NDAF model for X-ray flares

Neutrino-cooled Accretion Model with Magnetic Coupling for X-ray Flares in GRBs

Abstract

The neutrino-cooled accretion disk, which was proposed to work as the central engine of gamma-ray bursts, encounters difficulty in interpreting the X-ray flares after the prompt gamma-ray emission. In this paper, the magnetic coupling between the inner disk and the central black hole is taken into consideration. For mass accretion rates around   s, our results show that the luminosity of neutrino annihilation can be significantly enhanced due to the coupling effects. As a consequence, after the gamma-ray emission, a remnant disk with mass   may power most of the observed X-ray flares with the rest frame duration less than 100 seconds. In addition, a comparison between the magnetic coupling process and the Blandford-Znajek mechanism is shown on the extraction of black hole rotational energy.

accretion, accretion disks - black hole physics - gamma-ray burst: general - magnetic fields

1 Introduction

The launch of Swift satellite has led to tremendous discoveries of gamma-ray bursts (GRBs) (see Mészáros, 2006; Gehrels et al., 2009, for a review). A surprising one is that large X-ray flares are common in GRBs and occur at times well after the initial prompt emission (Romano et al., 2006; Falcone et al., 2007; Chincarini et al., 2007; Bernardini et al., 2011). The X-ray flare is an episodic phenomena showing a sudden brightness at the late afterglow stage. From the spectral and temporal analysis of X-ray flares, it is strongly suggested that X-ray flares may have a common origin as the prompt gamma-ray pulses and are related to the late time activity of the central engine (Bernardini et al., 2011; Romano et al., 2006).

For the energy reservoir of powering GRBs or X-ray flares, it is believed that they are produced through an ultra-relativistic jet, i.e., a neutrino annihilation-driven jet or a Poynting flux-dominated jet. For the former jet, neutrinos annihilate above the disk and form a hot fireball which subsequently expand to accelerate the jet by its thermal pressure (Popham et al., 1999; Di Matteo et al., 2002; Chen & Beloborodov, 2007). On the other hand, in the case of a Poynting flux-dominated jet, the jet derives its energy from the rotational energy of the central black hole (BH) via the Blandford-Znajek (BZ) process (Blandford & Znajek, 1977; McKinney & Gammie, 2004; Tchekhovskoy et al., 2008). Several mechanisms were proposed to explain the episodic phenomenon of X-ray flares, including fragmentation of a rapidly rotating core (King et al., 2005), magnetic regulation of the accretion flow (Proga & Zhang, 2006), fragmentation of the accretion disk (Perna et al., 2006), differential rotation in a post-merger millisecond pulsar (Dai et al., 2006), transition from thin to thick disk (Lazzati et al., 2008), He-synthesis-driven winds (Lee et al., 2009), the propagation instabilities in GRB jets (Lazzati et al., 2011), and the episodic, magnetically dominated jets (Yuan & Zhang, 2012). Apart from the energy and the episodic activity of X-ray flares, their evolution has also been investigated, which shows that the average energy released in the form of X-ray flares overlaid on the power-law decay of the afterglow of GRBs (Lazzati et al., 2008; Margutti et al., 2011).

In the present work, we will concentrate on another issue, i.e., whether or not the remnant disk after the prompt gamma-ray emission can power X-ray flares through neutrino annihilation. The luminosity of neutrino annihilation produced by the accretion disk is sensitive to the accretion rate. In the late stage of disk evolution, the accretion rate will probably be quite low (e.g., s) and the annihilation luminosity will drop sharply according to previous numerical calculations (e.g., Popham et al., 1999). In other words, the observed X-ray flares, if produced by the annihilation mechanism, will still require a relatively high accretion rate. We take the second flare of GRB 070318 as an example, which has an isotropic luminosity of erg s and a rest frame duration of 80.6 s (see Table 1). In order to produce such a luminosity by neutrino annihilation, an accretion rate around s is required for a rapidly rotating BH () due to our numerical calculations in Section 3, which is consistent with the results in Popham et al. (1999). Subsequently, the required remnant disk mass is around , which is obviously beyond what the progenitor can provide (e.g., Shibata & Taniguchi, 2008; Pannarale et al., 2011).

