6 Models with decreasing anisotropy

6.1 Background

In order to construct dynamical models with a decreasing anisotropy, i.e. with a tangentially anisotropic halo, no special inversion techniques exist, such that we have to rely on the general formulae of Dejonghe (1986) to invert the fundamental integral Eq. (1). A disadvantage is that these formulae are numerically unstable. Their usefulness is therefore actually restricted to analytical models. But this is not straightforward: a direct application of the inversion formulae to an arbitrary analytical augmented density $\tilde{\rho}(\psi,r)$, even if its looks rather simple, can result in a cumbersome mathematical exercise, because the inversion formulae are quite elaborate.

A useful strategy to construct models with a tangential halo without the need to cope with the complicated general formulae, is to profit from the linearity of the integral Eq. (1). In particular, it is very interesting to generate augmented densities $\tilde{\rho}(\psi,r)$, which can be expanded in a series of functions $\tilde{\rho}_k(\psi,r)$, which depend on r only through a power law,

$$\tilde{\rho}(\psi,r) = \sum_k \tilde{\rho}_k(\psi,r) = \sum_k f_k(\psi)~r^{-2\beta_k}.$$ (64)

Each of the augmented densities $\tilde{\rho}_k(\psi,r)$corresponds to a dynamical model with a constant anisotropy $\beta_k$. Combining the linearity of the integral Eq. (1) with the results of Sect. 4.1, we find that the distribution function corresponding to the density (64) reads 010

$$F({\cal{E}},L) = \sum_k F_k({\cal{E}},L),$$ (65a)

with

 

$$F_k({\cal{E}},L) = \frac{2^{\beta_k}}{(2\pi)^{3/2}}~ \frac{L^{-2\... ...\rm d}f_k}{{\rm d}\psi}~\frac{{\rm d}\psi}{({\cal{E}}-\psi)^{1/2-\beta_k}}\cdot$$ (65b)

Equivalently, the moments of the distribution function can be derived from the series expansion.

6.2 Hernquist models with decreasing anisotropy

6.2.1 Construction of the augmented density

For every potential $\psi(r)$, we can create an infinite number of functions $Z(\psi,r)$ which satisfy the identity $Z(\psi(r),r)\equiv1$. For the Hernquist potential, we can easily create such a one-parameter family of functions $Z_n(\psi,r)$,

$$Z_n(\psi,r) = \left[\psi~(1+r)\right]^n \equiv 1,$$ (66)

with n a natural number. If we multiply this family with the augmented density (27) of the constant anisotropy Hernquist models, we create a new two-parameter family of dynamical models, that will self-consistently generate the Hernquist potential-density pair,

$$\tilde{\rho}(\psi,r) = \frac{1}{2\pi}~ \frac{\psi^{4-2\bet... ...n}}{(1-\psi)^{1-2\beta_0}}~ \frac{(1+r)^n}{r^{2\beta_0}}\cdot$$ (67)

Defining a new parameter $\beta_\infty = \beta_0-\frac{n}{2}$, we can write this augmented density also as

$$\tilde{\rho}(\psi,r) = \frac{1}{2\pi}~ \frac{\psi^{4-2\bet... ...~ \frac{(1+r)^{2(\beta_0-\beta_{\infty})}}{r^{2\beta_0}}\cdot$$ (68)

Because we assumed that n is a natural number, we can expand the binomial in the nominator of the density (67), and write it in the form (64), with 010

 

$$f_k(\psi) \frac{1}{2\pi}~ \left(\begin{array}{c} n\\ k \end{array}\right)~ \frac{\psi^{4-2\beta_\infty}}{(1-\psi)^{1-2\beta_0}}$$ = (69a)

$$\beta_k \beta_0-\frac{k}{2},$$ = (69b)

with $0\leqslant k\leqslant n$. The reason why we chose $\beta _0$ and $\beta_\infty$ as parameters becomes clear when we look at the expression for the anisotropy corresponding to this family of density functions - for the moment being without bothering whether the density corresponds to a physically acceptable distribution function. By means of the formula (11), we obtain

$$\beta(r) = \frac{\beta_0+\beta_\infty~r}{1+r}\cdot$$ (70)

The anisotropy equals $\beta _0$ in the center and decreases to $\beta_\infty$ at large radii. Because n can in principle assume any natural number, this family of augmented densities hence corresponds to dynamical models which can grow arbitrarily tangential in the outer regions. In particular, by setting n=0we recover the models with constant anisotropy $\beta=\beta_0=\beta_\infty$ from Sect. 4.2.

6.2.2 The distribution function

Figure 6: Comparison of the distribution function corresponding to Hernquist model with increasing anisotropy with $\beta_0=\frac{1}{2}$ and $\beta _\infty =-2$, and the constant anisotropy model with $\beta =-2$. The distribution functions are represented as isoprobability contours in turning point space.

