Determining the dark matter mass density profile of dwarf

spheroidal galaxies using Jeans analysis

Dylan van Arneman

April - June 2018

Student number 10919414

Supervisor dr. Christoph Weniger

Examiner dr. Shin’ichiro Ando

Faculty Faculty of Science, University of Amsterdam

Institute ITFA

Report Bachelor Project Physics and Astronomy

Size 15 EC

Conducted between 03-04-2018 and 03-07-2018

Date of submission July 3, 2018

Abstract

One of the biggest mysteries in modern physics and astronomy is the nature of dark

matter. In this paper the dark matter content of the Carina, Draco, Fornax, Leo I & II,

Sculptor, Sextans and Ursa Minor dwarf spheroidal galaxies is inspected using spherical

Jeans analysis. For each of these galaxies the so-called astrophysical factor, or the J-

factor, gets computed using parametric models for the dark matter mass density and

the stellar number density. For determining the dark matter mass density an Einasto

parametric model is used whilst a Zhao-Hernquist model is used to determine the stellar

number density. The J-factor is calculated for several constant values of the anisotropy and

give results very similar to those computed using a parametric model for the anisotropy.

Finally, we also find that the resulting value of the J-factor is strongly dependent on the

chosen boundaries of the line-of-sight integral in the expression for the J-factor.

Contents

1 Introduction 3

2 Theory 4

2.1 The J-Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 The Jeans equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Density profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Method 9

3.1 Estimating the stellar number density . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Estimating the LOS velocity dispersion . . . . . . . . . . . . . . . . . . . . . 10

3.3 Calculating the J-factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Results 12

4.1 Stellar density results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.2 Jeans equation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.3 J-factor results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Discussion 15

6 Conclusion 18

7 Acknowledgements 19

8 Bibliography 19

9 Appendix 22

2

Dutch summary

Donkere materie is een van de grote nog onbeantwoorde vragen in de moderne natuur- en

sterrenkunde. Al vanaf de allereerste observaties van het fenomeen van donkere materie

zijn wetenschappers geconfronteerd met talloze vragen rond de aard van dit fenomeen. Wat

is donkere materie? Bestaat het uberhaupt wel? In dit onderzoek wordt er gekeken naar

donkere materie binnen bolvormige dwergsterrenstelsels. Als donkere materie inderdaad uit de

voorgestelde ’neutralino’ deeltjes zou bestaan, zou het bepaalde eigenschappen van ’bekende’

deeltjes moeten tonen. Een van deze eigenschappen is onder andere het verval van deze

deeltjes. Als neutralinodeeltjes inderdaad in andere deeltjes zouden kunnen vervallen, zou dit

waarneembaar moeten zijn. Een grootheid die te maken heeft met dit waarneembare signaal is

de J-factor. De J-factor is ook direct gekoppeld aan de massadichtheid van de donkere materie

binnen een bepaald sterrenstelsel. Door de J-factor te onderzoeken kan dit uiteindelijk leiden

tot meer inzicht in het fenomeen van donkere materie. Het doel van dit onderzoek is om

de resultaten van een eerder onderzoek, dat gebruik maakt van een andere analysemethode,

te reproduceren. In ons onderzoek wordt de donkere materie massadichtheid van bepaalde

dwergsterrenstelsels met behulp van computermodellen berekend. Met deze massadichtheid

kan de J-factor van het sterrenstelsel bepaald worden. Deze resultaten worden vervolgens

vergeleken met die van het eerdere onderzoek. Uit onze resultaten is te concluderen dat onze

methode de waarde van de J-factor goed kan reproduceren.

1 Introduction

One of the biggest open questions in modern physics and astronomy is dark matter (DM).

What exactly is dark matter? Does it behave like classical massive particles, or is it a new

type of matter? What other properties does it have? Does it even exist? These are just some

of the questions surrounding the nature of dark matter that physicists have been trying to

answer since the first observation of the dark matter phenomenon.

This project investigates the nature of dark matter and will hopefully contribute to future

discoveries on the nature of dark matter. In this report we focus on the properties of dark

matter within dwarf spheroidal galaxies (dSphs). We choose to look at dSph galaxies that

are satellite galaxies of the Milky Way (MW). It is thought that, similarly to regular galaxies,

dSph galaxies contain halos of dark matter [1]. If the dark matter particles within these halos

are in fact similar to particles described by the Standard Model, they should also exhibit

behaviour and interactions of ’known’ particles. An example of such an expected interaction

of these DM particles is decay and annihilation. If two DM particles within a halo were to

3

decay or annihilate with each other, they would produce observable gamma ray radiation [2].

Finding such gamma ray signals could prove the existence of dark matter, as well as give

further information on the nature of dark matter particles.

Satellite galaxies of the MW seem to be ideal astrophysical targets to find these gamma ray

signals, as they have low intrinsic background signals and are closer by than other potential

targets [1]. In contrast to ’regular’ galaxies, these galaxies are often smaller and appear to be

spheroidal. Such galaxies are called dwarf spheroidal galaxies.

In this report we look at eight different dSph galaxies (Carina, Draco, Fornax, Leo I&II,

Sculptor, Sextans and Ursa Minor), and investigate the physics related to the previously

mentioned gamma ray signals. The goal of this project is to find a quantity related to the

DM mass density within these dSphs called the J-factor or astrophysical factor. This quantity

will be found using Jeans analysis. Furthermore, we will investigate the dependence of the

J-factor on the kinetic structure of its respective galaxy by examining the anisotropy. This

will ultimately lead to more insight into the gamma ray signals. We will try to reproduce the

results of the work by [1], and will use a similar method to calculate the J-factors. The biggest

difference between this project and [1] is that we use constant values for the anisotropy, whilst

[1] uses a parametric version. In this project, the J-factors of each dSph will be calculated

for four different (constant) values of the anisotropy, and will ultimately be compared to each

other. We also investigate the influence of the anisotropy on the values of these J-factors, and

see if a constant anisotropy can give results similar to those found by a parametric anisotropy,

such as in [1].

This report is structured as follows: chapter 2 will give theory background information

about Jeans analysis and the J-factor, and its relation to gamma ray signals. Chapter 3 will

describe the method used to determine the J-factor and chapter 4 will present the results of

the J-factors for all eight dSphs, as well as their dark matter density profiles. Chapter 5 will

discuss these results and finally, chapter 6 will give the conclusion of these results, and will

give recommendations for future studies on this topic.

2 Theory

2.1 The J-Factor

As the main goal of this paper is to calculate the J-factors, it is important to understand what

the J-factor is and what information it can give about the nature of dark matter particles.

A proposed candidate for dark matter is the so-called neutralino particle [2]. If it is

assumed that dark matter in dSphs consists of neutralino particles, they could annihilate with

each other and produce gamma-rays. These gamma rays would be observable by telescopes,

4

and would have a flux given by [2][10]:

Φγ =Nγσv

8πM2χ

∫line of sight

ρ2(l)dl, (1)

where Nγ is a value dependent on the type of annihilation that took place, σv is the annihila-

tion cross-section, Mχ is the neutralino mass and ρ(l) is the DM mass density. It is important

to note that the first part of the right-hand-side of equation (1),

Nγvσ

8πM2χ

,

gives information on the properties of the DM particle, such as the mass, whereas the second

part of the equation, ∫line of sight

ρ2(l)dl,

is solely dependent on the density structure of the dSph galaxy.

In order to relate these expressions to more directly observable quantities, equation (1)

can be rewritten in terms of a differential flux measured within a solid angle ∆Ω [1]:

dΦγ

dEγ= ΦPP

J (Eγ)× J(∆Ω). (2)

This new equation is now dependent on the astrophysical factor, also known as the J-factor,

and on ΦPPJ (Eγ). All information on the particle properties is stored within the quantity

ΦPPJ (Eγ). The J-factor is given by a double integral along the line-of-sight (LOS) and over

the solid angle of the dSph:

J =

∫∆Ω

∫LOS

ρ2DM (l,Ω)dldΩ. (3)

The solid angle in equation (3) is given by [1]

∆Ω = 2π × (1− cosαint), (4)

where αint is the integration angle. For αint 1, this equation can be rewritten as:

∆Ω = πα2int. (5)

There exists an angle at which the value of the J-factor is minimally dependent on the

expression for the density that is used to calculate the J-factor [15]. This is ideal, since this

means that several different models for the DM density could be used to estimate similar

values of the J-factor. According to [15], 80% of the stars within a dSph can be found within

a few 100 pc (also known as the half-light radius rh) of the dSph’s radius. Therefore both

this paper as well as [1] assumes the integration angle to be optimal at αint = 2rh/d, where

5

d is the distance to the dSph. This angle is also known as the critical angle (αcrit) [15]. The

values for the critical angles used in this analysis have been extracted from [1].

