Lunar perturbations of artificial satellites of the earth

Abstract

Abstract. Lunisolar perturbations of an artificial satellite for general terms of the disturbing function were derived by Kaula (1962). However, his formulas use equatorial elements for the Moon and do not give a definite algorithm for computational procedures. As Kozai (1966Kozai ( , 1973 noted, both inclination and node of the Moon's orbit with respect to the equator of the Earth are not simple functions of time, while the same elements with respect to the ecliptic are well approximated by a constant and a linear function of time, respectively. In the present work, we obtain the disturbing function for the Lunar perturbations using ecliptic elements for the Moon and equatorial elements for the satellite. Secular, long-period, and short-period perturbations are then computed, with the expressions kept in closed form in both inclination and eccentricity of the satellite. Alternative expressions for short-period perturbations of high satellites are also given, assuming small values of the eccentricity. The Moon's position is specified by the inclination, node, argument of perigee, true (or mean) longitude, and its radius vector from the center of the Earth. We can then apply the results to numerical integration by using coordinates of the Moon from ephemeris tapes or to analytical representation by using results from lunar theory, with the Moon's motion represented by a precessing and rotating elliptical orbit. Elements for the Moon and Other

On Analytic Modeling of Lunar Perturbations of Artificial Satellites of the Earth

“…Since algebraic errors appear in the mathematics leading to equation (15) in [2], equations (15 to 19) there are false; but the form of (19) is desirable, and so the mathematics which follow will be aimed toward the objective of producing a similar form. Suppose m is even and / -777 is even, then Employing the identity cos(x -f) = sinx.…”

“…Now Giacaglia proceeded to average (2.6) over the mean anomaly of the satellite to produce the secular and long periodic parts of the disturbing function together with the short period corrections, and a solution is certainly possible by this route, but since the real advantage of (2.6) lies in the fact that it paves the way for simultaneous integration of the angular rates for the satellite and the Moon, we will defer averaging techniques to the next section where all averaging is performed on the outset directly from (1.2). Therefore, we will skip to (117) in [2], where the true anomalies of the satellite and the Moon are converted to their respective mean anomalies via the Hansen coefficients, which satisfy the equation If the mean motion n is approximated as n = no(l - §^f )> where 6a is the perturbation of the semimajor axis due to a above, and R^ is substituted for R in the Lagrange equations, then a complete integration of the system is possible provided a,a^,e,e^., I, Ik, i, -A/, 3/fc,w, Wfc, fi, Qk are ah held fixed and constant throughout the integration. Numerical tests show that this approximation is improved over the long term if the mean values for these constants are used instead of the initial value conditions.…”

“…Since this is an initial value problem, we can remove the constant part of this quantity. A second averaging is now performed on (z) with respect to the mean anomaly of the Moon (1 +e fc cos/ fe ) 2 The long period corrections are readily available as…”

“…It is the intent for the numerical examples to highlight the simplifying issues mentioned above. The first method to be presented will be called the "series expansion method" and follows the development of Kaula [1] and Giacaglia [2]. We will refer to this solution as the (SEM) solution.…”

On analytic modeling of lunar perturbations of artificial satellites of the earth

“…It is the intent for the numerical examples to highlight the simplifying issues mentioned above. The first method to be presented will be called the 'series expansion method' and follows the development of Kaula [5] and Giacaglia [2]. We will refer to this solution as the (SEM) solution.…”