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Search and detection of low frequency radio transients

Spreeuw, J.N.

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Citation for published version (APA): Spreeuw, J. N. (2010). Search and detection of low frequency radio transients.

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receptie. TRANSIENTS om 14:00 uur SEARCH AND plaatsvinden in De promotie zal tot het bijwonen tot het verdediging van verdediging van de openbare openbare van de de Agnietenkapel, 231 te Amsterdam Na afloop is er een mijn proefschrift op mijn proefschrift UITNODIGING FREQUENCY RADIO vrijdag 18 juni 2010 vrijdag 18 Johannes N. Spreeuw DETECTION OF LOW LOW OF DETECTION Oudezijds Voorburgwal Voorburgwal Oudezijds radio transients radio of low frequency of low Search and detection and Search Johannes N. Spreeuw

Search and detection of low frequency radio transients Johannes N. Spreeuw i i

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Search and detection of low frequency radio transients

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On-line version: Author contact: [email protected]

Front cover image: The supernova remnant G359.1-0.5 from the discovery observation of GCRT J1745-3009 (highlighted). Back cover image: A wide ﬁeld image of SgrA and surrounding sources, from a VLA-B observation on 1989 March 18 at 90 cm. The supernova remnant G359.1-0.5 (1 degree south of SgrA) is barely visible. GCRT J1745-3009 is not active.

This thesis was typeset in LAT X by theE author and printed by Ipskamp ISBN: 978-90-9024055-8

This research has been supported by The Netherlands Organisation for Scientiﬁc Research (NWO).

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Search and detection of low frequency radio transients

ACADEMISCH PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magniﬁcus prof. dr. D.C. van den Boom ten overstaan van een door het college voor promoties ingestelde commissie, in het openbaar te verdedigen in de Agnietenkapel op vrijdag 18 juni 2010, te 14:00 uur

door Johannes Norbertus Spreeuw geboren te Alkmaar

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Promotiecommissie

Promotor: Prof.dr.R.A.M.J.Wijers Overige Leden: Prof. dr. A.G. de Bruyn Prof. dr. L. B. F. M. Waters Prof. dr. R. P. Fender Dr. S. B. Markoﬀ Dr.R.A.D.Wijnands Dr. M. Wise Prof. dr. R. G. Strom

Faculteit der Wiskunde, Natuurwetenschappen en Informatica

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voor Sanne

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Contents

1 Introduction 1 1.1PreparingforLOFAR...... 1 1.2Sourceextractioninmaps...... 4 1.3 Scientiﬁc expertise from studying transients with ”classical”radio telescopes atlowfrequencies...... 9 1.4 Theoretical predictions for the detection of radio emission from extrasolar planetswithLOFAR...... 11 2 LOFAR’s Transients Key Project: Detailed description of the Source Extraction System 17 2.1Abstract...... 17 2.2LOFAR...... 17 2.3TheLOFARKeyProjects...... 18 2.4TheTransientsKeyProject(TKP)...... 18 2.5 Automated transient ﬁnding in the TKP pipeline ...... 19 2.6 A brief rationale behind the current design of the Source Extraction System . . 24 2.7 The TKP Python Source Extractor (PYSE) ...... 29

3 Zeroth order validation of TKP source extraction and source measurement code 55 3.1Abstract...... 55 3.2Description...... 55 3.3General...... 56 3.4Sourcefreemaps...... 56 3.5 Validation of the TKP background mean and background noise estimation . . . 58 3.6FalseDiscoveryRatealgorithm...... 72

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ii Contents

3.7Deblending...... 75 3.8 Determination of peak ﬂux densities and positions ...... 78 3.9Conclusions...... 84

4 A new perspective on GCRT J1745-3009 89 4.1Abstract...... 89 4.2 Introduction ...... 90 4.3Datareduction...... 91 4.4 The source on the opposite side of the supernova remnant ...... 96 4.5 Overview of ﬂux measurements of GCRT J1745-3009 ...... 97 4.6 Reanalysis of the 2002 discovery dataset ...... 99 4.7Discussion...... 105 4.8Conclusions...... 108 4.9Acknowledgements...... 109 5 Low frequency observations of the radio nebula produced by the giant ﬂare from SGR 1806-20: Polarimetry and total intensity measurements 113 5.1Abstract...... 113 5.2 Introduction ...... 114 5.3 Observation and data reduction ...... 114 5.4Results...... 119 5.5Discussion...... 124 5.6Conclusions...... 126 5.7Acknowledgements...... 126

6 Predicting low-frequency radio ﬂuxes of known extrasolar planets 129 6.1Abstract...... 129 6.2 Introduction ...... 130 6.3 Exoplanetary radio emission theory ...... 131 6.4Requiredparameters...... 134 6.5 Expected radio ﬂux for know exoplanets ...... 142 6.6Conclusions...... 149 6.7Acknowledgements...... 150 6.8Appendix...... 150

7 Samenvatting in het Nederlands 161 7.1LOFAR...... 161 7.2VoorbereidenopLOFAR...... 165

8 Epilogue 175

9 Dankwoord 177

10 Publication list 179

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CHAPTER 1

Introduction

1.1 Preparing for LOFAR

Although radio astronomy started at low frequencies (20.5 MHz), with the pioneering work of Karl Jansky (Kraus 1986), in the decades after the war the focus of radio astronomy shifted to higher frequencies, with higher angular resolution per unit of projected baseline and little ionospheric disturbance. Low-frequency radio astronomy in the US languished in the ’80s when Clarke Lake Radio Observatory was closed down (White et al. 2003). Technical developments, such as optical fibre, have opened the opportunity of achieving high angular resolution at low frequencies, by correlating data from very long baselines. In this way, the sensitivity limitation from noise by confusion of sources can be avoided. Another aspect that has influenced the design of low frequency arrays is the difficulty of designing dishes that are large and sufficiently stiff at the same time and can be built at reasonable cost. Collecting area and other considerations led to the first design concepts of LOFAR (Bregman 2000) in the late ’90s: an aperture synthesis instrument without steerable parts for the frequency range 10-300 MHz, comprised of 30 stations spanning baselines up to 300 km. Each station would consist of a few hundred dipoles. Over the years, the design for the Low Band Antenna (LBA) and High Band Antenna (HBA) has changed. The most recent images of both antenna types are shown in figures 1.1 and 1.2. Also the station layout and the array configuration was altered, particularly after a significant descope. Figure 1.3 shows an aerial photograph of the location of the inner core stations in the very heart of LOFAR on top of a large terp, i.e., an articial hill. The LOFAR core, near Exloo in the province of Drenthe, The

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2 Chapter 1

Figure 1.1: Low Band Antennas as they are presently deployed.

Netherlands, is presently (2010) being completed. There will be at least 18 core stations, 18 remote stations (in the Netherlands) and 8 international stations. These and other planned but yet not (fully) funded international stations are depicted in ﬁgure 1.4.

The research plan for my Ph.D. position was written down in 2004 when only prototype stations, such as ITS (Wijnholds et al. 2004) and THEA, the Thousand Element Array (Bij de Vaate et al. 2003), were deployed. At that time it was anticipated that the roll-out of LO- FAR would be completed by the end of 2006. Originally, I would spend the first two years of my thesis (2005, 2006) on the development of code for the detection of transient sources and apply this code in the last two years (2007, 2008) to actual LOFAR data. The LOFAR science case (De Bruyn et al. 2002) mentioned the search for variable sources and ”LOFAR as an All-Sky Monitor”. Fast radio transients were to be discovered by processing visibilities and/or maps from short integrations in real time. Those maps would together cover a large patch of the sky. They could, in fact, be image cubes, with frequency as a third dimension. All data were to be recorded in full Stokes, so in principle four maps could be made from any integration. The message was clear: the datastream was huge and only conpressed forms of data could be stored indefinitely. Any code written to find transient sources in the LOFAR datastream would have to be fast, efficient and fully automated, performing only tasks that were obviously necessary. Although these requirements were well understood, essential directions were unclear at the

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Introduction 3

Figure 1.2: A close-up of two High Band Antennas.

beginning of this project. For instance, the optimum way of finding transients was not obvious. The ”classic”way has always been to compare the fluxes of all sources in a number of maps. If there were only one source that was substantially brighter or fainter in one map than in all other maps while all the other sources had similar flux, one could be pretty confident of having found a transient source. However, in radio astronomy the maps come from (u, v) data and it was not yet investigated if there were certain techniques that could find transient sources directly in the visibilities. In particular, it was anticipated that the ionosphere would leave phase errors in the data and that perhaps after subtraction of consecutive datasets these errors would cancel out. Generally, there is no straightforward manner of subtracting consecutive sets of visibilities because the (u, v) coverage changes with time, so subtraction cannot be done without interpolation, unless the (u,v) coverage is complete. On the other hand, on very short timescales, on the order of seconds, one may argue that the (u, v) coverage does not change significantly and visibilities may be subtracted baseline by baseline. It turned out that this could yield acceptable results in a quiet ionosphere if ”second-order-differencing”is applied, but subtracting maps pixel by pixel may still work well (Miller-Jones 2006). Still desperately needed are tests in an unstable ionosphere. Things are much simpler if a proper model of the field is available. That model can be used to calibrate a time sequence of (u, v) datasets and also to subtract it from those datasets after calibration. The time sequence

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4 Chapter 1

Figure 1.3: A 2009 photograph of the location of the inner core stations in the very heart of LOFAR. The inner core stations will be built on top of a large ”terp”, i.e. an artiﬁcial hill. This ”super terp”is surrounded by a large ditch.

of dirty images from the residual (u, v) data will show any transient source well above the noise. In any case, it was obvious for the detection of transients, that routines for extraction and measurement of sources in images, dirty or cleaned, were needed.

1.2 Source extraction in maps

Another issue that was unclear at the start of my Ph.D. project was the necessity of writing new software for source extraction. Source extraction generally means the identification of sources in images by segmentation and labeling contiguous pixels (islands). Naturally, source measurement was also needed. By this we mean the calculation, for each island of pixels, of the flux (peak and integrated), position, size and shape. I soon found out that SExtractor (Bertin & Arnouts 1996) could process most FITS images in a second, meeting the required speed for LOFAR. An interesting aspect that needed to be investigated was the accuracy of its source measurements and error bars, since no fitting was done, all source parameters were derived from moments. In the end, SExtractor failed to meet our requirements with respect to optimum accuracy. The (barycenter) positions were

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Introduction 5

Figure 1.4: The bright spots indicate all LOFAR stations that are currently deployed or being deployed, funded and/or planned. Only 8 international stations are presently (2010) funded, 5 in Germany, 1 in France, 1 in the UK and 1 in Sweden.

less accurate than from fitting and there were systematic underestimates of the peak flux density. Both of these problems become prominent at high signal-to-noise levels, as shown in figures 1.5 and 1.6. At that point it became clear that new software needed to be developed, software that would meet the requirements with regard to speed and accuracy. Python was well on its way of becoming a popular programming language for scientific applications when Space Telescope Science Institute (STScI) adopted it for their processing routines. They developed fast routines for segmentation and labeling 1, the very basis of source extraction in images. I

1See http://stsdas.stsci.edu/numarray/numarray-1.5.html/index.html

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6 Chapter 1

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accidentally run into the simplest possible source extraction code, based on those routines, in a tutorial by two people from STScI (Greenﬁeld & Jedrzejewski 2007, paragraph 3.7.9). The most essential commands are listed here:

clipped = where(sci > threshold, 1, 0) (1.1) labels, num = nd−image.label(clipped) (1.2) f = nd−image.find−objects(labels) (1.3)

nd−image is the extension module to numarray written at STScI. The ﬁrst command selects all pixels above the threshold and gives them the value 1. The second labels groups of pixels based on connectivity. By connectivity we mean the level of contiguity. By default we use 4-connectivity which means that groups of pixels only connected at corners are considered separate. This is not the case when one uses 8-connectivity. The last command uses

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Introduction 7

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Figure 1.6: The peak pixel method for photometry on point sources does not reach the theoretical limit of accuracy and high s/n. If the beam is oversampled this eﬀect is lessened. This plot was constructed fromthesamedataasﬁgure2.4.

those labels to select 2D arrays, i.e., groups of pixels. These are the islands that can then be ”measured”, i.e. the peak, total ﬂux, position and shape parameters are determined. In fact, some of the measurement tasks were also included in nd−image:

positions = nd−image.center−of−mass(sci, labels, range(num)) (1.4)

This gives the barycenter positions of the islands. Of course, this is not the final word on the positions of the sources, or we would run into the same accuracy problems as SExtractor. However, the barycenter positions are good initial values for Gauss fitting. Moreover, it became clear that I had found an excellent framework for the extension of these algorithms and that the complete source extraction process could be written in Python. It was reassuring that Python as a programming language continued to develop. The nd−image routines were incorporated in Scipy and Numarray was further developed as Numpy. A complete, but not very detailed, overview of the source extraction process is depicted in figure 1.7. The first part of this thesis comprises a chapter on the description of all subprocesses in this scheme

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8 Chapter 1

Figure 1.7: An overview of the source extraction process in the TKP pipeline

and another chapter on their validation, i.e., the actual checks that the output from the TKP software is correct.

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Introduction 9

1.3 Scientiﬁc expertise from studying transients with ”classical”radio telescopes at low frequencies

Evidently, there was more we could do to prepare for transient searches with LOFAR besides the development of robust and eﬃcient software for source extraction. Perhaps observations of transient sources with ”classical”radio frequencies could reveal new techniques for detecting these sources. Luckily, two of those opportunities arose in the ﬁrst few months after the start of my appointment!

1.3.1 SGR 1806-20 On 2004 December 27 a Giant Flare (GF) of γ-rays from the soft-γ-ray repeater (SGR) 1806- 20 was detected by INTEGRAL, Swift and three other space missions in the third interplanetary network of GRB detectors (Hurley et al. 2005). It caught the attention of the general public as the brightest flash of radiation from beyond the solar system ever recorded. As- tronomers studied the afterglow of the explosion from this magnetar by follow-up observations at various wavelengths (Rea et al. 2005; Israel et al. 2005; Palmer et al. 2005; Schwartz et al. 2005; Fender et al. 2006). In particular, the flux from the radio nebula produced by the explosion (Gaensler et al. 2005a; Cameron et al. 2005; Taylor et al. 2005) was measured very frequently in 2005 January. These observations focused on total intensity measurements at various radio wavelengths between 0.24 and 8.5 GHz and on polarimetry at 8.5 GHz. Some polarimetry was done at lower frequencies, but without the proper correction for the leakages (Gaensler et al. 2005b). One chapter of this thesis describes a set of 19 WSRT observations pointed at SGR 1806-20. It involves polarimetry at 350, 850 and 1300 MHz. Also, the Stokes I flux from the radio nebula at 350 and 850 MHz was measured more accurately by observing the same field again in 2005 April/May. In this way, the background sources could be subtracted from the visibilities of the 2005 January observations. One of the main conclusions from these observations is that we do not see any compelling evidence for significant depolarization at low frequencies, but strong evidence for different polarization angles as shown in figure 1.8, indicating that we are probing substructure that is not appearing at high frequencies. This is actually a quite encouraging result for LOFAR. Imagine that LOFAR would have been fully deployed at the time of the GF: we would have known a great deal more about the evolution of the radio nebula and its magnetic fields, until a few weeks after the explosion.

1.3.2 GCRT J1745-3009 Another opportunity for studying a transient source at low frequencies arose in 2005 March when a paper about a peculiar transient source apppeared in Nature (Hyman et al. 2005). This truly amazing source showed ﬁve bursts at the Jy level each lasting about 10 minutes with a 77 minute recurrence during a 6h observation at 325 MHz on 2002 September 30/October 1. The source is located 1.25◦ south of the Galactic Center and just outside a supernova remnant. Theorists immediately started to investigate if the recurrence time could be attributed to a pe-

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10 Chapter 1

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Figure 1.8: The polarization angles of the radio nebula produced by the Giant Flare from SGR 1806-20 at 350, 850, 1300 and 8400 MHz

riod of rotation (Zhang & Gil 2005), revolution (Turolla et al. 2005) or precession (Zhu & Xu 2006). A nulling pulsar and an ’X-ray quiet, radio-loud’ X-ray binary were also suggested (Kulkarni & Phinney 2005) as well as an exoplanet and a flaring brown dwarf (Hyman et al. 2005). The discovery led to follow-up observations and re-examination of archival data at both 325 MHz and other bands. Those did not reveal a source (Zhu & Xu 2006; Hyman et al. 2005, 2006), with two exceptions (Hyman et al. 2006, 2007). Both of the redetections were single bursts, possibly due to the sparse sampling of these observations. The first redetection was possibly the decaying part of a bright (0.5 Jy level) burst that was detected in the first two minutes of a ten minute scan. The second redetection was a faint short ( 2 minute) burst that was completely covered by the observation. The average flux density during that burst was only 57.9 ± 6.6mJy/beam. This redetection also showed evidence for a very steep spectral index (α = −13.5 ± 3.0). In summary, the source was only detected at three epochs, separated by less than 18 months, all at 325 MHz, while the source was not detected in this band at 33 epochs over a period of more than 16 years (see Hyman et al. 2006, table 1) nor in any other

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Introduction 11

band, ever. We observed the field containing GCRT J1745-3009 with the WSRT at 325 MHz (2005 March) and at 1350 MHz (2005 May), but did not detect the source. My attention was drawn to a source on the opposite side of the supernova remnant that did not appear in a map that was made by concatenating four datasets from the late ’80s (see LaRosa et al. 2000, figure 11). The noise levels in this map were such that it should have been detected at the 10 − 20σ level, so a clear detection was expected. We investigated possibly transient behaviour of this source by reducing these and other observations from the ’80s in the VLA archive. We found that the source was actually clearly present in one of our maps with a sufficiently low noise level and good (u, v) coverage. We concluded that, most likely, the source was in fact concealed by a negative background peak and that the fidelity of the LaRosa et al. (2000) maps is questionable. This can be seen more clearly if one has a look at the larger 4◦ × 2.5◦ and 4◦ × 5◦ images (see LaRosa et al. 2000, figures 1 and 5). A large but unsuccessful effort was put into establishing the transient nature of this source. However, it is actually an important lesson for transient searches with LOFAR, in the sense that the fidelity of maps has to be judged by considering not only noise levels but also a few other citeria, like the presence of many contiguous pixels with negative values. As an exercise, I also re-reduced the discovery dataset of GCRT J1745-3009. After many iterations of self-calibration I attained noise levels in my maps substantially lower than mentioned in the discovery paper (Hyman et al. 2005) and I included those results in a paper on this source. The referee then encouraged me to also re-extract the lightcurve which led to a wealth of new information on the source. For instance, it turned out that all five bursts have the same shape, with a steep rise, a gradual brightening and a steep decay as shown in figure 1.9. We improved the accuracy of the recurrence period by an order of magnitude and we found no evidence for aperiodicity. This means that, in terms of possible models for this source, rotating systems are favoured and that models that predict symmetric bursts can be ruled out.

1.4 Theoretical predictions for the detection of radio emission from extrasolar planets with LOFAR

. Another aspect of successful transient searches with LOFAR, besides the development of software and scientiﬁc expertise is the devising of an observational strategy. A priori, this is a pretty complicated matter. First of all, there are known and unknown transients. For the latter group, an observational strategy can be devised if they belong to a class of known transient radio sources, by interpolating and extrapolating the observational characteristics of their class. For the former group, things could be simpler. However, since the sky is relatively unexplored at LOFAR frequencies, we are not sure what to expect from targeted LOFAR observations of known sources with established transient behaviour at higher frequencies and we still have to do quite a bit of uncertain (frequency) extrapolation. The known extrasolar planets (”exoplanets”) are targets for LOFAR although their transience

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12 Chapter 1

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has never been observed at any frequency. Only in the case of eclipsing exoplanets, was the optical flux from the star slightly variable, i.e., the exoplanet would block a small fraction of the starlight. However, variability of this kind cannot be extrapolated to radio frequencies. In fact, the emission process that may result in the first detection by LOFAR of radio emission from an extrasolar planet is of a completely different origin and similar to Jupiter’s radio storms: cyclotron emission from charged particles from a stellar wind that get trapped in the planet’s magnetosphere. In the case of Jupiter and, more generally, in the solar system beyond 1 AU, the kinetic flow power is much larger than the Poynting flux from the interplanetary magnetic field, but the latter is converted much more efficiently into radio power. In fact, it turned out, when we considered the parameters of all 197 known exoplanets (2007 January), that the conversion of magnetic Poynting flux into radio waves was the most promising mechanism for producing detectable low frequency emission. The results for this mechanism, called the magnetic energy model, are shown in figure 1.10. The equivalent of the Jupiter-Io interaction, called the unipolar model, did not yield any observable emission for this set of exoplanets. We conclude that radio emission from a few of the exoplanets that were known early 2007 could be detected by LOFAR.

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Introduction 13

101 ] 1 −

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Figure 1.10: Maximum emission frequency and expected radio flux for known extrasolar planets according to the magnetic energy model, compared to the limits of past and planned observation attempts. Open triangles: Predictions for planets. Solid lines and filled circles: Previous observation attempts at the UTR-2 (solid lines), at Clark Lake (filled triangle), at the VLA (filled circles), and at the GMRT (filled rectangle). For comparison, the expected sensitivity of new detectors is shown: upgraded UTR-2 (dashed line), LOFAR (dash-dotted lines, one for the low band and one for the high band antenna), LWA (left dotted line) and SKA (right dotted line). Frequencies below 10 MHz are not observable from the ground (ionospheric cutoff). Typical uncertainties are indicated by the arrows in the upper right corner.

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14 Chapter 1

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Bibliography

Bertin, E. & Arnouts, S. 1996, A&AS, 117, 393 Bij de Vaate, J., Wijnholds, S., & Bregman, J. 2003, in Phased Array Systems and Technol- ogy, 2003. IEEE International Symposium on, 359–364 Bregman, J. D. 2000, in Perspectives on Radio Astronomy: Technologies for Large Antenna Arrays, ed. A. B. Smolders & M. P. van Haarlem, 23 Cameron, P. B., Chandra, P., Ray, A., et al. 2005, Nature, 434, 1112 De Bruyn, A. G., Fender, R. P., Kuijpers, J., et al. 2002, Exploring the Universe with the Low Frequency Array - A scientific case Fender, R. P., Muxlow, T. W. B., Garrett, M. A., et al. 2006, MNRAS, 367, L6 Gaensler, B. M., Kouveliotou, C., Gelfand, J. D., et al. 2005a, Nature, 434, 1104 Gaensler, B. M., Kouveliotou, C., Wijers, R., et al. 2005b, GRB Coordinates Network, 2931, 1 Greenfield, P. & Jedrzejewski, R. 2007, Using Python for Interactive Data Analysis Hurley, K., Boggs, S. E., Smith, D. M., et al. 2005, Nature, 434, 1098 Hyman, S. D., Lazio, T. J. W., Kassim, N. E., et al. 2005, Nature, 434, 50 Hyman, S. D., Lazio, T. J. W., Roy, S., et al. 2006, ApJ, 639, 348 Hyman, S. D., Roy, S., Pal, S., et al. 2007, ApJL, 660, L121 Israel, G. L., Belloni, T., Stella, L., et al. 2005, ApJL, 628, L53 Kraus, J. D. 1986, Radio astronomy (Powell, Ohio: Cygnus-Quasar Books) Kulkarni, S. R. & Phinney, E. S. 2005, Nature, 434, 28 LaRosa, T. N., Kassim, N. E., Lazio, T. J. W., & Hyman, S. D. 2000, AJ, 119, 207 Miller-Jones, J. C. A. 2006, Another method of differencing in the uv-plane, Tech. rep., Uni- versity of Amsterdam Palmer, D. M., Barthelmy, S., Gehrels, N., et al. 2005, Nature, 434, 1107 Rea, N., Israel, G., Covino, S., et al. 2005, The Astronomer’s Telegram, 645, 1

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Schwartz, S. J., Zane, S., Wilson, R. J., et al. 2005, ApJL, 627, L129 Taylor, G. B., Gelfand, J. D., Gaensler, B. M., et al. 2005, ApJL, 634, L93 Turolla, R., Possenti, A., & Treves, A. 2005, ApJL, 628, L49 White, S., Kassim, N. E., & Erickson, W. C. 2003, in Society of Photo-Optical Instrumenta- tion Engineers (SPIE) Conference Series, Vol. 4853, Society of Photo-Optical Instrumen- tation Engineers (SPIE) Conference Series, ed. S. L. Keil & S. V. Avakyan, 111–120 Wijnholds, S. J., Bregman, J. D., & Boonstra, A. J. 2004, Experimental Astronomy, 17, 35 Zhang, B. & Gil, J. 2005, ApJL, 631, L143 Zhu, W. W. & Xu, R. X. 2006, MNRAS, 365, L16

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CHAPTER 2

LOFAR’s Transients Key Project: Detailed description of the Source Extraction System

2.1 Abstract

The limitations of a source extraction package (SExtractor) that meets the speed requirements for the Transients Key Project (TKP) are discussed. The necessity for the design of a new source extractor is emphasized. The main components of this new package for the Transients Key Project software pipeline are described. LOFAR maps can be processed in real time. The properties of the sources from those maps and their error bars can be compared with the content of databases and used for alerts and updates, also in real time. The eﬀects of correlated noise in radio maps are taken into account. Sources can be deblended and a False Discovery Rate (FDR) algorithm can be applied. The code is written in Python and meets present day standards for maintainability, ﬂexibility and modular programming.

2.2 LOFAR

LOFAR 1, the LOw Frequency ARray, is an innovative new radio telescope in Northwestern Europe, which will observe the radio sky in the frequency range 30–240 MHz. This frequency range covers the lowest energy extreme of the electromagnetic spectrum, that can be observed from the surface of the earth. Its construction will be completed this year (2010). The main part of this telescope is located in the Netherlands.

1see www.lofar.org

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Radio astronomy actually started at low frequencies (20.5 MHz) with the pioneering work of Karl Jansky (Kraus 1950). Jansky investigated the background noise plaguing transat- lantic short-wave communications for Bell Telephone Laboratories. He then serendipitously discovered the Milky Way and in particular the Galactic Centre, as a strong source of low frequency radio emission. At that time (1931), the sun was at a solar minimum, so Jansky did not discover the active sun as a strong radio source. In the decades after the war, radio astronomy developed mainly at much higher frequencies, where the spatial resolution per baseline length is better and where the ionosphere is much less of a problem. LOFAR correlates antennas separated by hundreds or even thousands of kilometers such that sources are separated which would otherwise be confused in blurred images. The jittering eﬀect of the ionosphere can be compensated by the application of algorithms that have been developed over the last years. These algorithms require substantial computer power, but that is now much less of a problem than in the previous century.

2.3 The LOFAR Key Projects

LOFAR science was originally divided in four ”Key Projects”: The Epoch of Reionization (EoR), Transient Sources, Deep Extragalactic Surveys and Ultra-High Energy Cosmic Rays (UHECRs). The focus of ”EoR”is on the early universe using the redshifted 21 cm line. The most distant radio galaxies, diﬀuse emission in galaxy clusters and star-forming galaxies are among the targets of the ”Surveys”Key Project. ”UHECRs”tries to answer one of the out- standing questions in astrophysics: ”Where do the highest energy particles come from and how were they made?”The Transients Key Project is described in some detail in the next section. As more antennas were deployed over the last years, the number of Key Projects grew; ”Cos- mic Magnetism”and ”Solar and Heliospheric Physics”were founded a couple of years ago.

2.4 The Transients Key Project (TKP)

The Transients Key Project (TKP) aims to study all variable and transient sources detected by LOFAR, including pulsars, gamma-ray bursts, X-ray binaries, radio supernovae, ﬂare stars and extrasolar planets. One of the main strategies of the TKP entails the continuous monitoring of a large area of sky, with the goal to detect many new transient events, and provide alerts to the international community for follow-up observations at other wavelengths. This mode of operation is called the Radio Sky Monitor (RSM) mode. In RSM mode maps are produced on timescales varying from 1s up to tens of seconds, minutes or even hours, when the ”classical”source confusion limit is reached, i.e., when the restoring beam cannot separate between adjacent sources. The integration time needed to reach that confusion limit depends strongly on the observing frequency and the size of the array (core or full array). However, when properly calibrated diﬀerenced images are used, it should in principle be possible to reach the thermal noise. The TKP will also piggyback on the observations of other KPs and observers, in order

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to trace and track transient sources in real time. Besides that, it will be monitoring a number of the known transient sources for long periods of time.

The TKP has been subdivided into ﬁve basic scientiﬁc working groups:

• Jet sources: AGN, GRBs, accreting white dwarfs, neutron stars and stellar-mass black holes

• Pulsars: classical radio pulsars, AXPs, RRATs

• Planets: solar system objects and exoplanets

• Flare stars: M, L, and T dwarfs and active binaries

• Serendipity: hitherto unexplored parameter space

2.5 Automated transient finding in the TKP pipeline 2.5.1 Successive images It is anticipated that most of the TKP target sources will be compact, so very likely unresolved. Consequently, these sources will be detected more easily in images than in visibilities. Transient phenomena involve variability in brightness at fixed positions on the sky which can be traced by comparing maps from different epochs, e.g., maps that were acquired in RSM mode. Hence, not only the fidelity of short exposure (snapshot) images but also the accurate and reliable processing of images from different epochs will be a cornerstone of the success of LOFAR and the TKP. When processing successive images, it is most essential that all sources in an image are measured in less time than the time interval between those images, to keep the delays from alerts as short as possible. This is also important for the timely freezing of the station ”Transient Buffer Boards”(TBBs), thereby saving valuable high time resolution data.

2.5.2 Pipeline concept and overview The processing of images or differenced images is actually only the start of the TKP pipeline. After sources have been either detected and measured or monitored, sources will be classified and the measurements will be archived in a database. The interaction of the monitoring process with the database of known and previously measured sources is shown prominently in figure 2.1. All of this is set up as an online system, able to send out immediate alerts when new sources appear or when known sources show interesting or unexpected behaviour in terms of brightness or polarization fluctuations. These alerts could result in a direct rescheduling of the observing program, the broadcasting of SMS messages or the sending of emails. Figure 2.1 also mentions a ”monitoring position list”. The idea is that the monitoring of faint sources should not be hindered by detection thresholds. The fluxes of these sources can be measured

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relatively straightforward, if they are unresolved, by converting their celestial coordinates to pixel coordinates and subsequently ﬁtting the clean beam to their positions in the images. Figure 2.2 takes into account the fact that, in general, we will not process images, but rather image cubes for each time interval corresponding to the sampling time of the visibilities. These cubes have extra dimensions with respect to plain images, i.e. the four Stokes values I, Q, U and V and spectral channel. If the ﬂuxes of sources are measured per spectral channel, a spectral index can be derived immediately.

2.5.3 Source Extraction and Measurement Principles

Source extraction involves the detection of sources above a certain threshold. A complete, but not very detailed, overview of the source extraction process is depicted in figure 2.3. Source measurement is the next step, a group of pixels is then described in terms of a model, usually a Gaussian with six free parameters. Monitoring of known sources generally entails fitting a Gaussian with less than four free parameters, because the position is fixed, if ionospheric refraction is properly accounted for. If it is unresolved, one free parameter (the peak flux) will suffice. It is worth noting that faint sources can only be monitored effectively if they are unresolved or if their shapes are known in terms of Gaussian parameters, i.e., axes and position angles. The reason is that a Gaussian fit with more than one free parameter is likely to fail if the peak flux is not higher than a few times the noise. Alternatively and possibly more effectively, instead of the images, the difference of two successive images can be analysed and inspected. This presumes that we can come up with an appropriate algorithm to handle ionospheric disturbance, such that sources are not moving around in successive images. If we succeed in doing this, the difference (pixel by pixel) of the two images will show the flux rise or decay of a transient source accurately. Constant sources will cancel out in the differenced image.

Requirements

I have listed above the main goals of the Transients Key Project as well as its prime target sources. As noted, these sources are most easily detected in images while the most straightforward way of tracing any transient behaviour is by measuring ﬂuxes in a sequence of images. From these facts we can derive the basic requirements for the source extraction and measurement code:

• Complete detection of all sources above a user given threshold and the rejection of all sources below that threshold.

• Measurement of source parameters, in terms of a Gaussian model, of the detected sources with the highest possible accuracy, i.e., reaching the theoretical limits.

• Likewise with respect to the monitoring of known sources, regardless of the threshold.

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Storage − residual imgs − uv data

LOFAR uv data residual imgs

Differencing Monitoring source detection (all) uvdiff and image image and diff bright? img img source detection (split) source detection rms of undetected

source params

all known coords detected? Database

Response view known pointing yes real source? classifier master no pointing

action update db

TBBext. reconfig freeze trigger

Figure 2.1: Interactions of the TKP pipeline processes with the database, taken from Law (2007)

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Start

In parallel "Differencing pipe" "Extraction & measurement pipe"

Image Retrieve list Monitoring differencing Other image cubes of monitoring positions from position list Differenced Pointing Table image cube Input image cube

Detect all Source Per-plane Extract sources sources in position transient from all channel 0 list Positions of list planes transients

Merge: produce master list of all positions to be measured Merge per-plane transient lists

Measure flux Potential action at each position on e.g. bright in each plane Working table source

Do not proceed until working table fully populated

Merge working Source table entries list

For each source

Is source a Transient new transient? list Yes

Flag for monitoring

Build spectrum & calculate derived quantities

Add to Pointing Table Pointing table

Asynchronous updates

Notify classifier Classifier of update

On-line pipeline finished. i Figure 2.2: Data ﬂow through the TKP pipeline, taken from Swinbank (2007), slightly modiﬁed. i

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Figure 2.3: An overview of the source extraction process in the TKP pipeline

• Complete processing of successive images in less time than the time interval between those images ( 1s) and the simultaneous processing of maps from stacked data, from logarithmically spaced time intervals. • Robustness. Millions of images will be processed and billions of source measurements will be performed. Of course, the processing of images should not stop at any point, but besides that there are requirements with respect to the robustness of the source measurements. Any least-squares algorithm, Gauss-Newton, L´evenberg-Marquardt or other, can diverge without crashing occasionally. If not taken care of this could lead to

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sending false alerts to the outside world. The TKP pipeline software can also provide the source parameters derived from moments. The parameters from ﬁtting should be compared with ”moments”before alerts are sent.

In theory, it was possible that some of these requirements could not be met at the same time. For instance, in the beginning it was thought that source measurement by least squares fitting would slow down the image processing too much. This turned out not to be the case, the time to fit one source turns out to be less than 20 ms. We did not encounter any other possible conflicts between requirements, except for the deblending algorithm. We chose a deblending algorithm that was fast rather than optimum in terms of separating sources that are very close together. This has the advantage that less time is wasted on the deblending of extended sources, which is useless. It was shown 2 that the simultaneous fitting of multiple Gaussians to an extended source causes serious delays in image processing. The speed requirement was, of course, ill defined because it was not kwown what computing power would be available at the time of the deployment of the LOFAR stations. Also, the number of maps and their sizes produced per visibility sampling time was not yet fixed. Presently (2010), it is anticipated that images will not be produced every seconds but rather every 10 seconds in the first year after the commissioning of LOFAR has been completed. It was clear that our code would be optimised for speed by others at a later stage. The design of the code was functional rather than complicated and we were reluctant to implement any piece of code of which the necessity was not immediately obvious. In Stokes I images sources are indicated by (large) positive pixel values. Source detection in total intensity images makes use of this. In Stokes Q, U and V images the presence of polarized sources is indicated by both negative and positive pixel values. Consequently, these images cannot be processed in the same manner and the source extraction code needs to be adjusted. These adjustments have not been implemented yet.

2.6 A brief rationale behind the current design of the Source Ex- traction System

Designing a new source extraction system was regarded as an option, but not as a necessity. This means that the implementation of another freely available source extraction package in the TKP pipeline was not ruled out from the beginning. We tested SExtractor (Bertin & Arnouts 1996) extensively because it can be run from a Unix shell and because of its speed. See Mohan & Röttgering (2005) for a comparison of the speed of SExtractor with SFIND (Miriad) and SAD (AIPS). At that time it was thought that the other packages were much slower because source parameters were derived by Gauss fitting. SExtractor, on the other hand, calculated source parameters from moments. Later, it turned out that Gauss fitting need not be a bottleneck in terms of speed. Also, in the beginning, SExtractor passed several tests with regard to accuracy of photometry and astrometry. In the end, however it failed to reach the theoretical limits, whereas Gauss fitting does reach them. The formulae below for post-fit

2A. Usov, private communication

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Figure 2.4: The barycenter method for calculating the position of a source does not reach the theoretical limit of accuracy and high s/n. If the beam is oversampled this eﬀect is lessened. For this test we pixellized a ﬁxed Gaussian and added correlated noise from random locations in a source free radio image after we scaled it in order to create a range of s/n ratios.

r.m.s. errors were taken from Fomalont (1999) and are commonly used in astronomy. σ(x0) and σ(A) are the errors associated with the position and peak ﬂux density, while σ√is the local r.m.s noise. μx is the ﬁtted width, it is equal to the semi-major axis divided by 2ln2, presuming the x-axis and semi-major axis are aligned.

σμ σ(x ) = x (2.1) 0 2A σ(A) = σ (2.2)

These formulae are actually simplifications; more accurate formulae were derived by Condon for correlated and uncorrelated noise (Condon 1997). Equation 2.1 states that the accuracy of astrometry increases linearly with signal-to-noise. That is not what we find from our tests when we use moments, cf. equations 2.50 and 2.51, see figures 2.4 and 2.5. This malfunction becomes apparent at high signal-to-noise and/or bad sampling of the synthesized beam. Equation 2.2 says that the standard error in photometry is about equal to the local rms noise. The peak pixel method in SExtractor finds the correct values at low signal-to- noise, with a standard deviation of about the rms noise, but severely underestimates peak flux

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ï ð ñ ò ñ ó ô ò õ ö ÷ ô õ ø ñ õ ó ù ú û ô û ñ ó ÷ ù õ ü ý ñ ú þ ÷ ü ¢ ô £ ý ÷ ô õ ¥ § © ô ù ó

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Figure 2.5: The peak pixel method for photometry on point sources does not reach the theoretical limit of accuracy andhighs/n. If the beam is oversampled this eﬀect is lessened. This plot was constructed from the same data as ﬁgure 2.4.

densities at high signal-to-noise. The origin of this discrepancy is immediately clear in noise free conditions: only when the source is centered on a pixel can we find the correct value, in all other cases we will underestimate the true value. There is another way of measuring peak flux densities of sources, which is called the ISOFLUX method in SExtractor. The isoflux method in SExtractor adds all pixel values in an island, from this volume the peak pixel value can be computed if the beam shape is known. As is shown in figure 2.5 this method does not seem to systematically over- or underestimate the true peak flux densities, but its accuracy is not optimal. Let me describe that method in somewhat more detail. The volume, V, under a circular Gaussian with peak, C, and HWHM axis, s, down to a threshold, T, is given by:

πs2 V = (C − T) (2.3) ln 2 The peak of the Gaussian can therefore be found by computing V from the sum of all the pixel values within the island.

ln 2 C = V + T (2.4) πs2

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The absolute accuracy of this method actually decreases when more pixels are used because we are computing a sum instead of an average. The relative error in C,i.e.ΔC/C, does decrease with signal-to-noise, as I will show below. 2 If N is the number of pixels of the island and πs = Ndep > N is the number of pixels in the synthesized beam, then the error in the peak ﬂux density ΔC can be expressed in terms of the error in the volume ΔV = Nσ

ΔV ln 2 N ΔC = = σ ln 2 (2.5) Ndep Ndep assuming that all the errors in pixels add up coherently if the total number of pixels is smaller than Ndep. Of course, we can express N in terms of the peak, threshold and HWHM axis, like this:

ln C N = N T (2.6) dep ln 2 We can combine equations 2.5 and 2.6, valid only for very marginal detections, less than 2σ:

C ΔC = σ ln (2.7) T If the number of pixels covers more than the synthesized FWHM beam the errors do not add up coherently anymore. We have to replace ΔV = Nσ by

2 ΔV = σ (N mod Ndep) + (N − (N mod Ndep))Ndep (2.8)

Of course, this is an approximation, because the transition between correlated and uncorrelated pixels is not so distinct. In general, for a detection with a high signiﬁcance we can approximate

ΔV = σ NNdep (2.9)

So the error in determining the peak ﬂux density using the ISOFLUX method in SExtrac- tor is given by

N ΔC = σ ln 2 (2.10) Ndep

This equation can be simpliﬁed somewhat by combining it with equation 2.6:

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C ΔC = σ ln 2 ln (2.11) T Clearly this error increases√ with C/T. The relative error does decrease, since limC→+∞ ΔC/C ∝ limC→+∞ ln C − ln T/C = 0. Another nuisance of this method is the fact that one needs to have a priori knowledge of the compactness of the source. Only the peak flux densities of unresolved sources can be determined in this manner. There is also an extra error in computing the volume which comes from the finite size of the pixels. This corresponds to the difference between exact integration and numerical integration. The size of this error can be computed using formulae for the trapezoidal rule in two dimensions. For one dimension, the error made by the trapezoidal integration is easy to compute, but for two dimensions the formula for that error is not easily available. We find that the typical fractional error converges to 1e-4 at very low thresholds. This fractional error is expressed relative to the peak of the Gauss. We derived this fractional error from our test runs in noise free maps with a pixel sampling of 4 pixels per FWHM beam, in both dimensions. The Gaussian was centered at position (0.65, 0.91), deliberately not on the center of a pixel, which corresponds to (0.5, 0.5). In principle, one could overcome the obstacle of decreasing accuracy with increasing C/T, by setting higher thresholds for bright sources. This, however, introduces an extra source of error unless the psf is heavily oversampled. This comes from the uncertainty in the lower limit of the numerical integration, assumed to be the threshold T. That lower limit is actually unknown; we just know that the lowest pixel values used to compute the volume are higher than T. Actually, we are integrating down to whatever the height of the Gauss is along the outer edges of the outer pixels of the source island. That error is equal to the derivative of the Gauss along the radial coordinate r times the pixel increment (=1), so it√ is smallest near the top of the Gauss and far away from the center. It is largest at r = ±s/ 2ln2= ±σr. We can compute that error like this: 2Cr ln 2 r2 ln 2 2T C ΔC =ΔT = exp(− ) ln 2 ln (2.12) s2 s2 s T The maximum error is given by C 2ln2 ΔC =ΔT = (2.13) s e That error is a large fraction of the peak. In our test runs we had 4 pixels per FWHM beam in both dimensions. This√ means s = 2 and the fractional error is about 36%! This occurs at thresholds near T = C/ e. The error from equation 2.12 has to be added in quadrature to the error computed from equation 2.7. If T is not much less than C, the number of pixels in the island is very limited. In this case T varies significantly as a fraction of C along the edges of the pixels. This means that formula 2.12 is only a rough estimate.

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I did not find a way to tweak the ISOFLUX method in SExtractor to make it perfom optimally with regard to photometry. Consequently, I conclude that there is no simple way to make the performance of SExtractor optimal with regard to photometry. This problem becomes apparent at high signal-to-noise. When using the maximum pixel method, the un- derperformance is obvious from figure 2.5. The deviations are smaller if the maximum pixel method is applied on images with an oversampled synthesized beam. The underestimates of the fluxes still occur, but at higher signal-to-noise. If SExtractor photometry is done with the ISOFLUX method, the relative errors decrease with signal-to-noise. However, the theoretical limit to the accuracy of photometry is an absolute error and the ISOFLUX method cannot reach this, although it does not systematically underestimate the peak flux density, like the maximum pixel method does. As explained in paragraph 2.7.5, it is possible to do a tweaked form of moment analysis. This will do almost optimum photometry without fitting, by solving a transcendental equation. The problem that remains, even with ”tweaked”moments, is the accuracy of astrometry. It is impossible to reach the theoretical limits at high signal-to-noise without fitting. Hence, the SExtractor package was excluded for use in the Transients Key Project software pipeline. The other main reason is that the SExtractor framework is such that making adjustments to the source code, e.g., changing the interface, is involved.

2.7 The TKP Python Source Extractor (PYSE) 2.7.1 General This document provides a detailed description of the TKP source extraction (SE) system, revision 1325. The level of detail is such that any astronomer experienced with programming in Python should be able understand and modify the code.

2.7.2 Basic Idea A ﬂowchart describing just the source extraction system is very simple, see ﬁgure 2.6.

threshold

Location of source source extraction Map pixels

Figure 2.6: The essence of source extraction.

The essential part in source extraction is provided for by a few simple python commands, originally coded at Space Telescope Science Institute (STScI)3:

3http://stsdas.stsci.edu/numarray/numarray-1.5.html/node98.html

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clipped = where(sci > threshold, 1, 0) (2.14) structuring−element = [[0, 1, 0], [1, 1, 1], [0, 1, 0]] (2.15) labels, num = nd.label(clipped, structuring−element) (2.16) f = nd.find−objects(labels) (2.17)

My interest for coding source extraction routines in Python was drawn by an example from a Python tutorial by Greenfield & Jedrzejewski (2007). The readiness of Python for source extraction was one of the key reasons to choose this programming language. The routines mentioned used in equations 2.16 and 2.17 were originally only in the stsci−python package from STScI, they were later also included in Scipy. I will briefly describe the 4 steps corresponding to equations 2.14 through 2.17. The input is the map sci with pixel values, so sci is a two-dimensional array. The pixel values above the user-defined threshold are set to 1 while the others are set to 0. The structuring element defines which groups of pixels are considered contiguous. In this case 4-connectivity is chosen, the default in the scipy.ndimage package (imported as nd), which means that groups of pixels only connected at corners are distinct. This is not the case when 8-connectivity is chosen. The structuring element for 8-connectivity is given by [[1, 1, 1], [1, 1, 1], [1, 1, 1]]. Labeling of the islands is done by the label command, which means that each group of pixels is given a value. The find−objects command provides a list of slices for cutting out the source islands from the map. Here is an example.

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sci = array([[1, 2, 2, 1, 1, 0], [0, 2, 3, 1, 2, 0], [1, 1, 1, 3, 3, 2], [1, 1, 1, 1, 2, 1]])

clipped = array([[0, 1, 1, 0, 0, 0], [0, 1, 1, 0, 1, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 1, 0]])

s = [[0, 1, 0], [1, 1, 1], [0, 1, 0]]

labels = array([[0, 1, 1, 0, 0, 0], [0, 1, 1, 0, 2, 0], [0, 0, 0, 2, 2, 2], [0, 0, 0, 0, 2, 0]])

sci[f[0]] = array([[2, 2], [2, 3]]) sci[f[1]] = array([[1, 2, 0], [3, 3, 2], [1, 2, 1]])

Please note that the ﬁnal source islands, sci[f[0]] and sci[f[1]] are rectangular with pixel values below the threshold for analysis. These pixels are to be masked when Gauss ﬁtting is done or when moments are computed.

2.7.3 One step back, construction of noise and background maps The scheme depicted in figure 2.6 is too simple in general for two reasons. First, there are often background levels in maps (positive or negative) that need to be subtracted before accurate flux measurements can be done. The cause of background levels in radio maps is due to missing spacings (Högbom 1974). This was also mentioned by Briggs, Schwab & Sramek (1999): ”An undersampled large-scale emission region may introduce large undulations in image intensity that are hard to remove”. These undulations can run diagonally across the image. In that case, accurate flux measurements are still possible because a background map can be computed which should be subtracted from the image.

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Background calculation

There are several ways to assess the background level in (part of) a map. The background level in a source free map is simply equal to the mean of all the pixel values, assuming it is does not vary across the map. When sources are present, it is approximated by the mode of the pixel values. This is an approximation because the mode can be shifted away from the true value, i.e., the mean background in the source free case, by pixels from weak sources and the outer pixels from strong sources. There is a direct way to find the mode, by constructing a histogram of pixel values. An exception may, however, come from maps that are confusion limited where the mode can be shifted by the presence of many sources. Another exception comes from extended sources that can dominate large part of a map. In this case the background cannot be computed reliably. The best thing to do in this case is to interpolate the background levels around this extended source or by applying a median filter that selects the most appropiate nearby background level. In other cases the overwhelming majority of all pixels are noise pixels and the mode is found at the peak of the histogram. Two difficulties remain. First the mode may be ill-defined because the histogram can be almost flat near the mode. Second, the mode is dependent on the bin size. There is no solution to the former issue, but the latter was addressed by Patat (2003), by the introduction of ”The Optimal Binning Technique”. Other tasks, like SFIND in Miriad, fit a Gaussian to the histogram in order to compute the noise mode and its standard deviation (Hopkins et al. 2002), without applying the Optimal Binning Technique. In principle we could use this approach. However, we want to trace variations of the background and noise across the image. In order to do so we would have to make histograms in small subimages. Patat (2003, paragraph 3) states that it is unlikely that the statistics in such a subimage will be good enough to derive the background and noise from a histogram. Bertin & Arnouts (1996) argue that the method developed by Bijaoui (1980) is probably the most unbiased but very noisy in small samples. They also note that this method requires excessive computing time. This is actually true for all methods that make use of histograms. We therefore refrain from any method that requires histogramming maps. Instead, we tweaked the method developed by Bertin & Arnouts (1996) for calculating the mean and standard deviation of the background noise. The original method was incorporated in the SExtractor source extraction package. It involves κ, σ clipping around the median until convergence. After each clipping, the median and standard deviation are recomputed. At the start all pixels are used to compute the median and standard deviation, after that the pixel range is set by the median ±3σ. Convergence is achieved when all pixels are within this range. We implemented κ, σ clipping slightly differently from SExtractor, by clipping ±nσ around the median instead of ±3σ.Also,σ, the standard deviation of the clipped distribution, is corrected for ”clipping bias”. This correction is needed because the rms of the clipped distribution underestimates the true noise rms. Both of these ”tweaks”are explained in paragraph 2.7.3. After convergence, the mode is estimated from the following formula:

mode = 2.5 · median − 1.5 · mean (2.18)

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if the distribution of pixel values is not too skewed. This requirement is quantiﬁed by

|mean − median|/sigma ≤ 0.3 (2.19)

If the distribution is strongly skewed we adopt the median, just as SExtractor does it. To my knowledge there are, however, no rigid justiﬁcations for the distinction made by the skewness criterion from equation 2.19. In fact the skewness check is a bit odd, since close to that skewness limit there is a step in the mode of size 0.45σ. This is easily seen by setting

|mean − median|/sigma = 0.3 (2.20)

which results in

mode = median − 0.45 · σ (2.21)

if the mean is larger than the median. Intuitively, one would favour a continuous transition from the skewed to the non-skewed regime. This has not been implemented yet because it is not trivial how the appropriate formula should be derived. Formula 2.18 differs from the usual empirical relationship (mode ≈ 3 · median − 2 · mean), see, e.g., Kenney & Keeping (1962), which holds for unimodal distributions of moderate asymmetry. We adopted formula 2.18 instead, because tests by Bertin & Arnouts (1996) proved it more accurate for clipped distributions. In very old versions of SExtractor there is a distinction in calculating the mode between crowded and uncrowded fields. If σ changes less than 20% during the process of κ, σ clipping, the field is considered not crowded and the mean of the clipped distribution is used to estimate the mode. Later the SExtractor code was changed to use formula 2.18 or the median to estimate the mode, depending on the skewness test from formula 2.19 4. The mean was no longer used to estimate the mode. The final background map is derived by interpolating the node values from the all the subimages that constitute the map. The size of the subimages is specified by the user. The interpolation can be of any order, for instance bilinear or bicubic spline.

Noise calculation SExtractor determines the background characteristics by κ, σ clipping. When this has con- verged, i.e. when all remaining pixels in a subimage are within 3σ from the median, the true background rms noise is calculated as the standard deviation of the clipped distribution. The boundary of 3σ in SExtractor is somewhat arbitrary. No pixels in a source free subimage should be clipped. However, if its size is large enough there will be pixels with values that diﬀer more than 3σ from the median. Naively, it seems that 3σ clipping works perfectly if the size of a source free subimage were small enough. In that case Gaussian statistics would predict that less than one pixel exceeded the 3σ boundary. On the other hand, if the subimage were too small, 3σ would be too large a boundary for clipping. This could result in source pixels remaining unclipped.

4See my discussion with E. Bertin at http://terapix.iap.fr/forum/showthread.php?tid=267

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From this reasoning it can be inferred that the ideal size of a subimage for 3σ clipping is 185, since 3σ corresponds to 0.27% and 185 · 0.27% = 0.5, i.e. ”half a pixel”would be clipped. If the number of independent pixels in the subimage were signiﬁcantly smaller or larger the determination of the rms noise would be biased. In a realistic source extraction system, the size of the subimage is chosen by the user, of course. To accomodate this, we decided to implement κ, σ clipping slightly diﬀerently from SExtractor by adjusting the limit for clipping based on the number of independent pixels Nindep in the subimage or the number that is left after a number of clipping iterations. The limit n · σ for clipping can be estimated because the number of source pixels is expected to be a minor fraction of the total number of pixels:

√ 1 n ≈ 2 · ErfcInv( ) (2.22) 2 · Nindep

Here ErfcInv is the inverse of the Complementary Error Function. Nindep is usually calculated as the total number of pixels N after zero or more clippings divided by the number of pixels Ndep in the synthesized beam. The clipping boundary is recomputed after each clipping iteration based on the number Nindep and standard deviation of the remaining pixels, using equation 2.22. If this were a plain standard deviation, as used by SExtractor, we would underestimate the true noise, so we implemented a correction for ”clipping bias”. This correction is easy to derive:

+∞ 1 −x2 σ2 = √ x2e 2σ2 dx (2.23) σ 2π −∞ 2 +D −x √1 2 2σ2 − x e dx σ2 = σ 2π D 2 (2.24) meas +D −x √1 2σ2 − e σ 2π D√ 2πErf( D√ ) σ2 = σ2 σ 2 √ 2 (2.25) meas − D 2πErf( D√ ) − 2 D e 2σ2 σ 2 σ σ2 Here, D is the clipping limit, meas is the measured variance of the clipped distribution and Erf is the Error Function. The ratio D/σ, i.e., the clipping limit in units of the true rms noise is simply equal to n as derived from equation 2.22. The size of the subimages correspond to the maximum area over which the noise and background are close enough to constant, from the perspective of the user. The more ambitious user will set it as small as possible in order to trace minor variations in noise and beackground level in the map but still large enough to do accurate statistics. ”large enough”implies that κ, σ clipping can separate source pixels from noise pixels. In order to do so it is necessary that the number of source pixels is a minor fraction of the total number of pixels. If the density of the sources in some part of a map is too high, some of the noise and background node values will be unreliable. In that case, these values can be replaced by the median of the surrounding

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background and noise node values. In determining the median, a median filter is applied to a two dimensional window of a size specified by the user. Alternatively, the user may set a subimage size so large that any of the subimages in the map will cover enough source free pixels. It should be clear to the user that if no precautions are taken, the noise and background will be overestimated near large concentrations of sources. The final noise map is derived in the same way as the background map, by the interpolation of the node values. It is not advised to use bicubic spline to interpolate the noise grid. Al- though the noise grid will only have positive values, a bicubic spline interpolation can result in negative(!) values in the final noise map if there are large differences between contiguous nodes. A bilinear interpolation of the noise grid, on the other hand, will never result in negative values.

2.7.4 Thresholding General For all types of thresholding there is the problem that a population of sources with fluxes just below the threshold will either be missed or overestimated in a time sequence of images. A consequence of this is that the average estimated flux of these sources over all positive detections will be too high. This is known as Malmquist bias or truncation bias. Apart from this problem, the observed distribution of source fluxes in an image can generally not be assumed to be representative of the true source distribution. In the unrealistic case where a complete population of sources is detected well above the threshold will the means of the observed and measured distributions agree. The shapes of the true and measured distributions will generally not agree, not even in the latter case. The problem of recovering the true distribution of fluxes that have been scattered by Gaussian noise was first addressed by Eddington (Eddington 1913, 1940). The average shift as a function of flux resulting from the convolution of the intrinsic distribution with Gaussian noise is referred to as Eddington bias. It was emphasized by Teerikorpi (2004, paragraph 3) that the bias itself is a threshold independent effect and that it vanishes not only if the intrinsic distribution is flat, but also when it is a linear function of the true flux. The reason for this is that the convolution of a linear function with a symmetric Gaussian does not change the function. ”Threshold independent”here means that the bias occurs at all flux levels, even well above the detection threshold, but only if one considers the sources that appear between certain flux limits. Again, the bias vanishes if one averages over a population that has been detected completely. The intrinsic distribution can be recovered from the observed distribution as a series expan- sion of the observed distribution (Eddington 1913, 1940; Teerikorpi 2004). In general, however, there is the practical problem of determining higher order derivatives of the observed distribution accurately. Chandrasekhar & Münch (1950) faced a similar problem when trying to derive the true distribution of the rotational velocities of stars from a sample of the apparent rotational velocities. It is much easier to derive moments of the true distribution than the true distribution itself. Strictly speaking, the latter is impossible in the case of Eddington bias since the accuracy in determining derivatives decreases with the order of the derivative. Hogg & Turner (1998) solved a slightly different problem: given a measured flux F with

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signal-to-noise r = F/σ, what is the most likely true flux S ML, assuming that it belongs to a class of sources with a known (true) flux distribution? If the (true) cumulative number of sources N increases down to lower fluxes as the flux to the power −q = d log N/d log S ,the most likely true flux S ML is given by the maximum of the conditional probability P(S |F): given a measured flux F, what is the most likely true flux. They use:

P(F|S )P(S ) P(S |F) = (2.26) P(F) 2 − (S −F) P(F|S ) ∝ e 2σ2 (2.27) − + P(S ) ∝ S (q 1) (2.28) 2 − + − (S −F) P(S |F) ∝ S (q 1)e 2σ2 (2.29)

The value S = S ML for which P(S |F) is maximal is then easy to ﬁnd:

+ S ML = 1 + 1 − 4q 4 1 2 (2.30) S 0 2 2 r This is equation 4 from Hogg & Turner (1998). From this equation these authors conclude that flux measurements done at signal-to-noise ratios of 4 or less are practically useless. From equation 2.30 one may be tempted to infer that, if one knew the intrinsic, unbiased, distribution of sources, it could be reconstructed by applying equation 2.30 to all measured fluxes. This statement is incorrect. If we were to apply equation 2.30 to all measured fluxes in an image than it is easy to show that the maximum likelihood concept is inappropiate, for instance by considering a bimodal intrinsic source distribution:

dN ∝ C δ(S − S ) + C δ(S − S ) (2.31) dS le f t le f t right right with

Cle f t > Cright (2.32)

where δ indicated the Dirac δ function. Now, the most likely true flux value for any measured flux will always be S le f t. This implies that the average of the most likely true fluxes also equals S le f t. This is not equal to the average true flux, (Cle f tS le f t +CrightS right)/(Cle f t +Cright). Hogg & Turner (1998) acknowledge that confidence intervals are more robust than maximum likelihood estimates, but refrain from using them because of a normalization problem. It is a pity that more authors have copied their method, see, e.g., Herranz et al. (2006); Wang (2004). If one wants to avoid flux biases, instead of deriving the most likely true flux for every mea- i sured flux F one should calculate the expected value S exp(Fmeas), like this:

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+∞ SP(S |Fi )dS +∞ i = 0 meas = | i S exp(Fmeas) +∞ SP(S Fmeas)dS (2.33) | i 0 0 P(S Fmeas)dS +∞ | i = ﬀ since 0 P(S Fmeas)dS 1. If the di erential source counts are well described by a power-law, dN/dS ∝ S −(q+1), we can write:

(S −Fi )2 +∞ − − meas +∞ S qe 2σ2 dS i = | i = 0 S exp(Fmeas) SP(S Fmeas)dS i 2 (2.34) +∞ − (S −Fmeas ) 0 −(q+1) 2σ2 0 S e dS For q > 0 even, the nominator can be expressed analytically, but not the denominator and vice versa for q odd. So either the nominator or the denominator have to be evaluated numerically, or both if q is not an integer. There is another complication, because the integral in the nominator diverges for q > 0. Evidently, the true fluxes cannot be described as a true power law down to zero flux, because the integrated flux would diverge and there is not an infinite amount of flux in the universe. This was also noted by Hogg & Turner (1998). Ideally, the power law as a means of describing the true source distribution should be replaced by a more realistic, normalizable, function. This is more accurate than setting a lower limit > 0 to the integrations in the nominator and denominator of equation 2.34. However, if the contribution from the faintest sources can be neglected, the latter approach will be sufficiently accurate.

Correcting individual fluxes according to equation 2.34 is appropiate, in the case that one knows the true flux distribution, the method of Hogg & Turner (1998) is not. Still, the average of all corrected fluxes, S exp, will differ from the average flux of the underlying distribution, S true, because the measured fluxes are a specific sample of all measurable fluxes. They are all above a threshold, T. In the limiting case, limT→−∞ S exp, we find:

F=+∞ S =+∞ | F=T S =0 SP(S F)dS P(F)dF lim S exp = lim +∞ (2.35) T→−∞ T→−∞ T P(F)dF +∞ = SP(S )dS = S true (2.36) 0 This is not found if the ﬂuxes are corrected as prescribed by equation 2.30. We have considered implementing a correction for Eddington bias, using equation 2.34, but it seemed more appropriate as a post processing step, because it requires assumptions about the detected sources. For the software validation, we will just have to check if the measured ﬂuxes are biased apart from Eddington and Malmquist (truncation) bias. When running these tests, we are in the advantageous position that we have perfect knowledge of the intrinsic

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distribution of sources that we insert in the image plane. We derive the theoretical, Eddington and Malmquist biased, mean measured ﬂux F by integrating over P(F) and comparing it with = M i / the actual measurements, Fmeas i=1 Fmeas M.

F=+∞ F=T FP(F)dF F = +∞ (2.37) P(F)dF T F=+∞ S =+∞ | = F = P(F S )P(S )dS dF = F T S 0 (2.38) F=+∞ S =+∞ | F=T S =0 P(F S )P(S )dS dF 2 F=+∞ S =+∞ − (S −F) F e 2σ2 P(S )dS dF = F=T S =0 2 (2.39) F=+∞ S =+∞ − (S −F) 2σ2 F=T S =0 e P(S )dS dF

If all the inserted sources had the same flux S ins, i.e., the flux distribution is a δ-function, Eddington bias vanishes, but Malmquist bias does not. We find:

P(S ) = δ(S − S ) (2.40) ins +∞ 2 (S ins−F) − 2 P(F|S )P(S )dS = P(F|S ins) ∝ e 2σ (2.41) 0 2 F=+∞ − (S ins−F) Fe 2σ2 dF F = F=T 2 (2.42) S F=+∞ − (S ins−F) ins 2σ2 S ins F=T e dF (S −T)2 √ − ins − 2σ2 T√S ins 2σe + 2πS insErfc( ) = √ 2σ − (2.43) 2πS Erfc( T√S ins ) ins 2σ

with Erfc the complementary error function. Consider the special case S ins = T = nσ, when one inserts sources at the nσ threshold. In that case we ﬁnd:

√ F 2 + n 2π = √ (2.44) S ins n 2π . = →∞ / → This ratio equals 1 16 for n 5. In the limit n , F S ins√ 1. But the absolute correction F − S ins does not vanish. For n →∞, F − S ins → σ 2/π. This shows why Malmquist bias cannot be ignored near thresholds, not even if these thresholds are very high with respect to the noise. This, of course, assumes that photometry can be done with an accuracy of about σ, the rms noise. If photometry is compromised by other, larger, errors, Malmquist bias may not become apparent.

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More generally, if we test the software by inserting a cumulative source distribution N(S ) ∝ S −q, with diﬀerential source counts dN/dS ∝ S −(q+1) in the image plane, the average measured ﬂux should be close to F, as given by:

2 F=+∞ S =+∞ − + − (S −F) F S (q 1)e 2σ2 dS dF = F=T S =−∞ F 2 (2.45) F=+∞ S =+∞ − (S −F) −(q+1) 2σ2 F=T S =−∞ S e dS dF

In general, F cannot be expressed in analytic form. For testing, it is much simpler and more accurate to take into account the discrete sampling of the inserted sources:

N = δ − i / P(S ) (S S ins) N (2.46) i=1 i 2 (S −T) √ i − ins T−S N 2σ2 i √ ins = 2σe + 2πS Erfc( ) = i 1 ins 2σ F √ − i (2.47) N T√S ins 2π = Erfc( ) i 1 2σ Equation 2.47 shows that the validation of ﬂux measurements is straightforward if all inserted sources are well above the detection threshold. This is the way the validation runs will be set up. For simplicity, we will insert sources with equal ﬂuxes in each map. This is described by equation 2.43 which reduces to

F = 1 (2.48) S ins when all sources are inserted well above the threshold.

Plain thresholding Once a background and noise map have been computed the background map is subtracted from the original (calibrated and cleaned) radio image. The noise map is multiplied by a user specified number to make a threshold map for finding sources. The values of pixels (after background subtraction) have to be higher than the local threshold level to be selected as source pixels. This is accomplished by equation 2.14, with ”threshold”not a number, but a 2-D array, the threshold map. In SExtractor, one may also set a fixed threshold in Jy/beam, independent of the local rms noise. We have dropped the option of a fixed threshold across the map because it did not seem useful. Imposing a nσ threshold√ implies that the fraction of false source pixels in the image will be smaller than Erfc(n/ 2), averaged over a very large number of images. For the NVSS (Condon et al. 1998), for example, which has a 2mJy/beam limit for catalogued sources and a close to uniform rms noise, one can calculate the fraction of false source pixels relative to the

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total number of pixels in the NVSS maps. That fraction is not easily translated to a number of false sources in the NVSS. In fact, two separate threshold maps may be applied simultaneously in the TKP software, the analysis threshold map and the detection threshold map. First we select all source pixels above the analysis threshold and then drop all islands that do not have peak pixels above the detection threshold. This is not supplied for in SExtractor or in SAD (AIPS), but Rengelink (1998) used it, too.

The False Discovery Rate (FDR) algorithm The criterion for controlling the False Discovery Rate was invented by Benjamini & Hochberg (1995). It was used for source detection in astronomical images a few years later (Miller et al. 2001; Hopkins et al. 2003). It was also implemented in the source detection task SFIND 2.0 (Hopkins et al. 2002) in the MIRIAD reduction package. The user should enter a maximum allowed fraction of ”false positives”, i.e., noise peaks erroneously interpreted as sources. The implemented False Discovery Rate (FDR) algorithm divides the radio map (with the background subtracted) by the noise map. This normalized map is then used to calculate the detection threshold, as explained in Appendix B of Miller et al. (2001). The noise map is multiplied by this threshold to make a threshold map for selecting source islands. On average the fraction of false source pixels will be lower than the user given maximum fraction. Again, as with plain thresholding, this refers to source pixels, not to sources, strictly speaking. Hopkins et al. (2002) have shown that the FDR algorithm is not as accurate with regard to sources as with regard to source pixels. In default mode, the FDR algorithm makes no use of a separate analysis threshold. Nonethe- less, this option is also available, like in SFIND. However, the maximum allowed fraction of false source pixels is not guaranteed when the analysis threshold is lower than the FDR threshold, as shown by Hopkins et al. (2002).

Deblending The lowest pixel values of the islands will be above the threshold for analysis while the peak pixel value will also be above the threshold for detection. If two or more sources are close enough, they will initially be detected as one source. The deblending algorithm implemented in the TKP pipeline uses subthresholds and connectivity as defined by the structuring element to separate sources analogous to SExtractor (Bertin 2006, paragraph 6.4). In SExtractor and in the TKP software, the user specifies the number of subthresholds exponentially spaced between the lowest and the highest pixel value. In both deblending algorithms, at each subthreshold level the structuring element is used to deblend the subislands from one of the lower subthresholds, although SExtractor uses 8-connectivity only. There are, however, differences in the codes. The most important difference is that in SExtrac- tor there is a subsequent procedure that descends back from the ”tips of the branches”down to the ”trunk”. At each junction threshold it checks if the flux above that level in that branch (subisland) is larger than some user given fraction of the total flux in the composite object. If

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this is the case and there is at least one more branch that also satisfies this condition at the same junction threshold, then those branches are identified as separate sources. The condition for a minimum flux in a branch ensures that noise spikes are not falsely identified as separate sources. In the TKP software, we have attempted to accomodate for this without a separate procedure. ”On the way up”there are simultaneous checks for connectivity and sufficient flux. Again, this is specified by the user as some fraction of the total flux. Unless there are at least two subislands (from the same split), both with sufficient flux above that subthreshold, nothing is done and the algorithm leaps to the next subthreshold. If the subislands are sufficiently bright and have peak values above the detection threshold they are identified as separate sources, while the residual pixels from the deblending of the ”parent”are discarded. The procedure continues up to the highest subthreshold or until some subthreshold is reached above which there is insufficient flux in all of the remaining subislands. It is worth noting that the user given number of subthresholds, DEBLEND−NTHRESH,isanim- portant parameter for deblending. If it is too small, it will slow down the source extraction process without achieving anything extra. If it is too large, the deblending algorithm may miss a source. It is easy to show that two equally high circular√ Gaussians with FWHM size 2s can make a saddle point if their separation is at least s 2/ ln 2 1.7s. If sources are to be separated by the use of connectivity, saddle points are required. Consequently, the deblending capacities of both the TKP software and SExtractor are rather limited. More refined algorithms for deblending, like the simultaneous fitting of multiple Gaussians were, of course, considered, but they tend to slow down the source measurement process significantly. It is hard to write code that can separate between blended and extended sources. Consequently, in the case of fitting multiple Gaussians, this results in wasting many cpu cycles on the deblending of extended sources. In the case of our deblending algorithm, far less processing time is wasted on this. Although the multiple Gaussian fit can separate sources that are closer together and although it is probably more accurate in individual cases, it seems to lack the robustness required for real time processing. Besides that, when fitting multiple Gaussians simultaneously, it seems hard to provide accurate error bars on the fitted parameters because the calculation of the error propagation is complex. Also, the reported error bars of the fitted parameters from the AIPS tasks ’IMFIT’, ’JMFIT’ and ’SAD’ seem to be underestimated, when multiple Gaussians are fitted simultaneously.

2.7.5 Source Measurement General Once islands of source pixels have been selected, they can be measured, as depicted in figure 2.7. Measurement in our context essentially means describing the source pixel values as well as possible by a Gaussian with six parameters: the peak flux density, right ascension, declination, semi-major axis, semi-minor axis and the position angle of the semi-major axis, east from local north. Those parameters can be derived either from moment analysis or from least squares fitting. In general seven quantities and their error bars are to be derived from each source: besides

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Source parameters and error bars

Source pixels source measurement

Residuals from Gauss fitting

Figure 2.7: Source measurement involves the computation of source parameters such as shape and peak ﬂux density for every island of pixels.

the six parameters mentioned, also the integrated flux density and its error bar. The integrated flux density is a dependent quantity, it can be derived from the peak flux density, the size of the axes and the size of the synthesized beam. Apart from these fourteen quantities, a measure of the quality of the fit, the fit residual, is also returned. The fit residual is actually a small 2-D array with the fit subtracted from the source pixels. The user may also request for a residual map, where, for each island, the computed Gaussian has been subtracted from the data. The computed Gaussian is sometimes derived from moment analysis if Gaussian fitting did not yield adequate results. The source measurement for the NVSS (Condon et al. 1998, paragraph 5.2.1) was performed with a maximum size for the fitted FWHM major axis equal to three times the size of the (circular) synthesized beam. If residuals were too high, up to four Gaussians were fitted simultaneously. It is essential to note that different choices have been made for the TKP pipeline. There is no upper limit to the fitted size. When residuals from a Gaussian fit to an island or subisland are high, there is no attempt for the simultaneous fitting of more than one Gaussian. This choice reflects one of the basic software requirements: speed. A consequence of these choices is that residuals, in units of the rms noise, will be higher than for the NVSS. However, if pixels at the location of extended sources were excluded, the residuals, in units of the rms noise, should be about equal to the NVSS residuals.

Moment analysis

Calculating the Gaussian model parameters There are seven ”moments”to be computed: the peak ﬂux density, the integrated ﬂux, the x and y position, the semi-major and the semi-minor axes and the position angle of the semi-major axis.

The peak ﬂux is the value of the maximum pixel, Pmax, with a correction for the fact that the peak is not located exactly at the center of a pixel. We can calculate an average correction,fudge−max−pix, hereafter fmp, assuming that the peak is located at a random position on the pixel and that the source is a point source, like this:

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y=0.5 x=0.5 2 2 (cos(θsb)x + sin(θsb)y) (cos(θsb)y − sin(θsb)x) fmp = exp(ln(2)[ + ])dxdy =− . =− . 2 2 y 0 5 x 0 5 msb Msb (2.49)

where Msb, msb and θsb are the semi-major and semi-minor axes and the position angle of the synthesized beam, respectively. The correction is usually of the order of a few %, such that 1.0 < fmp < 1.1. We adopt a peak flux density of C = Pmax · fmp if the source island consists of only one pixel. The only flaw of this method is that it does not take account of a detection bias: given that that pixel is above the detection threshold the source is more likely to be located close to the center of the pixel. If it were not, it would not have been detected. In order to take account of this effect, one has to make assumptions about the distribution of sources as a function of their flux. This has not been implemented. Rengelink (1998) adopted a fixed value for fmp = 1.06 for all the WENSS maps, although the size of the synthesized beam varied with declination.

The center position is calculated as follows:

∈S I x xbar = x = i i i (2.50) i∈S Ii ∈S I y ybar = y = i i i . (2.51) i∈S Ii where i∈S indicates a summation over all the pixels that belong to the source, i.e., the island with pixel values above the threshold for analysis and with the highest pixel value above the threshold for detection, disconnected from other islands. It can also be a subisland if the island has been deblended. The position angle of the semi-major axis, measured counterclockwise from the y-axis, is given by:

2xy tan 2θ = (2.52) x2 − y2

Of course, there are two solutions in the interval ] − π/2, 3π/2] for 2θ, namely 2θ and 2θ + π,soforθ the solutions are arctan(2xy/(x2 − y2))/2 and arctan(2xy/(x2 − y2))/2 ± π/2. If the ﬁrst solution has the opposite sign as xy, we choose that. If not, we select from the second solutions a value of θ in the interval ] − π/2,π/2]. The lengths of (half of) the axes are given by:

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⎛ ⎞2 semimajor2 M2 x2 + y2 ⎜ x2 − y2 ⎟ = = + ⎝⎜ ⎠⎟ + xy2 (2.53) 2ln(2) 2ln(2) 2 2 ⎛ ⎞2 semiminor2 m2 x2 + y2 ⎜ x2 − y2 ⎟ = = − ⎝⎜ ⎠⎟ + xy2 (2.54) 2ln(2) 2ln(2) 2 2

Note that both the semi-major and the semi-minor√ axis diﬀer from the description in the SExtractor manual (Bertin 2006) by a factor 2 ln(2). Formulas 2.53 and 2.54 underestimate the length of the axes because of the nonzero cutoﬀ at the threshold as set by the user. The easiest way to see this is by using a coordinate system in which the semi-major axis is aligned with the y-axis, so θ = 0. This means that also xy = 0. We can now compare x2 and y2 for zero and nonzero threshold. In the noise-free case, one has:

= = 2 2 y ymax x xmax 2 − x + y = = Cx exp( ln(2)[ 2 2 ])dxdy 2 = y 0 x 0 m M x = = 2 (2.55) y ymax x xmax − x2 + y y=0 x=0 C exp( ln(2)[ m2 M2 ])dxdy

with xmax and ymax such that ln T y2 x = m − C − (2.56) max ln 2 2 M ln T y = M − C (2.57) max ln 2

with T the threshold for source detection. The denominator in equation 2.55 can be shown to be πMm(C − T)/ ln(2). When integrating over x and then over y one finds m times a function that depends only on M, C and T.If M = m the integration is straightforward in polar coordinates and one finds πM2(C−T)/ ln(2). It then follows that this function must be πM(C − T)/ ln(2). √ For calculating the nominator, it is easiest to change variables, replacing y by u = y m/M and then transform to polar coordinates: x2 +u2 = r2, x = r cos(φ), u = r sin(φ). We then find:

m2 ln( T ) x2 = (1 + C ) (2.58) 2ln(2) C − T 1 and

M2 ln( T ) y2 = (1 + C ) (2.59) 2ln(2) C − T 1

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We correct for the nonzero threshold by dividing the size of the axes as derived from equations 2.53 and 2.54 in this way:

m mcorr = (2.60) + T / C − 1 ln( C ) ( T 1) M Mcorr = (2.61) + T / C − 1 ln( C ) ( T 1)

This results in mcorr > m and Mcorr > M. Note that the position angle of the semi-major axis is not affected by the cutoff at the threshold, as can be seen from equation 2.52. One can always rotate the (x,y) coordinate system such that xy = 0 and then rotate the coordinates back. Now, for xy = 0thecutoff does not play a role, so it does not play a role in any coordinate system. After the semi-major and semi-minor axes and the position angle have been determined, the peak flux density, C, could be determined like this:

θ Δ + θ Δ 2 θ Δ − θ Δ 2 = · (cos( ) x sin( ) y) + (cos( ) y sin( ) x) C Pmax exp(ln(2)[ 2 2 ]) (2.62) mcorr Mcorr

Here, Δx = x − xmax and Δy = y − ymax are the x and y oﬀsets from the center of the peak pixel, at xmax, ymax. If the source island consists of only one pixel, we do not apply equation 2.62. In that case we just keep C = Pmax · fmp. The integrated ﬂux, F, would then be simply calculated as follows: M m F = C · corr corr (2.63) Msbmsb

However, this approach is not 100% correct since both mcorr and Mcorr use C = Pmax · fmp instead of equation 2.62. But the latter equation, on the other hand, uses mcorr and Mcorr. This can be solved because there are in fact three equations for the three unknown variables, mcorr, Mcorr and C:

M2 M2 = m2 (2.64) corr corr m2 m2 m2 = (2.65) corr + T / C − 1 ln( C ) ( T 1) 2 = · ln(2) θ Δ + θ Δ 2 + θ Δ − θ Δ 2 m C Pmax exp( 2 [(cos( ) x sin( ) y) (cos( ) y sin( ) x) 2 ]) mcorr M

= · Pmax exp( 2 ) mcorr (2.66)

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The latter two equations can be combined into one transcendental equation for the peak ﬂux density:

1 + ln( T )/( C − 1) C = C T ln( ) 2 (2.67) Pmax m Once the peak flux density has been determined, the axes and the integrated flux follow from equations 2.60, 2.61 and 2.63. This is the optimum way for calculating the moments. From tests we found that the peak flux densities derived in this manner are almost as accurate as from Gauss fitting. Figure 2.5 shows that maximum pixel method is too coarse at high signal to noise. Now that we have an almost optimum method for determining peak flux densities without fitting, it may seem that Gauss fitting is no longer needed. This is not the case. For astrometry, we found that the barycenter position from the method described above, which we call ”tweaked moments”, is still outperformed by Gauss fitting. Plain moments are used as input for Gauss fitting, but both moments and fitted parameters should be catalogued. Occasionally, the Levenberg-Marquardt algorithm for fitting fails. Very rarely it converges to a runaway solution. In those cases the user of the catalogue should be able to revert to the results from moment analysis. Ideally, this should be ”tweaked moments”instead of plain moments, but at present ”tweaked moments”has not been implemented yet.

