Ph.D. Thesis (California Unrv.) 150 P HC $G050 CSCL 03D Unclas Cj/.3G 15536

(UASD-CE-135674) UN THE ESSABLISHHENT W73-32728 AYD EVOLUTION OP OHBIT-OBBIT RESONANCES Ph.D. Thesis (California Unrv.) 150 p HC $g050 CSCL 03D Unclas Cj/.3G 15536

on th ert&lim)~nnnt and i ?mIutlon OK Orbft-Orbit Renonanoaw

A dlrrort.tlon rimitto4 11, puthl metiml.ction of tho r.qulrolontm for the dsgrao of

Doctor e? Philomophy

Phyalco

hy Cl1.rlmu rlrmey Ydmr

Cowmittme In chrgmr

Proferror 9. J. Pule, Chairman

~rofanmorD. s. connoll

Prcfesmoz J. 8. Hart10

Augumt 1973 Augurt 1'. 3

Pvayonr LI Wac@ tht 8 Uak Neh am thin La MVW dma ontlroly alone but ir thr result or dlrCU#.lOnB ulth protrrmrr,

other grduato rtudrntr, Crionda an4 fully rho ark ubt you ara

datng and thcn attapt to prappla ulth your oprqur oxplunrticn. %#t PVBLICATXON OPTION lcrportdnt !a tho arPtlonr1 wpprt oC wlln and fully uhrn thi~890 I horoby romuvo all riqhta of publicatton, 1ncluULng wmnq and 1Graa dry up. tha right to roproduco thia tharir in any lorn, for I would arpocially 1Ike to thnk my adv1-r Dr. SUntOn ?eale a -rid of threo year. frcrm tho date ol nubhaion. for piv~nqu tu1pful advice attor I pot my rtuir toyothr and wain writing ard rnurltlnq and prduclnp rmothlnq of rem1 lntrrrrt.

Wr quit. civillrd approwh of luvlnq D. ta my om drvlcaa dn4

arrurlnp M a rtoady Inca. in orpacldlly approclar*1, Yathlo, Vtp

bora and took oar. ol Sarah, and typd two draft. of t?.lr thoair

and Mrrrir Walkor who poll~hrd It oll r1~)darorvo rocopr,itlcn.

Thla wrk war aupporcmd by tha Planotolqy Proqru, OtlIce of

Sprcr Solrnoe, MU, unbar Want N.O.R..-OS-O10-061.

iii lv VITA

-- Bern - On thm PrtabllJlment and hrolutlon of Orbi tar bl t R.roluncor 196P -- B.A., Univuoity of California, Santa Barba+a bu 1970-1971 -- Teaching Aafiistant, Department of PhyofC8, Wnivorofty of California, anta 8.rb.ra Chrlrr tlnnoy Yodar 1971 - M.A., Cnivarsity of California, Sant. Barbra In the rolu ryntm, thoro wilt rsvarrl oxamplor of 1971-1973 -- Research Amriotant, Univrroity of Caiffarnf. Wt. Barbara gravitational remonance botuoen tw or more oatolllteo or p-anets

in which a op.cl.1 anqle variable io oboorvd Lo librate. Celdroich

ha ouqgortod that in the care of planetary ratellltrs 8 tidally

inducad torque actlnq on the oatollltea myhav. p1.y.d an

oroential rolo in thr eot~bllotunantand aubrequent evolution of the

obaorvad ro8onanco8. Thio propooal Io thoroughly lnvestlqatd as

it applieo to the thrso re~nanceowren9 palro or aatelllter of

Saturn and is ohan to bo a plauolble mechanlnn for tholr establish-

mnt but Am loor ruccorrful, in the Tltal-Hyperlon case, In provid-

in9 I rorwnable tinu ma10 for tho dunping of rho mplltudo of libration . Tho mlution of tho problan io reach4 in threo rt.9eo. flrmt,

a thooretlcal derriptbn ol transltion ir dovelopd for a rbple tho dependent pndulum plur conrtant rppi1.d torque. The

evolution of tho rystan through tho vuiour ?hares U.r. porltlve rotation, ~atioeroMtien and libration) io described in term of

tha motion of tlia extranor or 'rooto. of tho AoOenhno va:'irbh In

V VI

c tlu complax plane. A tranaition phsb ia dofind and equation. of Hipkln that tho moon may hove baa trappd ln M orbitcorblt nution of theae mots are dujsod frcm which a lowest order eathato rommce with anothar planet in tha pst la -in& and found to of the Fobability for transition fra a rotation into libration is ba untenable. obtain4. '

Second, th ttm bodv grrvitatlonal interaction in expar.d.d md reduced to a one diaenaional tim indepndent Hamiltonian which aceurataly describes the nutior. .:I the resonance variable in th. absence of tides - if the satellites' inclination. and eccentricitieo are relatively amall and if the porturhtiona in the s~alwjor

UbS during each pkw or its evolution are also mall. me effect of the tide. io then intrhced by rodefining the orbital elanenta fa scch a way as to recover the H.miltonian tormulation, the wrtant difference bmng that it is now tine dopendent.

Tha theoretical azroach outlined Kor simple p.ndul\ap syrtcrmr io the;. applied to ecccr.tricity dopendant zesonancei. The de;mdonco of the probability for transition into libration la obtainoil as a function of the mean eccentricity and the mwhanim governing transition in various ltaits in di.cu6s.d. Tho damping of the amplitude of libration am a function of the tidal charge in th. orbital parmeters (prhcipally embejar axial is found via tha actlon integral.

Finally, tbs thooKetLCal lood.1 dwdoped applid t0 tho Saturn reaonanceo and found to agree with th. recent wrk of

Ulan, Greenberg and Sinelair. In addition, a propoMl by R. Q.

vii Pa90

3.4 hhavior of rhr rupplitudo or LlhrAtiOn ...... 186 Chpatuh...... 1 CtuptwFour ...... 206 1.1 Introduction ...... 1 4.1 The Batellite-Satellite Rero~nceof Saturn ...... -W 1.2. Theory of Ranrition for 81mple PmBolum 4.2 Tho Lunar-Planetary ROWMIICO HypOthorLr ...... 1a8 systQos ...... 11 4.3 Qn t h. Throe-SatOllItO ROrOMncO Of jupitu ...... 27' Chaptuw ...... 4a 2.1 Novtonian RHory for plansury Sywtemr ...... 48 ~deroncor...... 379 2.2 Analytlcal Developssnt of Disturbing Function .... 55 Appndix A: The Canonical eguationr of HDtion by tho Jacobeannothod ...... 2B2 2.3 classification Schemes ...... 65 Appondbt 81 RIs1uat:on of tho Action Intogral ...... 2d0 2.4 H.lraIton's Frmatioru ...... 69 AppondLr CI Solution 2.5 Eliaination of the Short Parlad 1mnu ...... 71 . .1 care ...... 331 2.6 Rduction of tho Tide-Free Haail??nian to e constantot tho htion ...... 87 2.7 Furthor Approximations and Stahility Analysie ...... 98 2.8 Effect of Tides ...... 107 2.9 Introducing tho Tidal Acceleration into the Tide-Prw miltonian ...... 113 2.10 wry...... 124 ChaptarThree ...... 129 3.1 Introduction ...... 129 3.2 Transition Thwq for simple -centricity- Dap.rdcnt ROSOMncOr f4 - ja + jlA1 + kdl Ik) - 2 ... 142 3.3 Transition Theory for simple ¢ricity * Rsw-or (6 (. jx + j'1' + ko) UhOTe ...... 150 ix x LIST CP TW LIST OP PIQVREII Tab1 e Pa9e

4.1.1 Table Of Z:UOMDCe frequwkciOS h.RO:ht.8 With 1.2.1 Pendulum atator ...... , , . . , ...... 14 the Zrl ccrmenwability ...... 211 1.2.2 hanrition phare diawmo 18 4.1.7 Data on six satellites of Saturn involved in ...... reaomnces...... 213 1.2.3a Diagram of i2Ve?NI 0 for p.:dulm like rymta, Subject to b conStbnt bpp1i.d Lrqur Which 18 4.1.3 Table lnclu6inq the rcl-vant frequencies of each of smetricintlmm...... 24 Saturn's satelliter Involv.8 In a tan body rewnmco . 214 2 1.2.N Diaqraa of verau. 4 for bmdulum plun torque 4.2.1 Table of relevant Hansens coefilcients xq A1.o 4 which le antisynetric in the , . , 26 included are the axplicit definition. of'& ...... inclination functions P (1) , . . . 250 e.m,p ...... 1.2.4 Poeltive rotation phane diagra ...... , 30 4.2.2 Orbital elment for Moon. mrth, Vemm an8 Feroury . , 260 1.2.5 Graph of the location of rolutionm of slrple -. pendulum plua torque in both polar cozrdlnate and 4.2.3 mmerical values of emof Laplace coefficient. complex momentun plane . . 35 ocsurring in th. expansion ...... 261 ...... 1.2.6 WPrms1tlon phase for pndulm plus torque where 4.2.4 M...ricrl values of the m&ximnn torquo on the Wwn b-conmtant , ...... , . 31 due to the given resonance ...... 262 2,l.l Orbital orlanation , , . , . , , . . , . , , . . . . . 51 4.2.5 Sculu elements of the planet...... 287 2.1.2 Elliptic variablrr . , . . . . , ...... , , 51 a 2,Z.l Sphoriccrl coordln4t.s of vecter r ...... 58 2.7.1 Phse epace curve ...... 103 3.2.1 Poaitiva rotation phm diagram8 for!kl- 2 ...... 144 3.3.1 Transition diagram in limlt 161 >> 1 foe k - +1 . , . , 159 L3.Z Traniition dia9r.r~in lhit IS^ >> 1 for k - -1 . . . . 160 3.3.3 Path traad "ut tr ~~I'~UAWIn poshtlve rotation phase for k m +l #e . , . . , , . . . . . , . , . . . 162 3.3.4 Ranrltjon directly into negativo robtion 20 k - -1 and 161 4 13.89. . . . . , , , ...... , . , . , . 165

3.9.9 Tmporary tranmitlon lnto 1nvurt.d Ilkation ph~s xi fot k - -1 and lEl * 13.89 . . . , ...... 166 Xll Pigur. p.9b

3.3.6 Graph or 161 P. x for th. sg.0~valw oi .- -112 .nB k - It...... 170 In tho rolar lystm thoro are sovet81 instances in vhlch tho 3.3.7 hinsition into libsation phase for 6 in ratio oC tho mun notions of a pair of ~tolli~esis vuy nearly a rar.ge (-0.2104 2 6 a -0.2722) and k - +1 ...... 171 Oinplo Craction. This kind of rOlAtIOnUhip is called a conmensura- 3.3.8 Tramition into negative rotation phase through a *ta-~!.orary" inverted libration phase for 6 in rang* bllity. Somr aanplea lncludo the three utslIite-retellite comen- (-0.2104 9 -0.2722) and k .+1 ...... 173 surabilitles of Saturn (Virum-Tethym, 2111 Enceradas-Dione, 2r.r 3.3 0 Graph of probabilitv for capture into libration Pc versus B/E Cor the k .+1, + 2 caaes...... 181 Titan-Hyperion, 413) and the 312 camoenaursbility of Neptune-Pluto. Cl

3.3.19 Graph of Pc vetsus 30/0cl for k '1, +3 C.l.0. . - * - 182 An oquivslent statrmont of a CoInJnenrurabi~ityis the relation 3 4.1 Miabatic aunpinq or om ir t.ie u.it 161 .. 1 .... 195 ja + jra' - 0, 3 :.1 Graph of the lnlttnl valuo of c at transition, d , here j and j' are intuJoes and *. and X' are t'c respectlvo aean versus the magnitude of 5 for lBl 5 ..... 203 .....'. longitude8 of tho p.ir of oatellires (or planets) . Tt.0 ohrlour 3.4.3 Pbt of the sine cf the amplltudo of libration versua the prmeter I1 - 2clt)/5) for oev08al extension of the above relatlon to U bodies is valurs of D above the c-itical value of Bcl ..... 204 N 3.4.4 Plot ot 4, verrua \c(t) for 161 2 205 . 1)"' ...... Z jnAn 0. (1 * 1.1) n-1 4.1.1 Plot of e and sin4 versua (1 - Zc(tJ/B) Cor tho Tlt ..Hypt:%n reaonan% ...... 23). Tho bomt known axamplo of a comenrurablllty of three bodiel 4.2.1 Plot of nmber of daya/syncdic nmth veraus esthtd aqe of each smple ...... 239 involve. the JI, JII, and JIII matellitem of Jupiter throuah tho

4.7.7 Vector diagkm ef the planet Wrth and lunar positions rolation with respect to the sun ...... 243

'JI .. 3A~~~+ ''J~ZI

Ohoervd -enmarabilities are not roatrictd to these orbit-

orblt typo.. Anothor type lnvolvss the ratio of the orbital period to the rotational perlod of either tho oar10 or dlCfwent bcdler. xiii 2 3

Respctive examples of this type are the spin-orbit interaction of ons-dimensional Hsmiltonlan might k derived as rn appraxinstion of

Mercury (Caldreich and Pealc, 1967) and the poanihie orbital this vuy :ow period behavior, and is the aubject of chapter two. carc~ensurabilitiesof artificial satellites with the earth's siderial If the erpanaion land Hamiltonian!) is valid for a range >f 4 which day (Rllan, 1967). inCludeEi both the rOhtiOM1 and lihatiOM1 phases of tbe IesoMnce These special relationships would not be nearly as intonating variable, thsr., of courne, 0 MY axecuts either rotations or 11- if thcy did mt ~vea physical basis for their existence. First of brrtions, depending on the parametera of the system. Althouq:~the ell, the ccwensurability relation is often not the phyaical libration Of the resOMnce variable can be aplained in terns of thm v8r.ablc which best descrfbms the obaervationa. Famnininq the mutual gravit~tionalinteractJon. btt*.een thc partners. it aems visual evidence amre closeiy, we find that in many canes there axista unlikely that a state of libratlon could have existed since the a sir?gie reaenance variable 0 which Lppears to librate about either earliest qtdgt36 of formation of the solar myatan. In 0th- mrds, mod '-1 OK nud[?W), ard, for the tvo-body Case, ha0 the fOrpl there rhould be nme mechanism or m*chanims by which the prtners

+ + {linear function of perihelion (0) I ,,~, 1 (1.1.2) evolved into their presently observed state. One lnter-qtinq fact and node (n) of asch body. that Roy and Ovenden (1954) have shown ia thAt the high frequency The mecbanim vhich wirtains this cmmensurability or KRBOM~O in of camenourabilities in tho solar rystem cannot be assignad to 4 ow known instarre involves a qraviutional interaction which i6 chance initial arrangement. fairly well understood (Hagihara, 1972. ~328-52). R study of the There appears to be two basic aolutionn to the tw qIeBtiona: astellitrsatellitc interaction of the tw-body resonance, after the 1) why ao many commensurabilitira? and 2) vhy w mny libratlng hro-hdy qravitatioml potential has be-n expanded in terms of the resonance variables? One p#Eibility is that a resonant or orbital elements biAc:i a*-cribs tho psition ot each body. reveals commensurable configuration is inherently more stable than a M.t + is that argument of a cosine funct:on in the expansion which sliqhtly off-resonant configuration. As illu6tratfon. con6idcr thm Woes very slowly varying for a nearly camennuate motion. hojan asteroids which in approximately the sme orbit aa mthuPore, this roaonant term in the expansion often acta as a Jupiter, cluntered in two Traps, 60° ahead end behind JUpitar p.odol?S?like potential. king the dominant factor coirtrolling the (ercun and Shook, 1964, pp. 250-ea). Their gravitational interaction orp long pariod motion of both satellites. This suggests that a with Jupiter tenda to maintain the one to one ronnenmrabillty. 4 5 mgine what umld happen to an asteroid with a nearly circular to the pull of each satellits. Already a skilar thwry, when orbit which is slightly largar or smaller than Jupiter’s. Within a applied to the spin-orblt resonatre of Mercury, has led to a ehort tise sprr: that astaroid would make a close approach to Jupiter. eatisfactory mtplsnation of evolution and capt\ue into libration

If close ehwgh, the gravitational force of Jupiter sould dominate Goldreich and Peale, 1967). In thi6 instance, captu~eis a?parently that of the sun, and radically change the orbit of this astaroid, caused by an saymatry in the tidal torque actin? on thn spin cf it the perhaps even r-inq frcm solar system by changing its orbit HmCUrY, 48 the VelOCity Of the LemOMIICe variable 4 vAni8ha6 and frcm elliptic to hyperbolic, er rmooving it through a collieion. In then chdng8e sign. An important difference in thc orbit-orbit type tkis -?la, the lifetime of any aateroid not in resonance with of resomnce is that the capture mechanism does not appear to depend Jupiter but havim nearly tha same orbit or a crossing orbit, vmuld in any importsnt way on the details of the tidal interaction itself, tend to be very short in comparison with the ape ef L?e solar systm. aa it does with the spin-orbit case. Recent moocrleal 6tdie8 of

Thus time, by Jupiter, endows the Trojans with a drvine relationship! Creenbarg (1972) an3 Sinclalr (1972) indicate that the tidal

A recent proposal by Ovenden (1972). based on rhis idea of maximum wchaniun &ea satisfactorily explain the Saturn resonances and that atability, is that the high frequency of amuenauabilities is a capture into libratlon is caused by the tidal torque acting through reflection of the evolution of the solar q8tm toward8 a ’Least the gravitational interaction, The existence of a sw ’ar tGKqua of

Interaction Action’ configuration, driven to its present state by any siynificance acting on the satellites of the major planets hss pur.ly conservative qravitatioM1 e. not been &rrmented with correspondina visual evidence of a secular

?he second possibility is thnt dissipative effects, which give chaw,e in their orbital perldls. TLs effect la apparently too rise to secu~artorques on the affected bodies, drive thsl towrdr mll to be meesurable at preaent, Still, an esthte of its a coramennuability with one or =re other bodies, and that something, magnitude has been infefld from the presrnt slre of orbit of the eithez in the nature of the r”SEipstiVe mechanimm or in ita inter- innermost satellite (Goldrsich. 1965). The best evidence for tidal action with the gravimtiona force, leads to transition into a friction cmoea froa observatie:is involvinq the poriod of our neon litration or a particular resonance Pariable. Gol(lreich (1965) has (hnk and MadoMld, 1960. p. 190). In fact, the present rate Of suggarted that the dissipative mechanisa operating in srtellite increaao lead8 to rmething of a paradox in the sqe of the earth- systems of the planets is the inelastic tidal re5w:-sb of Lhe planet moon system compared to tha age of the earth, asPmning a cor.mtant 6 'I dissipative mechanism. A aoOel proporal of R. Ripkin (in prose) is follow thsk evolution back in th. that the moon may hare been trapped in a -enmuability with Venw Attempting to understand how these three satellite-eatallite at a fixed radius for a 10- enough period of time to rewlve the re6oMnces of Saturn evolved, in the context of Goldreich's time scale paradox. Unfortunnttly, the proposal doea nut appear to hypothesis, was certainly one of the major gals of thi~thesis, be feasible because of various factors discussed in section 4.2. although the first problam attempted yam to determine the feami-

Perhapr the solution is that, in the past, the dissipation function bility of Hipkin's lunar reeonence hypothesis. Lots of the passed

ha^ been variable, as indicated by some paleontological widence before it was realized that each was governed by a onedimensional

(Panne1:a. MacClintak and Thanpson, 1968). Hamiltonian, although it turns out to be a much poor- approximation

At present, Goldreich's tidal emluticn hypothesis must find ita in Hipkin's lunar case. More time was spent determining how to bemt support thxough indirect evidence involving its consistency in introduce the tidal torque into the Hamiltonian eo as to proseme its sxplarnirq present observations and past history. Applied to each canonical character. The resultinq Hamiltonian is. of course, an satellitcsatellite resonance, it should lead to the results 1) that explicit function of tho the, and it is this axplicit depedence capture in- the presently observed resonance ia a reasonably which allows the systcm t' evolve. men more time elapsed before It prubable went. and 2) that the time fAat this event took place was was realized that no adequate analytical theory existed with which within the ago of the solar system, given a reasonable estimate of transition for even the slmplest pendulua systea was thorwghly the tidal tcrque. bllan (1969) has already shown for the Mhs- explained. In tho process, the scope of the thesis has brsordened

Tethya came-ability. given tidally induced torques acting on Eonsidsrably and Meit difficult to find some point to ana the each reaomnce mrtner, that the evolution of :.e orbital elements, affair and bring it to some conclusion. The exposition of this including the ampliMe of libration, could be followed backward in paper breaks down into three sxerciscsi 1) developlent of a one- tloa to determine the initial values of the elments at the time of dhcnsional Kmiltonian from the satellite-satellite interaction, capture. estimated to be aht2 x 10' years ago. Ur,fortunately, the 2) dwmlopent of transition theory for this pendulum-like appraxhtlons that Allan made for this Lase cannot k used for the miltonion, and 3) applicationn. remaining two casos. A major problm beforo us i8 to carry out a The developnent of an approximat3 description of the motion due sinilar analysis of the 0th- satellite-satellite rrsonancee and to the resomnce variable is a complex exercise usiw, for tha most

. - -- 9

pnrt, wall-*nOvn tuhniqnes of celestial mechanics and variational mots in ex~crLj-*he extrrsaes of the motion of x, while in *be the-

thwry. The first step is to reduce the interaction to a one- dependent CLSO, they at Ieast bound the motion of X. This method of attack and its associated ''picture'' have rather wide application and dimensit- -1 >: -1tonian of the form can rduce many difficult problem. involving transitions bctveen H(x,O,t) - 1/2h + c(t1)' + b;x.t)cor+. (1.1.3) distinct phases to a tractable form. Capture criteria are Ckptrr two takes the SpeciriL =1ymle of a sate1:itrsatellit.e specifically developed for tva kinds of eccenuicity-dependent gravitational interaction and outlines a procu.:-re for expanding the resonance variables. intataction in tam* of the orbital elemsnra. The variational The results are then applied in chapter four to the three equa:ions of option of a canonical set of elements are also derived. satellite-satellite resonances of Saturn and to Hipkia's lunar- A method for the elimination of the "sha-t-pariod" tenus in tho planetary resonance hy:.~thcsis. In the first exercise we fin6 that interaction is sketched. along with A qbalitative discussion of the the existence of a tidally lnbuced torme does successfully erplain aprroxinations involved in redwin? the system to one degree of the capture process. In fact, for two of the three wples freddcm. ?he tidal intersetlon is furthtr discuss& and a method is discussed, the resonance variable autzmatfcslly evolves into libra- proposed for introducing the tidal interaction into the tidcfree tion. But the hypothesis is less successful in resolving the Naniltonian. In addition, a discwsion of the other physical evolutionary time scale. The negative results of the second sitations far which the above Hemiltonian is a good approxirpation exercise -Cold have a sobering influence on those whmight over- of the =tion is yivcin. In chaptar three, .he analytic bohavior of esthte the importance of -esoMnce phenocleoa. Pi: illy, the the Hamiltonian n(x,4.t) is diecussa- in detail. First, tho three-satellite corplensurability of Jupiter. !* briefly discussed, shilaritles to and differe-es from a simple pendulm are discussed and a probable history of its evolution is given. for the tide-free case, and the pansible notions of the system for %e laaterial in chapter three concerning the * r criteria differant functional forms of bk) are found using elvmentary is different enough froa other approacbes to che proble r-ire analytical principlg. Then the motion of tha time-dcpendent systcsl, a lengthy introduction of ita om,mainl, '0 unders.ta?d ' nature is obuid through an investigation of the 1Dotin.i of the "roots" of of the approximations which will be used. First, *t :on 01 a a polyrmial in X. In the the-independent system, a pair of these simple pendulum with a the-dependcnt restorir7 It -. I?

' L 10 11

hmtigated. Attention is focrued on the 'root.". We dim= that

they are four in nuaber and ere points in the caplei plane. One pair of these roots bounds *be motion of the PoIIIentm variable, and The first exclrrple we shall examine is that of a simple pendulum each rout -4n be uniquely labeled. Wether the angle variable 6 governed by the Hamiltonian executes pcritive roations, librations, or negative rotations is Hip,#) - 1/2 p2 + b(t)cos+, (1.5 a) qualitatively detsrmined by the relative position of the roots, along with the specification of these roots which bound the motion. This where the emations of motion are given by

leads eventuslly to tho precise definition of the trarsition pane b) la terms of Lie mutual notion of the roots 4 d of the mentum variable. ?b complement this picture, the flrdt-order .quatiom of C) ration of each root are durivd, fra which t'e analytical pr3petti.s are deducd and a uaruition integral* le defined. Naxt, a constant

torque term is added to this simple pendulum Hamlltonirn and Ihe sion convention for the equar.lons D? i.ot

orbit interaction. In that interaction, the e-ivalent kinetic

energy tsrm is nega,tive-definite.

Initially, say, the pendulmo is raecuting positive rotatlons.

If the coefficient b(t) slowly increases in magritude with the, then

I. .. - .." I 12 13 fhe velocity of the penctulum M it passes wer the top will slowly set of roota corresponds to the value of + at the top and botmn of decrease [Best, 1968). Eventually, the pendulum will not have its SriWl. enough energy to pnss over the top, and thereafter will librate. An Thexdore. the four roots cm be cmpletely lakled ty determin- eXAminAtidn of the solution for the manentam variable p will euggest ing the value of 4 (either mod(*) or '2~))and the sign of (. : : an euuivalent picture of *.e transition. If b't) were constant, than at that root. Inspection of the equations of I tion reveals ttut vc could fini A solution :or p in terms of the tine by using the the first set corresponds to 4 - mod(w) and the second set to Miltmien to elhimat 3 0. .he -ez.ulting integral solution is: 4 - mod(2n), and chat they are campletely specified by the set of labels p,+, pw,, p,,,, and pa-. Pictoring R(pl in terns of these (1.2.2) mots we have

(1.2.4a) R(P) 1/4(pa+ - PI (Pn+ - P) (Pw- - P) (P - P2r-) I

And

me functlon R(p) is A quartic polynopial whore four roots -e Wen b) by

Physically p is alvays real. forcing R(p) to be 2 0. If all mota - *5m, ?.WH3. (1.2.3) the rwtL rre real, then the metion of p is bound 3etween either me wition of p is bounded by a pir of these rwts, with p oscillab- p2.-, pw-, or pw+, ph+, and corresponds to positive or negative inq hack and forth btween them with increasing time. Inspection of rotation, respectively. The motion is bod& between the tw) v- the -?tion of mtmn for p reveals that these turning pointr of p roots or two Zn-roots, or, .) librates only if the qrposite pair is (puima ail minima) occur whon complex. We shall adopt the convention that in the rotation Fh8-e + - 2nr, or + - (Zn + Ur, (all roots real) the w-xoots lie in'erior to the 2-roots, or b < 0.

JI being an intequ. In subsequent discussion, ndl(w) and mod(2w) Diaqranmmtically, we EA? reprnscnt *he three distinct states of the ab811 d. signate 4 equal to 2nn ad (2n + 1)w, reaperLively. If the pendulum -- positive rotation. negative roution, ad lmatiori -- by pgdu\\n is executing positive rotations ((4) > o), then the negative a graph of the relative locations of the roots ir. the cm-plex plane

I I 14 IS

4 I m

.-- 16 17

aucid to tin opporito 2v-?00tr and 3) it ondr whm p roturns to b) tho orlgln (tb rwl prt ol tha p, mct). TNm aotlon corrmrponds

rwghly ta the rowlotion in which i goor to coroO rovorsos sign as Iho qulitativo motion of oach can bo ouily dotomind, qivm tho tho pondulm moves b.cltwudr thrsuph the t. -tom and .?.in =os to siqm or qbodt and of tho root. If Ib(t)l incroaros with tiplo, and uro nou tho top. If inittally tho pondulm ~ocutosposltlve since wo have adopt4 tho comontlon Uut b(t) bo noqativo, thon tho roUtions and > 0, then figuro 1,Z.Z roprosontr tho aptlonr In a-roots mcve to-mrri Lha oriqin and tho 2a-roots amy lrm it. mua tha caaplox p plana. tho fluctuation in i cawd ty tho pdulum rorco grows am Tho change In tho -roots during this trrnrltlon phso can h trwltion to libration 18 approach&. On. intarosting obmorvation dPUt obULnd by Intmgratinq tha oqurtlon of motion. 'Ru remult is that tho a-roots azo stationary (- 0) whon 4 mod(r) or is dt - p - p,. But p itself is s minima whon it is at tho raOt p, and la l/2(p3f) - PtWl - 1;; dt2 (1 + cos4). (i.z.e) stationary. :l tho rwt vu0 not stationary when p - pur-thon lt i. mtico chat p:~) o from tho dollnition or transition pharo. ~h. a ~!sp!ouerciao to ahow that p .auld ~fforan infinite aceelor- - inteqraM is negative doflnlto 4 01, hrpiying that p:(t) ia stion at point. Tho arbmqous situation holds uhon p this %a* cwpativo or p,,(l) imaginary. 7'ho rap. of tho Initial value 4, is Yo could use (1.2.7s. b) to dotonuin. tho first ordv (in

' +u L +i S 3w. Th. rnglr + thon incrorsos, prrlng through th. srulsr tmhavior of tho rwta by integrating thm ovor one botton position of tho p.ndu1rm end firvlly roachinp a avxLarm at revolution and approximating the motion of x and + replacing b(t) 4, 37. 6+c, Mors l+c is a -11 pnritlvo angls of wich its mean value mor tho rmlutlon. In this instarno, tho - action intogral reprosonts a oiap1.r wthod to obuin tho soculer O(lbl'"' *. motions. 0 than rworms sign, and tho anqh docrumor ubtll it roach.. a noabove equations aro uniquoly woful in tho trrnrition phi.. + , minimno at v + 6+f(6+f 0 and O(lbl -3/z N))dt This phase will dofind Condition6 tht 1) it st - k by tho aut. TO siapiify OuI problan, lot's cmso to mnsmnt, ad tho instant tho w-mts coincido and booaw imaginary for aQo. d-nd that it bo comparatlvoly .null. Llro chan90 tho Intogration inithi values (pi,6i), 2) it centimos as tha motion of p in vulablo from t to 6. Tho roault is 19

4 I whore

It ehould ba .rph.rlxd that the akve is d path-lib lntuptrl in ipw + which the variation of in going frem Oi to 4f ie deterahad by th. transition phase diagram (fig. 1.2.2). Duriy trr?rltion K:tl

Slnm dH at changes from one to a value slightly above it. -dt - -db dt 6 - modfr), K(t) is very nearly a constmt, no matter how long a ththe pendulum spends near the tep durlnq tranaitlon. he

contribution to the integral is mall for 0 a w sirare tke integrand

vaniahes. Rccept for near the top, the motion of 4 is fast sorpard

to any change in H(t) or K(t), and the intcgral lrs w.11-bounded. Thue is a statiorrory solution for p 0, 4 - mod (v) or sod Ow), This phse will k ddined by *he :=idltions that and thare exists th. aingular po*sitllity of a sticking motion in it atarts at the instant the a-mots CoiMide and kc~u whlch p elowly approaches the top and “stlcks” there. Motions vuy imaginary for saw initial values neat thia singular went will bve very long trsnsition times, -re w choose the mlninnun possible value of 4i to equal implying H(t) could change appreciably as p moves htwmen the it covtinues as tho nution of p is first tomrds the p21-mot, compluc w-roots. “he question la, hou near? where 0 - 2r ar3 +(2a-) 0. and then back towards the point It turns out that thr set 02 sotions uhlch have a long wtmre i vanishes at a as;xina amile 6,) transition tinu are restrict& to an upnnentially aull set it continues as 0 rwaraes sign and iho nution of p is carrid Of 0([ble3’’ 97of initial value8 of #i. lb deonrtrate thls to the oppoaite 21-root where 4 ajain .qu&lr 2n but +G!n+) 4 01 assertion, lat’s drtsnnine the condition for rhlch tha chanoe in it end. when p returne to the origin (the r-1 part of the H(t) in the the inierval ta 6 t tb la of O(H(ta)). In addltion. pl-roo-) whcre 0 again vanirhea at lane angle 4f near 4 - w. 20 21

to its valuo at 0 - be, and approsfmat-. :ha int-and on the right hand side by

.. .. where K(t) - 1 + A(t). tinally, U(t) can k rsplacd by b(t1 in the -d~~i~(O1~O) fk ' + j#b dbSiw(01# < 0) I jtb(2H(t))1/2d.. integrand of tha right hand side of (1.2.10). (1 + K(tl~or+)~/~ 'c (1 + rtt)cor+)1/2 ta For A linearly changing b(t), ws can write (1.2.10) b(t) - b(t,) (1 + ), U.2.12) The intoma1 over + begin8 at an angle 0, for which 0 is positive,

continuer to the angle 4c where 4 vanishes and rovereas elm, d and 7 ..a the "slow tho" associetd with the change in bo. With Th. remlt adb as 0 m~eaback to the angle 8b. Th6 integrand on ths left -8. approxhations, both sida can h int.matd. is8 hand ride of (1.2.10) is large only for anales vary near 0 -modtw),

and it6 VAIW Will tend to be indepsndOnt of thu 1UtS And Ob AS

long as ehsy are not nearly equal to 0,. "herefore, the (1.2.13) calculation can be shpl'.iid by chooaing 4, - Ob, -ding coo# 60, can bo choson to be >> 68, Duch tbt the contribution fron the about + - 3n - 64 and changing the limits of intmgxation as 1- limit is relatively -11. Tha angle 69, is ths angle for followsY which 6 vanirhes or for which (1 + K(4,)cosQ) - 0. Thus 64, is dirwtly related to A(+=), and is approx-tely

Tb. -11 differences, 69c and bo,, are positive and 6#c < 66a 1/2 6 2 4, m A(4cl * (1.2.14) sine. 4 c 0,. loeu the top of the pendulva H is very nearly -1 to b(tr. Thin means that the function P(t) vary nearly eqwls the "he transition rime \ - ta mst k of ordu T if th. cbngo in -dud one to O(b-' g)during transition, implying that th. change b(t) is of O(b(tfi)); For the SAke of calculation, we shall daavnd natural prld, T, of in ?It) as the pendulum moves ovez tne top is of O(1b-l El'). A that b(tA) - 2b(ta) or that t,, - t,, - t. The first order rsthte can be obtsine3 by choosing Ktt) to b. equsl , which is the period of the Fondulm in 22 23

tho rull libration limit. - f)l - 1/21 U.2.15) b(tc) * 3/2 b(t,) . Tha vilue of A(+c) is Tho next rtep la to obtain an approxbnate ramlt for the 1.2.11) integral in (1.2.4.7). Clearly, tha only va~uemof ei rrhlch (1.2.16) corrs6pond to a long transition the arm near the value (31. - EOc). Thareforr com4/2 can ba 0xp.ndSd about 0 - 3n. Praa f1.2.13), thr and p:ter), which doponds on the initial value Oi, can k tzaneition the is rouqly proportional to (ln(60(tl+~ZA+ a2+(tl 12/3, approximated bp BO that the integrand doom tan8 to vanish If b(t) lncreasom

indefinitely. Since t chanqa much mors mlowly than 0 for + not tca near 3w - we can approxhta b(t) by ita initial value b(ta). (1.2.17) ~h.r.pprox*te solution tor p:(ec) isr

Observe tht (1.2.16) dopods on the value of b(to), not b(ta) or b(tb). Th. relationship betmran tCand ta can be found frop an inapestion of (1.2.10), in tho light of the approximations so far By contmctlon, 3n - Oi 6eC, !-plying that p,(Oc) is imuglrury. lnvokd. We ems tbt wch of the tams on the left hand side of 4bc can k ellminstd using (1.2.15). Solving for On - Oil,

1.2.10 are approximutely wual. This impliar, that the integrals

(obtaired from tho right hand aide) ovalvatad batwean t and t and b and tCaleo -1. Explicitly For large value6 of a, the set of valuem of Ol which land to a long uansition tiPU is exponentially mall cmpared to the full rang. J%(Z~(t)'/~dt5 Jtb(2~1(t)1/2dt . ahova rssult agraaa qualitatively with bat (19W) ta E of b1. The although he appear* to calcu!ate a qulto different pasmeter not Tha abwe integrals are approxhtd by the rlgbt hand midm of marly eo wall related to the initial co.ditionm. Pnyway, the (1.2.13!. Given $-ta-Zr, the important relrult.8 are: /,*d 1s wall defined importent ramult im that the tranaltlon integral a,.J-f- '

+<.A.

