Astronomy & Astrophysics manuscript no. neutrino˙cooling January 2, 2019
(DOI: will be inserted by hand later)
Neutrino-cooled accretion and GRB variability
Dimitrios Giannios
Max Planck Institute for Astrophysics, Box 1317, D-85741 Garching, Germany
Received / Accepted
Abstract. For accretion rates Ṁ ∼ 0.1 M⊙/s to a few solar mass black hole the inner part of the disk is expected to make a
transition from advection dominance to neutrino cooling. This transition is characterized by sharp changes of the disk properties.
I argue here that during this transition, a modest increase of the accretion rate leads to powerful enhancement of the Poynting
luminosity of the GRB flow and decrease of its baryon loading. These changes of the characteristics of the GRB flow translate
into changing gamma-ray spectra from the photosphere of the flow. The photospheric interpretation of the GRB emission
explains the observed narrowing of GRB pulses with increasing photon energy and the luminosity-spectral peak relation within
and among bursts.
Key words. Gamma rays: bursts – Accretion, accretion disks
1. Introduction
The commonly assumed model for the central engine of
gamma-ray bursts (hereafter GRBs) consists of a compact ob-
ject, most likely a black hole, surrounded by a massive ac-
cretion disk. This configuration results naturally from the col-
lapse of the core of a fast rotating, massive star (Woosley 1993;
MacFadyen & Woosley 1999) or the coalescence of a neutron
star-neutron star or a neutron star-black hole binary (for simu-
lations see Ruffert et al. 1997).
The accretion rates needed to power a GRB are in the range
Ṁ ∼ 0.01 − 10 M⊙/s. Recently, much theoretical work has
been done to understand the microphysics and the structure of
the disk at this very high accretion-rate regime (e.g., Chen &
Beloborodov 2007; hereafter CB07). These studies have shown
that while for accretion rates Ṁ ≪ 0.1M⊙/s the disk is advec-
tion dominated, when Ṁ ∼ 0.1M⊙/s it makes a sharp transition
to efficient neutrino cooling. This transition results to a thiner,
much denser and neutron rich disk.
Here I show that, for reasonable scalings of the magnetic
field strength with the properties of the inner disk, the advection
dominance-neutrino cooling transition results in large changes
in the Poynting flux output in the GRB flow. During this transi-
tion, a moderate increase of the accretion rate is accompanied
by large increase of the Poynting luminosity and decrease of
the baryon loading of the GRB flow. This leads to powerful
and “clean” ejections of material. The photospheric emission
from these ejections explains the observed narrowing of GRB
pulses with increasing photon energy (Fenimore et al. 1995)
and the luminosity-spectral peak relation within and among
bursts (Liang et al. 2004).
Send offprint requests to: giannios@mpa-garching.mpg.de
2. Disk transition to efficient neutrino cooling
In accretion powered GRB models the outflow responsible for
the GRB is launched in the polar region of the black-hole-disk
system. This can be done by neutrino-antineutrino annihilation
and/or MHD mechanisms of energy extraction. In either case,
the power output in the outflow critically depends on the physi-
cal properties of the inner part of the accretion disk. In this sec-
tion, I focus on the disk properties around the transition from
advection dominance to neutrino cooling. The implications of
this transition on the energy output to the GRB flow are the
topic of the next section.
Recent studies have explored the structure of accretion
disks that surround a black hole of a few solar masses for accre-
tion rates Ṁ ∼ 0.01 − 10 M⊙/s. Most of these studies focus on
1-D “α”-disk models (where α relates the viscous stress to the
pressure in the disk; Shakura & Sunyaev 1973) and put empha-
sis on the treatment of the microphysics of the disks connected
to the neutrino emission and opacity, nuclear composition and
electron degeneracy (Di Matteo et al. 2002; Korhi & Mineshige
2002; Kohri et al. 2005; CB07; Kawanaka & Mineshige 2007;
hereafter KM07) and on general relativistic effects on the hy-
drodynamics (Popham et al. 1999; Pruet at al. 2003; CB07).
These studies have shown that for Ṁ<
0.1 M⊙/s and viscos-
ity parameter α ∼ 0.1 the disk is advection dominated since the
large densities do not allow for photons to escape. The temper-
ature at the inner parts of the disk is T >
1 MeV and the den-
sity ρ ∼ 108 gr/cm3 which results in a disk filled with mildly
degenerate pairs. In this regime of temperatures and densities
the nucleons have dissociated and the disk consists of free pro-
tons and neutrons of roughly equal number. The pressure in the
disk is: P = Pγ,e± + Pb. The first term accounts for the pres-
sure coming from radiation and pairs and the second for that
http://arxiv.org/abs/0704.1659v1
2 Dimitrios Giannios: Neutrino-cooled accretion and GRB variability
of the baryons. In the advection dominated regime the pressure
is dominated by the “light particle” contribution (i.e. the first
term in the last expression).
For accretion rates Ṁ ∼ 0.1 M⊙/s, a rather sharp transition
takes place in the inner parts of the disk. During this transition,
the mean electron energy is high enough for electron capture
by protons to be possible: e− + p → n + ν. As a result, the
disk becomes neutron rich, enters a phase of efficient neutrino
cooling and becomes thinner. The baryon density of the disk in-
creases dramatically and the total pressure is dominated by the
baryon pressure. After the transition is completed the neutron-
to-proton ratio in the disk is ∼ 10. Hereafter, I refer to this
transition as “neutronization” transition.
The neutronization transition takes place at an approxi-
mately constant disk temperature T ≈ several×1010 K and is
completed for a moderate increase of the accretion rate by a
factor of ≈ 2 − 3. During the transition the baryon density in-
creases by ≈ 1.5 orders of magnitude and the disk pressure by
a factor of several (see CB07; KM07).
