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PlanetHuntingwithPython

RichardP.Nelson,GuillemAnglada-Escudé&GavinColemanSchoolofPhysics&Astronomy,QueenMaryUniversityofLondon

AbstractNASA’sKeplerspacecraftwaslaunchedin2009andspentapproximately4yearsstaringat150,000stars,searchingforplanetsorbitingaroundthemusingthetransitdetectionmethod.Themissionresultedinthediscoveryof4717exoplanetcandidates,ofwhich2303havebeenconfirmedasbonafideplanets,andanumberofwhicharefoundtoresideinmulti-planetsystems.Morerecently,NASA’sTESSplanethuntingmissionwaslaunchedandisnowtakingdata.Duringyourresearch,youwilllearntowritecomputerprogrammesinthePythonprogramminglanguage,andyouwilluseyournewlydevelopedskillstowriteroutinestoanalysedatafromtheKeplerandTESSmissionsusingsimplifiedalgorithms.YouwillanalyselightcurvesthathavebeendownloadedfromtheKeplerandTESSspacecraft,anddeterminethekeyphysicalparametersoftheplanetarysystemsincludingorbitalperiods,orbitalinclinationstothelineofsight,orbitalsemi-majoraxes(i.e.thedistancebetweenthestarandtheplanet),andthephysicalradiioftheplanets.Youwilllearnaboutfittingmodelstodata,andtechniquesforoptimisingtheaccuracyofthesemodels.Basedonyouranalysis,youwillbeabletodrawconclusionsabouttheobservedpopulationofexoplanets,includingwhetherornottheylookliketheplanetsinourSolarSystems,andexplorehowdynamicalinteractionsbetweenplanetsinmulti-planetsystemscanbedetectedinthedata.IntroductionThefirstextrasolarplanetdiscoveredorbitingaSun-likestarwas51Pegasib(Mayor&Queloz1995),andthisdiscoveryledtoMichelMayorandDidierQuelozbeingawardedtheNobelPrizeforPhysicsin2019.Duringthe24yearssincethisground-breakingdiscovery,thetotalnumberofconfirmedextrasolarplanetshasincreasedto4,0731withnewdiscoveriesbeingmadeonanalmostdailybasis.WenowknowthatthepopulationofexoplanetsinourGalaxyisverydiverse,andthatourSolarSystemdoesnotprovideanexampleofthemosttypicalarchitecture.ThelaunchoftheKeplerspacecraftinMarch2009heraldedadramaticimprovementinourknowledgeaboutexoplanets(Boruckietal2010).KeplerwaslaunchedintoanEarth-trailingheliocentricorbit,whichallowedittostare

1Numbertakenfromexoplanetarchive.ipac.caltech.eduinOctober2019.

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continuouslyatoneregionoftheskyforaperiodofapproximately4years2.Themissionusedthetransitdetectiontechniquetofindplanets.Thismethodlooksfortheperiodicdimmingofthelightfromastarasseenbyanobserverwhenaplanetpassesinfrontofthestarduringitsorbit(seeFig.1).

Figure1:Thisdiagramillustratesthephysicalprinciplesbehindthetransitdetectionmethod.Whenanorbitingplanetpassesinfrontofitshoststar,someofthelightfromthestarisblockedoutandthestarappearstodimslightly.Detectingtheperiodicdimmingofadistantstarmaythereforeindicatethepresenceofaplanet.

Forgeometricalreasons,mostplanetaryorbitsarounddistantstarsdonotcausetheplanettopassinfrontofthestarwhenobservedbytheKeplerspacecraft.Forarandomlyorientatedorbittheprobabilityofobservingatransitistypicallyabout1%,soforthisreasontheKeplerspacecraftmonitoredthebrightnessofapproximately150,000starsforthedurationofthemission,resultinginthediscoveryof4717planetcandidates,ofwhich2303havebeenconfirmedtobegenuineexoplanets.Notethattherearenumerouswaysinwhichdifferentastrophysicalphenomenacanmimicatransitingplanet,hencetheneedforadditionalobservationsthatareabletoconfirmtheplanetarynatureoftheKeplerplanetcandidatesystems.TheKeplerspacecraftproduceddataintheformoflightcurvesforeachoftheobservedstars–datafilesthatlistthetimeofobservationandthebrightnessofthestaroveranextendedperiodoftime.Theselightcurveswerethenprocessedtoremoveartefactsandotherunwantedfeaturesfromthedata(aprocessknownasdetrending)priortothedatabeingmadeavailableforscientiststoanalyse.SeeFig.2foranexampleofadetrendedlightcurve.Youwillusethesedetrendedlightcurvesinyouranalysis,andoneofthethingsthatyouwilllearnishowtodownloadthedatafilesforindividualsystemsfromtheKeplerdataarchivesothatyoucanchoosewhichsystemstoanalyse.2MoreinformationabouttheKeplermissioncanbefoundatkepler.nasa.gov

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Figure2.ThetoppanelshowsadetrendedlightcurveforthesystemKepler7b.They-axisshowsthefluxoflightfromthestarinarbitraryunitsandthex-axisshowsthetime(inunitscalledJuliandays).Theperiodicdipsinthelightcurvecorrespondtowhentheplanetispassinginfrontofthestar.Thelowerpanelshowswhatiscalledthephase-foldedlightcurve(blackcrosses)withamodelfittedtoit(redcurve).Thephase–foldedcurvecontainsallthedatainthetoppanel,butplottedsothatthemid-pointofeachtransitoccursattimet=0.

Aprimaryaimofthisprojectistoprovidehands-onexperienceofprogramminginPythonwithinthecontextofastronomicaldataanalysis.Pythonisbecomingacommonlyusedprogramminglanguageinmanyareasoflife,includinginthescientificworldwhereitisusedfordataanalysisandmathematicalmodelling,andinschoolsanduniversitieswhereitisbecomingthelanguageofchoicewhenteachingcomputerprogramming.Python’sincreasingpopularityarisesforanumberofreasons:itisfreetouse;itrunsunderalloperatingsystems(Windows,MacOSX,Linux);itsbasicstructureandsyntaxissimilartomanyotherlanguages;itcontainstheabilitytoperformmathematicalcalculationsandtoplotdataingraphicalformwithinaself-containedprogrammingpackage.Laterinthisdocumentyouwillbeprovidedwithasequenceofprogrammingtasksofincreasingcomplexitytohelpyoudevelopyourprogrammingskillsinincrementalstages.

