TheRetrogradeMotionofMarswithaSextant
BobCava,BobVanderbei,andLizSeibel,PrincetonNJ
This project started when one of us (Bob C.) came across a statue of Jan Hevelius in Gdansk Poland.
(Figure1).Heveliusmappedthemoonandstarsusingnakedeyeobservations350yearsago,sowhynot
tryitnow?Monsterangle-measuringinstrumentslikeHeveliususedarenotthingsthatyouseeforsale
nowadays,butthemodernequivalentisa nauticalsextant.Professionalqualityrealnauticalsextants
aretooexpensiveforjustfoolingaround,andsoaplasticoneandtwobooksabouthowtouseit(“How
touseplasticsextants”byDavidBurchand“TheSextantHandbook”byBruceBauer)werewhattoget
totrythiskindofthing.DuringatriptoFloridainearly2016,Bobandhisbrother-inlawmeasuredhigh
noon with the plastic sextant as a test and got their position on earth right to within about 3 miles,
showingthatitispossiblefornovicestouseasextantreasonablywell.Alittlelater,thissamebrotherin-law found a used, 1970s vintage professional metal sextant at a garage sale that cost less than the
newplasticsextantdidinthefirstplaceandsentituptoNewJersey.
Sowithsextantsinhand(Figure2)itwastimetodosomemeasuring.Theplasticsextantismuchflakier
opticallyandmechanicallythantherealsextant,butit’slighterandsowasmuchlesstiringtouseduring
themeasurements, which wereoften time-consuming. Also, the second-hand real sextant has smaller
mirrorsthantheplasticone,makingfainterstarshardertosee.Forcalibrationpurposes,Bobmeasured
thedistancesbetweentenpairsoffixedstarswithseparationsbetweenabout3degreesand25degrees
withbothsextantsandcomparedhisresultstotheknownangularseparations-theplasticsextantgot
theanglesrightto(RMSerrors)within0.05degreesonaverageandtherealsextantgottheanglesright
towithin0.02degreesonaverage,inneithercasewithanymeasurementoutliers.Themeasurements
seemedgood,and,intheend,theincreasedaccuracyoftherealsextantgenerallywasn’tworththearm
fatigueandeyestrainforwhathadtobedone,soBobmostlystuckwiththelightweight,plasticsextant;
therealsextantservedperiodicallyasasanitycheck.
The original idea was to mount the sextant on a lightweight tripod with a photo camera ball head to
eliminate fatigue and to steady things up, but because the measurements for the project involved
defining the plane of measurement every time for three points – observing eye, Mars, and reference
star - it turned out to be much easier to just find the orientation of the measurement plane by hand.
Andsincesextantsaregoodformeasuringanglesonrockingships,justhand-holdingonewasthebest
way to get good measurements anyway – “steadying things up” was not necessary and would have
probablybeenworseduetotherockingtechniqueneededtomakethemeasurements
Marsseemedlikeagoodchoiceforfollowingwiththesextantsbecauseitspropermotionagainstthe
fixed stars would be large, and adding to the fun would be that around opposition it would exhibit
retrogrademotion.ObservationsinvolvedmeasuringtheangularseparationsbetweenMarsandthree
to seven reference stars, and often Saturn (which moves too), and sometimes involved playing
peekaboowithcloudsthatcoveredsomeofthetargetobjectsforlongperiodsoftimeinthemiddleof
the measurements. Antares and beta and delta Scorpio, zeta Ophiuchus and beta Libra were used as
reference stars on many nights; and sigma and tao Scorpio and alpha Libra were used whenever they
werevisibleinspiteofthelightpollutionandgeneralhazeinPrinceton.Measurementsweremadeof
Mars’positionon68nightsoveraperiodof7monthstocollectthedata.Luckily,itwaspossibletocrawl
outofbedintheweemorninghourstotaketheearlymeasurementsfrominsideasunroom,(Itscoldin
NewJerseyinFebruary!)butsomeofthemeasurementsweremorechallenging–onbusinesstripsto
variousplaces,andonacruiseshipatoddhours,forexample.Eventuallythedramabegantounfold
duringmorenormalobservinghours,finallyendingaroundduskinthefall,whensomeofthefaintstars
weretoughtoseeinthetwilightandmosquitoshadbecomeafactor.
