An analytic model for rotational modulations in the photometry of spotted stars

25 Pages·2012·0.73 MB·English

by  Garth Stewart

Mon.Not.R.Astron.Soc.427,2487–2511(2012) doi:10.1111/j.1365-2966.2012.22124.x
An analytic model for rotational modulations in the photometry
of spotted stars
(cid:2)†
David M. Kipping
Harvard–SmithsonianCenterforAstrophysics,60GardenStreet,Cambridge,MA02138,USA
Accepted2012September12.Received2012September12;inoriginalform2012August6
D
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w
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ABSTRACT lo
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Photometricrotationalmodulationsduetostarspotsremainthemostcommonandaccessible d
e
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wetaryytoofs∼tu1d5y0s0te0l0lasrtaacrstivpirteys.eInntitnhgeKanepulenrp-reercae,dtehnetreednoowppeoxrtiustnsitpyrefcoirses,tactoinsttiicnauloaunsaplyhsoetsomof- from
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these modulations. Modelling rotational modulations allows one to invert the observations ttp
s
intoseveralbasicparameters,suchastherotationperiod,spotcoverage,stellarinclinationand ://a
differentialrotationrate.Themostwidelyusedanalyticmodelforthisinversioncomesfrom c
a
d
BuddingandDorren,whoconsideredcircular,greystarspotsforalinearlylimbdarkenedstar. e
m
Inthiswork,weextendthemodeltobemoresuitableintheanalysisofhighprecisionphotom- ic
.o
etry,suchasthatbyKepler.OurnewfreelyavailableFORTRANcode,macula,providesseveral up
improvements,suchasnon-linearlimbdarkeningofthestarandspot,asingle-domainanalytic .co
m
function,partialderivativesforallinputparameters,temporalpartialderivatives,dilutedlight /m
n
compensation,instrumentaloffsetnormalizations,differentialrotation,starspotevolutionand ra
s
predictionsoftransitdepthvariationsduetounoccultedspots.Throughnumericaltesting,we /a
find that the inclusion of non-linear limb darkening means macula has a maximum photo- rtic
le
metric error an order-of-magnitude less than that of Dorren, for Sun-like stars observed in -a
b
the Kepler-bandpass. The code executes three orders-of-magnitude faster than comparable stra
numericalcodesmakingitwellsuitedforinferenceproblems. ct/4
2
7
Keywords: methods:analytical–techniques:photometric–planetarysystems–starspots. /3
/2
4
8
7
/1
1
0
5
Henryetal.2000)hasledtoasurgeinthedesignandconstruction 5
1 INTRODUCTION 1
of precise photometric instruments. Notably, the Kepler mission 9
b
hasdetected2165eclipsingbinaries(EBs;Slawsonetal.2011)and y
1.1 Stellaractivity g
2321 planetary candidates (Batalha et al. 2012) with nearly con- u
e
A variety of cool stars with external convection envelopes have tinuous photometry at a precision of ∼50ppm (for V (cid:3) 12) per st o
been observed to exhibit magnetic activity similar to that of the long-cadenceexposure(29.4244min).Preliminaryanalysisofthe n
0
Sun (e.g. Kron 1947; Mullan 1974; Vogt 1975). These magnetic Keplertargetstars(over150000)hasrevealedthatthosewhichex- 3
A
fields, generated by cyclonic turbulence in the outer convection hibitperiodicmodulationgenerallyhaveamuchhigheramplitude p
zone, penetrate the stellar atmosphere forming starspots, plages, ofvariability(Basrietal.2011).Oneofthelegacyproductsofthe ril 2
0
networks,etc.(Berdyugina2005).Thestudyofthesemanifestations Keplermissionwillbeavastdatabaseofprecisecontinuouspho- 1
9
onotherstarsallowsforcrucialtestsofstellardynamotheory.For tometryandtheeffectiveexploitationofthisdatabasewillsurely
example,Skumanich(1972)firstsuggestedthatrotationplaysakey leadtodeepinsightsintostellaractivity.
roleingeneratingstellaractivity.
Sincethediscoveryofrotationallymodulatedbrightnessvaria-
tionsduetostarspots,photometryremainsthemostcommontech-
1.2 Starspots
niqueforstudyingstellaractivity.Inparticular,space-basedphoto-
metricinstrumentshaveprovidedmanyhighcadence,preciselight Starspotsareaverycommonsourceofphotometricvariabilityand
curves (e.g. Hipparcos; van Leeuwen et al. 1997). Recently, the haveadiversevaluetoastronomers,varyingfromfriendtofoe.The
detectionoftransitingextrasolarplanets(Charbonneauetal.2000; presenceofdarkstarspotsonthesurfaceofarotatingstarinduces
periodicphotometricvariabilityduetothestellarrotation.Ananal-
ysisoftheserotationalmodulationsallowsforadeterminationof
(cid:2)E-mail:[email protected] several basic properties of the star. The most accessible of these
†CarlSaganFellow. propertiesistherotationperiod,whichcanoftenbeinferredusing
(cid:4)C 2012TheAuthor
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2488 D.M.Kipping
a simple Lomb–Scargle periodogram or autocovariance analysis, modulationalone,asisthecasefortheKeplermission,isamore
andhasseveralastrophysicaluses.Forexample,therotationperiod challengingproblem.
may be used with gyrochronology to estimate the age of the star EclipsemappingofEBsallowsforgreatlyimprovedinversionsof
(Barnes2009).Anotherexampleisdemonstratedintherecentwork thespotcoveragethroughphotometryalone.Thisisusuallydoneby
ofHiranoetal.(2012)whoshowhowaspectroscopicVsinI∗,an numericallypixellatingthestarandapplyingthemaximum-entropy
estimateofthestellarradius(R∗)andtherotationperiodallowsone method(MEM)toinvertthemap(e.g.Collier-Cameronetal.1997).