Error bars The theoretical variances in multipole moments in the presence of correlated noise are not given in textbooks or anywhere on the web, as far as I can tell, so I tried to deduce these variances myself. The barycenter position (x, y) is given by equations 2.50 and 2.51. With all pixels i independent, the uncorrelated noise σi gives the following variance in x:

∂ i∈S Ii xi 2 2 i∈S Ii xi 2 i∈S Ii 2 σ(x) = (d jx) = (d j ) = ( dIj) (2.68) ∈S I ∂I j∈S j∈S i i j∈S j x ∈S I − ∈S I x x − x = j i i i i i 2 = j 2 ( 2 dIj) ( dIj) (2.69) ∈S ( i∈S Ii) ∈S i∈S Ii j j 2 2 ∈S(x j − x) σ σ 2 j j (n) 2 = = (x j − x) (2.70) ( ∈S I )2 ( ∈S I )2 i i i i j∈S

where we have assumed that the noise dIj = σ j = σ(n), the local rms background noise, does not vary signiﬁcantly over the source. This equation for the position variance is implemented in the SExtractor package (Bertin 2006, equation 32). If we want to take account of correlated noise, we’ll have to alter these equations slightly. Instead of

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σ 2 σ 2 2 (n) 2 (n) 2 σ(x) = (x j − x) = N (x j − x) (2.71) ( ∈S I )2 ( ∈S I )2 i i j∈S i i j∈S

where we have denoted the number of pixels in the source by N, we should write:

σ 2 2 (n) 2 2 σ(x) = N (x j − x) (2.72) ( ∈S I )2 i i j∈S

We want to replace the latter expression in terms of the peak ﬂux density C and the detection threshold T. It is relatively easy to show that

θ2 C (x − x)2 = m ln (2.73) j 16 ln 2 T j∈S θ2 C (y − y)2 = M ln (2.74) j 16 ln 2 T j∈S

where, again, we have assumed that the y-axis is aligned with the major axis of the elliptical gaussian. It is also possible to express N, the number of pixels, in terms of C and T if we neglect the ﬁnite size of the pixels and use the well known formula for the area of the ellipse to replace N:

πθ θ C N = M m ln (2.75) 4ln2 T Finally, the denominator in equations 2.71 and 2.72 can be replaced by the formula for the volume of a gaussian with peak height C and mimimum height T when, again, we neglect pixellation eﬀects:

πθ θ I = M m (C − T) (2.76) i 4ln2 i∈S If we compile the latter equations, we ﬁnd the theoretical limits for the accuracy of the barycenter method in the presence of partially correlated noise:

σ 2θ σ(n)2θ2 (n) m 2 C ≤ σ 2 ≤ m 3 C 2 ln (x) 2 ln (2.77) 4πθM(C − T) T 16 ln 2(C − T) T σ 2θ σ(n)2θ2 (n) M 2 C ≤ σ 2 ≤ M 3 C 2 ln (y) 2 ln (2.78) 4πθm(C − T) T 16 ln 2(C − T) T

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Of course, noise is not either completely correlated or completely uncorrelated. The approach I have used in section 2.6 in analysing the ISOFLUX method in SExtractor is to consider all Ndep pixels within the ”correlated area”completely correlated and outside that completely uncorrelated. Applying that here essentially translates into replacing the factor N 2 2 in equation 2.71 or the factor N in equation 2.72 by (N mod Ndep) +(N−(N mod Ndep))Ndep. 2 For high enough signal to noise (N mod Ndep) +(N−(N mod Ndep))Ndep can be approximated by NNdep. We will use this approximation from now on. This means that we will validate the results from moments analysis in the TKP software pipeline by comparing the diﬀerences between the measured positions and the true positions with the following theoretical position variances:

σ 2 θ θ θ σ 2 = (n) m b B 2 C (x) 2 ln (2.79) 16(C − T) θM T σ 2 θ θ θ σ 2 = (n) M b B 2 C (y) 2 ln (2.80) 16(C − T) θm T

where we have replaced Ndep by πθbθB/4, with θB and θb the correlation lengths along the minor and major axis of the ellipse, respectively. Generally, θb θm and θB θM are good approximations, but we will not use them to avoid loss of generality. It is non trivial to derive the variances of other quantities, like the peak, the axes and the position angle, in the same way. Instead, we try the same approach as Condon (1997) by relating all the variances to a generalized signal to noise, ρ:

θ θ − 2 ρ2 = 4 m M (C T) 2 2 (2.81) ln 2 θbθB σ(n) (ln C − ln T)

Now we can derive all variances from these equations:

σ(C)2 2 = (2.82) C2 ρ2 σ(y)2 σ(x)2 = 8ln(2) = 8ln(2) (2.83) θ2 θ2 M m σ(θ )2 σ(θ )2 = M = m (2.84) θ2 θ2 M m σ φ 2 (θ2 − θ2 )2 = ( ) M m 2 (2.85) 2 (θMθm)

The integrated ﬂux and its relative variance can then be obtained from the previously derived parameters and their variances, in the same way as from the ﬁtted parameters in the next paragraph:

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θ θ I = C m M (2.86) ΘmΘM σ(I)2 σ(C) θ θ σ(θ )2 σ(θ )2 = + B b [ M + m ] (2.87) 2 2 θ θ 2 2 I C M m θM θm

where the clean beam FWHM minor and major axes are denoted by Θm and ΘM, respectively. The ensemble averaged ﬁtted peak is biased (too high, see Refregier & Brown 1998, paragraph 3.2, for some background). The correction for this bias is shown in equation 2.96. For the peaks from the maximum pixel method, with the fudge factor from equation 2.49, or from the yet unimplemented ”tweaked moments”, we do not expect that any bias correction is necessary, but this remains to be veriﬁed.

Gauss ﬁtting

Calculating the Gaussian model parameters Gauss fitting can do more accurate astrometry than moments, as shown in figure 2.4. However, Gauss fitting may sometimes fail while moments can always be calculated. In all cases the moments are used as initial values for Gauss fitting. The actual fit on the data is performed by scipy.optimize.leastsq on the errorfunction, i.e. the difference of the Gauss and the data. This routine uses a modification of the robust Levenberg-Marquardt algorithm. It is just a wrapper around MINPACK’s LMDIF and LMDER algorithms (Moré 1977). These algorithms do not provide for contraints on the fit, contrary to the procedure for the NVSS catalogue (Condon et al. 1998). Consequently, the boundary condition that the fitted semi-major and semi-minor axes can never be less than the corresponding axes of the clean beam, as applied in the production of the NVSS catalogue, is not forced.

Error bars Formulae for the errors from Gauss ﬁtting were ﬁrst derived by Condon (1997):

σ(C)2 2 = (2.88) C2 ρ2 σ(y )2 σ(x )2 = 8ln(2) 0 = 8ln(2) 0 (2.89) θ2 θ2 M m σ(θ )2 σ(θ )2 = M = m (2.90) θ2 θ2 M m σ φ 2 (θ2 − θ2 )2 = ( ) M m 2 (2.91) 2 (θMθm)

and

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2 2 2 2 2 σ(α) = σ(x0) sin(φ) + σ(y0) cos(φ) (2.92) 2 2 2 2 2 σ(δ) = σ(x0) cos(φ) + σ(y0) sin(φ) (2.93)

Here σ indicates standard deviation and C, I,(α, δ), θM, θm and φ indicate the fitted peak flux density, integrated flux, position, FWHM major and minor axis and position angle, respectively. (x0, y0) is the fitted position in pixel coordinates. The y-axis is implicitly assumed to be aligned with the major axis of the elliptical Gaussian. ρ is a generalized ”signal to noise”, which is given by

π θ θ C2 ρ2 = M m (2.94) 8ln2 σ(n)2 if there is no pixel to pixel correlation of σ(n), the background rms noise. Condon (1997) also derived semi-quantitative formulae for the variances in the presence of correlated noise, which were generalized by Hopkins et al. (2003, equation 41) for a non-circular synthesized beam:

θ θ θ θ C2 ρ2 = M m + B 2 αM + b 2 αm [1 ( ) ] [1 ( ) ] 2 (2.95) 4θBθb θM θm σ(n)

Here, the area of noise correlation is assumed to have a Gaussian shape and θB and θb are its FWHM major and minor axes. Generally, the synthesized beam is a good estimate of the noise correlation area. (αM,αm) = (1.5, 1.5) for amplitude errors and (αM,αm) = (2.5, 0.5) for the errors on x0 and θm.(αm,αM) = (0.5, 2.5) for errors on y0, θM and φ. Equation 2.95 is formally exact only in the limit of high signal to noise, unfortunately (Condon 1997). Consequently, the validation of the correctness of the error bars on fitted parameters is only possible at high signal to noise. The positional variances in the NVSS (Condon et al. 1998, equation 25) differ from the equations above by a factor 2, taking into account the very dirty synthesized beam of VLA snapshots. Note that the catalogued value for the fitted peak, Cc, is corrected for bias from the local noise gradient:

σ(n)2 C = C − (2.96) c C The integrated ﬂux and its relative variance are given by:

θmθM I = Cc (2.97) ΘmΘM σ(I)2 σ(C) θ θ σ(θ )2 σ(θ )2 = + B b [ M + m ] (2.98) 2 2 θ θ 2 2 I Cc M m θM θm

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where Θm and ΘM indicate the minor and major FWHM clean beam axes, respectively.

For the NVSS (Condon et al. 1998, paragraph 5.2.5), optimal solutions are derived in the case where the source is unresolved in one or two dimensions. In these cases the variances of the peak flux density and the integrated flux are smaller, which is a consequence of the reduced number of degrees of freedom in the Gaussian fit. It is not necessary to redo the fit. The best estimates for both Cc and I can be derived by adjusting the results from the initial fit with all (=6) degrees of freedom (Condon 1997). We have not incorporated this in the TKP software pipeline, mainly because it is hard to make a clear distinction, from the results of a fit, between resolved, partially resolved and completely resolved sources. Of course, it is easy to make this distinction a posteriori, for instance if a later observation with a better resolution did not resolve the source. In that case, it is possible to rederive the optimum values for the peak flux density and the integrated flux as an image post processing step, using only the catalogued values for the source parameters. It is, however, unclear to me whether the solution for the position could be improved by redoing the fit with only 3 free parameters. Since in the error matrix (Condon 1997, equation 10) there are no source parameters that correlate or anticorrelate with position, I am inclined to state that a 3 parameter fit would not improve the accuracy of the position. Condon (1997) derived the equations for the variances in the presence of correlated noise in a heuristic manner. A rigorous approach was pursued by Refregier & Brown (1998). They found formulae for errors in elliptical Gaussian fits which involve the noise Auto Correlation Function (ACF). Their formulae are complex and cumbersome to implement because the noise ACF in general cannot be approximated easily by an analytic expression. Nonetheless, it is conceivable that the Condon formulae will not suffice in the long term and that indeed some measurement of the noise ACF will be needed to calculate the error bars accurately.

Deconvolution from the restoring beam

Deconvolved parameters The fitted axes and position angle are convolved with the restoring beam. As noted in paragraph 2.7.5, fitting in the TKP software is unconstrained, contrary to the NVSS (Condon et al. 1998) procedure. This means that the fitted axes may be smaller than the restoring beam axes. Consequently, deconvolution from the restoring beam is not always possible. The deconvolved axes and position angle are computed in AIPS (Greisen 2003) using the module DECONV.FOR. This module is available in the TKP pipeline. The equations for the deconvolved shape parameters, ϑm, ϑM and ϕ are:

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β2 = θ2 − θ2 2 + Θ2 − Θ2 2 ( M m) ( M m) − θ2 − θ2 Θ2 − Θ2 φ − Φ 2( M m)( M m)cos(2( )) ϑ 2 = θ2 + θ2 − Θ2 +Θ2 − β 2 m ( M m) ( M m) ϑ 2 = θ2 + θ2 − Θ2 +Θ2 + β 2 M ( M m) ( M m) 1 (θ2 − θ2 )sin(2(φ − Φ)) ϕ = arctan( M m ) +Φ θ2 − θ2 φ − Φ − Θ2 − Θ2 2 ( M m)cos(2( )) ( M m)

Here, we have used Θm and ΘM to denote the restoring beam FWHM axes. The clean beam position angle is indicated by Φ.

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Bibliography

Y. Benjamini, & Y. Hochberg, Controlling the False Discovery Rate: a Practical Approach to Multiple Testing, J.R. Stat. Soc. 57, 289-300, 1995. E. Bertin, & S. Arnouts, SExtractor: Software for Source Extraction, A&AS 117, 393–404, 1996. E. Bertin, SExtractor v2.5 User’s Manual, draft version, Institut d’Astrophysique & Obser- vatoire de Paris, 2006. A. Bijaoui, Sky Background estimation and Application, A&A 84, 81–84, 1980. D.S. Briggs, F.R. Schwab & R.A. Sramek, in Synthesis Imaging in Radio Astronomy II,ed. G.B. Taylor, C.L. Carilli, & R. A. Perley (San Francisco, CA: ASP), 301, 1999. S. Chandrasekhar & G. Münch, On the integral equation governing the true and the apparent rotational velocities of stars, ApJ 111, 142, 1950. J.J. Condon, Errors in Elliptical Gaussian Fits, PASP, 109, 166-172, 1997. J.J. Condon, W.D. Cotton, E.W. Greisen, Q.F. Yin, R.A. Perley, G.B. Taylor, & J.J. Broderick, The NRAO VLA Sky Survey, AJ 115, 1693, 1998. A.S. Eddington, On a Formula for Correcting Statistics for the Effects of a known Probable Error of Observation, MNRAS 73, 359-360, 1913. A.S. Eddington, The correction of statistics for accidental error, MNRAS, 100 354, 1940. E.B. Fomalont in Synthesis Imaging in Radio Astronomy II, ed. G.B. Taylor, C.L. Carilli, & R. A. Perley (San Francisco, CA: ASP), 301, 1999. P. Greenfield & R. Jedrzejewski, Using Python for Interactive Data Analysis, http://stsdas.stsci.edu/perry/pydatatut.pdf, 2007. E.W. Greisen, 2003, in Information Handling in Astronomy - Historical Vistas,ed.A. Heck (Astrophysics and Space Science Library, Kluwer Academic Publishers, Dordrecht, Netherlands), 285, 109. D. Herranz, J.L. Sanz, M. López-Caniego & J. González-Nuevo, A Bayesian Approach To

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Flux Correction In Extragalactic Source Detection,inThe 2006 IEEE International Sym- posium on Signal Processing and Information Technology,vol.1, 541-544, 2006. J.A. Högbom, Aperture Synthesis with a non-regular distribution of interferometer baselines, A&AS 15, 417 (1974). D.W. Hogg, & E.L. Turner, A Maximum Likelihood Method to Improve Faint-Source Flux and Color Estimates, PASP 110, 727-731, 1998. A.M. Hopkins, C.J. Miller, A.J. Connolly, C. Genovese, R.C. Nichol, & L. Wasserman, A New Source Detection Algorithm Using the False-Discovery Rate, AJ 123, 1086–1094, 2002. A.M. Hopkins, J. Afonso, B. Chan, L.E. Cram, A. Georgakakis, & B. Mobasher, The Phoenix Deep Survey: The 1.4 GHz MicroJansky catalogue, AJ 125, 465–477, 2003. J.F. Kenney & E.S. Keeping in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 50-54, 1962. J.D. Kraus, Radio Astronomy, Cygnus-Quasar Books, Powell, Ohio, 1986. C.J. Law, Outline of the Transients Detection Pipeline and its Database Interactions,TKP internal documentation, 2006. C.J. Miller, C. Genovese, R.C. Nichol, L. Wasserman, A. Connolly, D. Reichart, A. Hopkins, J. Schneider, & A. Moore, Controlling the False Discovery Rate in Astrophysical Data Analysis, AJ 122, 3492-3505, 2001. N.R. Mohan & H.J.A. Röttgering, Source Detection and Source Measurement for LOFAR Images, LOFAR internal documentation, 2005. J.J. Moré, The Levenberg-Marquardt algorithm: Implementation and Theory, Numerical Analysis, in Lecture Notes in Mathematics ed. G.A. Watson, 630, 1977. F. Patat, A robust algorithm for sky background computation in CCD images, A&A 401, 797- 807, 2003. A. Refregier, & S.T. Brown, Effect of Correlated Noise on Source Shape Parameters and Weak Lensing Measurements, http://arxiv.org/pdf/astro-ph/9803279v1, 1998. R.B. Rengelink, The Westerbork Northern Sky Survey: The Cosmological Evolution of Radio Sources,PhDThesis,Rijksuniversiteit Leiden, 1998. J. Swinbank, Transients Key Project: Pipeline Data Flow, TKP internal documentation, 2007. P. Teerikorpi, Influence of a generalized Eddington bias to galaxy counts, A&A 424, 73-78, 2004. Q.D. Wang, Correction for the flux measurement bias in X-ray source detection, ApJ 612, 159-167, 2004.

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CHAPTER 3

Zeroth order validation of TKP source extraction and source measurement code

3.1 Abstract

The source extraction and source measurement modules in the TKP software pipeline were validated by processing a large number of different maps with and without artificial point sources. To make those maps, the visibilities of a certain VLA observation were replaced by Gaussian noise. The point sources, with the size and shape of the appropriate restoring beam, were inserted in the images. We tested the background and noise estimators, the deblending algorithm, the False Discovery Rate algorithm and the peak flux density and position measurements.

3.2 Description

Zeroth order validation is the first step in the validation of the TKP source extraction and source measurement software. It comprises of a series of tests on artificial maps whose properties are completely controlled and known. Sources are therefore inserted directly in the image plane, to avoid the introduction of biases from a cleaning process. Although these biases are probably small and can be quantified, we decided to conveniently avoid them at this first stage in the validation process. The next step, first order validation, would involve the insertion of sources in visibilities. The maps from those visibilities would then be cleaned using some appropriate algorithm. Subsequently, the measured fluxes and positions would be compared with the corresponding values of the inserted sources. After that, second order validation would use real maps with cleaned sources, e.g., NVSS and WENSS maps, and

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compare the source parameters derived by the TKP software with the corresponding catalogues that were constructed using diﬀerent source extraction packages.

3.3 General

All of the output from the TKP source extraction and source measurement software, such as noise and background levels and source parameters are considered suﬃciently accurate if they are the ”best”estimators of the true values. However, ”best”estimator as such is an ill deﬁned concept. In general, since an extremely large number of maps will be processed by the TKP, we will want all of our estimators to be unbiased. This means that the sample estimators of a certain quantity averaged over the total number of samples, should be equal to that quantity over the complete ensemble. We also require the sample estimators to have the smallest possible mean squared error (M.S.E.). It is known from statistics that these two requirements cannot always be met simultaneously and I will discuss those cases when they occur.

3.4 Source free maps

The maps were made by adding Gaussian noise to the visibilities of the dataset from the discovery observation of GCRT J1745-3009 (Hyman et al. 2005), using the AIPS task ’UVMOD’. The actual data were not used, just the (u, v) coverage of this observation. Gaus- sian noise can be added to the visiblities in units of Jy/weight, by setting the adverb ’FLUX’ to the desired rms value. To avoid any complexity from the visibility weights, we changed them all to 1 using the AIPS task ’WTMOD’. This ensures that, for both of the polarization products, the pixel rms noise will be equal to the value of the adverb ’FLUX’ divided by the square root of the number of visibilities in the case of natural weighting. We aimed for an − rms noise value√ in Stokes I dirty images of 1 Jy beam 1,sowesettheadverb’FLUX’in UVMOD to 2 · 807417 to make√ the ﬁrst ”random”(u, v) dataset. 807417 is the total number of visibilities, the extra factor 2 is required to attain the desired noise level in Stokes I. We made another 9996√ (u, v) datasets,√ every new dataset was made by adding noise with a standard deviation of 807417 to −1/ 2 times the previous dataset. In this way, we made sure that all 9997 datasets had the same noise level. The dirty images were made with natural weighting by setting the adverb ’UVWTFN’ equal to ’NA’ in the AIPS task ’IMAGR’. A pixel size of 12” in both dimensions was chosen and an image size of 2562 pixels. The FWHM size of the dirty beam is 67.14” × 56.15”, so the dirty beam was sampled with more than 4 pixels in both dimensions. To check whether the distribution of the noise in both the visibilities and the maps is normally distributed, we made histograms of both the real and imaginary part of the visibilities for the ﬁrst dataset. We have also investigated if the cumulative noise of the visibilities in the datasets was still Gaussian. This was indeed the case. However, we found that deviations from normal distributions√ do occur when√ new datasets are made by adding noise to the previous dataset times 1/ 2 instead of −1/ 2.

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Box number 0 20 40 60 80 100 Box number 0 20 40 60 80 100

2000 2000

1500 1500

1000 1000 Number of pixels Number of pixels

500 500

-3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 JY/BEAM JY/BEAM Range = -3.9421E+00 to 4.7242E+00 JY/BEAM Range = -4.0885E+00 to 4.2806E+00 JY/BEAM Interval = 8.6663E-02 JY/BEAM Interval = 8.3692E-02 JY/BEAM

(a) First image (b) Last image

0 50 100

0 Kilo VIS (IM)

-2

-4

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-6 -4 -2 0 2 4 6 Kilo VIS (RE) Grey scale flux range= 0.0 136.0 COUNT

Figure 3.1: 3D Histogram of the last source free (u, v) dataset as produced by ’UVMOD’ (bottom) and histograms of pixel values (top).

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A 3D histogram of the real and imaginary part of the visibilities of the last source free dataset is shown at the bottom of figure 3.1. Histograms of the pixel values of the first and last image are also shown in that figure. This indicates that we have indeed produced Gaussian noise in our images. The noise will also be spatially correlated within the psf, as we desired.

3.5 Validation of the TKP background mean and background noise estimation 3.5.1 General For validating the correct determination of the background mean and noise, we divided each of the 9997 source free images into 7 × 7 subimages, of size 32 × 32 pixels. Of course, since each of the images has a size of 256 × 256 pixels, we could have used 8 × 8 subimages per image, but we wanted to avoid any edge eﬀects by excluding an edge of 16 pixels along all sides. So the total number of submaps processed is 9997 × 49 = 489853 and the total number of pixels processed is 5 · 108. These numbers are large enough to provide statistics that are accurate enough for validation. For source free maps, the plain mean is the true background level, so this gives us a way to assert the accuracy of the TKP background level determination, which is essentially a mode estimator. Without sources, the mean should be equal to the mode. In the presence of sources the mean background should be close to the mode of all the pixel values. Thus, two approximations are involved when sources are present. The mode is estimated and the mode itself is an approximation to the true source free mean background, because pixels from weak sources and the outer pixels from strong sources can shift the mode depending on the density of sources.

3.5.2 Source free maps We ran tests on these 489853 maps as well as on an equal number of maps of the same size with uncorrelated Gaussian noise. We also compared the TKP method with an alternative method commonly used: histogram fitting. For every subimage we made a histogram with 100 bins and fitted a 1-D Gaussian to it. Figures 3.2 and 3.3 show all the results graphically and table 3.1 contains the most important numbers. The main conclusions are that neither method shows any serious bias in determining the means or modes, but that the variance of the means is smaller when they are determined by histogram fitting. For uncorrelated noise this effect is most significant. It is remarkable that, in the case of correlated noise, the statistics of the subimages are better described by 30 than by 50 independent elements. The FWHM synthesized beam is covered by 20.56 pixels, so if every synthesized beam would correspond to an independent element, that would amount to 1024/20.56 = 49.8 independent elements. Evidently, the spatial correlation between pixels is significantly larger than the FWHM synthesized beam. In the TKP software, the user can enter the pixel correlation length in both dimensions. This is indeed a very important parameter, since it is used in the determination of the error

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59 59 2 2 50) 50) = = 0.020 0.020 0.040 0.040 (N (N variance variance Theoretical plain Theoretical plain . 2 489853 ) ) ∗ 2 2 N (Jy (Jy 0.048 0.038 0.036 0.052 2 Variance Variance Jy 1 1 4 σ . 2 489853 ∗ 23 -15 correlated noise correlated noise N -0.28 -0.049 489853 1Jy ∗ overall bias overall bias Significance of Significance of N √ 4 σ 1) − 489853 ∗ 5 6 N 2 − − 2( N σ 10 10 489853 ∗ · · N 7 9 . . √ 0.0067 -0.0042 5 9 − − (True-measured) (True-measured) Overall bias (Jy) Overall bias (Jy) 2 2 4 4 − − 10 10 1024) 1024) · · = = 8 8 0.0020 0.0020 . . 9 9 (N (N variance variance Theoretical plain Theoretical plain ) ) 2 2 (Jy (Jy 0.002 0.0046 0.0015 0.0044 Variance Variance 1 1 62 -16 0.57 -0.57 uncorrelated noise uncorrelated noise overall bias overall bias Significance of Significance of 5 4 5 3 − − − − 10 10 10 10 · · · · 5 9 . . 9 5 . . 2 9 3 2 − − Overall bias (Jy) Overall bias (Jy) (True-measured) (True-measured) . . 2 2 Table 3.1: Comparison of background means in source free maps as determined by TKP software and by histogram fitting. TKP TKP As a multiple of the standard deviation ofIn the Jy mean for the total number of maps processed: As a multiple of the standard deviation ofIn the Jy variance for the total number of maps processed: Method Method Table 3.2: Comparison of background variances in source free maps as determined by TKP software and by histogram fitting. 1 2 1 2 Histogram fitting Histogram fitting

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bars in ﬁtted source parameters. The product of the two correlation lengths (×π/4) should correspond to an area somewhat larger than the FWHM synthesized beam such that the number of independent elements matches the pixel statistics, similar to the Gaussians with N = 30 in Figure 3.3.

The distribution of the submap noise variances, s2 is theoretically given by a Pearson type III distribution 1, at least in the case of uncorrelated noise:

N N−1 2 Ns2 ( 2σ2 ) N−3 − f (s2) = (s2) 2 e 2σ2 (3.1) Γ N−1 ( 2 ) where Γ denotes the gamma function. A plot of this function for unit variance and 20 elements N−1 σ2 per sample is shown in figure 3.4. The mean of this distribution is N . This result should not surprise us since we know that s2 is a biased estimator of the true variance, σ2.The plain noise variances are compared with Pearson type III distributions in figure 3.5 for both correlated and uncorrelated noise. Both the theoretical curves as well as the actual noise N variances were corrected for biases by multiplying the abscissa values by N−1 . It is striking that the shape of the distribution of plain noise variances is best matched by a Pearson type III distribution with N > 60 although the distributions of the means in figure 3.3 were indicating that the statistics of the submaps were well described by N 30. Nevertheless, it turns out that solving N in the equation that corrects for the bias in the variances:

N σ2 = < s2 > (3.2) N − 1 where <> denotes the averaging over the samples, gives N 30 as before. In calculating N we have assumed that the true noise variance is equal to the variance of the complete ensemble of 489853 submaps:

σ2 2 stot (3.3) A similar assumption was actually made in determining the biases with respect to the true mean in the sense that the true mean was assumed to be equal to the mean of the complete ensemble. The fact that we ﬁnd N = 30 again from equation 3.2 should not be a surprise. Consider the following equation for the variance of the complete ensemble of submaps: σ2 2 =< 2 > + < 2 > − < >2=< 2 > + < 2 > −μ2 =< 2 > +σ 2 stot s x x s x s (x) (3.4) with μ is the average of all pixel values in all samples and σ(x)2 the ensemble variance of the means. Equation 3.4 is equivalent to equation 3.2 if

σ(x)2 = σ2/N (3.5)

1http://mathworld.wolfram.com/SampleVarianceDistribution.html

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However, from the ensemble statistics of the means we have already determined σ(x)2:

σ(x)2 σ2/30 (3.6)

It follows that we find N = 30 again. Summarizing, the bias of the noise variance is well corrected by N = 30, while the variance of the noise variance seems to correspond to a number of independent elements per sample at least twice as large. Histograms of measured variances calculated by the TKP software and by histogram fitting are shown in figures 3.6 and 3.7. The most important results from the validation runs are shown in table 3.2. The main conclusion is that the TKP method outperforms histogram fitting in terms of the mean squared error, i.e., the variance of the variance, but not with respect to the overall bias. The rms noise as calculated by the TKP software is significantly biased too low, but only by 0.2% in the case of uncorrelated noise and by 0.3% in the case of correlated noise. This will not compromise the overall performance of the pipeline in any way. The advantage of lower mean squared errors seems more important than the disadvantage of a larger, but still relatively small bias. In the TKP pipeline sources are detected above a threshold defined in terms of the local rms noise. A robust estimation of the local rms noise is therefore necessary since any severe underestimate of the local rms noise will cause false alerts, i.e., noise peaks will be falsely identified as sources. For the purpose of validating the TKP noise variance estimation we have made use of the following formula for the variance of the plain variance 2, s2:

− σ2 2 = N 1σ4 2 σ4 (s ) 2 2 (3.7) N N 2 σ(s2) σ2 (3.8) N The latter expression is used to determine the signiﬁcance of the overall bias.

3.5.3 Maps with sources With sources added to the maps, determining the rms and mean of the background becomes more challenging. When the number of source pixels approaches the number of noise pixels, distinguishing between them becomes increasingly diﬃcult. At some point it will be clear that determining the background characteristics will become impossible, e.g., in the case of a ”classical”confusion limited map (Hogg 2001; Condon 1974). Here, it is our goal to assess the proper operation of the TKP software for some reasonable concentration of sources in a map, i.e. when the number of source pixels is still a minor fraction of the total number of pixels. We added sources to the 9997 maps of the previous section, with ﬂuxes randomly distributed between 0 and 5 Jy. The source positions were also randomly chosen. We did three test

2From http://mathworld.wolfram.com/SampleVarianceDistribution.html

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Figure 3.2: Comparison of background level determination methods, for uncorrelated noise in the source free case. i i

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Figure 3.3: Comparison of background level determination methods, for correlated noise in the source free case. i i

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Software validation Chapter 3 2 50) = 0.040 0.040 0.040 Theoretical plain variance (N . 2 ) 489853 2 ∗ N (Jy 0.088 0.060 0.171 Variance 2 Jy 3 4 σ . 2 489853 ∗ -867 -444 N -1846 489853 overall bias 1Jy ∗ Signiﬁcance of N Background variance estimates √ 4 σ 1) − 489853 ∗ N 2 2( N σ 489853 ∗ N -0.25 -0.13 -0.53 √ (True-measured) Overall bias (Jy) 2 50) = 0.020 0.020 0.020 Theoretical plain variance (N ) 2 (Jy 0.058 0.052 0.072 Variance 1 -732 -361 -1523 overall bias Signiﬁcance of Background mean estimates -0.15 -0.31 -0.073 Overall bias (Jy) (True-measured) . 2 Table 3.3: Comparison of background characteristics in maps with sources as determined by TKP pipeline software. 100 200 400 As a multiple of the standard deviation ofIn the Jy mean for the total number of mapsAs processed: a multiple of the standard deviation of the variance for the total number of maps processed: 1 2 3 of sources Average number

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Figure 3.4: The distribution of sample variances is a Pearson type III distribution. This plot is for = σ2 = N−1 σ2 N 20 elements per sample and for unit variance, 1. The mean of this distribution equals N .

runs, one with 100 inserted sources per map, one with 200 and one with 400. The results for all three runs are shown in table 3.3 and figures 3.8, 3.9 and 3.10. It is evident that both the biases and mean squared errors become significant when the average number of sources per map is 400. Both the background level and the noise are severely overestimated. The number of pixels in a map, in units of the number of pixels in the FWHM synthesized beam, is 49 · 1024/20.56 = 2440. In units of truly independent pixels, as we have calculated above, this number is about 49 · 30 = 1470. Thus, 400 sources represents a fraction of 27% of the total number of independent pixels. Apparently this fraction is too high to determine the background characteristics accurately. It remains to be investigated where this error comes from. Is the mode shifted away from the true mean background or is the TKP mode estimator not accurate enough? In the former case histogram fitting will not give better results. In the latter case, it may be more accurate. Histogram fitting has the disadvantage that the bin size should be chosen carefully and that, in the presence of sources, only the highest part of the histogram should be used for fitting. Alternatively, one could decide to use only the negative part of the histogram for fitting. This reduces contamination by source pixels in Stokes I images. Note that negative pixel values can still have a small contribution from a source. Also, in radio maps there are often negative pixel values near sources from cleaning artefacts or negative sidelobes. Problems with accuracy remain if the histogram does not have a sharp peak. According to Patat (2003, paragraph 3) the statistics in small subimages are probably not good enough to derive background characteristics from histograms. Also, histogramming is a computationally expensive

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Figure 3.6: Comparison of noise variance determination methods, for uncorrelated noise in the source free case.

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Software validation Chapter 3

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Figure 3.7: Comparison of noise variance determination methods, for correlated noise in the source free case. i i

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Figure 3.8: TKP background level and noise estimates with 100 sources per 2562 pixel map. The FWHM beamsize is 20.56 pixels. i i

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Software validation Chapter 3

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Figure 3.9: TKP background level and noise estimates with 200 sources per 2562 pixel map. The FWHM beamsize is 20.56 pixels. i i

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Figure 3.10: TKP background level and noise estimates with 400 sources per 2562 pixel map. The FWHM beamsize is 20.56 pixels. i i

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Software validation Chapter 3

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Figure 3.11: The distribution of the number of falsely detected pixels divided by the total number of detected pixels with 100 inserted sources per 2562 pixel map. All three histograms are normalized equally. In order to represent, in this loglog plot, the bin with most counts, i.e., with zero falsely detected pixels, we have chosen to center the bins on the right bin edges. The left bin contains all fractions in the interval [0,1e-3]. All other bins have equal size on a log scale.

process. We expect that improvements could come from an asymmetric clipping algorithm, possibly similar to the one proposed by Ratnatunga & Newell (1984). Perhaps the simple adjustment of clipping only the pixels that are larger than the median+3σ and not the pixels that are smaller than the median−3σ will remove most of the bias if the algorithm is applied to properly cleaned maps where negative sidelobes are negligible.

3.6 False Discovery Rate algorithm

The False Discovery Rate algorithm was derived relatively recently (Benjamini & Hochberg 1995) and implemented in SFIND 2.0, the MIRIAD source extraction algorithm (Hopkins et al. 2002). The prime advantage of it is that the maximum number of falsely detected source pixels is expressed as a fraction of the total number of detected pixels. For the TKP, this can be translated into a maximum allowed fraction of false alerts for the discovery of new transients, which is a useful quantity. Without this algorithm, the threshold would have to be set as a multiple of the local rms noise (nσ), which can be converted to a number of false positives only after the total number of pixels processed has been determined. It is a slight disadvantage of the FDR algorithm that it refers to source pixels, not sources. Hopkins et al. (2002) pointed out that the FDR algorithm provides a less than strict constraint with

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respect to sources but that it is still a quite good estimator in that respect. These authors also note that FDR loses some of its ﬁdelity when contiguous pixels with values between the detection threshold and the (lower) analysis threshold are included. For simplicity, we have only considered source pixels in our tests and we have always set the analysis threshold equal to the detection threshold. It should be clear that the maximum allowed fraction refers to ensemble averages (Miller et al. 2001). This means that in the processing of individual maps the fraction of falsely detected source pixels can exceed the overall maximum fraction. Before we discuss the results from our tests it should be noted that it is not precisely α,the maximum allowed fraction of false positives, that is entered in the algorithm, but rather α/CN . The quantity CN expresses the degree of correlation between the pixels:

N 1 C = (3.9) N i i=1 where N is the total number of pixels in the image if all pixels are fully correlated. The usual way of processing radio maps is to set CN equal to the number of pixels in the FWHM synthesized beam. We have chosen a slightly larger area of pixels corresponding to the true correlation length. In this way, we again have that the number of independent pixels, as cal-

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Software validation Chapter 3

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Figure 3.13: The distribution of the number of falsely detected pixels divided by the total number of detected pixels with 400 inserted sources per 2562 pixel map. All three histograms are normalized equally. In order to represent, in this loglog plot, the bin with most counts, i.e., with zero falsely detected pixels, we have chosen to center the bins on the right bin edges. The left bin contains all fractions in the interval [0,1e-3]. All other bins have equal size on a log scale.

culated from the bias of the sample variances, times this area equals the total number of pixels in the map. There is a subtlety with respect to the definition of correctly and falsely detected source pixels. A correctly detected pixel can be associated with an inserted source. This means that the detected pixel must be above one of the source pixels. However, the inserted sources are Gaus- sians which, strictly speaking, extend out infinitely, although we have inserted them as 322 pixel arrays. It is unclear how ”associated”is defined in the tests run by Hopkins et al. (2002). We have chosen a 52 pixel area centered on the source to distinguish between falsely detected, unassociated pixels and correctly detected, associated pixels. More concretely, the code used for our tests makes use of the watershed−ift algorithm from Scipy. watershed−ift uses a watershed from markers algorithm as described by Felkel et al. (2002). Thus the association is determined by 25 equal markers centered on every inserted source. This number is somewhat arbitrary but it takes care of the few rare cases where the peak pixel of a source is missed due to a negative noise peak while adjacent pixels were in fact detected. It should be clear that watershed−ift also marks all the contiguous pixels as ”correct”pixels if they were detected. SFIND 2.0, which was tested by Hopkins et al. (2002), also discards all pixels from fits that did not converge. We did not copy that in our tests. We have done three test runs with a number of 100, 200 and 400 point sources inserted at random positions in each map. We again used our 9997 source free maps of 2562 pixels for each

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Table 3.4: Ensemble averaged fraction of falsely detected pixels Sources inserted α = 0.1 α = 0.01 α = 0.001 100 0.011 0.0040 0.0019 200 0.0060 0.0023 0.0011 400 0.0028 0.0010 0.00047

run. The rms noise in these maps is 1.0 Jy. The fluxes of the inserted sources were randomly distributed between 0 and 5 Jy. The results are presented in figures 3.11, 3.12 and 3.13 and in table 3.4. The main conclusion that can be drawn is that a fraction of less than 0.1% false positives is generally not achieved. Only in the case where many (400) sources are inserted are the FDR promises actually kept. Hopkins et al. (2002) did not run tests for α = 0.001, so this makes it hard to compare. If pixels that do not give converging fits are left out, as these authors did, the fraction of falsely detected pixels will likely drop. We tried to derive the cause of the discrepancy by also running tests on source free maps. For α = 0.1, 0.01 and 0.001 we found that the fraction of maps with any pixel above the FDR threshold was 2%, 0.7% and 0.17%, respectively. The latter fraction decreased by only 0.02% when we changed the correlation length to include all pixels in the maps. This makes it unlikely that the correlation length was chosen too small. Thus it remains unclear why the FDR algorithm gives poor results when the allowed FDR is set very low 3.

