A&A 369, 694-705 (2001)
DOI: 10.1051/0004-6361:20010131
H. C. Spruit - F. Daigne - G. Drenkhahn
Max-Planck-Institut für Astrophysik, Postfach 1317, 85741 Garching bei München, Germany
Received 21 April 2000 / Accepted 17 January 2001
Abstract
We consider possible geometries of magnetic fields in GRB outflows,
and their evolution with distance from the source. For magnetically
driven outflows, with an assumed ratio of magnetic to kinetic energy
density of order unity, the field strengths are sufficient for
efficient production of -rays by synchrotron emission in the
standard internal shock scenario, without the need for local
generation of small scale fields. In these conditions, the MHD
approximation is valid to large distances (
1019cm). In
outflows driven by nonaxisymmetric magnetic fields, changes of
direction of the field cause dissipation of magnetic energy by
reconnection. Much of this dissipation takes place outside the
photosphere of the outflow, and can convert a significant fraction
of the magnetic energy flux into radiation.
Key words: gamma-ray bursts - magnetic fields - radiation mechanisms: nonthermal - shock waves
Several models for cosmic -ray bursts (hereafter GRBs) make
use of rapidly rotating compact stellar-mass sources. Though many
details in each case are uncertain, the two more developed and popular
scenarios involve the coalescence of two compact objects (neutron star
+ neutron star or neutron star + black hole) and the collapse of a
massive star to a black hole (collapsar) (Mészáros & Rees
1992; Narayan et al. 1992; Woosley
1993; Paczynski 1998). Both lead to the
same system: a stellar mass black hole surrounded by a thick torus
made of stellar debris or of infalling stellar material partially
supported by centrifugal forces. An other interesting proposition
(Usov 1992; Kluzniak & Ruderman 1998; Spruit
1999) associates GRBs with highly magnetized millisecond
pulsars.
The energy release by such a source can be fed from various reservoirs. In the case of a thick disk + black hole system, the burst can be powered by the accretion of disk material by the black hole or by extracting directly the rotational energy of the black hole via the Blandford-Znajek mechanism. In the case of a highly magnetised millisecond pulsar the energy release comes from the rotational energy of the neutron star.
Luminosities as high as those of GRBs cannot be radiated in the close
vicinity of the source. The energy released must first drive a wind
which rapidly becomes relativistic. Its kinetic energy is then
converted into -rays (producing the prompt emission of the
GRB) at large radii via the formation of shocks, probably within the
wind itself (internal shock model) (Rees & Mészáros
1994; Daigne & Mochkovitch 1998). The wind is
finally decelerated by the external medium, which leads to a new shock
responsible for the afterglow emission observed in X-rays, optical
and radio bands (external shock model) (Katz 1994; Rhoads &
Paczynski 1993; Sari & Piran 1997; Wijers
et al. 1997).
The Lorentz factor
of the relativistic wind must reach high
values (
-103) both to produce
-rays
and to avoid photon-photon annihilation along the line of sight,
whose signature is not observed in the spectra of GRBs
(Goodman 1986, see also Baring 1995). This
limits the allowed amount of baryonic pollution in the flow to a very
low level and makes the production of the relativistic wind a
challenging problem. Only a few ideas have been proposed and none
appears to be fully conclusive at present.
However it is suspected that large magnetic fields play an important
role. This is obvious in models using a highly magnetised millisecond
pulsar, but in the case of a thick disk + black hole system a magnetic
wind is also probably more efficient than the initial proposition
where the wind is powered by the annihilation of
neutrino-antineutrinos pairs along the rotational axis (Berezinskii
& Prilutskii 1987; Goodman et al. 1987;
Ruffert et al. 1997). The magnetic field in the disk
could be amplified by differential rotation to very large values (
G) and a magnetically driven wind could then be emitted
from the disk (Thompson 1994; Mészáros & Rees
1997). under severe constraints on the field geometry
and the dissipation close to the disk, large values of the lorentz
factor can then be reached (Daigne & Mochkovitch
2000c,2000a).