2.1. Scalings of the disk properties with Ṁ
Although the numbers quoted in the previous section hold quite
generally, the range of accretion rates for which the neutroniza-
tion transition takes place depends on the α viscosity parame-
ter and on the spin of the black hole. For more quantitative
statements to be made, I extract some physical quantities of the
disk before and after transition from Figs. 13-15 of CB07 for
disk viscosity α = 0.1 and spin parameter of the black hole
a = 0.95. I focus at a fixed radius close to the inner edge of
the disk (for convenience, I choose r = 6GM/c2). The quanti-
ties before and after the transition are marked with the super-
scripts “A” and “N” and stand for Advection dominance and
Neutrino cooling respectively. At ṀA = 0.03 M⊙/s, the density
of the disk is ρA ≃ 3 · 109gr/cm3 and has similar number of
protons and neutrons, while at ṀN = 0.07 M⊙/s, the density is
ρN ≃ 9 · 1010gr/cm3 and the neutron-to-proton ratio is ∼ 10.
The temperature remains approximately constant for this range
of accretion rates at T ≃ 5 · 1010 K. A factor of ṀN/ṀA ≃ 2.3
increase in the accretion rate in this specific example leads to
the transition from advection dominance to neutrino cooling.
Around the transition the (mildly degenerate) pairs con-
tribute a factor of ∼ 2 more to the pressure w.r.t. radiation. The
total pressure is: P = Pγ,e± + Pb ≈ arT
4 + ρkBT/mp, where ar
and kB are the radiation and Boltzmann constants respectively
(Beloborodov 2003; CB07). Using the last expression, the disk
pressure before the transition is found: PA ≃ 6 · 1028 erg/cm3;
dominated by the contribution of light particles as expected for
an advection dominated disk. At the higher accretion rate ṀN,
one finds for the pressure of the disk PN ≃ 4 · 1029 erg/cm3.
Now the disk is baryon pressure supported.
From the previous exercise one gets indicative scalings for
the dependence of quantities in the disk as a function of Ṁ
during the neutronization transition: ρ ∝ Ṁ4 and P ∝ Ṁ2.3.
Doubling of the accretion rate during the transition leads to a
factor of ∼ 16 and ∼ 5 increase of the density and pressure of
the disk respectively.
Similar estimates for the dependence of the disk density
and pressure on the accretion rate can be done when the inner
disk is in the advection dominance and neutrino cooling regime
but fairly close to the transition. In these regimes, I estimate
that ρ ∝ P ∝ Ṁ (see, for example Figs. 1-3 in KM07).
Does this sharp change of the disk properties associated
with the neutronization transition affect the rate of energy re-
lease in the polar region of the disk where the GRB flow is
expected to form? The answer depends on the mechanism re-
sponsible for the energy release.
3. Changes in the GRB flow from the
neutronization transition
Gravitational energy released by the accretion of matter to the
black hole can be tapped by neutrino-antineutrino annihilation
or via MHD mechanisms and power the outflow responsible for
the GRB. We consider both of these energy extraction mecha-
nisms in turn.
The neutrino luminosity of the disk just after the neutron-
ization transition is of the order of Lν ∼ 10
52 erg/s and con-
sists of neutrinos and antineutrinos of all flavors. The fraction
of these neutrinos that annihilate and power the GRB flow de-
pends on their spatial emission distribution which, in turn, de-
pends critically on the disk microphysics. For Ṁ ∼ 0.1M⊙/s,
this fraction is of the order of ∼ 10−3 (Liu et al. 2007), pow-
ering an outflow of Lνν̄ ∼ 10
49 erg/s; most likely too weak to
explain a cosmological GRB. The efficiency of the neutrino-
antineutrino annihilation mechanism can be much higher for
accretion rates Ṁ>
1M⊙/s (e.g., Liu et al. 2007; Birkl et al.
2007) which are not considered here.
The second possibility is that energy is extracted by strong
magnetic fields that thread the inner part of the disk (Blandford
& Payne 1982) or the rotating black hole (Blandford &
Znajek 1977) launching a Poynting-flux dominated flow. The
Blandford-Znajek power output can be estimated to be (e.g.
Popham et al. 1999)
LBJ ≈ 10
50a2B215M
3 erg/s, (1)
where B = 1015B15 Gauss and M = 3M3M⊙. taking into
account that magnetic fields of similar strength are expected
to thread the inner parts of the disk, the Poynting luminosity
output from the disk is rather higher than LBJ because of the
larger effective surface of the disk (Livio et al. 1999). In con-
clusion, magnetic field strengths in the inner disk of the order
of B ∼ 1015 erg/s are likely sufficient to power a GRB via MHD
mechanisms of energy extraction.
3.1. Luminosity and baryon loading of the GRB flow as
functions of Ṁ
In this section, I estimate the Poynting luminosity of the GRB
flow for different assumptions on the magnetic field-disk cou-
pling. The mass flux in the GRB flow is harder to constrain
since it depends on the disk structure and the magnetic field
geometry on the disk’s surface. During the neutronization tran-
sition, the disk becomes thinner and, hence, more bound grav-
itationally. One can thus expect that a smaller fraction of Ṁ is
Dimitrios Giannios: Neutrino-cooled accretion and GRB variability 3
injected in the outflow. Here, I make the, rather conservative,
assumption that throughout the transition, the mass flux in the
outflow is a fixed fraction of accretion rate Ṁ.
How is the magnetic field strength related to the properties
of the disk? The magneto-rotational instability (hereafter MRI;
see Balbus & Hawley 1998 for a review) can amplify magnetic
field with energy density up to a fraction ǫ of the pressure in the
disk. This provides an estimate for the magnetic field: B2MRI =
8πǫP. This scaling leads to magnetic field strength of the order
of∼ 1015 Gauss for the fiducial values of the pressure presented
in the previous Sect. and for ǫ ≃ 0.2.
The Poynting luminosity scales as Lp ∝ B
MRI ∝ P ∝ Ṁ
with the accretion rate during the neutronization transition (see
previous Sect.). This leads to a rather large increase of the lu-
minosity of the GRB flow by a factor of ∼ 7 for a moderate
increase of the accretion rate by a factor of ≃ 2.3. Furthermore,
if we assume that a fixed fraction of the accreting gas is chan-
neled to the outflow, then the baryon loading of the Poynting-
flux dominated flow scales as η ∝ Lp/Ṁ ∝ Ṁ
1.3. This means
that during the transition the outflow becomes “cleaner” de-
creasing its baryon loading by a factor of ∼ 3.