3.7 Deblending

The performance of deblending algorithms naturally depends on the separation between sources, on the fluxes of the blended sources with respect to the detection threshold, but also on the ratio of the fluxes of two neighbouring sources if the algorithm requires saddle points. Saddle points emerge at some minimum separation. We are interested in the limiting case where the minimum between two peaks is not yet apparent. Instead, we have a flat plateau near the weakest source which can be found by setting the first and second derivatives of the blended source to zero:

2 2 − x − (x−D) Σ=Ce 2σ2 + Be 2σ2 (3.10) dΣ = 0 (3.11) dx d2Σ = 0 (3.12) dx2 We have assumed that the blended source is composed of two point sources with peak values C and B and separation D. In the case of point sources the sources they have the same

3See, however, the remark in paragraph 3.8

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width σ as the psf. When we work out these equations, we ﬁnd

√ D ± D2 − 4σ2 x± = (3.13) 2 √ √ √ √ 2 2 2 2 2 2 B D ± D − 4σ ∓ D D −4σ d ± d − 4 ∓ d d −4 ± = √ e 2σ2 = √ e 2 (3.14) C D ∓ D2 − 4σ2 d ∓ d2 − 4 With d ≡ D/σ. The functional dependence between d and B/C± has been depicted in figure 3.14. It is clear that the mimimum separation depends weakly on the flux ratio. A separation of 6 Gaussian widths is required for a flux ratio of 106 : 1. Higher flux ratios should not be expected, because this ratio exceeds the highest dynamic range ever achieved in radio astronomy (5 · 105 : 1 is the highest according to Walker (2006)). Validation of the deblending algorithm was run on 9800 of our 2562 pixel maps, with − 1Jybeam 1 rms noise and 64 pairs of sources per map. The pairs were inserted on a uniform grid to avoid any overlap between pairs. We chose a 10σ detection threshold and a 3σ analysis threshold. The flux of the weakest source was always 20 Jy and the flux ratios were logarithmically spaced between 1 and 1000. For every flux ratio 20 maps were processed successively, each map having a fixed separation between the two sources in each of the 64 pairs. The separations corresponding to each of the 20 maps were linearly spaced between 2 and 7 Gaussian widths. Since the psf is elliptical, we have chosen to align the sources in each

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Table 3.5: Performance of TKP deblending algorithm for diﬀerent values of DEBLEND−MINCONT 0.1 0.05 0.01 0.005 0.001 % extra sources deblended 6.9 14.8 31.6 37.5 43.7 % of incorrect deblendings per source 6.9 · 10−3 2.1 · 10−2 0.88 1.6 5.6 ﬁgure of merit 6.9 14.7 28.5 31.9 23.7

pair along their minor axes and use the Gaussian width along the minor axis of the ellipse, 1.987 pixels, as a unit for separation. The high detection threshold ensured that no detection could be a noise peak, which sim- plified the validation of the deblending algorithm. The user sets the deblending parameters DEBLEND−NTHRESH and DEBLEND−MINCONT as in SExtractor (Bertin 2006, paragraph 6.4) runs. We set DEBLEND−NTHRESH = 32, the default value in SExtractor. This determines the number of subthresholds between the smallest and largest pixel value of the composed object, logarithmically spaced. We ran tests for five different values of DEBLEND−MINCONT. This parameter ensures that only subislands with significant flux will be regarded as separate sources. It works like this: connectivity is determined at subsequent subthreshold levels, starting at the lowest level. When more than one subisland is found, the fluxes of the new-found ”branches”above the subthreshold are divided by the flux of the original blended source. At least two ratios have to be above DEBLEND−MINCONT before the corresponding subislands will be considered separate. Also, the peak values of those subislands have to be above the detection threshold. Pixels belonging to subislands that do not satisfy both of those conditions are neglected. The results for DEBLEND−MINCONT = 0.1, 0.05, 0.01, 0.005, 0.001 are shown in table 3.5. The ”% extra sources deblended”is the % of extra sources that are found relative to plain source extraction. Plain source extraction can, of course, identify the different components of a blended object if they are not connected at the level of the analysis threshold. Thus, plain source extraction already does some deblending. It should be noted that these and other numbers in table 3.5 depend on the specific range of flux ratios and separations of the inserted pairs of sources. The ”% of incorrect deblendings per blended source”is the rate at which blended sources are decomposed in more than two components, which is, of course, incorrect in these specific test runs. The ”figure of merit”applies a penalty for those incorrect deblendings, such that one incorrect deblending crosses off one correct deblending. If the figure of merit is defined in this way, we find that DEBLEND−MINCONT = 0.005 is best. This is also the optimum value found from SExtractor test runs (Bertin 2006, paragraph 6.4). It should be noted that DEBLEND−MINCONT sets a lower limit on the flux ratio of the faintest source to the brightest source in a composed object that can be deblended. That lower limit is somewhat larger than DEBLEND−MINCONT because the significance criterion of the deblending algorithm compares the flux of a subisland above the subthreshold with the total flux of the composed object. Without a deblending algorithm, the numbers of detected sources are depicted in figure 3.15. For DEBLEND−MINCONT = 0.05 and 0.005 we have depicted the numbers of detected sources

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divided by these numbers without a deblending algorithm. It is evident from figure 3.16 that if the flux ratios are very small, deblending is done very poorly for DEBLEND−MINCONT = 0.05. In figure 3.17 we see more complete deblending for these small flux ratios. It should be kept in mind, however, that this comes at the price of a larger number of erroneously deblended objects, with more than two components, as quantified in table 3.5.

3.8 Determination of peak ﬂux densities and positions

For the validation of the correct determination of fluxes and positions, we used 9980 of our 2562 pixel maps, with 1 Jy beam−1 rms noise. All 64 sources in a single map were inserted with equal flux on a regular grid. The values BACK−SIZEX and BACK−SIZEY that determine the size of the background window for mean background and background noise interpolation across the map, were set to 256, which results in a flat background across the map. Both the analysis threshold and the detection threshold were set to 6 times the rms noise, to avoid picking up noise peaks which would complicate the validation. It is interesting to do some checks on the number of pixels above a certain level. We have processed a total of 9980 · 2562 = 6.54 · 108 pixels in our validation runs. If all of these pixel were independent, then we could expect less than one pixel, 0.645 to be precise, above 6 times the rms noise, i.e., 6 Jy. We found that, in fact, there were a total of 147 pixel values from 90 maps above 6 Jy, with the highest pixel value between 8 and 9 Jy! Of course, the

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pixel values are correlated in each map, but even so, for any number of uncorrelated pixels per map the ratio of the actual to the expected number of very high pixel values remains equal to 147/0.645 = 228. This shows that, in fact, there are strong deviations from normal distributions in the pixel values, although the visibility noise was strictly Gaussian. As we know, the Fourier Transform (FT) of a Gaussian is also a Gaussian, so if the pixel distribution deviates from a normal distribution this must arise from the fact that we have applied a Fast Fourier Transform (FFT) with interpolation and limited sampling instead of applying a true FT to an infinitely extending and complete (u,v,w) space. In any case, this is something that needs to be accounted for when searching for transient sources. It involves a deeper understanding of the noise properties of maps, which is beyond the scope of this thesis. However, me may now have found an explanation for some unsatisfactory results in the FDR validation runs. When the rate of false positives was set very low (α = 0.001), the number of falsely detected pixels was slightly higher than allowed, also in the case of source free maps. It should be kept in mind that the FDR algorithm assumes that the background noise has a normal distribution. If this is not actually the case, FDR cannot be expected to work correctly. For the purpose of testing the accuracy of the measurements by the TKP software we have chosen the source fluxes to be linearly spaced between 10 and 1000 Jy. The lower limit of 10 Jy, 4σ above the detection threshold, was chosen to avoid any considerable truncation bias in the flux measurements. Also, the error bars that we use for the fitted parameters are given by the Condon formulae (Condon 1997). They are valid only at high signal-to-noise. The

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upper limit of 1000 Jy was chosen to cover a wide range of signal-to-noise ratios. We chose a flat flux distribution for the inserted sources to give equal weight to all signal-to-noise ratios in our statistics. In 10 of the 90 maps with noise peaks above 6σ, the number of detected sources exceeded 64. Those maps were not used for our validation statistics. In the other 80 maps, the slight overestimate of the mean background and noise caused by the 64 inserted sources, filtered out those noise peaks at the 6σ detection threshold. Figure 3.18 shows a histogram of the peak flux density measurements in the remaining 9970 maps, where the differences between the calculated peaks and the true peaks have been divided by their reported errors. It is immediately obvious that the distribution from our measurements is narrower than it should be. Most likely, the reported errors are overestimated. If the reported errors were 14% larger overall, the measured distribution would match the theoretical curve. It may be that the errors are smaller than expected since we have inserted sources in the image plane, which is somewhat artificial. More realistic would be to insert sources in the visibilities and then clean the images. This has the large disadvantage that the cleaning process is complicated and can introduce unknown biases, e.g., because it is unclear at what level cleaning should be stopped. Thus, this approach hinders a straightforward validation.

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Figure 3.19 shows the number density of the reported flux errors. They are very close to − the rms noise (1 Jy beam 1), as expected from theory (Condon 1997). For the validation of position measurements, we had to add some lines of code to the flux measurement validation script. The positions of the inserted sources were converted to celestial coordinates by new code, i.e. code that was independent from the conversion code in the TKP pipeline software. Also, since the order of the output source measurements was unknown and each measurement had to be coupled to the corresponding inserted source, code for correct reordering had to be applied. This, of course, would have been unnecessary if only one source were inserted per map. The results for the right ascension are depicted in figures 3.20 and 3.21. From figure 3.20 we, again, have to conclude that the reported Right Ascension errors are overestimated, by about 23%, since the distribution is narrower than a normalized Gaussian. The reported position variances, as implemented in the TKP software, are copied from the NVSS (see equation 25 in Condon et al. 1998). They are twice the theoretical values (Condon 1997), taking into account the very dirty synthesized beam of a VLA snapshot. Most likely, this is the cause of the overestimate, since the (u,v) coverage used to produce the artificial noise maps for these validation runs, corresponds to about 6h of VLA synthesis and not to a snapshot.√ The theoretical values, on the other hand, would be too small. A value in between 1 and 2 times the theoretical position standard deviation would be most appropriate. The declination errors are depicted in figures 3.22 and 3.23. From these graphs, similar

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Figure 3.21: Reported Right Ascension errors from 638080 source measurements in 9970 maps by TKP pipeline software as a normalized histogram.

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! ! $ & ( * , - / , 1 / - & * 4 4 * , / 7 8 - & 7 ! 8 7 8 1 8 7 & : ! $ & ( * * = ! & * 4 * !

Figure 3.22: Relative declination errors of 638080 sources in 9970 maps by TKP pipeline software as a normalized histogram and the theoretical curve, i.e., a standard normal distribution.

conclusions can be drawn as from the Right Ascension statistics. The reported declination errors are 40% too large. In other words, if the reported declination errors were decreased by 40% overall, the observed distribution in figure 3.22 would match the theoretical curve. Table 3.6 lists the outliers and biases of both the flux and position measurements. We have processed 638080 sources, so we expect our highest outlier at about the 4.7σ level. This is not what we see for the positions, indicating, again, that the position errors have been overestimated. The flux outliers are also smaller than expected, but not by so much. The positions are measured without any considerable bias. The fluxes, however, are systematically low. This is due to the overestimate of the background level from κ, σ clipping. Table 3.3 shows an overestimate in the mean background of 0.073 Jy when 100 sources are inserted. From this table, one would expect an overestimate in the mean background of about 0.046 Jy when 64 sources are inserted. Consequently, when the error in the background mean has been corrected, one expects an overall bias in the fluxes of only +0.013 instead of -0.033. Of course, a more exact approach would involve redoing the calculations of paragraph 3.5.3 for exactly the set of sources we have used here.

3.9 Conclusions

We have processed large numbers of maps with and without sources. The TKP source extraction and source measurement software met the requirements with regard to robustness: we

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d g j d n u m h f g d e f g h d j j d k l m n o h m f n d g g f g u

_ Y }

_ Y |

_ Y {

_ Y x

_ Y y

_ Y z

_ Y w y

_ Y w x

Y Z Y Y Y Y Y Z Y Y Y ] Y Z Y Y _ Y Y Z Y Y _ ] Y Z Y Y ` Y Y Z Y Y ` ] Y Z Y Y b Y

c d e f g h d j j d k l m n o h m f n d g g f g r j d t g d d u v

Figure 3.23: Reported declination errors from 638080 source measurements in 9970 maps by TKP pipeline software as a normalized histogram.

Table 3.6: Outliers and biases in TKP flux and position measurements highest highest overall significance negative positive bias of overall outlier outlier bias (σ) (measured - true peak flux)/reported error 4.3 -4.2 -0.033 26 (measured - true R.A.)/reported error 3.8 -3.8 0.0022 1.8 (measured - true declination)/reported error 3.6 -3.4 0.00084 0.67

did not encounter any halts. Speed requirements were met: these 256*256 pixel maps were processed in less than 1s per map. Errors in the determination of background and noise levels increased with the fraction of source pixels. Possibly, the application of an asymmetric clipping algorithm could improve the performance, especially towards confusion limited maps. The deblending of sources was optimum for DEBLEND−MINCONT = 0.005, which resulted in 37.5% extra sources deblended and an error rate of 1.6% per source. The FDR algorithm performed satisfactory down to a fraction of less than 1% false positives. The discrepancy for α = 0.001 can possibly be explained by non-Gaussianity in the distribution of the background pixels. Source positions were measured accurately. Source ﬂuxes were biased low. This reﬂects the overestimate of the background level which increases with the fraction of source pixels. Again, performance could possibly be improved by a more advanced clipping

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algorithm.

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Bibliography

Y. Benjamini, & Y. Hochberg, Controlling the False Discovery Rate: a Practical Approach to Multiple Testing, J.R. Stat. Soc. 57, 289-300, 1995. E. Bertin, & S. Arnouts, SExtractor: Software for Source Extraction, A&AS 117, 393–404, 1996. E. Bertin, SExtractor v2.5 User’s Manual, draft version, Institut d’Astrophysique & Obser- vatoire de Paris, 2006. J.J. Condon, Confusion and ﬂux-density error distributions, ApJ 188, 279 (1974). J.J. Condon, Errors in Elliptical Gaussian Fits, PASP 109, 166 (1997). J.J. Condon, W.D. Cotton, E.W. Greisen, Q.F. Yin, R.A. Perley, G.B. Taylor, & J.J. Broderick, The NRAO VLA Sky Survey, AJ 115, 1693 (1998). P. Felkel, M. Bruckwschwaiger & R. Wegenkittl, Implementation and complexity of the watershed-from-markers algorithm computed as a minimal cost forest,inThe EURO- GRAPHICS conference, Manchester, UK, 2001. D.W. Hogg, Confusion errors in astrometry and counterpart association, AJ 121, 1207 (2001). A.M. Hopkins, C.J. Miller, A.J. Connolly, C. Genovese, R.C. Nichol, & L. Wasserman, A New Source Detection Algorithm Using the False-Discovery Rate, AJ 123, 1086–1094, 2002. A.M. Hopkins, J. Afonso, B. Chan, L.E. Cram, A. Georgakakis, & B. Mobasher, The Phoenix Deep Survey: The 1.4 GHz MicroJansky catalogue, AJ 125, 465–477, 2003. S.D. Hyman, T.J.W. Lazio, N.E. Kassim, P.S. Ray, C.B. Markwardt, & F. Yusef-Zadeh, Na- ture 434, 50, 2005. C.J. Miller, C. Genovese, R.C. Nichol, L. Wasserman, A. Connolly, D. Reichart, A. Hopkins, J. Schneider, & A. Moore, Controlling the False Discovery Rate in Astrophysical Data Analysis, AJ 122, 3492-3505, 2001.

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F. Patat, A robust algorithm for sky background computation in CCD images, A&A 401, 797- 807, 2003. K.U. Ratnatunga, & E.B. Newell, The reduction of panoramic photometry. III - an asymmetric clipping algorithm, AJ 89, 176, 1984. A. Refregier, & S.T. Brown, Eﬀect of Correlated Noise on Source Shape Parameters and Weak Lensing Measurements, http://arxiv.org/pdf/astro-ph/9803279v1, 1998. R.B. Rengelink, The Westerbork Northern Sky Survey: The Cosmological Evolution of Radio Sources,PhDThesis,Rijksuniversiteit Leiden, 1998. P. Teerikorpi, Inﬂuence of a generalized Eddington bias to galaxy counts, A&A 424, 73-78, 2004. C. Walker, High Dynamic Range Imaging, presentation at the Tenth Summer Synthesis Imag- ing Workshop, Albuquerque, New Mexico, 2006.

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CHAPTER 4

A new perspective on GCRT J1745-3009

H. Spreeuw, B. Scheers, R. Braun, R.A.M.J. Wijers, J.C.A. Miller-Jones, B.W. Stappers & R.P. Fender Astronomy and Astrophysics, 502, 549 (2009)

4.1 Abstract

Reports on a transient source about 1.25◦ south of the Galactic Centre motivated these follow- up observations with the WSRT and the reinvestigation of archival VLA data. The source GCRT J1745-3009 was detected during a 2002 Galactic Centre monitoring programme with the VLA at 92 cm by five powerful 10-min bursts with a 77-min recurrence while apparently lacking any interburst emission. The WSRT observations were performed and archival VLA data reduced to detect GCRT J1745-3009 again at different epochs and frequencies, to constrain its distance, and to determine its nature. We attempted to extract a more accurate lightcurve from the discovery dataset of GCRT J1745-3009 to rule out some of the models that have been suggested. We also investigated the transient behaviour of a nearby source. The WSRT data were taken in the “maxi-short”configuration,using 10 s integrations, on 2005 March 24 at 92 cm and on 2005 May 14/15 at 21 cm. Five of the six VLA observations we reduced are the oldest of this field in this band. GCRT J1745-3009 was not redetected. With the WSRT we reached an rms sensitivity of 0.21 mJy beam−1 at 21 cm and 3.7 mJy beam−1 at 92 cm. Reanalysis of the discovery observation data resulted in a more accurate and more complete lightcurve. The five bursts appear to have the same shape: a steep rise, a more gradual brightening, and a steep decay. We found variations in burst duration of order 3%. We improved the accuracy of the recurrence period of the bursts by an order of magnitude: 77.012 ± 0.021 min. We found no evidence of aperiodicity. We derived a very steep spectral index: α = −6.5 ± 3.4. We improved the 5σ upper limits for interburst emission and frac-

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tional circular polarisation to 31 mJy beam−1 and 8%, respectively. Any transient behaviour of a nearby source could not be established. Models that predict symmetric bursts can be ruled out, but rotating systems are favoured, because their periodicity is precise. Scattering constraints imply that GCRT J1745-3009 cannot be located far beyond the GC. If this source is an incoherent emitter and not moving at a relativistic velocity, it must be closer than 14 pc.

4.2 Introduction

Reports of a peculiar radio transient, GCRT J1745-3009, about 1.25◦ south of the Galactic Centre (Hyman et al. 2005, 2006, 2007) and the suggestion that this may be the prototype of a new class of particularly bright, coherently emitting radio transients have led to speculation about its nature. In particular, the 77 minute recurrence of the Jy level bursts was attributed to a period of rotation (Zhang & Gil 2005), revolution (Turolla et al. 2005) and precession (Zhu & Xu 2006). A nulling pulsar and an ‘X-ray quiet, radio-loud’ X-ray binary have also been suggested (Kulkarni and Phinney 2005), as well as an exoplanet and a flaring brown dwarf (Hyman et al. 2005). The discovery has led to follow-up observations and re-examination of archival data at both 92 cm and other bands. Those did not reveal a source (Zhu & Xu 2006; Hyman et al. 2005, 2006), with two exceptions (Hyman et al. 2006, 2007). Both of the redetections were single bursts, possibly due to the sparse sampling of these observations. The first redetection was possibly the decaying part of a bright (0.5 Jy level) burst that was detected at the first two minutes of a ten minute scan. The second redetection was a faint short ( 2 minute) burst that was completely covered by the observation. The average flux density during the burst was only 57.9 ± 6.6mJy/beam. This redetection also showed evidence for a very steep spectral index (α = −13.5 ± 3.0). The source has only been detected at three epochs, separated by less than 18 months, all at 92 cm, while the source was not detected in this band at 33 epochs over a period of more than 16 years (see Hyman et al. 2006, table 1) nor in any other band, ever. We observed the field containing GCRT J1745-3009 using eight 10-MHz IFs in the 92 cm band because its possible association with the supernova remnant G359.1-0.5 would mean that this source is about as far as the Galactic Center. That, in turn, implies a substantial dispersion measure (DM) that will become apparent as a delay of several seconds between the highest frequency IF and the lowest. This would be measurable if the bursts had some sufficiently sharp feature. An observation at 21 cm was performed to make use of the lower Galactic confusion and high sensitivity of the WSRT. We reanalysed five archival VLA datasets taken between 1986 and 1989 and the 2002 discovery dataset. All of these except the last were pointed at SgrA. Two of them, both obtained in A-configuration, had not been imaged before with the proper three- dimensional image restoration techniques. The complete set of observations we reduced is specified in Table 4.1.

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4.3 Data reduction

4.3.1 General We used AIPS (Greisen 2003) for the reduction of all datasets.

4.3.2 The 92 cm WSRT observations on 2005 March 24 The WSRT 92 cm observations on 2005 March 24 started at UT 01:22 with the observation of the calibration source 3C295. We acquired data from the target field from 02:33 until 07:50 using 10s integrations, with eight 10-MHz IFs, consisting of 128 channels, each 78.125 kHz wide, separated 8.75 MHz from each other and centered on frequencies ranging from 315.4 to 376.6 MHz. RFI was excised from the spectral line data using the AIPS task ’SPFLG’, while remaining RFI was removed from the continuum data using the AIPS task ’TVFLG’. Calibration was done in four steps. First we determined the variation in system temperature as a function of time (and therefore also position on the sky), using the intermittent firing of a stable noise source. Next we performed a bandpass calibration using the AIPS task ’BPASS’. We applied the bandpass solution using the AIPS task ’SPLAT’, producing a continuum file with one channel per IF. After that, we performed an external absolute gain calibration using an assumed flux of 61.5 Jy for 3C295 in the lowest frequency IF, by running the AIPS tasks ’SETJY’ and ’CALIB’. ’SETJY’ was set to use the absolute flux density calibration determined by Baars et al. (1977) and the latest (epoch 1999.2) polynomial coefficients for interpolating over frequency as determined at the VLA by NRAO staff. Finally, we self- calibrated the data for time variations in the relative complex gain phase and amplitude. Theoretically, we should be able to reach a thermal noise level of 0.15 mJy/beam in a 5 hour integration, or at least the nominal beam confusion noise limit of 0.3 mJy/beam. However, we did not attain this sensitivity due to the limited uv-coverage, RFI, and the existence of bright diffuse emission in the field. The latter compromises both self-calibration and image quality. This could be remedied to some extent by excluding spacings below a certain limit (uvmin > some multiple of λ, the wavelength). We chose a uvmin of 1.0kλ to eliminate the bulk of the diffuse emission, which could not be deconvolved with the available uv-coverage. SgrA and Tornado are the dominant sources in the field, their sidelobes contributed significantly to the image noise level of 9.0 mJy beam−1 at the location of GCRT J1745-3009. These and other sources were deconvolved in an image with an asymmetrical cell size (10 × 60). We chose to do so because a symmetrical cell size would yield a very elongated synthesized beam, this would hamper the deconvolution process. We subtracted the clean components of all sources from the uv-data before imaging the residual data with a symmetrical cell size. To lower the noise from the sidelobes of the two poorly subtracted extended sources, this final residual image was made by imposing a more severe lower limit of 2.5kλ on the spacings, which re- sultedinanoiselevelof3.7mJybeam−1. That final image was made from only 7% of the recorded visibilities. In retrospect, it is possible that the self-calibration process was adversely affected by bandwidth smearing, particularly because SgrA and Tornado were located far from the phase tracking center. Bandwidth smearing could have been diminished by keeping many channels

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0 100 200 300

-29 40

-30 00 DECLINATION (J2000)

17 47 00 46 30 00 45 30 00 44 30 00 RIGHT ASCENSION (J2000)

Figure 4.1: The supernova remnant G359.1-0.5 with ”The Snake”to the northwest, from our reduction of the GCRT J1745-3009 discovery observation on 2002 September 30/October 1 with the VLA in CnB configuration. This observation revealed this transient, indicated by a circle, for the first time (see Hyman et al. 2005). Noise levels in this image vary from 5 to 13 mJy beam−1 across the image. A Gaussian fit to the unresolved GCRT J1745-3009 gives a peak flux density of only 116±14 mJy beam−1 because the five Jy-level bursts have been averaged over about 6h of observation. A Gaussian fit to the source to the northeast of the supernova remnant, indicated by the box, gives a peak flux density of 91±14 mJy beam−1. Correction for primary beam attenuation has been applied.

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93 93 3 per IF (MHz) per IF ging (kHz) ging (MHz) Specifications of these observations. 2 IFs 1 Table 4.1: antennas conf.) of of of chann. per IF pol.prod. for ima- for ima- time (h) + (yymmdd) ( 12 8603293 860805 VLA A4 861226 VLA B5 881203 11 VLA C6 890318 VLA 8 A7 020930 1 15 VLA B8 050324 VLA CnB 22 1 127 050514 1 27 WSRT WSRT 22 (21cm) 127 2 3.1 63 14 2 12 2 0.8 7 1 0.8 6 7 7 31 1 1.4 64 128 98 1 0.7 3.0 98 1 10.0 20.0 2.5 98 1 2 195 0.7 4 4 4.6 0.7 98 98 2.7 4.9 12813 6328 6.2 1.4 8.2 154 5.7 89 5.3 5.3 4.6 5.3 This is the nummer of antennas afterThis flagging. is the number of IFs afterThis flagging is averaged total over bandwidth the RR for and Stokes LL I polarisation imaging, products, we if added both RR are and available. LL bandwidth. No. Date Telescope Number Number Number Bandwidth Number of Ch.width Tot.BW On-source 1 2 3

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GCRT J1745-3009 Chapter 4 ) 54 27 6 4 9 33 36 14 × × × × × × × × × 44 44 105 ecember 26 accounted for this 1 1 3 beam) ( / 8 for the 1986 August 5 and D . However, many bright compact sources 1 1 − − 2 beam) (mJy / 8 2008), but set the AIPS task ’IMFIT’ to solve for and 69 mJy beam 1 − 2 beam) (mJy / conf.) (mJy + (Kaplan et al. 2008). Corrections for primary beam attenuation and bandwidth smearing have been applied 7 . 52 09 ◦ 30 − = ,δ Flux measurements at 92 cm (unless otherwise noted) for detections and nondetections of GCRT J1745-3009 at 15s . (yymmdd) ( that should be detectable inby these replacing maps, the are rms not noise due by to a higher the number, very in poor this uv way coverage giving of a this very crude observation. representation We of these missing sources. observations respectively, (much) lower than the indicated value of 100 mJy beam The formal rms noise levels in these two maps are 19 mJy beam Here we did not tiepeak flux the density clean as beam well fit as to position the inThis the position is residual from image. the Kaplan average et noise in al. the ( residual image. 1 2 3 17h45m5 = 3 8612267 VLA C8 050324 050514 -26 WSRT WSRT (21cm) 29 -0.3 5 100 0.2 4 0.2 4 68 148 45 8812036 890318 020930 VLA A VLA B VLA CnB -18 41 110 15 19 15 8 12 27 12 860329 860805 VLA A VLA B -49 -19 27 20 18 100 10 Table 4.2:α No. Date Telescope Peak flux density error on fit rms noise resolution where appropiate.

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per IF in ’SPLAT’. SgrA and Tornado were close to the half power beam width (HPBW). This also hampers self-calibration because the frequency dependence of the primary beam attenuation is much stronger near the HPBW than near the pointing center. It could have been ﬁxed to some extent by running self-calibration per IF, at the expense of signal to noise. These ﬂaws, the poor uv-coverage, the exclusion of many spacings and the Galactic plane contribution to the system temperature explains why the achieved noise level is still well above the thermal noise limit of 0.68 mJy beam−1 for this number of visibilities, imaging bandwidth and IFs (see table 4.1), for a circular 60 beam towards cold sky.

4.3.3 The 21cm WSRT observations on 2005 May 14/15 The 2005 May 14/15 observations at 21 cm started at UT 22:33 with the observation of the calibration source 3C286. We acquired data from the GCRT J1745-3009 field from 23:09 until 03:46 using 10 s integrations, with eight 20-MHz IFs, separated 17 MHz from each other and centered on frequencies ranging from 1265 to 1384 MHz. The calibration was done in the same way as for the 92 cm WSRT observation. The assumed flux for the calibrator source 3C286 in the lowest frequency IF was 15.6 Jy. Theoretically, the rms sensitivity of these observations could be as low as about 21 μJy beam−1, for a 4.6 h integration. However, as for the 92 cm WSRT data, we excluded short spacings to eliminate most of the diffuse emission, which was necessary for successful self-calibration. The rms noise level in the final residual image was about 210 μJy beam−1. That noise level is partly due to the loss of data: the exclusion of spacings below 2.5kλ and the excision of RFI. The total loss of visibilities up to the final image was as high as 55%. With this number of visibilities and with the imaging bandwidth and IFs as mentioned in table 4.1, the theoretical thermal noise limit is 45 μJy beam−1 for a circular 13 beam towards cold sky.

4.3.4 The 92 cm VLA discovery dataset of 2002 September 30/October 1 The specifics of the 2002 discovery dataset are shown in table 4.1. We started its reduction with the flagging of 4 of the 27 antennas. Also, we flagged individual spectral channels per baseline, per IF and per polarisation product for all or part of the observing time, using the AIPS task ’SPFLG’. We flagged small portions, of 1 minute or more, of data at the beginning and end of each scan using the AIPS task ’QUACK’. We also clipped data contaminated by RFI using the AIPS task ’CLIPM’. Next, we performed an external absolute gain calibration with an assumed flux of 25.9 Jy for 3C286 in the lowest frequency IF. This flux was determined by running the AIPS task ’SETJY’, using the absolute flux density calibration determined by Baars et al. (1977) and the latest (epoch 1999.2) VLA polynomial coefficients for interpolating over frequency. We determined gain phase and gain amplitude solutions for both the primary calibrator 3C286 and the phase calibrator 1711-251, using the AIPS task ’CALIB’. This task was run using all spacings for the primary calibrator and spacings longer than 1kλ for the phase calibrator. The AIPS task ’GETJY’ determines the flux of the secondary calibrator from those gain solutions and the flux of the primary calibrator. ’GETJY’ found a flux of 11.1 Jy for 1711-251 at the highest frequency IF (327.5 MHz). The gain solutions were interpolated using the AIPS task ’CLCAL’.

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Next, we used 3C286 to find a bandpass solution. In doing so, we applied the interpolated gain solutions from ’CLCAL’ for spacings longer than 500 wavelengths (uvmin > 0.5kλ). For one of the antennas no visibilities were recorded during the scan of 3C286. Hence, no bandpass solution could be found for this antenna and only 22 antennas were left for imaging. We applied the gain and bandpass solution to 20 of the total of 31 available channels using the AIPS task ’SPLAT’. Every two channels were averaged. Next, we performed 18 iterations of phase only self-calibration, using initial solution intervals of 5 minutes, gradually decreasing down to 1 minute. We used 195 kHz channels for imaging and a cellsize of 4. We used 85 512 × 512 pixel facets to cover the primary beam and no facets for outlier fields. We performed an amplitude and phase self-calibration and we produced the final model from the spectral averaged dataset. After that, we reran ’SPLAT’ on the line data, but this time without spectral averaging, selecting 21 × 97 kHz of the available channels. We phase self-calibrated the new dataset using the acquired model from the spectral averaged data. Next, we imaged and deconvolved our phase self-calibrated dataset using 61 facets to cover the primary beam and 22 facets for the outlier fields. This time we used 256 × 256 pixel facets with a pixel size of 10. We self-calibrated again, but this time we solved for amplitude and phase, using a solution interval of 1 minute. The total average gain was normalized in this process. We imaged and deconvolved 450 Jy of total flux from the amplitude and phase self-calibrated dataset to make our final model. Figure 4.1 shows the central facet of this model after correction for primary beam attenuation. We noticed that SgrA is by far the brightest source in the field and that it is near the half power beam point. We anticipated that the calibration of the uv data could be optimized by applying separate gain solutions to the clean components of the facet with SgrA, so we ran the AIPS runfile PEELR on the clean components of the facet of SgrA, solving for gain amplitudes and phases on a timescale of 10s. We subtracted the clean components from the peeled data using the AIPS task ’UVSUB’ and we determined the position of GCRT J1745-3009 in our final model using the AIPS task ’IMFIT’. We shifted the phase stopping centre to this position using the AIPS task ’UVFIX’ and we averaged all spectral channels using the AIPS task ’SPLIT’. We did a final edit using the AIPS task ’CLIP’ and set uvmin = 1.0kλ. We ran the AIPS task ’DFTPL’ on this final residual dataset to produce our lightcurves. We did not correct the output of ’DFTPL’ for primary beam attenuation because GCRT J1745-3009 was about 13 from the pointing center. Primary beam attenuation for this angular separation is only 1.8%. In retrospect, it turned out that both the amplitude and phase (A&P) self-calibration and the peeling of SgrA had negligible effect on the burst shapes in the final lightcurves. So the dataset could be reduced in a standard way, except perhaps for the large number of selfcal iterations and the exclusion of a rather large number of antennas, 5 of the 27 antennas being excluded for the entire observation.

4.4 The source on the opposite side of the supernova remnant

The source northeast of the supernova remnant G359.1-0.5, indicated by a box in figure 4.1 is resolved in VLA A configuration. From a combination of three VLA datasets, two in A configuration and one in B configuration, this source was detected with a peak flux density

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of 17.1±2 mJy beam−1 and an integrated flux of 47.6 mJy (see Nord et al. 2004, table 2, source 72). Apparently the synthesized beam of the combination of these datasets (12 × 7) resolves this source. As noted in the caption of figure 4.1, the peak flux density we derived from the 2002 discovery observation is 91±14 mJy beam−1. A large fraction of the difference with the integrated flux measurement by Nord et al. (2004) is probably caused by extended emission. Indeed, when we exclude the shortest spacings, uvmin = 1.0kλ,wefindamuch lower peak flux density of 73±5 mJy beam−1. The remaining difference may also come from extended emission that is picked up differently by these observations. However, the main reason that this source drew our attention is its absence in a high dynamic range image of the Galactic Centre at 92 cm with a noise level of about 5 mJy beam−1 and an angular resolution of 43 (see LaRosa et al. 2000, figure 11, hereafter called the LaRosa map). The datasets used for the LaRosa map were taken on 1986 August 5 (B conf., 8 antennas) and 1986 December 26 (C conf., 15 antennas), 1987 March 25 (D conf., 15 antennas) and 1989 March 18 (B conf., 27 antennas). Our reduction of the 1989 March 18 data shows the source at the ≥ 6σ level, a Gaussian fit gave a peak flux density of 53±7 −1 mJy beam . Here, the size of the synthesized beam is 27 × 14 while we set uvmin to 2.0kλ. This clear detection indicates that the non-detection of the source in the LaRosa map is probably not due to transience. More likely, the source is concealed in the LaRosa map by a negative background peak.

4.5 Overview of ﬂux measurements of GCRT J1745-3009

We hoped to redetect GCRT J1745-3009 with the WSRT, with some of the VLA observations mentioned in the previous section and with two additional A configuration observations from the VLA archive. We did not redetect the source, but we measured its flux at its position in all of the seven maps. Specifics of these observations are shown in table 4.1. Note that the on-source time for the two WSRT observations is comparable to the VLA observations, despite the limited time for which the WSRT can observe this low declination source. The reason for this is that the WSRT in general does not need to observe secondary calibrators. The results of the flux measurements at these epochs and at the time of the discovery are shown in table 4.2. For the seven nondetections, we fitted the restoring beam to the position reported by Kaplan et al. (2008). We have also imaged 10 minute subsets of the residual data from the five 1986-1989 observations to look for isolated bursts, but we found none. We merged our results from table 4.2 with those from a recent overview of observations since 1989 (see Hyman et al. 2006, table 1) together with the results from the second redetection (Hyman et al. 2007) to produce a plot of 5σ flux upper limits on quiescent emission from GCRT J1745-3009 in the 92 cm band (see fig. 4.2). In order to derive appropriate values, we scaled the 10-minute scan sensitivities mentioned (20 and 10 mJy beam−1 for the VLA and the GMRT respectively, after correction for primary beam attenuation) with the square root of the observing bandwidth, taking 6.2 MHz as the base. The sensitivities for complete observations were also scaled with the square root of the total on-source time. We note that the 1989 March 18 observation was already analysed by Hyman et al. (2006) and their reduction

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Ò É Å Æ ¾ ¼ Å ¿ À Á Â Ë Å Ê º » Ç Ó Ã Å Æ Ö Õ ½ » Ê Á Ç Å º È ¿ Ç Ï Å Ô Õ Õ × Î Á Æ º Ç

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Figure 4.2: Approximate detection thresholds (5σ noise levels) at the location of GCRT J1745-3009 of 41 Galactic Center observations at 92 cm over two decades. For the WSRT observation at 92 cm, the 10 minute scan sensitivity is not indicated, since the snapshot point spread function (psf) of a linear array does not allow to do this accurately. The observations in this plot start on 1986 March 29 and end on 2005 September 27.

led to slightly more constraining values, so we adopted these in figure 4.2. Here, we took account of the fact that the total bandwidth of that observation was actually 1.4 MHz instead of the 12.5√ MHz mentioned in their table 1. Consequently,√ √ we derived 5σ upper limits of 5 · 20 · 6.2/1.4 = 210 mJy beam−1 and 5 · 20 · 6.2/1.4/ 5.3 · 6 = 37 mJy beam−1,for those 10-minute scans and for that complete observation, respectively. The lowest noise level of all 92 cm observations, about 6 mJy beam−1 in a 2 minute interval, was achieved at the time of the second redetection, with the GMRT on 2004 March 20 (Hy- man et al. 2007). This is actually the only observation that could have detected bursts of this kind and only by making 2 minute scan averages. None of the observations included in figure 4.2 can detect the 2004 burst (Hyman et al. 2007) in 10 minute averages at the 5σ noise level. The WSRT 2005 May 14/15 5σ upper limit at 21cm (1.05 mJy beam−1) was less constraining than the VLA upper limit at that wavelength on 2005 March 25 (0.4 mJy beam−1,seeHyman et al. 2006). 21 cm observations are not included in figure 4.2.

