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INVESTIGATING THE ORIGIN OF THE PIONEER ANOMALY THROUGH JAVA COMPUTER MODELING

Permanent Link: http://ncf.sobek.ufl.edu/NCFE004796/00001

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Title: INVESTIGATING THE ORIGIN OF THE PIONEER ANOMALY THROUGH JAVA COMPUTER MODELING
Physical Description: Book
Language: English
Creator: Kiker, Kathleen
Publisher: New College of Florida
Place of Publication: Sarasota, Fla.
Creation Date: 2013
Publication Date: 2013

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Subjects / Keywords: Pioneer Anomaly
Computer Modeling
Astronomy
Genre: bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract: Analysis of the radio tracking data of Pioneers 10 and 11 over the last 30 years has indicated a small unmodeled acceleration of 8:74 + 1:33 x 108 cm/s2 that is directed towards the sun. This acceleration appears between 20 and 70 AU. Recent data has indicated that the acceleration may be decreasing with time at a rate of 2 x 1011 m/s2/yr and that it is consistent with a thermal recoil force caused by the Pioneer's on-board electrical systems. In this thesis I review the history of the anomaly and the various mechanisms proposed to explain it. I test several of these possibilities in a Java computer simulation and conclude that, while one of the gravitational theories produces a force large enough to account for the anomaly, it appears to be inconsistent with the orbits of the planets, and that the best explanation for the anomaly is most likely a thermal recoil force produced by the onboard systems, however, a MOND gravity force cannot be ruled out by this study.
Statement of Responsibility: by Kathleen Kiker
Thesis: Thesis (B.A.) -- New College of Florida, 2013
Electronic Access: RESTRICTED TO NCF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE
Bibliography: Includes bibliographical references.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The New College of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Local: Faculty Sponsor: Ruppeiner, George

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Permanent Link: http://ncf.sobek.ufl.edu/NCFE004796/00001

Material Information

Title: INVESTIGATING THE ORIGIN OF THE PIONEER ANOMALY THROUGH JAVA COMPUTER MODELING
Physical Description: Book
Language: English
Creator: Kiker, Kathleen
Publisher: New College of Florida
Place of Publication: Sarasota, Fla.
Creation Date: 2013
Publication Date: 2013

Subjects

Subjects / Keywords: Pioneer Anomaly
Computer Modeling
Astronomy
Genre: bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Analysis of the radio tracking data of Pioneers 10 and 11 over the last 30 years has indicated a small unmodeled acceleration of 8:74 + 1:33 x 108 cm/s2 that is directed towards the sun. This acceleration appears between 20 and 70 AU. Recent data has indicated that the acceleration may be decreasing with time at a rate of 2 x 1011 m/s2/yr and that it is consistent with a thermal recoil force caused by the Pioneer's on-board electrical systems. In this thesis I review the history of the anomaly and the various mechanisms proposed to explain it. I test several of these possibilities in a Java computer simulation and conclude that, while one of the gravitational theories produces a force large enough to account for the anomaly, it appears to be inconsistent with the orbits of the planets, and that the best explanation for the anomaly is most likely a thermal recoil force produced by the onboard systems, however, a MOND gravity force cannot be ruled out by this study.
Statement of Responsibility: by Kathleen Kiker
Thesis: Thesis (B.A.) -- New College of Florida, 2013
Electronic Access: RESTRICTED TO NCF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE
Bibliography: Includes bibliographical references.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The New College of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Local: Faculty Sponsor: Ruppeiner, George

Record Information

Source Institution: New College of Florida
Holding Location: New College of Florida
Rights Management: Applicable rights reserved.
Classification: local - S.T. 2013 K4
System ID: NCFE004796:00001


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INVESTIGATINGTHEORIGINOFTHEPIONEER ANOMALYTHROUGHJAVACOMPUTERMODELING ByKATHLEENKIKER ATHESIS SubmittedtotheDivisionofNaturalSciences NewCollegeofFlorida Inpartialfulfllmentoftherequirementsforthedegree BachelorofArtsofPhysics UnderthesponsorshipofGeorgeRuppeiner Sarasota,Florida May20,2013

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Acknowledgements IwouldfrstliketothankmycommitteeofDr.MarianaSendovaandDr.Don Colladayforoeringtheirguidanceandinsight,notjustonthisthesis,butinall myclasseswiththemaswell.Secondly,Iwouldliketothankmythesissponsorand advisorofthreeyears,Dr.GeorgeRuppeiner.Icouldnotaskforamorepatientand obligingmentor.I'dliketothankmyroommatesfortheirpatienceandsupport,and LawrenceLevine,whomightaswellhavebeenmyroommateforallthetimeIspent workingathishouse.AlthoughIcannotlistthemall,manythankstoallthefaculty, friends,family,andnovocollegiansofalldisciplinesthathaveaidedinmyacademic journeyoverthelastfouryears.

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Preface ThePioneeranomalyisasmall,unmodeledradialaccelerationthatappearedin thetrajectoriesofbothPioneer10and11inthedirectiontowardsthesun,witha magnitudeofroughly8 10 )Tj/T1_4 7.97 Tf(8 cm/s 2 .Inthisthesis,Iwilldiscussseveralsuggested explanationsfortheanomaly,andtestthesepossibilitiesusingaJavasimulationof thesatellite'sorbit.Inmyinvestigation,Iwillfocusonforcesaectingthespacecraft (notthoseaectingthesignal,suchasclockacceleration)andwillconsiderboth internal(thermalrecoil,gasleak)andexternal(solarwind,interplanetarydust,nonNewtoniangravity)sourcesofunmodeledforces. Inmyfrstchapter,Iwillgiveanoverviewoftheanomaly,howitfrstappeared, andthepropertiesofthespacecraft.InthesecondchapterIwilldiscussatgreater lengththephysicsbehindthetheoriesthatIwillbetestinginmyprogram.Namely, interplanetarydustasasourceofdrag,thethermalrecoilforce,andtwodierent theoriesofadjustedgravity,modifedNewtonianmechanics(MOND)andscalartensor-vectorgravity(STVG).Inmythirdchapter,Iwillbrieryexplainmyprogram, includingatestcasetodetermineitsaccuracy,anddiscusstheresultsoftheprogram inregardstotheimplicationsforthepossiblecausesoftheanomaly. Besidesinterplanetarydustdrag,whichwasseveralordersofmagnitudetoosmall, allthreeoftheremainingtheorieswerelargeenoughthattheycouldreasonably accountfortheanomaly.Futhermore,whentheMONDandSTVGtheorieswere testedontheorbitoftheEarth,theMONDtheorydemonstratedadierenceofjust onecmperyear,toosmalltoruleitoutasaviabletheory.However,whenSTVG wastested,theEarthcouldnotremaininastableorbitandwasejectedafterasingle orbit.Whileitisnotcertainwhetherthisisentirelyaproblemwiththetheoryor withtheprogram,however,itseemstoindicatethataSTVGforcelargeenoughto causetheanomalywouldperturbtheorbitsoftheouterplanets.

