Is the Galactic bulge devoid of planets?

1.1 MB·

by  Matthew T. Penny

DRAFTVERSIONNOVEMBER17,2016
PreprinttypesetusingLATEXstyleemulateapjv.12/16/11
ISTHEGALACTICBULGEDEVOIDOFPLANETS?
MATTHEWT.PENNY1,2,CALENB.HENDERSON3,4 ANDCHRISTIANCLANTON5,4
DraftversionNovember17,2016
ABSTRACT
Considering a sample of 31 exoplanetary systems detected by gravitational microlensing, we investigate
whether or not the estimated distances to these systems conform to the Galactic distribution of planets ex-
pected from models. We derive the expected distribution of distances and relative proper motions from a
6 simulated microlensing survey, correcting for the dominant selection effects that affect the planet detection
1 sensitivityasafunctionofdistance,andcompareittotheobserveddistributionusingAnderson-Darling(AD)
0 hypothesistesting. Takingtherelativeabundanceofplanetsinthebulgetothatinthedisk,f ,asamodel
bulge
2 parameter, we find that our model is only consistent with the observed distribution for f < 0.54 (for a
bulge

v p-valuethresholdof0.01)implyingthatthebulgemaybedevoidofplanetsrelativetothedisk. Allowingfora
o dependenceofplanetabundanceonmetallicityandhostmass,oranadditionaldependenceofplanetsensitivity
N on event timescale does not restore consistency for f = 1. We examine the distance estimates of some
bulge
eventsindetail,andconcludethatsomeparallax-baseddistanceestimatescouldbesignificantlyinerror. Only

6 bycombiningtheremovalofoneproblematiceventfromoursampleandtheinclusionofstrongdependences
1 ofplanetabundanceordetectionsensitivityonhostmass, metallicityandeventtimescaleareweabletofind
consistencywiththehypothesisthatthebulgeanddiskhaveequalplanetabundance.

]
P
E
1. INTRODUCTION tance to the planet hosts. This is particularly interesting for
. microlensingdetections, becausemicrolensingeventscanbe
h Theplanetformationprocessiscomplexandtheabundance
caused by stars at any distance between the Earth and the
p and architectures of planetary systems will likely depend on
- numerousfactors. Theprimeexampleofsuchadependence Galacticbulge,wheremostofthestarsthatactassourcesre-
o side. Therefore,thedistributionsofhostage,metallicityand
is the strong correlation of giant planet abundance with host
r kinematic population membership (e.g. disk or bulge) will
t star metallicity (e.g., Santos et al. 2004; Fischer & Valenti
s vary as a function of distance, and if the planet abundance
2005), and suggests that the formation of giant planets de-
a is affected by any of these parameters it may be possible to
[ pendsontheamountofsolidsintheprotoplanetarydisk(Ida
& Lin 2004). A correlation of giant planet abundance with constrainthisdependence. Measuringtherelativeabundance

2 hostmass(Johnsonetal.2010),thoughlesssecure(see,e.g., of planets between the disk and bulge is an especially inter-
v Lloyd2013),alsopointstoasimilarconclusion,oratleasta estingprospectbecauseitisextremelydifficulttoprobewith
7 anyothertechnique. Forexample, Clarksonetal.(2008)at-
dependenceontotaldiskmass. Itisalsopossiblethatextrin-
0 tempted to address this question using 16 planet candidates
sic factorssuch asthe presenceof a binarycompanion (e.g.,
8 discovered by the HST Sagittarius Window Eclipsing Extra-
Kaib et al. 2013), the density of nearby stars (e.g., de Juan
2 solar Planet Survey (SWEEPS), but because of a potentially
Ovelar et al. 2012) or the intensity of the ambient radiation
0 largefalsepositivefractiontheycouldnotdrawfirmconclu-
field(e.g.,Thompson2013)mayaffectplanetformationdur-
1. ingtheprotoplanetarydiskphaseorlaterbydynamicalinter- sions. Oursampleof31microlensingplanethostsisalready
0 actions.Inordertounderstandthemagnitudeofeachpossible significantly larger, and future microlensing samples will be
6 effectonplanetabundance,itwillbenecessarytosearchfor largerstillandsignificantlycleanerandeasiertomodel. Ad-
1 planets around stars that span a broad range of fundamental ditionally,observationsfromfuturespace-basedmicrolensing
: surveyswillenablethediscoveryoflargesamplesoftransit-
v propertiesandformationenvironments.
ingplanets(McDonaldetal.2014),whosepopulationscanbe
i Withtheadventofmodern,high-cadencegravitationalmi-
X determinedusingsimilarmethodstothoseofClarksonetal.
crolensing surveys it is now becoming possible to measure
(2008).
r theabundanceofroughlyNeptune-to-Jupiter-massplanetsat
a theorbitaldistancesatwhichsuchplanetsarethoughttoform Our aim is to investigate what information the distribu-
tion of planet host distances inferred by microlensing planet
(seeGaudi2012foradetailedreview).Sofar,studieshavefo-
searches might contain about the planet formation process.
cusedonmeasuringtheoverallabundanceandmass-function
The sample we have to work with is far from ideal for this
ofplanetsintheseorbits(Gouldetal.2010b;Sumietal.2010;
task. Ithasbeenconstructedthroughaheterogeneousmixof
Cassan et al. 2012; Shvartzvald et al. 2016), but microlens-
survey strategies that aim to maximize the number of planet
inglightcurvescombinedwithvariousauxiliaryobservations
detections, but that do not always allow an easy assessment
canalsoyieldeithermeasurementsorconstraintsonthedis-
of the detection sensitivity due to human intervention in the
[email protected] observing schedule and the large, heterogeneous network of
1DepartmentofAstronomy,OhioStateUniversity,140West18thAv- follow-up telescopes that are necessary to achieve signifi-
enue,Columbus,OH43210,USA cant planet sensitivity in many cases. This has made it ex-
2SaganFellow
tremely difficult to measure the abundance of microlensing
3Jet Propulsion Laboratory, California Institute of Technology, 4800
OakGroveDrive,Pasadena,CA91109,USA planetsfrommicrolensingsurveysconductedsofar: thesur-
4NASAPostdoctoralProgramFellow veysthemselveswereuntilrecentlynotabletoobserveathigh
5NASAAmesResearchCenter,MoffettField,CA94035,USA enoughcadencetofindmostplanetsblind,andfollow-upre-
2 PENNY,HENDERSON&CLANTON
sourcesneededtobedirectedinamannerthatwasindepen- precisionbetterthan20percent. Thismeansthatπ isusu-
rel
dentofthepresenceofaplanet. allyagoodindicatorofthelensdistance.Unfortunately,many
Theseconsiderationswouldseemtorenderourtaskhope- papers that estimate lens distances do not quote an estimate
lesswiththecurrentdataset. However,whilethepresenceor forπ anddonotcontainenoughinformationtocalculateit,
rel
notofapotentialplanetarysignatureinamicrolensingevent soinsteadwewillworkwiththelensdistanceestimatesD to
l
mayaffectthewayinwhichitisobserved,thereislittletono investigatethedistributionoflenslocations.
informationonthehostdistancethatcouldaffectthedecision Itispossibletomeasureπ andhenceobtainagoodesti-
rel
processofanobserver. Thisisnottosaythatthedetectability mateofthelensdistanceiftwopotentiallyobservablequanti-
ofaplanetinamicrolensingeventisindependentofitshost tiesaremeasured: θ andthemicrolensingparallax
E
distance, justthatthereisunlikelytobeanythingthatwould
π
makethedetectabilityasafunctionofhostdistancedifficult π = rel. (4)
E
θ
to model. This means that if we can construct a reasonable E
model of both planet detectability as a function of host dis- Adetailedderivationofmicrolensingmassanddistancemea-
tance and of the distribution of lens stars (i.e., the potential surementsusingstandardnotationisgivenbyGould(2000),
planet hosts) then we can reasonably compare the distribu- Gould&Horne(2013)andCalchiNovati&Scarpetta(2016).
