On the nature of the radial and cross track errors for artificial earth satellites

The analysis of the radial and cross track errors of artificial earth satellites is discussed in terms of the interference of two one-dimensional celestial mechanical wave trains. Resulting equations for these tracking errors describe the behavior of the uncertainties in the orbital parameters as oscillatory in nature, with a rapidly oscillating term, which is a function of the sum of the observed and computed orbital frequencies, modulated in amplitude by a slowly varying oscillation. This latter term is itself a function of either the difference between these orbital frequencies or between the values of the computed and observed right ascensions, depending upon whether it is the radial or cross track case under consideration. These results indicate that the cross track calculation describes the behavior of uncertainties in the right ascension of the ascending node and the inclination, while the radial calculation gives information on uncertainties in the semi-major axis, the eccentricity, and the argument of perigee. In addition, expressions for the radial and cross track oscillatory frequencies are obtained in terms of the orbital frequencies of the satellites. Data show that the time average of the radial and cross track errors in any case, will both approach zero.