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Subsections

   
4 A Local Group population model for the CHVC ensemble

Determining Local Group membership for nearby galaxies is not a trivial undertaking. The well-established members of the Local Group have been used to define a mass-weighted Local Group Standard of Rest, corresponding to a solar motion of $V_\odot = 316\rm\;km\;s^{-1}$towards $l=93^\circ,\;b=-4^\circ$ (Karachentsev & Makarov 1996). The $1\sigma$ velocity dispersion of Local Group galaxies with respect to this reference frame is about 60 km s-1 (Sandage 1986). A plot of heliocentric velocity versus the cosine of the angular distance between the solar apex and the galaxy in question, as shown in Fig. 10, has often been used to assess the likelihood of Local Group membership (e.g. van den Bergh 1994) when direct distance estimates have not been available. Local Group galaxies tend to lie within about one sigma of the line defined by the solar motion in the LGSR reference frame. Braun & Burton (1999) pointed out how the original LDS CHVC sample followed this relationship, although offset with a significant infall velocity. The all-sky CHVC sample has been plotted in this way in Fig. 10. Both the sample size and sky coverage are significantly enhanced relative to what was available to Braun & Burton. A systematic trend now becomes apparent in the CHVC kinematics. While the CHVCs at negative $\cos(\theta)$ (predominantly in the southern hemisphere) tend to lie within the envelope defined by the LGSR solar apex and the Local Group velocity dispersion, the CHVCs at positive $\cos(\theta)$ have a large negative offset from this envelope. Obscuration by Galactic H  may well be important in shaping this trend. Only by analyzing realistic model populations and subjecting them to all of the selection and sampling effects of the existing surveys can meaningful conclusions be drawn.

  \begin{figure}
\par {\includegraphics[width=16.6cm,clip]{ms2407f23.ps} }
\end{figure} Figure 23: Predicted distribution on the sky of detected synthetic clouds corresponding to the model #9 simulation of a CHVC population in the Local Group, contrasting LDS and HIPASS sensitivities; the parameters of the simulation are given in Table 4. The black dots correspond to objects that exceed the LDS and HIPASS detection threshold and are not obscured by Galactic H  I. The red dots are predicted additional detections if the HIPASS sensitivity were extended to the northern hemisphere. The superposed grey-scale image shows, as in Fig. 5, the observed H  column depths, following from an integration of observed temperatures over velocities ranging from $V_{\rm LSR} = -450\rm\;km\;s^{-1}$ to $+400\rm\;km\;s^{-1}$, but excluding all gas with $V_{\rm DEV} < 70\rm\;km\;s^{-1}$. The boundary between the LDS regime at $\delta >0\hbox {$^\circ $ }$ and the HIPASS regime at lower declinations is evident in the grey-scale image. A much higher CHVC concentration, relative to that currently observed in Fig. 5, is predicted in the direction of the Local Group barycenter at this increased sensitivity. (This figure is available in color in electronic form.)

Recently several simulations have been performed to test the hypothesis that the CHVCs are the remaining building blocks of the Local Group. Putman & Moore (2002) compared the results of the full N-body simulation described by Moore et al. (2001) with various spatial and kinematic properties of the CHVC distribution, as well as with properties of the more general HVC phenomenon, without regard to object size and degree of isolation. Putman & Moore were led to reject the Local Group deployment of CHVCs, for reasons which we debate below. Blitz et al. (1999) performed a restricted three body analysis of the motion of clouds in the Local Group. In their attempt to model the HVC distributions, Blitz et al. modeled the dynamics of dark matter halos in the Local Group and found support for the Local Group hypothesis when compared qualitatively with the deployment of a sample of anomalous-velocity H  containing most HVCs, but excluding the large complexes (including the Magellanic Stream) for which plausible or measured distance constraints are available. Assuming that 98% of the Local Group mass is confined to the Milky Way and M 31 and their satellites, Blitz et al. described the dynamics of the Local Group in a straightforward manner. Driven by their mutual gravity and the tidal field of the neighboring galaxies, the Milky Way and M 31 approach each other on a nearly radial orbit. The motion of the dark matter halos which fill the Local Group was determined by the gravitational attraction of the Milky Way and M 31, and the tidal field of the neighboring galaxies. All halos which ever got closer than a comoving distance of 100 kpc from the Milky Way or M 31 center were removed from the sample. Blitz et al. describe how their simulations account for several aspects of the HVC observations. We follow here the modeling approach of Blitz et al., but judge the results of our simulations against the properties of the CHVC sample, viewing the simulated data as if it were observed with the LDS and HIPASS surveys.

4.1 Model description

We use a test particle approach similar to that used by Blitz et al. (1999) to derive the kinematic history of particles as a function of their current position within the Local Group, but combine this with an assumed functional form (rather than simply a uniform initial space density) to describe the number density distribution of the test particles. The density function contains a free parameter which sets the degree of concentration of the clouds around the Milky Way and M 31. By determining a best fit of the models we are able to constrain the values of this concentration parameter, and thereby constrain the distance to the CHVCs. The fits depend upon the derived velocity field and the H  properties of the clouds.

  \begin{figure}
{\includegraphics[width=11.3cm,clip]{ms2407f24.ps} }
\end{figure} Figure 24:  Demonstration of the quality of the fits of the Galactic Halo models discussed in Sect. 5. The upper panels show the column-density fits; the lower panels show the size fits. The panels on the left represent one of the better fits, characterized by $\chi ^2 ({\rm size}) = 1.7$ and $\chi ^2 ({N_{\rm HI}}) = 0.7$; the panels on the right show one of the poorer fits in the acceptable category, characterized by $\chi ^2 ({\rm size}) = 4.9$ and $\chi ^2 ({N_{\rm HI}}) = 4.9$. The observed distributions are indicated by the red histograms, against which the simulations are judged. (This figure is available in color in electronic form.)