The emission of photons at large radii via the formation of shocks is
perhaps better understood than the central engine. It is believed that
a non-thermal population of accelerated electrons is produced behind
the shock waves and synchrotron and/or inverse Compton photons are
emitted. A magnetic field is then required. In the case of the
afterglow emission, the external shock is propagating in the
environment of the source, and the magnetic field has to be locally
generated by microscopic processes (Mészáros & Rees
1993; Wijers 1997; Thompson & Madau
2000). In the case of the prompt -ray emission
which is probably due to internal shocks within the wind, such a
locally generated magnetic field is also usually invoked (Rees &
Mészáros 1994; Papathanassiou & Mészáros
1996; Sari & Piran 1997; Daigne &
Mochkovitch 1998). It has also been pointed out that a
large scale field originating from the source could play the same role
(Mészáros & Rees 1993,1997; Tavani
1996).
The argumentation in this paper is organized as follows. In Sect. 2 we show that for typical baryon loading, the particle density in the outflow is so large that the MHD approximation is appropriate out to distances of the order 1020cm. This will turn out to be larger than the other relevant distances.
Due to the baryon loading, the GRB case is therefore different from the essentially baryon-free pulsar wind case (e.g. Usov 1994), where the MHD approximation breaks down much earlier, and plasma theories of large amplitude electromagnetic waves (LAEMW) are applied. The GRB case is actually simpler than the pulsar case in this respect.
The evolution of the magnetic field can therefore be dealt with in MHD
approximation. In Sect. 3 we discuss how the strength and
configuration of the field in the outflow depends on assumptions about
the central engine. This is done first without allowing for decay of
the field by internal MHD processes. In Sect. 4 we
show that the field strengths so obtained are sufficient to produce
synchrotron and/or inverse Compton emission in the standard internal
shock model, without the need for local generation of microscopic
magnetic fields. In Sect. 5 we then argue that internal
MHD processes are, in fact, likely to cause magnetic field energy
density to be released during the outflow. This may be a significant
contributor to the observed emission. The efficiency of conversion of
the primary energy of the central engine to observable -rays
can also be much higher than in internal shock models. The arguments
are summarized again in Sect. 7.
In this section, as well as throughout this paper, a prime denotes quantities measured in the rest frame of the outflowing matter, unprimed quantities are measured in the "laboratory'' frame (understood here as a frame at rest relative to the central object of the burst).
In order to maintain the necessary current there must be enough plasma
available. As an example, consider the case of a magnetized outflow
with magnetic field of alternating direction, as happens in a
pulsar-like model (Sect. 3.4). We approximate this as a
plane sinusoidal wave with a wave number k and a angular frequency
:
Plasma physical instabilities can set in at current densities much lower than (4). They will produce an anomalous resistivity in the plasma so that an electric field is present also in the rest frame of the plasma. The electric field due to this resistivity is small, however, compared to the magnetic field strength as long as the charged particle density is larger than the minimum density. The MHD approximation in this sense is then a good one on large scales, in spite of the presence of small scale plasma processes.
If the outflow has a finite duration ,
and a constant speed
in this time interval, it moves as a shell of width
.
When the shell (assumed spherical) has expanded to
a radius r, the particle density measured in the lab frame is
where M is the total mass ejected
and
the proton mass.
With (3) and (4), the minimum particle density
is
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(6) |
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(8) |
In baryon-free cases, the expected values of
may be much higher
(Usov 1994), as in the typical pulsar-wind scenarios (Kennel
& Coroniti 1984; Begelman & Li 1992;
Blackman & Yi 1998), and the critical radius for MHD
condition correspondingly smaller. Such low baryon loading seems
rather unlikely, however, for the currently proposed scenarios for GRB
engines. Outflows from accretion disks, merging stars, and supernova
envelopes are all intrinsically highly baryon loaded environments, and
have some difficulties reaching baryon loadings as small as
.
Even in the rapidly magnetized msec neutron star
scenario (Spruit 1999), emergence of magnetic fields from
the star at the required short time scales is likely to imply that
some baryonic mass is lifted together with the magnetic fields.