tionofactualplanethostdistancestothatweexpectfromthe Inallbutoneplanetarymicrolensingeventtodateθ hasbeen
E
model. Morespecifically,wecantestthehypothesisthatthe measuredfromfinitesourceeffects(Witt&Mao1994):these
distributionofactualplanethoststardistancesisdrawnfrom resolvethesourcestar, whichprovidesanangularrulerwith
the same distribution of host star distances that we produce whichtomeasureθ ,andhencealsoµ . Measuringπ is
E rel,⊕ E
fromourmodel. more difficult, and until recently has required detecting sub-
Inthispaperweconstructasimulatedsampleofmicrolens- tledistortionstothemicrolensingeventlightcurvethatresult
ing planet host stars with detected planets and compare the fromtheaccelerationoftheEarthinitsorbit(Gould&Loeb
resultant(expected)distributionofhostdistanceswiththeob- 1992). The chance of detecting these distortions is greatly
served distribution. We begin by describing how distances increasediftheeventhasalongtimescaleandiftheprojec-
are measured to microlensing planet hosts in section 2. In tionoftheEinsteinringfromthesourceontothesolarsystem
section3wedescribetheselectionoftheobservedmicrolens- is small. These conditions end up strongly favoring nearby
ingplanethostsampleandinsection4wedescribethemodel lenses. A much less biased way to measure π requires ob-
E
withwhichwebuildthecomparisondistribution. Insection5 servationofamicrolensingeventfromaspacecraftwellsep-
we present the results of our comparisons of the model and arated from the Earth (Refsdal 1966; Gould 1994) and has
observed distance distributions. In section 6 we discuss our recentlycometofruitionusingSpitzer(Yeeetal.2015;Udal-
resultsandtheirpotentialcauses,andinsection7wepresent skietal.2015c;CalchiNovatietal.2015;Streetetal.2016;
ourconclusions. Shvartzvald et al. 2015) and will be further advanced with
Campaign 9 of the K2 mission (Howell et al. 2014; Hender-
2. MEASURINGDISTANCES sonetal.2015).
Beforediscussingtheselectionofourobservedplanethost If it is not possible to measure the lens distance through
sampleitisnecessarytofirstintroducethemethodsbywhich measurementsofbothθEandπE,itispossibletoestimatethe
lens(host)distancesaremeasuredinmicrolensingevents. In distance through a Bayesian analysis by applying what con-
mostsinglepointlensmicrolensingeventstheonlyconstraint straintsyouhappentohavetogetherwithpriorsonthedistri-
onthelensdistanceisprovidedbytheeventtimescale butionoflensesandsourcesfromaGalacticmodel,aswellas
oftenunstatedbutimpliedflatpriorssuchastheabundanceof
t = θE , (1) planetsasafunctionofhoststarmass,metallicityandorbital
E µ architecture. In the absence of constraints on the lens dis-
rel,⊕
tance,theseBayesianestimateswilltendtoclusteratthepeak
whereθE istheangularEinsteinradiusandµrel,⊕ istherela- ofthepriorprobabilitydistribution. Forthisreason,mostau-
tivepropermotionbetweenthelensandsource,measuredin thors caution against using Bayesian distance estimates for
thegeocentricreferenceframe.6 TheangularEinsteinradius the kind of study we are conducting. As there is presently
nootherwaytoaccessdistantplanets, wechoosetocommit
(cid:112)
θE = κMπrel, (2) thissinandthroughoutremainwaryofitsconsequences(see
subsection6.2foradiscussionofitsimpact).
isdeterminedbythelensmassM andtherelativelens-source
parallaxπ ;κ=8.144masM−1isaconstant. 3. DATA
rel (cid:12)
Fromanobserversstandpoint,πrelisthefundamentalquan- We select our sample of planet hosts from all planetary
tity related the location of the lens, and is a measure of the microlensing events detected in seasons up to and including
proximityofthelenstothesource, 2014andwithapublicationdateof2015orearlier. Ouronly
cutistoexcludesystemswithmassratios(q M /M)larger
AU AU p
π = , (3) than0.03. Thiscutpreventsastrongdistanc≡ebiasthatexists
rel
D − D
l s forlargeplanetsthatorbitverylowmassstars,suchasOGLE-
where D and D are the lens and source distances, respec- 2011-BLG-420 (Choi et al. 2013). If the system has a mass
l s
tively. For most microlensing events, the source lies in the ratio that overlaps that possible for stellar binaries then it is
bulge, and it is reasonable to assume that D is known to a likely,first,thatitwillnotbeanalyzedindetail,andsecond,
s
thatunlessitsparallaxismeasured,itwillnotberecognized
6 The geocentric reference frame is an inertial frame moving with the asaplanet.
Earthatsomesignificantreferenceepoch, usuallychosentobetheepoch Table1liststhedistancesandrelativepropermotionsofthe
ofaprominentlightcurvefeature. microlensing planet hosts in our sample, and the method by
ISTHEGALACTICBULGEDEVOIDOFPLANETS? 3
Table1
Microlensingplanethoststars
Event Dl µrel θE logq (cid:96) b Method Reference
(kpc) (masyr−1) (mas) (◦) (◦)
O03-235 5.8+0.6 3.1±0.5 0.52±0.08 −2.4 +2.20 −3.69 Bayes 1,2
−0.7
O05-071 3.2±0.4 4.3±0.2 0.85±0.05 −2.1 −4.42 −3.79 πEθE 3,4
O05-169 4.1±0.4 7.7±0.8 1.00±0.22 −4.2 +0.68 −4.74 Bayes 5,6,7
O05-390 6.6±1.0 6.8±0.7 0.20±0.03 −4.1 −0.27 −2.36 Bayes 8
O06-109 1.5±0.1 4.3±0.1 1.50±0.04 −3.3 −0.21 −1.89 πEθE 9,10
M07-192 0.70+−00..2112 5.1±1.4 0.56±0.06 −4.4 +4.03 −3.39 πEθE 11,12,13
O07-368 5.9+0.9 3.5±0.6 0.53±0.08 −4.0 −1.65 −3.69 Bayes 14
−1.4
M07-400 5.8+0.6 8.2±0.5 0.32±0.02 −2.6 +2.38 −4.70 Bayes 15
−0.8
O08-092 8.1±2.0 3.2±0.2 0.34±0.02 −3.6 −4.75 −3.34 other 16
M08-310 7.7±2.0 5.1±0.3 0.15±0.01 −3.5 −4.09 −4.56 Bayes 17
O08-355 6.8±1.1 3.1±0.4 0.28±0.03 −1.9 −0.08 −3.45 Bayes 18
M08-379 3.3+1.3 7.6±1.6 0.88±0.19 −2.2 +0.38 −3.11 Bayes 19
−1.2
M09-266 3.0±0.3 5.9±0.3 0.98±0.04 −4.2 −4.93 −3.58 πEθE 20
M09-319 6.1+1.1 7.5±0.7 0.34±0.03 −3.4 +4.20 −3.01 Bayes 21
−1.2
M09-387 5.7±1.3 2.8±0.3 0.31±0.03 −1.9 −3.44 −4.09 Bayes 22
M10-328 0.81±0.10 5.7±0.7 0.98±0.12 −3.6 −0.16 −3.21 πEθE 23
M10-353 6.4±1.1 6.0±2.6 0.19±0.09 −2.9 +3.60 −2.90 Bayes 24
M10-477 2.3±0.6 10.3±0.8 1.38±0.11 −2.7 +0.05 −5.09 Bayes 25
O11-0251 2.6±0.6 4.3±1.6 0.75±0.28 −2.7 +0.67 +2.33 πEθE 26
M11-262 7.2±0.8 11.6±0.9 0.12±0.01 −3.4 −0.37 −3.92 Bayes 27
O11-265 4.4±0.5 2.9±0.3 0.42±0.04 −2.4 +2.70 −1.52 πEθE 28
M11-293 7.7±0.4 4.3±0.3 0.26±0.02 −2.3 +1.52 −1.65 Bayes 29,30
M11-322 7.6+1.0 5.0±1.7 >0.085 −1.5 +3.63 −2.81 Bayes 31
−0.9
O12-0026 4.1±0.3 3.5±0.3 0.91±0.09 −3.9 +0.19 +3.06 πEθE 32
O12-0406 5.0±0.3 3.0±0.3 0.53±0.05 −2.2 −0.46 −2.22 πEθE 33,34
O12-0563 1.3+0.6 6.4±0.6 1.36±0.13 −3.0 +3.31 −3.25 Bayes 35
−0.8
M13-220 4.5±2.0 12.6±0.9 0.46±0.03 −2.5 +1.50 −3.76 other 36
O13-0341 1.2±0.1 10.2±0.8 0.93±0.07 −4.3 −0.05 −1.68 πEθE 37
O13-0723 0.49±0.04 3.8±0.3 0.70±0.06 −4.2 −0.02 +2.83 πEθE 38
O14-0124 4.1±0.6 2.3±0.7 0.84±0.26 −3.2 +2.34 −2.92 πEθE 39
MOA-bin-1 5.1+1.2 8.8±1.4 0.77±0.11 −2.3 −0.11 −1.48 Bayes 40
−1.9
Notes. Lensdistanceestimates, propermotions, massratiosandgalacticcoordinatesforoursampleofmicrolensingplanethosts. Themethodcolumnis
explainedinthetext.EventnamesoftheformatSURVEY-YYYY-BLG-NNN(N)areabbreviatedasSYY-NNN(N)withO=OGLEandM=MOA,e.g.,OGLE-
2014-BLG-0124becomesO14-0124.Eventswithjointsurveynamesarelistedonlybythefirstname.
