Viscous stirring and dynamical friction for a planet embedded in a ring of planetesimals. The figure shows root mean square eccentricity (top panel) and inclination (bottom panel) of a ring of planetesimals with an embedded planet. There are 1000 planetesimals with mass 10²⁴ g and 1 planet with mass 10²⁶ g. The ring is centred at 1 au and the surface density is 10 g cm⁻². The planet is initially at 1 au. Shown are the results of our model (orange), a full N-body simulation with mercury (grey), and the semi-analytic model of Ohtsuki et al. (2002; black-dashed). Planetesimals and the planet are indicated with solid and dotted lines, respectively.