A Revised View of the Transient Radio Sky
D. A. Frail, S. R. Kulkarni, E. O. Ofek, G. C. Bower, E. Nakar
aa r X i v : . [ a s t r o - ph . H E ] O c t Draft of October 28, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
A REVISED VIEW OF THE TRANSIENT RADIO SKY
D. A. Frail , S. R. Kulkarni , E. O. Ofek , , G. C. Bower & E. Nakar Draft of October 28, 2018
ABSTRACTWe report on a re-analysis of archival data from the Very Large Array for a sample of ten longduration radio transients reported by Bower and others. These transients have an implied all-sky ratethat would make them the most common radio transient in the sky and yet most have no quiescentcounterparts at other wavelengths and therefore no known progenitor (other than Galactic neutronstars). We find that more than half of these transients are due to rare data artifacts. The remainingsources have lower signal-to-noise ratio (SNR) than initially reported by 1 to 1 . σ . This loweringof SNR matters greatly since the sources are at the threshold. We are unable to decisively accountfor the differences. By two orthogonal criteria one source appears to be a good detection. Thus therate of long duration radio transients without optical counterparts is, at best, comparable to that ofthe class of recently discovered Swift J1644+57 nuclear radio transients. We revisit the known andexpected classes of long duration radio transients and conclude that the dynamic radio sky remainsa rich area for further exploration. Informed by the experience of past searches for radio transients,we suggest that future surveys pay closer attention to rare data errors and ensure that a wealth ofsensitive multi-wavelength data be available in advance of the radio observations and that the radiosearches should have assured follow-up resources. Subject headings: catalogs — radio continuum: galaxies — surveys INTRODUCTIONA century ago, the study of variable stars was a lead-ing area of astronomy. With the increasing availabilityof large format optical detectors and inexpensive highspeed computing this sub-field, as witnessed by projectssuch as ASAS, OGLE, Catalina Sky Survey, the Palo-mar Transient Factory, PanSTARSS and SkyMapper, ismaking a come-back. Radio astronomy appears to bepoised for a similar growth. At meter wavelengths, com-mercially available signal processing chips make it fea-sible to image the entire primary beam of a dipole ora cluster of dipoles. These technological innovations lieat the heart of LOFAR (R¨ottgering et al. 2003), MWA(Lonsdale et al. 2009) and LWA (Ellingson et al. 2009).At centimeter wavelengths the “large number, small di-ameter” (LNSD) array approach (made possible by in-expensive signal processing, advances in commercial RFtechnology, innovative ideas in the design of small diam-eter telescopes and phased array focal planes) has nowbeen demonstrated to be a cost effective method of build-ing high speed mapping machines (Welch et al. 2009;Dewdney et al. 2009; Jonas 2009; Oosterloo et al. 2009).The LNSD approach has motivated a new generation ofradio facilities: Apertif/WSRT (Oosterloo et al. 2009);MeerKAT (Booth et al. 2009); ASKAP (Johnston et al.2008).We divide radio transients into four categories based National Radio Astronomy Observatory, P.O. Box O, So-corro, NM 87801 Caltech Optical Observatories 249-17, California Institute ofTechnology, Pasadena, CA 91125, USA Benoziyo Center for Astrophysics, Faculty of Physics, TheWeizmann Institute for Science, Rehovot 76100, Israel Astronomy Department, University of California, Berkeley,601 Campbell Hall Raymond and Beverley Sackler School of Physics & Astron-omy, Tel Aviv University, Tel Aviv 69978, Israel on two attributes. The first is the duration of the ba-sic phenomenon (shorter than or greater than a few sec-onds). The second is their location (within the Galaxyor extra-galactic). Roughly speaking the duration mapsto coherent versus incoherent emission and the locationto repeated versus cataclysmic events.Pulsars and related phenomenon (giant pulses, nullingpulsars, erratic pulsars, rotating radio transients, andmagnetars) are the dominant category of short durationradio transients at meter and centimeter wavelengths.There are no secure examples of short duration radiotransients that are located beyond the local Group. Flarestars and associated phenomena are prime examples oflong duration radio transients of Galactic origin.The focus of this paper is long duration transients ofextra-galactic origin. Known examples in this group aresupernovae (Weiler et al. 2010) and gamma-ray burst af-terglows (Gehrels et al. 2009). In both cases, the radioemission arises as the fast moving debris interacts withthe circumstellar matter. In Table 1 we summarize theareal density of radio-emitting supernovae (including thesub-classes) and GRB afterglows. Note the areal densityof “live transients” (transients present at any given in-stant of time) of both supernovae and GRB afterglows isless than 0.05 per square degree.In 2007, Bower et al. [hereafter, B07] reported on theanalysis of a single field observed every week as a part ofthe Very Large Array (VLA) calibration program. Theobservations were conducted at 4.8 GHz and 8.4 GHz andlasted 22 years. The 944 epochs and the weekly cadencemakes this data set a most valuable set to probe thedecimeter band for long duration transients at the sub-milliJansky level. These authors reported the discoveryof eight transients found in only one epoch (hereafter“single-epoch”; duration, 20 minutes < t dur < TABLE 1Long Duration Transient Populations
Class Rise Decay D Host Rate Ref.yr yr mag deg − Type II SNe 0.1-1 10 100 Mpc 16 0.04 Gal-Yam et al. (2006)Type Ib/c SNe 0.1 0.3 50 Mpc 14.5 5 × − Berger et al. (2003)SN1998bw-like 0.1 0.1 300 Mpc 18.4 3 × − Soderberg et al. (2006)Sw J1644+57-like 0.1 1 z ∼ . − Levinson et al. (2002)NS-NS mergers 0.1–1 0.1–3 800 Mpc 20.5 5 × − Nakar & Piran (2011)
Note . — Detectability distance and rates have been calculated assuming a single snapshot at aflux density threshold of 0.3 mJy. See section 5 for details. D is the distance at which the typicaltransient will have a specific flux of 0.3 mJy. Host is the apparent magnitude of a galaxy with − D . (hereafter, “multi-epoch” sources).Deep observations towards these sources were under-taken at optical, near-IR and X-ray bands. The most re-markable feature of the B07 sources is an absence of opti-cal and near-IR counterparts, despite deep searches (B07;Ofek et al. 2010). As noted by Ofek et al. (2010) allextra-galactic transients (regardless of the band at whichthe transient was discovered) have detectable opticalcounterparts, namely, their host galaxies. Furthermore,remarkably the areal density of transients live at anygiven time was estimated to be 1.5 deg − ( S > .
