Vertical Advection Effects on Hyper-accretion Disks and Potential Link between Gamma-ray Bursts and Kilonovae

TL;DR

This paper investigates how vertical advection in hyper-accretion disks influences their structure and radiation, suggesting a potential connection between gamma-ray bursts and kilonovae through gamma-ray photon escape.

Contribution

It introduces a revised model of hyper-accretion disks incorporating vertical advection, highlighting its effects on temperature, neutrino, and gamma-ray luminosities, and proposes a link to gamma-ray bursts and kilonovae.

Findings

Vertical advection lowers disk temperature compared to classic models.

Neutrino luminosity from the disk is reduced due to strong cooling.

Gamma-ray photons can escape in super-Eddington amounts, potentially powering gamma-ray bursts.

Abstract

Recent simulations on super-Eddington accretion flows have shown that, apart from the diffusion process, the vertical advection based on magnetic buoyancy can be a more efficient process to release the trapped photons in the optically thick disk. As a consequence, the radiative luminosity from the accretion disk can be far beyond the Eddington value. Following this spirit, we revisit the structure and radiation of hyper-accretion disks with mass accretion rates in the range $10^{-3}\sim 10~M_{\sun}~{\rm s}^{-1}$. Our results show that, due to the strong cooling through the vertical advection, the disk temperature becomes lower than that in the classic model without the vertical advection process, and therefore the neutrino luminosity from the disk is lower. On the other hand, the gamma-ray photons released through the vertical advection can be extremely super-Eddington. We argue that the large amount of escaped gamma-ray photons may have more significant contribution to the primordial fireball than the neutrino annihilation, and may hint a link between gamma-ray bursts and kilonovae in the black hole hyper-accretion scenario.

Equations

$$\dot{M}=-4\pi \rho HR{v}_R,$$

$$v_R=-\alpha c_{\mathrm{s}}\frac{H}{R}{f}^{-1}g,$$

$$P=\frac{\rho k_{\mathrm{B}}T}{m_{\mathrm{p}}}\left(\frac{1+3X_{\mathrm{nuc}}}{4}\right)+\frac{11}{12}a{T}^4+\frac{2\pi hc}{3}(\frac{3}{8\pi m_{\mathrm{p}}}\frac{\rho }{{\mu }_e}{)}^{\frac{4}{3}}+\frac{u_{\nu }}{3},$$

$$Q_{\mathrm{vis}}=Q_{\mathrm{adv}}+Q_z+Q_{\nu }.$$

$$Q_{\mathrm{vis}}=\frac{1}{4\pi }\dot{M}{\Omega }^{2}fg,$$

$$Q_{\mathrm{adv}}=-\xi v_R\frac{H}{R}T(\frac{11}{3}a{T}^3+\frac{3}{2}\frac{\rho k_{\mathrm{B}}}{m_{\mathrm{p}}}\frac{1+3X_{\mathrm{nuc}}}{4}+\frac{4}{3}\frac{u_{\nu }}{T}),$$

$$X_{\mathrm{nuc}}=\mathrm{Min}[1,20.13{\rho }_{10}^{-3/4}{T}_{11}^{9/8}\exp (-0.61/T_{11})],$$

$$Q_z=\overline{V_z}(u_{\mathrm{ph}}+u_{\nu }+u_{\mathrm{gas}}),$$

$$\overline{V_z}=\lambda c_{\mathrm{s}},$$

$$L_{\nu }={\int }_{R_{\mathrm{in}}}^{R_{\mathrm{out}}}4\pi R\cdot (Q_{\nu }+\overline{V_z}{u}_{\nu })\mathrm{d}R,$$

$$L_{\mathrm{ph}}={\int }_{R_{\mathrm{in}}}^{R_{\mathrm{out}}}4\pi R\cdot \overline{V_z}{u}_{\mathrm{ph}}\mathrm{d}R.$$

