€reprint from i—rth —nd €l—net—ry ƒ™ien™e vetters

e p ossi˜ilityX ™oEevolution of the wil—nkovit™h ™y™les —nd the

e—rthEmo on system

„—k—shi stoD wineo uum—z—w—D ‰ozo r—m—noD —nd „—k—fumi w—tsui

e˜str—™t

ƒol—r insol—tion v—ri—tion due to the gr—vit—tion—l p ertur˜—tion —mong the pl—net—ry ˜ o dies in the

sol—r systemD so ™—lled wil—nkovit™h ™y™le is widely ˜ elieved —s — m— jor ™—use of the ™lim—ti™ ™h—nge

su™h —s the gl—™i—lEintergl—™i—l ™y™les in u—tern—ryD —nd its typi™—l frequen™ies —re supp osed to ˜ e

™onst—nt during u—tern—ryF roweverD the p erio ds of the wil—nkovit™h ™y™les must h—ve ˜ een l—rgely

™h—nged in the longer time s™—le of ˜illion ye—rs following to the dyn—mi™—l evolution of the e—rthEmo on

systemF he™™eler—ted rot—tion—l velo™ity of the e—rth h—s ˜ een m—king the dyn—mi™—l ellipti™ityofthe

e—rth sm—ller —nd lengthening the m— jor p erio ds of ˜ oth wil—nkovit™h ™y™les —nd tid—l ™y™lesF ‡eh—ve

studied the rel—tion ˜ etween the frequen™ies of the wil—nkovit™h ™y™les —nd the rot—tion r—te of the

e—rth on the ˜—sis of the theoreti™—l —nd ™omput—tion—l —n—lysis on the e—rthEmo on system with sever—l

—ssumptions involvedF yur ™on™lusion is th—t this ™y™li™ity whi™h ™—n ˜ e re™orded in the sediments —re

mutu—lly rel—ted well —s — fun™tion of the dyn—mi™—l ellipti™ity —nd the —˜solute —geF ‡e —lso p erformed

some simple estim—tion —˜ out the ee™t of ™h—oti™ ˜ eh—vior of the pl—net—ry motion of the sol—r system

for the purp ose to ˜re—k down the illusion of the word ’™h—os4F ‡e use the se™ul—r v—ri—tion of the

fund—ment—l frequen™ies of v—sk—r @IWWHD IWWIA —nd xo˜ili et —lF @IWVWA —nd got the results whi™h imply

th—t the ee™t of ™h—os m—y˜emu™h sm—ller then we h—d exp e™ted ˜ eforeF „his f—™t implies th—t we

™—n est—˜lish the st—nd—rd time s™—le for me—suring the rel—tive —geD in other wordsD the l—p time ™lo ™k

or the ™hronometer for de™o ding the whole history of the e—rthD ˜y ™omp—ring the strip es in fsp —nd

other sediments of er™he—n or €roterozoi™ with — set of theoreti™—l wil—nkovit™h ™y™le —nd tid—l ™y™le

frequen™iesF rere we present the prelimin—ry referen™e mo del of the evolution of the wil—nkovit™h ™y™les

—nd tid—l ™y™les —nd —ttempt to est—˜lish the l—p time ™lo ™kwhi™h will ˜ e — p otenti—l devi™e for our

pro je™t to ™l—rify evolution—ry history of our e—rth9s environment˜—™kto Rq—F

sntro du™tion

ƒol—r insol—tion onto the surf—™e of the pl—net h—s ˜ een s—id to v—ry p erio di™—lly —nd ™—use long p erio d

R T

™lim—te ™h—ngeF „ime s™—le of this ™y™le is IH $ IH ye—rsD —nd it h—s ˜ een ™onsidered —s — p—™em—ker

of the gl—™i—lEintergl—™i—l ™y™les in u—tern—ry on the e—rth @r—ys et —lFD IWUTAF „his is ™—lled the

wil—nkovit™h ™y™le —nd its me™h—nism h—s ˜ een theorized qu—ntit—tively from so e—rly times ˜ e™—use

it ™onsists of ™l—ssi™—l ™elesti—l me™h—ni™s —˜ out p oint m—sses —nd rigid ˜ o dies @wil—nkovit™hD IWRIAF

W

rowever in view of the longer time s™—le @y @IH Aye—rsAD se™ul—r ™h—nge of the e—rthEmo on dist—n™e —nd

rot—tion—l velo ™ity of the e—rth must h—ve ™—used the gre—t ee™t on the wil—nkovit™h ™y™lesF „his

ee™t w—s qu—lit—tively shown ˜y‡—lker —nd —hnle @IWVTA up PXSq—D —nd qu—ntit—tively ™omputed

˜y ferger et —lF @IWWPA only up to HXSq—F es insisted ˜y fergerD it needs ™—reful h—ndling to tr—™e the

wil—nkovit™h ™y™les ˜—™k to the older er—D ˜ e™—use the relev—nt dyn—mi™s is not line—r th—t the v—ri—tion

of the wil—nkovit™h ™y™les h—ve ˜ een supp osed to ˜ e ™h—oti™ @ƒussm—n 8 ‡isdom IWWPD v—sk—r IWWHAF

xevertheless weh—ve — strong dem—nd to identify —nd de™o de the ™y™li™ity o˜served in the sediments

su™h —s ˜—nded iron form—tion of er™he—n or €roterozoi™F „he ™y™li™ity re™ognized in the sediments w—s

pro˜—˜ly ™—used ˜y the ™lim—te ™h—nge due to wil—nkovit™h ™y™les —s estim—ted from the sediment—tion

r—teF „herefore we h—ve ex—mined — p ossi˜ility to ™l—rify the evolution—ry history of the wil—nkovit™h

™y™lesF

„he wil—nkovit™h ™y™le is dened —s the long p erio d v—ri—tion of sol—r insol—tion on to the top of the

e—rth9s —tmosphereF pigure Q upmost right one is — typi™—l ex—mple of ™—l™ul—tion of the wil—nkovit™h I



™y™le whi™h represents the d—ily —ver—ge insol—tion —t TS xD summer solsti™eF „he m—ximum —mplitude

is up to —˜ out PH7 of the —ver—ge v—lue —™™ording to the ™h—nge of —xi—l tilt of the e—rthY o˜liquity —nd

pre™ession —ngleF „his os™ill—tion of the sol—r insol—tion is ™onsidered to trigger the gl—™i—l —nd intergl—™i—l

™y™les in u—tern—ryF „he p ower sp e™trum of this time series d—t— o˜t—ined ˜y the st—nd—rd pourier

tr—nsform—tion is shown in pigure Q upp er leftF epp—rently you ™—n see four sh—rp p e—ksF „he p e—ks of

IWkyr —nd PQkyr —re due to the os™ill—tion of pre™ession —ngleD —nd the p e—ks of RIkyr —nd SRuyr is due to

the os™ill—tion of o˜liquity @though the p e—k of SRuyr is r—ther we—kAF ‡hen — nonline—r e—rth system is

su˜ je™ted to the for™ing with the sp e™trum —s —˜ oveD the output resp onse of $ IHHuy p erio d whi™h w—s

origin—ted from the os™ill—tion of the e—rth9s e™™entri™itym—y ˜ e re™overed —s — mo dul—ted —mplitude

v—ri—tion of the two pre™ession ™omp onentsD @PQ  IWAa@PQ IWA % IHHuyF „he nonline—r surf—™e ™lim—te

system of the e—rth might serve —s — lter to extr—™t the origin—l or˜it—l for™ingF wil—nkovit™h ™y™le

™onsists of these four ™omp onents of IWD PQD RI —nd SR uy @we ™—ll them wpID wpPD woI —nd woP

resp e™tivelyAF rere wep—y our —ttention to these frequen™ies in the er™he—n or €roterozoi™F „he length

of these p erio ds @IWD PQD RI @—nd IHHA uyA —re thought to ˜ e ne—rly ™onst—nt during the short time s™—le

of u—tern—ryF

W

sn view of the longer time s™—le @IH ye—rsAD these typi™—l p erio ds of the wil—nkovit™h™y™lesmust h—ve

