BBlack hole accretion discs ∗ Jean-Pierre Lasota † Institut d’Astrophysique de Paris, CNRS et Sorbonne Universit´es,UPMC Universit´e Paris 06, UMR 7095, 98bis Boulevard Arago, 75014 Paris, France andNicolaus Copernicus Astronomical Center,Bartycka 18, 00-716 Warsaw, Poland
This is an introduction to models of accretion discs around black holes. After a presentation ofthe non–relativistic equations describing the structure and evolution of geometrically thin accretiondiscs we discuss their steady–state solutions and compare them to observation. Next we describein detail the thermal–viscous disc instability model and its application to dwarf novae for whichit was designed and its X–ray irradiated–disc version which explains the soft X–ray transients,i.e. outbursting black–hole low–mass X–ray binaries. We then turn to the role of advection inaccretion flows onto black holes illustrating its action and importance with a toy model describingboth ADAFs and slim discs. We conclude with a presentation of the general-relativistic formalismdescribing accretion discs in the Kerr space-time.
I. INTRODUCTION
The author of this chapter is old enough to remember the days when even serious astronomers doubted the existenceof accretion discs and scientists snorted with contempt at the suggestion that there might be such things as blackholes; the very possibility of their existence was rejected, and the idea of black holes was dismissed as a fancy ofeccentric theorists. Today, some 50 years later, there is no doubt about the existence of accretion discs and blackholes; both have been observed and shown to be ubiquitous in the Universe. The spectacular ALMA image of theprotostellar disc in HL Tau [46] is breathtaking and we can soon expect to see the silhouette of a supermassive blackhole in the center of the Galaxy in near infrared [57] or millimeter-waves [10].Understanding accretion discs around black holes is interesting in itself because of the fascinating and complexphysics involved but is also fundamental for understanding the coupled evolution of galaxies and their nuclear blackholes, i.e. fundamental for the understanding the growth of structures in the Universe. The chance that inflows ontoblack holes are strictly radial, as assumed in many models, are slim.The aim of the present chapter is to introduce the reader to models of accretion discs around black holes. Becauseof the smallness of black holes the sizes of their accretion discs span several orders of magnitude: from close to thehorizon up 100 000 or even 1 000 000 black-hole radii. This implies, for example, that the temperature in a discaround a stellar–mass black hole varies from 10 K, near the its surface, to ∼ K near the disc’s outer edge at 10 black-hole radii, say. Thus studying black hole accretion discs allow the study of physical regimes relevant also in adifferent context and inversely, the knowledge of accretion disc physics in other systems such as e.g. protostellar discsor cataclysmic variable stars, is useful or even necessary for understanding the discs around black holes.Section II contains a short discussion of the disc driving mechanisms and introduces the α –prescription used in thischapter. In Section III after presenting the general framework of the geometrically thin disc model we discuss theproperties of stationary solutions and the Shakura–Sunyaev solution in particular. The dwarf-nova disc instabilitymodel and its application to black-hole transient sources is the subject of Section IV. The role of advection in accretiononto black holes is presented in Section V with the main stress put on high accretion rate flows. Finally, Section VIand VII about the general–relativistic version of the accretion disc equations concludes the present chapter. Notations and definitions
The Schwarzschild radius (radius of a non-rotating black hole) is R S = 2 GMc = 2 . × M M (cid:12) cm , (1)where M is the mass of the gravitating body and c the speed of light. ∗ Chapter 1 in
Astrophysics of Black Holes – From fundamental aspects to latest developments , Ed. Cosimo Bambi, Springer: Astrophysicsand Space Science Library 440, (2016); DOI: 10.1007/978-3-662-52859-4. † [email protected] a r X i v : . [ a s t r o - ph . H E ] J un The Eddington accretion rate is defined as˙ M Edd = L Edd ηc = 1 η πGMcκ es = 1 η πcR S κ es = 1 . × η − . M M (cid:12) g s − , (2)where η = 0 . η . is the radiative efficiency of accretion, κ es the electron scattering (Thomson) opacity.We will often use accretion rate measured in units of Eddington accretion rate:˙ m = ˙ M ˙ M Edd . (3) Additional reading:
There are excellent general reviews of accretion disc physics, they can be found in references[9], [18] [26] and [55].
II. DISC DRIVING MECHANISM; VISCOSITY
In recent years there have been an impressive progress in understanding the physical mechanisms that drive discaccretion. It is now obvious that the turbulence in ionized Keplerian discs is due to the Magneto-Rotational Instability(MRI) also known as the Balbus-Hawley mechanisms [8],[7]. However, despite these developments, numerical simu-lations, even in their global, 3D form suffer still from weaknesses that make their direct application to real accretionflows almost infeasible.One of the most serious problems is the value of the ratio of the (vertically averaged) total stress to thermal(vertically averaged) pressure α = (cid:104) τ rϕ (cid:105) z (cid:104) P (cid:105) z (4)which according to most MRI simulation is ∼ − whereas observations of dwarf nova decay from outburst un-ambiguously show that α ≈ . − . ∼ K increase α to values ∼ .
