4 Correction for the Malmquist bias

The method of sosie galaxies (Eq. (5)) does not correct for statistical bias due to magnitude incompleteness (Malmquist bias). Two methods exist to detect unbiased galaxies: the method of normalized distances (Bottinelli et al. 1986) and the Spaenhauer diagram (Spaenhauer 1978) used by Sandage (1994). A detailed discussion about these methods is given by Teerikorpi (1997). In the case of galaxies which are sosie of a same calibrator, both diagrams are equivalent. In order to correct for the Malmquist bias we have first to determine the actual completeness limit of the apparent magnitudes.

4.1 Completeness limit in B and I apparent magnitudes

The knowledge of the limiting apparent magnitude informs us about galaxies missing at a given distance for a given absolute magnitude. The construction of the curve $\log N$ vs. apparent magnitude (N is the number of galaxies brighter than the considered apparent magnitude) is a classical way to estimate this limit. If the distribution of galaxies is homogeneous, the slope expected is 0.6. In Fig. 5 we show the completeness curves for our sample, in B- and I-band respectively. The limiting apparent magnitude is given where the curve begins to bend down.

Figure 5: Completeness curve for the B and I-band apparent magnitudes. The curve begins to bend down at $\approx$14 mag and $\approx$12.5 mag, respectively.

The limiting apparent magnitude is $m_{\rm lim}=14$ mag in B and $m_{\rm lim}=12.5$ mag in I. The slopes are $0.39\pm0.01$ and $0.37 \pm 0.01$, for B and I respectively. The result that the slope is smaller than the theoretical one has already been discussed (Paturel et al. 1994; Teerikorpi et al. 1998). In the following sections we will use the observed slope (0.4) and the limits 14 and 12.5 for B and I magnitudes.

4.2 Theoretical aspects of the Spaenhauer diagram

The Spaenhauer diagram for galaxies which are sosie of a given calibrator is a diagram of absolute magnitude against distance. From a practical point of view, we are drawing the absolute magnitude versus the radial velocity which is an estimate of the distance through the Hubble law, after correction for the infall of our Local Group (hereafter LG) towards Virgo. For this correction a preliminary infall velocity has been adopted (200 km s^-1). Eventually, it will be revised in forthcoming sections.

The construction of the Spaenhauer diagram requires three curves:

1.

The curve showing the cut due to the limiting apparent magnitude. We recall the equation of this curve:

$$M = m_{\rm lim} - 25 - 5 \log \frac{V}{H}$$ (6)

where H is the adopted Hubble constant and $m_{\rm lim}$ the limiting apparent magnitude, as defined in the previous section.

2.

The bias curve for a galaxy with a Gaussian absolute magnitude distribution $\mathcal{G}(M_{\rm o} \sigma)$ is given by the equation (Teerikorpi 1975):

$$M= M_{\rm o} + \sigma {\sqrt{\frac{2}{\pi}} \frac {{\rm e}^{-A^2}}{1+erf(A)}}$$ (7)

with

$$A= \frac {m_{\rm lim} -25 -5\log~V/H - M_{\rm o}} {\sigma \sqrt{2}}$$ (8)

and

$$erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x {\rm e}^{-t^2} {\rm d}t.$$ (9)

3.

The envelope of the distribution is given by the following equations demonstrated in the Appendix A:

$$M = M_{\rm o} \pm \frac{\alpha \sigma}{1.55}\log~V + k$$ (10)

where $\alpha$ is the slope of the completeness curve divided by 0.2 (e.g., for the theoretical slope of 0.6 we expect $\alpha= 3$). We assume that $\sigma$does not depend on the distance, but its value will be calculated at each step of the iterative process. k depends on the adopted Hubble constant and on the mean density of the considered galaxies. Its expression is:

$$k= m_{\rm lim} - M_{\rm o} + 5 \log H - \left(5+ \frac {\alpha \sigma}{1.55}\right) \log V_{0.05} - 25$$ (11)

where V_0.05 is the velocity at which the biased absolute magnitude is changed by 5 percent. This level is arbitrarily chosen as the beginning of the observable bias.

