Orbital averages and the secular variation of the orbits

Transcription

1 Obital avags and th scula vaiation of th obits Mauizio M. D Eliso Ossvatoio S.Elmo - Via A.Caccavllo 89 Napoli Italy Obital avags a mployd to comput th scula vaiation of th lliptical plantay lmnts in th obital plan in psnc of ptubing focs of vaious kinds. Thy a also usful as an aid in th computation of ctain complx intgals. An xtnsiv list of computd intgals is givn. Kywods: Unptubd motion ptubation quations thid-body ptubations oth foc laws obital avags loop intgals. Intoduction W mploy obital avags fo th analytical and numical dtmination of th scula pat of th vaiation of th lmnts a ω smi-majo axis ccnticity agumnt of pihlion of an lliptic obit du to ptubing focs. Th divations a dvlopd thoough an avaging pocss of th fist-od quations aising fom th mthod of vaiation of th abitay constants. W shall us th fomalism of complx vaiabls as w consid only th ptubations acting in th obital plan. As a bypoduct of ou wok w dduc som usful mthods fo th computation of ctain awkwad intgals latd to th gomty of th llips. I. UNPERTURBED MOTION W bgin with a shot viw in th complx notation of th pincipal fomulas and sults of th two-body poblm mployd in this wok. Th position of a plant on th obital plan w suppos lying on th complx plan is givn by th vaiabl = t wh = t = xt + iyt. Thn = bing th complx conjugat of. Th al and th imaginay pats of a complx numb a dnotd by R = + / and by I = i /. Thn R = R = Ii I = I = Ri. In pola coodinats = iθ = cos θ + i sin θ θ bing th tu longitud masud fom th abitay fixd axis x. Th function t will b known as soon as w found th tim dpndnc of θ so that t = [θt] xp iθt; w hav also ṙ = + i θ. If w wit µ = k M + m k M wh M is Sun s mass which w tak as unity k is Gauss gavitational constant th initial valu poblm is solvd if a known fou indpndnt intgals of th motion that can b found intoducing th following intgal opations on Z CZ I Z = 3 HZ R Z = 4 EZ I i Z =. 5 W so asily obtain th constant functions two al and on complx. Thy a. Aa intgal: CZ = Iṙ = i ṙ / = θ = c = al const. It follows: = dθ/c.. Engy intgal: HZ = ṙ / µ/ = h h = µ/a ṙ = µ / /a. 3. Eccnticity vcto: EZ = ṙ = iµ/c/ + xpiω th ccnticity vcto is a complx constant is th ccnticity and ω is th agumnt of pihlion. 4. Elliptic obit in tms of th tu longitud: Iṙ = c = iθ = c /µ iθ [ + cosθ ω] a is th smi-majo axis c = µa f θ ω is th tu anomaly iθ = iω if df = dθ. Elliptic obit in tms of th ccntic anomaly η: = η = a cos η + ib sin η a iω = = a cos η oigin of coodinats at th cnt of th llips dη = an. 5. Thid law: ct = T c = I T ṙ = I π = I η η = πab = T /a 3 = 4π /µ. 6. Oth dfinitions: b = a n µ/a 3 n is th man motion T = π/n is th piod of motion. Th man longitud is dfind as l nt. Z + µ 3 = = ṙ = ṙ II. ORBITAL AVERAGES In psnc of ptubations ach obital lmnt E i a c bcoms a function of tim and th ptubation

2 quations a fist-od quations fo th lmnts of th fom Ė i = g ṙ t. 6 To obtain th scula pat of th ptubations of th lmnts w avag with spct to tim th ight-hand sid of th ptubation quations and obtain so th scula valus E i. Fo this w nd to know th man valu of som obital vaiabls and functions ov th unptubd motion wh with th wod obital w man piodicity shaing th sam piod of th lliptical motion. By dfinition th tmpoal avag of a piodic function gt ov th piodicity intval T is gt av gt = T T gt. 7 If gt is a total divativ of a piodic function ht gt = dht ht h thn gt av = = T bcaus of th piodicity of motion. If g is a constant thn g = g. Th tmpoal avags can b also calculatd by mans of angula vaiabls in th ang [ π] mploying th following lations: T = n π = dη π a = df π a b = dl π 8 applid spctivly to th functions gt