Two temperature accretion around rotating black holes: Description of general advective flow paradigm in presence of various cooling processes to explain low to high luminous sources
aa r X i v : . [ a s t r o - ph . H E ] O c t Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 3 September 2018 (MN L A TEX style file v2.2)
Two temperature accretion around rotating black holes: Descriptionof general advective flow paradigm in presence of various coolingprocesses to explain low to high luminous sources
S. R. Rajesh ⋆ , Banibrata Mukhopadhyay † Astronomy and Astrophysics Program, Department of Physics, Indian Institute of Science, Bangalore 560012, India
ABSTRACT
We investigate the viscous two temperature accretion disc flows around rotating black holes.We describe the global solution of accretion flows with a sub-Keplerian angular momentumprofile, by solving the underlying conservation equations including explicit cooling processesselfconsistently. Bremsstrahlung, synchrotron and inverse Comptonization of soft photonsare considered as possible cooling mechanisms. We focus on the set of solutions for sub-Eddington, Eddington and super-Eddington mass accretion rates around Schwarzschild andKerr black holes with a Kerr parameter 0 . ffi cient phase andvice versa. Hence the flow governs much lower electron temperature ∼ − . K, in therange of accretion rate in Eddington units 0 . < ∼ ˙ M < ∼ ∼ . − . K. Therefore, the solution may potentially explain the hard X-raysand γ -rays emitted from AGNs and X-ray binaries. We then compare the solutions for two dif-ferent regimes of viscosity and conclude that a weakly viscous flow is expected to be coolingdominated, particularly at the inner region of the disc, compared to its highly viscous counterpart which is radiatively ine ffi cient. With all the solutions in hand, we finally reproduce the ob-served luminosities of the under-fed AGNs and quasars (e.g. Sgr A ∗ ) to ultra-luminous X-raysources (e.g. SS433), at di ff erent combinations of input parameters such as mass accretionrate, ratio of specific heats. The set of solutions also predicts appropriately the luminosityobserved in the highly luminous AGNs and ultra-luminous quasars (e.g. PKS 0743-67). Key words: accretion, accretion disc — black hole physics — hydrodynamics — radiativetransfer
The cool Keplerian accretion disc (Pringle & Rees 1972; Shakura & Sunyaev 1973; Novikov & Thorne 1973) was found to be inappropriate toexplain observed hard X-rays, e.g. from Cyg X-1 (Lightman & Shapiro 1975). It was argued that secular instability of the cool disc swells theoptically thick, radiation dominated region to a hot, optically thin, gas dominated region resulting in hard component of spectrum ∼ ∼ K and ∼ × K which confirms that cool, one temperature, pure Keplerian accretion solution is not unique. IndeedEardley & Lightman (1975) found that a Keplerian disc is unstable due to thermal and viscous e ff ects when viscosity parameter α (Shakura& Sunyaev 1973) is constant. Later Eggum et al. (1985) showed by numerical simulations that the Keplerian disc with a constant α collapses.Around eighties, therefore, the idea of two component accretion disc started floating around. For example, Paczy´nski & Wiita (1980)described a geometrically thick regime of the accretion disc in the optically thick limit, while Rees et al. (1982) introduced accretion torus in ⋆ [email protected] † [email protected] (cid:13) the optically thin limit. Moreover the idea of sub-Keplerian, transonic accretion was introduced by Muchotrzeb & Paczy´nski (1982), whichwas later improved by other authors (Chakrabarti 1989, 1996; Mukhopadhyay 2003). Other models were proposed by e.g. Gierli´nski et al.(1999), Coppi (1999), Zdziarski et al. (2001), including a secondary component in the accretion disc. On the other hand, Narayan & Yi(1995) introduced a two temperature disc model in the regime of ine ffi cient cooling resulting in a vertical thickening of the hot disc gas.Here the pressure forces are expected to become important in modifying the disc dynamics which is likely to be sub-Keplerian. Other modelswith similar properties were proposed by, e.g., Begelman (1978), Liang & Thompson (1980), Rees et al. (1982), Eggum, Coroniti & Katz(1988). Abramowicz et al. (1988) proposed a height-integrated disc model, namely “slim disc”, having high optical depth of the accretinggas at super-Eddington accretion rate such that the di ff usion time is longer than the viscous time. The model was further applied to study thethermal and viscous instabilities in optically thick accretion discs (Wallinder 1991; Chen & Taam 1993).Shapiro, Lightman & Eardley (1976) initiated a two temperature Keplerian accretion disc at a low mass accretion rate which is opticallythin and significantly hotter than the single temperature Keplerian disc of Shakura & Sunyaev (1973). The optically thin hot gas cools downthrough the bremsstrahlung and inverse-Compton processes and could explain various states of Cyg X-1 (Melia & Misra 1993). Similarly,the “ion torus” model by Rees et al. (1982) was applied to explain AGNs at a low mass accretion rate. However, the two temperaturemodel solutions by Shapiro, Lightman & Eardley (1976) appear thermally unstable. Narayan & Popham (1993) and subsequently Narayan& Yi (1995) showed that introduction of advection may stabilize the system. However, the solutions of Narayan & Yi (1995), while oftwo temperatures, could explain only a particular class of hot systems with ine ffi cient cooling mechanisms. They also described the hotflow based on the assumption of “self similarity” which is just a “plausible choice”. They kept the electron heating decoupled from thedisc hydrodynamical computations which merely is an assumption. Later on, the solutions were attempted to generalize by Nakamura etal. (1997), Manmoto et al. (1997), Medvedev & Narayan (2001), relaxing e ffi ciency of cooling into the systems, but concentrating onlyon specific classes of solutions. On the other hand, Chakrabarti & Titarchuk (1995) and later Mandal & Chakrabarti (2005) modeled twotemperature accretion flows around Schwarzschild black holes in the general “advective paradigm”, emphasizing possible formation of shockand its consequences therein. However, they also did not include the e ff ect of electron heating self-consistently into the hydrodynamicalequation, and thus the hydrodynamical quantities do not get coupled to the rate of electron heating (see also Rajesh & Mukhopadhyay 2009).In the present paper, we model a selfconsistent accretion flow in the regime of two temperature transonic sub-Keplerian disc (see alsoSinha, Rajesh & Mukhopadhyay 2009; a brief version of the present work, but around Schwarzschild black holes). We consider all thehydrodynamical equations of the disc along with thermal components and solve the coupled set of equations selfconsistently. We neitherrestrict to the advection dominated regime nor the self-similar solutions. We allow the disc to cool selfconsistently according to the thermo-hydrodynamical evolution and compute the corresponding cooling e ffi ciency factor as a function of radial coordinate. We investigate thatwhen does the disc switch from the radiatively ine ffi cient nature to general advective paradigm and vice versa.In order to implement our model to explain observed sources, we focus on the under-luminous AGNs and quasars (e.g. Sgr A ∗ ), ultra-luminous quasars and highly luminous AGNs (e.g. PKS 0743-67) and ultra-luminous X-ray (ULX) sources (e.g. SS433), when the lastitems are likely to be the “radiation trapped” accretion discs. While the first two cases correspond to respectively sub-Eddington and super-Eddington accretion flows around supermassive black holes, the last case corresponds to super-Eddington accretors around stellar mass blackholes.In the next section, we discuss the model equations describing the system and the procedure to solve them. Subsequently, we discuss thetwo temperature accretion disc flows around stellar mass and supermassive black holes, respectively in § §
