STABILITY OF THE RELATIVE EQUILIBRIA OF A RIGID BODY IN A J 2 GRAVITY FIELD

AAS -760 SABLY OF HE RELAVE EQULBRA OF A RGD BODY N A J GRAVY FELD Yu Wang, * Haichao Gui, and Shiji Xu h motion of a point mass in th J problm is gnralizd to that of a rigid bod in a J gravit fild. Diffrnt with th original J problm, th gravitational orbit-rotation coupling of th rigid bod is considrd in this gnralizd problm. h linar stabilit of th classical tp of rlativ quilibria of th rigid bod, which hav bn obtaind in our prvious papr, is studid in th framwork of gomtric mchanics with th scond-ordr gravitational potntial. Non-canonical Hamiltonian structur of th problm, i.., Poisson tnsor, Casimir functions and quations of motion, ar obtaind through a Poisson rduction procss b mans of th smmtr of th problm. h linar sstm matri at th rlativ quilibria is givn through th multiplication of th Poisson tnsor and Hssian matri of th variational Lagrangian. Basd on th charactristic quation of th linar sstm matri, th conditions of linar stabilit of th rlativ quilibria ar obtaind. With th stabilit conditions obtaind, th linar stabilit of th rlativ quilibria is invstigatd in dtails in a wid rang of th paramtrs of th gravit fild and th rigid bod. W find that both th zonal harmonic J and th charactristic dimnsion of th rigid bod hav significant ffcts on th linar stabilit. Similar to th attitud stabilit in a cntral gravit fild, th linar stabilit rgion is also consistd of two rgions that ar analogus of th Lagrang rgion and th DBra-Dlp rgion. Our rsults ar vr usful for th studis on th motion of natural satllits in our solar sstm. NRODUCON h J problm, also calld th main problm of artificial satllit thor, in which th motion of a point mass in a gravit fild truncatd on th zonal harmonic J is studid, is an important problm in th clstial mchanics and astrodnamics []. h J problm has its wid applications in th orbital dnamics and orbital dsign of spaccraft. his classical problm has bn studid b man authors, such as Rfrnc [] and th litraturs citd thrin. Howvr, nithr natural nor artificial clstial bodis ar point masss or hav sphrical mass distribution. On of th gnralizations of th point mass modl is th rigid bod modl. Bcaus of th non-sphrical mass distribution, th orbital and rotational motions of th rigid bod ar * PhD Candidat, Dpartmnt of Arospac Enginring, School of Astronautics, Bihang Univrsit, Bijing, 009, China. -mail: wang@sa.buaa.du.cn PhD Candidat, Dpartmnt of Arospac Enginring, School of Astronautics, Bihang Univrsit, Bijing, 009, China. -mail: guihaichao@gmail.com Profssor, Dpartmnt of Arospac Enginring, School of Astronautics, Bihang Univrsit, Bijing, 009, China. -mail: starsju@ahoo.com.cn

