Science, Faculty of
Physics and Astronomy, Department of
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Scott, William K.
Doctor of Philosophy - PhD
University of British Columbia
The discovery and monitoring of variations of radio flux density in compact radio sources has greatly increased our understanding concerning the nature of many astronomical objects. The exceptional radio sources SS433 and Cygnus X-3 are examples of objects that first stood out because of their dramatic radio variability. This thesis presents a catalogue of 146 radio-variables, discovered with the data from the Greenbank Northern Hemisphere 6 cm sky survey. In addition, one particular radio-variable, GT2318+620, is investigated in more detail by the use of VLA observations.
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C O M P A C T R A D I O S O U R C E V A R I A B I L I T Y A T C E N T I M E T E R W A V E L E N G T H S By Will iam K . Scott B . Sc. (Physics) University of Manitoba, 1985 M . Sc. (Physics) University of Manitoba, 1988 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF DO C T O R OF PHILOSOPHY in T H E FACULTY OF G R A D U A T E STUDIES DEPARTMENT OF PHYSICS AND ASTRONOMY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August> 1996 © William K . Scott, 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics and Astronomy The University of British Columbia 6224 Agricultural Road Vancouver B .C . Canada V6T 1Z1 Date: Abstract The discovery and monitoring of variations of radio flux density in compact radio sources has greatly increased our understanding concerning the nature of many astronomical objects. The exceptional radio sources SS433 and Cygnus X-3 are examples of objects that first stood out because of their dramatic radio variability. This thesis presents a catalogue of 146 radio-variables, discovered with the data from the Greenbank Northern Hemisphere 6 cm sky survey. In addition, one particular radio-variable, GT2318+620, is investigated in more detail by the use of V L A observations. ii Table of Contents Abstract ii Table of Contents iii List of Tables v List of Figures vi Acknowledgements xi 1 Introduction 1 1.1 Why study Variability? 1 1.2 Extragalactic Radio Source Variability 4 1.3 Galactic Radio Source Variability 8 1.4 Variability Searching Methodologies and Overview of the Thesis 12 2 The Greenbank Survey 14 2.1 Data Acquisition 14 2.2 Image Construction 17 2.3 The Greenbank Catalogues 20 3 Bayesian Theory 32 3.1 Introduction to Bayesian Inference 32 3.2 Bayesian Statistics and the Long-Term Variability Problem 36 3.3 Bayesian Statistics and Flux Density/Position Determination 38 iii 3.4 Why Bayesian? 42 4 Development of the Variable Source Catalogues 50 4.1 Development of the Long-Term Variable Source Catalogue 50 4.2 Development of the Short-Term Variable Source Catalogue 53 5 The Variable Source Catalogues 64 5.1 The Long-Term Variable Source Catalogue 64 5.2 The Short-Term Variable Source Catalogue 74 6 GT2318+620 93 6.1 A Brief History of GT2318+620 93 6.2 V L A Observations of GT2318+620 95 6.3 Editing, Calibration, and Image Construction with the V L A Data . . . . 98 6.4 Results From the V L A Data 101 7 Conclusions 110 Bibliography 112 A Solution to integrals of the form: JQ eax~^x2 dx 118 B Source Flux Density Prior Probability Distributions 119 iv List of Tables 3.1 l o g 1 0 Q ( x 2 I 1) for Various Values of y 2 47 5.1 A List of Long Term Variables 67 5.2 Sources Identified with NVSS/FIRST Sources 71 5.3 Variables Identified with Optical Sources 72 5.4 Daily Flux Densities for Selected Long-Term Variables 75 v List of Figures 1.1 A schematic diagram of SS433 (adapted from [Hjellming & Johnston 1986]). 3 1.2 Radio galaxy 0326+396 at 6 cm (from [Bridle 1982]) 5 2.1 Orientation of the seven feeds to the scan direction 15 2.2 The pattern traced by the seven beams on the celestial sphere over a three day period. Note that the horizontal scale is greatly exaggerated for clarity. The actual N /S scan rate was 10°/minute, while the E / W drift rate was 0.25 cos 5 °/minute 16 2.3 Combined 1986 & 1987 image for a = 0 h 20 m , 6 = 70° 19 2.4 Point-spread function for the 1986 Greenbank survey. Top: Declination dependence of the 1986 PSF H P B W in the North/South direction. Bot-tom: Declination dependence of the 1986 PSF H P B W in the East/West direction 21 2.5 Point-spread function for the 1987 Greenbank survey. Top: Declination dependence of the 1987 PSF H P B W in the North/South direction. Bot-tom: Declination dependence of the 1987 PSF H P B W in the East/West direction 22 2.6 Point-spread function for the combined-epoch Greenbank survey. Top: Declination dependence of the combined epoch PSF H P B W in the North/-South direction. Bottom: Declination dependence of the combined-epoch PSF H P B W in the East/West direction 23 vi 2.7 Background noise for the 1986 epoch maps as a function of declination, averaged over the complete R A range 24 2.8 Background noise for the 1987 epoch maps. Best estimate for the 1987 epoch noise is the product F • S. Top: Noise dependence on R A for the 1987 epoch maps, averaged over all declinations. For a fit of this curve to observational data, see [Condon et al. 1989]. Bottom: Noise dependence on declination for the 1987 epoch maps, averaged over the entire R A range. Boxes represent the average noise in each 10° declination strip 25 2.9 Background noise for the combined-epoch maps as a function of declina-tion, averaged over the complete R A range 26 3.1 x2 versus log 1 0(Variability Odds) for approximately 5300 sources from the GB6 catalog. The vertical line indicates an odds ratio of 105, above which a source is considered to be a long-term variable 46 3.2 Limiting cases of figure 3.1. Top: Figure 3.1 with flat prior probabilities and constant rms errors. Bottom: Figure 3.1 with flat prior probabilities and rms flux density errors given by equation 4.1 48 3.3 x 2 v s - log 1 0 Variability Odds for approximately 5300 simulated "constant" sources. Note the difference in scale between this graph and figures 3.1 and 3.2 49 4.1 Contour plots of the long-term variable with centre: R A = 3 h23m32.7 s, 5 = 28°32'24". Upper plot is taken from the 1986 epoch images, lower plot from the 1987 images. Each plot covers an identical 16| x 16| arcmin patch on the sky. Lowest contour on both plots is -5 mJy. Contour interval is 10 mJy 54 vii 4.2 HPBW's in the N /S direction for beams 1 through 4 as a function of declination 57 4.3 HPBW's in the N / S direction for beams 5 through 7 as a function of declination 58 4.4 Histograms illustrating the random telescope positional errors. The solid curves represent the best Gaussian fits to the data. Top: The N / S posi-tional error. For the best Gaussian fit to the histogram, a$ — 20". Bottom: The E / W positional error. For the best Gaussian fit to the histogram, CRA - 19" 59 4.5 Histogram of the errors in the daily flux density determinations for 18 V L A calibrators. The solid curve represents a Gaussian with a mean of zero and unity rms width 62 5.1 Distribution of long-term variables in Galactic ( l n , b n ) coordinates. In these coordinates 0° latitude corresponds to the galactic plane, while 0° longitude, 0° latitude corresponds to the direction of the Galactic centre. Those variables for whom optical identifications have been made are rep-resented by crosses, while all other long-term variables are represented by boxes. The upper and lower bounds of the G B surveys, corresponding to 5 — 75° and 5 = 0°, are also indicated on this map 73 6.1 November 4, 1988 V L A images of GT2318+620 (from [Taylor et. al. 1991]). Left: GT2318+620 at 20cm. Contours are -0.35, -0.25, 0.25, 0.35, 0.5, 0.75, 1.0, 1.5, 2.0, 2.5 and 5.0 mJy/beam. Angular resolution is 1.5". Right: GT2318+620 at 6 cm. Contours are-0.15, 0.15, 0.25, 0.35, 0.5, 0.75, 1.0, 2.0, 2.5, 3.5 and 5.0 mJy/beam. Angular resolution is 0.5". 94 viii 6.2 Left: Epoch 1 image of GT2318+620 at 20 cm before CLEANing . Right: Epoch 1 image of GT2318+620 at 20 cm after C L E A N i n g with M X . Con-tours for both images are -0.35, -0.25, 0.25, 0.35, 0.5, 0.75, 1.0, 1.5, 2.0, 2.5 and 5.0 mjy/beam. Angular resolution is 1.73" 103 6.3 Left: Epoch 1 image of GT2318+620 at 6 cm before CLEANing . Right: Epoch 1 image of GT2318+620 at 6 cm after CLEANing with M X . Con-tours for both 6 cm images are -0.15, 0.15, 0.25, 0.35, 0.5, 0.75, 1.0, 2.0, 2.5, 3.5 and 5.0 mjy/beam. Angular resolution is 0.5" 104 6.4 Epoch 2 images of GT2318+620. Left: 20 cm image. Contours are -0.35, -0.25, 0.25, 0.35, 0.5, 0.75, 1.0, 1.5, 2.0, 2.5 and 5.0 mjy/beam. Angular resolution is 1.73". Right: 6 cm image. Contours are -0.15, 0.15, 0.25, 0.35, 0.5, 0.75, 1.0, 2.0, 2.5, 3.5 and 5.0 mjy/beam. Angular resolution is 0.5" 105 6.5 Epoch 3 images of GT2318+620. Left: 20 cm image. Contours are -0.35, -0.25, 0.25, 0.35, 0.5, 0.75, 1.0, 1.5, 2.0, 2.5 and 5.0 mjy/beam. Angular resolution is 5.10" x 4.88". Right: 6 cm image. Contours are -0.15, 0.15, 0.25, 0.35, 0.5, 0.75, 1.0, 2.0, 2.5, 3.5 and 5.0 mjy/beam. Angular resolution is 1.73" 106 6.6 L band difference image; Epoch 1 minus epoch 2. Contours are 0.2, 0.3, 0.35, 0.4, 0.45 and 0.5 mjy/beam 107 B . l Top: Weighted source number density versus flux density. Individual data points are source counts taken from [Gregory et. al. 1995]. Solid curve is the best Gaussian fit to the data. Bottom: p(fci|M 2). Individual data points are the normalized source counts from [Gregory et. al. 1995]. . . . 120 ix B.2 Top: Estimates of the fraction of radio sources with V > VQ, (modified from [Gregory & Taylor 1986]). Bottom: Normalized probability density function for V . The individual data points were extracted from the cu-mulative plot above. Curve indicates the best two-component fit to the data 122 x Acknowledgements Firstly, I would like to thank Dr. Phil Gregory for his advice and encouragement, without which this thesis would have been impossible. In addition, I would like to thank my wife, Darlene, for her patience and encouragement, despite only catching the odd glimpse of me over this last year, and my parents, for their confidence in me over this "four year" degree. I am also grateful to the examining committee of Dr.'s Fahlman, Halpern, Hardy, and Measday, for their suggestions, and the time they gave to examine this thesis. Finally, the contribution of NSERC through a research fellowship during a part of this research, is gratefully acknowledged. xi Chapter 1 Introduction 1.1 Why study Variability? The discovery and study of time-varying phenomena has long played an important part in all scientific endeavour. Studying variability in an observable quantity gives insights into the processes underlying a physical system. As well, these variations can often provide constraints on models constructed to describe said systems. In astronomy in particular, the discovery and study of time-variable systems has been richly rewarding. An obvious example illustrating this point was the discovery of the period-luminosity relation of Cepheid variable stars. This relation has proved invalu-able as an extragalactic distance indicator. The importance of this relation was recently magnified with the measurement of a Cepheid variable in a galaxy in the Virgo cluster, al-lowing a more accurate determination of Hubble constant to be made [Pierce et al. 1994]. In the field of radio-astronomy, pulsars are an excellent example of the benefits to be gleaned from searching for and studying variable radio sources. Since their discovery by Jocelyn Bell and Anthony Hewish in 1967 [Hewish et al. 1968], pulsars have been used for many diverse purposes, such as probing the interstellar medium and as tests of general relativity in strong gravitational fields. What allows these objects to be applied to so many different fields is their variable radio emission, which is periodic, with extremely stable pulse periods (the pulse periods of some pulsars vary by less than one part in 10 1 5). Another example illustrating the astronomical "superstars" which a search for variable 1 Chapter 1. Introduction 2 radio sources can reveal was the serendipitous discovery of Cygnus X-3. Cygnus X -3, an X-ray binary star system, was first discovered to be a weak variable source in 1972, [Braes & Miley 1972]. Its flux density at that time fluctuated between 10 and 40mJy. Then, on September 2, of that year, it was caught flaring to a level of over 20 Jy at 10, 522 MHz ([Gregory 1972], [Gregory et al. 1972]). This unprecedented discovery initiated a flurry of activity amongst the astronomical community, resulting in twenty-one papers appearing in the October 23, 1972 issue of Nature Physical Science. Flares such as this are ideal examples of events most often thought of in connection with quasars. The variable radio source SS433 is another radio variable that has garnered intense interest. This object, located in the supernova remnant W50, was discovered to be a radio source with variable emission by [Ryle et al. 1978] and [Seaquist et al. 1978]. Since then, extensive observations in all wavelength ranges have revealed SS433 to be a binary star system continuously ejecting plasma at a speed of 0.26c. This plasma is ejected into twin "jets", which are precessing with a period of 162.5 days. SS433 is a rare example where the velocity and relative precessional geometry of a relativistic jet are unambiguous. It therefore allows for observation the evolution of synchrotron emitting plasma in relativistic jets directly (see figure 1.1). A final example illustrating the benefits of monitoring source flux density variations are the constraints these variations place on the size of the radio emitting region. Clas-sically, if a source is observed undergoing a flux density fluctuation of duration At, then this places an upper limit on the physical size of the emitting region of order cAt. If the emission region were larger than this, then even if all portions of a source varied their emission simultaneously, the signal travel time over the source would blur the variations. In the case of many extragalactic radio sources, this has led to an apparent discrep-ancy [Hunstead 1972], [Dennison & Condon 1981]. If the cosmological interpretation for Chapter 1. Introduction 3 / - — N Normal star Accretion disk Figure 1.1: A schematic diagram of SS433 (adapted from [Hjellming & Johnston 1986]). Chapter 1. Introduction 4 the redshift is accepted, then the distance derived from the redshift allows an infer-ence of the intrinsic source brightness to be made, based upon the apparent source brightness. When combined with the implied source diameters, this suggests source brightness temperatures greater than 10 1 2 K . For a source radiating incoherent syn-chrotron emission (the assumed emission mechanism) this exceeds the brightness tem-perature upper limit of 10 1 2 K due to cooling of the electrons by inverse Compton scatter-ing [Kellermann & Pauliny-Toth 1969]. To resolve this discrepancy astrophysicists were forced to consider a variety of new explanations [Jones & Burbidge 1973]. One possible explanation is described in more detail in the next section. In any case it is clear that the search for, and study of, flux density variations in radio sources has been an extremely rewarding undertaking. It has led to the discovery of new astrophysical phenomena and in some cases forced astrophysicists to rethink previously held assumptions. 1.2 Extragalactic Radio Source Variability It was 1965 before the first confirmed evidence for variability in extragalactic sources was provided by Dent [Dent 1965]. He reported a 40% increase in the flux density of quasar 3C 273 over a three year period. Now it is known that virtually all extragalactic radio sources show variability on time-scales from a few days to a few years, with fractional flux density changes ranging from a few percent up to around 100%. Even some of the standard flux density calibrators have recently been shown to have mild variations on the time scale of years to decades [Ott et al. 1994]. The standard morphology for these extragalactic radio sources consists of a central Chapter 1. Introduction 5 source (a quasar or radio galaxy) and some sort of extended double lobe structure sur-rounding this core. Many of these extragalactic sources also have long, narrow tubes of radio emission, or jets, connecting the central source to the outlying radio lobes. A prototypical example is the radio galaxy 0326+396 (see figure 1.2 below). 4 8 6 5 . IBB tlHZ 0326+396 VLA 4"6X5:6 Figure 1.2: Radio galaxy 0326+396 at 6 cm (from [Bridle 1982]). Their phenomenology is based on their identification with optical sources. The vari-ables which can be identified with galaxies, for the most part giant ellipticals, are called radio galaxies. Quasars are those radio sources coinciding with highly redshifted unre-solved sources referred to as QSO's (quasi-stellar objects). Their high redshift implies these sources are extremely luminous, and their optical spectrum usually shows a blue or ultraviolet excess compared to normal Galactic spectra. The flux density variations seen in radio galaxies and quasars typically have timescales on the order of months. A rela-tively rare subclass of quasars are the B L Lacertae (BL Lac) objects. These objects have very weak, or no spectral lines, and are much more highly variable (variability timescales on the order of days). However, the degree of variability for extragalactic sources tends to correlate less Chapter 1. Introduction 6 with source classification (the type of source), than with spectral shape. Steep spectrum sources (sources with a < —0.5 ; S oc ua, a = frequency, S = flux density) are frequently less variable than flat-spectrum sources (sources with a > —0.5) [Kesteven et al. 1977], [Edelson 1987]. As well, intensity variations that are intrinsic to a source should be separated from variations that are due to some extrinsic effect. Small flux density fluctuations (~ 2%) on time-scales of a day have recently been found in many flat-spectrum extragalactic radio sources [Heeschen 1984], [Simonetti & Cordes 1990]. If these fluctuations are due to some process intrinsic to the sources, then the standard light travel time argument outlined in Section 1.1 would imply unphysically high source brightness temperatures of up to 10 1 9 K [Quirrenbach et al. 1992]. To explain these observations, several extrinsic mechanisms are plausible. At the current time, the most plausible explanation attributes these low amplitude intraday variations to refractive interstellar scintillations caused by electron density irregularities in the interstellar medium [Rickett et al. 1984]. To describe larger amplitude fluctuations a variability mechanism intrinsic to the source is necessary. The standard model to explain intrinsic variations in extragalactic objects is the expanding-source van der Laan model [van der Laan 1966]. This model is an application of an earlier model [Shklovskii 1960] describing the expansion of a supernova remnant. In the van der Laan model, the radio outburst is triggered by an adiabatically expanding cloud of relativistic electrons emitting synchrotron radiation in a magnetic field coextensive with the cloud. In its simplest form, this model assumes the electrons are homogeneously distributed, initially have a power law spectrum, are produced in a short time and in a small space, and that the expansion occurs in three dimensions at constant velocity with magnetic flux being conserved. The cloud is initially optically thick, but becomes optically thin as it expands. With these assumptions, the van der Laan model has succeeded in describing some Chapter 1. Introduction 7 of the observed features in some radio outbursts. An example is Dent's description of the 1966-67 outburst in quasar 3C273 [Dent 1968]. In general however, a more realistic model is needed. For instance, to explain the larger-than-expected variations at lower fre-quencies, this model was generalized to include the effects of continued particle injection [Peterson & Dent 1973]. As well, in order to mimic relativistic jet behavior more closely, a generalization was proposed which only allows for cloud expansion in one dimension [Marscher & Gear 1985]. The fact that the plasma velocity in the jet is highly relativistic also helps resolve the theoretical difficulty outlined in section 1.1, i.e. the excessive brightness temperatures implied by the rapid flux density variations. For relativistic motion closely aligned along the line of sight (relativistic beaming), Doppler boosting amplifies the synchrotron emis-sion, and through time dilation effects reduces the variability timescales. This allows the emission region to be much larger that cAt, avoiding the problem of having too much energy being crammed into a small volume. Evidence for relativistic beaming is found in time-sequenced V L B I observations of A G N (Active Galactic Nuclei). The transverse velocities of jet components are seen to move at apparent speeds surpassing the speed of light (superluminal motion). For an example of this effect, see [Pearson et al. 1981]. In fact, there have been very few extragalactic sources where it has been clearly established that the jet components travel at velocities much less than the speed of light. This beaming also provides a possible scheme for unifying flat and steep spectrum quasars. It has been proposed that flat-spectrum quasars are those quasars whose rela-tivistic jets are aligned close to the line of sight [Orr & Browne 1982]. Since the variability timescale for Doppler boosted jets is sharply reduced, this would explain the high degree of variability seen in flat spectrum quasars. If a quasar has its jets aligned away from the line of sight, then its observed spectrum would be steep, with a much reduced degree of Chapter 1. Introduction 8 variability. If this model is correct however, the Doppler boosted fiat spectrum sources would be expected to have radio emission strongly dominated by the core. Conversely, the less variable steep spectrum sources should have lobe dominated radio emission. The observation of weak core highly variable sources (see for example [Duric et al. 1989]) has given credence to the competing view which holds that the flat and steep spectrum sources are intrinsically different [Rudnick et al. 1986]. 