5 A Galactic Halo population model for the CHVC ensemble

In the previous section we have outlined a physical model for self-gravitating, dark-matter dominated CHVCs evolving in the Local Group potential. While that model was quite successful in describing the global properties of the CHVC phenomenon, we noted that some aspects of the observed kinematic and spatial deployment were strongly influenced by the effects of obscuration by foreground Galactic H  I and that, furthermore, the sensitivity limitations of the currently available H  I survey material preclude tightly constraining the characteristic distances. In this section we consider to what extent a straightforward model in which the CHVCs are distributed throughout an extended halo centered on the Galaxy might also satisfy the observational constraints. We consider such a Galactic Halo model ad hoc in the sense that it lacks the physical motivation that the hierarchical structure paradigm affords the Local Group model.

We consider a spherically symmetric distribution of clouds, centered on the Galaxy. The radial density profile of the population is described by a Gaussian function, with its peak located at the Galactic center and its dispersion to be specified as a free parameter of the simulations. The H  I mass distribution is given by a power-law, the slope of which is a free parameter. Different values are allowed for the lowest H  I mass in the simulation. The H  I density distribution of an individual cloud is also described by a Gaussian function. The central volume density is the same for all clouds in a particular simulation. Given the H  I mass and central density of an object, the spatial FWHM of the H  I distribution follows. For the velocity FWHM we have simply adopted the thermal linewdth of an 8000 K H  I gas of 21 km s^-1.

Each simulated cloud is "observed'' with the parameters corresponding to the LDS observations, if it is located in the northern celestial hemisphere, but with the HIPASS parameters if it is located in the southern hemisphere. Clouds are removed from the simulation if they are too faint to be detected. To include the effects of obscuration by the Milky Way, the velocity field of the clouds must be specified. The population is considered in the Galactic Standard of Rest system, where it is distributed as a Gaussian with a mean velocity of $-50\rm\;km\;s^{-1}$ and dispersion of $110\rm\;km\;s^{-1}$. These values follow directly from the observed parameters summarized in Table 2 after correction for obscuration as in Fig. 2. Clouds with a deviation velocity (as defined in Sect. 2.2.1) less than  $70\rm\;km\;s^{-1}$ are removed. Additional clouds that pass the selection criteria are simulated until their number equals the number of CHVCs actually observed.

We performed the simulations with the following values for the four parameters that describe the distance, H  I mass, and spatial extent of the population.

• The spatial dispersion of the cloud population. Values range from  $10\rm\;kpc$ to $2\rm\;Mpc$; specifically we consider the values of 10, 15, 20, ...50, 60, 70, ...100, 150, 200, ...500, 1000, 2000 kpc;
• The slope of the H  I mass distribution, $\beta$. Values for $\beta$ were $-2.0, -1.8, \ldots, -0.8$;
• The lowest H  I mass, M₀, allowed in a simulation. Values for M₀ were 10², 10³, 10⁴, or  $10^5\;M_\odot$;
• The central gas density in the clouds, n₀. Values for n₀, which remained constant for a single run, were $3\times10^{-3}$, $1\times10^{-2}$, $3\times10^{-2}$, 0.1, and  $0.3\rm\;cm^{-3}$.

The only measured quantities which can usefully be compared to the models are the distributions of angular sizes and peak column densities. This is because the average kinematics in these simulations have already been defined to match the data by our choice of the mean velocity and its dispersion. Given four free model parameters for each simulation and only two distributions to determine the degree of agreement between simulations and observations, it is clear that the problem is under-determined. We can only hope to constrain the range of reasonable parameters in the four-dimensional parameter space.

In order to assess the degree of agreement between the simulation outcomes and the observations, we use a $\chi ^2$-test from Sect. 14.3 of Numerical Recipes, (Press et al. 1993). The size and column density distributions of the models and the data are compared. A simulation was considered acceptable if $\chi^2 ({\rm size}) < 5$ and $\chi^2 ({N_{\rm HI}})<5$. Figure 24 shows examples of the range of fit quality that was deemed acceptable for both the column density and size distributions.

Table 6 lists the parameter combinations that produce formally acceptable results, and shows that for each M₀value the acceptable solutions are concentrated around a line. The solutions range from nearby models, for which the central density is of the order of  $0.1\rm\;cm^{-3}$, the mass slope is -2.0, and the characteristic distance is several tens of kpc, to more distant models, having a central density of  $0.01\rm\;cm^{-3}$, a mass slope of -1.4, and characteristic distances of several hundreds of kpc. Since column density is simply the product of depth and density this coupling of distance to central density is easily understood.

