Frozen Orbit Design and Stability

Abstract

Apophis, an Aten-type asteroid, was considered a significant threat to Earth upon its discovery due to its potential impact with Earth in 2029. Although the collision has now been ruled out, Apophis will still perform an exceptionally close flyby of Earth that same year at a distance of 32,000 km from the surface. This event presents a unique opportunity for the investigation of rotational changes due to tidal forces and post-encounter ephemerides for planetary defence purposes. In fact, the upcoming OSIRIS-APEX mission will enter orbit around Apophis shortly after the 2029 flyby. Even though orbits around near-Earth asteroids like Apophis face challenges due to the asteroid's low gravity and the strong perturbations induced by solar radiation pressure, frozen orbits - a specialised orbit with constant eccentricity and argument of periapsis on average - can achieve orbital stability in such a complex dynamical environment.

Frozen orbits were successfully employed for some of the mission phases of the OSIRIS-REx mission, and past research has greatly focused on the investigation of frozen orbits around Apophis and other small bodies through the use of analytical and numerical methods. However, no prior research has addressed the design of frozen orbits that can survive the close approach in 2029 without orbital correction manoeuvres. The aim of this research is thus to investigate the stability of control-free frozen orbits around Apophis during the 2029 Earth flyby.

To fulfil this goal, both analytical and numerical methods are employed. The analytical analysis involves averaging of Lagrange's Planetary Equations including perturbations from solar radiation pressure and Apophis' zonal gravity up to degree four in combination with a Lyapunov stability analysis and a comparison to numerical simulations. Assuming an argument of periapsis and longitude of the ascending node of +-90 degrees, the analytical method identifies two main solution families: near-equatorial heliotropic/anti-heliotropic orbits and near-polar Sun-terminator orbits. However, the stability analysis predicts only half of the sampled solutions to be stable. The comparison to numerical simulations shows that both analytical techniques fail to identify stable, frozen orbits. The stability index correctly identifies stability for 66.67% of the results that reach the end of a numerical propagation without surface impact or orbital escape. More significantly, 42% of the results are identified as false positives. The variations in eccentricity and argument of periapsis for the solutions that reach the end of a 28-day simulation are approximately 0.69 and 600 degrees respectively for the
near-equatorial solutions and, at best, 0.89 and 215 degrees for the near-polar solutions, which are too large to be considered frozen orbits.

In the numerical analysis, the frozen orbit problem is defined as a multi-objective optimisation problem with two objectives: minimisation of the maximum variation in eccentricity and argument of periapsis. Trajectories with different orbital injection parameters are simulated to find the optimal initial state leading to a frozen orbit. First, the results are focused exclusively on the pre-flyby period with no constraint on surviving the flyby. The best solutions lead to a maximum variation in eccentricity and argument of periapsis of approximately 0.047 and 66 degrees respectively over a 28-day period. However, these orbits all eventually collide with the asteroid at the time of the flyby. Imposing a constraint on survival increases the maximum variation in eccentricity and argument of periapsis to ranges of 0.08-0.21 and 103-110.5 degrees in the pre-flyby period. The behaviour post-flyby is stable for some of these solutions but no longer corresponds to the frozen configuration. In both cases, the solutions are categorised under the near-circular, near-polar, Sun-terminator frozen orbit family, the same type of orbit employed for the frozen orbit phases of the OSIRIS-REx mission. Despite the limitations of this work, the numerical pre-flyby results exhibit robustness against uncertainties in modelling parameters and orbital injection inaccuracies.