Transit timing variations in WASP-10b induced by stellar activity?

2.4 MB·English

by  S. C. C. Barros

Mon.Not.R.Astron.Soc.000,1–17(2002) Printed17January2013 (MNLATEXstylefilev2.2)
Transit timing variations in WASP-10b induced by stellar activity?
S. C. C. Barros 1⋆, G. Boue´ 2,3,4, N. P. Gibson 5, D. L. Pollacco6, A. Santerne 1,
3 F. P. Keenan 7, I. Skillen8, R. A. Street9
1
0 1AixMarseilleUniversite´,CNRS,LAM(Laboratoired’AstrophysiquedeMarseille)UMR7326,13388,Marseille,France
2 2CentrodeAstrof´ısica,UniversidadedoPorto,RuadasEstrelas,4150-762Porto,Portugal
3AstronomieetSyste`mesDynamiques,IMCCE-CNRSUMR8028,ObservatoiredeParis,UPMC,77Av.Denfert-Rochereau,75014Paris,France
n 4DepartmentofAstronomyandAstrophysics,UniversityofChicago,5640SouthEllisAvenue,Chicago,IL60637,USA
a 5EuropeanSouthernObservatory,Karl-Schwarzschild-Str.2,85748GarchingbeiMunchen,Germany
J 6DepartmentofPhysics,UniversityofWarwick,CoventryCV47AL,UK

6 7AstrophysicsResearchCentre,SchoolofMathematicsandPhysics,Queen’sUniversityBelfast,UniversityRoad,Belfast,BT71NN,UK
1 8IsaacNewtonGroupofTelescopes,ApartadodeCorreos321,E-38700SantaCruzdelaPalma,Tenerife,Spain
9LasCumbresObservatory,6740CortonaDriveSuite102,Goleta,CA93117,USA

]
P
E
.
h Accepted2013January16,Received2013January15,inoriginalform2012November14
p
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o
ABSTRACT
r
t Thehot-JupiterWASP-10bwasreportedbyMaciejewskietal.(2011a,b)toshowtransittim-
s ing variations(TTV) with an amplitude of ∼ 3.5 minutes. These authors proposedthat the
a
[ observed TTVs were caused by a 0.1 MJup perturbing companion with an orbital period
of ∼ 5.23d, and hence, close to the outer 5:3 mean motion resonance with WASP-10b. To

1 test this scenario, we presenteightnew transitlight curvesof WASP-10b obtainedwith the
v
FaulkesTelescopeNorthandtheLiverpoolTelescope.Thenewlightcurves,togetherwith22
0
previouslypublished ones, were modelled with a Markov-ChainMonte-Carlotransit fitting
6
code.
7
Transit depth differencesreported for WASP-10b are thoughtto be due to star spot in-
3
. duced brightness modulation of the host star. Assuming the star is brighter at the activity
1 minimum, we favour a small planetary radius. We find R = 1.039+0.043R in agree-
0 p −0.049 Jup
ment with Johnsonetal. (2009) and Maciejewskietal. (2011b). Maciejewskietal. (2011a)
3
and Husnooetal. (2012) find no evidence for a significant eccentricity in this system. We
1
presentconsistentsystem parametersfora circularorbitand refine the orbitalephemerisof
:
v WASP-10b.
i OurhomogeneouslyderivedtransittimesdonotsupportthepreviousclaimedTTVsig-
X
nal, which was strongly dependenton 2 previously published transits that have been incor-
r
a rectlynormalised.Nevertheless,alinearephemerisisnot astatisticallygoodfittothetransit
timesofWASP-10b.Weshowthattheobservedtransittimevariationsareduetospotocculta-
tionfeaturesorsystematics.Wediscussandexemplifytheeffectsofoccultationspotfeatures
inthemeasuredtransittimesandshowthatdespitespotoccultationduringegressandingress
being difficult to distinguish in the transit light curves, they have a significant effect in the
measured transit times. We conclude that if we account for spot features, the transit times
ofWASP-10areconsistentwitha linearephemeriswiththeexceptionofonetransit(epoch
143)which isa partialtransit. Therefore,thereis currentlyno evidenceforthe existenceof
acompaniontoWASP-10b.OurresultssupportthelackofTTVsofhot-Jupitersreportedfor
theKeplersample.
Keywords: stars:planetarysystems–stars:individual(WASP-10)–stars:spots–techniques:
photometric
1 INTRODUCTION
ThediscoveryofWASP-10b,a 3MJuphot-Jupiterplanetina3.09
day orbit, was reported by Christianetal. (2009). Its host star is
⋆ E-mail:[email protected] a K5V type with Teff = 4675 ±100K, [M/H] = 0.03 ±0.2
2 S.C.C. Barroset al.
(Christianetal. 2009) and is relatively young, with an age of Thenew observations are described inSection 2. InSection
270±80 Myr(Maciejewskietal.2011a).Usingahighsignal-to- 3,wediscussourtransitmodel.Wepresenttheupdatedparameters
noise transit light curve, Johnsonetal. (2009) updated the stellar ofthesysteminSection4andinSection5discussourresultsand
densityandre-derivedthestellarmasstoM∗ = 0.75±0.04M⊙ show possible causes of spurious TTVs. Finally, we present our
and radius to R∗ = 0.698 ± 0.012R⊙. There has been some conclusionsinSection6.
