Spin-Orbit Synchronization and Singular Perturbation Theory

Abstract

In this study, we formulate a set of differential equations for a binary system to describe the secular-tidal evolution of orbital elements, rotational dynamics, and deformation (flattening), under the assumption that one body remains spherical while the other is slightly aspherical throughout the analysis. By applying singular perturbation theory, we analyze the dynamics of both the original and secular equations. Our findings indicate that the secular equations serve as a robust approximation for the entire system, often representing a slow-fast dynamical system. Additionally, we explore the geometric aspects of spin-orbit resonance capture, interpreting it as a manifestation of relaxation oscillations within singularly perturbed systems.

Figures (10)

• Figure 1: Illustration of the Deformation Invariant Manifold $\Sigma_{\epsilon_d}:=\{(\mathbf{x},\dot{\mathbf{x}},\ell_s,\epsilon_d)\to\boldsymbol{\delta}\mathbf{B}\}$. With the parameterization defined by $(\mathbf{x},\dot{\mathbf{x}},\ell_s,\epsilon_d)$, the vector field on $\Sigma_{\epsilon_d}$ follows from (\ref{['epd5']}).
• Figure 2: Vector field $\frac{\dot \omega}{n^2}=V(\tau n, \frac{\omega}{n},e)$ with constant $n$ and $e$. $V_{max}$ represents the maximum rate of variation of $\frac{\omega}{n}$ and $\tau_s^{-1}=\tan \psi$ denotes the time constant of a stable equilibrium.
• Figure 3: Orbits of the equation (\ref{['av3']}) on the invariant plane $e=0$. The orbits are labelled by the total angular momentum $\ell_{ T}$ by means of the nondimensional parameter $\epsilon^{-1}=\frac{\ell_{ T}^4}{ {\,\rm I}_\circ \mu c^2}$. The equilibria are on the horizontal line $\frac{\omega}{n}=1$: the green dots represent stable equilibria and the red dots represent unstable equilibria. The black dot at $\tilde{a}=\frac{9}{16}$, $\frac{\omega}{n}=1$ represents the single equilibrium that occurs for the special value $\epsilon=\frac{27}{256}$. For $\epsilon>\frac{27}{256}$ (small angular momentum) all the solutions lead to a collision.
• Figure 4: The graph of $\frac{\dot e}{e\tilde{c}}$ as a function of $\frac{\omega}{n}$ for values of $\tau n$ equal 1, 10, and 100. For $n=10$, $\frac{\dot e}{e\tilde{c}}$ has a zero, not easily seen in the Figure, at $\frac{\omega}{n}=5.26$.
• Figure 5: LEFT: The figure shows possible orbits on the eccentricity-semi-major axis ($\delta_a = \tilde{a} - \tilde{a}_e$) plane, $\delta_a = \text{constant}\ e^{\lambda_0/\lambda_e}$ with $\text{constant} = 1$ for various $\lambda_0/\lambda_e$ values. RIGHT: A graphical method to find the special value of $\tilde{a}_e$, where $\lambda_0/\lambda_e = 1$, as a function of $\tau \tilde{n}_e$.