The density fields which we use are a sum of two Gaussian distributions, centered on the Galaxy and M 31, respectively. As a free parameter we use the radial dispersion of these distributions. This free parameter has the same value for both the concentration around the Galaxy and that around M 31. The ratio of the central densities of the distributions at M 31 and the Galaxy must also be specified. We set this ratio equal to the mass ratio of the two galaxies. The Gaussian dispersions which are used in the models range from 100 kpc to 2 Mpc; i.e. the distributions range from very tightly concentrated around the galaxies to an almost homogeneous filling of the Local Group.

  \begin{figure}
\par {\includegraphics[width=17cm,clip]{ms2407f25.ps} }
\end{figure} Figure 25: Summary of the simulated spatial and kinematic properties of the CHVC ensemble, for one of the better fits of the empirical Galactic Halo model described in Sect. 5. The panels give the properties of the simulation in the same way as the panels in Fig. 9 describe the observed data. This Galactic Halo simulation is determined by the following parameters: $M_0 =
10^4\;M_\odot$, $n_0 = 3\times10^2\rm\;cm^{-3}$, $\beta = -2.0$, and $\sigma_{\rm d} = 30\rm\;kpc$. The quality of the fit is described by $\chi ^2 ({\rm size}) = 2.3$ and $\chi ^2 ({N_{\rm HI}}) = 2.6$. The thick-line histograms indicate the LDS (northern hemisphere) contributions to the total detections.

The CHVC kinematics are simulated by tracking the motions of small test particles within the gravitational field of the Milky Way, the M 31 system, and the nearby galaxies. Both the description of the tidal field that is produced by the nearby galaxies, and the properties of the Galaxy-M 31 orbit, are taken from the analysis of Raychaudhury & Lynden-Bell (1989), who studied the influence of the tidal field on the motion of the Galaxy and M 31, deriving the dipole tidal field from a catalog of galaxies compiled by Kraan-Korteweg (1986). The motions of the Galaxy and M 31 are determined in a simulation. In this simulation M 31 and the Galaxy are released a short time after the Big Bang. The initial conditions are tuned in such a way that the relative radial velocity and position correspond with the values currently measured.

  \begin{figure}
\par {\includegraphics[width=17.3cm,clip]{ms2407f26.ps} }
\end{figure} Figure 26: Summary of the simulated spatial and kinematic properties of the CHVC ensemble, for one of the better fits of the empirical Galactic Halo model described in Sect. 5. The panels give the properties of the simulation in the same way as the panels in Fig. 9 describe the observed data. This Halo simulation is determined by the following parameters: $M_0 =
10^5\;M_\odot$, $n_0 = 3\times10^{-3}\rm\;cm^{-3}$, $\beta = -1.4$, and $\sigma_{\rm d} = 200\rm\;kpc$. The quality of the fit is described by $\chi ^2 ({\rm size}) = 2.9$ and $\chi ^2 ({N_{\rm HI}}) = 2.5$. The thick-line histograms indicate the LDS (northern hemisphere) contributions to the total detections. Although this Halo model is fundamentally different from the Local Group models, the characteristic distance of the simulated CHVCs is similar.

We track not only the motions of M 31 and the Galaxy, but also the motions of a million test particles. The test particles are, together with M 31 and the Milky Way, released a short time after the Big Bang with a velocity equal to that of the Hubble flow. Initially, the test particles homogenously fill a sphere with a comoving radius of 2.5 Mpc. Their motions are completely governed by the gravitational field of M 31 and the Milky Way and by the tidal field of the nearby galaxies. The result of the simulation at the present age of the Universe is used to define the kinematic history as a function of current 3-D position within the Local Group for our simulated CHVC populations. For every 3-D position where an object is to be placed by our assumed density distribution, we simply assign the kinematic history from the test particle in the kinematic simulation which ended nearest to that 3-D position. The most important aspects of the kinematic history are merely the final space velocity vector as well as the parameters of the closest approach of the test particle to M 31 and the Milky Way, where the effects of ram pressure and tidal stripping will be assessed.

The parameters which determine the outcome of the simulation are the Hubble constant, H, the average density of the Universe, $\Omega_0$, the total mass of the Local Group, $M_{\rm LG}$, (= $M_{\rm M~31}$+ $M_{\rm
MW}$), and the mass-to-light ratio of the nearby galaxies, which make up the tidal field. The Hubble constant and the average density of the Universe not only set the age of the Universe, but also the initial velocities of the test particles. Further evolution is independent of these parameters. The evolution is set by the values of the tidal field and the masses of M 31 and the Galaxy. Values for all these parameters were taken from Raychaudhury and Lynden-Bell (1989), namely: $H = 50 \rm\;km\;s^{-1}$, $M_{\rm LG} =
4.3\times10^{12}\;M_\odot$, a mass-to-light ratio of 60, $\Omega_0
= 1$, and $M_{\rm M~31} / M_{ \rm MW} = 2$. Whereas the mutual gravitational attraction between the Milky Way and M 31 is described by a point-mass potential, the gravitational attraction of these galaxies on the test particles is described by an isochrone potential of the form

\begin{displaymath}\Phi_{\rm iso} (r) = - \frac{{G} \; M}{r_0 + \sqrt{{r_0}^2 + r^2}},
\end{displaymath} (4)