37 mJy).This density exceeds that of all other known radio tran-sient source populations by an order of magnitude (ormore); see Table 1. Ofek et al. (2010) thus argued thatthe absence of an optical counterpart means that B07transients have to be repeating sources of Galactic ori-gin, and proposed that B07 transients are old neutronstars (which naturally satisfy the requirement of beingoptically almost invisible).Given that the search for transient and strong vari-ables is one of the primary motivators for the next gen-eration radio facilities (described earlier) it is importantto critically investigate the B07 transients since this classnominally dominates over all other known classes of ra-dio transients (see Table 1). To this end, here we reporton a re-analysis of the original data of B07 ( § §
3) andrevisit the transient reported by Ofek et al. 2011 ( § § § OBSERVATIONS AND RE-ANALYSISThe data used by B07 arose from a calibrator pro-gram carried out during the period 1983–2005. All ob-servations were made in the standard continuum modewith 100 MHz of total bandwidth in each of two adjacent50-MHz bands (IFs) at center frequencies of 5 GHz and8.4 GHz and in both hands of circular polarization. SeeB07 for more details about the full data-set.For the re-analysis we selected, from the archive, onlythe raw data relevant to the transients reported in B07.This means the eight epochs from which the single-epochtransients were first found and the 3+8 data sets fromwhich the two multi-epoch transients were found. Datawere taken at other radio frequencies in about half of thecases. Some details of the single epoch and two-monthtransients can be found in Table 2). For the re-analysis we used AIPS (Greisen 2003). Thedata reduction and imaging followed the same path usedby B07 with a small exception. B07 employed AIPS forthe flagging and analysis of the single-epoch transients,and used the Miriad package (Sault et al. 1995) for imag-ing the two-month averages. We endeavored to makethe calibration and the flagging of UV data (AIPS task
TVFLG ) data for each epoch as uniform way. Followingthese steps we ran each raw visibility data set throughthe VLA pipeline (
VLARUN ).No flux density calibrator was observed during anyepoch of these test observations. Following B07, the fluxdensity of the phase calibrator (B1803+784) was fixed tobe 2.2 Jy (5 GHz) and 2.8 Jy (8.4 GHz). For those epochswith 22 GHz and 1.4 GHz observations the flux densityof the phase calibrator was taken to be 3 Jy and 2 Jy, re-spectively. It is evident from the strong variations in theradio light curves for B1803+784 that these mean val-ues are only approximate. Our reinvestigation confirmthat at least during the period 1981–1999 the variationwas less than 15%. Fortunately, an accurate flux densityscale is not crucial for our analysis since we report resultsin terms of the signal-to-noise.Following B07 we applied a Gaussian weighting to thevisibility data in order to limit the effects of bandwidthsmearing. This was done by applying a 150-k λ taperto all visibility data prior to imaging ( IMAGR ). For eachfield we required that a source be present in both fre-quency bands (IFs) with similar flux densities and withsimilar positions. Images were made with extra largefields-of-view. The wide field-of-view is necessary to re-duce the effect of side-lobes that can mimic sources innarrow fields. These final analysis images had a sizeof about 40-arcmin at 5 GHz data and 27-arcmin at8.4 GHz. For guidance, the full-width at half power forVLA antennas is 45-arcmin/ ν (GHz), or 9.3-arcmin at5 GHz and 4.3-arcmin for 8.4 GHz. Measurements of theVLA beam power response beyond the first null are givenin Cotton and Perley (2010), while the polynomial co-efficients needed to correct for the primary beam attenu-ation can be found in the AIPS task PBCOR . We do notethat these corrections are uncertain for large angular dis-tances from the beam center. All data were taken in theB1950 coordinate system. We stayed in the B1950 sys- U. Michigan Radio Astronomy Observatory database adio transients 3tem throughout calibration and imaging. FINDINGSBelow we offer a detailed report for each of the tensources reported in B07. Summarizing our results for theimpatient reader we find four of the eight single-epochtransients [RT 19840502 ( § § § § § § Miriad imaging yields an SNR of 7.4.For the remaining single epoch sources [RT 19920826,RT 19970205, RT 19970528, RT 19990504] our analysis(undertaken by DAF; see § σ . Since the sourceslie close to threshold of detection even a small shift ofa single σ has an exponential effect in their confidence.In effect, the reality of this group (taken as a whole) ap-parently depends on details of algorithms in AIPS and Miriad – the investigation of which is beyond the scopeof this paper. Below in § § §
4) we synthesize these findings andpresent our conclusions about the B07 transients.3.1.
RT 19840502
B07 report finding a transient close to the pointingcenter (13 ′′ ) with a primary beam-corrected flux densityof 448 ± µ Jy, or a SNR=6.1. We imaged the calibrateddata set and confirmed the presence of emission at thislevel at the reported position. However, we identifieda phase center artifact in the visibility data for the lefthand polarization of the upper IF band. The effect of thisartifact in the image plane is to create strong positive andnegative slidelobes, with the positive feature identified asa transient source (Figure 1). When the upper IF banddata are removed the resulting image is noise-like and thepeak flux density at the nominal position of the reportedtransient is 191 ± µ Jy. Additional observations weretaken during the same epoch at 15 GHz. The peak fluxdensity at the same position is − ± µ Jy.3.2.
RT 19840613
B07 report finding a transient coincident with a pos-sible host galaxy ( z = 0 . ± µ Jy, or a SNR=7.0.Our deconvolved image shows a source at that location.Gaussian fitting suggests that the source is resolved andthis conclusion is supported with an integrated flux den-sity (715 ± µ Jy) being clearly larger than the esti-mate of the peak flux density of 388 ± µ Jy. Imaging
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Fig. 1.—
Image of RT 19840502 marked by cross. The fea-ture at the center (0,0) is an artifact (“phase center”). Oncethe bad visibility data is removed the transient candidate isnot visible on the final image. Contours are displayed in stepsfrom − − . − the lower and upper IFs separately we find another dis-crepancy. The peak flux density in the lower IF is fourtimes weaker than the upper IF band.The likely source of the problem can be seen by com-paring the dirty image with the dirty beam (Figure 2).RT 19840613 appears to be an uncleaned side-lobe ofJ150123+781806, one of the brighter persistent sourcesdetected in 452 images made by B07 at 5 GHz. The puta-tive transient is 56 ′′ away from J150123+781806 to thenorthwest, close to a local maximum (10% of peak) inthe dirty beam at this location. This side-lobe artifact isstronger in the upper band but it is still present in thelower band.Deconvolution does not fully remove the side-lobe fromthe image and the effect is to produce a false transient.We investigated whether the artifact is due to short-timescale ( ∼
10 min) variability of J150123+781806 butafter dividing the visibility data in half (by time) andre-imaging, we found no evidence for variability. The ar-tifact may be due to some low level interference pickedup by some antennas baselines but we were not able toidentify the bad data. It is possible that RT 19840613 isa real transient that unfortunately lies on the side-lobeof the dirty beam, but its Gaussian fit parameters andits variation in the lower and upper sidebands suggestthat it is not a real source.Additional observations were made during this epochat 1.5 GHz. The data quality is good and the images haveno obvious artifacts. No source is visible at the transientposition. The peak flux density is 133 ± µ Jy.3.3.
RT 19860115
B07 report a transient with a primary beam-correctedflux density of 370 ± µ Jy, or a SNR=5.5. Some ofthe same issues with the image of the previously dis-cussed RT 19840613 were also seen for RT 19860115. In Frail et al.
