text:
\section{Introduction}\label{sec:intro

\subsection{Motivation}

The model we propose here is a natural extension of the well known spin-orbit problem of celestial mechanics. The spin-orbit model is an elementary, but not trivial, model to study the rotational dynamics of a satellite about its center of mass when it orbits around a planet. Here the planet acts as a point mass and the satellite is an extended body whose spin axis is perpendicular to the orbital plane. This model has the structure of a nearly integrable and periodically forced pendulum. It has attracted much attention not only for its accurate physical implications but also for its mathematical richness. Some pioneer papers are \cite{bel1966} for the conservative case and \cite{gol1966} for the dissipative case. This model contributes to explain the synchronization of the rotational motion of the Moon and its orbital motion around the Earth. In other words, the Moon is in a 1:1 spin-orbit resonance. This phenomenon is indeed very common in the solar system for natural satellites that are close enough to their respective planets, \cite{mur}. Besides, Mercury, as an orbiting body around the Sun, is locked in a 3:2 spin-orbit resonance. According to \cite{corlas2004}, in its chaotic evolution, Mercury could have reached large orbital eccentricities that made possible the capture into this higher order resonance. It is accepted that the phenomenon of capture into resonances is driven by dissipative torques, caused by internal frictions within the satellite, \cite{mac1964}. The concept of stability of a resonance in the conservative regime is linked to the concept of capture in the dissipative case and both can be related. In one hand, \cite{cel1990} studies the KAM stability in the conservative case, whereas  \cite{cel2009} proves the existence of quasiperiodic attractors for the dissipative problem, that bifurcate from the KAM tori of the conservative case. On the other hand, \cite{misort2020} proves the existence of an asymptotically stable solution in 1:1 resonance that is a continuation of a linearly stable odd periodic solution of the conservative case. The onset of chaos is another interesting feature of this problem. The oblateness of the satellite produces chaotic regions in the phase space that surround the libration regions of resonances. Chaotic zones can be very large due to overlapping of different resonances, \cite{chi1979}. A large eccentricity emphasizes this behavior, as in the case of Hyperion, \cite{wis1984}, \cite{wis1987}.

The Full Two-Body Problem (F2BP) deals with the dynamics of two extended bodies interacting gravitationally. It has been extensively investigated, especially in the last two decades, due to an increasing interest on binary systems. Due to its complexity, most of the studies are numerical explorations of particular cases, see \cite{fah2006} or \cite{comlem2014}. There are some works with a more analytical approach dealing with relative equilibria and stability, \cite{sch2002} and \cite{mac1995}. The spin-spin model is motivated mainly by \cite{boulas2009}, \cite{sch2009}, \cite{davsch2020} and \cite{batmor2015}. In one hand, \cite{boulas2009} is focused on the  evolution of the orbit and the spin axes of the bodies in the secular F2BP (averaging over fast angles). This paper points out that the mutual influence in the spin dynamics is contained in the terms of order $1/r^5$ of the expansion of the potential energy of the system, where $r$ is the distance between the bodies. On the other hand, \cite{sch2009} studies the relative equilibria and stability in the planar case, i.e., the spin axes of the bodies are perpendicular to the orbital plane, that is also a common equatorial plane. \cite{davsch2020} studies the observability of non-planar stable oscillations around the double synchronous equilibrium in binary asteroids. In \cite{sch2009} and \cite{davsch2020}, only terms up to $1/r^3$ of the potential energy are considered, so the resulting system is equivalent to two uncoupled spin-orbit problems. The planar spin-spin coupling was first studied in \cite{batmor2015}, making an analogous study as the classical paper \cite{gol1966} on the spin-orbit coupling. Particularly, \cite{batmor2015} studies the spin of the body 1, identified with two point masses slightly separated from each other (dumbbell model), that moves in a circular orbit around the body 2, an ellipsoid with uniform rotation. They focus on the case when the orbital motion is slow and the angular velocity of the body 1 becomes commensurable with the angular velocity of the body 2 (spin-spin resonance).

The model we propose in this paper deals with the complete coupled dynamics of the F2BP in the planar and ellipsoidal case. As usual in the spin-orbit problem, we also assume that the orbital motion takes place in Keplerian ellipses. This reduces the high dimensional phase space of the F2BP to a problem of two degrees of freedom (spins) plus time-dependence (orbit). For a small non-zero orbital eccentricity, it has the structure of a nearly integrable system of coupled pendula that is periodically forced. This setting is suited to study the phenomena related to spin-orbit and spin-spin resonances. Furthermore, the intrinsic dissipative nature of the capture into resonances supports the relevance of this model. The reason is that the most used family of dissipative torques, see \cite{mac1964}, is of order $1/r^6$, whereas the spin-spin coupling appears at order $1/r^5$. In addition to the questions related to the spin-orbit problem, this model of coupled oscillators opens new questions that were not possible to consider before. We will discuss this in \Cref{sec:discussion}.

\subsection{Setting of the model.}\label{sub:model}

Consider two homogeneous ellipsoids $\mathcal E_1$ and $\mathcal E_2$ with respective masses $M_j$, $j=1,2$, principal moments of inertia $\mathcal  A_j<\mathcal B_j< \mathcal C_j$ and corresponding principal semi-axes $\mathsf a_j>\mathsf b_j>\mathsf  c_j$. Assume that the orbital motion of the ellipsoids is the same as for two point masses, say, the centers of the ellipsoids describe coplanar Keplerian orbits of eccentricity $e\in[0,1)$ with a common focus at the center of mass of the system. Moreover, assume that the spin axis of each body is the principal axis associated to $\mathsf c_j$ and is perpendicular to the orbital plane.

\begin{figure}
	\centering
	\scalebox{0.5}{\includegraphics{_fig_spin-spin_orbit.pdf}}
	\caption{The planar spin-spin problem.}
	\label{fig:spin-spin}
\end{figure}

\begin{comment}
Then, the parameter $\epsilon= \frac{3}{2} \frac{\mathcal B-\mathcal A}{\mathcal C}$ measures the oblateness of the satellite in the orbital plane.
\end{comment}

Let us identify the orbital plane with the complex plane $\mathds C$. Consider the center of mass of the system fixed at the origin and let the center of each ellipsoid be $\vec r_j$, then, $M_1 \vec r_1+M_2 \vec r_2=0$. If we define the relative position vector $\vec r= \vec r_2-\vec r_1$ and choose the units of mass such that $M_1 +M_2 =1$, then, $\vec r_1=-M_2 \vec r$ and $\vec r_2=M_1 \vec r$. The orbital motion is defined by $\vec r$, which can be written as $\vec r=r \exp(i f)\in \mathds C$, where $r>0$ and $f$ are real functions of the time. Note that $\vec r$ describes an ellipse of eccentricity $e\in[0,1)$ and semimajor axis $a$ with focus at the origin, so, the polar coordinates $r$ and $f$ vary periodically with time, and are known by the Kepler problem. Let us take convenient units of time so that the period is $2\pi$. In the usual terminology, $f$ is called \textit{true anomaly} and the time $t$ is the \textit{mean anomaly}. There is a third useful angle $u$, the \textit{eccentric anomaly}, which is defined by the famous Kepler's equation
\begin{equation}\label{t}
t=u-e\sin u,
\end{equation}

\noindent and let us determine the Keplerian ellipse simply by
\begin{equation}\label{r}
r=a(1-e \cos u).
\end{equation}

\noindent Also, using the graphical definition of the eccentric anomaly and some geometrical relations of ellipses, we can write the position of $\vec r$ in terms of the eccentric anomaly as
\begin{equation}\label{f}
r\exp(i f)=a(\cos u-e + i \sqrt{1-e^2} \sin u).
\end{equation}

\noindent Note that for $t=0$ we assumed that $f=u=0$, and consequently, $f=u=\pi$ when $t=\pi$. The expressions \cref{f,r} relate the true and eccentric anomalies. Moreover, \Cref{t,r,f} define $u=u(t,e)$, $\frac{r}{a}=\frac{r}{a}(t,e)$ and $f=f(t,e)$ as analytic functions in both entries.

Recall Kepler's third law for the Two-Body Problem
\begin{equation}\label{kepler3}
G(M_1+M_2) \parentesis{\frac{T}{2\pi}}^2=a^3,
\end{equation}

\noindent where $G$ is the Gravitational constant and $T$ is the orbital period. In consequence, $G=a^3$ in our units. For our model to be completely non-dimensional and adequate to the scale of the system, we take convenient units of length such that $\mathcal  C_1+\mathcal  C_2=1$. In these units the semi-major axes $\mathsf{a}_j$ of the ellipsoids are of order $1$, whereas $a$ should be much larger. See \Cref{ap:units} for specific conversion of units.

Let $\theta_j$ be the polar angle of the principal direction associated to $\mathsf a_j$ with respect to the orbit's major axis.  See \Cref{fig:spin-spin}. The spin dynamics of the ellipsoids is modelled by the following coupled system of ordinary differential equations
\begin{equation}\label{spin_spin_dis}
\mathcal C_j \ddot \theta_j  = \mathcal T_j^\text{C}(t,\theta_1,\theta_2)+\mathcal T_j^\text{D}(t,\dot \theta_j),\qquad j=1,2,
\end{equation}

\noindent where $\mathcal T_j^\text{C}$ and $\mathcal T_j^\text{D}$ are respectively the \textit{conservative} and \textit{dissipative} torques acting on $\mathcal E_j$.

The \textit{conservative} torque is derived from the potential gravitational energy, see \Cref{sec:model}, and it takes the form
\begin{multline}\label{T_cons}
\mathcal T_j^\text{C}(t,\theta_1,\theta_2)= - \parentesis{\frac{a}{ r(t)}}^{3} \frac{\Lambda_j}{2} \sin (2\theta_j-2f(t))   \\
- \parentesis{\frac{a}{ r(t)}}^{5}
\sum_{(m_1,m_2)\in \Xi }
\frac{m_j \,\Lambda_{m_2}^{m_1}}{2}
\sin(2 m_1(\theta_1-f(t))+2m_2 (\theta_2-f(t))),
\end{multline}

\noindent where
\begin{equation*}
\Xi=\{(m_1,m_2)\in \mathds Z^2 : \ |m_1|+ |m_2|\le 2\}.
\end{equation*}

\begin{comment}
The convergence of the series \cref*{T_cons} is guaranteed because it is a Taylor expansion of an analytic potential as we will see in \Cref{sec:model}.
\end{comment}

\noindent In \cref{T_cons} we have ignored terms of order $\parentesis{{a}/{ r(t)}}^{n}$ with $n\ge7$. The parameters $\Lambda_j$ and $\Lambda_{m_2}^{m_1}$ are positive small quantities depending on the physical parameters of the bodies and on $a$. These parameters satisfy $\Lambda_{m_2}^{m_1}=\Lambda_{-m_2}^{-m_1} < \Lambda_j<3 \mathcal C_j$. Note that if all the constants $\Lambda_{m_2}^{m_1}$ in \cref{T_cons} vanish, the system \cref{spin_spin_dis} is formed by two uncoupled spin-orbit problems in $\theta_1$ and $\theta_2$. The coupling of the system is contained in the terms $({m_1},{m_2})$ of type $(\pm 1,\pm 1)$ and $(\pm 1,\mp 1)$, whereas the rest of them are high order spin-orbit terms.

On the other hand, the \textit{dissipative} torque $\mathcal T_j^D$ has different forms depending on the model. We will use a linear MacDonald torque \cite{mac1964}
\begin{equation}\label{macdonald}
\mathcal T_j^D(t,\dot \theta_j) = -C_{M,j} \parentesis{\frac{a}{ r(t)}}^6  \sin(2 \Delta t_j (\dot \theta_j - \dot f(t))) \approx -\delta_j \mathcal{C}_j\parentesis{\frac{a}{ r(t)}}^6 (\dot \theta_j - \dot f(t)),
\end{equation}

\noindent where $C_{M,j}$ are constants depending on the parameters of the bodies. Here we assumed that $|\Delta t_j (\dot \theta_j - \dot f(t))|\ll 1$ because the parameters $\Delta t_j$ and $\delta_j$ are very small positive numbers. This type of torque has been extensively used, taking as reference \cite{gol1966} or \cite{mur}, for example. According to \cite{efr2013}, to obtain \cref{macdonald}, the dissipation is modelled by assuming that there is a time delay between the deforming disturbance and the actual deformation of each body. That delay is a small fixed amount $\Delta t_j$ (time lag), leading to an angular lag of $ (\dot f(t,e) -\dot \theta_j )\Delta t_j$ (geometric lag). It is worth mentioning that there is no physical reason for both lags (or both $\delta_j$) to match.

