3 Isotropic models

3.1 Background

The simplest dynamical models are those where the augmented density is a function of the potential only, $\tilde{\rho}=\tilde{\rho}(\psi)$. For such models, the distribution function is only a function of the binding energy, i.e. the distribution function is isotropic. In this case, the integral Eq. (1) can be inverted to find the well-known Eddington relation

$$F({\cal{E}}) = \frac{1}{2\sqrt{2}\pi^2} \frac{{\rm d}}{{\r... ...{{\rm d}\psi}~ \frac{{\rm d}\psi}{\sqrt{{\cal{E}}-\psi}}\cdot$$ (12)

For such isotropic models, we do not use the general anisotropic moments (6), but define the isotropic moments as

$$\mu_{2n}(r) = 4\pi\int F({\cal{E}})~v^{2n+2}~{\rm d}v.$$ (13)

Similarly as for the anisotropic moments, we can derive a relation that allows to calculate the augmented isotropic moments from the augmented density $\tilde{\rho}(\psi)$. Indeed, they satisfy the relation (Dejonghe 1986)

$$\tilde{\mu}_{2n}(\psi) = \frac{(2n+1)!!}{(n-1)!!} \int_0^{\psi} (\psi-\psi')^{n-1}~\tilde{\rho}(\psi')~{\rm d}\psi'.$$ (14)

In particular, we obtain an expression for the velocity dispersion profile by setting n=1,

$$\sigma^2(r) = \frac{1}{\rho(r)} \int_0^{\psi(r)}\tilde{\rho}(\psi')~{\rm d}\psi'.$$ (15)

3.2 The isotropic Hernquist model

The isotropic model that corresponds to the potential-density pair (2ab) is described in full detail by Hernquist (1990). We restrict ourselves by resuming the most important results, for a comparison with the anisotropic models discussed later in this paper. The augmented density reads

$$\tilde{\rho}(\psi) = \frac{1}{2\pi}~\frac{\psi^4}{1-\psi}\cdot$$ (16)

Substituting this density into Eddington's formula (12) yields the distribution function

 

$$F({\cal{E}}) = \frac{1}{8\sqrt{2}\pi^3} \left[ \frac{\sqrt{{\cal{... ...cal{E}})^2} + \frac{3\arcsin\sqrt{{\cal{E}}}}{(1-{\cal{E}})^{5/2}} \right]\cdot$$ (17)

Combining the density (16) with the general formula (14) gives us the moments of the distribution function,

$$\tilde{\mu}_{2n}(\psi) = \frac{3\cdot2^{n+9/2}}{(2\pi)^{3/2... ...{\Gamma(n+5)}~ \psi^{n+4}~ {}_2F_1\left(5,1;n+5;\psi\right).$$ (18)

For all $n\geqslant0$, this expression can be written in terms of rational functions and logarithms. For example, for the velocity dispersions, we obtain after substitution of the Hernquist potential (2a),

$$\sigma^2(r) = r~(1+r)^3~\ln\left(\frac{1+r}{r}\right) -\frac{r~(25+52r+42r^2+12r^3)}{12(1+r)},$$ (19)

in agreement with Eq. (10) of Hernquist (1990). From an observational point of view, it is very useful to obtain an explicit expression for the line-of-sight velocity dispersion. For isotropic models, the line-of-sight dispersion is easily found by projecting the second-order moment on the plane of the sky, i.e.

$$\sigma_{\rm p}^2(R) = \frac{2}{I(R)}~ \int_R^\infty \frac{\rho(r)~\sigma^2(r)~r~{\rm d}r}{\sqrt{r^2-R^2}}\cdot$$ (20)

For the Hernquist model this yields after some algebra

$$I(R)~\sigma_{\rm p}^2(R) \frac{1}{24\pi~(1-R^2)^3}$$ =

$$\times \Bigl[3R^2~\left(20-35R^2+28R^4-8R^6\right)~X(R)$$

$$+ (6-65R^2+68R^4-24R^6)\Bigr] -\frac{R}{2}\cdot$$ (21)