1 ... . 24 25

eept for initia? valuer vary clow to the sticking motion. Eut

the functional apgroximntion u8dl for the integrand is only qood to O(lbl'" Therefore, wu can effectively ignore those ei vhich

have 4 lonq tranuition time, it all that is desired in a solution to

the integral accurate to first ordu. C: cour80, ir. any real

phyniCA1 synt0la gOVernnd by (1.2.1). theUrn arbltruy

fluctuations which wuld effcctlvely elhlrate tta Fcrslbility of a

sticklng uution and inhibit trsnsitlons which take an ucoptionoliy

long time.

A syetm more nearly related to the problm r.'. *.r. is that Jf dc a -endulum abject to a constant applid torque, ;ii:. Ttd Hamiltonian in this came is

(1.2.20)

We shall choose to ba positive such that if the prnndul\n,

initially executes positive rotatbno it will to slowed down by tho PIGUM 1.Z.b torque, ad 4 will eventually twerse sign. pot tha special case Diagram of i7versus $ for equation 1.2.20 where b is constant. where b(t) I const., Rfp,$,t) is a constant of tho motion. Pique (1) A value of R mch tht $ is, at IIQ~Otime t., a rotating 1.2.- i8 8 graph Of 4' VeINI 0 for 1) 4 ValUO Of H 8UCh thrrt 0 in, ~uirbie. A value ror vhich $ itbratea. (a or R at soma the, a rotating variable, and 2) a value of H for whlch

libratem. In the first case, tha graph reveals that t& path of

motion of $ into znro is the name pcrth it Lollown amy fron zero. Transition from rotation into libration cannot occur, except tor

the sinqukar event of a sticking mtlon. Obviounly. if capture I.

to occur, a non-tlme-spBnetric tom must be included in th*;.: 26 27

A Rmiltonian. Pigure 1.2.3b shews how NC~a tam bxraks the tima 42 aynnetty. Prom the prwlous ownole *. can dwhrce thst li is constant, Ib(t) must bo an increasing function of the the for

capture to occur (a'w mee Slnclalr, 1972). Alm, we axcept thnt

the criteria for capture rirl depend on the torque, the tunction btt), its derlvatlv~,and on tho initial. conditions. Incidantly, in spin-

orbit coupling, the the epmatry of 0.2.20) im broken by a

velocity dependent toroue (see Goldrmich and Peal., 1966, and 3.1.10-17).

:'he abva Hsmjltonisn lacks the simplicity necassarv to expresa I + 0 aa a function of p. Fortunately it can ba transformed to a new Hamiltonian H[x,+,tl which har the roquisite ehplicicy, defiwd ty

(Or. 2.9.12) I

FZGURF: 1.2.3b whore

~i.gr= or i2 4 ios a pendulum-like .ystm, subject to '2 a torque, which is asymmetric in the. ti is the kinotic energy of a ~ulumea it goas m.2: the top for the 1a.t time, dilo is and the paximum poreible value. 64' is tho kiqetic vrugy aitar it haa row-ad direction and again approachad the top. Capturr 0ccu'- if 6;' > ;:. Thr value or c(t) CM b chown such that # - .c in the absence of the ~ndulrraforce. The vrrlable x them twesents a fluctuatlon-&n i caused ly the pmaulum fOrC*.

Pcr e(:) and b(t) constant, the turning points in the notion of x occur at 6 - md (71, mod(21) , and we can solve for x as a function The denominators ars .,qual ts the value of (-3) evaluated at x - of t, as was done to& the simple p.ndrrhm8 root. The equations differ farhos &rived for the siqh pndultnu

u .2.22) in the first %em. ~ikethowe .quatione, th- morion of the rwte stationary whenever x xw, or x x and 0 o or 6 29, is - 279 - - respectively. As hfore, the s-roots lie interior to the 2s-roots.

Zlnlika thome OqUatLOna, howv*r, the roots are not spmaeuic about wain. the quartic pblyna~ialRlx) can be Zactored in tarof its the origin but about (-c (tl) Also, the rutJon of each -mir of w four mots, and the forn roots uniquely lab& by the val& of 4 . and In rwta im equal in wgnihde and opposite in direction about and sign(-0) for x equal to tht mot. Allowing c and b to k tima this wing p!.t (-c(t)) (see 1.2.23). dopedent does not change this nituation in the rotation phaee, Ut’s choose x to mebetween x,_ and xZnc (positive since the anximum and mnimm of x still occur et ood(u) or &(ZIT). mtbtion) If m ignore tbe second cem in qach mation, then x,,- These roots are wves tovard the right and XZnc towards the left, inplylng that the flwztuation, 6x (Dcf: 6% - x- - xmin), qroum 6s the eyetan, approaches transition. The othu pair of roots, burlden separating, has a secular notion towards the left of O(3 (see figure 1.2.4).

The wpations of amtion for c#eh of the roots are obtained in The transition phase begins, as before, when the tvo n-mote a manner ~M~O~USto that used marlir for the simple pendulum, coincide at the ti and thereafter become complex. The equations of except that the roots are dependent on three variables: c (t),b(t) , motion for the IT-robtscould be separated into their real and and Rt.6, t). The owations arm inagiwy parte, ht cbfe procodurs can be circumvented her. bY obrervinq tbt xn2 + c goes to zrro as H -> -b(t) and then becamss 30 31

inugiwy kme 1.2.23a). Thug thn tu1 and iraagiwy parts obey the

oquations :

, (1.2.25a)

2 We should imte that if x is camplax, +hen Dn I is real and 0 x Ls poeitive definite.

TM related Integral whlch determines tha value of Im xn at

time bf when x makes the second colncldence with Re xs 1s

PIGURB 1.2.4 OOsITNll ~~OIOPIOIBI

In arrom indicate the relative veloaity of each root. u .2.m

2 Rm the prwioue alewe sxpect that if Jh x,(f) is positive definite, than the w-roots are still aqinuy ot time tf and

revases sign, implying transition into libration beoccurred. But

if m2xl(f) 1s negatlve, it lruplles that the r-roots returned to the

roe1 ario bafore x reach4 Re x fr'm the right, and the pendulum executes negatlve rotations. Therefore Im x(f) - 0 correspond8 to the sticking motion. ti -> 0-) which separates the transition into

the libtotion phre from the transition into the nwative rotation

phase. Note that this occurs after the firrt sign reversal of C in 32 33 which i -2 O+. Thfs condftfon (that nnrW raniah) is not -1etelg The meaning of this statemant shall becaru clesrer as am procoed.

accurate as -11 presently b. demonstrated. % bpxtant point to The relative motion of x during tramition can be discwerod p.ke here ia that the above relation can still be wed to find to frm the simpler case where b(t) is constant. First of all, the lowest order the critical initial angle O,, which leadr to this sxplicit tin. dependenee of Hb,),t) can be derlved and is (from

sticking motion. l.Z.Zlc,d)r

The description of the transition phase is more complex than dc H(x.4.t) const. U.2.28) - - -dt 0o that defined earlier (figure 1.1.2). ?L+ important question to reaolva is the relative laation of % with reqmct to Re xu for the Tho constant in this oqutlon can be chose blch that the argrrunt plod of time that the u-mots are -lexx. We should keep in mM of the radical in (1.2.238) for the w-roots vanishes when 0 - Oi, &t the major goal is to define the appropriate 'transition The resulting equation for the n-roots is; integral' dich can be approximated to first order in the mall _. U.2.29) p.rametcrrs. There a~etwo -11 paraamtars in this aye- of pendulum plus constant torque3 1) the first parameter is tho ratio of and the w-roots becmo cinnplex for 4 P Oi. Th. minlmum initial the constant torque to the arwfnan penduIum torque and ~~11ii is angle is d(n) and shall be chosen for this discussion to equal n,

Tho initial angular velocity Oi mst be 2 08 otherwise 0 would bve U .2.27a) woviously vanished and reversed slgn. Rlso, if 4 vanishes at the 2) the socond parametar r8tio 02 relativo chnago is the the Ln b of -nt the s-roots coincide, Lhen 0, must be equal to n. Unless O(b'1'2 to the initfal valuo b(ti), and is -11 if 0, - n, the angle 0 must increase for t 'I ti until 4 reaches a muinnno $,, at which time 0 vanishes, and 4 thereafter do~rE8SeS.

wentually 4 returns to the value +i at a later tlme t, and the since tho aquatien of motion ~n x: already first of is in w-roots are thereafter real. Prm (fig. 1.2.3a), we find that whan '2 '2 'chose -11 psrswters, we sqMct tht the motion of x and $ can k (0 returns to the value +i,$ a10 returns to its Initial value $i. xaplacad their terovrdu motion calculating transition by in the In the complex x-plane, $ vanishes when x - Re xn, and reverses integral. Equivalently, the transition phase can b replaced by its sign as x mes to the right of Re X. After x moves to the rioht lovast order apprcximntion in defining the appropriate integral. 34 35 of Re x,, 4 docreasas. Kb aoa tht om0 9 ~etur~to the value x is still to the right of re 'xr sinco + < 0, and the motion of x is tr4ppcd ktvann I* and 21+ an8 has entered the negative rotatfon phaae. ?hie .Equence neglects the possibility of a sticking motion.

Pxactly ho* this mtion of x wuld appear in tho complex plme depends on thm relatlve magnitude MB direction of the prdulum torque at 6 ei aa compared to the constant applied torque. Th a) Diagram of stationary psitions relative mtlon of x with respect to Re xI for the time interval of "physical" pendulum. ti t t wlmn x1 is -lex, is found from the equation of j' ration of e zh variable, x and Re X,,I

- (1.2.30) X2lr ' XZ*. "he two velocities (and the tw torques!) azo equal for angles

(-1) and 4o (-21) given by the relation

sineo b-' (112.31) - dt . b) Equivalent picture of c) Equivalent picture of stable ClWly 1s' s 1 for these angles to SXiEt. Ru-8, the unstable stationary stationary solution. 60(2~) hro stationary solutions (9 - constant) of the system correspond to solution in co~~plurx- is -2r m:c\ that cos+, 5 *l. &sa tmu anyles. Figure 1.2.58 is a "physical" picture of these plane. $,(I?) is near I two seaticnary solutlons, wNle figures 1.2.5b,c arm their equivalent such that 6 -1. -c~rercntationsin the ccrmplex x-plane. Incidently, the distance PIGVRE 1.2.5 of each of the roots frum re X1! in figures 1.2.5b,c is obtained Location of stationary solutions of s-le pandulm plus torme. fra (1.2.231, and 36 31

and towards ~e xu othervise.

Careful consideration of these facta reveals that thara are two qualitatively distinct "transition phases" involvM the motion of

th. --roots and the variaale x. Pipe 1.2.6a rhas the relative a) Rere the hitiax anale Oi lies in the ranqe mtion of x during transition for the cam where Oi > 4,. along +o 6 #i 6 3n. The diagram on the left is a with the equivalent picture for the real pndulun. This diagram La description of transition in tha corplex x- very similar to the trans?tion pha8e of a eiaple timadependent plane, while the diaqram on the right is the pndulm rithout an applied torque (figure 1.2.2). On the other equivrlent description of th, motion of a hand, if +; lies in the range u 5 5 +o, then figure 1.2.6b is a physical pendulb. picture of the motion during transition. Th. next question is, which of these diagr-8 is important?

Frcm (1.2.29), 1n2xn(4) ir moat positive when $ reach08 ita maximm value at 0 - 0, and 0 vanishes. Since N - bcoaOc when 4 vanishes, ~1~x~(0~)i8 also given by b) Xn these diagrams, 0, lies in the range

n 6 +i 5 +,(*I. msei an13 4c are relatd by (1.2.23): PI- 1.2.6 (1.2.32)

The sticking motion i -> O+ must correspond to a motion described 38 39 by fiwa 1.2.a. in vhich the initial angle 4i (sticking) 1lam in the ranpe (r ei * 40W. -so, the pandulrra has not just (1.2.33) prsssd over the top prior to the coincidence of the a-roots. Recall that ’is the unrtabla equilifpium position. Since the right hand side of (1.2.32) is a lllaxLnum for 4c - #o(w), it follors that 4i lies in the range w 5 Qi 3W while 4f equals 3n to lowest older. the naximra amunt of the w-roots can move off the raal axie for Also, the contribution to Xm2x(f) fram the fi-at ifitcq:l tonla to the set of transitlons defined by figure 1 2.6b is given by the bs negative definite. sticking ratLon. Since 40 - b-l dt, IM’XISt.) i8 Of O(1b-l :+I2) The lowest order approximation to the second integral is and is effectircly seconu order in the -11 parameter associated identical to hat Cowd for the s‘mple pandulrrm (1.2.171 and is with . But the ma~bwmpossible value of - 4i) is :r, which correspond. to mtion given by figure 1.2.611. me mu- of lra2r i8 therefore of O& .

Thin swgests tht the followity .mxbations be epploysd to Thus the critical initial angle B,c which sapara+es transition inW find the first order notion lin b-’ 9f the w-tootst 1) neglect WrnSitiOM imolving figuro 1.2.61, for which #i is in range libration from escape into n*iative rotation satisfies tha relation

.D 00, since set of mtions are second order8 2) for P ei this of (1.2.39) the case fi 0. approximata the sticking mtion where -> 0- (and

Ca&ure libratic * occurs f..r Oi in range x - Re xn by the condition tht In x - 0 when x - Re X~I3) neglect to first ordar. into the is any affect connected with exceptionally 1o.q txansition times. With 1 L 4i 4ic, sinca the first integral propor;‘ona1 to thasa approximations, the description of the transition phase for waller thsn tb second integral proportional to 2 for 4 i in this the system of pendulrrm plus torque reduces to that for the simple range. On tba other hand, if 4i lies in the ranqe Qic 4i 3n. he negative rotation phase. pendulm (sea figura 1.2.2). %en the pendulum has escr-4 into If

The next step is to find the first order approximation to each +ic, am determined +y i1.2.35), ia Beater than ;n. the Lmplicaeion integral in (1.2.26). For the irat integrand, x + a equals [-+). is that the pev4ulum will inevitably onter the libration phase, independmt of the initial conditions. 41 40

'2 Often it happens that the a LO )i is unknown. In such an the pendulum over the top hfore # rwerses sign is Ol and is instance, a more valuable -1 would k D function which Je8crib.s equally distributed betwwn 0 and Ai2, where Ai2 is the maximun, the probaailit;. th-xt caphrre inte libre'ion will occur, given po8dblO value of 4;. In order to define Ai2, we had to analytically '2 probability densities for th. initial argle Oi of the system. The cuntinue the graph of $ versu8 $ to nrJative values of i2(dashed '2 -8t physically reaSOMble aS8iglBBOnt Of V.?. ,:A1 y -S the fOllOWing: line). Thus A'; is the difference in 0 betwee-. successive nlinima.

If YT eearure the vsiue, of 6 a+ : tar from transition (t + - m), The value a2; is the decrease In the kinetic energy over one then for a fixed valrie -'+(--I the angle #(e)would be equally revolution as measured at the top, caused by the term which breaks distributed ir :.e range +* @(--I 5 0' + Zn, where $* is the the-symmetry. Capture just occurs it arhltra,';. 'lnfortunately, it is not clear how this statement t-inslater in defini.ig the protability associated with A given valuuc of Ci at transition. For the speclal case b = mnstant, the Therefore, the probability of capture (Pc) is translation is that $i is equally cistributed fn * ye (1.2.36) 1 *1 3r.

Another praneter which ca. ASSUIU~to equally distributed h k 2 '2 The next step is to relate Im A,,, evcluated betwen ?miopiate in wme closed range for tNs case is the value of @ as the limits#, to each of he quantities appearing in (1.2.36). In order -?endulls raves over the top for the last time. Both Goldreich ad to accmplish this, the meaning of negative values of G2, hplied by Pede (1966) ard Sinciair (1972) adopt this ddfinition of the continuation of the graph below the $-rxis, must be axplain&. probability in th - respective studies in which the th.syr . try The quantity is xoken, in tJn former case by a veloclty-dependent torque, and

in the lat7e.r case by n timo-dependent coefficient b(t). This second case wctly correspond- to th- oxample being diw-ussed. is a measure of the docrease in the kinetic sr.-rgy below the value Sin-ze it i8 always desirable to make coctact with otbarm' results, --ro as measured from near the top. But ;:, to lowest order, equals tkia definition of probability density will be adopted here. r-(n), which equal81 €'ran figure 1.2.3b, :he value of i2 durim the last passage of 41 4)

rwtlon J.1 wa rhll L .d that cha o-tlorto 1 of ,.rpturo probrblllty

for porrlkl. Cooownco arDOChtd with tho Pha-tothym

oorrunrurrblllty rqr'oe 4th the nrurlcrl calculrtiorm of SlficloAr, (1.f J'l! Doforv uoncludlnq *hlr dlwurmlon, Char. 10 ona nxo int*rortlnq

lorwe WII rhll Invortlprto, r0lAt.d to trrnrltio, frm tho

porltlvo rotatron into tho noyrtlvo rotatlon pharo, nnl It 1s

dour1b.d by tho Collowlngi If w Look a* tlo Iptm af pondulun

PlUr tatquo err lrm tronrlrlon, at approxim*r!y qunl rlrr lntu- l;z (1 .1. JU t Val# kotoro and .?tor trannltlon, wo obrorvo that Char. La a

rwulrr chrnqo in the morn valuo of a, u: oquivrlontly, in tho

moan /alum of 4. Thrt 11, It w mamrr tho maan VA~U~ot fi ln tho

pdrltlvo rctatlon g~hrofu from CrrnrLtAon and Ilnd that le ir

rqul te, my, (-c:c 1) at tlmo ta lwhoro c, -> --I, rlron tho rrrult

of porformrnq I lmilar noarurrent in tho nogarlvo rctrtlon phoe

at tlw t,, (who;. -% -* +-) rhll ki

Uslaw (1.2.351 to olialruto tho wllclt le~mndonooon Lha protobillty 1s uhnro 41 11 thlr r+culrr chwa. Tho valuo ol hw bo datornlnrd uoinq 1) tho action lntogtrl and 31 tha condltlon (i(ta))por,rot,- (1. f ,491 -aka). *o tiw rvrrag. of 6 at th. ta 1.

Pc is soro IC !?? - 0, whllr P, approachor it. mxLauD valuo oi v-.&ty thn tho ratlo of tho ull rr~motorsof th. ayotm ir of 011). (1.2.41) Also, lt mhould ba p1r.t.d out that tho abvo formula door r.ot apply to the n. eclr.1 cam. 2 - 0 ninc. (1.2.35) in then lnvalld. Tn Cpplylnq tha rrcord condition, w find 44

t1.2.42) rapidly inaruror Cor -11 chryom in +h. rolattve roperation of tk. r-rootr. Thorelor. J can k calculrtd at trrnrltion to d.torn.no A. t!!s wnangular velccity docrurer md tho *vat- apprmohor tho r.au1.r ehmr in tho th-depondont functionr. At tho transition, (x) tends to inereare and PYT~teuard tho xw-rmt. ~hlr ti, Hi - b(ti) rnd J ir a conaequonce of e.0 fact tht the SyBtmI tend- to upend more tlp. near tho top or 1 pasition the pendulum mtion. I 0 jamxd4 ja"(-i)dg 7na(ti) of At Jpo.. rot. - - - 0 0 transition, the mar. val~eof x is - Re xn.

The next etsp Is to wrlurto th. action intogral J in th. positive rotation phase Car frca l~af18itiOnI

Jpo..rot. - $O..rot. (l.Z.43) Or Jpor.rot. -elbtti)\1'2 - 2nc(ti) 8 O.

Since the flucturatlr,la in 4 tom to vrnlrh tho Curthor t+ ryrtmn Mter tho pendulcm tam detho tranrltlon into noptive is from transition, 4 im approrinatoly conrtant, and J roatioti at ti- tC,J crn again bo calculated. por.rot. vanimhes mlnce it 4. a?praxiwtelly proportiom1 to (ad. 1;' A4 I Ii"(-i)d4 2nc(tf). (1.2.45) JnepI rot. - - The action intoqral is an adxahtic constrnt liWrn, 1925) ln mch phnse, am long as the inmtantanoous irocruency of the pendu:um unlorr tf - ti IS uccoptio~1lylargo, c(t,) 8 c(ti)' and ia larqa compared to the mlow chaw~esin the aystm indued by tho b(t,I 0 b(rL) 0 H(tfl, Tho rerult is torque and t?o time-dopendent coefficient b(t). In the ucaapleo (1.2.46) hi- dimrused, tho raatriction is violated only when the n-rootm are so :Icao thnt tho 1natsntano.is frequency 1s vrry .mil. Recall anb Jnog.rot. ir nonmro. Par Iron tranrltlon, thm moan value of a that the instantaneous fr.quency for the first ~eunplotendod to tonds to Ax in tho nogatlvo rotation phemo, and the value of blow up loqaritbxrdcally am a function of tha a-root reparation. In Jnw.rot, corrorpondr to 2nAx. Thus addition, we hsvo found that the rwta do not move vory arch during & * $jb(ti) !Ir2. (1.2.47) transition arcapt for the 6ituation vhoro the CraMltlon thir awcoptiorully long. Furth-re, thr Lnetrntaneous frequency Po? a rI.~rplop.ndu!.w, (-AX) corresponds to the dolay in *A4 46 48

evolution of (3 by th. p.n8~l~mpotential 2s. to uu constant applied

torque. In tha orbit-orbit interaction, the variable x is related to fluctrutions in the orbital elements aler and I Isee section 2.1). The devolopnent of a ons-dhenalonal ~lltonlancnnemnes more This mean. that there is a soculnr change in the orbital elements pnges than anyone in hie right nlnd would nnt to read. This associated with paasage through re-mnce, not cntir ly comectd to exorcise is, for thr aost part, a rehaehinp of old material to mke the tidal interrction. it suit our ow purposes, .\nd braakr dom into four stagrs. First -??le prpore of this investigation hnr ken to develop a is the ionnulation oi the many-body planetary problem in terms of a description of Waneltion along with ~3oyanalytical tool8 which dlrturblng functlon, acting on each planet, r*ich is distlnct Ira sh.1: prove useful when applied to the Hamiltonian governinq the the wre coI0Y)n potential function. In the plnnetnry problem, tha orbit-orbit interaction. Bofors we proceed to diecues trnnsition principal interaction of tho planets ic wi R the sun, nnd 1. the for the orbit-orbit case, the Hamiltonian ,hich approxinutea this nujor factor in deteminlnq thelr orbits. The planet-planot (or interaction will be derived. Mtellite-matellite) intaraction can '= developed ns a perturbation

on the two-body pl.tiet-sun or ratdlitr:-planst orbit. Instud of

coneiderlnq porturbstions on the COordiMtes, it is mra useful t@

fin4 th. perturbations of the tw-body "constants* of the motloii.

The serond stage is the axpansion of tha dlsturhlng function

in terns of the Keplerian elatoents of tho two-body orbit. It 1s

anuring that the resulting upnsion Ir of any use since it is .o

complex, but useful approximations can ba nore rendily iwoked with

the dlsturblng function in this fonn. The terns in the axpnslon are

claesifid, adthe relative importance of ench cl-rs la nlmcusaad.

We deternine the reFtrictions which must be @POscd on these

clnaoam such that a single hvpothetical resonance varintle vi11

dominate the long-tam hhsvior of the system. Alw, a procedure is 19 50 outlined for tho analytic elimination of the short-p.riod tern* ordu daivd independently by Lllan and bllnclalr is discussod, to $rrt:ll by order in pars of a "mall" axpansion parcneter. This s*rves confidence that no rarloum flwr otirt butnen bepinring and ed. tw purposes. First, it licitly show# that the effwt of much Thr beqinnlng, of course, 11 Nrwton'r laws of grrv:tation. toms on the resonance Is of seem12 order. We nee that there temr The gravitation81 force between two bodies Jeperdn on their hsve a minor influence en the tmrbdy resonances diacussed later in NSS, dupe, and the distance botween than. Most celestial WIem

(4.1)~on the othsr had, in the lunar resonance problem ~xcussod approach sphericity, allwin9 the gravltatlonal force f4 be in (4.-) the second-order nixing of short-puid terms is hportant, approxlwtd by that betwclcrn tu) point masses. Given a Get of n because the sun can bo a substantial lndiruct participant in the interacting point mssses, the forces acting on the l'th body are mn-oarth-planet resonance. Second, we me tht the tcms in the additive and individually dorivablr from A potentia11 expansion vhlch look like (1 mm of pendulum-like potentials, all (2.1 .I) ..I .. Jitt-cent angaes, is not entirely an rccldent of the axpnsion prccedwe -- a librrtjon of one of! those arilcrs is implicitly where 6, operates on the coordinates of mi and the potential Vi is posrible.

01ce sstidled tht th criginal interaction can be reduced to (2.1.2) one which involves only one angle variable, WH then show tht the system of equations can b. reduced, in most camas, to a aingle For cIQ bc4'-..1 th path of !?&ion each de~rlbosis a conic cazonical sot fx,+). Th. method umd is similar to the first step rectbn, either an ellipv, a hyperbola or a parahla. If the system in Delauney's wlut-on Lo th. sun-moon intuaction iBrown 1960, p. is bound, the ahpe and sirs of ths ellipse are spacifiod by its

140). The Hamiltonian derived is a constant of the motion in the eccentricity n and s.raima)or axis a, Its orlentcrtlon .JI resrdct aksence of any disstpativh tldil intorutlone. to a reforence frame Is giver, 'jy the mler anjlcs 2, I. w (cf. t..

Our lac? act is to interject the tj.9~irto tho tide-free 1.1.11. These symbolr and their 3 tinitions arb 9eculiar to

Hamiltonian just developd. Essentrally this is acccmplishcd by astrur-xay. R is the "longitude of tld ascending . 'L;' w is t. 3 subtrac'rrq out tho seals efPIcC on thcanonical VariAblOS. "argument of poricenter." whereas I is the 'inclinatlon". hlthugh

Finally. we ~~mari~ethe resul+.s in (1.1C) Also, the relation not a pb iical angle, anothor Irequeiitly used 'broken angle" is between cur Haniltonlar. and the si.,nd-orde. cquation of mtion 0, defined by .-- t2 91 52

am3 calld the 'lonpitud. of paricontar".

Tbpositlon of a body on this ollip~o,with rospect to the pericontor, can be spacifid by its tr.a 4-1y I, accontric a-ly

E (fig. 2.1.2), or maan a-ly H. Tho anpleo E and f are rolatod to tb distance r by

r - a(l - o * cos E) (2.1.4~1)

afl e') r- - b) 1 e - COO f PIOURP 2.1.1. ORBITAL ORZEI~TXO# The mun anaauly is defind I?,

M n(t - T) - E - e . ':OS t. (2.1.5)

In most annlytlcal wark, PI Is the aost useful anolpsly, 6lwo it is a linear function of 'ha tlmo ln th. absenco of perturbstlons. Tta

element T Is tho tiaw of perihelion parsage and im tho sixth consbnt

which fully sp.flCios the two-body systo. Instead of 7, asother

wiconta choico for tho sixth constant is , tho epoch, which Is detlned b>

ni-c-B (2.1.f

Tho constant n is the "moan twtlon" and Ib relatad to th. semimajor

axis by

23 uo - GCmo + rn) - n a , f2.1.7)

which is rompnizd as Keplor's Third Law.