4.3 Application to our sample

We construct the Spaenhauer diagram for each calibrator using our sample of sosie galaxies. Let us recall that the adopted parameters are the following: $m_{\rm lim}=14$ in B-band and 12.5 in I-band (Sect. 4.1). From Fig. 5, we use the observed values $\alpha= 0.4 / 0.2 = 2$ for both B and I magnitudes. We adopt provisionally a Local Group infall velocity of 200 km s^-1 In a first step we start with a preliminary value of H and $\sigma$. We adopted a starting dispersion $\sigma =0.7$ in B and $\sigma =0.6$ in I and a Hubble constant of 70 km s^-1 Mpc^-1 These values have no incidence on the final result. They only affect the number of iterations. We calculate the new H and $\sigma$ values from the unbiased galaxies of the "plateau'' (i.e. galaxies with a velocity less than V_0.05) The new estimates are re-injected in a new iteration, and so on. The calculation is stopped when H becomes stable. Figures 6 and 7 give the Spaenhauer diagrams in B and Ifor the 21 calibrators (two calibrators have no I-band magnitude). It is interesting to compare NGC 224 = M 31 and NGC 598 = M 33, the first two galaxies in Fig. 6, because they represent two opposite cases: one luminous galaxy (M 31) and one low luminosity one (M 33). The effect of the bias is clearly visible.

Figure 6: Spaenhauer diagrams in B magnitudes. For each calibrator we plot its sosie galaxies, with the predicted envelope (doted curve). The bias curve is show as a solid curve. The region of completeness is above the dashed curve. Finally, the unbiased region ("plateau'') is defined on the left of the vertical line. The name of the calibrating galaxy is indicated in the upperleft corner of each frame.

Figure 7: Spaenhauer diagrams in I magnitudes (same as Fig. 6).

4.4 The unbiased sample

From the Spaenhauer diagram we extract the galaxies with a velocity below V_0.05. This is done for B and I separately. A mean distance modulus is derived in B and I for each galaxy using Eq. (5): The weighted mean distance modulus is then:

$$<\mu>~=\frac{\sum{w_i \mu_i}}{\sum{w_i}}$$ (12)

where the weight ( $w_i = 1/\sigma_{i}^{2}$) of each individual distance modulus ($\mu_i$) is calculated from the mean error on B or I magnitudes using the relations:

$$\sigma_{B}^2 = \sigma_{B_{\rm o}}^{2} + a_{B}^{2} \sigma_{\log VM}^{2}$$ (13)

$$\sigma_{I}^2 = \sigma_{I_{\rm o}}^{2} + a_{I}^{2} \sigma_{\log VM}^{2}.$$ (14)

Here a_B=5.9 and a_I=7.7 are the adopted slopes of the Tully-Fisher relations in B and I, respectively. The mean errors $\sigma_{B_{\rm o}}$, $\sigma_{I_{\rm o}}$ and $\sigma_{\log VM}$ are taken from LEDA.

The mean error on the mean distance modulus is then:

$$\sigma_{<\mu>} = \frac{1}{\sqrt{\sum{w_i}}}\cdot$$ (15)

The list of 283 unbiased galaxies is given in Table 2 with the following parameters:
Column 1: PGC number from LEDA.
Column 2: Alternate name from LEDA.
Column 3: Right Ascension and declination for equinox 2000 in hours, minutes, seconds and tenth and degrees, arcminute and arcseconds.
Column 4: Radial velocity corrected for a preliminary infall velocity of the LG towards Virgo ( $V_{\rm infall}= 200$ km s^-1).
Column 5: Weighted mean distance modulus and its mean error. The merging of B and I bands is given in Fig. 8.
Column 6: Angular distance from the Virgo center. The center of Virgo is placed on M 87 at equatorial coordinates of J123049.2+122329.

 

 

Table 2: The sample of 283 unbiased galaxies. The full table is available in electronic form at CDS.

PGC	Alternate name	RA.DEC.2000	$V_{\rm vir}$	$<\mu>$	$\Theta$
		$\rm h,mn,s\ \ \deg \ '\ ''$	km s-1		$\deg$
0004596	NGC 452	J011614.8+310201	5078	$34.24\pm 0.14$	135.2
0005035	NGC 494	J012255.4+331025	5582	$34.17\pm 0.06$	132.8
0005268	NGC 523	J012520.8+340130	4881	$33.70\pm 0.11$	131.8
0010048	NGC 1024	J023911.8+105049	3522	$33.34\pm 0.08$	140.6
0014906	NGC 1558	J042016.1-450154	4278	$33.99\pm 0.07$	121.6
0016359	UGC 3207	J045609.8+020926	4494	$34.26\pm 0.12$	112.6
0018739	UGC 3420	J061601.8+755611	5353	$34.52\pm 0.42$	78.9
0021336	NGC 2410	J073502.5+324921	4769	$34.05\pm 0.40$	69.9
0026101	IC530	J091517.0+115309	4981	$34.64\pm 0.43$	47.7
...

We can now start the study of the local velocity field around Virgo from this unbiased sample.