# Analysis of Tidal Accelerations in the Solar System and in Extrasolar Planetary Systems
## 3 Results
### Extrasolar Planetary System TRAPPIST-1  
A well-investigated extrasolar planetary system is TRAPPIST-1, an M-dwarf with seven exoplanets [5,6], the largest extrasolar planetary system of terrestrial Earth-sized planets found so far. Since all planets are orbiting very close to each other and their host star, this system is of great interest and a challenging example for our approach concerning tidal interactions and consequences, especially for tidal heating. What distinguishes this system from all others is that the masses, semi-major axes, eccentricities, mean angular motions, radii, and inclinations are known with high accuracy, and this, quite remarkably, for all the seven terrestrial Earth-sized planets [6,11,12]. The central star is an ultra-cold red dwarf star [6,10] of spectral type M8V, with a size slightly larger than Jupiter, but 94 times Jupiter's mass, which is located 12 pc (about 41 light-years) away from the Sun [35]. There are extensive studies elaborating on the possible habitability of this system. Considerations about the equilibrium temperatures based on the irradiation by the host star [12] could allow for liquid water on the surfaces of up to six planets (TRAPPIST-1c, -d, -e, -f, -g, and -h) when assuming a very optimistic circumstellar habitable zone around the host star, or up to three planets (TRAPPIST-1e, -f, and -g), when assuming a conservative  
Figure 8: Tidal accelerations acting on TNO (136108) Haumea using the version of the simulation with simplified 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. circumstellar habitable zone, depending on definition, see, e.g., [36; 37; 38; 39], among many others. The models in [40; 41], considering the potential water loss from terrestrial planets around ultra-cold red dwarf stars during their pre-main sequence phase, conclude for the TRAPPIST-1 planets -b and -c that they may have lost as much as 15 Earth oceans of water; -g may have lost more than 20 Earth oceans and -e less than 3 Earth oceans of water. For TRAPPIST-1d, they predict a water loss of possibly less than one Earth ocean. Depending on the initial water content of these planets, they could have enough water to remain habitable when they enter the habitable zone in the main sequence phase of their host star TRAPPIST-1. The elaborate climate model in [42] predicts that TRAPPIST-1e has the highest probability for being an Earth-like ocean world. Figure 10 shows tidal accelerations for the innermost planet TRAPPIST-1b, which even exceed the values found for the Jovian moon (J1) Io, the most strongly tidally influenced object in the solar system. Equally, the planets on outer orbits show values in the same range (not shown). We may therefore assume strong to very strong tidal heating, and consequently siliceous volcanism may be present if we refer to the relations of Figure 6. This is in good agreement with results in other publications, e.g., [43], confirming an applicability of our model to extrasolar planetary systems. Additionally, the individual tidal influences of all other planets are also visible. The influence of TRAPPIST-1c alone would be enough to periodically exert tidal forces on TRAPPIST-1b up to a lower limit from which on tidal heating of low-melting liquids might already be considered. As M-dwarfs often possess extremely strong magnetic fields, another source for heat induction has to be considered [44]. Adding up stored accretion energy, radioactive, tidal, and induction heating, this gives an impression of the planets of TRAPPIST-1 as volcanic worlds.