- The document investigates the implications of a proposed metallicity-dependent initial mass function (IMF), which suggests the IMF varies based on the metallicity of a star formation environment. - Using observations of globular cluster Palomar 14 and open cluster M42, the author constrains an upper bound for metallicity dependence, resulting in a two-part power law IMF function that depends on metallicity. - However, the document concludes that current evidence is inadequate to prove a metallicity-dependent IMF, as measurements of cluster IMFs are complicated by issues like dynamics, binaries, and evolution over time.

1. An investigation of the metallicity dependent initial mass function
Charles Bergman
Term Project
Astronomy 25, Dartmouth College
March 7, 2014
Abstract
We investigate the implications of a metallicity dependent initial mass function discussed
in Kroupa (2007). Using observations of globular cluster Palomar 14 and open cluster M42,
we constrain an upper bound for metallicity dependence assuming an otherwise constant IMF,
resulting in a two-part power law function of m−.1−.55x for m < 0.3 solar masses, and
m−2.3−.65x for m > 0.3 solar masses. With this function we explore the potential implications
for the mass and luminosity distributions for clusters of differing metallicities, particularly in
the early universe.
1 Introduction
The initial mass function describes the distribution of initial masses for a population of main se-
quence stars as a probability distribution function. Accurately describing this function is critical
to astronomy, due to the close relationship between a star’s mass and various properties, including
color, luminosity, and metallicity. While several good approximations have been developed over
the years, beginning with Salpeter in 1955, significant uncertainty remains. One major point of
debate is whether and how the IMF varies depending on the star formation environment. Kroupa
(2007), examines the hypothesis based on current knowledge of stellar formation that the IMF is
dependent on the metallicity of the star formation environment. He determines that no statistically
significant evidence exists for metallicity dependence, and that therefore our current knowledge of
how stars form is wrong. Before exploring the implications of a metallicity dependent IMF, this
paper will examine the debate as it stands.
2 The canonical metallicity independent IMF
The current consensus regarding the initial mass function is that it is independent of metallicity.
”Canonical” mass functions are all rooted firmly in the Salpeter function (Salpeter 1955):
ξ(m)∆m = km−α
(1)
where α = 2.35. More recent propsals for the initial mass function modify the Saltpeter function
somewhat with certain mass ranges. For example, the Chabrier function (Chabrier 2003) proposes
1

2. a log-normal function
ξ(m)∆m = (0.158/m) exp[−(log(m) − log(0.08))2
/(2 × 0.692
)] (2)
for substellar masses, and a Saltpeter function with α = 2.3 for greater than solar masses. The
Kroupa (2007) function, on the other hand, has the following two-power law specification
α = 1.3, 0.08 < m < 0.5 (3)
α = 2.3, 0.5 < m (4)
Each of these functions map roughly similar distributions. The process by which we have arrived
at them is detailed below.
2.1 Mapping the Luminosity Function
The first stage of determining the IMF is to roughly determine the distribution of stellar luminosi-
ties. This can be done either through observations of the galactic disc, or of clusters. While clusters
have the advantage of relatively similar position, age and metallicity, dynamic considerations de-
tailed later in this paper render such observations questionable. For the galactic disc, their are two
main methods of finding a luminosity function. The first is a simple survey of all stars within a
given distance of the Earth. One advantage of this method is that most local stars have quite well
defined distances from the Earth. However, the low-mass faint end of the luminosity spectrum is
rather poorly constrained. Observations of solar mass stars are considered fairly complete within
20-30 pc, but the faintest dwarves may often go undetected at distances of 5 pc (Kroupa 2007). The
other method involves deep, pencil thin surveys of the galactic disc, to arrive at the ”photometic
luminosity function.” Luminosities are calculated using photometric parallax, relating a star’s lu-
minosity with its color. While this is less accurate than measurements of nearby stars, it has the
advantage of a much larger sample size, since these surveys can be conducted for numerous fields
of view.
2.2 Converting from luminosity to initial mass
After a luminosity function has been plotted, it becomes possible to derive the mass function using
the mass-luminosity relationships predicted by the main sequence. For most main sequence stars,
this can be roughly be estimated with
L = m3.5
(5)
(where L is in solar luminosities and m in solar masses). After constructing the mass function, one
must correct for the presence of binaries. This is not very problematic for local surveys, where
binary systems can usually be resolved, but it can be a major concern for galactic disc surverys.
The presence of unresolved binaries would result in many small mass stars being omitted from
consideration, as they are completely eclipsed by their brighter partners. The low mass end of
the function would therefore have a shallower slope than the actual distribution. Additionally,
given that the galactic disc as a whole is not extremely young, it becomes necessary to correct for
selective star death, due to the much shorter life span of large stars. After these corrections are
made, one should have a fairly good picture of what the initial mass function should be for the
2

