\documentstyle[paasms4,epsf,pub]{preprint}
\newcommand{\Ha}{H$\alpha$}
\newcommand{\Teff}{$T_{\rm{eff}}$}
\newcommand{\Hb}{H$\beta$}
\newcommand{\Myr}{M$_{\odot}$~yr$^{-1}$}
\newcommand{\Ms}{M$_{\odot}$}
\newcommand{\kms}{km~s$^{-1}$}
\newcommand{\vinf}{v$_{\infty}$}
\newcommand{\Zs}{Z$_{\odot}$}
\newcommand{\Md}{$\dotM$}
\newcommand{\Mlow}{$M_{\rm{low}}$}
\newcommand{\Mup}{$M_{\rm{up}}$}
\begin{document}
 
\title{EVOLUTIONARY SYNTHESIS MODELING OF 
RED SUPERGIANT FEATURES IN THE NEAR-INFRARED}
 
\author{Livia Origlia,\thanks{Current address: Osservatorio Astronomico di Bologna, 
        Via Zamboni 33, I-40126 Bologna, Italy; e-mail: origlia@astbo3.bo.astro.it}
Jeffrey D.\ Goldader, Claus Leitherer, and Daniel Schaerer\\
\\
Space Telescope Science Institute\\
\\
3700 San Martin Drive\\
\\
Baltimore, MD 21218\\
\\
e-mail: goldader@stsci.edu, leitherer@stsci.edu, schaerer@stsci.edu
\and
Ernesto Oliva\\
\\
Osservatorio Astrofisico di Arcetri\\
\\
Largo E.\ Fermi 5, I-50125 Firenze, Italy\\
\\
e-mail: oliva@arcetri.astro.it}

\pub{The Astrophysical Journal}
\recacc{21 July 1998}{29 September 1998}
\url{1293}
\maketitle
\begin{abstract}
We present evolutionary synthesis models applied to near-infrared spectral 
features observed in the spectra of young Magellanic Cloud 
clusters and starburst galaxies. 
The temporal evolution of the first and second overtones of CO at 2.29~\micron\ 
(2--0~bandhead) and 1.62~\micron\~(6--3 bandhead) 
and of the $(U-B)$, $(B-V)$ and $(J-K)$ colors are investigated.  
 
We find that the current evolutionary tracks of massive stars with 
sub-solar chemical composition in the red supergiant phase are not reliable
for any synthesis of the temporal evolution of infrared stellar features. 

The high sensitivity of the selected infrared features 
to the atmospheric parameters of cool stars allows us to place  
constraints on the temperature and the fraction of time 
spent in the red part of the Hertzsprung-Russell diagram 
by massive stars during their core-helium burning phase. 

We derive a set of empirically calibrated spectrophotometric models by 
adjusting the red supergiant parameters such that the properties of the 
observed templates are reproduced. 
\end{abstract}
 
\section{INTRODUCTION}
 
Since Tinsley's (1972) pioneering work, various  
evolutionary synthesis models of stellar populations in galaxies 
have been developed, using improved theoretical and empirical spectral 
libraries for better comparisons between predicted and observed 
spectrophotometric properties of galaxies. 
A comprehensive compilation of the available models 
was published by Leitherer et~al.\ (1996). 
 
Galaxies with dominant young stellar populations ($\la$100~Myr) are called
`starbursts' (Searle, Sargent, \& Bagnuolo 1973). An overview of the
properties of the stellar content of starburst galaxies can be found in
Leitherer (1996). Moorwood (1996) reviewed starburst galaxies with particular
emphasis on infrared (IR) aspects. 
 
Substantial theoretical and observational efforts have been made to 
calibrate suitable ultraviolet (UV) and optical diagnostics for studying 
the stellar content of starburst galaxies (Leitherer et~al.\ 1996).
Massive stars have only few and comparatively weak features in the optical 
spectral region. Furthermore, most absorption lines are blended with nebular 
emission lines if the stars are embedded in ionized gas. Notable exceptions 
are the strong hot-star wind features in the UV (Leitherer, Robert, \& Heckman
1995) and the Wolf-Rayet feature at 4650/4686~\AA, which has been used to define 
a subset of starbursts called Wolf-Rayet galaxies (Conti 1991). Wolf-Rayet stars 
are massive stars at the end point of their evolution (Abbott \& Conti 1987).
Attempts to utilize Wolf-Rayet features for constraining the stellar 
population have been made (e.g., Meynet 1995; Schaerer 1996; Schaerer \& Vacca 
1998) although uncertainties in their modeling remain.
 
