5 Influence of Virgo on log H

The histograms of $\log H$ derived from B and I are presented in Fig. 8. The visible result is that both photometric bands give the same mean.

The means $\log H (B{-}\rm band)= 1.8263 \pm 0.0085$ and $\log H (I {-}\rm band)= 1.8264 \pm 0.0075$ are obviously not significantly different from each other. This justifies that the distance moduli in B and I are combined as explained in the previous section. Similarly, the $\log H$ will be now the weighted mean of $\log H$ in B and I. In order to see the influence of Virgo, we plot now $\log H$ vs. $\Theta$. Three regions are defined (Fig. 9):

1.

Region I: This is the region in the direction of Virgo. It is defined by $\Theta < 60^{\circ}$. For $\Theta < 20^{\circ}$, the diagram divides in two branches. This is the trace of the frontside and backside infalls onto Virgo. For $20^{\circ} \leq \Theta < 60^{\circ}$ (hereafter region Ib) the velocity field appears quieter ($\log H$ does not differ from the unperturbed region III).

2.

Region II: This region is perpendicular to the direction of Virgo. It is defined by $60 \deg \leq \Theta < 140 \deg$. The mean $\log H$ does not differ from values of unperturbed regions (region III and region around $\Theta \approx 50 \deg$).

3.

Region III: Region opposite to the direction of Virgo. It is defined by $140 \deg \leq \Theta$.

Let us study in detail these regions with the target of fitting Peebles' model.

In region III the influence of Virgo is small on individual galaxies because they are far from the Virgo center but the infall of our Local Group may be not negligible. If the adopted LG-infall is too large the $\log H$ for this region will be too high (and vice versa). On the contrary, in region Ib (i.e., region I without the central region Ia of the Virgo center) one expects $\log H$ to diminish when the adopted LG-infall increases. This is confirmed by Fig. 10 where we calculated the mean $\log H$ in regions Ib and III for different LG-infall velocities. The error bar on each point is used to define internal uncertainties on $V_{\rm LG-infall}$ and $\log H$.

The intersection of the two curves of Fig. 10 gives accurately both the infall velocity of the LG and $\log H$. We obtain:

$$V_{\rm LG-infall} = 208 \pm 9.3 \rm ~km~s^{-1}$$ (16)

$$\log H = 1.809 \pm 0.010 \ \ \ (H \approx 64\rm ~km~s^{-1}~Mpc^{-1}).$$ (17)

The uncertainty on $\log H$ is the internal uncertainty. The total uncertainty is discussed in the conclusion. The present infall velocity $V_{\rm LG-infall}$ is comparable with the mean value, $222 \pm 15~\rm km~s^{-1}$, found in the recent literature (Mould et al. 2000; Theureau et al. 1997; Bureau et al. 1996; Hamuy et al. 1996). Although our present target is the derivation of the parameters of Peebles' model it is not possible to disentangle it from the calculation of the Hubble constant and of the LG-infall velocity.

Figure 8: Histograms of $\log H$ derived from B and I. The dashed line shows the B-band histogram and the solid line the I-band one. Both photometric bands give the same mean.

Figure 9: Mean $\log H$ derived from B and I against the angular distance to the Virgo center. This figure allows us to define three regions: region I in the direction of Virgo (also separated in Ia and Ib), region II perpendicular to the direction of Virgo and region III opposite to Virgo.

Figure 10: Determination of the LG infall velocity onto Virgo and of the corrected Hubble constant. The full curve represents how $\log H$changes in region III when LG infall velocity is changed. The dashed curve gives the same result for the region Ib. The LG infall velocity and the corrected $\log H$ are accurately determined at the intersection of both curves (vertical doted line).

In order to search for possible residual bias, we plotted the mean $\log H$ of each calibrator class versus the absolute magnitude of its calibrator, in Band I-band, (Fig. 11). A tendency to have large $\log H$ when the calibrator is faint might be present for the less luminous calibrator. The slope of this relation is $0.015 \pm 0.006$. This is barely significant at the 0.01 probability level (to be significant, the Student's t test requires t_0.01>2.57, while we observe t=0.015/0.006=2.5). If one removes the six less luminous calibrators (NGC 598, NGC 4496A, NGC 2090, NGC 3319, NGC 3351 and NGC 4639), the tendency disappears (t=0.4). In this case, the mean log of the Hubble constant becomes $\log H=1.815 \pm 0.007$ (instead of $1.826\pm 0.006$). Thus, the mean Hubble constant is not severely affected, but small $\log H$ is probably better.

Figure 11: Relation between the mean $\log H$ of each calibrator class versus the B- or I-absolute magnitude. There is no significant increase of H with the absolute magnitude. The small tendency is well within the error.

5.1 Peebles' model

Let us consider now the galaxies in the direction of Virgo (region Ia). The infall of each individual galaxy ( $V_{\rm g-infall}$) can be calculated from their distance (d), from their observed radial velocity ( $V_{\rm vir}$) corrected for the LG infall and from the angle ($\Phi$) between Virgo and the LG seen from this considered galaxy (see Fig. 2 for the definitions of r, d and $\Phi$):

$$V_{\rm g-infall}= \frac{H \ d - V_{\rm vir}}{\cos \Phi}$$ (18)

with,

$$\cos \Phi=\frac{d^2 + r^2 - d_{\rm Vir}^{2}}{2dr}\cdot$$ (19)

The adopted distance to Virgo is $d_{\rm Vir}=18~\rm Mpc$.

