Gyrostatic Frequency Shifting and Near-Singular Transitions in the Two-Stage Rotational Dynamics of an Impact-Struck Rubble-Pile Asteroid During Hyperbolic Planetary Flyby

The rotational evolution of a rubble-pile asteroid that sustains a near-perpendicular kinetic impact immediately prior to entering the gravitational influence zone of a planet is investigated within the gyrostat framework, wherein the total angular momentum components along the three principal axes are impacted by the constant angular momentum of internally spinning rotors fixed in the carrier body. A complete two-stage semi-analytical treatment is developed: the first stage characterizes the post-impact free rotation under time-varying principal moments of inertia driven by rubble-pile mass ejection, while the second stage analyzes the spin state during hyperbolic planetary approach under gravitational tidal torques. In the first stage, the spin-up law governing the angular velocity’s first component is shown to be independent of the gyrostatic bias first component to leading order, and the kinematic relation sinψ sinθ = const is established as an exact first integral of the leading-order reduced Stage-1 kinematics, unaffected by the gyrostatic extension. In the second stage, a second-order ordinary differential equation for the angular momentum second component is derived, whose homogeneous part oscillates at a gyrostatically shifted natural frequency; the correction is obtained in closed form for the squared frequency, Δ(ω^2)=ω_{gyro}^{2}-ω_{gyro}^{0}, and is positive throughout the spun-up phase. Numerical integration of the full nonlinear system confirms the analytical predictions and additionally reveals three dynamically significant phenomena absent from the classical formulation.