Constraints on Cold HI in the Halo of NGC 3079 from Absorption Measurements of Q0957+561
Judith A. Irwin , Lawrence M. Widrow , Jayanne English, PASA, 16 (1), in press.

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Contents Page: Volume 16, Number 1

Constraints on Fractal HI in a Dark Matter Halo

Density of Dark Matter at the Position of the Q 0957+561

We model the dark matter distribution in the galaxy as an isothermal spheroid

$\begin{displaymath} \overline{\rho} = \frac{\lambda(q) V_c^2}{4\pi G}\frac{1}{R^2 + z^2/q^2} \end{displaymath}$

(2)

where V_c is the circular rotation speed, R is the galactocentric radius in the plane of the disk, z is the coordinate perpendicular to the plane (

$r^2\,=\,R^2\,+\,z^2$ ), q is the flattening parameter (q<1 for an oblate halo), $\lambda(q)$ is a geometric factor equal to 1 for q=1 and rising to 6.8 for q=0.1. For

$V_c = 215 {\rm km\, s}^{-1}$ (Irwin & Seaquist 1991), R = 64 kpc, z = 0, and assuming that the r^-2 profile extends out well beyond 64 kpc, the mean surface density at the position of the quasar is then,

$\begin{displaymath} \overline{N} = \left\{ \begin{array}{ll} 3.3\times 10^{21} \... ...{22} \, {\rm\,cm^{-2}} & \mbox{for $q=0.1$} \end{array}\right. \end{displaymath}$

(3)

The Observational Parameters

For HI clouds in front of a uniform background continuum source, the brightness temperature, T_B, observed in a given beam and velocity channel is given by,

$\begin{displaymath} T_{B}\,=\,\left [T_c\,e^{-\tau}\,+\,T_s(1\,-\,e^{-\tau}) \right ]\,{\cal N}\,f_v\,f_b + \,T_c(1\,-{\cal N}\,f_v\,f_b) \end{displaymath}$

(4)

where T_c is the brightness temperature of the background continuum, T_s, $\tau$ are the spin temperature and optical depth, respectively, of the gas, f_b, f_v are the beam area and velocity filling factors, respectively, for a single cloud, and $\cal N$ is the number of clouds in the beam/velocity resolution element. Thus, the quantity,

${\cal N}\,f_v\,f_b$ is the filling factor for a resolution element. Since only a part of a cloud may fall within a beam, $\cal N$ can be less than 1. Alternatively, ${\cal N}<1$ can be thought of as the probability of finding a cloud in a given beam.

After continuum-subtraction,

$\begin{displaymath} \Delta\,T_{B}\,=\,<br /> (T_s\,-\,T_c) (1\,-\,e^{-\tau})\,{\cal N}\,f_v\,f_b \end{displaymath}$

(5)

This is the measured quantity in the continuum-subtracted cubes, in temperature units. If $T_c \gg T_s$ (which is the case for almost all lines of sight to the background continuum, Fig. 1 right), then a ratio map can be formed by computing

${{\Delta\,T_{B}}/{-T_c}}$ for each line of sight to the background continuum for any channel. In the optically thick and optically thin limits, this ratio is,

$\begin{displaymath} \frac{\Delta T_B}{-T_c} = \left\{ \begin{array}{ll} {\cal N}... ...b & \mbox{for $\tau < 1$\ (diffuse gas)} \end{array} \right. \end{displaymath}$

(6)

We have only an upper limit to

${\Delta\,T_{B}}$ which we take to be 3 x the rms map noise, identical for each channel. Thus the ratio map is the same for any channel and has the appearance of the reciprocal of the continuum map. This ratio map would be a map of optical depth under the more common assumptions of unity filling factors and optically thin gas.

The minimum value of the computed ratio map is 0.01. Thus, if the gas consists of fractal clouds, our strongest constraint is that

${\cal N}\,f_v\,f_b\,\le\,0.01$ (Condition 1). If such fractal clouds exist, then they would not be detected if the filling factor in a resolution element is less than this observationally determined limit. This could mean either that there were clouds in the beam/channel (

${\cal N} \ge 1$ ) but their filling factors were too low for detection, or that there were no clouds or too small a fraction of a cloud within the beam/channel for detection. The second possibility, we represent as

${\cal N}< 0.15$ . This is derived by requiring that the probability of there being a cloud in at least one of the 28 beams be > 99%. (N.B. This also assumes that the probability of detecting a cloud is equivalent in each of the 28 beams. In fact, this probability depends on the varying strength of the background continuum and thus this quantity may be revised.) We can now express the filling factors in terms of fractal parameters. The velocity and beam filling factors are dealt with separately since there are different conditions under which these two filling factors become unity.

The velocity filling factor in a channel of width, $\Delta v_{ch}$ , is the number of clouds, $\cal N$ , times the channel filling factor for a single cloud, f_v, i.e.

$\displaystyle {\cal N}\,f_v$	=	$\displaystyle min\left \{ {{\cal N}\,\left(\frac{v_c}{\Delta v_{ch}}\right)}, 1\right \}$	(7)
	=	$\displaystyle min\left \{{\cal N}\, \left(\frac{R_c}{R_}\right)^{(D-1)/2}\left(\frac{ v_}{\Delta v_{ch}} \right),1\right \}$	(8)

where Eqn. 8 has made use of Eqn. 1. The right term in the braces indicates the case in which the cloud or clouds ``fill" the velocity channel.

The area filling factor in a beam of radius, R_b, is the number of clouds, $\cal N$ , times the beam filling factor for a single cloud, f_b, i.e.

