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Transfer design between neighborhoods of planetary moons in the circular restricted three-body problem: the moon-to-moon analytical transfer method

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Abstract

Given the interest in future space missions devoted to the exploration of key moons in the solar system and that may involve libration point orbits, an efficient design strategy for transfers between moons is introduced that leverages the dynamics in these multi-body systems. The moon-to-moon analytical transfer (MMAT) method is introduced, comprised of a general methodology for transfer design between the vicinities of the moons in any given system within the context of the circular restricted three-body problem, useful regardless of the orbital planes in which the moons reside. A simplified model enables analytical constraints to efficiently determine the feasibility of a transfer between two different moons moving in the vicinity of a common planet. In particular, connections between the periodic orbits of such two different moons are achieved. The strategy is applicable for any type of direct transfers that satisfy the analytical constraints. Case studies are presented for the Jovian and Uranian systems. The transition of the transfers into higher-fidelity ephemeris models confirms the validity of the MMAT method as a fast tool to provide possible transfer options between two consecutive moons.

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Acknowledgements

Assistance from colleagues in the Multi-Body Dynamics Research group at Purdue University is appreciated as is the support from the Purdue University School of Aeronautics and Astronautics and College of Engineering including access to the Rune and Barbara Eliasen Visualization Laboratory. The authors also thank the anonymous reviewers for their thoughtful comments and suggestions. The paper is much improved as a result of their input.

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Correspondence to David Canales.

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Appendices

A Transformations between rotating and inertial frames

1.1 A.1 Transformation from the rotating to the inertial frame

The transformation of a state from the rotating frame to the inertial frame is straightforward. Assuming that the moons move on circular orbits (zero eccentricity) defined in terms of an epoch in the Ecliptic J2000.0 frame, the position (\(\bar{r}\)) and velocity (\(\dot{\bar{r}}\)) of the s/c are produced in the Ecliptic J2000.0 planet-centered inertial frame given \(\varOmega _{moon}\), \(i_{moon}\), \(a_{moon}\) and \(\theta _{moon}\). Note that the moon location is evaluated using \(\theta _{moon}=st+\theta _{0}\), where s is the angular velocity along the moon orbit given the period of the moon, t is the actual time in seconds, and \(\theta _{0}\) is the angle of the moon with respect to the ascending node line at the time of departure (\(t=0\)). Given the moon position (\(\bar{r}_{moon}\)) and velocity (\(\dot{\bar{r}}_{moon}\)) at a time t in the Ecliptic J2000.0 frame, the construction of the planet-centered conic employs \(\bar{h}_{moon}=\frac{\bar{r}_{moon}\times \dot{\bar{r}}_{moon}}{|\bar{r}_{moon}\times \dot{\bar{r}}_{moon}|}\), i.e., the angular momentum of the moon at the given time along the trajectory. The rotation matrix is represented by \(\mathbf{R}=[\hat{x}^T\ \hat{y}^T\ \hat{z}^T]\), such that \(\hat{x}=\frac{\bar{r}_{moon}}{\left| \bar{r}_{moon}\right| }\), \( \hat{z}=\frac{\bar{h}_{moon}}{\left| \bar{h}_{moon}\right| }\) and \(\hat{y}=\hat{z}\times \hat{x}\) define the instantaneous axes of the planet–moon rotating frame. Note that boldface denotes a matrix. Given that the s/c rotating position and velocity states (\(\bar{r}_{rot}\) and \(\dot{\bar{r}}_{rot}\), respectively) are computed relative to the barycenter, \(\mu \) is added to the x-state of the position vector to shift the origin to the planet, but the velocity components remain the same after a translation. Using the basic kinematic equation, the velocity in the inertial frame, expressed in the rotating basis, is computed as \(\dot{\bar{r}}_{in}=\dot{\bar{r}}_{rot}+\bar{s}_{rot}\times \bar{r}_{rot}\), where \(\bar{s}_{rot}=[0\ 0\ s]^T\). As a result, the position and velocity in the inertial basis are defined by \(\bar{r}=\mathbf{R}\bar{r}_{rot}\) and \(\dot{\bar{r}}=\mathbf{R}\dot{\bar{r}}_{in}\).