1.3 Galactic Radio Source Variability The first confirmed evidence for radio frequency variations in a Galactic source (out-side the solar system) predates the discovery of extragalactic variables. In 1961 a flux density decrease of approximately 1% per year was observed in the supernova remnant Cas A [Hogbom & Shakeshaft 1961]. This decrease had actually been predicted earlier [Shklovskii I960]. It is typical of the kind of low-level flux density variations seen in supernova remnants. For other Galactic variables, the degree of variability is often more extreme. These variables include pulsars, as well as a plethora of different types of "radio stars". The former, pulsars, are now almost universally assumed to be rotating neutron stars. The simplified model for a pulsar has collimated radiation being beamed from the magnetic poles of a neutron star. For those neutron stars with the magnetic dipole axis inclined to the axis of rotation, this will produce a pulse of radiation with every rotation. These pulses vary in shape and intensity and typically only occupy a small percentage (1-5%) of the period. However, as was mentioned in section 1.1, the periods themselves are remarkably stable, making neutron stars excellent clocks. Radio stars are the second type of Galactic radio variable. Since the radio emission Chapter 1. Introduction 9 from normal stars is far too weak to be detected on Earth (our sun excepted), radio stars tend to be unusual objects, and are frequently binary systems. The emission from variable radio stars can be broadly categorized as one of two types, thermal or non-thermal. A thermal emission process is one in which the electrons involved have a Maxwellian velocity distribution. The emission is completely unpolarized, and the observed spectral index is greater than -0.1. Thermal bremsstrahlung, or free-free emission, is the most common thermal emission process at centimetre wavelengths. A non-thermal emission process has one or more non-Maxwellian electron velocity distribution components. It is characterized by moderate to high brightness tempera-tures, and an observed optically thin spectral index which is generally steeper (-0.7 being a typical value). Synchrotron and gyrosynchrotron emission are examples of processes of this type. For synchrotron emission, a high degree of linear polarization can sometimes be observed [Seaquist et al. 1974], while gyrosynchrotron emission can sometimes show high degrees of circular polarization. Cataclysmic variables (novae and dwarf novae) make up most of the thermal Galactic radio variables, and vary on timescales of months to years. In these binary systems an explosion on the surface of a white dwarf primary generates an expanding shell of hot ionized gas. The observed radio flux from the novae can be explained as thermal bremsstrahlung emission from this shell as it expands [Hjellming et al. 1979]. Of the kinds of variable Galactic radio sources emitting non-thermal radiation, flare stars are probably the most numerous. Red dwarfs are the most common variety of star exhibiting these flares, the prototypical object being the binary system U V Ceti [Davis et al. 1978], [Bastian & Bookbinder 1987]. The flares occur on the timescale of minutes to hours and are often highly circularly polarized. The physical processes in these stars are related to close "radio-active" binaries that flare with even greater frequency and strength. Chapter 1. Introduction 10 Close binary systems can be classified into three major divisions. In all cases, radio variability (if any is observed) is initiated when a secondary component (main-sequence star, giant, or subgiant) fills its Roche lobe and mass streams onto the smaller primary star. If the primary is a white dwarf, then the system is classified as a cataclysmic variable. If the binary is an X-ray source, then the system is called an X-ray binary. In such a system the primary can be a white dwarf, neutron star, or black hole. Finally, an arrangement where the primary is a main-sequence star or cool subdwarf is termed a classical interacting binary. Of the classical binaries, Algol (f3 Persei) and j3 Lyrae were the first to be detected as radio sources, with Algol exhibiting variability on timescales of minutes to hours [Wade & Hjellming 1972]. Algol is particularly notable for its high level of flaring with quiescent intervals of weeks to months between bursts of activity. RSCVn's (a class of binaries typified by the binary RS Canis Venaticorum) are also notable for their intense, highly variable radio emission. The first member of this class observed emitting variable radio emission was A R Lac [Hjellming & Blankenship 1973]. R S C V n type binaries differ from classical binaries in their optical properties. Super-imposed on their eclipsing light curve they have a wave of amplitude up to 0.2 magnitudes that propagates slowly through the light curve as a function of time. This wave is thought to be associated with starspots (cool regions of the photosphere where magnetic flux tubes protrude). These starspots cover a large fraction of the stellar surface of one star in the binary. The radio emission properties for both classical and R S C V n binaries are, however, very similar, and are generally consistent with gyrosynchrotron emission. The variability is thought to arise from flares and the interaction of the ejected electrons with the complex intertwined magnetospheres in the binary system [Uchida & Sakurai 1983]. The last, and perhaps the most spectacular, type of radio star is the X-ray binary. Chapter 1. Introduction 11 Both Cyg X-3 and SS433 are binaries of this type. A member of this class of binary was first observed at radio frequencies in 1971 when variable radio emission was de-tected coming from Sco X - l , the brightest object (apart from the sun) in the X-ray sky [Hjellming & Wade 1971]. Since this discovery, programs for monitoring other X-ray binaries at radio frequencies have been initiated. It was expected that any radio emission found in these binaries would be highly variable. However, it turns out that radio-emitting X-ray binaries are rather rare [Nelson & Spencer 1988]. Some well known radio X-ray binaries include Sco X - l , SS433, Cyg X - l , Cyg X-3, GT0236+610, and Circinus X - l . Cyg X - l is notable since its primary star has been estimated to have a mass of nearly 10 solar masses, making Cyg X - l the first convincing astronomical evidence for the existence of a black hole. GT0236+610 was first identified as radio variable during a search for Galactic variables [Gregory & Taylor 1986]. This search was phase one of the "Galactic Radio Patrol", a project initiated by the author's supervisor, Dr. Gregory. The motivation for the patrol stemmed from his discovery, in 1972, of the flaring in Cygnus X-3 (cf. section 1.1). In essence, this thesis is phase two of the project. One feature is common to all these radio X-ray binaries. In each case the radio emission is almost certainly synchrotron radiation from relativistic electrons. This fact, the high degree of variability, and the similar morphologies between these binaries and the much larger extragalactic jet sources have underscored the possible link between these two classes of objects. In fact, Rees has shown that the jet structures in both SS433 and extragalactic sources are similar except for a scaling factor [Rees 1982]. More recently, superluminal motion was discovered in the X-ray binary GRS1915+105, and the X-ray nova GRO J1655-40, [Mirabel & Rodriguez 1994], [Tingay et al. 1995]. These "microquasars" appear to be scaled up versions of the radio jet sources SS433 and Cygnus X-3. Because of their relative proximity, these Galactic microquasars offer the best Chapter 1. Introduction 12 opportunities to gain a better understanding of the relativistic ejections seen in the cores of active galaxies. 1.4 Variability Searching Methodologies and Overview of the Thesis In order to search for radio variables two different methodologies can be adopted. A small number of specific sources can be targeted, or a great number of unbiased sources can be inspected by surveying a portion of the sky at several different epochs. The former case typically focuses on a class of sources that are suspect radio variables. Examples of this type of targeted survey include, for instance, surveys of X-ray binaries [Nelson & Spencer 1988] or surveys of flat spectrum AGN's [Valtaoja et al. 1992]. In the latter case, no particular emphasis is placed on any specific class of sources. Instead a portion of the sky is repeatedly surveyed in the hope that some of the radio sources contained therein will prove to be interesting radio-variables. This technique for finding variables has been employed at low frequencies (151 MHz) by [Riley 1993], at intermediate frequencies (1.4GHz) by [Rys & Machalski 1990], at high frequencies (4.76 GHz) in the Galactic plane by [Gregory & Taylor 1986] (the Galactic Radio Patrol), and by [Machalski & Magdziarz 1993] at a variety of different frequencies. In this thesis, a combination of these two methods is used. Maps constructed from a complete survey of the northern hemisphere in two different years, the Greenbank survey, are used to uncover long-term variables. These variables are then targeted for a more in-depth short-term variability study (variability timescale of days to one month) by analyzing the individual scans that were used to construct the individual epoch maps. A discussion of the Greenbank Northern Hemisphere sky survey is found in the next chapter. As well, an outline is given of the procedure used to convert this data into computer images. Using these images, catalogues of source flux density and position Chapter 1. Introduction 13 were made for the two different survey epochs, and this cataloguing procedure is also described. Bayesian inference is used in the identification of long-term variables, and in the de-termination of short-term flux densities. Hence, in chapter 3 a brief overview of Bayesian inference is given, as well as an outline of its application to the variability problem. Chapters 4 and 5 detail the development of both the long-term and short-term vari-ability catalogues. Chapter 4 describes the methodology used to uncover the short-term and long-term variables, while chapter 5 presents the results. Chapter 6 is a self-contained study of a possible Galactic variable source with jets, GT2318+620 [Gregory & Taylor 1986], made with high resolution observations from the V L A . The original intention of this thesis was to make further observations of some of the most interesting objects in the short-term variable catalogue. However, time constraints limited follow-up observations to this one previously discovered source. Finally, in chapter 7, a summary of all the results is presented. Chapter 2 The Greenbank Survey 2.1 Data Acquisition As previously mentioned, unbiased searches for radio variables require repeated surveys of a portion of the sky. In 1986, the possibility of repeated surveys of most of the Northern Hemisphere was opened up when a new, seven feed horn, 6 cm receiver system became available for use on the Greenbank 91 m transit radio telescope, located near Greenbank W V . As a continuation of the Radio Patrol Project, the author's supervisor, Dr. Gregory, proposed to use the system for variability studies. Independently, another group led by Condon of the N R A O proposed to use the same system to image and catalogue the Northern Hemisphere. The decision was made to combine these projects: N R A O took primary responsibility for production of the sky survey images, while U B C took responsibility for the source cataloguing and variability analysis. The result of this collaboration, the Greenbank Sky Survey [Condon et al. 1989] , [Condon et al. 1994a], was comprised of scans covering the entire Northern Hemisphere between 0° and 75° declination. The sky was surveyed at 4.85 GHz (6 cm wavelength) over a one month period in both 1986 and 1987. Variability information over both the short term (during the one month surveys) and the long term (between the 1986 and 1987 survey epochs) was obtained. A survey of the sky during a third epoch, 1988, was also attempted. However, the untimely collapse of the telescope during this run, and the large pointing inaccuracies that were a precursor to the telescope's demise, made these 14 Chapter 2. The Greenbank Survey 15 data unsuitable. As mentioned, the telescope itself used a 7 feed receiver. Each feed accepted both left and right circular polarizations, and the feeds were arranged in the hexagonal pattern shown in figure 2.1. Figure 2.1: Orientation of the seven feeds to the scan direction. Each survey consisted of a series of elevation scans during which the telescope was driven alternately north and south between 0° and 75° declination at a rate of 10°/minute. Because of the rotation of the Earth at a rate of 0.25 cos 5 °/minute (5 = declination), the angle between the scan direction and the meridian varied from 1.4° at the equator to 0.5° at 5 = 75°. The resulting scan pattern on the sky is depicted in figure 2.2. The separation between individual beam tracks was one H P B W (half-power beam Chapter 2. The Greenbank Survey 17 width). This was half the Nyquist sampling rate for map restoration, so on alternate days the feeds were offset 0.5 H P B W . After every pair of days a new set of tracks was started 7 H P B W east of the previous set. After 26 days this procedure provided full Nyquist sampling at 5 = 0° for both the northbound and southbound sets of scans, with higher declinations being oversampled by an amount ~ sec S. Thus, it is possible to make two independent images for each observing epoch. However, the final published images for each epoch consist of a combination of both the northbound and southbound scan data. Individual data points were recorded by integrating (averaging) the receiver volt-age outputs in 0.1 s increments. The resultant data, together with the associated tele-scope coordinates and times, were then stored on magnetic tape in FITS table format [Harten et al. 1988]. 2.2 Image Construction In order to transform the data into images, the N R A O Astronomical Image Processing System (AIPS) was used. The first steps were to separate the data into manageable-sized chunks, subtract baselevels, then convert it into pseudo (u,v) format. The (u,v) format is a data format designed for aperture-synthesis observations. It is comprised of complex visibility-amplitudes, together with the associated projected baseline lengths u and v. Pseudo (u,v) format is (u,v) format adapted for single dish observations. The system temperatures (after baselevel subtraction) are stored as the real part of the visibility amplitudes, while the baselevels themselves are stored as the imaginary parts. The right-ascension and declination are stored as the u and v baseline lengths. The single-dish data were stored in pseudo (u,v) format so that existing AIPS software, which was originally designed for aperture synthesis observations, can be used. Chapter 2. The Greenbank Survey 18 After sorting and baselevel subtraction, the recorded system temperatures were con-verted into flux densities by dividing them by the telescope gain. Bad data were then flagged, and preliminary maps were created in the sine projection by gridding and con-volving the data, then running additional checks for bad data. The maps were then calibrated for both intensity and position, and the corrections were applied to the map headers. For a more complete description of the process, see [Condon et al. 1989] and [Condon et al. 1994a]. The final maps cover an approximately 6 steradian strip in the sky between 0° and 75° declination, and over the complete R A range. Each map covers an 11° by 11° square on the sky, with 288 maps comprising the entire 6 steradian range. The 1987 data were collected between September 30 and November 1. During this time the proximity of the sun to the survey region forced the rejection of the 3 maps covering (12 h40m < a < 14 h40m), (0° < 8 < 5°) due to solar interference. The 1986 observations were made between Nov 7 and December 13. During this time, the sun was near a = 16h, 8 = —20°. This was far enough south that no 1986 map had to be rejected due to solar interference. Better maps were also made by combining the 1986 and 1987 epoch data. This reduced the noise by a factor of >/2- This was accomplished by combining the pseudo (u,v) data from both epochs, then remapping in the same manner as was done for each epoch separately. An example of a combined map is shown in figure 2.3. This map makes up a region of the sky centred o n « = 0 h 20 m , 8 — 70°. Almost all of the point sources seen on the map are radio-galaxies or quasars. The large ring-like structure in the upper part of the image is the supernova remnant CTA1 [Harris & Roberts 1960], [Sieber et al. 1979]. Both elongated structures south of the C T A l are emission from the ionized hydrogen region N G C 7822 [Bonsignori-Facondi & Tomasi 1979]. The point-source response for the various maps can be described by an elliptical Chapter 2. The Greenbank Survey 19 m Figure 2.3: Combined 1986 k 1987 image for a = 0 h 20 m , 5 = 70°. Chapter 2. The Greenbank Survey 20 Gaussian point-spread function (PSF) with HPBW's of approximately 3.3' and 3.6' in the east-west and north-south directions respectively. Gravitational deformation of the 91 m primary reflector caused the point-source response to change with declination. These variations are shown in figures 2.4, 2.5, and 2.6, for the 1986, 1987, and combined-epoch maps respectively. In the 1987 data there was also a dependence on R A of the point-source response function. This was due to heating of the primary reflector and caused the PSF to broaden by about 2% near the solar noon. The RMS map intensity fluctuation, a, is a quadrature sum of the RMS noise, crn, and the RMS confusion, ac, from Galactic emission and faint extragalactic sources. For the Greenbank telescope at 4.85 GHz with an integration time of 0.1 s, o is dominated by the noise (an ~ 5 mJy, ac « 1 mJy). The measured values for a averaged over all RA's for the 1986 and combined maps are shown in figures 2.7 and 2.9 respectively. For the 1987 maps, the noise also depends weakly on R A due to increased interference near the solar noon. This dependence, as well as the dependence of o on declination in the 1987 maps, is shown in figure 2.8. 