Testing black hole neutrino-dominated accretion discs for long-duration gamma-ray bursts

Abstract

Long-duration gamma-ray bursts (LGRBs) are generally considered to originate from the massive collapsars. It is believed that the central engine of gamma-ray bursts (GRBs) is a neutrino-dominated accretion flow (NDAF) around a rotating stellar-mass black hole (BH). The neutrino annihilation above the NDAF is a feasible mechanism to power GRB. In this work, we analyse the distributions of the isotropic gamma-ray-radiated energy and jet kinetic energy of 48 LGRBs. According to the NDAF and fireball models, we estimate the mean accreted masses of LGRBs in our sample to investigate whether the NDAFs can power LGRBs with the reasonable BH parameters and conversion efficiency of neutrino annihilation. The results indicate that most of the values of the accreted masses are less than 5 M_⊙ for the extreme Kerr BHs and high conversion efficiency. It suggests that the NDAFs may be suitable for most of LGRBs except for some extremely high energy sources.

accretion, accretion discs, black hole physics, neutrinos, gamma-ray burst: general

INTRODUCTION

Gamma-ray bursts (GRBs) are extremely energetic transient events and isotropically distribute over the sky. Depending on a separation at 2 s of their duration T₉₀, they are generally divided into two classes, i.e. long- and short-duration GRBs (LGRBs and SGRBs; see Kouveliotou et al. 1993). The percentage rates for LGRBs and SGRBs in the BATSE sample are about 75 per cent and 25 per cent (Sakamoto et al. 2008, 2011), respectively, and the LGRB fraction is further larger for other space detectors, such as Swift and Fermi (Zhang et al. 2012). The progenitors of LGRBs are usually considered as the collapse of massive stars with rapid spin, low metallicity, and stripped of their helium and hydrogen envelope (e.g. Woosley 1993; Kumar & Zhang 2015). In this circumstance, the stellar-mass Kerr black hole (BH) surrounded by a debris torus or disc forms quickly in the central parts of the stellar core. Moreover, SGRBs are related to the mergers of two neutron stars (NSs) or NS-BH binaries (e.g. Eichler et al. 1989; Narayan, Paczyński & Piran 1992; Nakar 2007). A similar accretion system may form in SGRBs due to the loss of the angular momentum and mass of the remnants.

Neutrino annihilation and Blandford–Znajek (BZ; see Blandford & Znajek 1977) mechanisms are proposed to power GRBs if there is a hyperaccretion disc in the centre of GRBs. A typical cosmological GRB requires the value of the accretion rate ranging from a fraction of solar mass per second to several solar masses per second. If the neutrino annihilation is the dominant cooling mechanism, a geometrically and optically thick disc is formed and a large number of free baryons appear in the inner region of the disc. The temperature and density are so high that photons are trapped and neutrino cooling becomes effective. This disc is named the neutrino-dominated accretion flow, which has been widely studied (NDAF, see, e.g. Popham, Woosley & Fryer 1999; Narayan, Piran & Kumar 2001; Di Matteo, Perna & Narayan 2002; Kohri & Mineshige 2002; Kohri, Narayan & Piran 2005; Gu, Liu & Lu 2006; Chen & Beloborodov 2007; Kawanaka & Mineshige 2007; Liu et al. 2007, 2008, 2012a,b, 2013, 2015d; Janiuk, Mioduszewski & Moscibrodzka 2013; Kawanaka, Piran & Krolik 2013; Li & Liu 2013; Xue et al. 2013; Hou et al. 2014a,b).

Another generally accepted model to power GRBs is the magnetar model (e.g. Usov 1992; Thompson, Chang & Quataert 2004; Metzger, Quataert & Thompson 2008; Mao et al. 2010; Metzger et al. 2011; Lü & Zhang 2014). The released energy via its spinning down can effortlessly provide the requirements of GRBs (e.g. Usov 1992; Wheeler et al. 2000), and this model may offer a plausible explanation for the shallow decay phases in some GRB afterglow light curves (e.g. Dai 2004; Zhang et al. 2006). No matter what kind of models, the puzzled central engines of GRBs are hidden by the electromagnetic radiation (e.g. Liu et al. 2015d), and the BH-NDAF systems may be probed directly by gravitational waves (e.g. Suwa & Murase 2009; Romero, Reynoso & Christiansen 2010; Sun et al. 2012) and MeV neutrino emission (e.g. Liu et al. 2015e).

