HUGE EJECTION OF ANTIELECTRON NEUTRINOS FROM MASSIVE ACCRETION DISKS

In the widely believed model of gamma-ray bursts (GRBs), the high-energy emission arises in the relativistic jets ejected from a central engine (Piran 2005). The formation of a relativistic jet has been investigated in two ways. The first way is owing to magnetic field. The rotating magnetized compact objects with matter accretion can generate strong toroidal magnetic fields driving highly magnetized plasmas into relativistic jets (Blandford & Znajek 1977; Balbus & Hawley 1998; Proga et al. 2003; Shibata et al. 2008; McKinney & Blandford 2009; Komissarov et al. 2009). Though of significant concern is that a strong toroidal field in the jets is unstable to the non-axisymmetric helical kink mode leading to rapid disruption (Bateman 1978), McKinney & Blandford (2009) had found that the supply of baryonic matter into a jet stabilizes the magnetohydrodynamical jet and constructs a long jet with a narrow angle. It is a subject in the origin of a relativistic jet by magnetic field that the energy density in front of an expanding jet can be produced superior to the rest-mass density.

The other formation mechanism of a jet is owing to neutrinos. The $\nu \bar{\nu }$-annihilation near accreting black holes can raise the energy deposition in the close vicinity of the black holes by the reactions $\nu + \bar{\nu } \rightarrow e^+ + e^- \rightarrow \gamma + \gamma$. The resulting e⁺e⁻-pair plasma–photon fireball can power a relativistic outflow of baryons (Cooperstein et al. 1986; Goodman et al. 1987; Jaroszynski 1993, 1996; Ruffert & Janka 1999; Salmonson & Wilson 1999; Asano & Fukuyama 2000; Madau & Thompson 2000; Miller et al. 2003; Belorborodov 2003; Birkl et al. 2007). However, it has been shown in the model of GRBs that most of the neutrinos are trapped in accreting matter falling into a black hole (Di Matteo et al. 2002; Chen & Belobororodov 2007; Kawanaka & Mineshige 2007). Nevertheless, it should be recognized first that the interaction of heavy nuclei with ambient free nucleons, which has been ignored in previous works, produces rich free neutrons and rare free protons in high dense matter. Second, the simplified assumption in previous works, the detailed balance in neutrino reactions, restricts the variations of thermodynamic quantities, ρ and T, and makes a particular thermal history of accreting matter. The rapidly transient matter in the massive accretion hardly satisfies the detailed balance. Third, the previous models of the massive accretion are based on the advection dominant flow (ADAF) and then the heating or cooling by advection is efficient while that is inefficient in standard accretion disk (STAD). The strong compression by the rapid deceleration of the flow at the middle region of the accretion disk largely heats up the accreting matter to be in the state of free nucleons. The gradual heating in STAD maintains the accreting matter in the domain of heavy nuclei. These different aspects in the massive accretion from previous works will produce the new thermal history of accreting matter and drive the neutrino ejection from the disk as a source of relativistic jets.

The massive accretion disk with the reactions of neutrinos has been studied in a number of works (Popham et al. 1999; Narayan et al. 2001; Kohri & Mineshige 2002; Kohri et al. 2005; Di Matteo et al. 2002; Lee et al. 2005; Chen & Belobororodov 2007; Kawanaka & Mineshige 2007). These models of accretion are based on the ADAF. At the outer region, r ⩾ 10²r_g, the main composition of the matter is the heavy nucleus and the fraction of free nucleons is small where the viscously heated fluid is cooled mainly by the rapid advection. At the middle region, 10r_g < r < 10²r_g, the accreting fluid is in turn heated mainly by the rapid compression. However, the magnetohydrodynamics of the accreting flow with self-consistent viscosity cannot behave like drastic radial motion (De Villiers & Hawley 2003; De Villiers et al. 2003, 2005; Hirose et al. 2004; Krolik et al. 2005), and then the significant cooling or heating by advection does not seem to be expected.

We consider here the massive accretion disk with Keplerian angular momentum, i.e., a STAD model proposed by Shakura & Sunyaev (1973) and Novikov & Thorne (1973) where the accretion disk is efficiently cooled by photons or neutrinos. The formation of collimated jets requires pre-collapsed matter of a massive star with sufficient angular momentum (Woosley & Janka 2005). The initial pre-collapsed matter with the freely falling time $t_{\rm{ff}}$ being less than the duration of γ-ray emission, t_γ ≈ 10 s, has the density ρ ⩾ G/t²_γ ≈ 10⁵ g cm⁻³. When the pre-collapsed matter with density ρ = 10⁵ g cm⁻³ has the angular momentum J ⩾ 10⁻¹J_K, at the radius r = 10⁹ cm, the accretion disk with Keplerian angular momentum may be formed at the radius r ⩾ 10⁷ cm, where J_K is the Keplerian angular momentum. The thermal history of accreting matter in ADAF is dissimilar to that in STAD. As a result, the neutrinos generated in ADAF are almost absorbed by rich nucleons heated by advection. The STAD holds the heat balance between the non-violent heating by viscosity and the neutrino cooling, and then maintains the temperature to be relatively cold over the wide radial range of the flow. The proton-poor disk with high density ejects huge antielectron neutrinos. It is shown here that the antielectron neutrino is a candidate for the source of relativistic jets.

In the previous works of ADAF, the free nucleons have been treated to be independent of heavy nuclei even in the dominant component of heavy nuclei, X_f < 0.5, where X_f is the mass fraction of the free nucleon X_f(ρ, T) (Qian & Woosley 1996; Meyer 1994). We investigate here the neutrino reactions to the evaporated free nucleons interacting with heavy nuclei, where the chemical potentials of neutrons and protons, μ_n and μ_p, are equal in the two phases of evaporated free nucleon gas and the condensed heavy nucleus (Lamb et al. 1981; Lattimer & Swesty 1991). It is shown that the accretion disk at all the locations in the region r_in ≈ r_g < r ⩽ 10²r_g, is cooled mainly by antielectron neutrinos, where r_in is the inner boundary radius of the viscously heating disk. The massively accreting matter in STAD is in the domain of heavy nuclei all over the flow accreting onto a black hole. Therefore, the emitting rates of neutrinos should be precisely expressed in the existence of heavy nuclei.

The reactions of leptons in the previous works of ADAF have been assumed to hold the detailed balances (Belorborodov 2003). In the neutrino-transparent matter, the rates of e⁻ and e⁺ captures by nucleons have been set to be equal, $\dot{n}_{e^-p}= \dot{n}_{e^++n}$. In the neutrino-opaque matter, the detailed balances have been now assumed to hold e⁻ + p ↔ n + ν_e and $e^+ + n \leftrightarrow p + \bar{\nu }_e$. These conditions of the detailed balance restrict the variations of ρ, T, and Y_e along the flow. As is presented here, these simplified assumptions of detailed balance are not adequate for the massive accretion with degenerated electrons. Though the reactions of electrons have been treated in the approximation of small degeneracy, η_e = μ_e/kT < 1 (Belorborodov 2003), we treat it in arbitrary degeneracy η_e since the electrons are degenerated in the massive accretion, η_e>4. We solve the lepton reactions by direct methods without equilibrium conditions.

The large amount of energy deposition by neutrinos could produce a relativistic jet. Qian & Woosley (1996) and Herant et al. (1994) had developed the explosion mechanism of a supernova driven by the neutrino heating. Qian & Woosley (1996) had shown that the mass outflow, "neutrino-driven wind," is produced when the sufficient flux of neutrinos, L_ν ⩾ 10⁵² erg s⁻¹, with high mean energy, $\bar{E}_{\nu } \ge$ 20 MeV, supplied from a newly born neutron star. If the number density of $\bar{\nu }_e$ is sufficiently large in the massive accretion disk, the degeneracy of $\bar{\nu }_e$, $\eta _{{\bar{\nu }}_e} = \mu _{{\bar{\nu }}_e}/{\rm k}T$, becomes large. The antielectron neutrinos with the high mean energy of $\bar{\nu }_e$ in comparison with the thermal energy of accreting matter, $\bar{E}_{{\bar{\nu }}_e} \gg {\rm k}T$, will be ejected from the accretion disk. When the outflow produced by $\bar{\nu }_e$ becomes a one-dimensional flow along the rotational axis of the disk, the required luminosity and mean energy of neutrinos may be L_ν < 10⁵² erg s⁻¹ and $\bar{E}_\nu <$20 MeV, since the decreasing rate of entropy density in the one-dimensional flow is less in comparison with that of the radial flow. The one-dimensional outflow driven by neutrinos may form a narrow beam since the rapid expansion of the flow due to the neutrinos acceleration enforces the larger reduction of pressure of the flow than that of the surrounding matter (Yokosawa 1982). We investigate here the luminosity, L_ν, and the mean energy, $\bar{E}_\nu$, of neutrinos ejected from a massive accretion disk.

The paper is organized as follows. In Section 2, the microphysics of neutrinos is described. The emissivity and the particle flux of each flavor of neutrinos are calculated on the basis of the stationary structure of the massive accretion disk. In Section 3, the relativistic model of an accretion disk is presented. The density and temperature are determined in the heat balance between the viscous heating and the neutrino cooling. In Section 4, the results of the massive accretion disk are shown. The effects of heavy nuclei on the chemical potentials of free nucleons are represented in Section 4.1. The degeneracy of accreting particles is depicted in Section 4.2. The profiles of the composition of accreting matter are given in Section 4.3. The structures of the disk are shown in Section 4.4. The emissivities and opacities of neutrinos are presented in Section 4.5. In Section 4.6, the luminosity and mean energy of neutrinos are given over a wide range of the mass scale and of the angular momentum of a black hole. In Section 5, we discuss the thermal instability of the accretion disk. We summarize the new results in this paper in comparison with previous works in Section 6. In Appendix A, the cross sections of neutrinos, opacities, emissivities, and the reaction rates of neutrinos are summarized. The neutrino transfer in a homogeneous disk and the dynamics of the rotating fluid in Kerr spacetime are presented in Appendices B and C.

The massive accretion disk producing GRBs has the state of matter with ρ = 10⁹ ∼ 10¹⁴ g cm⁻³ and T = 10¹⁰ ∼ 10¹¹ K which is the same state in the iron core collapse forming a neutron star (Bruenn 1985). The nuclear statistical equilibrium abundance of nuclei is nonzero in this state (Meyer 1993, 1994). Lattimer et al. (1985) have derived the thermally equilibrium state of the mixture; nuclei, nucleons, α-particles, electrons, and photons, where the nuclei are assumed to consist of a Wigner–Seitz cell with central nuclei surrounded by a nucleon vapor containing also α-particles. Because of the attractive nuclear forces, the nuclear component condenses into nuclei. Outside of these, we have a low-density "gas" of nucleons and α-particles. Thus, we have a two-phase system; the chemical potential μ must be continuous at the boundary between the two phases, both for neutrons and protons. Lattimer & Swesty (1991) have presented the chemical potential difference, $\hat{\mu }= \mu _n - \mu _p$, as a function of the inputs (ρ, T, Y_e). The number ratio of protons to neutrons in the vapor gas phase is expressed by

$$\begin{equation} \frac{n_p}{n_n} = e^{-{\hat{\mu }}/kT}. \end{equation} \tag{ 1 }$$

The leptons, e⁻, e⁺, and ν_i, react mainly to protons and neutrons in the vapor gas phase.

The rate of e⁻ capture on protons can be derived from the standard electroweak theory (Tubbs & Schramm 1975; Yueh & Buchler 1976; Bruenn 1985; Burrows 2001):

$$\begin{eqnarray} \dot{n}_{e^-p} &=& G^2 \frac{2^3}{h^3c^2} \big(g_V^2+3g_A^2\big)\left \lbrace \frac{g}{h^3}\int d^3p f_p(\epsilon)(1-f_n(\epsilon))\right \rbrace \nonumber \\ & &\times\left \lbrace \frac{g}{h^3}\int d^3p f_e(\epsilon +Q) (\epsilon +Q)^2 (1-f_{\nu _e}(\epsilon +Q))\right \rbrace,\quad \ \ \end{eqnarray} \tag{ 2 }$$

where the factor G² is the Fermi constant with the value G² = 5.18 × 10⁻⁴⁴ MeV² cm², g_V and g_A are the constants, g_V = 1 and g_A = 1.23, g is the spin degeneracy, f_i() is the Fermi–Dirac occupation function of an i-particle, f_i() = [1 + exp(( − μ_i)/kT)]⁻¹, and Q is Q = (m_n − m_p)c². The number density of the i-Fermi particle is given by

$$\begin{equation} n_i = \frac{g}{h^3}\int d^3p f_i(\epsilon). \end{equation} \tag{ 3 }$$

The energy of an i-particle, , is related to its rest mass, m_i, and momentum, p, as ² = m²_ic⁴ + p²c². The former part of the integral of the distribution function in the above expression (Equation (2)) is rewritten as

$$\begin{eqnarray} &&\frac{g}{h^3} \int d^3p f_p(\epsilon)(1-f_n(\epsilon)) \nonumber \\ &&\quad =\frac{g}{h^3} \int d^3p \frac{ \exp [(\epsilon - \mu _p)/kT)] - \exp [(\epsilon - \mu _n)/kT)]}{ \exp [(\mu _n - \mu _p)/kT)] -1}\nonumber\\ &&\qquad \times f_p(\epsilon)f_n(\epsilon) \nonumber \\ &&\quad =\frac{1}{ \exp [(\mu _n - \mu _p)/kT)] -1} \frac{g}{h^3} \int d^3p [f_n(\epsilon) - f_p(\epsilon) \nonumber \\ && \quad =\frac{n_n - n_p}{ \exp [\hat{\mu }/kT)] -1} = \eta _{pn}.\nonumber\\ \end{eqnarray} \tag{ 4 }$$

The quantity η_pn is the dynamic structure factor for the electron–proton interaction and is equal to n_p in the nondegenerate regime of a proton. The capture rate of e⁺ on neutrons is similarly expressed by

$$\begin{eqnarray} \dot{n}_{e^+n} &=& G^2 \frac{2^3}{h^3c^2} \big(g_V^2+3g_A^2\big) \eta _{np} \frac{g}{h^3}\int d^3p f_{e^+}(\epsilon -Q) (\epsilon -Q)^2\nonumber\\ &&\times (1-f_{\bar{\nu }_e}(\epsilon -Q)), \end{eqnarray} \tag{ 5 }$$

where the quantity η_np is

$$\begin{equation} \eta _{np}= \frac{n_p - n_n}{ \exp [-\hat{\mu }/kT)] -1}. \end{equation} \tag{ 6 }$$

The difference of chemical potential $\hat{\mu }=\mu _n -\mu _p$ is positive because all the nuclei concerned have an excess of neutrons. In the high dense and hot matter of the collapse of iron core, the difference is generally large, $\hat{\eta }\equiv \hat{\mu }/kT > 7.5$ (Bethe 1990). The fraction of protons in the gas phase becomes $Y_p =Y_ne^{-\hat{\mu }/kT} \approx e^{-\hat{\mu }/kT} < 5\times 10^{-4}$. The creation rate of electron neutrinos ν_e by the reaction e + p → ν_e + p is slow, and the absorption rate of $\bar{\nu }_e$ by $\bar{\nu }_e + p \rightarrow e^+ + n$ becomes little in the high dense and hot matter of the core collapse. When the core is slowly collapsed owing to the centrifugal force, the existence of nuclei is important for the reaction of neutrinos.

