The little we do know of the physical conditions in gamma-ray bursters makes
them conducive to the acceleration of high-energy cosmic rays, especially if
they are at cosmological distances. We find that, with the observed statistics
and fluxes of gamma-ray bursts, cosmological bursters may be an important
source of cosmic rays in two regions of the observed spectrum: 1. At the
very-high-energy end (E>10^{19} eV), where cosmic rays must be of extragalactic
origin. 2. Around and above the spectral feature that has been described as a
bump and/or a knee, which occurs around 10^{15} eV, and starts at about 10^{14}
eV. The occasional bursters that occur inside the Galaxy--about once in a few
hundred thousand years if burst emission is isotropic; more often, if it is
beamed--could maintain the density of galactic cosmic rays at the observed
level in this range. These two energy ranges might correspond to two typical
energy scales expected from bursters: one pertinent to acceleration due to
interaction of a magnetized-fireball front with an ambient medium; the other to
acceleration in the fireball itself (e.g. shock acceleration).Comment: 12 pages in Late

Milgrom, Mordehai

Usov, Vladimir

English

arXiv

M MILGROM

V USOV

Crossref

Gamma-ray bursters as sources of cosmic rays

The little we do know of the physical conditions in gamma-ray bursters makes them conducive to the acceleration of high-energy cosmic rays, especially if they are at cosmological distances. We find that, with the observed statistics and fluxes of gamma-ray bursts, cosmological bursters may be an important source of cosmic rays in two regions of the observed spectrum: 1. At the very-high-energy end (E>10^{19} eV), where cosmic rays must be of extragalactic origin. 2. Around and above the spectral feature that has been described as a bump and/or a knee, which occurs around 10^{15} eV, and starts at about 10^{14} eV. The occasional bursters that occur inside the Galaxy--about once in a few hundred thousand years if burst emission is isotropic; more often, if it is beamed--could maintain the density of galactic cosmic rays at the observed level in this range. These two energy ranges might correspond to two typical energy scales expected from bursters: one pertinent to acceleration due to interaction of a magnetized-fireball front with an ambient medium; the other to acceleration in the fireball itself (e.g. shock acceleration)

