Statistical Time-Resolved Spectroscopy: A higher fraction of short-period binaries for metal-rich F-type dwarfs in SDSS

0.17 MB·English

by  T. Hettinger

Draftversion May13,2015
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
STATISTICAL TIME-RESOLVED SPECTROSCOPY: A HIGHER FRACTION OF SHORT-PERIOD BINARIES
FOR METAL-RICH F-TYPE DWARFS IN SDSS
T. Hettinger
DepartmentofPhysicsandAstronomy,MichiganStateUniversity,EastLansing,MI48824, USA
5 C. Badenes
1 DepartmentofPhysicsandAstronomyandPittsburghParticlePhysics,AstrophysicsandCosmologyCenter (PITTPACC),University
ofPittsburgh,Pittsburgh,PA15260, USA
0
2

y J. Strader
a DepartmentofPhysicsandAstronomy,MichiganStateUniversity,EastLansing,MI48824, USA
M
S.J. Bickerton
2
KavliInstitute forthePhysicsandMathematics oftheUniverse(WPI), TodaiInstitutes forAdvanced Study,theUniversityofTokyo,
1
Chiba,277-8583,Japan

and
]
R T.C. Beers
S Dept. ofPhysicsandJINACenter fortheEvolutionoftheElements,Univ. ofNotreDame,NotreDame,IN46556, USA
. Draft version May 13, 2015
h
p ABSTRACT
-
o Stellarmultiplicityliesattheheartofmanyproblemsinmodernastrophysics,includingthephysics
r of star formation, the observational properties of unresolved stellar populations, and the rates of
t
s interacting binaries such as cataclysmic variables, X-ray binaries, and Type Ia supernovae. However,
a little is known about the stellar multiplicity of field stars in the Milky Way, in particular about the
[ differences in the multiplicity characteristicsbetween metal-richdisk starsand metal-poor halo stars.

Inthisstudyweperformastatisticalanalysisof∼14,000F-typedwarfstarsintheMilkyWaythrough
2
time-resolvedspectroscopywiththesub-exposuresarchivedintheSloanDigitalSkySurvey. Weobtain
v
absolute radialvelocitymeasurements throughtemplate cross-correlationof individualsub-exposures
2
6 with temporalbaselines varyingfromminutes to years. These sparselysampledradialvelocitycurves
9 are analyzed using Markov chain Monte Carlo techniques to constrain the very short-period binary
4 fraction for field F-type stars in the Milky Way. Metal-rich disk stars were found to be 30% more
0 likely to have companions with periods shorter than 12 days than metal-poor halo stars.
. Subject headings: binaries: close—binaries: spectroscopic—Galaxy: stellarcontent— stars: statis-
1
tics — surveys
0
5
1
: 1. INTRODUCTION ablessuchasstellarage,metallicity,anddynamicalenvi-
v ronment remains poorly understood. Moe & Di Stefano
Stellar multiplicity plays a crucial role in many fields
i (2013) find no significant trends with metallicity for O-
X of astronomy. Star formation and evolution, Galactic
and B-stars, but more work is needed for lower-mass
chemical evolution, nuclear astrophysics, and cosmology
r stars.
a are all influenced by our understanding of the multiplic-
The recent review by Duchˆene & Kraus (2013) sum-
itypropertiesofanunderlyingstellarpopulation. Binary
marizes the state of the art in multiplicity studies.
interactionsleadtophenomenaasdiverseascataclysmic
The fraction of systems with companions is known
variables, classical novae, X-ray binaries, gamma-ray
to be a strong function of stellar mass (Lada 2006;
bursts, and Type Ia supernovae. Stellar interactions are
Raghavan et al. 2010; Clark et al. 2012), and there are
alsothecauseoftheanomaloussurfaceabundancesmea-
hints that lower mass systems have smaller separations
suredin Ba stars,CH stars,andthe majority ofcarbon-
(Duquennoy & Mayor1991; Allen 2007; Raghavan et al.
enhanced metal-poor stars (Lucatello et al. 2005). The
2010). Studies of the Solar neighborhood also indicate
rates of these phenomena depend on the multiplicity
that lower metallicity stars are more likely to have stel-
properties such as the fraction of stars with compan-
lar companions (Raghavan et al. 2010).
ions and the distributions of separations and mass ra-
These results are based on heterogeneoussamples of a
tios. How these properties are in turn affected by vari-
fewhundredstarsatmost,oftendominatedbywidesys-
tems which will never become interacting binaries. The
[email protected]
[email protected] spectroscopic surveys that reach small periods are labor
[email protected] intensivebecauselargenumbersofradialvelocities(RVs)
[email protected]
are required to find the orbital solution of each target.
