Abstract the Mond External Field Effect on Dwarf

Home , Andromeda XIX, Crater (constellation), Leo II (dwarf galaxy), Leo I (dwarf galaxy), Sextans, Velocity dispersion

ABSTRACT

THE MOND EXTERNAL FIELD EFFECT ON DWARF SPHEROIDAL GALAXIES

by Benjamin David Blankartz

MOdified Newtonian Dynamics, or MOND, is an alternative to the dark matter paradigm which can be used to explain the missing mass problem. The External Field Effect(EFE) is a feature unique to MOND that has no counterpart in the dark matter paradigm. The EFE comes about when a gravitationally bound system is under the influence of an external gravitational field. We embed three Milky Way satellites, Fornax, Leo I and Crater II, as well as one Andromeda satellite, Andromeda XIX, in an external field produced by their host galaxy. We then model these systems kinetically. The velocity dispersion profiles are calculated statistically and then compared to observational data; acceleration profiles of each system are also presented. Fornax and Leo I show good agreement with observations in their velocity dispersion. We also examine structural similarities of the acceleration plots. THE MOND EXTERNAL FIELD EFFECT ON DWARF SPHEROIDAL GALAXIES

Thesis

Submitted to the Faculty of Miami University in partial fulﬁllment of the requirements for the degree of Master of Science by Benjamin David Blankartz Miami University Oxford, Ohio 2017

Adviser: Stephen Alexander, PhD Reader: Herbert Jaeger, PhD Reader: Jennifer Blue, PhD

THE MOND EXTERNAL FIELD EFFECT ON DWARF SPHEROIDAL GALAXIES

Benjamin David Blankartz

has been approved for publication by

College of Arts and Science and Department of Physics

Stephen Alexander, PhD

Herbert Jaeger, PhD

Jennifer Blue, PhD Contents

1 Introduction 1 1.1 Dark Matter ...... 1 1.2 ΛCDM ...... 2 1.3 MOND ...... 3 1.4 The MOND External Field Eﬀect ...... 4 1.5 Dwarf Spheroidal Galaxies ...... 4 1.5.1 Previous Work ...... 5 1.6 Focus ...... 7

2 Method 8 2.1 Dwarf Spheroidal Model ...... 8 2.1.1 Code Conditions ...... 9 2.1.2 Hermite Individual Time Step Integrator ...... 10 2.1.3 Implementing MOND in Isolation ...... 12 2.2 Implementing the MOND External Field Effect ...... 13 2.2.1 The External Field Effect ...... 13 2.2.2 The External Field Effect in our Model ...... 13 2.2.3 The External Field Effect in Practice ...... 14 2.3 Data Analysis Technique ...... 18 2.3.1 Velocity Dispersion Profiles ...... 18

iii 3 Results and Discussion 19 3.1 Model Conditions ...... 19 3.2 Acceleration Plots and Velocity Dispersion Comparision ...... 20 3.3 Velocity Dispersion Proﬁles ...... 26 3.4 Possible Weaknesses of Work ...... 30

4 Conclusion 32 4.1 Future Work ...... 33

Bibliography 34

iv List of Tables

3.1 Initial data for four dwarf spheroidal galaxies. Rh is the half-light radius, L is the

luminosity, D is the distance to the host galaxy, σobs is the observed bulk dispersion and ML is our best ﬁt mass to light...... 19

3.2 Mass data for host system. M is the unit solar mass...... 26

v List of Figures

1.1 Eight Milky Way Dwarfs and their dispersion proﬁles. The red line is the isolated MOND dispersion while blue is the Newtonian velocity dispersion. The black circles are observational data published in Walker 2009 [14] ...... 6

2.1 Case 1 ...... 15 2.2 Case 2 ...... 16 2.3 Case 3 ...... 17

3.1 Fornax Acceleration and Dispersion Profiles ...... 21 3.2 Leo I Acceleration and Dispersion Profiles ...... 22 3.3 Crater II Acceleration and Dispersion Profiles ...... 23 3.4 Andromeda XIX Acceleration and Dispersion Profiles ...... 24 3.5 Fornax velocity dispersion profile. Black markers are observational data from Walker 2009[14]...... 27 3.6 Leo I velocity dispersion profile. Black markers are observational data from Walker 2009[14]...... 28 3.7 Crater II ...... 29 3.8 Andromeda XIX ...... 29

vi Dedication

To my parents, Debbie and David. For all their work raising me and getting me to where I am today.

vii Acknowledgements

I would like to ﬁrst and foremost thank my adviser, Dr. Stephen Alexander for the years of academic and professional guidance while I have attended Miami University. I would also like to acknowledge Matthew Walentosky and Tristan Clark for laying the ground work to our research into MOND. Joshua Schussler, Justin Messinger and Alex Staron for helping with the simulation work. I would also like to thank Dr. Jens Muller along with the entire Miami HPC group for all the support they gave me. Finally, I would like to thank my family, friends and professors for the uncountable amount of help given throughout my time as a graduate student.