1.2 A.2 Transformation from the inertial to the rotating frame

The transformation of a state from the Ecliptic J2000.0 planet-centered inertial frame to the rotating frame follows a reverse rotation from the one in Appendix A.1. The rotation matrix is again defined by \(\mathbf{R}\). Given the orbital angular velocity of the moon, \(\bar{s}_{in}=\mathbf{R}\bar{s}_{rot}^T\), the state of the s/c as expressed in the rotating frame is evaluated as \(\bar{r}_{rot}=\mathbf{R}^T\bar{r}\) and \(\dot{\bar{r}}_{rot}=\mathbf{R}^T(\dot{\bar{r}}-\bar{s}_{in}\times \bar{r})\). Finally, to locate the state relative to the barycenter, \(\mu \) of the arrival planet–moon CR3BP system is added to the x position component. Recall that arcs rotated from the Ecliptic J2000.0 planet-centered inertial frame to the arrival rotating frame are scaled with the characteristic quantities of the arrival planet–moon CR3BP system.

Fig. 41
figure 41

Scheme that represents a sample formulation of the corrections problem to determine a transfer between moons in the coupled spatial CR3BP using the \(\tau \)-\(\alpha \) method introduced in Haapala and Howell (2015)

B Differential corrections in the coupled spatial CR3BP

To produce transfers between moons in the coupled spatial CR3BP, a multiple shooter serves as the basis for a differential corrections algorithm. In particular, differential corrections strategies based on multi-variable Newton methods are applied in this investigation to solve boundary value problems. The multiple-shooter scheme is inspired by the \(\tau \)-\(\alpha \) method introduced in Haapala and Howell (2015). This algorithm produces connections between unstable and stable manifold trajectories, allowing the departure or arrival location on the manifold trajectory in the periodic orbit to be free. A sample schematic of the multiple-shooter is illustrated in Fig. 41. Note that, in this example, both departure and arrival manifold trajectories are subdivided into 2 segments, but the number of segments could be any positive integer number. The times \(\tau _{d}\) and \(\tau _{a}\) correspond to a location in position and velocity at which the manifold trajectory departs from or arrives at the periodic orbit with respect to the point where the periodic orbit originates. The \(\alpha \)’s correspond to the time span along each segment that discretizes the unstable and stable manifolds as reflected in Fig. 41. The vectors \(\bar{x} _{PO_{d}}\) and \(\bar{x} _{PO_{a}}\) are the states at which the manifold trajectory departs or arrives at the periodic orbit, respectively. The states along the segments on the unstable manifold trajectory are denoted as \(\bar{x} _{d_{k}}\), where \(k=1, \dots , N_{d}\) and \( N_{d}\) is the number of segments that comprise such a trajectory. Similarly, the states along the segments on the arrival stable manifold trajectory are denoted as \(\bar{x} _{a_{k}}\), where \(k=1, \dots , N_{a}\) and \(N_{a}\) is the number of segments along the arrival trajectory. The variables \(t_{0_{d_k}}\) and \(t_{f_{d_{k}}}\) are the initial and final times corresponding to a departure segment, respectively, and \(t_{0_{a_{k}}}\) and \( t_{f_{a_{k}}}\) are the initial and final times on an arrival segment. Hence, the objective is the determination of a design variable vector, \(\bar{X}\), that satisfies the constraint vector, \(\bar{F}(\bar{X})=\bar{0}\), and ensures position continuity at the intersection between the departure and arrival arcs, but allows a discontinuity in velocity:

