symmetry, reduction and relative equilibria of a rigid body in the j2 problem

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Advances in Space Research 51 (2013) 1096–1109

Symmetry, reduction and relative equilibria of a rigid bodyin the J2 problem

Yue Wang ⇑, Shijie Xu

Department of Guidance, Navigation and Control, School of Astronautics, Beijing University of Aeronautics and Astronautics, Room B1024,

New Main Building, 100191 Beijing, China

Received 6 May 2012; received in revised form 11 October 2012; accepted 12 October 2012Available online 12 November 2012

Abstract

The J2 problem is an important problem in celestial mechanics, orbital dynamics and orbital design of spacecraft, as non-sphericalmass distribution of the celestial body is taken into account. In this paper, the J2 problem is generalized to the motion of a rigid bodyin a J2 gravitational field. The relative equilibria are studied by using geometric mechanics. A Poisson reduction process is carried out bymeans of the symmetry. Non-canonical Hamiltonian structure and equations of motion of the reduced system are obtained. The basicgeometrical properties of the relative equilibria are given through some analyses on the equilibrium conditions. Then we restrict to thezeroth and second-order approximations of the gravitational potential. Under these approximations, the existence and detailed proper-ties of the relative equilibria are investigated. The orbit–rotation coupling of the rigid body is discussed. It is found that under the second-order approximation, there exists a classical type of relative equilibria except when the rigid body is near the surface of the central bodyand the central body is very elongated. Another non-classical type of relative equilibria can exist when the central body is elongatedenough and has a low average density. The non-classical type of relative equilibria in our paper is distinct from the non-Lagrangian rel-ative equilibria in the spherically-simplified Full Two Body Problem, which cannot exist under the second-order approximation. Ourresults also extend the previous results on the classical type of relative equilibria in the spherically-simplified Full Two Body Problemby taking into account the oblateness of the primary body. The results on relative equilibria are useful for studies on the motion of manynatural satellites, whose motion are close to the relative equilibria.� 2012 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: J2 problem; Rigid body; Full two body problem; Non-canonical Hamiltonian structure; Reduction; Relative equilibria

1. Introduction

The J2 problem is an important problem in celestialmechanics, orbital dynamics and orbital design of space-craft (Broucke, 1994). In the J2 problem, the motion of apoint mass in a gravitational field truncated on the zonalharmonic J2 is studied. The J2 problem has its applicationsin the design of the sun synchronization orbits and the J2

invariant relative orbits in the spacecraft formations (Koon

0273-1177/$36.00 � 2012 COSPAR. Published by Elsevier Ltd. All rights rese

http://dx.doi.org/10.1016/j.asr.2012.10.030

⇑ Corresponding author. Tel.: +86 10 8233 9751.E-mail addresses: [email protected] (Y. Wang), starsjxu

@yahoo.com.cn (S. Xu).

et al., 2001; Schaub and Alfriend, 2001; Xu et al., 2012).This classical problem has been studied in many works,such as Broucke (1994) and the literatures cited therein.

Strictly speaking, neither natural nor artificial celestialbodies are point masses or have spherical mass distribu-tions. The motion of an extended body is different in manyrespects from that of a point mass. One of the generaliza-tions of the point mass model is the assumption that thebody considered is perfectly rigid. Because of the non-spherical mass distribution, the orbital and rotationalmotions of the rigid body are coupled through the gravita-tional field. The orbit–rotation coupling may cause qualita-tive effects in the motion. This concept was mentioned inmany previous works on the motion of rigid bodies or

rved.


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Y. Wang, S. Xu / Advances in Space Research 51 (2013) 1096–1109 1097

gyrostats interacting through the gravitational potential(Duboshin, 1958, 1972a,b, 1976; Barkin and Demin, 1982).

As for the orbit–rotation coupling, we must mentionthat there are many works on a spherically-simplifiedmodel of the Full Two Body Problem (F2BP), in whichone body is assumed to be a homogeneous sphere andthe gravitational field of the other body is truncated onthe second-order terms (Kinoshita, 1970, 1972a,b; Barkin,1979; Balsas et al., 2008, 2009), or the other body isassumed to be a general rigid body (Aboelnaga and Barkin,1979; Barkin, 1980, 1985; Beletskii and Ponomareva, 1990;Koon et al., 2004; Scheeres, 2006a), an ellipsoid (Scheeres,2004; Bellerose and Scheeres, 2008a,b), a symmetrical body(Vereshchagin et al., 2010), a dumb-bell (Gozdziewski andMaciejewski, 1999), or a model of two material segmentsand a central mass (Breiter et al., 2005).

There are also several works on the more general modelsof the F2BP, in which both bodies are non-spherical. InScheeres (2002), the sufficient conditions for Hill stabilityand instability, and for stability against impact werederived for two general bodies using basic principles fromthe N-body problem. In Scheeres (2009), the relative equi-libria and their stability properties of the planar F2BP werestudied with the gravitational potential expanded up to thesecond order in terms of moments of inertia of the bodies.In Jacobson and Scheeres (2011), equations of motion ofthe planar F2BP were derived with the gravitational poten-tial expanded up to the second order in terms of momentsof inertia, and the equations of motion were modified toaccount for mutual body tides and the solar tides. InMcMahon and Scheeres (submitted), the existence of stableequilibrium points, and the linearized and nonlineardynamics around equilibrium points in the planar F2BPwere investigated. In their problem, the primary bodywas assumed to be oblate, with the gravitational potentialexpanded up to the second order in terms of moments ofinertia. In Koon et al. (2004), the reduction process of anaxis-symmetrical body and a general body is discussed. InBoue and Laskar (2009), the secular motion of two rigidbodies, both of which are assumed to have three symmetryplanes, was investigated with the gravitational potentialexpanded up to the fourth order. In Maciejewski (1995),the Hamiltonian dynamics and relative equilibria of twogeneral rigid bodies were studied.

In the several models of F2BP mentioned above, theorbit–rotation coupling of the non-spherical bodies wasconsidered via the gravitational potential, which dependson the rotational motion of the non-spherical bodies. Theorbit–rotation coupling is the essential difference betweenthe F2BP and the classical two body problem.

The orbit–rotation coupling was also discussed in theworks on the motion of a rigid body or gyrostat in an iner-tial stationary central gravitational field (Wang et al., 1991,1992, 1995). In Wang and Xu (in press), the orbit–rotationcoupling of a rigid satellite around a spheroid planet wasespecially assessed, and it was found that the orbit–rotationcoupling is significant and should be considered for a

spacecraft orbiting around a small asteroid or an irregularnatural satellite around a planet.

