This work addresses the observability analysis for a cooperative range-based navigation system that uses a beacon with circular motion installed on-board the support platform to find the location of an underwater vehicle. First, the observability analysis is carried out without taking into account the ocean current that affects the vehicle. To that end, an augmented state-space realization is obtained in order to convert the nonlinear system into a linear time varying (LTV) system. Then, the observability analysis is performed considering constraints on the initial conditions due to the state transformation. Based on these results, a second observability analysis is carried out taking into account ocean current in the model. We show that, by knowing the initial beacon's position, the system is observable when the vehicle is moving in a trim trajectory with and without ocean currents. The obtained results include information about the initial conditions of the beacon and the ocean currents in the observability analysis. This result is important from a theoretical point of view, and represents a new way of implementing range-based navigation systems, without relying on deviation of the vehicle's mission or the need of another vehicle to ensure observability.