FIGURE 2.1.2. ELLIPTIC VARImLEa The general problem of three interacting bodies is rtilA unrolvad. 53 54

Jn a plrna*ary system, OM body ia much more nussivo than any of the b) others, prdmimtly dotemining the path af motion of all the other bodies. This fact suggests that a mro-ordor solution for each secondary dss vould bo an ellipse whose foms is at tho center of C) edis of the two-bdy systcnr (primary and secoruiary), the mutual Ri im known as tho dimturbing function of ti,. i'th body duo to th. interactions between tho secondaries boirq igmwd. The effect of j tho secondary interactions can be dwolopad as prkurbations on tho action of tho j'th body, and ham th. opposite aign from that zero-order ellipse, in which tho six constants just described comonly assipned 4 potential function. Its parts are called tho

!e.,e,I,c,fl,wJ bocane variables. 'Ihs method is known as "the Direct ad Indirect terms rospcutlvely. In the absence of any variation of arbitrary constants" (Brouwer and Clemnco, 1961a, pp. disturbing body sxcapt the primary, Ri - 0 and (2.1.8.) drmcrlbes the the i'th wlth the center msb the 273-307). 7hhr six difforential equation8 of the elements -0 first llotion of mass respect to of of taa-body mymtan. ordor In the. cappard io the three cqations of the coordinates, which are send order. The first step towards a solution is to acpard the potenticl function (2.1.1) in t- of the orbital elenants ef the i'th disturbed body and tho other n-2 disturbing bodies. To accomplish this goal, it is conveniont to choose the prinury mass as tho coordinate origin and subtract tha motion of the pr-ry bo)caused by tho disturbiw mass. l?10 ?omit of mch an opuation in the equation of mtion of the relative poaition vector ;,r

whore it Is understood tlut ;i - ;io. The two functionr Vt and Ri aret 56 55

In 9enora1, orbits are not c~hwand tlu rolatianshlp of 9 to the The disturbing function acting on an/ giuen secondary is the Buler angles is much mor* complicated. sm of the rndlvilual disturbing functions duo to othu secondaries. Tho next step is to perform th spherical harmonic expansion of

These may be other satellites, planets, or wen the sun itielf if tho th Lagendre polynmical to relate the v.ctcrs ;l ad ;2 to a conrmon qlven secondary be n "noon" of a planet. Our prrimry concer? is to reference frame (lbid.; through their spharic.1 cfmrdinates (r,e,$) understand the hro-My satellitce.atellite remnances of Saturn in (fig. 2.2.1): whlch the tnprtant perturbations involve a single two-My interaction between the partners of the reaonanco. Tho critical devoloFrnent involves tb axpansion of A-' - I;, - ~~1-l. he where inverse separation, b-l in term of rl, ra, and , th hngl? im-n

A rl and r2, im bl

(2.2.1) and the associated Legendre function is defined by

(Jackson, 1962,~.62), where r, is the greater and r< is the lessor C) of rl and r2. If thorbits are coplanar, Q is the diffuence in + + true logitudes of r1 and r2 whus tho.. supwrrript indicates the sunphx conjugate ovation. The spherical harmonics can be related to tho Ebler angles [n,I,f t w) (2.2.2a) 3- L1 - L2 through an explicit sxp.r.#ion in which tho u:gonometric relations

betwoen (e,+) and (n,I,f t w) are utilized (aula, 1966, pp. 30-35).

Another approach is to use the group properties of thq spherical L I r + w t n - f t O. b) harmonics undor rotitions (Iszak, 1964). The results are: if the orbits are a180 circulsr, thon @ reduces to @ - A1 - X2, where A is called the %man longitudt" and is defined by 51

Y b)

X ard k is sdover all non-negative facterials (cf. Allan, 1967).

The incllmtion function P- has M haportant -try property which relates the coefficiante of angles that differ only in sign:

Thereforc, the coefficients of tho angle ((1- Zp) (f + w) + dtl and its negative aro ieentical except for sign. Since (2.1.4a) involves products products of spherical harmonice with the - L and m, the sign of the product of coefficient8 will be the .same for angles which differ only in sign. The baaic properties of Pw(I) for null I are 59 60

~hofiwA1 expansion relates tha vue anaarly f and th distsnco r to tho mean a-ly M. From equations (2.1.1), (2.2.4a), and

(2.2.91) we see that the hurtion which mat be sxpandd is rs {;;I slcp i (tf)). Mnsen's coefficients are defined by the relatton:

This expanlion can be reducod to a cosine series frmn th. 8FmoUy

relations (2.2.7, 2.1.9). The axpansion oi fk+r.sm'acoefficients in a power series in e is The agnc.1on of the indirect pare of R is, frcm (2.1.81, fairly ~ClpliC6tadand can be found in Pluumer (1W. p. 44). Table (4.2.1) gives a few typical values to lowest ordu in e, While more extensive tabulations are published by Cayley (1961). The basic properties of xu te) are 9, t

Using these axpuuiom (1.2.1. 2.2.4, 2.2.%, 2.2.8) we find tho direct part of the disturbitq function R: (2.1.8) to be 61 62

the secular term of the disturbiM fmction is a plymanid series In power0 of a, I2 and e', and 'chis term haa no leading factors of e or I. 1 Obviously, the axpansion of R2 is a very mplicatd function

of the orbital parameter#, and wuld not be VCP useful if it wcre

mt the case that the perturbations produced by the disturbing

body are sinall, and, :urthermore, that the major portion of the

vufation of the factor multiplying the cosine function is

determined by the leading factors of e and I, as denwnstrated in

(2.6.1.2).

The expansion as outlined does have one defect. For the d< satellits-satellite interaction (or planet-planet) the ratio (-) a> is not much different from one. Instead of axpanding A-' uping (2.2.1), we can srpand it directly in a coJino series in 3. The The rectrlction on the sums in (2.2.13) are that L - Zp, m and q are reeult of this operation is constant (i.e. 0 io fixed). The leading terra ia of -1PlP2P142 oMer ohin, where is the smallest value of L ccnsistent ria

the arqwnent of the cosine and the range of values for P and p. Z and the Laplace coefficient b:,2(o) is obtained from the integral forms in c which contain e or I: are at least of order e' or 1

smaller than the leading tams which only contafn factors $1 a.

The important point, which will be dewmstrated later (Z.6.10,11),

is that any variation in C (due to a variation of the orbital b) parameters) is of or 1'1 SUmllar than the correspoiding Oh2 The drawback with this approach Is that 5 is 6 cmp1icat.d variation of the leading factors. This Implies thnt th. function of the Euler angles, and o is a function of the mesa variation of C with respect to x can be neqler '-. r ...I .:r.?y, amlies. Subsequent e*rpsnsions which reduce the disturbing 63 64

functions to a function of the ia,e,ft X,Q,n) comprise a Tha serier expanelon of bzb), which can b. used to m-rically very todioua Lxcrciae. aa anyone familiar with tho topic -11 kwn evaluate these cwfficients, is given by

(cf. Plunner, 1960, pp. 133-48). Ths resulting expression is formrlly b) identical to tho axpansion given by (2.2.12,13) in that tho leading

factor. of 0 and I in any given tam ?an bo factored out ad the 2 where the function F(B,E + j, j + lr a ) 1s the hypery.ometric series reruining sum of term can be 1- trqethor as the cooffjcient c.

The only difference would be that C is mt a pwer suies in a, 1'

and e', but a series in which the sum over a is exprossad as (cf. Plrrmner, 1960, p. 158). functions of hplace coeff1c:ents. The important point to enphasizo is that the formal sxpansions The first approach outllnd is umally rosorvd for the of tho satellite-satellite interaction, the luw-planetary intor- expansion ci the mn-rphuical goopotential acting on an artificial action, and tho interaction of an artlficial satellite vlth a Mn- ruteliite (cf. Allan, 1967). The ratio enters in tho expansion rpherical geopvtential lead to similar toms and that the behavlor of whue R is the eL-th's mean radius and a is the sainajor rutia of these systcma near or in a "resoMnce" where one of these term the satollrce, but Ch problo of convergence in powors of 1s dmiMtes the disturbing function is essentially the same. avoided kause of the furrtional d.p.ndsnc0 on the appropriate The noxt serios of ooerations Cnvolvos the introductfcn of a multipole -ant ir each term of the disturbing function. Rapid Hamiltonian whoas variables are functions of the orbital alamsnts mnvergmnce of tna axpansion is obtained, because the multipole rather than of t:14 coordinates and IPcrmentA. E'venturlly thfs -nts docreas0 rapidly with tha ordor of the -ne. Haniltoni6h 1s rduced to a single degree of freedom and a consrant A mixcure of the two procdurem can be appliod to tho of thm motion for the tido-free C~EO. To aid this process, let's rxp.nsion of tho disturbcng function of a planat actinq on the Dwn, discurs the various types of terms in R12 and the different types of with a definite reduction in do amount of labor uausily requited refaonnnco variables which can occur. (&m,1960, p 252). Higher order Laplace coeeficienta appear in tho expansion and are definod by

:1.2.15a) 65 66

the priod of disturbed or disturbing bodies.

2.3 aASSIp~TI~ b) Long Period Terms. Those tewwith arguments which do not ccntain 1, but do contain S or n, are in this category. he The art of classification is to impoee en order on a hadgepodge largest term in this class have hm porers of e or I in the of msterial, be it physical objects or ideas or terms of a disturb- coefficient of the cosine. ing function, using criteria which suggest a aeaninsful way of c) Secular Terms. The secular terns are those for rhich 0 IO. thinking about that material. Tho :::st schame presented classifies Only even powers of the e and I occur as factors. If the these tenns using the relative corresponding to the cosine HamAltonian is reduced t4 terns of this class, then the arqrmcnt as the principal critd-ion. Its purpose is to suggest action variabl~s(defined later) beccae constants and the which kinds of toms can be hanjled using standord perturbatii.I angle variables are linear functions of the time. technioues and which cannot. "hose terms which cannot be removed Thare are still terms whose cosine argrrment is not some multiple of ma? be ignorcd if thcy satisfy thn basic criterion that their the resonant angle and which do rot belona ta any of the classes juet corresponding coefficient ~f the cosine is mch muller thal: the defined but have Ionq periods. They are of two types; coe'ticicnt of thgiven 'rwwnant' tsrm. Otherwise tha system a) Those terns wltich "abet" satisfy the cmensurability cw.wt bo approxiaatd by a Hamiltonian having one degree of freedop. condition jlnl + j2n2 5 0, but which have difforent jl,j2 Incidently, a 'reso~woterm" is a tom 17 the expansion of R whose than the re-mance variable. If the integer pair is much cosine argument is nearly constant due to special valuer of the larger than the >es>nantpair, then its coefficient will be orbital anqular pdraaetercr. This term I5 distinct fza "secular" much crmcrller than the resonant coefficient, and for th16 terms vhose cosine arguments aro identically zero. We shall define reason is ignored. If this is not the case, these terns a "resonance variable' as *he argument of axh a resonant term. must be directly compared with the term (or terms) contafning The Hamiltonian contains an infinite number of terms which my the resonance variable. or may not affect the rewmnce. A useful classification in treat- b) If an angle differ6 from the remnant angle by a function of ing the "non-resonant'! t- is the following: the slow angle variables 0 or 0, then the period of that a) short Peria3 Tarms, These are tonus which explfcitly angle will be of the order of the slow variables near contain A and have peridle on tho order of lor lee. than) reconance. They can be neglected if either the coefficient 67 6a

of nrh tam# is mch ~7~118~tlw the remnant term cv if d) sdixed I typar q1 - q2 08 0 - 1 (I - 2pl) (fl + w1) - the perid of the slow variable ir relatively fast ccapared (I - 2r,) (f2 + + m(n, - n2)). Thc only auample is the to the libration period of the resonance variable. Miman-Tethys remnance (2XMi - 4ATe + si + nTe). Therqare seveal types of rasonance angles which sball La 0th- types are the mixed e-type and lnLxd type whose definitions discussed in detail. Below is a list of these types, aloccl with a are obvious. NO rati:-qlly occurring examples belong to either tpe. lint of rpecific exemples.

8) Purely SynCdiC: 1 - 2F1,, mi %,2 - O' 4 - n(Al - X2). "'tm hojan asterbid. librate about tJm Laqra73ian triangular points of Jupitar with a resonance

angle of the abovd type (Brown and Shook, l%4, ch. 9).

Cnfortumtuly, t% perturbation =xpansion of the type

developed n (2.1) cannot be used for the Trojan rebQnance problem.

b) s-1- (I TYPOI e - 2~~,~- mr q2 - or 4 - m(X1 - A2) + qp1. Most known resonances fall in,+ this clasa. The mceiadas-Dione (Am - 21Di + GDi), Ths Titan-Hyperion MAHy - 3ATi - awl, and the ~tuna-Pluto

(3Ap 2An J j resonances are well-known axamples. The - - P leading terms in these tmes of resonawes contain a fact- of elp1. AC~IM~~Y,the class of obrrerved e-types is restricted to t'a Iql - 1 case. C) si~nphI tyy: q1 - q2 - 01 I - 2p2 - mt 4 - fe - 2P1) (fl + MI) + mill - maax'. The?, are no krwrvn examples of this type. 70

(2.4.2) The P?iltonian which describes the motisn of a disturbed body can be written as

and we shall adopt tho conventjon that yo rq-laces u0 + pi in (2.4.1).

The H-J equations in these variables are where no 18 the Hamiltonian of the unputurbsd two-body .?ystav of Fhryand secondrry. (Note again that Ho is the negative of its urns1 counterpart in ordinary mechanics. Of course, the sign has no (2.4.3) significance except at trrdition.) The canonical elements of H ,.re the conjugata mDOenta and position coordmates. H is not a constant of the notion sirre the mordimtea of the dhturbing body are Th. above set has the acqantngs that Z is an approximate consbut of wxrnined ax.Aicitly in R. since H supposedly dominatrs R, the the amtior *w -type re-manc~, and l' is a Cons*ant for an I-t m. mtion of the disturbed body can be derrihd in tenon of the Also, !r ,rd - sm!l, then variatron of an instantMeou8 ellipse vhich detemaines its position and velortty. The simplest method for dexivim a canonical set of conjugate act'on and angle varialles is a nethod involvinj the or bo-h I' and 2 are very small quantities campared to L. Hamilton-Jacob1 equation (cf. Appendix A'. If R I 0, then all of th- variables except A are constants of Several sets or canonicr; variables have bran dp-ivd dX motion. Prcdn (2.2.3) and (2.4.1-3). the equation for is (Wagihara, 1970, pp. 526-555). Tho set chosen for this diKuslrion dA '"0 - - n I constant (2 4.5) is k- a8 the &ifid DdalIIley VUidbleS fL,r,Z ',&n>. Th. dt -aL angle variables bve already ken defined. The action variables are and 1 is a linear function of tkae. related to tho Kep1rui.m elcments by: The disturbing function co4:tains many terms other than +die

rewmnt :ern. It in important that tnrir effect on c3e resoranre 71

.-- 13 74 o:iniruted Ire, W.0 Acolltonhn wdor by order in tarma of a "aull" pair of conjuqnt. wt&n and truiab1.s of th fkrt My. Thr mansion jarmator [Brown nn8 Shook, 1964, ob. 6). "he coupling above dirturbinq tunatinn can br toraully s.purtd in- tw partar wfll to shovn to raullmr a factor of M -noion parumtor, ta by R - R, + Re. c.5.2) altbugh the parameter my bo dif%r.nt frm t-t which mrwa Rr ah11 inoluda tho rhwt puicdic tern and Re all tho rrudnlnq directly In t)u tw-body disturbing htion. Wut, th conditione toma. The Hamiltmian ir thartlore [2.4.1)1 whi:h deearD1ne *hather the procehra [:an be ap&.lId to tha long mid tarnu at0 n-mtd. At :his stage the Hamiltonian rill H + Ho(Ll + R, + I_ . (3.5.3) hwc been r.duc& to the beewar :ems, Lema which cvntain a Bonnally, R. can bm writton as nrlciplo or the r4270nawo angle, ud other t6.p. which canmt ba o1ircMtaf by a parfur&tlon expuion haaur t)ioir cosine

~9unmtaare vuy functlona tlpl.. BlMllyr M explicit slow of -. c tculation must 30 mado to see whether the cooiikimt of the resormce tor7 is much larqor thn the r.nulniw tonus. Xi tho 6, - jA + j'A' 4 {function of nodor and lotmitide or pricontu). b) litta umt.' effrt an the pcrrtrsrn oC tho resomnoo is ccmparatinly Tho WAII longltudr X in allo .rqurl to (2.1.6,2.2.3)r mdL, then LSey ?an be lgnord.

m raduso t?le prardure to ito s#sentialmr tho distrcbing fwrtion ot first kdy will bo uooentrrlly rwtrictad to th a Xt Colla that the mwn longltcda is an explicit Cuncticm of ita sinlio--body potmtiai of tb ion (rmn 2.2.12) aonjugato action mriable L. ti, x~kethis pclnt now to &wid confusfon later on.

3%. moxt #top is ta mako a H.rlIton-Jnccbi tranafomn.;lon on the and other effects such ma +hse due to the preronso of othr old Hamiltonian HfJ,w). with thc dmmnd thirt tho new HMi1xnf.n satellites, othu planeta. the sun, and 0.0prbary plmot's gmmratd, H(j,h, doon not contain the mhort period terms to firrt cblatoness will bo ignomd. The .ubscript- nobtion is here replaad odor. Tho transtornution frm tJw old oat of action-angli by Frfnod notation, where wprlnd ancl primd variables rofw to the variables can bo ucoay~lishedwith a generat?ng furctbn first and re-and tudier rgspectively. 12,~)is rhorthand for any ShmJ,; Mve no torplioit thDdopanderre, can be replaced by 6 In

the H-J aquation. Again, th4 mw trannfomtion cau ba pnifomped or.

8' with BhllU r8INltl.

Neither in II iirr In H' door the tL.. vdiable literally occur,

yet both H.nilto.~irruaSI. tim dop.ndont ~OCOUBO cf the aFpta:mce of

the "extunal varlnblen' belonging to their rerpr*tlvr psrtr.ers. A sialtaclwun +:ransfomtion will be wade on Ih. action-angle Explicitly, we Kind tht the equation EK mction is: vuiaules of the 8wwd body and the correspundir.g Rmiltonian H', oi H where R*, H'. fil ad 3. are dudid by rrlations similar to

(2.5.1-71. .a~tiwi~~GLf,iij J*w') in the above equation is a function

OK thold vub~blsshlongring ta the primd mtollite, &entually the rrght hnd ,sid¶ will ?n uxpuuhd In terms of the now variables of ho:h .;.telli:er.

:f +%e prrwbattcns dee te *hshort period terms inn. are mal.., tMn cte old Huailtor.ian can k expanded in the di:fc-en:s batweer thold d new Varlebler. A now gnuatlng The iAplicatlon OK +his last aquatien :2.5.10) 1s that the .?artLal Irmct-on, 9, can he ddi& which diKKrrr itan by the identity eaderivative ocsurriny in uch H-J nguatiln acts tua on th4 trdndomation explicit Cina dep.ndencm which my occur in the rerpucUvo

;-J;+s. 11.5.8) ransratinq funorions and on thQ "exterwl V~riAblcS"which occur

therein. cowme, s and 8' can c'loren ow% thnt Then, by (2.5.4), S is directly relatad to the diffnencu 65, 6w OK bo th* tine variable not occur rrplicitly Ln sitter qaneratlng fu-ctlon. defxrd doer noreeore 75

4.5.12)

(2.5.17) 79 BO

61 My contain tam# of O(Y2 ) like those in Rc. But if w axpudd uhee voc;.*) - j; ij'a', tho equations for fQ,6w) in t.raj of the now oarlablea, rn would the? (2.5.151 vaatlcbs ldentically. find, for oumplo, that such productr 8s 65- 6 '8 in tho expandod -are appmws to be an inconsirtcncg in that 8 drpnd~on tho 6n2 aquation for 6w would alm lead to a racc.4 orEw contributlon tc old action varia.ala J while, on tho right hand aide of (2.3.171, th~8P~~lar and long perid t.rms contained in di0 i Kc). the c~fflclcntof the cosine 1s axplicitly dependent on though 3 Tho new Hanlltonian il ir, to Ob ), ,J(n,n'). I? rullty, this only Wk.6 tha rmmtton batweon 8 ane 61 mro :m,~llcated. Explicitly (2.5.9b)r

vh8re the second order raralndur eRc And 6R, Are glven by O.S.ldb,c), and daHO, b@ are

W.c, tho unhrrod variablar are roplacd bf Weir burad cmntu-

parts -41 each of the raulnderr to obtaln an upreralon for

acnult. to Wu5.

Some elauntary conclusions CUI be dram about the qualitative

nature of these rrauindet6. each of thnm tams 1. of O(;," or y'c)

Obvkurly, the oquivalcnt mecond order puturbations of the primed W $*e thst the diffuenze brtrson f. and E in t!!o parating 2 wtulllf. due to the unprlad are alno ct O(r or U'JJ). XI thaw fur-tioi i. a sccond odor efioct. II, any L'P.86, tho diffamen 6J, aontrlbutiona am to k n~lipllde,then both u 8.9 )I' nult k 6u can bo derlval in an ertFtely consirtent mnner frcw tho generat- rdatively -11 cmparcd to p3 at. We shsll find in rertion 2.7 ing function qiver. by (2.5.171. %e o.4~apparent dravbsek in that 81 82 thnt In th. ~bit-~bit@le, trOU ~JIR of Cbv2) rill b. of tln hrncketd set of term8 to nSg1occ.d in order to rJuce chr t'srlltcnian to one dlrPeneiona1 form. the recr;lar an8 fnaonnnt pnrts of ii is .qual to ttw iol1wing, to

Thus, the second oear rwpling la a oqaratively minor effect ic o#~) &!isproblem.

I:oxt, ths raindm 6'~~11rmlves pzocucta UP short prriod term (2.5.20) in 4i rhich csn be r.rexpressd with a s1~1Seconin- fcrtor. Ita nrqmeat will involve the SM an3 diff.rm:e of the angha which occur in a given Iroiuct. If the two anglnr are Identical, the Tha first set ot tern on each ~fdoU.0. Egand 314agxw since will pr&u-.c: a socord order contrihzion to th. s.cular pnr+ of the only step hak4n tn rq2lace ty it. equivalent 6w (I.S.9). R. zf ?m sa or ditfezetae batweon two @wrt period terms in a The next tvo nets of terrr un each 2ie.e fa3 bo shown to bgreo to mlti310 of the IC.Y)MIKO aqwcnt, 'hen L% .hove proclucos a -1s' -. . O(W'/~) b! first rep1acir.g P; by I- 5j). Hose It la und4r.U rarvd order contribution to tho rasonence tern in ii. e, a siani1.r thnt t3e partial with rerpar:t to t ~t6only on t3u i and 1' ljne oC rnannlrg, CRc contarns only short period terms, since 6W. deprndcncer ocnuring In thr co.;lne nryunwt of aach tum. Tnking LT, etc., are a11 &art periodic, diln ii contninr on11 long )raid the SKOnd de: 01' tern8 on the left hard side, lst'l ZWitr it as md remnarse term&. On tha other honc, the rmt\irdus dRs and followal 6(g.).re mmrh Lika a2!ro, altbuch rhsa rruihiaru ccntain many mre tuau. Close i-tion of e(%at reveal# ehit only the products The product ea3 be axprenad as a coslw naries. So.. of thoro toms will presumably contaln cosine argur.int. idantisally equnl to aero U.e. secular) or aqua1 to tha remonnnce variible. mntnin recules nnd long pori& toms. Therefore, tha mecod order Th. partial with cespct to t uf the secular part fa idontically contribution to the secular adresonant tenw in ii are oonteid in imro. Thc effect of the partial on tha re8onnnCe tern in to nultiply It by :jn + 5'n'), Pur rommnce this frfquence ir of

O(W1/2), while thc pruduct of the disturbing functic.ns I# tl Oh2). 83 a4

differences SJ, ev, e*-, ate waily duived from the gmnerating

iunctioni fiMlly, t);e rathod tend. to cmphaaite th. point thet tho real pr0bl.a. m must 1W1ve is the long tom behavior of ii.

'bt meo&rations can k applied to L\e tbird mrt of term on each Methods for the o:ini~tion of the long pOtiod tonal for vfith ride in (2.5.20) to show that they a160 avee. F?ia socond j - j' are LOBS tiatlrbcrary. If tke motion of the %low" angle fornnr1a:ion is wed in sctlon 4.3 to find +-he se-%nd ordu varisblrs G,O am nearly linear -apt for a perfodlc perturbotion centribution causd by the ldlrect action of tho sun. due to the: iong 1mrfodlc tum in Rc, then this peridic mtlon of

"hue aro a crupls of drawbacks tc tha procedure outlined. 6) and n ank rwvd order by order using a rpodiCied version of

First, if M wi#h to dctemine th0 orbital 01-nta affiU?.te to th. procedure already dwelopd. Ttr first step im to separate Rc reccnd order. then a somnd H-J urnsfomtion mmt b. nudo on k, into three part.. 1) 8ecUlUr 2) 1or.g periodic? and 31 other tea. +rrr.scominp th. vrridles tram 5, G, e=., te 5. E o eliminate th. including threwMnc* tam. wcond order short praod toms in ii. Souord, the secular and renoMnt

:arms aze not remtricted te thHaPLltc?ian ii, but ala0 occur In tho Taking ou cue Eta (2.5.15), pn cbr. construct a pnrersting dtlterences CJ end 6w. This mans that in ccorpsrfns the obaenrrl. function SLp vbich will didMte, to first order, thse long long period bs!lAViOr of L Wltl 1- thOOr*uCAl rotion CakU1At.d to as period terms: s.cod order, w must inclu3e th. second order contribution in

Explicitly:

(Low miod) I fa (long motion of L

91s old Hamiltonian is, 01 before, urpr4d.d in the perturbations.

and recond order coupling my occur which affect. the resonance

tom. If all skrt periodic terms in H have been prwicusly 85 dfJniK¶t.d, and if Xrr only contains remmnt w, then the cougl- inq of the long period temm will produce hither malar or other long perid terms. Unless the remnnnce variable is also like ths long peridd t9ms being dUnated (j - j' - 01, the sacond ordor coupling will only affect tJu secular psrt 02 the Xari1toni.n.

It mist Be -hasized that if thq above ptCeeddute is to wrk, then each u nlst k 1) 1y3~ero.and 2) larqe enough such &Atthe

.perturbation -ion convarges. Instead of e, the effective expansion p.rmcte.r for the long puioct tern is 3, and it is t~s parawtar whi:h mat ba mall. The secular tents in the two-bdy Interaction usually A0 not satisfy this criterion. For -le, the mtions of tho planetary parlhelioa aru? rdes (which in sans cases 5 cumt be dofrned:) have prids 10 lsrger than the orbital period

[Prowar and Clanence, 1%1h, p. 46). Still, th procedure can be a--lied to at hast ttm of the eatellite-satellite resonmess of

Aturn. The reaeon is thst the canblned perturbatins caused by the p?anst's -,blawncss, the sun, and the lrgest satellite Titan, lead to a relacively Large notia-. in tha perkenter and node of the inner 2 mtallire wit% periods 10 larger than their orbit p.rLo8r (Joifreyr, 1953). [t stnuld be pointed aut that tha least satisfactory case is tlu Ti'aa-Hypwion resonar.ce. Ths progrde motlon of the pericenter of Titan is on3y abDut 0.5*/ye~, while 2sorbital period is a?proximately 15 days. "tie poricentcrr =tior. of mionio actually retrograde an6 caused entxely by the bpressed reeoiw.ce. In thio cam, we must sppoal to the fact that the corfticient of the long

.. 88

there my gli8t term8 which nearly mtirfy t!m -=ability

relation (1.1) for A different set of integers i5,5') than occurs in 0. Thbse vary long period term bre nqfected in our approximation.

m bve raeched the stage where each limiltonian tht de8o+ibe8 Implicit in the expansion criteria is the fact that the tva orbits cannot interuect very close approachsu. Then the the wtton of *.e appropriate partner of the rcsomnce can be or nuke magnitude explicitly reduced to one degree of freed.u. The two tide-free of the perturbation acting on rither ustellite can be cmparable to Rvxiltonians which govern the mdon of the first end mcond partners that or the Frimary. take tha forms We can see that the oca. 'ence of the angle variables in each Hamiltonian is restricted to the combination of enplea vhlch

cauprisee 0. The motion of tho action variables ie detrrmined by the

partial derivatives (acting cn the appropriate Hamiltonian) wit . reipect to their *:orresponding conjugate angle varlablas. Therefore their eqwtiona of motion mnt be proportional. Ex?licitly. 'the uh.re the aqle 4 is +Tiations of motion of *he firut psrtner are

s! = :A + j'h' + kZj + k'a' + iQ + i*p., win, pccnnd and unprimd variables refer to the first and 8eu0nd partners rssFectlvely.) Recall that the two-My interaction m8t scrti.fr tile criteria that it can be axpandd in pornrm of o or hplace coefficiests, aad that the hypothotical reaomnce dominates the nution mar a long time scale. This mans that all short period term. j 0. 3' f 0) aid long period tenon (j - j' - 0) are clearly, the varlat$ons of the action variable8 are proportional to as-1 negligible and that any tenam in the disturbing function wch other. Therefore, a nwv-iable x can be defined wNch which tuy have very long per+oda Wst have ?d.atiVdy Cmull sJmultaneou8ly eatisfie8 tba relations cceffit:ie?ts ccsnpared to those for U.e variable 0. For epuaple,

.. I I e9

L = jx + Lo (2.6.4; 7 = !cx + r # 0 (2.6.1) 5=iX+Z . 0- , whu. 0 - y + Y'. Still, neither H nor H' is a csnstant of the motion. Th. roason is tmt the above Huniltonians still cmtain an explicit Limo dopendemo because of the appoJrance of, say, the second partner's (2.6.2) variablss in ths first partner's Hamiltonian. Ttw, next step is to

establirh scme connection betwoen the two sets of varirbles .and

find a aow Wmiltunian which la (1 function of a single set of

Conjugate VAriAbles (X,@). To do t-IiS, WQ Y3U.t C-.?0fder the fOIlS Of

(2.6.3) tho coefficients 5 and $ and determtne how they AIC reiatd. The secular prt in 0Ach Hamiltonian corresponds to the 1 - 0 Tho tl~wangle variable8 oam bo redwed to o!n alr. Dmfim y tern and inrludea H0. Thb coofficr.nts A,, and A: CAn bo fOraSlly *S separated into two parts p = jh t k3+ in; Ao(X,X') - S(x) t V(X,X') * equcltton for im AA(X'rX) S'(x') + .*'(X',X).