Fan & Wei (2011) and Liu et al. (2015c) investigated the capabilities of the NDAFs to power SGRBs. As a result, the disc mass of a certain SGRB mainly depends on the output energy of the NDAF, jet opening angle, and characteristics of central BH. Almost all of SGRBs are satisfied with the reasonable disc masses, only a few approach or exceed the limits in simulations, ∼0.5 M_⊙ (e.g. Kluźniak & Lee 1998; Lee & Kluźniak 1999; Popham et al. 1999; Liu et al. 2012b), even with the extreme BH parameters. Compared with SGRBs, LGRBs need more massive accreted mass and more explosive energy, so we want to figure out whether the NDAFs could power LGRBs.

This paper is organized as follows. In Section 2, we introduce the NDAF and fireball models to connect the observational values and model parameters. Based on the collected LGRB data, we analyse the distributions of the isotropic radiation energy and jet kinetic energy and show the distribution of the accreted masses with different parameters of the BH-NDAF system in Section 3. The last section is devoted to the conclusions and discussion.

MODEL

The NDAF and fireball models for GRBs have been widely discussed for several decades. Fan & Wei (2011) proposed a method to estimate the mass accreted by the central BH of progenitors of SGRBs, where it is assumed that the outflow is driven from the neutrino-antineutrino annihilation steaming from an underlying NDAF model. Liu et al. (2015c) developed this method further and enlarged the number of SGRB events to which the method is applied. Here, we reuse the same methodology as in the former papers but applied to LGRBs. Generally, the mean output power from the central engine |$\dot{E}$| can be expressed as

\begin{eqnarray} \dot{E} \approx \frac{(1+z)(E_{\rm \gamma ,iso}+E_{\rm k,iso})\theta _{\rm j}^{2}}{2 T_{90}}, \end{eqnarray}

(1)

where z is the redshift, T₉₀ can roughly be considered as the duration of the activity of the central engine, E_k,iso is the isotropic kinetic energy estimated by the X-ray luminosity during the afterglow phase with the standard afterglow model, E_γ,iso is the isotropic equivalent energy radiated in gamma-ray band, and θ_j is the opening angle of the ejecta.

Meanwhile, |$\dot{E}$| is a fraction of the total neutrino annihilation luminosity |$L_{\nu \bar{\nu }}$| averaged in the whole accretion process, i.e.

\begin{eqnarray} \dot{E}=\eta L_{\nu \bar{\nu }}, \end{eqnarray}

(2)

where η is a dimensionless number that represents the conversion efficiency of the neutrino annihilation. (e.g. Eichler et al. 1989; Aloy, Janka & Müller 2005; Liu et al. 2012b, 2015c). In this paper, we take η = 0.2 and 0.5. The approximate analytical formula of neutrino annihilation luminosity we adopt here is proposed by Zalamea & Beloborodov (2011),

\begin{eqnarray} L_{\nu \bar{\nu }}\ ({\rm erg\ s^{-1}})&\approx & 5.7 \times 10^{52} x_{\rm ms}^{-4.8}({M_{_{\rm BH}}/\mathrm{M}_{\odot }})^{-3/2} \nonumber \\ &&\times \Bigg \lbrace \begin{array}{ll}0 & \hbox{$\dot{M} <\dot{M}_{\rm ign}$}\\ (\dot{M}/\mathrm{M}_{\odot }\ \rm s^{-1}) \rm ^{9/4} & \hbox{$\dot{M}_{\rm ign} < \dot{M} < \dot{M}_{\rm trap}$}\\ (\dot{M}_{\rm trap}/\mathrm{M}_{\odot }\ \rm s^{-1}) \rm ^{9/4} & \hbox{$\dot{M} > \dot{M}_{\rm trap}$}\\ \end{array}, \end{eqnarray}