In this work, we will investigate the neutrino-cooled disk with the magnetic coupling (MC) between the inner disk and the BH, where the magnetic field connects the accretion disk and the central BH (Li, 2002; Wang et al., 2002; Uzdensky, 2005). In such a model, the angular momentum and the energy are transported from the BH horizon to the accretion disk through a closed magnetic field. Accordingly, this transported energy will be released and therefore increase the radiation of the disk (Li, 2002). The MC effects on the structure and radiation of neutrino-cooled disks have first been studied by Lei et al. (2009). They showed that both the neutrino luminosity and the annihilation luminosity will increase significantly owing to the MC process. In the present work, we will focus on relatively low accretion rates s to study the possibility of the remnant disk to power X-ray flares. We would point out that, although such type of magnetic fields is one of the possible field geometries discussed by McKinney (2005, Figure 2) and has been studied by a few previous works, the existence of such magnetic fields remains a problem according to MHD simulations. We will discuss this issue in the last section.

The paper is organized as follows. Equations are presented in Section 2. The structure and neutrino radiation of the disk are calculated in Section 3. A comparison of our numerical results with the observed X-ray flares is shown in Section 4. Conclusions and discussion are made in Section 5.

2 Equations

For neutrino-dominated accretion, the disk is extremely hot and dense and the neutrino radiation can balance the viscous dissipation. The structure and radiation of such a disk have been widely investigated in previous works (Popham et al., 1999; Di Matteo et al., 2002; Gu et al., 2006; Chen & Beloborodov, 2007; Liu et al., 2007; Lei et al., 2009; Pan & Yuan, 2012; Liu et al., 2013). Some simulations showed that the accretion flow is very dynamic and the inner radius of the flow changes with time. In addition, the flow is subject to various HD and MHD instabilities so that it is non-asymmetric. In order to avoid the complexity of solving partial differential equations, the present work is still based on the assumption of a steady and axisymmetric accretion flow. In such case, the basic equations of the neutrino-cooled accretion disk including the MC effects may refer to Section 2 of Lei et al. (2009), and the relativistic effects of the spinning BH were shown in Riffert & Herold (1995).

For simplicity, we define the gravitational radius as , the dimensionless radius as , and the dimensionless spin parameter as . The disk is assumed to be Keplerian rotating, thus the angular velocity of the flow is expressed as

(1)

In the present work, we investigate the properties of the neutrino-cooled disk with magnetic field lines connecting the BH with the inner disk. Such a MC process may have substantial effects on the energy and angular momentum balance of the disk (Li & Paczyński, 2000; Li, 2002; Wang et al., 2002; Janiuk & Yuan, 2010; Kovács et al., 2011). In addition, Uzdensky (2005) showed that the inner part of the disk is magnetically coupled to the BH, but the magnetic field cannot be stable in the outer region. Here, we follow the assumptions of Wang et al. (2003) that the MC process is constrained by a critical polar angle , and the magnetic field varies as a power law with the disk radius. The magnetic torque exerted to the disk from the BH horizon can be expressed as

(2)

where is a critical polar angle. In the scenario (e.g., Wang et al., 2002, Figure 1), the MC process exists in the range . On the contrary, for the space with , the BZ process may occur and therefore some accreted materials may be pushed away, particularly in the inner region of the disk. In the present study, we will focus on the MC process and a constant accretion rate is assumed for simplicity. The critical angle is calculated by

(3)

where , , , and g cm s. is the ratio of the angular velocity of the disk (Eq. (1)) to the angular velocity at the horizon, where . In addition, under the equipartition assumption (e.g., McKinney, 2005; Lei et al., 2009), the magnetic field strength at the horizon can be estimated as .

The energy equation is written as

(4)

where is the viscous heating rate, including the contributions of the gravitational potential and the MC process . The quantities and are respectively the advective cooling rate and the neutrino cooling rate. Here, we neglect the radiation of photons since they are trapped in the disk. The neutrino cooling generally consists of the four processes: the electron-positron pair annihilation, the bremsstrahlung emission of nucleons, the plasmon decay, and the Urca process (e.g., Liu et al., 2007). In addition, for the equation of state, the total pressure consists of five terms, i.e., the gas pressure, the radiation pressure, the degeneracy pressure, the neutrino pressure, and the magnetic pressure. The detailed description of the neutrino cooling and the pressure can be found in some previous papers (e.g., Popham et al., 1999; Gu et al., 2006; Janiuk & Yuan, 2010). For the term of magnetic pressure, following Lei et al. (2009), the ratio of magnetic pressure to total pressure is assumed to be 0.1.

3 Numerical results

3.1 Disk structure

In our calculations we fix , , , , and . The numerical results of structure and radiation are shown in Figure 1, where the solid lines and the dashed lines represent the solutions with and without the MC process, respectively. We choose three accretion rates, i.e., , , and s for the study. Figures 1a, 1b, and 1c show respectively the radial profiles of the mass density, the temperature, and the neutrino cooling efficiency defined as the ratio of the neutrino cooling rate to the viscous heating rate .