References.(1)Bennettetal.2006;(2)Bondetal.2004;(3)Dongetal.2009b;(4)Udalskietal.2005;(5)Bennettetal.2015;(6)Batistaetal.2015;(7)Gould
etal.2006;(8)Beaulieuetal.2006;(9)Bennettetal.2010;(10)Gaudietal.2008;(11)Kubasetal.2012;(12)Gouldetal.2010a;(13)Bennettetal.2008;(14)
Sumietal.2010;(15)Dongetal.2009a;(16)Poleskietal.2014;(17)Janczaketal.2010;(18)Koshimotoetal.2014;(19)Suzukietal.2014;(20)Murakietal.
2011;(21)Miyakeetal.2011;(22)Batistaetal.2011;(23)Furusawaetal.2013;(24)Rattenburyetal.2015;(25)Bacheletetal.2012;(26)Kainsetal.2013;
(27)Bennettetal.2014;(28)Skowronetal.2015;(29)Batistaetal.2014;(30)Yeeetal.2012;(31)Shvartzvaldetal.2014;(32)Hanetal.2013b;(33)Tsapras
etal.2014;(34)Poleskietal.2014;(35)Fukuietal.2015;(36)Yeeetal.2014;(37)Gouldetal.2014;(38)Udalskietal.2015b;(39)Udalskietal.2015c;(40)
Bennettetal.2012
whichtheirdistancewasmeasured. Wherethereismorethan is described in the next section. In both cases we arbitrar-
one possible solution we use the solution favored by the au- ily adopt an error bar of 2.0 kpc in order to be able to run a
thors,orwheretheyareagnosticwechoosethesolutionwith bootstrapanalysisinsection6.
thelowestχ2. Oftenthechoiceofsolutionhasverylittleim-
pactofthedistanceestimate. Wherethemethodcolumnlists 4. MODEL
π θ , the host distance was measured by combining mea-
E E ForourmodelweadopttheexistingsimulationoftheKMT-
surementsofparallaxandfinitesourceeffects.Wheremethod
NetmicrolensingsurveybyHendersonetal.(2014),hereafter
islistedasBayes,D wasestimated usingaBayesiananaly-
l H14.ThesimulationassumesthattheGalaxyfollowstheHan
sis. ManyoftheseBayesianestimatesincorporateconstraints
& Gould (2003) model, and tracks whether stars belong to
fromAOimaging.
the bulge (which follows a Dwek et al. 1995 G2 model as
For two of the events included in our sample, the authors
described by Han & Gould 1995a, 2003) or the disk (which
did not compute Bayesian estimates of the lens distance:
followsaBahcall1986modelasdescribedbyHan&Gould
OGLE-2008-BLG-092(Poleskietal.2014)andMOA-2013-
1995b). Source and lens stars are drawn from the Galactic
BLG-220 (Yee et al. 2014). It is important to include these
model and assigned a weight proportional to their contribu-
events, because excluding events without parallax measure-
tion to the event rate, i.e., proportional to µ θ . Here,
ments would bias the sample toward more nearby lenses. rel,(cid:12) E
µ is the relative lens-source proper motion measured in
Poleskietal.(2014)arguethattheirlensisinthebulgedueto rel,(cid:12)
atheheliocentricframe,becausetheH14simulationsdonot
the small relative proper motion and perform a Monte Carlo
considertheEarth’sorbitalmotion. GiventheEarth’sinstan-
simulationtoestimateadistanceof8.1kpc,whichweadopt.
taneousvelocityof2πAUyr−1,theheliocentricandgeocen-
Yeeetal.(2014)useblendedfluxconstraintsandpropermo-
tric proper motions can differ significantly, but for the pur-
tionstoarguethattheirlensresidesinthedisk,butonlypro-
posesofthispaperitisrareforalargerelativepropermotion
videanapproximatedistanceupperlimitof6.5kpc;weadopt
inoneframetobesmallintheotherframe.Fortherestofthis
adistanceof4.5kpcwhichcorrespondsapproximatelytothe
paperthen,wewilltreatheliocentricandgeocentricvelocities
peak of the disk distance distribution in our model, which
asequivalentwhencomparingmodeltodata;thisisnotideal,
4 PENNY,HENDERSON&CLANTON
butwehavenootheroptionasthesimulationsdidnotoutput Bayesianestimates Model(diskandbulge)
Parallax&FS Model(diskhostsonly)
enoughinformationtoreconstructthegeocentricpropermo-
tion,anditisnotpossibletocomputetheheliocentricproper s
t
motionoftheobservedeventsunlessparallaxismeasuredor os 10
h N(<5kpc)=17
thesourceandlensstarareresolvedwithhigh-resolutionob-
t 8 model: 9.0 2.5
servations long after the microlensing event (see Yee et al. ne ±
2015 and Bennett et al. 2007; Bennett et al. 2015; Batista pla 6
et al. 2015 for discussions of each option, respectively). We
of 4
willthereforeuseµ torefertorelativepropermotionsfrom
rel r
here on, be they helio- or geo-centric, and assume that their be 2
m
differencedoesnotsignificantlyaffectourresults.
u 0
Aplanetisassignedtoeachlens,withmassMp andsemi- N 1 Observeddistribution
major axis a chosen from a grid, and observations of the Modeldistribution
planetary microlensing event are simulated. To match ob- e 0.8 Modeldiskhostsonly
v
servationalconstraintsonthemassfunctionofplanetswhere ati 0.6
microlensing is sensitive, we weight each event by a factor ul
M0.73 (Cassanetal.2012). Thesimulateddataarefitwitha m 0.4
p u
singlepointmassmicrolensingevent,andiftheχ2 ofthisfit C 0.2
exceedstheχ2ofthedatarelativetotheunderlyingplanetary
0
microlensing event by 160, the event is flagged as a planet
0 2 4 6 8 10
detection. Further details of the simulation can be found in
Planet host distance (kpc)
H14.
KMTNetplanstoobserveasmallnumberoffieldscontinu-
ouslyathighcadence,whichisadifferentobservingstrategy Figure1. Histogramandcumulativedistribution(blacklines)ofmicrolens-
ingplanethostdistanceestimates, shadedbymeasurementmethod. Light
thanwasusedtofindmostoftheplanetsinoursample. Most
grayshadingrepresentseventswithdistancemeasurementsmadebydetect-
werefoundusinglowercadencesurveyobservationsthatcov- ingparallaxandfinitesourceeffects,darkgrayrepresentsdistanceestimates
eredalargerregionoftheGalacticbulge,incombinationwith usuallymadeusingBayesiantechniques.Theexpecteddistancedistribution
follow-up observations of individual events, though some of fromourmodelisplottedwitharedsolidline;errorbarsshowthePoisson
intervalofthemodelforeachbin.Theexpectedcontributionfromdiskhosts
the later discoveries can be classified as survey-only detec-
isplottedwithadashedorangeline.
tions. Without using extremely complicated simulations, we
areforcedtoassumethattheobservingstrategyusedwillnot
fromthemodel. Wecanimagineasophisticatedprocedureto
significantlybiasthedistancedistributionofoursample.This
incorporate all possible information on each planet host and
mayormaynotbethecase,butwhatisperhapsmorelikelyto
assess the likelihood of a given model in a Bayesian frame-
affectthedistancedistributionistheareaoverwhichtheplan-
work. In the future, with more uniform samples of planet
etsaresearchedfor.The4fieldsurveyplannedforKMTNetis
hoststhislevelofdetailwillbeappropriate,butgiventhenon-
farsmallerthantheareaoverwhichoursampleisdistributed.
uniformity of our sample and the difficulty in understanding
However,H14alsosimulatedalargerareasurveyatlowerca-
itsselectioneffects,weelectasimplerfrequentisthypothesis
denceinordertooptimizethenumberoffieldsobserved. We
testingapproachasafirstattempt.
choose to use the widest, 13 field simulation that H14 used;
ThetoppanelofFigure1showsthedifferentialdistribution
thiscoversalargefractionoftheareacoveredbytheOGLE-
ofplanethostdistanceestimatestogetherwiththedistribution
IIIsurvey,withfieldsbetween(cid:96)=7and(cid:96)= 8andb= 2
− − expectedfromourmodel. Themodel(normalizedtothetotal
tob = 6aswellasacoupleoffieldsinthenorthernbulge.
