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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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Representative terms from entire chapter:
konus experiment
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
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.
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