Uncovering the planets and stellar activity of CoRoT-7 using only radial velocities

1.8 MB·

by  J. P. Faria

Astronomy&Astrophysicsmanuscriptno.AA_2015_27899 (cid:13)cESO2016
February4,2016
Uncovering the planets and stellar activity of CoRoT-7
using only radial velocities(cid:63)
J.P.Faria1,2,R.D.Haywood3,B.J.Brewer4,P.Figueira1,M.Oshagh1,5,A.Santerne1,andN.C.Santos1,2
1 InstitutodeAstrofísicaeCiênciasdoEspaço,UniversidadedoPorto,CAUP,RuadasEstrelas,PT4150-762Porto,Portugal
e-mail:[email protected]
2 DepartamentodeFísicaeAstronomia,FaculdadedeCiências,UniversidadedoPorto,RuaCampoAlegre,4169-007Porto,Portugal
3 Harvard-SmithsonianCenterforAstrophysics,60GardenStreet,Cambridge,MA02138,USA
4 DepartmentofStatistics,TheUniversityofAuckland,PrivateBag92019,Auckland1142,NewZealand
5 InstitutfürAstrophysik,Georg-August-Universität,Friedrich-Hund-Platz1,37077Göttingen,Germany
6 ReceivedDecember7,2015;acceptedJanuary27,2016
1
0
2 ABSTRACT

b
Stellaractivitycaninducesignalsintheradialvelocitiesofstars,complicatingthedetectionoforbitinglow-massplanets.Wepresent
e
amethodtodeterminethenumberofplanetarysignalspresentinradial-velocitydatasetsofactivestars,usingonlyradial-velocity
F observations.Insteadofconsideringseparatefitswithdifferentnumberofplanets,weuseabirth-deathMarkovchainMonteCarlo

3 algorithmtoinfertheposteriordistributionforthenumberofplanetsinasinglerun.Inanaturalway,themarginaldistributionsfor
theorbitalparametersofallplanetsarealsoinferred.ThismethodisappliedtoHARPSdataofCoRoT-7.Weconfidentlyrecoverthe

] orbitsofbothCoRoT-7bandCoRoT-7calthoughthedatashowevidenceforthepresenceofadditionalsignals.
P
Keywords. methods:dataanalysis–stars:planetarysystems–stars:individual:CoRoT-7–techniques:radialvelocities
E
.
h
p
1. Introduction Simultaneous photometric and RV observations have been
-
o successfullyusedtoconstrainactivity-inducedRVsignals.How-
Imagine we have at our disposal a set of spectroscopic obser-
r ever,thisapproachrequireseitherajointmodelforphotometric
tvations of an unknown star, and we can obtain precise radial-
s and RV variations, which can be statistical (e.g. Rajpaul et al.
avelocity (RV) measurements, using the cross-correlation func- 2015) or based on a description of the stellar features inducing
[tion (CCF) technique (Baranne et al. 1996; Pepe et al. 2002).
the signal (e.g. Lanza et al. 2010), or a conversion from pho-

2 Theso-calledlineprofileindicators(suchasthefullwidthathalf tometric variations to RV variations (e.g. Haywood et al. 2014,
maximum of the CCF, its bisector span, and associated quanti-
v hereafterH14).
ties, e.g. Figueira et al. 2013) and activity indicators, like the
5 Here we present a framework that models activity-induced
9logR(cid:48)HK (Noyes et al. 1984), are also usually available but are signalsascorrelatednoiseintheRVobservationsanddoesnot
4notnecessaryforouranalysis.Andwithoutotherinstrumentsat require simultaneous photometric observations. The method is
7hand,wecannotmeasurethestar’sphotometricvariations,asis basedonthefactthat,fordataofsufficientquality,itispossible
0thecaseformosttargetsinRVsurveys.
todistinguishifanoscillatingsignalhastheKeplerianshapethat
1. Armed with some statistical artillery, we aim to answer the isexpectedfromaplanet,orsomeotherapproximately-periodic
0following question: how many orbiting planets can be confi- shape as expected from stellar activity. Below we describe our
6dentlydetectedinthesedata,andwhataretheirorbitalparame- model and apply it to High Accuracy Radial Velocity Planet
1tersandminimummasses? Searcher(HARPS;Mayoretal.2003)observationsofCoRoT-7.
