- Email: [email protected]
A new model of gamma-ray burst
Chin. A&on. Astrophys. (1994)18/3,266-271 A translation of Acta Astron. Sin. (1994)36/1,14-19 Copyright @ 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0275-1062/94524.00+.00
Pergamon
A new model of gamma-ray burstt LU Tan
WE1 Da-ming Department
of Astronomy,
Nanjing
University,
Nanjing
210008
Abstract We propose a new model of y burst, based on the assumption that it originates in old neutron stars. We assume that the surface magnetic field of an old neutron star consists of two parts, one is a weak background dipole field, the other is strong, local, tube-like fields. This model can explain both the absorption features and the high-energy tail in the spectrum. Our model gives a good fit to the observed spectrum of GB 880205 Key
words:
neutron
stars--y
burst-radiative
transfer
1. INTRODUCTION Although much progress has been made in the study of 7 bursts, many problems still remain to be solved. Mazets’ team ~1 found absorption features in the low-energy portion (30-80 keV) in about 20% of the burst spectra which would require magnetic fields in excess of 1012 G, if they are to be explained by gyroresonance scattering by electrons. On the other hand, the SMM y-spectrograph found a high-energy tail121, with photon energies in excess of 10 MeV. In a strong magnetic field such energetic photons would be absorbed and transformed into electron-positron pairs, hence the presence of this tail requires that the field be less than 1012 G. Thus, the existence of the absorption features and the high-energy tail places contradictory constraints on the magnetic field strength. With the aim of overcoming this difficulty we propose here a new model. The 7 bursts come from old neutron stars; the surface magnetic field on these stars consists of two parts: one is the already attenuated dipole background, the other is tube-like, localized, strong fields131. This model can explain both the absorption features and the high-energy tail. In this paper we use this model to make a Monte Carlo fit to the spectrum of GB 880205.
2. MODEL
OF MAGNETIC
FLUX
TUBES
It is now generally believed that both 7 bursts and pulsars originate in neutron stars. But there is no observational connection whatever between the two, that is, the two cannot be identified with one another. Therefore, the two kinds of neutron star must be located in t Received
1993 May 15
7 Burst Model
267
different physical states or evolutionary stages. For the pulsars, there a clear cutoff line in the p - r; diagram141, the pulsars are all located above the line, hence the r-bursters must be below the line. The region below the cutoff corresponds to longer period and weaker magnetic fields, hence the y-bursters should be old neutron stars. Neutrons stars are born in supernova explosions when they possess velocities of several hundreds km/s. Old neutron stars have often reached the galactic halo and they show a relatively isotropic distribution in space15@l. Analysis of observed data of pulsars 171showed their dipole field to be continually decaying. However, researchls*gl also showed that there is the second kind of field which is weak but not decaying. Therefore, the old neutrons stars that are 7-bursters should have relatively weak dipole fields, and if they do have strong fields, these can only be local and are ineffective in modifying the weak dipole character of the global field. As is well-known, the sun has a weak dipole field, and, at the same time, much stronger, localized, tubular fieldsl’Ol. We assume that certain mechanism causes numerous localized, tube-like strong fields being formed on the surface of the star within a very short time scale. These fields are randomly distributed on the surface. At the same time, large quantities of high-energy electrons enter the magnetic flux tubes, generate synchrotron radiation and hence 7 bursts. The absorption coefficient for high-energy photons being absorbed by the magnetic field and then transformed into electron-positron pairs isl”1 P’
(1)
where Q is the fine structure constant, B, = m2c3/eh = 4.413~1O’~G is the critical field, E is the photon energy and 0 is the angle between the photon and the field. In order that the photon is not absorbed, the magnetic flux tube must be optically thin for the photon, that is, we require ~1 < 1, 1 being the diameter of the tube. If we take B sin@ = 1012 G, E = 10 MeV, then we require 1 < 10B4 cm. Thus, the existence of high-energy tail requires that the flux tubes be very thin. We have already used this model to calculate the second harmonic in the spectrum of GB 880205 and found a fit parameter n .I - 1021 cmm2, n being the electron number density inside the tube. We also calculated the depth of the third harmonic. In this paper, we shall use Monte Carlo method to make a fit to the spectrum.