When the dimension of the rigid body is very small incomparison with the orbital radius, the orbit–rotation cou-pling is not significant. In the case of an artificial Earth satel-lite, the point mass model of the J2 problem works very well.However, when a spacecraft orbiting around an asteroid oran irregular natural satellite around a planet, such as Phobos,is considered, the mass distribution of the considered body isfar from a sphere and the dimension of the body is not smallanymore in comparison with the orbital radius. In thesecases, the orbit–rotation coupling causes significant effectsand should be taken into account in the precise theories ofthe motion, as shown by Koon et al. (2004), Scheeres(2006b) and Wang and Xu (in press).

Here we generalize the J2 problem to the motion of a rigidbody in a J2 gravitational field. The gravitational field ofspheroid asteroids, dwarf planets and planets can be approx-imated as a J2 gravitational field, thus this generalized prob-lem is a good model for coupled orbital and rotationalmotions of a spacecraft orbiting a spheroid asteroid, or anirregular natural satellite around a dwarf planet or planet.

This problem can also be regarded as a generalization ofthe problem of a rigid body in a central gravitational fieldin Wang et al. (1991, 1992). The central gravitational fieldis generalized to the J2 gravitational field in this paper.Notice those works on the spherically-simplified model ofthe F2BP mentioned above, our generalized problem canbe regarded as a kind of restricted F2BP, and the primarybody is generalized from a homogeneous sphere to a spher-oid with the harmonic coefficient J2. Compared with theplanar F2BP in Scheeres (2009), Jacobson and Scheeres(2011), and McMahon and Scheeres (submitted), our prob-lem is not restricted on the equatorial plane of the primarybody. Notice that the problem in McMahon and Scheeres(submitted) is very similar to our problem; the differencesare that in their problem the motion is restricted on theequatorial plane of the primary body and the mass centerof the primary body is not fixed in the inertial space. Thisimplies the existence possibility of non-coplanar relativeequilibria in our problem, which are not considered inMcMahon and Scheeres (submitted).

In similar settings to our problem, the restricted attitudemotion of a rigid body in the J2 gravitational field of aspheroid planet has been considered by Maciejewski(1997) and Gozdziewski and Maciejewski (1998). In theirstudies, the orbit–rotation coupling is not considered, andthe attitude motion is studied on a predetermined orbit,in which the rigid body is treated as a point mass. There-fore, their problem is essentially different from our problemin this paper.

The equilibrium configuration exists generally amongthe natural celestial bodies in our solar system. It is wellknown that many natural satellites of big planets evolvedtidally to the state of synchronous motion (Wisdom,1987). The orbits of a group of them are almost perfectlycircular—they have eccentricities smaller that 0.01 and

1098 Y. Wang, S. Xu / Advances in Space Research 51 (2013) 1096–1109

inclinations smaller than 0.5 deg, such as Jupiter’s satel-lites: Metis, Adrastea, Amalthea; Galilean satellites Io,Europa, Ganymede and Callisto; Saturn’s satellites: Pan,Atlas, Prometheus, Pandora, Epimetheus, Janus, Encela-dus, Dione, Rhea; Uranus satellites: Ariel, Umbriel, Tita-nia and Oberon (Gozdziewski and Maciejewski, 1998).More importantly, the equilibrium solution is an importantqualitative property and plays a unique role in the phasespace of the system. In the present paper, we will concen-trate on the relative equilibria of the problem in the frame-work of geometric mechanics.

The geometric mechanics has been used in widely-ran-ged problems in celestial mechanics and astrodynamics.In the framework of geometrical mechanics, several power-ful techniques can be performed (Koon et al., 2004; Wangand Xu, 2012), such as the reduction of the symmetric sys-tem with the determination of the relative equilibria on thereduced system from a global point of view (Maciejewski,1995; Vera, 2008, 2009, 2010; Guirao and Vera, 2010b),and the energy-Casimir method for determining the stabil-ity of relative equilibria (Wang et al., 1991, 1995; Mondejaret al., 2001; Vera and Vigueras, 2004, 2006; Guirao andVera, 2010a). By using the tools of geometric mechanics,the reduced system of a symmetrical problem can beachieved by a reduction process. On the reduced system,the relative equilibria can be determined from a globalpoint of view (Mondejar et al., 2001; Wang and Xu,2012). Due to the advantages of geometric mechanics inthe determination of relative equilibria, we will adopt geo-metric mechanics in this paper, rather than the traditionalmethods in previous works on F2BP.

In the present paper, a Poisson reduction is applied onthe original system by means of the symmetry. After thereduction process, the non-canonical Hamiltonian struc-ture and the equations of motion of the reduced systemare obtained. Two basic geometrical properties of the rela-tive equilibria are given through some analyses on the equi-librium conditions. Then under different approximations ofthe gravitational potential, the existence and detailed prop-erties of the relative equilibria for each approximated sys-tem are investigated.

We also make comparisons with previous results on therelative equilibria in the F2BP (Kinoshita, 1972b; Aboelnagaand Barkin, 1979; Scheeres, 2004, 2006a; Bellerose andScheeres, 2008a,b; McMahon and Scheeres, submitted)and in the rigid body dynamics in a central gravitationalfield (Wang et al., 1991, 1992). The similarities and differ-ences between the relative equilibria in this paper and thosein previous works are discussed in details.

2. Configuration and phase space

As described in Fig. 1, we consider a small rigid body B

in the gravitational field of a massive axis-symmetricalbody P. Assume that P is rotating uniformly around itsaxis of symmetry, and the mass center of P is stationaryin the inertial space, i.e., P is in free motion without being

affected by B. The gravitational field of P is approximatedthrough truncation on the second zonal harmonic J2. Theinertial reference frame is defined as S = {e1, e2, e3} whoseorigin O is attached to the mass center of P. e3 is along theaxis of symmetry of P. The body-fixed reference frame isdefined as Sb = {i, j, k} whose origin C is attached to themass center of B. The frame Sb coincides with the principalaxes reference frame of B. The attitude matrix of the rigidbody B with respect to the inertial frame S is denoted by A,

