This work develops and analyzes a control algorithm for an unmanned aerial vehicle (UAV) to circumnavigate an unknown target at a fixed radius when the UAV is unable to determine its location and heading. Using a relationship between range-rate and bearing angle (from the target), we formulate a control algorithm that uses the range-rate as a proxy for the bearing angle and adjusts the heading of the UAV accordingly. We consider the addition of measurement errors and model the system with a stochastic differential equation to carry out the analysis. A recurrence result is proven, establishing that the UAV will reach a neighborhood of the desired orbit in finite time, and a mollified control is presented to eliminate a portion of the recurrent set about the origin. Simulation studies are presented to support the analysis and compare the performance against other algorithms for the circumnavigation task.