Probe lifetime around natural satellites with obliquity

The dynamics of a probe orbiting a moon can be significantly influenced by the non-coincidence between the moon’s equatorial and orbital planes. Thus, we performed a general analysis about the effects of the angle (obliquity) between the above-mentioned planes and of the angle (nodal phasing) between the nodal lines of the mother planet’s apparent orbit and the probe orbit on the lifetime of the probe. The lifetime, strictly correlated to the variations in eccentricity of the probe orbit, was evaluated starting from low values of the semi-major axis, moderate eccentricity, and high inclination to offer high ground spatial resolution and extend latitudinal coverage of the natural satellite. This investigation, carried out through numerical simulations, may be useful for identifying the optimal initial conditions of the probe’s orbit elements, leading to an important increase in the probe lifetime in missions devoted to the exploration of natural satellites.


Introduction
It is well known that the long-term dynamics of a probe moving around a natural satellite can be studied by performing a double average of the disturbing function. This function is written by considering the main perturbing effects acting on the probe, usually represented by the gravitational attraction of the planet around which the moon orbits (third-body effect), also called the mother planet (hereafter referred to as MP), and the principal gravitational asymmetries of the natural satellite (primary body). This approach, which can retain the "leading terms" related to the variations of the probe orbit elements, can be applied under the hypothesis that these orbit elements remain constant over the time intervals in which the aforementioned averages are taken (the orbital period of the probe and orbital period of the MP in its relative motion with respect to the natural satellite). Such a hypothesis is usually verified with a good level of accuracy, and consequently, the double-averaged approach enables an accurate estimation of the real dynamics of the probe. Concerning this, de Almeida Prado [1] and Broucke [2] described the equations related to the long-term dynamics of a probe under the third-body effect in the case of a circular orbit for the perturbing body. Domingos et al. [3] generalized the same issue to the case of an elliptical orbit for the perturbing body.
However, in the case of probes positioned over highaltitude orbits around moons (although very far from the MP, thus defining a hierarchical triple system) with a very massive MP (such as Jupiter or Saturn), the thirdbody perturbation can be strong enough to increase the amplitude of the periodic oscillations of the orbit elements (short-term effects) that are linked to the instantaneous position of the MP and averaged out (over the thirdbody period). In this case, the hypothesis assuming that the probe orbit elements are constant can induce nonnegligible errors in predicting the real dynamics of the probe, and it is necessary to appropriately modify the classical double-averaged approach [4].
Most of the main moons of the solar system (e.g., Io, Europa, Ganymede, Callisto, Titan, and Enceladus) are characterized by an MP that, in its relative motion with respect to the moon, quasi-lays on the moon's equatorial plane. This implies that the dynamics of the probe is not significantly influenced by the difference between the right ascension of the ascending node (RAAN) of the probe orbit and that of the disturbing body orbit [5], hereafter referred to as nodal phasing. Therefore, in many studies focused on the dynamical properties of a probe orbiting these natural satellites, the hypothesis of co-planarity between the orbital plane of the MP and the moon's equatorial plane has been adopted, thus neglecting the effect of nodal phasing (e.g., Refs. [6][7][8][9]). However, around natural satellites that do not have an orbital plane coinciding with their equatorial plane, the effects of this non-co-planarity (obliquity and nodal phasing) on the dynamics of the probe become important [10,11].
In particular, the lifetime of a probe is critical in missions focused on natural satellites because of possible probe-moon collision. In fact, this element is strictly linked to the variation in the probe's orbit eccentricity due to gravitational attraction of the MP. Thus, the aim of this study is to investigate the influence of obliquity and nodal phasing on the lifetime of a probe orbiting at low altitude (to allow high ground spatial resolution of remotely sensed images) and high inclination (to gain wide latitudinal coverage of the moon) around a natural satellite. The work is organized as follows: Section 2 describes the averaging processes that lead to equations describing the long-term dynamics of a probe under the third-body effect, in the general case of an inclined elliptical orbit for the MP; Section 3 investigates the lifetime of the probe in this general case; and Section 4 summarizes the results of this study.
