1969AJ 74. . 7 7 4K THE ASTRONOMICALJOURNALVOLUME74,NUMBER6AUGUST1969 surability oftheirmeanmotionwiththatJupiter, original listofHill,togetherwiththecoefficient suffer largeperturbationsinlongitude.TableIisthe used inthisinvestigation.Theyarerepresentativeof inequality anditsperiod. independent investigations(Zielenbach1968;Fiala next threehavealreadybeenthesubjectofother equality fortheseobjectsspantherangeofvalues.The the groupascoefficientsoflong-periodin- gathered, 64%werereducedtotheFK4systemby the fourminorplanetsusedinthisinvestigation. data whichwerecollected.Ofthe3250observations 1968; andO’Handley1967). have beenderived. were referredtocataloguesforwhichnozonecorrections application ofzonecorrections(Boss1937;Gyllenberg A LISToftwelveminorplanetswasproposedby 1948; Broscheetal.1964).Theremainingobservations attending YaleUniversityunder the auspicesofProfessional University incandidacyforthedegree ofDoctorPhilosophy. Development ProgramoftheU.S. NavalObservatory. The firstfourminorplanetslistedinTableIwere Table IIisashortsynopsisoftheorbitalelements * Thispaperisasummaryofdissertation presentedtoYale Table HIcontainsasummaryoftheobservational fThis investigationwasaccomplished whiletheauthorwas © American Astronomical Society • Provided by theNASA Astrophysics Data System G. W.Hill(1873)which,becauseofthecommen- 31 Euphrosyne 24 Themis 48 Doris 49 Pales 52 Europa 57 Mnemosyne 86 Semele 65 Cybele 10 Hygeia 90 Antiope 62 Erato 76 Freia Minor planet for thedeterminationofmassJupiteryieldscorrespondingreciprocalmasseswiththeirmean Europa, 1047.337±0.027.AsimultaneoussolutionforthemassofJupiterutilizingcombinedobservations error: 10Hygeia,1047.351±0.006;24Themis,1047.359rt0.010;31Euphrosyne,1047.372±0.00652 of allfourminorplanetsyieldsthevalue1047.360db0.004. Table I.Hill’slistofminorplanets. II. OBSERVATIONALMATERIAL An analysisoftheobservationsfourminorplanetssuggestedbyG.W.Hillasbeingespeciallysuitable The MassofJupiterandtheMotionFourMinorPlanets*! I. INTRODUCTION Coefficient 28 997 32 244 28 568 14 606 14 676" 12 956 13 655 11639 13 145 10 861 6 584 5 087 U. S.NavalObservatory,Washington,D.C. Period 103 121 104 92 98 yr 99 68 94 83 62 65 72 (Received 19May1969) W. J.Klepczynski 774 made withanequatorialtelescopewheretheminor planet isobserveddifferentiallywithrespecttosome rected forthecircularpartofannualaberration,areall reference star,andmeridiancircleobservationscor- comparable withanastrometricephemeris,itwas which tocomparethem. decided toapplythenecessarycorrectionsreduceall the observationstomeanequinoxandequatorof obtained foreachminorplanetcorrespondingtoan unknown wasobtainednumericallybydifferencingtwo adopted reciprocalmassforJupiterof1047.355. definitive setofosculatingelements(seeTableIV)was 1950.0 andgenerateanastrometricephemeriswith mass ofJupiterusedforthispurposeweretheadopted numerical integrationswhichutilizedthesamestarting to themassofJupiter.Thepartialderivativeforthis value of1047.355andacontrol1047.500. to theellipticelementswasaugmentedbyacorrection from ananalysisoftheobservationseachminor ing massofJupiter.Thetwovaluesthereciprocal coordinates butslightlydifferentvaluesforthedisturb- results ofthisinvestigationutilizingfourminorplanets planet areshowninTableV.Forcomparisontheresults O’Handley (1967)arealsogiven.Itisapparentthatthe derived byZielenbach(1968),Fialaand gations. Asexpected,thelargestmeanerrorsoccurin agree withtheresultsofthreeindependentinvesti- the long-periodinequality. those minorplanetshavingthesmallestcoefficientof Since photographicobservations,visualobservations III. ANALYTICSOLUTIONFORTHEMASSOFJUPITER Before seekingacorrectiontothemassofJupiter, When thesewereobtained,thesetofsixcorrections The valuesforthereciprocalmassofJupiterresulting planet motionSemi-axisEccentricity Inclination Minor Mean // 31 6343.1530.218 26.44 52 6523.0940.111 7.44 24 6413.1290.135 0.80 10 6393.1370.118 3?82 Table II.Orbitalcharacteristicsof minor planetsusedinthisstudy. 1969AJ 74. . 7 7 4K 2 2 /,o numerical partialderivatives,itwasdecidedtofitfour additional integrationstotheobservationsofeach Jupiter wasobtained.Thesumsofthesquares planet foreachcontrolvalueofthereciprocalmass reciprocal massofJupiter:1047.285,1047.320,1047.390, the fourminorplanetsusingfollowingvaluesfor versus massforeachminorplanetandaparabolafitted residuals (2v)obtainedbysubstitutingthesolutionfor and 1047.425.Adefinitivesetofelementsforeachminor mass whichmade'Evaminimumforeachminorplanet. the fiveintegrationsforeachminorplanet. tions ofconditionwereobtainedwithrespecttoeach the correctionstosixelementsbackintoequa- The valuessoobtainedduplicatedtheresults of theparabolaitwaspossibletoobtainvalue through thefivepoints.Bydifferentiatingequation by theanalyticsolution. in a.u. of thefourminorplanetsinordertosolvefortwenty- plane totheecliptic;eiseccentricity;asemimajoraxis longitude oftheascendingnode;iisinclinationorbital a> 3061036.7411044425.315594237.8283315856.865 l 164°4728Î603341°5050!527139°2148!802lll0020r093 l isthemeananomaly;