4 Results

We present the corrected velocity vs. distance relations for the very local velocity field ( R_2/3 < 8 Mpc) in Figs. 1 and 2. In Fig. 1 velocities are corrected only for the solar motion according to Yahil et al. (1977). We calculate the velocity dispersion without the Local Group members (the first 4 symbols marked as crosses), where certainly the internal dynamics masks the possible Hubble term which is small compared with the value of the velocity dispersion. The velocity dispersion is $\sigma_{\rm v}=42{\rm\ km\, s^{-1}}$ solved from galaxies between $R_{2/3}\in[1.0,8.0]{\rm\ Mpc}$. The expected Hubble law is given as a solid straight line and the $1\sigma$ dispersion as dotted lines.

In Fig. 2 we show the influence of the correction for the virgocentric motion. Now the velocity dispersion decreases down to $\sigma _{\rm v}=37$ kms^-1. The decrease in $\sigma_{\rm v}$ is not significant. This is expected because most of the galaxies are at large angular distance from the Virgo except IC 4182 having $\Theta=26.4\hbox{$^\circ$ }$. It is quite remarkable that after the correction this galaxy follows almost exactly the expected Hubble law.

Both Figs. 1 and 2 support the conclusion 4 of Sandage (1986). The observed random velocities get smaller as the uncertainties in the adopted distance indicator diminish.

Figure 3 shows the predictions for the velocity perturbations in Sandage's point-mass model for the case when the mass of the Local Group $M_{\rm LG} = 2~10^{12}~M_{\odot}$. For example, van den Bergh (1999) estimates that $M_{\rm LG} = 2.3\pm0.6~10^{12}~M_{\odot}$. Evans et al. (2000) give an upper limit $M_{\rm LG} = 2.4~10^{12}~M_{\odot}$. The velocities were deduced for three cosmologies using a numerical algorithm for the Tolman-Bondi calculations (Hanski et al. 2000). The solid curve shows the prediction for the now preferred Friedman model $\Omega _{\rm m}=0.3$ and $\Omega _\Lambda =0.7$. The dashed line corresponds to the classical flat model $\Omega _{\rm m}=1.0$ and $\Omega _\Lambda =0.0$. The dotted line is the prediction for a low mass-density universe $\Omega _{\rm m}=0.1$ and $\Omega _\Lambda =0.0$. $\Omega_{\rm m}$ is the mean density parameter of the matter and $\Omega_\Lambda$ is the density parameter induced by the $\Lambda$-term. One may note that these models differ little from each other, which shows that the value of the velocity deflection (and the zero-velocity distance) is essentially determined by the mass of the LG.

Only galaxy that shows any significant deviation from the expected Hubble law is IC 1613 but as pointed out in Sect. 2 its dynamical status is not well known. On the other hand M 31 deviates only little from the Hubble law (in fact it is within the $1\sigma$ limit).

Figure 1: Velocities corrected to the rest frame of the centroid of the Local Group according to Yahil et al. (1977). The origin of the distances has been changed to the centroid following Sandage (1986). The solid straight line is the Hubble law for H0=57 kms-1Mpc-1. The dotted lines give the $1\sigma$ dispersion calculated for galaxies between $1{\rm\ Mpc}$ and $8{\rm\ Mpc}$giving $\sigma = 42$ kms-1 Crosses refer to LMC, N6822 N224 and N598. The open circle is IC 1613. The error bars reflect the formal uncertainties in the Cepheid distances

Figure 2: As Fig. 1 except that now we have also corrected for the Virgocentric motions. The effect is small except for IC 4182. This is the only galaxy that is at a small angular distance from Virgo cluster. The velocity dispersion $\sigma _{\rm v}=37$ kms-1

Figure 3: As Fig. 2. Now we have added three theoretical predictions. Each of them is based on a point mass model assuming that the mass of the Local Group is $M_{\rm LG} = 2~10^{12}~M_{\odot}$. We examined three cosmologies: 1) the Friedman model with $\Omega _{\rm m}=0.3$and $\Omega _\Lambda =0.7$ (the solid curve), 2) the classical flat model with $\Omega _{\rm m}=1.0$and $\Omega _\Lambda =0.0$ (the dashed curve) and 3) a low mass density model with $\Omega _{\rm m}=0.1$ and $\Omega _\Lambda =0.0$ (the dotted curve)

Sandage (1986) discussed the position of the barycentre. He pondered whether it would be possible to reduce the scatter in the Hubble diagram by "changing the origin of the distances to the centroid as well". We have checked the behaviour of the velocity dispersion $\sigma_{\rm v}$ as a function of the position of the barycentre between Galaxy and M 31.

Recently Evans et al. (2000) claimed that the halo of M 31 is about as massive as the halo of Galaxy. In this case the centroid would be at the middle distance. Now, $\sigma_{\rm v}$ changes quite slowly in the region between R_1/2 and R_2/3. For example at R_1/2 the dispersion $\sigma_{\rm v}=44{\rm\ km\,s^{-1}}$and at R_1/4 it is $\sigma_{\rm v}=49$ kms^-1. It is thus not possible to make any statistically relevant claims about the exact position of the barycentre. With a larger sample of field galaxies, the position of the barycentre (and hence the mass ratio $M_{\rm M~31}/M_{\rm Galaxy}$) could perhaps be determined.

Finally we note that due to the lack of galaxies with Cepheid distances within $R_{2/3}=1{\rm\ Mpc}$ and $R_{2/3}=2{\rm\ Mpc}$it is not possible to determine the zero-velocity surface. It is though important to note that galaxies within $R_{2/3}=1{\rm\ Mpc}$ show practically no deflection as compared to the theoretical curves shown in Fig. 3, except possibly the "free-floating" IC 1613, which could fit some theoretical curve with $M_{\rm LG}<10^{12}~M_{\odot}$.