2 The homogenisation of Mg₂ indices of galaxies

  Three steps should be made to bring Mg₂ indices in a common, homogeneous system: (i) broadening correction (spectral resolution matching and velocity dispersion correction), (ii) aperture correction and (iii) dataset zero-point. The recipes for these corrections vary among the different authors and some of them give only corrected data.

We did not attempt here to standardize the measurements of globular clusters.

2.1 Broadening corrections

Both the spectral resolution of the observation and the intrinsic broadening of the spectral lines (velocity dispersion) affect the values of the Mg₂ measurements.

Corrections for these effect are small and often provided by the authors. In general the observations are preprocessed to match the Lick/IDS resolution and are usually normalized to 200 km s^-1 (see Gonzales 1993, and JFK95). Recently Worthey & Ottaviani (1997) provided an elaborate mapping of the Lick resolution which is about 8.4 Å at 5300 Å thus corresponding to instrumental $\sigma\,=\,200$ km s^-1. Hence, we did not make any further broadening correction than those already applied by the authors.

  

Figure 1: The distribution of the aperture corrections of Mg2 for 983 catalogued objects. The value $r_\mathrm{norm} = 0.595 \,h^{-1}$ kpc from JFK95 was used for normalization. Local Group objects and globular clusters are not included in the distribution

2.2 Aperture correction

Galaxies show radial (in general negative) gradients in the Mg₂ index. Thus, the derived central index depends on the size of the aperture used during the observations. Therefore Mg₂ index must be corrected for the effect of the increasing projected aperture size in the more distant galaxies which weakens their indices.

In their earlier works 7Sam have used the aperture corrected values of the observed index, (Mg$_2)_\mathrm{obs}$, given by

$$\displaystyle (\mathrm{Mg}_2)_\mathrm{corr} = (\mathrm{Mg}_2... ...athrm{Virgo}} {V_\mathrm{Coma}-V_\mathrm{Virgo}} + 0.94\right].$$ (2)

The correction was calculated using the group velocity, $V_\mathrm{Group}$, or heliocentric one if no group can be identified.

Perharps the better way to perform aperture corrections is proposed by JFK95:

$$\displaystyle (\mathrm{Mg}_2)_\mathrm{corr} = 0.04 \log\frac{r_\mathrm{ap}}{r_\mathrm{norm}} + (\mathrm{Mg}_2)_\mathrm{obs},$$ (3)

where $r_\mathrm{ap}$ is the physical radius sampled by that circular aperture from which one obtains the same (Mg$_2)_\mathrm{obs}$as through the actual aperture used. For a rectangular aperture of angular dimensions x and y radians, and a galaxy at distance d, the equivalent aperture is

$$\displaystyle r_\mathrm{ap} \approx 1.025\left(\frac{xy}{\pi}\right)^{1/2}d,$$ (4)

where the factor 1.025 is introduced by JFK95 to provide a better matching to detailed galaxy models.

For the normalization JFK95 have used a physical radius $r_\mathrm{norm} = 0.595 \,h^{-1}$ kpc, which is equivalent to an angular diameter of 3.4 arcsec for the distance of Coma cluster. We adopted this normalisation which corresponds to a mean correction in the whole sample of $-0.0008\,\pm\,0.0133$.

We have used distances based on flow-smoothed velocity which is defined as the velocity of the cosmologic flow associated with each galaxy ($H_0\,=\,70$ km s^-1 Mpc$^{-1}\/$is adopted throughout this paper).

The flow-smoothed velocity is computed by averaging the velocity of galaxies found in the neigborhood of any galaxy. This grouping of galaxies is done iteratively. At each step the size of the neighbouring region (in radius on the sky and in velocity interval) is computed from the distribution of the galaxies grouped at the previous step. The initial size goes from 0.5 Mpc for nearby galaxies to 3 Mpc for galaxies at the distance of 130 Mpc (9000 km s^-1), the initial velocity interval is 500 km s^-1. The velocities are taken from HYPERCAT (in turn updated from LEDA). The flow-smoothed velocity is then corrected for the deviations from the linear flow assuming the three-component velocity field model (Great Attractor, Virgocentric infall and Local Group Anomaly) described in Faber & Burstein (1988) and Burstein et al.(1989). Thus, for example, for the distance of Coma cluster the flow-smoothed velocity is 6903$\,\pm\,45$ km s^-1.

For Local Group galaxies the distances are taken from the literature, in particular from van den Bergh (1989).

2.3 Dataset zero-point correction

  Finally, even when much care has been taken to reduce all data to a common scale, systematic differences, at the level of $\approx 0.005$, still persist. It should be mentioned here that many of the authors of the original papers attempted to calibrate their Mg₂ measurements to the Lick/IDS system (in general to consistency with the $7\ Samurai$ data in some form), and therefore most of the offsets are fairly small. The intercomparison between datasets allow to detect and correct such zero-point differences.

We have determined these zero-points, together with a scaling of the error, in an iterative algorithm. At each step, for each galaxy, we have computed the error-weighted average Mg₂ (using the internal error) and we determined the mean and rms residual for each dataset after summing over all galaxies. The mean residual give the zero-point for the considered dataset, the rms is an estimate of the external error. The comparison between the external error and the mean internal error allows to re-scale the latter. The process is iterated after zeropointing and rescaling the internal error.

As a primary dataset at the beginning of the process the most comprehensive subsample from 7Sam is taken - the LICK dataset with 502 measurements for 272 galaxies. The algorithm used is in principle not stable and depends on the subset of objects in common between datasets. Hence, we interactively decided of the relevance of the corrections between each iteraction.

The corresponding values of the mean zero-point corrections, together with their errors and the mean rescaled internal errors, are given for each dataset in the list of the observational parameters (Table 2).

This standardisation could be done for 42 datasets totalizing 1060 galaxies. These datasets and galaxies are flagged in Table 2 and Table 4 respectively. The "standard homogeneous system'' here is the union of all datasets which have been intercompared.

 

Table 2: Observational parameters

 

 

Table 2: continued