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Figure 1:
Evolution of an initial dipole field by a radial outflow
(schematic). Top left: initial field configuration. Top right:
view on a larger scale, when the outflow, of finite duration
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Open with DEXTER |
A burst of duration
produces a shell of width
,
travelling outward at a speed
.
The observed radiation from
this shell is produced when it has expanded to a radius
.
This
thin shell carries with it magnetic field lines from the source, which
we call "trapped field lines''. For the rest of this section, we assume
a relativistic outflow,
.
If reconnection processes inside the shell can be neglected, the field
lines are frozen in the expanding shell. If
(in spherical coordinates), and the width of the shell is
constant, the components then vary with distance as
,
.
This is because the radial
component is divided over the surface of the spherical shell, while
the components parallel to the shell surface decrease with the
circumference. More formally, the induction equation
has the components
How many field lines are trapped in the outflow, and hence which fraction of the outflow energy is magnetic, depends on conditions near the source. We consider here the representative possibilities.
First consider a fireball expanding from the surface of a central
object (not specified further) of radius R and a dipolar magnetic
field of moment .
We assume that the magnetic field plays no role
in the driving of the outflow, but that it can confine plasma up to
its own energy density
.
Assume first that the expansion is kinematic, i.e. that the magnetic
field is weak and its backreaction on the flow can be neglected. The
number of field lines trapped in the outflow is equal to the flux of
field lines outside crossing the dipole's equator outside the radius
R:
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(12) |
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(14) |
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(16) |
In magnetic models for GRB engines, the magnetic field serves to
extract rotation energy from a rapidly rotating relativistic object.
The details of such magnetic extraction (especially three-dimensional
ones) are still somewhat uncertain, but basic energetic considerations
are simple. Rotation of the mass-loaded field lines induces an
azimuthal field component .
Let the distance from the rotating
object where this component is equal to the radial field be r0.
Except for cases with large baryon loading that are probably not
relevant for GRB engines, r0 is of the order of the Alfvén
radius, which can lie anywhere between the surface of the object Rand the light cylinder
.
The energy output Ltransmitted by the azimuthal magnetic stress (
)
is
then of the order
.
For a field
dominated by its dipole component
,
this yields a luminosity
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Figure 2:
Field configuration in quasi-spherical magnetic outflow
driven by a perpendicular rotator ("pulsar-like'' case)
(schematic). Left: view in the equatorial plane, with dots and
pluses indicating field lines into and out of the plane of the
drawing. Right: top view from the rotational pole. Bottom right:
same view on larger scale, at a later time ![]() |
Open with DEXTER |
The energy estimate (17) does not tell what the field
configuration in the outflow looks like. The possibilities for
dissipation of magnetic energy inside the outflow depend strongly on
this configuration, which in turn depends on the nature of the
magnetic field of the central engine. An important distinction is
whether the rotating magnetic field is axisymmetric or
nonaxisymmetric. In the nonaxisymmetric case, the outflow
contains magnetic fields varying on the rather small scale
,
the wavelength of the outgoing wave. In such a field,
internal dissipation turns out to be much more likely to be important
than in an axisymmetric field, where the length scale of the field is
of the order of the distance r. In the following, this is
illustrated with a few specific cases.
Estimate (17) only gives the total luminosity. Which
fraction of it is in the form of kinetic energy and which in magnetic
energy depends on more physics. At one extreme is Michel's
(1969) model, in which the outflow consists of cold
(pressureless) matter accelerated exclusively by the magnetic field.
In this case, the ratio
of
magnetic to kinetic luminosity is of the order
.
At large Lorentz factors, the energy is almost entirely in
electromagnetic form, in this model. This is probably a property of
the special geometry of the model, in which the flow is limited to the
equatorial plane. If outflow in other directions is considered, much
smaller values of
result (Begelman & Li 1994).
At the other extreme, consider a case in which most magnetic energy is
released inside the light cylinder, in the form of a dense pair plasma
(by some form of magnetic reconnection, for example). The result would
then be just like the hydrodynamic expansion of a simple (Goodman
1986; Paczynski 1986) pair plasma
fireball, and the resulting outflow would have
.
