Disk formation in an NFW halo

We want to relate the scale-length of an exponential disk to the properties of the halo in which it forms. The density distribution for the NFW potential (Navarro et al. 1997) is

$\begin{displaymath} \rho_{\rm halo}(r) = {\rho_0 \over (1 + r/a)^2 r/a}, \end{displaymath}$

(3)

where the halo scale-length, a, and the normalising density, $\rho_0$ , are constants. This density distribution is truncated at r₂₀₀ and the NFW concentration parameter is defined as c = r₂₀₀/a.

The angular momentum of the halo is conveniently specified in terms of the spin parameter

$\begin{displaymath} \lambda_{\rm h} = {J_{\rm h} \vert E\vert^{1/2} \over G M_{\rm h}^{5/2}}, \end{displaymath}$

(4)

where $J_{\rm h}$ is the angular momentum of the halo, E its total energy and $M_{\rm h}$ its mass. Tidal torques typically make the spin parameter about 0.05 (Steinmetz & Bartelmann 1995). We define a spin parameter for the disk, $\lambda _{\rm d}$ to make the specific angular momentum of the disk the same as that of the halo when $\lambda_{\rm d} = \lambda_{\rm h}$ , i.e. so that the specific angular momentum of the disk is

$\begin{displaymath} {J_{\rm d} \over M_{\rm d}} = {\lambda_{\rm d} G M_{\rm h}^{3/2} \over \vert E\vert^{1/2}}. \end{displaymath}$

(5)

Any ejection is likely to selectively remove gas with higher than average specific angular momentum, so that we should expect the spin parameter of a disk that formed from gas to be comparable to or smaller than the spin parameter of the halo.

The angular momentum of an exponential disk is determined by its scale-length, b, and the mass distribution of the halo in which it resides, so that specifying $\lambda _{\rm d}$ determines the exponential scale-length in terms of halo properties. For $b \ll a$ (i.e. $\lambda_{\rm d} \ll 1$ ), $b \propto \lambda_{\rm d}^{2/3} a$ . Figure 1 shows the ratio of disk scale-length to halo scale-length as a function of halo concentration parameter, for some reasonable values of the disk spin parameter. Physically reasonable values of these parameters give $b/a \simeq 0.1$ .

**Figure 1:** Disk scale-length vs halo concentration. The ratio of disk scale-length, b, to halo scale-length, a, is plotted as a function of halo concentration parameter, c, for some values of the disk spin parameter, $\lambda _{\rm d}$ . The value of the spin parameter is shown above each curve.
$\begin{figure} \centerline{\psfig{file=scale.ps,height=10cm,width=12cm,angle=270}}\end{figure}$

The small value for the disk scale-length raises the issue of whether the mass of the disk dominates the mass of the halo in any part of the disk. Figure 2 shows the ratio of disk mass to halo mass within radius r as a function of the scaled radius, r/b, for some typical values of b/a and a gas fraction of 20 percent (the halo concentration parameter is taken as 3.5 and the results are not sensitive to it). About 60 percent of the mass of the disk is contained within r = 2 b and about 90 percent within r = 4b. We see from the figure that the disk dominates the halo, except in the outermost part of the disk. Lowering the gas fraction and/or increasing the angular momentum of the disk can modify the degree of dominance, but for reasonable values of these parameters, we should expect at least the inner half of the disk (by mass) to be self-gravitating.

**Figure 2:** Disk mass over halo mass vs radius. This shows the ratio of the mass within r in an exponential disk to the mass within r in an NFW halo as a function of the scaled radius, r/b, where b is the disk scale-length. Curves are shown for several values of b/a, where a is the scale-length for the NFW halo (equation 3). A gas fraction of 0.2 and a halo concentration parameter of 3.5 were used for the figure.
$\begin{figure} \centerline{\psfig{file=diskd.ps,height=10cm,width=12cm,angle=270}}\end{figure}$

Ejection and other processes can mean that the mass ending up in the disk is substantially smaller than the mass of gas that first collapses. Immediately after collapse the gas must be supported either by pressure or by rotation. In a cold collapse the support will usually be by rotation, so that the gas quickly collects in the region where the disk is to form (the exception being for small halos, where heating by photoionisation can keep the gas pressure supported). Based on the argument above, such gas will be largely self-gravitating. Star formation, with the associated feedback, and other processes complicate this picture. Nevertheless, we can expect the gas and stars that form from it to be largely self-gravitating. This means that the gas cannot simply be regarded as collapsing in a dark matter halo for most cold (dwarf) collapses.

Self-gravitation can give the disk a substantially higher rotation velocity than the halo in which it forms. A rough estimate gives the boost to the rotation speed of the disk as about $v_{\rm rot, d} /v_{\rm rot,h} \simeq f_{\rm gas}/ \lambda_{\rm d}$ , for a gas fraction of $f_{\rm gas}$ . If the disk rotation speed is substantially higher than that of the halo, then the decline in the rotation curve at the edge of the disk may give the impression that there is no dark halo. Self-gravity will certainly modify the shape of the rotation curve near the centre of a galaxy (e.g. Gelato & Sommer-Larsen 1998).

The main issue here is that, if the gaseous component is largely self-gravitating, then gas processes must play a major role in determining the structure of the visible part of a galaxy. To date, treatments of gas processes, especially in cold systems, have been crude. Until we have a much better understanding of gas processes in dwarf galaxies we will not be able to construct a good theory of galaxy formation.

Next Section: Conclusion
Title/Abstract Page: Gas and Galaxy Formation
Previous Section: Dwarf (Cold) Galaxy Formation

Contents Page: Volume 16, Number 1