Only recently has such a systematic analysis been extended
to the near-IR range where red supergiants (RSGs) dominate. 
This spectral range is of particular
interest to the modeling of starburst spectra. 
RSGs are less evolved than Wolf-Rayet stars, and therefore 
their evolutionary status is thought to be better understood. 
The main features in the IR spectra of integrated stellar systems
are absorption lines due to neutral metals and molecules (e.g.,
Kleinmann \& Hall 1986; Origlia et~al.\ 1993, hereafter OMO93)
tracing red stellar populations, and the hydrogen
recombination lines of the Paschen and Brackett series, which mainly
provide information on the number of ionizing photons. Once the first RSGs have
formed, the near-IR spectra of starbursts show their characteristic
absorption features. Due to the fast technological improvements in IR array 
performances, near-IR spectra of stellar clusters and galaxies showing RSG
features are becoming available in increasing numbers (e.g., 
Lan\c{c}on \& Rocca-Volmerange 1992; Goldader et~al.\ 1995, 1997; 
Oliva et~al.\ 1995; Smith et~al.\ 1996; Oliva \& Origlia 1998, hereafter OO98).
 
When dealing with integrated populations, one of the most critical issues in 
properly classifying their stellar content is the disentanglement of age and 
metallicity effects (cf.\ e.g., the review by Renzini 1986 and, more recently, 
Worthey 1994). OMO93 discussed the behavior of the near-IR CO features at 
$\lambda$1.62 and $\lambda$2.29 in individual red stars with varying stellar 
parameters such as effective temperature, gravity, metallicity, and microturbulent 
velocity. The results of OMO93, combined with theoretical models for the dependence 
of these parameters on stellar age and mass, offer the opportunity to make 
predictions for the strength of RSG spectral features during the times when massive 
RSGs dominate in starbursts, i.e., from about 8 to 30~Myr.
 
In this paper, we couple the new near-IR spectral diagnostics of OMO93 
to evolutionary synthesis models based on those of 
Leitherer \& Heckman (1995, hereafter LH95 and Leitherer et~al.\ 1998, 
hereafter L98). The technique is outlined in \S~2. Results of the numerical 
simulations and the confrontation with observations are in 
\S~3. Empirical modifications to the RSG models which lead to near-IR 
features in better agreement with observations are discussed in \S~4.  
Our conclusions are presented in \S~5.  
 
\section{SYNTHESIS TECHNIQUE}
 
All models in this paper were constructed using an updated evolutionary 
synthesis code based on that of LH95. The new code is described in detail
by L98.  Briefly, the code now incorporates the most recent stellar evolutionary 
tracks from the Geneva group at $Z=0.04$, 0.020, 0.008, 0.004, and 0.001 
(Schaller et~al.\ 1992; Schaerer et~al.\ 1993). It has also been updated to
allow the use of isochrone synthesis.  Finally, the corrected grid of atmospheres 
compiled by Lejeune, Cuisinier \& Buser (1997) is used. This atmosphere grid 
reproduces the observed effective temperature {\it versus\/} color relation at 
all metallicities.

As with the LH95 code, the new version allows the star formation rate to be one 
of two cases. First, stars could form in an {\em instantaneous burst\/} from 
interstellar gas of mass $M_{\rm{tot}}$ with no subsequent star formation. 
Alternatively, the models allow for {\em continuous star formation\/}, 
at a constant level determined by the star formation rate in \Myr. The stellar 
population is distributed along the zero-age main sequence following an intial 
mass function (IMF) which we parameterize as
\begin{equation}\phi(M) = \frac{dN}{dM} =  C M^{-\alpha}
\label{eq:imf}
\end{equation}
between the upper and lower cutoff masses, \Mup\ and \Mlow, respectively.
The normalization constant $C$ is determined by the total stellar mass. 
$\alpha = 2.35$ corresponds to the classical Salpeter (1955) value.   

Each star is tracked in the Hertzsprung-Russell diagram (HRD) from
the zero-age main sequence until its death. Effective temperatures \Teff,
surface gravities $\log g$, and surface abundances are assigned during each
time step and appropriate model atmospheres are chosen. 

[Z/Fe]~=~0 at all metallicities{\footnote{Throughout this paper we define 
metallicity as the mass fraction of all elements heavier than helium. The sun 
has $Z=0.020$ in this definition.}} are adopted in both the model atmospheres 
and in the evolutionary models. The nebular continuum (bound-free, free-free,
and two-photon) is computed and added to the stellar continuum. Thus, dilution 
of stellar spectral features by the nebular continuum is accounted for.
  
For the present study we enhanced the code by implementing the 
theoretical equivalent widths of CO $\lambda$1.62 and $\lambda$2.29
from OMO93. Their models predict these indices 
as a function of \Teff, $\log g$, microturbulent velocity $\xi$,  
and metallicities between 1/100 and 3 times solar.
These indices are successful in reproducing the corresponding 
observed spectra of both single giant and supergiant stars (OMO93), and
integrated stellar clusters with known parameters in both our Galaxy 
(Origlia et~al.\ 1997) and the Magellanic Clouds (OO98). 
The equivalent widths were obtained at each time step from a 
multi-dimensional interpolation in the spectral grids.
 