[email protected]
2 Hettinger et al.
This leads to small sample sizes, which have only in- boring fibers on the CCD. After plate-wide comparisons
creased modestly in the past two decades, from 167 in of F-stars, RV correlations were corrected where possi-
Duquennoy & Mayor (1991) to 454 in Raghavan et al. ble. Corrections applied to the 104 RVs are as large as
(2010). Thedrivetocollectcompletesampleshaslimited 17km s−1 with a standarddeviationof2.2 km s−1. Not
previousspectroscopicstudiestotheSolarneighborhood allcorrelationscouldbeidentifiedautomaticallybecause
or specific stellar clusters,but neither of these strategies of multiple groups of correlatedshifts, opposite in direc-
can probe the full range of metallicities and ages span- tion,onsomeplates. Visualinspectionofplatescontain-
ningthefieldstarsoftheMilkyWay(MW)diskandhalo ingnumerousfalse binarydetections leadto the removal
components. Theselimitsbiastheinterpretationofdata of25platesincluding1155stars. Weurgeindividualsus-
againsttheglobalpropertiesof,andvariationwithin,the ingsub-exposurespectroscopyinSDSStoconsiderthese
MW field. Thus, we are motivated to take a statistical systematic shifts in the wavelengthsolutions.
approach with a sample of stars located throughout the Qualitycontrolconsistedoftheremovalof: starswith-
field in order to investigate their multiplicity properties out valid parameters in SSPP, fibers located on ‘bad’
withrespecttoage,[Fe/H],andcomponentmembership. plates,sub-exposureswithamedianpixelsignal-to-noise
With the advent of multiplexed spectroscopic surveys ratio (SNR) less than 20 or with fewer than 3000 un-
like SDSS (York et al. 2000) and LAMOST (Cui et al. flagged pixels, stars with time lags ∆T < 1800 s, stars
2012), we can use multiple RV measurements of thou- with less than three cleansub-exposures,and corruptor
sands of stars to study the properties of stellar multi- misclassifiedspectra(fromvisualinspectionofstarswith
plicity thataremorerepresentativeofthe entireGalaxy. the largest RV variation or non-characteristic T ). The
eff
SDSS Data Release 8 (Aihara et al. 2011) contains over final sample consists of 14,302 stars (16,894 fibers) with
1.8 million optical spectra from the original SDSS spec- as many as 47 sub-exposures, spanning up to nine years
trographs including over 600,000 stellar spectra. In of observations (Figure 1).
this work we employ a lesser known SDSS feature, the Our cleaned sample is characterized by metallicities
time-resolved dimension. To facilitate cosmic ray re- ranging from −3.41 ≤ [Fe/H] ≤ +0.52. To aid com-
moval, spectra were constructed through co-addition parisoninour analysis,the final sample wassub-divided
of several individual sub-exposures, typically 15 min- into three groups of equal size by cuts in metallicity at
utes in duration. Although under-utilized, the bene- [Fe/H] = −1.43 and [Fe/H] = −0.66 (Figure 1). The
fit of the sub-exposure domain is recognized in works majority of the stars have three or four sub-exposures
suchasBadenes et al.(2009)andBickerton et al.(2012). (median = 4), typically taken about 15 minutes apart.
Portions of the sky were also re-observed for calibra- The median time lag for a star is 2 hours, however more
tion and scientific purposes. These additional pointings, than three years between observations can be seen in
combined with the sub-exposures, yield a time dimen- more than 250 stars (Figure 1). SNRs for sub-exposures
sion where single stars have exposure coverage ranging lie in the range 20 < SNR < 84 with a median value of
from 3 sub-exposures up to over 40 sub-exposures, and 32.
time gaps from hours to nearly a decade. The tech-
niques employedhereinfollowthe time-resolvedworkby
2.2. Radial Velocities
Badenes & Maoz (2012) and Maoz et al. (2012).