Note that if $\mathcal T_j^\text{D}=0$, the system \cref{spin_spin_dis} has a Hamiltonian structure. The corresponding Hamiltonian has two degrees of freedom and time dependence and it is given by
\begin{equation}\label{ham}
H(\theta_1,\theta_2,p_{\theta_1},p_{\theta_2},t) = \frac{p_{\theta_1}^2 }{2 \mathcal C_1}+ \frac{p_{\theta_2}^2 }{2 \mathcal C_2} + \mathcal V(t, \theta_1,\theta_2),
\end{equation}
\noindent where
\begin{multline*}
\mathcal V(t, \theta_1,\theta_2) = -\frac{1}{4} \parentesis{\frac{a}{ r(t)}}^3\sum_{j=1}^2 \Lambda_j\cos (2\theta_j-2f(t)) \\-\frac{1}{4}\parentesis{\frac{a}{ r(t)}}^{5}
\sum_{(m_1,m_2)\in \Xi}
\Lambda_{m_2}^{m_1}
\cos(2 m_1(\theta_1-f(t))+2m_2 (\theta_2-f(t))).
\end{multline*}

\noindent Due to the explicit time dependence of the Hamiltonian, the energy of the system is not constant even though $\mathcal T_j^\text{D}\equiv0$. However, if $\mathcal T_j^\text{D}\equiv0$, the system \cref{spin_spin_dis} will be called \textit{conservative}, because no dissipative forces are involved in the physical derivation of the model. On the other hand, if $\mathcal T_j^\text{D}$ is not identically zero for all time, then we will call it \textit{dissipative}. The italic font will remark this point. In \Cref{sec:model} we will see also a purely conservative version of the model involving $(r,f,\theta_1,\theta_2)$ as unknown functions of time.

There are solutions of \cref{spin_spin_dis} that are especially relevant. Since the spin-orbit problem is a particular case of \cref{spin_spin_dis}, a solution satisfying $\theta_1(t+2\pi n_o)=\theta_1(t)+2\pi n_s$, with $n_s,n_o\in \mathds Z$, is called $n_s:n_o$ spin-orbit resonance of the ellipsoid $\mathcal E_1$. The same is true for $\mathcal E_2$. Spin-spin resonances arise when the spin rates of the two ellipsoids become commensurable. In \cite{batmor2015} these resonances were studied independently from the orbital rate. There are some solutions in which the ellipsoids are simultaneously in a spin-orbit and a spin-spin resonance. The simplest of these resonances is the double synchronous resonance of equation \cref{spin_spin_dis}, that is, solutions satisfying $\theta_j(t+2\pi )=\theta_j(t)+2\pi $, for both $j=1,2$. In other words, the spin of both ellipsoids synchronize with the orbital motion at the same time.

\subsection{Setting of our approach and results}

We are going to deal with the capture into the double synchronous resonance. In the same way as in \cite{misort2020}, in this paper we will approach this phenomenon from an analytical point of view. We will look for conditions resulting in the existence of a double synchronous solution of the \textit{conservative} model that can be continued to an asymptotically stable solution of the \textit{dissipative} model. In this context, the asymptotic stability of the solution represents the phenomenon of capture into the resonance: solutions in the vicinity of the asymptotically stable solution get closer and closer to it as $t\rightarrow +\infty$.

Let us take the change of variable $\Theta_j =2(\theta_j - f)$, such that the system \cref{spin_spin_dis} turns into
\begin{multline}\label{spin_spin_dis_mayus}
\mathcal C_j\ddot \Theta_j + \delta_j \mathcal C_j \parentesis{\frac{a}{ r(t)}}^{6} \dot \Theta_j
+\parentesis{\frac{a}{ r(t)}}^{3} \Lambda_j\sin \Theta_j \\
+\parentesis{\frac{a}{ r(t)}}^{5}
\sum_{(m_1,m_2)\in \Xi }m_j \,\Lambda_{m_2}^{m_1}
\sin(m_1\Theta_1+m_2 \Theta_2) = -2\mathcal C_j \ddot f(t)  .
\end{multline}

\noindent The system \cref{spin_spin_dis_mayus} models a couple of damped and forced pendula of variable length. Since $f(t+2\pi)=f (t)+2\pi$, then, double synchronous resonances correspond to solutions of \cref{spin_spin_dis_mayus} satisfying $\Theta_j(t+2\pi)=\Theta_j (t)$.

In \Cref{sec:model} we will make the derivation of the \textit{conservative} model from the Lagrangian of the physical system and obtain the expression of $\Lambda_j $ and $\Lambda_{m_2}^{m_1} $ in terms of physical parameters. In \Cref{sec:lin_stab}, we will deal with the \textit{conservative} system, say, \cref{spin_spin_dis_mayus} with $\delta_j=0$,
\begin{equation}\label{spin_spin_mayus}
\mathcal C_j\ddot \Theta_j
+\parentesis{\frac{a}{ r(t)}}^{3} \Lambda_j\sin \Theta_j
+\parentesis{\frac{a}{ r(t)}}^{5}
\sum_{(m_1,m_2)\in \Xi }m_j \,\Lambda_{m_2}^{m_1}
\sin(m_1\Theta_1+m_2 \Theta_2) = -2\mathcal C_j \ddot f(t)  ,
\end{equation}

\noindent and discuss the existence, uniqueness and linear stability of an odd $2\pi$-periodic solution. This solution is a continuation of the trivial solution $\Theta(t)\equiv 0$ for $e=0$. This will lead us to a region of linear stability in the space of parameters of the system. In this section we will use some properties of symmetric matrices and linear Hamiltonian systems with periodic coefficients. We are interested in the linear stability of the periodic solution of \cref{spin_spin_mayus} because it will allow us to find, by continuation, an asymptotically stable periodic solution for the \textit{dissipative} case \cref{spin_spin_dis_mayus} for $\delta_j>0$. This will be proved in \Cref{sec:asymp}, provided that $|\Lambda_{m_2}^{m_1}|$ and $|\delta_j|$ are small enough. In \Cref{sec:app} we will explain how to apply our results to real cases and use the Pluto-Charon system and the binary asteroid 617 Patroclus as two representative examples. We will also compare our estimates with some numerical experiments and with the spin-orbit problem. Finally in \Cref{sec:discussion} we will make a discussion about the model and our results.

\section{Derivation of the conservative spin-spin model}\label{sec:model

In this section we will compute the equations of motion of the ellipsoids with respect to the inertial frame with origin at the barycenter of the system. \Cref{sub:Lag} is devoted to find the equations of motion of the full system of four variables $( r,f,\theta_1,\theta_2)$, in terms of the gravitational potential energy $V=V(r,f,\theta_1,\theta_2)$. In \Cref{sub:kep} we fix the Keplerian orbit and obtain the final model in terms of physical parameters of the system.

\subsection{The planar Lagrangian model}\label{sub:Lag}

Let the Lagrangian of the system be $L=T-V$, where $T$ is the kinetic energy and $V$ the potential energy of the system. Recall that the positions of the bodies are $\vec r_1=-M_2 \vec r$ and $\vec r_2=M_1 \vec r$, where the relative position vector is defined by $\vec r= \vec r_2-\vec r_1=r \exp(if)$. Besides, for each body, the angle $\theta_j$ defines the orientation of the axis associated to $\mathsf a_j$. We are going to use $r$, $f$, $\theta_1$ and $\theta_2$, depicted in \Cref{fig:spin-spin}, as the Lagrangian variables of our system. The total orbital kinetic energy is given by
\begin{equation*}
T_{orb}=\frac{1}{2} (M_1 \dot{\vec r_1}^2 +M_2 \dot{\vec r_2}^2) = \frac{\mu}{2} \dot{\vec r}^2 = \frac{\mu}{2} (\dot r^2 + r^2 \dot f^2),
\end{equation*}

\noindent where $\mu=M_1M_2$ is the the reduced mass of the system (recall $M_1+M_2=1$). While the rotational kinetic energy is $T_{rot}=\frac{1}{2} \mathcal C_1 \dot \theta_1^2 + \frac{1}{2} \mathcal C_2 \dot \theta_2^2$. See \Cref{app:potential} for the derivation of the full expression of the potential energy of the system $V=V(r,f,\theta_1,\theta_2)$, equation \cref{V_full}. The Euler-Lagrange equations corresponding to the Lagrangian $L=T_{orb}(r,\dot r,\dot f) + T_{rot}(\dot \theta_1,\dot \theta_2)-V(r,f,\theta_1,\theta_2)$ are
\begin{equation}\label{spin-spin_V}
 \mathcal C_1\ddot \theta_1  = -\partial_{\theta_1} V ,\quad  \mathcal C_2\ddot \theta_2  = -\partial_{\theta_2} V,
\end{equation}
\begin{equation}\label{orbital_eqs}
\mu \ddot r = \mu r\dot f^2 - \partial_r V,\quad  \ddot f = -\frac{1}{\mu  r^2}\partial_{f} V  -2 \frac{\dot r \dot f}{r}.
\end{equation}

In \Cref{sub:V_planar} we give the expansion of the potential energy. In the case of ellipsoids, it has the form $V=\sum_{n=0}^\infty V_{2n}$, where $V_{2n}$ is proportional to $1/r^{2n+1}$. The first terms of the expansion are
\begin{equation*}
V_0=-\frac{GM_1M_2}{r}.
\end{equation*}
\begin{equation*}
V_2= -\frac{GM_2}{4r^3} \parentesis{q_1+3d_1\cos(2(\theta_1-f))}
-\frac{GM_1}{4r^3} \parentesis{q_2+3d_2\cos(2(\theta_2-f))},
\end{equation*}
\begin{equation}\label{V_4_ph_parameters}
\arrays{rl}{
	\displaystyle V_4=-\frac{3G}{4^3 r^5}  \{
	& 12q_1q_2 + \frac{15}{7} [\frac{M_2}{M_1}d_1^2+2\frac{M_2}{M_1}q_1^2 + \frac{M_1}{M_2}d_2^2+2\frac{M_1}{M_2}q_2^2]  \\
	&+d_1 M_2\left \{ [20\frac{q_2}{M_2} + \frac{100}{7}\frac{q_1}{M_1}]\cos (2(\theta_1-f)) +25 \frac{d_1}{M_1}\cos (4(\theta_1-f)) \right \}\\
	&+d_2 M_1 \left \{[20\frac{q_1}{M_1} + \frac{100}{7}\frac{q_2}{M_2}]\cos (2(\theta_2-f)) + 25 \frac{d_2}{M_2}\cos (4(\theta_2-f)) \right \}\\
	&+6d_1 d_2 \cos (2(\theta_1-\theta_2))+70d_1 d_2  \cos (2(\theta_1+\theta_2)-4f)
	\},
}
\end{equation}

\noindent where we defined the parameters
\begin{equation}\label{q_d}
d_j=\mathcal B_j- \mathcal A_j, \qquad
q_j=2\mathcal C_j- \mathcal B_j- \mathcal A_j.
\end{equation}

\noindent Note that $d_j$ is proportional to $C^{j)}_{22}$, whereas $q_j$ is proportional to $C^{j)}_{20}$, where $C^{j)}_{nm}$ are the usual coefficients in the expansion of the gravitational potential of the ellipsoid $\mathcal E_j$. The quantity $d_j/\mathcal C_j$ measures the oblateness of the section of the ellipsoid in the plane of motion, whereas, $q_j/\mathcal C_j$ measures the flattening with respect to the plane. If $\mathcal  A_j \le\mathcal B_j\le \mathcal C_j$, then, $q_j\ge d_j\ge 0$. Note that the term $V_0$ contains the dynamics of two point masses, $V_2$ the uncoupled spin-orbit dynamics and $V_4$ the spin-spin coupled dynamics between $\theta_1$ and $\theta_2$. The coupling terms appear in the last line of \cref{V_4_ph_parameters}.

\subsection{The Keplerian assumption and the spin-spin model}\label{sub:kep}

The complete dynamics of the system is given by \Cref{spin-spin_V,orbital_eqs}, with $V$ in \cref{V_full}. In this paper we impose that the orbital motion is Keplerian, i.e., we keep only $V_0$ in the orbital part \cref{orbital_eqs}. Besides, in the spin part \cref{spin-spin_V}, we truncate $V$ ignoring terms of order $1/r^7$ and higher, then $V\approx V_0+V_2+V_4$. The resulting system is
\begin{equation}\label{spin-sin_V_trun}
\mathcal C_1\ddot \theta_1  = -\partial_{\theta_1} (V_0+V_2+V_4) ,\quad  \mathcal C_2\ddot \theta_2  = -\partial_{\theta_2} (V_0+V_2+V_4),
\end{equation}
\begin{equation}\label{orbital_eqs_kep}
\mu \ddot r = \mu r\dot f^2 - \partial_r V_0,\quad  \ddot f = -\frac{1}{\mu  r^2}\partial_{f} V_0  -2 \frac{\dot r \dot f}{r}.
\end{equation}

\noindent Note that, since $\partial_{\theta_j} V_0=0$, the system \cref{orbital_eqs_kep} is now decoupled from \cref{spin-sin_V_trun}. Its solution is $r=r(t)$, $f=f(t)$ given by \Cref{t,r,f} and depends on the eccentricity of the orbit $e$ and its semi-major axis $a$.