3. Figure 1: Local (green) and photmoetric (red) stellar luminosity functions for the galactic disc. (Kroupa
2007)
galactic disc overall. Such a function, however, cannot tell us much about the intial mass function
of particular environments. That requires observation of clusters, which pose an entirely new set
of challenges detailed in section 4.
3 Theoretical support for the metallicity dependent IMF
3.1 Jeans Mass Arguments
According to Kroupa (2006), the hypothesis that metallicity effects the IMF is founded on two
arguments. The first is referred to as the “Jeans-mass argument.” The Jeans-mass is the amount of
mass that must be concentrated within a cloud with a radius of “Jean’s length” for gravitational col-
lapse to occur (Kroupa 2006). This mass is dependent on temperature and density by the following
relation:
M ∝ T3/2
ρ−1/2
(6)
(Larson 1998). Metal-rich environments should, according to proponents of this argument, allow
collapsing gas to cool more effectively due to larger quantities of dust. This means that the Jeans-
mass and, consequently, the initial mass of stars will be reduced. Additionally, the Jeans-mass
argument seems to explain observations of two-power law initial mass functions with a character-
istic mass somewhere between 0.08 and 0.5 solar masses. Bonnell et al. (2006) suggests this may
arise due to a coupling effect between dust and gas in the pre-stellar cloud, where the temperature
to density relation goes from cooling as
T ∝ ρ−0.25
(7)
3

4. to heating slightly as
T ∝ ρ0.1
(8)
. Higher dust content would tend to decrease this characteristic mass, shifting it towards smaller
masses. One problem Kroupa (2007) identifies with the Jeans-mass argument is that observations
seem to indicate a positive relationship between pre-cluster cloud density and stellar mass, when
an inverse relationship is predicted with Jeans mass theory (as denser clouds should reach Jeans
instability and collapse more readily, resulting in smaller fragments and lower stellar masses).
3.2 Sound Velocity and Accretion Rate Theory
The other argument, which avoids the aforementioned dense-cloud large-star problem, contends
that the Jeans mass has little effect on the stellar mass. Instead, the region of initial collapse simply
forms a hydrostatic core, which accretes the surrounding matter at a rate dependent on the sound
velocity (Kroupa 2007). Sound velocity is dependent on temperature. Therefore the relationship
between high dust content and cooling of the cloud shown earlier, should reduce the sound velocity
and thus the accretion rate. Thus the mass of the star once accretion completes should once again
be lower in high metallicity environments.
3.3 Concordance with Reionization Theory
Alongside theoretical star formation models, current understanding of the reionization process at
6 < z < 20 (Alvarez et al 2006), during the early universe also supports a metallicity dependent
IMF. During this epoch, the predominantly neutral gas within the universe which emerged after
recombination was once more reheated and ionized by high energy radiation. The theoretical
source of this radiation best supported by current evidence is a hypothetical population of very
old (Population III) stars (Alvarez et al. 2006). Such stars would have to be very luminous and
blue to produce sufficient UV radiation to reionize the universe. An initial mass function which
favors large stars at low metallicity explains this phenomena neatly, due to the ultra-low metallicity
expected in the early universe.
4 Inadequacy of current evidence for a metallicity dependent IMF
Given that two mutually-exclusive models of star formation and current theories of early universe
reionization predict a relationship between metallicity and the initial mass function, we would ex-
pect such a relationship to be fairly robust. In order to find tangible evidence for these predictions,
astronomers have performed luminosity function observations of clusters. Indeed, measuring clus-
ters may be the only way to find a relationship between the IMF and metallicity. Given that the
galactic disc as a whole has, to the best of our knowledge, a roughly solar metallicity, an IMF de-
rived using galactic disc observations would not be accurate for different metallicity environments
if the IMF is metallicity dependent. Use of clusters would seem to avoid most of the problems
associated with the galactic disc, as stars within clusters are tightly correlated in space, age and
metallicity. However, Kroupa (2007) contends that any measurement of the IMF using cluster ob-
servations, barring extremely exacting knowledge of cluster dynamics, faces too many problems
to be considered credible evidence of a metallicity dependent IMF.
4

