5 Models with increasing anisotropy

5.1 Background

Osipkov (1979) and Merritt (1985) developed an inversion technique for a special class of distribution functions that only depend on energy and angular momentum through the combination

$$Q\equiv {\cal{E}}-\frac{L^2}{2r_{\rm a}^2},$$ (42)

with $r_{\rm a}$ the so-called anisotropy radius, with the additional condition that

$$F({\cal{E}},L)=0 \qquad\quad {\rm for}\ Q<0.$$ (43)

Such models correspond to an augmented density of the form

$$\tilde{\rho}(\psi,r) = \left(1+\frac{r^2}{r_{\rm a}^2}\right)^{-1} f(\psi).$$ (44)

In this case, the fundamental integral Eq. (1) can be inverted in a similar way as the Eddington relation,

$$F({\cal{E}},L) = \frac{1}{2\sqrt{2}~\pi^2}~ \frac{{\rm d}}... ...\rm d}f}{{\rm d}\psi}~ \frac{{\rm d}\psi}{\sqrt{Q-\psi}}\cdot$$ (45)

The anisotropy $\beta(r)$ for the Osipkov-Merritt models can be found by means of formula (11),

$$\beta(r) = \frac{r^2}{r^2+r_{\rm a}^2}\cdot$$ (46)

These models are hence isotropic in the center and completely radially anisotropic in the outer regions. The parameter $r_{\rm a}$determines how soon the anisotropy turns from isotropic to radial. In particular, for $r_{\rm a}\rightarrow\infty$, Q is nothing else than the binding energy, and the Osipkov-Merritt models reduce to the isotropic models.

The Osipkov-Merritt models were generalized by Cuddeford (1991), who considered models which correspond to an augmented density of the form

$$\tilde{\rho}(\psi,r) = r^{-2\beta_0} \left(1+\frac{r^2}{r_{\rm a}^2}\right)^{-1+\beta_0} f(\psi).$$ (47)

These models reduce to the Osipkov-Merritt models if we set $\beta_0=0$. Within this formalism, the distribution function can be calculated in a similar way as for the Osipkov-Merritt models. The distribution functions have the general form

$$F({\cal{E}},L) = F_0(Q)~L^{-2\beta_0},$$ (48)

with the additional condition (43). The solution of the integral Eq. (1), valid for $\beta_0<1$, reads in this case 010

 

$$F({\cal{E}},L) = \frac{2^{\beta_0}}{(2\pi)^{3/2}}~ \frac{L^{-2\be... ...int_0^Q \frac{{\rm d}^m f}{{\rm d}\psi^m}~ \frac{{\rm d}\psi}{(Q-\psi)^\theta},$$ (49a)

where

$$1+{\rm int}~\left(\frac{1}{2}-\beta_0\right)$$                   m = (49b)

$$\theta {\rm frac}~\left(\frac{1}{2}-\beta_0\right).$$ = (49c)

The most interesting cases are those where $\beta _0$ is either integer or half-integer. For integer values of $\beta _0$, the general formula (49abc) reduces to

$$F({\cal{E}},L) = \frac{2^{\beta_0}}{2\sqrt{2}~\pi^2}~ \fra... ...m d}\psi^{1-\beta_0}}~ \frac{{\rm d}\psi}{\sqrt{Q-\psi}}\cdot$$ (50)

For half-integer values of $\beta _0$, the integral Eq. (1) is a degenerate integral equation, which can be solved without a single integration (Dejonghe 1986; Cuddeford 1991),

$$F({\cal{E}},L) = \frac{2^{\beta_0}}{(2\pi)^{3/2}}~ \frac{L... ...0}f}{{\rm d}\psi^{\frac{3}{2}-\beta_0}} \right]_{\psi=Q}\cdot$$ (51)

As for the Osipkov-Merritt models, the anisotropy of the Cuddeford models has a very simple functional form, which can be found through (11),

$$\beta(r) = \frac{r^2+\beta_0r_{\rm a}^2}{r^2+r_{\rm a}^2}\cdot$$ (52)

They hence have an anisotropy $\beta _0$ in the center, and become completely radially anisotropic in the outer regions. The anisotropy radius $r_{\rm a}$ is again a degree for how quick this transition takes place. In particular, for $r_{\rm a}\rightarrow\infty$, the Cuddeford models reduce to models with a constant anisotropy $\beta=\beta_0$. Because the range of values for $\beta _0$ for which the inversion (49abc) is mathematically defined is only restricted by $\beta_0<1$, distribution functions can in principle be calculated with any degree of anisotropy in the center, ranging from very radial to extremely tangential. Whether these distribution functions correspond to physically acceptable solutions depends on the positivity, however.