Let us now write $V_2$ and $V_4$ in a more convenient way. The quantity $M_j a^2$ is a sort of orbital moment of inertia of the body $\mathcal E_j$. Then, we can define
\begin{equation}\label{hat_d_q}
\hat d_j=\frac{d_j}{M_j a^2},\quad \hat q_j=\frac{q_j}{M_j a^2},
\end{equation}

\noindent so that $\hat d_j$ measures the equatorial oblateness of $\mathcal E_j$ with respect to the size of the orbit and $\hat q_j$ measures the flattening of $\mathcal E_j$ with respect to the size of the orbit.

Taking into account that in our units $G=a^3$, the terms $V_2$ and $V_4$ can be written in a compact way as
\begin{equation}\label{V_2}
V_2=-\frac{1}{4}\parentesis{\frac{a}{ r(t)}}^{3}  (\Lambda_0+\Lambda_1\cos (2\theta_1-2f(t))+\Lambda_2\cos (2\theta_2-2f(t)))
\end{equation}

\noindent and
\begin{equation}\label{V_4}
V_4=-\frac{1}{4}\parentesis{\frac{a}{ r(t)}}^{5}
\sum_{(m_1,m_2)\in \Xi}
\Lambda_{m_2}^{m_1}
\cos(2 m_1(\theta_1-f(t))+2m_2 (\theta_2-f(t)))
\end{equation}

\noindent where
\begin{equation*}
\Xi=\{(m_1,m_2)\in \mathds Z^2 : \ |m_1|+|m_2|\le 2 \},
\end{equation*}

\noindent and the following $\Lambda$ parameters are defined by
\begin{equation}\label{Lambda_j}
\Lambda_1=3 d_1M_2,\qquad \Lambda_2=3d_2M_1,
\end{equation}
\begin{equation}\label{Lambda_1_0}
\Lambda_0^1=\Lambda_0^{-1}=\frac{5}{56}(7\hat q_2+5\hat q_1) \Lambda_1, \qquad \Lambda_1^0=\Lambda_{-1}^0=\frac{5}{56}(7\hat q_1+5\hat q_2) \Lambda_2,
\end{equation}
\begin{equation}\label{Lambda_2_0}
\Lambda_0^2=\Lambda_0^{-2}=\frac{25}{32}\hat d_1 \Lambda_1, \qquad \Lambda_2^0=\Lambda_{-2}^0=\frac{25}{32}\hat d_2 \Lambda_2,
\end{equation}
\begin{equation}\label{Lambda_1_1}
\Lambda_1^1=\Lambda_{-1}^{-1} = \frac{35}{16} \hat d_1 \Lambda_2 = \frac{35}{16} \hat d_2 \Lambda_1,\qquad
\Lambda_1^{-1}=\Lambda_{-1}^1 = \frac{3}{16} \hat d_1 \Lambda_2 = \frac{3}{16} \hat d_2 \Lambda_1
\end{equation}
\begin{equation*}
\Lambda_0= q_1M_2+ q_2M_1,\qquad\Lambda_0^0= \frac{9}{4}\hat q_1 q_2 M_1 + \frac{15}{112}(\Lambda_1 \hat d_1+6\hat q_1 q_1M_2 + \Lambda_2 \hat d_2+6\hat q_2 q_2M_1).
\end{equation*}

\noindent With the last definitions we can write equations \cref{spin-sin_V_trun} as $\mathcal C_j \ddot \theta = \mathcal T_j^C$, where $\mathcal T_j^C=-\partial_{\theta_j} (V_2+V_4) $ are the \textit{conservative} torques of the spin-spin model shown in \cref{T_cons}. This can be checked with the expressions \cref{V_2} and \cref{V_4}. Note that $m_j\Lambda_{m_2}^{m_1}$ is in all cases proportional to the corresponding $\Lambda_j$. Then, the equations of the \textit{conservative} spin-spin model \cref{spin-sin_V_trun} can be written in terms of the physical parameters in the following symmetric way for $j=1,2$,
\begin{multline}\label{spin_spin_j}
0=   \ddot \theta_j +
  \frac{\lambda_j}{2} \Big \{   \parentesis{\frac{a}{ r(t)}}^{3} \sin(2 \theta_j-2f(t)) + \\
+  \parentesis{\frac{a}{ r(t)}}^{5} \Big [
\frac{5}{4} \parentesis{ \hat q_{3-j}+ \frac{5}{7}  \hat q_j} \sin(2 \theta_j-2f(t))
+\frac{25   \hat d_j}{8}  \sin(4 \theta_j-4f(t))  \\
+ \frac{3   \hat d_{3-j}}{8}  \sin(2 \theta_j-2\theta_{3-j})
+ \frac{35   \hat d_{3-j}}{8}  \sin(2 \theta_{3-j}+2\theta_j-4f(t))
\Big ] \Big\},
\end{multline}

\noindent where
\begin{equation*}
\lambda_j=\frac{\Lambda_j}{\mathcal C_j} = 3\frac{d_j}{\mathcal C_j}\frac{\mu}{M_j}.
\end{equation*}

It is worth mentioning that the terms with $\hat q_j$ and $\hat d_j$ in \cref{spin_spin_j} were missing in the model used in \cite{batmor2015} due to the dumbbell simplification for one of the bodies in the derivation of the equations. Not all the parameters appearing in \cref{spin_spin_j} are free because the following identities hold
\begin{equation}
\mathcal C_1+\mathcal C_2=1, \quad  \Lambda_1 \hat d_2=\Lambda_2 \hat d_1,\quad \Lambda_1 \hat q_2=\Lambda_2 \hat q_1.
\end{equation}

\noindent In consequence, our model depends on six independent parameters with physical meaning $(e;\mathcal C_1, \lambda_1, \lambda_2, \hat d_1,\hat q_1)$. Moreover, in \cref{spin_spin_j} we see that spin of the ellipsoid $\mathcal E_2$ is affected by the spin-spin coupling with a strength essentially given by $\hat d_1$, and vice versa.

\section{Linear stability of the double synchronous resonance in the conservative model}\label{sec:lin_stab

In this section we deal with the \textit{conservative} system with the notation in \cref{spin_spin_mayus}, that is more convenient for our purpose. The main result is \Cref{th_estabilidad}. It determines a region of linear stability of the double synchronous resonance in the space of parameters of the system.

\subsection{Existence of the odd $2\pi$-periodic solution}

The system \cref{spin_spin_mayus} can be written as
\begin{equation}\label{F}
\mathcal C \ddot \Theta  +F(t,\Theta)=0,
\end{equation}

\noindent where
\begin{equation*} \Theta=\vecdos{\Theta_1}{\Theta_2},	\quad \mathcal C=\matdos{\mathcal C_1}{0}{0}{\mathcal C_2},\quad \mathcal C_j>0,
\end{equation*}

\noindent and $F(t,\Theta)$ is the bounded function given by
\begin{multline}\label{F_Theta}
F(t,\Theta) =
\parentesis{\frac{a}{ r(t)}}^{3}  \vecdos{\Lambda_1\sin \Theta_1}{\Lambda_2\sin \Theta_2} + \\
+\parentesis{\frac{a}{ r(t)}}^{5}
\sum_{(m_1,m_2)\in \Xi } \vecdos{m_1}{m_2} \,\Lambda_{m_2}^{m_1}
\sin(m_1\Theta_1+m_2 \Theta_2)
	+2\ddot f(t) \vecdos{\mathcal C_1}{\mathcal C_2}.
\end{multline}

Note that equation \cref{F} is invariant under the change $(t,\Theta)\rightarrow (-t,-\Theta)$, since  $f(-t)=-f(t)$ and $r(-t)=r(t)$. Then, if $\Theta(t)$ is a solution of \cref{F}, so it is $-\Theta(-t)$. On the other hand, for $e=0$, we have $f(t)=t$ and $r(t)=a$, meaning that the system \cref{F} is that of two coupled free pendula. For this case, the trivial solution $\Theta(t)\equiv 0$ is a stable equilibrium. Then, for $e\ne0$, it is natural to look for the $2\pi$-periodic continuation of $\Theta(t)\equiv 0$  in the family of the odd solutions of \cref{F}, say, solutions satisfying $\Theta(-t)=-\Theta(t)$. This is equivalent to solve the Dirichlet problem
\begin{equation}\label{Dirichlet_e}
\left \{
\arrays{l}{
	\mathcal C \ddot \Theta  +F(t,\Theta)=0, \\
	\Theta(0)=\Theta(\pi)=0.
}
\right.
\end{equation}

It is well known from nonlinear analysis that the system \cref{Dirichlet_e} has at least one solution because $F(t,\Theta)$ is bounded. We can give a simple proof for this. Let $\Theta(t)=\vartheta(t,v)$ be the solution of \cref{F} satisfying initial conditions $\Theta(0)=0$, $\dot\Theta(0)=v\in \mathds{R}^2$. Solutions of the problem \cref{Dirichlet_e} are in correspondence with the solutions of the equation $\vartheta(\pi,v)=0$. From \cref{F}, we know that $\vartheta$ satisfies the following integral equation
\begin{equation}\label{integral_eq}
\vartheta(t,v) = v t - \int_0^t (t-s)\mathcal C^{-1}F(s,\vartheta(s,v))\d s.
\end{equation}

Let $||\cdot||$ be a norm in $\mathds R^2$, for instance, the maximum norm or the Euclidean one. We will employ the same notation for the corresponding induced matrix norm in $\mathds R^{2\times 2}$. Since there exists a positive number $M\ge ||\mathcal C^{-1}F(t,\Theta)||$, then
\begin{equation*}
||\vartheta(t,v) - v t || \le M\frac{t^2}{2},
\end{equation*}

\noindent for each $t\in \mathds R$. If we take $t=\pi$, then, $||\Phi(v)||\le M \pi/2$, with $\Phi(v)=v-\vartheta(\pi,v)/\pi$ and $v\in \mathds R^2$. Hence, we can apply Brouwer's fixed-point theorem to guarantee that $\Phi(v)$ has a fixed point for some $v_0$ satisfying $||v_0||\le M \pi/2$. For such point we have that $\vartheta(\pi,v_0)=0$, and the corresponding $\vartheta(t,v_0)$ satisfyies \cref{Dirichlet_e}.

\subsection{Uniqueness of the solution}

We know now that the Dirichlet problem \cref{Dirichlet_e} has a solution, however, it is not necessarily unique. For instance, if $\Lambda > 1$, there is not a unique solution for the free pendulum equation $\ddot x+\Lambda \sin x =0$, $x\in \mathds R$, with Dirichlet conditions $x(0)=x(\pi)=0$. See \cite{maw2004}. We would like to determine sufficient conditions on the space of parameters of the system such that there is uniqueness for the problem \cref{Dirichlet_e}.

