Modeling the Near-Infrared Luminosity Functions of Young Stellar Clusters

Abstract

We present the results of numerical experiments designed to evaluate the usefulness of near-infrared luminosity functions for constraining the Initial Mass Function (IMF) of young stellar populations. From this numerical modeling, we find that the luminosity function of a young stellar population is considerably more sensitive to variations in the underlying initial mass function than to either variations in the star forming history or assumed pre-main-sequence (PMS) mass-to-luminosity relation. To illustrate the potential effectiveness of using the KLF of a young cluster to constrain its IMF, we model the observed K band luminosity function of the nearby Trapezium cluster. Our derived mass function for the Trapezium spans two orders of magnitude in stellar mass (5 Msun to 0.02 Msun), has a peak near the hydrogen burning limit, and has an IMF for Brown Dwarfs which steadily decreases with decreasing mass. Comment: To appear in ApJ (1 April 2000). 37 pages including 11 figures, AAS: ver 5.0

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arXiv:astro-ph/9912384v1 17 Dec 1999
To appear in Astrophy. J. April 1, 2000 issue, Vol. 532
Modeling the Near-Infrared Luminosity Functions of
Young Stellar Clusters
August A. Muench1,2
Department of Astronomy, University of Florida, Gainesville, FL 32611 and
Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138
gmuench@cfa.harvard.edu
Elizabeth A. Lada
Department of Astronomy, University of Florida, Gainesville, FL 32611
lada@astro.ufl.edu
and
Charles J. Lada
Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138
clada@cfa.harvard.edu
ABSTRACT
We present the results of numerical experiments designed to evaluate the usefulness
of near-infrared luminosity functions for constraining the Initial Mass Function (IMF)
of young stellar populations. We test the sensitivity of the near-infrared K band lumi-
nosity function (KLF) of a young stellar cluster to variations in the underlying IMF,
star forming history, and pre-main sequence mass-to-luminosity relations. Using Monte
Carlo techniques, we create a suite of model luminosity functions systematically varying
each of these basic underlying relations. From this numerical modeling, we find that the
luminosity function of a young stellar population is considerably more sensitive to varia-
tions in the underlying initial mass function than to either variations in the star forming
history or assumed pre-main-sequence (PMS) mass-to-luminosity relation. Variations
in a cluster’s star forming history are also found to produce significant changes in the
1Smithsonian Predoctoral Fellow
2present address: Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Mail Stop 42, Cambridge, MA
02138 USA

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KLF. In particular, we find that the KLFs of young clusters evolve in a systematic man-
ner with increasing mean age. Our experiments indicate that variations in the PMS
mass-to-luminosity relation, resulting from differences in adopted PMS tracks produce
only small effects on the form of the model luminosity functions and that these effects
are mostly likely not detectable observationally.
To illustrate the potential effectiveness of using the KLF of a young cluster to con-
strain its IMF, we model the observed K band luminosity function of the nearby Trapez-
ium cluster. With knowledge of the star forming history of this cluster obtained from
optical spectroscopic studies, we derive the simplest underlying IMF whose model lumi-
nosity function matches the observations. Our derived mass function for the Trapezium
spans two orders of magnitude in stellar mass (5 > M⊙ > 0.02) and has a peak near the
hydrogen burning limit. Below the hydrogen burning limit, the mass function steadily
decreases with decreasing mass throughout the brown dwarf regime. Comparison of our
IMF with that derived by optical and spectroscopic methods for the entire Orion Neb-
ula Cluster suggests that modeling the KLF is indeed a useful tool for constraining the
mass function in young stellar clusters particularly at and below the hydrogen burning
limit.
Subject headings: infrared: stars — stars: low-mass, brown dwarfs — stars: luminosity
function, mass function — stars: pre-main sequence
1.Introduction
The development of sensitive, large format imaging arrays at near-infrared wavelengths has
made it possible to obtain statistically significant and complete samplings of the near-infrared
luminosity functions of very young embedded clusters. In principle such luminosity functions should
provide fundamental constraints on the stellar initial mass function (IMF)1of these clusters and
its possible variation in space and time. There are several advantages and disadvantages in using
young embedded clusters to study issues of the initial mass function (for a summary see Lada, Lada
& Muench (1998); Lada (1999)). The biggest advantage of studying the luminosity functions of
very young clusters is that the low mass stars in these clusters are brighter than at any other time
in their evolution. Consequently modern NIR detectors even on modest telescopes can completely
sample the entire IMF of nearby (D < 1 Kpc) embedded clusters down to and well below the
hydrogen burning limit. However, the youth of the clusters also makes interpreting the luminosity
functions of these clusters, in terms of an IMF, difficult, since most members are still in the pre-
main sequence phase of their evolution. For pre-main sequence stars, there is no unique mass to
1In all cases we will refer to the mass function not the mass spectrum of stars. The mass function is the number of
stars per unit volume per unit log mass. The Salpeter (1955) mass function would be a power-law ξ(log(m⋆)) = mΓ
with a index Γ = −1.35 in this definition.