Th. function s(x) contains no, the secular tOZmS dun to the

oblatancss of the FrLmary, and the secult~tenon due to a host of

other interactions, but not the secular part due to its remnmce

gartner. The s(llw holds for s'b'). The mixed functions v(c,c')

and (v'(x',x) are tho secular parts dorived from interaction with 91 92

thok roepectiva partners, and aro proportional to fkst Mor. Tkis proportionel to first order. Tht im,

folla fron the fact that thb dkect parts of the disturbing

funceans are proportional, and the indirect parts, to firat order, hve no secular term (2.1.k). BY inspetion of fZ.1.Bc). we find 8lncs th coefficient6 are proportional, the qmtioru of motion for that v(x,x') is related to v'k',x) by x and x' mnt alw be prDportiomlt

V' (X'A - + V(X,X'). t2.6.3)

<'kmtpsringsLp.ilor coefficients A.(X,X') and A;Lx'rX)r ys find tbt Y they will a1.o be propartlonal to first order, provide that the romonnnc~variable + is mt contained in tho indirect put of the disturhlng function. Prm (Z.Z.lO), the angles contaimd Jn th _. . indirut ptt are restricted to thm following sat,

~h.quation of motion for #(- Y + Y')

rms into the abore class of rewtmnce variables, d.p.admt OIL -h Miract part. Thin roquiros a slightly difr'sront approxiraatbn to

reduce the Ifmiltoninn to one dimensional fm. P*cbpt for cases

-h a8 this, hanv8?, lh mffiCimt6 A hex') Md A;,(X',X) mb I Thb mno of tha partial duivativbm with teapbct to x and ii em k 94 93

tern of a sbple e-type rew-e :; a 6 jX + jgA + kD). ran

f2,2.12), AILX) ia:

~h.function ~(n)in a polynmial seien in n, I' and e*, and toma

which contain fat* -s of e' or 1' are at least of order e' or I'

UOAller than the leading terms wh:ch contain factors of a. After

exp~d!.tyC(x1 and e(x) to first order in x, we have

the ordinary tinw dorivativa of G(xr6) nnd relating that -.A sum of partials. Dcplicitly, (2.6.111

The partials with respect to L, L' and P acting on each tom in E A slightly different spproxi~tionah11 mado to establish k a reduce thoae terms which depend on L, L' or l' by A factor of L, L' proportionality relationahip kt%oenA (x,x') and A'(%',%) for &e Y Y or l', resgectively. Since the largest tenna which depend cn r RPDi CAS*. Pran (2.2.10) And (2.2.11) We See that each part of multiply A golynolllhl in a of the same order AS the leading term, we facbrs, c 10) each disturbing function contain8 the leading dwpending see that each of the partials of C 2x0 of O(-). mat in the lkl LO eccentxicities and inclinationa W. have on the of the two bodies. expansion of e(O), the coofficiant of t!a ,!near tam in x is of already noted that -11 wwntricities or inclinations 2 for the o(+). Thus the variation of c(x) is of o(e ) smaller than the 0 Ikl *acticul iluctuation in ~(x)ia of 0(e2 1') wller thsn in or correspording .rariation of e Lx), hplyim that this leadiq r(x) or zb), respctively. This means that those weleading factor prdkminately detonaines the variation of AILX) with x. It

fACtorS dominate the total fluctuation in A (x). Y is a160 clear that the variation in x ot the le3diF factor9 in A (XI ve shall -licitly dmOnStrAt0 this assnrtion for the y 1 22 - determine the variation of A LX) with x to Ofe or I ) for the =re Y 95 ?6

see, pzopoZtioMlity conatant differ. gamsal wue. mike in the prwiOuS tha for each value of y. Th. principal sreeptionn for- tlm two Myinteraction are th. Since we have alxeady presumed that e and I are mil, tho y- 2 tern in sach Hamiltonian of ~(e~k~e~~k'!~~i~~~~i~~) synodic-type remnances. for which tha axpursion of th8 disturbing is

muller than the y 1 tea. If *I) I the f-nctlon tn powra of D is invalid anyny. or a synodic rewnnnce - rict miltmians to the Y 1 term, then the action variables x' and x are again proimrtional to occur, X i X', oc the satellites must k at approximately the same - Pactor Still, thie not enough, because the nhed radius. Ihua the whole concept of a resonnnce variable -rhlsh can bY the K$. is secular tsm v(x,x') a evolve alody frcsl rotation into libation is inapplicable to this in li ha. different proportionality constant case. Hanrar, synodic resonances involving three or more bodiee than the y - 1 tarm (cf. (2.6.5). (2.6.10)). It the SYS.~ is to b. dOSCribed by a ona dimensional Hamiltonian, then the mFxd secular CUI MtiBfy this evolutionary deacription. tern mnt satisfy one of the foliowing ctiteriar 1) The nixed torr The coafficimts Ayk,x') and A;(X',x) w1 b fodly v(x,x') is negligible conpsred to the umixed aacular tams and can -rated into their direct and indirect part.: -. ignored. 2) ~hsproportionality factors+- ia either vary largo

OT very small, such that the effect of one of tha eatelllte. on the

othdr is negligible. Of COUIB~, this bplics that the fluckwtions

in x or x' ate negligible (if K+ is 1x9. or onull, respectively1 U and the one dimensional form i8 inmediately obtained. It should be 0.5; : Lo pointed out that rho Enceladas Dione case satisfies the first

mentioned criterion. If this is tha case, then this HMilanian is

and the equation. of mution a-e l?~~.lore,th8 relation between A bx') and A'(x',X) io I Y

(2.6.12) 97

?a shall we the above mmiltonlan am an appraizmf.ion to the motion of a11 two--hdy reaonancea, whether or not hie indirect. put contributes. This is not tha ud of approximations. several more T)u noxt 6ts.p is to skr* that the secular tam in (2.6.13) can 8tep murt'k before Hamiltonian reuced to the taken the &eve is k uxpnnded in a rapidly decreasing polynomial series in x to J(xz),

3e.tr.d form. wi'hout increas!ng the order oL aFpro..ation already established

(roughly el. iqain, the secular part of the dlsturbinJ function E can be fomdly ueparated into ttm distikc piecesr ,,2 , , .)2

f8C.JlAr part) -n - - + E (2.' .1) - ZL~)0 -2~'(x12 + 'r (x).

~hrfirst two terms are the zero-order part of ii. Rcall t rt uo

and Y' are shorthand for uo + 11 and yo + v', respctive'j. The

paraneter K equals one unr*:ss the indirect prt contairn a con-

tribution to the reeonnncs. In that case K is given b.1 (2.6.10).

Thu function s (x) formall:, repreqants the sum of mecular P. perturbstions acting on both prtners of the re%na:.ce. There is no

problem expanding the first two terms, as lon?. a- zhe fractional

fluctustron in L9x) and L' (x) is small. k'hether s (XIcan ka P JX~EI+& is less certain. Recall that the secular part of the

satellite-satellite disturbing function vas a polynomial sarios in 22 a, e , e' , and I?', or, equivalently, a Folyncwial sariei m L,

hnd Z (cf. 2.2.131. Part of the motion of 4 is derived from tht

partial derivativa ,uith respect tc x of s W, and the effect of P ti.is derivative is to rtluc~any given term in 6 (x) by a factor of P L r or 2 (' 5.11~). "ha reason that this is importar.t is that if 99 100

(x) woro to contain t-• rithtrotorm of, my, oo', than 9 thomo toma would provide rmlativoly lnr9. oot.trrbutiune to tho =Cion of

J or a' If 01th~'a or o' wro owli, rorpaotivoly. 1Bnr of thr pommlblo qravitatlmal porturbrtiona havo aooulu toms ol Cha rbovo typo. A :i:tor cZ 00' In alwyr amroclato3 with a low ported tom vhcmo costee aryYID*nt contain. IbI - &'I. Tho conclumlon w oan draw

1s Lh.f 1 fY' is nJt gwlltntivJy dlfforont fror the suo-ordmr P torma awl can b* mq1m:tod without muioumly aflwtinq thm mtlon ol either x or $. Thr only acra;tien la tho cam lor arujlor 4 which da not contiin Tho man lmgitlde 61 olthar pirtnor. For thw an#om, tho zero-order put im lndrpndart ol n (that la. j - j' - I), and tho co-itritutlon frm 8 (XI1s unlguoly important. Iqnorlnq ih;m P case, v. ilnd the expansion oC th. mwulw put to be, to a pa! &.~proximtion,

(2.7 .Pa rhus (1.'. 2.1.7, 2.4.2)

. *. 101 102

o(& in th. smluput ia .qukvalont ta nogwtinq t- ci

O(u6x), vhlch :s oL thsame ordor as tho approxiantion UtpenaC tn

the function blx). Note that tho .bovo H.rai1tori.n is idantlctl to

tbt o! g.endul'.m plua torquo ucopt that tho puunrtu c hero Is a constant arid nut a 1iM.r lunctiori of the thc, and tho cocificicmt

bfx) dopond0 OD the action, not tho tho. Ths next major exorcise Therefcce. +.-.e equations o! motion can bo returned to canonical foa is to include thu of!oct o! tho t:dor, and make tho prar.oter tFnn tv rep:a:irq by R*. men if A- in t-drpondont, a nm- dnpordont. First, lot's oxanin. tho above tim1;toniart for Uio linear tire E is defined, caiatonco and atahility of libration fontera.

(2.7.1) Tho utirtenee and location of Libration centors can bo drterslind

elthur rnalytlrally or graphicallv from Cm phatie-sraco cwos such trat tho aquation9 o! notion of x ad 9 aro uti11 Cbnbnicrtl and generat.od : tho tido-Crac MdItonian HL*,+). A typical ghaen-space derivJ1o €rm ttu iIamiltonian, A>. The form for tlr tide-frm cwo in shown in fig. 2.7.1. Tho asymwtry akut tho (-axir La Hat.nil'.cnicr is, rhcn, caused by the nomontm dopo~onco,of the potent-81-like 13 ti term I' Alo, ohiarvo that In figure 2.7.1 ".ora oxirt n':ablo Ilkration cwters

at 4 qual to ktn wen ard odd m.;tiples of n, ad is tho roralt of

the momontun depondonco of tho potnntlal term. By the way, ttls

c..!k , kind of beh8v.wr is not penoral a& doponds on the fcrn UC the momentum dopondonco ad tho vali o of the rrrmotus of tho systm.

In tho limit in which tho w rnhm er%nh:eo vanlrhs6, ftf 1s

roducd to the Hamiltonian o! b ssple pcniu1u.n. In thls caw, etis

libration canters ar; at olthor won zodJ muLtiples of w, de*p:;di.w

on the rlgn ot b.

Tho analytical definition of a lihatfon center 1s 68 Zo:.lonr

Glvan thara ulsra a point (x ,4 1 .such that ,+ is zoro for all PP PP 103 1CI

tinm, and if thee awimt alord carves about (x ,+ generated by PP H(x t 6x,+ + 60) in rogion aurroundlng the temt point, then P 0 tha+ point is a libratlon centor. Inmpoctlon of (:.6.3, 7.6.1Sb,c)

mhovm thnt xp can be met oqual to sero, arid 0 !.E v .hew or :!w. P - Next expand the Hamiltonian in a mall rwion aboct thr tort pinti

Thl# ryusdrrtw form ddinem A cmic rrotion (Oakley, 1949, p. 1091, 2 and at1 olllpee im dellned by tho condi:ion H c 0. the x4 - RnR,, For n-llh-nt.:on center, the above co.dltion rducer to

For t 1- In-crrter the equivalent condltlcn in

. L (1,bxs) U* (2.7.10)

IF IbwI lm lsmm than on., then k mrt bo ponitive for a w- libration center and nqatlve for a Iw ccntmr. Tho mign of t

c?e:ermInea thm "ncmal" Libration center. am it did fcr the rlaplc

porr?ulum erunpler dimcursed in (1 1). If I&I im qreater tlvn one,

and ha the oppomitu rlqn of b, then thr mystam ca? Iltrate tri both

ita wrml ad invwtod mmLtions. But if j hrr tha mami slqn IbM am b, thn nlethcx Ilbratic:, ccnbr ib :tablo. TbIa bnuaual

pb.or,tmenon in c6ur.d by the momntm dnjw*ndance el tho potential.

w* coindetsnnine which rnao~i~cmvariahrm can wctdblt oither tuc

rtahle libration centarm or perhapn now by expxrltly cqardirq tho 106 105 cwlficiwt, uoing (2.2.121. Wal: that u, 0 and I typo rooomcoo imeatipatd in (3.3). - the funation bCx) hns ledling or elx1 ard re.p.cti~v.ly. factern XI'', At thio point y. nuk. a kid dipmaoLon CO tho Cuhjrct of tides ccngulnq similar and 1 typo rom~~eafor wkich 3 - i (2,2.1:, and thoir oocular dlrct on tb motions of uullftrn. A procduro 2.6.1), wo find that the Hutlltonianr for rach +re frmurlly tho *me. ir than outlind which introducer tho tid.1 efloct in- tha tide-tree Thorelore, bk) nmod on:y bu axpudd lor a ohQ1lr s.t.lpo ?O.O!III~O. Llnl1tani.n jurt dtrvolopd. I?.* ro1a:i.m stability of the resomos :an bo drtcratnd as a fwtion 3:' k. The coofficionta ol rh. etprnaion art) (2.2.12, 2.4.4,

2.6.1) * 107 1%

a tido on thmons But th moon'. rotation is aynchromus w:*a its.

2.8 mm or TIDB -bill1 lun motlon, uul Uu "ra4ial" tide raid on thmen nly wakly affoctm ita orbit. Thia offoct shrll k diacurrd latw. lb understand rha wchrnics of tldom, 1.t') rostriat ourrohc~ Th bra4 on the noon duo tm tho tide ralmml on tho eartk no: Lo a spwlflc axmple, tho oarth-noon aystm. &cause th. wn1s a only alfects tha wan wtlon, but the occantrIcIty and 1ncllnrc:an ~s Ktnito dlstance Cran the earth, thor4 atlmtn a prdiant in tho 1.- wll. IC th. lunu orbit 1s orcsntrlr, thon tho torque on ttr mr forco at tha utondd miltion of tho mrth. Tho earth io shrt (and oarth) is stronger at perigao thn at apoqoo, causing Um orti? tp;ierical, ht mt p.rCeclly rigid or elastic. Thua tha urth'8 to boeccl. -14 occenVLc a# lt ucpandr. This posltlvr. ctunqr Ln ttri *hap. ic distorted rntu s toott.11-like object vhlrh attwta to occontricity doprrda 06 the Wtn's routlor1 hinp fagtor thrn the I01;w the -8's appzent motion. Th. llvxLNa rorponse to the maon'* orbit81 rstlon. The riqn of this eftact MI rwerfo4 distorting Corca lags khid tho 4ppli.d Cero., as with any osoillat- be oncat synchronour roution with rho -on's motton 1s Pchx-vad. Tht. ing mystas in which thoro xs frlctiorsl loa. and in which +he Coro- "averagd" torquo aortal on tho nnon todr to k nc.mal to chi Inq Crwoncy is auch lora thn tho natural froquwtcy. 'Fh. nugnitudm uliptlc plan. and trndr CO increarr tho normal conpnont to t!ro of tho anergy disaiprtion my dopond on tb rolatlve diCLuonca rcliptlc plan. of tho lunar orbltal momntwn, withut chnqliv tho b.twen the Croquencios am8ociated with the applid Corco and the colaponont it tha plane. Thrrrfoo-• the lunar orbit A# driven tnward. MCWal frrquoffiles oC th. aCClctd Ldly In a coapliortd my ($aula, colncidmca wlth tha wllptlc plan., and tha 1ua-r nrblUl inc,llu- 1964). Tho quantitative .;Cat 8C s!dal frlution on tho urth'r tion io docreand. The reaultm oC a doullod caicblatlon (KI'.:~, rotrtioi is uncertsio. but thr guaL1::ative dCwt ia mll understood

196.0 show tkat tho fractional rat*. ot chanqe in a, a, and ! (IC* OY (Munk and M~(acW-.alC. 19t0, p. 134) tho saw mpnitude. k%plicitly, At pramnt, ttt oa t'i'a t,iUt .o) la ruDh famtu thm tho mn'a orbital mot-en, 8s: 12% '?..II.:* ~1 cr.rld ahed ol tha Urth- won uim. 81nc.c ..h ..CIL. snt rn '=t*r lr. r(lr yravltatianal for.3. io The nwtical cooff!clents in (?.8.1) wrn derived nndor the .yra~~tricalong tb) irrth-nee? uis, II rttarptn to roalign t!m- aamm@tion that Lh, Individual pha~a1-a asoociato4 w!th uch dIsrort.6 earth alony tht Uli, me rorulting twpu0 drspina %ha ftrquency In the tidal dlet~rbhppotonh.ia1 bve thv v(i1Uo. urch and accelerates t!m mlon in its orbit. Tho oorth alw ratson aaum 109 110

Sin:. torbitdcruroe, UI~amlrYjor uir mst drcroreo. The

rolation bomen tho tldnl chnpo 111 the ~unbrscent, lcity ad tht tn the smhjar axis can bo dorivd frm tho conrtercy of the lunar ortiltal mntm Xi

Taklnp tho tLRr derivatlvo of both 8:daR UO ClndI

11 Gt dt

*era TI rofers to tho rdlsl tldo conpnont. Ihr rtant plrrr

to rcoqnim is that tho induced chbnga ln is oppoiito t3 that dah awmd by the earth tidm. haa tho functional form aa -h.

earth tldn, cnc8pt tlrt it ia aultiplid by a factor of th, rqu&ro 4a h of tho lunar eccontricity. All othor things bo-ng w.1, is daT w:h oarller than T. Uula (19G4) indicates tht Lho s?culrr

charqm In o BUe to tho radial tido is abut tm-thirdn thy called

by the Urth tide, at tha prosent the. POI othcr srtolli-e .yiL.ms, 111 112

whloh roaululy ohnpos Ho (of. 2.4.1). his cug-pmts tht rwthod a9f-m 1.~9~% hloldreiah, €@at A -dt to >e than 1963). the of lntrodwinp the tid.1 otfect on the tldo-Lroe Nmiltonlan would supprtzng .immts Lor eithu case ill. not ab.olutaly oonviaaing, nw action varhbles which twnatmts erpscial.ly rinoe bth depand on the relatlw dirrlpltion in tho be to define &re in ths abmence of any othar Corcem wcept th cidom. primary akthdc eazellites, OZ which little is krmn. Fortu~tely, the questio? of whether the totrl rate of ohmge of the eocsntricity is kositiva or negabve is irrelevant for e-rypo resanances during transiticn .id after capture intu libration. Tlu tidal torrie acting on th. n(Yn motim IIKIUC~~a change in th. e:centxicity thr:ugh the ir.tervrntioi oC tho reso~nco. For an e-type came, this r*sonmce induced chanpe in o is auch largas than tho tidaLly inducd change.

This arsartlon will k nupprtd latmr. The bcic fact is tht the inclartle tidal rampana8 produces a secular chsnje in tlm si.tion variablca, dike the purely gzavitational I*rturbatlor. which tenda to causa only periodic variation. in the act ion 01.rictnts. 'hi. point has mt ken proven, and tho mjor thtorstica! supper'. for thts conclusion is "Poisson's theorem on Uuv invarnrbility 0' the srmirujor axes" and the taplace-Lagrange thaory of stdu per:urh.tiona (mgihara, 1972, pp. 1644-85).

Xu hsvo not di8cuss.d thm secular ckanie in ern ang:e variables focruaa the tial toRU at0 not qIditatiVbly diffumt fnrm those pmdncad by conservative gravitational i'oroes and nre mch smeller. By far tho largest change OCL.WS indirectly throuph tho implied rdar c-0 in L, 113 114

In order to introduce tides into the Haailtonicm, let's roturn to the original -1toni.n formrlation kforo it vat8 rducd ts on. dcgroe of fr- (2.4.31. First, *he soculnr tidal chngu in L, r and 2 shall bo forrally added to thpart due to mnservative ~-avitrtlomdinteractions already discusad, T3n result Ls

(2.9. la) elements Z, T and fi 4ro constants in SO aboenco of any gravitaticml porturbationa othu than tho tidoa. Their oxplicit

depondencc on the tidse ha6 boon eliminated, but the tide-free

H.miltenian dnpands on the unbrrrd action elaraants, rat on the where Rf in the tide-freo H.ai1roni.n for tho dishPbad body (2.4.1), barred variables. Wo can substitute the harrd variables for tho dr 6% and 2 ,Gand ,> are rates oi change of L, r and Suo to unharrod *mriables in lif, but the swuAar tidal wtions 1% etc.) ticfur %lone. for The equatiots ai motion thangle variable uo of the elmenta are a1.m functions of tho old elements, although they well approxinut& by the ti&-free equations of motion, because the ue approniwtely concant if the tidally inducd. changer in L, r tw- which aK1.c frcm inelastic tidal intoraction secular the a0 and Z aro relatively .mull. If this approximtiira ..a not ponsiblc, mch rPuller than the duo COnSclrvltiVe secular terms to thm the unbarred elements still can be succossi.rc1:, e>.-ri-uted in gravxtatio1d intoractionti. the miltonian by a procoss of iteration. niter &ding the sealar tidal change, next atop i8 tho to There is another problmn with ths barred actsx vnriat'oa in :n-rcf.uce .I now vf action variables, and sot E, F 1, &.Pined by tho that the equations of motion of tho angle variables deph! upon ..?e rela-ions: wtial derivatives with respect to the mrrcnponding untarrmd actrcn

elomonts. we can replace the unbarred partfa1 wit') the harred 11s 116

partial, wit'% reatricticn that yarti.1 dozivative d008 not the tba toma alter the Ilail-n. d"r 4.- dr m 'zh 0pgrox:mate tide1 acceleration -dt operate on -&or as it --c, 5 in F+. ~h equations of of th. Omn motion of a satellite, CAuSw1 by the inclastlc tide motion lor each variable are nou derivable frm the MpIe Hamiltanian. ralsd in the primary, is hllon, 196911 Ihe procdlue outlined In ssction 2.1-2.7 can k umd to rduce the xaniltoriri to one daqrer of freddrrm. But this ondimensional (2.9.4)

Ilaniltorian is no longer a coxstant of the mtion mince an explicit where Q in the diesipatlon function (Wcmnald, 1964) and R is the time dependence is retiined through tS0 tidal interaction intcoduced an* e5 radius of the primary. (henca -$ 2s A rapidly decrecaiw into the Wmiltonian. We c $11 see how this tima depndcnce occws by function of thn aemhrjor axia a. Since the fraction31 fluctuation mklng the time derlvetive ot Sif (J,W) I d'h in n near cocrrpcnrurabil1ty is wall, the change in d and --T &"T with transition will alw We cafi sxp3nd amaociated be slull. -dt. AS a function of x to first order (2.1.7, 2.4.2, 2.6.11~

6x mall that ae tractional fluctuation ..n x, p was cf ~(v''~). dJ 0 since +t always oeura in in aasociation we abmorb H~ with 5, em The change in L(x) asecciatad with Uanrition into libration or into first am tarma directly partial drrivetivs €+ the of in- the of reverse rotation is alw of 0(p1'*). rituitively, w can mea eat with respct the tima. to if the tidrl acceleration ?xeases apprsclably durirq transition, '=he secular pact Hcuniltnnian already axpandd of th has ken then this affect by itself codld l~sdto capture inbJ the librational dn a pow series in x. l'terefore, A simpler procedure T in Zor atate. In ordtr for the change in during tranaitlon to ba of introducing thk tidal interaction into the tide-free HMliltmdan im 1 O(IO- ot the conatant term, y muit tc of 0(10-~). ne wss rqlace all constants integration e-.) to the of (Lo,ro,Z0, with ratios of the rpre raassive partner in sach of the satellite- *4r their tidal CQUntUpartS {lao + !et) e*., h the mb-dbNlBhd 8atellite remlonccs klonging to Saturn ('+) are (Jeftreys, 1953) c tido-free Hamiltonian (2.7.11a).

The nmt Step 10 tO dlJtdM jUat hor these tbO-dwndOnt 117 118

fl?ncelsdea- LFXIO-~ grmtest in the zmro-ordar part of the HBDllto-71M. ?my the 4% dspendenco occurring in the coefficient of the poteririel term is of AxhYs 1.14~10-6 p.8 OO~)mller. Let's return to the rtamiltonisn i after ths 8ecular

.*an 2,4u10-4. Bart had been mpanded in a polynanial aerie. in x. Replacing the He constants of Integration by their corresponding time-de-Jendmnt tdma,

it app.era that for tha Titm-Hyp.rion case ohang. in &'T my the coeffi:;mta bf the axpanaion arm (2.7.2): b. an Lzpxtent factor during trrnsitia~. It dom~not for the shple reaeon that the tidal acceleration of mionis imi~.ific.nt ccapared to that of Titan. The re-nt Klucturation in tb. d%(xl re-for axis or 71-11 (d- ) 1s wvord by the mama retio of dtTi cj %parion vhlch is of OUO-~! rmy1;er than litan's. For most oases, we attall find thst dt can well appm*iAted by its FGvalue Earlier, them cumtants were manipulated to obtain a Hamiltonian near rh. colll.ncurabllir:~. which had the desired form. We ahall wv show thet these oparations It in possible that tlwr eff-tivs change in n durirrg transition are still legitimate. can La substantially 1n:ru:sd if the ratio of the tidal motions of % If etc., axe replecd by thmr DUM veluem, than eech of n ad n' ckys appraxhtely the conwneurebility iatio found in tho ebove coefficient. depandm only on the timr. Purely time- *.e resonewe variable 0. Rather surprimingly, this occurs in the dependent terms can be added to the Haniltoniar. vllhout changing *a-Tethys and Enceldaa-Diom camensurebilitios, incrfmsing the the eqwtionr or notion for either x or 6. Again, absorblnq the effective change in the runimejoz axis during tranuition by an order 1 term h0 ~i~A&) into the Wiltonjen is a lqitinsu oprrtion. of mgnitudm (section 3.1). This will not substantially affect the - Intrducing a raw time variable E, (2.7.41, related ta t by proc.88 of transition frox rotatim into libation, but it can leed to an appreciable ctmnge in the long-texm bohevior of the rmeonancm, (c.9.E) especially In th amplitude of lihatiun (Allen. 1969) as a iuaction or the age of the resonance. ir quaivalent to inMduaing a nonlinear tinr aad is also a

r)u erfbct of the time dqwdencu of the "conatMtab ia legitimate opaation (see 2.7.6-7). 119 120

L(x,tl n jx + jxdtd4 + Lo

Instead of expanding d"T in t- of x, wu mhll include the lowest

order time dnpendence and expand in

ob-e tInt tbe the derivatiq oi ab) is proportional to tho M (jx + 2-0 (t - to),. of the secular acceieretions of tlwr man notiow acting in thr To the first order tams fn (2.9.9) we mst add the term rsm Once variable. ExplicLt.ly, tha time derivative im - '

(3.9.10)

The next question we r'lall ruolw is th. aPf.ct of the . The depndence on x in the coefficient A-means that tho operation maentry de-wrdence, contaimd in &T , on thana Eoefficiurts. Tho of redefining the tinu (2.9.8) is no longer explicitly valld. In yroatrr-t depmndencc would smm to cccur in the lowest order thio case, the dependence on x will be ignored eo that ths aiaplest coefficient Ao. But thim amfficient has m effect on the equations form for the H.mi1tonL.n can k retained. Besides, it can be of motion for either x or 0 since the partial derivative with % dmaonstrat& thnt In those cases in *ich this depenasrre plays an reapact to x cannot legitbkely act on -, ew. Thus the funotion dt *rtdnt role in the rvolutionary history of a rasomnce, the (Ao AiA&l CUI Still h. abrorbad in without chlnging tb - contribution frcun this term is insignificant. equation8 of motion. The meffkdurt in which the mc.wnhm The time depr@ence in the coefficient of tha potential tam d.psndence is lsrgest im +huePore A=. A, depends -licitly on dn plays a relatively minor role for a and 1 type r-3~)nnnces. This the t:hl torques as they mrinride tenan ouch as ~dt-&x;tl . dn can k damnstrata! by making a tranrfonnation of variables frm T hpreasirig (x,tl as a :unction of t(x,t) we find (2.9.4)i 121 122

2 is of O(noao), or equivalently of O(Lo). unless the sealax tidal acce1eratior.a are c-nsurablo, thr fir& term in (2.9.14) is of

0(e2) amnllcr than that asaociatod with att). nesame redt applies (Z.9.12a) to the element 2. We have already sham that the x dcpeMence in

b(x) can be approximatail to 0(y1”) by the leading factors of r and

2, and the tidal chango in these elaU0ntS be just been Shorn to be

small compred with that related to c(x,t). We shall now shw that

the above Hamiltonian Lwtion agrees reaaonab?y well rjth the

second order equation of mtion derived by Si?c!air a;d aln ky Alh.

If the fractional fluctuation in r and 2 ore emall, then the

coefficient of the potential term b(x) can be erpanddl LQ first

order in X. The equations of motion for x and 0 are

-dn - -(bo + xbx)sinO, (2.9.15aJ dF

Taking the time derivative of anrl substituting for s,we have: dt ai dr 21t

’Iro O(y”’), -1s (-a - c(t,n)), and the hro t- in (2.9.16) aF dependent on x and .itend to cancel. The function includes the aF by (2.4.4, 2.3.9). By (2.8.1-2). the fractional rate of tidal change nsa+ly constant torque tern (2.9.10) plus a tern hich is Of each ui Me elements is of MpnitudO. Tho factor thc ”0 suspicimsly like a term. If we eliminate x in favor of i, cb-0 - AOYU 4 uls time variablw from to t (7.9.81, and replace bo by A,>,, etc., 124 123

the rentlt is:

2.10 StmmRr

After a long sequence of approximations, we find thmt the

satellitbsatellite interaction can approximated one- 12.9.171 be by the dimensional Hamiltonian where the factor P in the dissipative tam is defined by (2.10.1) (2.9.181 The equations of rotion for x and 0 are dx aH If dc(t) is -11 car--ed to d.pand.nca d! >H b-, than the tima of E' r, dt m = n=-- the torque will have little effect in the likatianal phase. In an8 it is understood that does not act on a(x,t). "he explicit addition, tha secular change in c(t,x) is very nearly ck). Th. -. relations of each of the vcriables in H to the parameters defining validity of the expansion is restricted to a time interval for each orbit follow. The angle varbble 1st which c(t)bx is smsll colapard to bo during transition. The first

three t- (with c(t,xl replnced by c(t)) constitute the egwtion &= jh+ j'h'+kZ+%'S' sin*i*rLt. (2.10.2) Sinclair (1972) d-ived in his nnerical calculation of transition The mat powerful restsiction on the integezs j, 1'. etc., derives

,wobabilitier for the MLoes-TOthys rewMnco. This restricted tram tha fact that the interaction is independent of *..e

equation is identica- to the Second simple perdulm example treated coordinate systci. Since each of the above angles is measured fran

in section (1.2). Allan neglezted to consider the effect of the f(t) a c-n reference, the 6m of the integers nust be zero. See

term on the problem of cirpture into libration, although he did (2.2.10) for further restrictions which apply in the two-body case.

implicitly include it and the i term in his theory on the evolution hloxt, the momentum x is related to the el-nts e, I (e and f

of the t4i-m systea afun transition. -11) and a by (2.6.1) and (2.9.2): 125 126

Z(x,t) -*12Lo 3 ix+Zo+jsdt,

L'tX,t) =!.;;a' = j(%)x+L,+{$ydt* eta.

Thus, the variable x is proportional to the fluctuation induced in

the elemnu a. e, and 1, e-., by tlm tom in the diaturbing

function which contains the angle 4. Observe that the fractional 2 fluctuation in L is of o(e2 or I ) avller mnwt of r or Z. ma The %-dependent tern in (2.10.6) results from the fact that the implies that the variation of blx) wibA x is principally determined tidal acceleration decreases as a function of the planet-satellite by its leading factors of T(x) and B(x) - if the inclination and separation.

eccentricity are -11. Finally, E is related to ordinary time t by Th. function b(x) is related to the $-depeadr7t term in th. - t 9 Aomt. disturb-ng function by (2.7.5~1, i.e.8 The most important restrictions and approximations imposed in (2.10.4) b(x) = !.ili>Al(X)* deriving 12.10.1) are the follwing. 1) The dirt.zbing fw-tion

can be expanded in '.ne ratio of the semiwjor axes, and is valid in where (Al tx)cis+) ia correspondlag tens 12.2.10.11~. the in R A- both the librational and rotatfonal prases. Therefore, the abze is the thM coefficient in tho Taylor expansion of * secular part Hamiltonian O.!.O.l) does not apply to a tw-hdy resonaxe of the of B and is approximately ;0.7.M), synodic type whic' is rasuicted to the 1x1 comiensurability. 2) (2.10.5) The fraztional fluctuations in the senimajor axes (01 in Land L')

caused by the perturbations are small. This allows us to axpi.d -%e parameter A quala one 9 the Mirect part of the dsturbing the secular part of H in a Taylor series in I But there are m xnctinn does not contribute to the remnance term. otharrise it restrictiors on the magnitude of the fraction fluc'uations in ir -iven by (2.6.10). either r or 1. 3) There exist no tams in ths disturtfng functian Tha function c(x,t) ia proportional to the accelruation of tho (such as the -.hart period tel-me) which cannot be r-ed or ipord ' motion of and ia (2 g.n,11): bec.wse oi. thcfr high frequelw

c(x,t) s crt) + P@< (2.10.6) 4) Another condition is necessary In sane cases (2.9), nareely, that th4 Wb1t.l hCllNtiOM ud OObU*&iOit~On k rOhC.'V*ly Wd1. Th. ~1.eutlnfy thm rolwant orlterh. rumon la that w irgoro tho rertriation thrt fractional fluctuation

In Lb) k -11 2onp.r.d ts **. :n! i Lx) or Efr). Cno COI)~.FU~I)OO is *hat ttie fluctuation in A~ .x, ;o o@, govrrnd by itr 0 1udir.g .actOrn of : and 0. Thl. gr~tlyrimpUfiefi tho functlcnrl fern of bW). horentriatior to -11 Incllnationr and

.~contricitiercar k lift-', but lt mnr th~t* ,0 tern in bb) ot C@ munt be ~llr'rydotominod to find the aooffioiont 11~. 0 Thin 1. u non-trlvial ..orcine in tho flrnt-order #p.nrIon of Ipfx).

