A&A 396, 693-703 (2002)
DOI: 10.1051/0004-6361:20021367
M.-A. Aloy1 - J.-M. Ibáñez2 - J.-A. Miralles3 - V. Urpin2,4
1 - Max-Planck-Institut für Astrophysik,
Karl-Schwarzschild-Str. 1, Postfach 1523,
Garching, 85748, Germany
2 - Departamento de Astronomía y Astrofísica,
Universidad de Valencia, 46100 Burjassot, Spain
3 - Departament de Física Aplicada, Universitat d'Alacant,
Ap. Correus 99, 03080 Alacant, Spain
4 - A.F. Ioffe Institute of Physics and Technology,
194021 St. Petersburg, Russia
Received 2 July 2002 / Accepted 30 August 2002
Abstract
We consider the transverse structure and stability
properties of relativistic jets formed in the course of the collapse
of a massive progenitor. Our numerical simulations show the presence
of a strong shear in the bulk velocity of such jets. This shear can be
responsible for a very rapid shear-driven instability that arises for
any velocity profile. This conclusion has been confirmed both by
numerical simulations and theoretical analysis. The instability leads
to rapid fluctuations of the main hydrodynamical parameters (density,
pressure, Lorentz factor, etc.). However, the perturbations of the
density are effectively decoupled from those of the pressure because
the beam of the jet is radiation-dominated. The characteristic growth
time of instability is much shorter than the life time of the jet and,
therefore, may lead to a complete turbulent beam. In the course
of the non-linear evolution, these fluctuations may yield to internal
shocks which can be randomly distributed in the jet. In the case that
internal shocks in a ultrarelativistic outflow are responsible for
the observed phenomenology of gamma-ray bursts, the proposed
instability can well account for the short-term variability of
gamma-ray light curves down to milliseconds.
Key words: magnetohydradynamics (MHD) - gamma rays: bursts - gamma ray: theory - ISM: jets and outflows - galaxies: jets
Catastrophic stellar events like massive stellar core collapse (see,
e.g., van Putten 2001; Mészáros 2001 and
references therein) or merging of a binary neutron star (Paszynski
1996; Eichler et al. 1989; Mochkovitch
et al. 1993) have been proposed to explain the
energetics of gamma-ray bursts (GRB). Nowadays, an increasing amount
of observational evidence stresses the association, at least, of some
GRB events with massive progenitors (Bulik et al.
1999; Bloom et al. 1999; Hanlon et al. 2000; Reeves et al. 2002). One of
the most attractive scenarios is the collapsar model (Woosley
1993; MacFadyen & Woosley 1999) which can produce
the required energy of a GRB even at cosmological distances. This
model involves the collapse of the central core of a massive, evolved
star to a newly-formed black hole (BH). The progenitor star can be,
for instance, a rotating Wolf-Rayet star
(1993). Hydrodynamic simulations of collapsars have been
performed for a 35
main-sequence star whose 14
helium core collapses to form 2-4
black hole. Provided that
the core has a sufficient amount of angular momentum, a massive
geometrically thick accretion disc of several tenths of a solar mass
can be formed around the BH. The BH accretes matter from the disc at
rates of the order of solar masses per second. The burst is generated
by a local energy deposition due to the annihilation of
coming from the accretion disk or/and due to the
release of the BH spin energy by means of magnetic fields. In either
case, the energy is preferentially released along the rotation axis,
close to the central engine and gives rise to a relativistically
expanding bubble of radiation and pairs. The duration of the burst is
given by the hydrodynamic time scale for the core of the star and its
Helium envelope to collapse or by the viscous evolution time for the
accretion disc, whichever is greater. In order to produce a
long-duration GRB with a complex pulse structure, accretion is argued
to proceed over a period of time comparable to the prompt phase of a
GRB. The energy released due to accretion is sufficient to drive a
collimated, baryon-dilute fireball that penetrates the outer layers of
the star forming a relativistic jet (Aloy et al. 2000).
