Space safety and hazard-reduction techniques for space-transportation vehicles are becoming critical in the advent of commercial crewed launches and suborbital intercontinental travel. One way to mitigate risks is to use advanced guidance strategies to centralise mission planning, and guidance, navigation and control. Convex optimisation can be suitable for this purpose, as it solves second-order-cone programming problems sequentially in polynomial time. As atmospheric re-entry contains a large number of non-convexities, an advanced optimal-control optimiser based on successive convex optimisation was developed and analysed with a golden-section line-search method to enhance its convergence. A direct-linearisation approach, although slightly sensitive to initial guesses, was seen to be robust when compared in different objective formulations. However, a hybrid approach using a lossless convexification performed better for most problems, although it relied on a regularisation condition, which was sometimes difficult to satisfy. The reachability capabilities of the SPHYNX lifting body during orbital and descent aborts were analysed and showed that around 14 minutes are available to perform an abort to an alternative landing site for a nominal mission. While there is significant room for improvement, the study shows that successive convex optimisation could be potentially used for mission planning and on-board applications as an optimal trajectory planner during realistic re-entry missions.@en