Master's Thesis Analysis of the NVSS Catalog and Number Counts
by Jonas Reckmann, B.Sc.
A Thesis submitted to Bielefeld University Faculty of Physics
in Partial Ful llment of the Requirements for the Degree of Master of Science on October 19, 2013
Supervisor: Prof. Dominik Schwarz
CONTENTS 3
Contents
1 Introduction 4
2 Radio Objects 5 2.1 Active Galactic Nuclei ...... 5 2.1.1 Radio Galaxies ...... 6 2.1.2 Quasars ...... 8 2.1.3 Seyfert Galaxies ...... 8 2.2 Starburst Galaxies ...... 9
3 The VLA and the NVSS 10 3.1 The Very Large Array ...... 10 3.2 The NRAO VLA Sky Survey ...... 11
4 Cosmology and Number Counts 13 4.1 Euclidean Number Counts ...... 13 4.2 Cosmological Volume Element ...... 13 4.3 Luminosity distance ...... 14 4.4 Radio source counts ...... 15
5 Optimal Data-based Binning 17
6 Preparing the Data 20 6.1 Division of the Catalog ...... 20 6.2 Setting Flux Limits ...... 21
7 Comparing Data Sets 22
8 Number Counts 29
9 Conclusion 35
10 Appendix 36 1 INTRODUCTION 4
1 Introduction
In the 1930s Karl Jansky detected the rst radio signals in the sky, coming from the constellation of sagittarius in the center of the Milky Way. Since then radio astronomy made its way to a major eld in astronomy. This development is also a result of the fact that radio astronomy allows observa- tions that can help to understand the early history of the universe. Due to their long wavelenghts radio waves do not get blocked by interstellar dust as waves in the optical spectrum would be. This allows radio telescopes to look signi cantly further away than optical ones. As the age of observed objects decreases the more distant they are, radio astronomy allows observations of the light of young galaxies. Hence, radio signals can promote our knowledge about early stages of the universe. The aim of this thesis is to develop an optimal way of presenting the features of the NVSS catalog data with histograms to work out a comparison of dierent parts of the catalog. Furthermore, a model is tted that describes these features to get an estimate of parameters that are relevant to describe radio galaxies and their distribution. This thesis starts with a short overview on the most important sources of radio waves in our universe. This is followed by some general information about the NVSS catalog and about the telescope it was obtained with. It will be continued with a brief introduction to the cosmology of number counts and the mathematical concept of optimal binning used in this thesis. This is followed by preparing dierent data sets and using the discussed method of binning to prepare them for further processing and to visualize them. Then the dierent data sets are compared to each other in order to see if they have notable dierences. In the last section a model of number counts will be tted to the data and evaluated how good it describes the data. 2 RADIO OBJECTS 5
2 Radio Objects
The look into the sky with radio telescopes reveals many dierent types of radio objects. In this thesis the focus lies on galaxies observed at radio wave- lenghts. The most general two categories that are observed are galaxies with an active galactic nucleus (AGN) and star forming galaxies. This Chapter gives a brief overview of the dierent types of galaxies that emit strong radio signals.
2.1 Active Galactic Nuclei Most of the stronger radio sources in the sky are galaxies with an AGN. The nucleus is a high concentration of mass in the central region of the galaxy, which is usually assumed to be a black hole. The nucleus generates an accretion disc due to the angular momentum of infalling material, which may be gas from the interstellar medium or stars in its vincinity. As nearby mass falls onto the nucleus gravitational energy is released. Around the accretion disc there is an opaque torus of in owing material which makes it impossible to observe the central regions of the galaxy from the sides. The last part in this simple model of an AGN are two jets of high energetic particles ejected from the core, perpendicular to the accretion disc. The mechanism that creates these jets is still unknown.
Figure 1: A simple model of an active galactic nucleus[BGS10]. 2 RADIO OBJECTS 6
2.1.1 Radio Galaxies
The most luminous radio galaxies can exceed power of 1038W [VKB88] in the radio frequency range. The strong radio emission from these galaxies comes from the jets, created in the nucleus, and structures that form when the jets start to dissipate, called lobes. The highly energetic and relativistic particles from the jets emit synchroton radiation as they interact with mag- netic elds along their path. Interaction with the surrounding interstellar and intergalactic medium creates radio hot spots along the jet and in the lobes.
