Full Spin-Spin Dynamics

• Full Spin-Spin Problem is characterized by two triaxial bodies whose rotational and orbital dynamics are mutually coupled by gravity.
• The formulation employs a Hamiltonian framework with potential expansion into spin-orbit and spin-spin terms, detailing resonance and stability criteria.
• Dissipative effects modeled via damping terms drive synchronization and capture into resonant states, aligning with observational benchmarks.

The full spin-spin problem refers to the rotational and orbital dynamics of two interacting rigid bodies—typically modeled as triaxial ellipsoids—subject to their mutual gravitational influence, without assuming a fixed Keplerian orbit. This configuration is central in celestial mechanics, especially for binary asteroid systems and planet-satellite pairs where both constituents display substantial triaxiality. Unlike the classic spin-orbit problem (one extended body, one point mass), the full spin-spin model incorporates not only the coupling between the spins of both bodies but also the feedback of rotational states on the orbital evolution, with possible dissipation.

1. Model Setup and Physical Assumptions

The problem centers on two homogeneous rigid bodies (mass $M_1$, $M_2$; moments of inertia $A_j$, $B_j$, $C_j$ for $j=1,2$) whose spin axes are both perpendicular to the orbital plane and coincide with their shortest principal axes. The bodies’ centers of mass trace trajectories in a common plane, and both rotational and orbital degrees of freedom are dynamical.

Three regimes are distinguished:

• Spin-orbit problem: One triaxial ellipsoid, one point mass, no spin-spin coupling.
• Keplerian spin-spin problem: Both bodies extended, fixed (Keplerian) coplanar orbit, spins coupled by higher-order terms in the gravitational potential.
• Full spin-spin problem: Both bodies extended, the orbit is dynamically coupled and evolves along with the rotational states; rotational and orbital motions are fully interdependent.

Dissipative effects (e.g., tidal dissipation due to non-rigidity) can be incorporated for one or both bodies via a linear damping term proportional to spin rate deviations.

2. Mathematical Formulation

The full system is described by a Hamiltonian:

$$H(p_r,p_f,p_1,p_2, r, f, \theta_1, \theta_2) = \frac{p_r^2}{2\mu} + \frac{p_f^2}{2\mu r^2} + \frac{p_1^2}{2C_1} + \frac{p_2^2}{2C_2} + V(\theta_1, \theta_2, r, f)$$

where $r$ (radius), $f$ (true anomaly) are orbital coordinates; $\theta_1, \theta_2$ are spin angles; $p_r, p_f, p_1, p_2$ their conjugate momenta; and $V$ is the gravitational potential.

Potential expansion:

$$V(\theta_1, \theta_2, r, f) = V_0(r) + V_\text{per}(\theta_1, \theta_2, r, f)$$

• $V_0(r) = -\frac{GM_1M_2}{r}$ (Keplerian)
• $V_\text{per}$ contains spin-orbit $V_2$ and spin-spin $V_4$ terms:

$$V_2 = -\frac{G M_2}{4 r^3}[q_1 + 3 d_1 \cos(2\theta_1-2f)] -\frac{G M_1}{4 r^3}[q_2 + 3 d_2 \cos(2\theta_2-2f)]$$

$$V_4 \supset -\frac{3G}{64 r^5} [6 d_1 d_2 \cos(2\theta_1-2\theta_2) + 70 d_1 d_2 \cos(2\theta_1 + 2\theta_2 - 4f) + \ldots]$$

Here $d_j = B_j - A_j$, $q_j = 2C_j - B_j - A_j$, encoding triaxiality.

Equations of motion follow from the Hamiltonian:

• Orbital: $\dot r = p_r/\mu$, $\dot f = p_f/(\mu r^2)$, $\dot p_r = p_f^2/(\mu r^3) - \partial_r V$, $\dot p_f = -\partial_f V$
• Rotational: $\dot\theta_j = p_j/C_j$, $\dot p_j = -\partial_{\theta_j} V$

Orbital dynamics and rotational dynamics are coupled via $V$, producing nonintegrable evolution except in special limiting cases.

3. Dissipative Effects

To model tidal/rotational dissipation, a linear damping term is introduced (MacDonald model):

$$C_j \ddot\theta_j + \frac{\partial V_2}{\partial\theta_j} + \frac{\partial V_4}{\partial\theta_j} = -\delta_j C_j \left(\frac{a}{r(t)}\right)^6 (\dot\theta_j - \dot f(t))$$

where $a$ is a reference (semi-major axis), and $\delta_j$ a dissipation parameter for each body. The dissipative component drives the system toward synchronous states, with complex transient evolution dependent on coupling strength, orbital parameters, and the magnitude of dissipation.