We adopt a simplified, analytical equation of state (EOS) for hot, dense matter given by Lattimer & Swesty (1991), in which the relevant thermodynamic quantities, i.e., n_n,  n_p,  n_α,  μ_n − μ_p,  A,  Z,  P_b, are represented as a function of the inputs (ρ, T, Y_e), where n_n,  n_p,  n_α are the number densities of the free neutron, free proton, and α-particle, μ_n − μ_p is the difference of chemical potentials between the free neutron and proton, A and Z are the mean mass number and proton number of heavy nuclei. We have constructed a three-dimensional table of these thermodynamic quantities as a function of the inputs (ρ, T, Y_e). One of the contour maps of the differential chemical potential, μ_n − μ_p, for Y_e = 0.3, is represented in Figure 1.

Zoom In Zoom Out Reset image size

The dominant reactions of neutrinos in the massive accretion are presented in Table 1. The cross sections of these neutrino reactions have been studied by Tubbs & Schramm (1975), Bruenn (1985), and Burrows (2001), which are listed in Table 2 (Appendix A). The main reactions of leptons are the electron and positron captures on nucleons:

$$\begin{equation} p+e \rightarrow n+\nu, \qquad n+e^+ \rightarrow p+\bar{\nu }_e. \end{equation} \tag{ 7 }$$

When the huge electron and positron pairs exist, n_± ⩾ n_e, the detailed equilibrium of e⁻ and e⁺ capture on nucleons may be established:

$$\begin{equation} \dot{n}_{e^- p} = \dot{n}_{e^+ n}. \end{equation} \tag{ 8 }$$

Then, four chemical potentials hold the equilibrium condition:

$$\begin{equation} \mu _{e^-} + \mu _p = \mu _{e^+}+\mu _n. \end{equation} \tag{ 9 }$$

In the previous works of neutrino reactions, the detailed balance (Equation (8)) is assumed in the neutrino-transparent matter and the detailed balances, e⁻ + p ↔ n + ν_e and $e^+ + n \leftrightarrow p + \bar{\nu }_e$, are assumed in the neutrino-transparent matter (Belorborodov 2003). However, in the high dense matter with degenerated electrons, the pair creation of e⁻ and e⁺ in the electromagnetic thermal bath is suppressed by the charge neutrality, which produces the condition n_± ≪ n_e. Thus, it is difficult for insignificant positrons to establish the detailed balances of e⁻ and e⁺ captures on nucleons.

Table 1. Neutrino Reactions

No.	Reaction Type	Reaction Process	References
1	e + p ⇌ νe + n	β-process	Burrows (2001)
2	$e^+ + n \rightleftharpoons \bar{\nu }_e + p$	Positron capture on neutron	Burrows (2001)
3	e + A ⇌ νe + A'	Electron capture on nuclei	Bruenn (1985)
4	νi + N → νi + N	Neutrino–nucleon scattering	Burrows (2001)
5	νi + A → νi + A	Neutrino–nucleus scattering	Bruenn (1985)
6	νi + e → νi + e	Neutrino–electron scattering	Burrows (2001)
7	$e+e^+\rightleftharpoons \nu _i+\bar{\nu }_i$	Electron pair process	Burrows (2001)
8	$N+N \rightleftharpoons N+N+\nu _i+\bar{\nu }_i$	Nucleon–nucleon bremsstrahlung	Raffelt (2001)
9	n + n → n + p + e + νe	URCA process	Shapiro & Teukolsky (1983)
10	$\gamma \rightarrow \nu _i + \bar{\nu }_i$	Plasma decay	Ruffert et al. (1996)

Download table as:  ASCIITypeset image

Table 2. Cross Section

No.	Neutrino Reaction	Cross Section
1	νe + n → e + p	$\sigma _{\nu _e, n}^a(\varepsilon _{\nu _e}) = \sigma _0\big(1+3g_A^2\big) \big(\frac{\varepsilon _{\nu _e}+Q}{2m_e c^2}\big)^2\big[1- \big (\frac{m_e c^2}{\varepsilon _{\nu _e}+Q}\big)^2\big]^{1/2}W_M$
2	$\bar{\nu }_e + p \rightarrow e^+ + n$	$\sigma _{\bar{\nu }_e, p}^a(\varepsilon _{\bar{\nu }_e}) = \sigma _0\big(1+3g_A^2\big) \big (\frac{\varepsilon _{\bar{\nu }_e}-Q}{2m_e c^2} \big)^2 \big[1-\big (\frac{m_e c^2}{\varepsilon _{\bar{\nu }_e}-Q}\big)^2 \big]^{1/2} W_{\bar{M}}$
3	νi + p → νi + p	$\sigma _{p}^s(\varepsilon _{\nu _i}) = \sigma _0 \big (\frac{\varepsilon _{\nu _i}}{2m_e c^2}\big)^2 \big[4\sin ^4\theta _W-2\sin ^2\theta _W+ \frac{1+3g_A^2)}{4}\big]$
4	νi + n → νi + n	$\sigma _{n}^s(\varepsilon _{\nu _i}) = \sigma _0 \big (\frac{\varepsilon _{\nu _i}}{2m_e c^2}\big)^2\big (\frac{1+3g_A^2)}{4}\big)$
5	νi + e → νi + e	$\sigma _e = \sigma _0\frac{1}{8}\big [(C_V+C_A)^2 + \frac{1}{3}(C_V-C_A)^2\big] \big(\frac{\varepsilon _{\nu _i}}{m_ec^2}+\frac{1}{2} \big)$

Notes. These cross sections are given by Burrows (2001). A convenient reference neutrino cross section is $\sigma _0 = \frac{4G_F^2(m_ec^2)^2}{\pi (h c/2\pi)^4}\simeq 1.705\times 10^{-44}{\rm cm}^2$. The term, g_A, is the axial-vector coupling constant (∼−1.26), θ_W is the Weinberg angle, and sin²θ_W ≃ 0.23, Q = m_nc² − m_pc² = 1.29332 MeV, and for a collision in which the electron gets all of the kinetic energy, $\varepsilon _{e^-}=\varepsilon _{\nu _e}+Q$. W_M is the weak magnetism/recoil correction and is approximately equal to ($1+1.1\varepsilon _{\nu _e}/m_nc^2$). C_V = 1/2 + 2sin²θ_W for ν_e and $\bar{\nu }_e$, C_V = −1/2 + 2sin²θ_W for ν_x and $\bar{\nu }_x$, C_A = +1/2 for $\nu _e,\, \bar{\nu }_\mu$, and $\bar{\nu }_\tau$, and C_A = −1/2 for $\bar{\nu }_e,\, {\nu }_\mu$, and ν_τ.

Download table as:  ASCIITypeset image

We shall evaluate the above detailed balances in the massive accretion. The number density of a pair of e⁻ and e⁺ in equilibrium with thermal photons is generally expressed as

$$\begin{equation} n_{e^{\pm }\mbox{th}} = \frac{3\zeta (3)}{2\pi ^2} \Bigl (\frac{kT}{\hbar c} \Bigr)^3 \langle 1-f(\eta _e)\rangle \langle 1-f(\eta _{e^+})\rangle, \end{equation} \tag{ 10 }$$

where the brackets indicate the mean values of the blocking factors in phase space and ζ(3) is the ζ function, ζ(3) = 1.202. The mean blocking is approximately represented with the mean energy of e⁻ and e⁺, $\bar{\epsilon }_{\pm }=7\pi ^4kT/180\zeta (3)$. Since the mean degenerate factor of electrons is taken in our massive accretion as $\bar{\eta }_e \approx 7$, the blocking factors are

$$\begin{eqnarray} \langle 1-f(\eta _e)\rangle & \approx & 1-\frac{1}{e^{(\bar{\epsilon } /kT)-\bar{\eta }_e}+1} \nonumber\\ &=& 2.08\times 10^{-2}, \qquad \langle 1-f(\eta _{e^+})\rangle \approx 1.\qquad \end{eqnarray} \tag{ 11 }$$

The number ratio of the pair to mean electron is

$$\begin{equation} \frac{n_{\pm }}{Y_e n_b}\approx 2.5 T^3_{11}\rho ^{-1}_{10}Y^{-1}_e\langle 1-f(\eta _e)\rangle. \end{equation} \tag{ 12 }$$

For the typical case of massive accretion, T = 3 × 10¹⁰ K, ρ = 10¹² g cm⁻³, and Y_e = 0.1, the pair of e⁻ and e⁺ is infinitesimal:

$$\begin{equation} \frac{n_{\pm }}{Y_e n_b}\approx 10^{-4}. \end{equation} \tag{ 13 }$$

The ratio of the capture rates, $\dot{n}_\beta /\dot{n}_{\bar{\beta }}$, is simply expressed by Equation (2)/Equation (5), and is evaluated by the typical quantities of the massive accretion in STAD, Y_p ≈ 10⁻⁴, Y_e ≈ 10⁻¹, Y_n ≈ 0.8, and $Y_{e^+} \approx 10^{-3}$, as

$$\begin{equation} \frac{\dot{n}_{e^-p}}{\dot{n}_{e^+n}} \approx \frac{\eta _{pn}n_e}{\eta _{np}n_{e^+}}\approx \frac{Y_pY_e}{Y_nY_{e^+}} \approx 10^{-2}. \end{equation} \tag{ 14 }$$

The reaction rate $n+e^+ \rightarrow p+\bar{\nu }_e$ is extremely frequent in comparison with p + e⁻ → n + ν_e.

In the neutrino-opaque matter, the detailed balance, ν_e + n ⇌ e⁻ + p, is also evaluated as follows. The difference in the reaction rate of the β-process is expressed as

$$\begin{eqnarray} &&\delta {\dot{n}}_\beta = \dot{n}_{e^- p} - \dot{n}_{\nu _e n}\qquad\qquad\qquad\qquad\qquad\quad\qquad \ \ \ \end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray} &=& \sigma ^a_{\nu _e, n}(\epsilon) \frac{gc}{h^3}\int d^3p[\eta _{pn} f_{e^-}(\epsilon +Q) (1-f_{\nu _e}(\epsilon +Q))\nonumber\\ && - \eta _{pn} f_{\nu _e}(\epsilon +Q) (1-f_{e}(\epsilon +Q))], \end{eqnarray} \tag{ 16 }$$

where $\sigma ^a_{\nu _e, n}(\epsilon)$ is the cross section of ν_e to n, ν_e + n → e⁻ + p (Burrows 2001),

$$\begin{equation} \sigma ^a_{\nu _e, n}(\epsilon) = G^2 \frac{2^3}{h^3c^2} \big(g_V^2+3g_A^2\big) (\epsilon +Q)^2. \end{equation} \tag{ 17 }$$

The ratio of the reaction rates is

$$\begin{equation} \frac{\dot{n}_{e^- p}}{\dot{n}_{\nu _e n}} \approx \frac{Y_{p}Y_e}{Y_{n}Y_{\nu _e}\langle 1-f_e(\epsilon +Q)\rangle } \approx 10^{2}. \end{equation} \tag{ 18 }$$

In the neutrino-opaque case of the typical massive accretion, the detailed balances, ν_e + n ⇌ e⁻ + p, are not established.

Thus, the changing rate of each lepton number in unit volume, $\dot{n}_l$, is directly determined through the following networks of reactions without equilibrium approximations:

$$\begin{eqnarray} \dot{n}_e &=& -\dot{n}_\beta +\dot{n}_{\bar{\beta }}-\dot{n}_A +\dot{n}_{\rm{URCA}}^+, \end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray} \dot{n}_{e^+} &=& -\dot{n}_{\bar{\beta }} -\dot{n}_{e e}, \end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray} \dot{n}_{\nu _e} &=& \dot{n}_\beta + \dot{n}_A + \dot{n}_{ee} +\dot{n}_{nn} +\dot{n}_{\rm{URCA}}^+ + \dot{n}_\gamma ^+, \end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray} \dot{n}_{\bar{\nu }_e} &=& \dot{n}_{\bar{\beta }} + \dot{n}_{ee} +\dot{n}_{nn} + \dot{n}_\gamma ^+, \end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray} \dot{n}_{\nu _x} &=& \dot{n}_{ee} +\dot{n}_{nn} + \dot{n}_\gamma ^+. \end{eqnarray} \tag{ 23 }$$

Here $\dot{n}_e,\, \dot{n}_{e^+},\, \dot{n}_{\nu _e},\, \dot{n}_{\bar{\nu }_e}$, and $\dot{n}_{\nu _x}$ are the net production rates of electrons, positrons, electron neutrinos, antielectron neutrinos, and heavy neutrinos. The heavy neutrino, ν_x, indicates the flavor of a heavy neutrino, ν_x = ν_μ, ν_τ. The changing rates, $\dot{n}_\beta,\, \dot{n}_{\bar{\beta }},\, \dot{n}_{ee},\, \dot{n}_A$, and $\dot{n}_{nn}$, are the net production rates of neutrinos by the following reactions; the β-process, e + p ⇌ ν_e + n, the positron capture on a neutron, $e^+ + n \rightleftharpoons \bar{\nu }_e + p$, the electron pair annihilation, $e+e^+\rightleftharpoons \nu _i+\bar{\nu }_i$, where ν_i and $\bar{\nu }_i$ mean the flavors of the neutrinos and their antineutrinos, i.e., ν_i = ν_e,  ν_μ,  ν_τ and $\bar{\nu }_i$ = $\bar{\nu }_e,\, \bar{\nu }_\mu,\, \bar{\nu }_\tau$, the electron capture on a nucleus, e + A ⇌ ν_e + A', and the nucleon–nucleon bremsstrahlung, $N+N \rightleftharpoons N+N+\nu _i+ \bar{\nu }_i$. The plus sign at the upper suffix denotes the production rates of neutrinos without inverse reactions of neutrinos, i.e., $\dot{n}_{\rm{URCA}}^+$ and $\dot{n}_\gamma ^+$ are the production rates by the URCA process, N + N → n + p + e + ν_e, and by the plasma decay, $\gamma \rightarrow \nu _i + \bar{\nu }_i$. The references and the numerical values of above reaction rates are summarized in Tables 1 and 5 (Appendix A).

The reactions of leptons are evaluated in the simplified disk with the thickness H(r) in which the matter is assumed to be homogeneously distributed in the vertical direction. In the stationary accretion disk, the time variations of lepton, $\dot{n}_l(=d{n}_l/d t)$, are calculated in the shell comoving with the accreting flow such as $\dot{n}_{l}=v^r\nabla _r n_{l}$. The changing rates of leptons in the unit number of baryon for each reaction of neutrinos, $\dot{Y}_l = \dot{n}_l /n_b$, are expressed in Table 5 (Appendix A). When the accreting matter moves along the distance, $\Delta r \approx 0.1r_{\rm{ms}} \approx 10^5(M_{\rm BH}/3\,{M}_\odot)$(cm) with the accreting velocity, v_accrt ≈ 10⁻³c ≈ 10⁷ (cm s⁻¹), where $r_{\rm{ms}}$ is the radius of a marginally stable orbit, the crossing time is Δt = Δr/v_accrt ≈ 10 (ms). In this time, the reactions of leptons are not always reach to the thermodynamic equilibrium. We expressively integrate the changing rate of leptons in time.

