The Influence of Viscosity on the Truncation of Accretion Disks Around Black Holes

CHINESE ASTRONOMY AND ASTROPHYSICS

ELSEVIER

Chinese Astronomy and Astrophysics Chinese Astronomy and Astrophysics3333(2009) (2009) 9–16 9–16

The Influence of Viscosity on the Truncation of Accretion Disks Around Black Holes  NIU Qiang National Astronomical Observatories/Yunnan Observatory, CAS, Kunming 650011

Abstract The influence of viscosity on the truncation radius of the accretion disks around the black holes for black hole X-ray binaries and low luminosity AGNs, as well as the changes between the high and low state in black hole X-ray binaries are studied in this paper using the accretion disk-corona model. From the previous studies it is indicated that the coronal structure is very sensitive to the viscosity. So we calculated in detail the coronal structures for a series of viscosity coefficients. In order to compare with the observational results, analytical fitting of the numerical results was made for the relations between the maximum evaporation rate and the viscosity coefficient α, M˙ /M˙ EDD ≈ 1.08α3.35 and between the truncation radius and the viscosity coefficient, R/Rs ≈ 36.11α−1.94. These results can be used to explain the transitions between the high and low spectral states and the changes in the truncation radius. In particular, these results are applied to two black hole X-ray binaries, XTE J1118+480 and GX 339-4, and to one AGN, NGC 4636. Key words: physical data and process: black hole physics— accretion, accretion disk—X-ray: binaries

1. INTRODUCTION Recently the black hole accretion theory has further developed. This theory provides a good theoretical basis for the understanding of many high-energy systems, such as AGNs, black hole X-ray binaries and γ-ray bursts [1,2] . This model can be divided into two classes according to temperature, the “cold” and the “warm” ones. The cold accretion model is Received 2007–05–07; revised version 2008–03–21 A translation of Acta Astron. Sin. Vol. 49, No. 3, pp. 243-250, 2008  [email protected] 

0275-1062/08/$-see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.chinastron.2009.01.004

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essentially the model of the standard thin disk (Shakura & Sunyaev disk, hereafter SSD)[3] , in which the temperature in the accretion flow is typically lower than 106 – 107 K. This model is mainly used for systems with accretion rates lower than the Eddington limit. The warm accretion model is essentially the advection-dominated accretion flow (ADAF) model [4,5,6] , in which the warm accretion flow might be a two-temperature flow, i.e. ions and electrons have different temperatures with the electron temperature in the inner accretion disk between 109 K and 1011 K. In systems with low accretion rates the accretion disk often consists of a thin disk in the outer region and an ADAF disk in the inner region. The truncation mechanism between the thin disk in the outer region to the ADAF disk in the inner region has become a focus of attention for many people. Meyer et al. [8] and Liu et al. [9] proposed the thin disk evaporation model, in which the hot corona described by ADAF is above the thin accretion disk. Because there is heat conduction between the hot corona and the cold thin disk, cold material can pass to the hot gas and enter the corona through the evaporation process. If the evaporation rate is lager than the accretion rate, then the whole thin disk becomes the hot corona, i.e. the transformation from the thin disk to the ADAF disk is accomplished. And the thin accretion disk is truncated at the radius of the transformation, called the truncation radius. Now the X-ray spectrum of a black hole X-ray binary with different states generally consists of two parts, a hot component that can be well fitted with the multi-color black body radiation for the standard thin disk [10,11] , and a power-law component. In this paper we shall mainly discuss the scenario of the low/hard state. The geometric structure of the accretion disk in the low/hard state is a standard thin disk formed of some accretion material under the gravity of the black hole and this thin disk is truncated at a certain radius to become a hot disk (ADAF). So far there is good evidence for the low/hard state model, for instance, from the observations of XTE J1118+480 [12,13] . In most previous studies the viscosity coefficient has been simply taken as a constant: α= 0.3. Mayer et al.[14] discussed the effect of a varying viscosity coefficient in the accretion disk + corona model, but they adopted only three values for the viscosity coefficient, 0.1, 0.2 and 0.3. Considering that the viscosity mechanism is not so clear [15] while the structure of the corona is very sensitive to the viscosity coefficient, so we shall study the effect of the viscosity coefficient on the corona structure by adjusting the viscosity coefficient. Compared with previous work, not only were viscosity coefficients over a much wider range (from 0.1 to 0.6) calculated, but also many more data points were taken in each calculation. Using data fitting, we obtained analytical relations between the maximum evaporation rate and the viscosity coefficient and between the smallest truncation radius and the viscosity coefficient. It is proved with numerical calculations that the viscosity coefficient of the gas in the corona strongly affects the accretion disk truncation and the spectral state change. In addition, from a comparison between the supermassive black hole and the stellar black hole, we quantitatively verified a previous prediction that the evaporation rate and the truncation radius are not related to the mass.