RV measurements were attained through cross-
2. MEASUREMENTS correlationofsub-exposureswithamastertemplatecon-
structed from 7207 sample-star, co-added spectra where
2.1. SDSS Observations and Sample Selection
the co-added SNR > 50. The spectra were de-shifted
F-typedwarfsarechosenforoursamplebecauseofthe using the redshift value assigned to the co-adds by the
large number of stars targeted by SDSS with repeat ob- SDSS pipeline, continuum-normalized, and averaged to-
servations, and their relatively mild variability and ac- gether.
tivity. Additionally, F-stars have main sequence (MS) Sub-exposureswereindependentlypreparedandcross-
lifetimes greater than 5 Gyr, allowing us to select MS correlated with the template. Spectra were continuum-
stars from both the younger disk and older halo. The normalized by dividing the spectrum with a highly
SloanStellarParameterPipeline(SSPP;Lee et al.2008) smoothed version of itself using a FFT smoothing al-
was developed to determine parameters for stellar spec- gorithm, and then cross-correlatedwith the template at
train the SDSS archive,including metallicity [Fe/H], ef- various integer pixel lags. Each spectrum had a cross-
fective temperature T , and surface gravitylogg. Sam- correlation function (CCF) that was fit with a smooth
eff
ple selection began with identifying science primary ob- spline interpolation. With spectral resolution of R ∼
jects from SEGUE-1 (Yanny et al. 2009) and SEGUE-2 2000,the peak lag in pixels translates to the spectrum’s
(Rockosi et al., in prep.) in the SSPP that were clas- redshift at 70 km s−1 pixel−1. The mean and standard
sified as an F-type star by the “Hammer” classification deviationofRVsforindividualstarsareshownintheFig-
code(Covey et al.2007). Tominimize theeffectsofstel- ure 2 distributions. The velocity dispersion of the mean
larevolutiononmultiplicity,weselectedonlydwarfstars RVsdecreaseswithincreasing[Fe/H],indicatingthatour
(logg ≥ 3.75). Stars with multiple fiber pluggings were [Fe/H]-groupssampleboththediskandhalocomponents
identified astrometrically and joined with the appropri- of the MW. The standard deviation of RVs within indi-
ate science primary fibers. vidual stars is larger for the metal-poor group; however,
After measuring stellar RVs (Section 2.2), systematics empiricallyestimateduncertaintiesalsoshowlargermea-
were revealed in the SDSS sub-exposure spectra. These surement errors for metal-poor stars. This underscores
correlations appear as similar shifts in RVs for many the importance of the use of proper error analysis in a
fiberslocatedonthesameplate,typicallyaffectingneigh- method such as ours.
Statistical Time-Resolved Spectroscopy 3
4 4
Metal-Poor
1d 1m 1y
Metal-Intermediate
3 Metal-Rich 3
N) 2 2
g(
o
l
1 1
0 0
−4 −3 −2 −1 0 1 3 4 5 6 7 8 9
[Fe/H] ∆t (log s)
Fig. 1.—Left: Metallicitydistributionfor14,302F-dwarfs. Right: Distributionofmaximumtimelagbetweenthefirstandlastexposure
ofastar.
4 4
Metal-Poor
Metal-Intermediate
3 3 Metal-Rich
N) 2 2
g(
o
l
1 1
0 0
−600 −400 −200 0 200 400 600 0 10 20 30 40 50 60
RV (km/s) std(RV) (km/s)
Fig.2.—Mean(left) andstandard deviation(right) ofradialvelocities withinastar. Variations inthe standarddeviation of velocities
areaffected, inpart,bythelargermeasurementuncertainties formetal-poorerstars.
2.3. Uncertainties Uncertaintiesaresub-pixel,fallingbelowthespectralres-
Itis wellknownthat uncertaintiesinCCF peaksmust olutionof70km s−1 pixel−1. ForexposureswithSNR<
be estimated empirically or through some Monte Carlo 25, uncertainties range from 3.0 to 8.0 km s−1, with a
method (e.g., Peterson et al. 1998). For this work we median value of 5.0 km s−1. Exposures with SNR > 40
determined RV uncertainties empirically by quantifying haveuncertainties inthe range1.9to 4.4km s−1, witha
the spread in measurements for spectra of similar qual- median value of 2.7 km s−1.
