We investigate , using the spherical Jeans equation , self-gravitating dynamical equilibria satisfying a relation \rho / \sigma _ { r } ^ { 3 } \propto r ^ { - \alpha } , which holds for simulated dark-matter haloes over their whole resolved radial range .
Considering first the case of velocity isotropy , we find that this problem has only one solution for which the density profile is not truncated or otherwise unrealistic .
This solution occurs only for a critical value of \alpha _ { \mathrm { crit } } = 35 / 18 = 1.9 \overline { 4 } , which is consistent with the empirical value of 1.9 \pm 0.05 .
We extend our analysis in two ways : first we introduce a parameter \epsilon to allow for a more general relation \rho / \sigma _ { r } ^ { \epsilon } \propto r ^ { - \alpha } ; second we consider velocity anisotropy parameterised by Binney ’ s \beta ( r ) \equiv 1 - \sigma _ { \theta } ^ { 2 } / \sigma _ { r } ^ { 2 } .
If we assume \beta to be linearly related to the logarithmic density slope \gamma ( r ) \equiv - \mathrm { d } \ln \rho / \mathrm { d } \ln r , which is in agreement with simulations , the problem remains analytically tractable and is equivalent to the simpler isotropic case : there exists only one physical solution , which occurs at a critical \alpha value .
Remarkably , this value of \alpha and the density and velocity-dispersion profiles depend only on \epsilon and the value \beta _ { 0 } \equiv \beta ( r = 0 ) , but not on the value \beta _ { \infty } \equiv \beta ( r \to \infty ) ( or , equivalently , the slope \mathrm { d } \beta / \mathrm { d } \gamma of the adopted linear \beta - \gamma relation ) .
For \epsilon = 3 , \alpha _ { \mathrm { crit } } = 35 / 18 - 2 \beta _ { 0 } / 9 and the resulting density profile is fully analytic ( as are the velocity dispersion and circular speed ) with an inner cusp \rho \propto r ^ { - ( 7 + 10 \beta _ { 0 } ) / 9 } and a very smooth transition to a steeper outer power-law asymptote .
These models are in excellent agreement with the density , velocity-dispersion and anisotropy profiles of simulated dark-matter haloes over their full resolved radial range .
If \epsilon = 3 is a universal constant , some scatter in \beta _ { 0 } \approx 0 may account for some diversity in the density profiles , provided a relation \rho / \sigma _ { r } ^ { 3 } \propto r ^ { - \alpha _ { \mathrm { crit } } } always holds .