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Subsections

4 Nuclear line-strength indices

The spectrum of a galaxy is produced by the sum of the spectral characteristics of its stellar content. The relative contribution given by each stellar type strongly depends on the wavelength considered. Buzzoni (1989, 1995) and Worthey (1992, 1994) have given estimates about the sensitivity of several indices to the metallicity or to the age of a stellar population. From these studies it is possible to infer that most of the line-strength indices in the Lick-IDS System are sensible to the galaxy metallicity and that only the H$\beta$ line-strength index is connected in a significant way to the galaxy age. Since we are investigating the effect of interaction on the stellar components of our sample galaxies, we decided to measure and calibrate a set of line-strength indices sensible to recent star Formation (SF), to which we refer as "blue'' indices, in addition to those in the standard Lick-IDS system ("red indices''). The details of the line-strength indices are described in the sections below.
  
Table 4: Observing parameters


\begin{tabular}
{lrrr}
\hline
 & Run 1 & Run 2 & Run 3 \\ \hline
Date of observa...
 ...ec] & 1.0 $-$\space 1.5 & 1.0 $-$\space 1.5 & $<$\space 2 \\ \hline\end{tabular}



  
Table 5: Journal of galaxies observations
\begin{tabular}
{rrrrrrlrrrrl}
\noalign{\hrule\medskip}
Ident. & Run & Slit& $t_...
 ...2 & 2 & 0.48 & RR~409a/b & 2 & 85
 & 130 & 1/1 & 0.23/0.23\\ \hline\end{tabular}


  
Table 6: Definition of the Lick-IDS indices
\begin{tabular}
{rrccr}
\noalign{\hrule\medskip}
Index & Spectral & $\lambda_{\r...
 ...$5387.500 & Atomic \\  & & & 5415.000$-$5425.000 & [\AA] \\ \hline \end{tabular}

4.1 Red and blue indices

We indicate as red 16 line-strength indices defined in the wavelength range between 4200 Å and 5500 Å. Atomic, $I_{\rm a}$, and molecular, $I_{\rm m}$, indices are defined by the following formulas (G93):
\begin{displaymath}
I_{\rm a} = \int_{\lambda_{\rm I_1}}^{\lambda_{\rm I_2}} (1 - {F(\lambda) \over
 C(\lambda)}) {\rm d}\lambda\nonumber \end{displaymath}   
and
\begin{displaymath}
I_{\rm m} = -2.5{\rm log}_{10}{\int_{\lambda_{\rm I_1}}^{\la...
 ...rm d}\lambda \over 
 (\lambda_{\rm I_2} - \lambda_{\rm I_1})}. \end{displaymath} (1)
The $C(\lambda)$ function represents the spectral continuum obtained by interpolating fluxes in two windows chosen near the two sides of the feature:
\begin{displaymath}
C(\lambda) = F_{\rm b}{\lambda_{\rm r}-\lambda \over \lambda...
 ...lambda-\lambda_{\rm b} \over \lambda_{\rm r} - \lambda_{\rm b}}\end{displaymath} (2)

\begin{displaymath}
F_{\rm b} = {\int_{\lambda_{\rm b_1}}^{\lambda{\rm b_2}} F(\...
 ...) {\rm d}\lambda \over 
 (\lambda_{\rm b_2}-\lambda_{\rm b_1})}\end{displaymath}

\begin{displaymath}
F_{\rm r} = {\int_{\lambda_{\rm r_1}}^{\lambda{\rm r_2}} 
F(...
 ...) {\rm d}\lambda \over 
 (\lambda_{\rm r_2}-\lambda_{\rm r_1})}\end{displaymath}

\begin{displaymath}
\lambda_{\rm r} = {(\lambda_{\rm r1} + \lambda_{\rm r2}) \ov...
 ...bda_{\rm b} = {(\lambda_{\rm b1} + \lambda_{\rm b2}) \over 2}. \end{displaymath}

The adopted spectral bandpasses of the red indices are taken from Burstein et al. (1984) and G93 and are detailed in Table 6. Spectral ranges involved by the indices measurements are shown in the Figs. 2a,b (shaded areas) superimposed on a set of stellar spectra. Different spectral types are selected in order to emphasize the variations of the line-strength indices as a function of the stellar surface temperatures.
  
