We present results from axisymmetric, time-dependent magnetohydrodynamic
(MHD) simulations of the collapsar model for gamma-ray bursts. Our main
conclusion is that, within the collapsar model, MHD effects alone are able to
launch, accelerate and sustain a strong polar outflow. We also find that the
outflow is Poynting flux-dominated, and note that this provides favorable
initial conditions for the subsequent production of a baryon-poor fireball.Comment: 4 pages, to appear in proceedings of "2003 GRB Conference" (Santa Fe,
  NM, September 8-12, 2003), needs aipprocs LaTeX class, movies are available
  at http://rocinante.colorado.edu/~proga

Armitage, P. J.

Begelman, M. C.

MacFadyen, A. I.

Proga, D.

English

arXiv

We present results from axisymmetric, time-dependent magnetohydrodynamic (MHD) simulations of the collapsar model for gamma-ray bursts. Our main conclusion is that, within the collapsar model,MHD effects alone are able to launch, accelerate and sustain a strong polar outflow.

We also find that the outflow is Poynting flux-dominated, and note that this provides favorable initial conditions for the subsequent production of a baryon-poor fireball

Proga, Daniel

MacFadyen, Andrew I.

Armitage, Philip J.

Begelman, Mitchell C.

Caltech Authors - Main

MHD Simulations of the Collapsar Model for
GRBs
Daniel Proga, Andrew I. MacFadyen†, Philip J. Armitage and Mitchell
C. Begelman
JILA, University of Colorado, Boulder, CO 80309-0440, USA
†California Institute of Technology, Mail Code 130-33, Pasadena, CA 9112
Abstract. We present results from axisymmetric, time-dependent magnetohydrodynamic (MHD)
simulations of the collapsar model for gamma-ray bursts. Our main conclusion is that, within the
collapsar model, MHD effects alone are able to launch, accelerate and sustain a strong polar outflow.
We also find that the outflow is Poynting flux-dominated, and note that this provides favorable initial
conditions for the subsequent production of a baryon-poor fireball.
INTRODUCTION
The collapsar model is one of most promising scenarios to explain the huge release of
energy in a matter of seconds, associated with gamma-ray bursts (GRBs; [1, 2, 3, 4, 5]).
In this model, the collapsed iron core of a massive star accretes gas at a high rate
( 1M

s1) producing a large neutrino flux, a powerful outflow, and a GRB. Although
the association of long duration GRBs with stellar collapse is now secure ([6, 7]), basic
properties of the central engine are uncertain. In part, this is because previous numerical
studies of the collapsar model did not explicitly include magnetic fields, although they
are commonly accepted as a key element of accretion flows and outflows.
We present a study of the time evolution of 2.5-dimensional, magnetohydrodynamic
(MHD) flows in the collapsar model. This study is an extension of existing models of
MHD accretion flows onto a black hole (BH; [8]). In particular, we include a realistic
equation of state, photodisintegration of bound nuclei and cooling due to neutrino emis-
sion. Our study is also an extension of collapsar simulations by [3], as we consider very
similar neutrino physics and initial conditions but solve MHD instead of hydrodynami-
cal equations.
MODELS
We begin the simulations after the 17 M

iron core of a 25 M

presupernova star has
collapsed and study the ensuing accretion of the 7 M

helium envelope onto the central
black hole formed by the collapsed iron core. We consider a spherically symmetric
progenitor model, but with spherical symmetry broken by the introduction of a small,
latitude-dependent angular momentum and a weak radial magnetic field. For more
CP727, Gamma-Ray Bursts: 30 Years of Discovery, edited by E. E. Fenimore and M. Galassi
© 2004 American Institute of Physics 0-7354-0208-6/04/$22.00
384
Downloaded 02 Oct 2007 to 131.215.225.176. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp
details, see [9]).
RESULTS
We find that after a transient episode of infall, lasting 0.13 s, the gas with l 