3. SPECTRUM
FIT
Compton scattering between the synchrotron-generated high-energy photons and the thermal electrons in the magnetic flux tube will occur. Observations showed the flux tubes to be optically thin for the continuum Compton scattering and optically thick for the resonance scattering. We have calculated the Compton scattering cross section for a strong fieldl”l and found that, at the fundamental resonance energy EB, the Compton scattering cross section is 10,000 times larger than that for Thomson scattering. Hence there is a high probability for photons with energy EB to collide with the electrons, and such a photon will undergo many collisions before it emerges from the magnetic flux tube. When such a collision takes place, the electron is first lifted from its Landau ground state to its first Landau excited
268
WE1 Da-ming & LU Tan
state, it then falls back to the ground state, releasing a scattered photon with an energy of about EB. When a photon with energy 2Eb collides with an electron, the latter is lifted to its second excited state, and since in a strong field, the probability of dropping to the first excited state is much greater than to the ground state li31, the result of the scattering is that the electrons are mainly scattered into the first excited state, generating again a photon of energy about En at the same time. The situation is similar for higher harmonics. When a photon with harmonic number n collides with an electron, the electron is mainly scattered to the n - 1 level, together with the generation of a photon of w EB. These photons formed from scattering will continue to collide with the electrons and the net result is a great increase in the number of photons of the first harmonic. This kind of multiple scattering is difficult to calculate with usual method, so we shall use Monte Carlo method below to simulate the transport of the photons. According to Liang’s calculation 1141,the synchrotron spectrum for optically thin thermal electrons is (2)
where E e = li(eB/mc)Tfl = ll.GBr~Z’~i keV is the characteristic energy, Bis being the field intensity in units of 10” G and Tii the temperature of the thermal electrons in units of mea. We take equation (2) as our source function for sampling the photons (Es, 6’0,tic), 60 being the azimuth, assumed to be uniformly distribute between 0 and 2r. To find the angle and energy distributions of the photons after scattering we need the redistribution function R(r’,$; z,c(), where p = ~0~8,~’ = case’, and c = (E is the non-dimensional energy of the photon. Prime indicates afEd/Ed2Tdmc2)1/2 ter scattering. Wasserman and Salpeterl15l first derived the expression for the redistribution function. In their derivation they divided the scattering process into two parts: first, a photon with energy E at angle $ is absorbed by the electron; then the electron is de-excited, generating a photon of energy E’ at angle 0’. Their result was
R(x',
e-;’
a/1~1~
r> -
P’P,
[w(cc)lp
- z12-i- (a/p)2
l
~“~(j.4 -
r’l’
(3)
where G(p)
Z-
-
$ (1 +
.x’ -
x
P’ -
P
cc’).
+(p’--
PL)*A.
A -
0.18B12CTll(keV)1-1’2.
(6)
3-
0.078(B/B,)[T,,(keV)]-u2,
(7)
X(cL)IP
-
xlr
--cIc1*
After scattering, the direction sampling function is
(8)
-y Burst Model
This is independent of x and p;-this function, the absorption and emission function after scattering is
dx’;x,
P
9
P’) =
R(x’,
I
R(x',
P
269
is because in the derivation were assumed to be bi-polar.
?,x,PI
p’;x,
p)dx'
c-s=
o/lrl* where H[z(p)/p,
of the redistribution The energy sampling
[X(P)lP -
(a/rY
21*+
X~2H[X(CL)/C1,~/ItCllI~- $1’ 00)
a/]/.~]] is the Voigt function.
