Satellite in a Circular Orbit About a Rotating Spherical Planet and Calculation of the Velocity Change Along the Ground Track

Jean C. Piquette

Retired physicist from the Naval Undersea Warfare Center and current Fellow of the Acoustical Society of America

An object in frictionless contact with and moving along a rotating spherical surface will experience Coriolis and centrifugal accelerations in the rotating frame. Although these are the only accelerations that appear explicitly in the equations of motion, assuming no other tangential physical forces are present, velocity changes of the object over a finite time interval cannot be correctly computed by integrating the sum of only these two accelerations over that time interval. This was proven in a recent publication [J.C. Piquette, “Velocity Change Calculation for an Object Moving on a Rotating Spherical Surface,” Phys. Educ. 31(1) (2015), art. num. 3, pp. 1-12]. It was found there that an unexpected additional acceleration, therein termed the “kinematic” acceleration, was also required to be integrated over the finite time interval in order to deduce the correct velocity change. Interestingly, a satellite in circular orbit about a spherical rotating planet satisfies everything required for the results of this previous work to apply. Hence, for example, the change in velocity of such a satellite, as seen in the rotating frame, cannot be determined by integrating over only the sum of the Coriolis and centrifugal accelerations. It was also found in the earlier work that the influence of the kinematic acceleration is dominant for high initial object speeds. The kinematic acceleration dramatically dominates both the Coriolis and centrifugal accelerations in the case of a satellite in circular near-Earth orbit, since such a satellite has a speed of about 18000 miles/hour. These conclusions also apply to the calculation of velocity changes along the ground track. To permit detailed understanding of the satellite’s motion along the ground track, the notion of a “shadow satellite” is introduced. The results and examples given here can be used in an undergraduate- or graduate-level classical mechanics course as modern space-age applications of classical mechanics that may be of high interest to students.