Since this integral is over the line-of-sight and not over the radius, the variable r needs

to be converted to the line-of-sight variable l [10]. This can be done by applying the law of

cosines, which is demonstrated in figure 1. In this figure, r is the distance from the center

of the galaxy to any point within the radius of the DM halo, d is the distance to the dSph

galaxy and l is an arbitrary point on the line-of-sight. Using the law of cosines,

r2 = d2 + l2 + 2dl cos θ, (6)

we can now substitute r within the expression for the DM density. ρDM (r) can now be

rewritten as ρDM (l, θ). The boundary for the LOS integral has been chosen to go from

l = d− δl to l = d+ δl, where δl has to be large enough to ensure that all the emitted gamma

rays are encapsulated within the integral.

Figure 1: A schematic representation of the observation of an arbitrary point within the DM

halo of a dSph galaxy. Here d is the distance from the observer to the center of the dSph

halo, r is the distance from the arbitrary point in the halo to the center of the halo, l is the

arbitrary point on the line-of-sight and θ is the angle between d and l. These variables can

be related to each other using the law of cosines.

If the differential gamma ray flux of a dSph galaxy were to be measured, only the J-factor

would be needed to determine the particle properties of the DM. An example of what one

could find by examining the J-factor is a new constraint on the annihilation cross section of

the proposed neutralino particle [4]. This is why it is interesting to compute and investigate

the J-factor. A similar expression for the decay of neutralino particles also exists and is called

the D-factor [4]. This report, however, only inspects the J-factor.

6

2.2 The Jeans equation

As noted in the previous section, the J-factor is dependent on the DM mass density of the

dSph galaxy. To relate the DM mass density to an observable quantity, one must first look

at the mass enclosed within a dSph galaxy. As the mass within a dSph is strongly dominated

by dark matter, the contribution of stars can be neglected in the equation for the mass [5].

Therefore, the expression for the enclosed mass at a given radius r, is given by

M(r) = 4π

∫ r

0ρDM (s)s2ds, (7)

where ρDM (s) is the DM mass density of the dSph.

To get the mass of the galaxy, we need to apply Jeans analysis. The goal of Jeans analysis

is to ultimately relate the mass density to an observable quantity, and to therefore be able

to compute the J-factor. The advantage of using dwarf spheroidal galaxies is that they are

spherically symmetric, so we can apply the Jeans equation [3]:

1

ν

d

dr(νv2

r ) + 2βani(r)v2

r

r= −GM(r)

r2, (8)

where ν(r) is the stellar number density, v2r (r) is the radial velocity dispersion, βani(r) is

the velocity anisotropy and M(r) is given by equation (7). The Jeans equation originates

from rewriting the collisionless Boltzmann equation in spherical coordinates and taking the

gravitational potential as the potential [3].

Now, using equation (8), we can relate the mass to the kinematic properties of the galaxy.

Though yet none of these quantities can be measured directly, they can be related to an

observable quantity: the projected line-of-sight (LOS) velocity dispersion, σp. This relation

is given by the following equation [1]:

σ2p(R) =

2

Σ(R)

∫ ∞R

(1− βani(r)

R2

r2

)ν(r)v2

r (r)r√r2 −R2

dr, (9)

where R is the projected line-of-sight radius, and Σ(R) is another observable quantity: the

surface brightness of the dSph. The surface brightness (also called the projected light profile)

is given by

Σ(R) = 2

∫ ∞R

ν(r)r√r2 −R2

dr. (10)

Using parametric models for the mass density, the stellar number density and the anisotropy,

it would be possible to solve equations (7) to (10). Then, the (parametrically) calculated val-

ues of the two observable quantities, σp(R) and Σ(R), could be fitted to actual data of dSphs

and could subsequently determine the best-fitting values for the parameters of each of the

models. In particular, the best-fitting parameters for the mass density could be used to solve

equation (3) and as a result calculate the J-factor.

7

The parametric models that will be used will be discussed in the following subchapter and

the fitting process will be discussed in chapter 3.

2.3 Density profiles

There are several parametric models that can be used to estimate the DM mass density, as

well as to estimate the stellar number density. Other models for anisotropy profiles could also

be used [11] for the Jeans analysis, but since this report only restricts itself to constant values

for the anisotropy, these profiles will be left out of this section.

When treating parametric density profiles, different ’families’ of models exist. Many of the

most commonly used profiles stem from the Hernquist density model [12]. In 1990, Hernquist

proposed a general expression to describe the behaviour of the mass density within a galaxy.

This model closely approximates and takes into account several empirically found laws, such

as the R1/4 law for elliptical galaxies and galactic bulges [12]. This parametric model was

further expanded by in 1996 by Zhao [13], and takes into account other observed galactic

phenomena such as galaxies having a cusped light profiles (suggesting a black hole in the

center of the galaxy) and non-spherical shaped density distributions [13]. This Zhao profile

is given by the following expression:

νZhao(r) =ν0

(r/rs)γ [1 + (r/rs)α](β−γ)/α, (11)

where ν0 is a normalization factor, rs is the scale radius, γ the inner slope, α is the transition

slope and β is the outer slope [4]. Another generalized version of this profile is called the

NFW model [14].

