Time delay (left panel) and frequency shift (right panel) due to the atmosphere of the occulting planet for an index of refraction profile given in Eq. (152a) with N₀ = 10⁻⁶. Red circles represent results of the numerical integration (i.e., in Eq. (157) for the left panel, and Eq. (158) with deduced from Eq. (156) for the right panel). The thick blue line corresponds to the analytical predictions (i.e., in Eq. (133) for the left panel, and Eq. (158) with deduced from Eqs. (137), (139), and (146) for the right panel). The dashed lines correspond to the difference between numerical and analytical results. The dotted lines represent the same difference setting Ω = 0 into analytical solutions. Above the horizontal thin black line, the difference between numerical and analytical results is dominated by numerical noise, while below, it is dominated by neglected second-order terms.