(3)

where |$\dot{M}$| is the mass accretion rate, x_ms = r_ms/r_g, |$r_{\rm ms}=\frac{1}{2}r_{\rm g} [3+Z_{2}-\sqrt{(3-Z_{1})(3+Z_{1}+2Z_{2})}]$| is radius of the marginally stable orbit, |$Z_{1}=1+(1-a_{\ast }^{2})^{1/3}[(1+a_{\ast })^{1/3}+(1-a_{\ast })^{1/3}]$| and |$Z_{2}=\sqrt{3a_{\ast }^{2}+Z_{1}^{2}}$| for 0 < a_* < 1. r_g = 2GM_BH/c² is the Schwarzschild radius, and |$\dot{M}_{\rm ign}$| and |$\dot{M}_{\rm trap}$| are the critical ignition accretion rate and the accretion rate when neutrino trapping event occurs in the inner region (e.g. Chen & Beloborodov 2007; Zalamea & Beloborodov 2011).

Hence, the mean accretion rate for the case can be defined as (e.g. Fan & Wei 2011; Liu et al. 2015c)

\begin{eqnarray} \dot{M} &\approx & 0.12\ \left[\frac{(1+z)(E_{\rm \gamma ,iso,51}+E_{\rm k,iso,51})\theta _{\rm j}^{2}}{\eta T_{90,\rm s}}\right]^{4/9} \nonumber \\ &&\times \, x_{\rm ms}^{2.1}\left(\frac{M_{\rm BH}}{\mathrm{M}_{\odot }}\right)^{2/3}\ \mathrm{M}_{\odot }\ \rm s^{-1}, \end{eqnarray}

(4)

where E_{k, iso, 51} = E_{k, iso}/(10⁵¹ erg) and E_{γ, iso, 51} = E_{γ, iso}/(10⁵¹ erg). Of course, we must ensure that the NDAF can be ignited, i.e. |$\dot{M}>\dot{M}_{\rm ign}$|⁠. Then we can estimate the accreted mass |${\sim } \dot{M} T_{90}/(1+z)$|⁠, i.e. (e.g. Liu et al. 2015c)

\begin{eqnarray} M_{\rm acc} &\approx & 0.12\left[\frac{(E_{\rm \gamma ,iso,51}+E_{\rm k,iso,51})\theta _{\rm j}^{2}}{\eta }\right]^{4/9} \left(\frac{T_{90,\rm s}}{{1+z}}\right)^{5/9}\nonumber \\ &&\times \ \ x_{\rm ms}^{2.1}\ \left(\frac{M_{\rm BH}}{\mathrm{M}_{\odot }}\right)^{2/3}\ {M_{{\odot }}}. \end{eqnarray}

(5)

Although the accreted masses can be estimated with the above equations and the observational data, it is still worth noting that there are some uncertainties. First, The hyperaccretion may result in the violent evolution of the BH's characteristics, which further leads to the evolution of the neutrino annihilation luminosity during the long duration. Fortunately, the extreme Kerr BHs and low mean accretion rates are considered in the central engines of LGRBs (e.g. Popham et al. 1999; Woosley & Heger 2006). Recently, we found that this evolution has little effect on the total energy of the neutrino annihilation for LGRBs (Song et al. 2015). Secondly, we replace the interval of the central engine with T₉₀ in this work. In fact, the duration of a GRB should be longer than the time interval of the activity of central engine if the radial expansion of the fireball is considered, and this effect is potentially more important in SGRBs than in LGRBs (Aloy et al. 2005; Janka et al. 2006). Thirdly, we do not consider the outflow from the disc, which may seriously influences the measurement of the disc mass (Liu et al. 2012b; Janiuk et al. 2013). Therefore, the accreted mass we calculated can be considered as a lower limit of mass of the original accretion matter.

