Dynamics of Artificial Satellites around Europa

Abstract

A planetary satellite of interest at the present moment for the scientific community is Europa, one of the four largest moons of Jupiter. There are some missions planned to visit Europa in the next years, for example, Jupiter Europa Orbiter (JEO, NASA) and Jupiter Icy Moon Explorer (JUICE, ESA). In this paper, we search for orbits around Europa with long lifetimes. Here, we develop the disturbing potential in closed form up to the second order to analyze the effects caused on the orbital elements of an artificial satellite around Europa. The equations of motion are developed in closed form to avoid expansions in power series of the eccentricity and inclination. We found polar orbits with long lifetimes. This type of orbits reduces considerably the maintenance cost of the orbit. We show a formula to calculate the critical inclination of orbits around Europa taking into account the disturbing potential due to the nonspherical shape of the central body and the perturbation of the third body.

1. Introduction

A planetary satellite of interest at the present moment for the scientific community is Europa, one of the four largest moons of Jupiter. There are some missions planned to visit Europa in the next years, for example, Jupiter Europa Orbiter (JEO, NASA) and Jupiter Icy Moon Explorer (JUICE, ESA). The dynamics of orbits around a planetary satellite, taking into account the gravitational attraction of a third body and the nonuniform distribution of the mass of the planetary satellite, has been studied by several authors, for instance, [1–5] including inside-cited references. The search for frozen orbits (orbits that keep the periapsis and the eccentricity almost constant) for planetary satellites is the subject of numerous papers, for instance, [6–8]. A fourth-order analytical theory is presented in [9] for the accurate computation of quasiperiodic frozen orbits.

In this work, we develop the disturbing potential in closed form up to the second order to analyze the effects caused on the orbital elements of the artificial satellite. The equations of motion are developed in closed forms to avoid expansions in power series of the eccentricity and inclination.

The approach for the development of the equations is based on [10–13]. In order to develop the long-period disturbing potential, the averaged method is applied. In this paper, the standard definition for average of periodic functions is applied with respect to eccentric anomaly (𝐸) and true anomaly (𝑓), using known equations from the celestial mechanics [14]. We analyze the disturbing potential effects on some orbital elements, such as eccentricity, inclination, argument of the periapsis, and longitude of the ascending node.

2. Mathematical Model

In this work, we consider the effects caused by the nonsphericity (𝐽₂,𝐽₃) of the central body (Europa) and the perturbation caused by a third body (𝑅₂, i.e., the planet Jupiter, assumed to be in circular orbit in the present case) in artificial satellites around Europa. We present an analytical theory using the averaged model, and applications were done by integrating numerically the analytical equations developed. In [15], the disturbing potential is developed including the action of the inclination of the perturbing body. Here, this inclination is neglected due to the fact that the orbit of Jupiter around Europa has small inclination, and so the disturbing potential is developed taking into account that Jupiter and Europa are on the same plane.

2.1. Disturbing Function

For the model considered in the present paper, it is necessary to calculate the 𝑅₂ term of the disturbing function due to the 𝑃₂ term. The disturbing potential 𝑅₂ can be written in the form [16]: 𝑅₂=𝜉²𝜇_𝐽𝑎_𝐽𝑟𝑎²𝑎_𝐽𝑟_𝐽³𝑃₂(cos𝑆), (1) where 𝑃₂ is the Legendre polynomial, 𝜇_𝐽 is the Jupiter’s gravitational parameter, 𝑟 is the modulus of the radius vector of the artificial satellite, 𝑟_𝐽 is the modulus of the radius vector of Jupiter, and 𝑆 is the angle between radius vectors. The artificial satellite is considered as a point mass particle in a three-dimensional orbit with osculating orbital elements: 𝑎 (semimajor axis), 𝑒 (eccentricity), 𝑖 (inclination), 𝑔 (argument of the periapsis), ℎ (longitude of the ascending node), and 𝑛 (the mean motion).

Here, 𝑎𝜉=𝑎_𝐽, (2) where 𝑎_𝐽 is the semimajor axis of Jupiter.

Using the relation between the angle 𝑆 and the true anomaly 𝑓 of the satellite, we get [11] cos(𝑆)=𝛼cos𝑓+𝛽sin𝑓, (3) where 𝑓 is the true anomaly of the artificial satellite.

