General Formulas of Effective Earth Radius and Modified Refractive Index

The effective earth radius can transform the curved ray path into a straight line, and the modified refractive index can transform the spherical ground into a plane. Using them can simplify the propagation modeling of electromagnetic waves in the atmosphere, so they are widely used in the calculation of long-range propagation of electromagnetic wave, and adopted by the ITU Recommendations and the IEEE Standards. In this paper, upon employing simple geometric methods, we provide general formulas for the effective earth radius and the modified refractive index, pointing out that they both originate from the use of conformal transformations. In turn, we prove that for both quantities the formulas contained in the ITU Recommendations and the IEEE Standards are special cases of the general formulas given in this paper. Our results provide a solid mathematical background to the concepts of effective earth radius and modified refractive index, and pave the way for their use beyond the propagation of electromagnetic waves in the earth atmosphere, e.g. to the propagation of other types of waves in generic media.