Perturbation Theory and Celestial Mechanics

Transcription

1 Copyght Petubaton Theoy and Celestal Mechancs In ths last chapte we shall sketch some aspects of petubaton theoy and descbe a few of ts applcatons to celestal mechancs. Petubaton theoy s a vey boad subect wth applcatons n many aeas of the physcal scences. Indeed, t s almost moe a phlosophy than a theoy. The basc pncple s to fnd a soluton to a poblem that s smla to the one of nteest and then to cast the soluton to the taget poblem n tems of paametes elated to the known soluton. Usually these paametes ae smla to those of the poblem wth the known soluton and dffe fom them by a small amount. The small amount s known as a petubaton and hence the name petubaton theoy. Ths pescpton s so geneal as to make a geneal dscusson almost mpossble. The wod "petubaton" mples a small change. Thus, one usually makes a "small change" n some paamete of a known poblem and allows t to popagate though to the answe. One makes use of all the mathematcal popetes of the poblem to obtan equatons that ae solvable usually as a esult of the elatve smallness of the petubaton. Fo example, consde a stuaton n whch the fundamental equatons govenng the poblem of nteest ae lnea. The lneaty of the equatons guaantees that any lnea combnaton of solutons s also a soluton. Thus, one fnds an analytc soluton close to the poblem of nteest and emoves t fom the defnng equatons. One now has a set of equatons whee the soluton s composed of small quanttes and the soluton may be made smple because of t. Howeve, the dffeental equatons that descbe the dynamcs of a system of patcles ae defntely nonlnea and so one must be somewhat moe cleve n applyng the concept of petubaton theoy. In ths egad, celestal mechancs s 9

2 a poo feld n whch to lean petubaton theoy. One would be bette seved leanng fom a lnea theoy lke quantum mechancs. Nevetheless, celestal mechancs s whee we ae, so we wll make the best of t. Let us begn wth a geneal statement of the appoach fo a consevatve petubng foce. 9. The Basc Appoach to the Petubed Two Body Poblem The fst step n any applcaton of petubaton theoy s to dentfy the space n whch the petubatons ae to be caed out and what vaables ae to be petubed. At fst glance, one could say that the ultmate esult s to pedct the poston and velocty of one obect wth espect to anothe. Thus, one s tempted to look dectly fo petubatons to as a functon of tme. Howeve, the nonlneaty of the equatons of moton wll make such an appoach unwokable. Instead, let us make use of what we know about the soluton to the two body poblem. fom Fo the two body poblem we saw that the equatons of moton have the & + Φ 0, (9..) whee Φ s the potental of a pont mass gven by GM Φ. (9..) Let us assume that thee s an adonal souce of a potental that can be epesented by a scala -ψ that ntoduces small foces actng on the obect so that We can then wte the equatons of moton as Φ >> Ψ. (9..3) & + Φ Ψ(, t). (9..4) Hee ψ s the negatve of the petubng potental by conventon. If ψ s a constant, then the soluton to the equatons of moton wll be the soluton to the two body poblem. Howeve, we aleady know that ths wll be a conc secton whch can be epesented by sx constants called the obtal elements. We also know that these sx obtal elements can be dvded nto two tplets, the fst of whch deals wth the sze and shape of the obt, and the second of whch deals 30