discussion about the planetary size, with Christianetal. (2009),
Dittmannetal.(2010)andKrejcˇova´etal.(2010)obtainingaplane-
taryradiusof1.28±0.09RJup whileJohnsonetal.(2009)derived 2 OBSERVATIONS
asmallerradiusof1.08±0.02RJup.Maciejewskietal.(2011b)
arguethatthediscrepancyinpreviousresultswasduestarspotin- We present further high precision transit observations of WASP-
ducedbrightnessvariability.Infact,WASP-10hasbeenshownto 10btakenwithFaulkesTelescopeNorth(FTN)andwiththeLiver-
havestellarvariabilityduetotherotationmodulationofspotswith poolTelescope(LT).
aperiodof11.91±0.05 d(Smithetal.2009)andsemi-amplitude TwofulltransitsofWASP-10bwereobservedonthe2008-09-
up to 10.1mmag. Starspots reduce the effective stellardisc area 15and2008-09-19withtheMeropecameraontheFTNusingthe
leadingtoanunderestimationofthestellarradiusandalargertran- SDSS-ifilter.TheMeropecameraconsistsofae2V,2048×2048
sitdepth.Thisvariationoftransitdepthwithspotcoverageexplains pixel-CCDandhasafieldofviewof4.7’×4.7’.Aperturephotom-
thedifferencesinthemeasuredplanetaryradius(e.g.Czeslaetal. etry was performed with the DAOPHOT package (Stetson 1987)
2009).Usingatransittakenclosetomaximumbrightnessthatcor- within the IRAF4 environment. The differential photometry was
respondstoaminimumspotcoverage,Maciejewskietal.(2011b) performedrelativetoatleast5comparison starsthatwerewithin
obtainedamaximumplanetaryradiusof1.03+−00..0073 RJup. theFTNfield-of-view.
Interestingly, Maciejewskietal. (2011a) reported a periodic Another six transits were observed with the fast CCD cam-
modulationoftransittimingvariations(TTVs)forWASP-10with era RISE (Steeleetal. 2008; Gibsonetal. 2008) mounted on the
anamplitudeof∼3.5±1.0minutes.Theysuggesteda0.1MJup 2.0m Liverpool Telescope on La Palma, Canary Islands. RISE
planet companion in theouter5:3 meanmotion resonant orbit as hasawidebandfiltercovering ∼ 500–700nmwhichcorresponds
thecauseofthemeasuredTTVs.Later,Maciejewskietal.(2011b) approximately to V+R. The pixel scale is 0.54 arcsec/pixel re-
confirmed the TTV solution and showed that only spots located sulting in a 9.4’ × 9.4’ field-of-view. Thanks to its frame trans-
close to the stellar limb would affect the transit times but they fer CCD, RISE has a deadtime of only 35 ms for exposures
wouldalsoresultintransitdurationvariations.Sincenotransitdu- longer than 1 second. To decrease systematic noise due to poor
rationvariationswerefoundintheirsample,theauthorsarguedthat guiding (Barrosetal. 2011) the telescope was defocussed to -
spot occultationfeaturesdidnot significantlyaffect themeasured 0.6mm and we used an exposure time of 9 seconds. However,
transittimesofWASP-10. in one of the observations (2010-11-10) we experimented with a
higherdefocussingof-1.0mmandlongerexposuretimeof34sec-
The possibility of finding additional planets by measuring
theireffectonthecentral transittimesofknowntransitingplanet onds.TheRISEdatawerereducedusingtheULTRACAMpipeline
(Dhillonetal.2007)followingthesameprocedure asforWASP-
systemswasproposedbyHolman&Murray(2005)andAgoletal.
21b (Barrosetal. 2011). Details of the observations are given in
(2005). They showed that for systems near mean-motion reso-
Table1.
nances this method can be sensitive to planets less massive than
The final new high precision transit light curves for WASP-
theEarth.ThishasledmanygroupstosearchforTTVsfromthe
10bareshowninFigure1alongwiththebest-fitmodeldescribed
ground.However,mostofthesearchesresultedinonlyupperlim-
inSection3.3.Weoverplotthemodelresidualsandtheestimated
its on the mass of possible companions (e. g. Miller-Riccietal.
uncertainties,whicharediscussedinSection3.2.
2008; Bean 2009; Gibsonetal. 2009, 2010). The few initial re-
Previously published WASP-10b light curves were included
ports of TTVs were unconfirmed later. For example, D´ıazetal.
inouranalysiswhichwerekindlymadeavailablebytheauthors.
(2008) found TTVs in OGLE-111b which were unconfirmed in
Detailsof these observations are given in Table 2. Forcomplete-
later studies (Adamsetal. 2010; Hoyeretal. 2011). At present,
nesswealsoshowthenewbest-fitmodeloverplotedontheprevi-
onlythreehot-JupitertransitingsystemsshowTTVsthatweremea-
ouslypublishedtransitlightcurvesinFigure2,whosereferences
suredfromground-baseddata:WASP-3(Maciejewskietal.2010),
forthesedataaregiveninTable2.
WASP-10 (Maciejewskietal. 2011a), and WASP-5 (Fukuietal.
Hereafter, we refer to each light curve by their epoch num-
2011).Meanwhile,Keplerdatahasallowedthediscoveryofmany
berrelativetotheephemeris presented inSection4.2.Theepoch
TTV systems (e.g. Kepler-9 (Holmanetal. 2010) and Kepler-11
numberisalsogivenintheobservationlogsandlightcurveplots.
