The radial±azimuthal instability of a hot two-temperature accretion disc with advection

Transcription

1 Mon. Not. R. Astron. Soc. 317, 737±742 (2000) The radial±azimuthal instability of a hot two-temperature accretion disc with advection Ding Shi-Xue, 1,2 Yang Lan-Tian, 1w Wu Xue-Bing 3 and Lu Ye 1,3 1 Department of Physics and Institute of Astrophysics, Huazhong Normal University, Wuhan , China 2 Department of Physics, Xiang Fan College, Xiang Fan , China 3 Beijing Astronomical Observatory, Chinese Academy of Sciences, Beijing , China Accepted 1999 December 8. Received 1999 November 17; in original form 1999 April 7 ABSTRACT The radial±azimuthal instability of a hot two-temperature accretion disc with advection is examined in this paper. We find that the inclusion of very little advection has significant effects on two acoustic modes for a geometrically thin, cooling-dominated two-temperature disc, but has no effect on acoustic modes for a geometrically slim, cooling-dominated twotemperature disc. We also find that, when azimuthal perturbations are considered, the stability properties of the disc are different from those in the pure radial perturbation case. An increase of the azimuthal wavenumber will stabilize the acoustic modes but make the viscous and thermal modes more unstable for a geometrically thin, cooling-dominated twotemperature disc. It makes the thermal mode more unstable and the acoustic mode more stable, but only affects the instability of the viscous mode for short-wavelength perturbations for a geometrically slim, cooling-dominated two-temperature disc. For a geometrically slim, advection-dominated two-temperature disc, the increase of the azimuthal perturbation makes the I- and O-modes more stable and the thermal mode more unstable, but has no effect on the viscous mode. Key words: accretion, accretion discs ± instabilities. 1 INTRODUCTION As a powerful and attractive theoretical model, accretion discs are widely believed to be sources of high-energy radiation for X-ray binaries and active galactic nuclei (AGNs). Since the standard thin disc model was constructed by Shakura & Sunyaev (1973), the stability of the disc has become an important area of study in the theory of accretion discs, but it has been unable to account for the hard X-ray spectra observed in black hole candidates such as Cyg X-1. Shapiro, Lightman & Eardley (1976, hereafter SLE) presented the optically thin, two-temperature accretion disc model for Cyg X-1 which can explain the observed X-ray spectrum from 8 to 500 kev. In a two-temperature disc, the iron temperature, T i,is as high as K, while the electron temperature, T e,is,10 9 K. Because these discs have much hotter plasmas than optically thick discs, the plasmas can evaporate from the discs for a certain range of accretion rate and viscosity parameter. In the last two decades, the SLE model has been widely studied and applied in the modelling of X-ray binaries and AGNs (e.g. Kusunose & Takahara 1984, 1988; Write & Lightman 1989; Wandel & Liang 1991; Luo & Liang 1994). However, the early work of Pringle (1976) and Piran (1978) indicated that the two-temperature disc model is still thermally unstable. The rapid growth of instability may result in w yanglt@ccnu.edu.cn the breakdown of the disc equilibrium, and make it unlikely to be the real configuration of the accretion flow. In the standard a-model, viscous heating balances radiative cooling. However, if the radiative cooling is not efficient, some energy will be advected inward, and the advection will be nonnegligible. Particularly in an optically thin disc, the radiative cooling rate is so low that most of the viscously generated energy is advected rapidly. Accretion disc models with advection have been studied for discs that are either optically thick or optically thin (Abramowicz et al. 1988; Kato, Honma & Matsumoto 1988; Narayan & Popham 1993; Narayan & Yi 1994, 1995a,b; Abramowicz et al. 1995; Chen et al. 1995; Chen 1995). In particular, advection may be quite important in a twotemperature disc, as calculated by some authors (Narayan & Yi 1995b; Esin et al. 1996). Recently, Wu (1997) performed a detailed linear study in order to investigate the stability of a hot two-temperature disc, considering the effects of both advection and thermal diffusion. The stabilities of the thermal mode, viscous mode and inertial±acoustic modes have been investigated, but in Wu's work only the radial perturbations were considered. A more realistic analysis of the stability of an accretion disc should include both the radial and azimuthal perturbations because of the strong shear of a viscous flow. Mckee (1990) has performed such an analysis for a radiation-pressure-dominated disc and found that the viscous instability, which always exists in a purely radial q 2000 RAS

2 738 S.-X. Ding et al. perturbation case, is absent when the azimuthal perturbations are considered. Wu et al. (1995a,b) also investigated the radial± azimuthal instability of standard thin discs and slim discs. They found that the consideration of azimuthal perturbations has significant effects on the stability properties of accretion discs. However, advection was not considered in these studies. Kato, Abramowicz & Chen (1996) performed an analytic stability analysis of the advection-dominated disc by considering purely radial short-wavelength perturbations. They suggested that the advection-dominated disc is thermally stable if it is optically thin but thermally unstable if it is optically thick. This result was confirmed by a subsequent study by Wu & Li (1996), who considered not only the stability of the thermal mode, but also the stability of the viscous and inertial±acoustic modes. However, Wu & Li (1996) also only considered purely radial perturbations. Stimulated by the above research, we have performed a detailed study of the radial±azimuthal instability of a hot two-temperature accretion disc with advection. In Section 2 we present the basic equations. In Section 3 we derive the perturbed equations and the dispersion relation. Stability analyses for different cases are performed in Section 4. Finally we discuss our results in Section 5. 2 BASIC EQUATIONS Throughout our work, we consider the non-self-gravitating disc. The effects of general relativity are introduced by the pseudo- Newtonian potential (Paczynski & Wiita 1980), C ˆ 2 GM ; R 2 R g where M is the mass of the central object, R ˆ r 2 z 2 1=2 ; and R g is the Schwarzschild radius. In cylindrical coordinates, the vertical integrated hydrodynamic equations that describe the timedependent structure of discs, including non-axisymmetric terms, can be written as follows (Wu et al. 1995a): S t 1 r S v r t rsv r 1 Sv f ˆ 0; 1 r f v r Sv r 2 Sr V2 2 V 2 K SV v r HP ˆ 22 F n ; f 2 Sr 3 V t Srv r 2 V r Sr 3 V V f ˆ 22r HP Snr 3 V f ; 3 C V S T t Sv T T r SV f 2 G T S t v S r V S f " r V 2 ˆ 4Sn 2 V 2 2 v r V 2 3 f 3 r f 1 v r V 2 f 2 2 v r V 2 Q 3 f 2 ; 4 where v r, v f and V are the radial, azimuthal and angular velocities, P, T and S are the total pressure, temperature and surface density, and C V and G 3 are the heat capacity per unit mass and a quantity associated with b, the ratio of gas to total pressure. For an optically thin, two-temperature disc, b ˆ 1; C V ˆ 3P=2rT; G 3 ˆ 5=3 and P ˆ rk T i T e =m p ; where T i, T e and m p are the ion and electron temperatures and proton mass. V K is the Keplerian angular velocity, given by V 2 K ˆ 1 r F n C : zˆ0 is the radial viscous force, which is often neglected in geometrically thin accretion discs but is perhaps not negligible in accretion discs with advection. It is given by (Papaloizou & Stanley 1986) 4 n r S rv r 3 r F n ˆ 2 2v r n r S ; 5 r where n r is the kinematic viscosity acting in the radial direction, and n is the viscosity acting in the azimuthal direction and is expressed as the standard a-prescription (Shakura & Sunyaev 1973). In this paper, we take n r ˆ hn; where h is a parameter and n ˆ ac s H: Here C s is the local sound speed defined by C 2 s ˆ P=r; where r is the density. H is the disc height given by H ˆ C s =V K : The term Q 2 on the right-hand side of equation (4) represents the radiative cooling. For an optically thin, one-temperature disc, the radiative cooling mechanism is usually taken as thermal bremsstrahlung. However, for a two-temperature disc, because T T e ; we can get Q 2 / S 7=5 T 1=5 i (Wu 1997). As in the work by Mckee (1990) and Wu (1997), the viscous terms involving derivatives of f are neglected in equations (2) and (3) for simplicity. They are normally much smaller compared with those of radial derivatives, and therefore their effects on the disc stability are expected to be not too significant. 3 PERTURBED EQUATIONS AND DISPERSION RELATION In order to study the stability properties, we consider perturbations of dynamical quantities S, V, v r and T. The radial and azimuthal perturbations are assumed to be of the form dv V ; ds S ; dv r v r ; dt T / exp i vt nf 2 kr Š; 6 where k is the radial perturbation wavenumber defined by k ˆ 2p=l and n is the azimuthal perturbation wavenumber which represents the number of azimuthal crests around a circumference of radius R. As in the work done by Wu & Li (1996) and Wu (1997), we also restrict our stability analyses to within the validity of the local approximation and vertically integrated equations, which requires r H. l H. 2pa H r : 7 We can see clearly that this inequality is satisfied for a geometrically thin accretion disc even if we set l/h to fall within a wide range, such as from 1 to 80, and for a geometrically slim disc when the perturbation wavelength is short and the viscosity coefficient a is sufficiently small (Kato et al. 1996; Wu et al. 1996). The range of l/h also moves to smaller values, such as 0.01±2, if a is about and H/r is about 0.6. Vertical hydrostatic balance can be realized if the time-scale associated with the

