TheRetrogradeMotionofMarswithaSextant
BobCava,BobVanderbei,andLizSeibel,PrincetonNJ
This project startedwhenoneof us (BobC.) came across a statue of JanHevelius inGdansk Poland.(Figure1).Heveliusmappedthemoonandstarsusingnakedeyeobservations350yearsago,sowhynottryitnow?Monsterangle-measuringinstrumentslikeHeveliususedarenotthingsthatyouseeforsalenowadays,but themodernequivalent isanauticalsextant. Professionalqualityrealnauticalsextantsaretooexpensiveforjustfoolingaround,andsoaplasticoneandtwobooksabouthowtouseit(“Howtouseplasticsextants”byDavidBurchand“TheSextantHandbook”byBruceBauer)werewhattogettotrythiskindofthing.DuringatriptoFloridainearly2016,Bobandhisbrother-inlawmeasuredhighnoonwith the plastic sextant as a test and got their position on earth right towithin about 3miles,showingthatitispossiblefornovicestouseasextantreasonablywell.Alittlelater,thissamebrother-in-law foundaused,1970svintageprofessionalmetal sextantatagaragesale thatcost less than thenewplasticsextantdidinthefirstplaceandsentituptoNewJersey.Sowithsextantsinhand(Figure2)itwastimetodosomemeasuring.Theplasticsextantismuchflakieropticallyandmechanicallythantherealsextant,butit’slighterandsowasmuchlesstiringtouseduringthemeasurements,whichwereoftentime-consuming.Also, thesecond-handrealsextanthassmallermirrorsthantheplasticone,makingfainterstarshardertosee.Forcalibrationpurposes,Bobmeasuredthedistancesbetweentenpairsoffixedstarswithseparationsbetweenabout3degreesand25degreeswithbothsextantsandcomparedhisresultstotheknownangularseparations-theplasticsextantgottheanglesrightto(RMSerrors)within0.05degreesonaverageandtherealsextantgottheanglesrighttowithin0.02degreesonaverage, inneithercasewithanymeasurementoutliers.Themeasurementsseemedgood,and,intheend,theincreasedaccuracyoftherealsextantgenerallywasn’tworththearmfatigueandeyestrainforwhathadtobedone,soBobmostlystuckwiththelightweight,plasticsextant;therealsextantservedperiodicallyasasanitycheck.Theoriginal ideawas tomount the sextantona lightweight tripodwithaphoto cameraballhead toeliminate fatigue and to steady things up, but because the measurements for the project involveddefining theplaneofmeasurementevery time for threepoints–observingeye,Mars, and referencestar - it turnedout tobemucheasier to just findtheorientationof themeasurementplanebyhand.Andsincesextantsaregoodformeasuringanglesonrockingships,justhand-holdingonewasthebestway to get goodmeasurements anyway – “steadying things up”was not necessary andwould haveprobablybeenworseduetotherockingtechniqueneededtomakethemeasurementsMarsseemedlikeagoodchoiceforfollowingwiththesextantsbecause itspropermotionagainstthefixed stars would be large, and adding to the fun would be that around opposition it would exhibitretrogrademotion.Observations involvedmeasuringtheangularseparationsbetweenMarsandthreeto seven reference stars, and often Saturn (which moves too), and sometimes involved playingpeekaboowithcloudsthatcoveredsomeofthetargetobjectsforlongperiodsoftimeinthemiddleofthemeasurements.Antares andbeta anddelta Scorpio, zetaOphiuchus andbeta Librawereusedasreferencestarsonmanynights;andsigmaandtaoScorpioandalphaLibrawereusedwhenever theywerevisibleinspiteofthelightpollutionandgeneralhazeinPrinceton.MeasurementsweremadeofMars’positionon68nightsoveraperiodof7monthstocollectthedata.Luckily,itwaspossibletocrawloutofbedintheweemorninghourstotaketheearlymeasurementsfrominsideasunroom,(ItscoldinNewJerseyinFebruary!)butsomeofthemeasurementsweremorechallenging–onbusinesstripsto
