Illumination Conditions Analysis for CubeSats in Rendezvous, Inspection and Docking Missions

Rendezvous, inspection and docking missions conducted with small satellites offer significant potential for in-orbit servicing, maintenance and assembly. Relative navigation in the terminal phase relies on optical-band cameras, whose performance is bounded on both sides by the illumination of the docking interface of the target: low illumination angles saturate the sensor, high ones leave the interface in shadow. That angle is set by the orbit and by the epoch, so the admissible geometry is a mission analysis constraint. This paper reduces it to a closed form and measures it on a case study of two 12U CubeSats docking in a circular Sun synchronous orbit at $500$ km of altitude. Written in a frame built on the orbit and on the Sun together, the illumination angle depends on two variables only, the elevation of the Sun over the orbital plane and the phase of the docking face along the orbit, with the approach direction entering as a shift of that phase. The relation is a right spherical triangle, and it gives the duration of the docking window in closed form, together with three conditions that can be checked before any propagation. The duration separates into three regimes in the Sun elevation, and in the intermediate one it is nearly constant, because the narrowing of the admissible arc and the retreat of the eclipse cancel. That plateau belongs to the in-track approach alone. Sixteen launch opportunities, sampling the local time of the ascending node with a step of one hour and thirty minutes, are propagated over one year and confirm the closed form to within six seconds. Four of them stay inside the intermediate regime all year and deliver about twenty-two minutes of docking window on every day. The requirement on the launch is therefore expressed as a band of Sun elevation rather than as a value of the local time of the ascending node, which is the form in which it survives a change of altitude, of sensor or of admissible range.