A ¼ ½i; j; k� ¼ix jx kx

iy jy ky

iz jz kz

264

375 2 SOð3Þ; ð1Þ

where the vectors i, j and k are expressed in the frame S, andSO(3) is the 3-dimensional special orthogonal group. A is thecoordinate transformation matrix from the frame Sb to theframe S. If W = [Wx, Wy, Wz]T are components of a vectorin frame Sb, its components in frame S can be calculated by

w ¼ AW : ð2Þ

We define r as the radius vector of point C with respect toO in frame S. The radius vector of a mass element dm(D) ofthe body B with respect to C in frame Sb is denoted by D,then the radius vector of dm(D) with respect to O in frameS, denoted by x, is

x ¼ rþ AD: ð3Þ

Therefore, the configuration space of the problem is the Liegroup

Q ¼ SEð3Þ; ð4Þ

known as the Euclidean group of three space that is the semi-direct product of SO(3) and R3 with elements (A, r). Noticethat the configuration space of the F2BP is SE(3)� SE(3),and it is reduced to SE(3) here by assuming that the massivebody P is in free motion without being affected by B.The ele-ments N of the phase space, the cotangent bundle T*Q, can bewritten in the following coordinates

N ¼ ðA; r; AP; pÞ; ð5Þ

where P is the angular momentum expressed in the body-fixed frame Sb and p is the linear momentum of the rigidbody expressed in the inertial frame S (Wang and Xu,2012). The hat map ^: R3 ! soð3Þ is the usual Lie algebraisomorphism, where so(3) is the Lie Algebras of Lie groupSO(3).The phase space T*Q carries a natural symplecticstructure x = xSE(3), and the canonical bracket associatedto x can be written in coordinates N as

ff ; ggT �QðNÞ ¼hDAf ;DAPgi � hDAg;DAPf i þ @f@r

� �T

@g@p� @g

@r

� �T@f@p; ð6Þ

for any f ; g 2 C1ðT �QÞ, h�; �i is the pairing betweenT*SO(3) and TSO(3), and DBf is a matrix whose elements

Fig. 1. A small rigid body B in the gravitational field of a massive axis-symmetrical body P.


are the partial derivates of the function f with respect to theelements of matrix B respectively (Wang and Xu, 2012).

The Hamiltonian of the problem H : T �Q! R is givenas follows

H ¼ jpj2

2mþ 1

2PT I�1Pþ V � sT�Q; ð7Þ

where m is the mass of the rigid body, the matrixI ¼ diagfIxx; Iyy ; Izzg is the tensor of inertia of the rigidbody and sT�Q : T �Q! Q is the canonical projection.

The gravitational potential of a rigid body in a gravita-tional field can be formulated as a volume integration ofthe potential of the mass elements over the rigid body.Through Taylor expansion and the definition of themoments of inertia of the rigid body, the volume integra-tion, i.e., the gravitational potential of the rigid body,can be written in the form of series and be truncated ona desired order. In Wang and Xu (in press), the gravita-tional potential V : Q! R up to the second order hasbeen obtained in terms of moments of inertia as follows:

V ¼ V ð0Þ þ V ð2Þ ¼ �GM 1mR� GM1

2R3

½trðIÞ � 3�RT I �Rþ em� 3emðc � �RÞ2�; ð8Þ

where G is the Gravitational Constant, and M1 is the massof the body P. The parameter e is defined as e ¼ J 2a2

E,where aE is the mean equatorial radius of P. c is the unitvector e3 expressed in the frame Sb. R = ATr is the radius

vector of the mass center of the rigid body expressed inframe Sb. Note that R = |R| and �R ¼ R=R.

3. Symmetry and reduction

The J2 gravitational field is axis-symmetrical with axis ofsymmetry e3. According to Wang and Xu (2012), the Ham-iltonian of the system is S1-invariant, namely the systemhas symmetry, where S1 is the one-sphere. Using this sym-metry, we can carry out a reduction, induce a Hamiltonianon the quotient T*Q/S1, and express the dynamics in termsof appropriate reduced variables, where T*Q/S1 is the quo-tient of the phase space T*Q with respect to the action ofS1.Notice that here we adopt the Poisson reduction inMarsden (1992), rather than the symplectic reduction inMarsden (1992) and Marsden et al. (2000). Therefore, thereduced system is a noncanonical Hamiltonian systemevolving on T*Q/S1, not a canonical system evolving onT*(Q/S1) the cotangent bundle of the quotient of the con-figuration space Q with respect to the action of S1. Theextra constraints on the phase space of the noncanonicalHamiltonian system are given by Casimir functions, whichwill be explained more detailedly later. The basic theoryabout the geometric mechanics and reduction process canbe found in Marsden (1992) and Marsden and Ratiu(1999).

According to Wang and Xu (2012), the reduced vari-ables in T*Q/S1 can be chosen as


z ¼ ½PT ; cT ;RT ;PT �T 2 R12; ð9Þwhere P = ATp is the linear momentum of the rigid bodyexpressed in the body-fixed frame Sb. The projection fromT*Q to T*Q/S1 is given by

WðA; r; AP; pÞ ¼ ½PT ; cT ;RT ;PT �T : ð10ÞAccording to Marsden and Ratiu (1999), there is a

unique non-canonical Hamiltonian structure on T*Q/S1

such that W is a Poisson map. That is to say, there is aunique Poisson bracket f�; �gR12ðzÞ such that

ff ; ggR12ðzÞ �W ¼ ff �W; g �WgT �QðNÞ ð11Þfor any f ; g 2 C1ðR12Þ, where f�; �gT �QðNÞ is the naturalcanonical bracket of the system given by Eq. (6).

According to Wang and Xu (2012), the Poisson bracketf�; �gR12ðzÞ can be written in the following form

ff ; ggR12ðzÞ ¼ ðrzf ÞT BðzÞðrzgÞ ð12Þwith the Poisson tensor B(z) given by

BðzÞ ¼

P c R P

c 0 0 0

R 0 0 E

P 0 �E 0

26664

37775; ð13Þ

where E is the identity matrix. This Poisson tensor has twoindependent Casimir functions. One is a geometric integralC1(z) = cTc � 1, and the other one is C2ðzÞ ¼ cT ðPþ RPÞ,the third component of the angular momentum with re-spect to origin O expressed in the inertial frame S. C2(z)is the conservative quantity produced by the symmetry ofthe system, as stated by Noether’s theorem.