3 wth the oentaton of that obt wth espect to a specfed coodnate system. A vey easonable queston to ask s how the pesence of the petubng potental affects the obtal elements. Clealy they wll no longe be constants, but wll vay n tme. Howeve, the knowledge of those constants as a functon of tme wll allow us to pedct the poston and velocty of the obect as well as ts appaent locaton n the sky usng the development n Chaptes 5, 6, and 7. Ths esults fom the fact that at any nstant n tme the obect can be vewed as followng an obt that s a conc secton. Only the chaactestcs of that conc secton wll be changng n tme. Thus, the soluton space appopate fo the petubaton analyss becomes the space defned by the sx lnealy ndependent obtal elements. That we can ndeed do ths esults fom the fact that the unfom moton of the cente of mass povdes the emanng sx constants of ntegaton even n a system of N bodes. Thus the detemnaton of the tempoal behavo of the obtal elements povdes the emanng sx peces of lnealy ndependent nfomaton equed to unquely detemne the obect's moton. The choce of the obtal elements as the set of paametes to petub allows us to use all of the development of the two body poblem to complete the soluton. Thus, let us defne a vecto ξ whose components ae the nstantaneous elements of the obt so that we may egad the soluton to the poblem as gven by ( ξ, t). (9..5) The poblem has now been changed to fndng how the obtal elements change n tme due to the pesence of the petubng potental -ψ. Explctly we wsh to ecast the equatons of moton as equatons fo dξ. If we consde the case whee the petubng potental s zeo, then ξ s constant so that we can wte the unpetubed velocty as d ( ξ, t) v. (9..6) Now let us defne a specfc set of obtal elements ξ 0 to be those that would detemne the patcle's moton f the petubng potental suddenly became zeo at some tme t 0 ξ t ). (9..7) 0 ξ( 0 The obtal elements ξ 0 epesent an obt that s tangent to the petubed obt at t0 and s usually called the osculatng obt. By the chan ule 3

4 d & + ξ. (9..8) Equaton (9..6) s coect fo ξ 0 and at t t 0, ξ ξ0 so f we compae ths wth equaton (9..6), we have & ξ 0. (9..9) Ths s sometmes called the osculaton conon. Applyng ths conon and dffeentatng equaton (9..8) agan wth espect to tme we get d & + ξ. (9..0) If we eplace d n equaton (9..4) by equaton (9..0), we get & + Φ + + ξ Ψ(, t) t t t. (9..) Howeve the explct tme dependence of ( ξ, t) s the same as ( ξ, t) so that 0 + Φ 0. (9..) t Theefoe the fst two tems of equaton (9..) sum to zeo and, togethe wth the osculatng conon of equaton (9..9), we have & ξ Ψ t & ξ 0 ξ, (9..3) as equatons of conon fo E. Dffeentaton wth espect to a vecto smply means that the dffeentaton s caed out wth espect to each component of the vecto. Theefoe [ ] s a second ank tenso wth components ]. [ Thus each of the equatons (9..3) ae vecto equatons, so thee s a sepaate 3

5 scala equaton fo each component of. Togethe they epesent sx nonlnea nhomogeneous patal dffeental equatons fo the sx components of ξ. The ntal conons fo the soluton ae smply the values of ξ 0 and ts tme devatves at t 0. Appopate mathematcal go can be appled to fnd the conons unde whch ths system of equatons wll have a unque soluton and ths wll happen as long as the Jacoban of (, v) 0. Complete as these equatons ae, the fom and applcaton ae somethng less than clea, so let us tun to a moe specfc applcaton. 9. The Catesan Fomulaton, Lagangan Backets, and Specfc Fomulae Let us begn by wtng equatons (9..3) n component fom. Assume that the Catesan components of ae x. Then equatons (9..3) become 6 6 x& x dξ dξ Ψ x 0,,3,,3. (9..) Howeve, the dependence of ξ on tme s bued n these equatons and t would be useful to be able to wte them so that dξ s explctly dsplayed. To accomplsh ths multply each of the fst set of equatons by x k and add the thee component equatons togethe. Ths yelds: 3 x 6 x& dξ k 3 x k Ψ x Ψ k k,, L,6. (9..) Multply each of the second set of equatons by ( x& k ) and add them togethe to get 3 6 x& ξ x d 0 k,, L,6. (9..3) k 33

6 Fnally add equaton (9..) and equaton (9..3) togethe, eaange the ode of summaton factong out the desed quantty ( ξ ) to get d 6 dξ 3 x k x& x x& k Ψ k, k,, L,6. (9..4) The ugly lookng tem unde the second summaton sgn s known as the Lagangan backet of ξ k and ξ and, by conventon, s wtten as 3 [ ξk, ξ ] x x& x x& k k. (9..5) The eason fo pusung ths appaently complcatng pocedue s that the Lagangan backets have no explct tme dependence so that they epesent a set of coeffcents that smply multply the tme devatves of ξ. Ths educes the equatons of moton to sx fst ode lnea dffeental equatons whch ae 6 [ ξ ξ ] dξ Ψ k, k,, L, 6. (9..6) k All we need to do s detemne the Lagangan backets fo an explct set of obtal elements and snce they ae tme ndependent, they may be evaluated at any convenent tme such as t o. If we eque that the scala (dot) poduct be taken ove coodnate ( ) space athe than obtal element (ξ ) space, we can wte the Lagangan backet as [ ] [ ] ξ ξ ξ ξ & & k,,. (9..7) Snce the patal devatves ae tensos, the scala poduct n coodnate space does not commute. Howeve, we may show the lack of explct tme dependence of the Lagange backet by dect patal dffeentaton wth espect to tme so that 34