2. CALCULATION MODEL A detailed description of the basic model and equations we adopted can be found in Ref. [16]. Consider the corona above the geometrically thin disk near the central black hole. In

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fact, this corona is an optically thin and very hot accretion flow located over a cold thin disk. As in the standard disk, gas in the corona rotates around the central object, and heat is produced through viscosity. Because the corona has a rather high temperature and under the corona there is a relatively cool disk, a large temperature gradient between the disk and the corona is generated, so that through the electrons the heat produced by the viscosity is very easily transmitted downward. When the temperature decreases from the higher value in the corona to the lower value in the chromosphere, downward heat transmission in the chromosphere becomes less effective, and material in the chromosphere is heated by the heat energy accumulated here and some material goes into the corona (this is called the process of material evaporation from the disk to the corona). This process boosts up the corona density so that energy balance can be reached by the heat energy transmitted from the gas energetic radiation above, and, on the other hand, pressure balance can also be established between the corona and the surface of the chromosphere. When gas evaporates from the cold accretion disk to the corona, it still keeps the original angular momentum and makes differential rotation around the central object as in the standard accretion disk. However, unlike the standard accretion disk, the ratio of the height H to the distance r is no longer less than unity. Gas in the corona is continually accreted onto the central object, causing the density to decrease in the corona so that the material on the boundary between the disk and the corona can no longer balance the heat conduction and so is heated and hence evaporated into the corona. This process results in a continuous evaporation of material from the cold thin disk below to the corona to reach a dynamic equilibrium. Thus material is accreted by the central object partly through the corona (the evaporation) and partly through the disk. Under the low accretion rate condition, evaporation in the region near the black hole is so very efficient as to exhaust the whole disk and the entire material flow passes through the corona to be continuously accreted onto the central black hole in the form of advection-dominated accretion flow (ADAF)[11] . The disk truncates at the radius where the evaporation rate is equal to the accretion rate. Our work in this paper is to study the variation of the truncation radius with different viscosity coefficients. The dynamical structure of the corona is described by the following equations: The state equation: Rρ T, (1) P = μ where the chemical composition of the corona is assumed to be the standard cosmic abundance (X = 0.75 and Y = 0.25) and the mean molecular weight is taken to be μ = 0.62. The continuity equation: d 2 2z (ρvz ) = ρvR − 2 ρvz . dz R R + z2 The kinematics equation in the z direction: ρvz

dP GM z dvz =− −ρ 2 . dz dz (R + z 2 )3/2

The energy conservation equation:   2   v γ P GM d + − 2 ρvz + F c = dz 2 γ−1 ρ (R + z 2 )1/2

(2)

(3)

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 2  γ P GM 2 3 v αP Ω − ne ni L(T ) + ρvR + − 2 2 R 2 γ−1 ρ (R + z 2 )1/2   2   v γ P GM 2z + − 2 − 2 ρvz + Fc , R + z2 2 γ −1 ρ (R + z 2 )1/2

(4)

where ne ni L(T ) is the cooling rate of the Bremsstrahlung and Fc is the heat conductivity given in Equation (5) (see Equation (49) in Ref.[16]: Fc = −κ0 T 5/2

dT , dz

(5)

where for the completely ionized plasma, κ0 = 10−6 g.cm.s−3 K−7/2 . For all the parameters the standard values are adopted and the CGS system is taken. It can be seen that four variables, P (z), T , Fc and m(z) ˙ ≡ ρvz are included in Equations (2)∼(5), so solutions can be found with four given boundary conditions[17] . At the lower boundary, z = z0 , T = 106.5 K and Fc = -2.73×106P while at the upper boundary, z = z1 , Fc = 0, vz2 = Vs2 ≡ P/ρ = RT /μ. In the numerical calculation, we start with some assumed values at the lower boundary, P0 and m ˙ 0 , Then, after integrating along the z direction, if the upper boundary conditions ˙ 0 are correct; otherwise, we start again are satisfied, then the assumed values of P0 and m with some other values of P0 and m ˙ 0 , until the upper boundary conditions are satisfied.

3. RESULTS OF CALCULATION For a black hole of mass M = 108 M , we calculated its corona structure for various values of the viscosity coefficient. The results are given in Table 1, which mainly illustrates how the evaporation rate in the accretion disk around the black hole varies with different viscosity coefficients. Table 1 contains the results for two black hole masses, 108 M and 10 M . The results for the former mass are the main ones, and the results for the latter mass are given just as comparison. We see that the truncation radii (in units of the Schwarzschild radius) in the two cases are basically equal and the evaporation rates (in units of the Eddington accretion rate) in the two cases are also equal. Therefore the results for the case of 108 M can be used to completely account for the status in X-ray binary systems. The dependence of the evaporation rate on viscosity can be seen in Fig.1. Obviously, the evaporation rate is very sensitive to the viscosity coefficient: it increases rapidly with increasing viscosity coefficient. The relation between the maximum evaporation rate and the viscosity coefficient α is presented in Fig.2a. It can be seen that the relation can be approximately expressed with Equation (6) below. The correlation coefficient here is R = 0.997. M˙ /M˙ EDD ≈ 1.08α3.35 .