Recent observations indicate that the rapid temporal decay of several
GRB afterglows is consistent with the evolution of a highly
relativistic jet after it slows down and spreads laterally rather than
with a spherical blast wave (Sari, Piran & Halpern 1999;
Halpern et al. 1999; Kulkarni et al. 1999; Rhoads 1999). Therefore,
formation of relativistic jets of baryon-clean material with bulk
Lorentz factors
102- 103 represents a mayor problem in
the collapsar model of GRB. Detailed simulations of formation and
propagation of a relativistic jet in collapsars have been performed by
Aloy et al. (2000). Assuming an enhanced efficiency of
energy deposition in polar regions (with a constant or varying
deposition rate), the authors obtained an ultrarelativistic jet along
the rotation axis, which is highly focused (with the half-opening
angle
)
and is capable of penetrating the star. The
simulations were performed with the multidimensional relativistic
hydrodynamic code GENESIS (Aloy et al. 1999) using a
two-dimensional spherical grid. A relativistic jet in this model forms
within a fraction of a second and exhibits the main morphological
elements of "standard'' jets: a terminal bow shock, a narrow cocoon,
a contact discontinuity, and a hot spot. The maximum Lorentz factor is
as large as
25-30 at break-out, and can be even greater after
the break-out (
44 at the end of simulations). This Lorentz
factor is only a bit smaller than the critical value
102-103 that one requires for the fireball model (Cavallo &
Rees 1978; Piran 1999).
According to current views, the GRB is made either as the jet encounters a sufficient amount of mass in circumstellar matter or by internal shocks in the jet (Rees & Mészáros 1992, 1994; Daigne & Mockkovitch 1998, 2000). Therefore, the properties of jets from collapsars are of crucial importance for understanding the mechanism of GRBs. For example, the non-uniformity across the jet can influence the structure of internal shocks and, hence, the gamma-ray emission from these shocks. A transverse gradient can also drastically change the temporal decay rate of afterglows, making the decay flatter or steeper depending on the transverse structure (Mészáros et al. 1998). Gamma-ray light curves also show a great diversity of time dependences ranging from a smooth rise and quasi-exponential decay, through light curves with several peaks, to variable light curves with many peaks, and substructure sometimes down to milliseconds. Hydrodynamic instabilities arising in the jet can be responsible for the fine time structure of GRBs in a row with a temporal variability caused by accretion onto the BH and the circumstellar interaction. This complex time dependence can provide clues for the understanding of the geometry and physics of the emitting regions. Apart from this, instabilities and their associated fluctuating hydrodynamic motions can lead to the generation of magnetic fields in a highly conductive e- e+ plasma. In the presence of turbulent magnetic fields, the electrons can produce a synchrotron radiation spectrum (Mészáros & Rees 1993; Rees & Mészáros 1994) similar to that observed (Band et al. 1993). The inverse Compton scattering of these synchrotron photons can extend the spectrum even into the GeV range (Mészáros et al. 1994).
In the present paper, we consider the stability properties of jets which are formed and propagate in collapsars. Large-scale hydrodynamic instabilities can be the reason for the observed morphological complexity of "standard'' jets, and this has motivated many analytical (see, e.g., Birkinshaw 1984, 1991, 1997; Hardee & Norman 1988; Zhao et al. 1992; Hanasz & Sol 1996) and numerical (Hardee et al. 1992; Hardee et al. 1998; Bodo et al. 1998; Micono et al. 2000; Agudo et al. 2001) studies of the stability properties of jets. Usually, the jet is considered as a beam of gas with one bulk velocity and constant density surrounded by a very narrow shear layer separating it from the external medium. One possible mechanism of destabilizing such jets is often attributed to the well-known Kelvin-Helmholtz instability which, in its classical formulation, is the instability of a tangential discontinuity between two flows, generally having different density (see, e.g., Landau & Lifshitz 1978; Chandrasekhar 1981). However, jets in collapsars show a complex structure with the presence of strong transverse shear and substantial density stratification (Aloy et al. 2000). Obviously, the stability properties of such sheared, stratified jets may well be different from those of jets with constant bulk velocity and density.
The outline of this paper is as follows. In Sect. 2, we represent the results of numerical calculations and discuss the transverse structure of a jet originating in a collapsar. In Sect. 3, we represent the linear stability analysis of a sheared, stratified jet by making use of a WKB-approximation. Finally, in Sect. 4, we discuss the possible role of the instability in the evolution of jets.
The simulations performed are the same as in Aloy et al. (2000). Here we reproduce the technical details for
completeness. The computational mesh is a two-dimensional spherical
grid (with coordinates ). In the r-direction the
computational grid has 200 zones spaced logarithmically between the
inner boundary and the surface of the helium star at
cm. Equatorial symmetry is assumed and the working angular
resolution is
close to the polar region (
)
and decreases logarithmically between
.
