A Statistical Comparison of Line Strength Variations in Coma and Cluster Galaxies at z0.3

Analysis

Spectral Indices and Errors

As a means of characterizing the spectra described in Section 2, we have measured 12 pseudo-equivalent widths for 9 different strong absorption features in each galaxy spectrum. The pseudo-equivalent widths, which we will call spectral indices, are computed using the ratio of the flux in a 30 - 40 Å  index passband to a continuum defined by connecting a straight line between the average flux levels in a red and blue side band, each also 30 - 40 Å  wide. This type of spectral index has been used extensively by the Lick group (Wood 1969; Faber 1973; Gorgas et al. 1993; González 1993; Worthey 1994); a list of all but the Balmer indices has been compiled in Worthey (1994). The four Balmer indices included here were defined in Worthey & Ottaviani (1996). Figure 1 gives a qualitative picture of the index definitions overlayed on three composites of galaxy spectra from the distant cluster data and Table 1 gives the wavelength definitions of the indices and the average 1 error for each index. The three data sets considered here were all collected with different telescope-instrument-detector combinations, and in an effort to make the most consistent error estimates possible, we have chosen to use pure photon statistics to determine the errors in the spectral indices, relying only on the number of counts per pixel in the spectra. The flux in each index band was computed as follows,

where is the summed flux for index i, is the flux at point k for index i, and is its poisson error bar. The cumlative error bar for an index passband was computed as follows,

The flux in each continuum band and associated errors were computed in exactly the same manner as for the index passbands.

A source of uncertainty in the comparison of index measurements between the three data sets (FOCAP, LDSS-1, and Hydra) is differences in continuum shape. As long as the slope of the continuum within each data set is constant, or changing in the same sense relative to the other data sets, over the wavelength range of the index (plus continuum bands), there will be no effect on the relative strengths of the index measurements. However, if the coninuum slopes are changing relative to one another in the three data sets, there can be systematic shifts in the index strengths between the data sets. In order to minimize that potential systematic shift, we have determined the relative spectral response functions of the three data sets and normalized all the data sets to the spectral response of the IPCS data from CS. To determine the transformation from the LDSS-1 data to the IPCS response, we ratioed the average spectra of ten galaxies common to both samples and made a smooth fit to that ratio. To go from Hydra to the IPCS response, we averaged all the E and S0 spectra from IPCS, and again for Hydra, ratioed those mean spectra, and fit the ratio.

A second source of uncertainty in comparing the data sets is the difference in physical dimension being sampled in the Coma and high redshift galaxies. While the the Coma data is looking only at the galaxy nuclei, the high redshift data is looking at a much more global aspect of the galaxies. The question to be addressed then is whether, within the uncertainties in the high redshift data, the differences we expect from line strength gradients within the galaxies will be detectable, and hence, produce a systematic offset between the data sets. Although there will always be pathological cases, we will confine our answer to the general trends observed in other studies. Perhaps the simplest and most extensive study of line strength gradients in early type galaxies is in the Ph.D. thesis of Jesus González (1993). He has not computed gradients for all the same indices as presented in this paper, so we will use his H gradients to compute the expected changes in our higher order Balmer lines H and H, and we will use his average Fe gradient to compute the expected changes in our average Fe index (his average Fe index contains the same individual Fe indices as ours plus others). González' average H gradient is,

and for Fe is,

At the distance of Coma (with ) 1''=0.34 Kpc, while in the distant clusters (z=0.31), 1''=6.07 Kpc. The Hydra fibres for the Coma data were 2'' in diameter, the LDSS-1 slit width was 1.''5, and the FOCAP fibre diameter was 2.''6. In Coma the 1'' radius is 0.34 Kpc, and taking the larger diamter of the FOCAP fibres, the high redshift observations extend out to 7.89 Kpc. This predicts a change of 0.050 in H and -0.964 in Fe, whereas the typical uncertainties in those indices are 1.430 and 2.124, respectively. Hence, the changes in the measured index values in the high redshift data relative to the Coma data would be undetectable due to the measurement uncertainties.

  
Figure 1: This figure shows qualitatively the index and continuum passbands for the twelve spectral indices defined in this paper. The passbands are overlayed on three composite spectra from the distant cluster sample of galaxies. The relative scale on the y-axis is in counts.

Table 1: Definitions of Spectral Indices

Passband Indices
Index	Index	Continuum (Å)		1 Errors (Å)
Name	Passband (Å)	Blue	Red	High-z	Coma
CN	4142.125 4177.125	4080.125 4117.625	4244.125 4284.125	1.6223	0.2134
CN	4142.125 4177.125	4083.875 4096.375	4244.125 4284.125	1.5911	0.4676
G4300	4281.375 4316.375	4266.375 4282.625	4318.875 4335.125	1.5784	0.3694
Ca4455	4452.125 4474.625	4445.875 4454.625	4477.125 4492.125	1.3575	0.5056
Fe4383	4369.125 4420.375	4359.125 4370.375	4442.875 4455.375	1.9135	0.6284
Fe4531	4514.250 4559.250	4504.250 4514.250	4560.500 4579.250	1.7801	0.6706
Fe5015	4977.750 5054.000	4946.500 4977.750	5054.000 5065.250	2.6780	0.4683
H	4084.750 4123.500	4042.850 4081.000	4129.750 4162.250	1.6582	0.2377
H	4092.250 4113.500	4058.500 4089.750	4116.000 4138.500	1.2007	0.2178
H	4321.000 4364.750	4284.750 4321.000	4368.500 4421.000	1.9689	0.2449
H	4332.500 4353.500	4284.750 4321.000	4356.000 4386.000	1.3432	0.1969
Fe(C)4668	4634.000 4720.250	4611.500 4630.250	4742.750 4756.500	2.4559	0.5715

Given the extensive descriptions of the spectral indices available in the literature and the fact that we have concentrated on a statistical analysis of the spectra as opposed to a detailed investigation of the stellar populations, we give only a brief discussion of the individual indices used here. For a full description of the spectral indices and their use in integrated light studies, the reader should refer to Worthey (1994), Worthey & Ottaviani (1996), and Jones (1996).

