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GPS Satellites Orbits: Resonance
The effects of perturbations due to resonant geopotential harmonics on the semimajor axis of GPS satellites are analyzed. For some GPS satellites, secular perturbations of about 4?m/day can be obtained by numerical integration of the Lagrange planetary equations considering in the disturbing potential the main secular resonant coefficients. Amplitudes for long-period terms due to resonant coefficients are also exhibited for some hypothetical satellites orbiting in the neighborhood of the GPS satellites orbits. The results are important to perform orbital maneuvers of GPS satellites such that they stay in their nominal orbits. Also, for the GPS satellites that are not active, the long-period effects due to the resonance must be taken into account in the surveillance of the orbital evolutions of such debris.
The period of the orbits of the GPS satellites is about 12 hours, and the main perturbations acting on their orbits are caused by the nonuniform distribution of the Earth’s mass, by the lunar and solar gravitational attractions and by the solar radiation pressure. In this paper, it is analyzed just some perturbations due to resonant terms of the geopotential coefficients. The resonance considered here is the 2 : 1 commensurability between the orbital period of the GPS satellites and the period of the Earth’s rotation. As it is pointed out by Hugentobler  the resonance leads a daily drift rate in semimajor axis of up to 7?m/day.
Lagrange planetary equations, describing the temporal variation of the orbital elements, are used here to analyze the orbital perturbations of the GPS satellites under the influence of resonant coefficients.
2. Disturbing Potential
The geopotential acting on Earth’s artificial satellite can be expressed as 
Here GM is the geogravitational constant, r is geocentric distance of the satellite, ae is the semimajor axis of the adopted Earth’s ellipsoid, a is the orbital semimajor axis, Cnm and Snm are the spherical harmonic coefficients. Pnm denotes the associated Legendre functions, are the satellite’s geocentric latitude and the longitude, n and m are, respectively, the degree and order of the harmonic coefficients.
The geocentric distance, the latitude, and the longitude of the satellite can be expressed in terms of the orbital elements and (2.1) becomes [2, 3] where and are, respectively, functions of the satellite’s orbital inclination i and eccentricity e, O represents the right ascension of the orbital ascending node, ? is the argument of perigee, M is the mean anomaly, and ?T is the Greenwich sidereal time. The function can be expressed as and the argument is given by
The functions and are presented as tables  and can be adapted to be used in computers.
3. Resonance in GPS Satellites
where is the Earth’s sidereal rotation.
For the GPS satellites where the commensurability is 2 : 1, there are bounds among the parameters
Taking into account that , (3.1) can be put as
since , we get
Considering harmonics of order and degree up to order 4, Table 1, presents some values for the parameters satisfying the resonance condition for the GPS satellites.
Taking into account in the summations the conditions and and putting 
In order to analyze the orbital perturbations of GPS satellites due to the resonant coefficients presented in (Table 1), the Lagrange planetary equations will be used. However, here this analysis will be concentrated initially in the secular effects of such perturbation on the orbital semimajor axis. Therefore, we have 
with being the mean motion of the satellite.
A numerical integration of (3.8) was performed for a period of one day using IGS/POE (The International GNSS Service Precise Orbital Ephemeredes) given on June, 07, 2007. Transformations were performed to get the corresponding orbital elements.
Figure 1 gives the variation due to the coefficients , which are responsible by the greatest variation in the semimajor axis that, in this case, corresponds to the satellite PRN 06. Figures 2, 3, and 4 present, respectively, the daily variations as function of the coefficients . Figure 5 represents the total daily variation due to the above considered coefficients.
Table 2 shows the maxima and minima values for the daily variation of the semimajor axis according to the resonant coefficients.
Figures 6 and 7 show the semimajor axis variation for the satellites PRN 02 and PRN 06 for a time interval of 200 days and taking into account the harmonic coefficients . During this period and for the considered initial condition, it can be observed variations of about 600?m and 680?m, respectively.
4. Long-Period Perturbations
The effects of the resonance are enhanced when long periods are considered. Table 3 and Table 4, Figures 8, 9, 10, 11, 12, and 13 show some simulations using hypothetical satellites and a particular method to study effects of resonance on the orbits of artificial satellites [6–9].
Table 3 contains the amplitude and period of the variations of orbital elements for hypothetical satellites considering low eccentricity, small and high inclination, and the influence of the harmonics and
Figures 8 and 9 represent, respectively, the temporary variation of the semimajor axis and of the eccentricity of artificial satellites of the GPS type in the neighborhood of the 2 : 1 resonance region when the harmonics and are considered. By Figure 10, assuming several values for the semimajor axes in the neighborhood of the 2 : 1 resonance, it can be observed distinct behavior for their temporary variations.
Figure 11 represents the temporary variation for the semimajor axis in the neighborhood of the resonance 2 : 1 considering the influence of the harmonics and for a satellite of the GPS type with inclination of about .
Table 4 gives the influence of the resonance due to the harmonics and considering different inclinations, including that of the GPS type satellite. It can be observed that the amplitudes of the variations are smaller when compared with those when the harmonics and are taken into account. The more the satellites approach the region that was defined as a resonant region, the more the variations increase.
Figure 12 represents the temporary variation of the semimajor axis of a satellite of the GPS type considering resonance due to the harmonics , , and .
Figure 13 represents the time variation of the orbital semimajor axis of a satellite with eccentricity about 0.01 and inclination about orbiting in a region near the 2 : 1 resonance. It is remarkable the oscillation of the semimajor axis in the region between by a = 26560.0?km and a = 26562.48?km. For instance, a variation of about 10?m in the initial semimajor axis (a = 26562.48?km) brings up a variation of about more than 10?km in its amplitude in a period of about 2000 days.
From the obtained results it can be seen that for the GPS satellites no negligible perturbations are provoked by resonant tesseral coefficients. The daily variation of the semimajor axis due to the is less than 4?m/day and those values found for a time interval of 200 days (~680?m) are compatible with the results presented by specialized literature (~7?m/day and ~1200?m) about this subject .
Taking into account the total effects of daily resonant perturbations, it can be observed that for some GPS satellites these effects are enhanced and for another satellites are attenuated but all of these effects are smaller than the amplitudes mentioned above. It was shown also that the effects of the resonance are very important for the analysis of long-period behavior of the GPS satellites orbits.
An important aspect to be considered is the necessity to perform orbital maneuvers of GPS satellites in such way that they stay in their nominal orbits. Also, for the GPS satellites that are not active, the long-term effects due to the resonance must be taken into account in the surveillance of the orbital evolutions of such debris.
This work was partially supported by CNPq under the contracts No. 305147/2005-6 and 300952/2008-2
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