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Document: April 4, 2003 - October 17, 2007
Typeset using LATEX style emulateapj v. 16/07/00
The Square Kilometer Array as a Radio Synoptic Survey Telescope:Widefield Surveys for Transients, Pulsars and ETI
Version 13.21October 17, 2007
Jim Cordes, Cornell University
ABSTRACT
This document explores some of the issues involved in searching for transients, pulsars, and ETIsignals with the assumption that steady sources are surveyed at the same time, including spectral linesand continuum from galaxies and AGNs. We present a survey metric that quantifies survey completenessand incorporates time-variable sources with arbitrary durations and event rates. We discuss the role ofpropagation effects in surveys. Some radio sources are intensely modulated, such as giant pulses from theCrab pulsar with structure as short as 0.4 ns, variations in active galactic nuclei on time scales of daysand longer, and GRB afterglows on day to month time scales. Apart from pulsars, we know about suchvariations from observations that target previously detected sources, as opposed to discovering them viatheir time variability in blind surveys. The transient or dynamic radio sky is rich enough to make itplausible that blind surveys will discover many sources of both known and unknown types. Schemes forconducting searches are outlined in the context of the Large-N-Small-D concept for the Square KilometerArray, though they are applicable to aperture array concepts with minimal changes. The particularscience addressed is for mid-range frequencies, e.g. 0.3 to 10 GHz. The Square Kilometer Array (SKA),properly designed, can provide the capabilities needed — high sensitivity, wide field of view, and flexibilityin processing the time and frequency domains — to serve as a radio synoptic survey telescope (RSST).An example synoptic cycle is presented that includes different scan rates for extragalactic and Galacticscience; it also includes options for non-survey projects and targets of opportunity triggered by othertelescopes. Further work is neeeded to confirm that surveys for transients, pulsars and ETI can be donesimultaneously with those for steady sources, such as galaxy HI surveys and continuum source surveys,including polarimetry to yield Faraday rotation measures. The example synoptic cycle underscores theneed for field-of-view expansion through use of multiple-pixel receivers in order to conduct a large-scalegalaxy survey in a reasonable amount of time. In addition, investigation is needed on whether follow-up monitoring observations, such as pulsar timing, can be encompassed in the survey mode or insteadrequire special pointings outside the survey paradigm. Finally, requirements for calibration and backendprocessing systems need to be identified.
1
Table of Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 Assumptions About the SKA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4 Search Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
5 Types of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
6 The Role of Multipath Propagation: Scintillations, Refractionand Broadening in Angle, Time, and Frequency . . . . . . . . . . . . . . 6
7 Pulsars and Magnetars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
7.1 General Pulsar and Magnetar Surveys . . . . . . . . . . . . . . . . . . . . . . . . . . 9
7.2 Galactic Center Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
7.3 Globular Cluster Surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
7.4 Extragalactic Pulsars and Magnetars . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
8 Transients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
8.1 Giant Pulse Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
8.2 Magnetar Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
8.3 Rotating Radio Transients (RRATs) . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
8.4 Extragalactic Fast Transients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
8.5 Maximal Giant Pulse Emission from Pulsars and Mergers . . . . . . . . . . . . . . . 17
8.6 Gamma-ray Bursts: Afterglows and Prompt Radio Emission . . . . . . . . . . . . . 18
8.7 Evaporating Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
9 SETI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
9.1 Asteroid/Comet Radars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
10 The Radio Synoptic Survey Telescope . . . . . . . . . . . . . . . . . . . . . 21
10.1 Figures of Merit for Raster-scan Surveys of Steady Sources . . . . . . . . . . . . . . 22
10.2 Figures of Merit for Raster-scan Surveys of TransientSources . . . . . . . . . . . . . 22
10.3 Completeness Coefficient of a Transient Survey . . . . . . . . . . . . . . . . . . . . . 23
10.4 Trading Field of View and Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . 23
10.5 The Synoptic Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
14 Requirements for Searching the Full Field of View . . . . . . . . . . 26
14.1 Sampling the Field of View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
16 Search Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
17 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Appendices
A General Survey Metrics (including transients) . . . . . . . . . . . . . . . . . . . . . 31
B Matched Filtering: Detection and Estimation . . . . . . . . . . . . . . . . . . . . . . 35
C Minimum Detectable Flux Density in Pulsar Surveys . . . . . . . . . . . . . . . . . 38
D Single-pulse vs. Periodic Pulse Detection for Highly Modulated Pulse Trains . . . . 39
E Scattering in the Galactic Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1. introduction
Science drivers for the SKA involve surveys on massive scales. These include surveys of steady sources,such as the billion-galaxy survey in HI to study dark energy and galaxy evolution and a Faraday rotationsurvey of continuum sources to study the evolution of cosmic magnetic fields. Time-variable sources are alsoa part of the key science case (Carilli and Rawlings 2004) through proposed surveys for pulsars and ETI.There is now growing recognition that the SKA should be geared for deliberate exploration of the unknown,including the time-variable radio sky. This follows from two facts. First, wide-field X-and-γ-ray telescopeshave been enormously successful discovery instruments for transient sources. Second, recent discoveriesat radio wavelengths have underscored what was already obvious: that the radio sky is equally rich intime-variable sources even though it has not yet been systematically surveyed.
In this article, we analyze the requirements for sampling radio transients taking into account the largerange of time scales (ns to years) and signal complexity in the time-frequency domain. We explicitlyconsider pulsars and ETI along with other known types of transients. We also speculate on particular newclasses of transients that may fill the overall phase space of amplitude, duration and frequency.
Implementation of the SKA for studies of time-variable sources necessarily must consider other proposedapplications — including the billion-galaxy survey and pointed observations of particular sources, targets ofopportunity, etc. — which place demands on a significant fraction of the available array time. Consequently,we consider implementation of the SKA as a synoptic survey telescope with which most or all of the proposedsurveys are done simultaneously. The advantages for survey throughput and community buy-in are obvious.However, such an implementation places great demands on processing and data-management requirements,some of which are discussed here.
2. motivations
While doing massive surveys for time-steady sources, the SKA can also do unprecedented deep surveysfor pulsars, transient radio sources, and signals from ETI. To do so places distinct constraints on theconfiguration and signal-processing capabilities of the SKA, which are discussed in this article.
Why search for transients? As Heraclitus might have said, “You don’t observe the same universetwice,” and in modern times we recognize the time domain as an important dimension in the overall phasespace of variables that characterizes the observable universe. Examples of transient radio signals are knownthat range from 0.4 ns and longer in time scale with apparent brightness temperatures from thermal to1042K. However, compared to the high-energy transient sky, we know next to nothing about the overallconstituency of the transient radio sky. A highlighted science area for the SKA is “Exploration of theUnknown” (Wilkinson et al. 2004), which includes the overall phase space opened up by the SKA andthe likely discovery of new classes of objects and phenomena. Another chapter in the SKA science book,“The Dynamic Radio Sky” (Cordes et al. 2004) discusses the anticipated payoff from an SKA design thatcombines widefield sampling of the sky with high sensitivity and flexibility in analyzing likely or hypotheticalevent signatures and time scales. One possibility is coherent emission from extra-solar planets (Lazio et al.2004) Finally, the SKA Memo “Discovery and Understanding with the Square Kilometer Array” includesa discussion of radio transients as a primary component of the SKA discovery space. Most known radiotransients have been found by making radio observations on targets selected from surveys made at otherwavelengths or energies, such as radio afterglows from gamma-ray burst (GRB) sources and the turn-onof periodic radio pulsations from a formerly quiescent magnetar, J1810-197 (Camilo et al. 2006). Recentexceptions include the discovery of the RRAT population (McLaughlin et al. 2006) and a transient sourcesin the Galactic-center (GC) direction (Hyman et al. 2006; Bower et al. 2006) found through archived VLAimaging observations of the GC. It is clear that blind surveys of the transient sky will yield new objectsand new classes of objects that will serve as laboratories for the basic physics of extreme states of matteras well as inventorying Nature’s proclivity for astrophysical complexity. Indeed, Nature is said to abhor avaccuum, and this appears to be true in parameter space as well as physical space.
Why search for more pulsars? Radio pulsars continue to provide unique opportunities for testingtheories of gravity, for detecting nano-Hz gravitational waves, and for probing states of matter otherwiseinaccessible to experimental science.
One of the five key projects identified for the SKA is the usage of pulsars for strong-field tests of gravity andgravitational wave detection (Kramer et al. 2004 and references therein). Such tests can provide answers toone of the questions posed in Connecting Quarks with the Cosmos: Eleven Science Questions for the NewCentury1: “Was Einstein right about gravity?” To do so requires timing of pulsars like the extraordinary1
National Academies Press, 2003, ISBN 0-309-07406-1
– 3 –
double-pulsar binary J0737-3039 (Lyne et al. 2004), which comprises a recycled pulsar with 23ms spinperiod and a canonical pulsar with 2.8s period in a 2.4-hr orbit. Additional such binaries remain to bediscovered, some with even smaller orbital periods, allowing correspondingly stronger tests of gravity.
A Galactic Census of Pulsars, including Magnetars: We envision a Galactic census of radio pulsarsthat aims to detect at least half of the active radio pulsars that are beamed at us. Taking beaming and theradio lifetimes of pulsars into account, the fiducial neutron star (NS) birth rate of 10−2 yr−1 implies ∼ 2×104
detectable pulsars in the Galaxy. The recent discovery of rotating radio transients (RRATs) by McLaughlinet al. (2006) through a reanalysis of the Parkes Multibeam pulsar survey suggests that the Galactic RRATpopulation is comparable in size to that of ordinary pulsars. The relationship between the two populationsis not clearly known, but the RRATs, though found through detection of single pulses, have been found,through re-observation, to be periodic sources with periods not unlike those of canonical pulsars. Theirdiscovery, however, illustrates the gain to be had in using algorithms that are less restrictive on signal type.Recently, two magnetars have been detected at radio wavelengths, with remarkably flat spectra extending toat least 100 GHz. Radio emission is episodic and bright, raising the possibility that the Galactic magnetarpopulation may be better sampled through radio rather than X-ray surveys. Furthermore, extragalacticmagnetars are detectable out to the Virgo cluster with the SKA.
The first reason for proposing a complete Galactic census is obvious: the larger the number of pulsardetections, the more likely it is to find rare objects that provide the greatest opportunities for use as physicallaboratories. These include binary pulsars as described above and also those with black hole companions;MSPs that can be used as detectors of cosmological gravitational waves; MSPs spinning faster than 1.5ms, possibly as fast as 0.5 ms, that probe the equation-of-state under extreme conditions; hypervelocitypulsars with translational speeds in excess of 103 km s−1, which constrain both core-collapse physics and thegravitational potential of the Milky Way; and objects with unusual spin properties, such as those showingdiscontinuities (“glitches”) and apparent precessional motions, including ‘free” precession in isolated pulsarsand binary pulsars showing geodetic precession. Free precession in neutron stars (which appears to occurin a few objects with long-precession periods) has eluded understanding because superfluid effects shouldrapidly damp precessional motion. Additional cases that occur under a range of conditions may help usunderstand the inner workings of NS.
The second reason for a full Galactic census is that the large number of pulsars can be used to delineate theadvanced stages of stellar evolution that lead to supernovae and compact objects. In particular, with a largesample we can determine the branching ratios for the formation of canonical pulsars and magnetars. Wecan also estimate the effective birth rates for MSPs and for those binary pulsars that are likely to coalesceon time scales short enough to be of interest as sources of periodic, chirped gravitational waves (e.g. Burgayet al. 2003).
The third reason is that a maximal pulsar sample can be used to probe and map the interstellar medium(ISM) at an unprecedented level of detail. Measurable propagation effects include dispersion, scattering,Faraday rotation, and HI absorption that provide, respectively, line-of-sight integrals of the free-electrondensity ne, of the fluctuating electron density, δne, of the product B‖ne, where B‖ is the LOS componentof the interstellar magnetic field, and of the neutral hydrogen density. The resulting dispersion measures(DM), scattering measures (SM), rotation measures (RM) and atomic hydrogen column densities (NHI)obtained for a large number of directions will enable us to construct a much more detailed map of theGalaxy’s gaseous and magnetic components, including their fluctuations. Radio data can be combined withmulti-wavelength data (e.g. Hα and NaI absorption measurements) for modeling the ISM.
Why search for ETI? Detection of a signal from another civilization in the Galaxy arguably wouldcomprise the greatest scientific discovery ever made. Many of the discovery issues are discussed in Tarter(2004) and references therein. A recent paper (Loeb & Zaldariagga 2007) points out that low-frequencyarrays now being built or planned (LOFAR, MWA/LFD, the low-frequency part of the SKA) will havesufficient sensitivity to detect Earth-equivalent leakage radiation to tens and hundreds of parsecs.
3. assumptions about the ska
We assume in this document that the SKA includes a core array that will allow wide-field surveys, consistentwith current nominal specifications. Antennas on long baselines from the core array are less easy to usebut may play a role in anti-coincidence spatial filters to filter out radio-frequency interference (RFI) andinstrumental interference.
Some of our calculations will be based on an architecture based on a large number of small-diameter dishantennas (LNSD) for specificity but our conclusions on surveys are not altered for sky looking, phased-arrayapproaches (aperture arrays).
4. search domains
Comprehensive censuses are needed to find the rare objects that we know exist within the known sourceclasses (e.g. pulsars in relativistic binaries) and that we suspect will be rare if they exist at all (e.g. radio-emitting ETI). A qualitative shift in how a survey is conducted takes place if one recognizes that there maybe only one target object in the entire sky. For large numbers, any subvolume of the sky is as good as anyother and it is generally true that covering more solid angle is better than integrating longer to fainter levelson a particular sky position. But with a singular, one-in-the-sky type object one must either choose targetdirections carefully or achieve high sensitivity on all sky positions.
Deep censuses for the three source classes suggest that the following regions on the sky need to be searchedwith high sensitivity and efficiency.
Pulsars:
1. Galactic plane: for young pulsars associated with supernova remnants, possiblemagnetar-like objects;
2. Intermediate latitudes: for millisecond pulsars and relativistic binaries (pulsars withother neutron star and black-hole [BH] companions);
3. Galactic center: pulsars in the star cluster orbiting the ∼ 4 × 106 M⊙ black hole, whichare difficult to detect because radio wave scattering from material in the GC regionbroadens the pulses at standard pulsar search frequencies;
4. Globular clusters;5. Nearby galaxies (<
∼ 1 Mpc)6. Galaxies out to ∼ 5 Mpc in searches for giant pulses like those typically seen from the
Crab pulsar in 1 hour to the Virgo cluster for plausible amplitudes of giant pulses.
Transient Sources:
1. Spatial domains similar to those for pulsars, with overlap for giant pulses from pulsarsand RRATs;
2. Local regions in the Galaxy (nearby planets and stars) require sampling of anapproximately isotropic distribution of sources;
3. Low frequencies for coherent sources;4. Fast transients: all sky5. Slow transients: all sky
SETI:
1. Targeted searches of nearby stars, especially those with suitable planetary systems;2. Blind surveys of the Galactic plane and other regions.
5. types of analysis
Signal detection inevitably boils down to matched filtering or an approximation thereof, as discussed inAppendix B. For the three broad source classes we consider, signal analysis necessarily considers thestructure of the signal in the frequency-time plane and processes that influence the detectability of thesignal.
Intrinsically, signals include the extremes of narrowband (spectral line) and narrow-time (pulselike) signals.Activities in the source may cause the signal’s frequency to drift in time or its emission time to be frequencydependent.
Extrinsic effects along the propagation path can also induce drifts of otherwise constant-frequency or narrow-time signals, such as source/observer acceleration or dispersive propagation through intervening plasma.Multipath propagation through the ISM also induce frequency-time structure via constructive and destruc-tive inteference. Faraday rotation produces quasi-periodic structure in the Stokes parameters Q and U .Radio frequency interference (RFI) is strongly episodic and frequency dependent. Finally, scattering ofradiation from telescope structure themselves can induce frequency-time structure.
Backend processing is specialized for the three broad source classes, though there is some overlap in thekinds of operations that are needed for detection and analysis of the sources.