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ObtainingphysicalinformationfromtransitdataFromFig.1wecanseethatareasonablemodelforaplanetarytransitistoassumethatanopaquecirculardiscofradiusRpmovesacrossauniformlyluminouscirculardiscofradiusRs,blockingsomefractionofthelightdetectedbyanobserver.Here,RpandRsaretheradiioftheplanetandstar,respectively.ObtainingtheplanetradiusfromthelightcurveExercise:Usingthesimplemodeldescribedabove,andshowninFig.1,foraplanetarytransit,obtainanexpressionforthechangeintheobservedluminosityofastarofradiusRswhenaplanetofradiusRptransitsinfrontofit.Inparticular,obtainanexpressionforΔL/LsintermsofRsandRp,whereΔL=Ls-LTandLsdenotesthestellarluminositywhentheplanetisnottransiting,andLTdenotestheobservedluminositywhentheplanetisatthemid-transitpoint.[Hint.Thinkaboutwhatfractionoftheemittedlightfromthestarisblockedoutbytheopaquedisc(i.e.theplanet)passinginfrontofit.ThesurfaceareaofthestarisπRs2andtheareaoftheplanetisπRp2.Theluminosityorbrightnessofauniformlyemittingdisc(i.e.thestar)canbewrittenasthefluxofradiationbeingemittedfromthesurfacemultipliedbythesurfaceareaofthedisc.Thefluxisdefinedtobetheamountofradiationemittedbythediscperunitareaperunittime.]Solution:Yourteacherhasthesolution.Trytoworkitoutforyourselfbeforeseekingassistance.Planet’sorbitalperiodObtainingtheplanet’sorbitalperiodfromatransitlightcurveiseasywhenthereisasingleplanettransitingastar.Itissimplythetimethatelapsesbetweensuccessivedipsinthelightcurve.WhenyoustarttoworkwiththeKeplerdata,youwillnoticethatthetimeslistedinthelightcurvedatafiles,andonthewebsitethathoststhedataarchive,areinstrangeunits–“BarycentricJulianDayminusaconstantoffset”(abbreviatedtoeitherBJDor[BJD–2454833]).Forcomplicatedhistoricalreasons,astronomersmeasuretimefromnoononthe1stJanuary4714BC.ThisdayisknownasJulianDayzero,andallsubsequentdaysareknownasJulianDay1,2,3,4,5,…Thenumber2,454,833mentionedabovecorrespondstonoonon1stJanuary2009andissubtractedfromtheJuliandateinordertomakethenumbersmoremanageable.HencethetimeslistedinKeplerlightcurvesarethenumberofdayssincenoonon1stJanuary2009.

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Orbitalsemi-majoraxisIfweassumethatthetransitingplanetsareincircularorbitaroundtheirstars,thenwecanobtainanexpressionforthesemi-majoraxisoftheorbit(thedistancebetweenthestarandplanetduringitsorbit)intermsofthemassofthestar(whichcanbeobtainedfromtheKeplerdataarchive)andtheorbitalperiod(whichcanbeobtainedfromthetransitlightcurve).Thisisgivenasanexercisebelowwithstepbystephints.Exercise:Whenaplanetisincircularorbitaroundastar,wecansaythereisaforcebalancebetweengravityandthecentrifugalforce.Hence,westartbyequatingthegravitationalforceactingbetweenthestarandplanettothecentrifugalforceassociatedwiththecircularorbitalmotionoftheplanet

𝐺𝑀$𝑀%

𝑎' = 𝑀%𝑣'

𝑎 wherevdenotesthevelocityofthecircularorbitandaisthesemi-majoraxis.MsisthemassofthestarandMpisthemassoftheplanet,andGisNewton’sgravitationalconstant.Manipulatethisequationsothatthesemi-majoraxis,𝑎,isexpressedintermsofeverythingelse.Yourteacherhasthesolutionsoaskifyouneedassistance.Thevelocity,𝑣,atwhichtheplanetmovesarounditscircularorbitisgivenbythedistancearoundthecircle(oritscircumference),C,dividedbythetimetaken,P,whereinourcasePistheorbitalperiod.(Remember:Velocityisdistancetravelleddividedbythetimetaken).Obtainanexpressionforvintermsofthesemi-majoraxis,𝑎,andtheorbitalperiodP.Yourteacherhasthesolutionsoaskifyouneedassistance.Nowcombineyourexpressionsforthesemi-majoraxis,𝑎,andthevelocity,v,togetanexpressionfor𝑎intermsof𝑀$ ,P,Gandπ.Yourteacherhasthesolutionsoaskifyouneedassistance.Toworkoutthevaluesofthesemi-majoraxis,𝑎,foreachexoplanetyouwillstudyinthisproject,youwillneedthemassofthestar,Ms.Wehaveprovidedthisinapdffile(Stellar_Mass_Radius.pdf)onyourmemorystickforeachofthesystemsyouwillanalyse.Thisfilealsocontainstheradiusofthestar.PleasenotethatthevaluesofMsandRsareexpressedinunitsoftheSolarmassandSolarradius,wheretheseareMSun=1.989x1030kgandRSun=7x108m.NotethatG=6.67x10-11m3kg-1s-2intheseunits,andthesemi-majoraxisyouwillobtainwillbeinmetres.YoumaywanttoconvertthisintoAstronomicalUnits(themeandistancebetweentheEarthandSun),where1AU=1.5x1011m.

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DeterminingthetransitimpactparameterFigure33showsthatthetransitimpactparameter,denotedbyb,isameasureofhowclosetothecentreofthestellardisctheplanetpassesatthemidpointofthetransit.bismeasuredinunitsofthestellarradius,R*.Notethatwerefertotheangleiastheinclinationoftheorbit,suchthatananglei=90ocorrespondstoaplanetthatpassesacrossthecentreofthestellardisc.Figure3showsthattheimpactparameter,b,thesemi-majoraxis,a,andtheinclinationoftheorbit,i,arerelatedbysimpletrigonometry.Figure3:Theobserverislocatedontheleftofthediagramandislookingrighttowardsthestar.Theplanetisassumedtobeatthemidpointofthetransit,anditsprojecteddistanceabovethecentreofthestarisgivenbybxR*.

Figure4showsafaceonviewofthepaththattheplanettakesacrossthestellardiscduringatransit,anddemonstratesthetrigonometricrelationdescribedabove.ThelengthofthechordthatcorrespondstothepathtakenbytheplanetacrossthestellardisccanbedeterminedfromPythagoras’theorem.

Figure4:Face-onviewofthepathtakenbyaplanetasitcrossesthestellardiscduringatransit.

Theequationforthelengthofthechordis

2𝑙 = 2,-𝑅∗ + 𝑅%1'−(𝑏𝑅∗)'

3ThediagramsusedinFigures3,4,and5wereobtainedfromhttps://www.paulanthonywilson.com/exoplanets/exoplanet-detection-techniques/the-exoplanet-transit-method/

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Thediagrambelowgivesa3-dimensionalviewofthegeometryoftheorbitofatransitingplanet.