ity. The median absolute deviation (MAD) is a ro-
bust measure of the variability of a sample and is re-
3. MULTIPLICITY
lated to the standard deviation by σ = 1.4826MAD,
whereMAD=median(|RVi−median(RV)|)(Leys et al. The probability of a star having a companion was
2013). All measurements were de-shifted into the rest determined through model comparison using a trans-
frame using the SDSS estimates of the co-add redshift, dimensional, hierarchical, Markov chain Monte Carlo
andplacedintobinsofsimilarmetallicity([Fe/H]±0.25) (MCMC)method. Twomodelswerecompared: asingle-
and signal-to-noise (SNR ± 2.5). Initial tests showed starmodelMs,andabinary-starmodelMb. Thehyper-
nocorrelationsbetweenmeasurementspreadsandeither parameterλ,indexesthemodelchoiceateachstepinthe
logg or T . Estimates for the uncertainty of RV mea- MCMCchain. Weevaluatedthehierarchicalmodelusing
eff
surements within a bin were calculated using MAD val- the Python package emcee, a MCMC ensemble sampler
ues. Here,itisassumedthatthemajorityofstarsdonot (Foreman-Mackey et al. 2013).
have detectable variability over the observed time base- Thesingle-starmodelMs,fitsastarwithnon-varying
line, and that effects from intrinsic variations in RV are RVs, parameterized by a systemic velocity V0. Because
minimized by adopting median values. After performing intra-plate systematics are known to exist, it is reason-
this process for all bins, a functional form for assign- able to assume inter-plate systematics exist as well. In
ing RV measurement uncertainties σ was fit with an light of this, (P −1) additional parameters psi, were in-
RV
inverse proportionality to SNR, and with a linear cor- cludedforeachstar,whereP isthenumberofplate-MJD
rection in [Fe/H]. The measurement uncertainty as a pluggings composing the star. These plate-shift param-
function of [Fe/H] and SNR is, in km s−1, eters allow all RVs from plate i to shift by some amount
psi, relative to the first plate P0. For the majority of
(−26.51[Fe/H]+50.52) starsP =1,noplate-shiftparametersarenecessary,and
σ ([Fe/H],SNR)= +1.23.
RV SNR Ms is a 1-parameter model.
(1) In the binary star model Mb, the sparsely sampled
4 Hettinger et al.
RVsarefitbyasinusoiddefinedbyfour-parameters: the (andwillbecharacterizedinfuturework). Anintermedi-
log of the semi-amplitude logA, the log of the period atecutatη >0.80isacompromisebetweentheselimits,
logP, the phase φ, and the systemic velocity V . We yieldingalargersampleofstars(406)withmodestmodel
0
assume circular orbits (eccentricity, e = 0), which is a constraints. The values of the binary fractions that we
safe assumption for tidally circularized, short-period or- derive below are insensitive, within the uncertainties, to
bits (P < 12 days; Raghavan et al. 2010), where we are the exact choice of cut in η. This implies that the RV
most sensitive. A small number of the binaries found variationsforourbinarydetectionsaresufficientlyabove
in this study may have longer periods and could have themeasurementuncertainties,andthatthebinaryfrac-
non-zero eccentricities, but this does not affect our re- tions reported are not biased due to differences in SNR
sults. Plate-shift parameters were also adopted in Mb or absorption features.
wherever P >1. Figure 4 shows the logP posteriors for each [Fe/H]-
Uninformative priors were used in the MCMC. The group, marginalized over all binary systems (η > 0.80).
model index λ, has a flat prior from 0 to 1, where The posterior distributions of logP for many of these
λ < 0.5 denotes Ms and λ ≥ 0.5 denotes Mb. The stars are complex: many are multimodal, affected by
semi-amplitudepriorislog-uniformfrom3km s−1,com- aliasingorotherissuesrelatedto the sparse,biasedtime
parabletothemeasurementuncertaintieswhereMs and sampling. One such effect is the increase in probabil-
Mb become degenerate,to 250km s−1, greaterthan the ity at logP = 4. Here the metal-rich and -intermediate
largest RV differences in the sample. The prior on the groups contain more stars than the metal-poor group
period is uniform in the range 4 ≤ logP (s) ≤ 7. The with ∆t ≃ 104 s. Systems with periods as short as this
lower limit logP (s) = 4.0 is equal to the orbital pe- areextremelyrare(Drake et al.2014),andourincreased
riodatwhichstellarcontactiscertainforlow-masscom- probabilityin this area may be due to overfitting. Addi-
panions. Above logP (s) = 7.0, RV amplitudes in bi- tionally, the gap at ∆t = 104.6 s = 12 h (Figure 1) may
narysystemsarecomparabletothe measurementuncer- affecttheestimateofaperiod. Wedeferamoresophisti-
tainties. Combined with the sparsity of the RV data, catedanalysisto afuture paper, butthese effects should
systems with periods longer than logP (s) = 7.0 are not alter the ability to rule out a single-star model. For
outside our range of sensitivity. Priors are also uni- now, Figure 4 illustrates that we are mainly sensitive to
form for the phase (0 ≤ φ ≤ 2π) and systemic velocity periodsintherange4<logP (s)<6,orlessthanabout
(−600 ≤ V (km s−1) ≤ 600). Markov chains were run 12days. Weemphasizethatamoredetailedanalysiswill
0
be necessary to estimate the true underlying logP dis-
independentlyoneverystarwithanensembleof200par-
allelchain“walkers”foratotalof2.4×106samples,then tribution in our sample.
