The sources of information for Earth orbiting space objects are passive or active observations by ground based or space based sensors. The majority of tracking and detection of new objects for contemporary applications is performed by ground based sensors, which are mainly optical or radar based. Challenges lie in the long times spans between very short observation arcs, large number of objects, brightness variations during observations, occlusions, crossing targets and clutter. Traditionally, tracking problems for space situational awareness have been approached through heuristics-based techniques such as Multiple Hypothesis Tracking (MHT). More recently, solutions derived from Finite Set Statistics (FISST) have been used, such as the Probability Hypothesis Density (PHD) filter, that describe the targets at the population rather than individual level. The recent mathematical framework for the estimation of stochastic populations combines the advantages of previous approaches by propagating specific information on targets (i.e. tracks) whenever appropriate, and by avoiding heuristics through its fully probabilistic nature. This paper presents the first application to space situational awareness problems of the novel filter for Distinguishable and Independent Stochastic Populations (DISP), a tracking algorithm derived from this framework, through a multi-object surveillance scenario involving a ground based Doppler radar and five crossing objects in all orbital regions. The preliminary results show that, despite the sensor's limited observability and constrained field of view, the DISP filter is able to detect the orbiting objects entering the field of view and maintain individual information on them, with associated uncertainty, even once they have left the field of view.