Figure5:3-dimensionalviewofthepathtakenbyatransitingplanetonitsorbit.

Fromthetriangledrawnintheorbitalplane,wecanseethatsin 9:'; = <

=(note

thatthediagramexaggerateshowclosetheorbitistothestar.Inalmostallcases,thedistancebetweenthecentreofthestarandthemidpointbetweenAandBiswellapproximatedbythesemi-majoraxisa).Assumingthattheplanetisonacircularorbit,thedurationofthetransitisgivenbytheexpression

𝑇?@A = 𝑃𝛼2𝜋

Whencombiningthiswiththeequationsabove,weobtaintheexpression

𝑇?@A = 𝑃𝜋 sin

EF

⎝

⎛,-𝑅∗ + 𝑅%1

'−(𝑏𝑅∗)'

𝑎⎠

⎞

Fromthisyoucanrearrangefor𝑏𝑅∗anddeterminethetransitimpactparameterbasedonmeasuringthedurationofthetransit,Tdur,fromthelightcurve.ExerciseObtainanexpressionfor𝑏𝑅∗fromtheaboveequationforTdur.UsingFigure4,obtainanexpressionfortheorbitalinclinationanglei.SolutionYourteacherhasthesolution.Trytoworkitoutforyourselfbeforeseekingassistance.

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Summary:Wenowhavesufficientinformationtobeabletoobtainthefollowinginformationfromthelightcurvesandstellarparameters(stellarmassandstellarradius)providedbytheKeplerdataarchiveforeachplanet-hostingstar:

• theplanet'sradius• theplanet’sorbitalperiod• theorbitalsemi-majoraxis• theimpactparameter• theorbitalinclination

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WritingprogrammesinPythonWenowprovideanintroductiontowritingcomputercodeinPython,withafocusonwhatyouwillneedtoanalyseKeplerdata.First,youwillbeintroducedtovariousprogrammingconceptsandPythoncommands,andthenyouwillbeprovidedwithasequenceofexercisesthatallowyoutoputthisnewinformationintopractice.Inadditiontofollowingwhatiswrittenbelow,youmayfinditusefultolookatsomeon-linePythontutorials.Ifyoutype“pythontutorial”intogoogle,thenyou’llfindliterallyhundredsofwebsitesthatofferinformationaboutprogramminginpython.Onethatwefoundusefulingettinganoverviewwhendevelopingthisprojectishttp://www.python-course.eu/course.phpAsitethatprovidesdocumentationonmanyaspectsofpythonishttp://scipy.orgIfyougetstuckandwanttoknowhowtodosomethinginPython,thengoogle“HowtodosomethinginPython”,wheresomethingcouldbe“plotagraph”,“readdataintoanarray”,oranyotherissueyou’rehavingproblemswith.TherearemanyversionsofPythonavailabletodownloadandinstallonyourcomputer.WestronglyrecommendusingAnaconda,whichmaybedownloadedfromthissite:https://www.anaconda.com/download/-windowsOnceyouhaveAnacondapythoninstalled,werecommendusingtheSpyderdeveloperenvironmentwhichwillappearasfollowswhenyoulaunchtheAnaconda-Navigator.ClickonthistolaunchSpyder.

PLEASENOTETHATPYTHONCOMESINTWODISTINCTFLAVOURS:PYTHON2andPYTHON3.WEWILLONLYUSETHEVERSIONPYTHON2.7.

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GettingstartedYouhavebeengivenaUSBmemorystickcontainingafolderPlanetHuntingWithPython2019.Beforegettingstartedwiththeprogrammingexamplesbelow,youareadvisedtocopythisPlanetHuntingWithPython2019folderandallofitscontentsontoyourcomputer.ThisfolderwillbewhereyoushouldsaveallofthePythonscriptsthatyoucreate.YouwillcreateandrunthepythonscriptsusingSpyder.Whenthishasbeenlaunchedyoushouldseethewindowshownbelow.

ThisiswhereyoutypeyourPythoncommands

ClickthegreenplaybuttontorunaPythonscript

ThisistheconsolewheretheoutputsfromyourPythonscriptsareprinted.Ifyourscriptrequiresinputfromtheuser,thenthisiswherethatinputshouldbetyped.

UsetheFilemenuatthetopofthescreentosaveyourscriptstothePlanetHuntingWithPython2019directory.ChooseSaveas…andnavigatetotherequireddirectory

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CreatingvariablesandarraysVariablesarenames(alwaysbeginningwithanupper-orlower-caseletter)thatyoucanassignavalueto.Youshouldthinkofavariableasalocationinthecomputer’smemorywhereavalueisstored(notethatthevaluecanbeanumber(realorinteger)orastring(i.e.aletteroraword)).Onceavariablehasbeenassignedaparticularvalue,itwillstorethatvalueuntilanothervalueisassignedtoit.Anexampleofassigningthevalue10toavariablesothatitisstoredasdecimalnumberisshownbelow(notethedecimalpoint):var=10.Arraysaredatastructuresthatstoreacollectionofvaluesthatarenormallyofthesametype(i.e.realnumbers,integersorstrings).Thiscollectionofnumberswilloftenhavesomerelationtoeachother.AnexamplerelevanttothisprojectisthatanarraycouldstoreallofthetimesassociatedwithaKeplerlightcurve,andanotherarraycouldstorethecorrespondingobservedluminosities/fluxesofthestar.Assigningvariablestoanarraycanbedoneasfollows:x=[1.,2.,3.,4.,5.]y=[1.,4.,9.,16.,25.]Thepreviousexampleswereofone-dimensionalarrays.Pythonalsoincludestwo-orthree-dimensionalarrays.Althoughwewon’tusethesehigher-dimensionalarraysinthisproject,anexampleofassignmenttoatwo-dimensionalarraywouldlooklikethis.z=[[1,2],[3,4]]Herethefirsttwoelementsofthearrayformthefirstcolumnandthesecondtwoelementsformthesecondcolumn,asshownbelow.Wecanthinkofarraysashavingsimilarstructurestomatricesinmathematics.Toaccessaspecificelementofaone-dimensionalarray,youwouldtypex[0]whichwouldreturnthevalue“1.”intheexampleabove,sincetheindexingofarraysinpythonstartsatzero.Similarly,accessinganelementofatwo-dimensionalarraycanbeachievedasfollows:z[0,0]whichwouldaccesstheelementinthefirstcolumnandfirstrow,givingavalueof1intheaboveexample.Aswewillseelater,arrayscanbecreatedautomaticallyinPythonbyreadingmultiplevaluesintoavariablenamefromadatafile,andyouwilldothisfrequentlyaspartofthisproject.