3 perturbations is longer than the dynamical time-scale. We shall study the stability properties for both thin and slim discs by numerical calculations within the above parameter range. Using the perturbed quantities in the basic equations (1)±(4), and considering the local approximation, we obtain the following equations: ~s ds S 2 i e dv r dv in ˆ 0; ~H V K r V K 8 2ie ~H ds S ~s 4 dvr 3 hae2 V K r 2 2 V ~ dv 2 ie ~H dt V K T ˆ 0; 9 i ~H 2 aeg ds ~H n S ~k2 dv r 2 V ~ V K r ~s ae2 dv V K i ~H 2 aeg dt ~H n ˆ 0; 10 T 2 ~s ds 5 q ag 2 S 2 aqg2 m ~H 2inga dvr V K r 2iag e dv 3 ~H V K 2 ~s dt 5 q ag 2 T ˆ 0; 11 A hot two-temperature accretion disc with advection 739 a 5 ˆ ae q Vg ~ e2 g q n ~Heag q 2 1 ~k 2 2 V ~ 2 2 V ~ q 2 1 h ~Hna 3 g 3 e 3 : 14 It is easy to see that when n ˆ 0; h ˆ 1; from dispersion equation (13), we get the same result as Wu (1997). It should be noted that we do not consider thermal diffusion in this paper, so we must let f equal zero in Wu (1997) when comparing the dispersion relations. The stability properties of two inertial±acoustic modes, thermal and viscous modes, can be obtained by analysing the four kinds of solutions of the dispersion relation. The instability of the inertial± acoustic mode was first addressed in viscous accretion discs by Kato (1978), and was later studied in detail by Blumenthal, Yang & Lin (1984). In the following, we will solve the dispersion relation numerically according to the different disc structures. The stability of advection-dominated discs will be analysed by assuming q! 1: where ~s ˆ s=v K ; and s ˆ i v 2 kv r nv ; here V ~ ˆ V=V K ; ~H ˆ H=r and e ˆ kh: In addition, g ˆ ~k 2 =2 V ~ 2 2 VŠ ~ and ~k ˆ k=v K ; where k is the epicyclic frequency defined by k 2 ˆ 2V 2V r V= Š: Here m is the Mach number defined by m ˆ jv r j=c s ; and q is the ratio of advective energy to viscous dissipated energy, namely T C V Sv r 2 G S Tv r ˆ qsn r V 2 : 12 If the disc is radiative cooling dominated, q is nearly zero; if it is advection dominated, q is nearly 1. The set of linear equations above can be solved by putting the determinant of the coefficients to zero, which leads to a fourthorder dispersion relation: a 1 ~s 4 a 2 ~s 3 a 3 ~s 2 a 4 ~s a 5 ˆ 0; where a i i ˆ 1; ¼; 5 is the coefficient given by a 1 ˆ 3 2 ; a 2 ˆ a e h q g 2 ; a 3 ˆ aeg 2 ae h q 2 2 i q m 2ha 2 e e2 3 2 ~k 2 ; a 4 ˆ 2i Va ~ 2 eg 3 q m ag q ae 2 2 h 2 iae 3 q m 8 3 h ae q ~k q 2 1 e2 ae 2 g ~k2 ~V 2 5 V ~ 5 2 ae4 4h ~Ha 2 e 3 gn 5 Hne 2 ~ ~k 2 2 V ~ 2 2 V ~ ; 13 4 NUMERICAL RESULTS In this section we will numerically solve the dispersion relation in equation (13). According to the different possible structures of a two-temperature disc, we solve the dispersion relation in the following three cases. 4.1 Geometrically thin, cooling-dominated two-temperature disc We take V ~ ˆ ~k ˆ 1; m ˆ 0:01 and h ˆ 1: According to the local restrictions, l/h is set from 1 to 80 for a geometrically thin disc (in this section we take H=r ˆ 0:01 : By solving the dispersion relation we get the following results. (i) Geometrically thin disc without advection: if it is purely radiative cooling by thermal bremsstrahlung that balances the viscous dissipation, then q ˆ 0: We take n ˆ 1; a ˆ 0:01: By solving the dispersion relation, we obtain the results shown in Fig. 1(a). The curves 1, 2 and 3 represent the viscous, thermal and acoustic modes respectively. Comparing this figure with fig. 1(a) in Wu (1997), we find that the thermal modes are always unstable and the acoustic modes are stable. However, the viscous modes are different. In our figure they are nearly neutral, and they are stable in Wu's figures. In addition, we see that the shape of the thermal mode curve is not the same as that of Wu's results. In shortwavelength perturbations our thermal mode is more unstable than his. (ii) Geometrically thin disc with very little advection: if the disc is dominated by radiative cooling but with very little advection, we get the results shown in Fig. 1(b). The solid and dashed lines correspond to q ˆ 0:01 and 0.02 respectively; other parameters are the same as in case (i). It is clear that the inclusion of very little advection has almost no effect on the thermal and viscous modes, but leads to the acoustic mode [Fig. 1(a) ± curve 3] bifurcating a outward-propagating acoustic mode (hereafter O-mode) and an inward-propagating acoustic mode (hereafter I-mode) [Fig. 1(b) ± curves 3 and 4]. Fig. 1(b) also gives the effect of advection increasing. When the advection increases, it increases the departure of two acoustic modes, the O-mode becomes more