2RSc
piles up outside the black hole and forms a thick torus bounded by a centrifugal barrier
near the rotation axis. Soon after the torus forms (i.e., within a couple of orbital times
at the inner edge), the magnetic field is amplified by the magnetorotational instability
(MRI, e.g., [10]) and shear. We have verified that most of the inner torus is unstable
to MRI, and that our simulations have enough resolution to resolve, albeit marginally,
the fastest growing MRI mode. The torus starts evolving rapidly and accretes onto the
black hole. Another important effect of magnetic fields is that the torus produces a
magnetized corona and an outflow. The presence of the corona and outflow is essential to
the evolution of the inner flow at all times and the entire flow close to the rotational axis
during the latter phase of the evolution. We find that the outflow very quickly becomes
sufficiently strong to overcome supersonically infalling gas (the radial Mach number in
the polar funnel near the inner radius is  5) and makes its way outward, reaching
the outer boundary at t  025 s. Due to limited computing time, our simulations
were stopped at t  028215 s, which corresponds to 6705 orbits of the flow near the
inner boundary. We expect the accretion to continue much longer, roughly the collapse
timescale of the Helium core ( 10 s), as in [3].
Figure 1 shows the flow pattern of the inner part of the flow at t  02437 s. The left
and middle panels show density and Bφ  maps, respectively. The two maps are overlaid
with the direction of the poloidal velocity. The right panel shows the flow domains of
different polarity of Bφ overlaid with the direction of the poloidal magnetic field. The
polar regions of low density and high Bφ coincide with the region of an outflow. We
note that during the latter phase of the evolution not all of the material in the outflow
originated in the innermost part of the torus – a part of the outflow is “peeled off” the
infalling gas at large radii by the magnetic pressure (see Fig. 2 in [9]). We imposed a
weak poloidal field of a given polarity at the outer boundary. This means that initially Bφ
changes sign only across the equator, which is a relatively unfavorable configuration for
subsequently producing Bφ reversals in the jet. Nevertheless, we find that the polarity of
Bφ changes with time. This is because the flow loses memory of the initial polarity by
the time it reaches the inner MRI-dominated regions where the jet is formed.
Sikora et al. (2003) [11] argue that Poynting flux-dominated jets with reversing
B-fields provide a natural and efficient way to dissipate energy via the reconnection
process. Poynting flux dominated jets have been found in previous numerical simulations
whereas jets with reversing B-fields appear as a relatively new result. Therefore, we have
reviewed results from [8] to check whether polar outflows generated during adiabatic
accretion onto BHs also exhibit reversing Bφ . Simulations reported by [8] are suitable to
study the flow pattern on relatively large length scales because they have been continued
for a few dynamical time scales at a distance of  103 BH radii.
Figure 2 shows the flow domains of different polarity of Bφ overlaid with the direction
of the poloidal velocity and magnetic field (left and right panel, respectively) for run D
385
Downloaded 02 Oct 2007 to 131.215.225.176. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp
FIGURE 1. Maps of logarithmic density, toroidal magnetic field, and toroidal magnetic field domains
with different polarity (white and grey regions) for Proga et al.’s [9] GRB simulation at t  02437 s (left,
middle, and right panel, respectively). The arrows in the left and middle panels indicate the direction of
the poloidal velocity while the arrows in right panel indicate the direction of the poloidal magnetic field.
The length scale is in units of the BH radius (i.e., r   rRS and z  zRS
in [8]. It appears then that reversing B-fields are not unique to our GRB simulations and
that domains with different polarity can be relatively large and long-lived.
CONCLUSIONS
We have performed time-dependent two-dimensional MHD simulations of the collapsar
model. Our simulations show that: 1) soon after the rotationally supported torus forms,
the magnetic field very quickly starts deviating from purely radial due to MRI and shear.
This leads to fast growth of the toroidal magnetic field as field lines wind up due to the
torus rotation; 2) The toroidal field dominates over the poloidal field and the gradient
of the former drives a polar outflow against supersonically accreting gas through the
polar funnel; 3) The polar outflow is Poynting flux-dominated; 4) The polarity of the
toroidal field can change with time; 5) The polar outflow reaches the outer boundary
of the computational domain (5 108 cm) with an expansion velocity of 0.2 c; 6) The
polar outflow is in a form of a relatively narrow jet (when the jet breaks through the
outer boundary its half opening angle is 5Æ); 7) Most of the energy released during
the accretion is in neutrinos, Lν  2 1052 erg s1. Neutrino driving will increase the
outflow energy (e.g., [12] and references therein), but could also increase the mass
loading of the outflow if the energy is deposited in the torus.
386
Downloaded 02 Oct 2007 to 131.215.225.176. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp
FIGURE 2. Maps of toroidal magnetic field domains with different polarity (white and grey regions)
overplotted with the direction of the poloidal velocity (left panel) and the direction of the poloidal
magnetic field (right panel) at the end of Proga & Begelman’s simulation of adiabatic accretion onto a
BH (run D in [8]). The length scale is also in units of the BH radius; the same as in Figure 1. Note
however the difference in the r  and z ranges.
ACKNOWLEDGMENTS
DP acknowledges support from NASA under LTSA grants NAG5-11736 and NAG5-
12867. MCB acknowledges support from NSF grants AST-9876887 and AST-0307502.
REFERENCES
1. Woosley, S. E., ApJ 405, 273 (1993).
2. Paczyn´ski, B., ApJ 494, L45 (1998).
3. MacFadyen, A., Woosley, S. E., ApJ 524, 262 (1999).
4. Popham, R., Woosley, S. E., Fryer, C., ApJ 518, 356 (1999).
5. MacFadyen, A., Woosley, S. E., Heger, A., ApJ 550, 410 (2001).
6. Hjorth, J., et al. , Nature 423, 847 (2003).
7. Stanek, K. Z., et al. , ApJ 591, L17 (2003).
8. Proga, D., Begelman, M. C., ApJ 592, 767 (2003).
9. Proga, D., MacFadyen, A. I., Armitage, P. J., Begelman M. C., ApJ 599, L5 (2003).
10. Balbus, S. A., Hawley, J. F., ApJ 376, 214 (1991).
11. Sikora M., Begelman M. C., Coppi P., Proga D., submitted to ApJ Letters (astro-ph/0309504) (2003).
12. Fryer, C. L., Mészáros, P., ApJ 588, L25 (2003).
387
Downloaded 02 Oct 2007 to 131.215.225.176. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp


MHD Simulations of the Collapsar Model for GRBs

Daniel Proga

Crossref

MHD Simulations of the Collapsar Model for
GRBs
Daniel Proga, Andrew I. MacFadyen†, Philip J. Armitage and Mitchell
C. Begelman
JILA, University of Colorado, Boulder, CO 80309-0440, USA
†California Institute of Technology, Mail Code 130-33, Pasadena, CA 9112
Abstract. We present results from axisymmetric, time-dependent magnetohydrodynamic (MHD)
simulations of the collapsar model for gamma-ray bursts. Our main conclusion is that, within the
collapsar model, MHD effects alone are able to launch, accelerate and sustain a strong polar outflow.
We also find that the outflow is Poynting flux-dominated, and note that this provides favorable initial
conditions for the subsequent production of a baryon-poor fireball.
INTRODUCTION
The collapsar model is one of most promising scenarios to explain the huge release of
energy in a matter of seconds, associated with gamma-ray bursts (GRBs; [1, 2, 3, 4, 5]).
In this model, the collapsed iron core of a massive star accretes gas at a high rate
( 1M s1) producing a large neutrino flux, a powerful outflow, and a GRB. Although
the association of long duration GRBs with stellar collapse is now secure ([6, 7]), basic
properties of the central engine are uncertain. In part, this is because previous numerical
studies of the collapsar model did not explicitly include magnetic fields, although they
are commonly accepted as a key element of accretion flows and outflows.
We present a study of the time evolution of 2.5-dimensional, magnetohydrodynamic
(MHD) flows in the collapsar model. This study is an extension of existing models of
MHD accretion flows onto a black hole (BH; [8]). In particular, we include a realistic
equation of state, photodisintegration of bound nuclei and cooling due to neutrino emis-
sion. Our study is also an extension of collapsar simulations by [3], as we consider very
similar neutrino physics and initial conditions but solve MHD instead of hydrodynami-
cal equations.
MODELS
We begin the simulations after the 17 M iron core of a 25 M presupernova star has
collapsed and study the ensuing accretion of the 7 M helium envelope onto the central
black hole formed by the collapsed iron core. We consider a spherically symmetric
progenitor model, but with spherical symmetry broken by the introduction of a small,
latitude-dependent angular momentum and a weak radial magnetic field. For more
CP727,Gamma-Ray Bursts: 30 Years of Discovery, edited by E. E. Fenimore and M. Galassi
© 2004 American Institute of Physics 0-7354-0208-6/04/$22.00
384
Downloaded 02 Oct 2007 to 131.215.225.176. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp
details, see [9]).
RESULTS
We find that after a transient episode of infall, lasting 0.13 s, the gas withl 

