The Euclidean Signed Distance Field (ESDF) is widely used in visibility evaluation to prevent occlusions and collisions during tracking. However, frequent ESDF updates introduce considerable computational overhead. To address this issue, we propose Eva-Tracker, a visibility-aware trajectory planning framework for aerial tracking that eliminates ESDF updates and incorporates a recovery-capable path generation method for target reacquisition. First, we design a target trajectory prediction method and a visibility-aware initial path generation algorithm that maintain an appropriate observation distance, avoid occlusions, and enable rapid replanning to reacquire the target when it is lost. Then, we propose the Field of View ESDF (FoV-ESDF), a precomputed ESDF tailored to the tracker's field of view, enabling rapid visibility evaluation without requiring updates. Finally, we optimize the trajectory using differentiable FoV-ESDF-based objectives to ensure continuous visibility throughout the tracking process. Extensive simulations and real-world experiments demonstrate that our approach delivers more robust tracking results with lower computational effort than existing state-of-the-art methods. The source code is available at https://github.com/Yue-0/Eva-Tracker.