viii Chapter 1

Introduction

1.1 Dark Matter

Midway through the 20th century, astronomers started using new technology to examine the kinematics of galaxies and stars within galaxies. This new access to the motion of stars within galaxies allowed for the testing of theories that described the dynamics of these systems. In 1933, astronomer Fritz Zwicky examined a clump of galaxies known as the Coma Cluster. He utilized the virial theorem, GM σ2 ≈ (1.1) R where M is the total mass, and R is a radius, and σ is velocity dispersion to make an estimate of the mass of the system by utilizing observations of the velocity. This led him to discover that the galaxies were moving too fast for the system to be gravitationally bound. Furthermore, he estimated that the mass required for the system to remain bound would be on the order of 400 times the visible matter; he coined the term ‘dunkle Materie’. This was one of the ﬁrst published indications that there may be a discrepancy between the theory and observation which was attributed to dark matter[1].

Almost thirty years after Zwicky’s proposal, astronomer Vera Rubin was examining the rotation curves of spiral galaxies. Rotation curves are the relationship between the radial velocity of an

1 object and its distance from the center of mass of the gravitating body. Assuming that a system is gravitationally bound and stable, the tangential velocity of the bodies should fall oﬀ as the radial distance increases. Rubin’s observations showed that the velocities did not fall oﬀ, but instead remained abnormally high. This was further proof for scientists that the gravitaional mass in galaxies were higher than the observationed mass would suggested. Since these discoveries, astronomers and physicists have been trying to unravel the mysteries of this missing mass known as unseen dark matter [2].

Beyond the initial observational problems, several other key discoveries point to a missing mass problem. Dwarf Spheroidal galaxies have been shown to have a large mass discrepancy[3]. Galaxy clusters also exhibit mass discrepancies, such as the in-falling of the Milky Way and Andromeda[4] as well as the X-Ray red shift of gas within individual galaxies in a cluster[5]. On cosmological scales, two main telling issues stand out: low baryonic mass density and large scale structural formation. Mass density estimates for the universe are signiﬁcantly lower than those derived from big bang nucleosynthesis[6]. Furthermore, formation rates of large scale structures due to gravitational attraction is much higher than should be possible with only observable matter[7]. These last two observations point towards two facts: dark matter must be non-baryonic and must be cold or not moving fast.

1.2 ΛCDM

The current theory of dark matter is known as Lambda Cold Dark Matter(ΛCDM), which is the culmination of years of observational, computational and theoretical research[8]. The Λ reﬂects the addition of General Relativity and by extension Dark Energy, to the theory of dark matter[9]. Dark Energy is a large amount of extra energy in the universe identiﬁed from observations of distant supernovae. The ’Cold’ in Cold Dark Matter refers to the fact that for dark matter to clump, the matter cannot be moving quickly. These theories combined give us our current estimates of the amount of baryonic matter, dark matter, and dark energy that exist in the universe.

2 1.3 MOND

Initial theories that aimed to correct for the ’missing mass problem’ proposed adding additional mass to the systems although this would not be the only option. Some people theorized that instead of adding additional mass to the system, another method for solving the missing mass problem would be to modify how gravity functions. Mordehai Milgrom proposed his theory of MOdified Newtonian Dynamics in 1983 [10]. The modification done by this theory affects Newtonian dynamics at low accelerations, thus keeping the rotation curves of galaxies flat. The modification that he proposes appears in the form1,

|gi | gn = µ gi , (1.2) ao where ao is a characteristic acceleration, µ is the matching function and gn is the Newtonian

acceleration. The term gi is what the actual acceleration would be after the MOND correction. The matching function can be several types of functions that behave with the limits:

 µ = 1, x >> 1 µ(x) =  (1.3)  µ = 0, x << 1 .  Several other matching functions could be used [11]; our research will utilize a simple function. The choice of matching function does little to change the results obtained. Finally,

m a = 1.2 × 10−10 , (1.4) o s2

is the characteristic acceleration that establishes the transition between Newtownian dynamics and Modiﬁed Newtonian dynamics. This has been established by ﬁtting functions to rotational velocity curves although there are other method to establish this value.

1Bold face variable are to be treated as vector notation.

3 1.4 The MOND External Field Eﬀect

When Milgrom first proposed his idea, an immediate issue stood out: why do we not experience MOND in our every day life? His solution to this is the external field effect. Planets, stars, star clusters and galaxies do not exist in isolation, they are influenced by the gravitational field of a nearby object which will change the dynamics of the system. This external field could help drive a system towards Newtonian dynamics even though the system itself should behave as if it was MONDian[10]. The equation,

|ge + gi | gn = µ (ge + gi) − gne (1.5) ao

where gn is the Newtonian acceleration, ge is the MOND external acceleration, gi is the the MOND

internal acceleration and gne is the Newtonian external acceleration. The external field effect has been a recent topic of interest, due to the fact that it is an effect that only occurs in MOND and is distinct from the tidal effect.