Pulsars:
– 5 –
1. Dedispersion with trial values of dispersion measure (DM);2. Matched filtering detection of single pulses of unknown width and shape;3. Fourier analysis (including harmonic summing); and4. Statistical tests and confirmation observations of candidate signals.
Transient Sources:
1. Fast transients: analyses similar to those for single-pulse searches for pulsars, withperhaps a generalized matched filtering procedure operating in the frequency-timeplane.
2. Slow transients: Processing of images obtained at requisite intervals, perhaps withspectral resolution to identify fine structure.
SETI:
1. Generalized frequency-time analysis similar to that needed for dispersed, broadbandpulses from pulsars but with much smaller channel bandwidths and frequency driftrates (ν) consistent with anticipated Doppler accelerations.
6. the role of multipath propagation: scintillations, refraction andbroadening in angle, time, and frequency
The ionized ISM causes angle-of-arrival variations and multipath propagation of radio waves that influencesurveys in several ways, all of which are highly frequency dependent. These include slow, broadband, refrac-tive scintillations (RISS) and fast, narrowband diffractive scintillations (DISS) that can render undetectablestrong sources and conversely make weak sources occasionally detectable. Pulses are broadened in time bymultipath by amounts that are severe at low frequencies and for long paths through the Galactic plane.Short period pulsars are strongly selected against in these directions. Narrowband signals, such as thosesometimes targeted in SETI, can be broadened by up to a few Hz at a frequency of 1 GHz on the longestpaths. Broadening effects (in time or frequency) usually will conserve flux density, though special geometriescan induce violation of flux conservation. DISS produces 100% modulations (1σ) at frequencies below thetransition frequency between strong and weak scintillation (typically ∼ 5 GHz for 1 kpc distances and upto 100 GHz for the Galactic center) while RISS shows ∼ 20% modulations.
Propagation effects strongly influence the choice of frequency and methodology in conducting surveys ofvarious kinds.
At radio wavelengths, multipath propagation is caused by density fluctuations in intervening plasmas,including the ionosphere, the interplanetary medium, the interstellar medium and, though not yet revealedin any measurements, the intergalactic medium (IGM). Galactic components other than the ISM per se,such as supernova remnants, HII regions, the Galactic center, and the ionospheres and interplanetary mediaof other stars and their planetary systems will also contribute to propagation phenomena. Observable effectsfrom multipath propagation include:
1. Intensity variations on time scales ∆td of seconds to years and frequency scales ∆νd ≪ ν (diffractivescintillation) and ∆νr ∼ ν (refractive scintillation). Intensity variations from propagation effects inintervening ionized media (supernova remnants, Galactic center, interstellar medium, intergalacticmedium, interplanetary media, ionospheres) occur on a variety of time and frequency scales. Thesewill apply to all sources of fast transients because the sources necessarily must be compact innature. Slower transients will show fewer such effects because they are likely to be larger in size,thus quenching diffractive interstellar scintillations (DISS). DISS can play a key role in the designof surveys for compact sources. Under general conditions, the intrinsic intensity of a point source ismodulated by a “gain” g that has a one-sided exponential probability distribution, fg(g) = e−gU(g),where U(g) is the unit-step function. Multiple trial strategies can exploit the fact that g sometimesboosts the signal strength above a predetermined threshold even though most of the time g < 1(Cordes & Lazio 1991; Cordes, Lazio & Sagan 1997).
The dynamic spectrum in Figure 1 shows strong DISS from a nearby pulsar. Figure 2 shows howthe characteristic DISS time scales with dispersion measure over a set of pulsars.
2. Angular broadening (“seeing”) of point sources on angular scales ranging from sub-mas to arcsecondsor more.
Fig. 1.— Left: Dynamic spectrum of pulsar B1133+16 at 0.43 GHz, showing diffractive ISS with 100% modulations (i.e.rms / mean) of the pulsed flux. The characteristic time and frequency scales are very strong functions of frequency and ofthe particular scattering along the line of sight to the pulsar.
Fig. 2.— Right: The characteristic DISS timescale at 1 GHz plotted against DM for a sample of pulsars. For objectswith multiple observations, the vertical bars designate the ±1σ variation of the mean value. The solid and dashed linesshow predicted DISS timescales for transverse source velocities of 10, 100 and 103 km s−1 using results discussed in Cordes& McLaughlin (2004). The DISS timescale varies with frequency as ∆td ∝ ν1.2 if the scintillation bandwidth has theKolmogorov scaling, ∆νd ∝ ν4.4, as appears consistent for some objects. For extragalactic sources, the DISS time scale willdiffer because the geometry of the scattering medium consists of a foreground region from the Galaxy and another regioncorresponding to material in the host galaxy (if any). Nonetheless, the order of magnitiude value of the time scale can beestimated using the value of DM expected from just the foreground material in the Galaxy. The points are from Bhat et al.(1999); Bogdanov et al. (2002); Camilo & Nice (1995); Cordes (1986); Dewey et al. (1988); Foster et al. (1991); Fruchter etal. (1988); Gothoskar & Gupta (2000); Johnston et al. (1998); NiCastro et al. (2001); and Phillips & Clegg (1992).
3. Temporal (pulse) broadening caused by multipath propagation. An example is shown in Figure 3.The characteristic time scale τd has been measured to from sub-ms to ∼ 1 sec. This time scale isproportional to the inverse of the diffractive bandwidth, τd ∝ ∆νd
−1. Figure 4 shows the pulsebroadening time vs. dispersion measure (DM) for a large sample of pulsars.
4. Spectral broadening of narrow spectral lines caused by temporal changes in geometrical path length;this effect is relevant only to SETI surveys (Cordes & Lazio 1991) because the amplitude is of order
∆νsb ∼ (∆td)−1 <
∼ 1 Hz except for very heavily scattered lines of sight.
DISS has characteristic time and frequency scales that scale roughly as ν4 and ν, respectively, below thetransition frequency and are of order MHz and minutes for a typical pulsar at 1 kpc distance. Pulsebroadening and spectral broadening are reciprocals of the characteristic DISS bandwidth and timescale,respectively, and thus scale as ν−4 and ν−1.
DISS and RISS are both well known in pulsar observations, while only RISS has been convincingly detectedfrom AGNs, owing to the quenching effects of finite source sizes for DISS. Pulsars are sufficiently compactto show DISS and plausible scenarios for ETI transmission suggests that they too will show DISS, unlessthey are too close to the Earth (<
∼ 100 pc at 1 GHz) for the strong scintillation regime to apply.
7. pulsars and magnetars
Detection of pulses from pulsars can exploit the periodicity of their emission but must contend with anumber of effects that attenuate Fourier amplitudes:
1. Instrumental response times longer than a pulse width; the narrowest pulse width ∼ 40 µs.
2. Differential arrival times caused by plasma dispersion; these can be removed, though there willbe residual smearing in post-detection dedispersion techniques or from dedispersing with anapproximate value of DM; these effects scales ∝ ν−3.
3. Temporal broadening caused by multipath propagation in the interstellar medium; this effect scalesstrongly with frequency (∝ ν−4).
– 7 –
1.23 GHz
1.52 GHz
2.40 GHz
Fig. 3.— Pulse profile vs. frequency for a pulsar with moderate value of dispersion measure (DM; Bhat et al. 2004). Thespread of differential arrival times from multipath propagation — which causes the asymmetric tail on the pulses — is muchstronger at lower frequencies owing to the frequency dependence of the index of refraction of the warm ionized interstellarmedium.
Fig. 4.— Right: Pulse broadening time τd and scintillation bandwidth ∆νd plotted against DM (see Cordes & Lazio 2002for references). The open circles represent measurements of τd while filled circles designate measurements of ∆νd. Arrowsindicate upper limits. The two quantities are related by 2πτd∆νd = C1 with C1 = 1.16. All measurements have been scaledto 1 GHz from the original frequencies assuming τd ∝ ν−4.4. The solid curve is a least squares fit and the dashed linesrepresent ±1.5σ deviations from the fit. Surveys for pulsars that involve a Fourier search for periodicities or matched-filteringfor dispersed, single pulses are strongly affected by pulse broadening, which conserves the pulse area, and thus reduces thepulse amplitude and reduces the number of detectable harmonics in the power spectrum. Continuum (imaging) surveys are,of course, less affected by the angular broadening that accompanies (indeed causes) pulse broadening, because it does notbroaden sources significiantly compared to telescope resolutions except on lines of sight through the Galactic center regionand lines of sight through the Galactic plane at low frequencies. Nearby, low-DM pulsars are unaffected by pulse broadening,but do show intensity scintillations when the scintillation bandwidth, ∆νd, is comparable to the receiver bandwidth.
4. Orbital motion is important even for data spans much smaller than the orbital period.5. Intrinsic variations, such as pulse nulling or other strong modulations, including giant-pulse emission
(see below).6. RRAT sources.7. Eclipses of binary pulsars.8. Interstellar scintillations, which modulate the received pulsar flux density on a variety of time and
frequency scales.
Galactic pulsars are distributed according to their birth from an extreme Population I distribution of starsand modified by the wide pulsar velocity distribution. The luminosity function of pulsars is a broad powerlaw because it is determined by beaming combined with intrinsic variation that depends on spin rate andmagnetic field. For pulsars we can factor Smin as (See Appendix C):
Smin =Smin1
hΣ, (1)
Smin1=
mSsys
(NpolB T )1/2, (2)
where Smin1is the minimum detectable flux density for a single harmonic in the power spectrum, m is
the threshold in units of sigma (rms flux density), Ssys is the system-equivalent flux density (for the entirearray in the case of the SKA) and hΣ is the harmonic sum commonly used in pulsar surveys. We define theharmonic sum as a dimensionless quantity,
hP(Nh) = N−1/2h
Nh∑
j=1
∣
∣
∣fj
∣
∣
∣; (3)
hP is equal to unity if the signal is an undistorted sinusoid but can be much larger than unity when manyharmonics (Nh) are significant. For heavily pulse-broadened pulsars, hP is maximized with Nh = 1 (thefundamental frequency component) and the amplitude will be exponentially suppressed.
For a gaussian pulse with width W (FWHM), the harmonic sum is maximized for Nh,max ≈ P/2W harmonics
(for small duty cycles, W/P ≪ 1) and is approximately hΣmax ≈ 12 (P/W )
1/2. Using nominal parameters
for Arecibo and the SKA2 (Ssys = 3.0 and 0.14 Jy, respectively, B = 400 MHz, and T = 300 s), we findthat
Smin1=(m
10
)
×
61 µJy
gθAO
2.8 µJy
gθfcSKA,
(4)
where we have included a factor gθ ≤ 1 that accounts for off-axis gain and for the SKA a factor fc equal tothe fraction comprising the fraction of the collecting area in a core array that is usable for blind searching.We have used a threshold of 10σ (m = 10).
The maximum detectable distance is Dmax = (Lp/Smin)1/2
where Lp is the “pseudo-luminosity” (fluxdensity times the square of the distance, as often used in the pulsar community):
Dmax =(m
10
)−1/2(
Lp
1 kpc2mJy
)1/2
×
4.0 kpc (gθhΣmax)1/2 AO
18.9 kpc (fcgθhΣmax)1/2
SKA.
(5)
To evaluate Dmax, we consider particular directions through the Galaxy and use an electron density model(NE2001, Cordes & Lazio 2002, 2003) to evaluate pulse broadening and its effects on the harmonic structureof the Fourier analysis used to search for periodicities. Figure 6 shows Dmax calculated for Arecibo and theSKA using this approach. We also include curves for the Green Bank Telescope (GBT) and the Parkes radiotelescope using the 1.4 GHz multibeam system. The figure demonstrates that for a wide range of valuesfor periods and Lp, the SKA can probe to the anticipated boundaries of the pulsar distribution, even aftertaking into account the high velocities of some objects. The SKA will provide a great leap in surveyingthe neutron-star population of the Galaxy. Also, for objects with Lp
>∼ 103 mJy kpc2 (of which there are a
few examples in the known Galactic population), the SKA can reach — in standard periodicity searches —Dmax ∼ 1 Mpc that includes the nearby galaxies M31 and M33 in standard periodicity searches.
7.1. General Pulsar and Magnetar Surveys
An analysis of radio pulsar detection indicates that:
• With an Arecibo-size antenna, 400 MHz bandwidth at L band, and 300-s dwell times, some pulsarscan be detected out to the far edge of the Galaxy on the opposite side of the Galactic center.However, many objects cannot be detected either because of 1/r2 effects (luminosity limited) orbecause the pulses are severely broadening by scattering (scattering limited).
• Greater sensitivity than Arecibo allows detection of pulsars that are luminosity limited with Arecibo.
• Surveys at frequencies higher than 1.4 GHz enable detection of objects that are scattering limitedat L band because the pulse-broadening time τd ∝ ν−4.
• Migration of surveys to higher frequencies must be traded against the generally lower flux densities(pulsar flux densities S ∝ ν−α with typical spectral indices −0.1 <
∼ α <∼ 3 at L band) and smaller
FOV and array beam sizes.2
For Arecibo these numbers are for the Arecibo L-band Feed Array (ALFA) system, with approximately 30 K systemtemperature and 10 K Jy−1 system gain. For the SKA, the value for Ssys follows from the nominal requirement ofAe/Tsys = 2 × 104 m2 K−1. Recent work (2007) suggests that Ae/Tsys will be reduced from this nominal value by afactor of 2 to 3 for the SKA.
– 9 –
Fig. 5.— Dmax vs. single-harmonic threshold for 7 pulse periods. At large values of Smin1, the slope is -1/2 because Dmax
is luminosity limited. For smaller values of Smin1and especially for short period pulsars, the curve is less steep because Dmax
is scattering limited owing to pulse broadening in the interstellar medium.
0.1
1
10
100
0.001 0.01 0.1 1 10
0.1
1
10
100
0.001 0.01 0.1 1 10
0.1
1
10
100
0.001 0.01 0.1 1 10
0.1
1
10
100
0.001 0.01 0.1 1 10
Fig. 6.— Dmax vs. spin period for pulsar detection for two directions through the Galaxy: (left): ℓ, b = 30, 0. (right):ℓ, b = 50, 5. In each case, the four frames correspond to different values of the ‘pseudo-luminosity’ Lp, which is the period-averaged flux density ×D2 (for a known pulsar, say). The distribution of Lp for pulsars is broad, covering several ordersof magnitude, because the emission is beamed. The colored, shaded regions shown from top to bottom in each frame arefor Arecibo, the GBT, Parkes, and the SKA. Top boundary of each shaded region: full sensitivity. Lower boundary of eachshaded region: partial sensitivities. For the SKA the partial sensitivity represents fcgθ = 0.25 (e.g. 25% of the collecting areain the core array at full on-axis gain (gθ = 1, or larger fraction combined with an off-axis gain factor gθ < 1. For Arecibo,the lower boundary is given by gθ = 0.5, i.e. sensitivity at the half-power point. Other survey parameters include ν = 1.4GHz, B = 400 MHz, time resolution = 64 µs, integration time = 300 s, and a 10σ threshold. An intrinsic pulse duty cycleof 0.05 is assumed. Propagation effects, which smear the pulse, are calculated using the electron density model NE2001. Fordistances >
∼ 5 kpc in directions through the Galactic plane (b = 0), Dmax is strongly influenced by pulse broadening fromscattering. The decrease in Dmax for smaller P occurs because pulse smearing has a greater effect on the assumed narrowerpulses of short-period pulsars.
• Magnetar radio emission has a remarkably flat spectrum extending to > 100 GHz (Camilo et al.2006). Though episodic, when “on” the emission has fairly steady pulse amplitudes that are verybright. From Figure 6 (bottom left panel), pulsars with Lp ∼ 10 mJy kpc2 are detectable to 0.1Mpc. Magnetars as bright as the two currently known radio emitters are therefore detectable acrossthe Galaxy. Radio surveys for magnetars are probably the most efficient method for a full magnetarcensus of the Galaxy.
These facts suggest that pulsar surveys with the SKA can be undertaken in a number of configurationmodes:
1. Full FOV surveys with full SKA/core sensitivity;
2. Full FOV surveys with partial SKA/core sensitivity (as a subarray);
3. Multiple FOV surveys with partial SKA/core sensitivity using multiple subarrays; each subarraywould survey one FOV’s worth of sky.