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Contents 1ThePioneerAnomaly 1 1.1ThePioneerMissions...........................1 1.2TheAppearanceoftheAnomaly....................3 1.3PossibleCausesofthePioneerAnomaly................6 1.3.1UnknownMasses.........................6 1.3.2DragForces............................7 1.3.3Non-NewtonianGravity.....................8 1.4PioneerAnomalyResolved?.......................9 2ModelingOrbits 10 2.1OrbitalMechanics.............................10 2.2AdditionalConventionalForces.....................15 2.2.1InterplanetaryDustDrag....................15 2.2.2ThermalRecoilForce.......................16 2.3NewTheoriesofGravity.........................17 2.3.1ModifedNewtonianDynamics..................17 2.3.2Scalar-Tensor-VectorTheory...................18 3ThePioneerSimulationinJava 20 3.1TheProgram...............................20 3.2TestProgram-OrbitoftheEartharoundtheSun...........23 3.3PioneerResults..............................25 3.3.1InterplanetaryDustDrag....................26 3.3.2ThermalRecoilForce.......................28 3.3.3MOND...............................31 3.3.4STVG...............................33

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35 4 Conclusion

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INVESTIGATINGTHEORIGINOFTHEPIONEERANOMALYTHROUGH JAVACOMPUTERMODELING KathleenKiker NewCollegeofFlorida,2013 ABSTRACT AnalysisoftheradiotrackingdataofPioneers10and11overthelast30yearshas indicatedasmallunmodeledaccelerationof8 :74 1:33 10 )Tj/T1_4 7.97 Tf(8 cm/s 2 thatisdirected towardsthesun.Thisaccelerationappearsbetween20and70AU.Recentdatahas indicatedthattheaccelerationmaybedecreasingwithtimeatarateof2 10 )Tj/T1_4 7.97 Tf(11 m/s 2 /yrandthatitisconsistentwithathermalrecoilforcecausedbythePioneer's onboardelectricalsystems.InthisthesisIreviewthehistoryoftheanomalyandthe variousmechanismsproposedtoexplainit.Itestseveralofthesepossibilitiesina Javacomputersimulationandconcludethat,whileoneofthegravitationaltheories producesaforcelargeenoughtoaccountfortheanomaly,itappearstobeinconsistentwiththeorbitsoftheplanets,andthatthebestexplanationfortheanomaly ismostlikelyathermalrecoilforceproducedbytheonboardsystems,however,a MONDgravityforcecannotberuledoutbythisstudy. Dr.GeorgeRuppeiner DivisionofNaturalSciences

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1ThePioneerAnomaly 1.1ThePioneerMissions OnMarch3rd1972,thePioneer10spacecraftwaslaunchedfromCapeCanaveralona missiontoJupiter.OnDecember3rd1973itpassedbyCallisto,Europa,Ganymede, andIo,beforecompletingitsy-byaroundJupiteronDecember4th,takingmeasurementsofitsmagneticeld,radiationbelts,atmosphere,andinterior,aswellastaking stunningpicturesoftheplanetanditsmoons;suchastheimagesofJupiterinFigure 1[1].Fromthere,itcontinuedonahyperbolicorbitclosetotheplaneoftheecliptic intotheoutersolarsystem,demonstratedinFigure2.Inallthistime,relatively fewmaneuversweremadeonthespacecraftthatwouldchangeitstrajectory 1 andit continuedsendingbackscienticandtelemetrydatatoEarthuntil2003. OnApril6th1973,Pioneer10wasfollowedbyPioneer11.Unlikeitspredecessor, Pioneer11usedaJupiterybyasaslingshottowardsSaturn,whereittooktherst everclose-uppicturesoftheplanet'supperatmosphere.LikePioneer10,itpassedby manyofthemoonsofJupiter,andmanymoremoonsofSaturn,onitspathtoward theplanet.Pioneer11nisheditsybyofSaturnonSeptember1st1979,whereit reachedescapevelocityfromthesolarsystemandfollowedahyperbolicorbitclose totheplaneoftheecliptictotheoppositesideofthesunasPioneer10.Thelast trajectoryadjustmentofPioneer11wasonMay26th1976andthelastsignalwas receivedinNovember1995. ThePioneercraftsweredesignedtolast.Theyhadarequiredminimumlifespan ofthreeyears,andahopethattheywouldsurviveatleastseven.Asithappened, Pioneer10continuedtosendasignalforalmost30years.Themainpowersource onboardPioneer10andPioneer11wasasetoffourradioisotopethermoelectric 1 OnlyfourmaneuversweremadethatseriouslyalteredthePioneer'strajectory;oneonMarch 7th,andoneonMarch24th,1972,bothtocorrectrocketinjectionerrorsandsetthecraftbackon astraightcoursetowardsJupiter,andtwomoretosetPioneer10upforanexperimentinvolvingIo 1

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Figure1:ImagesofJupiter,takenbyPioneer10 Figure2:TrajectoriesofPioneer11andPioneer10fromabovetheplaneofthe ecliptic.Bothcraftsarenowmorethan80AUawayfromthesun,wellbeyondthe orbitofNeptuneAU. generatorsRTGsmountedonanextensionboom.Theirprimarysourceoffuelwas Plutonium-238.ExcessheatfromtheRTGswasventedintospaceviaasetofsix heat-radiatingns.ThePioneercraftsheldanumberofscienticinstruments 2 and afullcommunicationsystem,withtheantennaemountedalongthecenterline.The Pioneerswereroughlyaxiallysymmetric,withtheirthreelongboomseachspaced 120 apart. 2 Forafulllist,seeFigure3 2

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Figure3:DesignofthePioneerspacecraft.Thepowersourceisontheendofalong boomwheretheeectsofitsradiativeheatareminimized. ThemaindierencebetweenthePioneersandtheothercraftsoftheirtimewas thatthePioneerswerespin-stabilized,meaningthattheyrotateaboutoneaxis,rather thanbeingstabilizedacrossallthree.Thishelpskeepasteadyorientationonthe othertwoaxesandmeansthatfewermaneuversarerequiredtorightthespacecraft, reducingthenavigationalnoise.Itispartiallybecauseofthisthatthevelocitiesofthe Pioneerscouldbemeasuredwithsuchaccuracy.Other,threeaxisstabilizedcrafts, suchastheVoyagers,requiredmoremaneuvers,whichmadeprecisecalculationof theiraccelerationsdicult. 1.2TheAppearanceoftheAnomaly TheNavigationalandtrackingdataofthePioneerswashandledbyNASA'sJet PropulsionLabJPL.TheyusedanS-BandDopplersignaltodeterminethevelocity, 3

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position,andorientationofthePioneers.InradioDopplertracking,asignalissent tothePioneersfromEarthwhereitiscoherentlytranspondedandreturned.The changeinfrequencyiscarefullymeasuredandusedtodeterminethevelocityofthe craft,bytheformula[2] f = f T v R c where f isthechangeinfrequency, f T isthetransmittedfrequency, v R isthe recessionvelocity,and c isthespeedoflight. Itisalsopossibletodeterminethecraft'spositionfromthisDopplerdataby measuringthediurnalvariationcausedbytheEarth'srotation.Anotherwayof determiningthecraft'svelocityistouserangingtechniques.Inranging,amodulated signalissenttothespacecraft.Thesignalwhichisreceivedbythespacecraft,and thenre-transmittedisthencomparedtoagroundduplicateonEarthtogivethe timedelay.SincetherangingandDopplerdataareindependentofeachother,they canbeusedtocheckforfrequencyerrorsinthedata. TogettheDopplerdata,NASAusedJPL'sDeepSpaceNetworkDSN.There, startingin1976,whenPioneer11wasstillonitswaytoSaturn,theystartedto takeperiodictrajectorydatafromthePioneerstomeasureanyunmodeledforces, sotheycouldputalimitonthemforfuturenavigationalpurposes.Theiranalysis alreadyincludedmodelsoftheeectsofplanetaryperturbations,radiationpressure, theinterplanetarymedia,generalrelativity,andbiasanddriftintheDopplersignal, aspartofthemodeledforcesactingonthecrafts. AsFigure4shows,theanomalybeganshowingupasearlyas1976;however,it wasn'tuntilthePioneersreachedadistanceof20AUfromthesunthattheanomaly wasrstlookedatasmorethanjustacuriosity.Upuntilthispoint,solarpressure fromthesunfarout-weighedthetinyanomaly.However,solarradiationpressure 4

