# Analysis of Tidal Accelerations in the Solar System and in Extrasolar Planetary Systems
## 3 Results
### The Solar System
This general observation makes no assumptions on the power sources of this volcanism. For larger \(R\), this may be more based on radioactive decay or stored accretion energy; for smaller \(R\), this points more toward the direction of tidal heating. This is indicated approximately by the dashed line in Figures 6 and 7 and was determined empirically by considering the volcanic activities of the objects in the solar system. Nevertheless, this boundary remains blurred and should not be considered too certain. Most planets with at least former volcanism known are below \(10^{-10}\) (m\({}^{2}\)/s\({}^{4}\)d) and above \(10^{-15}\) (m\({}^{2}\)/s\({}^{4}\)d) but have higher \(R\) values. The probably most interesting new objects exposed to higher gravitational forces found by this simulation can be seen in the upper left corner of the diagram with \(\{A_{total}\)-(\(\Delta A_{total}/\Delta t)\}_{max}\) larger than \(10^{-6}\) (m\({}^{2}\)/s\({}^{4}\)d) but \(R\) smaller than 200 km. New considerations may also be taken for all other objects shown, even if they are not inside of the boundaries as the objects already described. As mentioned above, for objects in extrasolar planetary systems, very often only a small fraction of their properties and parameters is available. Even orbital elements may be missing or vague. Thus, the application of our simulations is running into uncertainties. To investigate the effect of neglecting some orbital elements on the results of our model, we repeated our whole full simulation by setting \(e\), \(\Omega\), \(i\), \(\omega\), and \(M_{0}\) to zero so that only \(a\), \(n\), \(M\), and \(R\) were remaining, which often may be the only well-known parameters for objects in an extrasolar planetary system. The new results based on this simplified simulation are given in Figure 7. Only small deviations were found between the two versions (Figure 7 vs. Figure 6) of the simulation. This may be mostly caused by neglecting the eccentricity \(e\) (as been seen in pre-simulations when testing the algorithm, data not shown). The similarity of Figures 6 and 7 shows that the simplified version of the simulation with a smaller set of orbital elements does not change the resulting tidal accelerations significantly. Thus, the simplified version could be very applicable to estimate the potential significance of  
Figure 4: Tidal accelerations acting on comet C/1995 O1 (Hale-Bopp) 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. tidal influences in extrasolar planetary systems, especially the ones with poorly known or many unknown orbital elements. In general, the deviations are reasonably small, and therefore our approach may be considered stable enough to provide an estimate for the tidal interaction and the heating potential in extrasolar planetary systems, even if not all parameters are known. An example is given in Figure 8 for Haumea, where even with strong influences from its orbital planes, the course of the values changes, thereby remaining in the same order of magnitude (in comparison to Figure 3). Figure 5: Overview over the celestial objects showing the largest tidal accelerations in the solar system. The version of the simulation with full orbital elements is used. Color-coded is what is known about volcanism on these objects [3; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34]. Figure 6: Maximal tidal accelerations against the radius of celestial objects in the solar system and the TRAPPIST-1 planetary system. The version of the simulation with _full orbital elements_ is used.