Reflective-Sail Weak Stability Boundaries

• The paper introduces reflective-sail WSB structures that integrate optimal sail control into the Sun–Earth PCR3BP, transforming a passive stability boundary into an active escape mechanism.
• It derives a closed-form locally optimal pitch law that maximizes the instantaneous growth of Earth-relative Keplerian energy using solar radiation pressure.
• Numerical simulations demonstrate that increasing the sail lightness number reduces stable regions and yields trajectories with shorter time of flight and higher hyperbolic excess velocity.

Reflective-sail weak stability boundary structures are weak stability boundary configurations defined for the Sun–Earth planar circular restricted three-body problem when the spacecraft is an actively controlled ideal reflective solar sail. In this setting, the weak stability boundary is no longer a purely ballistic separatrix in a conservative flow; it becomes a controlled structure in a non-conservative dynamical system, organized by solar radiation pressure and a locally optimal sail-attitude law that maximizes the instantaneous growth of the Keplerian energy with respect to the Earth. The resulting structures identify periapsis initial conditions that remain weakly Earth-bound or become escape-prone over at least one Earth-centered revolution, and they are used to construct Earth-escape trajectories with shorter time of flight and higher estimated hyperbolic excess velocity than ballistic trajectories in the same Sun–Earth PCR3BP model (Fu et al., 26 Jun 2026).

1. Classical WSB geometry and the controlled generalization

Classical weak stability boundary theory was developed for gravity-only restricted three-body dynamics, where it distinguishes initial conditions leading to temporary capture, bounded motion, and escape. In the planar circular restricted three-body problem, a generalized WSB can coincide, under suitable conditions on mass ratio and energy, with a branch of the global stable manifold of the Lyapunov orbit about a Lagrange point; this gives WSB a precise separatrix interpretation rather than treating it as a purely heuristic capture boundary (Belbruno et al., 2012). In the elliptic restricted three-body problem, the same neighborhoods are also characterized by strong finite-time stretching and sensitivity, and can be analyzed through the flow map, the Cauchy–Green tensor, and FTLE/LCS constructions (Nolton et al., 2024).

Reflective-sail WSB structures extend that framework to a controlled, non-conservative system. The extension is not merely a perturbative embellishment of ballistic WSB: the sail continuously alters the Earth-relative energy and therefore changes the stable/unstable partition of periapsis phase space. A plausible implication is that the classical manifold-based intuition remains geometrically useful, but the relevant boundaries are now conditioned by both multi-body gravity and an explicit control law rather than by passive dynamics alone.

2. Controlled Sun–Earth PCR3BP with an ideal reflective sail

The model is the Sun–Earth planar PCR3BP in the rotating frame, with nondimensional units defined by the total Sun–Earth mass, the Sun–Earth distance, and $T_{\mathrm{SE}}/2\pi$, where $T_{\mathrm{SE}}$ is the Sun–Earth orbital period. The mass parameter is

$\mu = 3.0542\times 10^{-6}.$

The rotating-frame state is

$\mathbf{X}=[x,\ y,\ u,\ v]^T,$

with distances to the Sun and Earth

$$r_1=\sqrt{(x+\mu)^2+y^2}, \qquad r_2=\sqrt{(x+\mu-1)^2+y^2}.$$

The translational equations of motion with solar radiation pressure are

$$\begin{aligned} \dot x &= u, \ \dot y &= v, \ \dot u &= x+2v-\frac{(1-\mu)(x+\mu)}{r_1^3}-\frac{\mu(x+\mu-1)}{r_2^3}+a_{\mathrm{SRP}x}, \ \dot v &= y-2u-\frac{(1-\mu)y}{r_1^3}-\frac{\mu y}{r_2^3}+a_{\mathrm{SRP}y}. \end{aligned}$$

For an ideal planar reflective sail,

$$\begin{aligned} a_{\mathrm{SRP}x} &= \frac{\beta(1-\mu)}{r_1^3}\cos^2\alpha\big[(x+\mu)\cos\alpha-y\sin\alpha\big], \ a_{\mathrm{SRP}y} &= \frac{\beta(1-\mu)}{r_1^3}\cos^2\alpha\big[y\cos\alpha+(x+\mu)\sin\alpha\big], \end{aligned}$$

where $\beta\in\{0.01,0.03,0.05\}$ is the sail lightness number and $\alpha\in[-\pi/2,\pi/2]$ is the pitch angle (Fu et al., 26 Jun 2026).

