Lunisolar Invariant Relative Orbits

The present study deal with constructing an analytical model within Hamiltonian formulation to design invariant relative orbits due to the perturbation of J2 and the lunisolar attraction. To fade the secular drift separation over the time between two neighboring orbits, two second order conditions that guarantee that drift are derived and enforced to be equal.


INTRODUCTION
Within the last decades several projects have been started where the goal is to use multiple satellites working as one. Thus formation flying clusters is a relatively new concept in space science. The design of relative navigation and control systems of the spacecrafts in this cluster is certainly one of the most challenging topics. In order to ease the tasks of these subsystems, a proper reference trajectory must be conceived and a relative motion, which shows no drift even in presence of perturbation forces, could be a very attractive solution. Mentioned here the most important papers dealing with this problem, the dominant J 2 oblateness effect in perturbations is considered as the main destroyer to the formation configuration and J 2 invariant orbit is firstly defined by (Schaub and Alfriend, 2001) in mean orbital elements to minimize the amount of fuel to maintain. They introduced an analytical technique to derive a class of orbits that are termed 'invariant' orbits. Satellites placed into 'invariant' orbits are not immune to the disturbance force, but instead two satellites placed into these orbits will have the same drift and thus the cluster remains together. The 'invariant' orbits are created by using two first order conditions that determine the correct differences in semi-major axis, eccentricity and inclination. Zhang and Dai (2002) removed the drifts by adjusting the semi-axis of the follower satellite and obtained a similar conclusion. However, all the above works are based on the mean orbital elements instead of the Cartesian coordinates, which prevents the J 2 invariant orbit from being applied. Biggs and Becerra (2005) proposed a method to determinate the J 2 invariant orbit with the leader's orbit of zero inclination based on the targeting method in chaos dynamics. Subsequently, (Yan and Alfriend, 2006;Breger and How, 2006;Xu and Xu, 2007;Sabatini et al., 2009) investigated J 2 invariant relative orbits from the perspective of relative orbital elements and relative Cartesian coordinates, thereby achieving different types of invariant conditions through numerical searches. El-Salam et al. (2006) used the Hamiltonian framework to construct an analytical method to design invariant relative constellation orbits due to the zonal harmonics; up to the second order, assuming J 2 being of order 1. Rahoma and Metris (2012) constructed an analytical method using Hamiltonian framework to design invariant relative emphasized on secular oblateness perturbations due to the zonal harmonics J 2 , J 3 , J 4 and the third body effect, assuming J 2 being of order 1.
After this introduction, the model of the oblateness of the Earth, truncated beyond the second zonal harmonic J 2 and the lunisolar attractions is devoted. The Hamiltonian of the problem was constructed by consideringn 2 ; the mean motion of the Moon; as the small parameter of the problem. The normalized Hamiltonian of the system considered, F, after eliminating the short, intermediate and long period terms respectively using a perturbation technique based on Lie series, Lie transform and Bohlin's technique, is represented as (Radwan, 2002). The

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AJAS expressions for the time rate of change of the secular elements are obtained, second order conditions are established between the differences in moment an elements are equal on the average.

Problem Formulate
We consider the motion of artificial satellites in the Earth's gravity fields and perturbed by lunisolar attraction.
Let us assume, with sufficient accuracy, the following assumptions: • The geopotential is restricted to the J 2 part, like in the main problem of artificial satellite theory • Sun and Moon move in circular orbits in the fixed ecliptic plane. Then we can set u 2 = n 2 t + const and u 3 = n 3 t + const, where (u 2 , n 2 ) and (u 3 , n 3 ) are the argument of latitude and the mean motion of Moon and Sun respectively. Noting that the time appears explicitly through u 2 and u 3 , we augment the set by the addition of the pair k 2 = n 2 t + const, k 3 = n 3 t + const and their conjugatesK 2 and K 3 respectively. Lunisolar potential are truncated at the second Legendre polynomial term.

The Normalized Hamiltonian
Consideringn 2 as the small parameter of the problem, the orders of magnitude up to the second order of the involved parameters are defined as follows Where: where, µ = k 2 m 0 , is the gravitational const., m 0 , m 2 and m 3 is the mass of Earth, Moon and Sun respectively S j = sin i j and C j = cos i j , j = 2, 3 refer to Moon and Sun and i is the inclination: where, R is the radius of the Earth, α is the semi major axis of the satellite.
Using the Hamiltonian canonical equations of the motion, to write the secular drift in the argument of mean latitude (θ = l +g): and the secular drift rates of the longitude of the ascending node: And : To proscription two neighboring orbits from drifting apart, the average secular growth needs to be equal. Since the mean angle quantities l, g and h do not directly contribute to the secular growth, their values can be chosen at will. However, the mean moment a values L, G and H (and therefore implicitly a, e and i) must be carefully chosen to match the secular drift rates. To hinder the satellites from drifting apart over time, it must be matched all three rates. We impose the condition that the relative average drift rate of the angle between the radius vectors be zero i.e.: where, θ  and h  can be rewritten as, Equation (11)

Second Order Expansions Conditions
Using the second order expansions conditions described by (Rahoma and Metris, 2012) for i θ  and i h  from the mean value denoted by 0 to write the drift rate about reference orbital elements: x xo x xo x xo x xo x xo x xo x xo x xo x xo x xo x xo x xo x xo x xo x xo where, we make use of the fact that only, also supposing δθ that is the difference in mean latitude rates δL = L i -L 0 , δη = η i -η 0 , δ i = i ii 0 , x = L, η, i and x 0 = L 0 , η 0 , i 0 .
Note that this theory will lead to an analytical second order conditions on the mean orbit elements. To establish a more precise set of orbit elements satisfying Equation 9 and 10, either δL, δη or δi could be chosen and the remaining two moment a orbit element differences found through a numerical root solving technique.

AJAS
The required derivatives can be evaluated as: Equation (15) and (16) are two simultaneous nonlinear algebraic equations in three unknowns, namely δL, δη, δi. When one of these three unknowns is assumed known (say δi), these two equations can be solved as: