Fast Earth–Moon transfers with ballistic capture

Abstract

This contribution deals with fast Earth–Moon transfers with ballistic capture in the patched three-body model. We compute ensembles of preliminary solutions using a model that takes into account the relative inclination of the orbital planes of the primaries. The ballistic capture orbits around the Moon are obtained relying on the hyperbolic invariant structures associated to the collinear Lagrangian points of the Earth–Moon system, and the Sun–Earth system portion of the transfers are quasi-periodic orbits obtained by a genetic algorithm. The trajectories are designed to be good initial guesses to search optimal cost-efficient short-time Earth–Moon transfers with ballistic capture in more realistic models.

Appendix

We now introduce the sequence of transformations needed to convert states from the EM to the SE system.

Let $t_{\mathit{EM}}$ and $t_{\mathit{SE}}$, respectively, be the time of flight in the EM reference frame and the time of flight in the SE reference frame. First, we perform a clockwise rotation of $t_{1}=t_{\mathit{EM}}+\varphi_{0} ^{\mathit{EM}}$ around the EM $z$-axis and a translation of the origin from the EM barycentre to the Earth position, given by

$$ \begin{aligned} \begin{pmatrix} x_{1} \\ y_{1} \\ z_{1} \end{pmatrix} &= M \times \begin{pmatrix} x_{\mathit{EM}} \\ y_{\mathit{EM}} \\ z_{\mathit{EM}} \end{pmatrix} + (\mu_{\mathit{EM}}) \begin{pmatrix} \cos (t_{1}) \\ \sin (t_{1}) \\ 0 \end{pmatrix} \\ \begin{pmatrix} {\dot{x}}_{1} \\ {\dot{y}}_{1} \\ {\dot{z}}_{1} \end{pmatrix} &= {\dot{M}} \times \begin{pmatrix} x_{\mathit{EM}} \\ y_{\mathit{EM}} \\ z_{\mathit{EM}} \end{pmatrix} + M \times \begin{pmatrix} {\dot{x}}_{\mathit{EM}} \\ {\dot{y}}_{\mathit{EM}} \\ {\dot{z}}_{\mathit{EM}} \end{pmatrix} \\ & \quad {}+ (\mu_{\mathit{EM}}) \begin{pmatrix} -\sin (t_{1}) \\ \cos (t_{1}) \\ 0 \end{pmatrix} , \end{aligned} $$

(4)

where

$$ \begin{aligned} &M = \begin{pmatrix} \cos (t_{1}) & -\sin (t_{1}) & 0 \\ \sin (t_{1}) & \cos (t_{1}) & 0 \\ 0 & 0 & 1 \end{pmatrix} \\ &\text{and} \\ &{\dot{M}} = \begin{pmatrix} -\sin (t_{1}) & -\cos (t_{1}) & 0 \\ \cos (t_{1}) & -\sin (t_{1}) & 0 \\ 0 & 0 & 0 \end{pmatrix} , \end{aligned} $$

(5)

with the subscript $\mathit{EM}$ referring to the EM normalized synodical frame, and the subscript 1 referring to the normalized Earth-centred inertial frame. The dimensionless time $t_{1}$ of the EM system is such that it coincides with the anomaly angle and a complete revolution of the primaries corresponds to $2\pi$.

Then, a rotation around the line of nodes $\hat{N}$ of the angle $i=5.145^{\circ}$ is performed by

$$ \begin{pmatrix} x_{2} \\ y_{2} \\ z_{2} \end{pmatrix} = R \times \begin{pmatrix} x_{1} \\ y_{1} \\ z_{1} \end{pmatrix} \quad \text{and}\quad \begin{pmatrix} {\dot{x}}_{2} \\ {\dot{y}}_{2} \\ {\dot{z}}_{2} \end{pmatrix} = R \times \begin{pmatrix} {\dot{x}}_{1} \\ {\dot{y}}_{1} \\ {\dot{z}}_{1} \end{pmatrix} , $$

(6)

with the direction of the line of nodes given by $\hat{N}=(\cos ( \gamma_{0}),\sin (\gamma_{0}),0)$ in the SE synodical frame, and

$$ R = \begin{pmatrix} r_{11} & r_{12} & r_{13} \\ r_{21} & r_{22} & r_{23} \\ r_{31} & r_{32} & r_{33} \\ \end{pmatrix} $$