(Lissaueretal. 2011) ). However, a recent TTV study of the Ke-
plersamplerevealedthelackofTTVsinhot-Jupiters,contrasting
with the TTVs found for longer-period Jupiters or hot-Neptunes
(Steffenetal.2012).Theauthorssuggestedthatthisimpliedadif- 3 DATAANALYSIS
ferent formation and/or evolution history for hot-Jupiters. There-
3.1 Re-analyseoftheradialvelocityobservations
fore,itisveryimportanttoconfirmandunderstandthehot-Jupiter
TTVdetectionsthathavebeenreportedfromground-baseddata. ThediscoverypaperforWASP-10b(Christianetal.2009)favoured
ToconfirmandpossiblycharacteriseWASP-10b’scompanion an eccentric orbit for the planet e = 0.059+0.014 and ω =
−0.004
weobtainednewhighprecisiontransitlightcurvesofWASP-10b. 2.917+0.222).However,Maciejewskietal.(2011a)arguedthatec-
−0.245
SixtransitobservationsofWASP-10bwereobtainedwiththeRISE centricitymighthavebeenoverestimatedduetostellarvariability
fast CCD camera mounted on the Liverpool Telescope and two and favoured a circular orbit. Later, Husnooetal. (2012) showed
morewiththeMeropecameraontheFaulkesTelescopeNorth.We that the eccentricity detection was dependent on the first two ra-
combinedthenewobservationswithpreviouslypublished22light dial velocity (RV) SOPHIE observations that could have a dif-
curvesofWASP-10tohomogeneouslyderivethetransittimes. ferent zero point. Using a method of Bayesian model selection
TTVs inWASP-10binducedbystellaractivity? 3
1.00
FTN 20 FTN 21
0.99 (1.3) (1.7)
0.98
0.97
0.96
0.95
0.94
1.00
RISE 26 RISE 252
0.99 (2.0) (2.0)
0.98
0.97
0.96
x
d flu 0.95
e
s 0.94
ali
m 1.00
or RISE 254b RISE 265
N
0.99 (3.0) (3.0)
0.98
0.97
0.96
0.95
0.94
1.00
RISE 274 RISE 285
0.99 (2.2) (1.5)
0.98
0.97
0.96
0.95
0.94
−2 −1 0 1 −22 −1 0 1 2
Time (hours) Time (hours)
Figure1.Phase-foldednewlightcurvesforWASP-10.Fromtoptobottomandlefttorightinchronologicalorder;FTN2008September15and19,RISE2008
October04,RISE2010September03and09,RISE2010October13,RISE2010November10,RISE2010December14.Thelightcurvenameincluding
theepochnumberisprintedineachplot.Foreachlightcurve,wesuperimposethebest-fittransitmodel2(solidredline),showtheresidualsfromthismodel
atthebottomofeachtheplot,andgivetheestimatedrednoiseparametrisedbyβinsideparenthesis.Forsomelightcurveswealsoshowatransitmodelthat
includesaspotfeature(dashedgreenline).ThelightcurvesandmodelshavebeenshiftedtotheT0determinedwithmodel2.TheRISEdataarebinnedinto
45secondperiodsexceptforthe2010November10whichhadalongerexposuretime.Theindividuallightcurvesplottedhereareavailableinelectronicform
atCDS.
4 S.C.C. Barroset al.
1.00
Mercator −103 Tenegra −89 Tenagra −88
0.99
0.98
0.97
0.96
0.95
0.94
1.00
Hawaii 0 Slovak 36 Torun 56
0.99 (1.4)
0.98
0.97
0.96
x
u 0.95
d fl
e
alis 0.94
m 1.00
Nor Jena 123a Rozhen 123b Rozhen 124
0.99 (1.1) (1.1)
0.98
0.97
0.96
0.95
0.94
1.00
Rozhen 132 Slovak 133a Jena 133b
0.99 (1.7) (1.8)
0.98
0.97
0.96
0.95
0.94
−2 −1 0 1 −22 −1 0 1 −22 −1 0 1 2
Time (hours) Time (hours) Time (hours)
Figure2.Phase-foldedlightcurvesforWASP-10frompreviouslypublishedlightcurveswhicharedescribedinTable2.Othercommentsarethesameas
Figure1butrednoisevaluesequalto1wereomitted.
TTVs inWASP-10binducedbystellaractivity? 5
1.00
Rozhen 134 Rozhen 143 Slovak 144
0.99 (1.3) (1.8) (1.6)
0.98
0.97
0.96
0.95
0.94
1.00
Slovak 145 Kuiper 147 Tenegra 158
0.99 (2.3) (4.3)
0.98
0.97
0.96
x
u 0.95
d fl
e
alis 0.94
m
Nor C A 242 C A 243 C A 253
(1.8) (2.3) (2.0)
−2 −1 0 1 −22 −1 0 1 2
1.00
C A 254a Time (hours) Time (hours)
0.99 (1.2)
0.98
0.97
0.96
0.95
0.94
−2 −1 0 1 2
Time (hours)
Figure2–continued
6 S.C.C. Barroset al.
Table1.WASP-10bnewobservationslog.
Epoch Date telescope exptime numberexposures apertureradius
sec arcsec
20 2008-09-15 FTN 125 115 6.1
21 2008-09-19 FTN 125 109 7.0
26 2008-10-04 RISE 5 1580 8.6
252 2010-09-03 RISE 9 1195 8.1
254b 2010-09-09 RISE 9 1233 8.6
265 2010-10-13 RISE 9 1063 10.3
274 2010-11-10 RISE 34 339 11.4
285 2010-12-14 RISE 9 755 9.2
Table2.WASP-10bpreviousobservations log.Thereferences areA=Christianetal.(2009),B=Johnsonetal.(2009),C=Dittmannetal.(2010),D=
Krejcˇova´etal.(2010),E=Maciejewskietal.(2011a),andF=Maciejewskietal.(2011b).