1.5 Dwarf Spheroidal Galaxies

Dwarf spheroidal galaxies are low surface brightness galaxies [12] that are usually satellite to a much larger galaxy. Due to their low surface brightness, the bulk properties are usually all that can be used to examine these systems. A luminosity as well as half light radius are primary properties reported. Utilizing the luminosity, the mass can be estimated by means of a mass-to-light(ML or M/L) ratio, which is a ratio of dynamical matter to visible matter. Mass-to-light ratios can be estimated by examining the bulk velocity dispersion, stellar kinematics, or even the age of the system. The half light radius (rh) is the radius at which half of the total light of the system exists. Due to the low surface brightness, these galaxies are usually described as dark matter dominant which would also indicate MOND-like effects taking place within them. Furthermore, several dwarf galaxies and globular clusters exist in the third case, where the effects of the external field may become important for the furthest stars. This allows dwarf spheroidal galaxies to be a good

4 test bed for MOND.

1.5.1 Previous Work

In Alexander et al. 2017 [13] we examined MOND effects on several dwarf spheroidal galaxies in isolation. We compared their observed dispersion profiles to kinematic simulations using MOND. Furthermore, McGaugh has implemented a virial like theorem to estimate the bulk properties of dwarf spheroidal galaxies in MOND [15] [16] which also included examination of the External Field Effect. In Alexander et al. 2017, we utilized statistical models to examine the kinematics of galaxies in isolation. Figure 1.1 displays 8 Milky Way Dwarf Spheroidal Galaxies. We matched the bulk dispersion of the galaxies by varying the Mass-to-Light ratios, which are displayed on the top right. Immediately the Newtonian runs seem to come well under the observational data. The addition of MOND was able to not only allow for a lower Mass-to-Light, but also raised the entire dispersion profile.

5 MOND Newton Walker Observations

15 Carina Draco MOND M/L = 5 MOND M/L = 20 Newton M/L = 80 Newton M/L = 80 10

Fornax Leo I MOND M/L = 1.5 MOND M/L = 2 Newton M/L = 7 10 Newton M/L = 10

5 )

s /

m Sculptor

k Leo II ( MOND M/L = 3.5 MOND M/L = 4 n Newton M/L = 22 o 10 Newton M/L = 22 i s r

p s i

D 5

Sextans Ursa Minor MOND M/L = 6 MOND M/L = 31 Newton M/L = 120 10 Newton M/L = 180

0 0 500 1,000 1,500 2,000 2,500 3,000 500 1,000 1,500 2,000 2,500 3,000 Distance from Galactic Center (pc) Distance from Galactic Center (pc)

Figure 1.1: Eight Milky Way Dwarfs and their dispersion proﬁles. The red line is the isolated MOND dispersion while blue is the Newtonian velocity dispersion. The black circles are observational data published in Walker 2009 [14]

6 1.6 Focus

The aim of this thesis is to continue the work published by Alexander et al. 2017. We hope to build off the previous practices and techniques by extending them to cover the external field effect. We will focus on select dwarf spheroidal galaxies that have good dispersion data or are obviously under the external field effect. Fornax and Leo I both have larger observational data sets and lower ML ratios. Crater II is a massive yet dispersed system that is a prime candidate for a system dominated by the external field effect [15]. Finally, Andromeda 19 is similar to Crater II although it is a satellite of M31 [17].

7 Chapter 2

Method

2.1 Dwarf Spheroidal Model

The modeling of dwarf spheroidal galaxies is performed using an in-house N-Body code written in Fortran. This has been used previously in Alexander and Agnor 1998 [18] and Alexander et al. 2017 [13]. Traditional N-Body codes are not possible with MOND, so we have modiﬁed our original code in such a way that the bodies only interact with a spherically symetric mass distribution. Our model utilizes the Hernquist distribution(Equation 2.1) which we use to initially populate the stars based on a statistical method.[19] The Hernquist density proﬁle is,

M a ρ(r) = s , (2.1) 3 2π r(r + as) where M is the total mass, r is a radius and as is the scale length which is related to the the half-mass radius by √ rh = as(1 + 2) . (2.2)

We populate our stars within the system using a statistical method that returns a radial position of the star which follows the Hernquist proﬁle. Two random angles are then chosen so that the stars are spread evenly in space. Each star is then given a random velocity to reproduce the motion of stars in a pressure supported spherically symmetric system. We also have the ability to employ initial

8 circular and radial velocities. Note, that due to non-physicality of the Hernquist density profile as r → 0, we implement a constant density core when r < rc where rc = αas, and we set α = 0.1. The Hernquist density profile is also used to calculate a mass internal to each star, based on the fact that each individual star should only feel a force from the mass internal to that star due to the spherical symmetry . This is similar to finding the electric field through a Gaussian surface, where only the charge enclosed matters. This gives us a mass internal at some distance,

r2 M (r) = M . (2.3) int 2 (r + as)

Due to how we calculate the interior mass of the system, we are limited to situations where there is spherical symmetry. Utilizing this equation, we can easily vary the bulk properties of the system to create a wide range of spherical galaxy types.

