New Mass Measurement for Galaxy Clusters

NEW MASS MEASUREMENT FOR GALAXY CLUSTERS

USING POSITION AND RADIAL VELOCITY

______

A Thesis

Presented to

The College of Arts and Sciences

Ohio University

______

In Partial Fulfillment

Of the Requirements for Graduation

With Honors in Physics – Astrophysics

______

Kayla Jo Fultz

June 2010 ii

This thesis has been approved by the

Department of Physics and Astronomy and the College of Arts and Sciences

______Assistant Professor of Physics and Astronomy

______Dean, College of Arts and Sciences iii

Abstract

Galaxy clusters are the largest structures in the Universe, and the evolution of galaxy cluster mass profiles is a useful tool for constraining cosmological models. Most methods established for obtaining a mass profile of a cluster of galaxies assumes the clusters obey the virial theorem; however, the majority of clusters are observed to contain non-virialized substructures. Zaritsky outlined a timing argument for obtaining mass profiles (Zaritsky, 1989). The timing argument assumes that at a time t = 0, every galaxy in the cluster was concentrated at one point, and then simultaneously exploded outward. The current position of each galaxy with respect to the cluster center is determined only by Newtonian gravitation. Zaritsky applied this method to the local group and obtained reasonable mass profiles (Zaritsky, 1989).

We test this new method for mass measurement and compare our results to values obtained using virialized methods. We apply this timing argument to a sample of galaxy clusters of nearby redshift (from z~0.05 to z~0.2) taken from the Sloan Digital Sky Survey, using a 12 Mpc radius (a region larger than the typical infall radius) for each cluster. We chose clusters from a paper written by Popesso that contained published velocity dispersions for each cluster (Popesso, 2006). The profiles we acquire through the timing argument have a useful astronomical application because they rely only on infalling galaxies in the cluster, forgoing the virial theorem. We estimate a mass based on these profiles and use that mass to calculate a velocity dispersion in each cluster. Our velocity dispersions are compared to published values. Our comparison shows that this method for mass measurement gives reasonable velocity dispersions when applied to a large sample of galaxies. There is no clear systematic offset between our data set and the published data set, and many variables within this method leave room for large errors. iv

Table of Contents

Introduction ...... 1

Methods ...... 5

Data ...... 14

Analysis ...... 15

Conclusions ...... 25

References ...... 29

Appendix ...... 30 1

Introduction

Galaxy clusters are among the largest celestial structures in the universe. Each cluster contains hundreds of galaxies that, in turn, contain billions of stars. These superstructures have been of great interest to astronomers and have been key components to answering many questions about our universe. The studying the evolution galaxy clusters helps give us a better understanding of the nature of our universe.

The study of galaxy clusters has led to recent breakthroughs in the field of study of astrophysics. In studying the universe, we are concerned with two types of matter: baryonic and nonbaryonic. By strict definition, baryonic matter consists of matter made up of three quarks, which include protons and neutrons: the type of matter we can observe. For astronomy, we consider baryonic matter to include objects such as electrons since they constitute a very small portion of the mass in atomic nuclei. Nonbaryonic matter consists of neutrinos--tiny, elementary particles that travel very fast and usually travel through matter without interacting, or other particles that have yet to be discovered.

Studying the ratio of baryonic to nonbaryonic matter is essential for cosmologists interested in uncovering specifics about the Big Bang. Models and calculations focusing on the Big Bang require very precise values for the ratio of baryonic to nonbaryonic matter. A fundamental law if physics is the fact that matter can neither be created nor destroyed; therefore, if a value for the current ratio of baryonic to nonbaryonic matter is reached, it is held as a constant even back to the time of the Big Bang.

Galaxy clusters have been crucial to developing this ratio of matter. When scientists observed the Coma Cluster, they obtained an unusual mass-to-light ratio. This discrepancy led scientists to conclude that there was more matter than they were 2 observing. When they looked at the cluster through a different filter, they discovered a large amount of hot gas (e.g. Kellogg, Baldwin, & Koch). Since the gas was not producing visible light like the stars in the cluster, it was not readily visible until scientists looked for it specifically. Discoveries such as the large amounts of gas in the

Coma Cluster lead to better evaluations of the ratio of baryonic to nonbaryonic matter.