2RSc
piles up outside the black hole and forms a thick torus bounded by a centrifugal barrier
near the rotation axis. Soon after the torus forms (i.e., within a couple of orbital times
at the inner edge), the magnetic field is amplified by the magnetorotational instability
(MRI, e.g., [10]) and shear. We have verified that most of the inner torus is unstable
to MRI, and that our simulations have enough resolution to resolve, albeit marginally,
the fastest growing MRI mode. The torus starts evolving rapidly and accretes onto the
black hole. Another important effect of magnetic fields is that the torus produces a
magnetized corona and an outflow. The presence of the corona and outflow is essential to
the evolution of the inner flow at all times and the entire flow close to the rotational axis
during the latter phase of the evolution. We find that the outflow very quickly becomes
sufficiently strong to overcome supersonically infalling gas (the radial Mach number in
the polar funnel near the inner radius is 5) and makes its way outward, reaching
the outer boundary att  025 s. Due to limited computing time, our simulations
were stopped att  028215 s, which corresponds to 6705 orbits of the flow near the
inner boundary. We expect the accretion to continue much longer, roughly the collapse
timescale of the Helium core ( 10 s), as in [3].
Figure 1 shows the flow pattern of the inner part of the flow att  02437 s. The left
and middle panels show density andBφ maps, respectively. The two maps are overlaid
with the direction of the poloidal velocity. The right panel shows the flow domains of
different polarity ofBφ overlaid with the direction of the poloidal magnetic field. The
polar regions of low density and highBφ coincide with the region of an outflow. We
note that during the latter phase of the evolution not all of the material in the outflow
originated in the innermost part of the torus – a part of the outflow is “peeled off” the
infalling gas at large radii by the magnetic pressure (see Fig. 2 in [9]). We imposed a
weak poloidal field of a given polarity at the outer boundary. This means thatinitially Bφ
changes sign only across the equator, which is a relatively unfavorable configuration for
subsequently producingBφ reversals in the jet. Nevertheless, we find that the polarity of
Bφ changes with time. This is because the flow loses memory of the initial polarity by
the time it reaches the inner MRI-dominated regions where the jet is formed.
Sikora et al. (2003) [11] argue that Poynting flux-dominated jets with reversing
B-fields provide a natural and efficient way to dissipate energy via the reconnection
process. Poynting flux dominated jets have been found in previous numerical simulations
whereas jets with reversing B-fields appear as a relatively new result. Therefore, we have
reviewed results from [8] to check whether polar outflows generated during adiabatic
accretion onto BHs also exhibit reversingBφ. Simulations reported by [8] are suitable to
study the flow pattern on relatively large length scales because they have been continued
for a few dynamical time scales at a distance of 103 BH radii.
Figure 2 shows the flow domains of different polarity ofBφ overlaid with the direction
of the poloidal velocity and magnetic field (left and right panel, respectively) for run D
385
Downloaded 02 Oct 2007 to 131.215.225.176. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp
FIGURE 1. Maps of logarithmic density, toroidal magnetic field, and toroidal magnetic field domains
