CHAPTER 15 SPECIAL PERTURBATIONS

Transcription

1 CHAPTER 5 SPECIAL PERTURBATIONS [Ths chapte s unde developent and t a be a athe long te befoe t s coplete. It s the ntenton that t a deal wth specal petubatons, dffeental coectons, and the coputaton of a defntve obt. Howeve, t wll pobabl poceed athe slowl and wheneve the spt oves e.] 5. Intoducton Chapte 4 dealt wth the subject of geneal petubatons. That s, f the petubaton R can be epessed as an eplct algebac functon, the ates of change of the obtal eleents wth te can be calculated b eplct algebac epessons nown as Lagange s Planeta Equatons. B wa of eaple we deved Lagange s equatons fo the case of a satellte n obt aound an oblate planet, n whch the depatue of the gavtatonal potental fo that of a sphecall setc planet could be epessed n sple algebac fo. Lagange s equatons ae potant and nteestng fo a theoetcal pont of vew. Howeve, n the pactcal atte of calculatng the petubatons of the obt of an asteod o a coet esultng fo the gavtatonal feld of the othe planets n the sola sste, that s not how t s done. The petubng foces ae functons of te whch ust be coputed nuecall athe than fo a sple foula. Such petubatons ae geneall efeed to as specal petubatons. Whle long-establshed copute pogas, such as RADAU5, a be avalable to ca out the necessa athe long coputatons wthout the use havng to undestand the detals, t s the ntenton n ths chapte to ndcate n pncple how such a poga a be developed fo scatch. Jupte s b fa the geatest petube, but fo hgh-pecson wo t a be necessa to nclude petubatons fo the othe ajo planets, Mecu to Neptune. Pluto a also be consdeed. Howeve, t s now nown that Pluto s a good deal less assve than t was once estated to be, so t s a nce queston as to whethe o not to nclude Pluto. Besdes, Pluto s pobabl not the ost assve of the tansneptunan objects - Es s beleved to be a lttle lage and hence possbl oe assve. The an belt object Cees a be oe potant than ethe of these. The total ass of the eanng asteods s usuall consdeed neglgble n ths contet. It wll be evdent that an copute poga ntended to copute specal petubatons wll have to nclude, as suboutnes, pogas fo calculatng, da-b-da, the postons and dstances of each of the petubng planets to be ncluded n the coputaton. Copute pogas ae avalable to povde these. In what follows, t wll be assued that the eade has access to such a poga (I do! o s othewse able to copute the planeta postons, and we ove on fo thee to see how we calculate the planeta petubatons.

2 5. Obtal eleents and the poston and veloct vecto The s eleents used to descbe the obt of an asteod ae the fala a, e, T Because of the pecesson and nutaton of Eath, the angula eleents ust, of couse, be efeed to a patcula equno and equato, usuall chosen to be that of the standad epoch J000.0, whch eans h 00 TT on 000 Janua 0. (The J stands fo Julan Yea. The eleent T s the nstant of pehelon passage. If the obt s neal ccula, the nstant of pehelon passage s ll-defned, and f the obt s eactl ccula, t s not defned at all. In such cases, nstead of T, we a gve ethe the ean anoal M 0 o the ean longtude L 0 at a specfed epoch (see Chapte 0. Ths epoch need not be (and usuall s not the sae as the standad epoch efeed to n the pevous paagaph. Suppose that, at soe nstant of te (to be nown, fo easons to be eplaned late, as the epoch of osculaton, the helocentc eclptc coodnates of an asteod o coet n an ellptc obt ae ( X, Y, Z and the coponents of the veloct vecto ae ( X &, Y&, Z&. We have shown n Chapte 0, Secton 0.0 how to calculate, fo these, the s eleents a, e, T of the obt at that nstant. Convesel, gven the obtal eleents, we could evese the calculaton and calculate the coponents of the poston and veloct vectos. Thus an obt a equall well be descbed b the s nubes X, Y, Z, X&, Y&, Z&. That s to sa the coponents, at soe specfed nstant of te, of the poston and veloct vecto n helocentc eclptc coodnates. We could equall well gve the coponents, at soe nstant of te, of the poston and veloct vectos n helocentc equatoal coodnates: ξ, η, ζ, ξ&, η&, ζ& We saw n Secton 0.9 that et anothe set of s nubes, P, Q, P, Q, P, Q wll also suffce to descbe an obt. It s assued hee that the eade s fala wth all fou of these altenatve sets of eleents, and can convet between the. Indeed, befoe eadng on, t a be a useful eecse to pepae a copute poga that wll convet nstantl between the. Ths

