# Analysis of Tidal Accelerations in the Solar System and in Extrasolar Planetary Systems
## 2 Materials and Methods
### Simulation of Orbital Motion  
The orbital elements are the semi-major axis \(a\), the eccentricity \(e\), the longitude of the ascending node \(\Omega\), the inclination \(i\), the argument of periapsis \(\omega\), the mean anomaly at a given epoch \(M_{0}\), and the mean orbital motion \(n\). For the calculation of the three-dimensional distance vectors, the coordinates of all objects have to be transformed into a single Cartesian coordinate system. As the uniform system, the Cartesian heliocentric ecliptic system is chosen with the Sun at the origin and the orbital plane of the Earth as the x-y-plane with the \(x\)-axis pointing to the vernal equinox. [PERSON]'s equation is solved to obtain the polar coordinates of the elliptic orbit; we convert them to Cartesian coordinates and then rotate the orbital plane using the other orbital elements with respect to the ecliptic. Since the orbital elements of the moons are in reference to their central planet/object, one has to shift their coordinates by those of the central object. The mutual influence of the moons on their orbital motion in the solar system, especially for moons around gas giants, leads to a precession of the orbits and makes it necessary to introduce a new reference plane for some moons, called the Laplace plane. This plane has two additional orbital elements, two angles changing in time, to emulate this precession in a linear approximation. For the outer moons of the gas giants in the solar system, this precession is partly caused by the mutual influence of the outer moons of different planets. Some of the inner moons orbit close to the equatorial plane of their gas giant and therefore use this plane as reference, which also gives two angles as additional orbital elements. Sometimes, these additional angles are in reference to the equatorial plane of the Earth, and one must include an additional rotation about the angle of the obliquity of the Earth. The code for the simulation written in C++ can be found under the following link:  
[[https://www.kip.uni-heidelberg.de/biophysik/software/tidal_accelerations](https://www.kip.uni-heidelberg.de/biophysik/software/tidal_accelerations)]([https://www.kip.uni-heidelberg.de/biophysik/software/tidal_accelerations](https://www.kip.uni-heidelberg.de/biophysik/software/tidal_accelerations)) (accessed on 1 July 2021). The need C++11 standard and the library \"Eigen\" are necessary [13] for compiling. In order to find an appropriate time window for the simulation of the orbital motion and the resulting tidal accelerations, we choose conjunctions of the planets, since the mutual influence of the moons of different planets, as mentioned above, and therefore the tidal influence should then be maximal. In May 2000, the planets Mercury, Venus, Earth, Mars, Jupiter, and Saturn were located in a relatively small angular segment seen from the Sun. Therefore, it makes sense to simulate around this time point, because the planets are getting very close to each other and may have the highest accelerations on each other. To include the approaching of the planets to this situation, we start our simulation at 1.1.1950, 12 h UT. Around May 2492, almost the same conjunction will occur, so we choose a time period of 600 years for the simulation to include this conjunction as well. Since one can only simulate for discrete time steps, a reasonable value for each time step has to be determined. For this, we consider the orbital periods of the objects in the solar system. The planets have orbital periods of barely over three months for Mercury to 165 years for Neptune or rather 248 years for the dwarf planet Pluto. In contrast to that, the periods of the moons of the gas giants range from a quarter of a day for the inner ones up to a few years for the outer ones. Among the asteroids, (101955) Bennu orbits in about 1.20 years, while (4) Vesta takes about 3.63 years. TNOs have orbital periods of up to 497.5 years for (90377) Sedna.