Recent analysis of relativistically expanding shells of cosmological
gamma-ray bursts has shown that if the bursts are cosmological, then most
likely total energy (E_0) is standard and not peak luminosity (L_0). Assuming a
flat Friedmann cosmology (q_o = 1/2, Lambda = 0) and constant rate density
(rho_0) of bursting sources, we fit a standard candle energy to a uniformly
selected log N-log S in the BATSE 3B catalog correcting for fluence efficiency
and averaging over 48 observed spectral shapes. We find the data consistent
with E_0 = 7.3^{+0.7}_{-1.0} X 10^{51} ergs and discuss implications of this
energy for cosmological models of gamma-ray bursts.Comment: A five page LateX file that uses the Revtex conference proceedings
  macro aipbook.sty, and includes three postscript figures using psfig. To Be
  published in the Proceedings of the Third Hunstville Symposium on Gamma-Ray
  Bursts, eds. C. Kouveliotou, M.S. Briggs and G.J. Fishman (New York:AIP).
  Postscript version availible at http://nis-www.lanl.gov/~jsbloom/LOG_S.p

Bloom, Joshua S.

Fenimore, Edward E.

Zand, Jean in 't

English

arXiv

Joshua S. Bloom

Edward E. Fenimore

Jean in ’t Zand

Crossref

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6
THE CORRECTED LOG N-LOG
FLUENCE DISTRIBUTION OF
COSMOLOGICAL γ-RAY BURSTS
Joshua S. Bloom1,2, Edward E. Fenimore2, Jean in ’t Zand2,3
1Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138
2Los Alamos National Laboratory, Los Alamos, NM 87544
3Goddard Space Flight Center, Greenbelt, MD 20771
Recent analysis of relativistically expanding shells of cosmological
γ-ray bursts standard and not peak luminosity (L0). Assuming a flat
Friedmann cosmology (qo = 1/2, Λ = 0) and constant rate density
(ρ0) of bursting sources, we fit a standard candle energy to a uniformly
selected log N-log S in the BATSE 3B catalog correcting for fluence
efficiency and averaging over 48 observed spectral shapes. We find the
data consistent with E0 = 7.3
+0.7
−1.0 × 10
51 ergs and discuss implications
of this energy for cosmological models of γ-ray bursts.
INTRODUCTION
On the basis of strong threshold effects of detectors, Klebesadel, Fenimore,
and Laros (1982) concluded that GRB fluence tests were largely inconclusive.
As a result, nearly all subsequent number-brightness tests have used peak flux
(P ) rather than fluence (S). However, the standard candle peak luminosity as-
sumption that is required by logN -log P studies is unphysical. If, for instance,
bursts originate at cosmological distances and are produced by colliding neu-
tron stars (11) then one might expect that total energy would be standard
and not peak luminosity. Moreover, recent analysis of the time histories in
relativistically expanding shell models has found the required differences in
bulk Γ factor between different GRBs all but eliminates the possibility of a
standard candle luminosity in such models (9).
In this paper, we seek to eliminate the large threshold effects present in
log N -log S studies by correcting the observed number of bursts at a given
fluence by the trigger efficiency of the detector. In §I we use the calculated
trigger efficiency in PVO (6) and the catalogue of PVO events (4) to test the
correction algorithm. In §II we examine the fitting algorithm to the log N -log
S curve in BATSE 3b and find a standard candle energy for cosmological
gamma-ray bursts. In §III we discuss the implications of such an energy and
the distances implied by the fit.
I. PVO CONSISTENCY CHECK
The Pioneer Venus Orbiter (PVO) had a peak flux trigger system (sampled
on 0.25, 1.0, and 4.0 sec timescales) and was sensitive to bursts down to fluxes
c© 1996 American Institute of Physics 1
2
of 5× 10−6 erg cm−2. Despite a substantially lower fluence trigger sensitivity
range, PVO saw hundreds more bright bursts than BATSE due the relatively
long on-time and large sky-coverage of PVO. As the bright region of the
BATSE log N -log S curve seems to fit a -3/2 power law well, we would expect
that the entire log N -log S curve of PVO should show a similar behavior.
Using the PVO trigger efficiency, ǫ(S), from in t’ Zand & Fenimore (1996),
for each burst i with fluence Si in the PVO catalogue we take the expected
number of bursts with pre-detection fluence to be Ni,true = Ni,obs/ǫ(Si). We
then compare the derived log N(> S)-log S curve with a -3/2 power law
with arbitrary S-intercept as seen in figure (1b). Although at lower fluence
there appears to be a deviation from -3/2, the fit is good: with a Kolmogorov-