At temperatures less than 3000~K we set the indices
to the value at the lowest tabulated temperature, with all other parameters
held constant; at temperatures higher than 4500~K, we set the indices to 
zero.
These assumptions are reasonable since at temperatures higher than 4500~K 
the CO molecule is almost dissociated, 
hence the equivalent widths of the selected features become very small.
Similarly, the CO temperature dependence becomes negligible below 3600~K
(OMO93), and the equivalent widths of the selected lines do not change 
significantly at lower temperatures. 
 
A large grid of models was run exploring the following parameter space:
\begin{itemize}
\item Instantaneous burst and continuous star formation models.
\item Ages $<25$ Myr. 
\item Metallicities $Z=0.020$ (=~\Zs), $Z=0.008$ (=~40\% solar).
\item IMF with slopes $\alpha = 2.35$ and 3.3.
\item \Mup~=~100~\Ms; \Mlow~=~1~\Ms.
\item Microturbulent velocities of 3 and 5~\kms, typical for RSGs 
      (e.g., Tsuji et~al.\ 1994; OO98, and references therein).
\item Solar and non-solar [C/Fe], where C was depleted  
      by 0.5~dex. Carbon is 
      expected to be slightly depleted (e.g., Lambert et~al.\ 1984) 
      due to internal mixing in post-main-sequence stages.  
\end{itemize}

The changes in microturbulent velocity and [C/Fe] 
abundances were kept internal to the routine in which the values of the
indices were computed, and have no effect on other values calculated
by the code (e.g., colors).
 
The near-IR properties of stellar populations dominated by RSGs are rather
insensitive to variations of \Mup\ between 30 and 100~\Ms\ (cf.\ LH95). This
results from the small number of RSGs evolving from stars with progenitor
masses above $\sim30$~\Ms.
 
\section{RESULTS FROM STANDARD RSG MODELS}
 
In this section we highlight the key results of our simulations.  
We show models for the CO $\lambda$1.62 and $\lambda$2.29 
features and $(U-B)$, $(B-V)$ and $(J-K)$ colors and we 
compare the results to two sets of observational data: 
a sample of Large Magellanic Cloud (LMC) clusters and starburst galaxies.  

\subsection{Observational Templates}

We selected the 5 young LMC clusters NGC1818, NGC1984, NGC1994, NGC2004 and 
NGC2011 with measured CO features from OO98. 3~clusters from the OO98 sample 
were excluded. NGC1866 and NGC1987 are too old to harbor massive RSGs,
and NGC330 is in the SMC with a much lower metallicity. Optical colors are from 
van den Bergh (1981), extinction and the $(J-K)$ color are from Persson et~al.\ 
(1983). The uncertainties of the CO features and color are 0.3--0.5~\AA\ and 
$\le0.1$~mag, respectively, as discussed in the cited literature.
The same measurement errors apply to the starburst sample further below.
A range of ages has been assigned to each cluster according to the 
estimates from different authors (Hodge 1983; Elson \& Fall 1985, 1988; 
Girardi et~al.\ 1995; Cassatella et~al.\ 1996).  

The metallicity of individual young LMC clusters is not accurately known
(cf.\ e.g., the compilation by Sagar \& Pandey 1989) but on average values 
well below solar are inferred. For the selected clusters several independent 
estimates based on optical and infrared spectra (cf.\ Richtler, Spite \& Spite 
1989; Reitermann et~al.\ 1990; Jasniewicz \& Th\'evenin 1994; OO98) indicate 
values between one tenth and at most half solar. 

The selected sample of starburst galaxies with measured spectral features 
is the one observed by Oliva et~al.\ (1995). The galaxies are NGC253,
NGC1614, NGC1808, NGC3256, NGC4945, NGC7552, and NGC7714. We also added NGC1705 
to this sample, a well studied galaxy with colors from  Lamb et~al.\ (1985) and 
Quillen, Ramirez \& Frogel (1995). For these galaxies we also used $(U-B)$ and 
$(B-V)$ colors from Veron-C\`etty (1984) and Hamuy \& Maza (1987) and the $(J-K)$ 
color from Glass \& Moorwood (1985). The colors were corrected for extinction 
assuming an intrinsic $(J-H)$ in the range 0.6--0.7 (cf.\ e.g., Scoville et~al.\ 
1985) and the interstellar extinction law by Rieke \& Lebofsky (1985) for a 
homogeneous dust distribution. The de-reddened values are quite uncertain due to 
the amount of the correction and its aperture-size dependence, particularly in 
the case of the $(U-B)$ and $(B-V)$. Our main arguments will therefore be based 
on the equivalent widths of the near-IR features, which are reddening independent.
The presence of RSGs in the central region of these galaxies (Oliva et~al.\ 1995) 
indicates a burst age (as traced by RSGs themselves and by the Brackett lines) 
in the range between 10 and 100~Myr. The inferred light to mass ratios seem also 
to suggest that there is no significant contamination by an underlying, older 
stellar population, which would dilute the observed features. Hot dust, if 
present, could dilute the spectral features as well. OMO93 used the ratio of 
CO$\lambda$1.62/Si$\lambda$1.59 and CO$\lambda$1.62/CO$\lambda$2.29 to quantify 
the degree of dust dilution. Oliva et~al.\ (1995) applied this method to the 
starburst sample discussed in this paper and found no significant dust dilution 
in the central few arcseconds.

