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HIGH-ENERGY ASTROPHYSICS 329 A Statistical Analysis of Gamma-Ray Bursts Detected by the Konus Experiment on Venera 11 and 12 MAARTEN SCHMIDT California Institute of Technology and J.C. HIGDON Claremont Colleges ABSTRACT We discuss the advantages of using the V/Vma~ method to test well- defined samples of gamma-ray bursts for the spatial uniformity of their parent population. We have applied the V/V, test to gamma-ray bursts of duration longer than 1 second recorded by the Konus experiment aboard Venera 11 and 11 Based on a sample of 123 bursts, we find = 0.46 ~ 0.03, consistent with a uniform distribution in space. We urge that experimenters give careful attention to the detection limit for each recorded gamma-ray burst, and that quantitative data for burst properties and detection limits be published. INTRODUCTION As long as optical identifications of gamma-ray bursts are lacking QIurley et al. 1986), our best hope for establishing their distances and luminosities is through the statistics of their observed properties. In partic- ular, if evidence can be found for a departure from a uniform distribution in space, then the interpretation of this departure will provide a distance scale. The space distribution of sources contains indirect information about their distances. The distribution of gamma-ray bursts on the sly appears to be isotropic (Mazets et al. 1981a; Atteia et al. 1987), allowing only distances that are either small on a galactic scale or large on a cosmological scale. In the radial direction, the space distribution is reflected in the observed distn~ution of burst intensities. Size-frequengy distnbutions, be., 329

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330 AMERICAN AND SOVIET PERSPECTIVES the number of bursts detected per year greater than a given flux, have been much studied (e.g., Fishman 1979; Jennings and White 1980, Mazets and Golenetskii 1981, 1988; Jennings 1982, 1984, 1988; Higdon and Lingenfelter 1984, 1986~. The observed flattening of the size-frequency distribution at low fluxes, below that of a-1.5 power law indicative of a spatiality uniform distribution, has been interpreted as evidence for a source distribution confined to the galactic disk or halo (Fishman 1979; Jennings and White 1980; Mazets and Golenetskii 1981; Jennings 1982, 1984) or as evidence for cosmological distances (Pacynski 1986~. Burst detections are based on count rates rather than energy fluxes. The relation between the fluxes and the count rates Is affected by the distribution of spectral shapes and of burst durations, both of which are poorly known. Higdon and Lingenfelter (1986) showed that the deviation of the Konus size-frequency distribution at low energy fluxes from a-1.S power law is dominated by these effects. On the basis of similar arguments, Mazets (1986) suggested that the most appropriate fonn of data represen- tation is the size-frequency distribution in terms of the peak count rates Nma2 of the bursts. Employing the complete Konus data bases (Venera 11 to 14) of sources with duration larger than 1 s, Mazets and Golenetskii (1988) found that the cumulative distribution of peak count rates "shows full agreement with a -3/2 law. Deviations in the region Nma~ = 100 to 400 can undoubtedly be attributed to the loss of weak events near the detection threshold." Since under a-an law, 7J8 of all sources with Nma~ > 100 are in the range Nmar = 100 tO 400' the agreement with a-3/2 law appears to be based on only a small fraction of the observed bursts. The erects discussed here are related to variations in He detection threshold caused by variations in the background. These will unavoidably lead to a flattening of the size-frequency distnbution, even if the sources have a uniform space distribution The V/Vma~ test for uniformity of the space distribution takes into account the detection limit associated with each individual burst source. This test, which has several further advantages, is described below and then used on bursts recorded in the Konus expenment. TlIE V/VMAX TEST The ratio V/Vma~ characterizes for each individual burst in a survey its radial location within the volume of space in which it could have been observed above the detection limit. If the source population is spatially uniform, then He distn~ution of V/Vma~ will be uniform between 0 and 1. The mean value of V/Vma~ for such a sample, , is 1/2, with an r.m.s. error of (12n)~~/2, where n is the number of bursts In the sample. Values of smaller than 1/2 will result if He sources

