Tidal effects on the eccentricity of HD 209458b with one or two companions. a) The companion is a 0.1   M_J planet at 0.4 AU with e₂ = 0.4 as in Mardling (2007). The blue curve was obtained without considering the conservative effect of the tides ( in Eq. (14)). The red curve in a), and all the evolutions presented in the subfigures b)–d) take into account this effect. b) The mass of the companion is set to 0.608   M_J to recover the final excentricity of Mardling’s simulation. The red curve is the result of a numerical integration of the full secular equations that are exact in eccentricity. The green one is the analytical solution of the linearized problem. c) Same as b) except for the initial Q-value of HD 209458b, which is set to 15.15 to enable the visualisation of both the damping and the oscillation of the eccentricity. d) Same as c) with an additional 0.1   M_J companion at 1.0 AU with e₃ = 0.1.