STELLARMULTIPLICITYANALYSISWITHTIME-RESOLVEDSPECTROSCOPY
ANDMARKOVCHAINMONTECARLOSIMULATIONS
By
ThomasBarrettHettinger
ADISSERTATION
Submittedto
MichiganStateUniversity
inpartialentoftherequirements
forthedegreeof
AstrophysicsandAstronomy|DoctorofPhilosophy
2015
ABSTRACT
STELLARMULTIPLICITYANALYSISWITHTIME-RESOLVED
SPECTROSCOPYANDMARKOVCHAINMONTECARLO
SIMULATIONS
By
ThomasBarrettHettinger
ThisdissertationexaminesthemultiplicitypropertiesofstarsintheMilkyWayandtheir
relationshipwithmetallicity.Wepresentmethodsandtechniquesfordataminingindividual,
raw,sub-exposureinformationfromspectroscopicsurveysasastatisticalapproachtoper-
formingscienanalysesinthiseraofBigData.WealsodescribehowBayesianinference
andMarkovChainMonteCarlosimulationsworkinconjunctionwiththesesub-exposure
spectroscopytechniques.
Binaryinteractionsplayakeyroleinmanyastrophysicalprocesses,fromalteringsurfaces
abundances,toproducingsupernova.InChapter1,wegiveabriefintroductiontostellar
multiplicity,beginningwithadescriptionofthestarformationprocessandpossiblescenarios
forbinarystarformation.WediscusshowbinarystarsinteractthroughRoche-lobeovw,
andhowbinarysystemsleadtovariousastrophysicalphenomena.Weconcludethechapter
withalookatourcurrentunderstandingofmultiplicitypropertiesofstarsintheMilky
Wayasdeterminedempiricallyfromobservationsandsurveys,andwithadiscussionforthe
futureoutlookofmultiplicitystudies.
InChapter2wedescribeamethodologyformeasuringradialvelocityvariationsinstellar
sourcesusingsub-exposurespectrafrommerspectroscopicsurveys.Inparticular,we
describeacross-correlationtechniqueusedonspectrathatwereobservedaspartoftheSDSS
survey.
InChapter3wegiveabriefintroductiontoBayesianinferenceandtheuseoftheMCMC
pythonpackage
emcee
.Wedescribethemethodsusedfordetectingbinarityinstellarsources
fromsparselysampledradialvelocitycurves.
Chapter4containsthepeer-reviewedarticleHettingeretal.(2015)publishedinthe
AstrophysicalJournalLetters.InthisLetter,weemploythesub-exposureradialvelocity
measurementtechniquesandtheMCMCmethodsoutlinedinthisdissertationtoexamine
apopulationofF-typedwarfstarsintheMilkyWay.Thesamplewasdividedintothree
groupsbymetallicity,withthegoalofinvestigatingthemetallicitydependenceonmultiplic-
ityproperties.Weahigherfractionofshort-periodbinariesforthemetal-richdiskstars
thanthemetal-poorhalostars.
Finally,inChapter5,weextendtheworkofHettingeretal.(2015)toinvestigatepossible
constraintsontheseparationdistributionofbinariesintheF-dwarfpopulation.
Formymother,RitaMurray.
iv
ACKNOWLEDGMENTS
Firstly,Iwouldliketoexpressmysinceregratitudetomyadvisorsandcollaborators,Carles
Badenes,JayStrader,TimothyBeers,andStevenBickerton,fortheircontinuoussupport,
motivation,andimmenseknowledge.IespeciallythankCarlesfortakingmeonashisown
student,andcommittingtoworkwithmeremotelyacrossuniversities.Hisguidancewas
invaluable,andIamthankfultohavehadhimasmyadvisor.
Tomywife,MenglingHettinger,Iamdeeplygrateful.Asacolleague,sheprovidedme
withdirection,focus,andHertutoringinmycourseworkandhersupportwere
crucialtomysuccess.Icouldnothavechosenabetterpartnertosharetherestofmylife
experienceswith.IloveyouMengling.
Iwouldliketothankmyseniorgraduatestudents,CharlesKuehn,ChrisRichardson,
CarolynPeruta,andAaronfortheiradvice;mymatesAlexDeibel,Brian
Crosby,TomConnor,andRyanConnollyfortheirhelpwithbrainstormingandformy
addictiontoee;andIwouldliketothank
Fortheirworkinoutreachandcommunityservice,IthankJohnFrench,ShaneHorvatin,
andDaveBatchattheAbramsPlanetarium,aswellasHoraceSmithandLauraChomiuk
fortheirworkwithpubliceventsatthetelescope.
IamgratefultotheNSF,NASA,andtheDoEforfundingthemajorityofallastrophysics
research.Astronomyistheleastapplicableofthesciences,butitasksthebiggestquestions.
Wearefortunateenoughtohavethecapacitytoaskthesequestions,thereforewemust.
Ithankallmyfamilyandfriendsfortheircontinuingsupportandencouragement.Finally,
IwanttothankJasonLinehanforintroducingmetothenightsky,withoutwhomIwould
neverhavepursuedastronomy.
v
TABLEOFCONTENTS
LISTOFTABLES
....................................
viii
LISTOFFIGURES
...................................
ix
Chapter1ABriefIntroductiontoStellarMultiplicity
............
1
1.1Introduction....................................1
1.2StarFormationandtheOriginofBinaryStars.................2
1.2.1PhysicalPrinciples............................2
1.2.2PossibleBinaryFormationMechanisms.................6
1.2.2.1Capture.............................6
1.2.2.2PromptFragmentation.....................6
1.2.2.3DelayedBreak-up........................7
1.2.3QuestionsandFutureInvestigation...................8
1.3BinaryEvolutionandInteractions........................10
1.3.1PrinciplesoftheEvolutionofBinarySystems.............10
1.3.2BinaryInteractioninAstrophysicalPhenomena............12
1.3.2.1HotSubdwarfs.........................13
1.3.2.2ChemicallyPeculiarStars...................13
1.3.2.3SymbioticBinaries.......................14
1.3.2.4BlueStragglers.........................14
1.3.2.5ThermonuclearSupernovae..................15
1.3.2.6CoreCollapseSupernovae...................15
1.4EmpiricallyDerivedMultiplicityProperties...................16
1.4.1MultiplicityPropertiesandSurveyMethodology............16
1.4.2TrendsandCharacteristicsofMultiple-StarSystems.........18
1.4.2.1MultiplicityandMass.....................18
1.4.2.2MultiplicityandAge......................22
1.4.3Discussion.................................23
1.5Conclusion.....................................26
Chapter2Time-ResolvedSpectroscopy
.....................
28
2.1Introduction....................................28
2.2TheSloanDigitalSkySurvey..........................30
2.2.1Sub-Exposures..............................31
2.2.2SEGUEStellarParameterPipelineAndSampleSelection.......32
2.2.3PlateSystematics.............................34
2.3RadialVelocities.................................41
2.3.1ContinuumNormalization........................41
2.3.2SpectralTemplate............................44
vi
2.3.3Cross-Correlations............................45
2.4EmpiricalUncertainties..............................48
2.5
e=i
Variability...................................53
2.6Discussion.....................................53
Chapter3MarkovChainMonteCarlo
......................
55
3.1Introduction....................................55
3.2BayesianInferenceandMCMC.........................56
3.3
emcee
:TheMCMCHammer..........................57
3.3.1AnvariantEnsembleSampler.................57
3.3.2Using
emcee
................................58
3.4CorrectingSystematicsinSDSSSpectra....................63
3.5ModelingMultiplicityWithRadialVelocityCurves..............64
3.5.1Examples.................................68
Chapter4StatisticalTime-ResolvedSpectroscopy:AHigherFractionof
Short-PeriodBinariesforMetal-RichF-typeDwarfsinSDSS77
4.1Abstract......................................77
4.2Introduction....................................78
4.3Measurements...................................80
4.3.1SDSSObservationsandSampleSelection................80
4.3.2RadialVelocities.............................82
4.3.3Uncertainties...............................83
4.4Multiplicity....................................84
4.5Discussion.....................................89
Chapter5BinaryFractionsandSeparationDistributions
..........
92
5.1Introduction....................................92
5.2MCMCandPopulation-WideMonteCarlo...................93
5.3Discussion.....................................100
REFERENCES
......................................
103
vii
LISTOFTABLES
Table2.1SuspectPlatesinSDSS.........................39
Table2.2AbsorptionFeaturesinF-dwarfs....................42
Table3.1PriorLimitsforHettingeretal.(2015)MCMC............67
viii
LISTOFFIGURES
Figure1.1Dependencyofmultiplicityfractionwithprimarymassformainse-
quencestarsandVLMobjects.Valuesusedfromthereviewby
Duch^ene&Kraus(2013).........................21
Figure1.2Left:Orbitalperioddistributioninthesolarneighborhoodfrom
Raghavanetal.(2010).ThelimitforRLOFintheMSisindicated
bytheblackverticaldashedline,andtherangecorrespondingto
pre-CEsystemsisshadedingray.Theblog-normalfunctionis
showninblack.Thedashedredplotrepresentsamofunction
thatalsothedata.Right:Comparingthenumberofsystems
intheblog-normaldistribution(solidblack)andthemo
model(dashedred).Theperiodrangescorrespondingtolow-massX-
raybinaryprogenitors,stablehabitableplanetsaroundbinarystars,
andSNIaprogenitorsareshownwithhorizontalrulers.Theyellow
dash-dottedlinemarksthepre-outburstperiodofV1309Sco.....25
Figure2.1Distributionofthenumberofsub-exposures(top)andthetimelags
(bottom)fortheF-dwarfstarsfromtheHettingeretal.(2015)sam-
ple.Metallicityvaluesare[Fe
=
H]=

1
:
43and[Fe
=
H]=

0
:
66.33
Figure2.2DistributionofstellarparametersfortheF-dwarfstarsfromtheHet-
tingeretal.(2015)sample,includingmetallicity(top),etem-
perature(middle),andsurfacegravity(bottom).Metallicitycuto
valuesare[Fe
=
H]=

1
:
43and[Fe
=
H]=

0
:
66.............35
Figure2.3Scatterplotshowingthedistributionofmetallicityandsurfacegrav-
ityforF-typestarsintheSSPPDR9.Thebimodaldistributionof
[Fe
=
H]tracestheHaloandDiskcomponentsoftheMilkyWay....36
Figure2.4RVsforF-dwarfstarslocatedonplateplugging2085-53379beforeand
aftercorrectingtheplateforsystematicsub-exposureRed
pointsaresub-exposuresthathavelowSNR.FiberIDsaregivenfor
eacherontherightaxis.Sub-exposuresintheplateareordered
chronologically.Correctionstosystematicaresuccessfulon
thisplate.................................38
Figure2.5Distributionof10,264systematicRVestimatedforallplate
sub-exposuresintheF-dwarfsample..................39
ix
Figure2.6SameasFigure2.4forplateplugging3002-54844.Theleftcolumnof
eachshowsRVsbeforecorrectingtheplateforsystematic
andtherightcolumnshowsRVsaftercorrectingfor
Correctionsarenotsuccessfulonthisplateduetoanti-correlated
subsetsofsimilarlycorrelateders...................40
Figure2.7ContinuumnormalizationprocessappliedtoanF-typestarshowing:
(a)therawsub-exposurespectrum,(b)thespectrumwithselected
absorptionfeaturesmaskedout,(c)asmoothedversionofthespec-
trum,and(d)thecontinuum-normalizedspectrum........43
Figure2.8Fullcontinuum-normalizedF-typedwarftemplatespectrum(top)
withadetailedviewoftheblueendofthespectrum(bottom).Promi-
nentspectralfeaturesareannotated...................45
Figure2.9Exampleofthetemplatecreationprocessusing7spectra.Fromtop
tobottom:blueendandfullspectrumoftheinputco-addstellar
spectra,blueendandfullspectrumofthenormalized,rest-frame
inputspectra,blueendandfullspectrumoftheinputspectraresam-
pledtoacommonwavelengthsolution,blueendandfullspectrumof
thetemplate(averagedofresampledinputspectra)........46
Figure2.10Cross-correlationfunction(perpixel)forasinglesub-exposurespec-
trumofanF-typestar.Correlationvaluesarecalculatedatinteger
pixellags,andasplineinterpolationofthefunctionistothese
values.Thefunctionpeaksat

3
:
275pixels,or

229kms

1
.....47
Figure2.11Distributionofradialvelocities(minussystemicvelocity)forsub-
exposureswith[Fe
=
H]=

1
:
75

0
:
25andSNR=30

2
:
5.The
distributionhasamedianabsolutedeviationofMAD=3
:
04kms

1
,
indicatingmeasurementuncertaintiesof
˙
=4
:
51kms

1
.......49
Figure2.12Empiricaluncertaintiesforcross-correlationmeasurementsinF-type
dwarfstarsshowinguncertainties(a)asafunctionofSNRindepen-
dentlydeterminedforeach[Fe
=
H],(b)asafunctionofSNRwitha
commonconstantand(c)asafunctionofSNRand[Fe
=
H]as
inEquation2.8..............................51
Figure2.13Valuesofcot
m
fromEquation2.6asafunctionof[Fe
=
H]...52
Figure2.14Distributionofempiricallyassignedmeasurementuncertaintiesfor
theF-dwarfstarsfromtheHettingeretal.(2015)sample.Metallicity
valuesare[Fe
=
H]=

1
:
43and[Fe
=
H]=

0
:
66.........52
x
Figure2.15Distributionof
e=i
,theratioofthestandarddeviationofRVsto
thetypicalmeasurementuncertaintyvalues,fortheF-dwarfstars
fromtheHettingeretal.(2015)sample.Metallicitycutovaluesare
[Fe
=
H]=

1
:
43and[Fe
=
H]=

0
:
66...................54
Figure3.1Mockobservationswith
X
valuesdrawnrandomlyfromtherange
[0,1]and
Y
valuesdrawnfrom
Y
i
=0
:
8
X
i
+0
:
3.Simulatedmeasure-
mentuncertaintieshavebeenaddedwithvaluesdrawnfromanormal
distributionwith

=0
:
0,
˙
=0
:
1....................59
Figure3.2Top:valuesof
m
fromtheMCMCforthe60stepstakenbyall
100chainwalkers.Convergenceisreachedbystep60.Bottom:same
fortheparameter
b
............................60
Figure3.3Left:parameter-spacelocationofthe60stepstakenfor7of
100chainwalkers.Starsrepresenttheinitalpositionofeachchain
walker,andsubsequentstepsaredisplayedascirclesofdecreasing
radius.Right:acloserviewofthecenterofthechainstepdistribution.61
Figure3.4Linearsolutionsfor300randomlyselected(
m
,
b
)pairssampledfrom
theMCMCposteriordistribution....................61
Figure3.5Posteriorprobabilitydistributionfor:(a)thejoindistributionofpa-
rameters
m
and
b
,(b)parameter
m
,marginalizedoverparameter
b
,and(c)parameter
b
,marginalizedoverparameter
m
.Bluelines
indicatevaluesof
m
and
b
usedinthecreationofthemockdatapoints.62
Figure3.6Two-parameterjointprobabilitydistributionsforplate-plugging2085-
53379.Allvaluesareinkms

1
.Fullymarginalized,single-parameter
probabilitydistributionsoccupythesubplotsonthediagonal.Param-
eters
dy
i
(

i
inthetext)representthecorrectionstobeappliedto
the
i
thexposureontheplate......................65
Figure3.7Individualsub-exposurespectra(top)usedintheproductionofthe
coaddspectrum(bottom)forerID2939-54515-194.........69
Figure3.8Parametervalueprogressionforall200chainwalkersinthemultiplic-
ityMCMCforerID2939-54515-194.Chainsampleshavealready
beenthinned.Samplesearlierthanthereddashedlinewereremoved
fromanalysisduringtheburn-inprocess................71
Figure3.9Orbitsconstructedfrom200randomsamplesoftheMCMCposterior
distributionforberID2939-54515-194.................72
xi
Figure3.10Posteriorprobabilitydistributionsofparametersinthemultiplicity
MCMCforerID2939-54515-194...................73
Figure3.11ChangesinredshiftwithtimeforerID2960-54561-375.Absorp-
tionlinesfromnormalizedsub-exposuresareorderedchronologically
fromtoptobottom.Velocitiesineacharerelativetothe
rest-framewavelengthofCalciumK(left),CalciumH(middle),and
H

(right).Dashedverticallinesrepresentthemeanvelocityofthe
star.....................................74
Figure3.12Orbitsconstructedfrom200randomsamplesoftheMCMCposterior
distributionforberID2960-54561-375.................75
Figure3.13Posteriorprobabilitydistributionsofparametersinthemultiplicity
MCMCforerID2960-54561-375...................76
Figure4.1Left:Metallicitydistributionfor14,302F-dwarfs.Right:Distribu-
tionofmaximumtimelagbetweentheandlastexposureofa
star.....................................82
Figure4.2Mean(left)andstandarddeviation(right)ofradialvelocitieswithin
astar.Variationsinthestandarddeviationofvelocitiesared,
inpart,bythelargermeasurementuncertaintiesformetal-poorerstars.83
Figure4.3Distributionof

,thefractionofposteriorsamplesusingthebinary
model,forstars..............................86
Figure4.4Averagedprobabilitydistributionsoflog
P
forallbinarydetections
(
>
0
:
80).Thesedonotactualdistributionsofperiods,and
shouldonlybeusedasaguidetoprobetheregionofMCMCsensi-
tivity.TheshadedregionindicateswhereRochelobeovwand
contactbecomesrelevant.Thedashedlinemarksthecircularization
limitataperiodof12days.......................88
Figure4.5Short-periodbinaryfractionlimits,relativetothemetal-richgroup.
Binarycompaniondetectionsarebyacutin

,thefraction
ofposteriorsamplesusingthebinarymodel.Groupmedianvaluesof
[Fe
=
H]areused..............................89
xii
Figure5.1Distributionof
e=i
valuesforasimulatedpopulationbasedonobser-
vationsanduncertaintiesinthemetal-richpopulationofHettinger
etal.(2015).Modelsillustratevariationsinthe
e=i
distributionfrom
changesinseparationdistributionpowerlawindex

,whilekeeping
theshort-periodbinaryfraction
f
b
=0
:
03andthemassratio
distributionpowerlawindex

=0
:
0..................94
Figure5.2Distributionof
e=i
valuesasinFigure5.1,withavarying
f
b
,and


=

1
:
0and

=0
:
0.......................94
Figure5.3Distributionof
e=i
valuesasinFigure5.1,withavarying

,and

=

1
:
0and
f
b
=0
:
03.........................95
Figure5.4
e=i
distributionforthemetal-poorgroup.Model
e=i
distributionsare
shownusingvaluesof

and
f
b
randomlysampledfromtheMCMC
posterior.Thedashedlinerepresentsthebelowwhichbin
heightswerenotusedinthelikelihoodfunction............97
Figure5.5SamedistributionasinFigure5.4,forthemetal-intermediategroup.97
Figure5.6SamedistributionasinFigure5.4,forthemetal-richgroup......98
Figure5.7PosteriordistributionfortheMCMCrunofthemetal-poorgroup,
withparametersfortheshort-periodbinaryfraction
f
b
,andsepara-
tiondistribution

............................98
Figure5.8PosteriordistributionfortheMCMCrunofthemetal-intermediate
group,withparametersfortheshort-periodbinaryfraction
f
b
,and
separationdistribution

.........................99
Figure5.9PosteriordistributionfortheMCMCrunofthemetal-richgroup,
withparametersfortheshort-periodbinaryfraction
f
b
,andsepara-
tiondistribution

............................99
Figure5.10PosteriordistributionfortheMCMCrunofthemetal-richgroup,
withanextendedpriorlimitin

....................100
Figure5.11PreliminarydistributionofmaximumRVvariationforDR12APOGEE
targets...................................102
xiii
Chapter1
ABriefIntroductiontoStellar
Multiplicity
1.1Introduction
Stellarmultiplicityplaysakeyroleinstarformation(Krumholzetal.,2012;Bate,2014),
stellarevolution(Paxtonetal.,2015),thechemicalevolutionofgalaxies(Kobayashietal.,
2006),andthestudyofunresolvedstellarpopulations(Conroy,2013).Manyinteresting
phenomenainastrophysicsarerelatedtointeractingbinarysystems.Theseincludethe
post-commonenvelopebinaries(Schreiber&ansicke,2003):cataclysmicvariables,classical
novae,X-raybinaries,gamma-raybursts,andSNIaprogenitors.
Thecurrentunderstandingofbinarystatisticsislimitedduetoayinobtaining
populationsamplesthatarecompleteandunbiased.Sp,theshort-periodsystems
thatwillonedayleadtocommonenvelopeepisodesrequireexpensivespectroscopicobserving
campaigns.Completesurveyshaveonlybeenpossibleinthesolarneighborhood(Duquennoy
&Mayor,1991;Raghavanetal.,2010),andprovideonlyafractionoftheglobalmultiplicity
propertiesoftheMilkyWay.ThismotivatedHettingeretal.(2015)toinvestigateastatistical
approachformeasuringmultiplicitypropertiesofstarsthroughtheraw,sub-exposuresthat
compriseeachindividualspectruminmerspectroscopicsurveys.
1
Thischapteractsasabriefintroductiontobinariesandmultiple-starsystems.Webegin
bydiscussingthebasictheoryofstarformation,andbyreviewingthecontendingmechanisms
thataimtoexplainbinarystarformation.Wealsolookattheimportantrolethatbinaries
andbinaryinteractionsplayinourgalaxy.Wediscussthephysicalprinciplesinvolvedin
masstransferbetweencompanions.Additionally,welookatafewselectedexamplesof
interactingsystems,andthephenomenaresultingfrombinaryinteractions.Finally,we
discusswhatweknowaboutthemultiplicitypropertiesofstellarsystemsintheGalaxy,as
wehavelearnedfromobservationsusingavarietyoftechniques.Fromtheseobservations
weoutlinethetrendsseeninmultiplicityproperties,andwediscussareasforimprovement
wherethecurrentunderstandingofmultiple-starsystemsislimited.
1.2StarFormationandtheOriginofBinaryStars
Inthissection,wediscussthephysicalprinciplesofstarformation,whichwillallowusto
considerthepossiblemechanismsfortheformationofbinaryandmultiplesystems.Fora
moredetaileddiscussionontheoriginofbinarystars,werecommendthereviewbyTohline
(2002),andthereferencestherein.
1.2.1PhysicalPrinciples
Theformationofsinglestarsfromthecollapseofagravitationallyboundmolecularcloud
canbebrokenupintoasequenceofestages,originallyoutlinedbyShuetal.(1987).The
molecularcloudisinitiallysupportedagainstcollapsebyamagnetic(Crutcher,2012).
Inthestage(StageI),themagneticleaksoutoftheover-denseregionsofthecloud,
allowingthedenseregionstoformdensercloudcores.InStageII,acondensingcloudcore
2
passesthecriteriaforJeansinstability(discussedbelow)anddynamicallycollapsestowards
stellardensities,leadingtotheformationofaprotostar.Thesurroundingenvelopeofgas
anddustcontinuestofallontotheprotostar,initiatingaccretionandtheformationofa
circumstellardisk.AtStageIII,theinfallingmaterialweakensandthestellarwindsare
abletobreakout,creatingbipolarws.WithStageIV,weseeawideningoftheow
openingangle,andtheprotostarbecomesapre-mainsequence(PMS)star.Finally,Stage
Visreachedafterthenebulardiskdisappears,andthePMSstarcontinuestoevolvetoward
themainsequence.
Itisusefultolookatthephysicalparametersdescribingthecollapseofacloudcore.
Theprotostellarcloudischaracterizedbyitsradius
R
,meantemperature
T
,totalmass
M
,
meanmolecularweight

