Trajectory Stability and Signature Diagnostics for Comet-Based Interstellar Navigation

Abstract

Interstellar objects (ISOs) motivate a coupled mission-design and inference question relevant to spacecraft dynamics and control in extreme environments: if volatile-rich, rotating comet-like bodies were used for sustained deep-space navigation by exploiting pre-existing hyperbolic motion and in-situ propellant, what stability requirements arise under non-gravitational forcing, and what astrometric signatures might distinguish active stabilization from uncontrolled natural dynamics? We develop a stability-theoretic framework for trajectory tracking with jet-actuated correction, and show that high-speed transit geometry -- including debris-belt avoidance and encounter phasing -- tightly constrains feasible trajectories, making long-horizon tracking stability mission-critical. We model tracking residuals as the balance of disturbances and corrective action, and derive stability conditions across four levels: disturbance-energy stability, outer-loop contraction, actuator-memory stability, and rotation-mediated (Floquet) stability. The analysis implies residual diagnostics that can motivate empirical tests: under comparable forcing, effective stabilization is expected to strengthen short-horizon error correction, reduce event-conditioned persistence and variance clustering, regularize standardized innovations, and yield bounded post-shock recovery. More broadly, the framework provides a reference for deep-space guidance and control under nonlinear, multi-field disturbances and for planetary-defense concepts involving attitude shaping or impulsive kinetic impact.

Figures (2)

• Figure 1: Schematic geometry linking debris-belt avoidance, node placement, and inner-system vertical offset.(a) Inner-system side view ($x$--$z$, $|x|\le 6\,\mathrm{AU}$). Solid lines compare two slope choices ($7^\circ$ and $5^\circ$) with the node placed near $1.25\,\mathrm{AU}$---a habitable-zone anchor that jointly minimizes encounter range to Earth and Mars when their orbital phases are favorable (Remark \ref{['rem:recon_template']}); the dashed line shows a Sun-centered node for reference. The bracket at Jupiter indicates the reduction in vertical offset from lowering the inner-system slope. (b) Vertical offset magnitude $|z(r)|$ versus heliocentric distance for the same two slopes (node at $\sim 1.25\,\mathrm{AU}$). The shaded band marks an illustrative outer-system corridor ($10$--$45\,\mathrm{AU}$) for degree-scale trajectory shaping, and the bracket at $45\,\mathrm{AU}$ highlights the resulting separation in $|z|$ at Kuiper-belt scale.
• Figure 2: Stability signatures under identical stochastic forcing. Trajectory context: (a) 3D view and (b) ecliptic projection for gravity-only baseline (gray dotted), planned reference (black dashed), passive realization (red; $\mathbf{o}_t\equiv\mathbf{0}$), and active realization (blue; offset correction). The Kuiper belt reference at 50 AU and start/end markers are shown for orientation. Signature panels map to Tests O1--O4: (c) cumulated cross-track error $e_t^{(N)}=s_t^{(N)}-o_t^{(N)}$; (d) ACF of $e_t^{(N)}$; (e) rolling mean of $(u_t^{(N)})^2$ (12-step window) with AR(1) coefficient estimated on quiet-to-quiet segments; (f) event-period QQ-plots for standardized innovations (standardized by quiet-period passive $(\mu_{q,p},\sigma_{q,p})$); (g) normalized recovery metric $\bar{d}_\tau$ from event-peak shock times; (h) event-period KDEs against the quiet baseline. Amber bands denote event windows (elevated disturbance variance). The reference geometry is chosen to be consistent with the debris-avoidance/node-placement setup in Section \ref{['sec:trajectory_geometry']}, but the residual signatures arise from control, not from the trajectory shape.

Theorems & Definitions (21)

• Proposition 1: Minimum avoidance slope for opposite-side belt crossings
• proof
• Remark 1: Scope and margin
• Proposition 2: Encounter range decomposition
• proof
• Proposition 3: Node representation of vertical offset
• proof
• Remark 2: Consequence of entering above and leaving below
• Corollary 1: When slope reduction yields meaningful range reduction
• proof