1. This might solve the problem of discrepancies between theMRI-calculated and the observed value of α [11]. One has, however, to keep in mind that the simulations in questionhave been performed in a so-called shearing box and their validity in a generic 3D case has yet to be demonstrated.Another problem is related to cold discs such as quiescent dwarf nova discs [31] or protostellar discs [7]. For thestandard MRI to work, the degree of ionization in a weakly magnetized, quasi-Keplerian disc must be sufficiently highto produce the instability that leads to a breakdown of laminar flow into turbulence which is the source of viscositydriving accretion onto the central body. In cold discs the ionized fraction is very small and might be insufficient forthe MRI to operate. In any case in such a disc non-ideal MHD effects are always important. All these problems stillawait their solution.Finally, and very relevant to the subject of this Chapter there is the question of stability of discs in which thepressure is due to radiation and opacity to electron scattering. According to theory such discs should be violently(thermally) unstable but observations of systems presumed to be in this regime totally infirm this prediction. MRIsimulations not only do not solve this contradiction but rather reinforce it [24]. A. The α –prescription The α –prescription [53] is a rather simplistic description of the accretion disc physics but before one is offered betterand physically more reliable options its simplicity makes it the best possible choice and has been the main source ofprogress in describing accretion discs in various astrophysical contexts.One keeps in mind that the accretion–driving viscosity is of magnetic origin, but one uses an effective hydrodynam-ical description of the accretion flow. The hydrodynamical stress tensor is (see e.g. [34]) τ rϕ = ρν ∂v ϕ ∂R = ρ d Ω d ln R , (5)where ρ is the density, ν the kinematic viscosity coefficient and v ϕ the azimuthal velocity ( v ϕ = R Ω).In 1973 Shakura & Sunyaev proposed the (now famous) prescription τ rϕ = αP, (6)where P is the total thermal pressure and α ≤
1. This leads to ν = αc s (cid:20) d Ω d ln R (cid:21) − , (7)where c s = (cid:112) P/ρ is the isothermal sound speed and ρ the density. For the Keplerian angular velocityΩ = Ω K = (cid:18) GMR (cid:19) / (8)this becomes ν = 23 αc s / Ω K . (9)Using the approximate hydrostatic equilibrium Eq. (19) one can write this as ν ≈ αc s H. (10)Multiplying the rhs of Eq. (5) by the ring length (2 πR ) and averaging over the (total) disc height one obtains theexpression for the total torque T = 2 πR Σ νR d Ω d ln R , (11)where Σ = (cid:90) + ∞−∞ ρ dz. (12)For a Keplerian disc T = 3 π Σ ν(cid:96) K , (13)( (cid:96) K = R Ω K is the Keplerian specific angular momentum.)The viscous heating is proportional to to τ rϕ ( d Ω /dR ) [34]. In particular the viscous heating rate per unit volume is q + = − τ rϕ d Ω d ln R , (14)which for a Keplerian disc, using Eq. (6), can be written as q + = 32 α Ω K P, (15)and the viscous heating rate per unit surface is therefore Q + = T Ω (cid:48) πR = 98 Σ ν Ω K . (16)(The denominator in the first rhs is 2 × πR taking into account the existence of two disc surfaces.) Additional reading:
References [7], [8] and [22].
III. GEOMETRICALLY THIN KEPLERIAN DISCS
The 2D structure of geometrically thin, non–self-gravitating, axially symmetric accretion discs can be split into a1+1 structure corresponding to a hydrostatic vertical configuration and radial quasi-Keplerian viscous flow. The two1D structures are coupled through the viscosity mechanism transporting angular momentum and providing the localrelease of gravitational energy.
A. Disc vertical structure
The vertical structure can be treated as a one–dimensional star with two essential differences:1. the energy sources are distributed over the whole height of the disc, while in a star there limited to the nucleus,2. the gravitational acceleration increases with height because it is given by the tidal gravity of the accretor, whilein stars it decreases as the inverse square of the distance from the center.Taking these differences into account the standard stellar structure equations (see e.g. [48]) adapted to the descrip-tion of the disc vertical structure are listed below. • Hydrostatic equilibriumThe gravity force is counteracted by the force produced by the pressure gradient: dPdz = ρg z , (17)where g z is the vertical component (tidal) of the accreting body gravitational acceleration: g z = ∂∂z (cid:20) GM ( R + z ) / (cid:21) ≈ GMR zR . (18)The second equality follows from the assumption that z (cid:28) R . Denoting the typical (pressure or density) scale-height by H the condition of geometrical thinness of the disc is H/R (cid:28) dP/dz ∼ P/H , Eq. (17)can be written as HR ≈ c s v K , (19)where v K = (cid:112) GM/R is the Keplerian velocity and we made use of Eq. (18). From Eq. (19) it follows that Hc s ≈ K =: t dyn , (20)where t dyn is the dynamical time. • Mass conservationIn 1D hydrostatic equilibrium the mass conservation equation takes the simple form of dςdz = 2 ρ. (21) • Energy transfer - temperature gradient d ln Tdz = ∇ d ln Pdz . (22)For radiative energy transport ∇ rad = κ R P F z P r cg z , (23)where P r is the radiation pressure and κ R the Rosseland mean opacity. From Eqs. (22) and (23) one recoversthe familiar expression for the radiative flux F z = − σT κ R ρ ∂T∂z = − σ κ R ρ ∂T ∂z (24)( F z is positive because the temperature decreases with z so ∂T /∂z < τ (cid:39) / z = 0, F z = 0, T = T c , ς = 0 at the disc midplane; at the disc photosphere ς = Σ and T ( τ = 2 /
3) = T . For a detaileddiscussion of radiative transfer, temperature stratification and boundary conditions see Sect. III E.In the same spirit as Eq. (19) one can write Eq. (24) as F z ≈ σT c κ R ρH = 83 σT c κ R Σ , (25)where T c is the mid-plane (“central”) disk temperature. Using the optical depth τ = κ R ρH = (1 / κ R Σ, thiscan be written as F z ( H ) ≈ σT c τ = Q − , (26)(see Eq. 77 for a rigorous derivation of this formula). Remark 1.
In some references (for example in [18]) the numerical factor on the rhs is “4/3” instead of “8/3”.This is due to a different definition of Σ: in our case it is = 2 ρH , whereas in [18] Σ = ρH . //In the case of convective energy transport ∇ = ∇ conv . Because convection in discs is still not well understood(see, however, [22]) there is no obvious choice for ∇ conv . In practice a prescription designed by Paczy´nski [41]for extended stellar envelope is used [21] but this most probably does not represent very accurately what ishappening in convective accretion discs [11]. • Energy conservationVertical energy conservation should have the form dF z dz = q + ( z ) , (27)where q + ( z ) corresponds to viscous energy dissipation per unit volume. Remark 2.