1324

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AddingcommentstoprogrammesIfyouwanttoaddacommenttoyourscriptssothatyouorotherpeoplecanunderstandthemlater,youcanusethe#symbol.Thisissomethingthatwewouldstronglyencourageyoutodoasyouwriteyourprogrammes.Pythonignoreseverythingonalinepastthe#symbol.ThecommentcanbeonthesamelineasaPythoncommand:var=10#ThisisacommentHere,Pythononlyexecutesthevar=10command,andignores‘Thisisacomment’.Alternatively,thecommentcanappearonaseparateline:#Thisisacommentvar=10PrintingtotheterminalInPythonyoucanprinttexttotheterminal.Thisisdoneusingtheprintcommandasfollows:print‘Hello’Theprintcommandcanalsoprintthevaluesofavariableoranarray.Ifwehaveavariablecalled‘var’whichisassignedthevalue10,thenthefollowingPythoncodecommandwillinitialisethevariableandprintitsvaluetotheterminal:var=10printvarNotethatforvariablesandarraysnoquotationmarksarerequired.Quotationmarksareonlyrequiredforprintingstrings.Ifwewanttoprintmultiplevariablesorpiecesofinformationonthesameline,weusethefollowingprintcommand:var=10print‘Thevariablevarisequalto:‘,varThe‘comma’allowsmultiplepiecesofinformationtobeprintedusingasingleprintcommandasshowninthefigurebelow

Pythoncommands

Outputprintedinconsole

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ForloopsForloopsareusefulforrunningthesameblockofcoderepeatedly.Theyhaveaspecificsyntax(writtenformat).Belowisanexampleofaforloop:count=0foriinrange(0,50):

count=count+1print‘count=‘,count

Notethattheindentation(eitheraspaceortab)isrequiredforPythontoknowwhichcommandsformpartoftheloop.Wecallthecommandsthatareexecutedrepeatedlythebodyoftheloop.Allcommandswithinthebodyoftheloopmusthavethesameindentation.Theaboveloopwoulditerate50times(Pythonstartswiththeloopcounter,i,havingthevaluezero,sothecodewilliterate50timesfromzero),adding1eachtimetothevariablecountandthenprintingthevalueofcounttotheterminal.ifstatementsforconditionalexecutionofcodeifstatementsareveryuseful.Theyuseconditionalstatementsthatevaluatetoeithertrueorfalsetodeterminewhichpiecesofcodetorun.Theyhaveaspecificsyntaxthatmustbefollowedforthecodetoexecuteproperly.Belowisanexampleofanifstatement:var=5ifvar==10: print‘true’else: print‘false’Since5doesnotequal10,pythonwillskiptheprint‘true’statementandexecutethecodeassociatedwiththeelsecondition,whichinthiscasewillcauseittoprint‘false’totheterminal.Itispossibletoinsert‘elseif’conditions,whichcausethescripttoonlyruntheblockofcodeassociatedwiththefirstconditionalstatementthatitdeterminestobetrue.Seetheexamplebelow:var=5ifvar==10: print‘varisequalto10’elifvar==5: #“elif”represents“elseif”here print‘varisequalto5’else: print‘varisnotequalto5or10’Withvarbeingequalto5,thecodewillnowprint‘varisequalto5’becausethesecondifconditionissatisfied.Withintheifstatement,youcanaddasmany‘elif’statementsasyoulike,andtheelsestatementshouldalwayscomeattheend.

Checkforthecolonhere

CHECK:Checkforthecolonhere

CHECK:Checkfortheindentationshere

Checktheindentationshere.Spyderwillnormallydotheindentationforyou

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NestedloopsNestedloopsareusedwhenrepeatedlyexecutingablockofPythoncodethatdependsonthevaluesoftwoindependentvariables(thesevariablescanbethevaluesoftheloopcountersthemselves,orothervariablesthataredefinedandmodifiedwithinthebodiesoftheloops).Notethatoneneedstotakecareoftheindentationswhenimplementingnestedforloops.Anexampleofmultipleforloopsisgivenbelow:foriinrange(0,3): forjinrange(0,3): print‘i=’,i,’j=’,jTheabovecodewillrunthroughthewholesequenceofthe‘j’loopduringasingleiterationofthe‘i'loop,printingthevaluesofiandjtotheterminal,asshowninthediagrambelow:

Pythoncodefrompreviousexample

Outputfrompreviousexample

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CombiningforloopsandifstatementsOneneedstobecarefulabouttheindentationwhencombiningloopsandifstatements.Anexampleisshownbelow:foriinrange(0,10): ifi==4: printi elifi==8: printiThecodeherewouldstartrunningtheforloopstartingati=0,beforecheckingtheconditionalifandelifstatementsandrunningtheappropriatepieceofcode.Itwouldthenreturntotheforloopandcontinuetothenextiterationofi,repeatingthissequence10times.AskingtheusertoinputinformationattheterminalItissometimesnecessaryforascripttoasktheusertosupplyinformationthatisrequiredforitscontinuedexecution.Forexample,hereisascriptfordeterminingifthereisenoughspaceinaroomforthenumberofpeoplerequired.space=30number_of_people=input(‘Howmanypeoplewillbeintheroom?’)ifnumber_of_people<=space: print‘Thereisenoughspace’else: print‘WARNING:Notenoughspace’HerePythonwillstopatthelinecontainingtheinputcommanduntiltheuserrespondsbytypingintherelevantvalueattheconsole,asshowninthescreenshotbelow.Thisvalueisthenstoredinthevariablenumber_of_people.

Thecodepromptstheusertoinputinformationhere.Clickontheconsole,typetherequiredinformationandpresstheenterkey

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Readingdataintoarraysfromatextfile–introducingnumpyTheKeplerdatathatyouwilluseinyourprojectwillbecontainedintextfiles,andyourprogrammeswillneedtoreadthisdatabeforeanalysingit.OnefeatureofPythonthatwehavenotyetdiscussedisthefactthatmanyofthefunctionsthatPythonoffersarecontainedinlibraries/packagesthatneedtobeloadedduringaPythonsessionbeforetheycanbeused.Whenwritingascript,itisagoodideatoloadtherequiredlibrariesatthebeginningofthescript.Onelibrarythatdealswithmathematicalfunctionsandnumericaldataiscallednumpy.Theusefulfeaturethatwewillintroducehereisitsloadtextcapability,butitalsoprovidesalargenumberofmathematicalfunctionssuchassine,cosine,sqrt,etc.Belowisascriptthatwillreadindatafromthefile‘xy.txt’locatedinthedirectoryDATAwhichisinthePlanetHuntingWithPython2019folder,andthenprintthedataouttotheterminal.importnumpyasnp #Wecouldjustuse‘importnumpy’,butassigningitasnp #allowsforeasiertypinglaterinthescript.x,y=np.loadtxt(‘DATA/xy.txt’,unpack=True)#Specifythatloadtxtispartof #thenumpylibrary->np.loadtxtforiinrange(0,len(x)):#weuselen(x)hereaswedonotknowthelengthofx.#Thecommandlen(x)determinesthenumberof#elementsinthearrayxwhichhasbeencreatedwhen#readinginthedatafromthefileDATA/xy.txt

printx[i],y[i]#Sincexandyarearrays,the[i]arerequiredtoaccessthe#specificelementsofthatarray.

