# Analysis of Tidal Accelerations in the Solar System and in Extrasolar Planetary Systems
## 2 Materials and Methods
### Simulation of Orbital Motion
Comets are strongly diverging, 5.37 years for the Jupiter-family comet 10P/Tempel and 2640 years for (C/1995 O1) [PERSON]. As the asteroids in the main belt have at least a period of about one year, we choose a step size of 90 days in order to have at least four points on the orbit for these asteroids. Additionally, the four inner terrestrial planets, all asteroids, comets, and TNOs, are simulated using this step size, resulting in _600 years/0.25 years = 2400_ iterations. The first interesting moon according to its size is the Saturn moon Mimas with an orbital period of about one day. Again, we choose a quarter of this as our step size and iterate 2400 times, resulting in a total time period of 600 days. In this way, the simulations are carried out for the moons and the terrestrial central objects, Earth, Mars, Pluto, and the TNOs Eris and Haumea, to obtain a more detailed insight into the tidal influence of their moons. To include the conjunction around May 2000, we start the simulation on May 1999 for this smaller step size. As the orbital periods of the planets in the TRAPPIST-1 system are relatively close to each other with 1.5 to 18.8 days, we chose a smaller step size of 0.1 days. By going for 1800 iterations, we obtain almost 10 orbits for the outermost planet. Converting the distance vectors into tidal acceleration vectors using Equation (1), we obtain a vector set for every object pointing to the other influencing objects. Since for many celestial bodies, we have three values for the mass (mean, minimal, and maximal guessed mass), we obtain a triple of these tidal accelerations for every time step using these different mass values in calculation, respectively. If the mass is given for an object, we only use this single value so that we end up with identical tidal acceleration vectors instead of different vectors in the triple. Adding up these vectors related to the influencing objects for a certain time step and taking the norm results in the tidal acceleration \(A_{total}\). Note that this takes the three-dimensional orientation of the vectors into account, as the tidal influences of the many objects may (partially) enhance or reduce each other. In addition, we calculate the vector difference of the total acceleration vectors \(A_{total}\) by subtracting them from each other for two consecutive time steps. We call the norm of this vector difference \(\Delta A_{total}\). This represents the change of the tidal acceleration vectors, including the change in the orientation. Our idea is that this could give a rough estimation of a deformation of the object caused by the change in the tidal forces. Equation (1) can be approximated for \(R\ll r\) by  
\[A\approx\frac{2 GMR}{r^{3}} \tag{2}\]  
Using \(\Delta t\) for the step size in the simulation in days, mean angular motion \(n\), and approximating \((\Delta A/\Delta t)\) by \(An\), we obtain  
\[A\frac{\Delta A}{\Delta t}\ \approx\ A^{2}n\approx\frac{4G^{2}M^{2}R^{2}n}{r^{6}} \tag{3}\]  
Since parts of the simulation use different step sizes \(\Delta t\), this product also guarantees the comparability of the values with each other (e.g., moons often complete an orbit faster than a planet). Please note that for the computation of \(A\) and \(\Delta A\) for each time step in the simulation, as well as the added-up equivalents \(A_{total}\) and \(\Delta A_{total}\) for each celestial object and their product \(A_{total}\)-\((\Delta A_{total}/\Delta t)\), the full accuracy of Equation (1) is still used. Equation (3) is given to show the variables on the right side (\(M\), \(R\), \(r\), \(n\)) the product \(A_{total}\)-\((\Delta A_{total}/\Delta t)\) corresponds to. This might make it easier to compare our results to the viscoelastic models mentioned in the introduction section, e.g., [1; 2; 3], as one can comprehend how the variables (\(M\), \(R\), \(r\), and \(n\)) make up the product \(A_{total}\)-\((\Delta A_{total}/\Delta t)\). To reduce the amount of resulting data for each object, we apply a pre-screening to find 10 objects with the greatest influence.