Theoretical investigation of the perturbed artificial satellite problem using oblate continued fraction potential

In this research article, a new idea concerning the application of concept of the continued fractions to the expansion of the geopotential is treated. The perturbed motion of the satellite in the oblate gravitational field of the Earth using a potential of a continued fractions procedure is studied. To compare with the usual perturbation theory, the continued fraction series is truncated beyond the third term. The first oblateness term of the potential is retained. The equations of motion are constructed using the Lagrange planetary equations. The integrals and averages required for solving the problem is outlined as well as the different cases are highlighted. The perturbations in all orbital elements due to considered model of continued fractions are evaluated. The new findings of the manuscript contrast with classical oblate potential field, the semi-major axis, eccentricity, and inclination, nonzero averaged terms in perturbations are obtained. Extra nonzero averaged perturbing terms in the argument of perigee and ascending node are revealed in addition to the classical perturbations.


Introduction
The realistic calculations of orbital elements are not a deterministic one. No unique methods of integration can be relied on for complete regime of orbital elements without any bugs. In investigation about the most ideal track of calculations, one is always looking for new methods that fulfill this target or even to interpret some uninterpreted phenomena e.g. the secular decay of the semi-major axis of LAGEOS satellite by a rate of 1.1 mm/day. Many force models are advised and examined to interpret these phenomena. But unfortunately, most of these trials indeed fail to interpret such phenomenon. In that way one can suggest different potential forms to account for unpredictable phenomena.
The Earth's potential due to the sphericity of the central body is considered as the most important perturbations of the two-body problem. There are several ways to express the Earth's potential. This form includes the two-body part of the potential is the first term. A detailed discussion of the harmonic gravity terms can be found in Roy [1]  where J n are the zonal harmonics, C n,m , S n,m are the sectorial harmonics of the gravitational field, R ⊕ is the mean equatorial radius of the Earth, P n (sin φ) are the Legendre polynomials, P m n are the associated Legendre functions, φ is the satellite's geocentric latitude, and λ is the satellite's geocentric longitude. The leading term is due to the potential of the point mass. The dominant terms beyond the leading term are due to J 2 zonal harmonic which is the most important effect. Many interested researchers investigated the effect of these perturbations on the motion of satellites, e.g. Iorio [2], Renzetti [3], Abd El-Salam et. al. [4], Abd El-Bar and Abd El-Salam [5], Abd El-Salam and Abd El-Bar [6] and Elshaboury and Mostafa [7].
The problem of computing the perturbed orbit due to oblateness of the Earth has received considerable attention since the beginning of the space era to the date. It is worth noting to sketch here some important works [8,9]. Some of these studies of this problem by means of general perturbation theories are Struble [10], Arsenault et al. [11] and Sagovac [12].
Mihail Bărbosu et al. [13] considered a generalization of the famous Lennard-Jones potential to study the two-body problem associated to this potential. They investigated all possible situations created by the interplay among the constants of integration and the field parameters. They obtained the global flow on the zero energy manifold of the two-body problem given by the sum of the Newtonian potential and the two anisotropic perturbations corresponding to the generalized Lennard-Jones potential, and illustrated it in both 3D and 2D. This flow exhibits a great variety of orbits, a homoclinic one is included. All phase portraits are interpreted in terms of physical trajectories.
Abd El-Salam et al. [14] have used a scheme of continued fraction to reformulate the two-body problem, they obtained the integrals of motion in this model.
Finally, the literature is wealth on the modified gravitational potentials and their effects, and it is worth noting to highlight some recent as well as important works [15][16][17][18][19][20][21][22]. On the other hand, the continued fractions can be hopefully used in the field theory, quantum mechanics [23] and metric theories [24].
In this work, we will extend our continued fraction model that has been published in Abd El-Salam et al. [14] to include the perturbations due to the oblateness of the Earth. We will assume a continued fraction procedure to the usual potential up to second zonal harmonics. The method that seems to be promising technique is using new potential. In the mentioned reference [14], we have treated the problem of two bodies generally and no perturbations have been applied, it checked only some major concepts of the dynamics of unperturbed two-body problem, mainly, it has reviewed the integrals of motion, e.g. the conservation of mechanical energy, angular momentum, the eccentricity vector, etc. In the current manuscript, we aim to evaluate the explicit perturbations in the classical Keplerian orbital elements, which have not been evaluated in the previous article [14] due to the nonexistence of the perturbations.
After this brief introduction section, the rest of this paper is organized as follows. In Section 2, some relevant definition is introduced. In Section 3, the problem statement is outlined. In Section 4, the Lagrange planetary equations are introduced and are adopted to be used for the perturbing force. In Section 5, the required partial derivatives are computed. In Section 6, the equations of motion are constructed. In Section 7, the perturbation used is explained. In Section 8, the perturbations in the orbital elements are evaluated.