4, for both sub-Eddington,Eddington and super-Eddington accretion rates. Section 5 compares the disc flow of low Shakura-Sunyaev (1973) α with that of high α andthen between the flows around co and counter rotating black holes. Then we discuss the implications of the results with a summary in § For the present purpose, we set five coupled di ff erential equations describing the law of conservation in the sub-Keplerian optically thinaccretion regime. Necessarily the set of equations describes the inner part of the accretion disc where the gravitational potential energydominates over the centrifugal energy of the flow.Throughout, we express all the variables in dimensionless units, unless stated otherwise. The radial velocity ϑ and sound speed c s areexpressed in units of light speed c , the specific angular momentum λ in GM / c , where G is the Newton’s gravitational constant and M isthe mass of the compact object, for the present purpose black hole, expressed in units of solar mass M ⊙ , the radial coordinate x in unitsof GM / c , the density ρ and the total pressure P accordingly. The disc fluid under consideration consists of ions and electrons — thus twofluid / temperature system, apart from radiation. Furthermore, at the high temperature, the disc flow with ions / electrons behaves as (almost)noninteracting gas. (a) Mass transfer:1 x ∂∂ x ( x ρϑ ) = , (1) c (cid:13) , 000–000 whose integrated form gives the mass accretion rate˙ M = − π x Σ ϑ, (2)where the surface density Σ = I n ρ h ( x ) , (3) I n = (2 n n !) / (2 n + . , (4) n is the polytropic index which is equal to 1 / ( γ −
1) when γ is the ratio of specific heats, and half-thickness, based on the vertical equilibriumassumption, of the disc h ( x ) = c s x / F − / . (5)(b) Radial momentum balance: ϑ d ϑ dx + ρ dPdx − λ x + F = F = ( x − a √ x + a ) x [ √ x ( x − + a ] , (7)where a is the specific angular momentum (Kerr parameter) of the black hole. We also define a parameter β = gas pressure P gas total pressure P = γ − γ −
1) (e . g . Ghosh & Mukhopadhyay 2009) , (8)where γ may range from 4 / / P gas = P i (ion pressure) + P e (electron pressure), such that P = ρβ c kT i µ i m i + kT e µ e m i ! = ρ c s , (9)where T i , T e are respectively the ion and electron temperatures in Kelvin, m i is the mass of proton in gm, µ i and µ e respectively are thecorresponding e ff ective molecular weight, k the Boltzmann constant. We assume β (and then γ ) constant throughout the flow.(c) Azimuthal momentum balance: ϑ d λ dx = Σ x ddx (cid:16) x | W x φ | (cid:17) , (10)where following Mukhopadhyay & Ghosh (2003; hereinafter MG03) the shearing stress can be expressed in terms of the pressure and densityas W x φ = − α (cid:16) I n + P eq + I n ϑ ρ eq (cid:17) h ( x ) , (11)where α is the dimensionless viscosity parameter and P eq and ρ eq are the pressure and density respectively at the equatorial plane. Note thatwe will assume P eq ∼ P and ρ eq ∼ ρ in obtaining solutions.(d) Energy production rate: ϑ h ( x ) Γ − dPdx − Γ P ρ d ρ dx ! = Q + − Q ie , (12)where following MG03 Q + = α ( I n + P + I n ϑ ρ ) h ( x ) d λ dx , (13)which is the heat generated by viscous dissipation, and Q ie is the Coulomb coupling (Bisnovatyi-Kogan & Lovelace 2000) given in dimen-sionful unit as q ie = π ) / e n i n e m i m e T e m e + T i m i ! − / ln( Λ ) ( T i − T e ) erg / cm / sec s when q ie = Q ie c / ( h G M ) . (14)Here n i and n e denote number densities of ion and electron respectively, e the charge of an electron, ln( Λ ) the Coulomb logarithm. We alsodefine (MG03) Γ = + Γ − β − β , (15) Γ = β + (4 − β ) ( γ − β + γ − − β ) . (16) c (cid:13) , 000–000 (e) Energy radiation rate: ϑ h ( x ) Γ − dP e dx − Γ P e ρ d ρ dx ! = Q ie − Q − , (17)where Q − is the heat radiated away by the bremsstrahlung ( q br ), synchrotron ( q syn ) processes and inverse Comptonization ( q comp ) due to softsynchrotron photons, given in dimensionful form as q − = q br + q syn + q comp , when q − = Q − c / ( h G M ) . (18)Various components of the cooling processes may be read as (see Narayan & Yi 1995; Mandal & Chakrabarti 2005 for detailed description,what we do not repeat here) q br = . × − n e n i T / e (1 + . × − T e ) erg / cm / sec , q syn = π c kT e ν a R erg / cm / sec , R = x GM / c , q comp = F q syn , F = η − x a θ e ! η ! , η = p ( A − − pA ) , p = − exp( − τ es ) , A = + θ e + θ e , θ e = kT e / m e c , η = − + ln ( p ) ln ( A ) ! , x a = h ν a / m e c , (19)where τ es is the scattering optical depth given by τ es = κ es ρ h (20)where κ es = .
38 cm / gm and ν a is the synchrotron self-absorption cut o ff frequency determined by following Narayan & Yi (1995). Note thatwithout a satisfactory knowledge of the magnetic field in accretion disks, following Mandal & Chakrabarti (1995), we assume the maximumpossible magnetic energy density to be the gravitational energy density of the flow. As the total optical depth should include the e ff ects ofabsorption due to nonthermal processes, e ff ective optical depth is computed as τ e ff ≃ √ τ es τ abs (21)where τ abs = h σ T e (cid:16) q br + q syn + q comp (cid:17) GMc , (22)when σ is Stefan-Boltzmann constant.Now combining all the above equations we obtain d ϑ dx = N ( x , ϑ, c s , λ, T e ) D ( ϑ, c s ) , (23)where N = Γ + Γ − ϑ c s J − α c s x H I n + I n c s + ϑ ! − α I n + I n HJ + Γ − Γ − ϑ c s L + α H λϑ c s x ! + π Q ie ˙ M ϑ c s x / F − / , (24) D = − Γ Γ − c s ϑ + α c s I n + I n H c s ϑ − ϑ ! + Γ + Γ − ϑ c s ϑ − c s ϑ ! + α ϑ H (cid:18) H ϑ (cid:19) (25)and L = x − F dFdx ! , H = (cid:16) I n + c s + I n ϑ (cid:17) , J = c s L + λ x − F ! . (26)We know that around the sonic radius N = D = D = M c = ϑ c c sc = s − B + (cid:0) B − AC (cid:1) / A , (27)where A = Γ + − Γ − α ( I n + − I n ) , B = Γ + − Γ − α I n + I n ( I n + − I n ) , C = α I n + I n (1 − I n + ) . (28) c (cid:13) , 000–000 Also from N =
0, we can compute explicitly c sc as a function of sonic / critical radius x c . For a physical x c , what one has to adjust in orderto obtain a physical solution connecting outer boundary to black hole horizon through x c , c sc and then ϑ c can be assigned, discussed inAPPENDIX A in detail. Note that an improper x c may lead to an unphysical / imaginary c sc and ϑ c .Finally combining Eqns. (6), (10) and (17) we obtain dc s dx = c s ϑ − ϑ c s ! d ϑ dx + Jc s , (29) d λ dx = α x ϑ c s I n + I n c s ϑ − ϑ c s ! + α x ! d ϑ dx + c s − x α Jc s + ϑ ! , (30) dT e dx = (1 − Γ ) T e ϑ c s d ϑ dx + (1 − Γ ) T e Jc s + L ! + ( Γ − π ˙ M c s x / F / (cid:16) Q ie − Q − (cid:17) . (31)Hence knowing ϑ we can obtain other variables c s , λ , T e . Note that d ϑ/ dx is indeterminate (of 0 / x c . APPENDIX B discusses theprocedure to obtain d ϑ/ dx at x c .As the entropy increases inwards in advective flows (e.g. Narayan & Yi 1994; Chakrabarti 1996; MG03), there is a possibility ofconvective instability and then corresponding transport, as proposed by Narayan & Yi (1994). Dynamical convective instability arises whenthe square of e ff ective frequency ν ff = ν + ν r < ν BV is the Brunt-V¨ais¨al¨a frequency given by ν = − ρ dPdx ddx ln P /γ ρ ! (33)and ν r is the radial epicyclic frequency. In order to obtain the steady state solution, as