The Earth-relative Keplerian energy is defined as

$$E_2=\frac{1}{2}\left[(u-y)^2+(v+x+\mu-1)^2\right]-\frac{\mu}{r_2}.$$

Its sign retains the usual two-body interpretation: $T_{\mathrm{SE}}$0 corresponds to a bound Earth-relative orbit, whereas $T_{\mathrm{SE}}$1 corresponds to a hyperbolic Earth escape. The sail contribution to the energy derivative is

$T_{\mathrm{SE}}$2

with

$T_{\mathrm{SE}}$3

The control problem is to choose $T_{\mathrm{SE}}$4 at each instant so as to maximize $T_{\mathrm{SE}}$5. Writing $T_{\mathrm{SE}}$6, the maximization reduces to

$T_{\mathrm{SE}}$7

The paper derives a closed-form locally optimal pitch law $T_{\mathrm{SE}}$8, including boundary cases for $T_{\mathrm{SE}}$9. The double-degenerate case $\mu = 3.0542\times 10^{-6}.$0, for which $\mu = 3.0542\times 10^{-6}.$1 for any $\mu = 3.0542\times 10^{-6}.$2, did not occur in the reported simulations. The typical optimal history sweeps from about $\mu = 3.0542\times 10^{-6}.$3 to $\mu = 3.0542\times 10^{-6}.$4 during escape, consistent with classical solar-sail energy-gain strategies (Fu et al., 26 Jun 2026).

3. Levi–Civita regularization about the Earth

Because the Earth terms contain $\mu = 3.0542\times 10^{-6}.$5, near-Earth motion is numerically singular in the original variables. The paper therefore derives a Levi–Civita regularization centered at the Earth. The transformation introduces regularized coordinates $\mu = 3.0542\times 10^{-6}.$6 and fictitious time $\mu = 3.0542\times 10^{-6}.$7: $\mu = 3.0542\times 10^{-6}.$8 This removes the Earth singularity and stretches physical time near $\mu = 3.0542\times 10^{-6}.$9 (Fu et al., 26 Jun 2026).

In regularized form, the equations become

$\mathbf{X}=[x,\ y,\ u,\ v]^T,$0

where $\mathbf{X}=[x,\ y,\ u,\ v]^T,$1, $\mathbf{X}=[x,\ y,\ u,\ v]^T,$2, and $\mathbf{X}=[x,\ y,\ u,\ v]^T,$3 are transformed gravitational and pseudo-force coefficients, and $\mathbf{X}=[x,\ y,\ u,\ v]^T,$4 are the transformed SRP terms. The Jacobi quantity is also propagated through an auxiliary differential equation $\mathbf{X}=[x,\ y,\ u,\ v]^T,$5, since SRP makes it time-varying. The paper gives explicit algebraic expressions for all of these terms.

The regularization is not incidental. Reflective-sail WSB structures are defined from periapsis configurations and first-return tests around Earth, so numerical robustness near the secondary is essential to avoid spurious stable/unstable classification.

4. Definition and numerical construction of reflective-sail WSB structures

The periapsis parameterization follows standard WSB practice. Initial conditions are specified by Earth-relative initial eccentricity $\mathbf{X}=[x,\ y,\ u,\ v]^T,$6, initial periapsis distance $\mathbf{X}=[x,\ y,\ u,\ v]^T,$7, and Earth phase angle $\mathbf{X}=[x,\ y,\ u,\ v]^T,$8. The corresponding initial Earth-relative Keplerian energy is

$\mathbf{X}=[x,\ y,\ u,\ v]^T,$9

so $$r_1=\sqrt{(x+\mu)^2+y^2}, \qquad r_2=\sqrt{(x+\mu-1)^2+y^2}.$$0 implies $$r_1=\sqrt{(x+\mu)^2+y^2}, \qquad r_2=\sqrt{(x+\mu-1)^2+y^2}.$$1, as expected for an elliptic Earth-relative periapsis state (Fu et al., 26 Jun 2026).

