^{1}

^{*}

^{2}

For a satellite in an orbit of more than 1600 km in altitude, the effects of Sun and Moon on the orbit can’t be negligible. Working with mean orbital elements, the secular drift of the longitude of the ascending node and the sum of the argu-ment of perigee and mean anomaly are set equal between two neighboring orbits to negate the separation over time due to the potential of the Earth and the third body effect. The expressions for the second order conditions that guaran-tee that the drift rates of two neighboring orbits are equal on the average are derived. To this end, the Hamiltonian was developed. The expressions for the non-vanishing time rate of change of canonical elements are obtained.

Formation flying is a key technology enabling a number of missions which a single satellite cannot accomplish: from remote sensing to astronomy. The relative motion, which shows no drift even in presence of a large disturbance, could be a very attractive solution. To maintain the formation and constellation, the relative drifts due to the perturbation between the spacecraft should be carefully considered. Invariant Relative Orbits shows no drift between the spacecraft due to the perturbation even if in presence of a large disturbance.

The literature is wealth with works dealing with designing certain invariant relative orbits for spacecraft flying formations, and it seems worth to sketch some of the most relevant works. Schaub and Alfriend [_{2} invariant relative orbits for spacecraft formation flying applications. They designed relative orbit geometry using differences in mean orbit elements. Two constraints on the three momenta element differences are derived. Zhang and Dai [_{2} invariant orbit with the leader’s orbit of zero inclination based on the targeting method in chaos dynamics. Abd El-Salam et al. [

Our propose was to extend Schaub and Alfriend [

There are several ways to derive the equations of motion for any such system. We emphasized on the Hamiltonian structure for this system. The Hamiltonian formulation allows for additional conservative forces to be added to the Hamiltonian, thus the addition of complexity to the model can be incorporated with ease. Non-conservative forces can be added in the momenta equations of motion. The Hamiltonian equations of motion allows us to directly use control and simulation techniques.

Notations in the whole text, we use the well-known keplerian elements: the semi-major axis a, the eccentricity e, the inclination, the right ascension of ascending node, the argument of perigee, and the mean anomaly M. We also use the true anomaly f and an intermediary variable.

The Hamiltonian in the present framework can be written in the form

where is the force function due to the Earth’s gravitational potential, and p is the canonical momentum vector and the disturbing function due to the effect of perturbing body.

The actual shape of the Earth is that of an eggplant. The center of mass does not lie on the spin axis and neither the meridian nor the latitudinal contours are circles. The net result of this irregular shape is to produce a variation in the gravitational acceleration to that predicted using a point mass distribution. The Earth’s gravitational potential is usually expressed by the following expression (Vinti’s potential)

where is the equatorial radius of the Earth,

is the Earth’s gravitational parameter where is the gravitational constant;

are the geocentric coordinates of the satellite with measured east of Greenwich;

and are harmonic coefficients;

are associated Legendre Polynomials.

In the potential function, the terms with, and correspond respectively to zonal, tesseral and sectorial harmonics. The Earth gravitational potential can be rewritten, up to second order in, truncating the series at, as, Abd El-Salam et al. [

(2)

where and is the zonal harmonic coefficients.

The effect of the third body in the motion of an artificial satellite have became particularly interesting now, when space debris imposes a serious threat to space activities. These perturbations are the most important mechanism of delivering major Earth orbiting objects into the regions where the atmosphere can start their decay.

If it is assumed that the main body; Earth; with mass is fixed in the center of the reference system x-y. The perturbing body, with mass is in an elliptic orbit with semi-major axis, , eccentricity, and mean motion, given by the expression, and are the radius vectors of the satellite and (assuming ), and is the angle between these radius vectors. The disturbing function (using the tradition expansion in Legendre polynomials) due to the third body is given by, Domingos et al. [

where and

with

Using the Delaunay canonical-variables defined by

Mean anomaly

Argument of the Perigee

Longitude of ascending node

Considering as a small parameter of the problem, the orders of magnitude, up to the second order, of the involved parameters are defined as follows:, and let us define the dimensionless parameters as

The Hamiltonian, Equation (1) up to the second order, can now be expressed as a power series in as follows

where represents the unperturbed part of the problem, is the perturbation:

Now we need to eliminate the short as well as the long periodic terms of the satellite motion in addition to the short periodic terms of the distance perturbing body. Using the perturbation technique based on Lie series and Lie transform, Kamel [

with

where

Using the Hamiltonian canonical equations of the motion, to write, argument of mean latitude () is the sum of the mean anomaly and the argument of perigee (i.e.), as

with

and the secular drift rates of the longitude of the ascending node,:

with

In order to prevent two neighboring orbits from drifting apart, the average secular growth needs to be equal. Short period oscillations can be ignored here since these are only “temporary” deviations. The long period rates appear secular over a few weeks and they are.

Since the mean angle quantities and do not directly contribute to the secular growth, their values can be chosen at will. However, the mean momenta values and H (and therefore implicitly and) must be carefully chosen to match the secular drift rates. To keep the satellites from drifting apart over time, it would be desirable to match all three rates. We impose the condition that the relative average drift rate of the angle between the radius vectors be zero. This results in

Now and can be rewritten as

where the non-vanishing coefficients and are computed in Appendix I.

Let the reference mean orbit elements be denoted with the subscript “0”. The drift rate of a neighboring orbit can be written as a series expansion about the reference orbit element, here it is enough to keep the second order only, as

(13)

(14)

where we make use of the fact that and only, also supposing that is the difference in mean latitude rates,

and

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 Equations

(9) and (10), either or could be chosen and the remaining two momenta orbit element differences found through a numerical root solving technique. However, the analytical second order conditions provide reasonably accurate solutions to these two constraints equations and provide a wealth of insight into the behavior of Earth potential and third body effect invariant relative orbits.

The required derivatives can be evaluated as

and

where and with.

To enforce equal drift rates and between neighboring orbits, we must set and equal to zero in expanded Equations (13) and (14), yields

(15)

(16)

Equations (15) and (16) are two simultaneous nonlinear algebraic equations in three unknowns, namely. When one of these three unknowns is assumed known (say), these two equations can be solved as:

Multiplying Equation (15) by and Equation (16) by and then subtracting yields

where

.

Substituting Equation (17) into Equation (15) yields an algebraic equation of fourth degree in only in the form

where

.

The roots of the quartic Equation (18) can be written as

where

with

,

where

and

.

Substituting the four roots’s into Equation (17) yields the four constraints’s that guarantee the invariance of the relative motion of certain satellite constellation

Accurate modeling of relative motion dynamics for initial conditions close to the leader satellite is essential for flying formation. Therefore, the solutions of interest are restricted to a specific set of initial conditions that lead to periodic motion, such that the satellites do not drift apart. This paper showed an analytical expression to secular drift rates due to oblate Earth model, truncating its potential series at, and third body effect and set it equal between two neighboring orbits. It followed the same steps used before in Abd El-Salam et al. [

The first author wish to express his appreciation for the support provided by the French government under the No de dossier: 688028B, No affiliation: 194264/733177.

The authors gratefully thank referees for their helpful, suggestions and comments.