toinferthestellarinclinationangle,I∗. Inprinciple,atransitingplanetoffersthesameopportunityaswas
Employing spot-modelling codes allows for more information discussed earlier for measuring spin-orbit alignments. However,
than just the rotation period to be derived. For example, Walker eclipsemappingusingplanetshasseveraldrawbacks.Forexample,
etal.(2007)usedrotationalmodulationsalonetoinferthestellar themuchsmallerratioofradiimeansonlyathinstripofthestaris
inclination angle for κ1 Ceti. Here the authors also showed how actuallysampled.Thismeansthatinferencesaboutthenon-eclipsed
their measurement could be used to predict VsinI∗ and verified portion of the star, which affect the perceived transit depth for
their solution was consistent with a spectroscopic determination. example,mustbemadeusingtherotationalmodulations.Further, D
o
Furthermore,theauthorswerealsoabletoestimatethedifferential theeclipse-mappingtechniqueonlyprovidesasnapshotofthespot w
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rotationrateofκ1Ceti,whichwasfoundtobereasonablycloseto coverage at the instant of the transit. For long-period planets, or lo
a
Solar. even stars without transits at all, this technique cannot uniquely d
e
theIftraasntsairthloigshtstacturravnesiatipnpgeparlasnteotiwnhcricehaspeasosveesrotvheerdaudraartkiosntaorsfptohte, infLerikeevseonmbaasniycpsrteolblalermpsroinpearsttireosp,hsuycsihcsa,smthoederolltiantgiornotpaetiroiondm.odu- d from
spotcrossingevent(e.g.Rabusetal.2009).Detectingthesamespot- lationsduetostarspotscanbeapproachedusingeithernumericalor h
crossingeventintwoconsecutivetransits,whichwillhavemigrated analytictechniques.Numericaltechniquesaremorediverse,allow- ttps
along in longitude, allows the observer to infer a nearly coplanar ingonetocomputeanystarspotshape,fluxprofile,limbdarkening ://a
c
spin-orbit angle. This technique has so far been successfully ap- law,etc.onewishes.However,theycomeattheexpenseofmuch a
d
pliedtoseveralcasesincludingCoRoT-2b(Nutzman,Fabrycky& higher computation times. Analytic models are extremely quick em
Fortney2011),WASP-4b(Sanchis-Ojedaetal.2011)andKepler- toexecute,oftenoutpacingtheirnumericalcounterpartsbyorders ic
.o
17b (De´sert et al. 2011). In the case of WASP-4b, the result was of magnitude. Such models are challenging to derive though and u
p
verifiedusingthemoretraditionalspectroscopictechniqueknownas are limited in that they assume fixed properties of each spot; for .c
o
theRossiter–McLaughlineffect(McLaughlin1924;Rossiter1924). example,theirshapeisoftenassumedtobecircular.However,ro- m
/m
Modellingspot-crossingeventsremainsoutsideofthescopeofthis tationalmodulationsrepresentadisc-integratedsnapshotofastar n
work,butexploitingsuchphenomenaisgreatlyaidedbyincluding withunresolvedsurfacefeatures.Forthisreason,thesizeofaspot ras
ionuftobrmyaNtiuotnzmenacnoedteadl.in(2th0e11o)u.t-of-transitphotometrytoo,aspointed acnhdantgheetflhuexamcopnltirtuasdteaorfethhiegrhelsyucltoinrrgelraotteadtiosinnaclemaoltdeurilnatgioenitsh.eIrnwthiilsl /article
Incontrasttotheexamplesgivensofar,otherauthorsconsider paradigm,modellingelaborateshapesforthestarspotswithdozens -a
b
starspotstobeanuisanceratherthanatool,duetotheirdiffering offreeparametersisunlikelytogenerateamoremeaningfulmodel stra
goals.Forexample,the‘HuntforExomoonswithKepler’(HEK) thanasimplecircularassumption.Inaddition,inthecircularmodel, c
project(Kippingetal.2012)anticipatesthatstarspotcrossingswill oneshouldinterpretthespotsasreallyrepresentingadarkpatchor t/42
7
beasourceoffalsepositivesforexomoonidentificationduetothe clusterofsmallspotsonthesurfaceratherthanaperfectlycircular /3
morphologicalsimilaritieswithplanet–moonmutualevents.Cross- starspot. /24
8
referencingthetransitanomalywithrotationalmodulationsmaybe Thediverserangeofspotswhichcanbemodelledusingnumer- 7
/1
usedtotestwhethertheeventisconsistentwithastarspotornot icaltechniquesisthereforenotapracticaladvantageoveranalytic 1
0
(Sanchis-Ojedaetal.2012). models. With this advantage lost,we consideranalytic models to 55
1
Finally,evennon-occultedstarspotsareasourceoffrustrationin beinvariablythepreferentialtoolformodellingstarspotssincethey 9
b
somearenas.Inparticular,thesespotssubtlychangetheperceived willexecutewithdramaticallyquickercomputationtimes. y
g
transit depth. Since spots vary in both time and wavelength, they u
e
arethereforecapableofproducingtransitdepthvariations(TδV). 1.4 Currentanalyticmodels st o
Thispointishighlysalientforthosestudyingtheatmospheresof n
0
exoplanets,whoseeksmallchromaticvariationsinthedepth.The Analytic models of rotational modulations due to starspots have 3
A
studyofrotationalmodulationscanbeusedtocorrecttheresulting existedfordecades.ThefoundationalpapercomesfromBudding p
transmissionspectra,asshowninDe´sertetal.(2009)forexample. (1977),whodescribeananalyticmodelfortherotationalmodula- ril 2
Itisthereforeclearthatthestudyandinterpretationofrotation tion due to multiple non-overlapping circular starspots. An alter- 01
9
modulationsduetostarspotsiscrucialtoseveralareasofmodern native derivation of an essentially identical model is presented in
astronomical research. In this work, we aim to provide a revised Dorren(1987).
modelforsuchmodulationswhichcanaccountforseveralprevi- Thesemodelshavebeensuccessfullyappliedtonumerousstud-
ouslyignoredeffects. iesofstarspots.WehighlighttherecentworkoftheMOST space-
telescopeindetectingdifferentialrotationon(cid:5)Eridani(Crolletal.
2006b)andκ1Ceti(Walkeretal.2007).Inbothofthesecases,the
authorsmakeuseofaMarkovChainMonteCarlo(MCMC)algo-
1.3 Numericalversusanalyticmodels
rithmtoregressthedata,whichyieldsBayesianinferencesofthe
Invertingthebrightnessmodulationofastarintoaphysicalmapof parameter posteriorsand correlations. Thecode, called StarSpotz
the starspot coverage can be broached in several ways. The most (Crolletal.2006a),includesparalleltemperingtolocatetheglobal
successfultechniqueisDopplerimagingaugmentedbyprecisepho- minimumintheinevitablycomplexparameterlandscape.Bayesian
tometry(e.g.seeVogt&Penrod1983;Collier-Cameronetal.1994; inference techniques, such as MCMC, offer significant improve-
Tuominen,Berdyugina&Korpi2002).However,usingrotational ments in the statistical interpretation of modelling starspots, but
(cid:4)C 2012TheAuthor,MNRAS427,2487–2511
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come at the cost of being inherently computationally expensive. isaccountedfor.Inthissection,wealsoprovideawaytouseour