Strawman survey parameters: Assume an L band survey with 300-s dwell time covering the full FOV.We consider the FOV specification of 1 deg2 for the SKA that matches the actual FWHM of the 12-mdishes of the LNSD concept. If all Galactic longitudes and latitudes |b| ≤ 10 are surveyed3, the 7200 deg2
would require 25 days to complete. Higher-latitude surveys are also of interest to find millisecond pulsars,relativistic binary pulsars, and high-space-velocity pulsars. Surveying the entire sky (41,253 deg2) wouldrequire 143 days.
7.2. Galactic Center Survey
Pulsars in the Galactic center (GC) provide especially important opportunities for probing the magnetoionicmedium, the gravitational potential in the GC region, and the central black hole (Sgr A*) (e.g. Cordes &Lazio 1997; Pfahl & Loeb 2004). The GC in this context refers to the region ∼ 200 pc in diameter centeredon Sgr A* with approximately a 1 deg2 solid angle. Scattering is especially severe for pulsars in this region,as summarized in Appendix E.
Table 1
Sensitivity Calculations for Galactic Center Pulsar Periodicity SearchesNote: all calculations assume a 6 hour integration
Telescope Frequency BW SEFD Smin1 Smin1D2gc Smin1D
2gc 1.4 GHz
(GHz) (GHz) (Jy) (µJy) (mJy kpc2) (mJy kpc2)
GBT 9 2 20.0 21.5 1.6 64EVLA 9 2 12.4 13.3 0.96 40
12 2 12.4 13.3 0.96 7115 4 12.4 9.4 0.68 7820 4 12.4 9.4 0.68 139
300m South 9 2 3.0 3.2 0.23 9.512 2 3.0 3.2 0.23 1715 4 3.0 2.3 0.17 19
SKA 9 2 0.14 0.15 0.011 0.4512 2 0.14 0.15 0.011 0.8015 4 0.14 0.11 0.0077 0.8820 4 0.14 0.11 0.0077 1.6
Figure 7 shows Dmax vs P at two frequencies. At 0.4 GHz, scattering in the inner Galaxy and especiallyfrom the scattering screen in the GC itself causes Dmax to “saturate” at about 8.4 kpc because pulses3
This would imply that there are both Northern and Southern hemisphere core arrays!
– 11 –
0.1
1
10
100
0.001 0.01 0.1 1 10
0.1
1
10
100
0.001 0.01 0.1 1 10
0.1
1
10
100
0.001 0.01 0.1 1 10
0.1
1
10
100
0.001 0.01 0.1 1 10
Fig. 7.— Dmax vs. spin period for pulsar detection looking toward the Galactic center (ℓ, b = 0, 0) at two frequencies(left): 0.4 GHz; (right): 9 GHz. The plot format is the same as in Figure 6.
originating from beyond the GC scattering screen at this distance are temporally broadened by hundredsof seconds or more. Because such broadening ∝ ν−4, the situation at 9 GHz (right panel of Figure 7) ismuch different. Low-luminosity pulsars (e.g. Lp
<∼ 1 mJy kpc2) still cannot be seen into the GC region but
more luminous pulsars “break through” the barrier distance, as indicated for the Lp = 10 mJy kpc2 case.
Figure 8 summarizes the situation for the known population of pulsars. The points are P and Lp = S1400D2
for known pulsars from the latest ATNF pulsar catalog4, where we use the period-averaged flux density at1.4 GHz. The plotted curves show detectability of pulsars in periodicity searches at 9 GHz for pulsars atthe location of Sgr A* for the GBT and SKA along with a curve for the SKA at 15 GHz. Pulsars need to beabove the curves to be detected. The curves are calculated at the indicated frequencies and then referredback to 1.4 GHz assuming a ν−2 spectrum. From the plot, we make the following notes:
• Only a small fraction of known pulsars is detectable using the GBT at 9 GHz, whereas most of thenon-recycled pulsars are detectable with the SKA at 9 GHz.
• Searches at 15 GHz do not increase the fraction of detectable long-period pulsars but do allowdetection of MSPs.
• Pulsars displaced from Sgr A* along the LOS are easier to detect if they are closer to the Sun, whilethose further away will be harder to detect owing to the scaling of pulse broadening with offset ofthe pulsar from the scattering screen (Eq. E1).
Overall, our calculations suggest that the SKA can be a powerful probe of the pulsar population in the GCif it can be used to conduct surveys at 9 GHz or higher.
7.3. Globular Cluster Surveys
Globular clusters (GCs) are prolific factories for recycled pulsars (e.g. Ransom et al. 2005) and may providethe best opportunities — apart from the Galactic center (another “GC”) — for finding pulsars orbiting ablack hole. GCs require high sensitivity but do not place any demands on FoV. Current programs with theGBT and Arecibo reach many GCs, but total radio flux density measurements from imaging observationsand pulsar counts suggest that many more faint pulsars remain to be discovered. The more distant GCsalso require high sensitivity capabilities, particularly in the Southern hemisphere.4
http://www.atnf.csiro.au/research/pulsar/psrcat
Fig. 8.— Detectability of pulsars in the Galactic center is shown in this plot of “pseudo luminosity” Lp = S1400D2 vs.spin period, where S1400 is the period-averaged flux density at 1.4 GHz. Duty cycles for pulsars range from 0.01 to 0.7, sothe peak flux density can be a large multiple [i.e. 1 / (duty cycle)] of the period-averaged flux density. Plotted curves assumea 6-hr integration time with a threshold of 10σ for bandwidths specified in Table 1. The long dashed curves are for 9 GHz.Solid curves are for 15 GHz and the short dashed curve for the SKA is for 20 GHz. Plotted points use periods and valuesof Lp from the ATNF/Jodrell pulsar catalog. Scattering is assumed to be in a filled region centered on Sgr A* with 1/ecylindrical radius of 0.15 kpc, as included in the NE2001 model (Cordes & Lazio 2002). For pulsars at the location of Sgr
A*, the dispersion and scattering measures are DMGC = 1577 pc cm−3 and SMGC = 107 kpc m−20/3. The curves are basedon harmonic sums of the Fourier power spectra of dedispersed time series. They correspond to surveys at these frequenciesbut have all been scaled to the equivalent sensitivity at 1.4 GHz assuming that the pulsar flux densities scale as ∝ ν−2.Less-steep spectra will move the curves downward. Incoherent summing of power spectra obtained from multiple 6-hr datasets will also move the curves downward. The SKA curves assume that the full, nominal sensitivity of the SKA is available(Ssys = 0.14 Jy.
7.4. Extragalactic Pulsars and Magnetars
Extragalactic pulsars and magnetars are reachable with the SKA. Figure 6 (bottom-right panel) shows thatpulsars with fairly modest Lp can be detected to ∼ 100 kpc. Milky Way pulsars extend to Lp as large as104 Jy kpc2 at 1.4 GHz, implying Dmax ∼ 10 Mpc for most spin periods >
∼ 10 ms. Pulse broadening fromscattering within the host galaxies will inhibit detection of short period pulsars in some cases.
Sporadic magnetar emission (see below under transients) has very flat spectra in two cases (e.g. Camilo etal. 2006) with Lp ∼ 103.5 mJy kpc2 implying detectability to about 1.7 Mpc in a 300 s observation.
Dmax values plotted in Figure 6 are shown as a band with full-SKA, on-axis gain at the top of the band, andfor 25% of the full gain the band bottom, either because observations are off axis or because only a subsetof the SKA is useable for blind surveys. The numbers quoted above are for approximately the midpoint ofthe band and of course there are other uncertainties to be considered.
8. transients
We know little about the transient radio sky. Giant pulses from radio pulsars are prototypes for fasttransients along with solar bursts and flare stars, while sources such as microquasars and gamma-ray burst(GRB) afterglows exemplify longer-duration transients. We may use these sources as initial guides forspecifying blind-survey parameters. However, simple observational phase space arguments suggest thatinstantaneous coverage of a large fraction of the sky with appropriate sampling of the frequency-time planewill yield a rich variety of transient sources, including new classes of objects. Figure 9 shows the phase spaceof pulse width W against flux density. Lines of constant brightness temperature are calculated assumingthat W is the light-travel time across the source,
Tb =S
2k
(
D
νW
)2
= 1020.5 K SmJy
(
Dkpc
νGHzWms
)2
(6)
– 13 –
where SmJy is the peak flux density (mJy) at frequency ν (GHz) and D is the distance (kpc). For somesources W can be much smaller than the light travel time owing to relativistic compression and of courseother sources can vary much more slowly.
Type II
Type III
Jup DAM BD LP944-20
B0540-69
IDV ISS
GRBISS
Type II
Type III
Jup DAM BD LP944-20
B0540-69
IDV ISS
GRBISS
Fig. 9.— Time-luminosity phase space for radio transients. Left panel: frame showing already-known radio transients.Right panel: frame that shows hypothetical transients, including maximal giant-pulse emission from pulsars, prompt radioemission from GRBs, and radar signals used to track potentially impacting asteroids and comets (see text). Each frame showsa log-log plot of the product of peak flux Spk in Jy and the square of the distance D in kpc vs. the product of frequency ν
in GHz and pulse width W in s. The “uncertainty” limit on the left indicates that νW >∼ 1 as follows from the uncertainty
principle. Lines of constant brightness temperature T = SD2/2k(νW )2 are shown, where k is Boltzmann’s constant. Pointsare shown for the ‘nano-giant’ pulses detected from the Crab (Hankins & Eilek 2007), giant pulses detected from the Crabpulsar and a few millisecond pulsars, and single pulses from other pulsars. Points are shown for Jovian and solar bursts,flares from stars, a brown dwarf, OH masers, and AGNs. The regions labeled ‘coherent’ and ‘incoherent’ are separated by thecanonical 1012K limit from the inverse Compton effect that is relevant to incoherent synchrotron sources. Arrows pointingto the right for the GRB and intra-day variable (IDV) points indicate that interstellar scintillation (ISS) implies smallerbrightness temperatures than if characteristic variation times are used to estimate the brightness temperature. The growingnumber of recent discoveries of transients illustrates the fact that empty regions of the νW −SpkD2 plane may be populatedwith sources not yet discovered, such as those shown in the right panel. These recent discoveries include the “rotatingradio transients” (RRATs; McLaughlin et al. 2006), the Galactic center transient source, GCRT J1745-3009 (Hyman et al.2006), the bright, bursting magnetar XTE J1810-197 (Camilo et al. 2006), and recently discovered transients (labelled “RT”)associated with distant galaxies (Bower et al. 2007). The rightward-directed triangles used to mark the two RT sourcesindicate that the transient durations are only known to be longer than ∼ 20 min. A lone pulse from the source J0118−75(Lorimer et al. 2007) has a DM that is too large (for the source direction) to be accounted for by the Milky Way or by theSmall Magellanic Cloud, a few degrees away. In the figure the nominal distance of 500 Mpc advocated by Lorimer et al. isused, with an upward directed arrow signifying that it could be further. Equally arguable, however, is that the source isor is seen through a relatively nearby galaxy or that it is in a binary system in the Milky Way’s halo, with the large DMcontributed by a companion star. Dashed lines indicate the detection threshold for the full SKA for sources at distance of10 kpc and 3 Gpc. Dotted lines correspond to a 10% SKA, comparable to the Arecibo telescope. At a given νW , a sourcemust have luminosity above the line to be detectable. The curves assume optimal detection (matched filtering).
Slow transients are defined as those with time scales longer than the time it takes to image the relevantregion of the sky, either in a single pointing or as a mosaic or raster scan. Detection of such objects canbe accomplished simply through repeated mapping of the sky and thus does not require special capabilitiesbeyond those needed for imaging applications. GRBs are currently detected at >
∼ 100 µJy levels using theVLA at frequencies of 5 and 8 GHz. The full SKA could detect GRB afterglows to at least 100 times fainterlevels.
Fast transients, by contrast, require the same observing modes and post-processing as for pulsars. TheCrab pulsar is the most extreme known case in terms of showing temporal structure down to ∼ 0.4 ns scales(Hankins et al. 2007) and giant pulses that exceed 130 times the entire flux density of the Crab Nebula.Figure 10 shows an example of such a pulse, including the dispersion sweep in the frequency-time plane.Pulsars and giant-pulse emission may represent prototypes for coherent radiation from other high-energy
objects in which collimated particle flows can drive the necessary plasma instabilities. Examples includeprompt radio burst emission from GRB-type sources, perhaps even from gamma-ray quiet objects; flarestars, jovian-burst like radiation from planets, AGNs, and merging NS at cosmological distances (Hansen& Lyutikov 2001).
Using GRBs as a guide, it may be noted that the rate of GRB detection with gamma-ray instruments hasrelied more on instantaneous wide-field sky coverage than on sensitivity. The same statement holds for thedetection of prompt optical emission at mv = 9 in at least one case (Akerlof et al. 1999; Bloch et al. 2000).For this reason, we suggest that use of subarray modes to increase the instantaneous sky coverage — at lessthan full sensitivity — will be productive in surveying the transient radio sky.
The signal-to-noise ratio for a pulse after dedispersion and matched filtering is
S
N=
S(NpolBW )1/2
Ssys, (7)
where S is the peak flux density and W is the pulse width (FWHM). Requiring S/N > m = 10 andusing Npol = 2 polarizations using nominal parameters for Arecibo and the SKA (see footnote 2), we get aminimum detectable flux density
Smin,SP =(m
10
)
(
1 ms
W
)1/2(1 GHz
B
)1/2
×
21 mJy
gθAO
0.98 mJy
gθfcSKA,
(8)
corresponding to a minimum brightness temperature,
Tmin,SP =(m
10
)
(
1 ms
W
)5/2 (1 GHz
B
)1/2 (Dkpc
νGHz
)2
×
1021.8K
gθAO
1020.5K
gθfcSKA.
(9)
A burst of amplitude Spk from a known source at distance D = Dkpc kpc can be detected to a maximumdistance
Dmax = D
(
Spk
Smin,SP
)1/2
= Dkpc
(
Spk
1 Jy
)1/2(10
m
)1/2(W
1 ms
)1/4(B
1 GHz
)1/4
×
6.9 kpc g1/2θ AO
32 kpc (gθfc)1/2 SKA.
(10)
The SKA can see about 1.5 orders of magnitude fainter than Arecibo. However, the much greater advantageof the SKA over Arecibo will consist of the sky coverage and greater resilience against RFI. With suchcoverage we can expect the SKA to yield new discoveries over much of the phase space depicted in Figure 9.
8.1. Giant Pulse Detection
The Crab pulsar’s giant pulses (GPs) are currently the brightest known, with examples at 0.4 GHz ofSpk ≈ 150 kJy occurring at a rate of once per hour. At this frequency, the pulse width W ≈ 100 µs, butis intrinsically much narrower because scattering has broadened the pulse. At high frequencies, structureinternal to GPs is as short as 0.4 ns (Hankins & Eilek 2007) with amplitudes as high as 2.2 MJy at 9 GHz.With these numbers, Eq. 10 implies that the 0.4 GHz pulses can be detected with a 50 MHz bandwidth to0.7 Mpc for nominal values of other parameters at Arecibo and to 3.3 Mpc for the full SKA. The 0.4 nspulse is detectable to 0.3 Mpc for Arecibo and 1.4 Mpc for the full SKA. Undoubtedly, there are larger —but less frequent — pulses from the Crab pulsar (Lundgren et al. 1995) and there are GP emitters that arebrighter than the Crab pulsar (see below in the transients section). It is therefore clear that GPs can beseen to nearby galaxies and, with reasonable extrapolation to brighter sources, the large number of potentialGP emitters in the Virgo cluster (D ∼ 20 kpc) can be reached.
– 15 –
Fig. 10.— Plot of intensity against time and frequency, showing a single dispersed pulse as it arrives at different frequenciescentered on 0.43 GHz (Cordes et al. 2004). The right-hand panel shows the pulse amplitude vs. frequency while the bottompanel shows the pulse shape with and without compensating for dispersion delays. This pulse is the largest in one hour ofdata, has S/N ∼ 1.1 × 104, and a pulse peak that is 130 times the flux density of the Crab Nebula, or ∼ 155 kJy. Note thatthe segments at either end of the bandpass where the pulse arrival time is opposite the trend at most frequencies is causedby aliasing of the signal.
