# Analysis of Tidal Accelerations in the Solar System and in Extrasolar Planetary Systems
## 3 Results  
### The Solar System  
A typical example is shown in Figure 1, in which all the strongest of the tidal influences acting on the Jovian moon Ganymede can be seen. As expected, Jupiter has the biggest contribution to the tidal acceleration since it is the largest mass around. Second by contribution are the other Galilean moons, but they are already in the range of at least four orders of magnitude smaller. Their periodicities reflect very well the orbital resonances of these moons. Moreover, other Jovian moons appear in the objects causing larger fluctuations than expected. Figure 2 shows as another example the tidal influence acting on (4) Vesta being dominated by the Sun, and its influence is also in the range of three to four orders of magnitude higher than those by planets nearby the main belt and other asteroids within this belt. The results presented in these two figures indicate some general characteristics of our computed tidal accelerations and the expected tidal influences for celestial bodies in our solar system. Not only the Sun and planets influence (4) Vesta; the Moon and other asteroids such as (1) Ceres, (9) Metis, or (85) Io also generate tidal accelerations that change dramatically over time. Since all these objects including (4) Vesta itself are relatively small, light, and/or too far apart, the absolute values of \(A_{total}\)'(\(\Delta A_{total}\)/\(\Delta t\)) are too small to be interesting enough in the context of tidal heating. Nevertheless, this shows the potential of asteroids for tidal heating besides moons around large gas giants. One can see in Figure 2 that the course of \(\Delta A_{total}\) follows the course of \(A_{total}\) in general. This can be explained by the fact that the  
Figure 1: Tidal accelerations acting on Jovian moon Ganymede using the version of the simulation with full orbital elements; \(\Delta t\) = 0.25 days (d). \(A\) denotes the individual tidal accelerations caused by other objects. The ordering of the legend is the same as in the uppermost plot with the strongest contribution at the top and the weakest at the bottom. \(A_{total}\) denotes the norm of the vector addition of all individual contributions. \(\Delta A_{total}\) denotes the norm of the vector difference of \(A_{total}\) for two consecutive time steps. Results for the three-body system Haumea in the Kuiper belt as given in Figure 3 are very interesting in this context, because the orbital planes of the two moons around Haumea are tilted towards each other due to very different inclinations. In this graph (Figure 3), the effect of non-coplanar orbital planes can be nicely seen: The charts showing \(A_{total}\) and \(\Delta A_{total}\) differ clearly, because the orientations of the tidal acceleration vectors become important in such a tilted system. This contrasts with (4) Vesta as the effects of the moons in the Haumean system are strong enough to dominate \(\Delta A_{total}\). For objects orbiting close to their central object, it is interesting to check if \(A_{total}\) and \(\Delta A_{total}\) differ significantly. Objects orbiting close to their central object are often tidally locked, i.e., they are always showing the same side to the central object. In tidal lock, the change in the orientation of the tidal acceleration vector caused by the orbital motion contributes to \(\Delta A_{total}\), but this alone would not contribute to a deformation of the influenced and tidally locked object over time. If \(A_{total}\) and \(\Delta A_{total}\) differ, other influences caused by objects other than the central one are contributing, resulting in an actual deformation of the influenced and tidally locked object. In general, it can be said that among those celestial bodies investigated, TNOs in possession of moons exhibit exceptionally strong tidal influence. In, e.g., [3] a large potential for tidal heating within these TNOs is also described.