(6)

It can be seen from Equation (6) that the maximum evaporation rate rapidly increases with even a small increase of the viscosity coefficient. This result might be used to explain the change from the hard state to the soft state for X-ray binaries with different accretion rates, because the maximum evaporation rate represents the boundary between the main parts

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of the standard disk and the ADAF disk. If the accretion rate is equal to the maximum evaporation rate, then accretion exists everywhere in the disk, and the spectrum is in the Table 1 Calculated results for different values of the viscosity coefficient. M is the mass of the central object, α is the viscosity coefficient, M˙ is the evaporation rate, m ˙0 and P0 are the material flow density and the pressure at the lower boundary in the vertical direction respectively, and Rs and M˙ EDD are the Schwarzschild radius and the Eddington accretion rate respectively. R, m ˙ 0 and P0 are all in the CGS system. M (M ) α logR log B M = log(2πR2 . m0 ) 108 0.1 16.80 22.54 17.00 22.62 17.10 22.56 0.15 16.31 23.13 16.57 23.49 16.60 23.46 0.20 16.16 23.74 16.41 23.90 16.62 23.83 0.25 15.95 24.07 16.20 24.22 16.50 24.10 0.30 15.90 24.46 16.00 24.48 16.24 24.42 0.35 15.75 24.63 15.90 24.70 16.05 24.67 0.40 15.75 24.86 15.81 24.88 16.20 24.70 0.45 15.60 25.02 15.71 25.04 15.80 25.03 0.50 15.50 25.17 15.64 25.19 15.90 25.09 0.60 15.37 25.42 15.47 25.45 15.67 25.38 10 0.30 8.90 17.46 9.00 17.48 9.24 17.42

log P0 -1.993 -2.511 -2.846 -0.321 -0.801 -0.902 -0.463 -0.157 -0.827 1.217 0.594 -0.378 1.565 1.288 0.549 2.067 1.672 1.211 2.036 2.008 0.738 2.672 2.368 2.094 3.026 2.640 1.806 3.512 3.242 2.613 8.564 8.288 7.549

R/RS 2.139 ×103 3.390 × 103 4.268 × 103 7.001 × 102 1.260 × 103 1.350 × 103 4.916 × 102 8.714 × 102 1.400 × 103 3.022 × 102 5.373 × 102 1.072 × 103 2.693 × 102 3.390 × 102 5.865 × 102 1.906 × 102 2.693 × 102 3.804 × 102 1.906 × 102 2.189 × 102 5.373 × 102 1.349 × 102 1.739 × 102 2.139 × 102 1.072 × 102 1.471 × 102 2.693 × 102 7.948 × 102 1.001 × 102 1.586 × 102 2.693 × 102 3.390 × 102 5.865 × 102

B M/M Edd 2.484×10−4 2.987 × 10−4 2.260 × 10−4 9.660 × 10−4 2.245 × 10−3 2.119 × 10−3 3.963 × 10−3 5.717 × 10−3 4.866 × 10−3 8.417 × 10−3 1.192 × 10−2 8.978 × 10−3 2.066 × 10−2 2.164 × 10−2 1.911 × 10−2 3.091 × 10−2 3.607 × 10−2 3.390 × 10−2 5.397 × 10−2 5.466 × 10−2 3.615 × 10−2 7.450 × 10−2 7.948 × 10−2 7.659 × 10−2 1.053 × 10−1 1.106 × 10−1 1.077 × 10−1 1.884 × 10−1 2.042 × 10−1 1.718 × 10−1 2.066 × 10−2 2.164 × 10−2 1.911 × 10−2

soft state. If the accretion rate is less than the maximum evaporation rate, then the inner disk is truncated, instead, the ADAF is formed, and the spectrum is in the hard state. A certain value of α can be selected to obtain the corresponding maximum evaporation rate, and if the accretion rate is larger than the maximum evaporation rate, then the disk can be extended to the last stable orbit and the spectral state is soft. If the accretion rate is less than the evaporation rate, the material in the inner disk region is evaporated, and advection-dominated accretion flow (ADAF) is formed[18] . Therefore, Equation (6) is useful for the determination of the spectral state of black hole binaries and AGNs. Fig.2b presents the relation between the truncation radius (in units of the Schwarzschild radius) and the viscosity coefficient in the case where the accretion rate is equal to the maximum evaporation

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Fig. 1 The dependence of the evaporation rate on the distance from the black hole center, for 5 values of the viscocity coefficient α. For a given α, the evaporation rate increases with decreasing distance, reaching a maximum at several hundreds Schwarzschild radii, then declines.

rate. Analytically, the relation from data fitting is: R/Rs ≈ 36.11α−1.94

(7)

with linear correlation coefficient, R = 0.998.