The change of the neutrino density is also brought about by the leakage of neutrinos from the disk. The transfer of neutrinos in a plane-parallel disk is solved in Appendix B. The number flux density of each neutrino, ν_i, at the surface of the disk, $F_{n(\nu _i)}$, is expressed with the production rate of the neutrino, $\dot{n}_{\nu _i}^+$, the absorbing optical depth, $\tau _{\nu _i a}$, the scattering optical depth, $\tau _{\nu _i s}$, and the total optical depth, $\tau _{\nu _i}=\tau _{\nu _i a}+\tau _{\nu _i s}$, as

$$\begin{equation} F_{n(\nu _i)}=\frac{\dot{n}^+_{\nu _i}H}{1.5\tau _{\nu _i a}\tau _{\nu _i}+\sqrt{3}\tau _{\nu _i a} + 1}. \end{equation} \tag{ 24 }$$

The absorbing opacity, $\kappa _{\nu _i, a}$, and scattering opacities, $\kappa _{\nu _i, s}$, of each flavor of neutrinos, ν_i, are described as

$$\begin{eqnarray} \kappa _{\nu _e a} &=& \kappa ^a_{\beta } + \kappa ^a_{A} + \kappa ^a_{nn}, \quad \kappa _{\bar{\nu }_e a} = \kappa ^a_{\bar{\beta }} + \kappa ^a_{nn}, \quad \kappa _{{\nu }_x a} = \kappa ^a_{nn},\qquad \end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray} \kappa _{\nu _i s} &=& \kappa _n^s + \kappa _p^s + \kappa _A^s + \kappa _e^s, \end{eqnarray} \tag{ 26 }$$

where $\kappa _{{\nu }_x a}$ is the absorbing opacity of the heavy neutrino, ν_x = ν_μ, ν_τ; $\kappa ^a_{\beta },\, \kappa ^a_{\bar{\beta }},\, \kappa ^a_{A}$, and κ^a_nn are the absorbing opacities by the inverse β-process, antielectron neutrino capture on a proton, electron neutrino capture on a nucleus, and by the inverse nucleon–nucleon bremsstrahlung. The opacities, κ^s_n,  κ^s_p,  κ^s_A, and κ^s_e, indicate the scattering opacities of a neutrino, ν_i, with neutrons, protons, nuclei, and electrons. The explicit expressions of these opacities are represented in Table 3 (Appendix A).

Table 3. Opacity

No.	Neutrino Reaction	Opacity	Reference
1	νe + n → e + p	$\kappa ^a_\beta = \eta _{np}\frac{\int d^3p f_{\nu _e}(\varepsilon) \sigma _{\nu _e, n}^a(\varepsilon)[1-f_{e}(\varepsilon _{\nu _e}+Q)]}{\int d^3p f_{\nu _e}(\varepsilon)}$	Burrows (2001)
		$\approx 4.3\times 10^{-6}({\rm cm}^{-1})\rho _{12}Y_{np}T_{11}^2\frac{F_{4}^+(\eta _{\nu _e},\eta _e,q)}{F_2(\eta _{\nu _e})}$
2	$\bar{\nu }_e + p \rightarrow e^+ + n$	$\kappa ^a_{\bar{\beta }}\approx 4.3\times 10^{-6}({\rm cm}^{-1})\rho _{12}Y_{pn}T_{11}^2\frac{F_{4}^+(\eta _{\bar{\nu }_e},\eta _{e^+},-q)}{F_2(\eta _{\bar{\nu }_e})}$	Burrows (2001)
3	$\nu _i+\bar{\nu }_i + N+N \rightarrow N+N$	$\kappa ^a_{NN}\approx \frac{3C_A^2G_F^2n_bT^5\Gamma }{\pi ^2}\frac{2}{5\varepsilon _\nu }$	Raffelt (2001)
		$\approx 1.43\times 10^{-9}({\rm cm}^{-1})(X_n\rho _{12})^2T^2_{11}\frac{F_2(\eta _{\nu _i})}{F_3(\eta _{\nu _i})}\big (\frac{3}{2+0.862T_{11}}\big)^{1/2}$
4	νe + A → e + A'	$\kappa ^a_A = \frac{\alpha ^2}{4}\sigma _0 n_A e^{\eta _n -\eta _p -q^{\prime }} \frac{2}{7} N_p(Z)N_h(N)$
		$\big (\frac{kT}{m_e c^2} \big)^2 \frac{F_4^{q^{\prime }-}(\eta _{\nu _e},\eta _e)}{F_2(\eta _{\nu _e})}$	Bruenn (1985)
		≈1.2 × 10−6(cm−1)ρ12YAT211fA(η)
5	νi + N → νi + N	$\kappa ^s_N \approx 1.1\times 10^{-6}({\rm cm}^{-1})\rho _{12}T_{11}^2C_{s,N}Y_{NN} \frac{F_4(\eta _{\nu _i})}{F_2(\eta _{\nu _i})}$	Burrows (2001)
6	νi + e → νi + e	$\kappa ^s_e \approx 1.66\times 10^{-6}({\rm cm}^{-1}) Y_e\rho _{12} T_{11}^2 \frac{F_4(\eta _{\nu _i})}{F_2(\eta _{\nu _i})}$	Burrows (2001)
7	νi + A → νi + A	$\kappa ^s_A=\frac{a_0^2}{6} \sigma _0 n_A A^2 \big (\frac{kT}{m_ec^2} \big)^2 \frac{F_4(\eta _{\nu _i})}{F_2(\eta _{\nu _i})}$	Bruenn (1985)
		$\approx 2.3\times 10^{-7}({\rm cm}^{-1}) \rho _{12} T_{11}^2 A^2 Y_A \frac{F_4(\eta _{\nu _i})}{F_2(\eta _{\nu _i})}$

Notes. The absorption opacity, κ_a, and scattering opacity, κ_s, are expressed with unit cm⁻¹. F_k(η) is the kth Fermi integral, F_k(η) ≡ ∫^∞₀x^k(1 + exp(x − η))⁻¹dx. The complex integrals, $F_{4}^+(\eta _{\nu _e},\eta _e,q)$ and $F_4^{q^{\prime }-}(\eta _{\nu _e},\eta _e)$, are shown in Table 6. The doubly indexed quantity, Y_np, is the number fraction defined as Y_np = η_np/n_b, and Y_pn is Y_pn = exp(η_p − η_n)Y_np, where η_np is the dynamic structure factor for neutrino–nucleon interaction, $\eta _{np} = \int \frac{2d^3p}{(2\pi)^3}f_n(\varepsilon)[1-f_p(\varepsilon)]=\frac{n_p-n_n}{\exp {(\eta _p -\eta _n)} -1}$. Y_NN is $Y_{NN}=\frac{Y_N}{1+\frac{2}{3}\max {(\eta _N, 0)}}$, and C_s,N is C_s,N = 1 if N = n and C_s,N = 0.827 if N = p. The factor, G²_F, is the Fermi constant. Γ is the spin relaxation rate. A is the mean mass number of heavy nuclei. n_A is the number density of heavy nuclei, and Y_A is the fraction, Y_A = n_A/n_b. X_n is the mass fraction of neutrons. f_A(η) is $f_A(\eta)=e^{\eta _n -\eta _p -q^{\prime }} \frac{2}{7} N_p(Z)N_h(N) \frac{F_4^{q^{\prime }+}(\eta _{\nu _e},\eta _{e})}{F_2(\eta _{\nu _e})}$, where $q^{\prime }\approx \eta _n-\eta _p+\Delta, \quad \Delta =\frac{3\,{\rm MeV}}{kT}$. N_p(Z) and N_h(N) are N_p(Z) = {0 for Z < 20;  Z − 20 for 20 < Z < 28;  8 for Z>28} and N_h(N) = {6 for N < 34;  40 − N for 34 < N < 40;  0 for N>40}. The density ρ₁₂ and temperature T₁₁ are normalized values by 10¹² (g cm⁻³) and 10¹¹ K.

Download table as:  ASCIITypeset image

Thus, the fraction change of each flavor of neutrinos, $\dot{Y}_{\nu _i}$, and those of electrons and positrons, $\dot{Y}_{e}$, are expressed by

$$\begin{eqnarray} &&\dot{Y}_{\nu _i} = n_b^{-1}(v^r\nabla _r n_{\nu _i}) = n_b^{-1}\Bigl (\dot{n}_{\nu _i} - \frac{F_{n(\nu _i)}}{H} \Bigr), \end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray} &&\nonumber \dot{Y}_{e} = n_b^{-1}(\dot{n}_e + v^r\nabla _r {n_{e^- \rm{th}}}), \qquad \dot{Y}_{e^+} = n_b^{-1}(\dot{n}_{e^+} + v^r\nabla _r {n_{e^+ \rm{th}}}). \end{eqnarray}$$

The emissivity of each flavor of neutrinos is described as

$$\begin{eqnarray} q_{\nu _e} &=& q_\beta + q_{ee} + q_A + q_{nn} + q_{\rm{URCA}} + q_\gamma, \end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray} q_{\bar{\nu }_e} &=& q_{\bar{\beta }} + q_{ee} + q_{nn} + q_\gamma, \end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray} q_{\nu _x} &=& q_{ee} + q_{nn} + q_{\gamma }, \end{eqnarray} \tag{ 30 }$$

where $q_\beta, q_{\bar{\beta }}, q_{ee}, q_A, q_{nn}, q_{\rm{URCA}}$, and q_γ are emissivities by the β-process, the positron capture on neutrons, the electron pair annihilation, the electron capture on nuclei, the nucleon–nucleon bremsstrahlung, the URCA process, and by the plasma decay. These explicit emissivities are expressed in Table 4 (Appendix A). The energy flux density of each flavor of neutrinos at the surface of the disk, $F_{\varepsilon (\nu _i)}$, is then given by

$$\begin{equation} F_{\varepsilon (\nu _i)}=\frac{q_{\nu _i}H}{1.5\tau _{\nu _i a}\tau _{\nu _i} + \sqrt{3}\tau _{\nu _i a}+1}. \end{equation} \tag{ 31 }$$

Table 4. Emissivity

No.	Neutrino Reaction	Emission Rate	References
1	e + p → νe + n	$q_{\beta }(\nu _e)= \eta _{pn} \frac{g c}{h^3}\int _0^\infty d^3p \ \varepsilon f_{e}(\varepsilon +Q) \sigma _{\nu _e, n}^a(\varepsilon)$	Burrows (2001)
		$[1-f_{\nu _e}(\varepsilon)]$
		$\approx 7.4\times 10^{33} \rho _{12} T_{11}^6 Y_{pn} F_{5}^- (\eta _{\nu _e}, \eta _e,q)$
2	$e^+ + n \rightarrow \bar{\nu }_e + p$	$q_{\bar{\beta }}(\bar{\nu }_e) \approx 7.4\times 10^{33} \rho _{12}T_{11}^6Y_{np}F_{5^+}^-(\eta _{\bar{\nu }_e},\eta _{e^+},q)$	Burrows (2001)
3	$N+N \rightarrow N+N+\nu _i+\bar{\nu }_i$	$q_{NN}(\nu _i,\bar{\nu }_i) \approx 7.3\times 10^{30} (X_N\rho _{12})^2 T_{11}^{5.5}$	Burrows (2001)
4	e + A → νe + A'	$q_A(\nu _e) = b\sigma _0 \frac{\alpha ^2}{4} N_A \frac{2}{7} N_p(Z) N_h(N)\big(\frac{kT}{m_e c^2} \big)^5$	Bruenn (1985)
		$kT F_5^{q^{\prime }-}(\eta _{\nu _e},\eta _{e})$
		$\approx 2.06\times 10^{33} \rho _{12}Y_A \frac{2}{7} N_p(Z)N_h(N)$
		$T_{11}^6 F_5^{q^{\prime }-}(\eta _{\nu _e},\eta _{e})$
5	n + n → n + p + e + νe	$q_{URCA}(\nu _e)=2.7\times 10^{34} \rho _{12}^{2/3} T_{11}^8 F_Q(\eta _e,\eta _{\nu _e})$	Shapiro & Teukolsky (1983)
6	$\gamma \rightarrow \nu _i + \bar{\nu }_i$	$q_\gamma (\nu _e,\bar{\nu }_e) \approx 1.20\times 10^{33}\;({\rm erg\,cm}^{-3}\,{\rm s}^{-1})$	Ruffert et al. (1996)
		$T^9 F_{Q\gamma }(\eta _{\nu _e})$
		$q_\gamma (\nu _x) \approx 8.3\times 10^{30} T^9 F_{Q\gamma }(\eta _{\nu _x})$	Ruffert et al. (1996)
7	$e+e^+ \rightarrow \nu _i+ \bar{\nu }_i$	$q_{ee}(\nu _e,\bar{\nu }_e) \approx 1.24\times 10^{33} T_{11}^9f_{\nu _e}(\eta)$	Thompson et al. (2000)
		$q_{ee}(\nu _x,\bar{\nu }_x) \approx 5.49 \times 10^{32} T_{11}^9 f_{\nu _x} (\eta)$	Thompson et al. (2000)

Notes. The emission rates, q, are expressed with unit (erg  cm⁻³  s⁻¹). The complex integrals, $F_{5}^- (\eta _{\nu _e}, \eta _e,q),\, F_{5^+}^- (\eta _{\bar{\nu }_e}, \eta _{e^+}, q)$, and $F_5^{q^{\prime }} (\eta _{\nu _e})$, are shown in Table 6, where q is q = Q/kT. F_Q and $f_{\nu _i}$ for i = e,  x, are $F_Q(\eta _e, \eta _{\nu _e})= \langle 1- f_e(\varepsilon _e)\rangle \langle 1-f_{\nu _e} (\varepsilon _{\nu _e})\rangle$, and $f_{\nu _i}(\eta) = ({F_4(\eta _e) F_3(\eta _{e^+})+ F_4(\eta _{e^+}) F_3(\eta _e)}) \langle 1-f_{\nu _i} (\varepsilon)\rangle _{ee} \langle 1- f_{\bar{\nu }_i}\langle _{ee} /{2F _4(0)F_3(0)}$. $F_{Q\gamma } (\eta _{\nu _e})$ is $F_{Q\gamma } (\eta _{\nu _e}) =\gamma ^6 e^{-\gamma } \frac{1}{2} ((1+(\gamma +1)^2) \langle 1- f_{\nu _e} (\varepsilon) \rangle _\gamma \langle 1-f_{\bar{\nu }_e} (\varepsilon) \rangle _\gamma$, where γ and γ₀ are $\gamma \approx \gamma _0 \sqrt{\pi ^2/3 +\eta _e^2}$ and γ₀ = hΩ₀/(2πm_ec²) = 5.565 × 10⁻². The mean blocking factors $\langle 1- f_{\nu _i} (\varepsilon)\rangle _{ee},\ldots$ are described in Table 6.