gη gf gl. In th following w shall giv poptis mthods of calculation and avags of th obital functions w nd to know. W shall us th notation F av to indicat th avag foc xtd by th ptubing plant whil th notation F av is svd to th succssiv avag with spct to th obit of th ptubd plant. III. THE PLANETARY EQUATIONS Whn th motion is ptubd th quation of motion bcoms = µ + F 9 3 wh F = F ṙ t is th ptubing foc in th plan xy. Th foc F is cntal if F = g wh g is a al function of. It is assumd that F is of small magnitud as compad to th Kplian tm. Thfo th plant movs on a wakly ptubd lliptic obit. Th tim scals of vaiation of its lmnts a a fw ods of magnitud long than th obital piod. Hnc on might pfom th avaging of th quantitis of intst ov fast volution th man anomaly o any oth angula vaiabl accoding to th lations 8. Th plant will mov along a vaiabl obit which at vy instant t can b dscibd as an osculating llips in which th obital lmnts a supposd slowly changing with th tim. Mathmatically this concpt can b tatd with th mthod of vaiation of abitay constants that can b ducd to action on th gnic intgal of th motion E i = E i ṙ of th diffntial opato d/ Ė i = de i E i ṙ dṙ + E i d = F E i ṙ + F E i which mans to consid ach lmnt as vaiabl and to pfom th odinay tim divativs of th intgals of th motion with th convntion that = dṙ = F fo th ptubd motion vidncing so only th acclations poducd by th ptubing focs. Thus w find fom th intgals c = Iṙ a = µ ṙ + 3 = i c µ ṙ 4 th following xpssions of th plantay quations in th plan ċ = IF 5 ȧ = a µ RF 6 ė = i c F + ṙ IF 7 µ w w can put ṙ = i n a b + = i n a b + ē. 8 In th ight-hand sid of ths quations in th fistod of appoximation all th lmnts a considd as constants. Th quation fo ċ is usful in th tatmnt of cntal ptubing focs bcaus thn ċ = IF = whby th fist quation fo ė is simplifid to ė ė = + i ω = i c µ F = ė + i ω = i c F. 9 µ Equating spaatly th al and th imaginay pat w gt ė = c i F µ R = c F µ I ω = c if µ I = c F µ R.

3 A Thid-body ptubations 3 If th foc F is cntal thn F = K with K al constant. Fom this w dduc that ė = c F µ I = c IK = µ so that w can wit ė = i ω ω = c µ F = c K. 3 µ In this cicumstanc th ccnticity vcto otats unifomly about th oigin in th -plan with a constant lngth. In gnal w can wit by avaging ȧ = a µ T R ė = ic µ F = ic µ F i n πµ F d = n a π µ R i IF µ T F d 4 IF 5 in which th appa closd contou o loop intgals ov th unptubd llips. Fom Eq. 4 w s that if F is cntal thn ȧ dosn t hav scula tms bcaus th intgal is a pu imaginay numb. IV. SECULAR PERTURBATIONS A. Thid-body ptubations Suppos that anoth plant P is moving on a coplana obit aound th Sun in th hypothsis that th systm P P b non-sonant so that th spctiv man motions a non-commnsuabl. If w dnot with th position vctos of P and P and with µ th quantity k m bing m th mass of P th ptubing foc on P is givn by 3 5 F = µ Th fist tm is th dict foc of P on P whil th scond tm is th intial indict foc du to th choic of th Sun as oigin of th fnc fam. 5 Th dtmination of th scula ptubations quis a doubl avaging pocss: fist w avag th ptubing foc with spct to P and obtain thus F av aft w avag spct to P th ptubation quations fo ȧ ė aft th substitution of F with F av obtaining thus ȧ ė. This pocdu is allowd fo th fist-od ptubations bcaus dos not contain tms dpnding fom P. Notic that th indict tm of th foc givs a null contibut to any scula ptubation sinc 3 =. W hav by dfinition π F av = µ πa b 3 df = µ T T 3 7 and it follows Gauss thom bcaus with dµ = µ df π a b = µ T 8 w can wit F av = 3 dµ wh th intgal is takn along th llips of th distubing body in th diction of motion. So th poblm is ducd to that of dtmining th scula obital ffcts of a massiv ing whos lmntay