Milgrom, M

Usov, Yu V

CERN Document Server

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GAMMA-RAY BURSTERS AS SOURCES OF COSMIC RAYS
Mordehai Milgrom and Vladimir Usov
Department of condensed-matter physics, Weizmann Institute, Rehovot 76100 Israel
The little we do know of the physical conditions in -ray bursters
[1]
makes them conducive to the acceleration of high-energy cosmic rays
(CRs), especially if they are at cosmological distances. We nd that, with
the observed statistics and uxes of -ray bursts, cosmological bursters
may be an important source of cosmic rays in two regions of the observed
spectrum: 1. At the very-high-energy end (E > 10
19
eV ), where CRs must
be of extragalactic origin. 2. Around and above the spectral feature that
has been described as a bump and/or a knee, which occurs around 10
15
eV ,
and starts at E
0
 (1   3)  10
14
eV . The occasional bursters that occur
inside the Galaxy{about once in a few hundred thousand years if burst
emission is isotropic; more often, if it is beamed{could maintain the den-
sity of galactic cosmic rays at the observed level in this range. These two
energy ranges might correspond to two typical CR energy scales expected
from bursters: one pertinent to CR acceleration due to interaction of a
magnetized-reball front with an ambient medium; the other to accelera-
tion in the reball itself (e.g. shock acceleration).
Cosmic rays are thought to originate from more than one type of source, but
what exactly their sources are is still an unsettled issue (for recent reviews see e.g.,
refs [2] - [4]). Those with energies above  10
19
eV must be extragalactic, because
their gyroradius in galactic magnetic elds is larger than galactic size. Those of
lower-energy, which are of galactic origin, also seem to come from more than one
type of source
[4]
, as indicated by the following observations: 1. The change in the
spectral form above E
0
may witness a change in type of source, or at least in the
acceleration mechanism. 2. Supernova remnants (SNRs), which, on various grounds,
are thought to be adequate sources for the lower energy CRs, seem incapable
[5]
of
producing the higher-energy galactic CRs with E > 10
14
eV . It may be signicant
that this theoretical cuto at  10
14
eV marks the beginning of the observed change
in both the spectrum, and the elemental composition. After Axford
[4]
we designate
the former, sub-E
0
region GCRI, and the latter by GCRII. It is widely acknowledged
(e.g. refs [3, 4]) that the origins of both GCRII and extragalactic CRs (EGCRs) have
1
not been pinpointed yet. Below, we argue that -ray-burst sources (GRBSs) may be
suitable sources of both these components of CRs.
Milgrom and Usov
[6]
have recently pointed out that the directions of arrival of
the two most energetic CRs observed to date each agree, within the uncertainties,
with that of an energetic -ray burst (GRB). Inspired by this we have proposed that
ultra-high-energy CRs (E > 10
20
eV ) are accelerated in the same cataclysmic event
that produces the GRB (the CRs arriving with some delay, and positional disparity
due to wriggling of its trajectory by intervening magnetic elds). Waxman
[7]
, and
Vietri
[8]
have made a similar proposal. This has, naturally, induced us to consider
the contribution of GRBSs to the observed CR population at lower energies.
The isotropy in the directions of arrival of -ray bursts, together with the distance
distribution implied by their log N   log S relation, points strongly to a GRB origin
far outside the disk of the Milky Way. This has lead to two alternative pictures: one
in which GRBs are of cosmological origin
[9]   [12]
and one by which their sources are
distributed in a spherical, extended corona, surrounding the Milky Way
[13]   [15]
.
We adopt the cosmological scenario in the following discussion; the galactic-corona
alternative will be discussed, occasionally.
One has to consider two types of cosmic-ray contribution: 1. CRs at very high
energies that are hardly aected by magnetic elds in their putative mother galaxy,
and in our own, arrive here as extragalactic CRs. 2. The occasional GRBSs that
occur in the Milky Way (still in the cosmological scenario for GRBs) produce a ux
of high-energy CRs passing Earth on their way out to become extragalactic CRs in
other galaxies; these we ignore here. They also emit CRs of low enough energies to
be trapped in the Galaxy for an extended period and contribute to its CR budget.
We rst consider such trapped CRs from bursts inside the Galaxy.
The observed energy density of all CRs with E > 10
10
eV in the neighborhood
of the solar system is w
CR
' 0:5 eV cm
 3
. For maintaining this energy density, in
spite of CR escape, a relling of the galactic volume by CRs with a power Q
I
CR
'
310
40
erg sec
 1
is required (see e.g. [16]). As to GCRII, the power which is needed to
maintain the observed energy density, w
II
CR
of CRs with energy above E
0
is estimated
at
Q
II
CR
'
V
CR
w
II
CR

conf
(E
0
)
' Q
I
CR

E
0
10
10
eV

 ( 2)+
erg sec
 1
: (1)
Here V
CR
= R
2
g
h
CR
' 10
67
(h
CR
=h
rad
) cm
3
is the volume where CRs accumulate;
R
g
' 10 kpc is some measure of the radius of the Galaxy, h
CR
is the thickness of the
CR volume; h
rad
' 750 pc is the thickness of the radio disk;  ' 2:7 is the spectral
power of CRs from 10
10
eV to  E
0
(ref. [4]); so that w
II
CR
' w
CR
(E
0
=10
10
eV )
 ( 2)
.
We have taken an energy dependence of the connement time of the form
2
conf
(E) ' 10
7
 
h
CR
h
rad
!