,androtationalvelocity
!
.Themeandensityofthecloudis
ˆ
=
3
M
4
ˇR
3
:
(1.1)
Thetemperaturevarieswithdensityuponcompressionby
T
/
ˆ


1
;
(1.2)
where

istheadiabaticexponentofthegas.Thisexponentisafunctionofthecloud's
density,andplaysanimportantroleintheformationprocess.Keytimescalesinvolvedin
thecollapseincludethefree-falltime,
t

=
 
3
ˇ
32
G
ˆ
!
1
=
2
;
(1.3)
3
thesound-crossingtime,
t
s
=
R
c
s
;
(1.4)
andtherotationperiodofthecloud,
t
rot
=
2
ˇ
!
:
(1.5)
Additionally,fromKepler'sThirdLaw,thebinaryorbitalperiodis
P
=
 
4
ˇ
2
a
3
GM
tot
!
1
=
2
;
(1.6)
where
M
tot
isthetotalmassofthesystemand
a
isthesemi-majoraxis.Foracloudin
equilibrium,thevirialtheoremstatesthat
E
therm
+
E
rot
=

1
2
E
grav
;
(1.7)
wherethetotalthermalenergy
E
therm
,totalrotationalenergy
E
rot
,andtotalgravitational
potentialenergy
E
grav
,areas
E
therm
˘
3
R
MT
2

;
(1.8)
E
rot
˘
MR
2
!
2
5
;
(1.9)
and
E
grav
˘
3
GM
2
5
R
:
(1.10)
Theconstant
R
inEquation1.8isthegasconstant.Finally,foraslowlyrotatingcloud,the
4
massofavirializedcloudmustberelatedtothedensityandtemperature,fromEquations
1.8and1.10,like
M
equil
˘
5
:
5
 
R
T

!
3
=
2
ˆ

1
=
2
:
(1.11)
If,foragivensetofphysicalvalues,theleft-handsideofEquation1.7islessthanthe
right-handside,thecloudwillcollapse,onafree-falltimescale,duetoalackofkinetic
support.Whenthisistrue,
M>M
equil
(ignoringrotation),andtheJeansinstability
criterion(Jeans,1919)ismet.Inthiscase,
M
equil
isequivalenttotheJeansmass
M
J
.Stage
IIofstarformationbeginswhenacloudcoreacquiresamassthatexceeds
M
J
.
Theadiabaticexponent

playsanimportantroleinthecollapseofacloud.Thecondition
forcollapse,intheabsenceofrotation,requires


E
therm
j
E
grav
j
<
1
2
:
(1.12)
UsingEquations1.2,1.8,and1.10,werevealtherelationshipbetween

and

,

/
ˆ


4
=
3
:
(1.13)
Accordingtothisequation,if
<
4
=
3,theenergyratio
decreases
duringcollapse,meaning
thermalpressurecannotstopthefree-fallcollapseofacloudaslongasthecloudevolves
whileholdingthisrelation.Indeedthevaluesof

duringthecollapseofacloudcorewill
determinetheconditionswherefragmentationmayoccur(ifany)toproduceabinarysystem.
5
1.2.2PossibleBinaryFormationMechanisms
Thepossiblemechanismsforproducingbinarystarsfallintothreebroadcategories:capture,
promptfragmentation,anddelayedbreak-up.Wewilldiscussthesethreemechanisms
beginningwithcapture.
1.2.2.1Capture
Thecapturemechanismstatesthatmoststarsformintosingle-starsystems,andthatbinaries
areformedwhenacompanioniscapturedthroughgravitationalinteractionsafterformation.
Inorderfortwostarstobind,somefractionofenergymustbedissipated(Clarke,1992).
Thiscanbeaccomplishedthroughtransferofenergyinathree-bodysystem,butthese
interactionsarerareoutsideofthedenseenvironmentsofstarclusters.Thelowfrequency
oftheseinteractionsindicatesthatthecapturemechanismcannotbethemainchannelfor
binaryformation.Infact,thePMSstellarpopulationisrichinmultiplesystems,comparable
tomain-sequencestars,suggestingaprimarymechanismthatbeginstoworkbeforeStage
IVofformation(Mathieu,1994).
1.2.2.2PromptFragmentation
Inthepromptfragmentationmechanism,theinitialangularmomentumofthegascloud
causesittospontaneouslybreakintotwopieces,duringorjustafterthefree-fallphase.
Collapseofthemolecularcloudcanbedividedintotwocategories,homologousandnon-
homologous.
Inahomologouscollapse,thecloudismostlysphericalanduniformindensity,witha
cloudmasstlylargerthan
M
J
.Because

decreasesduringtheisothermalphaseof
thecollapse,thelocal
M
J
alsodecreases,andthecloudcloselyapproximatesapressure-free
6
spheroid.Virtuallyallofthemassreachestheatthesametime,andin
onefree-falltimethemassaccretionratebecomesveryhigh.
Ontheotherhand,acloudcancollapseinanon-homologousmanner.Thisoccursifthe
cloudisonlymarginallyJeansunstableandthedensityiscentrallycondensed.Theregions
ofhigherdensityhaveshorterfree-falltimes,sothecentralregionscanrunawayfromthe
lessdenseouterregionsofthecloud.Thiswillleadtoanextendedperiodofmassaccretion
ontothecore.
Three-dimensionalhydrodynamicalsimulationsstronglysupporttheargumentthat,if
fragmentationoccursatall,itdoessoaftertheinitialfree-fallphase,afterthecorehas
collapsedintoaquasi-equilibriumstate(Bate,1998,2012;Truelove
etal.,1998;Tsuribe&Inutsuka,1999).Simulationshavealsoshownthatfragmentation
canoccureasilyafterfree-fallforhomologouscollapse,butnon-homologouscollapseshave
generallynotproducedpromptfragmentation.Becauseofthelimitationsofcurrentsimula-
tions,furtherworkisneededtodetermineifthesefragmentationswillleadtobinarystars,
andatwhatfrequency.
1.2.2.3DelayedBreak-up
Aftercollapseinanon-homologousmanner,theresultingcorewilloftenbestableagainst
fragmentation,andcontainasmallfractionofthecloud'stotalmass.Asmassisaccreted
ontothecore,thecore'sangularmomentumwillincrease,alsoincreasingtheratioof
E
rot
j
E
grav
j
.
Additionally,adiskwillformifithasn'talready.Instabilitiesineitherthecoreorthedisk
mayleadtoabreakupoftheprotostarintoabinarysystem.Variousmethodsofbreak-up
havebeeninvestigated.
7
Lebovitz(1974,1984)revisedtheclassicalformulationofthetheoryofbinary
starformation(Lyttleton,1954),whereabar-likestructureisformedatthecorewhich
evolvesintoapear-shapedordumbbell-shapedstructure.Ifconditionsareright,thebar-
likestructuremaybreakintoseparatecomponents.Hydrodynamicalcodeshavebeenused
tostudywhetherbreakupoccursfromanaxisymmetric(Tohlineetal.,1985;
Pickettetal.,1996;Brown,2000).SuchhavenotleadtoMore
computationally-extensivestudiesoftheoriginallyproposed,non-axisymmetriccasehave
yettobecarriedout.
Analternativemethodofdelayedbreak-upinvestigatesthestabilityoftheaccretiondisk.
Sincemostoftheinfallinggaswillfallontotheaccretiondisk,thediskmaybecomeunstable
whenitacquiresasmuchmassasthecentralcore.Again,simulationsofarotatingdiskare
computationallyintensive,dueinparttothedynamicrangeofsizescalesandthenumber
ofrequiredtimesteps.Thesesimulationsfromlimitedscope,sometimesendingdue
toissueswithboundaryconditions,failingtofullyevolvethetantalizingclumpsinthedisk
(Woodwardetal.,1994;Laughlin&Rozyczka,1996).
1.2.3QuestionsandFutureInvestigation
FollowingtheargumentsofBoss(1988),thereisageneralagreementthatthepreferred
mechanismispromptfragmentation(Clarke&Pringle,1993;Bodenheimeretal.,1993).
Infraredobservationsofprotostarshavebeenrecordedforoveradecadenow(Dunham
etal.,2014)andobservedmultiplesystemsattheearlieststagesofformation(Pinedaetal.,
2015)thatfragmentationplaysacrucialroleinbinaryformation.
Thereisbroadagreementonthefollowingpoints(Tohline,2002).Captureinnota
strongcandidatefortheprimarybinaryformationprocess.Fromobservations,itisknown
8
thatstarsareboundasbinarysystemsbeforeStageIVofstarformation,andcaptureis
toot.Numericalsimulationshaverepeatedlyshownthatcloudsdonotfragment
duringthefree-fallcollapsephase.Instead,cloudscollapsetoabefore
fragmentation(ifany)occurs.Promptfragmentationworksimmediatelyafterthefree-fall
phaseifatfractionofthecloud'smassfallsontotheconwithin
ashorttime.Therefore,cloudsmayfragmentiftheybeginasauniformdensitycloudwith
morethanafewJeansmasses.Non-homologouscollapsedoesn'tseemtoproduceprompt
fragmentation.Axisymmetriccloudcoresdonotfragment,butwillsettleintoaspinning
bar-likestructure,andadiskwillformaroundthecore.Protostellardisksbecomeunstable
towardlong-wavelengthstructureswhenmasscontainedinthediskbecomescomparableto
themassofthecore.
Somequestionsontheformationprocessesofbinarystarsincludethefollowing.If
promptfragmentationistheprimarymechanism,howdomolecularcloudsformsuchthat
theycontaintherequiredfewJeansmasseswithoutinitiallycollapsing?Magneticmay
playatrole(Palauetal.,2013).Dothefragmentsofpromptformationleadto
binarysystems?Currentsimulationsareunabletocontinuemuchfurtherthantheinitial
instantoffragmentation,andmoreworkisneededtoevolvethesesystemsmore.With
thecollapseofaxisymmetriccoresintoellipsoidswithbar-modeinstabilities,willfurther
evolutionoftheinstabilitiesproducethebinariespredictedbytherevisedmodel
hypothesizedbyLebovitz(1974,1984)?Amethodwillhavetobefoundtoslowlyevolve
thealongasequenceofmoreandmoredistortedellipsoids.Finally,how
promisingarediskinstabilitiesasabinaryformationmechanism?Recentnumericalstudies
havesuggestedthatdiskfragmentationmayleadtobinarysystemsinPopulationIIIstars
(Stacyetal.,2010;Latif&Schleicher,2015)andlow-massstars(Stamatellosetal.,2012).
9
Giventhatdynamicalrangeisaproblemforthesesimulations,wewillneedtowaitformore
sophisticatedandetsimulationsandhardwaretoout.
1.3BinaryEvolutionandInteractions
Inthissectionwediscussthebasicprinciplesofbinaryevolutionandmasstransfer.Although
masstransfercanoccurthroughtheaccretionofwinds,wefocusourattentiononRoche-lobe
ovow.Wealsodiscussvarioussystemsandphenomenaresultingfromtheinteractionof
binaries,highlightingthekeyrolethatbinarysystemsplayintheevolutionandfatesof
stars,andinproductionofdramaticeruptions.
1.3.1PrinciplesoftheEvolutionofBinarySystems
Intheco-rotatingframeofabinarysystem,anepotentialcanbederived.Within
thistivepotential,epointscontainzeroepotential,theso-calledLagrangian
points.TheequipotentialpassingthroughtheLagrangianpoint,apointbetweenthe
twostars,theRocheLobe.Mattercanwfromonestartoanotherthroughthe
LagrangianpointinaprocesscalledRoche-lobeovw(RLOF).TheRocheloberadius
onlydependsontheorbitalseparation
a
,andthemassratio
q
=
M
k
=M
j
.Forastarwith
mass
M
j
,andacompanionmass
M
k
,theRoche-loberadiusisapproximatedby
R
L
=
0
:
49
q

2
=
3
a
0
:
6
q

2
=
3
+ln(1+
q

1
=
3
)
(1.14)
(Eggleton,1983).Forlargercompanionmassesand/orcloserseparations,
R
L
foragiven
stardecreases.Likewise,decreasingastar'sownmasswillalsodecreasethatstar's
R
L
.If
10
astar'sradiusisatleastaslargeasthestar's
R
L
,masstransfercanoccurthroughRLOF.
InthecaseofonestaritsRochelobe,thebinarysystemissaidtobesemi-detached.
IfneitherstartheirRochelobe,thesystemisadetachedsystem.And,ifbothstars
theirRochelobes,thesystemisreferredtoasacontactbinary,oracommon-envelope(CE)
binary.
AstheradiusofastarontheMainSequencedoesnotexpandbyalargeamount,itis
muchmorelikelythatRLOFbeginsafterthedonorstarleavesthemainsequence.Since
starsspendthemajorityoftheirlivesonthemainsequence,mostbinariesobservedinthe
skyhavenotyethadanystronginteractions,butmanywillinthefuture.
RLOFcanoccurinoneoftwomodes.Instablemasstransfer,mostofthemasstrans-
ferredoverisaccretedbythecompanionstar,endingwhenmostofthehydrogen-richen-
velopeofthedonorstarhasbeenremoved.Thisprocessresultsinahydrogen-depleted
heliumstar.Therecipientstarwillberejuvenatedifitwerestillonthemainsequence.
Otherwise,therecipientmaybypasstheredgiantphaseandexplodeasabluesupergiant
(Podsiadlowski&Joss,1989).
Thesecondmode,unstablemasstransfer,occurswhentherecipientisunabletoaccrete
allofthematerialtransferredover.Thetransferredmaterialbuildsupontherecipient,
causingtherecipienttoovitsownRochelobe,resultinginaCEsystem.Whenthedonor
losesmassadiabatically,theconvectiveenvelopewillexpandratherthanshrink.Meanwhile
themasstransferofthedonorstarcausesits
R
L
toshrink,leadingtoarunaway,ordynamical
masstransfer.FrictionwithintheCEleadstoaspiral-inuntiltheenvelopecanbeejected,
resultinginorbitalperiodsbetweenabout0.1dand10d.Insomecases,theorbitalenergy
maynotbegreatenoughtoejecttheenvelopeandabinarymergermayoccur,resulting
inasinglerapidlyrotatingstar.Themergerprocessandcommon-envelopeevolution,in
11
general,aretheleastunderstoodprocessesinmultiplicitystudies(Taam&Sandquist,2000;
Podsiadlowskietal.,2001).Ivanovaetal.(2013)presentareviewofthecurrentstandingof
CEevolutionandaguideformovingforward.
Masstransfercanbedrivenbytwomechanisms,eitherexpansionofthedonorstar,ora
lossofangularmomentumfromthesystem.
Asaresultofintrinsicstellarevolution,thedonorstarexpands,andmasstransferoccurs
ataratewhichdependsonthemassratioofthesystem.Ifthedonorstarisinitiallyless
massive,themasstransferwilloccuronanucleartimescaleasthestarevolves.Ontheother
hand,ifthedonorstarismoremassiveinitially,themasstransferwilldrivetheseparation
downinordertoconserveangularmomentum.Since
R
L
ismoredependentonseparation
thanmassratio,masstransferwilloccuratanacceleratedrateuntiltheminimumseparation
isreachedat
M
j
=
M
k
,andfurthermasstransferbeginstoincrease
R
L
.
Thesecondmechanismdrivingmasstransferdoessobydecreasingorbitalseparations
throughangularmomentumloss,eitherbygravitationalradiation,ormagneticbraking.
Gravitationalradiationbecomesimportantforonlytheshortestperiods(
P<
12hr).Mag-
neticbrakingisunderstoodtobeimportantbutthedetailsarestilluncertain.
1.3.2BinaryInteractioninAstrophysicalPhenomena
Interactionamongbinariesplayakeyroleinmanyareasofastrophysics,andunderstanding
theglobalpropertiesofbinarysystemscanhelpelucidatethephysicalprocessesinvolvedin
thecreationofvariousobjectsanderuptiveevents.Belowwebridiscussafewinstances
wherebinarityplaysatrole.Additionaltopicsnotdiscussedhereincludecata-
clysmicvariableeventsresultingfrommasstransferontoawhitedwarf,binarystarmergers
andtheproductionofgamma-raybursts,theofmasstransferontheformationof
12
blackholesvs.neutronstars,thevariousmethodsofmasstransferamongx-raybinaries,
andmore.
1.3.2.1HotSubdwarfs
Thesehot,compactobjectsarehelium-coreburningstarswithmasses0
:
5
M

,andare
depletedofnearlyallhydrogen.Thissingleclassofbinariesillustratesavarietyoft
typesofbinaryinteractions.StudiesbyHanetal.(2002,2003)haveconcludedthatthere
arethreeequallyimportantformationchannels,includingRLOF,CEevolution,andbinary
mergers.AsthesehotsubdwarfsareadominantsourceofUVradiation,ithasbeenproposed
thattheymaybeacontributingfactortotheUVupturnseeninellipticalgalaxies(Han
etal.,2007).
1.3.2.2ChemicallyPeculiarStars
BaandCHstarscontainanoverabundanceofcarbonandothers-processelementsintheir
atmospheres.Theexplanationfortheoriginoftheseoverabundanceswasresolvedbyin-
cludingmasstransferfromabinarycompanion.Inthesesystems,anAGBstarproduces
carbonands-processelementswhicharedredgedup,throughconvection,tothesurface.
Then,masstransferfromtheAGBstarontotheobservedcompanionpollutestheobserved
star,transformingthecompanionintotheBaorCHstarthatweseetoday.Indeed,ob-
servationshaveshownthatallBaandCHsystemscontainbinarycompanions(McClure&
Woodsworth,1990).Similarly,thereisaclassofcarbon-enhancedmetal-poor(CEMP)stars
residingintheMilkyWayhalo.CEMPradialvelocitystudiesalsoreturnbinarycompanion
fractions(atleastforthemostcommonsubclass,thes-processenhancedCEMP-sstars)
consistentwitha100%binaryfraction(Lucatelloetal.,2005).LiketheBaandCHstars,
13
masstransferfromanAGBstaristhefavoredexplanationfortheirexistence(Herwig,2005;
Snedenetal.,2008).AsCEMPstarstracesomeoftheoldeststarsintheGalaxy,theyact
asafossilrecordallowingtheexplorationoftheearlystagesofgalaxyformationandgalactic
chemicalevolution.
1.3.2.3SymbioticBinaries
Theseareinteractingsystemswhereagiantstar(S-type)oraMiravariable(D-type)transfer
massontoawhitedwarf.Bothtypesofsymbioticbinariesinvolvemasstransferinan
atypicalmanner.Theobservedorbital-perioddistributionforS-typesystems(
˘
10

1400d;
Miko lajewska,2007)isnoteasilyexplainedbybinarypopulationsynthesismodels,which
onlyinvolvestableorunstabletransfer.UnstabletransferresultinginaCEphasewould
produceshorterperiodsthanthosethatareobserved,andstablemasstransfertypically
leadstoawideningofthesystem.Thisissue,realizedbyWebbink(1986),mayhave
asolutionwithquasi-dynamicalmasstransfer.Thismaybeachievedifthemassratioof
thebinariesiscloseto1,andthemasstransferrateislargeenoughtoleadtoaCEphase
withouttspiral-in.D-typesymbiotics,ontheotherhand,proposeanotherissue
forcurrentbinaryevolutiontheories.Periodsontheorderof1000yearsseemtoowidefor
binariestobeinteracting.ItseemsthattheMiravariablesareabletotheirRochelobes,
notfromthestellarradiusitself,butratherthroughaslowwind,inwhatisreferredtoas
windRoche-lobeovw.
1.3.2.4BlueStragglers
Theseexceptionallyluminousandbluestarsappeartobeunevolvedandliveonthemain
sequenceforlongertimesthantheirtraditionalmainsequencecounterparts.Itappearsthat
14
bluestragglersarisefromavarietyoftbinaryinteractions.Indeed,bluestragglers
aredividedintotwopopulations,basedoncolors,withonepopulationlikelyresultingfrom
binarycollisions,andtheotherpopulationfrombinarymasstransfer(Piottoetal.,2004;
Luetal.,2010).
1.3.2.5ThermonuclearSupernovae
TypeIasupernovaeareusedasstandardizablecandlesfordistancemeasurements.They
haveprovidedtheindicationforanacceleratingUniverse,andtheyplayanimportant
roleinconstrainingcosmologicalparameters(Riessetal.,1998).Thereisbroadagreement
thatTypeIaexplosionsresultfromathermonuclearexplosionofacarbonoxygenwhite
dwarf.Currently,however,thereiscontinueddebateonwhichprogenitorsystem(s)lead
totheexplosion.Itisclearthough,thatbinaryinteractionsofsomekindarerequired.In
thesingle-degeneratescenario,thewhitedwarfgrowsinmassduetotheaccretionofmass
fromacompanionstarsuchasamainsequencestaroranevolvedstar(Whelan&Iben,
1973).Alternatively,TypeIaeventsmaybeexplainedthroughthemergeroftwowhite
dwarfs,knownasthedouble-degeneratescenario(Iben&Tutukov,1984).Oneapproach
todeterminingtherelativeimportanceofeachscenarioistocomparetheratesofType
Iaexplosionswithprogenitorratesbasedonbinarypopulationsynthesismodels.Abetter
understandingofthemultiplicitypropertiesofthestarformationprocess,andthesubsequent
evolutionofbinarysystemswouldbeinvaluableinthisregard.
1.3.2.6CoreCollapseSupernovae
Asamassivestarevolves,anironcoredevelops.Whentheironcorereachesacritical
mass,thecorecollapsesandbouncesbackdrivingashockthatultimatelyresultsina
15
supernovaexplosion.Asupernovacanbebasedontheobservedcharacteristics
oftheexplosionandtheelementsseeninthespectra.Thediversityinthecharacteristics
canbeunderstoodbyasequenceofincreasedmassloss,withTypeII-Lhavinglostafraction
oftheirH-richenvelopebeforeexploding,SNIbwithnoH-richenvelope,andSNIcwith
noH-richorHe-richenvelopes.Whilestellarwindsmayplayanimportantrole,binary
interactionsmustplayabiggerroleintheremovalofstellarenvelopesthroughmasstransfer
precedingcorecollapse(Podsiadlowskietal.,1992).
1.4EmpiricallyDerivedMultiplicityProperties
Inthissection,weexploreempiricallyderivedmultiplicitypropertiesandtrendsobserved
forstarsintheMilkyWay.Webeginwithadescriptionofmultiplicitysurveytechniques
andmethodologies,followedbywhatweknowaboutthemultiplicitypropertiesofvarious
populations,andhowmultiplicityvarieswithstellarmassandage.Weendthesectionwith
abriefdiscussionontheimplicationofcurrentobservationsasitpertainstoformationmech-
anismsandtheevolutionofbinarysystems.Valuesformultiplicityfractionsandcompanion
fractionsaretakenfromthereviewbyDuch^ene&Kraus(2013),wherethemultiplicity
propertiesofstellargroupsarederivedbycollatingworksfromacrossthediscipline.
1.4.1MultiplicityPropertiesandSurveyMethodology
Withregardstomultiplicity,themostobviouspropertiesofinterestforstellarpopulations
isthemultiplicityfraction(MF),thefractionofstarswithacompanion,andthecompanion
frequency(CF),theaveragenumberofcompanionspertarget.CFvaluescan,inprinciple,be
greaterthan100%,forcaseswhereapopulationhasmanysystemswithhigherordermultiple
16
systems(triple,quadruple,orhigher).Eithertheorbitalperiod
P
,ortheseparation
a
,for
abinarysystemwillbereportedbasedonthemethodofobservation.Perioddistributions
withinapopulationaredescribedbyeitherapowerlawform
f
(
P
)
/
P

,oralog-normal
representationwithparameters
P
and
˙
log
P
.The

=

1powerlaw
distribution,knownas

Opik'slaw(

Opik,1924),issometimesadopted.