Figure 2: The radio galaxy Cygnus A. The electrons in the jets travel great distances at nearly the speed of light and form hot spots in the radio lobes as they are collide with the intergalactic medium. [source: http://images.nrao.edu/110]
Many radio galaxies are highly symmetric regarding their straight jets and lobes, but some are asymmetric with bent jets. It is believed that this is due to the high velocities the galaxies have while moving through the intergalactic medium. Another common form of asymmetry in radio galaxies is a missing counter jet. This can be the result of relativistic beaming, where the jet moving in the direction of the observer appears much brighter than the other up to the point of an invisible counter jet. 2 RADIO OBJECTS 7
Radio galaxies are divided into two classes, the Fanaro-Riley Class I (FR I) and the Fanaro-Riley Class II (FR II)[FR734]. The jets in FR II sources move at supersonic speed and thus emitting most of their energy in the lobes, making them brighter in the outside regions. The speed of the jets in FR I sources becomes subsonic much earlier and most of their energy is emitted near the core region which results in more diuse structures and more core centered brightness distribution.
Figure 3: Top: FR I galaxy M84 with a strong radio emission at the center that decreases towards the outer regions. Bottom: FR II galaxy 3C175 with two bright radio lobes. Only one the right lobe has a jet that connects it to the center of the host galaxy[Sch06]. 2 RADIO OBJECTS 8
2.1.2 Quasars Quasi-Stellar Objects (QSOs) or quasars are very luminous high energy AGN with broad emission lines. As quasars normally outshine their host galaxies they appear to be point-like at optical wave-lengths. Quasars can be highly energetic radio sources and the size of the core can be measured. As these radio-loud quasars have a core, which has a diameter of a fraction of a parsec, they must have a very high brightness temperature. The high energies of quasars can be explained by the angle it is observed at. They could be AGNs with a jet pointed directly towards the observer.
2.1.3 Seyfert Galaxies Seyfert galaxies are spiral galaxies with a bright core and broad and nar- row emission lines. They have similar appearance to quasars but are much less luminous.The brightness of the core region of Seyfert galaxies can be compared to the brightness of a Milky-Way like galaxy. As they are faint radio sources they could be radio-quiet quasars viewed from another angle. Seyfert galaxies are distinguished by type 1 and type 2. Type 1 Seyfert galaxies have broad and narrow emission lines, while type 2 seyfert galaxies only show the narrow emission lines. 2 RADIO OBJECTS 9
2.2 Starburst Galaxies Starburst galaxies have a peculiarly high star formation rate, up to 100 times higher than normal galaxies like the milky way. To achieve this there has to be a large amount of cold gas occupying a small region of space. Typically this is the result of a merger between two gas-rich galaxies. Due to the high rate of star formation, starbursts normally contain a large amounts of dust, which absorbs the UV emissions of the numerous young stars and re-emits them in the infrared. This makes them highly luminous in the infrared part of the spectrum. The radio emission from starburst galaxies typically has power of 1033W [BGS10] and is the result of massive stars ionizing the interstellar medium with their solar winds and electrons accelerated to rela- tivistic speeds in supernova remnants. Radio emissions from normal galaxies usually have power about 1030W [BGS10].
Figure 4: The starburst galaxy M82. The dark spots are supernova remnants [BGS10]. 3 THE VLA AND THE NVSS 10
3 The VLA and the NVSS
3.1 The Very Large Array
Figure 5: The Very Large Array. [source: http://images.nrao.edu/Telescopes/VLA/298]
The Very Large Array (VLA) is a radio observatory located on the Plains of San Augustin, New Mexico. It was built in 7 years, inaugurated in 1980, upgraded in 2011 and renamed in 2012 to Karl G. Jansky Very Large Array . The VLA is an interferometer consisting of 27 radio antennas which are each 25 meters in diameter. The antennas are arranged on three 21km long rail tracks resulting in a y-shaped con guration.As the antennas can be moved along the rail tracks, the VLA can operate in four main and several hybrid con gurations resulting in dierent baselines. The frequency range of the VLA is 74 MHz to 50 GHz with a minimal angular resolution of 0.04 arcseconds at 43 GHz.
