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Co-Evolution of the Milankovitch Cycles and the Earth-Moon System

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Co-Evolution of the Milankovitch Cycles and the Earth-Moon System reprint from irth nd lnetry iene vetters e p ossiilityX oEevolution of the wilnkovith yles nd the erthEmo on system kshi stoD wineo uumzwD ozo rmnoD nd kfumi wtsui estrt olr insoltion vrition due to the grvittionl p erturtion mong the plnetry o dies in the solr systemD so lled wilnkovith yle is widely elieved s m jor use of the limti hnge suh s the glilEinterglil yles in uternryD nd its typil frequenies re supp osed to e onstnt during uternryF roweverD the p erio ds of the wilnkovith yles must hve een lrgely hnged in the longer time sle of illion yers following to the dynmil evolution of the erthEmo on systemF heelerted rottionl veloity of the erth hs een mking the dynmil elliptiityofthe erth smller nd lengthening the m jor p erio ds of oth wilnkovith yles nd tidl ylesF ehve studied the reltion etween the frequenies of the wilnkovith yles nd the rottion rte of the erth on the sis of the theoretil nd omputtionl nlysis on the erthEmo on system with severl ssumptions involvedF yur onlusion is tht this yliity whih n e reorded in the sediments re mutully relted well s funtion of the dynmil elliptiity nd the solute geF e lso p erformed some simple estimtion out the eet of hoti ehvior of the plnetry motion of the solr system for the purp ose to rek down the illusion of the word hos4F e use the seulr vrition of the fundmentl frequenies of vskr @IWWHD IWWIA nd xoili et lF @IWVWA nd got the results whih imply tht the eet of hos myemuh smller then we hd exp eted eforeF his ft implies tht we n estlish the stndrd time sle for mesuring the reltive geD in other wordsD the lp time lo k or the hronometer for deo ding the whole history of the erthD y ompring the strip es in fsp nd other sediments of erhen or roterozoi with set of theoretil wilnkovith yle nd tidl yle frequeniesF rere we present the preliminry referene mo del of the evolution of the wilnkovith yles nd tidl yles nd ttempt to estlish the lp time lo kwhih will e p otentil devie for our pro jet to lrify evolutionry history of our erth9s environmentkto RqF sntro dution olr insoltion onto the surfe of the plnet hs een sid to vry p erio dilly nd use long p erio d R T limte hngeF ime sle of this yle is IH $ IH yersD nd it hs een onsidered s pemker of the glilEinterglil yles in uternry on the erth @rys et lFD IWUTAF his is lled the wilnkovith yle nd its mehnism hs een theorized quntittively from so erly times euse it onsists of lssil elestil mehnis out p oint msses nd rigid o dies @wilnkovithD IWRIAF W rowever in view of the longer time sle @y @IH AyersAD seulr hnge of the erthEmo on distne nd rottionl velo ity of the erth must hve used the gret eet on the wilnkovith ylesF his eet ws qulittively shown ylker nd hnle @IWVTA up PXSqD nd quntittively omputed y ferger et lF @IWWPA only up to HXSqF es insisted y fergerD it needs reful hndling to tre the wilnkovith yles k to the older erD euse the relevnt dynmis is not liner tht the vrition of the wilnkovith yles hve een supp osed to e hoti @ussmn 8 isdom IWWPD vskr IWWHAF xevertheless wehve strong demnd to identify nd deo de the yliity oserved in the sediments suh s nded iron formtion of erhen or roterozoiF he yliity reognized in the sediments ws proly used y the limte hnge due to wilnkovith yles s estimted from the sedimenttion rteF herefore we hve exmined p ossiility to lrify the evolutionry history of the wilnkovith ylesF he wilnkovith yle is dened s the long p erio d vrition of solr insoltion on to the top of the erth9s tmosphereF pigure Q upmost right one is typil exmple of lultion of the wilnkovith I yle whih represents the dily verge insoltion t TS xD summer solstieF he mximum mplitude is up to out PH7 of the verge vlue ording to the hnge of xil tilt of the erthY oliquity nd preession ngleF his osilltion of the solr insoltion is onsidered to trigger the glil nd interglil yles in uternryF he p ower sp etrum of this time series dt otined y the stndrd pourier trnsformtion is shown in pigure Q upp er leftF epprently you n see four shrp p eksF he p eks of IWkyr nd PQkyr re due to the osilltion of preession ngleD nd the p eks of RIkyr nd SRuyr is due to the osilltion of oliquity @though the p ek of SRuyr is rther wekAF hen nonliner erth system is su jeted to the foring with the sp etrum s oveD the output resp onse of $ IHHuy p erio d whih ws originted from the osilltion of the erth9s eentriitymy e reovered s mo dulted mplitude vrition of the two preession omp onentsD @PQ IWAa@PQ IWA % IHHuyF he nonliner surfe limte system of the erth might serve s lter to extrt the originl oritl foringF wilnkovith yle onsists of these four omp onents of IWD PQD RI nd SR uy @we ll them wpID wpPD woI nd woP resp