We can prove uniqueness by a contradiction argument. Define the following matrix
\begin{equation*}
\mathcal{C}^{1/2} = \matdos{\sqrt{\mathcal C_1}}{0}{0}{\sqrt{\mathcal C_2}}
\end{equation*}

\noindent and its inverse $\mathcal{C}^{-1/2}=(\mathcal{C}^{1/2})^{-1}$. Let $\Theta^{(0)}(t)$ and $\Theta^{(1)}(t)$ be two non-identical solutions of \cref{Dirichlet_e}. Then, we can check that $y(t)=\mathcal{C}^{1/2}(\Theta^{(1)}(t)-\Theta^{(0)}(t))$ is a solution of the Dirichlet problem
\begin{equation}\label{y}
\left \{
\arrays{l}{
	\ddot y + A(t) y=0, \\
	y(0)=y(\pi)=0,
}
\right.
\end{equation}

\noindent with $A(t)$ a symmetric\footnote{In this paper we use properties of linear systems with symmetric coefficient matrices. $\mathcal C^{-1}\partial_\Theta F(t,\Theta)$ is not symmetric, but we obtain the desired structure using $\mathcal C^{1/2}$. See \cite{yak1975}.} matrix given by
\begin{equation}\label{A_int}
\mathcal{C}^{1/2}A(t)\mathcal{C}^{1/2}=\int_0^1 \partial_\Theta F(t, \Theta^{(\lambda)}(t)) \d \lambda,
\end{equation}

\noindent where $\Theta^{(\lambda)}(t)=\lambda \Theta^{(1)}(t) + (1-\lambda) \Theta^{(0)}(t)$ and
\begin{equation*}
F(t,\Theta) = \vecdos{F_1(t,\Theta)}{F_2(t,\Theta)}, \quad
\partial_\Theta F(t,\Theta)
=\matdos{\frac{\partial F_1}{\partial{\Theta_1}} }{\frac{\partial F_1}{\partial{\Theta_2}} }{\frac{\partial F_2}{\partial{\Theta_1}} }{\frac{\partial F_2}{\partial{\Theta_2}} }.
\end{equation*}

The statement of uniqueness is given in \Cref{propo_unicidad}. We can prove it by guaranteeing that \cref{y} has only the trivial solution. In the proof we are going to apply the following lemma to \cref{y} for a generic matrix $A(t)\in \mathds R^{d\times d}$. But first we need some definitions. Let $\langle \cdot, \cdot \rangle$ be the Euclidean inner product in $\mathds R^d$ and $||\cdot||$ its corresponding norm. Let $\mathds 1$ be the unit matrix in $R^{d\times d}$.

\begin{definition}\label{def:sym}
	Let $A_1,A_2\in \mathds R^{d\times d}$ be two symmetric matrices. We say that $A_1\le A_2$ if, for the corresponding quadratic forms, $\langle A_1 \, y , y  \rangle \le \langle A_2 \, y , y  \rangle$ for all $y\in \mathds R^d$.
\end{definition}

\begin{lemma}\label{lema_A}
	Assume that, for some $\gamma<1$, the matrix $A(t)\in \mathds R^{d\times d}$ is such that $A(t) \le \gamma \mathds 1 $ for each $t\in [0,\pi]$. Then, the only solution of $\ddot y + A(t) y=0$, $y\in \mathds R^d$, with Dirichlet conditions $
	y(0)=y(\pi)=0$ is the trivial one.
\end{lemma}

\begin{proof}
	Proceed by contradiction. Let $y(t)$ be a non-trivial solution of $\ddot y + A(t) y=0$,  $
	y(0)=y(\pi)=0$, then,
	\begin{equation*}
	\int_0^\pi \langle \ddot y (t), y (t) \rangle + \int_0^\pi \langle A(t) y (t), y (t) \rangle =0,
	\end{equation*}

	\noindent integrating by parts it follows that
	\begin{equation*}
	\int_0^\pi ||\dot y (t)||^2 = \int_0^\pi \langle A(t) y (t), y (t) \rangle .
	\end{equation*}

	Let $y_n(t)$ be the components of the vector $y(t)$. From the Sobolev inequality $\int_0^\pi|y_n(t)|^2\le \int_0^\pi|\dot y_n(t)|^2$, see \cite{zha} or \cite{misort2020}, we get that
	\begin{equation*}
	\int_0^\pi ||y (t)||^2 \le  \int_0^\pi \langle A(t) y (t), y (t) \rangle.
	\end{equation*}

	\noindent This contradicts the hypothesis $A(t) \le \gamma \mathds 1 $ for some $\gamma<1$. Then, $y (t)$ must be the trivial solution.
\end{proof}

Let us define the matrix $\tilde A(t,\Theta)=\mathcal{C}^{-1/2}  \partial_\Theta F(t,\Theta)\mathcal{C}^{-1/2}$,
\begin{multline}\label{A_t_Th}
\tilde  A(t,\Theta)=\parentesis{\frac{a}{ r(t)}}^{3}
\matdos{\frac{\Lambda_1}{\mathcal C_1} \cos \Theta_1}{0}{0}{\frac{\Lambda_2}{\mathcal C_2} \cos \Theta_2}
\\
+\parentesis{\frac{a}{ r(t)}}^{5}
\sum_{(m_1,m_2)\in \Xi } \matdos{\frac{m_1^2}{\mathcal C_1}}{\frac{m_1m_2}{\sqrt{\mathcal C_1\mathcal C_2}}}{\frac{m_1m_2}{\sqrt{\mathcal C_1\mathcal C_2}}}{\frac{m_2^2}{\mathcal C_2}} \,\Lambda_{m_2}^{m_1}
\cos(m_1\Theta_1+m_2 \Theta_2).
\end{multline}

We will use the maximum norm
\begin{equation*}
||y||=\max \{ |y_1|,|y_2| \},\quad y=\vecdos{y_1}{y_2},
\end{equation*}

\noindent and its induced norm in matrices
\begin{equation*}
||A||=\max \left \{|A_{11}|+|A_{12}|,|A_{21}|+|A_{22}| \right \},\quad A=\matdos{A_{11}}{A_{12}}{A_{21}}{A_{22}}.
\end{equation*}

\begin{theorem}\label{propo_unicidad}
	Assume that $e\in[0,1)$ and the parameters of the problem satisfy
	\begin{equation}\label{cond_un}
	1>\frac{1}{(1-e)^3}\max \left \{ \frac{\Lambda_1}{\mathcal C_1}  (1 + \alpha_1) , \frac{\Lambda_2}{\mathcal C_2}  (1 + \alpha_2) \right \},
	\end{equation}

	\noindent where
	\begin{equation}\label{alpha}
	\alpha_j \frac{\Lambda_j}{\mathcal C_j} = \frac{1}{(1-e)^2} \sum_{(m_1,m_2)\in \Xi }\parentesis{\frac{m_j^2}{\mathcal C_j}+\frac{|m_1m_2|}{\sqrt{\mathcal C_1\mathcal C_2}}}
	\Lambda_{m_2}^{m_1}.
	\end{equation}

	\noindent Then, there exists a unique solution of the Dirichlet problem \cref{Dirichlet_e}, denoted by $\Theta^*(t)$.
\end{theorem}

\begin{proof}
	Using the fact that $a/r\le 1/(1-e)$ by \cref{r}, equations \cref{cond_un} and \cref{alpha} imply that $1>||\tilde A(t,\Theta)||$ for all $(t,\Theta)\in \mathds R^3$, where we use the maximum norm. Furthermore, if $\rho (A)$ is the spectral radius of $A$, the well known inequality $ ||\tilde A(t,\Theta)||\ge \rho (\tilde A(t,\Theta))$ guarantees that $ \gamma \mathds 1\ge \tilde A(t,\Theta)$ for some $\gamma<1$. Then, $ \gamma \mathds 1\ge  A(t)$ for $A(t)$ defined in \cref{A_int}.  Now a direct application of \Cref{lema_A} finishes the proof.
\end{proof}

\begin{remark}
	Note that, as in the spin-orbit problem, there are two special cases for which $\Theta^*$ can be computed explicitly for some combination of parameters satisfying \cref{cond_un}. If $\Lambda_j=0$ and $\Lambda_{m_2}^{m_1}=0$ for $m_1m_2\ne0$, for each $e\in(0,1)$ the solution is the synchronous resonance of the uncoupled system
	\begin{equation}
	\Theta^*(t)=2(t-f(t,e))\vecdos{1}{1}.
	\end{equation}

	\noindent On the other hand, if $e=0$ the solution is $\Theta^*(t)=0$.
\end{remark}

\subsection{Linear stability of the solution}

Now we are interested in the stability properties of the solution $\Theta^*(t)$, which should be seen as $2\pi$-periodic and odd from now on. In the following we will find a region of parameters guaranteeing stability of the (scaled) linearized system of \cref{spin_spin_mayus} at the periodic solution $\Theta^*$, say,
\begin{equation}\label{variacional_Theta}
\ddot y + A(t) y=0,
\end{equation}
\noindent where we take the symmetric matrix $A(t)$ now defined by
\begin{equation*}
A(t)=\tilde A(t,\Theta^*(t))=\mathcal{C}^{-1/2}  \partial_\Theta F(t,\Theta^*(t))\mathcal{C}^{-1/2},
\end{equation*}
\noindent and $\tilde A(t,\Theta)$ was defined in \cref{A_t_Th}.

Recall from \Cref{sub:model} that the \textit{conservative} spin-spin model has a time-dependent Hamiltonian structure given by \cref{ham}. The variational equations associated to periodic solutions, like \cref{variacional_Theta}, are linear Hamiltonian systems with periodic coefficients. We will abbreviate them by LPH systems\footnote{The linear system $\mathcal C \ddot y + \partial_\Theta F(t,\Theta^*(t)) y=0$ is an LPH system in the general sense. However, for simplicity, we particularize the general theory to \cref{variacional_Theta}.}. These systems have some special properties that we will use in the following. For the general theory see \cite{yak1975} or \cite{eke}. For example, assume that $\varphi$ is a Floquet multiplier of an LPH system. Then, its inverse $\varphi^{-1}$, its complex conjugate $\bar \varphi$ and $\bar \varphi^{-1}$ are also  multipliers and have the same multiplicity as $\varphi$. This is stated in Corollary 6 of Chapter 1.1 of \cite{eke}. Let us point out two interesting consequences. First, a necessary condition for stability of an LPH system is that all its Floquet multipliers must have modulus 1. Second, an LPH system can never be asymptotically stable. In order to do continuation of periodic solutions to the \textit{dissipative} regime we will need the concept of strong stability for LPH systems.

\begin{definition}
	Let $A_0(t)\in \mathds R^{d\times d}$ be a fixed symmetric and $T$-periodic matrix. Assume that the there exists a number $\varepsilon>0$ such that the equation $\ddot y +A_*(t)y=0$ is stable for all $A_*(t)\in \mathds R^{d\times d}$ symmetric and $T$-periodic satisfying $\int_0^T||A_*(t)-A_0(t)||<\varepsilon$. Then, $\ddot y + A_0(t) y=0$ is strongly stable.
\end{definition}
In other words, if an LPH system is strongly stable, then, any  sufficiently small perturbation of it is stable. The perturbation should keep the Hamiltonian structure. Let us illustrate this with an example of the so-called Mathieu equation. Consider the $2\pi$-periodic equation
\begin{equation*}
\ddot x + \frac{1}{4} (1+\epsilon \cos t)x=0,\quad x\in \mathds R.
\end{equation*}
\noindent For $\epsilon=0$ it is stable, but not strongly stable, because we can always find a small number $\epsilon\ne 0$ such that the corresponding equation is not stable. This is called parametric resonance, see \cite{arn}.

Strong stability can be characterized with the Floquet multipliers of the system. For example, take an LPH system whose multipliers belong to the unit circle. If the multiplicity of all the multipliers is one, then the system is strongly stable. However, the converse is not true. M. Krein developed a theory to determine if a system is strong stable with further algebraic properties of the multipliers. For our purpose of making continuation of periodic solutions the following property is relevant.

\begin{proposition}\label{propo:1}
	Assume that $\ddot y + A(t)y=0$, with $A(t)\in \mathds R^{d\times d}$ symmetric and $T$-periodic, is strongly stable. Then, neither $1$ nor $-1$ are Floquet multipliers of the system.
\end{proposition}

We will not prove this property because it is a particular result of the general theory. Nonetheless, it can be inferred by the paragraph previous to Theorem 10 in Chapter 1.2 of \cite{eke}, that is the main result of Krein's theory.