TABLE 2Single Epoch and Two-Month Transient Candidates
Name Type Freq. FWHM/2 ∆ θ BeamCandidate (GHz) (arcmin) (arcmin) (arcsec) NotesRT 19840502 SE 4.9 4.6 0.22 6.0 Phase center artifact.RT 19840613 SE 4.9 4.6 3.3 5.7 Side-lobe of bright source.RT 19860115 SE 4.9 4.6 1.3 14.8 Side-lobe of bright source.RT 19860122 SE 4.9 4.6 4.6 14.5 Artifact. Lower IF is bad.RT 19870422 2M 4.9 4.6 8.0 12.8 Artifact. Bad pointing.RT 20010331 2M 8.5 2.6 4.4 1.5 No detection (see § D SNR: ; .RT 19970205 SE 8.5 2.6 4.4 1.4 B SNR: ; (CA)RT 19970528 SE 4.9 4.6 6.8 3.9 CnB
SNR: ; . (CA)RT 19990504 SE 4.9 4.6 8.9 18.9 D SNR: ; .RT 19970528 SE 8.5 2.6 6.8 1.3 CnB
No detection (BA)RT 19990504 SE 8.5 2.6 8.9 8.3 D No detection (BA)
Note . — Starting from the left the columns are as follows. The name of the transient asRT YYYYMMDD where YYYY is the UT year, MM is the month index and DD is the day num-ber at which the transient was first detected; the type of transient: single epoch (SE) or two monthaverage (2M); the center frequency in GHz; one half of the full-width at half-maximum of the primaryresponse beam in arc minutes; the offset of the transient from the phase center, also in arc minutes; thebeam size in arc seconds computed as geometric mean of the major and minor axes;. The last columnreport the array configuration, two SNRs (for sources detected and reported as such in B07) and somecomments. The left SNR (in italics ) are the SNR from B07 and the right SNR (in typewriter font)are SNR resulting from the work presented here (see § Figure 3 we show the dirty image along with the syn-thesized beam. The RT 19860115 appears to be an un-cleaned side-lobe of J150123+781806 and hence not a realtransient. RT 19860115 lies at the same angular distance(3.3 ′ ) and position angle (100 ◦ CCW CC) of a local max-imum in the side-lobe structure (25% of the peak beam).We were unable to identify the source of these strongside-lobes. As in the case of RT 19840613, we were ableto rule out that the strong side-lobes originated fromshort-term variability of J150123+781806.3.4.
RT 19860122
B07 report a transient with a primary beam-correctedflux density of 1586 ± µ Jy, or a SNR=6.4. Imagesmade with all the visibility data do show emission atthe position of RT 19860122 with the reported peak fluxdensity. However, there does appear to be some erro-neous visibility data resulting in low-level rings in theimage centered at the phase center. The bad data weretraced to the lower IF. If the data for the lower IF bandare removed there is no source at the reported transientposition (peak flux density of 303 ± µ Jy).3.5.
RT 19870422
This is one of two transients found by binning in-dividual epochs into two-month averages. B07 reporta 5 GHz transient with a primary beam-corrected fluxdensity of 505 ± µ Jy, or a SNR of 6.1. The sourceJ150050+780945.5 is positionally coincident with a bluehost galaxy ( z = 0 . RT 20010331
This is the second of the transients which were found bybinning individual epochs into two-month averages. B07report a 5 GHz transient with a primary beam-correctedflux density of 697 ± µ Jy, or a SNR=7.4. Separately,Croft et al. (2011) reported a marginal X-ray source atthis position.Eight single epochs were used to form the average.They are in YYYYMMDD format: 20010306, 20010314,20010321, 20010328, 20010403, 20010411, 20010418 and20010425. We calibrated and imaged the eight epochs ofobservations from 2001 March 6 to 2001 April 25 (all inB-configuration). While the four quiescent sources fromB07 can be seen in this deep image, we do not iden-tify a significant source at the position of RT 20010331.The brightest peak within the synthesized beam (natu-ral weighting) is 42 ± µ Jy (or 3.2 σ ). An independentreduction using the AIPS package by GB confirms theabsence of this source.The first problem lies with the primary beam correc-tions reported in B07. RT 20010331 lies 4.4 ′ from thephase center, close to the 10% response radius of the pri-mary beam. Our estimate for the rms noise of 13 µ Jylies within a few percent of the theoretical value. Thebeam-corrected rms noise in this case (at 10% responseadio transients 5
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Fig. 2.—
Top:
The dirty beam for the RT 19840613 field.This sub-image shows the northwestern side of the beam. Thepeak is located in the lower right hand corner. The contoursare 5, 10, 15, 20, 25 and 30% of the peak flux.
Bottom:
The dirty image toward RT 19840613. A cross marks thelocation of transient. Contours are 0.13, 0.23, 0.33, 0.43,0.53, 0.63 mJy beam − . Note that the transient candidateRT 19840613 lies at the same angle and position as a side-lobe from the bright source J150123+781806 located in thebottom right corner of this image. of the primary beam) would then be about 129 µ Jy, notthe 94 µ Jy given in B07. Rather than correcting at the10% radius, it appears that the flux density and noise inB07 were mistakenly corrected at the 14% power level.This multiplicative error has no impact on the signal-to-noise.In order to investigate this signal-to-noise discrepancybetween our image and B07, we split the data in vari-ous ways (separate epochs, months of March and Aprilepochs, adjacent IF bands) and re-imaged, looking for abright peak. None were found.The discrepancy between B07 and the work reportedapparently can be traced back to differences in the
Miriad and AIPS imaging packages. Our calibrated visibilitydata, when processed through
Miriad using nearly iden-tical imaging parameters as those in AIPS, gives a fluxdensity of 91 ± µ Jy (7 σ ). We have no explanation forthe discrepancy between the two results obtained fromAIPS and Miriad . It is worth noting that the peak fluxdensities of the persistent sources identified by B07 agreein these images to 0 . σ . For this paper we accept theanalysis given here.There was one epoch (2001 March 6) in which datawere also taken at 5 GHz. The peak flux density at theposition of RT 20010331 is − ± µ Jy. In summary,
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Fig. 3.—
Top:
The dirty beam for the RT 19860115 field.This sub-image shows the southwestern side of the beam.The peak of the beam is located in the upper right handcorner. The contours are 7.5, 12.5, 17.5, and further insteps of 5% up to 37.5% of the peak flux.