Opik'slawsuggestsa
scale-freeprocess,whereasalog-normaldistributionhasascalepreference,meaningthatthe
empiricallyderivedchoiceinparameterizationhasanimplicationforthebinaryformation
process.Massratios,
q
=
M
2
=M
1

1,canbeestimatedthroughuxcomparisons,and
arealsodescribedthroughapowerlawdistribution
f
(
q
)
/
q

.Eccentricities
e
,canbe
usedtoevaluatethedynamicalevolutionofasystem,butcanonlybeestimatedforshort
binarieswheretheorbitisfullymapped.Attheshortestperiods,tidaldissipationwill
leadtoacircularizationoftheorbit(Koch&Hrivnak,1981).Three-bodyinteractionscan
leadtohighlyeccentricorbitsthroughtheKozaimechanism(Kozai,1962),andcouldbea
mechanismformergersintheproductionofTypeIasupernovae(Thompson,2011).
Theobservationaltechniquesusedtodetectbinariesarevariedinimplementationandin
rangeofsensitivity.Afewofthemostcommontechniquesaredescribedhere.Visualbinaries
(VB)aresystemswhereboththeprimaryandsecondaryareseenthroughphotometry,and
theirprojectedorbitscanbemeasuredthroughperiodicmotionsonthesky.Whenonly
theprimarystarisseen,andthesecondaryisinferredthroughthemotionoftheprimary,
thesystemisreferredtoasanastrometricbinary.Visualandastrometricbinariesrelyon
precisepositionmeasurements,and/ortheabilitytoresolvethecompanion.Thus,these
observationaltechniquesarelimitedtowide,long-periodsystems.Spectroscopicbinaries
(SB)aresystemswheretheDopplershiftofone(single-lined)orboth(double-lined)ofthe
starscanbemeasuredthroughlineidenortemplatecorrelationsofstellarspectra.
17
Radialvelocities(RV)aremeasuredatmultipletimes,producingaRVcurvethatcanbe
usedtoestimateorbitalparameters.StrongchangesinRVareeasiertodetect,making
shorter-periodsystemsmorefavorabletodetectionfromspectroscopicmethods.Eclipsing
binariesrequireanearlyedge-onprojectionoftheorbit,butallowthedetectionofabinary
companionthroughvariationsintheprimary'sasthesecondarypassesinfrontofand
behindtheprimary.Theeclipsingtechniquehasthebofallowingadirectmeasurement
ofstellarradiiandstrongestimatesofstellarmasses,butunfortunatelyrequiresasp
inclinationoftheorbitfortheobservationstobemade.
Theoptimalstrategyforcompletingabinarysurveyistogatheracomplete,uniform,
volume-limitedsampleusingavarietyoftheavailableobservingtechniquesinordertoprobe
thedynamicrangeoforbitalperiods.Asamplesizeof100isrequiredtomeasureMFvalues
withaprecisionof5%.Togetasimilarprecisionfororbitalparametersforbinaries,an
evenlargernumberoftargetswillberequiredtogetatnumberofbinarysystems.
Thusfar,completeandvolume-limitedsurveysarerare,limitedtothesolar-neighborhood.
Ingeneral,surveyswillattempttocorrectforcompletenesswithsomeassumptionsabout
thepropertiesofstellarsystemsforwhichtheyarenotsensitive.
1.4.2TrendsandCharacteristicsofMultiple-StarSystems
1.4.2.1MultiplicityandMass
Solar-typestars(0
:
7

1
:
3
M

)areidealforstudyingmultiplicityonthemainsequencedue
totheirrelativelyhighfrequencyandbrightluminosities.Itisnotsurprisingthatthemost
completesurveysofmultiplicityarethoseofsolar-typestarsinthevolumeofspacearound
ourSun.Duquennoy&Mayor(1991)coordinatedthemodern,volume-limitedsurveyof
18
solar-typestarswithanalysisof164objectsoutto22pc,withnewRVmeasurementstaken
inadditiontothosefromotherworks.However,incompletenesswasstillanissue,leading
Raghavanetal.(2010)toimproveupontheworkwithadditionalobjectsforatotalof454
starsoutto25pc,withveryhighcompleteness.Additionalstudiesofsolar-typestarswere
conductedusingindividualdetectiontechniques,suchassurveysofspectroscopicbinaries
(J.L.Halbwachsetal.,2003),visualbinaries(Masonetal.,1998),andcommonproper
motioncompanions(Tokovinin,2011).FromDuquennoy&Mayor(1991)andRaghavan
etal.(2010),CF=62

3%andMF=44

2%forsolar-typestarsonthemainsequence.
Furthermore,itappearsstarsthatareslightlymoremassivethantheSunhaveahigherMF
thanthosejustunder1
M

,withvaluesMF=50

4%andMF=41

3%respectively.This
isatrendwewillcontinuetoseeaswelookatstarsthroughoutthemainsequence.These
modernsamplesofsolar-typestarsexclude

Opik'slaw,withperiodsshowingalog-normal
distributionwith
P
=250yrand
˙
log
P
(d)=2
:
3.Thedistributioninmassratiosfollows
adistribution(

=0
:
28

0
:
05)inratiosallthewaydownto
q
=0
:
1withamarginal
peakat
q>
0
:
95forshort-periodsystems.
Atthelower-massendofthemainsequence(0
:
1

0
:
5
M

),starsaremoreabundant,but
lessluminous.Observationsfromvariousdirectimaging,speckleinterferometry,andradial
velocitystudies,totaling166stars,havebeencollectedbyFischer&Marcy(1992)inoderto
performacompleteanalysisofMdwarfmultiplicity.Additionally,volume-limitedsurveys
oflow-massstarshavebeencarriedout(Reid&Gizis,1997;Delfosseetal.,2004;Dieterich
etal.,2012),includinganearlycompletesamplebytheRECONSconsortiumoutto10pc
(Henryetal.,2006).ThesenearlycompletesurveysgivemultiplicityvaluesofCF=33

5%,
MF=26

3%,andseparationdistributionparameters
a
=5
:
3AUand
˙
log
P
(d)=1
:
3
inalog-normaldistributionforseparationslessthan500AU.Likethesolar-typebinary
19
systems,short-periodM-dwarfbinariesarebiasedtowardhigh-
q
systems.Verylowmass
(VLM,
<
0
:
1
M

)starsareevenfainter,andhavealowerMF,makingcompletestudiesof
asigntnumberofbinarysystemsall-the-moreReviewsofmultiplicityinthis
sub-stellarregimeincludeBurgasseretal.(2007)andLuhman(2012).Estimatesbasedon
incompletesurveyssuggestMF
˘
20

25%.Thedistributioninseparationsismuchnarrower
thanthehigher-masscounterparts,andveryfewsystemswithseparations
a>
50AUare
known.ThemassratiodistributionforVLMstarsisheavilyskewedtowardsequal-mass
systems,withapowerlawindex

=4
:
2

1
:
0.
Issueswithstudiesofintermediate-mass(1
:
5

5
M

)andhigh-mass(
>
8
M

)stars
includetheirlargerdistances,lowernumbers,andveryhighluminositycontrastswiththeir
companions.Thismakesmultiplicitysurveysespeciallyformid-rangeperiods,
whereVBandSBtechniquesarelimited.Additionally,some
˘
30%ofA-dwarfstarsare
chemicallypeculiar,oftenbecauseofbinaryinteractions(Abt,1965).SBsearchesamong
intermediate-andhigh-massstarsincludetheworksofCarquillat&Prieur(2007)andChini
etal.(2012),andVBsearchesincludetheworkssuchasBalegaetal.(2011).Inadditionto
thesestudies,intermediate-andhigh-massstarshavebeenstudiedextensivelyin
theScorpius-CentaurusOBassociation(Kouwenhovenetal.,2007)andotherclustersand
associations(Sanaetal.,2009;Kiminkietal.,2012).Infact,onlyabout20%ofnearbyO
starsarefoundintheandmultiplicitypropertiesoftheareassumedtobesimilarto
thoseinassociations.Althoughincomplete,surveysconcludethatMF
>
50%(1
:
5

5
M

),
MF
>
60%(8

16
M

),andMF
>
80%(
>
16
M

).Theorbitalperioddistributionfor
intermediate-massstarsappearstobebimodalwithpeaksat
P
˘
10dand
a
˘
350AU,and
themassratiodistributionhasyettobefullycharacterized.Forhigh-massstars,theperiod
distributionmayhaveacomplexfunctionalform,possiblywithapopulationofshort-period
20
binaries(log
P
(d)
<
1)forsome30%ofallhigh-massstars,andapowerlawdistribution
extendingoutto10
4
AU.Forhigh-massstarswesee,again,arelativelymass-ratio
distribution(

=

0
:
1

0
:
6for
M>
16
M

)withapeakaround
q
=0
:
8.
Fromstudiesspanningtheentiremainsequencemassrange,itisclearlyseenthatthereis
astrong,monotonicdependenceofmultiplicityonmass,withMFincreasingforhighermass
primarystars.ValuesfromDuch^ene&Kraus(2013)formultiplicityfractionatvariousstellar
masseshavebeenplottedinFigure1.1.Higherordersystemscontainingmultiplestarsalso
Figure1.1:Dependencyofmultiplicityfractionwithprimarymassformainsequencestars
andVLMobjects.ValuesusedfromthereviewbyDuch^ene&Kraus(2013).
favorhighermassprimaries.Orbitalperioddistributionsforsolar-typeandlower-massstars
areunimodal,withthemedianseparationandwidthbothdecreasingforlower-masssystems.
Ontheotherhand,intermediate-andhigh-masssystemshavecomplexdistributionswith
strongpeaksatlog
P
(d)
˘
0

1.AsecondpeakforVBsisobservedforintermediate-
21
massstars,whileashallowpowerlawiscurrentlypreferredforhigh-massstars,although
moreworkisneededinthisarea.Thedistributioninmassratiosisrelativelyforall
massesdowntoabout0
:
3
M

,belowwhichthemassratiobecomesskewedtowardshigh-
q
systems.ThehypothesisofrandompairingofstarsfromanIMFtomakeupthebinary
populationisexcludedfromtheobservations.Eccentricityshowsverylittledependence
onmass,withadistributionforallsystemswithperiodsgreaterthan
˘
100d,and
circularizedeccentricitiesfororbitsshorterthan
˘
10d.
1.4.2.2MultiplicityandAge
PopulationII(PopII)starsaremetal-poorstarsthattracetheMilkyWayhalocomponent,
andgenerallyactasaprobeforinvestigatingtheformationprocessesatearlierchemical
times.Accordingtoananalysisof171highpropermotion,single-linesystems,theproperties
ofPopIISBsappeartobeverysimilartothoseofthemetal-richpopulation(Lathametal.,
2002).ForVBs(
a>
10AU),however,thefrequencyofmultiplesystemsislowerformetal-
poorerstars(ZapateroOsorio&Mart2004;Lodieuetal.,2009).Theoverallfrequencyfor
PopIIsystemsisCF=39

3%forprimariesintherange0
:
5

1
:
3
M

andCF=26

6%
forprimariesintherange0
:
1

0
:
6
M

(Jaoetal.,2009;Rastegaev,2010).Theperiod
distributionforPopIIstarsischaracterizedbyanarrowpeakaroundlog
P
(d)
˘
2

3
andatailoutto10
4
AU.Thereisalsoalackmultiplesystemswithperiodsshorterthan
P
˘
10d.Themassratiodistributionisnotwellconstrained,butappearstoberoughly
uniform.
Toinvestigateyoungerpopulations,studieshavefocusedonopenclusterswithages50
Myrto1Gyr(Patienceetal.,2002;2012),nearbyassociationswithages77-100
Myr(Brandekeretal.,2003;Evansetal.,2012),PMSstarswithages1-5Myr(Luhman,
22
2012),andevenprotostarsintheearlieststagesofevolution(Duch^eneetal.,2007;Connelley
etal.,2008).Thetotalfrequencyforsolar-typestarsinopenclustersisCF
˘
65%,witha
perioddistributionthatisbroadandunimodalandpossiblyindistinguishablefromthatof
stars.Wealsoseeatdistributioninmassratiosforopen-cluster,solar-typestars.A
uniformanalysisofnearbystellarassociationsisstillneeded,butthecharacteristicsofthe
multiplicitypopulationseemstobeconsistentwiththatoftheSurveysofPMSstars
havemeasuredCFvaluesofsolar-typestars(CF
˘
65

80%)tobetwiceashighasthose
ofthemainsequence.Single-starsystemsstillrepresentaquartertoathirdofallPMS
systems,suggestinganincreaseinthenumberofhigh-ordermultiplesystems.
Itisimportanttonotethatpopulationsotherthannearbystarshavenotbeen
studiedwellenoughtodrawstrongconclusionsonpropertiesofmultiplicity.ForSBs,there
arenotamongPMSstars,openclusterstars,PopIstarsandPopII
stars.ForVBs,thereappearstobeadichotomyofstellarmultiplicityproperties,withhigher
CFvaluesperdecadeofseparationamonglowerdensityenvironments,suchasassociations,
thanamongthedenserenvironmentsofclusters.Withineachenvironmentthough,there
doesn'tseemtobeanystrongtrendswithage.FortheolderPopIIstars,asmallerfrequency
ofwidebinariesisobserved,suggestingametallicitydependenceonbinaryformationor
dynamicevolutionontimescalesofGyr.Itisclearthatincompletenessandbiaseshavenot
yetallowedadetailedpictureoftheglobalvariationinmultiplicitypropertiesasafunction
ofage,andthereismuchmoreworktobedoneintheseareasofresearch.
1.4.3Discussion
Here,wereporttheimplicationsofcurrentobservationsonbinaryformationandevolution,
aswellastheissuesbroughtupforfutureresearch,asdiscussedbyDuch^ene&Kraus
23
(2013).MultiplicityisasmoothfunctionofprimarymassforboththeMSandearlier
phases,suggestingthatpromptfragmentationappearstobetheleadingbinaryformation
mechanism.Fragmentationappearstobemild,asseenbythelackofhigher-ordermultiple
systems,withaproductionof1to3corespercloud.Thisisingoodagreementwith
numericalsimulations(Bate,2012).
Thelowermultiplicityobservedinopenclusters,comparedtoloosestellarassociations,
hintsatthepossibilitythatmultiplicitypropertiesarenotuniversalamongbirthenviron-
ments.Unfortunately,currentobservationsdonotprobemultiplicityatanearlyenough
evolutionarystagetobeconclusive(dynamicalevolutionisoverafteronly1Myr,Marks&
Kroupa,2012).Infact,theinmultiplicitycouldbeexplainedthroughdynam-
ics,withassociationsrepresentingebinarydisruption,andclustersrepresenting
edisruption(Kroupa&Bouvier,2003).
Asstatedabove,post-CEbinariesplayacriticalroleinastrophysicalphenomena.Solar-
typebinariesundergoatleastoneCEepisodeiftheirinitialperiodisbelowlog
P
(d)
˘
2
:
8
(Davisetal.,2010).Inthisregime,smalldeviationsfromthelog-normalfunction(Raghavan
etal.,2010)canresultinlargeinthetotalnumberofpre-CEsystems,byasmuch
asafactorofthree.ThisisillustratedinFigure1.2wheretheRaghavanetal.(2010)period
distributioniscomparedwithanalternate,yetdata-consistentmodel.Thispoorhandleon
themultiplicitystatisticsforstarswithlog
P
(d)