Some sufficient conditions for strong stability of \cref{variacional_Theta} are given by the following Lyapunov-like stability criterion, from Test 4, in \cite{yak1975}, Chapter III, Section 7.

\begin{stability}
	The equation $\ddot y + A(t)y=0$, with $A(t)\in \mathds R^{d\times d}$ symmetric and $2\pi$-periodic, is strongly stable provided that, for all $x\in \mathds R^d \backslash\{0\}$,
	\begin{equation}\label{cond_lin_est}
	\int_0^{2\pi} \langle A(t)x,x \rangle \d t >0 \quad \text{and} \quad \int_0^{2\pi} \Tr (A(t))\d t <\frac{2}{\pi}.
	\end{equation}
\end{stability}

This stability test is the main tool for the proof of the next theorem.

\begin{theorem}\label{th_estabilidad}
	Assume that the parameters of the model satisfy the following conditions.
	\begin{equation}\label{cond_lin_est1}
	\frac{1}{\pi^2}>
	\frac{1}{(1-e)^3} \parentesis{\frac{\Lambda_1}{\mathcal C_1}+\frac{\Lambda_2}{\mathcal C_2}}
	+ \frac{1}{(1-e)^5} \sum_{(m_1,m_2)\in \Xi }\parentesis{\frac{m_1^2}{\mathcal C_1}+\frac{m_2^2}{\mathcal C_2}}
	\Lambda_{m_2}^{m_1},
	\end{equation}
	\begin{equation}\label{cond_lin_est2}
	\frac{1}{4\pi} > M := \frac{1}{(1-e)^3} \max \left \{\frac{\Lambda_1}{\mathcal C_1},\frac{\Lambda_2}{\mathcal C_2} \right \}
	+\frac{1}{(1-e)^5} \sum_{(m_1,m_2)\in \Xi }\max \left \{\frac{|m_1|}{\mathcal C_1},\frac{|m_2|}{\mathcal C_2} \right \} \Lambda_{m_2}^{m_1} + \frac{4e\sqrt{1-e^2}}{(1-e)^4},
	\end{equation}
	\begin{equation}\label{cond_lin_est3}
	\cos (2\pi^2 M ) \min \left \{\frac{\Lambda_1}{\mathcal C_1},\frac{\Lambda_2}{\mathcal C_2} \right \} > \max \left \{\alpha_1\frac{\Lambda_1}{\mathcal C_1},\alpha_2 \frac{\Lambda_2}{\mathcal C_2} \right \}
	,
	\end{equation}

	\noindent with $\alpha_j$ defined in \cref{alpha}. Then the solution $\Theta^*(t)$ is strongly linearly stable.
\end{theorem}

Note that the second condition of \cref{cond_lin_est} is guaranteed by \cref{cond_lin_est1}. The first condition of \cref{cond_lin_est} is a bit more complicated, but its proof is immediate by the following two lemmas.

\begin{lemma}\label{lema_bound_sol}
	The components of the solution $\Theta^*(t)$ satisfy the following bounds $|\Theta^*_j(t)|\le 2 \pi^2 M$, $|\dot \Theta^*_j(t)|\le 2 \pi M$ provided that $M\ge ||\mathcal C^{-1} F(t,\Theta^*(t))||$.
\end{lemma}

\begin{proof}
	Integrating the identity $\ddot\Theta^*(t)+\mathcal C^{-1} F(t,\Theta^*(t))=0$ and taking the first component,
	\begin{equation*}
	\dot\Theta_1^*(t) = \dot\Theta_1^*(t_0) -\int_{t_0}^t u_1 \, \mathcal C^{-1} F(s,\Theta^*(s)) \d s,
	\end{equation*}

	\noindent where $u_1$ is the row vector $(1, 0)$. Then, for $t\in[t_0,t_0+2\pi]$,
	\begin{equation*}
	|\dot\Theta_1^*(t)|  \le |\dot\Theta_1^*(t_0)| + \int_{t_0}^{t_0+2\pi} ||\mathcal C^{-1} F(s,\Theta^*(s))||\d s \le  |\dot\Theta_1^*(t_0)| +2\pi M,
	\end{equation*}

	\noindent where $||\cdot ||$ indicates a matrix norm induced by a norm in $\mathds R^2$. Since $\Theta_1^*(t)$ is $2\pi$-periodic, we can choose $t_0$ such that $ \dot\Theta_1^*(t_0)=0$. The same is applicable to $\Theta_2$ for a possibly different $t_0$, consequently, $|\dot\Theta_j^*(t)|  \le 2\pi M$ for all $t$. Furthermore, since $\Theta_1^*(0)=0$,
	\begin{equation*}
	\Theta_1^*(t) = \int_{0}^t \dot\Theta_1^*(s) \d s,
	\end{equation*}

	\noindent and, due to the odd symmetry of $\Theta_1^*(t)$, it is enough to consider $t\in[0,\pi]$. Then, $|\Theta_1^*(t)|\le 2\pi^2M$. The same is true for $\Theta_2^*(t)$.
\end{proof}

\begin{lemma}\label{lema_def_pos}
	The conditions \cref{cond_lin_est2} and \cref{cond_lin_est3} are sufficient so that  $A(t)=\tilde A(t,\Theta^*(t))\ge \gamma \mathds 1$ for some $\gamma>0$.
\end{lemma}

\begin{proof}
	The proof this lemma is based on the following fact. Considering the partial ordering of symmetric matrices given by \Cref{def:sym}, the conditions \cref{cond_lin_est2} and \cref{cond_lin_est3} imply that the term proportional to $1/r^3$ in \cref{A_t_Th} dominates the other term, that is proportional to $1/r^5$. Let us prove it. We can compute the derivatives of $f(t)$ using \Cref{t,r,f} and get
	\begin{equation*}
	\ddot f(t)=-\frac{2e\sqrt{1-e^2} \sin (u(t))}{(1-e\cos (u(t)))^4},
	\end{equation*}

	\noindent where $u$ is the eccentric anomaly. Using the maximum norm we see from \cref{cond_lin_est2} and \cref{F_Theta} that $1/(4\pi)>M\ge ||\mathcal C^{-1} F(t,\Theta^*(t))||$. Furthermore, from \Cref{lema_bound_sol} we know that $|\Theta^*_j(t)|\le 2 \pi^2 M$, then, we can see graphically that
	\begin{equation*}
	\cos \Theta_1^*(t) \ge \cos (2\pi^2 M ) > 0,
	\end{equation*}

	\noindent therefore,
	\begin{equation}\label{order_r3}
		\matdos{\frac{\Lambda_1}{\mathcal C_1} \cos \Theta_1^*(t)}{0}{0}{\frac{\Lambda_2}{\mathcal C_2} \cos \Theta_2^*(t)} \ge \cos (2\pi^2 M )  \min \left \{\frac{\Lambda_1}{\mathcal C_1},\frac{\Lambda_2}{\mathcal C_2} \right \}  \mathds 1.
	\end{equation}

	On the other hand, let us define
	\begin{equation*}
	B=-\parentesis{\frac{a}{ r(t)}}^{2}
	\sum_{(m_1,m_2)\in \Xi } \matdos{\frac{m_1^2}{\mathcal C_1}}{\frac{m_1m_2}{\sqrt{\mathcal C_1\mathcal C_2}}}{\frac{m_1m_2}{\sqrt{\mathcal C_1\mathcal C_2}}}{\frac{m_2^2}{\mathcal C_2}} \,\Lambda_{m_2}^{m_1}
	\cos(m_1\Theta_1^*(t)+m_2 \Theta_2^*(t)).
	\end{equation*}

	\noindent As we did in the Proof of \Cref{propo_unicidad}, we can take the maximum norm and obtain that
	\begin{equation*}
	\max\left \{\alpha_1\frac{\Lambda_1}{\mathcal C_1},\alpha_2 \frac{\Lambda_2}{\mathcal C_2} \right \} \ge ||B||\ge\rho(B)
	\end{equation*}

	\noindent where $\rho(B)$ is the spectral radius of $B$, then,
	\begin{equation*}
	 \max \left \{\alpha_1\frac{\Lambda_1}{\mathcal C_1},\alpha_2 \frac{\Lambda_2}{\mathcal C_2} \right \} \mathds 1 \ge B.
	\end{equation*}

	\noindent From this inequality, \cref{order_r3} and the definition \cref{A_t_Th} of  $\tilde A(t,\Theta)$, we prove that $\tilde A(t,\Theta^*(t))\ge \gamma \mathds 1$  with
	\begin{equation*}
	\gamma=\cos (2\pi^2 M )  \min \left \{\frac{\Lambda_1}{\mathcal C_1},\frac{\Lambda_2}{\mathcal C_2} \right \} - \max \left \{\alpha_1\frac{\Lambda_1}{\mathcal C_1},\alpha_2 \frac{\Lambda_2}{\mathcal C_2} \right \}>0.
	\end{equation*}
\end{proof}

Now we see that \Cref{lema_def_pos} implies the first condition of \cref{cond_lin_est} because $\langle A(t)x,x \rangle\ge \gamma ||x||^2>0 $.

\section{The synchronous resonance in the dissipative regime}\label{sec:asymp

Recall from \cref{spin_spin_dis_mayus} that the \textit{dissipative} spin-spin model takes the form of the system
\begin{equation}\label{spin_spin_dis_mayus_bis}
 \ddot \Theta +  \diag (\delta) D(t) \dot \Theta + \mathcal C^{-1}F(t,\Theta) = 0   ,\qquad \delta=\vecdos{\delta_1}{\delta_2},\enspace \delta_j\ge 0,
\end{equation}

\noindent with $D(t)=(a/r(t))^6$. We know from \Cref{th_estabilidad} that, for $\delta=0$, there exists an odd $2\pi$-periodic solution $\Theta^*(t)$, that is strongly linearly stable in the set of the parameters space satisfying the conditions given in \Cref{cond_lin_est1,cond_lin_est2,cond_lin_est3}.

The main result of this section is \Cref{teo_asint}. There we will see that the \textit{conservative} periodic solution $\Theta^*(t)$ can be continued in the presence of friction to an asymptotically stable periodic solution $\Psi^*(t,\delta)$. However, the odd symmetry of the solution is lost because \cref{spin_spin_dis_mayus_bis} is not invariant under the change $(t,\Theta)\rightarrow (-t,-\Theta)$ as in the \textit{conservative} case. The proof of \Cref{teo_asint} is mainly based on Theorem 2 in \cite{misort2020} and on classical results on continuation of periodic solutions summarized in the next proposition.

\begin{proposition}\label{propo:cod}
	Let $\mathcal F$ be a real analytic function $\mathcal F= \mathcal F(t,x,\zeta)$, such that $\mathcal F(t+T,x,\zeta)= \mathcal F(t,x,\zeta)$, with $t\in\mathds R$, $ x\in \mathds R^n$, $\zeta\in \mathds R^d$. Assume that the equation $\dot x = \mathcal F(t,x,0) $ has a $T$-periodic solution $x=p(t)$. \begin{enumerate}
		\item Suppose that $1$ is not a Floquet multiplier of the corresponding variational equation at $x=p(t)$,
		\begin{equation*}
		\dot y = \partial_x\mathcal F(t,p(t),0) y.
		\end{equation*}

		\noindent Then, for $\zeta\ne0$, with small enough norm $||\zeta||$, the equation $\dot x = \mathcal F(t,x,\zeta) $ has a $T$-periodic solution $x=p_c(t, \zeta)$ such that $p_c(t, 0)=p(t)$. Moreover, $p_c(t, \zeta)$ is an analytic function and it is unique of each $\zeta$.
		\item If additionally, $p(t)$ is asymptotically stable, then this is also true for $p_c(t, \zeta)$.
	\end{enumerate}
\end{proposition}

For the detailed proof of this proposition, see Theorems 1.1 and 1.2 in Chapter 14, \cite{cod}. Now we can state the main theorem.

\begin{theorem}\label{teo_asint}
	Assume that the parameters of the system satisfy the conditions in \Cref{th_estabilidad}. If $|\delta_j|$ are small enough, then there exists a function $\Psi^*(t,\delta)$, analytic in both entries, satisfying
	\begin{enumerate}
		\item[i)] $\Psi^*(t,0)= \Theta^* (t)$ for each $t\in \mathds R$.
		\item[ii)] $\Psi^*(t,\delta)$ is a $2\pi$-periodic solution of \cref{spin_spin_dis_mayus_bis}. Moreover, if $|\Lambda_{m_2}^{m_1}|$ are small enough, then, $\Psi^*(t,\delta)$ is asymptotically stable.
	\end{enumerate}
\end{theorem}

\begin{proof}
	Recall that the \textit{conservative} periodic solution $\Theta^* (t)$ is strongly linearly stable. \Cref{propo:1} guarantees that $1$ is not a Floquet multiplier of the variational equation at $\Theta^* (t)$. Then, we can apply the first item of \Cref{propo:cod} to make the analytic continuation of the periodic solution from the \textit{conservative} ($\delta_j=0$) to the \textit{dissipative} regime ($\delta_j> 0$). We conclude that there exists a unique analytic $2\pi$-periodic solution $\Psi^*(t,\delta)$ of \cref{spin_spin_dis_mayus_bis} such that $\Psi^*(t,0)= \Theta^* (t)$ for small enough $\delta_j$.

	Let us explain more in detail the proof that the continuation is asymptotically stable. If $\Lambda_{m_2}^{m_1}=0$ for all $({m_1},{m_2})\in \Xi$, then \cref{spin_spin_dis_mayus_bis} takes the form of two uncoupled \textit{dissipative} spin-orbit equations

	\begin{equation}\label{spin-orbit_j_dis}
	\ddot \Theta_j + \delta_j \parentesis{\frac{a}{ r(t)}}^6 \dot \Theta_j + \frac{\Lambda_j}{\mathcal C_j} \parentesis{\frac{a}{ r(t)}}^3 \sin \Theta_j =0.
	\end{equation}

	Besides, conditions in \Cref{cond_lin_est1,cond_lin_est2,cond_lin_est3} guarantee that, for $\Lambda_{m_2}^{m_1}=0$, the \textit{conservative} solution $\Theta^*(t)$ is strongly linearly stable. We can see the solution $\Theta^*(t)$ split in two components $\Theta_j^*(t)$, each of them is a solution of the \textit{conservative} spin-orbit problem \cref{spin-orbit_j_dis} with $\delta_j=0$. Now we can apply Theorem 2 in \cite{misort2020} that guarantees that each equation in \cref{spin-orbit_j_dis} has an asymptotically stable $2\pi$-periodic solution $\Theta_{j,\delta_j}^*(t)$ provided that $\delta_j\in (0,\bar \delta_j]$. Here $\bar \delta_j$ are small numbers quantified in \cite{misort2020}. Moreover, $\Theta_{j,\delta_j}^*(t)$ is the unique continuation of $\Theta_j^*(t)=\Theta_{j,0}^*(t)$.