coupld through th gravit fild. h orbit-rotation coupling ma caus qualitativ ffcts on th motion, which ar mor significant whn th ratio of th dimnsion of rigid bod to th orbit radius is largr. h orbit-rotation coupling and its qualitativ ffcts hav bn discussd in svral works on th motion of a rigid bod or grostat in a cntral gravit fild [] [5]. n Rfrnc [6], th orbit-rotation coupling of a rigid satllit around a sphroid plant was assssd. t was found that th significant orbit-rotation coupling should b considrd for a spaccraft orbiting a small astroid or an irrgular natural satllit orbiting a plant. h ffcts of th orbit-rotation coupling hav also bn considrd in man works on th Full wo Bod Problm (FBP), in which th rotational and orbital motions of two rigid bodis intracting through thir mutual gravitational potntial ar studid. A sphr-rstrictd modl of FBP, in which on bod is assumd to b a homognous sphr, has bn studid broadl b man scholars, such as Kinoshita [7], Barkin [8], Abolnaga and Barkin [9], Bltskii and Ponomarva [0], Schrs [], Britr t al. [], Balsas t al. [], Bllros and Schrs [4], and Vrshchagin t al. [5]. hr ar also svral works on th mor gnral modls of FBP, in which both bodis ar nonsphrical, such as th works b Macijwski [6], Schrs [7][8], Koon t al. [9], Boué and Laskar [0], McMahon and Schrs []. Whn th dimnsion of th rigid bod is vr small in comparison with th orbital radius, th orbit-rotation coupling is not significant. n th cas of an artificial Earth satllit, th point mass modl of th J problm works vr wll. Howvr, whn a spaccraft orbiting an astroid or an irrgular natural satllit orbiting a plant, such as Phobos, is considrd, th mass distribution of th considrd bod is far from a sphr and th dimnsion of th bod is not small anmor in comparison with th orbital radius. n ths cass, th orbit-rotation coupling causs significant ffcts and should b takn into account in th prcis thoris of th motion, as shown b Koon t al. [9], Schrs [], Wang and Xu [6]. For th high-prcision applications in th motions of a spaccraft orbiting a sphroid astroid, or an irrgular natural satllit orbiting a dwarf plant or plant, w hav gnralizd th J problm to th motion of a rigid bod in a J gravit fild in our prvious papr []. n that papr, th rlativ quilibria of th rigid bod wr dtrmind from a global point of viw in th framwork of gomtric mchanics. hrough th non-canonical Hamiltonian structur of th problm, th gomtric mchanics provids a sstmic and ffctiv mthod for dtrmining th stabilit of th rlativ quilibria, as shown b Bck and Hall [4]. h linar stabilit of th classical tp of rlativ quilibria, which hav bn alrad obtaind in Rfrnc [], will b studid furthr in this papr with th gomtric mchanics. hrough th stabilit proprtis of th rlativ quilibria, it is sufficint to undrstand th dnamical proprtis of th sstm nar th rlativ quilibria to a big tnt. h quilibrium configuration ists gnrall among th natural clstial bodis in our solar sstm. t is wll known that man natural satllits of big plants volvd tidall to th stat of snchronous motion [5]. Notic that th gravit fild of th big plants can b wll approimatd b a J gravit fild. h rsults on th stabilit of th rlativ quilibria in our problm ar vr usful for th studis on th motion of man natural satllits. W also mak comparisons with prvious rsults on th stabilit of th rlativ quilibria of a rigid bod in a cntral gravit fild, such as Rfrncs [] and [5]. h influnc of th zonal harmonic J on th stabilit of th rlativ quilibria is discussd in dtails. NON-CANONCAL HAMLONAN SRUCURE AND RELAVE EQULBRA h problm w studid hr is th sam as in Rfrnc []. As dscribd in Figur, w considr a small rigid bod B in th gravit fild of a massiv ais-smmtrical bod P. Assum that P is rotating uniforml around its ais of smmtr, and th mass cntr of P is stationar in

th inrtial spac. h gravit fild of P is approimatd through truncation on th scond zonal harmonic J. h inrtial rfrnc fram is dfind as S={,, } with its origin O attachd to th mass cntr of P. is along th ais of smmtr of P. h bod-fid rfrnc fram of th rigid bod is dfind as S b ={i, j, k} with its origin C attachd to th mass cntr of B. h fram S b coincids with th principal as rfrnc fram of th rigid bod B. Figur. A small rigid bod B in th J gravit fild of a massiv ais-smmtrical bod P. n Rfrnc [], a Poisson rduction was applid on th original sstm b mans of th smmtr of th problm. Aftr th rduction procss, th non-canonical Hamiltonian structur, i.., Poisson tnsor, Casimir functions and quations of motion, and a classical kind of rlativ quilibria of th problm wr obtaind. Hr w onl giv th basic dscription of th problm and list th main rsults obtaind b us thr, s that papr for th dtails. h attitud matri of th rigid bod B with rspct to th inrtial fram S is dnotd b A, A [, i j, k ] SO(), () whr th vctors i, j and k ar prssd in th fram S, and SO() is th -dimnsional spcial orthogonal group. A is th coordinat transformation matri from th fram S b to th fram S. f W [ W, W, W z ] a vctor prssd in fram S b, its componnts in fram S can b givn b w AW. () W dfin r as th position vctor of point C with rspct to O in fram S. h position vctor of a mass lmnt dm(d) of th bod B with rspct to C in fram S b is dnotd b D, thn th position vctor of dm(d) with rspct to O in fram S, dnotd b, is r AD. () hrfor, th configuration spac of th problm is th Li group Q SE(), (4) known as th spcial Euclidan group of thr spac with lmnts ( Ar, ) that is th smidirct product of SO() and. h lmnts Ξ of th phas spac, th cotangnt bundl Q, can b writtn in th following coordinats Ξ ( A, r; AΠˆ, p ), (5)