The ten-dimensional invariant manifold or symplecticleaf of the system is defined in R12 by Casimir functions

R ¼ fðPT ; cT ;RT ;PT ÞT 2 R12jcT c ¼ 1; cT ðPþ RPÞ¼ constantg ð14Þ

which is actually the reduced phase space T*(Q/S1) of thesymplectic reduction. The restriction of the Poisson bracketf�; �gR12ðzÞ to R defines the symplectic structure on this sym-plectic leaf.The equations of motion of the system can bewritten in the Hamiltonian form

_z ¼ fz; HðzÞgR12ðzÞ ¼ BðzÞrzHðzÞ ð15ÞWith the Hamiltonian H(z) given by Eq. (7), the explicitequations of motion are given by

_P ¼ P� I�1Pþ R� @V ðc;RÞ@R

þ c� @V ðc;RÞ@c

;

_c ¼ c� I�1P;

_R ¼ R� I�1Pþ Pm ;

_P ¼ P � I�1P� @V ðc;RÞ@R

:

ð16Þ

4. Relative equilibria

In this section, working on the reduced system, we inves-tigate the basic geometrical properties of the relative equi-

libria based on the equilibrium conditions first, and thenwe investigate the existence and detailed properties of therelative equilibria based on two approximations of thegravitational potential.

4.1. Basic geometrical properties

The relative equilibria are the equilibria of the reducedsystem. That is to say, at the relative equilibria, the coordi-nates of the reduced phase space, i.e., P, c, R and P, are allconstant. According to the equations of motion given byEq. (16), the equations that determine the relative equilib-ria are given by

I Xe �Xe þ Re �@V ðc;RÞ@R

��e

þ ce �@V ðc;RÞ

@c

��e

¼ 0; ð17Þ

ce �Xe ¼ 0; ð18Þ

Re �Xe þPe

m¼ 0; ð19Þ

Pe �Xe �@V ðc;RÞ@R

��e

¼ 0; ð20Þ

where the subscript e is used to denote the value at theequilibria. Here the angular momentum P and the angularvelocity I�1P in Eq. (16) are denoted by IX and X respec-tively. Actually, Eqs. (17) and (20) are the torque and forcebalance equations respectively, and Eqs. (18) and (19) de-scribe the geometry of the configuration of the relativeequilibria. The basic geometrical properties of the relativeequilibria will be investigated in this subsection based onEqs. (18) and (19).At the configurations of relative equilib-ria, for an observer fixed in a uniformly rotating frame thatis rotating around origin O in a constant angular velocityxe, the rigid body B is motionless. Therefore, xe is actuallythe angular velocity of the rigid body expressed in the iner-tial frame S, i.e., xe = AXe. According to Eq. (18), Xe isparallel to ce, therefore xe is parallel to in the inertialframe. Observed in a uniformly rotating frame that is rotat-ing around e3, the rigid body is motionless. Thus, we haveproved that

Proposition 1. At the relative equilibria, the rigid bodyrotates uniformly in a constant angular velocity xe that is

parallel to e3 in the inertial frame S. Observed in a uniformly

rotating frame around e3 with angular velocity xe, the rigid

body is motionless.

According to the second part of the above proposition, atthe relative equilibria, the radius vector re(t) of the masscenter C of the rigid body B will generate a cone in the iner-tial frame S with xe as its axis. When Re is perpendicular toce, this cone will degenerate into a plane. Therefore, the or-bit of the mass center of rigid body is a circle that is thebase of the cone, and the angular velocity of the orbit isxe, same as that of the rigid body. Notice that there isno priori reason that the center of the circular orbit coin-cides with origin O, and the orbital plane is perpendicular


to e3, i.e., parallel to the equatorial plane of body P. Thisconclusion is consistent with Eq. (19). Therefore, we haveproved the following proposition.

Proposition 2. At the relative equilibria, the orbit of the mass

center of the rigid body is a circle with its center located on e3

and the orbital angular velocity is equal to xe. The orbital

plane is parallel to the equatorial plane of the body P.

4.2. Zeroth-order approximation

The zeroth-order approximation of the gravitationalpotential is given by

V 0ðc;RÞ ¼ V ð0Þ: ð21ÞUnder this approximation, the gravitational potential is

actually the potential of a point mass in a central gravita-tional field. Using the zeroth-order potential Eq. (21), wecan write the torque and force balance equations Eqs.(17) and (20) as follows

IXe �Xe ¼ 0; ð22Þ

Pe �Xe �GM 1m

R3e

Re ¼ 0: ð23Þ

From the balance equations Eqs. (22) and (23), we knowthat the rigid body is in a torque-free rotational motion,and the orbital motion is a Keplerian circular orbit, sameas the zeroth-order approximated results in Wang et al.(1991). This is because in the case of this zeroth-orderapproximation of the gravitational potential, the gravitygradient torque is vanished and the orbital motion is theclassical restricted two body problem.

According to Eq. (22), we know that ce is parallel to aprincipal axis of the tensor of inertia I. By taking the innerproduct of both sides of Eq. (23) with Xe, we conclude thatRe � Xe = 0, i.e., Re is perpendicular to ce. Therefore, theorbit plane of the mass center of the rigid body is in theequatorial plane of body P, and the center of the circularorbit coincides with origin O. Using Eqs. (19) and (23),we can obtain

ðXe � ReÞ �Xe ¼GM 1

R3e

Re: ð24Þ

From Eq. (24) and the condition Re � Xe = 0, we get theclassical Kepler frequency formula in the restricted twobody problem

Xe ¼GM1

R3e

� �1=2

: ð25Þ

Thus we have proved the following proposition.

Proposition 3.

Under the zeroth-order approximation, at the relative

equilibria, the rigid body rotates uniformly around one of its

principal axes that is parallel to e3 in the inertial frame S inangular velocity Xe, whose norm is given by Eq. (25). The

orbit of the mass center of the rigid body is a circle in the

equatorial plane of body P with its center coinciding with

origin O, and the orbital angular velocity is equal to the

angular velocity of the rigid body.

4.3. Second-order approximation

The second-order approximation of the gravitationalpotential is given by

V 2ðc;RÞ ¼ V ð0Þ þ V ð2Þ: ð26ÞUsing the formulations of the zeroth and second-order

potential, we can write the torque and force balance equa-tions Eqs. (17) and (20) as follows

I Xe �Xe þ T2 ¼ 0; ð27ÞPe �Xe þ F2 ¼ 0; ð28Þwhere T2 and F2 are the gravity gradient torque and grav-itational force expressed in the body-fixed frame Sb

respectively.

