GravitationalWaveDetectionintheIntroductoryLabLiorM.Burko*

SchoolofScienceandTechnology,GeorgiaGwinnettCollege,Lawrenceville,Georgia30043

February14,2016;RevisedMarch22,2016I.IntroductionAlongtimeagoinagalaxyfar,farawaytwoblackholes,oneofmass36solarmassesandtheotherofmass29solarmassesweredancingtheirdeathwaltz,leadingtotheircoalescenceandtheemissionofgravitationalwavescarryingawaywiththemthreesolarmassesofenergy.Moreprecisely,ithappened1.3billionyearsagoatadistanceof410Mpc.WhenthewaveswereemittedthemostcomplexlifeformsonEarthwereeukaryots.AsthegravitationalwavespropagatedtowardtheEarthitchangedmuch.Fivehundredmillionyearsafterthewaveswereemitted,or800millionyearsago,thefirstmulticellularlifeformsemergedontheEarth.TheEarthsawtheCambrianexplosion500millionyearsago.Sixty-sixmillionyearsagotheCretaceous-Paleogeneextinctioneventcausedthedisappearanceofthedinosaurs.Thefirstmodernhumansappeared250,000yearsago.Independentlyoftheevent,onehundredyearsago,in1916,AlbertEinsteinconcludedthatgravitationalwavesexist.TheconstructionofLIGO,theLaserInterferometerGravitational-waveObservatory,commenced22yearsago.FiveyearsagoAdvancedLIGOinstallationstarted,andwasfinishedwithafullengineeringrunduringSeptember2015,justintimetocatchonSeptember14,2015thegravitationalwavesthatwereemittedbythetwoblackholes1.3billionyearsago.ThewavespenetratedthesouthernhemispherefromthegeneraldirectionoftheMagellanicclouds(althoughonecannotpinpointthedirectionwithmuchaccuracybecauseonlytwodetectorswereonline),traveledthroughtheEarthandwerecapturedfrombelowbythetwinLIGOdetectors,oneinLouisiananearLivingston,andtheotherintheStateofWashingtonnearHanford,therebyusheringintheneweraofgravitationalwaveastronomy.

*burko@ggc.edu

II.WhatAreGravitationalWaves?Gravitationalwavesareripplesinthecurvatureofspaceandtime,whichpropagateatthespeedoflight.Saythesourceofagravitationalfieldweretoundergoanabruptchange,e.g.,saytheSunweretosuddenlydisappear.HowwouldthataffecttheorbitoftheEarth?StudentsoftheintroductorycollegecourselearnthattheEarthwouldmoveuniformlyalongatangenttoitspreviousorbit,asrequiredbyNewton’sFirstLaw.ButwhenwilltheEarthdepartfromitsorbitalmotionandstartmovinguniformly?AccordingtoNewtoniangravitythatwouldhappeninstantaneously,butinrealitytheremustbesomedelaybetweenthedisappearanceoftheSunandthemomentatwhichtheEarthwouldstartmovinguniformly,astheinformationthattheSundisappearedcannottravelfasterthanlight.Infact,itisthoughtthatthechangeinthegravitationalfieldpropagatesasgravitationalwaves.Gravitationalwavesarethemechanismbywhichthechanginggravitationalfieldofachangingmassdistributionpropagates,looselyanalogoustheelectromagneticwaves,whicharethemechanismbywhichachangingelectricchargedistributiontellsremotelocationsthatthesourceischanged.Thedirectdetectionofgravitationalwavesdidmuchmorethanjustproveexperimentallytheconceptofgravitationalradiationandthatbinaryblackholesystemsexist,bigachievementsinthemselves;ItopenedanewwindowontotheUniverse,thatmanybelievewillallowustoseehithertounimaginedastrophysicalphenomenainadditiontomanyconventionalorspeculativeastrophysicalsystems.Fromthispointofview,thedetectionofgravitationalwavesisarguablyevenmoretransformationalthantherecent2012discoveryoftheHiggsbosonandiscomparabletotherevolutionthatoccurredwhenGalileofirstpointedatelescopetotheskies.WhenEinsteinfirstpredictedtheexistenceofgravitationalwaves(Einstein,1916)itwasbelievedthattheireffectwassominisculethattheywouldneverbediscovered,andlatertheirveryexistencewasthetopicofalivelydebateamongrelativists(forahistoryofgravitationalwavesresearchseeKennefick,2007).Bothbetterunderstandingof