If some but
not all energy is dissipated close to the source, intermediate values
of
would result. Since these important questions have not been
resolved yet, we keep
as a free parameter in the following.
Where necessary we assume that it typically has values or order unity.
Assume we have a perpendicular rotator, i.e. the rotating object has
its dipole axis orthogonal to the rotation axis. At the source surface
r0, the rotating field is then strongest at the equator, and one
expects the energy flux to be largest near the equatorial plane. With
each rotation, a "stripe'' consisting of a band of eastward and one of
westward azimuthal field moves outward (Fig. 2, see also
Coroniti 1990; Usov 1994). Assuming the
outflow to be relativistic, the width
of such a stripe in
the rest frame of the central object is
.
The
(absolute value of the) azimuthal flux
in each half-wavelength
is of the order of the poloidal flux outside the source surface r0,
.
For spherical expansion of this amount of
azimuthal flux, the field strength at distance r is then
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(18) |
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(19) |
Consider a collimated outflow along the axis of rotation
(Fig. 3). This might be achieved by magnetic models in which
the azimuthal field collimates the flow towards the axis by hoop
stress (Bisnovatyi-Kogan & Ruzmaikin 1976; Blandford
& Payne 1982; Sakurai 1985). In such a
model, one assumes an axisymmetric (about the rotation axis) poloidal
field, which is wound up into an azimuthal field wrapped around the
axis. Let the opening angle of the outflow be
(assumed
constant), and
the cylindrical radius. At the source surface
the poloidal and azimuthal field components
are equal, and in the absence of magnetic reconnection processes
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= | ![]() |
(21) |
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= | ![]() |
(22) |
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Figure 3:
Jet-like outflow of finite duration, magnetically
driven by an axially symmetric rotator. Left: configuration near
the source, showing how the field in the outflow is wound into a
toroidal (azimuthal) field. Right: large scale view after a time
long compared to the duration ![]() |
Open with DEXTER |
The internal shock model assumes that the initial distribution of the
Lorentz factor in the shell is highly variable. Rapid layers catch up
with slower ones leading to internal shocks propagating within the
relativistic shell. The hot material behind the shock waves is
radiating efficiently and produces the observed prompt -ray
emission of GRBs (Rees & Mészáros 1994; Kobayashi
et al. 1997; Daigne & Mochkovitch 1998;
Panaitescu & Mészáros 1999; Daigne &
Mochkovitch 2000b). We only summarize here the basic
assumptions of the model. For details see Daigne & Mochkovitch
(1998).
Consider two layers of equal mass (for simplicity) and of Lorentz
factor
and
(
)
which are
emitted on a timescale
.
They will collide at a radius
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(24) |
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(25) |
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(26) |
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(27) |
It is generally assumed (Rees & Mészáros 1994;
Papathanassiou & Mészáros 1996; Sari &
Piran 1997; Daigne & Mochkovitch 1998) that
behind the shock wave a fraction of the electrons come into (at least
partial) equipartition with the protons and become highly
relativistic. If a fraction
of the dissipated
energy goes into a fraction
of the electrons, their
characteristic Lorentz factor (in the comoving frame) will be
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(28) |
The magnetic field involved in the synchrotron radiation is usually
assumed to be generated locally by microscopic processes. Such a
process has been proposed by Medvedev & Loeb (1999).
(This is also assumed for the external shock propagating in the
interstellar medium and responsible for the afterglow; see however
Thompson & Madau 2000.) If this magnetic field is also
into equipartition with the protons and electrons, it will have
typical values of
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(29) |
If the GRB is powered magnetically, however, the outflow is naturally
magetized. (One good reason for assuming magnetic powering is that
alternatives like
annihilation are energetically
inefficient.) As shown in Sect. 3, the magnetic energy
content of the outflow is constant as long as internal dissipation of
magnetic energy can be neglected. For the three cases considered, the
ratio
of the (large scale-) magnetic to kinetic
energy density is
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(32) |
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(33) |
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O(1), | (34) |
The corresponding comoving magnetic field in all cases is
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(35) |
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(36) |
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(37) |
Synchrotron emission by accelerated electrons in a magnetic field
occurs at a typical energy (observer frame)
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(38) |
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(39) |
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(40) |
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(41) |
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(42) |
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(43) |
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(44) |
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(45) |
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(47) |
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(48) |
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(49) |
We consider now the other extreme case where only a small fraction of
the electrons are accelerated:
.
