seeing earth's orbit in the stars

7/28/2019 Seeing Earth's Orbit in the Stars

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SeeingEarthsOrbitintheStars:ParallaxandAberrationToddTimberlake,BerryCollege,MountBerry,GA

Duringthe17thcenturytheideaofanorbitingandrotatingEarthbecame

increasinglypopular,butopponentsofthisviewcontinuedtopointoutthatthe

theoryhadobservableconsequencesthathadnever,infact,beenobserved.1Why,

forinstance,hadastronomersfailedtodetecttheannualparallaxofthestarsthatmustoccurifEarthorbitstheSun?

Toaddressthisproblem,astronomersofthe17thand18thcenturiessoughtto

measuretheannualparallaxofstarsusingtelescopes.Noneofthemsucceeded.

Annualstellarparallaxwasnotsuccessfullymeasureduntil1838,whenFriedrich

Besseldetectedtheparallaxofthestar61Cygni.2

Buttheearlyfailurestodetectannualstellarparallaxledtothediscoveryofanew(andentirelyunexpected)

phenomenon:theaberrationofstarlight.Thispaperrecountsthestoryofthe

discoveryofstellaraberration.Itisaccompaniedbyasetofactivitiesandcomputer

simulationsthatallowstudentstoexplorethisfascinatinghistoricalepisodeand

learnimportantlessonsaboutthenatureofscience.3

Parallax

Holdupyourthumbinfrontofyourfaceatarmslength.Closeyourrighteyeand

viewyourthumbagainstthebackground.Nowopenyourrighteyeandcloseyour

left:youwillseeyourthumbappeartomoveagainstthebackground.Theapparentdisplacementofyourthumbisreallycausedbythedisplacementofyourobservinglocation(fromyourlefteyetoyourright).Thisphenomenon,knownasparallax,

playsanimportantroleinastronomy.

TheAncientGreekswereawarethatcelestialobjectsviewedfromdifferent

locationsonEarthmightappearinslightlydifferentlocationsrelativetothebackgroundstars.Theyusedthiseffect,knownasdiurnalparallax,toaccurately

estimatethedistancetotheMoon.Figure1illustratesthegeometryofthis

measurement.ForsimplicitywewillassumetheMoonliesintheequatorialplane

andobservationsaremadefromoppositesidesofEarthsequator(thepointsO1

andO2).ThetwoobserversseetheMoonatdifferentlocationsontheCelestialSphere(orthestarrybackground,tousemoremodernterminology).These

apparentlocationsareseparatedbyanangle2.Fromtrigonometry(andusingthe

smallangleapproximation)wefind:

=!"!

!", (1)

whereistheparallaxangleindegrees,bisthebaseline(inthiscasethediameter

ofEarth),anddisthedistancefromthecenterofEarthtothecenteroftheMoon.


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Notethatifyoucanmeasuretheparallaxangle,andyouknowthebaseline,youcan

determinethedistancetotheobject.ThisishowtheAncientGreeksdeterminedthe

distancetotheMoon.Buttheycouldnotdetectanyparallaxinthestars.Thiswas

easilyexplained:thestarswereveryfarawaycomparedtotheEarthsdiameter

(b


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beengreatlyexaggeratedinordertomaketheeffectvisibleinthesimulation,but

theplotillustratesthecorrectpatternofchangesinDeclinationresultingfrom

annualparallax.

In1674Hookepublishedhisresultsandclaimedtohavedetectedannualparallax.5

Althoughhegaveadetaileddiscussionofhiscarefulmeasurementprocedure,hepresentedonlyfourobservations.Problemswiththetelescopeandhisownhealth

preventedhimfromcontinuingthework.HookesdataisshowninFigure3.A

comparisonwiththeparallaxpredictioninFig.2showsthatHookesdataseemsto

matchtheexpectedpatternforannualparallax,buthiscontemporariesdidnotfindfourobservationsmadewithanunreliabletelescopeveryconvincing.Theracefor

parallaxwasnotyetover.

AberrationofStarlight

SeveralastronomersattemptedtofollowuponHookesmeasurement.G.D.Cassini

andJeanPicardmeasuredvariationsinthepositionofPolaris,butwithinconclusiveresults.AstronomerRoyalJohnFlamsteedthoughthehadmeasuredanannual

parallaxforPolarisuntilCassinipointedoutthatFlamsteedsdatadidnotfitthe

expectedpattern.6

In1725,SamuelMolyneuxandJamesBradleysetouttorepeatHookes

measurementsofGammaDraconisbyconstructingazenithtelescopeinMolyneuxs

mansionatKewnearLondon.Theyfoundthatthestarvarieditsposition,butnotin

thewayreportedbyHooke.Bradleyfolloweduponthisworkbymeasuringseveral

morestarsusingashorterzenithtelescopeinhisresidenceatWanstead.Fig.3

showsBradleysdataforGammaDraconisandAlkaid(EtaUrsaMajoris).Thedata

showthesinusoidalvariationexpectedforparallax,butthephaseis3monthsofffromthepredictionsshowninFig.2!

