our model assumes a constant time lag, and not a constant $Q'$ value, a history track computed with the $Q'$ “$e^2$ model” with a constant couple ($\qp,\,\qs$) is compared with a history track computed in our model with a constant couple $(k_2\dtp,\,k_2\dts)$ given by $(k_2\dtp=\frac{3}{2 n_{\mathrm{obs}} \qp} ,\,k_2\dts=\frac{3}{2 n_{\mathrm{obs}} \qs} )$ (See §\[sec:q\] and Eq.(\[QforExo\])). This ensures that - although our calculations are conducted with a *constant* $\Delta t$ - the quality factor computed with Eq.(\[QforExo\]) in the object at the present time is the same as the one used in the *$Q$ constant* model.

Calculations at low eccentricity
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We first compare the results of the two models on a system which has a zero measured eccentricity and is not inflated, namely TrES-1. Such a system does not require a substantial initial eccentricity for its observed properties to be reproduced and thus provides an opportunity to test the quasi circular limit, where the “$e^2$ model” used by @MFJ09 and our model should yield similar results. Fig.\[fig:tres1\] illustrates the results of the integration of the coupled internal/orbital evolution equations with our constant time lag model (solid curve) and with the “$e^2$ model” (dashed curve) for an initial eccentricity of 