Forthisreason,analyticmodelsarehighlyprizedduetotheirun- modeltocomputetheTδVduetounoccultedspots.InSection3,we
matchedcomputationalefficiency. compareourresultstothatoftheDorren(1987)modelandshow
DespitethesuccessesoftheBudding(1977)andDorren(1987) that the small-spot approximation used in this work is accurate
model,thereareseveralareasforimprovement.Firstandperhaps to within ∼100ppm for spots of angular size (cid:2)10◦. Further, the
mostcritically,themodelsarelimitedtoalinearlimbdarkeninglaw modelhas amaximumerrorwhich isanorder ofmagnitude less
whichisgenerallyapoordescriptionstellarspecificintensitypro- thanthatofBudding(1977)andDorren(1987)foranon-linearly
files.Claret(2000)remarkthatthemostaccuratelimb-darkening limbdarkenedSun-likestarobservedintheKeplerbandpass,with
functions are the quadratic and ‘non-linear’ laws, both of which spotsofangularsizebelow10◦.InSection4,wedemonstratean
arewidelyusedintheexoplanetcommunityforexample.Indeed, applicationtoapreviouslystudiedexample,MOSTobservationsof
recentlyNutzmanetal.(2011),whoalsomadeuseoftheDorren κ1CetibyWalkeretal.(2007)usingamultimodalnestedsampling
(1987)model,remarkedonhowextendingthemodeltonon-linear algorithm,MULTINEST(Ferozetal.2008;Feroz,Hobson&Bridges
laws would be a significant improvement. Given the dramatic in- 2009).DiscussionofthekeyhighlightsisprovidedinSection5. D
o
crease in photometric precision since the era of Budding (1977) w
n
totheKepler-era,thispointisnotjustpertinentbutimperativeto lo
2 THE MODEL a
address. d
e
moSdeeclo,nwdhlyic,hthweroeulcdulrereandtltyo aexfiustrsthenrospiganrtiifiacladnetriimvaptrivoevsemfoerntthine 2.1 Assumptions d from
regressionanalysis.Oneoftheobstaclesinachievingthisgoalis For clarity, we list the assumptions made throughout this work h
thattheBudding(1977)andDorren(1987)equationforthemodel below: ttp
s
flux is not a single-domain function and has multiple cases. This (i) all starspots are circular and lie on the plane of the stellar ://ac
meanspartialderivativeswouldhavetobetediouslyderivedforall a
surface; d
cases individually. Finally, if one sets the goal of deriving partial (ii) starspotsneveroverlaponeanother; em
derivatives,thenitwouldbeadvantageoustoobserverstoinclude ic
(iii) starspotsaresmallrelativetothestellarradius; .o
numerousintrinsiceffectsinthemodel,forwhichtheirrespective u
(iv) eachstarspotisgreyandhasauniformtemperature;and p
freeparameterscouldalsohavepartialderivativescomputed.Ex- .c
amplesincludedifferentialrotation,starspotevolution,dilutedlight (v) thestarisaspherewithprojectedcircularsymmetry(i.e.no om
gravitydarkening). /m
andinstrumentaloffsets. n
Wedonotclaimthatthesearenecessarilyphysicallytruestate- ras
1.5 Goalofthiswork mreaesnotsn.aTblheeafnudnacltlioownofofrtahesseelf-acsosnusmisptteinotnasniaslythtiactstohleuytioanrefobrromaoddly- /article
For reasons described in Sections 1.2 and 1.3, the principal goal ellingbothin-of-transitandout-of-transitstarspots. -a
b
of this paper is to provide a revised analytic model for rotational Outofallourassumptions,theonewhichismostlikelytoimpose stra
modulationsduetostarspots.Thisnewmodelhaswideapplications practicalrestrictionsontheapplicationofourmodelisthesmall- c
forbothstarswithandwithoutorbitingplanetsofferingnumerous spot approximation. One may reasonably question why such an t/42
7
advantagesovertheBudding(1977)andDorren(1987)algorithms. assumptionisindeedrequired.Byusingthesmall-spotapproxima- /3
Specifically,wehighlightthefollowingkeyfeaturesofthemodel tion,wemaytreatthesurfacebrightnessofthestartobeconstant /24
8
presentedinthispaper. under the disc of the starspot. This assumption, inspired by the 7
/1
small-planetapproximationusedformodellingtransitlightcurves 1
(i) Allows for N non-overlapping small starspots, assumed to 0
besmallrelativetoSthestellarradius. in Mandel & Agol (2002), leads to dramatically simpler expres- 551
sions.Forexample,thenon-linearlimbdarkeningmodelofMandel 9
(ii) Fullnon-linearlimbdarkeningofthestellarandspotsurface b
&Agol(2002)requireshypergeometricfunctionswhereasafterthe y
isincludedwithvectorscandd,respectively. g
authors apply the small-planet approximation the most computa- u
(iii) Differential rotation is included via a latitude-dependency e
includingtermsinsin2(cid:6)andsin4(cid:6). tionallyexpensivefunctionisarccosine.Inthecaseofatransiting st o
planet,onecanseethatthecircularsymmetryoftheplanetisasim- n
(iv) Starspotevolutionpermittedusingalinearmodel. 0
plerproblemthanthatoftheforeshortenedstarspot,suggestingan 3
(v) Umbra/penumbraeffectmaybegenerated. A
analyticnon-linearlimbdarkeningsolutionwithoutthesmall-spot p
qu(avrtie)rMoffisneststriunmKeenptalelrodfafstae)ts are allowed for (e.g. quarter-to- approximationwouldcertainlynotbecomputationallycheap. ril 20
Thesmall-planetapproximationislatershowntobedramatically 1
(vii) Mblendedlightdilutionfactorsareallowedfor(e.g.quarter- 9
moreaccuratethanthelinearlimbdarkeningassumptionofprevious
to-quartercontaminationinKeplerdata)
worksforalmostallfeasiblespotsizes.Weestimateamaximum
(viii) Oursolutionmaybeexpressedasasingle-domainanalytic
errorof100ppmforspotsofangularsizes(cid:2)10◦,andtypicallythe
function.
error is much smaller than this. More detailed estimators for the
(ix) Consequently,weareabletoprovidesingle-domainanalytic
accuracyofoursmall-spotmodelareprovidedinSection3.
expressionsforthepartialderivativesofthemodelfluxwithrespect
toallinputparameters,aswellastime(F(cid:6)).
(x) WealsoshowhowthemodelmaybeusedtopredictTδVdue 2.2 Definitions
tonon-occultedspots.
WedefinethehoststartohaveN starspotsonitssurfacelabelled
(xi) WemakefreelyavailablethenewalgorithminFORTRAN90 byk=1,2, ...,N −1,N .EacShspothasafixedangularradius
code,macula(seewww.cfa.harvard.edu/∼dkipping/macula.html). S S
α , which represents the solid-angle of the cone swept out from
k
In Section 2, we introduce the model and discuss the various the stellar centre to the stellar surface. Further, each starspot has
definitions and how differential evolution and starspot evolution a flux-per-unit-area contrast ratio, relative to the star, defined by
(cid:4)C 2012TheAuthor,MNRAS427,2487–2511
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f .Thisisessentiallyaproxyforthetemperatureofthespotand
spot,k
isassumedtobeuniformwithineachstarspot(butvariablebetween
spots).Settingf >1reproducesabrightfacula,ratherthana
spot,k
darkstarspot.