: Besides planets, other physical processes can induce varia-
v
itions in the radial velocities of a star. These include stellar os-
Xcillations, granulation, spots and faculae/plages, and long-term 2. ABayesianmodelforRVdata
rmagneticactivitycycles(see,e.g.Saar&Donahue1997;Santos
aetal.2010;Boisseetal.2011;Dumusqueetal.2011a).Someof In this section we present in detail our model for RV data and
these activity-induced signals can be mitigated or averaged out describethetwotypesofsignalsweconsider:planetarysignals
byadaptingtheobservationalstrategy(Dumusqueetal.2011b). andactivity-inducednoise.
Butsignalscausedbythepresenceofactiveregionsonthestel-
larsurfacecanshowperiodicitiesandamplitudessimilartothe
2.1. RVplanetarysignals
onesinducedbyrealplanetarysignals,andmaybehardertodis-
entangle.Indeed,thesesignalscanevenmimicplanetarysignals The dynamical evolution of a planetary system is governed by
(e.g. Figueira et al. 2010; Santos et al. 2014; Robertson et al. thegravitationalinteractionsofitsconstituentbodies.Formost
2015). systemswithmultipleplanetsonecanassume,toagoodapprox-
imation,thatthemutualplanetaryperturbationsarenegligibleon
(cid:63) Alldataandsoftwarepresentedinthisarticleareavailableonline timescales that are comparable to the duration of observations.
athttps://github.com/j-faria/exoBD-CoRoT7. The stellar RV perturbations that are due to a multiple-planet
Articlenumber,page1of7
A&Aproofs:manuscriptno.AA_2015_27899
systemcanthenbemodelledasalinearsuperpositionofKeple-
rianorbits. µP wP µK
Each Keplerian can be described with five RV observables:
the semi-amplitude K, the orbital period P, the eccentricity e,
anorbitalphaseφ,andthelongitudeoftheline-of-sight,ω.We
also consider a systemic velocity, vsys, of the centre of mass of P K e η1 η2 η3 η4
thesystem,whichcorrespondstoanRVzero-pointmeasuredby Np
HARPS.
φ ω
2.2. Gaussianprocessestomodelstellaractivity Npplanets
Even if activity signals cannot be easily described analytically vsys Σ s
(a complete description would require knowledge of the active
regiondistributionandevolution,temperaturecontrastandstel-
larparameters,suchasinclinationandlimbdarkening),wecan
makesomeassumptionsabouttheirform.Besidesassumingthe
signals will be continuous and smooth, stellar rotation induces vi
a periodicity or quasi-periodicity, as active regions evolve and ti vkiep
σ
cycleinandoutofviewonthestellarsurface. i
For the purposes of detecting planets using RV measure-
ments, the signals caused by stellar activity can then be seen Ndatapoints
ascorrelatedquasi-periodicnoise.Gaussianprocesses(GP)are
Fig.1.Representationoftherelationsbetweenparametersandobserva-
anincreasinglycommontooltodealwithcorrelatednoise(e.g.
tionsinourRVmodel,asaprobabilisticgraphicalmodel.Anarrowbe-
Robertsetal.2012).Intheirapplicationtoregressionproblems,
tweentwonodesindicatesthedirectionofconditionaldependence.The
GPscanbeseenaspriordistributionsoverfunctions,whichwill
circlednodesaretheparametersofthemodel,whosejointdistribution
be constrained by the data (e.g. Rasmussen & Williams 2006).
