4 Models with a constant anisotropy

   
4.1 Background

A special family of distribution functions that can easily be generated using the technique outlined in Sect. 2 corresponds to models with a density that depends on r only through a factor $r^{-2\beta}$, i.e.

$$\tilde{\rho}(\psi,r) = f(\psi)~r^{-2\beta}.$$ (22)

It is well known that such densities correspond to models with a constant anisotropy (i.e. Binney & Tremaine 1987), which can easily be checked by introducing $g(r)=r^{-2\beta}$ in the formula (11). For a given potential-density pair, a family of models with constant anisotropy can hence be constructed by inverting the potential as $r(\psi)$, and defining

$$f(\psi) = \tilde{\rho}(\psi)~\bigl(r(\psi)\bigr)^{2\beta}.$$ (23)

Notice that such models are not the only constant anisotropy models corresponding to a given potential-density pair, as argued in Sect. 1.5 of Dejonghe (1986). This family, however, is very attractive due to its relative simpleness. In particular, the corresponding distribution function is a power law in L, and can be found through an Eddington-like formula,

 

$$F({\cal{E}},L) = \frac{2^\beta}{(2\pi)^{3/2}}~ \frac{L^{-2\beta}}... ...ac{{\rm d}f}{{\rm d}\psi}~\frac{{\rm d}\psi}{({\cal{E}}-\psi)^{1/2-\beta}}\cdot$$ (24)

For the moments of the distribution function, the relation (7) can be simplified to

$$\tilde{\mu}_{2n,2m}(\psi,r) = \frac{2^{m+n}}{\sqrt{\pi}}~ \frac{\... ...-\beta)}~ r^{-2\beta} \int_0^\psi (\psi-\psi')^{m+n-1}~ f(\psi')~ {\rm d}\psi'.$$ (25)

In particular, the radial velocity dispersions reads

$$\sigma_{\rm r}^2(r) = \frac{1}{f(\psi(r))} \int_0^{\psi(r)} f(\psi')~{\rm d}\psi'.$$ (26)

   
4.2 Hernquist models with a constant anisotropy

   
4.2.1 The distribution function

Figure 1: The distribution function of the Hernquist models with a constant anisotropy, represented as isoprobability contours in turning point space. The distribution functions in solid lines represent a radial model with $\beta=\frac{1}{2}$ (left panel) and a tangential model with $\beta =-2$ (right panel). The dotted contour lines in both panels correspond to the isotropic Hernquist model.

Applying the formula (23) to the Hernquist potential-density pair (2ab) yields

$$f(\psi) = \frac{1}{2\pi}~ \psi^{4-2\beta}~(1-\psi)^{2\beta-1}.$$ (27)

Substituting this expression into the general formula (24) gives us the corresponding distribution function

 

$$F({\cal{E}},L) = \frac{2^\beta}{(2\pi)^{5/2}}~ \frac{\Gamma(5-2\b... ...}{2}-\beta}~ {}_2F_1\left(5-2\beta,1-2\beta;\frac{7}{2}-\beta;{\cal{E}}\right).$$ (28)

This expression reduces to the isotropic distribution function (17) for $\beta=0$, as required. Whether the expression (28) corresponds to a physically acceptable distribution function for a given value of $\beta$ depends on the condition that the distribution function has to be positive over the entire phase space.

It is no surprise that the distribution function is not positive for the largest possible values of $\beta$, because models where only the radial orbits are populated can only be supported by a density profile that diverges as r^-2 or steeper in the center (Richstone & Tremaine 1984). It turns out that the distribution function (28) is everywhere non-negative for $\beta\leqslant\frac{1}{2}$.

For all integer and half-integer values of $\beta$, the hypergeometric series in (28) can be expressed in terms of elementary functions. Very useful are half-integer values of $\beta$, because the energy-dependent part of the distribution function can then be written as a rational function of ${\cal{E}}$. For integer values of $\beta$, the hypergeometric series can be written as a function containing integer and half-integer powers of ${\cal{E}}$ and $1-{\cal{E}}$ and a factor $\arcsin\sqrt{{\cal{E}}}$, similar to the isotropic distribution function (17).

The limiting model $\beta=\frac{1}{2}$ is particularly simple. It has an augmented density that is a power law of potential and radius,

$$\tilde{\rho}(\psi,r) = \frac{1}{2\pi}~\frac{\psi^3}{r},$$ (29)

and the corresponding distribution function simply reads

$$F({\cal{E}},L) = \frac{3}{4\pi^3}~\frac{{\cal{E}}^2}{L}\cdot$$ (30)

This model is a special case of the generalized polytropes discussed by Fricke (1951) and Hénon (1973).

In Fig. 1 we compare the distribution functions of the radial model with $\beta=\frac{1}{2}$ and the tangential model with $\beta =-2$ with the distribution function of the isotropic Hernquist model. The distribution functions are shown by means of their isoprobability contours in turning point space, which can easily be interpreted in terms of orbits. Compared to the isotropic model, the radial model prefers orbits on the upper left side of the diagram, with an apocenter much larger than the pericenter, i.e. elongated orbits. The isoprobability contours of tangential models on the other hand lean towards the diagonal axes where pericenter and apocenter are equal, i.e. nearly-circular orbits are preferred.

4.2.2 The velocity dispersions

By means of substituting the expression (27) into the general formula (7), we can derive an analytical expression for all moments of the distribution function,

$$\tilde{\mu}_{2n,2m}(\psi,r) \frac{2^{n+m-1}}{\pi^{3/2}}~ \frac{\Gamma(5-2\beta)~\Gamma\left(n+\frac{1}{2}\right)~\Gamma(m+1-\beta)} {\Gamma(m+n+5-2\beta)~\Gamma(1-\beta)}$$ =

$$\times r^{-2\beta}~\psi^{m+n+4-\beta}~$$

$$\times {}_2F_1\left(5-2\beta,1-2\beta;m+n+5-2\beta;\psi\right).$$ (31)

We are mainly interested in the velocity dispersions, which can be conveniently written by means of the incomplete Beta function (Abramowitz & Stegun 1972),

$$\sigma_{\rm r}^2(r) = r^{1-2\beta}~(1+r)^3~ {\rm B}_{\frac{1}{1+r}}\Bigl(5-2\beta,2\beta\Bigr).$$ (32)

For all anisotropies $\beta<\frac{1}{2}$, the radial dispersion equals zero in the center of the galaxy, rises until a maximum and then decreases again towards zero for $r\rightarrow\infty$. The asymptotic behavior for $r\gg1$ is

$$\sigma_{\rm r}^2(r) \approx \frac{1}{5-2\beta}~\frac{1}{r} +\cdots$$ (33)

The expression (32) can be written in terms of elementary functions for all $\beta$ with $4\beta$ an integer. For the integer and half-integer values of $\beta$, the expression involves polynomials in r and a factor $\ln(1+1/r)$, very analogous to the expression (19) of the isotropic Hernquist model. For the quarter-integer values of $\beta$, it contains polynomials and square roots in r and a factor ${\rm arccotg}~\sqrt{r}$.

Figure 2: The velocity dispersion of the Hernquist models with a constant anisotropy. The upper and lower panels show the radial velocity dispersions $\sigma _{\rm r}(r)$ and the line-of-sight velocity dispersion $\sigma _{\rm p}(R)$. The profiles are shown for different values of the anisotropy parameter $\beta$: plotted are $\beta=\frac{1}{2}$, $\frac{1}{4}$, 0, $-\frac{1}{2}$, -2, -5 and the limit case $\beta \rightarrow -\infty$.

