A mathematical formulation for calculating precisely the effective emissivity ( epsilon _a) of a plane diffuse surface covered by a specular hemisphere is presented. Exact expressions are developed for the exchange factors governing radiation exchange between points on the plane surface after (multiple) reflection from the hemisphere, with allowance for an aperture at the top and a gap between surface and hemisphere. The resulting integral equation for epsilon _a is solved numerically. The results show that epsilon _a varies non-uniformly along a radius as different numbers of specular reflections come into play. The top aperture only slightly decreases values of epsilon _a below those for a complete hemisphere; the gap produces substantial and non-uniform changes. A prior closed-form expression for epsilon _a is shown to be a good approximation within one-third of the radius from the centre. The effects of temperature gradients across the surface are shown to affect epsilon _a considerably.