− number of hosts) under-predicts the number of hosts in each
Somewhatsurprisingly,wedidnotfindasignificantvariation
distancebinlessthan6kpc,whereasbeyond6kpcitgenerally
of the relative bulge to disk planet host ratio with field loca-
over-predicts. We have also plotted the contribution of disk
tion apart from the most outlying southern fields. With 13
hostsandnotethattheshapeofthedisk-onlyhostdistribution
fields,thecadenceofobservationwas32.5minutes.
couldpotentiallyofferabetterfittothedata.
For this simulation, H14 use a relatively sparsely sampled
Shown in the lower panel of Figure 1 is the same data in
planet mass-semimajor axis grid, with masses of 1, 10 and
cumulative form. We computed the Anderson-Darling test7
100M andloga = 0.15,0.40and0.65; however,ourearly
⊕ statisticA2 =6.661comparingthemodel8tothedata,which
work showed that our results were not changed significantly
hasap-valueof5.0 10−4. Comparingonlythediskhosts
when we used their more densely sampled fiducial simula- from the model yield×s A2 = 1.468, which has a p-value of
tionscoveringasmallerareaathighercadence.
0.18.TheADtestthereforerejectsthehypothesisthatthedis-
As we are interested in the role of metallicity, we assign
tributionofmicrolensingplanethostdistanceestimateswere
each lens star a metallicity according to the prescription of
drawnfromthesamedistributionasourmodel, butdoesnot
Clanton&Gaudi(2014).Metallicitiesfordiskstarsaredrawn
rejectthehypothesisthatthedataweredrawnfromoursimu-
from the distributions of [Fe/H] as a function of Galacto-
lateddistributionincludingonlydiskhosts.
centric radius and distance from the plane as measured by
APOGEE(Haydenetal.2014), andbulgestarsareassigned
5.1. APaucityofBulgePlanets?
ametallicitydrawnfromthesampleofbulgedwarfmetallic-
itiescollectedbyBensbyetal.(2013).
5. THEDISTANCEDISTRIBUTIONOFPLANETHOSTS 7 The Anderson-Darling test is similar to, but more sensitive than, the
Kolmogorov-Smirnovtest.
With our sample and model defined we can begin to in- 8Toconvertthediscretemodeldistributiontoacontinuousdistribution,we
vestigate whether the observed data match our expectations linearlyinterpolatedtheverywellsampledcumulativeweightdistribution.
ISTHEGALACTICBULGEDEVOIDOFPLANETS? 5
0.6 c Bayesianestimates Model(diskandbulge)
atistic 4 Dl distribution 0.5 tatisti ts 12Parallax&FS Model(diskhostsonly)
t s s
s o
t st h
gtes 3 p=0.001 0.4 ovte anet 9
arlin 2 p=0.01 0.3 mirn ofpl 6
D p=0.05 S r 3
- - e
on p=0.20 0.2 ov mb
rs 1 Anderson-Darling or u 0
de Kolmogorov-Smirnov og N 1
n P.A. M1.0[Fe/H]1.2 0.1 m
√A 0 ∝ l Kol ve 0.8
0 0.5 1 1.5 2 2.5 3 ati 0.6
Bulge relative planet abundance f ul Observeddistribution
bulge m 0.4
Modeldistribution
u
Figure2. The Anderson-Darling (solid) and Kolmogorov-Smirnov (dot C 0.2 Modeldiskhostsonly
dashed, rightaxis)teststatisticsplottedasafunctionoffbulge, theabun-
danceofplanetsinthebulgerelativetothedisk.Horizontallinesareplotted 0
atlabeledp-valuesfortheADtest. TheKStestaxisisplottedsuchthat
thereisanapproximatelylinearmappingbetweenADandKSp-valuesin 0 5 10 15
the0.001–0.2range. Lightlyshadedlinesweighttheplanetabundanceby Relative proper motion (mas yr 1)
−
thejointmassandmetallicitycorrelationfoundby(Johnsonetal.2010).
Ratherthanaskthebinaryquestionofwhetherornotthere Figure3. Thedistributionofrelativepropermotionsformicrolensingevents
withplanets.SeeFigure1forthedescriptionoftheplotelements.
are planets in the bulge, we can ask are there fewer planets
inthebulgethanthedisk? Wehaveweightedthebulgehosts
0.6 c
pdinliasoknuehrtosmsatosrdoaeunlndbdryebpauenlagateeddhdoitthsieotsnAarDellfatatecisvttoearttfovbatuhrligeoeu,asbthuveanldauabenuscnoedfaafnrbocueulgnoedf. tatistic 4 µrel distributionAnderson-Darling 0.5 statisti
We plot the results in Figure 2 as the dark blue solid line.9 s Kolmogorov-Smirnov t
A2 rises as fbulge increases, and crosses the p = 0.05 and test 3 P.A.∝Ml1.0[Fe/H]1.2 0.4 tes
p = 0.01 lines at fbulge = 0.32 and 0.54, respectively. The g p=0.001 ov
lKinSetaetstfbsutalgtiest=ic0b.e9h3a.vTehsissimreislualrtlys,ugbugtesctrsotshsaetsththeeapbu=nda0n.0c1e arlin 2 p=0.01 0.3 mirn
of planets in the bulge relative to the disk is less than 0.54 D p=0.05 S
- -
at3-sigma. Beforewecandrawthisconclusionhowever,we on p=0.20 0.2 ov
mustmorecriticallyexamineourassumptionsregardingboth rs 1 or
e g
thedataandourmodelandalsorejectanyotherpotentialex- d o
planations. An 0.1 m
√ ol
0 K
5.2. RelativeProperMotions 0 0.5 1 1.5 2 2.5 3
Bulge relative planet abundance f
Given the difficulty in making distance measurements of bulge
planethosts,thereareanumberofwaysthatthedistribution Figure4. AD and KS test statistics as a function of fbulge for the µrel
of distance estimates could be altered in a way that we have distribution.SeeFigure2foradescriptionoftheplotelements.
not modeled. Given that relative proper motions are mea-
suredroutinelyinplanetarymicrolensingeventsandthatstel- in the same frame, so it is possible that we are biasing the
larkinematicsarecorrelatedwithpopulation,wecanusethe result.
distributionofpropermotionmeasurementsasaconsistency Thelackofstatisticalpowerisborneout,withalargerange
check on our findings from the previous subsections. Fig- of f producing AD test statistics with large p-values.
bulge
ures3and4showtheresultsofthesameexerciseperformed However, theADstatisticdoesrisetowardsmallervaluesof
withtherelativepropermotionmeasurements. f reaching 2.704 at f = 0, which corresponds to
bulge bulge
The proper motion distribution should be much more ro- p = 0.039indicatingtensionwiththehypothesisofnobulge
busttocatastrophicerrorscomparedtothedistancedistribu- planets.However,thepropermotiondistributiondoesnotrule
tion, but due to the significant overlap of the proper motion outallofthespacewithf < 0.54thatisconsistentwith
bulge
distributionsofbulgeanddiskstars,itwillhavelessstatisti- thedistancedistribution.
calpowertodisentanglethecontributionsofeachpopulation.
Recallalsothatwearenotcomparingrelativepropermotions 5.3. OtherPhysicalExplanations?
