Math 2.0

Astronomy and astrophysics Definition

Lindblad resonance

Builds on Epicyclic frequencyAngular velocityResonance

Definition

Let a disc have circular angular velocity Ω(R)\Omega(R) and epicyclic frequency κ(R)\kappa(R). Let a perturbation have mm-fold symmetry, m≥1m \ge 1 an integer, and rotate rigidly with pattern speed Ωp\Omega_p. So it depends on the angle φ\varphi and the time tt only through m(φ−Ωpt)m (\varphi - \Omega_p t). A Lindblad resonance is a radius RR at which m(Ω(R)−Ωp)=±κ(R).m \left( \Omega(R) - \Omega_p \right) = \pm \kappa(R). With the sign ++ it is the inner Lindblad resonance; with the sign −- it is the outer Lindblad resonance. The radius where Ω(R)=Ωp\Omega(R) = \Omega_p is the corotation resonance.

Remarks

Meaning: a star on a nearly circular orbit at radius RR meets the crests of the pattern with frequency m∣Ω(R)−Ωp∣m \abs{\Omega(R) - \Omega_p}. At a Lindblad resonance this forcing frequency equals the natural frequency κ\kappa of radial oscillation, a resonance.

Example: take a flat rotation curve Ω=v0/R\Omega = v_0 / R, so κ=2 Ω\kappa = \sqrt{2} \, \Omega, and a two-armed pattern, m=2m = 2. The inner Lindblad resonance solves 2(Ω−Ωp)=2 Ω2(\Omega - \Omega_p) = \sqrt{2} \, \Omega, that is R=(1−1/2) v0/ΩpR = (1 - 1/\sqrt{2}) \, v_0 / \Omega_p. The outer one is at R=(1+1/2) v0/ΩpR = (1 + 1/\sqrt{2}) \, v_0 / \Omega_p, and corotation is at R=v0/ΩpR = v_0 / \Omega_p.

Non-example: an axisymmetric perturbation, m=0m = 0, does not rotate in angle, and the condition reduces to κ=0\kappa = 0; it gives no Lindblad resonance in a stable disc.

In planetary rings the pattern is the gravitational forcing of a moon, and Lindblad resonances in the rings excite spiral density waves.

Reading path

To understand this page from the ground up, read these 20 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 vectorconceptIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionResonance, concept, no evidence levelResonanceconceptVector 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 velocityconceptLindblad resonance, definition, no evidence levelLindblad resonancedefinition
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. Identity matrix object
  6. Matrix definition
  7. Resonance concept
  8. Vector space definition
  9. Velocity concept
  10. Square matrix definition
  11. Transpose definition
  12. Linear map definition
  13. Multilinear map definition
  14. Orthogonal matrix definition
  15. Alternating multilinear map definition
  16. Orthogonal group object
  17. Determinant definition
  18. Special orthogonal group object
  19. Angular velocity concept
  20. Lindblad resonance definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Lindblad_resonance
About this page
TagGSD51WAGRM
URIm2://GSD51WAGRM
Kinddefinition
WritingW0 stub
MSC85A05, 70F15
DefinesLindblad resonance
UsesEpicyclic frequency, Angular velocity, Resonance
Changed3 November 2026

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