# Analysis of Tidal Accelerations in the Solar System and in Extrasolar Planetary Systems
## 3 Results
### The Solar System
Multi-body TNO systems often consist of relatively large objects orbiting each other in short distance and have, in comparison with solar system planets and moons, quite uncommon values for their orbital elements. Nevertheless, as the accuracy or availability of orbital and physical data for TNOs are in general not very satisfying, especially for TNOs with moons, a (more precise) determination of these parameters could be very interesting for a more detailed evaluation of tidal heating. Figure 2: Tidal accelerations acting on main belt asteroid (4) Vesta using the version of the simulation with full orbital elements; \(\Delta t\) = 90 days. \(A\) 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. As the model gives a complete three-dimensional simulation evolving over time, rare and only occasionally occurring encounters of generally not-associated objects are more easily revealed. This is especially interesting for comets and some asteroids that travel across the inner and outer regions of the solar system due to their high eccentricities. In Figure 4, the flyby of the comet C/1995 O1 (Hale-Bopp) past Jupiter is shown. The tidal influence increases significantly as Hale-Bopp approaches Jupiter. The tidal accelerations are nevertheless small, and the considered parameters are many orders of magnitude below those of objects shown above. This may be mainly due to the small size of Hale-Bopp but could become interesting for bigger asteroids or comets. Figure 5 gives an overview of the celestial bodies showing the largest tidal accelerations in the solar system. \(\{A_{total}\cdot(\Delta A_{total}/\Delta t)\}_{max}\) is the maximal value found for the celestial object over all time steps. In Figures 6 and 7, the volcanism of differently sized objects is discussed in relation to the maximal value \(\{A_{total}\cdot(\Delta A_{total}/\Delta t)\}_{max}\). Celestial bodies marked by orange triangles show signs of at least former volcanism, meaning that these objects certainly had volcanism in the past and might even be faintly igneous today. Data of the space probe MESSENGER shows signs of siliceous flood volcanism and other volcanic features on Mercury [14,15]. Venus shows multiple signs for former and maybe still present volcanism. The Monitoring Camera of the space probe Venus Express [16], numerical simulations [17], and other investigations [18, 19, 20] find hints/evidence for volcanism. It is reasonably certain that Mars used to be volcanic in the past (e.g., Olympus Mons) and maybe still is today. Additionally, our Moon shows signs of former volcanism [21], and seismic measurements during the Apollo missions indicate present volcanism [22]. This subgroup is presented just for orientation, as one must consider for these huge objects that radioactive heating and stored accretion energy strongly contribute along with tidal interactions to volcanism. Figure 3: Tidal accelerations acting on TNO (136108) Haumea using the version of the simulation with full orbital elements; \(\Delta t=0.25\) days. \(A\) 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. As may be already seen from Figure 6, if \(\{A_{total}\)-(\(\Delta A_{total}/\Delta t)\}_{max}\) is larger than \(10^{-15}\) (m\({}^{2}\)/s\({}^{4}\)d) and \(R\) larger than 200 km, volcanism is known for all objects in the solar system.