Thecentreofastarspothasalongitude(cid:8) andlatitude(cid:6) .These
k k
twoanglesmaybecombinedintotheauxiliaryangle,β ,defined
k
as
βk =cos−1[cosI∗sin(cid:6)k+sinI∗cos(cid:6)kcos(cid:8)k], (1)
whereI∗ istheinclinationofthestar.Starspotsmaymigrateover
timefromareferencelocationduetothestellarrotation.Weassume
thatnomigrationoccursinlatitudebutlinearmigrationispermitted
inlongitudevia D
o
(cid:8)k =(cid:8)ref,k+ 2π(tP−∗,ktref,k), (2) Figure1. Anexampleofourlinearstarspotevolutionmodel.Weplotthe wnload
e
(cid:6)k =(cid:6)ref,k, (3) tsoizreigohftt)hmeastrakrsthpeoteinnduonfitisngorfeαssm,atxhaesinasftuannctttimoanx,oafntidmteh.eTshtaertgorifdelignreesss(.left d from
(cid:8)wrahde=iaren(cid:8)sPir∗en,fk,klo.isnWgthieteustditemreaessnfdohretrrteehf,ekthiksathtanmspaoordtbitifotyriaunrngydreoerufgreorceaondcceehattniomgeeinowcflhu2edπne √wh1e−rerc2n, 0ar≤e trhe≤li1mibs tdhaerkneonrimngaliczoeedffircaideinatls,coμord=inatceoso(cid:11)n th=e https://a
disc of the star. We employ the definition of a normalized limb c
latitude migration is trivial but the partial derivatives returned by a
the algorithm are only valid under the above assumption. Due to darkening coefficient, c0, as utilized by Mandel & Agol (2002), dem
differential rotation, this period varies for each spot and here we wherec0=1−c1−c2−c3−c4. ic
.o
assumeasimplelatitudedependenceforthedifferentialrotationof u
p
P 2.3 Solution .co
P∗,k = 1−κ2sin2(cid:6)EkQ−κ4sin4(cid:6)k, (4) A detailed derivation of the model presented here is provided in m/m
n
whereP istherotationperiodfortheequatorofthestarandκ and Appendices A, B and C. To summarize, the analytic solution for ra
EQ 2 s
sκt4arasrpeoctomefafiyceievnotlsveofvitaheadliinffeearregnrtioawltrho/tdaeticoanypmroofidleel.oIfntahdedaintigounl,aar ethxeprmesosdeedlaflsuxfromNSnon-overlappingcircularstarspotsmaybe /artic
⎡ ⎤ le
sααizmke(atxiv,)kia=Ik−1[(cid:10)t1H((cid:10)t1)−(cid:10)t2H((cid:10)t2)] Fmod =(cid:2)mM=1Um(cid:12)m⎣BmFF((αα=,β0),β) + BmB−m 1⎦, (11) -abstract/4
−Ek−1[(cid:10)t3H((cid:10)t3)−(cid:10)t4H((cid:10)t4)]. (5) whereUmistheinstrumentaloffsetofthemthdataset(anormaliza- 27/3
andusing tionfactorforeachKeplerquarter,forexample),Bm isablending /2
4
factorforeachdataset(orquarter)and(cid:12) isabox-carfunction 8
(cid:10)t1 =ti−tmax,k+ L2k +Ik, (6) definedby m 7/110
(cid:12)m(ti;Tstart,m,Tend,m)=H(t−Tstart,m)−H(t−Tend,m). (12) 551
L 9
(cid:10)t2 =ti−tmax,k+ 2k, (7) theInenthdeoafbthoevem,tThstdarat,tmaissett.hTehsetaFrt(oαf,tβh)efmunthctdioantaissegtivaenndbTyend,m is by g
u
(cid:7) (cid:8) e
(cid:10)t3 =ti−tmax,k− L2k, (8) F(α,β)=1−(cid:2)4 nn+cn4 −(cid:2)NS Aπk st on 0
(cid:10)wht4er=eαtim−ax,tkmiasx,tkh−eaLn2gku−larEksi,zeofthekthspotatareferencetim(9e) ×⎡⎣⎛⎝n=(cid:2)n0=40 4(cn−n+dnf4skp=o1t,k)ζ−2,k−ζ−nζ,+2k+42,k−+ζ+δn,+ζ2k+4,k,ζ−,k⎞⎠⎤⎦, 3 April 2019
t ,L isthe‘lifetime’ofthespot(technicallythefullwidthat
max,k k
full-maximum)andI andE aretheingressandegressdurationsof (13)
k k
thespot’sgrowthprofile.H(x)istheHeavisideThetastep-function. where δx,y is the Kronecker delta fucntion and c =
Anillustrativeexampleofourstarspotgrowth/decaymodelisshown {c0,c1,c2,c3,c4}T andd = {d0,d1,d2,d3,d4}T describethenon-
inFig.1. linear limb darkening coefficients of the stellar surface and spot
For simplicity, macula defines the reference times tref,k to be surface,respectively.ThefunctionAkdefinesthesky-projectedarea
equal to t , although one may change this definition without ofthekthstarspotandisgivenby
max,k
affectingthevalidityofthereturnedpartialderivatives.Finally,we A (α ,β )=R[cos−1[cosα cscβ ]
employthefour-coefficientnon-linearlimbdarkeninglawofClaret k k k k k
(2000),wherethespecificintensityofthestarisgivenby + cosβksinαk(cid:14)k−cosαksinβk(cid:15)k], (14)
(cid:2)4 whereweuse
I∗(r)=1− cn(1−μn/2), (10) (cid:14) =sinα cos−1[−cotα cotβ ], (15)
n=1 k k k k
(cid:4)C 2012TheAuthor,MNRAS427,2487–2511
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Figure2. Examplesoflightcurves(solid)andTδVs(dashed)forfourrandomlygeneratedscenariosusingmacula.Eachsimulationassumesfivespots rticle
withrandomproperties,includingrandomangularsizesbetween0◦and10◦andSun-likenon-linearlimbdarkening.Evenwithfivespots,thephotometric -a
b
behaviourcanbehighlycomplexduetospotevolution,whichisalsorandomlygeneratedinallfourcases. s
tra
(cid:15)k =(cid:13)1−cos2αkcsc2βk. (16) Nficoanti-oonccouflttehdesappoptasreanlstotraafnfesicttdtehpethtra(nCszietliandetiraelc.tl2y00v9ia);asno-acmalplelid- ct/427/3
Finally,equation(13)includestheζ function,whichwedefine TδV. This occurs because the planet transits a non-spotty region /2
4
aζs(x)=cosxH(x)H(cid:14)π −x(cid:15)+H(−x), (17) wblhoecrkeedmoouretbflyuxtheiseccolinpcseen.tTrahteedobasnedrvtehdustrmanosrietdoefptthheistodtaelfiflneudxaiss 87/110
2 δ = Fout-of-transit−Fin-transit. (18) 551
panrodvwideefsuormtheertydpefiicnaeleζx−a,km=plζes(βgken−erαakte)danbdyζm+a,cku=laζ(uβskin+gαrakn)d.Wome obs Fout-of-transit 9 by
Foranunspottedstar,thisyields g
inputparametersforfourrealizationsofafive-spotmodelinFig.2. u
αli→m0δobs=p2=δ, (19) est o
n
2.4 Generatingumbra/penumbra wherepistheratiooftheplanettostarradius,RP/R∗.Foraspotted 03
starwhosefluxvariesnegligiblyoverthetime-scaleofthetransit, A
p
S‘nuanksepdo’tsspcoatsn.Fmoarnmifaellsyt,wouitrhmuomdeblraassaunmdepsennaukmedbrnao,no-orvseirmlapplpyinags onema(cid:7)yshowthat (cid:8)(cid:16) (cid:18) ril 20
spots. However, our algorithm macula does allow one to place δ = 2− Fmod,0 (cid:17)δ , (20) 19
spotsontopofoneanothertoo.Doingsoallowsonetogenerate obs Fmod 1− Nk=S1 Aπk
umbra/penumbraeffectsviaasuperposition. whereF isgivenby
mod,0
To generate a single starspot with an umbra and penumbra of
angularradiiαumbraandαpenumbra,respectively,onesimplygenerates Fmod,0 =αli→m0Fmod. (21)
twospotsofthesesizes.Iftheumbrahasafluxcontrastoff
umbra One can see from the above that the effect is purely due to
andthepenumbrahasf ,thenthetwospotsgeneratedwill
penumbra blindlynormalizingthedatabythelocalbaseline,whichisaffected
have{α,f }equalto{α ,f }and{α ,f −f }.
spot p penumbra u penumbra umbra by rotational modulation. ‘Astrophysical detrending’ of a contin-
uous photometric time series, in this case using a spot-model to
detrend the data, would eliminate any apparent depth variations.
