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The equations for the variations of the Keplerian elements of the orbit of a spacecraft perturbed by a third body are developed using a single average over the motion of the spacecraft, considering an elliptic orbit for the disturbing body. A comparison is made between this approach and the more used double averaged technique, as well as with the full elliptic restricted three-body problem. The disturbing function is expanded in Legendre polynomials up to the second order in both cases. The equations of motion are obtained from the planetary equations, and several numerical simulations are made to show the evolution of the orbit of the spacecraft. Some characteristics known from the circular perturbing body are studied: circular, elliptic equatorial, and frozen orbits. Different initial eccentricities for the perturbed body are considered, since the effect of this variable is one of the goals of the present study. The results show the impact of this parameter as well as the differences between both models compared to the full elliptic restricted three-body problem. Regions below, near, and above the critical angle of the third-body perturbation are considered, as well as different altitudes for the orbit of the spacecraft.

Most of the papers on this topic consider the third-body perturbation due to the Sun and due to the Moon in a satellite around the Earth. This is the most immediate application of the third-body perturbation. One of the first studies was made by Kozai [

Later, Giacaglia [

Hough [

In the last years, several researches, based on both analytical and numerical approaches, studied the third-body perturbation. The majority of them concentrated on studying the effects of a perturbation caused by a third body using a double-averaged technique, like those of Šidlichovskyý [

Using the double-averaged analytical model in Domingos et al. [

The main idea behind the single-averaged technique is that it eliminates only the terms due to the short-time periodic motion of the perturbed body. The results are expected to show smooth curves for the evolution of the mean orbital elements for a long-time period when compared with the full restricted three-body problem. In other words, a better understanding of the physical phenomenon can be obtained, and it allows the study of long-term stability of the orbits in the presence of disturbances that cause slow changes in the orbital elements.

So, an interesting point would be a study to show the differences between those two averaged models when compared with the full elliptic restricted three-body problem. The idea of the present paper is to study this problem, but assuming that the perturbing body is in an elliptical orbit. Studies under this assumption are available in Domingos et al. (see [

So, our goal is to make a more complete study of this problem and to perform some tests to verify the differences in the results obtained by those two approximated techniques. A comparative investigation is made to verify the differences between the single-averaged analytical model with the model based on the double-averaged technique, as well as a comparison with the full restricted elliptic three-body problem that can provide some insights about their applications in celestial mechanics.

The assumptions used to develop the single-averaged analytical model are the same ones of the restricted elliptic three-body problem (planet-satellite-spacecraft). Our analysis for the evolution of the mean orbital elements will be based only on gravitational forces. The equations of motion are obtained from Lagrange’s planetary equations, and then we numerically integrated those equations. Different initial eccentricities for the perturbing body are considered.

The set of results obtained in this research performs an analysis of several well-known characteristics of the third-body perturbation, like (i) the stability of near-circular orbits, to investigate under which conditions this orbit remains near circular; (ii) the behavior of elliptic equatorial orbits; and (iii) the existence of frozen orbits, which are orbits that keep the eccentricity, inclination, and argument of periapsis constants.

A detailed study considering the “critical angle” of the third-body perturbation, which is an inclination that makes a near circular orbit that has an inclination below this value to remain near-circular, is made. This work is structured as follows. In Section

For the determination of the equations of motion, we started by assuming the existence of a central body, with mass

Illustration of the dynamical system.

Using the traditional expansion in Legendre’s polynomials (assuming that

For the case of elliptic orbits of the perturbing body, the parameters

The mean anomaly

Thus, the variations in the orbital elements of the perturbed body are obtained. To do this, we derived Lagrange’s planetary equations that describe the variations of the mean orbital elements of the spacecraft. The semimajor axis is constant, since the mean anomaly

Those equations show some characteristics of this system compared with similar researches (see [

Circular orbits exist, but they are not planar. It is immediate from the inspections of the time derivatives of the Keplerian elements. If the initial eccentricity is zero, then the time derivative of the eccentricity is also zero and the orbit remains circular, but the time derivative of the inclination is not zero, so the orbit does not stay planar.

Elliptic equatorial orbits are not stable. The equatorial case (

Those facts explained above also show that there are no frozen orbits under this analytical model, which would be orbits where the time derivatives of the inclination, eccentricity, and argument of periapsis are all zero.

Domingos et al. (see [

In this section, we show the effects of the nonzero eccentricities of the perturbing body in the orbit of the perturbed body. We numerically investigate the evolutions of the eccentricity and the inclination for a spacecraft within the elliptic restricted three-body problem Earth-Moon-spacecraft, as well as using the single- and the double-averaged models up to the second order. The spacecraft is in an elliptic three-dimensional orbit around the Earth, and its motion is perturbed by the Moon. To justify our study and to make it a representative sample of the range of possibilities, we used three values for the semimajor axis of the spacecraft: 8000 km, 26000 km, and 42000 km. They represent a

We focus our attention On the stability of near-circular orbits. Results of numerical integrations showed that this stability depends on

Thus, for our numerical integrations, the initial inclinations for the orbit of the spacecraft received values near the critical inclination (in the interval ^{o}, 60°, 70°, and 80°). The eccentricity of the perturbing body is assumed to be in the range

All of the cases considering the

Our numerical results are summarized in Figures

These figures show the evolution of the eccentricity as a function of time for low-inclination orbits. On the top of each figure is shown the corresponding eccentricity of the perturbing body (

