Math 2.0

Astronomy and astrophysics Definition

Oort constants

Builds on Galaxy rotation curveAngular velocityGalactic coordinate systemEpicyclic frequency

Definition

Let vc(R)v_c(R) be the circular speed of a disc galaxy at radius RR, as in the rotation curve. Let Ω(R)=vc(R)/R\Omega(R) = v_c(R)/R be the angular velocity, and let R0R_0 be the radius of the observer. The Oort constants are A=12(vcR−dvcdR)∣R0=−12R0 Ω′(R0),A = \frac{1}{2} \left( \frac{v_c}{R} - \frac{d v_c}{d R} \right)\Big|_{R_0} = -\frac{1}{2} R_0 \, \Omega'(R_0), B=−12(vcR+dvcdR)∣R0.B = -\frac{1}{2} \left( \frac{v_c}{R} + \frac{d v_c}{d R} \right)\Big|_{R_0}.

Remarks

They satisfy A−B=Ω(R0)A - B = \Omega(R_0) and A+B=−vc′(R0)A + B = -v_c'(R_0). The epicyclic frequency at R0R_0 is κ2=−4B(A−B)\kappa^2 = -4 B (A - B).

Take a star on a circular orbit in the plane, at distance dd and galactic longitude ll. To first order in d/R0d / R_0, its velocity relative to the local circular motion has radial part vr=A dsin⁡2lv_r = A \, d \sin 2l and transverse part vt=d (Acos⁡2l+B)v_t = d \, (A \cos 2l + B). This is how AA and BB are measured. Measured values in the solar neighbourhood are roughly A≈15A \approx 15 and B≈−12B \approx -12 km/s/kpc.

Example: for a flat rotation curve vc′=0v_c' = 0, so A=vc/(2R0)=−BA = v_c / (2 R_0) = -B. Non-example: a rigid rotation vc=ΩRv_c = \Omega R has vc′=vc/Rv_c' = v_c / R, so A=0A = 0 and B=−ΩB = -\Omega; there is no shear, and the radial velocities of nearby stars show no sin⁡2l\sin 2l pattern.

Reading path

To understand this page from the ground up, read these 21 pages in order. The foundations come first.

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Cross product, concept, no evidence levelCross productconceptDerivative, concept, no evidence levelDerivativeconceptEpicyclic frequency, concept, no evidence levelEpicyclic frequencyconceptEuclidean vector, concept, no evidence levelEuclidean vectorconceptGalactic coordinate system, concept, no evidence levelGalactic coordinate s…conceptGalaxy rotation curve, concept, no evidence levelGalaxy rotation curveconceptIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector 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 mapdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionOrthogonal group, object, no evidence levelOrthogonal groupobjectDeterminant, definition, no evidence levelDeterminantdefinitionSpecial orthogonal group, object, no evidence levelSpecial orthogonal gr…objectAngular velocity, concept, no evidence levelAngular velocityconceptOort constants, definition, no evidence levelOort constantsdefinition
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. Epicyclic frequency concept
  4. Euclidean vector concept
  5. Galactic coordinate system concept
  6. Galaxy rotation curve concept
  7. Identity matrix object
  8. Matrix definition
  9. Vector space definition
  10. Velocity concept
  11. Square matrix definition
  12. Transpose definition
  13. Linear map definition
  14. Multilinear map definition
  15. Orthogonal matrix definition
  16. Alternating multilinear map definition
  17. Orthogonal group object
  18. Determinant definition
  19. Special orthogonal group object
  20. Angular velocity concept
  21. Oort constants definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Oort_constants
About this page
TagRW5R29N12M
URIm2://RW5R29N12M
Kinddefinition
WritingW0 stub
MSC85A05
DefinesOort constants
UsesGalaxy rotation curve, Angular velocity, Galactic coordinate system, Epicyclic frequency
Changed3 November 2026

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