Analysis of the Systematic Errors in the Positions of BATSE Catalog Bursts

5 Pages·0.22 MB·English

by  Carlo Graziani

Analysis of the Systematic Errors in
the Positions of BATSE Catalog Bursts
C. Graziani1 and D. Q. Lamb2
1NASA Goddard Space Flight Center, Greenbelt, MD 20771
2Dept. of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637
We analyze the systematic errors in the positions of bursts in the
BATSE 1B, 2B and 3B catalogs, using a likelihood approach. We
use the BATSE data in conjunction with 196 single IPN arcs. We
assumecircularGaussianerrors,andthatthetotalerroristhesumin
6
9 quadrature of the systematic error σsys and statistical error σstat, as
9 prescribedbytheBATSEcatalog. Wefindthatthe3Bburstpositions
1 are inconsistent with the value σsys = 1.6◦ stated in the BATSE 3B
catalog.
n
a
J INTRODUCTION

4
2 The stated systematic error in the BATSE 3B catalog (1) burst locations
◦
is σsys = 1.6 . This value was estimated by taking the RMS deviation of
1 BATSE positions from knownpositions of 36 bursts, determined by the IPN,
v
WATCH, and COMPTEL (2). Unfortunately, many of these same known
3
positions were used to calibrate the BATSE burst locationsoftware,i.e., as a
4
1 guideindeterminingwhateffectstoincludeintheburstlocationalgorithm. In
1 order to calibrate properly the BATSE burst position errors, an independent
0 set of burst locations is needed. Such a set exists, in the form of 196 bursts
6
for which single IPN arcs exist (3). In this paper we analyze the systematic
9
errors in the positions of bursts in the BATSE 1B, 2B and 3B catalogs using
/
h these 196 bursts. Our analysis is based upon the likelihood approach.
p
-
o
ANALYSIS
r
t
s We assume that the systematic error σsys and statistical error σstat in po-
a sitionarecircularGaussians,to be addedin quadrature,asprescribedby the
:
v BATSE catalog. The circular approximation should be good, since roughly
i 2/3 3/4ofallBATSEburstshaveχ2 positionalcontoursthatarenearlycir-
X
−
cular (2), and the bursts with IPN arcs tend to be those with larger fluences.
r
a We further assume that the sky is flat (i.e., that the size of the total error
in the BATSE burst position is not too large), and that the IPN arcs have
zero width. The former is a good approximation, since the largest BATSE
◦ ◦
σ 12 90 among the bursts with IPN arcs. The latter is a good ap-
tot
≈ ≪
proximationsincethecharacteristicwidthofthe3-σ contoursoftheIPNarcs
◦
is a few arcminutes, which is much less than the smallest BATSE σ 1 .
tot
≈
c 1993 American Institute of Physics 1
(cid:13)
2
Figure 1. Configurations for a burst with a known position (left diagram)
and for a burst with a single IPN arc (right diagram).
Figure2. Probabilitydensityasa functionofσ for the MLconstantσ
sys sys
model. The vertical lines show the 1-σ interval.
±
Thelikelihoodfunctionforburstswithknownpositions(e.g.,twointersect-
ing IPN arcs) is given by
N N 1 1d2
L(σsys)=YLi ≡Y2πµ2 exp(cid:0)−2µi2(cid:1), (1)
i=1 i=1 i i
whereas the likelihood function for bursts with single IPN arcs is given by
Narc Narc 1 1a2
L(σsys)= Y Li ≡ Y √2πµ exp(cid:0)−2µi2(cid:1). (2)
i=1 i=1 i i
Here, d is the deviation of the BATSE position from the known position, a
i i
is the perpendicular deviation of the BATSE position from the IPN arc (see
Figure 1), and µ =0.43σ =0.43(σ2 +σ2 )1/2.
i tot sys stat
The likelihood function (σ ) allows exploration of any model that we
sys
L
may have for σ . Here we focus on three models: (1) σ = a constant,
sys sys
(2) σ = A(S/10−5 erg cm−2)α, and (3) σ = A(σ /1◦)α. The first
sys sys stat