The metal abundance of starbursts  is poorly known and the available estimates 
vary by up to an order of magnitude, depending on the element used to trace 
metallicity, the galaxy sampled region, the observational technique and the 
spectral synthesis modeling. As an example, for NGC253 values ranging between 
less than one tenth up to about solar have been proposed (cf.\ e.g. Webster \& 
Smith 1983; Carral et~al.\ 1994; Zaritsky, Kennicutt \& Huchra 1994; Arimoto 
et~al.\ 1997; Ptak et~al.\ 1997; OO98). Nevertheless, in our sample there at 
least two objects whose metallicity is better constrained in the range 0.3--0.5 
solar, namely NGC1705 (Storchi-Bergmann, Calzetti \& Kinney 1994) and NGC7714 
(Gonz\'alez-Delgado et~al.\ 1995; Garc\'{\i}a-Vargas et~al.\ 1997; OO98).
 
The star-formation histories in the LMC clusters are most likely close to 
the idealized case of an instantaneous burst, as suggested by their 
small sizes of a few pc. This makes them the simplest stellar systems for 
comparisons with models. Starburst galaxies are more complex, with typical 
size scales of $\sim$kpc. By averaging over such large regions, most of the 
time information is lost, and the star formation history can be approximated by 
a steady-state model. The relatively small color and equivalent width range observed 
for starbursts suggest a steady-state situation as well. As opposed to the LMC 
cluster sample, the starbursts were not selected by their ages. Therefore it 
would be difficult to understand why they fall into a very narrow parameter 
range, which, for a given metallicity, would indicate equal ages in instantaneous 
models. For simplicity and as two possible extreme situations in the plots we 
compared pure instantaneous bursts with stellar clusters and continuous 
star-formation models with galaxies.

\subsection{Spectroscopic Indices and Colors}

The temporal variation of the equivalent widths of the selected CO $\lambda$1.62 
and $\lambda$2.29 features predicted by our models for both an instantaneous burst 
and continuous star formation using a Salpeter IMF are plotted in Figs.~1 and 2 
for solar and 40\% solar metallicities, respectively. The corresponding values 
measured in the selected LMC clusters and starburst galaxies are plotted for 
comparison, as well.
\begin{figure}[tp]%1
\centerline{\epsfxsize=4in \epsfbox{p1293f1.eps}}
%\caption{\label{fig1}
\medskip
\noindent{\bf Figure 1.---}
The behavior of the CO spectroscopic features for instanteneous burst 
(left panels) and continous star formation models (right panels)
using a Salpeter IMF, solar metallicity, microturbulent velocity $\xi$ of 3 
and 5~km/s, solar [C/Fe] (dotted lines), one third solar [C/Fe] (dashed lines). 
The CO $\lambda$2.29 is related  to photometric CO index by the empirical 
relations (Kleinmann \& Hall 1986 and Origlia et~al.\ 1997):              
$$ {\rm [CO]_{spec}} = -2.5 {\rm ~log~}
          \left [ 1 - {{\rm W_{\lambda}}(2.29)\over 53 {\rm \AA}} \right ];~~~  
{\rm [CO]_{phot}} \simeq -0.6\times {\rm [CO]_{spec}} - 0.017.$$ 
For the instantaneous burst models we also show the values measured in a sample 
of young LMC clusters (cf.\ OO98), while in the continuous star formation 
models the rectangular boxes indicate the range of values ($\pm 1\sigma $) 
measured in a sample of starburst galaxies by Oliva et~al.\ (1995).  
\end{figure}
\addtocounter{figure}{1}
\begin{figure}[tp]%2
\centerline{\epsfxsize=4in \epsfbox{p1293f2.eps}}
\caption{\label{fig2}
As in Fig.~1, but for 40\% solar metallicity.}
\end{figure}

The $(U-B)$, $(B-V)$ and $(J-K)$ colors of the $Z=0.020$ and $Z=0.008$ models 
and the corresponding, de-reddened  values measured in the LMC clusters and 
starburst galaxies are plotted in Fig.~3. The relative large spread in color 
shown by the LMC clusters (a few tenths of magnitude) can be ascribed mainly to 
age and/or metallicity effects and some uncertainty in the reddening correction. 
In the case of starbursts the main uncertainty is the 
adopted reddening correction, as already mentioned above.      
\begin{figure}[tp]%3
\centerline{\epsfxsize=4in \epsfbox{p1293f3.eps}}
\caption{\label{fig3}
Colors of the standard burst (left panels) and continuous star formation 
(right panels) models with $Z=0.020$ (dotted lines) and $Z=0.008$ (dashed lines). 
As in Figures~1 and~2, the de-reddened values of a sample of LMC clusters and 
starburst galaxies (boxes) are also shown for comparison. The boxes indicate the 
range of values computed for the galaxies, assuming uncertainties in the adopted 
$A_V$ of $\pm0.5$~mag.}
\end{figure}