Epoch Date telescope exptime numexp filter ref
sec arcsec
-103 2007-09-01 Mercator 30 170 V A
-89 2007-10-15 Tenagra 125 125 I A
-88 2007-10-18 Tenagra 125 91 I A
0 2008-07-16 2.2mHawaii 50 194 sloanz’ B
36 2008-11-04 Slovak 120 R D
56 2009-01-05 0.6mTorun 50 254 R E
123a 2009-07-31 0.6mJena 70 144 R E
123b 2009-07-31 0.6mRozhen 120 116 R E
124 2009-08-03 0.6mRozhen 90 122 R E
132 2009-08-28 0.6mRozhen 90 99 R E
133a 2009-08-31 Slovak 165 R D
133b 2009-08-31 0.6mJena 55 203 R E
134 2009-09-03 0.6mRozhen 90 105 R E
143 2009-10-01 2.0mRozhen 15 548 V E
144 2009-10-04 Slovak 118 R D
145 2009-10-07 Slovak 114 R D
147 2009-10-14 Kuiper 10 579 I C
158 2009-11-17 Tenagra 76 76 R E
242 2010-08-03 2.2mCalarAlto 40 165 R F
243 2010-08-06 2.2mCalarAlto 40 213 R F
253 2010-09-06 2.2mCalarAlto 50 164 R F
254a 2010-09-09 2.2mCalarAlto 60 184 R F
theyfindtheeccentricitydetectionisnot significantandobtained a sinusoidal effect (semi-amplitude of 65m/s) in the measured
e=0.052±0.031.Furthermore,theeccentricityisverycorrelated RVs which should be accounted for when deriving the stellar
withtheoffsetbetweentheSOPHIEandFIESdata.Forallthese reflex velocity due to WASP-10b. Following the procedure of
reasonsweassumeacircularorbit.Abetterconstraint ontheec- Maciejewskietal. (2011a), we computed the Lomb-Scargle peri-
centricityrequiresfurtherRVobservations.Toestimatethestellar odogram of the RV residuals after removing the circular model
reflexvelocityforacircularorbitwere-analysedtheRVobserva- givenabove.ThisisshowninFigure3aswellasthestellarrotation
tionspresentedinChristianetal.(2009)usingthenewephemeris periodof11.91±0.05 ddeterminedbySmithetal.(2009)andthe
giveninSection4.2. falsealarmprobabilityat2σ(95.4%)confidencelimit.Wefindno
significantpeakatthestellarrotationperiod,andhencewecannot
Using an MCMC routine we fitted 5 parameters of a keple-
confirmthattheradialvelocitiesaremodulatedbyspotvariability.
rian model for the host star reflex motion: orbital period, central
Moreover,otherhoststarsthatshow1-2%photometricvariability
transittime(T0),RVsemi-amplitude(K),centre-of-massvelocity
havesmalleractivity induced jitterinthe measured RVsthan the
(γ),andanoffsetbetweenSOPHIEandFIESdata.Weimposeda
claimed65m/sforWASP-10.Forexample,HD189733bhasaRV
GaussianpriortotheorbitalperiodandT0usingthenewephemeris
jittersemi-amplitudeof15m/s(Boisseetal.2009)andCoRoT-7b
giveninSection4.2andobtainedK = 506.4±7.4ms−1,γ =
has a RV jittersemi-amplitude smallerthan 20m/s (Quelozetal.
−11427.9±5.3ms−1,andoffset= 112.0±1.2ms−1.Thisso-
2009).Therefore,weconcludethattheextrajitterobservedinthe
lutionhasmedianrmsof43m/swhichcompareswiththemedian
RVs of WASP-10 does not appear to be periodic. Including one
RV uncertainty of 20 m/s. Hence, theradial velocities haveextra
jitterparameterforeachdata setintheradial velocityfitwefind
jitterwhichcouldbeduetostellaractivity.
that the FIESdata jitteris less than 13ms−1 while the SOPHIE
Maciejewskietal.(2011a)arguedthatthestellaractivityhad
TTVs inWASP-10binducedbystellaractivity? 7
Table3.Estimatedrednoisevaluesparametrisedasβcalculatedaccording
7
Prot toWinnetal.(2008)
6 2 sigma
5
Epoch betaind betasimu
wer 4
Po 3 -103 1.0 1.0
-89 1.0 1.0
2
-88 1.0 1.0
1 0 1.0 1.0
0 20 1.0 1.3
10
Period (days) 21 1.5 1.7
26 1.5 2.0
36 1.4 1.4
Figure3.Lomb-ScargleperiodogramoftheRVresidualsaftersubtracting
56 1.0 1.0
acircularorbitforWASP-10b.Theverticallineshowstherotationperiod
123a 1.1 1.1
of11.91andthe horizontal line shows thefalse alarm probability at 2σ
123b 1.1 1.1
confidencelimit.