4 740 S.-X. Ding et al. Figure 1. Stability of a geometrically thin, cooling-dominated disc. Lines 1, 2, 3 and 4 represent the viscous mode, the thermal mode, the O-mode and the I- mode respectively. Parameters (V Ä, kä, h, HÄ, m) are taken to be (1, 1, 1, 0.01, 0.01). In (a) no advection is included q ˆ 0 ; and (n, a) are taken to be (1, 0.01); (b) shows the case with very little advection q ˆ 0:01 and the influence of increasing advection (solid line for q ˆ 0:01 and dashed line for q ˆ 0:02 ; parameters (n, a) are taken to be (1, 0.01); (c) shows the influence of increasing azimuthal perturbations (solid line for n ˆ 1 and dashed line for n ˆ 3 ; parameters (q, a) are taken to be (0.01, 0.01); (d) shows the same situation as (c) but in an inviscid disc; parameters (q, a) are taken to be (0.01, 0). Figure 2. Stability of a geometrically slim, cooling-dominated disc. The definitions of line 1, 2, 3 and 4 are the same as in Fig. 1. Parameters (V Ä, kä, h, HÄ, m) are taken to be (1, 1, 1, 0.6, 0.01). Panels (a) and (b) show the influence of increasing azimuthal perturbations (solid line for n ˆ 1 and dashed line for n ˆ 5 : Parameters (q, a) are taken to be (0.01, 0.001) for (a) and (0.01, 0) for (b). unstable and the I-mode becomes more stable. These results are the same as in fig. 2(a) of Wu (1997). The solid and dashed lines of Figs 1(c) and (d) correspond to n ˆ 1 and 3 respectively. From Fig. 1(c), we can see clearly that the inclusion of azimuthal perturbations changes greatly the stability properties of discs. With the increase of n, the thermal and viscous modes become more unstable, while the acoustic modes become stable. In Fig. 1(d), except for a trivial ~s ˆ 0; three solutions of equation (13) represent two acoustic modes and a thermal mode. The results of increasing n in Fig.1(d) are the same as those of Fig. 1(c). becomes more stable and the thermal mode is unstable but seem to remain stable to very short-wavelength perturbations. This is not the same as Wu's results. The growth rate of instability of the viscous mode is increasing at short wavelengths, but has almost no effect when the perturbation wavelength is longer than H. From Fig. 2(b), it is clear that the azimuthal perturbation enhances the thermal instability but makes the acoustic mode more stable. The viscous mode is nearly coincident with the x-axis and it is not influenced by azimuthal perturbations. 4.2 Geometrically slim, cooling-dominated two-temperature disc Some previous works have suggested that a hot optically thin disc may be not geometrically thin, but may be geometrically slim or thick (e.g. SLE) even if the advection term is ignored. Here we show the stability of a geometrically slim, two-temperature disc. In this section, we will discuss its stability in detail according to the slim disc structure and the contributions of azimuthal perturbation wavenumber n. We adopt V ~ ˆ ~k ˆ 1; ~H ˆ 0:6; h ˆ 1; m ˆ 0:01 and q ˆ 0:01 throughout this section. Other parameters are dependent on the detailed disc structure or are set as variables. In Fig. 2(a), we show the effect of considering azimuthal perturbations, in which a ˆ 0:001: We also study the effect of increasing azimuthal perturbations in the case without viscosity (i.e. a ˆ 0 in Fig. 2(b). The solid and dashed lines correspond to n ˆ 1 and 5, respectively, in both panels. Comparing Fig. 2(a) with fig. 2(a) in Wu (1997), we can see that acoustic and viscous modes are always stable. The thermal mode is unstable (Fig. 2a) when the perturbation wavelength is longer than H. In addition, with the increase of azimuthal perturbations, the acoustic mode 4.3 Geometrically slim, advection-dominated two-temperature disc The advection-dominated equilibrium of an optically thin, twotemperature disc has recently been constructed by several workers (e.g. Narayan & Yi 1995b; Chen et al. 1995; Nakamura et al. 1996). Such a disc is usually geometrically slim. For a