The mass internal to each star is used to calculate the acceleration from the center of mass using Newtonian Dynamics, GM g = − int r , (2.4) n r3

where Mint is the mass internal using equation 2.3, and r is the vector away from the center of

mass. gn will be used to signify the internal Newtonian acceleration. We can then use the HITS integrator to ﬁnd the motion of the objects in time.

2.1.1 Code Conditions

The base units used in the code are not the standard MKS. This is due to the large distances and high mass of objects. Instead, our units are parsecs, solar mass, and million years 1 which give us a pc3 value of G = 0.0045 My2 M for the gravitational constant. Furthermore, our bodies do not interact with each other directly and can not collide; this allows us to make a simpliﬁcation in the form of setting the mass of each star equal to 1M .

1 16 30 A parsec is ≈ 3.4 light years or ≈ 3.08 × 10 m. M is the unit ’solar mass’ ≈ 1.989 × 10 kg.

9 2.1.2 Hermite Individual Time Step Integrator

The Hermite Individual Time Step (HITS) integration scheme was proposed my Makino and Aarsseth in 1992 [20]. This is a fourth order integration method, that allows for a variable time step to increase both integration accuracy as well as speed. The HITS integrator requires a as well as aÛ, which are the total acceleration and ﬁrst derivative of acceleration 2 that act on the object that

is to be integrated. For our purposes a is simply gn from Equation 2.4. Starting with

GM a = − int r , (2.5) r3

we can take the derivative of a with respect to time. This may seem trivial at ﬁrst, but Mint has an r dependency that must be taken into account. We then calculated the jerk by taking the time derivative with the Hernquist interior mass which gives us

( " # ) GM 2a (r · v)r aÛ = − int v − 3 − s . (2.6) 3 2 r (r + as) r

The constant density core is imposed by ﬁrst calculating the core mass interior,

Mc Mc = r3 , (2.7) int 3 rc

where α is the scale of the core. Then, using Equation 2.4 again,

GMc a = int r , (2.8) 3 rc

we have the acceleration inside the core. The jerk is then

GMc aÛ = − int v . (2.9) 3 rc 2Also known as the jerk.

10 By knowing a and aÛ, we can implement the integration scheme, which is based on the Taylor Series3 and knowing x(t), v(t), a(t), aÛ(t). We choose a body to update and ﬁnd the next minimum time step, which is set in the program,

t = ti + ∆ti (2.10)

A prediction based on this next time step along with the Taylor series is used to ﬁnd the position and velocity,

1 1 x (t) = x(t ) + v(t )(t − t ) + a(t )(t − t )2 + aÛ(t )(t − t )3, (2.11) p i i i 2 i i 6 i i 1 v (t) = v(t ) + a(t )(t − t ) + aÛ(t )(t − t )2. (2.12) p i i i 2 i i

The predicted positions and velocities are applied to the previous a and aÛ equations to ﬁnd a predicted acceleration and jerk. Now, with the current value and predictions for the acceleration and jerk, a system of Taylor series can be written,

1 1 a (t) = a (t ) + aÛ(t − t ) + aÜ(t − t )2 + Ýa(t − t )3, (2.13) p i i i 2 i 6 i 1 aÛ (t) = aÛ(t − t ) + aÜ(t − t ) + Ýa(t − t )2. (2.14) p i i 2 i

These equations can then be solved simultaneously for aÜ and Ýa which can be plugged back into the predictions of position and velocity.

1 1 x(t) = x (t) + aÜ(t )(t − t )4 + Ýa(t )(t − t )5 (2.15) p 24 i i 120 i i 1 1 v(t) = v (t) + aÜ(t )(t − t )3 + Ýa(t )(t − t )4 (2.16) p 6 i i 24 i i

The corrected prediction for position and velocity can now be used to step the body forward in time. Once this is done, a new variable time step is calculated which is based on the acceleration of the object. The higher the acceleration, the ﬁner the precision needs to be for stepping an object forward in time. This is then repeated for all bodies until the simulation comes to completion at the

− 3 0 00 (x−a)2 n−1 (x−a)n 1 f (x) = f (a) + f (a)(x − a) + f (a) 2! ... + f (a) (n−1)!

11 maximum run time. We have successfully implemented this integration scheme as shown in both Alexander 1998 and et al. 2017 [18] [13].

2.1.3 Implementing MOND in Isolation

In order for the HITS integrator to run, we must have the Newtonian acceleration as well as jerk. We calculate these terms utilizing the methods mentioned above. Then, we pass the Newtonian acceleration into the MOND equation to ﬁnd a Mondian correction. Starting at equation 1.2 we can solve for the internal acceleration,

gi gn = µ gi , (2.17) ao and using our simple matching function,

x µ(x) = , (2.18) x + 1

we can solve for the internal acceleration or gi which is also Equation 14 in Alexander et al. 2017[13], " r # gn ao gi = 1 + 1 + 4 . (2.19) 2 gn

Knowing the acceleration, we can calculate the jerk by taking a time derivative which is Equation 15 from the same paper.

