1. Introduction
Black hole X-ray binary systems consisting of either a high-mass X-ray binaries (HMXB) such as “Cyg X-1” or a low-mass X-ray binaries (LMXB) such as “GX339-4” show two quasi-steady X-ray spectral states—hard state and soft state—and state transitions between them. X-ray satellites and other instruments have revealed the hidden nature of X-ray binaries, such as existence of the intermediate state, jet ejections and quasi-periodic oscillations [
1]. Theoretically, accretion flows in the hard state are explained by optically thin high-temperature accretion flows, “advection-dominated accretion flows (ADAFs)” [
2,
3]. On the other hand, the soft state is explained by the optically thick cool accretion disks, “Shakura Sunyaev disks (SSDs)” [
4]. To understand the structural change of accretion flows during state transitions; however, two-dimensional or three-dimensional numerical studies are necessary.
Hard-to-soft state changes has been studied by Machida et al. by performing three-dimensional MHD simulations considering radiative cooling by bremsstrahlung [
5]. They found that the magnetically supported mildly cool accretion disk is formed during the vertical contraction of the disk driven by the loss of internal energy due to radiative cooling.
Recently hydrodynamical (HD) simulations have been performed using the alpha viscosity. Das and Sharma carried out two-dimensional HD simulations taking into account bremsstrahlung and isotropic thermal conduction [
6]. By changing the accretion rate, they reproduced transitions from hard state to soft state and vice versa. Wu et al. carried out two-dimensional HD simulations including synchrotron and inverse-Compton scattering in addition to thermal bremsstrahlung [
7]. Moreover, they studied two-temperature structures of accretion flow, although they assume a simple equation of ion and electron temperatures. Based on their numerical results, they proposed two-phase accretion model for the intermediate state of black hole X-ray binaries. HD simulations with alpha viscosity, however, cannot simulate the formation of low beta (
) disks. Magnetohydrodynamical approach is preferable, because the origin of viscosity in accretion disk is believed to be the angular momentum transport driven by the magneto-rotational instability [
8,
9].
Thermal conduction is regarded as important in solar astronomy. Yokoyama and Shibata performed MHD simulations to investigate solar flares by including anisotropic thermal conduction [
10,
11,
12]. The heat released by magnetic reconnection above the chromosphere is transported downward to the chromosphere along the magnetic field line and evaporates them. Protostellar flares are also studied taking into account thermal conduction [
13]. Meyer and Meyer-Hofmeister and their collaborators intensively studied the role of thermal conduction in accretion disk system, such as dwarf novae and black hole accretion systems from stellar size to the size of active galactic nuclei [
14,
15]. Once we consider the coexistence of hot corona and cool accretion disk, inevitably evaporation occurs. We must include the effect of thermal conduction, when we study the accretion disk surrounded by hot corona.
We perform two-dimensional MHD simulations including thermal conduction to study the transition from hard state to soft state, in other words, to vertical contraction of the hot accretion flow by extracting the energy by radiative cooling. Our purpose is to study the effect of thermal conduction for the coexistence of hot and cool region. In
Section 2 we introduce the basic equations and initial and boundary conditions. In
Section 3 we present numerical results. In
Section 4 we discuss and compare the results with previous works.
4. Discussion
We solved the 2D MHD equations including the thermal conductivity. Previous works done by Das and Sharma, and Wu et al. are not MHD simulations but HD simulations using alpha viscous parameter [
6,
7]. We did not assume alpha viscosity. We assume that the angular momentum is transported mainly by the
element of the magnetic stress in the rotating gas. We define viscous alpha parameter
as
where “Domain” is the region of integration,
,
and
.
Figure 4a shows the time evolution of viscous parameter
. Black and red lines show the results of Model 1 and Model 2, respectively. Before radiative cooling is included (
), both calculations exhibit the magnitude is around
. This is a typical value used in hot accretion flows [
6,
7]. There is no large difference between our alpha parameter and ones of previous works. Hot accretion flows do not have large difference between models with thermal conduction or not. After radiative cooling is taken into account, the cool and dense accretion disk appears and
rises and drops in a short time. There is the maximum
. In the early stage of transition, the magnetic field changes the structure drastically and azimuthal magnetic field becomes strong by condensation [
5]. That is the reason parameter
shows rapid variation. The evolution of alpha parameter is not considered in previous works. Further analysis of the angular momentum transportation will be done in future.
