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Properties of long gamma-ray
bursts from massive compact
binaries
Ross P. Church1 , Andrew J. Levan2 , Melvyn B. Davies1
rsta.royalsocietypublishing.org
and Chunglee Kim3
1 Lund Observatory, Department of Astronomy and Theoretical
Research
Cite this article: Church RP, Levan AJ, Davies
MB, Kim C. 2013 Properties of long gamma-ray
bursts from massive compact binaries. Phil
Trans R Soc A 371: 20120230.
http://dx.doi.org/10.1098/rsta.2012.0230
One contribution of 15 to a Discussion Meeting
Issue ‘New windows on transients across the
Universe’.
Subject Areas:
astrophysics, stars
Keywords:
gamma-ray bursts, stars: binary, stars: black
holes, supernovae
Author for correspondence:
Ross P. Church
e-mail: [email protected]
Physics, Lund University, Box 43, 221 00 Lund, Sweden
2 Department of Physics, University of Warwick, Coventry,
CV4 7AL, UK
3 Department of Physics, West Virginia University, Morgantown,
WV 26506, USA
We consider the implications of a model for longduration gamma-ray bursts in which the progenitor
is spun up in a close binary by tidal interactions
with a massive black-hole companion. We investigate
a sample of such binaries produced by a binary
population synthesis, and show that the model
predicts several common features in the accretion
on to the newly formed black hole. In all cases,
the accretion rate declines as approximately t−5/3
until a break at a time of order 104 s. The accretion
rate declines steeply thereafter. Subsequently, there is
flaring activity, with the flare peaking between 104
and 105 s, the peak time being correlated with the
flare energy. We show that these times are set by the
semi-major axis of the binary, and hence the process
of tidal spin-up; furthermore, they are consistent with
flares seen in the X-ray light curves of some long
gamma-ray bursts.
1. Introduction
There is strong observational evidence for a correlation
between long-duration gamma-ray bursts and type Ibc
supernovae [1,2]. This fits with a model closely related
to that of Woosley [3], in which the burst forms through
the accretion of roughly a Solar mass of material onto
a newly formed black hole, though an accretion disc
that forms during the core collapse of a massive star.
The outstanding problem with this model is that the
outer parts of the core must retain sufficient angular
2013 The Author(s) Published by the Royal Society. All rights reserved.
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We use the binary population of Church et al. [7], within which we searched for double blackhole binaries that satisfy the criterion of Levan et al. [6]. In summary, this requires that the binary
be tight enough that, assuming tidal locking, the core is spun up sufficiently that its outer parts
form an accretion disc following core collapse. For each of these binaries, we expect a black hole
to form via fall back. After the supernova, a neutron star of mass approximately 2 M forms.
The remainder of the core’s mass is ejected in the explosion. Some of this ejecta is stalled by a
reverse shock and falls back; accretion of this material on to the newly formed neutron star causes
it to collapse into a black hole. To model the fall back, we use the hydrodynamic simulations
of MacFadyen et al. [8]. They show that accretion on to the central object proceeds at a constant
rate for 260 s, then declines as t−5/3 . We launch a set of particles from the exploding star, which
then move in the gravitational potential of the binary. We choose their initial velocities so that,
in the absence of the companion, they would fall back at the rate found by MacFadyen et al. [8].
Some particles are deflected by the companion and miss the newly formed black hole, falling back
instead into a disc.
To follow the resulting accretion disc, we use the model of Perna et al. [9]. We assume that
material falling into the disc self-collides and dissipates its excess angular momentum, hence
circularizing within a few orbital time scales. If it collides with other particles already in the disc
during this process, we merge them and circularize at the radius implied by the new specific
angular momentum. Viscous friction causes material at disc radius r, in a circular orbit of angular
frequency ΩKep , to move inwards at a rate given by ṙ = −rαΩKep h̃2 , where α = 0.1, and we assume
a constant ratio of disc scale height to radius, h̃ ≡ Hdisc /Rdisc = 0.1. When the material reaches the
black hole’s last stable orbit, it is accreted.
To test the sensitivity of our model to potential black-hole natal kicks, we calculated models
both with no kick and with kicks in the ±{x, y} directions in the orbital plane of 100 km s−1 . We
chose this velocity as the median of the distribution of Hansen & Phinney [10], scaled down by
a factor of 3, as expected for a 4.5 M black hole formed by fall back onto a 1.4 M neutron star.
Our results do not clearly distinguish kicked systems from unkicked systems.
