3 The method

To study the local Hubble flow at smaller distances, it becomes increasingly important to make relevant corrections to distances and radial velocities. The first correction, discussed by Sandage (1986), is due to the shift of the observer to the centre of expansion, which in the self-gravitating Local Group is not in our Galaxy, but presumably in the barycentre. By assuming the mass ratio of M 31 and Galaxy is $M_{\rm M~31}/M_{\rm G}\approx2$ Sandage set the barycentre to be on the line between M 31 and Galaxy at 2/3 of the distance to M 31. This is also our choice. We denote the distance from the barycentre by R_2/3.

The observed mean heliocentric velocities must also be corrected. We first correct the observed velocity to the value as it would be measured by an observer in our Galaxy being at rest relative to the centroid of the Local Group. This we do according to Yahil et al. (1977), which is also the preferred choice in the LEDA database. We also remind that in Ekholm et al. (1999a) the correction was also made according to Richter et al. (1987) but difference to the correction of Yahil et al. (1977) was quite small. The velocity correction used in reads:

$$% \Delta v = 295.4\sin l \cos b - 79.1\cos l \cos b - 37.6\sin b.$$ (1)

After the solar motion is thus corrected for, the infall to the Virgo Supercluster is the next disturbance. We correct the radial velocity for the virgocentric motion following Ekholm et al. (1999a):

 

$$% V_{\rm corr}=V_{\rm Yahil} \pm [v(d)_{\rm H}-v(d)]$$

$$\times \sqrt{1-\sin^2\Theta/d^2}+V_{\rm LG}^{\rm in}\cos\Theta,$$ (2)

where (-) is valid for points closer than the tangential point $d_{\rm gal}<\cos\Theta$ and (+) for $d_{\rm gal}\ge\cos\Theta$.

$\Theta$ is the angular distance of a galaxy from the centre of Virgo and d is its distance R from the centre normalized to the distance of Virgo ( $d=R/R_{\rm Virgo}$). Following Ekholm et al. (1999a) we take $R_{\rm Virgo}=21{\rm\ Mpc}$ and $H_0=57{\rm\ km\,s^{-1}\,Mpc^{-1}}$. The Hubble velocity at a distance d from the centre of Virgo is $v(d)_{\rm H}=V_{\rm cosm}(1)\times d$. The cosmological velocity of the Virgo cluster becomes $V_{\rm cosm}(1)=1200{\rm\ km\,s^{-1}}$. The predicted velocity v(d) is solved using the Tolman-Bondi model as described by Ekholm et al. (1999a).