eccentricity when using a $2^{\mathrm{nd}}$ order truncated expansion in eccentricity.

In particular, as discussed in the next section, a high eccentricity ($e\gtrsim 0.6$) cannot be maintained for a few 100 Myr to a few Gyr in a system like HD 209458 in agreement with the results of @MFJ09 (see Fig.\[fig:xo4\] below). This is in contrast with @IB09 who find that the radius HD 209458b can be matched and that the system can sustain a significant eccentricity up to the observed epoch. Such discrepancies between these two studies based on the same tidal model may reveal differences in the implementations of the tidal equations, or a difference in the calculation of interior structures or boundary conditions.

Effect of the truncation to $\mathbf{2^{\mathrm{nd}}}$ order in $\mathbf{e}$: Simulation results {#sec:comp}
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In this section, we present the comparison of the results of our complete model with the “$e^2$ model”. We have calculated evolutionary tracks of the tidal evolution for various transiting systems, coupling the internal evolution of the object either with our tidal model or with the “$e^2$ model” used in @MFJ09 and @IB09. In order to ensure a consistent comparison with these authors, we directly convert their set of tidal parameters. Since 