Bottom:
Dirtyimage toward RT 19860115. A cross marks the location ofthe transient. The contours are 0.15, 0.25, 0.35, 0.45, 0.55,0.65 and 0.75 mJy beam − . Note that the transient candidateRT 19860115 lies at the same angle and position as a side-lobefrom the bright source J150123+781806 located in the upperright corner of this image. we find no evidence to support that RT 20010331 is asignificant detection.3.7. Remaining Four Single Epoch Sources
RT 19920826
B07 report a transient with a primary beam-correctedflux density of 642 ± µ Jy, or a SNR=6.4. We con-firm a source at this position but with a slightly reducedSNR. Using natural weighting of the gridded visibilities,our measured peak flux density is 460 ± µ Jy, or 5.8 σ .Despite the lower significance, there is some confidencethat RT 19920826 is real since it appears in both IFs withcomparable flux densities. Further investigation by GBshows that the difference in the SNR between B07 andthe analysis here can be traced to differences in flaggingof the UV data (of two specific antennas).3.7.2. RT 19970528
B07 report a transient with a primary beam-correctedflux density of 1731 ± µ Jy, or a SNR=7.5. These ob-servations were taken during a time when some antennaswere being moved to the B configuration and so we ap-plied antenna position corrections (via AIPS task
VLANT )before calibration.RT 19970528 is 6.8 ′ from the center of the image,where the response of the antennas is only 16% of theirpeak. The uncorrected flux density is 270 ±
47. A pointsource search (AIPS task
SAD ) of the four million pix-els enclosed interior to a radius around this candidate,shows six other un-cataloged candidates with similarsignal-to-noise. The primary-beam corrected flux density Frail et al.1 . ± . . ± . SAD , JMFIT ) between 6.2, 7.4 and 7.1. (with-out any corrections discussed above). The correspondingSNRs for the image discussed above is 6.1, 5.1 and 6.8.The source is a bit “ratty” and this may explain the vari-ation in SNR. We therefore find this source to be a weakdetection.Additional observations were made during this epochat 8.5 GHz. The source lies close to the first null(6 . ± . ′ from the phase center) of the beam at thisfrequency. The attenuation by the primary beam is se-vere and hence the sensitivity is not sufficient to provideany strong spectral index constraints.3.7.3. RT 19990504
B07 report a transient with a primary beam-correctedflux density of 7042 ± µ Jy, or a SNR=7.3. Croft et al.(2011) report an X-ray source in the vicinity of the radiosource. Deep multi-wavelength data is consistent withthe X-ray source arising from a QSO but located fivearcseconds from the putative radio transient.We find the (uncorrected) peak flux density is 290 ± µ Jy (or an SNR of 5.7). RT 19990504 lies 8.9 ′ from thephase center, close to the first null (11 . ± . ′ from thephase center) where we would not expect to find sources.The polynomial expressions used in AIPS to correct forthe beam attenuation are increasingly inaccurate outsidethe 20% response radius, and they are not applicableclose to the null. The azimuthally averaged measuredvalue of the beam response is 1.8%, implying a flux den-sity correction factor of 55 × (Cotton and Perley 2010)or 16 . ± . SAD ) and 6.8 (
JMFIT )and a similar variation with the image reported here. Itis worth noting that the source may be extended andalso that a visual inspection of the annular region re-veal a number of similar sources. We conclude thatRT 19990504 is not a robust detection.Additional observations were made during this epochat 8.5 GHz. An image was made from these higher fre-quency data but, as expected, no source was found.3.7.4.
RT 19970205
This is the only single-epoch transient identifiedat 8.5 GHz by B07. They report a primary beam-corrected flux density of 2234 ± µ Jy, or a SNR=7.8.RT 19970205 is 4.4 ′ from the center of the image, wherethe response of the antennas is only 9.8% of their peak.There is an elongated source at this position with a peakflux density of 231 ± µ Jy (using natural weighting).There is some indication that the source is extended sincethe integrated flux density is about twice the peak fluxdensity.The peak flux density was likely underestimated byB07 since they did not fully correct for bandwidth ortemporal smearing effects. These data were taken in the BnA array with an integration time of 3 1/3 s and a50-MHz bandwidth. In this observing configuration, theeffects of temporal smearing with this dump time willlikely reduce the peak flux density by only a few percent.However, chromatic aberration is expected to be largerwith the effect of smearing a point source along the radialdirection (Perley et al. 1989).Accounting for the bandwidth effect, we measure apeak flux density of 3 . ± . ≈ Other Surveys
Ofek et al. (2011) presented a survey for 5 GHz radiotransients at low Galactic latitudes. Most of the datawere reduced in near-real time (2 hr delay) and transientcandidates were followed up with radio, visible light andX-ray instruments. The authors reported a single tran-sient candidate, J213622 .
04 + 415920 . . ± .
41 mJy (5.8 σ ) and with no obviousoptical counterpart.We have re-analyzed this data set and confirm thatthe SNR is 5.8. Considering that the search resultedfrom inspecting 1 . × independent beams ( § D) andassuming Gaussian statistics, we find that the probabilityof the highest event found in these many independentbeams is attributable to chance or noise is 3.6% (see § A).Therefore, we advocate that this candidate is not a realevent. THE REVISED B07 TRANSIENT RATEAs noted in § and the lack of quiescent op-tical counterparts. In § § σ be-low that reported in B07. The revised and B07 SNRs aregiven in Table 2. We note that an independent pipelinewritten in ParselTongue (Kettenis et al. 2006), a Pythoninterface to AIPS, and run on three of these four sourcesconfirms the lower SNR values found here (Bell 2011):RT 19920826 (SNR=6.0), RT 19970528 (SNR=4.4), andRT 19990504 (SNR=4.0).The lowering of SNR (from between 7 and 8 to between5 and 6) has a pernicious effect when the number of inde-pendent beams which were search is included. In § D weestimate this number to be n ≈ × . In § A we derivethe probability density function for the highest m valuesof n Gaussian random numbers . In Figure 4 we plot the Theory informs us that the statistics of beam values or equiv- adio transients 7density function for the highest value ( m = 1) and thefourth highest value ( m = 4). As can be seen from thisFigure 4, if the SNRs reported here are accepted then theglobal case for the remaining B07 transients is entirelyweakened. If, on the other hand, the SNRs reported inB07 are accepted then the four transients reported inB07 do argue for a new class of radio transients. Fig. 4.—
The probability density function of the highestvalue ( m = 1) and the fourth highest value ( m = 4) of apopulation of n = 9 × Gaussianly distributed randomnumbers with zero mean and unit variance. The probabil-ity density function plotted here is discussed in § A and thejustification for n can be found in § D. The dots representthe SNRs of the following single-epoch transients discussedin Table 2. (from left to right): RT 19905054, RT 19920826,RT 19970205 and RT 19970228. (Top): SNRs as reported inthis paper. (Bottom): SNRs as reported in B07. Note thatthe vertical location of the dots is arbitrary. For details ofthese density distributions see § A. The above approach of using a fixed threshold for allepochs does not result in optimal detection. In partic-ular, the threshold for a low resolution survey is lowerthan that for a higher resolution survey (since the lat-ter has a correspondingly larger number of synthesizedbeams). B07 addressed this problem by requiring thatthe probability of a false detection (PFD) in an individ-ual epoch was constant and less than N where N is thetotal number of images. With this approach, the expec-tation number of false detections is 1 for the entire survey.Applying the B07 method we find the following PFDs: alently pixel values of interferometric maps should follow Gaussiandistribution. RT 19920826 (log(PFD)= − . − . − . − . § (and in some ways or-thogonal to the above SNR based approach) is to lookat the angular distribution of the transient sources withrespect to the primary axis . Basic interferometry the-ory informs us that the dirty image is simply the Fouriertransform of the visibility data. As such the radiomet-ric noise in the dirty image should be independent ofthe angular offset from the phase center. In contrast,the point source sensitivity decreases as one goes awayfrom the pointing center and this is governed by the pri-mary beam response (assuming that the spectral resolu-tion of the survey is high enough that the delay beamis larger than the primary beam). Thus, once the min-imum SNR for detection is fixed, cosmic sources shouldbe concentrated towards the pointing direction whereasnoise spikes (masquerading as threshold point sources)should be uniformly distributed.In § B we derive the expected distribution of cos-mic sources as a function of the angular offset. InFigure 5 we plot the expected cumulative distribu-tion and also the angular offset of the four sourceswhich are not artifacts but whose SNR seems to be un-der dispute, namely RT 19920826 ( § § § § Fig. 5.—
The cumulative probability of finding a cosmicsource (but integrated from angular offset of infinity to zero)as a function of the angular offset with respect to the point-ing center. The angular offset is normalized in units of θ FWHM . The sources are represented by dots and are (fromleft to right): RT 19920826, RT 19970528, RT 19970205, andRT 19990504. The points are deliberately placed at cumu-lative probability of 1%. RT 19920826 is firmly within theregion where one naturally expects cosmic sources.