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Figure 4.3: The plot above shows the lightcurve from the discovery dataset of GCRT J1745-3009 with 30s sampling. This plot is setup in the same way as the lightcurve in the discovery paper except for the flux density measurements between bursts. For those nondetections Hyman et al. (2005) showed 3σ upper limits on interburst emission, we show the actual background flux density measurements. Also, we have folded the lightcurve at intervals of 77.012 minutes instead of 77.130 minutes. The first interval is shown in the bottom panel, starting at 20h50m00s on 2002 September 30 (IAT). The average of all the error bars shown is 74 mJy. The gaps are due to phase calibrator observations.

4.6 Reanalysis of the 2002 discovery dataset

4.6.1 Lightcurve The lightcurve that we extracted from the discovery dataset of GCRT J1745-3009 at the position derived in paragraph 4.6.3 is shown in ﬁgure 4.3. The bursts seem to have similar shapes: a steep rise, a gradual brightening and a steep decay, more consistent than the bursts shown in ﬁgure 1 of the discovery paper (Hyman et al. 2005). This lightcurve is twice as accurate as the original one. We also ran the AIPS task ’DFTPL’ with 5s sampling, this is the integration time for the recording of the visibilities in the discovery dataset. We found no compelling evidence for interburst emission, not even on the shortest (5s) timescale. We determined the recurrence interval between bursts by measuring the times of steepest rise for four of the bursts. Consecutive 1 minute chunks of data were selected by a sliding window. For each chunk of data we determined its average slope by weighted linear regression. The weights

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Figure 4.4: This plot illustrates how the times of steepest rise for four of the bursts are determined. Weighted linear regression is performed on successive one minute chunks of data. The chunks have a maximum of 55s of overlap time. Here the rising part of the ﬁrst burst is shown.

come from the reciprocal of the noise variances from ’DFTPL’. The time corresponding to the steepest positive slope was then calculated as the weighted average of the timestamps in the datachunk. For the first burst, this method is illustrated in figure 4.4. Weighted linear regression also calculates the error bars of the times of steepest rise from the error bars of the data points. The times of steepest rise and the corresponding error bars are shown in table 4.3. The times mentioned in that table are relative to 20h50m00s on 2002 September 30 (IAT). We then again applied the formulae for weighted linear regression to find the period between bursts and its 1σ error. We found a period of 77.012±0.021 min from the values in table 4.3. We have improved the error on the period by an order of magnitude (Hyman et al. 2006, paragraph 3 and caption of figure 3), but the period itself agrees with the previously determined period of 77.1 m ±15s. However, it is important to note that our method differs from the one used by Hyman et al. (2005). We have made no assumption with regard to the burst shapes in determining the period. The residuals with respect to that fit are 0.097, -0.114, 0.053 and -0.007 minutes for the first, second, fourth and fifth burst, respectively. The residual for the second burst is the largest,

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F E 4

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Figure 4.5: The plot above shows the ﬁve bursts from the discovery dataset of GCRT J1745-3009 with 30s sampling folded at intervals of 77.012 minutes. Time is relative to 20h50m00s on 2002 September 30 (IAT) (plus multiples of 77.012 minutes).

6.8s ”too late”with respect to the fit, this corresponds to 1.9σ, σ = 0.060 min, this is the error on the time of steepest rise of the second burst. We were also able to measure the times of steepest decay for four of the bursts in a similar manner, see table 4.4. For three bursts we could measure both the time of steepest decay and the time of steepest rise. In this way we found that the time between steepest rise and steepest decay varies. We found intervals of 8.29±0.08, 8.87±0.09 and 8.66±0.09 min for the second, fourth and fifth burst, respectively. So for the second burst the interval between steepest rise and steepest decay is 3.45% less than the weighted mean of those three intervals. The significance of this deviation is 3.0σ. We can use the derived period to fold the bursts in one plot, see figure 4.5. This plot shows that the bursts indeed have similar shapes.

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Table 4.3: Measurements of times of steepest rise for four bursts Burst Time of steepest 1σ error Slope 1σ error number rise (min) (min) (Jy/min) (Jy/min) 1 60.624 0.068 0.706 0.158 2 137.848 0.060 0.828 0.175 4 291.704 0.065 0.724 0.138 5 368.776 0.065 0.743 0.150

Table 4.4: Measurements of times of steepest decay for four bursts Burst Time of steepest 1σ error Slope 1σ error number decay (min) (min) (Jy/min) (Jy/min) 2 146.136 0.041 -1.125 0.146 3 223.489 0.057 -0.811 0.150 4 300.578 0.067 -0.717 0.150 5 377.439 0.066 -0.734 0.148

4.6.2 Implications for other observations

Now that we have determined the periodicity of the bursts more accurately, we can check if other short GC observations at 92 cm before and after the discovery observation should have detected GCRT J1745-3009. The observation closest in time was taken on 2002 July 21 (see Hyman et al. 2006, table 1). This was a 59.2 minute scan starting 1719.75 hours before the start of the bright part of the first burst in the discovery dataset. This corresponds to 1339.86 periods of 77.012 minutes. Consequently, the source should not have been seen during that short scan and this was indeed the case (Hyman et al. 2006). However, there is a large uncertainty in calculating burst times over an interval as large as 71 days. The error is 0.021min · 1339 = 28 min. From that uncertainty and Gaussian statistics, we calculated that the chance of having observed at least 5 minutes of bursting activity on 2002 July 21 was 74%, assuming that GCRT J1745-3009 were bursting as during the discovery observation. If GCRT J1745-3009 was indeed active on 2002 July 21, we can infer from the nondetection on that occasion that P, the recurrence interval between bursts is tightly constrained: 77.007min < P < 77.021min. The next observation closest in time was taken on 2002 June 24. Its duration was only 34.5 min, starting 1842.17 periods of 77.012 minutes before the start of the bright part of the first burst in the discovery dataset. During this observation we should have seen at least 6 minutes of a burst if we take into account the constraints on the period from the nondetection on 2002 July 21. From the fact that we did not detect emission on 2002 June 24 we may conclude that activity started after this 34.5 minute scan. The first suitably pointed 92 cm observation after the discovery observation was taken on 2003 January 20. The source was not detected, but the data were taken with the VLA in CD configuration. This implies that rms noise levels from 10-min scans are about 250 mJy

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beam−1 (see Hyman et al. 2006, and ﬁgure 4.2 in this paper). Thus it is likely that GCRT J1745-3009 could not have been detected at the 5σ level on 2003 January 20, even if an individual ten minute scan were spaced in time such that it completely covered a burst. It may be that the activity continued until the summer of 2003 when three 59 minute and four 34 minute GC observations were performed with the VLA in A conﬁguration. At least two of these scans are spaced in time such that if one covered the interval between two bursts, the other must have covered a complete burst. So we are sure that the recurrent bursting activity of GCRT J1745-3009 stopped before it was redetected on 2003 September 28. In summary, the bursting activity with a period of 77.012 minutes as seen during the discovery observation must have started after 2002 June 24 and may have continued until the summer of 2003. Unfortunately, we cannot constrain the timespan of a recurrently bursting GCRT J1745-3009 to less than a year.

4.6.3 Position and ﬂux measurements; spectral index determination

The most accurate position measurement, corresponding to the highest signal to noise ratio, can be achieved by selecting just the time intervals that cover the bursts. We found a peak flux density of 900 ± 23 mJy beam−1 and this J2000 position: α = 17h45m05.015s ± 0.045s,δ= −30◦0952.19 ± 0.52. This position of GCRT J1745- 3009 has not yet been corrected for ionospheric-induced refraction (see Nord et al. 2004, for some background). That correction, which is basically, but not exactly, a global position shift of all sources in the field, will significantly increase the uncertainty in the position of GCRT J1745-3009. Here, we just mention that in our maps the bright source SGR E46 is 0.33s west and 0.89 north of the NVSS (Condon et al. 1998) position. The NVSS catalogue mentions a positional accuracy of 0.45 in right ascension and 0.6 in declination for this source. We consider the actual uncertainty for the given position of GCRT J1745-3009 to be 5 in both right ascension and declination. Rms noise values in the map that constitutes our final model range between 5 and 13 mJy −1 beam . We also made a map from the same data, but without short spacings (uvmin = 1.0kλ). Noise levels then drop significantly, varying between 4 and 6 mJy beam−1 across the image. We removed the bursts and we made a cleaned image with the same spacings. The noise levels are somewhat higher now: between 5 and 7 mJy beam−1. In order to derive an upper limit on interburst emission we fitted the clean beam to the position measured above. We found a peak flux density of −0.6±6.4 mJy beam−1, after correction for primary beam attenuation (1.8%). This gives a 5σ upper limit on interburst emission of 31 mJy beam−1. This is more than twice as constraining as the original upper limit. Neglecting primary beam attenuation, we found a weighted mean flux of 103.5 ± 2.9mJy beam−1 from the output of the AIPS task ’DFTPL’ on the residual data with full (5s) sampling. We also ran ’DFTPL’ on this data for each of the five bursts and for each of the two IFs separately. We only selected times for which both IFs had fluxes and then calculated the natural logarithm of the ratio of the fluxes for each timestamp and the variance of that quantity. We then calculated the weighted mean of these logarithms for each burst. The spectral index and error bar for each burst are shown in table 4.5, using the average frequencies of IF1

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Table 4.5: Measurement of spectral index for each burst Burst α 1σ error α number (S ν ∝ ν ) 1-9.96.7 2-9.09.3 30.98.7 4-0.46.9 5 -12.3 6.9

(327.5000 MHz) and IF2 (321.5625 MHz). The spectral indices and error bars of the individual bursts do not seem inconsistent with Gaussian statistics, so we calculated the weighted mean spectral index as well: α = −6.5±3.4. This is not incompatible with the spectral indices found by Hyman et al. (2006, 2007, α = −4 ± 5andα = −13.5 ± 3.0), given the large error bars. The weighted mean of these three measurements is α = −9.4 ± 2.1.

4.6.4 Circular polarisation

We compared the lightcurves for left (’LL’) and right (’RR’) circular polarisation with 30s sampling. Although there are occasional ’LL’ and ’RR’ flux differences during the bursts larger than the sums of the respective error bars, this is also seen in between the bursts. There is no compelling evidence for circularly polarised emission during any particular phase of the burst cycle. On the other hand, we cannot exclude it completely, because we have insufficient signal to noise in Stokes V. From the residual data, we selected the times corresponding to the bursts and we made a Stokes V dirty image. We corrected for primary beam attenuation and fitted the clean beam to the position of GCRT J1745-3009 as we did in the previous paragraph to determine the upper limit for interburst emission . We measured a Stokes V of −20 ± 10 mJy beam−1. Using the total intensity averaged over the bursts, 900 ± 23 mJy beam−1, we found that the 5σ upper limit on the fractional circular polarisation, |V|/I, is 8%. Hyman et al. (2005) derived a weaker constraint of 15% on the fractional circular polarisation averaged over the bursts. Despite the lack of evidence for circularly polarised emission in the discovery observation, it has been detected in the data from the 2003 recovery observation (Roy et al. 2008). Here, only the last part a single burst was covered. From this detection and the fact that the average of Stokes V over a complete burst (almost completely) vanishes we infer that during an earlier part of the burst, Stokes V must have the opposite sign. In other words, if we can assume that the 2003 burst is similar to the 2002 bursts with regard to circularly polarised emission, there must be a sign change in the circular polarisation during the bursts.

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4.6.5 Maximum source size and maximum distance for incoherent emission All of the steep rising part of the bursts can be well approximated by a straight line. This is true even at the very beginning of the bursts, when the flux is at or just above the noise level. It can be seen in the lightcurve down to 10s sampling, but at full (5s) sampling we have insufficient signal to noise to trace any possible slope flattening down to the first 5s of the beginning of the bursts. The average slope of the bursts in table 4.3 is 0.75 Jy/min or 0.125 Jy/10s. This implies a flux doubling time of Δt = 10s at the beginning of the bursts, when the flux is 125 mJy. The maximum source size at that time is then 10 lightseconds, if we assume that the source is not moving at a relativistic velocity (see, e.g., Harris et al. 2006, for some background). We can use the maximum source size c · Δt to link the brightness temperature Tb(K) to the flux F and maximum distance D (see, e.g, Rybicki & Lightman 1979):

λ2Iν λ2F 2F D T = = = ( )2 (4.1) b 2k 2kπθ2 πk νΔt

where λ, Iν,ν,kandθ are the wavelength, the speciﬁc intensity, the frequency, Boltzmann’s constant and the angle subtended by the radius of the source, respectively. If we express the distance in pc, the ﬂux in Jy and the frequency in GHz, we get:

D T = 4.39 · 1011F( )2 (4.2) b νΔt

If synchrotron self-Compton radiation limits the brightness temperature to 1012K, the maximum distance for a source of size ten lightseconds and a ﬂux of 0.125 Jy emitting incoher- ently at 325 MHz is 14pc, assuming it is not moving at a relativistic velocity. Hyman et al. (2005) used the decay time of the bursts (conservatively estimated at 2 min.) to calculate a maximum distance of 70 pc. So we have improved this upper limit by a factor 5.

4.7 Discussion

Five of these upper limits on the flux of GCRT J1745-3009 come from the oldest observations of this field in the 92 cm band. This may provide interesting constraints on the feasibility of the double neutron star binary model (Turolla et al. 2005) in the near future. In this model, similar to J0737-3039, the period of recurrence of the 2002 bursts is explained by an orbital period of 77 minutes. The lack of activity for many years is explained by geodetic precession, which could have caused the wind beam of the most luminous pulsar not to intercept the magnetosphere of the other pulsar for decades. Zhu & Xu (2006) claim that the redetection in 2003 (Hyman et al. 2006) does not support this model. Their remark was, however, erroneously based on a geodetic precession period of 3yr, but this is actually 21 years1.The last redetection (2004 March 20) and the first observation (1986 March 29) are 18 years apart. Unfortunately this timespan is too short to test the double neutron star binary model, but not

1The ”characteristic time for changing the system geometry” as mentioned by Turolla et al. (2005) diﬀers from the period of geodetic precession by a factor 2π.

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if we redetect the system in the near future. More constraining are the results from population synthesis models (see, e.g., Portegies Zwart & Spreeuw 1996, ﬁg.2): fairly eccentric (0.3

−5 1.6 −5 3 4.4 τsc = 4.5 · 10 · DM · (1 + 3.1 · 10 · DM ) · λ (4.3)

−3 with the scattering time (τsc) in ms, the dispersion measure (DM) in cm pc and the observation wavelength (λ) in meters. From this relation we ﬁnd a dispersion measure of 925 cm−3pc. This would imply that GCRT J1745-3009 is located beyond the GC. For a check on consistency we compared this dispersion measure with the DM that can be found from the

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formula for the dispersion delay Δt (in seconds):

Δ = · · 1 − 1 t 4150 DM ( 2 2 ) (4.4) f1 f2

between the highest ( f2 = 327.50 MHz) and lowest frequency IF ( f1 = 321.56 MHz) using the times of steepest rise for four of the bursts. The delay we found was −0.94 ± 3.65s corresponding to a DM of −653 ± 2530 cm−3pc, consistent with the value above, but a very weak constraint. From the poorer quality of the exponential fits relative to the linear fits we are inclined to conclude that the shape of the tails of the observed bursts are dominated by tails in the intrinsic emission. It seems justified that the average decay time from the exponential fits (0.72 min) is merely an upper limit for the true scattering time. In general we can state that for scattering times corresponding to distances not far beyond the Galactic Center the intrinsic burst shape will not differ greatly from the observed burst shape, besides any unresolved variability on very short timescales. The reason for this is that the duration of the observed bursts is much longer ( 10 min) than any reasonable scattering time for sources near the GC. We can work out the original burst profile using theorems for Laplace transforms. The intrinsic emission I(t) is convolved with the scattering function ζ(t). This gives the observed burst O(t):

O = I ∗ ζ (4.5)

where * denotes convolution. For simple scattering, ζ is the product of the Heaviside step function Π and an exponential: −t ζ(t) =Π(t) · exp( ) (4.6) τsc

1 The Laplace transform of this product is equal to s+α , with s the transformed coordinate and α = 1/τsc. Now, using the theorems for Laplace transforms of convolved functions and derivatives we find: dO I · κ = α · O + (4.7) dt with κ a constant for normalization. If no emission is absorbed, it follows that κ = α. Thus, we could reconstruct the intrinsic, unscattered burst from the observed burst if we knew the scattering time τsc. If the observed burst is represented very accurately by three straight lines for the steep rise, the gradual brightening and the steep decay, the original burst must have the same slopes. It then follows that τsc = 1/α = 0, hence no scattering, unless there are faults, i.e. sudden ”jumps”, in the intensity of the intrinsic emission. So the breaks link scattering times and fault sizes. Without any assumptions on the possible degree of faulting in the intrinsic emission, we can find an upper limit for the scattering times using the end of the tails of the observed bursts. The slopes seem constant until the flux is essentially zero for at least three of the

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bursts. For the end of the tail of the second burst, which is relatively noisy, this is not so clear. Equation 4.7 then imposes an upper limit on the scattering time τsc from the condition that the intrinsic emission cannot be negative. This means that the scattering time must be smaller than the time resolution for which we can determine the slopes with confidence: 10s. This implies that GCRT J1745-3009 cannot be located far beyond the GC. From the NE2001 model of Cordes and Lazio (2003) we find a pulse broadening time of 9.93s at 325 MHz for a distance of 11 kpc in the direction of GCRT J1745-3009. We conclude from this discussion that the observed bursts depicted in figure 4.5 will closely resemble the intrinsic bursts. Models will need to explain the asymmetry of the bursts, the steep rise, the more gradual brightening and the steep decay and the breaks between them as well as the fact that the brightest emission is seen just before the steep decay.

4.8 Conclusions

We have derived new upper limits on the quiescent emission of GCRT J1745-3009 at seven epochs. Six observations were made in the 92 cm band and one in the 21 cm band. The 92 cm observation of GCRT J1745-3009 on 2005 March 24 with the WSRT was the second deepest until that time. Five of these seven epochs constitute the oldest set of 92 cm observations taken of the Galactic Center. The nondetections at those epochs do not provide evidence for the double neutron star binary model (Turolla et al. 2005) with a geodetic precession period close to 18 years. However, geodetic precession times could well be somewhat longer. We have reproduced the lightcurve of the discovery dataset of GCRT J1745-3009 more accurately and more completely than in the discovery paper. We see that the shapes of the five bursts are consistent: a steep rise, a gradual brightening and a steep decay. We have improved the 5σ upper limit on interburst emission from 75 mJy beam−1 to 31 mJy beam−1.Also,we further constrained the 5σ upper limit on the fractional circular polarisation from 15% to 8%. We determined the recurrence interval between bursts more accurately: 77.012 ± 0.021 min. We see no evidence for aperiodicity, but we do find that the duration of the bursts varies at the level of a few %. We derived a very steep spectral index, α = −6.5 ± 3.4. We have investigated scattering and we have shown that scattering times must be less than 10s. This implies that GCRT J1745-3009 cannot be located far beyond the GC. It also means that the shape of the observed bursts will differ little from the intrinsic emission. Models for GCRT J1745-3009 have to explain the asymmetry in the shape of the bursts and in particular the gradual brightening until the steep decay. Some of the suggested models (Turolla et al. 2005; Zhu & Xu 2006) predict symmetric bursts. The simplest interpretations of those models can now be ruled out, but it is conceivable that the asymmetry in the bursts could be achieved by adding some complexity to those models. Our results favour a rotating system, like the white dwarf pulsar (Zhang & Gil 2005), because that can explain the high level of periodicity we see. We have shown that it is very unlikely that this transient is an incoherent synchrotron emitter, because it would have to be closer than 14 pc, unless the emitting region is moving at a relativistic velocity. Although we now have more contraints on the properties of this source, we are still unsure about its basic model. A better understanding of its nature should come from more detections by long time moni-

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toring with high sensitivity and high angular resolution, to tackle the confusion limit and to reduce the number of possible optical counterparts. The next generation of radio telescopes, like LOFAR (see, e.g., Fender et al. 2006), will help to do so. The most pressing issue in revealing the nature of GCRT J1745-3009 is still the determination of its distance, which could be achieved by a new detection with suﬃcient bandwidth between sidebands, in order to measure the time delay from dispersion towards the Galactic Center. We have also investigated possible transient behaviour of a source on the opposite side of the supernova remnant G359.1-0.5 but we found no compelling evidence for variability.

4.9 Acknowledgements

We thank Tao An from Shanghai Astronomical Observatory for supplying us with an A- conﬁguration model of the radio galaxy 1938-155. This model was necessary to phase calibrate our 1986 March 29 observations. We thank the anonymous referee for encouraging us to reproduce the lightcurve from the 2002 discovery dataset of GCRT J1745-3009 after we reported low noise levels in a map from that dataset. The National Radio Astronomy Obser- vatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.

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Baars, J.W.M., Genzel, R., Pauliny-Toth, I.I.K., & Witzel, A. 1977, A&A, 61, 99. Chatterjee, S., Goss, W.M., & Brisken, W.F. 2005, ApJ, 634, L101. Condon, J.J., Cotton, W.D., Greisen, E.W., Yin, Q.F., Perley, R.A., Taylor, G.B., & Broderick, J.J. 1998, AJ, 115, 1693. Cordes, J.M., & Lazio, T.J.W. 2003, astro-ph/0301598, http : //rsd − www.nrl.navy.mil/7213/lazio/ne−model/. Fender, R.P. et al., in Proceedings of the VI Microquasar Workshop: Microquasars and Be- yond, 2006, 104. Greisen, E.W. 2003, in Information Handling in Astronomy - Historical Vistas,ed.A. Heck (Astrophysics and Space Science Library, Kluwer Academic Publishers, Dordrecht, Netherlands), 285, 109. Harris, D.E., Cheung, C.C., Biretta, J.A., Sparks, W.B., Junor, W., Perlman, E.S., & Wilson, A.S. 2006, ApJ 640, 211. Hyman, S.D., Lazio, T.J.W., Kassim, N.E., Ray, P.S., Markwardt, C.B., & Yusef-Zadeh, F. 2005, Nature 434, 50 Hyman, S.D., Lazio, T.J.W., Roy, S., Ray, P.S., Kassim, N.E., & Neureuther, J.L. 2006, ApJ, 639, 348. Hyman, S.D., Roy, S., Pal, S., Lazio, T.J.W., Ray, P.S., Kassim, N.E., & Bhatnagar, S. 2007, ApJ, 660, L121. Kaplan, D.L., Hyman, S.D., Roy, S., Bandyopadhyay, R.M., Chakrabarty, D., Kassim, N.E., Lazio, T.J.W., & Ray, P.S. 2008, accepted for publication in ApJ. Kulkarni, S.R., & Phinney, E.S. 2005, Nature, 434, 28. Mitra, D., & Ramachandran, R. 2001, A&A, 370, 586. Nord, M.E., Lazio, T.J.W., Kassim, N.E., Hyman, S.D., LaRosa, T.N., Brogan, C.L., & Duric, N. 2004, AJ, 128, 1646.

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LaRosa, T.N., Kassim, N.E., Lazio, T.J.W., & Hyman, S.D. 2000, AJ, 119, 207. Portegies Zwart, S.F., & Spreeuw, H.N. 1996, A&A, 312, 670. Reid, M.J. 1993, ARA&A, 31, 345. Roy, S., Hyman, S.D., Pal, S., Lazio, T.J.W., Ray, P.S., Kassim, N.E., & Bhatnagar, S. in Proceedings of ’Bursts, Pulses and Flickering: wide-ﬁeld monitoring of the dynamic radio sky’, 12-15 June 2007, Kerastari, Tripolis, Greece., p.9. Roy, S., Hyman, S.D., Pal, S., Lazio, T.J.W., Ray, P.S., Kassim, N.E., & Bhatnagar, S. in Proc. 25th meeting of ASI (2007), 2008, p. 25. Rybicki, G.B. & Lightman, A.P. in Radiative Processes in Astrophysics, Wiley, New York, NY, 1979, p 8, p 25-26. Turolla, R., Possenti, A., & Treves, A. 2005, ApJ, 628, L49 Zhang, B., & Gil, J., 2005, ApJ, 631, L143. Zhu, W.W., & Xu, R.X. 2006, MNRAS, 365, L16.

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CHAPTER 5

Low frequency observations of the radio nebula produced by the giant ﬂare from SGR 1806-20: Polarimetry and total intensity measurements

H. Spreeuw, B. Scheers & R.A.M.J. Wijers Astronomy and Astrophysics, 509A, 99 (2010)

5.1 Abstract

The 2004 December 27 giant flare from SGR 1806-20 produced a radio nebula that was detectable for weeks. It was observed at a wide range of radio frequencies. To investigate the polarized signal from the radio nebula at low frequencies and to perform precise total intensity measurements. We made a total of 19 WSRT observations. Most of these were performed quasi simultaneously at either two or three frequencies, starting 2005 January 4 and ending 2005 January 29. We reobserved the field in 2005 April/May, which facilitated an accurate subtraction of background sources. At 350 MHz, we find that the total intensity of the source is lower than expected from the GMRT 240 MHz and 610 MHz measurements and inconsistent with spectral indices published previously. Our 850 MHz flux densities, however, are consistent with earlier results. There is no compelling evidence for significant depolarization at any frequency. We do, however, find that polarization angles differ substantially from those at higher frequencies. Low frequency polarimetry and total intensity

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measurements provide a number of clues with regard to substructure in the radio nebula associated with SGR 1806-20. In general, for a more complete understanding of similar events, low frequency observations can provide new insights into the physics of the radio source.

5.2 Introduction

The 2004 December 27 flare from the Soft-γ-ray Repeater SGR 1806-20 was a major event in astronomy in a number of ways. First of all by the energy of the explosion: the brightest flash of radiation from beyond our solar system ever recorded. This is how it caught the attention of a larger audience. Secondly, because the flare provided new observational data about a known class of objects: magnetars, i.e., strongly magnetized neutron stars (see, e.g., Hurley et al. 2005). Also, it led to speculation about a possible link with γ-ray bursts (GRBs) (see, e.g., Tanvir et al. 2005). Theorists investigated the connection between the magnetic field and the explosion (see, e.g. Blandford 2005). Others focused on modeling the fireball and the afterglow (see, e.g., Nakar et al. 2005; Dai et al. 2005; Wang et al. 2005). Astronomers performed a number of follow-up observations at various wavelengths (Rea et al. 2005; Israel et al. 2005; Palmer et al. 2005; Schwartz et al. 2005; Fender et al. 2006). In particular, the flux from the radio nebula produced by the explosion (Gaensler et al. 2005a; Cameron et al. 2005; Taylor et al. 2005) was measured very frequently in 2005 January. These observations focused on total intensity measurements at various radio wavelengths and on polarimetry at 8.5 GHz. Some polarimetry was done at lower frequencies, but without the proper correction for the leakages (Gaensler et al. 2005b). We have performed accurate polarimetry at 350, 850 and 1300 MHz. Also, we were able to measure the Stokes I flux from the radio nebula at 350 and 850 MHz more precisely by observing the same field again in 2005 April/May. In this way, we could properly subtract the background sources from the (u, v) data of the 2005 January observations. We compare our measurements with those at nearby frequencies.

5.3 Observation and data reduction

5.3.1 General A total of 19 observations were performed in January, April and May of 2005. Four of these, on January 16, 20, 23 and 29 were alternating between 350 and 850 MHz. On January 7 and 10 scans at 1300 MHz were also included. On January 4 we observed at 350, 650 and 1300 MHz, but the 650 MHz data was not used. A summary is shown in table 5.1. We used AIPS (Greisen 2003) and ParselTongue (Kettenis et al. 2006) scripts for the reduction of all 19 datasets. The Westerbork Synthesis Radio Telescope (WSRT) was used for all observations. The WSRT is a linear array with 14 equatorially mounted 25-m dishes equipped with linear feeds. Its maximum baseline is 2.7 km. All datasets recorded four polarization products with 8 IFs. 3C286 was observed before the target and 3C48 after. RFI was excised from the spectral line data using the AIPS task ’SPFLG’. Calibration was done in four steps. First we determined the variation in system temperature

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Table 5.1: Summary of these 19 WSRT observations Epoch Days since Time on Frequency burst source (min) (MHz) January 4 7.6 54 1300 January 4 7.6 91 350 January 5 8.6 462 850 January 7 10.5 107 1300 January 7 10.5 107 350 January 7 10.5 107 850 January 10 13.6 78 1300 January 10 13.6 71 350 January 10 13.6 71 850 January 16 19.6 181 350 January 16 19.6 181 850 January 20 23.6 181 350 January 20 23.6 165 850 January 23 26.6 198 350 January 23 26.6 198 850 January 29 32.6 196 350 January 29 32.6 196 850 April 30/May 1 124.3 444 350 May 2 125.2 464 850

as a function of time (and therefore also as a function of position on the sky), using the intermittent firing of a stable noise source. Next we performed a bandpass calibration using the AIPS task ’BPASS’ using either 3C48 or 3C286 or both. We applied the bandpass solution using the AIPS task ’SPLAT’. After that, we performed an external absolute gain calibration using an assumed flux for 3C48 by running the AIPS tasks ’SETJY’ and ’CALIB’. ’SETJY’ was set to use the absolute flux density calibration determined by Baars et al. (1977) and the latest (epoch 1999.2) polynomial coefficients for interpolating over frequency as determined at the VLA by NRAO staff. Finally, we self-calibrated the data for time variations in the relative complex gain phase and amplitude. Polarization calibration was performed by running the AIPS task ’LPCAL’ on 3C48 and ’CLCOR’ to correct for the instrumental XY phase offset. Generally, we followed the scheme for data reduction of WSRT data in AIPS as outlined by Robert Braun 1, although we ran some AIPS tasks differently depending on frequency. Those differences mainly involved the details of polarization calibration. For instance, the leakage terms (”D terms”) of the WSRT IFs are channel dependent, as pointed out by Brentjens (2008, paragraph 3.2). We took account of this, by first averaging groups of 5 channels through the AIPS task ’SPLAT’. Next,

1See http://www.astron.nl/radio-observatory/astronomers/analysis-wsrt-data/analysis-wsrt-dzb-data-classic- aips/analysis-wsrt-d

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we ran ’UVCOP’ to make separate datasets from the averaged channels. After that, we ran ’LPCAL’ and ’CLCOR’ on each of these separately before applying the feed and XY instrumental phase offset corrections by again running ’SPLAT’. Before imaging Stokes Q and Stokes U and before merging the datasets from 5 channel averaging back together through ’DBCON’, we applied a ParselTongue script for ”derotation”to the residual data, i.e., the (u, v) data where all sources except the target were removed, by running the AIPS task ’UVSUB’. The original Aips++ glish script was kindly given to us by G. Bernardi; we modified and translated it to a Python/ParselTongue script on a channel by channel basis. The ”derotation”of the visibilities is absolutely necessary, since the rotation measure (RM) of SGR 1806-20 is large, 272 rad/m2 (Gaensler et al. 2005a). This means that the polarized signal would vanish if all IFs were imaged simultaneously. For the 350 MHz data, one really needs the derotation of the visibilities to be performed on a channel per channel basis, because the imaging of even one single IF would result in a severly cor- rupted measurement and underestimate of the fractional linear polarization. The uncertainty in this RM (10 rad/m2, see Gaensler et al. 2005a) is too large for accurate polarization angle measurements, especially at frequencies below 1 GHz. For this reason we determined the RM more accurately, by fitting the sin 2 · RM · λ2 spectrum of either Stokes U or Stokes Q to its measured values at the 8 wavelengths λ corresponding to the IFs near 350 MHz and 850 MHz. The contribution to this RM from the ionosphere is naturally included in this fit, at least the part that did not vary during the observation run. We checked the output of the AIPS task ’TECOR’ for any significant variations in the ionospheric Faraday rotation during every observing run. The ionospheric Faraday rotation computed by ’TECOR’ is considered accurate since it does not use a model for the ionosphere but actual data from the CDDIS archive. We did not apply the ionospheric corrections from ’TECOR’ to our data because it implicitly assumes that one has recorded data from circular feeds. It should be clear from table 5.1 that the maximum observing time is 7.7 h due to the low declination of the source. Hence, the (u,v) coverage is sparse for all observations, since linear arrays like the WSRT ideally have 12h runs. The worst coverage was at three epochs when we alternated between three frequencies.

5.3.2 Detailed desciption of the datasets Observations at 350 MHz The 350 MHz observations were performed on January 4, 7, 10, 16, 20, 23 and 29 and April 30/May 1 of 2005. The last observation was made to make an accurate subtraction of background sources possible. This mainly concerns the subtraction of the Luminous Blue Variable discussed in Supplementary Table 1 of Gaensler et al. (2005a). The time resolution of all observations, except the ﬁrst and the last was 30s. On January 4 and April 30/May 1 the sampling times of the visibilities was 60s. The bandwidth per IF was 10 MHz, separated 8.75 MHz from each other and centered on frequencies of 315.00, 323.75, 332.50, 341.25, 350.00, 358.75, 367.50 and 376.25 MHz. The IFs were split into 64 channels, each 156.25 kHz wide, except for the April 30/May 1 observation. For that observation, the IFs were split into 128 channels, each 78.125 kHz wide. We used an automated ﬂagger for

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the initial editing of our data: WSRT flagger2. 3C286 was included in the external gain calibration, along with 3C48. This was trivial, since 3C286 is unpolarized at this frequency. The assumed fluxes for 3C48 and 3C286 in the lowest frequency IF were 43.889 and 26.106 Jy, respectively. The April 30/May 1 observation has the best (u,v) coverage. After performing 10 iterations of self calibration on this dataset the rms noise in the final image was 2.5mJy/beam. Its clean components were used to solve for the gain phases and amplitudes of the other datasets using a rather sophisticated scheme. First, a deconvolution of each of the 2005 January datasets was done in order to subtract the central region containing the radio nebula and the LBV, using the AIPS tasks ’IMAGR’, ’CCEDT’ and ’UVSUB’. The residual data were calibrated on the April 30/May 1 model which had the clean components from the central region removed. The gain phase and amplitude solutions were then copied and applied to the original 2005 January datasets. It this way we made sure that the Stokes I flux from SGR 1806-20 would not be reduced by calibrating on a model from an observation months after the flare. As explained in section 5.3.2, amplitude self calibration could also reduce the Stokes Q flux. However, due to the large RM of the source and because we use 45 of the available 64 channels, the Stokes Q flux almost completely vanishes in a single IF at 350 MHz. Thus this problem does not occur, at least not before ”derotation”. PSR 1937+21 was observed in between SGR 1806-20 and 3C48 for polarization calibration. This polarization calibration technique is decribed in detail by Brentjens (2008, paragraph 3.2). Since the RM of this pulsar is positive, Stokes Q should be 90◦ ahead of Stokes U with increasing λ2, as noted by Brown & Rudnick (2009, paragraph 2.3).

Observations at 850 MHz (”UHF high”)

We observed SGR 1806-20 on January 5, 7, 10, 16, 20, 23, 29 and May 5 of 2005. The last observation was performed to make an accurate subtraction of background sources possible. The time resolution of all observations, except for the first and the last, was 30s. The sampling time of the visibilities on January 5 and May 5 was 60s. The bandwidth of the eight IFs is 10 MHz, they were separated exactly 10 MHz from each other and ranging from 805 to 875 MHz. Each IF was split in 64 channels with a width of 156.25 kHz, except for the May 1/2 data that were split into 128 channels of 78.125 kHz. The external gain calibration was performed using an assumed flux for 3C48 of 24.240 Jy for the lowest frequency IF. The 850 MHz were reduced in almost the same way as the 1300 MHz data. Only polarization calibration was performed slightly differently. Since the Stokes Q (and U) of 3C286 are not known for the ”UHF high”frequencies, when the task ’CALIB’ was run on this calibrator, it was set to solve for gain phases only and not for gain amplitudes.

2http://www.astron.nl/ renting/

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Table 5.2: Stokes I flux measurements at 350 and 850 MHz; clean components from the 2005 April 30/May 1 and May 1/2 observations were subtracted 350 MHz 850 MHz Epoch Days Stokes I 1 σ Stokes I 1 σ (2005 since flux dens. error flux dens. error date) burst mJy/beam mJy/beam mJy/beam mJy/beam Jan. 4 7.6 186 20 Jan. 5 8.6 157 10 Jan. 7 10.5 84 10 97 28 Jan. 10 13.6 78 10 50 22 Jan. 16 19.6 16 10 35 14 Jan. 20 23.6 7 10 21 8 Jan. 23 26.6 13 10 17 7 Jan. 29 32.6 1 10 22 9

Observations at 1300 MHz

We observed SGR 1806-20 at 1300 MHz on January 4,7 and 10 of 2005. The total intensity measurements have already been published (see Gaensler et al. 2005a), so we focused on the polarized signal. However, we did check that our Stokes I fluxes agreed with those previously published. On 2005 January 4 visibilities were recorded every 60s, on January 7 and 10 every 30s. The eight 20-MHz IFs were centered on frequencies of 1255, 1272, 1289, 1306, 1323, 1340 and 1357 MHz. Each IF was split in 64 channels with a width of 312.5 kHz. The external gain calibration was performed using an assumed flux for 3C48 of 17.388 Jy for the lowest frequency IF. 3C286 was also included in the external gain amplitude and phase calibration using an assumed flux of 15.550 at 1255 MHz. 3C286 is linearly polarized. We took account of this and of the usual ”AIPS for linear feeds”projection (R→X,L→Y) by placing the assumed Stokes Q flux of 3C286 (0.594 Jy at the lowest frequency IF) with a minus sign at the position of Stokes V in the AIPS SU table. For the other IFs we kept the same ratio between Stokes I and Stokes Q. In this way we could use 3C286 not only for fixing the instrumental XY phase offset, but also for external gain calibration. Self-calibration was run to solve for the gain phases only, since solving for the amplitudes could reduce the Stokes Q flux. The AIPS task ’CALIB’ cannot be set to run simultaneously on a Stokes I and Stokes Q model. Obviously, when ’CALIB’ is run on a Stokes I model, it implicitly assumes that Q=0. Consequently, the same model is used to derive the X gains from the XX visibilities as the Y gains from the YY visibilities, while in fact XX=I-Q and YY=I+Q, so different models should be used. When solving for gain phases only, the error made is generally considered acceptable.