5. 4.1 Young Clusters
Young clusters such as M42 would seem to be the ideal environment for determining the IMF, but
Kroupa (2007) identifies several problems with their use. Very young clusters on the order of a few
million years are likely to consist of many pre-main sequence stars. As a result the mass-luminosity
relation for these clusters is difficult to determine, and calculated masses may experience typical
deviations of as much as 50% (Kroupa 2002). There is additionally the problem that early clusters
have a high binary fraction, which would require corrections. The most problematic feature of
young clusters though is their extremely rapid evolution during the first few million years due
to residual gas expulsion. M42, while only 1 million years old, is thought to be 5 to 15 initial
crossing times old already (Kroupa 2007). This may result in substantial losses of the initial stellar
population, and if there was some mass segregation within the initial population, this will distort
the observed mass function. Since the initial mass segregation is nearly impossible to determine,
the precise nature of the corrections needed to arrive at an appropriate initial mass function are
similarly indeterminate.
4.2 Old Clusters
Old clusters pose their own host of problems. The most obvious is the significant evolution of
the mass function due to stellar death, which would tend to result in a mass function which favors
small stars more than initial mass function. Additionally, the presence of binaries and other mul-
tiple star systems remains a problem. One additional consideration which Kroupa (2007) believes
biases measurements of cluster IMFs is the problem of mass segregation. Following the initial
explusion of loose gas from a cluster in the early stages of development, the cluster continues to
evolve dynamically. The most troublesome product of this revirialization is the equipartition of
energy among stars in the cluster. As the rotational energies of stars in the cluster equalize, it is
expected that larger stars will drift towards the center of the cluster, while small stars will drift to
the outskirts, and may in fact leave the cluster entirely. As a result a calculation of the initial mass
functions for a cluster which fails to account for this will be significantly flatter than the actual
value. Baumgardt & Makino (2003) found that clusters which had aged to half their disruption
time have essentially flat mass functions, while some inital mass functions of very old clusters
actually had positive slopes. This is especially problematic for investigations of metallicity depen-
dence because low metallicity is strongly correlated with age, and is predicted to increase the IMF
slope. The predicted effects of metallicity and the effects of cluster dynamics can thus be easily
confused in old clusters.
4.3 Case study: the Arches cluster
As an example of how evidence for a variation in the IMF falls apart once cluster dynamics are
considered, Kroupa (2007) cites two studies of the Arches cluster, a cluster near the galactic center.
The cluster’s metalicity is roughly solar and typical of galactic disc clusters. However the temper-
ature of gas within the cluster is significantly warmer than is typical within the disc due to it’s
position near the center (30 K rather than 10 K). As a consequence the Jeans mass and accretion
theories of star formation predict a shift of the IMF towards larger stars, as one might expect for
a low metallicity population. The first of the two studies, Klessen, Spaans, Jappsen (2006), con-
5