Con guration A B C D Maximal antenna seperation 36 km 10 km 3.6 km 1 km
Table 1: The four main con gurations of the VLA and the maximal distance between two antennas. 3 THE VLA AND THE NVSS 11
3.2 The NRAO VLA Sky Survey
Figure 6: The NVSS Catalog in mollweide projection. This was generated with HEALpix. [http://healpix.sourceforge.net]
The NRAO VLA Sky Survey (NVSS) was conducted using the D con gu- ration and the DnC con guration, a hybrid con guration of the D and C con gurations, of the VLA. It is a continuum radio sky survey which covers around 82% of the sky. The survey was made at 1.4 GHz and is based on 217,446 partially overlapping images made from 1993 to 1997. The resulting catalog of radio sources contains 1,773,484 sources with a ux S ≥ 2.5mJy distributed in the sky north of a declination δ = −40°.
Table 2: The rst entries of the NVSS catalog. 3 THE VLA AND THE NVSS 12
Figure 7: The rms noise of the NVSS in dependance of the declination δ [CCG+98]. 4 COSMOLOGY AND NUMBER COUNTS 13
4 Cosmology and Number Counts
Number counts are a way to test the viability of dierent cosmological models and to compare them to each other.
4.1 Euclidean Number Counts
The emitted power of a source P at distance r is related to the measured energy ux S as P S = (1) 4πr2 Considering N sources in a sphere around an observer with emitted power greater P with a source density n ∞ N(> S) = dr4πr2n(> 4πr2S) (2) ˆ0 Changing the integration variable to r = pP/(4πS), dr = dP/(2pP/(4πS)) and ordering the terms results in
∞ √ 1 − 3 N(> S) = √ S 2 dP PN(> P ) (3) 4 π ˆ 0
− 3 As observations do not show N(> S) ∝ S 2 , it is clear that this calculation needs some adjustments.
4.2 Cosmological Volume Element To achieve a three-dimensional, homogeneous, and isotropic space there are more than one possible geometries. The rst one is three-dimensional at space, as used in 3.1, with line element
ds2 = dx2. The next is the surface of a four-dimensional sphere in Euclidean space with radius a, which has line element ds2 = dx2 + dz2, where z2 + x2 = a2. The last possible geometry is a surface in four- dimensional space with line element
ds2 = dx2 − dz2, 4 COSMOLOGY AND NUMBER COUNTS 14 where z2−x2 = a2 with a a positive constant. After rescaling the coordinates to x0 ≡ ax, z0 ≡ az and zdz = ∓x · dx the line elements can be combined
(x · dx)2 ds2 = a2 dx2 + k (4) 1 − kx2 with +1 spherical k = −1 hyperbolical 0 Euclidean Extending this to space-time geometry by including a term for the time yields (x · dx) ds2 = −g (x)dxµxν = dt2 − a2(t) dx2 + k (5) µν 1 − kx2 where a(t) is the Robertson-Walker scale factor. A change to spherical coordinates dx2 = dr2 + r2dΩ, dΩ = dθ2 + sin2 θdφ2 leads to
dr2 ds2 = dt2 − a2(t) + r2dΩ (6) 1 − kr and the diagonal metric
a2(t) g = , g = a2(t)r2, g = a2(t)r2 sin2 θ, g = −1. rr 1 − kr θθ φφ 00 The volume element then is
p a3(t)r2 sin θdrdθdφ dV = det gijdrdθdφ = √ (7) 1 − kr2 4.3 Luminosity distance At large distances the expansion of the universe can not be neglected and (1) needs to be modi ed. As light from the source arrives at the observer at t = t0, the area of the sphere drawn around the source and including 2 2 the observer is 4πr a (to). As a result the fraction of light that reaches the observer is not 1 as assumed in but 1 . Furthermore the rate and 4πr2 (1) 4πr2a2(t ) energy of the arriving photons dier from the rate0 of the emitting photons by each a factor of 1/(1 + z). With all these modi cations (1) becomes
P (8) S = 2 2 2 4πr a (t0)(1 + z) 4 COSMOLOGY AND NUMBER COUNTS 15 or P (9) S = 2 4πdL with the luminosity distance