\begin{figure*}
\includegraphics [width=8.5cm,clip=]{ds1429f2a.eps}

\includegraphics [width=8.5cm,clip=]{ds1429f2b.eps}

a)\hspace{8cm} b)\end{figure*} Figure 2: Wavelength range (shaded area) of line-strength spectral indices superimposed onto different stellar spectra. The shaded areas are intended to highlight the variations of the spectral features as a function of spectral types

Almost all these indices are good metallicity indicators, while they are only slightly sensitive to stellar age variations. The only relevant exception is the H$\beta$ index, that is the most widely adopted optical age indicator. Like all the Balmer absorptions, the H$\beta$ line appears very weak in the cold stellar types (M, K), while its intensity grows with temperature, reaching its maximum value in the spectra of A type stars. In Paper III, we will show some evidences that suggest that the H$\beta$ index is not very sensitive to the ages of stellar populations younger than 1 Gyr.

We indicate as blue three indices, not present in the Lick set, in the wavelength range $3750 < \lambda <$ 4200 Å, namely, H+K(CaII) and H$\delta$/FeI indices, defined by Rose (1984, 1985), and the $\Delta$(4000 Å) index defined by Hamilton (1985). The "blue'' part of a galaxy spectrum is much more sensitive to the stellar population age than the "red'' one (see Paper IV).

The H+K(CaII) index represents the ratio between the central intensity of the H(CaII)+H$\varepsilon$ line (a blend of the H(CaII)3968.5 Å with the Balmer H$\varepsilon$) and that of K(CaII)3933.7 Å line. In the same way, the definition of H$\delta$/FeI index is the ratio of the central intensity of the Balmer H$\delta$line with the average value obtained from the central intensities of two FeI lines, Fe4045 and Fe4063. The spectral windows adopted to identify the centre of each of these lines are reported in Table 7. These two indices are Balmer lines measures, and they are then good age indicators (just like H$\beta$); H$\delta$/FeI is sensitive also to the metallicity parameter for its dependence from the FeI lines.

  
Table 7: Definition of the "blue'' indices


\begin{tabular}
{rrcc}
\noalign{\hrule\medskip} 
Index & Spectral & $\lambda_{1}...
 ...\AA\ break & & 3750.0$-$3950.0 \\  & & & 4050.0$-$4250.0 \\ \hline \end{tabular}


The $\Delta$(4000 Å) index maps the 4000 Å break. It is defined as the ratio of the average fluxes (for frequency unit) measured in the spectral ranges [4050 Å-4250 Å] and [3750 Å-3950 Å]:
\begin{displaymath}
\Delta(4000~\mbox{\AA}) =\displaystyle\displaystyle\frac{F_{...
 ...x{\AA}]}
{F_{\nu}\left[3750~\mbox{\AA}-3950~\mbox{\AA}\right]}.\end{displaymath} (3)
The definition of this index needs a measure of fluxes per frequency units [Hz-1], while data are calibrated in counts per wavelength units [Å-1]. So, we need to multiply the ratio between fluxes/Å by the correction factor:
\begin{displaymath}
(\lambda_1/\lambda_2)^2 = (4150/3850)^2=1.162\nonumber\end{displaymath}   
where $\lambda_1$ and $\lambda_2$ represent the central wavelength of the two spectral bandpasses adopted for the $\Delta$(4000 Å) measure. Note that this index gives information about the stellar parameters of the turn off stars (and consequently about the mean age of the stellar population) (W92).

Spectral bandpasses of the indices have been corrected for the galaxy redshift z:

\begin{displaymath}
\lambda_i{\rm (corr)} = \lambda_i (z+1)\end{displaymath}

\begin{displaymath}
\Delta\lambda{\rm (corr)} = \Delta\lambda (z+1)\end{displaymath}

where $\lambda_i$ stands for a generic bound of the spectral windows and $\lambda_i{\rm (corr)}$ for its corresponding corrected value, $\Delta\lambda$ and $\Delta\lambda{\rm (corr)}$ indicate respectively the original and the corrected spectral width. The redshift value, z = v/c, is directly estimated from the spectra lines.