6. ducted extensive hydrodynamical calculations of the IMF for this cluster, and found an IMF which
sharply declined at masses below 7 solar masses, suggesting an abnormally top-heavy function
consistent with theory that higher gas temperature would result in larger initial masses. However,
a study immediately following, Kim et al. (2006), conducted N-body dynamical corrections for
the cluster, and found the 7 solar mass feature to be a localized bump. The overall IMF for stars
heavier than 1.3 solar masses was found to have an α of 2 to 2.1, quite consistent with canonical
Invariant IMFs, and contradicting the expectations of theoretical star formation models.
5 Analysis
5.1 Difficulties of a comprehensive cluster survey
The ideal investigation of the relationship between metallicity and the initial mass function would
consist of a comprehensive survey of clusters within environments of differing metallicity. For
example, within the Milky Way one might compare globular clusters within the galactic halo (ex-
pected to have a relatively low metallicity), with open clusters within the galactic disc. These
surveys would identify the luminosity distribution of stars within the cluster and attempt to deter-
mine the mass distribution using our understanding of the mass-luminosity relationship of main
sequence stars. This mass distribution would then be adjusted to account for the shorter life span
of more massive stars, which would tend to result in current mass distributions with higher con-
centrations of low mass stars than the initial mass distribution. Finally, one would compare these
distributions to current estimates of the initial mass function, such as the two-power law estimate
provided in (Kroupa, 2007), and determine if there was a systematic deviation evidenced by low
or high metallicity star populations. Such a deviation would support a link between metallicity
and the initial mass distribution of star populations, particularly if high metallicity resulted in an
abnormally steep slope to the initial mass function or low metallicity resulted in an abnormally
shallow slope, as predicted by the accelerated cooling and accretion theories of star formation.
Unfortunately, the technical hurdles in conducting this analysis are beyond the scope of this
project. In part this is due to the relative disparate nature of data relating to cluster mass distribu-
tions. Cluster metallicity is relatively well cataloged within various sources, and its determination
from color, while crude, is more consistent than the determination mass distributions. Initial mass
distribution calculations, however, require more advanced corrections. The principal method in-
volves the use of isochrones, lines on the Hertzprung-Russel diagram representing stars of the
same age. These lines are then “evolved back” to determine an initial mass distribution. However,
the choice of isochrones can lead to drastically different IMF estimates. For example, the use of
three different isochrones resulted in substantially different estimates for the initial mass function
of the Orion Nebula (Da Rio et al., 2012). Added to the difficulty of isochrone selection are the
difficulties described earlier in correcting for dynamic processes within clusters. Very few clusters
even within our own galaxy have had their dynamic history studied in sufficient depth that we may
quantify dynamic corrections with any degree of certainty. As a consequence, most studies are
content to assume a singular canonical mass function, and individual peculiarities, while discussed
in the context of particular clusters, are not comprehensively tabulated by any study I have been
able to locate.
6

7. Figure 2: Isochrones used in calculation of IMF for Orion Nebula Cluster. Different interpretations of
stellar evolution result in very different estimates (see Figure 4)
5.2 Making an informed hypothesis regarding metallicity dependence
What I have done instead is put forward a hypothetical initial mass function with a metallicity
factor, then explore the implications for clusters of different metallicity, in order to create some
predictions that might be testable in future surveys. To provide a plausible upper bound for the
importance of the metallicity factor, I use the difference between initial mass functions determined
independently for two Milky Way clusters, the Orion Nebula and Palomar 14 clusters, and set
an upper bound for the metallicity factor where it explains all of the variation between these two
estimates. Lower estimates for this parameter will also be explored, under the more plausible
hypothesis that only some of the difference between initial mass function estimates for these two
clusters can be explained by metallicity differences. The initial mass function will be treated as a
2-power law function with a cutoff somewhere between .1 and .5 solar masses. It will be assumed
(perhaps falsely) that all dynamic processes have been accounted for in these studies. Whether
this is the case is not too important, given that this analysis is focused on the consequences of a
metallicity dependent function rather than a precise quantification.
5.3 Globular Cluster Palomar 14
The cluster Palomar 14 is a globular cluster located 71.6 kpc from the galactic center (Harris,
96). As is true of most globular clusters, it is relatively old and low metallicity in comparison
to the galactic disc. The clusters has been estimated to have a metallicity of -1.62 (Harris, 96),
corresponding to an Fe/H ratio approximately 40 times lower than that of our sun. The estimated
slope of the initial mass function m−α
in the range 0.53-0.78 solar masses was α = 1.27 (Jordi
et al., 2009), significantly shallower than the Kroupa value of 2.3 for stars of mass > 0.5 solar
masses. The paper proposes this may be consistent with gas expulsion from a cluster which formed
7

