notes on the planetary motion

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1、NotesontheTheoryofPlanetaryMotionfanius@163.comNovember2013Thisarticleistosharesomeofmycalculationexerciseonthissubject,whichshouldbeaccessibleandfreelyusedbyeveryone.1EllipseinPolarCoordinateSystemLetMbethecentreofanellipse,andF1,andF2bethefoci.SetkF1Mk=kF2Mk=d.TakePfromtheellipse,andsetthesemi-cir

2、cumferenceof4PF1F2=s+d,wheresisaconstantbydenition.Itcanbecalculatedthats2d2theequationofellipseinpolarcoordination(;)hastheform=:s+dcosSetp=(s2d2)=s,ande=d=stoobtainthestandardformp=:(1.1)1+ecosHerepiscalledsemi-latusrectum,andetheeccentricity.Usingsomealgebraicgimmickwouldbeeasiertoobtain

3、thesemi-majoraxis.Apparently,bytakingps2d2=0,thesemi-majoraxisis+d,or+d=s.Usingd=es,and1+es+dppequate+es=s,onehassemi-majoraxiss=.Toobtainsemi1+epp1e2pminoraxis,onenoticesitslengthiss2d2=s1e2=p.1e2Inordertoobtaintheareaofellipse,it'sbettertomakeuseitsforminCartesianSystem.TakeMastheorigin,andl

4、etitssemimajoraxisbea,andsemiminorbeb.Theparametricformofanellipseisx=acos;(1.2a)y=bsin:(1.2b)ZaZ=22Thequarterareaofanellipsethereforeisydx=absind.NowZ00=221cos2absin=,andcos2d=0,thusthequarterareais,or204abintotal.Thatistosaytheareaofanellipseisp2(1e2)3=2undertheexpressionof(1.1).No

5、teife=0,namelydvanishes,thentheellipsedegeneratestoacircle,andtheareabecomess2,sthusbeingtheradius.Z212Theareaformulaunderpolarsystem,intuitively,isd.Thuswe20haveZ2d23=2=2(1e);(1.3)0(1+ecos)21whichmightbediculttoobtainbyregularintegrationtechnique.Thiscanbefoundinanyreasonablywell-written

6、treatiseofintegrationtheory.2VelocityandAccelerationNowlet=(t);=(t),and=(t)=ei,hencedescribesthemotionofaparticleinpole.Dierentiatewithrespecttot,oneobtainsthevelocity:

_=_ei+_iei;(2.1)or_=(_=)+i_.HereweusetheNewtoniannotation
ratherthatLeibnizdforderivative.Itisseenfrom(2.

7、1)thatthevelocitycouldbeendiscomposeddxintwodirections,eachorthogonaltotheother.NowweswitchtoCartesianSystemtohaveafreshviewon(2.1):x=cos;(2.2a)y=sin:(2.2b)UseChainRuletodierentiate(2.2):x_=_cos