Particular cases are the models that correspond to the most radial and tangential distribution functions. On the one hand, the limit case $\beta=\frac{1}{2}$ has the simple velocity dispersion profiles

$$\sigma_{\rm r}^2(r) = \sigma_{\rm t}^2(r) = \frac{1}{4}~\frac{1}{1+r}\cdot$$ (34)

Particular about this dispersion profile is that it assumes a finite value in the center. On the other side of the range for possible anisotropies, we can consider the limit $\beta \rightarrow -\infty$, which corresponds to a model with purely circular orbits. For such a model, the radial dispersion is of course identically zero, whereas the transverse velocity dispersion is the circular velocity corresponding to the Hernquist potential,

$$\sigma_{\rm t}^2(r) = v_{\rm c}^2(r) = \frac{1}{2}~\frac{r}{(1+r)^2}\cdot$$ (35)

In the top panel of Fig. 2 we plot the radial velocity dispersion profile for various models with a different anisotropy $\beta$. Both at small and large radii, the radial dispersion is a decreasing function of $\beta$, as expected.

The line-of-sight velocity dispersion for anisotropic models is found through the formula

$$\sigma_{\rm p}^2(R) = \frac{2}{I(R)}~ \int_R^\infty \frac{\rho(r)~\sigma_{{\rm los}}^2(r,R)~r~{\rm d}r}{\sqrt{r^2-R^2}},$$ (36)

where $\sigma_{{\rm los}}(r,R)$ is the velocity dispersion at the position r on the line of sight R in the direction of observer. It is a linear combination of the radial and transverse velocity dispersions in this point,

$$\sigma_{{\rm los}}^2(r,R) = \left(1-\frac{R^2}{r^2}\right)\sigma_{\rm r}^2(r) + \frac{R^2}{2r^2}~\sigma_{\rm t}^2(r).$$ (37)

We can equivalently write

$$\sigma_{\rm p}^2(R) = \frac{2}{I(R)}~ \int_R^\infty \left... ...c{\rho(r)~\sigma_{\rm r}^2(r)~r~{\rm d}r}{\sqrt{r^2-R^2}}\cdot$$ (38)

For the general Hernquist models with a constant anisotropy, the integration (38) cannot be performed analytically. But for all integer and half-integer $\beta$'s, $\sigma_{\rm p}^2(R)$ can be expressed in terms of polynomials and the function X, defined in Eq. (4). For example, for the limit model $\beta=\frac{1}{2}$ we obtain after some algebra

$$I(R)~\sigma_{\rm p}^2(R) \frac{1}{48\pi~(1-R^2)^3}$$ =

$$\times \Bigl[3(4-14R^2+35R^4-28R^6+8R^8)~X(R)$$

$$- (28-57R^2+68R^4-24R^6)\Bigr] +\frac{R}{4}\cdot$$ (39)

For the other limit model, the one with only circular orbits, we find

$$I(R)~\sigma_{\rm p}^2(R) \frac{R^2}{48\pi~(1-R^2)^4}$$ =

$$\times \Bigl[-(120-120R^2+189R^4-108R^6+24R^8)~X(R)$$

$$+ (154-117R^2+92R^4-24R^6)\Bigr] +\frac{R}{4},$$ (40)

in agreement with Hernquist (1990).

In the bottom panel of Fig. 2 we plot the line-of-sight dispersion profiles for a number of different values of $\beta$. The behavior of the individual profiles is analogous to the spatial dispersion profiles: except for the $\beta=\frac{1}{2}$ model, which has a finite central dispersion, the $\sigma_{\rm p}$ profiles start at zero in the center, rise strongly until a certain maximum and then decrease smoothly towards zero at large projected radii. The behavior for $R\gg1$ can be quantified if we introduce the asymptotic expansion (33) into the formula (38),

$$\sigma_{\rm p}^2(R) \approx \frac{8}{15\pi}\left(\frac{5-4\beta}{5-2\beta}\right)\frac{1}{R} + \cdots$$ (41)

The dependence of the line-of-sight dispersion as a function of the anisotropy is depends strongly on the projected radius: at small projected radii, $\sigma_{\rm p}$ decreases with increasing $\beta$, whereas for large radii, $\sigma_{\rm p}$ increases with increasing $\beta$. In other words, radial models have a larger central and a smaller outer line-of-sight dispersion than their tangential counterparts. This is a direct consequence of the weight of the radial and transversal velocity dispersions in the linear combination (37): at small projected radii, the radial dispersion contributes the dominant term, whereas for the outer lines of sight, the transversal dispersion term dominates.