3 a not be a tval tas, but I stongl ecoend dong so befoe eadng futhe. The faclt to convet nstantl between one set and anothe s an enoous help. To convet between eclptc and equatoal coodnates, ou wll need, of couse, the oblqut of the eclptc at that nstant - t vaes, of couse, wth te. The eade wll have notced the fequent occuence of the phase at that nstant n the pevous paagaphs. If the asteod wee not subject to petubatons fo the othe planets, t would etan ts obtal eleents foeve. Howeve, because of the planeta petubatons, the eleents a, e, T coputed fo X, Y, Z, X&, Y&, Z& o fo ξ, η, ζ, ξ&, η&, ζ& at a patcula nstant of te ae vald onl fo that nstant. The eleents wll change wth te. Theefoe n quotng the eleents of an asteodal obt, t s entel necessa to state cleal and wthout abgut the nstant of te to whch these eleents ae efeed. The unpetubed obt, and the eal petubed obt, wll concde n poston and veloct at that nstant. The eal and unpetubed obts wll ss o osculate at that nstant, whch s theefoe nown as the epoch of osculaton. The eleents a, e, T calculated fo a patcula epoch of osculaton a suffce fo the coputaton of an ephees fo wees to coe. But afte onths the obseved poston of the object wll stat to devate fo ts calculated ephees poston. It s then necessa to calculate a new set of eleents fo a late epoch of osculaton. Dependng on ccustances, obtal eleents a be ecalculated eve ea, o eve 00 das o eve 40 das o eve 0 das, o at soe othe convenent nteval. It wll be the pupose n what follows to do the followng. Gven that at soe nstant (.e. at soe epoch of osculaton the eleents ae a, e, T (o the poston and veloct vectos ae ξ, η, ζ, ξ&, η&, ζ&, how do we calculate the eleents at soe subsequent epoch, tang nto account planeta petubatons? As ponted out at the end of Secton 5., we shall need to now the postons and dstances of the ajo planets as a functon of te. We suppose that we have suboutnes n ou poga that we can call upon to calculate these data at an date. As entoned above, the equatons of oton can be wtten n equatoal o eclptc coodnates, though t s oe lel that, fo the postons of the ajo planets, we shall have avalable the postons n equatoal coodnates. 5. The equatons of oton Fst let us consde the oton of an asteod unde the gavtatonal nfluence of the Sun alone, gnong petubatons fo the othe planets. We tae the ass of the Sun to be M and the ass of the asteod to be. The foce on the asteod and, of couse, b GM Newton s thd law, the foce on the Sun s, whee s the dstance between the two bodes. The two bodes ae, of couse, n oton aound the coon cente of ass, whch, n the case of an asteod, s ve close to the cente of the Sun.

4 4 GM The acceleaton of the asteod towads the cente of ass s, and the acceleaton of the Sun towads the cente of ass s. If we efe the oton to the G ( M Sun as ogn, we see that the acceleaton of the asteod towads the Sun s. In vecto fo we a wte ths as G ( M & =, 5.. whee s a vecto dected fo the Sun towads the asteod, wth helocentc ectangula coponents (,,. These helocentc coodnates could be ethe eclptc coodnates, fo whch we have htheto used the sbols ( X, Y, Z ; o the could be equatoal coodnates, fo whch we have htheto used the sbols ( ξ, η, ζ. The sbols (,, wll be undestood hee to efe to ethe, at ou convenence. It s oe lel that we shall have avalable the equatoal athe than the eclptc coodnates. The decton cosnes of ae,,, and consequentl the ectangula coponents of equaton 5.. ae G( M & = 5.. G( M & = 5.. G( M & = 5..4 These ae the equatons of oton of the asteod wth espect to the Sun as ogn. The quanttes,,, ( = ae, of couse, functons of te. The soluton of these equatons descbe the ellptcal (o othe conc secton obts of the asteod and all the othe popetes that we have dscussed n pevous chaptes. If we ae usng eclptc coodnates ( X, Y, Z, the X-as s dected towads the Fst Pont of Aes, the Y-as s dected along the decton of nceasng eclptc longtude, and the Z-as s dected towads the noth pole of the eclptc. If we ae usng equatoal coodnates ( ξ, η, ζ, the ξ-as s dected towads the Fst Pont of Aes, the η-as s dected along the decton of 6 hous ght ascenson, and