Intheaboveexamplethedataisreadinandprintedoutlinebyline.Nowthatwehavethexandyarraysloadedin,let’sconsiderwhatwewoulddoifwewantedtocreateanewarrayzconsistingoftheelementsofthetwoexistingarraysmultipliedtogether(i.e.x[i]*y[i]forallvaluesofi).Anexampleisshownbelow:z=[] #initialisethearrayzwhichisemptyforiinrange(0,len(x)): #iterateforloopoverallelementsofx z.append(x[i]*y[i]) #Thiswillcalculatethevalueofx[i]*y[i]andappend#ittothelistofvaluesalreadycontainedinthe#arrayzprintz[i]

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PlottingdataItisalmostalwaysusefultoplotyourdata,andforthetaskofanalysingKeplerlightcurvesitisessential.InordertoplotdatainPythonweneedtoloadthelibrarymatplotlib.pyplot,whichprovidestheplottingfunctionsthatweneed.Belowisascriptthatwillloadinthedatafrom‘xy.txt’andthenplotit,asshownbythefigure.importnumpyasnpimportmatplotlib.pyplotasplt#Similartothenumpyexampleabove,#useplttomakethingseasierx,y=np.loadtxt(‘DATA/xy.txt’,unpack=True)plt.plot(x,y) #Thisplotsthedatatothecurrentfigure.Moreoptionscanbe#includedinthiscommandtodefinetheline-width,colour,#markersymbol,etc.Forexample,plt.plot(x,y,‘r.’)wouldplotthe#datausingreddotsasmarkers.plt.show()#Thisdrawsthefigureonthescreen,asshownbelow

Importantnote:WhengeneratingaplotinSpyder,bydefaultitwillembedtheplotinthesmallconsolewindow.Thisisnotwhatwewantbecauseweneedtobeabletomanipulatetheplotswegenerateinthisproject,andthatisnotpossiblewhentheplotisembeddedintheconsolewindow.ToallowSpydertogenerateaseparateplotwindowthatwecanmanipulatepleasedothefollowingsteps(noteitshouldonlybenecessarytodothisonceasSpyderwillrememberyou’vechangedthesettings):1).ClickonthespannersymbolthatopenstheSpyderpreferencesmenu2).ClickonIPythonconsole3).ClickontheGraphicstab4).UnderGraphicsbackendselectAutomatic,andthepressApplyandOK5).NowrestartSpyderandruntheplottingscript

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Otherusefulcommandsnp.sum(x)#givesthesumofallarrayelementsnp.mean(x)#givesthemeanofallarrayelementsifcount%10==0#the%symbolgivestheremainderwhencountisdividedby#10inthisexample.Sometimeswecallthisthemodulusof#dividing10intocount.Theremaindervalueisthen#comparedwith0todetermineiftheconditionintheif#statementistrueplt.xlabel(‘String’)#putsanaxislabelonthexaxis.Changexlabeltoylabelto#labeltheyaxislen(x)#givesthenumberofelementsinarrayxINT(x)#givesjusttheintegerpartofthenumberstoredinx.Couldalsoreplace#xwithanexpressionsuchasINT(x/y)wherexandyaretwonumbers.np.sin(x)#Calculatethesineofx(assumedtobeinradians).np.arcsin(x)#Calculatethearcsineofx.x**2#Calculatesxsquared.Notethatx**ycalculatesxtothepowerofy,where#yisanynumbernp.sqrt(x)#CalculatessquarerootofxPythonprogrammingexercisesBelowwehaveprovided11exercisesthatputintopracticetheprogrammingconceptsandcommandsdiscussedearlierinthisdocument.Yourteacherandvisitingprojectteachingassistant(ifpresent)havebeenprovidedwithsamplecodesthatsolveeachtask,sofeelfreetoaskforhintsifyougetstuck.

1. Writeaprogramthatoutputs‘HelloWorld’totheterminal

2. Writeaprogramthatoutputs‘HelloWorld’50timesusingaforloop

3. Writeaprogramwithaforloopthatiterates50timesandoutputs‘Hello

World’afterevery5iterations.

4. Writeaprogramwithaforloopwith50iterationsandoutputs‘Hello

World’everyoddvalueoftheloopcounterand,and‘GoodbyeWorld’

everyevenvalue.

5. Readinthetextfile‘xy.txt’andprintthedatatotheterminalin2columns

6. Plotthedatain‘xy.txt’withbluedots

7. Plotthedatain‘xy.txt’withreadcrosses

8. Plotthedatain‘xy.txt’asalineplot

9. Findtheindividualsumsofthe‘x’dataandthe‘y’dataandprintthemto

theterminal

10. Determinetheindividualmeansforthe‘x’and‘y’dataandprintthemto

theterminal

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11. Writeacodethatcreates3variablescalled‘day’,‘month’,‘year’.Getthe

codetoasktheusertoinputvaluesforeachvariable,andthenoutputthe

valuesistheform“Todaysdateis:”dd/mm/yyyy.Nowrunthecodeusing

todaysdateastheinput.

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Keplerdataanalysisexercises

LightcurvesfromtheKeplermissionarestoredinfileswhosenamesindicate

thereferencenumberofthestarintheKeplerInputCatalogue(KIC).This

cataloguewascreatedbeforethemissionlaunch,sothatastronomershada

sourceofinformationaboutthestarsthatKeplerwasgoingtosurvey.Thelight

curvesarepresentedinaforminwhichthechangeinfluxofthestar,relativeto

thefluxreceivedwhentheplanetisnottransiting,islistedagainsttime

measuredinJulianDaysminus2454833,suchthattimezerocorrespondsto12

noonon1stJanuary2009.Below,wehaveprovidedasequenceoftasksthatwill

allowyoutodeterminethekeyparametersofsometransitingplanets.

1. Writeaprogrammetoplotthelightcurve(fluxagainsttime)forthe

systemKIC006922244.Notethatthedatasetisstoredinthefile

KIC006922244.tblthathasbeendownloadedfromtheKeplerdata

archive(storedintheDATAdirectorythatyouwillfindinthe

PlanetHuntingWithPython2019folder).Thiscontains3columnsofnumbers

thatmustallbereadinbyyourprogramme.Thefirstcolumncontains

non-usefuldataandshouldbeignoredafterithasbeenreadin.Lateryou

willlearnhowtodownloadyourowndatafilesfromtheKeplerarchive,

andthesewillhavethesameformatasKIC006922244.tbl.Notethatthe

first3linesofthisfilealsoneedtobeignored,soyouneedtousethe

skiprows=3optioninthenp.loadtxtcommandthatyouusetoreadthe

datafile.

