Clohessy-Wiltshire Equations

Also known as Hill’s equations can be written as follows:


where we assume $t$ is the time between maneuvers, $v$ is the maneuver delta-v, and $a_{d}$ is a random disturbance representing unmodeled differential accelerations due to solar radiation pressure, atmospheric drag, plume effects, and $J_{2}$ and higher-order gravity terms. If the x, y, and z components of $^{LVLH}$ are altitude, downrange, and cross track, respectively, the above transition matrices are given by:





I wrote all of this mostly as $$ practice. It took a long time.

[!warning] WIP: Write the CW equations in the form of the Self Rescue Strategies for EVA paper which is a more ubiquitous form. Separate this note by paper