# Analysis of Tidal Accelerations in the Solar System and in Extrasolar Planetary Systems
## 1 Introduction  
Over the last decades, many objects in the solar system have been identified with liquid oceans powered by tidal heating. Well-known examples are (J1) Io (designation (J1) given in this paper for distinction from asteroid (85) Io), Europa, Enceladus, and Ganymede, among many more. Numerous models have been evolved for description and further analysis, containing various predictions for the interior structure of the examined celestial objects. Most of these models are based on vast and extensive theories of viscoelasticity considering different analytical and numerical models for different oscillation modes and layers forming the inner structure of the celestial body, see, e.g., [1, 2]. For more complex models distinguishing oscillation modes and causes for the tidal heat dissipation such as eccentricity, obliquity, and asynchronous spin to orbit rates, see, e.g., [3]. Thesestudies, among many others, provide a very precise determination of custom-fit models for each celestial object, achieving a detailed and deep understanding of the tidal heating processes involved. In principle, tidal heating or, in more general considerations, tidal interactions happen in all planetary systems known in the universe. An additional specialty in many extrasolar planetary systems, especially in those of the numerous M and K stars, is the very great proximity of planetary objects to each other [4; 5; 6]. This might already result in an influence the planetary objects have on each other. These additional effects based on multi-body contributions to tidal interactions were so far only rarely considered, especially since all relevant physical parameters required for complex modeling were not available for many extrasolar systems. Without denying the need for deep and detailed analyses based on complex theoretical approaches as, for instance, the successful viscoelastic multi-body theories mentioned above, it may be of advantage to have a rough calculation procedure whenever the measured data availability does not sufficiently satisfy the theories' requirements. This may offer a glimpse of potential systems of interest in advance by using a model in which mainly distances between the objects, their radii \(R\), the eccentricities of their orbits \(e\), masses, and other orbital elements are used. Here, we present a newly developed approach for the simulation of tidal accelerations in systems of celestial objects under _multi body conditions_, requiring a _minimalistic set of experimental physical parameters_. The model was first applied to our solar system as a \"gold standard\" to determine whether this is sufficient to identify candidates for tidal interactions. Our solar system is the best test candidate, because a comparison to precise measurements and models developed for the many objects in the solar system can be made. Often space probes have visited these objects, and accurate investigations have been performed so that the results can be used in some way as a standard to classify other systems so far not investigated in such great details. This may even allow an assessment of whether a certain value found in the model may favor liquid solvents or cryo- or even siliceous volcanism inside the object. In addition, some aspects of the evolution of the eccentricity \(e\) over time have been investigated in this approach. In a final step, this approach has been applied to an extrasolar planetary system, namely TRAPPIST-1 [5; 6]. All planets known for this system are included. We further present our results for a simplified version of our simulation, using a smaller set of orbital elements. We compare this simplified version to the original one and outline how this simplified version could be applicable to other extrasolar planetary systems for which usually only a smaller set of orbital elements is known with acceptable accuracy. This could give a first estimate of the potential significance of tidal influences in other less well-known extrasolar systems.