In summary, two different statistical tests, one basedon SNR and the other making use of the spatial signa- This test was recommended to us by J. Condon. We assume that all antennas are pointed in the same directionand this direction is both the pointing axis as well as the phasecenter.
Frail et al.ture provided by the primary beam, suggest that of theremaining four sources detected at threshold, only one,namely, RT 19920826 is a good detection. Thus a sim-ple interpretation of our re-analysis is that the rate ofB07 transients is considerably lower than that reportedby B07, perhaps an order of magnitude smaller.We acknowledge that the discrepancy between theanalysis presented in B07 and the analysis presented here(and re-investigated) is disturbing. In the previous sec-tion ( § Miriad ) and the specific methodused to compute SNRs. However, none of these expla-nations are satisfactory. We are continuing this investi-gation but at the present time we consider this topic tobeyond the scope of the paper.In contrast, the sources which we find as artifacts haveready explanations (see Table 2). One source is a resultof the file header containing a pointing direction of theprevious pointing. Another is due to a systematic associ-ated with local signals (RFI). These signals do not havethe natural fringe rate of cosmic sources and appear ascandidates close to the phase center. Two are side-lobesof a stronger sources. Suffice to say that such rare er-rors will be found if one inspects sufficiently large numberof beams! A REVISED LOOK AT THE TRANSIENT RADIOSKYIn Figure 6 we plot the areal densities of threeknown transients (SN1998bw-like, type-II RSN and SwiftJ1644+57-like) and that of two expected classes (NS-NSmergers and orphan afterglows of long duration GRBs)of transients. The areal density of these five classes isalso summarized in Table 1. We briefly discuss each ofthese five classes of transients below.The traditional extra-galactic radio transients are typeII radio supernovae. The rate that we present here isbased on the single radio SN detected in a blind radiosurvey by Gal-Yam et al. (2006) and it agrees with an in-dependent estimate by Lien et al. (2011). As illustratedby the example presented in Gal-Yam et al. (2006), themain advantage in the search for radio SNe in a blindsurvey is the unique view that it provides to the oth-erwise hidden population of heavily obscured SNe. Theobserved rate of other types of radio SNe (e.g. ordinarytype Ib/c SNe) are considerably lower than that of typeII supernovae and are not discussed further.Radio emission is expected from both classical longduration gamma-ray bursts (GRBs) as well as themore abundant but less luminous GRBs exemplified byGRB 980425 associated with the energetic SN Ic su-pernova, SN 1998bw (Galama et al. 1998; Kulkarni et al.1998). The radio emission is far brighter than that of or-dinary core collapse SN (II, Ib, Ic) and the increased vol-ume makes up for the intrinsically smaller birth rate. Therecent discovery of the energetic supernova SN2009bb(Soderberg et al. 2010) demonstrates that radio surveys Unfortunately, side-lobes are the exception to the expectationof Gaussian statistics for interferometric images. It is said that“the Central Limit theorem covers a large number of sins but notall sins.” can find such sources without resorting to high energy(gamma-ray) missions.Levinson et al. (2002) estimated the number of after-glows from classical GRBs and whose explosion axis isdirected away from us (“orphan” afterglows). The ex-pected rate depends strongly on the poorly constrained γ -ray beaming. On one hand this makes any rate predic-tion uncertain. On the other hand, even a non-detectionby the kind of survey that we discuss below will providean independent and unique constraint on the averageopening angle of long gamma-ray bursts, their true rateand total energy output (Rossi et al. 2008; Nakar et al.2002; Totani & Panaitescu 2002). The areal density inFigure 6 is derived using a typical opening angle of 10degrees.A surprising and apparently an important develop-ment in the field of radio transients took place just thisyear with the discovery of a radio transient associatedwith the nucleus of a modest size galaxy. The source,Swift J1644+57 was initially detected as a hard X-raytransient (Burrows et al. 2011). Subsequent follow upfound a bright, compact self-absorbed radio counterpart,localized at the center of a normal galaxy at z = 0 . -10 M ⊙ supermassive black holes inotherwise normal galaxies.The areal density in Figure 6 is calculated assumingan observed rate of 0.2 yr − Swift J1644+57-like eventsand a gamma-ray beaming factor of 10 (Zauderer et al.2011; Bloom et al. 2011). Nominally, Swift J1644+57-like sources appear to be the most frequent extra-galactictransients that will be found in radio transient searches.We acknowledge that the uncertainty of both the ob-served rate and the gamma-ray beaming is high and thetrue rate may be significantly different. Now we come to the most uncertain as well as poten-tially the most important extra-galactic radio transient– the merger of two neutron stars (or a black hole anda neutron star). It is generally accepted (or expected)that short hard bursts are on-axis explosions of thesemergers (Nakar 2007; Metzger & Berger 2011). As inlong duration GRBs, radio emission is expected by after-glow (on-axis or orphan). The rates are highly uncertainbecause there are very few observations of short hardGRBs. Thus there still continues to be a debate aboutthe geometry of these explosions (“jetted” or not). Next,while the expected radio emission is straightforward toestimate (subject to the usual parametric uncertaintiesof the energy fractions of relativistic electrons and mag-netic field) an additional uncertainty is the density of theambient gas (which is necessary for the production of theafterglow emission). Estimates based on theoretically predicted TDE ratesand luminosities (Giannios & Metzger 2011; Bower 2011;van Velzen et al. 2011) result in areal densities that vary bythree orders of magnitude. The rate that we predict here isconsistent with the upper range of these predictions. adio transients 9 −1 −4 −3 −2 −1 f ν [mJy] A r ea l D en s i t y ( > f ν ) [ deg − ] t y pe − II R S N S N w li k e N S − N S m e r ge r s S w i ft J + O r phan l ong − G R B Carilli+03 Croft+10deVries+04 Le v i n s on + / G a l − Y a m + B1B2 Frail+03 O G r ego r y & T a y l o r B ann i s t e r + Scott96
Fig. 6.—
Cumulative areal density of transients as a function of peak flux density for all major transient surveys. Most of thesurveys are upper limits and the allowed phase space is above and to the right of the L-shaped symbol. The three dark blue L’s(annotated as B2 for the 2-month transients from B07, B1 for the single-epoch transients from B07 and O for the lone transientreported in Ofek et al. 2011) are the upper limits derived as a result of the analysis presented here. These limits were derivedby assuming no detection (whence a Poisson upper limit of 3 at the 95% confidence level; see § C) and survey areas summarizedin Ofek et al. (2010).