˜ een l—rgely ™h—nged following to the dyn—mi™—l evolution of the e—rthEmo on systemF st m—y ˜ e sure

th—t the sol—r system h—ve˜eenevolving ™h—oti™—lly @‡isdom —nd rolm—nD IWWIY ƒussm—n —nd ‡isdomD

IWWPA —nd m—ny s™ientists insist th—t it is imp ossi˜le to predi™t the pre™ise or˜it—l elements over ™ert—in

V W

time s™—le su™h —s IH ye—rsD not to mention IH ye—rsF yn the other h—ndD the pro ™ess of dyn—mi™—l

evolution of the e—rthEmo on system is not ™h—oti™ ˜ut —˜solutely se™ul—rF ren™eD —lthough the ™h—oti™

pl—net—ry p ertur˜—tion existsD we noti™ed the ee™t of the pl—net—ry p ertur˜—tion is nonEsystem—ti™ —nd

ovit™h™y™les™—n˜e minor in the v—ri—tion of wpID wpPD woI —nd woP —nd the evolution of the wil—nk

tr—™ed ˜—™k to the —n™ient times when the e—rth w—s spinning mu™h f—sterF „he purp ose of the present

p—p er is @IA to dis™uss —˜ out some sp e™i—l —ssumptions needed for the ™—l™ul—tionD @PA to investig—te in

the ee™t of ™h—oti™ pl—net—ry motion —nd put the devi—tion error ˜—rs due to the ™h—oti™ ˜ eh—vior of

the sol—r system on the evolution p—ths of the wil—nkovit™h ™y™les ˜y ™onsulting the v—ri—˜le r—nge of

—mplitudes —nd frequen™ies of the fund—ment—l frequen™ies ™omputed ˜y v—sk—r @IWVVA —nd xo˜ili et —lF

@IWVWAD @QA to present some results of the p ossi˜le evolution di—gr—m of the wil—nkovit™h ™y™les on the

˜—sis of st—nd—rd ™—l™ul—tion—l result of the dyn—mi™—l evolution of the e—rthEmo on systemD —nd @QA to

provide — ™lue to rel—te the physi™—l ™lim—te mo del with the —™tu—l d—t— su™h—sintheperiodi™striped

˜—nds in ˜—nded iron form—tion of er™he—n or €roterozoi™ @pigure U —nd VAF

iqu—tion of motion

„he —nnu—lly —ver—ged equ—tion of motion of the rot—tion—l —xis of the pl—net @in this ™—seD the e—rthA

is derived from the iuler9s equ—tion of rigid ˜ o dy rot—tion viewing from the inerti—l ™oEordin—te system

@‡—rdD IWURY fillsD IWWHA

ds

a @s  nA@s  nA @IA

dt

where  is ™—lled the pre™ession—l ™onst—nt representing the m—gnitude of the gr—vit—tion—l torque o˜E

t—ined ˜y the equ—tori—l ˜ulge of the e—rth @present v—lue is SHFRR @—r™se™Gye—rAAF s a@s Ys Ys Aisthe

x y z

spin —xis unit ve™tor of the e—rthF n is the or˜it—l norm—l unit ve™tor of the e—rth —nd expressed ˜y P

or˜it—l in™lin—tion s —nd longitude of the —s™ending no de  —s

n a @sin s sin Y sin s ™os Y ™os s A @PA

‡e ™—n get the time v—ri—tion of the o˜liquity  —nd the pre™ession —ngle 0 of ‡—rd @IWURA —nd fills

@IWWHA from following rel—tionshipX

 

s

y

I I

 a™os @s  nAY 0 a sin @QA

sin 

—nd o˜t—in the sol—r insol—tion v—ri—tion su™h —s in pigure Q right side @™fF ferger 8 voutreD IWUV˜AF es

for the time series of or˜it—l elements sY D —nd e™™entri™ity e D longitude of p erihelion with resp e™t to

s

the xed vern—l equinox 6 Dwe use the solution of v—sk—r @IWVVA9s se™ul—r p ertur˜—tion theoryF

homin—nt f—™tors of the wil—nkovit™h ™y™les

es mentioned ˜ eforeD pre™ession—l ™onst—nt  whi™h represents the degree of gr—vit—tion—l torque

o˜t—ined ˜y the equ—tori—l ˜ulge of the pl—net ™—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 ol—r —nd the equ—tori—l moment of inerti— of the e—rthD n is the e—rth9s me—n

motion to the sunD e is the e™™entri™ity of the mo on9s or˜itD w —nd w is the m—ss of the mo on —nd

m m s

the sunD — —nd — is the length of the semim— jor —xis of the mo on9s or˜it —nd the e—rth9s or˜itD —nd i

m s m

is the in™lin—tion of the mo on9s or˜it —g—inst the or˜it of the e—rthF „he frequen™ies of the wil—nkovit™h

I

 9 ™os `snb @`b me—ns ™y™les —re determined ˜y the time v—ri—tion of nD D —nd the me—n o˜liquity

the time —ver—geAF n represents the orient—tion of the e—rth9s or˜it—l in™lin—tion whi™h is —e™ted ˜y

the gr—vit—tion—l p ertur˜—tion —mong other pl—netF ƒin™e we ™onsider th—t the ee™t of p ossi˜le ™h—oti™

motion of the sol—r system is sm—llD we —ssume th—t the os™ill—tion of the e—rth9s e™™entri™ity e is not

s

so dierent from the present —ge during these Rq—D —nd use the qu—siEp erio di™ v—ri—tion of e shown ˜y

s

v—sk—r @IWVVAF

e˜ out the lun—r or˜it—l elements e —nd i D ™—l™ul—tion—l results —re mu™h dierent —mong e—™h

m m

rese—r™hersF rere we utilize two kinds of ™omput—tion—l results of evolution of e —nd i Xe˜eet —lF

m m

@IWWPA —nd „ur™otte et —lF @IWUUAF em —nd i —re just equ—l to zero in the simple mo del of „ur™otte et —lF

m

@IWUUAD —nd e˜ e et —lF pre™isely ™—l™ul—ted the ™h—nge of them @pigure RAF rowever the —˜solute v—lues

of e —nd i —re origin—lly so sm—ll th—t they h—ve only slight ee™t on the ™h—nge of the pre™ession—l

m m

P

P P

yAF ƒimil—rly I e $ I in — go o d —pproxim—tion ™onst—nt  @ieFD I e —nd I sin i is —lmost unit

s m m

so —™tu—lly we ™—n ™onsider th—t the pre™ession—l ™onst—nt  is determined —lmost —ll ˜y the rel—tionship

g e

˜etween three v—ri—˜lesX dyn—mi™—l ellipti™ity of the e—rth D rot—tion—l —ngul—r velo ™ity of the e—rth

g

3 D —nd the e—rthEmo on dist—n™e — F „he ™h—nges of the e—rthEsun dist—n™e — D the m—sses w Yw —re

m s s m

not t—ken into —™™ount in this rese—r™hF

e˜ out the evolution of the me—n o˜liquity of the e—rth there —re m—ny hyp othesisX ™lim—te fri™tion

@‚u˜in™—mD IWWHAD sto ™h—sti™ —™™umul—tion pro ™ess @hones —nd „rem—ineD IWWQAD tid—l evolution @u—ul—D

IWTRAF sn this dis™ussion we only t—ke into —™™ount the tid—l evolution ee™t ™—l™ul—ted ˜ye˜e et —lF

@IWWPA ˜ e™—use other f—™tors —re r—ther v—gueF fut —™tu—lly the me—n o˜liquity —t Rq— of e˜ e et —lF

  

