Analytical Third Order Solution for Coupling Effects of Earth Oblateness and Direct Solar Radiation Pressure on the Motion of Artificial Satellites

Coupling effects of Earth oblateness and direct solar radiation pressure on the motion of an artificial satellite are evaluated. Secular and periodic terms are retained up to order three and two respectively, where the coefficient 2 J of the second zonal harmonic of the geopotential is considered of first order. The solution revealed the existence of secular terms at order three that arises from the couplings between terms, of lower orders, resulting from the solar radiation pressure.


Introduction
Analytical theories of celestial mechanics are usually more tractable when tackled within the domain of Hamiltonian mechanics, and fortunately most non-Hamiltonian systems of differential equations can be Hamiltonized by a simple technique [1] [2]. As for as canonical perturbation methods are concerned, the basic demand is a canonical transformation such that the new Hamiltonian has fewer degrees of freedom, which results in integrals of motion equal in number to the number of ingrate coordinated, and thus successive transformations reduce the system to quadratures. If the Hamiltonian is a periodic function of time, a further requirement would be the averaging of the Hamiltonian to eliminate the time.
If the Hamiltonian admits a Taylor series expansion in powers of a small parameter ε at 0 ε = , a generator of the transformation can obtain a series of ε up to any desired power.
Many works deal with the effects of solar radiation pressure, Musen (1960) [3] derived first order expansion for the rate of change in the osculating elements using the method of variation of vector elements. Kosai (1961) and Brouwer (1962) [4] [5] used Lagrange's planetary equations to find the first order solutions with the integration performed between the times of exit from, and entry into, the shadow. The resonance effects produced by the commensurabilities between the different mean motions provided good field for detailed theoretical studies (e.g. Musen, 1960 [3]; Brouwer, 1962 [5] and Hori, 1966 [6]). De Moraes (1981) [7] developed semi-analytical theory including the joint effects of direct solar radiation pressure and atmospheric drag for satellites of perigee heights between 500 and 900 Kms. The difficulties arising from solar radiation pressure are analyzed in three very useful and interesting expositions given by Kampos (1968) and Sehnal (1970 and1975) [8]- [10]. An interesting application is to use solar radiation pressure as means of spacecraft propulsion (solar sailing). The idea is described and its dynamics is studied by Mc Innes and Brouwn (1990) [11].
It has been repeatedly stated by many authors (e.g. Kampos, 1968;Ferraz-Mello, 1972 andAnselono et al., 1983) [8] [12] [13] that there are no secular or long periodic changes of the semimajor axis and no second order terms (or even, by some author, no secular terms whatever the order). But Geyling and Westerman (1971), Lala (1972) and Sehnal (1975) [10] [14] [15] drew attention to secular terms which appear at higher orders. McMahon, Jay W., (2011) [16] modeled the solar radiation pressure acceleration as a Fourier series which depends on the Sun's location in a body-fixed frame; a new set of Fourier coefficients are derived for every latitude of the Sun in this frame, and the series is expanded in terms of the longitude of the Sun. Lücking et al. (2012) [17] further explores a passive strategy based on the joint effects of solar radiation pressure and the Earth's oblateness acting on a high area-to-mass-ratio object. In 2001, Cook found that the most significant effect relating to solar radiation pressure is the changing cross-sectional area of the satellite projected to the Sun [18].
The present work is concerned with the effects of solar radiation pressure at higher orders to emphasize the effects of the couplings between them and with those resulting from the oblate gravity field of the Earth. The canonical equations of motion are formed in terms of the Delaunay elements augmented with the pair ( ) , k K where k is the mean longitude of the sun. The equations include the radiation pressure force and the geopotential up to 2 J . Two canonical transformations are made to eliminate the short and long period terms in succession respectively. The Hamiltonians and generator are assumed to be expandable as ( ) where the suffix r refers to terms arising from solar radiation pressure.   v being is the mean motion of the sun, 1 A is a constant, ε is the obliquity of the ecliptic and R is the equatorial radius of the Earth. When deriving the above equations the Sun is assumed moving in a circular orbit so that its mean longitude is const t + , and the direction and distance of the satellite from the Sun are considered similar to those of the Earth. Clearly 2 H ∆ represents the contribution of solar radiation pressure.

Short-Period Perturbations
Since 0 H depends only on L , the angle  will be a fast variable while the other angles are slow variables. Therefore, we shall perform two transformations to eliminate, in succession, the short and long period terms. Adopting the transformation techniques developed by Deprit (1969) and Kamel (1969) [1] [20], the identities for the short period transformation will be: The elements of the transformation are obtained from Performing the operations described in Equations (6) with the secular and periodic terms retained up to the third and second order respectively, we arrive, after some lengthy manipulations, at the following results. All variables are understood to be single primed but the primes will be dropped out for the sake of simplicity of writing. 2  2  1  22  12  32  3 3   3  2  3  2  esin  sin  esin  sin  2 3 At the subsequent orders (order 2 and 3) we will be concerned only with terms arising from the radiation pressure and those due to the coupling between oblateness and radiation pressure effects. We thus separate 2 W as where the suffixes g and r refer to gravity and radiation pressure, respectively. At the second order we have where: Regarding (6.3) we find that the manipulations at order 3 require evaluating where: where the Ψ i s ′ are functions of �́,́,́�, the subscripts h and k mean partial derivatives with respect to each, and    2  2 2  2  4  3   3  15  sin3  e  4  cos  4 2e where: ˆ, where we note that l (1) ,  ,̀( 1) consists only of terms arising from the oblateness effects and therefore need be taken into consideration.

Perturbations of Long Period
After the short-period terms have been eliminated the problem is now reduced to the system of canonical equations with Hamitonian * � ,́, h ,́;́,́,́,́�.
We now proceed to eliminate the long-period terms, i.e. those periodic in , h , ̀. ; Performing the processes outlined in Equations (16.1),…, (16.6) and (20) to find H * * and w * assuming no resonant conditions the following results follow (all variables are double primed)