3hasprofoundimplications.The
disagreementbetweentheoreticalSNIaratecalculationsfrombinarypopulationsynthesis
andobservationsisafactorofafew(Claeysetal.,2014),similartotheuncertaintyinthe
numberofCEsystemsinFigure1.2.Modelscannotbepastacertainpointwithout
abetterknowledgeoftheinitialmultiplicitystatistics.Similarargumentscanbemadefor
cataclysmicvariablesandnovae(Davisetal.,2012)andX-raybinaries(Podsiadlowskietal.,
24
Figure1.2:Left:OrbitalperioddistributioninthesolarneighborhoodfromRaghavanetal.
(2010).ThelimitforRLOFintheMSisindicatedbytheblackverticaldashedline,andthe
rangecorrespondingtopre-CEsystemsisshadedingray.Theblog-normalfunction
isshowninblack.Thedashedredplotrepresentsamofunctionthatalsothedata.
Right:Comparingthenumberofsystemsintheblog-normaldistribution(solidblack)
andthemomodel(dashedred).Theperiodrangescorrespondingtolow-massX-ray
binaryprogenitors,stablehabitableplanetsaroundbinarystars,andSNIaprogenitorsare
shownwithhorizontalrulers.Theyellowdash-dottedlinemarksthepre-outburstperiodof
V1309Sco.
2003).Inordertotlyexplorethestatisticsofstellarmultiplicityoverawiderange
ofstellarpropertieslikemetallicity,age,ordisk/halomembership,itisnecessarytouse
massivelymultiplexedspectrographs.
Worksinthefuturewillneedtobuildmorecompletesurveysofmultiplicityinavariety
ofstellarpopulations,includingveryyoungPMS,protostellar,intermediate-mass,andhigh-
massstars.Bigquestionsthataretobeaddressedincludethefollowing:Arethemultiplicity
propertiesresultingfromthestarformationprocessuniversalordotheydependonthenative
environment?Aretheresitinthemultiplicitypropertiesofhighest-mass
starsrelativetoallotherstars,ordothepropertiesfollowsmoothlyacrossthesemasses?
ThefrequencyandpropertiesofvisualbinariesappeartobesetbythePMSphase,butis
thisalsotrueforspectroscopicbinaries?
25
1.5Conclusion
Overdecadesoftheoreticalwork,andwiththeimprovementincomputationalpower,we
havereachedabroadagreementthatpromptfragmentationisthelikelyscenarioforbinary
starformation.Therelativeimportanceoftheothermechanismsofstarformationhasyet
beenfullyrealized.Thefuturewilllikelyseeimprovementintheresolutionandphysical
accuracyofsimulations.Wewanttoknowhowfrequent,anditwhatnumber,binaries
areformed,aswellastheirinitialseparationsandmassratios.Theseinsightswillhelp
constrainpopulationsynthesismodelsandstarformationmodels,ultimatelyallowingusto,
forexample,investigatehowclustersandassociationsdissolveandinteract,orhowfrequent
thevarietyofpotentialprogenitorsofnovaeandsupernovaemaybe.
Itisclearthatsingle-starsystemsproduceonlyalimitednumberofthephenomenawe
observeintheskies.Withtheintroductionofabinarycompanion,webegintoseeavariety
ofinterestingandvariedinteractions.Weseeinstancesofmasstransferbetweenobjectsof
allmasses,fromwhitedwarfsandblackholes,tosupergiantsandeverythinginbetween.
Binaryinteractionsproducespectacularexplosionslikenovae,supernovae,andgamma-ray
bursts.ThebinaryinteractionsinvolvedinchemicallypeculiarstarslikeBaandCEMPstars
illustratestheimportanceofunderstandingbinarypropertiesandtherolethatbinarityplays
inobservedsurfaceabundances.Withoutafullunderstandingoftheofmass
transferonobservedabundances,thetaskofmodelingtheprocessesthatoccurtoenrich
theGalaxywithelementsheavierthanhydrogenandheliumremainsdiThus,wewill
continuetoseestudiesfocusedonunderstandingthefrequencyandpropertiesofbinaries,
aswellasthewaysinwhichbinariescanpotentiallyinteract.
Fromobservations,we'vefoundasmoothtrendofincreasingmultiplicitywithmass.It
26
isstilltooearlytotellhowolderpopulationsmayfromyoungerpopulationsinterms
ofmultiplicity,andwhetherdynamicalevolutionand/orintrinsicformationprocessesplay
animportantrole,asmodernstudieshavenotyetbuiltupacompleteenoughsampleof
thesestars.Whetherstarformationischaracterizedbyuniversalmultiplicitydistributions
isanopenquestionandhasimportantbearingonstarformationtheoryandtheinitial
massfunctionetal.,2014).Thebestapproachforcompilingcompletesamplesis
tolookatmultiplicitywithavarietyoftechniquesincludingphotometryandspectroscopy.
Spectroscopyistimeexpensivehowever,andacquiringalarge,completedatasetisanarduous
task.Inthemeantime,weshouldbeutilizinglargemerspectrographsinordertodo
statisticalanalysesoflargecountsofstars,asacomplementtothemorerigoroustarget-
selectedsurveys.
27
Chapter2
Time-ResolvedSpectroscopy
2.1Introduction
Withtheincreasingnumberoflargesurveytelescopes,weareenteringaneraoflarge-data
astronomy(NationalResearchCouncil,2010).Withagreaterthroughputofcelestialobjects
observedandawiderspanofre-observationcadences,anewwindowofobservationhas
opened.Theaccessibilityofthistime-domainwindowisgrowingwiththerecentintroduction
ofprogramssuchasCRTS(Djorgovskietal.,2011),PTF(Lawetal.,2009),Pan-STARRS
(Kaiseretal.,2002),andDES(TheDarkEnergySurveyCollaboration,2005),andwith
upcomingprogramssuchasLSST(LSSTScienceCollaborationetal.,2009).Theincreased
accesstothetime-domaindimensionhasprovidedastronomersanadditionalavenueto
approachresearchinvariousareasincludingexoplanets,variablestars,microlensingevents,
novaeandsupernovae,andactivegalacticnuclei.Wewillalsolikelyseetheemergenceof
unknownphenomenathat,untilnow,havebeenunobservable.
Beforetheadoptionofmerspectrographs,obtainingspectrafortargetsinthesky
wasatime-consumingprocess.DuetothedispersionofthelightsourceacrosstheCCD,
longerexposuretimesareneededtoobtainhighsignal-to-noiseratios(SNR)inspectra.
Largerexposuretimesmeantthatgatheringvaluablespectraforapopulationofobjects
wouldbeinfeasible.Muerspectrographsintroducedparallelizationtotheobserving
28
process.Bundlesofhundredsorthousandsofopticalerscannowbepositionedinthe
togatherlightfrommultiplesourcessimultaneously,greatlyreducingthetime
togatherspectraforgroupsofobjects,andincreasingtheofobservingbyordersof
magnitude.Weareseeinganincreaseinthedevelopmentanduseofmerspectroscopic
instrumentswithLAMOST(Cuietal.,2012),BOSS(Smeeetal.,2013),Hectospec(Fab-
ricantetal.,2005),andothers,andwewillcontinuetoseedevelopmentwithinstruments
suchastheupcomingPFS(Sugaietal.,2012)andDESI(Levietal.,2013).
Withtheincreaseinthenumberofastronomicalsourcesandtheamountofcoverageof
sourcesateachobservation,itisimportanttoconsiderthemethodsusedfordataextraction.
OnegoaloftheworkofHettingeretal.(2015)wastoinvestigatetheusefulnessofamethod
foridentifyingradialvelocity(RV)variabilityinspectraobtainedfromthemerspec-
trographusedintheSloanDigitalSkySurvey(SDSS;Yorketal.,2000).Inthischapter
wepresent,indetail,themethodsadoptedinHettingeretal.(2015)duringthedevelop-
mentofapipelinedesignedforobtainingradialvelocitymeasurementsfromsub-exposure
spectratakenwiththeSDSSmerspectrograph.Section2.2givesabriefdescription
oftheSDSSsurvey,discussingthesub-exposurepropertiesinherenttotypicalmer
spectroscopicsurveys,andalsogivesacautionarywarningwithregardstothesystematic
uncertaintiesobservedintheSDSSsub-exposures.Section2.3detailstheprocessforobtain-
ingradialvelocitiesfromsub-exposurespectrausingtemplatecross-correlation.Wediscuss
empiricalestimatesoftheuncertaintiesinSection2.4.And,onemetricformeasuring
variabilityinasourceisdiscussedinSection2.5,deferringthedescriptionofthe
MCMCmethodofdetectingvariabilitytoChapter3.
29
2.2TheSloanDigitalSkySurvey
SDSSisanimagingandmerspectroscopicsurveyprogramusingthe2.5moptical
telescopeatApachePointObservatoryinNewMexico.SDSSbegandatacollectionin2000,
andhasoperatedthroughthepresentdayintheopticalandnear-infraredwithsurveys
focusedonnearbyanddistantgalaxies,supernovae,quasars,exoplanets,andstarsinthe
MilkyWay.
TheoriginalLegacysurveyfromSDSS-I(2000-2005)imagedmorethan8000deg
2
ofthe
skyineopticalandobtainedspectraofgalaxiesandquasars.Theoriginalspec-
trographwasaeropticalspectrograph,operatinginthe4000

A{9000

Awavelength
range,witharesolution
R
˘
2000andapixelscaleof70kms

1
pixel

1
.Uponcomple-
tionoftheLegacysurvey,SDSShadimagedover2millionobjectsandobtainedspectrafor
800,000galaxiesand100,000quasars.
SDSS-II(2005-2008)sawtheintroductionoftwonewsurveysusingtheoriginalcamera
andspectrograph.TheSupernovaSurvey(Friemanetal.,2008)scanned300deg
2
ofthe
sky,thousandsofsupernovaeandvariableobjects.Thesecondsurvey,theSloanEx-
tensionforGalacticUnderstandingandExploration(SEGUE;Yannyetal.,2009),targeted
240,000starsintheMilkyWay,creatingamapandprovidingadetailedpictureoftheage,
composition,anddistributionofstarsinourGalaxy.
AdditionalsurveyscompriseSDSS-III(2011-2014).UsingtheoriginalSDSSspectro-
graph,theSEGUEsurveywasexpandedwithSEGUE-2(C.M.Rockosietal.,inprepara-
tion),focusingontheMilkyWayHaloandadding120,000stars.TheMulti-objectAPO
RadialVelocityExoplanetLarge-areaSurvey(MARVELS)monitored11,000brightstars
withtheMARVELSspectrograph,lookingforthesignaturesofexoplanets,buthadminimal
30
success.Ahigh-resolutioninfraredspectrographwasaddedfortheAPOGalacticEvolution
Experiment(APOGEE;S.R.Majewskietal.,inpreparation)survey,aimedat100,000red
giantsacrosstheMilkyWay.SDSS-IIIalsocontainedtheBaryonOscillationSpectroscopic
Survey(BOSS;Dawsonetal.,2013),designedtomeasuretheexpansionrateoftheuniverse
throughmeasurementsofthespatialdistributionofluminousredgalaxies.WiththeBOSS
surveycameanupgradetotheSDSSspectrographwithnewCCDsandanincreasefrom
640to1000simultaneouserpluggings.
SDSS-IV(2014-2020)continuestodaywiththeextensionoftwoSDSS-IIIsurveys;APOGEE-
2andeBOSScontinuetosurveystarsintheMilkyWayandbaryonicoscillationsinthe
universe.Additionally,thenewMappingNearbyGalaxiesatAPO(MaNGA)surveywill
examinethedetailedinternalstructureof10,000nearbygalaxieswithintegralunits.
2.2.1Sub-Exposures
WhenacosmicraystrikesaCCDdetector,anartifactappearsasmanypixelsbecome
saturated.Forspectra,thesepixelsoftenleadtoincorrectlyreportedincreases
inatparticularwavelengths.Tofacilitatetheremovaloftheseartifactsfromspectra
inSDSS,theoriginaldataprocessingpipelineusesthemedianvaluesateachpixel
averagedovermultiplesub-exposurestakeninsuccession.Thisrequirementmeansthatall
singleobservationsofanastronomicalsourceactuallycontainseveralsub-exposurestaken
oversomeshortperiodintime.Figure3.7ofSection3.5.1providesanexample,foran
F-typedwarfstar,ofacoaddspectrumwithcosmicraysremoved,alongwiththeindividual
sub-exposurespectrausedtoconstructthespectrum.Sub-exposurespectra,althoughlower
inSNRandbesetwithcosmicrays,addatimedimensiontothedatathatcanbeusedfor
monitoringspectralvariability.
31
IntheLegacyandSEGUEsurveys,typicalspectraarecomposedofthreesub-exposures,
withtypicalexposuretimesofabout15minutes.Thispresentsthepotentialfordetecting
variabilityinspectrawhichisexpectedtooccurattimescalesontheorderofhours.Addi-
tionally,manyareasoftheskywerere-observedovertheyearsforcalibrationandscien
purposes,yieldingadditionalsub-exposuresandincreasingtherangeoftimescalesthatcan
beprobed.Withthesere-pointingsincluded,sub-exposurecountsperobjectinthesample
ofHettingeretal.(2015)rangefrom3to47,withbaselinesrangingfrom30minutestoover
9years(Figure2.1).
Becausecosmic-rayremovalmustbehandledbyallspectroscopicobservations,sub-
exposuresareanexpectedproductofallmulterprograms.Asweenteraneraoflarge
spectroscopicdatasets,thereisanincreasingpotentialforvariabilitystudiesthroughdata
miningofsub-exposurespectra.Thus,wearemotivatedtodevelopthetoolsandtech-
niquesforextractinginformationfromthesesub-exposures.Weencouragethedesignersof
futuremerprogramstoconsiderthechoiceinsub-exposurefrequencyandcadences,
inadditiontotargetselectionandscheduling,inordertomaximizescienpotential.
2.2.2SEGUEStellarParameterPipelineAndSampleSelection
WiththeintroductionoftheSEGUEsurveyinSDSS-II,aimedatinvestigatingthestellar
propertiesandcompositionoftheMilkyWaycomponents,therewasaconcertedto
buildapipelineforautomaticallydeterminingstellarparametersofstarsfromtheabundant
spectraprovidedbytheSDSSmerspectrograph.TheSEGUEStellarParameter
Pipeline(SSPP;Leeetal.,2008)wasdesignedtocarryouttheseautomatedtasks.The
SSPPprovides,usingavarietyoftechniques,fundamentalstellaratmosphericparameters
suchasmetallicity[Fe
=
H],etemperature
T

,andsurfacegravitylog
g
,aswellas
32
Figure2.1:Distributionofthenumberofsub-exposures(top)andthetimelags(bottom)
fortheF-dwarfstarsfromtheHettingeretal.(2015)sample.Metallicityvaluesare
[Fe
=
H]=

1
:
43and[Fe
=
H]=

0
:
66.
33
spectralAresourcesuchastheSSPPisinvaluableforincreasingsciengains
obtainedfromdataminingtechniquesappliedtolarge,merspectroscopicsurveys.
TheSSPPwasusedforidentifyingthesamplegroupsinHettingeretal.(2015).
StarsofspectraltypeFwereselectedusingthe(Hammermethod)spectral
providedbytheSSPP.Unevolved,mainsequencestarswereusedbyselectingdwarfswith
surfacegravitieslog
g

3
:
75.ThedecisiontouseF-typestarswasbasedonseveralfactors.
Thecombinationofhighfrequency,andtlybrightluminosity,yieldsastatistically
robustsamplewithdecentSNRvalues.TheintrinsicvariabilityinF-typeRVsduetosurface
activityisrelativelylowandwillhaveminimalimpactonmeasurementuncertainties.Also,
F-typestarshavemainsequencelifetimesgreaterthan5Gyr,allowingustoselectunevolved
starsfromboththeyoungerdiskandtheolderhalo.TheHettingeretal.(2015)samplewas
furtherdividedintothreemetallicitygroups,aimedattracingtheMilkyWaycomponents,
usingthestellarparametersprovidedbytheSSPP(Figure2.2).Figure2.3illustratesthe
[Fe
=
H]andlog
g
distributionofF-typestarsintheSSPPDR9dataset.Thebimodal[Fe
=
H]
distributionseparatestheHaloandDiskcomponents,andthetailsatlowervaluesoflog
g
identifythestarsthathaveevolvedawayfromtheMainSequence.
2.2.3PlateSystematics
ThespectrausedinHettingeretal.(2015)arefromtheSDSSLegacy,SEGUE-1,and
SEGUE-2surveys,capturedusingtheoriginalSDSSspectrograph.Theoriginalspectrograph
operatedbyplugging640opticalersintoholes(correspondingtoeachtarget'sposition
onthesky)ononeofthemanypre-drilledobservingplates.EachspectrumwasgivenanID
intheformoferusingtheplateIDnumber,theMoJulianDateofthe
observation,andtheerIDnumber.
34
Figure2.2:DistributionofstellarparametersfortheF-dwarfstarsfromtheHettingeretal.
(2015)sample,includingmetallicity(top),etemperature(middle),andsurfacegrav-
ity(bottom).Metallicityvaluesare[Fe
=
H]=

1
:
43and[Fe
=
H]=

0
:
66.
35
Figure2.3:ScatterplotshowingthedistributionofmetallicityandsurfacegravityforF-
typestarsintheSSPPDR9.Thebimodaldistributionof[Fe
=
H]tracestheHaloandDisk
componentsoftheMilkyWay.
36
AftermeasuringRVsforF-typestarswiththemethodsdetailedinSection2.3,correla-
tionsinRVvariationswerefoundforersresidingonthesameplate.Correlationswere
typicallyseenamongersthatwereadjacenttoeachotherontheCCD.Figure2.4illus-
tratesthesystematicissueswithanexamplefromplateplugging2085-53379.This
showsthemeasuredRVsofthefoursub-exposuresforeachoftheeightF-typedwarfsfound
onthisplate.ItisclearlyseenthattheRVsofthestarsdecreasesystematicallyfromthe
tosecondsub-exposures,decreasefurthermorefromthesecondtothirdsub-exposure,
andincreaseslightlyonthelastsub-exposure.
Tocorrecttheplatesystematics,weadoptedaMarkovchainMonteCarlo(MCMC)
methodtoestimatethesystematicshiftsfromsub-exposuretosub-exposure(Section3.4).
10,264ersub-exposureswereexaminedacross2453plates.TheRVcorrectionsapplied
tothesesub-exposuresaredistributedasseeninFigure2.5,withastandarddeviationof
2
:
2kms

1
andcorrectionsaslargeas17kms

1
.
Unfortunately,notallplatescouldbecorrectedusingtheMCMCmethods,becauseof
multiplecorrelatedsubsetsofersonthesameplate.Usingplate3002-54844(Figure
2.6)asanexample,wecanseewhysimpletechniquestocorrectsystematicshiftsarenot
successful.Inthisplate,theRVsfromtheF-dwarfstarsthatwereexposedononeareaofthe
CCD(Panela)arecorrelatedwitheachother,butanti-correlatedwiththeersexposed
onanotherareaoftheCCD(Paneld).Becauseofthis,asystematicshiftinRVforasingle
exposureonaplatedoesnotincreasethetotalvalueoftheMCMClikelihood(Equation
3.7).ItisshownfromthecorrectioncolumnofFigure2.6,thatthesystematicswerenot
resolved.Thisleadtotheinspectionandremovalof25plates(Table2.1),requiringmanual
inspectionofsub-exposureRVshifts.
Itisunclearatthispointwherethesourceofthesystematicshiftsinradialvelocityseen
37
Figure2.4:RVsforF-dwarfstarslocatedonplateplugging2085-53379beforeandafter
correctingtheplateforsystematicsub-exposureRedpointsaresub-exposuresthat
havelowSNR.FiberIDsaregivenforeacherontherightaxis.Sub-exposuresinthe
plateareorderedchronologically.Correctionstosystematicaresuccessfulonthis
plate.
38
Figure2.5:Distributionof10,264systematicRVestimatedforallplatesub-exposures
intheF-dwarfsample.
Table2.1:SuspectPlatesinSDSS
Plate-MJD
0888-523392252-535652683-541532890-544952940-54508
1665-529762252-536132701-541542899-545683111-54800
2042-533782393-541562839-544612900-545693166-54830
2053-534462670-541152856-544632905-545803187-54821
2055-537292682-544012861-545832911-546313207-54850
PlateswithsystematicuncertaintiesremovedfromHettingeretal.(2015),listedbyplate
numberandModiJulianDate.
onplatesintheSDSSsub-exposurescanbetracedto.Westronglyencourageindividuals
whowishtousesub-exposureinformationfromtheSDSSspectrographtobeawareofthese
issuesandtakethemintoconsiderationwhenanalyzingradialvelocities.
39
40
(a)
(b)
(c)
(d)
Figure2.6:SameasFigure2.4forplateplugging3002-54844.TheleftcolumnofeachshowsRVsbeforecorrecting
theplateforsystematicandtherightcolumnshowsRVsaftercorrectingforCorrectionsarenotsuccessfulon
thisplateduetoanti-correlatedsubsetsofsimilarlycorrelateders.
2.3RadialVelocities
RadialvelocitiescanbederivedfromspectrausingtheDopplershiftbycomparingthewave-
lengthsoftheobservedabsorptionfeaturesofthesource

o
,totherest-frameexpectations

e
.Thevelocity
v
,relativetothespeedoflightis
v
c
=

o


e

e
:
(2.1)
Whereappropriate,thisisaccomplishedthroughacross-correlationoftheobject'sspectrum
withatheoreticalorempiricallyderivedtemplate.Inthissection,wediscussthemethods
developedinthisworkfordeterminingRVsthroughtemplatecross-correlation,referencing
theimplementationofthesemethodsinHettingeretal.(2015).Weaddresstheprocesses
fornormalizingspectra,creatingaspectraltemplate,andperformingthecross-correlation
measurements.
2.3.1ContinuumNormalization
Cross-correlationrequiresthatspectrabecontinuumnormalized,suchthattde-
viationsfromunityareattributedtoabsorptionandemissionfeatures,ratherthanthe
blackbodycontinuum.Theprocessofnormalizingaspectrumisasfollows.
Amodelofthecontinuumisestimatedbygeneratingamoversionofthespectrum,
whichisagreatlysmoothedcopyofthespectrum.Thesmoothedcopyhasalloutlying
pixelscleanedandsetequaltothemedianvalue.Absorptionlinesmustbemaskedout
priortomodelingthecontinuum.Thepatchedregionstobemaskedarespexplicitly
withasetofwavelengthlimitswheretheabsorptionfeaturesareexpectedtoappearforthe
41
particularastronomicalsource.SeeTable2.2,forexample,foralistofthemajorregionsthat
weremaskedoutintheF-dwarfstars.Theentiretyofeachregionisreplacedwithapatch
Table2.2:AbsorptionFeaturesinF-dwarfs
WavelengthRangeFeature
3600

A{4000

ACaH,Kandothers
4070

A{4130

AH

4290

A{4360

AH

4830

A{4890

AH

6540

A{6580

AH

value,calculatedbytakingthemedianofallvaluesintheregionwherethevalues
aregreaterthanthemedianvalue.Inotherwords,theregionispatchedwithacontinuous
valueequaltoanaveragevalue,ignoringabsorptionfeatures.SeeFigure2.7foran
exampleofanF-typestarwiththeabsorptionregionspatchedout.Maskingabsorption
featuresthisway,onarelativelysimplespectrumsuchasthatfromanF-typestar,can
bedoneeasily,however,spectrawithmoreintenseormorecomplicatedfeatures,suchas
thoseofanM-dwarf,wouldrequireanalternativemethodforcontinuumnormalization(Ness
etal.,2015).Withtheabsorptionregionsmaskedout,asmoothedversionofthespectrum
isproducedbyremovinghighfrequenciesintheFourierTransformusingaFFTsmoothing
algorithm.Finally,theoriginalspectrumisdividedbythesmoothcontinuumtoobtain
thecontinuumnormalizedspectrum.Itisimportanttoperformthenormalizationprocess
correctly.Earlyattemptswithpoorcontinuumattheendsofthespectraresultedin
asymmetricdistributionsinmeasuredradialvelocities.
42
Figure2.7:ContinuumnormalizationprocessappliedtoanF-typestarshowing:(a)theraw
sub-exposurespectrum,(b)thespectrumwithselectedabsorptionfeaturesmaskedout,(c)
asmoothedversionofthespectrum,and(d)thecontinuum-normalizedspectrum.
43
2.3.2SpectralTemplate
TwocommonmethodsformeasuringtheRVofasourceare,idenionoflineindices,
andtemplatecross-correlation.Theformerusesanalgorithmtoidentifyabsorptionand
emissionfeaturesinspectrathatareexpectedtobepresent,andcomparesthemeasured
centroidsofthelineswiththeexpectedrest-framewavelengthsofthefeaturestodirectly
measureaDopplershift.Methodsforlocatingcentroidsandcorrectlyidentifyinglines
associatedwithparticularfeaturescanberelativelycomplex.Thelattermethod,using
templatecross-correlation(Tonry&Davis,1979),ismoresimplebutrequiresanaccurate
templatespectrumforeachsourcethatyouwishtomeasureRVsfrom.InHettingeretal.
(2015),allsourcesareF-typedwarfstars,andcanbewithasingletemplate.Oneb
ofperformingcross-correlationonalargedatasetofsimilarsources,isthatatemplatecan
beconstructedfromthedataitself.Amastertemplate(Figure2.8)wasconstructedusing
highqualityspectrafrom7207F-typedwarfstarsfromSDSSDR10.
FirstattemptsatmakingRVmeasurementswithtunedtemplatesyieldedlittle
changeinmeasuredvariationsinRV.Wecreatedseveraltemplatescomposedofsourceswith
similarstellarparameters,allowing[Fe
=
H],log
g
,and
T

tovary.Thestandard-deviationof
individualradialvelocitieswithinstarsremainedvirtuallythesameacrosstheentire[Fe
=
H]
range,regardlessofthechoiceintemplate.SimilarbehaviorwasobservedforRVvariations
asafunctionoflog
g
and
T