T2 ¼ Re �@V 2ðc;RÞ

@R

��e

þ ce �@V 2ðc;RÞ

@c

��e

¼ 3GM1

R5e

Re � IRe;

ð29Þ

F2 ¼ �@V 2ðc;RÞ

@R

��e

¼ �GM1m

R3e

Re

� 3GM 1

2R5e

ð1� 5ðce � �ReÞ2Þemþ ð1� 5ð�RxeÞ

2ÞIxx

h

þ ð1� 5ð�RyeÞ

2ÞIyy þ ð1� 5ð�RzeÞ

2ÞIzz

iRe

� 3GM 1

R5e

IRe �3GM 1emðce � ReÞ

R5e

ce: ð30Þ

By taking the inner product of both sides of Eq. (28)with ce, and from the condition Xe � ce = 0, we concludethat F2 � ce = 0, i.e.,

GM1m

R3e

Re � ce þ3GM1

2R5e

½ð1� 5ð�RxeÞ

2ÞIxx þ ð1� 5ð�RyeÞ

2ÞIyy

þ ð1� 5ð�RzeÞ

2ÞIzz þ ð3� 5ðce � �ReÞ2Þem�Re � ce

þ 3GM 1

R5e

IRe � ce ¼ 0: ð31Þ

4.3.1. Classical type of relative equilibria

First we assume a particular case when Re is parallel to aprincipal axis of the tensor of inertia I. According to Eq.(31), we have

GM1m

R3e

þ 3GM1

R5e

IR þ3GM 1

2R7e

½ðR2e � 5ðRx

eÞ2ÞIxx

�

þ ðR2e � 5ðRy

eÞ2ÞIyy þ ðR2

e � 5ðRzeÞ

2ÞIzz þ 3R2eem�

� 15GM1em

2R7e

ðRe � ceÞ2

�Re � ce ¼ 0; ð32Þ

where IR is defined by IRe = IRRe.


From Eqs. (27) and (29), we have T2 = 0 and

I Xe �Xe ¼ 0: ð33Þ

We know that ce is parallel to a principal axis of the ten-sor of inertia I. For a general rigid body B, this can meaneither Re � ce = 0 or Re � ce = 0. From the physical intui-tion, the case of Re � ce = 0 means that the mass centerof the rigid body is always located above the pole of bodyP, which is impossible in practical applications of a space-craft orbiting around an asteroid or an irregular naturalsatellite around a planet. Therefore we have Re � ce = 0,which is consistent with Eq. (32). Thus, the orbit plane ofthe mass center of the rigid body is in the equatorial planeof body P, and the center of the circular orbit coincideswith origin O. By taking the inner product of both sidesof Eq. (28) with Xe, we have Re � Xe = 0 that is consistentwith Re � ce = 0. From the fact that Re and Xe are parallelto two different principal axes of the tensor of inertia andEq. (19), we conclude that Pe is parallel to the third princi-pal axis of the tensor of inertia I.

Without loss of generality, we assign Re ¼ ½Re 0 0 �T ,ce ¼ ½ 0 0 1 �T , Xe ¼ ½ 0 0 Xe �T and Pe ¼ m½ 0 ReXe

0�T . Then Eq. (28) can be written as

Pe �Xe �GM 1m

R3e

Re �3GM1

2R5e

½�2Ixx þ Iyy þ Izz þ em�Re ¼ 0:

ð34Þ

Using Eqs. (19) and (34), we can obtain

ðXe�ReÞ �Xe ¼GM 1

R3e

Reþ3GM1

2R5e

�2Ixx

mþ Iyy

mþ Izz

mþ e

� �Re:

ð35Þ

From Eq. (35) and the condition Re �Xe ¼ 0, we get

Xe ¼GM1

R3e

þ 3GM1

2R5e

�2Ixx

mþ Iyy

mþ Izz

mþ e

� �� 1=2

: ð36Þ

Then the condition of existence of this relative equilib-rium can be obtained from Eq. (36) as follows

GM1

R3e

þ 3GM 1

2R5e

�2Ixx

mþ Iyy

mþ Izz

mþ e

� �> 0: ð37Þ

Notice that the ratio of the second term to the first termon the left side of Eq. (37) is order of J 2ðaE=ReÞ2. Therefore,the condition Eq. (37) can be satisfied in most applicationsexcept when aE=Re is large and J 2 is negative with a largeabsolute value, i.e., the body B is near the surface of thebody P and P is very elongated.

Similarly, if we assign Re ¼ 0 Re 0½ �T , we get

Xe ¼GM1

R3e

þ 3GM1

2R5e

Ixx

m� 2

Iyy

mþ Izz

mþ e

� �� 1=2

; ð38Þ

and the condition of existence of this relative equilibrium

GM1

R3e

þ 3GM 1

2R5e

Ixx

m� 2

Iyy

mþ Izz

mþ e

� �> 0: ð39Þ

If it is assumed that Re ¼ 0 0 Re½ �T , we can obtain

Xe ¼GM1

R3e

þ 3GM 1

2R5e

Ixx

mþ Iyy

m� 2

Izz

mþ e

� �� 1=2

; ð40Þ

and the condition of existence

GM1

R3e

þ 3GM1

2R5e

Ixx

mþ Iyy

m� 2

Izz

mþ e

� �> 0: ð41Þ

According to Eqs. (36), (38), and (40), the orbitalmotion of the rigid body is affected by the moments of iner-tia of the rigid body. This effect can be considered equiva-lently as a change of the oblateness of the central body inthe sense of the point mass model. This is the consequenceof the mutual coupling between the orbital and rotationalmotions, and is consistent with the conclusions in Wangand Xu (in press). Notice that in the case of Ixx ¼Iyy ¼ Izz, i.e., the mass distribution of the rigid body is asphere under the second-order approximation, the effectsof the moments of inertia in the orbital motion arevanished, that is consistent with the physical origin of thegravitational orbit–rotation coupling.

Therefore, we have the following proposition.

Proposition 4. Under the second-order approximation, thereexists a classical type of relative equilibria except when the

body B is near the surface of the body P and P is very

elongated. At this type of relative equilibria, the orbit of the

mass center of the rigid body is a circle in the equatorial

plane of body P with its center coinciding with origin O, and

the norm of the orbital angular velocity Xe is given by Eq.

(36), (38), (40). The rigid body rotates uniformly around one

of its principal axes that is parallel to in the inertial frame Sin angular velocity that is equal to the orbital angular

velocity Xe. The radius vector Re and the linear momentum

Pe are parallel to another two principal axes of the rigid

body. The condition of existence of this type of relative

equilibria is given by Eq. (37), (39), (41).

