Efficient design of direct low-energy transfers in multi-moon systems

Abstract

In this contribution, an efficient technique to design direct (i.e., without intermediate flybys) low-energy trajectories in multi-moon systems is presented. The method relies on analytical two-body approximations of trajectories originating from the stable and unstable invariant manifolds of two coupled circular restricted three-body problems. We provide a means to perform very fast and accurate computations of the minimum-cost trajectories between two moons. Eventually, we validate the methodology by comparison with numerical integrations in the three-body problem. Motivated by the growing interest in the robotic exploration of the Jovian system, which has given rise to numerous studies and mission proposals, we apply the method to the design of minimum-cost low-energy direct trajectories between Galilean moons, and the case study is that of Ganymede and Europa.

Appendices

Appendix 1

In the following, we derive the relationship between two planet-centered ellipses generated by one and the same state vector on the CI at two different orbital phases of the moon. Let $\mathbf{s} = (x,y,{\dot{x}},{\dot{y}})^T$ be a state vector on the CI in synodical coordinates. Denote by $\mathbf{S}_1=(X_1,Y_1, {\dot{X}}_1, {\dot{Y}}_1)^T$ and $\mathbf{S}_2=(X_2,Y_2, {\dot{X}}_2, {\dot{Y}}_2)^T$ two state vectors in the planet-centered frame obtained by $\mathbf{s}$ when the orbital phases of the moon (i.e., the planet-centered angles from the X-axis to the location of the moon) are $\alpha _1$ and $\alpha _2$, respectively. Also, let $\varSigma _1$ and $\varSigma _2$ be the two elliptical orbits, with focus at the planet, passing through $\mathbf{S}_1$ and $\mathbf{S}_2$, respectively. In Remark 1 it has been stated that $\varSigma _1$ and $\varSigma _2$ have the same shape (i.e., they have the same semimajor axis and eccentricity) and are related by a rotation of angle $\varDelta \alpha =\alpha _2-\alpha _1$, i.e., $\omega _2=\omega _1 + \varDelta \alpha$. We now provide a proof of the claim. Denoting $R(\alpha )=\left[ \begin{array}{rr} \cos \alpha &{} -\sin \alpha \\ \sin \alpha &{} \cos \alpha \end{array}\right]$ the rotation matrix of angle $\alpha$, we show that

• i) $(X_2,Y_2)^T = R(\varDelta \alpha )(X_1,Y_1)^T$

• ii) $({\dot{X}}_2,{\dot{Y}}_2)^T = R(\varDelta \alpha )({\dot{X}}_1,{\dot{Y}}_1)^T$.

i) Let us consider the trajectory passing through $\mathbf{s}$ at time $t=T$ in the rotating frame. In the same time units in the planet-centered frame, the trajectory is given by

$$\begin{aligned} \left[ \begin{array}{c} X\\ Y \end{array}\right] (t)=\kappa R(\beta (t))\left[ \begin{array}{c} x\\ y \end{array}\right] (t), \end{aligned}$$

(9)

where the constant $\kappa$ is the scaling factor from normalized to physical units and $\beta (t)=\beta +t$, being $\beta$ the relative phase of the rotating frame with respect to the planet-centered frame at time $t=0$. If at time $t=T$ the orbital phase of the moon is $\alpha _i$ (i = 1,2), then the phase at time $t=0$ is $\beta _i=\alpha _i-T$. Hence, in the planet-centered frame we consider two trajectories

$$\begin{aligned} \left[ \begin{array}{c} X_1\\ Y_1 \end{array}\right] (t)=\kappa R(\beta _1+t)\left[ \begin{array}{c} x\\ y \end{array}\right] (t), \end{aligned}$$

(10)

and

$$\begin{aligned} \left[ \begin{array}{c} X_2\\ Y_2 \end{array}\right] (t)=\kappa R(\beta _2+t)\left[ \begin{array}{c} x\\ y \end{array}\right] (t). \end{aligned}$$

(11)

Since the rotations form a group, $R(\beta _2+t)=R(\beta _1+t+\varDelta \alpha )=R(\varDelta \alpha )R(\beta _1+t)$. Thus,

$$\begin{aligned} \left[ \begin{array}{c} X_2\\ Y_2 \end{array}\right] (t)=R(\varDelta \alpha ) \left[ \begin{array}{c} X_1\\ Y_1 \end{array}\right] (t). \end{aligned}$$

(12)

The last expression, evaluated at $t=T$, gives i). Differentiation and time units rescaling yield ii). This proof neglects the distance between the center of mass of the planet-moon system and the center of the planet. In the cases under study this distance does not exceed a few tens of km.