2.5 TδVfromunoccultedspots
However,ground-basedobserversareusuallyonlyabletoobtaina
It is well known that an eclipsing body which occults a starspot smallamountofdataeithersideofatransiteventleavingnoalterna-
leavesasignificantimprintonthetransitprofile(Rabusetal.2009). tiveexcepttoblindlynormalizethedata.Infact,evenspace-based
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transit observations are almost always blindly normalized using ofspot-crossingeventsduetheparametricformoftheexpressions
polynomials, running medians or linear trends. macula therefore describingthearcsalongthestarspotrimandbulge.
offerstheopportunitytheastrophysicallydetrendphotometry. Sincethetwomodelsessentiallyonlydifferintheirtreatmentof
In the typical case of blindly normalized data, equation (20) limb darkening, one should expect them to be exactly equivalent
allows one to fit a set of TδV with a spot model using macula. forthecaseofauniformbrightnessstar,whichweeasilyverified
Alternatively,onemaywishtomakecausalpredictionsoftheTδV through numerical experiments. However, for the case of a limb
effectbaseduponout-of-transitrotationalmodulations.Westress darkened star, one might ask, for what spot size does our model
that the equation is only valid if the spots are unocculted. These significantly deviate away from that of Dorren (1987)? We will
TδVsmayoccurintimeduetorotationalmodulation(seeexamples investigatethisquestioninthefollowingsubsections.
inFig2),oreveninwavelengthduetothechromaticnatureofspots.
Indeed,correctingforthechromaticTδVeffectiscrucialinaccurate
interpretationofexoplanettransmissionspectra(e.g.De´sertetal. 3.2 Modelerrorduetooursmall-spotapproximation
2009). A more detailed discussion of the applications of TδVs is D
presentedinSection5.3 Letusdefinethemodelflux,aspredictedbythiswork,asFmod(as ow
aremaincpuultateddi.reTchtliysrfeetuatrunrsethiesfounn/cotfifonsw(δiotbcsh/δa)baletasloltitmhaets,otin,ewmhiacyh ubsyeDdothrrreonu(g1h9o8u7t)).aLseFtmuosd,fDu8r7t.hIefrwdeeafisnseumtheeamliondeealrfllyulximasbpdraerdkiecnteedd nload
star,thentheonlydifferencebetweenthemodelofDorren(1987) e
efeitwh)ervcahluoeo(sse)otofntio,tsuucshetahsethfeeattiumree(osr)poefrthraapnssijtumstininimpuutmo.nePa(rotriaal a(1n9d8t7h)adtoonfotth.iTshweroerfkories,dthiraetcwtceomaspsuarmiseonabsemtwalele-snptohteatwndomDoodrreelns d from
derivativesofthisfunctionarenotprovidedalthoughmaybeeasily h
computedsincemaculaevaluatesthepartialderivativesofA and for linear limb darkened stars yields the error in our model of ttp
Fmod. k aesrrsourmininoguarsmmoadlle-lspisot.Accordingly,onemaywritethattherelative s://ac
(cid:19) (cid:19) a
3MOCDOEMLPARISON TO THE DORREN (1987) (cid:10)Fmod =(cid:19)(cid:19)(cid:19)(cid:19)(cid:19)Fmod,D87−Fmod(c1Fm=odc,D38=7 c4=0;c2=uL)(cid:19)(cid:19)(cid:19)(cid:19)(cid:19). (22) demic.ou
p
.c
3.1 Overview Forsimplicity,weassumethelimbdarkeningofthespotandstar om
areequivalentandsetthespot-starcontrast,fspot,tobezero(ablack /m
n
As discussed earlier in Section 1.4, the most widely used ana- spot). Numerically evaluating (cid:10)Fmod over the domain of interest ra
lyticmodelsforstarspotmodellingcomefromBudding(1977)and revealstheerrorismaximizedwhenβ =α.Therefore,wedefine s/a
Dreoferrretno(c1o9m8p7a).riTsohnetmoothdeelDsoarrereens(s1e9n8ti7a)llmyioddeenltoicnallyafnrodmsohweree-woinl-l (cid:10)FTmmhoaedxf=un(cid:10)ctiFomnod(cid:10)(βFmm=oadxαg)r.owswithbothu1 andα,tendingtozero rticle-a
inforbrevity. whentheybothequalzero,asexpected.ForaSun-likestar(T = b
eff s
ThemaindifferencebetweenourmodelandthatofDorren(1987) 6000K,logg=4.5dex,[M/H]=0),Claret(2011)estimatethatthe tra
c
isthatourderivationassumesthestarspotissmall,i.e.sinα(cid:2)0.1, best-fittinglinearlimbdarkeningcoefficientintheKeplerbandpass t/4
whereasDorren(1987)didnot.Bymakingthisassumption,wehave isuL=0.5733.Onemaynowset(cid:10)Fmmoadxtosomedesiredtolerance 27/3
derived a full non-linear limb darkening treatment for starspots, level (e.g. the noise level of the data) and solve for α, i.e. the /2
4
whereastheDorren(1987)modelislimitedtoasimplelinearlimb maximumspotsizewhichleadstomodelerrorsbelowthetolerance 8
7
darkeninglawonly.Further,ourmodelisamenabletotheinclusion level.WeplotthisfunctioninFig.3(solidline). /1
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Figure3. Left-handpanel:comparisonofthemaximummodelerroroftheDorren(1987)linearlimbdarkeningassumption(dashed)versusthesmall-spot
approximationofthiswork(solid),foraSun-likestar.Asexpected,overtherangeofsmall-spotsizes,themodelpresentedinthisworkisconsiderablymore
accurate.Gridlinesmarkthepointatwhichourmodelisaccurareto50ppmatα=7◦.6.Right-handpanel:sameasleft-handpanel,exceptwezoomouttoa
greaterx-scale.TheDorren(1987)becomesmoreaccuratethanthemodelpresentedinthisworkforspotslargerthan57◦.6(markedwithgridlines).Atthis
point,theerrorinbothmodelsis105mmagandisarguablyunusableineithercase.Notethatthelocationoftheminimuminourmodelerrornear50◦ is
sensitivetothelimbdarkeningcoefficientsused.
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SinceaM =12starhasatypicalnoiseof∼50ppmperlong- the‘truth’,assumedtobeF itselfstartstobecomeerroneousat
Kep mod
cadence measurement (note that most Kepler targets are fainter highα.
than this), we estimate that the maximal error of our small-spot Nevertheless,itisclearthatourmodelismoreaccuratethanthe
approximationisbelowthatofatypicalKeplermeasurementerror Dorren(1987)modelforeven astarspotequivalent tothelargest
forstarspotsofangularradiusα(cid:2)7◦.6.Thiscorrespondstoaspot spot ever detected (α (cid:3) 30◦; Strassmeier 1999). Despite this, we
coverage of (cid:2)1.7 per cent. Note that the modal spot coverage of would urge observers to use a numerical approach for such large
stars in the Kepler sample is (cid:3)1 per cent (Basri et al. 2011). A spots since the model errors are significantly greatly than typical
1.7percentspotcoverageroughlycorrespondstoV =0.83on measurementerrors.Spotsofsizeα(cid:2)10◦shouldbewelldescribed
rng
fig. 4 of Basriet al. (2011), which can be seen to encompass the bytheanalyticmodelpresentedinthiswork.