Considering the disturbing body in a circular orbit, the coefficients 𝛼 and 𝛽 may be written in the form [11, 12]: 𝛼=cos𝑔cos−ℎ+𝑙_𝐽+cos(𝑖)sin𝑔sin−ℎ+𝑙_𝐽,𝛽=−sin𝑔cos−ℎ+𝑙_𝐽+cos(𝑖)cos𝑔sin−ℎ+𝑙_𝐽, (4) where 𝑙_𝐽 (=171.016 degrees) is the mean anomaly of the disturbing body.

The averaged method is applied to eliminate the short-period terms of the artificial satellite. The standard definition for single averaged is applied with respect to the eccentric anomaly 𝐸. This is done by using known equations from the celestial mechanics, which are 𝑓=√sin1−𝑒²sin(𝐸)(𝐸),𝑓=(𝐸)1−𝑒coscoscos−𝑒(𝐸),𝑟1−𝑒cos𝑎(𝐸),((𝐸=1−𝑒cos𝑑𝑙=1−𝑒cos))𝑑𝐸. (5)

Using the expressions given by (1)–(4), and taking into account the known relationships from the celestial mechanics mentioned above (5), we average the equations according to the eccentric anomalies to eliminate the short-period terms. Thus, we obtain the disturbing potential expanded up to the second order in a small parameter (𝜉).

Note that the average was calculated only to eliminate the terms of short period of the artificial satellite. We consider the orbit of Jupiter as circular and fixed in space, so we do not eliminate the mean anomaly of Jupiter, for which we use the numerical values given at JPL (http://naif.jpl.nasa.gov/naif/index.html).

The long-period disturbing potential (𝑅₂) can be written as 𝑅₂=15𝑛32^𝐽2𝑎²×𝑒²(cos(𝑖)+1)²cos2𝑔+2ℎ−2𝑙_𝐽+𝑒²×(cos(𝑖)−1)²cos2𝑔−2ℎ+2𝑙_𝐽−65×(cos(𝑖)−1)𝑒²+23(cos(𝑖)+1)cos2ℎ−2𝑙_𝐽+−2𝑒²(cos(𝑖))²+2𝑒²+6cos2𝑔5𝑒²+23−13+(cos(𝑖))², (6) where 𝑛_𝐽 (=2.0477×10⁻⁵?rad/s) is the mean motion of the disturbing body.

2.2. Nonsphericity of Europa

To analyze the motion of an artificial satellite around Europa, it is necessary to take into account Europa’s nonsphericity [1, 17–20]. Here, we also develop the equations of motion in closed form, but now the average is applied with respect to the true anomaly.

The zonal perturbation due to the oblateness is [19, 21] 𝑅_𝐽2𝜇=𝜖4𝑟³1−3cos(𝑖)²−3sin²(𝑖)cos2𝑓+2𝑔, (7) where 𝜖=𝐽₂𝑅²_𝐸, 𝜇 (=3202.7?km³/s²) is the gravitational constant of Europa, and 𝑅_𝐸 is the equatorial radius of Europa (𝑅_𝐸=1560.8?km).

The zonal perturbation due to the pear shaped is defined by [19, 21] 𝑅_𝐽3(𝑖)𝜖=−𝜇sin₁8𝑟⁴×3sin𝑓+𝑔−5sin3𝑓+3𝑔+5sin3𝑓+3𝑔×cos²(𝑖)−15sin𝑓+𝑔cos²(𝑖), (8) where 𝜖₁=𝐽₃𝑅³_𝐸.