Full text

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

1 Introduction

Gamma-ray bursts (GRBs) are the most energetic phenomenon occurred in the cosmological distance. The luminosity of GRBs can reach the order of around $10^{50}~{}{\rm erg}~{}{\rm s}^{-1}$. There are various models to explain the energy source of GRBs. The popular model for the generation of short GRBs is the mergers of compact binaries (e.g., Narayan et al., 1992; Nakar, 2007; Berger, 2014), either black hole-neutron star binaries or double neutron stars. And it is commonly believed that the collapse of a massive star can account for long GRBs (e.g., Woosley, 1993; Paczyński, 1998; Woosley & Bloom, 2006). In both cases, a dense accretion disk is expected to form, with extremely high temperature and accretion rates. The neutrino-antineutrino annihilation process is a possible mechanism to provide energy for the bursts. The other mechanism based on hyper-accretion disks around black holes is the well-known Blandford-Znajek (BZ) process (e.g., Blandford & Znajek, 1977; Lee et al., 2000). Alternatively, a rapidly rotating neutron star with strong magnetic fields can also be responsible for GRBs (e.g., Usov, 1992; Dai & Lu, 1998; Dai et al., 2006; Metzger et al., 2011; Lü et al., 2015).

In this paper we will focus on the hyper-accreting disks with extremely high accretion rates ($10^{-3}~{}M_{\odot}~{}{\rm s}^{-1}\lesssim\dot{M}\lesssim 10~{}M_{\odot}~{}{\rm s}^{-1}$), where the neutrino radiation may be a dominant cooling mechanism, and therefore such flows are usually named as neutrino-dominated accretion flows (NDAFs). Popham et al. (1999) pioneered to study the NDAFs in details, and proposed that the neutrino annihilation of a hyper-accreting black hole system can explain GRBs up to $10^{52}~{}{\rm erg}~{}{\rm s}^{-1}$. The hyper-accreting black hole system typically has accretion rates around $0.01\sim 10~{}M_{\odot}~{}{\rm s}^{-1}$. The NDAFs have been widely studied on the radial structure and neutrino radiation (e.g., Popham et al., 1999; Narayan et al., 2001; Di Matteo et al., 2002; Kohri & Mineshige, 2002; Kohri et al., 2005; Gu et al., 2006; Liu et al., 2007; Chen & Beloborodov, 2007; Kawanaka et al., 2013; Zalamea & Beloborodov, 2011; Xue et al., 2013; Cao et al., 2014; Xie et al., 2016), vertical structure and convection (e.g., Liu et al., 2010; Pan & Yuan, 2012; Kawanaka & Kohri, 2012; Liu et al., 2014, 2015a), and time-dependent variation (Janiuk et al., 2004, 2013). The density can reach $10^{8}-10^{12}~{}\rm g~{}cm^{-3}$ in the inner region of the disk, and the temperature can be up to $10^{10}-10^{11}~{}\rm K$. Such disks can be extremely optically thick, leading to the trapping of large portion of photons, which are carried along the radial direction and are eventually absorbed by the central black hole. Thus, the neutrino cooling is the dominant cooling process in the inner part.

Recently, whether the neutrino annihilation process can work as the central engine for GRBs has been investigated in more details (Liu et al., 2015b; Song et al., 2015, 2016). Such works revealed that the neutrino annihilation process can account for most GRBs. For some long-duration GRBs, the central engine is more likely to be the BZ mechanism rather than the neutrino annihilation since the former is much more efficient. On the other hand, if the X-ray flares after the prompt gamma-ray emission are regarded as the reactivity of the central engine, the neutrino annihilation mechanism may encounter difficulty in interpreting the X-ray flares. Even including a possible magnetic coupling (Lei et al., 2009) between the inner disk and the central black hole, Luo et al. (2013) showed that the annihilation mechanism can work for the X-ray flares with duration $\lesssim 100~{}{\rm s}$. However, the annihilation mechanism is unlikely to be responsible for those long flares with duration $\gtrsim 1000~{}{\rm s}$, even though the role of magnetic coupling is included. More recently, Mu et al. (2016) investigated the central engine of the extremely late-time X-ray flares with peak time larger than $10^{4}$ s, and suggested that neither the neutrino annihilation nor the BZ process seems to work well. Instead, a fast rotating neutron star with strong bipolar magnetic fields may account for such flares.