burned and thinned to 6×105 independent samples of
4.0
the posterior.
3.5 Metal-Poor
Evidence fordetection ofa companionstaris reflected Metal-Intermediate
by the relative probabilities of λ. We define the proba- 3.0 Metal-Rich
bility for the binary model, η as the fraction of samples 2.5
)
in the marginalized posterior having λ = Mb. We note (N 2.0
g
that the value of η is dependent on the choice of priors, o 1.5
l
and is sensitive to the treatment of the SDSS systemat- 1.0
ics. Moreover, a degeneracy arises as the RV curve of 0.5
a long-period, low-amplitude system becomes indistin- 0.0
guishable from a single-star system. With this mind, we
0.0 0.2 0.4 0.6 0.8 1.0
stressthatvaluesforη arenotabsoluteprobabilitiesofa η
systemhavingacompanion,butreflecttheabilityofthe
data to rule out models under the given prior. However, Fig.3.—Distributionofη,thefractionofposteriorsamplesusing
the [Fe/H]-groups can be compared, relatively, by con- thebinarymodel,forstars.
sideringthe fractionof systems whereη is largeandMs
is stronglydisfavored. The resultsare shownin Figure 3 4. DISCUSSION
We also investigated the e/i parameter proposed by
Geller et al. (2008) as a metric for identifying the stars In Figure 5 we show fb, the measured lower bound
for the fraction of stars with short-period companions
withlargeRVvariations. Wefindthatthee/iparameter
(P . 12 days) for each metallicity group, normalized to
singles out many of the same stars as our more sophis-
ticated MCMC-based inference. Our method not only themetal-richbinaryfraction. fb isalowerlimitbecause
of non-detections as a result of sparsely sampled RVs
takes into account deviations in RV from the mean, but
andhighorbitalinclinations,resultinginlowamplitudes.
also how well the data fit the expected periodicity of a
binary system. We see agreementin fb measuredfor all three choices in
η cutoff (0.65, 0.80, 0.95). With a cutoff of η = 0.80,
Analysisof the posterior,andvisualinspections ofthe
binary model fits, show that 681 stars with η > 0.65 values of fb for the metal-poor, -intermediate, and -rich
groups respectively are: 2.5%±0.2%, 2.8%±0.2%, and
are true spectroscopic binaries, though given the spar-
3.2%±0.3%. Since the observational biases that affect
sityofthe RVcurvesampling,therearesometimes large
binarydetectionaremostlyduetothesparsityoftheRV
uncertainties in the fitted values for specific model pa-
coverage,whichisn’tmetallicity-dependent,weconclude
rameters. Anothernaturalbreakpointisη >0.95;these
that the field F-type MS stars in our metal-rich sample
are209starsforwhichthedeterminationandanalysisof
are,ata2-sigmalevel,30%morelikelythanthoseinour
accurateindividualmodelparametersshouldbe possible
metal-poor sample to have close binary companions.
