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Astron. Astrophys. 326, 950-962 (1997)

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2. The model

The two historical models of Tinsley (1972) and Searle et al. (1973) computed, respectively, the photometric evolution of galaxies from isomass stellar evolutionary tracks and isochrones. When applied to the same stellar evolutionary model and with similar input data (stellar spectra library, bolometric corrections...), both methods should give equivalent results, so that output differences must arise from deficient algorithms. Refined algorithms recently improved to conserve the released energy and the stability of outputs without suffering any degradation by smoothing methods are presented here. The stellar library is improved in the visible and the UV and is coherently extended by using NIR spectra and colors of stars on a significant cover of the HR diagram. Evolutionary tracks of the Geneva and Padova groups may be used up to the beginning of thermal pulses and are completed with stellar models of the final phases up to the PAGB phase. The nebular emission, computed as in Guiderdoni & Rocca-Volmerange (1987) (hereafter GRV), is extended to the NIR and a new modeling of extinction in elliptical galaxies is proposed.

2.1. The integration algorithm

Although the principle of spectral evolutionary synthesis is simple, computational problems and erroneous results may be caused by unoptimized algorithms. The monochromatic flux of a galaxy at age t and wavelength [FORMULA] may be written


where [FORMULA] is the star formation rate (SFR) at time [FORMULA] in [FORMULA] per time and mass units, [FORMULA] the initial mass function (IMF) defined in the interval [FORMULA] and normalized to [FORMULA], and [FORMULA] the monochromatic flux of a star with initial mass m at wavelength [FORMULA] and at age [FORMULA] since the zero age main sequence (ZAMS) (null if [FORMULA] exceeds the lifetime duration). A simple discretization of both integrals leads however to oscillations of the emitted light (Charlot & Bruzual 1991). Mainly due to the rapid evolutionary phases such as TP-AGB or massive red supergiants, they can be solved with a sufficient time resolution requiring substantial computer times (Lançon & Rocca-Volmerange 1996). In fact, oscillations are observed at any time t that a stellar population of mass m moves off from a stellar phase before the subsequent population of mass [FORMULA] reaches the same evolutionary phase. Resulting oscillations present a real difficulty in simulating instantaneous bursts, while they are artificially smoothed with continuous star formation laws. To avoid that problem, one possibility is to discretize only one of the two integrals. For example (isomass method, discretization of the integral on mass):


where [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA] is sufficiently small that equivalent phases of consecutive masses overlap. An alternative is to discretize the other integral on time (isochrone method):


with [FORMULA] short enough, so that consecutive isochrones [FORMULA] have evolved little. Both algorithms have been checked by us to give identical results. In the following, we prefer, unlike in our previous models, the isochrone method partly because isochrones are directly comparable to color-magnitude diagrams of star clusters and also for computational reasons.

2.2. Stellar spectra and calibrations

2.2.1. The stellar library

Although the libraries of synthetic stellar spectra become more and more reliable, the physics of the cold stars dominating in the NIR ([FORMULA] to [FORMULA]), notably the blanketing effects that lead to color temperatures very different to the effective ones (Lançon & Rocca-Volmerange 1992), is at the moment not taken sufficiently into account to build synthetic spectra of galaxies. For this reason, as in our previous models, we prefer to adopt observational libraries when possible and synthetic spectra otherwise. After reduction of various photometric systems to the Glass filters, standard optical and infrared colors were derived by Bessel & Brett (1988) for stars later than B8V and G0III. We have used these colors to derive fluxes at mean wavelengths of the infrared filters for the stars of our UV-optical library and fitted cubic splines to these fluxes. Hotter star spectra are extended in the NIR with a blackbody at [FORMULA]. The color temperatures derived in Lançon & Rocca-Volmerange (1992) could be used instead, but it should make no significant difference since, at these temperatures the blackbody is used in the Rayleigh-Jeans wavelength range. A stellar library observed with a better resolution in the NIR with the FTS/CFHT is in preparation. In the mid-infrared ([FORMULA]), we use the analytic extension of Engelke (1992) for stars colder than 6000 K. For M giants, which strongly dominate in the NIR, we use the spectra of Fluks et al. (1994) with the temperatures they provide. Their good resolution in spectral types is essential since [FORMULA] increases very rapidly with decreasing temperature.

