Underestimating tidal heating {#sec:edot_2ndOrder}
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The key quantity arising from the coupling between the orbital evolution and the internal cooling history of a planet is the amount of energy dissipated by the tides in the planet’s interior, which may compensate or even dominate its energy losses. As a result, tides raised in an eccentric planet can slow down its contraction [@BLM01; @LBC09; @BCB10] or even lead to a transitory phase of expansion [@MFJ09; @ISB09]. Correctly determining the tidal heating rate is thus a major issue in the evolution of short-period planets. The often used formula is [@Kau63; @PC78; @JGB08] $$\dot{E}_{\mathrm{tides}}=
7\Kp e^2=\frac{21}{2}\frac{\kp}{Q} \left( \frac{G
\Ms^2}{\Rp} \right) \left( \frac{\Rp}{a} \right)^6 n e^2
\label{tidal_energye2}$$ (the $21\kp/2Q$ is rewritten $63/4Q'$ in @MFJ09). As already stated by @Wis08, although this formula gives a fair approximation of the tidal dissipation rate for the small eccentricity cases, typical in the solar system, it severely underestimates the tidal heating for moderate and high eccentricities. Fig.\[fig:edot\] illustrates the power dissipated in a pseudo synchronized planet as a function of the eccentricity. It shows that for $e\gtrsim 0.45$, the truncated formula used in @MFJ09 and @ISB09 underestimates the actual tidal dissipation rate *by more than one order of magnitude 