Statistical Time-Resolved Spectroscopy 5
former work does not make use of sub-exposure infor-
Metal-Poor
0.20 Metal-Intermediate mation (using only two RV epochs per star) and relies
DF Metal-Rich on the correctness of model values for the period distri-
P 0.15 bution, mass ratio distribution, and initial mass func-
d
e tion. The latter work, which uses photometric color
n
bi 0.10 deviations to infer companions, shows a modest metal-
m licity dependence on total binary fraction. Since their
o
C 0.05 method is not sensitive to period, the binary fractions
they report are strongly dominated by more common,
wider-period systems near the peak of a log-normal pe-
0.00
4.0 4.5 5.0 5.5 6.0 6.5 7.0 riod distribution (logP (s) = 10 for nearby, Solar-like
logP (s) stars; Raghavan et al. 2010). It is clear that conclusions
Fig.4.—AveragedprobabilitydistributionsoflogP forallbinary about binary fraction depend on a number of factors,
detections (η>0.80). These donot reflect actual distributions of especially the range of periods to which the search is
periods,andshouldonlybeusedasaguidetoprobetheregionof sensitiveandassumptionsmadeabouttheoverallperiod
MCMCsensitivity. TheshadedregionindicateswhereRochelobe
distribution.
overflowandcontactbecomesrelevant. Thedashedlinemarksthe
circularizationlimitataperiodof12days. Our MCMC analysis yields posterior probabilities in
parameter space, allowing for a more detailed study of
Our metal-rich and metal-poor samples mostly trace binary properties (e.g., period and separation distribu-
the MW disk and halo. Differences in the fraction of tions),whichwillbepresentedinfuturework. Thetech-
short-period systems can stem from differences in the niques in this work have direct applications for current
star-formationprocess,dynamicalinteractions after star and future multiplexed spectroscopic surveys.
formation, or some combination of the two.
Three-dimensional hydrodynamic models from
Machida et al. (2009) actually suggest a higher fre-
quency of binaries formed through cloud fragmentation
for metal-poor clusters, due to the decreased require-
ment of a cloud’s initial rotation energy to fragment.
Moreover,theirmodels yieldsystemswithshorterinitial
separations at lower metallicities. The increased fb
observedformetal-richstarsinthisworkcanmorelikely
be explained by dynamical processes than by formation
processes.
The observed differences in fb could be explained if
the clusters that yielded halo field stars had larger stel-
lar densities and/or gas densities than those of the disk.
Korntreff et al. (2012) explore the effects of gas-induced
orbital decay on period distributions in clusters. They
note that an increased density of gas in a newly formed
cluster will lead to a larger number of short-period sys-
temmergersshortlyafterformation. Parkeret al.(2009)
describehowclusterswithhigherstellardensitiesdestroy
wide binaries through dynamical interactions. An in-
crease in the destruction of high-mass, wide-binary sys-
tems leads to the ejection of former F-star secondaries
intothefield. Theseorphaned,single-starsystemswould
increase the total number of F-star systems in the halo
field, effectively decreasing the short-period binary frac-
tion measured. Observational evidence of these denser
cluster environments is needed to support these argu-
ments for a lower fb in the halo.
Additionally, some close binaries may also transfer
massandcovertthemselvesintobluestragglers(Lu et al.
2010). Evidence for an abundance of blue stragglers in WethankEwanCameron,DanMaoz,JeffreyNewman,
thehalohasbeenseen(Yanny et al.2000),andmaycon- ChadSchafer,andtherefereeforusefuldiscussions. T.H.
tributetothelowerfb observedinthemetal-poorgroup. and T.C.B. acknowledge partial support from grants
Also,Duchˆene & Kraus(2013)showadecreaseinfbwith PHY08-22648;PhysicsFrontierCenter/JINA,andPHY
age for Solar-type stars, although this result is based on 14-30152; Physics Frontier Center/JINA Center for the
visual binaries with wider periods, and is poorly con- Evolutionof the Elements (JINA-CEE), awardedby the
strained due to limited sample sizes. US National Science Foundation. Funding for SDSS-III
WenotethattherecentresultsofGao et al.(2014)and hasbeenprovidedbytheAlfredP.SloanFoundation,the
Yuan et al. (2014), using data fromSDSS, show a larger Participating Institutions, the National Science Founda-
binaryfractionformetal-poorthanmetal-richFGKstars tion, and the U.S. Department of Energy Office of Sci-
in the field. In addition to probing longer periods, the ence.
6 Hettinger et al.
1.1 0.036
1.0 0.032
0.9 0.029
)
0
richfb 0.8 0.026>0.8
/ η
fb (
0.7 0.023fb
η > 0.65
0.6 0.019
η > 0.80
0.5 η > 0.95 0.016
−2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5
[Fe/H]
Fig.5.— Short-period binary fraction limits, relative to the metal-rich group. Binary companion detections are defined by a cut in η,
thefractionofposteriorsamplesusingthebinarymodel. Groupmedianvaluesof[Fe/H]areused.
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