whr Π is th angular momntum prssd in th bod-fid fram S b and p is th linar momntum of th rigid bod prssd in th inrtial fram S [6]. h hat map ^: so() is th usual Li algbra isomorphism, whr so () is th Li Algbras of Li group SO(). h phas spac Q carris a natural smplctic structur brackt associatd to can b writtn in coordinats Ξ as SE(), and th canonical f g g f { f, g} ( Ξ) D f, D ˆ g D g, D Q A ˆ f AΠ A AΠ r p r p, (6) for an f, g C ( Q),, is th pairing btwn SO () and SO (), and DB f is a matri whos lmnts ar th partial drivats of th function f with rspct to th lmnts of matri B rspctivl [6]. h Hamiltonian of th problm H : Q is givn as follows H p V Π Π Q, (7) m whr m is th mass of th rigid bod, th matri diag,, zz is th tnsor of inrtia of th rigid bod and Q: Q Q is th canonical projction. According to Rfrnc [], th gravitational potntial V : Q up to th scond ordr is givn in trms of momnts of inrtia as follows: (0) () GMm GM V V V tr mm R R R R γ R, (8) whr G is th Gravitational Constant, and M is th mass of th bod P. h paramtr is dfind as JaE, whr a E is th man quatorial radius of P. γ is th unit vctor prssd in th fram S b. R Ar is th position vctor of th mass cntr of B prssd in fram S b. Not that R R and R R R. h J gravit fild is ais-smmtrical with ais of smmtr. According to Rfrnc [6], th Hamiltonian of th sstm is S -invariant, naml th sstm has smmtr, whr S is th on-sphr. Using this smmtr, w hav carrid out a rduction, inducd a Hamiltonian on th quotint Q/ S, and prssd th dnamics in trms of appropriat rducd variabls in Rfrnc [6], whr Q/ S is th quotint of th phas spac Qwith rspct to th action of S. h rducd variabls in Q/ S can b chosn as z Π, γ,r,p, (9) whr P =A p is th linar momntum of th bod B prssd in th bod-fid fram S [6] b. h projction from Q to Q/ S is givn b ˆ, A, r; AΠ p Π, γ,r,p. (0) hr is a uniqu non-canonical Hamiltonian structur on Q/ S such that is a Poisson map. hat is to sa, thr is a uniqu Poisson brackt {, } ( z ) satisfing [7] 4