Fig. 2

(a) The relation between the maximum evaporation rate and the viscosity coefficient (The straight

line in the figure is the linear fitting result.) (b) The relation between the truncation radius and the viscosity coefficient when the accretion rate is equal to the maximum evaporation rate

Fig.2b shows that a larger α corresponds to a smaller truncation radius. It is also seen from Fig.1 that, at the same radius, the evaporation rate increases with α. This result indicates that this variation with viscosity coefficient can be interpreted as different black hole accretion systems with the same accretion rate having different truncation radii. If the radius referred to is in units of the Schwarzschild radius and the accretion rate referred to is in units of the Eddington accretion rate, the results obtained in this paper are

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independent of the mass, so that our results are equally effective for both AGNs and black hole binaries.

4. APPLICATION OF CALCULATION RESULTS Our theoretical results in this paper can be used to explain the spectral state change and the disk truncation radius found in some observations. For the standard viscosity coefficient α = 0.3, the model predicts the spectral state ˙ EDD, and this is consistent with the observed mean statistical result to change at 0.02M [19] of Maccarone . It is found that although the mean accretion rate is 0.02, in fact, the ˙ EDD during the state change. This may range of the accretion rate is from 0.01 to 0.04M [20] be caused by different viscosities. Belloni et al. and Zdziarski et al.[21] also found that even for the same object at different explosion periods, during the spectral state change, its accretion rate is also different and that is also possibly caused by a viscosity change. Our theoretical results are able to explain other observational phenomena. The black hole X-ray binary, XTE J1118+480, was discussed in Ref.[13]. Its mass function is f (M ) = 6.00 ± 0.36 [23,24] . They took the black hole mass as M = 8M to fit the spectrum. Yuan et al.[13] M ˙ EDD found the disk truncation radius to be Rtr = 300Rs , the accretion rate, M˙ = 0.05M and α = 0.3. The model with a hot accretion flow in the inner part and a cold disk in the outer part was adopted for their fitting that is similar to ours in this paper. It is noted that our definition of the viscosity coefficient is 3/2 times theirs, thus α = 0.3 in their model is equivalent to α = 0.45 in ours. If α = 0.45 is adopted in our model, then the calculated truncation radius and accretion rate are consistent with their results within a certain error range (see Fig.1). This implies that the change of the viscosity rate is able to explain this observational phenomenon. Our model is also able to explain the spectral state change in the black hole X-ray binary, GX 339-4. Zdziarski et al.[21] mentioned that, for GX 339-4, state changes happened at different accretion rates. With assumption of M = 10M , they found that, on the two occasions when the spectrum was in the soft state, accretion rates ˙ EDD respectively. If we believe that the viscosity coefficient ˙ EDD and 0.25M were ∼ 0.07M can be different at different times, then this phenomena can be well explained. In our model α = 0.43 is adopted (calculated from Equations (8) and (9)), then the spectral state changes into the soft state at 0.07 M˙ EDD . Similarly, if α = 0.65 is taken, the spectral state changes into the soft state at 0.25 M˙ EDD . But this value of α is obviously too large. Our calculation results may also possibly be used for AGNs. For instance, the trunca˙ EDD[25] for NGC 4636. If α = ˙ = 0.018M tion radius is 300 Rs with the accretion rate of m 0.35 is adopted in our model, a satisfactory result can also be obtained. It is indicated from these results that using different viscosity coefficients may possibly explain the formation of different truncation radii.

5. CONCLUSIONS In this paper, the properties of the corona of the accretion disk are calculated for a set of values of the viscosity coefficient. Compared with the previous work, we have covered a larger range of α, and obtained more detailed results. From our numerical results, approximate

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analytical relations between the evaporation rate and the viscosity coefficient, and between the truncation radius and the viscosity coefficient are obtained. It is found that the viscosity coefficient in the coronal gas strongly affects the accretion disk truncation and the spectral state change. In addition, comparison between the supermassive black hole and stellar black hole is made and the previous result that the evaporation rate (in units of the Eddington accretion rate) and the truncation radius (in units of the Schwarzschild radius) are massindependent, is quantitatively verified. Furthermore, some observational characteristics of two black hole binaries, XTE J1118+480 and GX 339-4, and an AGN, NGC 4636, can be explained with our results as regards the truncation radius and the spectral state change. ACKNOWLEDGEMENTS We are grateful to all the members in the High Energy Astrophysics Group, Yunnan Observatory for their help. We also thank the referee for his/her pointing out some problems and giving suggestions on the first draft. References 1

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