Mathematical Problems in Engineering

Volume 2015, Article ID 382340, 14 pages

http://dx.doi.org/10.1155/2015/382340

## Study of Some Strategies for Disposal of the GNSS Satellites

^{1}Instituto Nacional de Pesquisas Espaciais (INPE), 1758 Avenida dos Astronautas, 12227-010 São José dos Campos, SP, Brazil^{2}Universidade Estadual Paulista (UNESP), 1515 Avenida 24A, Caixa Postal 178, 13500-970 Rio Claro, SP, Brazil

Received 3 June 2014; Accepted 15 August 2014

Academic Editor: Chaudry M. Khalique

Copyright © 2015 Diogo Merguizo Sanchez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The complexity of the GNSS and the several types of satellites in the MEO region turns the creation of a definitive strategy to dispose the satellites of this system into a hard task. Each constellation of the system adopts its own disposal strategy; for example, in the American GPS, the disposal strategy consists in changing the altitude of the nonoperational satellites to 500 km above or below their nominal orbits. In this work, we propose simple but efficient techniques to discard satellites of the GNSS by exploiting Hohmann type maneuvers combined with the use of the resonance to increase its orbital eccentricity, thus promoting atmospheric reentry. The results are shown in terms of the increment of velocity required to transfer the satellites to the new orbits. Some comparisons with direct disposal maneuvers (Hohmann type) are also presented. We use the exact equations of motion, considering the perturbations of the Sun, the Moon, and the solar radiation pressure. The geopotential model was considered up to order and degree eight. We showed the quantitative influence of the sun and the moon on the orbit of these satellites by using the method of the integral of the forces over the time.

#### 1. Introduction

Global navigation satellite systems are a general denomination for constellations of navigation satellites, such as GPS (USA), GLONASS (Russia), Galileo (Europe), and Beidou (China), mainly placed in the medium earth orbit (MEO) region. Up to July 2013, the GPS had 31 active satellites in orbits with altitude near 20,000 km and 55 degrees of inclination. The Galileo system, at the same epoch, had four active satellites, with approximately 23,000 km of altitude and nominal inclination of 56 degrees. The Beidou system is composed by satellites in MEO, geosynchronous orbit (GEO), near circular, and inclined (55 degrees) geosynchronous orbit (IGSO). At the previously mentioned epoch this system had 14 satellites, four satellites being placed in MEO, five satellites in GEO, and five placed in IGSO [1]. The GLONASS system has 31 active satellites with inclinations ranging between 63° and 65° and 19,129 km of altitude. We will not consider the GLONASS system in this work, as well as the Beidou satellites with near circular orbits, because our technique is mainly devoted for satellites with inclinations around 56°, in which the 2 : 1 resonance between the argument of the perigee and the longitude of the ascending node is effective and causes a significant growth of eccentricity of the satellites.

In addition to the 2 : 1 perigee-ascending node resonance, the GNSS satellites are subject to resonances caused by the commensurability between the orbital period of these satellites and the rotation of the Earth (spin-orbit resonance). The GPS satellites are in a “deep” 2 : 1 resonance with the rotation of the Earth (the orbital period of these satellites is half of the rotation period of the Earth), and this resonance is dominated by the terms and of the geopotential [2]. The Galileo satellites are in 17 : 10 resonance, and the MEO satellites of the Beidou system are in 13 : 7 spin-orbit resonance [1]. The harmonic of the geopotential also plays an important role in the GNSS, because it causes short periodic variations in the semimajor axis [3, 4]. In general, the tesseral harmonics induce perturbations with short period and low amplitude in the orbital motion of the satellites. However, these terms may produce effects of high amplitude and long period [5, 6].