The Lorentz factor of
the electrons then reaches very high values of 5000-50000 and
-rays can be produced directly by synchrotron emission:
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(50) |
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(51) |
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(52) |
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(53) |
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||
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(54) |
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(55) |
We then can conclude that (i) for an equipartition field
has to be of the same order as
in order for the synchrotron process is to produce
-rays at high efficiency (
). (ii)
For a passively expanded source field the case
is again excluded because of an extremely low efficiency
and a typical energy which is more in the X-rays range. The case
suffers the same limitations as
before, the efficiency is only
.
(iii)
For a magnetically driven quasi-spherical outflow and a magnetically
driven jet the situation is similar to the equipartition case: the
efficiency is very high (
for a
quasi-spherical outflow and
for a jet).
In conclusion for this section, a passive expanded source field is
probably too weak in most cases to produce a -ray burst
without a locally generated magnetic field, but in the two other cases
described in this paper, a magnetically driven quasi-spherical outflow
and a magnetically driven collimated jet, there is no difficulty to
produce
-rays without any need of a supplementary field. The
efficiencies of the radiative process in these magnetically driven
cases are comparable to those already calculated for an equipartition
field. In particular, as was already pointed out in Daigne &
Mochkovitch (1998), this efficiency is expected to be
higher if the
-rays are directly produced by synchrotron
emission (which is possible if only a fraction of the electrons are
accelerated behind the shock waves).
Internal release of magnetic energy can be important for the
-ray lightcurves if the time scale is sufficiently fast that
the release is significant before the outflow reaches the external
shock. It should not be too fast, however. If the release takes place
before the photospheric radius, i.e. in the optically thick part of
the outflow, the internal energy generated is not radiated away.
Instead, it is converted, through the radial expansion of the shell,
into kinetic energy.
Assume that the outflow is driven by the rotating magnetic field of the central, compact object, i.e. that the field strength estimates (20) or (23) apply. Depending on the field configuration at the source, the field in the flow can be nonaxisymmetric to a greater or lesser degree, and depending on the nature of the acceleration process it can be quasi-spherical or more collimated along the rotation axis. Consider again the two cases discussed in Sect. 3.
In the perpendicular rotator case, the field consists of "stripes'' of alternating polarity, in which the field energy can be released by reconnection. In the axisymmetric jet case, the field is unstable to a kinking process. The magnetic field in both cases is far from a minimum energy configuration (a potential field). The free energy it contains is available if it can be released on a sufficiently short time scale. The initial perturbations from which the MHD instabilities grow are likely to be present at significant amplitude, from the start of the outflow, except in the unlikely event that the field configuration of the source is highly symmetric. Thus we may assume that the ordered configurations of Figs. 2 and 3 significantly change to more disordered ones within an MHD instability time scale. These more disordered configurations then are subject to fast reconnection processes. Reconnection takes place on time scales governed by the Alfvén speed. It depends on plasma resistivity as well, but in practical reconnection configurations (as opposed to highly symmetric textbook examples like tearing mode instability), the resistivity enters only weakly. In the Sweet-Parker model for 2-D reconnection, for example (e.g. Biskamp 1993), it enters through the logarithm of the magnetic Reynolds number. In more realistic 3-D modes of reconnection, the basic geometry of reconnection (a "chain link'' kind of configuration) differs from the 2-D geometries. Also, the reconnection tends to be distributed over many current sheets instead of a few reconnection points (see Galsgaard & Nordlund 1996,1997, for recent numerical results). The reconnection rate is still weakly dependent on the resistivity in these 3-D configurations.
In the perpendicular rotator, the field in the outflow changes (in the
lab frame) on a length scale
.