4.2 The conversion to the Lick-IDS system

The Lick Group (Burstein et al. 1984; Faber et al. 1985; Burstein et al. 1986; Gorgas et al. 1993; W92; Worthey et al. 1994) has built fitting functions for 21 indices, based on a stellar spectral library of more than 400 stars. Their observations have been performed at the 3 m Shane Telescope (Lick Observatory) equipped with an Image Dissector Scanner, characterized by a resolution of 8.2 Å (FWHM). The present work is based on the measure of 16 red indices common to those studied by the Lick Group, for which we will use their standard fitting functions (Longhetti et al. 1997b, Paper III). In this context we need to transform our data into the "standard'' Lick-IDS system. For this purpose, we have observed a sample of 19 stars of different spectral types (Table 8),
  
Table 8: Stars from Lick Library


\begin{tabular}
{cccccccc} 
\hline 
HR & HD & $V$\space & $B-V$\space & $U-B$\sp...
 ...03 & & & & 0.61 & K5V & \\  & 64606 & & & & 0.47 & G8V & \\ \hline \end{tabular}


common to the Lick library. On their spectra, once degraded to the IDS resolution, we measured the 16 red indices. The comparison between our results with those of the Lick group (Faber et al. 1985; Gorgas et al. 1993; W92) shows an acceptable agreement between the two measurement systems for some indices (H$\beta$, Mgb, Fe5335, Ca4227, G4300, Fe5270), while systematic differences are observed for the others. Table 9 reports the average values of the differences found on the stellar sample between the two measurement systems.

G93 has already pointed out that residual differences are connected to the fact that while calibration of CCD data are really relative flux (i.e. corrected for the CCD response function constructed from observations of standard stars), IDS data are calibrated with a tungsten lamp as a reference source. As a consequence, IDS measurements contain the lamp contribution to the corresponding spectral features. Following G93 we have applied a shift to some indices measures in order to fully transform them in the Lick-IDS system. The adopted shifts for each index are listed in the Col. 4 of Table 9. The shifts have been applied only to the indices for which the (IDS-CCD) values have a systematic trend. G93 reports 8 "red'' indices (among the 21 of the Lick set) measured on a sample of 35 stars common to the Lick library. The average differences reported by G93 are compatible with ours within errors. For comparison, in Table 9 we report also the shifts adopted by G93. A remarkable exception is represented by Fe5406, which shows a systematic scatter between our and Lick data, unlike G93 data. Figures 3a,b show the comparison between our fully transformed data of the sample of 19 stars and the Lick-IDS data.

  
\begin{figure*}
\includegraphics [width=8.5cm,clip=]{ds1429f3a.eps}

\includegraphics [width=8.5cm,clip=]{ds1429f3b.eps}

a)\hspace{8cm} b)\end{figure*} Figure 3: Comparison between our CCD measure (after the complete transformation to the Lick system) and W92 IDS data on the common sample of 19 stars. In each diagram, the error bar refers to the average value obtained on the whole sample. The dimension of the error bar along the abscissa is of the same order as the symbols

4.3 Correction of line-strength indices for velocity dispersion

The observed spectrum of a galaxy can be regarded as a stellar spectrum (reflecting the global spectral characteristics of the galaxy) convolved with the radial velocities distribution of its stellar population. Therefore the spectral features in a galaxy spectrum are not the simple sum of the corresponding stellar features, because of the motions of the stars composing the galaxy. If we want to explain the measure of the indices in terms of the stellar composition of the galaxies, we need to correct them for the effects of the velocity dispersion. We have estimated the correction studying the behaviour of all the indices on a sample of about 80 stars. Stellar spectra (after they have been degraded to the Lick-IDS resolution) have been convolved with gaussian curves of various width in order to simulate different galactic velocity dispersions. We have considered velocity dispersions in the range $50~\rm km~s^{-1} < \sigma < 300~\rm km~s^{-1}$. The results of these convolutions are reported in Table 10. Corrections for the velocity field were finally applied to the indices for which the correction itself shows a monotonic variation as a function of the velocity dispersion. This was the case for H+K(CaII), H$\beta$, Mg2, Mgb, G4300, Fe5270, Ca4227, CN, CN2, Ca4427, Fe4531, Fe4668, Fe5015, Fe5335, H$\delta$/FeI indices. The $\Delta$(4000 Å) behaviour is not reported in Table 10 because this index is insensitive to the spectral broadening caused by velocity dispersion, since it refers to quite large bandpasses. Mg1, Fe4383 and Fe5406 indices have not been corrected since they show a fluctuating behaviour as a function of $\sigma$ and any tentative correction can introduce a further error.

Corrections are calculated as a linear interpolation of the data in Table 10 corresponding to the actual velocity dispersion of each galaxy. Since the spectral indices refer to the nuclear region the correction is derived assuming a value of the velocity dispersion in the same region. Kinematical data adopted for the whole sample are reported in Paper II.