issampledbytheMarkovchainMonteCarlo(MCMC)algorithm.The
Forourpurposes,weusetheGPtomodelthestochasticcompo- double circled node v represents the observed RVs. The filled nodes
i
nentofoursignal–thatis,thestellarnoise. represent deterministic variables: if these variables have parent nodes
AGPisdefinedbyitsmeanfunction,thedeterministiccom- (vkep and Σ), they are given by a deterministic function of those par-
i
ponentofthesignal,anditscovariancefunction,whichdefines ents;iftheydonothaveparents(t andσ),theyareassumedasbeing
i i
the overall behaviour of the functions under the GP distribu- givenandthusfixed.Thevariablesinsideboxesarerepeatedanumber
tion.Whenthecovariancefunctionisevaluatedattheobserved oftimes,asshowninthebottomleftcornerofeachbox.
times,thecovariancematrixisobtained.Fromthemanypossible
choicesforacovariancefunction,thequasi-periodickernelisthe
times(foreachofthe N planets).Anobserveddatasetiscom-
mostwidelyusedintheexoplanetliterature(e.g.H14,Grunblatt p
posed of N radial-velocity observations v, at times t and with
et al. 2015; Rajpaul et al. 2015), which results in a covariance i i
associated uncertainty estimates σ. The diagram in Fig. 1 is a
matrixoftheform i
representationofthejointprobabilitydensityfunction(PDF)for
Σij =η21exp−(ti2−ηt22j)2 − 2sin2η(cid:16)24π(tηi−3tj)(cid:17)+(cid:16)σ2i +s2(cid:17)δij. (1) apll(cid:16)Θth,e{vvia},ri(cid:8)avbkielpe(cid:9)s(cid:12)(cid:12)(cid:12):{ti},{σi},I(cid:17),
This represents some of our expectations for the activity-
inducedRVsignals:thecorrelationsdecayonatimescaleofη where we condition on the information I1. In the remainder of
2
daysand[Note1:thesuggested’to’changesthemeaningof the paper we will include {ti} and {σi} in I, because they are
the sentence.] have a periodic component with period η days. assumedtobefixed.Toeasethenotation,wealsogrouptheRV
The parameter η4 controls the relative importance of th3e peri- values in a proposition D = {vi}. The (cid:8)vkiep(cid:9) are obtained from
odicanddecayingcomponents.Theparameterη representsthe evaluatingthesumofKepleriansignalsattheobservedtimes.
1
amplitudeofthecorrelations.Thiscovariancematrixalsotakes ThejointPDFcanbefactoredandrearrangedtogiveBayes’s
additional uncorrelated noise into account, added quadratically theorem,
tothediagonal,whereσ arethereportedRVuncertaintiesand
i
sisafreeparameter. p(Θ|I)p(D|Θ,I)
p(Θ|D,I)= , (2)
p(D|I)
2.3. Thecompletemodel
providinganexpressionfortheposteriordistributionforallthe
Our complete model for the RV observations is shown in Fig.
parameters,conditionedontheobserveddata.Theposteriordis-
1, in the form of a probabilistic graphical model. The diagram
tribution contains all the information about the parameters that
relatesalltheparameters(seeTable1fortheirdescription)and
isavailabletous.Withthisdistributiontohand,wecanthenin-
observablesinthemodel,statingtheirinter-dependencies.From
ferhowmanyplanetsaresupportedbythedataandtheirorbital
thisgraphicalrepresentationwecanbuildanexpressionforthe
parametersandmasses.
jointprobabilityoftheparametersofinterestandthedata.
Let Θ be the vector of all the parameters in the model:
(cid:104) (cid:105) 1 Here, I encodes the assumptions considered when setting up the
Θ = N ,µ ,w ,µ ,{P,K,e,φ,ω},v ,η ,η ,η ,η ,s .Theno-
p P P K sys 1 2 3 4 problem,e.g.theformoftheGPkernelorthefactthatweignoreplanet-
tation means that the subset {P,K,e,φ,ω} will be repeated Np planetinteractions,etc.