Averaged over an orbital period, dissipation simplifies to:

$$C_j \ddot\theta_j + \left\langle \frac{\partial V_2}{\partial \theta_j} + \frac{\partial V_4}{\partial \theta_j} \right\rangle = -\bar\gamma_j (\dot\theta_j - \bar\mu_j)$$

with $\bar\gamma_j$, $\bar\mu_j$ functions of the orbital eccentricity.

4. Spin-Spin Resonances and Stability Analysis

A spin-spin resonance $\left(m_1:n_1, m_2:n_2\right)$ occurs when body 1 rotates $m_1$ times and body 2 $m_2$ times per $n_1$, $n_2$ orbits, respectively. Main attention is given to:

• $(1:1, 1:1)$ (double synchronous),
• $(3:2, 3:2)$,
• $(1:1, 3:2)$.

Linear stability is analyzed by expanding about resonant equilibria and examining the eigenvalues of the linearized system (after averaging and isolating the resonant angle):

• Spin-orbit problem, $(1:1,1:1)$: equilibrium stable if $e < \sqrt{2/5}$ (eccentricity threshold).
• Spin-spin problem: stricter threshold $e < 1/\sqrt{11}$ due to added coupling, reduced parameter space for stability. $(3:2,3:2)$ resonance is linearly stable for all $e$.

The dissipative setting modifies stability: with weak dissipation, equilibria become attracting (negative real parts in the spectrum), leading to gradual synchronization or capture into resonance states.

5. Numerical Investigation and Phase Space Structure

High-precision numerical integration (adaptive Runge-Kutta, energy conservation to $10^{-10}$) is used to explore the phase space:

• Poincaré sections (stroboscopic sampling at specific orbital phases) reveal the architecture of libration islands, separatrices, and chaos.
• Spin-spin coupling ($V_4$ terms) modifies the topology and area of libration islands compared to the spin-orbit case; destruction of higher order islands occurs in the full spin-spin model due to time-dependent orbital elements.
• Oscillatory evolution of semi-major axis and eccentricity arises from energy exchange between orbital and rotational modes.
• Dissipation causes convergence toward attractors (i.e., steady resonant configurations), with observable oscillations in both spin and orbital variables.

For real systems:

• Patroclus-Menoetius and Pluto-Charon are benchmarked, illustrating phase space structure alterations and justifying, in some limits, the Keplerian approximation.

6. Implications for Celestial Bodies and Observations

The full spin-spin model quantifies how observable rotational states (e.g., synchronous or higher-order spin ratios in minor planets and satellites) reflect underlying dynamical stability, triaxiality, and dissipation. Particularly, significant triaxiality in both bodies results in stricter requirements for resonant stability. The destruction of higher order islands implies that most observed systems are near primary resonances, and strongly coupled evolution must be considered for close, nearly equal-mass binaries.

Dissipative dynamics are critical for understanding long-term evolution and synchronization, aligning with observed timescales and states in asteroid and planetary satellites.

7. Summary Table: Models & Main Findings

Model	Coupling	Orbit Fixed	Key Findings
Spin-orbit (SO)	No	Yes	Decoupled, broad stable regime
Keplerian spin-spin (KSS)	Spin-spin	Yes	Reduced stability, spin coupling
Full spin-spin (FSS)	All (spin-orbit)	No	Greatest complexity, destruction of high-order islands, $a,e$ vary

Main resonances: $(1:1,1:1)$, $(3:2,3:2)$, $(1:1,3:2)$ Methods: Analytical averaging, linearized stability, eigenvalue analysis, numerical phase space mapping Implication: Resonant states and their stability sensitively depend on coupling magnitude, eccentricity, and dissipation, most acutely for triaxial, closely interacting binaries.

8. Conclusion

The full spin-spin problem elucidates the intricate interplay between rotational and orbital degrees of freedom when both celestial bodies are triaxial and interact gravitationally. Compared to simplified spin-orbit models, capture into and persistence in resonant rotational states are far more delicate—affected by coupling, orbital eccentricity, and dissipation. The analytic and computational methodologies enable quantitative prediction and physical interpretation of rotational-orbital evolution in binary asteroid systems, informing both observation and long-term dynamical forecasts.