Proceedings of the XII DINAME, Feb.26-Mar02 2007, Ilhabela - SP - Brazil
Edited by P.S Varoto and M.A. Trindade
INFLUENCE OF LUNI-SOLAR PERTURBATIONS ON SECULAR RATES DUE TO THE EARTH GRAVITY FIELD
Giorgio E. O. Giacaglia 1, Wendell de Queiroz Lamas 2
1 University of Taubaté, Dept. of Mechanical Engineering, Rua Daniel Danelli s/n, Taubaté, SP, Brazil
2 University of São Paulo at Lorena, Dept.
Abstract. Equations for long period perturbations due to the Moon and Sun are developed and applied to equations expressing their influence on the secular rates due to the Earth potential field. Analytical results represent with good approximation observed values for the amplitude and periods of the perturbed elements of a satellite . As an example we have computed these values for a particular GPS satellite
Keywords: Luni-soalr perturbations.Long period. Secular rates.
Nomenclature
Proceedings of the XII DINAME, Feb.26-Mar02 2007, Ilhabela - SP - Brazil
Edited by P.S Varoto and M.A. Trindade
G = gravitational constant
m = mass
n = mean motion
a= semi-major axis
C20 = second harmonic coefficient of the Earth
e = eccentricity
PRN03 = GPS satellite
RAAN = right ascension of ascending node
GPS = Global Positioning System
M = mean anomaly
f = frequency
ℓ, m, p, q = integers
N = gravitational factor
Greek Symbols
β = defined by Eq. 3
= longitude of perigee
Ω = longitude of ascending node
λ = mean longitude
d= perturbation in satellite variables.
Subscripts
M relative to Moon
relative to Sun
relative to Earth
2 second order perturbation
sec = secular rate
Introduction
Several works have been published dealing with third body perturbations on a satellite of the Earth and other celestial bodies. A complete review was published in 2003 (Giacaglia and Prado, 2003). We have recently developed theories dealing with satellites orbits of small eccentricity and small inclination (Giacaglia, 1975; Prado, 2003; Solórzano and Prado, 2004, Giacaglia and Schutz, 2008 a, 2008b, 2009). Some results obtained in the above papers and in Van de Graff (1977) have been used. In this work the use of classical Keplerian elements show no singularities in the equations of motion. We initially evaluate long period luni-solar perturbations and apply the results to the secular rates in longitude of perigee, longitude of ascending node and mean longitude produced by the Earth major oblateness. The results are in agreement with observations over a long period of time (Schutz and Giacaglia, 2009)
Lunar perturbations on artificial satellites, for general terms of the disturbing function, were derived several years ago, by Kaula (1962). However, his formulas use equatorial elements for the Moon and do not give a definite algorithm for computational procedures. In fact, as Kozai (1966) suggested, 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. His work was limited to simple truncated part of the disturbing function, although using ecliptic coordinates for the Moon. In the present work, we obtain the disturbing function for the Moon's perturbations using ecliptic elements for the Moon and equatorial elements for the satellite, to any order of approximation. Perturbations due to geopotential have been extensively obtained by several authors since the late fifties.
In case of small eccentricity and inclination Giacaglia (1975) developed a complete theory with the use of special variables suggested by Lagrange (apud Brouwer, 1962). This development, similar to subsequent works, presents a difficulty due to the resulting very cumbersome equations. These same variables were used in recent works by Solarzano and Prado (2003).
In a recent work by Giacaglia and Schutz (2008) it has been show that developing the differential equations for these variables by writing the right hand side in terms of classical Keplerian orbits, because of well- known D´Alembert Characteristics (Brouwer, 1962), the equations do not contain any singularity, except for retrograde equatorial orbits, a singularity also present when expressing these equations in Lagrange´s special variables, often called “non- singular variables”. This terminology is not quite appropriate because the singularities are actually eliminated in the differential equations.