The disk can support large-scale fields more powerful that
those generated by MRI. These fields may have been advected
with the matter during the core collapse of the star (or the bi-
nary coalescence) or are captured by the disk in the form a mag-
netic islands and brought in the inner parts of the disk (Spruit &
Uzdensky 2005). These large scale fields can arguably provide
much more promising conditions to launch a large scale jet.
Stehle & Spruit (2001) have shown that a disk threaded by
a large scale field becomes violently unstable once the radial
tension force of the field contributes substantially against grav-
ity. This instability is suppressed if the radial tension force is a
faction δ ∼ a few % of the gravitational attraction. Large-scale
magnetic fields with strength: B2LS = δ8πρcsvk ∝ (ρP)
1/2 can
be supported over the duration of a GRB for δ ∼ a few %. In
the last expression cs =
P/ρ stands for the sound speed and
vk is the Keplerian velocity at the inner boundary.
The last estimate suggests that large scale field strong
enough to power a GRB can be supported by the disk. The
output Poynting luminosity scales, in this case , as Lp ∝ B
(ρP)1/2. During the neutronization transition, the Poynting lu-
minosity increases steeply as a function of the accretion rate:
Lp ∝ (ρP)
∝ Ṁ3.2. This translates to a factor of ∼ 15 in-
crease of the luminosity of the jet for a modest increase by
∼ 2.3 of the accretion rate. Assuming that the rate of ejection
of material in the GRB flow is proportional to the mass accre-
tion rate, the baryon loading of the flow is found to decrease by
a factor of ∼ 6 during the transition (since η ∝ Lp/Ṁ ∝ Ṁ
2.2).
Before and after the transition the disk is advection dom-
inated and neutrino cooled respectively. When the disk is in
either of these regimes the disk density and pressure scale
roughly linearly with the accretion rate (at least for accretion
rates fairly close to the neutronization transition; see previous
Sect.), leading to Lp ∝ Ṁ and η ∼ constant. The Poynting lu-
minosity and the baryon loading of the GRB flow around the
neutronization transition are summarized by Fig. 1.
Although the Poynting flux output depends on assumptions
on the scaling of the magnetic field with the disk properties,
0.1 1
accretion rate (solar masses per second)
Poynting luminosity L
Baryon loading η
I II III
Fig. 1. Poynting luminosity and baryon loading (both in arbi-
trary units) of the GRB flow around the neutronization transi-
tion of the inner disk. In regions marked with I and III the inner
disk is advection and neutrino cooling dominated respectively.
In region II, the neutronization transition takes place. During
the transition, the Poynting luminosity increases steeply with
the accretion rate while the baryon loading of the flow is re-
duced (i.e. η increases).
the neutronization transition generally leads to steep increase
of the Poynting luminosity as function of the accretion rate and
to a “cleaner” (i.e. less baryon loaded) flow. Observational im-
plications of the transition are discussed in the next section.
4. Connection to observations
The mechanism I discuss here operates for accretion rates
around the neutronization transition of the inner disk and pro-
vides the means by which modest variations in the accretion
rate give magnified variability in the Poynting flux output and
baryon loading of the GRB flow. Since the transition takes
place at Ṁ ∼ 0.1 M⊙/s which is close to the accretion rates ex-
pected for the collapsar model (MacFadyen & Woosley 1999),
it is particularly relevant for that model. To connect the flow
variability to the observed properties of the prompt emission,
one has to assume a model for the prompt emission. Here we
discuss internal shock and photospheric models.
Episodes of rapid increase of the luminosity of the flow can
be viewed as the ejection of a distinct shells of material. These
shells can collide with each other further out in the flow lead-
ing to internal shocks that power the prompt GRB emission
(Rees & Mészáros 1994). For the internal shocks to be efficient
in dissipating energy, there must be a substantial variation of
the baryon loading among shells. This may be achieved, in the
context of the model presented here, if the accretion rate, at
which the neutronization transition takes place, changes during
the evolution of the burst. The accretion rate at the transition
decreases, for example, with increasing spin of the black hole
(CB07). Since the black hole is expected to be substantially
span up because of accretion of matter during the evolution of
the burst (e.g. MacFadyen & Woosley 1999), there is the possi-
4 Dimitrios Giannios: Neutrino-cooled accretion and GRB variability
bility, though speculative at this level, that this leads to ejection
of shells with varying baryon loading.
4.1. Photospheric emission
Photospheric models for the origin of the prompt emission have
been recently explored for both fireballs (Mészáros & Rees
2000; Ryde 2004; Rees & Mészáros 2005; Pe’er et al. 2006)
and Poynting-flux dominated flows (Giannios 2006; Giannios
& Spruit 2007; hereafter GS07). Here, I focus mainly to the
photosphere of a Poynting-flux dominated flow since it is di-
rectly applicable to this work.
In the photospheric model, the observed variability of the
prompt emission is direct manifestation of the central engine
activity. Modulations of the luminosity and baryon loading of
the GRB flow result in modulations of the location of the pho-
tosphere of the flow and of the strength and the spectrum of the
photospheric emission (Giannios 2006; GS07). In particular, in
GS07 it is demonstrated that if the increase of the luminosity of
the flow is accompanied by decrease of the baryon loading such
that1 η ∝ L0.6, the photospheric model can explain the observed
narrowing of the width of the GRB pulses with increasing pho-
ton energy reported by Fenimore et al. (1995). The same η-L
scaling also leads to the photospheric luminosity scaling with
the peak of the ν · f (ν) spectrum as Lph ∝ E
p during the burst
evolution in agreement with observations (Liang et al. 2004).