distibution of mass has th masu givn by Eq Fo th analytical dtmination of this foc w must appoximat th iational facto 3 = + cosθ θ 3/ = 3 + cosθ θ 3/ 3 3/ > 9 3 3/ > 3 wh 3/ is as 3/ but with and intchangd. W dvlop ths xpssions in pows of th atios / o /. This quis whn applid to plantay ptubations a gat numb of tms fo an accptabl convgnc but th had wok is don by comput algba. W can also wit 3/ i = i / i and xpand in pows of / o / only th scond facto. Ths two choics whn applid to th sam poblm a quivalnt but thy codify diffntly th infomation on th ptubing foc in thi succssion of tms. Fo ou puposs it is mo advantagous to us th fist option in th lat tatmnt of an lliptical ptubing obit whil fo th cicula obit w shall us th scond on that has th advantag to giv mo compact xpssions. B. Avag foc of a plant on a cicula obit Lt b a plant P of mass m in a cicula obit so that = a iθ = a in t wh n is th constant man motion. Consid th foc F on th point = iθ of th obit of an intnal plant P lying in th sam plan. With th notation α = /a < α = + α α cos ϕ 3 ϕ = θ θ dϕ = dθ 3 w hav a iθ F = µ iϕ a iθ = iθ 3 µ a Thfo avaging with spct to P Fav = πa b = µ π π π dϕ 3/ α F x dθ = π iϕ a a 3 a 3 π F x dϕ. 33 3/ α. 34

4 B Avag foc of a plant on a cicula obit 4 In this intgal w wit iϕ a a 3 π Fav = µ dϕ π α / α µ π dϕ iϕ α πa α + α cos ϕ + α α cos ϕ +... Fom th fomula π ±inϕ α = π cos nθ α. 35 = παn α 36 with α < n =... w s that to od α 5 in th numato only th following tms contibut to th total man foc Fav = µ πa + π α α3 4 α α α5 dϕ [ 38 α 564 α4 cos ϕ α cos ϕ α cos 3ϕ so that aft th intgation w hav F av = µ a α α + 3α 4 8 α 75 α α 3 8 a α α3 = µ = µ 8 a ] 75 a a 33 a It asily sn that also to high appoximations F av tains th sam gnal stuctu with th sam fist tm and mo tms with high odd pows of th atio /a. This avagd ptubing foc is cntal so that w can mploy th simplifid ptubation quation fo ω. As valu of th adius a of th cicula obit w tak th smi-majo axis of th tu lliptical obit of P bcaus th avag valu of th modulus of th adius vcto is a to od. W find wh = 3 a A 4 = 5 8 a B 4 = a A B a A +B+ AB a a AB 43 C 44 A a + a a B a a a. 45 A is al fo vy positiv valu of a whil B is al fo a outsid a dfinit intval that fo th Eath is wh th fomula is maninglss. W hav thn to od α 5 Fav = µ 75 8 a C a 3 A 3 a 5 B 46 and so ω = κ c µ F av 47 in which κ is a numical facto fo th convsion fom adian/day to acsc/cntuy givn by κ = π Thn calling that µ /µ = m th cntnnial pcssion at is ω = κ m c 8 a a A B + a a 4 C. 49 If w xpand Fav in pows of α w hav Fav = µ a 3 6 a 5 8 a th fist tm it is popotional only to th position vcto. Thus an xtnal plant xts an appoximatly lina pulsiv foc dictd away fom th cnt on a paticl locatd somwh na th cnt of th obit. W hav to push th appoximation as fa as th tm bcaus of th lativ gatnss of th atio /a fo th intnal plants Mas Eath Vnus and Mcuy. In th litatu 9 was obtaind with anoth mthod th appoximation s Eq. 38 Fav = µ a a 5 fo a numical valuation of th classical pat of th motion of Mcuy s pihlion. To this gad w not that th computation can b don analytically sinc w know that in this cas by Eqs w gt ω = κ c µ F av = κc µ i µ a C i 5 i with th obvious maning of th symbolism. W find so th valu of vy na to th xact valu obtaind by tating th poblm in its full gnality. Obviously this is only a coincidnc du to an almost xact compnsation among th vaious plants but it lavs th fals impssion that th omittd tms do not dstoy this xcllnt agmnt. With th mo complt xpssion of F av w obtain th valu of Th diffnc is du to th fact that w hav nglctd th llipticity of th obits of th distubing plants oth that th mutual inclination of obits which in this cas is ath significant.