A short chapter by Berger (1974) dealt with the coupling effects of the Moon-Earth perturbations on artificial satellites, although no mathematical formulation was given, quoting results presented in a previous work.
The book “Third Body Perturbations on Satellites” by Giacaglia and Prado (2003) presents in great details the present subject, also giving an extensive bibliography since 1959 (Blitzer, 1959) through 2003 (Prado, 2003) quoting 166 scientific articles on the subject.
In recent years, several works on the subject were developed by Giacaglia (1975) and Giacaglia and Schutz (2008 a) and by Schutz and Giacaglia (2008b). This last article shows that, under special circumstances, analytical results agree quite well with numerical analysis of a particular GPS Satellite (Schutz and Giacaglia, 2008b)
Current literature , to our knowledge, does not give a full analytical formulation on the influence of Moon and Sun on the satellite secular rates in longitude, perigee and node due to the geopotential . In this paper we shall limit our analysis to the influence of secular luni-solar perturbations on the secular rates of a satellite induced by the Earth gravity field.
Long Period and Secular Perturbations due to the Geopotential and the Lunar Gravitational Field on a an artificial satellite of the Earth, using ecliptic coordinates for the Moon and equatorial coordinates for the satellite, are developed. For solar perturbations the theory is applicable with obvious adaptations. It is shown that Lagrange Differential Equations, for a special set of coordinates, are non singular for small eccentricity and inclination (except retrograde equatorial orbits), even when developing the disturbing function in terms of classical Keplerian Elements. Equations for long period and secular perturbations due to the Moon and due to the Earth potential field are derived. We give the equations expressing the influence of the third body long period and secular perturbations on the secular rates due to the Earth potential field. Preliminary results show good agreement with observed values (Schutz and Giacaglia, 2009).
The question arising from the use of the longitude of the perigee and the longitude of the ascending node should take into account the D´Alembert Characteristics. If the derivative of the disturbing function with respect to the longitude of the perigee is not zero, it must be factored by at least the first power of the eccentricity. Similarly, if the derivative of the disturbing function with respect to the longitude of ascending node is not zero, then it is factored by at least the first power of sine of half the inclination.
Because of these considerations, no singularities will be present in the right hand members of Lagrange Equations. Singularities in the derivatives with respect to the eccentricity and the inclination are only apparent due, again, to D´Alembert Characteristics.
In case of no resonances between the motion of the satellite and that of the Moon, the elimination of short period terms (depending on the mean anomaly M of the satellite) reduce to the evaluation of definite integrals, that can be expressed all in closed form, both in the eccentricity and inclination.
The averaged long period and secular disturbing function does not depend on the mean longitude and it is an explicit function of time only through the coordinates of the Moon and Sun, considering the inclination of the Moon with respect to the ecliptic and de obliquity of the ecliptic to be constant, which is a good approximation.
The integration of the pertinent equations can be performed numerically by using as input lunar ecliptic coordinates – or, for that matter, equatorial coordinates, in which case the theory is greatly simplified – stored in memory. This will produce precise evaluation of the true lunar motion. However, such a method can be very expensive in time.
A good approximation can be obtained by considering the inclination, eccentricity and semi-major axis and the obliquity of the ecliptic to be constant and the angular coordinates of the Moon with respect to the ecliptic to be linear functions of time. Also, an expansion in power series of the eccentricity of the Moon orbit will converge rapidly owing to the small value of this eccentricity, about 0.05.
The disturbing function of the Moon in a primitive form is transformed into orbital elements and rotation of the lunar coordinates to an ecliptic frame of reference, improving the results obtained by Kaula (1961).
The differential equations of motion for the satellite contains the frequencies which are integer multiples of the mean motion of the perigee of the satellite and the Moon and Sun, longitude of the ascending node of the satellite and the Moon and the ecliptic mean longitude of the Moon and Sun. These frequencies give an indication of possible resonances.
