an unambiguous verification of imf...
TRANSCRIPT
The Effect of Dynamics on the Mass Function of Globular Clusters:
An Unambiguous Verification of IMF Variation
Siemens Competition for Math, Science, and Technology
Astrophysics Category
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INTRODUCTION
A star’s mass is one of its most significant characteristics that is used to trace and
determine its evolutionary path; hence, one of the most defining factors of a stellar population is
its mass function, which is the empirical description of the distribution of the masses of its stars.
Essential for comprehending both the observed results of star formation and the theoretical star
formation process, the Initial Mass Function (IMF) is the mass function of the stars in a single
molecular cloud immediately after formation (Kroupa 2002). Fundamental to formation
theories, the IMF and its suggested universality are most useful for evolutionary and population
synthesis modeling of galaxies and clusters; this distribution affects the subsequent evolution of
the star cluster through its metal productivity, supernova rates, and luminosity (Larson 2006).
For example, through the IMF, astronomers have the ability to gain insight and calculate the
stars’ luminosity function and radiation spectrum at any point in history. Over time, as stars age
and die and the star population evolves with mass segregation, the population’s mass function
also evolves and can be distinct from the IMF, becoming the observed Present-Day Mass
Function (PDMF).
Globular clusters (GCs) are dense, spherical star clusters. Since the stars in the globular
clusters are presumed to have been created and formed at approximately the same time due to
their significant proximity, they are assumed to contain similar chemical composition and
metallicity; their stars are abundant, single-aged, and at a fixed distance (Forbes 2009). This
single starburst is a result of the lack of gas in the cluster after the most massive stars explode as
supernova, pushing all the gas out of the cluster, thereby terminating star formation (Li, 2016).
Therefore, these clusters provide an appropriate and ideal environment for the measurement and
analysis of the mass function due to their compositional near-homogeneity.
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Nonetheless, as a result of the short relaxation time scales of the cores of the GCs relative
to their ages, GCs experience significant mass segregation as a result of dynamical evolution,
where their massive stars tend to congregate at the dense centers of the clusters and lower mass
stars migrate to the outskirts of the clusters as the clusters evolve (Hillenbrand and Hartmann,
2001; Kirby 2016). Therefore, only the PDMF of the star cluster is able to be directly measured,
and the IMF must be inferred with dynamical models that consider both mass loss and stellar
lifetimes. Furthermore, recent spectroscopy collected over the past decade indicates the existence
of variances in light element chemical abundances among stars within a single globular cluster;
analyzing the chemical inhomogeneity of GCs from spectroscopy and photometry data, the
existence of multiple generations in a single GC was suggested and proposed (Gratton et al,
2004). A second generation can be produced in a GC tens of hundreds of million years after the
first starburst when the cluster has the ability to absorb more gas from the asymptotic giant
branch (AGB) winds and ejecta (Trenti et al, 2015).
The presence of both mass segregation and the existence of multiple stellar generations in
a GC superficially makes it seem difficult for astronomers to calculate and analyze the IMF;
however, both properties provide opportunities to explore the effect of dynamical evolution on
the nature and evolution of the mass function of GCs in one of the most extreme star formation
environments that has not been previously addressed.
For simplicity, we consider two main generations in a GC. In a globular cluster with two
generations, G1 and G2, that have been dynamically relaxed many times, because both
generations experience the same stellar evolution and dynamics within a cluster, although the
PDMF differs from the IMF for both generations, we hypothesize that the IMF transforms and
evolves in the same way for both generations. Hence, given two mass function slopes 𝛼1 and
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𝛼2, corresponding to G1 and G2 respectively, dynamical evolution of the cluster does not affect
their relative ordering. If 𝛼1 > 𝛼2 at t=0, then 𝛼1 > 𝛼2 after the dynamical evolution occurs.
The same applies for 𝛼1 = 𝛼2 and 𝛼1 < 𝛼2. Through our work, we wish to demonstrate that the
relationship between the PDMF of the two generations is similar to the relationship between the
IMF of the two generations, allowing us to gain insight to the nature of the star formation of the
two generations without directly measuring the IMF of each generation itself.
Many observations in the past have suggested a universality of the IMF between different
star-forming regions, and evidence and verification of variation of the IMF is very rare and
important to the insight of stellar formation processes. Nevertheless, the different characteristics
between the two generations of a GC suggest that their PDMFs are distinct. Therefore, through
the results of our experiments, we aim to show that if the observed PDMFs from the stellar
census of the two generations are different, the similarity in the relationship of the present day
and initial mass functions allows possible evidence and verification of IMF differentiation.
METHODS AND PROCEDURE
A population’s mass function Φ(M) is an empirical formula that describes the number of
stars per unit mass such that Φ(M)dM is the number of stars with masses between M and M+dM.
Therefore, the mass function is represented as the derivative of the number of stars with respect
to mass, or the number of stars dN within a mass interval dM:
"#"$
= Φ(M)
Because GCs have negligible dark matter and the dynamical mass is equivalent to the
stellar mass, we can normalize the mass function such that 𝛷(𝑀)𝑑𝑀+, = 1.
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The mass function can be represented in the form of a Salpeter Power Law function such that it
is proportional to 𝑀-.where 𝛼is termed the slope of the mass function and k is an arbrtirary
constant (Kroupa, 2001):
Φ(M) = "#"$
= k ∙ 𝑀.
Thus, a linear relationship can be established to determine the value of 𝛼:
𝑙𝑛("#"$
) = ln(k)+ (𝛼) 𝑙𝑛(𝑀)
A cluster’s mass function shape is primarily identified and defined by its value of 𝛼. To
analyze the evolution and transformation of the mass function slopes 𝛼1 and 𝛼2 of G1 and G2 in
a specified GC, we create a microcosm of four different GCs. The Starlab Modular Software
Package and Environment for Collisional Stellar Dynamics and the kira general N-body
integrator with Hermite integration algorithms are used to simulate the dynamical evolution of a
series of five globular clusters of two generations over the course of 149.4 and 298.8 Myr
(Heggie, Hut, McMillan, 1996). The N-body code is written in N-body standard units with unity
length and time scales, and 149.4 and 298.8 Myr correspond to 10 and 20 time units,
respectively. We use a W0=8.5 King density profile to generate the three dimensional position
and velocity of each star within the cluster. Since our objective is to study and analyze the
evolution of the stellar mass functions of the two generations, the 4,000 stars in each model
(2,000 stars per generation) are assigned masses based on a Salpeter mass spectrum between 0.08
to 0.8 solar masses (M☉) with varying Power Law IMF slopes for the G2. Since in an ideal GC, it
takes around 108 – 109 years for a generation and population to dynamically relax, and the
difference in formation time for G1 and G2 is approximately 107 years, we only need to vary
slopes for G2 because we assume that both generations have been dynamically relaxed many
times, and thus the two generations are interchangeable (Kirby 2015).
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We create five different models to study the evolution of the stellar mass function. The
initial model parameters of the entire suite of simulations are summarized in Table 1. A
example diagram of the stellar positions of Model 3 is represented in Figure 1, respectively.
Table 1: Model Input Parameters Model Time of
Simulation (t = Myr)
Number of Particles for each generation (n = )
Input IMF Slope of G1 (𝜶1 =)
Input IMF Slope of G2 (𝜶2 =)
Model 1 149.4 2000 -2.0 -2.0 Model 2 149.4 2000 -2.0 -2.1 Model 3 149.4 2000 -2.0 -2.3 Model 4 149.4 2000 -2.0 -2.5 Model 5 298.8 2000 -2.0 -2.3
We set Model 1 as our control group, and we alter the IMF slopes of G2 for each model,
which serves as the independent variable. We first generate the first generation of stars in the
GC, and then we generate the second generation. We proceed by having both generations
undergo the same dynamical evolution in the presence of each other from t = Myr to the final
time. We simulate all five models on an 8-processor desktop computer at Caltech. Models 1-4
took approximately 2 months to complete, and Model 5 took approximately 4 months to
complete. After the simulations were completed, we obtain the initial and final positions of the
stars and collect their masses located in the square of side length 7 pc , -3.5 pc < X < 3.5 pc and -
3.5 pc < Y < 3.5 pc, with the origin defined as the cluster center as the cluster evolve; we do this
for each generation of each model. Using these mass distributions, through the IDL
Astronomical Programming Language, we create a probability distribution function (PDF) and
mass distribution histogram for each generation to graph and analyze the distribution of masses
in the specified region ("#"$
). Taking the natural logarithm of these frequencies, we were able to
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deduce a linear relationship between ln("#"$
) and ln(M/ M☉) to determine the empirical value of
the mass function slope of each generation for each model at both t=0 Myr and at the end of the
simulation. We use the Least Squares Regression Line function, “ladfit”, to exclude outliers and
approximate the slope 𝛼 of the scatter plot through a STY Stepwise Maximum Likelihood fitting
method, to obtain the final Mass Function slope of the generation (Sandage et al 1979). These
values allow us to observe and analyze the transformation and evolution of the mass functions of
each generation relative to the other.
RESULTS
For each globular cluster model generation, we take a sample of stars located in the
square of side length 7 pc, with its center at the cluster core (taken to be the average of the
positions of all the stars in the cluster), at time t = 0 Myr and after the simulation. Using these
stars in the interval, we obtain their stellar masses to graph the mass function Φ(M) through a
Figure 1: A diagram that portrays the initial positions and distribution of the stars in the sample of the model globular cluster for Model 3. Each red particle represents a star from Generation 2, and each black particle represents a star from Generation 1. Although the cluster spans for more than 4 pc in radius, as the sample incorporates more than 95% of the particles in the cluster, the sample is representative of the GC.
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histogram. Using Least Squares Regression, we obtain the mass function slope a for each
generation before and after the evolution of the GC by calculating the slope of the graph. We
also analyze the linearity of the data points through the correlation coefficient to determine the
accuracy of 𝛼 based on the LSRL approximation. We further represent the stars and their masses
with a cumulative histogram to analyze and analyze the change in the number of stars in the
interval and the effects of mass segregation in the clusters.
We consider an example of a complete analysis of GC Model 4, where both G1 and G2
are composed of 2,000 stars each, with a stellar IMF slope of
-2.0 and -2.5 for G1 and G2, respectively. The
extreme difference between the IMF slopes of G1
and G2 enables us to describe and enhance
observations seen in other models. At t = 0 Myr,
within the square sample space, we observe 1,931
stars from G1 and 1,948 stars from G2, thus
producing an average of 96.98% of the stars from
the entire simulated cluster, allowing us to have a
representative sample of the GC generations
(Table 2).
Figure 2 depicts the change in the stellar
positions of the stars in the globular cluster before and
after the evolution of the cluster with the application of
dynamical interactions; with the black dots representing the stars at t = 0 Myr and the purple dots
representing the stars at t = 149.4 Myr, we observe that over time, the cluster core and center of
Figure 2: A diagram that portrays the initial and final positions of the stars of both generations in the GC of Model 4. The black dots represent the stellar positions at t = 0 Myr and the violet dots represent the stellar positions at t = 10 Nbody time units (t = 149.4 Myr)
Model 4: Stellar Positions
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mass becomes significantly displaced from its original position in the simulation (from [-0.011, -
0.038] to [-0.552, 2.207]) .This is caused by the presence and orientation of the binary systems in
the cluster. Hence, to calculate the final mass function slopes 𝛼1 and 𝛼2 of the two generations,
we redefine the sample square center as the cluster core after it has moved as the average final
positions of all the stars in the cluster. Furthermore, we notice that not only does the cluster
center become displaced, but it also has more stars congregated at its center, while the cluster
Figure 3: Left: The initial (blue) and final (red) mass functions (histograms) of the stellar masses within the sample space for G1 (top) and G2 (bottom) of GC Model 4 based on a Power Law representation. Right: The initial (blue) and final (red) cumulative mass distributions of the stellar masses within the sample space for G1(top) and G2 (bottom). The distributions also depict the change in the total number of stars within the sample size.
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radius increases as stars become more widespread and distant around the outskirts of the cluster.
Therefore, there are less stars in the final sample space of the cluster as depicted by the
cumulative distribution of the masses of stars in the two generations (Figure 3), and we observe
only 1,292 G1 stars and 1,293 G2 stars in the final sample space of the cluster (Table 2).
This observation of the changes in the position of the stars in the cluster and the mass
distribution graphs (Figure 3) corroborate the effect of mass segregation in the GC as heavier
stars become significantly more concentrated at the cluster center than lower mass stars as a
result of two-body relaxation. Through the histograms and distribution graphs of G1 and
G2 in Figure 3, with blue representing the initial mass function we observe at t = 0 Myr and red
Figure 4: The linear representations of the initial (left) and final (right) mass functions of G1 of Model 4 obtained from the logarithm of the Power Law mass functions. By calculating the slopes of the linear functions through a LSRL, we obtain the mass function slopes 𝛼1(initial) = -2.066 and 𝛼1 (final) = -1.905 of G1 in the sample size. The correlation coefficient of the data points and the equation of the linear regression line are shown on each graph.
𝑙𝑛("#"$
) =0.8126665 + (-𝟐. 𝟎𝟔𝟓𝟒𝟗) 𝑙𝑛(𝑀)
𝑙𝑛("#"$
) =0.684362 + (-𝟏. 𝟗𝟎𝟒𝟓𝟔) 𝑙𝑛(𝑀)
r = -0.976
r = -0.962
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representing the final mass function at t = 149.4 Myr, we observe a notable decrease in lower-
mass stars within a 3.5 pc radius of the cluster center. This observation is enhanced when we
calculate the mass function slopes for before and after the simulation.
Mapping the mass histogram distributions for each of the two generations in the GC at t =
0 (Figure 3) and taking its logarithmic relationship to create a linear function (Figure 4), we
Figure 5: The linear regression lines of the initial (blue) and final (red) mass functions of G1 (thin lines) and G2 (thick lines) of Model 4 obtained from the logarithm of the Power Law mass functions. By calculating the slope of these lines, we obtain the mass function slopes of both generations before and after the simulation. The slopes are shown on the graph.
𝛼1 (initial) = -2.066 𝛼1 (final) = -1.905
𝜶2 (initial) = -2.510 𝜶2 (final) = -2.300
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calculate the function’s slope to acquire the IMF slope, obtaining a slope of 𝛼1 = -2.066 for G1
(-3.30% error from input slope for G1) and 𝛼2 = -2.510 for G2 (-0.40% error from input slope
for G2) (Table 3). Similarly, after the effect of dynamics and gravitational interactions on the
GC generations over the course of 149.4 Myr is applied, we recalculate the mass function slopes
(PDMFs) of the two generations and obtain 𝛼1 = -1.905 and 𝛼2 = -2.300 (Table 3). We portray
the change in the mass function slopes of the two generations in Figure 4, where the thick lines
represent slopes of G2 with blue as the initial function and red as the final function; the thin lines
represent the slopes of G1. Hence, because the mass function slope of each generation decreased
at the end of the simulation, we demonstrate that over time, with the effect of dynamical
evolution, the mass function of each generation becomes less bottom-heavy as the ratio of
higher-mass stars to lower-mass stars increases and evidence of mass segregation is observed.
Table 2: Number of stars in sample space and correlation of determination for linear logarithmic relationship Model Generation Number of
stars in sample (initial)
Number of stars in sample (final)
𝒓𝟐 for IMF slope
𝒓𝟐 for PDMF slope
Model 1 Model 1
G1 G2
1930 1930
1063 1060
0.91205 0.93123
0.95179 0.90497
Model 2 Model 2
G1 G2
1940 1941
1110 1136
0.92160 0.91012
0.90060 0.87610
Model 3 Model 3 Model 4
G1 G2 G1
1928 1927 1931
1285 1277 1292
0.92160 0.94673 0.95258
0.91203 0.94090 0.92544
Model 4 Model 5
G2 G1
1948 1924
1293 480
0.95063 0.93123
0.95453 0.83723
Model 5 G2 1924 514 0.95648 0.88548
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Analyzing the correlation coefficient of the data points in the linear function of the
relationship between log(dN/dM) and log(M) for each generation, we observe that the data points
are strongly correlated, and an average of 95.1% of the variance of log(dN/dM) is predicted from
the linear regression line. The coefficient of determination of each generation of each model at
the initial and final times of the simulations are shown in Table 2. Moreover, we observe less
linearity and a flattening of the mass function in the function at very low and very high masses,
thus suggesting an upper mass limit or a high mass break.
We perform the same process for all five models. The large number of stars in the
sample space from each generation prior to the simulation demonstrate that an average of
96.62% of the stars of the cluster are initially in the sample space, thus providing a representative
reflection of the cluster as a whole. The high values of the correlations of determination of each
generation before and after the simulation underscore the strong fit of the regression line that is
used to determine the slopes of the mass functions. Nevertheless, in Model 5, where the two
generations are initially created identically to Model 3, yet the cluster is simulated to
dynamically evolve twice as long to 298.8 Myr, we observe that the number of stars in the
sample significantly decrease to 480 stars for G1 and 514 for G2 (Table 2). This decrease in the
number of stars in the sample space suggests that the longer the cluster evolves, the more the
cluster radius expands while the more the final mass function slopes further decrease, making the
cluster more top-heavy with more massive stars sinking to the center while lower mass stars
expand beyond the boundaries of the sample space.
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Table 3: Summary of Results Model 𝜶1i 𝜶2i 𝜶2i-𝜶1i Relation (i) 𝜶1 f 𝜶2 f 𝜶2f-𝜶1f Relation (f) Model 1 -2.001 -2.001 0.000 𝛼1 ≈ 𝛼2 -1.698 -1.700 -0.002 𝛼1 ≈ 𝛼2 Model 2 -2.055 -2.124 -0.069 𝛼1 > 𝛼2 -1.718 -1.839 -0.121 𝛼1 >𝛼2 Model 3 -2.103 -2.335 -0.232 𝛼1 > 𝛼2 -1.916 -2.173 -0.257 𝛼1 > 𝛼2 Model 4 -2.066 -2.510 -0.444 𝛼1 > 𝛼2 -1.905 -2.300 -0.395 𝛼1 > 𝛼2 Model 5 -2.059 -2.287 -0.228 𝛼1 > 𝛼2 -1.669 -1.912 -0.243 𝛼1 > 𝛼2
Table 3 summarizes the initial and final mass function slopes of all five GC Models for
each generation. We calculate the difference of the initial mass function slopes of the two
generations for each model and compare it with that of their final mass function slopes. The
differences allow us to determine the relationship between the two generations before and after
the simulation and how the relationships are preserved despite the effect of the dynamics on the
clusters. Using the data obtained, we perform a Chi-Square Goodness-to-fit Test with a
significance level of 𝛼= 0.01 with four degrees of freedom to compare the initial and final
differences.
Chi-Square Goodness-to-fit Test with α = 0.01 with four degrees of freedom: 𝐻?: The differences in the PDMF slopes of G1 and G2 are consistent with the differences in the IMF slopes of the two generations (𝛼2i-𝛼1i = 𝛼2f-𝛼1f). 𝐻@: The differences in the PDMF slopes of G1 and G2 are not consistent with the differences in the IMF slopes of the two generations (𝛼2i-𝛼1i ≠ 𝛼2f-𝛼1f). For each model: Observed (o) = 𝛼2f-𝛼1f Expected (e) = 𝛼2i-𝛼1i Degrees of Freedom: 5 – 1 = 4 χB = ∑(𝑜 − 𝑒)B/𝑒 = 1.4643 P(χB>0)=χBpdf = 0.106707 ∴Since P > 0.01, we accept the null hypothesis and conclude consistency between the relationship of the two generations in the final and initial mass function slopes.
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Since the test confirms consistency between the relationship of the two generations both
before and after the simulation, through the five globular cluster models, we are able to
demonstrate that because the dynamical evolution of the globular cluster affects G1 in the same
way as G2, their mass functions evolve in the same manner. Given the two mass function slopes,
we are able to prove through these simulations that dynamical evolution does not affect their
relative ordering or mass function slope difference, and for models in which 𝛼1 = 𝛼2 or 𝛼1 < 𝛼2
before the simulation, the same results were acquired after the simulations.
DISCUSSION AND APPLICATIONS
The IMF is the foundation for determining the subsequent pathway for the stellar
evolution of clusters and populations; for many years, observational studies of the IMF have
suggested a universality of the IMF with very little evidence of variation. For example, in the
Milky Way (MW), the functional form of the IMF is the Kroupa IMF, with "#"$
∝ 𝑀-K.L for stars
in the range 0.08 < M/𝑀M< 0.5 and 𝑀-B.L for M > 0.5 M/𝑀M (Kroupa 2001).
Figure 6: From Armitage Observational statistics of the IMF slopes throughout galaxies suggest a universality in the IMF.
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Furthermore, it has also been previously suggested that the IMF is independent of the Jeans Mass
and the metallicity of the population (Elmegreen et al 2008). Despite the significant evidence for
the invariability of the IMF in a galaxy or cluster, many theoretical studies suggest otherwise due
to the possibility of fragmentation and thus the production of more massive stars, resulting in the
variation in the IMF (Bastian 2010). Recently, Kalirai et al. presented evidence for an IMF with
a smaller slope than that of the MW in the Small Magellanic Cloud (SMC), and used the
abundance of metal-poor stars in the SMC to corroborate the association between the metallicity
and the shape of the IMF and attribute this mass function difference to chemical abundances
within the cluster (Kalirai 2013). Nevertheless, the uncertainty in this data still remains fairly
large. Furthermore, the different slopes between Local Group dwarf galaxies and giant elliptical
galaxies prompt a new unambiguous verification of IMF differentiation (Elmegreen 2004).
Through our simulations, we are able to conclude that in a globular cluster, one of the
most extreme star formation environments, despite the presence of mass segregation and
dynamical influences, because the stars in G1 and G2 experience the same dynamical evolution,
the relationship of the two slopes of stars in Generation 1 and Generation 2 remained constant
throughout time, the stars in G1 and G2 experience the same dynamical evolution, as
demonstrated by our chi-square goodness of fit test analysis with significance level 0.01.
Therefore, in globular cluster environments, because both generations experience multiple
dynamical relaxation times, although the PDMF differs from the IMF, through our results, we
are able to conclude that the relationship of the PDMF slopes of both generations would give an
accurate representation and inference of the relationship of the IMF slopes of G1 and G2.
We can apply this relationship to propose evidence for the differences in the shape of the
IMF in a single stellar cluster. Some globular clusters, such as the GC 47 Tucanae (Kirby 2015),
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have inhomogeneous chemical abundances; furthermore, they have relatively constant iron
abundances and simple chemical compositions that enable astronomers to effectively identify the
presence of and distinguish between the different generations in the cluster. These globular
clusters provide an ideal environment in which the PDMF of the clusters’ generations can be
measured distinctly. Because numerous factors and galactic evidence suggest PDMF slope
differentiation in the cluster, such as the fact that the generations differ in light-element
abundances, and G1’s star-forming gas cloud is self-gravitating, while other generations have
clouds held together by G1’s preexisting stars, through our simulations and results, we can
conclude that PDMF differentiation also suggests IMF differentiation in a single star-forming
region, thus providing an novel, unambiguous verification of IMF differentiation.
CONCLUSION AND FUTURE WORK
The shape of the Initial Mass Function (IMF) dictates stellar evolution of an entire stellar
population. It drives characteristics of the population, such as the luminosity and color
distributions and stellar lifetimes. Through a series of N-body simulations from the Starlab
Software and Environment, we generated and simulated five different globular cluster models
over a course of a few hundred million years to observe the effect of kinematic and dynamic
interactions and evolution on the nature of the mass functions of two generations in each
globular cluster. We conclude that our initial hypothesis is warranted; although the mass
function evolves due to the dynamical evolution of mass segregation, since both generations
experience the same evolution, the differences in PDMF of the two generations would also
indicate differences in the IMF of the two generations, thus potentially providing a method of
verifying IMF variation in an extreme star formation environment of a globular cluster.
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Nevertheless, our GC models have room for improvement. To accurately determine the
full effect of the dynamical evolution on the mass function of the clusters, we need to extend the
time required for evolution of the stellar populations, ideally to even 12 Gyr, 1000 times longer
than our initial simulations, to create a more accurate and representative microcosm of an ideal
globular cluster. Based on the results of Model 5, by doubling the amount of time evolved, not
only do we see a significant drop in the number of stars in the sample size after the simulation,
we also observe a drastic decrease in the PDMF slope compared to other Models. To improve
our models and minimize the discrepancies between the difference of the final and initial mass
function slopes of the two generations, we would also seek to increase the number of particles in
the Nbody simulations to approximately 20,000 stars per generation. A supercomputer would be
required to complete this task.
Moreover, even though we utilized “ladfit” on IDL to remove outliers and removed a few
points of the heavier-mass stars to improve the regression fitting, our linear relationship to
determine the regression slope of the mass function fails to account fully for the slight curving of
the mass function towards the heavier mass stars, thus resulting in an inaccurate value for the
mass function slopes of the generations simulated. Accounting for this curve would provide a
more consistent relationship between difference of the IMF of the two generations and that of the
PDMF.
Furthermore, in an ideal GC model, because the first generation is more extended than
the second, the King radius needs to be larger for G1 with respect to G2. For the simulations in
this project, we assumed that the King radius was constant for both G1 and G2. Moreover, to
also reflect an ideal GC model, we could extend beyond the kira integrator with the addition of
the SeBa stellar evolution to model the evolution of the globular cluster by adding stellar
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evolution to the N-Body code, thus allowing the stars in the cluster to lose mass over the course
of their lifetime, undergo supernova, and disappear.
Through these efforts, we strive to develop and investigate a mathematical model to infer
and predict the evolution of a mass function given a set of parameters. Determining and
calculating the IMF of a stellar population can be extremely difficult; hence, through our
simulation results and conclusions, we demonstrate and prove IMF differentiation with mere
PDMF differentiation.
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REFERENCES
Armitage: Figure 6: Image
Bastian, N., & Goodwin, S. P. 2006, MNRAS, 369, L9
Bastian, N., Covey, K. R., & Meyer, M. R. 2010, ARA&A, 48, 339
Baumgardt, H., Makino, J., Hut, P., McMillan, S. & Portegies Zwart S.; 2003, Astrophys. J.
Lett. 589, L25-L28.
Elmegreen, B. G., Klessen, R. S., & Wilson, C. D. 2008, ApJ, 681, 365
Elmegreen, B. G., Klessen, R. S., & Wilson, C. D. 2008, ApJ, 681, 365
Gratton, R., Sneden, C., & Carretta, E. 2004, ARA&A, 42,385
Hillenbrand, L. A., & Hartmann, L. W. 1998, ApJ, 492, 540
Hut, P., 2003, in Astrophysical Supercomputing Using Particle Simulations, IAU Symposium
208, ed.: P. Hut and J. Makino (San Francisco: the Astronomical Society of the Pacific), pp. 331-
342.
Kalirai, J. S., et al. 2013, ApJ, 763, 110
Kalirai, J. S., Fahlman, G. G., Richer, H. B., & Ventura, P. 2003, AJ, 126, 1402
Kirby, E. (2015, August). Personal Interview.
Kirby, E. (2016, April). Email Interview.
Kroupa, P. 2001, MNRAS, 322, 231
Kroupa, P. 2002, Science, 295, 82
Larson, R. B. 2005, MNRAS, 359, 211
Li, Y., Klessen, R. S., & Mac Low, M.-M. 2003, ApJ, 592, 975
Sandage, A., Tammann, G.A., Yahil, A., 1979, ApJ, 232, 352
Trenti, M., Padoan, P., & Jimenez, R. 2015, ApJ, submitted, arXiv:1502.02670
21
The Effect of Dynamics on the Mass Function of Globular Clusters: An Unambiguous
Verification of IMF Variation
ABSTRACT
The masses of stars obey a law called the stellar initial mass function (IMF). It is the
fundamental mass distribution function that drives stellar evolution. Globular clusters (GCs) are
dense, spherical star clusters that contain at least two generations of stars that differ in chemical
composition and age. The IMF in GCs has been affected by dynamics, leading to the Present
Day Mass Function. In this report, we perform a series of N-body simulations through the
Starlab Software and Environment and its N-body integration program, the Kira Integrator, to
model the dynamical evolution of two stellar generations within a set of 5 globular cluster
models over a range of various IMF slopes from α = -2.0 to α = -2.5. We find that the slopes of
the star generations’ mass functions evolve similarly to one another; Generation 1 and
Generation 2 (G1 and G2) of the globular clusters experience the same dynamical evolution,
regardless of the effect of mass segregation and the presence of the multiple stellar populations.
We use these conclusions to provide insight to the verification of IMF variance in a star-forming
region from potential PDMF differentiation.