A trajectory launched from periapsis has stable motion if it completes at least one revolution around Earth, returns to the same Earth phase line $$r_1=\sqrt{(x+\mu)^2+y^2}, \qquad r_2=\sqrt{(x+\mu-1)^2+y^2}.$$2, and satisfies $$r_1=\sqrt{(x+\mu)^2+y^2}, \qquad r_2=\sqrt{(x+\mu-1)^2+y^2}.$$3 at the first return with

$$r_1=\sqrt{(x+\mu)^2+y^2}, \qquad r_2=\sqrt{(x+\mu-1)^2+y^2}.$$4

Otherwise it is classified as unstable. The 1-stable set is

$$r_1=\sqrt{(x+\mu)^2+y^2}, \qquad r_2=\sqrt{(x+\mu-1)^2+y^2}.$$5

Higher-order stable sets $$r_1=\sqrt{(x+\mu)^2+y^2}, \qquad r_2=\sqrt{(x+\mu-1)^2+y^2}.$$6 are defined recursively, with $$r_1=\sqrt{(x+\mu)^2+y^2}, \qquad r_2=\sqrt{(x+\mu-1)^2+y^2}.$$7 for $$r_1=\sqrt{(x+\mu)^2+y^2}, \qquad r_2=\sqrt{(x+\mu-1)^2+y^2}.$$8. The paper focuses on $$r_1=\sqrt{(x+\mu)^2+y^2}, \qquad r_2=\sqrt{(x+\mu-1)^2+y^2}.$$9. The complement $$\begin{aligned} \dot x &= u, \ \dot y &= v, \ \dot u &= x+2v-\frac{(1-\mu)(x+\mu)}{r_1^3}-\frac{\mu(x+\mu-1)}{r_2^3}+a_{\mathrm{SRP}x}, \ \dot v &= y-2u-\frac{(1-\mu)y}{r_1^3}-\frac{\mu y}{r_2^3}+a_{\mathrm{SRP}y}. \end{aligned}$$0 is the unstable set, and backward-stable sets $$\begin{aligned} \dot x &= u, \ \dot y &= v, \ \dot u &= x+2v-\frac{(1-\mu)(x+\mu)}{r_1^3}-\frac{\mu(x+\mu-1)}{r_2^3}+a_{\mathrm{SRP}x}, \ \dot v &= y-2u-\frac{(1-\mu)y}{r_1^3}-\frac{\mu y}{r_2^3}+a_{\mathrm{SRP}y}. \end{aligned}$$1 are obtained by backward integration (Fu et al., 26 Jun 2026).

For each $$\begin{aligned} \dot x &= u, \ \dot y &= v, \ \dot u &= x+2v-\frac{(1-\mu)(x+\mu)}{r_1^3}-\frac{\mu(x+\mu-1)}{r_2^3}+a_{\mathrm{SRP}x}, \ \dot v &= y-2u-\frac{(1-\mu)y}{r_1^3}-\frac{\mu y}{r_2^3}+a_{\mathrm{SRP}y}. \end{aligned}$$2 and each $$\begin{aligned} \dot x &= u, \ \dot y &= v, \ \dot u &= x+2v-\frac{(1-\mu)(x+\mu)}{r_1^3}-\frac{\mu(x+\mu-1)}{r_2^3}+a_{\mathrm{SRP}x}, \ \dot v &= y-2u-\frac{(1-\mu)y}{r_1^3}-\frac{\mu y}{r_2^3}+a_{\mathrm{SRP}y}. \end{aligned}$$3, the initial grid is

$$\begin{aligned} \dot x &= u, \ \dot y &= v, \ \dot u &= x+2v-\frac{(1-\mu)(x+\mu)}{r_1^3}-\frac{\mu(x+\mu-1)}{r_2^3}+a_{\mathrm{SRP}x}, \ \dot v &= y-2u-\frac{(1-\mu)y}{r_1^3}-\frac{\mu y}{r_2^3}+a_{\mathrm{SRP}y}. \end{aligned}$$4

The regularized equations are integrated forward under $$\begin{aligned} \dot x &= u, \ \dot y &= v, \ \dot u &= x+2v-\frac{(1-\mu)(x+\mu)}{r_1^3}-\frac{\mu(x+\mu-1)}{r_2^3}+a_{\mathrm{SRP}x}, \ \dot v &= y-2u-\frac{(1-\mu)y}{r_1^3}-\frac{\mu y}{r_2^3}+a_{\mathrm{SRP}y}. \end{aligned}$$5, with stopping conditions