BradleyknewofOleRmers1676estimateofthefinitespeedoflight,andeventuallyusedthisideatodeviseanexplanationforhisdata.Bradleyrealizedthat

ittakesthelightfromastarafinitetimetotravelthroughatelescopetube,andduringthistimethetubemovesslightlybecauseofthemotionofEarth.Bradleys

theoryisillustratedinFigure4forthecaseofastarthatliesdirectlyoverhead.The

starlightentersthetopofthetubeatp1.Itthentravelsadistancehbeforereaching

theeyepieceatp3.Ifcisthespeedoflight,thenittakesatimeh/cforthelightto

travelthisdistance.Duringthistimethetelescopehasmovedadistancevh/c,

wherevisthevelocityofEarthsorbitalmotion.7Asaresult,thetelescopecannotbeaimeddirectlyuptowardthestar.Itmustbetiltedslightlytowardthedirection

ofthetelescopesmotion.Thisphenomenonisnowcalledtheaberrationofstarlight.

ApplyingtrigonometrytoFig.4wefindthat

tan =!

!

, (2)whereistheangleoftilt.ForaninteractiveanimatedversionofFig.4seethe

StellarAberration2Dprogram.3


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Fig.4illustratestheapparentdisplacementofthestarduetoaberrationatasingle

moment in time. As the Earth moves around in its orbit, its velocity changes

direction and therefore the displacementof starsdue toaberrationwill change.8

Thepatternofapparentmovementdependsonthelocationof thestarin thesky.

These patterns are illustrated in the StellarAberration3D program.3

ThisprogramwasusedtoconstructplotsoftheapparentdeclinationofGammaDraconis

and Alkaid, usingBradleys theory ofstellar aberration, as shown inFigure 5. A

comparisonwithFig.4showsthatthepatternpredictedbyBradleystheoryfitshis

observationaldata.

Bradleysdataindicatesadisplacementof20.2secondsofarcforastarthatliesina

directionperpendiculartoEarthsmotion.UsingEquation2Bradleyfoundthatthespeedoflightmustbe10,210timesasgreatasEarthsorbitalspeed,solighttakes8

minutes,12secondstotravelfromtheSuntoEarth.9

BradleystheoryexplainedthemovementofPolarisobservedbyCassini,Picard,andFlamsteed,buthecouldnotgiveanexplanationforHookesmeasurementsofGammaDraconis.Hookeapparentlysawwhathewantedtoseeamidstastormof

instrumentalerror.Bradleyshowedthathisdatafittheaberrationtheorysowell

thatanydifference,whichmightbeduetoannualparallax,wasprobablylessthan

onehalfsecondofarc.ThismeantthatGammaDraconiswasmorethan400,000

timesfartherfromusthantheSun.

ItwasimpossibletomeasureparallaxusingHookesmethodwithoutfirstknowing

aboutaberration.AsEq.2shows,theangulardisplacementduetoaberrationdoes

notdependonthedistancetoastar.ButasEq.1shows,theparallaxangle

decreaseswithdistance.SinceGammaDraconisisrelativelybright,andthusprobablynearby,itmightbeexpectedtohaveoneofthelargerparallaxes.Evenso,BradleyhadshownthataberrationcompletelyswampedparallaxforGamma

Draconis,andformoredistantstarsthesituationwouldonlygetworse.Itwasonlybymeasuringtherelativemotionsoftwostars,closetogetherinourskybutfar

apartinspace,thatastronomerslikeBessel,Struve,andHendersonwouldfinally

measuretheannualparallaxofstarsinthe1830s.Thisdoublestarmethod,however,wouldnothaveledtothediscoveryofaberrationbecauseaberration

affectsbothstarsequally.10

Lessons

ThereareseverallessonsthatcanbelearnedfromthestoriesofHookeand

Bradley.11Hookesclaimtohavemeasuredparallaxshowsusthatreliable

conclusionscannotbedrawnfromasmallnumberofmeasurements.Aconclusive

demonstrationrequiresaseriesofmanymeasurements,allofwhichfitadefinite

patternliketheoneestablishedbyBradley.Bradleysstoryalsoshowsusthat

scientistssometimessucceedinunexpectedways.Bradleysoughttoconfirm

Earthsorbitalmotionbydetectingannualparallax.Instead,heconfirmeditby


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detectingandexplainingstellaraberration.Buthissuccesswasnotcomplete:a

parallaxmeasurementwouldhavegiventhedistancetothestar,whilea

measurementofaberrationprovidesnoinformationaboutstellardistances.