The library from the far-UV to the NIR respects the effective temperature of any spectral type all along the wavelength range. Anomalous stellar spectra and wrong identifications of spectral types in the published libraries may produce erroneous colors and spectra of galaxies. For this reason, optical spectra were selected from the library of Gunn & Stryker (1983) according to the following procedure: [FORMULA], [FORMULA] and [FORMULA] color-color diagrams for all the stars of the library were plotted. Least square polynomials were fitted to the points, and we only selected stars in good agreement with the fits. Effective temperatures were derived from [FORMULA] according to the calibration from Straiys (1992), except for M dwarfs, the temperatures of which were calculated from [FORMULA] according to Bessel (1995). In the far-UV (1230-3200 Å), stellar spectra are extracted from the IUE ESA/NASA librairies (Heck et al. 1984) and, after correction for extinction with the standard law of Savage & Mathis (1979), connected to the visible. [FORMULA]   is computed from the observed [FORMULA] taken from Lanz (1986) or Wesselius et al. (1982) and the [FORMULA] corresponding to the spectral type from Straiys (1992). Anomalous spectra near 2000 Å (especially O stars) due to the bump of the extinction curve have been eliminated, as well as those in strong disagreement with the slope of the UV continuum of Kurucz (1992) for the corresponding temperature. In the extreme-UV (220-1230 Å), we complete our spectra with Kurucz (1992) models for [FORMULA]. The models of Clegg & Middlemass (1987) are finally used at all wavelengths for stars hotter than 50 000 K. Our stellar library is available in the AAS CD-ROM or on our anonymous account.

2.2.2. Bolometric corrections

We compute the bolometric corrections from Fluks et al. (1994) spectra for M giants (getting thus a coherent set of spectra, temperatures and bolometric corrections), and adopt those given by Bessel (1995) for late-M dwarfs, Vacca et al. (1996) for very hot stars or else the values tabulated by Straiys (1992). The bolometric corrections that we compute from our spectra, since only a negligible flux should be emitted outside our wavelength range, are in good agreement with the above values from the literature, making us confident that our identification in [FORMULA] is correct and that the junctions between the UV, optical and NIR domains are valid.

2.3. Evolutionary tracks

Stars are followed from the ZAMS to the final phases (supernovae or white dwarfs according to their masses), including the TP-AGB, fundamental to model NIR spectra, and PAGB phases. To check the sensitivity of spectral synthesis to evolutionary tracks, we compare the solar metallicity tracks of Bressan et al. (1993) (hereafter "Padova") to those of Schaller et al. (1992) extended by Charbonnel et al. (1996) (hereafter "Geneva"). The "Padova" tracks overshoot for masses [FORMULA] and use a higher ratio of the overshooting distance to the pressure scale height and down to lower masses than Geneva tracks, which include overshooting above [FORMULA] only. Both sets use the OPAL opacities (Iglesias et al. 92), similar mixing lengths, helium contents (0.28 for Padova and 0.30 for Geneva) and mass loss rates. We do not consider other metallicities, since these tracks already lead to significant discrepancies (see 3.1.1) which make the comparison of observed and synthetic spectra uncertain. Both sets go up to the beginning of the TP-AGB for intermediate and low-mass stars and have been prolonged by TP-AGB using typical luminosities and evolutionary timescales from Groenewegen & de Jong (1993) for stars less massive than [FORMULA]. The PAGB models of Schönberner (1983) and Blöcker (1995), supported by observations of planetary nebulae (Tylenda & Stasiska 1994), are finally connected to the tracks.

Whatever the algorithm used, interpolation between tracks requires the identification of the corresponding points. The interpolation algorithm adopted here aims to conserve the released energy along any track. For Padova models, sets of evident equivalent points are selected on consecutive mass tracks. Considering now such points [FORMULA] and [FORMULA] of track A and the corresponding ones [FORMULA] and [FORMULA] of track B, we may build a new track [FORMULA] replacing track B with [FORMULA] and [FORMULA]. Intermediate points [FORMULA] are computed iteratively so that [FORMULA], where [FORMULA] is the energy emitted from u to v. Equivalent points are given for Geneva tracks except in the interval [FORMULA], for which the previous procedure has been used. For low mass stars, we use the tracks of Vandenbergh et al. (1983). Except when otherwise stated, we use Padova tracks with their complements to the latest phases and to low mass stars, because of their higher resolution in mass and time and because Geneva tracks may not be interpolated after 16 Gyr, since post-helium flash evolution is not available in the [FORMULA] track.

2.4. Nebular emission

Gaseous nebulae are assumed to be optically thick in Lyman lines, according to case B recombination, the most likely for isolated nebulae (Osterbrock 1989). As in our previous models, the ratio of line intensities in the hydrogen recombination spectrum is computed for a given set of electronic temperature and density ([FORMULA], [FORMULA]) of astrophysical interest. In the NIR, the Paschen and Brackett lines were computed, relative to Balmer lines, by Pengelly (1963) and Giles (1977). Other emission lines such as [FORMULA] ([FORMULA]), He I ([FORMULA]) and [FeII ] ([FORMULA]), detected in NIR spectra of galaxies, were added in a ratio observed in typical starbursts (Lançon & Rocca-Volmerange 1996). Main lines of starbursts were also added, following Spinoglio & Malkan (1992).

