This paper proposes a method for the feasibility analysis of orbital pursuit-evasion missions, by the geometric relationship between the reachable domains of the pursuer and evader with a single impulse. First, the orbital state propagation is expressed using the second-order state transition tensors (STTs). This allows for a rapid acquisition of the state point cloud under an initial impulsive over time, the boundary of which is the envelope of the reachable domain can be obtained rapidly using the convex hull algorithm. Then, a feasibility analysis method for pursuit-evasion missions is proposed, by analyzing whether the reachable domains of the pursuer and evader intersect with each other. A feasibility index is introduced to represent the success rate of pursuit, which is defined as the proportion of the area of the reachable domain intersection to that of the evader′s reachable domain. Finally, the effects of different mission parameters on the feasibility index are analyzed, including the initial relative position between the pursuer and evader, their maneuverability, and the flight time. An example of successful pursuit is presented, when the feasibility index is equal to 100%. The study demonstrates that the proposed second-order STTs method can provide a fast and accurate way for the computation of the reachable domain with a single impulse for both spacecrafts, as well as their intersection, serving as a quick design reference for orbital pursuit-evasion missions.