 —t Rq— —nd the present is ™os IV ™os PQXS % HXHQRD @IWWPA is —˜ out IV D —nd the dieren™e of ™os

whi™h is f—irly sm—llF Q

T

sn u—tern—ry time s™—le @IH ye—rsAD rot—tion—l —ngul—r velo ™ity of the e—rth 3 D e—rthEmo on dist—n™e

g e

— D —nd dyn—mi™—l ellipti™ity of the e—rth —re ™onsidered to ˜ e ne—rly ™onst—ntF rowever they —re

m

g

W

not ™onst—ntinIH ye—rs time s™—leD ˜ e™—use 3 h—s ˜ een ˜ e™oming sm—ller —nd — h—s ˜ een ˜ e™oming

m

l—rger ˜ e™—use of the tid—l fri™tion ˜ etween the e—rth —nd mo onF ren™e  ™h—nges with —geD —nd so do

the frequen™ies of the wil—nkovit™h ™y™lesF rere we expli™itly put three —ssumptions for determining the

evolution of  —nd ™onsider —˜ out the v—lidity of these —ssumptions in the next se™tionF

essumptions —nd their extentofv—lidity

IF gonserv—tion of the —ngul—r momentum of the e—rthEmo on systemX „he e—rthEmo on

system h—s ˜ een losing its —ngul—r momentum —long the evolution due to the tid—l torque from the sunF

„his ee™t is ™—l™ul—ted ˜y e˜ e et —lF @IWWPA —nd the results —re shown in pigure S @™AF es you ™—n seeD

the —ngul—r momentum of the e—rthEmo on system h—s ™h—nged no more th—n I7 in these Rq—D so this

—ssumption is quite go o d —s the rst —pproxim—tionF

PF hensity stru™ture of the e—rth9s interior h—s not ˜ een ™h—ngedX „hough the timing of

™ore form—tion of the e—rth is still not ™le—rD it is sure th—t the e—rth underwent the ™ore form—tion st—ge

quite e—rlier in its —™™umul—tion history @xewsomD IWWHAF yf ™ourse some events like fr—™tion—tion of the

™rust from the m—ntleormode™h—ngeofm—ntle™onve™tion from the one l—yer mo de to the twol—yer

mo de m—yh—ve™h—nged the density stru™ture of the e—rthD ˜ut we ™—n s—y th—t the time v—ri—tion of

the density stru™ture of the e—rth9s interior h—s ˜ een enough sm—ll within the pr—™ti™—lly v—ri—˜le r—nge

for our purp oseF

QF hyn—mi™—l ellipti™ity is prop ortion—l to the squ—re of the rot—tion—l —ngul—r velo ™ity

g e

P

of the e—rth is the most imp ort—nt qu—ntity in our dis™ussionF „here @3 A X hyn—mi™—l ellipti™ity

g

—re two ™on™epts —˜ out the o˜l—teness of the pl—netD one is the dyn—mi™—l ellipti™ity whi™h indi™—tes the

degree of gr—vit—tion—l —ttening of the pl—netD —nd the other is the geometri™—l —ttening f whi™histhe

g e

r—tio of the semim— jor —xis —nd semiminor —xis of the e—rthF ‡hen the —ngul—r velo™ity 3 is l—rgeD

g

g e

—nd f —re sm—llF —nd f —re l—rgeD —nd the pl—net is more o˜l—teF yn the ™ontr—ry when 3 is sm—llD

g

„he —™™ur—te determin—tion of the dyn—mi™—l ellipti™ity —s — fun™tion of rot—tion—l —ngul—r velo ™ity 3 in

the ™—se of the —™tu—l e—rth is quite ™ompli™—ted ˜ e™—use we should use gl—ir—uts9s theorem to ™—l™ul—te

g e

of the str—tied pl—net @h—rkov —nd „ru˜itsynD IWUVY henisD the equip otenti—l surf—™e —nd o˜t—in

g

g e

IWVTAF fut here we —ssume th—t the dyn—mi™—l ellipti™ity is ne—rly equ—l to the geometri™—l —ttening

g

P

f —nd prop ortion—l to the squ—re of rot—tion—l —ngul—r velo ™ity@3 AF „his —ssumption is just ™orre™t

in the ™—se of ™onst—nt density rot—ting ˜ o dy —nd justied well if the density stru™ture of the e—rth is

hydrost—ti™ @„ur™otte —nd ƒ™hu˜ertD IWVPY ƒt—™yD IWWPAF „he ee™ts of nonhydrost—ti™ st—te —nd the

v—ri—˜ility of the density stru™ture within the e—rth —re supp osed to ˜ e sm—llF fut of ™ourse they —re

one of the imp ort—nt su˜ je™ts of our future workF

e˜ ovewe put three ˜ old —sumptions to ™—l™ul—te the evolution—ry history of the wil—nkovit™h ™y™lesF

„hough the —˜ ove —ssumptions —re very roughD they m—y ˜ e essenti—llyl true in the light of ™ommon

sense of the geophysi™s of the line—r systemF roweverD we —re living in — typi™—l nonEline—r dyn—mi™

system | ™h—oti™ sol—r systemF felow we investig—te in — sensi˜ility of the wil—nkovit™h ™y™les to the

p ossi˜le ™h—oti™ motions of the pl—nets in this sol—r systemF R

ie™t of the p ossi˜le ™h—oti™ motions of the pl—nets

‡hen the st—˜ility of our sol—r system is dis™ussedD two o˜ je™tions often —riseF pirstD this pro˜lem

h—s ˜ een ™oming —round for to o long ye—rsD never getting to the n—l p oint to st—te ™le—rly whether

the system is st—˜le or notY the few denite results refer to m—them—ti™—l —˜str—™tions su™h—s x E˜ o dy

mo dels —nd do not re—lly —pply to the re—l sol—r systemF ƒe™ond the sol—r system is m—™ros™opi™—lly

W

st—˜le | —t le—st for — few IH ye—rs | sin™e it is still thereD —nd there is not mu™h p oint in giving —

rigrous —rgument for su™h —n intuitive protertyF fy the pro˜lem of the st—˜ility of the sol—r system we

me—n to underst—nd whether our sol—r system is st—˜le for its entire lifetime or notF ‡e —re ™on™erned

W IH

only with — nite @IH $ IH Aye—rs timesp—nD not with —n innite timesp—n whi™h h—s st—˜le solution

—s proved ˜y €oin™—reover — ™entury —goF rowever in spite of the eorts of m—ny s™ientists @‡isdom

—nd rolm—nD IWWIY ƒussm—n —nd ‡isdomD IWWPA no st—˜le @or p erio di™A solution is found —nd it even

seems they give up nding the st—˜le solution of the sol—r system —nd intend to show numeri™—lly —nd

syntheti™—lly the inst—˜ility of this sol—r system @xo˜ili et —lFD IWVWAF

sn v—sk—r @IWWHA@v—sk—rD IWWHA he writes two issues —˜ out the ™h—oti™ ™h—r—™teres of the sol—r systemX

@IA it is imp ossi˜le to ™ompute the ex—™t motion of the sol—r system over more th—n IHHwyr —nd the

solution over PHHwyr will ˜ e just — qu—lit—tive p ossi˜ility ˜ efore IHHwyrF @PA fund—ment—l frequen™ies

of the sol—r system g —nd s —re not ™onst—nt —nd slowly v—ry with timeF

i i

wil—nkovit™h frequen™ies —re determined ˜y two f—™torsF yne is the luniEsol—r pre™ession of whi™h

periods —re ™h—r—™terized ˜y the pre™ession—l ™onst—nt  D ieFD the e—rth9s rot—tion—l —ngul—r velo™ity

g e

without —sso ™i—tion with 3 D e—rthEmo on dist—n™e — D —nd the dyn—mi™—l ellipti™ity of the e—rth

m

g

the motion of other pl—netsF „he other f—™tor is ™—lled the pl—net—ry pre™ession whi™h represents the

movement —nd deform—tion of the e—rth9s or˜it—l pl—ne in the inerti—l ™o ordin—te systemF „he ™h—oti™

ee™t will —pp e—r in the wil—nkovit™h ™y™les through the l—tter f—™torF sn ferger et —lF @IWWPA@ferger

et —lFD IWWPA they showed the evolution of the m—in wil—nkovit™h p erio ds ˜—™k to PHHwyr in™luding the

v—ri—tion of the fund—ment—l seul—r frequen™ies g —nd s D —nd ™on™ludes th—t the imp—™t of the ™h—nges

i i

of g —nd s —re mu™h less th—n th—t of the v—ri—tion of the pre™ession—l ™onst—nt  F