Download table as:  ASCIITypeset image

Table 5. Fraction Change

No.	Neutrino Reaction	Fraction Change
1	νe + n ⇌ e + p	$\dot{Y}_e(\beta)= \frac{4\pi c}{(h c)^3} \int _0^\infty d^3p \ \sigma _{\nu _e, n}^a(\varepsilon)$
		$[ Y_{np} f_{\nu _e}(\varepsilon) [1-f_{e}(\varepsilon +Q)] - Y_{pn} f_{e}(\varepsilon +Q) [1-f_{\nu _e}(\varepsilon)]]$
		$\approx 9.0\times 10^2\;({\rm s}^{-1}) T_{11}[Y_{np}F_4^+(\eta _{\nu _e},\eta _e,q)-Y_{pn}F_4^-(\eta _{\nu _e},\eta _e,q)]$
2	νe + A ⇌ e + A'	$\dot{Y}_e(A)=b\frac{\alpha ^2}{4}\sigma _0 Y_A \frac{2}{7} N_p(Z) N_h(N)$
		$\big(\frac{kT}{m_e c^2} \big)^5 \big (e^{\eta _n -\eta _p -q^{\prime }} F_4^{q^{\prime }+}(\eta _{\nu _e}, \eta _e) - F_4^{q^{\prime }-}(\eta _{\nu _e}, \eta _e) \big)$
		$\approx 2.4\times 10^{2}\;({\rm s}^{-1}) T_{11}^5 Y_A \frac{2}{7} N_p(Z)N_h(N)$
		$\big(e^{\eta _n -\eta _p -q^{\prime }} F_4^{q^{\prime }+}(\eta _{\nu _e},\eta _e) - F_4^{q^{\prime }-} (\eta _{\nu _e}, \eta _e) \big)$
3	n + n → n + p + e + νe	$\dot{Y}_e(\rm{URCA})\approx \frac{q_{\rm{URCA}}}{\bar{\varepsilon }_{\nu _e} n_b}$
		$\approx 3.3\times 10^{3}\;({\rm s}^{-1})\rho _{12}^{-1/3}T_{11}^7 \frac{F_2(\eta _{\nu _e})}{F_3(\eta _{\nu _e})} \langle 1-f_e(\varepsilon)\rangle _{nn} \langle 1-f_{\nu _e}(\varepsilon)\rangle _{nn}$
4	$\bar{\nu }_e + p \rightleftharpoons e^+ + n$	$\dot{Y}_{e^+}(\bar{\nu }_e p) \approx 9.0\times 10^2\;({\rm s}^{-1}) T_{11}[Y_{pn}F_4^+(\eta _{\bar{\nu }_e},\eta _{e^+},-q)-Y_{np}F_4^-(\eta _{{\bar{\nu }} _e},\eta _{e^+},-q)]$
5	$N+N \rightarrow N+N+\nu _i+\bar{\nu }_i$	$\dot{Y}_{\nu _i}(NN)\approx \frac{q_{\nu \bar{\nu }}(nn)}{2\bar{\varepsilon }_{\nu _i}n_b} \approx 1.3\;({\rm s}^{-1}) X_N^2\rho _{12}T_{11}^5\frac{F_2(\eta _{\nu _i})}{F_3(\eta _{\nu _i})}\big(\frac{3}{ 2+0.862T_{11}}\big)^{1/2}$
6	$e+e^+ \rightleftharpoons \nu _i+\bar{\nu }_i$	$\dot{Y}_{\nu _e}(ee^+)\approx 36.3\;({\rm s}^{-1})T_{11}^8 f_{3,\nu _e}(\eta)$
		$\dot{Y}_{\nu _x}(ee^+) \approx 31.2\;({\rm s}^{-1})T_{11}^8 f_{3,\nu _x}(\eta)$
7	$\gamma \rightarrow \nu _i + \bar{\nu }_i$	$\dot{Y}_{\nu _e}(\gamma)\approx 1.4\times 10^2\;({\rm s}^{-1}) F_\gamma (\eta)\rho _{11}^{-1} T^8_{11}$
		$\dot{Y}_{\nu _x}(\gamma)\approx 1.0\;({\rm s}^{-1}) F_\gamma (\eta)\rho _{11}^{-1} T^8_{11}$

Notes. The parameter b is $b=4\pi \big(\frac{m_ec}{h}\big)^3$. The complex integrals, $F_4^{q^{\prime }+}(\eta _{\nu _e}, \eta _e)$ and $F_4^{q^{\prime }}(\eta _{\nu _e}, \eta _e)$, are shown in Table 6. $f_{3,\nu _i}(\eta)$ and $F_{Q\gamma }(\eta _{\nu _e})$ are $f_{3,\nu _i}(\eta)=(F_3(0))^{-2}\lbrace F_3(\eta _e)F_3(\eta _{e^+}) \langle 1-f_{\nu _i}(\varepsilon)\rangle _{ee}\langle 1-f_{{\bar{\nu }}_i}(\varepsilon) \rangle _{ee} - F_3(\eta _{\nu _i})F_3(\eta _{{\bar{\nu }}_i}) \langle 1-f_{e}(\varepsilon)\rangle _{\nu _i {\bar{\nu }}_i}\langle 1-f_{e^+}(\varepsilon)\rangle _{\nu _i {\bar{\nu }}_i} \rbrace$ and $F_{Q\gamma }(\eta _{\nu _e})=\gamma ^6 e^{-\gamma }\frac{1}{2} ((1+(\gamma +1)^2)\langle 1-f_{\nu _e}(\varepsilon)\rangle _\gamma \langle 1-f_{\bar{\nu }_e}(\varepsilon)\rangle _\gamma$. The mean blocking factors $\langle 1-f_{\bar{\nu }_e}(\varepsilon)\rangle _\gamma,\ldots$ are described in Table 6. X_N is the mass fraction of nucleons.

Download table as:  ASCIITypeset image

Here, the spectra of leptons are assumed to be Fermi–Dirac distributions with the local gas temperature, T. When the density, n_l, and temperature, T, are given, the chemical potential of the lepton, μ_l, is numerically determined through the Fermi integral,

$$\begin{equation} n_l(\mu _l) = 4\pi g \Bigl (\frac{kT}{hc}\Bigr)^3 \int _0^\infty \frac{x^2}{1+\exp (x-\mu _l/kT)} dx, \end{equation} \tag{ 32 }$$

where g is the spin degeneracy.

The ejection rate of neutrinos from the massive accretion disk depends significantly upon the temperature, T, and density, ρ, of the accreting matter. It is important precisely to determine the heat balance of the disk. In order to evaluate the proper heating rate by viscosity in the rotating spacetime, we introduce a set of local observers who rotate with the Keplerian, circular orbits (orbiting frame; Yokosawa & Inui 2005). The local energy conservation is rewritten in the orbiting frame. To avoid the infinite density at the inner boundary of the standard accretion disk proposed by Novikov & Thorne (1973), we resolve the transport equation of the angular momentum with the different boundary condition from the case of Novikov & Thorne (1973). The scale height of the disk and the heating rate by viscosity depend on the pressure and internal energy density of accreting matter with degenerate electrons and heavy nuclei. We shall represent here the stationary massive accretion disk unified in the rotating spacetime.

We introduce the Kerr spacetime and choose units with G = c = 1. The metric of the spacetime in terms of the Boyer–Lindquist time, t, and any arbitrary spatial coordinates, x^j, has the form

$$\begin{equation} ds^2 = -\alpha ^2dt^2 + g_{jk}(dx^j+\beta ^jdt)(dx^k+\beta ^kdt). \end{equation} \tag{ 33 }$$

The metric coefficients, α,  β^j, and g_jk, are the lapse, shift, and three-metric functions. These functions are given by

$$\begin{eqnarray} \alpha &=& \Bigl (\frac{\Sigma \Delta }{A}\Bigr)^{1/2}, \quad g_{rr}=\frac{\Sigma }{\Delta }, \quad g_{\theta \theta }=\Sigma, \quad g_{\varphi \varphi }=\frac{\sin ^2{\theta }A}{\Sigma }, \nonumber \\ \beta ^\varphi &=& {-}\omega = -\frac{2Mar}{A}, \ \ \beta ^j=g_{jk}=0, \ \ {\rm for\ all\ other}\ j{\rm \ and}\ k, \nonumber \end{eqnarray}$$

where M is the mass of a black hole and the functions, Δ,  Σ,  A, are defined by

$$\begin{eqnarray} \Delta &=& r^2 - 2Mr + a^2, \quad \Sigma =r^2+a^2\cos ^2{\theta }, \nonumber\\ A & = & (r^2+a^2)^2-a^2\Delta \sin ^2{\theta }. \nonumber \end{eqnarray}$$

We introduce a set of local observers who rotate with the Keplerian, circular orbits (Yokosawa & Inui 2005). Each observer carries an orthogonal tetrad. For the Keplerian orbiting observer with a coordinate angular velocity of Ω, its world line is r = constant, θ = constant, φ = Ωt + constant. The observer in the locally non-rotating frame (LNRF; Bardeen et al. 1972) who rotates with the angular velocity, ω, measures the linear velocity of a particle moving in a Keplerian orbit as $v_{(\varphi)}=\alpha ^{-1} \sqrt{g_{\varphi \varphi }}(\Omega - \omega)$. Its Lorentz factor is γ = (1 − v²_(φ))^−1/2. We adopt the Keplerian orbiting, observer's frame ("orbiting frame") with the set of its basis vectors:

$$\begin{eqnarray} {\bf e}_{\hat{t}} &=& \gamma e^{-\nu }\left (\frac{\partial }{\partial t}+\Omega \frac{\partial }{\partial \varphi }\right), \nonumber\\ {\bf e}_{\hat{\varphi }} &=& \gamma \left(e^{-\psi } \frac{\partial }{\partial \varphi }+ v_{(\varphi)}e^{-\nu }\left(\frac{\partial }{\partial t}+\omega \frac{\partial }{\partial \varphi }\right)\right), \end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray} {\bf e}_{\hat{r}} &=& e^{-\mu _1}\frac{\partial }{\partial r}, \quad {\bf e}_{\hat{\theta }}= e^{-\mu _2}\frac{\partial }{\partial \theta }, \end{eqnarray} \tag{ 35 }$$

where $e^\nu,\, e^\psi,\, e^{\mu _1},\, e^{\mu _2}$ are defined by $e^\nu =\alpha,\, e^{2\psi }=g_{\varphi \varphi },\, e^{2\mu _1}=g_{rr},\, e^{2\mu _2}=g_{\theta \theta }$, respectively.

The corresponding orthogonal basis of one-forms is

$$\begin{eqnarray} &{\vec{\omega }}^{\hat{t}} = \hat{\alpha }dt - v_{(\varphi)}\gamma e^\psi d\varphi, \quad {\vec{\omega }}^{\hat{\varphi }} = -\Omega \gamma e^\psi dt + \gamma e^\psi d\varphi \nonumber \\ &{\vec{\omega }}^{\hat{r}} = e^{\mu _1}dr, \quad {\vec{\omega }}^{\hat{\theta }} = e^{\mu _2}d\theta, \end{eqnarray} \tag{ 36 }$$

where $\hat{\alpha }$ is defined by $\hat{\alpha }=\alpha \gamma (1+ v_\omega v_{(\varphi)})$.

The shear stress in the Boyer–Lindquist coordinate is defined by

$$\begin{equation} \sigma _{\alpha \beta } \equiv \frac{1}{2}\big(u_{\alpha ; \mu } h^{\mu }_{\beta } + u_{\beta ; \mu } h^{\mu }_\alpha\big) - \frac{1}{3}\theta h_{\alpha \beta }, \end{equation} \tag{ 37 }$$

where θ is the expansion, θ = u^α_α, and h_μν is the projection tensor, h_μν = g_μν + u_μu_ν. The shear stress in the orbiting frame is expressed by

$$\begin{equation} \sigma ^{\hat{r}\hat{\varphi }} = e_\alpha ^{\hat{r}}e_\beta ^{\hat{\varphi }} \sigma ^{\alpha \beta }= e^{\hat{r}}_r \big(e^{\hat{\varphi }}_\varphi \sigma ^{r \varphi } + e^{\hat{\varphi }}_t \sigma ^{r t}\big) = \frac{1}{2}\gamma ^2 e^{-\mu _1}e^{\psi -\nu } \frac{\partial \Omega }{\partial r}, \end{equation} \tag{ 38 }$$

where $e_\alpha ^{\hat{a}}$ is the Boyer–Lindquist components of the basis of one-forms (Equation (36)), ${\vec{\omega }}^{\hat{a}} = e^{\hat{a}}_\alpha dx^\alpha$. The shear stress, $\sigma ^{\hat{r}\hat{\varphi }}$, in the orbiting frame is finite at the inner boundary of the accretion disk, r = r_g, for a = 1 even though the radial difference in the orbiting frame is infinitesimal, $\partial \Omega /\partial X^{\hat{r}}= \partial \Omega /e^{\mu _1}\partial r \rightarrow 0$ for r → r_g, where r_g is the gravitational radius of a black hole. It is also finite at the critical radius, r ≈ 2r_g, where the differential rotation of the accreting matter measured in the LNRF (Bardeen et al. 1972) or by the zero angular momentum observers (ZAMOS; Thorne et al. 1986) is nothing, ∂(Ω − ω)/∂r = 0 at r ≈ 2r_g, The shear stress, $\sigma ^{\hat{r}\hat{\varphi }}$, is consistent with that given by Novikov & Thorne (1973).

The production rate of the viscous heat in the orbiting frame is assumed as

$$\begin{equation} \dot{\epsilon }=-t_{\hat{i}\hat{j}}\sigma ^{\hat{i}\hat{j}}, \end{equation} \tag{ 39 }$$

where $t_{\hat{i}\hat{j}}$ is the stress tensor. In the "α-model" of the viscosity, the stress tensor is expressed by $t_{\hat{i}\hat{\varphi }}=\alpha _{\rm vis} P_{\rm th}$, where P_th is the thermal pressure and α_vis is a parameter with α_vis = 0.1 − 0.01.

