A second order Laplace-Lagrange theory

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1、Astron.Astrophys.327,416–427(1997)ASTRONOMYANDASTROPHYSICSAsecondorderLaplace-LagrangetheoryappliedtotheuraniansatellitesystemApostolosA.ChristouandCarlD.MurrayAstronomyUnit,QueenMaryandWestfieldCollege,MileEndRoad,LondonE14NS,UK(A.Christou@qmw.ac.uk,C.D.Murray@qmw.ac.uk)Received18December1996/A

2、ccepted11June1997Abstract.Computeralgebraisusedtoderiveanalyticalexpres-AnattemptbyLaskar&Jacobson(1987)tofitthetheorysionsfortheelementsofasecularperturbationtheoryofthedescribedbyLaskar(1986)toVoyagerandEarth-basedobser-secondorderinthemasses.Thelinearpartofthistheoryisin-vations,thelatterspan

3、ningapproximatelyoneorbitalperiodofcorporatedintothestandardLaplace-LagrangesolutionwhichUranus,necessitatedtheintroductionofempiricalcorrectionsisthenappliedtothemotionoftheuraniansatellites.Thisen-totheirderivedseculareigenfrequencies,possiblyduetotheablestheeffectsofmeanmotionnear-resonances

4、inthissystemunmodelledeffectofthesenear-resonances.Thefirstanalyticaltobemodelled.TheresultsareinverygoodagreementwithtreatmentofthesephenomenawasperformedbyMalhotraetthefilteredoutputofanumericalintegrationbyMalhotraetalal(1989)(hereafterreferredtoasMFMN)inwhichtheeffects(1989).Theadditionalcomp

5、lexitiesofstudyingtheoutersolarofthe2:1Umbriel-Titaniaandthe3:2Titania-Oberonnear-systemusingasimilarapproacharediscussed.resonancesontheLaplace-Lagrangemodelwerequantified.Theresultwasfoundtobeinverygoodagreementwithanu-Keywords:celestialmechanics–satellites:satellitesofUranusmericalintegration

6、describedinthesamepaper.However,there–methods:analyticalstillremainedadiscrepancyexceedingthe1%level(abench-marksetbytheirresultsfortheinclinationvariations)thatcanprobablybeattributedtotheeffectofweakernear-resonances.Thesubjectofthispaperisthedescriptionandrigorous1.Introductiontestingofamode

7、lthatisbasedonageneraltheoryofthesec-ondorderinthemasses.ThiswasdescribedindetailbyDuriezThesatellitesystemoftheplanetUranusisanidealtesting(1979)andwassubsequentlypresentedinamoresystematicgroundforsecularperturbationtheoriessinc