i i

„hen wh—t —˜ out the evolution—ry p—ths of the wil—nkovit™h ™y™les over the whole Rq— in the e—rth9s

historyc sf this sol—r system is tot—lly ™h—oti™ we ™—nnot m—n—ge to re—™h the ™le—r ™on™lusion —˜ out

the evolution of the wil—nkovit™h ™y™lesD nothing to s—y the pl—net—ry motionsF fut even if soD weknow

there —re some p erio di™ environment—l ™h—nges in er™he—n of €roterozoi™ e—rth9s surf—™e whi™h indi™—te

the existen™e of insol—tion v—ri—tions —t th—t timeF st is indeed signi™—nt to get the evolution—ry history

of the wil—nkovit™h ™y™les —s long —s the e—rthEmo on system ˜y using —ll d—t— —nd knowledge o˜t—ined

for nowF

u—siEp erio di™ motion of the or˜it—l pl—ne

H H H

yr˜it—l elements eY 6 Y s Y  —re expressed ˜y the p—r—meters e Ye Y# Y# Y0 Y0 whi™h represents the

j j j

j j j

qu—siEp erio di™ motions of the sol—r system —s followsX @v—sk—rD IWWHA

n

ˆ

p p

I6 a I@# t C 0 A @SA e exp e exp

j j j

j

n

ˆ

p p

s

H H H

sin e exp exp I a I@# t C 0 A @TA

j j j

P

j S

H H H

is listed in the t—˜les of v—sk—r @IWVVAF „he ex—™t m—them—ti™—l denition Y0 Y0 Y# Y# €—r—meters e Ye

j j j

j j j

of so ™—lled ’™h—os4 v—ries widely —mong the rese—r™hersD ˜ut it ™—n ˜ e summ—rized th—t ’slightly sm—ll

dieren™e of the initi—l ™onditions ™—use the unexp e™tedly l—rge dieren™e of the output results4F sn

H H

D—nd D frequen™ies # Y# this ™—se it ™—n ˜ e tr—nsl—ted into the phr—se th—t the m— jor —mplitude e Ye

j j

j j

V H

is not ™onst—nt during the time s™—le of over IH ye—rs @iFeFD not p erio di™ the initi—l ™onditions 0 Y0

j

j

—nd predi™t—˜leA —nd we ™—n h—rdly know their ex—™t v—luesF „his ˜rief interpret—tion —˜ out the term

’™h—oti™ sol—r system4 is su™ient for us ™lim—te rese—r™hers to dis™uss —˜ out the ™lim—te ™h—nge of

surf—™e systems of the e—rthF es for the p—r—meters of or˜it—l elements eY 6 Y s Y  —nd the ™h—r—™teristi™

H H H

—mplitudes e Ye D frequen™ies # Y# D —nd the initi—l ph—ses 0 Y0 D only wh—t weh—ve is the present

j j j

j j j

v—luesF „he sux j @j aIY  YnD—nd n in the list of v—sk—r @IWVVA is VHA h—s of ™ourse their physi™—l

me—ningD some of them —re strongly su˜ je™t to ™h—nges —nd some of them —re notF

t—l frequen™ies # over PHHwyr —nd numeri™—l „—˜le IF veft three ™olumnsX me—n v—lue of the fund—men

lower estim—tes  of the size of the ™h—oti™ zones @—fter v—sk—r @IWWHAAF # —re the v—lues of the

# H

fund—ment—l frequen™ies of the line—r p—rt of the se™ul—r systemF ‚ight three ™olumns —re the me—n

e

p

—mplitudes of e D their —˜solute v—ri—˜le r—nges Qe —nd the rel—tive v—ri—˜le r—nges of the prop er

p p

e

p

mo desF ell v—lues in the right three ™olumns —re rough estim—tion from pigure V —nd W of v—sk—r @IWWIA

@g $ g Ys $ s A —nd pigure R of xo˜ili et —lF F @IWVWA @g $ g Ys $ s AF

I R I R S V T V

# # #

H

Q e

p

e Qe

p p

@—r™se™GyA @—r™se™GyA @—r™se™GyA

e

p

g SFVTHRT SFSW HFIH IFP HFP HFIU

I

g UFRTHRI UFRSS HFHIQ IFH HFP HFPH

P

g IUFRTSHW IUFQH HFIU IFH IFH IFH

Q

g IVFIIQVI IUFVS HFPH HFU HFU IFH

R

g RFIPVTT RFPRVVP HFHHHHP HFHHSHTS HFHHHHPS HFHHSH

S

g PQFRUPVH PVFPHQ HFHHIH HFHHPQIT HFHHHHHS HFHHPP

T

PFWVWVH QFHVWSP HFHHHHU HFHHHSWS HFHHHHQ HFHSH g

U

g HFTSPUH HFTTTWV HFHHHHQ HFHHHHVS HFHHHHHQ HFHQS

V

s ESFPHHVU ESFSW HFHV HFU HFU IFH

I

s ETFSUHPU EUFHH HFPQ IFH IFH IFH

P

s EIVFURRSQ EIVFVV HFHT IFH HFP HFPH

Q

s EIUFTQRTI EIUFVH HFIP IFH HFP HFPH

R

HFH HFH HFH HFH HFH HFH s

S

s EPSFTUQRS EPTFQQHPH HFHHHHV HFHHHHUIS HFHHHHIS HFPI

T

EPFWPVVS EQFHHSTQ HFHHHHT HFHHHHVQ HFHHHHHR HFHRV s

U

EHFTVPUU EHFTWIWS HFHHHHI HFHHHHQRP HFHHHHHIS HFHRR s

V

es mentioned in ferger 8 voutre @IWVUA m— jor sp e™tr—l p e—ks of the mo dern wil—nkovit™h ™y™les —re

determined ˜y the fund—ment—l frequen™ies g —nd s D the pre™ession—l ™onst—nt D —nd time —ver—ged

i i

 @ferger —nd voutreD IWVUAF  —nd  h—ve ˜ een ™h—nged se™ul—rly with the dyn—mi™—l evolution o˜liquity

of the e—rthEmo on system @ h—s ˜ een de™re—sed —nd  h—s ˜ een in™re—sedAF ren™e the ™h—oti™ ˜ eh—vior

of the sol—r system will —pp e—r in the wil—nkovit™h ™y™les through the v—ri—tion of the —mplitudes —nd

the frequen™ies of the g —nd s F st is f—irly di™ult to pre™isely know the ™h—oti™ v—ri—˜le r—nge of the

i i

fund—ment—l frequen™iesD ˜ut in v—sk—r @IWWHA he showed the lower estim—tion of the p ossi˜le ™h—oti™ T

zonesofthemF ƒo we utilize them —nd ™—l™ul—te the v—ri—˜le r—nge of the typi™—l wil—nkovit™h p e—ks

wpID wpPD woI —nd woP @sto et —lFD IWWQA during these Rq—F

H

F e™™ur—te —nd ™omplete ™—l™ul—tion of the —mplitudes e „he trou˜le o ™™urs in the —mplitudes e Ye

j j

j

H

le—ds to quite ™ompli™—ted eigenv—lue pro˜lem @even in the ™—se of — line—r se™ul—r p ertur˜—tion —nd e

j

theoryA so we utilized the ™—l™ul—tion—l results of the —mplitude of the prop er mo des in xo˜ili et —lF

@IWVWA —nd v—sk—r @IWWIAF u—lit—tive prop erties of the motion of the p eriheli— —nd the no des of the

outer pl—nets —re des™ri˜ ed well ˜y — line—r se™ul—r p ertur˜—tion theory @xo˜ili et —lFD IWVWA ˜ e™—use of

their l—rge m—ssesF yn the other h—nd the —n—logous plots in v—sk—r @IWWIA seems to indi™—te th—t the

dyn—mi™s of the inner pl—nets is mu™h less regul—r th—n th—t of the outer pl—netsF †—ri—˜le r—nges of the

fund—ment—l frequen™ies —nd the —mplitudes of prop er mo des —re listed in „—˜le IF