Galaxy clusters were also vital in the discovery of dark matter. Dark matter is a very active topic of study for modern astrophysicists. Dark matter was originally proposed by astrophysicist Fritz Zwicky. Zwicky was studying galaxy clusters and the movement of galaxies within these clusters. Based on his observations, Zwicky concluded that there was more mass present in the system than could actually be observed for the galaxies to behave as they did (Zwicky, 1933).

Further observations of galaxy clusters proved the existence of dark matter by study of the phenomenon known as gravitational lensing (e.g. Schneider, Ehlers, & Falco,

1992). A gravitational lens occurs when light from a very distant source is bent around a massive object. In most cases, the massive object is a galaxy cluster. By studying the severity of such lensing effects, scientists can calculate the amount of matter in a galaxy cluster and, more specifically, derive an estimate for the amount of dark matter within a given cluster.

To qualify as a galaxy cluster, the structure must contain at least 50 galaxies that are gravitationally bound to one another. Galaxy clusters vary in size, number of galaxies, and distance from our solar system.

When astronomers discuss distances of astronomical objects, they refer to a celestial structure’s redshift, denoted by the letter “z.” There are two types of redshift we 3 observe in the universe. There is a redshift that corresponds to an object’s movement within the Universe, for instance, galaxies gravitationally bound within a cluster and rotating about its center will have light waves shifted depending on if they’re moving towards us or away from us; however, we live in an expanding universe, so all other celestial objects constantly appear to be moving away from our solar system. Because the

Universe is expanding, these objects appear to move away from us, the observer, and the electromagnetic waves emitted by these objects get shifted. Specific colors of light correspond to distinct wavelengths: red has a longer wavelength while blue has a shorter wavelength. Because in an expanding universe these objects are moving away from us, the light emitted becomes “stretched,” and so the phenomenon is called redshift.

Redshifts can be used to calculate distances to celestial objects. This relation was first observed by Edwin Hubble. Hubble discovered that the speed at which an object moves away from our solar system could be directly related to the distance from that object to us. These two values are related by a cosmological constant called Hubble’s constant. Theoretical analysis and study of galaxy clusters are constantly updating the value of Hubble’s constant.

Determining the mass of a galaxy cluster is a complex task. Traditionally, astronomers use the virial theorem to estimate cluster masses. The virial theorem states that twice the kinetic energy of a virial object is equal to the negative of the object’s potential energy. For galaxy clusters, the virial theorem is thought to hold out to a virial radius. Current estimates of cluster masses must be done assuming the cluster obeys the virial theorem. This method does not hold for nonvirialized substructures within the cluster. 4

The purpose of this project is to determine a lower-mass limit of a selection of galaxy clusters without using the virial theorem. To accomplish this, we utilize a timing argument outlined by Zaritsky (Zaritsky, 1989). This timing argument assumes each galaxy is an infalling galaxy; this means that the galaxy is assumed it is not yet gravitationally bound by the cluster. This argument completely ignores what we know about cosmology and our expanding universe. The timing argument uses pure Newtonian mechanics and the basic laws of gravitation to calculate a minimum cluster mass necessary to have the galaxy at its current distance within the cluster. According to this argument, the galaxy and cluster are on a path towards one another, or an elliptical orbit with eccentricity of 1. Basically, the timing argument goes by the incorrect assumption that at a time t = 0, the galaxy and the cluster were at the same position, and then they simultaneously moved apart, and now are slowly gravitating back towards one another.

Their current positions relative to one another are based solely on gravitation.

Since this method ignores many cosmological factors, it is surprising that it is a reliable technique; however, this timing argument was used by Zaritsky in 1989 to study objects within the Local Group, the small group of galaxies near to the Milky Way. In

Zaritsky’s application, the timing argument yielded remarkably accurate mass estimates.

This project will be the first time this technique has been applied to a large survey of galaxy clusters. In order to determine if our mass estimates using the timing argument are good estimates, we will compare our values to Popesso’s published velocity dispersions

(Popesso, 2006).

Methodology

The first step in this project was determining a sample of galaxy clusters. Clusters were chosen from the online Abell Northern Catalog (Abell, Corwin, & Olowin, 1989).

There are hundreds of clusters included in just this one catalog, so we had to choose some parameters to help narrow down our sample size. Galaxy clusters chosen were required to have a redshift z ≤ 0.2. Secondly, we wanted clusters with a reasonable richness class.