To eliminate the short-period terms of the potential given by (7) and (8), the averaged method is applied with respect to the true anomaly using known equations from the celestial mechanics, which are 𝑎𝑟=1+𝑒cos𝑓1−𝑒², (9) and we also used the area integral in the form [10]: 1𝑑𝑙=√1−𝑒²𝑟²𝑎²𝑑𝑓. (10)

Using the expressions given by (7)–(10), we average the equations with respect to the true anomaly to eliminate short-period terms. Thus, considering the equatorial plane of Europa as the reference plane, we obtain the disturbing potential (𝑅_𝐽2,𝑅_𝐽3): 𝑅_𝐽21=−4𝜖𝑛²−2+3sin²(𝑖)1−𝑒^23/2,𝑅_𝐽33=−8𝑒𝜖₁𝑛²sin(𝑖)−4+5sin²(𝑖)sin𝑔1−𝑒^25/2𝑎. (11)(12)

As mentioned above, only the harmonic coefficients 𝐽₂ and 𝐽₃ are considered. Their numerical values [22] are given in Table 1. In the case of Europa, there is no high-order model for the spherical harmonics, only 𝐽₂ is measured with some accuracy. The 𝐽₂ term was calculated during the passage of the Galileo spacecraft. The 𝐽₃ term is used by some authors with an approximate value.

Table 1 

Numerical values for 𝐽₂ and 𝐽₃.

3. Results

The disturbing potential of the orbital motion of artificial satellites orbiting Europa taking into account the gravitational attraction of a third body (𝑅₂) and the nonuniform distribution of mass (𝐽₂,𝐽₃) of the planetary satellite can be written in the form: 𝑅=𝑅₂+𝑅_𝐽2+𝑅_𝐽3. (13)

To analyze the effects of the potential on the orbital elements of the orbit of the spacecraft, (13) is replaced by the Lagrange planetary equations [23], and a set of four nonlinear differential equations is numerically integrated using the software Maplesoft. Polar orbits are very important because they allow a better coverage of the surface of Europa when compared to other types of orbits. Orbits around Europa are generally of short duration, around 160 days [2, 3], so a study with a duration of around 300 days covers a time long enough for several types of missions. Figures 1 to 8 show the behavior of a polar orbit using the analytical equations developed. The diagram 𝑒 versus 𝑔 (Figures 1 and 5) shows the regions of libration for several values of the eccentricity.

Figure 1 

Time evolution (300 days) of the eccentricity 𝑒 and the argument of the periapsis 𝑔. Initial conditions: ℎ=90∘ and 𝑔=270∘.

Figures 1 to 4 show the results for orbits with a semimajor axis equal to 2000?km. Figures 5 to 8 show the results for orbits with a semimajor axis equal to 2341?km. The main difference among these figures is that the magnitude of the orbit with a semimajor axis of 2341?km has smaller libration amplitude than the orbit with a semimajor axis of 2000?km as shown in Figures 1 and 5. Performing this search, we found orbits with long lifetimes (frozen orbits, see [6, 8, 19]) that librate around an equilibrium point of 𝑔=270^∘, for example, with inclination 𝑖=90^∘ and a semimajor axis in the interval 2000≤𝑎≤3000?km.

As a result, the behavior of 𝑒 versus 𝑔 librates around an equilibrium point of 𝑔=270^∘ with small amplitudes. The set of initial conditions for orbits with long lifetimes is given by 𝑎=2341?km, 𝑒=0.005, 𝑖=90^∘, 𝑔=270^∘, and ℎ=90^∘. These initial conditions can be used by an Europa’s orbiter to study the planetary satellite with lower cost of station keeping.

Figures 2 and 6 show the diagrams 𝑒 versus 𝑖. Comparing these figures, we see that for the orbit where 𝑎=2341?km, the eccentricity ranged from 0.005 to 0.0064, while when considering 𝑎=2000?km, the eccentricity varies from 0.005 to 0.024. Figures 3 and 7 display the diagrams ℎ versus 𝑖. In this case, the two figures show practically the same behavior. Figures 4 and 8 represent the diagrams 𝑔 versus 𝑖. Note that for the orbit where 𝑎=2341?km, the 𝑔 term ranges from 264^∘ to 276^∘, while for the situation where 𝑎=2000?km, the 𝑔 term varies from 230^∘ to 310^∘. Note that the inclination was well behaved in all cases analyzed, ranging between 82^∘ and 98^∘. Therefore, the orbit with 𝑎=2341?km shows an important characteristic, where the orbital elements have small variations compared with the orbit where 𝑎=2000?km, which may be considered for future missions for artificial satellites around Europa.

Figure 2 

Diagram 𝑒 versus 𝑖. Time evolution 300 days. Initial conditions: ℎ=90∘ and 𝑔=270∘.