„—˜le PF emplitudesD p erio ds —nd frequen™ies of the m— jor terms of the ™lim—ti™ pre™ession —nd o˜liquity

on the presente—rthF ‡e to ok m— jor four terms for ™lim—ti™ pre™ession —nd three terms for o˜liquityF

we—n —mplitude e is —fter t—˜les of ferger 8 voutre @IWVUAD —nd e Y e is ™—l™ul—ted ˜y

me—n m—x

min

   

e e e

p p p

using in „—˜le I —s e a e I —nd e a e F f is — IC

me—n m—x me—n me—n

min

e e e

p p p

simple sum of —rgument @exFD in the ™—se of ™lim—ti™ pre™ession xoFI termD f @SRXTVVTIA a g C k a

me—n S

RXPRVVP C SHXRQWV a SRXTVVTIAF f Yf is ™—l™ul—ted ˜y using  in „—˜le I —s f a f @I  A

m—x # me—n #

min min

P%

F —nd f a f @I C  AF €eriods —re equ—l to

m—x me—n #

f

me—n

glim—ti™ €re™ession @presentA

f f f period

me—n m—x

min

—rgument e e e

me—n m—x

min

@ye—rA @—r™se™GyA @—r™se™GyA @—r™se™GyA

I PQTWU g C k HFHIVTHV HFHIVSIS HFHIVUHI SRFTVVTI SRFTVVSW SRFTVVTQ

S

P PPQVS g C k HFHITPUS HFHIQHPH HFHIWSQH SUFVWRUW SUFVVIUW SUFWHUUW

P

Q IVWUU g C k HFHIQHHU HFH HFHPTHIQ TVFPVWUW TVFHVWUW TVFRVWUW

R

R IWIQP g C k HFHHWVVV HFH HFHIWUUU TUFUQWUW TUFSTWUW TUFWHWUW

Q

y˜liquity@presentA

period e e e f f f

me—n m—x me—n m—x

min min

—rgument

@ye—rA @—r™se™A @—r™se™A @—r™se™A @—r™se™GyA @—r™se™GyA @—r™se™GyA

I RIHTR s C k PRTPFPP IWTWFUV PWSRFTT QIFSSWUW QIFRWWUW QIFTIWUW

Q

P QWUHT s C k VSUFQP TVSFVT IHPVFUV QPFTQWUW QPFSIWUW QPFUSWUW

R

SQUSR s C k TPWFQP RWUFIT UTIFRV PRFIHWSW PRFIHWSI PRFIHWTU Q

T

†—ri—˜le r—nges of the wil—nkovit™h frequen™ies

„here —re two f—™tors whi™h —e™t the —mplitudes —nd frequen™ies of the wil—nkovit™h ™y™lesF yne is

™—lled the ™lim—ti™ pre™ession e sin3 ~ D where3 ~ is the longitude of p erihelion with resp e™t to the moving

vern—l equinox @fergerD IWUTAF fe™—use e™™entri™ity e is sm—ll num˜er the m—ximum —mplitude of the

™lim—ti™ pre™ession is —lso sm—llF enother f—™tor is o˜liquity  F sn the ™—se of presente—rthDv—ri—˜le



r—nge of o˜liquity is $ I so the order of —mplitudes is y @IHH@—r™se™Gye—rAA $ y @IHHH@—r™se™Gye—rAAF

y˜liquity is in™luded in the sol—r insol—tion ™—l™ul—tion —s the term sin  sin j9j where 9 is l—titude on

the e—rth @wil—nkovit™hD IWRIAF sn „—˜le P weshowed the p ossi˜le v—ri—˜le r—nges of the m— jor terms U

„—˜le QF emplitudesD p erio ds —nd frequen™ies of the m— jor term of the ™lim—ti™ pre™ession —nd o˜liquity

on the e—rth of Qq—F gontents of e—™h ™olumns —re the s—me —s pigure PF xoti™e th—t the —mplitudes

of o˜liquity —re lower th—n the present v—lues due to the l—rge pre™ession—l torqueF

glim—ti™ €re™ession @Qq—A

period f f f

me—n m—x

min

—rgument e e e

me—n m—x

min

@ye—rA @—r™se™GyA @—r™se™GyA @—r™se™GyA

I TPRV g C k HFHIVTHV HFHIVSIS HFHIVUHI PHUFRPSRR PHUFRPSRP PHUFRPSRT

S

P TISP g C k HFHITPUS HFHIQHPH HFHIWSQH PIHFTQITP PIHFTIVTP PIHFTRRTP

P

Q SVTQ g C k HFHIQHHU HFH HFHPTHIQ PPIFHPTTP PPHFVPTTP PPIFPPTTP

R

R SVUV g C k HFHHWVVV HFH HFHIWUUU PPHFRUTTP PPHFQHTTP PPHFTRTTP

Q

y˜liquity@Qq—A

period e e e f f f

me—n m—x me—n m—x

min min

—rgument

@ye—rA @—r™se™A @—r™se™A @—r™se™A @—r™se™GyA @—r™se™GyA @—r™se™GyA

I UHQP s C k TIIFPT RVWFHI UQQFSI IVRFPWTTP IVRFPQTTP IVRFQSTTP

Q

P TWWI s C k PIPFVQ IUHFPU PSSFRH IVSFQUTTP IVSFPSTTP IVSFRWTTP

R

Q UQPV s C k ISTFPQ IPQFRP IVWFHR IUTFVRTRP IUTFVRTQR IUTFVRTSH

T V

of ™lim—ti™ pre™ession —nd o˜liquityF ‡hen we des™ri˜ e the —mplitudes of the wil—nkovit™h ™y™les —s e

—nd the ™ontri˜ution of luniEsol—r pre™ession —nd the pl—net—ry p ertur˜—tion —s e —nd e resp e™tivelyD

 p

e

e p

—pp—rently e a e e D so the r—tio is just equ—l to the r—tio Frerek in the ™olumn of —rgument

 p

e e

p



 where  is the time —ver—ge v—lue of o˜liquityF et present  aPQXR so k aSHXRQWV equ—ls to  ™os

@—r™se™Gye—rAF sn „—˜le P e h—s the s—me v—lue —s in ferger @IWVUA —nd the v—ri—˜le r—nges of

me—n

the —mplitudes —re shown ˜y e —nd e determined from the —mplitude v—ri—tion of the prop er

m—x

min

mo des in „—˜le IF ƒimil—rly f is the simple sum of the —rgument frequen™ies @egF g C k A—ndthe

me—n S

r—nge estim—ted ˜y using v—sk—r @IWWHA9s  @„—˜le IA is shown ˜y f —nd f F€erio ds —re equ—l

# m—x

min

to P%af F

me—n

„—˜le P shows the ™—se of present —ge when  aSRXWTHH @—r™se™Gye—rAF ‡e ™—n estim—te the —mplitudes

—nd frequen™ies v—ri—tion in —n™ient times su™h like er™he—n or €roterozoi™ in — simil—r w—yF „—˜le Q

shows the r—nges of m— jor terms of ™lim—ti™ pre™ession —nd o˜liquity —t th—t timeF e™™ording to the

™—l™ul—tion—l results of the dyn—mi™—l evolution of the e—rthEmo on systemD  aPIRXVVQV @—r™se™Gye—rA

—t Qq— @e˜ e et —lFD IWWPAF „here is one more p oint to noti™e hereF emplitude of o˜liquity os™ill—tion is

—pproxim—tely prop ortion—l to the re™ipro ™—l of k @‡—rdD IWURAF ‡hen k is l—rge @ieFD gr— vit—tion—l torque

sA tends to ˜ e xed on the inst—nt—neous on the equ—tori—l ˜ulge of the pl—net is l—rgeA me—n spin —xis @

or˜it—l norm—l nD —nd o˜liquity —ppro—™hes ™onst—nt in spite of the movementof nF „his is the re—son

why e Y e Y e of o˜liquityin„—˜le Q —re mu™h sm—ller th—n those in „—˜le PF ever—ge

me—n m—x

min

o˜liquity h—s —lso ˜ een s—id to in™re—se due to the lun—r —nd sol—r tideF e™™ording to e˜ e et —lF @IWWPAD

 