with different polarity (white and grey regions) for Proga et al.’s [9] GRB simulation att  02437 s (left,
middle, and right panel, respectively). The arrows in the left and middle panels indicate the direction of
the poloidal velocity while the arrows in right panel indicate the direction of the poloidal magnetic field.
The length scale is in units of the BH radius (i.e.,r   rRS andz  zRS
in [8]. It appears then that reversing B-fields are not unique to our GRB simulations and
that domains with different polarity can be relatively large and long-lived.
CONCLUSIONS
We have performed time-dependent two-dimensional MHD simulations of the collapsar
model. Our simulations show that: 1) soon after the rotationally supported torus forms,
the magnetic field very quickly starts deviating from purely radial due to MRI and shear.
This leads to fast growth of the toroidal magnetic field as field lines wind up due to the
torus rotation; 2) The toroidal field dominates over the poloidal field and the gradient
of the former drives a polar outflow against supersonically accreting gas through the
polar funnel; 3) The polar outflow is Poynting flux-dominated; 4) The polarity of the
toroidal field can change with time; 5) The polar outflow reaches the outer boundary
of the computational domain (5108 cm) with an expansion velocity of 0.2 c; 6) The
polar outflow is in a form of a relatively narrow jet (when the jet breaks through the
outer boundary its half opening angle is 5Æ); 7) Most of the energy released during
the accretion is in neutrinos,Lν  21052 erg s1. Neutrino driving will increase the
outflow energy (e.g., [12] and references therein), but could also increase the mass
loading of the outflow if the energy is deposited in the torus.
386
Downloaded 02 Oct 2007 to 131.215.225.176. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp
FIGURE 2. Maps of toroidal magnetic field domains with different polarity (white and grey regions)
overplotted with the direction of the poloidal velocity (left panel) and the direction of the poloidal
magnetic field (right panel) at the end of Proga & Begelman’s simulation of adiabatic accretion onto a
BH (run D in [8]). The length scale is also in units of the BH radius; the same as in Figure 1. Note
however the difference in ther  andz ranges.
ACKNOWLEDGMENTS
DP acknowledges support from NASA under LTSA grants NAG5-11736 and NAG5-
12867. MCB acknowledges support from NSF grants AST-9876887 and AST-0307502.
REFERENCES
1. Woosley, S. E., ApJ405, 273 (1993).
2. Paczýnski, B., ApJ494, L45 (1998).
3. MacFadyen, A., Woosley, S. E., ApJ524, 262 (1999).
4. Popham, R., Woosley, S. E., Fryer, C., ApJ518, 356 (1999).
5. MacFadyen, A., Woosley, S. E., Heger, A., ApJ550, 410 (2001).
6. Hjorth, J.,et al. , Nature423, 847 (2003).
7. Stanek, K. Z.,et al. , ApJ591, L17 (2003).
8. Proga, D., Begelman, M. C., ApJ592, 767 (2003).
9. Proga, D., MacFadyen, A. I., Armitage, P. J., Begelman M. C., ApJ599, L5 (2003).
10. Balbus, S. A., Hawley, J. F., ApJ376, 214 (1991).
11. Sikora M., Begelman M. C., Coppi P., Proga D., submitted to ApJ Letters (astro-ph/0309504) (2003).
12. Fryer, C. L., Mészáros, P., ApJ588, L25 (2003).
387
Downloaded 02 Oct 2007 to 131.215.225.176. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp


ApJ 588, L25

ApJ 599,

submitted to ApJ Letters

MHD simulations of the collapsar model for GRBs

MHD simulations of the collapsar model for GRBs

Abstract

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