majorityofperiodicvariables.Giventheconservativeassumptions
ofusingarelativelybright12thmagnitudestar(mostKeplertargets
4 AN EXAMPLE APPLICATION TO κ1 CETI
arefainter),thefactthatweassumeonlyasinglespotisresponsible
fortheentirespotcoverage(makingthespotaslargeaspossible) 4.1 MOSTobservationsofκ1Ceti Do
andthefactthattheerrorderivedisthemaximalerrorratherthan w
n
the typical error, we conclude that the large majority of spotted κ1CetiisarelativelynearbyG5dwarf9.1pcfromtheSolarsystem. lo
stars within the Kepler sample are appropriately modelled by the Thestarisnotableforhavingafairlyrapidrotationalperiodof∼9d ad
e
expressionsinthiswork. atinodnsfoorfκbe1iCngetaiibnr2ig0h0t3Sruenve-lailkeedstthaerparteVsen=ce4o.8f4t.wMoOstSarTspoobtssewrviath- d from
rotationperiodsof8.9and9.3d(Rucinskietal.2004).However, h
3.3 Modelerrorduetoalinearlimbdarkeninglaw thissingledatasetwasinsufficienttouniquelydeterminethelati- ttps
assumption tudesofthespotsandthusthedifferentialrotationcoefficient,κ2 ://a
c
couldnotbemeasured(theauthorsdidnotconsiderthefourth-order a
d
For larger spots, the modelling error becomes larger due to our coefficientκ4). em
small-spot assumption and thus an observer may opt to use the Subsequently,MOST observedκ1 Cetitwomoretimesin2004 ic
Dorren(1987)modelinstead.However,wepointoutthatthismodel and 2005 in order to gather enough data that a unique solution .ou
p
assumesalinearlimb-darkeninglawwhichissomewhatunphysical could be inferred. Indeed, these data, reported by Walker et al. .c
o
in itself. The question therefore arises, at what point is the error (2007),wasarguedbytheauthorstobesufficienttolocateasingle m
in assuming a large spot with linear limb darkening better than minimum. The authors made use of the StarSpotz (Croll et al. /mn
assumingasmallspotwithnon-linearlimbdarkening? 2006a) algorithm to regress the data, which in turn employs the ras
sumWpetihoanv,easasluremaidnygcthaelcsutlaarteidsptherefeecrtrloyrddeusecrtiobetdhebysmaalilnl-esaproltimasb- Bthuedmdiondge(l1o9f7D7)ormreonde1l9f8o7r).stSatrasrpsoptostz(nlootceatthesatththeisgliosbeaqlumivianliemntumto /article
darkeninglaw((cid:10)Fmod).Wemaysimilarlydefineanerrorinassum- usingparalleltemperingandderivesparameterposteriorsusingthe -a
b
ianngoanl-ilnineeaarrlilmawbdarkeninglawwhenthestarisreallydescribedby MCTMheCphteocthonmiqeturey.spanthreedatasets,exhibitdifferentialrotation strac
(cid:10)Fmod,D87 =(cid:19)(cid:19)(cid:19)(cid:19)Fmod,DF8m7o−d Fmod(cid:19)(cid:19)(cid:19)(cid:19). (23) t(ah2nu0ds0s7re)ev.qeFunoirrseptdhoetcssoeonrvseieadrseo3rnaysb,rlet(hrecaondmgaitpnaugmtafatrkiooemnfaoαlrkeafn=foidr5te◦.a9bl5ytetWosta1ol6fk◦.ne7ro6te)otannalldy. t/427/3/24
8
Forastarwiththesameproperties asusedinthepreviousex- ourmodelherebutforanalternativeregressingtechnique. 7
ample(Teff =6000K,logg=4.5dex,[M/H]=0),Claret(2011) /110
−eins0tit.mh0e2a2tKe7e,thp−alet0rt.h0be8a3bn9ed}spt.a-fiWsstteianargelsno{ocan1s-,sliucnm2e,aerca3li,mbclba4}cdka=rskpe{on0ti.nw3g9it9ch9o,ethf0fie.c4si2ae6mn9tes, N4.e2stMeduslatimmpoldinagln(Sekstileldinsgam20p0l4in)giswaitMhoMnUteLTCIaNrlEoSTmethodwhich 5519 by
g
limbdarkeningasthestar,aswasdoneforthepreviousexample. putsthecalculationoftheBayesianevidenceinacentralrole,but u
e
Pfulontcttiinogntohfeβf,unwcetifionnd(cid:10)thFemomd,aDx8i7mfaolresrervoerroalccrueraslizaattβion=sαo/f2α.Tahsuas aislsgoepnreordaullcyecsopnossitdeeriroarbilnyfemreonreceesffiasciaebnyt-tphraondMucCt.MNeCstmedesthaomdpslifnogr st on
wαefIondreFtfiihgne.eS3(cid:10),uwnF-emmliopakdxl,eoDt8li7tmh=ibs(cid:10)dfuaFnrkcmetoindo,iDnn8ga7l(coβoneg=ffiwαcii/teh2n)(cid:10)t.sFcmmoomadxpaustaedfubnyctCiolnaroeft ccshooomsmwpeoudltointghgiactathltehaepBipralyiicmeasptiialoennmsee,vnMitdauetnkiochneeroojeffeta,hPemamorkdeietnhlsoofidnt.r&FeoqLruiiedrxedsalema(p2fla0ec,0t6oin)r 03 April 2
(2000).Thefigurerevealsthattheerrorinassumingasmall-spot of∼100fewerposteriorevaluationsthanthermodynamicintegra- 01
9
is substantially smaller than the error in assuming a linear limb tionwithMCMC.Afulldiscussionofnestedsamplingisgivenin
darkeninglawforα ≤10◦,asoneshouldexpect.Asanexample, Skilling (2004) and Feroz et al. (2008) and for brevity we direct
sunspotstypicallyhaveangularsizes(cid:2)5◦ andforα =5◦ wefind thoseinterestedtotheseworks.
(cid:10)Fmax=10ppmwhereas(cid:10)Fmax =237ppm,i.e.ourmodelis Multimodal nested sampling is an implementation of the tech-
mod mod,D87
morethananorderofmagnitudemoreaccurate.Forspotsofsize nique to efficiently search parameter space under the assumption
α=10◦wefind(cid:10)Fmax=162ppmversus(cid:10)Fmax=981ppm. thatoneormoremodesmayexistinthedata.Ferozetal.(2008,
mod mod
Wefindthatthesignificanterrorinassumingalinearlimbdarken- 2009)describemultimodalnestedsamplingindetail,inparticularin
inglawdoesnotbecomeabetterapproximationthanthesmall-spot regardtotheirpubliclyavailablealgorithmMULTINEST.MULTINEST
modeluntilα>57◦.6,forSun-likelimbdarkening.Afterthispoint, is used by the HEK project (Kipping et al. 2012) to compare the
the error in our model rapidly tends to infinity and becomes un- Bayesianevidenceofaplanetversusplanet-with-moonregression.
tenable. The exact locations of the various turnovers and minima WewillheredemonstratetheuseofMULTINESTwithourstarspot
inFig.3dependuponthelimbdarkeningparametersused.Also, modelfortheκ1CetiMOSTphotometry.Currently,MULTINESTdoes
thereliabilityofthe(cid:10)Fmax functionworsensforlargeα since notmakeuseofthelikelihoodpartialderivativesandsothepartial
mod,D87
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derivativeswereturnedoffinourimplementationofmacula.Since AsrevealedinTable1,theagreementbetweenthederivedsystem
weonlyfitasinglemodelthroughthedata,thereisnouseofthe andspotparametersofκ1Cetiisexcellentwithmarginaldifferences
Bayesianevidencehereandthusweemploytheconstantefficiency betweentheestimates.TheresidualsofthefitsinFigs4–6closely
modeofMULTINESTatatargetefficiencyof1percentwith4000live match those of Walker et al. (2007). This therefore shows that
points. maculacoupledwithMULTINESTiscapableofmatchingtheresults
ofStarSpotz.