5.2 Hernquist models with increasing anisotropy

5.2.1 The distribution function

For the Hernquist potential-density pair (2ab), the augmented density corresponding to the Cuddeford formalism is readily calculated. We obtain

$$f(\psi) = \frac{1}{2\pi} \left[ 1+\lambda\left(\frac{1-\p... ...^{1-\beta_0} \frac{\psi^{4-2\beta_0}}{(1-\psi)^{1-2\beta_0}},$$ (53)

where we have set $\lambda=1/r_{\rm a}^2$. Combining this expression with the general Cuddeford solution (49abc) we can obtain distribution functions that self-consistently generate the Hernquist potential-density pair, and which have an arbitrary anisotropy in the center and a completely radial structure in the outer regions. In order to represent physically acceptable dynamical models, it is necessary that these distribution functions are positive over the entire phase space, i.e. $F({\cal{E}},L)\geqslant0$ for $0\leqslant Q\leqslant1$. Before trying to actually calculate the distribution functions, it is useful to investigate which region in the $(\beta _0,\lambda )$ parameter space corresponds to physically acceptable distribution functions.

First of all, it is obvious that the models with $\beta_0>\frac{1}{2}$ will not correspond to non-negative distribution functions: the distribution function is already too radial for $\lambda =0$ (Sect. 4.2.1), and will become even more radial for larger $\lambda$. We can therefore limit the subsequent discussion to $\beta_0\leqslant\frac{1}{2}$. Now consider such a fixed value $\beta _0$, and consider all Cuddeford models corresponding to this central anisotropy. For $\lambda\rightarrow0$, the Cuddeford model reduces to the model with constant anisotropy $\beta _0$, which is physically acceptable (Sect. 4.2.1). For $\lambda \rightarrow \infty$, the distribution function will only consist of radial orbits, for which the distribution function is not positive. It can therefore be expected that, for a given value of $\beta_0\leqslant\frac{1}{2}$, a range of $\lambda$'s is allowed, starting from 0 up to a certain $\lambda_{{\rm max}}$.

 

 

Table 1: The range of anisotropy radii which give rise to a positive distribution function of the Cuddeford type, consistent with the Hernquist potential-density pair. For a given value of $\beta _0$, this range corresponds to $0\leqslant \lambda \leqslant \lambda _{{\rm max}}$, or equivalently, to $r_{a,{\rm min}}\leqslant r_{\rm a}\leqslant \infty$.

$\beta _0$	$\lambda_{{\rm max}}$	$r_{{\rm a, min}}$
$\leqslant-1.500$	0.000	$\infty$
-1.375	1.764	0.753
-1.250	3.598	0.527
-1.125	5.550	0.424
-1.000	7.582	0.363
-0.875	9.680	0.321
-0.750	11.83	0.291
-0.625	14.02	0.267
-0.500	16.23	0.248
-0.375	18.51	0.232
-0.250	20.57	0.220
-0.125	22.61	0.210
0.000	24.42	0.202
0.125	25.87	0.197
0.250	26.70	0.194
0.375	26.42	0.195
0.500	24.00	0.204

Next, we have to investigate how $\lambda_{{\rm max}}$ varies with $\beta _0$, i.e. which anisotropy radii are allowed for a given central anisotropy? Distribution functions with a strong central tangential anisotropy and a small anisotropy radius are likely to be negative. Indeed, consider the orbital structure of such a galaxy. Because the outer regions of the galaxy $(r\gg r_{\rm a})$ are strongly radially anisotropic, the vast majority of the stars there must be on nearly radial orbits. These stars also pass through the central regions, where they will contribute to the central density and radial velocity dispersion as well. The smaller the value of $r_{\rm a}$, i.e. the larger the value of $\lambda$, the stronger the contribution of stars on such nearly radial orbits. In order to create a core where the anisotropy is tangential, a large number of stars hence have to be added which move on tightly bound nearly circular orbits. But we are limited from keeping on adding such stars, because we cannot exceed the spatial density of the Hernquist profile, which has only a fairly weak r^-1 divergence. We therefore expect that no Cuddeford models will exist beyond a certain minimal $\beta _0$ (except for the degenerate case of the constant anisotropy models, which have no radial anisotropy at large radii). Moreover, it can be expected that for models with a tangential central anisotropy, the range of anisotropy radii is more restricted than for models with a radial or isotropic central anisotropy, i.e. that $\lambda_{{\rm max}}(\beta_0)$ is a increasing function of $\beta _0$.

Figure 3: The region in $(\beta _0,\lambda )$ space corresponding to a positive distribution function of the Cuddeford type, consistent with the Hernquist potential-density pair.