Cluster richness is denoted by a number from zero to five. A cluster with a richness class of zero would have few galaxies, while a cluster with richness class five would contain many galaxies. For our calculations, we wanted a cluster with at least a richness class of two in order to get the best possible data.

For each cluster chosen from the Abell Northern Catalog, we recorded the cluster’s position and redshift. The position was given in the form of right ascension

(hours:minutes:seconds) and declination (degrees:arcminutes:arcseconds.) Right ascension describes an object’s longitudinal position while declination describes an object’s latitudinal position.

From these parameters, we selected 30-40 clusters. Next we needed data on individual galaxies within the cluster. To obtain this, we utilized the Sloan Digital Sky

Survey, or SDSS. The SDSS is a survey of the sky using multiple filters as well as spectroscopic redshift surveys using an optical telescope in New Mexico. The survey will cover 25% of the entire night sky. Objects covered by the SDSS average a redshift of around z = 0.1. 6

To obtain spectra from the SDSS, we first had to make sure our sample of galaxies fell within the area covered by the survey. After that, we used the spectroscopy query form on the SDSS website. The query form allows the user to enter position constraints. The position constraints on the SDSS form wanted the right ascension and declination in the form of degrees. From the Abell catalog, our right ascension was in hours, minutes, and seconds, and our declination was in degrees, arcminutes, and arcseconds. Conversion is a simple process, shown in equations 1 and 2:

αd = (hours + minutes/60 + seconds/3600) * 15 Eq. 1

δd = degrees + arcminutes/60 + arcseconds/3600 Eq. 2

where αd is the right ascension in the form of degrees, there are fifteen hours in one degree of arc, and δd is the declination in the form of degrees.

After the positions of the galaxy clusters were converted, we had to determine a box size for the spectroscopy query form. A galaxy cluster is not a small point object in the sky like a star would be. A galaxy cluster contains many galaxies and each galaxy itself contains many stars. The box size in the spectroscopy form would gather data within the box parameters. In order to determine what box size to use for each galaxy, we utilized a program called distance. This and all other programs mentioned are part of the

IMCAT package, written by Nick Kaiser and Douglas Clowe. Distance uses the redshift of the cluster to determine the size of the object on the sky. It returns a value in 7 kiloparsecs per arcsecond. For the sake of reference, one parsec (pc) is equal to approximately 3.1 x 1016 meters, or about 3.3 light-years. To be safe, we want a box size of about ten Mpc, and we used distance to translate a ten Mpc radius to degrees on the sky.

Since our number from distance is in units of kpc/arcsecond, we need to translate a 10 Mpc radius to kpc. There are 10,000 kpc in 10 Mpc. When we divide 10,000 kpc by the number returned from distance, we are left with a radius for our cluster in terms of arcseconds. From equation 2, we have that 1 degree = 3600 arcseconds, and we can translate our radius into degrees to match our cluster’s right ascension and declination.

In our spectroscopic query form, it called for a minimum and maximum right ascension and declination, calling it a box size. The maximum box area was 10 deg2, which cut down our sample size. Approximately twenty of our clusters created a radius on the sky greater than allowed for a 10 deg2 area.

Once we had our box size for each cluster, we ran a spectroscopic search for each cluster. The query form allowed us to select the data we wanted: we selected right ascension, declination, and redshift. We also limited our search to return only objects with the classification galaxy. This meant that each data file we received from the SDSS would contain the right ascension, declination, and redshift of any galaxy contained within our box size. We did not set a limited number of objects for the SDSS to return, so we were certain to get all galaxies within the box constraints. The SDSS gave us the data in the form of CSV files. CSV stands for comma separated variables, and it is an easy format to import into data analyzing software, such as Microsoft Excel. Upon downloading CSV files for each cluster, the files are imported into Microsoft Excel. 8

Now we have a list of galaxies, their positions, and their redshifts for each cluster.

For the timing argument, we need the distance of each galaxy to the cluster center. This is not as simple as subtracting the cluster right ascension and declination by the galaxy right ascension and declination. We have to consider that right ascension and declination give us two-dimensional coordinates for celestial objects, but that these coordinates do not take into consideration the three-dimensional location of the galaxy with regards to the cluster. To obtain the actual distance from the galaxy to the cluster, we employ the use of spherical trigonometry. Using the spherical law of cosines, we are able to derive an equation for the distance from the galaxy to the cluster, shown in equation 3:

-1 r = cos [sinδg sinδc + cosδg cosδc cos(δg – αc)] Eq. 3

where r is the distance from the cluster to the galaxy in degrees, δg is the galaxy declination, δc is the cluster declination, and αc is the cluster right ascension. This equation was typed into Excel, and a distance from galaxy to cluster was determined for each galaxy in the cluster.