Recent simulations have made significant progress on the super-Eddington accretion process, including the presence of strong outflows (Ohsuga et al., 2005; Ohsuga & Mineshige, 2011; Yang et al., 2014) and the radiation-powered baryonic jet (Sa̧dowski & Narayan, 2015). In addition, some simulations (e.g., Sa̧dowski & Narayan, 2016) show the anisotropic feature of radiation. More importantly, the simulation of Jiang et al. (2014) revealed a new energy transport mechanism in addition to the diffusion, which is named as the vertical advection. Their simulation results show that, for the super-Eddington accretion rate $\dot{M}=220L_{\rm Edd}/c^{2}$, the radiative efficiency is around 4.5%, which is comparable to the value in a standard thin disk model. The physical reason is that, a large fraction of photons can escape from the disk before being advected into the black hole, through the vertical advection process based on the magnetic buoyancy, which dominates over the photon diffusion process.

In this work, following the spirit of the vertical advection, we incorporate the vertical advection process into NDAFs and revisit the structure and radiation of hyper-accretion disks around stellar-mass black holes. The remainder is organized as follows. The basic physics and equations for our model are described in Section 2. Numerical results and analyses are presented in Section 3. Conclusions and discussion are made in Section 4.

2 Basic Equations

In this section we describe the basic equations of our model. We consider a steady state, axisymmetric hyper-accretion disk around a stellar-mass black hole. The well-known Paczyński-Wiita potential (Paczyński & Wiita, 1980) is adopted, i.e., $\Psi=-GM_{\rm BH}/(R-R_{\rm g})$, where $M_{\rm BH}$ is the black hole mass and $R_{\rm g}=2GM_{\rm BH}/c^{2}$ is the Schwarzschild radius. The Keplerian angular velocity can be expressed as $\Omega_{\rm K}=(GM_{\rm BH}/R)^{1/2}/(R-R_{\rm g})$. We use the usual convention of quantities to describe the accretion disk: the half-thickness of the disk is $H=c_{\rm s}/\Omega_{\rm K}$, where $c_{\rm s}=(P/\rho)^{1/2}$ is the isothermal sound speed, with $P$ being the pressure, and $\rho$ the density. We adopt the standard Shakura-Sunyaev prescription for the kinematic viscosity coefficient, i.e. $\nu=\alpha c_{\rm s}H$.

The basic equations that describe the accretion disk are the continuity, azimuthal momentum, energy equation, and the equation of state. The continuity equation is

[TABLE]

where $v_{R}$ is the radial velocity. With the Keplerian rotation assumption $\Omega=\Omega_{\rm K}$, the azimuthal momentum equation can be simplified as (e.g., Gu et al., 2006; Liu et al., 2007):

[TABLE]

where $f=1-j/\Omega_{\rm K}R^{2}$, $g=-{\rm d}\ln\Omega_{\rm K}/{\rm d}\ln R$, and $j$ represents the specific angular momentum per unit mass accreted by the black hole. The equation of state takes the form (e.g., Di Matteo et al., 2002)

[TABLE]

where the four terms on the right-hand side are the gas pressure, radiation pressure of photons, degeneracy pressure of electrons, and radiation pressure of neutrinos, respectively.

The energy equation is written as

[TABLE]

The above equation shows the balance between the viscous heating and the cooling by radial advection, vertical advection, and neutrino radiation. The viscous heating rate $Q_{\rm vis}$ and the advective cooling rate $Q_{\rm adv}$ for a half-disk above or below the equator are expressed as (e.g., Di Matteo et al., 2002; Gu et al., 2006)

[TABLE]

[TABLE]

where $T$ is the temperature, $s$ is the specific entropy, $u_{\nu}$ is the neutrino energy density, and $\xi$ is taken to be $1$. $X_{\rm nuc}$ is the mass fraction of free nucleons (e.g., Kohri et al., 2005):

[TABLE]

where $\rho_{10}=\rho/10^{10}~{}\rm g~{}cm^{-3}$ and $T_{11}=T/10^{11}~{}\rm K$. The quantity $Q_{\nu}$ is the cooling rate due to the neutrino radiation. We adopt a bridging formula for calculating $Q_{\nu}$ as shown in Di Matteo et al. (2002) and Liu et al. (2007).

