the Solar system, i.e. of Jupiter. For coplanar and circular orbits, a dimensionless version of Eq.(\[evol\_a\]) reads: $$\label{adotozero}
\dot{\tilde{a}}=- \frac{1}{\tilde{a}^7}\left[1 -\frac{\os}{n}\right],$$ where $\tilde{a}=a/a_0$, 0 indices refer to initial values and time is counted in units of $$\tau = \frac{1}{6} \frac{\Mp^2}{M_\mathrm{S}(M_\mathrm{S}+\Mp)}(\frac{a_0}{\Rp})^8 \frac{\Rp^3}{G\Mp \kp\dtp},$$ where $M_\mathrm{S}$ is the satellite mass. $\tau$ is the typical timescale of tidal evolution of the semi-major axis. Injecting angular momentum conservation $$J_\mathrm{tot}=\frac{\Mp M_\mathrm{S}}{\Mp+M_\mathrm{S}} a^2n +\sum_\ii C_\ii \omega_\ii$$ into Eq.(\[adotozero\]), and integrating over time yields $$\label{exactsole0}
\int_1^{\frac{a(t)}{a_0}} \frac{-\tilde{a}^7 \dd \tilde{a}}{1 -\tilde{a}^{3/2}[\frac{\omega_{\mathrm{p},0}}{n_0}-\beta(\sqrt{\tilde{a}}-1)]}=\frac{t}{\tau},$$ where $\beta =(M_\mathrm{S}\Mp a_0^2)/((\Mp+M_\mathrm{S})C_\mathrm{p})$. Note that this integral cannot be performed down to $a=0$ because the satellite first crosses the corotation radius where the integrand tends to infinity, which is an unstable equilibrium state for the system [@Hut80]. Furthermore, this result is not limited to the case of a planet-satellite system and can be directly used to compute the inspiral time of a close-in exoplanet (setting p$\,\rightarrow\star$ and S$\,\rightarrow\,$p).

For the Io-Jupiter system, taking $t=-4.5\,\times10^9$yr and $a(t)$ equal to the Roche limit in Eq.(\[exactsole0\]) yields $k_\mathrm{J}\Delta t_\mathrm{J}\lesssim5\times 10^{-3}$s. Therefore, for the actual Io-Jupiter system, Eq.(\[QforJ\]) implies $Q'_\mathrm{J}\gtrsim 1\times10^6$, slightly smaller than the value derived by @GS66. As discussed by these authors, our upper limit on $\dtp$ must 