These figures show the evolution of the inclination as a function of time for near critical inclinations. On the top of each figure is shown the corresponding eccentricity of the perturbing body (

These figures show the evolution of the eccentricity as a function of time for near critical inclination spacecraft orbits. The first, second, and third rows correspond to the results for eccentricity of the perturbing body (

These figures show the evolution of the inclination as a function of time for high-inclination spacecraft orbit. On the top of each figure is shown the corresponding eccentricity of the perturbing body (

These figures show the evolution of the eccentricity as a function of time for high-inclination orbits. The first, second, and third rows correspond to the results for eccentricity of the perturbing body (

Figure

The use of second-order averaged models is not recommended for eccentricities of the primaries of 0.3 or larger. A larger expansion is required in this situation. In general, both averaged models have results that are much closer to each other than closer to those of the full model. It means that both averages tend to give the same errors, at least for eccentricities of the primaries up to 0.2. For eccentricity of 0.3 and above, the averaged models begin to show different behaviors, and those differences increase with this eccentricity. Regarding the comparison between both averaged models, the results depend on the value of the initial inclination.

The increase of the semimajor axis makes the spacecraft to stay more time at shorter distances of the Moon, so the third-body perturbations are stronger. This fact accelerates the dynamics of the system, and the period of the oscillation of the eccentricity is shorter when compared with the low-altitude orbits. It is visible that high-altitude orbits have oscillations of eccentricity increased in amplitude with the increase of the eccentricity of the primaries and the averaged models are not able to predict this property. They have results that decrease in quality with the increase of the eccentricity of the primaries. The amplitude of the oscillations increased about ten times with respect to the previous lower orbits due to the increase of the perturbation effects. The period of the oscillations is also reduced as an effect of the increase of the perturbations. The increase of the eccentricity of the primaries has the same effect of increasing the amplitudes and reducing the period of oscillations. The evolutions of the inclinations are not shown here because they remain constant for all of the situations considered in Figure

Figures

It can be noticed that near the time 20000 canonical units, medium-altitude orbits have presented some deviations between the results of the full and the averaged models, in particular for the higher inclinations. This is a result of the increase of the semimajor axis from low to medium orbits, which places the spacecraft in an orbit that is much more perturbed by the third body. Again, those models begin to give results that are also not so accurate when the eccentricity of the primaries increases. It is visible that the two-second order averaged models have results that are very closer to each other with similar deviations from the full model. So, in this range of inclinations and altitudes, both averaged models have about the same quality of results. The increases of the semimajor axis accelerate the dynamics also in this situation. Note that the time scale of the axis is different for low-altitude earth orbits and the deviations of the averaged models occur very early.

The third columns of Figures

Figures

The single-averaged model for low-altitude orbits, in an average over the time, gives results that are closer to those of the full model. This fact is very clear for eccentricities of the primaries 0.2. It is also noted that, for

For medium altitudes, when

Note that the strong changes in inclination and eccentricity occur earlier as a result of the stronger perturbations resulting from the increase of the semimajor axis of the orbit. The increase of the eccentricity of the primaries has the same effect, showing that an elliptic orbit for the perturbing body causes more perturbation on the spacecraft, when compared with circular orbits with the same semimajor axis, because the distance between the bodies decreases at the periapsis. The correspondent increase of that distance at the apoapsis does not compensate the previous aspect. For values of

For high-altitude orbits, both averaged models have results that are excellent for

The equations of motion of a spacecraft perturbed by a third body using a single-averaged technique are developed, considering an elliptic orbit for the disturbing body.

Looking at the overall behavior, the results are according to the literature: short oscillations in the inclination and eccentricity for initial inclinations below the critical value, which increases fast in amplitude around the critical value, then the typical behavior of having the inclinations remaining constant for a long time, and then returning very fast to the critical value, to increase fast again to its original values. The eccentricity shows an opposite behavior and increases when the inclination decreases and vice versa.

The results also showed that circular orbits exist, but frozen orbits do not exist under this model. The circular orbits, in general, do not keep the inclination constant, as happened in the double-averaged model.

The increase of the semimajor axis causes an increase of the third-body perturbation, and this fact accelerates the dynamics of the system. It was noticed that the period of the oscillations of the inclination and the eccentricity are shorter when compared with lower-altitude orbits. The increase of the eccentricity of the primaries, considering the semimajor axis constant, has the same effect of accelerating the dynamics. So, elliptic orbits for the disturbing body have the effect of increasing the perturbation, if all other elements remain the same.

It was also showed that inclined orbits are less perturbed by a third body, since the mean distance between the spacecraft and the Moon is larger than in planar orbits.

A comparison with the double-averaged technique up to the second order and with the full restricted elliptic problem was made, and it showed some other facts. The second-order averaged models are very accurate when the perturbing body is in a circular orbit. This accuracy decreases with the increase of the eccentricity of the primaries. The use of second-order averaged models is not recommended for eccentricities of the primaries of 0.3 or larger, and a better expansion, including more terms, is required in this situation. In general, both second-order averaged models have results that are much closer with each other than closer to those of the full model. It means that both averages tend to give similar errors, when compared with the full model.

The authors wish to express their appreciation for the support provided by CNPq under Contracts nos. 150195/2012-5 and 304700/2009-6, by Fapesp (contracts nos. 2011/09310-7 and 2011/08171-3), and by CAPES.