Dimension-ratio-based observability analysis of asteroid photometric model and optimization strategy for completely observable satellite detection

The physical parameters of asteroids are obtained through the inversion of the asteroid photometric model. However, due to limitations in observation orientation, asteroid lightcurves exhibit insensitivity to certain errors in physical parameters. This insensitivity leads to multiple virtual solutions when inverting the asteroid photometric model. To theoretically evaluate the performance of this inversion, we propose a dimension-ratio-based observability analysis of the asteroid photometric model. Firstly, we employ the traditional observability analysis method to generate candidate vectors representing errors in physical parameters. Subsequently, we select unobservable vectors corresponding to very small chi-square values from these candidate vectors. The unobservable vectors span into an unobservable subspace, meaning that large errors within this unobservable subspace result in minimal disturbances in lightcurves, indicating low estimation accuracy for this unobservable subspace. Finally, the dimension-ratio, which is the ratio of the dimension of unobservable subspace to the total number of physical parameters, is used for the observability degree. Building upon this, in response to the issue that ground-based detection hardly acquires completely observable physical parameters, we propose an optimization strategy for completely observable satellite detection, utilizing the hippopotamus optimization algorithm to optimize the satellite orbit. In this optimized orbit, the physical parameters of the asteroid become completely observable. Simulation results demonstrate that the distribution of virtual solutions is consistent with our theoretical analysis. Furthermore, a unique solution available for estimating the physical parameters of the asteroid exists in the optimized satellite orbit.