We should remember that one of the results of Kolmogorov (1953) celebrated work on quasi periodic motions, is that for large enough integers the denominator above can become smaller than any given quantity, which the basis for showing that perturbations techniques based on successive approximations, as for instance, Canonical Methods or Lie Series Methods, cannot converge to the true solution. In this respect, we must rely on Poincaré (1989) statement about asymptotic series “stop the series at a low degree of approximation”.
Approximate values for the Moon and Sun angular variables are used to estimate the secular rates of the satellite perigee and ascending node. For a satellite with eccentricity 0.007, inclination 60o and semi-major axis of 26.750 km we have the secular motion in longitude of perigee is approximately 24.85 x 10-3degree/day and in longitude of the ascending node is approximately 33.26 x 10-3 degree/day, both negative, so we can estimate the amplitude associated to any given set of the 8 integers entering the frequencies of the solution. Exact evaluation of the integral leading to the mean eccentricity and mean inclination should be from an initial time of observation up to any given successive time, and this requires a transformation from osculating to mean elements, a task to be developed in a future work.
By considering lunar and solar perturbations, in the absence of resonances, secular perturbations due to the Moon are obtained by eliminating all angular variables in the disturbing function. The equations of motion are greatly simplified, even because the angular variables appearing explicitly in the equations are considered to be affected only by secular variations.
Simple equations are found for the secular rates in longitude of perigee, ascending node and mean longitude of the satellite
For a real analysis of perturbations of an artificial satellite we consider perturbations due to the Earth gravity field, noting that up to geosynchronous heights the dominant term is due the Earth oblateness. Secular perturbations in eccentricity and inclination are zero, only being affected by short period and long period terms. As a first approximation, the secular rates may be computed by a simple quadrature, by keeping constant the right-hand members of the pertinent differential equations. Direct additions of secular rates due to the Moon (and / or Sun) and the geopotential are obtained.
When no resonances occur, the long period perturbations are defined by simply eliminate the dependence of the Disturbing Function from the mean longitude. The period of rotation of the Earth, about 24 hours, is considered a long period.
The final task has been to take into consideration the influence of the third body on the secular rates resulting from the Earth potential field. This is done by developing variational equations for the secular rates of the satellite angular variables, where the eccentricity and inclination are affected by long period perturbations due to the Moon and Sun. The corresponding equations are easily constructed.
Perturbations Evaluation
It is noted that Moon perturbations are factored by
(1)
This energy when compared with the central Newtonian attraction of the Earth gives a ratio of 1.2 x 10-7 for low satellites and 3.18 x 10-5 for geosynchronous satellites. For the Sun it is found that
(2)
The satellite Keplerian negative energy is
(3)
The relative size of the perturbing force function wrt the main Keplerian central attraction is given by
(4)
For low satellites ( 90 min),. (5)
For high satellites ( h),. (6)
It is seen that perturbations from the Moon and the Sun in a low satellite are comparable in magnitude. In the above range of periods, with low values of the eccentricity, the dominant part of the disturbing function of a satellite is due to the Earth oblateness (C20) and lunar (and solar) perturbations are about second order with respect to this. In cases of higher satellites, depending on the values of semi-major axis and eccentricity, the situation might even be reverted, so that for a full evaluation of the lunar perturbations, truncation of the corresponding disturbing function may not be advisable.
Long Period Perturbations
The integration of the pertinent equations can be performed numerically by using as input lunar and solar ecliptic coordinates or equatorial coordinates stored in memory, in which case the theory is greatly simplified. This will produce precise evaluation of the true lunar motion. However, such a method can be very expensive in time. Using the ecliptic as a reference, a good approximation can be obtained by considering the inclination, eccentricity and semi-major axis of the Moon and Sun and the obliquity of the ecliptic fixed values and the corresponding angular values to be linear functions of time, neglecting accelerations of these elements. Also, an expansion in power series of the eccentricity of the Moon and Sun will converge rapidly owing to the small value of the eccentricity of the Moon orbit.