Isolated and non-isolated dark matter haloes and the Navarro, Frenk and White profile

We compare the conclusions reached using the coarse-graining technique employed by Henriksen for a one degree of freedom (per particle) collisionless system to those presented in a paper by Binney based on an exact one degree of freedom model. We find agreement in detail, but in addition we show that the isolated 1D system is self-similar and therefore unrelaxed. Fine graining of this system recovers much less prominent wave-like structure than in a spherically symmetric isotropic 3D system. The rate of central flattening is also reduced in the 1D system. We take this to be evidence that relaxation of collisionless systems proceeds ultimately by way of short wavelength Landau damping. N-body systems, both real and simulated, can be trapped in an incompletely relaxed state because of a break in the cascade of energy to small scales. This may be due to the rapid dissipation of the small-scale oscillations in an isolated system to the existence of conserved quantities such as angular momentum, or to the failure in simulations to resolve sub-Jeans length scales. Such a partially relaxed state appears to be the Navarro, Frenk and White (NFW) state and is to be expected especially in young systems. The NFW core is shown to be isolated. In non-isolated systems, continuing coarse-grained relaxation should be towards a density core in solid body rotation.