Another ’family’ of density profiles is the Einasto profile. [14] argues that a model in

which the logarithmic slope of the density varies continuously with the radius leads to a

better description of the given observational data of galaxies, when compared to the NFW

or Zhao profiles. [14] claims that the following expression for the density provides the best

overall fits:

ρEinastoDM (r) = ρ−2 exp

− 2

α

[(r

r−2

)α− 1

], (12)

where ρ−2 is once again a normalization constant, r−2 is the scale radius and α is the loga-

rithmic slope.

Another advantage to using the Einasto profile for fitting the kinematic data is that it

only has three free parameters, as opposed to Zhao’s five free parameters. This generally

makes it easier to fit to data and reduces the overall computing time.

The anisotropy in equation (8), βani(r), is defined as [1]:

βani(r) ≡ 1− v2θ/v

2r , (13)

8

where v2θ is the velocity dispersion in the θ (polar) direction, and thus gives information on

how the stellar velocity is distributed over the different directions. According to [3], there

is no actual valid observational nor theoretical reason for assuming βanis to be constant or

isotropic. As mentioned in the introduction however, we do take constant values for βanis.

Whilst [1] uses a Baes & van Hese parametric profile for the anisotropy, we use the following

four values for βanis to compute the J-factor: βanis = -1, -0.5, 0 and 0.5.

3 Method

3.1 Estimating the stellar number density

The fits mentioned in chapter 2.2 rely on the two observable quantities: the LOS velocity

dispersion σp(R), and the surface brightness Σ(R). Firstly, Σ(R) must be fitted and deter-

mined, as it appears in the equation for the velocity dispersion (equation (9)) and is also

needed to calculate ν(r), which also appears in equation (9). Since there is no direct model

for surface brightness, we can use the Zhao-Hernquist density profile discussed in section 2.3

for the stellar density ν(r). We can then plug this parametric version of the stellar density

into equation (10) and calculate a modeled version of the surface brightness. This modeled

version of the surface brightness can now be fitted to actual measured values of the surface

brightness.

This fit will be done by minimizing the χ2 related to the likelihood function: [1]

exp(−χ2

)=

Nbins∏i=1

exp

[−1

2

(Σdata(Ri)− Σmodel(Ri))2

σ2Σ(Ri)

]. (14)

By taking the natural logarithm of equation (14) and minimizing this quantity using the

python iMinuit algorithm [7] [8] we can find the ideal parameters for the Zhao-Hernquist

stellar density profile. The inspected ranges for the parameters are given in table 1. These

ranges have not been based off of any theoretical priors but were found by manually adjusting

or re-sizing the ranges until a minimum had been found.

Parameter range Units

100 ≤ ν0 ≤ 250× 103 [kpc−3]

0.01 ≤ rs ≤ 20 [kpc]

−5 ≤ α ≤ 50 unitless

−5 ≤ β ≤ 105 unitless

−5 ≤ γ ≤ 50 unitless

9

Table 1: The inspected parameter ranges used for to fit the Zhao stellar density profile to the

surface brightness. All of these ranges were found by manually adjusting the ranges until a

minimum had been found in each dSph galaxy.

The values used for Σ(Ri)data have been extracted from [6], which determined the surface

brightness of the dSphs by counting the amount of stars within an elliptical aunulus. Before

this data can be used however, it must be adapted slightly. Since this project assumes that

the dSphs are completely spherical and not elliptical, the distance from a counted bin to

the center of the dSph (Ri) must be converted in order to stay consistent with our spherical

assumption. This is done by swapping Relliptical with Rcircular =√ab, where a and b are

the semiminor and semimajor axes of the ’elliptical’ galaxy respectively. The values of the

semiminor and major axes can be computed by using the ellipticity of the galaxy, which is

also given in [6]. Another change that needs to be made to the data is a correction to the

units. Due to the units of Σ(Ri) of [6] being in no./′2, these have to be changed to no./kpc2

by multiplying Σ(Ri) by a factor of(

206264.860×d

)2where d is the distance to the galaxy in kpc.

This conversion of units also brings along an added uncertainty to the observed data, which

will be used in our likelihood function. Whereas [1] only uses the Poisson error for σΣ(Ri), we

adjust this by making σΣ(Ri) = ∂Σ(Ri)∂d × δd, where δd is the uncertainty on the distance to

the galaxy. The values for d and δd have been taken from [9].

