Hill's equations

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The hill's equations (after George William Hill (1838 to 1914)) describe track changes to a satellite within the co-rotating reference system. They can be used to calculate which further course (path and speed) a satellite will take when changing its speed.

They are the solution of the coupled system of equations:

• ${\displaystyle {\ddot {x}}+2\omega {\dot {z}}=b_{x}}$
• ${\displaystyle {\ddot {y}}+\omega ^{2}y=b_{y}}$
• ${\displaystyle {\ddot {z}}-2\omega {\dot {x}}-3\omega ^{2}z=b_{z}}$

Orbital equations

• ${\displaystyle x(\omega ,t)=\left({x_{0}-2{\frac {{\dot {z}}_{0}}{\omega }}}\right)+2{\frac {{\dot {z}}_{0}}{\omega }}\cos \omega t+\left({6z_{0}+4{\frac {{\dot {x}}_{0}}{\omega }}}\right)\sin \omega t-\left({6z_{0}+3{\frac {{\dot {x}}_{0}}{\omega }}}\right)\omega t}$

• ${\displaystyle z(\omega ,t)=\left({4z_{0}+2{\frac {{\dot {x}}_{0}}{\omega }}}\right)+{\frac {{\dot {z}}_{0}}{\omega }}\sin \omega t-\left({3z_{0}+2{\frac {{\dot {x}}_{0}}{\omega }}}\right)\cos \omega t}$

Velocity equations

• ${\displaystyle {\dot {x}}(\omega ,t)=-3{\dot {x}}_{0}-6\omega z_{0}-2{\dot {z}}_{0}\sin \omega t+\left({6\omega z_{0}+4{\dot {x}}_{0}}\right)\cos \omega t}$

• ${\displaystyle {\dot {z}}(\omega ,t)=\left({3\omega z_{0}+2{\dot {x}}_{0}}\right)\sin \omega t+{\dot {z}}_{0}\cos \omega t}$

Examples

Radial maneuver

Change in orbit of a satellite with radial change in speed

A radial maneuver results in an ellipse with a ratio of 1: 2 .

Initial conditions :
Position: Speed: ${\displaystyle (x;z)=(0;0)}$
${\displaystyle ({\dot {x}};{\dot {z}})=(0;\Delta v)}$

Orbital equations :

• ${\displaystyle x=2{\frac {\Delta v}{\omega }}\left({\cos \omega t-1}\right)}$
• ${\displaystyle z={\frac {\Delta v}{\omega }}\sin \omega t}$

Tangential maneuver

Change in orbit of a satellite with tangential change in speed

A tangential maneuver results in a cycloidal path.

Initial conditions :
Position: Speed: ${\displaystyle (x;z)=(0;0)}$
${\displaystyle ({\dot {x}};{\dot {z}})=(\Delta v;0)}$

Orbital equations :

• ${\displaystyle x=4{\frac {\Delta v}{\omega }}\sin \omega t-3\Delta v\cdot t}$
• ${\displaystyle z=2{\frac {\Delta v}{\omega }}\left({1-\cos \omega t}\right)}$
• ${\displaystyle {\dot {x}}_{1}=-3{\dot {x}}_{0}+4{\dot {x}}_{0}\cos \omega t}$

After half a revolution, the satellite moves seven times ${\displaystyle \Delta v}$ in the opposite direction in the co-rotating reference system :

• ${\displaystyle {\dot {x}}_{1}\left({t={\frac {T}{2}}}\right)=-3\Delta v-4\Delta v=-7\Delta v}$

Hohmann maneuvers

Execution of the Hohmann crossing with two maneuvers

Two tangential maneuvers are performed at the Hohmann transition .

See also: Hill's differential equation (three-body problem)