{, } ( z) {, } ( Ξ ), () f g f g Q for an f, gc ( ), whr {, } ( Ξ ) is th natural canonical brackt givn b Eq. (6). Q According to Rfrnc [6], th Poisson brackt {, } ( z ) can b writtn as follows: with th Poisson tnsor Bz () givn b { f, g} ( z) f B( z ) g, () z z Πˆ γˆ Rˆ Pˆ γˆ 0 0 0 Bz () ˆ, () R 0 0 E ˆ P 0 E 0 whr E is th idntit matri. his Poisson tnsor has two indpndnt Casimir functions. On is a gomtric intgral C () z γ γ, and th othr on is C () z γ Π RP ˆ, th third componnt of th angular momntum with rspct to origin O prssd in th inrtial fram S. C () z is th consrvativ quantit producd b th smmtr of th sstm. h tn-dimnsional invariant manifold or smplctic laf of th sstm is dfind in Casimir functions ˆ which is actuall th rducd phas spac Q/ S b Π, γ,r,p γ γ, γ Π RP constant, (4) of th smplctic rduction. h rstriction of th Poisson brackt {, } ( z ) to dfins th smplctic structur on this smplctic laf. h quations of motion of th sstm can b writtn in th Hamiltonian form z {, z H()} z () z B() z H() z. (5) With th Hamiltonian H () z givn b Eq. (7), th plicit quations of motion ar givn b V( γ,r) V( γ, R) Π Π Π R γ, R γ, P R R Π, m V ( γ,r) P P Π. R γ γ Π Basd on th quations of motion Eq. (6), w hav obtaind a classical tp of rlativ quilibria of th rigid bod undr th scond-ordr gravitational potntial in Rfrnc []. At this tp of rlativ quilibria, th orbit of th mass cntr of th rigid bod is a circl in th quatorial plan of bod P with its cntr coinciding with origin O. h rigid bod rotats uniforml around on of its principal as that is paralll to in th inrtial fram S in angular vlocit that is qual to th orbital angular vlocit Ω. h position vctor R and th linar momntum P ar paralll to anothr two principal as of th rigid bod. z (6) 5

Whn th position vctor R is paralll to th principal as of th rigid bod i, j, k, th norm of th orbital angular vlocit Ω is givn b th following thr quations rspctivl: GM GM zz 5 R R m m m GM GM zz 5 R R m m m GM GM zz 5 R R m m m h norm of th linar momntum P is givn b: / / /, (7), (8). (9) P mr. (0) Figur. On of th classical tp of rlativ quilibria. With a givn valu of R, thr ar 4 rlativ quilibria blonging to this classical tp in total. Without of loss of gnralit, w will choos on of th rlativ quilibria as shown b Figur for stabilit conditions 0, 0,, 0, 0,, R 0 0, 0 mr 0, 0 0 zz Π γ R P Ω. () Othr rlativ quilibria can b convrtd into this quilibrium b changing th arrangmnt of th as of th rfrnc fram S b. LNEAR SABLY OF HE RELAVE EQULBRA n this sction, w will invstigat th linar stabilit of th rlativ quilibria through th linar sstm matri using th mthod providd b th gomtric mchanics [4][8]. Conditions of Linar Stabilit h linar stabilit of th rlativ quilibrium z dpnds on th ignvalus of th linar sstm matri of th sstm at th rlativ quilibrium. According to Rfrnc [4], th linar sstm matri Dz of th non-canonical Hamiltonian sstm at th rlativ quilibrium z can b calculatd through th multiplication of th Poisson tnsor and th Hssian of th variational Lagrangian without prforming linarization as follows: Hr th variational Lagrangian F F z is dfind as D z B z z. () 6

i i F z H z C z. () According to Rfrnc [4], th rlativ quilibrium of th rigid bod in th problm corrsponds to th stationar point of th Hamiltonian constraind b th Casimir functions. h stationar points can b dtrmind b th first variation condition of th variational Lagrangian F z 0. B using th formulations of th Hamiltonian and Casimir functions, th quilibrium conditions ar obtaind as: Π γ 0, GM m γ ˆ R R γ Π RP 0, R ˆ V (4) Pγ 0, R P ˆ γr 0. m As w pctd, th rlativ quilibrium in Eq. () obtaind basd on th quations of motion is a solution of th quilibrium conditions Eq. (4), with th paramtrs and givn b i zz mr,. (5) B using th formulation of th scond-ordr gravitational potntial Eq. (8), th Hssian of th variational Lagrangian F z is calculatd as: 0 0 GM m V ˆ ˆ RR 5 P R R γ R F z V ˆ V 0 ˆ P γ γ R R 0 Rˆ γˆ m. (6) h scond-ordr partial drivats of th gravitational potntial in Eq. (6) ar obtaind as follows: V GM m 4 5 R γ R γr γ R RR γ R, (7) V GMm R R GM 5 R GM 5 R RR γγ 5GM 5 RR RR mγ RγR Rγ. R RR 5R R tr m 5γ R 7RR (8) 7