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HIGH-ENERGY ASTROPHYSICS 331 are galactic and sampled to distances greater than the scale height of their parent population, or if sources at cosmological distances are evaluated using Euclidean geometry. A larger than 1/2 might indicate cosmological distances and evolutionary effects, such as have been found for quasars (Schmidt 1968~. The V/Vma~ test can be applied to gamma-ray bursts as follows (Schmidt et al. 1988~. Consider a burst with peak count Cp over an integration time T. Let the limiting count that would trigger the detection of this burst be Cam. In Euclidean space in which the count for a given source vanes with distance as r~2 and volume varies as r3, the ratio of the volume V out to the source distance r, to the volume Vma~ out to distance rma~ at which the source would produce a count Coin is V/Vma2 = (Cp/Ciim) I (1) Note that Me distance r, which Is unlmown for gamma-ray bursts, does not appear in the expression for V/Vma~. The limiting count Chin is usually set at a multiple of the noise associated with the background rate B. In the Konus experiment, the noise is evaluated Tom a reference count that includes both background and some burst signal. In this case, the following more general formulation of the V/Vma~ test should be applied. Based on the specific algorithm used to trigger burst detection, derive for each recorded gamma-ray burst a reduction factor Rmin, defined such that if the amplitude of the burst is multiplied by Rmin, it would just marginally be detected. In this case, it iS easy to show that V/Vma~ = R3m~n (~2) There are several advantages associated with the V/Vma~ test, all related to the fact that it treats sources in a sample Individually, rather than as an ensemble as Is done in the size-frequency distribution N(>Cp). We mention three examples where the V/Vmax test accommodates the experimental situation accurately. None of these cases can be handled properly by the size-frequengy distribution N(>Cp). 1. Usually C'im is Penned in terms or fine noise associated win the background, so C`im = k(C`,kgr`~/2. Since the background shows considerable variations, C'im will be different for different bursts. V/V takes this into account for each source. 2. If burst trigger detection requires that nvo detectors or experiments observe the burst, then each of the two observations produces a V/Vma~ ratio. The requirement of two detections means that the smaller of He two Vma2 volumes is relevant, therefore the larger of the two VJVma~ values should be used. This situation applies to the BATSE experiment aboard , ~. . ~. ~a.

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332 AMERICAN AND SOVIET PERSPECTIVES GRO, where the burst recording trigger will demand detection by two or more detectors. 3. If burst detection requires that either of two detectors observe the burst and it is observed by both detectors, then the smaller of the two V/Vma2 values should be used. This applies to the two time integration bins Mat are employed in parallel in the Konus experiment. DERIVATION OF V/VMAX FROM THE KONUS DATA On Venera 11 and 12, each Konus detector system, covering the nominal energy band from 0.05 to 0.15 MeV, consists of six scintillator detectors aligned along the axes of a Cartesian coordinate system (Mazets and Golenetskii 1981~. These detectors operate in a triggered mode. In a paper discussing burst detection on the later Venera 13 and 14 flights, Mazets et al. (1983) indicated that the count rate of each detector is monitored through analog circuits with time constants of 0.25, 1.5, and 30 seconds. When the count rate of either the 0.25 