In contrast with accretion discs, stellar envelopes have dF z /dz = 0The α prescription does not allow deducing the viscous dissipation stratification ( z dependence), it just saysthat the vertically averaged viscous torque is proportional to pressure. Most often one assumes therefore that q + ( z ) = 32 α Ω K P ( z ) , (28)by analogy with Eq. (15) but such an assumption is chosen because of its simplicity and not because of somephysical motivation. In fact MRI numerical simulations suggest that dissipation is not stratified in the way aspressure [22]. • The vertical structure equations have to be completed by the equation of state (EOS): P = P r + P g = 4 σ c T + R µ ρT, (29)where R is the gas constant and µ the mean molecular weight, and an equation describing the mean opacitydependence on density and temperature. B. Disc radial structure • Continuity (mass conservation) equation has the form ∂ Σ ∂t = − R ∂∂R ( R Σ v r ) + S ( R, t )2 πR , (30)where S ( R, t ) is the matter source (sink) term.In the case of an accretion disc in a binary system S ( R, t ) = ∂ ˙ M ext ( R, t ) ∂R (31)represents the matter brought to the disc from the Roche lobe filling/mass losing (secondary) companion of theaccreting object. ˙ M ext ≈ ˙ M tr , where ˙ M tr is the mass transfer rate from the companion star. Most often oneassumes that the transfer stream delivers the matter exactly at the outer disc edge, but although this assumptionsimplifies calculations it is contradicted by observations that suggest that the stream overflows the disc surface(s). • Angular momentum conservation ∂ Σ (cid:96)∂t = − R ∂∂R ( R Σ (cid:96)v r ) + 1 R ∂∂R (cid:18) R Σ ν d Ω dR (cid:19) + S (cid:96) ( R, t )2 πR . (32)This conservation equation reflects the fact that angular momentum is transported through the disc by a viscousstress τ rϕ = R Σ νd Ω /dR . Therefore, if the disc is not considered infinite (recommended in application to realprocesses and systems) there must be somewhere a sink of this transported angular momentum S (cid:96) ( R, t ). Forbinary semi-detached binary systems there is both a source (angular momentum brought in by the mass transferstream form the stellar companion) and a sink (tidal interaction taking angular momentum back to the orbit).The two respective terms in the angular momentum equation can be written as S j ( R, t ) = (cid:96) k πR ∂ ˙ M ext ∂R − T tid ( R )2 πR . (33)Assuming Ω = Ω K , from Eqs. (30) and (32) one can obtain an diffusion equation for the surface density Σ: ∂ Σ ∂t = 3 R ∂∂R (cid:26) R / ∂∂R (cid:104) ν Σ R / (cid:105)(cid:27) . (34)Comparing with Eqs. (30) one sees that the radial velocity induced by the viscous torque is v r = − R / ∂∂R (cid:104) ν Σ R / (cid:105) , (35)which is an example of the general relation v visc ∼ νR . (36)Using Eq.(10) one can write t vis := Rv visc ≈ R ν ≈ α − Hc s (cid:18) HR (cid:19) − . (37)The relation between the viscous and the dynamical times is t vis ≈ α − (cid:18) HR (cid:19) − t dyn . (38)In thin ( H/R (cid:28)
1) accretion discs the viscous time is much longer that the dynamical time. In other words,during viscous processes the vertical disc structure can be considered to be in hydrostatic equilibrium. • Energy conservationThe general form of energy conservation (thermal) equation can be written as: ρT dsdR := ρT ∂s∂t + v r ∂s∂R = q + − q − + (cid:101) q, (39)where s is the entropy density, q + and q − are respectively the viscous and radiative energy density, and (cid:101) q isthe density of external and/or radially transported energy densities. Using the first law of thermodynamics T ds = dU + P dV one can write ρT dsdt = ρ dUdt + P ∂v r ∂r , (40)where U = (cid:60) T c /µ ( γ − T = T c , using Eq. (30) and the thermodynamical relations from Appendix (for β = 1) one obtains ∂T c ∂t + v r ∂T c ∂R + (cid:60) T c µc P R ∂ ( Rv r ) ∂R = 2 Q + − Q − c P Σ + (cid:101) Qc P Σ , (41)where Q + and Q − are respectively the heating and cooling rates per unit surface. (cid:101) Q = Q out + J with Q out corresponding to energy contributions by the mass-transfer stream and tidal torques; J ( T, Σ) represent radialenergy fluxes that are a more or less ad hoc addition to the 1+1 scheme to which they do not belong since itassumes that radial gradients ( ∂/∂R ) of physical quantities can be neglected.The viscous heating rate per unit surface can be written as (see Eq. 16) Q + = 98 ν ΣΩ (42)while the cooling rate over unit surface (the radiative flux) is obviously Q − = σT . (43)In thermal equilibrium one has Q + = Q − . (44)The cooling time can be easily estimated from Eq. (44). The energy density to be radiated away is ∼ ρc s (seeEqs 228 and 232), so the energy per unit surface is ∼ Σ c s and the cooling (thermal) time is t th = Σ c s Q − = Σ c s Q + ∼ α − Ω − K = α − t dyn . (45)Since α < t th > t dyn and during thermal processes the disc can be assumed to be in (vertical) hydrostaticequilibrium.For geometrical thin ( H/R (cid:28)
1) accretion discs one has the following hierarchy of the characteristic times t dyn < t th (cid:28) t vis . (46)(This hierarchy is similar to that of characteristic times in stars: the dynamical is shorter than the thermal(Kelvin-Helmholtz) and the thermal is much shorter than the thermonuclear time.) C. Self-gravity
In this Chapter we are interested in discs that are not self-gravitating, i.e. in discs where the vertical hydrostaticequilibrium is maintained against the pull of the accreting body’s tidal gravity whereas the disc’s self-gravity can beneglected. We will see now under what conditions this assumption is satisfied.The equation of vertical hydrostatic equilibrium can be written as1 ρ dPdz = − g = ( − g z − g s ) = g z (cid:18) g s g z (cid:19) =: − g z (1 + A ) , (47)therefore self-gravity is negligible when A (cid:28)
1. Treating the disc as an infinite uniform plane (i.e. assuming the surfacedensity does not vary too much with radius) one can write its self gravity as g s = 2 πG Σ, whereas the z -componentof the gravity provided by the central body is g z = Ω K z (Eq. 18). Therefore evaluating A at z = H one gets A H := g s g z (cid:12)(cid:12)(cid:12)(cid:12) H = 2 πG ΣΩ K H . (48) A H is related to the so-called Toomre parameter [56] Q T := c s Ω πG Σ , (49)widely used in the studies of gravitational stability of rotating systems, through A H ≈ Q − T . We will therefore expressthe condition of negligible self-gravity (gravitational stability) as Q T > . (50)Using Eqs. (19), (10) and (57) one can write the Toomre parameter as Q T = 3 c s G ˙ M , (51)or as function of the mid-plane temperature T = 10 T Q T ≈ . × α T / m ˙ m , (52)where m = M/ M (cid:12) . This shows that hot ionized ( T (cid:38) ) discs become self-gravitating for high accretor massesand high accretion rates. Discs in close binary systems ( m (cid:46)
30) are never self-gravitating for realistic accretionrates ( ˙ m < m = 10 a hot disc will become self-gravitating at R/R S ≈ m ∼ − . In general, geometrically thin, non–self-gravitating accretion discs around supermassive black holes havevery a limited radial extent. Additional reading:
References [12], [19],[20], [35], [43] and [56].