Figure 4: Comparison of the distribution function corresponding to Hernquist models of the Cuddeford type and Hernquist models with a constant anisotropy. The distribution functions are represented as isoprobability contours in turning point space. The solid lines correspond to the Cuddeford distribution functions, with the parameters $\beta _0$ and $\lambda$ displayed in the down left corner of each diagram. The dotted lines represent the distribution functions of the corresponding Hernquist models with a constant anisotropy $\beta _0$.

By numerical evaluation of the integral in Eq. (49a), we calculated $\lambda_{{\rm max}}(\beta_0)$ for a number of values for $\beta _0$ (Table 1). The region in parameter space where the Cuddeford-Hernquist models are physical is shown in Fig. 3. Notice that all models with $\beta_0\leqslant-\frac{3}{2}$ and $\lambda>0$ are negative at some point in phase space and are thus unphysical: the Hernquist potential-density pair can support no (non-degenerate) distribution functions of the Cuddeford type with a central anisotropy $\beta_0\leqslant-\frac{3}{2}$.

We are primarily interested in those models where the distribution function can be expressed in terms of elementary functions. This is of course possible for all half-integer values of $\beta _0$, because the calculation of the distribution function involves no integrations. Also for the integer values of $\beta _0$, the distribution function can be calculated analytically, through the formula (50). Because of the limited region in $(\beta _0,\lambda )$ space where Cuddeford models are non-negative, this leaves us with four models with analytical distribution functions, corresponding to $\beta_0=\frac{1}{2}$, 0, $-\frac{1}{2}$ and -1. The most simple of them is the case $\beta_0=\frac{1}{2}$, for which we obtain

$$F({\cal{E}},L) = \frac{1}{4\pi^3}~\frac{1}{L}~ \frac{3Q^2+... ...^2-5Q+2)} {\sqrt{1+\lambda\left(\frac{1-Q}{Q}\right)^2}}\cdot$$ (54)

It is straightforward to check that this distribution function remains positive for $0\leqslant\lambda\leqslant24$, in agreement with the numerical result in Table 1. For $\beta_0=0$, we recover the Osipkov-Merritt model,

$$F({\cal{E}},L) = \frac{1}{8\sqrt{2}\pi^3} \left\{ \frac{3\arcsin\... ...2}} + \sqrt{Q}~(1-2Q) \left[\frac{8Q^2-8Q-3}{(1-Q)^2}+8\lambda\right] \right\},$$ (55)

in agreement with Hernquist (1990). For the two other cases, $\beta_0=-\frac{1}{2}$ and $\beta_0=-1$, the distribution function can also be written in terms of elementary functions, but the expressions are somewhat more elaborate.

In Fig. 4 we show the distribution function of the Cuddeford type for four different models. The models on the top row have a radial central anisotropy, whereas those in the bottom panels have a tangential anisotropy in the center. The left and right column correspond to two different values of the anisotropy radius. The dotted distribution functions on the background are the distribution functions with a constant anisotropy $\beta _0$.

The character of the Cuddeford models can directly be interpreted from these figures. Compared to the constant anisotropy models, the Cuddeford models have a much larger fraction of stars on radial orbits, visible for both models with radial and tangential central anisotropy. The most conspicuous feature of each of the Cuddeford distribution functions is that the right part of the (r_-,r₊) diagram is completely empty, i.e. at large radii only the most radial orbits are populated, which is necessary to sustain the radial anisotropy. The boundary of the region in turning point space beyond which no orbits are populated can be calculated by translating the equation Q=0 in terms of the turning points r_- and r₊.

$$\frac{r^2_-~\psi(r_-)}{1+\lambda~r_-^2} = \frac{r^2_+~\psi(r_+)}{1+\lambda~r_+^2}\cdot$$ (56)

When we substitute the Hernquist potential (2a), we can actually calculate the range of allowed orbits,

$$\leqslant r_-\leqslant r_{{\rm c, max}}$$                                 0 (57)

$$\leqslant r_+\leqslant \frac{(1+r_-)+\sqrt{1+2r_-+r_-^2+4\lambda r_-^3}}{2\lambda r_-^2},$$ r_- (58)

where $r_{{\rm c, max}}$ represents the radius of the largest allowed circular orbit for a given $\lambda$,

$$r_{{\rm c,max}} = \frac{3^{1/3}+(9\sqrt{\lambda}+\sqrt{81\l... ...sqrt{\lambda}~(9\sqrt{\lambda}+\sqrt{81\lambda-3})^{1/3}}\cdot$$ (59)

Obviously, the larger $\lambda$, the more restricted the range of allowed orbits, because the transition to radial anisotropy occurs at smaller radii for large values of $\lambda$. This can be seen when comparing the left and right panels of Fig. 4.