Regardless of the uncertainty whether neutron starmergers are the sources of short GRBs or not, a substan-tial sub- and mildly relativistic outflow is expected to beejected during the merger. Nakar & Piran (2011) esti-mate radio emission from these outflows. The areal den-sity in Figure 6 is calculated based on their estimates ,assuming a NS-NS merger rate of 300 Gpc − yr − andthat any merger ejects 10 erg of a mildly relativisticoutflow. We note that Nakar & Piran (2011) suggestedthat RT 19870422 was the radio emission from the re-mains of a neutron star merger. However, as noted in § WAY FORWARD: NEW SURVEYSThere are sound reasons to continue the explorationof the dynamic radio sky. Radio searches are an idealway to discover core-collapse supernovae embedded in orbehind dusty regions. The discovery of SN 2009bb showsthat large radio searches can find urgently needed addi-tional examples of nearby low luminosity GRBs. Next,the many rewards of radio follow up observations of The rate density of such mergers is poorly constrained. Itranges between 10 to 10 Gpc − yr − for NS-NS mergers (Phinney1991; Narayan et al. 1991; Kalogera et al. 2004; Abadie et al.2010) Swift 1644+57 (accurate localization, energetics, beam-ing, outflow velocity) show the tremendous diagnosticpower of radio observations of this entirely new class ofextragalactic transients.As exciting as these developments are, the searchfor new classes of radio transients has involved sev-eral false starts. The euphoria that followed thediscovery of a highly dispersed (and therefore ar-gued to be of extragalactic origin) millisecond burst(Lorimer et al. 2007) was rapidly diminished by the dis-covery of many such bursts, presumably of terrestrialorigin (Burke-Spolaor et al. 2011); but see Keane et al.(2011). Similarly, a long-duration transient found byLevinson et al. (2002) and Gal-Yam et al. (2006) waslater traced to a glitch in the VLA on-line data takingsystem (Ofek et al. 2010), apparently affecting 0.29% ofall FIRST survey pointings (Thyagarajan et al. 2011).Finally, our re-analysis (see § −1.5 −1 −0.5 0 0.5 12526272829303132 RSN−IIRSN−Ib/c NS−NSSN1998bw TDEOrphan GRB log ( τ [yr]) l og ( L ν [ e r g c m − s − H z − ] ) D e t e c t ab ili t y d i s t an c e @ . m Jy [ M p c ] Fig. 7.—
Phase space diagram showing the predicted radio lu-minosity versus evolutionary time scales for several types of longduration radio transient populations. Transparent zones indicatesource populations which are typically optically thin, while greyzones indicate source populations that are expected to be opticallythick before maximum light, evolving to an optically thin phase atlater times. A similar optical version of this figure can be found inRau et al. (2009) and a more comprehensive figure which includesshort duration events is in SKA Memo 97. the implied areal rate of B07 transients would be compa-rable to the recently established class of Swift J1644+57transients. However, unlike any other long duration ra-dio transient (to wit, supernovae; active stars; tidal dis-ruption events; gamma-ray burst afterglows, beamed orotherwise) the B07 transients are remarkable for the ab-sence of a quiescent optical counterpart.The event RT 19920826 survives two independent tests.As such it useful to speculate on the origin of this tran-sient. The absence of an optical (B07) and near IR qui-escent source could mean one of two origins. The eventis extra-galactic in origin but the host galaxy is faintenough not to have been detected (as does happen for afew percent of long duration GRB host galaxies) or thatit is offset from a host galaxy (as in the case for a fewshort hard bursts; Berger 2009). Alternatively, the eventis a Galactic neutron star and we have to then appeal toan optically invisible Galactic (neutron star) population(Ofek et al. 2010).Clearly, a new survey which can net a dozen of suchsources (but brighter!) would help resolve the origin. Forthe Galactic hypothesis one expect no quiescent counter-part even at HST sensitivity whereas detectable galaxies,in the majority, are expected for the extra-galactic hy-pothesis.There are great gains in the discovery of new classesof radio transients but at the same time the path totrue success is littered with false starts or easy specu-lations. The way forward must incorporate the lessonslearnt from the false starts. We elaborate on this conclu-sion below.To start with we believe that the field of radio tran-sients (at least in the decimeter band) is sufficiently ma-ture that any new survey which just sets an upper limitrelative to the known population is of marginal value.Future surveys have to be sufficiently deep and coverlarge enough sky that success (i.e. detection of a few tomany transients) is assured. In our opinion, this means that a survey should be designed to find at least one ormore Swift J1644+57-like transient (Figure 6).Next, timely and multi-wavelength follow-up is es-sential. For example, the transients reported inGal-Yam et al. (2006), Gregory & Taylor (1986) andBannister et al. (2011a,b) have plausible origins as su-pernovae and Swift J1644+57-like sources (see Figure 6).However, the lack of timely follow-up or deep multi-wavelength followup of these events preclude us fromcoming to a definitive conclusion.Finally, the search should be restricted to sources witha high level of significance. This certainly means payingattention to the large number of beams searched. How-ever, at low thresholds and with large number of beams(cf §
3) it would be prudent to set thresholds beyond merestatistical considerations . A threshold of 9 or even 10 σ may be appropriate. Alternatively, an immediate verifi-cation of a transient by deeper observation or a confir-mation by observations at other wavelengths would allowdetection of transients closer to threshold.We start with a discussion of two recent developments.Bell et al. (2011) undertook an ambitious program sim-ilar in spirit to B07, namely the investigation of fieldssurrounding VLA calibrators. Sources with duration be-tween 4 and 40 d were searched for. The total integrationtime was 435 hr. No transient source in the GHz rangewith flux greater than 8 mJy was found. The authorsplace an upper limit to the areal density of 0.032 deg − .Assuming S − / scaling this areal density is 4.4 deg − and is well above the B07 rate.The FIRST survey imaged the sky in an hexagonalgrid, in which each position in the survey footprintwas observed on average 3–4 times (Becker et al. 1995).Thyagarajan et al. (2011) used this fact to constructlight curves of sources detected in individual FIRST sur-vey snapshots. They identified 1627 variable candidateswith variability exceeding 5 σ . This effort is probably thelargest variable and transient survey ever carried out.One disadvantage of such a survey for transient identifi-cation is that the co-added images are not much deeperthan a single epoch image. This make it hard to tell ifan apparent transient source is really a transient or justa variable source that exceeded the detection thresholdin one of the epochs.The limitations discussed above lead us to suggesta new EVLA survey specifically tailored to systemati-cally explore the radio sky. Following that we reviewa far more ambitious survey – the VAST key project onASKAP. For the discussion below we will adopt the ratessummarized in Table 1. The rates are specified to a fluxdensity of 0.3 mJy and are extrapolated to higher fluxdensities as N( > S ) ∝ S − / .6.1. EVLA Survey