The energy–momentum tensor of the viscous fluid with the heat flux is described in the orbiting frame as (Misner et al. 1973; Novikov & Thorne 1973; Yokosawa 1995)

$$\begin{eqnarray} &&T^{\hat{\alpha }\hat{\beta }} =(\rho +\epsilon)u^{\hat{\alpha }}u^{\hat{\beta }} + Ph^{\hat{\alpha }\hat{\beta }} + t^{\hat{\alpha }\hat{\beta }} + q^{\hat{\alpha }}u^{\hat{\beta }} + u^{\hat{\alpha }}q^{\hat{\beta }}, \end{eqnarray} \tag{ 40 }$$

where ρ,  , and P are the rest-mass density, internal energy density, and the total pressure, and $q^{\hat{\alpha }}$ is the heat flux carried by neutrinos and photons. By using the covariant derivative, $\vec{\nabla }$, the local energy conservation, $\vec{u}\cdot (\vec{\nabla }\cdot {\rm \bf T})=0$, is represented as

$$\begin{eqnarray} &&\frac{\partial \epsilon u^{\hat{0}} + q^{\hat{0}}}{\partial x^{\hat{0}}} + \frac{1}{\hat{\alpha }} \nabla _{\hat{i}} \cdot \hat{\alpha }(\epsilon { u}^{\hat{i}} + { q}^{\hat{i}}) + P\frac{\partial u^{\hat{0}}}{\partial x^{\hat{0}}} + \frac{P}{\hat{\alpha }} \nabla _{\hat{i}} \cdot \hat{\alpha }{ u}^{\hat{i}} \nonumber\\ &&\quad + \sigma ^{\hat{\alpha }\hat{\beta }}t_{\hat{\alpha }\hat{\beta }} = 0. \end{eqnarray} \tag{ 41 }$$

The mass conservation, $\vec{\nabla }\cdot (\rho \vec{u})=0$, is written as

$$\begin{eqnarray} &&\frac{\partial \rho u^{\hat{0}}}{\partial x^{\hat{0}}} + \frac{1}{\hat{\alpha }} \nabla _{\hat{i}} \hat{\alpha }\rho { u}^{\hat{i}} = 0. \end{eqnarray} \tag{ 42 }$$

The conservation of angular momentum is clearly expressed in the frame fixed at the distant stars (Thorne et al. 1986). The Euler equation in the φ-direction is expressed in the Boyer–Lindquist coordinate as ${\rm \bf h}^{\varphi }\cdot (\nabla \cdot {\rm \bf T})=0$, where h^φ is the projection vector in the φ-direction, h^φ = h^φβ. It is explicitly described as

$$\begin{eqnarray} &&\frac{\partial (\rho +\epsilon +p) u^{0}u_{\varphi }}{\partial t} + \frac{1}{\alpha } \nabla _{i} \big (\alpha (\rho +\epsilon + P){u}^{i}u_{\varphi }+ t^i_\varphi + u_{\varphi }{ q}^{i} \big)\nonumber\\ &&\quad + \frac{\partial \alpha p}{\alpha \partial x^{\varphi }} = 0. \end{eqnarray} \tag{ 43 }$$

We are interesting in the gross properties of the disk. For simplicity, we consider the steady disk with an axially symmetry. As in the standard theory of the thin accretion disk (Novikov & Thorne 1973), we consider the height-averaged quantities. Putting the integrated stress as $W= \int _{0}^H t_{\hat{r}\hat{\varphi }}dz=\int _{0}^H \alpha _{\rm vis}P_{\rm th}dz$ and introducing the flux density at the surface of the disk, $F = q^{\hat{z}}(z= H)$, and the enthalpy, h = P + , the energy equation is expressed by

$$\begin{eqnarray} &&F+ \sigma _{{\hat{r}}{\hat{\varphi }}}W + \left(\frac{h}{\hat{\alpha }} \nabla _{\hat{r}} \hat{\alpha }u^{\hat{r}} + u^{\hat{r}} \nabla _{\hat{r}} \epsilon \right) H = 0. \end{eqnarray} \tag{ 44 }$$

The flux density F is evaluated with the total flux density of neutrinos, $F= \sum _{\nu _i} F_{\epsilon (\nu _i)}$, where ν_i means each flavor of neutrinos, $\nu _i=\nu _e, \bar{\nu }_e, \nu _\mu, \bar{\nu }_\mu, \nu _\tau, \bar{\nu }_\tau$. The third term in Equation (44) is the heating by the convection, and the fourth term is the variation of the internal energy due to the movement of the matter along the flow. Since the completely degenerated matter cannot be cooled by the convection, the enthalpy in Equation (44) is changed by the thermal component of the enthalpy, h → h_th = P_th + _th.

From Equation (43), the conservation of angular momentum is expressed by

$$\begin{equation} \nabla _{i} \cdot \big (\alpha (\rho +\epsilon + P){u}^{i}u_{\varphi }+ t^i_\varphi + u_{\varphi }{q}^{i} \big)=0. \end{equation} \tag{ 45 }$$

The angular momentum is transported by the viscous stress tⁱ_φ and by the neutrino flowing qⁱ. Introducing the total mass flux, $\dot{M}$, we described the transport equation of the angular momentum as

$$\begin{equation} \frac{\partial }{\partial r}\left (-\frac{\dot{M}}{2\pi }u_\varphi + \sqrt{\Delta } \gamma e^\psi W\right) - 2ru_\varphi F =0. \end{equation} \tag{ 46 }$$

It has the solution:

$$\begin{equation} W= \frac{\dot{M}}{2\pi }\Omega \Pi, \end{equation} \tag{ 47 }$$

where Π is the function with the value of unity at the great distance from the hole:

$$\begin{eqnarray} &&\Pi = \frac{{u}_\varphi + C}{\sqrt{{\Delta }} {\gamma } e^{\psi }{\Omega }} + \frac{2\pi fg}{{\Omega }\dot{M}}. \end{eqnarray} \tag{ 48 }$$

The term, fg, is due to the neutrino flowing. The functions, f and g, are

$$\begin{eqnarray} f&=& {-}\frac{\dot{M}}{2\pi }\int ^\infty _r \frac{2{u}_\varphi \tilde{\sigma }_{\hat{r}\hat{\varphi }}}{g{\Delta }{{\gamma }}^2 e^{2\psi }} ({u}_\varphi + C) d\varpi, \nonumber\\ g &=& \frac{1}{\sqrt{\Delta }\gamma e^{\psi }}\exp \biggl [ -\int ^\infty _r \frac{2 {u}_\varphi \tilde{\sigma }_{\hat{r}\hat{\varphi }}}{\sqrt{{\Delta }} {\gamma } e^{\psi }} d\varpi \biggr ], \end{eqnarray} \tag{ 49 }$$

$$\begin{eqnarray} \tilde{\sigma }_{\hat{r}\hat{\varphi }} &=& {\sigma }_{\hat{r}\hat{\varphi }} \left(1 + \left(1+ \frac{\epsilon }{P}\right) \frac{\sqrt{\Delta } u^{\hat{r}}}{2 \sigma _{\hat{r}\hat{\varphi }} \alpha _{\rm vis}\varpi ^2} \right.\nonumber\\ &&\left. \times \left(\frac{\partial \ln {\sqrt{\Delta } u^{\hat{r}}}}{\partial \varpi } + \frac{1}{1+ P/\epsilon } \frac{\partial \ln \epsilon }{\partial \varpi }\right)\right). \end{eqnarray} \tag{ 50 }$$

Here C is an integral constant. Novikov & Thorne (1973) adopted it as $C=-{u}_\varphi (r_{\rm{ms}})$ since they assumed no viscous stress can act at $r=r_{\rm{ms}}$. The numerical simulations of the accretion disk with viscosity show that the accreting matter continuously flows into a black hole through the boundary radius $r_{\rm{ms}}$ (Yokosawa 1995; De Villiers & Hawley 2003; De Villiers et al. 2003, 2005; Hirose et al. 2004; Krolik et al. 2005). We set the constant, C, such as the accreting matter continuously flows through the radius, $r_{\rm{ms}}$:

$$\begin{eqnarray} &&C= - {u}_\varphi (r_{\rm{ms}}) + \frac{4\pi H}{\dot{M}}\sqrt{{\Delta }(r_{\rm{ms}})} {\gamma }(r_{\rm{ms}}) e^{\psi }(r_{\rm{ms}})\alpha _{\rm vis}{P_{\rm th}}(r_{\rm{ms}}).\nonumber\\ \end{eqnarray} \tag{ 51 }$$

The energy equation (Equation (44)) and the transport equation of the angular momentum (Equation (47)) determine the stationary, thermally equilibrium structure of the accretion disk.

The thickness of the disk, H, in a dynamically equilibrium state is evaluated as follows. The equation of motion of the fluid, h^k_iT^j_k;j = 0, for the steady, axially symmetric, and circularly rotating fluid can be expressed in the total differential form:

$$\begin{equation} \frac{dp}{\rho +P}=\gamma ^2\big({-}d\nu +v_{(\varphi)}^2d\psi -v_{(\varphi)}v_\omega d\ln \omega\big) \equiv -dU. \end{equation} \tag{ 52 }$$

The equipressure surfaces are given by the equation U = constant. The quantity U = U(p) is equal in the Newtonian limit to the total potential (gravitational plus centrifugal potential) expressed in the units of c² (Abramowicz et al. 1978). In the neighborhood of the equatorial plane, the potential, U, is expanded with z/r(≪1):

$$\begin{equation} U = U_0(r,0) + \frac{1}{2}\Omega ^2_0 z^2, \end{equation} \tag{ 53 }$$

where Ω²₀ is

$$\begin{equation} \Omega ^2_0 = \frac{{\rm G}M}{r^3}\Phi (r). \end{equation} \tag{ 54 }$$

The function, Φ(r), becomes unity far away from a hole. Its explicit expression of Φ(r) is shown in Appendix C. The thickness of the disk, H, is approximately given by the scale height:

$$\begin{equation} H = \sqrt{2\frac{P}{\rho }}\Omega _0^{-1}. \end{equation} \tag{ 55 }$$

The total pressure, P, and the total internal energy density, , of the accreting matter are given by

$$\begin{eqnarray} P &=& P_b +P_e + a_rT^4\left(\frac{11}{12} + \frac{7}{8}\tau \right), \nonumber\\ \epsilon &=& \epsilon _b + \epsilon _e +a_rT^4\left(\frac{11}{4} + \frac{21}{8}\tau \right). \end{eqnarray} \tag{ 56 }$$

The third terms on the right-hand sides of both the expressions represent the contributions of the radiation and neutrinos in thermally equilibrium with relativistic electron positron pairs, where a_r is the radiation constant and τ is the neutrino opacity, $\tau =\Sigma _{\nu _i}(\tau _{\nu _i}/2 + 1/\sqrt{3})/ (\tau _{\nu _i}/2 + 1/\sqrt{3} +1/3\tau _{\nu _i a})$ (Popham & Narayan 1995). The baryonic matter consists of free nucleons, α-particles, and heavy nuclei. Its pressure, P_b, is given by the EOS for the given density, ρ, temperature, T, and the fraction of electrons, Y_e; P_b = P_b(ρ, T, Y_e) (Lattimer & Swesty 1991). The values of P_b are almost the same as n_bkT over the phase plane, 10¹¹(g cm^-3) < ρ < 10¹³ (g cm⁻³), 10¹⁰(K) < T < 10¹¹(K), and 0.1 < Y_e < 0.5. Thus, the thermal component of the internal energy of baryonic matter, $\epsilon _{b,\rm th}$, is set as $\epsilon _{b,\rm th} = 3/2n_b kT$. The pressure and the internal energy of electrons, P_e and _e, are expressed approximately by (Lattimer & Swesty 1991)

$$\begin{eqnarray} P_e &= & \frac{\mu _e}{12\pi ^2}\left(\frac{\mu _e}{\hbar c}\right)^3\biggl [1 + \biggl (\frac{kT}{\mu _e}\biggr)^2 \biggl (2\pi ^2 -3 \biggl (\frac{m_ec^2}{kT}\biggr)^2 \biggr)\nonumber\\ && + \biggl (\frac{kT}{\mu _e}\biggr)^4 \pi ^2 \biggl (\frac{7}{15}\pi ^2 - \frac{1}{2} \biggl (\frac{m_ec^2}{kT} \biggr)^2 \biggr) \biggr ], \end{eqnarray} \tag{ 57 }$$

$$\begin{eqnarray} \epsilon _e &= & \frac{\mu _e}{4\pi ^2} \biggl (\frac{\mu _e}{\hbar c} \biggr)^3 \biggl [ 1+ \biggl (\frac{kT}{\mu _e} \biggr)^2 \biggl (2\pi ^2 - \biggl (\frac{m_ec^2}{kT} \biggr)^2 \biggr)\nonumber\\ && + \biggl (\frac{kT}{\mu _e} \biggr)^4 \biggl (\frac{7}{15}\pi ^2 - \frac{1}{2} \biggl (\frac{m_ec^2}{kT} \biggr)^2 \biggr) \biggr ]. \end{eqnarray} \tag{ 58 }$$

The first terms in both the square brackets provide the pressure and internal energy corresponding to the completely degenerated state. The second and third terms depend on the temperature. In the state of massive accretion disks, the degeneracy factor of electrons has the value in the range 2 < η_e < 15 (see Figure 6). We adopt the above expressions without the first term as thermal pressure and thermal internal energy of electrons, $P_{e, \rm{th}}$ and $\epsilon _{e, \rm{th}}$.

We have solved the stationary equations of the accreting flow, Equations (44) and (47), and the fraction changes of leptons (Equation (28)) measured in the orbiting frame, $\dot{Y}_l=dY_l/dx^{\hat{0}}$, from the outer boundary, $r_{\rm out}=10^2r_{\rm{ms}}$, to the inner boundary, $r_{\rm in}=r_{\rm{ms}}$. At the outer boundary, we adopt the components of neutrinos, $Y_{\nu _i}$, in quasi-equilibrium abundance in the given (ρ, T, Y_e); the quasi-equilibrium abundance, $Y_{\nu _i}$, satisfies the conditions, $\dot{Y}_{\nu _i} \approx 0$ for each neutrino ν_i. The fraction of electrons, Y_e, which satisfies the stationary conditions, Equations (44) and (47), is taken to be the most large value within Y_e ⩽ 0.5. The accreting velocity, $v(r) \equiv u(r)^{\hat{r}}/u(r)^{\hat{0}}$, is derived from the mass conservation, $\dot{M} = \mbox{constant}$, for the given density, ρ(r), and height, H(r). The time interval of the accreting fluid moving the distance, $dx^{\hat{r}}$, is taken as $dx^{\hat{0}} = dx^{\hat{r}}/v(r)$. We introduce the normalized values of the radius, $\tilde{r}$, accretion rate, $\dot{m}$, and the mass of a black hole, m, defined as $\tilde{r} \equiv r/r_g$, $\dot{m} \equiv \dot{M}/(1\,M_\odot$ s⁻¹), and m ≡ M_BH/M_☉. The accretion rate, $\dot{m}$, and the angular momentum of a black hole, a, are selected as $\dot{m} = 0.01 \sim 1$ and a = 0 ∼ 1. The typical parameters for the model of GRBs are such as $a=0.9, \dot{m} = 0.1$, and m = 3 (MacFadyen & Woosley 1999). The viscous parameter is set to be α_vis = 0.05.

4.1. Effects of Heavy Nuclei on the Massively Accreting Matter

The loci of accreting matter in the ρ–T plane for $\dot{m} = 0.01,\, 0.1$, and 1 are shown in Figure 1. The contours of the difference in chemical potentials between a neutron and a proton, μ_n − μ_p, provided by EOS are overplotted. The profile of the contour changes distinctly at the phase boundary between the matter with heavy nuclei and the uniform gas of evaporated free nucleons and α-particles without heavy nuclei (Lamb et al. 1981; Lattimer & Swesty 1991). The loci of the accreting matter are put in the domain with coexistence of heavy nuclei. The massive accretion disks contain non-ignorable abundance of heavy nuclei.

The locus of the typical accretion disk (STAD) with ${\dot{m}} = 0.1$ and those of ADAFs, ADAF1 for ${\dot{m}} = 0.1$ (Di Matteo et al. 2002) and ADAF2 for ${\dot{m}} = 0.2$ (Chen & Belobororodov 2007), are also plotted on the ρ–T plane in Figure 2. The matter of STAD takes the state with relatively low temperature and high density. The quasi-radially falling matter in ADAF is heated up by the strong compression due to rapid deceleration, and the heated thermal energy is confined within the neutrino-opaque disk. The geometrically thin disk (STAD) produces high dense and neutrino-transparent matter in which the frequent neutrino reactions are enhanced.