 " #  1  r a 2g gÛ · g a  gÛ = gÛ 1 + 1 + 4 o − n n n + o  . (2.20) i n q 2 2  gn 1 + 4 ao gn gn   gn    The matching function is the simplest form of a function set that satisﬁes the requirements laid out in Section 1.3. We will use this function throughout our work, although other options are possible.

12 2.2 Implementing the MOND External Field Eﬀect

2.2.1 The External Field Eﬀect

The equation that describes the MOND External Field Eﬀect is given by Famaey & McGaugh(2012)[11],

|ge + gi | gn + gne = µ (ge + gi) , (2.21) ao

where gn is the Newtonian acceleration, ge is the MOND external acceleration, gi is the the MOND

internal acceleration and gne is the Newtonian external acceleration.

2.2.2 The External Field Eﬀect in our Model

The Newtonian external acceleration is,

−GM g = MW , (2.22) ne D2

10 where MMW = 6.5 × 10 M is the mass of the Milky Way and D is the distance to the center of the satellite from the host galaxy. The MOND correction for External Field is then

ge gne = µ ge . (2.23) ao

Due to the large distance between the host galaxy and satalite galaxy, we model the host external ﬁeld as a point mass and due to the small size of the dSph, we can treat the ﬁeld as if it were uniform across the satellite system. This has been done before by Derakhshani 2014 [21], as well as Haghi 2010[22] and will help to reduce a complicated three dimension problem to one dimension.

Furthermore, we make the assumption that |ge + gi| ≈ |ge| + |gi |. This approximation has been used in Famaey 2007 [23] to simplify computations while still oﬀering insight.

13 Using these as a starting place, we can ﬁnd a gi as well as gÛi. Starting with equation 2.21 we can solve for gi utilizing some algebra and the quadratic formula,

" r # gn (gn + gne) (gn + gne) ao gi = + 1 + 4 − ge . (2.24) gn 2 2 (gn + gne)

Then with ﬁnesse and a steady mind, we can take the time derivative to ﬁnd the jerk,

s ! gÛn (gn + gne) (gn + gne) 4ao gÛi = + 1 + − ge gn 2 2 (gn + gne) ( " s ! # gn 1 4ao 2ao + gÛn · gn 1 + 1 + − 2 2 (g + g ) q 4a gn n ne 1 + o (g + g ) (gn+gne) n ne s !) 1 (gn + gne) (gn + gne) 4ao − + 1 + − ge . (2.25) gn 2 2 (gn + gne)

2.2.3 The External Field Eﬀect in Practice

The External Field Effect can alter the internal dynamics of a satellite when it is in the gravitational field of some host. This effect will lead to certain limiting MOND cases.

• Case 1: ge < gi < ao The internal dynamics are governed by MOND.

• Case 2: gi < ao < geThe internal dynamics of the system are governed by Newtonian Dynamics

• Case 3: ge ' gi < ao The system is under the eﬀects of MOND with the external ﬁeld partially present.

• Case 4: gi << ge < ao This is a simplified case where the external dynamics dominate while the system remains Mondian. This case allows a simplification that utilizes a modified gravitational constant of the form: G = G ao 4 e f e ge

14 Figure 2.1: Case 1

1e+04

100

1 ) o a ( 0.01 n o i t a r e l e

c 0.0001 c A gni giso 1e-06 gne ge geff 1e-08 gi

1e-10 0 500 1,000 1,500 2,000 2,500 Radius(pc)

Figure 2.1 shows the acceleration for Case 1. This is the isolated case, where the giso plot follows the corrected MOND acceleration or gi. The line for gi shows a plot of our Equation 2.24, which is the internal acceleration for the system. The close match comes about due to the external

MOND ﬁeld(ge) being much lower than the internal acceleration which is an extreme boundary condition. 4This case, though interesting is primarily used as a simpliﬁcation for mathematics purposes such as in McGaugh’s 2016 paper on Crater II [15].

15 Figure 2.2: Case 2

100

10 gni giso 1 gne ge geff )

o gi a ( 0.1 n o i t a r e l

e 0.01 c c A

0.001

0.0001

1e-05 0 500 1,000 1,500 2,000 2,500 Radius(pc)

In Case 2, shown in Figure 2.2, the external field dominates throughout the entire system and is much higher than ao. Because of the External Field Effect, the system is not under the effects of MOND and thus the corrected acceleration follows the Newtonian acceleration plot. This shows that the boundary condition of an external dominated system is also in agreement with the purpose of the External Field Effect.

16 Figure 2.3: Case 3

gni giso 1 gne ge geff ) o

a gi ( n o i t

a 0.1 r e l e c c A

0.01

0.001 0 500 1,000 1,500 2,000 2,500 Radius(pc)

Case 3 which is shown in Figure 2.3 is interesting due to the inner regions acting as if they are in isolation while the outer regions have the External Field Eﬀect present. This is due to the

external gravitational ﬁeld(ge) being on the same order of magnitude as the internal gravitational

field(giso). Now the distance to the center of the galaxy plays a part in whether or not the EFE has an effect. Notice that the point where this transition happens is around the point where the internal acceleration goes below the external field corrected for MOND.