model has one free parameter (σ ); the second and third models have two
sys
free parameters (scale factor A and power law index α).
RESULTS
Fitting the constant σ model to the 18 BATSE 3B bursts with IPN
sys
positions,wefinda maximumlikelihood(ML) valueσsMysL =1.85◦+−00..2282◦◦. This
◦
isconsistentwiththevalueof1.6 quotedintheBATSE3Bcatalog(1),which
was found using these bursts and some others.
Fittingtheconstantσ modeltothe196BATSE3BburstswithIPNarcs,
sys
we find a ML value σsMysL =3.7◦+−00..2242◦◦. This is inconsistent (at the 10σ level!)
3
Figure 3. Comparison of expected and observed cumulative distributions of
the normalized deviation D = a /σi for the burst samples and data as
| i tot|
labeled, assuming the ML models of σ as labeled.
sys
◦
with the value of 1.6 quoted in the BATSE 3B catalog, as shown by the
probabilitydistributioninσ (seeFigure2)andcomparisonoftheexpected
sys
and observed distribution of normalized deviations D = a /σi assuming
◦ | i tot|
σ =1.6 (see Figure 3).
sys
4
Fitting the second model, in which σ =A(S/10−5 erg cm−2)α, we find
sys
Figure 4. Upper left-hand and lower and upper right-hand panels: 1-σ, 2-σ,
and3-σcontoursinthe(A,α)-planefortheburstsamplesanddataaslabeled,
assuming the σ -dependent model. Lower left-hand panel: same as Figure
stat
2, except for the σ -dependent model.
stat
ML values A = 3.8 0.3 and α = 0.18+0.06. The probability P(α
0) = 1.6 10M−L3, showi±ng that theMdaLta p−refer −th0e.05power-law model over th≥e
× ◦
ML model with constant σ =3.7 .
sys
Fitting the third model, in which σ =Aσα , we find ML values A =
sys stat ML
3.8 0.2 and α = 0.32+0.06 (see Figure 4). The probability P(α 0) =
4.7±10−5, showMiLng that t−h0e.08data strongly prefer the power-law mod≤el over
× ◦
the ML model with constant σ =3.7 .
sys
These results imply that σ is strongly correlated with S and σ . For
sys stat
◦ ◦ ◦ ◦
example, σ (σ = 0.1 ) = 2 while σ (σ = 10 ) = 8 . This is il-
sys stat sys stat
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lustrated in Figure 5, which shows the ML S-dependent and σ -dependent
stat
models.
Figure 5. Left-hand panel: The ML S-dependent model. Right-hand panel:
The ML σ -dependent model. The points are not data points, but rather
stat
are the ML constant σ model for the given interval in S or σ .
sys stat
It is also interesting to examine the character of σ for subsets of the 3B
sys
catalog,in particularthe 1B sample ofbursts inwhich evidence forrepeating
wasfound. Fitting the three models to the “old”1B positions,wefind σ =
sys
◦
4.0 fortheMLconstantσ model,andthatthe datadonotrequestamore
sys
complicated model(see Figures 3 and 4). In contrast, fitting the 3B positions
forthe1Bsampleofbursts,wefindthatthedatarequesttheσ -dependent
stat
model at the 95% confidence level [e.g., P(α 0) = 1.6 10−2] (again see
≤ ×
Figures 3 and 4).
These results imply that for the 1B sample of bursts, the new (3B) σ ’s
sys
are smaller than the old (1B) ones for bursts with S >4 10−6 erg cm−2 or
◦ ×
with σ < 1.6 . However, for the 80% of bursts with lower fluences or
stat
≈
larger σ ’s, they imply that the new σ ’s are larger than the old ones.
stat sys
We are grateful to Kevin Hurley for providing us with the 3rd catalog of
IPN arcs, without which this analysis would not have been possible. We also
wishtothankGeoffPendletonandMichaelBriggsfordiscussingBATSEburst
location issues with us.
REFERENCES
1. C. A.Meegan, et al., electronic catalog (grossc.gsfc.nasa.gov).
2. G. N.Pendleton, privatecommunication (1995).
3. K. Hurley,et al., Third IPN Catalog (1995).