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A Minimal Four-Thruster System for Comet-Based Interstellar Navigation

Bo Pieter Johannes Andrée

Abstract

Interstellar comets arrive with key ingredients for deep-space platforms already in place: volatile inventories convertible to propellant, natural rotation providing continuous attitude variation, and hyperbolic trajectories that carry them through the inner Solar System and back out to interstellar space. Rather than constructing spacecraft from scratch, we ask what \emph{minimal modification} is required to steer such a body along a controlled trajectory. The answer is surprisingly modest. By relaxing full six-degree-of-freedom control to forward-cone steering -- sufficient for practical navigation -- we show that \emph{four thrusters suffice}: one primary jet and three secondary jets at $120^\circ$ intervals. The secondary jets synthesize continuous in-plane steering, while the primary jet provides low-bandwidth attitude shaping: as the body rotates, the primary-jet torque direction sweeps predictably over a cycle, enabling out-of-plane steering via phase-scheduled firing. We formalize reachability under bounded-curvature constraints, characterize the rotation-mediated steering envelope, discuss enabling requirements including non-solar power at large heliocentric distances, and identify operational regimes and observable signatures implied by active trajectory control. The setting of a nutating axis is briefly considered and conjectured to preserve core results. The findings contribute to the broader effort of understanding the dynamics and control of small-body missions and offer a reference architecture relevant to long-horizon deep-space exploration and to potential planetary-defense concepts.

A Minimal Four-Thruster System for Comet-Based Interstellar Navigation

Abstract

Interstellar comets arrive with key ingredients for deep-space platforms already in place: volatile inventories convertible to propellant, natural rotation providing continuous attitude variation, and hyperbolic trajectories that carry them through the inner Solar System and back out to interstellar space. Rather than constructing spacecraft from scratch, we ask what \emph{minimal modification} is required to steer such a body along a controlled trajectory. The answer is surprisingly modest. By relaxing full six-degree-of-freedom control to forward-cone steering -- sufficient for practical navigation -- we show that \emph{four thrusters suffice}: one primary jet and three secondary jets at intervals. The secondary jets synthesize continuous in-plane steering, while the primary jet provides low-bandwidth attitude shaping: as the body rotates, the primary-jet torque direction sweeps predictably over a cycle, enabling out-of-plane steering via phase-scheduled firing. We formalize reachability under bounded-curvature constraints, characterize the rotation-mediated steering envelope, discuss enabling requirements including non-solar power at large heliocentric distances, and identify operational regimes and observable signatures implied by active trajectory control. The setting of a nutating axis is briefly considered and conjectured to preserve core results. The findings contribute to the broader effort of understanding the dynamics and control of small-body missions and offer a reference architecture relevant to long-horizon deep-space exploration and to potential planetary-defense concepts.
Paper Structure (58 sections, 21 theorems, 59 equations, 4 figures, 4 tables)

This paper contains 58 sections, 21 theorems, 59 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Motivation: Comets as Interstellar Vehicles
  3. Scope of minimality.
  4. Theoretical Framework
  5. Frames and kinematics
  6. Body frame $\mathcal{B}$ (geometry).
  7. Trajectory frame $\mathcal{I}$ (navigation).
  8. Rotation map and inertial force.
  9. Steady-spin approximation.
  10. Reachability: Formal Definition
  11. The Four-Thruster Configuration
  12. Operating Regimes: Cruise versus Encounter
  13. In-Plane Control: The 120$^\circ$ Placement Theorem
  14. Optimality.
  15. Out-of-Plane Control: The Large Thruster Requirement
  16. ...and 43 more sections

Key Result

Theorem 1

Let $\{\mathbf d_i\}_{i=1}^n\subset\mathbb R^2$ be unit thrust directions and let $u_i\ge 0$ (unidirectional thrust). Define the achievable set i.e., the conic hull of the directions. Then:

Figures (4)

  • Figure 1: Directional force coverage from nonnegative jet combinations. Blue arrows: individual jet directions; gray arrow: example resultant. Green: reachable directions; red: unreachable. Two jets at $90^\circ$ yield a wedge (left). Three jets at $120^\circ$ achieve full planar coverage via pairwise combinations (middle). Four jets at $90^\circ$ also yield full coverage but are redundant relative to the $120^\circ$ triad (right).
  • Figure 2: Example forward-cone steering trajectory. A 3D path over 14 hours ($\approx$2 rotations) with steering constrained within a forward cone. Markers indicate spin phases at which pitch-axis torque reaches extrema (squares; $\cos\omega t = \pm 1$) and roll-axis torque reaches extrema (circles; $-\sin\omega t = \pm 1$), illustrating how sign-controllable out-of-plane authority becomes available at predictable phases.
  • Figure 3: Minimal four-thruster configuration for practical forward-cone steering.(a) Three-dimensional view showing the spherical body, three small thrusters (green, T1--T3) at $120^\circ$ intervals in the equatorial plane used to synthesize transverse (in-plane) thrust vectors for heading control, one large thruster (red, T4) providing low-bandwidth attitude authority for long-horizon reorientation, the rotation axis $\omega$ (blue), and forward velocity $v_0$ (orange). (b) Top view emphasizing the $120^\circ$ spacing of the small thrusters, which enables bidirectional planar thrust synthesis under unidirectional constraints. (c) Conceptual control authority: instantaneous in-plane steering (green) and slower out-of-plane steering enabled by attitude shaping (red). (d) Large-thruster torque direction versus rotation phase, illustrating phase-dependent attitude authority that sweeps a transverse torque plane over one period.
  • Figure 4: Cone-limited navigation with regime-linked jet signatures (schematic). Left: 2D heliocentric trajectory with four operating points (1--4); Kuiper belt radius ($\sim$50 AU) shown for scale. Right: corresponding jet-pattern icons for a $120^\circ$ three-jet system. (1) warm standby (R1), (2) elevated warm bias (R1/R2 readiness), (3) warm differential steering (R2), and (4) retargeting with axial-jet authority (R3) with possible thrust-neutral trim (R2a). Icon lengths/opacity indicate relative duty. The mapping from thrust allocations to net force is governed by Proposition \ref{['prop:warm_zero_force']} and Proposition \ref{['prop:differential_decomposition']}; fuel/brightness asymmetries under lowering versus raising follow Proposition \ref{['prop:lower_vs_raise']}; and thrust-neutral reallocation follows Proposition \ref{['prop:thrust_neutral_unique']}. Jet directions in the schematic are simplified as radially outward for visual clarity; the actual mounting geometry (tangential for T1--T3, axial for T4) is shown in Figure \ref{['fig:config']} and discussed physically in Section \ref{['sec:nutation_architecture']}.

Theorems & Definitions (44)

  • Remark 1: Space versus frame
  • Remark 2: Realism of the steady-spin assumption
  • Definition 1: Instantaneous controllability (wrench-level)
  • Definition 2: Practical forward-cone reachability (trajectory-level)
  • Theorem 1: Minimum thrusters for full planar force-direction coverage
  • proof
  • Proposition 1: Insufficiency of two thrusters
  • Proposition 2: Minimality of three thrusters
  • Theorem 2: Torque requirement for spin-axis reorientation
  • Proposition 3: Four thrusters suffice for forward-cone reachability
  • ...and 34 more