D. Stationary discs
In the case of stationary ( ∂/∂t = 0) discs Eq. (30) can be easily integrated giving˙ M := 2 πR Σ v r , (53)where the integration constant ˙ M (mass/time) is the accretion rate .Also the angular momentum equation (32) can be integrated to give − πR Σ v r (cid:96) + 2 πR Σ ν Ω (cid:48) = const. (54)Or, using Eq. (53), − ˙ M (cid:96) + T = const., (55)where the torque T := 2 πR Σ ν Ω (cid:48) ; (for a Keplerian disc T = 3 πR Σ ν Ω K ).Assuming that at the inner disc radius the torque vanishes one gets const. = − ˙ M (cid:96) in , where (cid:96) in is the specificangular momentum at the disc inner edge. Therefore˙ M ( (cid:96) − (cid:96) in ) = T (56)which is a simple expression of angular momentum conservation.For Keplerian discs one obtains an important relation between viscosity and accretion rate ν Σ = ˙ M π (cid:34) − (cid:18) R in R (cid:19) / (cid:35) . (57)From Eqs. (57), (42), (43), and the thermal equilibrium equation (44) it follows that σT = 83 σT c τ = 38 π GM ˙ MR (cid:34) − (cid:18) R in R (cid:19) / (cid:35) . (58) FIG. 1: The observed temperature profile of the accretion disc of the dwarf nova Z Cha in outburst. Near the outburst maximumsuch a disc is in quasi-equilibrium. The observed profile, represented by dots (pixels), is compared with the theoretical profilescalculated from Eq. (58) and represented by continuous lines. Pixels with
R < . R L correspond to to the surface of theaccreting white dwarf whose temperature is 40 000 K. The accretion rate in the disc is ≈ − M (cid:12) y − . [Figure 6 from [23]]. This relation assumes only that the disc is Keplerian and in thermal ( Q + = Q − ) and viscous ( ˙ M = const. )equilibrium. The viscosity coefficient is absent because of the thermal equilibrium assumption: in such a state theemitted radiation flux cannot contain information about the heating mechanism, it only says that such mechanismexists. Steady discs do not provide information about the viscosity operating in discs or the viscosity parameter α .To get this information one must consider (and observe) time-dependent states of accretion discs.From Eq. (58) one obtains a universal temperature profile for stationary Keplerian accretion discs T eff ∼ R − / . (59)For an optically thick disc the observed temperature T ∼ T eff and T ∼ R − / should be observed if stationary,optically thick Keplerian discs exist in the Universe. And vice versa, if they are observed, this proves that such discsexist not only on paper. In 1985 Horne & and Cook [23] presented the observational proof of existence of Kepleriandiscs when they observed the dwarf nova binary system ZCha during outburst (see Fig. 1).
1. Total luminosity
The total luminosity of a stationary, geometrically thin accretion disc, i.e. the sum of luminosities of its two surfaces,is 2 (cid:90) R out R in σT πRdR = 3 GM ˙ M (cid:90) R out R in (cid:34) − (cid:18) R in R (cid:19) / (cid:35) dRR . (60)For R out → ∞ this becomes L disc = 12 GM ˙ MR in = 12 L acc . (61)In the disc the radiating particles move on Keplerian orbits hence they retain half of the potential energy. If theaccreting body is a black hole this leftover energy will be lost (in this case, however, the non-relativistic formula ofEq. 61 does not apply – see Eq. 178.) In all the other cases the leftover energy will be released in the boundary layer,if any, and at the surface of the accretor, from where it will be radiated away.0 FIG. 2: The observed temperature profile of the accretion disc of the dwarf nova Z Cha in quiescence. This one of the mostmisunderstood figures in astrophysics (see text). In quiescence the disc in not in equilibrium. The flat temperature profile is exactly what the disc instability model predicts: in quiescence the disc temperature must be everywhere lower than the criticaltemperature, but this temperature is almost independent of the radius (see Eq. 89) . [Figure 11 from [58]].
The factor “3” in the rhs of Eq. (58) shows that radiation by a given ring in the accretion disc does not come onlyfrom local energy release. Indeed, in a ring between R and R + dR only GM ˙ M dR R (62)is being released, while 2 × πR Q + dR = 3 GM ˙ M R (cid:34) − (cid:18) R in R (cid:19) / (cid:35) dR (63)is the total energy release. Therefore the rest GM ˙ MR (cid:34) − (cid:18) R in R (cid:19) / (cid:35) dR (64)must diffuse out from smaller radii. This shows that viscous energy transport redistributes energy release in the disc. E. Radiative structure
Here we will show an example of the solution for the vertical thin disc structure which exhibit properties impossibleto identify when the structure is vertically averaged. We will also consider here an irradiated disc – such discs arepresent in X-ray sources.We write the energy conservation as : dFdz = q + ( R, z ) , (65)where F is the vertical (in the z direction) radiative flux and q + ( R, z ) is the viscous heating rate per unit volume.Eq. (65) states that an accretion disc is not in radiative equilibrium ( dF/dz (cid:54) = 0), contrary to a stellar atmosphere.For this equation to be solved, the function q + ( R, z ) must be known. As explained and discussed in Sect. III A theviscous dissipation is often written as q + ( R, z ) = 32 α Ω K P ( z ) . (66)1Viscous heating of this form has important implications for the structure of optically thin layers of accretion discsand may lead to the creation of coronae and winds. In reality it is an an hoc formula inspired by Eq. (15). We don’tknow yet (see, however, [11]) how to describe the viscous heating stratification in an accretion disc and Eq. (66) just assumes that it is proportional to pressure. It is simple and convenient but it is not necessarily true.When integrated over z , the rhs of Eq. (65) using Eq. (66) is equal to viscous dissipation per unit surface: F + = 32 α Ω K (cid:90) + ∞ P dz, (67)where F + = (1 / Q + because of the integration from 0 to + ∞ while Q + contains Σ which is integrated from −∞ to+ ∞ (Eq. 12).One can rewrite Eq. (65) as dFdτ = − f ( τ ) F vis τ tot , (68)where we introduced a new variable, the optical depth dτ = − κ R ρdz , κ R being the Rosseland mean opacity and τ tot = (cid:82) + ∞ κ R ρdz is the total optical