As dscribd b Eqs. (7), () and (5), at th rlativ quilibrium z, w hav 0, 0, 0, 0, R,0,0, 0, 0 0, mr, 0 Ω 0, 0,, Π zz, γ, R, R, P, mr. hn th Hssian of th variational Lagrangian F zz and rlativ quilibrium z can b obtaind as: 0 0 GM m V ˆ ˆ R 5 R P R R γ R F z V ˆ V ˆ 0 P γ γ R R ˆ 0 R γˆ m. z at th h scond-ordr partial drivats of th gravitational potntial in Eq. (9) at th rlativ quilibrium z ar obtaind through Eqs. (7)-(8) as follows: whr V γ R GM m γα 4 R (9), (0) 5 5tr 5mαα mγγ, () V GMm GM = αα 5 R R R 5 tr m α is dfind as α 0 0. h Poisson tnsor Bz ( ) at th rlativ quilibrium z can b obtaind as: whr β is dfind as β ˆ ˆ ˆ ˆ zzγ γ Rα mrβ ˆ ( ) γ 0 0 0 Bz R ˆ, () α 0 0 E mr ˆ β 0 E 0 0,, 0. n Eqs. (9)-(), w hav 0 0 0 α ˆ 0 0, 0 0 0 0 β ˆ 0 0 0, 0 0 0 0 γ ˆ 0 0, 0 0 0 0 0 αα 0 0 0, () 0 0 0 0 0 0 0 0 0 0 0 0 γγ, γα 0 0 0. (4) 0 0 0 0 hn th linar sstm matri Dz of th non-canonical Hamiltonian sstm can b calculatd through Eqs. (), (9) and (). hrough som rarrangmnt and simplification, th linar D z can b writtn as follows: sstm matri 8

D z GMm GM GM ˆ ˆ 5 4 ˆ ˆ zzγ γ 0 mr tr m 4 R R α α 0 R γˆ ˆ γ 0 0 R ˆ ˆ α 0 γ m V V ˆ mr ˆ β γ γ R R (5) As statd abov, th linar stabilit of th rlativ quilibrium z dpnds on th ignvalus of th linar sstm matri of th sstm Dz. h charactristic polnomial of th linar sstm matri Dz can b calculatd b Ps () dt s D z. (6) h ignvalus of th linar sstm matri Dz ar roots of th charactristic quation of th linarizd sstm, which is givn b dt s Dz 0. (7) hrough Eqs. (5) and (7), with th hlp of Matlab and Mapl, th charactristic quation can b obtaind with th following form: 4 6 4 zz 0 4 0 s ( m s A s A )( m s B s B s B ) 0, (8) whr th cofficints A, A 0, B 4, B and B 0 ar functions of th paramtrs of th sstm: GM,, R,, m,, and zz. h formulations of th cofficints ar givn in th Appndi. According to Rfrnc [4], th non-canonical Hamiltonian sstms hav spcial proprtis with rgard to both th form of th charactristic polnomial and th ignvalus of th linar sstm matri Dz : Proprt. hr ar onl vn trms in th charactristic polnomial of th linar sstm matri, and th ignvalus ar smmtrical with rspct to both th ral and imaginar as. Proprt. A zro ignvalu ists for ach linarl indpndnt Casimir function. Proprt. An additional pair of zro ignvalus ists for ach first intgral, which is associatd with a smmtr of th Hamiltonian b Nothr s thorm. Notic that in our problm, thr ar two linarl indpndnt Casimir functions, and th two zro ignvalus corrspond to th two Casimir functions C () z and C () z. h rmaining tn ignvalus corrspond to th motion constraind b th Casimir functions on th tndimnsional invariant manifold. W hav carrid out a Poisson rduction b mans of th smmtr of th Hamiltonian, and prssd th dnamics on th rducd phas spac. h additional pair of zro ignvalus according to Proprt has bn liminatd b th rduction procss. hrfor, our rsults in Eq. (8) ar consistnt with ths thr proprtis statd abov. According to th charactristic quation in Eq. (8), th tn-dimnsional linar sstm on th invariant manifold dcoupls into two ntirl indpndnt four- and si-dimnsional subsstms undr th scond-ordr gravitational potntial. t is worth our spcial attntion that this is not 9