dL = a(t0)r(1 + z). (10)
4.4 Radio source counts
Radio surveys receive at a xed frequency ν from the variable frequency of the source ν(1 + z). It can be assumed that most sources have radio spectra of the form P (ν) ∝ ν−α (11) where α is called the spectral index. The ux received from a source at redshift z that emits a power P (ν)dν between frequencies ν and ν + dν is
P (ν)dν (12) S(ν) = 2 α−1 4πdL(z)(1 + z) as the observed frequency is xed this reduces to
P (13) S = 2 α−1 . 4πdL(z)(1 + z) The observed number of sources with power greater than S is
∞ 4πr2a3(t)dr n(> S) = dP N(P, t) √ (14) 2 ˆ0 ˆ 1 − Kr with N(P, t)dP the number of sources per proper volume with power be- tween P and P + dP at time t. Assuming the time dependency can be parameterized as a(t)β N(P, t) = N(P ) (15) a0 and with 1 + z = a0/a(t) (14) becomes
∞ 4πr2(1 + z)−β−3dr n(> S) = dP N(P ) √ . (16) 2 ˆ0 ˆ 1 − Kr The upper limit of the r integration is determined by (13) as
P a2r2(1 + z)1+α < . (17) 0 4πS 4 COSMOLOGY AND NUMBER COUNTS 16
The integral over r can be converted into an integral over z with the power series 1 a H r = z − (1 + q )z2 + ..., (18) 0 0 2 0 where q0 is the decceleration parameter, to 1 a H dr = dz 1 − (1 + q )z + ... . (19) 0 0 2 0 The upper limit changes to r PH2 1 z < 0 1 − (α − q ) + ... . (20) 4πS 2 0 With these equation (16) becomes
∞ " 1 # 2 2 1 3 3 PH 2 0 N(> S) = √ 3 N(P )P dP · 1 − (5 + β + 2α) + ... . 3 4πS 2 ˆ 4 4πS 0 (21) Equation (21) is only valid for large ux values. 5 OPTIMAL DATA-BASED BINNING 17
5 Optimal Data-based Binning
The histogram is the simplest density estimator. Even though there are better methods its simplicity makes the histogram a widely used method of data visualization and data analysis. A problem that is regularly encoun- tered with histograms is the choice of the right amount of bins. If there are too few bins one may miss some important features of the data, but if there are too many the noise becomes easily indistinguishable from the data. Un- fortunately, the question to the right amount of bins in a histogram is not an easy one to answer. The easiest and often used ways to estimate the number of bins are rules of thumb. Following the square-root choice. the amount of bins should be equal to the square root of the number of data points √ M = N.Another possibility would be Sturge's formula M = dlog N + 1e, wich makes an assumption on the data distribution. However the latter es- timation of the optimal number of bins does not give good results when the data is not normally distributed. To obtain a histogram of the ux of sources in the NVSS-catalog with an optimal amount of bins, the algorithm, presented in Kevin H. Knuth. Optimal Data-Based Binning for Histograms[Knu06]was used. Within this paper the histogram is considered to be a piecewise-constant model of the density function of the data. With M bins over the width of the histogram V with equal bin-width v, probability density hk for the region of each bin and probability mass πk = hk · v, the likelihood to nd a data point dn in the region of the kth bin is
p(dn | πk,M,I) = hk. (22)
I is the prior knowledge including the range of the data and the bin align- ment. (22) can be rewritten to
M p(d | π ,M,I) = π . (23) n k V k For N independant Data points this becomes M p(d | π,M,I) = ( )N πn1 πn2 . . . πnM−1 πnM (24) V 1 2 M−1 M with d = {d1, . . . dN } and π = {π1, π2, . . . , πM−1}. 5 OPTIMAL DATA-BASED BINNING 18
The posterior probability of the histogram model is proportional to the product of the likelihood and the priors (Bayes' Theorem).