2.3 The Greenbank Catalogues Once the data were converted into images, the next step was to use these images to construct a catalogue of source flux densities and positions. In addition to being of use to the astronomical community in general, new long-term (and possibly short-term) variable sources could be uncovered by comparing the catalogues from the different epochs. Unfortunately, this task was made difficult by the sheer volume of the data. Each image is typically composed of more than 300 sources whose flux densities are greater than 5cr. To catalogue all the sources individually would have been a monumental and Chapter 2. The Greenbank Survey 21 T 1 r -i 1 r 1986 20 40 D e c l i n a t i o n (degrees) 60 80 1 ~i 1 r T T _L 1986 60 80 0 20 40 D e c l i n a t i o n (degrees) Figure 2.4: Point-spread function for the 1986 Greenbank survey. Top: Declination dependence of the 1986 PSF H P B W in the North/South direction. Bottom: Declination dependence of the 1986 PSF H P B W in the East/West direction. Chapter 2. The Greenbank Survey 22 -| 1 r -i 1 r -i r 3 k 2 k 1987 3 k 2 k _L 20 40 60 D e c l i n a t i o n (degrees ) -i r T T 1 r T i 1 r T 1987 _| l i _L 20 40 60 D e c l i n a t i o n (degrees) 80 80 Figure 2.5: Point-spread function for the 1987 Greenbank survey. Top: Declination dependence of the 1987 PSF H P B W in the North/South direction. Bottom: Declination dependence of the 1987 PSF H P B W in the East/West direction. Chapter 2. The Greenbank Survey 23 1 I L 1986+1987 20 40 60 D e c l i n a t i o n (degrees) 20 40 60 D e c l i n a t i o n (degrees) 80 80 Figure 2.6: Point-spread function for the combined-epoch Greenbank survey. Top: Dec-lination dependence of the combined epoch PSF H P B W in the North/South direction. Bottom: Declination dependence of the combined-epoch PSF H P B W in the East/West direction. Chapter 2. The Greenbank Survey 24 20 40 D e c l i n a t i o n (degrees) 60 80 Figure 2.7: Background noise for the 1986 epoch maps as a function of declination averaged over the complete R A range. Chapter 2. The Greenbank Survey 25 CO 1.1 1 .05 -1 -4 -1987 20 40 D e c l i n a t i o n (degrees) 60 80 Figure 2.8: Background noise for the 1987 epoch maps. Best estimate for the 1987 epoch noise is the product F • S. Top: Noise dependence on R A for the 1987 epoch maps, averaged over all declinations. For a fit of this curve to observational data, see [Condon et al. 1989]. Bottom: Noise dependence on declination for the 1987 epoch maps, averaged over the entire R A range. Boxes represent the average noise in each 10° declination strip. Chapter 2. The Greenbank Survey 26 6 r -5 r -4 r -3 r -0 1986+1987 _L 60 80 20 40 D e c l i n a t i o n (degrees) Figure 2.9: Background noise for the combined-epoch maps as a function of declination, averaged over the complete R A range. Chapter 2. The Greenbank Survey 27 extremely time-consuming task. Therefore, an automatic source cataloguing routine was developed using D A O P H O T software [Stetson 1987] in the I R A F environment. Although I R A F (Image Reduction and Analysis Facility) is primarily a package designed for optical astronomy, it was used in preference to AIPS, the more traditional radio-astronometric software, because of the inability of AIPS to handle large numbers of sources automati-cally. This source cataloguing routine was used to catalogue the sources found in the 1986 and 1987 images separately, as well as the sources from images constructed by combining the 1986 and 1987 data. The two single epoch catalogues will henceforth be referred to as the GB86 and GB87 catalogues respectively, while the combined epoch catalogue will be referred to as the GB6 (Greenbank 6 cm) catalogue. The only major procedural difference in each case was the specification of the noise level. For the GB87 source catalogue, the noise for each map was specified by multiplying the average noise level in each declination strip by a function dependent upon the right ascension. For the GB86 and GB6 catalogues, the noise level was specified by a continuous function of declination (cf. Figures 2.7 and 2.9). There was no appreciable noise dependence as a function of right ascension in these latter two cases. The first step in this cataloguing procedure involved the use of the D A O P H O T routine D A O F I N D to detect unresolved point sources in each of the images. This routine was able to differentiate point sources from extended emission and noise peaks, and attempted to resolve blended pairs of point sources. Next, D A O P H O T routines were used to fit all the point sources to an approximate elliptical Gaussian model of the PSF. The free parameters for each fit included the position, baselevel, and amplitude of the Gaussian. Since D A O P H O T was designed for C C D images, a fitting of the PSF to unresolved point sources would normally provide accurate source strength measurements. However, for the GB86, GB87, and GB6 images Chapter 2. The Greenbank Survey 28 there is no true PSF because of the seven different feed horns involved, together with gain and pointing variations between successive scans. Much more reliable flux densities were determined by fitting each individual source to an elliptical Gaussian, and allowing all of the parameters of the fit (baselevel, position coordinates, amplitude, major/minor axis' H P B W , and major axis position angle) to vary for each source. Unfortunately, the density of radio sources in an image is often sufficiently high that the area required for an elliptical Gaussian fit often includes other confusing sources. To reduce the confusion to a tractable level, D A O P H O T routines were used to subtract off the PSFs of detected sources. This was done in two stages. In the first pass all sources stronger than 100 mJy (for which sidelobes may be visible above the noise) were fit to the PSF and subtracted from the image. This first PSF extended to a radius of 800" and included near-in sidelobes as tabular corrections. The result of this subtraction produced what was called "the first difference image". In the second pass, a smaller PSF was used (radius = 320") to fit the weaker sources left in the first difference image (any source stronger than 3.9 times the noise). These sources were then also subtracted out, producing the "second difference image". A l l the PSFs from the first and second passes were then sorted into groups, with all the members of a group having non-overlapping PSFs. Then each group in turn was added back to the second difference image. This resulted in de-confused images, on which new elliptical Gaussian fits to each source could be made. Another problem arose because residuals from the first pass PSF subtractions were occasionally picked up in the second pass and spuriously labelled as sources. To eliminate these "ghosts" various automatic editing criteria were used: -Any source within 4' of a stronger source was rejected -For unresolved sources, the peak flux and fitted flux should have approximately Chapter 2. The Greenbank Survey 29 the same value. Hence, if there was a large difference between the fitted flux and the peak flux of a source, the source was rejected -If the fitted H P B W of a source was either too large or too small, the source was rejected -For sources stronger than 1 Jy, an exclusion radius was specified wherein all sources were rejected unless they had a strength of greater than 10% of the strong source peak flux. This criterion was established to eliminate sidelobes from appearing as sources in the catalogue -For the five strongest sources, a larger individualized source exclusion radius was implemented. These sources are the Crab Nebula, M87 (3C 274), W51 (3C 40), Cyg A, and Cas A. In addition, to try to minimize the number of noise features identified as sources, all "sources" with flux densities less than 5a were excluded from the final source list. As a last check for spurious sources, an interactive routine was developed whereby sources could be manually added or subtracted from the catalogue. This was done by visual inspection of each image, with the location of the catalogued sources overlaid. For each image in the GB87 and GB6 catalogues, it was almost always necessary to delete a small number of weak "sources" found in strips of noisier data. For the GB86 catalogue, no final checks for spurious sources were made, since this catalogue is being used strictly for variability studies, and there are no plans to publish this catalogue as a separate entity. However, sources tagged as variables by using the GB86 epoch catalogue were individually inspected (cf. Chapter 4), and this prevented any possible spurious results from propagating into the variable list. The next step was to make corrections for any systematic effects. For all catalogues, Chapter 2. The Greenbank Survey 30 several small flux density biases were added/subtracted to all the sources to compensate for i): an overestimate of the source strengths due to elliptical Gaussian fitting in the presence of noise, and ii) an underestimate of the source strengths due to median baseline subtraction in the presence of a source (see [Gregory & Condon 1991] for a more complete description of these biases). Both of these effects were small ( « 1 mJy and 2.5mJy respectively for a 25mJy source). The final step in this process involved comparison of the flux densities and positions from the catalogues to more accurate measurements from other surveys. Differences in flux density/position were divided by the quadrature sums of the uncertainties (which were dominated by the uncertainties in the G B catalogue flux densities/positions) to yield normalized uncertainties. If there are no systematic biases, and the error estimates are accurate, then these normalized errors should have a Gaussian distribution with zero mean and unity rms width. For the GB87 catalogue, flux densities were compared to the corresponding flux den-sities from a set of steep-spectrum sources from the B3 source list [Kulkarni et al. 1990]. Source positions were compared to a set of strong sources with accurate positions ex-tracted from the University of Texas 365 MHz survey [Douglas et al. 1973]. These com-parisons verified that the GB87 catalogue flux densities and source positions had reason-able error estimates and no remaining systematic biases. Similarly, for the GB6 catalogue, source flux densities were verified by comparison with more accurate values measured by the MPIfR 100 m radio telescope located near Bonn [Forkert & Altschuler 1987], [Maslowski et al. 1981]. Positional accuracy was ver-ified by comparison with sources from the FIRST survey [Becker et al. 1994], and a set of compact sources with accurately measured V L A positions [Patnaik et al. 1992]. For the GB86 catalogue, flux densities were verified by comparison with the GB87 catalogue. After correcting for median baseline subtraction and fitting in the presence Chapter 2. The Greenbank Survey 31 of noise (see above), a small bias of 1.5 mJy in favour of the GB87 data remained. This bias was added to all sources in the GB86 catalogue. The final GB87 source catalogue [Gregory & Condon 1991] contains 54,579 compact sources stronger than 5a ( « 25mJy). The GB6 catalogue [Gregory et.al. 1995] con-tains 75,162 sources, and a noise level approximately y/2 lower than the single epoch catalogues. Chapter 3 Bayesian Theory 3.1 Introduction to Bayesian Inference Identification of long-terra variables by comparison of the GB86 and GB87 source cata-logues required some method of quantifying the degree of variability. For those sources determined to be highly variable, pseudo (u,v) data (individual daily scan data) were ex-tracted for the purpose of generating daily "light curves". This would indicate whether these long-term variables were also short-term variables. To generate these light-curves a set of parameter estimating routines was needed in order to fit the data to the expected beam profiles. Bayesian Probability Theory provides a method for performing both of these tasks. Since most readers will not be familiar with Bayesian methods, this section will give a brief tutorial in basic Bayesian concepts. In section two of this chapter, these concepts will be applied to the problem of identifying long-term variables. Section three of this chapter will discuss the Bayesian parameter estimation problem. Finally, the last section of this chapter will detail why it is preferable to apply Bayesian inference to these problems, rather than the more traditional "frequentist" statistical concepts. To begin the tutorial, it is instructive to go back to the foundations of Bayesian inference, i.e., the familiar sum and product rules of probability theory, p(A\C) + p(A\C) = l and p{A,B\C) = p(A\B,C) • p(B\C) 32 Chapter 3. Bayesian Theory 33 = p(B\A,C)-p(A\C) (3.2) where A, B, and C are understood to be propositions, A means "negation of proposition A " , p(A\C) denotes the probability of proposition A being true given that proposition C is true, and the juxtaposition A, B inside the parenthesis means proposition A A N D proposition B. By equating the right-hand-sides of equation 3.2 and solving for p(A\B, C) Bayes' theorem is trivially obtained, Now if: Proposition A = H — a proposition asserting the truth of some hypothesis, Proposition B = D — a proposition asserting the value of the data, Proposition C = I = a proposition stating some background information about the way H and D are related, then equation 3.3 becomes: m D t I ) = m m n . ,3,) The quantity on the left-hand-side is then read as "the posterior probability of hy-pothesis H, given the data D , and background information 7". On the right-hand side p(H\I) is the prior probability of the hypothesis. The quantity p(D\H,I) is referred to as the likelihood function for the hypothesis. Finally, p{D\I) is often called the global likelihood function. From the sum and product rules it can be shown that p(D\I) = Y,Pmi)-p(D\HitI) (3.5) i where the sum is over all Hi in the assumed hypothesis space. Thus, in this case, it is clear that p(D\I) just assumes the role of a normalization constant. However, in the model comparison problem it will assume a more important role. Chapter 3. Bayesian Theory 34 Frequently however, instead of having a discrete set of different hypotheses {Hi}, there exists a single model M with the different hypotheses represented by the possible values of a continuous free parameter 9. Using this model M as background information, Bayes' theorem 3.4 becomes: with the continuum analog of the global likelihood function, equation 3.5, being written as p(D\M) = Jp(e\M) •p(D\9,M)d9. (3.7) Equation 3.6 can be generalized even further by considering the case of a model with many free parameters. A further complication arises when, of these free parameters, only a subset is of interest. If, for instance, a model has two parameters 9 and 4>, and only the value of 6 is of any interest, then from the sum and product rules: p(9\D,M) = jp{9,ct>\D,M)d(j> = pTJ^jjrtM|M)-p(Z?|M,M)d0. (3.8) This procedure is called marginalization (the parameters of no interest are marginal-ized out). The global likelihood p(D\M) is now calculated by integrating over both parameters, and is just a special case of marginalization in which all the parameters are integrated out. Equation 3.8 describes the solution to the problem of parameter estimation for a single model. Often however, there exists more than one parameterized model to explain the data, and a choice between the models must be made, based on their ability to predict the data. This is the model comparison problem. If these different models are specified by Mi, and the background information / is taken to be the proposition that one of the Chapter 3. Bayesian Theory 35 models under consideration is true, then for each model Bayes' theorem can be written as: mv,I)=mwpji. , 3 , , However, p(D\Mi, I) = p(D\Mi), since the proposition: (Mu I) = "Model Mi is true A N D one of the models is true" is equivalent to the proposition (Mi) = "Model Mi is true, with the global likelihood function, p(D\M{), being calculable using equation 3.7. The denominator in equation 3.9 is again just a normalization constant. Calculation of it can be avoided by taking the ratios of the probabilities of the two models. For instance, using equation 3.9, the odds ratio Oij = pj^'p'}] in favour of model Mj over model Mj would be written as: p(Mj\T)p(D\Mt) _ p(Ml\I) U i i ~ p(MJ\I)p(D\Mj) - piMAlf* ^ The first factor is just the ratio of the priors. The second term, B^, is the ratio of global likelihoods, and is called the Bayes' factor. It automatically favours simpler models (models with fewer parameters) unless the data justify the additional complexity. Thus the Bayes' factor provides a quantification of "Ockham's razor". For a good explanation of how this arises in theory, see [Gregory & Loredo 1992]. Now if the odds ratio in favour of each model over one given model (say model M{) is calculated, then equation 3.9 can be expressed in terms of these odds ratios, i.e. p(Ml\I)p(D\Ml) p(Ml\D,I) P(D\I) p(M 1 | / )p (£> |M 1 ) ] P{Mj\I)p(D\Mj) Chapter 3. Bayesian Theory 36 " & ( 3 ' n ) 3 where the sum is over all possible models, and (of course) O n = 1. 3.2 Bayesian Statistics and the Long-Term Variability Problem An immediate application of these odds ratios is found in the long term variability prob-lem. This problem can be expressed in the following manner. Two measurements of a source's flux density, Si and 52, with Gaussian error distributions, oi and 02, are ob-tained at times t\ and t2- Based on these measurements, what is the probability of the source being a variable? To answer this question, consider two models: Model Mi = Flux density of the source is constant, Si = 5 2 = k Model M 2 = The source is variable, with Si = ki at i i , and 52 = k2 at t2l together with the background information that asserts that one of these two models must be true. The question of variability can then be quantified by determining the probability for model M 2 . From equation 3.11 this probability is given by: w i t h ov=*zi??lT;{ <"•») p{Mi\I)p(D\Mi) Thus, to calculate p(M2\D, I), all that is needed are expressions for the global likelihoods, p(D\M\) and p(D\M2), as well as the prior probabilities for the respective models. For the purpose of this thesis, no prior knowledge concerning the incidence of source variability will be assumed. Hence, the prior probabilities for both Mi and M 2 will be | , Chapter 3. Bayesian Theory 37 reducing equation 3.13 to a ratio of global likelihoods. These global likelihoods can be calculated from equation 3.7. For instance, for the constant model M\. p(JD\Mx) = Jp(k\Mx) -p(D\Muk)dk. (3.14) The prior probability for the average flux density, p{k\M\), can be estimated from radio source counts as a function of flux density (cf. Appendix B). To calculate the likelihood function, p(D\Mi, k), it is assumed that the difference be-tween the data and the model is just noise, with the noise having a Gaussian distribution. Thus: p(D\Muk) = p(Sl\M1,k)-p{S2\M1,k) Hence, the global likelihood, equation 3.14, becomes: p{D\Mx) = (3.15) Formally, equation 3.15 is integrated from zero to oo (or more realistically, to the flux density which saturates the telescope receivers). However, as a practical matter, the integration need only be carried up to a flux density 5 m a x , where for flux densities > Smax the contribution to the integral is negligible. To calculate the global likelihood function for the variable source model, p(D\M2), equation 3.7 can be generalized to account for two free parameters instead of one, i.e.: p{D\M2) = f [ p{k1,k2\Mi)-p{D\M2,k1,k2)dk1dk2. (3.16) Chapter 3. Bayesian Theory 38 The prior probability for the two source flux density measurements in this equation can be written as a product of priors: p(ki,k2\M2)=p(k1\M2)p{k2\kuM2). (3.17) Again, evaluation of both of these priors is detailed in Appendix B. The likelihood function in equation 3.16 can be derived using similar arguments as were used to derive the constant model likelihood function. Hence, equation 3.16 can be rewritten as: « p { ( - i ) E p ^ ) 2 } < * A . (3-18) Substituting the global likelihood expressions for the constant and variable models, equations 3.15 and 3.18, into equation 3.13, yields an expression for the odds ratio in favour of the variable model over the constant model: a 2,1 J0Sm"-I0Sm" p(^\Mz)r,(ki\ku M 2 ) • exp { B ) [ ( ^ ) 2 + ( <72 / | dk\ dk2 St"'-p(k\Mi)^[{-\) | dk .19) This equation can be evaluated numerically for all sources in the GB6 source list. 3.3 Bayesian Statistics and Flux Density/Position Determination Another application of Bayes' theorem is found in the determination of the daily flux densities and source positions. In general, the most straightforward method of doing this is to fit each day's data around the target sources to a two-dimensional model for the beam, and extract the amplitude of the fit. Unfortunately, however, this is not feasible. While the sampling in the N / S direction is adequate (the sky was sampled every 0.1 seconds, implying sampling every 1' in declination) the sampling in the E / W direction is Chapter 3. Bayesian Theory 39 very poor (recall from section 2.1 that for any scan, the separation between the individual beam tracks is one H P B W , or approximately 3.1'). Even knowing the shape of the beam accurately, the uncertainty in the source position, together with the paucity of E / W data and the relatively low signal to noise ratio, makes two-dimensional fits to low-to-moderate strength sources extremely unstable. Much better results are obtained by dividing the two-dimensional fits into a number of one-dimensional fits. Each of the individual beams in any one scan can be fitted to a one-dimensional beam profile. The peaks of these fits can then be used as data points for a second one-dimensional fitting, essentially fitting the averaged E / W profile of the beams. Mathematically, this problem can be expressed in the following way: Consider the data generated by K parallel tracks from radio telescope beams passing in the vicinity of a source. A "track" in this context refers to the path any one telescope beam traces on the sky as the telescope is driven northward or southward. Each track (denoted by the index j) generates Nj data points, with each data point consisting of a North-South.position Xij, and an amplitude at that position, y^. Superimposed on the "true" amplitudes from beam j is background noise, characterized by a Gaussian distribution of width a = Sj. Given these initial conditions, the problem is to fit each of the K data tracks to a model for the appropriate telescope beam. To first order, each beam can be modelled by a Gaussian, whose width cr,, is assumed to be known. Hence, each of the K data tracks must be fit to a Gaussian of unknown amplitude A,-, and common centre X0. For a frequentist, a typical algorithm to solve this problem would be to fit each track to a Gaussian, allowing the amplitude and centre of each Gaussian to vary as free parameters. This will result in K amplitude and centre estimates. However, while the amplitude estimate for any one track is essentially independent of the other tracks, the common centre is unique, and should be the same regardless of which track is being fitted. A possible solution to this problem would be to decide on the "best" value for Chapter 3. Bayesian Theory 40 XQ by taking a weighted average of all the fitted centres, then, refit all of the scans to Gaussians again, only this time holding the centres fixed at XQ. From a Bayesian point of view however, this problem is exactly solvable. The starting point is Bayes' theorem: p(AlA2 • • • Aj, X.