# Design of Ganymede-Synchronous Frozen Orbit around Europa

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## Abstract

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## 1. Introduction

## 2. Dynamic Model

#### 2.1. Dynamical Model of Europa Probe

_{J}, Y

_{J}, Z

_{J}] and Jovian equator are shown in black. The X

_{J}-axis points to the Jupiter equinox, the Z

_{J}-axis is aligned with the Jupiter rotation axis, and the Y

_{J}-axis completes the right-handed frame. The Europa-centric inertial (ECI) coordinate system [X

_{E}, Y

_{E}, Z

_{E}] and Ganymede-centric inertial coordinate system [X

_{G}, Y

_{G}, Z

_{G}], which also represent the revolutions of Europa and Ganymede around Jupiter, are illustrated in red and blue, respectively. The X

_{E}-axis is in the intersection line between the Jovian equator and Europa’s revolution plane, the Z

_{E}-axis is normal to Europa’s revolution plane, and the Y

_{E}-axis satisfies the positively oriented frame. In this case, the angle between X

_{J}and X

_{E}is defined as the longitude of the ascending node Ω

_{E}of Europa’s revolution orbit with respect to Jupiter, and the angle between the fundamental planes of these two systems is the inclination i

_{E}of Europa’s revolution orbit. It should be noted that Europa’s equator and Europa’s revolution plane around Jupiter are assumed as coplanar since the axial tilt between these two planes around Jupiter is 0.0965°. Similarly, the X

_{G}-axis coincides with the intersection between the Jovian equator and Ganymede’s revolution plane, the Z

_{G}-axis is normal to Ganymede’s revolution plane, and the Y

_{G}-axis completes the right-hand frame. The angle between X

_{J}and X

_{G}defines the longitude of the ascending node Ω

_{G}of Ganymede’s revolution orbit around Jupiter. Additionally, the angle between the fundamental planes of these two systems is the inclination i

_{G}of Ganymede’s revolution plane.

_{E}, μ

_{J}, μ

_{G}are the gravitational parameters of Europa, Jupiter, and Ganymede, respectively;

**r**represents the position vector of the probe in an Europa-centric inertial coordinate system, r = ||

**r**||;

**r**

_{JP}and

**r**

_{EJ}are the vectors from Jupiter to the probe and from Europa to Jupiter, r

_{JP}= ||

**r**

_{JP}||, r

_{EJ}= ||

**r**

_{EJ}||, respectively;

**r**

_{GP}and

**r**

_{EG}are the vectors from Ganymede to the probe and from Europa to Ganymede, r

_{GP}= ||

**r**

_{GP}||, r

_{EG}= ||

**r**

_{EG}||, respectively.

_{2}represents the coefficient of Europa’s J

_{2}non-spherical gravitation; R

_{E}represents Europa’s radius; φ is the latitude of the probe regarding to the Europa.

_{EG}(the distance between Europa and Ganymede). Meanwhile, angle β defines the angle between the distances r

_{EJ}(the distance between Europa and Jupiter) and r

_{JG}(the distance between Jupiter and Ganymede). Consequently, distance r

_{GP}, from Ganymede to the probe, is determined by distances r, r

_{JE}, and r

_{JG}, as well as the angles α and β, which can be calculated in a simple trigonometric way as follows:

_{1}and K

_{2}are given in Equations (35)–(39) of Appendix A.

_{G}− f

_{E}. The position vectors in Equations (5)–(8) are derived as follows:

_{EG}oscillates significantly, which makes the mean relative motion of Ganymede around Europa significantly different from that in other cases due to the apparent motion of Jupiter around Europa. The long-term orbit evolution behaviors of the Europa probe under the proposed accurate model and simplified model are investigated in the next section.

#### 2.2. Legendre Expansion

_{EG}in Equation (6).

_{G,a}is given in Equation (16). Considering the magnitudes of distances r and r

_{EG}, the expansion is truncated to zero order, first order, and second order, respectively.

_{G,b}in Equation (12). Therefore, the potential term R

_{G,a,2}dominates the Ganymede gravitation perturbation on the long-term evolution of Europa probe.

_{EG}. From the definition in Equation (6), the distance r

_{EG}is determined by the motions of Europa and Ganymede, whose orbits are not exactly circular and coplanar. In this situation, it is difficult to analytically expand the distance r

_{EG}to high order. Consequently, only the mean motion of the Europa probe is considered rather than the accurate model. Then, applying the simplified model in Equations (10) and (11), both the mean motion of the Europa probe and the mean relative motion of Ganymede around Europa are considered. The Legendre expansion is employed again towards the Europa–Ganymede distance r

_{EG}. Since the magnitudes of Europa’s and Ganymede’s semi-major axes are approximated, the expression of the distance r

_{EG}is truncated up to the fifth order (about 0.1 of order of magnitude) to guarantee the accuracy.