124 1.1 1.0
132 1.0 1.0
data jitter is 41 ± 17ms−1. The latter is much higher than the 133a 1.4 1.7
expected jitterfor SOPHIEsupporting the result of Husnooetal. 133b 1.7 1.8
(2012)aboutthepossibleoffsetofthetwofirstSOPHIEpointsrel- 134 1.3 1.3
143 1.5 1.8
ativeto thesubsequent observations. Thus, we experimented dis-
144 1.4 1.6
cardingthefirsttwoSOPHIEpointsfromthefitwhichresultedin
145 1.9 2.3
amuchlowerestimatedjitterfortheSOPHIEdata=13±18ms−1
147 2.4 4.3
compatiblewiththeexpectedvaluesforthisinstrument.Thiswas
158 1.0 1.0
adopted as our final solution for the RVs with the remaining pa- 242 1.5 1.8
rametersbeing:FIESjitter8±13ms−1,K = 540±11ms−1, 243 1.8 2.3
γ = −11461±13ms−1,andoffset= 84±18ms−1.Ourfinal 252 2.0 2.0
solutionforthestellarreflexvelocityisinagreementwiththevalue 253 1.7 2.0
presentedbyChristianetal.(2009).TheexpectedRVsignalofthe 254a 1.0 1.2
possiblecompanionof14m/sisbelowthemedianRVuncertainty, 254b 3.0 3.0
265 2.7 3.0
asalsopointedoutbyMaciejewskietal.(2011a).
274 1.8 2.2
285 1.3 1.5
3.2 Photometricerrors
Anaccurateestimateofthephotometricerrorsisimportant toob-
fitting analyses we fit all the light curves simultaneously. How-
tainreliablesystemparameters.Webeginbyscalingtheerrorsof
ever, here we also present the individual light curve fits, mostly
each light curve so that the χ2 of the best fitting model is 1.0.
red for comparison with previous results and call the latter model 1
Thisisespeciallyimportantinourcasebecausewearecombining
andtheformermodel2.WeusedtheMandel&Agol(2002)tran-
datafromdifferentinstrumentswhichwerereducedwithdifferent
sitmodelparametrisedbythenormalisedseparationoftheplanet,
pipelines.
Exoplanettransitobservationsareoftenaffectedbycorrelated a/R∗, ratio of planet radius to star radius, Rp/R∗, orbital incli-
red noise which must be taken intoaccount in erroranalyses. To nation, i, and the transit epoch, T0, of each light curve. A circu-
larorbitwasadoptedasdiscussedabove.Foreachlightcurve,we
estimatethetime-correlatednoisefactorβwefollowedtheproce-
dureofWinnetal.(2008).Thisconsistsofcomparingthestandard alsoincludetwoextraparameters(FoutandTgrad)toaccountfor
alinearnormalisation. Weused the quadratic limbdarkening co-
deviationoftheresidualsofthebestfitsolutionfordifferenttime
efficients(LDCs)deduced fromthe tablesof Howarth (2011) for
bins.Foreachlightcurve,werescaledtheerrorsbytheestimated
T = 4675K,logg= 4.4 and [M/H] = 0.03 (Christianetal.
β which isgiven in Table3. Thisprocedure does not distinguish eff
2009).Mostofthelightcurvesareofinsufficientqualitytoallow
betweensystematicnoiseandrealstellarvariability.Alldepartures
forthefittingoftheLDCs.Tobeconsistentinthetreatmentofall
fromthetransitshapeareconsideredasnoise.Forexample, when
thelightcurveswefixedthelimbdarkeningcoefficientstothethe-
wefitsimultaneously alllight curves(model 2), variations of the
oreticalvalues.Althoughthismayleadtoaslightunderestimation
transitshapewillalsobeconsideredasnoise.Therefore,thevalues
of the errors, it was preferred toavoid spurious correlations. The
ofβforthesimultaneousfitarehigherthanfortheindividualfits.
valuesoftheLDCsusedforeachfilteraregiveninTable4.
Iftheshapevariationsarerealthisprocedurewilloverestimatethe
The MCMC chains were constructed following the
errors. However, if they are indeed caused by systematic noise it
Metropolis-Hastings algorithm. At each step of the chain
willallowamoreaccurateerrorestimation.
each parameter is perturbed by a jump function, which is a
random numbersampled fromaGaussiandistributionwithmean
3.3 Transitmodelandfittingprocedure zero and standard deviation equal to the respective parameter
uncertainty. The new parameter set is accepted with probability:
To determine the planetary and orbital parameters, we fitted the P = min 1,exp −∆χ2 where ∆χ2 isthedifference inthe
light curves of WASP-10b with a transit model coupled with an (cid:16) (cid:16) 2 (cid:17)(cid:17)
Markov Chain Monte Carlo (MCMC) routine, a procedure simi- χ2 of subsequent parameters sets. Thejump functions are scaled
lar to Barrosetal. (2011) for WASP-21b. In our previous transit by a common factor in order to ensure that the percentage of
8 S.C.C. Barroset al.
theonetakeninthereddestfilter,andhencethatshouldbelessaf-
Table4.Limbdarkeningcoefficientsusedforeachfilter.
fectedbystellarvariability.Therefore,atotalof30×4+3=123
parameterswerefittedsimultaneously.WecomputedsevenMCMC
filter linearcoefa quadraticcoefb
chainseachof1500000pointsanddifferentinitialparameters.
sloanz’ 0.391 0.210
I 0.421 0.192
R 0.602 0.133
rise 0.612 0.146
V 0.763 0.036 4 RESULTS
4.1 Model1
accepted steps lies between 20% and 30%. Further details about The individual light curve fittedparameters are given in Table 5.
thefittingprocedurecanbefoundinBarrosetal.(2011).Forboth Transit times were converted to Barycentric Dynamical Time
modelstheinitial20%ofeachMCMCchainthatcorrespondedto (TDB)usingtheIDLcodeskindlymadeavailablebyEastmanetal.
