement with the spectrum compared to most stars Irwin et al. (2009), Morales et al. (2009a), Carter et al.
(χ2 7, see Section 3.3). Photometry for this star may (2011), Irwin et al. (2011), Kraus et al. (2011), Doyle
beνi(cid:39)naccurate, which would have affected our derived ra- et al. (2011), Orosz et al. (2012a), Orosz et al. (2012b),
Properties of M Dwarfs 9
4000
TThhiiss wwoorrkk
0.6 LLMMEEBBss +0.4 +0.4
3800
us (R)Sun00..45 +0.2 grande (K) 33460000 +0.2
Radi 00..23 Fit to our sample -+00..20[Fe/H] T Casaeff3200 +0.0[Fe/H]
al 0.1 Fit to LMEBs 3000 -0.2
u
sid 0.05 -0.4 400
Re 0.00 al 300
c -0.05 du 200 -0.4
Fra 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Resi 1000
Mass (M ) -100
Sun
3000 3200 3400 3600 3800 4000
T this work (K)
Figure 4. Mass–radiusdiagramforstarsinoursample(redcircles) eff
andthosefromlow-masseclipsingbinaries(LMEBs,bluestars). A
typicalerrorbaronourmeasurementsisshowntotheleft. Stars
inoursamplearecolor-codedbytheirmetallicity. Thefittoboth
11 +0.4
samples is shown as a dashed line. The bottom panel shows the
fractionalresidualbetweenthesetwofits. Moredetailscanbefound
inSection5.2. g) 10 +0.2
a
m
Heteałml.i(n2i0a1k4)e.t al. (2012), Bass et al. (2012), and Torres m (bol 9 +0.0H]
e/
We fit the M -R relations for both our stars and the F
∗ ∗ [
LMEBs with second order polynomials. We show the 8 -0.2
fractional difference between these fits in Figure 4. There
is a notable discrepancy at the masses M∗ (cid:38) 0.65M(cid:12). al 0.05
Bsheolwowin0.S6e5c0ti.o6n5M8.(cid:12)4,amgoredeeml-iennfterirsedbemttaesrsetshafonr5o%ur.sAamspwlee esidu 0.00 -0.4
R -0.05
better reproduce the LMEB mass-luminosity relation
across the sample. Thus the disagreement is most likely 8 9 10 11
m this work (mag)
due to (expected) errors in the mass-luminosity relation bol
from Delfosse et al. (2000), which was based on only 2-3
objects with masses in this range. Figure 5. ComparisonbetweenT (left)andm (right)from
eff bol
this work to those derived in Casagrande et al. (2008). Dashed
lineindicatesequality. Thedifferencebetweenthemeasurementsis
5.3. Temperatures and Bolometric Magnitudes Based on
showninthebottompanelofeachfigure. Pointsarecolor-coded
the Infrared Flux Method byourmetallicities. SeeSection5.3formoreinformation.
Casagrande et al. (2008) extended the infrared flux
method for FGK dwarfs from Casagrande et al. (2006)
Further, while this assumption could change our F val-
to M dwarfs with a method they called the multiple bol
ues, it has no effect on our T determinations. MOITE
optical-infrared technique (MOITE). There are 19 stars eff
also assumes that the PHOENIX model T scale is accu-
in our sample with parameters from Casagrande et al. eff
rate. While the more recent CIFIST version reproduces
(2008). We found significant systematic differences be-
the T scale from LBOI determinations, the older ver-
tween MOITE derived parameters and our values for eff
sions (i.e., the best available to Casagrande et al. (2008))
these 16 stars, which we show in Figure 5. Our T are
eff differsystematicallyfromLBOIderivedT s(Mannetal.
on average 140 24K hotter, with increasing disparity at eff
lower T and h±igher metallicity. Our bolometric magni- 2013b).
eff Casagrandeetal.(2008)arguedthatMOITEisinagree-
tudesare0.020 0.05maghigher. Foranyindividualstar
the difference i±n the bolometric magnitude is small, but mentwithLBOIdeterminations. However,thiswasbased
on just nine stars, many of which have since had their
taken together the systematic offset is significant. The
parameters revised by more precise LBOI measurements.
MOITE values for T and F convert to θ values that
eff bol Further, at least four of these stars have problematic
were on average 8.1 1.5% larger than ours.
IRFM and MOITE±assume that a star can be approx- 2MASS magnitudes (saturated or near saturated). Since
MOITEdependsonreliableinfraredmagnitudes, itcould
imated as a blackbody beyond 2µm. While this is
reasonable for warmer stars, M d(cid:39)warfs have significantly not be directly applied to these stars. We conclude that
our T and θ determinations are more reliable, and that
more flux in the NIR than predicted by a blackbody eff
the MOITE values are more likely systematically in error.