The main difference from previous works is that the vertical advection term $Q_{z}$ is taken into account in our work, which is written as

[TABLE]

where $u_{\rm ph}$ is the energy density of photons, $u_{\nu}$ is the energy density of neutrinos, $u_{\rm gas}$ is the energy density of the gas. In our calculation the third term $\overline{V_{z}}u_{\rm gas}$ is dropped since the escaped gas through the magnetic buoyancy can be negligible. The quantity $\overline{V_{z}}$ is the averaged velocity of the vertical advection process, which can be simply written as

[TABLE]

where $\lambda$ is a dimensionless parameter. By comparing the typical vertical velocity $\overline{V_{z}}$ in the simulation results (Figure 14 of Jiang et al., 2014) and the theoretical estimate of sound speed, we take $\lambda=0.1$ for our numerical calculations. The vertical advection term describes the released photons and neutrinos due to the magnetic buoyancy, which can dominate over normal diffusion process.

Equations (1)-(4) can be solved if the parameters $M_{\rm BH}$, $\dot{M}$, $\alpha$, and $j$ are given. We consider a stellar-mass black hole with $M_{\rm BH}=3M_{\odot}$. The viscous parameter $\alpha=0.02$ is taken from the simulation results (Hirose et al., 2009). The specific angular momentum $j=1.83cR_{\rm g}$ is just a little less than the Keplerian angular momentum at the marginally stable orbit, i.e., $l_{\rm K}|_{3R_{\rm g}}=1.837cR_{\rm g}$. Our study focuses on the solutions in the range $3\sim 10^{3}~{}R_{\rm g}$.

3 Numerical Results

In this section we present our numerical results and the analyses of the physics behind these results. The calculation reveals that the vertical advection process has essential effects on the structure and radiation of the disk. First, the radial profiles of mass density and temperature are investigated for two cases, i.e., with and without the vertical advection process. The radial profiles of density $\rho$ are shown in Figure 1, where the solid (dashed) lines correspond to the results with (without) the vertical advection process. The five typical mass accretion rates are $\dot{M}=10^{-3}$, $10^{-2}$, $0.1$, $1$, and $10~{}M_{\odot}~{}{\rm s}^{-1}$, which are shown by different colors. It is seen from Figure 1 that, for a certain radius $R$ and a given $\dot{M}$, the density $\rho$ in the disk with vertical advection process is significantly higher than that without such a process, particularly for low accretion rates such as $\dot{M}=10^{-3}~{}M_{\odot}~{}{\rm s}^{-1}$. The radial profiles of temperature $T$ are shown in Figure 2, where the explanation of different color and types of lines is the same as in Figure 1. It is seen that the temperature of the disk with the vertical advection is generally lower than that without the advection.

The physical understanding of the above difference in density and temperature is as follows. Since a large fraction of the trapped photons can be released through the vertical advection process, the radiative cooling through such a process is efficient. As a consequence, the temperature $T$ together with the total pressure $P$, the vertical height $H/R$, and the radial velocity $v_{R}$ will decrease, whereas the mass density $\rho$ will increase. For relatively low accretion rates such as $\dot{M}=10^{-3}~{}M_{\odot}~{}{\rm s}^{-1}$, the radiative cooling of neutrinos is quite inefficient, so the effects of vertical advection may be more significant, as indicated by Equation (4).

Figure 3 shows the radial profiles of vertical scale height of the disk. It is seen that the relative height $H/R$ of the disk with vertical advection is significantly thinner than that without the vertical advection, particularly for low accretion rates. The physical reason is mentioned above, which is related to the decrease of temperature and pressure. The decrease of vertical height also implies the decrease of sound speed and therefore the decrease of radial velocity, as inferred by Equation (2).

For a typical mass accretion rate $\dot{M}=0.1~{}M_{\odot}~{}{\rm s}^{-1}$, Figure 4 shows the radial profiles of energy components, where the three dimensionless factors are $f_{z}=Q_{z}/Q_{\rm vis}$ (red line), $f_{\rm adv}=Q_{\rm adv}/Q_{\rm vis}$ (green line), and $f_{\nu}=Q_{\nu}/Q_{\rm vis}$ (blue line). It is seen that in the most inner region ($R\lesssim 4R_{\rm g}$) the radial advection is the dominant cooling mechanism. For the outer part of the disk ($R\gtrsim 20R_{\rm g}$) the energy transport through the vertical advection becomes dominant. For the region $4R_{\rm g}\lesssim R\lesssim 20R_{\rm g}$ the neutrino cooling may dominate over the other two mechanisms. The red line implies that the photon radiation through the vertical advection process can reach a large fraction of the total released gravitational energy, and therefore the photon luminosity can be extremely super-Eddington. We will investigate the corresponding photon luminosity in Figure 6.