where M is the total mass of the galaxy and r0 is a characteristic radius; r0 is set such that the rotation velocity as derived from the potential equals the measured one at the edge of the unwarped H  disk, i.e. $V_{\rm circ}^{\rm MW} (12 \rm\;kpc) = 220\;km\;s^{-1}$ and $V_{\rm circ}^{\rm M~31} (16 \rm\;kpc) = 250\;km\;s^{-1}$. Figure 11 shows the average velocity field superposed on density contours for a Gaussian distribution of the test particles, characterized by a dispersion of $200\rm ~kpc$. The ellipsoidal turn-around surface of the Local Group can be seen where the velocity vectors approach zero length at radii near 1.2 Mpc. The velocity field is approximately radial at large distances from both M 31 and the Galaxy, but becomes more complex at smaller radii.
  \begin{figure}
\par\includegraphics[width=7.5cm,clip]{ms2407f27.ps}\end{figure} Figure 27: Histogram of object distances for one of the best-fitting Local Group models. The detected objects in model #9 of Table 4 are plotted in the histogram. A broad peak in the distribution extends from about 200 to 450 kpc, with outliers as far as 1 Mpc. The filled circles along the top of the figure are the distance estimates for individual CHVCs of Braun & Burton (2000) and Burton et al. (2001), based on several different considerations. While few in number, they appear consistent with this distribution.


  \begin{figure}
\includegraphics[width=7.6cm,clip]{ms2407f28.ps}\end{figure} Figure 28: Comparison of a model Local Group CHVC population with the population of Local Group galaxies. The distribution of H  masses of one of our best-fitting Local Group models, model #9 of Table 4, is plotted as a thin-line histogram after accounting for ram-pressure and tidal stripping and as a thick line after also accounting for Galactic obscuration and the finite sensitivities of the LDS and HIPASS observations. The Local Group galaxies (excluding M 31 and the Galaxy) tabulated by Mateo (1998) are plotted as the hatched histogram after summing the H  mass with the stellar mass assuming $M/L_B
=3~M_\odot/L_\odot$. The diagonal line has the slope of the H  power-law mass function, $\beta =-1.7$.

Before we can simulate the way in which a Local Group population of clouds would be observed by a HIPASS- or LDS-like survey, we have to set the H  properties of the test clouds. To do so, we assume that the H  clouds are isothermal gas spheres, with each such cloud located inside a dark-matter halo. Given the temperature of the gas and the potential in which it resides, the density profile follows from the relation

 \begin{displaymath}n(r) = n_0 \cdot \exp
\left(
- \frac{m_{\rm HI} } { {k} T_{\rm eff} }
\left[
\Phi(r) - \Phi(0)
\right]
\right),
\end{displaymath} (5)

where $m_{\rm HI}$ is the mass of the hydrogen atom, $\Phi(r)$ is the potential at a distance r from the cloud center, and $T_{\rm
eff}$ is an effective gas temperature. Since in addition to the thermal pressure there is also rotational support of a gas cloud against the gravitational attraction of the dark-matter halo, an effective temperature is used which incorporates both processes. In general, the average energy of an atom equals   $\frac{1}{2} kT$ per motional degree of freedom, so we have defined the effective temperature such that

 \begin{displaymath}\frac{3}{2} {k} T_{\rm eff} =
\frac{3}{2} {k} T_{\rm kin} + \frac{1}{2} {m}_{\rm HI}
V^2_{\rm circ},
\end{displaymath} (6)

where $T_{\rm kin}$ is the gas kinetic temperature, taken to be 8000 K, and $V_{\rm circ}$ is a characteristic rotation velocity.

The description of the gravitational potential of the dark matter halo follows that of Burkert (1995), who was able to fit a universal function to the rotation curves of four different dwarf galaxies. The shape of the function is completely set by the amount of dark matter in the core, M0. The potential, which is derived from the rotation curve, has the form

\begin{eqnarray*}\Phi(r) -\Phi(0) & = & -\pi \; {G} \; \rho_0 \; r_0^2 %
\left\{...
...} \right)
\cdot \ln \left( 1 + (r / r_0)^2 \right) \; \right\},
\end{eqnarray*}


where the core radius, r0, and the central density, $\rho_0$, are set by the relations

\begin{eqnarray*}r_0 & = & 3.07 \; \left( \frac{M_0}{10^9\;M_\odot}
\right)^{3/...
...( \frac{M_0}{10^9\;M_\odot}
\right)^{-2/7} \rm\;g\;cm^{-3}. \\
\end{eqnarray*}


The circular-velocity rotation curve has a maximum value

\begin{displaymath}V_{\rm circ}^{\rm max} = 48.7 \; \left( \frac{M_0}{10^9 ~ M_\odot}
\right)^{(2/7)}\rm\;km\;s^{-1}.
\end{displaymath} (7)

We use $V_{\rm circ}^{\rm max}$ as a parameter for $V_{\rm circ}$ in Eq. (6). Because the total mass corresponding to the given potential is infinite, the dark matter mass is characterised either by the core mass, M0, or by the virialized mass of the halo, $M_{\rm vir}$. We adopt a total dark-matter mass of $M_{\rm vir}$ for each object. According to Burkert (1995), these are related by $M_{\rm vir} = 5.8\;M_0$.

Although we could use Eq. (5) directly to determine the predicted 21-cm images of a CHVC given an H  mass, we instead chose to approximate the corresponding column-density distribution by a Gaussian, in order to enable faster evaluation of the simulation. Two parameters specify the Gaussian, namely the central density, n0, and the FWHM, derived as follows. The volume-density distribution given by Eq. (5), can be closely approximated with an exponential form with a matched scale-length, $h_{\rm e}$, defined by $n(h_{\rm e})=n(0)/e$. The column density distribution can then be expressed in an analytic form, containing a modified Bessel function of order 1 (e.g. Burton et al. 2001). This analytic representation of the column-density distribution is then approximated by a Gaussian of the same halfwidth from, ${\it FWHM}=2.543 h_{\rm e}$. Knowing the mass of the H  gas cloud and its FWHM, the central density of the Gaussian can be determined.