Smirnov (KS) probability of 40% that the corrected distribution comes from a
-3/2 power law. We derive this KS statistic by finding the maximum distance
between the corrected and -3/2 distributions (in linear space) down to S =
10−4.5 erg cm−2, the fluence at which ǫ(S) falls to 50%. By comparing the
distributions down to lower fluences, we find an even better KS fit. This
is understandable because as we add more bursts to the distribution, small
deviations at high fluences contribute less to the KS distance parameter. Note
that although the fit is acceptable, a -3/2 slope is not necessarily required by
the corrected data. We thus conclude that the trigger efficiency determination
algorithm from in t’ Zand and Fenimore (1996) is sound, at least in PVO.
II. Deriving the log N-log S Curve
A. BATSE Trigger Efficiency
One subtly worth noting is that BATSE trigger efficiencies are model depen-
dent, ie. they depend on the choice of E0 and cosmology, since it is necessary
to know a priori the true underlying distribution of bursts that passes by the
detector.
3
FIG. 1 The log N -log S curve
for PVO. a) The uncorrected
curve for 293 events in the PVO
catalogue which shows signif-
icant departure (KS ≃ 0.0)
from the -3/2 power law (shown
as solid line) expected from
BATSE observations. b) The
corrected curve (KS = 0.40 at
ǫ(S) = 0.50) using the PVO
trigger efficiency from in t’ Zand
& Fenimore (1996). Any de-
viation from a -3/2 power-law
at low fluences we attribute to
an incomplete understanding of
the trigger efficiency.
In fact, the derivation of the PVO ǫ(S) assumed an underlying -3/2 distri-
bution. Petrosian and Lee (1996) have constructed trigger efficiencies us-
ing bivariate correlation. While this method does not assume a particular
cosmology, it does require that GRB brightness and duration are inherently
uncoupled. Our method does not have this requirement and we make no as-
sumptions about the bursts other then they are cosmological in origin. The
BATSE trigger efficiencies could be calculated for any E0 (q0 = 1/2, Λ = 0)
and two are depicted in figure (2a). Note that the efficiency is nearly unity
for the several orders of magnitude in fluence. The corrected log N -log S
curve for BATSE is depicted in figure (2b) for two values of E0. Interestingly,
the two distributions are nearly identical for most of the fluence range. In
addition, it is clear that the bend from -3/2 in log N -log S is true.
B. Standard Candle Energy Fits
The observed fluence of a source depends strongly on the spectrum, and
since the observed spectral shape depends on the distance to the object, the
intrinsic spectrum of a GRB object must be used. In addition, the normal-
ization and the spectral shape vary over the duration of the burst, adding to
the uncertainty in analysis.
Following a similar analysis as in Fenimore and Bloom (1995), we take as
our baseline spectra averages over the GRB spectra fit by Band et al. (1993).
Each such baseline burst has associated with it an observed fluence, Si, and
an observed spectral shape, φi(E).
4
a b
FIG. 2: a) The BATSE trigger efficiency for different E0 and b) the corrected
log N -log S curve for 830 BATSE bursts. Along with the uncorrected log N -
log S curve, we depict a corrected curve corresponding to an assume standard
candle energy of E0 of 10
52 ergs (dot-dash line) and a corrected curve where
the effect of redshift on the baseline spectra is removed (dash line).
Since each Band et al. (1993) burst spectrum is averaged over the duration
of the burst, we assume in the following analysis that the spectral shape is
constant, that is, φ(E, ts) ≃ N(ts)φi(E). The fluences, Si [ergs cm
−2], are
available for 48 of the Band et al. (1993) bursts in BATSE 3B (10).
The observed spectral shape, φi(E), will not necessarily come from a burst
at z ∼ 0 especially if E0 is large. Therefore, for a given E0, Si, and φi(E)
we first solve for the redshifts, zi, of the baseline events associated with each
spectral shape. The standard candle energy, E0, is given by,
E0 = 4πR
2
i,z
∫ ∞
0
N(ts)dts
∫ 2000
30
Eφi
(
E
1 + zi
)
dE (1)
where N(ts) is the normalization of the spectrum (units of ergs keV
−1) as
a function of time at the source. The comoving distance, Ri,z, is defined in
eq. [2] of Fenimore & Bloom (1995). The energy range used in calculating
E0 in eq. (1) is taken as 30-2000 keV, since we later compare E0 to standard
candle peak luminosity found in the same energy band.
The observed fluence of the ith baseline burst in the energy range 50-300
keV is given by,
Si =
∫ ∞
0
N(tobs)dtobs
∫ 300