Some relevant definitions
Definition: [continued fraction] An infinite continued fraction in F is a sequence, called sequence of partial quotients, whose domain is N ≥ 0.

Problem statement
In this paper, we will apply definition for continued fraction to the potential function of a perturbed twobody problem to compute the perturbing effects in the orbital elements. We will truncate the continued fraction series at some relevant terms depending on rigorous computation steps. The potential function of a perturbed two-body problem could be written as where μ ||r−R ⊕ || is the potential of the unperturbed problem, R ⊕ is the equatorial radius of the Earth, r is the mutual distance between the bodies and μ = 0.39860 × 10 6 km 3 /s 2 . The c s to be adjusted later, and R represents as usual the perturbation in celestial mechanics, here it can be interpreted as the terms of the continued fraction expansion beyond the first one.
Here we can design another form of (2) depending on the concept of the continued fraction series (1) as retaining the first three terms only of (3) we get which can be written as Setting α = c 1 + c 2 and retaining the first two terms in the binomial expansion yields Finally, it could be written in the form then Equation (6) can be written as could be expressed in terms of the generating function of Legendre polynomials as Therefore using (8), the continued fraction potential (7) can be written in the form where J n are the oblateness (dynamical shape) parameters; their numerical values are found easily from any geophysical data of the Earth. With J 2 ∼ = 10 −3 , we can assume that J 2 as small parameter of the problem and is being of order 1. The angle θ = π 2 − δ is the co-declinations, i.e. Equation (9) can be written in the form The included Legendre polynomials are given by P 2 (sin δ) = 1 2 (3sin 2 δ − 1). Define the Keplerian orbital elements (a, e, I, ω, , f ), where a is the semi-major axis, e is the ececcntricity, I is the angle between the orbital plane and the fundamental plane (here is the equatorial plane), called the orbital inclination, ω is the angle measured on the orbit from the ascending node to the satellite, is the angle measured on the fundamental plane from the vernal point of equinox to the ascending node (the point of the intersection between the fundamental plane and the orbit plane), and f is the angle measured on the orbit from the major axis to the position of the satellites, it is called the true anomaly, and it is called the argument of perigee. Adopting the notation F ij = if + jω using sin δ = sin I sin(f + ω), the Legendre polynomials P 2 (sin δ) can be written in terms of orbital elements as P 2 (sin δ) Using the above-mentioned notations, the potential (10) can be expressed in terms of orbital elements as (11) can be rewritten as

Lagrange planetary equations
Lagrange was the first to derive the rates of change of the osculating orbital elements in a system of six ordinary differential equations, known as the Lagrange planetary equations. He studied the planetary motion around the Sun and disturbed by a small perturbation due to another gravitational attraction of the planets. See Roy [1], Brouwer and Clemence [25] and Taff [9].
where M is the mean anomaly of the satellite, it is the angular distance from the perigee of a fictitious body moving in a circular orbit, and n is the mean motion of the satellite which is the angular speed required by the satellite to complete one orbit.