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J.P.Fariaetal.:UncoveringtheplanetsandstellaractivityofCoRoT-7usingonlyradialvelocities
2.4. Determiningthenumberofplanets assumedauniformpriorbetween[Note2:Ibelieve’between’
isbettersuitedherethanthesuggested’of’]10and40days.
Tocalculatethejointposteriordistribution,followingEq.(2),we
Tosamplethejointposteriordistribution,weusedthealgo-
need the three terms on the right-hand side: the prior p(Θ|I), rithmproposedbyBrewer(2014).Theparticulardifficultyhere
likelihood p(D|Θ,I),andevidence p(D|I).
–since N isnotknown–isthatthesamplingalgorithmneeds
Forthelikelihood,thechoicethatreflectsmostgenuinelyour to jump bpetween candidate solutions with different numbers of
stateofknowledgeisamultivariateGaussiandistribution2,with
mean (cid:8)vkep(cid:9) and covariance matrix Σ. The complete covariance planets.Brewer(2014)proposedamethodthatusesbirth-death
matrix cian be obtained, deterministically, from the values of MtheardkioffvucsihvaeinnMestoendtesaCmarplloin(gMfCraMmCew)morokv(eBsrteowienrfeertNapl.,2w0i1th1i)n.
η ,η ,η ,η ,σ,and s,accordingtoEq.(1).Thelog-likelihood
1 2 3 4 i Thisway,onecanestimatethevalueoftheevidencewhileim-
isthengivenby: proving the mixing in complex posteriors (affected by multi-
modalityandphasetransitions).Brewer&Donovan(2015)ap-
1 1 N
lnp(D|Θ,I)=− rTΣ−1r− lndetΣ− ln2π, (3) pliedthismethodtoRVdataofνOphandGliese581,although
2 2 2
theseauthorsdidnotincorporateGPsintheiranalysis.
whereristhevectorgivenbyv −vkep foralldatapoints.
i i
The priors used for all the parameters are listed in Table 1.
3. ApplicationtoHARPSdata
Most of them are the same as those used by Brewer & Dono-
van(2015)andwerechosentorepresentuninformativeorvague
We apply the method described in the previous section to
priorknowledge.Fortheorbitalperiodsandsemi-amplitudeswe
HARPS observations of CoRoT-7. The planet CoRoT-7b was
assign hierarchical priors that are conditional on the hyperpa-
first announced by Léger et al. (2009) and was the first super-
rameters,µ ,w ,andµ ,respectively(seeTable1).Thisreflects
P P K Earth with a measured radius. Its orbital period is estimated
our belief that knowing the parameters of one planet provides fromthetransitsintheCoRoTlightcurveasbeing P = 0.854
a small amount of information about the parameters of another b
days (Léger et al. 2009). A second non-transiting planet, with
planet. P = 3.69 days, was detected in a follow-up RV campaign
c
(Queloz et al. 2009) and a more disputed detection of a third
Table1.Meaningandpriordistributionfortheparametersinthemodel.
planetarysignalwasreportedbyHatzesetal.(2010).
Insomecaseswesampleonthelogarithmoftheparameter.
Owingtothehighactivitylevelsofthehoststar,thissystem
has since generated a wealth of discussion, resulting in differ-
hyperparameters
ent estimates for the masses of the planets (Lanza et al. 2010;
N numberofplanets U(0,10)
p Boisse et al. 2011; Ferraz-Mello et al. 2011; Pont et al. 2011;
µ medianorbitalperiod log C(5.9,1)
P Hatzesetal.2011).SimultaneousobservationsfromCoRoTand
w diversityorbitalperiods U(0.1,3)
P HARPSwereobtainedin2012tohelpsettletheseissues(Barros
µ meansemi-amplitude log C(0,1)
K etal.2014;H14).TheseobservationswereanalysedbyH14with
planetparameters anRVmodelthatissimilartooursbutconsideringinformation
P orbitalperiod log L(logµ ,w ) fromthesimultaneousphotometricobservations.