It is seen that, for the outer region with , the solid and dashed lines are identical to each other, which implies that the MC effects in this region are negligible. On the contrary, for the inner region with , the solid lines and the dashed lines are apparently separate, which indicates that the MC effects are substantial. For the inner region, Figure 1a shows that the density with MC is obviously larger than that without MC. The physical understanding is as follows. The MC process can transfer angular momentum from the BH to the inner disk. As a consequence, the additional transferred angular momentum will work as a barrier to prevent the flow from radial acceleration, and therefore the accreted matter will accumulate in this region and the mass density will significantly increase. On the other hand, for s, it is shown by Figure 1a that the solid line drops inwards even faster than the dashed line. The reason is that, for large , the disk will become optically thick to neutrinos. Thus, most of the generated neutrinos are trapped in the disk instead of escaping away. The total pressure, which includes the neutrino pressure, will therefore increase significantly and the disk will probably become geometrically thick. Accordingly, the mass density will drop sharply due to the increased vertical height and radial velocity. The neutrino trapping can be also indicated by Figure 1c, where the solid line steeply drops inwards for  s, indicating that most neutrinos are trapped in the disk rather than being radiated.

3.2 Annihilation luminosity

In the scenario of hyper-accretion disks, the GRBs are powered by the neutrino and anti-neutrino annihilation above the surface of disks. Since the X-ray flares and the gamma-ray emission have the same power origin, the flares may still be powered by the neutrino annihilation. The disk luminosity is calculated from the marginal stable orbit to the outer boundary . The neutrino luminosity from the disk is calculated as

(5)

The total annihilation luminosity is calculated by the integration over the whole space outside the disk, following the method in previous works (e.g., Ruffert et al., 1997; Popham et al., 1999).

Figure 2 shows the variations of the neutrino luminosity () and the annihilation luminosity () with the mass accretion rate . Similar to Figure 1, the solid lines and the dashed lines represent the solutions with and without the MC process, respectively. The upper solid (dashed) line corresponds to (), and the lower solid (dashed) line corresponds to (). It is seen that, both the neutrino luminosity and the annihilation luminosity with the MC process are significantly larger than those without the MC process. In particular for relatively low accretion rates such as s, is larger than by up to four orders of magnitude. The physical reason is that, in addition to the gravitational energy, the MC process can efficiently extract the BH rotational energy into neutrino radiation. Moreover, the apparent difference between the lower solid line and the lower dashed line implies that it is quite possible for a remnant low-mass disk to power X-ray flares.

3.3 Efficiency of BH rotational energy extraction

We compare the efficiency of BH rotational energy extraction between the MC process and the BZ mechanism. The former efficiency is defined as

(6)

For the BZ mechanism, the magnetic field lines, which are dragged in by the accretion disk, accumulate around the BH horizon and then get twisted by the BH space-time, which enable the extraction of BH rotational energy. The efficiency of the BZ process was discussed in Tchekhovskoy et al. (2011),

(7)

where is the dimensionless magnetic flux threading the BH, is a numerical constant related to the magnetic field geometry (we adopt here), and is set to be 1. The above equation shows that the BZ efficiency is relevant to the dimensionless magnetic flux . The value of , however, is quite uncertain and may be related to the accretion rate. It can be regarded as the ability for the accretion disk to drag magnetic fields into the BH horizon. Recent simulations show that can be as large as several tens and the disk can be depicted as a magnetic arrest disk. At a high , a large amount of magnetic flux is transported to the center and the efficiency can be larger than 1. Here we simply assume a constant magnetic flux, , which is close to the simulation result (Tchekhovskoy et al., 2011).

Equations (6) implies that is independent of the accretion rate . The variation of and with the spin parameter is shown in Figure 3. It is seen that, for , can be significantly larger than the efficiency of normal accretion process , and can be even larger than 1 for extremely spinning BH. Such a result indicates that the MC process is quite an efficient mechanism to extract the BH rotational energy. The efficiency decreases sharply with decreasing for . The reason is that the angular velocity of BH will be less than that of the disk at ISCO for , and therefore the MC process will become weak for BH spin below this critical value. It is also seen that, for , is several times larger than , whereas for , is significantly larger than due to a sharp decrease of with decreasing . In other words, for fast spinning BH systems, MC may be more powerful than BZ on the extraction of rotational energy, whereas for slow spinning BH systems, BZ is likely to be more powerful.