\D, M) = ^ • - • A " M M p ] ^ A 2 ' • • A " *» M ) (3.20) with the model M describing the data in each track as a one-dimensional Gaussian with amplitude Aj and common centre XQ. Assuming the priors for the amplitudes and the center are independent and uniform, the posterior probability distribution (the left-hand-side of equation 3.20) becomes directly proportional to the likelihood function, p(A,A2 • • • Aj, X0\D, M) oc p(D\AYA2 • • • Aj, X 0 , M). (3.21) The likelihood function can be evaluated as was done in section 3.2. Under the assump-tion of a Gaussian noise distribution it can be written as a product of Gaussians: m M A t - A ^ M ) oc nilexp|(-l) ^-f{AJ^Xl"X^J - n o p f H ) ! ^ - ^ ; ^ ) ' } <-> with / ( ^ , X 0 , ^ ) = ^ e x p j ( - l ) j = V y . (3.23) Substituting equation 3.22 into 3.21, and assuming the prior values for the amplitudes range between zero and a maximum value Amax, the unnormalized probability for any value of XQ being the "true" centre can then be determined by marginalizing over all the Chapter 3. Bayesian Theory 41 i = i 1 / 0 = n ^ o | ^ - , M ) (3.24) Equation 3.24 says that the unnormalized posterior probability distribution for X0 is simply the product of the p(X0\Dj, M) 's , the posterior probability distributions for the XQ. An analytic expression can be derived by solving the integral with the aid of equation A . l , of appendix A i.e.: 2 Ni p(X0\Dj,M)= / exp Jo I V z Jo = e 7 j / exp (ajAj - <iA cx /3~ 1 / 2 • e4/3J [erf(by) - erf (a,)] (3.25) (3.26) iv,-where O;J = ^ i = i Pi = Vij and bo __Oj_ 2 A / £ Substituting equation 3.26 into 3.24 gives the unnormalized posterior probability for the centre X0, K p ( X 0 | D , M ) « n /?71 / 2 • e^I [erf(bj-) - erf (a,-)] (3.27) Chapter 3. Bayesian Theory 42 The probability of the k data track, 1 < k < K, having amplitude Ak — A is determined similarly, by marginalizing equation 3.22 over all the other amplitudes and X0: where Xu and XL are the upper and lower limits for the prior on the centre. The integral over the exponent in this equation is just the posterior probability for the kth data track having amplitude A^. The product of the p(X0\Dj, M)'s inside the integral provides for a coupling with the other data tracks due to their common centre. 3.4 Why Bayesian? Traditionally, probability is identified with long-run frequencies of occurrence, either in repeated experiments or in an ensemble of "identically prepared" systems. This view of probability is the basis for most statistical procedures currently in use in science. However, the reliance on long-run behavior for frequentist statistical evaluation makes the connection to real life problems tenuous. Further, there are often several different methods available to answer any statistical question, with no compelling criteria for choosing among them. By comparison, to a Bayesian, probability is regarded as a real-number-valued mea-sure of the plausibility of a proposition. In this measure, 0 represents the certainty of falsehood, and 1 represents the certainty of truth. This definition has a more obvious Chapter 3. Bayesian Theory 43 connection with what is commonly thought of "probability". In addition, unlike frequen-tist statistics, with the Bayesian approach there is a well-defined procedure to solve any problem, with the starting point always being Bayes' theorem. For a more complete discussion of these points, the reader is referred to [Loredo 1990]. In this thesis, Bayesian inference was used to resolve two different problems. In the first of these problems, the long-term variable/nonvariable model assessment problem, Bayesian inference offers several advantages over the "traditional" approach. Firstly, Bayesian inference allows the use of prior information to be incorporated into the problem. When the amount of data in a problem is large, the addition of prior information may not significantly alter the result. However, when the amount of data is small, as is the case in this model comparison problem, the incorporation of prior information into the problem is particularly important. This prior information amounts to previous knowledge of the number density of radio sources as a function of flux density, and how, statistically, radio sources vary from one epoch to the next. In any Bayesian odds ratio calculation, the complete set of alternatives to any hy-pothesis must be specified beforehand. The odds ratio calculation then compares all of the alternatives, penalizing those alternatives having more parameters with a built-in Ockham's penalty. For the long-term variability problem, there are only two alternatives, with the constant source model having one parameter, and the variable model having two. Hence, even though the ratio of likelihood functions always favours the variable model, this is balanced by an Ockham's penalty always favouring the simpler, constant model. By contrast, in frequentist model assessment (%2 goodness-of-fit test, Kolmogorov-Smirnov test, etc.), no alternatives to the model being tested are considered. An in-herently bad model can be made to "pass" any goodness-of-fit test simply by adding more parameters to the model. As well, even when a parameterized model is adequate, Chapter 3. Bayesian Theory 44 frequentist goodness-of-fit tests require the assumption that the best fit parameter values are the true values. This is certainly not always the case. Finally, without alternative hypotheses, it is much harder to quantify what the results of any goodness-of-fit test mean, since it is unclear what alternatives these results should be compared to. For the long-term variability problem, a comparison of frequentist model assessments versus Bayesian odds ratio calculations is shown in figure 3.1. On the abscissa of this graph is plotted the log 1 0(OddsRatio) for the variable source model versus the constant model (equation 3.19). On the ordinate axis is plotted x2 f ° r the two, single-epoch, source flux density values. In general, x2 f ° r a n v series of measurements is defined by equation 4.2, but for the case of only two measurements is equivalent to ( A S / a ) 2 , where A S is the absolute value of the flux density difference between the two epochs, and a is the the sum in quadrature of the rms flux density errors from each epoch. Approximately 5300 sources from the GB6 catalogue are represented on this graph, corresponding to sources with RA's between 0 h and l h 2 0 m . A l l sources to the right of the line corresponding to an odds ratio of 105 are defined to be long-term variables (cf. section 4.1). From this graph it is evident that this criterion for long-term variability is more conservative than the frequentist "5a" criterion, which corresponds to all sources with |AS|/<7 > 5, or identically, x2 > 25. In table 3.1 is listed various values of x2, together with \og10Q(x2 | 1), the logarithm of the probability that the observed chi-square will exceed the value x2 by chance when the specified model is correct. In this case, the specified model is the "constant source model", which has only one free parameter. Since an odds ratio of 105 corresponds to a X2 of approximately 30, this implies that the false positive rate for the long-term variable catalogue is < 1 : 10 7 4 . The spread in odds ratios for any fixed value of x2 seen in figure 3.1 is primarily due to the presence of the prior probability on source variability, p(k2\k\, M 2 ) , in equation Chapter 3. Bayesian Theory 45 3.19. For a fixed value of x 2 , those sources having the higher fractional variability index (cf. Appendix B), actually have a lower odds ratio. This is because sources with higher fractional flux density variations are less probable, a priori, and thus require a greater amount of supporting evidence in order to be believed, than do sources with smaller fractional flux density changes. In the limiting case of flat prior probabilities, with all source flux density measure-ments having identical rms errors, then the plot of x 2 versus log 1 0(OddsRatio) reduces to a straight line. This is depicted in the upper graph of figure 3.2. Another interesting limiting case to consider is one with flat prior probabilities, but this time maintaining the true source rms flux density errors (given by equation 4.1). This is depicted in the lower graph of figure 3.2. In this case, for a given fixed value of x2i there is again a spread in odds ratios. Those sources having a larger AS (and also larger a) have higher odds ratios than do sources with the same x 2 but a smaller AS (and smaller a). An understanding of this effect can be obtained by considering the following argument. The value of x2 is only a measure of the misfit of the data to the constant model. The Bayesian odds ratio considers the relative fit of the variable versus the constant model and automatically includes an Ockham's factor. In addition to being dependent upon the relative number of parameters in the two competing models, the Ockham's factor is also dependent upon the amount of prior model parameter space that gets ruled out by the data. The variable model has one additional parameter and for a given x 2) less parameter space gets ruled out by the data for larger values of a. Finally, in figure 3.3 is plotted x 2 versus log 1 0(OddsRatio) for an identical number of simulated "constant" sources. These sources were generated by adding random Gaussian noise to constant sources having the same number density distribution as the sources in the upper graph. For only a handful of these simulated constant sources were variability odds ratios greater than 10:1 calculated, and for all sources the variability odds ratios Chapter 3. Bayesian Theory 46 were less than 100:1. The second problem addressed by Bayesian inference was the parameter estimation problem, or more specifically, the determination of source flux densities and positions from the daily scan data. For this problem, the principle advantage of the Bayesian approach was the ability to marginalize over unwanted parameters. In many of the parameter fits attempted, the number of data points only slightly exceeded the number of unknown parameters (cf. section 4.2). As a result, the traditional frequentist fitting routines tended to give results that were extremely sensitive to both the noise and the initial guesses for the parameters. By marginalizing, the number of unknown parameters is effectively reduced, resulting in more stable and reliable parameter estimates. L o g l O ( O d d s ) Figure 3.1: x2 versus log 1 0(Variability Odds) for approximately 5300 sources from the GB6 catalog. The vertical line indicates an odds ratio of 105, above which a source is considered to be a long-term variable. Chapter 3. Bayesian Theory 47 Table 3.1: log 1 0 Q(x 2 | 1) for Various Values of x 2 x 2 log 1 0 Q(x 2 1 1) 5.0 -1.6 10.0 -2.8 15.0 -4.0 20.0 -5.1 25.0 -6.2 30.0 -7.4 35.0 -8.5 40.0 -9.6 45.0 -10.7 50.0 -11.8 Chapter 3. Bayesian Theory 48 Figure 3.2: Limiting cases of figure 3.1. Top: Figure 3.1 with fiat prior probabilities and constant rms errors. Bottom: Figure 3.1 with flat prior probabilities and rms flux density errors given by equation 4.1 Chapter 3. Bayesian Theory 49 Figure 3.3: x2 vs. log 1 0 Variability Odds for approximately 5300 simulated "constant" sources. Note the difference in scale between this graph and figures 3.1 and 3.2. Chapter 4 Development of the Variable Source Catalogues 4.1 Development of the Long-Term Variable Source Catalogue The first step towards uncovering long-term variables in the Greenbank data was to use the GB6 catalogue as a template, and look for corresponding sources in the GB86 and GB87 source catalogues. Any source from the single epoch sources catalogues located within f» 70" of a GB6 source position was considered to be "corresponding" to that GB6 source. Unfortunately however, in many cases there is no source analogous to a GB6 source in either or both of the single epoch source lists. There are several possible reasons for this. The most obvious reason is the y/2 higher, low source flux density cut-off of the sin-gle epoch catalogues. For the GB87 source catalogue the limiting source flux density is approximately 25mJy (the 5a level), while for the GB86 catalogue source flux densi-ties down to ~ 20 mJy (4a) are available. Since the GB6 catalogue lowest source flux density limit ( « 18mJy) is lower than both the GB86 4a limit, and the GB87 5a flux density limit, many low strength GB6 sources are simply too weak to be included in the single epoch source lists. For some variable sources, this problem may even be exacer-bated. A source with strength well above the flux density limits in one epoch could quite conceivably vary, and fall well below them in the other epoch. In addition, a source may be located in.a region of bad or absent data in one of the 50 Chapter 4. Development of the Variable Source Catalogues 51 two epochs. In the latter case (absent data), obviously no estimates of the flux density can be ascertained. In the former case, the noise level may be too high to obtain stable fits and accurate flux density estimates for low strength sources. In any event, unless the source is located in a region of absent data in one of the epochs, it is important to obtain some estimate of the source flux density in both epochs in order to estimate a source's long-term variability. To that end, a list was made for both epochs of all the sources found in the GB6 catalogue, but without a corresponding source in that epoch's source catalogue. The positions of these sources were superimposed on the appropriate single epoch image, and the sources that fell on bad data areas were flagged. No long-term variability information can be deduced for these sources. The remainder of the sources on these lists were fit to elliptical Gaussians to try to establish an estimate of the sources' flux density. Unfortunately, almost all of these fits were to low strength sources, with a very low signal to noise ratio. Hence, satisfactory fits could not always be obtained. Indeed, in the instance of a variable source whose flux density was very low in one epoch, any attempted fit will simply be a fit to the noise. For a fit to be deemed "satisfactory" it had to pass several criteria. Firstly, the centre of the fit and the position of the source from the GB6 catalogue were required to coincide to within three pixels distance (approximately 2 arcmin). In addition, the same conditions used to accept/reject fits in the GB86, GB87, and GB6 catalogues were applied to these fits. These conditions guarantee a reasonable value for the fitted H P B W , and reasonable agreement between the peak and fitted flux density values. The "fitted flux density" corresponds to the amplitude of the Gaussian fit to the source. The "peak flux density" of a source corresponds to the source's maximum image flux density value. It is only intended as crude estimate of a source's flux density when a Chapter 4. Development of the Variable Source Catalogues 52 Gaussian fit to a source was unsatisfactory. It was determined by taking the maximum pixel value of the nine pixels centred on the source's GB6 position, and then subtracting the background sky level. Using the nine pixels around the GB6 position allows for possible differences in source position between the source on the GB6 image and the single-epoch image. When the peak flux density was negative, it was set to zero. Using the peak flux density instead of the fitted flux density slightly overestimates the flux density for faint sources. This is because the peak position corresponds to the largest noise peak around the source position. For the purposes of studying source variability, this will result in an underestimate of a source's variability. The background sky level used to calculate the peak flux density was determined by taking the mean pixel value in an annulus around the source. Originally the sky value had been determined from the background sky level calculated as a parameter in the elliptical Gaussian source fit. However, when the fits were unstable anomalous sky values were often produced, and this in turn produced anomalous peak flux densities. The rms uncertainty as in the source flux densities is given by the sum in quadrature of two terms [Gregory & Condon 1991]: where a « 5 m Jy is the uncertainty due to the effects of noise plus confusion (cf. Chapter 2), and 7 ~ 0.047 (cos S)1^2 is an intensity proportional error coefficient. 7 was determined by comparing the flux densities of 81 strong (> 500 mJy) sources from the two single epoch catalogues. It is conservative in that it assumed none of the sources was intrinsically variable over the time period in question. The dependence of 7 on the declination 5 is the result of the (cos<5)-1 sampling density variation in the G B surveys. At higher declinations the greater sampling density "averages out" the intensity proportional error. Armed with both single epoch flux densities plus error estimates, equation 3.19 was [o2 + (75) 211/2 (4.1) as ~ Chapter 4. Development of the Variable Source Catalogues 53 used to calculate the odds of variability for all GB6 sources that did not fall on bad data. Given the odds, the probability of a source being variable can be calculated from equation 3.12. Those sources having an odds ratio greater than 105 were sorted according to right ascension, and are listed in the next section. An odds ratio cutoff of 105 was chosen primarily from thesis time constraints. Eventually, there are plans to reduce this cutoff to 104 or lower, increasing the number of variables by approximately a factor of ten. Sample contour plots of one of the new variables found in this list is shown in figure 4.1. 