4.3.2. Non-classical type of relative equilibria

Here we assume the case when Re is not parallel to aprincipal axis of the tensor of inertia I . From Eqs. (27)and (29), we have

I Xe �Xe ¼3GM1

R5e

IRe � Re: ð42Þ

By taking the inner product of both sides of Eq. (42)with Xe, we have

ðIRe � ReÞ �Xe ¼ IRe � ðRe �XeÞ ¼ 0: ð43ÞTherefore, IRe lies in the plane spanned by Re and Xe. In

the same way we get

I Xe � ðRe �XeÞ ¼ 0: ð44Þ


I Xe also lies in the plane spanned by Re and Xe. Thenthe plane spanned by Re and Xe is parallel to a principalplane of the tensor of inertia. From Eq. (19), Pe is parallelto the principal axis, which is perpendicular to the principalplane spanned by Re and Xe.

Without loss of generality, we assume that Pe is parallelto the principal axis j

Re ¼ ½Rxe 0 Rz

e �T; ce ¼ ½ cx

e 0 cze �

T;

Xe ¼ Xe½ cxe 0 cz

e �T; Pe ¼ mXe½ 0 Rx

ecze � Rz

ecxe 0 �T :

Then Eq. (27) can be written as follows

X2ec

xec

zeðIzz � IxxÞ ¼

3GM 1

R5e

RxeR

zeðI zz � IxxÞ: ð45Þ

Here a general rigid body with ðIzz � IxxÞ–0 is consid-ered, therefore

X2ec

xec

ze ¼

3GM 1

R5e

RxeR

ze: ð46Þ

Re is not parallel to a principal axis of the rigid body,thus Rx

eRze–0. From Eq. (46) and Rx

eRze–0, we know that

ce is not parallel to a principal axis either, and

Re � ce ¼ cxeR

xe þ cz

eRze ¼ cx

eRxe þ

3GM1

R5eX

2ec

xe

RxeðRz

eÞ2–0:

Therefore, the orbit of the mass center of the rigid body isa circle with its center located on e3 but not coinciding withorigin O, and the orbital plane is parallel to but not in theequatorial plane of body P. Fig. 2 illustrates the geometryof this non-classical type of relative equilibria. In Fig. 2,cz

e and cxe are the coordinates of the vector e3 on the axes

k and i respectively; Rze and Rx

e are the coordinates of the vec-tor r on the axes k and i respectively. The force balanceequation (28) can be written into two equations

Fig. 2. A non-classical type of relative equilibria under the se

mX2e ðcz

eÞ2 � 3GM1

R5eX

2e

ðRzeÞ

2

!Rx

e �GM 1m

R3e

Rxe

� 3GM 1

2R7e

R2etrðIÞ � 5ðRx

eÞ2Ixx � 5ðRz

eÞ2Izz

h

þ ðR2e � 5ðcx

eRxe þ cz

eRzeÞ

2ÞemiRx

e �3GM 1

R5e

IxxRxe

� 3GM 1emðcxeR

xe þ cz

eRzeÞ

R5e

cxe ¼ 0; ð47Þ

mX2eððcx

eÞ2�3GM 1

R5eX

2e

ðRxeÞ

2ÞRze�

GM 1m

R3e

Rze�

3GM1

2R7e

�½R2etrðIÞ�5ðRx

eÞ2Ixx�5ðRz

eÞ2IzzþðR2

e�5ðcxeR

xeþcz

eRzeÞ

2Þem�

�Rze�

3GM1

R5e

IzzRze�

3GM1emðcxeR

xeþcz

eRzeÞ

R5e

cze¼ 0: ð48Þ

With the orbital angular velocity Xe given, the existencecondition of this non-classical type of relative equilibria isequivalence to the solvable condition of the algebraicequations Eqs. (46)–(48) in practical parameter ranges ofRx

e and Rze. Here the practical parameter range means the

possible range in which the solution is likely to exist.The practical parameter ranges can be estimated in themodel of the restricted two body problem by using theorbital angular velocity Xe, and masses of the bodies Pand B, as explained more detailedly later in the numericalstudies.

4.3.2.1. Case studies. However, it is difficult to analyze theexistence of relative equilibria through theoretical studiesof this system described by nonlinear algebraic equations.Thus, we will consider a representative example and makenumerical studies. We consider the case of a satellite thathas the same mass distribution parameters and orbital per-iod as Phobos moving around Mars. The parameters ofMars and Phobos are as follows (Wang et al., 1992;Konopliv et al., 2011)

cond-order approximation of the gravitational potential.


GM1 ¼ 4:283� 1013 m3=s2; m ¼ 1:082� 1016 kg;

Ixx ¼ 5:500� 1023 kg m2; Iyy ¼ 4:718� 1023 kg m2;

Izz ¼ 6:481� 1023 kg m2; Xe ¼ 2:282� 10�4 s�1;

J 2 ¼ 0:001831; aE ¼ 3:396� 106 m; e ¼ 2:112� 1010 m2:

With the orbital angular velocity Xe given, to investigatethe existence of the non-classical type of relative equilibria,we need to identify the solvable condition of the algebraicequations Eqs. (46)–(48) in the domain of Rx

e and Rze, which

are two independent variables in Eqs. (46)–(48).From Eq. (46), we get expressions as follows without

loss of generality

cxe ¼

1

2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

4� ð3GM1

X2eR5

e

RxeR

zeÞ

2

s !12

;

cze ¼

1

2þ

ffiffiffi1

4

r� 3GM1

X2eR5

e

RxeR

ze

!20@

1A

12

: ð49Þ

So cze > cx

e > 0 and we assume that Rxe > 0 and Rz

e > 0without loss of generality, as shown by Fig. 2. Accordingto Wang et al. (1992), we can make a reasonable conclusionthat the offset of the orbital plane from the equatorial planeof body P is necessarily small, thus the angle between re(t)and the equatorial plane of body P, denoted by h, is alsosmall. We also have

h ¼ h1 þ h2; h1 > 0; h2 > 0; ð50Þwhere h1 is the angle between re(t) and i; h2 is the angle be-tween e3 and k. Therefore, h1 and h2 are both necessarilysmall. Then cz

e is near 1, cxe is almost zero, Rx

e is close tothe value of the orbital radius in point mass model that isabout 9 � 103 km, and Rz