Following these calculations, we had all of the data we needed to make our mass estimates using the timing argument. Zaritsky derived a series of equations for determining mass estimates in his paper using the timing argument. These are shown in equations 4 and 5.

r = (1 – cosθ)2 Eq. 4 vt sinθ (θ – sinθ)

M = r3 ( - sin)2 Eq. 5 t2 (1 – cos)3 G

where r is the distance from the galaxy to the cluster, θ is the orbital position of the galaxy, v is the radial velocity of the galaxy, t is the age of the universe at the position of the galaxy, and M is the mass of the cluster.

We don’t know the value of θ, the orbital angle, for each galaxy, and there is no way to directly derive the orbital angle. Furthermore, we cannot solve for θ; the bottom term in equation 4 and top term in equation 5 prevent this. To solve for M, we set up an algorithm to cycle through all possible orbital angles. Since we are dealing with infalling galaxies, the orbital angle would be less than 360° because the galaxies have not yet made one full rotation. We set up our computer program to cycle through θ= 180° up to θ

= 360°. The algorithm was created in such a way that the cycle would stop once the lowest possible value of the mass, M, was achieved.

We added this algorithm to a computer program called timingprog. Timingprog’s primary function was to take a known redshift and determine the age of the universe at that particular redshift. Since many of the celestial objects we observe are millions of light-years away, by the time the light from the object reaches us, it is millions of light- years old. Effectively, we are looking back in time millions and sometimes billions of years. 10

Timingprog is a program which utilizes simple cosmology in order to determine an age of the universe for our given redshifts. We need the age of the universe, t, to solve equations 4 and 5 and determine mass profiles for our clusters. Once we added our algorithm to the program, timingprog would give us a lower-mass limit for each cluster.

Timingprog requires a certain file type and format as an input. We narrowed down our Excel spreadsheets until each cluster spreadsheet only contained information on the galaxy redshift and the distance from each galaxy to the cluster center. We saved these as text (.txt) files and attempted to run them through mcat, a program designed to alter the format of the text files so that timingprog could read them properly. Mcat did not work correctly the first several times we tried to input our text files. After some searching, we discovered that Excel saved our text files with a return character that mcat could not recognize. We changed these strange characters to regular returns through a series of

Unix commands. After this alteration, we were able to run our text files through mcat, resulting in a series of cat (.cat) files.

With the changes we had made to timingprog, inputting the cat files returned a series of mass estimations. A mass estimation was calculated using each galaxy in the cluster, resulting in a series of estimates for the cluster.

After running the cat files through timingprog, we noticed something was off with our mass estimates. Our estimates appeared to be systematically offset from the profiles we were expecting. We went back through our steps carefully to determine where a mistake was made, and discovered that our error had occurred at the very beginning of our project. 11

When initially selecting our clusters from the Abell Northern Catalog, we forgot to take into consideration that the coordinates for the clusters in this catalog were given in terms of their coordinates in the year 1950 (J1950.) Because the Earth’s axis is tilted, it gradually changes its orientation over time in a phenomenon known as precession. This means that a celestial object’s right ascension and declination will gradually change over time. We used the coordinates of the clusters based on their 1950s positions in the sky, but we used data for their galaxies based on the more recent SDSS values. This discrepancy was the cause of the systematic error in our mass estimations.

To correct for this, we had to go back to step one of our project. Since we had the names of our clusters, we utilized the NASA Extragalactic Database (NED) to find the coordinates of our clusters from the year 2000. NED is an online database containing information on extragalactic objects, including object position, redshift, images, abstracts, and papers. Since we used our cluster coordinates when taking data from the SDSS, we had to re-download our CSV files from the SDSS with our new coordinates in place.

Again, we had to use equations 1 and 2 to convert our right ascension and declination for each cluster in terms of degrees for use in the SDSS spectroscopic query form.

This time, rather than use distance to determine what would give us a 10 Mpc radius, we simply used a ± 2.2 degree box size in our spectroscopic query form for the

SDSS. Since the maximum box area was 10 degrees2, this ± 2.2 effectively gave us a maximum box size for each cluster. Our calculations and analysis will eliminate objects that are not part of the cluster, so the large box size will not be a problem.