As found above, the magnetic flux tube is very thin, so we can treat it approximately as a cylinder. Cylindrical coordinates (see Fig. 1) are used in all the calculations. We derive the position of the photon after being scattered m times to be
where [ is a random energy
number
and o(E,,,_~) is the scattering
cross section
corresponding
to
E,,,_ 1.
For the second and higher harmonics we can approximately substitute for scatteringl’sl. The photon path length through the flux tube is
gyro absorption
(13) where (rc, $0) are the initial coordinates of the photon and 8s is the angle between the initial photon and the magnetic field. The photon fluxes at the second and third harmonics are then
F,(E,
e> -
JlJr f(E,e) H
F,(E,
e) -
J: Jr f(E,e)
l
where H is the thickness The spectrum fit was by Monte
l
H
of the emitting
exp{--na(E2N)
l
l
exp{-ndE,)S)
rdrh,
l
l
rdrdp,
(14)
(15)
region.
of GB880205 was fitted in three parts. In the 5-35 keV interval, the Carlo; in 35-65keV, we used equations (13)-(15); for the rest, we used
270
WE1 Da-ming
&
-31 0
LU Tan
’
*
*
’
’
I
Fig. 2
*
s
s .’
2 log Energy(keV)
The fitted spectrum
The solid (dashed)
(without) Fig. 1
*
’
r
’
*
I
3
of GB880205.
line is the spectrum
with
the effect of high harmonic scattering
The geometry of photon scattering
equation (2). Results of the fit are shown in Fig.2. We considered two cases: one, we considered only the scattering at the first harmonic and neglected the effect of scattering at the higher harmonics, and the result is shown by the dashed line. Two, photons generated by scattering at the higher harmonics were added to the first harmonic and become subject again to Compton scattering; the result is shown by the solid line. In the fitting, we took the following parameter values: kTir = 5 keV, B = 1.7x1012G.
4.
CONCLUSIONS
In this paper we propose a model of old neutron star with strong local magnetic field. This model can explain both the gyro absorption features and the high-energy tail in the 7 burst spectrum, thus solving an important problem in this topic. In our spectrum fit we used equation (2) as energy input function. Of course, other forms of input can also be used, such as a power law (many burst spectra do show a power law in the high-energy portion). However, whatever form of input we use, our calculation will remain effective. One basic assumption in this paper is that there are many magnetic flux tubes on the
y Burst Model
271
surface of the neutron star and the field intensity, electron number density, temperature etc. can differ greatly on either side of the tube. We are currently studying the question as to how such strong local fields can be formed. ACKNOWLEDGMENT
We thank Professor LI Xiao-qing for support and helpful discussion.
References 111
Mazets E. P. et al., ApSS, 1981,80,
[21
Matz S. M. et al., ApJ, 1985, 288, L37
[31
Lu T., G amma-ray Bursts: Observation, Analyses and Theories, eds., Cheng Ho, Ft. I. Eg stein, E. E. Fenimore
(41 151 161 [71
Zhao Yongheng, et al., A&A, 1989,223,147
I’31 [91 PO1 WI WA
7
Shklovskii I. S., Mitrofanov I. G., MNRAS, 1985, 212, 545 Meegan C. A. et al., Nature, 1992,355,143 Stollmarm G. M., A&A, 1987,178,143 Lu Tan, Progress in Astronomy (in Chinese), 1988,6, Kulkarni S. Ft., ApJ, 1986,306,
265
L85
Li Xiao-qing, Publ. Purple Mountain Observatory, 1989,8,
1
Erber T., Rev. Mod. Phys., 1966,38,626 Wei Da-ming, Lu Tan, Ni Chen-ping, Ping Jia-hm, CAA 1994, 17(3), 265 = AApS 1994, 13(2), 108 Daugherty J. K., Ventura J., A&A, 1977,61,
P31 1141 1151
Wasse-
I161
Fenimore E. E. et al., ApJ, 1988, 335, L71
723
Liang E. P., Nature, 1982,299,321 I., Salpeter E., ApJ, 1980, 241, 1107
Pergamon
A new model of gamma-ray burstt LU Tan
WE1 Da-ming Department
of Astronomy,
Nanjing
University,
Nanjing
210008
Abstract We propose a new model of y burst, based on the assumption that it originates in old neutron stars. We assume that the surface magnetic field of an old neutron star consists of two parts, one is a weak background dipole field, the other is strong, local, tube-like fields. This model can explain both the absorption features and the high-energy tail in the spectrum. Our model gives a good fit to the observed spectrum of GB 880205 Key
