factor (substituting $f$ by 1 in Eq.\[finc\] gives the mean flux received at the substellar point). The atmospheric models were computed with $f=1/2$ as described by @BHA01.

Tidal Model {#sec:tidalmodel}
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We consider the gravitational tides raised by both the host star and the planet on each other and follow the traditional “viscous” approach of the [*equilibrium tide*]{} theory [@Dar08]. The secular evolution of the semi-major axis $a$ can be calculated exactly (e.g. @Hut81; @NL97; @LCC07; @CL10; see appendix \[appendix\] for the derivation of these equations for any value of the eccentricity and obliquity) $$\begin{aligned}
\frac{1}{a}\frac{\dd a}{\dd t}=\frac{4\,a}{GM_{\star}\Mp}
&\Big\{&\Kp&\left[N(e)\,x_\mathrm{p}\,
\frac{\op}{n}-N_a(e)\right]  \nonumber\\
&+& \Ks&\left[N(e)\,x_\star\,
\frac{\os}{n}-N_a(e)\right]\Big\}\ , \quad
\label{evol_a}\end{aligned}$$ with $$\label{n_e}
N(e) = \frac{1+\frac{15}{2}e^2+\frac{45}{8}e^4+\frac{5}{16}e^6}{(1-e^2)^{6}}$$ and $$\label{na_e}
N_a(e)=\frac{1+\frac{31}{2}e^2+\frac{255}{8}e^4+\frac{185}{16}e^6+\frac{25}{64}e^8}{(1-e^2)^{15/2}},$$ where $G$ is the gravitational constant, $\op$ is the planet’s rotation rate, $\Mp$ and $\Rp$ its mass and radius, $\ep$ its obliquity (the angle between the equatorial and orbital planes), with $x_\mathrm{p}=\cos
\ep$, and $n$ the orbital mean motion. $$\Kp = \frac{3}{2} \kp \dtp  \left( \frac{G
\Mp^2}{\Rp} \right) \left(\frac{\Ms}{\Mp} \right)^2 \left( \frac{\Rp}{a}
\right)^6 n^2 \ ,$$ where $\kp$ and $\dtp$ are the potential Love number of degree 2 and the constant time lag for the planet. The stellar parameters correspond to the same definitions, by 