Zoom In Zoom Out Reset image size

The mass fraction of free nucleons produced by the photodisintegration process, X_nuc = T^9/8₁₁ρ^−3/4₁₂exp(−0.61/T₁₁), is presented by Qian & Woosley (1996). In the previous work (Di Matteo et al. 2002), the free nucleons have been evaluated by X_nuc. The locus of the typical accretion disk (STAD) is almost resemble with that of the mass fraction, X_nuc = 10⁻². If the mass fraction of free nucleons is evaluated by X_nuc, the fraction of free neutrons becomes very small. The amount of free neutrons in the massive accretion disk can be provided by the evaporation process from the heavy nuclei in high dense matter (Lattimer & Swesty 1991). The phase boundaries between the matter with heavy nuclei and the uniform gas of evaporated free nucleons without heavy nuclei in the states, Y_e = 0.5, 0.05 (Lamb et al. 1981), are also plotted in Figure 2. The matters in ADAFs pass over the boundaries. If the neutrino reactions in the domain of heavy nuclei are introduced and if the detailed balances are relaxed in ADAFs, other loci of ADAFs may be represented in the ρ–T plane.

The free nucleons could also be produced by the photodisintegration if there were enough photons to disintegrate the nuclei (Burbidge et al. 1957; Meyer et al. 1992). The photon-to-baryon ratio ϕ_γb is expressed as

$$\begin{equation} \phi _{\gamma b} =\frac{n_\gamma }{\rho N_A} =0.034 \frac{T_{11}^3}{\rho _{12}}, \end{equation} \tag{ 59 }$$

where N_A is Avog${\acute{a}}$dro's number. The constant lines of the ratio for ϕ_γb = 0.5 and 0.01 are shown in Figure 2. All the loci of the accreting flows are put at the domain of insufficient photons with ϕ_γb < 0.1.

4.2. Degeneracy of Accreting Particles

The profiles of degeneracy factors of electrons η_e(=μ_e/kT) and neutron–proton difference η_np(=(μ_n − μ_p)/kT) are shown in Figure 3. The degeneracy of electrons η_e in the massive accretion is strong, η_e ≈ 5 ∼ 10, though the electrons are not always in the completely degenerated state. In the previous works in ADAF (Chen & Belobororodov 2007; Kawanaka & Mineshige 2007), it is mild, η_e ≈ 1 ∼ 3. The degeneracy factors of neutron–proton difference are also strong, η_np ≈ 6 ∼ 12, in the typical case of the accretion with $\dot{m} = 0.1$ and a = 0.9. Since the number ratio of the free neutron to the free proton is proportional to the degeneracy factor, n_n/n_p ≈ exp(η_np), the increase of the degeneracy η_np from 6 at the outer region to 12 at the inner side of the disk is corresponding to the variation of number ratio such as n_n/n_p ≈ 4 × 10² → 10⁵. The degeneracy factor of $\bar{\nu }_e$ reaches to $\eta _{\bar{\nu }_e} \rightarrow 5$ at the inner boundary of the disk. The increases in η_np and η_e are caused by the contraction of accreting matter with relatively low temperature. The rapid decreases of η_np and η_e at the inner side of the disk are brought about by the rapid decrease of Y_e.

Zoom In Zoom Out Reset image size

The thermal equilibrium approximation in the β-process is not always in high degree of accuracy for the massively accreting matter. While the degeneracy factors of η_np and η_e change in similar profiles along the flow, the thermal equilibrium approximation, η_e = η_np + Q/kT, is not satisfied, but the degeneracy factor η_np becomes superior to η_e at the inner side of the accretion disk.

4.3. Composition of Accreting Matter

The profiles of the composition of accreting matter are shown in Figures 4 and 5. The case of the low accretion rates, $\dot{m} = 0.005 \sim 0.1$ for a = 0.9, is shown in Figure 4. On the other hand, Figure 5 shows the profiles for massive accretion rates, $\dot{m}=0.05 \sim 1$ for a = 0. The profiles of compositions of baryons (free neutrons and free protons) and of leptons (electrons, positrons, electron neutrinos, antielectron neutrinos, and heavy neutrinos) have been represented at first. The characteristic property of the accreting matter is the remarkable discrepancy in the fractions between free neutrons and protons. The fraction of free neutrons becomes dominant at the inner side of the disk, Y_n = 0.7 ∼ 0.8, while that of free protons decreases to an extremely minor composition, Y_p = 10⁻⁴ ∼ 10⁻⁵. When the gas consists of free neutrons, protons, and electrons, the minimum ratio of n_p/n_n in the thermally equilibrium state is n_p/n_n = 2.6 × 10⁻³ at ρ = 7.8 × 10¹¹ (g cm⁻³) (Shapiro & Teukolsky 1983). In the high dense matter with heavy nuclei, the free protons become extremely infinitesimal.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The high density and the sufficient time in the reactions of leptons change the fraction of electrons, Y_e, which gradually decreases along the accreting flow and reaches to Y_e = 0.01 at the inner boundary, $r=r_{\rm{ms}}$. The fraction of electron neutrinos $Y_{\nu _e}$ rapidly decreases along the flow, while the fraction of antielectron neutrinos $Y_{\bar{\nu }_e}$ gradually increases to be $Y_{\bar{\nu }_e} \approx 0.02$ at the inner disk. The fraction of heavy neutrinos $Y_{\nu _x}$ increases in proportional to the density of accreting matter. The heavy neutrinos are produced mainly by the reactions, $e+e^+ \rightleftharpoons \nu _i + \bar{\nu }_i$ and, $N+N \rightleftharpoons N + N + \nu _i + \bar{\nu }_i$. In massively accreting cases, $\dot{m} \ge 0.1$ for a = 0.9, the heavy neutrinos are not negligibly rare and their fraction exceeds the electron neutrinos at the inner side of the disk, $Y_{\nu _x}= 10^{-3} \gg Y_{\nu _e} = 10^{-5}$.

When the accretion rate $\dot{m}$ increases, the fraction of protons decreases, $Y_p \propto \dot{m}^{-1}$, though the fraction of neutrons at the inner side of the disk remains almost constant in the wide range of accretion rate; Y_n ≈ 0.7 ∼ 0.8 for $\dot{m} = 0.01 \sim 1$. Similarly, the fractions of electron neutrinos $Y_{\nu _e}$ and that of positrons $Y_{e^+}$ are inversely proportional to $\dot{m}$, while the fraction of antielectron neutrinos $Y_{\bar{\nu }_e}$ remains almost constant, $Y_{\bar{\nu }_e} \approx 10^{-2}$. On the contrary, the fraction of heavy neutrinos increases as $Y_{\nu _x} \propto \dot{m}$. At the inner side of the disk with $\dot{m} \ge 0.1$, the heavy neutrinos become a non-ignorable component in the network of the lepton reactions.

4.4. Flow Structures of Accretion Disks

The flow profiles, density ρ(r), temperature T(r), accreting velocity v(r), and thickness H(r), are plotted in Figures 6 and 7. Figure 6 shows the distinctive flow structures for the variation of accretion rate, $\dot{m}=0.01 \sim 1$, where the central mass is fixed as m = 3. Figure 7 shows the flow profiles depending on the mass scale of a black hole, m = 1 ∼ 10. The remarkable feature of the accreting flow is the profile of the temperature. The temperature becomes nearly constant along the flow, T ≈ 3 × 10¹⁰ K, and little changes for the wide range of accretion rate, $\dot{m}=0.01 \sim 1$. This characteristic property is caused the optically thin disk for the antielectron neutrinos. The main emissivity of $\bar{\nu _e}$ is given by the reaction of positron capture on neutrons whose emissivity is sensitive to temperature, $q_{\bar{\beta }}(\bar{\nu _e}) \propto \rho T^6\;({\rm erg\,cm}^{-3}\,{\rm s}^{-1}$) (Burrows 2001). The optically thin antielectron neutrinos adjust the temperature balanced between heating and cooling of the disk. The height of the geometrically thin disk is also little changed, $H(\tilde{r}) \approx 0.1 r_{g}(\tilde{r})^{5/4}$, for $\dot{m}=0.01 \sim 1$, The accreting velocity, v, is uniform along the flow, v ≈ 10⁻³c. The typical dynamic time of the accretion disk is Δt ≈ 10²r_g/v ≈ 10 (s). The density of accreting matter reaches to a very high value at the inner disk, ρ ≈ 10¹³ ∼ 10¹⁴ (g cm⁻³), for $\dot{m} \ge 0.1$.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Figure 6 shows that the central density at r = r_in depends on the mass of a black hole as ρ(r_in) ∝ m⁻² and the thermally equilibrium temperature is inversely proportional to the mass, T(r_in) ∝ m^−1/3. The profiles of density $\rho (\tilde{r})$ and temperature $T(\tilde{r})$ are expressed approximately as $\rho (\tilde{r}) \approx 2\times 10^{14} {\tilde{r}}^{-2.5} \dot{m} m^{-2}$ g cm⁻³ and $T(\tilde{r}) \approx 5\times 10^{10} \tilde{r}^{-1/4} m^{-1/3}$ K. When the mass of a black hole increases owing to the accreting flow, the central density and temperature around a black hole rapidly change with time. The emission rate of antielectron neutrinos is then $q_{\bar{\beta }}(\bar{\nu _e}) \propto \rho T^6\;({\rm erg\,cm}^{-3}\,{\rm s}^{-1}) \propto m^{-4}$ (Burrows 2001). Even if the emitting region of neutrinos is $V_{\rm emit}(\bar{\nu _e}) \propto m^3$, the neutrino flux will decrease with time after the birth of a black hole when the accretion rate is nearly constant.

The vertical structure of a massive accretion disk is supported mainly by baryons. The compositions of the pressure, P_b,  P_e, and P_r, and of the thermal pressure, P_th and $P_{e,\rm th}$, for the typical case of the accretion are shown in Figure 8. Even though the electrons are degenerated, the inner disk is little supported by the electrons but mainly supported by baryons (free neutrons), P_b ≫ P_e. The accreting free neutrons are continuously heated by viscosity and thus its pressure P_b becomes superior to P_e at the region r < 30r_g. The electron pressure P_e itself decreases at the inner region of the disk where the electron fraction Y_e rapidly decreases. The thermal part of the electron pressure $P_{e,\rm th}$ is comparable with the degenerated part of P_e. The contribution of the radiation pressure P_r to the total pressure P is negligibly small, P_r ≪ P.

Zoom In Zoom Out Reset image size

4.5. Emissivities and Opacities of Neutrinos

The emissivities of each flavor of neutrinos $q_{\nu _i}$ and that ejected by each type of neutrino reactions for the typical accretion disk are shown in Figure 9. The energy flux density of each flavor of neutrinos for the typical accretion disk $F_{\nu _i}$ is shown in Figure 10. The antielectron neutrino acts as the most efficient source of emissivity. The emissivity of antielectron neutrinos $q_{\bar{\nu }_e}$ is provided mainly by the positron capture on a neutron, $q_{\bar{\beta }}$; $e^+ + n \rightarrow \bar{\nu }_e + p$. At the inner side of the disk, the emissivity by heavy neutrinos $q_{\nu _x}$ becomes efficient, which is provided by the reactions of the nucleon–nucleon bremsstrahlung q_nn and the electron pair process q_ee. The emissivity of electron neutrinos $q_{\nu _e}$ is 1 order of magnitude less than $q_{{\bar{\nu }}_e}$ since the proton fraction Y_p is infinitesimal. At the outer side of the disk, $r \ge 20r_{\rm{ms}}$, the pair production, q_ee; $e+e^+ \rightarrow \nu _e+\bar{\nu }_e$, and the nuclei reaction, q_A, e + A → ν_e + A', sufficiently contribute to the total emissivity. At the inner side of the disk, the URCA process $q_{\rm{URCA}}$ is efficient. The emissivity of neutrinos rapidly increases along the flow. The total emissivity is expressed as $q = q_{\bar{\nu }_e}+ q_{\nu _e}+ q_{\nu _x} \approx 10^{35}\tilde{r}^{-4.25}$ erg cm⁻³ s⁻¹.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The flux density of antielectron neutrinos at the surface of the disk $F_{\bar{\nu }_e}$ rapidly increases along the accreting flow while that of electron neutrinos $F_{\nu _e}$ maintains almost the constant value. The ratio of the flux density of $\bar{\nu }_e$ to that of ν_e becomes large to be $F_{\bar{\nu }_e} / F_{\nu _e} \approx 10^{3} \sim 10^{4}$ at the inner side of the disk, since the electron neutrinos are almost absorbed by dense neutrons. The flux density of ν_x rapidly increases in the vicinity of the inner boundary and reaches to be comparable with $F_{\bar{\nu }_e}$. The total flux density is expressed as $F(r) = F_{\nu _e} + F_{\bar{\nu }_e} + F_{\nu _x} \approx 3.8 \times 10^{40} \tilde{r}^{-2.9}$ erg cm⁻² s⁻¹ in the typical case of the accretion.

The opacity and the optical depth of each flavor of neutrinos are shown in Figure 11. The absorbing opacity of electron neutrinos κ_a(ν_e) is by far larger than those of other flavors of neutrino, $\kappa _a(\nu _e) \gg \kappa _a(\bar{\nu }_e) \ge \kappa _a(\nu _x)$. The electron neutrino ν_e is absorbed mainly by dense free neutrons through the reaction ν_e + n → e + p, while the antielectron neutrino $\bar{\nu }_e$ is little absorbed by rare free protons, $\bar{\nu }_e + p \rightarrow e^+ + n$. The scattering opacities of $\bar{\nu }_e$ and ν_e are in the same order except near the inside boundary where the chemical potential of $\bar{\nu }_e$ is some larger than that of ν_e, $\ \mu _{\bar{\nu }_e} > \mu _{{\nu }_e}$, which brings about the larger scattering opacity of $\bar{\nu }_e$; $\ \kappa _s (\bar{\nu }_e) > \kappa _s(\nu _e)$. The total optical depths of $\bar{\nu }_e$ and ν_e are in the same order of magnitude and become very large at the inner disk, $\tau _{\bar{\nu }_e} =\tau _{\bar{\nu }_e a} +\tau _{\bar{\nu }_e s} \ge \tau _{\nu _e}=\tau _{\nu _e a} +\tau _{\nu _e s} \sim 10^2$. However, there is a remarkable difference in the effective optical depth among them, $\tau ^{\rm eff}_{\nu _e} \ge 30 \gg \tau ^{\rm eff}_{\bar{\nu }_e}(\sim 0.1 \sim 10^{-2})$, where the effective optical depth is defined as $\tau ^{\rm eff}_{\nu _i} = ({\tau ^a_{\nu _i} \tau ^s_{\nu _i}})^{1/2}$. Most of ν_e are absorbed in the disk and their energy is deposited in the accreting matter. On the other hand, the antielectron neutrinos $\bar{\nu }_e$ carry out the thermal energy of the accreting matter from the disk. The effective optical depth of ν_e is approximately described as $\tau ^{\rm eff}_{\nu _e}\approx 6\times 10^3 \bar{r}^{-2} (m)^{-1}\dot{m}$. It becomes over unity at the radius, $r_{\nu _e} = 14r_g(m/3)^{-1/2}(\dot{m}/0.1)^{1/2}$, where the neutrino sphere (disk) of ν_e is formed. The neutrino sphere (disk) of $\bar{\nu }_e$ or ν_x is not formed for $\dot{m} \le 0.1$. The total optical depths of $\bar{\nu }_e$ and ν_e become over unity, $\tau _{\nu _e}, \tau _{\bar{\nu }_e} \ge 1$, at the radius, $r \approx 24 r_g (m/3)^{-1/2} (\dot{m}/0.1)^{1/2}$.