7 [ ξ, ξ] & & & + + &, (9..8) o e-aangng the ode of dffeentaton we get & [ ξ, ξ] & & &. (9..9) Usng equaton (9..6) and Newton's laws we can wte ths as v v [, ] Φ Φ ξ ξ 0. (9..0) Remembe that we wote ( ξ v, t) so that the coodnates x and the tme devatves x& depend only on the set of obtal elements ξ and tme. Thus the Lagangan backets depend only on the patcula set of obtal elements and may be computed once and fo all. Thee ae vaous pocedues fo dong ths, some of whch ae tedous and some of whch ae cleve, but all of whch ae elatvely long. Fo example, one can calculate them fo t T o so that M E ν 0. In adon, whle one can fomulate 36 values of [ξ k, ξ ] t s clea fom equaton (9..5) that [ ξk, ξ ] [ ξ, ξk ]. (9..) [ ξk, ξk ] 0 Ths educes the numbe of lnealy ndependent values of [ξ k, ξ ] to 5. Howeve, of these 5 Lagange backets, only 6 ae nonzeo [see Taff l p.306, 307] and ae gven below. / [, Ω] na ( e ) sn [a, Ω] [e, Ω] na [a, o] [e, o] na [a,t ] 0 na( e e( e na( e a e( e) 35 ) )cos / / ) / cos, (9..)

8 whee n s ust the mean angula moton gven n tems of the mean anomaly M by n(t T ). (9..3) M 0 Thus the coeffcents of the tme devatves of ξ ae explctly detemned n tems of the obtal elements of the osculatng obt ξ 0. To complete the soluton, we must deal wth the ght hand sde of equaton (9..6). Unfotunately, the patal devatves of the petubng potental ae lkely to nvolve the obtal elements n a complcated fashon. Howeve, we must say somethng about the petubng potental o the poblem cannot be solved. Theefoe, let us assume that the behavo of the potental s undestood n a cylndcal coodnate fame wth adal, azmuthal, and vetcal coodnates (, ϑ, and z) espectvely. We wll then assume that the cylndcal components of the petubng foce ae known and gven by Then fom the chan ule R Ψ Ψ I ϑ Ψ ℵ z. (9..4) Ψ Ψ Ψ + ϑ ϑ Ψ z + z. (9..5) The patal devatves of the cylndcal coodnates wth espect to the obtal elements may be calculated dectly and the equatons fo the tme devatves of the obtal elements [equatons (9..6)] solved explctly. The algeba s long and tedous but elatvely staght fowad and one gets 36

9 da ( e ) n de ( e ) na d dω dt / [ na ( e ) ] ℵ / [ na ( e ) sn ] d o ( e ) nae 0 n dω / / [ Resn ν + Ia( e ) / ] / [ Rsn ν + I(cos E + cosν) ] cos( ν + o) ℵ sn( ν + o) sn [ a( e ν + I a( e ) dω + )] Rcosν / [ ( e ) sn ( ) ] / dω d / [ + ( e ) sn ( o )] e [ + ( e ) ] R na. (9..6) An altenatve set of petubaton equatons attbuted to Gauss and gven by Taff (p.3l4) s 37