depth. f ( τ ) is given by: f ( τ ) = P (cid:16)(cid:82) + ∞ P dz (cid:17) (cid:16)(cid:82) + ∞ κ R ρdz (cid:17) κ R ρ . (69)As ρ decreases approximately exponentially, f ( τ ) is the ratio of two rather well defined scale heights, the pressureand the opacity scale heights, which are comparable, so that f is of order of unity.At the disc midplane, by symmetry, the flux must vanish: F ( τ tot ) = 0, whereas at the surface, ( τ = 0) F (0) ≡ σT = F + . (70)Equation (70) states that the total flux at the surface is equal to the energy dissipated by viscosity (per unit timeand unit surface). The solution of Eq. (68) is thus F ( τ ) = F + (cid:32) − (cid:82) τ f ( τ ) dττ tot (cid:33) , (71)where (cid:82) τ tot f ( τ ) dτ = τ tot . Given that f is of order of unity, putting f ( τ ) = 1 is a reasonable approximation. Theprecise form of f ( τ ) is more complex, and is given by the functional dependence of the opacities on density andtemperature; it is of no importance in this example. We thus take: F ( τ ) = F + (cid:18) − ττ tot (cid:19) . (72)To obtain the temperature stratification one has to solve the transfer equation. Here we use the diffusion approxi-mation F ( τ ) = 43 σdT dτ , (73)appropriate for the optically thick discs we are dealing with. The integration of Eq. (73) is straightforward and gives: T ( τ ) − T (0) = 34 τ (cid:18) − τ τ tot (cid:19) T . (74)The upper (surface) boundary condition is: T (0) = 12 T + T , (75)where T is the irradiation temperature, which depends on r , the albedo, the height at which the energy is depositedand on the shape of the disc. In Eq. (75) T (0) corresponds to the emergent flux and, as mentioned above, T eff total flux ( σT = Q + ) which explains the factor 1/2 in Eq (75). The temperature stratificationis thus : T ( τ ) = 34 T (cid:20) τ (cid:18) − τ τ tot (cid:19) + 23 (cid:21) + T . (76)For τ tot (cid:29) τ = 2 / T (2 /
3) = T eff .Also for τ tot (cid:29)
1, the temperature at the disc midplane is T ≡ T ( τ tot ) = 38 τ tot T + T . (77)It is clear, therefore, that for the disc inner structure to be dominated by irradiation and the disc to be isothermalone must have F irr τ tot ≡ σT τ tot (cid:29) F + (78)and not just F irr (cid:29) F + as is sometimes assumed. The difference between the two criteria is important in LMXBssince, for parameters of interest, τ tot (cid:38) − in the outer disc regions. F. Shakura-Sunyaev solution
In their seminal and famous paper Shakura & Sunyaev [53], found power-law stationary solutions of the simplifiedversion of the thin–disc equations presented in Sects. III A, III B and III D. The 8 equations for the 8 unknowns T c , ρ , P , Σ, H , ν , τ and c s can be written asΣ = 2 Hρ (ı) H = c s R / ( GM ) / (ıı) c s = (cid:115) Pρ (ııı) P = R ρTµ + 4 σ c T (ıv) τ ( ρ, Σ , T c ) = κ R ( ρ, T c )Σ (v) ν ( ρ, Σ , T c , α ) = 23 αc s H (vı) ν Σ = ˙ M π (cid:34) − (cid:18) R R (cid:19) / (cid:35) (vıı)83 σT c τ = 38 π GM ˙ MR (cid:34) − (cid:18) R R (cid:19) / (cid:35) . (vııı)Equations (ı) and (ıı) correspond to vertical structure equations (21) and (19), Eq. (vıı) is the radial Eq. (57), whileEq. (vııı) connects vertical to radial equations. Eq. (ııı) defines the sound speed, Eq. (ıv) is the equation of state and(vı) contains the information about opacities. The viscosity α parametrization introduced in [53] provides the closureof the 8 disc equations. Therefore they can be solved for a given set of α , M , R and ˙ M .Power-law solutions of these equations exist in physical regimes where the opacity can be represented in the Kramersform κ = κ ρ n T m and one of the two pressures, gas or radiation, dominates over the other. In [53] three regimes havebeen considered:3 a. ) P r (cid:29) P g and κ es (cid:29) κ ff b. ) P g (cid:29) P r and κ es (cid:29) κ ff c. ) P g (cid:29) P r and κ ff (cid:29) κ es .Regimes a. ) and b. ) in which opacity is dominated by electron scattering will be discussed in Sect. V. Here we willpresent the solutions of regime c. ), i.e. we will assume that P r = 0 and κ R = κ ff = 5 × ρT − / c cm g − . (79)The solution for the surface density Σ, central temperature T c and the disc relative height (aspect ratio) are respectivelyΣ = 23 α − / m / R − / ˙ M / f / g cm − , (80) T c = 5 . × α − / m / R − / ˙ M / f / K , (81) HR = 2 . × − α − / m − / R / ˙ M / f / , (82)where m = M/ M (cid:12) , R = R/ (10 cm), ˙ M = ˙ M / (10 g s − ), and f = 1 − ( R in /R ) / . FIG. 3: Stationary accretion disc surface density profiles for 4 values of accretion rate. From top to bottom: ˙ M = 10 , , and 10 gs − . m = 10M (cid:12) , α = 0 .
1. The continuous line corresponds to the un-irradiated disc, the dotted lines to an irradiatedconfiguration. The inner, decreasing segments of the continuous lines correspond to Eq. (80). Dashed lines describe irradiateddisc equilibria (see Sect. IV C) [Figure 9 from [16]].
Although for a 10 M (cid:12) black hole, say, Shakura-Sunyaev solutions (80), (81) and (81) describe discs rather far fromits surface ( R (cid:38) R S ) the regime of physical parameters it addresses, especially temperatures around 10 K are ofgreat importance for the disc physics because it is where accretion discs become thermally and viscously unstable.This instability triggers dwarf nova outbursts when the accreting compact object is a white dwarf and (soft)
X-raytransients in the case of accreting neutron stars and black holes.It is characteristic of the Shakura-Sunyaev solution in this regime that the three Σ, T c and T eff radial profiles varyas R − / . (This implies that the optical depth τ is constant with radius – see Eq. vııı.) For high accretion ratesand small radii the assumption of opacity dominated by free-free and bound-free absorption will break down and thesolution will cease to be valid. We will come to that later. Now we will consider the other disc end: large radii.One sees in Fig. 3, that for given stationary solution ( ˙ M = const. ) the R − / slope of the Σ profiles extends downonly to a minimum value Σ min ( R ) after which the surface density starts to increase. With the temperature droppingbelow 10 K the disc plasma recombines and there is a drastic change in opacities leading to a thermal instability.