th dcoupling btwn th frdoms of th rotational motion and th orbital motion of th rigid bod, sinc th orbit-rotation coupling is considrd in our stud. Actuall, th four-dimnsional subsstm and s ar th thr frdoms of th orbital and rotational motions within th quatorial plan of th bod P, and th othr thr frdoms, i.. orbital and rotational motions outsid th quatorial plan of th bod P, constitut th si-dimnsional subsstm. h linar stabilit of th rlativ quilibria implis that thr ar no roots of th charactristic quation with positiv ral parts. According to Proprt, th linar stabilit rquirs all th roots to b purl imaginar, that is s is ral and ngativ. hrfor, in this cas of a consrvativ sstm, w can onl gt th ncssar conditions of th stabilit through th linar stabilit of th rlativ quilibria. According to th thor of th roots of th scond and third dgr polnomial quation, that th s in Eq. (8) is ral and ngativ is quivalnt to A 4A0 0, A 0, A0 0; mzz mzz (9) B4 B B4 B4B B0 m 0, 7 m m 4 7 m m B 0, B 0, B 0. 4 0 W hav givn th conditions of linar stabilit of th rlativ quilibria in Eqs. (9) and (40). Givn a st of th paramtrs of th sstm, w can dtrmin whthr th rlativ quilibria ar linar stabl b using th stabilit critrion givn abov. Cas Studis Howvr, th prssions of cofficints A, A 0, B 4, B and B 0 in trms of th paramtrs of th sstm ar tdious, sinc thr ar larg amount of paramtrs in th sstm and th considrd problm is a high-dimnsional sstm. t is difficult to gt gnral conditions of linar stabilit through Eqs. (9) and (40) in trms of th paramtrs of th sstm. W will considr an ampl plant P, which has th sam mass and quatorial radius as th 4 Earth, but has a diffrnt zonal harmonic J. hat is GM.986005 0 m / s and 6 a 6.784 0 m. Fiv diffrnt valus of th zonal harmonic J ar considrd E (40) J 0.5, 0., 0, 0.8, 0.. (4) h orbital angular vlocit is assumd to b qual to.655 0 s with th orbital priod qual to.5 hours. With th paramtrs of th sstm givn abov, th stabilit critrion in Eqs. (9) and (40) can b dtrmind b thr mass distribution paramtrs of th rigid bod: m, and, whr and ar dfind as zz zz,. (4) h ratio m dscribs th charactristic dimnsion of th rigid bod; th ratios and dscrib th shap of th rigid bod to th scond ordr. hr diffrnt valus of th paramtr m ar considrd as follows: 0

5 0,5 0 7,5 0 m, (4) which corrspond to a rigid bod with th charactristic dimnsion of ordr of 00m, 0km and 000km rspctivl. n th cas of ach valu of m, th paramtrs and ar considrd in th following rang,, (44) which hav covrd all th possibl mass distributions of th rigid bod. Givn th mass distribution paramtrs of th rigid bod, th orbital radius R at th rlativ quilibrium can b calculatd b Eq. (7). hn th stabilit critrion in Eqs. (9) and (40) can b calculatd with all th paramtrs of th sstm known. h linar stabilit critrion in Eqs. (9) and (40) is calculatd for a rigid bod within th rang of th paramtrs Eqs. (4) and (44) in th cass of diffrnt valus of th zonal harmonic J. h points, which corrspond to th mass distribution paramtrs guaranting th linar stabilit, ar plottd on th plan in th 5 cass of diffrnt valus of m and J in Figurs ()-(7) rspctivl. n our problm, th gravitational potntial in Eq. (8) is truncatd on th scond ordr. According to th conclusions in Rfrnc [9], onl th cntral componnt of th gravit fild of th plant P is considrd in th gravit gradint torqu, with th zonal harmonic J nglctd. hat is to sa, th attitud motion of th rigid bod in our problm is actuall th attitud dnamics on a circular orbit in a cntral gravit fild from th point viw of th traditional attitud dnamics with th orbit-rotation coupling nglctd. o mak comparisons with th traditional attitud dnamics, w also plot th classical linar attitud stabilit rgion of a rigid bod on a circular orbit in a cntral gravit fild in Figurs ()-(7), which is givn b: 0, 4, (45) 0. h classical linar attitud stabilit rgion givn b Eq. (45) is consistd of th Lagrang rgion and th DBra-Dlp rgion [0]. h Lagrang rgion is th isoscls right triangl rgion in th first quadrant of th plan blow th straight lin 0, and DBra-Dlp rgion is a small rgion in th third quadrant blow th straight lin 0. Notic that at th rlativ quilibrium in our papr, th orintations of th principal as of th rigid bod ar diffrnt from thos at th quilibrium attitud in Rfrnc [0], and thn th dfinitions of th paramtrs and in our papr ar diffrnt from thos in Rfrnc [0] to mak sur that th linar attitud stabilit rgion is th sam with that in Rfrnc [0].