3.2 Estimating the LOS velocity dispersion

After the stellar density ν(r) has been determined using the method discussed in the previous

section, we can start to solve the Jeans equation (eq. (8)). First, we model M(r) by plugging

the Einasto DM mass density profile into equation (7). We are only interested in looking at

the DM inside the galaxy, so we compute M(r) up to r = 10 kpc using 500 bins. This will

be the drop off limit for all calculations involving M(r). Since this calculation is only using

500 bins, the exact value of M(ri) will be determined using an interpolation.

Next, depending on whether a constant or parametric anisotropy β(r) is used, we can

solve equation (8) by two ways: the first being the analytic solution for a constant value of

β. We can compute ν(r)v2r (r) for constant β using [3]:

ν(r)v2r (r) =

1

r2β

∫ ∞r

r′2βν(r′)GM(r′)

r′2dr′, (15)

where G is the gravitational constant. If we had chosen to use a parametric version of β(r),

we had to have solved equation (8) as a first order differential equation, and plug the solution

into equation (9). Since this project only investigates constant values of β, we only use the

second method to cross-check some of the results.

10

Finally, we can plug the resulting curve for ν(r)v2r (r) into equation (9) to calculate σ2

p(R).

Once the LOS velocity dispersion has been computed, we can use the following likelihood

function to fit the computed value with the observed value of the velocity dispersion [1]:

L =

Nbins∏i=1

(2π)−1/2

∆σi(Ri)exp

[−1

2

(σobs(Ri)− σp(Ri)

∆σi(Ri)

)2], (16)

where ∆σi =

√∆σ2

obs(Ri) +(

0.5 [σp(Ri + ∆Ri)− σp(Ri −∆Ri)]2)

is the error associated

with this likelihood function. In order to ensure that every bin (and thus every value of Ri)

will be included in the fitting process, an interpolated version of σp is used.

The ranges used to fit the Einasto DM profile to the kinematic data can be found in

table 2. Unlike the parameters for the Zhao-stellar density fit, these parameter ranges were

based off of suggestions by [4] and [1]. [4] suggests to constraint the scale radius so that it

is greater than or equal to the the scale radius found for the Zhao-stellar density fit. This

is because it is assumed that the dSph DM halo is at least as big as the stellar population

of the dSph. This assumption would also lead to smaller uncertainties on the best-fitting

parameter value [4]. Another constraint that [4] suggests is for α ≥ 0.12, as this would also

lead to smaller uncertainties in the found values for the parameters. This suggestion however

was not applied, as the algorithm was unable to minimize properly when this constraint was

present.

Parameter range Units Comments

1.5 ≤ log10(ρ−2) ≤ 13 ρ−2 in [M kpc−5]

rs ≤ r−2 ≤ 250 [kpc] Only two out of the 32 fits had an r−2 ∝ 102.

0.008 ≤ α ≤ 48 unitless

Table 2: The inspected parameters ranges for fitting the Einasto DM density profile to the

kinematic data of the dSph. Note that the range on r−2 is dependent on the value found for

rs in the Zhao stellar density fit.

Similarly to section 3.1, the χ2 related to this likelihood function is minimized using

the iMinuit [7] algorithm. This will estimate the best-fitted parameters for the Einasto DM

density.

3.3 Calculating the J-factor

Once the best-fitted values for the Einasto density profile have been found, these can be used

to calculate the J-factor. Plugging in the Einasto model into equation (3), the J-factors can be

11

computed. Both the critical angle, as well as the line-of-sight is dependent on the galaxy that

is being inspected. As mentioned in section 2.1, the critical angle is dependent on the angular

size of the galaxy and is taken from [1]. Moreover, the LOS integration boundaries have to

be large enough in order to include all of the galaxy’s emission in the integral. Therefore we

choose δl to be 2 kpc.

4 Results

4.1 Stellar density results

After fitting the modeled stellar density to the surface brightness data it is clear that all eight

galaxies have similarly shaped surface brightness curves. The main difference between the

curves of the different galaxies is the starting points, the highest being 16000 stars/kpc2 and

the lowest being 3000 stars./kpc2. The point at which the surface brightness asymptotically

goes to 0 (if the background signal is removed) seems to be in agreement with the radial size

of its respective galaxy. This is a good way to check the correctness of the results, as there

should be no stars outside of the galaxies. Two examples of the stellar density curves can

be seen in figure 2. The rest of the results for the remaining galaxies can be found in the

appendix.

