Ruler,Pencil,Paper

MaterialsNeeded—

•Ruler,pencil,paper

Objectives—

AnIntroductiontoParallax

—OR—

JustHowFarAwayIsThatStar,Anyway?

•Understandthenatureofparallaxandthesourceofthiseffectinobservationalastronomy.

•Observehowparallaxchangeswithdistanceandapplythistomeasuringtherelativedistancesofstars.

•Introducetheconceptoftheastronomicaldistanceladder,ofwhichparallaxdistancesarethefirstrung,andwhichallowstoestimatedistancestothefarcornersoftheobservableuniverse.

Introduction—Parallaxisageometricaleffectthatcanbeusedtoobtainadirectmeasurementofthedistancetoanobject.Adriverandherpassenger,forexample,mayfallpreytothiseffectwhenarguingaboutacar’sspeed.Ifthecarusesaneedle–typespeedometer,wheretheneedleismountedslightlyinfrontofitsdialscale,thetwoarelikelynottomakethesamereadingwhencheckinghowfastthecarismoving. Thedrivermayseethatthecarisgoing60mph(i.e.,theneedleappearsprojectedagainstthe60markonthedial),butthepassengermaythinkthatthecarismovingatsomeotherspeed.Thisisduetothegapbetweenneedleanddialandtothefactthatthedriverandpassengerobservethegagefromdifferentdistancesandviewingangles.

Butparallaxisn’tjustasourceofarguments:asthisactivitywillshow,it’sapowerfultoolformeasuringdistances,somethingthat

Fig.1.—TychoBrahe,oneofthegreatobservationalastronomersofanyera,andthemostfamousbrawlingastronomerever.Helosthisnoseinaduelatage20,andworeafakefortherestofhislife.TheportraitisbyanunknownartistandreproducedwithpermissionfromtheWikipediaCommons.

isincrediblydifficultinastronomy.Thefirstrecordedinstanceoftheuseofparallaxinastronomycamein1577,whenTychoBrahe(seeFigure1)usedittofindthedistancetoacomet.WhilewenowhaveothertechniquesformeasuringthedistancestoobjectsintheSolarSystem,parallaxremainstheonlywaytodirectlymeasurethedistancetoobjectsoutsideofourSolarSystem—particularlytostars.Inthisactivityyouwillfirstbuildupyourintuitionaboutwhatparallaxisandhowitdependsondistance,andthenapplythistoobservationsofstars1.

PartI:Whatisparallax?

1.Predictwhatyourfingerwouldappeartodorelativetothebackgroundifyouweretoputitabout3–4inchesfromyourfaceandcloseoneeyeatatimewhilewatchingit.

2.Nowgoaheadanddothis.Wasyourpredictioncorrect?Comment.

3.Predict howtheapparentmotionofyourfingerwouldchangeifyoumovedyourfingertwiceasfarfromyourface?

4.Nowdothis.Wasyourpredictioncorrect?Comment.

5.IfyouhadamazingStretchArmstrongarms,istherealimittohowfaryoucouldmoveyourfingerandstillseeitappeartomove?Ifso,howfar awaydoyouthinkthatwouldbe?(To getanideaofthisdistance,havesomeonefarawayfromyouholduptheirfinger.)

6.Whatisitaboutoureyesthatallowsustoseethisapparentmotion?

PartII:Whathappenswhenyourfingerisastar?

7.ThepointlabeledEinFigure2representstheEarthinitsorbitinJanuary.UsearuleranddrawalinefromtheEarthtothebackgroundstarsgoingthroughStarA.MarktheEarth–StarA–Sunangle.Thisiscalledtheparallax.

8.DescribewhatyouthinkwillhappentothatangleifweweretodowhatwejustdidbutforStarB.

9.TestyourpredictionbyusingFigure2.Commentonyourresults.Wereyoucorrect?

10.Now,findwheretheEarthwillbeinsixmonths.

TheimaginarylinethatrunsfromtheEarth’spositioninJanuarythroughtheSuntotheEarth’spositioninJuly,isourbaseline.Ingeneral,thisistheseparationbetweenthepositionsfromwhichtwomeasurementsaremade,anditisperpendiculartothedirectiontotheobjectwhosedistancewearemeasuring.

