On Friday, April 13, 2029 the asteroid Apophis will have a close approach to the Earth, coming closer than the GEO orbit belt. The flyby will drastically change the Apophis heliocentric orbit and its spin state, but some debate remains about what changes may occur on the surface and interior of this asteroid due to the tidal forces from the Earth. In order to better understand the effect of the flyby, there are multiple spacecraft missions that are being planned to observe the asteroid, both prior to and after its close flyby. One mission proposal in particular, ESA’s Ramses mission, will also be able to observe the asteroid from a close proximity vantage point, potentially as close as a few kilometers, as the asteroid travels through its Earth close approach. In this paper we discuss the detailed dynamics and possible mission design trajectory of a spacecraft close to Apophis, fully accounting for the tidal effects of the Earth around closest approach of the hyperbolic trajectory. 

Initial analysis of how to maintain and control a spacecraft relative to Apophis during its hyperbolic flyby has been analyzed previously, where several different approaches were identified to maintain close proximity through the flyby [1]. Among the most viable of these approaches is to place the spacecraft in a neighboring hyperbolic orbit that will naturally drift relative to the asteroid, or to  maintain a fixed location relative to the asteroid using active thrusting. When the detailed observability of the surface is taken into account, it becomes important to also place the spacecraft so that it can observe the surface during and after the close approach. This problem can be analyzed using the Tschauner–Hempel (TH) equations evaluated along a hyperbolic orbit [1]. These equations provide an accurate and flexible approach to evaluate the stationkeeping and observation strategies that may be relevant for an observer in proximity of Apophis during this flyby. In [1] a limited analysis was given of several different approaches to stay in proximity to Apophis during its flyby, however the discussion did not consider what observation strategies might be most scientifically useful or how the relative geometry of the asteroid to the Earth will influence the changes on the body. 

In this presentation we analyze in greater detail the relative proximity dynamics of a spacecraft and the asteroid Apophis as it travels through its Earth close approach. We find a large spectrum of possible motions relative to Apophis in the time around closest approach, within ±8 hours of perigee. These relative orbits can be designed to be ballistic, meaning that the spacecraft control system can be utilized to perform stationkeeping control to an ideal trajectory. Utilizing the closed form of the TH equations we are able to lay out the basic design principles for these relative trajectories, accounting for the tidal effect of the Earth. We will show that, depending on the desired observation perspective, it is possible to design a pathway that can accommodate the scientific observation goals. Specifically, we will show that it is possible to find ballistic motions that linger relatively slowly over the lit side of Apophis.

A companion paper submitted by H.-S. Wang will explore the propagation of navigation uncertainty through such relative trajectories, and the effect of errors and uncertainties for hovering implementations. 

References
[1] D.J. Scheeres, “Proximity Operations About Apophis Through Its 2029 Earth Flyby,” The Journal of the Astronautical Sciences (2022) 69:1514–1536. https://doi.org/10.1007/s40295-022-00360-w