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Figure4:EarlydatashowingthePioneerunmodeledacceleration. decreasesas r )]TJ/F30 7.9701 Tf 6.587 0 Td [(2 ,soatdistancesofover20AU,itscontributiontothePioneer'svelocity isatinyaccelerationdirectedawayfromthesun,lessthan5 : 3 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(8 cm/s 2 .Dueto thelackofexternalforcesandtheveryfewattitudeadjustmentsneededtomaintain asignalwiththeEarth,thiswastheperfecttimetobeginaconcentratedeortinto solvingtheanomaly.MuchmoreaccuratedatawastakenfromthePioneersfromthen untiltheystoppedtransmitting.Thiswasaveragedandanunmodeledacceleration ofaround8 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(8 cm/s 2 wasfound. InaneorttodeterminewhethertheanomalywasanerrorintheODPorbital dataprogramnavigationalcodeoranactualforceactingonthecraft,thedatafrom thePioneerswascheckedusinganotherprogramcalledCHASMP.Thisprogramused thesamephysicalconstantsmassesoftheplanets,gravitationalconstant,etc.,buta completelydierentalgorithmtodeterminethevelocityandposition.Iftheerrorwas inthecode,theCHASMPanalysisshouldshownoanomalousacceleration.However, CHASMPalsoreturnedanunmodeledaccelerationtowardsthesun,withavalueof 5

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: 65 0 : 03 10 8 ,agreeingwithODP'sresult 3 [3]. Theanomalyrstbecamethesubjectofscienticinterestin1994whenitwas broughtupinaconferenceonLowEnergyAntiprotons,asanexampleofaphenomenonthatwasunexplainedbyNewtonianGravity.Notlongafter,NASAput togetherthePioneerExplorerCollaborationtoinvestigatetheanomaly.TheCollaborationpublishedtheirinitialndingsin1998,givingavaluefortheunmodeled accelerationofaround8 : 5 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(8 cm/s 2 [3],laterfollowingupwithamoredetailed analysisin2002[4].Inthislaterpaper,datawastakenfromPioneer10between1987 and1989,andPioneer11between1987and1991,andanalvalueof8 : 74 1 : 33 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(8 cm/s 2 wasgiven,afterallknownsystematicshadbeentakenintoaccount.Whilethe paperdiscussedmanypossiblecauses,itwastheirbeliefandmanyothers'thatthe Pioneeranomalycamefromsomesystematicerrorcausedbythedataanalysis,or thespacecraftitself.However,untilonewasfound,interestturnedtomoreexotic explanationsandtheprospectofnewphysics. 1.3PossibleCausesofthePioneerAnomaly TherehavebeenmanypossiblecausesgivenforthePioneerAnomaly,fromSolar WindtoDarkMatter.However,mostoftheproposedcausestintooneofthree dierentcategories: 1.3.1UnknownMasses Thersttypeofexplanationisthatsomepreviouslyunknownmassisapplyinga regular,NewtoniangravityforceonthePioneers.KuiperBeltobjectswereone suchsuspect.TheseobjectsarefaroutbeyondtheorbitofPlutoandmayexert somesmallgravitationalforceonthePioneercrafts.However,mostestimatesof theKuiperBelt'smassputtheaccelerationcontributiontothePioneersatjusta 3 ThedierenceCHASMP'serrorisduetothefactthatitusedabatchleastsquarestoverthe wholeorbit,ratherthanlookingforvariationswithdierentdistances 6

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fractionoftheknownanomaly.Furthermore,astudyintotheeectoftheKuiper BeltonthePioneersshowedthatitdidnotaccountfortheirdecreaseinacceleration. [5]Anothersuggestionisthat'DarkMatter'suchasheavyneutrinosintheform ofsphericalhalosaroundthesuncouldaddtheextragravitationalforceneeded. However,theundisturbedorbitsofMarsandJupitercreateaconstraintonthesize ofanygravitationalforceactingonthesatellites,soitisquestionablewhetherthere existsagravitationalsolutionlargeenoughtoaccountfortheanomaly,butsmall enoughnottoaecttheorbitsoftheouterplanets[6]. 1.3.2DragForces Anothertypeofexplanationgivenisthatthereissomeunknownmediumcausing adragforcetobeappliedonthecraft.Interplanetarydustisthemainsourceof dragconsidered.However,sofartheknowndensityofinterplanetarydustatthe relevantdistancesisseveralordersofmagnitudetoosmalltoaccountforthePioneer's unmodeledacceleration[7]. Another,moreexotic,kindofdragforceinvolvesaformofdarkmattercalled 'mirrormatter'whichcouldtheoreticallyprovideadragforcesucientlylargeto causetheanomaly[8].Thistheorysuggestthatifanumberoftheorderof10 5 Hydrogenmirrorparticleswerespreadroughlyevenlyinasphereofradius100AU, thiscouldaccountfortheanomalousacceleration.Thiswouldsuggestadensityof mirrorparticles,muchgreaterthantheirnormalcounterparts.Whilethismodel maywork,itwouldnotbepossibletoprovewithoutanothermissiontotheouter solarsystem. 7

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1.3.3Non-NewtonianGravity Manytheorieshavebeenputforwardtoanswertheanomalybysuggestingchanges totheknownmodelofgravity 4 [9][10][12].However,thelargestproblemwithallof themisthatanysuchmodicationwouldshoweectsonotherobjectsasfarout asthePioneers,suchastheouterplanetsandasteroids.Sometheorieshavebeen abletoshowevidencethattheirmodeltswiththeseotherbodies,butmanydonot, andtherehavebeenquestionsraisedastowhetheragravitationalsolutiontothe anomalyisevenpossible[6]. Aswellasthesethreetypes,thereisanotherpossiblecausethathasbeenproposed.ItwasrstwassuggestedinthePioneerCollaboration'srstpaperonthe anomaly.Theretheysuggestedthataclockaccelerationoftheform a P = a t c ,where a P isthePioneeracceleration, a t isaclockacceleration,andcisthespeedoflight, wouldexplainthediscrepanciesintheDopplerdata. Inaddition,severalproposalshavebeenmadeforanewspacecrafttobesentinto theoutersolarsystemwiththesolepurposeofinvestigatingtheanomaly[13].This wouldnallylaytoresttheissueofwhetheritwasasystematicerror,whetherthere wereunknownmassesinspace,orwhetherthetheoryofNewtoniangravityneeded tobexed.Severalattemptswerealsomadetoconrmtheanomalywithother, existingspacecraft,however,forvariousreasons,noneoftheseweresuccessful 5 .It wassuggestedthateithertheCassiniortheNewHorizonsspacecraftscouldalsobe usedtoinvestigatetheanomaly[14][15].However,intheend,nosuchmissionwas evernecessary. 4 Higher-ordercorrectionstotheNewtonianpotential,scalartensorextensiontothestandard gravitationalmodel,interactionofthespacecraftwithalong-rangescalareldcoupledtogravity, andaninversetimedependenceforthegravitationalconstant G arejustafew. 5 TheVoyagerswerenotspinstabilized,whiledatafromtheGalileoandUlysseswascompromised bySolarwindandmalfunctionsoftheattitudecontrolsystems[14]. 8