The time scale
of magnetic energy release scales with the Alfvén
crossing time over this length scale. In a comoving frame
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(56) |
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(57) |
The relativistic Alfvén speed is (e.g. Jackson
1999)
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(58) |
Let the ratio of magnetic energy flux (Poynting flux) to kinetic
energy flux be :
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(59) |
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(60) |
The magnetic dissipation radius, the distance
from the
source where magnetic energy release becomes important, is thus of the
order
For the case of an axisymmetric outflow along the rotation axis, the
situation is a bit different. The length scale L of the magnetic
field is now the jet radius,
,
where
is the
opening angle of the jet. Since this is measured perpendicular to the
flow, it is the same in the lab and the comoving frame. In the
comoving frame, the time for an Alfvén wave to communicate over this
distance is
,
in the lab
frame
.
This is less
than the time for the flow to reach a distance r,
if
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(63) |
On the other hand, magnetic energy release in the inner core so
defined already starts very close to the source (where the length
scale
is small). Thus only a fraction of the dissipation
will happen in the optically thin, observable regions.
The conclusion from the above is that in the case of a perpendicular
rotator, there is a well defined "magnetic dissipation radius''
cm where most of the magnetic energy is
dissipated. For a purely axisymmetric outflow along the rotation axis,
on the other hand, only a fraction of the magnetic energy can be
released, unless the opening angle is less than
.
This
fraction probably dissipates close to the source, and not necessarily
in the optically thin region where it could contribute to the observed
emission. In intermediate cases, where both an axisymmetric and a
nonaxisymmetric component are present, the magnetic field in the
outflow changes direction on the length scale
,
without completely changing direction. In such cases, the amount of
the magnetic energy that can be released in directions outside the
central core
is of the order
,
where
is the nonaxisymmetric
part of the field. This is a significant fraction of the total
magnetic energy unless the field is nearly axisymmetric. The
axisymmetric jet case, though attractive as a computable model, is
thus rather singular with respect to the question of magnetic
dissipation which we address in this paper.
The reconnection radii derived above need to be compared with the
radius
of photosphere in the outflow. If
is larger than
,
the dissipation of magnetic energy
takes place mainly in the optically thin regime, and the dissipated
energy is radiated away locally. If on the other hand
,
the energy released from the initially ordered field
configuration increases the internal energy of the plasma. The radial
expansion of the flow converts this energy into kinetic energy of
outflow. Though this may be useful in obtaining large Lorentz factors,
it also implies that the magnetic energy left in the flow by the time
it passes through the photosphere is small if
.
The net output of a magnetically driven fireball with
magnetic dissipation taking place mostly in the optically thick regime
would be essentially the same as a standard non-magnetic fireball,
with the attendant problem of a low efficiency of the production of
radiation by internal shocks.
The photospheric radius in a steady relativistic outflow has been
derived by Abramowicz et al. (1991). Assume a total
amount of mass M is ejected with a constant Lorentz factor ,
at a rate
which is is constant in the time interval
,
(and zero outside this interval). The radius at which the
optical depth of the expanding shell reaches unity is then
,
where
is the
opacity. This a factor
smaller than for photons
propagating through a non-moving shell. This is a consequence of the
expansion of the shell. The optical depth of a moving shell of
constant (in time) density is a Lorentz invariant, hence the same for
photons moving along the direction of the flow and those moving in the
opposite direction (assuming also that the opacity is
energy-independent). In a radial outflow, however, a photon sees a
different history of mass density depending on its direction.
The result is widely used, e.g. Rees & Mészáros (1994)
and Beloborodov (2000), and is somewhat important for
the quantitative estimates below. Since its correctness is sometimes
questioned, we rederive it here in a different way from Abramowicz
et al. (1991), by working in the lab frame. Let a
photon be released at the inner edge
of the expanding
shell at time t=0. With the aid of a Minkowski diagram, one finds
that the photon exits at the outer edge of the shell at time
and radius
where
is the
shell thickness. To compute the optical depth traversed by the
photon, we note that by the assumption of a constant outflow rate the
density
in the lab frame is only a function of r,
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(64) |
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= | ![]() |
|
= | ![]() |
(65) |
The optical depth is a decreasing function of time as the shell
expands and
increases. Define now the photospheric
radius
as the value of
for which
.