of previous work (MG03, Mukhopadhyay 2003), primarily we need to fix the appropriatecritical radius x c (in fact the energy at the critical radius which is not conserved in the present cases) and the corresponding specific angularmomentum λ c of the flow. The detailed description of the procedure to obtain physically meaningful values of x c and λ c , to be determinediteratively, is given in APPENDIX A. As the flow is considered to be of two temperatures, at x c an appropriate electron temperature T ec alsoneeds to be determined; also discussed in APPENDIX A. Note that one has to adjust the set of values x c , λ c , T ec appropriately / iteratively toobtain self-consistent solution connecting outer boundary and black hole event horizon through x c . Depending on the type of accreting systemto model, we then have to specify the related inputs: ˙ M , M , γ and a . Important point to note is that unlike former works (e.g. Chakrabarti& Titarchuk 1995, Chakrabarti 1996, MG03) here x c changes with the change of ˙ M , because the various cooling processes considered hereexplicitly depend on ˙ M . Finally, we have to solve the Eqn. (23) from x c to inwards — upto the black hole event horizon, and to outwards— upto the transition radius x o where the disc deviates from the Keplerian to the sub-Keplerian regime such that λ/λ K = λ K being thespecific angular momentum of the Keplerian part of the disc). Figure 1 shows how the ratio λ/λ K varies as a function of radial coordinate fordi ff erent a . Note that higher a , which corresponds to a lower disc angular momentum (Mukhopadhyay 2003), reassembles the Keplerian partadvancing with a smaller size of the sub-Keplerian disc. On the other hand, for a lower a the inner edge of the Keplerian component recedes.The fact of moving in and out of the inner edge of the disc reassembles respectively the soft and hard state of the black hole (e.g. Gilfanov etal. 1997). Hence it is naturally expected to link with the spin of the black hole.However, the important point to note is that there is no selfconsistent model to describe the transition region where λ/λ K =
1. Therefore,the transition of the flow from the Keplerian to sub-Keplerian regime does not appear smooth. This is mainly because the set of equationsused to model the sub-Keplerian flow is not valid to explain the cold Keplerian flow, unless an extra boundary condition is imposed at theouter edge of the sub-Keplerian disc. However, in the present paper we do not intend to address the transition zone; rather we prefer to stickwith the sub-Keplerian flow. Narayan et al. (1997) imposed boundary conditions at both the ends of accretion flows along with at the criticalradius to fix the problem, at the cost of more input parameters than the parameters chosen in the present work. But still the transition of theflow from the Keplerian to sub-Keplerian regime remains undefined. Later Yuan (1999) discussed how the solutions vary with the change ofouter boundary conditions influencing the structure of an optically thin accretion flow.Below we discuss solutions in various parameter regimes to understand properties of the accretion disc around, first, stellar mass( M =
10) and then super-massive ( M = ) black holes. c (cid:13) , 000–000 λ / λ K log(x) Figure 1.
Variation of ratio of disc specific angular momentum to corresponding Keplerian angular momentum as a function of radial coordinate, when solid,dashed, dotted, dot-dashed lines correspond to the cases with a = − . , , . , .
998 respectively. Other parameters are ˙ M = M = α = . Primarily we concentrate on two extreme regimes: (1) sub-Eddington and Eddington limits of accretion, (2) super-Eddington accretion.Furthermore, at each case of accretion rate, we focus on solutions around nonrotating (Schwarzschild) and rotating (Kerr with a = . ff ect the disc dynamics and then the cooling e ffi ciency vary overthe disc radii. The cooling e ffi ciency f is defined as the ratio of the energy advected by the flow to the energy dissipated, which is 1 for theadvection dominated accretion flow (in short ADAF; Narayan & Yi 1994, 1995) and less than 1 for the general advective accretion flow (inshort GAAF; Chakrabarti 1996; Mukhopadhyay 2003; MG03) in general. Therefore, f directly controls the ion and electron temperatures ofthe disc. Far away from the black hole where the gravitational power is weaker, the angular momentum profile becomes Keplerian and thusthe disc becomes (or tends to become) of one temperature in the presence of e ffi cient cooling. We first consider flows around static black holes where the Kerr parameter a =
0. Figure 2 shows the behavior of flow variables as functionsof radial coordinate for ˙ M = . , . ,
1; throughout in the text we express ˙ M in units of Eddington limit. The sets of input parameters forthe model cases described here are given in Table 1. Figure 2a c (cid:13) , 000–000 log(x) f −4 (d) log(x) ν ff log(x) ϑ log(x) l og ( ρ ) Figure 2.
Variation of dimensionless (a) radial velocity, (b) density, (c) cooling factor, (d) square of convective frequency, as functions of radial coordinatefor sub-Eddington and Eddington accretion flows. Solid, dashed, dotted curves are for ˙ M = . , . , a = α = . M =
10; see Table 1 for details.
Table 1: Parameters for accretion with α = .
01 around black holes of M =
10, when the subscript ‘c’ indicates the quantity at the criticalradius and T ec is expressed in units of m i c / k ˙ M a γ x c λ c T ec Sub-Eddington, Eddington accretors0.01 0 1.5 5.5 3.2 0.00010.01 0.998 1.5 3.5 1.7 0.00010.1 0 1.4 5.5 3.2 0.0001640.1 0.998 1.4 3.5 1.7 0.0001531 0 1.35 5.5 3.2 0.0002251 0.998 1.35 3.5 1.7 0.0002122Super-Eddington accretors10 0 1.345 5.5 3.2 0.00018156510 0.998 1.345 3.5 1.7 0.0004432100 0 1.34 5.5 3.2 0.00038678100 0.998 1.34 3.5 1.7 0.00055verifies that a higher radial velocity corresponds to a lower mass accretion rate of the flow ( ∼ .
01) which results in less possibleaccumulation of matter in a particular radius attributing to a lower disc density (Fig. 2b). This hinders the bremsstralung process to cool theflow. However, at around x =
30 the centrifugal barrier dominates and brings the velocity ϑ down, particularly for ˙ M = .
01, which finallymerges with that of higher ˙ M -s. On the other hand, a lower ˙ M corresponds to a gas dominated hot flow, which is radiatively less e ffi cient andquasi-spherical in nature. As a result ϑ is high, as seen in Fig. 2a. E ffi ciency of cooling is shown in Fig. 2c. Naturally a low ˙ M correspondsto a radiatively ine ffi cient flow rendering f > ∼ . x =
30. At x <
30, the dominance of centrifugal barrier slows down the infall whichincreases the residence time of matter in the disc before plunging into the black hole. This allows matter to have enough time to radiate by thesynchrotron process and inverse Comptonization due to synchrotron soft photons, rendering f → M = .
01, the disc is essentially radiatively ine ffi cient, upto x ∼
30, and therefore the electron temperature never goes down. However,the density sharply increases in the vicinity of the black hole (Fig. 2b) which favors e ffi cient cooling at a high temperature.Therefore, although far away from the black hole a sub-Eddington flow appears to be radiatively ine ffi cient, from x =
30 onwards it c (cid:13) , 000–000 log(x) l og ( T ) (d)0.5 1 1.5 2 2.5 3 3.501234 log(x) l og ( T ) (b)0.5 1 1.5 2 2.50123 log(x) l og ( T ) (f)0.5 1 1.5 2 2.5−34−32−30−28−26 log(x) l og ( Q i e ) ,l og ( Q b r ) ,l og ( Q sy n ) ,l og ( Q c o m p ) (c)0.5 1 1.5 2 2.5 3 3.5−38−36−34−32−30 log(x) (a)0.5 1 1.5 2 2.5−34−32−30−28−26−24 log(x) (e) Figure 3.