8.2. Magnetar Detection
The magnetar J1810-197 has been detected with peak single-pulse amplitudes ∼ 10 Jy and pulse widths∼ 0.15 s. Given its distance of 3.3 kpc (Camilo et al. 2006), single pulses from magnetars like J1810-
197 are detectable to Dmax = D (Spk/Smin,SP)1/2 ≈ 0.2 Mpc for Arecibo and 0.94 Mpc for the full SKA
using Equation 8. These distances are on-axis values, so a blind survey will reach less far by 2−1/2 at thehalf-power point of the beam. However, the assumed 0.4 GHz bandwidth can easily be increased to 1 to2 GHz because magnetar emission has a flat radio spectrum, allowing high-frequency (5 to 8 GHz, say)observations where the sky background is minimized. This increases Dmax by a factor of 51/4 = 1.5, or0.3 Mpc for Arecibo and 1.4 Mpc for the full SKA.
Magnetar emission is also periodic. With P = 5.5 s for J1810-197, a T = 10 hr synchronous average will
increase Dmax by a factor N1/4p = (T/P )1/4 = 9 to 2.7 Mpc for Arecibo and 12.7 Mpc for the SKA.
As with giant pulses, it is very practical to search for magnetars in nearby galaxies with Arecibo and tonearly the Virgo cluster with the full SKA. Given that only two magnetars have been detected in radioemission, it is highly likely that brighter magnetars exist. A tenfold increase in peak flux density yields afactor of three increase in Dmax. If the full SKA cannot be used for blind searching, this greater brightnesscan counteract the lower gain, thus yielding approximately the same distance estimate.
8.3. Rotating Radio Transients (RRATs)
The recent discovery of RRATs (McLaughlin et al. 2006) is a dramatic illustration of how flexible processingof the frequency-time plane and significant sky coverage can reveal new sources. It is not yet clear whetherthe RRATs represent merely a new empirical class of radio pulsar or whether they are a new physical classof neutron star, as with magnetars. When investigated closely, most of the RRATs appear to be very similarto the much more steady pulsar population.
A census of RRATs requires long dwell times on a good fraction of the the Milky Way, both in and outsidethe disk. Surveys with Parkes and Arecibo using multibeam systems are revealing new RRATs, but witha high degree of incompleteness. It will take a survey with large product of ΩsT (solid angle surveyed andobservation time per sampled direction) to provide a sufficient sample, requiring a large-FoV system.
Interstellar scattering will limit the detectability of single-pulse detections. Figure 11 shows pulse broadeningfrom scattering as a function of distance for 0.4 GHz observations (left panel). Broadening conserves pulsearea, so peak detection is clearly reduced for distant single-pulse emitters. In the right-hand panel, themaximum detectable distance vs. peak pseudo-luminosity (SpkD
2) shows that the free-space predictionfor maximum detectable distance breaks down owing to scattering. The panels in Figure 12 shows similarcurves for 1.4 GHz.
8.4. Extragalactic Fast Transients
Lorimer et al. (2007) have reported detection of a 30 Jy pulse at 1.4 GHz that has DM ≈ 375 pc cm−3. Itwas seen in a direction near but well outside the contours of the Small Magellanic Cloud (SMC). There areinsufficient electrons in diffuse regions in the Milky Way and SMC to account for the DM. This may requirea cosmological distance of 500 Mpc or greater if the DM is accounted for by the diffuse IGM. However, thesource could reside in or be seen through a much nearer galaxy, implying a more modest luminosity. Thesource might also be in a binary system with where there is dense plasma.
Whatever the distance, denoting it as D, the maximum detectable distance of equivalent transients usingEq. A14 is 4.7D for nominal values of other parameters (10σ detection, 1 GHz bandwidth) and for a fullSKA of sensitivity. For the 500 Mpc assumed distance, this becomes 2.4 Gpc.
8.5. Maximal Giant Pulse Emission from Pulsars and Mergers
Fig. 11.— Left: A sequence of pulses vs. distance calculated as described in the text for 0.43 GHz and the directionindicated. Distances are logarithmically spaced. Asymmetric broadening from multipath propagation conserves pulse area asit gets progressively larger with increasing distance. The plotted amplitudes take into account broadening but do not includeinverse-square effects. Broadening becomes severe for distances greater than about 6 kpc for the particular frequency anddirection used in the calculation.Right: Dmax vs. peak luminosity Lp = L/fdc at 0.43 GHz for three intrinsic pulse widths, Wi. At low luminosities for
which sources cannot be detected very far away and Dmax ∝ L1/2p as expected from the inverse-square-law scaling of flux
density. Dmax varies more slowly with increasing luminosity at large distances because scattering broadens the pulse, makingit harder to detect.
– 17 –
Radio emission from pulsars is a tiny fraction of the spindown energy loss rate, E. This is true for theaverage emission from objects with large values of E, such as the Crab pulsar, but is not true for olderlong-period pulsars that can have Lr/E ∼ 1 to 10% (e.g. Arzoumanian, Chernoff & Cordes 2002). Inaddition, very large giant pulses are sometimes seen from the Crab pulsar that carry significant power forshort periods of time. For example, the largest giant pulse seen from the Crab pulsar is 2.2 MJy at 9GHz with an (unresolved) width of 0.4 ns (Hankins & Eilek 2007). This corresponds to an instantaneous
radio luminosity Lr ∼ 1037.0(Ωr/4π) erg s−1 that can be compared to E ∼ 1038.7 erg s−1, where Ωr is the
radio-beam solid angle. While beaming may make Lr/E very small even for this case, the key point is thatthe radio emission can be a signficant perturbation on the particle flow in the magnetosphere.
With this motivation, we calculate the maximal giant-pulse emission using E as an upper bound combinedwith a radio efficiency ǫr ≡ Lr/E. This is
Lr,max = ǫr
(
4π
Ωr
)
(
E
4π∆νr
)
= 107.92 Jy kpc2 ǫrE38∆ν−1GHz
(
4π
Ωr
)
(11)
where ∆νr is the emission bandwidth.
For nominal parameters, this radio luminosity is only about two orders of magnitude larger than seen in thelargest giant pulses from the Crab pulsar. With small beaming solid angles and small radiation bandwidths,much larger instantaneous luminosities can be obtained within the energy budget of the pulsar.
Objects probably exist that have values of E ∝ PP−3 much larger than that of the Crab pulsar, such as theCrab itself when born with P ∼ 10 ms compared to its present-day 33 ms or, more extremely, objects bornas high-field MSPs. Similar radiation may occur from merging NS-NS binaries, whose magnetospheres willbe reactivated once their light cylinders interact (e.g. Hansen & Lyutikov 2001). In the last moments of the
merger, the effective E will approximate that of a similarly magnetized NS of maximal spin rate. Burstsfrom such objects may be much larger than those seen from isolated radio pulsars. We may therefore expecttransient, one-shot bursts to emanate from merging sources at cosmological distances slightly prior to anygamma-ray burst that is also produced. However, reception of the transient may occur after the GRB ifdispersion delays along the path (in the host galaxy, in the IGM, and in the Galaxy) are significant (seereferences in Hansen & Lyutikov).
8.6. Gamma-ray Bursts: Afterglows and Prompt Radio Emission
Fig. 12.— Left: Dmax vs. Lp for single pulses for an inner-Galaxy direction using survey parameters for the Parkesmultibeam pulsar survey (1.4 GHz). Right: Same but for the Arecibo pulsar survey at 1.4 GHz using ALFA. Dmax is largerby about a factor of two for Arecibo compared to Parkes at low luminosities because of the difference in overall sensitivity.Breaks in the curves at higher luminosities are different for the two telescopes because scattering broadening itself is distantdependent; also, dispersion smearing across channels is larger for the Parkes multibeam survey because of the wider channelbandwidths (3 MHz vs. 0.4 MHz).
Afterglows: Gamma-ray bursts detectable with current gamma-ray instruments occur at a rate ∼ 600yr−1 in an isotropic sky distribution. Carilli, Ivison and Frail (2003) estimate that 10-25% of these aredetectable above 100 µJy in afterglows that last ∼ 0.1 yr. Frail et al. (2006) show that afterglow peak fluxdensities are weakly redshift (z) dependent, as expected from the compensation of inverse-square effects byspectral and temporal redshifts (Lamb & Reichart 2000; Ciardi & Loeb 2000). Using 10% and assumingthat GRB afterglows occur with a flux density distribution with αs = 2 that extends down to 1 µJy, theimplied total rate is 102 times larger than for those above 100 µJy, or
nΩ(GRB) = 0.15 yr−1 deg−2. (12)
For ΩFOV = 1 deg2 and no subarrays or multiple FOV, x = ΩFOV/Ωs = 10−4.6. The number of eventsoccurring in the FOV in a dwell time of 30 d is 0.012 implying that NFOV = 84 would bring the detectionrate to about one event per pointing. Alternatively, subarrays could be used if the flux-density distributionis shallower than αs = 2. It is also possible that radio afterglows occur much more often than the GRBrate, especially if beaming causes gamma-ray burst counterparts to be less frequent, as in orphan afterglows(e.g. Gal-Yam et al. 2006).
Frail et al. (2006) show that the peak flux densities at 8.5 GHz of afterglows (from long bursts) are nearlyredshift independent for z >
∼ 1. Peaks occur at roughly 5 days after the burst in the rest frame of the source.The range of flux densities is from about 50 µJy to over 1 mJy and, in the observed sample, narrows ingoing to higher redshift. We model this as a wedge in log Sν − log z space that has a constant, lowervalue of 1 µJy and an upper value log Sν = 4.2 − 0.95(log z/0.1) for 0.1 ≤ z ≤ 10. We plot this wedgein the transient phase-space plot by using Sνd2
L, where dL(z) is the luminosity distance. The lower boundis substantially smaller than in the observed sample presented by Frail et al. because we assume, due tobeaming and to the gas density that the shocks propagate into, that peak flux densities cover a much widerrange than has been detected so far. In addition, we have broadened the range of νW beyond that explicitlypresented by Frail et al.
Prompt Emission: Prompt radio emission from GRBs (e.g. Palmer 1993) will necessarily be from acoherent emission process if the time scale is comparable to that of the short GRBs (seconds or less) butnot necessarily for long GRBs (seconds to minutes). However, their detectability is problematic at the veryleast from the point of view of solid-angle coverage. GRBs themselves occur ∼ 1 day−1 using hemisphericdetectors, so any radio instrument would need comparable sky coverage to detect events at a similar rate.Except at very low frequencies (<
∼ 0.3 GHz), where dipole arrays can give ∼ 1 sterad coverage, radio arrayswill yield very low rates if they are limited to the FoV of, say, the primary beam of a parabolic antenna.However, it is possible that prompt radio bursts are very bright, in which case one could use antennas toindependently observe different sky positions. With an N = 4000 array of 6m antennas, ∼ 104.1 deg2 couldbe sampled, yielding a rate ∼ 0.6 day−1.
Suppose that GRBs (whether short or long in duration) have accompanying radio emission. We know thatGRBs are infrequent and isotropically distributed. A radio survey can just as well stare at any part of thesky rather than scan the sky. However, we assume that any prompt radio bursts would be searched for aspart of a synoptic survey campaign that does scan the sky. For ∼ 1 s bursts, radio emission necessarilymust be coherent in order to be detectable.
In order to evaluate Eq. A12 we assume the following for population and SKA parameters: Ωpop = 4π,W ≈ 1 s, fsky = Ωs/Ωpop = 0.7, fc ≡ SKAnom/SKA ≤ 1, Ts ≫ W , Ωb = (π/4)θ2
b = (π/4)(1.17λ/D)2 ≈3.2 deg2 (νGHzD10)
−2. These nominal values imply that Np pointings must be made to cover the fraction
fsky of the sky:
Np =Ωs
Ωi=
fskyΩpop
NsaNFoVΩb= 104.0 fsky (νGHzD10)
2
NsaNFoV(13)
The event rate per object η is very low, since most GRBs are non-repeating and are associated withobjects that merge on time scales ∼ 100 Myr or with hypernovae with several Myr lifetimes. Thus we haveηW ≪ ηTs ≪ 1 so case I of Eq. A19 applies:
J(η, W, Ts, τ) =ητW 3/4
1 − e−ηTs≈ W 3/4
Np≈ 10−4 W 3/4NsaNFoV
fsky (νGHzD10)2 , (14)
where we have used τ = Ts/Np.
We also need to calculate Dmax,1 for which we need a radio luminosity. In true luminosity units (erg s−1)the GRB peak luminosity is fiducially Lγ = 1051Lγ,51 erg s−1. We assume that the true radio luminosity issome multiple ǫr of this: Lr = ǫrLγ . Remarkably, over many kinds of astrophysical objects (stars, pulsars,AGNs), we find ǫr ranging from about 10−8 (Crab pulsar) to 10−3 (blazars). Here we use a fiducial value
– 19 –
ǫr = 10−5ǫr,−5 and a radio emission bandwidth ∆νr = 1 GHz ∆νr,GHz. We then calculate the pseudo-luminosity L used in Eq. A14 in units of Jy kpc2 by calculating the radio flux from the GRB assuming (onlyfiducially) isotropic emission and multiplying it by distance squared; we multiply the pseudo-luminosity incgs units by (Jy / cgs flux units)(kpc/cm)2 to obtain pseudo-luminosity in Jy kpc2. This gives a stupendouslylarge pseudo-luminosity,
L = 1015.9 Jy kpc2
(
ǫr,−5Lγ,51
∆νr,GHz
)
(15)
corresponding to a peak flux density
Spk = 102.9 Jy
(
ǫr,−5Lγ,51
∆νr,GHz
)(
3 Gpc
dL
)2
. (16)
Beaming may influence the radio luminosity estimate as it does the γ-ray luminosity. Substitution of Linto Eq. A14 implies that radio bursts of this amplitude are detectable throughout the entire universe, i.e.Dnom,1 > Dpop. In this case, the completeness coefficient defined in Appendix A becomes
Cs ≈ fskyJ(η, W, Ts, Np) ≈ 10−4 W 3/4NsaNFoV
(νGHzD10)2 . (17)
The interpretation is that only one out of ∼ 104 bursts will be seen in a survey that uses the 3.2 deg2 FoVof a 10m diameter antenna. Much wider instantaneous FoV is needed. However, much less sensitivity willalso suffice, so subarrays can be used. Another implication is that pre-SKA observations can be made usingsmall antennas, including dipole arrays.
Is it reasonable to expect prompt radio emission from GRBs? Photon reprocessing may limit radio brightness(Macquart 2007) in some geometries but an empirical approach is the best way to answer the question. Whatother cosmic gamma-ray emitters might also produce radio emission of some kind? We can answer thesequestions by investigating the ratio of radio to gamma-ray luminosities for sources in different classes. Allknown classes of gamma-ray sources contain members that are radio emitters, including solar-type stars,pulsars, soft-gamma repeaters, GRBs (radio afterglows) and blazars. Individual objects, of course, providecounter-examples, such as the Geminga pulsar; however, the most plausible explanation for the lack of radioemission from Geminga is that the radio beam does not intersect our line of sight.
Flux densities are large enough that they may be detectable with wide-field, low-gain antennas. For examplean antenna with gain
G =4πAe
λ2(18)
and system temperature Tsys has a system equivalent flux density (SEFD or Ssys, ν in GHz):
Ssys =Tsys
κ=
8πkTsys
λ2G= 1.16 × 107 Jy
ν2
G
(
Tsys
30 K
)
, (19)
yielding minimum detectable flux density (m = minimum S/N, B = bandwidth in GHz, τ = integrationtime in seconds):
Smin =mSsys
(NpolBτ)1/2
= 103.41 Jyν2
G
(m
10
)
(
Tsys
30 K
)(
2
NpolBτ
)1/2
. (20)
Thus a small antenna that is sampled with high time resolution over a large bandwidth may be capable ofdetecting the most extreme events (e.g. Green et al. 1996; Morales et al. 2005).