4.2 Development of the Short-Term Variable Source Catalogue As well as being of intrinsic interest in themselves, the long-term radio variables dis-covered in this thesis are prime candidates to be short-term variables. In fact, in many cases their long-term variability may have been the result of a short-term "flare event" in one of the two data epochs. Therefore, many of these variables were subjected to closer scrutiny by using the individual telescope beam data (pseudo (u,v) scan data) to obtain their daily flux densities. The first step was to go back to GB6 radio sky images and individually inspect each of the sources in the long-term variable list. Only those sources that are fairly isolated in the sky were analyzed for short-term variability. In this context, "fairly isolated" means that no other source of any significant intensity is found within a rectangular region centred on the target source, approximately 16' (26 pixels) wide in the E / W direction, and 40' (60 pixels) long in the N / S direction. This window defines the region from which the raw pseudo (u,v) data were extracted for each source. Since all fitting routines assume only the presence of a single source in the data, any other sources in the region will result in anomalous estimates of the background level and/or erroneous estimates of the Chapter 4. Development of the Variable Source Catalogues 54 Figure 4.1: Contour plots of the long-term variable with centre: R A = 3 h23m32.7 s, S = 28°32'24". Upper plot is taken from the 1986 epoch images, lower plot from the 1987 images. Each plot covers an identical 16| x 16 | arcmin patch on the sky. Lowest contour on both plots is -5mJy. Contour interval is lOmJy. Chapter 4. Development of the Variable Source Catalogues 55 target source flux density and position. The size of the window is a compromise between two competing concerns. A small data window allows close neighbouring sources to be excluded from the data, making more of the long-term variables available for the short-term variability study. However, the window also has to be large enough to ensure an adequate amount of data is acquired for each source. In the N /S direction the major factor dictating the data window size is the determi-nation of accurate baselevels. Although a running median baseline had been subtracted from all the scans, this may have left "bowls" around the stronger sources (cf. section 2.3). Hence, for each scan a more accurate zero baselevel was established by subtracting a least-squares linear fit from the scan data. The data used for this fit included all points north or south of a 16' strip in the centre of the window. In addition, to prevent any large noise spikes from unduly affecting the fit, any data point with flux density > 50 mJy (approximately 5cr) was excluded from the baselevel calculation. In the E / W direction the window size is dictated by the second Gaussian fitting (the E / W beam profile fit). Because the beams are spaced one H P B W apart, that meant only two, or at most three, beam tracks pass over the source. It is therefore important to have data from at least two other beam tracks, one on each side of the source, to help constrain the E / W fit. Therefore, a data window of at least 5 H P B W (approximately 26 pixels) wide was used. The next step in the process was to fit each of the data tracks to a Gaussian profile. In theory, equation 3.27 can be used to determine the source declination, and then equation 3.29 utilized to obtain amplitude estimates for each track. However, several complications arise which prevent a straightforward application of these equations. The first complication is the inadequate knowledge of the shape of each of the seven different beams. To a good first approximation, each beam can be modelled by an ellipti-cal Gaussian of unknown H P B W . However, while an estimate of the average beamwidth Chapter 4. Development of the Variable Source Catalogues 56 in the N /S direction (as a function of declination) can be estimated from the PSF, fig-ure 2.4 or figure 2.5, this is not accurate enough to obtain reliable fits. An accurate determination of the H P B W for each of the individual beams is required. To that end, 117 strong, isolated sources were selected at a variety of different de-clinations. After extracting the scan data around these sources and subtracting off the baselevel, those data tracks having a peak of > 100 mJy were binned according to the beam which generated the track, and declination. The amplitude for each track as a function of the H P B W was then estimated using equation 3.29. The HPBWs which resulted in the best amplitude fits (the fits having the narrowest probability distribution for the amplitude) are shown in figures 4.2. and 4.3. The second complication is the presence of a random N /S pointing error in the tele-scope which varied from scan to scan. This pointing error is depicted in the upper histogram plotted in figure 4.4. This histogram was generated from the data tracks sur-rounding nine strong sources with well-known positions from a list of V L A calibrators. Represented on the abscissa of this histogram is the difference between the peak of the P(XQ\DJ, M) 'S (the best fitted declination estimates from individual data tracks), and the known source declinations. Only data tracks with maximum amplitudes greater than 5x the noise level are represented on this histogram. If pointing errors are small, this his-togram should approximate a Gaussian with centre zero and a standard deviation of 2", the average standard deviation of the p(X0\Dj, M) probability distributions for strong sources. However, while a Gaussian fit to the histogram does indeed indicate no net systematic offset, the standard deviation of the histogram indicates a significant random declination pointing error, as ~ 20" » 2". This random error in the single scan declina-tion estimates is not much larger than the declination error for strong sources from the single epoch catalogs (between 6" and 20", depending on the source declination), even though the latter are based upon data from many scans. Chapter 4. Development of the Variable Source Catalogues 57 Beaa 1 -2 4 -i—|—I—I—I—|—I—I—i—|—I—I—I—|—I—r 21—I I I I I I I i i I 1 I I I ' ' • 0 20 40 60 80 Declination (degrees) Beai 2 3r-- i — | — i — i — i — | — i — i — i — | — i — i — i — | — r 21 i I i i i I i i i I i i i L 20 40 60 Declination Idegreesl 80 Bean 3 4 S 3 k i—|—i—i—i—|—I—I—r 21—i—I—i—i—i I i i i I i i ' I I I 0 20 40 60 Declination (degrees) Beai 4 5 1 — i — | — i — i — i — | — i — i — i — | — i — r J—i i I i i i I i i i_ 20 40 60 Declination (degrees) Figure 4.2: HPBW's in the N /S direction for beams 1 through 4 as a function of decli-nation. Chapter 4. Development of the Variable Source Catalogues 58 Figure 4.3: HPBW's in the N /S direction for beams 5 through 7 as a function of decli-nation. Chapter 4. Development of the Variable Source Catalogues 59 30 20 -10 -~i—r 1 1 I 0 L J — i — I — l _ i — L — H 1 1 I 1 1 li ii 1 1 I 1 1 li il i I I l>-l— i i I i i - 2 - 1 0 1 2 N/S P o s i t i o n a l E r r o r (arcmin) 20 15 10 -5 -- i — i — i — i — | — i — i — i — i — r T i i r -| 1 1 1 r -1 0 1 E/W P o s i t i o n a l E r r o r (arcmin) Figure 4.4: Histograms illustrating the random telescope positional errors. The solid curves represent the best Gaussian fits to the data. Top: The N / S positional error. For the best Gaussian fit to the histogram, as = 20". Bottom: The E / W positional error. For the best Gaussian fit to the histogram, CTRA = 19". Chapter 4. Development of the Variable Source Catalogues 60 As a result of these pointing errors equation 3.27 cannot be used to determine the source declinations. Instead the declination was determined by taking a weighted average of the p(X0\Dj, M)'s , calculated from equation 3.26. The error estimates in these dec-lination probabilities were modified to include the effects of the random pointing offset, i.e.: erf = a 2 + (20")2. In order to exclude most of the fits to data tracks containing only noise, only those tracks containing a data point of amplitude greater than 3x the noise a were included in the average. After estimating the source declination, the peak of each data track was determined via equation 3.29. Again though, because of the random pointing offset, this equation has to be modified by removing the term coupling any one data track to all the other data tracks. Once the amplitudes of all the tracks in a scan were determined, these were used, together with the RA's of the data tracks, as data points for a Gaussian fitting in the E / W direction. This fitting determines the R A of the source, as well as the source intensity at the time of the telescope beams' passage over the source. Unfortunately however, the same complications present in the N / S fitting arise in the E / W fitting. Following a procedure similar to the one done for the N /S fittings, a histogram was generated of the error in the E / W position determinations (in arcmin) for the same nine V L A calibrators. This histogram is depicted in the bottom of figure 4.4. Although there appears again to be no systematic offset, the standard deviation of the histogram, 19", is again greater than the average error estimate determined from the fitting alone, 11". As a result of this, an additional [(19)2 — ( l l ) 2 ] 1 / 2 = 16 arcseconds was added in quadrature to all E / W position estimates. In addition, equations 3.27 and 3.29 were modified, as before, to uncouple the positional dependence between different scans. Because the number of data points in the E / W direction is very small, it is essential to have a more accurate determination of the average E / W beamwidth. Knowing only Chapter 4. Development of the Variable Source Catalogues 61 the average E / W beamwidth of all seven beams is sufficient, since each data point in the E / W direction was generated from a different beam. The average E / W H P B W was determined using 18 constant strength calibrators from the V L A calibrator list. Daily flux densities for these calibrators were determined via equation 3.29 for different values of the E / W H P B W (neglecting the coupling probabil-ities again). The H P B W which resulted in the best daily flux density estimates for the calibrators is 2.54' + f(5), where f(5) is a declination dependence estimated from the E / W , PSF declination dependence (cf. figure 2.4 or figure 2.5). Convolution of the scan data with a restoring function resulted in the E / W F W H M for the image PSF, seen in figures 2.4 and 2.5, being much broader than the E / W H P B W . Averaging over declina-tion, the E / W H P B W is 2.6', which is close to the approximate value of 2.8' quoted in [Condon et al. 1989]. Verification of the daily flux densities determined with this value of the H P B W was obtained by plotting a histogram of the difference between the known calibrator flux density and the fitted daily flux densities, all scaled by the expected rms error in this difference. If there are no additional systematic effects, and the flux density error estimates are reasonable, this histogram should approximate a Gaussian with zero mean and unity rms width. Figure 4.5 shows that this is indeed the case. With the estimate for the E / W H P B W , together with the estimate of the N /S H P B W , and equation 3.29, daily flux densities were determined for all the fairly isolated sources in the long-term variable list. In addition, the right ascension and declinations were determined using equation 3.27. In general, these positions are slightly more accurate than the positions quoted in either single-epoch catalogue, but less accurate than the GB6 positions. However, they were computed before the GB6 catalogue was available, and are included in this thesis for completeness. To determine the probability of the long-term variables also being short-term vari-ables, ideally a Bayesian equation similar to equation 3.19 can be derived. However, Chapter 4. Development of the Variable Source Catalogues 62 T 1 1 1 1 1 1 1 1 1 1 1 r Del taS/S igma Figure 4.5: Histogram of the errors in the daily flux density determinations for 18 V L A calibrators. The solid curve represents a Gaussian with a mean of zero and unity rms width. Chapter 4. Development of the Variable Source Catalogues 63 time constraints made this unfeasible. Therefore, in this case, a frequentist approach was adopted, and a least-squares fit to the weighted mean of each epoch was evaluated by calculation of x2'-N r x 2 = E Si- [i 2 (4.2) where TV is the number of daily flux density measurements of the epoch in question, /j, is the weighted mean of those measurements, and Si, Oi are that epoch's daily flux densities and rms errors respectively. The goodness-of-fit of the data to this constant model was estimated by determining the probability that the observed chi-square for a correct model should exceed x2 by chance, Q(x2 \ v)- This probability is a function of v, the number of degrees of freedom. Since the fit is to a constant model, v — TV — 1. A value of Q < 10~3, indicates that the data are not consistent with a constant model, implying short-term variability. The coordinates for all the isolated long-term variables, together with the daily flux densities, and log 1 0 Q for each epoch (with more than one flux density measurement), are listed in order of increasing right ascension in table 5.4. Of the 146 long-term variables, 104 are included in this table. For 43 of these variables, log 1 0 Q for one or both epochs was < —3. Chapter 5 The Variable Source Catalogues 5.1 The Long-Term Variable Source Catalogue In table 5.1 is a list of all 146 sources from the GB6 catalogue with odds > 105 favouring the variable source model over the constant source model. The first two columns list the GB6 catalogue right ascension and declination in B1950 equatorial coordinates, together with the rms uncertainties. In the next two columns are listed the source flux densities and rms uncertainties from the GB86 and GB87 catalogues. In the fifth column is listed the log 1 0(Odds) for the variable source model over the constant source model. For comparison with a frequentist approach, in column six is listed the absolute value of the flux density difference between the two epochs (AS) , scaled by the expected rms uncertainty in that difference, (a = [c| 8 6+<J| 8 7] 2). When looking at this column, it must be remembered that for strong sources, the uncertainty in the flux density is proportional to the source strength (cf. equation 4.1). Hence, the largest possible value for AS ja is limited to 7 - 1 , where 7 is the proportionality constant. A much better "feel" for the degree of variability is obtained by looking at the odds ratio. Finally, in the last column are listed any additional notes: C—Source is found in a confused region and/or a source of comparable or greater flux density is nearby. As a consequence, no short-term variability information was determined for this source L = Source has been identified with one of the long-term variables uncovered during 64 Chapter 5. The Variable Source Catalogues 65 the Gregory/Taylor 6cm survey of the Galactic plane [Gregory & Taylor 1981], [Gregory & Taylor 1986] S= Source has been identified with one of the short-term variables uncovered during the Gregory/Taylor 6 cm survey of the Galactic plane [Gregory & Taylor 1981], [Gregory & Taylor 1986] B = Source has been identified with a source from the 20 cm, high resolution, V L A FIRST survey [Becker et al. 1994] F = Source has been identified with a highly variable source from a survey of faint sources at 2.7 and 4.75 Ghz [Forkert & Altschuler 1987] N= Source has been identified with a source from the high resolution, 1.4 GHz, N R A O V L A sky survey (NVSS) [Condon et al. 1994b] V = Source has been identified with a V L A calibrator <S= Source also exhibits short-term variability (log 1 0 Q < —3 for either or both epochs). There are several notable absences from this table. For example, neither Cygnus X - l , Cygnus X-3, GT2318+620, GT0236+610, or the Galactic superluminal source GRS1915+105 are present. However, all of these sources have very low quiescent flux density levels, and are located in regions of considerable confusion. None of these sources appeared in any of these source catalogues. SS433, however, did appear in both single epoch catalogues, although its flux density measurements (519 ± 25mjy in 1986 and 544 ± 26mJy in 1987) did not indicate source variability. For 15 of the long-term variables, identifications were made with V L A calibrators. These calibrators have all been previously well-studied. Chapter 5. The Variable Source Catalogues 66 Long-term variables from table 5.1 having counterparts in either the NVSS or FIRST survey, are listed in table 5.2. The first two columns of this table list the GB6 catalogue right ascension and declinations in B1950 coordinates, together with the rms uncertain-ties. The next two columns list the B1950 coordinates for the counterpart sources from the NVSS/FIRST survey, with rms uncertainties (for sources from the FIRST survey, the error in both coordinates represents the radius, in arcseconds, of a 90% confidence region). The final column lists the NVSS/FIRST 20cm source flux densities and rms uncertainties. For five of the long-term variables, optical identifications were located by search-ing for coincidences between the GB6 positions, and positions of optically identified sources from the Rys/Machalski survey [Rys & Machalski 1990] or the Einstein Slew Survey [Elvis et al. 1992]. One of the variables identified is a Galactic source, while the other four variables are either extragalactic, or probably extragalactic. A l l five variables are listed in table 5.3. The first two columns of this table list the GB6 catalogue right ascension and declinations in B1950 coordinates, together with the rms uncertainties. The next two columns list the corresponding Galactic coordinates. The final column gives the optical identification. Finally, figure 5.1 shows the distribution of all the long-term variables in Galac-tic coordinates. Since some of the variables are expected to be of Galactic origin, a higher concentration of variables along the Galactic plane should result. However, this is counteracted by the confusion in the Galactic plane due to HII regions, resulting in the relatively random distribution seen in figure 5.1. The paucity of variables along the right-hand edge of the distribution is not thought to be physically significant. Chapter 5. The Variable Source Catalogues Table 5.1: A List of Long Term Variables 67 Right Ascension Declination S86 ( mJy) S87 ( mJy) log 0 2 , i AS/a Notes 0 0 44.4 ± 0.6 24 42 58 ± 9 140 ± 8 59 ± 6 11.18 8.04 S 0 2 30.5 ± 0.7 54 11 54 ± 7 184 ± 8 259 ± 10 5.64 5.74 0 5 55.5 ± 0.8 68 20 40 ± 5 270 ± 9 406 ± 12 15.46 8.94 C 0 7 56.6 ± 0.5 10 42 4 ± 9 694 ± 33 285 ± 15 24.79 11.35 N V 0 8 48.7 ± 0.9 70 28 48 ± 5 789 ± 22 624 ± 18 6.21 5.87 0 11 21.1 ± 0.5 18 53 56 ± 8 160 ± 9 272 ± 14 8.07 6.86 N 0 11 23.6 ± 0.7 51 18 37 ± 8 50 ± 5 118 ± 6 12.69 8.47 0 16 53.7 ± 0.9 73 10 48 ± 5 1377 ± 35 1712 ± 44 6.54 5.98 <SV 0 34 21.8 ± 0.6 18 15 32 ± 9 52 ± 6 147 ± 9 14.61 9.01 «SN 0 44 6.1 ± 0.7 56 41 28 ± 6 208 ± 8 322 ± 12 11.01 7.72 S 0 45 4.0 ± 0.5 24 18 42 ± 8 373 ± 18 250 ± 12 5.50 5.74 0 45 27.5 ± 1.1 68 10 8 ± 7 49 ± 4 102 ± 5 9.57 7.51 s 1 25 6.1 ± 0.6 48 45 41 ± 7 594 ± 23 288 ± 12 26.96 11.74 s 1 38 37.3 ± 0.6 29 27 24 ± 8 209 ± 11 121 ± 7 8.04 6.87 c 1 43 38.7 ± 1.0 65 21 8 ± 7 70 ± 5 118 ± 6 7.09 6.52 1 45 59.0 ± 0.6 1 14 4 ±10 83 ± 9 171 ± 11 5.61 6.10 N 2 14 47.1 ± 0.8 54 9 18 ± 8 66 ± 5 124 ± 6 8.50 7.10 N 2 24 41.3 ± 0.8 67 7 33 ± 5 1120 ± 33 1767 ± 52 21.78 10.49 sv 2 34 55.6 ± 0.5 28 35 16 ± 8 2250 ± 99 3376 ±149 6.94 6.29 V 2 44 22.8 ± 0.8 19 58 36 ±12 35 ± 6 88 ± 7 5.11 5.99 N 2 48 12.9 ± 0.7 56 4 2 ± 6 352 ± 13 252 ± 10 6.55 6.09 SN 2 51 32.1 ± 0.6 39 19 29 ± 8 262 ± 12 408 ± 18 8.42 6.88 2 52 1.0 ± 0.8 63 39 1 ± 6 212 ± 8 141 ± 6 9.09 7.08 2 56 5.7 ± 0.8 51 26 30 ± 8 119 ± 6 61 ± 5 8.38 7.08 «SN 2 56 12.7 ± 0.5 5 28 57 ± 9 139 ± 10 241 ± 13 6.27 6.22 S N 2 58 45.8 ± 0.7 53 20 5 ± 7 220 ± 9 358 ± 14 12.96 8.33 2 59 49.1 ± 0.8 68 10 4 ± 5 942 ± 27 1491 ± 43 23.03 10.77 S 3 7 37.3 ± 0.6 38 3 31 ± 8 356 ± 16 760 ± 32 24.78 11.31 s 3 14 5.5 ± 0.7 56 33 6 ± 7 277 ± 11 197 ± 8 6.22 5.97 L N 3 20 29.8 ± 0.7 46 3 43 ± 8 72 ± 5 133 ± 7 7.71 6.81 S 3 23 32.7 ± 0.6 28 32 24 ± 9 192 ± 10 41 ± 5 35.41 13.40 s 3 23 43.2 ± 0.5 15 11 6 ± 9 332 ± 16 191 ± 10 9.13 7.24 C N 3 29 13.9 ± 1.8 54 25 8 ±17 50 ± 5 4 ± 5 7.22 6.97 s 3 33 37.5 ± 0.8 59 38 16 ± 7 171 ± 7 118 ± 6 5.46 5.70 N 3 54 17.1 ± 0.6 41 40 4 ± 8 71 ± 6 131 ±. 7 7.00 6.56 S N 4 8 52.3 ± 0.6 8 35 38 ± 9 56 ± 7 198 ± 11 21.38 10.65 <SN Chapter 5. The Variable Source Catalogues 68 Right Ascension Declination S86 ( mJy) S87 ( mJy) log 0 2 , i AS/o Notes 4 22 12.3 ± 0 . 5 0 29 15 ± 9 712 ± 34 1118 ± 53 7.10 6.41 V 4 37 30.9 ± 0 . 5 14 32 24 ± 9 175 ± 10 347 ± 17 13.82 8.73 4 42 19.5 ± 0 . 5 7 10 13 ± 9 505 ± 25 271 ± 14 12.12 8.22 S 4 44 42.5 ± 0 . 8 63 26 58 ± 6 399 ± 13 606 ± 20 14.82 8.78 N 4 54 26.2 ± 0 . 5 6 40 41 ± 9 849 ± 40 435 ± 21 15.09 9.06 C V 4 59 7.2 ± 0 . 6 38 45 31 ± 8 172 ± 9 289 ± 13 10.12 7.53 SN 5 42 53.8 ± 0 . 6 40 42 53 ± 8 256 ± 12 176 ± 9 5.10 5.54 C N 5 48 33.1 ± 1.0 68 57 27 ± 6 79 ± 5 136 ± 6 9.87 7.47 N 6 10 4.2 ± 0 . 9 61 40 17 ± 7 58 ± 5 102 ± 6 5.72 6.03 N 6 42 37.0 ± 0 . 5 5 34 45 ± 9 277 ± 15 452 ± 22 7.25 6.52 C 6 48 46.5 ± 1 . 5 47 1 40 ±16 63 ± 5 4 ± 5 11.72 8.37 c 7 16 5.1 ± 0 . 6 33 12 43 ± 8 247 ± 12 364 ± 16 5.67 5.79 N F 7 26 24.3 ± 0 . 9 67 25 2 ± 6 292 ± 9 171 ± 7 21.02 10.38 cSN 7 33 16.7 ± 0 . 6 7 41 29 ± 9 105 ± 8 192 ± 11 6.26 6.27 C 8 29 24.7 ± 0 . 5 18 42 16 ± 8 511 ± 24 795 ± 37 7.24 6.46 C V 8 40 11.9 ± 0 . 6 42 26 25 ± 7 412 ± 17 583 ± 24 5.72 5.76 N 8 44 8.6 ± 0 . 6 46 20 36 ± 7 188 ± 9 269 ± 12 5.31 5.62 c 8 51 53.8 ± 1.0 71 57 58 ± 5 144 ± 6 208 ± 7 8.97 7.01 «SN 8 59 22.1 ± 0 . 8 68 9 8 ± 5 577 ± 17 751 ± 22 7.11 6.24 C N V 9 12 23.8 ± 1.2 73 55 34 ± 6 109 ± 5 161 ± 6 7.33 6.46 N 9 24 52.3 ± 1.1 73 17 12 ± 5 105 ± 5 206 ± 7 26.63 11.65 C N 9 25 8.6 ± 0 . 6 44 59 4 ± 7 163 ± 8 256 ± 11 8.09 6.78 N 9 32 33.1 ± 0 . 5 9 28 38 ± 9 348 ± 17 213 ± 12 6.91 6.40 9 54 13.0 ± 0 . 