E
10
10
eV

 
yr (2)
and  = 0:4  0:2 (refs [17] - [19]). Actually, very little is known about the conne-
ment time for energies outside the range 10
10
  10
12
eV , where it can be measured
approximately from the abundance of radioactive secondaries among the CRs. In this
range one result is
[17]
 ' 0:6 . More generally, it was argued
[18][19]
that in a wide
range of energies, from  10
9
eV to  10
17
eV , the mean value of  is about 0:30:1.
If we use  = 0:3 and E
0
= 3  10
14
eV we have from eq.(1)
Q
II
CR
' 5 10
38
erg sec
 1
: (3)
While the value of h
CR
is very uncertain, and can be anywhere between  h
rad
and
 2R
g
(e.g. ref. [16]), our estimate of Q
II
CR
does not depend on it.
What fraction of such a CR luminosity is reasonable of GRBSs to supply? To get
an idea we estimate the long-time-mean luminosity in -rays produced by GRBSs
inside the Galaxy. From the typical distances (as inferred from observed statistics),
and from the uxes of GRBs, it is estimated (e.g. refs [1][20][21]) that, if the -rays
are emitted isotropically from their sources, an average GRBS outputs  410
51
ergs
in -rays; and that if GRBSs occur in galaxies with rates proportional to the galactic
mass, about one GRB occurs in a galaxy like ours once every  4 10
5
years. If the
-ray emission is beamed the luminosities are smaller, in proportion, and the rate
of occurrence in a galaxy is higher, in inverse proportion, to the beam solid angle,
which might be rather small. By these estimates the mean power that is released
in CRs in the Galaxy from bursts inside it is L
CR
' 3  10
38
erg sec
 1
, where  is
some typical ratio of CR-to--ray total energy output. Thus if  is of order unity
the CR luminosity produced by the GRBs in the Galaxy may tally with what is needed
to maintain the observed CR ux in GCRII by eq.(3). In the galactic-corona-origin
picture similar arguments require  > 10
4
.
To achieve the observed degree of isotropy of GCRII is not much more dicult
than with other type of sources, say SNRs; even though GRBSs inject CRs in short,
far-between bursts in space and time. Especially so as that the connement time may
be much larger than the repetition time of the GRBSs in the Galaxy.
Consider now the possible contribution of GRBSs to the observed EGCR. The
reckoning here is dierent: we now compare the integrated ux of -ray bursts with
that of CRs of energy E > 10
19
eV . We assume that intergalactic magnetic elds
are small enough so that such CRs are not accumulated in the IGM. The integral
energy ux of EGCRs is I
CR
(E > 10
19
eV ) ' 4  10
 10
erg cm
 2
s
 1
above 10
19
eV
and I
CR
(E > 10
20
eV ) ' 0:610
 10
erg cm
 2
s
 1
above 10
20
eV (ref [22]). The energy
ux in GRBs is I