Variation of (a) dimensionless energy of Coulomb coupling (thicker line), bremsstrahlung (dotted line), synchrotron (solid), inverse Comptonizationdue to synchrotron photon (dashed line) processes in logarithmic scale, (b) corresponding ion (solid) and electron (dotted) temperatures in units of 10 K, asfunctions of radial coordinate for ˙ M = .
01. (c), (e) Same as (a) except ˙ M = . , M = . , a = α = . M =
10; see Table 1 for details. turns out to be a radiatively e ffi cient advective flow with f much less than unity. However, for ˙ M = . , M = .
01 and hence the bremsstrahlung e ff ect starts playing role in radiation mechanisms at much outer radii. This decreases the ion-electrontemperature di ff erence at the transition radius x o . However, as the flow advances the synchrotron and corresponding inverse Compton e ff ectsdominate attributing to strong radiative loss. This renders f < ∼ . x =
10. Further in, a strong radial infall, in absence of any centrifugalbarrier compared to a low ˙ M case, does not permit matter to radiate enough, rendering f upto 1. This is particularly because the strongadvection decreases the residence time of the flow before plunging into the black hole and thus renders a weaker ion-electron coupling. Thisin turn hinders the transfer of energy from the ions to electrons attributing ions to remain hot, while electrons continue to be cooled downfurther by radiative processes.However, Fig. 2d shows that either of the cases do not exhibit convective instability (see, however, Narayan, Igumenshchev & Abramow-icz 2000, Quataert & Gruzinov 2000) upto very inner edge, evenif the radiatively ine ffi cient flow deviates to a radiatively e ffi cient GAAF. Ata very inner edge, discs with ˙ M = . , M ( = .
01) the system is radiatively ine ffi cient, relative to that of higher ˙ M -s ( = . , M ( = ffi cient bremsstrahlungradiation at high density. In the vicinity of black hole T e ∼ K in the flow with ˙ M = .
01 when f →
0, as explained above, in the contraryto the cases with ˙ M = . , f → T e sharply decreases. Note that the accretion disc around a stellar mass black hole is arrestedby significant magnetic field. This results in dominance of the synchrotron e ff ect over the bremsstrahlung as the flow advances. We consider the rotating black holes with a = . ffi cient) upto, e.g., x ∼
100 (see the c (cid:13) , 000–000 log(x) f log(x) ν ff log(x) ϑ log(x) l og ( ρ ) Figure 4.
Same as Fig. 2, except a = . outer radius in Fig. 4), the flow cools down significantly before deviating to the sub-Keplerian regime. Figure 4a shows that the velocityprofiles for all ˙ M -s are similar to each other, in absence of a strong centrifugal barrier. However, f , while very small at x ∼ f → . x ∼
30 for ˙ M = .
01, when the density is lowest (see Fig. 4b).However, as the flow approaches to the black hole the synchrotron emission increases, and hence the system acquires enough softphotons which help in occurring the inverse Compton process. As a result the flow cools down further. When ˙ M = .
01 the cooling processat the very inner edge of the accretion disc is dominant due to relatively high residence time of the flow, compared to that of a higher ˙ M ,rendering high T e and then f → M > ∼ .
1, on the other hand, the flow is strongly advective and then unable to cool down before plunging into theblack hole. Figure 5 shows the profiles of cooling processes and ion-electron temperatures. Basic nature of the profiles is pretty similar tothat of Schwarzschild cases, except all of them advance in.
The “radiation trapped” accretion disc can be attributed to the radiatively driven outflow or jet. This is likely to occur when the accretionrate is super-Eddington (Lovelace et al. 1994, Begelman et al. 2006, Fabbiano 2004, Ghosh & Mukhopadhyay 2009), as seen in the ultraluminous X-ray (ULX) sources such as SS433 (with luminosity ∼ erg / s or so; Fabrika 2004). In order to describe such sources, themodels described below are the meaningful candidates. We consider ˙ M = , A high mass accretion rate significantly enhances density, upto two orders of magnitude compared to that of a low ˙ M , which severely a ff ects f and finally temperature profiles. The profiles of velocity shown in Fig. 6 are quite similar to that of sub-Eddington and Eddington cases.Because of similar reasons explained in § M ( = M flowwill have relatively more gas and then quasi-spherical structure compared to that of a higher ˙ M ( = f →
0. For a lower ˙ M , at x ∼
50, the energy radiated due to bremsstrahlung process becomes weaker than the energy transferredfrom protons to electrons through the Coulomb coupling (see Fig. 7), which increases f (see Fig. 6c). Subsequently, the synchrotron processbecomes dominant (see Fig. 7), reassembling f →
0. However, very close to the black hole a strong advection does not allow the flow, c (cid:13) , 000–000 log(x) l og ( T ) (d)1 1.5 201234 log(x) l og ( T ) (b)0.5 1 1.5 201234 log(x) l og ( T ) (f)0.5 1 1.5 2−32−30−28−26 log(x) l og ( Q i e ) ,l og ( Q b r ) ,l og ( Q sy n ) ,l og ( Q c o m p ) (c)1 1.5 2−34−32−30−28 log(x) (a)0.5 1 1.5 2−30−28−26−24 log(x) (e) Figure 5.
Same as Fig. 3, except a = . independent of ˙ M , to radiate e ffi ciently rendering f → x <
10, as shown inFig. 6d.Figure 7 shows that discs remain of one temperature at the transition radius. As the flows advance with a sub-Keplerian angular momen-tum, T p profile deviates from that of T e . For ˙ M = ffi cient bremsstrahlung radiation all theway. In the vicinity of the black hole, an e ffi cient cooling reassembles a sharp downfall of T e . As λ decreases in the case of a higher a , like low ˙ M cases (see Mukhopadhyay 2003), the transition region advances an order of magnitudecompared to that of Schwarzschild black holes. Similar to the cases of low ˙ M , as shown in Fig. 8a, any centrifugal barrier smears out.However, unlike the flow around a static black hole, here the disc with ˙ M =
10 remains stable upto very close to the black hole. The reason isthat a high a corresponds to a larger inner edge of the disc and thus the residence time of matter in the disc is higher. As a result the radiativeprocesses keep cooling and then stabilizing the flow upto very inner edge.Figure 9 shows that although a high ˙ M exhibits a one temperature transition zone due to extremely e ffi cient cooling processes, par-ticularly due to bremsstrahlung radiation, as ˙ M decreases the Keplerian disc itself becomes of two temperatures before deviating to thesub-Keplerian zone, unlike that of the Schwarzschild case. This is mainly because a flow with a high a brings the Keplerian disc further inwhere the transport of angular momentum increases leading to the decrease of the residence time of matter which does not allow an e ffi cientcooling. However, the basic behaviours of various cooling processes is pretty similar to that around a static black hole. As of stellar mass black holes, here also we concentrate on two extreme regimes: (1) sub-Eddington and Eddington limits of accretion, (2)super-Eddington accretion; focusing on both nonrotating (Schwarzschild) and rotating (Kerr with a = . c (cid:13) , 000–000 log(x) f −3 (d) log(x) ν ff log(x) ϑ log(x) l og ( ρ ) Figure 6.
Variation of dimensionless (a) radial velocity, (b) density, (c) cooling factor, (d) square of convective frequency, as functions of radial coordinate forsuper-Eddington accretion flows. Solid, dashed curves are for ˙ M = ,
100 respectively. Other parameters are a = α = . M =
10; see Table 1 for details. log(x) l og ( T ) (b)1 1.5 20.511.522.5 log(x) l og ( T ) (d)1 1.5 2 2.5−36−34−32−30−28−26−24 log(x) l og ( Q i e ) ,l og ( Q b r ) ,l og ( Q sy n ) ,l og ( Q c o m p ) (a)1 1.5 2−35−30−25 log(x) l og ( Q i e ) ,l og ( Q b r ) ,l og ( Q sy n ) ,l og ( Q c o m p ) (c) Figure 7.
