As the geostationary Earth orbits (GEO) belt becomes more crowded it is increasingly difficult to acquire slots for new satellites. Consequently, many organizations choose to collocate their spacecraft in the same slot. Also for missions which a single satellite cannot accomplish, as global position satellite system (GPS), the needed of formation flight began.
The formation flight concept is the use of several small satellites, which work together in a group (or constellation) to accomplish the objective of one larger, usually more expensive, satellite. This increases the likelihood of mission success in the event of a malfunction Hughes  . Formation flights have invariant relative 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  in addition to the studies of Tschauner and Hempel  . 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  , Abd El-Salam et al.  passing through Li and Li  until Abd El-Salam and El-Saftawy  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  also, discussed the J2 invariant relative orbits with the effect of lunisolar attraction.
In this paper, we extend the works of Schaub and Alfriend  and Abd El-Salam et al.  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.
2. 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 J2 (The second geopotential zonal harmonic) up to the second order, and using Lie-Deprit-Kamel perturbation method Kamel  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.  .
With, , = SinI (I is the inclination of the orbit), μ is the gravitational parameter of the planet and zero order quantities defined as:
where and are the Delaunay elements ( , , ), is the equatorial radius of the Earth, and J2, J4 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 is the sum of the mean anomaly and the argument of perigee (l1 + l2). 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, , and the secular drift rates of the longitude of the ascending node, can be calculated, i.e. .
So, using Equation (1) in Equation (2), the result can written in the form:
With, and are published by Abd El-salam et al.  and given by:
where and are function of the action variable and introduced in Appendix.
To prevent the satellites from drifting apart over time, the average secular growth needs to be equal. So, it would be desirable to match all three rates and between the satellites in each formation. So and of all satellites in the formation should be equal.
Denoting the reference means orbit elements with the subscript “0”. Using Taylor expansion for the drift rate and of a neighboring orbit “i” about the reference orbital elements, retaining the terms up the second-order derivatives, can be simplified as:
With = CosI, similarly:
where is the difference between the drift rates of the argument of mean latitude of the reference orbit and one of the neighboring orbits, and .
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 and Equation (8') by and subtracting we will get:
with the coefficients ’s are:
and the derivatives of ’s and ’s are in the appendix. Solving Equation (9), for , we get:
Substituting the last results in Equation (7') we get the quartic equation:
where the coefficients ’s are functions of ’s and given by:
Solving the resulting equation we will get four roots of :
and four roots of :
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, , ,
, , , ,
3. 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 (which we redefined as for simplicity), as in Equation (10), and four solutions for as in Equation (10'). Here, we will present the plots of these solutions, which we obtained in the last section (Equations (10) and (10')) to compare the effect of J4 on the conditions of invariance for the formation. Before we introduce the graphs, it is important to mention that in all figures, we provided a set of curves in each condition for the inclination of each member of the formation with respect to the leader orbit as example. By means that the follower satellites’ orbits, in the formation, will be inclined by a range of 2˚ with respect to the leader satellite orbit, of course we can extend this range. Also, we will introduce a comparison between the effects of in the formation.
The variation of the formation relative to the leader orbit in is related to the variation in eccentricity for the followers through:
i.e. the variation in the follower’s orbit in ( ) is scaled by in their variations in eccentricities . It is important to note that the scale function is always negative for circular and elliptical orbits. In our case the eccentricity of the leader orbit, , is zero, then is always positive.
While the variation in the , for the followers, is scaled by for the variation of the semi-major axis through the relation . It is important to note that the scale function is always positive, but maybe positive, negative or zero.
3.1. The First Solution (Circular Formation)
The first solution of Equations (10) for and (10') for for the orbits of the followers with respect to the leader orbit, for inclination range [−2˚, +2˚] in case of J2 and the net effects of J2 and J4 can represent in the following curves.
Figure 1(a) and Figure 1(b) show the first choice of the invariant relative conditions, in the equatorial-circular case. In this choice, and get Zero for all and values. That is mean, the formation for the followers orbit will be in the same orbital plane of the leader satellite (In Plane Circular Formation) with their eccentricities and semi-major axis was scaled by and of the leader satellite. The scale function , for this solution, is equal zero whatever the choosing the value of . Also the scale function never equal zero, then 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 J4 has no significant variation in the formation.
3.2. The Second Solution
Figure 2(a) and Figure 2(b) show the second choice of the invariant relative formation, in the equatorial-circular case. In this solution, the choice of eccentricites and semi-major axis for the followers orbit is not affect the formation in case of J2 effect whatever choosing the inclination for folowers orbits. But the J4 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.
Figure 1. (a) The formation under the effect of J2 only for the 1st. solution in the circular equatorial of the leader orbit; (b) The formation under the net effect of J2 and J4 for the 1st. solution in the circular equatorial of the leader orbit.
Figure 2. (a) The formation under the effect of J2 only for the 2nd solution in the circular equatorial of the leader orbit; (b) The formation under the net effect of J2 and J4 for the 2nd solution in the circular equatorial of the leader orbit.
The effects of J4 are changing slightly while the choosing of eccentricies is not for choosing the semi-major axis.
3.3. The Third Solution
Figure 3(a) and Figure 3(b) show the third choice of the invariant relative formation, in the equatorial-circular case. In this solution, the value of the function which gives the limits for the eccentricities through the inqualety .
The solution for this inqualety has only one positive value 0.839936. With respect to choosing the semi-major axis, it is not depend on choosing the inclination, by mean that the formation will be in plane of the leader one.
In the case of the formation, under the effects of J2 and J4, the function is modified the limit of the eccentricities to be 0.774904.
3.4. The Fourth Solution
Figure 4(a) and Figure 4(b) show the fourth choice of the invariant relative formation, in the equatorial-circular case. In this choice, the formation for defferent inclination is distributed about the leader orbit (with ) and the function is increasing for positive while it is decreasing for negative . For choosing the semi-major axis, it increases by increasing the semi-major axis of the leader orbit whatever choosing .
In this formation choics, the effect of J4 is significant for choosing the eccentricities for the followers as it clear from the first of Figure 4(b). In the first of
Figure 3. (a) The formation under the effect of J2 only for the 3rd solution in the circular equatorial of the leader orbit; (b) The formation under the net effect of J2 and J4 for the 3rd solution in the circular equatorial of the leader orbit.
Figure 4. (a) The formation under the effect of J2 only for the 4th solution in the circular equatorial of the leader orbit; (b) The formation under the net effect of J2 and J4 for the 4th solution in the circular equatorial of the leader orbit.
Figure 4(b), the function is positive and increasing while is always negative. And , in our case, must be positive or zero. For that this solution is not mathematically accepted.
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 is decreasing by increasing the semi-major axis.
The problem formulated using the oblate Earth model, truncating its potential series at J4 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 or and the semi-major axis of the leader satellite orbit, at different .
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 is not affect the choosing the semi-major axis (In plane with different eccentrisities).
The third choice, under the effects of J2 and J4, the value of the function which gives the limits for the eccentricities through the inequality . The solution for this inqualety has only one positive value 0.839936. With respect to choosing the semi-major axis, it does not depend on choosing the inclination, by mean that the formation will be in plane of the leader one.
The fourth choice, the function is positive and increasing while is always negative. And , in our case, must be positive or zero. For that, the eccentricities of the followers must be negative and that is not mathematically accepted.
Also, the second of Figures 4(b) shows that this choice is not continuous for orbits with semi-major axis greater than 2.5 earth radii. And the function is decreasing by increasing the semi-major axis.
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 Abd El-Salam, F.A., El-Tohamy, I.A., Ahmed, M.K., Rahoma, W.A. and Rassem, M.A. (2006) Invariant Relative Orbits for Satellite Constellations: A Second Order Theory. Applied Mathematics and Computation, 181, 6-20.
 Abd El-Salam, F.A. and El-Saftawy, M.I. (2012) Second Order Constraints in the Theory of Invariant Relative Orbits Including Relativistic and Direct Solar Radiation Pressure Effects. Indian Journal of Science and Technology, 5, 1-14.
 El-Saftawy, M.I., Ahmed, M.K.M. and Helali, Y.E. (1998) The Effect of Direct Solar Radiation Pressure on a Spacecraft of Complex Shape, II. Astrophysics and Space Science, 259, 151-171.