The required partial derivatives
In what follows, we will compute the partial derivatives required to solve the system (13)- (18). Now consider the potential function given by (12) we obtain the following non-vanishing results:

Integrals and averages
To evaluate the integrals of these quantities with respect to l, therefore integrals of the form (a/r) i cos nf and (a/r) i sin nf follows directly.

For i ≥ 2
Let where i, n are integers and i ≥ 2.
Using the relations, see Ahmed [26] = η −2,2 (1 + e cos f ), = a r and dl = η −1,1 df 2 we can write (32) in the form The integrals in (33) can be written as (34) It is clear that for i < 2, the above expression will not lead to closed expressions, we have Then substituting equation (35) into (34) gives

Averages of i cos nf and i sin nf
The averages (38) and (39) will be evaluated for different i and n.

For i = 0
The average values may be evaluated using the formula [27,28]: The following special cases are of particular interest

For i = 1
The averages may be evaluated using the relation where the I c 0,n are defined by Equation (40).

For i ≥ 2
A direct consequence of Equations (38) and (39) is that (i)

For n ≤ i − 2
The required averages are obtained from Equations (38) and (39).
The following special cases are of particular interest

Perturbations in the orbital elements
After some lengthy computations and using the obtained averages in the previous section and its sections, we can obtain the following final expressions for the perturbations in the orbital elements

Perturbation in the semi-major axis
The semi-major axis suffers no perturbations in classical unperturbed potential, while when considering the continued fraction potential, we obtained a nonzero averaged perturbation. 11,11 (2 + e 2 ) sin 2ω (44)

Perturbation in the eccentricity
The eccentricity suffers no perturbations in classical unperturbed potential, while when considering the continued fraction potential, we obtained a nonzero averaged perturbation.

Perturbation in the inclination
The inclination suffers no perturbations in classical unperturbed potential, while when considering the continued fraction potential, we obtained a nonzero averaged perturbation.

Perturbation in the argument of perigee
The argument of perigee suffers a definite periodic perturbation in the case of the classical unperturbed potential, but when considering the continued fraction potential, we obtained extra nonzero averaged perturbing terms.

Perturbation in the longitude of the ascending node
The longitude of the ascending node suffers a definite periodic perturbation in the case of the classical unperturbed potential, but when considering the continued fraction potential, we obtained extra nonzero averaged perturbing terms.

Perturbation in the mean anomaly
The mean anomaly suffers a definite secular perturbation in the case of the classical unperturbed potential, but when considering the continued fraction potential, we obtained extra nonzero averaged perturbing terms.

Conclusion
We can point out the original contribution of this idea as follows. The geopotential is suited to accept the continued fraction procedure. Since the continued fractions continues infinite time of fractions schemes, the series is truncated beyond the third-order term as well as the first oblateness term of the potential Lagrange planetary equations is modified with new terms arising from the adopted continued fractions scheme. The equations of motion of a satellite move in a continued fraction potential field are derived. To solve these equations, the required integrals and averages are outlined, and different cases are highlighted using Ahmed [26]. The perturbations in all orbital elements due to considering the model of continued fractions are computed.
The new results to highlight are that the semi-major axis, eccentricity, inclination are suffering perturbations when considering the continued fraction potential in contrast to classical oblate potential. The argument of perigee also suffers a definite perturbation in the case of the classical unperturbed potential, but when considering the continued fraction potential, extra nonzero averaged perturbing terms is obtained. All these findings are new in comparison with the last contribution to the authors [16]. In that article, the authors have treated the problem of two bodies generally and no perturbations have been applied, it checked only some major concepts of the dynamics of unperturbed twobody problem, mainly, it has reviewed the integrals of motion, e.g. the conservation of mechanical energy, angular momentum, the eccentricity vector.

Disclosure statement
No potential conflict of interest was reported by the author(s).