This is a good time to mention Case 4(ge f f ), which does not show good agreement with either of the previous cases. We do notice that after a certain radius in Case 3, it has better agreement that the isolated acceleration(giso).

17 2.3 Data Analysis Technique

Position and velocity data for each object in the simulation are output at a set write step. This data can then be analyzed for kinematic information of the system. Due to Dwarf Spheroidal Galaxies being pressure supported, a rotation curve is not a valid method for describing bulk kinematics. Instead, we can use a statistical approach to examine the velocity dispersion of the galaxy. Velocity dispersion is simply the standard deviation of the observed velocity,

p σ = < v >2 − < v2 > (2.26) which gives us an idea about the ’randomness’ of the motion in the system. Using the Virial theorem Eq. 1.1, observations of motion of stars can give an insight to the mass of these satellite systems. The same can be done with simulations, and estimate of the mass can be made based on simulating the motion of stars in a system. By observing the total velocity dispersion of the system, a rough estimate of the mass can be calculated. If a more meaningful information is needed concerning the kinematics of a galaxy, a line of sight velocity dispersion proﬁle can be calculated. The line of sight velocity dispersion is the motion of the stars back and forth into the observed plane of the object. Line of sight velocity dispersion is what observers report and is also what our data analysis utilizes.

2.3.1 Velocity Dispersion Proﬁles

Velocity dispersion proﬁles are created by binning starts based on their radial distance then calcula- √ ting the dispersion of each bin. The total number of bins is N (where N is the number of bodies), √ which are then populated with N/ N stars. Each bin covers a set annular area that encompasses the calculated number of stars per bin, starting from the inner most star until all stars are enclosed. This usually leads to a higher concentration of bins towards the center, with the diﬀerence in the boundaries of the bins increase also increasing with radius. This can leave some of the outer bins covering several hundred parsecs.

18 Chapter 3

Results and Discussion

3.1 Model Conditions

We hope to expand on our previous work which focused on examining MOND in isolated dwarf spheroidal galaxies to now include the External Field Effect. We first examine the acceleration profiles for observed satellite systems and we then analyze the velocity dispersion profiles for these systems. Our models focus on the three Milky Way dwarf spheroidal galaxies: Fornax, Leo I, and Crater II; and one Andromeda dwarf: Andromeda XIX. The observed properties of the four galaxies we have selected are in Table 3.1. The reported value of σ is from observations. Our choice of mass to light is primarily from our previous work in the fitting of bulk velocity dispersion.

Table 3.1: Initial data for four dwarf spheroidal galaxies. Rh is the half-light radius, L is the luminosity, D is the distance to the host galaxy, σobs is the observed bulk dispersion and ML is our best ﬁt mass to light. dSph Rh LD σobs ML Ref Name (pc) L (kpc) (km/s) Fornax 792±58 2.04 × 107 147 11.7±0.9 1 [16] Leo I 298±29 5.50 × 106 258 9.2±0.4 2 [16] Crater II 1066 1.60 × 105 120 2.7±1.0 2 [16] [24] And XIX 1799±52 3.30 × 105 116 5.7±1.6 2 [16]

19 3.2 Acceleration Plots and Velocity Dispersion Comparision

We present over the next few pages the acceleration proﬁles along with a comparison of velocity dispersion proﬁles for the four dSph using the methods described previously. The accelerations

plots of gi is the fully corrected for acceleration that our code uses to model the test galaxies, giso is the MOND acceleration if the system was in complete isolation, and ge is the MOND acceleration from the external host galaxy. gni and gne are the raw Newtonian internal and external acceleration 1 calculated using the Hernquist proﬁle . In addition, ge f f is Case 4, which is a simpliﬁed method

utilizing an estimate for the External Field Effect. Note that gne and ge are straight lines due to a non-varying field strength. The velocity dispersion plots compare MOND in isolation with an external field as well as the Newtonian cases. 1Equation 2.1

gni 1 giso gne ge geff gi ) o

a 0.1 ( n o i t a r e l e c c 0.01 A

0.001

0.0001 0 500 1,000 1,500 2,000 2,500 R(pc)

8 ) y m / c p ( n 6 o i

s EFE r e ISO p s

i NEWT D 4

0 0 500 1,000 1,500 2,000 2,500 Radius(pc)

Figure 3.1: Fornax Acceleration and Dispersion Proﬁles

100

gni 10 giso gne ge geff gi ) o 1 a ( n o i t a r e l e c

c 0.1 A

0.01

0.001 0 500 1,000 1,500 2,000 2,500 Radius(pc)

8 ) y m / c p ( n 6 o EFE i s

r ISO e

p NEWT s i D 4

0 0 500 1,000 1,500 2,000 2,500 Radius(pc)

Figure 3.2: Leo I Acceleration and Dispersion Proﬁles

gni giso gne ge geff 0.01 gi ) o a ( n o i t a r e l e c c A

0.0001

0 500 1,000 1,500 2,000 2,500 Radius(pc)