5 C Avag foc of a plant in an lliptical obit 5 W consid now th situation in which th obit of P is intnal to that of P. With β = + β β cos ϕ β = α = a / < 53 w hav at th od β 3 Fav i = µ π π 3 = µ π 3 µ π 3 µ π 3 π π π dϕ 3/ β β iϕ dϕ β iϕ β dϕ β iϕ β / β + β cos ϕ + β 4 β /4 β dϕ = µ 3 β 4 4 β = µ a 4 3 4a = a 3 µ 3 4a = µ µ By dvloping in pows of β w find a. 54 F i av = µ µ a Now w hav m µ /µ ω = κ c Fav i = κ 3 iθ µ 4 m c 56 a wh iθ a = a A + B a A B a a AB with A B dfind as bfo so that D 57 ω int = κ3 4 m c D. 58 C. Avag foc of a plant in an lliptical obit Lt us suppos th obit of P lliptical. Thn F av dpnds also fom th mutual gomtical disposition of th two obits i.. fom th angula distanc ω ω of th spctiv pihlia. If P is xtnal with ψ = f f + ω ω γ = / w hav F = µ = µ 3 + cos ψ 3/ µ = 3 + γ γ cos ψ µ + 9 3/ 3 4 γ γ γ cos ψ γ cos ψ +... = µ j H j cos jψ j = wh th cofficints H j a pow sis in γ givn with ν = 3/ by with H = + ν γ + ν γ H = ν γ + ν ν γ 3 + ν ν γ H = 3 ν γ + ν ν γ 4 + ν ν γ H 3 = ν = ν ν + ν =! ν ν + ν ! By dvloping th abov xpssion of F aft th substitution cos jψ = j j + j j j j 65 w can wit th foc as th sum wh F µ = F + F + F + F F = j F = j F 4 = j h j j j+3 h j j j+5 h 4 j 3 j j+7 F = j F 3 = j F 5 = j h j j 67 j+3 h 3 j j 68 j+5 h 5 j j 3 69 j+7 and so on. Th cofficints h i j i h =... a ational numbs. Aft th avag with spct to P Fav will hav tms popotional to ē k a j+n k i j k a j+n k i j. wh n = Fom this w a lad to th following conclusions as gad to Fav: All tms go to zo as th atio i j k n. a i+j a j+n k fo Fav is a foc of th cntal typ that coincids with = with th xpssion of Fav alady found fo th cicula obit. F av givs th contibuts of od whil all th succssiv tms giv contibuts containing th poduct of pows of. Fav and Fav 3 Fav 4 and Fav 5 and in gnal Fav i and Fav i+ giv contibuts of th sam od of gatnss so that thy must b calculatd always togth.

6 C Avag foc of a plant in an lliptical obit 6 As gad to ȧ th dmonstation that at th fist od this lmnt dos not hav scula tms aising fom th plantay ptubing foc h is immdiat only fo th cntal pat Fav. Th oth focs Fav i will giv littl positiv and ngativ contibuts that in th long un countact thmslvs. Now w inst Fav in th ptubation quations and w gt fo th scula pat ȧ = a n π µ R Fav d 7 ė = ic µ F av and dividing by ė i n π µ ic + i ω = µ F av i n π µ IFav 7 IF av 7 so that w must find Fav and loop intgals of th fom i j d i j d 73 I i j+ I i+ j. 74 Considations of th sam typ can b mad whn th obit of P is intnal. Thn w hav th foc F i µ = F O + F I + F II + F III F O = j F II = j F IV = j k o j k II j j+3 j k IV j j+3 j F I = j F III = j j+5 j F V = j k I j j+3 j k III j k V j 76 j+5 j 77 j+7 3 j 78 and so on wh kj N a ational numbs. Thn th typical tms of Fav i a k a j+k i j+n i ē k a j+k i. 79 j+n i All tms go to zo as th atio i j k n. a i+j a j+n k fo fo Fav O is a foc of th cntal typ that coincids with = with th xpssion of Fav i alady found fo th cicula obit. F I av givs th contibuts of od whil all th succssiv tms giv contibuts containing th poduct of pows of. Fav II and Fav III Fav IV and Fav V and in gnal Fav N and Fav NI giv contibuts of th sam od of gatnss so that thy must b calculatd always togth. At last w must find F i av and loop intgals of th fom i d j+n i I j+n 3 i d 8 j+n i+ I. 8 j+n V. APPLICATIONS Aft th fomal dvlopmnt of th plantay ptubing foc w a lft with th pactical application of th fomulas. Fo th scula ptubations in th plantay poblm w hav som possibilitis ach dpnding fom th conct poblm at hand. So aft th choic of th od of appoximation w can pocd fist symbolically and thn w shall hav th chaactistic stuctu of ach of th paticula tms considd and aft th succssiv numical dtmination w shall hav th contibution to th scula vaiation of th sam tm. All this it is possibl bcaus w a in a lina nvionmnt: fist-od ptubations intgations of a sum of lmntay tms al and imaginay pats dtminations. W can vify th wok don with a dict dtmination of ȧ and ė. Fo this it is quid th numical computation of th doubl intgals wh a m π a b b i m 4 π a a b b ṙ = iµ c π π π π RF df df 8 ṙ I + c 3 3 df df At last w gt fo a cntuy ȧ sc = 3655 ȧ 85 ė ė sc = 3655 R 86 ė ω sc = κ I. 87 Fo a numical vification of th mthod applid to th motion of th pihlion w pfd to consid th Eath instad of th usual Mcuy bcaus th coplanaity of th obits involvd is bst vifid fo th fom plant. Th sults fo th Eath to od α 5 a givn in th following tabls wh w hav mployd th plantay lmnts givn in th Appndix fo th poch Januay. In th scond tabl lativ to th Eath s pihlion in acsc/cntuy th fist column is fd to th full classical appoximation whil th oths giv spctivly th contibutions computd by ou mthod of th cicula by Eq. 5 and lliptical pats and thi

