# Clohessy-Wiltshire equations

The Clohessy-Wiltshire equations describe a simplified model of orbital relative motion, in which the target is in a circular orbit, and the chaser spacecraft is in an elliptical or circular orbit. This model gives a first-order approximation of the chaser's motion in a target-centered coordinate system. It is very useful in planning rendezvous of the chaser with the target.[1]

$\ddot{x} = 3n^2x+2n\dot{y}$

$\ddot{y} = -2n\dot{x}$

$\ddot{z} = -n^2z$

$n = \sqrt{\frac{\mu}{a^3}}$

For intuition, in low earth orbit, $\mu = 3.986E14 \frac{m^3}{s^2}$ and $a = 6378137m + 415000m = 6793137m$, so $n=0.0011 s^{-1}$.

## References

1. ^ "Clohessy-Wiltshire equations" (PDF). University of Texas at Austin. Retrieved 12 September 2013.