Analytical Solution for Formation Flying Problem near Equatorial-Circular Reference Orbit

The relative motion between multiple satellites is a developed technique with many applications. Formation-flying missions use the relative motion dy-namics in their design. In this work, the motion in invariant relative orbits is considered under the effects of second-order zonal harmonics in an equatorial orbit. The Hamiltonian framework is used to formulate the problem. All the possible conditions of the invariant relative motion are obtained with different inclinations of the follower satellite orbits. These second-order conditions warrantee the drift rates keeping two, or more, neighboring orbits from drifting apart. The conditions have been modeled. All the possibilities of choosing mean elements of the leader satellite orbit and differences in momenta between leader and follower satellites’ orbits are presented.

tive orbits for their satellites to ensure that they will not separate over time.
The invariant relative orbits have been studied for a long time, as the earlier work of Clohessy and Wiltshire [2] in addition to the studies of Tschauner and Hempel [3]. These models introduced conditions on the initial relative position and velocity so that the relative orbits result to be periodic, which are closed orbits. Recently, Schaub and Alfriend [4], Abd El-Salam et al. [5] passing through Li and Li [6] until Abd El-Salam and El-Saftawy [7] in which they discussed the invariant relative orbits due to the influence of the perturbative effects of the asphericity of the Earth, the relativistic corrections and the direct solar radiation pressure. Rahoma [8] also, discussed the J 2 invariant relative orbits with the effect of lunisolar attraction.
In this paper, we extend the works of Schaub and Alfriend [2] and Abd El-Salam et al. [5] model by introducing an atlas for the curves of invariant relative orbits' conditions. This atlas will be presented using Mathematica program to calculate and plot graphics of the initial conditions of invariant relative orbits. Those graphics will be shown as curves in 2D; in the case of the orbit of the leader satellite is equatorial.

Hamiltonian System
There are several ways to derive the equations of motion for any such system. We emphasized on the Hamiltonian structure of this system. The Hamiltonian formulation allows additional conservative forces to add to the Hamiltonian, thus the addition of complexity to the model can be incorporated with ease. Non-conservative forces can add in the momenta equations of motion. The Hamiltonian equations of motion allow us to directly use control and simulation techniques.
After expressing the Hamiltonian, as a series in power of J 2 (The second geopotential zonal harmonic) up to the second order, and using Lie-Deprit-Kamel perturbation method Kamel [9] to eliminate, in successive, the short and long periodic terms, the transformed Hamiltonian, H ** , for different orders 0, 1, and 2, are obtained by El-Saftawy et al. [10]. where, r is the equatorial radius of the Earth, and J 2 , J 4 are the second and fourth geopotential zonal harmonic respectively.
The problem of designing invariant relative orbits for spacecraft flying formations is outlined as follows: 1) Compute the secular drift of the longitude of the ascending node and the sum of the argument of perigee and mean anomaly.
2) These secular drift rates are set equal between two neighboring orbits.
3) Having both orbits drift at equal angular rates on the average, they will not separate over time due to the influence of the perturbative effects of the asphericity of the Earth up to the desired order of magnitude (or the accuracy) to the equations of motion.
Using the canonical equations of motion, Since the argument of mean latitude θ is the sum of the mean anomaly and the argument of perigee (l 1 + l 2 ). Evaluating the derivatives yields the sum of the argument of perigee and the mean anomaly rate of changes. Follows, the rate of change of mean latitude, θ  , and the secular drift rates of the longitude of the ascending node, 3 l  can be calculated, i.e.
With  = CosI, similarly:    2  2  2  2  3,  3,  3,  1,1  1  2, 0  2  1  2  1  3  2   2  2  2  2  2  3,  3,  3,  2  2  1,1  4,2  2  2  2  3  3  3   2  2  3, 3, 2 2,1 2 2 1 2 is the difference between the drift rates of the argument of mean latitude of the reference orbit and one of the neighboring orbits, and  is the difference between the drift rates of the ascending node of the reference orbit and one of the neighboring orbits, and Now, the conditions satisfying the invariance property for the relative orbits are: Substituting the included derivatives of the last two equations into Equations (5) and (6) and after the needed mathematical manipulations we will get: Multiplying Equation (7') by with the coefficients i a 's are:   International Journal of Astronomy and Astrophysics where the coefficients i b 's are functions of i L 's and given by: Solving the resulting equation we will get four roots of and four roots of   27

Modeling of Invariant Relative Orbit Conditions for near Equatorial-Circular Case
When we choose the leader orbit to be circular equatorial then, we will obtain four solutions for

The First Solution (Circular Formation)
The first solution of Equations (10) for e δ and (10') for a δ for the orbits of the followers with respect to the leader orbit, for inclination range [−2˚, +2˚] in case of J 2 and the net effects of J 2 and J 4 can represent in the following curves. δ . Also the scale function ( ) mag a never equal zero, then a δ must equal zero. That can me conclude that the follower satellite's must be in the same orbit of the leader one.
In this case, the effect of J 4 has no significant variation in the formation. effects change the choosing of eccentricities slightly while the choosing the semi-major axis still unaffected whatever chosing the inclinations for the followers orbits. As we see in the vertical axis. The semi-magor axis and eccentricites of the followers orbits must be greater than those for the leader one.      Also, the second of Figure 4(b) shows that this choice is not continuous for orbits with semi-major axis greater than 2.5 earth radii. And the function ( ) mag a a δ is decreasing by increasing the semi-major axis.

Conclusions
The problem formulated using the oblate Earth model, truncating its potential series at J 4 to the equations of motion, and then the canonical equations of motion and the Hamiltonian formed. In order to keep the relative motion invariable, eight-second order conditions between the differences in the semi-major axis a and the inclination I are obtained. These conditions guarantee that the drift rates of neighboring orbits are equal on the average. The resulting orbits require less control and maintenance fuel. Then we studied the curves of these conditions in equatorial-circular case. The plots of these cases are presented as relations between δη or 1 L δ and the semi-major axis of the leader satellite orbit, at different I δ .
In the first choice, we can conclude that the follower satellite's must be in the same orbit of the leader one (on orbit formation).
In the second choice, whatever the choosing the inclination for the followers it