8. Figure 3: Plot of the initial mass function for stars of 0.5 < m < 0.8 solar masses in Palomar 14 (Jordi et
al., 2009)
with a normal, canonical initial mass function. However, a significantly shallower mass function
is also consistent with the theoretical explanation that low metallicity environments as found in
clusters of this type result in more massive stars generally. Therefore, we will hypothesize that
this shallow slope is a natural product of some metallicity factor to the initial mass function. The
slope of the initial mass function below the cutoff is not explored in depth in this study. However,
citing studies of GCs NGC 6397 and M4 (Richer et al. 2004, 2008), the authors estimate a slope
of α = −1, which we will use here. This is a notable deviation from the canonical IMF which
assumes the number of stars of given mass continues increasing at low values, albeit at a lower
rate. Palomar 14 has significantly fewer low mass stars than predicted by the Kroupa initial mass
function.
5.4 The Orion Nebula Cluster
The Orion Nebula, alternately classified as M42 or NGC 1976, is a highly active star-forming
region relatively close to the Earth. A definitive metallicity could not be found, however, Orazi
et al. (2007) conducted a pilot study and found that 6 of 7 stars surveyed had metallicity values
very close to that of our sun, so a metallicity of 0 is a plausible estimate. The value of the slope
calculated in Da Rio et al (2012) for the initial mass function above the cutoff of the two-power
law is subject to various estimates depending on which isochrones are used. For this analysis we
will use the value determined from the D’Antona & Mazzitelli (1998) isochrone, which results
in a slope value of α = 2.3 (Da Rio et al., 2012). This is consistent with the Saltpeter and
Kroupa canonical values, and results in the largest spread between α values, consistent with our
goal of using these estimates for upper bounds. For initial masses below the cutoff, there are again
multiple estimates. Using the data determined with the Baraffe et al. (1998) isochrones, which
8

9. Figure 4: Initial mass function calculations for the Orion Nebula Cluster using three different isochrones.
Plots on the left use a Chabrier function while those on the right use a two-part power law function (Da Rio
et al., 2012). Shaded areas enclose the 90% confidence interval for each fit.
maximizes the spread between clusters and is most consistent with the theoretical predictions that
high metallicity will allow for more low mass stars, we find an α = −0.12. This would indicate
a much greater prevalence of low mass stars than under the initial mass function for the Palomar
14 cluster, and closer conformity to the Kroupa IMF. Data from this isochrone also suggests the
characteristic mass lies at approximately .3 solar masses, which we will use as the characteristic
mass for our model IMFs. The Jeans mass theory would suggest that the characteristic mass of
a true metallicity dependent IMF should also be dependent on metallicity. However, this analysis
presumes a constant characteristic mass, due to insufficient information on the characteristic mass
in Palomar 14.
6 Calculations
If one assumes that metallicity relates to the slope of the initial mass function by a simple power law
relationship, it follows that the initial mass function with a metallicity parameter can be described
as m−α+βx
, where x is the metallicity and β is some constant. For the high-mass branch of the
IMF, we can assume that α = 2.3, consistent with our data on the initial mass function of the
Orion Nebula with approximate metallicity of 0. If our data on Palomar 14 is consistent with this
function, it follows that −2.3 − 1.62β = −1.27 This gives an approximate value of -.65 for the
constant β above the characteristic mass. Repeating this calculation for the low-mass branch, we
find −1 + −1.62β = −0.12, and β is -.55. Our final form of the initial mass function is thus
m−.1−.55x
(9)
9