Variation of (a) dimensionless energy of Coulomb coupling (thicker line), bremsstrahlung (dotted line), synchrotron (solid), inverse Comptonizationdue to synchrotron photon (dashed line) processes in logarithmic scale, (b) corresponding ion (solid) and electron (dotted) temperatures in units of 10 K, asfunctions of radial coordinate for ˙ M =
10. (c) Same as (a) except ˙ M = M = a = α = . M = (cid:13) , 000–000 log(x) f −3 (d) log(x) ν ff log(x) ϑ log(x) l og ( ρ ) Figure 8.
Same as Fig. 6, except a = . log(x) l og ( T ) (b)0.5 1 1.511.522.53 log(x) l og ( T ) (d)0.5 1 1.5 2−32−30−28−26−24−22 log(x) l og ( Q i e ) ,l og ( Q b r ) ,l og ( Q sy n ) ,l og ( Q c o m p ) (a)0.5 1 1.5−32−30−28−26−24−22 log(x) l og ( Q i e ) ,l og ( Q b r ) ,l og ( Q sy n ) ,l og ( Q c o m p ) (c) Figure 9.
Same as Fig. 7, except a = . (cid:13) , 000–000 log(x) f −3 (d) log(x) ν ff log(x) ϑ log(x) l og ( ρ ) Figure 10.
Variation of dimensionless (a) radial velocity, (b) density, (c) cooling factor, (d) square of convective frequency, as functions of radial coordinatefor sub-Eddington and Eddington accretion flows. Solid, dashed, dotted curves are for ˙ M = . , . , a = α = . M = ; see Table 2 for details. The under-luminous AGNs and quasars (e.g. Sgr A ∗ ) had been already described by advection dominated model, where the flow is expectedto be substantially sub-critical / sub-Eddington with a very low luminosity ( < ∼ erg / s). Therefore the present cases, particularly of ˙ M < ∼ . Table 2 lists the sets of input parameters for the model cases described here. Naturally a disc around a supermassive black hole will havemuch lower density compared to that around a stellar mass black hole. Therefore, the cooling processes, particularly the bremsstrahlungradiation which is density dependent, are expected to be ine ffi cient leading to a high f . However, the velocity profiles shown in Fig. 10a arevery similar / same to that around a stellar mass black hole. Figure 10c shows that f → M = . M increases, the density increases and thus the bremsstrahlung radiation increases, as shown in Fig. 11, which leads to the transitionof radiatively ine ffi cient flow to GAAF. When ˙ M = ff ect is very high resulting an GAAF with f much smaller thanunity upto very close to the black hole. Figure 10d shows that close to the black hole there is a possible convective instability for all ˙ M -s.This is because a strong advection of matter close to the black hole hindering cooling processes which results in f →
1. This reassembles apossible convective instability at the inner edge. For a higher ˙ M , the density is high which favours convection and thus brings the convectiveinstability earlier in, at a relatively outer radius. c (cid:13) , 000–000 log(x) l og ( T ) (d)0.5 1 1.5 2 2.5 3 3.501234 log(x) l og ( T ) (b)0.5 1 1.5 2 2.501234 log(x) l og ( T ) (f)0.5 1 1.5 2 2.5−30−25−20 log(x) l og ( Q i e ) ,l og ( Q b r ) ,l og ( Q sy n ) ,l og ( Q c o m p ) (c)0.5 1 1.5 2 2.5 3 3.5−40−35−30−25 log(x) (a)0.5 1 1.5 2 2.5−35−30−25−20 log(x) (e) Figure 11.
Variation of (a) dimensionless energy of Coulomb coupling (thicker line), bremsstrahlung (dotted line), synchrotron (solid), inverse Comptonizationdue to synchrotron photon (dashed line) processes in logarithmic scale, (b) corresponding ion (solid) and electron (dotted) temperatures in units of 10 K, asfunctions of radial coordinate for ˙ M = .
01. (c), (e) Same as (a) except ˙ M = . , M = . , a = α = . M = ; see Table 2 for details. Table 2: Parameters for accretion with α = .
01 around black holes of M = , when the subscript ‘c’ indicates the quantity at the criticalradius and T ec is expressed in units of m i c / k ˙ M a γ x c λ c T ec Sub-Eddington, Eddington accretors0.01 0 1.5 5.5 3.2 0.00010.01 0.998 1.5 3.5 1.7 0.00010.1 0 1.4 5.5 3.2 0.0001780.1 0.998 1.4 3.5 1.7 0.000231 0 1.35 5.5 3.2 0.00024931 0.998 1.35 3.5 1.7 0.000295Super-Eddington accretors10 0 1.345 5.5 3.2 0.00042710 0.998 1.345 3.5 1.7 0.0006100 0 1.34 5.5 3.2 0.0003874100 0.998 1.34 3.5 1.7 0.00059The temperature profiles shown in Fig. 11 are pretty similar to what we obtain in stellar mass black holes what we do not explainhere again. However, note that unlike stellar mass black holes, only the bremsstrahlung radiation is e ff ective in cooling the flow around asupermassive black hole, particularly for ˙ M = The specific angular momentum of the black hole is chosen to be a = . c (cid:13) , 000–000 log(x) f log(x) ν ff log(x) ϑ log(x) l og ( ρ ) Figure 12.
Same as Fig. 10, except a = . of the flow, shown in Fig. 12. As discussed in § a corresponds to a smaller λ which in turn decreases ϑ at a particular radiusof the inner edge of the disc, when the inner edge is stretched in, compared to that around a Schwarzschild black hole. This results in theincrease of residence time of the flow in the sub-Keplerian disc before plunging into the black hole. Therefore, the bremsstrahlung processkeeps cooling and then stabilizing the flow upto very inner edge. Important point to note, as a consequence, is that the disc flow around arotating black hole is convectively more stable compared to that around a nonrotating black hole. This is particularly because the densitygradient of the inner (e.g. the vicinity of x =
2, which is the event horizon for a nonrotating black hole) flow around a rotating black hole isless stepper compared to that around a nonrotating black hole, and hence the flow is convectively more stable in the former case.Figure 13 shows that basic features of the temperature profiles are similar to the cases of static black holes. However, the transitionregion reveals that for a lower ˙ M , the Keplerian flow exhibits the inverse Comptonization via synchrotron photons. As the flow advances witha sub-Keplerian angular momentum, the residence time decreases and thus inverse Comptonization decreases, resulting a hotter flow. Ultra-luminous accretors with a high kinetic luminosity ( ∼ − erg / s) radio jet have been observed in the highly luminous AGNsand ultra-luminous quasars (e.g. PKS 0743-67; Punsly & Tingay 2005), possibly in ULIRs (Genzel et al. 1998) and narrow-line Seyfert 1galaxies (e.g. Mineshige et al. 2000). Therefore, the following cases could be potential models to explain such sources. Figures 14 and 15 show that the basic flow properties are pretty similar to that around stellar mass black holes, except in the present casesthe centrifugal barrier smears out. This is because a high black hole mass corresponds to a low density of the flow and thus a fast infall. Thisalso results, unlike stellar mass black holes, in an ine ffi cient synchrotron radiation even at the inner edge of the disc. Figures 16 and 17 repeat the same story of that around stellar mass black holes, but with the smeared centrifugal barrier, as described abovefor static black holes. However, due to decreasing density, the overall cooling e ff ects decrease keeping the disc hotter, particularly for ˙ M = c (cid:13) , 000–000 log(x) l og ( T ) (d)0.5 1 1.5 201234 log(x) l og ( T ) (b)0.5 1 1.5 201234 log(x) l og ( T ) (f)0.5 1 1.5 2−30−25−20 log(x) l og ( Q i e ) ,l og ( Q b r ) ,l og ( Q sy n ) ,l og ( Q c o m p ) (c)0.5 1 1.5 2−34−32−30−28−26−24−22 log(x) (a)0.5 1 1.5 2−30−25−20 log(x) (e) Figure 13.