Figure 3 

Diagram ℎ versus 𝑖. Time evolution 300 days. Initial conditions: ℎ=90∘ and 𝑔=270∘.

Figure 4 

Diagram 𝑔 versus 𝑖. Time evolution 300 days. Initial conditions: ℎ=90∘ and 𝑔=270∘.

Figure 5 

Time evolution (300 days) of the eccentricity 𝑒 and periapsis 𝑔. Time evolution 300 days. Initial conditions: ℎ=90∘ and 𝑔=270∘.

Figure 6 

Diagram 𝑒 versus 𝑖. Time evolution 300 days. Initial conditions: ℎ=90∘ and 𝑔=270∘.

Figure 7 

Diagram ℎ versus 𝑖. Time evolution 300 days. Initial conditions: ℎ=90∘ and 𝑔=270∘.

Figure 8 

Diagram 𝑔 versus 𝑖. Time evolution 300 days. Initial conditions: ℎ=90∘ and 𝑔=270∘.

Frozen orbits are very important for real missions because they require small amounts of fuel for orbital maintenance. In this situation, the cost is low to keep orbits that pass at the same altitude for a given latitude, benefiting the users with this regularity. In other words, this type of orbit maintains an almost constant altitude over any point on the surface of the central body. Since a satellite in polar orbit with a low altitude (about 100?km) around Europa collides in a short time period [2, 3], it is necessary to look for polar orbits in which the satellite presents a longer lifetime. Indeed, we found a region of initial semimajor axis where the polar orbits survive longer than that found in the literature [2, 3]. A semimajor axis of 2341?km is a good location for a spacecraft because it implies in an altitude around 768?km from the surface of Europa, and the orbit remains for a period of time longer than 300 days, as shown in Figure 5. We found near circular frozen polar orbits whose semimajor axis is in the interval between 2000 and 3000?km. This is high enough to avoid a strong effect from the non-Keplerian terms of the gravitational potential of Europa, but still not too high to cause problems in the observation due to the large distance between the spacecraft and Europa. The region found for the semimajor axis is due to the coupling of the perturbations of Jupiter and the nonspherical shape of the central body. It is worth noting that the orbits found survive for more than 300 days, since they are frozen (see Figure 5).

The dynamics of orbits around planetary satellites, taking into account the gravitational attraction of a third body and the nonuniform distribution of mass of the planetary satellite, was studied in [19]. Lifetimes for these orbits are computed through the single- and double-averaged method. Comparison between the results obtained by the single- and double-averaged method was presented. Reference [19] also shows that the single-averaged model is more realistic than the double-averaged model. Considering the single-averaged method, [19] found unstable polar orbits where the satellite does not impact the surface of Europa for at least 200 days. Here we find, using the single-averaged model, stable polar orbits (frozen) around Europa where the satellite has longer lifetimes. It is important to emphasize that the value of the semimajor axis found here is different from the one presented in [19].

4. Critical Inclination

In [24], a formula to calculate the critical inclination of the case of a lunar orbit is presented taking into account the principal perturbations suffered by a satellite in a low orbit around Europa, which are the terms due to 𝐽₂ and 𝐶₂₂. In the case of a lunar orbit, the perturbation of the third body could be neglected for some analyses (see [6]). In the case of Europa, the disturbing body cannot be neglected in the dynamics because of the strong perturbation of Jupiter, the opposite of a lunar orbit where it is possible to neglect the perturbation of the Earth for some analysis. Then, we consider the two major perturbations for an orbit around Europa taking into account the influence of the third body and the nonuniform distribution of mass of the central body to develop a formula to calculate the critical inclination of these orbits.