From the comparison of models and observational templates we find that the
observed CO features and $(J-K)$ colors can only be reproduced by models with 
metallicity very close to solar. Theoretical curves at 40\% solar metallicities
systematically predict too weak CO features and blue $(J-K)$ colors for all 
objects but NGC1818 and NGC1984 which, however, are known to have metallicities 
$\le$20\% solar.

The discrepancy between models and observations becomes even more severe when 
the latter are compared with continuous star formation models, which significantly 
underestimate (by up to a factor of two) the equivalent widths of the observed CO 
features and the $(J-K)$ color. 

Both the 40\% and solar metallicity models can reasonably fit the observed 
$(B-V)$ colors, while the behavior of the $(U-B)$ color is more complex, being 
redder than observed in the case of an instantaneous burst and bluer than observed 
in the case of continuous star formation.
 
\subsection{Why Does the Standard RSG Model Fail?}

By comparing theoretical predictions and observations of both 
LMC clusters and starburst galaxies with sub-solar metallicities, 
we have demonstrated that the near-IR features cannot be reproduced 
with models at appropriate metallicities.  

Different assumptions for the IMF do not change the conclusion, as one can see 
in Fig.~4 where we compare models with both a Salpeter and a Miller-Scalo IMF 
(approximated as a power-law with index 3.30). The results are very similar at 
both $Z=0.020$ and $Z=0.008$. The negligible dependence on the IMF simply follows 
from the fact that RSGs provide both the line flux and the adjacent continuum. 
Therefore the IR indices sample a relative small stellar mass interval. This is 
very different from, e.g., stellar-wind lines in the UV or the Wolf-Rayet feature 
at $\lambda$4686~\AA\ (cf.\ Leitherer et~al.\ 1995; Schaerer \& Vacca 1998).
\begin{figure}[tp]%4
\centerline{\epsfxsize=4in \epsfbox{p1293f4.eps}}
\caption{\label{fig4}
Comparison between burst models with standard Salpeter (continuous lines) 
and Miller-Scalo (dashed lines, approximated by a power-law slope of 3.30) 
IMFs at $Z=0.02$. Differences in the IMF have very little effect on the 
strengths of both the spectroscopic indices and colors.}
\end{figure}

Different assumptions for the model of atmospheres do not change the result, 
either. When we built the OMO93 code we performed tests to investigate how the 
choice of the adopted model of atmosphere for given stellar parameters influences 
the computed equivalent widths. No significant dependence on the adopted model 
atmospheres was found. The main reason is simply the fact that the main source of 
opacity in late type stars is the H$^-$ ion. We also performed test calculations 
with our L98 code, which has the capability to model cool stars with Kurucz or 
Lejeune models, or with blackbodies. All three choices uniformly give the same 
result: too weak RSG features. As we will demonstrate below, the dominant effect 
is the failure of the evolutionary models which predict incorrect RSG parameters.
The failure is so severe that the details of the model atmospheres become negligible.

To understand why the model spectra differ from those of real stellar clusters 
and galaxies, we should explore how RSGs affect the integrated light. Before the 
first RSGs form, the near-IR spectrum will be dominated by emission from ionized 
gas.  The UV spectrum is dominated by the photospheric emission from O-B stars.  
Once these stars begin to turn off the main sequence, they become massive RSGs 
and dominate the near-IR spectrum in the range of ages between 8--30~Myr 
(involving stars of mass $\sim25$--10~M$_{\sun}$). Actually the evolution of 
intermediate mass RSGs ($\sim5$--15~M$_{\sun}$) is more complex since these 
stars also have blue excursions towards higher temperatures which may last a 
large fraction ($\ge$50\%) of the total core-helium burning phase, particularly 
at low metallicities.

During the RSG phase the integrated colors of the evolving 
stellar population should become red and the absorption 
features typical of cool stars deep, but how much red and 
deep depends on their average temperature and blue-loop duration, 
which in turn depends on many factors (metallicity, mass loss rate etc.)