5 5 the ζ-as s dected towads the noth celestal pole. The Eath wll be on the X- o ξ- as n Septebe (not Mach. Now let us ntoduce a thd bod, a petubng planet, such as, pehaps, Jupte. We ll suppose that ts ass s, that ts dstance fo the Sun s and ts dstance fo the asteod s (see fgue XV.I, n whch S s the Sun, A s the asteod, and P s the petubng planet. Ths s now a thee-bod poble and a geneal soluton n tes of algebac functons s not possble, and t has to be solved b nuecal coputaton. P A X Y S M FIGURE XV.I In addton to the acceleatons of the asteod towads the Sun and the Sun towads the asteod descbed on page, used n developng equatons 5..-4, we now have also to consde the acceleatons of the asteod and the Sun towads the petubng planet, as ndcated n fgue XV.II.

6 6 FIGURE XV.II

7 7 The -coponents of these ae and, and so the addtonal acceleaton of A, elatve to the Sun, n the X-decton s, and ths has now to be added to the ght hand sde of equaton 5..: = ( M G & & 5..5 Nethe G no M ae nown to geat pecson, but the poduct GM s nown to ve geat pecson. Indeed n coputatonal pactce we ae use of the Gaussan constant a 0 GM =, whee a 0 s the astonocal unt of length. Ths constant has denson T and s equal to the angula veloct of a patcle of neglgble ass n ccula obt of adus au aound the Sun, whch s adans pe ean sola da. Theefoe n coputatonal pactce, equaton 5..5 s geneall wtten as = ( & &, 5..6 n whch the unts of ass, length and te ae, espectvel, sola ass, astonocal unt, and ean sola da. Recall that s the ass of the asteod whose obt we ae coputng, and s the ass of the petubng planet, and that the ogn of coodnates s the cente of the Sun. Sla equatons appl to the - and -coponents: = ( & &, 5..7 = ( & & If we add the petubatons fo all the ajo planets fo Mecu (M to Neptune (N, these equatons becoe, of couse, = = N M ( & & 5..9 and sla equatons n and.

8 8 In the case of an asteod o a coet, t a be pessble to neglect n ths equaton (.e. set = 0, but not, of couse,. We shall do that hee, so the equaton of oton n becoes N = = & &, 5..0 M wth sla equatons n and. The,.,, etc., ae nuecal data, whch have to be suppled b ndependent coputatons (suboutnes fo all the planets. As stated at the end of the pevous Secton, we suppose that we have suboutnes n ou poga that we can call upon to calculate these data at an date. We also ponted out that the equatons of oton ae vald fo ethe eclptc o equatoal coodnates, although the coodnates of the planets ae oe lel to be avalable s equatoal athe than eclptc coodnates. The ae all functons of te, so that, n effect, we have to develop nuecal ethods fo ntegatng equatons of the fo, whee f (t s not an algebac epesson, but athe a table of nuecal values. & = f (t. 5.. d& That s to sa = f (t. 5.. dt We suppose that we now & at the epoch of osculaton. Then we can fnd & at an subsequent date b an standad technque of nuecal ntegaton, such as Spson s o Weddle s Rules, o Gaussan quadatue, o b a Runge-Kutta pocess. Thus we now have a table of & as a functon of te: & = g(t 5... That s to sa d = g(t dt We ntegate a second te, untl we ave at both and & at soe subsequent epoch of osculaton (pehaps 00, o 40, das nto the futue. Repeat wth the and coponents, so we eventuall have a new set of (,,, &, &, & fo a late epoch, and hence also of a, e, T.