The nebular continuum emission coefficients in the infrared are taken from Ferland (1980) for H I and He II. H I coefficients may be used instead of He I in the NIR (Ferland 1995). Two-photon emission coefficients are taken from Brown & Mathews (1970) but are negligible in the NIR.

The number of ionizing photons is a fraction f of the number of Lyman continuum photons computed from our spectral library, while the rest is assumed to be absorbed by dust. We take [FORMULA], in agreement with the values obtained by DeGioia-Eastwood (1992) for H II regions in the LMC.

2.5. Extinction

The extinction by dust which affects the SED of galaxies depends on the spatial distribution of dust and stars and on its composition, narrowly related to the metallicity Z of the ISM. The optical depth [FORMULA] is related as in GRV to the column density of hydrogen [FORMULA] and the metallicity.

The metallicity evolution [FORMULA] is computed from Woosley & Weaver (1995) SNII models without the instantaneous recycling approximation. Since the extinction is described as a function of the global metallicity, we neglect the yields of SNIa and intermediate and low mass stars. Models A are used for initial masses [FORMULA] and B for [FORMULA] following Timmes et al. (1995). From ([FORMULA]) and ([FORMULA]) Woosley & Weaver (1995) models, we may approximate the net yield [FORMULA] in solar masses as follows:


Dust effects may be approximated in the simplest cases of the phase function, respectively isotropy and forward-scattering. Calzetti et al. (1994) proposed to model scattering effects, with a combination of isotropic and forward-only scattering accounting for anisotropy, for mixed dust and sources by replacing [FORMULA] by the following effective depth


where the albedo [FORMULA] is from Natta & Panagia (1984), and the weight parameter [FORMULA] is derived by the authors from a Henyey-Greenstein phase function. We adopt their expression instead of the one of GRV, which corresponds to the case [FORMULA].

The geometry for the disk extinction is modelled by a uniform plane-parallel slab as in GRV. The resulting face-on optical depth for Sa-Sc spirals at 13 Gyr is about 0.55 in the B -band. Assuming the same geometry, Wang & Heckman (1996) deduced the face-on optical depth of a sample of disks from far-UV to far-infrared ratio and found [FORMULA], where [FORMULA] is the classical Schechter parameter of luminosity functions and [FORMULA] the corresponding depth, in good agreement with our values.

To model the extinction in spheroids, we must specify the distribution of stars and dust. Since it is more appropriate to describe the inner regions of ellipticals, which are the more affected by extinction, we prefer to use a King model for stars rather than a de Vaucouleurs profile. The distribution of dust in spheroids is poorly known. Fröhlich (1982) proposed to describe the density of dust as a power of the density of stars: [FORMULA], and found [FORMULA] for two ellipticals of the Coma cluster. Witt et al. (1992) and Wise & Silva (1996) suggest [FORMULA]. Values [FORMULA] would lead to strong color gradients in ellipticals that are not observed (Silva & Wise 1996). Tsai & Mathews (1995) obtain from the X-ray profile of three ellipticals that the distribution of gas is proportional to the square root of the starlight profile. Assuming a constant dust to gas ratio in these galaxies, we find once again [FORMULA]. We keep this value in the following and estimate the amount of extinction for the parameters of the model (b) of Tsai & Mathews (1995) which corresponds to a [FORMULA] galaxy. We finally suppose that the galaxy geometry has not changed with time. Introducing the core radius [FORMULA] and the outer radius [FORMULA], the light density at a distance r from the center is


if [FORMULA] and 0 otherwise. The ratio of the dust dimmed global flux F of the spheroidal galaxy to the direct flux [FORMULA] is


where z and R are the cylindrical coordinates, [FORMULA], [FORMULA] and [FORMULA]. k is computed from the central optical depth [FORMULA] derived from the central column density of hydrogen [FORMULA]. At any radius, we have


where K is derived from the total mass of hydrogen [FORMULA] of the galaxy. For a helium mass fraction of 28 % in the interstellar medium, we have


where [FORMULA] is the gas fraction, [FORMULA] the initial mass of gas of the galaxy and [FORMULA] the mass of the hydrogen atom. We finally obtain for the parameters of the model (b) of Tsai & Mathews (1995)


Although the central optical depth may be very high in the early phases of evolution of spheroidal galaxies, the extinction of the overall galaxy is only about 0.4 magnitudes in the B -band at maximum and is negligible nowadays. Since neither the geometry of stars and dust, nor the quantities and properties of dust in past E/S0 are known, the previous calculations should be considered simply as a reasonable attempt to give an order of magnitude of the effect of extinction in these galaxies.

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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998