The simple model for the connection of the GRB flow to
the properties of the central engine presented here predicts that
L ∝ Ṁ2.3...3.2 and η ∝ Ṁ1.3...2.2 during the neutronization tran-
sition. The range in the exponents comes from the different as-
sumptions on the disk-magnetic field connection (see Sect. 3).
This translates to η ∝ L0.6...0.7 which is very close that assumed
by GS07 to explain the observed spectral and temporal proper-
ties of the GRB light curves.
Although the launched flow is Poynting-flux dominated, it
is conceivable that it undergoes an initial phase of rapid mag-
netic dissipation resulting to a fireball. The photospheric lu-
minosity and the observed temperature of fireballs scale as
Lph ∝ η
8/3L1/3, Tobs ∝ η
8/3L−5/12 respectively (Mészáros &
Rees 2000). Using the scaling η ∝ L0.6...0.7 found in this work
and identifying the peak of the photospheric component with
the peak of the emission Ep one finds that Lph ∝ L
1.9...2.2 and
Ep ∝ L
1.2...1.4. The last scalings suggest that the photospheric
emission from a fireball can further enhance variations in the
gamma-ray luminosity while Lph and Ep follow the Liang et al.
relation. Still dissipative processes have to be considered in the
fireball so that to explain the observed non-thermal spectra.
5. Conclusions
In this work, a mechanism is proposed by which moderate
changes of the accretion rate at around Ṁ ∼ 0.1 M⊙/s to a
few solar mass black hole can give powerful energy release
episodes to the GRB flow. This mechanism is directly applica-
ble to the collapsar scenario for GRBs (Woosley; MacFadyen
1 In GS07, the parameterization of the baryon loading of the flow is
done by the magnetization σ0 that is related to η through η = σ
& Woosley 1999) and can explain how moderate changes in the
accretion rate result in extremely variable GRB light curves.
This mechanism operates when the inner part of the ac-
cretion disk makes the transition from advection dominance to
neutrino cooling. This, rather sharp, transition is accompanied
by steep increase of the density and the pressure in the disk
(CB07; KM07). This leads to substantial increase of the mag-
netic field strength in the vicinity of the black hole and conse-
quently boosts the Poynting luminosity of the GRB flow by a
factor of ∼ 7 − 15. At the same time, assuming that the ejec-
tion rate of material scales linearly with the accretion rate, the
baryon loading of the flow decreases by a factor ∼ 3 − 6. This
results in a luminosity-baryon loading anticorrelation.
The changes of the characteristics of the GRB flow can be
directly observed as modulations of the photospheric emission
giving birth to pulses with spectral and temporal properties
similar to the observed ones (GS07). The photospheric inter-
pretation of the prompt emission is in agreement with the ob-
served narrowing of the pulses with increasing photon energy
(Fenimore et al. 1995) and the luminosity-peak energy corre-
lation during the evolution of GRBs (Liang et al 2004). The
Amati relation (Amati et al. 2002) is possibly result of the fact
that more luminous bursts are on average less baryon loaded.
Acknowledgements. I wish to thank H. Spruit for illuminating discus-
sions on the disk-magnetic-field coupling.
References
Amati, L., et al. 2002, A&A, 390, 81
Balbus, S. A., & Hawley, J. F. 1998, Rev. Mod. Physics, 70, 1
Beloborodov, A. M. 2003, ApJ, 588, 931
Birkl, R., Aloy, M. A., Janka, H.-T., Müller, E. 2007, A&A, 463, 51
Blandford, R. D., & Payne, D. G. 1982, MNRAS, 199, 883
Blandford, R. D., & Znajek, R. L. 1977, MNRAS, 179, 433
Chen, W.-X., & Beloborodov, A. M. 2007, ApJ, 657, 383 (CB07)
Di Matteo, T., Perna, R., & Narayan, R. 2002, ApJ, 579, 706
Fenimore, E. E., in ’t Zand, J. J. M., Norris, J. P., Bonnell, J. T., &
Nemiroff, R. J. 1995, ApJ, 448, L101
Giannios, D. 2006, A&A, 457, 763
Giannios, D., & Spruit, H. C. 2007, A&A, in press,
arXiv:astro-ph/0611385 (GS07)
Kawanaka, N., & Mineshige, S. 2007, ApJ, in press,
arXiv:astro-ph/0702630 (KM07)
Kohri, K., & Mineshige, S. 2002, ApJ, 577, 311
Kohri, K., Narayan, R., & Piran, T. 2005, ApJ, 629, 341
Liang, E. W., Dai, Z. G., & Wu, X. F. 2004, ApJ, 606, L29
Liu, T., Gu, W.-M., Xue, L., & Lu, J.-F. 2007, ApJ, in press,
arXiv:astro-ph/0702186
Livio, M., Ogilvie, G. I., & Pringle, J. E. 1999, ApJ, 512, 100
Mészáros, P., & Rees, M. J. 2000, ApJ, 530, 292
Pe’er, A., Mészáros, P., & Rees, M. J. 2006, ApJ, 642, 995
Popham, R., Woosley, S. E., & Fryer, C. 1999, ApJ, 518, 356
Pruet, J., Woosley, S. E., & Hoffman, R. D. 2003, ApJ, 586, 1254
Rees, M. J., & Mészáros, P. 1994, ApJ, 430, L93
Rees, M. J., & Mészáros, P. 2005, ApJ, 628, 847
Ruffert, M., Janka, H.-T., Takahashi, K., & Schaefer, G. 1997, A&A,
319, 122
Ryde, F. 2004, ApJ, 614, 827
Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 337
Spruit, H. C., & Uzdensky, D. A. 2005, ApJ, 629, 960
Stehle, R., & Spruit, H. C. 2001, MNRAS, 323, 587
Woosley, S. E. 1993, ApJ, 405, 273
http://arxiv.org/abs/astro-ph/0611385
http://arxiv.org/abs/astro-ph/0702630
http://arxiv.org/abs/astro-ph/0702186
	Introduction
	Disk transition to efficient neutrino cooling
	Scalings of the disk properties with "705FM
	Changes in the GRB flow from the neutronization transition
	Luminosity and baryon loading of the GRB flow as functions of "705FM
	Connection to observations
	Photospheric emission
	Conclusions

  For accretion rates Mdot~0.1 Msun/s to a few solar mass black hole the inner
part of the disk is expected to make a transition from advection dominance to
neutrino cooling. This transition is characterized by sharp changes of the disk
properties. I argue here that during this transition, a modest increase of the
accretion rate leads to powerful enhancement of the Poynting luminosity of the
GRB flow and decrease of its baryon loading. These changes of the
characteristics of the GRB flow translate into changing gamma-ray spectra from
the photosphere of the flow. The photospheric interpretation of the GRB
emission explains the observed narrowing of GRB pulses with increasing photon
energy and the luminosity-spectral peak relation within and among bursts.