$\displaystyle {\cal N}\,f_b$	=	$\displaystyle \, min\left \{{\cal N}\, {\left({\frac{R_c}{R_}}\right)}^{(D-2)}... ...({\frac{R_c}{R_b}}\right)^2, {\left({\frac{R_b}{R_}}\right)}^{(D-2)} \right \}$	(9)
	$\textstyle \,=\,$	$\displaystyle min\left \{{\cal N}\, \left({\frac{R_c}{R_}}\right)^{D} \left({\frac{R_}{R_b}}\right)^2, \left({\frac{R_b}{R_*}}\right)^{(D-2)} \right \}$	(10)

where we assume $D\,<\,2$ . The term,

$\left({R_c}/{R_*}\right)^{(D-2)}$ , is the area filling factor for cloudlets in a single fractal cloud and the right term in the braces indicates the case in which the cloud or clouds fill the area of the beam (except for the filling factor of the cloudlets).

These two expressions, which assume no-overlap of clouds or cloudlets, along with Condition 1 allow us to place constraints on the fractal parameters, R_c and D.

Results for Fractal Clouds

Fig. 2 Left shows the results, assuming that the clouds fill the velocity channel (

${\cal N}\,f_v\,=\,1$ ), which puts a limit on the number of clouds in a beam/velocity resolution element, i.e.

${\cal N}\,=\,1/f_v$ . The two solid curves denote Condition 1, i.e. $f_b \,= 0.01$ with the more horizontal curve corresponding to the case in which the clouds do not fill the beam and the more vertical curve corresponding to the case in which they do. The number of clouds in the resolution element varies along the solid curve. If fractal clouds exist and occur within a beam, they would not have been detected if they have parameters which fall below these curves. The dashed line indicates the condition,

${\cal N}\,=\,0.15$ . Above this curve, fractal clouds would not have been detected because there is a low probability that the cloud is within the beam. Thus, we can rule out parameter space which falls between these curves; this region is denoted by the bold horizontal lines in the figure.

In addition, the velocity resolution is quite fine (2.6 km s^-1). Suppose that there are two clouds within the beam, i.e. separated by

$\,\hbox{\raise 0.5 ex \hbox{$<$}\kern-.77em \lower 0.5 ex \hbox{$\sim$}$\,$}$ 95 pc. For these clouds to be within the same beam/channel element, they must be separated in velocity by < 2.6 km s^-1. Yet observed velocity dispersions in disk gas are of order 10 km s^-1, independent of radius, and may be higher in an extended dark matter disk, given the requirement of disk stability, (see discussion in PCM). If this is the case, then it is unlikely that these clouds would be separated in velocity by less than 2.6 km s^-1 which suggests that

${\cal N}\,\le\,1$ . Under these circumstances, we can also rule out the region of parameter space marked by the thin horizontal lines under the

${\cal N}\,=\,1$ (dotted) curve.

Fig. 2, Right, shows the results for the case in which the clouds do not fill the velocity channel (

${\cal N}\,f_v\,<\,1$ ). The solid curves denote the case,

${\cal N}\,=\,1$ , and the dotted curves denote

${\cal N}\,=\,0.15$ . Thus, we can rule out parameter space between these curves, shown by horizontal bold lines. Again, if

${\cal N}\,\le\,1$ , then the region marked by thin horizontal lines can also be ruled out.

**Figure 2:** **Left:** Fractal parameters for the case in which the clouds fill velocity space. The solid curves denote the observational limit when the clouds do not fill the beam (more horizontal curve) and when they do (more vertical curve). The dashed curve represents ${\cal N}=0.15$ above which a cloud would not be in the beam and the dotted curve represents ${\cal N}=1$ . **Right:** Fractal parameters for the case in which the clouds do not fill velocity space. The solid curves denote the observational limit (as at left) for the case, ${\cal N}=1$ , and the dotted curves represent ${\cal N}=0.15$ .
$\begin{figure} \begin{center} \centerline{\hbox{ \psfig{figure=fig_2left.ps,ang... ...gure=fig_2right.ps,angle=-90,height=7cm,width=6.5cm} }}\end{center} \end{figure}$

We can now compute the mean column density in a beam,

$\overline{N({\rm HI})}\,=\,{\cal N}\,{\left( R_c/R_b \right)}^2\,N({\rm HI})_c$ , for fractal parameters which fall below our observational limit and compare this to the theoretically expected value for a dark matter halo (Eqn. 3). The results indicate that it is possible to ``hide" all the dark matter in fractal clouds. For example (Fig. 2, left), D=1.6 and

log(R_c/R_*)=5.7 are (just) below the observational limit. These parameters yield a value of

$\overline{N({\rm HI}) }\,=\,f_H\,1.3\times 10^{22}$ cm^-2. Then the mean column density of HI + H₂ gas in fractal clouds is

$\overline{N_f}\,=\,1.3\times 10^{22}$ cm^-2 which (cf. Eqn. 3) implies that virtually all the dark matter could be hidden in such clouds.

Results for Diffuse HI

It is interesting to contrast the above results with what would be obtained if we make the more common assumptions that the gas is optically thin and the filling factors are all 1. In this case, the observational limit can be written as

$\tau\,\le\,0.01$ (Eqn. 6) which gives

$\overline{N({\rm HI}) }\,\le\,4.7\times 10^{16}~T_s$ cm^-2, illustrating the high sensitivity of these observations. For $T_s\,=\,3$ K, a comparison with the theoretical value for spherical and flattened halos (Eqn. 3) implies that HI cannot constitute more than $\sim$ 10^-5 of the dark matter, if it is in the form of cold diffuse gas.