10. Figure 5: Mesh plot displaying calculated relationship between the IMF and metallicity using the hypoth-
esized metallicity parameter for m¿ 0.3 solar masses. The number density of stars of given mass within a
cluster of given metallicity is 10z times the number density of solar mass stars within that cluster.
for m < 0.3 solar masses, and
m−2.3−.65x
(10)
for m > 0.3 solar masses
7 Results and Implications
7.1 Metallicity and Initial Mass Relations
Figure 5 displays a (non-normalized) mesh plot relating the initial mass function for stars above
0.3 solar masses to metallicity. At a metallicity of 0 the slope is rather similar to the Saltpeter
function, as −α+βx = −2.3. Super-solar metallicities (extreme Population I stars) should exhibit
an even steeper number function decline with increasing mass. Strongly sub-solar metallicities
(Population II stars such as those in globular clusters) have a significantly shallower slope for
the IMF, indicating a greater proportion of larger stars. At metallicity of -2 (roughly the lowest
metallicity for Milky Way globular clusters) the model predicts function m-1. Under this condition
it’s expected that the mass distribution (as opposed to the number distribution) of stars within the
cluster will be completely flat. The IMF itself is predicted to flatten out at a metallicity of -3.5.
This metallicity exceeds any current galactic cluster, but falls well below ultra metal-poor stars
such as SM0313, with a metallicity of -7.4 (Keller 2014). Clusters at metallicities below -3.5 were
therefore likely to be quite prevalent during the reionization epoch, supporting the idea that during
this time period the initial mass function might have favored large stars over smaller ones.
10

11. 7.2 Metallicity and Luminosity Relations
Having determined the main sequence initial mass function for any particular metallicity, we can
draw conclusions regarding the expected luminosity profiles of these clusters. To do so, we can
employ the typical main-sequence solar mass to solar luminosity relation: L = m3.5
By multiply-
ing this by the initial mass function, we can construct a function describing the total contribution
of stars of a particular mass to the cluster’s luminosity
L∆m = m−α+βx
m3.5
(11)
. Using the our hypothetical initial mass function, this converts neatly to
L∆m = m3.4−.55x
(12)
for m < 0.3 solar masses, and
L∆m = m1.2−.65x
(13)
for m > 0.3 solar masses. As might be expected from the invariant IMF, large stars dominate
luminosity at all metallicities, barring extremely high super-solar metallicities of 2 or greater. This
is much higher than any known cluster. The degree to which they dominate however increases
dramatically at very low metallicity. In solar metallicity environments, 10 solar-mass stars are
expected to contribute 10-20 times more to cluster luminosity than solar-mass stars. At a very
low metallicity of around -4, however, 10 solar-mass stars are predicted to produce 100 to 1000
times more light than solar-mass stars. Young population III star clusters which formed during the
reionization era can therefore reasonable be expected to be far bluer than even the most energetic
star forming regions today, and produce the large quantitues of UV radiation required to reionize
the universe.
7.3 Further study
The impending completion of the James Webb Space Telescope will likely put Population III
clusters from the reionization epoch within our reach in the near future. However, it is unlikely
that observations sufficiently accurate to determine an initial mass function for such clusters will
result, due to the difficulty in distinguishing individual clusters, much less stars, at such a distance,
as well as the problems with cluster IMF measurment detailed in Section 4. Acquiring spectra,
however, may lead us to reject the invariant initial mass function even if determining the actual
IMF is beyond our reach. Spectra indicating a relative prevalence of high-mass stars (e.g O6, B5
and B0) 2 or 3 magnitudes greater than current young clusters would be strong evidence to support
a top-heavy early IMF, which can be explained quite well by a metallicity dependence factor.
Any breakthrough in determining the IMF for modern clusters will mostly likely come from more
accurate and advanced simulations of cluster dynamics, rather than more advanced telescopes. If
the proper corrections for mass segregation in globular clusters can be determined with reasonable
confidence, then it should be possible to get an accurate measurement of globular cluster IMFs.
Comparing these functions with the galactic disc and open clusters should then allow determination
of the metallicity dependence factor.
11

12. Figure 6: Mesh plot displaying calculated relationship between the luminosity function and metallicity
using the hypothesized metallicity parameter for m¿ 0.3 solar masses. Luminosity contribution at given
mass is 10z times the contribution of solar mass stars
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