Hill equations ($\ddot{x}-2n\dot{y}-3n^{2}x=0,\ddot{y}+2n\dot{x}=0$)

The Hill equations describe linear relative motion near a reference circular Kepler orbit. Let that orbit have radius $a$ and mean motion $n=\sqrt{GM/a^3}$. In a rotating reference frame centred on a reference test particle, let $x$ point radially outward, $y$ forward along the orbit, and $z$ normal to its orbital plane. Neglect mutual gravity and other perturbing forces, and require separations small compared with $a$.

Linearizing the central gravitational acceleration and including the Coriolis force givesThe planar pair was the starting form above; the third equation describes vertical simple harmonic motion. In spacecraft relative-motion applications these are also called the Clohessy-Wiltshire equations. Axis conventions vary, so the radial coordinate must be identified before comparing formulas.^[ref]

A directly checkable planar solution isFor $C=0$ the relative motion is a bounded epicyclic oscillation; for $C\ne 0$ its mean radial offset causes a secular along-track drift, the local Keplerian shear. Mutual gravity adds encounter forcing, and large separations or long accumulated errors require the nonlinear orbital model.

The planar Hill equations describe small displacements about a circular Kepler orbit in a frame rotating at angular frequency $n$. With $x$ radial and $y$ along the orbit, their unforced solution isDirect differentiation gives $\ddot{x}-2n\dot{y}-3n^{2}x=0$ and $\ddot{y}+2n\dot{x}=0$.

The $A,B$ terms describe epicyclic motion with tangential amplitude twice the radial amplitude. The constant radial offset $C$ causes an accumulating along-track displacement through Keplerian shear, while $D$ is an initial along-track offset. This separation makes the solution useful for comparing nearly neighbouring orbits and for identifying which motion requires additional encounter forcing.

The oscillatory integration constants determine the orbital eccentricity vector, the constant radial offset determines the semi-major axis, and the constant along-track offset fixes the mean longitude. The radial and tangential epicycle amplitudes have ratio $1:2$.