A moderately ambitious survey with the EVLA can re-sult in great progress. This survey has two virtues. One,the EVLA offers excellent spatial resolution. Next, theEVLA is fully commissioned and is working to specifica-tions. Separately we caution that the discussion of statistics assumesthat the underlying statistics are Gaussian to a very high degreeof precision. As noted in the text, the sidelobes of strong sourcesadd an additional source of non-Gaussian noise. adio transients 11Specifically consider a 100 square degree survey under-taken in the 2–4 GHz band. An integration time of 85 sresults in a sensitivity of 0.3 mJy (10- σ ). A single epochcovering 100 square degrees would require 50 hours. Ascan be seen from Figure 7 (taken from Table 1) such asurvey would have to explore a variety of time scales toprobe the emerging classes of transient. Fifteen epochscould reasonably cover the range of a week to years.Noting the great importance of multi-wavelength imag-ing data, such a survey would sensibly focus on regions ofsky where considerable multi-wavelength data (includingradio) exists. One such region is, for example, the SDSSequatorial stripe (Hodge et al. 2011). Furthermore, thehigh instantaneous sensitivity of the EVLA makes rapidfollow up of newly transients possible.After the first four epochs the reference images will betwice as deep as the survey field. A single new epochwould then yield about ten Swift J1644+57-like sourcesand four supernovae. Ten such images may find a newexample of an SN 1998bw-like event, several clear exam-ples of orphan afterglows and have an excellent chanceof finding the first examples of neutron star mergers. Wenote that these different classes of objects have differentcharacteristics, both in duration (Figure 7) and also inhost magnitudes and location with respect to host galaxy.Therefore, it is possible to distinguish between differenttypes of objects.As note earlier, rapid verification of a transient (eitherby additional and deeper radio observations or observa-tions at other wavelengths) can reduce the requirementfor a high detection threshold. This would then requireclose rapid reduction – well within the reach of moderncomputers. 6.2. VAST (ASKAP)
The Variable & Slow Transient (VAST) is an approvedkey project of ASKAP . The VAST-Wide survey aimsto survey in the 1.2 GHz band about 10,000 square de-grees every day for 2 years. With 40-s integrations the expected single-epoch rms is 0.5 mJy (VAST Memo known transient sources, future synopticradio imaging surveys are expected to yield substantialnumbers of exotic transients. Such surveys will also pro-vide the definitive test for the B07 population.DAF thanks Jim Condon and Alicia Soderberg for im-portant discussions early on in this project. We thankSteve Croft for a most careful reading of the paper andJ. Condon for making several insightful suggestions.The National Radio Astronomy Observatory is a facil-ity of the National Science Foundation operated undercooperative agreement by Associated Universities, Inc.SRK thanks the Department of Astronomy, Universityof Wisconsin at Madison for their hospitality. EOO issupported by an Einstein fellowship and NASA grants.SRK’s research in part is supported by NASA and NSF.This research has made use of data from the University ofMichigan Radio Astronomy Observatory which has beensupported by the University of Michigan and by a seriesof grants from the National Science Foundation, mostrecently AST-0607523. This research has made use ofNASA’s Astrophysics Data System. REFERENCESAbadie, J., et al. 2010, Classical and Quantum Gravity, 27,173001Bannister, K. W., Murphy, T., Gaensler, B. M., Hunstead, R. W.,& Chatterjee, S. 2011, MNRAS, 412, 634Bannister, K. W., Murphy, T., Gaensler, B. M., Hunstead, R. W.,& Chatterjee, S. 2011, Erratum, July 2011Becker, R. H., White, R. L., & Helfand, D. J. 1995, ApJ, 450, 559Bell, M. E. 2011, “The Low Frequency Array and the Transientand Variable Radio Sky”, PhD thesis, Southampton University.Bell, M. E., Fender, R. P., Swinbank, J., et al. 2011, MNRAS,415, 2Berger, E. 2009, ApJ, 690, 231Berger, E., Kulkarni, S. R., Frail, D. A., & Soderberg, A. M.2003, ApJ, 599, 408Bloom, J. S., et al. 2009, arXiv:0902.1527Bloom, J. S., et al. 2011, Science, 333, 203Booth, R. S., de Blok, W. J. G., Jonas, J. L., & Fanaroff, B. 2009,arXiv:0910.2935Bower, G. C., Saul, D., Bloom, J. S., Bolatto, A., Filippenko,A. V., Foley, R. J., & Perley, D. 2007, ApJ, 666, 346Bower, G. C. 2011, ApJ, 732, L12Bower, G. C., Whysong, D., Blair, S., Croft, S., Keating, G., Law,C., Williams, P. K. G., & Wright, M. C. H. 2011,arXiv:1107.1517Burke-Spolaor, S., Bailes, M., Ekers, R., Macquart, J.-P., &Crawford, F., III 2011, ApJ, 727, 18 Hodge, J. A., Becker, R. H., White, R. L., Richards, G. T., &Zeimann, G. R. 2011, AJ, 142, 3Johnston, S., Taylor, R., Bailes, M., et al. 2008, ExperimentalAstronomy, 22, 151Jonas, J. L. 2009, IEEE Proceedings, 97, 1522Kalogera, V., et al. 2004, ApJ, 601, L179Keane, E. F., Kramer, M., Lyne, A. G., Stappers, B. W., &McLaughlin, M. A. 2011, MNRAS, 415, 3065Kettenis, M., van Langevelde, H. J., Reynolds, C., & Cotton, B.2006, Astronomical Data Analysis Software and Systems XV,351, 497K¨ording, E., Rupen, M., Knigge, C., Fender, R., Dhawan, V.,Templeton, M., & Muxlow, T. 2008, Science, 320, 1318Kulkarni, S. R., Frail, D. A., Wieringa, M. H., et al. 1998, Nature,395, 663Law, N. M., Kulkarni, S. R., Dekany, R. G., et al. 2009, PASP,121, 1395Lazio, J. 2008, Astronomische Nachrichten, 329, 330Levan, A. J., et al. 2011, Science, 333, 199Levinson, A., Ofek, E. O., Waxman, E., & Gal-Yam, A. 2002,ApJ, 576, 923Lien, A., Chakraborty, N., Fields, B. D., & Kemball, A. 2011,arXiv:1107.0775Lonsdale, C. J., Cappallo, R. J., Morales, M. F., et al. 2009,IEEE Proceedings, 97, 1497Lorimer, D. R., Bailes, M., McLaughlin, M. A., Narkevic, D. J.,& Crawford, F. 2007, Science, 318, 777Metzger, B. D., et al. 2010, MNRAS, 406, 2650Metzger, B. D., & Berger, E. 2011, arXiv:1108.6056Nakar, E. 2007, Phys. Rep., 442, 166Nakar, E., Piran, T., & Granot, J. 2002, ApJ, 579, 699Nakar, E., & Piran, T. 2011, Nature, 478, 82Narayan, R., Piran, T., & Shemi, A. 1991, ApJ, 379, L17Nissanke, S. M., Sievers, J. L., Dalal, N., & Holz, D. E. 2011,arXiv:1105.3184Sault, R. J., Teuben, P. J., & Wright, M. C. H. 1995,Astronomical Data Analysis Software and Systems IV, 77, 433 Thyagarajan, N., Helfand, D. J., White, R. L., & Becker, R. H.2011, arXiv:1107.5901Ofek, E. O., Breslauer, B., Gal-Yam, A., Frail, D., Kasliwal,M. M., Kulkarni, S. R., & Waxman, E. 2010, ApJ, 711, 517Ofek, E. O., & Frail, D. A. 2011, ApJ, 737, 45Ofek, E. O., Frail, D. A., Breslauer, B., Kulkarni, S. R., Chandra,P., Gal-Yam, A., Kasliwal, M. M., & Gehrels, N. 2011,arXiv:1103.3010Oosterloo, T., Verheijen, M. A. W., van Cappellen, W., et al.2009, Proceedings of Wide Field Astronomy Technology for theSquare Kilometre Array (SKADS 2009). 4-6 November2009. Chateau de Limelette, Belgium.Phinney, E. S. 1991, ApJ, 380, L17Perley, R. A., Schwab, F. R., & Bridle, A. H. 1989, SynthesisImaging in Radio Astronomy, 6,Rau, A., et al. 2009, PASP, 121, 1334Rossi, E. M., Perna, R., & Daigne, F. 2008, MNRAS, 390, 675R¨ottgering, H., de Bruyn, A. G., Fender, R. P., et al. 2003, Texasin Tuscany. XXI Symposium on Relativistic Astrophysics, 69Scott, W. K. 1996, PhD Thesis,
