[*qualitatively*]{} wrong tidal evolutions. The same arguments for the evolution of the eccentricity show that tides raised on a pseudo synchronous planet lead to a *growth* of the eccentricity for $$e\geqslant \sqrt{\frac{7}{66}}\approx0.326$$ and not to a *decrease*. This is illustrated by Fig.\[fig:phasespace\] that shows the pseudo synchronization curve (solid), the $\dot{a}=0$ curve (dashed) and the $\dot{e}=0$ curve (dotted) for the full model (top panel) and the truncated one (bottom panel), in the $\omega/n$ vs $e$ phase space. As demonstrated before, the pseudo synchronization curve crosses the $\dot{a}=0$ and $\dot{e}=0$ lines in the $2^{\mathrm{nd}}$ order model (Fig.\[fig:phasespace\]b) whereas it does not when solving the complete Hut equations (Fig.\[fig:phasespace\]a). *As a result, with the truncated model, a pseudo synchronized planet can erroneously enter the zone of the phase space where its tides act to increase both the semi-major axis and the eccentricity.* While this behavior is not observed with the constant phase lag model because it assumes that the star is slowly rotating ($\os/n\ll1$) and that the planet is near synchronization ($\op/n\approx1$) - placing them in the $\dot{a}<0$ and $\dot{e}<0$ zone of the phase space - this formal demonstration sets clear limits on the domain of validity of the quasi circular approximation.

