Study on the estimation of Euler angles for Macau Science Satellite-1

The Euler angle estimation is a calibration method for vector data measured by the magnetometer on a satellite. It is used to find the relative rotation between the coordinate system of the magnetometer and the satellite (usually determined by Star Imagers). Before launch of the low-orbit, low-inclination Macau Science Satellite-1 (known as MSS-1), we simulated the estimation of Euler angles by using the magnetic measurements of the in-orbit Swarm and China Seismo-Electromagnetic Satellite (noted as CSES), with various data combinations. In this study, 11 data sets were designed to analyze the estimation results for the MSS-1 orbit by using a joint estimation method of the geomagnetic field model parameters and Euler angles. For the model results, we found that all the spatial power spectral lines showed behavior consistent with that of the CHAOS-7.8 model at low degrees (corresponding to large-scale magnetic signals). The spectra of models without global data coverage deviated much more (by a maximum of ~10⁴ nT²) from those of the CHAOS-7.8 model at higher degrees. For models with global data coverage and with various data combinations, the spectral lines were distributed similarly. Moreover, the models with accordant power spectral distributions demonstrated different Euler angle estimations. As more vector data at higher latitudes were included, the estimated Euler angles varied monotonically in all three directions. The models with vector data in the same latitude range showed similar Euler angle results, regardless of whether the poleward scalar data were included. The largest value difference was found between the models using vector data within ±40° latitudes and those using vector data within ±60° latitudes, which reached to ~28″. Therefore, we concluded that the inversion of the spherical harmonic Gauss coefficients in our tests was mainly affected by the spatial coverage range of the data, whereas the estimation of Euler angles largely depended on the latitude range where the vector data could be obtained. These results can be used for future in-flight data testing. We expect the estimation of Euler angles to improve as other methods are adopted.