We can calculate the distribution function of these models by applying the recipe (65b) to each of the components (69ab). We obtain after some algebra

 

$$F({\cal{E}},L) \frac{2^{\beta_0}}{(2\pi)^{5/2}}~ \Gamma(5-2\beta_\infty)~ L^{-2\beta_0}~{\cal{E}}^{5/2-2\beta_\infty+\beta_0}$$ =

$$\times \sum_{k=0}^n~ \left(\begin{array}{c} {n}\\ {k} \end{array... ...\frac{k}{2}-2\beta_\infty+\beta_0)}~ \left(\frac{L}{\sqrt{2{\cal{E}}}}\right)^k$$

$$\hspace*{-0.5cm}\times {}_2F_1\left(5-2\beta_\infty,1-2\beta_0;\frac{7}{2}-\frac{k}{2}-2\beta_\infty+\beta_0;{\cal{E}}\right).$$ (71)

This family of models is restricted by the condition $\beta_0\leqslant\frac{1}{2}$, because for higher values of $\beta _0$the distribution function becomes negative. For all half-integer and integer values of $\beta _0$ (and therefore also of $\beta_\infty$), this expression can be written in terms of elementary functions, very analogous with the distribution functions of the constant anisotropy models: the expression contains integer and half-integer powers of ${\cal{E}}$ and $1-{\cal{E}}$ and a factor $\arcsin\sqrt{{\cal{E}}}$. The models characterized by $\beta_0=\frac{1}{2}$ are of a particular kind. The hypergeometric functions in Eq. (71) disappears for $\beta_0=\frac{1}{2}$, such that the distribution function can be written as a finite power series of $\sqrt{{\cal{E}}}$ and L.

An interesting characteristic of these models is revealed when we look at the asymptotic behavior of the distribution function at large radii, i.e. for ${\cal{E}}\rightarrow0$. The term corresponding to k=n will contribute the dominant term in the sum (71), such that we obtain

 

$$F({\cal{E}},L) \approx \frac{2^{\beta_\infty}}{(2\pi)^{5/2}}~ \fr... ...}-\beta_\infty\right)} L^{-2\beta_\infty}~{\cal{E}}^{5/2-\beta_\infty} + \cdots$$ (72)

This expansion is at first order independent of $\beta _0$, such that all models with the same $\beta_\infty$ will have a similar behavior at large radii. In particular, all distribution functions corresponding to a particular $\beta_\infty$ will at large radii behave as the Hernquist model with constant anisotropy $\beta=\beta_\infty$. This is illustrated in Fig. 6, where we compare the distribution function of a model with a radial core and a tangential halo with the constant anisotropy model that has the corresponding tangential anisotropy. At small radii, the difference between both distribution functions is obvious: the former one has more stars on radial orbits, whereas the latter prefers to populate circular-like orbits. At large radii, however, the isoprobability contours of both models agree very well.

Finally, notice that there is no analogue for this behavior at small radii: not all models with a fixed $\beta _0$ will have a similar behavior for ${\cal{E}}\rightarrow1$, i.e. at small radii.

6.2.3 The velocity dispersions

Figure 7: The radial (upper panel) and line-of-sight (lower panel) velocity dispersion profiles of Hernquist models with a decreasing anisotropy. All models have the same tangential outer anisotropy $\beta _\infty =-2$, but they have a different central anisotropy parameter $\beta _0$: plotted are $\beta_0=\frac{1}{2}$, 0, $-\frac{1}{2}$, -1 and -2 (black line).

In order to calculate the velocity dispersion profiles of the models of this type, we have various possibilities. We can either calculate the dispersion for each of the n terms (69a) through formula (26) and sum the results, or directly apply the general recipe (8ab) on the expression (68). In either case, we obtain an expression very akin to the expression (32) of the models with constant anisotropy,

$$\sigma_{\rm r}^2(r) = r^{1-2\beta_0}~(1+r)^{3+2\beta_\infty... ... {\rm B}_{\frac{1}{1+r}}\Bigl(5-2\beta_\infty,2\beta_0\Bigr).$$ (73)

This expression can be written in terms of elementary functions for all $\beta _0$ with $4\beta_0$ an integer (and hence also $4\beta_\infty$ an integer).

Not as a surprise, the asymptotic expressions for $\sigma_{\rm r}^2(r)$for $r\gg1$ read

$$\sigma_{\rm r}^2(r) \approx \frac{1}{5-2\beta_\infty}~\frac{1}{r} +\cdots$$ (74)

i.e. they are similar to the corresponding expansions of the constant anisotropy models with $\beta=\beta_\infty$. This behavior is illustrated in the upper panel of Fig. 7, where we plot the radial velocity dispersion profile for a set of models with varying $\beta _0$ and a fixed $\beta_\infty$. At small radii, the models have different profiles (those with the most radial anisotropy have the largest values of $\sigma_{\rm r}$), but at large radii, they all converge towards a common asymptotic expansion.

The calculation of the line-of-sight velocity dispersion is also similar to the case of constant anisotropy. It is found that $\sigma _{\rm p}(R)$ can be written in terms of elementary functions for all integer and half-integer values of $\beta _0$, and that the asymptotic behavior for $R\gg1$ reads

$$\sigma_{\rm p}^2(R) \approx \frac{8}{15\pi}\left(\frac{5-4\beta_\infty}{5-2\beta_\infty}\right)\frac{1}{R} + \cdots,$$ (75)

which is at first order independent of $\beta _0$. An illustration is shown in the bottom panel of Fig. 7.