Additional reading : We have assumed that accretion discs are flat. This might not be true in general becauseaccretion discs might be warped. This has important and sometimes unexpected consequences; see e.g [28] [40] and[45], and references therein.4
IV. DISC INSTABILITIES
In this section we will present and discuss the disc thermal and the (related) viscous instabilities. First we willdiscuss in some detail the cause of the thermal instability due to recombination.
A. The thermal instability
A disc is thermally stable if radiative cooling varies faster with temperature than viscous heating. In other words d ln σT d ln T c > d ln Q + d ln T c . (83)Using Eq. (77) one obtains d ln T d ln T c = 4 (cid:34) − (cid:18) T irr T c (cid:19) (cid:35) − − d ln κd ln T c . (84)In a gas pressure dominated disc Q + ∼ ρT H ∼ Σ T ∼ T c . The thermal instability is due to a rapid change ofopacities with temperature when hydrogen begins to recombine. At high temperatures d ln κ/d ln T c ≈ − d ln κ/d ln T c ≈ −
10, and in theend cooling is decreasing with temperature. One can also see that irradiation by furnishing additional heat to thedisc can stabilize an otherwise unstable equilibrium solution (dashed lines in Fig. 3).This thermal instability is at the origin of outbursts observed in discs around black-holes, neutron stars and whitedwarfs. Systems containing the first two classes of objects are known as Soft X-ray transients (SXTs, where “soft”relates to their X-ray spectrum), while those containing white-dwarfs are called dwarf-novae (despite the name thatcould suggest otherwise, nova and supernova outbursts have nothing to do with accretion disc outbursts).
B. Thermal equilibria: the S -curve We will first consider thermal equilibria of an accretion disc in which heating is due only to local turbulence,leaving the discussion of the effects of irradiation to Section IV C. We put therefore T irr = (cid:101) Q = 0. Such an assumptioncorresponds to discs in cataclysmic variables which are the best testbed for standard accretion disc models. Thethermal equilibrium in the disc is defined by the equation Q − = Q + (see Eq. 41), i.e. by σT = 98 ν ΣΩ (85) FIG. 4: Thermal equilibria of a ring in an accretion discs around a m = 1 . cm; accretion rate 6 . × g / s. The solid line corresponds to Q + = Q − . Σ min is the critical (minimum) surface densityfor a hot stable equilibrium; Σ max the maximum surface density of a stable cold equilibrium. ν is a function of density and temperature and in the following we will use the standard α –prescription Eq. (7). The energy transfer equation provides a relation between the effective and the disc midplanetemperatures so that thermal equilibria can be represented as a T eff (Σ) – relation (or equivalently a ˙ M (Σ)–relation).In the temperature range of interest (10 (cid:46) T eff (cid:46) ) this relation forms an S on the (Σ , T eff ) plane as in Fig. 4. Theupper, hot branch corresponds to the Shakura-Sunyaev solution presented in Section III F. The two other branchescorrespond to solutions for cold discs – along the middle branch convection plays a crucial role in the energy transfer.Each point on the (Σ , T eff ) S -curve represents an accretion disc’s thermal equilibrium at a given radius, i.e. athermal equilibrium of a ring at radius R . In other words each point of the S -curve is a solution of the Q + = Q − equation. Points not on the S -curve correspond to solutions of Eq. (41) out of thermal equilibrium : on the left ofthe equilibrium curve cooling dominates over heating, Q + < Q − ; on the right heating over cooling Q + > Q − . Itis easy to see that a positive slope of the T eff (Σ) curve corresponds to stable solutions . Indeed, a small increase oftemperature of an equilibrium state (an upward perturbation) on the upper branch, say, will bring the ring to a statewhere Q + < Q − so it will cool down getting back to equilibrium. In a similar way an downward perturbation willprovoke increased heating bringing back the system to equilibrium.The opposite is happening along the S -curve’s segment with negative slope as both temperature increase anddecrease lead to a runaway. The middle branch of the S -curve corresponds therefore to thermally unstable equilibria.A stable disc equilibrium can be represented only by a point on the lower, cold or the upper, hot branch of the S -curve. This means that the surface density in a stable cold state must be lower than the maximal value on thecold branch: Σ max , whereas the surface density in the hot stable state must be larger than the minimum value onthis branch: Σ min . Both these critical densities are functions of the viscosity parameter α , the mass of the accretingobject, the distance from the center and depend on the disc’s chemical composition. In the case of solar compositionthe critical surface densities are Σ min ( R ) = 39 . α − . . R . m − . g cm − (86)Σ max ( R ) = 74 . α − . . R . m − . g cm − , (87)( α = 0 . α . ) and the corresponding effective temperatures ( T + designates the temperature at Σ min , T − at Σ max ) T +eff = 6890 R − . M . K (88) T − eff = 5210 R − . M . K . (89)The critical effective temperatures are practically independent of the mass and radius because they characterize themicroscopic state of disc’s matter (e.g. its ionization). On the other hand the critical accretion rates depend verystrongly on radius: ˙ M +crit ( R ) = 8 . × α − . . R . M − . g s − (90)˙ M − crit ( R ) = 2 . × α . . R . M − . g s − . (91)A stationary accretion disc in which there is a ring with effective temperature contained between the critical valuesof Eq. (89) and (88) cannot be stable. Since the effective temperature and the surface density both decrease withradius, the stability of a disc depend on the accretion rate and the disc size (see Fig. 3). For a given accretion rate astable disc cannot have an outer radius larger than the value corresponding to Eq. (86).A disc is stable if the rate (mass-transfer rate in a binary system) at which mass is brought to its outer edge( R ∼ R d ) is larger than the critical accretion rate at this radius ˙ M +crit ( R d ).In general, the accretion rate and the disc size are determined by mechanisms and conditions that are exterior tothe accretion process itself. In binary systems, for instance, the size of the disc is determined by the masses of thesystem’s components and its orbital period while the accretion rate in the disc is fixed by the rate at which the stellarcompanion of the accreting object loses mass, which in turn depends on the binary parameters and the evolutionarystate of this stellar mass donor. Therefore the knowledge of the orbital period and the mass-transfer rate shouldsuffice to determine if the accretion disc in a given interacting binary system is stable. Such knowledge allows testingthe validity of the model as we will show in the next section.
