and by more than a factor $10^3$ for $e\gtrsim 0.7$*, an eccentricity value often advocated by these authors to explain the highly inflated planets (see §\[sec:comp\]).

From a mathematical point of view, the fact that a truncation to $2^{\mathrm{nd}}$ order in eccentricity yields such discrepancies is due to the presence of $(1-e^2)^{-\mathrm{15}/2}$ factors in the equations for the tidal dissipation. As already stated by @Wis08, for moderate to large eccentricity, this function is poorly represented by the first terms of its polynomial representation. Indeed, the first terms of the energy dissipation rate are given by: $$\frac{\dot{E}_{\mathrm{tides}}}{7\Kp e^2}=
1+\frac{54 }{7}e^2+\frac{1133}{28} e^4+\frac{31845}{224} e^6+\frac{381909}{896} e^8+O\left(e^{10}\right).
\label{tidal_energye10}$$ The dissipation rate calculated up to $e^{10}$ is plotted in Fig.\[fig:edot\] (dotted curve), where it can be compared with the exact result. It is clear that, *for $e\gtrsim0.4$, the polynomial developments of the tidal evolution equations must be done to a much higher degree that done in previous studies, or complete calculations such as the ones done in @Hut81 must be used.* The same argument holds for the evolution of the semi-major axis and the eccentricity. Since Eqs.(\[evol\_a\]) and (\[evol\_e\]) also contain $(1-e^2)^{-\mathrm{p}/2}$ factors, the decrease of $a$ and $e$ is severely underestimated at even moderately large 