2 Long-term dynamics under the effect of a distant MP Let us consider a hierarchical triple system formed by a probe, a natural satellite (moon), and its MP. The probe moves around the moon (primary body), which in turn orbits around the MP (disturbing body). For this system, the relative motions of the probe and MP with respect to the moon can be studied in a right-handed X, Y , Z reference system with its origin in the center of mass of the moon, Z axis coincident with the polar axis of the moon, and X and Y pointing in fixed directions on the equatorial plane of the moon. In particular, the X-axis is taken as coincident with the nodal line of the MP orbit so that the RAAN of the MP orbit (Ω d ), assumed to be constant, is equal to zero (Fig. 1). This hypothesis is well verified, considering that the operational life of a probe around a natural satellite is usually far smaller than the rotational period of the nodal line of the perturbing body. Consequently, hereafter, the nodal phasing is intended to be ∆Ω = Ω − Ω d = Ω, with Ω = RAAN of the probe orbit (angle, measured in the equatorial plane of the moon, between the X-axis and the nodal line of the probe orbit). In this reference system, assuming µ d as the planetary constant of the MP and S as the angle between the position vector of the probe  • the MP orbit is outer with respect to the probe orbit; • the probe-moon distance (r) is far less than the MPmoon distance (d), so that the traditional expansion in Legendre polynomials can be truncated at second order. Then, the disturbing function (U d ) owing to the gravitational attraction of the MP can be written as In Eq. (2), f and f d are the true anomalies of the probe and MP, respectively, and − cos ω cos Ω sin ω d (1) with respect to the positions of both the probe (over its orbital period) and MP (over the orbital period of the MP in its apparent motion with respect to the moon), then introducing this double-averaged potential into the Lagrange planetary equations (LPE). The first average, performed with respect to the orbital period of the probe (T ), can be solved analytically using the eccentric anomaly of the probe (E). To this end, we use the following relationships: (1) as a function of E, and thus we obtain where a and a d are the semi-major axes of the orbits of the probe and MP, respectively; e and e d are the eccentricities of the orbits of the probe and MP, respectively; and M is the mean anomaly of the probe. The second average, performed with respect to the orbital period of the MP (T d ), can be computed analytically by considering the true anomaly of the MP. In this case, we have where M d denotes the mean anomaly of MP. Equation (4) indicates that the double-averaged disturbing function is composed of a part that is independent of the eccentricity of the probe and a part that depends on e 2 . By replacing Eq. (4) in the LPE, the equations governing the longterm variations of eccentricity, inclination, RAAN, and argument of the pericenter of the probe orbit can be expressed as Eq. (5): − 20 cos(2i) sin 2 ω] where n is the probe's mean motion and cot α = 1/ tan α = cos α/ sin α. Equations (5) are mutually matched. By integrating this system of equations numerically, step by step, it is possible to determine the long-term variations of the probe orbit elements. Having averaged out the mean anomaly of the probe implies that the semi-major axis is constant, on average. This can easily be highlighted by observing the form of the LPE or, equivalently, by expressing the perturbation equations in the canonical form using the Delaunay elements.
When i d = 0 • , the RAAN of the perturbing body is not defined, and the problem is axisymmetric with respect to the polar axis of the moon. The RAAN of the probe, which has no influence on its dynamics, represents a cyclic variable, leading to the conservation of its conjugate momentum, the Z-component of the angular momentum µa(1 − e 2 ) cos i = constant [2]. When the temporal evolution of the RAAN of the probe orbit is small in the time interval of interest, a fixed mean value for this orbit element may be considered in Eq. (5) to find a first-guess solution for the secular dynamics of the probe. Imposing the same condition (for example, Ω = 0) in the disturbing function (Eq. (4)) would theoretically represent an approximation (that implies only a modification of the equation related to the variation in inclination). In fact, as pointed out by Naoz et al. [12], the elimination of the RAAN in the disturbing function (or in the Hamiltonian) would mean that the Z-component of the angular momentum of the probe is a constant, whereas this should occur only for i d = 0 • .