astrophysics,specificallyaboutcompactobjectsthatmoveatrelativisticspeedsandunprecedentedprogressintechnologyallowedus,afteracenturyoftheoryandhalfacenturyofexperimentalsearchestofinallydetectgravitationalwaves.III.Howcanthedetectionofgravitationalwavesbeincludedintheintroductorycourse?Recentphysicsbreakthroughsarerarelyincludedintheintroductoryphysicsorastronomycourse.Generalrelativityandbinaryblackholecoalescencearenodifferent,andcanbeincludedintheintroductorycourseonlyinaverylimitedsense.(SeediscussionofgravitationalwavesinThePhysicsTeacherinRubboetal,2006;Larsonetal,2006;andSpetz,1984.SeealsoRubboetal,2007.)However,wecandesignactivitiesthatdirectlyinvolvethedetectionofGW150914,thedesignationofthegravitationwavesignaldetectedonSeptember14,2015(Abbottetal,2016a),therebyengagethestudentsinthisexcitingdiscoverydirectly.Theactivitiesnaturallydonotincludetheconstructionofadetectororthedetectionofgravitationalwaves.Instead,wedesignittoincludeanalysisofthedatafromGW150914,whichincludessomeinterestinganalysisactivitiesforstudents.Thesameactivitiescanbeassignedeitherasalaboratoryexerciseindataanalysisorasacomputationalprojectforthesamepopulationofstudents.Theanalysistoolsusedherearesimpleandavailabletotheintendedstudentpopulation.Itdoesnotincludethesophisticatedanalysistools,whichwereusedbyLIGOtocarefullyanalyzethedetectedsignal.However,thesesimpletoolsaresufficienttoallowthestudenttogetimportantresults.Onemayarguethattheinclusionofcomputationalprojectsintheintroductorycalculus-basedcourseisbeneficialtomanystudents,asitallowsstudentstobecomefamiliarwithaveryimportanttoolinthetoolkitoftheworkingphysicist,inadditiontootherbenefits(RedishandWilson,1993;ChabayandSherwood,2006).Itisthechoiceoftheinstructorwhethertorequireaspecificcomputerlanguage(apopularchoiceisPython),orletthestudentschooseanycomputerlanguagetheyalreadyknow(oruseaspreadsheetsuchasExceliftheydonotknowanycomputerlanguage.Inaddition,manycalculationscanbecarriedoutonplatformswidelyavailableintheintroductorylaboratory,

