Some expansions for an electromagnetic wave propagating through a spherically symmetric refracting medium

Three expansions are derived for an electromagnetic wave propagating through a spherically symmetric refracting medium. The first enables one to compute the phase path length for any vacuum zenith angle when a model for the index of refraction is given. The second enables one to compute the refracted zenith angle corresponding to any vacuum zenith angle. The third enables one to compute the various order effects that the refracting medium has on the received frequency. Although the three expansions are valid for any spherically symmetric refracting medium, they are particularly useful for a wave propagating through the ionosphere because of the fact that they split the terms according to increasing powers of inverse transmitter frequency and therefore, converge rapidly for signals of 150 mhz or higher. A satellite simulation computer program was developed and the first two expansions checked for consistency. It was found that they disagreed the most at large zenith angles. The difference, however, was less than 0.0003% at 85 deg. In addition, the third expansion was used to find the various order contributions to the received frequency. For small zenith angles around 5 deg, the second order contribution (second order in the expansion parameter; third order in inverse frequency) was only about 0.001 as great as the first order and at large zenith angles around 85 deg, the second order contribution was about 0.01 as great as the first order. The transmitter frequency was taken to be about 150 Mhz.