.Forsimplicity,weusedasingletemplatecomposedofspectra
fromstarsofvaryingstellarparameters,usingallavailablescienceprimaryerswitha
co-addSNR
>
50.
Stepsforcreatingatemplatespectrumareasfollows.Allinputco-addspectraare
continuum-normalized(Section2.3.1).Next,theco-addspectraarede-shiftedintotherest-
44
Figure2.8:Fullcontinuum-normalizedF-typedwarftemplatespectrum(top)withadetailed
viewoftheblueendofthespectrum(bottom).Prominentspectralfeaturesareannotated.
frameusingthe\zBest"redshiftvaluesassignedtotheco-addsintheSDSSpipeline.It
shouldbenotedherethatwhiletheSDSSpipelineprovidesRVestimatesfortheco-add
spectrumofeacher,thesewereobtainedfromanaverageoftheindividualsub-exposures,
therebyaveragingoverthechangesinRVbetweensub-exposures{thechangesthatwe
areseekingtomeasure.Onceallofthespectraareshiftedtorest-framewavelengths,each
spectrumisbyathird-orderB-splineandresampledtoacommonwavelengthsolution.
Finally,allco-addspectraareaveragedbytakingthemeanateachwavelengthinthe
resampling.Figure2.9illustratesthetemplatecreationprocessforasmallsetofinput
co-adds.
2.3.3Cross-Correlations
RadialvelocitiesobtainedfromtheDopplershiftaremeasuredbythewavelength
shift,inpixels,ofthesourcewhichmaximizesthecross-correlationcoentofthesource
andtemplate.Thisprocessisdetailedhere.Foreverystarinthesample,eachsub-exposure
45
Figure2.9:Exampleofthetemplatecreationprocessusing7spectra.Fromtoptobottom:
blueendandfullspectrumoftheinputco-addstellarspectra,blueendandfullspectrum
ofthenormalized,rest-frameinputspectra,blueendandfullspectrumoftheinputspectra
resampledtoacommonwavelengthsolution,blueendandfullspectrumofthetemplate
(averagedofresampledinputspectra).
46
spectrumisnormalized(Section2.3.1)andcleanedofcosmicraysandothersuspectpixels.
Next,thetemplatespectrumisresampledsothatthesourceandtemplatespectrasharethe
samewavelengthvalueateachpixel.Thecorrelationcotperpixel,
C
=
1
N
N
X
i
=1
y
1
;i

y
2
;i
(2.2)
iscalculatedusingthecleanedsourcespectrumandthetemplatespectrum,repeatedwith
integerpixelshiftsofthesourcespectrumfrom

20to+20pixels.Thisprocessproduces
across-correlationfunction(CCF)thathasamaximumvalueatthebestestimateforthe
Dopplershiftofthesource.AnexampleCCFcanbeseeninFigure2.10.
Figure2.10:Cross-correlationfunction(perpixel)forasinglesub-exposurespectrumofan
F-typestar.Correlationvaluesarecalculatedatintegerpixellags,andasplineinterpolation
ofthefunctionistothesevalues.Thefunctionpeaksat

3
:
275pixels,or

229kms

1
.
ThevalueobtainedforthepixellagcanbeconvertedtoaDopplershiftinkms

1
withaconversionthatisdependentontheresolutionofthespectrum.ForSDSSspectra,
thisconversionis70kms

1
pixel

1
.Toobtainsub-pixelprecision,asmoothB-spline
47
interpolationisperformedontheCCF,yieldingabetterestimatefortheDopplershift.
2.4EmpiricalUncertainties
Uncertaintiesincross-correlationlagmeasurementsmustbeestimatedempiricallyorthrough
someMonteCarlomethod.ForanexampleofaMonteCarlomethod,seePetersonetal.
(1998),wherecross-correlationsareperformedovermanyiterations,takingrandom
tionsinpixelandselectingarandomsub-setofpixelsateachiteration.Hettingeretal.
(2015)employedanempiricalmethod,lookingatthespreadinRVmeasurementsforspectra
ofsimilarquality.Fromthesecomparisons,wederivedafunctionthatassignsuncertainties
basedontheSNRand[Fe
=
H]ofeachsub-exposure.Measurementuncertaintiesareexpected
toincreaseforstarswithlowermetallicity,asthesestarshavelesspronouncedabsorption
features,therebyreducingthesignalof,andbroadeningthepeakof,theCCF.Thefollowing
describesthestepstakentoderivetheempiricaluncertainties.
Tobegin,themeanRVforastarissubtractedfromallsub-exposuremeasurementsof
thestar,resultinginnewRVswithameanvalueof0
:
0kms

1
.Similarmeasurementscan
nowbecomparedforallsub-exposuresfromallstarsbyidentifyingspectrawithsimilar
[Fe
=
H]andSNRvalues;initialtestsshowednotcorrelationbetweenmeasurement
uncertaintyandotherstellarparameterssuchas
T

andlog
g
.Onceallmeasurements
areseparatedintosetsoflike[Fe
=
H]andSNR,anempiricalestimateforthemeasurement
uncertaintyforeach[Fe
=
H]-SNRsetiscalculatedusingthemedianabsolutedeviation,or
MAD(Leysetal.,2013).TheMADvalueisthemediandeviationofallvaluesfromthe
medianvalue,
MAD=median(
j
RV
i

median(RV)
j
)
;
(2.3)
48
andisrelatedtotheempiricalestimateofthemeasurementuncertaintyby
˙
=1
:
4826MAD
:
(2.4)
OneexampleofanempiricaluncertaintyestimateisshowninFigure2.11.Allsub-exposures
usedinthis[Fe
=
H]-SNRsethave[Fe
=
H]=

1
:
75

0
:
25andSNR=30

2
:
5.Thisprocess
isrepeatedforall[Fe
=
H]-SNRsetsthathaveatleast800measurements.
Figure2.11:Distributionofradialvelocities(minussystemicvelocity)forsub-exposures
with[Fe
=
H]=

1
:
75

0
:
25andSNR=30

2
:
5.Thedistributionhasamedianabsolute
deviationofMAD=3
:
04kms

1
,indicatingmeasurementuncertaintiesof
˙
=4
:
51kms

1
.
Interpolationofthevaluesobtainedfortheempiricaluncertaintyestimatesofall[Fe
=
H]-
SNRsetsyieldsafunctionalformdescribingthemeasurementuncertaintywithrespectto
[Fe
=
H]andSNR.First,a1-dimensionalsolutionistoall[Fe
=
H]-SNRsetssharingthe
49
same[Fe
=
H]valueusingtheform
˙
[Fe
=
H]
(SNR)=
m
[Fe
=
H]
SNR
+
b;
(2.5)
wheretheuncertainty
˙
[Fe
=
H]
,isinverselyproportionaltoSNRwithconstants
m
[Fe
=
H]
and
b
.Thisisperformedindependentlyforeachofthenine[Fe
=
H]valuesusedinthesets.The
medianvalue
b
=1
:
235isadoptedandusedtoallsolutions,
˙
[Fe
=
H]
(SNR)=
m
[Fe
=
H]
SNR
+1
:
23
:
(2.6)
TheresultsfromthisareshowninFigure2.12(b).Alinearrelationshipbetween[Fe
=
H]and
theconstant
m
isdescribedby
m
([Fe
=
H])=

26
:
51[Fe
=
H]+50
:
52
;
(2.7)
andisillustratedinFigure2.13.Altogether,therelationshipbetweentheempiricalestimate
ofthemeasurementuncertaintyandthesub-exposure[Fe
=
H]andSNRtakestheform
˙
([Fe
=
H]
;
SNR)=
(

26
:
51[Fe
=
H]+50
:
52)
SNR
+1
:
23
:
(2.8)
Figure2.12(c)illustratestheformoftheempiricaluncertaintieswithvaluesplotted
inlinesofconstant[Fe
=
H].Hettingeretal.(2015)adoptedEquation2.8toassignuncertainty
valuestoeverysub-exposuremeasurementinthesample.Thedistributionoftheassigned
uncertaintiesisshowninFigure2.14.
50
Figure2.12:Empiricaluncertaintiesforcross-correlationmeasurementsinF-typedwarfstars
showinguncertainties(a)asafunctionofSNRindependentlydeterminedforeach[Fe
=
H],
(b)asafunctionofSNRwithacommonconstantand(c)asafunctionofSNRand
[Fe
=
H]asinEquation2.8.
51
Figure2.13:Valuesofcot
m
fromEquation2.6asafunctionof[Fe
=
H].
Figure2.14:DistributionofempiricallyassignedmeasurementuncertaintiesfortheF-dwarf
starsfromtheHettingeretal.(2015)sample.Metallicityvaluesare[Fe
=
H]=

1
:
43
and[Fe
=
H]=

0
:
66.
52
2.5
e=i
Variability
Aftersub-exposureRVsaredeterminedandtheuncertaintiesarewellcharacterized,ametric
mustbechosentodescribethevariabilityofthesource.Onesuchmetric,describedbyGeller
etal.(2008)andMillimanetal.(2014),detectsvariabilityinRVmeasurementsbycomparing
theRVdeviations(externalvariation,
e
)tothetypicalmeasurementuncertainty(internal
variation,
i
),or
e=i
.Thisway,starswithSNR,andthereforeentmeasurement
uncertainties,areweightedappropriately,preventingstarswithlargeRVuncertaintiesfrom
beinginterpretedastruevariables.The
e=i
metricisnedastheratioofthestandard
deviationofthestar'sRVstothemeanofthemeasurementuncertainties.Valuesof
e=i
tlylargerthan1
:
0indicateanincreasinglikelihoodthatastar'svariabilityisnot
duetomeasurementuncertainties,butrathertotheintrinsicchangesinthestar'sRV.
Onedisadvantageofthe
e=i
method,however,isthatitdoesnotuseaphysicalmodel
todescribethesystem.Forinstance,RVvariabilityofastarinthepresenceofacompanion
bodyisexpectedtoshowaperiodicbehavior.Thislackofspyiswhatledusto
adopttheMCMCmethodsdetailedinSection3.5.Itisreassuringthough,that
examinationofthe
e=i
valuesofthestarsintheHettingeretal.(2015)sample(Figure
2.15)yieldsaconclusionconsistentwiththatgatheredfromtheMCMCmethod,withboth
methodsindicatingahigherfractionofshort-periodbinariesinthemetal-richcomponentof
theMilkyWay.
2.6Discussion
Timeresolved-spectroscopyisnotlimitedtoradialvelocitymeasurements.Theprocesses
discussedinthischaptercanbemotolookatchanges,forexample,intheemissionand
53
Figure2.15:Distributionof
e=i
,theratioofthestandarddeviationofRVstothetypical
measurementuncertaintyvalues,fortheF-dwarfstarsfromtheHettingeretal.(2015)
sample.Metallicityvaluesare[Fe
=
H]=

1
:
43and[Fe
=
H]=

0
:
66.
absorptionfeaturesofstarssuchasthoseduetosurfaceactivityofcoolstarsorpulsationof
RRLyraestars.Futuredevelopmentsintelescopesandinstrumentsmayevenleadtouseful
studiesofshort-timescalevariabilityofactivegalacticnuclei.Werecommendthatfuture
observingprogramsconsiderthesub-exposurepropertiesofmerspectrographs,as
theyaddscienvalueinregardstothetime-domaindimensionofthedata.Withcareful
planningoftheexposuretimes,counts,andfrequenciesadoptedintargetselection,the
scienreturnobtainedfromsurveyprogramscanbemaximized.
54
Chapter3
MarkovChainMonteCarlo
3.1Introduction
Manyproblemsinastronomyandastrophysicsaresolvedusingcomplicated,costlymodels
withlargenumbersoffreeparameters,oftencombinedwithlowsignal-to-noiseobservations.
Becauseofthis,therehasbeenanincreasedadoptionofprobabilisticanalysis,suchas
Bayesianinference.Additionally,withtheincreaseinpowerandciencyofmodernday
computers,numericalmethodslikeMarkovchainMonteCarlo(MCMC)methodsarebeing
usedtosolveproblemsthatwouldhavebeenpreviouslyunsolvable.Hoggetal.(2010)include
moreargumentsforwhyonewouldwanttoadoptnumericaldataanalysistechniques,and
additionallyprovideinstructionsforusingMCMCtechniquesformodelstodata.
Inthischapter,wedescribehowMCMCmethodscanbeusedinanastrophysicalcontext,
referringtotheiruseinHettingeretal.(2015)asanexample.Webeginwithabrief
introductiontoBayesianinferenceinSection3.2.InSection3.3,weintroducethe
emcee
PythonpackageandexplainhowanensembleMCMCalgorithmisusedtoexpectation
valuesofparametersofinterestfromamarginalizedsamplingoftheposterior.InSection3.4,
weillustratetheuseofthe
emcee
packageinidentifyingsub-exposuresystematicsinSDSS
spectra.Section3.5adescriptionoftheprocessfordeterminingstellarvariabilityfrom
theRVcurveofastarusingahierarchicalMCMCmethod.
55
3.2BayesianInferenceandMCMC
WebeginwithabriefintroductiontoBayesianinference.Bayesianinferencederivesa
posteriorprobabilitythatobservations
D
,canbedescribedbysomemodelwithavectorof
parametersasitrelatestothelikelihoodfunctionofthemodelandapriorprobability
ofthemodelparameters.Sp,Bayesianinferencecomputestheposteriorprobability
fromBayes'theorem:
p

j
D
)=
1
Z
p
(
D
j

p

:
(3.1)
Here,
p

j
D
)istheposteriorprobability,whichsptheprobabilitythatasetofmodel
parameterscanbeinferredfromthedata,subjecttothelikelihoodandprior.In
otherwords,itdescribestheprobabilitythatahypothesisiscorrect,aftertheobservations
arecollected.Thelikelihood
p
(
D
j
describesthecompatibilityoftheobservationswiththe
modelparameters.Thepriorprobability
p
spthepreviousestimate,ifany,that
themodelparametersarecorrect,beforeanyknowledgeofthedataistakenintoaccount.
Themodelevidence
Z
,isanormalizationfactorwhichremainsconstantforallchoicesof
parametervalues,andcanthereforebeignoredforourpurposes.
Aposteriorprobabilitydensityfunction(PDF)givestheposteriorprobabilityforall
ofpossiblemodelparameters,havingmaximumvaluesforsetsofparameterswhichare
mostlikelytoyieldtheobservations.ComputingthePDFisoftenduetocomplex
likelihoodfunctions.MCMCmethodscanbeadoptedtoapproximatethePDF,numerically,
bysamplingfromtheposteriorwithaclassofalgorithms.WiththesamplingfromthePDF,
MCMCmethodseasilyallowmarginalizationovernuisanceparameters(parametersthatare
requiredbythemodel,butareoflessinterest)toretrievetheprobabilitiesandexpectations
forvaluesofanyparameterofinteresttotheproblem.
56
MarkovchainMonteCarlomethodsareaclassofalgorithmsdesignedtosamplefroma
probabilitydistribution.TheydosobyconstructingaMarkovchainthatallowsawalkerto
movebetweenstatesinparameterspacewithsometransitionprobability.Ateachstepin
thechain,somesetofparametersarecomparedwiththedatathroughalikelihoodfunction.
Thetransitionprobabilitiesbetweenstepsarerelatedtotherelativelikelihoodvaluesofeach
state.MCMCchainshavethepropertythatinthelimitthatthechaintakesan
numberofsteps,thedensityofstatessampledrepresentsthePDFforthemodel.Whenthe
chainhasreachedthenumberofstepswherethisisapproximatelytrue,thechainissaidto
haveconverged.Stepscanbeselectedrandomly,providingasampleofthePDF.In
thenextsection,weshowhowanMCMCalgorithmcanbeusedtothebestchoiceof
modelparametersfromasetofobservations.
3.3
emcee
:TheMCMCHammer
3.3.1AnvariantEnsembleSampler
AcommonlyusedMCMCmethodistheMetropolis-Hastings(MH)method.TheMH
methodproposeschainstepsbasedonsomedistribution(suchasamultivariateGaussian)
centeredonthecurrentchainposition.Thisrequiresanumberoftuningparameterswhich
scalesas
N
(
N
+1)
=
2,where
N
isthedimensionalityofthemodel.Cmanytun-
ingparametersiscostlyandrequiresmanyburn-insteps,especiallyforhighlyanisotropic
densities.ThePythonpackage
emcee
(Foreman-Mackeyetal.,2013)addressesthisissue
byimplementingthevariantensemblesamplingalgorithmproposedbyGoodman&
Weare(2010).
emcee
usesanensembleofparallelchainwalkersthattakestepsinseries.
Beforeeachwalkertakesastep,astepproposalisdrawnusingastretchmove.Thatis,a
57
randomwalkerisselectedfromtheremainingensemble,andastepisproposedalongthe
vectorconnectingthetwowalkers.Thisprocessisinvariant,meaningthealgorithm
performsequallywellunderalllineartransformations.Thebofusingthestretchmove
algorithmisthatonlytwotuningparametersarerequired,regardlessofdimensionalityof
themodel.Thisallowsanensembleofchainwalkerstoexploreanisotropicdensitiesvery
tly.Additionally,
emcee
implementsmulti-threadingbyrunningchainwalkersas
separatethreads,greatlyincreasingCPUwhenthecodeisrunonamulti-core
machine.Becauseofthewithanisotropicdensitiesandtheeaseofparallelization,
Hettingeretal.(2015)uses
emcee
fortestingbinarymodelsontheF-dwarfRVcurves.
3.3.2Using
emcee
Asanintroductiontothe
emcee
packageand,moregenerally,theuseofMCMCmethods
forinference,wepresentasimpleexampleproblem.Formoreinformationonusing
emcee
,
pleaseconsultForeman-Mackeyetal.(2013).Inthisexampleproblem,wehavesimulated
30observations(
X
i
,
Y
i
)fromtherelationship
Y
i
=0
:
8
X
i
+0
:
3
;
(3.2)
withsimulatedmeasurementuncertaintiesin
Y
foldedinbydrawingfromanormaldistribu-
tionwith
˙
Y
=0
:
1(Figure3.1).WewishtouseBayesianinferenceandthe
emcee
package
todeterminethebestvaluesfortheparametersofsomehypotheticalmodel.
Webeginbysuggestingahypotheticalmodel
y
(
x
)=m
x
+b
;
(3.3)
58
Figure3.1:Mockobservationswith
X
valuesdrawnrandomlyfromtherange[0,1]and
Y
valuesdrawnfrom
Y
i
=0
:
8
X
i
+0
:
3.Simulatedmeasurementuncertaintieshavebeenadded
withvaluesdrawnfromanormaldistributionwith

=0
:
0,
˙
=0
:
1.
forwhichwearetryingtodeterminethebestvaluesof
m
and
b
.TheMCMCensemble
sampleriscreatedin
emcee
byspecifyingthenumberofchainwalkers(say100),thedi-
mensionalityofthemodel(inourcase,2),andafunctionusedtocalculatetheposterior
probabilityateachstep.Forthisproblemwehavetheposteriorprobabilitytobe
p

j
D
)=
p

p
(
D
j

;
(3.4)
withalikelihood
p
(
D
j
=
N
Y
i
=1
1
˙
Y
p
2
ˇ
exp
2
4

(
Y
i

y
(
X
i
))
2
2
˙
2
Y
3
5
;
(3.5)
andaprior
p
=
(
1
;
0
;
(

10
<m<
10)and(

10
<b<
10)
otherwise
:
(3.6)
We'veadoptedauniformpriorforboth
m
and
b
.Formostsimpleproblems,anuninfor-
mativepriorisagoodchoice,assumingmostoftheprobabilitydensitylieswithinthelimits
oftheprior.Sincethepriorismultipliedbythelikelihood,ifnoinformationabouttheprior
isgiven,theposteriorwillbeequaltothelikelihood,andthereforethepeakintheposterior
59
willbeequaltothemaximumlikelihood.
Aftertheensemblesamplerisweinitializethestartingpositionsforthe100
walkers.Asimplechoiceforstartingpositionsisarandomdistributioninparameterspace
bythepriorlimits.Wecannowruntheensemblesamplerchainforsomenumberof
steps
S
.Thetotalnumberofstepsneededforarobustsolutiondependsonthecomplexity
oftheproblemandthestartingpositionsofthewalkers.Wewouldlikeenoughstepssuch
thatthechainsconverge,andwewantenoughstepssoastoproduceafewindependent
representativesamplingsoftheposterior.Wecanseefromthechainhistory(Figure3.2
andFigure3.3)thatallchainsconvergeinabout60steps.Therefore,wewantatleast60
stepsperchain,whileburningandremovingthe60stepsfromthedistribution.
Additionally,wemaywanttothinthesampletoreduceautocorrelation.Becausechain
Figure3.2:Top:valuesof
m
fromtheMCMCforthe60stepstakenbyall100chain
walkers.Convergenceisreachedbystep60.Bottom:samefortheparameter
b
.
walkersoftentakestepsproposedintheirvicinity,thereissomecorrelationamongadjacent
steps.Tomitigatethiswewillkeepevery3rdstepfromawalker,anddiscardthe
remainder.Figure3.4showsthesolutionsfor300randomlyselected(
m
,
b
)valuessampled
60
Figure3.3:Left:parameter-spacelocationoftherst60stepstakenfor7of100chain
walkers.Starsrepresenttheinitalpositionofeachchainwalker,andsubsequentstepsare
displayedascirclesofdecreasingradius.Right:acloserviewofthecenterofthechainstep
distribution.
fromtheMCMCposterior.Figure3.5depictstheposteriordistributionsforourmodel
parameters.S(a)isthejointPDFofthetwovariables
m
and
b
.Ifweignoreone
Figure3.4:Linearsolutionsfor300randomlyselected(
m
,
b
)pairssampledfromtheMCMC
posteriordistribution.
parameterandtakethehistogramfortheother,weobtainthemarginalizedPDFforasingle
variable.Marginalizingoveranuisanceparameterpropagatestheoftheuncertainty
inthenuisanceparameterintothePDFofthemarginalizedparameter.Themarginalized
PDFsfor
m
and
b
arealsodepictedFigure3.5.
Wearriveatoursolution.UsingthemeanvaluesofthemarginalizedPDFs,our
61
Figure3.5:Posteriorprobabilitydistributionfor:(a)thejoindistributionofparameters
m
and
b
,(b)parameter
m
,marginalizedoverparameter
b
,and(c)parameter
b
,marginalized
overparameter
m
.Bluelinesindicatevaluesof
m
and
b
usedinthecreationofthemock
datapoints.
62
Bayesianinferenceyieldsthevalues
m
=0
:
77witha68%credibleinterval0
:
71
<m<
0
:
83,
and
b
=0
:
30witha68%credibleinterval0
:
27
<b<
0
:
34.
3.4CorrectingSystematicsinSDSSSpectra
AsdiscussedinSection2.2.3,RVmeasurementsofstarsfromtheSDSSwereobservedto
havesystematicuncertainties.Eachplate-pluggingcontains640ers(
M
ofwhichare
F-typestars)thatwereexposedsimultaneously,
N
times,foratotalof
N