suchasthePASCOCapstoneprogram.).Iamtypicallyinfavorofthelatter,asIbelievethatclasstimeinaphysicscourseshouldnotbedevotedtoteachingprogramming.However,Ihaveseenmanyinstructorsusetheformerapproachwithmuchsuccess.ThecomputationalprogramsIassignintheintroductorycourse(includingthepresentone)canallbedoneusingspreadsheets,suchthatstudentswithoutbackgroundinprogrammingcanstilldoalltheassignments.Whenusingaspreadsheetthestudentcancalculateanumericalderivativebydividingthedifferenceinthequantityofinterestbetweenthetimestepaftertheoneinquestion(typically,onecelllower)andtheoneprecedingit(typically,onecellhigher),anddividebythesameforthetimedifference.Thecomputationcanthenbecopiedfortheentirecolumn(skippingthefirstandlastcells,forwhichtheprocedureisilldefined),producingthenumericalderivativeatalltimesteps(exceptthefirstandthelast).Forexample,ifColumnAincludes,say,thedataforthetime,andcolumnBincludesthedataforpositioninthecorrespondingcells,wecancalculatethespeedincolumnCbyenteringinthecellC2theformula(B3-B1)/(A3-A1).Thisformulacanthenbefilledintheremainingofthecolumn.IV.Suggestionsforthedesignofclassactivitiesa)PresentationofthewaveformOnecanaccessonlinethedataofthegravitationalwavesignalanduseitforanalysis.Asthesedataaresomewhatnoisy,Iprefertousethenumericalrelativitywaveform,whichmatchestheobservedsignalverywell.Onemayofcourseusetheactualdetecteddatainsteadofthenumericalrelativitywaveform,atthecostofhavingtodealwithmorenoise.TheimpressiveagreementofthenumericalrelativitywaveformandtheobserveddatacanbeseeninFigure1.Thenumericalrelativitywaveformisatheoreticallyconstructedwaveformthatwascomputedusingthephysicalparametersthatcameoutofadetailedcomputersearchoftheobserveddata,lookingforatheoreticalwaveformthatbestmatchesthedata.Alsoavailableonlinearetheresiduals,thedifferencesbetweenthenumericalrelativitywaveformandthedetectedsignal.Interestedinstructorsmayincludetheresidualsintheactivity,or

evenincludeabonusexercisetocalculatethecrosscorrelationintegralofthedetectedsignalandthenumericalrelativitywaveform.Alldataareavailableathttps://losc.ligo.org/events/GW150914/.(Allresultspresentedherewereobtainedfromthedatafilehttps://losc.ligo.org/s/events/GW150914/GW150914_4_NR_waveform.txt.)Theamountbywhichabodyisdistorted(stretchedorcompressed)relativetoitsreferencelengthisknownasstrain.Analogously,therelativestretchingorcompressionofspacewhenagravitationalwavepassesthroughthedetectorisalsocalledstrain.Plottingthestrainhasafunctionofdetectortime,weobtainFigure1.

Figure1:TheGWstrainhforGW150914(multipliedby10!")asafunctionofthetimetinseconds(fromthenumericalrelativitywaveform).Thetimeisshiftedsothatmaximumstrainisatt=0s.Theinsetshowsasectionofthedatainthemainfigure(inblue),togetherwiththeobservedH1data(inred),i.e.,thedatafromtheHanforddetector.ThefeaturesoftheL1data,thedatafromtheLivingstondetector,aresimilartothoseoftheH1data.

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ThewaveformshowninFig.1includestwomainparts:first,atearliertimes,theso-called“chirppart”,andatlaterpartsthering-downpart.Thechirppartofthewaveformissonamedbecauseboththeamplitudeandfrequencyincreaseasafunctionoftime,thefeaturesofabird’schirp.Whilestudentsoftheintroductorycourseseeoscillationswithvaryingamplitude(say,inthecontextofdampedharmonicoscillations),theydonotnormallyseevaryingfrequencies.Thischirppartofthewaveformcanbeagoodlaboratoryorcomputationalexerciseinanalyzingdatawithavaryingfrequency,specificallytheemphasisonalocalcalculationoffrequencyasopposedtoaglobalcalculation.Itisinstructivetoletthestudentsseparatethesignalintoitstwomainparts,andemphasizethateachparthasdifferentpropertiesandthereforerequiresdifferentanalysistools.b)BasicrelationssatisfiedbygravitationalwavesfromacoalescingbinaryFortheanalysisofthegravitationalwavesignalwewillneedsomebasicrelationsthattheparametersofthewavessatisfy.Thefollowingdiscussionisajustificationoftheserelationsbasedonscalingarguments.Thefullequationsrequireadetailedsolutionusinggeneralrelativity,whichisbeyondthescopeofthisPaperandtheleveloftheintendedaudience.ConsiderasystemoftotalmassMandoftypicalsizeR,whosemomentofinertiascaleslike!!!.Anunchangingmassdistributioncannotradiate.Gravitationalwavesarethereforegeneratedbyatime-changingmomentofinertia.Auniformlymovingmassalsocannotradiate,becauseforaco-movingobserveritisseenasstatic.Asthepresenceofradiationmustbeobserverindependent,nootherinertialobservercanmeasureanyradiationeither.Onemustthereforehaveanacceleratingmassdistributiontohaveradiation.Morespecifically,thewavesdependonthesecondtimederivativeofthemomentofinertia.(Ananalogousargumentappliesalsoforacceleratingelectricchargesthatemitelectromagneticwaves.)