4.2 Jeans equation results

After getting the values for the best fitting parameters for the Zhao stellar density profile

for each dSph, these can now be used to determine the stellar density ν(r). The resulting

expression for ν(r) will be filled into equation (8) and can be solved for the radial velocity

dispersion. Since this project only looks at constant anisotropy, we only restrict ourselves to

using the analytic solution (eq. (15)) of the Jeans equation. In order to allow the code to run

as smoothly as possible, M(r) is calculated from 0 to 10 kpc, using the Einasto DM density

profile and with a total of 500 bins. By choosing M(r) to be defined up to r = 10 kpc, the

radial velocity will drop off to zero at r = 10 kpc instead of at r = ∞. This greatly reduces

the time it takes to run the fitting algorithm, but in return also decreases the accuracy of

the calculated values of the radial velocity. To further reduce the computing time, only four

values of the anisotropy β (ranging from −1 to 0.5) have been inspected.

As mentioned before, once the Jeans equation has been solved for v2r , this expression can

now be plugged into the equation for the LOS velocity dispersion (eq. (9)). Finally, the result

of the LOS velocity dispersion can be used to minimize the likelihood function related to the

LOS velocity dispersion (eq. (16)). For the χ2 minimization of equation of equation (16), the

12

(a) Stellar density of the Carina dSph.

(b) Stellar density of the Leo I dSph.

Figure 2: The resulting surface brightness for both the Carina dSph as well as the Leo I dSph

galaxy. The green curve is the projected light profile (surface brightness) found by inputting

the best fitting parameters into the Zhao density profile, and the red dots are data points

from the measured stellar density of each galaxy. Note that at some point for R, both the

green curves asymptotically go to zero. This point roughly coincides with the radius of the

dSph.

parameter ranges mentioned in section 3.2 have been used. The resulting line-of-sight velocity

dispersion for the Carina and Leo I dSphs have been plotted as a function of projected radius

for four different values of the anisotropy and can be seen in figure 3. The rest of the velocity

dispersion plots for the other dSphs can be found in the appendix.

13

(a) LOS velocity dispersion of the Carina dSph.

(b) LOS velocity dispersion of the Leo 1 dSph.

Figure 3: The line-of-sight velocity dispersion plots for both the Carina as well as the Leo

I dSph galaxies. Each of the curves represent the velocity dispersion computed using Jeans

analysis. The blue dots correspond to measured values of the velocity dispersion. Note that

the curves for β = 0.5 in both (a) and (b) have a higher starting point than the rest of the

curves.

14

4.3 J-factor results

Now that the best fitting parameters for the Einasto density profile (eq. (12)) have been

determined for each of the dSphs and for each value of β, these can now be inserted into

equation (3) to calculate the J-factor. The resulting J-factors for Carina and Leo I can be

seen in figure 4. In this figure, the light and dark blue areas represent the results of [1]

(whose results this project is trying to replicate), where the darker blue section represents the

results at 68% C.L. and the light blue section represents the results at 95% C.L. The red dots

represent the J-factors that were estimated using the Jeans analysis of this project. Similar

figures for the remaining six dSph galaxies can be found in the appendix.

5 Discussion

One of the first things to notice when inspecting the calculated J-factors (see fig. 4 or the

appendix) is the lack of error bars on the computed J-values. The biggest problem with the

results that were introduced in the previous section, is that they do not have uncertainties.

We were unable to calculate the uncertainties, as it would have taken too much computing

time (roughly eight hours per J-factor). This lack of error bars on the results makes it difficult

to asses the behaviour of the J-factors and their relation to the anisotropy.

As mentioned earlier, the results of the projected light profiles can be cross-checked, to

see if they indeed asymptotically move towards zero at the edge of their respective galaxy. All

eight dSphs seem to do this, indicating that all of the estimated light profiles are accurate.

Moreover, when the line-of-sight velocity dispersion plots are compared to those acquired by

[1], they appear to be very similar in shape. Only the curve for β = 0.5 seems to consistently

behave differently by starting at a higher point than the rest of the velocity curves. The

general behaviour of the curves is very similar to that of figure 1 from [1]. In all, a constant

value for the anisotropy seems to be able to describe the LOS velocity dispersion almost as

well as an anisotropy profile (like in [1]) would have.

As can be seen in figure 4 and the figures in the appendix, there is no general trend that

the J-factors seem to follow when it comes to the anisotropy. In figure 4(a), the value of

the J-factor seems to be increasing slightly as a function of β, whereas in figure 4(b) the

J-factor seems to be slightly parabolic or sinusoidal. There is also no clear value of β that

always guarantees a resulting J-factor that lies within the error margins (the blue areas in

the graphs) of the J-factors computed by [1]. Only Carina, Draco and Leo I have all four of

their computed values inside the 68% zone of [1], whilst the other galaxies have at least one

value outside of the 68% zone. Even more noteworthy is the results of Leo II (figure 12(b) in

the appendix), where all of the computed values of the J-factor are outside of the 95% zone.

