Spacecraft Formation Flying

4.7 Equations of Relative Motion under the Influence of $J_2$

The equations of motion of a satellite under the influence of gravitational and thrust perturbations can be written in frame $ℐ$ as (see our discussion of orbital perturbations in Section 2.5)

$\ddot{r}=-{\nabla }_r\mathcal{V}+u$

  (4.89)

where $r$ is the position vector, $\mathcal{V}=\mathcal{U}-\mathcal{R}$ is the total gravitational potential (the nominal ...