p(π,M | d,I) ∝ p(π | I)p(M | I)p(d | π,M,I) (25) where p(M | I) is a uniform density, which is absorbed into the proportion- ality constant and
M " M−1 !# Γ( 2 ) X − 1 p(π | I) = π π . . . π 1 − π 2 , (26) Γ( 1 )M 1 2 M−1 i 2 i=1 which is Jerey's prior for the multinomial likelihood[Jef61]. With (24) and (26) equation (24) becomes
n − 1 N M−1 ! M 2 M 1 1 1 M Γ( ) n1− n2− nM−1− X p(π,M | d,I) ∝ 2 π 2 π 2 . . . π 2 1 − π V Γ( 1 )M 1 2 M−1 i 2 i=1 (27) to get the posterior probability for the number of bins M, equation (25) is integrated over all possible values of π1, π2 . . . πM−1.
a a N M 1 2 M Γ( ) n − 1 n − 1 p(M | d,I) ∝ 2 dπ π 1 2 dπ π 2 2 ... 1 M 1 1 2 2 V Γ( 2 ) ˆ ˆ 0 0
aM−1 1 nM−1− n − 1 ... dπ π 2 (a − π ) M−1 2 (28) ˆ M−1 M−1 M−1 M−1 0 with M−2 X aM−1 = 1 − πk. (29) k This can be solved considering the general integral
a k n − 1 k 2 bk (30) Ik = dπkπk (ak − πk) ˆ0 where +and 1 . bk ∈ R bk > 2
b +n + 1 1 I = a k k 2 B(n + , b + 1) (31) k k k 2 k with the Beta function 1 Γ(n + 1 )Γ(b + 1) k 2 k (32) B(nk + , bk + 1) = 1 . 2 Γ(nk + 2 + bk + 1) 5 OPTIMAL DATA-BASED BINNING 19
Using the recusion ak = ak−1 − πk−1to write
b +n + 1 1 I = (a − π ) k k 2 B(n + , b + 1). (33) k k−1 k−1 k 2 k Integrating (28) gives us the posterior probability
N M−1 M Γ( M ) Y 1 p(M | d,I) ∝ 2 B(n + , b + 1). (34) V Γ( 1 )M k 2 k 2 k=1
This can be simpli ed by de ning 1 and 1 bM−1 = nM − 2 bk−1 = bk + nk + 2
M N Γ( M ) QM Γ(n + 1 ) p(M | d,I) ∝ 2 k=1 k 2 . (35) V 1 M M Γ( 2 ) Γ(N + 2 ) For easier optimization the proportionality constant is neglected and the logarithm of equation (35) taken.
M 1 log p(M | d,I) = N log M + log Γ − M log Γ − ... 2 2
M M X 1 ... − log Γ N + + + log Γ n + + K. (36) 2 k 2 k=1 To get the optimal number of bins equation (36) has to be maximized with respect to M. The Matlab algorithm presented in [Knu06] uses a Brute-Force method to nd the M that maximizes (36). This algorithm was converted to a Python script for this thesis. 6 PREPARING THE DATA 20
6 Preparing the Data
6.1 Division of the Catalog
For the analysis data with galactic latitude |b| < 10 was taken out of the set to mask the Milky Way. The remaining set was divided by the VLA con gurations used to obtain the data and then divided by right ascention to obtain the western half at α < 12h and the eastern half at α ≥ 12h.
RA α < 12h RA α ≥ 12h Con guration D D-West D-East Con guration DnC DnC-West DnC-East
Table 3: The four data subsets.
Set Number of Sources NVSS Catalog 1773484 NVSS without |b| < 10 1477311 D-West 526699 D-East 531828 DnC-West 208311 DnC-East 210473 Table 4: The amount of sources in the dierent sets of the NVSS catalog.
