Fig. 8

Rotational state of Tiangong-1 and atmospheric density gradient torque. The left model presents the normal attitude of Tiangong-1 during its active operational life. The Ox-axis of the body-fixed system points in the forward direction, and the Oz-axis points to the zenith. The right model presents the rotational state of Tiangong-1 during the joint observation. The angular momentum \({\varvec{H}}\) precesses in space around the normal of the orbital plane \({\varvec{N}}\) with the angle \(\theta _H = 23.1^\circ \pm\,2.5^\circ\). The precession speed ranges from approximately \(-\,280^\circ \,{{\rm day}}^{-1}\) (November 2017) to \(-160^\circ \,{{\rm day}}^{-1}\) (March 2018). The shadow shows that the atmospheric density \(\rho\) decreases with increasing orbital altitude. Therefore, the atmospheric force \({{\rm d}}{\varvec{f}}_1\) at the end of the satellite nearest to the Earth is greater than \({{\rm d}}{\varvec{f}}_2\) at the far end, which generates a torque with the same direction of its self-spin