	Let us consider \cref{spin-orbit_j_dis} as a system of two equations. This system has an asymptotically stable $2\pi$-periodic solution  $\Psi^*(t,\delta)=(\Theta_{1,\delta_1}^*(t),\Theta_{2,\delta_2}^*(t))^\mathsf{T}$ such that $\Psi^*(t,0)= \Theta^* (t)$. If $|\Lambda_{m_2}^{m_1}|$ are small, we can see \cref{spin_spin_dis_mayus_bis} as a perturbation of the system \cref{spin-orbit_j_dis} and apply the second item of \Cref{propo:cod}. In this way we guarantee that $\Psi^*(t,\delta)$ has a $2\pi$-periodic continuation for $\Lambda_{m_2}^{m_1}\ne0$ that is asymptotically stable if $|\Lambda_{m_2}^{m_1}|$ are small enough.
\end{proof}

Note that for asymptotic stability we require not only that $|\delta_j|$ should be small, but also $|\Lambda_{m_2}^{m_1}|$. We would like to erase this condition on the coupling parameters $\Lambda_{m_2}^{m_1}$. However, from a theoretical point of view, this is certainly difficult to address in general since we deal with systems of differential equations. Let us explain this point. The variational equation of \cref{spin_spin_dis_mayus_bis} near $\Psi^*(t,\delta)$ is
\begin{equation}\label{lin_dis}
\ddot \eta +  \diag (\delta) D(t) \dot \eta + \mathcal C^{-1}\partial_\Theta F(t,\Psi^*(t,\delta))\eta = 0   , \quad \delta=\vecdos{\delta_1}{\delta_2}.
\end{equation}

For $e\ne0$, \cref{lin_dis} is a linear $2\pi$-periodic system of two equations of second order. In \cite{misort2020}, asymptotic stability was proved for the spin-orbit problem taking advantage of the following fact. Any second order periodic equation $\ddot x + a_1(t)\dot x + a_0(t) x =0$, $x\in \mathds R$, $a_n(t)=a_n(t+T)$, can be converted into a Hill's equation $\ddot \chi + \alpha(t)\chi =0$, $\alpha(t)=a_0(t)-\frac{1}{4}a_1(t)^2- \frac{1}{2}\dot a_1(t)$, by the change of variables $\chi(t)=x(t)\exp (\frac{1}{2}\int_0^ta_1(s) \d s)$. See \cite{mag}. We can see the \textit{dissipative} problem ($a_1(t)\ne0$) as a perturbation of the \textit{conservative} one ($a_1(t)=0$). Assume that $\ddot x + a_0(t) x =0$ is strongly stable, then $\ddot \chi + \alpha(t)\chi =0$ is stable. Since it is a Hill's equation (also a LPH system), the modulus of the Floquet multipliers of $\ddot \chi + \alpha(t)\chi =0$ is 1. Now we undo the change of variables and conclude that the modulus of the Floquet multipliers of $\ddot x + a_1(t)\dot x + a_0(t) x =0$ is smaller than 1, therefore, it is asymptotically stable. However, it is not clear how to perform an analogous procedure in \cref{lin_dis}. The main obstacle is the non-commutativity of matrices due to the asymmetric nature of the \textit{dissipative} problem ($\delta_1\ne\delta_2$). Actually, if we follow the same steps, we end up with a system of equations that is no longer periodic for $\delta_1\ne\delta_2$. The numbers $\delta_j$ depend on several parameters of the bodies and we do not see any good physical reason to impose both dissipative parameters to be equal. In fact, if $\delta_1\ne\delta_2$, in principle the \textit{dissipative} spin-spin model cannot be considered conformally symplectic as the spin-orbit problem. See \cite{calceldel}. From this discussion, we conclude that this it is necessary a deeper theoretical study, but it is beyond the scope of this paper.

On the other hand, let us see that for $e=0$, the solution of \cref{lin_dis} is asymptotically stable. The solution given by \Cref{teo_asint} is $\Psi^*(t,\delta)\equiv 0$. Taking $y=\mathcal C^{1/2} \eta$, the corresponding variational equation is
\begin{equation}\label{lin_dis_cons}
\ddot y + \diag (\delta) \dot y + A y =0,
\end{equation}

\noindent where $A$ is the symmetric constant matrix given by
\begin{equation*}
A=
\matdos{\xi_1}{\sigma}{\sigma}{\xi_2}
=\matdos{\frac{\Lambda_1}{\mathcal C_1} }{0}{0}{\frac{\Lambda_2}{\mathcal C_2} }
+
\sum_{(m_1,m_2)\in \Xi } \matdos{\frac{m_1^2}{\mathcal C_1}}{\frac{m_1m_2}{\sqrt{\mathcal C_1\mathcal C_2}}}{\frac{m_1m_2}{\sqrt{\mathcal C_1\mathcal C_2}}}{\frac{m_2^2}{\mathcal C_2}} \,\Lambda_{m_2}^{m_1}
.
\end{equation*}

\noindent Note that, by conditions \cref{cond_lin_est2} and \cref{cond_lin_est3}, $A$ is a positive definite matrix. See \Cref{lema_def_pos}. The characteristic polynomial of equation \cref{lin_dis_cons} is
\begin{equation*}
p(\omega)=\omega^4 + (\delta_1+\delta_2 )\omega^3+(\xi_1+\xi_2+\delta_1\delta_2 )\omega^2+ (\xi_1\delta_2+\xi_2\delta_2)
\omega + \det A.
\end{equation*}

Equation \cref{lin_dis_cons} is asymptotically stable if and only if all the roots of $p(\omega)$ have negative real parts. This can be checked with the Routh-Hurwitz criterion, see \cite{gan1959}. According to it, all the roots of the polynomial have negative real parts if and only if the associated Hurwitz determinants of the polynomial are strictly positive, say,
\begin{equation*}
D_1=\delta_1+\delta_2 ,\quad D_2=\delta_1^2\delta_2+ \delta_2^2\delta_1+ \xi_1\delta_1 + \xi_2 \delta_2,
\end{equation*}
\begin{equation*}
D_3=D_1^2\sigma^2+\delta_1\delta_2(D_1(\xi_1\delta_2+\xi_2\delta_2)+(\xi_1-\xi_2)^2),\quad D_4=D_3 \det A.
\end{equation*}

\noindent Since $A$ is positive definite, we get asymptotic stability for all $\delta_1$ and $\delta_2$ such that both are non-negative and at least one is different from zero.

\section{Applications}\label{sec:app}

Recall from the end of \Cref{sec:model} that our model depends on six independent physical parameters $(e;\mathcal C_1, \lambda_1, \lambda_2, \hat d_1,\hat q_1)$, where $e$ is the orbital eccentricity, $\mathcal C_j$ the moment of inertia of $\mathcal E_j$ with respect to the $\mathsf{c}_j$-axis, $\lambda_j=\Lambda_j/\mathcal C_j$ is the oblateness of $\mathcal E_j$ in the plane of motion,  and $\hat d_j$ and $\hat q_j$ are, respectively, the oblateness and the flatness of $\mathcal E_j$ with respect to the size of the orbit.

We have two type of estimates. The first type in \cref{cond_un} guarantees uniqueness of the synchronous resonance in the \textit{conservative} regime. The second one in \Cref{cond_lin_est1,cond_lin_est2,cond_lin_est3} guarantees linear stability of the same solution. Our estimates depend on certain values $\alpha_j$ in \cref{alpha}. To write them in terms of the physical parameters, we use the definitions in \Cref{Lambda_j,Lambda_1_0,Lambda_2_0,Lambda_1_1}, then
\begin{align}
\label{sum1}
\sum_{(m_1,m_2)\in \Xi } \frac{m_j^2}{\mathcal C_j}
\Lambda_{m_2}^{m_1} = &
\lambda_j
\parentesis{
	\frac{25}{4}\hat d_j + \frac{25}{28}\hat q_j + \frac{19}{4}\hat d_{3-j} + \frac{5}{4}\hat q_{3-j}
},\\
\label{sum2}
\sum_{(m_1,m_2)\in \Xi } \frac{|m_1m_2|}{\sqrt{\mathcal C_1\mathcal C_2}}
\Lambda_{m_2}^{m_1} =& \frac{19}{4} \sqrt{\frac{\mathcal C_1}{\mathcal C_2}}
\lambda_1 \hat d_2 = \frac{19}{4} \sqrt{\frac{\mathcal C_2}{\mathcal C_1}}
\lambda_2 \hat d_1 ,
\\
\label{sum3}
\sum_{(m_1,m_2)\in \Xi } \frac{|m_j|}{\mathcal C_j}
\Lambda_{m_2}^{m_1} = &
\lambda_j
\parentesis{
	\frac{25}{8}\hat d_j + \frac{25}{28}\hat q_j + \frac{19}{4}\hat d_{3-j} + \frac{5}{4}\hat q_{3-j}
}.
\end{align}

\noindent Now we are ready to apply our estimates to specific cases.

\subsection{Real systems}

In one hand, the Pluto-Charon binary is the largest known system that is in double synchronous resonance. The physical parameters of the system relevant for the spin-spin model are shown in \Cref{tab:real}. Pluto is almost twice the size of Charon, contains the 89\% of the mass and the 97\% of the body moment of inertia ($\mathcal C_j$) of the system. Besides, the size of the orbit is quite large ($a=27.2$) compared to the sizes of the bodies. This results in very small values of $\hat d_j$ of order $10^{-7}$, which means this is a certainly weak spin-spin coupling. The orbit has a very small eccentricity $e=0.0002$. Recall that the double synchronous resonance of the circular case ($e=0$) is the trivial solution $\Theta(t)\equiv0$, both for the \textit{conservative} case \cref{spin_spin_mayus} and the \textit{dissipative} case \cref{spin_spin_dis_mayus}. The asymptotic stability of the solution for any value of the dissipative parameters is easily guaranteed, as it was shown at the end of \Cref{sec:asymp} using equation \cref{lin_dis_cons}. For the real eccentricity, the solution $\Theta^*(t)$ of \cref{spin_spin_mayus} oscillates very close to zero and our estimates guarantee the uniqueness and linear stability of  solution. Furthermore, \Cref{teo_asint} shows the existence of an asymptotically stable solution $\Psi^*(t,\delta)$ of the \textit{dissipative} model provided that $\delta_j$, $\hat d_j$ and $\hat q_j$ are small enough. Unfortunately, this last result is not quantified in this paper for the real parameters.

\begin{table}
	\begin{center}
		\begin{tabular}{|c||cccccc|cc|}
			\hline
			System   &  $M_j$ & $\mathsf{a}_j$ &$\mathcal C_j$    &  $ \lambda_j$ & $\hat d_j$         & $\hat q_j$ & $a$ & $e$ \\ \hline  \hline \hline
			Pluto &  0.89    & 1.65   & $0.97$ & $ 3.3\cdot 10^{-5} $  & $1.5\cdot 10^{-7}$ & 	$1.2\cdot 10^{-6}$ & \multirow{2}{*}{$27.2$} & \multirow{2}{*}{$2.0 \cdot 10^{-4}$} \\[0.5ex]
			Charon  & 0.11 & 0.84    & $0.03$ &  $2.4\cdot 10^{-3}$     & $3.5\cdot 10^{-7}$ & $8.2\cdot 10^{-7}$  & & \\	\hline   \hline
			Patroclus &  0.56    & 1.7   & $0.60$ & $ 0.11 $  & $2.6\cdot 10^{-4}$ & 	$1.2\cdot 10^{-3}$ & \multirow{2}{*}{$18.2\pm 0.5$} & \multirow{2}{*}{$0.02\pm0.02$} \\[0.5ex]
			Menoetius  & 0.44 & 1.6    & $0.40$ &  $0.14$     & $2.2\cdot 10^{-4}$ & $9.9\cdot 10^{-4}$  & &  \\	\hline
		\end{tabular}
		\caption{Real physical parameters for two binary systems. For Pluto and Charon, we take the largest values of $\lambda_j$, $\hat d_j$ and $\hat q_j$ obtained from data in \cite{kho2020}. The parameters of Patroclus and Menoetius are obtained from data in \cite{davsch2020} and the orbital parameters from \cite{mar2006}.}
		\label{tab:real}
	\end{center}
\end{table}

On the other hand, the Trojan binary asteroid 617 Patroclus is a system whose components are of similar size, mass and moment of inertia. See the physical parameters of its components, Patroclus and Menoetius, in \Cref{tab:real}. Each body has a diameter of around one hundred kilometres, almost ten times smaller than Charon. Patroclus and Menoetius have a more oblate ellipsoidal shape than Pluto and Charon and the size of the orbit in this case ($a=18.2\pm0.5$) is smaller. In consequence, the corresponding dynamical parameters $\lambda_j$, $\hat d_j$ and $\hat q_j$ are several orders of magnitude larger. The orbital eccentricity is not measured with enough precision, $e=0.02\pm 0.02$. With our estimates, we are able to guarantee the uniqueness of the solution $\Theta^*(t)$ of \cref{spin_spin_mayus} for eccentricities up to $e=0.04$. However, we fail to guarantee linear stability even for $e=0$. The main reason is that the stability test given by the conditions \cref{cond_lin_est} is not fine enough for such large values of $\lambda_j$. In the following subsection we will explain what is the range of parameters that is covered by our study.