A central Schwarzschild BH of mass
provides the
gravitational field that couples the system. Self-gravity of
the star is neglected, i.e., we consider only the gravitational
potential of the BH. Our equation of state (EoS) includes the
contributions of non-relativistic nucleons treated as a mixture of
Boltzmann gases, radiation, and an approximate correction due to
e+e--pairs as described in Witti et al. (1994).
Complete ionization is assumed, and the effects due to degeneracy are
neglected. We advect nine non-reacting nuclear species which are
present in the initial model: C12, O16, Ne20,
Mg24, Si28, Ni56, He4, neutrons and protons.
![]() |
Figure 1: Snapshot of the logarithm of the rest-mass density after an evolution time of 5.24 s. The color scale indicates the value of the logarithm of the rest-mass density (in CGS units). The solid arc marks the limit of the exponentially decaying atmosphere while the dashed one marks the surface of the collapsing star. The numbers around the box represent distances in units of 1010 cm. |
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In a consistent collapsar model a jet will be launched by any process
which gives rise to a local deposition of energy and/or momentum, as
e.g.,
-annihilation, or magneto-hydrodynamic processes. We
reproduce such a process by depositing energy at a prescribed rate
homogeneously within a
cone around the rotation axis. In
the radial direction the deposition region extends from 200 km to
600 km. We have investigated a constant energy deposition rate
erg s-1, of
erg s-1, and
a varying deposition rate with a mean value of 1050 erg s-1.
The constant rates bracket the expected deposition rates of collapsar
models, while the varying rate mimics, e.g., time-dependent mass accretion
rates resulting in time-dependent
-annihilation
(MacFadyen & Woosley 1999).
We have endowed the star with a Gaussian atmosphere, which at
passes over into an external uniform medium with a
density 10-5 g cm-3 and a pressure
10-8 p(R). The
computational domain is extended to
with 70 additional
zones. At the end of the simulation, a relativistic jet has broken out
from the surface of the star and propagates through the added
atmosphere.
We will focus in the following on the model C50 of Aloy et al. (2000). The logarithm of the rest-mass density and the
Lorentz factor after an evolution time of 5.240 s are displayed in
Figs. 1 and 2.
![]() |
Figure 2: Snapshot of the Lorentz factor after an evolution time of 5.24 s. The color scale indicates the value of the Lorentz factor at every point of our computational domain. The solid arc marks the limit of the exponentially decaying atmosphere while the dashed one marks the surface of the collapsing star. The number around the box represent distances in units of 1010 cm. Note that the horizontal scale has been enlarged by a factor of 3 in order to display more details of the stratified structure of the fluid in the jet. |
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In Fig. 3,
![]() |
Figure 3:
Lorentz factor as a function of the radius for three
different angles (
![]() |
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Figure 4 shows the radial dependence of the rest-mass
density for a few selected polar angles.
The density shows an even
stronger dependence on
than does the Lorentz factor. For
instance, the density contrast between the axis (where the density is
extremely low,
10-3 - 10-4 g/cm3) and
is 4-6 orders of magnitude. This is in large contrast to
the relatively small average change of the density with r which does
not exceed one order of magnitude when r varies from
cm to R. The main reason for this change is the fact that the
radial grid stretches logarithmically and, therefore, as r grows,
there is a loss of numerical resolution that may suppress small-scale
structures. Like the Lorentz factor, the density shows a very
remarkable small-scale structure approximately with the same length
scale as
.
However, fluctuations of the density are larger (up
to a factor of
102-103). The reason for this behavior of the
density is that the beam is radiation dominated according to our EoS
(see Sect. 2) because the average temperature of the beam
is
K. In this radiation-dominated flow, the
relative variation of the density, resulting from variations of
temperature and of pressure, can be written as
![]() |
(1) |
![]() |
(2) |
![]() |
Figure 4:
Rest-mass density as a function of the radius for three
different angles (
![]() |
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In Fig. 5,
![]() |
Figure 5:
Plots of four physical quantities as a function of the polar
angle, ![]() ![]() |
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We model the jet from collapsars by an infinitely long plasma cone
with a half-opening angle
.
From simulations, the typical
value of
is somewhat around 6-10
.
