36. The Hubble constant at Fornax

1. Distance - The two panels of Figure 30 show a comparison of the Virgo and Fornax clusters of galaxies drawn to scale, as seen projected on the sky. The comparison of apparent sizes is appropriate given that the two clusters are at approximately the same distance from us. In the extensive Virgo cluster (right panel), the galaxy M100 can be seen marked ~ 4 degrees to the north-west of the elliptical-galaxy-rich core; this corresponds to an impact parameter of 1.3 Mpc, or 8% of the distance from the LG to the Virgo cluster. The Fornax cluster (left panel) is more centrally concentrated than Virgo, so that the back-to-front uncertainty associated with its three-dimensional spatial extent is reduced for any randomly selected member. Roughly speaking, converting the total angular extent of the cluster on the sky (~ 3 degrees in diameter (7)) into a back-to-front extent, the error associated with any randomly chosen galaxy in the Fornax cluster, translates into a few percent uncertainty in distance; and that uncertainty in distance will soon be reduced when the two additional Fornax spirals are observed with HST in the coming year.

2. Velocity - Here, we note that the infall-velocity correction for the Local Group motion with respect to the Virgo cluster (and its associated uncertainty) becomes a minor issue for the Fornax cluster. This is the result of a fortuitous combination of geometry and physics. We now have Cepheid distances from the Local Group to both Fornax and Virgo. Combined with their angular separation on the sky this immediately leads to the physical separation between the two clusters proper. Under the assumption that the Virgo cluster dominates the local velocity perturbation field at the Local Group and at Fornax, we can calculate the velocity perturbation at Fornax (assuming that the flow field amplitude scales with 1/ R_Virgo and characterized by a R^-2 density distribution (Schechter 1980). From this we then derive the flow contribution to the measured line-of-sight radial velocity, as seen from the Local Group. Figure 33 shows the distance scale structure (left panel) and the velocity-field geometry (right panel) of the Local Group-Virgo-Fornax system. Adopting an infall velocity of the Local Group toward Virgo of +200 km/sec (10) with an uncertainty of ± 100 km/sec, the flow correction for Fornax is only -44 ± 22 km/sec.

   Figure 33. Relative geometry (left panel), and the corresponding velocity vectors (right panel) for the disposition and flow of Fornax and the Local Group with respect to the Virgo cluster. The circles plotted at the positions of the Virgo and Fornax clusters have the same angular size as the circles minimally enclosing M100 and NGC 1365 in the two panels of Figure 30.
   

3. H₀ at Fornax, and its Uncertainties - Correcting to the barycentre of the Local Group (-90 km/sec) and compensating for the -44 km/sec component of the Virgocentric flow, derived above, we calculate that the cosmological expansion rate of Fornax is 1,321 km/sec. Using our Cepheid distance of 18.0 Mpc for Fornax gives H₀ = 73 (± 7)_r [± 20]_s km/sec/Mpc. The first uncertainty (in parentheses) includes random errors in the distance derived from the PL fit to the Cepheid data, as well as random velocity errors in the adopted Virgocentric flow, combined with the distance uncertainties to Virgo propagated through the flow model. The second uncertainty (in square brackets) quantifies the currently identifiable systematic errors associated with the adopted mean velocity of Fornax, and the adopted zero point of the PL relation (combining in quadrature the LMC distance error, a measure of the metallicity uncertainty, and a generous estimate of the possible differences in the true modulus that might be generated from adopting different stellar photometry packages). Finally, we note that according to the Han-Mould model (Han & Mould 1990), the so-called ``Local Anomaly'' gives the Local Group an extra velocity component of approximately +73 km/sec towards Fornax. If we were to add that correction our local estimate, the Hubble constant would increase to H₀ = 77 km/sec/Mpc.

Given the highly clumped nature of the local universe and the existence of large-scale streaming velocities, there is still a lingering uncertainty about the total peculiar motion of the Fornax cluster with respect to the cosmic microwave background restframe. Observations of flows, and the determination of the absolute motion of the Milky Way with respect to the background radiation suggest that line-of sight velocities ~ 300 km/sec are not uncommon (Coles & Lucchin 1995). The uncertainty in absolute motion of Fornax with respect to the Local Group then becomes the largest outstanding uncertainty at this point in our error analysis: a 300 km/sec flow velocity for Fornax would result in a systematic error in the Hubble constant of ~ 20%. We shall however be able to look from afar, and revisit this issue, following an analysis of more distant galaxies made later in this section.