To get the amount of H  mass in the cloud, we adopt the relations between the dark-matter mass and baryonic mass for galaxies. For normal, massive galaxies there is approximately ten times as much dark matter as there is baryonic matter. Chiu et al. (2001) show that this ratio depends on the total mass. Whereas the mass spectrum of the dark-matter halos in their simulation has the form $n(M_{\rm
dark}) \propto M_{\rm dark}^{-2}$, the baryonic mass spectrum has the form $n(M_{\rm HI}) \propto M_{\rm HI}^{-1.6}$. The difference in slope is due to the ionizing extragalactic radiation field. The lowest mass halos are simply unable to retain their ionized baryonic envelopes, which have a kinetic temperature of 104 K. The slopes of both the baryonic and the dark-matter distributions completely determine the mass dependency of the ratio between dark matter and baryonic matter. Furthermore, given the fact that the ratio equals 10 for objects with a baryonic mass of  $10^9\;M_\odot$, the ratio is set for all masses. Although the simulation of Chiu et al. gives a value of -1.6 for the baryonic mass spectrum slope, we explore a range of values for the H  mass spectrum. The most appropriate value is then obtained by fitting the models to the observations.

For the sake of simplicity, we assume that the baryonic mass of each simulated cloud is entirely in the form of H I. In fact, a significant mass fraction will be in the form of ionized gas. It is likely that the mass fraction of ionized gas will increase toward lower masses such that below some limiting mass the objects would be fully ionized. A realistic treatment of the ionized mass fraction was deemed beyond the scope of this study. However, we do comment further on the implications of this simplifying assumption where appropriate.

The definitions of the velocity and density fields of the test objects in the Local Group, together with their H  properties, resulted in simulated CHVC populations which could be sampled with the observational parameters of an LDS- or HIPASS-like survey. The free parameters, defined above, were allowed to take the following values:

Figure 12 shows the basic cloud properties as a functions of H  mass. As indicated above, the dark-matter fraction as function of mass is determined by $\beta $, the slope of the H  mass spectrum, so three curves are shown in each panel, corresponding to $\beta =-1.2$, -1.6, and -2.0, respectively. The typical object size and internal velocity dispersion increases only slowly with H  mass, from about 1.5 to 4 kpc, and 10 to 20 km s-1, respectively, between $M_{\rm HI}=10^5$ and $10^8~ M_{\odot}$. The central H  volume density varies much more dramatically with H  mass, as does the peak column density. Note that the peak column densities of simulated clouds only exceed $N_{\rm HI}>10^{19}$ cm-2 for $M_{\rm
HI}>10^{5.5}~M_\odot$ and $\beta $ in the range -1.6--2.0. It is critical that peak column densities of this order are achieved in long-lived objects, since this is required for self-shielding from the extragalactic ionizing radiation field (e.g. Maloney 1993; Corbelli & Salpeter 1993).

In order to compare the simulation results with the observational data in the most effective way, and thereby constrain the model parameters, we created a single CHVC catalog from the HIPASS and LDS ones. Thirty-eight CHVCs at $\delta \ge 0^\circ$ were extracted from the de Heij et al. (2002) LDS catalog, and 179 at $\delta <
0^\circ$ from the Putman et al. (2002) HIPASS one. A large concentration of faint sources (Group 1 noted above) with an exceptionally high velocity dispersion is detected toward the Galactic south pole. Because this overdensity may well be related to the nearby Sculptor group of galaxies (see Putman et al. 2002), the 53 CHVCs at $b < -65^\circ$ were excluded from comparison with the simulations.

A simulation was run for each set of model parameters. We chose a position for each object, in agreement with the spatial density distribution of the ensemble, but otherwise randomly. The velocity is given by the velocity field described above. The H  mass of the test cloud is randomly set in agreement with the given power-law mass distribution between the specified upper mass limit and a lower mass limit described below. The physical size and linewidth of each object follow from the choice of $\beta $. Once all these parameters were set, we determined the observed peak column density and angular size. Objects in the northern hemisphere were convolved with a beam appropriate to the LDS, while those at $\delta<0\hbox{$^\circ$ }$ were convolved with the HIPASS beam. Simulated objects were considered detected if the peak brightness temperature exceeded the detection threshold of the relevant survey, i.e. the LDS for objects at $\delta >0\hbox {$^\circ $ }$, and HIPASS otherwise. Furthermore, in order for a test object to be retained as detected its deviation velocity was required to exceed $70\rm\;km\;s^{-1}$ in the LSR frame, and its Galactic latitude to be above $b = -65^\circ$ (for consistency with the exclusion of Group 1 from our CHVC sample noted above). In addition, as we describe below, each simulated cloud should be stable against both tidal disruption and ram-pressure stripping by the Milky Way and M 31. We continued simulating additional objects following this prescription until the number of detected model clouds was equal to the number of CHVCs in our observed all-sky sample.


  \begin{figure}
\includegraphics[width=11cm,clip]{ms2407f29.ps}\end{figure} Figure 29: External appearance of a model Local Group CHVC population. The spatial distribution of one of our best-fitting Local Group models, namely model #9 of Table 4, is shown projected onto a plane as in Fig. 20. The grey dots correspond to objects which have been disrupted by ram-pressure or tidal stripping during a close passage. The red and black dots correspond to are objects which should survive these processes, with the color of the dot distinquishing objects by H  mass. The red dots indicate clouds more massive than $M_{\rm HI}=3\times10^6~M_\odot$; the black dots indicate CHVCs falling below this mass limit. (This figure is available in color in electronic form.)

