2 The construction of anisotropic models

A general discussion on the inversion of the fundamental Eq. (1), and hence on the construction anisotropic distribution functions for a given spherical potential-density pair, is presented by Dejonghe (1986). The key ingredient of the inversion procedure is the concept of the augmented mass density $\tilde{\rho}(\psi,r)$, which is a function of potential and radius, such that the condition

$$\tilde{\rho}(\psi(r),r) \equiv \rho(r)$$ (5)

is satisfied. The augmented mass density is in fact equivalent to the distribution function $F({\cal{E}},L)$: with every augmented density $\tilde{\rho}(\psi,r)$ we can associate a distribution function $F({\cal{E}},L)$ and vice versa. There exist various transition formulae between these two equivalent forms of a dynamical model, amongst others a formalism that uses combined Laplace-Mellin transforms.

Besides providing a nice way to generate a distribution function for a given potential-density pair, the augmented density is also very useful to calculate the moments of the distribution function. The anisotropic moments are defined as

$$\mu_{2n,2m}(r) = 2\pi\iint F({\cal{E}},L)~v_{\rm r}^{2n}~v_{\rm t}^{2m+1}~{\rm d}v_{\rm r}~{\rm d}v_{\rm t},$$ (6)

where $v_{\rm t}\equiv\sqrt{v_\theta^2+v_\phi^2}$ is the transverse velocity. One can derive a relation that links the higher-order moments to the augmented mass density $\tilde{\rho}$, when written explicitly as a function of $\psi$ and r,

 

$$\tilde{\mu}_{2n,2m}(\psi,r) = \frac{2^{m+n}}{\sqrt{\pi}}~ \frac{\... ...m}}{({\rm d}r^2)^{m}} \left[ r^{2m}~\tilde{\rho}(\psi',r) \right] {\rm d}\psi'.$$ (7)

In particular, the radial and transverse velocity dispersions can be found from the density through the relations, 010

  

$$\sigma_{\rm r}^2(r) \frac{1}{\rho(r)} \int_0^{\psi(r)} \tilde{\rho}(\psi',r)~{\rm d}\psi',$$ = (8a)

$$\sigma_{\rm t}^2(r) \frac{2}{\rho(r)} \int_0^{\psi(r)} \frac{{\rm d}}{{\rm d}r^2} \left[r^2~\tilde{\rho}(\psi',r)\right] {\rm d}\psi'.$$ = (8b)

By means of these functions, we can define the anisotropy $\beta(r)$ as

$$\beta(r) = 1-\frac{\sigma_{\rm t}^2(r)}{2\sigma_{\rm r}^2(r)}\cdot$$ (9)

We will in this paper consider augmented densities which are separable functions of $\psi$ and r, and we introduce the notation

$$\tilde{\rho}(\psi,r) = f(\psi)~g(r).$$ (10)

For such models, the anisotropy can be directly calculated from the augmented density as

$$\beta(r) = 1-\frac{1}{g(r)}~\frac{{\rm d}}{{\rm d}r^2} \left[r^2~g(r)\right],$$ (11)

as a result of the formulae (8ab), (9) and (10).