 9 IW —t Qq— so k aPIRXVVQV  ™os IW aPHQXIUTTP @—r™se™Gye—rAF

sn pigure I —nd P we illustr—ted the v—ri—˜le r—nges of —mplitude —nd frequen™y in the m— jor sp e™tr—l

pe—ks of „—˜le I —nd PF rorizont—l —xis denotes the frequen™y @f A —nd verti™—l —xis denotes the —˜solute

v—lue of the —mplitude @jejAF ‡h—t we noti™e here is the distinguishment of e—™h sp e™tr—l p e—ksD th—t

isD r—nges of the frequen™y v—ri—tion do not overlie e—™h otherF ‚—nges of the —mplitude v—ri—tion —re

l—rger the the frequen™y v—ri—tionD ˜ut r—ther mo der—te ex™ept wpI whi™h v—ries from H to PHH7 of

the origin—l v—lueF rowever in gener—lD we ™—n ™on™lude th—t the ee™t of ™h—oti™ ˜ eh—vior of the sol—r

system to the —mplitudes —nd frequen™ies of the wil—nkovit™h ™y™les m—y not ˜ e so imp ort—nt th—n h—d

exp e™tedD —nd we ™—n distinguish the three m— jor sp e™tr—l p e—ks wpID wpP —nd woI whi™h we re™ognize

in the s—mples from the o ™e—n ˜ ottom sediments or p ol—r i™e ™ores of the u—tern—ry @woP is origin—lly

sm—ll —nd not sure to ˜ e found in sediment—tion re™ordsAF end moreoverD the dieren™e ˜ etween the

wil—nkovit™h frequen™ies of the present —nd the —n™ient times @su™h —s er™he—n or €roterozoi™D in pigure

I —nd PA is —pp—renteven weif™onsiderthe™h—oti™ ee™tF

‚esults —nd his™ussions

felow we show the ™—l™ul—tion—l results of the evolution of e—rth9s rot—tion—l —ngul—r velo ™ity 3 —nd

the e—rthEmo on dist—n™e — ˜ye˜e et —lF @IWWPA —nd „ur™otte et —lF @IWUUA in pigure SF sn pigure S

m

@—A 3 is repl—™ed ˜y vyh@vength of h—yAF „he dieren™e ˜ etw een two results in pigure S @—A@˜A@™A@dA@eA

is ™—used ˜y the dieren™e of the metho d of mo deling of the tid—l me™h—nismF „ ur™otte9s metho d is —

simple —nd —n—lyti™—l oneD —nd e˜ e9s metho d is ™ompletely numeri™—l in™luding the nonE—xisEsymmetri™—l

distri˜ution of ™ontinents —nd o ™e—nsD —nd ee™t of the sol—r tideF

„hen we ™—n ™—l™ul—te the motion of the rot—tion—l —xis of the e—rth ˜y solving the equ—tion of motion

@IA —nd o˜t—in —n™ient wil—nkovit™h ™y™lesF pigure Q lower left ones —re one of the resultsD showing W



the p ower sp e™trum of d—ily —ver—ge insol—tion time series of summer solsti™e —t TS x—t QHq—F „he

s—mpling dur—tion is four million ye—rs —nd integr—tion time step is one thous—nd ye—rsF yf ™ourse we

™—n dire™tly integr—te the equ—tion of motion @IA —nd digitize the p e—k frequen™ies m—nu—llyD ˜ut typi™—l

frequen™ies of the wil—nkovit™h ™y™les —re known to ™onsists of line—r ™om˜in—tions of the fund—ment—l

frequen™ies of pl—net—ry p ertur˜—tion g —nd s —nd pre™ession—l ™onst—nt  @ferger —nd voutreD IWVUAF

i i

sn this p—p er we ™on™entr—ted on the four p e—k frequen™ies @wpID wpPD woID woPA des™ri˜ ed in „—˜le

R —nd tr—™ed their evolution ˜—™kto Rq—F „he results —re summ—rized in pigure S @eAF „he m— jor

periodsofwil—nkovit™h ™y™les wpID wpPD woI —nd woP shift shortened —s going up timeF ‡e —ssume

the ™onst—n™y of the fund—ment—l frequen™ies —s mentioned ˜ eforeF ƒin™e we —re ™on™entr—ting on the

dis™ussion of the frequen™y dom—in @frequen™ies of the wil—nkovit™h ™y™lesA we should t—ke ™—re only

—˜ out the frequen™ies of the resultsD not —˜ out the ph—ses @on the ™ondition th—t the initi—l v—lues —re

’mo der—te4 or physi™—lly ’re—son—˜le4 onesAF

I

p e—k —rgument resume 3

period

I IUXVSCSHXRR

3 wpI g C 

R

QTHHQTH IVWUU

RXPRWCSHXRR I

wpP g C  3

S

QTHHQTH PQTWU

I IVXVVCSHXRR

3 woI s C 

Q

QTHHQTH RIHTR

I PTXQQCSHXRR

3 woP s C 

T

QTHHQTH SQUSQ

„—˜le RF yrigin of the wil—nkovit™h frequen™ies wpID wpPD woI —nd woPF  is pre™ession—l ™onst—ntF

‡hole v—lues —˜ ove —re of the presentD —nd the unit of p erio d is ye—rF „he f—™tor QTHH  QTH denotes the

™onversion of unit from r—di—n to —r™se™F

„here is —nother p oint here to noti™eF sn the di—gr—ms of pigure Q leftsiedD p ower of the o˜liquity

term @woI —nd woPA of older er— is —pp—rently mu™hlower th—n th—t of the present —geF „his me—ns

th—t the —mplitude of the time series d—t— of o˜liquity os™ill—tion w—s depressedF y˜liquity os™ill—tion

is ™—used ˜y the motion of inst—nt—neous or˜it—l norm—l n whi™h gives the gr—vit—tion—l torque on the

equ—tori—l ˜ulge of the pl—netF ‡hen the pre™ession—l ™onst—nt  equ—ls to zeroD the pl—net never get

—ny gr—vit—tion—l torque —nd spin —xis s rem—ins ™onst—nt in the inerti—l sp—™eF yn the other h—nd

when  3I @ieFD m—gnitude of the gr—vit—tion—l torque re—™hes to innityAD time —ver—ge of spin —xis

@pre™ession—l ™enterA is xed on the or˜it—l pl—ne —nd o˜liquity —ppro—™hes ™onst—ntv—lue in spite of

the movement of the or˜it—l pl—neF w—gnitude of gr—vit—tion—l torque is mu™h l—rger —t Qq— th—n the

present so pigure Q lower left ones ™orresp ond to the l—tter ™—seF „his is the re—son whythepower of

woI —nd woP ˜ e™omes low in pigure Q lower left onesF ƒtri™t m—them—ti™—l formul—tions —re given in

equ—tion @IHA to @QPA of ‡—rd @IWURAF

„he di—gr—m with the horizont—l —xis of the —˜solute —ges @pigure SA is strongly dep endentonthe

mo del of tid—l dissip—tion me™h—nism ˜ etween the e—rth —nd the mo onF ‡e ™—n —void this pro˜lem ˜y

repl—™ing the —˜solute —ge with vength of h—y @vyhA —s the horizont—l —xisD ˜ e™—use vyh is essenti—lly

— fun™tion only of the density stru™ture of the e—rthD —nd the dyn—mi™—l ellipti™ity or the rot—tion—l sp eed

of the e—rthF „herefore vyhEp erio ds plot dep ends only on the rot—tion—l —ngul—r velo ™ity 3 F pigure T

e ™—n o˜t—in — simil—r result if we t—ke the e—rthEmo on is the plot using vyh˜y the horizon t—l —xisF ‡

dist—n™e —s the horizont—l —xis inste—d of vyhF

sn pigure U we showed the f—nded sron porm—tion of QXQq— whi™h w—s found in eustr—li—F „he

˜l—™k @—™tu—lly redA —nd white strip ed ˜—nds on it —re p erio di™ —nd mo dul—tedD indi™—ting the presen™e IH

of environment—l v—ri—tion with plur—l p erio dsF ren™e they —re ˜ elieved to form either or ˜ oth ˜ythe

lun—r tides —nd the wil—nkovit™h ™y™lesF sf they were ™—used ˜y wil—nkovit™h ™y™lesD the rel—tive —ge

