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

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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 in 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 &centricity * 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 ....
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