Math 2.0

Mechanics of particles and systems Definition

Clohessy–Wiltshire equations

Builds on Kepler orbitRotating reference frameCoriolis forceOrdinary differential equation

Definition

Let μ>0\mu > 0 and let a target move on a circular Kepler orbit of radius RR, with mean motion n=μ/R3n = \sqrt{\mu / R^3}. Use the rotating frame centred at the target. In it, xx points radially outwards, yy points along the direction of motion, and zz is normal to the orbital plane. Let (x,y,z)(x, y, z) be the position of a chaser in this frame, and assume it is small compared with RR. The Clohessy–Wiltshire equations are the ordinary differential equations x′′−3n2x−2ny′=0,y′′+2nx′=0,z′′+n2z=0.x'' - 3 n^2 x - 2 n y' = 0, \qquad y'' + 2 n x' = 0, \qquad z'' + n^2 z = 0. They come from the exact relative equations of motion by keeping only the terms of first order in (x,y,z)(x, y, z).

Remarks

The terms 2ny′2 n y' and 2nx′2 n x' come from the Coriolis force, and the term 3n2x3 n^2 x combines the centrifugal force with the tidal part of gravity. The out-of-plane motion is a harmonic oscillation with the orbital period. The in-plane motion combines an oscillation of the same period, a constant offset and a secular drift along yy.

Example: x(t)=x0x(t) = x_0, y(t)=y0−32nx0ty(t) = y_0 - \frac{3}{2} n x_0 t, z(t)=0z(t) = 0 is a solution. Check: x′′=0x'' = 0 and y′=−32nx0y' = -\frac{3}{2} n x_0, so x′′−3n2x−2ny′=−3n2x0+3n2x0=0x'' - 3 n^2 x - 2 n y' = -3 n^2 x_0 + 3 n^2 x_0 = 0; and y′′+2nx′=0y'' + 2 n x' = 0. A chaser below the target, x0<0x_0 < 0, thus drifts ahead of it, as a lower circular orbit is faster.

Non-example: x(t)=x0≠0x(t) = x_0 \ne 0 with yy constant and z=0z = 0 is not a solution, since the first equation would give −3n2x0=0-3 n^2 x_0 = 0.

The same equations are also called Hill's equations, after Hill's work on the motion of the Moon.

Reading path

To understand this page from the ground up, read these 28 pages in order. The foundations come first.

Show the path as a graph
Cross product, concept, no evidence levelCross productconceptDerivative, concept, no evidence levelDerivativeconceptEuclidean space, object, no evidence levelEuclidean spaceobjectEuclidean vector, concept, no evidence levelEuclidean vectorconceptIdentity matrix, object, no evidence levelIdentity matrixobjectKepler orbit, concept, no evidence levelKepler orbitconceptMatrix, definition, no evidence levelMatrixdefinitionOrdinary differential equation, concept, no evidence levelOrdinary differential…conceptVector space, definition, no evidence levelVector spacedefinitionVelocity, concept, no evidence levelVelocityconceptClassical mechanics, concept, no evidence levelClassical mechanicsconceptSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionMomentum, concept, no evidence levelMomentumconceptInertial frame of reference, concept, no evidence levelInertial frame of ref…conceptOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionNewton's laws of motion, statement, no evidence levelNewton's laws of moti…statementOrthogonal group, object, no evidence levelOrthogonal groupobjectAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionSpecial orthogonal group, object, no evidence levelSpecial orthogonal gr…objectAngular velocity, concept, no evidence levelAngular velocityconceptRotating reference frame, concept, no evidence levelRotating reference fr…conceptFictitious force, concept, no evidence levelFictitious forceconceptCoriolis force, concept, no evidence levelCoriolis forceconceptClohessy–Wiltshire equations, definition, no evidence levelClohessy–Wiltshire eq…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cross product concept
  2. Derivative concept
  3. Euclidean space object
  4. Euclidean vector concept
  5. Identity matrix object
  6. Kepler orbit concept
  7. Matrix definition
  8. Ordinary differential equation concept
  9. Vector space definition
  10. Velocity concept
  11. Classical mechanics concept
  12. Square matrix definition
  13. Transpose definition
  14. Linear map definition
  15. Momentum concept
  16. Inertial frame of reference concept
  17. Orthogonal matrix definition
  18. Multilinear map definition
  19. Newton's laws of motion statement
  20. Orthogonal group object
  21. Alternating multilinear map definition
  22. Determinant definition
  23. Special orthogonal group object
  24. Angular velocity concept
  25. Rotating reference frame concept
  26. Fictitious force concept
  27. Coriolis force concept
  28. Clohessy–Wiltshire equations definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Clohessy%E2%80%93Wiltshire_equations
About this page
TagHRR414Q2V2
URIm2://HRR414Q2V2
Kinddefinition
WritingW0 stub
MSC70M20, 70F05
DefinesClohessy–Wiltshire equations
UsesKepler orbit, Rotating reference frame, Coriolis force, Ordinary differential equation
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.