theburn-inphasewerediscardedandtheremainingpartsmerged (2010). The weighted mean of the fitted parameters is: a/R∗ =
into a master chain. The best fit parameter was estimated as the 11.910±0.024, Rp/R∗ = 0.15869±0.00019, and i = 88.67
medianofitsprobabilitydistributionandthe1σlimitsasthevalue degreesandthedispersionofinclinationisequalto0.13 degrees
at which the integral of the distribution equals 0.341 from both asexpectedfromtheprior.
sides of the median. To test convergence, the Gelman&Rubin Comparing the values of a/R∗ for all light curves we con-
(1992) statistic was computed for each fitted parameter and was clude that individual deviations from the mean value of the nor-
found to be less than 0.9% from unity for all the parameters in malisedseparation of theplanet aresmallerthan 2σ except for2
bothmodels. transits. These are epoch 20 which was observed in 2008-09-15
withFTNandepoch147(Dittmannetal.2010).Bothhavea/R∗
higherthanthemeana/R∗bymorethan3σ.Bothofthesetransits
3.3.1 Model1 showsignsofcorrelatednoiseeitherduetosystematicsorstellar
variabilityandwillbediscussedinsection5.2.
The existence of an additional planet or moon affects the tran-
Asexpected,duetospotinducedbrightnessvariabilityofthe
sit timings of the transiting planet, but can also affect the or-
bital parameters of the system. Hence, in previous analyses of host star,theparameter Rp/R∗ showsawidespread. Twoof the
lightcurves(243,254a)showasignificantlyhigher(3σ)valueof
the transit times of WASP-10, the authors opted for fitting each
Rp/R∗,whilethreelightcurves(0,133,158) show asignificantly
light curve individually (Dittmannetal. 2010; Maciejewskietal.
lowervalue.Therefore,weconfirmtransitdepthvariationsreported
2011a,b).However, in somecases they wereforced tofixthein-
byotherauthorsandtheneedtoincludetheextradf parameterin
clinationofthesystemduetothepoorqualityof individual light
theglobal fit.Forcompleteness we alsotested amodel withfree
curves.Here,wefollowthesameprocedureprimarilyforcompari-
inclination and concluded that the inclination is constant within
sonwithpreviousmethods.Theresultswillalsotestpossiblevari-
theuncertainties.Hence,weconcludethattherearenosignificant
ationsofthelightcurveshape.Thus,wefittedeachlightcurvein-
shape variations apart from the transit depth, except for transits
dividuallyusingtheMCMCmethodexplainedabovewith500000
epoch20and147.
pointsperchain.Fivechainswerecombinedtoobtainthefinalre-
Besides testing variations of transit shape, model 1 is also
sult.Insteadoffixingtheinclinationwechoosetoimposeaprior
useful for direct comparison with previous transit times. This al-
oftheform:
lowstobetterisolatethecausesforthedifferencesintheestimated
(i−i0)2, (1) transittimes.Thefirstthreetransits(-103,-89,-88)werepreviously
σ2 combinedtogetherwiththeWASPdatainasingleephemeris,and
i
hence,theycannotbedirectlycompared.Regardingtheotherprevi-
whereweassume i0 = 88.66,σi = 0.12, takenfromthesimul- ouslypublishedtransittimes,ourresultsarewithin1σofprevious
taneousfit(model2).Thisallowstoaccountfortheuncertaintyof
resultsexceptforepochs36and147forwhichthedifferenceisre-
theinclinationinthefit.Werefertothisasmodel1.
spectively−2.0±0.94and0.67±0.47minutes.Thedisagreement
canbeexplainedbyfailuretoaccountforthebaselinefunctionin
thepreviousanalysisofthesetransits.Inourmodelalineartrendis
3.3.2 Model2
fittedsimultaneouslywiththetransitmodel.Whenwedonotfita
Formodel2weassumethatapossibleadditionalplanethasnosig- trendwerecoverthepreviouspublishedresults.T0correlateswith
nificanteffectonthetransitshapeduringthetimespanoftheobser- thebaselinefunctionbecausethelatteraffectsthesymmetryofthe
vations.Therefore,weperformedasimultaneousMCMCfittoall lightcurve.Therefore,includingthebaselinefunctioniscrucialto
30lightcurves,fittinggloballya/R∗,Rp/R∗,andiandindividu- obtainreliableT0s
allyT0,FoutandTgrad.Weremindthereaderthatdeviationsfrom Ingeneral,weobtainslightlylargererrors(∼1.6×)thanpre-
aconstantshapeareconsideredtobeonlyduetosystematicnoise viousstudies.Foreachcasethisisduetooneormoreofthefol-
orintrinsicstellarvariabilityandhencearetreatedasnoise.Specif- lowingdifferences fromtheprevious results:inclusion of abase-
ically,deviationsoflimbdarkeningparametersfromthetabulated linefunctioninthefit,inclusionoftherednoisefactor,and/orthe
valuesarealsoconsideredasnoise.However,sincethebrightness treatmentoftheinclination.Wearguethatfixingorartificiallycon-
ofWASP-10hasbeenshowntovarysignificantlyduetostarspots straining the inclination will cause underestimation of the transit
(Smithetal. 2009; Maciejewskietal. 2011a) a factor df was in- timeerrors.