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1.4PioneerAnomalyResolved? Formanyyears,theanomalyremainedunsolved.Manyexplanationswereproposed, butnoneseemedtoconclusivelytthedata.However,in2011,newdatawascompiled thatshowedthattheanomalymightinfactbedecreasingslightlywithtime,instead ofconstantasoriginallythought[17].Thisnewdatawastakenoveraperiodtwice aslongasbeforeandshowedthattheaccelerationwasdecayingtemporallyatarate of2 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(11 m/s 2 /yr.Thisdiscoveryopenedthedoorforpreviouslydisregarded theories,includingmanyoftheearlysystematicexplanations,tobereexamined. OneofthesewastheideathatthethermaldecayoftheRTGsmightbethecause oftheanomaly.Thistheorywasoriginallydismissed,inpart,becauseitpredicted asmall, slightlydecreasing acceleration.In2012,Turyshevetal.[16]publisheda paperthatnallyexplainedtheanomaly.Init,theymodeledathermalrecoilforce totboththePioneerdataandtheirownindependentmodelsofthespacecraft's heatemission.Theyfoundnostatisticallysignicantdierencebetweenthetwo results,andfurthermore,theyfoundthatoncethethermalrecoilforcewastaken intoaccount,therewasnoanomalousaccelerationremaining.Thishasbecomethe acceptedsolutiontotheanomaly. ItseemsnowthatthemostlikelyexplanationforthePioneerAnomalyisanisotropic radiationthatis,radiationthatisnotsymmetricinalldirectionsfromtheironboardequipment.Inthisthesis,IwilluseaJavasimulationtoexaminethistheory alongwithseveraloftheearlierexplanationsandcomparetheresultsofthesemodels totheknowndata. 9

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Figure5:DemonstrationofKepler'srstandsecondlaws.Thetwosegments,taking equaltimes,wouldhaveequalarea.ByKepler'srstlaw,oneofthetwofocimust bethesun. 2ModelingOrbits 2.1OrbitalMechanics ThebehaviorofplanetarymotionismostconciselydescribedbyKepler'sthreelaws [18]; 1.Allplanetsmoveinellipticalorbits,withthesunatonefocus. 2.Alinethatconnectsaplanettothesunsweepsoutequalareasinequaltimes. 3.Thesquareoftheperiodofanyplanetisproportionaltothecubeofthe semimajoraxisofitsorbit, T 2 = ka 3 ; where T istheperiod, a isthesemimajoraxis,and k isaconstantthatisthesame foranysatelliteorbitingagivenmasscenter. WhiletheselawswererstdeducedbyKeplerfromobservingthemotionofthe planetsaroundthesun,theyapplytotheorbitsofothersatellitesaswell.Theycan beusedtotracksatellitesinorbitandpredictthemotionofspacecraft. "Kepler'sProblem"referstoasystemoftwoparticlesunderacentralinverse squareforce.Inthiscase,thecentralforceactingistheforceofgravity 10

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Figure6: d ofaparticleinmotion.Takenfrom[18]. ~ F G = )]TJ/F32 11.9552 Tf 9.299 0 Td [(GMm r 2 ^ r; where G isthegravitationalconstant6 : 67 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(11 m 3 kg )]TJ/F30 7.9701 Tf 6.586 0 Td [(1 s )]TJ/F30 7.9701 Tf 6.586 0 Td [(2 M isthelargermass inthiscase,thesun, r isthedistancebetweenthetwoparticles,and m isthemass oftheorbitingbody.Theequationofaparticlemovingundertheinuenceofsuch aforce,isgivenby[18] r = Z L=r 2 dr r 2 E + k r )]TJ/F33 7.9701 Tf 16.964 4.707 Td [(L 2 2 r 2 + constant; where L istheangularmomentum, = mM= m + M isthereducedmass, E isthe energy,and k isaconstant, GMm .Thesolution,forachoiceoforiginsuchthat r is minimumwhen =0,is cos = L 2 k 1 r )]TJ/F18 11.9552 Tf 11.956 0 Td [(1 1 q 1+ 2 EL 2 k 2 : Ifwedeneconstants: = s 1+ 2 EL 2 k 2 ; 11

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Figure7:Examplesofthefourtypesoforbits,withtheireccentricities. = L 2 k ; thissimpliesto r =1+ cos ; whichistheequationforaconicsection,where istheeccentricityand2 iscalled thelatusrectumoftheorbit.When0 << 1,thisyieldsKepler'srstlaw. Theeccentricitycanbeusedtodeterminetheshapeoftheorbit,asshownin Figure5.Therelationshipbetweeneccentricity,energy,andangularmomentumcan tellussomethingaboutasatellite'senergyfromitsorbit.Namely,thattheenergy, theshape,andtheeccentricityoftheorbitareconnectedasbelow: E = V min Circle =0 V min 0Hyperbola > 1 12

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where E isthetotalenergy, istheeccentricity,and V min = )]TJ/F32 11.9552 Tf 9.299 0 Td [(k 2 2 L 2 ; where isthereducedmass, k is GMm ,and L istheangularmomentum. V min is theleastenergyabodycanhaveandremaininastableorbit. Inacircularorbit,gravityistheforcethatprovidesthecentripetalacceleration ofbodiesinorbitaroundthesun, ~ F C = )]TJ/F32 11.9552 Tf 9.299 0 Td [(mv 2 r ^ r = )]TJ/F32 11.9552 Tf 9.298 0 Td [(mr 4 2 T 2 ^ r; where v isthevelocityofthesatellite, m isitsmass, r isthedistancefromthesun, andthesecondexpressionisgivenintermsoftheperiodofonerotation T .Boththe forceandtheaccelerationaredirectedinward,towardsthesun. SettingthisequaltoequationandputtingeverythingintermsofAUand years,thisgivesavalueforGMcalledthestandardgravitationalparameterof4 2 Thisequationcanalsobeforvelocitytogiveanexpressionforthespeedofasatellite inacircularorbit v = r MG r : Thisequationholdsonlyformotioninacircularorbit.Foranellipticalorbit, motionismorecomplicated.Togetthevelocity,rstconsidertwopoints,oneat apogeeandtheotheratperigee.Thetotalenergyofthesatelliteisequaltothesum ofitspotentialandkineticenergies; E T = E G + E K = )]TJ/F32 11.9552 Tf 9.299 0 Td [(GMm r + 1 2 mv 2 13

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Sinceenergyisconserved,thisquantitymustbethesamethroughoutthesatellite's orbit,so )]TJ/F32 11.9552 Tf 9.299 0 Td [(GMm r 1 + 1 2 mv 2 1 = )]TJ/F32 11.9552 Tf 9.298 0 Td [(GMm r 2 + 1 2 mv 2 2 whichistrueforanytwopointsinanorbit.Cancelingoutthecommonfactor,m, gives, v 2 1 2 )]TJ/F32 11.9552 Tf 13.151 8.088 Td [(v 2 2 2 = GM r 1 )]TJ/F32 11.9552 Tf 13.151 8.088 Td [(GM r 2 : SincethevelocityvectorsareperpendiculartothedirectionofForceataphelionand perihelion,conservationofangularmomentumtellsusthat v 1 r 1 = v 2 r 2 and 1 2 r 2 2 )]TJ/F32 11.9552 Tf 11.955 0 Td [(r 2 1 r 2 2 v 2 1 = GM r 1 )]TJ/F32 11.9552 Tf 13.151 8.088 Td [(GM r 2 : Puttingthisbackintoequationandrecallingthat r 1 + r 2 =2 a when r 1 and r 2 arethepointsatperihelionandaphelion, E 1 = GM 2 a )]TJ/F32 11.9552 Tf 11.955 0 Td [(r 1 2 ar 1 )]TJ/F32 11.9552 Tf 13.151 8.088 Td [(GM r 1 = )]TJ/F32 11.9552 Tf 10.494 8.088 Td [(GM 2 a ; so,byconservationofenergy,foranypointontheorbit v 2 2 )]TJ/F32 11.9552 Tf 13.151 8.088 Td [(GM r = )]TJ/F32 11.9552 Tf 10.494 8.088 Td [(GM 2 a ; andthevelocitycanbewritten, v 2 = GM 2 r )]TJ/F18 11.9552 Tf 13.296 8.087 Td [(1 a ; where v isthevelocity, GM isthestandardgravitationalparameter, r isthedistance 14