Again, we imported our CSV files into Microsoft Excel and used the spherical law of cosines (equation 3) to determine a distance (in degrees) from each galaxy to the 12 cluster center. These numbers, along with each galaxy’s redshift, were converted into text files, altered to remove the strange return character, and copied into a cat file.

With these latest mass estimations, we could compare our profiles to mass profiles generated.

To further analyze the accuracy of our mass profiles, we needed to generate mass estimations based on accepted methods. To accomplish this, we needed velocity dispersions for each cluster. We ran literature searches through NED, but were only able to find velocity dispersions for a couple of our galaxy clusters. Since this was a necessary step in our procedure, we decided to reevaluate our cluster sample.

From NED, we discovered a paper written by Popesso et al from 2006, which contained a large list of published velocity dispersions from clusters covered in the

SDSS. Velocity dispersions, the average velocity about the mean velocity, are required in order to compare this timing method with other accepted methods of mass measurement.

We chose a new list of clusters based on the list from Popesso’s paper. Since all clusters in this paper were taken from the SDSS, their redshifts were well within our decided range of z ≤ 0.2. We chose a sample of 27 clusters at random from this paper, keeping a list of the cluster names in Excel. Next, we went to the NED website and obtained the right ascension, declination, and redshift for each of our clusters, making sure our coordinates for the clusters were recent. We once again used converted our positions with equations 1 and 2, and then downloaded our CSV files using a ±2.2 degree box size from the SDSS.

We imported the CSV files into our new Excel spreadsheet. Using the spherical law of cosines (equation 3), we determined the distance (in degrees) from each galaxy to 13 the cluster center. We took these numbers, along with the galaxies’ redshifts, and created text (txt) files. With a Unix command, we removed the unusual return character from our text file, and then ran the files through imcat to transform them into cat files.

Before we ran these files through timingprog, we wanted to separate the infalling galaxies from the galaxies that are already gravitationally bound to the cluster. To accomplish this, we use a computer program called plotcat to graph our cat files.

Data

A list of clusters was chosen at random from Popesso et al (2006). From their extensive list, we chose 27 clusters to be analyzed in our research. The values for cluster redshifts were originally taken from the NASA Extragalactic Database (NED.) Later these redshifts were switched for the redshifts taken from Popesso (Popesso, 2006). The clusters’ positions, the right ascension and declination of their centers, were taken from

NED. The data on galaxy positions and redshifts was taken from the Sloan Digital Sky

Survey. The list of galaxy clusters, cluster centers, and cluster redshifts are given in Table

1. The cluster centers are given by the cluster’s right ascension and declination in units of degrees.

RA center Dec center Cluster (deg) (deg) z A1205 168.3417 2.5111 0.0754 A1218 169.7042 51.7097 0.0779 A1221 169.9625 62.6764 0.2124 A1236 170.6875 0.4589 0.101847 A1302 173.3417 66.4072 0.1165 A1346 175.2917 5.6894 0.0975 A1364 175.9125 -1.7608 0.1058 A1366 176.2375 67.4225 0.117 A1368 176.2375 51.2558 0.1291 A1376 176.5625 -1.0778 0.1176 A1406 178.3125 67.8886 0.1178 A1407 178.3875 -1.745 0.1358 A1424 179.3875 5.0383 0.0768 A1501 183.5333 63.2222 0.131 A1516 184.7375 5.2392 0.0769 A1539 186.5792 62.5567 0.1712 A1559 188.2708 67.1078 0.1071 A1564 188.7375 1.8414 0.0792 A1579 189.4625 65.8419 0.2005 A1599 190.6875 2.8094 0.0855 A1646 193.9375 62.1628 0.1068 A1692 198.0667 -0.9319 0.084175 A1767 204 59.2119 0.070302 A1882 213.6625 -0.3325 0.1367 A1937 218.6083 58.2653 0.1385 A2082 232.675 3.4467 0.0862 A2149 240.4083 53.8786 0.0679

Table 1. Table gives the list of cluster names, cluster center’s right ascension and declination in units of degrees, and cluster redshift.

Analysis

We first analyzed the cluster using plotcat. We plotted each individual galaxy’s redshift verses its distance from the cluster center in degrees. The resulting graph contained a spread of data points with the cluster galaxies creating a dense spread. Figure

1 is a good example of such a plot.