$$\begin{aligned} \dot x &= u, \ \dot y &= v, \ \dot u &= x+2v-\frac{(1-\mu)(x+\mu)}{r_1^3}-\frac{\mu(x+\mu-1)}{r_2^3}+a_{\mathrm{SRP}x}, \ \dot v &= y-2u-\frac{(1-\mu)y}{r_1^3}-\frac{\mu y}{r_2^3}+a_{\mathrm{SRP}y}. \end{aligned}$$6

or crossing the line $$\begin{aligned} \dot x &= u, \ \dot y &= v, \ \dot u &= x+2v-\frac{(1-\mu)(x+\mu)}{r_1^3}-\frac{\mu(x+\mu-1)}{r_2^3}+a_{\mathrm{SRP}x}, \ \dot v &= y-2u-\frac{(1-\mu)y}{r_1^3}-\frac{\mu y}{r_2^3}+a_{\mathrm{SRP}y}. \end{aligned}$$7 with $$\begin{aligned} \dot x &= u, \ \dot y &= v, \ \dot u &= x+2v-\frac{(1-\mu)(x+\mu)}{r_1^3}-\frac{\mu(x+\mu-1)}{r_2^3}+a_{\mathrm{SRP}x}, \ \dot v &= y-2u-\frac{(1-\mu)y}{r_1^3}-\frac{\mu y}{r_2^3}+a_{\mathrm{SRP}y}. \end{aligned}$$8, plus a maximum physical integration time $$\begin{aligned} \dot x &= u, \ \dot y &= v, \ \dot u &= x+2v-\frac{(1-\mu)(x+\mu)}{r_1^3}-\frac{\mu(x+\mu-1)}{r_2^3}+a_{\mathrm{SRP}x}, \ \dot v &= y-2u-\frac{(1-\mu)y}{r_1^3}-\frac{\mu y}{r_2^3}+a_{\mathrm{SRP}y}. \end{aligned}$$9.

The computed structures show a systematic shrinkage of stable regions as $$\begin{aligned} a_{\mathrm{SRP}x} &= \frac{\beta(1-\mu)}{r_1^3}\cos^2\alpha\big[(x+\mu)\cos\alpha-y\sin\alpha\big], \ a_{\mathrm{SRP}y} &= \frac{\beta(1-\mu)}{r_1^3}\cos^2\alpha\big[y\cos\alpha+(x+\mu)\sin\alpha\big], \end{aligned}$$0 increases. At $$\begin{aligned} a_{\mathrm{SRP}x} &= \frac{\beta(1-\mu)}{r_1^3}\cos^2\alpha\big[(x+\mu)\cos\alpha-y\sin\alpha\big], \ a_{\mathrm{SRP}y} &= \frac{\beta(1-\mu)}{r_1^3}\cos^2\alpha\big[y\cos\alpha+(x+\mu)\sin\alpha\big], \end{aligned}$$1, the number of stable initial periapses decreases from 583,388 for ballistic PCR3BP $$\begin{aligned} a_{\mathrm{SRP}x} &= \frac{\beta(1-\mu)}{r_1^3}\cos^2\alpha\big[(x+\mu)\cos\alpha-y\sin\alpha\big], \ a_{\mathrm{SRP}y} &= \frac{\beta(1-\mu)}{r_1^3}\cos^2\alpha\big[y\cos\alpha+(x+\mu)\sin\alpha\big], \end{aligned}$$2 to 455,739 for $$\begin{aligned} a_{\mathrm{SRP}x} &= \frac{\beta(1-\mu)}{r_1^3}\cos^2\alpha\big[(x+\mu)\cos\alpha-y\sin\alpha\big], \ a_{\mathrm{SRP}y} &= \frac{\beta(1-\mu)}{r_1^3}\cos^2\alpha\big[y\cos\alpha+(x+\mu)\sin\alpha\big], \end{aligned}$$3, 294,659 for $$\begin{aligned} a_{\mathrm{SRP}x} &= \frac{\beta(1-\mu)}{r_1^3}\cos^2\alpha\big[(x+\mu)\cos\alpha-y\sin\alpha\big], \ a_{\mathrm{SRP}y} &= \frac{\beta(1-\mu)}{r_1^3}\cos^2\alpha\big[y\cos\alpha+(x+\mu)\sin\alpha\big], \end{aligned}$$4, and 237,878 for $$\begin{aligned} a_{\mathrm{SRP}x} &= \frac{\beta(1-\mu)}{r_1^3}\cos^2\alpha\big[(x+\mu)\cos\alpha-y\sin\alpha\big], \ a_{\mathrm{SRP}y} &= \frac{\beta(1-\mu)}{r_1^3}\cos^2\alpha\big[y\cos\alpha+(x+\mu)\sin\alpha\big], \end{aligned}$$5. Similar monotonic decrease occurs for other $$\begin{aligned} a_{\mathrm{SRP}x} &= \frac{\beta(1-\mu)}{r_1^3}\cos^2\alpha\big[(x+\mu)\cos\alpha-y\sin\alpha\big], \ a_{\mathrm{SRP}y} &= \frac{\beta(1-\mu)}{r_1^3}\cos^2\alpha\big[y\cos\alpha+(x+\mu)\sin\alpha\big], \end{aligned}$$6, confirming that the optimally controlled sail generally facilitates Earth escape (Fu et al., 26 Jun 2026).

