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Through the averaged equations we revisit theoretical and numerical aspects of the
strong resonance that increases the eccentricity of the disposed objects of GPS and
Galileo Systems. A simple view of the phase space of the problem shows that the
resonance does not depend on the semi-major axis. Therefore, usual strategies of
changing altitude (raising perigee) do not work. In this problem we search for a set of
initial conditions such that the deactivated satellites or upper-stages remain at least for
250 years without penetrating in the orbits of the operational satellites. In the case that Moon's perturbation is not significant, we can identify, in the phase space, the regions
where eccentricity reaches maximum and minimum values so that possible risks of
collision can be avoided. This is done semi-analytically through the averaged system of
the problem. Guided by this idea, we numerically found the

Broadly speaking, the GPS, GLONASS, and Galileo

[

In a constellation of a navigation system, the members must be kept under precise requirements of functionality. However, after some time, they have to be deactivated since some level of these requirements cannot be fulfilled for long time. The destination of these deactivated objects is a problem since they must be moved into some disposal regions in order to preclude collisions with operational members of the constellation. While these vehicles can be designed a priori to transport additional propellant (at some nonnegligible cost) to be used in some planned maneuvers to insert them in the disposal regions; the same is not true for the upper-stage. In some cases (block IIF of GPS system), due to design restrictions, this upper-stage cannot be easily guided to the disposal region. It must perform several operations after the satellite is injected in the constellation. All these operations change its final parameters [

The above strategy of keeping small eccentricity can generate some additional problem: after some time, the disposed vehicles will accumulate and a graveyard of these objects will be created. Therefore, a risk of collisions amongst themselves is a crucial problem, since the products of these extra collisions are almost untrackable fragments that may offer more risks to the operational elements of the constellation.

According to Jenkin and Gick [

In this sense, we briefly started the investigation of some initial conditions that can cause large increase of the eccentricity, for a minimum time interval, considering also different initial inclinations of the Moon’s orbit (see Appendix

As we want to highlight some theoretical aspects, it is instructive to write the main disturbing forces in terms of the orbital elements.

In this section we obtain the averaged disturbing function of the Sun. Following the classical procedure [

Expanding (

In order to write cos(

Geometry of the problem.

Considering classical relations of the two-body problem, we write cos(

In order to get rid of the short period variations, we have to obtain the averaged system and the rigorous procedure is to apply the classical von-Zeipel or Hori’s method [

Performing a second and similar average with respect to the mean anomaly of the Sun, we get

For the oblateness, the disturbing function truncated at second order of

For close satellites, usually the oblateness is the dominant part. In this case, the main frequencies of the system are given by

The ratio of these two frequencies is

Note that for

In order to see the effects of the resonances which affect GPS and Galileo satellites, the osculating equations of a satellite will be integrated. For the moment, as disturbers, we consider only the Sun and the oblateness (the complete Cartesian equations involving the remaining disturbers will be given in Section

Time evolution of the eccentricity (a) and the critical angle (b). Initial conditions:

Same as Figure

Let us pay more attention to the case

_{i}_{i}_{L}

Level curves of Hamiltonian (

Time evolution of the osculating eccentricity (top) and the osculating critical angle (bottom) for a disposal GPS satellite. Note that the minimum of the eccentricity occurs when _{T} (22,324 km), _{L}

In the previous section, we considered only the effects of the Sun and of the oblateness. Moreover, in the presence of the resonance, the main effects are governed by the long term variations, so that we eliminated the short period terms. From a theoretical point of view, this averaged system is quite efficient to highlight the basic dynamics that affects the eccentricity of the GPS and Galileo satellites. However, for a more complete and realistic study, we need to include more disturbers.

In this section we want to find some special initial conditions such that the satellites can remain stable for at least 250 years with very small eccentricity without causing any risk of collision to the operational elements of the constellation. The strategy to search these particular initial conditions is guided from the theoretical approach described in the previous section.

In this section we integrate the osculating elements of a disposal satellite of the Galileo system under the effect of the Sun, Moon, and the oblateness.

The equations for the osculating elements (exact system), including Moon, are

_{L}

As we said before, we take 500 km above of the nominal altitude of the constellation. The initial elements are fixed to _{T}

Black dots: represent (_{L}

Same of Figure

Time evolution of the eccentricity (a) and the critical angle (b) obtained from integration of (_{L}_{L}_{L}

Same as Figure

Here let us consider the GPS system. Again we take

Up to now, we have not considered tesseral and sectorial harmonics. Since GPS satellites have orbital period near to 12:00 h, the inclusion of such harmonics must be examined when drawing the figures of Section

_{nm}

Figure

the position vector of the satellite.

inclination.

geocentric latitude.

longitude.

longitude of node.

According to Figure

Let us define

Therefore,

Proceeding in the similar way, we obtain

Note that the zonal terms (

Therefore considering only

Once we have introduced

To express

The effects of these additional terms are shown in Figure

Figures

With the averaged equations, we showed the dynamics of the

Same of Figure

Same of Figure _{L}

Geometry of the problem.

Time evolution of eccentricity of GPS satellite. Initial conditions:

Same of Figure

Same of Figure _{L}

Black dots represent _{L}

Same of Figure _{L}

Time evolution of eccentricity (a) and critical angle (b) of a GPS satellite with _{L}

Here we give the complete expression of the averaged disturbing function up to first order in eccentricity of the third body:

Note that all the cosines in the above relations have

Following the same strategy to obtain Figures

The authors thank CNPQ, FAPESP, and FUNDUNESP. An anonymous referee is gratefully thanked for very useful comments.