Figur. Linar stabilit rgion in th cas of J 0.5 and m5 0. Figur 4. Linar stabilit rgion in th cas 7 of J 0.5 and m5 0. Figur 5. Linar stabilit rgion in th cas of J 0.5 and m 5 0. Figur 6. Linar stabilit rgion in th cas of J 0. and m5 0. Figur 7. Linar stabilit rgion in th cas 7 of J 0. and m5 0. Figur 8. Linar stabilit rgion in th cas of J 0. and m 5 0.

Figur 9. Linar stabilit rgion in th cas of J 0 and m5 0. Figur 0. Linar stabilit rgion in th 7 cas of J 0 and m5 0. Figur. Linar stabilit rgion in th cas of J 0 and m 5 0. Figur. Linar stabilit rgion in th cas of J 0.8 and m5 0. Figur. Linar stabilit rgion in th 7 cas of J 0.8 and m 5 0. Figur 4. Linar stabilit rgion in th cas of J 0.8 and m 5 0.

Figur 5. Linar stabilit rgion in th cas of J 0. and m5 0. Figur 6. Linar stabilit rgion in th 7 cas of J 0. and m5 0. Figur 7. Linar stabilit rgion in th cas of J 0. and m5 0. 4

Som Discussions on th Linar Stabilit From Figurs ()-(7), w can asil achiv svral conclusions as follows: (a). Similar to th classical linar attitud stabilit rgion, which is consistd of th Lagrang rgion and th DBra-Dlp rgion, th linar stabilit rgion of th rlativ quilibrium of th rigid bod in our problm is also consistd of two rgions locatd in th first and third quadrant of th plan, which ar th analogus of th Lagrang rgion and th DBra-Dlp rgion rspctivl. his is consistnt with th conclusion b iidó Román [5] that for a rigid bod in a cntral gravit fild thr is a linar stabilit rgion in th third quadrant of th plan, which is th analogu of th DBra-Dlp rgion. Howvr, whn th plant P is vr longatd with J 0., for a small rigid bod thr is no linar stabilit rgion; onl in th cas of a vr larg rigid bod with m5 0, thr is a linar stabilit rgion that is th analogu of th Lagrang rgion locatd in th first quadrant of th plan. (b). For a givn valu of th zonal harmonic J (cpt J 0. ), whn th charactristic dimnsion of th rigid bod is small, th charactristic dimnsion of th rigid bod hav no influnc on th linar stabilit rgion, as shown b th linar stabilit rgion in th cass of 7 m5 0 and m 5 0. n ths cass, th linar stabilit rgion in th first quadrant of th plan, th analogu of th Lagrang rgion, is actuall th Lagrang rgion. Whn th charactristic dimnsion of th rigid bod is larg nough, such as m5 0, th linar stabilit rgion in th first quadrant of th plan, th analogu of th Lagrang rgion, is rducd b a triangl in th right part of th first quadrant of th plan, as shown b Figurs (5), (8), () and (4). n th cas of J 0.8, also th linar stabilit rgion in th third quadrant of th plan, th analogu of th DBra-Dlp rgion, is rducd b th larg charactristic dimnsion of th rigid bod, as shown b Figur (4). (c). For a givn valu of th charactristic dimnsion of th rigid bod, as th zonal harmonic J incrass from -0.8 to 0.5, th linar stabilit rgion in th third quadrant of th plan, th analogu of th DBra-Dlp rgion, pands in th dirction of th boundar of th DBra- Dlp rgion, and cross th boundar of th DBra-Dlp rgion at J 0. For a small valu of th 7 charactristic dimnsion of th rigid bod, such as m 5 0 and m5 0, as th zonal harmonic J incrass from -0.8 to 0.5, th linar stabilit rgion in th first quadrant of th plan, th analogu of th Lagrang rgion, kps qual to th Lagrang rgion. Whras for a larg valu of th charactristic dimnsion of th rigid bod m 5 0, as th zonal harmonic J incrass from -0.8 to 0.5, th linar stabilit rgion in th first quadrant of th plan, th analogu of th Lagrang rgion, is shrunk b th influnc of th zonal harmonic J. CONCLUSON For nw high-prcision applications in clstial mchanics and astrodnamics, w hav gnralizd th classical J problm to th motion of a rigid bod in a J gravit fild. Basd on our prvious rsults on th rlativ quilibria, th linar stabilit of th classical tp of rlativ quilibria of this gnralizd problm is invstigatd in th framwork of gomtric mchanics. h conditions of linar stabilit of th rlativ quilibria ar obtaind basd on th charactristic quation of th linar sstm matri at th rlativ quilibria, which is givn through th 5