With
and solving
for
we get
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(66) |
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(67) |
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(69) |
In addition to the baryons, pairs could also contribute to the opacity. The photospheric radius (68) then does not apply, because it includes only the constant opacity of scattering on the electrons associated with the baryonic mass. A general property of GRB fireball models is that the photosphere lies well outside the region where pairs contribute to the opacity. We summarize the argument here.
Assume a steady wind in which pairs dominate the kinetic energy flux
and the opacity, and in which the kinetic energy of the pairs is a
fraction
of the total luminosity
.
Let
K be the temperature at the pair
photosphere (due to the steep dependence of the pair density on
temperature, this value does not depend much on conditions in the
outflow). Ignoring the opacity due to baryonic matter, the radius of
the pair photosphere
is (Usov 1994):
This value is much smaller than the photospheric radius (68). Thus, for typical baryon-loaded GRB parameters pairs annihilate before they reach the optically thin domain.
Magnetic fields may well be the main agent tapping the
rotational/gravitational energy in the central engines of GRB (for
reviews see Mészáros 1999). Alternatives like the
production of pair plasma fireballs by neutrino annihilation have
turned out to have a rather small efficiency of conversion of
gravitational energy (Ruffert et al. 1997). A fireball
powered by magnetic fields ("magic hydrodynamics'') can in principle
produce -rays in the same way as in field-free mechanisms,
namely by internal shocks in the optically thin part of the flow. A
well-known problem with the internal shock mechanism, however, is the
low efficiency of conversion to
-rays Daigne & Mochkovitch
1998), of the order of a few percent. This is hard to
avoid, since efficient dissipation requires that a significant
fraction of the energy in the outflow is in relative motion between
parts moving at different velocities. The required large variations in
lead to dissipation close to the source, however, in the
optically thick regime. The dissipation taking place further out in
the optically thin regime is due to smaller velocity differences that
carry less energy. Ways to circumvent this limitation have been
invented (Kumar & Piran 2000; Beloborodov
2000; Lazzati et al. 1999). Even in
these schemes, however, it seems unrealistic to expect the
internal shocks to dissipate more than the external shock, hence the
conflict between observed gamma-ray fluences and actual estimations
of the energy dissipated by the external shock based on the
observations of a few afterglows remains.
A magnetically powered outflow naturally carries a magnetic field with it ("Poynting flux''). This raises the question whether dissipation of this internal magnetic energy in the outflow can perhaps produce the observed radiation with a better efficiency.
We have addressed this question by considering a few possible scenarios for magnetic fields in GRB outflows. In the first, the "passive scenario'', the magnetic field is assumed not to be responsible at all for powering the outflow, but only advected passively by an outflow produced by something else. We find that the maximum field strengths possible in this case are small compared with equipartition with the kinetic energy of the outflow, but still potentially significant for the effective production of synchrotron and/or inverse Compton emission in the internal shock model.
As magnetically-driven models we consider the case of a quasispherical outflow produced by a rotating non-axisymmetric magnetic field, and the case of a jet-like outflow along the axis of a rotating axisymmetric field. The quasispherical case is like the models produced for the Crab pulsar (Coroniti 1990; Gallant & Arons 1994). It has been developed for a completely baryon-free, pure pair-plasma outflow by Usov (1994). The jet case is similar to magnetohydrodynamically driven wind models (Blandford & Payne 1982; Sakurai 1985).
The magnetic field in all these cases is confined in an outward moving
shell of width ,
where
is the duration of the flow. If
internal dissipation is ignored in the outflow, the total magnetic
energy in the shell is constant, and the field strength B varies
with distance as r-1 (B and
measured in the observer
frame). The configuration of the magnetic field in the shell is
different in each case (see Figs. 1-3). If the
central engine is time-dependent, for example in the form of a series
of sub-bursts, magnetic shells like this are produced by each
sub-burst, and the magnetic flux and energy in the shells is renewed
with each burst (i.e. not limited by the magnetic flux of the central
object).
