CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves n a epleran ellpse (or hyperbola f ts netc energy s large enough) that can be descrbed by the sx orbtal elements a, e,, Ω, ω, T, or any equvalent set of sx parameters If the potental s a lttle dfferent from GM / r, say ( GM / r R), the orbt wll be perturbed, and R s descrbed as a perturbaton As a result t wll no longer move n a perfect epleran ellpse Perturbatons may be perodc or secular For example, the elements such as a, e or may vary n a perodc fashon, whle there may be secular changes (e changes that are not perodc but constantly ncrease or decrease n the same drecton) n elements such as Ω and ω (That s, the lne of nodes and the lne of apsdes may monotoncally precess; they may advance or regress) In some stuatons t may be possble to express the perturbaton n terms of a smple algebrac formula An example would be a partcle n orbt around a slghtly oblate planet, where t s possble to express the potental algebracally The am of ths chapter wll be to try to fnd general expressons for the rates of change of the orbtal elements n terms of the perturbng functon, and we shall use the orbt around an oblate planet as an example In other stuatons t s not easly possble to express the perturbaton n terms of a smple algebrac functon For example, a planet n orbt around the Sun s subect not only to the gravtatonal feld of the Sun, but to the perturbatons caused by all the other planets n the solar system These specal perturbatons have to be treated numercally, and the technques for dong so wll be descrbed n chapter 5 4 Contact Transformatons and General Perturbaton Theory (Before readng ths secton, t may be well to re-read secton 0 of Chapter 0) Suppose that we have a smple problem n whch we now the hamltonan H 0 and that the Hamlton-Jacob equaton has been solved: S S H 0 q,, t 0 4 q t
Now suppose we have a smlar problem, but that the hamltonan, nstead of beng ust S H 0 s H H 0 R, and K H t Let us mae a contact transformaton from (p, q ) to (P, Q ), where K K Q& and P& In the orbtal context, followng Secton 0, we P Q dentfy Q wth α and P wth β, whch are functons (gven n Secton 0) of the orbtal elements and whch can serve n place of the orbtal elements The parameters are constants wth respect to the unperturbed problem, but are varables wth respect to the perturbng functon They are gven, as functons of tme, by the soluton of Hamlton s equatons of moton, whch retan ther form under a contact transformaton α & and β& 4a,b Perturbaton theory wll show, then, how the α and β wll vary wth a gven perturbaton The conventonal elements a, e,, Ω, ω, T are functons of α, β, and our am s to fnd how the conventonal elements vary wth tme under the perturbaton R We can do that as follows Let A be an orbtal element, gven by A A ( α, β ) 4 Then A & α& β& 44 By equatons 4a,b, ths becomes A & 45 But and 46a,b A & 46 â A
That s A & 47 Ths can be wrtten, n shorthand: R A& { A, A } α, β 48 A Here the symbol { A, s called the Posson bracet of A, A wth respect to α, A } α, β β (In the language of the typographer, the symbols (), [] and {} are, respectvely, parentheses, bracets and braces; you may refer to Posson braces f you wsh, but the usual term, n spte of the symbols, s Posson bracet) Note the property { A, A, } A } α, β { A α, β 4 The Posson Bracets for the Orbtal Elements A wored example s n order From equatons 47 and 48, we see that the Posson bracets are defned by The A are the orbtal elements {, }, A A α β 4 For our example, we shall calculate {Ω, } and we wrte out the sum n full: { Ω, } 4 Refer now to equatons 07 and 9, and we fnd { Ω, } 0 0 0 0 0 4 α α / α Fnally, referrng to equatons 00 and, we obtan
4 { Ω, } 44 GMm a( e ) sn Proceedng n a smlar manner for the others, we obtan { a, T} { e, T} a, 45 GMm a( e ), 46 GMme {, ω } 47 GMm a( e ) tan In addton, we have, of course, {, Ω} { Ω, }, { T, a} { a, T}, { T, e} { e, T} and { ω, } {, ω} All other pars are zero 44 Lagrange s Planetary Equatons We now go to equaton 48 to obtan Lagrange s Planetary Equatons, whch wll enable us to calculate the rates of change of the orbtal elements f we now the form of the perturbng functon: a a&, 44 GMm T a( e ) e e&, 44 GMme T me GMa ω &, 44 GMm a( e ) sn GMm a( e ) tan ω e ω&, 444 me GMa e GMm a( e ) tan
