The apparent conflict between log N-log S curve and isotropic celestial distribution of the gamma ray bursts is discussed. A possible selection effect due to the time profile of each burst is examined. It is shown that the contradiction is due to this selection effect of the gamma ray bursts

Nishimura, J.

Yamagami, T.

English

NASA Technical Reports Server

OG 1.2-10
SS
RELATIONSHIPS BETWEEN LOG N-LOG S AND CELESTIAL
DISTRIBUTION OF GAMMA-RAY BUSRTS
Jun Nishimura and Takamasa Yamagami
Institute of Space and Astronautical Science
Komaba, Meguro, Tokyo, Japan
ABSTRACT
We discuss the apparent conflict between log N-log S curve
and isotropic celestial distribution of the gamma-ray
bursts. We examined a possible selection effect due to the
time profile of each burst, and showed the contradiction is
due to this selection effect to the gamma-ray bursts.
I. INTRODUCTION
The apparent contradiction between the isotropic latitude
distribution and the bending of log N-log S curve of gamma-
ray bursts has been discussed for many years. To resolve
this contradiction, an effect of intrinsic luminosity
distribution of the bursts (T.L.Cline and U.D.Desai: 1976),
a large scale halo model ( M.C.Jennings: 1984) etc. have
been suggested so far. Based on the discussion in the
reference (J.Nishimura and T.Yamagami: 1985), we showed, in
disk model, latitue distribution of burst sources is
directly related to the shape of log N-log S curve without
referring to an explicit form of intrinsic luminocity
function of the burst source. Then, we examined a possible
selection effect in the data of Venera 11 and 12. (E.D.Mazet
et al: 1981,1982). After analyzing the Venera data, we
found a selection effect is arised from the different time
profile of each burst. The contradiction is resolved if we
take account of this selection effect.
2. RELATION BETWEEN LOG N-LOG S AND LATITUDE DISTRIBUTION
OF GAMMA-RAY BURST IN THE DISK MODEL
Based on a general discussion of disk model for the spatial
distribution of burst sources, we show latitude
distribution of burst sources has a one to one
correspondence with the form of log N-log S distribtuion
without referring to the explicit form of intrinsic
luminosity distribution and the form of distribution of the
burst sources in height from the disk plane. In this
treatement, we assume that the spatial distribution of the
burst sources is given by a form of G(Z)dZ , where Z is the
height of the burst source position from the disk plane. No
azimuthal asymmetry is included. The following arguments,
however, holds even in case some azimuthal asymmetry
exists, if we redefine G(Z) as an average over azimuthal
angle, or a function of the azimuthal angle.
• 
OG 1.2-10 
55 
RELATIONSHIPS BETWEEN LOG N-LOG S AND CELESTIAL 
ISTRIBUTION OF GA MA-RAY B SRTS 
Jun Nishimura and Takamasa Yamagami 
Institute of Space and Astronautical Science 
Komaba, Meguro, Tokyo, Japan 
BSTRACT 
We discuss the apparent conflict between log N-log S curve 
and isotropic celestial distribution of the gamma-ray 
bursts. We examined a possible selection effect due to the 
time profile of each burst, and showed the contradiction is 
due to this selection effect to the gamma-ray bursts. 
1. INTRODUCTION 
The apparent contradiction between the isotropic latitude 
distribution and the bending of log N-log S curve of gamma-
ray bursts has been discussed for many years. To resolve 
this contradiction, an effect of intrinsic luminosity 
distribution of the bursts (T.L.Cline and U.D.Desai: 1976), 
a large scale halo model (M.C.Jennings: 1984) etc. have 
been suggested so far. Based on the discussion in the 
reference (J.Nishimura and T.Yamagami: 1985), we showed, in 
disk model, latitue distribution of burst sources is 
directly related to the shape of log N-log S curve without 
referring to an explicit form of intrinsic luminocity 
function of the burst source. Then, we examined a possible 
selection effect in the data of Venera 11 and 12. (E.D. azet 
et al: 1981,1982). After analyzing the Venera data, we 
found a selection effect is arised fro  the ifferent ti e 
profile of each burst. The contradiction is resolved if we 
take account of t is selection effect. 
2 • RELATION BET EEN LOG N-LOG S AND LATITUDE DISTRIBUTION 
F GA MA-RAY RST I  T E IS  ODEL 
Based on a general discussion of disk model for the spatial 
distribution of burst sources, we show latitude 
distribution of burst sources has a one to one 
correspondence with the form of log N-log S distribtuion 
without referring to the explicit form of intrinsic 
luminosity distribution and the form of distribution of the 
burst sources in height from the disk plane. In this 
treate ent, we assume that the spatial distribution of the 
burst sources is given by a form of G(Z)dZ , where Z is the 
height of the burst source position from the disk plane. No 
azimuthal asymmetry is included. The following arguments, 
however, holds even in case some azimuthal asymmetry 
exists, if we redefine G(Z) as an average over azimuthal 
angle, or a function of the azimuthal angle. 
https://ntrs.nasa.gov/search.jsp?R=19850025605 2020-03-20T17:55:35+00:00Z
OG 1.2-10
56
Also, an integral intrinsic luminosity distribution in each
burst sources is assumed to be
f(L/4 _ ),
where L is the total luminocity of a burst source. No
dependence of the form of the intrinsic luminocity
distribution on the position of the source is assumed here.
Under these assumptions, we derive a simple relation between
celestial distribution at latitude b per sterad., N1(>S,b),
and observed frequency N(>S,b) as
NI(>S,b) = N2( S /sin2b )/2 _ sin3b ,
where
N2(S) = -dSN(>S)/dS. (I)
This relation does not include any explicit forms of spatial
distribution and intrinsic luminocity function , and then
observed log N-log S curve can be directly compared with the
observed latitude distribution of the burst sources. Any
effects to celestial distribution due to the spatial
distribution of the burst sources are included in the form
of log N-log S curve. (J.Nishimura and T.Yamagami: 1985)
The latitude distribution derived by using eq.(1) from the
observed log N-log S curve by Venera, PVO and others is
shown in Fig.1. There is a large deviation from isotropic
< -5 2latitude distribution for S 10 erg/cm , which contradicts
the observed evidence of located bursts of Venera 11 and 12.
. , ., ., _, • ,- /-2,10' N(>S)=No/(S/So ) I+S/S0)I "
eV.F
&, M N
10' b ,,-_ • S!L "..,7.c -1-....."" .
!
I c%oo J" • I
I lJ|l
I0-' I0" I0-' I0" I0" I0" I0' IO' Io'
S (erg/cm t) P..kC_,..,.(ct,/0.25,,c)
Fig.1. Latitude distribution of Fig.2. Scattered plot of the burst
burst sources derived by observed by Veuera [or
eq(1) and log N - log S fluence S vs Peak count P.
curve.
 . -10 
 