Someremaininganomaliesintheresidualsareevidentandone
4.3 Priors
may be tempted to input more spots to fit these out. However,
InordertomakeafaircomparisontotheWalkeretal.(2007)result, Walker et al. (2007) specifically caution against such a process
wemakethesameassumptionsastheoriginalauthors.Accordingly, arguing that the anomalies correlate to moon-light contamination
weassumethesamenumberofspotsforeachdataset,i.e.twospots andotherinstrumentaleffects.
for2003, three spotsfor 2004and two spotsfor 2005. Thespots Althoughitisnotthefocusofthisworktoexploretheinterpa-
areassumedtobenon-evolvingoverthecourseofeachdatasetand rametercorrelationsandoptimalfittingstrategies,weherebriefly D
o
havealifetimewhichensurestheyonlyexistwithinasingledata comment on this issue. Our fits reveal the strongest correlations w
n
set,aswasassumedbyWalkeretal.(2007).Wealsoassumefspot,k= betweenαvaluesassociatedwiththesamedataset,i.e.theampli- loa
0.22forallkandthatthedifferentialrotationprofileisquadraticis tudesofthesignalcomponentsarecorrelated.Wealsofindthatthe d
e
nthaetusrtear(ia.er.eweequfiixvaκle4n=t a0n)d.Ffionlalollwy,alimlinbedaarrlkaewninggovfeorrntehdebspyoutLan=d eexqhuiabtoitrimaluptuerailodco,rthreelasttieollnasr,irnecsluinltaitnigonfraonmdtthheeinudnicveirdtuaainltlyatiitnudthees d from
0.6840.Usingtheseassumptions,wehavethesamenumberoffree differentialrotationdetermination. h
parameters(27)aswasusedbyWalkeretal.(2007). ttps
The 27 parameters are seven reference longitudes, (cid:8)0,k, seven ://ac
reference latitudes, (cid:6) , seven angular radii, α , one equatorial a
0,k 0,k d
rotation period, PEQ, one differential rotation coefficient, κ2, one 5 DISCUSSION em
Rstaetlhlaerrinthcalinnaltaiboenlatnhgeleo,fIfs∗eatsndbtyhrmee=ins1t,ru2m,e3n,twaleofufsseetmter=ms2,0U0m3,. 5.1 Performance ic.ou
p
2004, 2005 for each year. Since each year has unique starspots, To test the speed of macula, we generated 1000 time stamps of .co
we do use the spot labels k = 1, 2, 3, 4, 5, 6, 7 but instead asinglesyntheticinputdatasetforarandomchoiceofthestar’s m/m
use 2003 1,2003 2,2004 1,2004 2,2004 3,2005 1 and 2005 2. parameters.Inallcases,fullnon-linearlimbdarkeningisemployed. n
These labels more clearly identify the spots associated with each Wegenerateasinglespotwithrandomparametersandcallmacula ras
/a
yearandfollowthelabellingnotationofWalkeretal.(2007).We 100000 times to evaluate the typical execution time. Every call rtic
adopttheuniformpriorsforall27parameterswiththesamerange inputtedrandomstarandspotparametersinordertoobtainareliable le
asthatofWalkeretal.(2007). averageexecutiontime.Allsimulationsarerunonasingle-thread -ab
ofaIntelCorei72.9GHzprocessorwithmaculacompileding95 stra
4.4 Results usiWnghtehnemoapctiumliazaitsiocnalflleadg,-oOn3e.may instruct the code whether to ct/42
7
TheglobalmaximumaposteriorimodelfitispresentedinFigs4– computethepartialderivatives.Withderivativesturnedoff,macula /3
6 for the data sets in 2003, 2004 and 2005, respectively. Table 1 requires0.59μsperdatapoint.Turningderivativesonyields6.09μs /24
8
presentstheposteriorsofthebest-fittingmodeandcomparesthem perdatapoint.Therefore,theactofturningonderivativesleadsto 7/1
sidebysidewiththeresultsreportedbyWalkeretal.(2007). a slowing down of the code by a factor of (cid:3)10.3, for a single- 10
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Figure4. Maximumaposterioritwo-spotmodelfittothe2003MOSTdataofκ1Cetiusingtheanalyticmodelpresentedinthiswork.Regressionperformed
usingMULTINESTinconjunctionwiththe2004and2005data.Residualstothefitareoffsetby0.94.Figuremaybedirectlycomparedtofig.4ofWalkeretal.
(2007),whereonecanseeanessentiallyindistinguishableresult.
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Figure5. Maximumaposterioritwo-spotmodelfittothe2004MOSTdataofκ1Cetiusingtheanalyticmodelpresentedinthiswork.Regressionperformed ttp
s
(u2s0in0g7)M,UwLhTIeNrEeSoTnienccaonnjsuenecatinoneswseinthtiathlley2i0n0d3istainndgu2i0sh0a5bdleatrae.sRulets.idualstothefitareoffsetby0.97.Figuremaybedirectlycomparedtofig.5ofWalkeretal. ://ac
a
d
e
m
ic
.o
u
p
.c
o
m
/m
n
ra
s
/a
rtic
le
-a
b
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tra
c
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Figure6. Maximumaposterioritwo-spotmodelfittothe2005MOSTdataofκ1Cetiusingtheanalyticmodelpresentedinthiswork.Regressionperformed by
g
usingMULTINESTinconjunctionwiththe2003and2004data.Residualstothefitareoffsetby0.97.Figuremaybedirectlycomparedtofig.6ofWalkeretal. u
e
(2007),whereonecanseeanessentiallyindistinguishableresult. s
t o
n
spot model. Note that these times include the small overhead of does(althoughSection5.3showshowradialvelocityvariationsare 03
generatingrandomsystemparameterstoo. easilygeneratedfromtheoutputsofmacula).Further,theauthors Ap
Increasingthespot-number,wefindthattheno-derivativescall usedaslower2.33GHzIntelCoreDuoprocessorfortheirbench- ril 2
scaleslinearlywithNS.However,thederivativescallexhibitssu- marktests.Nevertheless,itisclearthatthedifferenceincomputation 01
perlinear,yetsubquadratic,scalingofN1.74,orroughlyN7/4.For speedsisthreeordersofmagnitude,makingmaculaapowerfultool 9
S S
thisscaling,doublingthenumberofspotsincreasestheCPUtime ininverseproblems.
byafactorof3.34.
Duetoitsanalyticnature,maculaperformssignificantlyfaster
thannumericalcodesmadeavailableintheliterature.Forexample, 5.2 Benefits
Boisse, Bonfils & Santos (2012) presented their numerical algo-
Theanalyticmodelforstarspotspresentedherehasseveraladvan-
rithm SOAP and report that generating 10000 time stamps of a
tageswhichwelistbelow.
single synthetic starspot requires less than 40s (but presumably
closetothisvalue).ThisindicatesSOAPrequires∼4msperdata (i) An analytic algorithm for modelling photometric rotational
point,comparedtomaculawhichrequires0.6μsperdatapoint,i.e. modulationduetomultiplecircular,greystarspots,performingthree
maculaisaround6800timesfasterthanSOAP.Thereareseveral orders-of-magnitudefasterthancomparablenumericalcodes.
points which make a direct comparison somewhat unfair though. (ii) Reproduces light curves with a maximum model error an
maculadoesnotcomputeradialvelocityvariations,whereasSOAP order-of-magnitudelessthanthatofthepreviousBudding(1977)
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Table1. ResultsfromfittingtheMOSTdataofκ1Cetiusingthemacula tionangle.Anexampleofthistypeofregressionistheanalysisof
modelpresentedinthisworkandtheMULTINESTalgorithm.Column2shows Walkeretal.(2007)forκ1Ceti,wheretheinclinationanglederived
the68.3percentcrediblerangederivedbyWalkeretal.(2007)(takenfrom fromrotationalmodulationaloneandananalyticmodelforstarspots
column6ofTable3ofthatwork). yieldsaresultfullyconsistentwiththespectroscopicVsinI∗value.