P P
K semi-amplitude E(µK) We emphasise that, in the following, we analyse the full
e eccentricity B(1,3.1) set of RV observations. In summary, the star was observed
φ orbitalphase U(0,2π) with HARPS in 2009 and 2012, with a total of 177 public RV
ω longitudeofline-of-sight U(0,2π) measurements.Theaverageerrorbaronthesemeasurementsis
2 ms-1 (whichincludesphotonandinstrumentalnoise)andthe
GPandnoiseparameters
RVdispersionis10 ms-1,overthecomplete3-yeartimespan.
η amplitudeofcovariance LU(0.1,50)
1
A common procedure in (current) RV studies is to fix the
η aperiodictimescale LU(1,100)
2
numberofplanetsandsampletheposteriorfortheremainingpa-
η correlationperiod U(10,40)
3
rameters(andpossiblycalculatetheevidence)inastep-by-step
η periodicscale LU(0.1,10)
4 approach. If we fix N = 2, we recover the orbital parameters
s extrawhitenoise log C(0,1) p
ofthetwoknownplanets,CoRoT-7bandCoRoT-7c,withinthe
vsys systematicvelocity U(minvi,maxvi) errorsreportedinH14andBarrosetal.(2014).Themethodwe
presentinthispaperis,however,muchmoregeneralandallows
Notes.Symbolmeaning:U(·,·)–Uniformpriorwithlowerandupper
forthefullposteriorforN tobeobtainedfromonerun.Wenow
limits;C(·,·)–Cauchypriorwithlocationandscale(thesedistributions p
proceedwiththisgeneralmethod.
were truncated for numerical reasons); L(·,·) – Laplace prior (some-
timescalleddoubleexponential)withlocationandscale;E(·)–Expo- We ran our algorithm on the full set of RVs, using the
nentialpriorwithmean;B(α,β)–Betapriorwithshapeparameters,α priors in Table 1, and obtained 16248 effective samples from
andβ,anapproximationtothefrequencydistributionofeccentricities the joint posterior distribution3. The evidence for our model is
proposedbyKipping(2013);LU(·,·)–Log-uniformpriorwithlower log(p(D|I)) = −530.9.Theresultingmarginalposteriordistri-
andupperlimits. butionforN isshowninFig.2.
p
Theposteriordistributionfor N isoneofthemainoutputs
p
FormostoftheparametersoftheGPcovariancekernel,we ofourmethod.Buttoactuallydecideonwhatisthenumberof
assigned log-uniform priors in sensible ranges. For η , the pa-
3
rameterthatcanbeinterpretedasthestellarrotationperiod,we 3 The computer used to run the simulations was equipped with an
Intel(cid:13)R CoreTM i5-4460CPUrunningat3.20GHzand4GBofRAM.
2 Fig.1showsthatallweknowaboutthedistributionofthe{v}areits Therunningtimetoobtain50000samples(fromdiffusivenestedsam-
i
firstandsecondmoments.ThemultivariateGaussianfollowsfromthe pling’stargetmixturedistribution)wasfourdays.Sincethecomputa-
principleofmaximumentropywhenthedistributionisconstrainedto tionalcostofthealgorithmisdominatedbytheinversionofthecovari-
havingaspecifiedcovariancematrix(e.g.Cover&Thomas2006). ancematrix,itisexpectedtoscaleroughlywithN3.
