# 2D MHD Simulations of the State Transitions of X-Ray Binaries Taking into Account Thermal Conduction

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Methods

#### 2.1. Basic Equations

#### 2.2. Initial Conditions

#### 2.3. Grids and Boundary Conditions

## 3. Results

#### 3.1. Results of Model 1

#### 3.2. Results of Model 2

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Results of Model 1. (

**a**) Distribution of number density at $t/{t}_{0}=10$ before cooling term is included. (

**b**) Distribution of number density at $t/{t}_{0}=30$. (

**c**) Vertical distribution of number density at three different time and at $r/{R}_{0}=0.8$. Black line shows initial condition at $t/{t}_{0}=0$. Blue line shows result at $t/{t}_{0}=10$. Red line shows result at $t/{t}_{0}=30$. (

**d**) Distribution of temperature at $t/{t}_{0}=10$. (

**e**) Distribution of temperature at $t/{t}_{0}=30$. (

**f**) Vertical distribution of temperature. Color scale is the same as (c). (

**g**) Distribution of magnetic energy density ${\left(B/{B}_{0}\right)}^{2}/8\pi $ at $t/{t}_{0}=10$. (

**h**) Distribution of magnetic energy density at $t/{t}_{0}=30$. (

**i**) Vertical distribution of magnetic energy density. Colors have the same meaning as (

**c**) and (

**f**).

**Figure 2.**Results of Model 2. Same quantities and conditions of visualization as Figure 1.

**Figure 3.**Results of Model 1 and Model 2 at $t/{t}_{0}=30$. Dashed lines represent the result of Model 1. Solid Lines represent the results of Model 2. These panels show the vertical distributions of the nine physical quantities averaged between $r/{r}_{0}=0.75$ and $r/{r}_{0}=0.85$. Black lines indicate positive values. Red lines indicate negative values, that is, mean the absolute values of quantities. (

**a**) Number density. (

**b**) Temperature. (

**c**) Pressure. (

**d**) Radial component of velocity. (

**e**) Azimuthal component of velocity. (

**f**) Vertical component of velocity. (

**g**) Radial component of magnetic field. (

**h**) Azimuthal component of magnetic field. (

**i**) Vertical component of magnetic field.

**Figure 4.**Evolutions of the averaged magnetic stress and integrated luminosity. Black lines show the results of Model 1 without thermal conduction. Red lines show the results of Model 2 including thermal conduction. (

**a**) Averaged viscous alpha parameter ${\alpha}_{\mathrm{MHD}}=\langle \left({B}_{r}/{B}_{0}\right)\left({B}_{\phi}/{B}_{0}\right)/4\pi \rangle /\langle p/{p}_{0}\rangle $. Integration is done in the domain of $0.4<r/{R}_{0}<1.2$, $0<\phi <2\pi $ and $-0.3<z/{R}_{0}<0.3$. (

**b**) Integrated luminosity normalized by the Eddington luminosity. Integration is done in the domain of $r/{R}_{0}<10$, $0<\phi <2\pi $ and $-10<z/{R}_{0}<10$ excluding the region of $\sqrt{{r}^{2}+{z}^{2}}/{R}_{0}<0.3$. Colors denote same meaning as (

**a**).

**Figure 5.**(

**a**) Distributions of radiative cooling rate $-{Q}_{\mathrm{br}}/{Q}_{0}$ of Model 2 at $t/{t}_{0}=30$. ${Q}_{0}={\rho}_{0}{V}_{0}{}^{3}/{R}_{0}=5.1\times {10}^{16}\mathrm{erg}/\left({\mathrm{cm}}^{3}\cdot \mathrm{s}\right)$ is normalization unit. (

**b**) Distributions of thermal conduction rate ${Q}_{\mathrm{hc}}/{Q}_{0}$ at $t/{t}_{0}=30$.

**Figure 6.**Distributions of the Field length ${\lambda}_{\mathrm{F}}/{R}_{0}$ of Model 2. (

**a**) Result at $t/{t}_{0}=10$ (

**b**) Result at $t/{t}_{0}=30$.

**Figure 7.**Time evolution of mass and volume of Model 1 and Model 2. Results are obtained by the integration in the domain of $0<r/{R}_{0}<2$, $0<\phi <2\pi $ and $-1<z/{R}_{0}<1$ except for the region of $\sqrt{{r}^{2}+{z}^{2}}/{R}_{0}<0.3$. These are normalized by initial total mass and volume of the above domain, respectively. (

**a**) Model 1. Blue line shows the mass of gas whose temperature range is $T/{T}_{0}<0.1$. Green line shows the mass of $0.1<T/{T}_{0}<10$. Red line shows the mass of $10<T/{T}_{0}$. Black line shows sum of three components. (

**b**) Model 2. Same as (

**a**). (

**c**) Model 1. Time evolution of volume of each temperature range same as (

**a**). Colors mean same as (

**a**). (

**d**) Model 2. Same as (

**c**).

Variable | Quantity | Unit | Value |
---|---|---|---|

$r$, $z$ | Length | ${R}_{0}=10{r}_{\mathrm{g}}$ | $3.0\times {10}^{7}\mathrm{cm}$ |

${v}_{r}$, ${v}_{\phi}$, ${v}_{z}$ | Velocity | ${V}_{0}={V}_{\mathrm{K}}$ | $7.4\times {10}^{9}\mathrm{cm}/\mathrm{s}$ |

$t$ | Time | ${t}_{0}=2\pi {R}_{0}/{V}_{\mathrm{k}}$ | $2.5\times {10}^{-2}\mathrm{s}$ |

$\rho $ | Density | ${\rho}_{0}$ | $3.8\times {10}^{-6}{\mathrm{g}/\mathrm{cm}}^{3}$ |

$n$ | Number Density | ${n}_{0}={\rho}_{0}/\mu {m}_{\mathrm{H}}$ | $4.5\times {10}^{18}{\mathrm{cm}}^{-3}$ |

$T$ | Temperature | ${T}_{0}=\mu {m}_{\mathrm{H}}{V}_{\mathrm{K}}{}^{2}/{k}_{\mathrm{B}}$ | $3.4\times {10}^{11}\mathrm{K}$ |

$p$ | Pressure | ${p}_{0}={\rho}_{0}{V}_{\mathrm{k}}{}^{2}$ | $2.1\times {10}^{14}{\mathrm{erg}/\mathrm{cm}}^{3}$ |

${B}_{r}$, ${B}_{\phi}$, ${B}_{z}$ | Magnetic field | ${B}_{0}=\sqrt{{\rho}_{0}{V}_{\mathrm{k}}{}^{2}}$ | $1.4\times {10}^{7}\mathrm{G}$ |

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**MDPI and ACS Style**

Nakamura, K.E.; Machida, M.; Matsumoto, R.
2D MHD Simulations of the State Transitions of X-Ray Binaries Taking into Account Thermal Conduction. *Galaxies* **2019**, *7*, 22.
https://doi.org/10.3390/galaxies7010022

**AMA Style**

Nakamura KE, Machida M, Matsumoto R.
2D MHD Simulations of the State Transitions of X-Ray Binaries Taking into Account Thermal Conduction. *Galaxies*. 2019; 7(1):22.
https://doi.org/10.3390/galaxies7010022

**Chicago/Turabian Style**

Nakamura, Kenji E., Mami Machida, and Ryoji Matsumoto.
2019. "2D MHD Simulations of the State Transitions of X-Ray Binaries Taking into Account Thermal Conduction" *Galaxies* 7, no. 1: 22.
https://doi.org/10.3390/galaxies7010022