In order to better identify the possible cause of the mismatch between predicted 
and observed IR features, we computed near-IR indices and colors of {\em individual 
stars\/} predicted by the evolutionary models instead of considering an entire 
population. The mass range of the relevant contributors to the IR indices and colors
at ages $\le30$~Myr is roughly 25 to 8~\Ms. Fig.~5 shows the predicted CO indices of 
a 25, 20, 15, 12, 10 and 8~\Ms\ star for solar and 40\% solar metallicity.
\begin{figure}[tp]%5
\centerline{\epsfxsize=4in \epsfbox{p1293f5.eps}}
\caption{\label{fig5}
Instantaneous burst models for the CO features, taking into account only the 
contribution of RSGs with single masses: 25, 20, 15, 10 and 8 \Ms, respectively, 
at $Z=0.02$ (left panels) and $Z=0.008$ (right panels). The shaded regions 
correspond to the blue loops. }
\end{figure}

The \Zs\ models (cf.\ Fig.~5, left panels) for 25, 20 and 15~\Ms\ reach the RSG stage 
at $\simeq6.5$, 8.5 and 12~Myr, respectively, when the equivalent widths start 
to peak. About 1~Myr later the RSGs explode as supernovae, and the CO feature 
is no longer observed. The duration of the RSG phase increases with decreasing 
main-sequence mass as indicated by the broader equivalent width peak at 15~\Ms\ as 
compared to that at 25 and 20~\Ms. A qualitative change of this behavior occurs for 
stars with masses smaller than 12~\Ms. The 8--12~\Ms\ models at \Zs\ have two peaks,
separated by a more than 1~Myr long gap of zero equivalent width. The gap is
caused by a blue loop in the evolutionary models, when temperatures above
10,000~K are reached, too high to produce any CO feature.  Since RSGs have a 
blue excursion lasting about 50\% of the entire RSG lifetime, no CO features 
are predicted during about 50\% of the RSG lifetime for masses below about 15~\Ms.

Next we consider stars with 0.4~\Zs\ (cf.\ Fig.~5, right panels). The resulting 
models produce lower equivalent widths than at solar metallicity. Three main 
reasons can explain such a behavior: at lower $Z$ the effective temperature
of RSGs is higher, blue loops start at larger masses ($\sim15$~\Ms), and 
they last longer (about 20\% more). The offset in time between the indices of 
stars having the same mass but different $Z$ is due to slower evolution at 
lower $Z$ (Maeder 1990).

In an instantaneous burst, integration over all mass ranges will lead to the
prediction that the presence or absence of blue loops significantly affects
the line equivalent widths and the $(J-K)$ color of a RSG population, while it 
does not severly contaminate the optical colors. A clear example of this 
prediction is the discontinuity at about 14~Myr in the instantaneous burst 
models of the infrared features, where the equivalent widths of the absorption 
lines drop and the $(J-K)$ color turns blue for a short period of time (see 
Figs.~1 and 2). This discontinuity is also present in the solar models but it 
is less obvious since blue loops are more pronounced at low metallicity.

Cases for continuous star formation are affected to a much smaller
degree: the main contributor to the line equivalent width has a main-sequence
mass of 15~\Ms\ and above, and such stars do not have blue loops.

As a result of the warm temperatures and very pronounced blue loops at sub-solar 
$Z$ in the Geneva models, the equivalent widths of the absorption features and the 
$(J-K)$ color cannot approach the values observed in the clusters. Mayya (1997) 
already reached similar conclusions for the $(J-K)$ color. OO98, using different 
models based on Padova evolutionary tracks (Bertelli et~al.\ 1994), also found 
that at low $Z$ the models predict too warm temperatures for the RSGs.

Stars on blue loops, which are still cooler than the more massive blue supergiants 
in an earlier evolutionary stage, significantly contribute to the $(U-B)$ color (mainly 
to the B~luminosity) in an instantaneous burst. This makes the $(U-B)$ redder. 
The continuous star formation models predict much bluer $(U-B)$ than the 
instantaneous ones since the blue-loop stars are less dominant. Moreover, the 
presence of the blue loops may also account for the apparent reddening of the 
$(U-B)$ color when metallicity decreases since the blue loops become progressively 
more pronounced.
  
While Meynet (1993) and Mayya (1997) suggested that perhaps changing the 
mass-loss rate could provide a solution, we have taken a more general approach.  
In an attempt to provide guidance to the stellar evolution groups, we now derive
empirical limits on the RSG temperatures and the fraction of the core-helium 
burning lifetimes the massive stars spent as RSGs (vs.\ time spent in blue loops, 
or as blue supergiants).

\section{RESULTS FROM THE MODIFIED MODELS}
 
\subsection{Theoretical Background}
 
There are a number of plausible reasons why the theoretical 
predictions in the near-IR do not match the observed RSG populations.
 