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betweenthetwobodies,and a isthesemimajoraxis.ThisiscalledtheVisViva equation[19],andwhilethisderivationisforanellipticalorbit,theequationholds forallkindsoforbits,includinghyperbolic. Forasatellitetoescapethesolarsystemandcontinueonahyperbolictrajectory, itmusthavetotalenergythatislargerthanzero 6 Theescapevelocityfromthesunwhichisapproximatelytheescapevelocityof oursolarsystem,atlarge r is,inAU/year, v escape = r 2 GM r =8 : 886 r 1 r ; where M isthemassofthesun, G isthegravitationalconstant,and r isthedistance tothesatellite.Ifasatelliteisnotclosetoaplanet r islargeanditsvelocityexceeds thisescapevelocity,itwillleavethesolarsystemonahyperbolicpath. 2.2AdditionalConventionalForces 2.2.1InterplanetaryDustDrag Becauseorbitalproblemstakeplaceinthenearperfectvacuumofspace,dragforces cansometimesbeneglected.However,whenmeasuringthetinyaccelerationsneeded fortrackingsatellites,typesofdragsuchasinterplanetarydustandsolarwindbegin tohaveasignicanteectonthesatellite'sacceleration. Theformofanaccelerationduetoadragforceofanykindofinterplanetary mediumis ~a d ~r = )]TJ/F32 11.9552 Tf 9.298 0 Td [(K d ~r v 2 s A m ^ v; where K d istheeectivereection/absorption/transmissioncoecientofthesurface 6 Ifthesatellite'stotalkineticandpotentialenergyisequaltozero,itwillescapefromthesolar systemona parabolic trajectory,whichisthemostenergy-ecientescapecourse. 15

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beinghitwhere K d is1forabsorption,2forreection,and0fortransmittanceof thedust, r isthedensityofthemedium, v s istherelativevelocityofthecraftto themedium, A isthecross-sectionalareaofthesurfacebeinghit,and m isthemass ofthespacecraft[20].Adragwillalwaysproduceaforceinthedirectionoppositeto ~v Modelsofthearrangementandamountofinterplanetarydustvarygreatly.However,thereisageneralconsensusthatitmusthavearoughdensity[23] IPD < 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(24 g/cm 3 2.2.2ThermalRecoilForce Inadditiontotheseexternalforcestherecanbeseveralforcesinternaltothespacecraftthatmayaectitsacceleration. Onesuchforceisthethermalrecoilforce,whichiscausedbytheheatgiveno byon-boardsystems.Thethermalrecoilforceis, F t = )]TJ/F18 11.9552 Tf 10.494 8.088 Td [(2 3 1 c Z ~qdA wherecisthespeedoflight, ~q istheheatux, ~q = )]TJ/F32 11.9552 Tf 9.298 0 Td [(k ~ r T ,and dA isasurface elementoftheemitter'sexterior[24].Aslongastheradiantintensityontheexterior surfaceandthegeometryoftheemitterareknown,thethermalrecoilforcecanbe calculated[25]. Foragivendirection,thisforcegivesanaccelerationof[16] a th = Q mc ; where Q = R ~qdA isthethermalpower, m isthemassofthesatellite, c isthespeedof light,and isadimensionlessconstantcalledthe"eciencyfactor",whichdepends onthegeometryandorientationoftheheatemittingobject. 16

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Figure8:aAsingleheat-emittingsurfaceelementwhichisintegratedoverforthe totalradiativeheat.bIftheheatemittedissymmetricalinalldirections,the accelerationtermswillsumtozero. Ifasatelliteisemittingheatsymmetricallyinalldirections,thethermalrecoil forceshouldnotaectthetrajectory.However,evenverysmalldierencesinthe amountofheatbeingradiatedindierentdirectionsmayaectthesatellite'sacceleration. 2.3NewTheoriesofGravity 2.3.1ModiedNewtonianDynamics Onemodicationtothetheoryofgravityisknownas'ModiedNewtonianDynamics', orMOND.ThistheorymakesacorrectiontoNewton'ssecondlaw, F = ma ,by makingtheassumptionthattheinertiaofanobjectdoesnotdependdirectlyonits acceleration,butrathersomemoregeneralfunctionofit[26], F = a=a 0 m a 17

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where a 0 isafundamentalacceleration,similarto c ,and isafunctionsuchthat x =1when x>> 1,and x = x when x<< 1.Thefunction isleftunspecied [27]. Given a 0 <
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where G istheregulargravitationalconstant, M isthemassofthelargebody, r is thedistancebetweenthem,and r and r arefunctionstobesolvedfor[10]. 19

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3ThePioneerSimulationinJava 3.1TheProgram Inordertoincreasethespeedoftheprogramandreducethedicultyoftheproblem, Ihavemadeseveralsimplicationsinmyconsiderationoftheproblem.Firstly,for simplicity,theonlyobjectsIconsiderarethePioneerspacecraftandtheSun.Ido notincludethegravitationalpulloftheplanets,whichwouldbesmallcomparedto thepulloftheSun 7 .Inaddition,themassoftheSunis7 : 9564 10 27 timesmore massivethanthePioneers,sothereducedmasscanbeapproximatedastheSun's mass.Finally,Ineglectedanydeviationfromtheplaneandconsideredonlythe2D problem,with x and y componentsofvelocitybutno z Forconvenience,myprogramwillberuninastronomicalunitsofAU,years,and SolarMasses,where GM =4 2 .Thisnecessitatesseveralchangesofunitsbetween theconstantsfoundintheliteraturebeforetheyareinputintotheprogram.Such changesofunitswillbeoutlinedbelow. TheprogramIusedwasamodiedversionofthe'Planet'and'PlanetApp'programsavailablefromopensourcephsyics 8 [21].Theprogramimplements'AbstractSimulation',aclassthatcontrolstheanimationandimplementstheabstractmethod "dostep".Anarray'states'isinitializedwithvaluesof x 0 v x 0 y 0 v y 0 t x v x y and v y ,where x 0 v x 0 y 0 ,and v y 0 arethecomponentsofpositionandvelocitywithno additionalforces,and x v x y ,and v y arethevelocityandpositionwithanadditional forceapplied.Asecondarray'rates',givestheirrateofchangeintime. Aftereachtimestep,thenew'state'arrayiscalculatedfromtheoldstatearray usingthe'rates',withafourth-orderRunge-Kuttaalgorithm.Givenasomefunction v = v u andanODE, dv=du = f u;v ,thefourthorderRungeKuttaalgorithm 7 Forexample,at30AU,theaccelerationcausedbytheEarth'spullonthePioneeris,atmost, 1 : 4099 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 AU/yr 2 ,whiletheSun'sis0.04386AU/yr 2 8 Theopensourcephysicslibrarycanbedownloadedbyanyonefrom: http://www.opensourcephysics.org/ 20

