Evolution of the semi-major axis of a 1   M_⊕ planet orbiting a 0.04   M_⊙ BD. The solid colored lines are for different initial semi-major axis. The dashed dotted line represent the radius of the BD and the dotted line the corotation radius. The habitable zone is also plotted using two different conditions of habitability (solid line: the planet receives an incoming flux of 400 W/m², dashed line: 240 W/m², see Sect. 4.1 for more details). The bold dashed line represents the Roche limit.