tokens: values of the parameters varied here, and we refer the reader to Appendix A for the parameters that remain constant in our simulations, including the initial orbital elements for the planets. We tested the three different prescriptions of the general relativity introduced in Sect. 2.3 . They gave very similar results so that in the following we compare the other effects with respect to the simulations performed using the prescription of Kidder (1995) . The planetary reference values were taken to be representative of the Earth; in particular, the quantity k 2,p Δ τ p is equal to 213 s ( Neron de Surgy Laskar 1997 ). The stellar reference values were chosen to be representative of fully convective M dwarfs ( Bolmont et al. 2015 ). For all the simulations we performed, we calculated the transit timing variations ( Agol Fabrycky 2018 ) as follows: (i) for each transit we find the time of the transit mid-time by performing an interpolation to find the precise time a given planet crosses a reference direction. This corresponds to the “Observed transit time” O ; (ii) we evaluate the “Calculated transit times” C by performing a linear fit 2 of the transit times calculated in step i) over the total number of transits; (iii) we calculate the difference O – C to obtain the TTVs as a function of the epoch (or transit number). To quantify the impact of each additional effect on the simulated TTVs, we compute the difference between the TTVs calculated taking into account an additional effect and the TTVs obtained for a Newtonian N -body integration. Table 1 Reference values for the parameters we vary in this study. Notes. We assume that all planets in the system have the same potential Love number of degree-2 k 2 f ,p , fluid Love number k 2,p , and time lag Δ τ p . 4 Influence of each effect on the TTVs We performed a set of six simulations of the TRAPPIST-1 system to test the impact of the additional effects listed in Sect. 2 , and we compare each with a Newtonian N -body simulation. One after another, we explored the effect of each parameter using the reference values in Table 1 and general relativity. We tested the influence of the dissipative part of the planetary tide by assuming for all planets, with all the other parameters set to zero. We repeated the operation for the non-dissipative part of the planetary tide (through k 2,p , equal for all planets), the rotational flattening of the planets (through k 2 f ,p , equal for all planets), the non-dissipative part of the stellar tide (through k 2,⋆ ), the rotational flattening of the star (through k 2 f ,⋆ ), and for general relativity. Figure 1 shows the results for planets b to d. The top panels of Fig. 1 show the transit timing variations for the three planets for the seven simulations, and the bottom panels show the difference between the TTVs and the TTVs corresponding to the pure N -body simulation. The different additional effects have a very limited impact on the shape of the TTVs, but computing
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