The cut out region of the Milky Way contained 16.7% of the NVSS cat- alog. 71.7% of the modi ed catalog was observed with the D-Con guration of the VLA and the eastern hemisphere has 0.5% more sources than the western hemisphere in the modi ed catalog. 6 PREPARING THE DATA 21
6.2 Setting Flux Limits If Knuth's algorith is used with the raw NVSS data it does not nd a global maximum or a local maximum over a feasible range of bins. In gure 8 a plot for equation (36) with the full NVSS data used is shown. The plots of the subsets show similar behaviour .
2.7e+06
2.6e+06
2.5e+06
2.4e+06
2.3e+06
2.2e+06 log p(M|d,I)
2.1e+06
2e+06
1.9e+06
1.8e+06
1.7e+06 0 100 200 300 400 500 600 700 800 900 1000 1100 M
Figure 8: Equation (35) for the full NVSS data.
To get an optimal number of bins for the NVSS data one has to set a ux limit. Setting a maximal ux value only worsens the divergence while setting a minimum value induces convergence for the algorithm. These numbers were obtained by slowly increasing the ux while observing the convergence behavior of the algorithm.
Data set minimum Flux Loss Bins Full NVSS 18mJy 79.9% 86 D 16mJy 79.8% 86 DnC 16mJy 76.8% 52 D-West 14mJy 75.6% 52 D-East 13mJy 73.7% 73 DnC-West 11mJy 69.6% 43 DnC-East 12mJy 70.7% 40 Table 5: Lower ux limits for the dierent data sets, data points lost and optimal number of bins. 7 COMPARING DATA SETS 22
7 Comparing Data Sets
To compare the dierent data sets the ux limit of the full Catalog, 18mJy was chosen. The resulting optimal bin numbers and loss of data compared to the full sets are shown in table 6. The bin lengths was set equal by choosing the median bin number of the two compared sets for the one with the higher number and adjusting the other accordingly.
D DnC D-West D-East DnC-West DnC-East Bins 86 47 67 59 42 38 Loss 80.2% 79% 80.3% 80.1% 79% 79%
Table 6: Optimal bins at a ux limit of 18mJy and the resulting data loss.
The resulting histograms were divided by the area the dierent sets cover in the sky. For the error of the number of sources in a bin a poisson error √ ∆N = ± N was used 7 COMPARING DATA SETS 23
10000 D DnC
1000
100
10 N/A
1
0.1
0.01 10 100 1000 10000 100000 1e+06 S[mJy]
Figure 9: D and DnC con guration histograms.
5000 D DnC
4500
4000
3500
3000
2500 N/A
2000
1500
1000
500
0 0 200 400 600 800 1000 S[mJy]
Figure 10: D and DnC con guration histograms without logarythmic scale.
The D and DnC sets are not very dierent from each other. In gure 10 the most notable bin is around 180mJy, where the D set has 100 sources per steradian more than the DnC set. Most of the other dierences in gure 10 are most likely from rounding errors in determining the bin lenghts. In gure 9 from 2.5Jy to 4Jy the DnC set uctuates around the D set with the most notable value at 3Jy where the D set has double the source density of the DnC set with 10 sources per steradian. 7 COMPARING DATA SETS 24
10000 D-West D-East
1000
100 N/A
10
1
0.1 10 100 1000 10000 100000 1e+06 S[mJy]
Figure 11: D-West and D-East histograms.
7000 D-West D-East
6000
5000
4000 N/A
3000
2000
1000
0 0 200 400 600 800 1000 S[mJy]
Figure 12: D-West and D-East histograms without logarythmic scale.
The D-East set has a constantly higher source density than the D-West set at low ux values as seen in gure 12. 7 COMPARING DATA SETS 25
10000 DnC-West DnC-East
1000
100 N/A
10
1
0.1 10 100 1000 10000 100000 1e+06 S[mJy]
Figure 13: DnC-West and DnC-East histograms.
8000 DnC-West DnC-East
7000
6000
5000
4000 N/A
3000
2000
1000
0 0 200 400 600 800 1000 S[mJy]
Figure 14: DnC-West and DnC-East histograms without logarythmic scale.