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sometimescalledRK4is[22]; k 1 = hf u n ;v n k 2 = hf u n + 1 2 h;v n + 1 2 k 1 k 3 = hf u n + 1 2 h;v n + 1 2 k 2 k 4 = hf u n + h;v n + k 3 v n +1 = v n + 1 6 k 1 + 1 3 k 2 + 1 3 k 3 + 1 6 k 4 + O h 5 where h istheintervalstep,and O x isaLandausymbol 9 Thesevaluesarethenappendedtothreegraphswherethedataisstoredanddisplayed.Therstgraphshowstheanimationofthesatellite'smovement.Thesecond graphshowsthevelocitywithrespecttotime,andthethirdgraphshowsthesimulatedspeedsubtractedfromthe"modeled"speed.Theprogramwillrstberunwith noadditionalforcesbesidesNewton'sgravityforce.Iwillcallthisthe"modeled" accelerationanditwillbeusedasabaselinewithwhichtocomparetheothers. Afulloutlineoftheprogramisgivenbelow: Pioneer Importtheopensourcephysicslibraryframes CreateanewclassPioneerthatusesmethodsinODE Initializeforce InitializeGMtofourpisquared Initializearraystatewithninevariables: x;vx;y;vy;t;xnew;vxnew;ynew;vynew CreatenewobjectodeSolverfromODESolverusingRK4method Createnewgraphframe1fromPlotFrame 9 If x or n arepositivefunctionsand f x and f n areanyfunctions,where n isaninteger and x issomevariable,thenif f = O ,then j f j
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Createnewgraphframe2fromPlotFrame Deneafunctiondostep Whendostepiscalled, CallstepinodeSolver Setspeedtotheabsolutevalueofthevelocity Setspeednewtotheabsolutevalueofthenewvelocity Plotspeedagainsttimeinframe1 Plotspeednewagainsttimeinframe1 Plotthedierencebetweenspeedandspeednewagainsttimeinframe2 DeneafunctionInitialize WhenInitializeiscalledwithanarrayinput Copytheinputarraytothearraystate CallinitializegetstepSizeinodeSolver DeneafunctionchooseForce WhenchooseForceiscalledwithinput'forcenum' Setforcetoforcenum DeneafunctiongetRate WhengetRateiscalledwithinputarraysstateandrate, Setrtothesquarerootofxsquaredplusysquared InitializeadditionalforcetermsAccxandAccytozero SetthevaluesofAccxandAccydependingonthevalueof'force' Settherateofxtovx SettherateofvxtoGMxdividedbyrcubed Settherateofytovy SettherateofvytoGMydividedbyrcubed Settherateofxnewtovxnew SettherateofvxnewAccx Settherateofynewtovynew SettherateofvynewAccy DenefunctiongetState WhengetStateiscalled Returnthearraystate PioneerApp Importtheopensourcephysicslibrariescontrolsandframes CreateanewclassPioneerAppthatusesmethodsinAbstractSimulation Createanewgraphframe3fromPlotframe 22

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CreateanewobjectsatellitefromPioneer DeneafunctionDostep WhenDostepiscalledbycommandwindow,orAbstractSimulation Calldostepinsatellite Graphxandypositionevery50steps Tellgraphtodisplaythetime Every100steps;Printthevelocityxandycomponentsandthetime DenefunctionInitialize WhenInitializeiscalledbythecommandwindow GetstepsizefromodeSolver Setx,y,vx,vy,andforcenumfromuserinputs CallInitializeinsatellitewithinitialvalues: x;vx;y;vy; 0 ;x;vx;y;vy CallchooseForcewithinputforcenum Setmessageinframe3tot=0 DenefunctionReset Whenresetiscalledbythecommandwindow Resetallvaluesincontrolwindowtodefaults CallInitialize TheexactJavacodeusedisgiveninAppendixI. 3.2TestProgram-OrbitoftheEartharoundtheSun Inordertotesttheaccuracyofmyprogram,Irstusedittomodelaknownexample; theorbitoftheEartharoundtheSun.Usingaslightalterationtothecode,outlined below, DeneafunctionDostepinEarthApp WhenDostepiscalledbycommandwindow,orAbstractSimulation, CalldostepinpioneerparTellgraphtodisplaythetime Setenergytohalfofthespeedsquaredplusfourpisquareddividedbyr Ifthepositionhaschangedinsignsincethepreviousstep, Setdistanceequaltovalueofxwhenyiszerousingquadraticinterpolation Settimeequaltovalueoftwhenyiszerousingquadraticinterpolation Keeptrackofthenumberoforbits Printthenumberoforbits,time,distance,andtheenergy Iusedtheprogramtocalculatethemodeledperiodoftheorbitanddistancefrom thesun,andthencomparedthesetotheknownlengthofayearandtheperihelion 23

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Figure9:Earthtestcaseatthebeginningandafterseveralorbits.Thedirectionof theinitialvelocityvectorisindicatedwithanarrow.TheSunisattheorigin. andapheliondistancesoftheEarth.IinitiallysettheEarthatperihelionadistance of0.983747AUsothatalloftheinitialvelocityisinthe y direction.Usingvalues fromNASAswebsite[28]andequation,thisgivestheinitialvelocityoftheEarth atperiheliontobe6.386193AU/yr,inadirectionshowninFigure9. Theprogramwasrunovertenorbitalperiods,withvarioussizesoftimestep. Ihadtheprogramprintoutthetime,thedistance,andtheenergyeverytimethe simulatedplanetcrossesthe x axis. Theapproximatetimeandpositionwhere y =0weredeterminedusingaquadratic interpolationmethodtodeterminethepointofcrossing x = y 1 y 2 y 0 )]TJ/F32 11.9552 Tf 11.955 0 Td [(y 1 y 0 )]TJ/F32 11.9552 Tf 11.955 0 Td [(y 2 x 0 + y 0 y 2 y 1 )]TJ/F32 11.9552 Tf 11.955 0 Td [(y 0 y 1 )]TJ/F32 11.9552 Tf 11.955 0 Td [(y 2 x 1 + y 0 y 1 y 2 )]TJ/F32 11.9552 Tf 11.955 0 Td [(y 0 y 2 )]TJ/F32 11.9552 Tf 11.955 0 Td [(y 1 x 2 : Thetimegivenbytheprogramfortenorbitsshouldbeexactlytenyears.The periheliondistanceandtheenergyshouldnotchange.Theperiheliondistanceafter 10orbits,inAU,thetime,inyears,andthedierenceinenergybetweentherstand 24

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tenthorbitsareshowninTable1,alongwiththepercentageerrorfromtheexpected results. Table1:ResultsofEarthTest dtPerihelionDistanceError d=d 0 TimeError t=t 0 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(1 N/A 10 N/AN/AN/A 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(2 0.98374551.5377 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(6 10.000195121.95118 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(5 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(3 0.98374701.1414 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(11 10.000195111.95110 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(5 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(4 0.98374704.5707 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(14 10.000211322.11317 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(5 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(5 0.98374704.7851 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(14 10.000191661.91664 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(5 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(6 N/A 11 N/AN/AN/A dtEnergyDierenceError E=E 0 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(1 N/AN/A 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(2 3.065 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(5 1 : 55259 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(6 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(3 3.062 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(10 1 : 55126 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(11 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(4 1.002 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(12 5 : 07558 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(14 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(5 7.994 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(13 4.04967 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(14 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(6 N/AN/A Whilethesizeofdtdoesnotchangetheerrorofthetime,thereisaclearincrease inaccuracywithdecreasingtimestepinthecalculationofposition,whichlevelsoat dt =10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(4 .Atthispoint,increasingthetimestepnolongersignicantlyincreasesthe accuracy,however,thetimetakentocompletethesimulationcontinuestoincrease. Therefore,inordertomaximizeforbothaccuracyandtime,Iwilluseatimestepof dt =10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(4 inallfuturesimulations,unlessotherwisestated. 3.3PioneerResults Tocheckthefunctionalityoftheprogram,itwasrstrunwithoutanyadditional forceterms,andtheenergywasplotted. Theinitialvelocitywasfoundusingashootingmethod.Takingtwopoints x 1 ;y 1 ;t 1 and x 2 ;y 2 ;t 2 fromFigure2,Iusedtrialanderrortorstguessedan 25

