Low-altitude, near-polar and near-circular orbits around Europa☆
Introduction
Low-altitude, near-polar orbits, are very desirable for scientific missions to study natural satellites, such as Europa, one of the moons of Jupiter. However, previous researches, like Scheeres et al., 2001, Lara and Russell, 2006, Paskowitz and Scheeres, 2006 showed that an artificial satellite in a low-altitude, near-polar orbit impact with Europa’s surface in a short time period. Frozen orbits around the Moon, natural satellites, or asteroids is of current interest because several space missions have the goal of orbiting around such bodies (see for instance Park and Junkins, 1995, Elipe and Lara, 2003, Folta and Quinn, 2006, Carvalho et al., 2010a and references therein).
The dynamics of orbits around a planetary satellite, taking into account the gravitational attraction of a third-body and the non-uniform distribution of mass of the planetary satellite, has been studied by several authors. Kozai (1963) makes a study on the motion of an orbit around a planetary satellite (Moon) that found stable and unstable orbits without integration of the equations of motion. In Paskowitz and Scheeres (2005b) the average problem is applied twice to reduce the original problem, a system with three degrees of freedom, to an integrable system with one degree of freedom. Carvalho et al., 2009a, Carvalho et al., 2009b shows the dynamics of an artificial satellite around the Moon, taking into account the non-uniform distribution of mass of the Moon and the perturbation caused by a third-body in elliptical orbit. It is also presented an approach to study frozen and sun-synchronous orbits. Applying the double-averaged method Russell and Brinckerhoff (2009) shows evolution of the eccentricities and inclinations for long-period orbits and discuss them for the dimensioned systems at Ganymede, Europa, Titan, Enceladus, and several other planetary moons. In Abad et al. (2009) an analytic model is presented to find frozen orbits for lunar satellites, where only the potential due to J2 and J7 zonal harmonics are taken into account. In Lara et al. (2010) the averaged problem up to the sixth order is used to obtain information on the dynamics close to Enceladus. In San-Juan (2010), an analytical theory based on Lie-Deprit transformation is used to locate family of frozen orbits for asteroids and natural satellites. The average is performed through Lie transformations and two different transformations are applied.
In this work we consider the effects caused by the non-sphericity (J2, J3, C22) of Europa and the perturbation of the third-body (Jupiter), assumed to be in elliptical orbit, on the orbital motion of an artificial satellite. We present an analytical theory using the averaged model and the applications were done by integrating numerically the analytical equations developed here. A special study is made for the case of frozen orbits (Scheeres et al., 2001, Elipe and Lara, 2003, Lara and Russell, 2006), that are orbits that keeps the eccentricity, inclination and argument of the periapsis of the orbit simultaneously almost constant, to make the satellite to pass by a given latitude with the same altitude. We fix a parameter to get frozen orbits when new terms are added to the disturbing potential. Previous researches (Scheeres et al., 2001, Lara and Russell, 2006, Paskowitz and Scheeres, 2006) show that low-altitude, near-polar orbits around Europa are unstable and have short lifetimes. In order to analyze the influence of the short period terms we make a study with respect to the terms of short-period, taking into account the problem without averaging the equations of motion. A comparison between the single and double averaged models is presented, as suggested in Paskowitz and Scheeres (2005a). Considering low altitude orbits, the perturbation of the third-body (R2) should be taken into account, because this perturbation creates a strong effect in the determination of frozen orbits with polar inclination, as we can verify in next sections. However, we can find in the literature works which take into account different sets of perturbations, thus, (a) R2 + J2 Scheeres et al., 2001, Lara and San-Juan, 2005, San-Juan, 2010, (b) R2 + J2 + C22 De Saedeleer, 2006, Carvalho et al., 2010a, (c) R2 + J2 + J3 Paskowitz and Scheeres (2005a), (d) R2 + J2 + J3 + C22 Paskowitz and Scheeres, 2005b, Lara and Russell, 2006, Carvalho et al., 2010b, Tzirti et al., 2010. In this paper, we analyze the influence of different perturbations (R2 + J2 + J3 + C22) to search for orbits with longer lifetime.
The paper has six sections. Section 2 is devoted to the disturbing function, showing the terms due the non-sphericity of Europa and the Hamiltonian system. Applications taking into account the long-period disturbing potential using the double-averaged method are presented in Section 3. In Section 4, applications taking into account the unaveraged disturbing potential are presented while, in Section 5, an approach taking into account the averaged models is presented. Section 6 shows the conclusions.
Section snippets
Equations of motion
The equations for the disturbing potential due to the third-body has been developed in Carvalho et al. (2010a), where the potential is developed up to the fourth order with expansion in Legendre polynomials. Here, the potential is considered up to the second order in Legendre polynomials to perform the applications. We use expansions in the eccentricity and in the mean anomaly to replace the disturbing potential in the Lagrange planetary equations.
Section 2.1 provides the disturbing potential
Frozen orbits
The single-averaged method is applied in Eq. (21) to eliminate the short-period (l) terms, in order to analyze the effect of the disturbing potential on the orbital elements. Periodic terms are calculated, and the results are replaced in the Lagrange’s planetary equations (Kovalevsky, 1967) and integrated numerically using the software Maple. The coefficients of the gravity field of Europa are given in Table 1.
The long-period (g, h) disturbing potential can be written in the form
The unaveraged potential for polar orbits
Due to the third-body perturbations, high-inclination orbits around planetary satellites are known to be unstable (Scheeres et al., 2001, Lara and Ferrer, 2005, Paskowitz and Scheeres, 2005a), therefore, candidates for Europa science orbit are among the unstable orbits. In general, we found that polar orbits are of short lifetimes. However, it is important to consider the short-period terms in the dynamics to search for long lifetime orbits (Paskowitz and Scheeres, 2005a, Lara and Russell, 2006
The long-period disturbing potential
In this section the disturbing potential is divided in two parts. First, an approach taking into account the double-averaged problem to eliminate the mean anomaly and the longitude of the ascending node of the satellite is done, like in Paskowitz and Scheeres, 2005a, Lara and Russell, 2006. In the second part, the single-averaged method is applied to eliminate the mean anomaly of the satellite. In this case, the disturbing potential has two degrees of freedom. After that, the two models are
Conclusions
In this paper, the dynamics for the orbits around a planetary satellite, taking into account the gravitational attraction of a third body in an elliptical orbit and the non-uniform distribution of mass of the planetary satellite was studied. The applications were presented for the Europa satellite, considering terms factored by R2, J2, J3, and C22. Diverse analyses were done, using the long-period disturbing potential, obtained through the averaged models, using the unaveraged system and
Acknowledgements
This work was accomplished with support of FAPESP under the Contract No. 2011/05671-5, 2007/04413-7 and 2006/04997-6, SP-Brazil, CAPES and CNPQ (300952/2008-2). A. Elipe acknowledges financial support from the Spanish Government (Project # AYA2008-05572). The authors also would like to thanks Dr. P. Willis and Dr. B. De Saedeleer for their valuables suggestions and corrections that improved the quality of this paper. They did a very detailed review of the present paper and we have incorporated
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