In gure 14 it can be seen that he DnC-West set has lower source density than the DnC-East set at lower ux values up to 300mJy. 7 COMPARING DATA SETS 26
10000 DnC-West D-West
1000
100 N/A
10
1
0.1 10 100 1000 10000 100000 1e+06 S[mJy]
Figure 15: D-West and DnC-West histograms.
8000 DnC-West D-West
7000
6000
5000
4000 N/A
3000
2000
1000
0 0 200 400 600 800 1000 S[mJy]
Figure 16: D-West and DnC-West histograms without logarythmic scale.
It can be seen in gure 16 that the DnC-West set has a signi cantly higher source density at the lowest ux values. These dierences are mostly gone after the rst few bins. 7 COMPARING DATA SETS 27
10000 DnC-East D-East
1000
100 N/A
10
1
0.1 10 100 1000 10000 100000 1e+06 S[mJy]
Figure 17: D-East and DnC-East histograms.
7000 DnC-East D-East
6000
5000
4000 N/A
3000
2000
1000
0 0 200 400 600 800 1000 S[mJy]
Figure 18: D-East and DnC-East histograms without logarythmic scale.
Up to 200mJy the DnC-East set has a signi cantly higher source density than the D-East set, as can be seen in gure 18. In gure 17 it can be seen that from 250mJy onwards the D-East and DnC-East sets similar with two exceptions. The rst one is around 1Jy where the DnC-East set has nearly double the source density with 70 sources per steradian. The second one at 3Jy, is the same as in gure 9, where DnC-East set has 10 sources per steradian and the D-East set 5 sources per steradian. 7 COMPARING DATA SETS 28
10000 NVSS D DnC D-West DnC-West D-East DnC-East
1000
100 N/A
10
1
0.1 10 100 1000 10000 100000 1e+06 S[mJy]
Figure 19: All sets with their optimal number of bins and respective minimal uxes.
Figure 19 shows that the slope of all sets is very similar up to the uc- tuations at high ux values due to the low source density. 8 NUMBER COUNTS 29
8 Number Counts
In this section the NVSS data is used with Equation (21). For the range of the data used in this thesis galactic evolution can be neglected [CCG+98] and thus β = −3 is chosen [Wei08]. With this equation (21) becomes
∞ " 1 # 2 2 1 3 3 PH0 2 (37) N(> S) = √ 3 N(P )P dP · 1 − (1 + α) + ... . 3 4πS 2 ˆ 2 4πS 0 This equation is of the form
− 3 − 1 f(x) = a · x 2 (1 − b · x 2 ) (38)
Equation (38) was tted to the NVSS data using the nonlinear least squares method. To get an initial value for a,
− 3 g(x) = ae · x 2 (39) was tted to the data using the linear least squares method. Equation (39) is similar to the euclidean number counts equation (3). To test the restrictions of equation (37) this process was done with the NVSS catalog data with ux values S > 1Jy and a second time for S > 100mJy.
1Jy 100mJy a 1.12541 · 106 ± 20932.2 949585 ± 12940.5 Table 7: Coe cients of the linear least squares ts.
1J 100mJy a 1.61441 · 106 ± 69199.8 1.67256 · 106 ± 19274.5 b 0.008646 ± 0.000786 4.91515 ± 0.07615 Table 8: Coe cients of the nonlinear least squares ts. 8 NUMBER COUNTS 30
100000 Full NVSS g(x>1000)
10000
1000
100 N/A
10
1
0.1 10 100 1000 10000 S [mJy]
Figure 20: Fit of equation (39) for S > 1Jy.
500 Full NVSS g(x>1000)
400
300 N/A
200
100
0 0 500 1000 1500 2000 2500 3000 3500 4000 S [mJy]
Figure 21: Fit of equation (39) for S > 1Jy without logarithmic axis.
The t of the euclidean number counts equation for S > 1Jy seems to be decent for the NVSS data. 8 NUMBER COUNTS 31
100000 Full NVSS f(x>1000)
10000
1000
100 N/A
10
1
0.1 10 100 1000 10000 S [mJy]
Figure 22: Fit of equation (38) for S > 1Jy.