7 55 37 17 ± 6 1704 ± 60 2270 ± 80 5.58 5.63 <SNV 9 54 57.6 ± 0 . 8 65 48 26 ± 5 908 ± 28 1417 ± 43 19.54 9.97 sv 10 5 37.6 ± 0 . 6 0 44 48 ±10 149 ± 11 253 ± 14 5.32 5.81 c 10 27 14.6 ± 1.1 74 57 22 ± 5 273 ± 8 202 ± 7 8.56 6.80 s 10 34 53.9 ± 0 . 6 33 14 12 ± 9 63 ± 6 138 ± 8 10.42 7.79 10 48 20.6 ± 0 . 6 20 43 27 ± 9 95 ± 7 179 ± 10 8.20 7.00 11 3 24.3 ± 0 . 5 28 28 59 ± 8 537 ± 24 274 ± 13 17.08 9.56 s 11 3 47.6 ± 0 . 5 17 29 44 ± 9 123 ± 8 207 ± 11 6.18 6.15 s 11 24 50.6 ± 0 . 7 57 6 44 ± 6 370 ± 14 597 ± 21 15.52 9.02 11 25 22.6 ± 0 . 7 59 41 36 ± 6 270 ± 10 393 ± 14 9.50 7.18 11 36 41.0 ± 0 . 8 53 17 35 ± 8 75 ± 5 123 ± 6 5.27 5.76 c 11 43 43.9 ± 0 . 7 59 5 9 ± 6 514 ± 18 674 ± 23 5.24 5.49 11 50 43.4 ± 0 . 7 25 25 44 ±10 91 ± 7 38 ± 5 5.71 6.22 s 12 17 30.6 ± 0 . 7 1 43 38 ±13 123 ± 10 11 ± 8 13.87 8.94 s 12 26 48.8 ± 0 . 8 55 39 16 ± 7 68 ± 5 127 ± 6 8.92 7.23 s 12 58 2.2 ± 0 . 7 44 8 50 ± 9 62 ± 5 129 ± 7 9.81 7.58 Chapter 5. The Variable Source Catalogues Right Ascension Declination S86 ( mJy) S87 ( mJy) log 0 2 ) i AS/a Notes 12 58 12.4 ± 0 . 7 14 32 59 ±11 59 ± 6 154 ± 9 12.78 8.51 C 13 4 32.5 ± 0 . 6 6 43 55 ±10 151 ± 10 263 ± 14 7.06 6.52 S 13 8 8.4 ± 0 . 5 32 36 47 ± 8 1892 ± 82 1131 ± 49 11.66 7.98 c v 13 8 38.0 ± 0 . 5 32 49 35 ± 8 1085 ± 47 688 ± 30 9.07 7.11 c 13 24 28.4 ± 0 . 6 44 50 41 ± 8 105 ± 6 162 ± 8 5.18 5.66 s 13 24 37.3 ± 0 . 5 22 26 8 ± 8 407 ± 19 849 ± 39 19.77 10.22 13 30 47.1 ± 0 . 5 27 40 47 ± 8 464 ± 21 161 ± 9 34.65 13.24 13 43 18.8 ± 0 . 5 7 21 22 ± 9 307 ± 16 186 ± 11 6.52 6.26 14 17 42.3 ± 0 . 6 38 35 45 ± 8 381 ± 17 871 ± 37 29.22 12.21 14 32 8.9 ± 0 . 6 42 16 19 ± 7 177 ± 9 353 ± 15 19.56 10.15 14 35 50.1 ± 0 . 6 30 14 57 ± 9 73 ± 6 132 ± 8 5.70 6.04 B 14 38 1.8 ± 0 . 5 5 44 17 ± 9 249 ± 14 142 ± 10 6.81 6.44 15 9 58.3 ± 1.2 10 35 41 ±20 9 ± 6 75 ± 7 7.42 7.01 15 53 21.3 ± 0.5 11 20 8 ± 8 368 ± 18 636 ± 30 10.28 7.60 c 16 0 6.0 ± 0 . 6 24 18 13 ± 9 70 ± 6 136 ± 8 6.80 6.53 c 16 0 7.0 ± 0 . 5 24 26 48 ± 8 165 ± 9 253 ± 12 5.37 5.72 c 16 1 20.5 ± 0 . 5 11 13 58 ± 8 363 ± 18 831 ± 39 22.52 10.86 s 16 15 22.4 ± 0 . 7 36 34 50 ±10 1 ± 5 71 ± 6 14.52 9.18 c 16 24 2.2 ± 0 . 7 41 25 19 ± 8 73 ± 6 137 ± 7 7.80 6.86 c 16 28 57.3 ± 0 . 6 10 58 26 ± 9 126 ± 9 209 ± 11 5.28 5.77 16 55 34.0 ± 0 . 7 53 26 27 ± 7 105 ± 6 174 ± 8 8.88 7.12 16 56 39.5 ± 0 . 6 47 42 19 ± 7 1025 ± 40 1420 ± 55 5.95 5.82 c v 17 0 4.0 ± 0 . 5 8 1 18 ± 9 254 ± 14 117 ± 9 12.91 8.51 17 0 19.4 ± 0 . 8 68 34 22 ± 5 341 ± 11 435 ± 13 5.46 5.57 s 17 1 1.4 ± 0 . 8 65 58 23 ± 6 202 ± 7 269 ± 9 5.51 5.63 17 1 37.4 ± 0 . 6 50 21 42 ± 7 187 ± 8 274 ± 11 6.69 6.20 17 10 38.9 ± 0 . 6 6 3 44 ±11 137 ± 9 64 ± 7 5.41 6.05 17 19 23.5 ± 0 . 6 35 45 10 ± 8 511 ± 22 874 ± 37 12.84 8.35 c v 17 22 6.8 ± 0 . 7 61 8 48 ± 6 151 ± 6 321 ± 11 33.37 12.95 s 17 24 24.5 ± 1.0 32 21 12 ±14 0 ± 5 57 ± 6 8.79 7.60 c 17 39 18.1 ± 0 . 6 43 49 52 ± 7 146 ± 8 234 ± 10 8.18 6.84 17 39 28.6 ± 0 . 6 52 13 14 ± 7 2316 ± 85 1133 ± 42 30.62 12.44 sv 17 45 57.3 ± 0 . 8 67 4 53 ± 5 248 ± 8 321 ± 10 5.20 5.48 17 47 29.6 ± 0 . 6 43 22 42 ± 7 241 ± 11 367 ± 15 7.91 6.68 17 54 58.9 ± 0 . 8 45 23 47 ±10 88 ± 6 34 ± 5 8.15 7.11 17 57 18.2 ± 0 . 6 28 47 58 ± 9 72 ± 6 169 ± 9 14.43 8.94 18 4 41.2 ± 0 . 6 11 59 27 ± 9 93 ± 7 167 ± 10 5.60 5.99 18 6 23.6 ± 0 . 6 39 4 32 ± 8 107 ± 7 169 ± 9 5.46 5.80 c 18 7 30.4 ± 0 . 6 1 36 10 ±11 175 ± 11 84 ± 9 6.15 6.32 s Chapter 5. The Variable Source Catalogues Right Ascension Declination S86 ( mJy) S87 ( mJy) log 0 2 , i AS/a Notes 18 7 48.5 ± 0 . 6 29 9 35 ± 9 146 ± 8 80 ± 6 6.62 6.41 C 18 19 42.1 ± 0 . 6 39 41 18 ± 7 740 ± 31 1216 ± 50 11.99 8.06 c 18 30 7.0 ± 0 . 5 1 15 55 ± 9 529 ± 26 335 ± 18 6.37 6.15 c 18 36 31.6 ± 0 . 5 9 24 48 ± 9 308 ± 16 175 ± 10 8.52 7.05 18 37 59.5 ± 0 . 6 24 54 48 ± 9 67 ± 6 156 ± 9 12.45 8.40 18 49 13.7 ± 0 . 5 0 31 36 ± 9 1004 ± 48 632 ± 31 7.38 6.53 c 18 51 10.3 ± 0 . 6 48 52 4 ± 7 219 ± 10 351 ± 14 10.93 7.73 s 18 53 3.4 ± 0 . 5 2 47 2 ± 9 137 ± 10 280 ± 15 10.82 7.89 c 18 54 4.6 ± 0 . 5 6 6 14 ± 9 267 ± 14 149 ± 10 7.82 6.83 s 19 8 25.4 ± 0 . 6 48 29 29 ± 7 598 ± 23 423 ± 17 6.47 6.06 c 19 22 15.0 ± 0 . 6 28 32 4 ± 9 140 ± 8 47 ± 5 16.71 9.55 s 19 26 49.2 ± 0 . 7 61 11 17 ± 6 474 ± 16 618 ± 21 5.32 5.52 C V 19 43 58.9 ± 0 . 5 22 52 37 ± 8 152 ± 9 269 ± 13 9.62 7.42 s c 20 51 43.4 ± 0 . 7 52 50 13 ± 7 170 ± 8 106 ± 6 7.42 6.57 s 20 54 33.7 ± 0 . 7 61 10 28 ± 6 307 ± 11 414 ± 14 6.26 5.94- s 20 56 45.3 ± 1.5 33 53 36 ±20 58 ± 6 12 ± 5 5.20 6.15 c 21 0 34.2 ± 0 . 6 46 50 13 ± 7 229 ± 10 137 ± 7 9.96 7.47 s 21 12 12.8 ± 0 . 6 52 15 22 ± 7 254 ± 10 360 ± 14 . 6.47 6.07 c 21 13 38.6 ± 0 . 6 36 33 23 ± 8 71 ± 6 136 ± 8 7.64 6.82 s 21 15 43.5 ± 0 . 7 0 0 37 ±11 219 ± 13 31 ± 8 27.65 12.00 c 21 42 54.5 ± 1.0 4 9 5 ±17 1 ± 7 97 ± 9 11.90 8.49 c 21 50 38.7 ± 0 . 5 12 26 53 ± 9 397 ± 19 264 ± 14 5.17 5.61 22 6 34.4 ± 1.0 61 18 11 ± 8 47 ± 4 92 ± 5 6.38 6.38 s 22 9 53.4 ± 0 . 5 32 53 51 ± 8 630 ± 28 333 ± 15 16.59 9.42 F 22 26 37.0 ± 1.2 9 10 12 ±20 102 ± 8 0 ± 6 17.52 9.96 c 22 30 24.1 ± 0 . 6 26 39 52 ± 9 128 ± 8 71 ± 6 5.11 5.80 22 30 37.5 ± 0 . 7 62 34 8 ± 6 370 ± 13 480 ± 16 5.14 5.44 c 23 6 12.8 ± 0 . 5 9 30 0 ± 9 630 ± 30 383 ± 19 8.42 6.94 c 23 9 18.3 ± 0 . 6 25 48 48 ± 9 154 ± 9 90 ± 7 5.48 5.90 c 23 9 27.9 ± 0 . 6 45 27 33 ± 7 386 ± 16 597 ± 24 9.76 7.32 23 26 33.1 ± 0 . 5 8 17 34 ± 9 188 ± 11 311 ± 16 6.71 6.33 23 31 40.9 ± 0 . 5 7 19 40 ± 8 1702 ± 80 933 ± 44 12.90 8.42 Chapter 5. The Variable Source Catalogues 71 Table 5.2: Sources Identified with NVSS/FIRST Sources GB6 Coordinates NVSS/FIRST Coordinates/Flux Densities Right Ascension Declination Right Ascension Declination 0 7 56.6 ± 0.5 10 42 4 ± 9 0 7 56.63 ±0.02 10 41 45.8 ±0.3 0 11 21.1 ± 0.5 18 53 56 ± 8 0 11 21.39 ±0.02 18 54 0.9 ±0.3 0 34 21.8 ± 0.6 18 15 32 ± 9 0 34 22.17 ±0.02 18 15 33.2 ±0.3 1 45 59.0 ± 0.6 1 14 4 ±10 1 45 59.29 ±0.02 1 14 6.5 ±0.3 2 14 47.1 ± 0.8 54 9 18 ± 8 2 14 47.60 ±0.03 54 9 15.8 ±0.3 2 44 22.8 ± 0.8 19 58 36 ±12 2 44 22.94 ±0.03 19 58 10.2 ±0.4 2 48 12.9 ± 0.7 56 4 2 ± 6 2 48 13.43 ±0.04 56 4 0.4 ±0.3 2 56 5.7 ± 0.8 51 26 30 ± 8 2 56 5.58 ±0.03 51 26 19.2 ±0.3 2 56 12.7 ± 0.5 5 28 57 ± 9 2 56 12.21 ±0.02 5 29 11.0 ±0.3 3 14 5.5 ± 0.7 56 33 6 ± 7 3 14 4.54 ±0.04 56 33 0.7 ±0.3 3 23 43.2 ± 0.5 15 11 6 ± 9 3 23 43.65 ±0.02 15 10 59.6 ±0.3 3 33 37.5 ± 0.8 59 38 16 ± 7 3 33 37.02 ±0.04 59 37 57.5 ±0.3 3 54 17.1 ± 0.6 41 40 4 ± 8 3 54 15.97 ±0.03 41 39 58.1 ±0.4 4 8 52.3 ± 0.6 8 35 38 ± 9 4 8 51.09 ±0.02 8 35 30.0 ±0.3 4 44 42.5 ± 0.8 63 26 58 ± 6 4 44 42.38 ±0.04 63 26 55.2 ±0.3 4 59 7.2 ± 0.6 38 45 31 ± 8 4 59 6.80 ±0.03 38 45 39.0 ±0.3 5 42 53.8 ± 0.6 40 42 53 ± 8 5 42 53.61 ±0.03 40 42 56.3 ±0.3 5 48 33.1 ± 1.0 68 57 27 ± 6 5 48 33.42 ±0.06 68 57 16.2 ±0.3 6 10 4.2 ± 0.9 61 40 17 ± 7 6 10 4.60 ±0.04 61 40 1.6 ±0.3 7 16 5.1 ± 0.6 33 12 43 ± 8 7 16 4.47 ±0.02 33 12 42.8 ±0.3 7 26 24.3 ± 0.9 67 25 2 ± 6 7 26 23.45 ±0.06 67 25 6.4 ±0.3 8 40 11.9 ± 0.6 42 26 25 ± 7 8 40 11.53 ±0.03 42 26 20.0 ±0.3 8 51 53.8 ± 1.0 71 57 58 ± 5 8 51 53.10 ±0.07 71 57 56.1 ±0.3 8 59 22.1 ± 0.8 68 9 8 ± 5 8 59 23.04 ±0.05 68 9 16.3 ±0.3 9 12 23.8 ± 1.2 73 55 34 ± 6 9 12 22.35 ±0.07 73 55 45.9 ±0.3 9 24 52.3 ± 1.1 73 17 12 ± 5 9 24 51.79 ±0.07 73 17 12.0 ±0.3 9 25 8.6 ± 0.6 44 59 4 ± 7 9 25 9.13 ±0.03 44 59 10.0 ±0.3 9 54 13.0 ± 0.7 55 37 17 ± 6 9 54 14.32 ±0.04 55 37 16.3 ±0.3 14 35 50.1 ± 0.6 30 14 57 ± 9 14 35 49.50 ± 0.5 30 15 3.8 ±0.5 S20 (mJy) 100.5 163.2 123.3 90.2 150.0 35.8 198.6 89.6 149.9 273.1 218.1 109.1 51.6 112.9 439.1 60.2 543.1 99.9 101.1 230.5 58.7 1432.4 65.4 563.9 247.7 150.1 174.1 3125.6 63.9 ± ± ± ± ± ± ± ± ± ± 11.9 ± 4.5 2.2 2.5 5.0 19.2 2.8 11.2 4.5 4.5 4.7 2.8 ± ± ± ± ± ± ± ± ± ± ± 29.5 1.4 24.7 10.9 6.6 3.6 ±138.0 ± 3.2 Chapter 5. The Variable Source Catalogues Table 5.3: Variables Identified with Optical Sources Right Ascension Declination l 1 1 b u Optical Identification 0 7 56.6 ± 0 . 5 10 42 4 ± 9 107.0 -50.6 III ZW 2 3 23 32.7 ± 0 . 6 28 32 24 ± 9 159.5 -22.9 U X Ar i 7 16 5.1 ± 0 . 6 33 12 43 ± 8 185.0 19.9 BSO 13 8 8.4 ± 0 . 5 32 36 47 ± 8 85.7 83.3 B L 14 17 42.3 ± 0 . 6 38 35 45 ± 8 69.8 68.4 BSO Notes: BSO = Blue stellar object (most likely a quasar) B L = B L Lac object U X Ar i = RS CVn variable star U X Arietis [Gibson et al. 1975] Chapter 5. The Variable Source Catalogues 73 Figure 5.1: Distribution of long-term variables in Galactic (l 1 1 , b 1 1) coordinates. In these coordinates 0° latitude corresponds to the galactic plane, while 0° longitude, 0° latitude corresponds to the direction of the Galactic centre. Those variables for whom opti-cal identifications have been made are represented by crosses, while all other long-term variables are represented by boxes. The upper and lower bounds of the GB surveys, corresponding to 5 = 75° and S = 0°, are also indicated on this map. Chapter 5. The Variable Source Catalogues 74 5.2 The Short-Term Variable Source Catalogue On the following pages are the 1986 and 1987 daily flux densities measured for sources from the list of long-term variables (table 5.1). For each source in this table, the right ascension and declination are listed in B1950 equatorial coordinates. These were cal-culated using equation 3.27. Below the coordinates are listed the daily flux densities plus error estimates (586> and corresponding times (Ts6,T$7). The flux densities are quoted in mJy, while the times are given as month/day. Finally, for each epoch with more than one flux density measurement, the degree of short-term variability is assessed by the calculation of log 1 0 Q (cf. section 4.1). Chapter 5. The Variable Source Catalogues Table 5.4: Daily Flux Densities for Selected Long-Term Variables 75 <x= 0: 0:43.6±0.7 5=24:42:46±7 T86 S86 T87 S87 11/ 8 85±11 11/15 114±11 9/30 33±8 10/ 1 58±9 10/ 8 87±11 11/2 119±32 l o g Q 8 6 = -1.2 l o g Q 8 7 = -3.8 a= 0: 7:57.1±0.5 5=10:42: 9±6 T86 S86 T87 Ss7 11/21 619±38 11/22 750±48 10/14 230±21 10/15 243±16 10/18 252±17 10/19 210±18 logQge = -1-5 logQ 87 = -0.4 Ci= 0: 2:29.2±0.8 8= 54:12: 6±5 T86 S86 T87 S87 11/23 231±18 10/16 263±20 11/24 224±16 10/17 223±18 12/ 1 169±18 10/24 243±18 12/ 2 252±17 10/25 240±15 logO.86 = -2.1 log Q87 = -0.3 0: 8:49.3±0.8 5= 70:28:59±3 T86 S86 T»7 S87 11/10 704±44 9/30 663±42 12/11 790±46 10/ 2 550±33 12/11 743±48 10/ 3 531±39 10/ 4 523±40 10/ 4 666±41 10/ 5 586±34 10/ 5 587±38 10/ 7 622±42 10/26 590±35 10/28 502±36 10/30 564±40 10/30 676±39 11/ 1 567±40 logQ 86 = -0.4 l0g<287 = -1.8 a= 0:11:21.3±0.5 5=18:53:50±7 T86 S86 T87 S87 11/8 123±12 11/25 150±11 9/30 205±18 10/ 1 267±16 10/18 271±22 10/19 237±18 i o g g 8 6 = -i .o logQs? = -1.4 a= 0:11:23.7±0.8 <J=51:17:25±6 T86 S 8 6 T 87 Ss7 12/6 51±13 12/11 37±7 10/4 119±14 10/ 5 132±13 10/12 107±12 10/13 115±11 10/30 139±13 logQse = -0.5 logQ 87 = -0.4 Chapter 5. The Variable Source Catalogues 76 a— 0:16:54.3±0.8 5= 73:10:56±3 T86 S86 T87 S87 11/21 1271±76 10/14 1407±96 11/21 1193±79 10/14 1798±89 11/22 1357±71 10/15 1866±98 11/23 1452±87 10/16 1904±98 11/23 1444±77 10/16 1719±108 11/24 1443±77 10/17 1883±100 11/24 1357±79 10/18 1941±101 11/25 1544±89 10/18 2153±115 11/25 1425±77 10/19 1721±90 11/28 1735±96 10/19 1943±102 10/21 2041±110 l0g<386 = -2.7 l o g Q 8 7 = -4.2 a= 0:34:22.3±0.6 £=18:15:24±8 T 86 Ss6 T87 S87 11/11 51±9 12/ 8 40±12 10/ 5 114±10 10/ 6 134±15 10/10 142±11 10/30 188±18 10/31 102±10 logQ 86 = -0.3 l o g Q 8 7 = -3.6 0:44: 6.2±0.6 8= 56:41:29±4 T 86 S86 T 87 S87 11/13 218±19 10/ 5 287±19 11/21 293±23 10/ 7 266±17 11/22 224±17 10/12 297±21 12/ 6 183±13 10/13 310±19 10/14 341±21 10/15 275±24 10/26 411±32 logQse = -3.3 logQ 87 = -2.9 0:45:31.4±1.1 8--=68:10:12±5 T86 S86 T87 S87 11/18 23±6 10/ 9 122±13 11/21 59±12 10/12 85±15 11/22 57±12 10/12 135±12 11/24 83±12 10/13 92±10 12/ 6 71±8 10/13 94±13 12/ 6 36±8 10/14 103±12 12/11 42±10 10/15 54±13 10/17 123±11 10/31 99±13 11/ 2 101±13 log<586 = -5.9 logQ 87 = -3.3 a= 0:45: 4.3±0.4 5=24:18:34±5 T86 Ss6 T 8 7 Sg7 11/ 8 362±23 11/ 9 353±24 12/ 3 369±28 12/ 8 443±30 10/ 6 252±19 10/20 256±18 10/21 210±15 10/30 224±19 l o g Q 8 6 = -1.0 logQ 87 = -0.8 a= 1:25: 5.1±0.5 <^=48:45:35±4 T86 Ss6 T87 Ss7 11/ 8 709±39 11/ 9 594±40 11/10 644±44 11/28 684±37 12/ 3 575±39 12/ 8 610±34 10/ 3 224±343 10/ 4 253±19 10/ 5 344±24 10/ 6 316±25 10/20 258±20 10/21 358±21 10/28 243±18 10/29 306±23 l o g Q 8 6 = -1.0 l o g g 8 7 = -4.1 Chapter 5. The Variable Source Catalogues 77 1:43:38.5±0.8 5--=65:21:4±4 T86 T87 S87 11/ 7 55±8 9/30 108±11 11/ 8 69±8 10/ 1 126±11 11/ 9 99±13 10/ 2 113±12 11/10 80±12 10/ 3 134±14 12/ 5 88±14 10/ 4 99±11 12/ 8 74±10 10/ 5 106±11 10/ 6 134±14 10/ 7 118±12 10/ 9 101±47 10/26 112±11 10/28 101±12 10/29 145±15 11/1 114±13 = -1.2 = -0.5 a= 1:45:58.7±0.5 5= 1:14: 3±7 T 86 S86 T 87 S87 11/25 86±22 11/28 80±10 12/ 3 133±15 10/18 170±24 10/19 142±15 10/20 182±15 10/21 161±14 l o g Q 8 6 = -1.9 logQ 87 = -0.5 a= 2:14:46.5±1.1 5=54: 9:17±6 T87 S87 11/10 53±16 11/11 54±22 11/13 76±14 12/ 5 98±13 10/ 4 140±13 10/24 121±13 10/25 120±12 10/29 97±16 l o g Q 8 6 = -0.9 l o g Q 8 7 = -0.7 cv= 2:24:40.9±0.6 5=67 7:43±3 T86 S86 T87 S87 11/23 1258±71 10/16 1355±80 11/24 1112±61 10/17 1579±82 11/25 1207±67 10/18 1831±101 11/25 1297±75 10/18 1736±106 11/26 1138±79 10/19 1455±86 11/26 1176±66 10/20 1738±92 11/28 1192±65 10/22 1875±100 11/29 1264±72 10/23 1745±97 11/30 1227±70 10/31 1646±97 12/ 3 1140±69 logQse = -0.2 l o g Q 8 7 = -3.7 a= 2:34:55.8±0.4 5=28:35: 7±5 T86 S86 T87 S87 11/ 7 2228±117 11/23 2395±123 11/24 2415±127 12/ 7 2468±138 12/10 2316±126 9/30 3319±186 10/ 1 3239±164 10/16 3711±189 10/17 3708±195 logQ 86 = -0.2 logQ 87 = -0.9 a= 2:44:21.7±0.7 5=19:58: 1±10 T 86 S86 Tg7 S87 11/13 34±7 11/21 49±10 11/22 40±9 12/ 5 43±9 10/ 7 65±9 10/14 79±15 10/15 78±10 10/26 76±11 logQ 8 6 = -0.2 logQ 87 = -0.1 Chapter 5. The Variable Source Catalogues 78 2:48:13.2±0.5 5=56: 4: 3±4 T86 S86 T 87 S87 11/ 7 367±25 9/30 187±16 11/ 8 404±28 10/ 1 204±18 11/ 9 437±25 10/ 2 232±19 11/15 440±25 10/ 3 207±17 12/ 7 349±21 10/ 8 212±19 12/ 8 306±24 10/ 9 228±20 12/10 321±24 10/28 251=1=18 11/1 273±17 = -4.2 log Q87 = -•2.0 a= 2:52: 0.5±0.7 <5=63:39:20±4 T86 S86 T87 S87 11/14 207±18 11/15 236±77 11/17 227±16 11/18 194=1=15 12/ 4 210±14 12/ 6 176=L15 12/ 8 220±15 10/ 8 154±16 10/ 9 144±13 10/10 157±14 10/11 119=1=12 10/12 157±15 10/13 141±14 10/14 131±17 10/15 165±18 logQse = -0-6 l o g Q 8 7 = -0.5 a= 2:56:12.9=1=0.4 6= 5:28:35±10 T86 S86 T87 S87 11/8 183±18 11/9 91±13 11/13 125=1=15 10/ 2 223±16 10/ 3 197±17 10/ 6 197=1=17 10/ 7 202±19 10/26 232±20 10/28 200±16 11/ 1 243=1=21 i o g g 8 6 = -3.7 logQs? = -0.4 a= 2:51:31.5=1=0.5 5=39:19:57±4 T 86 Ss6 T87 Ss7 11/10 260±19 11/11 254±16 11/23 318±21 11/24 279d=21 10/ 4 467±27 10/ 5 510±35 10/16 490±29 10/17 453=t32 10/29 457±26 logQse = -1.1 l o g Q 8 7 = -0.2 a= 2:56: 4.9±0.6 0=51:26:19=1=5 T 86 Ss6 T 8 7 S87 11/21 99=1=19 11/22 103±10 11/28 118=1=44 11/29 162=1=14 11/30 113±14 12/3 114=bll 12/4 118=1=15 12/6 112=1=13 10/12 51=1=8 10/14 134=1=14 10/15 91=1=12 10/20 84=4=11 10/21 92±10 10/22 4=1=13 10/23 40±14 i o g g 8 6 = -i .2 l o g Q 8 7 =-11.7 2:58:44.6=1=0.6 5= 53:20:10=b4 T 86 S86 T 87 S87 11/ 7 329±22 10/ 1 324±23 11/23 221=1=19 10/16 298±30 11/26 275=1=18 10/18 395±26 12/ 1 253±16 10/19 298±21 12/ 2 232±17 10/24 316=1=21 12/ 9 270±18 12/10 239±16 log<386 = -2.4 logQ87 = -1.4 Chapter 5. The Variable Source Catalogues 79 <x= 2:59:48.5±0.6 5=68:10: 7±3 T87 S87 11/ 7 946±52 11/ 9 887±56 11/29 936±52 11/30 991±56 12/ 1 910±51 12/ 1 851±54 12/ 2 788±53 12/ 2 1044±57 12/ 3 975±63 12/10 979±58 10/ 1 1403±80 10/ 3 1608±89 10/20 1502±100 10/22 1587±85 10/23 1219±76 10/24 1503±90 10/24 1522±80 10/25 1585±83 10/25 1534±84 10/28 1731±99 11/ 1 1741±95 l o g Q 8 6 = -1.2 logQ 87 = -3.1 a= 3:14: 4.2±0.7 5=56:33: 5±4 Tg6 Sg6 T87 S87 11/26 252±21 12/ 1 299±19 12/ 9 253±20 12/11 278±18 10/16 196±15 10/18 186±15 10/19 201±18 10/24 163±13 10/25 207±14 logQse = -0.6 l o g Q 8 7 = -0.8 a= 3:23:33.8±0.7 5=28:32:30±7 T86 Ss6 T 87 S87 11/ 8 25±6 11/26 413±25 12/11 75±12 10/ 3 14±8 10/18 93±13 10/19 90±13 10/28 53±19 11/ 1 27±11 logQse =-50.5 logQ 87 = -8.1 a= 3: 7:37.6±0.4 5=38: 3:17±4 T86 Sg6 T87 Sg7 11/10 243±18 11/11 524±30 11/23 473±28 11/24 462±31 10/ 4 890±50 10/ 5 747±50 10/ 6 805±48 10/16 732±41 10/17 729±42 10/29 890±52 l o g Q 8 6 =-20.5 l o g Q 8 7 = -1.6 a= 3:20:28.5±0.7 5=46: 3:19±7 Tg6 S86 Tg7 S87 11/ 7 106±13 11/16 110±91 11/17 "96±10 12/ 7 41±13 12/10 113±13 10/ 1 120±14 10/10 110±12 10/11 104±11 logQge = -3.2 logQ 8 T = -0.2 a= 3:29:16.3±1.9 5=54:24:13±9 Tg6 Sg6 Tg7 S87 11/21 93±14 11/24 33±10 11/29 23±7 11/30 22±10 12/1 19±11 12/ 2 4±9 10/15 39±13 10/22 12±11 10/23 4±8 10/24 1±5 10/25 50±11 logQse = -4.9 l o g Q 8 7 = -3.7 Chapter 5. The Variable Source Catalogues 80 a= 3:33:36.1±0.7 5=59:38:16±5 T 86 S86 T 87 Sg7 11/ 8 190±15 11/ 9 206±15 11/11 168±16 11/29 182±17 10/ 2 141±12 10/ 3 107±18 10/ 4 125=1=13 10/ 5 98±12 10/6 153=1=15 10/22 114±13 10/24 125=1=14 10/25 116=1=13 10/28 108=1=12 10/29 102=1=14 11/ 1 98±63 l o g Q 8 6 = -0.4 l o g Q 8 7 = -0.9 a= 4: 8:52.4±0.5 5= 8:35:36±7 T 86 Ss6 T 8 7 S 8 7 11/23 29±8 11/24 68=1=10 11/27 75±16 12/ 3 53±18 10/16 165=1=13 10/17 195=1=15 10/20 254±17 10/21 189=1=15 l o g Q 8 6 = -2.2 logQ 87 = -3.3 a= 4:37:31.2±0.4 5=14:32:28=1:6 T 86 Ss6 T 87 S87 11/17 139=1=13 11/22 96±70 11/23 155=1=14 11/24 146=1=11 10/10 312±25 10/11 283±22 10/16 388±24 10/17 301±23 logQ 8 6 = -0.1 l o g Q 8 7 = -2.1 a= 3:54:17.2±0.7 5=41:39:59=1=6 T86 S 8 6 T87 S 8 7 11/8 162=1=14 11/ 9 126=1=15 11/22 58±15 10/ 2 114=1=13 10/ 3 137±14 10/15 108=1=12 10/28 125±19 11/1 169=1=15 logQse = -5.7 log<5s7 = -1-7 a= 4:22:12.0±0.4 5= 0:28:51±5 T86 S86 T 87 S 8 7 11/18 697=1=39 11/21 716=1=59 11/22 760±43 12/ 6 771=1=50 12/ 9 696±42 10/12 1222=1=70 10/13 1176±65 10/14 995±54 10/15 983=1=55 l o g Q 8 6 = -0.2 l o g Q 8 7 = -2.1 a= 4:42:20.0±0.4 5= 7:10:18=1=5 T 86 S86 T 87 S87 11/25 354±31 11/29 490=1=31 11/30 533=1=38 12/ 9 588±33 12/11 516=4=33 10/18 272=1=20 10/19 243±17 10/22 303±19 10/23 271=1=19 10/31 289=1=23 logQse = -5.3 l o g Q 8 7 = -0.7 Chapter 5. The Variable Source Catalogues 81 a= 4:44:41.6±0.7 5=63:26:58±3 T86 S86 T87 Ss7 11/17 350±24 11/21 425±27 11/22 493±31 11/24 364±25 12/ 6 382±27 12/ 8 355±23 10/ 8 632±41 10/10 623±38 10/11 593±37 10/14 611 ±33 10/15 563±39 10/16 612±35 10/17 574±35 l o g Q 8 6 = -2.7 l o g Q 8 7 = -0.1 a= 5:48:32.6±1.1 5=68:57:12±5 T 86 Ss6 T 87 S87 11/18 86±14 11/22 101±12 11/23 72±10 12/ 5 88±81 12/ 6 62±7 10/10 134±15 10/11 176±17 10/12 139±17 10/13 129±13 10/14 125±14 10/15 108±12 10/15 159±18 10/16 146±12 10/17 109±14 l o g Q 8 6 = -1.2 logQsr = -1-6 a= 7:16: 4.6±0.5 5=33:12:27±5 T86 S86 T 87 Ss7 11/11 273±20 11/22 235±18 11/23 269±16 10/ 5 353±25 10/ 6 384±23 10/16 423±30 10/17 422±25 10/29 418±25 logQse = -0.6 l o g Q 8 7 = -0.7 a= 4:59: 6.7±0.5 5=38:45:55±4 T86 S86 T87 Ss7 11/9 178±14 11/11 127±14 11/22 175±14 11/23 170±17 12/ 8 207±15 10/ 2 240±16 10/ 3 213±24 10/ 6 263±19 10/16 390±28 10/17 350±23 10/28 202±18 11/ 1 272±24 logQse = "2-5 l o g Q 8 7 = -8-9 ol= 6:10: 5.4±0.8 <J=61:40:24±5 T86 S86 T87 Ss7 11/ 7 68±10 11/24 54±10 11/27 53±8 12/ 8 89±11 12/ 9 62±10 12/10 29±25 9/30 59±32 10/ 1 153±14 10/18 97±10 10/19 100±14 10/20 137±14 10/21 109±10 10/24 109±12 10/25 108±11 logQse = -1.1 l o g Q 8 7 = -1.9 a= 7:26:24.7±0.8 5=67:25: 3±4 T 86 S86 T 87 S 8 7 11/21 303±21 11/22 335±21 11/22 217±17 11/23 273±20 11/23 263±20 11/24 293±22 10/12 179±16 10/14 174±16 10/15 149±18 10/16 166±15 10/16 171±16 10/17 160±13 10/17 182±15 10/19 177±17 10/21 203±18 logQse = -3.4 l o g Q 8 7 = -0.2 Chapter 5. The Variable Source Catalogues 82 a= 8:40:11.6=1=0.6 5=42:26:35=1=5 T 8 6 Sse T87 S 87 10/10 588±35 10/11 488±34 10/24 554±32 10/25 558±35 logQ 8 7 = -0.7 8:51:54.2±0.9 5= 71:57:52±4 T86 S86 T87 S87 11/ 7 189±17 9/30 244±19 11/ 8 147±14 9/30 297±21 11/ 8 203=1=17 10/ 2 225=1=24 11/10 164±14 10/ 3 305±21 11/29 137±13 10/23 235±20 11/30 150±14 10/24 202±15 12/ 1 121±11 10/24 252=1=20 12/ 8 163±15 10/25 229±18 12/ 8 142±14 10/25 198±16 10/28 195±15 11/ 1 203±26 log Qm = -2.8 logQ 8 7 = -4.6 9:12:23.3±1.1 5= 73:55:26±4 T86 S86 T87 S87 11/12 151±13 10/ 4 164±16 11/13 109±13 10/ 5 156=1=14 11/13 125±13 10/ 7 117±17 11/14 113=1=13 10/ 9 147±15 11/14 109=1=15 10/29 176=1:15 11/15 139±16 10/29 162±14 11/16 127±13 10/31 166=1=15 12/ 4 130±14 12/ 5 109±13 12/11 100±10 logQse = -0.9 logQs7 = -0.7 9:32:34.4±0.4 5= = 9:28:54=1=6 Tg6 S 86 T87 S87 11/ 7 367±26 9/30 170±19 11/ 8 366±25 10/23 207±17 11/29 348=1=28 10/27 175=1=16 12/ 8 300±22 12/ 9 302±23 logQse = -1.0 log <287 = -0.6 a= 9:25: 9.0=1=0.6 5=44:58:56=1=6 T 8 6 S 86 T87 Ss7 11/12 148=1=17 12/11 143±11 10/ 4 262±16 10/ 5 236±18 10/20 241±18 10/21 225±18 10/29 264±23 l o g Q 8 6 = -0.1 l o g Q 8 7 = -0.3 • a= 9:54:13.6=1=0.5 5=55:37:34±4 T 8 6 S 86 T87 S 87 11/23 2125±109 11/25 2395±123 11/26 2529±137 12/ 1 1525±103 12/10 2231=1=114 10/16 2112=1=119 10/17 1741=1=95 10/18 2255±122 10/19 2472±138 10/24 2362±124 10/25 1876=1=94 l o g g 8 6 = -9.0 log<587 = -5.4 Chapter 5. The Variable Source Catalogues 83 9:54:58.0±0.6 5=65:48:36±3 T87 S 87 11/ 8 927±60 10/18 1520±85 11/25 1039±58 10/20 1347±79 11/28 