Zoom In Zoom Out Reset image size

4.6. Luminosity and Mean Energy of Neutrinos

The total luminosities L_ν and the mean energies $\bar{E}_\nu$ of neutrinos measured at infinity are plotted in Figure 11, where the mass of a black hole m, angular momentum a, and the accretion rate $\dot{m}$ are changed over the range such as m = 1 ∼ 10, a = 0 ∼ 1, and $\dot{m} = 0.02 \sim 1$. We evaluate the luminosity L_ν and mean energy $\bar{E}_{\nu }$ as follows. The luminosity measured at the infinity is expressed in the Boyer–Lindquist coordinate as

$$\begin{equation} L_{\nu _i}=\int q_{\nu _i}^z(r) dS_z, \end{equation} \tag{ 60 }$$

where $q_{\nu _i}^z(r)$ is the z-component of the flux density vector at the radius r, where the z-direction is the vertical one to the disk, and dS_z is the z-component of the surface element vector, dS_z = α⁻¹2πrdr (Thorne et al. 1986). The component, $q_{\nu _i}^z$, is connected to the local surface density in the orbiting frame $F_{\nu _i}$ such as

$$\begin{equation} q_{\nu _i}^z=e^{\hat{t}}_t F_{\nu _i}, \end{equation} \tag{ 61 }$$

where $e^{\hat{t}}_t$ is the t-component of the orbiting basis ${\vec{\omega }}^{\hat{t}}$ expressed in Equation (36), $e^{\hat{t}}_t = \alpha \gamma (1+v_\omega v_{(\varphi)})$. Thus the luminosity observed at the infinity is expressed by

$$\begin{equation} L_{\nu _i}=2\pi \int _{r_h}^{r_{\rm max}} \gamma (1+v_\omega v_{(\varphi)}) F_{\nu _i} rdr. \end{equation} \tag{ 62 }$$

The mean energy of neutrinos emitted from the accretion disk $\bar{E}_{\nu _i}$ for each type of neutrino is defined as follows. When the neutrinos are thermally equilibrium in the accreting matter, the local mean energy of neutrino $\bar{\epsilon }_{\nu _i}(r)$ is expressed by the matter temperature T(r) and the degeneracy factor of neutrino $\eta _{\nu _i}(r)$:

$$\begin{equation} \bar{\epsilon }_{\nu _i}(r) = kT(r) \frac{F_{3} (\eta _{\nu _i}(r))}{ F_{2} (\eta _{\nu _i}(r))}, \end{equation} \tag{ 63 }$$

where $F_k(\eta _{\nu _i})$ is the kth Fermi integral, F_k(η) ≡ ∫^∞₀(1 + exp(x − η))⁻¹x^kdx, for k = 2, 3. The number of neutrinos emitted from the shell (ring) over the width of radius, r ∼ r + dr, per unit time is given by

$$\begin{equation} dN_{\nu _i}(r) = \frac{dF_{\nu _i}(r)}{\bar{\epsilon }_{\nu _i}(r)}, \end{equation} \tag{ 64 }$$

where $dF_{\nu _i}(r)$ is the energy flux of neutrinos from the shell (ring) with the width dr. The total number of neutrinos for each type of neutrinos is described as $N_{\nu _i}=\int _{r_h}^{r_{\rm max}} dN_{\nu _i}(r)$. Thus, the mean energy of neutrinos for each type of neutrinos is defined as $\bar{E}_{\nu _i}=L_{\nu _i}/N_{\nu _i}$.

The total luminosity for all types of neutrinos is $L_\nu = \sum _i L_{\nu _i}$. The mean energy in all types of neutrinos is simply defined by $\bar{E}_\nu = L_\nu / \sum _i N_{\nu _i}$. We set the quantities of neutrinos within $r_{\rm{ms}}$ to be the same as the values at the boundary, $r=r_{\rm{ms}}$. The maximum radius of the integral is taken to be r_max = 10²r_g.

The diagram of the luminosity of neutrinos, L_ν, versus the mean energy, $\bar{E}_\nu$, in Figure 12 shows the characteristic properties as follows. (1) The luminosity L_ν and the mean energy $\bar{E}_\nu$ are proportional to the specific angular momentum of a black hole, L_ν ∝ a and $\bar{E}_\nu \propto a$. When m = 3 and $\dot{m}=0.1$, the ratio of the luminosity for a = 1 to that for a = 0 is L_ν(a = 1)/L_ν(a = 0) ≈ 3.5, and the ratio of mean energy is $\bar{E}_\nu (a=1)/\bar{E}_\nu (a=0) \approx 2.5$. (2) The luminosity is proportional to the accretion rate, $L_\nu \propto \dot{m}$, but it is almost independent of the mass scale of a black hole, L_ν ≈ constant, for m = 1 ∼ 10. (3) The mean energy of neutrinos $\bar{E}_\nu$ is proportional to the accretion rate while it is inversely proportional to the mass of a black hole, $\bar{E}_\nu \propto \dot{m}^{0.8} m^{-1.07}$. If the luminosity of each flavor of neutrinos $L_{\nu _i}$ and its mean energy $\bar{E}_{\nu _i}$ are observed (Halzen & Hooper 2002; McDonald et al. 2004), the physical parameters of a central black hole will be made clear.

Zoom In Zoom Out Reset image size

The typical accretion disk provides the luminosity, L_ν ∼ 10⁵² erg s⁻¹, and the mean energy, $\bar{E}_\nu \ \sim 20$ MeV, which is several times larger than the thermal energy of a particle in the disk. This high mean energy $\bar{E}$ is caused by the large degeneracy, $\eta _{\bar{\nu }_e} \approx 5$, at the inner disk. The energy flux, L_ν ∼ 10⁵² erg s⁻¹, with the mean energy, $\bar{E}_\nu \ \sim 20$ MeV, emitted from a disk could produce the outward flow in the vertical direction to the disk.

We have studied the massive accretion disk in a simplified model in which the disk is assumed to be stationary and its vertical structure is treated by the one-zone model with the thickness H. These simplifications might restrict the properties of the accretion disk. We first discuss the transient thermal stability of the disk.

5.1. Transient Thermal Instability

The scattering optical depth of antielectron neutrinos is $\tau _s({\bar{\nu }}_e) \approx 10^2$ for the typical accretion disk. Then, the diffusion time of the neutrinos is t_diff ≈ τ_sH/c ≈ 10 (ms). One of the characteristic features of γ-ray emission is the short spike-like emissions with the duration time Δt ⩽ 10 ms. When the sufficiently rapid increase of heat δh is generated in the disk, the excess of heat is deposited within the disk in the duration of t_diff, which could drive the transient thermal instability of the accretion disk. We discuss the transient thermal stability by a perturbation analysis.

When the scattering optical depth is large, τ_s ≫ 1, and the absorption optical depth is negligible, τ_a ≪ 1, the neutrinos randomly walk in the disk. The time in which a neutrino is trapped in the disk is expressed as t_diff ≈ Nl/c ≈ (l/c)(H/l)² ≈ (H/l)(H/c) ≈ τ_st_cross, where N is the collision number of a neutrino in the disk, l is the mean free path, and t_cross is the crossing time over the scale height H. The time required for a neutrino to escape from the disk is τ_s times larger than the crossing time over the scale height. We assume a perturbation δh in the disk. When τ_s ≪ 1, the change of the cooling rate in the unit column of the disk is then expressed as δQ⁻ = δF_ν = δh/t_cross = δ(q_νH). When τ_s ≫ 1, we can express the perturbation of the cooling rate as δQ⁻ = δh/t_diff = δ(q_νH)/τ_s. Since the electron neutrinos ν_e are almost absorbed within the disk, we express its perturbed cooling rate as $\delta {Q}_{\nu _e}^- = \delta ({q_{\nu _e}H} / (1.5\tau _{\nu _e a}\tau _{\nu _e} +\sqrt{3}\tau _{\nu _e a}+1))$. The perturbed cooling rates by $\bar{\nu }_e$, ν_x, and ν_e for any optical depth can be represented as

$$\begin{eqnarray} \delta {Q}^{-} &=& \delta {Q}_{\bar{\nu }_e}^- + \delta {Q}_{\nu _x}^- + \delta {Q}_{\nu _e}^- = \delta \biggl (\frac{q_{\bar{\nu }_e}H}{1+\tau _s(\bar{\nu }_e)} \biggr) + \delta \biggl (\frac{q_{\nu _x}H}{1+ \tau _s (\nu _x)}\biggr)\nonumber\\ && + \delta \biggl (\frac{ q_{\nu _e}H }{1.5\tau _{\nu _e a}\tau _{\nu _e} +\sqrt{3}\tau _{\nu _e a}+1} \biggr). \end{eqnarray} \tag{ 65 }$$

The change of heating rate is δQ⁺ = δQ_vis + δQ_adv where δQ_vis and δQ_adv are changes by viscous heating and by advection. We evaluate the thermal stability for the perturbation of temperature δT with the restriction of constant surface density Σ,

$$\begin{equation} n_{Q} = \biggl (\frac{\partial \log {{Q}^+}}{\partial \log {T}}\biggr)_\Sigma - \biggl (\frac{\partial \log {Q}^-}{\partial \log {T}} \biggr)_\Sigma. \end{equation} \tag{ 66 }$$

The value of the displacement, Q⁺(T + δT) or Q⁻(T + δT), for Σ = constant, is numerically calculated and then the value of differential, $n_{Q^+} = ({\partial \log { Q^+}} /{\partial \log {T}})_\Sigma$ or $n_{Q^-} =-({\partial \log { Q^-}} / {\partial \log {T}})_\Sigma$, is obtained.

The result for the net index of the differential, $n_{Q} = n_{Q^+} + n_{Q^-}$, is shown in Figure 13, where the parameters of the accretion disk are set as $\dot{m}=0.5, m=3, a=0.9$, and α_vis = 0.1. The sign of n_Q becomes positive when $\tau _s(\bar{\nu }_e) \ge 1$. The sign of $n_{Q^-}$ also becomes positive when $\tau _s(\bar{\nu }_e) \ge 10$, since the opacity of $\bar{\nu }_e$ so rapidly increases for the increase of temperature, $({\partial \log {\tau _s(\bar{\nu }_e)}} / {\partial \log {T}})_\Sigma \gg 1$, that the differential of the cooling rate by temperature becomes negative, $({\partial \log { Q^-}} / {\partial \log {T}})_\Sigma \propto {\partial }_T (q_{\bar{\nu }_e}H/(1+\tau _s(\bar{\nu }_e)))_\Sigma < 0$. For the region where the scattering optical depth of $\bar{\nu }_e$ is larger than unity, $\tau _s(\bar{\nu }_e) \ge 1$, the accretion disk becomes thermally unstable at least within the duration of the diffusion time t_diff.

Zoom In Zoom Out Reset image size

We next discuss on the dynamics of nucleons interacting with neutrinos. In the typical case of the accretion disk, the effective optical depth of electron neutrinos τ_eff(ν_e) exceeds unity at the radius, r ⩽ 10r_g. The ratio of the flux density of ν_e to that of $\bar{\nu }_e$ becomes infinitesimal, $F_{\nu _e} / F_{\bar{\nu }_e} \approx 10^{-3} \sim 10^{-4}$, which seriously restricts the efficiency of the annihilation of ν_e and $\bar{\nu }_e$ at the outer side of the disk. This results in the difficulty in the formation of a fireball by the pair annihilation between ν_e and $\bar{\nu }_e$. However, the effective optical depths of $\bar{\nu }_e$ and ν_x within the radius r ⩽ 10r_g are less than unity, $\tau _{\rm eff}({\bar{\nu }_e}), \tau _{\rm eff}({\nu _x}) < 1$, and their scattering optical depths are greater than unity, $\tau _s({\bar{\nu }_e}), \tau _s({\nu _x}) \ge 1$. The interactions of $\bar{\nu }_e$ and ν_x with the nucleons might push the nucleon matter outward and could drive the convection or neutrino driving wind or jets. Since the mean energy of neutrinos exceeds the rest-mass energy of electrons, $\bar{E}_\nu \approx 20\,{\rm Mev} \gg m_ec^2$ for the accretion with a ⩾ 0.9, $\dot{m} \ge 0.1$ and m ⩽ 3, the collisions between neutrinos and electrons efficiently transfer the momentum and energy from neutrinos to electrons (Thompson et al. 2000). The intensive wind or jets from the disk might be produced. The one-dimensional flow driven by neutrinos will become fast in comparison with the radial flow in a supernova explosion.

The intense flux of antielectron neutrinos is ejected from the disk but the electron neutrinos absorbed in the accreting matter fall into a black hole, which means that the lepton number decreases in the free world apart from the event horizon. In the case of the typical accretion disk with $\dot{m}=0.1$, the ejection rate of neutrinos is $\dot{N}_\nu \approx 10^{57}$ s⁻¹. The baryon number of a star with the mass M = 1M_☉ is N_b ≈ 10⁵⁷. The lepton number which is 1 order of magnitude larger than the baryon number falling into a black hole decreases in an event of a GRB.

We have studied the stationary accretion disk with neutrino loss and evaluated the effects of neutrinos on the structure of the disk. The dynamics of the neutrino-driven wind, the dynamical behaviors at the inner disk, and the thermal instability of the massive accretion disk should be farther more precisely investigated by the non-stationary numerical methods of general relativistic hydrodynamics.

We summarize the new results in this paper in comparison with previous works as follows.