lso, an i tegral i trinsic lu inosity istribution i  each 
urst urces  ed   
/4 n , 
here  i  t e t tal l inocity f a urst urce. o 
ependence f t  for  f t  i trinsic l inocity 
istribution  e osition f  urce  ed ere. 
nder t ese sumptions, e rive a si ple r lation et en 
elestial istribution t l tit de b er st rad., 1(>S,b), 
a  served fr uency (>S,b) s 
1 (>S,b)  2(  2'b /  n iT13 b , 
here 
 2 ( S)   ( > S ) / dS.  1 ) 
his relation does not include any e plicit for s of s atial 
istribution d i trinsic l inocity f nction, a  t n 
bserved l  -log  rve   irectly pared ith t  
bserved l tit de istribution f t  rst s urces. ny 
effects to celestial istribution due to the s atial 
istribution f  urst urces re luded    
f l  -log  rve. ( . ishimura d . am gami: 85) 
he tit de istribution erived  sing .(1) fr  t  
bserved l  -log S c rve by enera, P  and t ers is 
sho n i  ig.1. here is a lar e eviation fro  is tropic 
l tit de istribution f r S  5 er /c 2 , hich ntradicts 
e served idence f ated ursts f enera  d . 
10' 
1O· 
O' 
10' • 
Ii 
t 10' -
..: 
t,; 
"-j 1 
" 
1<1' 
f 1<1 . 
Fig.l. 
(>S)-No/(S/So)(I / )1/2 10 ., 
--
" 
" 
10 t .. 
• 
-
. ,-
~ - '-
• 
• 
", 
I; ~. 
" . 
" 
- -
- -
--
-
• V.f' 
of 
a  
"S 
.W.P 
• • ~ " i· V; P- • r--b-30' -
bleO' 
b.gO· 
I--L..L_-,-l _--,-I,---L- L--...J-I 
1(1" 10" 10" 10" 10' tlf' 
S (erg/em') 
Latitude distribution of 
burst sources derived by 
eq(1) and log N - log S 
r e. 
u 
~ 10 
.. 
.. 
CIl 
10 
S~~ 0 'r- 0 0 
-
'-' t~~' • -I-' -co ':_~ 0 • • e .-~. (!l,0 • • 
, 0Jao'p' e' -- r--• • • ~ 
• 
-
, J 10 
10' 10' 10' I()' 
Fig.2. Scattered plot of the bursl 
observed by enera for 
c e   VB  nt . 
OG 1.2-10
S7
3. SELECTION EFFECT IN THE BURST OBSERVATION
From the arguments in section 2, it is quite possible that
the contradiction arises from a selection effect for smaller
bursts below I0 -s erg/cm . The selection effect may arise
from the different time profile of each burst, and we
examine the effect by classifying the bursts according to
their time profiles. We tentatively classify the bursts with
respect to their time profiles as :
VF : Single peak burst with FWMH of its peak is less
than I/4 sec.
F : Single peak burst with FWMH betweem I/4 to 5 sec.
N : Sigle peak burst with FWMH of larger than 5 sec.
S : Burst lasting for more than 15 sec without clear
pea]<.
MP : Burst with many spiky peaks.
The bursts obseved by Venera 11 and 12 are plotted in Fig.2,
referring to their peak count, P, and the fluence S.
Clearly two types of detection threshold are observed,
i.e., for peak count, P, and for fluence , S, of a burst.
These threshold values depend on the type of time profile of
each burst. Quantitatively, the deviation from S in log N-
log S starts at around ;
S = 7.8xi0-7 erg/cm2 for VF type
S = 5.0xi0-6 erg/cm2 for F type
S = 7.0xi0-6 erg/cm2 for N type
S = 4.0xi0-5 erg/cm2 for S type
S = 4.8xi0-5 erg/cm2 for MP type.
For logiN-log P, the curve starts to bend at aroundp = 00ct/cm2. sec.,
for all types of the bursts except for VF type bursts,
-6 2
which almost corresponds to the fluence of 10 erg/cm .sec.
Higher thresholds of the fluence for S and MP type bursts
are due to the detection criteria for a burst, in which a
burst is selected by the increase of counts exceeding a
certain value within a fixed duration. While , the