$$\begin{aligned} \bar{X}= & {} \left( \begin{array}{c} \tau _{d}\\ \tau _{a}\\ \sum _{k=1}^{N_{d}}\bar{x}_{d_{k}}(t_{f_{d_{k}}})\\ \sum _{k=1}^{N_{d}}\alpha _{d_{k}}\\ \sum _{k=1}^{N_{a}}\bar{x}_{a_{k}}(t_{0_{a_{k}}})\\ \sum _{k=1}^{N_{a}}\alpha _{a_{k}} \end{array}\right) , \end{aligned}$$
(29)
$$\begin{aligned} \bar{F}= & {} \left( \begin{array}{c} x_{a_{1}}(t_{0_{a_{1}}})-x_{d_{N_{d}}}(t_{f_{d_{N_{d}}}})\\ y_{a_{1}}(t_{0_{a_{1}}})-y_{d_{N_{d}}}(t_{f_{d_{N_{d}}}})\\ z_{a_{1}}(t_{0_{a_{1}}})-z_{d_{N_{d}}}(t_{f_{d_{N_{d}}}})\\ \sum _{k=2}^{N_{a}}(\bar{x}_{a_{k}}(t_{0_{a_{k}}})-\bar{x}_{a_{k-1}}(t_{f_{a_{k-1}}}))\\ \sum _{k=2}^{N_{d}}(\bar{x}_{d_{k}}(t_{0_{d_{k}}})-\bar{x}_{d_{k-1}}(t_{f_{d_{k-1}}}))\\ \bar{x}_{d_{1}}(t_{0_{d_{1}}})-\bar{x}_{PO_{d}}\\ \bar{x}_{a_{N_{a}}}(t_{f_{a_{N_{a}}}})-\bar{x}_{PO_{a}} \end{array}\right) =\bar{0}. \end{aligned}$$
(30)

Note that x, y and z in \(\bar{F}\) correspond to the position states in the arrival rotating frame in which position continuity is ensured between the last segment of the unstable manifold trajectory and the first segment of the stable manifold trajectory.

Numerical strategies are required to determine trajectories that satisfy the constraints in Eq. (30). To solve for \(\bar{X}\), a Taylor series (truncated to first order) is expanded on \(\bar{F}(\bar{X})=\bar{0}\) about the free-variable initial guess, \(\bar{X}_0\). Given the relative lengths of the design and constraint vectors, the problem is solved by determining the minimum norm solution that satisfies \(\bar{F}(\bar{X})=\bar{0}\) within a given tolerance. Then, \(\mathbf{DF}=\left[ \frac{\partial F_i}{\partial X_j}\right] \) is the Jacobian matrix and, consequently, it is necessary to solve the problem using an iterative corrections scheme. Since, as previously mentioned, departure trajectories are rotated onto the arrival moon plane, the former become time dependent and the Jacobian matrix is, thus, solved numerically by means of a central difference approximation. The general goal behind central difference is to numerically evaluate differential equations by approximating the slope of the solution at discretized points in time. As a result, each term inside the Jacobian matrix is computed numerically as

$$\begin{aligned} \frac{\partial F_i}{\partial X_j}=\frac{F_i(X_j+h)-F_i(X_j-h)}{2h}, \end{aligned}$$
(31)

where h represents a small perturbation value. To conclude, using the presented methodology, transfers in the coupled spatial CR3BP are successfully corrected given an adequate initial guess obtained from the 2BP-CR3BP patched model.