Now, we can plot $V_{\rm g-infall}$ vs. r (Fig. 12) to check directly Peebles' model (Eq. (1)). From the previous relation it can be seen that the errors on x- and y-axis are strongly correlated. The dominant error comes from d. The asymptotic velocity infall is zero at large r distances. The location of the LG is represented by a large open square. We calculated $\gamma$ by minimizing the dispersion on $V_{\rm g-infall}$. For this calculation we limited r between 3 Mpc and 30 Mpc in order to avoid the Virgo center, where there is a very large uncertainty on $V_{\rm g-infall}$ due to the small value of r. The best result is obtained for $\gamma =1.9$. This result is quite comparable with the value adopted by Peeblees ($\gamma =2$) but the uncertainty, estimated visually, is large (about 0.2). The application to the Local Group infall leads to $C_{\rm Virgo}=2800$. In order to highlight the tendency, we plot the mean x and y values of individual points in four x-boxes: (3 to 7) Mpc, (7 to 13) Mpc, (13 to 21) Mpc and (21 to 31) Mpc. These mean points (open circles) are represented with their observed scatter divided by the square root of the number of points in each box.

Figure 12: Direct determination of the parameters of Peebles' model. The adopted model is represented with the solid curve. Dashed curves correspond to a change of $\gamma$ by $\pm 0.2$. Without infall the points should be distributed around the horizontal line. The position of the Local Group is represented with a large open square. Mean points are represented with their uncertainty (open circles) in order to show better the expected tendency.

It is interesting to see in detail which galaxies are exactly in the direction of the Virgo center. If we consider galaxies with $\Theta<15 \ \deg$ and $\mu \leq 32.50$only 9 galaxies remain. They are presented in Table 3 following increasing radial velocities. For each galaxy we give the observed Hubble constant H. In this table two parameters are independent: the radial velocity and the distance modulus.

 

 

Table 3: Detail of the central region of Virgo. Galaxies with NGC name have their distance from the Cepheid PL relation.

PGC	NGC	$V_{\rm vir}$	$\mu$	$\Theta$	H
0041934	NGC 4548	587	$30.94 \pm0.37$	2.4	38
0042833		917	$31.86 \pm0.17$	5.1	39
0043798		1089	$32.26 \pm0.25$	11.7	38
0042741	NGC 4639	1097	$31.70 \pm0.18$	3.1	50
0043451	NGC 4725	1360	$30.81 \pm0.15$	13.9	94
0041823	NGC 4536	1846	$30.93 \pm0.11$	10.2	120
0041812	NGC 4535	2029	$30.98 \pm0.31$	4.3	129
0041024		2069	$31.29 \pm0.23$	4.7	114
0042069		2342	$31.47 \pm0.11$	1.8	119
Mean			$31.36 \pm0.17$
Adopted			$31.3 \pm0.17$

The main feature from this table is that the apparent Hubble constants are sorted according to velocities. This means that the distance does not intervene very much. The natural interpretation is that all these galaxies are almost at the same distance (distance of Virgo). The Hubble constant reflects only the infall velocity. Large velocities correspond to galaxies in front of Virgo and falling onto Virgo, away from us. On the contrary, galaxies with small velocities are beyond Virgo and falling onto it in our direction. Indeed, the four galaxies with small velocities have a mean distance modulus of $\mu=31.69\pm0.27$ while the four galaxies with high velocities have a larger mean distance modulus of $\mu=31.17\pm0.13$. The difference is significant at about $3 \sigma$. This confirms clearly the interpretation. From the table one can conclude that the distance modulus of Virgo is $\mu=31.4\pm0.2$. However, it is difficult to give a mean radial velocity because of the strong perturbation of the velocity field. Amazingly, it is better to measure radial velocities out of the center to obtain a good mean velocity of a cluster. If one adopts the velocity of 980 km s^-1 (see the discussion in Teerikorpi et al. 1992), and both the LG-infall velocity and the Hubble constant found in this paper, the distance modulus of the Virgo cluster is $\mu=31.3$. This gives a coherent system of parameters.

We can also discuss the region perpendicular to the direction of Virgo (region II). The weighted mean Hubble constant in this region is nearly the same as the one found in region III (i.e., $\log H=1.827\pm0.007$). In the direction of the Fornax cluster one can repeat what we have done in the direction of Virgo, but the number of galaxies is smaller. The center of Fornax is assumed to be $\rm RA(2000)=03h42m$ and $\rm DEC(2000)=-36 \ \deg$. In Table 4 we summarize the results. It appear that galaxies PGC 12390 and PGC 10330 can be considered as backside galaxies falling onto Fornax towards us (small radial velocity, large distance, small apparent Hubble constant). Galaxies PGC 13255 and PGC 13602 are roughly at the position of Fornax, while PGC13179 and PGC 13059 are in front of Fornax (large radial velocity, small distance and large apparent Hubble constant). From the table one can conclude that the distance modulus of Fornax is $\mu=31.7\pm0.3$. It is not possible to measure how the infall velocity changes with the distance to the center of Fornax. Nevertheless, if one still adopt $\gamma =1.9$one can determine the parameter C for Fornax. Indeed, the observed infall velocity is roughly 270 km s^-1 at a distance of 6 Mpc of the center. This leads to $C_{\rm Fornax}=1350$.

 

 

Table 4: Detail of the central region of Fornax. Note that the angular separation $\Theta$ is counted with respect to the direction of the Fornax center.

PGC	$V_{\rm vir}$	$\mu$	$\Theta$	H
0012390	767	$31.98 \pm0.19$	14.5	31
0010330	1249	$32.73 \pm0.33$	14.4	36
0013255	1279	$31.92 \pm0.17$	11.1	53
0013602	1299	$31.51 \pm0.05$	6.0	65
0013179	1423	$30.96 \pm0.13$	2.1	91
0013059	1670	$31.31 \pm0.11$	3.4	91
Mean		$31.74 \pm0.25$