variousplaces,andonacruiseshipatoddhours, forexample. Eventuallythedramabegantounfoldduringmorenormalobservinghours,finallyendingaroundduskinthefall,whensomeofthefaintstarsweretoughtoseeinthetwilightandmosquitoshadbecomeafactor.Itfeltgoodtofinishtheobservingproject,andthatcouldhavebeentheendofit.ButthenBobrealizedthatthedatacouldprobablybeanalyzedquantitatively–thatgiventhedatesandthepositionsofMars,someone(butnothim)couldprobablyextractquantitativeinformationabouttheorbitofMarsfromtheobservations.This iswhereLizandBobVcame in.Lizdidthefirst roundofdataplottingandanalysis(SeeFigure3forexample),andthenBobV.analyzedthedatatodetermineMars’orbitthroughafittoanorbitalmodel.Asinput,Bobtookthedatesoftheobservationsandtheangularseparationstothereferencestars,andthendeterminedMar’spositionineclipticcoordinatesforall68observingsessionsusingalinearleast-squaresalgorithm. (SeeFigure4.)HethenfedthedatesandpositionsofMars intoahighlynonlinearleast-squares regressionmodelwhereheassumed thatMars travels ina circularorbit inclined to theplaneoftheearth’sorbit.Thefittingparametersweretherelativeradiiof theorbitsof theearthandMars, the angle ofMar’s orbital plane with that of earth (aka the ecliptic latitude), and the eclipticlongitudeoftheascendingnode.Theresultingleast-squares-fittedorbitandtheobservationsareshowninFigure5.ThenumbersBobgotforMars’orbitalparametersare:Mars’orbitSemimajoraxis=1.530au, Inclination=2.1degrees,andLongitudeofAscendingNode=48.5degrees.Comparethesetotheknown values: Semimajor axis = 1.524 au, Inclination = 1.85degrees, LongitudeofAscendingNode=49.56degreesandthey lookprettygood,right?Toobadwewerea fewhundredyearstoo latetobefirsttofigureitout.Noticebycomparingtheobservedandcalculatedpositionsonaparticulardate(seetheasterisksinFigure5)thatthemodelfitisnotperfect.Wethinkthattheobservationsareoveralongenoughperiod, 7months, that the ellipticity ofMars’ orbit should likely be taken into account – butfittingthedatausingKepler’slawmakesthingsmorecomplicatedandwehaven’ttriedityet.Thiswasafunproject,notintheleastbecauseitmotivatedoneofustogoouttoobserveonmanynicenightswhenhemighthavebeenasleepordoingsomethingboringinsteadofwatchingsomethingcoolhappeninginthesky.Also,analyzingthedatatogetherwaslotsoffun.ThedataclearlyshowthatMarsreallydoesmovea lot andevengoesbackwardsquite abitwith respect to the fixed starsduringanopposition-somethingonedoesn’talwayspayattentionto.Finally,whatcouldmorefunthanfoolingaroundwith unfamiliar equipment like sextants, taking somemeasurements on beautiful nights, andthenfittingthedatatodetermineaplanetaryorbit?Atleastoneofus(BobC.)can’tthinkofanythingthatwouldbebetter.ByfollowingMarswithasextantforsevenmonths,andthenfiguringoutitsorbitbyfittingthedata,wesuregaineda lotofrespectforthe ingenuityandgritofearlyastronomers likeHevelius.
Figure1Theinspirationfortheproject.Oneoftheauthors(BobC.)inGdanskPolandwithastatueofJanHevelius.
Figure2Thetwosextantsusedtomakethemeasurements.Theplasticsextantisontheleftandthe“real”one,asecond-handprofessionalsextantfromthe1970s,isontheright.
Figure3.Thefirststepsinthedataanalysis.MeasuredseparationsofMars(indegrees)tosomeofthenearbyfixedstars(andSaturn),plottedversuslocalcalendardate.Annoyingclouds,haze,themoon,outof town trips, and lightpollution created somegaps in thedata, and the separation toSaturnwasn’tmeasureduntilabouthalfwaythroughtheproject.
Figure 4How the position ofMarswas determined. The circles show the sextant-measured angularseparationsofMars from7 reference stars (September32016hadanice clearnight.)Mars is at theintersectionofallthecircles,andthereferencestarsinScorpio,OphiuchusandLibraareshownasstarsymbols.Longitudeshownonthehorizontalaxis,altitudeontheverticalaxis,ineclipticcoordinates.
Figure5TheretrogrademotionofMarswithasextant.2016apparition.Observations:yellowpointsandline,fitteddata:purplepointsandline.MarsstartedattheupperrightofthispathinFebruaryanddepartedonthe lower leftof thepath inSeptember.Theretrogrademotion,resulting inanS-shapedpath,isclearlyseen–oppositionisinthemiddleoftheretrogradesegment.Partsoftheconstellationsinthearea(Scorpio,LibraandOphiuchus.)areshownbybluelines,withthestarsasopencircles.ThefitwasusedtodeterminetheorbitofMars.
