Comparison of the analytical expression for the obliquity envelope with secular simulations. In panel (a), we show the final semimajor axis on the x-axis, and the final obliquity on the y-axis. Panel (b) shows the final obliquity as a function of the initial mutual inclination. In panel (c), we show the initial mutual inclination between the planets as a function of the final semimajor axis of the HJ. The analytical results are shown as solid black lines. The results from our secular simulations are shown as red and blue squares. The results shown in blue only take into account quadrupole-order perturbations, and assume that the inner planet is a test particle. For the results shown in red, the test-particle approximation is dropped and octupole-order perturbations are also included. We can see that the analytical envelope is in good agreement with both sets of secular simulations.