10.11. Anisotropy profiles \(\beta_{\rm ani}(r)\)

Possible keyword choices for the anisotropy profile to be set in the file gLIST_HALOES_JEANS (no named variable):

Keyword Anisotropy \(\beta_{\rm anis}(r)\) \(f(r)\) to solve Eq. (6.12) \(=\nu(r)\bar{v_r^2}(r)\) Kernel \({\cal K}(u,u_a)\) to solve Eq. (6.14) \(=I(R)\sigma_p^2(R)\) Refs (for profile and/or solution)
kCONSTANT \(\beta_0\) \(r^{2\beta_0}\) \(\frac{\sqrt{1-u^{-2}}}{1-2\beta_0}+\frac{\sqrt{\pi}}{2}\frac{\Gamma(\beta_0-1/2)}{\Gamma(\beta_0)}u^{2\beta_0-1}\) \(\left(\frac{3}{2}-\beta_0\right)\left[1-I\left(\frac{1}{u^2},\beta_0+\frac{1}{2},\frac{1}{2}\right)\right]\) with \(I(x,a,b)=\) incomplete Beta function Mamon and Lokas (2005,2006)
kBAES \(\frac{\beta_0 + \beta_\infty (r/r_a)^\eta}{1+(r/r_a)^\eta}\) \(r^{2\beta_0}\left[1+\left(\frac{r}{r_a}\right)^\eta\right]^{2(\beta_\infty-\beta_0)/\eta}\) No analytical Kernel Baes & van Hese (2007)
kOSIPKOV \(\frac{r^2}{r^2+r_a^2}\) (special case of kBAES) \(\frac{r_a^2+r^2}{r_a^2}\) \(\displaystyle\frac{(u^2+u_a^2)(u_a+1/2)}{u(u_a^2+1)^{3/2}}\tan^{-1}\sqrt{\frac{u^2-1}{u_a^2+1}}\) \(-\displaystyle\frac{\sqrt{1-u^{-2}}}{2(u_a^2+1)}\) Osipkov (1979), Merritt (1985), Mamon and Lokas (2005, 2006)