7 A Oth foc laws 7 sum. Th diction Thoy is fd to complt publishd calculations which howv mak fnc to a slightly diffnt poch. Plant Thoy A = Total B B-A Mcuy Vnus Mas Jupit Satun Uanus Nptun Total Scula motion of th Eath pihlion W hav a ath good agmnt btwn th two sts of sults. Th discpancis a mainly du to: to having nglctd th non-coplanaity of plantay obits to having usd lmnts latd to diffnt pochs 3 th od of appoximation considd. As a futh xampl w consid th scula advanc of th pihlion of Mas. H Thoy fs to th computation of Doolittl. 3 Plant Thoy A = Total B B-A Mcuy Vnus Eath Jupit Satun Uanus Nptun Total Scula motion of th Mas pihlion with th sam stictions as in th pvious tabl. In conclusion a woth noting th significant coctions to th scula motion of th pihlion du to th psnc of llipticity of th ptubing plants and to th nglct of th lativ inclinations of th obits. It is also vidnt that Vnus and th Eath fo thi spctiv poximity to Mcuy and Mas would qui th intoduction of mo high-od tms in th dvlopmnt of thi distubing foc. A. Oth foc laws W xamin now th ffcts of som oth foc laws 45 bginning fom: Gnal lativity GR. Th gnal lativistic coction to th gavitational law in th fist appoximation can b obtaind intoducing modifications to th classical quation of th motion. If w put β = ṙ /c wh c is th spd of light in vacuum AU/day and st γ β GR quis th following coctions to t m : 6 t t γ t + β = t + α 88 µ µ γ µ + β = µ + α 89 ṙ γ β = α 9 to th od β wh th last is a pu gnal lativistic ffct bcaus involvs th adial distanc and wh w hav put β ṙ /c = µ/c = α/ with α µ/c dfining th gavitational adius of th mass M. W mak ths substitutions in th quation of motion by opping th zo suffixs + µ 3 + α + µ + α 3 α 3 = 9 and to th Oα w hav: + µ 3 + α + α + 3α = 9 + µ 3 = 6µα 4 = F. 93 Th ptubing foc is cntal with 4 = / b 3 so that ω = κ c µ F = κ 3 α c b 3 = κ 3 α n a b c b 3 3 α n = κ c a 94 Fo th cntnnial motion w hav ω = κ T 3 α n c a = κ 6 π α c a. 95 W s fom th abov dvlopmnts that th vaiations with th spd of th plant of tim mass and adial distanc giv a spctiv contibution of /3 /6 and / to th pihlion pcssion. Almost invs-squa law. Lt us suppos that th gavitational law gos as +ϵ wh < ϵ. Thn by xpanding in pows of ϵ +ϵ ϵln and w hav th ptubing cntal foc F = µ ϵln. 97 Th scula pihlion s motion is givn by ω av = κ c Ln F = κϵnab µ iθ ω = κ ϵ nab iω πab = κ nϵ π π π Ln if df 98 Ln if df 99

8 A Oth foc laws 8 Now so Ln Lna + cos f 5 4 cos f ω κnϵ Lt us calculat ϵ fo a tntativ xplanation of th nonclassical pihlion shift of Mcuy. W find 4.95 = ϵ = ϵ = Wb s Law. Now w bifly study a poposd altnativ to th Nwton s law Wb s law 7 and apply it to Mcuy pihlion. This law it is intsting oth than fo histoical asons also bcaus it intoducs additional tms containing th tmpoal divativs of to th invs-squa law. By quating th two xpssions of ṙ w find + ic = iµ c + iµ c iθ 3 and taking th al pat = µ c R i iθ = µ c sinθ ω = µ sin f c 4 d = µ cosθ ω = µ cos f 5 bcaus fom th aa intgal d = c Wb s law is [ F = µ d df. 6 + c ] c d 3 7 wh c is th spd of light. Substituting fo th divativ and intoducing th tu anomaly f = θ ω w hav F = µ µ c c µ cos f + cos f if c 8 and w find F av = πab π µ µ c c cos f + µ c cos f if df = µ 9 c b3 a complx constant. Th scula pihlion motion is givn by ω = κ c µ F n µ av = κ c a and numically fo Mcuy ω n µ = κ c a = 4.3. B. Th luna aps As last application of th mthod of th avags w apply thm to th divation at od m 3 of th pat of th motion of th luna pig indpndnt fom th ccnticity wh m = n /n is th atio of th man motions of th Sun and th Moon in th hypothsis of th main luna poblm th Sun in a cicula obit in th sam plan of Moon s obit. Th ptubing foc F = µ 3 3 with = a iθ = a il bcoms with µ /a 3 = n 3 and nglcting th sola paallax F = n + 3 a cosθ θ +... n n + 3 n cosθ θ a = n + 3 n iθ iθ θ + iθ θ = n + 3 n iθ = n + 3 n il. 3 In th unptubd motion at od w hav fo th Moon s obit in tms of th man longitud l = a il 3 a + a ē il. 4 In th fist appoximation by solving th ptubation quations w find th vction givn by th following tms 8 δ = 45 6 a ē m il a m l l. 5 In th scond appoximation w put = + δ in th quation ė = i { } Iṙ F + ṙ IF 6 µ with ṙ = n/dl. W cannot us F av in this quation bcaus in ṙ a psnt tms containing l. W find at od th following constant tms i 3 n a 3 Iṙ F = i 4 m m3 n 7 i n ṙ IF = i45 a3 6 m3 n 8 and piodical functions of l l that giv a zo contibution to th doubl avag. Thus ė = π π 3 ė dl dl = i 4 π 4 m m3 n so that fom Eq. 3 ė = ω = m m3 n.