˜etween strip es ™—n ˜ e known ˜y theoreti™—lly estim—ting the p erio ds of wil—nkovit™h ™y™les —t th—t

timeD using the results we o˜t—ined —˜ oveF sn pigure V there showed — photogr—ph of ƒtrom—toliteF

„hus we ™—n est—˜lish the st—nd—rd me—surement of rel—tive —geD ieFD the l—p time ™lo ™k for de™o ding

the history of the e—rth ™omp—ring the strip es on fsp with theoreti™—l wil—nkovit™h ™y™le frequen™iesF

ƒin™e the wil—nkovit™h ™y™les h—ve˜eenpl—ying —n signi™—nt role in the ™lim—te ™h—nge on the e—rthD

wh—t we did in this dis™ussion shows —nother p ossi˜ility of the ™oEevolution of the wil—nkovit™h ™y™les

—nd the e—rthEmo on systemF

wil—nkovit™h ™y™les h—ve ˜ een evolving following to the dyn—mi™—l evolution of the e—rthEmo on systemF

‡e put three —ssumptions to ™—l™ul—te —n™ient wil—nkovit™h ™y™lesD —nd investig—ted the sensi˜ility of

the ™y™les to the p ossi˜le ™h—oti™ motion of the sol—r systemF yur results using these —ssumptions show

the p ossi˜ility of shorter p erio ds of wil—nkovit™h ™y™les —t —n™ient timesD —nd est—˜lish the prelimin—ry

referen™e mo del of ™oEevolution of wil—nkovit™h ™y™les —nd tid—l ™y™lesF sn further rese—r™hes we will

m—ke — det—iled studies on the devi—tion due to these —ssumptionsD —nd ˜uild the more pre™ise mo del of

the evolution of the wil—nkovit™h ™y™lesF

e™know ledgmentsF „he —uthors h—ve gre—tly ˜ enetted from sever—l stimul—ting —dvi™e —nd en™our—geE

ment from hrF fFpFgh—o of qo dd—rd ƒp—™e plight genterD xeƒeF het—iled —nd ™onstru™tive reviews ˜y

uo oiti w—sud— of „okyo wetrop olit—n niversityh—ve ™onsider—˜ly improved the present—tion of this

p—p erF „his p—p er w—s presented —t the wultisphere snter—™tionD ivolution —nd ‚hythm ƒymp osium on

he™em˜ er R —t i—rthqu—ke ‚ese—r™h snstituteD niversityof„okyoD IWWPF II

‚eferen™es

e˜ eD wFD wizut—niD rFD „—mur—D ‰FD —nd yo eD wF @IWWPA „id—l evolution of the lun—r or˜it —nd the

o˜liquity of the e—rthD €ro™F sƒeƒ vun—r €l—netF ƒympFD PSD PPT{PQIF

fergerD eFvF @IWUTA y˜liquity —nd pre™ession for the l—st SHHHHHH ye—rsD estronF estrophysFD SID IPU{IQSF

fergerD eFvF @IWUVA e simple —lgorithm to ™ompute longEterm v—ri—tions of d—ily or monthly insol—E

tionD gontri˜ution de l9snstitut d9estronomie et de qeophysiqueD niversite g—tholique de vouv—inD

vouv—inEl—ExeuveD xoFIVF

fergerD eFvF —nd voutreD wFpF @IWVUA yrigine des frequen™es des elements —stronomiques interven—ntd—ns

”

le ™—l™ul de l9insol—tionD in ƒ™iF ‚epFD IWVUGIQD snst estron qeophys qF vem— itreD nivF g—tholique

de vouv—inD vouv—inEl—ExeuveD RS{IHTF

fergerD eFvFD voutreD wFpFD —nd v—sk—rD tF @IWWPA ƒt—˜ility of the —stronomi™—l frequen™y over the e—rth9s

history for €—leo ™lim—ti™ studiesD ƒ™ien™eD PSSD STH{STTF

fillsD fFqF @IWWHA „he rigid ˜ o dy o˜liquity history of w—rsD tF qeophysF ‚esFD WSD IRIQU{IRISQF

henisD gF @IWVTA yn the ™h—nge of kineti™—l p—r—meters of the e—rth during geologi™—l timesD qeophysF

tF ‚F estronF ƒo™FD VUD SSW{STVF

honesD vF —nd „rem—ineD ƒF @IWWQA ‡hy do es the e—rth spin forw—rdcD ƒ™ien™eD PSWD QSH{QSRF

r—ysD tFD sm˜rieD tFD —nd ƒh—™kletonD xF @IWUTA †—ri—tions in the e—rth9s or˜itX p—™em—ker of the i™e —gesD

ƒ™ien™eD IWRD IIPI{IIQPF

—D wFD r—m—noD ‰FD w—sud—D uFD —nd w—tsuiD „F @IWWQA enother p ossi˜ilityX ™oE stoD „FD uum—z—w

evolution of the wil—nkovit™h ™y™les —nd the e—rthEmo on systemD qeophysF ‚esF vettFDto˜esu˜mitE

tedF

u—ul—D ‡FwF @IWTRA „id—l dissip—tion ˜y solid fri™tion —nd the resulting or˜it—l evolutionD ‚evF qeophysFD

PD TTI{TVSF

v—sk—rD tF @IWVVA ƒe™ul—r evolution of the sol—r system over IH million ye—rsD estronF estrophysFD IWVD

QRI{QTPF

v—sk—rD tF @IWWHA „he ™h—oti™ motion of the sol—r systemX e numeri™—l estim—te of the size of the ™h—oti™

zonesD s™—rusD VVD PTT{PWIF

wil—nkovit™hD wF @IWRIA u—non der ird˜estr—hlung und seine enwendung —uf d—s iiszeitpro˜lemD†olF

IQQ of uonigli™h ƒer˜is™he e™—demie €u˜li™—tionDuonigli™h ƒer˜is™he e™—demieF

xewsomD rFiF @IWWHA e™™retion —nd ™ore form—tion in the e—rthX ividen™e from siderophile elementsD in

yrigin of the i—rthD yxford niversity €ressD xew ‰orkD PUQ{PVVF

xo˜iliD eFwFD wil—niD eFD —nd g—rpinoD wF @IWVWA pund—ment—l frequen™ies —nd sm—ll divisors in the

or˜its of the outer pl—netsD estronF estrophysFD PIHD QIQ{QQTF

‚u˜in™—mD hF€F @IWWHA w—rsX gh—nge in —xi—l tilt due to ™lim—tecD ƒ™ien™eD PRVD UPH{UPIF IP

ƒt—™yDpFhF@IWWPA€hysi™s of the i—rth @„hird iditionAD fro okeld €ressD fris˜—neD eustr—li—F

ƒussm—nD qFtF —nd ‡isdomD tF @IWWPA gh—oti™ evolution of the sol—r systemD ƒ™ien™eD PSUD ST{TPF

„ur™otteD hF —nd ƒ™hu˜ertD qF @IWVPA qeodyn—mi™s | eppli™—tion of gontinuum €hysi™s to qeologi™—l

€ro˜lemsD tohn ‡iley 8 ƒonsD ƒ—nt— f—r˜—r—F

„ur™otteD hFD gisneD tFD —nd xordm—nnD tF @IWUUA yn the evolution of the lun—r or˜itD s™—rusD QHD PSR{PTTF

‡—lkerD tFgFqF —nd —hnleD uFtF @IWVTA vun—r no d—l tide —nd dist—n™e to the wo on during the

€re™—m˜ri—nD x—tureD QPHD THH{THPF

‡—rdD ‡F‚F @IWURA glim—ti™ v—ri—tions on w—rsD ID estronomi™—l theory of insol—tionD tF qeophysF ‚esFD

UWD QQUS{QQVTF

‡isdomD tF —nd rolm—nD wF @IWWIA ƒymple™ti™ m—ps for the x E˜ o dy pro˜lemD estronF tFD IHPD ISPV{