4.4 Error estimation in line-strength indices

Formally, the definition of an index is a ratio between fluxes integrated over particular bandpasses. The estimate of the error is then the propagation of the relative uncertainties in the fluxes used to calculate its value. If uncertainties of relative fluxes in different bandpasses are independent, the error is:
\begin{displaymath}
\sigma(I) = \sqrt{{\partial I \overwithdelims () \partial f_...
 ...{\partial I \overwithdelims () \partial f_3}^2 \sigma_{f3}^2 } \end{displaymath} (4)
where I = I(f1, f2, f3) indicates the dependence of the index on the fi flux measured in the three bandpasses. The derivatives of the formulas (1) and (2) give the expected uncertainty for atomic ($\sigma(I_{\rm a})$) and for molecular indices ($\sigma(I_{\rm m})$):
\begin{displaymath}
\sigma^2(I_{\rm a})={F_{\rm I1,I2} \overwithdelims () C_{\rm...
 ... \overwithdelims () \lambda_{\rm r}-\lambda_{\rm b}}^2
\Biggl] \end{displaymath} (5)

\begin{displaymath}
\sigma(I_{\rm m}) = {2.5\times 10^{0.4I_{\rm m}} \over 2.3026 (\lambda_{\rm I2} 
- \lambda_{\rm I1})}
 \sigma(I_{\rm a})\end{displaymath} (6)

\begin{displaymath}
\lambda_{\rm I} = {(\lambda_{\rm I1} + \lambda_{\rm I2}) \over 2}\end{displaymath}

\begin{displaymath}
\lambda_{\rm r} = {(\lambda_{\rm r1} + \lambda_{\rm r2}) \over 2}\end{displaymath}

\begin{displaymath}
\lambda_{\rm b} = {(\lambda_{\rm b1} + \lambda_{\rm b2}) \over 2}\end{displaymath}

\begin{displaymath}
C_{\rm I} = C(\lambda_{\rm I}) \hspace*{5.8cm}({\rm see}\ (2))\end{displaymath}

\begin{displaymath}
F_{\rm I1,I2} = \int_{\lambda_{\rm I1}}^{\lambda_{\rm I2}}F(...
 ...}^{\lambda_{\rm I2}} {F^2(\lambda) \over 
 \sigma^2(\lambda)} }\end{displaymath}

\begin{displaymath}
\sigma^2(F_{\rm b1,b2}) = {F_{\rm b}^2 \over \int_{\lambda_{...
 ... b2}}
 {F^2(\lambda) \over \sigma^2(\lambda) } {\rm d}\lambda}i\end{displaymath}

\begin{displaymath}
\sigma^2(F_{\rm r1,r2}) = {F_{\rm r}^2 \over \int_{\lambda_{...
 ...m r2}}
 {F^2(\lambda) \over \sigma^2(\lambda) } {\rm d}\lambda}\end{displaymath}

where $F(\lambda)$ is the spectral flux and $\sigma^2(\lambda)$ the flux variance as a function of wavelength. The flux variance is calculated from the wavelength calibrated frames, transforming the ADU counts into the corresponding photons counts. In the present work, the transformation is given by:

\begin{displaymath}
I{\rm (col,row)[photons]} = {\rm ADU} \dot
( I{\rm (col,row)[counts]}) \end{displaymath}

with $1 {\rm ADU} = 1.4{\rm e}^-$. Summing the Read Out Noise (RON) characterizing the CCD, we calculate the variance frame:

\begin{displaymath}
\sigma^2({\rm col,row})=I{\rm (col,row)[photons]}+\sigma^2_{\rm RON}\end{displaymath}

where $\sigma^2_{\rm RON}=6.5^2=42.25$. From this frame we extracted a 1D spectrum with the same parameters adopted to extract the fully calibrated spectrum of the corresponding object. This represents the $\sigma^2(\lambda)$ in the previous formulas. In fact, this variance describes the uncertainty that characterizes the measurements of flux for each spectrum, i.e. it represents the statistical error of the flux measurement pixel by pixel, which is present even if the photometric system is perfectly reproducible.

  
\begin{figure}
\centering

\includegraphics [width=8.5cm,clip=]{ds1429f4.eps}\end{figure} Figure 4: Comparison between our measures (indicated as "(our)'') and those of G93 (indicated as "(Gonzalez)'') on a common sample of 5 template galaxies (Legenda: asterisk= NGC 7626, open square= NGC 7562, full square= NGC 7785, open circle= NGC 7619 and full circle=NGC 584). In both systems, data are not corrected for velocity dispersions
Figure 4 shows the comparison between our and G93 data, on a subset of 6 red indices, including the H$\beta$ and Mg and Fe indices which will be used to reconstruct the star formation history of the galaxies. Our line-strength indices are in agreement, within the errors, with those of G93.