4.5 NEWT 4 EFE ISO 3.5 ) y

m 3 / c p ( n 2.5 o i s r e p

s 2 i D 1.5

0.5

0 0 500 1,000 1,500 2,000 2,500 Radius(pc)

Figure 3.3: Crater II Acceleration and Dispersion Proﬁles

23 10

gni giso 1 gne ge geff ) o

a gi ( n o i t a 0.1 r e l r e c c A

0.01

0.001 0 500 1,000 1,500 2,000 2,500 Radius(pc) 5 EFE ISO NEWT

4 ) y 3 m / c p ( n o i s r e p s 2 i D

0 0 500 1,000 1,500 2,000 2,500 Radius(pc)

Figure 3.4: Andromeda XIX Acceleration and Dispersion Proﬁles

24 Fornax (Figure 3.1) and Leo I (Figure 3.2) have good agreement between the External Field Eﬀect and the isolated accelerations for small radii. As the radius increases, the EFE and isolated cases tend to diverge. Figure 3.3 shows Crater II exhibiting an issue with the isolated acceleration

at all radii. This is most likely due to a very high external acceleration that is still lower than ao. Due to low accelerations throughout, the galaxy is well within the boundaries of the external field effect. Furthermore, in Figure 3.4 the non Milky Way dwarf Andromeda XIX exhibits similar behavior to Fornax and Leo 1, where at a smaller radius there is little difference in acceleration, but they diverge at large radii. The comparison of Crater II and Andromeda XIX acceleration plots is interesting due to the fact that some astronomers suggest that these dwarf spheroidal galaxies are similar in structural properties such as being very diffuse[15]. The fact that these two galaxies are structurally similar yet have different acceleration demonstrates a difference in systems completely dominated by the External Field Effect or those where the EFE gradually takes over. Finally, it is worthwhile to note two things: the ge f f or Case 4 tends to follow the shape of ge, and the Newtonian case tends to be an order of magnitude lower than the accelerations given by MOND. The velocity dispersion comparison have interesting properties. Fornax and Leo I have kinematic properties consistent with what we have assumed. The External Field Effect case starts out similar to isolated case, but the EFE is lower as the radius increases. The lowering or "curviness" of the EFE cases is more pronounced for Fornax than in Leo I. This could be attributed to an external field strength of Fornax double that of Leo I. Thus the internal acceleration is lower than the external acceleration at lower radii. Comparing velocity dispersion profiles of Crater II and Andromeda XIX leads to a noticeable issue immediately: the external field profile seem to be low with a large percentage of stars having escaped from the galaxy. The separation of the isolated and external case may be due in part to the External Field Effect dominating throughout the entire system. The diffuse nature of the stars in the system could be due to the External Field Effect pushing the system towards Newtonian dynamics. This should not be the case though, since the External Field is still below the characteristic acceleration ao.

25 3.3 Velocity Dispersion Proﬁles

We now vary the the external field by setting the bounds of the host galaxy mass to published limits, which we list in Table 3.2. Varying the external field in this manner will give us an idea of how the External Field Effect modifies the internal dynamics. Also in this section we compare our calculated dispersion profiles with observations if such observations exist. At this time there is no observed dispersion profile for Crater II or Andromeda XIX. We do present observational data for Fornax and Leo I.

Table 3.2: Mass data for host system. M is the unit solar mass. Host Min Mass Max Mass Ref 10 10 Name (×10 M ) (×10 M ) Milky Way 3.3 6.6 [25] Andromeda 12 19.2 [26]

26 Fornax

min max Walker 2010 12 ) y

m 10 / c p ( n o i s r e

p 8 s i D

4 0 500 1,000 1,500 2,000 2,500 Radius(pc)

Figure 3.5: Fornax velocity dispersion proﬁle. Black markers are observational data from Walker 2009[14].

16 Min Max Walker

12 ) y m / c p (

n 10 o i s r e p s i

D 8

4 0 500 1,000 1,500 2,000 2,500 Radius(pc)

Figure 3.6: Leo I velocity dispersion proﬁle. Black markers are observational data from Walker 2009[14].

28 2

1.8 max min 1.6 )

y 1.4 m / c p ( n 1.2 o i s r e p s i

D 1

0.8

0.6

0.4 0 500 1,000 1,500 2,000 2,500 Radius(pc)

Figure 3.7: Crater II

min max

1.5 ) y m / c p ( n 1 o i s r e p s i D

0.5

0 0 500 1,000 1,500 2,000 2,500 Radius(pc)

Figure 3.8: Andromeda XIX

29 Here we have the velocity dispersion profiles for our four test galaxies. Fornax and Leo I (shown in Figures 3.5 and 3.6) both have published dispersion profiles; Crater II and Andromeda XIX(shown in Figures 3.7 and 3.8) do not have published dispersion profiles. That being said, the observational data sets of Fornax and Leo I have large scatter and error bars. Fornax has the most complete observational data set with N = 200 contributing to its data set. We chose to vary the mass of the host objects while keeping everything else the same, in order to create a noticeable difference in the profiles as the external field varies. We found that as the external field strength is increased, the dispersion profile tends to be lower further from the center. This may be related in part to the original aim of the external field effect: to drive systems that should be in a MOND regime towards Newtonian dynamics. The profiles for Crater II and Andromeda XIX are more rough in appearance due to resolution issues. Fornax and Leo I are more compact and thus they have a higher bin density than Crater II and Andromeda XIX. Even with the roughness of the data, the curved nature of the dispersion profile are still present.

