Left: optical depth variation of β (Eq. (6)), plotted in green for the Doppler profile; this is the fundamental CEP function describing the average probability for escape from a uniform slab with optical depth τ across its full width (see text). Also plotted are escape probability approximations for a uniform, static sphere (Eq. (13)) and LVG approximations for spherical (Eq. (14)) and plane-parallel (Eq. (15)) geometries. In the LVG cases, the appropriate velocity gradient replaces the ratio Δυ/ℓ in the definition of optical depth. Right: the ratio of each escape probability approximation and the fundamental slab function.