geometrically slim, advection-dominated two-temperature disc, we take V ~ ˆ ~k ˆ 1; ~H ˆ 0:6; h ˆ 1; m ˆ 0:01 and q ˆ 0:99: Figs 3(a) and (b) show the same problem as Figs 2(a) and (b) but for the advection-dominated case. From Fig. 3(a), we see that the effects of azimuthal perturbations on the thermal and acoustic modes are nearly the same as in Fig. 2(a). The only difference is that the acoustic mode splits into two modes and the azimuthal perturbation does not affect the viscous mode in advection-dominated discs. Comparing Fig. 3(a) with fig. 3(a) in Wu (1997), we see that the O-mode in Wu's paper is unstable if the perturbation wavelength is fairly large, but it is always stable in our study. These differences come from the fact that Wu considered purely radial perturbations and we deal with not only radial but also azimuthal perturbations. This means that azimuthal perturbations are able to

5 A hot two-temperature accretion disc with advection 741 Figure 3. Stability of a geometrically slim, advection-dominated disc. The definitions of lines 1, 2, 3 and 4 are the same as in Fig. 1. Parameters (V Ä, kä, h, HÄ, m) are the same as in Fig. 2. Panels (a) and (b) show the influence of increasing azimuthal perturbations (solid line for n ˆ 1 and dashed line for n ˆ 5 : Parameters (q, a) are taken to be (0.99, 0.001) for (a) and (0.99, 0) for (b). stabilize the acoustic mode. The effect of azimuthal perturbations in Fig. 3(b) is exactly the same as in Fig. 2(b). 5 DISCUSSION The stability of a hot two-temperature accretion disc with advection has been studied, taking into account both radial and azimuthal perturbations. The main results are as follows. (i) When n ˆ 0; h ˆ 1; our result is the same as in the work by Wu (1997) if thermal diffusion is not considered. (ii) If azimuthal perturbations are included, the stability of a disc is different from that in the case of purely radial perturbations. Whether for standard discs or for slim discs, the thermal instability is the dominant instability at present. Moreover, with increasing azimuthal perturbations, the thermal mode becomes more unstable, while the acoustic mode become more stable and the growth rate of the viscous instability increases except for the case of an advection-dominated disc. The azimuthal wave crest will appear to drift backwards with respect to the rotating disc, and then enhance the thermal instability (Mckee 1990). (iii) From Fig. 1(b), we see that inclusion of very little advection has nearly no effect on the thermal and viscous modes, but leads to the bifurcation of two acoustic modes. Fig. 1(b) also gives the effect of advection increasing. When the advection increases, it increases the departure from two acoustic modes. These conclusions are the same as those resulting from fig. 2(a) of Wu (1997). (iv) Figs 1(d), 2(b) and 3(b) show the solutions in the case without viscosity (i.e. a ˆ 0 : Except for a trivial solution ~s ˆ 0; the dispersion relation (13) gives three solutions. Two of them represent the acoustic modes, which are always stable; the other one represents the thermal mode, which is always unstable. With increasing n, the acoustic modes become more stable, while the thermal mode becomes more unstable. Mckee (1990, 1991) concluded that the coupling of radial and azimuthal perturbations can eliminate the viscous instability. However, in our study we note that the viscous instability still exists, whether for standard discs or for slim discs, in the case of small azimuthal wavenumber or large viscosity. Such a difference results from the fact that the perturbation to angular velocity was not considered in Mckee's study, where he obtained a secondorder dispersion relation only. Another advantage of our work is that we can examine the stability of acoustic modes by analysing the fourth-order dispersion relation. In summary, our present research indicates that the radial± azimuthal instability always exists in the two-temperature disc with advection, and the azimuthal perturbation wavenumber n is important for considering the instabilities of each mode. There are two kinds of instabilities. One is the acoustic mode instability. These stability properties may become important in helping to explain the quasi-periodic oscillation (QPO) phenomenon in systems such as Galactic black hole candidates (Chen & Taam 1995; Manmoto et al. 1996). The other type of instability is the thermal mode instability, which probably exists in some unstable systems such as the inner regions of AGNs, X-ray binaries and cataclysmic variables. 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