\subsection{Stability diagrams in the space of parameters}

Note that all the terms appearing in \Cref{sum1,sum2,sum3} are positive. Since $\hat q\ge \hat d$, and, in order to reduce the parameters in the upper bounds for the expressions in \Cref{sum1,sum2,sum3}, we can take $\hat d_j=\hat q_j$. In this way, we reduce the independent parameters to five $(e;\lambda_1,\lambda_2,\mathcal C_1, \hat q_1)$. Note now that, to take $\hat q_1=0$ is equivalent to break the coupling of the system, resulting in two independent spin-orbit problems.

We will consider two special cases with three free parameters. In one hand, the case of identical bodies, that we compare with the asteroid 617 Patroclus. Here the parameters are $e$, $\lambda_j=\lambda$ and $\hat q_j=\hat q$. On the other hand, the case when $\mathcal E_1$ is twice the size of $\mathcal E_2$, that we compare with the Pluto-Charon system. Here we consider the same density and the free parameters are $e$, $\lambda_2$ and $ \hat q_1$, whereas the dependent parameters are $\lambda_1=2^{-3}\lambda_2$ and $\hat q_2=2^{-5}\hat q_1$.

\Cref{fig:both} shows regions in the space of parameters for which there is uniqueness and linear stability of the double synchronous resonance according to our theoretical estimates. We see that we cover the Patroclus-Menoetius system (top panels) only for the uniqueness of the solution but not for the linear stability. In contrast, the Pluto-Charon system (bottom panels) is covered for linear stability as well. We can compare the diagrams of $\hat q=0$ and $\hat q_1=0$ with the theoretical estimates obtained in \cite{misort2020}, shown in \Cref{fig:stab_spin_orbit}. We see that, although the uniqueness region is similar, the stability region (in yellow) is considerably larger in \Cref{fig:stab_spin_orbit} than those in \Cref{fig:both}. This shows that the mathematical techniques used in \cite{misort2020} are much finer than in this paper. In \cite{misort2020} we used generalized Lyapunov criteria using $L^p$-norms, with $p\in[1,\infty]$, see \cite{zha}, and upper and lower solutions to bound the amplitude of the solution. Instead, in this paper we use the stability test given by \cref{cond_lin_est}, that is of type $L^\infty$, and a rougher bound for the amplitude of the solution in \Cref{lema_bound_sol}. Since the model is quite new, here we initiate the analysis with a simpler approach. Besides, the mathematical tools are not as well developed for systems of equations as for standard second order scalar equations.

\begin{figure}
	\centering
	\scalebox{0.9}{\includegraphics{_fig_stab_both.pdf}}
	\caption{Stability diagrams in the $(e,\lambda)$-plane of the synchronous resonance of the spin-spin model. Top: both bodies are equal. Bottom: one body is double the size of the other. The double synchronous resonance is unique under the dashed lines (right) and linearly stable under the black lines (left) for the indicated value of $\hat q$. In the left we see zoomed views of the stable regions. The more yellow is the region indicates that stability is guaranteed for larger values of $\hat q$. The gray regions in the right are unstable for the uncoupled system (spin-orbit), i.e., with $\hat q=0$.}
	\label{fig:both}
\end{figure}

\begin{figure}
	\centering
	\scalebox{0.9}{\includegraphics{_fig_zona_seguridad_spin_orbit.pdf}
	}
	\caption{Stability diagram in the $(e,\lambda)$-plane of the spin-orbit in \cite{misort2020}, Figure 3.}
	\label{fig:stab_spin_orbit}
\end{figure}

\begin{figure}
	\centering
	\scalebox{0.5}{\includegraphics{_fig_stab_several3.pdf}
	}
	\caption{Stability diagrams in the $(e,\lambda)$-plane in the case of equal bodies. The six plots in the left show the unstable region in gray for different values of $\hat q$. The image in the right shows the six diagrams superimposed. Darker tones of gray indicate more overlapping between unstable regions.}
	\label{fig:stab_several}
\end{figure}

We see in \Cref{fig:both} that an increase in the value of $\hat q$ results in a global reduction of the regions that we estimated theoretically, both for stability and uniqueness regions. This behavior can be compared with the numerical plots in \Cref{fig:stab_several}. We focus only on the case of equal bodies. Here we see how the instability region changes when we increase $\hat q$. There are some interesting phenomena.

\begin{enumerate}
	\item For $\hat q=0$ there is only one bifurcation point for the unstable solution in the $\lambda$-axis at $(e,\lambda)=(0,0.25)$. However, for $\hat q>0$, it becomes two bifurcation points at $(0,\lambda_{(1)})$ and $(0,\lambda_{(2)})$, with $0<\lambda_{(1)}<\lambda_{(2)}<0.25$. This opens a small window of stability at the points $(e,\lambda)$ with $e$ close to $0$ and $\lambda\in (\lambda_{(1)},\lambda_{(2)})$.
	\item For $\hat q=0$, apart from the instability region bifurcating from the $\lambda$-axis, there is another one bifurcating from the $e$-axis at $(e,\lambda)\approx(0.682,0)$. The existence of such bifurcation was studied in \cite{misort2020}. However, for $\hat q>0$, it looks that the last bifurcation point moves to the right, at the same time that the two instability regions merge into a single one. This shows that turning on the coupling has a stabilizing effect of the synchronous resonance for large $e$ and small $\lambda$. This holds up to a critical $\hat q\in (0.05,0.1)$ for which another unstable region bifurcates from the $e$-axis. This region merges with the large one at some $\hat q\in (0.1,0.2)$. This leaves an island of stability for large $e$ and small $\lambda$.
	\item In the right panel of \Cref{fig:stab_several} we see that there are some regions (the darkest ones), that remain unstable, not very affected by changes in $\hat q$. Instead, the lighter regions show more susceptibility to change their stability when $\hat q$ changes.
\end{enumerate}

In \Cref{fig:stab_several} we have taken large values of $\hat q$, compared to the real values in \Cref{tab:real}. From its definition in \cref{hat_d_q} and \cref{q_d} we see that $\hat q\le 1/a^2$ for equal bodies ($M=0.5,\mathcal C=0.5$). In order to be consistent with the Keplerian orbit approximation, $a$ should be quite larger than 1, that gives the scale of the objects. For example, $a$ of order $10$ would give an upper estimate of $\hat q$ of order $10^{-2}$. In consequence, for more realistic parameters, we should not consider the appearance of the additional instability region bifurcating from the $e$-axis from large $\hat q$.

\section{Discussion}\label{sec:discussion}

In this paper we have proposed a simplified mathematical model for the rotational dynamics in the Full Two-Body Problem. This model is a straightforward continuation of the spin-orbit problem. In consequence, we hope it will be of interest for physical applications as well as for theoretical studies. We have approached the problem from a theoretical point of view, but always keeping what we think is the essence of the physical problem: the dissipative effects are fundamental to explain the universe we observe today. In this sense, the spin-spin model not only broaden the scope of the spin-orbit problem in a higher dimensional phase space, but also contributes to fill the gap between the conservative and the dissipative effects considered in the spin-orbit problem. More precisely, if the \textit{dissipative} torque (of order $1/r^6$) is important in the evolution of a satellite, then, we should consider also the spin-spin interaction (of order $1/r^5$). Of course this two effects are more important when the bodies are closer to each other. In fact, in the spin-spin model the strength of the terms of order $1/r^5$ is given by parameters that compare the shape of the bodies with the size of the orbit, say, $\hat d_j$ and $\hat q_j$. In contrast, the spin-orbit problem only regards the equatorial oblateness of the satellite $d_j/\mathcal C_j$. It is reasonable to think that the different types of interactions, say, point-point, spin-orbit and spin-spin, must have their own specific relevance in different ranges of parameters. This shows that the non-Keplerian behavior of the full Lagrangian model \cref{spin-spin_V}, \cref{orbital_eqs}, should be investigated more deeply. Here the full expansion of the potential energy, given in \cref{V_full}, may also play a role. Moreover, as \cite{davsch2020} shows, non-planar oscillations around solutions of the planar problem can be studied and are of practical interest.

In the present research, we have made a brief theoretical study that allowed us to point out the importance of the double synchronous resonance and compare it with the synchronous resonance of the spin-orbit problem. Particularly, in a similar way than \cite{misort2020}, we determine sufficient conditions for the existence of an asymptotically stable periodic solution (capture into resonance). Besides, note that our estimates do not pretend to be optimal at all. Instead, we illustrate a way to extend to the spin-spin model the tools used for the spin-orbit model, as well as to compare them. Furthermore, in this sense we have included some numerical diagrams of linear stability in \Cref{fig:stab_several} that show us how the spin-spin interaction alters the schemes of the spin-orbit model.

We have applied our study to two real systems in double synchronous resonance. In one hand, Pluto and Charon are representative of a large binary with one body much larger than the other one, see \cite{dob1997}. On the other hand, the binary asteroid 617 Patroclus is an archetype of a small system of similar components, see \cite{davsch2020}, \cite{mar2006}. Here we propose a way how to make an effective comparison between different systems. Note that the convenient choice of units and parameters helps to clarify the comparison. As we expected, the best candidates to apply the spin-spin model are binary asteroids. They are very abundant in the solar system, e.g., about 15\% of the near-Earth asteroids are thought to be binaries. For a detailed discussion on the applications of the general spin-spin model and its full Lagrangian version, we refer to \cite{batmor2015} and the bibliography therein. With our study on the double synchronous resonance we hope to contribute to the study of the spin-spin resonances made in \cite{batmor2015}. Whereas they focus on the synchronization of both spins for slow circular orbital motion ($\dot f \ll \dot \theta_j$), we consider the full synchronization including the orbit with arbitrary eccentricity. According to \cite{davsch2020}, most of the equal mass binaries are expected to be in the double synchronous state. In \cite{mur}, Section 4.14, they provide a formula for a critical mass ratio of the components for this state to be possible. We want to remark also that, apart from the application to binary asteroids and large natural satellites, the spin-spin interaction can be relevant for artificial satellites whose rotation state along an orbit is important. For instance, communication satellites in equatorial orbits or even spacecraft exploring small bodies.