etivelyAF rere wepy our ttention to these frequenies in the erhen or roterozoiF he length of these p erio ds @IWD PQD RI @nd IHHA uyA re thought to e nerly onstnt during the short time sle of uternryF W sn view of the longer time sle @IH yersAD these typil p erio ds of the wilnkovithylesmust hve een lrgely hnged following to the dynmil evolution of the erthEmo on systemF st my e sure tht the solr system hveeenevolving hotilly @isdom nd rolmnD IWWIY ussmn nd isdomD IWWPA nd mny sientists insist tht it is imp ossile to predit the preise oritl elements over ertin V W time sle suh s IH yersD not to mention IH yersF yn the other hndD the pro ess of dynmil evolution of the erthEmo on system is not hoti ut solutely seulrF reneD lthough the hoti plnetry p erturtion existsD we notied the eet of the plnetry p erturtion is nonEsystemti nd ovithylesne minor in the vrition of wpID wpPD woI nd woP nd the evolution of the wilnk tred k to the nient times when the erth ws spinning muh fsterF he purp ose of the present pp er is @IA to disuss out some sp eil ssumptions needed for the lultionD @PA to investigte in the eet of hoti plnetry motion nd put the devition error rs due to the hoti ehvior of the solr system on the evolution pths of the wilnkovith yles y onsulting the vrile rnge of mplitudes nd frequenies of the fundmentl frequenies omputed y vskr @IWVVA nd xoili et lF @IWVWAD @QA to present some results of the p ossile evolution digrm of the wilnkovith yles on the sis of stndrd lultionl result of the dynmil evolution of the erthEmo on systemD nd @QA to provide lue to relte the physil limte mo del with the tul dt suhsintheperiodistriped nds in nded iron formtion of erhen or roterozoi @pigure U nd VAF iqution of motion he nnully verged eqution of motion of the rottionl xis of the plnet @in this seD the erthA is derived from the iuler9s eqution of rigid o dy rottion viewing from the inertil oEordinte system @rdD IWURY fillsD IWWHA ds a @s nA@s nA @IA dt where is lled the preessionl onstnt representing the mgnitude of the grvittionl torque oE tined y the equtoril ulge of the erth @present vlue is SHFRR @rseGyerAAF s a@s Ys Ys Aisthe x y z spin xis unit vetor of the erthF n is the oritl norml unit vetor of the erth nd expressed y P oritl inlintion s nd longitude of the sending no de s n a @sin s sin Y sin s os Y os s A @PA e n get the time vrition of the oliquity nd the preession ngle 0 of rd @IWURA nd fills @IWWHA from following reltionshipX s y I I aos @s nAY 0 a sin @QA sin nd otin the solr insoltion vrition suh s in pigure Q right side @fF ferger 8 voutreD IWUVAF es for the time series of oritl elements sY D nd eentriity e D longitude of p erihelion with resp et to s the xed vernl equinox 6 Dwe use the solution of vskr @IWVVA9s seulr p erturtion theoryF hominnt ftors of the wilnkovith yles es mentioned eforeD preessionl onstnt whih represents the degree of grvittionl torque otined y the equtoril ulge of the plnet n e expressed s 2 3 Q P Q Q g e Q w Qn s m P P P P P I e I sin i @RA C I e a s m m P3 g w P s m where e nd g re the p olr nd the equtoril moment of inerti of the erthD n is the erth9s men motion to the sunD e is the eentriity of the mo on9s oritD w nd w is the mss of the mo on nd m m s the sunD nd is the length of the semim jor xis of the mo on9s orit nd the erth9s oritD nd i m s m is the inlintion of the mo on9s orit ginst the orit of the erthF he frequenies of the wilnkovith I 9 os `snb @`b mens yles re determined y the time vrition of nD D nd the men oliquity the time vergeAF n represents the orienttion of the erth9s oritl inlintion whih is eted y the grvittionl p erturtion mong other plnetF ine we onsider tht the eet of p ossile hoti motion of the solr system is smllD we ssume tht the osilltion of the erth9s eentriity e is not s so dierent from the present ge during these RqD nd use the qusiEp erio di vrition of e shown y s vskr @IWVVAF e out the lunr oritl elements e nd i D lultionl results re muh dierent mong eh m m reserhersF rere we utilize two kinds of omputtionl results of evolution of e nd i Xeeet lF m m @IWWPA nd urotte et lF @IWUUAF em nd i re just equl to zero in the simple mo del of urotte et lF m @IWUUAD nd e e et lF preisely lulted the hnge of them @pigure RAF rowever the solute vlues of e nd i re originlly so smll tht they hve only slight eet on the hnge of the preessionl m m P P P yAF imilrly I e $ I in go o d pproximtion onstnt @ieFD I e nd I sin i is lmost unit s m m so tully we n onsider tht the preessionl onstnt is determined lmost ll y the reltionship g e etween three vrilesX dynmil elliptiity of the erth D rottionl ngulr velo ity of the erth g 3 D nd the erthEmo on distne F he hnges of the erthEsun distne D the msses w Yw re m s s m not tken into ount in this reserhF e out the evolution of the men oliquity of the erth there re mny hyp othesisX
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