Same as Fig. 11, except a = . α AND AROUND COROTATING AND COUNTER ROTATINGBLACK HOLES
So far we have restricted to a typical Shakura-Sunyaev viscosity parameter α = .
01, for corotating black holes. Now we plan to explore alower α , as well as a counter rotating black hole to understand any significant change in the flow behaviour. α = . and α = . α naturally decreases the rate of energy-momentum transfer between any two successive layers of the fluid element and increasingthe residence time of the flow in the sub-Keplerian disc. This also recedes the Keplerian-sub-Keplerian transition region further out. This ismainly because a low value of α can not keep continuing the outward angular momentum transport e ffi ciently in the Keplerian flow belowa certain radial coordinate. Therefore, the disc flow can not remain Keplerian and becomes sub-Keplerian at a larger radius, compared to aflow of high α .We know, on the other hand, that increasing residence time increases the possibility of completing various radiative processes in the discflow, before the infalling matter plunges into the black hole. Therefore, the flow is expected to appear cooler with smaller f . Hence, for thepurpose of comparison, we consider a flow with ˙ M = .
01 around a supermassive black hole of M = , e.g. Sgr A ∗ , which is radiativelyine ffi cient and hot for α = . α , the size of the sub-Keplerian disc is about fivetimes for α = . α = .
01. Inside x =
17 the low α disc flow becomes cooler very fast, rendering f → x >
10 (see Fig.18c). Therefore, the flow sharply transits from radiatively ine ffi cient in nature to GAAF. As a consequence, the low α flow remains stable,as shown in Fig. 18b, all the way upto the event horizon. As the sub-Keplerian flow of a smaller α extends further away where the influenceof black hole is very weak, T e and T i merge (see Fig. 18d) before the flow crosses the transition radius, unlike the flow with α = .
01 when T i > T e there. c (cid:13) , 000–000 log(x) f log(x) ν ff log(x) ϑ log(x) l og ( ρ ) Figure 14.
Variation of dimensionless (a) radial velocity, (b) density, (c) cooling factor, (d) square of convective frequency, as functions of radial coordinatefor super-Eddington accretion flows. Solid, dashed curves are for ˙ M = ,
100 respectively. Other parameters are a = α = . M = ; see Table 2 fordetails. As we already discussed that the model with a low mass accretion rate around a supermassive black hole is a potential case in explaining theobserved dim source Sgr A ∗ . On the other hand, the ultra-luminous X-ray sources presumably correspond to the models with a high massaccretion flow around a stellar mass black hole. Therefore, in order to compare the flow properties between co and counter rotating blackholes, we choose these two extreme cases.Qualitatively, the flows with similar initial conditions around co and counter rotating black holes of same mass are similar, as shownin Figs. 19, 20 for a = ± .
5. However, the sub-Keplerian disc size around the black hole with a = − . ff ective angular momentum (Mukhopadhyay 2003) of the system. Hence the radial velocity is almost an order of magnitude higher,particularly at the inner edge, for a = − . We model the two temperature accretion flow, particularly around black holes, combining the equations of conservation and comprehensivecooling processes. We consider self-consistently the important cooling mechanisms: bremsstrahlung, synchrotron and inverse Comptoniza-tion due to synchrotron photons, where ions and electrons are allowed to have di ff erent temperatures. As matter falls in, hot electrons coolthrough the various cooling mechanisms, particularly by the synchrotron emission when the magnetic field is high. This is particularly thecase for the flow around stellar mass black holes where the magnetic field may also act as a boost in transporting the angular momentum.However, in the present paper, we do not consider such processes in detail, rather stick with the standard α -prescription.By solving a complete set of disc equations we show that in general the disc system exhibits GAAF. However, in certain circumstancesGAAF becomes radiatively ine ffi cient, depending on the flow parameters and hence e ffi ciency of cooling mechanisms. Transitions fromGAAF to radiatively ine ffi cient flow and vice versa are clearly explained by the cooling e ffi ciency factor f , shown in each model cases.While the previous authors, who proposed ADAF (Narayan & Yi 1994, 1995), especially restricted with flows having f = ffi cientcooling), here we do not impose any restriction to the flow parameters to start with and let the parameter f to determine self-consistentlyas the system evolves. Therefore, our model is very general whose special case may be understood as a radiatively ine ffi cient advectiondominated flow.We have explored especially the optically thin flows incorporating bremsstrahlung, synchrotron and inverse Comptonization processes.In Fig. 21 we show the variation of the e ff ective optical depth as a function of disc radii for two limiting cases. While flows around rotating c (cid:13) , 000–000 log(x) l og ( T ) (b)1 1.5 20.511.522.5 log(x) l og ( T ) (d)0.5 1 1.5 2−35−30−25−20 log(x) l og ( Q i e ) ,l og ( Q b r ) ,l og ( Q sy n ) ,l og ( Q c o m p ) (a)1 1.5 2−40−35−30−25−20 log(x) l og ( Q i e ) ,l og ( Q b r ) ,l og ( Q sy n ) ,l og ( Q c o m p ) (c) Figure 15.
Variation of (a) dimensionless energy of Coulomb coupling (thicker line), bremsstrahlung (dotted line), synchrotron (solid), inverse Comptonizationdue to synchrotron photon (dashed line) processes in logarithmic scale, (b) corresponding ion (solid) and electron (dotted) temperatures in units of 10 K, asfunctions of radial coordinate for ˙ M =
10. (c) Same as (a) except ˙ M = M = a = α = . M = ;see Table 2 for details. log(x) f log(x) ν ff log(x) ϑ log(x) l og ( ρ ) Figure 16.
Same as Fig. 14, except a = . (cid:13) , 000–000 log(x) l og ( T ) (b)0.5 1 1.511.522.53 log(x) l og ( T ) (d)0.5 1 1.5 2−30−25−20 log(x) l og ( Q i e ) ,l og ( Q b r ) ,l og ( Q sy n ) ,l og ( Q c o m p ) (a)0.5 1 1.5−30−25−20 log(x) l og ( Q i e ) ,l og ( Q b r ) ,l og ( Q sy n ) ,l og ( Q c o m p ) (c) Figure 17.
Same as Fig. 15, except a = . log(x) f log(x) ν ff log(x) l og ( ϑ ) log(x) l og ( T ) (d) Figure 18.
Comparison between solutions for high and low α : Variation of (a) velocity, (b) square of convective frequency, (c) cooling factor, (d) ion (upper setof lines) and electron (lower set of lines) temperatures, a functions of radial coordinate, when solid lines correspond to α = .
01 and dashed lines correspondto α = . M = . M = , a = (cid:13) , 000–000 log(x) f −3 (b) log(x) ν ff log(x) ϑ log(x) l og ( T ) (d) Figure 19.
Comparison between solutions for co and counter rotating stellar mass black holes: Variation of (a) velocity, (b) square of convective frequency,(c) cooling factor, (d) ion (upper set of lines) and electron (lower set of lines) temperatures, as functions of radial coordinate, when solid lines correspond to b = . b = − .
5. Other parameters are ˙ M = M = α = . log(x) f log(x) ν ff log(x) ϑ log(x) l og ( T ) (d) Figure 20.
Same as Fig. 19, except ˙ M = . M = . c (cid:13) , 000–000 -5-7-9-11 321 l og ( τ e ff ) log(x) Figure 21.
Variation of the e ff ective optical depth as a function of radial coordinate. Solid ( a =
0) and dotted ( a = . M = M =
100 and dashed ( a =
0) and dot-dashed ( a = . M = , ˙ M = .