Introduction
The commonly assumed model for the central engine of
gamma-ray bursts (hereafter GRBs) consists of a compact ob-
ject, most likely a black hole, surrounded by a massive ac-
cretion disk. This configuration results naturally from the col-
lapse of the core of a fast rotating, massive star (Woosley 1993;
MacFadyen & Woosley 1999) or the coalescence of a neutron
star-neutron star or a neutron star-black hole binary (for simu-
lations see Ruffert et al. 1997).
The accretion rates needed to power a GRB are in the range
Ṁ ∼ 0.01 − 10 M⊙/s. Recently, much theoretical work has
been done to understand the microphysics and the structure of
the disk at this very high accretion-rate regime (e.g., Chen &
Beloborodov 2007; hereafter CB07). These studies have shown
that while for accretion rates Ṁ ≪ 0.1M⊙/s the disk is advec-
tion dominated, when Ṁ ∼ 0.1M⊙/s it makes a sharp transition
to efficient neutrino cooling. This transition results to a thiner,
much denser and neutron rich disk.
Here I show that, for reasonable scalings of the magnetic
field strength with the properties of the inner disk, the advection
dominance-neutrino cooling transition results in large changes
in the Poynting flux output in the GRB flow. During this transi-
tion, a moderate increase of the accretion rate is accompanied
by large increase of the Poynting luminosity and decrease of
the baryon loading of the GRB flow. This leads to powerful
and “clean” ejections of material. The photospheric emission
from these ejections explains the observed narrowing of GRB
pulses with increasing photon energy (Fenimore et al. 1995)
and the luminosity-spectral peak relation within and among
bursts (Liang et al. 2004).
Send offprint requests to: giannios@mpa-garching.mpg.de
2. Disk transition to efficient neutrino cooling
In accretion powered GRB models the outflow responsible for
the GRB is launched in the polar region of the black-hole-disk
system. This can be done by neutrino-antineutrino annihilation
and/or MHD mechanisms of energy extraction. In either case,
the power output in the outflow critically depends on the physi-
cal properties of the inner part of the accretion disk. In this sec-
tion, I focus on the disk properties around the transition from
advection dominance to neutrino cooling. The implications of
this transition on the energy output to the GRB flow are the
topic of the next section.
Recent studies have explored the structure of accretion
disks that surround a black hole of a few solar masses for accre-
tion rates Ṁ ∼ 0.01 − 10 M⊙/s. Most of these studies focus on
1-D “α”-disk models (where α relates the viscous stress to the
pressure in the disk; Shakura & Sunyaev 1973) and put empha-
sis on the treatment of the microphysics of the disks connected
to the neutrino emission and opacity, nuclear composition and
electron degeneracy (Di Matteo et al. 2002; Korhi & Mineshige
2002; Kohri et al. 2005; CB07; Kawanaka & Mineshige 2007;
hereafter KM07) and on general relativistic effects on the hy-
drodynamics (Popham et al. 1999; Pruet at al. 2003; CB07).
These studies have shown that for Ṁ<
0.1 M⊙/s and viscos-
ity parameter α ∼ 0.1 the disk is advection dominated since the
large densities do not allow for photons to escape. The temper-
ature at the inner parts of the disk is T >
1 MeV and the den-
sity ρ ∼ 108 gr/cm3 which results in a disk filled with mildly
degenerate pairs. In this regime of temperatures and densities
the nucleons have dissociated and the disk consists of free pro-
tons and neutrons of roughly equal number. The pressure in the
disk is: P = Pγ,e± + Pb. The first term accounts for the pres-
sure coming from radiation and pairs and the second for that
http://arxiv.org/abs/0704.1659v1
2 Dimitrios Giannios: Neutrino-cooled accretion and GRB variability
of the baryons. In the advection dominated regime the pressure
is dominated by the “light particle” contribution (i.e. the first
term in the last expression).
For accretion rates Ṁ ∼ 0.1 M⊙/s, a rather sharp transition
takes place in the inner parts of the disk. During this transition,
the mean electron energy is high enough for electron capture
by protons to be possible: e− + p → n + ν. As a result, the
disk becomes neutron rich, enters a phase of efficient neutrino
cooling and becomes thinner. The baryon density of the disk in-
creases dramatically and the total pressure is dominated by the
baryon pressure. After the transition is completed the neutron-
to-proton ratio in the disk is ∼ 10. Hereafter, I refer to this
transition as “neutronization” transition.
The neutronization transition takes place at an approxi-
mately constant disk temperature T ≈ several×1010 K and is
completed for a moderate increase of the accretion rate by a
factor of ≈ 2 − 3. During the transition the baryon density in-
creases by ≈ 1.5 orders of magnitude and the disk pressure by
a factor of several (see CB07; KM07).