ISQVF

h—rkovD †FxF —nd „ru˜itsynD †F€F @IWUVA €hysi™s of €l—net—ry snteriorsD€—™h—rtD „u™sonF IQ

pigure g—ptions

pigure IF ƒp e™trum di—gr—m of the v—ri—˜le r—nges of ™lim—ti™ pre™ession —nd o˜liquity —t present

@h—t™hed —re—AF rorizont—l —xis denotes the frequen™y @—r™se™Gye—rA —nd verti™—l —xis denotes the —˜solute

v—lue of —mplitudeF emplitude of ™lim—ti™ pre™ession is dimensionless —nd th—t of o˜liquity is —r™se™F

xoti™e th—t the v—ri—˜le r—nge of frequen™ies —re sm—ll enough to distinguish e—™h p e—ksF †—ri—˜le r—nges

of —mplitude is r—ther l—rge esp e™i—lly wpI @PHH7 of the me—n v—lueAF

pigure PF ƒp e™trum di—gr—m of ™lim—ti™ pre™ession —nd o˜liquityof Qq— —goF rorizont—l —nd verti™—l

s™—le is the s—me —s pigure I ˜ut —ll p e—ks —re shifted tow—rd the high frequen™y regionD ˜ e™—use the

gr—vit—tion—l torque on the equ—tori—l ˜ulge of the e—rth w—s mu™h l—rger —t this —geF xoti™e th—t the

—mplitudes of o˜liquity —re lower th—n the presentv—lues due to the l—rge pre™ession—l torqueF

pigure QF „ime series d—t— —nd p ower sp e™trum di—gr—m of the d—ily —ver—ge sol—r insol—tion v—ri—tion



@TS xD summer solsti™eAF @‚ight sideA time series d—t— —t presentD Iq—D Pq—D Qq—F nit of verti™—l

P

—xis is ‡am F ‰ou ™—n see the s—me envelop w—ve ™—used ˜y the os™ill—tion of e™™entrisity eF @veft

sideA p ower sp e™trum di—gr—m of the right d—t—F et the present one we noti™e on the p e—k of —˜ out

IWky @wpIAD PQky @wpPAD —nd RIky @woIAF wpP is — dou˜let of two sh—rp sp e™tr—l p e—ks owing to the

mo dul—tion of sever—l eigen frequen™ies of or˜it—l p—r—metersF fut in pr—™ti™e these p e—ks —re o˜served —s

IV

— single —nd ˜ro—d p e—k in the p ower sp e™trum of the time series of geologi™—l d—t— su™h —s  y —nom—lyD

˜ e™—use this dou˜let will ˜ e tot—lly smo othed out ˜y v—rious pro ™esses of surf—™e ™lim—te system on the

e—rth @r—ys et —lFD IWUTAF ell these p e—ks —re shifted tow—rd the short high frequen™y region —s going

˜—™k the —geD ˜ e™—use the pre™ession—l ™onst—nt  were mu™h l—rger —t —n™ient timesF woreover the

—mplitude of the o˜liquity term @woI —nd woPA —re suppressed in the di—gr—m of older er—F st is —lso

˜ e™—use of the strong gr—vit—tion—l torque from the ne—r mo onF

pigure RF yr˜it—l in™lin—tion @with resp e™t to the e—rth9s e™lipti™A —nd e™™entri™ity of the mo onD me—n

o˜liquity of the e—rth —nd the tot—l —ngul—r momentum of the e—rthEmo on system —fter e˜ e et —lF @IWWPAF

@—A sn™lin—tion of the mo on i @degreeA @˜A e™™entri™ityofthemoon e @nonEdimension—lAD @™A we—n

m m

QR P P

o˜liquity @degreeAD @dA the —ngul—r momentum of the e—rthEmo on system @IH ug  m as AF sn the mo del

of „ur™otte et —lF @IWUUA e a i a HD —nd the me—n o˜liquity is not t—ken into —™™ount @they only

m m

™onsidered the se™ul—r ™h—nge of the rot—tion—l —ngul—r velo ™ity of the e—rthD no ™onsider—tion —˜ out the

pre™ession or o˜liquityAD —nd the tot—l —ngul—r momentum is kept ™onst—ntF IR

pigure SF „r—nsition di—gr—m of the wil—nkovit™h ™y™les in™luding the dyn—mi™—l evolution of the e—rthE

mo on systemF @—A vength of h—y @vyhD presentv—lue equ—ls to the unityAD @˜A hist—n™e ˜ etween the

e—rth —nd the mo on @r—dius of the e—rth ‚ AD @™A hyn—mi™—l ellipti™ity of the e—rth @nonEdimension—lAD

i

@dA €re™ession—l ™onst—nt  @—r™se™Gye—rAD @eA ivolution p—ths of the m— jor p erio ds of the wil—nkovit™h

™y™les wpI @solid lineAD wpP @d—shed lineAD woI @dotEd—sh lineAD the unit is ye—rF „he solid lines in the

™olumn of @—A@˜A@™A@dA —re the result using the tid—l mo del of e˜ e et —lF @IWWPA —nd the dotted lines

—re using „ur™otte et —lF @IWUUAF sn @eA the four thi™k lines —re from e˜ e et —lF @IWWPA —nd the four

thin lines —re from „ur™otte et —lF @IWUUAF

pigure TF ƒ—me —s the pigure SD ˜ut the horizont—l —xis is vength of h—y @vyhA inste—d of —˜solute

—geF sn this ™—se there is no dieren™e ˜ etween e—™hmodelsF

pigure UF „he ˜—nded iron form—tion from gre—v—villeD eustr—li—F „he ˜l—™k @—™tu—lly redA ˜—nds

™onsist of oxidized iron sedimentD the white ˜—nds ™onsist of ™hertF e˜solute —ge of the whole ro ™k is

QXQq—F

pigure VF „he strom—tolite from €il˜—r—D eustr—li—F e˜solute —ge of the whole ro ™kis QXQq—F „hese

will ˜ e some signi™—ntkeys to de™o de the history of the surf—™e environment of the e—rthF IS 2 Period (Kyr) Insolation (W/m ) 100 30 20 10 7 5 Mp2 6 Present 540 Mp1 Mo1 4 500

2 Mo2 460

6 -1Ga 540

4 500

2 460

6 -2Ga 540

4 500

2 460

6 -3Ga 540

4 500

2 460

02468101201235 4 -4 Frequency (10 Hz) Age (100Kyr) 0.03 Climatic precession

0.02

0.01

0 56 60 64 68 Frequency (arcsec/year)

3000 Obliquity

2000

1000

0 24 26 28 30 32 Frequency (arcsec/year) 0.03 Climatic precession

0.02

0.01

0 208 212 216 220 Frequency (arcsec/year)

800 Obliquity

600

400

200

0 177 179 181 183 185 Frequency (arcsec/year) (a) Inclination of the Moon’s orbit 6

5

0.08 (b) Eccentricity of the Moon’s orbit 0.06 0.04 0.02 3.40 3.38 3.36 (c) Angular Momentum 3.34 24 (d) Mean Obliquity 22 20 18 (b) Earth-Moon Distance 1.0 (e) Length of Day 60 0.9 0.8 0.7 50 0.6 0.5 60 (f) Earth-Moon Distance 40

50 0.015 40 0.015 0.010 0.010

0.005 (g) Dynamical Ellipticity 0.005 300 (c) Dynamical Ellipticity 200 300 100 (h) Precessional Constant 200 0 40000 (i) Evolution of Milankovitch Cycles Mo1

30000 Mp2 100 20000 Mp1 10000 (d) Precessional Constant 0 0 -4 -3 -2 -1 0 Mo2 Time [Ga] 50000 (e) Evolution of Milankovitch Cycles 40000 Mo1 30000 Mp2 20000 Mp1 10000 0 0.5 0.6 0.7 0.8 0.9 1.0 Length of Day (Present = 1) (a) Inclination of the Moon’s orbit 6

5

0.08 (b) Eccentricity of the Moon’s orbit 0.06 0.04 0.02 24 (c) Mean Obliquity 22 20 18 3.40 3.38 3.36 (d) Angular Momentum 3.34 -4 -3 -2 -1 0 Time [Ga]