9 A Analytical Mthods 9 Th m 3 tm is that found fo th fist tim numically by Claiaut and algbaically by D Almbt in thi spctiv thois of luna motion and this solvd an appant poblmatic aspct of th luna obit vidncd sinc th publication of th Pincipia of Nwton. VI. ORBITAL AVERAGES - B A. Analytical Mthods Som avags a immdiat. So ṙ = T T = T = bcaus this is a contou intgal ov a closd obit. In gnal fo th analyticity of th intgands w hav n ṙ = n. Fom th obital xpssion of ṙ w hav at onc = iθ = iω if = if = n = n 3 n π n+ = inω inf df = n =.... πab 4 Many avags w must comput a of th typ m ±n m ±n 5 with m n positiv intgs. W can limit ouslvs to tak m bcaus vy combination of can b ducd to on of th pcdnt typ mploying th quality = / : m n = m m m ±n = m ±n m = m ±n m. 6 An impotant popty of ths avag is that thy a of th fom m ±n = K m 7 wh K is al bcaus w hav m ±n = imω πab π π g ±imf df 8 wh g is a piodical vn function of f and th intgal of th imaginay pat of g ±imf is zo bcaus this function is odd. In paticula if F is of cntal typ F = g w hav F = K. It is also vidnt that in gnal m ±n = imω imf m±n m a m±n. 9 Th calculations of an avag will b don by adopting th mo convnint vaiabl fo th situation at hand. If n = it will b usd th ccntic anomaly m = am imω π cos ηcos η +i sin η m dη. π If m = it will b convnint mploy th ccntic anomaly fo n > and th tu anomaly fo n <. So w hav fo n and fo n n = π π n+ dη = an cos η n+ dη πa π n = π df π πab = an 3 + cos f n df. n πb n 3 Fo m + n + > th computation with spct to th tu anomaly quis th intgal imω πab π imf m+n+ df 3 with = a / + cos f that it can b don by mans of th patd us of Cauchy intgal fomula fo th divativs. If w substitut in th intgal if s cos f s + df ds s is 3 w obtain [ ] a m+n+ πiab s = s m+n+ 3 [s ps q] m+n+ wh s = is th unitay cicl cntd at th oigin and wh + p = q = p q =. 33 a th solutions of th quation in s s + s + =. 34 Th pol within th cicl s = is p of od m + n +. Thn by Cauchy intgal fomula s = w can wit with m n = ab fs = fs ds s p k = πi f k p 35 k! [ a ] m+n+ f m+n+ p m + n +! 36 sm+n+. 37 s q m+n+

10 A Analytical Mthods This fomula fails fo m + n + < and in this cicumstanc w sot to th calculation of inω π [ ] m+n+ + cos f df inf πab a 38 fo m n + >. Som sults: n n + n n = n 39 n n n n + = n 4 a n { π n n + = inω inf n = df = ab πab n > = b = a b = + 3 a 8b = a 3 b 9 46 = a 47 = a / 48 3 = a 3 3/ 49 4 = 5 = + 5 a 4 5/ a 5 7/ 7 = a 7 / 9 = a 9 5/ 53 = a 9/ 54 = iθ = if iω 55 cos f = cosθ ω = R iω = 3 a 56 sin f = sinθ ω = I iω =. 57 Somtims it is possibl to obtain th sam sult mo asily by mans of a clv us of alady known lations. So fom th immdiat avags = π dη = a 58 πa ṙ = T T = T = πi T = ni 59 w hav fom th obital xpssion of ṙ = ic µ = a + cn µ = Fom th xpssion m n = iω π a π with m n w find ṙ. 6 a [ a cos η+ia sin η a ] m [a cos η] n+ dη 6 = a + 6 = a = 3 a 64 = a = 5a = 3a = 7a = 9a = a = 7 = a 7 3 = 73 4 = b = a b = 3 a b = a b = 3 a b 3 78 = a b 7 79 = 5 a 8 5 = 8 7 = 3 4a 5 8 7/ 9 = 5 + 4a /

