Gravitational dependence of contact angle hysteresis for a solid defect pinning a liquid-vapor interface

We study the case of contact angle hysteresis, when a solid defect is quasistatically pulled out and pushed into a liquid, in the presence of gravity. Pie solve the problem analytically in an approximation which, in the case of a truncated parabolic pinning force, becomes exact in the limit of slow variations in the height of the liquid-vapor interface. The case of a Gaussian pinning force is solved by a geometrical construction. We show that the extension of the defect, and not the capillary length, is the determining length scale when the extension of the defect is much smaller than the capillary length.