M
F-typesub-
exposuresperplugging.Forsomeplate-pluggings,groupsofershadpositivelycorrelated
RVchanges.Toquantifythesystematicshifts,weemployedaMCMCchainwith
emcee
.
Everyplate-pluggingwascheckedindependentlywithasingleensemblesampler.The
samplersconsistedof
N
parameters

i
,oneforeachplateexposure.Theseparameters
representtheoptimalcorrectionthatshouldbeappliedtothe
i
thRVofall
M
starsonthe
plate.ThechoiceinthelikelihoodfunctionseekstoreducethescatterofRVsaboutthe
stars'meanvelocities,withtherequirementthattheappliedtoexposure
i
isidentical
forallstarsontheplate.Thelog-likelihood
1
is
ln
p
(
D
j
=

1
2
M
X
j
N
X
i
2
6
4
ln(
˙
2
j;i
)+

Y
j;i


i

h
Y
i
j

2
˙
2
j;i
3
7
5
;
(3.7)
where
Y
j;i
and
˙
j;i
aretheRVanduncertaintyforthe
i
thexposureofthe
j
thstar,and
h
Y
i
j
istheweightedmeanvelocityofthe
j
thstar.Weareassumingherethatthecontribution
ofscatterduetointrinsicvariabilityinthestarsisnegligible.Forthepriorweuseuniform
1
Topreventissueswithnumericalprecision,MCMCsamplers,suchas
emcee
,usethesummationof
log-likelihoodsratherthantheproductoflikelihoods.
63
distributions,
ln
p
=
(
0
;

;
(

50kms

1
<
i
<
50kms

1
)forall
i
otherwise
:
(3.8)
Thesamplerforeachplateuses200chainwalkers,running600stepseach,andthinned
byafactorof3.The100stepswereremovedasaburn-in.Figure3.6depictsthe
resultingPDFforanexampleplate-plugging,2085-53379.Thisplatehas8F-dwarfstars
with4sub-exposureseach.ThecorrectionstotheradialvelocitiesareillustratedinFigure
2.4.
3.5ModelingMultiplicityWithRadialVelocityCurves
TheworkofHettingeretal.(2015)examinestheRVcurvesofalloftheF-dwarfstarsinthe
sampletodeterminetheprobabilitythatasystemhasabinarycompaniondetection.Once
againweuse
emcee
,creatingatrans-dimensional,hierarchicalMCMCensemblesampler.
Thissamplerconsistsoftwomodels:asingle-starmodelwithasoleparameter,andabinary-
starmodelwithfourparameters.Ahyperparameter

,servesasanindexforchoosingone
ofthemodelsateachstepinthechain.Aftermarginalizingoverallmodelparametersfor
bothmodels,themarginalizedPDFof

inferstheposteriorprobabilitythatastar'sRV
curvesupportsashort-periodbinarydetection.
Thesingle-starmodel
M
s
,adoptsanon-varyingRVcurve,parameterizedbyasystemic
velocity
V
0
,
M
s
(
t
)=
V
0
:
(3.9)
Inthebinary-starmodel
M
b
,theRVcurveisbyasimplesinusoidnedbyfour
64
Figure3.6:Two-parameterjointprobabilitydistributionsforplate-plugging2085-53379.All
valuesareinkms

1
.Fullymarginalized,single-parameterprobabilitydistributionsoccupy
thesubplotsonthediagonal.Parameters
dy
i
(

i
inthetext)representthecorrectionstobe
appliedtothe
i
thexposureontheplate.
65
parameters:thelogofthesemi-amplitudelog
A
,thelogoftheperiodlog
P
,aphase
˚
,and
asystemicvelocity
V
0
,
M
b
(
t
)=
A
sin

2
ˇ
t
P
+
˚

+
V
0
:
(3.10)
Thismodelassumescircularorbits(eccentricity=0).Atshortperiods(
P<
12days),where
ourdataismostsensitive,circularorbitsarenearlyubiquitous(Raghavanetal.,2010),as
orbitsbecomecircularizedduetotidalforces.
InadditiontotheparametersdescribingtheRVbehavioroftheprimarystar,wein-
cludedanumberofparameterstomitigatetheofpossibleinter-platesystematics,
discussedhere.AsmentionedinSection2.2.3,intra-platesystematicswerefoundtohave
beenshiftingtheRVsoflargegroupsofers.Theseissueswereaddressedwith
MCMCmethods,butitremainstothesesystematicsbetweenplates,because
erplugpositionswereoftenchangedbetweenplateobservations.Itisreasonabletoas-
sumethatsystematicsmaybepresentfromplatetoplate,soadditionalparameters
!
p
are
usedtomodelpossibleshiftsinRVbetweenplateobservations.Thisworksbyallowing
allRVsofanindividualstartakenonplate
p
,toshiftsystematicallyrelativetothest
plate(
p
=1).Byintroducingthesefreeparameters,binarymodelsthattheRVcurve
basedsolelyonplate-to-plateerencesarepenalized,whilemodelsthatshowstrongRV
variationswithinaplatearelessFormosters,RVdataisdrawnfromasingle
plateobservation,andthereforedonotcontainany
!
p
parameters.Otherers,whichhave
P>
1plateobservations,willhaveatotalof
P

1plate-shiftparameters.
Withtheplate-shiftparametersandthetwoRVmodelsthelog-likelihoodused
66
inthe
emcee
samplerbecomes
ln
p
(
D
j
=

1
2
N
X
i
2
6
4
ln(
˙
2
i
)+

Y
i

!
p

M
(
t
i
)

2
˙
2
i
3
7
5
;
(3.11)
where
M
(
t
)isequivalentto
M
b
(
t
)whenever

selectsthebinarymodel(


0
:
5),and
M
s
(
t
)otherwise.
2
Y
i
istheRVofthestaratexposure
i
withameasurementuncertainty
˙
i
.
Thechoiceinpriordistributionsfortheparameters(Table3.1)wereconsideredcarefully.
Intheregimewheretheperiodgrowslongerortheamplitudedecreasesinmagnitude,the
Table3.1:PriorLimitsforHettingeretal.(2015)MCMC
ParameterLowerLimitUpperLimit

0.01.0
˚
02
ˇ
!
i
(kms

1
)-2020
V
0
(kms

1
)-600600
log
A
(kms

1
)0.482.40
log
P
(s)4.07.0
PriorlimitsadoptedinHettingeretal.(2015)fortheparametersintheMCMCensemble
sampler.Priorprobabilitieswereuniformlydistributedwithintheseranges.Parameters
includethemodelselectorindex

,orbitalphase
˚
,plateshiftparameters
!
i
,systemic
velocities
V
0
,thelogofthesemi-amplitudelog
A
,andthelogoftheorbitalperiodlog
P
.
binary-starmodelgivesthesamelikelihoodvaluesasthesingle-starmodel.Therefore,the
limitsonthepriorprobabilitiesshouldnotbearbitrarilylarge,butshouldinsteadbeset
atareasonablevaluesthatarerepresentativeofthesensitivityofthesparselysampleRVs.
Forsystemswithperiodslog
P
(s)
>
7
:
0,theexpectedRVamplitudesdecreasetolevels
belowthemeasurementuncertaintyofourdata.Asexpected,earlyMCMCtrialsfoundno
2
Unfortunately,
emcee
doesnothandlebinarydatatypesasamodelparameter,soweset

asaparameter
withuniformpriorbetween0and1,using0.5asthedividingpointfordeterminingwhichmodeltouseat
anystep.
67
tprobabilityfortheselargeperiodsinthe
M
b
model.Alowerlimitlog
P
(s)=4
:
0
isused;thisistheorbitalperiodat
a
=1
R

,forwhichstellarcontactbetweentwostars
iscertain.Forthesemi-amplitude,thelowerlimitontheprioris3kms

1
,comparableto
themeasurementuncertaintiesintheRVs.Theupperlimitontheprioris250kms

1
,a
valuegreaterthanthemaximumRVamplitudeexpectedfromanF-dwarfsystem.Systemic
velocitiesallowedarethoselessthantheescapevelocityoftheMilkyWay.
TheMCMCsamplerswereexecutedindependentlyforallstarsinthesampleusing200
chainwalkers,takingatotalof2
:
4millionsteps.Thechainswereburnedtoconvergence,
andthinneddowntoasizeof600,000samplesperstar.Foradescriptionofthe
andadiscussionoftheresults,pleaserefertoSection4.4andSection4.5.Instead,wewill
showheretheresultsforselectstars,anddiscusssomefeaturesoftheposteriors.
3.5.1Examples
TheexamplestarisabinarycandidatewiththeerID2939-54515-194.Thisstar
isaF-typedwarfstarfromtheHettingeretal.(2015)sample,withstellarparameters
[Fe
=
H]=

1
:
50andlog
g
=4
:
3.Twelveindividual,rawsub-exposures(
h
SNR
i
=40),
comprisethecoaddspectrum(Figure3.7).
68
69
Figure3.7:Individualsub-exposurespectra(top)usedintheproductionofthecoaddspectrum(bottom)forerID2939-
54515-194.
Sub-exposuresweretakenonthreeseparatenightswithinaweekforabaselineof143
hours.Themeasuredsub-exposureRVsbyasmuchas33
:
7kms

1
,indicatingthat
thestarisalikelybinarycandidate.
UsingthesettingsfromSection3.5,anensembleofMCMCchainwalkerscomparedthe
RVswiththebinary-andsingle-starmodels,ultimatelythebinarymodeltobethe
mostprobable.Figure3.8reportstheparametervaluesforallchainwalkersasafunctionof
stepnumber.Allparametersconvergedbystep1000(3000beforethinning),afterwhichwe
seevirtuallynosampleswith
<
0
:
5,indicatingastrongprobabilityofthebinarymodel
overthesingle-starmodel.We'veplottedtheradialvelocitycurvefor2939-54515-194in
Figure3.9,alongwithpotentialorbitsconstructedfrom200randomlydrawnsamplesfrom
theposterior.Weseethetypicalaliasingofthelikelyorbitalperiodsthatoneexpectsto
inferfromasparselysampledRVcurve.Thegapsincoverageallowformodelswithorbits
thathaveshorterperiods,oftenatharmonicfrequencies.Becauseofthis,wemaynotbe
abletospecifytheexactperiodwithbuttheMCMCinferencedoesallowusto
ruleoutthesingle-starmodel.Thiscanbeseen,forexample,whenwemarginalizeover
log
P
(andallotherparameters)andlookatthethemarginalizedPDFforthe

selection
parameter(Figure3.10,bottom-right).Allvaluesof

are

0
:
5,indicatingat
preferencefor
M
b
.ExaminingtheposteriorPDFsinFigure3.10,weseeotherfeatures.
Theperiodaliasingappearsagaininthemarginalizedposteriorforlog
P
asmultiplepeaks.
Additionally,arelationshipamong
V
0
,
˚
,andlog
A
exists.Tothedatawithmodels
havingextremevaluesof
V
0
,themodelsmustalsohavelargeramplitudeswithaphase
shift.Also,thesemodelscanonlyworkwithlongerperiods,explainingtherelationship
seenbetweenlog
P
andlog
A
.Largeramplitudesaresupportedonlybylongerperiods.
WithbetterRVcoverage,theregionofparameterspacethatcanproduceviableorbitsto
70
Figure3.8:Parametervalueprogressionforall200chainwalkersinthemultiplicityMCMC
forerID2939-54515-194.Chainsampleshavealreadybeenthinned.Samplesearlierthan
thereddashedlinewereremovedfromanalysisduringtheburn-inprocess.
71
Figure3.9:Orbitsconstructedfrom200randomsamplesoftheMCMCposteriordistribution
forerID2939-54515-194.
thedatadiminishes,allowingforamoretestimateoforbitalparameters.Our
nextexamplehasbettercoverageanddemonstratesacasewheretheorbitalparametersare
spmoretly.
Thenextstar,erID2960-54561-375,isanF-typestarlike2939-54515-194,butit
hasaslightlylowersurfacegravitywithlog
g
=3
:
6.Becausethisstar'ssurfacegravity
isbelowtheconservativefordwarfstarsinHettingeretal.(2015),it
wasnotusedinanyoftheanalyses.However,thisstarisamongthestarswiththemost
constrainedvalues,duetothelargevariationsinRV,combinedwithadequatesampling.
UnlikemostoftheotherF-typestarsmeasuredwithourcross-correlationtechnique,star
2960-54561-375hasmeasuredvelocitiesthataregreaterthanthedispersionvelocityofthe
SDSSspectra(70kms

1
pixel

1
),meaningchangesintheabsorptionfeatureshaveshifts
thataregreaterthanapixel,andarevisiblebyeye.WehighlightthisinFigure3.11,where
wehaveplottedaclose-upviewofthreeabsorptionfeatures.Theeightsub-exposuresare
orderedchronologicallyfromtoptobottom,andplottedasafunctionofredshiftvelocity.
72
Figure3.10:PosteriorprobabilitydistributionsofparametersinthemultiplicityMCMCfor
erID2939-54515-194.
73
Figure3.11:ChangesinredshiftwithtimeforerID2960-54561-375.Absorptionlines
fromnormalizedsub-exposuresareorderedchronologicallyfromtoptobottom.Velocities
ineacharerelativetotherest-framewavelengthofCalciumK(left),CalciumH
(middle),andH

(right).Dashedverticallinesrepresentthemeanvelocityofthestar.
74
Thatis,eachpixel'swavelengthinthespectrumcorrespondstoameasuredredshift,had
thatpixelcontainedthecentroidoftheabsorptionfeatureforthatparticularline(e.g.,Ca
K).Tocomputethis,weconverteachpixelwavelengthtoavelocityfromtheDopplershift
(Equation2.1)usingthespabsorptionfeatureastherest-framewavelength.Usingthe
meanvalueofthestar'smeasuredRVs(dashedverticalline)asaguide,shiftsintheline
centroidsovertimebecomeapparent.
Lookingatrandomlyselectedorbitsfromtheensemblesampler(Figure3.12),wesee
astrongpreferenceforamodelwithaperiodof
P
=7
:
94days,followedbymodelswith
harmonicperiods.Intheposterior(Figure3.13),weseethepeaksinthemarginalizedprob-
Figure3.12:Orbitsconstructedfrom200randomsamplesoftheMCMCposteriordistribu-
tionforerID2960-54561-375.
abilitiesforallparameters,correspondingtothebestmodel.Wealsoseestrongprobability
concentratedinasecondmodelwithashorterperiodat
P
=1
:
14days.
75
Figure3.13:PosteriorprobabilitydistributionsofparametersinthemultiplicityMCMCfor
erID2960-54561-375.
76
Chapter4
StatisticalTime-Resolved
Spectroscopy:AHigherFractionof
Short-PeriodBinariesforMetal-Rich
F-typeDwarfsinSDSS
ThisChaptercontainsanexpandedversionofthepeer-reviewedarticleHettingeretal.
(2015)publishedintheAstrophysicalJournalLetters
4.1Abstract
Stellarmultiplicityliesattheheartofmanyproblemsinmodernastrophysics,includingthe
physicsofstarformation,theobservationalpropertiesofunresolvedstellarpopulations,and
theratesofinteractingbinariessuchascataclysmicvariables,X-raybinaries,andTypeIa
supernovae.However,littleisknownaboutthestellarmultiplicityofstarsintheMilky
Way,inparticularaboutthecesinthemultiplicitycharacteristicsbetweenmetal-
richdiskstarsandmetal-poorhalostars.Inthisstudyweperformastatisticalanalysisof
˘
14,000F-typedwarfstarsintheMilkyWaythroughtime-resolvedspectroscopywiththe
77
sub-exposuresarchivedintheSloanDigitalSkySurvey.Weobtainabsoluteradialvelocity
measurementsthroughtemplatecross-correlationofindividualsub-exposureswithtemporal
baselinesvaryingfromminutestoyears.Thesesparselysampledradialvelocitycurvesare
analyzedusingMarkovchainMonteCarlotechniquestoconstraintheveryshort-period
binaryfractionforF-typestarsintheMilkyWay.Metal-richdiskstarswerefoundto
be30%morelikelytohavecompanionswithperiodsshorterthan12daysthanmetal-poor
halostars.
4.2Introduction
Stellarmultiplicityplaysacrucialroleinmanyofastronomy.Starformationandevo-
lution,Galacticchemicalevolution,nuclearastrophysics,andcosmologyareallby
ourunderstandingofthemultiplicitypropertiesofanunderlyingstellarpopulation.Binary
interactionsleadtophenomenaasdiverseascataclysmicvariables,classicalnovae,X-ray
binaries,gamma-raybursts,andTypeIasupernovae.Stellarinteractionsarealsothecause
oftheanomaloussurfaceabundancesmeasuredinBastars,CHstars,andthemajorityof
carbon-enhancedmetal-poorstars(Lucatelloetal.,2005).Theratesofthesephenomena
dependonthemultiplicitypropertiessuchasthefractionofstarswithcompanionsandthe
distributionsofseparationsandmassratios.Howthesepropertiesareinturnby
variablessuchasstellarage,metallicity,anddynamicalenvironmentremainspoorlyun-
derstood.Moe&DiStefano(2013)nottrendswithmetallicityforO-and
B-stars,butmoreworkisneededforlower-massstars.
TherecentreviewbyDuch^ene&Kraus(2013)summarizesthestateoftheartinmulti-
plicitystudies.Thefractionofsystemswithcompanionsisknowntobeastrongfunction
78
ofstellarmass(Lada,2006;Raghavanetal.,2010;Clarketal.,2012),andtherearehints
thatlowermasssystemshavesmallerseparations(Duquennoy&Mayor,1991;Allen,2007;
Raghavanetal.,2010).StudiesoftheSolarneighborhoodalsoindicatethatlowermetallicity
starsaremorelikelytohavestellarcompanions(Raghavanetal.,2010).
Theseresultsarebasedonheterogeneoussamplesofafewhundredstarsatmost,often
dominatedbywidesystemswhichwillneverbecomeinteractingbinaries.Thespectroscopic
surveysthatreachsmallperiodsarelaborintensivebecauselargenumbersofradialvelocities
(RVs)arerequiredtotheorbitalsolutionofeachtarget.Thisleadstosmallsamplesizes,
whichhaveonlyincreasedmodestlyinthepasttwodecades,from167inDuquennoy&Mayor
(1991)to454inRaghavanetal.(2010).Thedrivetocollectcompletesampleshaslimited
previousspectroscopicstudiestotheSolarneighborhoodorspstellarclusters,but
neitherofthesestrategiescanprobethefullrangeofmetallicitiesandagesspanningthe
starsoftheMilkyWay(MW)diskandhalocomponents.Theselimitsbiastheinterpretation
ofdataagainsttheglobalpropertiesof,andvariationwithin,theMWThus,weare
motivatedtotakeastatisticalapproachwithasampleofstarslocatedthroughoutthein
ordertoinvestigatetheirmultiplicitypropertieswithrespecttoage,[Fe
=
H],andcomponent
membership.
WiththeadventofmultiplexedspectroscopicsurveyslikeSDSS(Yorketal.,2000)and
LAMOST(Cuietal.,2012),wecanusemultipleRVmeasurementsofthousandsofstarsto
studythepropertiesofstellarmultiplicitythataremorerepresentativeoftheentireGalaxy.
SDSSDataRelease8(Aiharaetal.,2011)containsover1.8millionopticalspectrafromthe
originalSDSSspectrographsincludingover600,000stellarspectra.Inthisworkweemploy
alesserknownSDSSfeature,thetime-resolveddimension.Tofacilitatecosmicrayremoval,
spectrawereconstructedthroughco-additionofseveralindividualsub-exposures,typically
79
15minutesinduration.Althoughunder-utilized,thebofthesub-exposuredomain
isrecognizedinworkssuchasBadenesetal.(2009)andBickertonetal.(2012).Portions
oftheskywerealsore-observedforcalibrationandscienpurposes.Theseadditional
pointings,combinedwiththesub-exposures,yieldatimedimensionwheresinglestarshave
exposurecoveragerangingfrom3sub-exposuresuptoover40sub-exposures,andtimegaps
fromhourstonearlyadecade.Thetechniquesemployedhereinfollowthetime-resolved
workbyBadenes&Maoz(2012)andMaozetal.(2012).
4.3Measurements
4.3.1SDSSObservationsandSampleSelection
F-typedwarfsarechosenforoursamplebecauseofthelargenumberofstarstargetedby
SDSSwithrepeatobservations,andtheirrelativelymildvariabilityandactivity.Addition-
ally,F-starshavemainsequence(MS)lifetimesgreaterthan5Gyr,allowingustoselect
MSstarsfromboththeyoungerdiskandolderhalo.TheSloanStellarParameterPipeline
(SSPP;Leeetal.,2008)wasdevelopedtodetermineparametersforstellarspectrainthe
SDSSarchive,includingmetallicity[Fe
=
H],etemperature
T

,andsurfacegrav-
itylog
g
.SampleselectionbeganwithidentifyingscienceprimaryobjectsfromSEGUE-1
(Yannyetal.,2009)andSEGUE-2(Rockosi,C.M.etal.,inprep.)intheSSPPthat
wereasanF-typestarbythe\Hammer"code(Coveyetal.,2007).
Tominimizetheofstellarevolutiononmultiplicity,weselectedonlydwarfstars
(log
g

3
:
75).Starswithmultipleerpluggingswereidenastrometricallyandjoined
withtheappropriatescienceprimaryers.
AftermeasuringstellarRVs(Section4.3.2),systematicswererevealedintheSDSSsub-
80
exposurespectra.ThesecorrelationsappearassimilarshiftsinRVsformanyerslocated
onthesameplate,typicallyneighboringersontheCCD.Afterplate-widecom-
parisonsofF-stars,RVcorrelationswerecorrectedwherepossible.Correctionsappliedto
the10
4
RVsareaslargeas17kms

1
withastandarddeviationof2
:
2kms

1
.Notall
correlationscouldbeidenautomaticallybecauseofmultiplegroupsofcorrelatedshifts,
oppositeindirection,onsomeplates.Visualinspectionofplatescontainingnumerousfalse
binarydetectionsleadtotheremovalof25platesincluding1155stars.Weurgeindivid-
ualsusingsub-exposurespectroscopyinSDSStoconsiderthesesystematicshiftsinthe
wavelengthsolutions.
Qualitycontrolconsistedoftheremovalof:starswithoutvalidparametersinSSPP,
erslocatedon`bad'plates,sub-exposureswithamedianpixelsignal-to-noiseratio(SNR)
lessthan20orwithfewerthan3000upixels,starswithtimelags
T<
1800s,
starswithlessthanthreecleansub-exposures,andcorruptorspectra(from
visualinspectionofstarswiththelargestRVvariationornon-characteristic
T