Before carrying out each simulation with a power-law distribution of H  masses we began by determining an effective lower H  mass limit, M0, to the objects that should be considered. This was necessary to avoid devoting most of the calculation effort to objects too faint to be detected in any case. As a first guess we took M0=0.5 M1. A sub-sample of twenty objects in the mass range M0-M1 was simulated which were deemed stable to both disruption and stripping and were detectable with the relevant survey parameters. Generating twenty detectable objects typically required evaluating of order 500 test objects. Given the total number of test objects needed to generate this observable sub-sample and the slope of the H  mass distribution function, it is possible to extrapolate the number of required test objects in other intervals of H  mass belonging to this same distribution. The predicted number of test objects in the interval 0.67 M0 to M0 was simulated. If at least one of these was deemed detectable, then the lower H  mass limit was replaced with 0.67 M0 and the procedure outlined above was repeated. This process continued until no detectable object was found in the mass interval 0.67 M0 to M0. Tests carried out with better number statistics, involving a sub-sample size of 180 objects and requiring a minimum of nine detections in the lowest mass interval, demonstrated that this procedure was robust.

Figure 13 shows the distance out to which simulated CHVCs of a given H  mass can be detected with the HIPASS survey. Both the H  linewidth and spatial FWHM are dependent on the dark-matter fraction, as illustrated in Fig. 12, so separate curves are shown for $\beta =-1.2$, -1.6, and -2.0. A limiting case is provided by $\beta =-1.2$ which is extremely dark-matter dominated for low H  mass. In this case the objects are so spatially extended (about 5 kpc FWHM) and have such a high linewidth (about 60 km s-1 FWHM) that they fall below the HIPASS detection threshold for log( $M_{\rm HI}) < {\sim}6.4$. More plausible linewidths and spatial FWHM follow for $\beta = -1.6$ and -2.0. Such objects are sufficiently concentrated that they can still be detected, even when highly resolved in the HIPASS data.

The clouds are regarded stable against ram-pressure stripping if the gas pressure at the center of the cloud exceeds the ram pressure, $P_{\rm ram} = n_{\rm halo} \cdot V^2$, for a cloud moving with velocity V through a gaseous halo with density $n_{\rm halo}$. Because both the gaseous halo density and the cloud velocity are the highest if the distance of the cloud to the galaxy is the smallest, the stability against ram-pressure stripping was evaluated at closest approach. We therefore kept track of the closest approach of each test-particle to the Galaxy and to M 31, while simulating the velocity field of the Local Group. We used a density profile for the Galactic halo which is an adaptation of the model of Pietz et al. (1998), derived to explain the diffuse soft X-ray emission as observed by ROSAT. Whereas their model is flattened towards the Galactic plane, we simply use a spherical density distribution, in which the radial profile is equal to the Galactic plane density profile of Pietz et al. The density at a distance r from the Galactic center is given by

\begin{displaymath}n(r) = n_0 \cdot \left(
\frac{\cosh(r_\odot / h)} {\cosh(r / h)}
\right)^2,
\end{displaymath} (8)

where $n_0=0.0013~{\rm cm}^{-3}$ is the central density, h=12.5 kpc is the scalelength of the distribution, and $r_\odot=8.5$ kpc is the radius of the solar orbit around the Galactic center. According to this model, the total mass in the Galactic halo is $1.5\times10^9\;M_\odot$. To describe the halo around M 31, we use the same expression and the same parameter values except for n0, for which we use a value twice the Galactic one. Figure 14 shows an example of the calculated distance at which a cloud of a particular H  mass will be stripped. The clouds in this example are assumed to have a relative velocity of  $200\rm\;km\;s^{-1}$ with respect to the Galactic halo.

A cloud will be tidally disrupted if the gravitational tidal field of either the Galaxy or M 31 exceeds the self-gravity of the cloud. We consider a cloud stable if

 \begin{displaymath}\frac{M_{\rm dark}}{\sigma^2} \ge
\left\vert \frac{\rm d^2}{{\rm d} r^2} \Phi_{\rm iso} (r) \right\vert
\cdot \sigma,
\end{displaymath} (9)

where $\sigma$ is the spatial dispersion of the Gaussian describing the H  distribution in the cloud, $M_{\rm dark}(r\le\sigma)$ is the core mass of the dark matter halo, and $\vert{\rm d}^2 \Phi_{\rm iso} (r) / {\rm d} r^2\vert$is the tidal force of either the Galaxy or M 31. Solving the equation for r shows that only the least massive clouds with the lowest dark-matter fractions are likely to suffer from tidal disruption. If the slope of the H  mass distribution is as steep as -2.0, then clouds with an H  mass less than $10^5\;M_\odot$ are tidally disrupted at distances of about 60 kpc, as shown in Fig. 15. For $M_{\rm
HI} > 2\times10^5\;M_\odot$, or $\beta>{-}2$, the clouds are stable. Changing the radius at which Eq. (9) is evaluated from $1\sigma$ to $2 \sigma$, does not dramatically change this result.

