3D analytical theory of the perturbed single-synchronous state. Application to the post-impact Didymos-Dimorphos system

TL;DR

The paper develops a 3D analytical perturbation framework for the perturbed single-synchronous state in the full two-body problem, introducing linear averaging and nonlinear Hamiltonian normal-form methods. It provides closed-form linear solutions and high-order nonlinear solutions that describe the coupled orbital and rotational dynamics of a binary with a fast-rotating primary and a synchronously rotating secondary, applicable to Didymos-Dimorphos after DART impact. Key contributions include a detailed linearization with explicit frequencies, a Lie-based normal-form approach yielding near-integrals of motion, and a practical post-impact application showing how resonances govern transitions to quasi-regular or chaotic behavior, with direct relevance to data-fitting and interpretation of observations. The results highlight the importance of resonances in determining stability and demonstrate that the nonlinear theory substantially improves accuracy over the linear model, offering a generic framework for similar binary systems with arbitrary parameter choices.

Abstract

We develop the 3D generalization of the planar analytical theory presented in Gaitanas et. al., 2024, which deals with states slightly perturbed from the exact `single-synchronous equilibrium state' (SSES) of the full two-body problem. The SSES corresponds to two non-spherical gravitationally interacting bodies, settled in nearly circular relative orbit, with rotation axes normal to the orbital plane, rapid rotation of the primary and synchronous rotation of the secondary. In the present paper we remove all simplifying assumptions of our previous work Gaitanas et. al., 2024, and show how to compute analytical solutions describing a 3-dimensional perturbation of the system from the SSES in the framework of two distinct theories, called `linear' and `nonlinear'. Linear theory stems from averaging the equations of motion over the primary's rapid rotation angle. This maps the SSES to an equilibrium point of the averaged system, around which analytical solutions can be computed by linearization of the equations of motion. In nonlinear theory, instead, we compute a high order normal form for the Hamiltonian of motion through a sequence of canonical transformations in the form of series. Resonances between the basic system's frequencies appear in the nonlinear theory as small divisors. We show that, close to resonances, the nonlinear theory leads to a partially integrable model, sufficient to analytically describe the evolution of the relative orbit, but only of some of the Euler angles of the system. As a basic application, we compute analytical solutions representing various possible Didymos-Dimorphos post-impact orbital and rotational states. In this case, all analytical formulas here proposed are of direct utility in fitting algorithms exploiting available time series of post-impact observational data.}}

Figures (19)

• Figure 1: Coordinate system setup of the F2BP. Both bodies can move along 3 directions and spin around 3 axes, while gravitationally interacting.
• Figure 2: Top: schematic representation of the SSES. The primary has two of its principal axes lying in the orbital plane and rotates around its third principal axis with spin frequency equal to $\dot{\phi}_1+\dot{\theta}=\nu_1$. The relative orbit is circular, with frequency $\dot{\theta}=\nu_\theta$. Bottom: an SSES created 'by rotation', i.e., by averaging the Hamiltonian (\ref{['Hfullform']}) with respect to the 'fast' angle $\phi_1$.
• Figure 3: Our adopted pre-impact kinetic state of the binary system 65803 Didymos. The functions $r(t),\theta(t),z(t)$ (upper row), $\theta_{1x}(t),\theta_{1y}(t),\phi_{1}(t)$ (mid row) and $\theta_{2x}(t), \theta_{2y}(t), \phi_2(t)$ (lower row) are numerically integrated in time for $G = 0.0864989$, $M_1 = 5.150418$, $M_2 = 0.0392693$, $I_{1x} = 0.260108$, $I_{1y} = 0.267618$, $I_{1z} = 0.337960$, $I_{2x} = 8.20454 \cdot 10^{-5}$, $I_{2y} = 8.88782 \cdot 10^{-5}$, $I_{2z} = 1.1899 \cdot 10^{-4}$, which we refer to as 'parameter set 1' in Section \ref{['sec:dart']}. The units of measurement are in [hr], [km], [kg$^{\ast}$], where 1 [kg$^{\ast}] = 10^{11}$[kg]. The system is very close to an exact SSES.
• Figure 4: Schematic representation of an oblique collision between the impactor and the secondary asteroid. In the chosen coordinate system, the impactor's velocity $\vec{\upsilon}_D$ is decomposed into a tangential component (lying in the $X_R-Y_R$ plane) and a normal component (along the $Z_R$ axis), with an inclination $\gamma$. During the collision it is assumed zero net torque on the secondary.
• Figure 5: Comparison between the numerical (blue) and the linear (orange) solution for the parameter set 1, post DART impact for $\beta = 1$. The impactor's inclination, relative velocity and mass are $\gamma = 9^o$, $\upsilon_D = 22121.6$ [km/hr] and $M_D = 5.79434 \cdot 10^{-9}$ [kg$^{\ast}$] respectively. The primary's initial orientation is set to $\delta \theta_{1x0} = \delta \theta_{1y0} = 0.0025^o$, corresponding to the maximum amplitude resulted by the numerical solution.