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initialvelocity,checkthepositionattime t = t 2 ,andadjusttheinitialveloctyuntil thesecondpointwasreachedwithreasonableaccuracy. Figure10showsthePioneer'sorbitandenergybeforeanyforcetermsareadded toNewton'sgravity,whileTable2belowshowstheenergyandpositionat5dierent pointsintime,inastronomicalunits, Table2:PositionandEnergyofPioneer TimeEnergyxpositionypositionDistancefromSun 53.42394448.37325.96050.950 103.42394462.47123.53065.172 153.42394476.34021.07179.195 203.42394490.05518.56893.079 253.423944103.65915.997106.861 Theenergyremainsconstantupto13signicantguresandispositive,which showsthattheobjectisinahyperbolicorbitandthattheprogramisrunningcorrectly.Thepositionistakentobethe'expected'pathofthePioneers,andthefollowingsimulationswillbecomparedtothistodeterminehowfartheyhavedeviated 'ocourse'. Theactualanomalywasfoundtobe5 : 82 0 : 89 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(6 AU/yr 2 ,withvelocities likethoseshowninFigure11. 3.3.1InterplanetaryDustDrag Forthedensityofinterplanetarydust,Iusedavalueof =10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(24 g/cm 3 whichis, inAstronomicalunits,1 : 68 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(18 SolarMasses/AU 3 .Thecoecientwas2for reection.Assumingthelargestpossibleincidentsurfacearea,fromFigure12,this wouldgiveavalue A =13.5m=9 : 024 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(11 AU.Themassofthesatelliteafter fuelusageis250kg,or1 : 256 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(28 SolarMasses[7]. Usingtheseconstants,Iranthesimulationwithanadditionalforcetermproportionalto v 2 inthedirectionoppositethevelocity. 26

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Figure10:EnergyofthePioneerwithnoadditionalforcetermadded.Inastronomical units. ThesimulatedvelocitiesareplottedinFigure13below.Theredline(notseen here)isthevelocityofthesatellitewithnoforceactingonitbesidesNewtonian gravity.Thegreenlineshowsthevelocitywiththeadditionofthe'extra'force.In thiscase,thedierenceinvelocitieswastoosmalltodierentiatebetweenthetwo lines.Figure14showsthedierencebetweenthevelocities,plottedagainsttime, whichIshallcallthe"unmodeled"velocity.Thisgivesanunmodeledaccelerationon 27

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Figure11:TheobservedDopplervelocitiesofthePioneerAnomaly. theorderof10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(10 AU/yr 2 ,muchsmallerthanourconsiderations.AsTable3shows, interplanetarydustaccountsforonlyasmalldeviationfromthepredictedcourse, muchsmallerthanthosereportedfortheactualPioneers. Table3:PositionaftertheInclusionofPlanetaryDust TimeyearsDist.fromSunAUDierenceAUDierencekm/year 550.9499-1.0750 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(8 -1.6081-0.32162 1065.1718-3.4848 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(8 -5.2132-0.52132 1579.1945-7.2899 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(8 -10.9055-0.72703 2093.0786-1.3763 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(7 -20.5889-1.02944 25106.8606-2.1357 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(7 -31.9492-1.27797 3.3.2ThermalRecoilForce FortheThermalRecoilforceIconsidertwosourcesofheat,theradiothermalgenerators,whicharelocatedontheendsofthelongbooms,andthemainelectrical 28

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Figure12:GeometryofthePioneers. systems,whichareonthecentralpanelofthespacecraft.Forthethermalpower intheelectricalsystems,Iusethedecitbetweentheelectricalpowerdrawnand thepowerused,asgivenby[16].FortheRTGs,IusethedecayofPu 238 ,fromthe equation, Q RTG t =[2 )]TJ/F30 7.9701 Tf 6.587 0 Td [( t )]TJ/F33 7.9701 Tf 6.587 0 Td [(t 0 = ] Q RTG t 0 ; where Q RTG isthethermalpoweroftheRTGs, t isthetime, isthehalflifeofPu 238 t 0 issomeinitialtime,and Q RTG t 0 isthepowerat t 0 29

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Figure13:Thevelocitiesinthecasewherenoadditionalforcewasaddedinred andthecasewhereanadditionaldragforceduetointerplanetarydustisappliedin green.AlltimesareinyearsandallvelocitiesinAU/yr. Figure14:Therelativevelocitybetweenthesystemwithnoforceaddedandthe systemwithanadditionaldustdrag.InyearsandAU/year. Forthevaluesof inEquation,[16]suggestedvaluesof =0.0406,forthe electricalsystems,and =0.0104,fortheRTGs. ThepowerisgiveninunitsofWatts,whichisJ/sorkgm 2 /s 3 .Toconvertthis toastronomicalunits,then,theconversionfactoris, 1W=kgm 2 = s 3 =7 : 0594 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(31 SolarMassesAU 2 = year 3 : 30

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Figure15:Unmodeledvelocityofthesystemwithanadditionalthermalrecoilforce. InyearsandAU/years. Figure15showstheunmodeledvelocity,whichgivesanaccelerationofaround )]TJ/F18 11.9552 Tf 9.299 0 Td [(2 : 8 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(6 AU/yr 2 .Thisiswithinafactorof2oftheknownPioneeranomaly.Table4shows thatthemodeledsatellitetravelledaround1000kmlessperyearthanpredicted, startingat30AU,increasingbyroughly1000kmeveryveyears,showingaroughly constantacceleration. Table4:PositionaftertheInclusionofThermalRecoil TimeyearsDist.fromSunAUDierenceAUDierencekm/year 550.9499-3.64112 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(5 -5447-1089.4 1065.1718-1.44039 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(4 -21548-2154.8 1579.1945-3.20537 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(4 -47952-3196.8 2093.0786-5.63528 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(4 -84303-4215.1 25106.8606-8.70596 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(4 -130239-5209.6 3.3.3MOND Because inequationcanbeanyconstant,avaluecanbechosensuchthatthe magnitudeofthisaccelerationisthesameasthatoftheanomaly.Foratransition radiusof, a 0 =6 : 8 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(10 m/s 2 =4 : 5266 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(6 AU/yr 2 [26],wecanchooseaconstant =1 : 286to't'theanomaly.Figure16showstherelativevelocitywithtime,and 31

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Figure16:UnmodeledvelocityofthesystemwithaMONDgravityforce.Inyears andAU/years. Table5showsthedistancetravelledandthedistanceocourse,bothofwhichare consistentwiththeanomaly,asexpected. Table5:PositionaftertheInclusionofaMONDgravityforce TimeyearsDist.fromSunAUDierenceAUDierencekm/year 550.9498-3.64112 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(5 -5447-1089.4 1065.1717-1.44039 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(4 -21548-2154.8 1579.1942-3.20537 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(4 -47952-3196.8 2093.0781-5.63528 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(4 -84303-4215.1 25106.8597-8.70596 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(4 -130239-5209.6 Inordertotestthevalidityofthetheory,IrerantheEarthTestprogramwith thisadditionalforceadded,toseeiftherewasameasurablechangeintheplanet's orbit.Ifoundthatovertenyears,theperiheliondistancedeviatedfromtheknown valueby7 : 170 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(13 AUafter10yearsasopposedtotheregularaccuracyofthe programatdt=10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(4 ,of4.50 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(14 AU.Thisworksoutas0.107movertenyears, orroughly1cmperyear.ComparedtoalltheforcesactingontheEarth,thisisnot asignicantcontributionandnotenoughtoruleoutaMONDgravityforceasthe 32

