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

Articles in Press

Document Type : Research Paper

Authors
A. H. Elneklawy 1
T. S. Amer 2
I. M. Abady 3
1 Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafr El-Sheikh 33516, Egypt
2 Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt
3 Department of Mathematics and Computer Science, Faculty of Science, Suez University, Suez 43518, Egypt
10.22055/jacm.2026.49494.6025
Abstract
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 sinψsinθ = 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(ω2) = ω2gyro - ω20, 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: a progressive phase drift between the gyrostatic and classical solutions is accumulated; near-singular transitions at the zero-crossings of the gyrostatic denominators, at which the reduced description becomes singular while the full nonlinear solution remains regular; and a measurable variation of the total angular momentum magnitude produced by the external tidal forcing and modulated by the internal rotor bias. The results establish gyrostatic frequency shifting as a quantitatively predictable and observationally accessible signature of internal rotor angular momentum in small-body spin dynamics, with direct implications for the interpretation of close-flyby light curve data and for planetary-defense mission planning.
Keywords
Asteroid gyrostat rotation
Rubble-pile asteroid
Nonlinear dynamics
Rotational dynamics
Subjects

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