8.7. Evaporating Black Holes
Black hole evaporation may cause radio pulses whose spectrum and duration are related to the evaporationtime tevap ≈ (Mbh/108 gm)3 s. If a fraction of the rest energy is radiated then Lbh = ǫMbhc
2/tevap.
9. seti
From a signal detection point of view, SETI has many similarities to the detection problem for pulsars andtransients with broad-band spectra. Radio signals from ETI will propagate dispersively and will scintillateand be otherwise affected by scattering (e.g. pulse broadening and spectral broadening). To the extent thatthey are intermittent, the competeness coefficient of a SETI survey can be expressed identically to thosecalculated for natural transients.
The particular details depend on the signal classes of ETI signals. In one extreme they may be broadbandbut pulsed and in another they may be narrowband and intrinsically steady. Intermediate cases wouldinclude a carrier modulated at a low effective bit rate.
A particularly important parameter for SETI is the sky density nΩ,ETI and the density-rate of ETI transients,nΩ,ETI, analogous to nΩ considered earlier for transients. Given that pulsars and transient radio source (e.g.Crab giant pulses, stellar flares, etc.) have already been found while ETI signals have not, it is reasonableto assume that nΩ,ETI and nΩ,ETI are both very small. There may be only a few ETI sources in theGalaxy, in which case the large number of directions, frequency channels and time windows to be searchedis enormously bigger than the number of expected detections. Accordingly, false-positive detections must beconsidered with care. We call this survey regime the hyper-sparse regime. The mean distance of transmittersfrom the Earth in this regime is of order the distance of the Sun from the Galactic center (∼ 8 kpc).
As pointed out elsewhere (Cordes & Lazio 1991; Cordes, Lazio and Sagan 1997; Lazio, Tarter and Backus2002), interstellar scintillations play a key role in the small source-number regime. At water-hole frequencies,directions at low Galactic latitudes and source distances >
∼ 1 kpc will all show strong diffractive scintillations(DISS), which modulate the signal by 100% (rms) with an exponential distribution. More often thannot, DISS will make a source of steady flux less detectable, while occasionally, it will boost an otherwiseundetectable signal above threshold. The long tail of the exponential distribution implies that a SETI surveyof transmitters in the small source-number regime is better done with a number of shorter observations ratherthan a single long observation if the total on-source time is held fixed. The multiple observations must bespaced in time by more than one characteristic scintillation time, which can be seconds to hours for pulsarsand will be approximately ten times longer for ETI sources, which are expected to have space velocitiesmuch smaller than the runaway pulsar population. This conclusion is independent of whether the signal isnarrow or broad band.
For the SKA, we need to consider both raster-scanning and staring type ETI surveys. Raster-scans natu-rally lend themselves to multiple trials on particular sky directions if the raster scan is repeated. Staringobservations should be repeated with a gap between them to allow scintillations to decorrelate.
As discussed below, the widefield search requirements for SETI are identical to those of pulsars. If thesignal is modulated on time scales <
∼ 1 sec, then pulse broadening is an issue for objects at large distancesin the Galactic plane.
9.1. Asteroid/Comet Radars
One of the brightest transmissions of our civilization is the planetary radar from the Arecibo telescope,which operates at 2.3 GHz with Pradar ∼ 1 Mwatt transmitted into a bandwidth ∆νradar ∼ 1 Hz. As arguedby (e.g.) Helmers 1996, similar radars may be used by other civilizations in their mitigation of impactthreats from asteroids and comets. Assuming that such signals are only slowly modulated, the pseudoluminosity is
Lradar =Lradar
Ωradar∆νradar= 1.6 Jy kpc2 Pradar,MW∆ν−1
radar,Hzθ−2radar,arcmin, (21)
where θradar,arcmin is the 1D beam size of the transmitter in arcmin.
10. the radio synoptic survey telescope
Surveys that involve time-variable sources like those we have discussed need to take into account the eventrates and durations as well as sensitivity requirements. Transients cover a very wide dynamic range in timescale and we have distinguished between slow and fast transients. Slow transients are those for which thesky may be sampled by raster scanning, covering a total solid angle Ωs in a time Ts and then repeating thescan. Figure 13 shows a schematic view. The dwell time per sky position is τ = (Ωi/Ωs)Ts. Slow transientsare also those for which pulse broadening from multipath propagation is unimportant. Fast transients arethose that cannot be well sampled through raster-scan imaging surveys, unless they are very frequent andthere is tolerance for a low completeness coefficient. Transients shorter than about one day qualify as fastif an all-sky survey is contemplated. Rare, fast transients are better sampled through staring observationsof large solid angles.5
5
In terms of number of detected events, it is equivalent to cover a large solid angle through raster scanning and small dutycycle per sky position, or to cover a smaller solid angle with continuous time coverage. If the goal is to characterize the angulardistribution of the transient population, then raster scanning is appropriate. However, if the event rate of particular sources isdesired or if the source population is restricted to a small solid angle (e.g. the Galactic center region), then staring observationsare needed.
– 21 –
10.1. Figure of Merit for Raster-scan Surveys of Steady Sources
For blind searching, the rate of sky coverage Ω (deg2 s−1) needs to be maximized while also achieving thedesired search depth, which we characterize as the maximum detection distance Dmax. Survey yield is anobvious metric and it involves the product of source number density ns, search volume Vmax = 1
3ΩsD3max. If
the survey of the solid angle Ωs is conducted in a time Ts, the resulting search volume yields a combinationof parameters identical to that obtained by calculating survey speed, SS = Ωi/τ , but raised to a differentpower. These two approaches lead to (Appendix A) the figure of merit “FoMSS” (which can be read as afigure of merit for either steady sources or for survey speed):
FoMSS = B
(
NFoVΩFoV
Nsa
)(
fcAe
mTsys
)2
, (22)
where (as before) NFoV is the number of fields of view (or pixels) for each antenna, ΩFoV is the solid anglefor each FoV, Nsa is the number of subarrays into which the array is divided (assumed equal in size andpointed in non-overlapping directions), fc is the fraction of the total effective area Ae usable in the survey,m is the threshold S/N in the survey, and Tsys is the system temperature. This expression is consistentwith the simple form often used for survey speed, BΩFoV(Ae/Tsys)
2, but makes manifest other variablesrelevant to SKA surveys.
Fig. 13.— (Top:) Schematic view of a raster scan where each small square indicates the instantaneously sampled solid angle.The star symbol indicates the occurrence of a transient of duration W at the indicated sky position. (Bottom:) Timeline forthe sampling of the event that begins at time te1 during a part of the raster scan when the sky position of the transient isnot in the instantaneous solid angle being sampled; the next sky position, that of the transient source, is sampled while thetransient is still occuring (shaded rectangle); the transient persists while the scan samples the next sky position.
10.2. Figure of Merit for Raster-scan Surveys of Transient Sources
FoMSS is relevant to surveys of sources that are homogeneously distributed within their spatial domain,that are standard candles, and that are time steady. It is not a good metric for transient sources. InAppendix A we present a more general metric that does apply to transients,
FoMTS = FoMSS × K(ηW, τ/W ) (23)
K(a, x) =(
1 + x2)−1/2
[
1 − e−a(1+x2)1/2]4/3
(24)
where a ≡ ηW is the product of event rate per source η and event duration W (which is a measure of the
overlap of events in time) and x ≡ τ/W = Ωi/ΩW , with Ωi = NsaNFoVΩFoV. The quantity Ω = Ωs/Ts isthe mean rate at which solid angle is surveyed. The factors included in K(a, x) account for the integrationtime possibly being determined by the transient duration (W ) rather than the raster-scan dwell time τ andthe probability that a source is pointed at when the event occurs, assuming Poisson statistics. The functionK(a, x) is shown in Figure A16.
10.3. Completeness Coefficient of a Transient Survey
Transient sources by definition are “on” and hence detectable for only a fraction of the time, which maybe small. We therefore need to consider how a survey samples a population of transient sources in boththe spatial and temporal domains. In Appendix A we define a completeness coefficient Cs as the ratio ofnumber of objects that are detected to the number of objects that could be detected during the scan timeTs. For a homogeneous population of sources distributed over solid angle Ωpop and out to a distance Dpop,this takes on the form
Cs = min
[
Ωs
Ωpop, 1
]
min
[
(
Dmax
Dpop
)3
, 1
]
Pt(η, W, τ)
Pt(η, W, Ts), (25)
where Pt is the temporal capture factor (the probability that at least one event occurs when the telescopeis pointed at a bursting source), given by (for Poisson statistics)
Pt(η, W, τ) = 1 − e−η(W 2+τ2)1/2
. (26)
In Appendix A we identify three regimes for Cs that depend on the event rate η and on whether the eventsare fast or slow. Previously (§8.6) we estimated Cs for a putative population of very bright, prompt burstsfrom GRBs to demonstrate the need for wide FoV. Similar estimates need to be made for other transientpopulations, such as giant pulses and RRATs, in designing surveys for the SKA and for other telescopes.
10.4. Trading Field of View and Sensitivity
The sensitivity and FoV requirements for transients can be discussed in terms of the event rate for particularclasses of sources. Figure 14 shows minimum detectable flux density plotted against instantaneous solidangle. For single-reflector, single-pixel telescopes, we have a simple relationship,
Sν,min =mSsys√
2Bτ=
8kTsys
ηAπ√
2BT
(
mΩi,1
λ2
)
, (27)
where the unity subscript implies a single aperture and we have assumed that Ωi,1 = (λ/D)2 (i.e. unitycoefficient). The integration time, T , may or may not be related to the dwell time on a given source position.The factor m, as before, is the threshold in units of the rms noise level for a bandwidth B and integrationtime T . The solid line in Figure 14 shows this relationship. For arrays with NA antennas each having amulti-pixel receiver with NFoV pixels, we have
Ωi = NFoVNsaΩi,1 (28)
Sν,min =1
NaNFoV
8kTsys
ηAπ√
2BT
(
mΩi
λ2
)
. (29)
We have allowed for the use of subarrays with the factor Nsa. This increases the field of view at the expenseof higher threshold. We show points in the figure for the full SKA outfitted with a single-pixel system(SKA/SP) and with a 32-pixel system (SKA/PAF). We also show a point for an SKA built around anaperture array, which provides ∼ 1 sr FoV. The slanted dashed line shows the relationship for subarrays,the ultimate configuration comprising NA subarrays of one antenna each. The figure delineates the variouspermutations of strong/weak and rare/common transients. Vertical dashed lines indicate the solid anglerequired to capture sources of indicated event rates in a reasonable dwell time (see caption). In the figurewe keep the integration time, T , fixed, as appropriate for short-duration transients with W ≪ τ , i.e. Wshorter than the dwell time per position, in which case T = W .
To have the temporal capture probability Pt ∼ 1 we require, for low-rate, short-duration transients (W ≪τ),
Ωi>∼
1
(η/Ωs)Ts. (30)
In the figure we have considered event rates η/Ωs of 0.1 s−1 deg−2 with a one-hour scan time and1 day−1 deg−2 and 1 day−1 hemisphere−1 with scan times of one day for the three vertical, dashed lines
– 23 –
Fig. 14.— Plot of minimum detectable flux density (Sν,min) vs. instantaneous solid angle Ωi. The solid diagonal line refersto single-reflector telescopes all having — for simplicity in presentation — the same center frequency (1 GHz), operatingbandwidth (0.3 GHz), system temperature (25 K) aperture efficiency (60%), and intergration time (1 s). The dashedextension of the “single reflector” line indicates that the implied reflector diameter < 6λ. The “Full SKA” line indicatesthe sensitivity for 4400 12-m antennas. “FoV expansion” implies an increase in solid-angle coverage through use of dishes +phased-array feeds (PAFs) or through an aperture array (AA). The subarray line indicates the tradeoff between sensitivityand instantaneous FoV for single-pixel systems. Vertical dashed lines indicate the solid angle needed to detect events atthe indicated rates assuming a one day total exposure time. The ellipse in the bottom-right corner indicates the fiducialamplitude of hypothetical prompt radio GRB emission (c.f. Eq. 16). The location of the GRB ellipse indicates that dipoleantennas may yield sufficient solid angle coverage but will be too insensitive to detect prompt GRB emission for nominalvalues of the parameters.
from left to right. The rightmost line, for example, is relevant to the case of prompt radio pulses fromGRBs, should they exist.
From the figure we can make a number of conclusions. First, if there are hyper-strong events (such ascoherent radio pulses from GRBs), a wide FoV system is necessary for a reasonable detection rate, butmodest collecting area may suffice for detection. However, rare weak events require high sensitivity as wellas wide FoV. Field of view expansion of single reflectors allows probing of the upper right-hand corner asdoes an aperture array. But it is also true that wide FoV may be achieved at moderate sensitivity throughthe use of subarrays. Since the low-event-rate radio sky is largely unexplored, we should entertain thepossibility of using the subarray approach as well as develop FoV expansion approaches.
10.5. The Synoptic Cycle: An Example
The observing cycle for a given survey consists of a raster scan of a region of sky Ωs in a time Ts, yieldinga dwell time per sky position τ = TsΩi/Ωs. This cycle can repeated many times, allowing detection of agrowing number of transient sources, both periodic ones like pulsars and magnetars, and one-shot objectslike GRBs. At the same time, signal-to-noise ratio is built up for steady sources, such as HI in galaxies,continuum sources for AGN surveys and Faraday rotation measurements, etc. Because multiple surveysare potentially doable, and each is demanding on total telescope time, all effort should be made to do thesurveys simultaneously (commensally).
To accomplish multiple surveys, a hierarchical approach to scanning rates (or cadences) is probably needed.A fast rate is appropriate for the extragalactic sky in order to sample GRB afterglows while also buildingS/N on HI galaxies. A slower rate is needed for the Galactic disk to provide adequate time series durationsfor pulsar surveys. Staring observations of the Galactic center source will allow deep pulsar and transientsurveys from the star cluster. Finally, guest-investigator experimental or one-time observations need to beaccomodated along with target-of-opportunity observations that will arise.
An example scenario includes both fast and slow scanning observations and staring observations. Thisexample sums to about 10 days per cycle, which would be repeated as needed:
1. Fast scan of the extragalactic sky: large-scale galaxy HI survey, Faraday rotation survey, AGNsurvey, and transients
(a) “Full sky” survey (80% of 4π) using a 1 deg2 FoV single pixel system(b) Ts = 5 days to cover one scan of the sky(c) τ ≈ 10 s dwell time per sky position(d) Smin ≈ (gθfc)
−115 µJy at 10σ where fc is the fraction of a full SKA available and gθ ≤ 1 isthe gain relative to the on-axis gain
(e) Field-of-view expansion through multiple feed clusters or phased-array feeds will increasethe sensitivity for fixed Ts; aperture arrays would also provide an increase. A thirty-beamphased-array feed, for example, would yield 300 s dwell time
(f) Subarrays will reduce the sensitivity but can cover more instantaneous solid angle and reduceTs.
2. Slower scan or staring observations on deep extragalactic fields e.g. 1 day
3. Slower scan for the Galactic plane: pulsars, masers, transients,etc. e.g. 1 day scan of the innerGalaxy (180 deg in longitude) in a ±1 deg swath in Galactic latitude, yielding 240 s per pointing
(a) Minimum contiguous dwell time needed for pulsar surveys that use Fast Folding Algorithms
or Fourier transforms of a contiguous time series combined with harmonic summing (100 to1000 s typical); single-pulse searches do not place strong requirements on contiguous blocks.
(b) Pulsar timing: frequent re-observations are needed for long-term monitoring; a 10-day cadenceis acceptable.
4. Staring observations: e.g. 12 hr on the Galactic center
5. Break out for targeted observations by individual investigators: 10% of the time?
6. Break out for targets of opportunity: e.g. GRB triggers, blazar observations, etc. 5% of thetime?
7. Calibration allowance
Additional comments on this scenario are as follows:
– 25 –
1. HI detection of galaxies at z ∼ 1 requires many hours of integration time, which would build upslowly unless there is field-of-view expansion.
2. Pulsar surveys can accumulate S/N through incoherent summing of power spectra from non-contiguous data segments, with due allowance for acceleration of the source or observer. Coherentsums across multiple days are probably too demanding computationally, but with requirements thatdepend on the cadence.