or 1.5 s circuit exceeds the count rate of the 30 s circuit by Or, a burst recording trigger pulse is generated. Mazets (1987) has confirmed that this strategy also applied to Venera 11 and 11 When this happens, the time and amplitude analyzers switch to the module most favorably oriented to view the gamma-ray burst and produce a digital record. These burst counts, plotted with 0.25 s resolution for 34 s and then with 1 s resolution for 32 s more, are displayed in the Konus catalog together with the mean background counts, measured before and after the burst planets et al. 1981a, b, c3. These times profiles were the source for our investigation of the Konus bursts. The digital data cannot be used directly to perform the V/Vma~ test, since the Konus burst detectors are analog devices which operate as resistor- capacitance circuits. These produce ~ voltage which, following a sharp pulse, decays exponentially with time constant T. If the digital burst count rate is D(t) and the analog output count Is A(t,T), then A(t,T) = / D(t'>)e T aft' -00 (3) For the present sample of bursts, the analog trigger criterion affects signifi- cantly the determination of Cp. The averages of the ratio of the peak A(t,l) to the digital peak are 0.64 and 0.82, for T = 0.25 and 1.5 s, respectively. According to Mazets et al. (1981a), the detection of gamma-ray bursts is triggered when the count exceeds the background by Or. In practice, the reference analog count (for T = 30 s) measures a pseudo background P(t) that includes both background (B is the background rate) and a contribution A(t,30) from the burst signal,

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HIGH-ENERGY ASTROPHYSICS P(t) = 30B + A(t, 30) The analog output S(t,T) for either of the time constants T = 1.5 s contains the burst signal Aft, and background BT, S(t, T) = A(t, T) ~ BT 333 (4) = 0.25 sorT (5) The trigger mechanism checks for the excess E(t,~ of the signal Set, over the pseudo background, E(t,T) = S(t,T)-(T/30)P(t) (6) The r.m.s. uncertainty of E(t,l) is composed of the noise associated with each of the two terms in eq. (6~. Since P(t) is derived over an effective integration time of as long as 30 s, its contribution to the statistical noise in E(t,~ is negligible compared to that of S(t,T). In evaluation the statistical noise in S(t,T), the experiment ignores the contn~ution of A(t,l) in eq. (5), and approximates the noise contribution of BT by that of ~/30) P(t). If the excess count E(t,T) exceeds the pseudo background by kit, then A(t, T)-(T/30)A(t, 30) > k{BT + (T/30)A(t, 30~/2 (7) In order to early out the V/Vmar test, we now ask by what factor it(t) the burst amplitude should be multiplied to yield an excess of 6 OCR for page 329
334 AMERICAN AND SOVIET PERSPECTIVES initial spike (several of the profiles show ondy the descending branch of a sharp initial spike). We have assigned a value of 0.9 lo V/Vma~ ~ each of these cases. The average value of V/Vma~ for the 123 bursts is = 0.46 ~ 0.03. If we exclude the sources GB 790215, 790323, and 800127a and b which are suspected by Atteia et al. (1987) to be solar flares, < V/VmaX > is essentially unchanged at 0.46. The mean value VlVma~ is statistically indistinguishable from 0.50. We conclude that a uniform distn~ution in space of the parent population of gamma-ray bursts from which the Konus bursts are drawn is consistent with our statistical evaluation. CONCLUSION We have employed the V/Vma~ test on a set of gamma-ray bursts detected with the Konus experiment aboard Venera 11 and 12. Unlike size-Dequency distributions, the V/Vma~ test is insensitive to variations in detection limit and in instrumental sensitivity. Hence, every burst for which the detection limit is known can be used in the test. We find that the 123 bursts of duration greater than 1 second, detected at 60 above background, have a compatible with a uniform distn~ution in Euclidean space. Therefore, the radial distribution of the Konus bursts gives us no clue to their distance scale. The application of the V/Vmaz test to gamma-ray bursts in this case is complex, because the Konus experiment