Plasma inside
the jet moves with a velocity
with respect to the ambient medium; r,
,
are the
spherical coordinates with
,
,
being the corresponding unit vectors. As seen from the
behavior of the velocity profile (Fig. 5), we can
distinguish two different regions within the jet: the central core
where the flow is ultrarelativistic, and the surrounding transition
layer where the velocity decreases from relativistic to
non-relativistic values. In the core region, the Lorentz factor is
large,
3, and may even reach values as high as
25-30(see Fig. 3). It is convenient to distinguish between
these two regions in our analysis because of the difficulty of
describing both relativistic and non-relativistic flows within the
framework of the same analytical formalism. Basically, the thickness
of the transition layer is smaller than the radius of the core. Since
our main interest is the processes in the core region we will model
the effect of the transition layer in terms of the boundary
conditions. Note that outside this transition layer there exists an
extended region where the flow is non-relativistic but we neglect the
influence of this region because of its low energy.
In relativistic hydrodynamics the continuity, momentum, and energy
equations read (see, e.g., Weinberg 1972)
Our analysis of stability is based on the linearized set of Eqs. (3)-(5). Small perturbations will be
marked by the index 1; for unperturbed quantities subscripts will be
omitted with the exception of vector components. We consider the
instability which arises on a time scale much shorter than the
characteristic evolution time scale of the jet. Therefore, we can
adopt a quasi-steady approximation neglecting the time dependence of
unperturbed quantities. In this approximation, the linearized
continuity, momentum and energy equations are
In a conical jet with a small opening angle, the characteristic
radial length scale of unperturbed quantities is of the order of the
spherical radius and is much greater than the transverse length scale.
Therefore, we can treat the radial and -dependences of
the perturbations separately. If we consider perturbations with
the radial wave vector, k, satisfying the condition
For our purposes, it will be convenient to use another form of the
Eq. (15). Making the substitution
![]() |
(19) |
In the core region where the temperature is high and the pressure is
determined by radiation, the sound speed, ,
is of the order of
and, hence, the acoustic frequency is comparable to that
of the light,
kc. Since the turnover time scale, |V'|-1,
characterizing the rate of hydrodynamical processes associated with
shear is also very short, we can expect that typical frequencies in
our model are sufficiently large. Therefore, we can try to obtain the
solution of Eq. (17) in a high frequency limit when
,
or
The behavior of q is determined by the -dependence of
the jet velocity and density. We consider the case closely related
to the calculations represented in Sect. 2. These simulations
indicate that the density profile within the jet core can be fitted
with a sufficient accuracy by a simple exponential dependence,
![]() |
(22) |
The velocity profile can be modeled by
the parabolic dependence,
![]() |
(23) |
Since
is large in the jet core, the contribution of the
last four terms on the r.h.s. of Eq. (21) which
are proportional to
or
is small compared to
the second term (
)
if we assume that
in the main
fraction of the core region where instability arises. After these
simplifications, we obtain the following expression for q2 in the
jet core
It is convenient to represent a WKB-solution from both sides of the
turning point in the form
To obtain the eigenvalues and eigenfunctions of Eq. (17) one needs the boundary conditions for
perturbations. One of the boundary conditions is obvious: since
p1 is a physical quantity it should be finite everywhere including
the jet axis and, hence, f should vanish at x=0. If the turning
point is not very close to 0, then usually it is sufficient to choose
an exponentially decreasing solution at x < x*. It is known (see,
e.g., Landau & Lifshitz 1981) that the solution
(29) at x > x* matches an exponentially decreasing
solution beyond the turning point (at x < x*) if
and
where C is constant. Hence, the solution at x>x* satisfying the
true boundary condition at x=0 has the form
![]() |
(30) |
![]() |
(31) |
![]() |
(32) |
![]() |
(35) |
![]() |
(36) |
![]() |
(37) |
![]() |
(38) |
The characteristic growth length, ,
can be estimated, for
,
as
![]() |
(40) |
We have treated numerically and analytically the instability that can
arise in jets from collapsars. The instability is caused by a combined
action of shear, which is unavoidable in such jets, and an extremely
high compressibility associated with a relativistic sound speed. It
turns out that jets from collapsars are more unstable than, for
example, standard extragalactic jets because of the relativistic
compressibility. The fact of instability itself does not depend on a
particular shape of the velocity profile: the instability can arise
for any dependence .
However, the growth time is shorter
for flows with a stronger shear. Note that only non-homogeneous
perturbations in the radial direction (
)
can be unstable.
