STABILITY OF EQUILIBRIA POINTS FOR A DUMBBELL SATELLITE WHEN THE CENTRAL BODY IS OBLATE SPHEROID

The main aim of the present work is to study the positions of the equilibria points and their stability in the frame work of satellite approximation. The signi cant implication is that the motion around these points is unstable in the linear sense. The principle of angular momentum conservation is used as a tool to reduce the degree of freedom of the dynamical systems of equations. The positions of the relative equilibria are explicitly found as well as necessary and su cient conditions for stable motion in the linear sense are stated.


Introduction
The dumbbell satellite in its most simple structure is composed of two point masses connected by a massless non extensible link. It moves around an object whose gravity central eld holds a mutual gravitational attraction with the masses described by the Newton's universal law of gravitation. The problem of dumbbell satellite is considered a special case of a tethered satellite problem which formally has the same structure of the dumbbell satellite with an extensible link between the two point masses. The dynamics of a dumbbell or tethered satellites have been extensively treated in the literature. We can highlight some signicant contributions related to existence and stability of equilibrium points as well as periodic and bifurcations solutions of these problems in the sequel. Celletti and Sidorenko (2008) investigated the dumbbell satellite's attitude dynamics when the center of mass moves on a Keplerian trajectory. They found a stable relative equilibrium position in the case of circular orbits which disappears as far as elliptic trajectories are considered. In circular orbits they replaced the equilibrium position by planar periodic motions, which are proved to be unstable with respect to out-of-plane perturbations. Wong and Misra (2008) examined the planar dynamics of a variable length multi-tether system at the second Sun-Earth Lagrangian point. They determined a closed form solution of the system under some simple tether length functions. They also obtained numerical results for tether pitch libration under more complex tether length functions. In addition they showed that the linear controller can accurately control the spiral motion via numerical simulations. The in-plane periodic solutions of a dumbbell satellite system in elliptic orbits were obtained via bifurcation with respect to the orbital eccentricity, and their trajectories of the searched periodic solutions were projected on the van der Pol plane by Nakanishi et al. (2011) Zhang et al (2012) applied coincidence degree theory to establish the criteria on the existence of periodic solutions for a tethered satellite system in an elliptical orbit. They presented the uniqueness of periodic solutions for the tethered satellite in a circular orbit. They also addressed the conditions on the global asymptotic stability of the equilibrium states for the tethered satellite system in accordance with the Lyapunov stability theory and Barbashin-Krasovski theory. A simplied model of an orbital cable system equipped with an elevator when the cabin performs periodic "shuttle" motions is studied by Burov et al. (2012), under the assumption that the elevator mass is small compared with the dumbbell mass. They used Poincare´s theory to determine the conditions for the existence of families of system periodic motions analytically depending on the arising small parameter and passing into some stable radial steady-state motion of the unperturbed problem as the small parameter tends to zero. They also proved that, for suciently small parameter values, each of the radial relative equilibria generates exactly one family of such periodic motions. Moreover they studied the stability of the obtained periodic solutions in the linear approximation. Vera (2013) gave the sucient conditions for the existence of periodic solutions of a rigid dumbbell satellite placed in the equilateral equilibrium L4 of the restricted three-body problem via averaging theory. The relevance of eccentric reference orbits on the dynamics of a tethered formation and the stability of the formations when a massive cable model is included in the analysis of a multi-tethered satellite formation is discussed and studied by Avanzini The above equations represent an autonomous nonlinear dynamical system for the motion of the dumbbell in satellite approximation. This equations can be rewritten in the form where φ = 2Θ , a 2 = n 1 n 2 and n i = m 3−i l/m s , i = 1, 2

Stability of motion around equilibria points
In this section we nd the equilibrium points and study the stability of motion around these points in the linear sense for the dynamical system given by Eqs. (2). Hence we impose that r = r e , θ = θ e and φ = φ e at equilibrium points. Therefor we will have the below conditions.
from Equation (3b) we imply that φ e = nπ , n ∈ Z. To linearize equations of motion given by System (2), we assume that the perturbations of the conguration variables from their equilibrium points are denoted by . Consequently the linearized system will be controlled by Eq.(4) represents an autonomists linear dynamical system with six degree of freedom and its characteristic equation is (7) It is clear that from Eq. (7) the linearized system has three conjugate pairs of the eigenvalues the rst two are equal to zero and the remaining four may be real or pure imaginary or mixed between real and pure roots according to the sign of the quantities of b 1 and b 3 . These roots will be ruled by Hence there are two types of equilibrium points rst when Θ = mπ, in this case the longitudinal axis of dumbbell is elongated in the radial of mass center and faced the attractor center by one of its ends and cos φ e = 1. While in the second type Θ = (2m + 1) π 2 and the longitudinal axis of dumbbell is elongated in the tangent to the orbit of the center of mass and cos φ e = −1. In the rst type we have b 1 > 0 and b 3 < 0 , therefore the roots of characteristic equation are two roots each of them equals zero and two real conjugate roots as well as two pure imaginary conjugate roots. While in the second type b 1 may have negative or positive value and b 3 > 0 and the characteristic equation has also two equal roots with zero value and two conjugate real roots but the other two roots may be real conjugate or pure imaginary conjugate roots. therefor we obtain unstable motion for all cases.