Figure 1. Plotcat graph of A1221. The x-axis displays the distance from the cluster center in degrees while the y-axis displays the galaxy redshift.

The galaxy cluster displayed in Figure 1 has a redshift of z = 0.1103, and a dense 16 line of galaxies is readily visible along this redshift. At a radius of approximately one degree, the cluster galaxies become fewer and less defined. In clusters like A1221 (Figure

1), the cluster galaxies are easily selected out using the select option on plotcat. We studied the plots of each cluster individually and chose the cluster galaxies by eye.

We also encountered some problems with the data while using plotcat to study the clusters. Sometimes the dense spread of the cluster did not start until at least one degree away from the cluster center. In these instances, the cluster appeared as though it was shifted along the x-axis. This offset was most likely caused by incorrect values for the position of the cluster center. In about half of our cluster samples, the data was simply sparse with no real defined cluster structure. An example of such a cluster is shown in

Figure 2.

Figure 2. Plotcat graph of A1346. The x-axis displays the distance from the cluster center in degrees while the y-axis displays the galaxy redshift.

Figure 2 displays the cluster A1346, which has a redshift of z = 0.1021. There is quite a spread of galaxies along this redshift, not at all dense and defined like in Figure 1.

For clusters like A1364, the selection process is much more difficult, leaving more room for human error.

After the cluster galaxies are selected out, we used the program called getisolatedobjects to subtract the cluster galaxies from the main cat file, leaving only noncluster galaxies. This left us with two distinct cat files: one for cluster galaxies and one for noncluster galaxies. These cat files were run through timingprog, which contained 18 our timing argument equations (Equations 4 and 5.) The output files contained two numbers: the galaxies’ distance from the cluster and the mass profile of the cluster.

These files were individually imported into XMGrace, a plotting program. The galaxies we chose to be cluster galaxies from plotcat were imported separately from galaxies we subtracted and determined to be noncluster galaxies. Ideally, we wanted the cluster galaxies to form a mass profile separate from noncluster galaxies. We wanted noncluster galaxies to fall well above the mass profile. These galaxies we assume are part of the Hubble flow, meaning they are simply a part of the expansion of the Universe and not gravitationally bound to our cluster.

To analyze our cluster mass profile, we used a program called gennfw3d. This stands for “generate nfw 3-D.” An NFW (Navarro-Frenk-Wright) profile relies on a simulation in order to generate a mass profile, used in simulating profiles of dark matter halos. We plot these NFW profiles on top of our cluster galaxies and noncluster galaxies in XMGrace. For an NFW profile to be a good fit, we want it to run parallel and slightly above the profile created by the cluster galaxies. The NFW program contains several settings. We run four NFW profiles for each galaxy cluster, selecting r200 (estimating the mass of the halo) at four different values: 1000, 1500, 2000, and 2500. A sample of galaxy clusters where the cluster galaxies and noncluster galaxies are separate and an

NFW profile can be fit are shown in Figure 3. 19

Figure 3. Figure displaying three galaxy clusters and their corresponding mass profiles. The black symbols represent cluster galaxies, the red represent non-cluster galaxies. The green, blue, yellow, and purple lines represent NFW models with r200 of 1000, 1500, 2000, and 2500, respectively.

Figure 3 shows cluster mass profiles for three galaxy clusters. The black symbols represent galaxies we selected from plotcat that we believe are part of the cluster. The red symbols represent noncluster galaxies. The green, blue, yellow, and purple lines represent 20

NFW models with r200 values of 1000, 1500, 2000, and 2500, respectively. The ideal

NFW profile would be situated parallel to and slightly above the cluster galaxies.

The cluster A1221 (top cluster in Figure 3) displays a good separation of cluster galaxies to noncluster galaxies. The noncluster galaxies are much higher on the plot, clearly separated from the cluster galaxies, which is what we expected to happen.

Galaxies giving higher mass profiles such as these are not part of the cluster, but part of the Hubble flow.

Figure 4 shows three galaxy cluster mass profiles that did not turn out as nice as the three in Figure 3. In these clusters, cluster galaxies and noncluster galaxies do not have a clear separation. 21

Figure 4. Figure displaying three galaxy clusters and their corresponding mass profiles. The black symbols represent cluster galaxies, the red represent non-cluster galaxies. The green, blue, yellow, and purple lines represent NFW models with r200 of 1000, 1500, 2000, and 2500, respectively.