(7)

with

$$ \begin{aligned} r_{11} &=\cos (i)+\bigl[1-\cos (i)\bigr] \cos^{2}(\gamma_{0}) \\ r_{12} &=r_{21}=\bigl[1- \cos (i)\bigr]\cos ( \gamma_{0})\sin (\gamma_{0}) \\ r_{13} &=-r_{31}= \sin (i)\sin (\gamma_{0}) \\ r_{22} &= \cos (i)+\bigl[1-\cos (i)\bigr]\sin^{2}( \gamma_{0}) \\ r_{23} &=-r_{32}=-\sin (i)\cos (\gamma_{0}) \\ r_{33} &= \cos (i). \end{aligned} $$

In Eq. (6), the subscript 2 refer to an Earth-centred reference frame that is inclined with respect to the $x_{1}$–$y_{1}$ plane.

A scaling transformation is applied to go from the units of the EM system to the units of the SE system, along with a translation of the origin from the Earth to the barycentre of the SE system:

$$ \begin{aligned} &\begin{pmatrix} x_{3} \\ y_{3} \\ z_{3} \end{pmatrix} = \dfrac{d_{\mathit{EM}}}{d_{\mathit{SE}}} \begin{pmatrix} x_{2} \\ y_{2} \\ z_{2} \end{pmatrix} + (1-\mu_{\mathit{SE}}) \begin{pmatrix} \cos (t_{\mathit{SE}}) \\ \sin (t_{\mathit{SE}}) \\ 0 \end{pmatrix} \\ &\,\,\text{and} \\ &\begin{pmatrix} {\dot{x}}_{3} \\ {\dot{y}}_{3} \\ {\dot{z}}_{3} \end{pmatrix} = \dfrac{d_{\mathit{EM}}\omega_{M}}{d_{\mathit{SE}}\omega_{E}} \begin{pmatrix} {\dot{x}}_{2} \\ {\dot{y}}_{2} \\ {\dot{z}}_{2} \end{pmatrix} + (1- \mu_{\mathit{SE}}) \begin{pmatrix} -\sin (t_{\mathit{SE}}) \\ \cos (t_{\mathit{SE}}) \\ 0 \end{pmatrix} \end{aligned} $$

(8)

The subscript 3 refer to a rescaled reference frame with axis parallel to the axis $x_{2}$, $y_{2}$ and $z_{2}$, but with origin at the barycentre of the SE system. The Moon’s average distance to Earth and the Earth’s average distance to the Sun are denoted by $d_{\mathit{EM}}$ and $d_{\mathit{SE}}$, respectively, and are given in kilometers. The scaling of time is given by $t_{\mathit{SE}}=(t_{\mathit{EM}}/\omega_{M})\omega_{E}$, with $\omega_{M}=2.6617 \times 10^{-6}$ rad/s and $\omega_{E}=1.99095\times 10^{-7}$ rad/s.

Finally, the coordinates and velocities in the SE synodic reference frame are obtained by a rotation of $t_{\mathit{SE}}$ around the SE $z$-axis:

$$ \begin{aligned} &\begin{pmatrix} x_{\mathit{SE}} \\ y_{\mathit{SE}} \\ z_{\mathit{SE}} \end{pmatrix} = T \times \begin{pmatrix} x_{3} \\ y_{3} \\ z_{3} \end{pmatrix} \\ &\text{and} \\ &\begin{pmatrix} {\dot{x}}_{\mathit{SE}} \\ {\dot{y}}_{\mathit{SE}} \\ {\dot{z}}_{\mathit{SE}} \end{pmatrix} = {\dot{T}} \times \begin{pmatrix} x_{3} \\ y_{3} \\ z_{3} \end{pmatrix} + T \times \begin{pmatrix} {\dot{x}}_{3} \\ {\dot{y}}_{3} \\ {\dot{z}}_{3} \end{pmatrix} , \end{aligned} $$

(9)

where

$$ \begin{aligned} &T = \begin{pmatrix} \cos (t_{\mathit{SE}}) & \sin (t_{\mathit{SE}}) & 0 \\ -\sin (t_{\mathit{SE}}) & \cos (t_{\mathit{SE}}) & 0 \\ 0 & 0 & 1 \end{pmatrix} \\ &\text{and} \\ &{\dot{T}} = \begin{pmatrix} -\sin (t_{\mathit{SE}}) & \cos (t_{\mathit{SE}}) & 0 \\ -\cos (t_{\mathit{SE}}) & -\sin (t_{\mathit{SE}}) & 0 \\ 0 & 0 & 0 \end{pmatrix} . \end{aligned} $$

(10)