Figure6:ThisfigureshowsthelightcurvecontainedinthedatafileKIC006922244.tbl

Usefulbuttons

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Figure7:Azoominonthefirst10transits

2. FindthetransitsignalforKIC006922244andestimateitsperiod.Note

thattheplottingwindowthatpopsupwhenperformingtask1abovehas

buttonstoallowyoutozoominondifferentregionsoftheplot(i.e.press

themagnifyingglass–andexpandtheplotbydraggingthecursoracross

therequiredarea).Furthermore,whenyoumovethecursorintheplot

windowthepositionofthecursorarrowshouldbeindicatedinthelower

partofthewindowframe,andthisisausefulfeaturefordeterminingthe

timeofatransitmid-pointandotherparameters.Itisworthexploring

whatthesebuttonsdoasyouwillusethemfrequentlyinthisproject.To

findtheperiod,weadvisefindingaregionofthelightcurvewithoutany

breakscontaining10transits(notethatwhenyouzoominonthelight

curveyouwillseemanysmallbreaksinthedatacorrespondingtotimes

whenthespacecraftwasdownloadingdataorwasfiringitsthrusters).

Findthetimeofmid-transitforthefirstandlastofthesetransits(make

sureyouzoominoneachoftheseandusethecursorvaluestogetan

accurateestimate),andthendividethetimeintervalbetweenthefirst

andtenthtransitbythenumberoftransitsminus1(i.e.9inthis

example).Youshouldnowhaveanestimateoftheorbitalperiod.

3. Writeaprogrammetophase-foldthelightcurvesothatalltransitsinthe

datalieontopofoneanother,andplotthefluxwithrespecttothephase

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(Note:Hintsaboutgeneratingaphasefoldedlightcurvearegivenin

theappendixattheendofthisdocument!Westronglyrecommend

thatyoureadthis.Ifyougetreallystuckthenyourteacherhasthe

Pythoncodeneededtodothephasefolding).Phasefolding

correspondstojustshiftingthetimecoordinateofthedatabythecorrect

amountsothatthetransitsalloccuratpreciselythesametime.You

shouldarrangeyourdatasothatthephase-foldedlightcurvehasthemid-

pointofthetransitsatphase0.0,withthelightcurvebeforeandduring

ingressoccurringwithanegativephase(withminimumvalue=-(orbital

period)/2)andegressoccurringwithpositivephase(wherethe

maximumtimeshownis+(orbitalperiod)/2-seethefigurebelow).The

phaseshouldberepresentedinunitsofJuliandayseithersideofzero.

Youcanaddthefollowinglinetoyourscripttosortthedatabefore

plottingit,wheret2isanarraythatcontainsthenewshiftedtime

coordinateforalloftheoriginalvaluesoftherelativeflux:

t2,flux=zip(*sorted(zip(t2,flux)))#Thisusesthet2arraytosortbotharrayst2andflux.i.e.#itsortsthemwithrespecttotimeNote:Thequalityofyourphase-foldedcurvewilldependverysensitively

onyourestimateoftheorbitalperiod.Youalsoneedtoknowthetimeof

thefirsttransitmid-pointtoperformthephasefolding.Intheexample

shownbelow,togetthephase-foldeddataintheleftpanelweestimated

theperiodtobe3.522XwheretheXrepresentsadigitthatyouwillneedto

determineforyourself.i.e.Youwillneedtohaveanestimateaccurateto

fourdecimaldigitstogetareasonablephasefoldedcurve.Therightpanel

showswhathappensiftheorbitalperiodestimatechangesbyjust-0.0005

JulianDays.

Figure8:Theleftpanelshowsthephase-foldedlightcurvewithanaccurateestimatefortheorbitalperiod.Therightpanelshowswhathappenswithaslightlyinaccurateestimateoftheperiod.

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4. Whensatisfiedthattheperiodisaccurate,obtainthefollowing

parametersfromthephasefoldedplot(usingthecursorandprinted

valuesintheplotwindowwillbeveryusefulhere!):

a. Maximumdepthofthetransit(notingthatthisshouldcontainan

approximateaverageofthenoiseinthedata).

b. Time(orphase)thatthetransitbegins,whentheplanetjust

beginstopassinfrontofthestar(ingressstarttime)

c. Timethatthetransitfirstreachesmaximumdepth,whenthe

wholeplanetisinfrontofthestar(ingressendtime)

d. Timethatthetransitreachesitsmidpoint.

5. Nowconstructapiece-wisefunctiontoactasamodelthatyouwillfitto

thedatausingtheparametersdeterminedintask4.Youcanassumethe

transitissymmetricaboutthemidpoint.Themodelshouldincludea

horizontalliney=0outsideofthetransit,sincethatisthemeanvalueof

thedatathere.Themodelfunctionbetweenthestartofingressandthe

endofingressshouldberepresentedasastraightlinewithan

appropriateslope.Thetimeintervalbetweentheendofingressandthe

beginningofegress,correspondingtomaximumtransitdepth,shouldbe

representedasahorizontallinewithy=minimumflux.Plotthephase-

foldedKeplerdataandthenthemodelonthesameplot(youneedtodo

theplottinginthisordersothatthedataplotdoesnotobscurethe

model).Plottingthemodelontopofthedataisachievedbyissuing

consecutiveplt.plotcommands.SeeFigure10onthenextpageforan

examplehowyourmodelanddatashouldlookwhenplotted.Asintask3,

youshouldaddthefollowinglinetoyourscripttosortthedata

time3,flux=zip(*sorted(zip(time3,flux)))#Thisusesthetime3arrayto

#sortbotharraystime3and

#flux.i.e.itsortsthemwith

#respecttotime

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Figure9:Plotshowingthemodel(redline)overplottedonthedata(bluedots)

6. Nowdetermineanerrorestimateforyourmodelusingthefollowing

equationfortheChi-squaredtest:

𝜒' =1𝑁N

(𝐹P − 𝐹Q)'

𝜎'

S

PTF

whereFiisthedata,Fmisthefluxpredictedbyyourpiece-wiselinear

function(themodel)correspondingtothetime/phaseassociatedwithFi,

andσ2isthevarianceofthedataaboutitsmean.Althoughnotcompletely

accurate,whencalculatingthemeanandσ2foryourdatayoushoulduse

alldatapoints,includingthosecorrespondingtothetransits,sincein

practicethismakesonlymodestdifferencecomparedwithremovingthe

contributionforthetransits.Rememberthatthevarianceisreallyjust

estimatingthenoiseinyourdata,orthelevelofscatteraboutzero.A

modelfitthatisconsistentwithbeingwithinthenoiseofthedatashould

haveaχ2valuelessthan1.