A common misconception is that the sail facilitates escape everywhere in periapsis phase space. The numerical WSB maps show that this is false. In some sectors, especially in the second quadrant of the $$\begin{aligned} a_{\mathrm{SRP}x} &= \frac{\beta(1-\mu)}{r_1^3}\cos^2\alpha\big[(x+\mu)\cos\alpha-y\sin\alpha\big], \ a_{\mathrm{SRP}y} &= \frac{\beta(1-\mu)}{r_1^3}\cos^2\alpha\big[y\cos\alpha+(x+\mu)\sin\alpha\big], \end{aligned}$$7 plane around Earth, initial conditions that are unstable in the ballistic model become stable under sail control. The controlled SRP can therefore either erode or create bounded regions, depending on local geometry and state.

5. Escape-set synthesis from backward-stable and forward-unstable sets

The paper uses reflective-sail WSB structures to construct Earth-escape trajectories. The key object is the escape set

$$\begin{aligned} a_{\mathrm{SRP}x} &= \frac{\beta(1-\mu)}{r_1^3}\cos^2\alpha\big[(x+\mu)\cos\alpha-y\sin\alpha\big], \ a_{\mathrm{SRP}y} &= \frac{\beta(1-\mu)}{r_1^3}\cos^2\alpha\big[y\cos\alpha+(x+\mu)\sin\alpha\big], \end{aligned}$$8

This is the intersection of the backward 1-stable set in the ballistic Sun–Earth PCR3BP with the forward unstable set under reflective-sail dynamics (Fu et al., 26 Jun 2026).

The logic is operational. Before deployment, the spacecraft follows ballistic dynamics and must not already be escaping in backward time. At periapsis, the sail is deployed and the locally optimal law is activated, so the post-periapsis trajectory must be forward-unstable and leave the Earth. Backward stability in the ballistic problem is computed through the PCR3BP symmetry

$$\begin{aligned} a_{\mathrm{SRP}x} &= \frac{\beta(1-\mu)}{r_1^3}\cos^2\alpha\big[(x+\mu)\cos\alpha-y\sin\alpha\big], \ a_{\mathrm{SRP}y} &= \frac{\beta(1-\mu)}{r_1^3}\cos^2\alpha\big[y\cos\alpha+(x+\mu)\sin\alpha\big], \end{aligned}$$9

which converts backward integration into forward propagation with sign reversal.

Trajectory construction therefore has two pieces. The backward segment is propagated ballistically and retained for $\beta\in\{0.01,0.03,0.05\}$0, corresponding to up to one Earth-centered revolution before periapsis. The forward segment is propagated under sail dynamics until either $\beta\in\{0.01,0.03,0.05\}$1 LU or the trajectory reaches $\beta\in\{0.01,0.03,0.05\}$2 and $\beta\in\{0.01,0.03,0.05\}$3. It is then truncated at the last crossing of a prescribed Earth-centered escape radius

$\beta\in\{0.01,0.03,0.05\}$4

If the first-return instability is detected through $\beta\in\{0.01,0.03,0.05\}$5, integration continues until the escape criteria are met, with a secondary maximum integration time equivalent to $\beta\in\{0.01,0.03,0.05\}$6. Earth-impacting cases $\beta\in\{0.01,0.03,0.05\}$7 are discarded (Fu et al., 26 Jun 2026).