produced digital time profiles, but triggered on exponential time-averaged counts produced by an analog circuit. Also, the reference count contained some burst signal. The V/Vma~ test can handle this complex case. In contrast, size-frequency distn~utions of fluxes or peak counts would have been powerless in completely modeling this experiment. It is obvious that the V/Vma~ test can only be used to its full advantage, if the required data are available, including the detection limit. Therefore, we urge that for each observed gamma-ray burst not only the peak count Cp be published, but also information about the detection limit, either in the form of the limiting count Coin, or the minimum reduction factor Rmin. We would Lee to acknowledge discussions with G. Hueter, and a communication from E.P. Mazets which was most helpful in interpreting the published Konus data. This work was supported by a grant from the President's Fund of the California Institute of Technology. REFERENCES Atteia, J.L-, er al. 1987. A second catalog of gamma-ray bursts: 1978-1980 locations from the interplanetary network. Astrophysical Journal Supplement Series 64: 305-38Z l

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HIGH-ENERGY ASTROPHYSICS 335 Fishman, GJ. 1979. Galactic distribution models of gamma-ray burst sources. Astrophysical Journal 233: 851-856. Higdon, J.C, and R.E. Lingenfelter. 1984. S~ze-frequency distributions of gamma-ray bursts from nuclear runaway on neutron stem Secreting interstellar gas. Pages 568-577. In: Woosley, S.E. (eddy. High Energy Transients in Astrophysics American Institute of Physics, New York Higdon, J.C, and IRE. Lingenfelter. 1986. Gamma-ray bunt size-frequency distributions: Spectral and temporal selection effects. Astrophysical Journal 307: 197-204. Hurley, K, I:L Cline, and R Epstein. 1986. Error boxes and spatial distnbution. Pages 33-38. In: Liang, E.P., and V. Petrosian (eds.~. Gamma-Ray Bursts. American Institute of Physics, New YorL Jennings, M.C. 1982. The Galaxy as the origin of gamma-ray bursts. II. The effect of an intrinsic burst luminosity distribution on log N(>S) versus log S. Astrophysical Journal 258: 11~120. Jennings, M.C 1984. The gamma-ray burst spatial distribution log N(>S) versus log S and N(>S,l,b) vs. S. Pages 412-421. In: Woosley, S.E. (ed.~. High Energy Transients in Astrophysics. American Institute of Physics, New York Jennings, M.C 1988. Intrinsic and artificial bias in the Konus cumulative number distribution. Astrophysical Journal 333: 700-718. Jennings, M.C, and R.S. White. 1980. The Galaxy as the origin of gamma-ray bursts. Astrophysical Journal 238: 110-121. Mazets, E.P. 1986. Observational properties of cosmic gamma-ray bursts. In: Proceedings 19th International Cosmic Ray Conference (I~ Jolla) 9: 415~30. Mazets, E.P. 1987. Private Communication. Mazets, E.P., and S.V Golenetskii. 1981. Cosmic gamma-ray bursts Astrophysics and Space Science Reviews 1: 205-266. Mazets, E.P., and S.V. Golenets~i. 1988. Observations of cosmic ga~nma-ray bursts. Astrophysics and Space Physics Renew 6: 281-311. Mazets, E.P., et al. 1981a. Catalog of cosmic gamma-ray bursts from the Konus experiment data, Part I and II. Astrophysics and Space Science 80: 3~3. Mazets, E.P., et al. 1981b. Catalog of cosmic gamma-ray bursts from the Konus e~cpenment data, Part III. Astrophysics and Space Science 80. 85-117. Mazets, E.P., et al. 1981c. Catalog of cosmic ga~nma-ray bunts from the Konus experiment data, Part IV. Astrophysics and Space Science 80: 11~143. Mazets, E.P., et al. 1983. Energy spectra of the cosmic gamma-ray bursts. Pages 3~53. In: Burns, M.L, and ASK Harding, and R Ramaty feds). Positron-lilectron Pairs in Astrophysics. American Institute of Physics, New York. Paczynski, B. 1986. Gamma-ray bursters at cosmological distances. Astrophysical Journal (Lettem) 308: L43-L46. Schmidt, M. 1986. Space distribution and luminosity functions of quasi-stellar radio sources. Astrophysical Journal 151: 393407. Schmidt, M., J.C Higdon, and G. Hueter. 1988. Applications of the V/Vma2 test to gamma-ray bursts. Astrophysical Journal (Lettered 329: L85-L87.