15

(a) J-factors of the Carina dSph as a function of the anisotropy β.

(b) J-factors of the Leo I dSph as a function of the anisotropy β.

Figure 4: The J-factors of the Carina and Leo I dSphs. The red dots represent the J-

factors calculated using the method of this project, whilst the (light and dark) blue sections

represent the J-factor as determined by [1], at different confidence levels. In (a) only half of

the computed J-values are within the 68% range, whereas all of the computed J-values in (b)

are within the 68% range.

This could possibly indicate that a constant anisotropy actually does not properly compute

the J-factor.

If we look at which value of β returns the closest value to the results of [1] for each galaxy,

we find that β = −1 tends to give a result that is closest to that of [1]. In four out of the

eight galaxies, β = −1 gives the closest result. The value of β that gives the least accurate

result is β = 0.5. These results however should be viewed with skepticism, as they do not

have uncertainties. It could very well be possible that the error bars of the J-factors bring

16

the computed J-values closer (or further away) to the results found by [1].

Something that is very important to mention, is that the calculated J-factor is heavily

dependent on the line-of-sight boundaries in the J-factor integral (eq. (3)). The line-of-

sight integral must be large enough to incorporate most, if not all, of the emission from the

inspected galaxy. If we choose a value of δl that is too small, we find J-factors that are in

strong disagreement with those found by [1]. For example, if we choose δl = 0.5 kpc we

find results that fall far outside the 95% C.L. results of [1]. We found that at most, these

results could differ by a factor of two (excluding possible error margins). Therefore, to get

the optimal results, one must choose a value of δl that is large enough to include all of the

dSph’s emitted gamma rays. δl = 2 kpc seems to be a good choice for the boundary, as the

value of the J-factor does not change significantly if δl is increased beyond this point. This

means that all of the emission has already been included in the integral.

Several other things that could have affected the final results is the likelihood function that

was used to fit the stellar density profile. As mentioned in section 2.3, the uncertainty on the

surface brightness had been estimated using error propagation of the error on the distance to

the respective galaxy δd. This may have lead to an overestimation of the uncertainty, which

does not correspond to the likelihood function from equation (14). This means that the best

fitted values of the Zhao parameters may be wrong. To test this, we ran the the fit using

the same likelihood function, but taking the Poisson error as the uncertainty instead. The

resulting Zhao parameters only differed by a small factor. A comparison of the results of the

two different errors for fitting Fornax data can be found in table 3. The parameters found

through both methods do not seem to differ by a significant amount, so it is actually unlikely

that it would have caused a big change in the value of the J-factors.

Using Poisson error Using the ’distance’ error Using both errors Units

ν0 = 14393 ν0 = 13302.62 ν0 = 14180.16 kpc−3

rs = 1.167 rs = 1.121 rs = 1.210 kpc

α = 2.465 α = 2.478 α = 2.435

β = 7.607 β = 7.906 β = 7.924

γ = −0.2358 γ = −0.1758 γ = −0.2189

Table 3: A comparison of the resulting best fitting parameters for the Zhao stellar density

using different errors in the likelihood function of equation (14). This comparison inspects

the results of the Fornax dSph. Judging based off of this comparison, the error chosen for

this likelihood function does not seem to impact the result by a significant amount.

Another issue that could have affected the fit for the Einasto profile, is the amount of

17

bins used to calculate M(r). These bins remained constant in calculating both v2r (r) as well

as σ2p(R). In order to keep the computing speed as high as possible, a bin number of 500 had

been chosen. This means that the ultimate values of v2r (r) and σ2

p(R) were greatly dependent

on how well the program was able to interpolate the 500 values. In certain galaxies, it was

not possible to interpolate σ2p(R) to the first given value of Ri, so this first data point had

to be omitted from the fitting process. This could have lead to a wrongly fitted value of the

three Einasto parameters, resulting in an incorrect J-value. It would be interesting to inspect

the influence of the amount of bins on the results in a future study.