The value of the inclination where 𝑑𝑔/𝑑𝑡=0 is called critical inclination. Now, applying the double averaged in (6) to eliminate the mean anomaly of Jupiter to simplify the equations, we find a simplified formula to calculate the critical inclination of orbits around Europa. We get 𝑅_2𝐷𝐴1=−𝑎16²𝑛²_𝐽×2+3𝑒²−9𝑒²cos²(𝑖)−6cos²(𝑖)+15𝑒²cos²(𝑖)cos2𝑔−15𝑒²cos2𝑔. (14)

Considering the effect of the third body (𝑅_2𝐷𝐴) given by (14) and the oblateness of Europa (𝐽₂) given by (11), replacing it in the Lagrange planetary equations and solving the equation 𝑑𝑔/𝑑𝑡=0, we found a formula for the critical inclination. We get cos𝑖_𝑐1=±5×√5−2𝜖𝑛²−𝑎²𝑛²_𝐽+𝑎²cos2𝑔𝑛²_𝐽×−𝑎²𝑛²_𝐽+5𝑎²cos2𝑔𝑛²_𝐽−2𝜖𝑛^21/2×−2𝜖𝑛²−𝑎²𝑛²_𝐽+𝑎²cos2𝑔𝑛²_𝐽⁻¹, (15) and replacing the initial conditions, we obtain?(1)𝑎=1685?km:?(a) in the case of a prograde orbit, the solution found is 𝑖_𝑐=47.8^∘, ?(b) in the case of a retrograde orbit, the solution found is 𝑖_𝑐=132.2^∘, ?(2)𝑎=2000?km:?(a) in the case of a prograde orbit, the solution found is 𝑖_𝑐=43.9^∘, ?(b) in the case of a retrograde orbit, the solution found is 𝑖_𝑐=136.1^∘, ?(3)𝑎=2341?km:?(a) in the case of a prograde orbit, the solution found is 𝑖_𝑐=41.6^∘, ?(b) in the case of a retrograde orbit, the solution found is 𝑖_𝑐=138.4^∘.

Considering a low-altitude orbit, for example, 𝑎=1685?km and neglecting the perturbation of the third body, we obtain the classical result for the critical inclination. Making 𝑛_𝐽=0 in (15), we get 𝑖_𝑐=63.4^∘ and 𝑖_𝑐=116.6^∘. Now neglecting the perturbation due to the nonspherical shape of Europa, we get the classical result for the perturbation of the third body (𝜖=0 in (15)) 𝑖_𝑐=39.2^∘ and 𝑖_𝑐=140.8^∘.

Our results are restricted to the influence of the gravitational attraction of Jupiter and Europa and of the nonuniform distribution of mass of Europa on the motion of artificial satellites with low and medium orbits. The solar gravitational attraction was not considered, and for orbits with large semimajor axis, the influence of the gravitational attraction of the Io and Ganymede satellites must be analyzed.

5. Conclusions

An analytical theory has been developed where the disturbing potential was obtained in closed form to avoid expansions in power series of the eccentricity and inclination. We have considered in the dynamics the influence of the third body (𝑅₂) in a circular orbit and the nonuniform distribution of mass (𝐽₂,𝐽₃) of the planetary satellite. Numerical simulations were performed to analyze several orbits around Europa with the equations developed.

We applied the single-averaged method to eliminate the mean anomaly of the artificial satellite and keep the mean anomaly of Jupiter in the dynamics, considering its orbit fixed in the space. We found a region of a semimajor axis between 2000<𝑎 (km) <3000, where polar orbits around Europa have longer lifetimes. Polar orbits that librate around an equilibrium point for different amplitudes which depends on the semimajor axis were found. As a result, it is shown that the behavior of 𝑒 versus 𝑔 librates around an equilibrium point for 𝑔=270^∘ with small amplitudes. We show two cases. The first one has an orbit with a semimajor axis equal to 2341?km, and it presents less libration amplitudes than the second case, that is, an orbit with 𝑎=2000?km. Integrations were performed by a time period of 300 days. It was found that the coupled perturbations help to control the increase of the eccentricity and to keep the orbits that librate around the equilibrium point.

We show a formula to calculate the critical inclination of orbits around Europa considering the terms 𝑅₂ (third body) and 𝐽₂ (nonsphericity of the central body). The critical inclination for orbit with a semimajor axis equal to 2341?km is 𝑖_𝑐=46.1^∘ for the prograde case and 𝑖_𝑐=138.4^∘ for the retrograde case.

Acknowledgments

The authors are grateful to São Paulo Research Foundation (FAPESP) under the Contract nos. 2011/05671-5, 2011/09310-7, and 2011/08171-3, SP, Brazil, National Council for Scientific and Technological Development (CNPq), Brazil, for Contracts nos. 304700/2009-6 and 3003070/2011-0, and CAPES.