Bradleystheoryofaberrationalsoshowsusthatgoodscientificexplanationscan

drawtogetheruntestedhypothesestoexplainsomethinginawaythatgivessupporttoallofthehypotheses.NeithertheorbitofEarthnorthefinitespeedof

lighthadbeenindependentlyconfirmedbefore1725(althoughNewtonsPrincipia

hadleftlittledoubtaboutEarthsorbit),butBradleyshowedthathisdatawere

easilyexplainedifbothofthesehypothesesweretrue.

Finally,thesestoriesshowthatassigningcreditforadiscoveryistrickybusiness.Hookeclaimedtomeasureparallax,buthisclaimisnowdiscredited.Flamsteed

measuredaberrationbutthoughthewasmeasuringparallax.Cassinirecognized

thatFlamsteedwasmeasuringsomethingotherthanparallax,butitwasBradleywhoexplainedhisdataandthereforeiscreditedwiththediscoveryofaberration.

TheactivitiesandcomputersimulationsinRef.3,whicharedesignedtoguide

studentsthroughanexplorationofthishistoricalepisode,illustratetheseimportant

lessonsaboutthenatureofscience.

Figures

Figure1:Parallaxofthemoon,viewedfromoppositesidesofEarth.NOTTOSCALE!

2b

d

O1

O2

M1

M2

EarthMoon

Celestial

Sphere


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Figure2:PredictedvariationinDeclinationresultingfromtheannualparallaxofGammaDraconis(magenta)andAlkaid(blue).Theamplitudeofthevariationis

greatlyexaggeratedsoastobevisibleinthesimulation.


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Figure3:Observednorthwarddeviationofstars.ThedatapointsshowHookes

dataforGammaDraconis(blacktriangles)andBradleysdataforGammaDraconis

(magentasquares)andAlkaid(bluetriangles).

0"

5"

10"

15"

20"

25"

30"

35"

40"

0" 0.25" 0.5" 0.75" 1"

NorthwardDevia.on(''

Time)of)Year)(years)from)Vernal)Equinox)


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Figure4:Illustrationoftheaberrationeffect,showingthetelescopetubeTatthree

differenttimes.Becausethetelescopemovesadistancevh/cduringthetimeit

takeslighttotraveldownthetube,thetubemustbetiltedinthedirectionofmotioninorderforthelighttomovealongtheopticalaxisofthetube.SoastaratSwill

appeartobeatS.


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Figure5:PredictedvariationinDeclinationresultingfromaberrationforGammaDraconis(magenta)andAlkaid(blue).Theamplitudeofthevariationisgreatly

exaggeratedsoastobevisibleinthesimulation.

1ChristopherM.Graney,TeachingGalileo?GettoKnowRiccioli!WhataForgotten

ItalianAstronomerCanTeachStudentsAboutHowScienceWorks,Phys.Teach.50,18-21(2012).2AlanHirshfeld,Parallax:TheRacetoMeasuretheCosmos(WHFreeman,2001).3Aself-containedJavaprogramcontainingallofthesimulationsisavailableat

www.opensourcephysics.org/items/detail.cfm?ID=12029.The

activityhandoutsareavailableassupplementarydocumentsonthesamepage.4HookeobservedgammaDraconisindaylight,aswellasatnight.Hemayhave

beenthefirstpersontoobserveastarthroughatelescopeduringtheday.5RobertHooke,AnAttempttoProvetheMotionsofEarthbyObservations(London,

1674).Availableonlineat

http://www.roberthooke.com/motion_of_the_earth_001.htm.6

M.E.W.Williams,FlamsteedsAllegedMeasurementofAnnualParallaxforthePoleStar,J.Hist.Astron.10,102-116(1979).7ThetelescopealsomovesduetoEarthsrotation,buttherotationalvelocityis

about64timessmallerthantheorbitalvelocity,sotoagoodapproximationwecanignoretheeffectofrotation.InanycasetheeffectBradleywastryingtoexplainhad

anannualperiod.


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8Notethatafixedaberrationanglewouldbeundetectable.Foradiscussionofthis

point,andhowaberrationrelatestospecialrelativity,seeThomasE.Phipps,Jr.

Relativityandaberration,Am.J.Phys.57,549-551(1989).9JamesBradley,ALettertoDr.EdmondHalleyAstronom.Reg.&c.givingan

AccountofanewdiscoveredMotionoftheFixdStars,Phil.Trans.35,637-661(1729).10IdliketothankCharlieHolbrowofColgateUniversityforthisinsight.11MichaelHoskin,StellarAstronomy:HistoricalStudies(ScienceHistoryPublications,

1986),pp.29-36.

seeing earth's orbit in the stars

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