7. Repeatsteps4-6foramaximumof10timestoobtainanimprovedmodel

forthedata,wherethebestfitisobtainedwhenχ2hasitsminimumvalue.

8. Usingtheyourbestfitparameters,writeaprogramme(orextendyour

existingprogramme)todeterminethefollowingplanetparameters(you

25

canusethestellarparameterslocatedinthefileStellar_Mass_Radius.pdf

whichmaybefoundintheDATAfolder):

a. Planetradius(inunitsofR_Earth)

b. OrbitalPeriod(days)

c. Semi-majoraxis(AU)

d. Transitimpactparameter

e. Planetorbitalinclination

Thevaluesyouobtainshouldbesimilartothese:

Planetradius=15.038Earthradii(Earth’sradiusisapproximately6.37x106m)

Period=3.5224

Semi-majoraxis=0.04712AU

Impactparameterb=0.66256

Inclination=84.5677degrees

9. NowperformthesameanalysisfortheotherKeplerandTESSsystemsfor

whichdatafilesarepresentinthedirectoryDATA:KICXXXXXX.tbl(whereXXXXXXrepresents8359498,11853905,6922244,2571238,10418224,

9631995,7950644,5881688)andTICXXXXXXXX.dat(whereXXXXXXXXX

represents100100827,271893367,266980320)

26

Suggestionsforfurtherindependentresearch

Wehavetriedtoprovideastructuredsetofexercisestoguideyouthroughthe

processofobtainingphysicalinformationaboutplanetarysystemsfromKepler

lightcurves.Herearesomesuggestionsforhowyoumightextendyourwork.

1.DownloaddatafromtheKeplerarchiveandanalyseamuchlargernumberof

systems(seebelowfordetailedinstructionsonhowtodownloaddatafromthe

Keplerarchive).Usetheplanetparametersthatyouobtaintodetermine

informationabouttheplanetpopulationthatisorbitingtheKeplertargetstars.

Forexample,determinethefrequencydistributionofplanetswithdifferentradii.

Howmanygiantplanetsarethere(e.g.Jupiters)versussmallerNeptune-sized

bodies?Whicharemorecommon–largerorsmallerplanets?Howarethe

planetsdistributedasafunctionoftheirorbitalperiods?

2.Foreachsystemyouhaveexamined,implementanautomatedsearchthrough

thedifferentmodel-fitparametervaluesinordertoobtainabest-fitsolution(we

suggestthatyouuseasinglevaluefortheorbitalperiodandvarytheother

modelparametersduetothesensitivitytochangingtheperiod).Here,youwill

needtoimplementasequenceofnestedforloopsthatscanthrougharangeof

parametervalues.Foreachindependentsetofparametervaluesyoushould

applytheChi-squaredtestdescribedearlier,andselectthemodelwiththe

lowestChi-squaredvalue.

3.Downloadlightcurvesforsomesystemsknowntohavetwoplanets.Useyour

previouslydevelopedcodestofindtheperiodandparametersofthemost

obvioustransitsignal.Writearoutinetomanipulatethelightcurvedataand

removethisdominanttransitsignal(youcandothisbysimplyaddingthe

negativeofthemodel-fittothedatawhichthenremovesthetransitsignal,or

youcanremovethedominanttransitsbyreplacingthedataduringeachtransit

withdatacopiedfromthetimebetweentransits),andnowapplyyour

programmestofindtheorbitalperiodandsystemparametersofthe2ndplanet.

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4.Forasingleplanetorbitingastar,theorbitalperiodshouldnotchange.Write

aprogrammetoexaminewhetherornottheorbitalperiodischangingforthe

systemsthatyouhaveexamined(ifchangesareoccurringthentheseareknown

asTransitTimingVariations–orTTVsforshort).Onewayofdoingthisisto

createaplotoftheoriginaldetrendedlightcurvedataandtosuperimposeonita

plotofthemodelfitappliedtothiswholetimeseriesratherthantothephase-

foldedlightcurve.Byeyeinspectionshouldthentellyouiftheperiodremains

constantsincethemodelshouldfitalltransits.Seeifyoucanfindawayof

automatingthesearchforTTVs,andplottheTTVsversustimetoseeifthereis

anystatisticallysignificantevidenceforrealTTVs.Choosesome2-planet

systemswheretheremaybegravitationalinteractionsbetweentheplanetsthat

couldperturbtheirorbitsandinduceTTVs.Canyoufindevidenceofplanet-

planetdynamicalinteractionswhicharesystematicallychangingtheorbital

periods?

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TodownloadlightcurvedatafromtheNASAexoplanetarchivegoto

http://exoplanetarchive.ipac.caltech.edu

Undertakethefollowingstepstoselectthesystemsthatyouwishtodownload

datafor.Ouradviceistoonlydownloadsystemsforwhichthereareconfirmed

planets,andforwhichtheplanettransitsignalismuchbiggerthanthenoise.

Thisnormallymeansthatyouwillselectplanetswithfairlylargeradiisothatthe

transitscaneasilybeseenbyeyeinplotsofthelightcurves.

ThefirststepistocreatealistofKeplersystemsthatyouwishtodownloadthe

datafor.ClickonthebuttonindicatedonthepreviousfigureshowingtheNASA

archivewebpage.Thiswillloadaninteractivetable:

Clickhere

29

Youwillnoticethatthetablecontainsinformationonmanysystems,andalso

containsinformationthatisnotrelevantwhenselectingalistofplanets(it

scrollsrightwardsalongway!).Theredboxescontainingwhitecrossesatthe

topofeachcolumncanbeusedtoremovethosecolumns.Wesuggestcullingthe

databyremovingthefollowingcolumns(movingfromlefttoright):Disposition

UsingKeplerData;DispositionScore;NotTransit-LikeFalsePositiveFlag;Stellar

EclipseFalsePositiveFlag;CentroidOffsetFalsePositiveFlag;Ephemeris

Match…;TransitEpoch;ImpactParameter;TransitDuration;TransitDepth;

EquilibriumTemperature;InsolationFlux;TCEPlanetNumber;TCEDelivery;

StellarEffectiveTemperature;RA;Dec;Kepler-band.

NowweneedtoaddtwocolumnsbyclickingontheSelectColumnsbuttonatthe

topleft(seepreviousfigure).ScrolldownandselecttheNumberofPlanets

optionandtheStellarMassoptionandclickontheUpdateSelectionbuttonin

thedropdownmenu(seefigurebelow).