3.4 Possible Weaknesses of Work

In Modified Newtonian Dynamics, it is difficult to model systems computationally due to the non- linear nature. This non-linear nature causes issues with conservation of energy for direct body-body interactions. Beyond this, we have no method for feedback into the model; both of these lead to a more static model. We also chose to make a simplification of the vector addition in Equation 2.21. This allowed us to avoid computational complexity, though if left us more "in the spirit of MOND" than following the technical theory. Other possible issues inherent to this study could be categorized as external shortcomings. Observations of dwarf spheroidal galaxies are challenging unto themselves, due to their nature of being low surface brightness. Some galaxies are reported using a single digit star count; this is not meant to be a criticism of the observer but an acknowledgment of the difficultlies in identifying and categorizing dSph. We chose our first two Milky Way satellites due in part to their better-than- average data set. Crater, though it does not have published dispersion data, it is interesting due

30 to its structural conﬁguration. Andromeda XIX follows this same structure, although more recent data of the galaxy may bring to light other issue. Martin et al. (2016) published a paper reporting

Andromeda XIX to possibly have rh nearly three times larger than the value used in this paper [27]. Furthermore, due to the incredibly diﬀuse nature of Andromeda XIX, the galaxy is most likely has a high elipticity. This leads us to a ﬁnal key point, our systems are limited to spherically symmetric cases. Most astronomical clusters of objects are not perfectly spherical in nature. Dwarf spheroidal galaxies are at times described as nearly spherical, but still have a small amount of asymmetry. Due to our integration method, we are limited to completely spherical systems. We avoid modeling tidally disturbed systems so that our code can remain valid.

31 Chapter 4

Conclusion

The dark matter problem has been a well known issues in astrophysics for the last century. With ever increasing observational techniques, astronomers have noticed what seems to be missing gravitational mass, which has led to the theory of ΛCDM. Mordehai Milgrom proposed a theory of Modified Newtonian Dynamics which aimed to solve the missing mass problem without needing additional mass. The effects of MOND should be revealed whenever the local acceleration is low enough, though this is not what we have observed. To accommodate this fact, the External Field Effect was proposed, where the external gravitational fields of a system could change the internal dynamics. We have expanded our initial work dealing with Modified Newtonian Dynamics in isolated systems to now include the external field effect. A modification has been made to our code allowing us to model dwarf spheroidal galaxies that experience an external field from their host system. We chose four test case galaxies and presented their acceleration profiles, velocity dispersion profiles and then a comparison between the External Field Effect and Isolated MOND. We notice a radial difference in the acceleration of the Isolated and EFE cases; also systems that should be completely in the EFE regime show the External Field Effect throughout. The examination of the velocity dispersion profiles show an interesting structure: as the strength of the external field increases, the dispersion profile is shifted downwards. Our comparison of the isolated and external field cases showed interesting kinematic properties for Leo I as well as Fornax; the lowering of the dispersion

32 profile was not outside the observational range. These results exposed differences in how Modified Newtonian Dynamics behaves depending on if the External Field is taken into account. While some aspects of our work are limited, we believe that this is a solid foundation of continued exploration of MOND. We hope to continue this work, so that it may be a contribution towards solving the missing mass problem.

4.1 Future Work

Our original paper published examining the isolated MOND cases included a much wider set of dwarf spheroidal galaxies. Future work should focus on reexamining these systems while taking into account he External Field Effect. New Andromeda satellite galaxies are also being constantly discovered as observational techniques improve. Globular clusters may also be interesting systems to study, as they tend to be lower mass systems with a nearby gravitational influence. Our computational approach could also be expanded so as to be more physically accurate. Feedback or active galactic nuclei or AGN could be included in our code, and is something several dark matter simulations have be trying to include. AGN is an umbrella that includes feedback into the galactic system due to baryon processes such as the lifetime of stars. Along this same line, handling gas dynamics computationally may give us the opportunity to examine non-spherical systems. This would not only improve our accuracy, but also expand the amount of systems that we can model. Finally, by having the opportunity to handle anisotropic systems, we could include tidal effects in our system models. Systems under the External Field Effect may also be tidally disrupted. The most immediate aim to improve upon our galaxy model would be to handle the external field as a vector. For now, we handle the MOND correction as vector magnitudes. We have explored recursive method for calculating the MOND correction and hope to at some point implement these methods. In addition, some of the more diffuse systems may need a distance dependency when implementing they External Field Effect.

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