5 Ω &, 445 GMm ( e ) sn a R a( e ) R T& 446 GMm a GMme e 45 Moton Around an Oblate Symmetrc Top In Secton 5 we developed an expresson (equaton 56) for the gravtatonal potental of an oblate symmetrc top (eg an oblate spherod) Wth a slght change of notaton to conform to the present context, we obtan for the perturbng functon Gm( C A) z R 45 r r Ths s the negatve of the addtonal potental energy of a mass m at a pont whose cylndrcal coordnates are (r, z) n the vcnty of a symmetrc top (whch I ll henceforth call an oblate spherod) whose prncpal second moments of nerta are C (polar) and A (equatoral) Ths s correct to order r/a, where a s the equatoral radus of the spherod Let us magne a partcle of mass m n orbt around an oblate spherod eg an artfcal satellte n orbt around Earth Suppose the orbt s nclned at an angle to the equator, and the argument of pergee s ω At some nstant, when the cylndrcal coordnates of the satellte are (r, z), ts true anomaly s v Exercse (n geometry): Show that z / r sn sn ( ω v) Havng done that, we see that the perturbng functon can be wrtten ( sn sn ( ω )) Gm( C A) R v 45 r Here, r and v vary wth tme, or what amounts to the same thng, wth the mean anomaly M Wth a (nontrval) effort, ths can be expanded as a seres, ncludng a constant (tme ndependent) term plus perodc terms of the form cos M, cos M, cos M, etc If the sprt moves me, I may post the detals at a later date, but for the present I gve the result that, f the expanson s taen as far as e (e we are assumng that the orbt of the satellte s not strongly eccentrc), the constant (tme-ndependent) part of the perturbng functon s R Gm( C A) ( e )( sn ) 45 a
6 Now loo at Lagrange s equatons, and you see that the secular parts of a &, e& and & are all zero That s, although there may be perodc varatons (whch we have not examned) n these elements, to ths order of approxmaton (e ) there s no secular change n these elements On the other hand, applcaton of equaton 445 gves for the secular rate of change of the longtude of the nodes GM ( C A) Ω & ( e )cos 454 7 / M a The reader wll no doubt be releved to note that ths expresson does not contan m, the mass of the orbtng satellte; M s the mass of the Earth The reader may also note the mnus sgn, ndcatng that the nodes regress To obtan the factor ( e ), readers wll have to do a lttle bt of wor, and to expand, by the bnomal theorem, whatever expresson n e they get, as far as e Let a be the equatoral radus of Earth Multply top and bottom of equaton 454 by a 7/, and the equaton becomes 7 / GM ( C A) Ω& a ( e ) cos 455 a Ma a Here M s the mass of Earth (not of the orbtng satellte), a s the sem maor axs of the satellte s orbt, and a s the equatoral radus of Earth [If we assume Earth s an oblate spherod of unform densty, then, accordng to example of Secton 0 of Chapter of our notes on Classcal Mechancs, C 5 Ma In 7/ GM ( C A) that case, equaton 455 becomes Ω& a ( e )cos But 5 a C a the densty of Earth s not unform, so we ll leave equaton 455 as t s] For a nearly crcular orbt, equaton 455 becomes ust 7 / GM ( C A) Ω& a cos 456 a Ma a Ths tells us that the lne of nodes of a satellte n orbt around an oblate planet (e C > A) regresses From the rate of regresson of the lne of nodes, we can deduce the dfference, C A between the prncpal moments of nerta, though we cannot deduce ether moment separately (If we could determne the moment of nerta from the rate of regresson of the nodes whch we cannot how well can we determne the densty dstrbuton nsde
7 Earth? See Problem 4 n Chapter A of our Classcal Mechancs notes to determne the answer to ths It wll be found that nowledge of the moment of nerta places only wea constrants on the core sze and densty) GM ( C A) Numercally t s nown for Earth that the quantty s about 04 rad a Ma s, or about 0 degrees per day Thus the rate of regresson of the nodes of a satellte n orbt around Earth n a near-crcular orbt s about Ω& a 0 a 7 / cos degrees per day We can refer to equatons 444 and 45 to determne the rate of moton of the lne of apsdes, ω& After some algebra, and neglect of terms or order e and hgher, we fnd or, f we multply top and bottom by a 7/, ( C A) G ω& (5cos ), 457 7/ 4a M 7 / ( C A) GM a ω& (5cos ) 458 4Ma a a Thus we fnd that the lne of apsdes advances f the nclnaton of the orbt to the equator s less than 6 o and t regresses f the nclnaton s greater than ths In ths secton, I have demanded a far amount of wor from the reader n partcular for the expanson of equaton 45 Whle the wor requres some patence and persstence, t s straghtforward, and the resolute reader wll be able to wor out the expanson n terms of the mean anomaly and the tme, and hence, by mang use of Lagrange s planetary equatons, wll be able to predct the perodc varatons n a, e and For the tme beng, I am not gong to do ths, snce no new prncples are nvolved, the am of the chapter beng to gve the reader a start on how to start to calculate the changes n the orbtal elements f one can express the perturbng functon analytcally For the effect of the perturbaton of a planetary orbt by the presence of other planets, we have to solve the problem numercally by the technques of specal perturbatons, whch, I hope, some tme n the future, may be the subect of an addtonal chapter