Asteroid detection polar equation calculation and graphical representation

Context. The observability of an asteroid from the ground depends on the distance to the Sun and the observer, on the phase angle, on the object shape, and on its surface reflectivity properties. Several magnitude systems have been proposed in the past decades to model the visual magnitude of the object based on these parameters. Aims. Independently of the magnitude system, there is a three-dimensional representation of the geometrical locus of equal visual magnitude when this value is constrained for a given asteroid. We called this the detection polar curve, or just, the detection polar. We derived a process in order to represent it graphically. Methods. We analysed the shape of this geometrical locus for the H, G magnitude system and determined its applicability to the representation of the detectability of an asteroid in its trajectory. We thus calculated the asteroid detection polar equation, as well as the threshold values that change the type of asteroid detectability solution. Results. The resulting detection polar is discussed, the synodic orbit visualisation tool is introduced, as are examples of how it can be used to analyse the graphical representation of an asteroid trajectory, and to represent the detection polar for a given limiting visual magnitude.