Translational-Rotational Motion of Earth Artificial Satellite (EAS) in Hill's Gravity Field

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1 Translaional-Roaional Moion of Earh Arificial Saellie (EAS) in Hill's Graviy Field Asemessova alamkas,a, Bekov Askar,b, Shinibaev Maksu, Ussipbekova Dinara Republic of azakhsan, Almay, Sapaev Sree, azakh Naional Technical Universiy afer I Sapayev Republic of azakhsan, Almay, 5 Shevchenko Sree, Join Sock Company Naional Cener of Space Researches and Technologies a naucha@mailru, b bekov@mailru *Asemessova alamkas eywords: Hill's graviy field, proofmass, a funcion of ime, he Moon, Moon moion heory Absrac The mehod presened below makes i possible o obain an approximae soluion o he problem of ranslaional-roaional moion of proofmass in Hill's graviy field, as explici funcions of ime Inroducion The problem of sudying he Moon's moion araced and keeps aracing aenion of many researchers [-6] In course of creaing he heory of he Moon's moion, GV Hill suggesed wo mehods of building he inermediae orbi of he Moon [7] The firs mehod was described in works published in [8] The dynamic meaning of he problem was in he following simplificaions: ) The Sun moves away o an infinie disance ) Simulaneously increasing is mass infiniely ) In he relaive geocenric moion, he Sun moves along epler's circular orbi Based on his formalizaion, Hill replaced he power funcion of an exac bodies problem wih a simplified one This mehod of building an inermediae orbi was subsequenly used in by EV Brown in order o finalize building of a complee analyical heory of lunar moion The second mehod of building an inermediae orbi of he Moon was proposed in 897, and was subsequenly called by BM Schigolev [9] "he second inermediae Hill's orbi" Here Hill considers i necessary ha he inermediae orbi conains perigee and ascending node secular movemen To his end, he proposed a simplified power funcion of he following form U vr ( v v) z, () r here is he produc of muliplying graviaional consan by he sum of he Earh and he Moon masses; and are properly seleced consan mulipliers If he perurbing and he perurbed bodies are on he same side from he cenral body, hen () corresponds o realiy and gives a good approximaion The defec occurs when he perurbing and he perurbed bodies are on opposie sides of he cenral body, bu i is miigaed by he fac ha he Sun a his poin is far away from he Moon Differenial equaions of proofmass orbial moion in Hill's variables In Hill's problem, he Moon is considered a passively graviaing body, and he cenral body is Earh, he perurbed body is he Moon and he perurbing body is he Sun, ie, he Moon's moion is considered in he "Earh-Moon-Sun" sysem In our problem, we consider orbial moion in he "Earh - proofmass- Moon" sysem, and he proofmass, AES model, is considered as he passively graviaing body In he Oxyz geocenric coordinae sysem, differenial equaions of proofmass orbial moion

2 have he form d x x d y y d z z vx, vy, vz () r r r They allow he area inegral dy dx x y C () and he energy inegral dx dy dz ( U h), () whereas C is he area inegral consan, h is he energy inegral consan In Hill's variables, equaions () ake he form: d w w, (5) / d w ( s ) d s d s,, (6) d w C whereas w 6, vc 6 ( v v) C,, (7) C here: is he projecion of radius-vecor r on he Oxy plane, is rue longiude, s - is laiude angen,, cons, w is Hill's variable If we resric ourselves o he orbis of small inclinaion o he Oxy plane, s, s, hen (5) allows reducion approximaely wdw d, (8) Hw w w whereas H is he consan of inegraion defined by he relaion H hc (9) Changing consans and H, we obain he following classificaion of ypes of moion: I Recilinear moion, H = II Parabolic moion, H III Ellipic moion, H IV Hyperbolic moion, H Inegraing differenial equaions (5) and (6) in case of ellipic moion This case corresponds o he following parameers values, H Equaion (8) akes he form wdw d () w w Hw Subradical polynomial has hree posiive l, l, l and one negaive l roos Le's arrange hem in descending order For acual movemens, he subradical polynomial mus be posiive Previously, in [] i was found ha i is posiive wihin wo inervals А) w, В) w