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Figure17:Unmodeledvelocityofthesystemwithascalar-tensor-vectormodied gravity.InyearsandAU/years. causeofthePioneeranomaly. 3.3.4STVG Table6:PositionaftertheInclusionofaSTVGforce TimeyearsDist.fromSunAUDierenceAUDierencekm/year 550.9499-2.76909 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(3 -41425-8285 1065.1719-1.02992 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(3 -154074-15407 1579.1945-2.17502 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(3 -325002-21667 2093.0786-3.64362 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(3 -545078-27254 25106.8606-5.39769 10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(3 -807483-32299 TheaccelerationcausedbySTVGisgivenbyequation.Theliterature[10] suggeststheuseoftheequations r = inf )]TJ/F18 11.9552 Tf 11.955 0 Td [(exp )]TJ/F32 11.9552 Tf 9.298 0 Td [(r=r bar b= 2 r = inf )]TJ/F18 11.9552 Tf 11.955 0 Td [(exp )]TJ/F32 11.9552 Tf 9.299 0 Td [(r=r bar b for r and r ,andtheconstants: 33

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Figure18:aSimulationoftheEarth'sorbit,withaddedSTVGforce.bEnergy oforbitwithSTVGadded.Bothwererunwithatimestep10 )]TJ/F30 7.9701 Tf 6.586 0 Td [(5 inf =10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(3 inf =47AU, r bar =4 : 6AU, b =4 : 0, wouldworkfortheanomaly. Usingthesevaluesinthesimulationgaveanaccelerationofaround1 : 83 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(5 AU/yr 2 .TheunmodeledvelocityisplottedinFigure17.Table6showsthedeviation fromexpecteddistanceofthePioneer,whichissomewhatlargeinthiscase.However, itisstillontherightorderofmagnitudetoexplaintheanomaly. However,asFigure18shows,whentheSTVGforcewasappliedtotheEarth,it hadadramaticeectontheplanet,forcingitoutoforbit.Totrytoaccountfor programerrorasthecauseofthis,Iincreasedthetimestepto10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(5 ,butthesame thingoccurred.Itisstillpossiblethatthisisanissuewiththeprogram,butthis wouldseemtoindicatethataSVTGforceofthissizeisnotconsistentwiththeknown orbitsoftheplanets. 34

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4Conclusion ThemagnitudeofthePioneeranomalywasfoundby[4]tobe8 : 74 1 : 33 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(8 cm/s 2 ,whichis5 : 82 0 : 89 10 )]TJ/F30 7.9701 Tf 6.587 0 Td [(6 AU/yr 2 .ThePioneerstravelledthousandsof kilometerslessthanpredicted,decreasingwithtimeasconsistentwithanearly constantacceleration. Interplanetarydustdidnotprovidealargeenoughaccelerationtoexplainthe anomaly,beingseveralordersofmagnitudetoosmall,andSTVG,whilelargeenough, didnottwiththeorbitsoftheplanets.However,boththethermalrecoilforceand theMONDgravityforcewereoftherightsizeandformtoaccountforthisanomaly. Furthermore,theMONDgravityforceseemedtobeconsistentwiththeorbitsofthe planets,atleastinthecaseoftheEarth.Therefore,Iwasnotabletoruleitoutas apossiblecauseoftheanomaly. However,ifthePioneeranomalyisproventobecausedentirelybyathermalrecoil force,asmanysuspectitis,thiswouldhaveasignicantimpactonbothMONDand SVTGtheoriesofgravityandindeedmanyothers,providingonlyfurtherproof againstthesetheoriesandperhapsthenalnailintheconofalternategravity theories. Noneofmysimulatedresultsgaveanaccelerationthatwasexactlythevalueof thePioneeranomaly,however,thiswastobeexpected,astherearemanysmallforces andeectsactingonthespacecraftthatIdidnottakeintoaccount.Atbest,these simulationsgiveanorder-of-magnitudeideaofwhichoftheproposedexplanations arereasonabletoexplaintheanomaly.Attheveryleast,itdemonstratestheability tosimulateaproblemsuchasthisusingafairlysimpleprogram. Furtherimprovementstotheprogramcouldimprovetheaccuracyofthesimulations.Withmorework,Ibelieveitispossibletogettheprogramtowhereit isaccurateenoughtocalculatethesizeofthePioneeranomalyexactlyfromthese eects.Ifitwererunonlarger,fastercomputers,amoreaccuratemethodofdier35

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entiatingcouldbeusedwithasmallertimestepandmoreeects,suchasthepull oftheouterplanets,couldbeconsidered.Inthefuture,thesameprogramsthatI've usedtoanalyzethePioneeranomalycouldbeappliedtootherastronomicalproblems withonlysmallalterations. References [1]R.O.Fimmel,J.VanAllen,andE.Burgess, Pioneer:FirsttoJupiter,Saturn, andbeyond (NASA,WashingtonD.C.,1980). [2]G.Brooker, SensorsandSignals (Sydney,2006)Chap.14,p445. [3]AndersonJD,LaingPA,LauEL,LiuAS,NietoMMandTuryshevSG, Phys.Rev.Lett. 81 (1998). [4]J.D.Anderson,P.A.Laing,E.L.Lau,A.S.Liu,M.M.Nieto,andS.G. Turyshev,Phys.Rev.D. 65 (2002). [5]O.BertolamiandP.VieiraClass.Quant.Grav. 23 (2006). [6]LorenzoIorio,Found.Phys.D. 37 (2007). [7]SlavaG.TuryshevandViktorT.Toth,LivingRev.Relativity 13 (2010). [8]R.Foot,R.R.Volkas,Phys.Lett.B 517 (2001). [9]OBertolamiandJParamos,Class.QuantumGrav. 21 (2004). [10]JRBrownsteinandJWMoat,Class.QuantumGrav. 23 (2005). [11]J.W.Moat,JCAP 0505 (2005) [12]OrfeuBertolami,ChristianG.Bohmer,TiberiuHarko,FranciscoS.N.Lobox, Phys.Rev. 75 (2007). 36

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[13]SlavaG.Turyshev,MichaelMartinNieto,andJohnD.Andersonin TheXXII TexasSymposiumonRelativisticAstrophysics,Stanford,2004 ,editedbyP. Chen,E.Bloom,G.Madejski,andV.Petrosian. [14]AndreasRathke,IAC-05-A3.4.02, PioneerAnomaly:Whatcanwelearnfrom futureplanetaryexplorationmissions? (2005). [15]AndreasRathkeandDarioIzzo,JournalofSpacecraftandRockets 23 (2006). [16]SlavaG.Turyshev,ViktorT.Toth,GaryKinsella,Siu-ChunLee,ShingM.Lok, andJordanEllis,Phys.Rev.Lett. 108 (2012). [17]SlavaGSGTuryshev,ViktorTVTToth,JordanJEllis,CraigBCBMarkwardt, PhysRevLett 107 (2012) [18]MarionThornton ClassicalDynamicsofParticlesandSystems (1995)Chap.8, p299-309. [19]T.Logsdon OrbitalMechanics:TheoryandApplications (NewYork,1998)Chap. 2,p30. [20]MichaelMartinNietoandSlavaGTuryshev,Class.Quant.Grav. 21,2004 [21]HarveyGould,JanTobochnik,andWolfgangChristian,Addison-Wesley IntroductiontoComputerSimulationMethods (2006)Chap.5,p115-117 [22]W.H.Press,B.P.Flannery,S.A.Teukolsky,W.T.Vetterling NumericalRecipes inFortran77:TheArtofScientifcComputing (Cambridge,1995)Chap.16, p704 [23]MichaelMartinNietoandSlavaGTuryshev,andJohnD.Anderson,Phys.Lett. B 613 (2005) [24]ViktorT.TothandSlavaG.Turyshev,Phys.Rev.D 79 (2009). 37

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[25]SlavaG.TuryshevandViktorT.Toth,LivingRev.Relativity 13 (2010) [26]M.Milgrom,AstrophysicalJournal 270 (1983) [27]M.Milgrom,MonthlyNoticesoftheRoyalAstronomicalSociety, 426 (2012) [28]EarthFactSheet:http://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html 38

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