Theluminosityofanywavedependsonthesquareoftheamplitude.(Theenergyofthewavemustbepositiveeventhoughthewaveoscillatesbetweenpositiveandnegativevalues,andthereforemustdependonanevenpoweroftheamplitude).Conservationofenergythendictatesthattheluminositydropswithdistanceasthesquareofthedistance,andthereforetheamplitudemustdecaylikethereciprocalofthedistancetotheobservationpoint,!.Inwhatfollowsweconsideronlythescalingrelationsanddonotworryaboutconstantordimensionlessfactors.Replacingatimederivativebythefrequency!,wefindthatℎ~!!!!! !.UsingKepler’sthirdlawforabinarysystem,or!~ ! !! ! !,wefindthatℎ~!!

!" .Infact,thisscalingrelationisconventionallywritten(afterusingKepler’slawagainandintroducingthechirpmass)asℎ~ !

! MC5/3f2/3.Whentheconstantfactorsareworkedout,theamplituderelationbecomes

ℎ = !!! !!! !

!! !"! !(GMC)5/3,(1)

wherewewritethedistancebetweenthedetectorandthesourceastheluminositydistanceDL,and!isthefrequencyofthewaves.Here,cisthespeedoflightandGisNewton’sconstant.SimpleargumentsfortheamplitudeequationintermsofdimensionalanalysisaregiveninSchutz(2003).

Next,westartwiththeluminosity!~4!!! !!ℎ!,inasimilarwaytohowluminosityoflightiswritten.(Theluminositymustdependonanevenpowerof!becausetheenergyofthewavesmustbepositiveforbothpositiveandnegativefrequencies.Adetailedcalculationshowsthattheluminositydependsonthesquareofthefrequency.)UsingKepler’slawandtheamplituderelation,wefindthat!~!! !!.Ontheotherhand,theluminosityLequals(thenegativeof)therateofchangeofthetotalenergyofthesource.Thetotalenergyisthegravitationalpotential

energy,or!~−!! !,andtherefore!~!!

!! !.Requiringthat! = −!wefindthat!~ − !!

!! .

DifferentiatingKepler’sthirdlaw,!~ ! !! ! !,therateofchangeof

thefrequency!~!! !

!! ! !.Substitutingourresultfor!,wefindthat!~ !! !

!!! !.UsingKepler’slawagain,andreintroducingthechirpmass,we

writeourresultas!~ MC5/3f11/3.Whentheconstantfactorsareincluded,wefindthat

! = !"!!! !

!! !!! ! (GMC)5/3.(2)

Thisequationfortherateofchangeofthefrequencyisknownasthechirp.c)AnalysisofthedataArmedwithEqs.(1)and(2),wecannowstartanalyzingthewaves.i)FindingthefrequencyAssumethatthechirppartofthewaveshastheform

ℎ ! = ! ! sin ! ! ×! ,whereboththeamplitude!andtheangularfrequency!arefunctionsofthetime!.Asimpleanalysistoolistoassumethatthechangesintheparameters!and!areslowcomparedwiththemeasurementtimestep.Thatis,wemayassumethataslongasthechangeintheparameterisslowonthescaleofthetimeincrement,ateachvalueofthetimewecantreatthewaveformasifithadconstantparametersinthefirstapproximation.Therefore,wecandifferentiatethestrainexpressionabovetwiceinthisapproximationtoget