Our main focus is the energy transport through vertical advection process. Figure 5 shows the radial profiles of the dimensionless cooling rate due to the vertical advection, i.e., $f_{z}=Q_{z}/Q_{\rm vis}$, where five typical accretion rates are adopted. It is seen that $f_{z}$ generally decreases with increasing $\dot{M}$. The physical reason is that the neutrino cooling is less important for relative low accretion rates. For the highest accretion rate with $\dot{M}=10~{}M_{\odot}~{}{\rm s}^{-1}$, the red line shows that there exists a big bump in the inner region ($\lesssim 20R_{\rm g}$). The physics for this bump is that the neutrino cooling is again less significant since the inner disk is optically thick to the neutrinos.

Finally, we calculate the neutrino luminosity and the photon luminosity of the accretion disk. The neutrino luminosity is derived by the integration of the whole disk:

[TABLE]

where $L_{\nu}$ includes the contributions from the direct neutrino radiation $Q_{\nu}$ and the vertical advection process on neutrinos $\overline{V_{z}}u_{\nu}$. Actually, the latter is negligible except for extremely high accretion rates $\dot{M}\gg 1~{}M_{\odot}~{}{\rm s}^{-1}$. The variation of neutrino luminosity with mass accretion rates is shown in Figure 6, where the red solid line corresponds to the neutrino luminosity with the vertical advection process, whereas the red dashed line corresponds to the neutrino luminosity without the process. It is seen that the red solid line is under the red dashed line, which means that the neutrino luminosity with the vertical advection is lower. The physical reason is as follows. The neutrino radiation is more sensitive to the temperature than the density. As shown by Figure 2, the disk temperature is lower for the case with the vertical advection. Thus, the corresponding $Q_{\nu}$ and $L_{\nu}$ in our cases are lower than those without the vertical advection.

The variation of photon luminosity $L_{\rm ph}$ is shown by the blue solid line in Figure 6, where $L_{\rm ph}$ is calculated by

[TABLE]

It is seen that $L_{\rm ph}$ is in the range $10^{50}\sim 10^{53}~{}{\rm erg}~{}{\rm s}^{-1}$ for $0.001\leqslant\dot{m}\leqslant 10$, which is more than ten orders of magnitude higher than the Eddington luminosity. The released photons are mainly in the gamma-ray band according to the thermal radiation of the inner disk with $10^{10}{\rm K}<T<10^{11}{\rm K}$, as shown by Figure 2. The huge amount of gamma-ray photons escape from the optically thick disk through the vertical advection process, which is much more efficient than the diffusion process. Such an extremely high photon radiation should have observational effects. We will have a discussion on that in next section.

4 Conclusions and discussion

In this work, we have studied the structure and radiation of hyper-accretion flows around stellar-mass black holes by taking into the role of vertical advection process. By the comparison of our results with the classic NDAF solutions, we have shown that the density is higher, the temperature is lower, and the vertical height is thinner in our solutions. The physical reason is that a large fraction of photons can escape from the optically thick disk through the vertical advection process. As a consequence, the neutrino luminosity from the disk is decreased. Thus, even without calculating the neutrino annihilation luminosity, we can conclude that the annihilation mechanism cannot be responsible for the long-duration GRBs and X-ray flares.

We would point out that outflows are not taken into consideration in the present work. However, outflows are believed to generally exist in accretion flows. Recent MHD simulations have shown that outflows exist both in optically thin flows (Yuan et al., 2012a, b) and optically thick flows (Jiang et al., 2014; Sa̧dowski & Narayan, 2015, 2016). From the observational view, Wang et al. (2013) reveals that more than 99% of the accreted mass escape from the accretion flow by outflows in our Galactic center. Based on the energy balance argument, Gu (2015) shows that the outflow is inevitable for the accretion flows where the radiative cooling is far below the viscous heating, no matter the flow is optically thin or thick. Thus, outflows may work as another process to help the trapped photons to escape (Shen et al., 2015), and will also have effects on the structure and neutrino radiation of the accretion flow. Such a mechanism is not included in the current work.