10 da de esn ν n( e ) ( e / ) na / a( e R + n sn ν ( e R + na / ) / ) e ( e / / d o ( e ) cos ν ( e ) sn ν a( e ) + R + nae nae I a( e ) sn( Ω + o) cot ℵ L d cos( Ω + o) ℵ L dω csc sn( Ω + o) ℵ L dt0 ( e ) cos ν ( e )sn ν a( e ) + + R + I n a a e na e a( e ) I a ) I. (9..7) whee πa L. (9..8) P These elatvely complcated foms fo the soluton show the degee of complexty ntoduced by the nonlneaty of the equatons of moton. Howeve, they ae suffcent to demonstate that the poblem does ndeed have a soluton. Gven the petubng potental and an appoxmate two body soluton at some epoch t 0, one can use all of the two body mechancs developed n pevous chaptes to calculate the quanttes on the ght hand sde of equatons (9..6). Ths allows fo a new set of obtal elements to be calculated and the moton of the obects followed n tme. The pocess may be epeated allowng fo the cumulatve effects of the petubaton to be ncluded. Howeve, one usually eles on the ognal assumpton that the petubng foces ae small compaed to those that poduce the two body moton [equaton (9..3)].Then all the tems on the left hand sde of equaton (9..6) wll be small and the moton can be followed fo many obts befoe t s necessay to change the obtal elements. That s the mao thust of petubaton theoy. It tells you how thngs ought to change n esponse to known foces. Thus, f the souce of the petubaton les n the plane defnng the cylndcal coodnate system (and the plane defnng the obtal elements) ℵ wll be zeo and the obtal 38

11 nclnaton () and the longtude of the ascendng node wll not change n tme. Smlaly f the souce les along the z-axs of the system, the sem-mao axs (a) and eccentcty (e) wll be tme ndependent. If the changes n the obtal elements ae suffcently small so that one may aveage ove an obt wthout any sgnfcant change, then many of the petubatons vansh. In any event, such an aveagng pocedue may be used to detemne equatons fo the slow change of the obtal elements. Utlty of the development of these petubaton equatons eles on the appoxmaton made n equaton (9..3). That s, the equatons ae essentally fst ode n the petubng potental. Attempts to nclude hghe ode tems have geneally led to dsaste. The poblem s bascally that the equatons of classcal mechancs ae nonlnea and that the obect of nteest s (t). Many small eos can popagate though the pocedues fo fndng the obtal elements and then to the poston vecto tself. Snce the equatons ae nonlnea, the popagaton s nonlnea. In geneal, petubaton theoy has not been tebly successful n solvng poblems of celestal mechancs. So the cuent appoach s geneally to solve the Newtonan equatons of moton dectly usng numecal technques. Awkwad as ths appoach s, t has had geat success n solvng specfc poblems as s evdenced by the space pogam. The ablty to send a ocket on a complcated taectoy though the satellte system of Jupte s ample poof of that. Howeve, one gans lttle geneal nsght nto the effects of petubng potentals fom sngle numecal solutons. Poblems such as the Kkwood Gaps and the stuctue of the Satunan ng system offe ample evdence of poblems that eman unsolved by classcal celestal mechancs. Howeve, n the case of the fome, much lght has been shed though the dynamcs of Chaos (see Wsdom 0 ). Thee emans much to be solved n celestal mechancs and the basc nonlneaty of the equatons of moton wll guaantee that the solutons wll not come easly. Fomal petubaton theoy povdes a nce adunct to the fomal theoy of celestal mechancs as t shows the potental powe of vaous technques of classcal mechancs n dealng wth poblems of obtal moton. Because of the nonlneaty of the Newtonan equatons of moton, the soluton to even the smplest poblem can become vey nvolved. Nevetheless, the maoty of dynamcs poblems nvolvng a few obects can be solved one way o anothe. Pehaps t s because of ths non-lneaty that so many dffeent aeas of mathematcs and physcs must be bought togethe n ode to solve these poblems. At any ate celestal mechancs povdes a challengng tanng feld fo students of mathematcal physcs to apply what they know. 39

12 Chapte 9: Execses. If the sem-mao axs of a planets obt s changed by a, how does the peod change? How does a change n the obtal eccentcty e affect the peod?. If v and v ae the veloctes of a planet at pehelon and aphelon espectvely, show that (-e)v (+e)v. 3. Fnd the Lagangan backet fo [e, Ω]. 4. Usng the Lagangan and Gaussan petubaton equatons, fnd the behavo of the obtal elements fo a petubatve potental that has a pue - dependence and s located at the ogn of the coodnate system. 40