RESULTS

Since no LGRB has a positively identified origin of accretion or magnetar models, in order to test the ability of the NDAF model, all LGRBs with the data of T₉₀, z, E_γ,iso, E_k,iso, and θ_j should be collected. There are some selection criterions should be stressed. First, in the collapse events, we can consider that all Type Ib/c supernovae (SNe) have an engine-driven BH/NS-NDAF (e.g. Liu et al. 2015e). If the ejections from NDAFs direct to the observers, various kinds of GRB light curves should be detected. So the LGRBs associated with SNe should be included, which are marked in Table 1. Secondly, the ultralong GRBs (ULGRBs) may derive from the blue supergiants (e.g. Nakauchi et al. 2013), which is different from the origin of ‘normal’ LGRBs. For the disc model, they cannot be explained by neutrino annihilation mechanism but BZ mechanism (e.g. Nathanail & Contopoulos 2015). So these sources are not included. Thirdly, GRB 060614 is a typical short-long GRBs, which may originate from an event of compact objects merger (e.g. Gehrels et al. 2006; Yang et al. 2015). It should not be considered.

Table 1.

Collected data of LGRBs. Coloumn (1): LGRB name, Column (2): duration, Column (3): redshift, Column (4): the isotropic radiated energy, Column (5): the isotropic kinetic energy, Column (6): the jet opening angle θ_j, Column (7): observatory, Column (8): references for E_γ,iso, E_k,iso, and θ_j.