e is small in comparison with thedimension of the satellite. This estimation of parameterranges of Rx

e and Rze is important for the following numer-

ical studies.We calculate the value of the left side of Eq. (48),

denoted by W, in the estimated ranges of Rxe and Rz

e, andwe give the 3-D graphical representation of W in Fig. 3.We find that W is non-zero when Rz

e–0, i.e., Eq. (48) can-not be satisfied when Rz

e–0. Obviously, Rze ¼ 0 means it is

the classical type of relative equilibria described in Propo-sition 4. Therefore, the non-classical type of relative equi-libria cannot exist in this case.Wang et al. (1991) showedthat in a central gravitational field, only the classical typeof relative equilibria in Proposition 4 can exist under thesecond-order approximation. Therefore, we only need todiscuss the relative effects of the second degree componentof the gravitational field with respect to the central compo-nent. The relative magnitude of the second degree compo-nent with respect to the central component is determinedby e but not affected by M1. Given the mass M1, e is deter-mined by the average density and coefficient J2 of the bodyP. Therefore, we need to consider different values of J2 andthe average density.First we discuss different values of J2

with the average density equal to that of Mars. We calcu-

late W in the estimated ranges of Rxe and Rz

e and give thegraphical representation in Fig. 4 in the cases ofJ 2 ¼ 0;0:1;0:2;0:4 respectively. We find that in allthese cases W is non-zero when Rz

e–0, i.e., the non-classicaltype of relative equilibria cannot exist. In Fig. 4, it is shownthat the surface of W is going down with Rx

e decreasingwhen J2 is negative with a small absolute value, zero andpositive, whereas with a negative J2, whose absolute valueis large enough, the surface is going up with Rx

e decreasing.The surface is more close to zero plane with a larger abso-lute value of the negative J2, and the surface may cross thezero plane in the case of a negative J2 with a large absolutevalue. Therefore, we further discuss the relative equilibriain the case when J2 is negative with a large absolute value.

We choose the case of J2 = �0.5, and we calculate thevalue of left side of Eq. (47), denoted by U, and W in theestimated ranges of Rx

e and Rze. The graphical representa-

tions of U and W are given in Fig. 5. From Eq. (49), Rxe

and Rze satisfy the following inequality

3GM1RxeR

ze=ðX2

eððRxeÞ

2 þ ðRzeÞ

2Þ5=2Þ < 1

2: ð51Þ

Surfaces in Fig. 5 are incomplete in a certain region dueto this constraint. Fig. 5 shows that the interaction of sur-faces of U and W is always below the zero plane whenRz

e–0, i.e., Eqs. (47) and (48) cannot be satisfied at the sametime when Rz

e–0. Therefore, the non-classical type of rela-tive equilibria cannot exist. The values of J2 discussedabove have covered the practical range of the oblatenessof bodies in celestial mechanics and astrodynamics. Thus,for the body P with the average density of Mars, thenon-classical type of relative equilibria cannot exist.Through the numerical results, we know that a negativeJ2 with a large absolute value is helpful to the existenceof non-classical type of relative equilibria.

Then we will discuss different average densities of thebody P in the case of J2 = �0.5. We calculate W in the esti-mated ranges of Rx

e and Rze and give the graphical represen-

tation in Fig. 6 when the average density of the body P isequal to 0.1, 0.5, 1 and 2 times of Mars’ average densityrespectively. We find that only in the cases of 0.1 and 0.5times of Mars’ average density, the surfaces cross the zeroplane, which implies that Eq. (48) can be satisfied whenRz

e–0. That is to say, the existence of non-classical typeof relative equilibria is possible only in the cases of 0.1and 0.5 times of Mars’ average density.

We calculate the value of U and W in the estimatedranges of Rx

e and Rze in the cases of 0.5 and 0.1 times of

Mars’ average density with J2 = �0.5, and the graphicalrepresentations are given in Figs. 7 and 8 respectively.We find that in the case of 0.5 times of Mars’ average den-sity, the interaction of the surfaces U and W is below thezero plane when Rz

e–0, i.e., Eqs. (47) and (48) cannot besatisfied at the same time when Rz

e–0. Therefore, thenon-classical type of relative equilibria cannot exist.Whereas in the case of 0.1 times of Mars’ average density,the interaction of the surfaces U and W crosses the zero

Fig. 3. The value of W in the case that J2 and the average density of the body P are equal to Mars.

Fig. 4. The value of W with the average density of body P equal to that of Mars. The seven surfaces, from up to bottom, are drawn in the cases ofJ2 = �0.4, �0.2, �0.1, 0, 0.1, 0.2, 0.4 respectively.


plane when Rze–0, which implies that Eqs. (47) and (48) are

satisfied at the same time when Rze–0. That is to say, in the

case of J2 = �0.5 and 0.1 times of Mars’ average density,the non-classical type of relative equilibria does exist.

Therefore we have proved the following proposition.

Proposition 5. Under the second-order approximation, when

the body P is elongated with a negative J2 whose absolute

value is large enough (such as J2 = �0.5), and the average

density of P is low enough (such as 0.1 times of Mars’), there

exists another non-classical type of relative equilibria. At this

type of relative equilibria, the orbit of the mass center of the

rigid body is a circle with its center located on e3 but notcoinciding with origin O, and the orbital plane is parallel to

but not in the equatorial plane of body P. The rigid body

rotates uniformly around e3 in the inertial frame S in angular

velocity that is equal to the orbital angular velocity Xe. The

linear momentum Pe is parallel to a principal axis of the rigid

body, whereas neither ce nor the radius vector Re is parallel

to a principal axis of the rigid body, but the plane spanned by

ce and Re is parallel to a principal plane of the rigid body,which is perpendicular to Pe.

4.4. Some discussions on the relative equilibria

Obviously, the relative equilibria we have obtainedabove correspond to the synchronous motion of a satellitearound a spheroid plant. As stated in Section 1, many nat-ural satellites of big planets evolved tidally to the state of

Fig. 5. The value of U and W with the average density of body P equal to that of Mars in the case of J2 = �0.5.

Fig. 6. The value of W in the case of J2 = �0.5. The four surfaces, from up to bottom, are drawn in the cases of the average density equal to 0.1, 0.5, 1 and2 times of Mars’ respectively.


synchronous motion. Therefore, our results on relativeequilibria are useful for the studies on the motion of natu-ral satellites, the motion of which are close to the relativeequilibria.