't MY k pornlble to lmprove tha orbor af approxhtion b) O(vp) if the irrllr.ct put of th. eirturblng Pimtlon doas rut contain a +dependant tom. Alw, 1 Po mar and/or angular ramntm of one partner In very much larger thn that of thr rorond, then a kttor approxhtion uybo obtaimd by n.9l.otinq '-he r>t..on of tb f:rrc putnor due to th. nocon;. Jitfortunrtely, the ordw of approxlmtion MY n?t be net by tho urpanrjon parrutor. but by thoro very long parlod am. which had to t* Ignord beoauae thly could not k r-rS by any tschnlqw) nuch OB BI'm'r nuthod. Although we hovr urd the #pacific rxrraple of tho ratellltr- o.co11ito qravltatlonal Intoraction. the rune mr)dr can k op1ey.d on pIavitatlona1 renonancer involvlnq moxe than two utellitor. Of course, the mny-baly roaorancr nirt ratirfy tho

SAM gnus1 rortriction. outlinad for tho tm-wly CAle. In addltion, pr~~It~tIorulrewnancer with the pwpol.wtia1 can k rimllarly rducod LO th- one dimanrlonsl Hunl1toni.n farm If QJY 129 130

From thio picture, 4 "tranmitlon phase" Wig. 1.2.2) WJP defined UI

rxtm. 3 * 1 I~ZIO'Y which our attcntior. wan focustd cn the raoticn of the interior

We found that their motion was wll-dofind ecopt for fa veri meall sot of psniblo nwrtlonr near the sticking motaon. The mine outhdr were mnployad en XI, with tho foIAoulw

renultsi 1) Wc fomd Chat itm traniltlon ahcu uta. apprcr.mto: tho

ram@as that CaLind fur I to firs: ordu in etas crnrll irawttr).cm

b-' dt r.nd b-"l dt . 2) A .:.-ftiral angle Jir warn der:v4 vhtch

dsfind the rticklng mtlon in which ttw petdulm urnt (wu the top

far tho last tho, rovereed .iign, and again slowly apprii?:ho3 the

top, trnetng to "itlck" there. This particular motion icpnrate.3

tranmltlon. uhlch led to 1ibzatIon of the pendnl\lr from trans! tlo-c

rsich led to roverma rotation. 3) Also, a Frobability fx ca~';rs waa mom a study of 1. a picture ot uanaiticm was dsvelopd which derlvoci, g'ven arbitrary kinotic enarcy ef the prdulurr on I? vent inrolvc*i the motiw of tho turning poinu or mots of (~(p)- 0) over the top for the lart time. obuimd from the Int.sgr.1 relution for x. Elpliaitly, 2%. other major dcvelopmnt involved the orbit-ort.1: lntlrsction

Md ho* It related to those Hi+U pondulm SUmPlOi. A OM-

dimonm;onQ Hauilton?an was clerivd that is D vr'id aI*'rc,ximtion, to The siinificant ro-dts vera?: 1) The roots ware labold in t' O(&, 02 t4.s very lonq puled motion, inclu3ing the ra:uIar rotation phase according to tho value'of 4 at that rod (either Lo mtlon due to the ties1 inflrence. "him third Hamilmr~lan (:II) ham apd(n) or mad(2r;) and to t!m aign of p(- -+I. 2) The relativa tho form (see section 2.10 for Safinl.: Lon of tamm) I porltlv of thcse roota In zht? complu plana malltatlvaly 1111 H@,+,E, 1/13 t ccX,EJl'- + blxiCcS4r (3.1-1.B) dotuaind whather thi systm axacutod rotatimu or 1Lksations

(fig.:2.1a-c1. 3: Trtansitivn invol*red tho motion of tho lntuior who;. it is undermtodl that the pwtia1 dorlvatlve vit'., rewe-t tc x root. towards tho arigln nwi 3on mt a1.q t!bc imnginary axi-. don# not nparete on tho x dependence in c(-:,t) (mectlo2 2.9 . Tls: 131 132

efCocts am tho B-lu cbngo Ln tfn rwts #ID tho ~ysta~approaehue the trmrition are rrlwant, n rho11 dolrr wlr axp1ic.t 0.1.m drtDdMtion until rwtlon 3.4, along vLU other topica relbtad tn

tho rcular crrn':i in tha elementi in each phase, inclu3jq the C) amplitude of IiLnaticn. The prlnclpal analytical twla w.11 k

1) action integral an4 2) averaged equrtlonm or mtlon fo* ttm

turninp polntri or X.

There are swersl step1 VD can parlorn preNrator)* to an

utplicit ox~alnrtionol transition, monp them idnq thdari'iatinn

of th. equation. of motion, and a rpo.c!irc labling meter-* tn

identify rwta. To Curther rianpllfy the 3roblsn, tn #hall ndopt --?e

follovinp conventions, rhlch In M vay rmstrict. tho gurlcra1;ry of .he prohlan. 4) The "mrml" libration cmntcr (oht.Ind fra b:x: - b(3)) LB at mod(Zn), or ewlvalent:y, I; 3 0. IC, for a V-VD~

I.OO)).IKe VUl4blD 4. tho WrornS; cln1.or Ia 4 m*l :w), than m

nw ranonanco varietle c4n bo d.!In*d I9 - 4 t n , lor thmicli th. normal center 1s at nAf2n). b) Tho 4 $a constructd euah that the .rdal torque

BWUlUlp docZcDDm i. ThlB aarnm that 4 IS p¶S-tiV*

kfore th. corroacn.umbi1lt.j 11 erttJ~11st.od. It lollms tht, in the poe!tfve rotation phro far rim traneir:m, (le function c(~,i)ie 1.m. than Eero un E I# grmrter than aoro . :3‘. 133

For th rlmpla rtyp. raim~anoo, the function bb) ha. tho tom (7.7.ea. 2.4.4)i

21 13.1.3) i +I$ - ‘&kS,t. (3.1.6)

Wlth thlr rat of trrnrforutlone, thc eqqrutlonr of motiim are.

bl 13b 137

b) 138 139

&l.lhI

The paztial du.vativer of Do X. rind 8. x, :an be o mind from

(3.1.9h) nftar %prratln;l theme qutiona into thoir real nnd hwinary parta. Thia procaduro w:ka -11 tor tho lkl - 2 can., but not for the !kl - 1 cnaa. A bottrt wthod for thia lnttu -10 i8 to separate :he quartlo polyrpanicrl (R1P.e xv + i In x,) - 0) into ita rul nnd imrqtnuy yarta, an8 frat then. rdotlonshipa d0-d~ the prrtinl duLvativrs as function* of the roal nnd iruglnsry pnrts of *,.

PII~OIO pru:erdim, w shall consider thJ oftoot of approxmating c;x,t) by c(0.t) during tha transition phrao. n, estmtc the ef'ect o!' tha action vuiablo in c(x,tl on the probability of cnpturQ into libration, lmt's agproxhtm b(x) by b(o) 1 r, EO thrt the penduiun motion ja npproxiruteiy that or 11. Bine. liar in tha rnnge w :: 4iG 1 3r, and d C, w CM The dewription of tho tranaition p!nra will bo qculitativaly doduco that oepture occurs only 111 :I 4 0. ~ho&ova cqbation sfmilu to ttat fer 1, to lowest omlor in tha .null puslnur (3.1.18) ia idontical to tho rolation rrhich definaa tlr mt.ickinp e" 1fi9. 1.2.2). ~h.equation of motion of tho imwinuy part motion for TI, if tho factor Ir roplncd by (ZpUl. "turofora of x,,, during transition, ia tha probability tor cnphrre cnn bo ~diatolydarlvrr ly raylactnp

tho .a~fnctor in (1.2.3al. The r9Nlt Is (3. I.. 15I 141 140

aquation which dororlka the mpin-orblt Ln’;.ractLon, Whlle the rlght hnd mid* of tNa quation le one poB3iblo Corn for ttuc alwnhpl- d.p.ndrnt -quo acting on the aifoctrd pla~t.Using tnls orpation.

boldraich and Pula (1966) drrlvd e prohbillty tor r.I:tU*I idtnticsl

W (3.1.19).

ia. fcr the sinpie-e cam 13.1.3, ~.10.6),

U.1.22)

3?m loft hand side of thin aquation la Lorop.lly identical to tho 142 143

brtwwn the mallez pair of roots xz,,- and xt-. Insprtion of the

3.2 TIUIISXTX010 MYPm 3- -1CXTY- equation of motion !!or x rweals that tha mbt nogrtire vai~oof x

D10-m --I 16 jX + j'Xl + k8) OC~ursat b I mod(2n). Thus wa c4n dduce x3rr- < x,,- TIie relative - lkl I 2 location of xn+, x2,+ can k diwovurod by ii,spaction of their Because of its rmthrutiorl rhplicity, tlu 1k1 - 2 cam will k dlCfarenceI -inad first. %a uplkit solution of th. roots as iunctionr of c and !I is prttmcllluly sbple, since b(x) is linear ln x. The explicit solutions for each root, along with their ~dantlficatlon, for k - -2, x 2++ > IC,* In the positive rotatlon phss. If k I +2, ere I then x,+ > at2,,+ far from tranrltion C.4. >> /bxl). eut AOU transition, there roots myinterchsnqe their re1nt:ve posltionr.

We rhould polnt out that in the po?itlve rotatlon ptase, *sic valur -1 of x for which blx) vanish.. (or e -* 0) equals k ard wit k less

than xZn- whan k < 0 and greater than x,- whsn k ' 0 lndeperdent of

the rrugnlt de of k From an Lnsp.ct1on oE (3.1.91, *ncan dnduco that for lkl - 2, s w and a 2n rwt are qua: It and only if they equal k-'. Canverssly, If a (1 (or 21) root aquala k-', then thue

mst exist At learnt one 2W (Or 8) IOOt Which equals k-'. Thweforo,

I3.2.L) w can conclude that 1) for k - -2, no root qrula I#-' in nither t-7. poritIvo rotation phase or the transition ptuissr 2) if ). - 1, then the roots x ard xln+ Interchange where bkl vanistea. nt Thla information is nrillclent ts construct. the poarlble

2 rotation phase di,agrms for each ceae Uig. 3.2.ld-c). InclCenUy, Since R is of OW2c ) in the posithe rotation phse, fu fro th principal qualitatLve dlffursnce in the poaitsvii rotation ph.4 transition, both the roots x2,, and x,,- are of o($ while xZn+ ~d dtegrams for this rxanple and I1 is that in I1 (flg:1.3.ia) the ,, are of 0(-2c). In the positive rotation pbse, x is hndd 144 145

midpoints. of the pir of I and 2r mota co'nclde, while in this

example their re.p.ctive midgoints, Mn end M3,, are offaet by an mnt(-2kB). There exists only one pssible diawam for k - 2 (fig. 3.2.la), while there are tu0 posrlbllltles for k - +2 a) Positive rotation phase for k -2. M and HZn are the mid, Anti - (fig. 3.2.lb.c). Figures 3.2.18,b are qualitatively slmilar to the or t,= I and 2a pairs or roots, respoliveiy.

diagram for the simple pendulum (fig. l.2.h). 13 addition, the Im x t transition phare for these diagrams wlll involve the --roots which coincide, and then develop an Lauglnary conponent.

Pipure 3.2.1~ suggests thac Lheee wistr a radicarly different

form of transitlon directly into the libration ptrne. in which the

roots x,,- and x2,+ exchsnge their relative poiitions on the real b) Positive rotation phase for k - +3. Here xZr+ and xr+ are to wis. The equctionr of motion of these roots (3.2.2) show that the iett or I( - +1/2. this autoaatic transltlon is an allowed motion 01 the root.. Im x Whether it orcurs depond8 sxplicitly on the value of the par.wters f I I c, H, and B at transitlon. SubtrMtlng xn- and xInt at

coincidence we find:

If k - -2 the above cartnot vanish sinco B is neative. Transposing (2k6) in the above rpatlon and squaring, we fin3 that H murt

satisfy the relation

. PIGvl(t 3.2.1 POSITIVE ROTATION PIRSE DIAQWIS FOR ]kl - 2. at coinc ldence,

The condition th8t ae,p.rates these c*o Cypcr ot tr'a-sitiorr ir 147 146

w-roots first coincide and th tb3 tf that x - R. X.. The result fer ttm two .-roots coincide at -t that Xw- m.d x to the - 2a+ 2 ir coincide. This rupirmnent is satisfied if A - 0, II - 28 and c kS - -1/2. It turns out that this information is not mfficient to uniquei.? sepcify tke pareatas. Pnothar relation can be obtained

by rraluating the action in the positive rotation phase for the 0.2.9) spacial case where x,- - x2.+ - xw+ - 1/2 (see 8. 19,ZO). The rem;'. By construction, 18n2xll(i) - 0. The condition which sepsratcs A! transition into libration from transition into rlagbtiV8 rotatLon is

that mzxw(f) equals zero to first order in the all Farmeter 6-1 nc Recall that this definer the stickinq notion where 4 -> 0-. -dt . If ! B 1 2 1/8 and k - +Z, the system autonatically ~AJKOS a transition If 1m2xn(f) 0, than the imaginary cOrmDnents are nonzero am frm the positive rotation phase the libration phase. into _.For x -> xn+ and 0 reverses sian again. Tha conclusion is that the IS! < 1/8, we expect that capture into libration depends on the system has successfully entered the libration phase. But if the initibl conditionrr, much as it did for 11. hansition of this type integral is negative, the impllcatlon is that the systm has rnde teqins when the w-roots coincide and Wve into the colaplax plane. the transition into the negative rotation phase. The i-roots are canplur if A K 0. Prom inspection of f3.2.U), The integration variable in (3.2.11) can be chanard from t to che real and imaginary cmponents of xu+ are easily identified and X. Then the condition that the integral vanirhes will det?nnine the are initial value of X. xic, which leads to tho stickan? mctit. t -> 0-. (3.2.7a) Re xnt = -(c + kB) Unfortunately, it turns out that xis explicitly depindr on the value

of the ?cots at transition, which in turn depends, in a co!apiicated mxwf- 4: . b) way, on the paranetor 6. Th. dependance on 8 must Lm determined

using the action J. Inatead of pursuing thls cou-se, we shall 2 detormlne an approrkate condition, accurate in the mall fluctuation zll xn dc 1/2 -dt - #e x .. XI. 0.2.9) limit (i: cq 1) and delay finding a pore a-ate relationship until

Th. next step is to iniegrate both rldes between the time ti that tha the next section. L .<.<\ '% ._ :49 140

~n instrxctive tranuf-tion is to Observe that x - Re X" io r.quirment.that I"_ - x,,+ - s2.+ at transition, or 6 - -VS. Th. relate4 to 4 by beformula predicts that Pc - 1 vhen 6 = -wZ/16. Therefore, the applicatility of the above formula must bo restricted to values of 0.2.10) 6b le1 much muller tha!i 1/8 in order to keep 6x and hence pmal! -e . For the lkl - 2 case, the "amall fluctuation limit" is dsfined by Ru transition criterion is thon the condition that 161 c< 1/8. Conversely, the "large fluctuation

limit' is defined by the condition that 161 2 1/8. At the end of

the following section a mrs accurate probability estimate is

(see To lowest order, OW) i w and Sic 1i.s in the range w 5 +ic 3". derived Pig. 3.3.9,lO). Thus tor k - -2, the systrr carapletoly svadeo capture into tha There is a more rierious question concerning the validity of libation phase. Obsarve that this relation is similar to (1.2.26) the Hamiltonian in the limit *ere bW vanishes. This problem ir -. if the parameter in (1.2.26) is replaced by ke. If thm connected with the existence of terms proportional to e in em fluctuation in b(x) is snsll, then tho motion is nearly that of a interaction which can cause very litrge fluctuations in 0 bS n

slmple pon4uP.m. Given this approximation, tho probability Pc for approaches zero, and is discu8sed mre fully at the end of section 3.3. indication from that that the aumtic capture can bo imnediately Obtained by replacing by kein The analysis is transition machanim 2 case seriously affected U.2.40). Furthermore, M can find a ganncual rrault ap:,licable in for the k - is bi the mall fluctuation limit if we replace -k6 by . "ha such tern. prubability for capture for a systco 4th an action-dependent potential is:

The restrictions are that Pc - 2 IBI-~/%~a 2/r. F- the e type reggMnee, condition that Pc - 1 (i.e. autcmatic transition) exactly corresponds to the 'et =.. ->. 150 151

separate. th. tw types of behaviar? For lkl - 2, tha condition wan that three roots equl 1/2 (3.2.6). The final question is, how does the systcrm enter the inverted libration phase - if it can? As with previous excmrples, the starting point of this discussion shall The case in which bW is propxdonal to the firnt paver of e b. an investigation of the mtjon of the syntm in the positive is particularly interesting because of both its -tic behavior an8 rotation phase. thfact that the majority of the naturally occurring resonames are Far from transition, the mtion of the real variable x in the of this type. It includes th Enceladas-Dione (a, - qi+ ODi) and cwnplex plane must be bunded on the left by a 21-root (x2,,-) and on the Titan-Hyperion (-4XRy + 3XTi + 1 -1es. hq of tricks JKY the rig.lt a rr-root (at,,-), while the magnitude of these roots is used on this case shill be more grulitative than the rather 8 of OC;). The relative magnitude and position of the raining war straightfomud method applied on the simple pendulum and Ikl - 2 of roots can be t?iacovered fran an inupection of the equation cases. The principal difficulty is that although the quartic (R(xr) - 0). For lkl 0 1, we find equation Ink) - 0) CM be solved for its roots, thm solutions are too coq1icatt-d to derive trwn tham ths position of the rcats in (3.3.1 1 th. cslplox plane as functions of the parsmeters C, 6 and H (see Since H is of 0(1/2 c2), the first tern in (3.3.1) is -11 crly if An. C). Fortunately, there are other vays to answer the important x is approxinutely equal to either 0 or -2c. The first case questions. Our past expercence with th. k - 2 cane should suggest correspondn to th. relatively -11 bounding roots, x2,,- and xn-. that this example should behave similarly in the -11 fluctuation If c is -11 compared to 1, thsn the pair of roots of Ot-2~)is 1-dt. That is, capture into libration MY occur, depending on the ala real, since the secod term in (3.3.1) is then nqative. But initial conditions, if k is positive, while it in evaded if k is if (-2kc) > 1, then both terms in fJ.3.1) are positive. The negative. Th. first really interesting question is, how does this conclusion must be that if (-2kc) >> 1, then the large roots are systam behave in the large fluctuation limit? Prom analogous complex. Thus, in the positive rutation phase, where -c >> 1, the behavior fak - 2, w q?ect to find that for k - +1 the systclm large mots are rea? if k L -1 and col~plaxif k - +-, vhile in the autcnatically enters the libration phase if the paremetee 6 is negative rotation phase, tho converse is true. larpe enough. A second question of concern is, what condition If the rwts are ceal, then their lahliw in the positive 153 152

which is -11 if /Re ",I >> ($2'3 rotation phase nust be sbilar to that of the simple pondulum. ?bet . For the case where all roots ar- real prlor- to entering the is, the large roots are xs+ and x2.+, and the two r-roots are transitton phase, we expect that transition is initilted by the interior to the 2s-roots, the as in 11. This description alays coincidence cl the interior n-roots. If rcots xn+ and x:+ applies to the k - -1 case and vould appear to lead to untnteresting the remain complex through transition, then sarmcthing a~lwousto the behavior during tramition in that it alwp enters the negative automatic transition from rotation into libration found for the roution phase. Althnqh this is true, as rhall presently be k = +2 case must occur. Paca11 that this phenmemn happened when damnstratd, this pecullar pendulurn can, fer a time, librate in an in the large fluctuation limit both c(t) and ? were relatively invert& fashion. large parameters. For k - +l, thase large mots are camplex, if -c >> 1, They lb discover the equivale-rt phenomenon for the Ikl 1 case, we can bd identitied by solving for the roots in the positive - crhll asme transition occurs %hen 1.1 >> 1. The maximcm fluctuation rotation phase far Cypiral values of c, R and 8, and then-nubetitut- of x must be of OU), since b(x:a. (-la + l>'I2 0. This inplies .nu fAem into (3.1.91, which defined the 2r and 1-roots, respectively. that the Hamiltonian can be anproximated by neglecting the x2 term 'rhe result of this exucise is that these ccmplsx rr?ts are both for values of lcl >> 1, since the real part of the comp~uroots is .-roots and, i.. the positive rotation phase, can be labeled xu+ of information that Jcld useful latsr of 0(-2c). Physically, this means that the variation in the mean and flw+.Another piece be longitude of either resonance partner is mall compared to the wuld b to have same idea of the amgni%de of the imaginary parts varition in the pericenter of the unprW partner. The result is of these "exterior' x-mots. We -t the irasginacy part to be

-11 c0apar.d to the real part (this shall be rigorously d.ponstrAted latu). If this is so, then we can substitute If one of the turning points in the motion of x occurs at x - k, x - Re xn+ iIm xu directly In the relation deZining the 8-roots then h equals kc. This condition defines the value of h for which (2.1.9), aXp.nding the facta ('h, + 1)1'2 and Ww8th5l OLt th0 b(x) periodically vanishes whenever x equals k. The values of h for lowest-order imginary caqronant. We find Ia xu is, approximately, which the system either librates or rotates can be determined by

(3.3.21 substituting ack for h in (2.3.3) and solving for cos$. The result

ia 154 155

In thb posiL.ive rotation phase, the roots x+, X-can be equavalently

0.3.4) labeled x,,- and xZnc, reapoctimly. Since the labeling im unique until h - ka, we mot conclude that the systcl. muat rotate up to tha where a Ir an arbitrary ionl loss plrcmwter. The phymical critical value of h - ka. solution muat ba such tht R 1 for all alMvalue. of x. If Thr ‘ ..viral solution for x, asarming that the pramcter c is the para er o is < 1. then it eppomrs UIllt the rlqht hand aide of coneto. ?n particularly uimle in that the inteqral is (3.3.3) can t orI both positive and negative values, with the P JW :*c arcsine of a linear function of X. *e solrtim bplicatlon that the syetr has either rotations or libtations of 531 .* .+on of t is: amplitude > 90.. Dater we shall discover that only the fomr is

al1ovacl.I But ii a > 1, then the lefc hand 51de camt chanqe sign f3.3.7)

ao x varies. si Ice Q > 0, the turning points in the notion of x hat me frequency of the lDntion equals 1.1 and becomes mst occur a: cos9 - k. l’ha irp1icat:ion is that thio syetsn lihrat Obsruve -11 ab 1.1 + 0. But transition is already presumed to occur about the In-centar for k - +1 and about the --center for k - -1 the secular behavior the system as a with amplatude s 93.. when 1.1 >> 1. Therefore, of

unction c cm be obtained from tl ction integral J, since J lb b0tt.r undbrSUnd the b.haViOr Of this UnUW-1 POttdUlrol, of is let’s reconsider the intqral solution of x: adiaba’.i;ally conserved when tho ?rcillation frequency is large mapared to the changes in the syeta’s parametars.

(3.3.5a) In the positive rotation phase, the appropriate integral is:

21 Jpos.rot. - Io xdo. (3.3.~

The first etep in evaluating this integral is to express x as a

runction of 4 using (3.3.3) I ad is a quadratlc polgm?al in - L~stadof 0 quartic polynaaial.

Tht is, there are only two bauding roota, and X-, givm by 136 157

At tranrltlon, h - kc, trrplying khat a l/'%

Thr nert @Cap La lo wrluito J In thm ilbrrtlonl phorm:

(3.3.151 mplitde of libration 0, as a funstiw ol ths guuwtu G. The axex- in tI.9 motion of +10,, - 4, - are ubtain.d from t.hs c;ndition that tha argments of th. rdlcal in (3.3.9) vanish. This cndition reUuces to thr following relation for +,I

2c .;inOD -g . (3.3.171 or *in#,, dwrcus linurly with o(t).

The fbllrving der :ription vf transition can ha d.duc.4 frm thse facts. For k - *le the mot XI- approache. .. 1 as c spproa0h.s thvalue, L/Zd. Thh sxns th.t the function blxi rhioh Am poportiom1 to I-% + 1)"' ,Ls mllmr in magnitudt at + -,~(n) than at 4 rdl(2n). At wansition, when c 1/2B. th. fwwtlon h(x) bJ Here tha root *'I- -1s *l, while th. Cunctlon Ib(x) I, van1nh.s at c = WdIw). In addition, thm mot a,,- chnpes from a 9 to a vanimhoa 4t O mod(2w), wbi' a for I. 1/2lE I, the p.nduArrm - I 2- root. 1ibra:es with nuximua anplitude ern - 90. about the Zr :enter. Bipdra 3.3.1 rhm this soqwnce of wenti for Ur k - +1 casa, while Clg.rro 3.3.2 s-s the equivalent mwpsnce Cor the k -1 0110, hi.h fa 1.1 librates about tha w cmtv.

e/ the my, it is ne bccldent in f&purs. 3.3.1.Z nat the path tracd out by the pondulm in mch phase I¶n circle 0' c?nst.nt U'- rdius. Oreanbug (1972b) observd thin phmnomenon In his anblysis of a mailu problrs. rurtbexxm, hm found the angular veloclty, as muaurod frep th. oentrs uf tht c:rcle, la conatant ud -1 to 1.1. To pmve these amsutl?ns let'm consldn fipurs 3.3.3 which il

.I diaqram of th. path tracod out by the pendulm in tie pomitfvs rotatlnn ~'hme. The center of the fiuure Im at D dhre d - CO la given by 161

\ !/I

4--,kE2 ac (J.?.ZO, x#- n+ XW- i Tho dlatanc. r Crca the contsr ol tha fiqurm to tha point P Is

rdatd to b(x), C and I, tho bngh lrulurd ftcr th. Cmtu

0 txi tha polnt I), by -.ha law et corinmml

xu- ; 1 A63

Th fact tturt 'I :r b.llnur lunation of +hr CLN lollon dlrrtly

from th law oC tlnar. bplioltlyl

PICURE 3.1.3 pat2 trud out by penjultsl mor.te9 by (3.3.3) in pwitiva rotation phrw~i. r k - +l tam. ?ha paramator d is the aldpoine betwen the dnlmum and rr*La\lla valurn (I.3.24) o! b(x). V is tk. angla mda by vactor wlch rarp.ot to :?a origin at D whllr P an tha anyla amda by vrotor Glx) with raapct to ttm origin at 0. 165 164 lachanlro.

so fnr rt. hrve 4 quaiitativr piawe or hol, th. pendu:m khves in both tho ml? ud large fluctuation liaitr to. tb

Ikl - 1 ck. me nact .Up. of the drrelopewnc rill k L(- determine the cri-ical vnluem of the pusnuterm whiah soprate th. tw typr of htuvior and to develop 6 nmre acouratn probability argument.

First w shall dlmcusa the cen@lete qualitative hhvlor for tho k - -1 case. since it can k easily ddirud fra the prwious arguments.

Par frw t-amition, the relative porltion or the four roots vi11 be as ehown in figure 3.3.4.. & th. nysta wolves, two posslbllItIos for the transition phraa uxlmt. If tho left bonndinp -4’ rOot x2,- doos mt reach tho V~LO-1 pr1w to rh. coincidence of , b) Diaprun of the w’.lon of x ,Notion d’slnp period vheci i just artor w-rc,ot I colwide. rworsom si9n. tho .ootr, than trurrltion !All involve ?he tcrporary motion of tho --roots off the roe1 axla. As wlth tlut k - t1 case, thr variable x wlll mepast th. pD1I.t whoro x - At xu. Thruiter tha coapononta flr xu uil: amme back mwud eh. ral axis, ruchlng

It before x roturns to th vrlu. r(. x”. ThIa W.IU that tho system hr. onterd the negative rotation phase wlthout any pomslbility of

11kating (see fig. 3.34b,o). Th. othu poaslbility 1s M.t th. , a) YqatIve rotathin jherr. mot xlr- does rerch the value -iprior to the colncldonae of UU interior .-roots. The perdulm thon wins libratiq about tha in. rtd position. kwentually th. r-roots will coinclde and b.a

.kt mover prt ttn porition uh.rr it quala Ro xu, thr cooCClc;.~nC

of tho pondulm ?ot*nti.l b(xl which 11 proportional to (x *.

rlm incroasem, aftu + ?ovorHr oipn. Aftor I( ruches the rwt

xz.,+, tho variatlo x thon nnvoo back Cowd the posltio.. Ro xu.

I4oanwhllo thr iruqlnary component sf xu ham v.nloh.6 and aqah tho a) Positivr rotation phme .Path of notion in polar coordis- in eocnplax plana. aka ( IbW I ,4) in tho porielor syrtan ontoro Lhe nqatlvo rotation plume. rotations. Tho rpoclal sizuation which tioparatoa these two twaet

hhsvior in thr fol!owingt Lot tho root xla- qual -1 at tho saw

Instant tbt ttm tu) lnterlor n-mot. colricldo. From th. action

int.pr.1 (~pp.8.231 tho vrluoo uf tho parhautmro for this spocicl

cboo aroi

t) TIlporary transition into imratod libration phnso.

