Author: Marc C. Canellas and Sergei M. Kopeikin
Date: September, 2013
Flybys have been utilized in interplanetary spaceflight since the 1970’s. However, despite the experience and expertise regarding their use, the source of anomalous tracking data for multiple spacecraft as they flew by the Earth in the 1990’s and 2000’s, known as the flyby anomaly, remains unknown. This paper proposes some foundational techniques of setting up planetocentric reference frames to analyze the relative orbital motion of celestial bodies and flyby maneuvers of spacecraft, which could better examine the source of the flyby anomaly. Beginning with a geometric description of two orbits in the barycentric (center of mass) reference frame of the solar system, we performed two transformations to create the planetocentric reference frame: translating the coordinate frame onto the planet in circular orbit and then setting the coordinate frame to rotate with the angular revolution of velocity of the planet about the center of mass. Gegenbauer polynomials along with Taylor series approximations were used extensively to simplify the equations and to reduce them to polar coordinates. To validate the approximations and assumptions, we used the resulting equations to calculate the position of Mars in the Earth’s polar coordinate planetocentric reference frame and compared it to that from the Jet Propulsion Lab’s (JPL) HORIZONS data, which is the same data used by the JPL for their own radar astronomy, mission planning, and spacecraft navigation. The results show that the approximations have sufficient accuracy to map circular and elliptical orbits in reference to a rotating coordinate frame centered on the orbiting body. These planetocentric equations form the foundation for future research to construct higher fidelity reference frames which would take corrections for relativistic effects into account. The higher fidelity reference frames will uncover whether matching the geocentric frame to the barycentric frame is sufficient to describe flyby trajectories. Furthermore, these equations would allow scientists to develop more practically-adequate approaches for calculations of the orbital motion of spacecraft.