## 3. Long-Term Evolution and Analysis

#### 3.1. Double-Averaging Method

^{4}, while the revolution periods of Jovian and Ganymede’s apparent motions around Europa are about 3 × 10

^{5}s and 6 × 10

^{5}s, respectively. Therefore, the motion of the probe can be considered as a fast variable. Consequently, the secular effect of Ganymede gravitation can be obtained by eliminating the short-period term oscillations with a double-averaging method. The first averaging is performed regarding the mean motion of the probe to obtain the mean variations of the orbital elements in one orbital period:

_{A}represents the periods of Jovian and Ganymede’s apparent motions around Europa. The Jovian apparent motion period T

_{A}equals the revolution period of Europa around Jupiter. The period of Ganymede’s apparent motion T

_{A,G}is the synodic period of the relative motion of Ganymede to Europa as shown in Equation (15), where ${n}_{\mathrm{E}}=\sqrt{{\mu}_{\mathrm{J}}/{a}_{\mathrm{E}}^{3}}$ and ${n}_{\mathrm{G}}=\sqrt{{\mu}_{\mathrm{J}}/{a}_{\mathrm{G}}^{3}}$. It should be noted that the orbit of Ganymede’s apparent motion around Europa is not a conic curve, and hence the right part of Equation (15) is invalid for Ganymede’s mean motion.

#### 3.2. Mean Motion of the Probe around Europa

_{G,a,2}, an accurate Europa–Ganymede distance r

_{EG}is first derived with the averaging method in Equation (14). The average potential of the probe under Ganymede’s gravitation in one orbit period is given as follows:

_{1}and K

_{2}are provided in Appendix A.

_{EG}is time-varying significantly, which makes it difficult to derive Ganymede’s mean apparent motion analytically using the double-averaging method in Equation (18). In this situation, the average potential for the Europa probe under Europa’s non-spherical gravitation and Jovian and Ganymede’s third-body gravitation is given as:

_{E}, i = 93°, Ω = 0°, and ω = 0°, unless otherwise specified in the following simulations. Eccentricity e = 0.1 is selected, and the evolution duration is set as 150 days. The orbital evolutions of the eccentricity, inclination, longitude of the ascending node and argument of periapsis are given in Figure 3. Blue and red curves represent the oscillating and mean orbits, respectively. Note that the semi-major axis remains constant and is not propagated.

#### 3.3. Sensitivity Analysis

_{E}. The average rates of change of the eccentricity and inclination are shown in Figure 4. The initial conditions of a near-circular orbit are considered by setting e = 0.001, ω = 35°. The average rates of change are propagated using the numerical integration method. From Figure 4a, it can be seen that the oscillation amplitude was four orders of magnitude smaller than the value of the rate of change, which suggests that Ganymede’s gravitation perturbation on the eccentricity is weaker compared to those of Europa’s non-spherical gravitation and Jovian gravitation. According to the literature [11,13], the eccentricity evolution mainly depends on the inclination and argument of periapsis.

_{E}determinates the value of the average rate of change. For a Ganymede-synchronous orbit, the initial phase angle is fixed with respect to both Europa and Ganymede for each Ganymede-apparent revolution period. In this situation, the average rate of change of the inclination is constant, which drifts the inclination continuously. Four zero-rate change conditions are obtained when the initial phase angle Ω − u

_{E}approximates −92.5°, −3°, 87.5°, and 177°. These conditions keep the inclination constant under Ganymede’s gravitation with a Ganymede-synchronous orbit.

_{E}are studied. The initial orbital conditions of the probe are e = 0.001 and ω = 35°. The calculation results are shown in Figure 5. Similarly, the average rates of change are propagated using the numerical integral method. According to Figure 5a, for the average rate of change of the longitude of the ascending node, the oscillation amplitude is eight orders of magnitude smaller than the value of the average rate of change. This means that Ganymede’s gravitation has little effect on the longitude of the ascending node. Meanwhile, the literature [11,13] shows that the semi-major axis and inclination determine the rate of change of the longitude of the ascending node when considering Europa’s non-spherical gravitation and Jovian gravitation perturbations. Therefore, to generate the Ganymede-synchronous frozen orbit, it can be assumed that the evolution of the longitude of the ascending node is decoupled with the initial phase angle Ω − u

_{E}to simplify the design of the initial orbital conditions.

_{E}is (−180°, −90°) and (0°, 90°) and negative regions where the initial phase angle Ω − u

_{E}is (−90°, 0°) and (90°, 180°). Between these regions, the contour line of di/dt = 0 deg/sec is obtained (define this line as DIZL). Due to the fact that for the synchronous frozen orbit the initial phase angle Ω − u

_{E}and inclination keep constant, the rate of change of the inclination is fixed. Therefore, the initial condition should locate on the DIZL to achieve the frozen behavior.

_{E}. The initial phase angles Ω − u

_{E}of the DIZL are about −91°, −1°, 89°, and 179°. Therefore, when a near-circular Ganymede-synchronous orbit is considered, the initial argument of periapsis can be chosen arbitrarily.

## 4. Design of a Ganymede-Synchronous Frozen Orbit

#### 4.1. Conditions of Synchronous Frozen Orbit

_{EG}approaches the minimum, the intersection angle of the orbital plane and the Europa–Ganymede vector should be a constant. Consequently, the geometry between the orbit and Ganymede is fixed. This Ganymede-synchronous condition can be described as in Equation (27). The rate of change of the longitude of the ascending node should be equal to the drift rate of the elongation in the inertial coordinate system to achieve Ganymede-synchronous behavior.