If 161 < 13.89, the pondt:lua bohvar sonrlly dwinp transition.

pot le1 z 13.89. ttn awtan temporarily ontoro tho invutd libration phroo, \ow ths tho IC - -1 care hr Lman disposed of, wa OW-1 cencontrat8 on tho rubtlo behavior of the k - +l cas* during tranrition. Vith jurt one mo?o pi.:. of information, YO CM outlln.

th. carplotr qualltatlvo behavior of th. pondulm durlng t:ansltlon c) huuition into nagatir. rotation. for tho k - *l C4ael. Wo arsrrt bnd will prove tatorl th&: th. function +(a) 13.1.8.) vanirhos whm tho utorlor u-~ooCs, xu+ and

x:+, roach tho rob; arir and cclncido. Rocall that +(TI o:rurred in

.I. - .~(r~(ejyfpv r-rooto we t vo 169

a rauh rllowd value of lBl for vhjch the equation tJ.3.2E) is satisfied. This critical valuo for 6 and xn c*n lu dotumilld by maxlairlng (3.3.26) with rrrpct to xn. The results =e

5 601 - - -0.2722) x~~~- - 0.8333. (3.3.79.) Ra me abve equation, the only all& motion aftor ooimidonoe 3K

is fOr th tu0 t00tB X5+ adq+ b S-ate. (h. t00k (X,,+) Incidently, the rrminlng values of the prrutorw for this specLa1

toward. the left bowding root x,,- hila thm otbrr (";+I mwen caw are (B.21) I

torarda the x 1 *1 position on tho real 8x1.. Pmour omliu 1 X2.,- -1/2 . b) H 9 - diecussion v. expat tht vhon th. root x& .I +1, it changes froa 7 For lBl < 16c,l, the valum >f x,, where tha axtorlor w-mots reach a ? te a 2n root (x2,,*) and throaftor docreasw. We also know that thv rul axis $8 in th. ranpe Q 6 xn 1, while tya 3urc.d zmro in it-) for c - -1~2lies in the ranqa (1/2 x, *I. Clurly, aL the

critical value id 6 .I Bel, the thrao n-roots co~ncide. lor valur. of 161 * lBcll :ha implication ia that tho uturicr n-roots never do Leach the real axis during subsoqusnt evolutior. of the rystm. (3.3.18) Thtrcefoors tranrltlon is qualitatively similar to the carllrr dis- Pigure 3.3.6 is a graph of lBl vusus xw+ for ths mpwi.1 value of cusn.d approxination. ~hssitua3on is mor0 wmplur for ltrl < IE,~~. c -1/1 and k +l. ais special value results Itan evalwlting - - As * eyst.lo ovolves, the roots Yn- And xst mV. twuds coincid-e. the .rctir*n intaqral at the instant x,,+ and xi+ ooincide (8.23). In If thard root. coincide bmfore x;, - *l, Shn the symtan can either addition. rh. psi! In of the left houndin3 rcet in-is graphd as I tmaporarily rvslre into an invertd libration yham or ?.main in the function of 6, using (C.2a) and (B.lba). Observe that the position positive rotatlon phse as s)wn in fig. 3.3.7-8. Which occurs of xn- ir to the left of th. nseond xro of #(o). Thio ia to ba depends on th#i motion of the hginary CmML8of the interior rxp.ccod tbmotion xn- io toward tho position whuo since of w roots durlqr the transition pbre. If tho Imagilary cmimnents (xI,c) vanishes. (note that this will occur frr a amre pomitive first weof? the rul axis and ttan return while x is to the right *u~luecf L' than c -1/2). According to figure 3.3.6, there exists - xV, then tho wtion of x ir rvrntully trap@ batvohn X,,+ of Re 170 1 / ,' 171

30 [ I I I I I I I I 1 25 I' - 20 - /'/ 'W ' - Transition is initiated by .Th.p.naUlm toward th0 coincidonco of interior n- top 'I position where 8 vaniatas at 4m 15 - 0 /'/ - mots and continuea as and revoraes direction. Tho / they move off real axls. angle tLsn increaye9 to the / The mtion of x is first mwhm 4; where C again - - towards x ,,-, next pnrds vanishes and reverses diroction IO Ro xWf, n8ar where 6 van- and then mvcs toWard the /' i6hes (near top n position) bottom n position. After / then bgck through Re x paasinq through the T roosition, // 55- ,/ where 4 aqaln revtrres"fign. b doer;ases the valic (-6;) -. . Thlos motion lnvolvoa wan- where it again revarstm o/ I I I I I t I I sieion from positive rota- direction. Fir.sl?,, it mv~s IJ tion phase to yet another toward the r yasltlon and : fC12345678910 "tanperary" posltlve rota- vanlshes for thc fourth tire, X tlon phase. cmpleling one revolution in ..

c- +- BIGLlftE 2 3.6 , xzw- +I = (x:*-x2w+) Qaph of 181 versua x,,+ for tho special value of - . i,z 1 aad k - +l, obhinod fram tho (#(r) - 0) 6;wtLon. pzoll Second traitsition, in whish Hero, wthe pndulw me6 into thr systan automatically libration phase as bLu) tends the line (-1, value solid ow finds tho of .tho root. evolve6 into tho libra- to vanish. The ~sximum tion phase as the xn+ rcot amplitzde of libratlon is in xw+ and x:+ at coincid-• aa a function of 1 B I. ~h. equals +I, then rSVctfI96 the range 90. 5 C, s le$*. direction, kaming a 2.- dash line (---I givos tho location of the second aero of mot. tho (#b) 01 equation at cLt) -1/2, ubi10 Lho hair- - - FIGURE 3.3.7 dct line (--) rapresanta the location of the xl" root TRANSITION ISTO LIBRCI?. -A PWSE WR B at c(t) n -1/2. During dsepwnt wvolution, the dash IN "HE RlUOOE 1-0.2104 2 0 *-0.2722) AMI k - +1. and daosh-s-t curves move toward coincidence. 172 and x;+. Theroforo. the pudulm hsmado tho transition froa positive rotation into invartd libration. But if these uolponentr nover return to the roal axis, tho motion of x 1s trapped botwoen x~~-and x;+. Although tho cwo trace8 out by tho pondulm is -0 conplax, ad 6 rweraes slgn four ths during one rotation, it still a) Transition in which intorior ,Tho pnoul~moa maris top mcecutos positivo rGtationa. In either case, subsequent ovolution . n-roots ilrst move orf, then a-position (X uhero 4 rwerson roturn to real anis, trappinq slgn, next &jng throuqh tho of the pondulm tendr increase x',+ decrease B(-x:+ 1)1'2 .r> 0) to (and t x botwson x and X* . This bottom a-position (x:+) and is a stato 1) inver€& again toward. acd throLoh the until it reacher the value 1. If the systm has wdo tho tranrlt~on libration. top n-paition. li.it~o of libration t* about tLe into invartd libation phae, then as Beoroases the pmt~m -- th. Icl contor is SO~. svontuelly antors tho ~ativorotation phae (soe fig. 3.3.8b). But iI the ayitan has remained in tho positlvo rotation phase, it will

WP-mtiCally OntOZ tho libration pb-8 with WlitdOOf 1AbrdtiOn s 180.. mother or not the p.ndulm e%ocut.s these exotic mtions during uanaitior. dspentq, intuitively, on the pnramter 6. If ita

.lagni*.xle ia too small. than X,+ equals +l, than docruses and changes labels prior to the coincidmca of the interior n-runts. . b) HUO x* - +1 and becomes .The Xbrdtion amplftud0 %a condftion which separate8 the ordinary transition from tis tm- 6 Zn-r&. At thlu point increases to 90. and ther.aftor tho syBteln automatically aocutes negativo rotations. stage typo just diacuamd is for q+= 1 &en x,,- and x,,+ co:.ncido. enters ths negative rotation ph.0. From the ection integral (a.231, we fi.d that the valuoa of the presstar. ares BC2 - -0.21041 c -0.47051 PIGURE 3.3.8 0.3.30) TPANSITION IHIY,NEGATIVE UOTATION PPME WUXGH n

"-PAW* 1-W LIBRATION PKRSE POX 6 IN RANGE H - 0.1407 . (-0.2104 6 161 6-0.27221 k - +A. 175 174

In m,m that Uwte arm thrw dirtinat modes hatn found of b) transition depending on the magnitude of the parameter 6. If

ID] < l~~,l.th.n tramition is qualitatively similar to II in that The second relation (3.3.31b) can k used to determine tha it i.molvas the coincldonce of interior 8-root& For 6 in the range part!al with rmspect to H ol the r-1 put of x,,. The roault in: {IDc2 I s ID/ s IB-~I),transition is a two-staged nechsnismt the flrst invol-es the coincidence of the interior u-roots, after which (3.3.32a) the syatau evolves either into the inverted libration phase or a colaplax positive rotation phase. In tha second stage tho pendulum where n and E are evolvso directly into either the negative rotation phase or the libration phase, respectively. During this stage b(x) Lends to vanish (i.0. ek) -> 0). l'he final mode involvea direct tranaition from positive rotation into libration it 1111 2 l~~,l.NOW tht tha The firmt relation can be used to determine tha parthl of the Sualitative behavior for the k - +I case has been thoroughly imaginary psrt in term of the partial of tha real put of x~.KO discussed, the next step is to back up 801. of our assertlone and to find develop a more accurate pxohbility esthete than that developed in arm xu 2 n(l nl - c section 3.2. -I - alc Im xw:.. - nP + E2 First let's obtain a more cu)licit form for the equations of ootion of the res1 end iaagiwy part. of the %,,-roots (3.1.9). Tho Thuefoce the equations of nution for the cornplax prrts of xw take partials oC the real and imigiP;ry parts with reapact to H can k the fOrm\l dmtenoined as explicit ftinctions of tha .?araantars a, 6, H and the r-root parrs by nepzrtirg the equation [Rae p,, +lZm p,) - 0) into its real 4-A bmgiwy components. Aitsr 8010 simple manipulations of these taro components, the fol'aving relations are obtaid: -_-.. - I In P, - -(xu 177 176

approxlwatu¶ by W. should point out that the above atpitions are valid only if

B2rl 6. Otherwise 0.3.27) governs tho m-.ion of the real room. (3.3.35) Inrtdently, the condition that Imzn vanishes implfer that +(u) which #hilar to the aquation of motion for the k 2 case. The also VMiiheS. can slmply dexonstrated substituting is - This be by H garmeter rl a slowly varying quantity. Therefore the factor cmlujtd at x - xWinto (3.3.31a) and observing that the resulting is ie also slowly varyiiig unless In vory i ' -ly vanishes. e.xmbsion 1s consistent with #tu) - 0 (3.3.27b). n1 - 1) Therefore, the transition integral can defined bafora. and It appears as the system evolves towards -%ansitton (for k - +l), be as tht the hgiMrY part of the exterior '1-roore ffrit gradually evaluated to lowest order bv replacing the slowly varylnq Faraverere

t) nnd xu by their values at coincidence. Afcu integratinp the increases (since they are 'W28 iRa X~+I-''~ for (-c) >> I), reach a Re above equation we have maximum and then decreaso towards zero. For E 2 >r n and n < 1, the rotion of Im xu is dofinitely towards aero. Th. valuo of tho -. paramotor n vhan the xu+ mots coineidr, after using 0.3.28) to e1imi:ate the B dependence, is By canrtructlon, j2a - 0. 91s time. let's cham@the integration variable from t to x, keeping in mind that t 2(-x,+ + 1) is n I, - (3.3.34) monotonically incrmasinq function. The right P.nd nlde of (3.3.36) (+xu+ - 1/11 then becomes since xl, lies in thr range f$s xl+ 11, n is bounded btmen-.i and naro, l~vlngita largnt value *an tho #rea *-roots coincide. (3.3.37) On0 can also show that the --mootor n is greater than or equal to 1

when thu interior u-mots coincide. Thia Is to bo axpestod, sinco Tho above is understood to be the path integral defined by the otherwise tho irpagimry part of x,,+ could not move off the real ais transition phase dlagrsla (fig. 1.2.2). At coincidence, R(x) is *mile ths V&iabh x is m th. left ot ?e XI+ (sea 3.3.3oa). During th. -1 trmaition pbse ti... coincidence of interior n-rootsl)

tho param..az E is of O(b" 3 and is -11 compared to n. Therefore the equation govarninq tho motion of x,+ during transition can be -- 179

to eliminate the specific dependence in Pc on 4ic. Prors (3.3.36) we find:

0.3.39)

P- thm transition phace diagram (fig. 1.2.2). we find that xf - Re xvt md xi lien batwean rZU,and Re xu. Although this integral can be explicitly found in toma of arcsine functions, this step WiAl be deferrod until later.

Recall that if th- integral is positive, the systm has (3.3.391 can be Used to evaluate sac\ of the term@ in (3.3.42) by dethe transition into the libration ph~e.Wlt if it negative is substituting the appropriate limits and tho value of 4 in the the converse is true. The value of xi for which the weintegral former aquation. We find vanishes (x ) separates these wants. Recall ic ttm that the measure was in terms thu value as m2x, 0) 1, 3s u + arcsinbr (3.3.43s) prcbability defined of of i2 the !i - -2 syStarn went Weth. top fOK the last the, and W8S directly related to tha function m2xUevaluated between fixed limits b) (1.2.36-38). Specifically,

(3.3.40)

Therefore Pc is given by the Of th t- is OVdWtsd in that part OL til0

transition pbse where 4 -e 0. Thcorrespondir hits in tha variable x are x, and xi=. Again the apprr: .mth Lqtepral fa

(3.3.40) over x is understood. The reldtion vt.ich defined xic can be

. .. . c SS. 181 180

This fomxla is valid only fc- tha k - 1,2 ewes with d in tha 6-e IO s 6 11. Pc vanishes if k -1,-2.

If the 2u-roots were wtrically placed abaut tim coinciding

--roots. then 8 ani the arcsin of 8 mwld vmi.h, iqplyw that P= would also vanish. But this case corresponds to bCxl = constant for which cqture in +- libration nmr occurs. The porition of the roots at transition is iaplicitly a function of the parameter 6. There-

fore, the intsresting relationship is how Pc depends on E. Again,

tir action intsqrd can be used to uniquely detecmine the parameters of the system as a functic+.n, of 6 (8.16-18). Pigure 3.3.9 is a plot of Pc veroua (R/B,,) for both tha k - 1 and k - 2 cases. Observe that the -11 fluct*ationlimit :or Pc where it is proportioral to

~6~’” (3.2.12) is 01 ly valid for Pc 2 0.1 for the k - 2 case, and Pc 5 0.5 tor the k - 1 caw. hothec interesting fact directly derivable frola fB.18) 18 that Pc - 1/2 when B - Bc2 and decreases to zero ea 6 approaches Bel. Observe that the probability that the pendulum MY temporarily entez the inverted libration phase o( - +1) ia reasonably la-ga. A =re revbaling graph (fig. 3.3.10) is a plot of P veraus th mean eccentricity e far frola transition or, FlcvRe 3.3.9 e equivalently, (5).Era (3-1.3), this ratio ia cl maph of probability for capwe ink l*ation. Pc versus 8/Bcl for +h k - +lo +Zrcases.

0.3.45) 182 183

Note how dramatically Pc decreases ad a function of e for the k - +l case. P is lesa than 0.1 for e no more than 3ecL. Fsrthennore.

the tw-stage transition occurs only if e lies in the narrow range

0.0 ecl a eo 1. OScl (3.3.46)

aefore turning to questions concerning tha secular behavior of 0.6 the e).z+em betore and after transition, there io a serims problen concernins the opplicabilit) >f the theory of transition develnpd pc for the e-type resonances, which must be examined. Recall that in 0.4 developing the one-dimensiona. Hanlltw'an which deocrlhes the

resonance interaction, the seccnd rrder effect of non-resonant and

0.2 non-secular tcnns on the i--traction was of Ofr') smaller than the

first order terms. These ;il'ded an infinity of *.ems for which

0 the coefficients of their xespective cosiw arqumonts yare 1.0 1.4 1.8 2.2 2.6 3.0 I proportional to the first irwer of the eccentricity c. If e were

vzry mall, then such terms muld tend to produce very large

Cluctuations in the motion of the perihelloc, 62. But, as YO have

seen, a major part of the notior. of the resona ce variable 6 is due

to the motion 6 e very In fact, e mat very FIQlRB 3.3.10 of if is -11. nearly vanish if transition involves the autanatk entry into Q1198 OP Pc ao/ecl EVR k - +1, + 2 CASE(. libration ptise. The questlon iq, could these other terms

poportionLi to e effectively inhibit this autam4tic transition

mechar...m? To get a qua1itati':e i4ea of thek affst, let's add

the following tem to the Wiltonian e?uationr

I' '-. 104 105

1) 1/'2 (x + + bkla + w+

_-- %re uetva lhLtD Ln -ah infoartion donorrnhq tlu rblabatio bsplrq of +m La t..bLly obtainrd, ud thy 4.g.ri4 on

whmrthor th. lupnitdr ol 0 is b* or ** than 1. Thr Iomor case Io Th. mst Intoxertlng khpv~orof any pondulla-llke lyrte oddly enouqh, th. rhplrrt, an4 arcy of the Fnportant romlta have q0vrrn.o by a thdmpondent lmmiltonian 1. the dmplng of the alrudy km obtalnod. libation mplitudo #m. Th. baplng In tha ratellite-sate ite lor tho care whore 101 >> I, Urnrltlon Ir warned by th remmnces to b. dirusrd In tho next chpter 1. prmser tly wend approximate ~Iltonlanhfa,#,t) (3.3.3). Rcbl1 that "tr~.irltIonm tha limit. of measur-t. Thin im elm t ue of tho mchnism which tnvo1v.r thm racular owlution of the myrtm from pomitLvo rotrtlon 1s +he root causa ol th. ntppord daoplng, th. tldal torque. Ar w dlroctly into llbrrtion (4- 90') without the posrltI1lry of tho hvrr already awn. them appear8 tu k a direct link betwen th. ryrtm enterlw th nqatlve rotatron pbno. Thlr ununul Mchnlm rats ol chnqe in th. libration .I0plltude wlth a glvon change In im dlrtInpuirhd by two faor.. Unilka the rImplo pndulm tJu c(t.x) an4 the value of the pu.nutu B at tranrltlon. Th. mrt Inrtanun8ous Lroquency rsrociard wlth th. pondulun laotion rmulnr Intoresting case 1. the k - +1, .-type roomnco, and mast ol our Clnlto and vorlrr mwothly durlnp uansltion. Zn dclltlon, the ofhrt oh11 k concontratd on fully underrtmdlng thir axamplo. rccentrIclty (and the lunation bk)) 8end ta VrnlDh durlnp Wanmition. Incidontly, two of th. throe rescmncoo among Saturn'n Thlr oven ocmur when th. p.rm8t.r ctt) 1/2#, Thr llbratlon utrtllltw (dlscunrd In 4.1) arm e-typa wlth k - +1. But 'he - uplltude thorrafter doarearea rapidly ar ott) * 0-. pcpllcitly Him.-Tethyr reao1).noe ir b &a4 I typ. (I... blw) ir proportlorul (3.3.171 to I, 4,k)~~~(x)). It happons that th. vrriation ol IT.(*) wlth x 1s or a(-- % +) ar1l.r mnth variation OK (x). meretore, UT. tha khvior of thi8 puticu1.r .~~~plecloroly rairlcr tht ol an -typo. mu., w oeo fully jurtlfld in rertriotlng thim dluurrion to thir one case. ?urth.rrore, wa ah11 noghat the k dependent term ar.cc1at.d with the arynawtry of tho applied tidal torque and armma here that c(t,x) 5 cttl. In th. aourre of thlr investigation, w how it shall Indlcata alfectr ch. damping of Incidently, th. ro&tlon lr valid In both tha poDitiVm 190 189 reation and libr8tlon pharer, indicating that (d tendr to vanirh 0.4.6) M C -b 4 - of COUEMI thir 18 th. (IDIp.Ctd bOh8ViW.

Tha eccentricity 1s related to th. action variable by th. Tho puwtor 6 11 a funation ol UW mmn orbltal el-ntr of the following rolatlon (3.1.3): reBonance partnotr walwtd in the porltive routlon phae far

from tranrltion (3.1.5). Put ch orbIt.1 elauentn avrllabla to ur

are thore tht the ryetan porresrrr 4t th. prrrent tlm In itr evolution. What we rmuld llko to know lr how tho above llmlutlon whmo eo is the wsn eccentricity in tha poritlve rotation phu, fu on thlr (3.4.6) approxlwtlcn’r valldlty tranrlater in tema of the Crcn trnnsition. In tha libration phase, tha nvorage rmntrlcity pmrameter 6, evaluate4 vlth thr pr*rmnCly obrorvd man orblcal (P 1/2fawx + e&) 1. imruaely prO~tlon8lto c(t), the rxclat elanuntr. reiation bdnq (3.3.18) I We have already reen that in 4n e-typ. ramun.e, any chnge

in the mcantricity caul& by UI. app1i.d torque ir of ~(e-’) lupa la.4.4) than rhllar changer ln the ranlm&jor ulr of either plrtnar. Thorefore, the important quertian 1. hac doer tha parameter p rcale on thu other hnd, tha ftuctuation de i (e- edn) rminr - am a Cwtion oc (e(t)). R*O.II tht in derlvlng tha dlneno:onleer co.uhnt in the llbratlon ph~ao,and oqunlr leo. 8y the way, th. form for the Hmlltonlan (#*e. 3.3.4-6) which darerlbd tha mtlon bmvior of env0 and 6e 1s rwerrd in th. poritive rotation phi.. for an o-typa remonanae, n war dlvl6.d by tlu Caotor Thr avaragu eccentrlcity 1s thon -1 to eo, while do ir givon by

(3.4.7) (3.4.)) sime th. ctsriioimt or th. pnduira tern ir piopoKtlOMl to el’!, TILLSbobvior directly follows it- 0.3.18-19), and 8 oueN tho p8ruut.r 6 tends to MU. llk. ?or tha k +I care, Fnrpcctlon of the agproprlte diagrm dercribirtg tranrition Ulg. thir hp1i.r tht the relatlon htumen 8, ovalwtd in th pcsltlvr 3.J.l). rotation phra (e0), and tlN UM plrmwtu ovaluatd wtth th. Th. validity of the abocro result. end into tho libration prerent moan value of OM,) lrt ph113. .-til

e 191 192

WO hv. rddd u) rubtIbCt.6 a tOlCD pZwthM1 b in g(I,t)

(3.4.61 6-0 wa want to .rophsino that Lt is tha fluctuation in Y ud not its nctul valuo which contslbutor to tho damping of f. Tho Thus, by (3.4.4,6,6), hCl,+,t) la a valid applslimution fm prmwtsr X is a dLwnrionlsas naqatlve4eflnitr constant of Ob:),

hich moms it is mall canpard to on.. Thorrfora the tam (3.4.9) le,, le,, I O(1J. 2 (1 + x(4) is -1 to one to O(so). mis in ripnificant sinco ror tho ~ncoladus-~io~resonance, Takinq tho tlm avuaqs of (3.4.10), w find 10norl * IO, which ~ulutht tho complete tidal avolution of thia

-le can k Ilmtex..lnod fruu the siaplif1.d Wltonian.

Ruthemoro, it appears that its prsrrantly -11 amplitwia of Ths .itam avaraqas of Y and x2 arm obtalnod from 0,3.?). Th. results Libration of 0(1*) can only k oxplaincd as the result of tidal 6r. duping via the rachanism just outlinad.

The last question connected with thin topic 181 now dcaa tho 0.4.13.) dissipative term in 5 (x,t) affect tho dmpinq in tho limit

lei >. 17 111 ais sane it happenn that the contribution or this term is us~lyobtained by taking the tiate avuago of th equation (2) 1/2 fbz + b4 b) of motion for h over one libration period. Th. equat!on of motion for b is: Thua, tha avuaqo oquation of motion of h(t) aftu chnpinp the indopondant vrrhbl. from t to tha dimmnsionlosa p.rr=atsr c(t) 1st

The tidal. torquo is (2.10.611

Intqratinp ttu abovs aquation ud choosing tha int.pration ConatMt 10 that h m ko whul c - 1/20, w find

x I -"- z 0. &(O.t)dc 194 193

SAC. th. frMt.ionn1 fluctuetion in hb) is -11, x can h.

0.4.15) r8plac.d by its WUI value w in tha function bW. Tha resultlnq Wltonlan :e thm identical to example I1 of shpl8 pnduJmr plus Going imck to th relation which Bofinea ain9, in term of h constant applid torque discusead In rectior. 1.2. Tharefore, in tha (3.3.9.17), we find that libration phase, H(x,#,t) for the k - t1 e-typ. resonance hi

H(x,#,t) 1/2(x + ~(t))~t B(c(t) 4 l)1’2Fom6 ,

while the equation of motion of H Is8 A. iom~66 1 18 -11, th. errect or the x-dmmt in th. tidal torque is minimal. dC (3.4.19) s!dt - SlXdt + Ctt)) + 1/2,dt(c(t) t l)-l‘%oe+. At tha oppoeit. exxtt~o., th magnitude of 6- Lor the Uimr Oboerve tht r)l. dgndent variable can k changed from t to c(t). Came iS % IO4. “hi8 fact, COUpld with it8 pre8Wtlg We can eithu take the the avera9e of the above oquation or lag8 libratlon amplitude (97.1 indicateu that a ms also -11 at uee the action integral to detwmine the recular khavior vith traneicon. For this case, a different approximation can k invoked of H to detrdne its 8scular behavior which depends on the fact that th. c(t). ”ha action ht.gKa1 rapresentr the simplest approach for If the x-dependant tom in c(x,t) is 1nclud.d. then f1uctw:ion in x is always small, capmod to one, if Bo is mall. this example. This foLlo*s from the observation Wt the maximum fluctuation in x it appare that the avera9.d equatlon of motion must be u#d and that nwrical inteqration is rmquird to Ilr.4 H as a function of is or o(IgI1”) in the entali riucttation limit. nurt Step io to considu tha time average OF 6 in the libntion phse: Clx,t) (Allan, 1969) - The evaluation of the action integral in the libtation plare is (4- - (x) - (0) - (bx(x)co*#l (3.4.17) a etraLghtforuard ox8raIee Md, zathu than repeat it hWWr w

This tbaverape identically ~ani.h.sin the ab.enam of a tidal shall at-arve that it aptems with the reeult Qf 8ost (1%9)8 dc torq-~e and umtappraxiutely vanieh in th adirhtio sense if the changes induced in bXW are slow cenpued with the libration trvnq. Therefore to o(Iel’’2) VB find,

(x, - C(t) (3.4.18) 195

(3.4.12.)

The aapliMe of libratlon am a function of ck), is

obtainad by .valuating H *re 4 vsnlahoa. w. find Uut can LU dlrrctly relatd to th. prmotu KI

0.4.23) 197 196

Using (3.0.20.23, Om is grqghd in Pig. 3.4.1 as a function of torques oL Mh8and %thy8 approxlaately ob~~tha same C-NI- ability relation as the resomnce vaxiable. Th. effect of thil is to greatly magnify the ratio of the x-dgndent term 4nd th.

effective torque, acting on the rasonance variable. %,dt In the limit of mall librationo the simple padulm can be If 6, happens to be ol OU), then no approxhation appear. to approximated by a harplonic omillator. Thadiabatic conitant feo k available. It happane that for tho Titan-Hyperion reMMnce, 2 a harmonic oscillator fa proportional to #mlb((X)l or rrhich we shall amine, has 8, - 0.058 and +mll)(~ow)= 36". 'Po trwt this cam we shall use the action integral to determlne the

secular bohavior bf the sy8t.a aa a function of c(t). The action

Thin meam that adiabntic damping in the small likation lbit (when integral in tha libration ph8m tokes rhe Collowinq form.

Is1 (< 1) is an sxtremaly slow process compart~to the adiabatic Jlib. w6 -k"J(-kx + 1)dp + k"*$ (3.4.25) For - wing found for the approximation obtained when 161 >> 1. the

WAYS-Tethys Zase we find that 9, eq. a 130. wnen tho function b(x1 We know that (-kc + 11 is positive definite since eCx)r(-kc + 1)i'2 was apymxlnrtely fitn timss larooc thn at present. This implies and e(%) in real. For a libration, the initial and final values of

L'..c the dimemaionless p.rllaotaK 6, 'LBS SppraXbately S3 th8 @ are identical. Therefore thr term $d4 vanlshes. A aero ucplicit larger than 6-. Including the -ntm dependence in the tidal form for the above intogral is: torque increases the rate of damping aa a function of the inclination of )?bas much that the function blx) was about three tlnas mp.ller than at present (Allan, 1969). In any case, the tidal evolution of Mimaa-tethys can be adequately detdnd using this appraximtion since the implied magnitude of 60 at tranaition is Since the Hamiltonian 11 nymmtric about 4 - nn (n klng an of O(ld2). intogar), the miJohl0 and aaxh4UEp1itrrde are equal anc opposite Again, we should obsume that this cintribution to tho dnoping ti.& bn - 0, - Corrsspnding to 9 equal b 9=, th. from the dissipative tezm orqaified in the Miras-TethyS ation variable x aquala xa where xm is defined by the conditlon resonance by the unusual situation that the ratio of the tidal 199 200 6, - 0 or

0.4.27)

In fi9. 3.3.7, thre aro two diitinct anglos, bm and $;, for which If th liboation canter im at $ - mdOn), then xm lie8 i vanishel. Observe that as th ayst6n evolves towardm tranoition botwoen tim M roots r2.,(i > 0) and xZa+(i < 0). we can change into tho libration phase, tbo anglo 4; increase. to 90.. Theruftu the intogration variable f. m 4 to x in (3.4.26). The result is 4; no longor corrasponds to a value for whlch 4 oanlmhes. On the other hsnd, Om doel correspond to a real libration amplitude in the

ljkation phase. The important qremtion 101 Which wlution (i) corresponds to thfs "noml"solution? 2%. right hand #Ida of

(3.4.a) (3.4.31) is poeitive definite and the abvo equation must bo valid m iunction cam obtaiw frmp 3 J@ atprosad for all poBOibl. value. of 6. In the limit >' 1. the function c(t), at transition, oquals 1/26 which is presunod to bo a large as a function of just x,H,6 and c baa 8.6). nogative dofinite nurnkr. Thorefore, for tho +1 cam, we Ih. mots x2"- and x2,+ can ba oxpressed as hurtions of k - should choose the + sign in (3.4.31). In the limit << 1, R.6 and E. (App. C). Ths limit xm 2an bo exprosmod in t- of -c(t) in tho libration phaso. Again the + wlution of (3.4.31) c and H by using tho iiamiltonian evaluated at s,f, xn agree8 with the expected bohavior. Tho "norm11" wlutIone for H - 1/2(x, c) + E(-% + l)w2c086m (3.4.29) (-xm + 1) and cos+m are: to elbinate tho ~004~dopendance occurring in 3.4.27. Ra~lthao tuo equations, ro find that pmG ~bp+ c) satisfies the idloving equation:

(3.4.30) Th. action integral can k aolvd in tMBof standard elliptic integrals, but its fmis Cncdingly cclmplan. A sLnpler Tb. solution far he tunction (-lac,,,+ 1) for th. lkl - 1 case is: procdurs io to numorically evi:uate the int-41 for a given value 201 202

of c and a test value of 11. Tbrn R can be varied wtil the that these aoluti0ns my or my not be strble against -11 maerically calculated value ot Jlib agree. with its appropriate perturbtlons. If dynamically stable, these stationary solutions

initial value. JILb,is easily calculated at rxansition and is were deBigMtd as libration centers. "he system of simple

found to be (8.30,31)1 pendulum plus constAnt applied torque suggested that librations ver9

pornriblo only if tho magnitude of tlie applied tidal rorque is rrrvller

than mslrimum value of the pendulum torque or is "tidally *table."

Finally, in investigating example 211 in which the coefficient

Figure 3.4.2 is a plot of the initial value c(ti1 wrms the b(x) of the pendulum term ua8 momantm dependent, we found that if asgnitude of 6. The break fn the curve eaat Bel -0.2722. It the system made a p.mnei.t transition into libration, the remlta from that fact that transition for IScll involve. the mngnitude of b(x) had to tend to increase thereaftu. Sinco it in

coincidence of hn w-rmts while for 161 > 16cll, it involves the the slow change in the parameter c(t) which indirectly C~USCS the vanishing of bk). Figure 3.4.3 is a graph of sial$m versus tho magnitude of bWto adiabatically incream or decrease, we shall paranater (1 - 2c(t)/B! ror several values of the parameter use the terns "hdisbatically etable" and "adiabatically Umtable" 181 18c11. This graph clearly supports the analytic in alluding to thie kind of behavior in the naxt chapter. approximation developed in the limit 181 >> 1. Note that the initial slope of the curves approaches the straight line generated by ploting 2c(t)/6 versus (1 - 2c(t)/6). Furthennore, the value of c(t) for which the slope begins to flatten out is approximately equal to b11'3- Figure 3.4.4 is a graph of 4m versus (c(t) + 11''' for valws of 161 2. The iapor-nt observation is atthe QuNe rapidly approaches the limiting c-e generated by the approximatior. obtained sinen 161 << 1.