Articlenumber,page3of7
A&Aproofs:manuscriptno.AA_2015_27899
theposteriordistributionissensitivetothepriordistributions,in
3000
s particularastohowsmallwebelieveK mightbe.
e
mpl 2500 pp((32)) ≈4.49 Thejointposteriordistributionsfortheorbitalperiods,semi-
a amplitudes,andeccentricitiesofthesignalsareshowninFig.3,
s
or 2000 where the samples for all Keplerians were combined. The fig-
eri ure shows a 2-dimensional histogram of the posterior samples,
ost 1500 wherethecolourmaprepresentsbincountsandissetinaloga-
p
of 1000 rithmicscale.
p(2)
er p(1) ≈∞ The two detected planets are seen as overdensity regions at
mb 500 Pb = 0.85 days and Pc = 3.69 days. Their amplitudes and ec-
Nu centricitiesarewellconstrained.Thereisaclearposteriorpeak
0
0 1 2 3 4 5 6 7 8 9 10 aroundtwodayswithamplitudeandeccentricitymostlyuncon-
strained.Theposterioralsoshowsasmallerpeakat9days,the
Numberofplanets
periodreportedbyHatzesetal.(2010)asapossiblethirdplanet
(seealsoTuomietal.2014).
Fig.2.PosteriordistributionforthenumberofplanetsN .Thecounts
p MarginalposteriordistributionsfortheparametersoftheGP
arenumberofposteriorsamplesinmodelswithagivennumberofplan-
and the extra white noise parameter s are shown in Fig. 4. The
ets. The two ratios of probabilities between models with 1, 2, and 3
planetsishighlighted;notethatp(0)= p(1)=0. posterior for η3 is particularly interesting as it provides a con-
straint to CoRoT-7’s rotation period, obtained exclusively from
the RVs. Our inferred value for the stellar rotation period of
1] 10 2in2.a3g0r+−e16e0.m1.108endta(ysseiesToabbtalein2e)dwsoitlheleyafrrloiemrethsteimRVatetsimweh-sicehrieusseadndthies
-s
m CoRoTlightcurve(Légeretal.2009;Lanzaetal.2010;H14).
1
[
de Alsointerestingisthejointbehaviourofη2,η3,andη4.For
u higher values of η , the periodic component of the covariance
plit 0.1 function loses imp4ortance relative to the decaying component,
m
mi-a 0.01 ηG2PgsemtsosomthaslltehreaRndVηs3obneacotimmeesscuanlceoonfstηrain≈ed3.Idnayths.isBcuatsew,hthene
2
e
S η isoforderunity(meaningtheperiodiccomponentispresent),
4
thedecayingtimescaleishigherandη isconstrainedaround22
3
0.8
days. The values of η in this situation (20-30 days) are closer
2
0.7 tothestellarrotationperiodandarealsoconsistentwiththeav-
y 0.6 erage lifetime of active regions measured in the CoRoT 2012
cit 0.5 photometry (20.6 ± 2.5 days, H14). Our results therefore vali-
tri 0.4 datetheapproachtakenby,e.g.H14andGrunblattetal.(2015),
n
ce 0.3 of modelling the activity-induced RV variations with a GP that
c
E 0.2 hasthecovariancepropertiesofthelightcurve.
0.1 Considering only the posterior samples with Np = 2, Table
0.0 2liststhemedianvaluesofsomeorbitalparametersforthetwo
0.1 1 10 100 1000 planets, and the maximum likelihood RV curves are shown in
Period[days] Fig.5.Ourestimatesfortheorbitalparametersareinagreement,
withintheuncertainties,withtheonesobtainedbyH14.
Fig.3.Jointposteriordistributionforthesemi-amplitudes(toppanel)
andtheeccentricities(bottompanel)togetherwiththeorbitalperiodsof
4. DiscussionandConclusions
theKepleriansignals.