Compact Source Variability atCentimeter Wavelengths , University of British ColumbiaSoderberg, A. M., et al. 2006, Nature, 442, 1014Soderberg, A. M., et al. 2010, Nature, 463, 513Sylvestre, J. 2003, ApJ, 591, 1152Thyagarajan, N., Helfand, D. J., White, R. L., & Becker, R. H.2011, arXiv:1107.5901Totani, T., & Panaitescu, A. 2002, ApJ, 576, 120de Vries, W. H., Becker, R. H., White, R. L., & Helfand, D. J.2004, AJ, 127, 2565van Velzen, S., Koerding, E., & Falcke, H. 2011, arXiv:1104.4105Weiler, K. W., Panagia, N., Sramek, R. A., et al. 2010,Mem. Soc. Astron. Italiana, 81, 374Welch, J., Backer, D., Blitz, L., et al. 2009, IEEE Proceedings,97, 1438Zauderer, B. A., et al. 2011, arXiv:1106.3568de Zotti, G., Massardi, M., Negrello, M., & Wall, J. 2010,A&A Rev., 18, 1
APPENDIX
A. PROBABILITY DENSITY FUNCTION OF M TH MAXIMUM
In this section, our goal is to compute the probability density function of the m th highest value of h j , j = 1 , , ...n .Let p ( h ) be the probability density function of h j with P ( h ) = R h −∞ p ( h ) dh being the cumulative function. We denotethe m th highest value by H m . Thus the maximum of the series of measurements is H and the minimum value is H n .Let ρ ( H m ) be the probability density function of h = H m . This means that at least one of the measurements lies isin the range [ H m , H m + dH m ]. The probability density for this event is p ( H m ). Next, then n − m measurements mustlie below this range and m − H m is P ( H m ) andthe probability for a value to higher than H m is is 1 − P ( H m ). This is now a binomial distribution with n − H m is ρ ( H m ) = np ( H m ) × ( n − n − m )!( m − P ( H m ) n − m h − P ( H m ) i m − . (A1)The first combinatorial factor of n accounts for the possibility that H m can occupy any position in the sequence. Thesecond combinatorial factor, n − C m − account for the combinations satisfying the condition that n − m values liebelow H m and m − H m . For both the maximum ( m = 1) and minimum ( m = n ) Equation A1 simplifiesto that expected from basic considerations.Now let us consider the specific case where h follows Gaussian statistics: p ( h ) = 1 √ π exp( − h /
2) (A2)where h is normalized in units of σ . The probability that an event is extreme or lies within the range ± h is φ ( h ) = Z + h − h p ( h ) dh = 2 √ π Z h √ exp( − z ) dz = erf( h/ √ . (A3)The probability that an event is extreme in only one sense, maximum or minimum, and lies outside the range [ −∞ , h ](say) is thus P ( h ) = 12 h (cid:16) h √ (cid:17)i = 1 −
12 erfc (cid:16) h √ (cid:17) (A4)where erfc( x ) = 1 − erf( x ).adio transients 13Consider the case where n ≫ or more) and m is small, say 10. Then we can approximate n − ≈ n , n − ≈ n , ..., n − m + 1 ≈ n . Furthermore using the approximation, (1 − x/n ) n ≈ exp( − x ), we find in the limitingcase where H m is greater than a few [so that 1 − P ( H m ) ≪ ρ ( H m ) = n ( m − p ( H m ) exp h − ( n − m )2 erfc (cid:16) H m √ (cid:17)ih n (cid:16) H m √ (cid:17)i m − . (A5) B. DISTRIBUTION OF SOURCES WITHIN THE PRIMARY BEAM
Provided that there is sufficient spectral resolution, the response of an interferometer to a source follows the antennaresponse function (“primary beam”). We will assume that the response function is azimuthally symmetric andspecified by g ( θ ) where θ is the angular offset from the pointing axis.Let the areal density of sources with flux density greater than S be a power law, say, N ( > S ) ∝ S α . For Euclideangeometry and most reasonable luminosity functions, α = − /
2. Next, we note that detection is really finding sourcesat a given SNR and above. Fortunately the noise distribution for an interferometric image is uniform. Thus a sourcewith a given flux density will have an SNR, S , that scales with the primary beam response, S ∝ Sg ( θ ). The numberof sources above a certain SNR and contained outside an angular radius of θ is n ( > S ; > θ ) ∝ Z ∞ θ πθ (cid:16) S g ( θ ) (cid:17) α dθ (B1)For the specific case of a Gaussian beam, g ( θ ) ∝ exp h −
12 ( θ/θ ∗ ) i (B2)where the traditional “full width at half maximum” (FWHM) is θ FWHM = p ln(256) θ ∗ . Substituting Equation B2into Equation B1 we obtain n ( > S ; > θ ) ∝ (cid:16) S /g ( θ ) (cid:17) α (B3)Half the sources will be detected outside the radius θ h = p − ln(4) /α θ ∗ . For α = − / θ h = p ln(16) / θ ∗ ≈ . θ ∗ . The expression θ h = p / θ FWHM ≈ . θ FWHM is more useful. A plot of n ( > S ; > θ ) can be found in Figure 8. Fig. 8.—
The expected distribution of sources above a threshold SNR as a function of θ/θ
FWHM from the pointing aixs andassuming α = − /
2. The response curve is normalized by insisting that the integral of the curve (from θ = to θ = ∞ ) is unity.Fifty percent of the sources are within 0 . θ FWHM (marked by a pentagram) and 97% within 0 . θ FWHM (marked by a cross;at this radius the primary beam gain is only 0.1 relative to that on-axis).
C. UPPER LIMIT FOR A NON DETECTION (POISSON)
It is not unusual to find no source after undertaking a survey. We wish to determine an upper limit to the numberof sources that were being searched. The number detected is given by Poisson statistics. The probability of finding r sources is then given by p ( r ) = λ r r ! exp( − λ ) (C1) We assume that the phase center coincides with the pointingaxis of the primary antenna. λ is the Poisson parameter and equal to h r i .Our goal is to determine the maximum value of λ given a non-detection. As the value of λ is increased the probabilityof detection, by which we mean the probability of detecting one or more events, also increases. This probability is p (1) + p (2) + ... which we note is 1 − p (0). This probability can be set to the desired confidence level, P and thence P = 1 − p (0) = 1 − exp( − λ ) . (C2)The reader with a stronger physical bent may find the complement, p (0) = 1 − P , more appealing. Regardless, we find λ = [3 , . , .
9] at a confidence level of [95%, 99%, 99.9%].
D. NUMBER OF INDEPENDENT BEAMS
Here we compute n , the number of independent beams for the VLA data that went into the analysis of B07 andOfek et al. (2011). For B07, a circular region with a radius of two times the half-power radius was searched for eachepoch. The number of independent beams per epoch is the ratio of that area to the area of the synthesized beam in thatparticular epoch. For individual epochs, this value ranged from as small as 10 to as large as 10 . The total numbersof independent beams for the 5 and 8.4 GHz data are 9 . × and 4 . × , respectively. The smaller number ofindependent beams for 8.4 GHz are due to the tapering of the visibility data, increasing the typical synthesized beamsize at 8.4 GHz.For Ofek et al. (2011) the search was made for transients in 4.65 ′ radius circular region. The FWHM of the synthesizedbeam is ≈ n = 1 . ×7