In the courae of this develomnt, oariuua kind. of s! nLL1ty huve been mentioned or implied. In sw3yir.g the mighborhood of statiorrpry aolutions of the Ultonian (Section 2.71, we !iifezred 204

1 .o I\\

0.8 SIN +rn 0.6

0.4

4p=-103

C 5 10 15 20 25

PIGORB 3.4.2

Graph of th. initial valw of c at transition, eo, veraw th. magnitude of B 1~1,~2. PlQlRe 3.4.3

plot of the aine of the ampli-0 OK libration versus the parameter (1 - 9)tor ssveral vauea ot B hm the critical value of Bel. 205 206

There exist three examples of two-body resonance interactions in

the ten-satellite systm of Saturn. Observations of #koas 9u (- 3.11) and Tethys (4.94) reveal that the ratios of their mean Re - rotions is Ztl and that conjunction of the two satellites tad. to

librate about the midpoint of their deswith Mlplitude &.Sa. An investigation of their mutual gravitational interaction (Tisacrand,

1896) shows that this phemenon can bo explained as a graVitstiOM1

resonance in which the rngle # 5 QAHi - ;A Ta + i7 Mi ".;, 1Wat.s

about Lero vith msxirmna amplitude 97' and with 6 peiiod of 70.7R

years. The coefficient of the term in the expansion uf the disturb-

ing function specifically responsible €or the observed behavior is 300 t proportional to the product of the itrclinations: IMiITe. Thus the 04 I I I I 0 5 IO 15 20 rf resonance Can be classed as a mixed 9 type (2.3). [c(t)+l]"Z The con+unctions of Znceladue b.99) and Dime (6.73) are obaerved to librate about the paricenter of the innew satellite vith

period of approximately twelve years (0 f iEn- ZAD + ul m 1. The amplitude of libration is very -11, quoted values rnrqino *KOD

20' (toldreich, 1965) to 1.5. (Sinclalr, 1972). '~k t'ns of

thr, resonant prrturbations in the mean !ongiuAes of LI $ir 8x8

even smaller8 1.4' in Enceladus end 0.9' '-. dime -ILruuvt Id

Claaence, 1961, p.113). This means tbt the librat>- the

rawnance variable iB principally governed by rrm - .,L of the 207 208

9 P.ricwmr of EKaludYa8. age of the 803~systtap (.Ir 4 x 10 years). A lower bound on th

Th. pair of satel' :es fartheot roPacd from Saturn io almo dissipation ?unction Q-' wan calculated by integratirq 0.9.4) and involved in an e-* resonance. In tzds cam, conjunction of Titan draandi.ng that the closest satellite WLpas) uam at the planet's

f2J.481 and mion CI4.63) likatr abut the apocenter of the wface 4 X lo9 years ago. The tidal dissipation function Q-' is

Outer satellite with mplitude of 36* and period equal to .~r2 yeare. defineC by the relation

The cocmensurability ratio is 314 and the re-e variable is: :# - uTi- 4xHy + $1. ?ne me1 aspect of this resonance phaneoemn is that it tmda whore Eo is the IWIX~~~XOenergy .tared in ulo tidal distortion a. ' to heap +-hc satellites as far apart an .yossible at conjunction. mis the integral is wer one complete cycle. behavior is oca as enhancing stability among the participating 2) A thcretica! alculetion of the dimsipation function was satellites. We should ala0 Psntion that the massem OZ the satellites attompted for Jupiter and Saturn and was found to be in rough CUI ba detenoined froa a knowledge of the p riod of likatior. sd the agrecrment with itm lower bod Goldrelch and Ester, 19661. Q-' ratio of the libration amplitude of the mean iongitde of each 5 for Saturn is estimatso to be ?I 1.5' x 10 . Acceptinq this estfmate Mtellite Wefferier, 1953). of Q and assuming that it a&?lies to all the other satellites. we This io the set of infomntion available concerning these find, for example, that Titan, wnich is the most massive of Saturn'e satellites. Any speculation concerning the existence Gf appreciable satellites, has increased the radius :f its brbit by only 1/4t over tidally-inducad torques acting on any of Saturn's satellited ha. the age of the solar Sp5t8JBl not cs yet hen su.vrtad with visual evidence of either a secular We should mention that L .irtg -his value of Q an3 (2.9.4\, the change of orbital periods or, in the case of resonances, a dis- tidal deceleration of t19 ma motim of Mha is 1.4 x set', Flat-t of the centu of libration e*ay Ptolll mcdln) or d(2a). oz equivalently, 0.04' century-'. Tho magl1;tude of the reTtrl.vInCe Goldreich suggested that tidal torques are ac:i,lg on the sal ' torque can be cstbted fram the period qf libration U e., 70.76 yfn. and offered tu0 argraanta to support his thesis: 1) The axi for Mi-Te!. We find that the ratio of ths tidal tcrque acting on so wr,y resonances canwt be a chance affair a:d aust bo due tc scm~ Mhsto the parameter 6 is .~r Buaus' the torques acting on machanirm. iie suggested that significant tidal evolution of tho ssch body tend to cancel in the remonance var'able, the ratio of inner satellites of Saturn ud Jupiter must have taken place over the

.. ... ,,>-. , II .+ ., . I.. , . 21 3 210 b(x) to the sum of tidal torques in P 12.9.10) is an ordu of 2) The motion of the perihelion 0 is prograde while that of Lhe node nuynihdo larger. Sinco rho torques anting on the 0th- satellite P is retrograde. This kind of behatior is caused by the secular t-

rew-es are weaker, tha ratio is even largol for the : syste. in the disturbing function. 3) The motions of 8' and SI' of the

Therefore, the hypothetical tidal evolution of thcse satellitas outer satellite are muallor than the corresponding motions of 0 and should be well descritd by tM, theory dsveloped in chapters one and n. three. Recall that its quantitative accuracy is set by the parame+. We should observe that all the resonances listed in Table 4.1.1 are "adiaabol:cally sable" U.e., thr ._la1 torque tends to ser ~k:ly

increase the coefficient b(x) in the libratio., 3hasc and decrease As mentioned e.-.?** the effect of dissipative tides raisd by a given sa'.ellita on its 2riPa.y is to cause a torque parallel to its the libration amplitude). This patholgical result is related tr,

.&ar velocirj. This is tne if the spin of the planet and the the tvo body re'anance. We have mt r?jected any tidally unstable resonance variablt,s cutright. stronqest tfdally unstable orbital wtion of ita satalXta are In the ~dwdirection ad the The planet's rotation wid is shxtar tha.1 tre satelLi*.a's orbifal resonance nust have at least three leading factors of e and I. For

parind. Tnis torque tends to increase the size of the orbit and example, the angle 9 - .I - 21 - 0 + 26' has a leading factor cf eeq2. decrcase its perid. One expects that in a many-satellite system, It is adiabatically unstatle if, in the libration phase the tidally decrease in e greater than the corresponding increase in aftu a the, OOPB pairs vi11 approach a camensurability and +hn induced is e*2. In the three Myinteraction, both kinds of resonance evolve throrgh a succession of related ~~BOMIIC~S.TableQ.l.'l is a list of the strongest res.mnces associated with a 221 variables appear with comparable coefficient- btx) (see Table 4.2.4 carrmenaurabilrty. (Unprimd variables refez to the inner for spesitic -plea).

nae*.lite, primed variablea :o me outer body.) Table 4.1.2 lists th:; pertinent data mu available on the masaes

They have bem ordered in the 8w sequence in which the pair -A orbital elements of these satellites. ThIo information shall k wuld ncounter them under ehe following aiswptionsi 1) "ha tida, called upon during the tourse of this discussion and is collected accelexd*-,on of the inner satellite is at least twice tkt of the here for convenience.

c.Jtar me. Th.t !J, the tidal torque acting on the inner satellite Table 4.1.3 lists the obaenred angular Crepencies of these

deternines the siyn of the torqre acting on the resonance -.-ariable. satellites, the libration period of the resonance and the observed associated the secclar wtion 4 and fi. The Therefore, in the ahexe 09 a resomnce $ tends to decrtose periods with oi 211 alJ

TAlLI 4.1.: 31

7- I i Sstollitea

4.1.2 11s 216 nyperion uraple far UU 413 ~nuabilltyindicates that thm first resonance variable onaount.r.d ir tha grerently obrorvd

examplo. 'rhls is duo entirely to Uu lugs negative (retrograde) metlon of SIw roaulting froa tlu resonance term. adore v. proooed ph moa oxpliottly Uut tha hpressrd mtlnn of 4 teldr to blw up am we should ark if this W situation gr.Vailod Ot transition for e,, * 0 only Lor ttu lkl - 1 came. If e0 11 vuy -11 at tcanritlon, tho other two oxan~ples. then the motion of bras may bo toth larqe ud rqtroqrado NCh tht

In thlibration phw, an adiabeticaliy atable resonance tdr tSe firrt re8onance vuiablo encounterd is the k +Ie-tyje. to causa a rotrogrde mtion of eithu puihelion or lrod8r depndlng The mixed-1 typo can also lead to a large rewogrde mtlon on whether the angle variable is an o or I typo, rerpactively. of Ul + n') if one but net the other inclination is vay mall. mu. This can be smby inmpaction of the appropriate Vrtion of so, wo expact that tho Lmpresrocl notion of the noden ie O(II aullu retion for P or n. In tho care OK an *-type im: 1~4.3~, than th. similar uotion of the porlhelion of thm llghtmr satollito. 3.1.7b): Thir man. that the e-type (k - f1) romnawe varlable can still ba the Lirrt mcounterd unlerr Io Ir very much rullu than eo. (4 -1.2) Profa Table 4.1.3 we can see that the motion# of S an4 9, in

which th #ocular term pradoniruter, tend to rqwl 4nd oppcmlto, *hare w iuamw 6 for an e-typo resonance variable ir given by bo ud that the retrograde motion of th. node of tha Innor satallite le Wutu than that of the Outoc one. (Spoclfi.ally, inmpect the Hinrr-Tithyr case to aaaure ~uraolfthat th. above rutment is

true.) Cf this ir tha cane, we rkuld detumine the critical value

of hrO rush that th. imprerred retrograde motion of 4 (Or for that Ilyttet, 0') ir graatar thn twice ?Ju retxoprrde nmtlon of tho node

ll Of thhr IMOZ r.t*lllt*.

Thm pzoront value of th. oocontrioity for niMr ir

gcc.ptiOMlly lt~g.-4K.d to tht fOC OthU iMe .atellit*. (coay.re ki- o.oza wlch tho nut lar9eet ecconulcity found wng 219 217 the inno satellites: em - .00444). madlythe xuson lios in rtelliter is presently oapturd. We shall dlacovu lator *t a the rslative irPportance of the tide raised by aswitk tlrr radial rusOMbl* emtlnbato for e at transition ia * lo4. Om tide raisad on nhaa by Saturn and their opposite efCcts on tho We now see that the oarliar dracripcion of tidal wolutlon eccentricity (see 2.8 for a diacussionl. Anotha; possibility Is through an orderd a.qurnce of wll-apacul reaomco variables only

eat exi wan driven to a largo value Mrouqh a prr7icJsly rrtabliahed applies if th eccentricitlea of oitlw partner of ths reaomnce are e-typo resonance with, MY, Fnceladua which since has hen diarJptad. not too amall. Otherwise, it may happen that a varlable NCh as

In either case, it means that m should look at the e-type resonance {A - 2.4’ + 3) is enoounterod first. Wksthor or not capture occurs our iiwolving tho pericenter to Tethys to dotonnine the mutimmn Value Of shall b. next t*>pic. But before w procod, wa should observe

such that the variable {+ 2ATe OTa) encwntOrr(l tht th one dinwnsioml delso carefully constructed MY fail eo - AMi - - is m f-rst. Using (4.1.3) and nettiny C - -1 (which 1s approxiwtrly undar certain circunstancei. correct), we rind tha* tho krpressd angular frequency (aTe) Consider the hpllcation that tho rohtlve ordrr of tl-a

(211Mil,, 5 -73O0/V is resonances my be inlrrchangod dopendir.9 on the parmeters eo, Io, ob. It my happen that tvo remnance fcoquencler may nearly wu-

(4.1.4) lap. It is no lonqer necessarily true that the reaonancs systs can k de.crib.6 by a one dimensional HslPiltOnlAn. #ornully one upocts

that an e-tyFe remnance (i.0. b(xl * a) is stronger than, say, a This valw is uvawly mall ami shall receive more cQwnt late?. mlxd I typo fi.e. b(x) 11’). Thon, hopetully, a reasonable For the Encehd~n-DiOM-1. thS largest pcssibl. rOtrogrd0 approximation would b. to ignore th. nFrd I type. But If eo is very mtion results frcrm *Am resonance with the porihelion 0 of the innu mall, then the doninant resonance may k tho mlxd I type. satellite. mting the saw appruxhations, but this time retting dZ Another possibility is that eo is R amall that tho correspondlnp (+)res -1 to (-2 x 1~2.~7.).uo find that ir reaomnce vrriable ia well-spacd fm any I type resonrnca txansition. If WI value ia lara than the critical value, then tho (A.l.5) aystep will automntically enter the libration phase with 4n initial

amplitude of YO.. Since tho rswnmce is tidally s*able, the moan

;!an the first resowe encountorad was tho one in which th. pair of value of thn eccentricity will theruafter lncreame. A1 C!w ncnn

r- 219 220 value of e increases, the mymnitde of the imprasscri retrcgr.de aPt-on oL the wihelion narst decrease. Eventually it vi11 overlap ulth an I-em resonance. Zt my bappen that this :: type dirrupte whore the dependence of tho inclinatlona on tha action variable x tho eStab1ish.d re00mce. dopending on their relat:.ve strength.. LD (2.4.41 2.6.11: mactly what nay transpire in either case would reqriire a ripororis

exmination of a two-reaonmce variable syrtmn subject to a cons-ant 'MI rsplied torque. - I (-x + 1)1'2 IO Dooi er*hr case have relevance tc the previounly discusred Mi examples7 In the Mimr-Tethyo care, metho value of e for 'To arch thaso two vexriblea overlap (4.1.4) and the present value of (4.1.7) hITe. ne find that the lattnr is of OUO) largor. Po1 this it appears that tho two overlap, I example, if rssonances the nixd %i ~h.m.9 ratio - ?I 3. ~husth variation in I~Wwith x ia type still predominantly determines the fluctuations In tha mean "re ($1 ChD smller than a corrnspnding cbnoe in I~~. his also langitudes. )I and 1'. On the other hand, if we make a suuilar Applies to any seculsr change in the inclination6 after libration coolprrison of the variables (hm - ZXDi + 0) and is established. The factor C can bo eqzessed in tams of kplace (zx, - 4xD, + si + ~7~1,W(I find that the coeriicient of ths o~fficients(2.2.15) and nulnorically *valuated (Tisserand, M1. 14, *type is 0110') groatar than the mLxod I type. 9 p. 100, 1896)t It sems unlikoly that e is or was ovu as .mall as 10- . If OTe this Le true, then hs-TetBys evolvecl through the sequence found (4.1.8) 2 in ~ablo4.1.1. The first resDMIIce ancountorad was the I res~n~co. Reamably because of unfavorable initial conditionr at transition, The probability of captum is determind by ths diaensionlers which occur8 a factor in pendulum-lib tern of the system evolved past this reaonanco Md later approachd the 1:' garamator as th. The approprbte garmeter in this case is resonance. "ha correspondiq~resonance tom in the disturbing thr thiltonlan (:.1.51.

tunctlon R has the forat i I1 a22

(4.1.11) The ma50 ratio wcurring in A- Sa Ir while &1. %-PI3 I f2I3. $ a - 0.9922 aTa - 0.9922 aT,W) ‘Te 50 om Evaluating B with the wesent valuea of the orbital parmeterr, M find

BmW - - 1.02 lo4. C4.1 .lo) Calculating 0 using tho above values, we find

Compare this with the critical value 6,1 a - 0.27 for which tho syatam 6, - -4.4 x lo-’. (d .1.12) automatically enters the libration ghse. Imidantly, the Since the value of the parmter Bo ia quite bmll ?onpared to one, incliMtlon of Whs would have kd to ix 11 for thi8 to occur. ’ !rb (3.2.12) CUI b. usad to apprtxhata the probability Pe for capture presmt libration wlitude (97.) ant thr .mull value of 6- into liboatlon. indicate chat 6, evaluated at transition, will aLoo b. mall.

%refore the approxinution rhveloped in section 3.4 ‘.n L?IO limlt (4.1.13; leal << 1 can b~ applied to this rem~nce. Neqlefting effect of r-dsp.ndent in tidal the the term thr -to that b(x) is proprtianal to (-I + 1l1/’(+ + 1)’” and that torque (2.10.6). we find that. transition occurred w’an b& 0.2bo, 1 Thus the mas6 ratlo of the inner to the ontez satellite is 4 p when mean inclination of Mimas wac one-tifth its present or the bx(0) euuels 1/2(1 + &) 161 here 6 aquds Bo. Slnclair Pound due. Allan (1969) included the effects of the x-dependnnt tam and through nwrical calculation that Pc ‘. 4\, which ia in agrecrmant determ?ned the value of the scakajor awis ud the inclination fqc with the ahve analytic rellult. The probability for LIw first mteiiite 180. nuasrically. His each +m - results are1 remmnce encounter can be e8tbateA by ccmparing the appropriate

function b(x) for M Xz Z~SOI~MC~uAth that for an 11’ type. For the k - +2 case bCx) is: 223 224

Frum Jefferies (lSU), the factor C .ppreth.t.iy equals - 0.753 2 For the 1' resonance, bx(0) quala 2161. The ooeffieimt ClI 1 CUI while the coefficient C' belonging to R' equals S-119 . The

be detcrnined by comparing tho lowest order contributions in the relevant parameter B for this type is1 2 disturbing function to the I Sna 11' resonance variable. #e find

tht c(12) - 1/2cIIX*). (mtet ti, obtain this result set idn - 4, m 0 4, p1 - 1 and pt - o for the I' tem. ~orttte 11' tern 2 2 set m - 3.1 ey the way, sicce CtI ) is positive, the I reaomnce Tho pumatar X is obtained frolp 2.6.10 and equals 1.59. The -86 m variable liLpates about the mcd(.r) peaition. Finally, the relati% ratio of the satellites is 2 0 &. caicuiating e,. w find %i ktween mc(12) and Pc(II') can be found by caDplring the equations

(4.1.9. 4.1.13. and 4.1.141. Spaclfirallyr

Since fAe critical value BC1 a - 0.27, and 1801 mBt have been

(4 I 1.15) ~arguthan l~,l, we can concld. that the Byitan automatica1:y entered the libration phase with nuxla~umamplitude of 90.. Since

sinclair-s estimte for th~scas. is * PI. 16-1 >a 1, the relevant approximation of tne t+amiltonisn applies 8 Incidently, Allan found that transition ocaurrsd 5 2.2 x 10 yeur (3.3.3). Using Binclaic's quoted value for the libration amplitlde ago uslng (1.9.4) and Goldreich's esthate of the dissipation (4m-m..l:5'), the changr in e since traneltion is obtalned fra function (9;' - 1.S x 10-~$,*is resonance a-s to have been (3.4.1). ope rind 9 established well within thn aye of the wlm syste. (* 4 x 10 yurm). (4.1.19) Th next case we shal: diacusa is th re8onance holving

Bnceladus and Dim. Tha zelevant put of the disturbing funotion The relation between the initial and the present values of the

is; parameter c(t) occurring in the Hanlltonhn equation 0.4.4) is given by 225 226

-de l4 I .20) c 41 0. . (4.1.23) Tha cbwe in c(t) is dircrotlp related to the change in the cammrurability relation by U.9.9). mplicitlyt Since the resonance only wakly aficts the motion of ths - longitudes in the limit 181 >> 1, we find, tor -le, that the change in the man notion of Enceladus since transition, s,is the fu!lowing. (4 .1.21)

T is the time since trensitioa. Of course c(T) - C-. The cksnge in the commwucdility relation OM be mpzoased in

of t;w for ea + terms present average value (eaw, - 1/2(eaax edn)) where k(T) equals c(T) - co - 41 c(T). This corresponds to a change using (3.4.8, 4.1.20). ~olrrwiy, in the amhjor -1s Aaa of Enceldus given ky (using the

approximate rclationr 5 i - f %) 1

Ae, Ae, 0.021 a,[T). (4.1.25) rt#re we have ash the (and seainujor axes) - that mean wtions have since hami cbwd little transition. Obseroe that the right Using Goldreich'a estimate for Q w find that the Wansition into slde tend8 em to vanish as increases. The Jmplication is that the 90° libration occurs when approxiante coPmnsurability tendm toward an exact one as th sy#tllra evolves. Goldreich vas apparently the fkst to make this observation (1965). wain, thii appears to b. an estimate vfehin the age of the solar

The naxt step is to det~rrtnehmW mch ehe mean motion ud systso. the .alujoI axis of Plnceladw &ye changed since transition. Titan and HyperLon are also presently engaged in an e-type

Assuming ~JIO same dissipation function for both satellites we find resonance. The zelevant part of th Siaturbtng function 1st 228

Incidently, value ol corresponding to EC1 27 J-.: the eIlu - ecl - 0.057. Id .l. 29) A. c is positive, this resonance reiable libeaten about tha d(r) Hy inste of the dU+)position. “he maen of Titan is OUd) Tha above value for 8,is certsinly close enough to the critical greater than that of syperion which impliea that 1) th. Udal value Cor automatic transition that M should expect that it did evolution of the reaowce is caused almost entirely by the tidal ocm. This deof tzansitbn CM tm inferred Iran the following torque acting on Titan and 2) ehe effect of the resonance on the argurmnt. orbit of Titan is almost nil. Assuming for the -nt that the limit 131 >> 1 can be applied

We Pa-tioned in section 2.9 that the x-deparadmt m associated to determine the mean value of the eccontricity at transition, we d\ (x) with the variation in - with x is small although a naive obtain the result tTi estmte of its effect seemed unusually large. The resoon the (4.1.30) estimate was wrong is that we assumed that the fluctucltlons in the torques were O@ ignored the fact that one tidally tidal of an8 - eHy(”cn’)sin(4d - 36.1. 0 driwfipartnar sight be nuch DoTe massive than th other. Th. This limit rapresents the fastest possible drmping of Om for the

pulleat possible change in the eccentricity of Hyperion. The

actual change in ew aince transition must be greater than the to tha tidal evolution of this resomnce can be safe:y ignored. above value. Thus this value (4.1.30) roprersnts an abpolute The parameter 0- for this case is maximum for th8 value of so at transition. After canparing w 6th (4.1.30), we can infer that capture into 90. libration ecl HY occurred. Unfortunately, we also see that nelther approxiaation to

the miltonian (i.e. 161 >> 1 and 161 m 1) can tm rigorously applied ta determine tho evolution of the wet=. Bvt we can Bmp - 0.0575. attampt to match tho solutions found for the tw ZLmits & &-in an estimato on the ewlutlon of the systm slnce tranmition. The .

219 230

(4-1.34)

Mditionally, th. ahange in eh. cxmmnmrability relatton is

Th. fractional change in %i can be relatud to M. prqsent value of

BHy wing M.L33,34,3S) I

Th. chang. in the eemimjor axis of Timis:

(4.1.39) 231 232

mthor surprisinply, tk above value for eo apteem r.amMbtv well I I I I I I w with Greenberg's (1972) estbate of 0.015. % change in ~ihn'a 7t semimajor axim up to the time F, a , ' the value tor T, can determind with axvent. sldlax to those applied in the'plceladus-

Dione case. The result irr

Thus the age of the remnancm and the total change in Titan's

sa dimajar axis are8

d.sTI a ~.ofa~~(noPt)~T a s x lolo years. (4.1.40) SIN

weenberg estimated that transition occurred 4 x 10" pars a*. I I - 15' 0.2 t I A mre accurate estimnte of hth the aoe and the change in the I orbi'al parameters can k obtained via the action integral. ?o 01 I I I II I I I I G" nuke use of the action integral, we must first reduce its 2cW dependence to a single unknown parameter. if we evaluate 4, H and '-P e at 6 equal to 0, - 36O where 4 vanishes at time T (corresponding to the present), M find 233 234

The explicit depedana on xm aa it WMS in E OM k elininatad The change in the parawtar c(t) since transition ,s related h the in fmor of e,,.. Ttrc raarlt is ohng. in %i by

The correspondinq age T is 6 X 10'' ;wars which is in reasonable

agreement with out earlier hexistie calculation. The evolution of (4 .1.431 ths paraneters as a function of cit) CM be found using the .(Me

procedure as outli?el in (3.4). Figure 4.1.1 is a plot of czle average

eccentricity (: 1/2(e- - emin)) and tha amplitude of libration as a function of the parameter (1 2c (t)/B). (4.1.44) - lhe raasonablenuss of these calculationn is naturally conditioned b. the initial arnrmgtio.,s. Clearly the mat crucial

of these id ths dependebe of the tides on a Constant 9 which is

the s~lhfor all satellites. If MFws could have risen from the seas of hturn, meladus an8 Titan would Pave bean nearly mtion-

less hpectators to the event. Only a 6\ and a 1/41 chaw. cur

Th action intagral Jlib CM b. americally ullculatad as a occur, respectively, in the orbits o€ Enceladus .and Titan. Thus function of eo aad (4.1.45) minimisad. The resrrlti,rg palws fox "significant" tidsl evolution is lbitd to tha closest satellites. the puyater.. are; Perhaps one way out is to say thet 9 b8 an anpUt~% dependence (let's guess that it's proportional to tho helght of the eo - O.O22C, 60 - -5.85 , au tides raised on Saturn's surface). Because of its greater mass, n might met that the Q for Titan is siqnlficmtly larger than it c - -2.62 , Crn) - 10.1. is for other satellites. I%e amplftude of** tide is roughly 235 236

obey (2.9.4), indopedent of the vdue of Q.

In th. Rbas-Tethys case the values of ths or!-it.L parameters

ma significantly affected by the fact that the x depe.*cnt temn is Ti& Reight eye)3. (4.1.42) an wrtant factor durrng evolution. The relative damping due to

Cmpaxing the tide hdqht raid by Titdn .ad by the nsrt this term is large only becsrrse the two torguos tend to cancel sn dn I ,% 8 strongest case, nthys, we find the colmaansvability relation (41.141;. If12 >>I-\, then ,dklI ’ dt.8. the x-dep qent term is much less 02 a factor. The evolution of Tide Height by Titan 1. the resoitance is then accuratel.1 described by the solution obtained Tide Peight b * Tethys 14.1.43) for tb limit ID) << 1. of course this mear~sthat the initial fmpl uhich we might conclude that it ia at 1w.t psaible that inclinations were lese than predicted by Allan. Going back and %itan is significantly larger and thrt ths age esthte for thia repeating the calculation we muld find that the prohbility for nsomnce is within rwsomble bo- . But if Titan’s tidal torquo capture for the If‘ rewmnce is increased to 7-81, while thzt fer is greater, then why not %thyso- Thr tide height of Tethys is the I’ resonance is increazed to but101. Contrary to about four times greater than that for ribs. The probllnn Is that expectations, the ac sorance is dec eased significantly tJm two tidal torques nearly cancel in the commensurability relation by about one fifth. .tuation holds in the Enceladus-

Dione cawe ii (4.1.44) with Mimes just barely winning the battle. If -d”T were just a dh few percent larger, the resonance variable becomes tidally meye already noted that the dissipative tern plays a minor role rmstsblel during capture and -tolution in the lkpit 161 >> 1. mt the W. should antion thnt the prohbility fur capture is cancella2icn o* the tidal torques does affrst the cammnaurabillty increased slightly C I*) by -h. % depedent teaa in IIltt d”T d”r gk,t). relation in the original probla. JW~evar,if - >, - , dtEn dtDi it ala, happens that the ages and the capture probabflitiea are the rge of the resonance is decreased by a factor of 0.28. Thus, significantly affected by the assumption that both tidal torques the age of the Enc radus-Dione resonance mybe as small as 239

8 4 x 10 yaars. Although we've generated aan~-e concerning tho ages and wlutior. vaxigg. remaancos dfncumed, M ffd god Qf tha that The won, like the satellites of Saturn and Jupiter, is spiraling crrrkrs bard find. are to This must be accepted as M werise away from the earth due to a tidally-iduced torque. If a simple which dwns*J.ates that Coldreich's hypothesis proba'fy is correct tidal del is invcked, ad if the present value cf the t*&l but that it generates =re -lee it solves. than acceleration is used to determine a constant dissipation function or

Q-numbol, then several investiqations have ctmtht the won was

within the Roche Limit less tlun two 'Jillion ycarm ago (see Y rcich,

1966). As the earth-moon systen appears tn ke much older, edm .lng

else mat be invoked to resolve the tine-scale paradox. Thc most

plausible solution is that crnplu factors intldence the tidal torque

and that, contruy to expectations, the erargy dissipation factor may

have been ccnniderably less in the pest. Pannella, YdcClintock and

Thompson (1968) examined tho tidally induced F.riu3icititP in the

dcily qrowLn structures of variouc type3 of shells of widely different

ages. The implication they drew (fig. 4.1.1) is that the tidal

torque was both variable and also probably less in the past, k-eyond

approximately 70 million years ago.

R. j. Wiplcin made the novel suggesl.ion that perlaps tb tine-

sca,e paraclox could be resolved if the moon were t.c&r)Pea for an

appreciable time in the past in a resonance with Venus. Goldreich

had already shcwi that partners of a resonance. subject to tidal

torques, tended to maintain their near comnennuabilities throuph a39 24 0

8 tranrlor of ayulu nocwntu from OM prtmr co rnothor. T)Uroforr,

hypothotloal lurur-VonurLrn ~O.OM.IYO could trrnrfu .nguiu

rumonto from tho lunar orbit into tho much lugor orbit or Vwnur.

Hipkin roarend thab buch r pfocosr would noqllglbly 4Cfack thr V~nurlrnorbit, mi would ~Ifoetivalytrap cJto lunar orhit at a flad

radium over tho llfrtlmo of tnb roronrnco. Ho 11 proerntly #rkAng

on this prablr, baricrlly uaing the I.D. approach am wtllnod hero,

ucopt ChrC hlr m?utlun lr rpparontly mch mor0 prM1ao. It rhould k mntlonod th.L in hto solution land horo 4100) thr planohry orbitr

are rrrwd CO bo both clrculrr md coplaw rlnco othorrlrr thr

rxprnrlon and ovalwtlon uf tho rrlovant eorfflclontr &coma

Irlordhataly dlfliouit. nut In calculating tho effect ol rocond ad hlghor ordrr coupLlw, HlpPLn door take LntO account tho coupllnq 01

toma in tho oxpsnmlon proportlorul to the rcrrntrlclty of rno of t!w

~CWMMO partnorm e?r_lor. rrttlnq tho planmtrry occb&rlcitIor to

uro (rr.10, private coawnlcntlonl. V. rhrli Clnd lator tht tho IXGURS 4.2.1 opproxlmation OC olrrulu plrnrtary orbltr 'tfrctr the nuhmvaluo

of tho rrsomnt torque. On tho other hand, eb .tMr ol tho tidal tcm!~o In tho part la not wll ~mteblldwdaithrr. RurmCors, th firme stop rhould bo to calculrto tho rrlovut gravlUtlona1 mrqur with appoximtlonr whloh rilpllfy tho calaulrtlon am auch a~ porarblo, ad ocmpwo thr wpnltud# of tho rnsonant torqum with tho pramont vrluo Cor tk. t1d.l torquo. IC tho rU.1 torquo Is mch

proator than a (11v.n rororunt torquw, 0r11thoto 11 no nod to 241 242

'urthar reflru t!m calculation. Th. kgliaatbn la tht tho tidal .voluelon of tha lunar orbit emld net b. arroatnd by th. phon crsonont Intarutlon. b)

Venus 18 chosrn a. Uw mort liktLy putnor in any ramo~nao birturbing lunctlon Aating on tha Moon by th. #un. b.cr*~~oit inducor lupe purturbatioiio Jn thm maan motion of th. mon duo to ita rslatlvo narrrora to tho outh. Although Jupltu lr C) a ahmre ~a~ivothan Venu8, a coun:orbalancinq factor of (-1 a, liqura 4.2.2 rhovr all tho rolwant rrdluu vectorr and aplrr. lb sntus in tho devslopmont of tho d1n:urbing tunctlon, whore tho rbplify tha oxpanrlon, wo rhll mako rhr opproxhtronr tht all Integer c Io the ratio of tho a@.: moan mofron ol tho mon to tho orblto am coplanu and, ucrpt Cor tho lunu orblt, ai0 circulu. 8ynd(lc wsn omtion of the d1iturbi;y plAIIOt. The ratio 0 ir - 14 In uklltlon, the motlon of tha oarth about the barycontar rhll bo for Jupiter and - 20 for Uenur. A *:rbial oalculation .hour that nq1rCt.d In uch or the dlorurblng functlon. lhir lutlon can bo

inaludd in th. oxpanoion by olpmdlnp tho redwant vwtoro abut th Hlpkin, lo Wrcury, rhlcb hi tho :tpullest ratio c ctf any planot hryC.ntOZ Of th.1 outh-n 8y8t.n fP1-K. IWO), but th. wtd f- 4). Hipkin.' original arpment. ~pplldonly to rosonancom of th. error Introductd by noglwtinp it io of oi-"e?4 e& and I. -11 oynodic typo, bct as w bvn reen, :h. simple o-typo rhould alro k c-rd to other approxirutlonr to bo Invoked la*er. Pinully, ttu mnsldored, b.csu8e of cayt'ae ccnrld.rotions if for no othrr rwmn. intora-tlon of tho mwn on rh. planetary prxtnrr wlll k Ipnord l?~emtW follor*.d rn cote:vitiing the first and rrond odor bocruso or tho planet'r rolrtivaly lu9.r ma## and angu1.r MwRttm contributions of the di8turbing func Lions acting on th. nwn ir th ~incrtho intotaction ktuoon tb p111n.t ant moon wlll tend to prucduro outllnod in o.ctioa8 1.2 0.M 2.5. ?&tura:ly, tho flrrt mncorvo angular momentum, any change in thu lunar anqular mentum 8t.p is tho expansion of rho rs:avatit dirt*xbing functlonr. Th.y aroi prducd by thphnot will bo balrnred by an oqd and opporlta o18turbing Punctlon Acting on the )coon by a Planot. chnge in tho plrnot'r mm8ntren by tlw mon. Thmolora the rrtio of

tha Cractlona:. chngo lur

e 24 3 a44

Tho above ratio indicrtoi that th. portuibatlor ut thm planat's crbltal

olmnts by the nann is cplto nqliglblo. Therefore, v. ody need

d+twrlne tha first and recnnd order mntrlbutions of the prwiwnly

scntronod distfrblnu funttlann. Startlnq with tha lurur-solu die-

turbinq function, R(0 + 3 1, let's first urprnc the dlrect part A-' (2.3.1).

