The Boltzmann and Jeans equations.

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Presentation transcript:

The Boltzmann and Jeans equations. ASTC22. Lecture L11 The Boltzmann and Jeans equations. The collisionless Boltzmann equation and the Jeans equations

Ludwig Boltzmann (1844-1906) (Sir) James H. Jeans (1877-1946)

The phase-space: space made of configuration space (x,y,z) and velocity space (vx,vy,vz)

Boltzmann equation. It is essentially a conservation equation in phase-space: Df(x,v,t)/Dt = 0

Df(x,v,t)/Dt = 0

This leads immediately to the usual continuity equation of fluids. The next step:

The Jeans equation: velocity dispersion as “pressure” in stellar systems Dv/Dt = -grad (potential) - (1/n) grad (pressure) (this is a typical Lagrangian way to write the equations of motion for a fluid with pressure).

2 V

~0 Integrate once

(read chapter 3 about the results...) Stars and gas Just gas Just stars

Isothermal spheres all have energy E in the Boltzmann factor & velocity dispersion sigma=const.