1 Introduction

The spectroscopic study of the blue and near-UV region around $\lambda$4000 Å has proven to be a useful tool to investigate the stellar populations of composite stellar systems. Obviously, this spectral region is specially suited to detect the presence of young stars and therefore, to study star formation histories. Furthermore, in a pioneering work, [40, Morgan (1959)] already showed that the difference in intensity of the continuum level on the two sides of H$\zeta$ ($\lambda3889$ Å) in galactic globular clusters correlated well with metal abundance.

Although different absorption line-strength indices have been defined in this spectral range to understand the stellar composition of early-type objects (e.g., [22, Faber 1973]; [7, Burstein et al. 1984]; [48, Rose 1984, 1994]; [58, Tripicco 1989]; [67, Worthey et al. 1994], hereafter W94; [32, Jones & Worthey 1995]; [66, Worthey & Ottaviani 1997]; [62, Vazdekis & Arimoto 1999)], some of them are quite dependent on spectral resolution (and therefore velocity dispersion) and, in many cases, their use requires relatively high signal-to-noise ratios. In this sense, an interesting spectral index that avoids these problems is the $\lambda$4000 Å break. We have already demonstrated [14, (Cardiel et al. 1998b)] that this discontinuity can be measured with a relative error of $\sim 10$% with a signal-to-noise ratio per Å $\sim 1$. Thus, the break is well suited to be measured in faint objects or at low surface brightnesses. However, this advantage, due to the large wavelength interval employed in its definition, also translates into an important drawback: many absorption lines are included in the break bandpasses. Therefore, the behaviour of the break is expected to be complex.

In this work we are using the definition adopted by [5, Bruzual (1983)], who defined this spectral index as the ratio of the average flux density, $F_\nu$ (erg s^-1 cm^-2 Hz^-1), in two bands at the long- and short-wavelength side of the $\lambda$4000 Å discontinuity, in particular  

$${D}_{4000} = \frac{(\lambda_2^{-}-\lambda_1^{-})}{(\lambda_... ...int_{\lambda_1^{-}}^{\lambda_2^{-}}} F_\nu \; {\rm d}\lambda},$$ (1)

where $(\lambda_1^{-},\lambda_2^{-},\lambda_1^{+},\lambda_2^{+})$= (3750, 3950, 4050, 4250) Å. Except for the combination of $\nu$ and $\lambda$, (due to the measurement method employed by Bruzual, in which the break was obtained from galaxy spectra plotted as $F_\nu$versus $\lambda$), and for not being in logarithmic units, this definition resembles that of a color. Because of this, the D₄₀₀₀ can be considered as a pseudo-color. However, it must be pointed out that, compared to the classical color indices in this spectral range (like U- B), the 4000 Å break is much less sensitive to extinction by dust, and hence, it is more valuable to investigate the stellar content of galaxies. In particular, using the average interstellar extinction curve from [50, Savage & Mathis (1979)], the D₄₀₀₀ can be corrected from internal reddening (and galactic extinction for objects at zero redshift) using the following expression

$${D}_{4000}^{\rm corrected}={D}_{4000}^{\rm observed}\; 10^{-0.0988\; { E(B-V)}}.$$ (2)

It is interesting to note that the break definition is still the same for redshifted objects, where the integral limits must be properly modified, and that absolute fluxes are not required. As a working expression, the 4000 Å break can be rewritten as  

$${ D}_{4000} = \frac{{\displaystyle \int_{4250\,(1+z)}^{4050... ...,(1+z)}^{3950\,(1+z)}} \lambda^2 f_\lambda \; {\rm d}\lambda},$$ (3)

where $f_\lambda$ is the flux density per unit wavelength (in arbitrary units), and z is the redshift of the object being measured.

Up to date, many authors have employed the D₄₀₀₀ to study the stellar composition and star formation history of early-type galaxies (e.g. [38, McClure & van den Bergh 1968]; [54, Spinrad 1980, 1986]; [5, Bruzual 1983]; [35, Laurikainen & Jaakkola 1985]; [28, Hamilton 1985]; [19, Dressler 1987]; [21, Dressler & Shectman 1987]; [31, Johnstone et al. 1987]; [33, Kimble et al. 1989]; [20, Dressler & Gunn 1990]; [47, Rakos et al. 1991]; [16, Charlot et al. 1993]; [15, Charlot & Silk 1994]; [17, Davidge & Clark 1994]; [52, Songaila et al. 1994]; [3, Belloni et al. 1995]; [18, Davidge & Grinder 1995]; [12, Cardiel et al. 1995, 1998]; [1, Abraham et al. 1996]; [29, Hammer et al. 1997]; [2, Barbaro & Poggianti 1997]; [36, Longhetti et al. 1998]; [46, Ponder et al. 1998)]. The reliable analysis of the break measurements rests on the comparison of the data with the predictions of stellar population models (e.g. [65, Worthey 1994]; [6, Bruzual & Charlot 1996)]. So far, such predictions are computed by using either model atmospheres, or stellar libraries with a poor coverage of the atmospheric parameter space, especially in metallicity.

  

Figure1: Spectrum of the star HD 72324 (G9 III) in the region around the $\lambda$4000 Å break. As a reference, we have also plotted the most intense (${\rm EW} \gt 200$ mÅ) Fraunhofer lines from the sun (using the tabulated data from [34, Lang 1974] --original source [39, Moore et al. 1966]--), together with the Balmer lines and two CN molecular bands (the central bandpass of the CN3883 index defined by [43, Pickles 1985], and the absorption band of the S(4142) index employed by [51, Smith et al. 1997)] which can be found in this spectral range. The contribution of the atomic metallic lines, especially from Fe I and Mg I, becomes very important bluewards $\lambda$4000 Å

In this paper we present an empirical calibration of the $\lambda$4000 Å break as a function of the main atmospheric stellar parameters (namely effective temperature, surface gravity and metallicity) in an ample stellar library which covers an appropriate range of parameters to study relatively old stellar populations. One of the main advantages of using fitting functions to describe the behaviour of spectral indices is that they allow stellar population models to include the contribution of all the required stars, through a smooth interpolation in the space defined by the fitted stellar parameters. The usefulness of this approach has been demonstrated by the successful inclusion of similar fitting functions in recent evolutionary synthesis models (e.g. [65, Worthey 1994]; [63, Vazdekis et al. 1996]; [4, Bressan et al. 1996]; [6, Bruzual & Charlot 1996)].

It is important to keep in mind that the empirical calibration is only a mathematical representation of the break behaviour as a function of atmospheric stellar parameters, and that we do not attempt to obtain any physical justification of the derived coefficients.

We briefly review the previous works devoted to understand the D₄₀₀₀ in Sect. 2. The star sample is given in Sect. 3. The observations and data reduction are described in Sect. 4, whereas Sect. 5 contains a description of the error analysis. In Sect. 6 we show the behaviour of the measured D₄₀₀₀ values as a function of the stellar atmospheric parameters. The fitting strategy and the resulting empirical function are presented in Sects. 7. Finally, in Sect. 8 we give a summary, providing a public FORTRAN subroutine written by the authors to facilitate the computation of the D₄₀₀₀ using the fitting function presented in this paper. Sections 4 and 5 are rather technical, due to the inclusion of a lengthy explanation of the data and error handling. We suggest the reader not interested in such details to scan Tables 1 and 2, and skip those sections.