Notethatwealsoreproducethisresultusingmaculainthiswork.
Parameter Walkeretal.(2007) Thiswork
Inaddition,itmaybepossibletomeasurestarspotevolutionusing
PIE∗Q(◦(d)) 85.77.48––86.38.15 8.6708.51+−+−0011....00331189 tchoeoRlloinstateatairros,nmawolhdmiecolhdemuhlapavtlioeoynbeedaenbnaylmymsoaescstuuclsoianm.gmmoancluyloabaserervneodttloimeixtehdibtiot
κ2 0.085–0.096 0.0868+−00..00002255 suchbehaviour. Therealsoexistsevidence forspotsonhotstars,
U2003 1.0003–1.0017 1.00105+−00..0000004307 such as the rapidly rotating B star HD 174648 (Degroote et al.
U2004 1.0129–1.0150 1.01916+−00..0000016113 2011).Indeed,maculawillalsobeapplicableforbrightspotson
αmaxU,220000351(◦) 1.1010.2693––11.10.08561 1.1010.474791+−+−0000....0000006600205584 h(2o0t1m1a)stsoiveexsptlaarisn,soubcsheravsatthioonsesporfolpatoeseOd-tbyypCeaanntdieelalorl&yBB-rtayipthewsataitres Down
α(cid:8)(cid:8)(cid:6)mrrreeeafffx,,,,22220000000033331122((((◦◦◦◦)))) 52.96.NN85––//AA63.41.88 −65311.1.900.3586+−.+−+−70011+−00....11..44113343..8833 miolnadcapmskduafietnrctoebgudmyl.thaCTTeoahδlRicVssooosmnTpat.airlnooyundpoeueu;crshemsipgiphthrofleyotdoruimctshteeieoftrundylseftofoeofrrrmsgtphrinoaecauTetni-δodbVn-absoaaesftdeardonotyboasbttieisomrenvreaavtlsaottparieoimernsips-. loaded from h
(cid:6)ref,20032(◦) 32.9–39.8 35.9+−12..90 It may also be useful in testing whether observed TδVs are con- ttps
α(cid:8)mreafx,,2200004411((◦◦)) 7.7N3–/A8.09 570..9523+−+−0000..11..990078 soibsltaetnetnwesisthwsittahrspproectsesvseiorsnus(Csaormteer&othWerinhnyp2o0t1h0e)s.is,e.g.planetary ://acad
α(cid:6)mreafx,,2200004412((◦◦)) 149..434––1167.8.31 1173.1.92+−+−2200....009846 emic.ou
(cid:8)ref,20042(◦) N/A 154.8+−11..00 5.3.2 Astrophysicaldetrending p.c
(cid:6)ref,20042(◦) −47.6to−43.2 −46.9+−11..22 Duetotheanalyticnatureofmacula,thecodeisquicktoexecute om
αmax,20043(◦) 11.60–13.53 14.26+−00..4581 andthereforemanyfindusesinastrophysicaldetrendingofphotom- /mn
α(cid:8)(cid:8)(cid:6)(cid:6)mrrrreeeeaffffx,,,,,2222200000000004545531311(((((◦◦◦◦◦))))) 9.752454..NN-94−––//AA761820...4128 −7−9691.30.9622.9306.+−.1+−+−6+−0011+−00....1132..771157..1722..8121 etnadmotearistlyarer,e.leasmnFmrucdocoohirhvndeefgeaoxlsisratnhmrostiortsrpratauhlnretmoasi,oirtetamtnshnt,aioetolaonnPlmriacDtloarfiCedmdnlu-tedMoeltasdratiAuiinblolPgeuandttmaidoplmgaunryeoeosirdbsteioeterhrlvlesmueitqnsauegtorhidsfrepo,eKofadaetn.saspt.WarlkoeIsnntrhproihoiplwsyspetsnhdrifpyecotossarriiallmcgynasnnisilneogiltygd--, ras/article-abstrac
αmax,20052(◦) 7.94–8.52 8.35+−00..1175 groundedmodel,suchasmacula,offersaviablealternativedueto t/42
(cid:8)ref,20052(◦) N/A 64.9+−11..55 itsfastexecutiontime. 7/3
(cid:6)ref,20052(◦) 42.9–50.1 47.3+−11..89 /248
7
/1
1
5.3.3 Radialvelocityvariationsduetostarspots 0
and Dorren (1987) for a Sun-like non-linear limb darkened star 5
5
observedintheKeplerbandpass,forspotsofangularsize(cid:2)10◦. Webrieflyremarkthatmaculamaybeusedtopredictradialvelocity 19
(iii) Modelaccountsforspotcontrast,non-linearlimbdarkening, variationsduetostarspotsviatheFF(cid:6)methoddescribedinAigrain, by
differentialrotationandstarspotevolution. Pont&Zucker(2012).Here,theauthorsproposethatradialvelocity gu
e
as(wive)llIanscMludbelsenbdaesedlilnigehntodrimluatiloiznaftaiocntoprsartaomaiedteirnsKfoerplMerdaantaalysesitss., vSapreicaitfiiocnasllcya,nthbeeaurethlioarbslyarpgrueedtihctaetdthferoflmuxflmuxulvtaiprilaietidobnys(itFs)daelroivnae-. st on
(v) ComputesTδVsduetounoccultedspots. tive in time reveals the radial velocity variations. macula returns 03
to(avlil)mPaordteiallpdaerraivmaetitveerssoafntdhetimmoed,ealnflduxmiasyprboevitduerdnewditohnr/eosfpfeacst AboptpheFndmioxdFan).d∂Fmod/∂t forallinputtimes(derivationprovidedin April 2
desired(seeAppendicesDandEforderivations). 01
(vii) CodeisfreelyavailableasaFORTRANroutine,macula,lo- 9
catedatwww.cfa.harvard.edu/∼dkipping/macula.html.
5.3.4 Correctingtransmissionspectra
Ifaplanettransitsacrossastar,theatmosphereoftheplanetcan
5.3 Potentialapplications reveal chromatic variations in the transit depth due to molecular
absorption.Inthisway,transitmeasurementscanrevealthe‘trans-
5.3.1 Rotationalmodulationmeasurements
missionspectrum’ofanexoplanet.Ifthehoststarhasunocculted
We foresee several possible applications of macula. First, mea- starspots,onewouldexpecttoseechromaticvariationsinthedepths
suring the rotational modulation of variable stars may be used to purely from spots too, introducing a confounding signal. Further,
determinetherotationperiod,whichmayinturnconstraintheages transitdepthmeasurementsareoftenscatteredbothinwavelength
ofstarswithgyrochronology(Barnes2009).Starsmonitoredwith andintimemeaningthatrotationalmodulationcanalsointroduce
highsignal-to-noiseratiocontinuousphotometry,suchasthatfrom spurious depth variations. Correcting for starspots is therefore a
Kepler,mayalsorevealdifferentialrotationandthestellarinclina- majorchallengeinstudyingtheatmospheresofextrasolarplanets.
(cid:4)C 2012TheAuthor,MNRAS427,2487–2511
MonthlyNoticesoftheRoyalAstronomicalSociety(cid:4)C 2012RAS