1. Dwarf nova and X-ray transient outbursts • Local view: the limit cycle6
FIG. 5: Local limit cycle of the state of disc ring at 10 cm during a dwarf nova outbursts. The arrows show the direction ofmotion of the system in the T eff (Σ) plane. The figure represents results of the disc instability model numerical simulations.As required by the comparison of the model with observations the values of the viscosity parameter α on the hot and coldbranches are different. [Figure adapted from [36]] Let us first describe what is happening during outbursts with a disc’s ring. Its states are represented by a pointmoving in the Σ − T eff plane as shown on Fig. 4 which represents accretion disc states at R = 10 cm (the accretingbody has a mass of 1 . (cid:12) ). To follow the states of a ring during the outburst let us start with an unstable equilibriumstate on the middle, unstable branch and let us perturb it by increasing its temperature, i.e. let us shift it upwardsin the T eff (Σ) plane. As we have already learned, points out of the S -curve correspond to solutions out of thermalequilibrium and in the region to the right of the S -curve heating dominates over cooling. The resulting runawaytemperature increase is represented by the point moving up and reaching (in a thermal time) a quasi–equilibriumstate on the hot and stable branch. It is only a quasi –equilibrium because the equilibrium state has been assumed tolie on the middle branch which corresponds to a lower temperature (and lower accretion rate – see Eq. 58). Tryingto get to its proper equilibrium the ring will cool down and move towards lower temperatures and surface densitiesalong the upper equilibrium branch (in a viscous time). But the hot branch ends at Σ min , i.e. at a temperature higher(and surface density lower) than required so the ring will never reach its equilibrium state. Which is not surprisingsince this state is unstable. Once more the ring will find itself out of thermal equilibrium but this time in the regionwhere cooling dominates over heating. Rapid (thermal-time) cooling will bring it to the lower cool branch. There,the temperature is lower than required so the point representing the ring will move up towards Σ max where it willhave to interrupt its (viscous-time) journey having reached the end of equilibrium states before getting to the righttemperature. It will find itself out of equilibrium where heating dominated over cooling so it will move back to theupper branch.Locally, the state of a ring performing a limit cycle on the Σ– T eff plane, moves in viscous time on the stable S -curvebranches and in a thermal time between them when the ring is out of thermal equilibrium. The states on the hotbranch correspond to outburst maximum and the subsequent decay whereas the quiescence correspond to moving onthe cold branch. Since the viscosity is much larger on the hot than on the cold branch, the quiescent is much longerthan the outburst phase. The full outburst behaviour can be understood only by following the whole disc evolution(Figs. 7 & 8). C. Irradiation and black–hole X-ray transients
We will present the global view of thermal-viscous disc outbursts for the case of X-ray transients. The maindifference between accretion discs in dwarf novae and these systems is the X-ray irradiation of the outer disc in thelatter. Assuming that the irradiating X-rays are emitted by a point source at the center of the system, one can writethe irradiating flux as σT = C L X πR with L X = η min (cid:16) ˙ M in , ˙ M Edd (cid:17) c , (92)where C = 10 − C , η is the radiative efficiency (which can be (cid:28) . M in the accretion rate atthe inner disc’s edge. Since the physics and geometry of X-ray self-irradiation in accreting black-black hole systems7 FIG. 6: Example S-curves for a pure helium disk with varying irradiation temperature T irr. The various sets of S-curvescorrespond to radii R = 10 , 10 and 10 cm. For each radius, the irradiation temperature T irr is 0 K, 10 000 K and 20 000 K. α = 0 .
16. The instable branch disappears for high irradiation temperatures. [From [32]. Reproduced with permission fromAstronomy & Astrophysics, c (cid:13)
ESO] is still unknown, the best we can do is to parametrize our ignorance by q constant C that observations suggest is ∼ − . Of course one should keep in mind that in reality C might not be a constant [17].Because the viscous heating is ∼ ˙ M /R there always exists a radius R irr for which σT > Q + = σT . If R irr < R d ,where R d is the outer disc radius, the outer disc emission will be dominated by reprocessed X-ray irradiation andthe structure modified as shown in Sect. III E. Irradiation will also stabilize outer disc regions (Eq. 84 and Fig. 6)allowing larger discs for a given accretion rate (see Fig. 3).Irradiation modifies the critical values of the hot disc parameters:Σ +irr = 72 . C − . − α − . . R . M − . g cm − (93) T irr , +eff = 2860 C − . − α . . R − . M . K (94)˙ M +irr = 2 . × C − . − α . . R . M − . g s − . (95)As we will see in a moment, irradiation also strongly influences the shape of outburst’s light-curve. * Rise to outburst maximum During quiescence the disc’s surface density, temperature and accretion rate are everywhere (at all radii) on the coldbranch, below their respective critical values Σ max ( R ), T − eff and ˙ M − crit ( R ). It is important to realize that in quiescencethe disc is not steady: ˙ M (cid:54) = const. Matter transferred from the stellar companion accumulates in the disc and isredistributed by viscosity. The surface density and temperature increase (locally, this means that the solution movesup along the lower branch of the S –curve) finally reaching their critical values. In Fig. 7 this happens at ∼ cm.The disc parameters entering the unstable regime triggers an outburst. In the local picture this corresponds to leavingthe lower branch of the S -curve. The next ‘moment’ (in a thermal time is represented in the left panels of Figure 7.8 FIG. 7: The rise to outburst described in Sect. IV C. The upper left panel shows ˙ M in and ˙ M irr (dotted line); the bottom leftpanel shows the V magnitude. Each dot corresponds to one of the Σ and T c profiles in the right panels. The heating frontpropagates outwards. The disc expands during the outburst due to the angular momentum transport of the material beingaccreted. At t ≈ . ∝ T c ∝ R − / ). [From [15]. Reproduced with permission from Astronomy & Astrophysics,c (cid:13) ESO]
This is when a large contrast forms in the midplane temperature profile and when a surface-density spike is alreadyabove the critical line. The disc is undergoing a thermal runaway at r ≈ × cm. The midplane temperature risesto ∼ R in ≈ × cm so the inwards propagating front quickly reaches the inner disc radius with no observable effects. Itis the outwards propagating heating front that produces the outburst by heating up the disc and redistributing themass and increasing the surface density behind it because it is also a compression front.One should stress here that two ad hoc elements must be added to the model for it to reproduce observed outburstsof dwarf novae and X-ray transients. • Viscosity.