Finally, we think that the theoretical interest of the model is large, even beyond the phenomena already observed in the spin-orbit problem. For example, in the spin-orbit problem we can apply the notion of KAM stability because KAM tori confine regions in the phase space. However this does not happen in the spin-spin model due to the increase in the phase space dimension (two degrees of freedom and time dependence). In fact, it is expected that Arnold diffusion takes place in this case. In general, the weak coupling and the Hamiltonian character of the system makes it suitable to apply perturbative techniques. Particular questions may be investigated, such as chaos by overlapping of resonances, stochastic phenomena, normally hyperbolic manifolds, scattering maps, among other phenomena, see \cite{chi1979}.

redpajama_set_name: RedPajamaArXiv


Ever since February 16 2005 when he motored away from New York City at precisely 6:04 am.
Ever since an incident with an incredible, life-changing cucumber salad while dining with his friend Jean-Louis. But more on that in a moment.
We're at the top of the South Island in New Zealand. Hubert approaches my campervan while we're both camped in Collingwood, Golden Bay. It's the last major town (population 235) before the road tapers into a narrow 26km long sand bank, Farewell Spit. Where seal pups learn to swim and where suicidal (or confused?) whales beach themselves in large numbers frequently.
Farewell Spit, top of the South Island in New Zealand.
Hubert introduces himself and stops to admire my campervan's antique dresser I picked up from a charity store a few weeks earlier. Hubert's French, it figures.
His glasses are bold and round and an unforgettable deep acrylic red. He has an old, severe-looking scar brimming from under his chin, running down his chest. I don't ask.
He tells me about his wife Lorraine who has a campervan a little bigger than mine, but just as unconventional. He says she'd like my dresser.
But back to the cucumber salad.
They move onto the couscous.
Jean-Louis asks Hubert a question (best imagined in a French accent), "Now that your younger daughter Jessica is finishing her college, what are you going to do?" He'd never considered his answer before now.
As they talked, Hubert and Jean-Louis established that if he were to sell everything he had, he'd have enough money to travel with his sidecar for ten years.
It was a common sense decision, says Hubert, that he took while drinking his coffee.
Three months later, on February 16, 2005 at 6:04 am, he was on his way to the Arctic Circle on the first leg of his 'Ten Years on the Road'.
At the corner of Mercer and Prince, leaving New York City. (Photo used with permission from Hubert).
It's been twelve years on the road now. In 2008, Hubert discovered some retirement money he was entitled to from France and the USA. He has no plan or reason to stop now.
Parked in my new campsite later that evening, I examine the black and white printed business card he gave me. The timeless ride dot com. Goodbye evening.
Hubert's website appears to be a series of lovingly hand-coded html pages. I can almost smell 2005.
"Warning: You are entering this web site at your own risk. Danger of contagion!!!" Choose from over 52 languages. "New to the site? Start here." I click.
"Who is Hubert?" A link with a French accent jumps off the page at me. "Open it if you dare to know the truth." I click again.
"Click on the photo if you can handle the truth!!!" I can handle it Hubert! I'm hooked.
I have the choice to find out even more about Hubert by date, by occupation, by sidecar, by support team, by philosophy, by favourite photos, by the "most" rides and by Q&A.
Hubert was born and raised in Paris where he lived for 37 years. He moved to Los Angeles and then to New York. He's been married 3 times, he has 3 daughters and held numerous occupations including real estate broker, sidecar messenger and owner of a graphic design business.
When he upgraded to his current Ural 2WD sidecar in 2008, Hubert gifted his original BMW motorbike and Ural sidecar to his younger daughter Jessica. She drives it daily around California.
His blog is a colourful photo journal, captioning his adventures and the people he meets along the way.
But I'll describe him no more.
Hubert is best discovered through his own information architecture, caption craft and exclamation marks. This is one French rabbit hole you will want to go down. You're welcome.
We exchange emails. I tell him he's an inspiration. He thanks me for my photograph and sends me pictures of his wife's campervan. He tells me he's already published a photo of my partner and I and our antique dresser in his Golden Bay collection.
You can support Hubert by contributing a gallon of gas, or write about him on his own website in his guest book:"write it all, don't withhold anything!".


Kapandji-Morhange and expert Pierre Bourdy will hold the auction "Rock Collection" at Drouot on April 10th. It retraces the evolution of rock from the mid-1950s through the 1990s with images of concerts, studio sessions and press photos. Most of the photographers are members of the French music press, including Richard Bellia, Bruno Ducourant, Tony Frank, Claude Gassian, Roger Kasparian, Jean-Pierre Leloir, Youri Lenquette, Renaud Monfourny, Jean- Louis Rancurel, Christian Rose, Dominique Tarlé and Pierre Terrasson.


Men are from Mars and women are from…Venus? Even though this statement is often referred to in jest, this basic premise can be applied to neurological functioning. A recently published study has even shown that gender based differences in the brain could influence epilepsy.
Many people tend to think that epilepsy issues in women are primarily related to pregnancy or seizures due to hormone fluctuations. However, there are many more issues. Women with epilepsy have a higher incidence of polycystic ovary syndrome (PCOS), early menopause, decreased bone density, fertility difficulties, and are more prone to depression and menstrual disorders.
Our Women in Mind program was developed to help support and educate women and girls with epilepsy about the unique health concerns that they may face.
Click here to view our Women in Mind program webpage for information and resources to assist women with epilepsy. This webpage is frequently updated with new study results.
Note: for information about the unique health concerns that men with epilepsy may face, click here to read our blog post about this issue.
The BC Epilepsy Society is hosting a large all-day conference about epilepsy on Saturday, November 21st. It is called the Family Education Day About Epilepsy. The event will go from 9:00 am to 5:00 pm at the Chan Auditorium at BC Children's Hospital.
There will be 9 different sessions on a variety of topics with over 12 different presenters. These range in topics from treatments, mental wellness, learning and memory issues, transition planning to adult health care services, and health literacy. It will also include talks and stories from people who are affected by epilepsy. Prepare to be inspired!
4th Annual Cycle Trek: Thank You for Your Support!
One of our board members, Steve Fruitman, recently did a two day, 200 km, bike ride from White Rock to Cultus Lake and back to help raise funds for the BC Epilepsy Society. He started this fundraising initiative in 2012. A big thank you to Steve and all the donors! You can still make a pledge or donation in support of this event. Click here to make a pledge.
When you have epilepsy, you are constantly put into the position of educating people about it. This can get frustrating at times. In a new article on The Mighty website a person with epilepsy addresses this and gives some tips about how to deal it with. It's a great article to share with friends, family, and other important people in your life.
Tara Klassen is a very talented epilepsy researcher at UBC. She is currently working to find out what medications work best for people with different types of epilepsy. The goal is to make sure that people do not have to try many different medications before they find the one that controls their seizures. Click here to watch a video of her explaining her research and the concept of personalized medicine. PS: She has also written an article about this for our upcoming Fall 2015 newsletter.
Our specially produced DVD about epilepsy for teachers has been described as "absolutely phenomenal". A nurse even told us she learns something new about epilepsy every time she watches it. This DVD is available for loan from our office, for purchase for $10, and has been donated by us to school district libraries across the province. It is a great DVD to have if you are a parent of a child with epilepsy. This way you can easily loan it to school staff to watch. The DVD has a 7 minute and 22 minute version on it. Click here to watch the 7 minute version on YouTube.
The BC Epilepsy Society is a non-profit, charitable organization, which provides education, advocacy, and support.


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At North Georgia Truss Systems LLC, experience is the keystone of our foundation. North Georgia Truss Systems was formed to provide quality designed trusses to the construction industry. We have served the Metro-Atlanta and North Georgia Market for over 19 years through the custom design, manufacturing and prompt delivery of the highest quality trusses available within the industry. Our owners have over 30 years experience in the manufacturing of custom-built trusses and construction.
Our staff believes in commitment. We are committed to providing our customers with quality truss products, on time and at a fair price. At North Georgia Truss Systems, we take commitment very seriously and will strive to continually meet our customers expectations.
Quality is important. At North Georgia Truss Systems, we buy our lumber from only the top quality mills in the country. We choose our lumber by hand, one piece at a time. Each piece is inspected to pass our high-quality expectations. This minimizes defects and gives each builder trusses that will last the life of the project.


text:
Articles written by: Ed Farnsworth
Editor-at-large at PSP. His work has also appeared at Philly.com, TheCup.us, Society for American Soccer History, The Encylopedia of Greater Philadelphia, Centro de Investigaciones de Historia y Estadística del Fútbol Español, EPYSA.org, and in the 2011 MLS All-Star Game program. Email him at [email protected]. Follow him on Twitter at @FarnsworthPSP.
Philadelphia soccer and the 1918 Spanish Flu epidemic
by Ed Farnsworth × on March 19, 2020 at 11:30 am × 5 comments
A look at soccer in Philadelphia during the 1918 Spanish Flu epidemic.
The life — and murder — of a U.S. Soccer president
by Ed Farnsworth × on July 26, 2017 at 11:01 am × 14 comments
In September 1953, Elmer Schroeder, U.S. soccer's first American-born president, was brutally murdered in his West Philadelphia apartment. No one was ever convicted for his murder. PSP's Ed Farnsworth writes about the life and death of a man who was an American soccer pioneer in more ways than one.
Daily news roundups
Picault signed, Bedoya/Pontius vs Blake in USA-JAM friendly tonight, league news, more
by Ed Farnsworth × on February 3, 2017 at 9:52 am × 32 comments
Union add a speedy winger. Union teammates face off in tonight's USA-Jamaica friendly. League news. For and against Trump travel ban. A thank you.
Footy on the Telly, Feb. 3-9
by Ed Farnsworth × on February 3, 2017 at 7:20 am × 0 comments
Your weekly listings of live soccer on TV, online, and on satellite radio.
Roster News
Union sign Fabrice "Fafa" Picault
by Ed Farnsworth × on February 2, 2017 at 6:23 pm × 21 comments
Picault is a Union trialist no more.
What Medunjanin will bring and other Union bits, league news, more
by Ed Farnsworth × on February 2, 2017 at 10:36 am × 42 comments
Insight into new signing Haris Medunjanin and other Union news. SoB election results. Steel open preseason training. League news. more.
Union sign Medunjanin, St. Pauli says Picault to join the Union, more news
More roster moves for the Union and other Union bits. League news. More.
USL - Philadelphia Union II
Bethlehem Steel announce 2017 schedule
by Ed Farnsworth × on January 31, 2017 at 2:44 pm × 2 comments
The Steel will face Harrisburg City Islanders and Pittsburgh Riverhounds, as well as FC Cincinnati and Toronto FC II, three times in 2017.
KYW Philly Soccer Show: Josh Yaro
Josh Yaro joins Greg and Mike. After that, the guys discuss the Onyewu signing.
Union sign Bosnian midfielder Haris Medunjanin
by Ed Farnsworth × on January 31, 2017 at 11:49 am × 101 comments
The left-footed 31-year-old central midfielder has experience in both attacking and defensive midfield roles.

redpajama_set_name: RedPajamaCommonCrawl


Alison Millar regularly represents vulnerable adults and abuse victims in civil liberties cases.
The school is a co-educational state boarding school which educates pupils aged 7 – 18.
The abuse team at Leigh Day is currently representing a number of former pupils of the school who claim that they suffered physical and sexual abuse by staff.
BBC documentary programme, Panorama, featured the school in a programme broadcast in November 2013. The programme revealed the existence of documents in the National Archives which gave details of a cover up of child abuse by seven perpetrators at the school between the 1950s and 1980s.
Two former staff at the school, Colin York and Allen Meredith, were jailed for serious sexual offences in September 2011 and 1995 respectively. York, an ex-assistant housemaster, was convicted of charges including sexual abuse during the 1970s.
Meredith admitted to Panorama that he had abused boys during his employment at the Royal Alexandra and Albert School and was sacked in 1985 before being employment by another state boarding school, the Royal Caledonian in 1989. He was jailed for ten years for serious sexual offences.
In May 2016 Philip Batten, a former housemaster and English teacher at the school, was sentenced to 17 years' imprisonment after being found guilty of a range of charges, including child cruelty and serious sexual assault.
The National Association of People Abused in Childhood runs a free confidential helpline on 0808 801 0331.
If you would like to speak to the abuse team about a compensation claim relating to alleged abuse at the Royal Alexandra and Albert School please speak to Denise on 020 7650 1241 or fill in our enquiry form and someone from the team will get back to you.


text:
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Children's Healthcare of Atlanta saved my little's brother life. He was born with a mass (ccam) on his lung and we didn't know if he was going to live or die when he was born. He had to get it removed and what was supposed to be a 3-4 day stay turned into weeks. He was in the ICU for over a week and a half, then in a regular room for another couple of weeks weaning him off of all the medications. I saw my parents sad and scared, not able to eat or sleep, not even able to relax for one moment, it was tough. When Javier finally came home, it was great! I wanted to give back and was too young to volunteer or just visit other kids in the hospital so me and my dad snuck in to the cafeteria on Christmas to feed folks. I knew the parents would have wristbands and that's who I wanted to target and pay it forward I wanted to at least bring a happy moment and a smile to them. It's hard enough being stuck in the hospital but over Christmas has to be really bad so I do it on Christmas Eve which was my nana's favorite holidays. My nana died 6 months after Javier was born so we say he has a personal angel above. Help me bring a smile to some families this Christmas!
I am going to host a few fundraisers from now until Christmas.
My first will be Sunday, Sept. 30 from 2 – 6 at The Wimbish House in Atlanta during Atlanta Streets Alive. I will be hosting a pop-up candy shop. Please stop by and treat yourself to some candy while helping families at CHOA!
My second event will be a Block Party on Sunday, November 11th from 4-6pm at 1131 Bald Eagle Trace in Hoschton, GA. I will have a bounce house, reptile petting zoo, soccer darts, basketball, snacks and more! Thank you to Astro Jump of NW Atlanta for the generous donations of activities and to Reigning Reptiles for bringing our reptile friends!


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