!! = − ℎℎwhereanoverdotdenotesthetimederivative.Studentsusethisrelationtogetanumericalestimateofangularfrequencyandhowitchangeswithtime.Noticethatwedonotneedtofindaphaseconstant,soitisassumedwithoutlossofgeneralitytoequalzero.Toobtainthesecondderivativestudentscaneitherwriteashortcodethatcalculatesthenumericalderivativeforacomputationalproject,useaspreadsheetsuchasEXCELasdescribedabove,orusesoftwarecommonlyavailable

intheintroductorylaboratorysuchasPASCOCapstone.Thelatterallowsfindingthe“acceleration”ofanoscillatingsignal.InFig.2weplottheangularfrequencyasafunctionofthetime,aftersmoothingoutthenoisewithasimpleaveragingalgorithm.Inordertoaverageoutnoisethestudentcanaddanaveragingroutinetothecode,oraddacolumntotheEXCELspreadsheetthatfindsthesimpleaverageforacellforacertainrangeofcellsaroundit.IfusingCapstone,thestudentcanuseabestfitcurveasthenoisesmoothingalgorithmorusethebuilt-incurvesmoothingfunction.Theaveragingdonehereisforintervalsof25ms.Theunsmootheddatafortheangularfrequencymayincludeisolatedpointsatwhichthevaluesareverydifferentthaninneighboringpoints.Suchpointsmayberemovedfromthedataset,whichmakesthesmoothingprocesssimpler.Inwhatfollows,allourcalculationsaredoneinthedetectorframe.(Thetwoblackholemassesgiveninthefirstparagrapharegiveninthesourceframe.)WethereforecomparewiththeLIGOvaluesforthedetectorframe.

Figure2:Frequencyasafunctionofthetime.Theincreasingfrequencyisshownfromabout20Hztoover150Hzduring8

cyclesofthebinary.

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ii)TheChirpmassTheleading-orderdependenceofthewaveformevolution(i.e.,theevolutionintheearlypartsofthewaveform)onthemassesofthebinaryblackholesisthroughtheso-calledchirpmass.Specifically,theindependentmassesofthebinaryblackholescannotbeeasilydeduced,asmoreinformation(whichincludesrelativisticeffectswhicharebeyondthescopeofthepresentPaper)isneededinordertofindtheindividualmasses.Itiscalledthechirpmassbecauseitistheparameterresponsibleforthechirp-likesound(increasingfrequencyandamplitude)thatisheardwhenagravitationalwavesignalfromabinarymergerisconvertedtosoundwaves.Thechirpmassisdefinedas

MC=!!!! !/!

!!!!! !/!,

wherem1andm2arethemassesofthetwomembersofthebinary.Itisthisparticularcombinationoftheindividualmassesthatcanbefoundfromtheleadingorderevolutionofℎ ! .Asthewaveformdependsatfirstonlyonthechirpmass,thewaveformdoesnotdependonanyothercombinationofthemasses,andforthatreasonitisausefulparametertointroduce.Thefrequencyevolutionofthewave,!,canbeusedfromEq.(2).Thisequationcannextbeinvertedtogivethechirpmassintermsofthecalculated!and!,whichcanbeconstructedfromtheGW150914data.(Thatis,havingfound! ! thestudentcannowcalculatethenumericalderivativetofind!.)InFig.3weplotthechirpmass,aftersomesmoothingdoneasabove.Noticethatthechirpmassiswelldefinedusingthesesimpleexpressionsonlyduringtheearlypartoftheinspiral.Wethereforereadthechirpmassasabout30solarmasses.Infindingthechirpmassonecanagaineitherwriteashortnumericalcode,useaspreadsheet,oralternativelyusePASCOCapstoneorasimilarprogram.Atearlytimesthechirpmassisnearlyconstant,butthenstartsoscillatingwildly.Indeed,ourcalculationofthechirpmassisonlyintendedtobeaccurateintheearlypartsoftheinspiral,whenthetwoblackholesarestillfarfromeachother.Astheygetcloserrelativisticcorrectionsbecomeimportant,andthesimplerelationsthatweusebecomelessandlessaccurateapproximations,asthebehaviorofthechirpmassshows.