Toinvestigateifotherphysicalfactorscouldcausetheob-
9Forcomparisonwehavealsoplottedtheless-sensitiveKStest;byplot- served distribution of distance estimates, we considered toy
tingthesquarerootoftheADtestitispossibletoplottherangeoftheKS
teststatisticsuchthatthep-valuesforeachtestaremappedapproximately modelsoftheplanetabundancef asafunctionofhostmetal-
one-to-oneovertherangep = 0.001to0.2. NotethatbecausetheKStest licity[Fe/H],hostmassM andplanetsemimajoraxisa:
islesssensitivethantheADtest,itwilltypicallynotrejectahypothesisas
stronglyandhencegivelargerp-values. f 10α[Fe/H], f Mβ, f aγ, (5)
∝ ∝ ∝
6 PENNY,HENDERSON&CLANTON
DlAD DlKS µrelAD µrelKS distributioncouldhaveaneffectonthedistancedistribution.
0.6
4 Metallicity 0.5 Second, H14 only explore a coarse grid of semimajor axis,
3 0.4 so it is not clear if a smooth distribution of semimajor axis
would change the result. Because the Einstein ring radius
2 0.3
statistic 01 000...126 statistic sprtElraondneegtpleaynbpudensadokanendDcealroaonnudnsdemmsiecimproaalrjeaontrisoiannxgsi’sosfwse1onrusElid,tivaaifdtfyeecpttoetnphdleaennpeclatesnoeisft
ngtest 34 Mass 00..45 novtest hboeGtswtivdeeeisnntabtnhocatethrdhaiodsstiratilbmvuaetsilosonca.intydsmuervtaelyliscihtyav,ewfeocuanndaaskcohrorwelastuiocnh
arli 2 0.3 mir acorrelationwouldchangeourfindingsfromsubsection5.1.
on-D 1 00..12 ov-S With no correlation with mass or metallicity (α = β = 0),
√Anders 034 Semimajoraxis 000...456 Kolmogor wJAoeDhnftoseousnntsdeot9na5l.th(p2ee0rDc1e0ln)atmnldiamsµsi-rtmseleodtfaisl0tlr.i0cibi2tuyt≤-iopnlafsn.beuWtlgaeebiug≤nhdti0ann.g3c2ebtyfrreotnhmde
(α=1.2,β =1.0)producesaqualitativelysimilarmodeldis-
2 0.3 tributiontoourfiducialmodel,butthenumberofdiskhostsat
0.2
1 distancesof3–5kpcisenhancedattheexpenseofbulgeplan-
0.1
0 etsbetween6and9kpc. Thisenhancementofplanetsinthe
-5 -4 -3 -2 -1 0 1 2 3 4 5 inner disk is due to the APOGEE radial metallicity gradient
Abundancepowerlawslope
risingabovethemeanmetallicityofthebulgeinthisregion.
Figure5. ADandKSteststatisticsasafunctionofpowerlawslopepa- TheADandKSstatisticsasafunctionoff includingthe
bulge
rametersfortoymodels(Equation5)oftheplanetabundanceasafunction
massandmetallicitycorrelationsareplottedinFigures2and
ofmetallicity(top)hostmass(middle)andsemimajoraxis(bottom)forboth
thedistanceestimateDl(blue)andrelativepropermotionµrel(red)distri- 4withlightershades. The95percentconfidencelimitsinthis
butions. case are 0.06 f 0.32. The 1 percent upper lim-
bulge
≤ ≤
its on f increase from 0.54 in theno-correlation case to
where bulge
d3N 0.7withtheobservedcorrelations. Notethatinthetwosce-
f = , (6) narios, f has a slightly different interpretation; with no
d[Fe/H]dMda bulge
mass or metallicity correlations, f can be interpreted as
bulge
following a long tradition of using power laws to model the theratioofplanetsperstarinthebulgerelativetothatinthe
distributionofexoplanetabundances(e.g.,Stepinski&Black disk,butwiththecorrelations,f isonlythecontribution
bulge
2000; Lineweaver & Grether 2003; Fischer & Valenti 2005; tothisratiothathasnotbeenaccountedforalreadybymass
Johnsonetal.2010).Figure5plotstheADandKSteststatis- andmetallicitycorrelations.
ticsasafunctionofeachparameterα,β andγ. Ifavalueof
α, β or γ different from our fiducial value of 0 significantly
6. DISCUSSION
reducesthevalueoftheADteststatistic,thenthiswouldin-
dicatethatratherthanadifferentabundanceofplanetsinthe So far, while acknowledging potential problems with the
diskandbulge,adependenceofplanetabundanceonintrinsic distanceestimates,wehavetreatedthemasiftheywerecor-
stellar properties or system architecture may be responsible rect and investigated the consequences. Now the time has
for the discrepancy that we see between the model and the come to examine the distance estimates more critically and
observationaldata. askiftheythemselvesarecausingtheapparentlackofplan-
Avalueofα =0.9improvesthemodel’smatchtothedis- ets in the bulge. We must also examine the validity of our
tanceestimatedistribution,butonlytoavalueofA2 =5.263 methodsandtheaccuracyofourmodel.
with p = 2.2 10−3. Interestingly, negative values of An obvious criticism of our work so far is that we have
α 1 are reje×cted by the AD test with p 0.01 for the not accounted for the uncertainties in the distance estimates
µ ≤d−istribution as well as the D distribution≤. Were there a (or proper motions) when computing p-values. If we were
rel l
strong negative correlation of planet abundance with metal- to do so, the p-values would increase. To compute accurate
licity, the fraction of stars in the low-metallicity tail of the p-valueswewouldneedtobootstrapresamplefromthepos-
bulge’s broad metallicity distribution would vastly overpro- terior probability distributions of each distance estimate, but
duceplanetscomparedtothedisktotheextentthatitwould thesedistributionsarerarelymadeavailablewiththeirpapers
evenaffecttherelativepropermotiondistributionofplanetary and are sometimes not even plotted. As an inferior substi-
microlensingevents. tutetotheposteriors(especiallywhensomearebimodal),we
Allowingnon-zerovaluesofeitherβandγdoesnotsignif- assumed that the quoted uncertainties on each measurement
icantlyreduceA2 (thoughimplausiblystrongnegativecorre- were Gaussian (with asymmetric error bars being treated as
lationsdoslightlyincreaseA2), implyingthattheseparame- two half Gaussians with equal probability), and resampled
tersdonotsignificantlyaffectthedistanceestimatedistribu- theDl distribution104 timeswithfbulge = 1; distancesthat
tion. Notehowever,forboththeseparametersourmodelhas were drawn as negative were redrawn until they were posi-
someshortcomings. First,forboththediskandbulgeH14do tive. 95 percent of the resamplings had a p-value less than
notsimulate eventswith hoststarsmore massivethan 1M . 1.7 10−3, compared to the p-value we measured for the
Thiscutoffisreasonablefortheolderbulge,wherestarsmo(cid:12)re actu×al sample of 5.0 10−4; and only 3 of the resamplings
×
massive would have evolved off the main sequence, but for had p > 0.01. This strongly suggests, but is not conclusive
theyoungerdiskmoremassivemainsequencestarswillstill givenoursimplistichandling,thatproperconsiderationofthe
be present. If there were a dependence of planet abundance posteriordistributionswouldnotexplaintheapparentlackof
onhostmass,therelativelylongerleverarmofthediskmass planetsinthebulge.
ISTHEGALACTICBULGEDEVOIDOFPLANETS? 7
DlAD DlKS µrelAD µrelKS 0.6 which the H14 survey-only simulations would not capture.
34 tE 00..45 Applyingat+1weightingalsobringsthemedianofthemodel
2 0.3 event timescEale distribution to 39.5 d (instead of 23.5 d for
1 0.2
0 0.1 the unweighted distribution) which is more in line with the
nderson-Darlingteststatistic 0123423401234 |s−1|u/s0s|| 0000000000000000................1234563456123456 mogorov-Smirnovteststatistic mpardiippnmiriarnessooDrtpgcedahvvr.oelmieeioeasdvmcvTnpeoeetetibrehitetmaoaniesirmbnsnete.l|hrnemeuiveAimstgfe0icifghd|nopsae.h-tfrlcsrmfeohttAaoevimbaorngteehgefpghmpdanot-lanthmiewebfieenfpyagercatlsegraa,vaalanttouawlmiinaivrotsfiiewhipenvibcdilnecpaleeivehpotn,tveiowlwoeerwassxnntenhheoptrdeiseeuclotosavslhbscefdneiasecinam|sliislttinnmoisun4−oogslgp0netaalo1rrtdypwieiv/o.frteiostonnthmr|hvso−at,irahttd1tetneh→ehptpyheeurE−aeerbpinires2mnlleyeiaisdsnpsnwthioteaemneeeaictdondsss-t
√A 01 00..12 Kol becauseofpotentialdegeneracywithstellarmassbinaries;al-
34 q 000...456 ctearunsaetivfoelllyo,wp-luapneotsbsweritvhatliaorngset|esnd−to1/osn|lymacyovbeer tmheissheidghbeer--
2 0.3
1 0.2 magnification portions of lightcurves. Finally, an inverse
0 0.1 power-law with mass ratio q(cid:46)−0.5 could explain the differ-
-5 -4 -3 -2 -1 0 1 2 3 4 5
Abundancepowerlawslope encesbetweenthedataandthemodel,butthisseemsimplau-
siblebecausewealreadyaccountforthemeasuredplanetary
Figure6. As Figure 5, but for microlensing parameters that could affect
planetdetectability. massfunctionofCassanetal.(2012),andthiswouldrequire
that the mass function be roughly twice as steep. Further-
6.1. Problemswiththemodel? more, a slope twice as steep as that of Cassan et al. (2012)
is more than 3-sigma inconsistent with the constraints on
Itiscertainthatourmodelisinaccurateatsomelevel. The
the slope of the planetary mass function reported by Clan-
Han&Gould(2003)GalacticmodelusedbyH14isrelatively
ton & Gaudi (2016), derived from a joint analysis of results
old by now in comparison to many new results on the struc-
from microlensing, radial velocity, and direct imaging sur-
tureandkinematicsofthebulgeanddiskovertheintervening
veys. Finally,wealsotriedincreasingthe∆χ2 thresholdfor
time(e.g.,Clarksonetal.2008;Natafetal.2010;McWilliam
themodelandfoundnosignificantchangeintheteststatistics
& Zoccali 2010; Wegg & Gerhard 2013). Additionally, the
untilsmallnumberstatisticsinthemodelbecameanissue.
diskofHan&Gould(2003)doesnothaveacentralhole,so
one might consider disk stars near the Galactic center to be-
6.2. Problemswiththedistanceestimates?
longtothebulge. Buttheeffectofalackofadiskholemight
be cancelled by not also accounting for stars that were born Itisalsopossiblethatthedistanceestimatesweusearethe
inthediskandlatersweptupbythebar. Despitetheseprob- cause of the disagreement between the data and the model.
lems,itisdifficulttoimaginehowthevariousinaccuraciesin Asmentionedinsection2,theauthorsthatcomputeBayesian
the model could combine to form an at least factor-of-three estimatesofthedistanceandmasswouldcautionusnottouse
error(forour95percentconfidencelimitsaccountingforthe them for statistical studies. However, we should also cast a
Johnsonetal.2010,massandmetallicitycorrelations)inthe criticaleyeupontheparallaxmeasurementsthataregenerally
relativemicrolensingeventratesbetweenthediskandbulge thoughttobeaccurate.
thatwouldbenecessarytoexplainthedisagreementbetween The constraints that shape the posterior distribution of the
modelanddata. Bayesian distance estimate come primarily from the mea-
ItispossiblethattheH14simulationsdonotaccuratelyre- surements of the angular Einstein radius and proper motion
flecttheprocessbywhichmostoftheplanetswerefound,and thatareroutineformostplanetarymicrolensingevents.10 θ
E
thereforealterthedistributionofdistances. Thesimplestway andµ arederivedfrommeasurementsofthesourceradius
rel
such an error may manifest itself is by an additional depen- crossing time t , t and the source color. t can essentially
∗ E ∗
denceofplanetdetectionefficiencyontheparametersofthe be“readoff”fromthedurationoffinitesourceeffectsinthe
microlensingevents. Wecanagainusetoypower-lawmodels lightcurve,andthoughthereissomesensitivitytothechoice
totesthowanunaccountedforterminthedetectionefficiency of limb darkening parameters, it should be determined very
behaves. Theeventtimescalet andimpactparameteru are robustly given sufficient coverage over the finite source fea-
E 0
the most likely culprits, but we also test s and q, and a de- tures. t and the source color can be smoothly degenerate
E
rived parameter s 1/s, which describes the approximate withblending,whichmeansthatuncertaintiesshouldaccount
positionofthepl|ane−taryc|austicsandhasaminimumats=1 for this degeneracy; however, discrete degeneracies between
wheremicrolensingismostsensitive;theresultsareshownin different models in the q–s plane can cause catastrophic er-
Figure6. rorsinµ andθ . Bayesiandistanceestimatescanincorpo-
rel E
Anadditionalweightingoft∼1 improvesthematchforthe ratethemultimodalposteriorsthatsuchadiscretedegeneracy
E
D distribution(p = 0.023), butatthecostofincreasedten- wouldproduce,butouranalysiscannot. Someeventsinthe
l
sion with the µ distribution (p = 0.013), and would not observed sample are affected by such a discrete degeneracy
rel
completely solve the problem. Such a weighting is a poten- (e.g.,MOA-2007-BLG-192Bennettetal.2008),buttheyare
tially plausible explanation for the lack of planets at large intheminority.
D because bulge-bulge lensing events tend to have shorter
l
timescales, but longer timescales allow for more time to or- 10 AnalysesthatusetE constraintsratherthanµrel areusingthesame
ganize follow-up observations of high-magnification events, informationbecauseµrel=θE/tE.
8 PENNY,HENDERSON&CLANTON
According to our model, the distribution of distances and the same direction as the relative proper motion vector.
as a function of θ is monotonic, and roughly linear in Oftenonlytheso-calledparallelcomponentofthevector(in
E
log(θ /mas) with a slope of 10 kpc dex−1 and an r.m.s. thedirectionparalleltothedirectionoftheEarth’saccelera-
E
∼
scatter of 1.3 kpc in each 0.1 dex bin (see also Figure 7 tion,whichcausesanasymmetricdistortiontothelightcurve)
∼
below); the dominant contributor to this scatter is the host is well measured, with the perpendicular component (which
mass distribution. Other Galactic models used as priors in causesasymmetricdistortion)beingdegeneratewiththeim-
Bayesian distance estimates will likely have similar proper- pact parameter and blending. In addition, orbital parallax is
ties. Often θ measurements are limited by a systematic er- alsodegeneratewithlensorbitalmotion(Batistaetal.2011;
E
rorintheconversionofthesourcecolortoanangularsource Skowronetal.2011)andsourceorbitalmotion(xallarap,e.g.,
size(e.g.,Yooetal.2004)withamagnitudeof 0.05–0.1in Poindexter et al. 2005), which can cause similar apparent
∼
ln(θ /mas). This implies that with only a θ measurement, changes to the source trajectory. The magnitude of the par-
E E
it should be possible to localize the distance of most lenses allaxdistortionsandtheirduration(coveringthewholemag-
to within 2 kpc or less. Therefore, we conclude that while nifiedpotionofthelightcurve)canalsomakethemvulnerable
∼
theBayesianestimatesofmassarelikelytobeunreliable,and tocorruptionbylong-termsystematictrendsinphotometry.
are likely to conform to our prior on the mass, the Bayesian The events with D < 2 kpc are OGLE-2006-BLG-109,
l
distanceestimatesarelikelytobemuchmoreuseful. MOA-2007-BLG-192, MOA-2010-BLG-328, OGLE-2012-
The precision of the Bayesian distance estimates is cer- BLG-0563, OGLE-2013-BLG-0341 and OGLE-2013-BLG-
tainlyenoughtoconductthepresentstudy. Evenifonewere 0723; we will use abbreviated names (see the notes for Ta-
suspicious of the exact form of the distance distribution that ble1)foreventsinoursamplefromhereon. Wehavehigh-
Bayesian estimates use as priors, with such uncertainties a lightedthemonθ -µ ,θ -D andµ -D diagramsinFig-
E rel E l rel l
more crude test asking if the number of hosts with distances ure7,whichvisuallycomparesourhostsampletoourmodel.
less than 5 kpc (beyond which bulge hosts should begin to Model points were drawn with replacement from the model,
contribute) is consistent with the model assuming binomial and displaced randomly from their actual values (in order
statistics, should be robust. Performing this test, the model to better visualize point densities) by log(θ /mas) 0.1,
E
predicts 9.0 2.5 hosts closer than 5 kpc, and the model log(µ /masyr−1) 0.1 and D 0.25 kpc, where±the
weighted by±the Johnson et al. (2010) mass and metallicity signinredlicatestheran±geofaunifolrm±distribution. ±
abundancetrendspredicts9.4 2.6hosts. Thereare17hosts Recalling that θ and µ should be relatively robustly
± E rel
closerthan5kpcinoursample,implyingthatourfiducialand measured,intheµ -θ panelofFigure7thenearbyhostsdo
rel E
mass-metallicitytrendweightedmodelsareinconsistentwith not look to be especially unusual; they cluster toward larger
thedataat3.2-σand3.0-σrespectively. θ , but are generally within the cloud of model points, with
E
The final ingredient to examine is the parallax distance O06-109(labeleda)beingthemostextreme. However,inthe
measurements. AscanbeseeninFigure1,allofthe12hosts D-θ andD-µ plots,atleastthreeofthenearbyhosts(b,
l E l rel
ineventswithparallaxmeasurementsarewithin5kpc,which c and f, or M07-192, M10-328 and O13-0723, respectively)
isnotsurprisinggiventhatthereisastrongbiastowardnearby standoutfromthecloudofmodelpoints;thesethreehostsare
lenses as discussed in section 2. Of the 12 hosts, 4 are be- theoneswithin1kpc. O06-109liesattheedgeofthemodel
tween 4 and 5 kpc, 5 are closer than 2 kpc, and 3 are closer cloudintheD-µ plotandO13-0341lieattheedgeinthe
l rel
than 1 kpc. By itself, this may not be too surprising given D-θ plot. O12-0563, being a Bayesian distance estimate,
l E
theobservationalbiases. However,ifincontrasttotherestof unsurprisinglylieswithinthecloudofpoints.
the paper we assume that the fiducial model is accurate, the Themajorityofthescatterinθ inthemodeliscausedby
E
numberofhostsatextremelysmalldistances(<2kpc)isex- therangeoflensmasses. Thiscanbeseenbycomparingthe
tremelysurprising. Giventhenumberofhostsinoursample, θ scatterintheD-θ plotbetweenthefiducialredandblue
E l E
our model would predict 0.26 hosts in the 0–1 kpc bin and points and the black black points which are the same real-
1.24 in the 1–2 kpc bin, compared to the three hosts in each izationwithθ plottedasifeverystarhadthesamemassof
E
(includingtheBayesianestimatedhosts).Inotherwords,ifall 1M . Fromthisitcaneasilybeseenthatthesmalldistance
(cid:12)
thehostsinthefirsttwobinsareinthecorrectbins,themodel estimates imply extremely small host mass estimates. The
wouldpredictthatthesamplewouldcontain61.9planethosts, hostmassesinourmodelrangefrom0.08to1M . Theau-
(cid:12)
twicethenumberthatareactuallyinthesample!Movingfour thorsofpapersonM07-192,M10-328andO13-0723estimate
or five outof the six events with Dl < 2 kpc to the 6–8 kpc hostmassesof0.084,0.11and0.031M(cid:12),respectively(Kubas
range(wherethereisthelargestdeficitofhostsrelativetothe etal.2012;Furusawaetal.2013;Udalskietal.2015b).These
model),wouldremovealltensionwiththemodel. eventshaverelativelynormalangularEinsteinradiiandrela-
So, could the parallax distance estimates be wrong? As tive proper motions, so the low mass and distance estimates
described in section 2, these require both parallax and finite arebeingdrivenentirelybytheparallaxmeasurements.Given
source measurements, but as discussed earlier in this sec- that at least some of the events closer than 2 kpc occupy a
tion, the finite source measurements, which yield θE should region of parameter space that our model predicts should be
be fairly robust. Until recently, the only way to measure the empty,itisworthlookingateachoneindividuallyinmorede-
microlensing parallax π was via orbital parallax.11 Orbital tailtoseeifwecanplausiblymovethemtoalargerdistance
E
parallax is an often subtle effect that arises from the depar- andbringtheminlinewithourmodel’sexpectations.
ture of the apparent trajectory of the source from linear mo- O06-109(a)wasanextremelycomplexeventwithfivedis-
tioncausedbytheEarth’saccelerationinitsorbit. π isac- tinct features from two planets (Gaudi et al. 2008). Parallax
E
tually a vector quantity (cid:126)π , with magnitude π = π /θ andfinitesourceeffectscombinedtogivemassanddistance
E E rel E
estimates of 0.51M and 1.51 kpc. The event also had ob-
(cid:12)
11Orviaterrestrialparallax(Holz&Wald1996;Gouldetal.2009),but servable terrestrial parallax (with a ∆χ2 12 Bennett et al.
∼
thisisextremelyrare(Gould&Yee2013)andwasnotmeasuredexclusively 2010), and H = 17.2 magnitudes of blended light coinci-
inanyofoursample.
ISTHEGALACTICBULGEDEVOIDOFPLANETS? 9
Figure7. Toprowandbottomrightpanel:ProjectionsoftheθE-µrel-Dlparameterspaceforarealizationofourmodel(smallredpointsforbulgehostsand
bluefordiskhosts)andourdatasample(largeblackpointsforparallaxandfinitesourcedistancemeasurementsandgraypointsforBayesianestimates).Events
inthesamplewithdistanceestimatesDl <2kpcareindividuallylabeled. IntheθE-Dlpanel,smallblackpointsshowwhatθEwouldbeifallhostshadthe
samemass(1M(cid:12);hostsinthemodelhavemassesbetween0.08and1M(cid:12)).Bottomleft:Cumulativedistributionofrelativepropermotionsfordetectedplanet
hostsinourmodelforallhosts(gray),hostsbetween1and2kpc(red)andhostscloserthan1kpc(black).
dent with the source in AO imaging matches essentially ex- timatethattheaprioriprobabilityoftheparallaxsignalbeing
actlythatwhichisexpectedfromtheparallaxsolution. Given caused by xallarap is 0.32 percent, accounting for the distri-
that terrestrial parallax can not be mimicked by xallarap or butionofbinaryperiods,butnotaccountingfortheincreased
orbitalmotionandthebrightandunmistakableblendedlight, stellar density in the bulge relative to the nearby disk. They
weconcludethatthereisverylittlechancethattheO06-109 guess that the increased density in the bulge would increase
distanceestimateissignificantlyinerror. this probability by a factor of about 7, but do not rigorously
M07-192(b)wasahigh-magnificationeventwithpoorcov- computethisnumber. Giventhedifficultyincomputingsuch
erageovertheeventpeak,andonlyfourdatapointscoverthe a probability, it might be reasonable to assume that it might
planetary deviation. As such, there were many models (16) actuallybesignificantlylarger.