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119 119

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Figure 5.1: Comparison between the 240, 350, 610 and 1300 MHz fluxes of the radio nebula associated with SGR 1806-20. The 1300 MHz fluxes were published previously (Gaensler et al. 2005a). The fluxes at 350 MHz and 1300 MHz are approximately equal on 2005 January 7, they coincide in this plot. For the 350 MHz flux on 2005 January 29, instead of the flux, the 3σ upper limit on the flux is depicted in order to enable a more appropriate vertical scale.

5.4 Results

5.4.1 Total intensity measurements

The total intensity flux measurements at 350 MHz were done by fitting a Gaussian of the same shape and size as the restoring beam to the (fixed) location of SGR 1806-20 in the Stokes I images. This was done by the AIPS task ’IMFIT’. We used the position from Gaensler et al. (2005a) (α = 18h08m39.343s,δ= −20◦2439.8) for the fits. The results are summarized in table 5.2. The error bars are conservative estimates from measurements of the residuals of bright sources in the field. The actual rms noise in these images is much lower, around 3mJy/beam, which is about the same as the error from ’IMFIT’.

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The Giant Flare from SGR 1806-20 Chapter 5

æ ç é ë ì í ç î ï ç ñ ó í ó ô õ ñ ÷ ç ó ù é í

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Figure 5.2: Determining the rotation measure of SGR 1806-20 by fitting the sinusoidal Stokes U spectrum. Here, the fit was made to the values of Stokes U on 2005 January 04 at the wavelengths corresponding to the 8 IFs near 350 MHz after the visibilities were ”derotated”by an angle corresponding to an RM of −272 ± 10 rad/m2. We used gnuplot to fit the function A · sin(2 · RM · λ2 + θ) for three free parameters A, RM and θ. The correction to the RM from this fit is −18.76 ± 1.82 rad/m2. The reported reduced χ2 is 0.69.

Table 5.3: RM measurements of SGR1806-20. Epoch Days Frequency Measured 1 σ (2005 since (MHz) RM error date) burst (rad/m2) (rad/m2) Jan. 4 7.6 350 253.24 1.82 Jan. 5 8.6 850 253.14 12.43 Jan. 7 10.5 350 253.65 1.05 Jan. 10 13.6 350 261.74 2.04

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121 121 1 2 χ Reduced ) ◦ σ ) error ( ◦ Polarization 1 σ Polarization 1 beam) fraction (%) error (%) angle ( / σ 1 Polarimetric measurements of SGR 1806-20 2 U + beam) error (mJy / 2 Q Table 5.4: Jan. 4Jan. 4Jan. 5Jan. 7.6 7Jan. 7.6 7 8.6 10.5 350 10.5 1300 850 350 850 2.68 0.71 2.22 2.30 1.71 0.81 0.66 0.28 0.64 0.52 1.44 0.47 1.41 2.73 0.46 1.76 0.44 0.20 0.82 103 0.74 31 96 69 39 44 26 7 0.70 38 1.57 9 0.85 0.33 0.65 Jan. 7 10.5 1300 1.90 0.37 2.29 0.48 50 7 1.06 Epoch Days since Frequency (2005) burst (MHz) (mJy Jan. 10Jan. 10 13.6 13.6 350 1300 1.14 1.31 0.87 0.40 1.46 3.65 1.13 1.38 36 69 47 12 0.79 1.28 Poor ﬁt. 1

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The Giant Flare from SGR 1806-20 Chapter 5

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5.4.2 Polarimetry General

Polarimetry was performed on 2005 January 4, 5, 7 and 10. Although all of our observations recorded full Stokes, we anticipated that it would not be possible to detect the polarized signal from SGR 1806-20 on later dates, since the total intensity drops rapidly. Also, we did not expect polarization fractions to exceed the values given by Taylor et al. (2005, table 2).

Determining the RM of SGR 1806-20

As noted before, the rotation measure (RM) as measured by (Gaensler et al. 2005a, 272 ± 10 rad/m2) has a rather large error bar which translates into a polarization angle uncertainty at 1300 MHz of 30.5◦. At 850 MHz this is even 71.4◦. Naturally, the RM should be determined more accurately before polarization angles are to be measured. This can be done by plotting Stokes U or Q ﬂuxes of SGR 1806-20 as a function of frequency

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123 123

T U

T w U

T U

T U U

}

w U

} ~

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Figure 5.4: Comparison between the polarization angles at 350, 850, 1300 and 8400 MHz

and fit for the RM. We are in the advantageous position that these WSRT observations were performed with eight IFs. Over a wide span of frequencies there are many turns of Stokes U (or Q) since its spectrum is sinusoidal as a function of λ2.Thiseffect is largest at low frequencies: at 1300 MHz, there is less than one cycle of A · sin(2 · RM · λ2 + φ), at 850 MHz there are almost two cycles and at 350 MHz there are 23 cycles. It is evident that the most accurate measurement can be made at the lowest frequency, if there is sufficient signal to noise. Fortunately, we could detect polarized signal at 350 MHz from all three observations on 2005 January 4,7 and 10 after an initial ”derotation”of our visibilities using the RM from Gaensler et al. (2005a, 272 rad/m2). This initial derotation prevents diminution of the polarized signal in a single IF. At 850 MHz this initial derotation was not necessary. The noise levels at that frequency were such that detecting a polarized signal was only possible on 2005 January 5 and 7, but the latter observation yielded a very poor constraint on the RM, so we left it out. The 1300 MHz data also gave very poor constraints on the RM, thus in determining the weighted mean RM we ignored those, too. For the other observations, we plotted Stokes U per IF and solved for the RM (850 MHz) or the correction to the RM (350 MHz), as illustrated in figure 5.2. The results are shown in table 5.3. It turned out that

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the noise levels in all of the Stokes Q maps were much higher than in the Stokes U maps, so we did not use them. In determining the weighted mean RM we also took into account the measurement by Gaensler et al. (2005a, 272 ± 10 rad/m2) From this set of five measurements we derived an RM of 255.01 ± 0.83 rad/m2. It should be clear that, with regard to the 350 MHz RM measurements, the fits give the same reduced χ2 for both the positive and the negative correction to the initial ”derotation”. We removed those ambiguities by considering the Stokes U measurements near 850 MHz data on 2005 January 5. The fit to that data gave an RM of 253.14 ± 12.43 rad/m2 which made all of the positive RM solutions to the 350 MHz data very unlikely ( 3.0σ level for January 4 and 7). It is evident that the contribution of the ionosphere to the RM, RMion, is included in all fits. For the 2005 January 4, 5, 7 and 10 observations, RMion as reported by the AIPS task ’TECOR’, is the range 2.1 ± 0.4rad/m2. Consequently, the interstellar RM is given by 2 RMint = 255.01 − 2.1 = 252.91 ± 0.92 rad/m .

Polarization fractions and position angles We were able to measure the fractional linear polarization on all of the four epochs mentioned in paragraph 5.4.2. At 850 MHz, we were not able to measure polarization on 2005 January 10. For the other occasions, the measured polarized fluxes, P = Q2 + U2, fractions and their error bars are listed in table 5.4. The latter two quantities are depicted in figure 5.3. The overall conclusion is that there is no compelling evidence for any significant depolarization at any frequency. Only the polarization fraction at 1300 MHz on January 4 is low compared to the 8.4 GHz measurements, but this fraction was determined from our worst fit, i.e., the fit with the highest reduced χ2. The polarization angles and their uncertainties are also listed in table 5.4. The observations at 850 and 1300 MHz gave the most accurate position angles, with typical uncertainties of order 10◦. They are depicted in figure 5.4. Here, we see compelling evidence for significantly different polarization angles with respect to the 8.4 GHz observations from Taylor et al. (2005), particularly on January 5 at 850 MHz and on January 7 at both 850 and 1300 MHz.

5.5 Discussion

5.5.1 Total intensity measurements It is clear from ﬁgure 5.1 that SGR1806-20 is much dimmer at 350 MHz than what would be expected from the GMRT observations at 240 and 610 MHz (Cameron et al. 2005). In principle the Luminous Blue Variable, 14 to the east of SGR 1806-20 (see the Supplementary Information to Gaensler et al. 2005a) should be easily distinguishable from the Soft Gamma Repeater in the GMRT images, even at 240 MHz. The FWHM beamsize reported at that frequency is 12 × 18 (Chandra 2005b). This makes it hard to understand the discrepancy. In principle the discrepancy cannot originate from the inclusion or exclusion of extended emission. The GMRT data were corrected for this (Chandra 2005a,b). We excluded short

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spacings (< 1kλ) from our 350 MHz WSRT observations. This was actually a necessity since these were daytime observations and solar interference would otherwise compromise our calibration (see also Brentjens 2008, end of paragraph 3.2). Also, it is possible that the LBV radio nebula is variable and that it was much brighter on 2005 April 30/May 1 than on some occasions in 2005 January. We ran the AIPS task ’IM- FIT’ on the map from our 2005 April 30/May 1 observation and we found a peak flux density of 138 ± 1mJy/beam and an integrated flux of 189 ± 2mJy/beam at the location of the LBV. The NVSS (Condon et al. 1998) image of this field shows this source at the 15 mJy level. This would indicate that the LBV has a spectral index of about -1.8, which is almost the index for thermal radio radiation. It should be noted that, at the times of the latest observations in January 2005, when the radio nebula was relatively dim, there is no evidence for negative residuals in our maps that could be caused by the subtraction of the LBV. This indicates that, most likely, the LBV had the same brightness at the times of at least some of the 2005 January measurements as on 2005 April 30/May 1. Variability at radio wavelengths of the radio nebulas from LBVs has been known for quite some time (see, e.g., Abbott et al. 1981). For the P Cygni nebula variability at timescales of days was established at cm wavelengths (Skinner et al. 1996). These authors report a 50% increase in flux in less than two days on one occasion during three months of observations on every other day. It is unknown how these variations translate to lower frequencies. We therefore cannot completely exclude that the LBV was brighter at the time of the 2005 April 30/May 1 observation than on some occasions in January 2005. Also, the spectral index derived above does not agree with any of the spectral indices of the four LBVs observed by Duncan & White (2002) at 3 and 6 cm. Two of those spectral indices are close to that of a spherically symmetric radially expanding stellar wind (+0.6, see Panagia & Felli 1975; Wright & Barlow 1975). However, at these wavelengths, those systems may well be described as optically thin, which may not be the case at the frequencies we are considering. The WSRT 850 MHz Stokes I measurements are not inconsistent with the 840 MHz MOST data published earlier (Gaensler et al. 2005a), given the rather large noise levels in the data from both telescopes. The last MOST observation was taken 15 days after the Giant Flare (GF). Consequently, the 850 MHz WSRT observations after 2005 January 10 cannot be compared with other observations in this band. The last three of the January 2005 observations at 850 MHz were less contaminated by RFI than the first four, which resulted in smaller error bars on the fluxes. There is evidence (> 2σ level) for a deviation from a power-law decay from about 15 days after the GF, analogous to the 4.8 GHz observations by Gelfand et al. (2005, paragraph 2). These authors also mention a gradual rebrightening from about 25 days after the GF, as a result of swept up ambient material. We can also see that in the WSRT 850 MHz data, but the evidence for this is less compelling, since the sampling of these observations is sparse in time. Consequently, it is shown only in one of our observations, on 2005 January 29, 32.6 days after the GF.

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5.5.2 Polarimetric measurements In figure 5.3 we compare the polarization fractions as listed in table 5.4 with the measurements at 8.4 GHz by Taylor et al. (2005, Table 2). In figure 5.4 we have done the same for the polarization angles. It is clear that the observations at 8.4 GHz are much more accurate. Still, we do not see any significant discrepancies in the polarization fractions. Our observations reveal larger polarization position angles than the 8.4 GHz observations. Most compelling are the observations on 2005 January 5 at 850 MHz and on January 7 at both 850 and 1300 MHz. The error bar on the polarization angle at 350 MHz on January 4 is rather large, but this measurement and the 850 MHz measurement on January 5 show the largest differences with the 8.4 GHz observation, about 85◦. At these times, the polarization angles from the 8.4 GHz observations suggest that the magnetic field in the emitting plasma is aligned preferentially along the axis of the radio source, on average (Gaensler et al. 2005a). Thus, the January 4 and 5 polarization angles at 350 and 850 MHz indicate that the magnetic field in the emitting plasma that causes linearly polarized radiation at these low frequencies is close to perpendicular to the axis of the radio source, within 20◦. Possibly a different substructure in the radio nebula is being probed. It seems hard to explain this feature without a complex model of the radio source.

5.6 Conclusions

It is striking that depolarization at low frequencies is absent. Also, we have shown that low frequency polarimetry of SGR 1806-20 provides hints with respect to the detailed substructure of the radio nebula which cannot be derived from the extrapolation of high frequency measurements. Models for the radio nebula need to take into account a distinct source of linearly polarized low frequency radiation with magnetic fields in the emitting plasmas aligned quite differently from the fields that are associated with radiation at high frequencies.

5.7 Acknowledgements

We thank Michiel Brentjens, James Miller-Jones and Gianni Bernardi for helpful discussions about polarization calibration. We thank Eric Greisen for providing background information about many AIPS tasks. The Westerbork Synthesis Radio Telescope is operated by AS- TRON (Netherlands Foundation for Research in Astronomy) with support from the Nether- lands Foundation for Scientiﬁc Research (NWO). This research was supported by NWO Vici grant 639.043.302 (HS and RAMJW) and by NWO NOVA project 10.3.2.02 (BS).

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CHAPTER 6

Predicting low-frequency radio ﬂuxes of known extrasolar planets

J.–M. Grießmeier, P. Zarka & H. Spreeuw Astronomy and Astrophysics, 475, 359 (2007)

6.1 Abstract

Close-in giant extrasolar planets (“Hot Jupiters”) are believed to be strong emitters in the decametric radio range. We present the expected characteristics of the low-frequency magnetospheric radio emission of all currently known extrasolar planets, including the maximum emission frequency and the expected radio flux. We also discuss the escape of exoplanetary radio emission from the vicinity of its source, which imposes additional constraints on detectability. We compare the different predictions obtained with all four existing analytical models for all currently known exoplanets. We also take care to use realistic values for all input parameters. The four different models for planetary radio emission lead to very different results. The largest fluxes are found for the magnetic energy model, followed by the CME model and the kinetic energy model (for which our results are found to be much less optimistic than those of previous studies). The unipolar interaction model does not predict any observable emission for the present exoplanet census. We also give estimates for the planetary magnetic dipole moment of all currently known extrasolar planets, which will be useful for other studies. Our results show that observations of exoplanetary radio emission are feasible, but that the number of promising targets is not very high. The catalog of targets will be particularly useful for current and future radio observation campaigns (e.g. with the VLA, GMRT, UTR-2 and with LOFAR).

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6.2 Introduction

In the solar system, all strongly magnetised planets are known to be intense nonthermal radio emitters. For a certain class of extrasolar planets (the so-called Hot Jupiters), an analogous, but much more intense radio emission is expected. In the recent past, such exoplanetary radio emission has become an active field of research, with both theoretical studies and ongoing observation campaigns. Recent theoretical studies have shown that a large variety of effects have to be considered, e.g. kinetic, magnetic and unipolar interaction between the star (or the stellar wind) and the planet, the influence of the stellar age, the potential role of stellar CMEs, and the influence of different stellar wind models. So far, there is no single publication in which all of these aspects are put together and where the different interaction models are compared extensively. We also discuss the escape of exoplanetary radio emission from its planetary system, which depends on the local stellar wind parameters. As will be shown, this is an additional constraint for detectability, making the emission from several planets impossible to observe. The first observation attempts go back at least to Yantis et al. (1977). At the beginning, such observations were necessarily unguided ones, as exoplanets had not yet been discovered. Later observation campaigns concentrated on known exoplanetary systems. So far, no detection has been achieved. A list and a comparison of past observation attempts can be found elsewhere (Grießmeier et al. 2006a). Concerning ongoing and future observations, studies are performed or planned at the VLA (Lazio et al. 2004), GMRT (Majid et al. 2006; Winterhalter et al. 2006), UTR2 (Ryabov et al. 2004), and at LOFAR (Farrell et al. 2004). To support these observations and increase their efficiency, it is important to identify the most promising targets. The target selection for radio observations is based on theoretical estimates which aim at the prediction of the main characteristics of the exoplanetary radio emission. The two most important characteristics are the maximum frequency of the emission and the expected radio flux. The first predictive studies (e.g. Zarka et al. 1997; Farrell et al. 1999) concentrated on only a few exoplanets. A first catalog containing estimations for radio emission of a large number of exoplanets was presented by Lazio et al. (2004). This catalog included 118 planets (i.e. those known as of 2003, July 1) and considered radio emission energised by the kinetic energy of the stellar wind (i.e. the kinetic model, see below). Here, we present a much larger list of targets (i.e. 197 exoplanets found by radial velocity and/or transit searches as of 2007, January 13, taken from http://exoplanet.eu/), and compare the results obtained by all four currently existing interaction models, not all of which were known at the time of the previous overview study. As a byproduct of the radio flux calculation, we obtain estimates for the planetary magnetic dipole moment of all currently known extrasolar planets. These values will be useful for other studies as, e.g., star-planet interaction or atmospheric shielding. To demonstrate which stellar and planetary parameters are required for the estimation of exoplanetary radio emission, some theoretical results are briefly reviewed (section 6.3). Then, the sources for the different parameters (and their default values for the case where no measurements are available) are presented (section 6.4). In section 6.5, we present our estimations for exoplanetary radio emission. This section also includes estimates for planetary

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magnetic dipole moments. Section 6.6 closes with a few concluding remarks.

6.3 Exoplanetary radio emission theory

6.3.1 Expected radio flux In principle, there are four different types of interaction between a planetary obstacle and the ambient stellar wind, as both the stellar wind and the planet can either be magnetised or unmagnetised. Zarka (2007, Table 1) show that for three of these four possible situations intense nonthermal radio emission is possible. Only in the case of an unmagnetised stellar wind interacting with an unmagnetised body no intense radio emission is possible. In those cases where strong emission is possible, the expected radio flux depends on the source of available energy. In the last years, four different energy sources were suggested: a) In the first model, the input power Pinput into the magnetosphere is assumed to be proportional to the total kinetic energy flux of the solar wind protons impacting on the magnetopause (De- sch & Kaiser 1984; Zarka et al. 1997; Farrell et al. 1999; Zarka et al. 2001b; Farrell et al. 2004; Lazio et al. 2004; Stevens 2005; Grießmeier et al. 2005, 2006b, 2007) b) Similarly, the input power Pinput into the magnetosphere can be assumed to be proportional to the magnetic energy flux or electromagnetic Poynting flux of the interplanetary magnetic field (Zarka et al. 2001b; Farrell et al. 2004; Zarka 2004, 2006, 2007). From the data obtained in the solar system, it is not possible to distinguish which of these models is more appropriate (the constants of proportionality implied in the relations given below are not well known, see Zarka et al. 2001b), so that both models have to be considered. c) For unmagnetised or weakly magnetised planets, one may apply the unipolar interaction model. In this model, the star-planet system can be seen as a giant analog to the Jupiter-Io system (Zarka et al. 2001b; Zarka 2004, 2006, 2007). Technically, this model is very similar to the magnetic energy model, but the source location is very different: Whereas in the kinetic and in the magnetic model, the emission is generated near the planet, in the unipolar interaction case a large-scale current system is generated and the radio emission is generated in the stellar wind between the star and the planet. Thus, the emission can originate from a location close to the stellar surface, close to the planetary surface, or at any point between the two. This is possible in those cases where the solar wind speed is lower than the Alfvén velocity (i.e. for close-in planets, see e.g. Preusse et al. 2005). Previous studied have indicated that this emission is unlikely to be detectable, except for stars with an extremely strong magnetic field (Zarka et al. 2001b, 2004; Zarka 2006, 2007). Nevertheless, we will check whether this type of emission is possible for the known exoplanets. d) The fourth possible energy source is based on the fact that close-in exoplanets are expected to be subject to frequent and violent stellar eruptions (Khodachenko et al. 2007b) similar to solar coronal mass ejections (CMEs). As a variant to the kinetic energy model, the CME model assumes that the energy for the most intense planetary radio emission is provided by CMEs. During periods of such CME-driven radio activity, considerably higher radio flux levels can be achieved than during quiet stellar conditions (Grießmeier et al. 2006b, 2007). For this reason, this model is treated separately. For the kinetic energy case, the input power was first derived by Desch & Kaiser (1984),

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who found that it is given by

∝ 3 2. Pinput,kin nveﬀRs (6.1)

In eq. (6.1), n is the stellar wind density at the planetary orbit, veff is the velocity of the stellar wind in the reference frame of the planet (i.e. including the aberration due to the orbital velocity of the planet, which is not negligible for close-in planets), and Rs denotes the magnetospheric standoff distance. The magnetic energy case was first discussed by Zarka et al. (2001b). Here, the input power is given by

∝ 2 2 Pinput,mag veﬀB⊥Rs (6.2)

In eq. (6.2), veff is the velocity of the stellar wind in the reference frame of the planet, B⊥ if the component of the interplanetary magnetic field (IMF) perpendicular to the stellar wind flow in the reference frame of the planet, and Rs denotes the magnetospheric standoff distance. For the unipolar interaction case (Zarka et al. 2001b), the input power is given by

∝ 2 2 Pinput,unipolar veﬀB⊥Rion (6.3) Eq. (6.3) is identical to eq. (6.2), except that the obstacle is not the planetary magnetosphere, but its ionosphere, so that Rs is replaced by Rion, the radius of the planetary ionosphere. CME-driven radio emission was ﬁrst calculated by Grießmeier et al. (2006b). In that case, the input power is given by

∝ 3 2. Pinput,kin,CME nCMEveﬀ,CMERs (6.4) Eq. (6.4) is identical to eq. (6.1), except that the stellar wind density and velocity are replaced by the corresponding values encountered by the planet during a CME. A certain fraction of the input power Pinput given by eq. (6.1), (6.2), (6.3) or (6.4) is thought to be dissipated within the magnetosphere:

Pd = Pinput (6.5)

Observational evidence suggests that the amount of power emitted by radio waves Prad is roughly proportional to the power input Pinput (see, e.g. Zarka 2007, Figure 6). This can be written as:

Pradio = ηradio Pd = ηradio Pinput (6.6)

As Pd cannot be measured directly, one correlates the observed values of Pradio with the calculated (model dependent) values of Pinput. Thus, one replaces Pinput by Pradio on the left- hand side of the proportionalities given by (6.1), (6.2), (6.3) and (6.4). The proportionality constant is determined by comparison with Jupiter. The analysis of the jovian radio emission allows to deﬁne three values for the typical radio spectrum: (a) the power during average conditions, (b) the average power during periods of high activity, and (c) the peak power

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(Zarka et al. 2004). In this work, we will use the average power during periods of high 11 activity as a reference value for all four cases, with Pradio,J = 2.1 · 10 W. The radio ﬂux Φ seen by an observer at a distance s from the emitter is related to the emitted radio power Pradio by (Grießmeier et al. 2007):

4π2m R3P Φ= Pradio = e p radio . 2 2 (6.7) Ωs Δ f eμ0Ωs M Here, Ω is the solid angle of the beam of the emitted radiation (Ω=1.6 sr, see Zarka et al. Δ Δ = max 2004), and f is the bandwidth of the emission. We use f fc (Grießmeier et al. 2007), max where fc is the maximum cyclotron frequency. Depending on the model, Pradio is given by max eq. (6.1), (6.2), (6.3) or (6.4). The maximum cyclotron frequency fc is determined by the max maximum magnetic ﬁeld strength Bp close to the polar cloud tops (Farrell et al. 1999):

eBmax μ M M max = p = e 0 ≈ . fc 3 24 MHz 3 (6.8) 2πme 4π2m R e p Rp

Here, me and e are the electron mass and charge, Rp is the planetary radius, μ0 is the magnetic permeability of the vacuum, and M is the planetary magnetic dipole moment. M and Rp denote the planetary magnetic moment and its radius relative to the respective value for Jupiter, 27 2 e.g. M = M/MJ, with MJ = 1.56 · 10 Am (Cain et al. 1995) and RJ = 71492 km. The radio ﬂux expected for the four diﬀerent models according to eqs. (6.1) to (6.7) and the maximum emission frequency according to (6.8) are calculated in section 6.5 for all known exoplanets.

6.3.2 Escape of radio emission To allow an observation of exoplanetary radio emission, it is not suﬃcient to have a high enough emission power at the source and emission in an observable frequency range. As an additional requirement, it has to be checked that the emission can propagate from the source to the observer. This is not the case if the emission is absorbed or trapped (e.g. in the stellar wind in the vicinity of the radio-source), which happens whenever the plasma frequency 1 ne2 fplasma = (6.9) 2π 0me

is higher than the emission frequency at any point between the source and the observer. Thus, the condition of observability is

max < max, fplasma fc (6.10)

max max where fc is taken at the radio source (e.g. the planet), whereas fplasma is evaluated along the line of sight. As the density of the electrons in the stellar wind n decreases with the distance to the star, this condition is more restrictive at the orbital distance of the planet than further

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out. Thus, it is sufficient to check whether condition (6.10) is satisfied at the location of the radio-source (i.e. for n = n(d), where d is the distance from the star to the radio-source). In that case, the emission can escape from the planetary system and reach distant observers. In section 6.5, condition (6.10) is checked for all known exoplanets at their orbital distance. Note however that, depending on the line of sight, not all observers will be able to see the planetary emission at all times. For example, the observation of a secondary transit implies that the line of sight passes very close to the planetary host star, where the plasma density is much higher. For this reason, some parts of the orbit may be unobservable even for planets for which eq. (6.10) is satisfied.

6.3.3 Radiation emission in the unipolar interaction model An additional constraint arises because certain conditions are necessary for the generation of radio emission. Planetary radio emission is caused by the cyclotron maser instability (CMI). This mechanism is only eﬃcient in regions where the ratio between the electron plasma frequency and the electron cyclotron frequency is small enough. This condition can be written as fplasma 0.4, (6.11) fc

where the electron cyclotron frequency fc is deﬁned by the local magnetic ﬁeld eB fc = (6.12) 2πme

and fplasma is given by eq. (6.9). Observations seem to favor a critical frequency ratio close to the 0.1, while theoretical work supports a critical frequency ratio close to 0.4 (Le Quéau et al. 1985; Hilgers 1992; Zarka et al. 2001a). Fundamental O mode or second harmonic O and X mode emission are possible also for larger frequency ratios, but are much less efficient (Treumann 2000; Zarka 2007). To avoid ruling out potential emission, we use the largest possible frequency ratio, i.e. fplasma ≤ 0.4. fc The condition imposed by eq. (6.11) has to be fulfilled for any of the four models presented in section 6.3.1. For the three models where the radio emission is generated directly in the planetary magnetosphere, n decreases much faster with distance to the planetary surface than B, so that eq. (6.11) can always be fulfilled. For the unipolar interaction model, the emission takes place in the stellar wind, and the electron density n can be obtained from the model of the stellar wind. In this case, it is not a priori clear where radio emission is possible. It could be generated anywhere between the star and the planet. In section 6.5, we will check separately for each planetary system whether unipolar interaction satisfying eq. (6.11) is possible at any location between the stellar surface and the planetary orbit.

6.4 Required parameters

In the previous section, it has been shown that the detectability of planetary radio emission depends on a few planetary parameters:

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• the planetary radius Rp • the planetary magnetic moment M

• the size of the planetary magnetosphere Rs

• the size of the planetary ionosphere Rion

• the stellar wind density n and its velocity veﬀ

• the stellar magnetic ﬁeld (IMF) B⊥ perpendicular to the stellar wind ﬂow in the frame of the planet • the distance of the stellar system (to an earth-based observer) s • the solid angle of the beam of the emitted radiation Ω The models used to infer the missing stellar and planetary quantities require the knowledge of a few additional planetary parameters. These are the following:

• the planetary mass Mp

• the planetary radius Rp • its orbital distance d • the planetary rotation rate ω

• the stellar magnetic ﬁeld (IMF) components Br, Bφ

• the stellar mass M

• the stellar radius R

• the stellar age t In this section, we brieﬂy describe how these each of these quantities can be obtained.

6.4.1 Basic planetary parameters As a ﬁrst step, basic planetary characteristics have to be evaluated:

• d,ωorbit and s are directly taken from the Extrasolar Planets Encyclopaedia at http://exoplanet.eu, as well as the observed mass Mobs and the orbital eccentricity e. Note that in most cases the observed mass is the “projected mass” of the planet, i.e. Mobs = Mp sin i,wherei is the angle of inclination of the planetary orbit with respect to the observer. • For eccentric planets, we calculate the periastron from the semi-major axis d and the orbital eccentricity e: dmin = d/(1 − e). Thus, the results for planetary radio emission apply to the periastron.

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• For most exoplanets, the planetary mass is not precisely known. Instead, usually only the projected mass Mp sin i is accessible to measurements, where i is the inclination of the planetary orbit with respect to the observer. This projected mass is taken from http://exoplanet.eu√ and converted to the median value of the mass: = · 1 = / · ≈ . median(Mp) Mobs median( sin i ) 4 3 Mobs 1 15 Mobs.

• For transiting planets, Rp is taken from original publications. For non-transiting planets, the planetary radius is Rp not known. In this case, we estimate the planetary radius based on its mass Mp and orbital distance d, as explained in appendix 6.8.1. The radius of a “cold” planet of mass Mp is given by

/ α 1 3 1/3 Mp Mp Rp(d = ∞) = ≈ 1.47RJ (6.13) 2/3 2/3 1 + Mp + Mp Mmax 1 Mmax

with α = 6.1 · 10−4 m3 kg−1 (for a planet with the same composition as Jupiter) and Mmax = 3.16 MJ.Again,Mp and Mmax denote values relative to the respective value for 27 Jupiter (using MJ = 1.9 · 10 kg). The radius of an irradiated planet is then given by

γ R (d) R(d) T p = p = · + . eq = ∞ 1 0 05 (6.14) Rp(d ) Rp(d = ∞) T0

where Teq is the equilibrium temperature of the planetary surface. The coeﬃcients T0 and γ depend on the planetary mass (see appendix 6.8.1).

6.4.2 Stellar wind model

The stellar wind density n and velocity veff encountered by a planet are key parameters defin- ing the size of the magnetosphere and thus the energy flux available to create planetary radio emission. As these stellar wind parameters strongly depend on the stellar age, the expected radio flux is a function of the estimated age of the exoplanetary host star (Stevens 2005; Grießmeier et al. 2005). At the same time it is known that at close distances the stellar wind velocity has not yet reached the value it has at larger orbital distances. For this reason, a distance-dependent stellar wind models has to be used to avoid overestimating the expected planetary radio flux (Grießmeier 2006; Grießmeier et al. 2007). It was shown (Grießmeier et al. 2007) that for stellar ages > 0.7 Gyr, the radial dependence of the stellar wind properties can be described by the stellar wind model of Parker (1958), and that the more complex model of Weber & Davis (1967) is not required. In the Parker model, the interplay between stellar gravitation and pressure gradients leads to a su- personic gas flow for sufficiently large substellar distances d. The free parameters are the coronal temperature and the stellar mass loss. They are indirectly chosen by setting the stellar wind conditions at 1 AU. More details on the model can be found elsewhere (e.g. Mann et al. 1999; Preusse 2006; Grießmeier 2006).

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The dependence of the stellar wind density n and velocity and veff on the age of the stellar system is based on observations of astrospheric absorption features of stars with different ages. In the region between the astropause and the astrospheric bow shock (analogs to the heliopause and the heliospheric bow shock of the solar system), the partially ionized local interstellar medium (LISM) is heated and compressed. Through charge exchange processes, a population of neutral hydrogen atoms with high temperature is created. The characteristic Lyα absorption (at 1216 Å) of this population was detectable with the high-resolution observations obtained by the Hubble Space Telescope (HST). The amount of absorption depends on the size of the astrosphere, which is a function of the stellar wind characteristics. Comparing the measured absorption to that calculated by hydrodynamic codes, these measurements allowed the first empirical estimation of the evolution of the stellar mass loss rate as a function of stellar age (Wood et al. 2002; Wood 2004; Wood et al. 2005). It should be noted, however, that the resulting estimates are only valid for stellar ages ≥ 0.7 Gyr (Wood et al. 2005). From these observations, (Wood et al. 2005) calculate the age-dependent density of the stellar wind under the assumption of an age-independent stellar wind velocity. This leads to strongly overestimated stellar wind densities, especially for young stars (Grießmeier et al. 2005; Holzwarth & Jardine 2007). For this reason, we combine these results with the model for the age-dependence of the stellar wind velocity of Newkirk (1980). One obtains (Grießmeier et al. 2007): −0.43 , = + t . v(1AU t) v0 1 τ (6.15)

The particle density can be determined to be −1.86±0.6 , = + t . n(1AU t) n0 1 τ (6.16)

11 −3 7 with v0 = 3971 km/s, n0 = 1.04 · 10 m and τ = 2.56 · 10 yr. For planets at small orbital distances, the keplerian velocity of the planet moving around its star becomes comparable to the radial stellar wind velocity. Thus, the interaction of the stellar wind with the planetary magnetosphere should be calculated using the effective velocity of the stellar wind plasma relative to the planet, which takes into account this “aberration effect” (Zarka et al. 2001b). For the small orbital distances relevant for Hot Jupiters, the planetary orbits are circular because of tidal dissipation (Goldreich & Soter 1966; Dobbs-Dixon et al. 2004; Halbwachs et al. 2005). For circular orbits, the orbital velocity vorbit is perpendicular to the stellar wind velocity v, and its value is given by Kepler’s law. In the reference frame of the planet, the stellar wind velocity then is given by ff = 2 + 2. ve vorbit v (6.17)

Finally, for the magnetic energy case, the interplanetary magnetic field (Br, Bφ)isre- quired. At 1 AU, the average field strength of the interplanetary magnetic field is Bimf ≈ 3.5 nT (Mariani & Neubauer 1990; Prölss 2004). According to the Parker stellar wind model

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(Parker 1958), the radial component of the interplanetary magnetic ﬁeld decreases as

− d 2 Bimf,r(d) = Br,0 . (6.18) d0

This was later confirmed by Helios measurements. One finds Br,0 ≈ 2.6nTandd0 = 1AU (Mariani & Neubauer 1990; Prölss 2004). At the same time, the azimuthal component Bimf,ϕ behaves as − d 1 Bimf,ϕ(d) = Bϕ,0 , (6.19) d0

with Bϕ,0 ≈ 2.4 nT (Mariani & Neubauer 1990; Prölss 2004). The average value of Bimf,θ vanishes (Bimf,θ ≈ 0). From Bimf,r(d)andBimf,ϕ(d), the stellar magnetic field (IMF) B⊥(d) perpendicular to the stellar wind flow in the frame of the planet can be calculated (Zarka 2007): ⊥ = 2 + 2 | α − β | B Bimf,r Bimf,ϕ sin ( ) (6.20)

with Bimf,ϕ α = arctan (6.21) Bimf,r and v β = arctan orbit . (6.22) v

We obtain the stellar magnetic ﬁeld B relative to the solar magnetic ﬁeld B under the assumption that it is inversely proportional to the rotation period P (Collier Cameron & Jianke 1994; Grießmeier et al. 2007):

B P = , (6.23) B P

− where we use P = 25.5dandtakeB = 1.435 · 10 4 T as the reference magnetic ﬁeld strength at the solar surface (Preusse et al. 2005). The stellar rotation period P is calculated from the stellar age t (Newkirk 1980): 0.7 ∝ + t , P 1 τ (6.24)

where the time constant τ is given by τ = 2.56 · 107 yr (calculated from Newkirk 1980). Note that first measurements of stellar magnetic fields for planet-hosting stars are just becoming available from the spectropolarimeter ESPaDOnS (Catala et al. 2007). This will lead to an improved understanding of stellar magnetic fields, making more accurate models possible in the future.

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6.4.3 Stellar CME model

For the CME-driven radio emission, the stellar wind parameters n and veff are effectively replaced by the corresponding CME parameters nCME and veff,CME, potentially leading to much more intense radio emission than those driven by the kinetic energy of the steady stellar wind (Grießmeier 2006; Grießmeier et al. 2006b, 2007). These CME parameters are estimated by Khodachenko et al. (2007b), who combine in- situ measurements near the sun (e.g. by Helios) with remote solar observation by SoHO. Two interpolated limiting cases are given, denoted as weak and strong CMEs, respectively. These two classes have a different dependence of the average density on the distance to the star d. w s In the following, these quantities will be labeled nCME(d)andnCME(d), respectively. w For weak CMEs, the density nCME(d) behaves as w = w / −2.3 nCME(d) nCME,0 (d d0) (6.25)

= w = w = = . · 6 −3 where the density at d0 1AUisgivenbynCME,0 nCME(d d0) 4 9 10 m . For strong CMEs, Khodachenko et al. (2007b) ﬁnd

s = s / −3.0 nCME(d) nCME,0 (d d0) (6.26)

s = s = = . · 6 −3 = with nCME,0 nCME(d d0) 7 1 10 m ,andd0 1AU. As far as the CME velocity is concerned, one has to note that individual CMEs have very diﬀerent velocities. However, the average CME velocity v is approximately independent of the subsolar distance, and is similar for both types of CMEs:

w = s = ≈ / . vCME vCME vCME 500 km s (6.27) Similarly to the steady stellar wind the CME velocity given by eq. (6.27) has to be corrected for the orbital motion of the planet: MG v ﬀ = + v2 . (6.28) e ,CME d CME In addition to the density and the velocity, the temperature of the plasma in a coronal mass ejection is required for the calculation of the size of the magnetosphere. According to Khodachenko et al. (2007b,a), the front region of a CME consists of hot, coronal material (T ≈ 2 MK). This region may either be followed by relatively cool prominence material (T ≈ 8000 K), or by hot ﬂare material (T ≈ 10 MK). In the following, the temperature of the leading region of the CME will be used, i.e. TCME = 2MK.

6.4.4 Planetary magnetic moment and magnetosphere For each planet, the value of the planetary magnetic moment M is estimated by taking the geometrical mean of the maximum and minimum result obtained by diﬀerent scaling laws. The associated uncertainty was discussed by Grießmeier et al. (2007). The diﬀerent scaling

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laws are compared, e.g., by Farrell et al. (1999)1, Grießmeier et al. (2004) and Grießmeier (2006). In order to be able to apply these scaling laws, some assumptions on the planetary size and structure are required. The variables required in the scaling laws are rc (the radius of the dynamo region within the planet), ρc (the density within this region), σ (the conductivity within this region) and ω (the planetary rotation rate).