3 Nex, le's look a he firs of moion inervals In he inerval w, he following ransformaion of expression (9) o he Legendre's normal form [] is valid * w dh d, () k sin h whereas Wih w, h, sin h w, ik k i sin h w, h,,, k,,, k, k, i, () *, k is he module of ellipic inegral of he s kind, h is an inermediae variable: h Le's inroduce ino () and () Jacobian funcions and use sandard developmens [], hen, keeping in developmens by powers of O ( k ) he module of ellipic inegral of he s kind, we have he following expressions for,, w, u : ( k k ) ( k k )cos u k cos u, () * ( w kw k w) u ( kw k w)sin u k w sin u, () w ( w kw k w) ( kw k w)cos u k w cos u, (5) whereas is he complee ellipic inegral of he s kind, u ( u ku k u) ( ku k u)sin k u sin k u cos, (6) whereas is ime Consan coefficiens,w, u are wrien in [] Expressions () and () define polar coordinaes, via (5) as explici funcions of ime for he proofmass in inerval w The same mehod can be applied in inerval w Roaional moion of he proofmass relaive o he cener of mass in case of A B mc Le he proofmass be fixed in he cener of mass x C yc zc and he main momens of ineria be relaed via A B mc, m cons, hen he full se of differenial equaions shall have form []: dp dq dr nqr n, npr n,, r cons, (7) arccos, sin sin, sin cos, cos, (8) p q m arcg,, n,, (9) m R p cos, q sin, r, () d d d r q, p r, q p () Differenial equaions (7), () and () allow he following firs inegrals p q C n, (), ()

4 p q C ( ) r n, () r r cons (5) In hese equaions ( p, q, r) is he graviaional consan, is he disance o he cener of mass of he Earh, R is he normalized angular velociy vecor; he fixed coordinaes sysem Cxyz is associaed wih he moving coordinaes of he sysem Cx yz according o he following able: Table Moving coordinaes of he sysem х у z x y z where,,,,, are direcion cosines;,, are Euler's angles Excluding from he firs inegrals and equaion p q ( ) values p q,, we obain d (6) a b b b Nex, le's reduce (6) o Legendre's normal form [] Subradical polynomial is posiive in wo inervals: ), ; ) Le's consider he second inerval, arranging polynomial roos in descending order In accordance wih [7] we have here d a, k, k, (7) k sin / ( ), ik k i wih, Le's inroduce noaions n a and move from (7) o Jacobian ellipic func- ions From (8) le's find u sin sin n,,, k,,, b n i, wih,, (8) k k cosq k cosq k cos6q, n, (9) q consequenly, he nuaion angle is: arccos[ k k cosq k cosq k cos6q] () From (9) le's find he precession angle k ) q k sin q k sin q k sin 6q () ( From () le's find he proper roaion angle ( k ) q k sin q k sin q k sin 6q ()

5 Thus, in he inerval le's find Euler's angles as funcion of ime wih accuracy O ( k ) Euler's kinemaic equaions and he Poisson's equaions considering ransferable angular velociy of proofmass cener of graviy in orbial movemen Le's inroduce he following coordinae sysems O * xyz, pu he sar O * o he cener of he Earh's mass, align axis O * z wih is axis of roaion, and axes x, y complemen he sysem o he righ and remain moionless, hen le's place he cener of proofmass O in he reference poin of wo coordinae sysems Oxyz and Ox yz, wih ha le's poin axis z o he coninuaion R O* O, and le's poin axes x, y so ha hey complemen he sysem o he righ, and poin y en normal o he orbi, x e along he angen o he orbi, z - along e R R, x, y, z should be direced along he main cenral axes of ineria of he proofmass x z X Z O * r R dm Y x e O e n e R Figure Coordinaes of he sysem y y z Le's define posiion of he coordinae sysems Oxyz and direcing cosines: Ox yz using he following able of Table Table of direcing cosines х у z e x e n y e R z Then he srengh funcion afer discarding members of order O ( ) shall have he form whereas - is he proofmass U m, () R Direcing cosines are relaed o Euler's angles by he following equaions 5