that could fit the data well, but these all had similar plane- M10-328(c)wasalow-magnificationeventwitha 5day
∼
tarymassratios,withtheonlysignificantdifferencebeingtwo long planetary deviation that was poorly covered due to bad
classes of models with θ differing by a factor of 2 (Ben- weather, except for the caustic exit over a cusp (Furusawa
E
∼
nettetal.2008);evenifthesolutionwiththesmallerθ were etal.2013). Theauthorsfindaxallarapmodelthatfitsbetter
E
chosen,thedistancetothehostwouldbelessthan2kpc,but thantheparallaxmodelby∆χ2 5, butarguethatthepar-
therelativepropermotionwouldbereducedto2–3masyr−1. allax interpretation is most likely∼because an unconstrained
Bennett et al. (2008) found xallarap models that fit the data, xallarapfitwithaperiodfixedto1yryieldsanorbitalorien-
butarguedthatthesehadalowprobabilityrelativetothepar- tationthatiscoincidentwiththeEarth’srelativetotheevent
allaxmodel. FromAOimaging,Kubasetal.(2012)findthat line of sight. We agree that this would seem to rule out xal-
thereisaK = 19.2starblendedwiththesource,butdonot larapasanimpostorforparallax. Additionally, thelargean-
s
measure its color very precisely. Kubas et al. (2012) do not gularEinsteinradiusoftheeventwouldimplythatifthehost
question the parallax measurement of Bennett et al. (2008) were in the bulge, it would be 1M , and therefore bright
(cid:12)
∼
sointerprettheblendedlightasbeingduetoa0.08-M star unless it were a stellar remnant. It therefore seems very un-
(cid:12)
nearby,buttheK magnitudetheymeasureiscompatiblewith likelythatwecouldplausiblyplacethishostinthebulge.The
s
a0.7-M starat 7kpc,andwouldonlycausesometension onlyremaininghopeofdoingsoisthatthepoorcoverageof
(cid:12)
∼
withtheir0.37magJ K errorbar.Bennettetal.(2008)es- theplanetaryanomalyandthecombinationofacuspcrossing
s
−
10 PENNY,HENDERSON&CLANTON
withorbitalmotiononthesinglecausticexitresultinalarger low-mass would go unnoticed, and there would be nothing
than reported uncertainty on the angular Einstein radius or a interestingtowriteapaperaboutthem. However,thisbiasis
misestimationofit. lessprominentforplanetarymicrolensingeventsasallplanets
O12-0563 (d) was a high-magnification event that was are publishable, though there could be a delay in publishing
densely covered over the peak (Fukui et al. 2015). Surpris- “lessinteresting”oneswithoutparallaxmeasurements,which
ingly, despite the source not crossing the central caustic, the wouldbiasplanetinbinarydetectionstosmallerdistances. It
authors claim they are able to measure θ to be 1.36 mas isperhapsmorelikelythatplanetsinbinarieswouldsuffera
E
to a precision of 10 percent. While formally the authors biasthatworksintheoppositedirection,duetothepossibility
∼
measure parallax, they argue that the detection is likely due thatparallaxeffectsinadditiontothetriplelensnature,would
to systematics, and so we do not count it among the events maketheeventsmoredifficulttomodelandhencedelaypub-
withparallaxdetections. AOimagingshowsblendedlightof lication.
K = 17.7. The authors do not consider a xallarap model. One might think that there is another possible bias that
s
Thelargeθ =1.36masoftheeventwouldbyitselfexclude could result in us only detecting planets in binary systems
E
the possibility of a bulge lens, unless it were a neutron star whentheyarenearby. Thatis, ifthebinarystarpreventsthe
orblackhole. Evenifθ wereoverestimated, itwouldneed formation or survival of planets on wider orbits, then it will
E
tobeseverelyoverestimatedtoallowustomovethelensinto only be possible to detect the surviving, closer-in planets in
thebulge. events with small physical Einstein ring radii r . However,
E
O13-0341(e)isatriplelenseventwithaverysmallplane- r (cid:112)D(D D), and so there are lenses with small
E l s l
tarysignal(thedip)thatbyitselfmightbeoverlooked(Gould phys∝ical Einstein−ring radii near their sources in the bulge
et al. 2014). However, the well-covered crossing of a large, as well as near us the observer; there should be many more
binary-starcentral-causticcrossingisalsobetterfitifaplanet planetsdiscoveredinbinarysystemsinthebulgethaninthe
withthesamepropertiesaswouldcausethedipisincludedin nearby disk. We are therefore forced to ask the question: is
themodel.Alowamplitudebumpduringanearlierobserving there something going wrong with the modeling of parallax
seasonidentifiesthebinaryasawidebinary,andsotheplanet in triple lens microlensing events (and maybe in binary star
orbits one of the stars. The parallax signal in the lightcurve events too)? Given the complexity of triple lens models it
is strong. The authors do not mention if they considered a seems plausible that something could be going wrong, but it
xallarapmodel,butthelargepropermotionandlargeangular isdifficulttoguessatwhatthiscouldbe,especiallyinthecase
Einsteinradiusplacetheeventinaregionofparameterspace ofO13-0341wheretheparallax,angularEinsteinradiusand
thatisalmostentirelyoccupiedbydiskstars. relativepropermotionseemtotellaconsistentstory.
O13-0723 (f) is a second triple lens planetary event dis- It is possible to test the predictions of the parallax mod-
coveredin2013. Italsoseemstohaveastrongparallaxsig- els to some degree. They predict the direction of the rela-
nal (Udalski et al. 2015b). Again, the planet orbits only one tive source-lens proper motion, and after 10 years the lens
ofthebinarystars,howeverinthiscasetheargumentforthis andsourcemayseparateenoughforthisd∼irectiontobemea-
scenario over a circumbinary planet is based upon an argu- sured. An expedited method to test the validity of the par-
ment that the parameters of the circumbinary model require allax models would be to model only the ground-based data
finetuning. Theauthorsdonotmentioniftheyconsidereda for upcoming binary microlensing events that are observed
xallarapmodel,butunlikeO13-0341,thelocationoftheevent bySpitzer(e.g.,Zhuetal.2015b;Udalskietal.2015c)orK2
intheµrel-θE planeisoccupiedbybothbulgeanddiskstars Campaign9(Howelletal.2014;Hendersonetal.2015),and
andsuchamodelshouldbeconsidered.12 then test the orbital parallax prediction with the much more
While both O13-0341 and O13-0723 seem to have strong robust Spitzer or K2 satellite parallax measurements. Udal-
parallax signals, it is unsettling to say the least that the ski et al. (2015c) performed this test for a planetary event
only two planets discovered in binary star microlensing andfoundthattheorbitalandSpitzer-measuredsatellitepar-
events with caustic crossings also have extremely small host allaxes agreed, but the binary event modeled by Zhu et al.
distance (and mass) estimates.13 There have been other (2015b) did not have a detectable orbital parallax signature,
binary microlensing events with small mass and distance andalasSpitzerdidnotobservethepasteventsforwhichwe
estimates, e.g., OGLE-2009-BLG-151/MOA-2009-BLG-232 areworriedabouttheirparallaxsolutions.
(0.39 kpc, 0.025M total mass), OGLE-2011-BLG-0420 While we can offer no convincing evidence to do so, po-
(cid:12)
(1.99 kpc, 0.034M Choi et al. 2013), OGLE-2012-BLG- tentially two of the six hosts with D < 2 kpc (M07-192
(cid:12) l
0358 (1.73 kpc, 0.024M Han et al. 2013a), OGLE-2013- and O13-0723)couldplausibly bemoved tolarger distances
(cid:12)
BLG-0578 (1.16 kpc, 0.16M Park et al. 2015) and MOA- if xallarap is mimicking parallax in either event or if there
(cid:12)
2011-BLG-149 (1.07 kpc, 0.16M Shin et al. 2012). For wereerrorsintheparallaxmodelingofO13-0723. Itisworth
(cid:12)
thesebinariesitiseasytodismisstheunusuallysmallmasses noting that Poindexter et al. (2005) found that 5 out of 22
and distances as a publication bias: were they further away, single lens events with parallax-like signals that they mod-
parallax would not be measurable, which would mean their eled were more likely to have been caused by xallarap. We
might therefore expect that 20 percent of our 12 parallax
∼
12 Notethatsincesubmission,analternative,two-bodylensmodelwith events (i.e., 2–3) would actually be caused by xallarap. In
∼
orbitalmotionhasbeenfoundforO13-0723,withlowerχ2 andaparallax fact,O07-368shouldalsobeaddedtoourparallax-likesam-
thatimpliesadistancesixtimesfurtheraway(Hanetal.2016).However,this ple, because it has a reasonable parallax solution, but Sumi
newmodelisnotwithoutproblems,asduringashortanomaly,itcontainsan et al. (2010) find that it is better fit and more likely caused
evenshortercausticcrossinginadatagapwhoseentrybeginsimmediately
byxallarap. IftheparallaxsolutionofO07-368hadbeenbe-
afterdatatakingends.Thiscanoccurifamodelsearchroutineistrappedin
alocalratherthanaglobalminimum. lievedblindlywithoutconsideringthepossibilityofxallarap,
13Thereisoneotherplanetaryeventinabinarysystem,O08-092(Poleski the event would also have been in our sub-sample with sus-
etal.2014), buttheeventwaslow-magnificationandnoneofthecaustics piciously small distance estimates less than 2 kpc. If we re-
associatedwiththebinarystarwascrossed.