• 1.  
  Effects of heavy nuclei on the massively accreting matter. The heavy nuclei ignored in previous works play an important role in the massive accretion disk. In the high dense matter with ρ = 10⁹ ∼ 10¹³ g cm⁻³, the heavy nuclei interact with ambient nucleons. The fraction of free protons interacting with heavy nuclei is extremely small, Y_p = 10⁻⁴–10⁻⁵, while that in the free nucleons without heavy nuclei is Y_p = Y_n = Y_e = 10⁻¹–10⁻² (Chen & Belobororodov 2007; Kawanaka & Mineshige 2007; Lee et al. 2005). This difference decides the opacity of neutrinos in the dense region of accreting matter. In previous works, the neutrinos become opaque at the inner disk, while $\bar{\nu }_e$ in the heavy nuclei are transparent all over the disk. The huge neutrinos extract the thermal energy (liberated gravitational energy) in the disk and may play an important role in the formation of relativistic jets emitting γ-ray bursts. On the other hand, the neutrinos absorbed in the accreting matter fall into a black hole.
• 2.  
  Accretion disk with Keplerian angular momentum. Previous works on the massive accreting matter with neutrino loss are based on ADAF (Popham et al. 1999; Chen & Belobororodov 2007; Kawanaka & Mineshige 2007). Since the formation of collimated jets requires pre-collapsed matter of a massive star with sufficient angular momentum, we have investigated the STAD with Keplerian angular momentum. The thermal history of accreting matter in ADAF is dissimilar to that in STAD. The strong compression by the rapid deceleration of ADAF at the middle region of the accretion disk largely heats up the accreting matter to be in the state of free nucleons. The gradual viscous heating and efficient cooling by antielectron neutrinos in STAD maintains the accreting matter at relatively low temperature, T ≈ 3 × 10¹⁰ K over a long range of accreting radius which produces very high dense matter, ρ ≈ 10¹² ∼ 10¹³ g cm⁻³, around a central black hole.
• 3.  
  Detailed balances in neutrino reactions. The reactions of leptons in the previous works of ADAF have been assumed to hold the detailed balances (Belorborodov 2003). These conditions of the detailed balances restrict the variations of ρ, T, and Y_e along the flow. However, these simplified assumptions of the detailed balances are not adequate for the massive accretion with degenerated electrons. We have solved the lepton reactions by the direct methods without equilibrium conditions and shown the profiles of lepton fractions along the flow. The electron fraction becomes very small at the inner disk, Y_e < 0.01.
• 4.  
  Mean energy of ejected neutrinos. The mean energy of neutrinos ejected from the disk is large, $\bar{E}_{\nu } \approx 20$ MeV, in comparison with previous results, $\bar{E}_{\nu } \approx 8$ MeV (Lee et al. 2005), because the neutrinos, $\bar{\nu }_e$, in STAD are generated in high dense matter with large degeneracy, $E_{\bar{\nu }_e} \approx \eta _{\bar{\nu }_e} k$T, $\eta _{\bar{\nu }_e} \approx 5$. The production condition of "neutrino-driven wind" (Qian & Woosley 1996) is satisfied in the case of STAD while it is difficult for ADAF.
• 5.  
  Transient thermal instability. We have investigated the transient thermal instability of the massive accretion disk cooled by $\bar{\nu }_e$, where the scattering optical depth is very large, $\tau _{s}(\bar{\nu }_e) \approx 10^2$. The accretion disk becomes thermally unstable at least within the duration of the diffusion time, t_diff ≈ 10 ms, which is the characteristic period of γ-ray emission for the short spike-like emissions.

We thank Professor T. Yoshida for helpful discussions and for very useful comments that improved the presentation of the paper.

The cross sections of neutrinos, opacities, emissivities, and reaction rates used in our model are listed in Tables 2–6. The numerical functions used in the reaction rates are listed in Table 6. Here the spectra of leptons are assumed to be Fermi–Dirac distributions with the local gas temperature T. The blocking in the lepton phase space is calculated by the approximate expressions given by Ruffert et al. (1996).

Table 6. Integrals and Blocking Factors

No.	Integrals and Blocking Factors
1	$F_{k}^+(\eta _{\nu },\eta _{e},q) = \int _0^\infty \frac{x^{k-2}\Theta (x+q-m)(x+q)^2}{1+\exp {(x-\eta _{\nu })}}(1-f_e(q+x))dx=\int _0^\infty \frac{x^{k-2}\Theta (x+q-m)(x+q)^2dx}{(1+\exp {(x-\eta _{\nu })})(1+\exp {(\eta _{e}-q-x)})}$
2	$F_{k}^-(\eta _\nu,\eta _e,q)=\int _0^\infty \frac{x^{k-2}\Theta (x+q-m)(x+q)^2}{(1+\exp {(x+q-\eta _e)})}(1-f_\nu (x))dx = \int _0^\infty \frac{x^{k-2}\Theta (x+q-m)(x+q)^2dx}{(1+\exp {(x+q-\eta _e)}) (1+\exp {(\eta _\nu -x)})}$
3	$F_k^{q^{\prime }-}(\eta _{\nu }, \eta _e) = \int _0^\infty \frac{(x+q^{\prime })^2 x^{k-2}}{\exp {(x+q^{\prime }-\eta _e)} +1}(1-f_{\nu }(x))dx = \int _0^\infty \frac{(x+q^{\prime })^2 x^{k-2} dx}{(\exp {(x+q^{\prime }-\eta _e)} +1)(\exp {(\eta _{\nu }-x)} +1)}$
4	$F_k^{q^{\prime }+}(\eta _{\nu },\eta _e) = \int _0^\infty \frac{x^{k-2}(x+q^{\prime })^2}{1+\exp {(x-\eta _{\nu })}}(1-f_e(q^{\prime }+x))dx = \int _0^\infty \frac{x^{k-2}(x+q^{\prime })^2 dx}{(1+\exp {(x-\eta _{\nu })}) (1+\exp {(\eta _e -q^{\prime } -x)})}$
5	$\langle 1-f_{e}(\varepsilon)\rangle _{nn} \approx \big\lbrace 1+\exp {\big[\eta _{e}-\frac{F_5(\eta _n)}{F_4(\eta _n)}\big]} \big\rbrace ^{-1}$
6	$\langle 1-f_{\nu }(\varepsilon)\rangle _{nn} \approx \big\lbrace 1+\exp {\big[\eta _{\nu }-\frac{F_5(\eta _n)}{F_4(\eta _n)}\big]} \big \rbrace ^{-1}$
7	$\langle 1-f_{\nu _i}(\varepsilon)\rangle _{ee} \approx \big\lbrace 1+\exp {\big[\eta _{\nu _i}-\frac{1}{2}\big (\frac{F_5(\eta _e)}{F_4(\eta _e) }+\frac{F_5(\eta _{e^+})}{F_4(\eta _{e^+})}\big) \big]} \big \rbrace ^{-1}$
8	$\langle 1-f_{e}(\varepsilon +q^{\prime })\rangle \approx \big \lbrace 1+\exp {\big [ \eta _e -\frac{F_5(\eta _{\nu _e})}{F_4(\eta _{\nu _e})} -q^{\prime }\big ]} \big \rbrace ^{-1}$
9	$\langle 1-f_{\nu _i}(\varepsilon)\rangle _\gamma \approx \big (1+\exp {\big [\eta _{\nu _i}-\big(1+\frac{1}{2} \frac{\gamma ^2}{1+\gamma }\big)\big ]}\big)^{-1}$
10	$\langle 1-f(\eta _{e^+})\rangle \approx \big\lbrace 1+\exp {[-(\bar{\varepsilon }^+ - \eta _{e^+})]} \big \rbrace ^{-1}$, $\bar{\varepsilon }^+ =\frac{7\pi ^4}{180\zeta (3)}kT$.

Notes. The quantity Θ(x) is the unit step function, Θ(x) = 0 if x < 0 and Θ(x) = 1 if x ⩾ 0. The quantity ζ(3) is the ζ(n) function with n = 3, ζ(3) = 1.202, and m is m = m_ec²/kT.

Download table as:  ASCIITypeset image

The accretion disk cooled by neutrinos is geometrically very thin. The calculating position is limited for r>1.885r_g, where 1.885r_g is the radius with null shear stress for a = 1. The radial velocity of the accreting flow in the orbiting frame is not relativistic. We neglect the relativistic effects on the transports of neutrinos. The analytic formulae of neutrino's transfers in the orbiting frame are derived by using the simplified treatment of the radiative transfer given by Popham & Narayan (1995). The $\hat{z}$-coordinate is taken to be perpendicular to the equatorial plane. Introducing the two streams, with intensities I⁺, I⁻ moving at an angle $\cos \theta =1/\sqrt{3}$ relative to $+\hat{z}$ and $-\hat{z}$, defines the net flux density j and the total intensity $\bar{j}$ such as ${j}=(I^+ - I^-)/2\sqrt{3}$ and $\bar{j}=(I^+ + I^-)/2$. The transfers of neutrinos are then described by the production rate of neutrinos $\dot{n}^+$, absorption opacity κ_a, scattering opacity κ_s, and the total opacity κ = κ_a + κ_s as

$$\begin{eqnarray} \frac{d{j}}{d\hat{z}}&=&{-} \kappa _a\bar{j} + \dot{n}^+ \end{eqnarray} \tag{ B1 }$$

$$\begin{eqnarray} \hspace{-1.5pc}\frac{d\bar{j}}{d\hat{z}} &=&{-} 3\kappa {j}. \end{eqnarray} \tag{ B2 }$$

The boundary condition at the center of the disk is j = 0. To derive the simplest solution, let us assume that $(\dot{n}^+ -\kappa _a\bar{j}), \kappa$ are independent of $\hat{z}$ (Hubeny 1990). Equations (B1) and (B2) can be integrated to give

$$\begin{eqnarray} j(\hat{z})&=&\big(\dot{n}^+_c - \kappa _a\bar{j}_c\big)\hat{z} \end{eqnarray} \tag{ B3 }$$

$$\begin{eqnarray} \bar{j}(\hat{z}) &=&\bar{j}_c - \frac{3}{2}\kappa \big(\dot{n}_c^+ - \kappa _a\bar{j}_c\big)\hat{z}^2, \end{eqnarray} \tag{ B4 }$$

where $\dot{n}^+_c$ and $\bar{j}_c$ are the values of $\dot{n}^+$ and $\bar{j}$ at the center of the disk, $\hat{z}=0$. The boundary condition at the surface of the disk is I⁻ = 0, i.e., $\bar{j}=\sqrt{3}{j}$. Applying this boundary condition, we find

$$\begin{equation} \bar{j}_c=\frac{\sqrt{3}+1.5\tau }{1+\sqrt{3}\tau _a+1.5\tau \tau _a}\dot{n}_c^+H, \end{equation} \tag{ B5 }$$

where τ_a is the absorbing optical depth, τ_a = κ_aH, and τ is the total optical depth, τ = κH. The escaping flux density of any sort of neutrino ν_i is then given by

$$\begin{equation} F_{n(\nu _i)}={j}_{\nu _i}(H)=\frac{\dot{n}^+_{\nu _i}H}{1.5\tau _{\nu _i a}\tau _{\nu _i}+\sqrt{3}\tau _{\nu _i a} + 1}, \end{equation} \tag{ B6 }$$

where $\tau _{\nu _i}$ and $\tau _{\nu _i a}$ are the total optical depth and the absorbing optical depth of ν_i, and $\dot{n}^+_{\nu _i}$ is the production rate of ν_i per unit volume.

The dynamically equilibrium structure of the disk is given by the Euler equation. The equation of motion of the fluid, h^k_iT^j_k;j = 0, for the steady, axially symmetric, and rotating ideal fluid can be expressed in the total differential form:

$$\begin{equation} \frac{dp}{\rho +\epsilon +P}=\gamma ^2\big({-}d\nu +v_{(\varphi)}^2d\psi -v_{(\varphi)}v_\omega d\ln \omega\big) \equiv dU. \end{equation} \tag{ C1 }$$

The equipressure surfaces are given by the equation, U=constant. In the Newtonian limit the quantity U = U(p) is equal to the total potential (gravitational plus centrifugal ones) expressed in the units of c² (Abramowicz et al. 1978). We solve the disk structure in terms of cylindrical coordinates ϖ, z, and φ instead of radial coordinates which are combined through the relations

$$\begin{eqnarray} &&r \equiv \sqrt{\varpi ^2+z^2}, \quad \theta \equiv \cos ^{-1}{(z/r)}. \end{eqnarray} \tag{ C2 }$$

If no pressure force exerts in the ϖ-direction, Equation (C1) determines the velocity of the geodesic circular orbit, v_(φ). Let us introduce the new variables, $\tilde{a},\,\tilde{m},s, c$ defined by $\tilde{a}\equiv a/r, \quad \tilde{m}\equiv M/r$, s ≡ sin θ,  c ≡ cos θ. The following functions are defined by

$$\begin{eqnarray} \Sigma ^{\prime } &=& 2rs(1- \tilde{a}^2c^2), \quad \Delta ^{\prime } = 2rs(1-\tilde{m}), \nonumber \\ A^{\prime } &=& 2r^3s \lbrace 2+\tilde{a}^2(1+\tilde{m}(1+c^2)- \tilde{a}^2c^2) \rbrace, \nonumber \\ A_v &=& 2\psi _{, \varpi } = \frac{A^{\prime }}{A} - \frac{\Sigma ^{\prime }}{\Sigma } + \frac{2c^2}{rs}, \nonumber \\ B_v &=& -v_\omega \frac{\partial \ln {\omega }}{\partial \varpi } = \frac{s}{r} - \frac{A^{\prime }}{A}, \nonumber \\ C_v &=& 2\nu _{, \varpi }= \frac{\Sigma ^{\prime }}{\Sigma } +\frac{\Delta ^{\prime }}{\Delta } -\frac{A^{\prime }}{A}.\nonumber \end{eqnarray}$$

The circular velocity is then given by

$$\begin{equation} v_{(\varphi)} = \frac{-B_v \pm \sqrt{B_v^2+A_vC_v}}{A_v}. \end{equation} \tag{ C3 }$$

This velocity at the equatorial plane is expresses as

$$\begin{eqnarray} v_{(\varphi)} &=& \frac{\Theta _1(r)}{r^{1/2}},\nonumber\\ \Theta _1(r) &\equiv& \frac{1-2aM^{1/2}r^{-3/2} + a^2r^{-2}}{(1- 2Mr^{-1}+ a^2r^{-2})^{1/2} (1+aMr^{-3/2})}. \end{eqnarray} \tag{ C4 }$$

The vertical force is described by the gradient of the equipotential, ∇_zU. The first order of the expansion of ∂U/∂z in z/r(≪1) gives the approximate expression such as

$$\begin{eqnarray} &\frac{\partial U}{\partial z} &\approx -\frac{Mz}{r^3}\Phi, \end{eqnarray} \tag{ C5 }$$

$$\begin{eqnarray} \Phi (r) &=& \frac{1-\omega /\Omega }{D}\left (\frac{\gamma \Psi }{\Theta }\right)^2 \biggl \lbrace 1 + \frac{1+v_{(\varphi)}^2}{v_{(\varphi)}v_\Omega } \Bigl [ 1+\Bigl (\frac{a}{r}\Bigr)^2 \Bigr ] \nonumber\\ &&- \Bigl (1+ \frac{ \omega }{\Omega } + \frac{1}{v_{(\varphi)} v_\Omega }\Bigr) \frac{\acute{A}}{\Psi } + \frac{1-M/r}{v_{(\varphi)} v_\Omega D} \biggr \rbrace, \nonumber \\ \Theta (r)&=& 1+ \frac{aM^{1/2}}{r^{3/2}}, \quad D=1-\frac{2M}{r} + \Bigl (\frac{a}{r}\Bigr)^2,\nonumber\\ \Psi &=& 1 + \left(\frac{a}{r}\right)^2 \left(1 + 2\frac{M}{r}\right), \nonumber \\ \acute{A} &=& 2 + \Bigr (4- 3\frac{M}{r}\Bigr)\Bigl (\frac{a}{r}\Bigr)^2 + \Bigl (\frac{a}{r}\Bigr)^4, \nonumber\\ v_\Omega &= &\Omega e^{\psi -\nu }=v_{(\varphi)} + \omega e^{\psi -\nu }. \end{eqnarray} \tag{ C6 }$$