4.2 Results of the Local Group simulations

Before searching for a global best fit, we determined the range of parameter values over which at least a moderately good representation of the observed data was possible with the simulated data. To quantify the degree of agreement between the simulated size-, column-density and velocity-FWHM distributions with the observations, we used a $\chi ^2$-test taken from Sect. 14.3 of Numerical Recipes (Press et al. 1993). The size-, column-density, and velocity-FWHM distributions of a simulation were considered reasonable if $\chi^2
({\rm size})<3$, $\chi^2 ({N_{\rm HI}})<5$, and $\chi^2 ({\it FWHM})<5$. The incorporation of the spatial and kinematic information was done by comparing the modeled (l,b), $(l,V_{\rm GSR})$, and $(V_{\rm GSR},b)$ distributions with those observed. We used the two-dimensional K-S test described in Sect. 14.7 of Numerical Recipes to make this comparison. The fits were considered acceptable if $\tilde\chi^2 (l,b)$, $\tilde\chi^2 (l,V_{\rm GSR})$, and $\tilde\chi^2
(V_{\rm GSR},b)$ where all less than 0.3.

Table 3 shows which part of the parameter space produces moderately good fits. The best fits have a Gaussian dispersion between 150 and $250\rm\;kpc$, an upper H  mass cut-off between 106.5 and $10^{8.0}\;M_\odot$ and a slope of the H  mass distribution of -1.7 to -1.9.


 

 
Table 3: Tabulation of the range of parameters entering the Local Group models described in Sect. 4. Three free parameters were specified for each simulation, namely the maximum allowed H  mass, M1 (in units of $M_\odot $), the slope of the H  mass distribution, $\beta $, and the Gaussian dispersion of the cloud population, $\sigma _{\rm d}$ (in units of kpc). The simulated CHVC populations were subjected to the same observational constraints as pertain to the LDS and HIPASS surveys. The simulated spatial and kinematic deployments, as well as the simulated size-, column density- and velocity- FWHM distributions, were compared with those observed. The table shows which Gaussian density dispersions produce acceptable results for each of the combinations of M1 and $\beta $. A field is blank if the simulation returned no acceptable fit for the given parameter combinations. Fits were deemed acceptable if $\chi^2~{\rm size} < 3$, $\chi ^2{N_{\rm HI}} < 5$, $\chi ^2\ {\it FWHM} < 5$, $\tilde\chi^2{(l,b)} < 0.3$, $\tilde\chi^2{(l,V_{\rm GSR})} < 0.3$, and $\tilde\chi^2{(V_{\rm GSR},b)} < 0.3$.
  M1= $10^{6}\;M_\odot$ $10^{6.5}\;M_\odot$ $10^7\;M_\odot$ $10^{7.5}\;M_\odot$ $10^{8}\;M_\odot$ $10^{8.5}\;M_\odot$ $10^9\;M_\odot$
$\beta =-1.2$              
-1.4              
-1.6              
-1.7   $\sigma_{\rm d}=150\ldots200$ kpc 150 $\;\ldots\;$250 150 $\;\ldots\;$200 150 $\;\ldots\;$200    
-1.8     $150\ldots250$ $150\ldots250$ $150\ldots250$    
-1.9       $200\ldots250$ $200\ldots250$    
-2.0              


Each simulation contains a relatively small number of detectable objects, namely the same number of objects as in our all-sky CHVC catalog. Therefore the $\chi ^2$ values are prone to shot-noise. By performing a larger number of simulations for a specific combination of free parameters, we are able to determine a more representative value of $\chi ^2$ for each model. The most promising combinations of parameter values, i.e. the entries in Table 3, were repeated 35 times to reduce the shot-noise, and the average results and their dispersions are shown in Table 4. The range of resulting fit quality due purely to this shot-noise is illustrated in Fig. 16, which shows model data with the highest and lowest $\chi ^2$ values from a sequence of 35 simulations.

The best overall fits are fairly well-constrained to lie between $\sigma_{\rm d}=150$ and $200\rm\;kpc$, with an upper H  mass cut-off of about 107 to $10^{7.5}\;M_\odot$ and a slope of the H  mass distribution of -1.7 to -1.8. Comparison with Fig. 12 suggests that populations of these types have sufficiently high peak H  column densities that they can provide self-shielding to the extragalactic ionizing radiation field for $M_{\rm HI}>10^{5.5}\;M_\odot$. The results of simulations #9 and #3 from Table 4 are shown in Figs. 17 and 18, respectively. A single instance of each simulation has been used in the subsequent figures that had $\chi ^2$ values consistent with the ensemble average.


 