Our calculations are based on the relation $\overline{V_{z}}=\lambda c_{\rm s}$ with $\lambda=0.1$. As mentioned in Section 2, the value for $\lambda$ is adopted following the simulation results for $\dot{M}=220L_{\rm Edd}/c^{2}$ (Jiang et al., 2014). In our case, the mass accretion rate is higher for more than ten orders of magnitude. Then, a key question may exist whether the vertical advection due to the magnetic buoyancy can also work for such hyper-accretion systems. In our opinion, the radiation pressure is always dominant up to $\dot{M}\lesssim 0.1~{}M_{\odot}~{}{\rm s}^{-1}$ or for the outer part of even higher accretion rates. Thus, such a mechanism seems to be an efficient process. On the other hand, even for the case that the parameter $\lambda$ is significantly smaller than $0.1$ in the hyper-accretion case, such as several orders of magnitude smaller, the released gamma-ray photons may still be extremely super-Eddington and the potential application is significant.

In this work we have assumed $\alpha=0.02$ according to the simulation results of Hirose et al. (2009). However, other simulations may provide different values for $\alpha$. As shown by Yuan & Narayan (2014), such a value may be related to the magnitude of net magnetic flux in the simulations. The values of $\alpha$ may also have significant effects on the energy transport of the vertical advection. As Equations (2) and (6) imply, $v_{R}$ is proportional to $\alpha$ and $Q_{\rm adv}$ is proportional to $v_{R}$ and therefore $\alpha$. Thus, we can expect that, for a larger value of $\alpha$, the advective cooling rate $Q_{\rm adv}$ can significantly increase and therefore the cooling rate due to the vertical advection $Q_{z}$ will decrease according to the energy balance of Equation (4). Nevertheless, the luminosity related to the radial integration of $Q_{z}$ will still be hyper-Eddington even though $Q_{z}$ may be lower than $Q_{\rm adv}$ for a large range of radii.

Our results of extremely super-Eddington luminosity of gamma-ray emission can also be generally applied to short GRBs. It is commonly believed that short GRBs originate from the merger of compact objects, i.e., the black hole-neutron star binary or the binary with double neutron stars. More importantly, the merger is a significant source of gravitational wave event. Thus, the released high-energy photons may have significant contribution to an electromagnetic counterpart for the gravitational wave event, such as kilonovae. Kilonovae have been widely studied in recent years (e.g., Li & Paczyński, 1998; Metzger & Berger, 2012; Jin et al., 2013; Yu et al., 2013; Gao et al., 2015; Kasen et al., 2015; Kawaguchi et al., 2016; Metzger, 2016; Fernández et al., 2016). In our case, a new picture is shown by Figure 7. It is seen from this figure that, either the merger of a black hole and a neutron star or the merger of two neutron stars may result in a gravitational wave event and a black hole hyper-accretion disk. According to our study in this work, most gamma-ray photons escaping from the direction perpendicular to the equatorial plane together with the neutrino annihilation contribute to the thermal fireball, and the remanent escaped photons diverge from other directions to trigger non-thermal emission due to diffusion into ambient environment. Obviously, such a progenitor of kilonovae is quite different from the origin from disk wind (e.g., Metzger & Berger, 2012; Kasen et al., 2015) or magnetar wind (e.g., Yu et al., 2013; Gao et al., 2015). In addition, a faint gamma-ray thermal component may exist owing to the large amount of escaped thermal gamma-ray photons from the disk, which may have potential contribution to the thermal component of the prompt gamma-ray emission (Abdo et al., 2009; Zhang et al., 2016). Moreover, the neutrino annihilation mechanism or the BZ mechanism based on the accumulation of magnetic fields through the hyper-accretion process will work as the main central engine for the GRB. In summary, in our scenario of black hole hyper-accretion with vertical advection process, short GRBs, kilonovae, and gravitational wave events can be naturally blended together (e.g., Fernández & Metzger, 2016).

This work was supported by the National Basic Research Program of China (973 Program) under grants 2014CB845800, the National Natural Science Foundation of China under grants 11573023, 11473022, 11573051, 11633006, and 11333004, the National Program on Key Research and Development Project of China (Grant No. 2016YFA0400704), the Key Research Program of Frontier Sciences of CAS (No. QYZDJ-SSW-SYS008), and the CAS Open Research Program of Key Laboratory for the Structure and Evolution of Celestial Objects under grant OP201503.