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luminosity relation and consequently one must rely on using evolutionary models to understand
the underlying IMFs of embedded clusters.
Various groups have modeled the luminosity functions of young clusters using realistic stellar
mass functions and appropriate mass-luminosity relationships (e.g., Zinnecker et al., (1993); Strom,
Strom & Merrill (1993); Fletcher & Stahler (1994a); Lada & Lada (1995); Meageth (1996)).
Zinnecker, McCaughrean & Wilking (1993) were the first to present model KLFs of very young
clusters. For their model clusters they adopted a “coeval” star formation history in which all the
stars were formed at a single instant of time. Moreover, they assumed blackbody radiation to derive
bolometric corrections and assumed a single form for the IMF. Consequently, their models were not
very realistic and they did not attempt to fit or directly compare their models to observed KLFs.
Lada & Lada (1995) (LL95, hereafter) improved on this work by developing evolutionary models
for the KLFs of young clusters ranging in age from 106to < 107yrs, using empirically determined
bolometric corrections and allowing for non-coeval or continuous star formation in the clusters.
Moreover, they directly compared their models to observed KLFs of young clusters. However,
similar to the Zinnecker et al. work, Lada & Lada assumed a single underlying IMF (i.e., the Miller
& Scalo (1979) field star IMF) and employed the published pre-main sequence evolutionary tracks
of D’Antona & Mazzitelli (1994). In addition, their model KLFs were constructed for stars having
masses between 0.1 and 20 M⊙, since the existing PMS tracks did not extend below 0.1 M⊙. Thus
their results are only valid as long as there are no, or at least very few, brown dwarfs in these
clusters. Their modeling efforts produced several interesting results. For example, Lada & Lada
found that for a fixed IMF the shape of a cluster KLF depends primarily on the duration of star
formation and age of the cluster and that in general the KLFs broaden with age.
It is somewhat difficult to evaluate the success of the early modeling work to constrain the IMF
of embedded clusters because: 1) these early models typically employed one set of PMS tracks with
stellar masses greater than 0.1 M⊙; 2) the models only considered one fixed IMF and, 3) the ages
and age spreads of the clusters were not well constrained. Fortunately, improvements in these areas
have recently been made. For example, PMS evolutionary tracks have been extended to objects
with masses well below the hydrogen burning limit. In addition, improved age estimates for several
clusters such as the Trapezium (Hillenbrand 1997) and IC 348 (Herbig 1998) have been made.
Given these advances, we have extended the earlier modeling work and have developed a
new suite of KLF evolutionary models. These new models are based on Monte Carlo techniques
and are a considerable improvement over the earlier calculations. In particular, the new models
incorporate improved bolometric corrections and new PMS tracks, thereby including objects with
masses below the hydrogen burning limit. In this paper, we describe our new models and present
our results from this modeling effort. In addition, we investigate the utility of using near-infrared
luminosity functions for interpreting and constraining the IMF of young embedded stellar clusters.
Since the observed luminosity function of a young cluster depends not only on the underlying IMF,
but also the pre-main sequence mass-to-luminosity relation and the star forming history of the
clusters, it is important to understand how changes in the pre-main sequence mass-to-luminosity