5.2.2 The velocity dispersions

Figure 5: The radial (upper panels) and line-of-sight (lower panels) velocity dispersion profiles of the Hernquist-Cuddeford models. The different curves in the two left panels correspond to models with the same anisotropy radius $\lambda =1$ (i.e. $r_{\rm a}=1$), but with a different central anisotropy parameter $\beta _0$: plotted are $\beta=\frac{1}{2}$, 0, $-\frac{1}{2}$and -1. The dotted curves are the dispersion profiles of the (hypothetical) completely radial Hernquist model, which corresponds to $\beta _0=1$. The two panels on the right-hand side contain the dispersion profiles of models with the same central, slightly tangential, anisotropy parameter $\beta_0=-\frac{1}{2}$, but with a varying anisotropy radius. The various curves correspond to $\lambda =0$ (black solid line), 0.2, 1, 3, 8 and $\lambda _{{\rm max}}\equiv 16.23$. Again, the dotted curves are the dispersion profiles of the (hypothetical) completely radial Hernquist model, which corresponds to $\lambda \rightarrow \infty$.

In order to calculate the radial velocity dispersion associated with models of the Cuddeford type, we use the general formula (8a). After some manipulation, we obtain

$$\sigma_{\rm r}^2(r) = \frac{r^{1-2\beta_0}(1+r)^3}{(1+\lamb... ...'{}^2)^{1-\beta_0}{\rm d} r'}{r'{}^{1-2\beta_0}(1+r')^5}\cdot$$ (60)

In general, this integral needs to be evaluated numerically, but for the four models with integer and half-integer values of $\beta _0$, it can be performed analytically. For example, for the Osipkov-Merritt model $\beta_0=0$, we find

$$\sigma_{\rm r}^2(r) = \frac{r~(1+r)^3}{1+\lambda r^2} \ln\left(\f... ...t) - \frac{r~(25+52r+42r^2+12r^3)-\lambda~r~(1+4r)} {12~(1+r)~(1+\lambda r^2)},$$ (61)

which reduces to the isotropic dispersion (19) for $\lambda =0$. For the other integer and half-integer values of $\beta _0$, the radial dispersion can also be expressed in terms of algebraic functions and logarithms, but the expressions are somewhat more elaborate.

In the top panels of Fig. 5 we plot the radial velocity dispersion profiles for Hernquist-Cuddeford models, for varying $\beta _0$ and varying $\lambda$ (left and right panels respectively). The behavior of $\sigma_{\rm r}$ as a function of $\beta _0$ is predictable. At small radii, the different models have a different behavior, with the largest dispersion for the most centrally radial models. At large radii they all have a similar, purely radial, orbital structure, and as a consequence their dispersion profiles all converge towards a single profile. This limiting profile is the radial velocity dispersion profile that corresponds to the (hypothetical) model with a completely radial orbital structure, which we can obtain by either setting $\beta=1$ in the expression (32), or setting $\beta _0=1$ in the expression (60),

$$\sigma_{\rm r}^2(r) = \frac{1}{12}~ \frac{1+4r}{r~(1+r)}\cdot$$ (62)

For a fixed central anisotropy, the behavior of the radial dispersion as a function of the anisotropy radius also follows a simple trend: the $\sigma_{\rm r}$ profiles increase with increasing $\lambda$, and the curves are all bounded by two limiting profiles: on the one hand the dispersion profile (32) of the constant anisotropy model (obtained by setting $\lambda =0$), and on the other hand the hypothetical dispersion profile (62) of the purely radial model (which corresponds to $\lambda \rightarrow \infty$). Dispersion profiles with large $\lambda$ will more quickly lean towards the purely radial profile than models with small $\lambda$, because the transition to a strongly radial anisotropy occurs at $r\sim r_{\rm a}=1/\sqrt{\lambda}$.

The bottom panels of Fig. 5 show the line-of-sight velocity dispersion of the Hernquist-Cuddeford models. These profiles had to be calculated numerically. The dependence of the line-of-sight dispersion upon $\lambda$ and $\beta _0$ can be easily interpreted. In particular, the line-of-sight dispersion profiles of the Cuddeford models tend towards the line-of-sight dispersion profile of the hypothetical purely radial Hernquist model, which reads

$$I(R)~\sigma_{\rm p}^2(R) = \frac{R}{8}+\frac{1}{96R} -\frac{1}{48\pi~(1-R^2)^2} \Bigl[R^2~(20-29R^2+12R^4)~X(R) + (2-7R^2+4R^4)\Bigr].$$ (63)