995±60 10/21 1399±79 11/29 849±51 10/23 1390±81 11/30 1093±60 10/24 1393±79 11/30 985±53 10/25 1227±68 12/ 1 2068±136 10/27 1359±89 12/ 2 823±57 10/28 1305±70 12/ 9 1116±61 12/ 9 1064±59 12/10 1100±60 =-14.6 l o g Q 8 7 = -0.5 ft=10:34:53.0±0.6 o=33:14: l l±5 T 86 S86 T 8 7 S87 11 /7 51±9 11/22 70±10 11/23 79±11 12/ 6 69±11 12/ 8 52±15 9/30 168±14 10/ 3 199±16 10/14 139±16 10/17 183±13 logQ86 = "0.6 logQ 87 = -1-3 a=U: 3:24.4±0.5 5=28:29:16±5 T 86 Ss6 T 87 Sg7 11/17 170±62 11/28 533±34 12/ 3 500±34 10/10 236±19 10/11 285±18 10/20 261±17 10/21 314±20 logQse = -6-0 logQ 87 = -1-5 a=10:27:13.4±1.4 5=74:57:14±4 T86 S 86 T87 Ss7 11/ 8 202±18 11/28 279±20 11/29 272±21 11/30 312±24 12/ 2 268±20 12/ 3 230±21 12/ 9 251±21 12/10 337±22 10/21 225±18 10/24 216±19 10/25 233±19 10/27 214±20 10/28 178±17 logQ 86 = -4-1 logQ 87 = -0.7 a=10:48:19.9±0.5 <S=20:44: 9±6 T86 S86 T 87 S87 11/28 99±12 12/ 7 96±9 10/12 179±13 10/13 170±17 10/20 204±15 10/21 190±21 logQ 8 6 = -0.1 l o g Q 8 7 = -0.3 a = l l : 3:47.5±0.5 5=17:29:59±7 T 8e S 8 6 T 87 S 8 7 11/19 124±12 11/28 117±11 12/ 7 158±13 10/12 159±19 10/13 135±12 10/20 211±17 10/21 194±18 logQse = -1-4 l o g Q 8 7 = -3.0 Chapter 5. The Variable Source Catalogues 84 g=l l :24:50.2±0.7 5=57: 7: 3±4 T86 Sg6 TW Ss7 11/15 340±29 10/14 481±36 11/16 356±29 10/31 626±35 11/22 377±22 12/ 4 374±23 12/ 5 392±23 12/ 6 353±23 log<386 = -0-l l o g Q 8 7 = -2.4 a=l l :43:45.4±0.6 5=59: 5:18±4 T 8 6 S 86 T 8 7 S 8 7 11/22 535±31 11/25 514±29 11/28 568±32 12/ 6 529±34 12/ 7 494±33 10/12 654±36 10/13 621±40 10/14 746±44 10/19 590±42 10/20 703±43 10/21 738±42 10/30 731±41 logQse = -0.2 l o g Q 8 7 = -1-4 a=12:17:29.5±l . l 5= 1:46:18±13 T 86 S 8 6 T 87 S 8 7 11/ 9 316±22 11/12 20±16 11/14 24±13 10/26 39±10 11/ 1 23±13 l o g Q 8 6 =-32.0 logQs7 = -0.5 a= 11:25:22.4±0.6 5= =59:41:49±4 T 86 S86 T 87 S87 11/15 294±19 10/10 380±29 11/16 258±17 10/11 360±24 11/17 278±20 10/14" 338±23 11/18 319±20 10/16 376±24 11/22 263±22 10/17 367±26 11/24 282±19 10/31 398±24 12/ 4 243±17 12/ 5 265±18 12/ 6 259±18 logQge = -0.8 log Qs7 = -0.2 a=l l :50:39.6±0.9 5=25:26: 9±10 T86 S86 T 87 S 8 7 11/18 72±8 11/25 83±9 10/10 36±8 10/11 68±11 10/19 20±7 logQse = -0.4 l o g Q 8 7 = -3.0 a=12:26:49.2±0.7 5=55:39:15±6 T 86 S86 T 87 S87 11/12 55±12 11/17 62±8 11/18 69±13 12/12 97±11 10/ 2 139±12 10/ 4 178±17 10/ 5 99±15 10/10 125±12 10/11 94±13 10/13 165±13 11/1 116±12 logQse = -1-5 l o g Q 8 7 = -4.3 Chapter 5. The Variable Source Catalogues 85 a=12:58: 2.8±0.8 5=44: 8:54±6 T86 S86 T87 Ss7 11/ 7 45±8 11/ 8 36±8 11/10 81±27 11/19 49±9 12/ 7 63±10 12/ 8 67±9 9/30 99±10 10/10 112±12 10/12 102±11 10/27 135±12 logQse = -1.1 l o g Q 8 7 = -1.0 cv=13:24:24.2±0.7 5=44:50: 1±5 T86 S 86 T 8 7 S 87 11/10 98±14 11/23 123±18 9/30 178±12 10/ 3 196±16 10/16 109±14 10/19 164±13 11/1 156±14 logQge = -0.6 log<387 = -3.5 a=13:30:47.5±0.6 5=27:40:36±6 T 8 6 Ss6 T 8 7 S 87 11/ 8 496±31 11/24 454±27 9/30 158±13 10/16 149±14 10/17 179±20 10/27 180±18 logQse = -0.5 l o g Q 8 7 = -0.4 or=13: 4:35.1±0.6 5= 6:44: 8±9 T 8 6 Ss6 T87 Ss7 11/10 171 ±20 11/14 64±9 12/10 179±16 10/ 7 178±16 10/26 215±17 logQse =-11.5 l o g Q 8 7 = -0.9 a=13:24:37.6±0.5 5=22:26:18±5 T86 S86 T 8 7 S 87 11/14 475±26 12/ 6 458±26 12/10 455±27 10/ 7 788±51 10/14 822±50 10/26 742±45 logQge = -0.1 l o g Q 8 7 = -0.3 ot=13:43:22.4±l.l 5= 7:21:14±8 T 8 6 Ss6 T87 S87 11/22 342±26 Chapter 5. The Variable Source Catalogues 86 tt=14:17:42.5±0.4 5=38:35:57±4 T86 S86 T 8 7 S87 11/19 374±25 12/ 1 311±22 12/ 2 398±24 12/ 7 443±29 12/ 8 396±29 9/30 762±45 10/10 727±49 10/12 783±52 10/13 871±49 10/24 740±44 10/25 842±57 10/27 918±56 11/ 1 947±58 logQgs = -2-4 l o g Q 8 7 = -1.9 a=14:35:49.9±0.6 5=30:15: 3±7 T 8 6 Ss6 T87 Ss7 11/10 60±15 11/23 92±10 11/24 93±11 12/10 54±8 9/30 134±16 10/ 2 112±10 10/16 125±11 10/17 110±15 l o g Q 8 6 = -2.4 logQsr = -0.2 cv=15: 9:58.7±1.1 5=10:34:23±18 T86 S86 T 8 7 Ss7 11/10 33±12 12/ 1 13±9 12/ 2 35±12 10/24 61±15 10/25 64±21 logQs6 = -0.6 log<587 = 0.0 cv=14:32: 9.5±0.6 5=42:16:17±6 T 8 6 S 86 T 8 7 S 87 11/29 207±14 11/30 160±16 10/ 9 369±22 10/23 381±22 10/25 466±28 10/26 338±24 logQse = -1-6 l o g Q 8 7 = -2.3 a=14:38: 2.2±0.4 5= 5:44: 6±6 T 8 6 S 86 T 8 7 S 87 11/10 217±16 12/ 1 203±17 12/ 2 200±20 12/10 231±20 9/30 135±17 10/ 2 198±14 10/24 194±17 10/25 182±18 logQse = -0.2 l°g<287 = -1.6 a=16: 1:19.5±0.4 5=11:14: 2±5 T 8 6 S 86 T 8 7 S 87 11/9 411±28 11/10 408±31 11/16 358±22 10/ 2 516±35 10/ 3 602±35 10/27 970±54 11/ 1 973±58 log<586 = -0.6 l o g Q 8 7 =-16.8 Chapter 5. The Variable Source Catalogues 87 cv=16:28:56.7±0.5 <5=10:58:27±7 T86 T87 S87 11 /7 129±12 11/8 127±10 11/30 115±14 12/ 8 125±13 10/ 1 136±12 11/2 187±15 logQ 8 6 = -0.1 l o g Q 8 7 = -2.1 a=17: 0: 3.2±0.5 5= 8: 2: 3±7 T 8 6 S86 T87 S87 11/28 275±23 12/ 1 221±17 12/ 3 206±20 10/21 68±13 10/22 107±10 10/25 133±13 10/26 103±10 logQge = -1.2 logQ 87 = -2.3 . 16:55:36.5±0.7 5= :53:26:28±5 T86 S86 T87 S87 11/11 105±12 10/ 4 151±16 11/12 175±16 10/ 5 188±14 11/29 132±14 10/24 185±15 11/30 153±12 10/26 229±17 logQge = -2.7 l o g Q 8 7 = -2.0 17: 0:19.7±0.7 6= 68:34:27±3 T 86 S86 T 87 S87 11/11 341 ±22 10/ 2 465±30 11/12 379±28 10/ 4 443±27 11/14 303±23 10/ 5 394±31 11/14 423±26 10/ 7 358±31 12/ 5 460±31 10/ 7 526±34 12/ 5 454±27 10/ 8 416±33 12/10 366±28 10/ 9 440±25 10/11 439±29 10/27 430±29 10/31 470±31 10/31 485±31 logQse = -4.7 log Q87 = -1.5 a= 17: 1: 0.5±0.7 5= 65:58:26±4 T86 S86 T87 S87 11/11 218±19 10/ 4 245±19 11/12 254±19 10/ 5 277±22 11/14 225±20 10/ 7 238±20 11/17 247±21 10/ 8 227±20 11/18 195±165 10/ 8 182±21 12/ 7 204±71 10/ 9 214±16 10/ 9 217±21 10/10 271±25 10/11 272±19 10/14 238±26 10/27 220±20 10/27 274±103 10/31 207±16 logQ 8 6 = -0.1 log<287 = -1.4 a=17: 1:37.3±0.7 <J=50:21:53±6 T86 S86 T87 Ss7 11/14 248±19 11/22 300±21 10/15 297±22 10/16 267±23 .10/18 310±20 110/31 264±20 l o g Q 8 6 = -1-2 logQ 87 = -0.5 Chapter 5. The Variable Source Catalogues 88 17:10:36.5±0.5 8= = 6: 3:14±8 T 8 6 S86 T 87 §87 11/19 122±18 10/12 79±11 11/23 134±12 10/14 75±12 12/ 7 112±11 10/17 91±10 10/18 62±11 l o g Q 8 6 = -0.4 logQ 87 = -0.6 a= 17:39:17.5±0.6 8= :43:49:48±5 T 8 6 S86 T87 11/ 7 142±12 9/30 278±21 11/ 8 198±15 10/ 1 289±17 11/22 153±12 10/14 290±17 11/ 2 290±22 logQse = -2-0 l0g<287 = 0.0 a= 17:45:58.1±0.6 8= =67: 4:54±3 T 8 6 S86 T 8 7 S87 11/ 8 295±21 9/30 277±21 11/ 9 364±24 10/ 1 337±22 11/ 9 279±18 10/ 2 284±23 11/10 305±22 10/ 2 318±22 11/11 278±21 10/ 3 257±21 11/12 279±19 10/ 4 293±22 11/14 247±17 10/ 5 309±20 12/ 5 326±23 10/ 6 296±19 12/ 8 299±20 10/ 7 299±20 10/31 287±22 11/1 350±23 11/ 2 312±23 logQse = -2.0 l o g Q 8 7 = -0.7 a= 17:22: 6.2±0.8 5=61 8:56±4 T86 S86 T87 S87 11/21 165±15 10/15 282±20 11/22 172±16 10/16 256±18 11/24 174±37 10/19 318±22 11/28 113±18 10/21 324±26 12/ 3 248±18 10/22 276±24 12/10 173±16 l o g Q 8 6 = -4.6 l o g Q 8 7 = -0.9 a=17:39:28.2±0.6 5=52:13:55±10 T86 S86 T87 S87 11/23 1900±109 12/ 1 4044±494 12/ 2 2159±114 12/10 1856±127 10/14 1227±69 10/17 1021±66 10/18 1212±65 10/25 1185±69 10/26 1160±62 10/29 989±76 logQse = -4.0 logQ 87 = -1.2 a= 17:47:29.3±0.5 8= :43:22:47±6 T86 S86 T87 S87 11/ 9 245±18 10/ 2 329±22 11/10 224±18 10/ 3 301±22 10/12 371±25 10/13 276±22 10/14 326±23 11/1 279±22 logQse = -0.4 logQ 87 = -1.4 Chapter 5. The Variable Source Catalogues 89 cv=17:54:58.7±0.6 5=45:24:43±7 Tg6 S86 T87 S 8 7 11/21 109±13 11/22 126±13 12/ 1 136=1=12 12/ 2 79±13 10/14 67±10 10/15 36±7 10/25 50±9 10/26 66±8 i o g g 8 6 = -2.1 l o g Q 8 7 = -1-8 a=18: 4:41.3±0.5 <5=11:59:31=L5 T86 S86 T 8 7 Ss7 11/ 7 96±10 11/ 8 104±13 12/ 8 87=L8 9/30 164±13 10/ 1 156=1=15 11/2 171±15 logQse = -0.3 l o g Q 8 7 = -0.1 a=18:36:32.0±0.4 5= 9:24:47±5 T 8 6 Ss6 T 8 7 S 8 7 11/ 7 298±20 11/ 8 298±18 12/ 8 282±21 9/30 158=1=13 10/ 1 124±11 10/23 154±32 10/24 157±12 10/29 174±246 11/2 160±14 l o g Q 8 6 = -0.1 l o g Q 8 7 = -0.6 a=17:57:18.0±0.5 5=28:47:45±8 T 8 6 S 86 T 8 7 S 87 11/14 48±9 11/23 77±10 12/ 5 102±13 10/ 7 185±15 10/17 189±16 10/18 152=1=15 10/31 190±19 l o g Q 8 6 = -2.7 l o g Q 8 7 = -0.6 a=18: 7:30.3±0.4 6= 1:35:59±7 T 8 6 S 86 T 8 7 S 8 7 11/9 173±16 11/10 164±16 11/11 147±15 11/12 169±20 10/ 2 122±15 10/3 91±18 10/ 4 128=t20 10/ 5 95±11 10/ 6 66±11 11/1 140=1=16 i o g g 8 6 = -0.2 l o g Q 8 7 = -3.0 a=18:38: 0.7±0.5 5=24:54:42±7 T 8 6 S 86 T 8 7 S 8 7 11/11 52±9 11/12 71±11 11/28 62±10 12/ 3 103±13 10/ 4 148±12 10/ 5 180±16 10/21 142±15 10/22 149±14 log(5 8 6 = -1.9 l o g Q 8 7 = -0.5 Chapter 5. The Variable Source Catalogues 90 a=18:51: 9.7±0.6 5=48:52:16±5 T86 S86 T 8 7 S 87 11/ 9 168±15 11/10 261±18 12/ 7 187±17 10/ 2 307±19 10/ 3 279±21 10/12 454±31 10/13 353±26 10/28 284±20 11/ 1 278±22 l o g Q g e = -3.6 l o g Q g ? = -4.9 a=19:22:15.5±0.6 5=28:31:58±7 T86 S 86 T87 S87 11/21 122±13 11/22 178±13 12/ 1 115±13 12/ 2 158±15 10/14 52±10 10/15 65±10 10/25 20±7 10/26 73±11 l o g Q g e = -2.8 l o g Q 8 7 = -4.5 a = 20:54:32.9±0.5 5= =61:10:45±3 T86 S86 T s 7 S 87 11/ 7 298±24 9/30 353±23 11/ 8 260±21 10/ 1 343±23 11/ 9 403±26 10/ 2 440±28 11/10 346±27 10/ 3 385±26 11/29 339±21 10/20 451±30 12/ 8 372±24 10/22 415±26 10/23 395±29 10/25 388±29 10/27 451±31 10/28 483±28 10/30 525±34 10/31 447±33 11/1 425±31 l o g Q s e = -3.7 l o g Q 8 7 = -4.3 a=18:54: 5.8±0.4 5= 6: 5:59±5 T86 Ss6 T87 S87 11/ 9 286±18 11/10 274±26 12/ 2 279±23 12/ 5 302±20 10/ 2 159±13 10/ 3 106±13 10/25 163±20 10/26 139±17 10/28 94±12 11/ 1 179±17 l o g Q g e = -0.1 l o g Q 8 7 = -4.5 a=20:51:44.4±0.6 5=52:50: 5±5 T86 S86 T87 S87 11/8 152±13 11/23 180±18 11/24 178±13 11/25 195±16 12/ 1 134±12 12/2 120±14 10/ 1 HOztlO 10/16 101±10 10/17 133±12 10/18 97±21 10/19 99±14 10/24 127±11 10/25 147±18 11/ 1 117±14 l o g Q s e = -3.0 l o g Q 8 7 = -0.8 a=21: 0:34.1±0.5 5=46:50:15±5 T 8 6 S 86 T 8 7 S 87 11/11 231±18 11/21 214±18 11/22 260±16 12/ 6 223±17 10/4 113±H 10/ 5 135±12 10/6 117±10 10/14 123±11 10/15 133±11 10/29 190±16 l o g Q s e = -0.6 l o g g 8 7 = -2.7 Chapter 5. The Variable Source Catalogues 91 a=21:13:40.4±0.7 5=36:33:28±6 T86 Sg6 T87 S87 12/ 5 51±11 12/ 7 89±13 10/12 126±13 10/13 83±10 10/24 183±14 10/25 202±13 l o g Q 8 6 = -1.6 l o g Q 8 7 =-13.4 a=22: 6:34.4±1.0 5=61:18: 8±7 T 8 6 Sg6 T 8 7 Ss7 11/21 25±12 11/22 54±9 11/23 48±8 12/ 4 28±9 12/ 9 76±8 10/ 9 71±10 10/10 82±13 10/11 73±14 10/15 115±13 10/16 96±11 10/17 154±14 11/ 2 60±12 • l o g Q 8 6 = -3.5 logQ 8 7 = -5.7 a=22:30:25.5±0.5 5=26:39:27±6 T86 S86 T 8 7 S87 11/15 97±13 11/17 l l l i l l 11/25 164±15 11/26 117±14 12/ 3 115±15 12/10 103±10 10/ 9 66±26 10/18 86±9 10/19 102±13 11/ 2 92±12 l o g Q 8 6 = -1.8 logQgr = -0.2 a=21:50:39.1±0.4 5=12:26:48±6 T86 S86 T87 S87 12/ 1 236±19 10/ 4 263±17 10/ 5 281±18 10/ 6 264±21 10/24 299±18 10/25 276±22 10/30 260±17 logQ 8 7 = -0.2 a=22: 9:54.0±0.4 5=32:53:49±5 T86 S 86 T87 Ss7 11/ 7 651±38 11/ 8 704±37 12/ 5 613±34 12/ 7 564±39 12/ 7[ 552±37 9/30 358±22 10/ 1 325±21 10/12 293±21 10/13 340±21 logQse = -1-6 l o g Q 8 7 = -0.8 o;=23: 9:28.7±0.5 5=45:27:24±4 Ts6 S86 T 8 7 Ss7 11/13 417±27 11/14 416±27 11/23 354±26 11/24 424±26 10/ 7 572±32 10/16 590±42 10/17 600±37 10/19 557±36 10/26 508±35 logQ 8 6 = -0.7 logQ 8 7 = -0.4 Chapter 5. The Variable Source Catalogues a=23:31:40.3±0.5 6= 7:19:39±5 a=23:26:34.0±0.8 5= 8:17:20±6 T86 Ss6 11/ 9 168±14 11/10 213±179 l o g Q 8 6 = -0.1 Tg7 S87 10/24 343±21 T86 Ss6 11/16 1808±98 12/ 3 1856±96 12/ 5 1588±142 logQse = -0.5 T87 Ss7 10/ 8 987±54 11/ 2 831±58 logQ 87 = -1.3 Chapter 6 GT2318+620 6.1 A Brief History of GT2318+620 As mentioned in the introduction, the original intention of this thesis was twofold; to uncover possible variable sources in the Greenbank survey, and then make further V L A observations of some of the more promising variables. High resolution V L A observations would give an indication of morphologies, as well as establishing accurate positions for the variables. Unfortunately, the compilation of the variable source catalogue consumed all the available time, allowing only limited follow-up investigations of any variables. However, the author did carry out follow-up V L A observations on the interesting source GT2318+620. This section will provide the motivation for looking at GT2318+620 by detailing the history of the source prior to V L A observations by the author. In sections 6.2 and 6.3 the author's observations and subsequent analysis will be presented. GT2318+620 first attracted attention when it was classified as a possible variable source in the Gregory/Taylor Galactic plane survey [Gregory & Taylor 1986]. In three 6 cm survey observations carried out over a 12 day period in August 1981 the source in-creased in flux density from lOmJy to 40mjy. In addition, at that time it was noted that this source lay within the error box of the X-ray source 4U2316+61 [Forman et al. 1978]. In order to establish a distance estimate to the source, a 21 cm neutral hydrogen absorption spectrum was taken in the direction of GT2318+620 with the Westerbork 93 Chapter 6. GT2318+620 94 Synthesis radiotelescope [Taylor et al. 1991]. By comparing this spectrum to a spec-trum of hydrogen emission in the same direction ([Weaver & Williams 1973]), a distance to the source of between 3kpc and 6kpc was inferred. This distance estimate places GT2318+620 within our galaxy. To determine an accurate position for GT2318+620, as well as ascertaining the mor-phology of the source, high spatial-resolution observations were made of GT2318+620 with the NRAO Very Large Array (VLA). Observations were obtained at A = 20 cm and A = 6cm with the VLA in the "A" configuration (the highest resolution configuration). The images constructed from these observations are reproduced in figure 6.1. At 20 cm 20 15 • 1 0 3 i 05 ) ) 00 62*00' 55 50 45 l o 1 — I - -1 — i — i i i _ c o &> t • f •rail ° O W * o -0 r J " -i 0 . OS ° » , «•%> -0 V1 • 1 1 1 1 23h lemin 26s 24 23 22 21 Right a scens ion 20 1 9 23h 18mln 23.2$ 22.8 22.6 22.4 Right a scens ion 22.2 22.0 Figure 6.1: November 4, 1988 VLA images of GT2318+620 (from [Taylor et. al. 1991]). Left: GT2318+620 at 20 cm. Contours are -0.35, -0.25, 0.25, 0.35, 0.5, 0.75, 1.0, 1.5, 2.0, 2.5 and 5.0 mJy/beam. Angular resolution is 1.5". Right: GT2318+620 at 6 cm. Contours are -0.15, 0.15, 0.25, 0.35, 0.5, 0.75, 1.0, 2.0, 2.5, 3.5 and 5.0 mjy/beam. Angular resolution is 0.5". the total flux density of the source was 3 7 ± 2 m J y with the unresolved core having a flux density of 6.3 ± 0.3 mJy. At 6 cm the unresolved core had a flux density of 7.7 ± 0.3mJy. The coordinates established for the core of GT2318+620 from these images were Chapter 6. GT2318+620 95 a = 23h 18m 22.57 ± 0.02s, 5 = 62° 01' 7.3 ± 0.15" (1950 coordinates). A search for an optical counterpart to the radio source revealed a 20 t h magnitude object well within 3a of these coordinates on the red Palomar Sky Survey Plate. Subsequently, a tentative identification of this object as a late-type main sequence dwarf was made. If the identifications and conclusions drawn about GT2318+620 are correct, then GT2318+620 joins a small class of radio-emitting X-ray binaries with jets on scales greater than an arcsecond. Others of this class include Sco X - l , SS433, GRS1915+105, and GRO J1655-40. Additional V L A observations could provide a confirmation of the radio variability, as well as searching for changes in jet morphology. Any observed changes in the jet structure would provide a "smoking gun" to GT2318+620's identification as a Galactic jet source. Small time-scale changes in morphology on the scale of arcminutes, or even arcseconds, are impossible for extragalactic sources due to their much larger intrinsic size (although sub-arcsecond changes in extragalactic sources have been observed). 6.2 V L A Observations of GT2318+620 GT2318+620 was observed in July 1991 with the V L A in the "A" configuration and in November 1991 with the V L A in the "B" configuration. In the A configuration the V L A has a maximum antenna separation of 36.4 km. In this configuration, the synthe-sised beamwidth is the narrowest, allowing for the highest resolution images. The B configuration is slightly less extended, with a maximum antenna separation of 11.4 km. In the A configuration, ten hours of observation time in the L, C, X , and U frequency bands were obtained (corresponding to wavelengths of 20, 6, 3.6, and 2 cm respectively). In the L band, the total time spent on source was roughly 110 minutes, while for each of the other bands just over 70 minutes was spent on source. The additional time spent on the L band observations was to compensate for the decreased sensitivity due to the Chapter 6. GT2318+620 96 smaller bandwidth used (see the discussion below). In all cases a 30 second visibility averaging time was used. The remainder of the time allotment was spent in telescope move times, and observing calibrators. These calibrators are used to determine the complex gains applied to the correlated output from all antenna pairs (cf. Section 6.3). For amplitude calibration, the radio source 3C286 was utilized, while to calibrate the phase, the calibration source 2229+695 was used. Unfortunately, the presence of a strong double source (integrated flux ~ 140 mJy) 2.9 arcminutes to the southeast complicates the observations. If not dealt with, the sidelobes from this source would seriously degrade the imaging of GT2318+620 by sig-nificantly raising the rms noise fluctuation level. The strategies used to handle this confusing source were dictated by the effects of bandwidth smearing (observing with a finite bandwidth). The principles upon which imaging with a synthesis instrument are based are, strictly speaking, only valid for monochromatic radiation. Accepting radiation with a finite band-width introduces chromatic aberrations which manifest themselves as radial smearing and a loss in peak response of a source. These effects are proportional to the fractional band-width [8v/v), multiplied by the distance (in synthesised beamwidths) of a source from the delay tracking centre (9/9syni\,). Hence, a confusing source can be dealt with in either of two different methods. If a confusing source is relatively nearby, an attempt can be made to image both the target and confusing source, then subtract the confusing source and its sidelobes from the interesting region. To do this successfully, the effects of bandwidth smearing in both the target and confusing source must be limited by reducing the bandwidth and/or displacing the delay tracking centre to a position intermediate between the two sources. Unfortunately, reducing the bandwidth also reduces the expected V L A sensitivity, but Chapter 6. GT2318+620 97 that is often a trade off that must be made. If a confusing source is relatively distant, then the best strategy is to avoid looking at it. This is done by using a large bandwidth, and keeping the delay tracking centre away from the confusing source. In this way, the response of the confusing source will be strongly attenuated by chromatic aberration. In the case of the GT2318+620 observations, the distance to the confusing source (in synthesized beamwidths) was smallest in the L band. Hence, for the L band observations, it was decided to adopt the former strategy, and attempt to eliminate the effects of the confusing source by imaging both the target and confusing source, then later subtracting out the confusing source. Consequently, for observations in this band, a smaller band-width (12.5 MHz) was chosen, and the delay tracking centre was moved one arcminute away from GT2318+620 towards the confusing source. At this distance, loss of peak response due to bandwidth smearing in GT2318+620 was still less than 5% . For the C, X , and U bands, the distance between GT2318+620 and the confusing source was large enough that it was possible to adopt the second strategy. Hence, in each of these bands a 50 MHz bandwidth was used, and the delay tracking centre was positioned to coincide with GT2318+620's position. With the V L A in the B configuration, 5 hours of observation time were obtained. Data were taken in the C and L bands, with approximately 100 minutes on source in each band, and a visibility averaging time of 30 seconds. For flux density calibrators, both 3C286 and 3C48 were used, while 2352+495 was utilized as the phase calibrator. Although the distance from GT2318+620 to 2352+495 was farther than the distance from GT2318+620 to 2229+695 (the A configuration phase calibrator), in this case the former was used in preference to the latter because of its much higher L band flux density. Chapter 6. GT2318+620 98 To deal with the confusing source the same strategies were employed in the B config-uration observations as were used in the A configuration observations. In the L band a 25 MHz bandwidth was used, and the delay tracking centre was moved one arcminute from GT2318+620 towards the confusing source. For the C band observations, the band-width was set to 50 MHz, and the delay tracking centre was positioned at GT2318+620's coordinates to attenuate any effects from the confusing source. 