GRB	T90	z	Eγ, iso	Ek,iso	θj	Observatory	Ref.
	(s)		(1051erg)	(1052erg)	(deg)
970508	14.0 ± 3.6	0.8349	5.5 ± 0.6	0.99 ± 0.14	21.63 ± 1.67	BATSE	1, 2
971214	31.23 ± 1.18	3.418	210.5 ± 25.8	8.48 ± 0.97	>5.54 ± 0.23	BATSE	1, 2
980613	42.0 ± 22.1	1.0964	5.4 ± 1	1.22 ± 0.38	>12.57 ± 0.58	BeppoSAX	1, 2
980703	76.0 ± 10.2	0.966	60.1 ± 6.6	2.41 ± 0.63	11.21 ± 0.81	BATSE	1, 2
990123	63.30 ± 0.26	1.600	1437.9 ± 177.8	20.28 ± 1.85	4.93 ± 0.43	BATSE	1, 2
990510	67.58 ± 1.86	1.619	176.3 ± 20.0	13.16 ± 1.12	3.36 ± 0.21	BATSE	1, 2
990705	32.0 ± 1.4	0.84	256.0 ± 20.3	0.34 ± 0.12	5.33 ± 0.41	BATSE	1, 2
991216	15.17 ± 0.09	1.02	535.4 ± 59.4	36.64 ± 1.79	4.57 ± 0.72	BATSE	1, 2
000210	9.0 ± 1.4	0.846	169.3 ± 14.1	0.50 ± 0.12	>6.84 ± 0.28	BeppoSAX	1, 2
000926	1.30 ± 0.59	2.0387	279.7 ± 99	9.97 ± 3.75	6.16 ± 0.31	BeppoSAX	1, 2
010222	74.0 ± 4.1	1.4769	857.8 ± 21.7	22.79 ± 2.48	3.20 ± 0.13	BeppoSAX	1, 2
011211	51.0 ± 7.6	2.1418	67.2 ± 8.6	71.32 ± 0.22	6.38 ± 0.40	BeppoSAX	1, 2
020405	40.0 ± 2.2	0.6899	72.0 ± 9.2	4.6 ± 1.29	7.81 ± 0.93	BeppoSAX	1, 2
021004	77.1 ± 2.6	2.3304	55.6 ± 7.2	8.35 ± 1.45	12.67 ± 4.51	HETE	1, 2
030329a	33.1 ± 0.5	0.1685	15.1 ± 0.1	|$12.36_{-0.06}^{+0.05}$|	3.78 ± 0.05	HETE	3, 4, 5
050315	96 ± 10	1.95	49 ± 15	|$512.40_{-65.58}^{+45.30}$|	|$4.35_{-0.52}^{+0.46}$|	Swift	4, 5, 6
050318	32 ± 2	1.4436	16.9 ± 1.7	|$11.26_{-0.69}^{+0.87}$|	2.18 ± 0.40	Swift	4, 5, 7
050319	139.4 ± 8.2	3.2425	44 ± 18	|$77.90_{-28.70}^{+20.50}$|	|$2.18_{-0.40}^{+0.29}$|	Swift	4, 7
050505	63 ± 2	4.27	|$444_{-112}^{+80}$|	|$237.83_{-49.20}^{+98.41}$|	|$1.66_{-0.17}^{+0.34}$|	Swift	4, 5, 8
050525Ab	8.8 ± 0.5	0.606	23.2 ± 0.6	|$142.78_{-43.31}^{+40.58}$|	2.12 ± 0.47	Swift	4, 5, 9
050820A	128.0 ± 106.9	2.6147	|$970_{-140}^{+310}$|	|$537_{-95}^{+80}$|	|$6.6_{-0.3}^{+0.5}$|	Swift	10
050904	183.6 ± 13.2	6.295	|$1325.5_{-400.7}^{+678.2}$|	|$88.4_{-44.2}^{+86.3}$|	1.95 ± 0.29	Swift	5, 10
050922C	4.1 ± 0.7	2.1992	|$99.3_{-12.7}^{+13.0}$|	|$27.3_{-2.5}^{+2.8}$|	1.8 ± 0.3	Swift	11
060124	298 ± 2	2.297	420 ± 50	|$578.87_{-12.66}^{+110.79}$|	|$3.04_{-0.23}^{+0.52}$|	Swift	4, 5, 12
060206	5.0 ± 0.7	4.05	47.8 ± 207.9	386.76 ± 93.02	2.01 ± 0.06	Swift	5, 11
060210	220 ± 70	3.9133	353 ± 19	|$1113.29_{-94.72}^{+105.39}$|	|$1.20_{-0.12}^{+0.17}$|	Swift	4, 5, 7
060418	52 ± 1	1.4901	|$100_{-20}^{+70}$|	|$0.12_{-0.01}^{+0.03}$|	|$22.5_{-2.5}^{+0.9}$|	Swift	10
060526	258.8 ± 5.4	3.21	25.8 ± 2.6	|$15.58_{-0.21}^{+0.24}$|	3.61 ± 0.06	Swift	4, 5, 7
060605	15.2 ± 2.3	3.773	28.3 ± 4.5	177.15 ± 13.12	1.55 ± 0.06	Swift	4, 5, 13
060714	108.2 ± 6.4	2.71	182.2 ± 25.3	250.46 ± 248.11	1.15 ± 0.06	Swift	5, 11
060908	18.0 ± 0.8	1.8836	44.1 ± 1.8	|$2017.68_{-504.42}^{+2522.09}$|	|$0.46_{-0.06}^{+0.29}$|	Swift	4, 5, 9
061121	80 ± 21	1.3145	272 ± 18	|$83.32_{-4.88}^{+18.90}$|	|$1.83_{-0.17}^{+0.34}$|	Swift	4, 5, 9
070125	63.0 ± 1.7	1.5477	|$957.6_{-87.4}^{+106.4}$|	|$6.43_{-0.17}^{+0.9}$|	13.2 ± 0.6	Swift	10
080319B	45.6 ± 0.4	0.9371	1440 ± 30	|$4.9_{-0.1}^{+3.2}$|	|$7.0_{-0.1}^{+0.7}$|	Swift	10
081008	162.2 ± 25.0	1.967	|$99.8_{-23.1}^{+23.4}$|	|$134.7_{-17.3}^{+18.3}$|	1.3 ± 0.4	Swift	11
081203A	96.3 ± 11.8	2.1	|$377.5_{-179.2}^{+174.2}$|	|$344.6_{-230.1}^{+224.4}$|	1.0 ± 0.6	Swift	11
090323	133.1 ± 1.4	3.568	3300 ± 130	|$116_{-9}^{+13}$|	|$2.8_{-0.1}^{+0.4}$|	Fermi	14
090328	57 ± 3	0.7354	96 ± 10	|$82_{-18}^{+28}$|	|$4.2_{-0.8}^{+1.3}$|	Fermi	14
090423	10.3 ± 1.1	8.23	100 ± 30	|$340_{-140}^{+110}$|	|$1.5_{-0.3}^{+0.7}$|	Fermi	15
090618	105.5 ± 1.7	0.54	|$256.6_{-92.2}^{+92.9}$|	|$37.1_{-12.3}^{+12.2}$|	3.5 ± 1.3	Swift	11
090902B	19.328 ± 0.286	1.8829	3200 ± 40	|$56_{-7}^{+3}$|	3.9 ± 0.2	Fermi	14
090926A	20 ± 2	2.1062	1890 ± 30	6.8 ± 0.2	|$9_{-2}^{+4}$|	Fermi	14
091127a	7.1 ± 0.2	0.49034	|$14.9_{-1.9}^{+1.8}$|	|$48.4_{-5.3}^{+5.9}$|	2.7 ± 0.4	Swift	11
100418A	8 ± 2	0.6235	|$0.99_{-0.34}^{+0.63}$|	|$3.6_{-0.7}^{+1}$|	20.9 ± 0.5	Swift	16
120326A	11.8 ± 1.8	1.798	32 ± 1	14.0 ± 0.07	4.6 ± 0.2	Swift	16
120404A	38.7 ± 4.1	2.876	90 ± 40	|$13.3_{-2.0}^{+3.5}$|	3.1 ± 0.3	Swift	16
120521C	26.7 ± 0.4	6.0	190 ± 80	|$22_{-14}^{+37}$|	|$3.0_{-1.1}^{+2.3}$|	Swift	15
130427Aa	162.83 ± 1.36	0.338	|$808.9_{-56.5}^{+49.6}$|	|$157.7_{-11.0}^{+9.7}$|	3.8 ± 0.3	Swift	11

