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Bienvenue au

Cours d’astrophysique III : Dynamique stellaire et galactique

Semestre automne 2011

Dr. Pierre North Laboratoire d’astrophysique

Ecole Polytechnique Fédérale de Lausanne Observatoire de Sauverny

CH – 1290 Versoix

http://lastro.epfl.ch

Wave mechanics of differentially rotating disks

a) Kinematic density waves

b) The dispersion relation for tightly wound spiral arms

c) Local stability of differentially rotationg disks

d) Kinetic theory

Wave mechanics of differentially rotating disks (cont.)

Isochrone potential Model I of the Galaxy

Ω-κ/2 is almost constant on a wide range of radii!

n=0 m=1

\ n=1, m=2

n=2 m=3

Ωp = Ω-κ/2 independant from R

The appearance of elliptical orbits in a frame rotating at Ωp = Ω-κ/2

\ n=1, m=-2

Relative positions of elliptical orbits: 1)  Random 2)  Coaxial => bar 3)  Shifted axes => spiral 4) Shifted axes except in the centre => barred spiral

Source: F. Combes, Pour la Science No 337, nov. 2005

Wave mechanics of differentially rotating disks Dispersion relation for tightly wound spiral arms

a) Tight-winding approximation: WKB (Wentzel-Kramers-Brillouin) approximation ∆R << R ; ∆R = 2π/k => |kR| >> 1.

Real spirals: |kR| ≈ 7-11

b) Potential of a tightly wound spiral: - surface density perturbation Σ1(R, ϕ, t) = H(R, t) exp(i[mϕ+f(R,t)]) – Taylor expand f – obtain Φ1 as a function of Σ1

c) The dispersion relation for fluid disks: - Euler’s equ. for 2D disk + equ. of state p = K Σd

γ – linearize Euler’s equ. to get vR1, vϕ1 and the Σda response of the disk – relate the response Σda to the total density Σa through the polarization function, and impose Σa = Σda – use the WKB approx. to obtain analytical local solutions for density waves – get P and impose =1

=> Dispersion relation

d) Establish the dispersion relation for stellar disks:

Source: Frank H. Shu, The physics of astrophysics, vol. II, Gas dynamics, chap. 11, p. 146

Local stability of differentially rotating disks

Swing amplification: from a leading to a trailing structure. « There are more things in heaven and earth, Horatio, than dreamt of in your philosophy » (similarly, the behaviour of even idealized galaxies may be much richer than the Lin & Shu scenario)

Source: Binney & Tremaine 2008, Galactic Dynamics, 2nd edition