The coolest temperatures predicted by evolutionary models for RSGs of fairly 
high mass ($M > 10$~\Ms) are quite uncertain. The major uncertainty stems from 
the treatment of the external convective layers in these stars. As discussed by 
Maeder (1987), the treatment of non-adiabatic convection with the usual mixing 
length theory (MLT) leads to supersonic convective velocities, which is in 
contradiction to assumptions of this theory. To remedy this situation one 
includes (cf.\ Schaller et~al.\ 1992) the contribution of the acoustic flux to 
the energy transport and the turbulent pressure in the hydrostatic equilibrium. 
Furthermore, a density scale height is adopted to prevent density inversions.
The difference between this treatment and the usual MLT is illustrated by Maeder \& 
Meynet (1987, their Fig.~9) and by Chiosi et~al.\ (1992). In view of these 
uncertainties, the predictions for both the effective temperatures of RSGs and 
their dependence on metallicity have to regarded as considerably uncertain. 
This most likely explains the difficulties of evolutionary models to correctly 
reproduce the radii and temperatures of massive RSGs. Future progress will be 
required to reliably model the extreme conditions in these convective envelopes.
 
Another crucial difference is the amount of time spent as a RSG during the 
core-helium burning phase. At solar metallicity a supergiant star spends a large 
fraction of its core-helium burning phase in the red near the coolest position 
it will reach on the HRD. As metallicity decreases, more and more of the 
core-helium burning lifetime is spent at significantly higher temperatures 
($\ge 10^4$~K), where these objects will presumably be classified as blue 
supergiants. It is well known that the transition between blue and red supergiants 
is particularly sensitive to different model ingredients (internal mixing, 
mass loss, opacities etc.; e.g., Maeder \& Meynet 1988; Ritossa 1996). As 
discussed in detail by Langer \& Maeder (1995) no current set of evolutionary 
models is able to reproduce correctly the {\em variation\/} of the observed 
ratio of blue to red supergiants with metallicity.
  
\subsection{Models with Empirically Adjusted RSG Parameters}
 
Given the above situation, we explored a set of models where the stellar 
effective temperature and the fraction of time spent by a massive star in the 
red part of the HRD during the core-helium burning phase are free parameters.
This should allow us to better quantify the role played by these parameters
in the overall spectral properties of the selected indices and to provide feedback 
constraints on the evolutionary tracks of massive stars. We ran models at $Z=0.008$ 
fixing the fraction of time during the core-helium burning phase spent in the 
red (i.e., as RSG) at 10\%, 50\% and 98\% and the RSG temperature at 3500~K, 
4000~K, and 4500~K, regardless its mass. The temperature and lifetime adjustments 
were performed as follows: for all time steps on the evolutionary track after 
the step corresponding to 90\%, 50\%, or 2\% of the total core-helium lifetime, 
the temperature was set to 3500~K, 4000~K, or 4500~K.

\begin{figure}[tp]%6
\centerline{\epsfxsize=4in \epsfbox{p1293f6.eps}}
\caption{\label{fig6}
Modified models for the CO $\lambda$1.62 index, where we have empirically 
adjusted the RSG temperature and lifetime during the core-helium burning phase.
The temperature is displayed at the top of the panels while the fraction of time 
spent as a RSG is displayed on the right side.  In each panel, we show the 
instantaneous burst (dotted lines) and continuous star formation (dashed lines) 
models at $Z=0.008$, solar [C/Fe] relative abundance and assume a Salpeter IMF.
The de-reddened values of young LMC clusters and starburst galaxies (shaded 
regtangles) are also plotted for comparison.}
\end{figure}
\begin{figure}[tp]%7
\centerline{\epsfxsize=4in \epsfbox{p1293f7.eps}}
\caption{\label{fig7}
As in Fig.~6, but for a [C/Fe]=--0.5.}
\end{figure}
\begin{figure}[tp]%8
\centerline{\epsfxsize=4in \epsfbox{p1293f8.eps}} 
\caption{\label{fig8}
As in Fig.~6, but for the CO $\lambda$2.29 index.}
\end{figure}
\begin{figure}[tp]%9
\centerline{\epsfxsize=4in \epsfbox{p1293f9.eps}}
\caption{\label{fig9}
As in Fig.~8, but for a [C/Fe]=--0.5.}
\end{figure}
\begin{figure}[tp]%10
\centerline{\epsfxsize=4in \epsfbox{p1293f10.eps}}
\caption{\label{fig10}
As in Fig.~6 but for the $(U-B)$ colors.}
\end{figure}
\begin{figure}[tp]%11
\centerline{\epsfxsize=4in \epsfbox{p1293f11.eps}}
\caption{\label{fig11}
As in Fig.~6, but for the $(J-K)$ colors.} 
\end{figure}
The main results are shown in Figs.~6--11 for the CO features with both solar 
and sub-solar [C/Fe], $(U-B)$ and $(J-K)$ color models, for both an instantaneous 
burst and continuous star formation. The behavior of the CO feature at $\lambda$1.62 
indicates that the observed values of young LMC clusters can be reproduced by 
40\% solar metallicity models only in a limited range of parameters. Temperatures 
$<4000$~K and fraction of time spent in the red during the core-helium burning 
phase of at least $\sim50$\% are required. These can be regarded as upper and lower 
limits, respectively, when models with carbon depletion are considered. For the 
galaxies, similar temperatures and lifetimes are required. It is also interesting 
to note that the discontinuity around 14~Myr progressively disappears when 
increasing the time spent in the red, also in better agreement with the modeling 
based on the Padova evolutionary tracks (cf.\ OO98). This is a clear indication 
that the discontinuity can be simply an artifact due to the blue loops and has 
nothing to do with the actual temporal evolution of the integrated stellar 
population.