words:
neutron
stars--y
burst-radiative
transfer
1. INTRODUCTION Although much progress has been made in the study of 7 bursts, many problems still remain to be solved. Mazets’ team ~1 found absorption features in the low-energy portion (30-80 keV) in about 20% of the burst spectra which would require magnetic fields in excess of 1012 G, if they are to be explained by gyroresonance scattering by electrons. On the other hand, the SMM y-spectrograph found a high-energy tail121, with photon energies in excess of 10 MeV. In a strong magnetic field such energetic photons would be absorbed and transformed into electron-positron pairs, hence the presence of this tail requires that the field be less than 1012 G. Thus, the existence of the absorption features and the high-energy tail places contradictory constraints on the magnetic field strength. With the aim of overcoming this difficulty we propose here a new model. The 7 bursts come from old neutron stars; the surface magnetic field on these stars consists of two parts: one is the already attenuated dipole background, the other is tube-like, localized, strong fields131. This model can explain both the absorption features and the high-energy tail. In this paper we use this model to make a Monte Carlo fit to the spectrum of GB 880205.
2. MODEL
OF MAGNETIC
FLUX
TUBES
It is now generally believed that both 7 bursts and pulsars originate in neutron stars. But there is no observational connection whatever between the two, that is, the two cannot be identified with one another. Therefore, the two kinds of neutron star must be located in t Received
1993 May 15
7 Burst Model
267
different physical states or evolutionary stages. For the pulsars, there a clear cutoff line in the p - r; diagram141, the pulsars are all located above the line, hence the r-bursters must be below the line. The region below the cutoff corresponds to longer period and weaker magnetic fields, hence the y-bursters should be old neutron stars. Neutrons stars are born in supernova explosions when they possess velocities of several hundreds km/s. Old neutron stars have often reached the galactic halo and they show a relatively isotropic distribution in space15@l. Analysis of observed data of pulsars 171showed their dipole field to be continually decaying. However, researchls*gl also showed that there is the second kind of field which is weak but not decaying. Therefore, the old neutrons stars that are 7-bursters should have relatively weak dipole fields, and if they do have strong fields, these can only be local and are ineffective in modifying the weak dipole character of the global field. As is well-known, the sun has a weak dipole field, and, at the same time, much stronger, localized, tubular fieldsl’Ol. We assume that certain mechanism causes numerous localized, tube-like strong fields being formed on the surface of the star within a very short time scale. These fields are randomly distributed on the surface. At the same time, large quantities of high-energy electrons enter the magnetic flux tubes, generate synchrotron radiation and hence 7 bursts. The absorption coefficient for high-energy photons being absorbed by the magnetic field and then transformed into electron-positron pairs isl”1 P’
(1)
where Q is the fine structure constant, B, = m2c3/eh = 4.413~1O’~G is the critical field, E is the photon energy and 0 is the angle between the photon and the field. In order that the photon is not absorbed, the magnetic flux tube must be optically thin for the photon, that is, we require ~1 < 1, 1 being the diameter of the tube. If we take B sin@ = 1012 G, E = 10 MeV, then we require 1 < 10B4 cm. Thus, the existence of high-energy tail requires that the flux tubes be very thin. We have already used this model to calculate the second harmonic in the spectrum of GB 880205 and found a fit parameter n .I - 1021 cmm2, n being the electron number density inside the tube. We also calculated the depth of the third harmonic. In this paper, we shall use Monte Carlo method to make a fit to the spectrum.