2.1. Scalings of the disk properties with Ṁ
Although the numbers quoted in the previous section hold quite
generally, the range of accretion rates for which the neutroniza-
tion transition takes place depends on the α viscosity parame-
ter and on the spin of the black hole. For more quantitative
statements to be made, I extract some physical quantities of the
disk before and after transition from Figs. 13-15 of CB07 for
disk viscosity α = 0.1 and spin parameter of the black hole
a = 0.95. I focus at a fixed radius close to the inner edge of
the disk (for convenience, I choose r = 6GM/c2). The quanti-
ties before and after the transition are marked with the super-
scripts “A” and “N” and stand for Advection dominance and
Neutrino cooling respectively. At ṀA = 0.03 M⊙/s, the density
of the disk is ρA ≃ 3 · 109gr/cm3 and has similar number of
protons and neutrons, while at ṀN = 0.07 M⊙/s, the density is
ρN ≃ 9 · 1010gr/cm3 and the neutron-to-proton ratio is ∼ 10.
The temperature remains approximately constant for this range
of accretion rates at T ≃ 5 · 1010 K. A factor of ṀN/ṀA ≃ 2.3
increase in the accretion rate in this specific example leads to
the transition from advection dominance to neutrino cooling.
Around the transition the (mildly degenerate) pairs con-
tribute a factor of ∼ 2 more to the pressure w.r.t. radiation. The
total pressure is: P = Pγ,e± + Pb ≈ arT
4 + ρkBT/mp, where ar
and kB are the radiation and Boltzmann constants respectively
(Beloborodov 2003; CB07). Using the last expression, the disk
pressure before the transition is found: PA ≃ 6 · 1028 erg/cm3;
dominated by the contribution of light particles as expected for
an advection dominated disk. At the higher accretion rate ṀN,
one finds for the pressure of the disk PN ≃ 4 · 1029 erg/cm3.
Now the disk is baryon pressure supported.
From the previous exercise one gets indicative scalings for
the dependence of quantities in the disk as a function of Ṁ
during the neutronization transition: ρ ∝ Ṁ4 and P ∝ Ṁ2.3.
Doubling of the accretion rate during the transition leads to a
factor of ∼ 16 and ∼ 5 increase of the density and pressure of
the disk respectively.
Similar estimates for the dependence of the disk density
and pressure on the accretion rate can be done when the inner
disk is in the advection dominance and neutrino cooling regime
but fairly close to the transition. In these regimes, I estimate
that ρ ∝ P ∝ Ṁ (see, for example Figs. 1-3 in KM07).
Does this sharp change of the disk properties associated
with the neutronization transition affect the rate of energy re-
lease in the polar region of the disk where the GRB flow is
expected to form? The answer depends on the mechanism re-
sponsible for the energy release.
3. Changes in the GRB flow from the
neutronization transition
Gravitational energy released by the accretion of matter to the
black hole can be tapped by neutrino-antineutrino annihilation
or via MHD mechanisms and power the outflow responsible for
the GRB. We consider both of these energy extraction mecha-
nisms in turn.
The neutrino luminosity of the disk just after the neutron-
ization transition is of the order of Lν ∼ 10
52 erg/s and con-
sists of neutrinos and antineutrinos of all flavors. The fraction
of these neutrinos that annihilate and power the GRB flow de-
pends on their spatial emission distribution which, in turn, de-
pends critically on the disk microphysics. For Ṁ ∼ 0.1M⊙/s,
this fraction is of the order of ∼ 10−3 (Liu et al. 2007), pow-
ering an outflow of Lνν̄ ∼ 10
49 erg/s; most likely too weak to
explain a cosmological GRB. The efficiency of the neutrino-
antineutrino annihilation mechanism can be much higher for
accretion rates Ṁ>
1M⊙/s (e.g., Liu et al. 2007; Birkl et al.
2007) which are not considered here.
The second possibility is that energy is extracted by strong
magnetic fields that thread the inner part of the disk (Blandford
& Payne 1982) or the rotating black hole (Blandford &
Znajek 1977) launching a Poynting-flux dominated flow. The
Blandford-Znajek power output can be estimated to be (e.g.
Popham et al. 1999)
LBJ ≈ 10
50a2B215M
3 erg/s, (1)
where B = 1015B15 Gauss and M = 3M3M⊙. taking into
account that magnetic fields of similar strength are expected
to thread the inner parts of the disk, the Poynting luminosity
output from the disk is rather higher than LBJ because of the
larger effective surface of the disk (Livio et al. 1999). In con-
clusion, magnetic field strengths in the inner disk of the order
of B ∼ 1015 erg/s are likely sufficient to power a GRB via MHD
mechanisms of energy extraction.
3.1. Luminosity and baryon loading of the GRB flow as
functions of Ṁ
In this section, I estimate the Poynting luminosity of the GRB
flow for different assumptions on the magnetic field-disk cou-
pling. The mass flux in the GRB flow is harder to constrain
since it depends on the disk structure and the magnetic field
geometry on the disk’s surface. During the neutronization tran-
sition, the disk becomes thinner and, hence, more bound grav-
itationally. One can thus expect that a smaller fraction of Ṁ is
Dimitrios Giannios: Neutrino-cooled accretion and GRB variability 3
injected in the outflow. Here, I make the, rather conservative,
assumption that throughout the transition, the mass flux in the
outflow is a fixed fraction of accretion rate Ṁ.
How is the magnetic field strength related to the properties
of the disk? The magneto-rotational instability (hereafter MRI;
see Balbus & Hawley 1998 for a review) can amplify magnetic
field with energy density up to a fraction ǫ of the pressure in the
disk. This provides an estimate for the magnetic field: B2MRI =
8πǫP. This scaling leads to magnetic field strength of the order
of∼ 1015 Gauss for the fiducial values of the pressure presented
in the previous Sect. and for ǫ ≃ 0.2.
The Poynting luminosity scales as Lp ∝ B
MRI ∝ P ∝ Ṁ
with the accretion rate during the neutronization transition (see
previous Sect.). This leads to a rather large increase of the lu-
minosity of the GRB flow by a factor of ∼ 7 for a moderate
increase of the accretion rate by a factor of ≃ 2.3. Furthermore,
if we assume that a fixed fraction of the accreting gas is chan-
neled to the outflow, then the baryon loading of the Poynting-
flux dominated flow scales as η ∝ Lp/Ṁ ∝ Ṁ
1.3. This means
that during the transition the outflow becomes “cleaner” de-
creasing its baryon loading by a factor of ∼ 3.