C Differential corrections for transfers between spatial periodic orbits in the 2BP-CR3BP patched model

The objective is the determination of a design variable vector, \(\bar{X}\), defined by the arrival moon epoch at arrival, \(\theta _{4_{m}}\), the departure conic time-of-flight, \(t_{d}\), and the arrival conic time-of-flight, \(t_{a}\), that satisfies the constraint vector, \(\bar{F}(\bar{X})=\bar{0}\), to ensure position continuity at the intersection between a departure and arrival conic (or departure and arrival plane):

$$\begin{aligned} \bar{F}(\bar{X})=\left\{ \begin{array}{c} x_{d}-x_{a}\\ y_{d}-y_{a}\\ z_{d}-z_{a} \end{array}\right\} =\bar{0},\ \bar{X}=\left\{ \begin{array}{c} \theta _{4_{m}}\\ t_{d}\\ t_{a} \end{array}\right\} . \end{aligned}$$
(32)

To solve for \(\bar{X}\), a Taylor series (truncated to first order) is expanded on \(\bar{F}(\bar{X})=\bar{0}\) about the free-variable initial guess, \(\bar{X}_0\): \(\bar{F}(\bar{X}_0)+\mathbf{DF}(\bar{X}_0)(\bar{X}-\bar{X}_0)=\bar{0}\). The vector \(\bar{X}_0\) is defined with the values \(\theta _{4_{m}}\), \(t_{d}\) and \(t_{a}\) obtained from the projection of \(\sigma \) onto the arrival moon plane (initial guess). The Jacobian matrix \(\mathbf{DF}\) of this particular problem corresponds to:

$$\begin{aligned} \mathbf{DF}=\left[ \begin{array}{c c c} \frac{\partial (x_{d}-x_{a})}{\partial \theta _{4_{m}}}&{} \frac{\partial (x_{d}-x_{a})}{\partial t_d}&{}\frac{\partial (x_{d}-x_{a})}{\partial t_a}\\ \frac{\partial (y_{d}-y_{a})}{\partial \theta _{4_{m}}}&{} \frac{\partial (y_{d}-y_{a})}{\partial t_d}&{}\frac{\partial (y_{d}-y_{a})}{\partial t_a}\\ \frac{\partial (z_{d}-z_{a})}{\partial \theta _{4_{m}}}&{} \frac{\partial (z_{d}-z_{a})}{\partial t_d}&{}\frac{\partial (z_{d}-z_{a})}{\partial t_a} \end{array}\right] . \end{aligned}$$
(33)

Given that the arrival moon trajectories are rotated toward the Ecliptic J2000.0 planet-centered inertial frame, the DF matrix is computed numerically by means of a central difference approximation, which is introduced in Appendix B. Note that this differential corrections scheme is only required to design direct moon-to-moon transfers between spatial periodic orbits in the 2BP-CR3BP patched model.

D Transformations between rotating and inertial frames in a higher-fidelity ephemeris model

For computing the rotations between the inertial and rotating frames in Appendix A, both moon orbits are assumed circular. Nevertheless, in a higher-fidelity ephemeris model, the orbits are no longer circular. Given the moon position (\(\bar{r}_{m}\)) and velocity (\(\dot{\bar{r}}_{m}\)) at a certain epoch t in the Ecliptic J2000.0 frame, an instantaneous rotating frame is constructed in an inertial planet-centered frame using the formulation introduced in Appendix A. Nevertheless, the characteristic time and length ’pulsate’ and this fact must be incorporated when expressing states in dimensionless form, since the position and velocity of the planet and the moon with respect to the barycenter vary over time. Note that the instantaneous characteristic length and time are denoted \(\tilde{l}_*\) and \(\tilde{t}_*\), respectively.

