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This paper proposes a novel method to efficiently construct the aerocapture attainability set based on convex optimization. Using dimensionality reduction, constructing the attainability set is equivalent to solving a set of discrete points on its boundary. As solving each of the boundary points is a typical nonlinear optimal control problem, a sequential convex programming method is adopted. The efficiency and accuracy of the proposed algorithm is demonstrated by high-fidelity numerical results. This is the first time that the configuration of the aerocapture attainability set is precisely described by the state variables at atmospheric exit. Since the quantification of the set is significant for assessing the feasibility of performing an aerocapture maneuver, the proposed method can be used as a reliable tool for systematic design for aerocapture mission.