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Jeans Isothermal Sphere

What you are seeing: the Jeans-equation solution for an isothermal, self-gravitating sphere has ρ∝r−2\rho \propto r^{-2}, M(<r)∝rM(\lt r) \propto r, and a flat circular-velocity curve vc=2σv_c = \sqrt 2 \sigma. This is the canonical "flat rotation curve" of an isothermal halo.

Figure 1. Rotation curve flat at 2σ\sqrt 2 \sigma.
σ (km/s)200

WHAT TO TRY

  • Raise the velocity dispersion sigma: the circular velocity v_c = sqrt(2) sigma rises with it and the flat rotation curve sits higher. A hotter halo spins its tracers faster.
  • Note the rotation curve stays flat at all radii: rho ~ r^-2 and M(<r) ~ r conspire to give a constant v_c. This is the textbook reason rotation curves do not fall off.
  • Read v_c against sigma: the fixed sqrt(2) ratio is the signature of the singular isothermal sphere, whatever the halo mass.