Objective Small space debris is difficult to observe from the ground and poses a significant threat to space missions. Observing the small debris from satellite has become a new solution. Compared with ground-based observation, the relative position between the satellite and the debris changes much more rapidly and significantly. The shape of space debris is usually complex and most debris are in a state of spin, making the luminosity characteristics of space debris vary greatly in the field of view of the satellite. Therefore, it is necessary to analyze the luminosity characteristics of arbitrary polyhedral and spinning space debris observed from space-based telescopes.

Methods A method of obtaining the apparent magnitude curve of space debris observed from a satellite is proposed. An observation model for space debris observed from satellite is established (Fig.1). Based on the positions of the Sun, the satellite, and the space debris in the geocentric inertial coordinate system and the attitude of the debris, the sun vector and satellite vector are acquired in the debris body-fixed coordinate system. By defining the normal vector of each surface of the debris and then transforming it into the debris body-fixed coordinate system, the solar incidence angle and the observation angle from the satellite can be calculated. The effective reflective area of each surface is then computed. Using Davies scattering model, the total irradiance from the Sun reflected by the space debris onto the entrance pupil of the satellite is obtained. According to the apparent magnitude formula, the apparent magnitude of the debris is calculated. The apparent magnitude curve is generated by defining a sequence of attitudes of the debris.

Results and Discussions   Three space rendezvous scenarios between the satellite and the debris are designed. The phase angle ranges from 50° to 90° in scenario 1, from 4° to 16° in scenario 2, and from 65° to 12° in scenario 3. Three kind of spin rate of the debris are analyzed. Apparent magnitude curve simulation is completed for debris with shapes including flat plate, cube, octahedron and asymmetric arbitrary polyhedral object. All the debris have an outer envelope size of 10 cm and the primary material is aluminum. The simulation results in scenario 1 show that plate debris is invisible at some specific attitude when the spin velocity is 0°/s and turn to alternate brightness and darkness when the debris is in the state of rotation (Fig.4). Block-shaped debris is always visible in different attitudes and turns to apparent magnitude fluctuation when the debris is in the state of rotation (Fig.5-Fig.7). The simulation results for scenario 2 show that the apparent magnitude of cubic and octahedral debris shows almost no fluctuation (Fig.10-Fig.11). The simulation results in scenario 3 show that for centrally symmetric debris such as the cube and octahedron, the apparent magnitude shows obvious fluctuation when phase angle is greater than 30°, while it shows almost no fluctuation when phase angle is less than 20° (Fig.13, Fig.15-Fig.16). With the increase in spin velocity, the apparent magnitude fluctuation frequency increase and the amplitude remain unchanged.

Conclusions A method of obtain apparent magnitude curve of arbitrary polyhedral and spinning space debris observed from satellite is proposed. The apparent magnitude curve of plate, cube, octahedron and asymmetric arbitrary polyhedral debris in three different scenarios are simulated. The apparent magnitude fluctuation frequency is related to shape and spin velocity of the debris. The apparent magnitude fluctuation amplitude is related to shape of the debris and the phase angle. The apparent magnitude curve of centrally symmetric spinning debris manifests as a single sine wave superimposed on the apparent magnitude curve of static debris. The apparent magnitude curve of asymmetric spinning debris manifests as mutiple sine waves superimposed on the apparent magnitude curve of static debris. The smaller the phase angle, the smaller the amplitude of the magnitude fluctuations for the centrally symmetric spinning debris. For asymmetric complex-shaped spinning debris, the amplitude of apparent magnitude fluctuations is primarily determined by the shape of the debris. Therefore, feature of the apparent magnitude curve can be used to analyze the symmetry of debris configuration. For wide field of view debris monitoring optical payload, it isi more recommended to observe the debris at a large phase angle to obtain more photometric characteristics. For debris pointing and tracking optical payload, observation at a small phase angle should be prioritized to achieve more stable tracking condition. The apparent magnitude curve calculation method and the simulation result can provide a reference for space-based optical telescope design and space debris attitude analysis.