Notes. The evidences for LGRBs associated with observable supernovae (SNe) according to the following scale (Hjorth & Bloom 2012; Melandri et al. 2014):

^aStrong spectroscopic evidence.

^bA clear light-curve bump as well as some spectroscopic evidence resembling a GRB-SN.

References:

(1) Lloyd-Ronning & Zhang (2004); (2) Bloom, Frail & Kulkarni (2003); (3) Zhang, Zhao & Zhang (2011); (4) Liang et al. (2008); (5) Lu et al. (2012); (6) Amati (2006); (7) Japelj et al. (2014); (8)Hurkett et al. (2006); (9) Nava et al. (2012); (10) Cenko et al. (2010); (11) Wang et al. (2015a); (12) Romano et al. (2006); (13) Ghirlanda et al. (2012); (14) Cenko et al. (2011); (15) Laskar et al. (2014); (16) Laskar et al. (2015).

Our sample contains 48 LGRBs as shown in Table 1. All data are taken from the literatures and Swift data archives. They are discovered and measured by the different telescopes including BeppoSAX, BATSE, HETE, Swift and Fermi. The isotropic radiated energy in the prompt emission phase E_γ,iso, redshift z, and the duration T₉₀ are directly available from measurements. E_k,iso and θ_j can be estimated by the X-ray afterglow phase with the standard afterglow model. Moreover, due to the restricted observations of LGRB afterglows, it is difficult to obtain the accurate jet opening angles. In Table 1, the lower limits of the jet opening angles in some LGRB data are shown, which cause the calculated accreted masses lower. In this sample, we collect the data of T₉₀, E_γ,iso, E_k,iso, and θ_j with errors because the results may depend on them.

Fig. 1 shows the distributions of the accreted masses for the different typical mean BH masses and spins and conversion efficiencies, which are respectively set to M_BH/M_⊙ = 3, 5, and 10, a_* = 0.9, 0.95, and 0.998, and η = 0.2 and 0.5 corresponding to Figs 1 (a–i). It needs to be emphasized that the BHs in the centre of the collapsars should be rotating very rapidly, thus the spin parameters are set larger than 0.9 (e.g. Popham et al. 1999; Woosley & Heger 2006). It is obvious that larger central BHs will require more accreted mass. Conversely, the accreted mass is negatively correlated with the mean BH spin and conversion efficiency. In these cases, the mean spin parameters are definitely more effective on the values of the accreted masses than the mean BH masses. For the cases with a_* = 0.9, 0.95, and 0.998, M_BH = 3 M_⊙, and η = 0.2, about 67 per cent, 79 per cent, and 96 per cent LGRBs’ accreted masses are less than 8 M_⊙, respectively. If we consider a_* = 0.998, M_BH = 3 M_⊙, and η = 0.5, all the values of the accreted masses are less than 7 M_⊙. Obviously, if the accreted mass is larger than 7 M_⊙, the mean BH mass must be much larger than 3 M_⊙.