The size of the planetary core rc and its density ρc The density proﬁle within the planet ρ(r) is obtained by describing the planet as a polytropic gas sphere, using the solution of the Lane-Emden equation (Chandrasekhar 1957; S´anchez-Lavega 2004): ⎛ ⎞ sin π r ⎜ πM ⎟ Rp ρ = ⎝⎜ p ⎠⎟ . (r) 3 (6.29) 4Rp π r Rp

The size of the planetary core rc is found by searching for the value of r where the density ρ(r) becomes large enough for the transition to the liquid-metallic phase (S´anchez-Lavega 2004; Grießmeier et al. 2005; Grießmeier 2006). The transition was assumed to occur at a density of 700 kg/m3, which is consistent with the range of parameters given by S´anchez-Lavega (2004). For Jupiter, we obtain rc = 0.85 RJ. The average density in the dynamo region ρc is then obtained by averaging the density −3 ρ(r) over the range 0 ≤ r ≤ rc. For Jupiter, we obtain rc ≈ 1800 kg m .

The planetary rotation rate ω Depending on the orbital distance of the planet and the timescale for synchronous rotation τsync (which is derived in appendix 6.8.2), three cases can be distinguished: 1. For planets at small enough distances for which the timescale for tidal locking is small (i.e. τsync ≤ 100 Myr), the rotation period is taken to be synchronised with the orbital period (ω = ωf ≈ ωorbit), which is known from measurements. This case will be denoted by “TL”. Typically, this results in smaller rotation rates for tidally locked planets than for freely rotating planets.

2. Planets with distances resulting in 100 Myr ≤ τsync ≤ 10 Gyr may or may not be subject to tidal locking. This will, for example, depend on the exact age of the planetary system, which is typically in the order of a few Gyr. For this reason, we calculate the expected characteristics of these “potentially locked” planets twice: once with tidal locking (denoted by “(TL)”) and once without tidal locking (denoted by “(FR)”). 3. For planets far away from the central star, the timescale for tidal locking is very large. For planets with τsync ≥ 10 Gyr, the eﬀect of tidal interaction can be neglected. In this case, the planetary rotation rate can be assumed to be equal to the initial rotation rate ωi, which is assumed to be equal to the current rotation rate of Jupiter, i.e. ω = ωJ with −4 −1 ωJ = 1.77 · 10 s . This case will be denoted by “FR”.

1containing a typo in the sixth equation of the appendix.

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Note that tidal interaction does not perfectly synchronise the planetary rotation to its orbit. Thermal atmospheric tides resulting from stellar heating can drive planets away from synchronous rotation (Showman & Guillot 2002; Correia et al. 2003; Laskar & Correia 2004). According to Showman & Guillot (2002), the corresponding error for ω could be as large as a factor of two. On the basis of the example of τ Bootis b, Grießmeier et al. (2007) show that the eﬀect of imperfect tidal locking (in combination with the spread of the results found by diﬀerent scaling laws) can lead to magnetic moments and thus emission frequencies up to a factor 2.5 higher than for the nominal case. Keeping this “error bar” in mind, we will nevertheless consider only the reference case in this work.

The conductivity in the planetary core σc Finally, the conductivity in the dynamo region of extrasolar planets remains to be evaluated. According to Nellis (2000), the electrical conductivity remains constant throughout the metallic region. For this reason, it is not necessary to average over the volume of the conducting region. As the magnetic moment scaling is applied relative to Jupiter, only the relative value of the conductivity, i.e. σ/σJ is required. In this work, the conductivity is assumed to be the same for extrasolar gas giants as for Jupiter, i.e. σ/σJ = 1.

The size of the magnetosphere The size of the planetary magnetosphere Rs is calculated with the parameters determined above for the stellar wind and the planetary magnetic moment. For a given planetary orbital distance d, of the different pressure contributions only the magnetospheric magnetic pressure is a function of the distance to the planet. Thus, the standoff distance Rs is found from the pressure equilibrium (Grießmeier et al. 2005): ⎡ ⎤ / ⎢ μ f 2M2 ⎥1 6 = ⎣⎢ 0 0 !⎦⎥ . Rs(d) 2 2 (6.30) 8π mn(d)veff(d) + 2 n(d)kBT

Here, m is the proton’s mass, and f0 is the form factor of the magnetosphere. It describes the magnetic field created by the magnetopause currents. For a realistic magnetopause shape, a factor f0 = 1.16 is used (Voigt 1995). In term of units normalized to Jupiter’s units, this is equivalent to ⎡ ⎤ 1/6 ⎢ 2 ⎥ ⎢ M ⎥ Rs(d) ≈ 40RJ ⎢ ⎥ , (6.31) ⎣ 2 2ñ(d)kBT ⎦ n˜(d)vff(d) + 2 e mveff,J

5 −3 withn ˜(d) = n(d)/nJ, veff(d) = veff(d)/veff,J, nJ = 2.0 · 10 m ,andveff,J = 520 km/s. Note that in a few cases, especially for planets with very weak magnetic moments and/or subject to dense and fast stellar winds of young stars, eq. (6.30) yields standoff distances Rs < Rp,whereRp is the planetary radius. Because the magnetosphere cannot be compressed to sizes smaller than the planetary radius, we set Rs = Rp in those cases.

6.4.5 Additional parameters The other required parameters are obtained from the following sources:

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• The stellar ages t are taken from Saffe et al. (2005, who gave ages for 112 exoplanet host stars). Saffe et al. (2005) compare stellar age estimations based on five different methods (and for one of them they use two different calibrations), some of which give more reliable results than others. Because not all of these methods are applicable to all stars, we use the age estimations in the following order or preference: chromospheric age for ages below 5.6 Gyr (using the D93 calibration), isochrone age, chromospheric age for ages above 5.6 Gyr (using the D93 calibration), metallicity age. Note that error estimates for the two most reliable methods, namely isochrone and chromospheric ages, are already relatively large (30%-50%). In those cases where the age is not known, a default value of 5.2 Gyr is used (the median chromospheric age found by Saffe et al. 2005). This relatively high average age is due to a selection effect (planet detection by radial velocity method is easier to achieve for older, more slowly rotating stars). As the uncertainty of the radio flux estimation becomes very large for low stellar ages (Grießmeier et al. 2005), we use a minimum stellar age of 0.5 Gyr.

• For the solid angle of the beam, we assume the emission to be analogous to the domi- nating contributions of Jupiter’s radio emission and use Ω=1.6 sr (Zarka et al. 2004).

6.5 Expected radio ﬂux for know exoplanets

6.5.1 The list of known exoplanets Table 1 2 shows what radio emission we expect from the presently known exoplanets (13.1.2007). It contains the maximum emission frequency according to eq. (6.8), the plasma frequency in the stellar wind at the planetary location according to eq. (6.9) and the expected radio flux according to the magnetic model (6.1), the kinetic model (6.2), and the kinetic CME model (6.4). The unipolar interaction model is discussed in the text below. Table 1 also contains values for the expected planetary mass Mp, its radius Rp and its planetary magnetic dipole moment M. For each planet, we note whether tidal locking should be expected. Note that http://exoplanet.eu contains a few more planets than table 1, because for some planets, essential data required for the radio flux estimation are not available (typically s, the distance to the observer). The numbers given in table 1 are not accurate results, but should be regarded as refined estimations intended to guide observations. Still, the errors and uncertainties involved in these estimations can be considerable. As was shown in Grießmeier et al. (2007), the uncertainty on the radio flux at Earth, Φ, is dominated by the uncertainty in the stellar age t (for which the error is estimated as ≈ 50% by Saffe et al. 2005). For the maximum emission frequency, max M fc , the error is determined by the uncertainty in the planetary magnetic moment ,which is uncertain by a factor of two. For the planet τ Bootes b, these effects translate into an uncertainty of almost one order of magnitude for the flux (the error is smaller for planets around stars of solar age), and an uncertainty of a factor of 2-3 for the maximum emission

2Table 1 is only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/475/359.

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frequency. This error estimate is derived and discussed in more detail by Grießmeier et al. (2007). The results given in table 1 cover the following range:

• The maximum emission frequency found to lie beween 0 to almost 200 MHz. However, max > all planets with fc 70 MHz have negligible ﬂux. Also note that any emission with fc ≤ 5 to 10 MHz will not be detectable on Earth because it cannot propagate through the Earth’s ionosphere (“ionospheric cutoﬀ”). For this reason, the most appropriate frequency window for radio observations seems to be between 10 and 70 MHz.

• The radio ﬂux according to the magnetic energy model, Φsw,mag lies between 0 and 5 Jy (for GJ 436 b). For 15 candidates, Φsw,mag is larger than 100 mJy, and for 37 candidates it is above 10 mJy.

• The ﬂux prediction according to the kinetic energy model, Φsw,kin is much lower than the ﬂux according to the magnetic energy model. Only in one case it exceeds the value of 10 mJy.

• The increased stellar wind density and velocity during a CME leads to a strong increase of the radio flux when compared to the kinetic energy model. Correspondingly, ΦCME,kin exceeds 100 mJy in 3 cases and 10 mJy in 11 cases. • Table 1 does not contain flux estimations for the unipolar interaction model. The reason is that the condition given by eq. (6.11) is not satisfied in any of the studied cases. This is consistent with the result of Zarka (2006, 2007), who found that stars 100 times as strongly magnetised as the sun are required for this type of emission. The approach taken in section 6.4.2 for the estimation stellar magnetic fields does not yield such strong magnetic fields for stars with ages > 0.5 Gyr. Stronger magnetic fields are possible (e.g. for younger stars). Strongly magnetised stars (even those without known planets) could be defined as specific targets to test this model.

• The plasma frequency in the stellar wind, fp,sw, is negligibly small in most cases. For a few planets, however, it is of the same order of magnitude as the maximum emission frequency. In these cases, the condition given by eq. (6.10) makes the escape of the radio emission from its source towards the observer impossible. Of the 197 planets of the current census, eq. (6.10) is violated in 8 cases. If one takes into account the uncertainty of the stellar age (30-50%, see Saffe et al. 2005), an uncertainty of similar size is introduced for the plasma frequency: In the example of τ Bootis, the age uncertainty translates into a variation of up to 50% for the plasma frequency. With such error bars, between 6 and 14 planets are affected by eq. (6.10). None of the best targets are affected.

• The expected planetary magnetic dipole moments lie between 0 and 5.5 times the magnetic moment of Jupiter. However, the highest values are found only for very massive planets: Planets with masses M ≤ 2MJ have magnetic moments M≤2MJ. For planets with masses of M ≤ MJ, the models predict magnetic moments M≤1.1MJ.

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The results of table 1 confirm that the different models for planetary radio emission lead to very different results. The largest fluxes are found for the magnetic energy model, followed by the CME model and the kinetic energy model. This is consistent with previous expectations (Zarka et al. 2001b; Zarka 2006; Grießmeier et al. 2006b; Zarka 2007; Grießmeier et al. 2007). The unipolar interaction model does not lead to observable emission for the presently known exoplanets. Furthermore, the impact of tidal locking is clearly visible in the results. As it is currently not clear which of these models best describes the auroral radio emission, it is not sufficient to restrict oneself to one scaling law (e.g. the one yielding the largest radio flux). For this reason, all possible models have to be considered. Once exoplanetary radio emission is detected, observations will be used to constrain and improve the models. Table 1 also shows that planets subject to tidal locking have a smaller magnetic moment and thus a lower maximum emission frequency than freely rotating planets. The reduced bandwidth of the emission can lead to an increase of the radio flux, but frequently emission is limited to frequencies not observable on earth (i.e. below the ionospheric cutoff). The results of table 1 are visualized in figure 6.1 (for the magnetic energy model), figure 6.2 (for the CME model) and figure 6.3 (for the kinetic energy model). The predicted planetary radio emission is denoted by open triangles (two for each “potentially locked” planet, otherwise one per planet). The typical uncertainties (approx. one order of magnitude for the flux, and a factor of 2-3 for the maximum emission frequency) are indicated by the arrows in the upper right corner. The sensitivity limit of previous observation attempts are shown as filled symbols and as solid lines (a more detailed comparison of these observations can be found in Zarka 2004; Grießmeier et al. 2006a; Grießmeier 2006). The expected sensitivity of new and future detectors (for 1 hour integration and 4 MHz bandwidth, or any equivalent combination) is shown for comparison. Dashed line: upgraded UTR-2, dash-dotted lines: low band and high band of LOFAR, left dotted line: LWA, right dotted line: SKA. The instruments’ sensitivities are defined by the radio sky background. For a given instrument, a planet is observable if it is located either above the instrument’s symbol or above and to its right. Again, large differences in expected flux densities are apparent between the different models. On average, the magnetic energy model yields the largest flux densities, and the kinetic energy model yields the lowest values. Depending on the model, between one and three planets are likely to be observable using the upgraded system of UTR-2. Somewhat higher numbers are found for LOFAR. Considering the uncertainties mentioned above, these numbers should not be taken literally, but should be seen as an indicator that while observation seem feasible, the number of suitable candidates is rather low. It can be seen that the maximum emission frequency of many planets lies below the ionospheric cutoff frequency, making earth-based observation of these planets impossible. A moon-based radio telescope however would give access to radio emission with frequencies of a few MHz (Zarka 2007). As can be seen in figures 6.1, 6.2 and 6.3, this frequency range includes a significant number of potential target planets with relatively high flux densities. Figures 6.1, 6.2 and 6.3 also show that the relatively high frequencies of the LOFAR high band and of the SKA telescope are probably not very well suited for the search for exoplanetary radio emission. These instruments could, however, be used to search for radio emission generated by unipolar interaction between planets and strongly magnetised stars.

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101 ] 1 −

Hz 10−1 2 −

Wm 10−3 26 − 10 −5 = 10 [Jy

Φ 10−7

1MHz 10 MHz 100 MHz 1GHz f

Figure 6.1: Maximum emission frequency and expected radio flux for known extrasolar planets ac- cordingtothemagnetic energy model, compared to the limits of past and planned observation attempts. Open triangles: Predictions for planets. Solid lines and filled circles: Previous observation attempts at the UTR-2 (solid lines), at Clark Lake (filled triangle), at the VLA (filled circles), and at the GMRT (filled rectangle). For comparison, the expected sensitivity of new detectors is shown: upgraded UTR-2 (dashed line), LOFAR (dash-dotted lines, one for the low band and one for the high band antenna), LWA (left dotted line) and SKA (right dotted line). Frequencies below 10 MHz are not observable from the ground (ionospheric cutoff). Typical uncertainties are indicated by the arrows in the upper right corner.

6.5.2 A few selected cases

According to our analysis, the best candidates are:

• HD 41004 B b, which is the best case in the magnetic energy model with emission above 1 MHz. Note that the mass of this object is higher than the upper limit for planets (≈ 13MJ), so that it probably is a brown dwarf and not a planet.

• Epsilon Eridani b, which is the best case in the kinetic energy model.

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101 ] 1 −

Hz 10−1 2 −

Wm 10−3 26 − 10 −5 = 10 [Jy

Φ 10−7

1MHz 10 MHz 100 MHz 1GHz f

Figure 6.2: Maximum emission frequency and expected radio flux for known extrasolar planets according to the CME model, compared to the limits of past and planned observation attempts. Open triangles: Predictions for planets. All other lines and symbols are as defined in figure 6.1.

• Tau Boo b, which is the best case in the magnetic energy model with emission above the ionospheric cutoﬀ (10 MHz). • HD 189733 b, which is the best case in both the magnetic energy model and in the CME model which has emission above 1 MHz. • Gliese 876 c, which is the best case in the CME model with emission above the ionospheric cutoﬀ (10 MHz). • HD 73256 b, which has emission above 100 mJy in the magnetic energy model and which is the second best planet in the kinetic energy model. • GJ 3021 b, which is the third best planet in the kinetic energy model. To this list, one should add the planets around Ups And (b, c and d) and of HD 179949 b, whose parent stars exhibit an increase of the chromospheric emission of about 1-2% (Shkol-

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101 ] 1 −

Hz 10−1 2 −

Wm 10−3 26 − 10 −5 = 10 [Jy

Φ 10−7

1MHz 10 MHz 100 MHz 1GHz f

Figure 6.3: Maximum emission frequency and expected radio flux for known extrasolar planets according to the kinetic energy model, compared to the limits of past and planned observation attempts. Open triangles: Predictions for planets. All other lines and symbols are as defined in figure 6.1.

nik et al. 2003, 2004, 2005). The observations indicate one maximum per planetary orbit, a “Hot Spot” in the stellar chromosphere which is in phase with the planetary orbit. The lead angles observed by Shkolnik et al. (2003) and Shkolnik et al. (2005) were recently explained with an Alfvén-wing model using realistic stellar wind parameters obtained from the stellar wind model by Weber and Davis (Preusse 2006; Preusse et al. 2006). This indicates that a magnetised planet is not required to describe the present data. The presence of a planetary magnetic field could, however, be proven by the existence of planetary radio emission. Al- though our model does not predict high radio fluxes from these planets (see table 1), the high chromospheric flux shows that a strong interaction is taking place. As a possible solution of this problem, an intense stellar magnetic field was suggested (Zarka 2006, 2007). In that case, table 1 underestimates the radio emission of Ups And b, c, d and HD 179949 b, making these planets interesting candidates for radio observations (eg. through the magnetic energy model or the unipolar interaction model). For this reason, it would be especially interesting

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to obtain measurements of the stellar magnetic ﬁeld for these two planet-hosting stars (e.g. by the method of Catala et al. 2007). Considering the uncertainties mentioned above, it is important not to limit observations attempts to these best cases. The estimated radio characteristics should only be used as a guide (e.g. for the target selection, or for statistical analysis), but individual results should not be regarded as precise values.

6.5.3 Statistical discussion It may seem surprising that so few good candidates are found among the 197 examined exoplanets. However, when one checks the list of criteria for “good” candidates (e.g. Grießmeier et al. 2006a), it is easily seen that only a few good targets can be expected: (a) the planet should be close to the Earth (otherwise the received flux is too weak). About 70% of the known exoplanets are located within 50 pc, so that this is not a strong restriction. (b) A strongly magnetised system is required (especially for frequencies above the ionospheric cutoff). For this reason, the planet should be massive (as seen above, we find magnetic moments M≥2MJ only for planets with masses M ≥ 2MJ). About 60% of the known exoplanets are at least as massive as Jupiter (but only 40% have Mp ≥ 2.0MJ). (c) The planet should be located close to its host star to allow for strong interaction (dense stellar wind, strong stellar magnetic field). Only 25% of the known exoplanets are located within 0.1 AU of their host star. By multiplying these probabilities, one finds that close (s ≤ 50 pc), heavy (Mp ≥ 2.0MJ), close-in (d ≤ 0.1 AU) planets would represent 8% (≈ 15 of 197 planets) of the current total if the probabilities for the three conditions were independent. However, this is not the case. In the current census of exoplanets, a correlation between planetary mass and orbital distance is clearly evident, with a lack of close-in massive planets (see e.g. Udry et al. 2003). This is not a selection effect, as massive close-in planets should be easier to detect than low-mass planets. This correlation was explained by the stronger tidal interaction effects for massive planets, leading to a faster decrease of the planetary orbital radius until the planet reaches the stellar Roche limit and is effectively destroyed (Pätzold & Rauer 2002; Jiang et al. 2003). Because of this mass-orbit correlation, the fraction of good candidates is somewhat lower (approx. 2%, namely 3 of 197 planets: HD 41004 B b, Tau Boo b, and HD 162020 b).

6.5.4 Comparison to previous results A first comparative study of expected exoplanetary radio emission from a large number of planets was performed by Lazio et al. (2004), who compared expected radio fluxes of 118 planets (i.e. those known as of 2003, July 1). Their results differ considerably from those given in table 1: • max As far as the maximum emission frequency fc is concerned, our results are considerably lower than the frequencies given by Lazio et al. (2004). For Tau Bootes, their maximum emission frequency is six times larger than our result. For planets heav- ier than Tau Bootes, the discrepancy is even larger, reaching more than one order of

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magnitude for the very heavy cases (e.g. HD 168433c, for which they predict radio emission with frequencies up to 2670 MHz). These differences have several reasons: Firstly, Farrell et al. (1999) and Lazio et al. (2004) assume that Rp = RJ. Also, these works rely on the magnetic moment scaling law of Blackett (1947), which has a large exponent in rc. This scaling law should not be used, as it was experimentally disproven ∝ 1/3 (Blackett 1952). Thirdly, these works make uses rc Mp . Especially for planets with large masses like τ Bootes, this yields unrealistically large core radii (even rc > Rp in some cases), magnetic moments, and emission frequencies. Note that a good estimation of the emission frequency is particularly important, because a difference of a factor of a few can make the difference between radiosignals above and below the Earth’s ionospheric cutoff frequency.

• The anticipated radio flux obtained with the kinetic energy model Φsw,kin is much lower than the estimates of Lazio et al. (2004). Typically, the difference is approximately two orders of magnitude, but this varies strongly from case to case. For example, for Tau Boo b, our result is smaller by a factor of 30 (where the difference is partially compensated by the low stellar age which increases our estimation), for Ups And b, the results differ by a factor of 220, and for Gliese 876 c, the difference is as large as a factor 6300 (which is partially due to the fact we take into account the small stellar radius and the high stellar age).

• As was mentioned above, the analysis of the jovian radio emission allows to deﬁne three terms for the typical radio spectrum: (a) the power during average conditions,(b) the average power during periods of high activity, and (c) the peak power (Zarka et al. 2004). When comparing the results of our table 1 to those of Lazio et al. (2004), one has to note that their table I gives the peak power, while the results in our table 1 were obtained using the average power during periods of high activity. Similarly to Farrell et al. (1999), Lazio et al. (2004) assume that the peak power caused by variations of the stellar wind velocity is two orders of magnitude higher than the average power. How- ever, the values Farrell et al. (1999) use for average conditions correspond to periods of high activity, which are less than one order of magnitude below the peak power (Zarka et al. 2004). During periods of peak emission, the value given in our table 1 would be increased by the same amount (approximately a factor of 5, see Zarka et al. 2004). For this reason, the peak radio ﬂux is considerably overestimated in these studies.

• Estimated radio ﬂuxes according to the magnetic energy model and the CME model have not yet been published for large numbers of planets. This is the ﬁrst time the results from these models are compared for a large number of planets.

6.6 Conclusions

Predictions concerning the radio emission from all presently known extrasolar planets were presented. The main parameters related to such an emission were analyzed, namely the plan-

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etary magnetic dipole moments, the maximum frequency of the radio emission, the radio flux densities, and the possible escape of the radiation towards a remote observer. We compared the results obtained with various theoretical models. Our results confirm that the four different models for planetary radio emission lead to very different results. As expected, the largest fluxes are found for the magnetic energy model, followed by the CME model and the kinetic energy model. The results obtained by the latter model are found to be less optimistic than by previous studies. The unipolar interaction model does not lead to observable emission for any of the currently known planets. As it is currently not clear which of these models best describes the auroral radio emission, it is not sufficient to restrict oneself to one scaling law (e.g. the one yielding the largest radio flux). Once exoplanetary radio emission is detected, observations will be used to constrain and improve the model. These results will be particularly useful for the target selection of current and future radio observation campaigns (e.g. with the VLA, GMRT, UTR-2 and with LOFAR). We have shown that observation seem feasible, but that the number of suitable candidates is relatively low. The best candidates appear to be HD 41004 B b, Epsilon Eridani b, Tau Boo b, HD 189733 b, Gliese 876 c, HD 73256 b, and GJ 3021 b. The observation of some of these candidates is in progress.

6.7 Acknowledgements

We thank J. Schneider for providing data via “The extrasolar planet encyclopedia” (http://exoplanet.eu/), I. Baraffe and C. Vocks for helpful discussions concerning planetary radii. We would also like to thank the anonymous referee for his helpful comments. This study was jointly performed within the ANR project “La détection directe des exoplanètes en ondes radio” and within the LOFAR transients key project (TKP). J.-M. G. was supported by the french national research agency (ANR) within the project with the contract number NT05-1 42530 and partially by Europlanet (N3 activity). P.Z. acknowledges support from the International Space Science Institute (ISSI) within the ISSI team “Search for Radio Emissions from Extra-Solar Planets”.

6.8 Appendix

6.8.1 An empirical mass-radius relation

For the selection of targets for the search for radio emission from extrasolar planets, an esti- Φ max mation of the expected radio ﬂux and of the maximum emission frequency fc is required. For the calculation of these values, both the planetary mass and the planetary radius are required (see, e.g. Farrell et al. 1999; Grießmeier et al. 2007; Zarka 2007). However, only for a few planets (i.e. the 16 presently known transiting planets) both mass and radius are known. In the absence of observational data, it is in principle possible to obtain planetary radii from numerical simulation, e.g. similar to those of Bodenheimer et al. (2003) or Baraﬀeetal.

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(2003, 2005), requiring one numerical run per planet. We chose instead to derive a simpliﬁed analytical ﬁt to such numerical results.

The accuracy of the ﬁt

The description presented here is necessarily only preliminary, as (a) numerical models are steadily further developed and improved, and as (b) more transit observations (e.g. by the COROT satellite, which was launched recently) will provide a much better database in the future. This will considerably improve our understanding of the dependence of the planetary radius on various parameters as, e.g. planetary mass, orbital distance, or stellar metallicity (as suggested by Guillot et al. 2006).

What accuracy can we accept? Within the frame of the models presented in section 6.3, an increase in Rp by 40% increases the expected radio flux by a factor of 2, and the estimated maximum emission frequency decreases by 40%. More generally, for a fixed planetary mass, Φ 7/3 max −1 is roughly proportional to Rp and fc is approximately proportional to Rp . Thus, it appears that the assumption of a single standard radius for all planets leads to a relatively large error. Comparing this to the other uncertainties involved in the estimation of radio characteristics (these are discussed in Grießmeier et al. 2007), it seems sufficient to estimate Rp with 20% accuracy.

What accuracy can we expect? Several eﬀects limit the precision in planetary radius we can hope to achieve:

• The definition of the radius: The “transit radius” measured for transiting planets is not exactly identical to the standard 1 bar radius. The differences are of the order of about 5-10%, but depend on the mass of the planet (Burrows et al. 2003, 2004). Because we compare modelled radii and observed radii without correcting for this effect, this limits the maximum precision we can potentially obtain.

• The abundance of heavy elements: The transiting planet around HD 149026 is substantially enriched in heavy elements (Sato et al. 2005). Models (Bodenheimer et al. 2003) yield a smaller radius for planets with a heavy core than for coreless planets of the same mass (more than 10% diﬀerence for small planets). For a planet with unknown radius, a strong enrichment in heavy elements cannot be ruled out, as this case cannot be distinguished observationally from a pure hydrogen giant (i.e. one without heavy elements).

For these reasons, we conclude that an analytical description which agrees with the (numerical) data within ∼20% seems sufficient. For such a description, the error introduced by the fit will not be the dominant one. To get a better result, it is not sufficient to improve the approximation for the radius estimation, but the more fundamental problems mentioned above have to be addressed.

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An analytical mass-radius relation The inﬂuence of the planetary mass A simple mass-radius relation, valid within a vast mass range, has been proposed by Lynden-Bell & O’Dwyer (2001) and Lynden-Bell & Tout (2001): 1/3 1 Mp R = (6.32) p 1/3 2/3 4 πρ 1 + Mp 3 0 Mmax

The density ρ0 is depends on the planetary atomic composition. Eq. (6.32) has a maximum in Rp when Mp = Mmax (corresponding to the planet of maximum radius). Fitting Jupiter, Saturn and the planet of maximum radius (Rmax = 1.16RJ, see Hubbard 1984), we obtain −3 Mmax = 3.16MJ and ρ0 = 394 kg m for a Jupiter-like mixture of hydrogen and helium −3 (75% and 25% by mass, respectively). For a pure hydrogen planet, ρ0 = 345 kg m . With there parameters, eq. (6.32) ﬁts well the results for cold planets of Bodenheimer et al. (2003). In this work, this estimation is used for the radii of non-irradiated (cold) planets.

The influence of the planetary age It is known that the radius of a planet with a given mass depends on its age. According to models for the radii of isolated planets (Baraffe et al. 2003), the assumption of a time-independent planetary radius leads to an error ≤11% for planets with ages above 0.5 Gyr and with masses above 0.5MJ. In view of the uncertainties discussed above, this error seems acceptable for a first approximation, and we use

Rp(Mp, t) ≈ Rp(Mp). (6.33)

The influence of the planetary orbital distance It is commonly expected that planets sub- jected to strong stellar radiation have a larger planetary radius than isolated, but otherwise identical planets. This situation is typical for “Hot Jupiters”, where strong stellar irradiation is supposed to delay the planetary contraction (Burrows et al. 2000, 2003, 2004). Clearly, this effect depends on the planetary distance to its star, d, and on the stellar luminosity L.In the following, we denote the radius increase by r, which we define as

Rp(Mp, d) r = . (6.34) Rp(Mp, d = ∞)

Herein, Rp(Mp, t, d) denotes the radius of a planet under the irradiation by its host star, and Rp(Mp, t, d = ∞) is the radius of a non-irraditated, but otherwise identical planet. Different exoplanets have vastly different host stars. A difference of a factor of two in stellar mass can result in a difference of more than order of magnitude in stellar luminosity. For this reason, it is not sufficient to take the orbital distance as the only parameter determining the radius increase by irradiation. Here, we select the equilibrium temperature of the planetary surface as the basic parameter. It is defined as (Bodenheimer et al. 2003) − 1/4 = (1 A)L , Teq 2 2 2 (6.35) 16πσSBd (1 + e /2)

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where the planetary albedo is set as A = 0.4 in the following. The stellar luminosity L is calculated from the stellar mass according to the analytical fit given by Tout et al. (1996) for the zero-age main sequence. In this, we assumed solar metallicity for all stars. Finally, σSB denotes the Stefan-Boltzmann constant. Using the data of Bodenheimer et al. (2003), we use the following fit for the radius increase r: γ Teq r = 1 + 0.05 . (6.36) T0 This form was selected because it has the correct qualitative behaviour: It yields a monotonous decrease of r with decreasing equilibrium temperature Teq (increasing orbital distance d). In the limit Teq → 0(d →∞)wefindr → 1. As an alternative to a continuous fit like eq. (6.36), Lazio et al. (2004) use a step-function, i.e. assume an increased radius of r = 1.25 for planets closer than a certain distance d1 = 0.1 AU only. However, this treatment does not reduce the number of fit-constants. Also, for planets with orbital distances close to d1, the results obtained with a step-function strongly depend on the somehow arbitrary choice of d1. A continuous transition from “irradiated” to “isolated” planets is less prone to this effect. The numerical results of Bodenheimer et al. (2003) show that the ratio r also depends on the mass of the planet: for small planets, r = 1.4 for an equilibrium temperature of 2000 K, whereas for large planets, r ≤ 1.10. For this reason, we allow to coefficients T0 and γ to vary with Mp: ct,2 Mp T0 = ct,1 · (6.37) MJ and cγ,2 cγ,1 γ = 1.15 + 0.05 · . (6.38) Mp

With the set of coefficient ct,1 = 764 K , ct,2 = 0.28, cγ,1 = 0.59MJ and cγ,2 = 1.03, we obtain an analytical fit to the numerical results of Bodenheimer et al. (2003). The maximum deviation from the numerical results is below 10% (cf. Fig. 6.4). For comparison, Fig. 6.4 also shows the value of r for the transiting exoplanets OGLE- TR-10b, OGLE-TR-56b, OGLE-TR-111b, OGLE-TR-113b, OGLE-TR-132b, XO-1b, HD 189733b, HD 209458b, TrES-1b and TrES-2b as small crosses. Here, r is calculated as the ratio of the observed value of Rp(Mp, d) and the value for Rp(Mp, d = ∞) calculated according to eq. (6.32). As the mass of all transiting planets lies between 0.11MJ ≤ Mp ≤ 3.0MJ, one should expect to find all crosses between the two limiting curves. For most planets, this is indeed the case: only for HD 209458b, r is considerably outside the area delimited by the two curves. Different explanations have been put forward for the anomalously large radius of this planet, but so far no conclusive answer to this question has been found (see e.g. Guillot et al. 2006, and references therein). As numerical models cannot reproduce the observed

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radius of this planet, one cannot expect our approach (which is based on a ﬁt to numerical results) to reproduce it either. For the other planets, Fig. 6.4 shows that our approach is a valid approximation.

Figure 6.4: Radius increase r as a function of the equilibrium temperature Teq according to eq. (6.36), (6.37) and (6.38). Upper line: radius increase r due to irradiation for planets of mass Mp = 0.11MJ. Lower line: Upper line: radius increase r due to irradiation for planets of mass Mp = 3.0MJ.Open symbols: results of the numerical calculation of Bodenheimer et al. (2003) for planetary masses Mp = 0.11MJ (diamonds) and Mp = 3.0MJ (triangles). Crosses: radius increase for observed transiting planets (see text).

6.8.2 Tidal locking? For the estimation of the planetary magnetic dipole moment in section 6.4.4, we require the planetary rotation rate. This rotation rate rotation greatly depends on whether the planet can be considered as tidally locked, as freely rotating, or as potentially locked. In this appendix, we discuss how we evaluate the tidal locking timescale τsync, which decides to which of these categories a planet belongs.

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The value of the planetary rotation ω depends on its distance from the central star. For close-in planets, the planetary rotation rate is reduced by tidal dissipation. In this case, tidal interaction gradually slows down the planetary rotation from its initial value ωi until it reaches the ﬁnal value ωf after tidal locking is completed. It should be noted that for planets in eccentric orbits, tidal interaction does not lead to the synchronisation of the planetary rotation with the orbital period. Instead, the rotation period also depends on the orbital eccentricity. At the same time, the timescale to reach this equilibrium rotation rate is reduced (Laskar & Correia 2004). Similarly, for planets in an oblique orbit, the equilibrium rotation period is modiﬁed (Levrard et al. 2007). However, the locking of a hot Jupiter in a non-synchronous spin-orbit resonance appears to be unlikely for distances ≤ 0.1 AU (Levrard et al. 2007). For this reason, we will calculate the timescale for tidal locking under the assumptions of circular orbits and zero obliquity. In the following, the tidal locking timescale for reaching ωf is calculated under the following simplifying assumptions: prograde orbit, spin parallel to orbit (i.e. zero obliquity), and zero eccentricity (Murray & Dermott 1999, Chapter 4). The rate of change of the planetary rotation velocity ω for a planet with a mass of Mp and radius of Rp around a star of mass M is given by (Goldreich & Soter 1966; Murray & Dermott 1999): ⎛ ⎞ ⎜ ⎟ 2 6 dω 9 1 ⎜GMp ⎟ M Rp = ⎝⎜ ⎠⎟ , (6.39) α 3 dt 4 Qp Rp Mp d

where the constant α depends on the internal mass distribution within the planet. It is deﬁned α = / 2 by I (MpRp), where I is the planetary moment of inertia. For a sphere of homogeneous α / α ≤ / density, is equal to 2 5. For planets, generally 2 5. Qp is the modiﬁed Q-value of the planet. It can be expressed as (Murray & Dermott 1999) 3Q = p , Qp (6.40) 2k2,p

where k2,p is the Love number of the planet. Qp is the planetary tidal dissipation factor (the larger it is, the smaller is the tidal dissipation), deﬁned by MacDonald (1964) and Goldreich & Soter (1966). The time scale for tidal locking is obtained by a comparison of the planetary angular velocity and its rate of change: ω − ω τ = i f . (6.41) sync ω˙

A planet with angular velocity ωi at t = 0 (i.e. after formation) will gradually lose angular momentum, until the angular velocity reaches ωf at t = τsync. Insertion of eq. (6.39) into eq. (6.41) yields the following expression for τsync: ⎛ ⎞ ⎜ R3 ⎟ M 2 6 τ ≈ 4α ⎜ p ⎟ ω − ω p d sync Qp ⎝ ⎠ ( i f) (6.42) 9 GMp M Rp

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6 The importance of this eﬀect strongly depends on the distance (τsync ∝ d ). Thus, a planet in a close-in orbit (d 0.1 AU) around its central star is subject to strong tidal interaction, leading to gravitational locking on a very short timescale. α ω ω In the following, we brieﬂy describe the parameters ( , Qp, i and f) required to calculate the timescale for tidal locking of Hot Jupiters.

Structure parameter α For large gaseous planets, the equation of state can be approximated κ = α α = / 2 by a polytrope of index 1. In that case, the structure parameter (deﬁned by I MpRp) is given by α = 0.26 (Gu et al. 2003).

Tidal dissipation factor Qp For planets with masses of the order of one Jupiter mass, one finds that k2,p has a value of k2,p ≈ 0.5 for Jupiter (Murray & Dermott 1999; Laskar & Correia 2004) and k2,p ≈ 0.3 for Saturn (Peale 1999; Laskar & Correia 2004). The value of k2,p = 0.5 ≈ will be used in this work. With eq. (6.40), this results in Qp 3Qp. 4 6 For Jupiter, one finds the following range of allowed values: 6.6 · 10 Qp 2.0 · 10 (Peale 1999). Several estimations of the turbulent dissipation within Jupiter yield Qp-values larger than this upper limit, while other theories predict values consistent with this upper limit (Marcy et al. 1997; Peale 1999, and references therein). This demonstrates that the origin of the value of Qp is not well understood even for Jupiter (Marcy et al. 1997). Extrasolar giant planets are subject to strongly different conditions, and it is difficult to constrain Qp. Typically, Hot Jupiters are assumed to behave similarly to Jupiter, and values . · 5 ≤ ≤ . · 6 in the range 1 0 10 Qp 1 0 10 are used. In the following, we will distinguish three different regimes: close planets (which are tidally locked), distant planets (which are freely rotating), and planets at intermediate distances (which are potentially tidally locked). The borders between the “tidally locked” and the “potentially locked” regime is calculated by τ = = 6 setting sync 100 Myr and Qp 10 . The border between the “potentially locked” and the τ = = 5 “freely rotating” regime is calculated by setting sync 10 Gyr and Qp 10 . Thus, the area of “potentially locked” planets is increased.

Initial rotation rate ωi The initial rotation rate ωi is not well constrained by planetary formation theories. The relation between the planetary angular momentum density and planetary mass observed in the solar system (MacDonald 1964) suggests a primordial rotation period of the order of 10 hours (Hubbard 1984, Chapter 4). We assume the initial rotation rate to be −4 −1 equal to the current rotation rate of Jupiter (i.e. ω = ωi = ωJ) with ωJ = 1.77 · 10 s .

Final rotation rate ωf As far as eq. (6.42) is concerned, ωf can be neglected (Grießmeier 2006).