(Rajpurohit et al. 2013). Hence MOITE T values tend
eff
to be systematically too low, with growing disparity at
5.4. Metallicities and Temperatures from Rojas-Ayala
cooler temperatures where stars deviate more strongly
et al. (2012) and Neves et al. (2014)
from the Rayleigh-Jeans law. Our F determinations
bol
also assumed a blackbody, but we only invoked this at Gaidos & Mann (2014) compared the metallicities de-
λ>10µm,wheremodelssuggestthisisasafeassumption. rivedfromourcalibration(Mannetal.2013a,2014,based
10 Mann et al.
0.7
4000 +0.4
a T (K)eff3500 +0.2 us (R)Sun000...456 ++00..24
al di 0.3
ojas-Ay 3000 +0.0Fe/H] Ra 0.2 +0.0e/H]
R [ d [F
-0.2 Ra 0.05 −0.2
esidual -1231000000000 -0.4 Rad−Fit)/ −00..0005 −0.4
R -200 (
-300
3000 3500 4000 9 8 7 6 5
Our T (K) M (mag)
eff K
Figure 6. Comparison of T from our analysis and T from Figure 7. Top: R∗ asafunctionofabsoluteKS-bandmagnitude.
eff eff Thebest-fittothedataisshownasabluedashedline(seeEqua-
Rojas-Ayalaetal.(2012)forthe59overlappingstars. Pointsare
color-codedbyourmetallicitydeterminations,althoughour[Fe/H] tion4andTable1). MK andradiusbothdependonthedistance,
sotheerrorsarecorrelated. Henceweshowaone-sigmaerrorellipse
values are consistent with those from Rojas-Ayala et al. (2012).
in the top-left which indicates the typical 1σ errors for a typical
FurtherdetailscanbefoundinSection5.4.
point(MKS (cid:39)6.6,R∗(cid:39)0.35). Bottom: fractionalresidualtothe
fit. Allpointsarecolor-codedbymetallicity.
on NIR spectra) to those from Rojas-Ayala et al. (2012,
6. EMPIRICALRELATIONS
also based on NIR spectra), Neves et al. (2013, based
on absolute V, K magnitude), and Neves et al. (2014, We calculated empirical relations between observable
briazseedtheoinr fihnigdhi-nrgess,oltuhteiomneoapntimcaeltaslplieccittrya)d.iffTeroenscuemsmarae- (Teh.ge.s,eMrKelSa)tioannsdainretrivnasliicd(feo.gr.,0.R1∗R)(cid:12)ste(cid:46)llaRr∗pa(cid:46)ram0.e7tRer(cid:12)s.
0o.f003.±230,.003.2,80,.0a8n±d00.0.528,,arnedsp0e.c0t6iv±e0ly.0.2Adllexo,fwthitehseχm2νevtahloudess (4s1p0e0cKtr)a,latynpdes 0K.67–<M[F7,e/4H.6]<<0M.5K. SH<ow9e.v8e,r2,7b0e0ca<uTseeffo<f
used wide binaries to calibrate their metallicity measure- sample selectio−n biases, the range of metallicities is signif-
ments. The small systematic offset between our [Fe/H] icantly smaller for stars of spectral types later than M4
values and those from Neves et al. (2013) and Neves et al. (mostly slightly metal-poor) and for the late-K dwarfs
(2014) may be a reflection of small systematic differences (mostly metal-rich). There are also only 15 stars in total
in the metallicities of the primaries of these calibrators with spectral types M5-M7 and only three stars with
(Hinkel et al. 2014). The low χ2 values may be due to [Fe/H]< 0.5. These sample biases should be considered
significant overlap in binary calibνration samples, and/or when app−lying these formulae.
overestimated errors for one or more method. For all relations we found the best-fit parameters of
T valuesfromNevesetal.(2014)werecalibratedusing polynomial functions with the least-squares minimization
eff
the Casagrande et al. (2008) T scale, which we already algorithm MPFIT (Markwardt 2009). The number of
eff
discussed inSection5.3 andfound tobeproblematic. We parameters (polynomial order) was determined by an F-
compared our T values to those from Rojas-Ayala et al. test;aprobabilityof>95%thatthefitisanimprovement
eff
(2012) in Figure 6. Rojas-Ayala et al. (2012) estimated is required to increase the fit order. Fits for R∗ and M∗
Teff based on their measurement of H2O-K2 index, which can be found in Table 1, for Teff (from color) in Table 2,
they interpolated onto BT-SETTL models. Because we and bolometric corrections (BC) in Table 3.
also used BT-SETTL models the methods are not com-
pletely independent. However, Rojas-Ayala et al. (2012) 6.1. Radius-Absolute Magnitude
used a different model grid (AGSS; with Asplund et al. In Figure 7 we show the relation between absolute K
S
(2009) relative abundances), and based their results on band magnitude (M ) and R . We derived the best-fit
Ks ∗
NIR instead of optical spectra, so this comparison is still of the form:
illuminating. The mean difference is 28 14K, which is
not significant, but there are some notabl±e trends. Rojas- Y =a+bX+cX2+..., (4)
Ayala et al. (2012) appear to overestimate T for the
eff where X is the absolute K band magnitude (M ), Y
warmeststars. Theχ2ν oftheTeffdifferencewas2.4,which is the stellar radius (R∗), and a, b, and c were pKaSrame-
is particularly high considering the methods are based ters allowed to float to provide a better fit (reported in
on a similar set of models. Rojas-Ayala et al. (2012) Table 1).
estimated the errors in their Teff values just from the The fit yielded an RMS scatter of only 2.9%, which
measurement error in their derived H2O-K2 values, and is remarkably tight given that typical errors in our radii
did not factor in errors in the models or shape errors in were 3-4%. The scatter is low because errors in M and
K
the spectra (Newton et al. 2014). These are exaggerated R are correlated in a direction similar to the trend (see
∗
by the fact that changes in M dwarf Teffs have a weaker the ellipse in Figure 7), a result in turn of both being
effect on NIR spectra than optical. Hence errors from driven largely by error in the distance. For example,
Rojas-Ayala et al. (2012) are probably underestimates. imagine a star with a measured distance 4% closer than