Actually the measurement of a line-strength index does not need an absolute flux calibrated spectrum, since only flux ratios enter in its definition. The estimate of the error calculated adopting (6) and (7) then corresponds to an upper limit of the real uncertainty that characterizes our index measurement. In fact, it does not take into account the consequences of the gaussian filter applied to our data before measurements of the index since the data quality is enhanced by this treatment to the spectral resolution cost. A more realistic way to calculate the uncertainty that affects our indices measures could then be to refer to a quantity that represents how the corresponding spectral feature is visible within a noisy continuum.

We substitute Eqs. (1) and (2) with their approximations obtained using constant average values of the fluxes in the integrals:
\begin{displaymath}
I_{\rm a} =
(\lambda_2 - \lambda_1) (1- {f_{\rm I} \over f_{\rm C}})\end{displaymath} (7)

\begin{displaymath}
I_{\rm m} = -2.5
{\rm log}_{10}{f_{\rm I} \overwithdelims () f_{\rm C}} \end{displaymath} (8)
where $f_{\rm I}$ is the flux average value in the central bandpass (indicated by $\lambda_1$and $\lambda_2$), and $f_{\rm C}$ is the average continuum flux calculated with (3) at $\lambda_{\rm I} = (\lambda_1 + \lambda_2)/2$.

In this way, Eqs. (6) and (7) become:
\begin{displaymath}
\sigma^2(I_{\rm a}) = (\lambda_2 - \lambda_1)^2 {f_{\rm I} 
...
 ...\overwithdelims ()
 \lambda_{\rm r} - \lambda_{\rm b}}
\Biggl ]\end{displaymath} (9)

\begin{displaymath}
\sigma^2(I_{\rm m}) = (1.0875)^2 \Biggl [ {\sigma(f_{\rm I})...
 ...overwithdelims () 
\lambda_{\rm r} - \lambda_{\rm b}}\Biggl ]. \end{displaymath} (10)
We then need to know only the average value of flux uncertainties, $\sigma(f_{\rm I})$, $\sigma(f_{\rm r})$ and $\sigma(f_{\rm b})$, substituting the variance function $\sigma^2(\lambda)$. These average values can be calculated as the dispersion of the flux around its average value in the corresponding bandpass. Errors computed following the procedure outlined above are reported in Tables 11, 12 and 13.

A remarkable exception is the error estimate for the H+K(CaII) and H$\delta$/FeI indices. Their definitions are based on the flux values corresponding to specific single pixels, and so their uncertainties are calculated propagating the measure of the errors on the flux extracted from the variance spectra (referred before to $\sigma^2(\lambda)$) at those pixels, decreased by a factor of about 2.8 (which corresponds to the square root of the number of the pixels involved in the gaussian filter windows).

4.5 Comparisons with line-strength nuclear indices for galaxies in the G93 sample

Table 11 reports our nuclear line-strength indices on the 5 "template'' galaxies belonging to the G93 sample. The Table lists also the differences between our results and those achieved by Worthey (1996, private communication), on the common subset of 16 "red'' indices. Worthey IDS data are of poorer quality than our CCD ones, and than they are characterized by greater uncertainties (see also G93).

The comparison is made using line-strength indices before the correction for the velocity dispersion, since this latter has been obtained by G93 with a procedure different from that described in Sect. 4.3. Our corrections for velocity dispersion are nearly identical to those of G93 for H$\beta$ and Mg indices, but our procedure led to corrections which are of a few percent smaller than in G93 for the Fe lines-strength indices. In Paper II we show that our estimate of velocity dispersions for template galaxies are in very good agreement with G93: $< (\sigma_{\rm G93} - \sigma_{\rm our}) \gt\ = 9\pm9$. These differences will be taken into account during the comparison of the two set of data in Paper III.

Data reported in Tables 12 and 13, relative to the indices measurements in our sample galaxies, have not been corrected for a possible contamination from the emission components. Paper II will address the issue of an analysis of these emission lines and a forthcoming paper that of a study of the absorption lines that takes into account their contamination due to filling.


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