Math 2.0

Astronomy and astrophysics Definition

Stellar dynamics

Builds on Newton's law of universal gravitationVlasov equationPoisson's equation

Definition

Stellar dynamics is the study of self-gravitating systems of NN point masses, called stars, with NN large. The exact model is the NN-body problem. Stars of masses mim_i at positions xi(t)∈ℝ3x_i(t) \in \R^3, i=1,…,Ni = 1, \dots, N, obey Newton's law of gravitation x¨i=−∑j≠iGmj(xi−xj)∣xi−xj∣3,\ddot x_i = -\sum_{j \ne i} \frac{G m_j (x_i - x_j)}{\abs{x_i - x_j}^3}, where GG is the gravitational constant. The collisionless model describes the system by a mass density f(t,x,v)≥0f(t, x, v) \ge 0 in phase space. This density solves the Vlasov equation ∂tf+v⋅∇xf−∇xΦ⋅∇vf=0,\partial_t f + v \cdot \nabla_x f - \nabla_x \Phi \cdot \nabla_v f = 0, where the potential Φ\Phi solves Poisson's equation ΔΦ=4πGρ\Delta \Phi = 4\pi G \rho with ρ(t,x)=∫ℝ3f(t,x,v) dv\rho(t, x) = \int_{\R^3} f(t, x, v) \, dv.

Remarks

In stellar dynamics the Vlasov equation with a gravitational potential is usually called the collisionless Boltzmann equation. The collisionless model is a good approximation for times shorter than the two-body relaxation time. For a galaxy this time is much longer than the age of the universe, and for a globular cluster it is shorter than its age.

Example: a single star in the fixed potential Φ(x)=12ω2∣x∣2\Phi(x) = \frac{1}{2} \omega^2 \abs{x}^2 of a homogeneous sphere moves on an ellipse centred at the origin. Any ff that depends only on 12∣v∣2+Φ(x)\frac{1}{2}\abs{v}^2 + \Phi(x) is a steady solution of the Vlasov equation in this potential.

Non-example: in a gas cloud, particles collide often and exchange momentum. Its dynamics is described by the Navier–Stokes equations, not by a collisionless model, and is not part of stellar dynamics.

Reading path

To understand this page from the ground up, read these 46 pages in order. The foundations come first.

  1. Derivative concept
  2. Differential equation concept
  3. Euclidean space object
  4. Euclidean vector concept
  5. Function concept
  6. Interval concept
  7. Ordinary differential equation concept
  8. Poisson's equation concept
  9. Vlasov equation concept
  10. Velocity concept
  11. Fréchet derivative definition
  12. Ordinary differential equation definition
  13. Cross product definition
  14. Function of several variables concept
  15. Vector-valued function concept
  16. Classical mechanics concept
  17. Lipschitz condition definition
  18. Limit of a function of several variables concept
  19. Momentum concept
  20. Solution of an ordinary differential equation definition
  21. Partial derivative concept
  22. Inertial frame of reference concept
  23. Continuity in R^n definition
  24. Differentiable function of several variables definition
  25. Initial value problem definition
  26. Partial derivative definition
  27. Newton's laws of motion statement
  28. Differentiable on an open set definition
  29. Gradient definition
  30. Higher partial derivatives definition
  31. Picard–Lindelöf hypotheses definition
  32. Iterated derivative definition
  33. Picard operator definition
  34. C^k function characterised by continuous derivatives statement
  35. Integral equation form of an initial value problem statement
  36. An iterate of the Picard operator is a contraction statement
  37. Vector field definition
  38. Picard–Lindelöf theorem statement
  39. Conservative vector field definition
  40. Classical mechanics definition
  41. Momentum definition
  42. Potential energy definition
  43. Angular momentum definition
  44. Central force definition
  45. Newton's law of universal gravitation statement
  46. Stellar dynamics definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Stellar_dynamics
About this page
TagSPC4AGXNDD
URIm2://SPC4AGXNDD
Kinddefinition
WritingW0 stub
MSC85A05, 70F10
DefinesStellar dynamics
UsesNewton's law of universal gravitation, Vlasov equation, Poisson's equation
Changed3 November 2026

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