6 Conclusion

In this project we used Jeans analysis to determine the J-factor of eight different dSph galaxies

and tried to reproduce the results of [1]. For the Jeans analysis, we chose to use an Einasto

parametric model to compute the DM mass density and a Zhao-Hernquist parametric model to

determine the stellar number density of the dSph. Furthermore, we investigated the influence

of a constant value for the anisotropy parameter on the result of the Jeans analysis and the

subsequent J-factor. Due to the results having no uncertainties, it is difficult to determine

the correctness of these results. We found that there appears to be no evident trend linking

the anisotropy β to the behaviour of the J-factor in any of the dSph galaxies. Furthermore,

although the results were mostly in agreement with [1], we have not found there to be any

’correct’ value of β that completely reproduces the results of [1]. In most of the galaxies,

we found that β = −1 seems to give results that are closest to the results of [1]. With the

exception of the Leo II galaxy, we have found that all galaxies have at least one value of

the J-factor that is in agreement with the results of [1]. A future study could perhaps look

deeper into the results of Leo II, and investigate this discrepancy between the results. We

recommend that if this analysis were to be redone in a future project, to include uncertainties

in the results.

Something that does appear to influence the resulting J-factor significantly however, is

the choice for the line-of-sight integral. If the range on the line-of-sight integral is too small,

a significant amount of the emission from the dSph may be left out of the calculation. This

leads to a value of the J-factor that strongly disagrees with that calculated by [1]. We found

that δl = 2 kpc is an optimal choice for the line-of-sight boundary, as it delivers results that

are mostly in agreement with the results of [1].

All in all, it can be concluded that if the correct line-of-sight boundaries are used, a

constant anisotropy can give J-factors very similar to those computed using a parametric

anisotropy profile.

18

7 Acknowledgements

I would like to thank my supervisor dr. Christoph Weniger for guiding me through this

project and helping me for the past three months. I would also like to congratulate him and

his family, as they have recently had a daughter. Thank you for taking time out of your busy

schedule to help me on this project.

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9 Appendix

Here all of the results for the remaining dSphs are presented. These results are based off of

the exact same analysis as those of section 4.

22

(a) Stellar density of the Draco dSph.

(b) Stellar density of the Fornax dSph.

Figure 5: The resulting surface brightness for both the Draco dSph as well as the Fornax

dSph galaxy. Similarly to figure 2, the green curve is the projected light profile corresponding

to the Zhao density profile, and the red dots are data points from the measured stellar density

of each galaxy.

23

(a) Stellar density of the Sculptor dSph.

(b) Stellar density of the Leo II dSph.

Figure 6: The resulting surface brightness for both the Sculptor dSph as well as the Leo II

dSph galaxy.

24

(a) Stellar density of the Sextans dSph.

(b) Stellar density of the Ursa Minor dSph.

Figure 7: The resulting surface brightness for both the Sextans dSph as well as the Ursa

Minor dSph galaxy.

25

(a) LOS velocity dispersion of the Draco dSph.

(b) LOS velocity dispersion of the Fornax dSph.

Figure 8: The line-of-sight velocity dispersion plots for both the Draco as well as the For-

nax dSph galaxies. The different coloured curves represent the LOS velocity dispersions for

different values of β. Note that the red curve starts at a much higher value than the other

curves.

26

(a) LOS velocity dispersion of the Sculptor dSph.

(b) LOS velocity dispersion of the Leo II dSph.

Figure 9: The line-of-sight velocity dispersion plots for both the Sculptor as well as the Leo

II dSph galaxies.

27

(a) LOS velocity dispersion of the Sextans dSph.

(b) LOS velocity dispersion of the Ursa Minor dSph.

Figure 10: The line-of-sight velocity dispersion plots for both the Sextans as well as the Ursa

Minor dSph galaxies. In each of the figures, the line-of-sight velocity dispersion for different

values of the anisotropy have been plotted. The blue dots correspond to measured values of

the velocity dispersion.

28

(a) Estimated J-factors of the Draco dSph.

(b) Estimated J-factors of the Fornax dSph.

Figure 11: The J-factors for the Draco and the Fornax dSph galaxies. In each of the figures,

the J-factor for different values of the anisotropy have been plotted. The light and dark blue

areas in the plots represent the values found by [1], at different confidence levels.

29

(a) Estimated J-factors of the Sculptor dSph.

(b) Estimated J-factors of the Leo II dSph.

Figure 12: The J-factors for the Sculptor and the Leo II dSph galaxies. In each of the figures,

the J-factor for different values of the anisotropy have been plotted. The light and dark blue

areas in the plots represent the values found by [1], at different confidence levels. We were

unable to calculate a J-value for Sculptor at β = −1. Note that all J-values of Leo II are

outside of the 68% zone.

30

(a) Estimated J-factors of the Sextans dSph.

(b) Estimated J-factors of the Ursa Minor dSph.

Figure 13: The J-factors for the Sextans and the Ursa Minor dSph galaxies. In each of the

figures, the J-factor for different values of the anisotropy have been plotted. The light and

dark blue areas in the plots represent the values found by [1], at different confidence levels.

31