Figure3:Thechirpmassasafunctionoftime.Weonlyreadthechirpmassfromtheearlypartoftheevolution.Intheearlypartof

themergerthechirpmassisnearlyconstant,anditstartsoscillatingwidelyafterrelativisticcorrectionsbecomeimportant.

iii)EvaluatingthetotalmassThetotalmassofthebinaryblackholesystemcanbecrudelyestimatedfromthechirpmass.Ifweassumethatthetwoblackholeshavemassratio! = !! !!,onecanusethedefinitionofthechirpmassandsolveforthetotalmass:Firstsubstitute!! = !!!inthedefinitionforthechirpmass,andfindthat!! = 1 + ! ! ! !! ! MC.Next,write

! = !! +!! = 1 + ! !!,andfindthatthetotalmass! = !!! !/!

!!/! MC

>2! !MC.Noticethatonedoesnotneedtoknowthemassratioinorder

tofindthisinequality(as!!! !/!

!!/! asafunctionofqisboundedfrombelowby2! !),eventhoughthemassratioisneededinordertofindthetotalmassitself(andnotjustalowerbound).However,determiningthemassratiorequiresrelativisticeffectsthatarebeyondthescopeofthisPaper.Notethatthelowerboundisachievedwhenthetwomassesareequal,andforunequalmassesthetotalmassisalwaysgreater.Specifically,! > ~70solarmasses.

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iv)DeterminationofthemassandspinrateofthefinalblackholeNext,weturnattentiontothering-downpartofthewaves.Inspectionrevealsitisaconstantfrequencysignalofexponentiallydecayingamplitude.Agoodquestiontoaskthestudentsiswhatkindofsystemdoesasignalsuchasthatsuggest.Itmaybeusefultorefrainuntilthispointfromcallingitthe“ring-down”part,asthenamesuggeststheanswertothequestion.Ifsuchaquestionisposed,instructorscandividethesignalintotheearlypartandlatepart,say,insteadofthechirppartandring-downpartwhenthesignalisfirstintroduced.Whenabellringsitemitssoundinacharacteristicspectrumoffrequencies.Anidealbell,whichringsonforeverwithnolosses,wouldemitsoundinfrequenciesknownasitsnormalmodes.Anyrealbellringsdownbecauseofthelossofenergytosoundwaves,anditsfrequenciesareknownasquasi-normalmodes.Blackholesarelikebells:whenperturbed,theytooemitradiationintheformofgravitationalwavesinfrequenciesoftheirquasi-normalmodes.Infact,blackholesarepoorbells,astheydon’tgettoringmanytimesbeforetheysettledowntoastationary,quiescentstate.Thespectrumofblackholequasi-normalmodesiscomplex,andthefrequenciesdependonlyonthemassandspinangularmomentumoftheblackhole.Inordertoprovethatthesourceisablackhole,multiplequasi-normalmodefrequenciesneedtobemeasured.However,ifweassumethatthering-downphaseoftheradiationisgivenbytheslowestdecayingquasi-normalmodeofaspinningblackholethatisperturbedfromitsstationarystate,wecancalculatethemassandspinangularmomentumofthefinalblackhole.Aswecanobserveinthedataonlyonemode(thatis,onefrequencyandonetimeconstantforthedecayrate),weassumethatthismodeistheslowestdampedmodeoftheblackhole’squasi-normalspectrum.Thechirpwaveformisfollowedbythequasi-normalring-downofthefinalsingleblackhole.Whenthefinalblackholeisfirstcreateditishighlydistorted,andemitsgravitationalwavesthatreducethisdistortionuntilitgettoitsstationaryandquiescentstate.Indeed,thewaveformforGW150914includesanexponentialdecayofthesignalwithconstantfrequencyafterthemerger.(Inpractice,theparametersforthering-downpublishedinAbbott,B.P.etal(2016a)arefoundfrom