 
Table 4: Average chi-square values and their dispersions for the 35 runs which were performed for each model of the Local Group deployment of CHVCs. Each model was specified by the indicated three parameters, namely the Gaussian dispersion of the cloud population, $\sigma _{\rm d}$, the maximum allowed H  mass, M1, and the slope of the H  mass distribution, $\beta $. Multiple runs yielded better estimates of each $\chi ^2$, reducing the sensitivity to the relatively small number of objects in each individual simulation. The output of each model was sampled with the observational parameters of the LDS and HIPASS surveys and compared with the observed data.
Model $\sigma _{\rm d}$ M1 $\beta $ $\chi^2({\rm size})$ $\chi^2({N_{\rm HI}})$ $\chi^2({\it FWHM})$ $\tilde\chi^2 (l,b)$ $\tilde\chi^2(l,V{\rm GSR})$ $\tilde\chi^2(V{\rm GSR},b)$
# (kpc) $({M}_\odot$)              
1 150 106.5 -1.7 2.0 $\pm$ 0.6 1.9 $\pm$ 0.3 4.0 $\pm$ 0.4 0.29 $\pm$ 0.03 0.25 $\pm$ 0.02 0.29 $\pm$ 0.03
2 150 107.0 -1.7 2.2 $\pm$ 0.5 2.3 $\pm$ 0.4 3.5 $\pm$ 0.4 0.27 $\pm$ 0.03 0.23 $\pm$ 0.03 0.28 $\pm$ 0.03
3 150 107.5 -1.7 2.6 $\pm$ 0.6 2.7 $\pm$ 0.4 3.9 $\pm$ 0.6 0.25 $\pm$ 0.04 0.21 $\pm$ 0.03 0.25 $\pm$ 0.03
4 150 108.0 -1.7 3.0 $\pm$ 0.6 3.0 $\pm$ 0.5 3.9 $\pm$ 0.5 0.24 $\pm$ 0.03 0.21 $\pm$ 0.03 0.24 $\pm$ 0.02
5 150 107.0 -1.8 2.0 $\pm$ 0.5 2.4 $\pm$ 0.5 4.5 $\pm$ 0.6 0.27 $\pm$ 0.04 0.23 $\pm$ 0.02 0.28 $\pm$ 0.03
6 150 107.5 -1.8 2.4 $\pm$ 0.5 2.6 $\pm$ 0.5 4.1 $\pm$ 0.5 0.25 $\pm$ 0.04 0.21 $\pm$ 0.03 0.24 $\pm$ 0.02
7 150 108.0 -1.8 2.7 $\pm$ 0.6 2.7 $\pm$ 0.4 4.0 $\pm$ 0.6 0.24 $\pm$ 0.03 0.21 $\pm$ 0.03 0.24 $\pm$ 0.03
8 200 106.5 -1.7 2.4 $\pm$ 0.5 2.9 $\pm$ 0.3 4.9 $\pm$ 0.6 0.29 $\pm$ 0.04 0.28 $\pm$ 0.02 0.30 $\pm$ 0.02
9 200 107.0 -1.7 2.3 $\pm$ 0.5 2.9 $\pm$ 0.3 4.1 $\pm$ 0.5 0.28 $\pm$ 0.04 0.24 $\pm$ 0.02 0.29 $\pm$ 0.03
10 200 107.5 -1.7 2.9 $\pm$ 0.6 3.0 $\pm$ 0.5 4.8 $\pm$ 0.7 0.26 $\pm$ 0.04 0.22 $\pm$ 0.03 0.26 $\pm$ 0.03
11 200 108.0 -1.7 3.0 $\pm$ 0.5 3.1 $\pm$ 0.5 4.9 $\pm$ 0.6 0.26 $\pm$ 0.04 0.22 $\pm$ 0.03 0.25 $\pm$ 0.03
12 200 107.0 -1.8 2.2 $\pm$ 0.5 3.6 $\pm$ 0.5 4.2 $\pm$ 0.5 0.28 $\pm$ 0.04 0.24 $\pm$ 0.03 0.29 $\pm$ 0.03
13 200 107.5 -1.8 2.5 $\pm$ 0.6 3.7 $\pm$ 0.4 3.8 $\pm$ 0.5 0.27 $\pm$ 0.04 0.22 $\pm$ 0.03 0.27 $\pm$ 0.03
14 200 108.0 -1.8 2.7 $\pm$ 0.5 3.7 $\pm$ 0.5 3.6 $\pm$ 0.5 0.26 $\pm$ 0.04 0.22 $\pm$ 0.02 0.26 $\pm$ 0.04
15 200 107.5 -1.9 2.3 $\pm$ 0.5 4.0 $\pm$ 0.5 4.8 $\pm$ 0.7 0.26 $\pm$ 0.04 0.22 $\pm$ 0.03 0.27 $\pm$ 0.04
16 200 108.0 -1.9 2.4 $\pm$ 0.5 4.0 $\pm$ 0.5 4.3 $\pm$ 0.8 0.26 $\pm$ 0.05 0.22 $\pm$ 0.02 0.26 $\pm$ 0.04
17 250 107.0 -1.7 2.3 $\pm$ 0.6 4.3 $\pm$ 0.6 4.8 $\pm$ 0.7 0.28 $\pm$ 0.04 0.24 $\pm$ 0.02 0.29 $\pm$ 0.03
18 250 107.0 -1.8 2.2 $\pm$ 0.5 5.2 $\pm$ 0.6 4.1 $\pm$ 0.5 0.26 $\pm$ 0.04 0.23 $\pm$ 0.02 0.27 $\pm$ 0.03
19 250 107.5 -1.8 2.5 $\pm$ 0.6 5.1 $\pm$ 0.6 3.8 $\pm$ 0.5 0.25 $\pm$ 0.03 0.22 $\pm$ 0.02 0.26 $\pm$ 0.03
20 250 108.0 -1.8 2.7 $\pm$ 0.4 5.3 $\pm$ 0.7 3.6 $\pm$ 0.6 0.25 $\pm$ 0.04 0.22 $\pm$ 0.02 0.25 $\pm$ 0.03
21 250 107.5 -1.9 2.3 $\pm$ 0.5 5.6 $\pm$ 0.6 4.8 $\pm$ 0.6 0.26 $\pm$ 0.03 0.22 $\pm$ 0.02 0.26 $\pm$ 0.03
22 250 108.0 -1.9 2.5 $\pm$ 0.6 5.6 $\pm$ 0.6 4.3 $\pm$ 0.6 0.25 $\pm$ 0.03 0.22 $\pm$ 0.02 0.25 $\pm$ 0.03


Both of these Local Group simulations succeed reasonably well in reproducing the observed kinematic and population characteristics of the CHVC sample as summarized in Fig. 9. The CHVC concentrations, named Groups 2, 3, and 4 above, while not reproduced in detail, have counterparts in the simulations which arise from the combination of Gaussian density distributions centered on the Galaxy and M 31, together with the Local Group velocity field, population decimation by disruption effects, and the foreground H  obscuration. A notable success of these simulations is their good reproduction of the smoothed velocity field, including both the numerical values and the location of minima and maxima.