6 coscos sin sin cos, cossin cossin cos, sin sin, cossin sin coscos, sin sin coscoscos, sin cos, sin sin, sin cos, cos () inemaic equaions ake he form: p cos e, q sin e, (5) r e Poisson's equaions, aking ino accoun e, have he form: d d d r q e, p r e, q p e, d d d r q e, p r e, q p e, (6) d d d r q, p r, q p From (6) and (5) we shall obain: e, e, d e (7) Inegraion of differenial equaions of ranslaional-roaional moion of he proofmass in Hill's graviy field (inervals w and ) Le he proofmass make ranslaional-roaional moion in Hill's graviy field, hen in accordance wih [] differenial equaions shall have he form: d x x d y y d z z vx, vy, vz, r r r (8) dp dq dr A ( C B) qr M x B ( A C) pr M y, C ( B A) pq M z Now, assuming ha he proofmass moves along ellipic orbi ype wih a sligh inclinaion o he main plane (Oxy), we have he soluion for he firs hree differenial equaions in form () and (), (5) (in he w inerval): ( k k ) ( k k )cos u k cos u, (9) * ( w kw k w) u ( kw k w)sin u k w cos u, () u ( u ku k u) ( ku k u)sin k u sin k u cos, () Le us consider he roaional moion of he proofmass in relaion o he cener of masses in he inerval in case A B mc In case e we ge Euler's angles (), () and () arccos [ k k cos k cos k cos6 ] ; () 6 ( k ) k sin k sin k sin ; () 6

7 ( k ) k sin k sin k sin 6 () Now le's consider ha, hen e e * n ( w kw k w) ( kw k w)cos k w cos 7, (5) and le's use (7), ie, e ( ), e, e (6) Having inegraed (6), we have ( k k )sin k cos ; (7) ( k k )sin k cos ; (8) arccos [( v kv k v) ( v kv k v)cos ] (9) Consan coefficiens,, v are wrien in [] The soluion obained provides he possibiliy o define he effec of he orbial moion of proof mass cener on is roaional moion: ( k k )sin k cos ; ( k k )sin k cos ; arccos [( v kv k v) ( v kv k v)cos ], here,,, obviously, he value of hese incremens depends on iniial condiions of proofmass moion REFERENCES [] olb, EW and MS Turner, 99 The Early Universe Wesview Press, pp: -8 [] Guh, A, 998 The Inflaionary Universe Basic, pp: - [] Turner, MS, 7Quarks and he Cosmos Science, 5: 59-6 [] Frieman, J, MS Turner and D Huerer, 8 Dark Energy and he Acceleraing Universe Annual Reviews of Asronomy and Asrophysics, 6: 85- [5] Barcelo, C, S Liberai, S Sonego and M Visser, 8Fae of Graviaional Collapse in Semiclassical Graviy Physical Review, 77(): 7-79 [6] Susskind, L, 8 The Black Hole War: My Bale wih Sephen Hawking o Make he World Safe for Quanum Mechanics Lile Brown, pp: -9 [7] Shchigolev, BM, 95 Inermediae orbis in a hree-body problem Bul Sernberg Asronomical Insiue Moscow Universiy, : 59-9 [8] Hill, GW, 878 Researchesin he Lunar heory Journal of Mahemaics, pure and applied,: 5- [9] Shchigolev, BM, 96 Abou inermediae orbi in Hill's hree-body problem Pub Sernberg Asronomical Insiue Moscow Universiy, 8: 9-98 [] Shinibaev, MD e al, 999 Ellipic ype of body movemen in second fla Hill's orbi In he Proceedings of he 999 Pracical Conference"Auezov Readings-", Shymken, Vol, pp: -5 [] orn, ZG and T orn, 97Mahemaical reference book for Scieniss and Engineers Moscow: Nauka, pp: 7 [] Aksenov, EP, 986 Special funcions in celesial mechanics Moscow: Nauka, pp: [] Shinibaev, MD, Translaional-roaional moion of a rigid body in saionary and nonsaionary Earh's graviy field Almay: Gylym, pp: