An Unambiguous Verification of IMF Variationmintfreshprofile.weebly.com/uploads/4/4/2/1/44218553/... · 2018. 11. 21. · IMF of the two generations, allowing us to gain insight to

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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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.