Figure 1.

Distributions of the accreted masses M_acc for different typical mean BH masses M_BH and spins a_* and conversion efficiencies η.

According to these results, the accreted mass of most LGRBs are less than the simulation results of the disc mass, ∼5 M_⊙ (e.g. MacFadyen & Woosley 1999; Popham, Woosley & Fryer 1999; Zhang, Woosley & MacFadyen 2003). Of course, there may still exist some LGRBs without the extreme Kerr low-mass BH in the centre or high efficiency, which need invoking alternative energy extraction mechanisms, such as BZ mechanism (e.g. Blandford & Znajek 1977; Lee, Brown & Wijers 2000b; Lee, Wijers & Brown 2000a; Yuan & Zhang 2012; Kawanaka et al. 2013; Hou et al. 2014a; Liu et al. 2015a).

CONCLUSIONS AND DISCUSSION

The goal of this work is to verify whether the NDAFs can account for the total energy of the prompt emission and afterglow phases in LGRBs. We collected data and calculated the distributions of the mean accreted masses for the reasonable states of the BH hyperaccretion system. As a result, most of LGRBs can be satisfied with the NDAFs, except for some extremely high energy cases. Additionally, one should stress that the accretion rate and the BH mass and spin we adopted are the mean quantity throughout the accretion process. Recently, we studied the effects of the evolution of the BH-NDAF system on the neutrino annihilation energy (Song et al. 2015), which has slight influences on the current results.

As mentioned above, increasing the neutrino emission rate or enhancing annihilation efficiency are the effective ways to improve the ability of NDAF model. For example, Liu et al. (2015b) presented that the vertical convection can suppress the radial advection, which effectively increase the neutrino emission rate. Moreover, Liu et al. (2010) investigated the vertical structure of the NDAF model. We noticed that the half-opening angle of the disc is actually very large, ∼80°. As mentioned in Birkl et al. (2007), the BH gravity can affect the traces of the neutrinos, especially for the neutrinos launched from the inner region of the disc, so they considered that the thin disc geometries are more efficient to raise the energy of the neutrino annihilation. But once the disc is extremely geometrically thick, the annihilable efficiency could be greatly enhanced due to the neutrinos trapped in a narrow space, even though some neutrinos falling into the BH.

BZ mechanism is another popular candidate for the energy sources of GRBs (e.g. Blandford & Znajek 1977; Lee et al. 2000b,a). Moreover, Yuan & Zhang (2012) proposed that the episodic magnetic reconnection from the disc, which is very similar to the events at the solar surface, can produce GRBs and subsequent flares. Additionally, the closed magnetic field lines connecting a BH with its surrounding disc can transfer the angular momentum and the energy, which may significantly enhance the neutrino luminosity (e.g. Lei et al. 2009; Luo et al. 2013). It should be emphasized that magnetar model is more dynamic than ever since ULGRBs and super-luminous SNe were observed (e.g. Wang & Dai 2013; Gao et al. 2015; Metzger et al. 2015; Wang et al. 2015b,c).

Furthermore, X-ray flares as the common features in GRB afterglows (e.g. Gehrels et al. 2004; Burrows et al. 2005; Chincarini et al. 2007) are not considered in this framework. They may have the similar origin with GRBs and ask for more extreme dynamic conditions and more massive accreted mass of the systems. Besides, the long ‘plateau’ phases (shallow decay phases) with more than thousands of seconds are generally observed in the X-ray afterglows (e.g. Zhang et al. 2006), which may require the energy injection from the central engine (e.g. Dai & Lu 1998; Zhang et al. 2006). Thus, the NDAF model has to confront more rigorous challenges.

We thank Bing Zhang, Ye Li, Shu-Jin Hou, and Hui-Jun Mu for beneficial discussion, and the anonymous referee for very useful suggestions and comments. This work was supported by the National Basic Research Program of China (973 Program) under grant 2014CB845800, the National Natural Science Foundation of China under grants 11274200, 11333004, 11373002, 11473022, 11573023, and U1331101, the Fundamental Research Funds for the Central Universities under grant 20720160024. TL acknowledges financial support from China Scholarship Council to work at UNLV.