Stability of motion around relative equilibria points
The Lagrangian of the dynamical system given by Eqs. (1) where p = p θ /m s If we use the same natation in the pervious section, the conditions of equilibrium points in this case are controlled by 4.1. Linearized equation of motion. Now our attention is directed to linearize Eqs. (10) to obtain simplied expression which can be handled more easily. For this purpose we impose that the perturbations of the conguration variables from their equilibrium points are denoted by (y 1 , y 2 ) where (ṙ,φ) ≡ (y 3 , y 4 ). Therefore the linearized system will be governed by It is obvious that Eq.(12) represents an autonomists linear dynamical system with four degree of freedom and its characteristic equation is Eq.(15) represents four roots and these roots will be given by (16) As aforementioned in the pervious section we have two types of relative equilibrium points. rst occurs when Θ = mπ while second appear when Θ = (2m + 1) π 2 where (m = 0, 1, 2, 3, .....). The system stability depends on the roots of Eq.(16). If all roots are pure imaginary we will have periodic stable solution in the proximity of relative equilibrium points. But the solution will be unstable if any of the roots are real or complex number.

Stability conditions. It is obvious that from Eq.(16) the conditions of obtaining stable motion are
In general the collection of these conditions is the guarantee to obtain stable motion if whole of them are satised together. While if one of the pervious stated conditions is not achieved then we have at least one root, that is not pure imaginary root. In this case the motion will be unbounded and this in turn leads to unstable motion. Therefore we determine the necessary and sucient conditions for stable motion around the relative equilibrium points through the following two theorems for each type of motion. Theorem 1. In the frame work of the rst type of dumbbell satellite motion the necessary and sucient condition for stable motion in the vicinity of relative equilibrium points is B + c 1 < 0 Proof Theorem 1 In rst type motion [cos φ e = 1, c 2 < 0 , c 3 c 4 < 0] and we have three cases: 1. If c 1 > 0 Then |c 1 + c 2 | < |c 1 − c 2 |, from Eq.(17b) |B| < √ D. If D > 0 we have two conjugate real roots and the other two are conjugate pure imaginary roots whatever B is negative or positive. But if D < 0 we have four complex roots every two of them are conjugate whatever B is also negative or not.
2. If c 1 = 0 There are two equal roots with zero value and the remaining two roots are conjugate pure imaginary or conjugate real according to B is negative or positive respectively.
3. If c 1 < 0 Then |c 1 + c 2 | > |c 1 − c 2 |, from Eq.(17b) |B| > √ D and D > 0. Hence we have four pure imaginary roots if B is negative. But if B is positive we have four real roots every two of them are conjugate. Then the conditions for stable motion in this case are c 1 < 0 and B < 0. Consequently the necessary and sucient conditions is B + c 1 < 0 Theorem 2. In the frame work of the second type of dumbbell satellite motion the necessary and sucient condition for stable motion in the vicinity of relative equilibrium points is B − c 1 < D Proof Theorem 2 In second type motion [cos φ e = −1, c 2 > 0, c 3 c 4 < 0] and we also have three cases: 1. If c 1 < 0 If D > 0 we have two conjugate real roots and the other two are conjugate pure imaginary roots whatever B is negative or positive. But if D < 0 we have four complex roots every two of them are conjugate whatever also B is negative or not.
2. If c 1 = 0 There are also two equal roots with zero value and the remaining two roots are conjugate pure imaginary or real conjugate according to B is negative or positive respectively.
3. If c 1 > 0 If D > 0 we have four pure imaginary roots every two of them are conjugate when B is negative. But if D < 0 we have four complex roots every two of them are conjugate whatever B is negative or not. Hence conditions for stable motion in this case are c 1 > 0, D > 0 and B < 0 there for the necessary and sucient condition is B − c 1 < D

Conclusions
In this paper the equilibria points for the dumbbell satellite motion in satellite approximation are found. We found that there are two types of motion around these points. In rst motion the symmetric axis of the dumbbell satellite is elongated in the radial of the common center of mass for the dumbbell and faced the attractor center by one of its ends. In second type the symmetric axis of the dumbbell is perpendicular to the radial of mass center. In general we found that the motion is not stable for each type of motion in the linear sense. After that we use the Routh reduction to reduce the degree of freedom from six to four. For the reduced system we also nd the equilibria points which in this case called relative equilibria points. The stability motion in the linear sense also around these points are studied. Finally the necessary and sucient conditions for stable motion around an equilibria points are constructed for every type of motion. This work has been partially supported by MICINN/FEDER grant number MTM2011Ð22587.