The CO at $\lambda$2.29 is a less sensitive thermometer than the CO at $\lambda$1.62 
but its behavior is fully consistent with the trends shown by the latter (cf.\ 
Figs.~8 and 9). The analysis of the colors supports the temperature/RSG lifetime
conclusions from the spectroscopic features.  Only in the same temperature/RSG 
lifetime regime which can account for the indices, do we approach the $(J-K)=0.8-1$ 
needed to account for the observations of clusters and galaxies (cf.\ Fig.~11).  

The $(U-B)$ color (cf.\ Fig.~10) becomes slightly bluer both increasing the lifetime 
and decreasing the temperature of RSGs. This behavior is the confirmation that 
blue loops being cooler than the bluest supergiants but significantly warmer than 
the RSGs have the net effect to make the $(U-B)$ redder and the $(J-K)$ colors bluer.
Going to lower temperatures and longer lifetimes in the RSG phase cause the models 
to have only very blue and very red stellar populations dominating the luminosity 
in the UV to blue and in the infrared domain, respectively, in better agreement 
with the observed features.
 
Fractions of time spent in the red during the core-helium burning phase larger 
than 50\% imply a ratio of blue to red supergiants less than 1. Direct estimates 
of this ratio from observed color-magnitude diagrams (CMDs) are very rare and 
severly affected by statistical effects. Nevertheless in the few published CMDs 
of young LMC clusters there is evidence that this ratio is truly $\le$1 even at 
relative low metallicities (e.g., Chiosi et~al.\ 1995 for NGC330 and Balona \& 
Jerzykiewicz 1993 for NGC2004 and NGC2100). More efforts should be made to 
obtain a larger sample of reliable CMDs of young LMC clusters to provide the ratio 
of blue to red supergiants with much higher statistical significance.  
 
As shown by Mayya (1997) the observed Ca~II triplet, $(J-K)$ color and photometric 
CO index of a sample of young LMC clusters are better reproduced by models which 
enhance the mass loss by a factor of two with respect to the standard RSG model. 
However, the photometric CO index is still less than observed, and the UV-optical 
colors become redder than the observed ones. This indicates that mass loss cannot 
be the single cause for the discrepancy and other crucial parameters most likely 
play a major role, as discussed in Sect.~4.1. Unless more details in the evolution 
of RSGs become clearer, any diagnostics which involve temporal evolution of the 
IR features is still too premature. For example the predicted decreases of the 
Ca~II triplet equivalent width and the CO photometric index in the age range 
between 12 and 100~Myr is most probably an artifact of too-pronounced blue loops 
in the Geneva evolutionary tracks.
  
\section{CONCLUSIONS}    
 
The key results of this study are:
\begin{itemize}
\item Evolutionary tracks of RSGs can reasonably account for the observed 
optical features but fail to reproduce the observed $(U-B)$ color and the IR 
spectra of stellar clusters and starburst galaxies at {\em sub-solar metallicity\/}.
They do not produce sufficiently cool supergiants with long enough core-helium 
burning phase in the red part of the HRD.   
\item The high sensitivity of the selected spectroscopic indices to the physical 
parameters of cool stars make the present simulations a powerful tool to place 
constraints on the temperature and fraction of time spent in the red: at 40\% 
solar metallicity, the former should be less than 4000~K, and the latter at 
least 50\%. 
\item Synthesis models for the burst evolution which use IR diagnostics of 
massive red stars with sub-solar metallicity are still unreliable due to 
deficiencies in stellar evolution models.
\item Models with empirically adjusted RSG parameters allow a reasonable fit 
for both the observed UV to blue and IR features in young stellar clusters and 
starburst galaxies.
\item The full model set, with original RSG parameters at \Zs\ and with 
empirical adjustments at 40\% \Zs\, is available in electronic form 
at {\tt http://www.stsci.edu/ftp/science/ starburst}.  
\end{itemize}
 
\acknow
 
Livia Origlia gratefully acknowledges financial support from the ST~ScI Visitor 
Program. Funding for Jeff Goldader was provided by NASA grant GO-06672.01-95A 
from the Space Telescope Science Institute, operated by the Association of 
Universities for Reasearch in Astronomy, Inc., under NASA contract NAS5-2655. 
Claus Leitherer received financial support from the Visiting Professor Program 
of the University of Bologna. Salary support for Daniel Schaerer was provided by 
the ST~ScI Director's Discretionary Research Fund. The authors acknowledge the 
anonymous referee for the helpful comments and suggestions.

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\end{document}