3. Diffractive and refractive interstellar scintillations will modulate compact sources to varying degreesand with a wide range of correlation times. To optimize detection, multiple passes on the same skyposition should be uncorrelated with respect to DISS and RISS (Cordes & Lazio 1991).
11. requirements for searching the full field of view
The present specifications for the SKA include a minimum instantaneous field of view (FOV) of 1 deg2
at 1 GHz. In the Large-N-Small-D (LNSD) concept for the SKA, antennas would be configured so as toyield significant collecting area on small, intermediate and large baselines. Blind surveys of large total solidangles require full-FOV sampling, which is feasible only for a subarray comprising the innermost antennas.We call this subarray the core array and characterize it as a circular distribution of na antennas withdiameter bc. The core array contains a fraction fc ≡ na/Na of the total number of antennas. For a centrallyweighted (parabolic) distribution of collecting area, the array beam for the core array is
θb ≈ 1.27λ
bc≈ 78.6 arcsec (νGHzbc,km)−1 . (31)
For comparison the primary beam for an antenna diameter D = 12 m D12 is6
θp ≈ 1.17λ
D≈ 1.68 deg (νGHzD12)
−1 . (32)
To pixelize the FOV requires
Npix ≈ 0.85
(
bc
D
)2
≈ 103.8 (bc,km/D12)2 pixels, (33)
where bc = 1 km is used as a fiducial value. In the LNSD Whitepaper (2002), about 25% of the overallsensitivity is contained inside an array of this size. A related quantity is the core array filling factor, theratio of collecting area to area occupied by the antennas,
ff = na(D/bc)2 ≈ 0.14
( na
103
)
(
D12
bc,km
)2
. (34)
To image or otherwise pixelize the FOV requires channelization of the total bandwidth B into Nν = B/∆νchannels. To have the ‘delay beam’ exceed the FOV requires θdb ≈ cbc∆ν > θp or ∆ν < 9.5 MHzD12 νGHz b−1
c,km.Another constraint on channel bandwidth derives from the need to dedisperse the signal when searching forpulsars or narrow pulses. The dispersion delay across ∆ν for an object with dispersion measure DM is
∆tDM = 8.3µ sDM ∆νν−3GHz, (35)
for DM in pc cm−3 and ∆ν in MHz. Values of DM extend to at least DMmax = 2000 pc cm−3, so limitingpulse smearing to be less than 100 µs at 1 GHz requires ∆ν < 6 kHz (∆tDM/100 µ s)(DMmax/2000 pc cm−3)−1ν3
GHz.Thus, channel bandwidths narrow enough for dedispersion more than suffice for allowing full FOV pixeliza-tion.
11.1. Sampling the Field of View
There are two ways to pixelize the FOV: (1) use an explicit hardware beam-former or (2) combine acorrelator and a Fourier-transform (FT) operation to form snap-shot images. In either case, we considerFX approaches that first channelize the signals from each antenna. On a mathematical basis, the twoapproaches are equivalent via the Wiener-Kinchine and van Cittert-Zernike theorems. On a practical basis,however, the two approaches diverge in terms of implementation and complementarity with imaging science.6
I assume parabolic illumination of the primary that reaches 25% amplitude at the edges, i.e.
g(r) = K +
"
1 −
„
2λr
D
«2#p
with K = 0.25 and p = 1 (c.f. Rohlfs & Wilson (1996), Table 6.1).
Direct beam forming: Let the channelized baseband voltage be εpν(xn, ti) for the n-th antenna at locationxn, for polarization p, baseband frequency ν and time ti. Channelization occurs in an ‘F’ stage requiring∆t = Nνδt = NνB−1 Nyquist samples. The ti are in steps equal to ∆t. From na antennas we form thecomposite voltage for beams centered on directions kj , j = 1, Npix,
εpν(kj , ti) =
na∑
n=1
εpν(xn, ti)e−ikkj ·xn . (36)
To form each beam given the phase factors requires na complex multiply/adds per polarization, channel, andtime step. Phase factors need not be calculated as often as the beam voltages, so we ignore those operationsin calculating the data rate. The number of operations is then naNpolNν per time step, corresponding toan operations rate for a single beam,
Nb,1 = naNpolNν/∆t = fcNaNpolB ≈ 1012.24 op s−1
(
fc
0.5
)(
Na
4400
)(
Npol
2
)(
B
400 MHz
)
, (37)
where we have used a bandwidth of 400 MHz as a reasonable choice for surveys conducted in the 1 to 2GHz range and consistent with the overall guideline specification of 20% bandwidth. The data rate for allbeams needed to pixelize the FOV is
Nb = NpixNb,1 = fcNaNpolNpixB. (38)
For Na = 4400 antennas and a core array comprising a fraction fc = 1/2 of these antennas, a bandwidthB = 400 MHz and the number of pixels calculated previously, we have
Nb = 1016.0 op s−1
(
fc
0.5
)(
Na
4400
)(
Npol
2
)(
bc,km
D12
)2 (B
400 MHz
)
. (39)
Beam-forming through gridding and FFT of sampled wavefield: An alternative approach recognizesthe efficiency of the FFT in forming beams. The sampled fields εpν(xn, ti), can be interpolated onto auniform grid and then Fourier transformed using an efficient algorithm. The number of grid points in onedimension of the 2D grid is, for the core array, NFFT = bmax/bmin = bc/µD, where µ ≈ 2 is the minimumantenna spacing in units of the antenna diameter. Ignoring gridding computations, the operations rateneeded for the 2D FFT is
NfbaNops/calc (NFFT log2 NFFT)2NpolB (40)
Fig. 15.— Schematic view of the primary beamor FoV (large cone) and its pixelization with syn-thesized beams, with three highlighted in red.
– 27 –
where “fba” stands for “fast beam algorithm” and Nops/calc ≈ 5 is the number of operations needed tocalculate each Fourier coefficient. For typical numbers, this rate is about a factor of ten less than for directbeam forming. The ratio of the two rates is
Nfba
Nb
≈ Nops/calc
µ2ζna
[
1
2log2
na
ff− log2µ
]2
. (41)
For Nops/calc = 5, µ = 2, ζ = 0.85, ff = 0.14 and na = 103, the operations ratio is 1/20.
Beam-forming through correlation: An alternative approach is to correlate all signals from all antennasof interest and combine them at appropriate time intervals into snapshot images. Using the channelized,complex baseband voltage previously defined, εpν(xn, ti), we calculate auto-and-cross correlations betweenthe n-th and m-th antennas,
Cmnpp′ν(ti) = 〈εpν(xn, ti)ε∗p′ν(xm, ti)〉∆t, p, p′ = 1, 2, (42)
where ∆t is the integration time. For pulsar searching, ∆t ≈ 100 µs is adequate. The number of compu-tations per correlation value is ∆t/δt = ∆tB/Nν per polarization, frequency channel and integration time.At each integration time the number of correlations to be computed is
Nc =1
2na(na − 1)NpolNν . (43)
Then the operations rate for computing correlations (including cross products, so Npol = 4) is
Nc = NcB/Nν =1
2na(na − 1)NpolB
≈ 1
2(fcNa)
2NpolB
= 1015.6 op s−1
(
fc
0.5
)2(Na
4400
)2(Npol
4
)(
B
400 MHz
)
. (44)
To pixelize the sky requires an N×N Fourier transform where N ≈ (4Npix/π)1/2
= 87bc,km/D12, or roughly
(2N log2 N)2 operations per ∆t, or about 20 Gop s−1, which is negligible compared to Nc.
Comparison of Direct and Correlation Beam Forming: The correlation method requires a lowerrate of operations than the direct beam method for nominal parameters. Note that the operations rate wascalculated for Npol = 4 for the correlation method and half that for beam forming. The correlation methodalso can be used to image outside the FOV defined formally as the FWHM of the primary antenna beam.Moreover, a correlator will be used with all antennas of the SKA, including those on VLBI baselines, forimaging. Thus a correlation approach with native dump times not unlike those currently used (∼ 100 µs)for imaging applications is an economical way to enable time-domain intensive surveys.
Direct beam-forming, on the other hand, can develop modularly to allow targeted surveys first, and thenfull-FOV blind surveys as cost/performance of hardware increases. Direct beam-forming also allows greatermitigation of RFI through placement of nulls in appropriate directions and through use of less-coarsequantization than correlators are likely to use.
It is useful to compare the operations rates for the two methods more directly and in terms of the sensitivityspecification for the SKA, [Ae/Tsys] = 2 × 104m2 K−1. The ratio of operations rates is
Nb
Nc
=2
fcNa
(
bc
D
)2
=π
2
η
[Ae/Tsys] Tsys
(
b2c
fc
)
, (45)
where η is the aperture efficiency; this ratio is independent of dish diameter. We can relate bc to fc
given a configuration for the SKA. Using Figure 3.4 of the USSKA LNSD Whitepaper (SKA Memo,http://www.skatelescope.org/pages/page astronom.htm, no. 2; see also no. 9), which shows sensitivityvs. baseline for the scale-free configuration, we can relate the fraction of the SKA’s collecting area to thebaseline:
bc,km ≈ 105(fc−0.25), 0.05 <∼ fc ≤ 1. (46)
For direct beam forming to be computationally favorable requires that Nb/Nc < 1 or
1010fc
fc<
100.2
(η/0.72)
(
[Ae/Tsys]
2 × 104m2 K−1
)(
Tsys
18 K
)
, (47)
which implies fc ≤ 0.10 in order for direct beam forming to be favorable. This result is independent ofdish diameter but depends on configuration and system temperature. A more compact core array obviouslyreduces the operations rate of direct beam forming because the rate scales as b2
c .
12. search processing
Here we consider the processing that takes place on the baseband voltage defined in Eq. 36 for the j-th beam.
We simplify the notation to εpν,j(ti) ≡ εpν(kj , ti). Pulsar analysis increasingly involves coherent dedispersionof the baseband voltage, which corrects voltage phases for dispersive propagation through the interstellarmedium. At present, blind surveys using coherent dedispersion are computationally too expensive, while onthe time scale of the SKA, such searches may become feasible for subsets of the parameter search space. Itis likely also that search schemes for signals with time-frequency signatures more complex than those seenfrom pulsars, and on shorter time scales, may be entertained.
We focus here on post-detection analyses for blind surveys that involve searches in dispersion measure aswell as spin period (for periodic sources). Let the intensity be defined as
Ipν,j(ti) = |εpν,j(ti)|2 (48)
for the j − th beam using direct beam-forming or the equivalent pixel calculated from visibilities. For somesurveys, two polarizations might be summed early in the processing while in others, where polarization isof interest, they would be kept separate.
Dedispersion: Frequency channels are combined according to a set of trial dispersion measures,
DMℓ, ℓ = 1, NDM. (49)
Typically, the number of dispersion measure values is comparable to the number of spectral channels,NDM ≈ Nν , though more precisely the number is determined by the spacing required to not degrade theS/N of pulses of a particular width. For large DM and low frequencies, the actual time resolution willbe limited by pulse broadening from scattering, thus relaxing the need for finely-spaced values of DM(McLaughlin & Cordes 2003). The dedispersed time series for the j-th beam and the ℓ-th trial value of DMis
Ip,j(ti, DMℓ) =∑
ν
Ipν,j(ti − tν [DMℓ]), (50)
where tν(DMℓ) is the dispersion-delay correction needed for frequency channel ν,
tν(DM) = −4.2 × 103 DM ν−2GHz µs. (51)
(Note that we use ν to represent a frequency index at baseband but also let it label the corresponding RF).The corrections can be precalculated. For each time step there are Nν summations per trial DM. Thus fora dwell time T on a given sky position, the total number of operations for dedispersing is NDMNνT/∆t perbeam. Over the FOV, this implies a full-Stokes processing rate
Ndedis = NpixNpolNνNDM/∆t ≈ 1014.4 op s−1
(
bc,km
D12
)2 (Npol
4
)(
Nν
103
)(
NDM
103
)(
∆t
100 µs
)
(52)
Note that the number of DM channels required is also affected by pulse broadening from scattering. Forlarge DM, corresponding to path lengths >
∼ 5 kpc and frequencies <∼ 1.5 GHz, pulse broadening limits the
achievable time resolution, so the search grid of DM need not be as fine as otherwise. The processing ofSKA data can thus be optimized according to direction and frequency. Nonetheless, the processing ratederived here is typical of what is needed.
Periodicity Search: The time series for each trial DM is Fourier transformed and the resultant powerspectrum is investigated for harmonics associated with pulses of different periods and duty cycles. Thresholdtests on the harmonic sum yield candidate signals that are further analyzed by revisiting the time series andsynchronously averaging it with the candidate spin period. For a dwell time T = Nt∆t, an Nt-point FFTis computed. Compared to dedispersion, the number of operations per unit time is negligible. Harmonicsumming and candidate diagnostics require similar numbers of operations.
Periodic Sources in Binaries: Current capabilities allow modest searching for binary signatures whilealso searching in period and DM. Binary motion even over short dwell times of minutes can smear pulsesunless compensation is made. The simplest analysis is an acceleration search if T <
∼ 0.15Porb, requiring∼ 102 to 103 trial accelerations. Alternative analyses can be made for T ≫ Porb that involve searchingfor orbital sidebands of the spin harmonics of pulsars. The most difficult search problem is for objects inhighly eccentric orbits, in which case a 5-parameter orbital parameter space must be searched. Dependingon which kind of orbital search is done and in conjunction with the type of P -DM search, the analysis caneasily exhaust the available computational resources.
Single-pulse Analysis: Dedispersed time series can be searched for strong individual pulses that can bemissed in a periodicity search. Pulsars with broad, power-law amplitude distributions such as the Crab
– 29 –
pulsar and its ‘giant’ pulses and the recently-discovered RRATs (McLaughlin et al. 2006) are examples.The search algorithm is very inexpensive, entailing threshold tests after matched filtering, approximated byhierarchical smoothing, as discussed by Cordes & McLaughlin (2003) and implemented in McLaughlin &Cordes (2003) and Cordes et al. (2004).
Generalized Transient Analysis: One can hypothesize the existence of signals with time-frequencystructure more intricate than that of dispersed pulses. Examples include flare star emissions and variousclasses of ETI signals (leakage or deliberately transmitted signals). To search for such signals requirescustomized analyses of the frequency-time plane [e.g. Ipν,j(ti)] that aim to identify signals according tospecified templates. Polarization signatures are also part of the search space. Such analyses would be muchmore costly than dedispersion and periodicity searches but presumably would unveil new source classes.
SETI Analyses: Searches for narrow spectral lines (<∼ 1 Hz) are the most common form of analysis,
though this has been generalized to modulated and drifting spectral lines that include pulsed, narrowbandsignals (ref). Mathematically, a modulated, drifting spectral line is identical to a modulated and dispersedpulse and search algorithms can thus be similar in that they involve matched filtering of the signal in thefrequency-time (f-t) plane. ETI signals may be both intrinsically modulated and scintillated.
13. summary
The work presented here includes sensitivity calculations for known kinds of time-variable objects and hypo-thetical transient sources. With adequate sensitivity, i.e. a good fraction of the original SKA specificationAe/Tsys = 2×104 m2 K−1, the SKA can do a complete census for pulsars in the Milky Way and in globularclusters, and it can detect pulsars and magnetars as periodic sources to local-group galaxies and potentiallyto the Virgo cluster. The same is true for giant pulses from extragalactic pulsars. Similarly, other knownkinds of transients (GRB afterglows, flares from stars and brown dwarfs, etc.) will be inventoried to a greatdeal of completeness. With adequate flexibility, hypothetical transients can also be investigated, such asprompt radio bursts from GRB sources, evaporating black hole emission, and ETI emission. Such flexibilityincludes the ability to process the frequency-time plane with a wide range of resolutions and the means forachieving wide fields of view.
A generalized survey metric is presented that applies to transient as well as steady sources. For steadysources, it reduces to the survey speed. For populations of bursting sources, an additional factor takes intoaccount that the integration time may be determined by the burst duration and the probability that thesource is pointed at during the burst.