).The
sampleconsistsof14,302stars(16,894ers)withasmanyas47sub-exposures,spanning
uptonineyearsofobservations(Figure4.1).
Ourcleanedsampleischaracterizedbymetallicitiesrangingfrom

3
:
41

[Fe
=
H]

+0
:
52.Toaidcomparisoninouranalysis,thesamplewassub-dividedintothree
groupsofequalsizebycutsinmetallicityat[Fe
=
H]=

1
:
43and[Fe
=
H]=

0
:
66(Figure
4.1).Themajorityofthestarshavethreeorfoursub-exposures(median=4),typically
takenabout15minutesapart.Themediantimelagforastaris2hours,howevermorethan
threeyearsbetweenobservationscanbeseeninmorethan250stars(Figure4.1).SNRsfor
sub-exposureslieintherange20
<
SNR
<
84withamedianvalueof32.
81
Figure4.1:Left:Metallicitydistributionfor14,302F-dwarfs.Right:Distributionofmaxi-
mumtimelagbetweentheandlastexposureofastar.
4.3.2RadialVelocities
RVmeasurementswereattainedthroughcross-correlationofsub-exposureswithamaster
templateconstructedfrom7207sample-star,co-addedspectrawheretheco-addedSNR
>
50.Thespectrawerede-shiftedusingtheredshiftvalueassignedtotheco-addsbytheSDSS
pipeline,continuum-normalized,andaveragedtogether.
Sub-exposureswereindependentlypreparedandcross-correlatedwiththetemplate.Spec-
trawerecontinuum-normalizedbydividingthespectrumwithahighlysmoothedversionof
itselfusingaFFTsmoothingalgorithm,andthencross-correlatedwiththetemplateatvar-
iousintegerpixellags.Eachspectrumhadacross-correlationfunction(CCF)thatwas
withasmoothsplineinterpolation.Withspectralresolutionof
R
˘
2000,thepeaklagin
pixelstranslatestothespectrum'sredshiftat70kms

1
pixel

1
.Themeanandstandard
deviationofRVsforindividualstarsareshownintheFigure4.2distributions.Thevelocity
dispersionofthemeanRVsdecreaseswithincreasing[Fe
=
H],indicatingthatour[Fe
=
H]-
groupssampleboththediskandhalocomponentsoftheMW.Thestandarddeviationof
RVswithinindividualstarsislargerforthemetal-poorgroup;however,empiricallyestimated
uncertaintiesalsoshowlargermeasurementerrorsformetal-poorstars.Thisunderscoresthe
82
importanceoftheuseofpropererroranalysisinamethodsuchasours.
Figure4.2:Mean(left)andstandarddeviation(right)ofradialvelocitieswithinastar.Vari-
ationsinthestandarddeviationofvelocitiesareinpart,bythelargermeasurement
uncertaintiesformetal-poorerstars.
4.3.3Uncertainties
ItiswellknownthatuncertaintiesinCCFpeaksmustbeestimatedempiricallyorthrough
someMonteCarlomethod(e.g.,Petersonetal.1998).ForthisworkwedeterminedRV
uncertaintiesempiricallybyquantifyingthespreadinmeasurementsforspectraofsimilar
quality.Themedianabsolutedeviation(MAD)isarobustmeasureofthevariabilityof
asampleandisrelatedtothestandarddeviationby
˙
=1
:
4826MAD,whereMAD=
median(
j
RV
i

median(RV)
j
)(Leysetal.,2013).Allmeasurementswerede-shiftedinto
therestframeusingtheSDSSestimatesoftheco-addredshift,andplacedintobinsof
similarmetallicity([Fe
=
H]

0
:
25)andsignal-to-noise(SNR

2
:
5).Initialtestsshowed
nocorrelationsbetweenmeasurementspreadsandeitherlog
g
or
T

.Estimatesforthe
uncertaintyofRVmeasurementswithinabinwerecalculatedusingMADvalues.Here,it
isassumedthatthemajorityofstarsdonothavedetectablevariabilityovertheobserved
timebaseline,andthatfromintrinsicvariationsinRVareminimizedbyadopting
83
medianvalues.Afterperformingthisprocessforallbins,afunctionalformforassigningRV
measurementuncertainties
˙
RV
waswithaninverseproportionalitytoSNR,andwitha
linearcorrectionin[Fe
=
H].Themeasurementuncertaintyasafunctionof[Fe
=
H]andSNR
is,inkms

1
,
˙
RV
([Fe
=
H]
;
SNR)=
(

26
:
51[Fe
=
H]+50
:
52)
SNR
+1
:
23
:
(4.1)
Uncertaintiesaresub-pixel,fallingbelowthespectralresolutionof70kms

1
pixel

1
.For
exposureswithSNR
<
25,uncertaintiesrangefrom3
:
0to8
:
0kms

1
,withamedianvalue
of5
:
0kms

1
.ExposureswithSNR
>
40haveuncertaintiesintherange1
:
9to4
:
4kms

1
,
withamedianvalueof2
:
7kms

1
.
4.4Multiplicity
Theprobabilityofastarhavingacompanionwasdeterminedthroughmodelcomparison
usingatrans-dimensional,hierarchical,MarkovchainMonteCarlo(MCMC)method.Two
modelswerecompared:asingle-starmodel
M
s
,andabinary-starmodel
M
b
.Thehyper-
parameter

,indexesthemodelchoiceateachstepintheMCMCchain.Weevaluatedthe
hierarchicalmodelusingthePythonpackage
emcee
,aMCMCensemblesampler(Foreman-
Mackeyetal.,2013).
Thesingle-starmodel
M
s
,astarwithnon-varyingRVs,parameterizedbyasystemic
velocity
V
0
.Becauseintra-platesystematicsareknowntoexist,itisreasonabletoassume
inter-platesystematicsexistaswell.Inlightofthis,(
P

1)additionalparameters
ps
i
,were
includedforeachstar,where
P
isthenumberofplate-MJDpluggingscomposingthestar.
Theseplate-shiftparametersallowallRVsfromplate
i
toshiftbysomeamount
ps
i
,relative
totheplate
P
0
.Forthemajorityofstars
P
=1,noplate-shiftparametersarenecessary,
84
and
M
s
isa1-parametermodel.
Inthebinarystarmodel
M
b
,thesparselysampledRVsarebyasinusoidby
four-parameters:thelogofthesemi-amplitudelog
A
,thelogoftheperiodlog
P
,thephase
˚
,andthesystemicvelocity
V
0
.Weassumecircularorbits(eccentricity,
e
=0),whichisa
safeassumptionfortidallycircularized,short-periodorbits(
P<
12days;Raghavanetal.,
2010),wherewearemostsensitive.Asmallnumberofthebinariesfoundinthisstudy
mayhavelongerperiodsandcouldhavenon-zeroeccentricities,butthisdoesnotour
results.Plate-shiftparameterswerealsoadoptedin
M
b
wherever
P>
1.
UninformativepriorswereusedintheMCMC.Themodelindex

,hasapriorfrom
0to1,where
<
0
:
5denotes
M
s
and


0
:
5denotes
M
b
.Thesemi-amplitudeprior
islog-uniformfrom3kms

1
,comparabletothemeasurementuncertaintieswhere
M
s
and
M
b
becomedegenerate,to250kms

1
,greaterthanthelargestRVinthe
sample.Thepriorontheperiodisuniformintherange4

log
P
(s)

7.Thelowerlimit
log
P
(s)=4
:
0isequaltotheorbitalperiodatwhichstellarcontactiscertainforlow-mass
companions.Abovelog
P
(s)=7
:
0,RVamplitudesinbinarysystemsarecomparableto
themeasurementuncertainties.CombinedwiththesparsityoftheRVdata,systemswith
periodslongerthanlog
P
(s)=7
:
0areoutsideourrangeofsensitivity.Priorsarealso
uniformforthephase(0

˚

2
ˇ
)andsystemicvelocity(

600

V
0
(kms

1
)

600).
Markovchainswererunindependentlyoneverystarwithanensembleof200parallelchain
\walkers"foratotalof2
:
4

10
6
samples,thenburnedandthinnedto6

10
5
independent
samplesoftheposterior.
Evidencefordetectionofacompanionstarisbytherelativeprobabilitiesof

.
Wetheprobabilityforthebinarymodel,

asthefractionofsamplesinthemarginal-
izedposteriorhaving

=
M
b
.Wenotethatthevalueof

isdependentonthechoiceof
85
priors,andissensitivetothetreatmentoftheSDSSsystematics.Moreover,adegeneracy
arisesastheRVcurveofalong-period,low-amplitudesystembecomesindistinguishable
fromasingle-starsystem.Withthismind,westressthatvaluesfor

arenotabsolute
probabilitiesofasystemhavingacompanion,buttheabilityofthedatatoruleout
modelsunderthegivenprior.However,the[Fe
=
H]-groupscanbecompared,relatively,by
consideringthefractionofsystemswhere

islargeand
M
s
isstronglydisfavored.The
resultsareshowninFigure4.3
Figure4.3:Distributionof

,thefractionofposteriorsamplesusingthebinarymodel,for
stars.
VariouscheckswereimplementedtoensuretherobustnessofourMCMCmethod.The
rangeofthelimitsonthepriorswereincreasedtosearchfortposteriorprobability
densityat,forexample,largerperiods.Withanincreasedrangeinsemi-amplitudesand
periods,notincreaseinprobabilitywasseenatvaluesexcludedinthecurrent
priorlimits,oursensitivitytoshort-periodsystemswithlog
P<
12(d).We
implementedasecondmethodforcomparingthesingle-starandbinary-starmodels.A
posteriorBayesfactor(POBF)wascalculatedonindependentMCMCruns,usingeachof
thetwomodels.UsingtheratioofthePOBFvaluesasametricforidentifyinglikely
binarycompanions,wesawgoodagreementwithstarspredictedtohavecompanionsvia
86
ourhierarchicalmethod.Additionally,weperformedvisualinspectionofmorethan500
starswiththehighestvaluesof

,aswellasstarswiththelowestvaluesof

.Asexpected,
thosestarswithhighvaluesof

showedtRVvariationsexpectedfromabinary
companion,andstarswiththelowestvaluesof

hadlittle-to-noRVvariations.Forthose
starswithlittleRVvariations,thedistributionsinthemarginalizedposteriorsofthe
M
b
parameterswereacrosstheentirerangesspbythepriors.
Wealsoinvestigatedthe
e=i
parameterproposedbyGelleretal.(2008)asametricfor
identifyingthestarswithlargeRVvariations.Wethatthe
e=i
parametersinglesout
manyofthesamestarsasourmoresophisticatedMCMC-basedinference.Ourmethodnot
onlytakesintoaccountdeviationsinRVfromthemean,butalsohowwellthedatathe
expectedperiodicityofabinarysystem.
Analysisoftheposterior,andvisualinspectionsofthebinarymodelshowthat
681starswith
>
0
:
65aretruespectroscopicbinaries,thoughgiventhesparsityoftheRV
curvesampling,therearesometimeslargeuncertaintiesinthevaluesforspmodel
parameters.Anothernaturalbreakpointis
>
0
:
95;theseare209starsforwhichthedeter-
minationandanalysisofaccurateindividualmodelparametersshouldbepossible(andwill
becharacterizedinfuturework).Anintermediatecutat
>
0
:
80isacompromisebetween
theselimits,yieldingalargersampleofstars(406)withmodestmodelconstraints.The
valuesofthebinaryfractionsthatwederivebelowareinsensitive,withintheuncertainties,
totheexactchoiceofcutin

.ThisimpliesthattheRVvariationsforourbinarydetections
aretlyabovethemeasurementuncertainties,andthatthebinaryfractionsreported
arenotbiasedduetoencesinSNRorabsorptionfeatures.
Figure4.4showsthelog
P
posteriorsforeach[Fe
=
H]-group,marginalizedoverallbinary
systems(
>
0
:
80).Theposteriordistributionsoflog
P
formanyofthesestarsarecomplex:
87
manyaremultimodal,byaliasingorotherissuesrelatedtothesparse,biasedtime
sampling.Onesuchistheincreaseinprobabilityatlog
P
=4.Herethemetal-
richand-intermediategroupscontainmorestarsthanthemetal-poorgroupwith
t
'
10
4
s.Systemswithperiodsasshortasthisareextremelyrare(Drakeetal.,2014),and
ourincreasedprobabilityinthisareamaybeduetoovAdditionally,thegapat

t
=10
4
:
6
s=12hr(Figure4.1)maytheestimateofaperiod.Wedeferamore
sophisticatedanalysistoafuturepaper,buttheseshouldnotaltertheabilityto
ruleoutasingle-starmodel.Fornow,Figure4.4illustratesthatwearemainlysensitiveto
periodsintherange4
<
log
P
(s)
<
6,orlessthanabout12days.Weemphasizethata
moredetailedanalysiswillbenecessarytoestimatethetrueunderlyinglog
P
distribution
inoursample.
Figure4.4:Averagedprobabilitydistributionsoflog
P
forallbinarydetections(
>
0
:
80).
Thesedonotactualdistributionsofperiods,andshouldonlybeusedasaguide
toprobetheregionofMCMCsensitivity.TheshadedregionindicateswhereRochelobe
ovowandcontactbecomesrelevant.Thedashedlinemarksthecircularizationlimitata
periodof12days.
88
4.5Discussion
InFigure4.5weshow
f
b
,themeasuredlowerboundforthefractionofstarswithshort-period
companions(
P<
˘
12days)foreachmetallicitygroup,normalizedtothemetal-richbinary
fraction.
f
b
isalowerlimitbecauseofnon-detectionsasaresultofsparselysampledRVs
andhighorbitalinclinations,resultinginlowamplitudes.Weseeagreementin
f
b
measured
forallthreechoicesin

(0.65,0.80,0.95).Withaof

=0
:
80,valuesof
f
b
for
themetal-poor,-intermediate,and-richgroupsrespectivelyare:2
:
5%

0
:
2%,2
:
8%

0
:
2%,
and3
:
2%

0
:
3%.Sincetheobservationalbiasesthattbinarydetectionaremostlydue
tothesparsityoftheRVcoverage,whichisn'tmetallicity-dependent,weconcludethatthe
F-typeMSstarsinourmetal-richsampleare,ata2-sigmalevel,30%morelikelythan
thoseinourmetal-poorsampletohaveclosebinarycompanions.
Figure4.5:Short-periodbinaryfractionlimits,relativetothemetal-richgroup.Binary
companiondetectionsarebyacutin

,thefractionofposteriorsamplesusingthe
binarymodel.Groupmedianvaluesof[Fe
=
H]areused.
Ourmetal-richandmetal-poorsamplesmostlytracetheMWdiskandhalo.in
89
thefractionofshort-periodsystemscanstemfromencesinthestar-formationprocess,
dynamicalinteractionsafterstarformation,orsomecombinationofthetwo.
Three-dimensionalhydrodynamicmodelsfromMachidaetal.(2009)actuallysuggesta
higher
frequencyofbinariesformedthroughcloudfragmentationformetal-poorclusters,due
tothedecreasedrequirementofacloud'sinitialrotationenergytofragment.Moreover,their
modelsyieldsystemswithshorterinitialseparationsatlowermetallicities.Theincreased
f
b
observedformetal-richstarsinthisworkcanmorelikelybeexplainedbydynamical
processesthanbyformationprocesses.
Theobservedcesin
f
b
couldbeexplainediftheclustersthatyieldedhalo
starshadlargerstellardensitiesand/orgasdensitiesthanthoseofthedisk.Kornetal.
(2012)exploretheofgas-inducedorbitaldecayonperioddistributionsinclusters.
Theynotethatanincreaseddensityofgasinanewlyformedclusterwillleadtoalargernum-
berofshort-periodsystemmergersshortlyafterformation.Parkeretal.(2009)describehow
clusterswithhigherstellardensitiesdestroywidebinariesthroughdynamicalinteractions.
Anincreaseinthedestructionofhigh-mass,wide-binarysystemsleadstotheejectionof
formerF-starsecondariesintotheTheseorphaned,single-starsystemswouldincrease
thetotalnumberofF-starsystemsinthehaloelydecreasingtheshort-period
binaryfractionmeasured.Observationalevidenceofthesedenserclusterenvironmentsis
neededtosupporttheseargumentsforalower
f
b
inthehalo.
Additionally,someclosebinariesmayalsotransfermassandconvertthemselvesintoblue
stragglers(Luetal.,2010).Evidenceforanabundanceofbluestragglersinthehalohasbeen
seen(Yannyetal.,2000),andmaycontributetothelower
f
b
observedinthemetal-poor
group.Also,Duch^ene&Kraus(2013)showadecreasein
f
b
withageforSolar-typestars,
althoughthisresultisbasedonvisualbinarieswithwiderperiods,andispoorlyconstrained
90
duetolimitedsamplesizes.
WenotethattherecentresultsofGaoetal.(2014)andYuanetal.(2015),usingdata
fromSDSS,showalargerbinaryfractionformetal-poorthanmetal-richFGKstarsinthe
Inadditiontoprobinglongerperiods,theformerworkdoesnotmakeuseofsub-
exposureinformation(usingonlytwoRVepochsperstar)andreliesonthecorrectnessof
modelvaluesfortheperioddistribution,massratiodistribution,andinitialmassfunction.
Thelatterwork,whichusesphotometriccolordeviationstoinfercompanions,showsa
modestmetallicitydependenceontotalbinaryfraction.Sincetheirmethodisnotsensitive
toperiod,thebinaryfractionstheyreportarestronglydominatedbymorecommon,wider-
periodsystemsnearthepeakofalog-normalperioddistribution(
log
P
(s)=10fornearby,
Solar-likestars;Raghavanetal.2010).Itisclearthatconclusionsaboutbinaryfraction
dependonanumberoffactors,especiallytherangeofperiodstowhichthesearchissensitive
andassumptionsmadeabouttheoverallperioddistribution.
OurMCMCanalysisyieldsposteriorprobabilitiesinparameterspace,allowingforamore
detailedstudyofbinaryproperties(e.g.,periodandseparationdistributions),whichwillbe
presentedinfuturework.Thetechniquesinthisworkhavedirectapplicationsforcurrent
andfuturemultiplexedspectroscopicsurveys.
WethankEwanCameron,DanMaoz,reyNewman,ChadSchafer,andthereferee
forusefuldiscussions.T.H.andT.C.B.acknowledgepartialsupportfromgrantsPHY08-
22648;PhysicsFrontierCenter/JINA,andPHY14-30152;PhysicsFrontierCenter/JINA
CenterfortheEvolutionoftheElements(JINA-CEE),awardedbytheUSNationalScience
Foundation.FundingforSDSS-IIIhasbeenprovidedbytheAlfredP.SloanFoundation,
theParticipatingInstitutions,theNationalScienceFoundation,andtheU.S.Department
ofEnergyofScience.
91
Chapter5
BinaryFractionsandSeparation
Distributions
5.1Introduction
AsmentionedinSection1.4.3,thenumberofsystemsthatwillexperienceaCEphase
dependsonthedetailsoftheseparationdistributionandthefractionofstarsthathave
short-periodcompanions.Understandingbinaryfractionsandseparationdistributionswill
impactthepredictedmergerratesandsupernovarates(Badenes&Maoz,2012).These
propertiesalsoshedlightontothedetailsofthestarformationprocessandbinaryformation
process,andspeaktothedynamicalhistoryofstellarassociationsandclusters.
InHettingeretal.(2015),weinvestigatedtheshort-periodbinaryfractionofF-type
dwarfstarsintheMilkyWayhaloanddisk.Inthischapter,weextendtheworktoinclude
aninvestigationoftheseparationdistribution.AnMCMCsamplerhasbeenadoptedincon-
junctionwithpopulation-wideMonteCarlosimulations,inordertoconstraintheseparation
distributionsthatareconsistentwiththedata.WewilllookatthemethodologyinSection
5.2,andconcludewithadiscussionoftheresultsandlimitationsinSection5.3.
92
5.2MCMCandPopulation-WideMonteCarlo
InourapproachtoinvestigatetheseparationdistributionofbinariesintheF-dwarfsample,
wecombinedaMCMCsamplerwithpopulation-wideMonteCarlosimulations.Weassume
thatthebinaryseparations,inourrangeofsensitivity(0
:
01AU
<a<
0
:
16AU),are
distributedlikeapowerlawwithanindexparameter

,
f
(
a
)
/
a

:
(5.1)
Apowerlawdistributionshouldbedecentapproximationtotheoften-usedlog-normaldis-
tribution,forthenarrowrangeofseparationstowhichwearesensitive.
InourMCMC,wehavetwofreeparametersdescribingthepopulationofstars,the
separationdistributionindex

,andthefractionofsystemsthathaveclose-separationbinary
companions
f
b
.Weinclude
f
b
,asthereisadegeneracyin

and
f
b
,thedistribution
ofRVvariationsamongthestarsinthepopulation.Figures5.1and5.2illustratehowchanges
ineither

or
f
b
thedistributionof
e=i
valuesforapopulationofstars.
Thedistributionofmassratiosinapopulationalsohassigtimplicationsonthe
starformationandbinaryformationprocesses,andtheexpectedratesforvariouspost-CE
systems.Unfortunately,changesinmassratiosdonothaveastrongimpactontheobserved
RVvariations(seeFigure5.3).Thestatisticalapproachforestimatingmultiplicityproperties
fromsparselysampledRVsislimitedinthisregard.Determiningmassratiodistributionswill
likelybelimited,forsometime,totargetedcampaignsconsistingofdetailedspectroscopic
andeclipsingobservations.
SincetheparametersinourMCMCsamplerdapopulationofstars,theentire
populationmustbesimulatedateverystepinthechainandcomparedwiththedata.This
93
Figure5.1:Distributionof
e=i
valuesforasimulatedpopulationbasedonobservations
anduncertaintiesinthemetal-richpopulationofHettingeretal.(2015).Modelsillustrate
variationsinthe
e=i
distributionfromchangesinseparationdistributionpowerlawindex

,
whilekeepingtheshort-periodbinaryfraction
f
b
=0
:
03andthemassratiodistribution
powerlawindex