Clicktoremovecolumn

Clicktoaddcolumn

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You should now have a table that looks like this

Nowweneedtocullthedatafurthertoobtainasetofplanetsforwhichyour

searchandparameterfittingalgorithmswillwork.Systemsforwhichthetransit

signalisnotmuchlargerthanthenoiseneedmuchmoresophisticatedroutines

thantheoneswehavediscussedalready.Textboxesareprovidedatthetopof

Typeinheretoselectdata

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eachcolumntoselectdataaccordingtouserdefinedcriteria.Firsttype“>100”

intotheTransitSignal-to-Noisetextbox,asshowninthefigureabove.Thetable

willautomaticallyupdate.Nowmakethefollowingselections(afterwhichyour

tableshouldhaveabout97systemslistedasshowninthefigurebelow).

NumberofPlanets:1

ExoplanetArchiveDisposition:CONFIRMED

OrbitalPeriod:>3.5

PlanetaryRadius:>3

StellarRadius:<2

Nowchoosewhichsystemyouwanttodownloadthedatafor.Makeanoteofthe

StellarRadiusandtheStellarMassasthesewillberequiredwhendetermining

planetparametersfromthelightcurves.

Hoveryourcursoroverthe“i”symbolnexttotheKepIDnumberthatyouwanttodownloaddatafor.Apop-upmenushouldappearwhenyouhoveroverthe“i”symbol.Scrolldownandclickontheoption“KeplerDVTimeSeriesandPeriodogram”.Aplotshouldnowappear(asshownbelow).TheexamplebelowisforKepID11250587.

Hovercursorhere

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Nowwewanttoplotthedetrendedlightcurve(thisisacleaned-upversionofthelightcurveprovidedbytheKeplermission)whichcanbeobtainedbyclickingontheYAxisColumnoption,asshownabove,choosingLC_DETRENDfromthemenu,andthenpressingtheredrawbutton,givingthefollowingplot.

Todownloadthedetrendedlightcurveintheformofadatafile,withthesameformatthatwasusedintheKeplerprogrammingexercisesprovidedearlierinthisdocument,clickonthebuttonindicatedintheabovediagram.Thiswilldownloadafilecalledplot.tblthatcontainsthelightcurveinatextfilewiththree

Clickheretoredraw

Clickhere

Clickheretodown-loaddata

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columnsandathree-lineheader.Younowhaveadatafiletoanalyseusingyourpreviouslydevelopedalgorithmsandprogrammes.ChangeitsnametomatchtheKICnumberandyou’rereadytogo.Todownloaddataonsystemscontainingmorethanoneplanet,repeattheabovesteps,exceptchooseNumberofPlanets=2toextractdataon2-planetsystems.Arecommended2-planetsystemtodownloadisKepler117.ToobtainmoreTESSdatacontactRichardNelsonatQMULwhocansupplydataformoresystems.ReferencesMayor,M.,Queloz,D.,AJupiter-masscompaniontoasolar-typestar,1995,Nature,378,355-359Borucki,W.J.etal,KeperPlanet-DetectionMission:IntroductionandFirstResults,2010,Science,327,977InformationabouttheKeplermission:https://kepler.nasa.gov/index.cfmNASAExoplanetArchive:http://exoplanetarchive.ipac.caltech.eduExoplanetDataExplorer(usefulforgeneratingplotsofexoplanetpropertiesandexploringthestatisticsoftheexoplanetpopulation):http://exoplanets.orgExtrasolarPlanetEncyclopedia(similartotheExoplanetDataExplorer):http://exoplanet.euAreferencesitefordocumentationonpythoncommandsandlibraries:http://scipy.org(Clickonthedocumentationbuttonattopofthepage)

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Appendix–HintsonhowtophasefoldthelightcurvesBelowweprovideasequenceofstepsforphasefoldingtheKeplerlightcurvessothatweobtaindatainthesameformatasthatplottedintheleftpanelofFig.9.Figure12showsazoom-inofthelightcurveforKIC006922244,theexampleusedearlierinthisdocument.Itshowsthetimeofthefirsttransit,indicatedbytheblackarrow.

Figure10ThisimageshowsazoominoftheKIC006922244datashowingthetimeofthefirsttransit

Step1:Determinethetimeofthefirsttransit.Letusrefertothisast_transit1.Alsodeterminetheperiodwithwhichthetransitsoccur.WeusethesymbolPtodenotetheperiod.Step2:Wenowwanttoshiftthelightcurvedatatotheleftinsuchawaythatthefirsttransitoccursatatimet=P/2.i.eThetimeofthefirsttransitshouldnowoccuratatimethatcorrespondstohalftheorbitalperiod.Thisstepcanbeimplementedinpythonbysimplysubtractingtheappropriatenumberfromallofthetimevaluescontainedinthelightcurve.TheresultofdoingthisisshowninFigure13.

Figure11ThisfigureshowstheKIC006922244dataafterthetimecoordinatehasbeenshiftedsothatthefirsttransitoccursatt=P/2,wherePistheperiodassociatedwiththetransits.Theblackarrowsindicatethatsuccessivetransitsoccuratt=P/2,3P/2,5P/2,etc…Theredverticallinesindicatethepositionsintimethatcorrespondtooneperiod,twoperiods,threeperiodsetc.

35

Step3:Wenowneedtousemodulararithmetictoshiftsectionsofthelightcurvesothattheyaresuperimposedononeanother.Thisistheessenceofphasefoldingthelightcurve.Considertwodecimalnumbers,AandB.InmanycomputinglanguagesthecommandMOD(A,B)willgivetheremainderofdividingAbyB.Inotherwords,ifwewriteA=NxB+R,whereNisaninteger,thenMOD(A,B)=R.ItshouldbeclearfromlookingatFigure13thatift[i]containsthetimesassociatedwitheachofthepointsontheshiftedlightcurve,thenapplyingMOD(t[i],P)toallpointswillcreateaphasefoldedlightcurvewiththemid-pointofthetransitoccurringatatimet=P/2.TheresultofimplementingthisinpythonisshowninFigure14.Importantnote:Inpython,wetype(A%B)andnotMOD(A,B)toobtaintheremainderofdividingAbyB.

Figure12ThisimageshowstheKIC006922244dataafterithasbeenphasefolded,withthetransitsoccurringattimet=P/2,wherePistheperiod

Step4:NowwesimplyneedtoshiftthelightcurvebyanamountP/2tocentrethetransitattimet=0.WedothisbysubtractingP/2fromallofthetimevaluesinthedata.TheresultofdoingthisisshowninFigure15.Rememberthatphasefoldingonlyworksifyouhaveanaccuratefigurefortheperiod!

Figure13ThisfigureshowsthedataforKIC006922244afterphasefoldingandshiftingsothatthemidpointsofallthetransitsoccurattimet=0.

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PythoncodeforphasefoldingYourteacherhasthesolution.Trytoworkitoutforyourselfbeforeseekingassistance.