Itfeltgoodtofinishtheobservingproject,andthatcouldhavebeentheendofit.ButthenBobrealized
thatthedatacouldprobablybeanalyzedquantitatively–thatgiventhedatesandthepositionsofMars,
someone(butnothim)couldprobablyextractquantitativeinformationabouttheorbitofMarsfromthe
observations.ThisiswhereLizandBobVcamein.Lizdidthefirstroundofdataplottingandanalysis
(SeeFigure3forexample),andthenBobV.analyzedthedatatodetermineMars’orbitthroughafitto
anorbitalmodel.
Asinput,Bobtookthedatesoftheobservationsandtheangularseparationstothereferencestars,and
thendeterminedMar’spositionineclipticcoordinatesforall68observingsessionsusingalinearleastsquaresalgorithm.(SeeFigure4.)HethenfedthedatesandpositionsofMarsintoahighlynonlinear
least-squares regression model where he assumed that Mars travels in a circular orbit inclined to the
planeoftheearth’sorbit.Thefittingparametersweretherelativeradiioftheorbitsoftheearthand
Mars, the angle of Mar’s orbital plane with that of earth (aka the ecliptic latitude), and the ecliptic
longitudeoftheascendingnode.Theresultingleast-squares-fittedorbitandtheobservationsareshown
inFigure5.ThenumbersBobgotforMars’orbitalparametersare:Mars’orbitSemimajoraxis=1.530
au,Inclination=2.1degrees,andLongitudeofAscendingNode=48.5degrees.Comparethesetothe
known values: Semimajor axis = 1.524 au, Inclination = 1.85 degrees, Longitude of Ascending Node =
49.56degreesandtheylookprettygood,right?Toobadwewereafewhundredyearstoolatetobe
firsttofigureitout.Noticebycomparingtheobservedandcalculatedpositionsonaparticulardate(see
theasterisksinFigure5)thatthemodelfitisnotperfect.Wethinkthattheobservationsareoveralong
enough period, 7 months, that the ellipticity of Mars’ orbit should likely be taken into account – but
fittingthedatausingKepler’slawmakesthingsmorecomplicatedandwehaven’ttriedityet.
Thiswasafunproject,notintheleastbecauseitmotivatedoneofustogoouttoobserveonmanynice
nightswhenhemighthavebeenasleepordoingsomethingboringinsteadofwatchingsomethingcool
happeninginthesky.Also,analyzingthedatatogetherwaslotsoffun.ThedataclearlyshowthatMars
really does move a lot and even goes backwards quite a bit with respect to the fixed stars during an
opposition-somethingonedoesn’talwayspayattentionto.Finally,whatcouldmorefunthanfooling
around with unfamiliar equipment like sextants, taking some measurements on beautiful nights, and
thenfittingthedatatodetermineaplanetaryorbit?Atleastoneofus(BobC.)can’tthinkofanything
thatwouldbebetter.ByfollowingMarswithasextantforsevenmonths,andthenfiguringoutitsorbit
byfittingthedata,wesuregainedalotofrespectfortheingenuityandgritofearlyastronomerslike
Hevelius.
Figure1Theinspirationfortheproject.Oneoftheauthors(BobC.)inGdanskPolandwithastatueof
JanHevelius.
Figure2Thetwosextantsusedtomakethemeasurements.Theplasticsextantisontheleftandthe
“real”one,asecond-handprofessionalsextantfromthe1970s,isontheright.
Figure3.Thefirststepsinthedataanalysis.MeasuredseparationsofMars(indegrees)tosomeofthe
nearbyfixedstars(andSaturn),plottedversuslocalcalendardate.Annoyingclouds,haze,themoon,out
of town trips, and light pollution created some gaps in the data, and the separation to Saturn wasn’t
measureduntilabouthalfwaythroughtheproject.
Figure 4 How the position of Mars was determined. The circles show the sextant-measured angular
separations of Mars from 7 reference stars (September 3 2016 had a nice clear night.) Mars is at the
intersectionofallthecircles,andthereferencestarsinScorpio,OphiuchusandLibraareshownasstar
symbols.Longitudeshownonthehorizontalaxis,altitudeontheverticalaxis,ineclipticcoordinates.
Figure5TheretrogrademotionofMarswithasextant.2016apparition.Observations:yellowpoints
andline,fitteddata:purplepointsandline.MarsstartedattheupperrightofthispathinFebruaryand
departedonthelowerleftofthepathinSeptember.Theretrogrademotion,resultinginanS-shaped
path,isclearlyseen–oppositionisinthemiddleoftheretrogradesegment.Partsoftheconstellations
inthearea(Scorpio,LibraandOphiuchus.)areshownbybluelines,withthestarsasopencircles.Thefit
wasusedtodeterminetheorbitofMars.