3. A typical example
In figure 1, we show the results for a typical system. It has a pre-existing black-hole companion
with a mass of 8.24 M ; the newly formed black hole has a total mass of 4.29 M . The semimajor axis at the point of supernova is 4.01 R . In figure 1a, we plot the position of material at
the point when it falls back into the orbital plane. As can be seen, material returning at early
times is largely unaffected by the companion and falls back straight onto the exploding star.
Material falling back at later times has travelled further from the exploding star, and hence
is more strongly deflected by the companion. Eventually, the material misses the star’s Roche
lobe and is lost from the system. At late times, some material falls back on to the companion
black hole.
These features can also be seen in the accretion history (figure 1b). At early times, the accretion
rate is only slightly lower than the single-star case, owing to the time scale for material to accrete
through the disc. After roughly 104 s, the edge of the accretor’s Roche lobe prevents further
material from being accreted, causing a break. This time scale is set by the semi-major axis of
the binary, and hence ultimately the requirement for it to be close enough to spin the exploding
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2. Methods
2
rsta.royalsocietypublishing.org Phil Trans R Soc A 371: 20120230
momentum to form a disc, whereas the strong stellar winds prevalent in high-mass stars
are expected to brake the core’s rotation and spin it down. In this work, we investigate the
implications of a model where the star is spun back up by a tidal interaction with a massive
companion [4–6].
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(b) 0.01
1051
Ṁ / M (s−1)
y/R
2
1049
10−6
1047
0
−2
−4
−8
10−8
−6
−4
−2
x/R
0
2
4
1045
100
1000
10 000
105
t (s)
Figure 1. A typical spin-up gamma-ray burst model. (a) Positions at which particles enter the discs. The filled circle (coloured
red in the online version) is the initial position of the exploding star; the nearby dots (red in the online version) are particles that
fall back into its Roche lobe. The filled circle (blue in the online version) is the black-hole companion; nearby dots (blue in the
online version) fall back into its Roche lobe. Other, pale dots are material that falls back outside both Roche lobes. Solid lines
show stellar orbits; dotted circles show the initial Roche-lobe radii. (b) Accretion rates on to the two stars. The (red) solid line
shows accretion on to the exploding star, the (blue) dotted line accretion on to the companion. The (grey) dashed line shows
the single-star case. The right-hand axis shows the accretion luminosity, assuming that a constant 10 per cent of the rest mass
of the accreted material is released as energy. (Online version in colour.)
5.5
0.10
4.5
4.0
1052
- (ii)
1051
10−3
-(iii) 1050
10−4
1049
10−5
3.5
3.0
- (i)
0.01
Mflare/M
a/R
5.0
(b)
no kick
−x
+x
−y
+y
3000
10 000
tbreak
30 000
10−6
10 000
Eflare (erg)
(a)
1048
20 000
50 000
105
tflare (s)
Figure 2. Correlations in our results. Black crosses denote models without natal kicks; open squares and circles (green in the
online version) are models with kicks in the −x and +x directions; filled squares and circles (brown in the online version)
have kicks in the −y and +y directions. (a) Binary semi-major axis a as a function of accretion break time, tbreak . (b) Flare
masses, Mflare , and energies, Eflare , as a function of the flare time, tflare . We assume that 10 per cent of accreted mass is converted
into energy. The upper limit symbols (pink in the online version) represent the isotropic energy in late-time flares from three
gamma-ray bursts: (i) GRB 050502B, (ii) GRB 070107 and (iii) GRB 070318. (Online version in colour.)
star up. After the break, material that has fallen back into a disc around the companion black hole,
and material that has remained in the disc around the exploding star, accrete to produce a flare at
around 3 × 104 s.
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10−4
4
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6
3
energy injection rate (erg s−1)
(a) 8
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4. General features
4
105 s
References
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We see a break in the accretion rate between
and
in all binaries in our sample. These
times are consistent with the typical times for the end of the plateau, found by Evans et al. [11] in
canonical X-ray light curves and light curves with a single, steepening break. We predict that later
breaks correlate with less steep pre-break declines. This is because both processes are regulated
by the binary’s semi-major axis at explosion, a. Figure 2a shows that there is a positive correlation
between a and the accretion break time, tbreak ; wider binaries have later flares. This is simply
understood: in a wider binary, it takes longer for the material falling back to have travelled far
enough from the exploding star to be affected by the companion. Hence, the characteristic time
scales of all the features in the accretion history are determined by the separation, which in turn is
constrained by the requirement for it to be close enough to spin the exploding star up. Similarly,
the correlation shown in figure 2b is also caused by the binary size. Later flares—from wider
binaries—contain less mass because the rate at which mass falls back decreases with time. This
relationship appears to fit the three long bursts with late-time X-ray flares in the sample of Curran
1
of the sky.
et al. [12]. We match their isotropic energies if the bursts are beamed to roughly 30
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