Figure 4 is a great example of some of the problem clusters we discovered while attempting to fit mass profiles. A1646 and A1406 are clearly very sparse clusters, and have little to no clear separation between the cluster galaxies and noncluster galaxies. 22

A1692 contains many galaxies, but again there is no separation between what we believe to be cluster galaxies and noncluster galaxies.

From these NFW profiles, we can determine a value for M200, the mass value corresponding to the r200 values. We studied each of our mass profiles and decided which NFW profile best fit the data, running parallel to and slightly above our cluster galaxies. Once we chose one, we looked at the data for that particular r200 profile and selected the M200 value that corresponded to that particular r200 value. Once we have this M200 for each cluster, we can use a conversion derived by Evrard et al in 2008, shown in Equation 6 to find a velocity dispersion for each cluster. These velocity dispersions will be compared to the ones published by Popesso in 2006.

15 α σ(M, z) = σ15 * [h(z) * M200 / 10 Mʘ] Eq. 6 where h(z) = Hz / 100 Eq. 7

A list of the velocity dispersions we derived using Equation 6 compared to the published velocity dispersions is given in Table 2, along with the r200 and M200 values used in these calculations.

σ(M,z) Published σ cluster r200 (kpc) M200 (Mʘ) (km/s) (km/s) A1205 1510 4.13E+14 722.220831 865 +/- 73 A1218 1210 1.42E+14 505.2203096 364 +/- 75 A1221 810 1.05E+14 458.1232674 289 +/- 132 A1236 1660 1.83E+14 551.8526469 533 +/- 59 A1302 1360 1.61E+14 529.6024392 691 +/- 80 A1346 1260 1.50E+14 515.159468 709 +/- 54 A1364 910 1.16E+14 473.3099257 553 +/- 59 A1366 1660 1.86E+14 555.7786754 691 +/- 70 A1368 1510 4.35E+14 741.6401007 735 +/- 92 A1376 1110 1.38E+14 503.1981967 461 +/- 204 A1406 1310 1.57E+14 525.1707003 337 +/- 97 A1407 1460 4.26E+14 736.8947794 561 +/- 142 A1424 1460 4.01E+14 715.1854806 662 +/- 45 A1501 1710 1.93E+14 564.5508401 496 +/- 57 A1516 1310 1.51E+14 514.6341616 705 +/- 71 A1539 1260 1.51E+14 517.4074702 510 +/- 60 A1559 1510 4.25E+14 732.872904 863 +/- 124 A1564 1660 1.79E+14 545.4898423 633 +/- 57 A1579 1010 1.26E+14 486.7545263 286 +/- 86 A1599 1460 4.05E+14 718.6547359 322+/- 38 A1646 1160 1.41E+14 506.0345692 573 +/- 88 A1692 1460 4.05E+14 718.3828039 561 +/- 64 A1767 1960 9.65E+14 959.7900848 884 +/- 55 A1882 1510 4.40E+14 745.5474316 733 +/- 99 A1937 1210 1.51E+14 520.149337 223 +/- 50 A2082 1710 1.84E+14 551.1372992 380 +/- 111 A2149 1260 1.45E+14 506.8233635 330 +/- 46

Table 2. Table contains each cluster in our project, along with the matching r200 and M200 values. These were used to calculate σ(M,z). The final column contains the published velocity dispersions.

To check if we have a systematic offset (our values consistently higher or lower than the published results), we plot our calculated velocity dispersions versus the published dispersions and plot a 1:1 line through the data. The resulting plot is shown in

Figure 5. 24

Figure 5. Figure displays our calculated velocity dispersions versus the published dispersions.

From this plot, it appears our data is scattered roughly equally below and above the 1:1 line. This means that there is no systematic offset.

Conclusions

From a study of Table 2, it is clear that our method of determining a cluster mass works sporadically. In a few cases it gives reasonable velocity dispersions, and in other cases the velocity dispersions are different from published results. In most cases, the cluster velocity dispersions appear to follow the general trend of the published dispersions. Figure 5 shows that there is no systematic offset to our method of mass 25 estimation. Our velocity dispersions are equally higher and lower than the published results.

There are a number of reasons for these discrepancies. The study in Popesso et al used clusters from the SDSS, as did our study. The SDSS is not a uniform survey.