Appendix 2

This appendix contains the algebraic computation of the intersection between two ellipses with one common focus. This represents the general, non-degenerate situation of the problem dealt with in this paper. Let us assume that the two Keplerian orbits are non-identical ellipses with $e_1,e_2 \ne 0$.¹ The point(s) of intersection between the two curves are obtained by developing the condition expressed in Eq. 3 with Eq. 4:

$$\begin{aligned} p_1 + p_1e_2 \cos (\theta _1 - \varDelta \omega ) = p_2 + p_2 e_1 \cos \theta _1. \end{aligned}$$

(13)

Here $p_i = a_i(1-e_i^2)$ ($i=1,2$) is the semilatus rectum of the ellipse. Substitution of $\cos (\theta _1 - \varDelta \omega )$ with $\cos \theta _1 \cos \varDelta \omega + \sin \theta _1 \sin \varDelta \omega$ provides

$$\begin{aligned} a + b\cos \theta _1 = c \sin \theta _1, \end{aligned}$$

(14)

where $a = p_1-p_2$, $b = p_1 e_2 \cos \varDelta \omega - p_2 e_1$ and $c= -p_1 e_2 \sin \varDelta \omega$. Taking the square of Eq. 14 yields

$$\begin{aligned} (b^2+ c^2) \cos ^2 \theta _1 + 2ab \cos \theta _1 + a^2-c^2 = 0, \end{aligned}$$

(15)

or

$$\begin{aligned} k_1 \cos ^2 \theta _1 + 2k_2 \cos \theta _1 + k_3 = 0, \end{aligned}$$

(16)

with $k_1 = b^2 + c^2$, $k_2 = ab$, $k_3 = a^2 - c^2$. Equation 16 is a second-degree algebraic equation in $\cos \theta _1$. It has real solutions if and only if

$$\begin{aligned} \lambda = k_2^2 - k_1k_3 \ge 0. \end{aligned}$$

(17)

When $\lambda > 0$, there are two distinct intersections, A and B (see Fig. 8 right). Their true anomalies $\theta _{1A}$ and $\theta _{1B}$ in the perifocal reference frame of ellipse 1 are given by

$$\begin{aligned} \cos \theta _{1A}= & {} \frac{-k_2 + \sqrt{\lambda }}{k_1}, \end{aligned}$$

(18)

$$\begin{aligned} \sin \theta _{1A}= & {} \frac{a + b\cos \theta _{1A}}{c}, \end{aligned}$$

(19)

$$\begin{aligned} \cos \theta _{1B}= & {} \frac{-k_2 - \sqrt{\lambda }}{k_1}, \end{aligned}$$

(20)

$$\begin{aligned} \sin \theta _{1B}= & {} \frac{a + b\cos \theta _{1B}}{c}. \end{aligned}$$

(21)

Equations 19 and  21 become indefinite if $c=0$ (the two apse lines either coincide, $\varDelta \omega = 0$, or are oppositely oriented, $\varDelta \omega = 180^{\circ }$). In this case, Eq. 14 becomes

$$\begin{aligned} a + b\cos \theta _1 = 0, \end{aligned}$$

(22)

which provides

$$\begin{aligned} \cos \theta _{1A} = \cos \theta _{1B}= -a/b, \end{aligned}$$

(23)

with

$$\begin{aligned} \sin \theta _{1A}= & {} \sqrt{1-\cos ^2 \theta _{1A}} \end{aligned}$$

(24)

$$\begin{aligned} \sin \theta _{1B}= & {} -\sin \theta _{1A}. \end{aligned}$$

(25)

Eventually, once $\theta _{2A}$ and $\theta _{2B}$ have been determined by either Eqs. 18–21 or Eqs. 23–25, Eq. 4 yields the values $\theta _{2A}$ and $\theta _{2B}$ of the true anomaly of A and B in the perifocal reference frame of ellipse 2. When $\lambda = 0$, A and B coincide, implying that the two ellipses are tangent to each other (see Fig. 8 center). Eq. 17 can be solved for the two values of $\varDelta \omega$ for which this occurs:

$$\begin{aligned} \cos \varDelta \omega = \displaystyle \frac{p_1^2e_2^2 + p_2^2e_1^2 - (p_1-p_2)^2}{2e_1e_2p_1p_2}. \end{aligned}$$

(26)

Inverting the cosine yields the quantities $\tau$ and $2\pi -\tau$ defined previously (Eq. 8).