11 A Analytical Mthods with iθ a = a A + a + a 84 A iθ a = a + B a a 85 B a = iθ a iθ a 86 + = a A B a A + B + AB a AB 87 iθ a = iθ a a + iθ a 88 + = a A + B a A B a a AB 89 A a + a a B a a a. 9 Loop intgals of th typ f and f ov an llips in th complx plan a psnt whn th xpssion to b avagd contains ṙ o. This is accomplishd by using th idntity ṙ = in th avags f ṙ and f / togth with th xpssions T = π n ṙ = iµ c + iµ c ṙ = µ + a 9 = + ṙ R = = ic + iµ c + iµ c. 9 Ths a asily computd whn is is possibl to us in th intgand th ccntic anomaly in obit s paamtization. W put = a cos η + ib sin η a iω 93 = a cos η 94 = a sin η ib cos η iω dη. 95 It is immdiat to s that whn th foc F is cntal th intgal F d is a pu imaginay numb bcaus F d = g d = π [ b a g ] sin η a sin η iab cos η+a b dη 96 and sinc g is an vn function of η th tms containing sin η sin η a zo in th intgation. In gnal w hav with n lativ intg m n a m+n+ m+ m 97 m n d a m+n+ m m. 98 Som xampls: = = T T d + T 99 = π i a b = π i a b d = = 3 π i a b and = n i a b. Cuv ctification. Fom µ = µ a 3 = π µ π a n πa 3 dη π = a cos η dη = llips lnght. 4 Som oth intgals: = πi + 5 n = nπi n = 3. 6 [a n ] n nπia = n 3/ + n+ n n+. 7 In ths xpssions w may assum that ω = = i.. that th smi-majo axis of th llips lis on th al axis. d = π i a b 8 d = π i a 3 b 9 4 d = 4 π i a5 b d = 8 π i a7 b d = 4 π i a b 3 d = 5 π i a3 b 3 4 d = 4 π i a 4 b d = 5 4 π i a5 b 4 5 d = πia b ē 6 4 d = πia 4 b ē 7 n = bcaus T [ n n+ ṙ = n + ] T =. 8

12 A Analytical Mthods Loop intgals of th fom n m n d m 9 can asily b tansfomd to an obital avag in th tu anomaly w alady hav considd. Fo xampl n T m = = iµ c = iµ c n π m ṙ = c π π n m + df n+ i µ df + m c n ṙ df m π n df. m Th indpndnt vaiabl f of ach of ths intgals can b tansfomd if mo convnint in η by mans of th lation df = b/ dη. This is an intsting by-poduct of th calculus of obital avags: th possibility of altnat computations of ctain intgals. Sinc an angula avag of a finit two-body xpssion can b don in two ways with f η as indpndnt vaiabls w can choic th most convnint fo th calculation and automatically w hav th sult of th cosponding intgal in th oth vaiabl. W dnot as dual th two colatd xpssions. In pactic th mthod is th following: givn th intgal of an xpssion divd fom a twobody obital function w can intpt it as an avag in f o η and aft th computation mad with th mo convnint vaiabl w hav also th valu of th dual intgal in η o f obtaind by using th lations df = b dη dη = df. b W giv h only th simplst xampl which non quis any intgation at all. With α > β β/α = < π dθ α + β cos θ = α π π dθ + cos θ = α a dθ = πa α π = α = πα 3/ α β. 3 Last th Gn s thom on th complx plan f dx dy = f 3 i D D wh D is th simply connctd domain boundd by th unptubd lliptic obit D givs immdiatly fo ach computd valu of a lin intgal th valu of a sufac intgal ov th domain and vicvsa givn a function g w fist intgat it g d = f 4 and aft w calculat th lin intgal of f. Som xampls: dx dy = D D dx dy = = π a b i D 5 dx dy = D 3 = π. 6 All ths sults a puly gomtical without fnc to th undlying dynamical poblm w mploy as solution dvic. VII. DISCUSSION W hav givn a ath consistnt numb of xampls of application of th mthod of th obital avags so it is now possibl to aw conclusions about its pos and cons. Fom th thotical point of viw it ducs th scula ptubation of two gavitationally intacting bodis in complx gomtic situations to a succssion of v small simpl focs ach of which povids a contibution that can b computd xactly. Having availabl a lital xpssion dpns ou knowldg about th vaious factos that hlp poduc th final sult. An impotant point to mphasiz is that with th symbolic fomulas th wok can b don onc and fo all bcaus in actual cass will suffic placing th vaious symbols with th numical valus of th obital paamts. W hav also