=
_
NS
0
' 3 10
 10
erg cm
 2
s
 1
, where
_
N ' 10
3
yr
 1
is the total
observed burst rate and S
0
' 10
 5
erg cm
 2
is a typical uence per burst, as derived
from the BATSE survey (e.g., ref [1]). From these estimates we can see that the
3
ux of EGCRs with E  10
19
eV may be produced by GRBSs if their luminosities
in -rays and CRs of energies  10
19
eV are similar:   1 at E  10
19
eV . The
fact that the estimated, integrated ux of CRs above 10
20
eV is rather smaller than
that above 10
19
eV does not indicate that the spectrum is so steep at the source, as
the higher-energy CRs suer eective degradation of energy on their way. In fact,
observations are consistent with an E
 2
energy spectrum that imparts equal energies
to equal logarithmic ranges. (See also the discussion by Waxman
[7]
who reaches
similar conclusions on this point).
The observed isotropy in the directions to GRBSs
[1]
, would naturally explain the
near isotropy of EGCRs, if these indeed are produced by GRBSs. There may be an
added isotropization eect due to Larmor bending of the trajectories of the lower-
energy EGCRs by intergalactic magnetic elds. (The limits on such elds are so
lax as to be consistent with either an important such eect, or with hardly any.)
However, CRs with the highest energies, say E > 510
19
eV , cannot, typically, come
from very large distances, because their energy is quickly degraded on travel
[23]
.
Their positional distribution will than reect that of the nearer-by GRBSs{at these
energies the CR apparent position has a good memory of the direction of the source
(e.g. ref. [23]). If GRBSs occur in galaxies, the CR distribution will reect any
anisotropy in the distribution of nearby galaxies (distance less than 200-300 Mpc).
Indeed, such an anisotropy for E > 410
19
eV , correlated with the local supercluster
has been found recently
[24]
. GRBSs within this distance are also far-between in time
(a few per year). Thus we could see E > 10
20
eV CRs to come in groups spread in
time over periods of a few months to a few years (see ref. [6]), and concentrated in
an area of the sky around the GRBS position. This could be tested in future with
the proposed 5000 km
2
CR detector, the Auger observatory, with which a few tens
such CRs are expected for these strongest GRBS.
What are the typical CR energies expected from GRBSs, and what are the relative
eciencies, , of CR to -ray productions, which enter our deductions of the expected
CR uxes from -ray observations?
There are two energy scales that might characterize the energy of cosmic rays
accelerated in GRBSs; one is  
2
0
mc
2
, where  
0
is the bulk Lorentz factor of a rela-
tivistic wind, and m the mass of the particle; the other may be dened, for example
by the electric potential produced by rotating magnetic elds so that it depends on
the magnetic-eld strength, and the rotation frequency. The rst scale enters as fol-
lows: A common feature of all acceptable models of cosmological GRBSs is that a
relativistic wind is involved in the -ray radiation process. The bulk Lorentz fac-
tor of such a wind is constrained to be more than 10
2
  10
3
(refs [21],[25]). A very
strong magnetic eld may exist in the plasma that ows away from GRBSs
[26]  
[27]
. It was argued by Meszaros and Rees
[28]
that the X-ray and -ray emission of
bursts is generated far from the bursters, at distances of r
d
 10
15
  10
16
cm, in
the process of interaction between relativistic winds with  
0
 10
2
and an external
medium (e.g., an ordinary interstellar medium, or plasma that is ejected from the
4
predecessor of the burster). Such a process has been considered numerically
[29]
, and
it was shown that the energy that is lost by a relativistic magnetized wind because
of its interaction with an external medium is distributed in the following way. About
half of this energy is in ultra-relativistic heavy particles, protons and nuclei that are
knocked by the wind front. In the rest frame of the front the particles are reected,
more-or-less elastically, from the front. Thus, the mean Lorentz factor of reected
heavy particles in the frame of the burster is   
2
0
. The other half of the wind-energy
losses is distributed, more-or-less evenly, between low-frequency oscillations of elec-
tromagnetic elds and both high-energy electrons and their high-frequency (X-ray
and -ray) emission. The total -ray eciency may reach 20   30%. Therefore, it is
estimated that the energy released in CRs is at least twice that in -rays:   2. For
this lower limit the estimated mean power released in CRs in the Galaxy from local
GRBSs is L
CR
' 6 10
38
erg sec
 1
, which is comparable to Q
II
CR
within uncertainties
of their estimates.
The mean energy of CRs knocked by the magnetized-reball front would be the
typical energy of the bump,  10
15
eV per nucleus, if  
0
 10
3
, which is much in
keeping with what is expected; but,  
0
may be as high as 10
5
(ref. [28]) giving CR
energies of up to  10
19
eV per nucleus. Hence, the occasional GRBSs that occur in
the Milky Way might explain both the energetics, and the typical energy of GCRII.
Our limited knowledge of the processes involved does not suce do determine the
expected spectrum of CRs.
The second energy scale enters because CRs may be generated by mechanisms
other than through knocking by the relativistic interface between the outowing
gas and an external medium, as described above. For example, by one class of
cosmological-origin models, GRBs are produced in rotating disk-like objects result-
ing from mergers of a neutron-star binaries (e.g. ref. [30]), or by neutron stars
formed by accretion-induced collapse
[27]
. The magnetic eld, B
S
, at the surface of
such objects may be as high as  10
16
G (refs [26], [27]), and their angular velocity

  10
4
s
 1
. The potential dierence between the surface of such an object, at radius
R  10
6
cm, and innity is
[31]
'
max
=


2
B
S
R
3
2c
2
 10
23



10
4
s
 1

2

B
S
10
16
G

R
10
6
cm

3
V : (4)
Charged particles that ow away from the surface may be accelerated, in principle,
up to the energy E
max
' e'
max
, enough to explain EGCRs of all observed energies.
Alternatively, particles may be accelerated by relativistic shocks that may be formed
in an unsteady relativistic wind
[32]
up to similar energies (see e.g. refs [6, 7, 8]).
For very-high-energy CRs (termed EGCR) to originate in coronal GRBSs requires
a CR to -ray ratio   1, but existing physical GRBS models (e.g. ref. [15]) are
incapable of producing CRs with such high energies.
5
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7


Gamma-ray bursters as sources of cosmic rays

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