1.1 D.1 Transformation from the inertial to the rotating frame

Given the position (\(\bar{r}_{p-s/c}\)) and velocity (\(\dot{\bar{r}}_{p-s/c}\)) of the s/c with respect to the central body in dimensional units in the inertial frame, the base point is shifted to the barycenter: \(\bar{r}_{B-s/c}=\bar{r}_{p-s/c}+\bar{r}_{B-p}\) and \(\dot{\bar{r}}_{B-s/c}=\dot{\bar{r}}_{p-s/c}+\dot{\bar{r}}_{B-p}\). Here, the subscript ’B’ denotes the barycenter. Given that the planet is the central body, \(\bar{r}_{B-p}=\mu \bar{r}_{p-m}\), where \(\mu \) is the mass ratio for the CR3BP system and \(\bar{r}_{p-m}\) is the location of the moon relative to the planet. Also, \(\dot{\bar{r}}_{B-p}=\mu \dot{\bar{r}}_{p-m}\), where \(\dot{\bar{r}}_{p-m}\) is the velocity of the moon with respect to the planet. The position state of the s/c in the rotating frame is expressed as \(\bar{r}_{rot}=\mathbf{R}^T \bar{r}_{B-s/c}\). Using the basic kinematic equation, the velocity in the rotating frame, expressed in the rotating basis, is computed as \(\dot{\bar{r}}_{rot}=\mathbf{R}^T(\dot{\bar{r}}_{B-s/c}-\frac{1}{\tilde{l}_*^2}\bar{h}_{m}\times \bar{r}_{B-s/c}-\frac{V_r}{\tilde{l}_*}\bar{r}_{B-s/c})\). The component \(V_r\) is the dimensional, instantaneous radial velocity of the moon with respect to the planet: \(V_r=\frac{\bar{r}_{p-m}\cdot \dot{\bar{r}}_{p-m}}{|\bar{r}_{p-m}|}\). Finally, the dimensional states are scaled by the instantaneous characteristic quantities to remove pulsation from the rotating coordinates.

1.2 D.2 Transformation from the rotating to the inertial frame

Given the position (\(\bar{r}_{rot}\)) and velocity (\(\dot{\bar{r}}_{rot}\)) of the s/c with respect to the central body in dimensional units in the rotating frame, and the angular momentum vector in the rotating frame \(\bar{h}_{rot}=\mathbf{R}^T\bar{h}_{m}\), it is possible to compute the velocity in the inertial frame, expressed in the instantaneous rotating basis: \(\dot{\bar{r}}_{in}=\dot{\bar{r}}_{rot}+\frac{1}{\tilde{l}_*^2}\bar{h}_{rot}\times \bar{r}_{rot}+\frac{V_r}{\tilde{l}_*}\bar{r}_{rot}\). Therefore, the position and velocity in the inertial basis are defined by \(\bar{r}_{p-s/c}=\mathbf{R}\bar{r}_{{rot}}-\bar{r}_{B-p}\) and \(\dot{\bar{r}}_{p-s/c}=\mathbf{R}\dot{\bar{r}}_{in}-\dot{\bar{r}}_{B-p}\). Finally, the dimensional states are scaled by the characteristic quantity reflecting the rotating frame in the CR3BP system (not the instantaneous quantity).

Fig. 42
figure 42

Available successful configurations for the intersection at \(\theta _{d_{Int}}\). Note that yellow regions correspond to a phase of departure from Titania in its orbit where a direct transfer to the arrival arc cannot occur

E Feasibility analysis for successful transfers for the application between Titania and Oberon

The feasibility analysis for the transfers between halo orbits near Titania and Oberon is represented in Fig. 42. Note that the analysis is presented for the \(\theta _{d_{Int}}\) configuration because it is the one that supplies the minimum-\(\varDelta v_{tot}\) configuration. The regions that are light yellow correspond to initial departure epochs where a feasible transfer is not encountered. However, it is also apparent in Fig. 43 that the case of \(\theta _{d_{Int}}+\pi \) is also interesting given that it supplies successful configurations for all possible \(\theta _{0_{Tit}}\). For this case, the configuration value is between the upper and lower limits of Eq. (21) (blue and red lines, respectively).

Fig. 43
figure 43

Evaluation of constraint Eq. (21) at \(\theta _{d_{Int}}+\pi \)

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Canales, D., Howell, K.C. & Fantino, E. Transfer design between neighborhoods of planetary moons in the circular restricted three-body problem: the moon-to-moon analytical transfer method. Celest Mech Dyn Astr 133, 36 (2021). https://doi.org/10.1007/s10569-021-10031-x

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