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Spiral Density-Wave Dispersion

What you are seeing: the Lin-Shu dispersion relation for tightly-wound spiral density waves: ν2=κ2−2πGΣ∣k∣+k2cs2\nu^2 = \kappa^2 - 2\pi G \Sigma |k| + k^2 c_s^2. Toomre's Q<1Q \lt 1 means ν2<0\nu^2 \lt 0 at some kk: axisymmetric instability.

Figure 1. Dispersion curve ν2(k)\nu^2(k); dips below zero when Q<1Q \lt 1.
σ1.50
κ1.50
2πGΣ3.0

WHAT TO TRY

  • Lower the surface-density term or raise sigma until Toomre Q drops below 1: nu^2 goes negative at some wavenumber and the disk is axisymmetrically unstable. The dispersion curve dips under zero.
  • Tune kappa (epicyclic frequency): it sets the long-wavelength stabilization. More rotational shear means a more stable disk at large scales.
  • Read Q live: one number decides whether a self-gravitating disk fragments or stays smooth, balancing pressure and rotation against self-gravity.