3. SPECTRUM
FIT
Compton scattering between the synchrotron-generated high-energy photons and the thermal electrons in the magnetic flux tube will occur. Observations showed the flux tubes to be optically thin for the continuum Compton scattering and optically thick for the resonance scattering. We have calculated the Compton scattering cross section for a strong fieldl”l and found that, at the fundamental resonance energy EB, the Compton scattering cross section is 10,000 times larger than that for Thomson scattering. Hence there is a high probability for photons with energy EB to collide with the electrons, and such a photon will undergo many collisions before it emerges from the magnetic flux tube. When such a collision takes place, the electron is first lifted from its Landau ground state to its first Landau excited
268
WE1 Da-ming & LU Tan
state, it then falls back to the ground state, releasing a scattered photon with an energy of about EB. When a photon with energy 2Eb collides with an electron, the latter is lifted to its second excited state, and since in a strong field, the probability of dropping to the first excited state is much greater than to the ground state li31, the result of the scattering is that the electrons are mainly scattered into the first excited state, generating again a photon of energy about En at the same time. The situation is similar for higher harmonics. When a photon with harmonic number n collides with an electron, the electron is mainly scattered to the n - 1 level, together with the generation of a photon of w EB. These photons formed from scattering will continue to collide with the electrons and the net result is a great increase in the number of photons of the first harmonic. This kind of multiple scattering is difficult to calculate with usual method, so we shall use Monte Carlo method below to simulate the transport of the photons. According to Liang’s calculation 1141,the synchrotron spectrum for optically thin thermal electrons is (2)
where E e = li(eB/mc)Tfl = ll.GBr~Z’~i keV is the characteristic energy, Bis being the field intensity in units of 10” G and Tii the temperature of the thermal electrons in units of mea. We take equation (2) as our source function for sampling the photons (Es, 6’0,tic), 60 being the azimuth, assumed to be uniformly distribute between 0 and 2r. To find the angle and energy distributions of the photons after scattering we need the redistribution function R(r’,$; z,c(), where p = ~0~8,~’ = case’, and c = (E is the non-dimensional energy of the photon. Prime indicates afEd/Ed2Tdmc2)1/2 ter scattering. Wasserman and Salpeterl15l first derived the expression for the redistribution function. In their derivation they divided the scattering process into two parts: first, a photon with energy E at angle $ is absorbed by the electron; then the electron is de-excited, generating a photon of energy E’ at angle 0’. Their result was
R(x',
e-;’
a/1~1~
r> -
P’P,
[w(cc)lp
- z12-i- (a/p)2
l
~“~(j.4 -
r’l’
(3)
where G(p)
Z-
-
$ (1 +
.x’ -
x
P’ -
P
cc’).
+(p’--
PL)*A.
A -
0.18B12CTll(keV)1-1’2.
(6)
3-
0.078(B/B,)[T,,(keV)]-u2,
(7)
X(cL)IP
-
xlr
--cIc1*
After scattering, the direction sampling function is
(8)
-y Burst Model
This is independent of x and p;-this function, the absorption and emission function after scattering is
dx’;x,
P
9
P’) =
R(x’,
I
R(x',
P
269
is because in the derivation were assumed to be bi-polar.
?,x,PI
p’;x,
p)dx'
c-s=
o/lrl* where H[z(p)/p,
of the redistribution The energy sampling
[X(P)lP -
(a/rY
21*+
X~2H[X(CL)/C1,~/ItCllI~- $1’ 00)
a/]/.~]] is the Voigt function.
As found above, the magnetic flux tube is very thin, so we can treat it approximately as a cylinder. Cylindrical coordinates (see Fig. 1) are used in all the calculations. We derive the position of the photon after being scattered m times to be
where [ is a random energy
number
and o(E,,,_~) is the scattering
cross section
corresponding
to
E,,,_ 1.