01. See Tables 1 and 2 for details. black holes appear slightly thinner compared to the corresponding cases of static black holes, in general τ e ff < ∼ × − . This verifies our choiceof optically thin flows throughout. However, for the present purpose, when the main aim is to understand disk dynamics in the global, viscous,two-temperature regime, we have ignored inverse Comptonization due to bremssstrahlung photons, if any. This may be important in cases ofvery super-Eddington accretion flows which we plan to explore in future, particularly, in analysing the underlying spectra.The temperature of the flow depends on the accretion rate. If the accretion rate is low and thus the flow is radiatively ine ffi cient, then thedisc is hot. Such a hot flow is being attempted to model since 1976 (Shapiro, Lightman & Eardley 1976) when it was assumed that locally Q + ∼ Q − and thus f →
0. While the model was successful in explaining observed hard X-rays from Cyg X-1, it turned out to be thermallyunstable. Rees et al. (1982) proposed a hot ion torus model avoiding f to unity. In the similar spirit Narayan & Yi (1995) proposed the hottwo temperature solution in the assumption of f → ff erential equations. Based on some simplistic assumptions they showedthe importance of advective cooling. Moreover, a single temperature description does not allow them to include all the underlying physicsnecessary to describe the cooling processes. In the due course, Mandal & Chakrabarti (2005) proposed a two temperature disc solution wherethe ion temperature could be as high as ∼ K. However, they particularly emphasized on how does the shock in the disc flow enablecooling through the synchrotron mechanism, without carrying out a complete analysis of the dynamics. The present paper describes, toour knowledge, the first comprehensive work to model the two temperature accretion flow self-consistently by solving the complete set ofunderlying equations without any pre-assumptive choice of the flow variables to start with.The generality lies not only in its construction but also its ability to explain the under-luminous to ultra-luminous sources, stellar massto supermassive black holes. Table 3 lists the luminosities for a wide range of parameter sets, obtained by our model. It reveals that for a verylow mass accretion rate ˙ M = . L ∼ erg / sec, which indeed issimilar to the observed luminosity from a under-luminous source Sgr A ∗ . In other extreme, for ˙ M =
100 around a similar black hole, L ∼ erg / sec, similar to what observed from the highly luminous AGNs like PKS 0743-67. On the other hand, when the black hole is considered tobe of stellar mass, then at a high ˙ M = L ∼ erg / sec which is similar to the observed luminosity from ULX sources(e.g. SS433). c (cid:13) , 000–000 Table 3: Luminosity in erg / sec˙ M M γ L . . .
01 10 . .
35 5 ×
100 10 .
34 10 .
01 10 1 . .
35 7 ×
100 10 1 .
34 10 In general, an increase of accretion rate increases density of the flow which may lead to a high rate of cooling and thus decrease ofthe cooling factor f . Hence, f is higher, close to unity reassembling radiatively ine ffi cient flows, for sub-Eddington accretors, and is lower,sometimes close to zero, for super-Eddington flows. Actual value of f in a flow also depends on the behaviour of hydrodynamic variableswhich determine the rate of cooling processes. Naturally, as the flow advances from the transition region to the event horizon, f variesbetween 0 and 1. However, if the black hole is considered to be rotating, the flow angular momentum decreases and thus the radial velocityincreases. This in turn reduces the residence time of the sub-Keplerian flow hindering cooling processes to complete. This flow is thenexpected to be hotter and hence f to be higher compared to that around a static black hole. Therefore, the system may tend to be radiativelyine ffi cient, evenif its counter part around a static black hole appears to be an GAAF. However, this also depends on the value of α , as shownin Fig. 18. A low value of α increases the residence time of matter in the disc which helps in cooling processes to complete, rendering aradiatively ine ffi cient flow to switch over to GAAF. This feature may help in understanding the transient X-ray sources.In all the cases, the ion and electron temperatures merge or tend to merge at around transition radius. This is because, the electrons arein thermal equilibrium with the ions and thus virial around the transition radius, particularly when ˙ M > ∼
1. As the sub-Keplerian flow advances,the ions become hotter and the corresponding temperature increases, rendering the ion-electron Coulomb collisions weaker. The electrons, onthe other hand, cool down via processes like bremsstrahlung, synchrotron emissions etc. keeping the electron temperature roughly constantupto very inner disc. This reveals the two temperature flow strictly.Important point to note is that we have assumed throughout the coupling between the ions and electrons is due to the Coulomb scattering.However, the inclusion of possible nonthermal processes of transferring energy from the ions and electrons (Phinney 1981, Begelman &Chiueh 1988) might modify the results. However, as argued by Narayan & Yi (1995), the collective mechanism discussed by Begelman &Chiueh (1988) may dominate over the Coulomb coupling at either a very low α or a very low ˙ M . Instead, the viscous heating rate of ionsis much larger than the collective rate of nonthermal heating of electrons, unless α is too small what we have not considered in the presentcases. Therefore, the assumption to neglect nonthermal heating of electrons is justified.Now the future jobs should be to understand the radiation emitted by the flows discussed here and to model the corresponding spectra.This will be the ultimate test of the model in order to explain observed data. APPENDIX A: DISCUSSION OF BOUNDARY VALUES
We have four coupled nonlinear di ff erential equations (6), (10), (12), (17) to be solved for ϑ , c s , λ , T e ; equations also involve ρ and P .To eliminate ρ and P , we use mass transfer equation (1) and equation of state (9). Therefore, in total we have five di ff erential equationssupplemented by an equation of state. Hence, we need five boundary conditions to start integration. Equation (1) can be integrated to obtain˙ M already given in Eqn. (2), which is supplied as an input parameter. Similarly, integrating Eqn. (10) we can obtain angular momentum flux˙ M ( λ − λ in ) = − π x | W x φ | , (A1)where λ in is the specific angular momentum at the inner edge of the disc, to be fixed by no torque inner boundary condition. Note that λ in λ c (see, e.g., Chakrabarti 1996).We therefore need the initial values of ϑ , c s , λ and T e to solve the set of equations. When we impose the condition that the flow mustpass through a critical radius x c (around a sonic radius) where D = ϑ and c s at x c are related by a quadratic equation of Mach numbergiven by Eqn. (24).For the continuity of d ϑ/ dx , N = x c . Therefore, from N = c s at x c can be computediteratively (using bisection method), which in turn fixes ϑ at x c as well, provided λ is known at that radius.Now we need to set appropriate values of x c and corresponding specific angular momentum λ c . This is fixed iteratively by invoking thecondition that the critical point to be saddle-type. This can be seen as follows. First we impose that λ in = λ c . Then by fixing the value of λ c wefind that if x c is greater than a certain critical value x cc , then the type of x c changes from saddle-type to O -type which matter never can passthrough. Figure A1a shows how the type of critical point and then solution topology change with a slight increase of x c . On the other hand,as x c decreases from x cc which corresponds to the energy at x c ( E c ) increases, the sub-Keplerian disc decreases in size. This advances theKeplerian disc. The reason is that increasing E c corresponds to decreasing x c and then increasing centrifugal energy ( λ / x | c ) which keeps c (cid:13) , 000–000 -1-2 21 l og ( ϑ ) log(x)(a) -1-2 21 l og ( ϑ ) log(x)(b)-1-2 21 l og ( ϑ ) log(x)(b) Figure A1.
Comparison of the variation of radial velocity as a function of radial coordinate (a) between solutions with x c = . x c = . λ c = .
2, (b) between solutions with λ c = . x c = . x c = .