The disk can support large-scale fields more powerful that
those generated by MRI. These fields may have been advected
with the matter during the core collapse of the star (or the bi-
nary coalescence) or are captured by the disk in the form a mag-
netic islands and brought in the inner parts of the disk (Spruit &
Uzdensky 2005). These large scale fields can arguably provide
much more promising conditions to launch a large scale jet.
Stehle & Spruit (2001) have shown that a disk threaded by
a large scale field becomes violently unstable once the radial
tension force of the field contributes substantially against grav-
ity. This instability is suppressed if the radial tension force is a
faction δ ∼ a few % of the gravitational attraction. Large-scale
magnetic fields with strength: B2LS = δ8πρcsvk ∝ (ρP)
1/2 can
be supported over the duration of a GRB for δ ∼ a few %. In
the last expression cs =
P/ρ stands for the sound speed and
vk is the Keplerian velocity at the inner boundary.
The last estimate suggests that large scale field strong
enough to power a GRB can be supported by the disk. The
output Poynting luminosity scales, in this case , as Lp ∝ B
(ρP)1/2. During the neutronization transition, the Poynting lu-
minosity increases steeply as a function of the accretion rate:
Lp ∝ (ρP)
∝ Ṁ3.2. This translates to a factor of ∼ 15 in-
crease of the luminosity of the jet for a modest increase by
∼ 2.3 of the accretion rate. Assuming that the rate of ejection
of material in the GRB flow is proportional to the mass accre-
tion rate, the baryon loading of the flow is found to decrease by
a factor of ∼ 6 during the transition (since η ∝ Lp/Ṁ ∝ Ṁ
2.2).
Before and after the transition the disk is advection dom-
inated and neutrino cooled respectively. When the disk is in
either of these regimes the disk density and pressure scale
roughly linearly with the accretion rate (at least for accretion
rates fairly close to the neutronization transition; see previous
Sect.), leading to Lp ∝ Ṁ and η ∼ constant. The Poynting lu-
minosity and the baryon loading of the GRB flow around the
neutronization transition are summarized by Fig. 1.
Although the Poynting flux output depends on assumptions
on the scaling of the magnetic field with the disk properties,
0.1 1
accretion rate (solar masses per second)
Poynting luminosity L
Baryon loading η
I II III
Fig. 1. Poynting luminosity and baryon loading (both in arbi-
trary units) of the GRB flow around the neutronization transi-
tion of the inner disk. In regions marked with I and III the inner
disk is advection and neutrino cooling dominated respectively.
In region II, the neutronization transition takes place. During
the transition, the Poynting luminosity increases steeply with
the accretion rate while the baryon loading of the flow is re-
duced (i.e. η increases).
the neutronization transition generally leads to steep increase
of the Poynting luminosity as function of the accretion rate and
to a “cleaner” (i.e. less baryon loaded) flow. Observational im-
plications of the transition are discussed in the next section.
4. Connection to observations
The mechanism I discuss here operates for accretion rates
around the neutronization transition of the inner disk and pro-
vides the means by which modest variations in the accretion
rate give magnified variability in the Poynting flux output and
baryon loading of the GRB flow. Since the transition takes
place at Ṁ ∼ 0.1 M⊙/s which is close to the accretion rates ex-
pected for the collapsar model (MacFadyen & Woosley 1999),
it is particularly relevant for that model. To connect the flow
variability to the observed properties of the prompt emission,
one has to assume a model for the prompt emission. Here we
discuss internal shock and photospheric models.
Episodes of rapid increase of the luminosity of the flow can
be viewed as the ejection of a distinct shells of material. These
shells can collide with each other further out in the flow lead-
ing to internal shocks that power the prompt GRB emission
(Rees & Mészáros 1994). For the internal shocks to be efficient
in dissipating energy, there must be a substantial variation of
the baryon loading among shells. This may be achieved, in the
context of the model presented here, if the accretion rate, at
which the neutronization transition takes place, changes during
the evolution of the burst. The accretion rate at the transition
decreases, for example, with increasing spin of the black hole
(CB07). Since the black hole is expected to be substantially
span up because of accretion of matter during the evolution of
the burst (e.g. MacFadyen & Woosley 1999), there is the possi-
4 Dimitrios Giannios: Neutrino-cooled accretion and GRB variability
bility, though speculative at this level, that this leads to ejection
of shells with varying baryon loading.
4.1. Photospheric emission
Photospheric models for the origin of the prompt emission have
been recently explored for both fireballs (Mészáros & Rees
2000; Ryde 2004; Rees & Mészáros 2005; Pe’er et al. 2006)
and Poynting-flux dominated flows (Giannios 2006; Giannios
& Spruit 2007; hereafter GS07). Here, I focus mainly to the
photosphere of a Poynting-flux dominated flow since it is di-
rectly applicable to this work.
In the photospheric model, the observed variability of the
prompt emission is direct manifestation of the central engine
activity. Modulations of the luminosity and baryon loading of
the GRB flow result in modulations of the location of the pho-
tosphere of the flow and of the strength and the spectrum of the
photospheric emission (Giannios 2006; GS07). In particular, in
GS07 it is demonstrated that if the increase of the luminosity of
the flow is accompanied by decrease of the baryon loading such
that1 η ∝ L0.6, the photospheric model can explain the observed
narrowing of the width of the GRB pulses with increasing pho-
ton energy reported by Fenimore et al. (1995). The same η-L
scaling also leads to the photospheric luminosity scaling with
the peak of the ν · f (ν) spectrum as Lph ∝ E
p during the burst
evolution in agreement with observations (Liang et al. 2004).
The simple model for the connection of the GRB flow to
the properties of the central engine presented here predicts that
L ∝ Ṁ2.3...3.2 and η ∝ Ṁ1.3...2.2 during the neutronization tran-
sition. The range in the exponents comes from the different as-
sumptions on the disk-magnetic field connection (see Sect. 3).