For the second and higher harmonics we can approximately substitute for scatteringl’sl. The photon path length through the flux tube is
gyro absorption
(13) where (rc, $0) are the initial coordinates of the photon and 8s is the angle between the initial photon and the magnetic field. The photon fluxes at the second and third harmonics are then
F,(E,
e> -
JlJr f(E,e) H
F,(E,
e) -
J: Jr f(E,e)
l
where H is the thickness The spectrum fit was by Monte
l
H
of the emitting
exp{--na(E2N)
l
l
exp{-ndE,)S)
rdrh,
l
l
rdrdp,
(14)
(15)
region.
of GB880205 was fitted in three parts. In the 5-35 keV interval, the Carlo; in 35-65keV, we used equations (13)-(15); for the rest, we used
270
WE1 Da-ming
&
-31 0
LU Tan
’
*
*
’
’
I
Fig. 2
*
s
s .’
2 log Energy(keV)
The fitted spectrum
The solid (dashed)
(without) Fig. 1
*
’
r
’
*
I
3
of GB880205.
line is the spectrum
with
the effect of high harmonic scattering
The geometry of photon scattering
equation (2). Results of the fit are shown in Fig.2. We considered two cases: one, we considered only the scattering at the first harmonic and neglected the effect of scattering at the higher harmonics, and the result is shown by the dashed line. Two, photons generated by scattering at the higher harmonics were added to the first harmonic and become subject again to Compton scattering; the result is shown by the solid line. In the fitting, we took the following parameter values: kTir = 5 keV, B = 1.7x1012G.
4.
CONCLUSIONS
In this paper we propose a model of old neutron star with strong local magnetic field. This model can explain both the gyro absorption features and the high-energy tail in the 7 burst spectrum, thus solving an important problem in this topic. In our spectrum fit we used equation (2) as energy input function. Of course, other forms of input can also be used, such as a power law (many burst spectra do show a power law in the high-energy portion). However, whatever form of input we use, our calculation will remain effective. One basic assumption in this paper is that there are many magnetic flux tubes on the
y Burst Model
271
surface of the neutron star and the field intensity, electron number density, temperature etc. can differ greatly on either side of the tube. We are currently studying the question as to how such strong local fields can be formed. ACKNOWLEDGMENT
We thank Professor LI Xiao-qing for support and helpful discussion.
References 111
Mazets E. P. et al., ApSS, 1981,80,
[21
Matz S. M. et al., ApJ, 1985, 288, L37
[31
Lu T., G amma-ray Bursts: Observation, Analyses and Theories, eds., Cheng Ho, Ft. I. Eg stein, E. E. Fenimore
(41 151 161 [71
Zhao Yongheng, et al., A&A, 1989,223,147
I’31 [91 PO1 WI WA
7
Shklovskii I. S., Mitrofanov I. G., MNRAS, 1985, 212, 545 Meegan C. A. et al., Nature, 1992,355,143 Stollmarm G. M., A&A, 1987,178,143 Lu Tan, Progress in Astronomy (in Chinese), 1988,6, Kulkarni S. Ft., ApJ, 1986,306,
265
L85
Li Xiao-qing, Publ. Purple Mountain Observatory, 1989,8,
1
Erber T., Rev. Mod. Phys., 1966,38,626 Wei Da-ming, Lu Tan, Ni Chen-ping, Ping Jia-hm, CAA 1994, 17(3), 265 = AApS 1994, 13(2), 108 Daugherty J. K., Ventura J., A&A, 1977,61,
P31 1141 1151
Wasse-
I161
Fenimore E. E. et al., ApJ, 1988, 335, L71
723
Liang E. P., Nature, 1982,299,321 I., Salpeter E., ApJ, 1980, 241, 1107