As the demands for aerospace missions continue to grow, the autonomy and reliability of launch vehicles become increasingly critical. When faced with engine failure scenarios, assessing the remaining capability of a launch vehicle is essential to determine whether it can proceed with the mission or must abort to orbit. This paper focuses on the method of determining target orbit reachable set of a launch vehicle abort mission during ascending flight with engine failure. A multi-mode computation method for target orbit reachable set is proposed, which decomposes the problem into several modes and establishes corresponding mode-based optimal control problems. This allows the reachable orbital elements of the launch vehicle to be obtained by sequentially solving a series of optimal control problems. Furthermore, a trajectory planning method based on successive convex optimization is designed, which can efficiently solve the optimal control problem in real-time. Numerical experiments demonstrate that the proposed method can obtain the reachable orbital elements in case of engine failures, and its convergence and real-time performance are sufficient to meet the requirements of critical real-time applications.