hanssd th pow of complx analysis to obtain ou sults in an lgant way. This also paadoxically constituts th majo awback of th mthod: it woks und conditions of coplanaity o at most in situations of almost coplanaity but w bliv that in its scop it allows to obtain intsting sults spcially in th calculation of th ffcts of gavitational focs aising fom altnativ thois to gnal lativity. In conclusion w think that this mthod psnt a usful woking tool that can povid valuabl svics to th sach in a wid fild of study. VIII. APPENDIX Tabls of th plantay obital lmnts.? a oundd to th fouth dcimal. Jan a b ω ad. Th figus Mcuy Vnus Eath Mas Jupit Satun Uanus Nptun

13 3 Jan m = µ /µ c n Mcuy Vnus Eath Mas Jupit Satun Uanus Nptun Elctonic adss: s.lmo@mail.com Battin R.H. An Intoduction to th Mathmatics and Mthods of Astodynamics. Rvisd Edition Aiaa Education Sis 999. Lang S. Complx Analysis. Fouth Ed. Sping. 3 Moulton F.R. An Intoduction to Clstial Mchanics. nd Ed. At. 97 p.35 Dov Publications Muay C.D. Dmott S.F. Sola Systm Dynamics. C.U.P Bouw D. Clmnc G. Mthods of Clstial Mchanics. Acadmic Pss Boccaltti D. Pucacco G. Thoy of Obits. Vol. Sping Vashkovyak M.A. and Vashkovyak S.N. Foc function of a slightly lliptical Gaussian ing and its gnalization to a naly coplana systm of ings. Sola Systm Rsach Volum 46 Numb Pags Ioio L. Obital Ptubations Du to Massiv Rings. Eath Moon and Plants Onlin Fist 8 May. 9 Pic M.P. Rush W.F. Nonlativistic contibution to Mcuy s pihlion pcssion. Am. J. Phys Bumbg V.A. Evdokimova L. S. and Skipnichnko V.I. Scula ptubations in gnal plantay thoy. Clstial Mchanics and Dynamical Astonomy Volum Numb Pags Migaszwski C. Gozdziwski K. A scula thoy of coplana non-sonant plantay systm. Mon. Not. R. Aston. Soc. Volum 388 Issu pags Clmnc G.M. Rviws of Modn Physics Doolittl E. Th scula vaiations of th lmnts of th fou inn plants computd fo th poch 85. G.M.T. Tansactions of th Amican Philosophical Socity Nw Sis Vol. No Bayshv Y. and Tikopi P. Pdictions of Gavity Thois. Astophysics and Spac Scinc Libay Volum 383 Fundamntal Qustions of Pactical Cosmology Pags Lin-Sn Li Paamtizd post-nwtonian obital ffcts in xtasola plants. Astophysics and Spac Scinc Onlin Fist 8 Apil. 6 Ny P. Elctomagntism and Rlativity. Hap & Row Nw Yok Rosva Mcuy s Pihlion fom Lvi to Einstin. Clandon Oxfod D Eliso M.M. Th ccnticity vcto of th Moon. Chins J.Phys. 3 9 Stwat M.G. Pcssion of th pihlion of Mcuy s obit. Am. J. Phys Van Lahovn C. and Gnbg R. Chaactizing multi-plant systms with classical scula thoy. Clstial Mchanics and Dynamical Astonomy Onlin Fist 3 Apil Apil. Bombadlli C. Bau G. and Plaz J. Asymptotic solution fo th two-body poblm with constant tangntial thust acclation. Clstial Mchanics and Dynamical Astonomy Volum. Lin-Sn Li Influnc of th gavitational adiation damping on th tim of piaston passag of binay stas. Astophysics and Spac Scinc Volum 334 Numb Pags 5-3 Numb 3 Pags Laa M. Palacin J.F. and Russll R.P. Mission dsign though avaging of ptubd Kplian systms: th paadigm of an Encladus obit. Clstial Mchanics and Dynamical Astonomy Volum 8 Numb Pags -. 4 Dulliv A.M. Evolution of Almost Cicula Obits of Satllits und th Action of Noncntal Gavitational Fild of th Eath and Lunisola Ptubations. ISSN 955 Cosmic Rsach Vol. 49 No. pp. 78. Pliads Publishing Ltd.. 5 Sidlmann K.P. Ed. Explanatoy Supplmnt to th Astonomical Almanac. p.36 Tabl 5.8. Univsity Scinc Books Mill Vally Califonia 99.