BuildingontheworkofBrewer&Donovan(2015),wehavede-
velopedasimplemethodtoestimatethenumberoforbitingplan-
planets orbiting CoRoT-7, and answer our initial question, we etsaroundactivestars,usingonlyRVmeasurements.Weapplied
needtoclarifywhatwemeanby“confidentlydetected”.Based thismethodtoHARPSobservationsofCoRoT-7andconfidently
onascalesuggestedbyJeffreys(1998)(seealsoKass&Raftery detectCoRoT-7bandCoRoT-7c,whilefindingweakerevidence
1995), some authors (e.g. Tuomi 2011; Feroz et al. 2011) have for two additional signals. In this framework, there is no need
proposedthat,toclaimadetectionofNpplanets,theprobability tousephotometricobservations,informationfromtransitdetec-
of Np should be 150 times greater than the probability of Np − tions,orauxiliaryactivityindicators.
1. This criterion requires ’strong’ (Jeffreys 1998) evidence for The posterior distribution for N shows evidence for the
p
detecting a planet, thus considering false positives to be much presenceofextrasignals,eveniftheydonotmeetourdetection
worse than false negatives. By applying this rule to our results, criteria. We note that the effects of considering a uniform prior
wechooseNp =2asthenumberofplanetsconfidentlydetected forNp(thusgivingconsiderablepriorweighttolargenumbersof
inourdataset. planets)andahierarchicalpriorforthesemi-amplitudes(which
We emphasise that having confidently detected two planets changes how the Occam’s razor penalty is taken into account)
isadifferentmatterfromknowingthatthenumberofplanetsis canbeimportantandwillbestudiedinthefuture.
actually two. According to the posterior distribution, it is more Ourchoicesforthelikelihood,covariancefunction,andpri-
likelythattherearefourplanets.Butourdetectioncriterionde- orsprovidethemodelwitharobustnessagainstpossibleoutliers
pends on the ratio of probabilities for consecutive values of N whichissimilartomostanalysesofRVdata.Ifwehadreasonto
p
(seeFig.2),notontheprobabilityvaluesthemselves.Ofcourse, suspect the presence of outliers, it would be straightforward to
Articlenumber,page4of7
J.P.Fariaetal.:UncoveringtheplanetsandstellaractivityofCoRoT-7usingonlyradialvelocities
s = 0.88+0.42
0.54
−
η = 9.31+1.53
1 1.31
−
16
1 12
η
8
η = 3.28+14.15
2 1.14
−
80
60
2
η
40
20
η = 22.21+9.79
40 3 −6.40
32
η3 24
16
η = 0.93+3.25
10 4 −0.36
8
6
4
η
4
2
0.5 1.0 1.5 2.0 8 12 16 20 40 60 80 16 24 32 40 2 4 6 8 10
s η1 η2 η3 η4
Fig.4.Marginalised1-and2-DposteriordistributionsfortheparametersoftheGPandtheextrawhitenoise.ThesamplesforallvaluesofN
p
werecombined.Thetitlesaboveeachcolumnshowthemedianoftheposteriorandtheuncertaintiescalculatedfromthe16%and84%quantiles.
Thesolidlinesarekerneldensityestimationsofthemarginaldistributions.
extendourmodel,e.g.byscalingeacherrorbarwithacommon In the analysis of H14, the authors modelled the out-of-
valueorwithindividualvalueswhosecommonpriorisdefined transit photometry using a GP (as an interpolator) and applied
hierarchically. the FF’ method of Aigrain et al. (2012) to obtain the RV sig-
nalthatisduetoactivity.Bycomparingtheevidenceofmodels
containingthisactivitysignalpluszero,one,two,orthreeplan-
Steppingbacktoappreciateourresults,wefindthatthereis
ets,theyassertedthatthetwo-planetmodelwasthemostproba-
alotofinformationcontainedintheRVs,bothaboutplanetary
ble.Tomodelallthequasi-periodicsignalsinthedata,H14in-
and (arguably more importantly) activity signals, and that this
cludedaGP(withthecovariancepropertiesoftheCoRoTlight
information can be recovered. The GP provides a flexible and
curve)aspartoftheRVmodel.Thiswasjustifiedbecause“the
accommodatingmodelforactivity-inducedsignals,allowingus
FF’methodislikelytoprovideanincompleterepresentationof
to infer the planetary masses and orbital parameters with more
activity-induced RV variations” (H14). We also note that these
realisticuncertainties.