A synoptic cycle is presented that demonstrates how survey and non-survey science may be conducted. Inprinciple, most or all the key survey science identified to date — HI in galaxies, Faraday rotation, pulsars,and transients — can be accomodated. Doing so places great demands on backend processing, which needsfurther investigation.
I thank the International SKA Science Working Group for providing a forum for implementations of SKAscience. For useful discussions over the last several years, I thank R. Bhat, S. Chatterjee, T. Hankins, M.Kramer, J. Lazio, D. Lorimer, M. McLaughlin, R. Schilizzi and P. Wilkinson. This work was supported bythe U.S. National Science Foundation through grants to Cornell University.
APPENDIX
a. general survey metrics (including transients)
Here we present survey metrics based on the amount of volume surveyed. It generalizes Appendix A ofMemo 85 to cases where sources are time variable.
Assume euclidean space and that a source population of interest has number density ns(x). Here x isstandard coordinate space. Sources are described by a parameter vector p of properties (such as luminosity,pulse width, period and rate, etc.) and the survey is described by vector q of properties such as the solidangle sampled, system parameters (SEFD, bandwidth), and the time Ts used to raster-scan the solid angle,etc. For given p and q, a source can be seen to maximum distance Dmax. The probability density functionof source properties is fp(p). For time variable sources, there is a probability of detection Pt determinedby the event rate for sources that emit bursts (whether periodic or not) and by the probability that a givensource is pointed at during an event. We are interested in the total number of objects in the populationNobj and in the number of objects that are detected in a survey Ndet(q).
Scanning parameters: If Ts is the time used to scan a total solid angle Ωs using a system that instanta-neously samples a solid angle Ωi = NsaNFoVΩb where Ωb is the solid angle of the single-dish antenna powerpattern, NFoV is the number of pixels in the feed system, and Nsa is the number of subarrays into whichthe full array is partitioned, with consequent reduction of sensitivity per subarray. The number of pointingsrequired to do the survey is Np = Ωs/Ωi and the dwell time per pointing (the time spent on any given skyposition) is τ = Ts/Np = (NsaNFoVΩb/Ωs)Ts.
Maximum detectable distance: Using a pseudo-luminosity that can be defined empirically as Lp = D2S,for a source of flux density S at distance D, the maximum distance at which the source could be detectedis Dmax =
√
Lp/Smin, where the minimum detectable flux density Smin depends strongly on source type(continuum, spectral line, steady, time variable, etc.), spatial distribution, and survey parameters. Inparticular, Smin depends on the on-source integration time. We therefore write Dmax = Dmax(p,q). Forsteady sources, the integration time is simply the dwell time, τ and Dmax ∝ τ1/4.
Temporal factor: Intermittent sources introduce two additional factors that we consider together. Forsimplicity we assume intermittency is in the form of bursts of duration W with events from a given sourceoccuring at a a Poisson event rate,7 η.
The integration time is generally not equal to the dwell time. For W ≪ τ , the integration time is W whilein the opposite extreme it is τ . For a gaussian burst and a gaussian beam that is continuously scanned, weapproximate the effective integration time as τeff = Wτ/
√W 2 + τ2, which equals W/
√2 when τ = W .
A second factor is a temporal probability factor that is simply the probability that at least one event occursin the time interval τ when the source is looked at. Because of edge effects in continuously scanning mode,the time aperture for detecting an event is approximately
√W 2 + τ2, yielding a probability
Pt(η, W, τ) = P≥1 = 1 − e−η(τ2+W 2)1/2
. (A1)
Source numbers: With these definitions, the number of objects in the source population that could bedetected in the total survey time Ts (of one synoptic cycle) is
Nobj =
∫
dpfp(p)
∫ ∫
population
dΩdD D2ns(x)Pt(η, W, Ts) (A2)
where x is a function of distance and direction. The number of detected sources in a time span τ — thetime during which a given source is actually pointed at — is
Ndet =
∫
dpfp(p)
∫ ∫
survey
dΩdD D2ns(x)Pt(η, W, τ). (A3)
Note that η and W are contained in the source vector p while survey time spans, etc. are contained in q.
Homogeneous standard-candle sources: In the following we will consider populations that are ho-mogenous within their domain, i.e. we assume standard candles with a uniform source density ns withinsome population solid angle Ωpop out to a distance Dpop. The figures of merit we derive moreover embodythe assumption that Dpop always exceeds the maximum distance reached. If not, the scaling laws will bedifferent.7
This is a good model for aperiodic sources and also for periodic sources, such as pulsars that emit giant pulses, which occurat discrete times tied to the rotation of a neutron star but whose amplitudes are highly sporadic. The Crab pulsar’s giantpulses, for example, are consistent with a Poisson event rate.
– 31 –
Figure of Merit for Steady Sources (Survey Speed): Steady sources or variable sources with eithervery long event durations or very large event rates (η(W 2+τ2)1/2 ≫ 1) have Pt = 1. For these, Dmax ∝ τ1/4
and the number of detected sources is
Ndet =1
3nsΩsD
3max ∝ (Bτ)3/4
(mTsys/Ae)3/2. (A4)
A convenient combination of parameters arises
N4/3det ∝ Bτ
m(Tsys/Ae)2∝ B
(
NFoVΩFoV
Nsa
)(
fcAe
mTsys
)2
. (A5)
We therefore define a survey figure of merit for steady sources as
FoMSS = B
(
NFoVΩFoV
Nsa
)(
fcAe
mTsys
)2
, (A6)
where
ΩFoV = field of view, the size of one single-dish pixel
NFoV = number of fields of view (i.e. the number of pixels in a multiple-feed cluster orphased-array feed)
Nsa = number of subarrays into which the collecting area is divided, assumed to be equal sized,with consequent reduction in gain and increase in instantaneous field of view
B = bandwidth
m = threshold signal-to-noise ratio
Ae = effective area of the entire array
fc = fraction of Ae that is usable in the survey, e.g. antennas in the core array
Tsys = system temperature.
The expression for FoMSS is consistent with that derived directly from survey speed, defined as SS = Ωi/τ ,often put in the form FoV(Ae/Tsys)
2B (sometimes without the bandwidth factor). The expression heremakes explicit for SKA applications that only the core array is usable for some surveys and that the surveyspeed depends on the significance level, m. It also shows the explicit dependence on number of fieldsof view and subarrays. It is clear that Nsa = 1 maximizes survey speed. However, there are cases whereinstantaneous sky coverage may override sheer survey speed, such as the case where very rare, fast transients
are sought (see below). Volume surveyed and the number of detected sources scale as FoMSS3/4.
Figure of Merit for Transient Sources: The volume surveyed for transient sources needs to take intoaccount the possibly altered integration time and the temporal probability factor. The former reducesDmax from what it is for steady sources. We therefore write Dmax(τeff) = Dmax(τ)(τeff/τ)1/4. The effectiveintegration time is τeff = τW/(W 2 + τ2)1/2.
The number of detected transient sources then scales as
Ndet(TS) = Ndet(SS)(τeff
τ
)3/4
Pt(η, W, τ). (A7)
The transient-source FoM then follows:
FoMTS ∝ N4/3det (TS) ∝ N
4/3det (SS)
(τeff
τ
)
P4/3t (η, W, τ), (A8)
or
FoMTS = FoMSS × K(ηW, τ/W ) (A9)
K(a, x) =(
1 + x2)−1/2
[
1 − e−a(1+x2)1/2]4/3
(A10)
where a ≡ ηW is the product of event rate per source η and event duration W , which is a measure of theoverlap of events in time, and x ≡ τ/W = Ωi/ΩW . The quantity Ω = Ωs/Ts is the mean rate at whichsolid angle is surveyed.
Figure A16 shows K(a, x) for several values of a. For x ≪ 1 and a ≪ 1, K(a, x) ≈ a4/3, so low-rate sourceswill have small values of K that imply an effectively small value for the overall metric. The function peaksat xmax ≈ a−1 for small a with a value Kmax ≈ x−1
max. Since the survey volume — and hence the number of
Fig. A16.— Plot of the temporal factor K(a, x) defined in Eq. A10 for several values of a. The vertical lines denote valuesfor x = ΩiTs/(ΩsW ), where Ωi is the instantaneously sampled solid angle, Ωs is the total solid angle surveyed in time Ts,and W is the transient duration. We assume W = 1 s in both cases. The leftward line applies to an extragalactic surveyusing the 7-beam ALFA system with 3.5 arcmin beam widths at Arecibo that surveys 30% of the sky in 2000 hr. The samex is achieved for an SKA system with single-pixel feeds having 1 deg beams that survey 80% of the sky in 5 days. The linelabelled “SKA(PAF)” is for the same SKA survey but using a phased-array feed with 100 beams.
detected sources — scales as K3/4, low values of K (i.e. rare events that are poorly sampled in solid angle)must be compensated by a large source density to make source detection probable.
Source Completeness Coefficient: We first define a source completeness coefficient, Cs ≤ 1, as theratio of the expected number of sources that are detected (in the mean) to the number of sources in thepopulation that display one or more temporal events (if variable) during a survey scan:
Cs =Ndet
Nobj. (A11)
Cs measures not just the volume that a survey samples, but also the likelihood that a source is “on” duringthe dwell time that it is pointed at. As defined, it takes into account that Dmax may not reach the fullpopulation distance Dpop or that the survey solid angle Ωs may be smaller than the population solid angle.
Consider a population of sources whose spatial distribution is uniform within a solid angle Ωpop, with allsources having the same luminosity, event rate and event duration. The completeness coefficient becomes
Cs = min
[
Ωs
Ωpop, 1
]
min
[
(
Dmax
Dpop
)3
, 1
]
Pt(η, W, τ)
Pt(η, W, Ts)(A12)
where
Dmax =
(
Lp
Smin
)1/2
= Dmax,1τ1/4eff (A13)
where Dmax,1 is the distance to which a source with pseudo-luminosity Lp can be seen with a one-secondintegration time:
Dmax,1 =
(
Lp
mSsys
)1/2
(NpolB)1/4
= 180 kpc
(
Lp
1 Jy kpc2
)1/2(10
m
)1/2 (B
1 GHz
)1/4 (Npol
2
)1/4(SKA
SKAnom
)1/2
, (A14)
– 33 –
where “SKA” is the value of Ae/Tsys and SKAnom = 2 × 104 m2 K−1 is a fiducial value for Ae/Tsys (one“SKA unit”).
For reference, the system equivalent flux density is
Ssys =
(
2kTsys
Ae
)
=
(
2k
SKA
)
= 0.138 Jy
(
SKAnom
SKA
)
. (A15)
Also useful is the number of antennas corresponding to one SKA unit:
Na = 8842D−212
(
0.7
η
)(
Tsys
35 K
)(
SKA
SKAnom
)
. (A16)
Recent work (2007 Oct) is considering designs with SKA ∼ 0.3SKAnom, requiring 2652 antennas with 35 Ksystem temperature that trades FoV against sensitivity by considering survey speeds.
For the case where Dmax < Dpop for any τ of interet, the scaling law in Eq. A12 becomes
Cs = min
[
Ωs
Ωpop, 1
](
Dmax,1
Dpop
)3
J(η, W, Ts, τ) (A17)
where
J(η, W, Ts, τ) =τ
3/4eff Pt(η, W, τ)
Pt(η, W, Ts)=
(
Wτ
(W 2 + τ2)1/2
)3/4[
1 − e−η(W 2+τ2)1/2
1 − e−η(W 2+T 2s )1/2
.
]
(A18)
Four regimes can be identified for the function J :
J(η, W, Ts, τ) =
ητW 3/4
1 − e−ηTs, W ≪ τ ≪ η−1 =⇒ Pt ≪ 1
Case I: Low-rate, fast transients;
η Wτ3/4
1 − e−ηTs, τ ≪ W ≪ η−1 =⇒ Pt ≪ 1
Case II: Low-rate, slow transients;
W 3/4, W ≪ τ & τ ≫ η−1 =⇒ Pt → 1Case III: High-rate, fast transients;
τ3/4, τ ≪ W & W ≫ η−1 =⇒ Pt → 1Case IV: High-rate, slow transients or steady sources.
(A19)
Event Completeness Coefficient: To quantify completeness in counting events instead of countingsources, we define
Ce =Ndet,e
Ntotal,e, (A20)
where Ndet,e is the number of events detected during the dwell time τ and Ntotal,e is the total number that
occur in the survey time Ts. Per source, the mean number of events detected ∼ η(W 2 + τ2)1/2 while themean number that occur ∼ η(W 2 + T 2
s )1/2. This yields an event completeness coefficient
Ce = min
[
Ωs
Ωpop, 1
]
min
[
(
Dmax
Dpop
)3
, 1
]
(
W 2 + τ2
W 2 + T 2s
)1/2
. (A21)
Staring Observations: The completeness coefficients were derived with a raster scan of the sky in mindbut they can also be applied to “staring” observations. After all, this is just a raster scan with one pointing,where τ = Ts and Ωi = Ωs. In this case we have
Cs ≡ Ce = min
[
Ωs
Ωpop, 1
]
min
[
(
Dmax
Dpop
)3
, 1
]
(A22)
and when Dmax ≤ Dpop for all τ of interest,
Cs = Ce = min
[
Ωs
Ωpop, 1
](
Dmax,1
Dpop
)3 [Wτ
(W 2 + τ2)1/2
]3/4
. (A23)
Comments and Examples:
1. For transient sources, the completeness coefficients are normalized by the number of sources thatactually burst during the survey time, Ts.
2. If prompt GRB events at radio wavelengths occur at roughly the same rate as events in thegamma-ray band (∼ 1 day−1), then a completeness coefficient >
∼ 0.01 is needed to detect a fewevents per year from this population.
3. Suppose there is only one pulsar/black-hole binary in the entire Milky Way and that it isintermittent, either intrinsically or because of interstellar scintillation. We would want a survey ofthe Galaxy to cover the source position multiple times to provide a good chance that the source isdiscovered.
4. SETI: similarly suppose there are only a few transmitting civilizations in the Galaxy and thatdetectable signals are intermittent. Sky coverage needs to be fast enough that the likelihood is highof catching ETI sources when they are on, taking into account that we do not know the nature oftheir intrinsic intermittency and that scintillation effects, as for pulsars, are likely to play a role formost sources in the Galaxy.
b. matched filtering: detection and estimation
Source detection in surveys invariably makes use of matched filtering or some approximation to it. Amatched filter (MF) by definition maximizes the signal-to-noise ratio of the test statistic used to definedetection. MFs are used for identifying sources in images, spectral lines in spectra, pulses in time series,and more complex events in spaces of higher dimensionality. Familiar examples include identification ofdispersed pulses in the ν − t plane and spectral lines with variable Doppler shifts from acceleration of thesource or the observer. The MF is usually parametric; when values of parameters are not known, the MFbecomes a family of filters whose performance is optimized over a grid of values.
We discuss here
1. Matched filtering in one-dimensional data sets (time series and spectra)
2. Matched filtering of dispersed pulses
(a) Coherent dedispersion(b) Post-detection dedispersion as an approximation to MF
3. Pulse broadening
4. Detection of periodic dispersed pulses
(a) Rail filter (folding) combined with dedispersion(b) Harmonic summing as an approximation to MF
5. Faraday rotation
General Principles: Let the observable be an intensity-like quantity (i.e. intensity vs. time, interfero-metric visibility, etc.) that is a function of a vector of independent variables x that generally will includetime, frequency, location on the sky, etc:
I(x) = Spkp(x) + n(x), (B1)
where Spk is the peak flux density, p(x) is the “shape” of the signal in x, and n is additive noise. Weassume n(x) has zero mean and has white-noise-like properties, as expressed in the correlation function〈n(x)n(x′)〉 = Wnσ2
nδ(x − x′), where Wn is the multidimensional characteristic scale of the noise in x.When x is just the time domain, for example, Wn is roughly the sample interval in a discrete, nyquist-sampled time series. Matched filtering maximizes the signal-to-noise ratio of the cross-correlation functionC(y) =
∫
dx I(x)T (x + y) of the measured intensity with a template T (x) when the template is the sameshape as the signal, T (x) = p(x). When the template is not known a priori, an approximate template or afamily of test templates may be used. This generally yields sub-optimal S/N and, when multiple templatesare used, the number of statistical trials is increased, requiring a higher threshold in order to maintain afixed false-alarm probability.