thenumericalrelativitywaveform,aswedohere.)InFig.4weshowthepartofthesignalafterthemerger(positivevaluesoftime).Thisisagoodexampleofthebenefitinplottingasemi-logarithmicgraph,asthestraightenvelopeshowstheexponentialdecayofthesignal,andtheconstantfrequencyiseasilyvisible.Thefrequencyanddecayratecanbereadfromthegraph,orinsteadcanbedeterminedfromthefitfunctionofprogramssuchasPASCOCapstone.Specifically,fittingtoadampedsinefunctionoftheformℎ ! = !!!!/!sin 2!"# + !! wedeterminetheparameterstobe! = 4.4×10!!sec,and! = 236Hz.ThesevaluesagreenicelywiththevaluesoftheLIGOexperiment,! = (4.0 ±0.3)×10!!sec,and! = 251 ± 8Hz(Abbottetal,2016b).Figure5showsthebestfitofthedataforthering-downportionofthewave,whichwedidusingPASCOCapstone.Amoresophisticatedapproach,alsoavailableinPASCOCapstone,involvestheFastFourierTransformofthedecayingoscillations,butisgenerallytoosophisticatedfortheintendedstudentpopulation.Asthering-downisassumedheretobeamonochromaticsignal(singlemode),thesimplerfittingtoadecayingsinefunctionissufficient.

Figure4:Aplotofthelogarithmofthesignalasafunctionoftimeforthering-downpartofthewaveform.Theexponentialdecay

rateandtheconstantfrequencyarevisibleinthesemi-logarithmicrepresentation.

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ring-downpartoftheGW150914signal.Thefrequency!andtimeconstant!cannextbeusedtofindthemass!! andspecificangularmomentum!ofthefinalblackhole.Here,! = ! ! !!! ,where!istheblackhole’sspinangularmomentum.Thespecificangularmomentum0 ≤ ! ≤ 1,andisaconvenientdimensionlessvariable.Weusehereasimplemethodthatisbasedonsimplefitfunctionsfortheblackhole’squasi-normalmodes.Specifically,letthequalityfactor! = !"#.Then,

!!! ×2!"!! = !! + !! 1 − ! !!

and! = !! + !! 1 − ! !! ,

wherethefitparametersfortheprincipalleastdampedmodeare!! = 1.525, !! = −1.157, !! = 0.129,and!! = 0.700, !! = 1.419, !! = −0.499(Bertietal,2009).Thesefitequationsweredevelopedtoreproducethefrequencyandqualityfactortobetterthan1%withtheleastnumberofparametersfortheexpectedrangeofthespecificangularmomentum.Thesefitequationsarenotderivedfromphysicalprinciples,andarenotevenaphenomenologicalfit.Instead,theyarethesimplestmathematicalequationsthatreproducetheknowndatatoagivenaccuracylevel.(Thevaluesthattheequationsfit,however,arederivedfromaclearphysicalmodelforthequasi-normalring-down.)Whilethismayperhapsnotprovidesomestudentswithastrongphysicalmotivationforusingthese