Figure 4b shows the evolution of the luminosity normalized by the Eddington luminosity
. Here
is the proton mass and
is the Thomson cross section. We calculate the luminosity as
Where the “Domain” means the region as
,
and
. Black and red lines show the results of Model 1 and Model 2, respectively, as well as
Figure 4a. Despite the appearance of the intermediate region, there is no difference between Model 1 and Model 2. Since mainly the cool accretion disk plays a role of the radiation of bremsstrahlung, we cannot distinguish the two models.
Figure 5a shows the radiative cooling rate of Model 2 at
. Region where radiative cooling is strong is localized in the cool accretion disk whose density is high. There is the temperature minimum at the midplane. Therefore, the radiative cooling rate at midplane become smaller than accretion flows around midplane.
Figure 5b shows the thermal conduction rate of Model 2 at
. Thermal conduction works at the large temperature gradient. In this region
, the vertical magnetic field is greater than the radial magnetic field (
Figure 3d,f). Heat is transported from the intermediate region to the cool accretion disk effectively.
To compare the efficiency of thermal conduction and radiative cooling, we computed the Field length
without including the heating term [
24,
25]. When the Field length is longer than the scale of the medium, thermal conduction dominates radiative cooling. On the other hand, when the Field length is shorter than the scale of the medium, radiative cooling dominates thermal conduction. Our numerical results show the magnetic field changes temporally and spatially in the
plane. Since magnetic turbulence develops in the disk, we simply assume that heat conduction isotropic in the
plane. The Field length computed without including the heating term is
Figure 6a,b show the distribution of the Field length
of Model 2 at
and
, respectively. In the coronal region at
, since the Field length is longer than the size of the simulation region, thermal conduction dominates radiative cooling. However, in the region
, since radiative cooling becomes dominant over heat conduction, hot accretion flows can condensate by the cooling instability. On the other hand, thermal conduction suppresses the cooling instability in the intermediate region at
.
Figure 7 shows the evolutions of mass and volume whose medium is in three temperature ranges in the cylinder defined as
,
and
except for the sphere
. Three temperature range are
(denoted by red line),
(denoted by green line) and
(denoted by blue line), these temperature ranges are supposed to be the hot corona, the intermediate region and the cool accretion disk, respectively.
Figure 7a,b show that the intermediate region remains by thermal conduction against radiative cooling. Although the mass of intermediate region is small compared to cool accretion disk, the volume of intermediate region is large compared to other temperature region (
Figure 7c,d). These situations and sandwich structure of three components is helpful to produce the hard X-ray spectral component by inverse-Compton processes. Post process calculations by a Monte Carlo simulation for X-ray spectrum will be presented in subsequent papers.
In this study, we considered radiative cooling by bremsstrahlung. Synchrotron radiation and inverse-Compton scattering can change the structure of accretion disks in the equatorial region and the intermediate region. Das and Sharma pointed out the possibility that synchrotron radiation and inverse-Compton scattering have same dependency on the density as bremsstrahlung [
6]. We think that increase of the density may compensate the lack of radiative cooling. When we consider radiative cooling by synchrotron radiation and inverse-Compton scattering, the formed cool accretion disk has lower temperature and higher number density than the results of our present study. Inverse-Compton scattering reduces the thickness of the intermediate region. On the other hand, thermal conduction suppresses the shrinks of the intermediate region. We will perform the parameter survey of density in future.
We set the upper limit of temperature to evaluate the coefficient of thermal conductivity, since we solve one-temperature MHD equations. We suppose that the upper limit of temperature is electron temperature in two-temperature accretion flows [
2]. It is also useful to avoid the unusual thermal conductivity in high-temperature and low-density corona. As is well known, ADAFs has the two-temperature structures. The electron temperature is suppressed to be lower than the ion temperature since the energy exchange rate between ions and electrons by the Coulomb coupling is low due to the low density and the short free-fall time scale. It is preferable to simulate the two-temperature MHD to consider the state transition when the intermediate region exists. Wu et al. consider the two-temperature HD models; however, they do not calculate the two energy equations for ions and electrons independently [
7]. We are seeking to update our codes which solve two-temperature MHD equations.
The state transition is driven by the variation of the mass accretion rate. However, we do not include the mass supply calculation in our codes. We studied the various initial condition of density,
. Instead of the mass supply, we choose the critical initial condition of the density which can make the hot accretion flow condense. It is easy to obtain the results of the MHD simulation of the state transition. Machida et al. and Wu et al. adopt this concept [
5,
7]. However, it is preferable to implement the subroutine of the mass supply to achieve the cycle of the state transition in simulations by controlling the mass accretion rate [
6]. This difficulty should be solved in future.