Some models for Gamma-Ray Burst spectra result in anisotropic emission. Here the effects of anisotropy on the log N-log S curve are considered

Pizzichini, G.

English

NASA Technical Reports Server

OG 1.2-4
41
THE EFFECT OF ANISOTROPIC EMISSION ON THE
LOG N-LOG S CURVE OF GAMMA-RAY BURSTS
G. Pizzichini
Istituto TESRE/CNR
Via de' Castagnoli 1
40126 Bologna, Italy
ABSTRACT
Some models for Gamma-Ray Burst spectra result in
anisotropic emission. We consider here the effects of
anisotropy on the log N-log S curve.
i. Introduction. Several authors have recently proposed anisotropic
emission mechanisms for Gamma-Ray Bursts. Synchrotron emissivity (I) is
maximum when the angle between the observer and the average magnetic
field direction is @ = _/2, while 7-_ pair production and annihilation
(2) and the more complex model of Hameury et al. (3) have maximum
emissivity for 8 = 0.
Assuming a random distribution for the direction of maximum emissivity of
a Gamma-Ray Burst, we have computed the effects of anisotropy on the two
log N-log S curves given by Jennings (ref. (4), fig. 8) for halo models.
The fact that these two curves already take into account an intrinsic
luminosity distribution for the events is not in conflict with
considering a further dependence of the observed burst intensity on the
angle between the observer and, for example, the magnetic field axis.
2. Method and results. We define as "original" the log N-log S curve that
we would obtain if we could observe all events in the direction of
maximum emissivity, and "averaged" the log N-log S which results from
taking into account anisotropic emission. We also define as "original"
and "apparent" the burst intensity in the direction of maximum emissivity
and in the direction of the observer, respectively. Then the "averaged"
log N-log S curve for an apparent burst intensity S is the weighted mean
over all angles between 0 and _/2 of the "original" log N-log S curve,
computed, for each angle, at the "original" intensity which will, at that
angle, produce an apparent intensity S in the direction of the observer.
In our case, the weight is simply sin 8. More details and the results
obtained using a very simple log N-log S function are given in ref. (5).
As already stated, we have used as "original" log N-log S the curves
given for halo models in fig. 8 of ref. (4) and the angular dependence
for emissivity of refs. (I), (2) and (3). The results are given in figs.
1 andS. The "a[_raged" log N-log S curve has not been renormalized at S
= I0 ergs cm ±n order to show the decrease in number of observed
events per year due to anisotropic emission. The fact that all events at
8>_/2, with the possible exception of a few just beyond 8 = n/2, would
not be observed, will further lower the averaged curves by a factor of =
2 (not included here).
As it might be expected, in all cases the "averaged" log N-log S
curve is lower-and smoother than the original one, but only the "aver-
aged" log N-log S curve obtained using the model of Hameury et el. (3)
differs sensibly from the corresponding "original" curve in the normali-
zation and, in fig. 2, also in the shape.
In fact, the angular dependence of the emerging flux calculated by
 . -4 
 