5. Other parameters are same as that inFig. 2 for ˙ M = . the flow Keplerian until inner region. However, in principle the solution of the model equations is possible to obtain for any value of x c from x cc to the marginally bound orbit x b .Once x c is fixed at x cc , we have to obtain the best value of λ c . By increasing the value of λ c beyond a certain critical value λ cc at aparticular x c , we again find a transition from saddle-type to O -type critical point. Figure A1b shows how the type of critical point and thensolution topology change with a slight increase of λ c . On the other hand, decreasing λ c from λ cc will tend the disc to more Bondi-type. Nowfor λ cc we again have to obtain a new value of x cc following the procedure outlined above and thereafter corresponding λ cc . This needs to becontinued iteratively until a specific combination of critical radius x cc and corresponding specific angular momentum λ cc , lying in a narrowrange, is obtained which leads to a physically interesting large sub-Keplerian accretion disc, when matter infalling from a largest possibletransition radius to a black hole event horizon through a saddle-type critical point. However, in principle the solution of the model equationsis possible to obtain for a range of λ c such that λ cc > λ c > T e at x c ( T ec ) to be assigned. Choice of T ec depends on the observed nonthermal radiationwhich restricts the value of T e in general. But this restriction can only provide an order of magnitude of T e . An exact value of T ec should beobtained iteratively from a plausible range of T e at x c so that the values of x cc and λ cc obtained following the above mentioned procedureconverge. APPENDIX B: COMPUTATION OF DERIVATIVE OF VELOCITY AT THE CRITICAL RADIUS
We first recall the derivative of velocity from Eqn. (20) d ϑ dx = N ( x , ϑ, c s , λ, T e ) D ( ϑ, c s ) . (B1)At the critical radius d ϑ dx = . (B2)Therefore, we apply l’Hospital’s rule and obtain d ϑ dx = DDx [ N ( x , ϑ, c s , λ, T e )] DDx [ D ( ϑ, c s )] = dNdx + dNd ϑ d ϑ dx + dNdc s dc s dx + dNd λ d λ dx + dNdT e dT e dxdDd ϑ d ϑ dx + dDdc s dc s dx . (B3) c (cid:13) , 000–000 Now combining with Eqns. (26), (27), (28) we obtain d ϑ dx = N + N d ϑ dx D + D d ϑ dx , (B4)where N = dNdx + dNdc s Jc s + dNd λ c s − x α Jc s + ϑ ! + dNdT e ( Γ − π ˙ M c s x / F / (cid:16) Q ie − Q − (cid:17) (1 − Γ ) T e Jc s + G ! , (B5) N = dNd ϑ + dNdc s c s ϑ − ϑ c s ! + dNd λ α x ϑ c s I n + I n c s ϑ − ϑ c s ! + α x ! + dNdT e (1 − Γ ) T e ϑ c s , (B6) D = dDdc s Jc s (B7) D = dDd ϑ + dDdc s c s ϑ − ϑ c s ! . (B8)Finally cross-multiplying in Eqn. (B4) we obtain a quadratic equation D d ϑ dx ! + ( D − N ) d ϑ dx ! − N = d ϑ dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c = N − D ± p ( D − N ) + D N D , (B10)where upper and lower signs correspond to wind and accretion respectively. ACKNOWLEDGMENTS
This work is partly supported by a project, Grant No. SR / S2HEP12 / REFERENCES
Abramowicz, M. A., Chen, X., Kato, S., Lasota, J.-P., Regev, O. 1995, ApJ, 438, L37.Abramowicz, M. A., Czerny, B., Lasota, J. P., & Szuszkiewicz, E. 1988, ApJ, 332, 646.Begelman, M. C. 1978, MNRAS, 184, 53.Begelman, M. C., King, A. R., & Pringle, J. E. 2006, MNRAS, 370, 399.Bisnovatyi-Kogan, G. S., & Lovelace, R. V. E. 2000, ApJ, 529, 978.Chakrabarti, S. K. 1989, ApJ, 347, 365.Chakrabarti, S. K. 1996, ApJ, 464, 664.Chakrabarti, S. K., & Titarchuk, L. G. 1995, ApJ, 455, 623.Chen, X., & Taam, R. E. 1993, ApJ, 412, 254.Coppi, P. S. 1999, ASP Conference Series, 161, 375.Eardley, D. M., & Lightman, A. P. 1975, ApJ, 200, 187.Eggum, G. E., Coroniti, F. V., & Katz, J. I. 1985, ApJ, 298, 41.Eggum, G. E., Coroniti, F. V., & Katz, J. I. 1988, ApJ, 330, 142.Fabbiano, G. 2004, RMxAC, 20, 46.Fabrika, S. 2004, ASPRv, 12, 1.Ghosh, S., & Mukhopadhyay, B. 2009, RAA, 9, 157.Gierli´nski, M., Zdziarski, A. A., Poutanen, J., Coppi, P. S., Ebisawa, K., & Johnson, W. N. 1999, MNRAS, 309, 496.Gilfanov, M., Churazov, E., & Sunyaev, R. 1997, LNP, 487, 45.Genzel, R., et al. 1998, ApJ, 498, 579.Liang, E. P. T., & Thompson, K. A. 1980, 240, 271.Lightman, A. P., & Shapiro, S. L. 1975, ApJ, 198, 73.Lovelace, R. V. E., Romanova, M. M., & Newman, W. I. 1994, ApJ, 437, 136.Mandal, S., & Chakrabarti, S. K. 2005, A&A, 434, 839.Manmoto, T., Mineshige, S., & Kusunose, M. 1997, ApJ, 489, 791.Matsumoto, R., Kato, S., Fukue, J., & Okazaki, A. T. 1984, PASJ, 36, 71.Medvedev, M. V., & Narayan, R. 2001, ApJ, 554, 1255. c (cid:13) , 000–000 Melia, F., & Misra, R. 1993, ApJ, 411, 797.Mineshige, S., Kawaguchi, T., Takeuchi, M., & Hayashida, K. 2000, PASJ, 52, 499.Muchotrzeb, B., & Paczynski, B. 1982, AcA, 32, 1.Mukhopadhyay, B. 2002, ApJ, 581, 427.Mukhopadhyay, B. 2003, ApJ, 586, 1268.Mukhopadhyay, B., & Ghosh, S. 2003, MNRAS, 342, 274; MG03.Nakamura, K. E., Kusunose, M., Matsumoto, R., & Kato, S. 1997, PASJ, 49, 503.Narayan, R., Kato, S., & Honma, F. 1997, ApJ, 476, 49.Narayan, R., Igumenshchev, I. V., & Abramowicz, M. A. 2000, ApJ, 539, 798.Narayan, R., & Popham, R. 1993, Nature, 362, 820.Narayan, R., & Yi, I. 1994, ApJ, 428, 13.Narayan, R., & Yi, I. 1995, ApJ, 452, 710.Novikov, I. D., & Thorne, K. S. 1973, in Black Holes, Les Houches 1972 (France), ed. B. & C. DeWitt (New York: Gordon & Breach), 343.Paczynsky, B., & Wiita, P. J. 1980, A&A, 88, 23.Pringle, J. E., & Rees, M. J. 1972, A&A, 21, 1.Punsly, B., & Tingay, S. J. 2005, ApJ, 633, 89.Quataert, E., & Gruzinov, A. 2000, ApJ, 539, 809.Rajesh, S. R., & Mukhopadhyay, B. 2009, New Astronomy (to appear); arXiv:0908.3956.Rees, M. J., Begelman, M. C., Blandford, R. D., & Phinney, E. S. 1982, Nature, 295, 17.Shakura, N., & Sunyaev, R. 1973, A&A, 24, 337.Shapiro, S. L., Lightman, A. P., & Eardley, D. M. 1976, ApJ, 204, 187.Sinha, M., Rajesh, S. R., & Mukhopadhyay, B. 2009, RAA (to appear).Thorne, K. S., & Price, R. H. 1975, ApJ, 195, 101.Wallinder, F. H. 1991, A&A, 249, 107.Yuan, F. 1999, ApJ, 521, L55.Zdziarski, A. A., Grove, J. E., Poutanen, J., Rao, A. R., & Vadawale, S. V. 2001, ApJ, 554, 45.c (cid:13)000