The classical type of relative equilibria under the second-order approximation in our problem is similar to the rela-tive equilibria in Wang et al. (1991). Also, our classical typeof relative equilibria is similar to the classical type of rela-tive equilibria in several works on the spherically-simplifiedmodel of the F2BP, such as the stationary motion inKinoshita (1972b), the long-axis equilibria and short-axisequilibria in Scheeres (2004), Bellerose and Scheeres(2008a,b), and the locally central point in Scheeres(2006a). Our results have generalized these previous resultsby taking into account the oblateness of the primary body.

It is worth our special attention that the classical type ofrelative equilibria in our problem is especially similar to theequilibrium points in McMahon and Scheeres (submitted),in which the oblateness of the primary body was also con-sidered. The differences are that in their problem the masscenter of the primary body is not fixed in the inertial space,and the motion is restricted on the equatorial plane of theprimary body. Therefore, the non-classical type of relativeequilibria in our problem was not considered by them andcannot exist in their problem.

Aboelnaga and Barkin (1979), Wang et al. (1992) andScheeres (2006a) have shown that there exists non-classical(non-Lagrangian) type of relative equilibria (or called non-locally central point in Scheeres, 2006a) in the motion of arigid body in a central gravitational field due to the orbit–

Fig. 7. The value of U and W in the case of J2 = �0.5 with the average density of the body P equal to 0.5 times of Mars’.

Fig. 8. The value of U and W in the case of J2 = �0.5 with the average density of the body P equal to 0.1 times of Mars’.


rotation coupling of the rigid body. However, Wang et al.(1991) showed that this kind of non-classical (non-Lagrangian) type of relative equilibria cannot exist in acentral gravitational field under the second-orderapproximation.

The non-classical type of relative equilibria in our prob-lem is due to the combined effects of the second zonal har-monic of the central body and the orbit–rotation couplingof the rigid body, and can exist even under the second-orderapproximation. Therefore, our non-classical type of relativeequilibria is a distinct phenomenon, different from the non-classical type of relative equilibria in Aboelnaga and Barkin(1979), Wang et al. (1992) and Scheeres (2006a).

Our non-classical type of relative equilibria shouldbelong to the non-Lagrangian relative equilibria of twogeneric rigid bodies discussed in Maciejewski (1995). Asstated by Maciejewski (1995), the non-Lagrangian relative

equilibria generically exist in the motions of two genericrigid bodies, whereas the Lagrangian relative equilibriacan exist only in exceptional cases when the bodies possesssome kind of symmetry and the mass distributions satisfycertain conditions.

In the case of J2 = 0, the gravitational field is central.Therefore, through the numerical studies we have also ver-ified the conclusion in Wang et al. (1991) that in a centralgravitational field non-classical (non-Lagrangian) type ofrelative equilibria cannot exist under the second-orderapproximation.

5. Conclusions

For new applications in celestial mechanics and astrody-namics, we have generalized the J2 problem to the motionof a rigid body in a J2 gravitational field. The relative equi-


libria of this generalized problem are investigated in theframework of geometric mechanics.

Based on the geometric structure of the original system,the non-canonical Hamiltonian structure of the reducedsystem is obtained through a Poisson reduction. Equationsthat determine the relative equilibria are obtained on thereduced system. The basic geometrical properties of the rel-ative equilibria are obtained through some analyses basedon the equilibrium conditions. It is found that at relativeequilibria the rigid body rotates uniformly around e3 inthe inertial space in angular velocity that is equal to theorbital angular velocity. The orbit of the mass center is acircle parallel to the equatorial plane of body P with itscenter located on e3.

Under the zeroth-order approximation of the gravita-tional potential, it is found that at relative equilibria theorbit of the mass center is a Keplerian circular orbit inthe equatorial plane of body P with its center located atorigin O, and ce is parallel to a principal axis of rigid body.

Under the second-order approximation of the gravita-tional potential, there exists a classical type of relative equi-libria except when the rigid body is near the surface of thecentral body and the central body is very elongated. At thistype of relative equilibria, the circular orbit is in the equa-torial plane of body P with its center located at origin O. ce,Re and Pe are parallel to three principal axes of the rigidbody respectively. We also find that the orbital motion ofthe rigid body is affected by the moments of inertia of therigid body, which can be considered equivalently as achange of the oblateness of the central body in the senseof the point mass model. Another non-classical type of rel-ative equilibria can exist only when P is elongated enoughand has a low density. At this non-classical type of relativeequilibria, the orbital plane is parallel to but not in theequatorial plane of the body P, Pe is parallel to a principalaxis of the rigid body, and the plane spanned by ce and Re

is parallel to a principal plane of the rigid body, which isperpendicular to Pe.

Our classical type of relative equilibria under the sec-ond-order approximation is similar to the classical typeof relative equilibria in Kinoshita (1972b), Wang et al.(1991), Scheeres (2004, 2006a), and Bellerose and Scheeres(2008a,b). Our results have generalized these previousresults by taking into account the oblateness of the primarybody. Our classical type of relative equilibria is especiallysimilar to the equilibrium points in McMahon and Sche-eres (submitted), in which the oblateness of the primarybody was also considered. In their problem, the motion isrestricted on the equatorial plane of the primary body.Therefore, the non-classical type of relative equilibria wasnot considered in their problem.

Our non-classical type of relative equilibria is mainlydue to the combined effects of the second zonal harmonicof the central body and the orbit–rotation coupling ofthe rigid body, and can exist under the second-orderapproximation. This non-classical type of relative equilib-ria is a distinct phenomenon, different from the non-classi-

cal type of relative equilibria in Aboelnaga and Barkin(1979), Wang et al. (1992) and Scheeres (2006a), which isdue to the orbit–rotation coupling of the rigid body andcannot exist under the second-order approximation.We also have verified the conclusion in Wang et al.(1991) that in a central gravitational field non-classical typeof relative equilibria cannot exist under the second-orderapproximation.

Our results on the relative equilibria are very useful forthe studies on the motion of many natural satellites, whosemotion are close to the relative equilibria. Further ques-tions concerning the bifurcation and stability of the relativeequilibria are of interest and worthy of detailed studies inthe future.

Acknowledgements

The authors thank the associate editor Pascal Willis andthe two anonymous reviewers for their helpful suggestionsand comments to improve this paper. This work is sup-ported by the Innovation Foundation of BUAA for PhDGraduates and the National Natural Science Foundationof China (11172020).

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symmetry, reduction and relative equilibria of a rigid body in the j2 problem

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