Over an ensemble, the average cross-correlation function (CCF) is 〈CIT (y)〉 = CpT (y). Additive noise inthe signal produces a CCF variance
σ2C = 〈(δC)2〉 =
∫ ∫
dx dx′ 〈n(x)n(x′)〉T (x)T (x′) = Wnσ2n
∫
dxT 2(x). (B2)
– 35 –
The signal-to-noise ratio of the test statistic is the CCF maximum divided by σC :(
S
N
)
CCF
=
(
Spk
σn
)∫
dx p(x)T (x + y)|max[
Wn
∫
dxT 2(x)]1/2
. (B3)
We can compare this with the optimal signal-to-noise ratio for matched filtering where T (x) ≡ P (x),
(
S
N
)
MF
=
(
Spk
σn
)
[∫
dx p2(x)]1/2
√Wn
≡(
Spk
σn
)(
Wp
Wn
)1/2
, (B4)
which defines Wp. The primary lesson of matched filtering is that detection algorithms can yield S/N ofthe test statistic that is no better than that for the exact MF.
Examples: Figure B17 shows examples of signal types in the frequency time plane. One basic point isthat, phenomenologically and topologically, a dispersed or otherwise frequency-swept signal can appear asa drifting spectral line and vice versa. An important difference of course is that the drift rates of frequency-swept pulses are typically much larger than the variable Doppler drift rates of spectral lines.
Pulse and Spectral Line Detection: For pulse detection in a time series, Wp is the characteristic widthof the pulse and Wn is the characteristic time scale of the noise, comparable to the sample interval indigitized data that is Nyquist sampled. By inspection, the S/N in the time series, Spk/σn, increases by a
factor (Wp/Wn)1/2 in the MF test statistic, in accord with the expected√
N law. For a spectral line, Wp
and Wn are characteristic frequency scales.
Dispersed Pulses: When a pulse propagates dispersively, its arrival time varies with frequency. In thefrequency-time plane, the signal shape I(t, ν) has a template that depends on the dispersion measure, DM,and on the temporal pulse width. “Post-detection” dedispersion sums over ν taking into account dispersive
Fig. B17.— Schematic views of six types of signals as seen in the ν − t plane. From left to right, top to bottom wehave: (a) a single cell with resolutions ∆ν and ∆t in an observational unit that spans total bandwidth B and total time T ;(b) a drifting event with instantaneous widths Wν and Wt and total spans Be and Te. For a uniform drift rate ν we haveWν/Wt = Be/Te = ν. Events of course may have curved trajectories in the ν − t plane. (c) a time-steady spectral line; (d)a broadband pulse; (e) a drifting spectral line with steady amplitude; (f) a dispersed pulse, including curvature of the pathaccording to the cold-plasma dispersion law. Drifting structures as in (b), (e) and (f) may occur from processes other thanthose mentioned (variable Doppler shift, dispersive propagation), including refraction effects, stellar and Jovian bursts, andgravitational shifts.
delays. The remaining step of matched filtering involves smoothing of the resulting time series with a filterthat matches the pulse shape, p(t).
A more exact method is “coherent” dedispersion, where the signal model is for the electric field rather thanintensity and thus involves phase as well as amplitude. Coherent dedispersion involves correction of thephase wrap of the signal that corresponds to the frequency-dependent time delays in the post-detectionmethod. As with the post-detection method, the coherent dedispersion MF is a one-parameter filter (DM);signal detection then involves smoothing with p(t), as before.
In survey applications, DM is not known a priori, so it becomes a parameter in a family of MFs. Inapplications on objects with DM known to arbitrary precision, coherent dedispersion may be viewed asexact while post-detection dedispersion is only an approximation of varying convergence to the exact case.
Pulse broadening: Multipath propagation from scattering and refraction in the ISM causes intrinsic pulseshapes pi(t) to be convolved with an asymmetric function pMP(t) to produce p(t) = pi(t)∗pMP(t). As above,the matched filter for detection is the measured shape p(t).
Periodic Dispersed Pulses: Periodicity introduces the period P into the overall MF. For a train of pulseswith fixed P and equal amplitudes, the MF is
∑
j p(t− jP ). The filter output over one cycle corresponds to
the “folded” pulse shape often computed in pulsar applications after smoothing by p(t) to obtain maximalS/N. Search applications require a family of such rail filters with parameter P . An approximate method— widely used — consists of Fourier transformation of the time series followed by summing of harmonicsfor trial periods P and trial numbers of harmonics. The resulting S/N is sub-optimal because harmonicsumming typically does not include phase information but can approach the MF result to within a factor2−1/4.
Orbiting pulsars: Orbital motion causes received pulses to be unequally spaced. For orbital periods muchlonger than the data span of interest, a single parameter is needed — the accelearation — in the MF, whichis essentially a rail filter with unequally spaced pulses. To search for objects with short orbital periods, afive Keplerian parameters need to be included in the MF. Approximate methods include standard Fourieranalysis on short data sets and summing of resultant spectra or analyzing orbital sidebands in the Fouriertransform of the entire time series (ref).
Precessing pulsars: Slowly precessing pulsars will mimic orbiting pulsars, to some extent. Objectsthat have complex, triaxial precession, may tumble and will defy a MF approach to an entire time series.Detection may need to rely on single-pulse detection.
ETI carrier signals: One type of proposed ETI signal is a simple carrier that drifts in frequency fromaccelerated motion by a drift rate ν. The MF takes into account the line width and drift rate. If the signalis transient with duration Te, that too must be included in the MF.
Faraday rotation: Rotation of the plane of polarization by an angle χ = λ2RM, where RM is therotation measure. Detection of the linearly polarized flux density is accomplished by derotating the quantityL = Q + iU in the complex plane by a phase φ = 2χ with RM as an unknown parameter.
Detection Sensitivities: Minimum detectable flux densities can be defined for the schematic cases shownin Figure B17. The rms flux density in a single ν − t cell is
σ1 =Ssys√2∆ν∆t
. (B5)
We consider matched filters that average (rather than sum) over the ν − t plane so that the minimumdetectable flux density can be written as
Smin = mσ1N−1/2νt , (B6)
where Nνt is the effective number of samples averaged and m is the detection threshold in units of the rmsMF output for the null case.
For the cases shown in Figure B17 and using definitions for characteristic widths Wν , Wt, bandwidth B
– 37 –
and time span T as shown in the figure, we have
Nνt =
1 single cell Figure B17 (a)
WνT
∆ν∆tsteady spectral line Figure B17 (c), (e)
BWt
∆ν∆tbroadband pulse Figure B17 (d), (f)
WνTe
∆ν∆t≡ BeWt
∆ν∆ttruncated line or pulse Figure B17 (b).
BWtNp
∆ν∆tperiodic pulse train (Np pulses).
(B7)
Note that Nνt and thus the detection threshold does not depend on drift rate ν, whether from Doppler driftor from dispersion delays.
c. minimum detectable flux density in pulsar surveys
Minimum detectable flux density: For periodicity surveys we calculate the minimum detectable flux densityas a function of pulse period by considering the signal to noise ratio of the harmonic sum. It is useful to firstcalculate the single-harmonic threshold, Smin1, the minimum detectable flux density for a single harmonic.This quantity is independent of any pulsar parameters and is approximately
Smin1 =mSsys√
2BT, (C1)
where Ssys is the system temperature (or system equivalent flux density, SEFD) expressed in Jy; B is thetotal bandwith; T is the total integration time; and m is the threshold in numbers of standard deviations,which we adopt (conservatively) to be m = 10. We assume that two polarization channels are combinedin surveys. The S/N of the harmonic sum will maximize for a number of harmonics Nh that depends onthe pulse duty cycle. This depends on the intrinsic duty cycle and on propagation and instrumental effectsthat broaden the pulse. For the GC, pulse broadening from multipath propagation dominates these effects.The minimum detectable flux density for a sum of Nh harmonics is roughly
Smin = N−1/2h Smin1 (C2)
but is better written in terms of the harmonic sum (see below),
Smin = h−1Σ Smin1. (C3)
We calculate Smin numerically, as follows:
1. Choose a value for pseudoluminosity Lp(ν) at the frequency of interest, ν.
2. Generate the pulse shape for an intrinsic duty cycle = 0.03 (note that this value becomes irrelevantbecause scattering dominates the observed duty cycle) with an amplitude that gives Lp(ν)
3. Calculate the FFT of the intrinsic pulse shape
4. Use the NE2001 model to calculate:
(a) DM for the particular direction (ℓ, b and distance)(b) SM (scattering measure) along the same path
5. Calculate the Fourier transforms of the functions that describe
(a) dispersion smearing over a channel bandwidth(b) pulse broadening from scattering(c) post-detection integration (e.g. the integration time in a correlator or spectrometer)
6. Multiply all Fourier factors and calculate trial harmonic sums hP, defined as a sum over FFT
magnitudes (fj) at the harmonic frequencies, j = 1, 2, . . .:
hP(Nh) = N−1/2h
Nh∑
j=1
|fj | (C4)
for different Nh, finding the one that maximizes hP; as defined, the Nh that maximizes hP
maximizes the S/N of the harmonic sum. (Note the N−1/2h factor in hP accounts for the scaling
with Nh of the noise in the rms harmonic sum.)
7. By using a zero crossing algorithm (ZBRENT in Numerical Recipes), find the minimum detectableperiod for the given Lp by solving hP(Pmin)/D2
max − Smin1 = 0.
8. Scale the pseudo-luminosity to 1.4 GHz by assuming a spectral index α = 2 (defined such that fluxdensity ∝ ν−α):
Lp(1.4 GHz) = Lp(ν) (ν/1.4 GHz)α
. (C5)
For imaging surveys, propagation effects are much less an issue because angular broadening of sources inthe GC is only about 1 arc sec. For the EVLA in the A configuration at 1.4 GHz, pulsars in the GC areresolved but not so much that the survey is desensitized.8 In an imaging survey, the minimum detectableflux density is equal to Smin1 defined above.
d. single-pulse vs. periodic pulse detection for highly modulated pulse trains
The recent discovery of the “RRAT” objects via single-pulse detection raises the question of under whatcircumstances single-pulse detection is more sensitive than a many-pulse, matched filter approach (e.g. theperiodic rail filter or the Fourier transform + harmonic summing approach that approximates the rail filter).
Consider a time series of length T that includes Np = T/P pulse periods and for which the pulse amplitudesare heavily modulated. Further consider an extreme case where only fNp of the pulses are “on” with identicalamplitude a while all others are “off”. As the first step, let the time series be smoothed by a filter thatmatches the pulse shape of an individual pulse.
Single-pulse detection: The S/N for a single pulse is(
S
N
)
1
=a
σ1, (D1)
where σ1 is the rms of additive noise after smoothing.
Many-pulse detection: We consider three approaches.
First, “fold” the pulses by convolving the time series with a periodic train of delta functions spaced by theperiod P . The maximum is S = N−1
p
∑
j aj and the ensemble average is 〈S〉 = fa, yielding(
S
N
)
Np
=fa√
Np
σ1. (D2)
For the folding method to be superior to the single-pulse method, we require(
S
N
)
Np
>
(
S
N
)
1
=⇒ fa√
Np
σ1>
a
σ1. =⇒ f > fmin = N−1/2
p =
(
P
T
)1/2
. (D3)
Thus the fraction of pulses that must be “on” is a decreasing function of the length of the time series in unitsof the pulse period. However, the actual number of required on pulses fNp =
√
Np grows with time-serieslength, but only as the square root of the number of periods.
The second many-pulse approach is Fourier transformation of the entire time series followed by harmonicsumming. This yields the same results as the folding method to within a constant of order unity.
A third approach usable in some circumstances involves recognition that pulse amplitudes are sporadic, sothat choosing the brightest pulses may yield a better folded average. Suppose we threshold the data at alevel of mtσ (with mt small) yielding N ′
p = ftNp pulse peaks that we average. Assuming that these are all“real” pulse amplitudes (not noise fluctuations), ft = f and we now get an average 〈St〉 = a (rather thanfa) and
(
S
N
)
N ′
p
=√
N ′p
(
S
N
)
1
(D4)
If, as a detection criterion for the sum, we require (S/N)N ′
p> m then we need
N ′p >
(
m
mt
)2
(D5)
pulses above threshold in order for the average to be superior to the single pulse detection. For example,if mt = 3 in the initial thresholding and m = 10, we require N ′
p = (10/3)2 ∼ 30 pulses above threshold forthis strategy to be competitive.8
We note, however, that background extragalactic souces are broadened to about 1 arcmin and are thus highly selected againstin imaging surveys of the GC.
– 39 –
e. scattering in the galactic center
The scattering diameter (FWHM) of Sgr A* at 1 GHz is 1.35 arcsec. OH/IR masers and one of the GCtransients have similar (in some cases substantially larger) angular sizes within a region approximatelycentered on Sgr A* that is about 0.5 × 0.25 in size.
In the simplest model — a thin screen at distance ∆ from Sgr A* along the line of sight of length D ≈ 8.5 kpc— the observed angular size of Sgr A*, θ0 = 1.35 arcsec, implies a pulse broadening time
τd =
(
D
∆
)(
1 − ∆
D
)(
Dθ20
8 ln 2c
)
≈ 570 sec
(
D
8.5 kpc
)2(∆
0.1 kpc
)−1(θ0
1.35 arcsec
)2
ν−4, (E1)
where ν is in GHz. The primary unknown is the GC-screen distance, ∆, but this has been constrainedthrough a wide variety of measurements to be approximately 100 pc. These measurements include scatteringdiameters of OH/IR stars and a transient source, along with Sgr A*; free-free absorption measurements onSgr A*; and number counts of extragalactic sources. Lazio & Cordes (1998) adopted a filled region thatsurrounds Sgr A* as a model more physically realistic than a screen. In LC98, the region is centered on SgrA* and is an ellipsoidal Gaussian with 1/e radius of 0.15 kpc in the (x-y) plane of the Galaxy and 0.075kpc in the z direction.
In the NE2001 model (Cordes & Lazio 2002,2003), the region was modified to numerically account for theangular sizes of all objects in the region while also including the effects of scattering from the remainderof the Galactic disk. The GC region in NE2001 is thus centered on (x, y, z) = (−0.01, 0,−0.02) kpc andthe in-plane and z-direction length scales are 0.145 kpc and 0.026 kpc, respectively. The internal electrondensity is 10 cm−3 and the fluctuation parameter is F = 0.6 × 105 (F relates the fluctuations in electrondensity to the local mean electron density).
Fig. E18.— Pulse broadening time τd plotted against distance from the sun. The red curve is for a line of sight passingthrough Sgr A* while the dashed lines are for directions that pass 2 and 5 above Sgr A*. The curves are integrations ofthe Galactic electron density model NE2001, which includes a compact, intense scattering component that is responsible forthe sharp rise in τd at D ≈ 8.5 kpc. The two vertical scales apply to radio frequencies of 1 and 10 GHz and the scaling withfrequency is assumed to be τd ∝ ν4. This scaling is consistent with that seen for the angular broadening and correspondsto the case where the phase structure function is quadratic, most likely reflecting the fact that scattering is dominated bylength scales much smaller than the inner scale of the electron density fluctuations.
The predicted angular diameter and pulse broadening time using the NE2001 model for a pulsar at thelocation of Sgr A* at 1 GHz are θ0 = 1.2 arcsec and τd = 2300 sec. The difference between the predictedangular size and that observed for Sgr A* is less than 10% and is within the fitting error of the model. The
predicted pulse broadening time is larger than for the thin-screen model for easy-to-understand reasons.Given a slab of scattering material that would scatter radiation at an angle9 θs, the observed angular size islarger the closer the slab is to the observer. For a thick medium, the weighting of each slab scales as (∆/D)2.A different weighting holds for pulse broadening, (∆/D)(1−∆/D). Consequently, a thick scattering regionwill contribute more to pulse broadening from slabs with small ∆ than they will to the angular broadening.
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9
Note that the scattering angle is not the same as the observed scattering size. The two are related by θ0 = (∆/D)θs.
– 41 –
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