Math 2.0

Geophysics Definition

Earth's rotation

Builds on Angular velocityRigid body

Definition

Let 𝒯\mathcal{T} be a terrestrial reference frame: an orthonormal frame attached to the Earth's crust, with origin at the centre of mass. Let 𝒞\mathcal{C} be a celestial reference frame: an orthonormal frame whose axes are fixed relative to distant extragalactic sources. For each time tt, let R(t)∈SO(3)R(t) \in SO(3) map the coordinates of a vector in 𝒯\mathcal{T} at time tt to its coordinates in 𝒞\mathcal{C}. The rotation of the Earth is the function t↦R(t)t \mapsto R(t). Its angular velocity vector ω(t)∈ℝ3\omega(t) \in \R^3, in celestial coordinates, is defined by R˙(t) R(t)Tv=ω(t)×vfor all v∈ℝ3.\dot R(t)\, R(t)^{T} v = \omega(t) \times v \quad \text{for all } v \in \R^3.

Remarks

Example: a uniform rotation about the third axis with rate Ω\Omega is R(t)=(cos⁡Ωt−sin⁡Ωt0sin⁡Ωtcos⁡Ωt0001)R(t) = \begin{pmatrix} \cos\Omega t & -\sin\Omega t & 0 \\ \sin\Omega t & \cos\Omega t & 0 \\ 0 & 0 & 1 \end{pmatrix}, and one checks that R˙RTv=(0,0,Ω)×v\dot R R^T v = (0, 0, \Omega) \times v, so ω=(0,0,Ω)\omega = (0, 0, \Omega). For the Earth, Ω≈7.292115⋅10−5\Omega \approx 7.292115 \cdot 10^{-5} rad/s. Non-example: the orbital motion of the Earth around the Sun is a translation of the origin, not part of R(t)R(t). In practice R(t)R(t) is factored into precession–nutation, the Earth rotation angle, and polar motion, whose parameters are published by the International Earth Rotation and Reference Systems Service. The model treats the crust as a rigid body; the frame 𝒯\mathcal{T} is defined so that the crust has no net rotation relative to it.

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Cross product, concept, no evidence levelCross productconceptDerivative, concept, no evidence levelDerivativeconceptEuclidean vector, concept, no evidence levelEuclidean vectorconceptIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionRigid body, concept, no evidence levelRigid bodyconceptVector space, definition, no evidence levelVector spacedefinitionVelocity, concept, no evidence levelVelocityconceptSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionOrthogonal 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 velocityconceptEarth's rotation, definition, no evidence levelEarth's rotationdefinition
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.
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  1. Cross product concept
  2. Derivative concept
  3. Euclidean vector concept
  4. Identity matrix object
  5. Matrix definition
  6. Rigid body concept
  7. Vector space definition
  8. Velocity concept
  9. Square matrix definition
  10. Transpose definition
  11. Linear map definition
  12. Orthogonal matrix definition
  13. Multilinear map definition
  14. Orthogonal group object
  15. Alternating multilinear map definition
  16. Determinant definition
  17. Special orthogonal group object
  18. Angular velocity concept
  19. Earth's rotation definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Earth%27s_rotation
About this page
TagCKJ91Z78AW
URIm2://CKJ91Z78AW
Kinddefinition
WritingW0 stub
MSC86A30, 70E15, 70F15
DefinesEarth's rotation
UsesAngular velocity, Rigid body
Changed3 November 2026

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