Performance is quantified by

$\beta\in\{0.01,0.03,0.05\}$8

and by the estimated hyperbolic excess velocity at $\beta\in\{0.01,0.03,0.05\}$9,

$\alpha\in[-\pi/2,\pi/2]$0

The analysis retains solutions with $\alpha\in[-\pi/2,\pi/2]$1 days as practically interesting.

6. Performance, interpretation, and broader implications

For all 20 escape sets considered across $\alpha\in[-\pi/2,\pi/2]$2, $\alpha\in[-\pi/2,\pi/2]$3, and the three values of $\alpha\in[-\pi/2,\pi/2]$4, the $\alpha\in[-\pi/2,\pi/2]$5 Pareto fronts shift toward the upper-left as $\alpha\in[-\pi/2,\pi/2]$6 increases: shorter time of flight and higher hyperbolic excess velocity than the ballistic case. The improvement is continuous over the tested sail lightness numbers: $\alpha\in[-\pi/2,\pi/2]$7 is slightly better than ballistic, $\alpha\in[-\pi/2,\pi/2]$8 is better still, and $\alpha\in[-\pi/2,\pi/2]$9 is the best among the examined cases (Fu et al., 26 Jun 2026).

The choice of escape radius affects the measured metrics. For the example $$E_2=\frac{1}{2}\left[(u-y)^2+(v+x+\mu-1)^2\right]-\frac{\mu}{r_2}.$$0, the median TOF and 90th-percentile TOF are 137 and 189 days for $$E_2=\frac{1}{2}\left[(u-y)^2+(v+x+\mu-1)^2\right]-\frac{\mu}{r_2}.$$1 LU, 161 and 292 days for $$E_2=\frac{1}{2}\left[(u-y)^2+(v+x+\mu-1)^2\right]-\frac{\mu}{r_2}.$$2 LU, and 181 and 580 days for $$E_2=\frac{1}{2}\left[(u-y)^2+(v+x+\mu-1)^2\right]-\frac{\mu}{r_2}.$$3 LU. Even so, the qualitative superiority of sail-assisted escape over ballistic escape remains unchanged.

A representative high-$$E_2=\frac{1}{2}\left[(u-y)^2+(v+x+\mu-1)^2\right]-\frac{\mu}{r_2}.$$4 solution is reported for $$E_2=\frac{1}{2}\left[(u-y)^2+(v+x+\mu-1)^2\right]-\frac{\mu}{r_2}.$$5, $$E_2=\frac{1}{2}\left[(u-y)^2+(v+x+\mu-1)^2\right]-\frac{\mu}{r_2}.$$6, and $$E_2=\frac{1}{2}\left[(u-y)^2+(v+x+\mu-1)^2\right]-\frac{\mu}{r_2}.$$7 LU. On the Pareto front, and within a 150-day window, the selected trajectory attains $$E_2=\frac{1}{2}\left[(u-y)^2+(v+x+\mu-1)^2\right]-\frac{\mu}{r_2}.$$8 days and

$$E_2=\frac{1}{2}\left[(u-y)^2+(v+x+\mu-1)^2\right]-\frac{\mu}{r_2}.$$9

at $T_{\mathrm{SE}}$00 LU. Along this trajectory, the backward ballistic segment executes one Earth-centered revolution, then the sail is deployed at periapsis, the locally optimal pitch law is applied, and $T_{\mathrm{SE}}$01 rises from negative to positive, marking the transition from bound Earth-relative motion to escape (Fu et al., 26 Jun 2026).

The broader significance is methodological as much as dynamical. Classical WSB theory has strong invariant-manifold and sensitivity interpretations in gravity-only three-body problems [(Belbruno et al., 2012); (Nolton et al., 2024)]. Reflective-sail WSB structures show that the same organizational role can persist in a controlled non-conservative setting, provided the control is built directly into the definition of stability and instability. This suggests a controlled analogue of WSB geometry: not a passive capture boundary, but a phase-space map of where low-thrust photonic control amplifies or suppresses escape.

The paper also delimits its scope. It treats the Sun–Earth planar PCR3BP with an ideal reflective sail and locally optimal instantaneous energy-growth control. Shadowing, higher-fidelity ephemerides, and additional perturbations are identified as future extensions. A plausible implication is that reflective-sail WSB structures will remain useful as a reduced-order design scaffold even when higher-fidelity models are introduced, because they expose the periapsis regions and control regimes in which escape is naturally favored or unexpectedly inhibited (Fu et al., 26 Jun 2026).