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relations, star forming history and the underlying IMF affect the resulting KLF. Therefore, we have
systematically varied each of these three basic underlying inputs and have evaluated the sensitivity
of the KLF to each input. Finally, we illustrate the effectiveness of using the KLF of a young cluster
to constrain its IMF by modeling the KLF of the Trapezium cluster for which detailed knowledge
of its star formation history is known from optical spectroscopic studies (Hillenbrand 1997).
2.Modeling the Luminosity Function
The observed luminosity function (LF) for a young cluster depends on the pre-main sequence
mass to luminosity (M-L) relation, the star forming history (SFH) of the cluster and the cluster’s
underlying initial mass function (IMF). We constructed model K-band LFs (KLFs) for a suite of
synthetic young clusters using varying PMS M-L relations, SFHs and IMFs. We first specified a
functional form for the SFH and the IMF for each synthetic cluster. The ages and masses of the
individual cluster members were then sampled from these adopted functions using a Monte Carlo
rejection method algorithm. We derived the monochromatic magnitudes for each model star from its
age and mass by using theoretical pre-main sequence evolutionary models and empirical bolometric
corrections taken from the literature. For each synthetic cluster, we binned the resulting magnitudes
in half magnitude bins to compare to observed cluster luminosity functions. Our standard cluster
contained 1000 stars and for each SFH and IMF we produced 100 independent luminosity functions.
We computed the mean luminosity function from these 100 realizations, and report the one sigma
standard deviation of the computed mean.
For the models presented in this paper, we assumed a constant star formation rate over the
SFH of the cluster. We parameterized the SFH using a ”mean age”, τ, and an ”age spread”,
∆τ. For example, a coeval cluster will have no age spread and ∆τ/τ = 0.0. A cluster with the
largest possible age spread would have ∆τ/τ = 2.0 with star formation starting 2×τ years ago and
continuing to the present. Figure 1 illustrates these definitions. Our model SFH then approximates
any SFH to first order by using the most common age of the members and a rough age spread.
In our standard model, stars have masses from 80 to 0.02 M⊙. For all the models considered
here, the youngest (1 − 5 × 105years) cluster members with masses greater than 5 M⊙will have
already reached the Zero Age Main Sequence (ZAMS). For these massive stars, we converted their
age and mass to bolometric luminosity and effective temperature using a theoretical ZAMS derived
from Schaller et al., (1992). Between 1 and 5 M⊙, we used pre-main sequence evolutionary models
from either Palla & Stahler (1993) or Bernasconi (1996a). For stars below 1 M⊙ and brown
dwarfs, we used PMS evolutionary models from D’Antona & Mazzitelli (1994,1998). Appendix B
describes how these three regimes were combined. Using the bolometric luminosity and effective
temperatures derived from these stellar models, we converted to an absolute magnitude using the
formula:
Mλ= Mbol,⊙− 2.5 × log(L/L⊙) − BCλ(Teff) (1)

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Our empirical bolometric corrections were tabulated as functions of effective temperature and were
taken from the literature. In addition, we assumed Mbol,⊙ = 4.75. We list the sources of the
bolometric corrections in Appendix A.
For the purposes of this paper, we will only present an analysis of the K-band (2.2 µm)
luminosity functions for our model stars, though any monochromatic luminosity functions could be
calculated. Note that our models did not include the effects of excess infrared flux due to unresolved
binaries or circumstellar accretion disks around PMS stars. However, by choosing relatively wide
(half magnitude) bins for the luminosity function, we can minimize the variable effects of infrared
excess flux. The model KLFs for our young clusters did not include the effects of differential
interstellar extinction common in young star forming regions. We did not include protostars in our
luminosity function models since the contribution of proto-stellar objects to the luminosity function
of most star forming regions should be small (Fletcher & Stahler 1994a,b). Finally, we defined our
input mass function as a single star mass function.
3.Experiments
3.1. Evolutionary Pre-Main Sequence Models
The evolution of pre-main sequence (PMS) stars across the HR diagram and onto the main
sequence is not observationally well constrained. Details of PMS evolution rely heavily upon the-
oretical PMS models. These theoretical PMS models vary in their predictions depending on the
numerical methods and theoretical assumptions used in their creation. Since these PMS models
are used to convert from a stellar age and mass to a monochromatic magnitude, the resulting lu-
minosity functions will depend to some degree on the PMS evolutionary models which are chosen.
To evaluate how PMS models with different input physics, chemical abundances or effective mass
ranges affect the shape and form of a model luminosity function, we constructed and compared
model luminosity functions calculated assuming different PMS models.
For these experiments, we fixed the initial mass function to have a log-normal distribution:
ξ(log(M
M⊙))=c1 ∗ exp(−c2 ∗ (log(M
M⊙) − c3)2)), (2)
where c1 is a normalization constant, c2 equals 1/(2log(σ)2), c3 equals log(M
mass of the distribution, and σ is the variance of this mean. The mean and variance that we
adopted correspond to the field star mass function given by Miller & Scalo (1979) (MS79, here
after), having constants of c2 = 1.09 and c3 = −1.02 or a mean mass of 0.0955 M⊙. We produced a
suite of model clusters with a range of mean ages from 0.2 to 15 Myrs and age spreads from coeval
to twice the mean age of the model cluster. For the purposes of evaluating the effects of using
different input PMS models, we only directly compared KLF models having identical star forming
histories.
M⊙) or the mean log