cluded to account for out-of-transit flux variability between each Despitethedifferencefrompreviousresultsbeingsmall,the
lightcurve.Thisfactorhasbeensetarbitrarilytozerofor ourbest new transit times are in much better agreement with a linear
quality light curve (Epoch0, Johnsonetal.(2009)) whichis also ephemeris. The χ2 of a linear fit to the transit times is 91.95 for
TTVs inWASP-10binducedbystellaractivity? 9
7
2 sigma 1 0 36 56 123a123b124 132 133b133a134 143 144 145 147 158 242 243 253 254a
6 Prot minutes) 0
5 1 sigma Time dif (−1
4
wer −2
Po 5
3
4
12 Error ratio23
1
0
0.0 0.1 0.2 0.3 0.4 0 0 36 56 123a123b124 132 133b133a134 143 144 145 147 158 242 243 253 254a
Frequency (cyc/Pb)
Figure4.Lomb-Scargleperiodogramofresidualtransittimesfrommodel Figure5.Comparisonbetweenourresultsandpreviouslypublishedones.
1aftersubtractingalinearephemeris.Theverticaldash-dottedlineshows Weshowthetimedifferenceinthetoppanelandtheerrorratiointhebottom
therotationperiodof11.91.ThedashedlinesshowtheTTVfrequencies panel. Model 1 comparison is shown as black stars and model 2 as red
proposedbyMaciejewskietal.(2011a)andthehorizontal lineshowsthe circles.
falsealarmprobabilityat1and2σconfidencelimit.
fore,weconfirmthatourchosenzeropointisvalid.However,this
28degreesoffreedom,i.e.χ2 =3.28whichcompareswith13.8 zeropointmightnotcorrespondtotherealmaximumbrightnessof
red WASP-10,andhencetheplanettostarratiocouldstillbeoveresti-
foundbyMaciejewskietal.(2011a).However,thep-valueisstill
low9.777×10−9andhencethenullhypothesiscanbediscarded mated.
InFigure5weplotthedifferencebetweentheourtransittimes
at5.7σ.Fourofthenewtransittimesdeviatemorethan3σfrom
derivedwithmodel1(blackstars)andmodel2(reddots)andprevi-
a linear ephemeris. These are epochs 20, 21, 143, and 147. Both
ouslypublishedtransittimes.Wealsoshowtheratiooftransittime
epoch21andepoch147showtrendsinthelightcurvethatcanbe
errors.Thehighestdifferenceintimesisfoundfortransits36,132
affectingthetransittimes.
and147.Forepochs36and147thisisduetothebaselinefunction
TotestthemodelfortheTTVsproposedbyMaciejewskietal.
not been accounted forin previous fits, asmentioned before. For
(2011a) we fitted the sinusoidal model given by equation 1 of
Maciejewskietal.(2011a)forfrequencies:fttv =0.183cyclP−1 epoch132thedifferenceiswithintheerrorsanditisduetoepoch
and fttv = 0.175cyclP−1, where P is1the orbital period of 132beingapartialtransitandhenceverysensitivetothedetailsof
2 the model fit. The transit timesderived from model 1 and model
WASP-10b.ThepreferredsolutionpresentedbyMaciejewskietal.
(2011a)forfrequencyfttvhadap-valueequalto3.4×10−4which 2are consistent within 1σ.Thebiggest differences intheresults
1 fromthetwomodelsareinpartialtransits.Inmodel2,theglobal
suggeststhatthemodelcouldberejectedatthe3.6σ level.How-
shapeofthetransitdefinedbythecompletetransitsconstrainsthe
ever,weremindthereaderthatthissinusoidalmodelisonly used
shapeofpartialtransits,thisallowsabetterconstrainonthebase-
to estimate the most probable TTV frequency to be used later in
linefunctionthatcorrelateswiththetransittimes.
the 3-body simulations. Therefore, the p-value test cannot be di-
From the bottom panel of Figure 5 it is also clear that our
rectly applied. Nevertheless, we found that the sinusoidal model
estimatederrorsarehigher thanprevious ones. Thisisduetothe
fittothenew transittimesisworse, asitresultsinamuchlower
p-value4.1×10−7forbothfrequenciesfttvandfttv.Mostimpor- inclusionofrednoiseinouranalysis.Formodel2theuncertainties
1 2 ofT0sofpartialtransitsaresmallerthanformodel1becausethe
tantly,forthenewresultsweobtain3.5timessmallerTTVsemi-
amplitudes:Attv =0.48±0.18andAttv =0.53±0.14. shape of the transit in model 2 is almost fixed by the other light
1 2 curves.Incontrast,someT0uncertaintiesofcompletetransitsare
In Figure 4 we show the Lomb-Scargle periodogram of
higherbecausedifferencesfromthecommonshapeareconsidered
the residual transit times of model 1 after subtracting a lin-
red-noisewhichisaccountedforasexplainedinSection3.2.This
ear ephemeris. Notice that the TTV frequencies proposed by
isthe key difference between the models and implies that in this
Maciejewskietal.(2011a)havedisappearedwhichagreeswiththe
casemodel 2 willproduce much more reliableresults, aswill be
muchloweramplitudesfoundforthesinusoidalmodelfit.
discussedlater.