=0
:
0.
Figure5.2:Distributionof
e=i
valuesasinFigure5.1,withavarying
f
b
,and

=

1
:
0
and

=0
:
0.
94
Figure5.3:Distributionof
e=i
valuesasinFigure5.1,withavarying

,and

=

1
:
0
and
f
b
=0
:
03.
requiredustoadoptasingle,easilycalculatedmetric,tobeusedforeachstarinthe
population.The
e=i
metric(Section2.5)providesameasurementofthevariabilityofa
star'sRV,andwillchangeinmagnitudedependingonthesystem'sbinaryseparation,orbital
orientation,andexposurecoverage.
Tosimulatean
e=i
distributionforthepopulation,werunapopulation-wideMonte
CarlosimulationateachstepintheMCMCchain.ThisisaccomplishedbysimulatingRV
measurementsforeverystellarsysteminthepopulation60times,calculating
e=i
foreach
system,andnormalizingthe
e=i
distributionsoitcanbecomparedwiththedata.
Forindividualstellarsimulations,abinarycompanioniseitherabsentorpresent,based
onthevalueofthepopulation-wide
f
b
parameteratthatstepintheMCMCchain.Next
aseparationischosenrandomlyfromapowerlawdistributionsetbythe

parameter
atthatstepintheMCMCchain,withtheseparationwithinthelimitsofour
sensitivity(0
:
01

0
:
16AU).PrimarymassesaredrawnfromanIMF(Kroupa,2001),and
secondarymassesaredrawnfromadistributionfrom10%to100%oftheprimary
95
mass.RVmeasurementsaresimulatedusingthesameobservingtimesthatthestarwas
originallyobservedat.Finally,measurementuncertaintiesarefoldedin,usingtheempirically
estimatedRVuncertaintiespreviouslyderivedforthatparticularstar(Section2.4).
Toimprovethestatisticsofthemodelathigh
e=i
values,thepopulationissimulated60
timesateverystepintheMCMCchain.Theresultinarelativelysmoothmodel,representing
atypical
e=i
distributionforthepopulationofstars,givenasppairofvaluesfor

and
f
b
.ThelikelihoodfunctionintheMCMCcomparesthehistogramof
e=i
valuesforthe
population,calculatedfromthedata,withthehistogramof
e=i
valuesfromthemodelusing
Poissonprobabilitiesateachbin.Thus,ateachstepintheMCMC,onepossiblepopulation
ofstarsissimulatedrobustlytoproducea
e=i
curve,andthatcurveiscomparedtothe
histogramoftrue
e=i
valuespreviouslycalculatedfromthedataset.Thiscomparisonallows
theMCMCtoacceptandrejectfutureproposalsfor

and
f
b
.
TheMCMCsamplerwasrunonallthreemetallicitygroupsinHettingeretal.
(2015).InFigures5.4,5.5,and5.6,weshowthe
e=i
distributionsofthemetal-poor,-
intermediate,and-richgroups.Overplottedineachisarandomselectionofmodels,
constructedfrom

and
f
b
parametersdrawnfromtheposterior.Itshouldbenotedthatthe
MCMClikelihoodfunctiononlycomparedbinheightsforbinswith
e=i>
3
:
0.Thereasoning
isthatourempiricallydeterminedestimatesforRVuncertaintieshadlimitedaccuracy.This
resultsinsomedeviationsinthecalculated
e=i
values.Sincethemajorityofstarshave
low
e=i
values,smalldeviationsintheassigneduncertaintiesleadtolargedeviationsinthe
binheights.Thusthelikelihoodfunctionwouldbedominatedbyinthelow-
e=i
values,andwouldhavereducedsensitivitytothehigh-
e=i
binarysystems.Theposterior
distributionsfor

and
f
b
aredepictedinFigures5.7,5.8,and5.9.Theresultsfromthe
MCMCrunarediscussedinthefollowingsection.
96
Figure5.4:
e=i
distributionforthemetal-poorgroup.Model
e=i
distributionsareshown
usingvaluesof

and
f
b
randomlysampledfromtheMCMCposterior.Thedashedline
representsthebelowwhichbinheightswerenotusedinthelikelihoodfunction.
Figure5.5:SamedistributionasinFigure5.4,forthemetal-intermediategroup.
97
Figure5.6:SamedistributionasinFigure5.4,forthemetal-richgroup.
Figure5.7:PosteriordistributionfortheMCMCrunofthemetal-poorgroup,withparam-
etersfortheshort-periodbinaryfraction
f
b
,andseparationdistribution

.
98
Figure5.8:PosteriordistributionfortheMCMCrunofthemetal-intermediategroup,with
parametersfortheshort-periodbinaryfraction
f
b
,andseparationdistribution

.
Figure5.9:PosteriordistributionfortheMCMCrunofthemetal-richgroup,withparame-
tersfortheshort-periodbinaryfraction
f
b
,andseparationdistribution

.
99
5.3Discussion
Unfortunately,thereislittleadditionalinformationtobedrawnfromtheposteriorofthese
MCMCruns.Theshort-separationbinaryfractionsallowedbythedata(2%{8%)are
consistentwiththevaluesobtainedinHettingeretal.(2015)throughthehierarchicalMCMC
method.
Thesepopulation-wideMCMCrunsfavoredlargervaluesof

,uptothepriorlimit
imposedat

=6.PreviousMCMCrunswithamuchlargerrangeinallowed

values
heldthatlargevalues(
>
20)achievethesameposteriordensityasmodestvalues(

˘
6;
Figure5.10).Thisislikelyduetothemaximumseparation(
a
=0
:
16AU)thatwasimposed
Figure5.10:PosteriordistributionfortheMCMCrunofthemetal-richgroup,withan
extendedpriorlimitin

.
fromourof
f
b
.Forany
f
b
,arbitrarilylargevaluesof

simplyconcentratebinaries
at
a
'
0
:
16AU.Atlarger

,allbinaryfractionswithinareasonablerangeproduceasimilar
e=i
distribution.
Asexpected,atlowervaluesof

,wedoseeacorrelationwith
f
b
,whereasmallervalue
100
of
f
b
isrequiredforseparationdistributionsfavoringshorterperiods.Withhigherquality
data,thisregionofparameterspacewouldbemoretightlyconstrained.
Acrossthe[Fe
=
H]groups,thevaluesof

and
f
b
arebroadlyconsistentwithoneanother.
Anexceptionis,forthemetal-intermediategroup,thereisincreaseddensityintheposterior
atlowervaluesof
f
b
.Thisislikelyduetothelackofobjectswith8
<e=i<
12inthis
[Fe
=
H]group.Also,theMCMCseemedtoignorethetwohighest
e=i
objects.
Thedetectionandperiodsensitivityofourmethodaredeterminedbythesurvey
parameters.ThelowspectralresolutionoftheSEGUEdatarequiresstringentqualitycuts
toensurethatsub-pixelRVshiftscanbedetectedateachepoch.Thetotalnumberofstars
inHettingeretal.(2015)whichsurvivedqualitycutsamountstoonly
˘
15
;
000amongthe
˘
250
;
000SEGUEtargetsasF-type.TypicalRVerrorsinSEGUEspectraare
˘
4kms

1
,limitingthelongestperiodsandseparationsthatcanprobed.
TheAPOGEEsurveyhasahigherresolution(
R
˘
20
;
000)andsmallerRVuncertainties
(
˘
0
:
5kms

1
).Applyingourtechniquestothisdatasetwillresultinamuchhigherdetec-
tion,andwillrequirelessstringentqualitycuts.PreliminaryworkwithAPOGEE
datayieldsapromisingdistributionofmaximumRVvariations,withthousandsoflikely
binarieswithlog
P
(d)ashighas3(Figure5.11).
Wewouldliketoseestatisticaltime-resolvedspectroscopytechniquescontinuetobeem-
ployedinthefuture.WorkontheAPOGEEsetofstarshasbegunalready,andanextension
tootherspectralclasseswithinSEGUEcanbeaccomplishedwithlittlemotothe
originalpipeline.Furtherdowntheroad,weexpectsimilartechniquesdescribedinthisdis-
sertationwillbeappliedtoothercurrentandupcomingmerspectroscopicmissions
suchasLAMOST,DESI,andPFS,whichareexpectedtocollectmillionsofstellarspectra
inthenextfewyears.
101
Figure5.11:PreliminarydistributionofmaximumRVvariationforDR12APOGEEtargets.
102
REFERENCES
103
REFERENCES
Abt,H.A.1965,ApJS,11,429
Aihara,H.,etal.2011,ApJS,193,29
Allen,P.R.2007,ApJ,668,492
Badenes,C.,&Maoz,D.2012,ApJ,749,L11
Badenes,C.,Mullally,F.,Thompson,S.E.,&Lupton,R.H.2009,ApJ,707,971
Balega,Y.Y.,Dyachenko,V.V.,Maksimov,A.F.,Malogolovets,E.V.,Rastegaev,D.A.,
&Romanyuk,I.I.2011,AstronomischeNachrichten,332,978
Bate,M.R.1998,ApJ,508,L95
|.2012,MNRAS,419,3115
|.2014,MNRAS,442,285
Bickerton,S.,Badenes,C.,Hettinger,T.,Beers,T.,&Huang,S.2012,inProceedingsof
theInternationalAstronomicalUnion,Vol.7,NewHorizonsinTime-DomainAstronomy,
ed.E.R.Hanisch,&R.Seaman,289{290
Bodenheimer,P.,Ruzmajkina,T.,&Mathieu,R.D.1993,inProtostarsandPlanetsIII,
ed.E.H.Levy&J.I.Lunine,367{404
Boss,A.P.1988,CommentsonAstrophysics,12,169
Brandeker,A.,Jayawardhana,R.,&Najita,J.2003,AJ,126,2009
Brown,J.D.2000,Phys.Rev.D,62,084024
Burgasser,A.J.,Reid,I.N.,Siegler,N.,Close,L.,Allen,P.,Lowrance,P.,&Gizis,J.2007,
ProtostarsandPlanetsV,427
Carquillat,J.-M.,&Prieur,J.-L.2007,MNRAS,380,1064
Chini,R.,V.H.,Nasseri,A.,Stahl,O.,&Zinnecker,H.2012,MNRAS,424,
1925
Claeys,J.S.W.,Pols,O.R.,Izzard,R.G.,Vink,J.,&Verbunt,F.W.M.2014,A&A,563,
A83
Clark,B.M.,Blake,C.H.,&Knapp,G.R.2012,ApJ,744,119
Clarke,C.1992,inAstronomicalSocietyofthePcConferenceSeries,Vol.32,IAU
Colloq.135:ComplementaryApproachestoDoubleandMultipleStarResearch,ed.H.A.
McAlister&W.I.Hartkopf,176
104
Clarke,C.J.,&Pringle,J.E.1993,MNRAS,261,190
Connelley,M.S.,Reipurth,B.,&Tokunaga,A.T.2008,AJ,135,2526
Conroy,C.2013,ARA&A,51,393
Covey,K.R.,etal.2007,AJ,134,2398
Crutcher,R.M.2012,ARA&A,50,29
Cui,X.-Q.,etal.2012,ResearchinAstronomyandAstrophysics,12,1197
Davis,P.J.,Kolb,U.,&Knigge,C.2012,MNRAS,419,287
Davis,P.J.,Kolb,U.,&Willems,B.2010,MNRAS,403,179
Dawson,K.S.,etal.2013,AJ,145,10
Delfosse,X.,etal.2004,inAstronomicalSocietyofthePConferenceSeries,Vol.318,
SpectroscopicallyandSpatiallyResolvingtheComponentsoftheCloseBinaryStars,ed.
R.W.Hilditch,H.Hensberge,&K.Pavlovski,166{174
Dieterich,S.B.,Henry,T.J.,Golimowski,D.A.,Krist,J.E.,&Tanner,A.M.2012,AJ,
144,64
Djorgovski,S.G.,etal.2011,ArXive-prints
Drake,A.J.,etal.2014,ApJ,790,157
Duch^ene,G.,Bontemps,S.,Bouvier,J.,Andre,P.,Djupvik,A.A.,&Ghez,A.M.2007,
A&A,476,229
Duch^ene,G.,&Kraus,A.2013,ARA&A,51,269
Dunham,M.M.,etal.2014,ProtostarsandPlanetsVI,195
Duquennoy,A.,&Mayor,M.1991,A&A,248,485
Eggleton,P.P.1983,ApJ,268,368
Evans,T.M.,etal.2012,ApJ,744,120
Fabricant,D.,etal.2005,PASP,117,1411
Fischer,D.A.,&Marcy,G.W.1992,ApJ,396,178
Foreman-Mackey,D.,Hogg,D.W.,Lang,D.,&Goodman,J.2013,PASP,125,306
Frieman,J.A.,etal.2008,AJ,135,338
Gao,S.,Liu,C.,Zhang,X.,Justham,S.,Deng,L.,&Yang,M.2014,ApJ,788,L37
Geller,A.M.,Mathieu,R.D.,Harris,H.C.,&McClure,R.D.2008,AJ,135,2264
105
Goodman,J.,&Weare,J.2010,Commun.Appl.Math.Comput.Sci.,5,65
R.F.2012,JournalofAstrophysicsandAstronomy,33,29
Han,Z.,Podsiadlowski,P.,&Lynas-Gray,A.E.2007,MNRAS,380,1098
Han,Z.,Podsiadlowski,P.,Maxted,P.F.L.,&Marsh,T.R.2003,MNRAS,341,669
Han,Z.,Podsiadlowski,P.,Maxted,P.F.L.,Marsh,T.R.,&Ivanova,N.2002,MNRAS,
336,449
Henry,T.J.,Jao,W.-C.,Subasavage,J.P.,Beaulieu,T.D.,Ianna,P.A.,Costa,E.,&
Mendez,R.A.2006,AJ,132,2360
Herwig,F.2005,ARA&A,43,435
Hettinger,T.,Badenes,C.,Strader,J.,Bickerton,S.J.,&Beers,T.C.2015,ApJ,806,L2
Hogg,D.W.,Bovy,J.,&Lang,D.2010,ArXive-prints
Iben,Jr.,I.,&Tutukov,A.V.1984,ApJS,54,335
Ivanova,N.,etal.2013,A&ARev.,21,59
J.L.Halbwachs,M.Mayor,S.Udry,&F.Arenou.2003,A&A,397,159
Jao,W.-C.,Mason,B.D.,Hartkopf,W.I.,Henry,T.J.,&Ramos,S.N.2009,AJ,137,
3800
Jeans,J.H.1919,Problemsofcosmogonyandstellardynamics
Kaiser,N.,etal.2002,inSocietyofPhoto-OpticalInstrumentationEngineers(SPIE)Con-
ferenceSeries,Vol.4836,SurveyandOtherTelescopeTechnologiesandDiscoveries,ed.
J.A.Tyson&S.W154{164
Kiminki,D.C.,etal.2012,ApJ,747,41
Kobayashi,C.,Umeda,H.,Nomoto,K.,Tominaga,N.,&Ohkubo,T.2006,ApJ,653,1145
Koch,R.H.,&Hrivnak,B.J.1981,AJ,86,438
KornC.,Kaczmarek,T.,&Pfalzner,S.2012,A&A,543,A126
Kouwenhoven,M.B.N.,Brown,A.G.A.,PortegiesZwart,S.F.,&Kaper,L.2007,A&A,
474,77
Kozai,Y.1962,AJ,67,591
Kroupa,P.2001,MNRAS,322,231
Kroupa,P.,&Bouvier,J.2003,MNRAS,346,343
106
Krumholz,M.R.,Klein,R.I.,&McKee,C.F.2012,ApJ,754,71
Lada,C.J.2006,ApJ,640,L63
Latham,D.W.,Stefanik,R.P.,Torres,G.,Davis,R.J.,Mazeh,T.,Carney,B.W.,Laird,
J.B.,&Morse,J.A.2002,AJ,124,1144
Latif,M.A.,&Schleicher,D.R.G.2015,MNRAS,449,77
Laughlin,G.,&Rozyczka,M.1996,ApJ,456,279
Law,N.M.,etal.2009,PASP,121,1395
Lebovitz,N.R.1974,ApJ,190,121
|.1984,ApJ,284,364
Lee,Y.S.,etal.2008,AJ,136,2022
Levi,M.,etal.2013,ArXive-prints
Leys,C.,Ley,C.,Klein,O.,Bernard,P.,&Licata,L.2013,JournalofExperimentalSocial
Psychology,49,764
Lodieu,N.,ZapateroOsorio,M.R.,&MartE.L.2009,A&A,499,729
LSSTScienceCollaborationetal.2009,ArXive-prints
Lu,P.,Deng,L.C.,&Zhang,X.B.2010,MNRAS,409,1013
Lucatello,S.,Tsangarides,S.,Beers,T.C.,Carretta,E.,Gratton,R.G.,&Ryan,S.G.
2005,ApJ,625,825
Luhman,K.L.2012,ARA&A,50,65
Lyttleton,R.1954,Bull.Amer.Math.Soc.60(1954),497-500DOI:http://dx.doi.
org/10.1090/S0002-9904-1954-09847-6PII,0002
Machida,M.N.,Omukai,K.,Matsumoto,T.,&Inutsuka,S.-I.2009,MNRAS,399,1255
Maoz,D.,Badenes,C.,&Bickerton,S.J.2012,ApJ,751,143
Marks,M.,&Kroupa,P.2012,A&A,543,A8
Mason,B.D.,Henry,T.J.,Hartkopf,W.I.,tenBrummelaar,T.,&Soderblom,D.R.1998,
AJ,116,2975
Mathieu,R.D.1994,ARA&A,32,465
McClure,R.D.,&Woodsworth,A.W.1990,ApJ,352,709
Miko lajewska,J.2007,BalticAstronomy,16,1
107
Milliman,K.E.,Mathieu,R.D.,Geller,A.M.,Gosnell,N.M.,Meibom,S.,&Platais,I.
2014,AJ,148,38
Moe,M.,&DiStefano,R.2013,ApJ,778,95
NationalResearchCouncil.2010,NewWorlds,NewHorizonsinAstronomyandAstrophysics
(TheNationalAcademiesPress)
Ness,M.,Hogg,D.W.,Rix,H.-W.,Ho,A.,&Zasowski,G.2015,ArXive-prints
S.S.R.,Clark,P.C.,Hennebelle,P.,Bastian,N.,Bate,M.R.,Hopkins,P.F.,
Moraux,E.,&Whitworth,A.P.2014,ProtostarsandPlanetsVI,53

Opik,E.1924,Publ.Obs.Tartu,25,6
Palau,A.,etal.2013,ApJ,762,120
Parker,R.J.,Goodwin,S.P.,Kroupa,P.,&Kouwenhoven,M.B.N.2009,MNRAS,397,
1577
Patience,J.,Ghez,A.M.,Reid,I.N.,&Matthews,K.2002,AJ,123,1570
Paxton,B.,etal.2015,ArXive-prints
Peterson,B.M.,Wanders,I.,Horne,K.,Collier,S.,Alexander,T.,Kaspi,S.,&Maoz,D.
1998,PASP,110,660
Pickett,B.K.,Durisen,R.H.,&Davis,G.A.1996,ApJ,458,714
Pineda,J.E.,etal.2015,Nature,518,213
Piotto,G.,etal.2004,ApJ,604,L109
Podsiadlowski,P.,&Joss,P.C.1989,Nature,338,401
Podsiadlowski,P.,Joss,P.C.,&Hsu,J.J.L.1992,ApJ,391,246
Podsiadlowski,P.,Rappaport,S.,&Han,Z.2003,MNRAS,341,385
Podsiadlowski,P.,Rappaport,S.,King,A.R.,D'Antona,F.,&Burderi,L.,eds.2001,
AstronomicalSocietyofthePConferenceSeries,Vol.229,EvolutionofBinaryand
MultipleStarSystems;AMeetinginCelebrationofPeterEggleton's60thBirthday
Raghavan,D.,etal.2010,ApJS,190,1
Rastegaev,D.A.2010,AJ,140,2013
Reid,I.N.,&Gizis,J.E.1997,AJ,113,2246
Riess,A.G.,etal.1998,AJ,116,1009
Sana,H.,Gosset,E.,&Evans,C.J.2009,MNRAS,400,1479
108
Schreiber,M.R.,&ansicke,B.T.2003,A&A,406,305
Shu,F.H.,Adams,F.C.,&Lizano,S.1987,ARA&A,25,23
Smee,S.A.,etal.2013,AJ,146,32
Sneden,C.,Cowan,J.J.,&Gallino,R.2008,ARA&A,46,241
Stacy,A.,Greif,T.H.,&Bromm,V.2010,MNRAS,403,45
Stamatellos,D.,Whitworth,A.P.,&Hubber,D.A.2012,MNRAS,427,1182
Sugai,H.,etal.2012,inSocietyofPhoto-OpticalInstrumentationEngineers(SPIE)Confer-
enceSeries,Vol.8446,SocietyofPhoto-OpticalInstrumentationEngineers(SPIE)Con-
ferenceSeries,0
Taam,R.E.,&Sandquist,E.L.2000,ARA&A,38,113
TheDarkEnergySurveyCollaboration.2005,ArXivAstrophysicse-prints
Thompson,T.A.2011,ApJ,741,82
Tohline,J.E.2002,ARA&A,40,349
Tohline,J.E.,Durisen,R.H.,&McCollough,M.1985,ApJ,298,220
Tokovinin,A.2011,AJ,141,52
Tonry,J.,&Davis,M.1979,AJ,84,1511
Truelove,J.K.,Klein,R.I.,McKee,C.F.,Holliman,II,J.H.,Howell,L.H.,Greenough,
J.A.,&Woods,D.T.1998,ApJ,495,821
Tsuribe,T.,&Inutsuka,S.-i.1999,ApJ,523,L155
Webbink,R.F.1986,HighlightsofAstronomy,7,185
Whelan,J.,&Iben,Jr.,I.1973,ApJ,186,1007
Woodward,J.W.,Tohline,J.E.,&Hachisu,I.1994,ApJ,420,247
Yanny,B.,etal.2000,ApJ,540,825
|.2009,AJ,137,4377
York,D.G.,etal.2000,AJ,120,1579
Yuan,H.,Liu,X.,Xiang,M.,Huang,Y.,Chen,B.,Wu,Y.,Hou,Y.,&Zhang,Y.2015,
ApJ,799,135
ZapateroOsorio,M.R.,&MartE.L.2004,A&A,419,167
109