Another different effect between MC and BZ is that, the BZ process can directly transfer the rotational energy into the jet, whereas the MC process can only transfer the rotational energy into the disk, and the jet power is also related to another process, i.e., neutrino annihilation. Thus, for the efficiency of powering the jet, the BZ process will probably be much more efficient. For a comparison, the dotted line shows the efficiency of radiation due to neutrino annihilation () for a typical accretion rate   s. It is seen that is significantly larger than for any . Thus, the BZ process is more efficient to power a jet than the MC process. It is also possible for the BZ process to work as the central engine to power the X-ray flares.

4 Comparison with observations

In order to compare our numerical results with the observations of X-ray flares, we compile a sample of 21 GRBs with 43 flares as shown in Table 1, which includes all the flares with available redshift in Table 1 of Chincarini et al. (2010). The width of the flares is calculated in the rest frame: , where is the redshift, is the observed width, and is the width in the rest frame. The isotropic energy of a flare can be estimated from the fluence : , where is the luminosity distance. Thus, the average, isotropic luminosity can be obtained as .

A comparison of our numerical results with the observations is shown in Figure 4, which includes all the flares in Table 1. Similar to Figures 1 and 2, the solid lines represent the results with MC, whereas the dashed lines represent the results without MC. The upper and lower solid (dashed) lines correspond to the disk mass   and 0.05 , respectively. The four theoretical lines are calculated by together with the relationship between and obtained in Section 3.

Figure 4 shows that most of the flares locate above the upper dashed line, which means that for the case without MC, it will generally require a remnant disk mass significantly larger than 0.5 , which may be unpractical. However, it is seen that most of the flares exist between the two solid lines, which indicates that if MC works, a reasonable remnant disk with is able to power nearly all the plotted flares. As mentioned in Section 1, taking the second flare of GRB 070318 as an example, the model without MC requires . On the contrary, if the MC effects are taken into account, the luminosity of erg s corresponds to s (according to our numerical calculations shown by the lower solid line in Figure 2). Therefore, the rest frame duration 80.6 s only requires for powering this flare.

5 Conclusions and discussion

In this paper, we have studied the neutrino-cooled disks by taking into account the MC process between the central BH and the inner disk. We have shown that, for mass accretion rates around s, the luminosity of neutrino annihilation can be enhanced by up to four orders of magnitude due to the MC effects. As a consequence, the remnant disk with   may power most of the observed X-ray flares with the rest frame duration less than 100 seconds.

We would point out that, for a few X-ray flares with extremely long duration, the neutrino-cooled disk cannot work as the central engine, even though MC is included. For example, GRB 050724 has three flares with a redshift of 0.258. The third flare of this source has a rest frame width of seconds and the luminosity erg s. According to our numerical results in Figure 2, the accretion rate should be around s for the case including MC. Thus, it requires a remnant disk , which is obviously unphysical. On the other hand, X-ray flares with peak time less than and larger than 1000 s may have different origin (Margutti et al., 2011). The mechanism for powering the flares with peak time larger than 1000 s is worthy for further studies, but is beyond the scope of the present paper.

Another issue we would like to stress is related to the configuration of the large-scale magnetic field in accretion disks. Although the MC process has been studied by quite a few previous works, such type of magnetic fields, however, has not been found in MHD simulations. Thus, it remains unclear whether the MC process can occur between the inner disk and the central BH. On the other hand, some simulations showed that the BZ mechanism can be a solution to the GRB’s central engine (e.g., Tchekhovskoy et al., 2008). In the scenario of the Poynting flux-dominated jet, the efficiency of extracting the BH rotational energy mainly depends on the magnetic flux being dragged in (Tchekhovskoy et al., 2011; McKinney et al., 2012). The theoretical analysis showed that it requires a geometrically thick disk to transport a large mount flux into the center (Lubow et al., 1994; Rothstein & Lovelace, 2008; Beckwith et al., 2009; Cao, 2011). Simulations also confirmed that a thick disk can efficiently transport magnetic flux (McKinney et al., 2012). For a neutrino-cooled disk, neutrinos play a vital role to release the dissipation heat and the disk is likely to be geometrically thin (e.g., Shibata et al., 2007). Then, for the Poynting flux-dominated jet, it remains a problem whether the accretion flow can accumulate adequate magnetic fields to the inner region to power the jet.

In this work, the flow is assumed to be steady and the mass accretion rate is a free parameter. In other words, for a given accretion rate, we will obtain a corresponding solution. On the other hand, the simulations of Tchekhovskoy et al. (2011) found a correlated variation between the accretion rate and the magnetic flux . In this spirit, a varying strength of MC process may also have effects on the variation of accretion rate. Such a study requires further time-dependent calculations.