  CN & CN - These two indices both straddle the CN bandhead at 4216 Å, and they share the same index passband, but have a different blue continuum band. The index response is nearly flat for dwarf stars of all temperatures and giants with T4500 K, but shows gravity and metallicity sensitivity below T4500 K.

  G4300 - This index measures the G-band of CH at 4300 Å  and shows mostly gravity and temperature sensitivity.

  Ca4455 - This index measures the only strong Ca I feature in this region of the spectrum. It is mostly sensitive to temperature and metallicity.

  Fe4383, Fe4531 & Fe5015 - These indices all measure prominent Fe absorption features. They are mostly sensitive to temperature and metallicity, but also show gravitiy sensitivity for G stars.

  H & H - These two indices are broad and narrow definitions for the H Balmer line, respectively. Balmer lines are generally temperature sensitive in all stars.

  H & H - Similar to H above, these are broad and narrow definitions for the H Balmer line. Again, these indices show mostly temperature sensitivity, although there is significant metallicity sensitivity in this definition of the H index.

  Fe(C)4668 - Originally denoted Fe4668, this index measures the strong absorption at 4668 Å. There is an Fe feature there, but it has been shown that the strength of the absorption is mostly a function of the molecular Carbon bands at the same location (Tripicco & Bell 1995). Accordingly, the index is sometimes now denoted C4668. It shows strong metallicity sensitivity, as well as gravity sensitivity.

Principal Component Analysis

We have carried out a principal component analysis (PCA) of the distant cluster and Coma data in order to quantify the line strength distributions in the nearby and distant cluster galaxies. A principal component analysis transforms the observable axes, in this case, the spectral line strengths, which are necessarily interdependent, e.g. all partially temperature sensitive, into a new set of axes, or components, which are mutually orthogonal. This is accomplished by choosing each new axis to maximize the total variance along that axis. Hence, the PCA acts as a means of data compression by bringing together in a single new axis correlations between several different observables, so that the most significant few principal components will describe all of the real variations in the data while the rest of the components will contain mostly uncorrelated noise. Each new axis is a linear combination of the observable axes and is associated with an eigenvalue, which is a measure of the fraction of the variation in the observables accounted for in that new axis.

For the PCA, in order to ensure a more meaningful comparison between the data sets, we have affected two further changes to the galaxy data. First, we have restricted both the Coma and high redshift samples to include only the morphologically classified E and S0 galaxies. This leaves 40 galaxies in the high reshift sample and 183 in the Coma sample. Second, we have degraded the Coma data to the same signal-to-noise as the high redshift data. This was done by adding Poisson noise and gaussian read noise to the Coma spectra with the IRAF task MKNOISE until the RMS scatter in the wavelength region divided by the average flux over that range was the same as in the high redshift data.

Of the 12 spectral indices measured, there are three pairs of indices where each index in the pair is a different measurement of the same feature. Those pairs are H and H, H and H, and CN and CN, which each cover the CN bandhead at 4216 Å. There are also three different iron features, Fe4383, Fe4531, and Fe5015. A single index is formed for each of the above sets of like indices by forming the weighted average of the indices in each set, so that out of the twelve original measurements, seven indices are input into the PCA code. The weighted averages were computed as follows,

where is the summed index value j for galaxy i, is the individual index k for galaxy i, and is its associated error. The errors for the new, summed indices were computed as follows,

It is then those seven summed indices which are transformed, through the PCA, into seven new, orthogonal axes. For reasons described in the next section, the PCA was run in two incarnations: first, on the high redshift, clean Coma, and noisy Coma samples indepentdently, producing three sets of new axes, or eigenvectors, which are each different linear combinations of the original spectral indices, and second, on a combined sample of high redshift and noisy Coma galaxies, producing a single set of eigenvectors describing the entire sample. The combined run was conducted with a random sample of 40 Coma galaxies selected to have the same morphological mix as the high redshift sample. This was done so that the principal components would not be artificially dominated by the Coma spectra due to the larger number of Coma galaxies. Table 2 shows the eigenvalues from the run of the separate samples, and Table 3 shows the eigenvectors from the combined run. With the new set of independent axes we will now look at how to relate them to the physical state of the cluster galaxies.

Table 2: Eigenvalues for the High Redshift, Noisy Coma, and Coma Samples

PC	High-z	Noisy Coma	Coma
1	2.1649(30.9)	3.3646(48.1)	3.4427(49.2)
2	1.5421(22.0)	0.8662(12.4)	1.1842(16.9)
3	1.1843(16.9)	0.8314(11.9)	0.6807(9.7)
4	0.9724(13.9)	0.6616(9.5)	0.5605(8.0)
5	0.4547(6.5)	0.5540(7.9)	0.4751(6.8)
6	0.3725(5.3)	0.4334(6.2)	0.3572(5.1)
7	0.3091(4.4)	0.2887(4.1)	0.2995(4.3)

© Copyright Astronomical Society of Australia 1997