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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 [

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.

The geopotential acting on Earth’s artificial satellite can be expressed as [

Here _{e}_{nm}_{nm}_{nm}

The geocentric distance, the latitude, and the longitude of the satellite can be expressed in terms of the orbital elements and (_{T }

The functions

Resonance is associated with small divisors. For artificial Earth satellites whose orbital periods are in commensurability with the period of the Earth’s rotation resonance can occur when [

For the GPS satellites where the commensurability is 2 : 1, there are bounds among the parameters

Taking into account that

Considering harmonics of order and degree up to order 4, Table

Resonant parameters.

Degree | Order | ||
---|---|---|---|

3 | 2 | 1 | 0 |

4 | 4 | 1 | 0 |

2 | 2 | 1 | 1 |

4 | 2 | 2 | 1 |

Taking into account in the summations the conditions

In order to analyze the orbital perturbations of GPS satellites due to the resonant coefficients presented in (Table

Special care must be taken to compute

A numerical integration of (

Figures

Drift rates in semimajor axis due to the resonant geopotential coefficient 32.

Drift rates in semimajor axis due to the resonant geopotential coefficient 44.

Drift rates in semimajor axis due to the resonant geopotential coefficient 22.

Drift rates in semimajor axis due to the resonant geopotential coefficient 42.

Figure

Drift rates in semimajor axis due to all resonant geopotential coefficients, Table

Table

Maximum values.

Resonants coefficients | Maximum drift rates m/day |
---|---|

−4.0 | |

1.5 | |

1.5 | |

0.002 |

Figures

Semimajor axis variation (PRN 02) due to the resonant geopotential coefficient 32, for a time interval of 200 days.

Semimajor axis variation (PRN 06) due to the resonant geopotential coefficient 32, for a time interval of 200 days.

The effects of the resonance are enhanced when long periods are considered. Table

Amplitude and period of perturbations due to the 2 : 1 resonance:

Orbital elements | Amplitude | Period | ||||
---|---|---|---|---|---|---|

t(days) | ||||||

0.01 | 4° | 12 km | 0.014 | 8.7 ^{-4} | 2500 | |

0.01 | 4° | 12.5 km | 0.0145 | 8.6 ^{-4} | 7000 | |

0.01 | 4° | 4.5 km | 0.0065 | 3.6 ^{-4} | 2000 | |

0.01 | 4° | 3 km | 0.0045 | 2.8 ^{-4} | 1000 | |

0.01 | 4° | 6.5 km | 0.0085 | 4 ^{-4} | 1700 | |

0.01 | 4° | 1.4 km | 0.0023 | 1.3 ^{-6} | 500 | |

0.05 | 55° | 1.8 km | 0.005 | 2 ^{-3} | 3200 | |

0.005 | 55° | 2.75 km | 0.006 | 2.8 ^{-3} | 3900 | |

0.005 | 55° | 1.2 km | 0.0045 | 1.4 ^{-3} | 2000 | |

0.005 | 55° | 900 m | 0.0033 | 10^{-3} | 1500 | |

0.005 | 55° | 1.8 km | 0.0055 | 1.9 ^{-3} | 2500 | |

0.05 | 63.4° | 7.5 km | 0.003 | 10.8 ^{-3} | 1000 | |

0.05 | 63.4° | 4 km | 0.0015 | 5.7 ^{-3} | 1000 | |

0.05 | 87° | 7 km | 0.0037 | 13.9 ^{-3} | 2000 | |

0.05 | 87° | 3.5 km | 0.002 | 10.3 ^{-3} | 1900 |

Amplitude and period of perturbations due to the 2 : 1 resonance:

Orbital elements | Amplitude | Period | ||||
---|---|---|---|---|---|---|

t (days) | ||||||

_{o} | 0.01 | 4° | 120 m | 0.0004 | 1.78 ^{-3} | 500 |

_{o} | 0.01 | 4° | 245 m | 0.0009 | 3.9 ^{-3} | 1000 |

_{o} | 0.01 | 4° | 110 m | 0.00045 | 1.78 ^{-3} | 500 |

_{o} | 0.01 | 4° | 100 m | 0.0003 | 1.15 ^{-3} | 250 |

_{o} | 0.005 | 4° | 100 m | 0.0009 | 4.6 ^{-5} | 500 |

_{o} | 0.005 | 4° | 700 m | 0.006 | 2.4 x 10^{-4} | 1500 |

_{o} | 0.005 | 55° | 1.9 km | 0.0085 | 6.8 ^{-4} | 16000 |

_{o} | 0.005 | 55° | 250 m | 0.0017 | 1.13 ^{-3} | 16000 |

_{o} | 0.005 | 55° | 30 m | 0.00055 | 2.3 ^{-4} | 1500 |

_{o} | 0.005 | 55° | 47 m | 0.00035 | 3.43 ^{-5} | 500 |

_{o} | 0.005 | 87° | 110 m | 0.00085 | 2.29 ^{-4} | 1800 |

_{o} | 0.005 | 87° | 45 m | 0.00035 | 8.59 ^{-5} | 800 |

_{o} | 0.005 | 87° | 20 m | 0.00014 | 4 ^{-5} | 500 |

Eccentricity versus time,

_{0}

Time variation of semimajor axis considering the harmonics

Time variation of the semimajor axis considering the harmonics

Table

Figures

Figure

Table

Figure

Figure

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

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