We find that the MHD approximation (in the sense that the electric field in the fluid frame is small compared with the magnetic field) is safe out to large distances from the source, of the order 1019cm or more. This is due to the relatively large amount of baryons in the outflow compared with, say, a pulsar wind situation. This simplifies the discussion of magnetic energy dissipation at least conceptually, since the energy release can be studied without detailed discussions of plasma processes (though they may enter again in the discussion of what produces the radiation).
The field strength in a magnetically driven outflow depends on the
extent to which internal MHD processes have been able to dissipate
magnetic free energy in the flow. In the absence of such processes,
the magnetic energy density is typically of the same order as the
kinetic energy density in magnetically driven flows, and the field is
well-ordered (large scale). In the internal shock model, such fields
are sufficient to produce synchrotron and/or inverse Compton emission
in the -ray range. The synchrotron case is favoured by a
higher efficiency during the whole course of the burst. There are now
some observational evidences that the magnetic field required for the
afterglow emission represents a small fraction of the equipartition
value whereas the prompt
-ray emission via internal shocks is
possible only with more intense fields close to the equipartition
value (Galama et al. 1999). In the standard picture where
the afterglow is due to the forward shock propagating in the external
medium where no large scale field is present, it could mean that the
generation of a local magnetic field behind the shock wave is
inefficient. It is then possible that this locally generated field is
also very small behind internal shocks within the shell but we show
here that the large-scale field present in a magnetically driven
outflow has no difficulty in most cases to supply the strength
necessary for synchrotron and/or inverse Compton emission.
The rate of energy release through MHD processes like instabilities
and fast reconnection is governed by the Alfvén speed. In the rest
frame of the outflow, the Alfvén speed is of the order of the speed
of light if the kinetic and magnetic energy fluxes are similar. In
outflows produced by a perpendicular rotator, the magnetic field
changes sign on the small length scale ,
and most of the
magnetic energy can be dissipated by reconnection. The typical radius
at which the energy release takes place in this case is
1012cm for standard GRB parameters (1052erg, 3s,
). The photospheric radius, on the other hand, is small
(mostly because of the relativistic effect discussed by Abramowicz
et al. 1991),
1010cm. We can thus be
confident that dissipation of energy stored in the magnetic field of
the outflow occurs in the optically thin regime.
In the case of an outflow produced by a purely axisymmetric rotating
magnetic object such changes of sign do not occur. Energy can still be
released by kink instabilities, but for causality reasons these are
not effective outside a narrow cone of angle
near the
rotation axis (Sect. 5). The expected amount of
magnetic dissipation in the optically thin regime is quite limited for
such an axisymmetric field. A purely axisymmetric configuration,
however, is a singular case, a priori unlikely for magnetic fields
produced in a transient object like the central engine of a GRB. In
the more likely case that a nonaxisymmetric field component is present
as well, the energy in this component can be released by reconnection
in nearly the same way as in the case of a perpendicular rotator.
Magnetic energy dissipation in the optically thin regime is probably
not as simple as the shock dissipation in the internal shock model. In
particular, the way in which nonthermal electron distributions are
produced still needs to be investigated. It is likely that nonthermal
radiation is again produced, as in the internal shock model, but the
shape of the radiation spectrum may be more difficult to compute, as
well as the typical time scale of the radiative process, which should
be short compared to the expansion time scale (see (46):
at
1013cm) to have a high radiative efficiency.
The main attraction of of GRB radiation produced by magnetic
dissipation in a magnetically driven outflow is the efficiency with
which the energy flux from the central engine can be converted into
observable radiation. This efficiency is limited only by the ratio of
Poynting flux to total energy flux in the flow. In the magnetically
driven cases considered, this ratio can be close to unity. In the
internal shock scenario, on the other hand, only a fraction
0.1
of the energy is dissipated in the optically thin region. Since the
bulk kinetic energy of the burst is dissipated in the afterglow, the
internal shock model predicts the afterglows to dominate the energy
output, which is probably inconsistent with current observations. In
magnetic dissipation models such as those discussed here, the energy
emitted in the afterglow can in principle be arbitrarily small
compared with the prompt emission. It is not yet clear, however, how
much of the magnetically dissipated energy can be in the form of
-rays.