threshold for the peak count is due to the fact that the
burst with small fluence less than 4xi0 -7 erg/cm 2 is not
detected. This seriousely affects the detection of a short
duration burst of VF type. Considering these detection
criteria, it is better to use log N-log P instead of using
log N-log S to see the consistency with the observed
celestial distribution of the bursts.
In Fi_.3, observed log N-log P curve is shown down to 10-6
erg/cm sec. by using Venera and PVO data (R.W.Klebesadel et
al: 1985). Here we do not include VF type bursts. Slight
deviation from P-3/2is observed near 10-6 erg/c_.sec.
We approximate the log N-log P in Fig.3 as
N(>P) = No/[(Po/P)(I+P/P0_/2],
where we put No = 6240/cm2 .yr. and Po = 4x10_rg/cm 2 .sec.
We then derive the latitude distribution from this log N-
log P using eq.(1), and show them in Fig.4. The result
seems to be consistent with the observed latitude
• 
.. 
OG .2-10 
57 
. ELECTION E FECT I  HE URST BSERVATION 
From the arguments in section 2, it is quite possible that 
t e ontradiction rises fr  a selection fect r aller 
ursts below 10-5 rg/cm. he selection ffect ay rise 
fro  t e ifferent ti e rofile f each urst, and e 
examine the effect by classifying the bursts according to 
their ti e profiles. We tentatively classify the bursts ith 
respect to their ti e profiles as : 
VF ingle peak burst ith F MH of its peak is less 
t an 1/4 ec. 
F Single peak burst ith FWMH betweem 1/4 to 5 sec. 
 igle peak urst ith F H f larger t an 5 sec. 
 urst l sting f r ore t an 5 s c ithout lear 
peak. 
P urst ith any spiky peaks. 
he ursts obseved by enera 11 and 12 are lotted i  ig.2, 
referring to t eir peak count, P, and the fluence S. 
learly two types of detection threshold are observed, 
i.e., for peak count, P, and for fluence, S, of a burst. 
hese t reshold alues epend  t  t e f ti e rofile f 
each burst. uantitatively, the deviation fro  S in log -
log S starts at around ; 
S 7.8x10-7 erg/c 2 f r F type 
S 5.0x10-6 erg/c 2 f r F type 
S = 7.0x10-6 erg/c 2 for N type 
S = 4.0x10-5 erg/c 2 for S type 
S = 4.8x10-5 erg/c 2 for MP type. 
For log N-log P, the curve starts to bend at around 
P = 100ct/cm 2 .sec., 
for all types of the bursts except for VF type bursts, 
hich al ost corresponds t  the fluence of 10- 6 erg/c .sec. 
igher thresholds of the fluence for Sand MP type bursts 
r  due t  t e etection riteria f r a urst, i  hich a 
burst is selected by the increase of counts exceeding a 
ertain alue ithin  d ration. hile,  
t reshold f r t e peak count is due t  t e fact t at t e 
urst ith s all fl nce l ss t an 10- 7 er /c 2 i  t 
detected. This seriousely affects the detection of a short 
duration burst of VF type. onsidering these detection 
riteria, it is etter t  use log -log P i stead of using 
log -log S t  see the consistency ith the observed 
lestial istribution f t  ursts. 
In Fi~.3, observed log N-lo~ P curve is shown down to 10- 6 
erg/e  .sec. by using enera and PVO data ( . .Klebesadel et 
al: 1985). ere e do not include VF type rsts. light 
via tion fro  - 3/2is served ear -  r  / crr{ . sec. 
e approximate the log -log P in Fig.3 as 
N( >P) = o/[ (Po/P) (1+p/POy/2], 
here we put No = 6240/cm2 .yr. and Po = 4x10- 7erg/cm2 .sec. 
e then derive the latit de istribution fro  t is log -
log P using e .(1), and sho  the  i  ig.4. The result 
se s t   nsistent ith t  served latit G 