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ed. R. O. Pepin, J. A. Eddy, & R. B. Merrill, 293–320 Parker, E. N. 1958, Astrophys. J., 128, 664 Pätzold, M. & Rauer, H. 2002, Astrophys. J., 568, L117 Peale, S. J. 1999, Ann. Rev. Astron. Astrophys., 37, 533 Preusse, S. 2006, PhD thesis, Technische Universität Braunschweig, ISBN 3-936586-48-9, Copernicus-GmbH Katlenburg-Lindau Preusse, S., Kopp, A., Büchner, J., & Motschmann, U. 2005, Astron. Astrophys., 434, 1191 Preusse, S., Kopp, A., Büchner, J., & Motschmann, U. 2006, Astron. Astrophys., 460, 317 Prölss, G. W. 2004, Physics of the Earth’s Space Environment (Berlin: Springer-Verlag) Ryabov, V. B., Zarka, P., & Ryabov, B. P. 2004, Planet. Space Sci., 52, 1479 Saffe, C., Gómez, M., & Chavero, C. 2005, Astron. Astrophys., 443, 609 Sánchez-Lavega, A. 2004, Astrophys. J., 609, L87 Sato, B., Fischer, D. A., Henry, G. W., et al. 2005, Astrophys. J., 633, 465 Shkolnik, E., Walker, G. A. H., & Bohlender, D. A. 2003, Astrophys. J., 597, 1092 Shkolnik, E., Walker, G. A. H., & Bohlender, D. A. 2004, Astrophys. J., 609, 1197 Shkolnik, E., Walker, G. A. H., Bohlender, D. A., Gu, P.-G., & Kürster, M. 2005, Astrophys. J., 622, 1075 Showman, A. P. & Guillot, T. 2002, Astron. Astrophys., 385, 166 Stevens, I. R. 2005, Mon. Not. R. Astron. Soc., 356, 1053 Tout, C. A., Pols, O. R., Eggleton, P. P., & Han, Z. 1996, Mon. Not. R. Astron. Soc., 281, 257 Treumann, R. A. 2000, in Geophysical monograph series, Vol. 119, Radio Astronomy at Long Wavelengths, ed. R. Stone, K. W. Weiler, M. L. Goldstein, & J.-L. Bougeret, 13–26 Udry, S., Mayor, M., & Santos, N. C. 2003, Astron. Astrophys., 407, 369 Voigt, G.-H. 1995, in Handbook of atmospheric electrodynamics, ed. H. Volland, Vol. II (CRC Press), 333–388 Weber, E. J. & Davis, Jr., L. 1967, Astrophys. J., 148, 217 Winterhalter, D., Kuiper, T., Majid, W., et al. 2006, in Planetary Radio Emissions VI, ed. H. O. Rucker, W. S. Kurth, & G. Mann (Austrian Academy of Sciences Press, Vienna), 595–602 Wood, B. E. 2004, Living Rev. Solar Phys., 1, 2, URL: http://www.livingreviews.org/lrsp- 2004-2, accessed on 23 February 2007 Wood, B. E., Müller, H.-R., Zank, G. P., & Linsky, J. L. 2002, Astrophys. J., 574, 412 Wood, B. E., Müller, H.-R., Zank, G. P., Linsky, J. L., & Redfield, S. 2005, Astrophys. J., 628, L143 Yantis, W. F., Sullivan, III., W. T., & Erickson, W. C. 1977, Bull. Am. Astron. Soc., 9, 453 Zarka, P. 2004, in ASP Conference Series, Vol. 321, Extrasolar planets: Today and Tomorrow, ed. J.-P. Beaulieu, A. Lecavelier des Etangs, & C. Terquem, 160–169 Zarka, P. 2006, in Planetary Radio Emissions VI, ed. H. O. Rucker, W. S. Kurth, & G. Mann (Austrian Academy of Sciences Press, Vienna), 543–569 Zarka, P. 2007, Planet. Space Sci., 55, 598 Zarka, P., Cecconi, B., & Kurth, W. S. 2004, J. Geophys. Res., 109, A09S15 Zarka, P., Queinnec, J., & Crary, F. J. 2001a, Planet. Space Sci., 49, 1137 Zarka, P., Queinnec, J., Ryabov, B. P., et al. 1997, in Planetary Radio Emissions IV, ed.

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H. O. Rucker, S. J. Bauer, & A. Lecacheux (Austrian Academy of Sciences Press, Vienna), 101–127 Zarka, P., Treumann, R. A., Ryabov, B. P., & Ryabov, V. B. 2001b, Astrophys. Space Sci., 277, 293

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CHAPTER 7

Samenvatting in het Nederlands

7.1 LOFAR

Op dit moment (2010) wordt de nieuwe radiotelescoop LOFAR (LOw Frequency ARray) afgebouwd. Het ontwerp van LOFAR is drastisch anders dan van vorige radiotelescopen, zoals bijvoorbeeld de Very Large Array (VLA) in de Verenigde Staten en de Westerbork Synthese Radio Telescoop (WSRT) in Nederland. Die telescopen bestonden allemaal uit schotelantennes, zoals in figuur 7.1. De vorm van de LOFAR antennes verschilt wezenlijk van de ”klassieke”schotelvorm. In respectievelijk figuren 7.2 en 7.3 staan de lage (30-80 MHz) en hoge (120-240 MHz) frequentie LOFAR antennes afgebeeld. Ze zijn dus gevoelig voor het frequentiegebied rond de FM band, dat u gebruikt als u naar de (auto) radio luistert. Eén van de redenen dat deze simpele en goedkope antennes voldoen is dat ze alleen bij frequenties onder de 240 MHz gebruikt worden. Hogere frequentie radiogolven vliegen er gewoon doorheen. Het omgekeerde is niet waar: de ”klassieke”schotelantennes zijn ook gevoelig voor radiogolven met ”LOFAR frequenties”. Maar schotelantennes hebben het nadeel dat ze veel duurder zijn. Er is een hele stijve constructie nodig om te zorgen dat hij zijn paraboolvorm houdt en niet inzakt onder invloed van zijn eigen gewicht. Een ander nadeel is dat ze alleen gevoelig zijn voor het kleine deel van de hemel waar ze op gericht zijn. LOFAR antennes zijn niet eens te richten, we kunnen de antennes niet vanuit een controle- toren mechanisch kantelen. Dat is geen enkel bezwaar, want ze zijn gevoelig voor bijna de hele hemel. Zelfs vlak boven de horizon is hun gevoeligheid niet verwaarloosbaar, hoewel ze natuurlijk recht naar boven (het zenith) het gevoeligst zijn. Door het op verschillende

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manieren combineren van de signalen van de verschillende antennes kun je met LOFAR antennes verschillende gebieden van de hemel tegelijk zien. Je kunt met LOFAR zoveel gebieden tegelijk bekijken dat je de facto altijd bijna de hele hemel ziet! Iets dergelijks is nog nooit eerder gepresteerd. Dat het nu pas kan heeft vooral te maken met de beschikbaarheid van glasvezel en computerkracht. In figuur 7.4 ziet u zo’n kaart die met een teststation van LOFAR (ITS) is gemaakt. De gevoeligheid van dit teststation is niet zo groot, vandaar dat er maare´ én bron (hemellichaam) te zien is, namelijk Jupiter in actieve toestand. Normaal zijn de objecten Cas A (een supernovarest in onze melkweg) en Cygnus A (een melkwegstelsel) de helderste objecten aan de noordelijke hemel, maar die bevinden zich ten tijde van deze waarneming te dicht bij de horizon om op deze kaart zichtbaar te worden. Wat wel hetzelfde is gebleven ten opzichte van de oude schotelantennes is het principe van apertuur synthese, hetgeen ruwweg inhoudt dat verschillende telescopen samenwerken om de verschillende structuren van hemellichamen in kaart te brengen. Dit is echt een belangrijk punt:e´ én grote telescoop met hetzelfde oppervlak als meerdere telescopen samen heeft wel dezelfde gevoeligheid, maar niet hetzelfde vermogen om, in dezelfde tijd, hemellichamen van verschillende afmetingen in kaart te brengen. De kern van de telescoop staat in Nederland, in Drenthe bij Exloo op een groot open veld van 2 bij 3 km. In het midden van dit veld is een enorme terp aangelegd met een sloot eromheen, waar een aantal LOFAR stations bij elkaar staan. Figuur 7.5 laat een foto uit 2009 van deze terp zien. Ieder LOFAR station bestaat uit 96 Lage Frequentie antennes en 48 Hoge Frequentie Antennes. In totaal staan er 18 van zulke stations in het veld bij Exloo en 18 op andere plaatsen in Nederland. Er zullen ook tenminste 8 stations worden gebouwd in het buitenland. E´´ en, bij Effelsberg, in Duitsland, is er zelfs al gereed. Een overzicht van de plaatsen in Europa waar stations worden neergezet of gepland zijn, ziet u in figuur 7.6.

Er is jaren aan gewerkt aan de totstandkoming van deze telescoop, daarnaast is er ongeveer e140 mln aan besteed, maar wat gaan we ermee doen? LOFAR biedt een onge¨eve- naarde gevoeligheid bij deze lage frequenties. Dat geeft op veel verschillende manieren mogelijkheden om ontdekkingen in het heelal te doen. We kunnen bijvoorbeeld signalen opvan- gen uit het vroege heelal, d.w.z. toen het heelal nog twintig keer zo jong was als nu. We kunnen ook de vorming van melkwegstelsels en sterren beter leren begrijpen. Verder weten we nog heel weinig over magneetvelden in de kosmos, zelfs over magneetvelden in ons eigen melkwegstelsel weten we nog relatief weinig. LOFAR gaat dat ook helpen verbeteren. Weer een ander onderzoeksgebied betreft elementaire deeltjes met een heel hoge energie die door het heelal vliegen: hoe worden ze gemaakt en wat zijn hun eigenschappen. Ook kunnen we met LOFAR meer leren over uitbarstingen van onze eigen zon. Maar dat zijn niet de onderwerpen van dit proefschrift, dit proefschrift over het detecteren van lage frequentie veranderlijk radiobronnen en dan met name bronnen die in relatief korte tijd van helderheid veranderen, dus bijvoorbeeld in een paar uur of een paar dagen. Verandering van helderheid in korte tijd kan komen door een explosie, maar het betekent in ieder geval dat het gebied waar de straling gemaakt wordt niet enorm groot kan zijn, niet zo groot als een heel melkwegstelsel, bijvoorbeeld. Die veranderlijke radiostraling komt van relatief kleine objecten zoals (zware) sterren, neutronsterren en zwarte gaten in het midden van een melkwegstelsel. Maar het kan ook gaan over planeten rond andere sterren dan de zon, zgn. exoplaneten. Die

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Figure 7.1: Een aantal van de schotelantennes van de Very Large Array (VLA) nabij Socorro, New Mexico, Verenigde Staten. Op de achtergrond de maan. (Credit: Image courtesy of NRAO/AUI)

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Figure 7.2: De Lage Frequenties Antennes van LOFAR zoals ze momenteel bij Exloo (Drenthe) staan opgesteld. Ze zijn vooral gevoelig bij frequenties tussen de 30 en 80 MHz.

planeten kunnen, als ze een magneetveld hebben, geladen deeltjes uit de sterrewind laten draaien (gyreren) rond hun magnetische veldlijnen. Het gevolg hiervan is dat ze een bepaald soort radiostraling, namelijk cyclotronstraling, gaan uitzenden. LOFAR is gevoelig voor de frequenties waarbij deze straling het sterkst is (en nog door de ionosfeer kan komen). Hier gaat heel hoofdstuk 6 van dit proefschrift over. Dit is een artikel uit 2007 waarbij we van alle bekende exoplaneten, dat waren er toen ruim 200, onderzocht hebben of hun helderheid voldoende zou zijn om door LOFAR gedetecteerd te kunnen worden. Dat bleken er slechts enkele te zijn, maare´ én detectie zou al iets unieks zijn, dat is nog nooit eerder gedaan. Overi- gens is er een heel goed voorbeeld van een planeet rond onze eigen ster, de zon, die sterke lage frequentie radiostraling uitzendt, namelijk Jupiter. Jupiter zal ook uitvoerig met LOFAR bestudeerd worden, maar we weten al relatief veel van deze planeet en zijn magneetveld. Deze kennis komt niet alleen door waarnemingen met de vorige generaties radiotelescopen, maar ook door de ruimtevaartuigen Pioneer en Voyager die er vlak langs zijn gevlogen en metingen hebben verricht. Jupiter is ook een voorbeeld van een veranderlijke radiobron. De helderheid van Jupiter in de kaart van figuur 7.4, was in de opnamen 8 minuten ervoor en 8 minuten erna, die ieder ook tweeëneenhalve minuut duurden, heel verschillend. Zelfs in een seconde kan de helderheid van Jupiter al aanzienlijk veranderen als we kaarten maken met een klein frequentie bereik.

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Figure 7.3: Hoge Frequenties Antennes van LOFAR van dichtbij bekeken.

7.2 Voorbereiden op LOFAR

7.2.1 Software

LOFAR is een instrument dat veel meer data produceert dan welke vorige telescoop dan ook zodat we niet goed uit de voeten kunnen met de hard- en software die voor bijvoorbeeld de Westerbork Radio Synthese Telescoop (WSRT) gebruikt wordt. Het correleren (vermenigvuldigen) van de signalen van de verschillende stations wordt voor LOFAR door software op een supercomputer (Blue Gene/P, gemaakt door IBM) gedaan. Voor de WSRT gebeurde dit op hardwareniveau, maar voor LOFAR zou dat onvoldoende ﬂexibel zijn. De stationscorrelator, die de signalen van de verschillende antennes in een LOFAR station com- bineert, bestaat trouwens w´el uit hardware. Dit is speciale hardware die geprogrammeerd kan worden. Uit Blue Gene komen de gecorreleerde signalen van de verschillende LOFAR stations. Deze output moet verder bewerkt worden voor er kaarten van kunnen worden gemaakt. Het be- werken bestaat uit het verwijderen van slechte data, veroorzaakt door interferentie van bijvoorbeeld radio en tv stations, middeling over verschillende frequentiekanalen en calibratie. Iedere seconde worden er kaarten gemaakt per frequentieinterval, per gebied van de hemel,

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elk nog eens voor vier verschillende polarisaties. Dat is dus een zeer grote hoeveelheid kaarten die iedere seconde verwerkt moet worden! Zo groot dat het niet mogelijk is om al deze kaarten langere tijd op te slaan, althans niet zonder ze eerst fors te comprimeren in tijd en/of frequentie. Toch hebben we, om veranderlijke radiobronnen te kunnen vinden, wel steeds iedere seconde nieuwe kaarten van de hemel nodig. Dit hebben we opgelost door speciale software die iedere kaart verwerkt en verbinding legt met een database. We zijn namelijk niet ge¨ınteresseerd in ieder pixel van een kaart, maar alleen in de bronnen die erin staan, hoe helder ze zijn en waar ze staan aan de hemel. De software die dat doet, dus die alle bronnen in een kaart ”meet”, heet bronextractie software (Engels: source extraction software). De helderheden en posities van elke bron in elke volgende LOFAR kaart worden in real time, dus steeds binnene´ ´en seconde, vergeleken met een database. In deze database staan de posities en helderheden van bronnen die vroeger zijn waargenomen. Als uit die vergelijking duidelijk naar voren komt dat er een verschil is in helderheid van een bron t.o.v. een eerdere waarneming of dat het om een nieuwe bron gaat, wordt er een ”alarm”afgegeven. Dit houdt in dat astronomen op de hoogte worden gesteld. In veel gevallen wordt LOFAR automatisch gereconﬁgureerd zodat er heel snel met extra gevoeligheid op deze nieuwe of veranderlijke bron kan worden ”ingezoomd”, zodat we er meer aan kunnen ontdekken. De beschrijving van alle componenten van de bronextractie software vindt u in hoofdstuk 2. Deze code heb ik niet alleen geschreven, maar ook uitvoerig getest, met behulp van statistis- che technieken. Het testen of software ook echt doet wat het moet doen, noemen we validatie. De validatie van alle componenten van mijn code heb ik beschreven in hoofdstuk 3.

7.2.2 Wetenschap SGR 1806-20 In de aanloop naar LOFAR, kunnen we nog meer doen dan ervoor zorgen dat onze hard- en software in orde is. Dat kunnen we doen door met onze ”klassieke”radiotelescopen bij zo laag mogelijke frequenties veranderlijke bronnen waar te nemen. Die mogelijkheden deden zich al snel na het begin van mijn proefschrift (november 2004) voor. Op 27 december 2004 was er een grote uitbarsting van het object SGR 1806-20. SGR staat voor ”Soft Gamma Re- peater”, dat zijn neutronensterren met een sterk magneetveld (magnetars) die herhaaldelijk heldere flitsen laag-energetische gammastraling uitzenden. SGR 1806-20 was dus al bekend bij astronomen, maar deze grote uitbarsting, die werd veroorzaakt door een reconfiguratie van zijn magneetveld, maakte hem bekend bij het algemene publiek. Want deze explosie gaf de meest energetische flits van gammastraling van buiten het zonnestelsel die ooit was gereg- istreerd! Die flits werd door de maan weerkaatst en verstoorde zelfs de aardse ionosfeer. Radiotelescopen nemen dit vanzelfsprekend niet waar, maar wel de radiostraling afkomstig uit de wolk van plasma die zich door de explosie rond de magnetar gevormd heeft. Deze radiostraling wordt net als bij de lage frequentie radiostraling van Jupiter (cyclotronstraling) veroorzaakt door electronen die versneld worden in magneetvelden. Alleen gaat het hier om de relativistische variant van cyclotronstraling, zgn. synchrotronstraling. Dat komt omdat de electronen in het schokfront van de explosie snelheden hebben gekregen in de buurt van de lichtsnelheid. Aan de eigenschappen van die synchrotronstraling kunnen we allerlei dingen

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afleiden. Het was al bekend, uit eerder onderzoek, dat de straling van de wolk rond SGR 1806-20 lineair gepolariseerd was. Die polarisatie wordt veroorzaakt doordat de beweging van de electronen die de straling uitzenden aan het lokale magneetveld in het plasma gekop- peld is. De polarisatierichting zegt dus iets over de orientatie van het magneetveld in het plasma, geprojecteerd op de hemel. Met de Westerbork Synthese Radio Telescoop (WSRT) hebben we in januari, april en mei van 2005 in totaal 19 waarnemingen van deze plasmawolk gedaan bij 350, 850 en 1300 MHz. Dit staat beschreven in hoofdstuk 5. Wat we ontdekt hebben is dat, bij lage frequenties, de polarisatiehoek heel anders is dan bij hoge frequenties. We hebben ook afgeleid dat de polarisatiegraad bij lage frequenties niet wezenlijk anders is dan bij hoge frequenties. De combinatie van deze twee ontdekkingen is heel opmerkelijk. Bij lage frequenties kan de polarisatiehoek inderdaad anders zijn, maar dan is dat meestal in combinatie met een sterke depolarisatie. Dat is het geval waarin de polarisatiehoek van het radiosignaal flink verdraaid wordt, door een verschijnsel dat Faraday Rotatie heet. Het netto effect van de verschillende lagen plasma die de radiostraling uit de verschillende gebieden van de stralingshaard langs verschillende wegen passeert, is dan dat er weinig gepolariseerd licht overblijft. Omdat dit nu niet het geval is, lijkt het erop dat de straling afkomstig is uit ander plasma, met een anders georiënteerd magneetveld, dan de hoge frequentie straling. Het is dus misschien zo dat je naar een ander deel van de wolk ”kijkt”. Hoe het deel van de wolk, dat de lage frequentie straling uitzendt, er dan precies uitziet, weten we niet. Daarvoor heb je een gedetailleerd model van de wolk nodig en meer en nauwkeuriger waarnemingen, bij verschillende frequenties en tijden, om de evolutie van de wolk te kunnen volgen.

GCRT J1745-3009 In maart 2005 verscheen een opzienbarend artikel in Nature over de ontdekking van een zeer krachtige veranderlijke radiobron, genaamd GCRT J1745-3009, ruim een graad ten zuiden van het centrum van onze Melkweg. De waarneming was gedaan bij 90 cm golflengte met de Very Large Array (VLA), op 30 september/1 oktober 2002 als onderdeel van een zoek- tocht naar veranderlijke bronnen in de buurt van het Galactisch Centrum (GC). Het artikel verscheen pas tweeëneenhalf jaar later omdat de auteurs, onder aanvoering van Scott Hyman, er zeker van wilden zijn dat wat ze gevonden handen echt was. Wat ze zagen was een hemellichaam dat ze niet konden oplossen, net ten zuiden van de randen van een supernovarest, dat vijf maal steeds ongeveer 11 minuten zeer krachtig straalde met tussenpozen van zo’n 66 minuten waarin deze bron totaal niet gedetecteerd kon worden. Jammer genoeg duurde de waarneming maar zes uur, anders hadden ze waarschijnlijk nog meer uitbarstingen kunnen waarnemen. Deze uitbarstingen leken dus regelmatig met een periode van 77 minuten. Een dergelijke veranderlijke bron was nog nooit eerder waargenomen. Hyman en zijn medewerkers hebben de tientallen waarnemingen vanaf 1989 van het Galactisch Centrum bij 90 cm in het VLA archief doorgespit, maar vreemd genoeg was GCRT J1745-3009 al die keren niet te zien, geen uitbarstingen dus. Na de publicatie van het Nature artikel bleek de bron wel twee keer te vinden te zijn in het archief van een Indiase telescoop, de Giant Meter Radio Telescope (GMRT). In 2003 en 2004 waren zwakkere eenmalige uitbarstingen van deze bron te vinden, maar nooit is GCRT J1745-3009 bij een andere golflengte dan 90cm gedetecteerd.

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Dat is echt heel vreemd. Het is zo’n bijzondere bron dat ik denk dat het Scott Hyman nog de Nobelprijs voor de Natuurkunde voor deze ontdekking kan krijgen! Ik heb geprobeerd om met de Westerbork Synthese Radio Telescoop (WSRT) deze bron opnieuw te detecteren, bij 90 cm en bij 21 cm golﬂengte. Dat is jammer genoeg niet gelukt. Ik heb ook drie van de vier alleroudste 90 cm waarnemingen van het GC met de VLA, uit 1986 en 1988, gereduceerd om GCRT J1745-3009 terug te vinden. Dat heeft helaas ook niet tot nieuwe detecties geleid, wel tot mooie kaarten van het centrum van onze Melkweg. Wat w´el heeft geleid tot nieuwe inzichten, is het opnieuw reduceren en analyseren van de VLA data uit 2002 waarmee Scott Hyman GCRT J1745-3009 heeft ontdekt. Dit staat in hoofdstuk 4 beschreven. Ik kwam erachter dat alle uitbarstingen precies hetzelfde verloop hadden, met eerst een snelle toename van de helderheid, dan een langzame toename en tenslotte een snel verval, dit alles in de 11 minuten die de uitbarstingen duurden. Ik heb ook ontdekt dat de vijf uitbarstingen niet allemaal precies even lang waren, de lengte varieerde een paar procent. De uitbarstingen waren wel strikt periodiek, er zat steeds 77.012 ± 0.021 minuten tussen het begin van twee opeenvolgende uitbarstingen. Ik vond dat de spectrale index, die aangeeft hoe helder de bron is als functie van frequentie, sterk negatief is, in zekere mate in overeenstem- ming met eerdere conclusies door Hyman en zijn medewerkers, op basis van de uitbarsting die in 2004 met de GMRT is waargenomen. Maar de hamvraag is: wat is GCRT J1745-3009 nu precies voor een hemellichaam? Zoals gezegd kunnen we deze bron niet oplossen, we zien GCRT J 1745-3009 als een puntbron. Zijn het twee neutronsterren die om elkaar heen draaien in 77 minuten? Of is het een preced- erende pulsar met een precessie periode van 77 minuten? Een probleem is dat beide systemen waarschijnlijk nogal schaars in de Melkweg aanwezig zijn. Dat hoeft ze niet uit te sluiten, maar wat ze wel uitsluit is mijn ontdekking dat de uitbarstingen asymmetrisch in de tijd zijn, terwijl (de simpelste vorm van) deze modellen symmetrische uitbarstingen voorspellen. Er is ook gesuggereerd dat GCRT J1745-3009 een totaal nieuw soort systeem is, een zogenaamde witte dwerg pulsar, die elke 77 minuten een uitbarsting heeft. Dit zou de asymmetrie van de uitbarstingen alsmede de exacte periodiciteit kunnen verklaren. Maar dat model kan niet makkelijk verklaren waarom de uitbarstingen alleen bij 90 cm zijn waargenomen of waarom de spectrale index zo extreem is. Exoplaneten zijn ook genoemd, maar zouden eigenlijk bij wat lagere frequenties zo helder moeten zijn. Zo zijn er voor elk model wel voors en tegens, geen enkel model kan alle eigenschappen van GCRT J1745-3009 makkelijk verklaren. We moeten eerst meer soortgelijke systemen ontdekken, bijvoorbeeld met LOFAR, of meer detecties hebben van GCRT J1745-3009.

7.2.3 Waarneemstrategie voor exoplaneten

Wat is nu de meest optimale manier om met LOFAR zoveel mogelijk veranderlijke bronnen te vinden? Om die vraag te beantwoorden moet je veranderlijke bronnen in klassen en dan, afhankelijk van de karakteristieken van die klassen, per klasse een strategie bepalen. Dat is nog niet zo gemakkelijk omdat je niet precies weet hoe de veranderlijkheid van bijvoorbeeld een accreterende neutronenster bij r¨ontgenfrequenties zich vertaalt naar lage frequentie radiogolven. Maar, stel dat je redelijk kunt voorspellen wat de helderheid en mate van veran-

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derlijkheid van een bepaald type bronnen bij LOFAR frequenties zal zijn, dan kan je daar je waarneemstrategie op afstemmen en meer van dit soort bronnen vinden. Echter, het is misschien niet het interessantst om zoveel mogelijk veranderlijke bronnen te vinden, maar meer om veranderlijke bronnen van een nieuw type te vinden, zoals Scott Hyman is gelukt met GCRT J1745-3009 (zie hierboven). In deze categorie vallen de exoplaneten, dat zijn planeten die rond andere sterren dan de zon draaien. Er zijn nu enkele honderden exoplaneten bekend, die bij de frequenties waarbij ze ontdekt zijn (voornamelijk zichtbaar licht) niet of nauwelijks veranderlijk zijn. Maar het is aannemelijk dat er een aantal exoplaneten op Jupiter zullen lijken en dus ook krachtige en veranderlijke radiostraling zullen uitzenden, zoals ik heb beschreven in paragraaf 7.1. Helaas is het nog nooit gelukt om met de huidige generatie radiotelescopen radiostraling van een exoplaneet waar te nemen, ondanks vele uren waarneemtijd. Als dit met LOFAR wel lukt, zou dit een historische ontdekking zijn! Het probleem is alleen dat als je Jupiter op de afstand van de dichtstbijzijnde ster plaatst, hij al te zwak is om te worden waargenomen, zelfs met LOFAR lukt dat niet. We zeggen dan dat de intrinsieke helderheid van systemen die lijken op de combinatie zon-Jupiter te laag is. Gelukkig zijn er genoeg exoplaneten waarvan we kunnen vermoeden dat hun intrinsieke helderheid vele malen hoger is dan Jupiter, bijvoorbeeld omdat deze exoplaneet veel dichter bij de ster staat dan Jupiter bij de zon. Behalve genoeg intrinsieke helderheid is het natuurlijk ook belangrijk voor een geschikte exoplaneet dat hij niet te ver bij ons vandaan staat, anders lukt het ook niet. Jean-Mathias Griesßmeier, Philippe Zarka en ik hebben alle exoplaneten waarvan op 13 januari 2007 voldoende bekend was (197 stuks) doorgerekend en een ranglijst gemaakt van de geschiktste kandidaten om met LOFAR te kunnen worden gedetecteerd. Dit staat beschreven in hoofdstuk 6. We hebben geconcludeerd dat er van die 197 exoplaneten een paar helder genoeg zullen zijn om te worden gedetecteerd met LOFAR. Dat is natuurlijk weinig, maar het is eigenlijk ook wel hoopgevend omdat er steeds meer exoplaneten ontdekt worden. Vandaag de dag (2 mei 2010) zijn er al 453 exoplaneten bekend! Als we alle exoplaneten zouden doorrekenen die er sinds 13 janauri 2007 zijn bijgekomen, zouden we waarschijnlijk weer een aantal goede kandidaten voor een detectie met LOFAR vinden.

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Figure 7.4: Een kaart van de hele hemel gemaakt met een teststation van LOFAR, het Initial Test Station. Het oosten is links, het noorden is boven. Misschien kunt u vaag de cirkel ontwaren die de horizon aangeeft. Het bijzondere aan deze kaart is dat er maare´ én (zeer heldere) bron in het zuiden te zien is. Dit is een veranderlijke bron, namelijk Jupiter tijdens een radio ”storm”op 8 februari 2004. Als Jupiter en de zon niet actief zijn, zijn Cassiopeia A (een supernovarest in onze melkweg) en Cygnus A (een melkwegstelsel) de helderste bronnen aan de noordelijke hemel, bij deze frequenties (20-30 MHz). Deze opname is ’s nacht gemaakt, maar ook overdag is de zon meestal niet de helderste radiobron aan de hemel. De maan is bij deze golflengten helemaal onzichtbaar. De hemel ziet er dus heel anders uit bij radiogolflengten dan bij zichtbaar licht (het optisch), waar ons oog gevoelig voor is.

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Figure 7.5: Foto uit 2009 van de ”superterp”in het binnengebied van de kern van LOFAR, waarop de binnenste kernstations van LOFAR geplaatst zullen worden. Deze terp wordt omringd door een sloot.

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Figure 7.6: De heldere plekken laten zien waar er LOFAR stations staan, worden gebouwd of gepland zijn. Slechts voor 8 buitenlandse LOFAR stations is momenteel geld beschikbaar: 5 in Duitsland, 1 in Frankrijk, 1 in het Verenigd Koninkrijk en 1 in Zweden.

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Figure 7.7: Een kaart van de supernovarest G 359.1-0.5, een graad ten zuiden van het centrum van onze Melkweg. Deze kaart is gemaakt van de data van de waarneming, op 30 september/1 oktober 2002 met de Very Large Array bij 90 cm golﬂengte, waarmee de veranderlijke bron GCRT J1745-3009 ontdekt is. In deze kaart is GCRT J 1745-3009, net ten zuidwesten van de supernovarest (het oosten is links). Hij staat in het midden van het heldere schijfje dat ik heb aangebracht om de positie aan te geven.

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CHAPTER 8

Epilogue

The source extraction (SE) software is complete and its components have been validated, at least in the case of artificial sources inserted in maps with correlated noise. The software is fast and efficient, well documented and transparent. It is ready for use in the TKP pipeline for the processing of sources in real time. However, if time permits, it is advisable to run tests on dirty and cleaned maps with some sources inserted in the visibilities. Also, it would be good to run it on a series of maps such as the WENSS or NVSS images and compare its output with the contents of the WENSS and NVSS catalogues. In any case, when this software is run on actual LOFAR maps, it is very likely that some adjustments of the code could improve its performance. For example, if the code is run on confusion limited maps, the background level is overestimated. Hence, the fluxes are underestimated. It may be that a more advanced asymmetric clipping algorithm, will reduce a large part of the bias in those cases. The False Discovery Rate (FDR) algorithm is an essential part of the SE process, because it controls the rate of false alerts. However, it assumes Gaussian statistics for the background noise. Some output of the validation runs indicate that, even in the case when the noise in the visibilities is strictly Gaussian, the noise in the image plane is not. The origin of this is that we are not dealing with a true Fourier Transform, but with either a ”Direct Fourier Transform”1 or with a Fast Fourier Transform (FFT). The result of this is that the number of noise outliers above some high threshold level, say 7σ, is much higher than Gaussian statistics would predict. Peter Jonker suggested to me that the FDR algorithm could be improved by fitting some distribution to the histogram of the noise pixel values. The FDR algorithm could then take account of deviations from Gaussianity. This is an interesting proposition. With regard to understanding the nature of the enigmatic radio source GCRT J1745-3009, we need either more detections of this source or discoveries of similar systems. In both

1Called DFT by radio astronomers. Other scientists use DFT for ”Discrete Fourier Transform”, an invertible operation on regularly spaced data.

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cases, many hours of observing time are probably necessary. With LOFAR it remains to be investigated to what extent observations at low elevations, which is entailed when monitoring the Galactic Center from these latitudes, can be properly calibrated. Chances are high that LOFAR will find similar sources at higher Galactic latitudes, which will provide us with the clues we are looking for. A WSRT-LFFE observation of GCRT J1745-3009 on 2009 June 23 remains to be reduced. With regard to the WSRT observations of SGR 1806-20 in 2005 January, Ger De Bruyn suggested that the error bars on the polarization angles could be improved by using the special RM synthesis software package. This year, LOFAR will start monitoring a large part of the low frequency radio sky and can track the evolution of radio nebulas from the beginning when explosions similar to the Giant Flare from SGR 1806-20 occur. Continous observing at different LOFAR frequencies will enable us to differentiate between different models for the nebula, since all four polarization products are always recorded. Probably these observations have to be supplemented by observations at higher frequencies by other radio telescopes in order to probe different substructures. LOFAR observations will also test our models for the radio emission of extrasolar planets. If we are able to detect bursts with LOFAR from many more exoplanets than we predicted, we will have to reconsider our assumptions. Likewise, if we detect none at all, this will also lead to adjustments for the modeling of radiation mechanisms in these systems. It will take a while before we are able to draw some firm conclusions because the search for these bursts takes a considerable amount of observing time.

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CHAPTER 9

Dankwoord

Pfoe, het zit erop! Wat een interessante, maar ook zware jaren waren het. Ik ben begonnen met het stellige voornemen om al in mijn eerste jaar mijn eerste artikel gepubliceerd te krijgen. Dat werd vier jaar later waardoor het de laatste twee jaar allemaal erg hectisch werd. Nu ik dit schrijf (4 mei 2010) heb ik er al een postdoc positie van zeven maanden bij het K.N.M.I. op zitten en op zoek naar een nieuwe baan. Het is achteraf allemaal niet zo gelopen als de bedoeling was. Ik heb op een gegeven moment per ongeluk mijn schijf gewist zodat dertien maanden werk verloren ging. Het kostte me toch wel een paar maanden om dat recht te zetten. Daarnaast was LOFAR pas veel later klaar dan gepland (opening 12 juni 2010) en ik heb op geen enkele manier LOFAR data gebruikt voor dit proefschrift. Oorspronkelijk zou LOFAR in 2006 af zijn en zou ik de software die ik in de eerste twee jaar van mijn proefschrift zou schrijven, in de laatste twee jaar gebruiken voor LOFAR kaarten. Voor die LOFAR data is WSRT data in de plaats gekomen die in eerste instantie niet tot iets publicabels kon leiden. Pas na veel werk kwam er iets moois uit, maar daar moest ik eerst nog een ﬂink aantal andere datasets voor uit het VLA archief halen en reduceren. Het heeft mij wel diep inzicht gegeven in de reductie van data uit interferometers en ik ben nu echt een AIPS freak. Ik heb Eric Greisen nog regelmatig lastiggevallen met fouten die ik tegenkwam in AIPS routines En ik vond het geweldig om, met 90cm data van de VLA, kaarten van het centrum van onze Melkweg te maken. Dat was echt leuk. In het schrijven van software had ik ook veel plezier en het zat niet tegen, dat liep wel volgens schema. Mijn introductie in de radiosterrenkunde begon eigenlijk al voor mijn proefschrift, in juni 2004 toen ik bij de zomerschool van ASTRON ben aangenomen. Aansluitend ben ik nog

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twee maanden junior scientific assistant geweest. In die vijf maanden heb ik veel geleerd van Ger de Bruyn, Michiel Brentjens, Stefan Wijnholds en Jan Noordam. Ik wil jullie dan ook hartelijk bedanken voor jullie hulp en uitleg! In november 2004 begon mijn aanstelling als promovendus bij de UvA. Ralph, heel erg bedankt dat je het risico hebt aangedurfd om zo’n vreemde vogel als mij aan te nemen en dat je mij wilde begeleiden. Het heeft tot een proefschrift geleid waar ik zelf eigenlijk wel trots op ben, omdat het een afgerond geheel is. Ik heb veel van je kunnen leren in tijden dat je het zelf ook enorm druk had. Dat zal nog wel even zo blijven nu je een ERC grant hebt gehad. Ik hoop dat je hier veel veranderlijke radiobronnen mee zal ontdekken! Dit onderzoek zal in de komende jaren een enorme versnelling krijgen, helaas weet ik niet of ik daar deel aan mag nemen, ik heb nu alleen het begin meegemaakt. John, heel erg bedankt voor de prettige samenwerking en voor wat ik allemaal van je heb kunnen leren. Het was heel belangrijk om iemand met een informatica achtergrond in de groep te hebben. Bart, ik was heel blij dat we jarenlang samen konden werken. Ik heb het heel gezellig met je gehad en we hebben enorm gelachen, vooral bij de zomerschool in Al- buquerque. Wat hebben we daar ook veel geleerd en wat hebben we goede discussies gehad. Sanne, jeetje lieverd, wat waren het een heftige jaren. Veel te zwaar natuurlijk, vooral van- wege het omgaan met Yaëls handicap, alle ziekenhuisbezoeken en het extreem slechte slapen. Jij hebt me gestimuleerd om terug te keren in de sterrenkunde en daar ben ik je heel dankbaar voor. Anders had ik nog op de afdeling Planning & Control van Woonzorg gewerkt, dat was zoiets als een comateuze toestand. Jij hebt me ook altijd gesteund de afgelopen jaren en vele zondagen alleen met Yaël doorgebracht omdat ik probeerde een beetje op schema te blijven. Dat vond ik ook héél erg stoer van je. Gelukkig hebben we ons er doorheen geslagen en kan ik er nu weer meer voor je zijn. Dit proefschrift is voor jou!

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CHAPTER 10

Publication list

PUBLICATIONS

A. Refereed journals ”Low frequency observations of the radio nebula produced by the giant flare from SGR 1806-20-Polarimetry and total intensity measurements”, with B. Scheers and R.A.M.J. Wijers, A&A 509A, 99 (2010). ”A new perspective on GCRT J1745-3009”, with B. Scheers, R. Braun, R.A.M.J. Wijers, J.C.A. Miller-Jones, B.W. Stappers and R.P. Fender, A&A 502, 549 (2009). ”Predicting low-frequency radio fluxes of known extrasolar planets”, with J.-M. Grießmeier and P. Zarka, A&A 475, 359 (2007). ”The galactic merger-rate of (ns, ns) binaries. I. Perspective for gravity-wave detectors.”, with S.F. Portegies Zwart, A&A 312, 670 (1996). B. Conference proceedings ”Predictions for Radio Emission from Extrasolar Planets”, with J.-M. Grießmeier and P. Zarka in Proceedings of Bursts, Pulses and Flickering: wide-field monitoring of the dynamic radio sky, 12-15 June 2007, Kerastari, Tripolis, Greece. ”The LOFAR Transients Key Project”, with R.P. Fender and several co- authors in Proceedings of the VI Microquasar Workshop: Microquasars and Beyond, September 18-22, 2006, Como, Italy. ”Expected Characteristics for Exoplanetary Radio Emission”, with J.-M. Grießmeier and P. Zarka in European Planetary Science Congress, 18-22 September 2006, Berlin, Germany.

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