6.3 Editing, Calibration, and Image Construction with the V L A Data In order to construct images of GT2318+620 from the raw data, the data first had to be edited and calibrated. After this, "dirty images" of GT2318+620 could be constructed, and then "cleaned" to reduce sidelobes from the dirty synthesized beam. The techniques used to perform these tasks were essentially the same for all the V L A images, and so will be described in general below. A l l of this analysis was done within the AIPS environment. The first step was to assess the data quality, and edit out any bad data. For any interferometric array equipped with approximately linear devices, the correlated output (visibilities) from any pair of antennas, V^-, is related to the "true" output, Vy, by Vij = GijVij, with Gij being the complex antenna pair gain. The initial visibility gain settings are adjusted so that, for unresolved calibrators, all the visibilities which are output from any pair of antennas should have roughly the same values. Visibilities associated with an antenna that have consistently high or low amplitudes, or have consistently large amplitude rms's, may indicate a flaky antenna. A l l affected data (both source and calibrator) for any suspect antenna was noted and flagged out. For example, if all the phase calibrator visibilities pertaining to antenna six during a calibration scan were anomalously high, then those visibilities, together with all the GT2318+620 antenna six visibilities relying on that phase calibration scan, were flagged out. Chapter 6. GT2318+620 99 After editing, the next step was to calibrate the antenna gains. The absolute flux densities of the primary calibrator (either 3C286 or 3C48) are well known, and hence they can be used to calibrate the amplitude of the antenna gains. If an array is operating well, these amplitudes should not vary by more than a few percent over the timespan of the observations. The phase of the antenna gains can vary more rapidly, thus much more frequent observations of the phase calibrator were needed to determine the phase as a function of time. Once the amplitude and phase of the antenna gains were determined for the calibrators, the antenna gains for GT2318+620 were calculated by interpolation from the calibrator gains. Finally, as a last check on the gain solutions, the antenna gains were applied to all the calibrator source visibilities, and the resultant visibilities Vij were again inspected for anomalies. Any discrepant visibilities, together with any affected visibilities, were edited out. The visibilities were then used to construct images. A fundamental result of ra-dio interferometry is the existence of a Fourier transform relationship between the sky brightness, 7", and the visibilities: where («, v) are the components of the baseline vector between any two antennas and (l,m) are the components of a vector describing a position on the celestial sphere, [Clark 1989]. Hence given the visibilities V(u, w), an image of the sky brightness can be constructed by performing the direct Fourier transform. When the number of visi-bilities is large, a direct Fourier transform is too slow, and a fast Fourier transform can be used. In that instance the visibility data are first gridded before the fast Fourier transform algorithm is used. In practice however, at any one time only visibilities in the (u, v) plane corresponding to each individual baseline in the array can be observed. Then, as the earth rotates, (6.1) Chapter 6. GT2318+620 100 the u and v components of each baseline change, tracing out tracks in the (u, v) plane. In this way a much greater (u,v) range of visibilities can be sampled. However, this sampling still provides far from complete coverage of the (w, v) plane. Even for sources with reasonably good («, v) coverage, the images resulting from the Fourier transform are very "dirty" (have significant sidelobe effects). To compensate for this, various algorithms exist which are capable of "cleaning" the dirty image. To see how this is done, consider a generalization of equation 6.1: roo roo ID(l,m)= / S(u,v)V{u,v)exp2^ul+vm^ dudv (6.2) J — O O / — O O with ID(l,m) denoting the dirty image, and S(u,v) describing the sampling function. This sampling function may be expressed in terms of a series of two-dimensional Dirac 5-functions. Using the convolution theorem: ID = I * B, (6.3) where the * denotes convolution and roo roo B{l,m) = S(u,v)exp27li{ul+vm) dudv (6.4) J—ocJ — OO is the synthesized beam corresponding to the sampling function S(u,v). Equation 6.3 says that the dirty image is the true intensity distribution convolved with the synthesized beam. In order to deconvolve equation 6.3 an iterative algorithm was devised by Hogbom [Hogbom 1974]. Variants of this algorithm are at the heart of most of the AIPS image cleaning routines (the maximum entropy method being the only other alternative). The algorithm proceeds roughly as follows: (1) Find the strength and position of the brightest point in the dirty image. (2) Subtract from the dirty image, at the position of the peak, the dirty beam Chapter 6. GT2318+620 101 multiplied by the peak strength and a damping factor 7 (7 < 1). Add the delta-function point source to an accumulating point source model. (3) Go to (1) unless the highest peak is below some user specified level. (4) Convolve the accumulated point source model with an idealized " C L E A N " beam. This C L E A N beam is typically an elliptical Gaussian fitted to the dirty beam. (5) Add back the residuals of the dirty image to the "CLEAN"ed image. Variants of this algorithm include the Clark algorithm [Clark 1980] and the Cotton-Schwab algorithm [Schwab 1984]. The former is used by the AIPS task A P C L N , while the latter is implemented in the AIPS task M X . See [Cornwell & Braun 1989] for a more detailed description of the Hogbom algorithm and these variants. Examples of dirty and CLEANed images are shown in the next section. 6.4 Results From the V L A Data In total, V L A observations of GT2318+620 were made in three different epochs. These include the original A configuration 1988 observations [Taylor et al. 1991] (henceforth referred to as the epoch 1 observations), the A configuration July 1991 observations (epoch 2), and the November 1991 B configuration observations (epoch 3). For each epoch, all the editing and calibration of the visibilities was done in the same way, following the procedure outlined in the previous section. However, for mapping and C L E A N i n g several different AIPS tasks were used. Originally, the epoch 1 data were mapped and CLEANed with the AIPS tasks U V M A P and A P C L N . However, it was later determined by the author that mapping and C L E A N i n g of the data with the AIPS task M X provided improved images, with a Chapter 6. GT2318+620 102 lower RMS noise level. Hence, M X was used to construct and C L E A N all of the epoch 2 and 3 images, and the epoch 1 data was remapped with M X . Examples of this mapping and cleaning are shown in figures 6.2 and 6.3. On the left side of these figures are epoch 1 images without any C L E A N i n g done, and on the right are the CLEANed images (CLEANing done with M X ) . The images on the right side should be compared with the original U V M A P / A P C L N e d images, figure 6.1. It is apparent that for the L band, much of the extended emission from the southern lobe of the U V M A P / A P C L N e d image is just an artifact of incomplete CLEANing . As well, the RMS noise in the original L band image was 0.1 mJy/beam, while in the MXed image the noise was reduced to a level of 0.06 mJy/beam. A smaller reduction in the noise RMS of the epoch 1, C band images was seen; from 0.07 mJy/beam to 0.06 mJy/beam. For the epoch 2 and epoch 3 observations, the resultant images for the L and C bands are shown in figures 6.4 and 6.5. Based on the L and C band images from the three epochs, and the flux densities derived from these images, it was not deemed worthwhile to process the X and U band data at this time. As mentioned in section 6.1, a "smoking gun" for the identification of GT2318+620 as a Galactic source would be any morphological changes in the structure of the source from one epoch to another. Looking at the images of GT2318+620 in figures 6.2 through 6.5, no gross changes in morphology are apparent. In both epochs 1 and 2 the jet appears to have roughly the same length and orientation. The only morphological difference between the two epochs was the presence of a "knot" in the northern jet of the epoch 1, L band image. To determine if this was a real feature, the epoch 1 image was subtracted from the epoch 2 image. The resultant difference image is depicted in figure 6.6. At the position corresponding to the knot there is a slight enhancement in this difference image. As well, there is a second stronger feature in the difference image, just south of Chapter 6. GT2318+620 103 Figure 6.2: Left: Epoch 1 image of GT2318+620 at 20 cm before CLEANing . Right: Epoch 1 image of GT2318+620 at 20 cm after CLEANing with M X . Contours for both images are -0.35, -0.25, 0.25, 0.35, 0.5, 0.75, 1.0, 1.5, 2.0, 2.5 and 5.0 mjy/beam. Angular resolution is 1.73". Chapter 6. GT2318+620 104 Epoch 1, A =6 cm E p o c h l j x =6 cm Figure 6.3: Left: Epoch 1 image of GT2318+620 at 6 cm before CLEANing . Right: Epoch 1 image of GT2318+620 at 6 cm after C L E A N i n g with M X . Contours for both 6 cm images are -0.15, 0.15, 0.25, 0.35, 0.5, 0.75, 1.0, 2.0, 2.5, 3.5 and 5.0 mJy/beam. Angular resolution is 0.5". Chapter 6. GT2318+620 105 Epoch 2, A =20 cm Epoch 2, A =6 cm RKJKT ASCEKSON (B19SO) aiiiu si w iu ai iu iu u nQHT ASCENSION (B1»M) Figure 6.4: Epoch 2 images of GT2318+620. Left: 20 cm image. Contours are -0.35, -0.25, 0.25, 0.35, 0.5, 0.75, 1.0, 1.5, 2.0, 2.5 and 5.0 mJy/beam. Angular resolution is 1.73". Right: 6 cm image. Contours are -0.15, 0.15, 0.25, 0.35, 0.5, 0.75, 1.0, 2.0, 2.5, 3.5 and 5.0 mJy/beam. Angular resolution is 0.5". Chapter 6. GT2318+620 106 Epoch 3, A =20 cm Epoch 3, A =6 cm Figure 6.5: Epoch 3 images of GT2318+620. Left: 20cm image. Contours are -0.35, -0.25, 0.25, 0.35, 0.5, 0.75, 1.0, 1.5, 2.0, 2.5 and 5.0 mjy/beam. Angular resolution is 5.10" x 4.88". Right: 6 cm image. Contours are -0.15, 0.15, 0.25, 0.35, 0.5, 0.75, 1.0, 2.0, 2.5, 3.5 and 5.0 mjy/beam. Angular resolution is 1.73". Chapter 6. GT2318+620 107 the position of the knot. This second feature corresponds to the northern edge of the central core on the epoch 1 and 2 maps. However, when compared to the background noise in the difference image, neither of these features appears strong enough to make any conclusive statement about a change in jet structure. Epoch 1 Minus Epoch 2, A =20 cm «201 *6 - I I I . o 4 I I I -I 0 W DECLINATION (B1SS0) 0 ^ if ^ ^ ' O 00 — o t -D C 3 00 45 _ p 0 I I . I I I rj i i i Figure 6.6: L band difference image; Epoch 1 minus epoch 2. Contours are 0.2, 0.3, 0.35, 0.4, 0.45 and 0.5 mJy/beam For the epoch 3, L band image a morphological comparison is difficult due to the much larger synthesized beamwidth. However, even here the orientation of the jet is similar to the orientation seen in the other two epochs. For the epoch 3, B configuration, C band image, the U-V coverage is similar to the A configuration L band coverage. This allows morphological comparison with the A configuration L band images to be made (keeping in mind the difference in wavelengths). Again however, the orientation and overall length of the jet appear similar to the earlier epochs. Chapter 6. GT2318+620 108 From the MX'ed maps determinations of the flux densities were made. Both the peak flux density of the unresolved core, and the total integrated flux density were measured. These are summarized below. Integrated Flux Densities Epoch 1 Epoch 2 Epoch 3 L Band 20 ± 2 mJy 16 ± 2 mJy 19.7 ±0.4 mJy C Band 9 ± 2 mJy 8 ± 2 mJy 13 ± 2 mJy Peak Flux Densities Epoch 1 Epoch 2 Epoch 3 L Band 6.6 ±0.2 mJy 6.3 ±0.1 mJy 9.2 ±0.9 mJy C Band 4.7 ±0.1 mJy 4.8 ±0.1 mJy 6.3 ±0.1 mJy In looking at the integrated flux densities there is no significant change from one epoch to another. The L band values are also consistent with the total flux density value of 15 ± 2 mJy, measured at 21 cm with the Westerbork radio telescope [Taylor et al. 1991]. There is, however, a large discrepancy between the epoch 1, L band value of 20 ± 2mJy above, and 37 ± 2mjy from [Taylor et al. 1991]. As explained earlier, this can be attributed to artifacts being included as extended emission in the latter image. As for the peak flux densities of the core, again it appears as if no significant amount of variability was measured. Virtually no change was seen between the epoch 1 and epoch 2 flux densities. For the third epoch, the flux densities were notably larger. However this was to be expected, since the larger epoch 3 beam encompassed more of the extended emission, resulting in a larger core peak value. Thus, in conclusion, the epoch 2 and epoch 3 observations cannot support the claim of GT2318+620's identification as a Galactic variable (although, of course, they do not Chapter 6. GT2318+620 109 rule out the possibility either). No morphological or significant flux density changes were seen. This behavior is more consistent with an "ordinary" extragalactic jet source (radio galaxy or quasar), than a Galactic X-ray binary. Chapter 7 Conclusions The original objectives of this thesis were to produce a catalogue of highly variable radio sources from the Greenbank survey data, then to investigate the more promising variables in more detail. While the first objective was achieved, time constraints limited follow-up investigations to the previously discovered source, GT2318+620. With respect to the first objective, sources in the GB6 catalogue were checked for long-term variability by comparison of the 1986 and 1987 flux densities. Bayesian inference was used to determine the odds of variability. The result was a catalogue of 146 long-term radio variables, each having odds of greater than 105 of being variable. Twenty-eight of these variables were identified with sources in the NVSS survey, and one with the FIRST survey, providing accurate position estimates. Fifteen variables were identified with well-studied V L A calibrators, and optical identifications were made for 5 of the variables. For 104 of these variables, short-term variability information in the form of daily flux densities was also obtained. These daily flux densities were determined by using Bayesian techniques to fit one-dimensional beam profiles to the individual scan data. An assessment of short-term variability was made by a x 2 , goodness-of-fit test to a model with a constant flux density in each epoch. Forty-three sources had a minimum x2 inconsistent with this model, implying short-term variability. With respect to the second objective, follow-up V L A observations of GT2318+620 in two different epochs were made. No gross morphological changes in the structure of the source, or significant differences in the flux density levels, were seen. These observations 110 Chapter 7. Conclusions 111 do not support (nor rule out) GT2318+620's identification as a Galactic variable source. In terms of the future, there are many directions this work can be taken. The most obvious extension is to follow-up on one of the original objectives, and make further investigations of the most interesting short/long term variables in this catalogue. For those variables whose positions are known imprecisely, snapshot V L A observations could provide accurate coordinates. Once the coordinates of a source have been determined within arcsecond accuracy, attempts at optical identifications can be made, as well as identifications with sources in other frequency bands. As for the long-term variable catalogue, the author intends to expand the catalogue to include all sources with odds of variability > 104. At this level, the number of false positive identifications will still be acceptably low, while the total number of long-term variables identified will increase by approximately a factor of ten. Finally, improvements can also be made in the procedure to evaluate short-term vari-ability. 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Ser. 8:1 (1973) Appendix A Solution to integrals of the form: J 0 B eax-Px2 dx To solve integrals of the form J0B eax &x2 dx, complete the square in the exponential giving: Jo dx = e 4 Jo = 0--1/2 / e"" 'a = P~ -1/2 /•—oo J 6 Joo = P~ -1/2 - e4" = P~ -1/2 - e4^ da; / • - O O _ 2 fb _ 2 dy — e y dy — e y dy J a. Joo [erf (b) - erf (a)] (A.l) where a a b = ^~trr and the error function, erf(x), and the complementary error function, erfc(x), are defined in the usual way as: and V7T JO 2 7^ dt f°° 2 erfc(x) = —= / e _ t dt. 118 Appendix B Source Flux Density Prior Probability Distributions Source counts as a function of flux density from the GB6 catalogue [Gregory et.al. 1995] were used to estimate the variable model, epoch 1, flux density prior probability, p{k\\M2), and the constant model, average flux density prior probability, p(A;|Mi). Sources were counted and grouped into flux density bins. The number density of sources per stera-dian per Jy, n(S), was then corrected for population law bias [Murdoch et al. 1973], and a plot of S5/2n(S) vs. S was fit to a Gaussian (upper plot in figure B . l ) . For a static Euclidean universe with no source evolution, this plot would be a horizontal line. A plot of p(ki\M2) vs. S was obtained by dividing by the S5^2 weighting factor and nor-malizing (lower plot in figure B . l ) . To calculate p(k\Mi), the procedure was identical except for the final normalization. Before normalizing p(k\Mi), all average flux density prior probabilities < 18 mJy were set to zero to account for the 18 mJy flux density lower limit in the GB6 catalogue from which the sources were selected (flux densities from this catalogue represent an average of the two single epoch flux densities). For normalization of p(ki\M2), prior flux density probabilities down to zero were allowed, since conceivably a source could have negligible flux density in one epoch, then be caught flaring in the second epoch, and thus have an average flux density greater than the 18mJy GB6 cutoff. To determine the prior probability for the second measurement of a source's flux den-sity at epoch t2, given an initial measurement at epoch t\, a simplifying assumption could be made that the two flux density measurements are independent, ie: p(k2\k\,M2) = p(k2\M2) = p[ki\M2). However, for all but the most highly variable sources, this is a bad 119 Appendix B. Source Flux Density Prior Probability Distributions 120 10° 10' 10* 10' 10' F lux Dens i t y (mJy) 10° 10' 10* 10 3 10' F lux Dens i t y (mJy) Figure B . l : Top: Weighted source number density versus flux density. Individual data points are source counts taken from [Gregory et. al. 1995]. Solid curve is the best Gaussian fit to the data. Bottom: p(ki\M2). Individual data points are the normalized source counts from [Gregory et. al. 1995]. Appendix B. Source Flux Density Prior Probability Distributions 121 assumption. For instance, if a source is measured to have a flux density of 1 Jy in epoch t\, in general it would be expected to have a flux density close to 1 Jy in epoch t2 as well. In other words, the two flux density measurements are strongly correlated. To quantify this coupling, a cumulative graph of number of sources vs. degree of variability was used (from [Gregory & Taylor 1986]). This graph was generated from a survey of 1274 discrete radio sources at 5 GHz. On this graph (upper plot in figure B.2) are plotted the number of sources with V >VQ (in %), vs. Vo, where V is a fractional variability index defined by: T / - 'S 'max Sm\n V = - - y — . (B.l) ' J m a x i ' - ' m m To determine p(k2\k\, M2), this curve was binned, and converted from a cumulative dis-tribution function to a number density function (number of sources vs. V). It was then fit to a two component variability model: a steep exponential to mimic the rapid drop in source numbers for V < .35, together with a flatter linear component which dominates at large V. Normalizing this curve generated a probability density function for measuring a source with variability index V (lower plot in figure B.2), from which p(k2\k\, M2) can be calculated for any k\. Appendix B. Source Flux Density Prior Probability Distributions 122 l O O r Figure B.2: Top: Estimates of the fraction of radio sources with V > V0, (modified from [Gregory & Taylor 1986]). Bottom: Normalized probability density function for V. The individual data points were extracted from the cumulative plot above. Curve indicates the best two-component fit to the data.
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Physics
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Compact radio source variability at centimeter wavelenghts
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