Articlenumber,page5of7
A&Aproofs:manuscriptno.AA_2015_27899
a andcanberecoveredwiththeflexibilityoftheGP,ifwebetter
70 70 accountfortheuncertaintyassociatedwithit.
60 60 We should finally note two important properties of the
50 50 CoRoT-7system,whichmadeitidealforthisanalysis.First,the
m/s] 40 40 amplitudesoftheplanetsignalsaremuchhigherthanthemean
V[ 30 30 error bar of the HARPS observations, regardless of the stellar
R activitycontamination.Second,thetimesamplingoftheobser-
20 20
vationsisalmostidealforthedetectionofshort-periodplanets,
10 10 andisverydifficulttoobtainaspartofatypicalRVsurvey.We
0 0
highlightheretheimportanceoffurthertests,usingotherwell-
4780 4830 4880 5940 5950 5960
studieddatasetsofactivehoststarsaswellassimulateddatasets,
BJD-2450000[days]
forassessingthelimitsofapplicabilityofourmethod.
b RV-GP-planetc c RV-GP-planetb Nevertheless,theimportanceofthisworkisnotonthespe-
20 20
cific application to CoRoT-7, but instead on providing a sim-
15 15
ple and fast method to infer the number of planetary signals in
10 10
the presence of stellar activity. With small modifications, this
s] 5 5
m/ methodcanbeusedtosearchforplanetsaroundstarswithdif-
0 0
[ ferentactivitylevels.
V -5 -5
R
-10 -10 Acknowledgements. The work presented here grew directly from a collabo-
-15 -15 ration that started at the Astro Hack Week 2015. We thank Andrew Collier
Cameronforinsightfuldiscussions.Weacknowledgetheexcellentopen-source
-20 -20
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Python packages made available to the community (in particular DAFT and
George).ThisworkwassupportedbyFundaçãoparaaCiênciaeaTecnologia
orbitalphase
(FCT)throughtheresearchgrantUID/FIS/04434/2013andthegrantPTDC/FIS-
AST/1526/2014. JPF acknowledges support from FCT through the grant ref-
Fig.5.Panela:RVmeasurementsofCoRoT-7from2009and2012and erenceSFRH/BD/93848/2013.RDHgratefullyacknowledgesagrantfromthe
thetwo-planetbest-fitmodel(blackcurve).Notethedifferentscalesin JohnTempletonFoundation.Theopinionsexpressedinthispublicationarethose
theabscissaeontheleftandrightpartsoftheplot.Panelsbandcshow of the authors and do not necessarily reflect the views of the John Temple-
thephasedRVcurvesaftersubtractingeachplanetsignalandtheGP. ton Foundation. BJB is supported by a Fast Start grant from the Royal So-
ciety of New Zealand. PF and NCS acknowledge support by FCT through
Investigador FCT contracts (IF/01037/2013 and IF/00169/2012, respectively),
Table2.ParameterestimatesfromourworkandfromH14.Weconsider andPOPH/FSE(EC)byFEDERfundingthroughtheprogramProgramaOp-
allmodelsthathaveN = 2andshowthemarginalposteriormedians, eracional de Factores de Competitividade - COMPETE. PF further acknowl-
p
togetherwiththe16%and84%quantiles. edges support from FCT in the form of an exploratory project of reference
IF/01037/2013CP1191/CT0001.MOacknowledgesresearchfundingfromthe
DeutscheForschungsgemeinschft(DFG,GermanResearchFoundation)-OS
units Thiswork H14 508/1-1. AS is supported by the European Union under a Marie Curie Intra-
European Fellowship for Career Development with reference FP7-PEOPLE-
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b −0.55
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