Wecomputealinearephemerisusingthenewestimatedtransit
timesformodel2:
4.2 Model2
The individual transit times and the out-of-transit flux variability Tt(TDB)=T(0)+EP, (2)
derivedfromasimultaneousfitofthe30lightcurvesofWASP-10 where P = 3.09272932 ± 0.00000032 and T0 =
aregiveninTable6. 2454664.038090±0.000048.Timeresiduals(orTTVs)fromthe
We conclude that three light curves 242, 243, 254a linearephemerisaregiveninTable6andshowninFigure6.The
(Maciejewskietal. 2011b) have a significantly lower df (at 3σ ) RISEdataareshowasstars,thenewFTNdataassquares,andthe
thanthechosenzeropoint.Inournotationanegativedfmeansthe newresultsbasedonpreviouslypublishedlightcurvesascircles.
star is fainter which corresponds to a larger transit depth. Some Theχ2ofthelinearephemerisis86for28degreesoffreedom
light curves have a higher df, but this is non-significant. There- and the p-value is 5.7×10−8 so we conclude that the linear fit
10 S.C.C. Barrosetal.
Table5.WASP-10MCMCfittedparametersformodel1(individualfit).TimesareinTDB.
Number a/R∗ Rp/R∗ inc(degrees) T1−4(hours) T0(days)
-103 12.02±0.21 0.1509±0.0034 88.67±0.12 2.198±0.043 4345.48630±0.00061
-89 12.02±0.17 0.1620±0.0030 88.65±0.12 2.219±0.033 4388.78602±0.00042
-88 11.13±1.06 0.1626±0.0046 88.67±0.12 2.411±0.245 4391.88193±0.00416
0 11.91±0.06 0.1575±0.0003 88.70±0.10 2.236±0.005 4664.03808±0.00005
20 12.35±0.13 0.1557±0.0011 88.64±0.12 2.143±0.017 4725.89175±0.00026
21 12.01±0.14 0.1570±0.0014 88.67±0.12 2.212±0.023 4728.98414±0.00038
26 11.15±0.41 0.1545±0.0021 88.72±0.11 2.392±0.097 4744.45206±0.00160
36 11.96±0.18 0.1641±0.0023 88.66±0.12 2.235±0.033 4775.37693±0.00051
56 11.93±0.15 0.1602±0.0017 88.65±0.12 2.233±0.027 4837.23105±0.00042
123 11.93±0.16 0.1572±0.0019 88.66±0.12 2.227±0.029 5044.44435±0.00044
123 11.90±0.16 0.1580±0.0026 88.68±0.12 2.238±0.031 5044.44407±0.00041
124 11.86±0.12 0.1559±0.0013 88.67±0.12 2.239±0.020 5047.53701±0.00033
132 11.67±0.51 0.1582±0.0057 88.66±0.12 2.281±0.116 5072.27832±0.00182
133 11.85±0.15 0.1598±0.0019 88.67±0.12 2.249±0.027 5075.37131±0.00041
133 12.11±0.21 0.1511±0.0024 88.67±0.12 2.181±0.038 5075.37145±0.00063
134 11.75±0.23 0.1623±0.0043 88.66±0.12 2.274±0.051 5078.46453±0.00080
143 11.76±0.14 0.1576±0.0024 88.68±0.11 2.265±0.028 5106.29690±0.00038
144 11.62±0.24 0.1629±0.0035 88.67±0.12 2.304±0.049 5109.39109±0.00080
145 11.50±0.26 0.1649±0.0034 88.67±0.12 2.334±0.055 5112.48544±0.00088
147 12.41±0.13 0.1601±0.0013 88.65±0.12 2.142±0.019 5118.66785±0.00030
158 12.00±0.15 0.1529±0.0018 88.66±0.12 2.206±0.026 5152.68920±0.00041
242 11.83±0.11 0.1608±0.0010 88.67±0.12 2.256±0.016 5412.47863±0.00024
243 11.81±0.09 0.1616±0.0009 88.68±0.12 2.264±0.013 5415.57161±0.00020
252 12.03±0.14 0.1550±0.0024 88.68±0.12 2.205±0.026 5443.40659±0.00035
253 11.80±0.09 0.1592±0.0009 88.68±0.11 2.260±0.013 5446.49881±0.00018
254 11.86±0.08 0.1610±0.0005 88.62±0.11 2.245±0.007 5449.59138±0.00007
254 11.86±0.12 0.1581±0.0014 88.66±0.12 2.245±0.020 5449.59120±0.00031
265 11.71±0.17 0.1614±0.0026 88.66±0.12 2.281±0.035 5483.61165±0.00054
274 11.92±0.10 0.1596±0.0008 88.67±0.12 2.236±0.014 5511.44560±0.00021
285 11.48±0.53 0.1573±0.0038 88.69±0.12 2.323±0.122 5545.46688±0.00205
4
3
) 2
n
i
m
( 1
r
a
e
n
i 0
l
C
−
O −1
−2
−3
−100 0 100 200 300
Epoch
Figure6. Model2estimatedtransitresidualsofWASP-10baftersubtractingalinearephemeris.Thetransittimeswerederivedfromasimultaneousfitofall
thelightcurves.Previouslypublishedfittedlightcurvesareshownascircles.ThenewRISEdataareshownasstarswhilethenewFTNdataassquares.
isnot aagood fitat5.4σ confidence level.However, onlyepoch for27degreesoffreedomandthep-valueincreasesto0.00176and
143deviatessignificantly(at5.8σ)fromthelinearephemeris.This hencewecanonlyrejectthenullhypothesesatthe3.1σlevel.
transitistheonlyoneobtainedwiththe2.0mRozhentelescopeand
isapartialtransit.Ifwediscardepoch143theχ2 decreasesto53 For these new transit times derived with model 2 we also
testedthesinusoidalmodelproposedbyMaciejewskietal.(2011a)
forthetwoproposedfrequenciesasdoneintheprevioussectionfor