500 Full NVSS f(x>1000)
400
300 N/A
200
100
0 0 500 1000 1500 2000 2500 3000 3500 4000 S [mJy]
Figure 23: Fit of equation (38) for S > 1Jy without logarithmic axis.
Equation (38) tted to the NVSS data results in a curve that is slightly to high at all points. 8 NUMBER COUNTS 32
100000 Full NVSS g(x>1000)
10000
1000
100 N/A
10
1
0.1
0.01 10 100 1000 10000 S [mJy]
Figure 24: Fit of equation (39) for S > 100mJy.
500 Full NVSS g(x>100)
400
300 N/A
200
100
0 0 500 1000 1500 2000 2500 3000 3500 4000 S [mJy]
Figure 25: Fit of equation (39) for S > 100mJy without logarithmic axis.
The t of the euclidean number counts equation to the NVSS data is too low up to 1Jy. 8 NUMBER COUNTS 33
10000 Full NVSS f(x>100)
1000
100
10 N/A
1
0.1
0.01
0.001 10 100 1000 10000 100000 1e+06 S [mJy]
Figure 26: Fit of equation (38) for S > 100mJy.
500 Full NVSS f(x>1000)
400
300 N/A
200
100
0 0 500 1000 1500 2000 2500 3000 3500 4000 S [mJy]
Figure 27: Fit of equation (38) for S > 100mJy without logarithmic axis.
The t of equation (38) for S > 100mJy ts the NVSS data well up to 3Jy where it starts to have signi cantly higher values than the data. 8 NUMBER COUNTS 34
For ux densities greater than 10mJy radio loud AGNs dominate the population[MS06]. Assuming these AGNs have an energy of 26 W [MS06] 10 Hz and with the value of the Hubble constant taken as km·s−1 , the H0 = 67.80 MP c source density n and spectral index α can be derived from the ts.
S > 1Jy S > 100mJy n 0, 002157Mpc−3 ± 0, 000092Mpc−3 0, 002235Mpc−3 ± 0, 000026Mpc−3 α −0.9096 ± 0, 0082 50, 3977 ± 0, 796 Table 9: Derived values for the source density and the spectral index.
Even though the t to the data with S > 100mJy seems to be better, the calculated spectral index is too large compared to the spectral index of α ≈ 0.7 used in [CCG+98]. 9 CONCLUSION 35
9 Conclusion
The procedure of nding the optimal amount of bins for a histogram revealed that the NVSS catalog seems to have a lot of random noise in the ux range of the faint sources, as that caused the algorithm to diverge when these were included. This problem is likely to be reduced in the future radio source catalogs using new radio telescopes like the upcoming Square Kilometer Array. The comparison of the dierent parts of the catalog with the help of the histograms revealed that there is a level of distinction in between the sets. However, many of the dierences between the sets of the NVSS data might have occurred because of rounding errors in the bin lenghts and bin numbers. But there are some non-similarities that are unlikely to be a rounding error. These non similarities might be due to small dierences in the distribution of galaxies in the sky or can be related to the dierent VLA con gurations used to make the snapshots for the NVSS catalog. It might be interesting to see, if some of these dierences will occur in future surveys to see wether these non similarities are real or a technical issue. The equation for radio source counts that was tted to the histogram of the full NVSS data in chapter 8, might need an even higher lowest ux limit or higher order terms to give more satisfying results. Maybe another method to t the equation or dierent starting values for the parameters should have been chosen. 10 APPENDIX 36
10 Appendix
The core code of the algorithm to nd the optimal number of bins in python: REFERENCES 37
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I would like to thank the following people for all their exceptional help, patience and support they have given me over the time I needed to complete this thesis:
Dominik Schwarz
Julia Wilkenloh
Edith Hemmersmeier Statement of Authorship
Except where reference is made in the text of this thesis, this thesis contains no material published elsewhere or extracted in whole or in part from a thesis presented by me for another degree or diploma. No other persons work has been used without due acknowledgement in the main text of the dissertation. This thesis has not been submitted for the award of any other degree or diploma in any other tertiary institution.
Bielefeld, October 19, 2015