#### 4.2. Preliminary Design of a Ganymede-Synchronous Frozen Orbit

Algorithm 1 Design Method for a Ganymede-Synchronous Frozen Orbit |

Input: Semi-major axis a, eccentricity e; Output: Inclination i, initial phase angle Ω − u_{E}, argument of periapsis 1: Assign $o{e}_{0}=\left[a,e\right]$ as the initial conditions; 2: Assign $\frac{d\Omega}{dt}=\frac{du}{dt}$ as the expected rate of change of the longitude of ascending node; 3. Main |

4: Inclination $i\leftarrow \frac{d\Omega}{dt}$; 5: DIZL and DWZL $\left(\frac{di}{dt},\frac{d\omega}{dt}\right)\leftarrow \left(a,e,i\right)$; 6: Initial phase angle and argument of periapsis $\left(\Omega -{u}_{\mathrm{E}},\omega \right)\leftarrow \left(\frac{di}{dt},\frac{d\omega}{dt}\right)$; 7: end |

_{E}, the obtained inclination is given as i = 78.842°.

_{E}is nearly constant when the eccentricity is small. In this situation, initial phase angle Ω − u

_{E}= 87.46° was chosen in the preliminary design phase. This orbit was considered as an expected orbit. The method of maintaining this orbit with the accurate model is discussed in the following section.

#### 4.3. Orbit Maintenance with Accurate Model

_{E}, e = 0.001, i = 78.842°, Ω = 120.7575°, and ω = 133.07°. In order to achieve a collinear behavior of ${u}_{\mathrm{E}}-{e}_{\mathrm{G}}=0\xb0$, the initial date MJD was selected as $\mathrm{MJD}=62504.384619$. The SK period was set as one synodic period of the Europa–Ganymede revolution T

_{A,G}. Fifteen numbers of SK maneuvers were performed to evaluate ∆V consumption. The evolution of the orbit is shown in Figure 11, and ΔV consumptions of the inclination (blue line), longitude of the ascending node (black line), and argument of periapsis (red line) are given in Figure 12.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

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**Figure 3.**Evolution behavior of a probe around Europa, for a = 1.2 R

_{E}, e = 0.1, i = 93°, Ω = 0°, and ω = 0°. Subplots (

**a**–

**d**) are the behaviors of the eccentricity, inclination, longitude of the ascending node and argument of periapsis, respectively.

**Figure 4.**Rates of change of the eccentricity (

**a**) and inclination (

**b**), for a = 1.2 R

_{E}, e = 0.001, i = 93°, and ω = 35°.

**Figure 5.**Rates of change of the longitude of the ascending node (

**a**) and the argument of periapsis (

**b**), for a = 1.2 R

_{E}, e = 0.001, i = 93°, and ω = 35°.

**Figure 6.**Rates of change of the inclination with respect to the eccentricity in subplot (

**a**) and inclination (

**b**) in subplot (

**a**), for a = 1.2 R

_{E}.

**Figure 7.**Rates of change of the inclination with respect to the argument of periapsis for e = 0.1 (

**a**), e = 0.05 (

**b**), e = 0.01 (

**c**), and e = 0.001 (

**d**).

**Figure 11.**Evolutions of orbital elements, for a = 1.2 R

_{E}, e = 0.001, i = 78.842°, Ω = 120.7575°, and ω = 133.07°. Subplots (

**a**–

**d**) show the evolutions of the eccentricity, inclination, longitude of the ascending node and argument of periapsis, respectively.

Parameter | Value |
---|---|

μ_{J} | 126,686,534.9218 km^{3}/s^{2} |

μ_{E} | 3202.74 km^{3}/s^{2} |

μ_{G} | 9887.83 km^{3}/s^{2} |

J_{2,E} | 0.0004355 |

R_{E} | 1560.8 km |

a_{E} | 671,100 km |

e_{E} | 0.0094 |

i_{E} | 0.465° |

a_{G} | 1,070,587.5 km |

e_{G} | 0.00195 |

i_{G} | 0.135° |

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**MDPI and ACS Style**

Huang, X.; Yang, B.; Li, S.; Feng, J.; Masdemont, J.J.
Design of Ganymede-Synchronous Frozen Orbit around Europa. *Mathematics* **2023**, *11*, 41.
https://doi.org/10.3390/math11010041

**AMA Style**

Huang X, Yang B, Li S, Feng J, Masdemont JJ.
Design of Ganymede-Synchronous Frozen Orbit around Europa. *Mathematics*. 2023; 11(1):41.
https://doi.org/10.3390/math11010041

**Chicago/Turabian Style**

Huang, Xuxing, Bin Yang, Shuang Li, Jinglang Feng, and Josep J. Masdemont.
2023. "Design of Ganymede-Synchronous Frozen Orbit around Europa" *Mathematics* 11, no. 1: 41.
https://doi.org/10.3390/math11010041