50
Eφi
[
1 + zr
1 + zi
E
]
dE, (2)
where N(tobs) is the observed normalization of the spectrum.
For a given standard candle energy, E0, we numerically determine the red-
shift (1 + zi) of the ith baseline burst using eqs. (1, 2) and letting zr = zi.
Note that (1 + zi)
∫
N(ts)dts =
∫
N(tobs)dtobs.
5TABLE 1. Best Fit Distribution of E0 = 7.0 × 10
51 ergs
Fluence Rangesa(50-300 keV) ∆N [Sj to Sj+1]
Bin Number(j) Sj Sj+1 Observed
b Predicted 1 + zj
1 2.16e-07 3.82e-07 51 38.4 3.88
2 3.82e-07 5.85e-07 42 50.5 3.24
3 5.85e-07 7.55e-07 42 36.3 2.84
4 7.55e-07 1.13e-06 46 63.0 2.64
5 1.13e-06 1.43e-06 37 36.8 2.36
6 1.43e-06 2.00e-06 48 49.7 2.22
7 2.00e-06 2.80e-06 39 44.3 2.04
8 2.80e-06 4.05e-06 44 41.1 1.89
9 4.05e-06 6.20e-06 37 37.2 1.74
10 6.20e-06 1.36e-05 40 44.7 1.60
11 1.36e-05 6.60e-05 41 32.3 1.41
a In ergs cm2
b Bursts with Cmin/Cmax > 1 on the 256 or 1024 ms timescale in BATSE 3b.
Instead of assuming a spectral shape at the source, we use an average
over baseline spectra to compute the number of expected observed bursts,
∆Nexp[Sj to Sj+1] in some fluence range [Sj , Sj+1]:
∆Nexp[Sj to Sj+1] =
4π
NBAND
NBAND
∑
i=1
∫ R(Sj+1)
R(Sj)
ǫ[Si(r)]
ρ0
1 + zr
r2dr. (3)
where NBAND = 48 is the number of baseline spectra used and ρo is the rate
density of bursts per comoving volume. The quantity Si(r) is the predicted
fluence (using eqs. [1, 2]) of the ith baseline burst if it was at a distance r.
This distance corresponds to a redshift 1 + zr.
We construct 11 fluence bins (in BATSE channels 2+3 corresponding to
approximately 50-300 keV) of roughly equal number of bursts. We select
bursts with Cmin/Cmax > 1 on either the 256 or 1024 ms timescale, then
find a minimized χ2 between the number of predicted bursts and observed
by varying E0. For 9 degrees of freedom we find an acceptable χ
2 = 14.7
corresponding to a standard candle E0 = 7.3
+0.7
−1.0 × 10
51 ergs. Table (1) gives
the bin ranges, number of observed bursts per bin, number of predicted bursts
for the best fit energy, and their implied redshifts.
III. CONCLUSIONS
Our fit of E0 = 7.0
+0.7
−1.0 × 10
51 [30-2000 keV] ergs seems a plausible number
on the basis that GRBs last on the average 10 sec and L0 = 4.6×10
50 erg s−1
from logN -log P studies (2). However, this E0 implies a rather large efficiency
of energy conversion to γ-rays (∼ 10%) if the bursting mechanism is colliding
neutron stars (Mtotal ≃ 2.8M⊙). Nevertheless, this result would seem to help
resolve the “no-host” problem (cf. Fenimore 1993 et al.). Interestingly, that
the dimmest bursts (S ≃ 5×10−8 erg cm−2) are required to be at a redshift of
1+ z ≃ 6.4 given this E0, would seem to rule out several cosmological models
that require GRB progenitors to be within galaxies (although see Lu et al.
6
1996). This surprisingly high redshift is due to the correct blueshifting of the
baseline spectra back to the source in eq. (1). If we neglect this factor, we
obtain a smaller, more tenable redshift of the dimmest bursts (1 + z = 5.2).
Whatever the conclusion about the models, we note two important results.
First, the bend in the log N -log S curve in BATSE is real, not an artifact of
strong threshold effects. This implies that we are seeing either a truncated
spatial distribution of GRBs (as in Galactic models) or an effect due to the
expansion of the universe. The bend might also be caused by a combination of
rate density or number density evolution, and a study of their possible effects is
certainly warranted. Secondly, with the availability of Monte Carlo modeling
of trigger efficiencies, log N -log S tests need no longer be inconclusive.
REFERENCES
1. Band et al. 1993 Ap.J.,413, 201.
2. Fenimore, E.E., and Bloom, J.S., 1995 Ap.J. 235, 29-35.
3. Fenimore et al. 1993 Nature, 366, 40.
4. Fenimore et al. 1996, in preparation.
5. Higdon, J.C., Lingenfelter, R.E. 1986 Ap.J., 307, 197.
6. in t’ Zand, J., and Fenimore, E. E., 1996, these proceedings.
7. Klebesadel, R.W., Fenimore, E.E., and Laros, J.G. 1982 in AIP Conf. Proc. 115
(ed. S. Woosley), 429-433.
8. Lu, L., Sargent, W.L.W, Womble, D.S., Barlow, T.A. 1996 Ap.J., 457, L1.
9. Madras, C., Fenimore, E.E. 1996, in preparation.
10. Meegan et al. 1996, Ap.J.S., in press.
11. Paczynski, B. 1990 Ap.J., 363, 218.
12. Petrosian, V., and Lee,T., 1996, these proceedings.


The corrected log N-log fluence distribution of cosmological γ-ray bursts

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The Corrected Log N-Log Fluence Distribution of Cosmological Gamma-Ray Bursts

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