PI- 0.2.2 vector dirgran of the planrt. earth and lunu poaitioru with raspeat to tho sun. 246

7hm an#. 0, lr jurt the angle mde by the vector D with roipoct to ch. rofarncr trma vectx In the orbi-1 plane, 9 Wig. 4.2.2). This mgle Is related to A and Ae through tho following rolationsr P

Ran tho riyllllutxy prgyrtios rst.bllrhd In soctlon * 1 th. abwe rill collapso U, a coslno rerio; Tho wpwitl:l can k r*plscd '.** 247 248

which vani8hem for (I: - m) odd. Given P - 2, ImI is rastsict& to the v.1u.m (0,Z). For a aynodlc type re80nanro (q - 01, the II - 2 tum i: the above axp4nrion corre8wr :s to twice the coaaensutahllity ratio, or a double ai010 reMnar.ce. The dimturblng function contain8

A ginen anqlr in saies has fLwd m,q (:I 8). tormn the and - %BO a factor of $Ip for a glven 4, and would h mimed to thq 0 n9rP which hvr the same 0 are &e mre obvious if II MW -tion eiqhth(ls.8) an4 fortieth';*40) pouue for Nercucy and Venu8, vaiable. p, intrduca?.. im respectively. The lowest order contrikrtion to the sinqlb awl0

p-J-s. reeonanre is contair.ed in tho L 3 tern of'tJ'te wension. A rouqh

compariwn of the cciefffcients of +he single ~nddcubla 4ng:e terms md the variable 8 i* elMnated. Sinoe th. resonance pattnOr8 of ia given by weatest interest are inferior to the earth, chose A, - 'Le. Tho rrsult of the- oporotions LI +ad

f~lculstionof thr bbve ratio Lrdicates tlat it is within an order of mgnltude of unity for both planets. Therefore, both tema shall k explicitly calculated for th*i synodic cwe. For the strongest poemible e-type reromce, q - '1. as [ral - C.Z for the I - 2 tcm,

the lowst order ccmtributlon fGr the ee'ype corresponds tf3 *h t4.2.6) siwl. angle cam. TW ratio (>) implier that the R :! term should give . A, - the Tte bW8t order contrihtion to .iwn angle can be vrrtten B - ddnant con-xibution. Th., inclination funation Fi,m,i-ra (0) (2.2.5) down dirrtly, usiig the explicit, larect orde. sxpansion of the 250

'%e ~aplffiecoefficientn CM Le ovaluatd us!ng the equivalent jrolymnial axpr?.ion (2.2.111. Wore cctldat.ing the above, lrt'a determima tha impnrtant Eaconl order con'ributjons.

T?I~ indirect influence of the wn CII a giren resonance canhot b. n.glected because af its rblatively great ~~l)sn,Unfortunately, the effect of the sun on the lunar orbit is 80 larye that an ordiwy perturbtion expansion of the disturbing furct:.or. in powera of the 251 252 mas, oquivalwhay, in c-q*s slowly or .-s of 3 , .* nore {W,J) rsprssrnte tho conpleto sat of conjugeta actlofi and anqlo n> Since obloct is only dotdne shook, 1964, -2. Thrh). our to vatiablms. The Zrmctlons bL9 , bw, , etc., arm th. short porldlic the plausibility of Hipkin's hypothesiu, h calculation lowst of thr first ordu pertwbtions of tho canonical variable6 and arc obtained coupling 6uffi:Lont. Wc found In soction 2.5 chat tho order vi'l ba fram tho epproprirta gonorating function through tlu relations (2.S.91: sa:o,d order contz ibutlon tm a given tern m 6 (2.6.2' rccurrd in $gv AS(J,%) 3 bJ = - 33(J,Fi) tRs, tt.0 coupling of shozt periodic tonu in Rsr and Woo. the coupling x* (4.2.11) of short periodic terms rn the urportwbd Hlmiltonlan Ho. "he rno gonorating function s LT,~is relat M to tho appropriato short tunsticn case, -t 6Ra, in is of 6Pmlp 3 ) and dR6(8 +B). priodic part (Re) of each individual disturbing functiob (2.5.7) "h perturbation in only incnlvea the pcrturbatbn hioh 6H0 6L *, acting on the wn. If ach Rs is forrprlly express.C an a transcon-

We is c-sd ~f bL IO + B ud 6L J (p -' ). ForUlly, ach of B dantal 80C-O. In 0 (2.5.4a). tteso Cum8 is 12.5.14)i -.-- R~ a HC.\eei'b, etc., \? 61,- 4 ~~~(d2L(g-p)+~L(1~-.D);iL(t~>)+h2L(0+D)). then the gencat!.ng function. arc f2.5.17) t (4.2.9)

This allovr us to write down bn0 Ma bR6 in twos of th. above fond ocpansioss for S and Re, and dotrmfno which contZIbutloru

L-o aignificonti 253 251

Given an angle 4, thsecond coupling correspondhq 4,P' ordor to 0,. + es,* will be restricted to a single term. That is, the

indhOS Qa, 9-8 and J" are fix&. Th. S.Cond odla couplinp Corres-

ponding to + 0,. will inwlve SOVOZA~ tom, which can be rentricted to the tenue of lwert order by ~inlnizingthe number of &Ho = factors of & 4 ep which occur in the axpansion, consintent with whero e the given nngla. The ratio of the relative magnitue% of second ordu

coupling of gerturhations of the lunnr orbit vith that of the earth's

orbit in 6n is roughly Fa~~llthat Y is a function of L (cf. 2.2.9). %r.iore the partial dariv-tive with resgct L eaplicitly acta on tr.qwncy v in to the (4.2.15) the above equations.

There ia th ?bvious roquir-t that each of th. above angle6 rnnrming tlwt the sum of bplaca coufficientn is approxbtely the SAM. a~~eratio tends to hold contributions frQp Os,, $sm, or ?,,. has nonzero fraque?cy. which is derived :ram the Th. for 6H0.

Given that *h ltgmssrtant contributions invol- -I frquencic? of O(nz ), short periodic nature of E*. &awe of tha assumption of coplanar, the approximate ratio circular (=Copt E) orbits, the only action variables vNch rntu in is tlu A~OW ue L), rb , L,, ~nthr caae oi synodic remmce, L~. .a. Z.ic J the set CM be rastzicted still further with the asamption that the

IW orbit 1s CumlA?. The explicit ang:teS which Ocain the Thus it aepurs that tlu parawbations in the asr'Lh' pc-.it(m above are (4.2.2.31 give the principal second order contribution tc, 4 given anqle In iic (p * p 1, which auggsrts tbt the second odor contrik~tionrbe 259 256 restricted +o 3 tat these UIPI. Yet another reawn L that only th. coupling dua b the firat ordo t.po. of the wlar-1- diaturblng function have been considered. Th. secod order tori4 of the solar- lunar disturbing function coupled with the first order perturbations "e 2 of the earth will produce a contrikrtion of O((-) I. But thio is "b the same order d8 those tmau involving first order coupling of the lunar cooldlnrrtes. If thrue approxinntiono are acceptedd, then the principal contribution tc a given angle +,,, is 4.P

(4.2.20) U.1.17al the contributions crm 6Rs for each of the arqles already considered

are 1

L b) (4.2.2la)

(4.2.10) 257 258

The factcr A- in the second orduc coeiflcient in the axpansion of the zero order ifamiltcnian (2.7.2dj. Since the remnance has litti8

effect on tho planets involved, A- is appmimtely given by:

(4.2.23)

2 The tidal torque tM 'A- &? acting on 6 ia (2.9.10.17):

Th. intege? p 1s neyative, implying that each of the bracketed tarnu, (4.2.24) which uo sums or uiphce codiicients, is positive. cornprring th above results with the direct part determind ealioz, we nee thst an, ha oldest detMiMt1on of + is that of spencer Jonea. bot!! contributions hsv. tho same sign for each angle.

The next step in to colpair the lunar tidal torque with thy puximwn torque due to resonance intrrsction vi*ii the given planet. Recent fnvestigatiaIs Newton (1969), van Plandern (1910). If the torque due to the action of the planet is conspicrously largex by than tre tidal torque, the tentative conrlusion is thst the resonance Morrison (1971) and Oestervinter and Cohen (1972) fndlcate tha: abut tvice Jcnes value a amre roasonnblc estimate. is 'tidally stable.' For this comparison, the rscorwl order equation Spencer is of motion for the angle variable 6 0.9.17) in most uaerul. This The functicn A1cos# is a tam in the disturblw function related equation in appraxbatoly to the ces~ll~ce,and is thr sm of the contributions Cram the direct and indirect parts. Explicitly,

bl .2.22 1

Another interesting question is hor Lon? ago the re6onances me 259 260

U .2.27) - Planet Semimajor Xean rr:aos Sccen- I ncllnz- Ax43 notion n Ratio tricity tlon to )Sotelute.or o..V.) 10-7,ec-~ Mp/I:Q(ldEc1ip:ic 4 I-.- . .- . - . . - . . . is calculated wing Spencer Jonea value for '3 (4.2.29) =m*qP *to 5! 0.3871 8.396 0.165 0.206 7'0' since it ap-s that the =re roeentdeterminstion applies* ody to --I------.. --- -. . - . .. 0.7233 3.230 2.44 O.OC.68 305 3' UW present value ad rat what it nuy have been in the distant past. iiLFTh-Q ~ ~ - ._ ._ 1.974 3.04 0.0163 Table 2 contains the relwantparamsters nede to CcllCUlate -. iiuin b 0 2.57 29.37 0.0376 0.0549 5 .1-2 ' Table 3 the n-ice1 OF & rb1mant 811138 A-Al. has valuer of Laplace coafficients. Table 4 has the numerical evaluations of AID and A~~.the estitantad rge oe the particular resomnCa, and an waluation of their tidal stability. &aa = .00845n, = .113ne (sromnde). Ths rewmarce angle 6, has kc- anmtructed w that it AOP $fib =-.00401n, =-O.0536re (retromade). sculrrly ducroases in the absence of th remonance. Therefore the

analysis of capture and thr stability developed in chaptar three (3) jirectlp sppliem. At the end of aection 3.4, we mention that thue lABLE 4.2.2

are varioua types OF stability, the most important here being the "tidal" and "adialmtic' stability. Rscall that adiabatic stability

im governed by whethar tho rmgnitude of Al is incressad or docreased

by the long tom tidal interaction Such that the amplitude of

1aa:ion deersagen with time. FOX -type remmnces, the leading

factor im ek), uhue 262

TABLI! 4.2.3

-teal values of mnua of Uplace Eoafflcfants occurrhg in the expansLon. 264 263

L'nfortunately, there are same serious flaws in either the For arll 1ibratlOM Gctt)) *t - to'. since g is positive appmxhtiona or th.3 suppsed effect of a given type resonance which (4.2.24). e increases or decreases, 3epadfng on whether q is .U drastically ch9e the results so far obtained. Rlso. ths quemtion of mwative 01 positive, rsspectively. Thus +2,-l,p is adiabstirally the likelihood of capture into libration needs to be answered. mt stable while 40,1,p is not. The "tid.1 stabillty" of a given first, let's -ine the approximations -re clssely. resonance is dotexmined by whether tho tidal torque is greater We could discuss the effect of second and higbrl order terms (mst&ie) or less than (stable) the maxh amplitude of tb which have been neglected. Although these terms FAY be Sizable, it resonance torque. it unlikely that they will critically ckanqe the ordor of lnaunituda We should ob-0 thnt the adiabatic stability of tho syndic of the coef:!cient A1. It appears at the qrossest approximtion is resonances is not geverned as much by whether b(x) increases or the circularity of rlanetary orbit although on the surface it seems dec--eases, since this is a rdatlvdy -11 effect, but by the to be fairly reasonable, at least for Venus and the earth. After all, as:maet.ry of the tidal Urque. This ametry arises fram the fact the present eccentricity of the sxth Ls - 0.017 while tlut of Venus that the torqrle is a npidly decreasing function of the planet- is - 0.007 which are both quite mall. Wexcury's ecce: ricity is satellite distance. To lonest order, this aeyuaetry will add a i -.2). But in the past :&mer and Clemence, 1961b). these term to the right hand 3ide of tho second order equation of =tion eccentricitice have varfed considorably due tv the long period for + (2.9.17). This term is perturbations of one planet on another (Table 4.2.5). The zmon

itself is indirectly aiOected by such perturlations, especially by (4.2.29) the long period fluctuation of the ear'h's eccentricity. These

ns the coefficient D is negative, this asyanstry ~liosa fluctuation. have periods ranging frm 50.000 years to approximately tu0 million years. The libration of all the planetary-lunar dissipative maclmnisn which tends to damp Out oscillations. The periods

results in Table 4.2.4 indicate marginal tidal stability for Sam of resonances are all rouyhly given

th resonances. This qualitat'-'dy agrees with Hipkin's results.

The rtrongest adiabatically stable resonance involves Yenu3 and the resonanca variable #2,-l,lg. Mercury also has surprisingly strong So the iaportant question is, how ooes the fact that the planetary rmsonances. 265 266 orbits are eccentric have lonp priod fluctuations affect the coefficient Al?

There are tw separate effects. The first is due to the short period averaqing, in which tne eccentricity is Veated as a

'sonctant". To eathate how it. changes 1,a rough estimate of tho Next, expand ti)' to secord order in e. Us0 use the fact that s >z It0 Mansen coefficient relwmt to either planet is roquird. l%a simplify the coefficient.. The result ia definition rf x: 12.2.9): a (e) is

Ii. 2 .3l) Next express the above in toran of Pl, accurate to second der in et

Since -ne of the rase-e variables contains the p.rihclLon of either planetar. resonance partnex, the relwant coofficiant is

le). 'Ib w. lurte yo (a), relations which connect r and f 4th set n nre needed. The radius r is related to a and t bf

E=?- 1 (4.2.32a) 0 -e cosf * while I is relata8 to H by the equation of cents2 &mart, 19S31, In the 1u.m - 1 -ietery dL iturbing function, the lain- exponas' of which is, to lowest oder, thr KB . rrf. is approrimntely lpl, where lpl >> 1. m.erefora

p the above equation for xo,t(e). ~hrrs,to lowest f z i<+?esin!:. b) set c . u. order, we can deduce that taking the eccenbicities inL account will

multiply by the following factor; (4.2.33)

data in Table 5 taken from an article Browor van apanding the sine and cosine of Iet~inbl,using Wu301 function The is by and Woarkom (1953). gives maximno variation the expansions, we find Wight. p- 198, 1C6'J), &d the of eccentricity

F- 267 260

du. to th. Interplrnatwy puturbatlonr jurt dlrmrrrd blw roe POUVQad Clwnar, 1Nlb) wiry th. Uplaae-t.puanps appow(nrtian.

It rpprrr that th. Cluccurlonr Ln bW for Yenur mat b. of 4t 1r.t an order of rugnltudr and MY aetually ravorse thm mLpn sf bk).

Morcury, on thn othnr hnd, (luctwua only by II factor of two or

three for th. ringlo rnqle ryndctlc rwrwnce and thr .-type rrmonance.

Thin ls~pllr~ qt the potentlrl rrsocrrtd vlch th. aero-

mocenulclty rpptoxlrutlon 11 now rpllt wwng rwaral “rid.-hnd”

Crrqurrcier which dllcer by th4 wtlonr of th. pr-rbellanr of the rkth ~ndtho Olrturblng planet. Tho motl3nr of tho Innrr plmeory

porlhellonir ire or ordrr 5 I 10) me. of rra p.r eontuy, 9;

C-rr thbr vlth tha mwimm vrlwi of 4 that the rrmnonce can

II\BLLE 4.2.S wlthrtand richout boinq dl#ruptatr 269 270

(4.2.40)

1C 1. The pramotu B 0.1.5) can k relrtd to wllcu :! 3 deLlnod Lunation. ir w identify )r*m;1ceI’l am Al ad Mo a7 np,. ?~llcl:ly,

1 where u+ hrvm mlt1pll.d by (\oxx/”o#c) and hwe urod -5 . a Tho atrongert poraible adiabetlcally rtable rrwmnce b.lm9~tc

Vmnur. Prom Table 5 we find

Calculating 8, with prorent value. lor the lumr orbital rl.mont, w rind ttmt

(4.2.43)

For aro .-typo reDoonance hare b(x) la propcrtional to e, 8 tendl to ncmle like e-3. Therefor;, IC th. lunar etcnnuicity mre ten tima amllrr at tranrltion than it La at prermt., the probali:.Iity Cor

capture would he increard to 4 J8.

In +h. cam or tho synotic remwc., captur apyurs to tc

m1n:y du4 to the )-dapdmnt tam aaro~,lr’.e4 with the tihl torque.

Rn emtinuto Por Pc can bo obtained frm 13.1.l4,15)1 272 271

orrcm notion, uxl thusfore the sclolnujor -28, is fLrd IC t!m nuon

ir trqped in one of there resonances, but t.b moan ra3lum aa:hully

(4.2.44) tends to increase. The short period avoragr~of r to ]own*: order :n

4 is

(4.2.4 5 I

The present value of e> put.. M ul'por burd on the lifethe of ar

.-typo ceoonance. Thii fraction im qualit&tlvely -11 CC~JWXC~to

the hmge in ab due to the tidal accmrerrtion over the 1i:ctfrSo of heprobability :lor capture j-s 'I. 0.5 x lo-', an8 unlike thr prsviouo th. earth-moon ryrtm. Thur, only the ryncdic type ieronani-e case, thxe appc.trs to he na way to rubstantially increare Lt. The offactively trap thn racon at a Pirod rad!d*. But the rynodic conclusI~~ito bc dram is thnt capture ia a highly unlikely want iC rem1wcem are, at beat., muglnally stab10 to tidal dirrupt:ori, .#it oniy tidal Ccucr-; are imlvd. subjwt to long period di8ruptive torquar due ts thr *~ccu~.I~*

The 1. nq tern varirtiLns in tho pondulm torqum armci8t0d with intecplanetary perturhtlonr an1 the anrociatwf "secular. cutipe An a qiven remnanm MY cailre tmprary capture. Since the variation the lunu moan motlon, and hwo a very low probtbilitv of c.ir%we. in the c~mfficrc-tb(x) is a significant fraction of its man valve, Wm we forced to conclude that it. Is unllktly that the ?Don war wur w shou:< expait that th? probability -Cor t-rary capturs lr much tralped in an orbit-orblt resonance with a planet. lrcqer than tho :w~~br~rr-alc?lat.od for prlmnent capture. But the mere la one last questlon to ask. C>uld the orbital tr1mmc.s ncxmm cine +'ut tho llnrotlun can laat IPof O(105 yro), the tLu have been mlgnll'lcar~tlychnnged nL'.Iier by pamerge thtolgh Cr rile t* LyI.: 'BY the p~rltds.saurciated vieh the so-called secular renonancr, or by temrorary capture? Ttre ctanye in t-70 man value of vciatlons of tho planats. Theye is one final arqumnt to k the remcntm variable due to paorbge throl.gh re601~nt:eIs -- laveid against tour earlier remlta. It ir that only the synodic crppraxhtrly given by U.2.47)1 type :@mnances tend to "look" the amvn at a flxd radi125 witbut e?p-ec15bl> aff feting t.ta other orbital parwaters. Recalr that the -sta: is .?-tjFe r+so'ianc+s Increnvo e if twre is a tidal tcrqur. The -- The theory developed in tho first thrro chqitrrc has nrt rlth varying 6UCCO*S whrn applied to rpcirlc examples. -,a pri-icqrl prc+blmir that whereas the Hamlltonian invokud Is aie-~l~o:,r,wra', the, real world is not. There always aesned to ha a !onpl,cBtir.: : it

MII ride-hnd fraquenclrr in the caz- of lumr ad plarrury ro:ronance. In the -pie of the tm-body ros3nmc4, It was tlid p*!iciblllty that Kunotlno in their evolatlon tvc, reswwxx vaz rabies aawciatd with the &vne c-nrurabllity might overlap arn? deitroy thcr slmple one-dimnalonal dorcription oP the phrnmena. Vhotbrr nu:h complex pheranena wlll yiuld their sGcrets so roaCIly as the

Mntljonian w dorived with the analytical tor-ls developed to dercriba traneition la best lePt to future invemtlqation.

Thore is D mcre 6pscLfLc umplo of 1) ratollltr oso~~ncelr.

V..,Lch the nasnszity of a multl-roronance-var!jblc t! ory e0 uplrin lath crytuv and its rreront libration Mplitde soins uMvoiP.abJe.

?Iris is tho throo Mi T-jracn relation satisfl*d by :he true@ i.,rcr GalLloan ratellitou of JuFIter. 'a~x- 3A~~~'. za~~~~' Olagihara, 1972). 'I?*following uhsorvattons aImut t'ylm relatlon are especially lntcratntlng. 1) WO libsation mplit Id# has Iran o?>#ervnd, 2) tho utn:litem ala0 nearly :atisfy a 211 ccrmci.s*nabili~y

Lrrtwaen tho innu pix (JI and JIJ) and lhc outer palr WJImid JXII I

3) at the prorent the, a tldal trrrquo wldch acts :rlnclpally OR

c- 271 27b

Ulr correapmding coaine argument (2.5.2.1). It .o happons thas &lulllut Ism X.gih.ra, J973) has dovelopad an 4MlytiC t.?~.oryalong tha aame lima as outlCn.8 in chapter thyee, and ha linda that the

dnrimnt contrlbuttuns involve tho coup1:ng of e-typ Annie vsridbles which nearly s4tisfy tie nwr ccmRrnsJra.~llLtlnsmonq ?.he inr,u

and outer pairs of s~~trllit~:~.Wm chid A~SOpoi:.t out that th

synodic lrmquency occurs as th. dltference 02 tw *-type froquencies.

~ptcifrcally, thy arc

o, - ob - ax + 3iX1 - 1~~~. The frequrncy trcurrinp in tho srcnd order coupling of than. dolPinant twms la

3 m

We should mte thAt the true fracyoncy .tssoclaC.d vith Vie rrlrtrd

8-ti-p anple variablc diffus from the rhaby LII amrant (2)'FCO.:) and is a natural result 02 the perturbitlor uprn:cxm. 5w

lluctuationm . which cay ocw in v due ts the remwance rippou to bo rartr1ct.d to the nnm notlonr. Anothwr pers:bilrty is that il vld

e cor@tam did cont:tkrte a luge retrograde mot~iinto 2, Ltm

abrence in v may indk4te a b?enkdonn in the pertwhtion nethJd. The difforarices of the mo4n notirmr in the tern (nX - 2n,I) art: a77 270

mscause OC capture wnsidoratlons the icond 60~0.the most plaumibble.

Whether the above r-ke have any relwance mst awit a nore

rig-. examination of thin type of pr~Slan.

thr of the pornsibis lmplicationn of ptevioua rawrk. is tbt th. xntesrctlon of ow or mxe e-t>pe vorkUes rlth the wm?lc xariablr

Vila c?xi.al In both CAD-.W~ ard suhrsgclmt 5alrpiz-g 0. the rei~~~nce.

It thesw variable8 ware cignif :curt, then $he Impressed rctroqrada cliu, nntibn of, ray, munt haoc havery largo in the past (greator *

2 Pram (4.1.3) we find t ,as laryr than 27OC.1/y.er). That IF I uer~) as 11-2, the haprossed retrograde wCim of f is st.111 ti O(dtgrces,; day). Hy qwss is that the critical wolut-on which I rplaIns Vu present malt librations appears ta be tied tci an 1p.t rmctiort involving synodic ard e-t)pa anylr variab!rs. How thome vari.rbLsi cam6 to overlap can be explaired by either of the fo113*ing tm

8cenariasr

1) Ramition lnte, the tketbedy syncdie reeonawe occurr*l

fir*.t. Th. sya:~.than Sr'0lV.d through the .-type CesOMnCe.

Tt.is varhbles' camplw interaction led to a rapid dampiny

of the arp;itudo of lLbration OF the synodic variable.

2) Fxqt, a pair of satol1ites established an *type reIOMfiCe. Slb9equent.y the systun evolved toward the 8:modic ccmn-

e.wab1lity. scrneha thn arIgIMl ,-type rrHJMnce we.

dicNFtd &.ill allOW-ng CAplWe Info the StnodiC tC8 OCCUT. P. CoLdreioh and 8. J. Pealo, "Spln-Qbit Couplln? with Solar Iystm," Utron, J. 2,425-28 (19661. P. Goldreich and S, J. Pmla, "Sph Clrblt Coupling in the Solar Eyitm 111 ?ce Rcltmant Ro-ation ef Venue," PitJon. J. 22 6624 (1957).

P. Goldruich an3 8. nota:, "Q in the lolax Symtm.' Iernis 2, 375-89 (1960). R. R. Ell.ln, "Evolutlon of #Ilms-Tethys ComsnsurabFlity," Aatmn. J. 33, 497-506 (1969). R. J. Gramberg, C C. Coureelnan 111, arul Shaplro, "Crl>lt-Orbit Reoor.ar.ce Csptucu in the Solar Sy6tmta." %:..once 2,74'1-9 (1972a) R H. Bas-., "On the Po-.lun of Charged PaftiCloS in it Slightly Om;.& I:inumidsl Potcatla1 NIIve," Phyaica 0, 182-7fi (1968). R. J. r reonbrrq, "PoLuticr. of Orbit4r'rit Rcsonmceo i? the :alar Systrm," Thesis, MI? flC7X). H. 3rn, Atomic PbysLzz, fBlackie C ¶on, Lo.don, 1!)4S), 3rd d., pp. 108-110. Y. Ht.g:hara, Cc1oi:tial ??cdanica I WIT ?reas, Cuubridqc, 1973). Y. t%gitanra, Clertlal Medanicn wnrr Preas, abridge, 1972). Vel. 2. J. E, IQzel and T. R. Y'A?jcf, ScJencc e,201 15'69;. 0. t?-smer ard t. M. C:cmcrye, Planets and Sstcllites I11 Nniverslty C. of ?,I=~IW, Chlcag?). 'hap. 3, elited by Xuiper and 'Yiddlehurnt, R. thpkin, "Orbit-Orbit Couplmq and the Hia:oty of the Earth- loon tiystm," Phys. Earth Vlanet. Irt. 4 (in prqsti, 19:'Or we Kar;:a, 3. 9rouwr dnd A. J. J. van Wcerkom, Aatr. i'qmrs Am. Ephan hut. 1970. Aln.s.r.ac z, Part 2 (195C). I. G. IZ8ak. J Ge0pW.n. Rts. 69. 2621 (1$64).

J. C. .Lickeon, tlaasical electrodynrnics (Uiley, New Yark, 1962). p. 101.

If. JrCfrios, "On t:ie Master of Satu17's Satell-tes,' Man. ?at. R. Am-. Eoc 113, 81-96 (1953).

F. T,. Jonee., 'The Roc.at1c.n of tte Earth and th: Secular Accelerations of the Sun, Mocn ard Planets," Ilon. Not. Roy. 4stron. sa:. 541-8 (1939). H. H. Cuight, Tables a' Integral8 and Other )*lthanatical Data C!sc!lillan, NmYC)Ikr, 865:. W. 3. hula, "'2idal CjHSd.FdtiOn by -lid WvDr'.CtiOIl and the Roou! tity Orbi-al LVolutLon," RWS. Caophym. 2, No. 4 (19641. T. C. van Plandun, "The Secular Accelrration of the Noon." Ar-n. J. ~2,-- 657-8 (1970). W. M. Kairla, ThWry of Satellltr Gra- LBlairdrtll, %!a1tb'Qp,Nf1.i.5.~ 1966,, P. Goldruich, "An Pxplt.Mtion of the Frequent Occurzence of W. M. aula, "Dynmical ASpct# of Lunbl Oriyin," &YE. Ccophys. Comenturable Mean Mot..one in the Wl8x Syate," Mon. Not. R. Aatr. Phy~.2, 217-28 (1971). Soc. e.15941 (1965:. Spa<* G. J. P. MacDmald, "Tidal Friction," Xcv. Gwphys. 2, 467-541 (1964). ------,"On the Eccentricity of Satellite C bite fn the SOiU s>sem,' YCR. Sot. R. Aatr. Soc. 2,25748 (i963). 281 L. v. slorrf-, "W.a Scular accelarationr oi the Hoon'e OcbJbl Xotior. ani1 tha eatth's zotatior.,' Tho Xeon 2. 25344 (1972).

W. W,kotd G. J. P. Maaonald, The Rotatioa of a- (CmLoidgq University Prem, Herr Y*;rk, 1960).

E. 0. Dakley, A~lytlcrmnctry thrnss and Noble, Iosw York, 1959).

C. Oesterrinter and C. J. Cohen, "New Orbital Elemintl for Moon and Planeta,- Clostlal :lwtymlcs 2, 317-95 (1972).

R. R. Newton. "Srular ~CCebC4tiOneof thc Fa'rrrth .ai Moon." Science, 3, 825-31 09i5). vhor. li ir a solut!on of the H-J -tion :. c. Parnella, MscClir.tock and 3. 1. Tholnpson, "P~laontolOgiCal Evidence ot Varitions in Length af Syndic Month Since Late Cam'rri.n." science (1968).

S. J. POiile and 7. Gold. HktUCS 206, 1241 (1965). Since the disturbin, furiot,on R is considered wll colpparad to H0' R. c. Plumor, An Introiuctary Treatise on Oynaniczl Astruraz (30vsr. NEW Ycrk, 1960). arA em b. expeded in tams of ai, BI, and t. $at. H' - R. ?hs A. Roy and Ovandnn, %I!. Aatr. Suc. E. n. W. Not. R. 1)4. 2.4-41 .s[uations of wtion of thm new variable6 become (195:).

A. 1. Sinclair, "On tlm Origfn of thL Ccarennurahllitior Wmg tho tA.3) Satallites of Saturn," *on. Not. R. Soc. 160, 163-87 (1972). w. v. mart, hstronany Spherical (Cambridge University Press, bndon. In *..e lmit R - 0, (ai,Bil aro conrtsn%e her.fore srt R - 0 b, 1944) , 4th ad. detsmina (aL,@il. In cpheric-1 cooxdlrratoo tJm --body P. ~userMd,hait. de Hdcanigua Colests N (Oauthier-Vil:ars, Paris, 1896). ;mi:.tonian H ir

Ho '.a the total- mugy of the two-body ryutm ruxl can be choscl: to *The mcerisl prenentsi here is drawn fcw an ax)rco;tion Ly E. Y. Brm in PIANEI'ARY TiiFORY, Ch. 4. -- 204 28 3

The conatants ro, e, are at ow disposal. Choore the valuns

Fo 8 *OdpCrfcwD r(l - 0) I boL 0 h.11)

Bi are definad by relation (n.lb)r

h.7)

The only form of S*(ai,qi) Por which (A.7:. is separable is

fA.8)

(A-9)

Smparats tha renmlning t- in h.7) into thoso uhlch depond on e 2 and thore which depend 01:r, and @quat. each to a conatant i2. The solutlon for S is 286 288 287

The transformation is deflnd by th. robtion. (A.17)

Tho integral in the 6) equation can k reducal by a similar lA.21) tra:mf-tion to Again, let R 0 to detednr tha new net. Since H' - 0 If H - 0, we can danand

(A.22)

Nor w can write dom the original S*bi,Bi) and use tA.15) to

eliminate t. The result ia

LA.19)

There ia c%iously quite a bit of freedm in chooming the new mm equation of amtion for thset of orbital elmonte fa,b,~,c,O,n) varlablea. One choice for wi is CM be daivud frop the above and b.3) aa a purely algebraic exercise. wz = h Thm above set ia not &e most uaeful *or our purpoaea. Make w: * G-R (A.2Qa) another H - J transC6rmtion on [oi,Bil to a n.* eat (Ji,wi> and Ws 8 12 dgund that the new Hamiltonian aatisfy For which the conjugate action variables are

(A. 20) 290

-TION OP THE &TION IWITGRAL

The fact that the action J is an ad'abatic constant as long as instantaneous frequency fast compred to the slow cmes This set constitutes the rell-bprm DslaM.y -tee of elements. the is induced in a Hamiltonian system, a means obtaining Several similar sets of conjugate variable8 can be obtained by provi3:s of the secular motions and of the as a function c. Ye ara rearranganent of the angle variables in b.23). A modified set used of H no+= of especially intere in evrluating Jpos.rot at transition. Tran- 4- this paper is "ed sition in the 8mall fluctuatian lindt involves the coincidence of

the twu interior n-roots, while in the larqe fluctuation limit --he

condition is that bfx,) vanishes. Each of these wi'l be calculated

separately.

In the positive rotation phase, the action is chosen to vanish.

The explicit intagrand we evaluate is

The +nteqrand (-kx t 1) is positive definite for all physical values

of x, implying that the associated integral is positive definite.

Changing tho integration variable Erm x to 0, and intevating over

the range x2"- < x s x,_ where xZn- and xl- are the left and right

bounding roots respectively, we obtain