the object or with the tidal frequency but also with the amplitude of the tidal potential. For example, $\qs$ values inferred from the circularization of close FGK binary stars [@MM05], may be lower than the actual $\qs$ encountered in star-planet systems [@OL07]. Consequently, the range of values considered here for both $\qs$ and $\qp$ should be seen as mean values and be re-evaluated when considering specific and/or atypical systems (XO-3, HAT-P-2 or CoRoT-Exo-3 for example).

Effect of the truncation of the tidal equations to $\mathbf{2^{\mathrm{nd}}}$ order in $\mathbf{e}$: Analytical analysis.  {#sec:2ndOrder}
==========================================================================================================================

Following the initial studies of @JGB08, all the studies exploring the effect of tidal heating on the internal evolution of “hot jupiters” (@MFJ09, @ISB09) have been using a tidal model assuming a constant $Q$ value during the evolution. Moreover, in all these calculations, the tidal evolution equations are truncated at the second order in eccentricity (hereafter referred to as the “$e^2$ model”), even when considering tidal evolution sequences with non-negligible values of $e$ at earlier stages of evolution. Although such a $e^2$-truncated model is justified for planets and satellites in the solar system [@Kau63; @GS66], it becomes invalid, and thus yields incorrect results for $a(t)$, $e(t)$ and $\dot{E}_{\mathrm{tides}}$ 