OG 1.2-10
58
• Venrra1|,12DJta
b5_.,_'_P0=4X'0 -Terg!p-S/2 I0' • Cm 2 S CC
h-60"._ No=6730/('m_year
o_, I0' :>,":'_ _ I0'
.3.. _ \f, .\ _ I \
-",o, \ .. j
i0'
,o, .\ \
i ) (erlt/¢m%e¢) I' (_rl/cm'tee)
Fig.3. Log N - log S curve. Fig.4. Latitude distribution
VF Type is exclude(], derived eq(;) and log N -
log P curve.
distribution. X2 fitting with observed latitude
distribution deviding the latitude as Ibl < 30 ° and 30 ° <
Ibl < 90 ° gives the value of X 2- = 0.09.
4. DISCUSSIONS
The selection effect for the gamma-ray bursts is analyzed on
the bases of a different time profile of each burst. It is
found that threshold for fluence depends seriousely on the
type of each burst, but, for peak counts does not depend
on their type except for VF type bursts. We thus use log N-
log P instead of log N-log S and derive latitude
distribution of burst sources. The result is consistent
with the observed latitude distribution.
The apparent conflict between log N-log S curve and latitude
distribution of the burst source is resolved by considering
the selection effect for burst observations.
REFERENCES
Cline,T.L.and Desai,U.D.: 1976 Astrophys. Space Sci. 42, 17.
Jennings,M.C.: 1984, High energy Transient in Astrophysics,
ed. S.E.Woosely, (New York, AIP.Press, P.412.
Klebesadel,R.W. et al : 1985 Submitted to APJ Suppl.
Matzets,E.P. et al : 1981 Astrophys. Space Sci. 80, 3
1982 Astrophys. Space Sci. 82, 261
Nishimura,J. and Yamagami,T.: 1985 submitted to APJ
OG 1.2-10 
58 
10' 
... 
~. lCt' \ 
r. 
.\ 
• 
z 10' \ 
• \ 
• \ 
10' e\ 
IO;0'~10TL.--L-"a·' . iUi'---IO'--
J> (nl/Cm'sn) 
Fig.3. Lng N - log S curve. 
VF Type is excluded. 
\..0 
,., 
CJ 
>. 
10' 
10' 
h' 0', • -7 ' 
b 10·,\'Po=4Xl0 erg/cm·'st.~c 
h ' 60' .':-\ N 0 =6 7 30 / c m 2 yea r 
" " 90' <:~~~\ 
~\ 
N 10' 
E 
\ 
\ u ....... ,..... 
.0 
. 
"' Z 
10' 
\ 
\ 
JO' _.A..." ..... l. ..... ~_, .~I ....... _J.--A.U.""". __ .• 1-.a. .... l ........ __ l .... ... l ........ 
la' 10' 10' 10' 10' 10' 
Fig.4. Latitude distribution 
derived cq(l) and log N -
log P curve. 
distribution. X2 fitting with obs rved latitude 
Ibl < 30° and 30° < distribution deviding the latitude as 
ibl < 90 0 gives the value of X2. = .09. 
4. DISCUS IONS 
The s lection effect for the gamma-ray bursts is nalyzed on 
the bases of a differen  time profile of each burst. It is 
found that threshold for fluence depend  seriousely on the 
type of each burst, but, for peak counts does not depend 
on their type except for VF type bursts. We thus use log N-
log P instead of log N-log S and derive latitude 
distribution of burst sources. The result is consistent 
with the obs rved latitude distribution. 
The apparent conflict between log N-log S curve and latitude 
distribution of the burst source is resolved by considering 
the s lection effect for burst observations. 
REFER NCES 
Cline,T.L.and Desai,U.D.: 1976 Astrophys. Space Sci. 42, 17. 
Jennings,M.C.: 1984, High nergy Transient in Astrophysics, 
ed. S.E.Woosely, (New York, AIP.Press, P.412. 
Klebesadel,R.W~ et al 1985 Submitted to APJ Suppl. 
Matzets,E.P. et al 1981 Astrophys. Space Sci. 80, 3 
1982 Astrophys. Space Sci. 82, 261 
N shimura,J. and Yamagami,T.: 1985 submitted to APJ 


Relationships between log N-log S and celestial distribution of gamma-ray bursts

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