Although it is commonly believed that three dimensional studies of the stability of relativistic jets will include additional instable modes (in many cases even more unstable that the axisymmetric ones), the recent work of Hardee & Rosen (1998), has pointed the fact that shear leads both to an enhancement of the axisymmetric modes and a suppression of the asymmetric modes. Hence, a fully three dimensional study would not yield to radically different results from the ones that we obtain here.
In the main fraction of the jet volume, the instability grows very
rapidly. If we estimate the average Lorentz factor as 10 and
the half-opening angle as
then the growth time (39) is of the order 0.01 r/c. This is much
shorter than the life time of the jet even at
.
Therefore,
we can expect that shortly after jet formation the instability will
generate well developed turbulent motions with substantial
fluctuations of the Lorentz factor, density, pressure, etc. This
conclusion is in good agreement with the results of numerical
simulations (see Figs. 3 and 4). During the jet's propagation, fluctuations can
become noticeable at a relatively early evolutionary stage and, hence,
at a small distance from the formation region because the growth time
is sufficiently short even in the inner region of the collapsar. Of
course, the initial amplitude of fluctuations in the jet is quite
uncertain. In our case, the initial fluctuations are triggered by
numerical reasons, but it is quite likely that they mimic the
irregularities of the process of accretion onto the central black
hole. Usually the time
required for an instability to
amplify the amplitude of perturbations to a noticeable value is longer
than the growth time,
,
by some factor which is typically
(5-10), so
.
After this time, the
fluctuations have grown by a factor
102-104 compared to
their initial value which probably is sufficient to become
significant. In the region where
is shorter than the
propagation time scale, which can be estimated as
r/c,
fluctuations reach noticeable values. On the contrary, fluctuations
seem to be insignificant in regions where
.
We can
define the radius where the instability starts to manifest itself as
that where the condition
is
fulfilled. Substituting
,
we obtain that
![]() |
(41) |
Note that the radial wave vector of unstable perturbations, k,
should satisfy rather restrictive conditions. First, k has to be
sufficiently large for the applicability of the local approximation
(see Eq. (9)). On the other hand, condition (28) which is necessary for the existence of the
turning point, implies that
![]() |
(43) |
All this allows one to speculate that the calculated fluctuations of
parameters within the jet are physical (i.e., not simply numerical
artifacts) and reflect the presence of a very strong shear-driven
instability. We can expect that inhomogeneities caused by this
instability will produce shocks in the course of their non-linear
evolution when faster fluctuations try to overtake slower ones or when
fluctuations moving in the positive and negative radial directions
collide. If this is the case then internal shocks might be more or
less randomly distributed and oriented within the jet forming
filamentary structures. It is often supposed that shocks in a
ultrarelativistic wind or jet are responsible for GRBs themselves
whereas the impact against the ambient matter of this wind produces an
external shock which likely produces the observed afterglows (Rees &
Mészáros 1992, 1994). Shocks can convert a portion
of kinetic energy into a non-thermal gamma/X-ray transient emission
which is usually ascribed to particle acceleration by
shocks. Typically, the efficiency of this conversion is not high,
1-2%, but it can be much greater (
20-40%) for the
interaction of fluctuations with very different Lorentz factors
(Kobayashi et al. 1997; Kobayashi & Sari
2001). Since in our model the calculated fluctuations move with
substantially different Lorentz factors we can expect a highly
efficient transformation of their kinetic energy into radiation. The
proposed instability can also accounts for the rapid variability of
the gamma-ray light curves, which lasting from tens to hundreds of
seconds, exhibit variability sometimes down to milliseconds (Fishman
& Meegan 1995). Likely, the most rapid temporal variability
associated with the shear-driven instability has a time scale of the
order of the growth time (39). At the surface
of a collapsar, for example, this time scale is as short as
10-3 s. Since the jet is highly inhomogeneous and the Lorentz
factor varies strongly during the jet's propagation, a slower
variability could also be represented in the gamma-ray light curves.
Another remarkable inference from the considered model is that the turbulent motions caused by the instability may also be important for the electron-proton energy exchange and, particularly, for the generation of the magnetic field in jets from collapsars.
Acknowledgements
This research was supported in part by the Russian Foundation of Basic Research and Deutsche Forschungsgemeinschaft (grant 00-02-04011). V.U. thanks Ministerio de Educación, Cultura y Deporte of Spain for the financial support under the grant SAB1999-0222. M.A.A. acknowledges the EU-Commission for a fellowship (MCFI-2000-00504). M.A.A. thanks E. Müller and P.E. Hardee for their enlightening discussions.