fitequations(andindeed,oneshouldprefertobeabletophysicallymotivateequationsthatoneuses),itisperhapsanopportunitytopointoutthatwhenonedoesresearch,sometimesonedoeswhatonecando.Instructors,whofeelthatthelackofphysicalmotivationfortheequationsandthefitparametersarereasonstonotusethem,mayomitthispartofthediscussion,andhavethestudentsonlyfindthefrequencyandtimeconstantofthering-down.Theseinstructorscanthendiscusshowthemassandspinofthefinalblackholecanbefoundfromdataavailableintables,whichwerederivedfromsolidphysicalmodels.Theauthor’sexperiencewithhisownstudentsisthatfindingthemassandspinofthefinalblackholeisoneofthefavoritepartsformanystudents.Bethatasitmay,theseequationsare,however,extremelyeffectiveforourpurposes.Wecannowsolvethissetoftwoequationsforthetwounknown!! and!,andfindthat!! = 67 !!"#and! = 0.67.Inthiscalculationweuse! !! = 0.248×10!!"skg-1,and!!"# = 1.99×10!"kg.ThevaluesreportedintheLIGOpaperare!! = 67 ± 4 !!"#and! = 0.67!!.!"!!.!".Sincewefoundabovethatthetotalmassbeforethemergerwas>~70!!"# ,weinferthatatleast3!!"#wereradiatedintheformofgravitationalwaves.Takingthemergertimetobe0.2s,theaveragepoweremittedis3×10!!ergs-1.v)DistancetothesourceIfthisactivityisassignedtostudentsofanintroductoryastronomyorcosmologycourse,wheretheconceptsofluminositydistanceandcosmicexpansionareintroduced,theactivitycanbeextendedtofindtheluminositydistance.Forstudentsoftheintroductoryphysicscoursethecalculationoftheluminositydistancedcanbeskipped.Onestartswiththeamplitudeequation(1),andcombinesitwiththefrequencyevolutionequation(2).Onecanthensolvefor!! intermsofonlyquantitiesthataredirectlymeasurablefromthewaveform,specifically,

!! = !!"!!

!!!!!.

Thelastresultisanextremelyimportantone,asitshowsthatthedistancefromthesourcecanbemeasureddirectlyfromthewaveform.Thisisauniquecaseforastronomy,asthedistancefromasourceobservedwithlightcannotbeinferredfromonlythesignal,andother

methodsneedtobeusedinordertoestimatethedistance(theso-calleddistanceladder).E.g.,therewasatfirstafiercedebateaboutthedistancetoquasarsbeforetheirredshiftwasmeasured,asthedistancecouldnotbeinferreddirectlyfromtheobservations.ThedistancetothesourceofGW150914,400MPc,ismuchlongerthanthetypicaldistancebetweengalaxies,~1Mpc,whichitselfisabout50timesthesizeofasinglegalaxy.Ifthestudentsarefamiliarwithcosmologicalexpansionandknowtheredshift–distancerelation,theycancheckthatthesourceisatacosmologicaldistance,specificallytheredshift! = 0.1.Insummary,thedetectionofgravitationalwaveswithLIGOprovidesphysicsorastronomyinstructorswithanopportunitytoengagestudentsoftheintroductorycoursewiththisveryexcitingandimportantdiscoveryusingsimpledataanalysistoolsthatareavailabletothestudents.Thisresearchhasmadeuseofdata,softwareand/orwebtoolsobtainedfromtheLIGOOpenScienceCenter(https://losc.ligo.org),aserviceofLIGOLaboratoryandtheLIGOScientificCollaboration.LIGOisfundedbytheU.S.NationalScienceFoundation.ReferencesAbbott,B.P.etal(2016a),ObservationofGravitationalWavesfromaBinaryBlackHoleMerger,Phys.Rev.Lett.116,061102

Abbott,B.P.etal(2016b),TestsofGeneralRelativitywithGW150914,arXiv:1602.03841

Berti,E.,Cardoso,V.,andStarinets,A.O.(2009)Quasinormalmodesofblackholesandblackbranes,ClassicalandQuantumGravity26,163001

Chabay,R.andSherwood,B.(2008),Computationalphysicsintheintroductorycalculus-basedcourse,Am.J.Phys.76,307

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Erratum: Gravitational Wave Detection in the Introductory Lab [The Physics Teacher 55, 288 (2017)]

Lior M. Burko, Georgia Gwinnett College

December 6, 2018

There is a typo in the amplitude relation, Eq. (1). The correct equation is

ℎ = !"!

#$"(%&)!/&(()")'/&.

Notice that this correction involves only constant factors, and therefore the derivation, which was done using scaling relations, remains unaffected.