Indirect methods for fuel-minimal rendezvous with a large population of temporarily captured orbiters

Figure 1.  Simulated TCOs orbits from the database calculated in ^[11], plotted in the inertial Earth-Moon (EM)-barycentric reference frame

Figure 2.  The distribution of TCOs capture times less than 3 years

Figure 3.  Distributions of the periapsis distances for all 16,923 temporarily captured orbiter, with respect to the Earth (perigee), Moon (perilune), $ L_1 $ (peri-$ L_1 $), and $ L_2 $ (peri-$ L_2 $)

Figure 4.  Fixed control structure

Figure 5.  $ \Delta v $ (m/s) vs. total thrusting time $ \tau $ (hours)

Figure 6.  Rendezvous points (small white dots) for TCO #1. Also shown are vectors indicating the direction of the velocity of the spacecraft and direction of the Sun for each rendezvous point

Figure 7.  Rendezvous points (in white) for TCOs #1-100 (right). These are still only a small portion of all the rendezvous points $ \mathbf{q}_{rdvz} $ we consider for the 16,923 minimoons

Figure 8.  Example of cloud extremal construction. In plain, our constructed trajectory. In dashed, the resulting cloud extremal. The Earth (left) and Moon (right) are shown as black circles, and the CR3BP equilibrium points $ L_i $ are plotted for reference as x's

Figure 9.  Control structure for constructing cloud extremals

Figure 10.  The Halo orbit discretized into 100 equally spaced (in terms of time) points

Figure 11.  Delta-v distribution for the 12,558 cloud elements

Figure 12.  Transfer time distribution for the 12,558 cloud elements

Figure 13.  Initial Sun true anomaly distribution for the 12,558 cloud elements

Figure 14.  Positions $ \mathbf{s}_{cloud} $ of cloud points $ \mathbf{q}_{cloud} $ for the 12,558 cloud elements, shown at different zoom levels

Figure 15.  Distribution of TCO cloud distances (LD, position only). Left: all TCOs, Right: truncated for distances less than 0.5 LD only

Figure 16.  Distribution of TCO cloud distances (non-dimensional, position and velocity)

Figure 17.  Initial results: Distribution of Δv for 1,338 computed transfers

Figure 18.  Initial results: Rendezvous with TCO #9. The Moon's orbit is shown as the ellipse around the Earth (gray dot). The thin grey path is the orbit of the TCO, starting from its capture point (triangle) to its escape point (square). The circle on the TCO orbit marks where the TCO was when the craft departed, and the star is the point of rendezvous. The dark gray path is the spacecraft's trajectory. It's three boosts are marked as white dots (including the final rendezvous boost)

Figure 19.  Initial results: Rendezvous with minimoon #14. The legend is the same as for Figure 18

Figure 20.  Initial results: Distribution of $ \Delta v $ for the best computed transfers for 93 minimoons

Figure 21.  Initial results: Position components of $ \mathbf{q}_{rdvz} $ versus $ \Delta v $

Figure 22.  Initial results: Velocity components of $ \mathbf{q}_{rdvz} $ versus $ \Delta v $

Figure 23.  Initial results: CR3BP Energy at $ \mathbf{q}_{rdvz} $ versus $ \Delta v $

Figure 24.  Initial results: CR3BP Energy at $ \mathbf{q}_{cloud} $ versus resulting $ \Delta v $

Figure 25.  Initial results: Transfer times versus $ \Delta v $

Figure 26.  Initial results: Continuation distance versus $ \Delta v $

Figure 27.  Initial results: Continuation final Sun true anomaly difference versus $ \Delta v $

Figure 28.  Large scale results: Distribution of $ \Delta v $ for the 3,000 TCOs

Figure 29.  Large scale results: Rendezvous with minimoon #9,046

Figure 30.  Large scale results: Rendezvous with minimoon #16,844

Figure 31.  Original (left) and refined (right) rendezvous with minimoon #5

Figure 32.  Refined results: distribution of $ \Delta v $ improvement for 93 TCOs

Figure 33.  Original (left) and refined (right) rendezvous with TCO #40

Figure 34.  Original (left) and refined (right) rendezvous with TCO #9,046