 

 

Table 6: Results of the models described in Sect. 5, in which the CHVCs are viewed as forming an extended halo population, centered on the Galaxy. Each simulation is determined by four free parameters, namely the central H  I density of the clouds, n₀, the slope of the H  I mass distribution, $\beta$, the lowest H  I mass in each simulation, M₀, and the dispersion of the spatial Gaussian that defines the distance, $\sigma _{\rm d}$, of the cloud population. The simulations were sampled with the observational parameters of the LDS and HIPASS surveys and compared with the CHVC sample. The table shows which distance dispersions produce acceptable results for each of the combinations of n₀, $\beta$, and M₀. A field is blank if there is no good fit for the given parameter combinations. A simulation is considered successful if $\chi ^2_{\rm size} < 5$ and $\chi ^2_{N_{\rm HI}} < 5$. The central cloud density, n₀, is shown horizontally above each section of the table, in units cm^-3; the slope of the H  I mass distribution is listed vertically on the left of the table: M₀ ranges from $10^2\;M_\odot$ for the top table to $10^5\;M_\odot$ for the bottom table. Distances are in kpc. The model is not tightly constrained, because of degeneracies in parameter combinations.

Simulations with M0=102:
	$n_0 = 3\times10^{-3}\rm\;cm^{-3}$	$1\ \times\ 10^{-2}\rm\;cm^{-3}$	$3\ \times\ 10^{-2}\rm\;cm^{-3}$	$1\ \times\ 10^{-1}\rm\;cm^{-3}$	$3\ \times\ 10^{-1}\rm\;cm^{-3}$
$\beta=-1.0$
-1.2	$\sigma_{\rm d}=150\ldots300$ kpc	$35\ldots45$	$10\ldots20$
-1.4	$200\ldots350$	$45\ldots100$	$15\ldots40$	15
-1.6	$200\ldots350$
-1.8	500
-2.0

Simulations with M0=103:
	$n_0 = 3\times10^{-3}\rm\;cm^{-3}$	$1\ \times\ 10^{-2}\rm\;cm^{-3}$	$3\ \times\ 10^{-2}\rm\;cm^{-3}$	$1\ \times\ 10^{-1}\rm\;cm^{-3}$	$3\ \times\ 10^{-1}\rm\;cm^{-3}$
$\beta=-1.0$
-1.2		$70\ldots100$
-1.4	$\sigma_{\rm d}=100\ldots400$ kpc	$45\ldots100$	$15\ldots40$	15
-1.6	200	50, 90	$15\ldots25$	$10\ldots15$
-1.8			$10\ldots15$	$10\ldots15$
-2.0			10	$10\ldots15$

Simulations with M0=104:
	$n_0 = 3\times10^{-3}\rm\;cm^{-3}$	$1\ \times\ 10^{-2}\rm\;cm^{-3}$	$3\ \times\ 10^{-2}\rm\;cm^{-3}$	$1\ \times\ 10^{-1}\rm\;cm^{-3}$	$3\ \times\ 10^{-1}\rm\;cm^{-3}$
$\beta=-1.0$
-1.2	$\sigma_{\rm d}=100\ldots250$ kpc
-1.4	$150\ldots450$	$50\ldots100$	$50\ldots70$
-1.6	400	$45\ldots100$	$40\ldots50$
-1.8		$40\ldots60$	$30\ldots45$
-2.0		$40\ldots45$	$30\ldots45$

Simulations with M0=105:
	$n_0 = 3\times10^{-3}\rm\;cm^{-3}$	$1\ \times\ 10^{-2}\rm\;cm^{-3}$	$3\ \times\ 10^{-2}\rm\;cm^{-3}$	$1\ \times\ 10^{-1}\rm\;cm^{-3}$	$3\ \times\ 10^{-1}\rm\;cm^{-3}$
$\beta=-1.0$
-1.2	$\sigma_{\rm d}=250$ kpc
-1.4	$250\ldots450$	150
-1.6	$150\ldots300$	$100\ldots150$
-1.8	$150\ldots200$	$100\ldots150$
-2.0		$90\ldots150$

Overviews of two of the best-fitting models of the Galactic Halo type are given in Figs. 25 and 26. Figure 25 shows a cloud population with 30 kpc dispersion, while the population in Fig. 26 has a dispersion of 200 kpc. These figures can be compared with Fig. 9, showing the situation actually observed. Despite there being almost a factor of ten difference in the average object distance for these two models, they produce similar distributions of observables, which are to a large extent determined by the effects of obscuration. Relative to the observed CHVC sample shown in Fig. 9, the density distributions of these models are more uniformly distributed on the sky. The average velocity fields are also more symmetric about $b=0\hbox{$^\circ$ }$, lacking the extreme negative excursion toward $(l,b)=(125^\circ,-30^\circ)$ seen in the CHVC population, that produces a large gradient in the $(V_{\rm GSR},b)$ plot.