One aspect of the observed CHVC distributions which can not be reproduced accurately by our simulations is the distribution of observed linewidth. The model objects are assumed to contain only the warm component of H  with a minimum linewidth corresponding to a 8000 K gas. The profiles are then further broadened by the contribution of rotation as indicated in Eq. (6). The actual objects are known to have cool core components (e.g. Braun & Burton 2001; Burton et al. 2001) which can contribute a significant fraction of the H  mass and consequently lead to narrower observed line profiles. This shortcoming of the model distributions is illustrated in the central panel of Fig. 16. The narrow-linewidth tail of the observed distributions can never be reproduced by the models. The best-fitting models can only succeed in reproducing the median value and high-velocity tail of the distribution.

In order to better isolate the effect of foreground obscuration from the intrinsic distribution properties of the simulation itself we show the unobscured version of model #9 in Fig. 19. Comparison of Figs. 17 and 19 illustrates how the foreground obscuration from the H  Zone of Avoidance modifies the distribution of object density. The location of apparent object concentrations are shifted and density contrasts are enhanced. The comparison also reveals that the large gradient in the smoothed GSR velocity field is an intrinsic property of the Local Group model and not simply an artifact of the Galactic obscuration. Substantial negative velocities (<-100 km s-1) are predicted in the direction of M 31 (which effectively defines the Local Group barycenter), while slightly positive velocities are predicted in the anti-barycenter direction, just as observed.

A better appreciation of the physical appearance of these Local Group models is provided in Fig. 20, where two perpendicular projections of the model #9 population are displayed. The (x,y)plane in the figure is the extended Galactic plane, with the Galaxy centered at (x,y)=(0,0) with the positive z axis corresponding to positive b. The intrinsic distribution of objects is an elongated cloud encompassing both the Galaxy and M 31, which is dominated in number by the M 31 concentration. The objects that have at some point in their history approached so closely to either of these galaxies that their H  would not survive the ram-pressure or tidal stripping are indicated by the filled black circles. Cloud disruption appears to have been substantially more important in the M 31 concentration than for the Galaxy. The objects that are too faint to have been detected by the LDS or HIPASS observations, depending on declination, are indicated by grey circles. The bulk of the M 31 sub-concentration is not detected in our CHVC sample for two reasons:  (1) these objects have a larger average distance than the objects in the Galactic sub-concentration, and (2) the M 31 sub-concentration is located primarily in the northern celestial hemisphere, where the lower LDS sensitivity compromises detection. We will return to this point below.

Those objects which are obscured by the H  distribution of the Galaxy are indicated in Fig. 20 by open red circles. Somewhat counter-intuitively, the consequences of obscuration are not concentrated toward the Galactic plane, but instead occur in the plane perpendicular to the LGSR solar apex direction $(l,b)=(93^\circ,-4^\circ)$. This can be understood by referring back to our discussion in Sect. 2.2.1 and the illustration in Fig. 1. Obscuration from the position- and velocity-dependent H  Zone of Avoidance is most dramatic when the kinematic properties of a population result in overlap with $V_{\rm LSR}$ = 0 km s-1, since this can occur over a large solid angle, while the Galactic plane is relatively thin.

The various processes which influence the observed distributions are further quantified in Table 5. Matching the detected sample size of 163 CHVCs above $b = -65^\circ$, required the simulation of some 6300 objects in the case of models #9 and #3. About three quarters of the simulated populations were classified as disrupted due to ram-pressure or tidal stripping; while 80% of the remaining objects were deemed too faint to detect with the LDS (in the north) or HIPASS (in the south). Obscuration by Galactic H  eliminated about one half of the otherwise detectable objects. The total H  masses involved in these two model populations were $4.3\times10^9$ $M_\odot $and $6.4\times10^9$ $M_\odot $, respectively. In both cases, about 75% of this mass had already been consumed by M 31 and the Galaxy via cloud disruption, leaving only 25% still in circulation, although distributed over some 1200 low-mass objects.


 

 
Table 5: Model statistics for the best-fit Local Group models. The total numbers of objects and their H  masses are indicated for models #9 and #3 (see Table 4) together with how these are distributed within different categories. A detected sample of 163 objects was required in all cases after obscuration by the Galaxy and excluding the anomalous south Galactic pole region at $b,-65\hbox {$^\circ $ }$. The table also lists the number of objects classified as disrupted by ram-pressure or tidal stripping as well as those too faint to be detected by the LDS or HIPASS observations.
Fate of input CHVCs model #9 model #3
  number $M_{\rm HI}$ number $M_{\rm HI}$
  of CHVCs ( $10^8~ M_{\odot}$) of CHVCs ( $10^8~ M_{\odot}$)
Total number 6281 43 6310 64
Disrupted by ram or tide 4759 31 5178 50
Too faint to be detected 1220 6.5 831 4.4
Detectable if not obscured 302 5.4 301 10
Unobscured by H  ZoA 173 3.3 172 6.5
Unobscured, not at SGP 163 3.2 163 6.4


The crucial role of survey sensitivity in determining what is seen of such Local Group cloud populations is also illustrated in Figs. 21 and 22. The red symbols in these figures indicate objects detectable with the relevant LDS or HIPASS sensitivities, while the black symbols indicate those that remain undetected due to either limited sensitivity or obscuration. If these models describe the actual distribution of objects, then the prediction is that future deeper surveys will detect large numbers of objects at high negative LSR velocities in the general vicinity (about $60\times60^\circ$) of M 31. To make this prediction more specific, we have imagined the sensitivity afforded by the current HIPASS survey in the south extended to the entire northern hemisphere. Figure 23 illustrates the prediction. A high concentration of about 250 faint newly detected CHVCs is predicted in the Local Group barycenter direction once HIPASS sensitivity is available.


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