Spectra of certain galaxies may have been observed for longer periods of time than others, making these spectra more thorough than ones observed for less time.

When studying the mass profiles for these clusters, or even just studying their positions and redshifts from plotcat, there is an obvious discrepancy in the richness of the clusters. Some clusters, like ones in Figure 4, are very sparse, and only a few galaxies on plotcat corresponded to the given cluster redshift. Richer clusters are more conducive to our study because they give us a greater data set.

There was no error propagation in this study. This is because the error propagated through z would have been very small compared to the human error present in selecting cluster galaxies from plotcat and attempting to fit NFW profiles by eye.

In selecting galaxies from plotcat, cluster galaxies were often unclear and judgment calls had to be made. This could be due to any number of factors. The redshift of the cluster could be off, as was the case in A1221. In A1221, the redshift for the cluster given by NED was a full +0.1 off from the more recent redshift given by

Popesso’s 2006 paper. If the Popesso paper used older z values for some of the clusters, then these redshifts could be inaccurate.

The cluster centers were taken from NED, and some of these centers were slightly off from what the data was telling us. In some instances, our plotcat data showed that cluster galaxies did not start until a radius of ~ 1 degree away from what NED claimed 26 was the cluster center. Cluster centers are important because we used these centers when taking data from the SDSS. If the centers were significantly incorrect, we could have missed large portions of the cluster when taking spectra from the SDSS.

The other large source of error came in selecting an NFW profile line to correspond to the cluster data. In many cases the profile was not a clear, simple answer.

In choosing an NFW line, we tended to focus on lines that clearly kept our cluster galaxies under the fit. In these instances, often noncluster galaxies would be under the line, as well. We still chose these lines with the caveat that selection from plotcat was not always so clear, and some of these galaxies that were labeled as not part of the cluster could, in fact, have been cluster galaxies.

Further study and analysis in this project could include refining the positions of the cluster centers. More clusters could be added to the survey in order to have a larger data pool. Richer clusters could be included, as well as clusters with larger spectra from the SDSS. Our study compared the clusters to four NFW lines. More NFW lines could be added to the study in order to further refine our mass estimates and velocity dispersions.

The timing argument is a method that was shown to generate reasonable mass profiles for the Local Group by Zaritsky in 1989. By assuming our galaxy and cluster were in the same position at t = 0, and then exploded outward, we can generate a series of equations that use Newtonian gravitation to determine their current positions. There are many variables that could contribute to error in this timing method. We took data from the SDSS, which may not give equal weight to all objects. We chose cluster galaxies by eye through use of plotcat, and many clusters did not have a clear distinction between cluster and noncluster galaxies. From this selection of galaxies, we plotted our cluster 27 galaxies, noncluster galaxies, and 3-D NFW mass profiles. Again by studying each plot closely and making educated guesses, we chose the NFW profile we thought best fit each cluster. We ended up with velocity dispersions that were hit-or-miss in their relation to published results, and had no clear systematic offset to try and explain the difference.

References

Abell, G., Corwin, H., Olowin, R. Astrophysical Journal Supplement Series, vol. 70, May 1989, p. 1-138.

Clowe, D., Kaiser, N., http://www.ifa.hawaii.edu/~kaiser/imcat/content.html

Kellogg, E., Baldwin, J. R., Koch, D. 1975 in Astrophysical Journal, vol. 199, July 15, 1975, pt. 1, p. 299-306.

NASA Extra-Galactic Database (NED), http://nedwww.ipac.caltech.edu/

Popesso, P. et al 2006 in Astronomy and Astrophysics 445, 29-42 (2006)

Schneider, P., Ehlers, J., Falco, E. Gravitational Lenses, XIV, 560 pp. 112 figs.. Springer-Verlag Berlin Heidelberg New York, 1992.

Sloan Digital Sky Survey (SDSS), http://www.sdss.org/

Zaritsky, D., and Olszewski, E. W. 1989 in The Astrophysical Journal, 345: 759-769, 1989 Octoboer 15

Zwicky, F., 1933 in Helvetica Physica Acta, Vol. 6, p. 110-127

Appendix

Appendix. Contains the mass profiles generated for each cluster in our survey. The black plusses denote cluster galaxies, and the red plusses are non-cluster galaxies. The green, blue, yellow, and purple represent NFW profiles with r200 values of 1000, 1500, 2000, and 2500.