multiplication of th Poisson tnsor and Hssian matri of th variational Lagrangian. With th stabilit conditions, th linar stabilit of th rlativ quilibria is invstigatd in a wid rang of th paramtrs of th gravit fild and th rigid bod b using th numrical mthod. h stabilit rgion is plottd on th plan of th mass distribution paramtrs of th rigid bod in th cass of diffrnt valus of th zonal harmonic J and th charactristic dimnsion of th rigid bod. Similar to th classical attitud stabilit in a cntral gravit fild, th linar stabilit rgion is consistd of two rgions locatd in th first and third quadrant of th plan rspctivl, which ar analogus of th Lagrang rgion and th DBra-Dlp rgion rspctivl. Both th zonal harmonic J and th charactristic dimnsion of th rigid bod hav significant influncs on th linar stabilit. Whn th charactristic dimnsion of th rigid bod is small, th analogu of th Lagrang rgion in th first quadrant of th plan is actuall th Lagrang rgion. Whn th charactristic dimnsion of th rigid bod is larg nough, th analogu of th Lagrang rgion is rducd b a triangl and this triangl pands as th zonal harmonic J incrass. For a givn valu of th charactristic dimnsion of th rigid bod, as th zonal harmonic J incrass, th analogu of th DBra-Dlp rgion in th third quadrant of th plan pands in th dirction of th boundar of th DBra-Dlp rgion, and cross th boundar of th DBra- Dlp rgion at J 0. Our rsults on th stabilit of th rlativ quilibria ar vr usful for studis on th motion of man natural satllits in our solar sstm, whos motions ar clos to th rlativ quilibria. ACKNOWLEDGMENS his work is supportd b th nnovation Foundation of BUAA for PhD Graduats, th Graduat nnovation Practic Foundation of BUAA undr Grant YCSJ-0-006 and th Fundamntal Rsarch Funds for th Cntral Univrsitis. APPENDX: HE COEFFCENS N CHARACERSC EQUAON h plicit formulations of th cofficints in th charactristic quations Eq. (8) ar givn as follows: m A 5 zz R m 9mzz R mzz 9zz R m R 5 7 4 zz 4 R m zz m R 9 R m R m, ( A.) A 5 0 9 * 0 m R mr m zz R m R R m 6R m R m 8R m 6R m 7 4 5 zz zz R m 6m 6 6, (A.) zz zz zz zz B4 5R m R m R m R m 4R m R m 5 4 5 zz R 5 7 5 R m zz 9 m 9 R m R m 9 R m, (A.) 5 zz zz zz 6

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