Repeatquestion7usingstarAatthisnewpositioninitsorbit.DrawonFigure3belowwhereStarAwillappeartobeinJanuaryandthensixmonthslater.

11.Extendourobservationsoveranumberofyears.HowwillStarAappeartomoveagainstthebackgroundstars?

12.HowaboutstarBoverthesamenumberofyearscomparedtothemotionofStarA?

13.TheapparentmotionofstarsasseenfromEarthrelativetoabackgroundofmoredistant,fixedstars,isknownasstellarparallax.Now think backto yourexperiment inpartI.WhatisitabouttheEarththatcorrespondstoblinkingyoureyes?

Fig.2.—Viewof Earth,Sun, two nearbystars,and a bunchofdistantstars.

Fig.3.— WherewouldStarAappearrelativetothebackgroundinJanuary?Sixmonthslater?

14.Belowisasetofparallaxobservationsofdifferentstars.Rankthemfromnearesttofarthest.Explainyourlogic.Nearest Farthest.

PartIII:Whathappenswhenyoureyesarefar,farapart?

15.WeusedFigure2toexaminetheparallaxangleofStarsAandB.Let’sfocusonStarAfornow.WhatwouldhappentotheparallaxangleofAifwemeasureditfromMarsinsteadofEarth?

16.Considerthisconversationbetweentwostudents:

Student1:IthinkthatifwemeasuredtheparallaxofastarfromMars,theanglewouldbelargerthanifwemeasureditfromEarthbecauseMarshasamuchlargerorbit.Thiswouldcausethestartomoveananglecomparabletothatofitsorbit.

Student2:IfwemeasuredtheparallaxfromMars,theanglewouldhavetobesmallerbecauseMarsisfartherfromtheSunsothestarwouldalsohavetobefartherawayfromMars.

Withwhomdoyouagree?Explain.

17.NowtestyourpredictionsaswellasthoseofStudent1andStudent2.Dothisby repeatingwhat wedidinquestion7,butusetheorbitofMarsinFigure2.Whatareyourresults?Whywouldalongerbaseline bedesirable?

Bonus:

1.WhymighttheargumentbetweenpassengeranddriverbeworseinaBritishcar?

2.WhatothertechniquescanweusetomeasurethedistancestoobjectsintheSolarSystem?Whywon’ttheseworktomeasurethedistancetoeventheneareststar?

3.Successfulstellarparallaxmeasurementsareafairlyrecentaccomplishment.Why?

PartIV:WhathappenswhenyourfingerisREALLYfaraway?

Eventhenearest star,Alpha Centauri,ismorethan200,000timesfurtherawaythanthediameteroftheEarth’sorbit.ThismeansthattheshiftinangleweobserveinAlphaCentauriislessthan1secondofarc,orlessthanthethicknessofahairseenacrossalargerooma.Itwasnotuntilthemid–19thcenturythat astronomers wereable tomeasuresuchsmallparallaxes.Inrecentyears,goodparallaxmeasurements,previouslylimitedtoafewthousandstars,havebeenextendedtogreaterdistancesbythescientificsatelliteHipparcos.Butultimately,evenfromspace,parallaxdistancescanonlybemeasuredforarelativelysmallnumberofobjectsthatarerelativelynearby.

Afulldiscussionofhowweestimatedistancestoobjectsforwhichwecannotobtainparallaxmeasurementsisbeyondthescopeofthisactivity.Butitisimportant to notethatultimatelyallofthedifferenttechniquesweuselinkbacktoparallaxmeasurements,whichiswhyparallax is oftenreferred to as thefirststepintheastronomicaldistanceladder(seeFigure4).

Forasuiteofactivitiesabouttheexpand-inguniverseandthedistanceladder,seeThistext and Figure 4areadaptedfrom“Theladderofastronomicaldistances”(TAP704–8)handoutavailablethere.

aYoumayhaveheardastronomersuseparsecswhendiscussingdistances.Aparsecisthedis-tanceatwhichastarwouldshiftby1secondofarctoandfroastheEarthmovesrounditsorbit,andit’sequaltoroughly3light–years.

Fig.4.—FromtheInstituteofPhysics,