Effects of the obliquity and nodal phasing
The lifetime of a probe is a key element of missions devoted to the observation of moons, and this element can be directly correlated to the evolution of the orbit eccentricity (as probe-moon collision occurs when the altitude of the pericenter of the probe orbit equals the radius of the moon). For this reason, we performed a numerical investigation into this issue, considering the general case of an inclined elliptical orbit for the MP (Eq. (5)) and adding the well-known secular variations of RAAN and ω due to the moon's oblateness (J 2 is the first zonal harmonic of the moon's gravitational field, and R P is the mean equatorial radius of the moon): · sin i d sin Ω(5e 2 cos(2ω) + 3e 2 + 2) − 20 cos(2i) sin 2 (ω)] A broad range of numerical simulations have been executed considering the Jupiter-Europa system. Accordingly, Figs. 2-5 report the lifetime of a probe in a low-altitude and high-inclination orbit, obtained by integrating the system composed of Eq. (6). However, to generalize the results and provide references for practical applications, fictitious values for Jupiter's orbit inclination were also considered: i d = 0 • (Fig. 2, which is close to the real value); i d = 30 • (Fig. 3); i d = 60 • (Fig. 4); and i d = 90 • (Fig. 5). With regard to the probe orbit elements, the following initial conditions were implemented: a = 1.1R P , e = 0.01 and i = 65 • , 75 • , 85 • , and 95 • . Regarding the nodal phasing and the argument of the pericenter, the simulations were extended to their entire definition ranges (for all figures, Ω d = 0 • , so that ∆Ω = Ω). For the other parameters involved, the following data provided by the Jet Propulsion Laboratory (JPL) of NASA [13] were implemented: a d = 671,100 km, Fig. 2 Lifetime of a probe around Europa, assuming that Jupiter's relative orbit is on the equatorial plane of Europa.    The strips are horizontal, to highlight the noninfluence of the nodal phasing, and become thinner as the probe's orbit inclination increases. Paskowitz and Scheeres [8] and Cinelli et al. [5] highlighted how the probe lifetime around Europa is strongly influenced by the initial value of the argument of the pericenter of the probe orbit. In particular, in Ref. [5], the two narrow strips were not perfectly horizontal, but oscillated around the values reported above as a function of the nodal phasing (Ω). This is because of the non-perfect co-planarity between Jupiter's relative orbit and Europa's equatorial plane, considered in that work. In the same work, the effect of the ellipticity of Europa equator was evaluated (C 22 harmonic), showing how it increases the amplitude of the oscillations of the strips.
In Fig. 3 (i d = 30 • ), the effect of nodal phasing becomes significant, and consequently, the narrow strips oscillate significantly as a function of Ω. In particular, at i = 65 • , the strips oscillate over a wider range of values of ω and are more asymmetric with respect to the cases at higher inclination. The longest lifetime increases considerably with respect to the case at i d = 0 • , especially for i = 65 • , where they exceed 400 terrestrial days.
In Fig. 4 (i d = 60 • ), the influence of the nodal phasing is strong, and the strips are considerably asymmetric. At i = 65 • and 75 • , for any ω, it is possible to find an appropriate value of Ω that maximizes the lifetime. The longest lifetime exceeds 300 terrestrial days. Vertical strips at high lifetime are also present (at approximately Ω = 0 • -30 • and Ω = 150 • -180 • ) to highlight the important dependence on Ω.