 CT F ISOTROPIC I SION  E 
 - G  RVE F MA-RAY RSTS 
. izzichini 
tit to SRE/CNR 
ia e' astagnoli 1 
126 ologna, t l  
STRACT 
So e odels for a ma-Ray urst spectra 
isotropic ission. e nsider ere t  
isotropy  t  l  -log  rve. 
result in 
ff cts f 
1. tr duction. everal thors ve ently posed isotropic 
e ission echanisms f r a ma-Ray ursts. nchrotron issivity (1) i  
axi um hen t  gle t en t  server  t  erage agnetic 
fi l  irection is 6  'IT/2, hile y-y air r duction and a nihilation 
(2) and t e ore co plex odel f ameury t a1. (3) have axi um 
i sivity f r 6  o. 
ssuming a rando  istribution f r t e irection f axi um e issivity f 
 a ma-Ray urst, e ave puted t  ffects f isotropy  t  t o 
l  -log S c rves i en by Je nings (r f. ( ), fi . 8) f r alo odels. 
he f ct t at t se t o rves lr dy t e i t  count  i trinsic 
l inosity istribution r t  ents  t i  nflict ith 
nsidering a f rther ependence f t  served urst i t nsity  t  
gle t en t  server d, f r ample, t  agnetic i l  is. 
2. ethod and r sults. e efine as " riginal" t e l  -log S c rve t at 
e ould btain  e uld bserve ll ents   irection f 
axi um e issivity, and "a eraged" t e l  -log S hich r sults fro  
t i g i t  count isotropic ission. e l  efine s riginal" 
 parent" t e urst i t nsity i  t  irection f axi um issivity 
and i  t e irection f t e bserver, respectively. hen t e "averaged" 
l  -log  c rve f r a  parent urst i t nsity  i  t  eighted ean 
ver ll gles t en   'IT/  f t  riginal" l  -log  rve, 
c puted, f r each gle, t t e riginal" i t nsity hich ill, t t at 
gle, r duce  parent i t nsity  i  t  irection f t  server. 
I  r case, t e eight is si ply si  e. ore etails and t e r sults 
btained si g a ery si ple l  -log  f ction r  i en i  r f. ( ). 
s l ady t ted, e ave sed  riginal" l  -log   rves 
i en r alo odels  .  f f.    gular pendence 
f r issivity f r fs. (1), (2) and ( ). he r sults r  i en i  fi s. 
1 an<!l. The "a~2raged" log N-log S curve has not been renormalized at S 
 10 er s c  i  rder t  sho  t e decrease i  nu ber f observed 
ents er ear e t  isotropic ission. he f t t at ll ents t 
6> 'IT/2, ith the possible exception of a fe  j st beyond e  'IT/2, ould 
ot  bserved, i l rther l er t  eraged rves   tor f ~ 
 t l ded ere). 
s  ight  pected, i  ll ses t  " eraged" l  -log  
rve  l er and other t n t e riginal ne, t nly t  ver-
a ed" l  -log S c rve tained si g t  odel f ameury t al. (3) 
iffers nsibly fr  t  rresponding riginal" r e i  t  rmali-
ti n d, i  fi . , l o i  t  s ape. 
In fact, the angular dependence of the e erging flux calculated by 
https://ntrs.nasa.gov/search.jsp?R=19850025601 2020-03-20T17:55:28+00:00Z
0GI.2-4
42
Carrigan and Katz for y-y pair production and annihilation is not
strongly collimated, while the maximum emissivity of synchrotron
radiation is at 8 = 7/2, which also has the highest weight in the
average. On the contrary, the emissivity in the model of Hameury et al.
is strongly peaked at @ = 0, where the weight goes to zero. We also note
that in fig. 2 the "averaged" log N-log S for this model is much
straighter than the "original" curve. In order to fit the experimental
data well with the "averaged" curve, we would have to use an "original"
log N-log S which is bent more and is bent at a higher burst intensity.
_A
104-_.___ , , , , , , , , ,f
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Fig. 1 Solid line: "original" log N-log5S curve from ref. (4) fig. 8,
halo model with e = -1._ _ = 6x10
Dot-dashed line: corresponding "averaged" log N-log S curve for
angular dependence of _he emissivity derived from either ref. (2)z
or ref. (I) with VLT between 2 and 200. They are practically
superimposed.
Dashed line: "averaged" log N-log S curve for the model of Hameury
et al. (3).
O  1.2-4 
 
a rigan a  K tz f  y-  p ir p duction a  a nihilation is n t 
strongly colli ated, hile the maximum e issivity of synchrotron 
radiation is at e = 7T/2, which also has the highest weight in the 
average. On the contrary, the e issivity in the odel of Ha eury et al. 
is strongly peaked at e = 0, here the eight goes to zero. e also note 
t t in fi . 2 th  " eraged" lo  -Iog S f r t i  odel i  uch 
straighter than the "original" curve. In order to fit the experimental 
data well with the "averaged" curve, we would have to use an "original" 
log N-Iog S which is bent more and is bent at a higher burst intensity. 
"-
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Fig. 1 Solid line: "original" log N-log_SS curve from ref. (4) fig. 8, 
halo odel ith a. = -1.0, Z;; = 6x10 . 
Dot-dashed line: corresponding "averaged" log N-Iog S curve for 
angular dependence of fhe emissivity derived from either ref. (2) 
or ref. (1) with vL T between 2 and 200. They are practically 
superimposed. 
Dashed line: "averaged" log N-log S curve for the model of Hameury 
et a1. (3). 
4 8 OG i. 2-4
104 , , , , , ,
102 _...
Is'-- . %'%'_' % '_' "%
e3 "\ ,\
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100 "', \h
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10-8 10-6 104
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Fig. 2 Same as fig. i, but with _ = -0.5 and _ = 7x10 -4.
References
i. Liang E.P. (1982), Nature 299, 321.
2. Carrigan B.J. and Katz J. I. (1984), Astronomy Express l, 89.
3. Hameury J.M. et al. (1984) preprint, accepted for publication in the
Astrophysical Journal, 1985 June Ist issue.
4. Jennings M.C. (1982), Astrophys. J. 25-9, Ii0.
5. Pizzichini G. (1984), Nuovo Cim. 7C, 591.
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eferences 
1. Liang E.P. (1982), Nature 299,321. 
-4 
 xl0 • 
2. Carrigan B.J. and Katz J. r:-(1984), Astronomy Express 1, 89. 
 1.  
. ameury J. . t l. (1984) reprint, accepted f r ublication i  t  
strophysical Journal, 1985 June 1st issue. 
4. Jennings M.C. (1982), Astrophys. J. 258, 110. 
5. Pizzichini G. (1984), Nuovo Cim. 7C, 591. 


The effect of anisotropic emission on the log N-log S curve of gamma-ray bursts

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