The Role of Tidal Forces in the Long-term Evolution of the Galilean System

Abstract

The Galilean satellites of Jupiter are called Io, Europa, Ganymede and Callisto. The first three moons are found in the so-called Laplace resonance, which means that their orbits are locked in a $2:1$ resonant chain. Dissipative tidal effects play a fundamental role, especially when considered on long timescales. The main objective of this work is the study of the persistence of the resonance along the evolution of the system when considering the tidal interaction between Jupiter and Io. To constrain the computational cost of the task, we enhance this dissipative effect by means of a multiplying factor. We develop a simplified model to study the propagation of the tidal effects from Io to the other moons, resulting in the outward migration of the satellites. We provide an analytical description of the phenomenon, as well as the behaviour of the semi-major axis of Io as a function of the figure of merit. We also consider the interaction of the inner trio with Callisto, using a more elaborated Hamiltonian model allowing us to study the long-term evolution of the system along few gigayears. We conclude by studying the possibility of the trapping into resonance of Callisto depending on its initial conditions.

APPENDIX. TIDAL CONTRIBUTION IN THE DELAUNAY VARIABLES

In this appendix we detail the computations of Eqs. (3.23) described in Section 3. Our goal is to transform the equations of the secular contribution due to the tidal effects acting on Io provided by [3]

$$\displaystyle\frac{\dot{a}}{a}=\frac{2}{3}c\left(1-\left(7D-\frac{51}{4}\right)e^{2}\right)$$

(A.1)

$$\displaystyle\frac{\dot{e}}{e}=-\frac{1}{3}c\left(7D-\frac{19}{4}\right)$$

into their formulation in Delaunay variables. We recall that the parameters $c$ and $D$ are defined as follows:

$$\begin{aligned}\displaystyle c=&\frac{9}{2}\frac{k_{0}}{{\cal Q}_{0}}\frac{m_{1}}{m_{0}}\left(\frac{R_{0}}{a_{1}}\right)^{5}n\\ \displaystyle D=&\frac{{\cal Q}_{0}}{{\cal Q}_{1}}\frac{k_{1}}{k_{0}}\left(\frac{R_{1}}{R_{0}}\right)^{5}\left(\frac{m_{0}}{m_{1}}\right)^{2},\end{aligned}$$

(A.2)

where $m_{\alpha}$ are the masses, $k_{\alpha}/{\cal Q}_{\alpha}$ the tidal ratios, $r_{\alpha}$ the radii of Io and Jupiter, and $n$ is the mean motion of Io. The parameter $c$ depends on time, and we set its constant part as

$${\texttt{c}_{c}}=\frac{9}{2}\frac{k_{0}}{{\cal Q}_{0}}\frac{m_{1}}{m_{0}}R_{0}^{5},$$

(A.3)

so that

$$c={\texttt{c}_{c}}\frac{n}{a_{1}^{5}}.$$

(A.4)

Given the definition of the Delaunay variables

$$L_{i}=\mu_{i}\sqrt{GM_{i}a_{i}},\quad P_{i}\simeq\frac{1}{2}L_{i}e_{i}^{2},$$

(A.5)

and recalling that we enhance the tidal effects by multiplying Eqs. (A.2) by a factor $\alpha_{P}$, we have the following expression for the derivative (omitting the sub-index $i=1$ to lighten up the notation):

$$\begin{aligned}\displaystyle\frac{{\rm d}L}{{\rm d}t}=&\mu\sqrt{GM}\frac{\dot{a}}{2\sqrt{a}}=\\ \displaystyle=&\frac{\mu\sqrt{GM}\sqrt{a}}{3}c\left[1-\left(7D-\frac{51}{4}\right)e^{2}\right]=\\ \displaystyle=&\frac{L}{3}{\texttt{c}_{c}}\frac{n}{a^{5}}\left[1-\left(7D-\frac{51}{4}\right)e^{2}\right]\\ \displaystyle=&\frac{L}{3}{\texttt{c}_{c}}\frac{\partial H_{kep}}{\partial L}\frac{1}{a^{5}}\left[1-\left(7D-\frac{51}{4}\right)2\frac{P}{L}\right]=\\ \displaystyle=&\frac{L}{3}{\texttt{c}_{c}}\frac{c_{1}}{L^{3}}\left(\frac{\mu^{2}GM}{L^{2}}\right)^{5}\left[1-\left(7D-\frac{51}{4}\right)2\frac{P}{L}\right]=\\ \displaystyle=&\frac{1}{3}\frac{c_{1}c_{2}}{L^{13}}{\texttt{c}_{c}}\left[L-2\left(7D-\frac{51}{4}\right)P\right].\end{aligned}$$

(A.6)

As we mentioned in Section 2.1, we choose normalised units so that $Gm_{0}=1$ and $a_{1}=1$. In light of this, $c_{1}c_{2}\simeq 1$ and we can write

$$\frac{{\rm d}L}{{\rm d}t}=\left[\frac{1}{3}\alpha_{P}{\texttt{c}_{c}}L-\frac{2}{3}\alpha_{P}{\texttt{c}_{c}}\left(7D-\frac{51}{4}\right)P\right]\frac{1}{L^{13}}.$$

(A.7)

Similarly, we have

$$ \begin{aligned}\displaystyle\frac{{\rm d}P}{{\rm d}t}=&\frac{1}{2}\frac{{\rm d}L}{{\rm d}t}e^{2}+Le\dot{e}=\\ \displaystyle=&\frac{1}{2}\frac{{\rm d}L}{{\rm d}t}e^{2}-\frac{1}{3}Le^{2}\alpha_{P}{\texttt{c}_{c}}\frac{n}{a^{5}}\left(7D-\frac{19}{4}\right)=\\ \displaystyle=&\left[\frac{1}{2}\frac{{\rm d}L}{{\rm d}t}-\frac{1}{3}L\alpha_{P}{\texttt{c}_{c}}\frac{c_{1}c_{2}}{L^{13}}\left(7D-\frac{19}{4}\right)\right]e^{2}=\\ \displaystyle=&\left[\frac{1}{2}\frac{{\rm d}L}{{\rm d}t}-\frac{1}{3}L\alpha_{P}\frac{{\texttt{c}_{c}}}{L^{13}}\left(7D-\frac{19}{4}\right)\right]\frac{2P}{L}=\\ \displaystyle=&\left[-\frac{1}{3}\alpha_{P}{\texttt{c}_{c}}L\left(14D-\frac{21}{2}\right)-\frac{2}{3}\alpha_{P}{\texttt{c}_{c}}P\left(7D-\frac{51}{4}\right)\right]\frac{P}{L^{14}}.\end{aligned}$$

(A.8)

By setting

$$\begin{aligned}\displaystyle f_{1}=&\frac{1}{3}\alpha_{P}{\texttt{c}_{c}},\\ \displaystyle f_{2}=&\frac{2}{3}\alpha_{P}{\texttt{c}_{c}}\left(7D-\frac{51}{4}\right),\\ \displaystyle f_{3}=&\frac{1}{3}\alpha_{P}{\texttt{c}_{c}}\left(14D-\frac{21}{2}\right),\end{aligned}$$

(A.9)

we finally get

$$\begin{gathered}\displaystyle\frac{{\rm d}L}{{\rm d}t}=\frac{f_{1}L-f_{2}P}{L^{13}},\\ \displaystyle\frac{{\rm d}P}{{\rm d}t}=-\frac{f_{3}L+f_{2}P}{L^{14}}P,\end{gathered}$$

(A.10)

i. e., the formulation of Eqs. (3.23).