Due to the variety of orbits and resonances involved, the GNSS is a complex system, which turns the creation of a definitive strategy to dispose the satellites of this system into a hard, even impracticable, task. In this way, each constellation of the system adopts its own disposal strategy; for example, in the GPS system, the disposal strategy consists in changing the altitude of the nonoperational satellites to 500 km above or below their nominal orbits [7]. Due to the increase of debris around the Earth and the potential instability of orbits with inclination around 56 degrees [8, 9], several studies [10–14] have been made to develop alternative strategies to dispose the satellites of the GNSS in a safer way. From the analysis of the initial conditions which lead to the previously mentioned instability caused by the 2 : 1 perigee-ascending node resonance, some works suggest moving the disposed satellites to regions such that the growth of the eccentricity does not allow the disposed satellites to invade the region of the operational satellites, at least for some acceptable time. This strategy may present some inconveniences, such as an accumulation of objects in the disposal region. On the other way, there are some works that suggest transferring the disposed satellites to appropriate orbits such that the new initial conditions lead to an increase of the eccentricity. The main goal is to lower continuously the periapsis radius of the disposed satellites, provoking their reentry in the atmosphere of the Earth. A positive characteristic of this approach is the decrease in the number of space debris, so reducing the collisional risk between inactive objects. Nowadays this number is not so high, but the continuous insertion of new satellites in this region certainly will change this scenario. Although the increase of the eccentricity may lead to a possible increase in the collisional risk between disposed and active objects, this risk can be minimized with a proper choice of initial conditions, which can accelerate the decay of these satellites. Therefore, the time that the satellite needs to reenter in the atmosphere is an important question related to the cost and success of this strategy. The effectiveness of disposal strategies for the GNSS satellites created so far can be found in [1].

In this work, we focus on the use of the strategy of eccentricity growth under the effect of the 2 : 1 perigee-ascending node resonance, which leads to the atmospheric reentry, and we compare this strategy with the direct discard using a Hohmann type transfer. This comparison is shown in terms of the velocity increment necessary to perform the atmospheric reentry. For the strategy that uses the resonance, grids of initial conditions were generated to evaluate the maximum eccentricity reached by the satellites after 250 years considering their nominal altitudes and also considering an apoapsis radius of 10,000 km above and below their nominal apoapsis radius. Testing different initial altitudes makes sense, since our purpose is to lower the periapsis of the orbit to drive the satellites to enter in the atmosphere within some prefixed time interval. We will see that these tests also show the “strength” of the resonance for different values of the semimajor axis of the orbit of the satellite, since, from the mathematical point of view, the 2 : 1 perigee-ascending node resonance always exists, no matter what is the altitude of the satellite.

As an additional study, we showed the quantitative influence of the sun and the moon on the orbit of these satellites by using the method of the integral of the forces over the time [15, 16]. The advantage of this method is that we can measure the total variation of the velocity caused by the moon and the sun without disregarding other perturbations; in this case, the perturbations of the geopotential and the radiation pressure of the sun which could interfere in the orbit of the satellite and, consequently, in the value of the perturbation of the sun and the moon. In the case of the sun, the total variation of the velocity is measured as a function of the perigee of the sun, and we show that this is an important element to take into account when planning the disposal of the GNSS satellites. For the moon, the most important element is its inclination.

#### 2. The Resonance and Its Effects

In order to explain how the resonance 2 : 1 perigee-ascending node can affect the orbits of the GNSS satellites, initially we consider only the main disturbers, namely, the oblateness of the earth and the gravity of the sun. As usually, the oblateness is the dominant part of the disturbing function of a satellite in the MEO, and then we can use the single averaged form of the oblateness [13] to define the resonances which involve the perigee and the ascending node of the satellites in MEO. The expression for the single averaged oblateness is given by where is the mean motion of the satellite and is the mean equatorial radius of the Earth.

In this case, the main frequencies of the system are given by The ratio between and is For integer we have the special resonances which do not depend on the semi-major axis. These resonances usually affect the eccentricity [17]. For we have for or . Another classical resonance occurs when , so that , and this resonance affects the GLONASS satellites, but those are not considered in the present work.

The closer the inclination of the GNSS satellite is to 56.06°, the stronger will be the effect of the resonance on the eccentricity [13]. In order to see the effects of this resonance, the osculating equations of motion of a satellite will be integrated. As we mentioned before, only for this part of this work, as perturbations, we consider just the sun and the oblateness of the earth (the complete Cartesian equations involving the remaining perturbations will be given in the next section). Figure 1 shows the effects of the resonance on the eccentricity and on the resonant angle. Note that an initial small eccentricity reaches a significant increase.