          Astron. Astrophys. 351, 920-924 (1999)

## 4. Total mass of the clouds in the Local Group

The accretion rate depends on the spatial density of the clouds ( ), their mean velocities ( ) and the cross section of galaxies to trap a cloud ( ). These three factors govern the dynamics: where is the mass of the accreting galaxies plus their neighbour satellites; the main components are the Milky Way and M31.

BL99 take the second and third factors as constants but this is too simplistic. The three factors are respectively:

• Density. The density can be expressed simply via: where M is the total mass of the Local group (constant, assuming that the local group does not exchange mass with any external reservoir) which includes the mass of the clouds and the galaxies ( ). Any ionized intercloud medium would make an extra contribution to the mass of the Local Group; although we do not include it explicitly here, its gravitational behaviour would be akin to that of the clouds. V is the physical volume of the Local Group, which is constant (we can assume that the Hubble expansion is neutralized on the scale of the Local Group since the cluster is gravitationally bound and collapse, if any, is very slow).
• Velocity. The velocity depends on the masses of the accreting galaxies. When the cloud approaches a centre of accretion, the average velocity of the cloud with mass is: where is the distance to the accreting galaxy of mass and , the velocity with respect to this object. depends on the mean energy of the clouds; if they have escape velocity, for a virialized system; etc. Masses of other objects have a negligible effect when the cloud is quite close to a galaxy, so, roughly, the velocity is proportional to . In our case, there are two accreting centres and the relation is more complicated. Eq. (3) applies near the mass . Near the mass , the average velocity of the clouds is: where is the distance to the accreting galaxy of mass and . Therefore, the velocity is proportional to .

In the region dominated by the barycentre rather than each individual galaxy, we can take as the equivalent mass of the centre of attraction. The first term is due to the galaxies ( ) and the second is the mass of the clouds which are inside the spherical region centred on the barycentre, and radius the distance to it. Since the distance is a variable, C (the fraction of the mass in clouds in that region: between 0 and 1) is variable, but we can take an average value for C to tipify infall of the clouds. The continuous increment of kinetic energy during infall is the sum of terms due to the galaxies and the clouds; hence, the total increment is the sum of two terms: one proportional to and another one proportional to .

Near the galaxies, making the reasonable assumption that and are constants, i.e. the ratio of the masses of the two galaxies does not change and the velocity of the galaxies with respect the barycentre is negligible compared with that of the clouds, the mean infall velocity ( ) will be proportional to , i.e. . The mean velocity is, in general, proportional to where C is an averaged quantity along the path of the clouds and lies between 0 and 1 ( is proportional to , since in the proximity of the galaxy i, , while on the rest of the path , but this is solved by dividing the expression for by the average ; so ).

The velocity may be slightly reduced by interaction with the disc, via collision of the high-velocity gas with the stationary disc gas (see calculations in Tenorio-Tagle et al. 1987, 1988; Comeron & Torra 1992, 1994). If drag forces dominate, the velocity becomes proportional to the square root of the column density. Benjamin & Danly (1997) found that intermediate-velocity clouds with low column densities may well reach terminal velocity, but HVCs should be hardly slowed down. As their time of interaction with the disc is relatively short, the effects of drag are negligible and the velocity can be calculated without taking them into account.

• Collision cross section of galaxies for the clouds. This is proportional to the square of the disc radius of the galaxy. In an essentially two-dimensional disc, such as that of a spiral galaxy, we can assume a dependence of the radius on the mass of form . This is in fact found observationally. In the plot of the radii vs. the masses of a sample of galaxies (Campos-Aguilar et al. 1993), a relation is fitted. Since M31 is an galaxy and the Milky Way is an , we can use this relationship for these objects and take the cross section to be proportional to , i.e. proportional to .

Therefore, the accretion rate follows: where K is a positive constant.

The factor falls with time thereby reducing the accretion rate for the Milky Way, but increases yielding an increase with time. Whether the net rate increases will depend on the ratio of the masses of the major galaxies to the integrated mass available for accretion in the clouds.

If the accretion rate is increasing or at least constant, as argued in the previous section, the minimum limit implied for the fractional mass of the HVCs in the Local Group can be evaluated and compared with observational parameters. The condition for the rate to be increasing with the time at the present epoch is that its derivative be greater than zero:     This gives us a minimum fraction of mass for the clouds within the Local Group which is between 40% and 50%, depending on the value of C (between 0 and 1). Although this result is approximate, it is qualitatively plausible: a low fraction of mass in clouds would not yield increasing accretion as the mean cloud density would fall more quickly than could be compensated by the increasing gravitational attraction of the accreting galaxies.

This result is in very fair agreement with the dynamical estimates of the total mass of the local group,  (Byrd et al. 1994), in which well over the half of mass is not in the baryonic masses of detectable galaxies. The HVCs would constitute much of the non-galactic mass of the Local Group, they would be at least a significant part of the dark matter in the Local Group. One can speculate that the dark matter (the difference between the total mass of the cluster and the sum of the masses of the individual galaxies) in all clusters of galaxies is due in significant degree to such clouds.

We can infer that the minimum mass of the clouds in the Local Group is  , i.e. around 5000 clouds with an average mass of  . Since the Milky Way mass, from its rotation curve within the inner 15 kpc, is around  (Honma & Sufue 1996), M31 has around twice this mass, and the rest of the galaxies also contribute a little, we can postulate that  is the maximum mass of the clouds given the above value of the total mass for the Local Group.

The clouds are not homogeneouly distributed, and most of them should be at a distance of Mpc. The mean column density expected from them would be: where f is the fraction of gaseous hydrogen ( according to BL99) and is Avogadro's number. The result is cm-2, whose order of magnitude is in agreement with observations (Wakker 1991). The column density extends over the range cm cm-2, and the observed average would be somewhat lower than cm-2 for km s-1; although this difference might be due to the non-inclusion of intermediate velocity clouds as well to scatter in the parameters used ( , f, d).    © European Southern Observatory (ESO) 1999

Online publication: November 16, 1999 