This translates to η ∝ L0.6...0.7 which is very close that assumed
by GS07 to explain the observed spectral and temporal proper-
ties of the GRB light curves.
Although the launched flow is Poynting-flux dominated, it
is conceivable that it undergoes an initial phase of rapid mag-
netic dissipation resulting to a fireball. The photospheric lu-
minosity and the observed temperature of fireballs scale as
Lph ∝ η
8/3L1/3, Tobs ∝ η
8/3L−5/12 respectively (Mészáros &
Rees 2000). Using the scaling η ∝ L0.6...0.7 found in this work
and identifying the peak of the photospheric component with
the peak of the emission Ep one finds that Lph ∝ L
1.9...2.2 and
Ep ∝ L
1.2...1.4. The last scalings suggest that the photospheric
emission from a fireball can further enhance variations in the
gamma-ray luminosity while Lph and Ep follow the Liang et al.
relation. Still dissipative processes have to be considered in the
fireball so that to explain the observed non-thermal spectra.
5. Conclusions
In this work, a mechanism is proposed by which moderate
changes of the accretion rate at around Ṁ ∼ 0.1 M⊙/s to a
few solar mass black hole can give powerful energy release
episodes to the GRB flow. This mechanism is directly applica-
ble to the collapsar scenario for GRBs (Woosley; MacFadyen
1 In GS07, the parameterization of the baryon loading of the flow is
done by the magnetization σ0 that is related to η through η = σ
& Woosley 1999) and can explain how moderate changes in the
accretion rate result in extremely variable GRB light curves.
This mechanism operates when the inner part of the ac-
cretion disk makes the transition from advection dominance to
neutrino cooling. This, rather sharp, transition is accompanied
by steep increase of the density and the pressure in the disk
(CB07; KM07). This leads to substantial increase of the mag-
netic field strength in the vicinity of the black hole and conse-
quently boosts the Poynting luminosity of the GRB flow by a
factor of ∼ 7 − 15. At the same time, assuming that the ejec-
tion rate of material scales linearly with the accretion rate, the
baryon loading of the flow decreases by a factor ∼ 3 − 6. This
results in a luminosity-baryon loading anticorrelation.
The changes of the characteristics of the GRB flow can be
directly observed as modulations of the photospheric emission
giving birth to pulses with spectral and temporal properties
similar to the observed ones (GS07). The photospheric inter-
pretation of the prompt emission is in agreement with the ob-
served narrowing of the pulses with increasing photon energy
(Fenimore et al. 1995) and the luminosity-peak energy corre-
lation during the evolution of GRBs (Liang et al 2004). The
Amati relation (Amati et al. 2002) is possibly result of the fact
that more luminous bursts are on average less baryon loaded.
Acknowledgements. I wish to thank H. Spruit for illuminating discus-
sions on the disk-magnetic-field coupling.
References
Amati, L., et al. 2002, A&A, 390, 81
Balbus, S. A., & Hawley, J. F. 1998, Rev. Mod. Physics, 70, 1
Beloborodov, A. M. 2003, ApJ, 588, 931
Birkl, R., Aloy, M. A., Janka, H.-T., Müller, E. 2007, A&A, 463, 51
Blandford, R. D., & Payne, D. G. 1982, MNRAS, 199, 883
Blandford, R. D., & Znajek, R. L. 1977, MNRAS, 179, 433
Chen, W.-X., & Beloborodov, A. M. 2007, ApJ, 657, 383 (CB07)
Di Matteo, T., Perna, R., & Narayan, R. 2002, ApJ, 579, 706
Fenimore, E. E., in ’t Zand, J. J. M., Norris, J. P., Bonnell, J. T., &
Nemiroff, R. J. 1995, ApJ, 448, L101
Giannios, D. 2006, A&A, 457, 763
Giannios, D., & Spruit, H. C. 2007, A&A, in press,
arXiv:astro-ph/0611385 (GS07)
Kawanaka, N., & Mineshige, S. 2007, ApJ, in press,
arXiv:astro-ph/0702630 (KM07)
Kohri, K., & Mineshige, S. 2002, ApJ, 577, 311
Kohri, K., Narayan, R., & Piran, T. 2005, ApJ, 629, 341
Liang, E. W., Dai, Z. G., & Wu, X. F. 2004, ApJ, 606, L29
Liu, T., Gu, W.-M., Xue, L., & Lu, J.-F. 2007, ApJ, in press,
arXiv:astro-ph/0702186
Livio, M., Ogilvie, G. I., & Pringle, J. E. 1999, ApJ, 512, 100
Mészáros, P., & Rees, M. J. 2000, ApJ, 530, 292
Pe’er, A., Mészáros, P., & Rees, M. J. 2006, ApJ, 642, 995
Popham, R., Woosley, S. E., & Fryer, C. 1999, ApJ, 518, 356
Pruet, J., Woosley, S. E., & Hoffman, R. D. 2003, ApJ, 586, 1254
Rees, M. J., & Mészáros, P. 1994, ApJ, 430, L93
Rees, M. J., & Mészáros, P. 2005, ApJ, 628, 847
Ruffert, M., Janka, H.-T., Takahashi, K., & Schaefer, G. 1997, A&A,
319, 122
Ryde, F. 2004, ApJ, 614, 827
Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 337
Spruit, H. C., & Uzdensky, D. A. 2005, ApJ, 629, 960
Stehle, R., & Spruit, H. C. 2001, MNRAS, 323, 587
Woosley, S. E. 1993, ApJ, 405, 273
http://arxiv.org/abs/astro-ph/0611385
http://arxiv.org/abs/astro-ph/0702630
http://arxiv.org/abs/astro-ph/0702186
	Introduction
	Disk transition to efficient neutrino cooling
	Scalings of the disk properties with "705FM
	Changes in the GRB flow from the neutronization transition
	Luminosity and baryon loading of the GRB flow as functions of "705FM
	Connection to observations
	Photospheric emission
	Conclusions