We thank Raffaella Margutti, Shujin Hou, Da-Bin Lin, and Mou-Yuan Sun for beneficial discussions, and the referee for helpful suggestions. This work was supported by the National Natural Science Foundation of China under grants 11073015, 11103015, 11222328, and 11233006.
Figure 1: Radial profiles of the density , the temperature , and the ratio of the neutrino cooling to the viscous heating for , 0.05, 0.5 s. The solid lines and the dashed lines represent the solutions with and without the MC process, respectively.
Figure 2: Variations of the neutrino luminosity () and the annihilation luminosity () with the mass accretion rate. The upper and lower solid lines correspond to and with the MC process, respectively, whereas the upper and lower dashed lines correspond to and without the MC process, respectively.
Figure 3: Variation of the efficiency with the spin parameter . The solid line shows the efficiency of the energy transfer from the rotating BH to the disk by the MC process. For a comparison, the dashed line shows the efficiency of the energy extraction by the BZ mechanism with . The dotted line shows the efficiency of radiation due to neutrino annihilation for a typical accretion rate   s.
Figure 4: A comparison of our numerical results with the observations in the diagram. The upper and lower solid lines correspond to the remnant disk mass and with the MC process, respectively, whereas the upper and lower dashed lines correspond to and without the MC process, respectively. The data for the 43 flares are shown in Table 1.
GRB z (erg s) (s)
050730 3.967 1.0210 0.4929 8.6370 1.0268
050730 3.967 1.3644 0.3684 22.2871 1.2684
050730 3.967 0.5679 0.2609 19.8108 2.7783
050908 3.344 0.1648 0.0520 26.6805 3.6602
060115 3.53 0.2332 0.0960 18.4106 3.0464
060210 3.91 5.5296 0.9617 12.3625 0.5295
060210 3.91 2.9090 0.5818 12.2607 0.7128
060418 1.489 2.6893 0.2241 10.0442 0.2812
060512 0.4428 0.0041 0.0010 54.8933 5.4062
060526 3.221 12.0001 14.6252 5.8043 0.4264
060526 3.221 6.0765 2.6736 8.9552 2.1559
060526 3.221 4.9712 1.1600 11.8218 1.1372
060526 3.221 1.6612 0.4345 17.0339 1.2556
060604 2.68 2.3969 0.6453 8.6957 0.5978
060604 2.68 2.1283 0.3502 5.9511 0.4076
060607A 3.082 2.2870 0.7277 3.8951 0.6369
060607A 3.082 2.6306 0.3617 18.4713 0.5879
060707 3.425 0.2153 0.1846 7.8192 2.6667
060714 2.711 4.0362 0.5259 13.3657 1.2665
060714 2.711 2.8113 0.9618 2.2096 1.1048
060714 2.711 4.1165 1.0400 3.7726 0.5120
060714 2.711 3.1626 0.5750 5.6858 0.2156
060729 0.54 0.2463 0.0265 28.3766 1.6234
060814 0.84 0.2704 0.0419 21.1957 1.3587
060904B 0.703 0.5598 0.0336 46.0951 0.5285
060908 1.8836 0.0066 0.0066 52.4345 31.1416
060908 1.8836 0.0070 0.0098 62.0752 19.8710
070306 1.4959 0.6855 0.1143 17.3885 1.3622
070318 0.836 0.0116 0.0128 15.7407 22.4401
070318 0.836 0.0296 0.0043 80.5556 11.0022
070721B 3.626 0.6013 0.8718 26.3078 63.3593
070721B 3.626 2.7102 1.8763 2.5292 2.1185
070721B 3.626 0.8519 0.3139 5.8798 2.1617
070721B 3.626 0.1109 0.0907 52.3130 44.6822
070724A 0.457 0.0067 0.0058 22.1688 2.8826
070724A 0.457 0.0018 0.0009 58.8195 14.8250
071031 2.692 2.7287 1.0720 16.5764 1.1376
071031 2.692 1.0844 0.2575 11.9177 1.3272
071031 2.692 0.6206 0.1494 14.0574 1.7064
071031 2.692 0.4104 0.0410 74.7833 2.7086
080210 2.641 1.4834 0.3350 9.8050 0.4944
080310 2.42 0.8247 0.0654 47.3684 3.2164
080310 2.42 1.3890 0.1543 17.4561 2.0468
Table 1: Observed flares with available redshift

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