In Fig. 5 (i d = 90 • ), the longest lifetime is very high (over 500 terrestrial days), reaching a maximum for values of Ω around 0 • and 180 • (vertical strips). All cases present these two vertical strips. Considering the Jupiter-Europa system with i d = 90 • and including the J 2 effect, Fig. 6 presents the case of a = 1.2R P and e = 0.1. Here, the strips are not perfectly vertical, and the longest lifetime is even longer than 5 terrestrial years.
The variation in eccentricity is directly correlated to the third-body perturbation (Eq. (5)), while the moon's oblateness, causing secular variations in RAAN and argument of pericenter, has an indirect effect on its temporal evolution. Thus, to highlight the contribution of the gravitational attraction of Jupiter to the probe lifetime, Europa's oblateness effect was also removed. Accordingly, Fig. 7 shows the lifetime obtained by  Fig. 7, there are two horizontal narrow strips, separated by an interval of 180 • , that provide the longest lifetime (now located around ω = 143 • and 323 • ). This case is related to the phase diagram shown in Fig. 2 in Ref. [2]. In this diagram, the level curves obtained by exploiting the two prime integrals of motion for this axisymmetric problem are reported (the disturbing function is constant in the averaged problem, and the Z-component of the angular momentum is constant). The values ω = 143 • and 323 • are, indeed, in the (e cos ω, e sin ω) plane, very close to the dynamical separatrix that divides the diagram into libration and circulation regions for the argument of the pericenter. The same level curves can also be found in Fig. 1 in Ref. [14] (where the Mercury-Sun system was considered). In that work, the level curves were reported in the (e, ω) plane for i = 90 • . The temporal evolutions of eccentricity and argument of the pericenter have also been described in Ref. [5], considering both the non-perfect co-planarity between Jupiter's relative orbit and Europa's equatorial plane, and the effects of the second-order and degree gravity field of Europa.
The results for i d = 0 • and 30 • in Fig. 7 (upper plots) are similar to those including the effect of Europa's oblateness, with the central values of the strips varying slightly. For i d = 60 • , the differences are more marked. In fact, the lifetime extends over almost five years, and a vertical strip appears around Ω = 180 • (the other strips are not symmetric with respect to it). For i d = 90 • , the results are still similar to those with Europa's oblateness, but the lifetime is longer, and the strips around Ω = 0 • and 180 • are far larger. Additional numerical simulations were performed considering the Saturn-Enceladus system. The system of Eq. (6) was integrated, adopting fictitious values for the third-body inclination to generalize the results and provide references for real applications. For Enceladus, the data reported in Ref. [15] were used (J 2 = 5.4352 × 10 −3 ).   As the evidence shows, although the lifetime is obviously different, from a dynamical point of view, the results are qualitatively the same as those related to the Jupiter-Europa system. In addition, note that, as in the case of a probe orbiting Europa [5], inside all the horizontal and oscillating strips presented in this section, many specific values of Ω at the longest lifetime were obtained.

Conclusions
This study investigated the lifetime of a probe in a low-altitude and high-inclination orbit around a natural satellite. This issue was addressed by considering the moon's oblateness and the disturbing attraction of the mother planet in the gravity-gradient approximation (second-order approximation for the thirdbody disturbing function), considering the general case of an inclined elliptical orbit for the disturbing body. Numerical simulations highlighted how the lifetime of the probe is influenced by both the third-body inclination (obliquity) and the angle (nodal phasing) between the nodal lines of the mother planet and probe orbits. The influence of the nodal phasing is moderate for low inclinations (i d ) of the mother planet orbit, becoming significant at i d = 30 • and strong at i d = 60 • and over. In the (Ω, ω) plane, the longest lifetime is located inside strips, and the absolute maximum lifetime occurs when the mother planet's orbit is perpendicular to the equatorial plane of the natural satellite, with values of nodal phasing close to 0 • and 180 • (vertical strips).

Funding note
Open access funding provided by Sapienza University of Rome within the CRUI-CARE Agreement.