First, if the increase in viscosity were due only to the rise in the temperature through the speedof sound ( ν ∝ c s , see Eq. 9) the resulting outbursts would have nothing to do with the observed ones. Toreproduce observed outbursts one increases the value of α when a given ring of the disc gets to the hot branch.Ratios of hot–to–cold α of the order of 4 are used to describe dwarf nova outburst. Although in the outburstmodel the α increase is an ad hoc assumption, recent MRI simulations with physical parameters correspondingto dwarf nova discs show an α increase induced by the appearance of convection [22]. • Inner truncation . Second, as mentioned already, the inner disc is assumed to be truncated in quiescence andduring the rise to outburst. Although such truncation is implied and/or required by observations, its physicalorigin is still uncertain. The inner part of the accretion flow is of course not empty but supposed to form a˙ M = const. ADAF (see Sect. V).In our case (Fig. 7), the heating front reaches the outer disc radius. This corresponds to the largest outbursts.Smaller-amplitude outbursts are produced when the front does not reach the outer disc regions. In an inside-outoutburst the surface-density spike has to propagate uphill, against the surface-density gradient because just beforethe outburst Σ ∼ R . – roughly parallel to the critical surface-density. Most of the mass is therefore contained inthe outer disc regions. A heating front will be able to propagate if the post-front surface-density is larger than Σ min – in other words, if it can bring successive rings of matter to the upper branch of the S -curve. If not, a cooling front will appear just behind the Σ spike, the heating front will die-out and the cooling front will start to propagate inwards(the heating-front will be ‘reflected’). X–ray transient outbursts are always of inside-out type. In dwarf novae both inside-out and outside-in outbursts are observed and resultfrom calculations[31]. FIG. 8: Decay from outburst peak. The decay is controlled by irradiation until evaporation sets in at t ≈
170 days ( ˙ M in =˙ M evap ( R min )). This cuts off irradiation and the disc cools quickly. The irradiation cutoff happens before the cooling frontcan propagate through most of the disc, hence the irradiation-controlled linear decay ( t ≈ −
170 days) is not very visiblein the lightcurve. T irr (dotted line) is shown for the last temperature profile. [From [15]. Reproduced with permission fromAstronomy & Astrophysics, c (cid:13) ESO]
The difficulty inside-out fronts encounter when propagating is due to angular-momentum conservation. In orderto move outwards the Σ-spike has to take with it some angular momentum because the disc’s angular momentumincreases with radius. For this reason inside-out front propagation induces a strong outflow . In order for matter tobe accreted, a lot of it must be sent outwards. That is why during an inside-out dwarf-nova outburst only ∼
10% ofthe disc’s mass is accreted onto the white dwarf. In X-ray transients irradiation facilitates heating front propagation(and disc emptying during decay – see next section).The arrival of the heating front at the outer disc rim does not end the rise to maximum. After the whole disc isbrought to the hot state, a surface density (and accretion rate) ‘excess’ forms in the outer disc. The accretion ratein the inner disc corresponds to the critical one but is much higher near the outer edge. While irradiation keeps thedisc hot the excess diffuses inwards until the accretion rate is roughly constant. During this last phase of the rise tooutburst maximum ˙ M in increases by a factor of 3:˙ M max ≈ M +irr ≈ . × C − . − R . d, m − . g s − . (96)Irradiation has little influence on the actual vertical structure in this region and T c ∝ Σ ∝ R − / , as in a non-irradiated steady disc. Only in the outermost disc regions does the vertical structure becomes irradiation-dominated,i.e. isothermal. * Decay Fig. 8 shows the sequel to what was described in Fig. 7. In general the decay from the outburst peak of anirradiated disc can be divided into three parts: • First, X-ray irradiation of the outer disc inhibits cooling-front propagation. But since the peak accretion rateis much higher than the mass-transfer rate, the disc is drained by viscous accretion of matter. • Second, the accretion rate becomes too low for the X-ray irradiation to prevent the cooling front from propa-gating. The propagation speed of this front, however, is controlled by irradiation. • Third, irradiation plays no role and the cooling front switches off the outburst on a local thermal time-scale. The peak luminosity is ∼ M +irr ( R d ); and the for the disc to be unstable the mass-transfer rate must be lower than the critical rate:˙ M tr < ˙ M +irr ( R d ). ‘Exponential decay’ In Fig. 8 the “exponential decay” the phase lasts until roughly day 80-100. At the outburst peak the accretion rateis almost exactly constant with radius; the disc is quasi-stationary. The subsequent evolution is self-similar: the disc’sradial structure evolves through a sequence of quasi-stationary ( ˙ M ( r ) = const ) states. Therefore ν Σ ∼ ˙ M in ( t ) / π and the total mass of the disk is thus M d = (cid:90) πR Σ dR ∝ ˙ M in (cid:90) rν dr. (97)At the outburst peak the whole disc is wholly ionized and except for the outermost regions its structure is verywell represented by a Shakura-Sunyaev solution. In such discs, as well as in irradiation dominated discs, the viscositycoefficient satisfies the relation ν ∝ T ∝ ˙ M β/ (1+ β ) . In hot Shakura-Sunyaev discs β = 3 / β = 1/3 (Eq. 92). During the first decay phase the outer disc radius is almost constant so that usingEq. (97) the disc-mass evolution can be written as:d M d d t = − ˙ M in ∝ M β d (98)showing that ˙ M in evolves almost exponentially, as long as ˙ M β in can be considered as constant (i.e. over about a decadein ˙ M in . ‘Exponential’ decays in the DIM are only approximately exponential.The quasi–exponential decay is due to two effects:1. X-ray irradiation keeps the disc ionized, preventing cooling-front propagation,2. tidal torques keep the outer disc radius roughly constant. ‘Linear’ decay The second phase of the decay begins when a disc ring cannot remain in thermal equilibrium. Locally this corre-sponds to a fall onto the cool branch of the S -curve. In an irradiated disc this happens when the central object doesnot produce enough X-ray flux to keep the T irr ( R out ) above ∼ K. A cooling front appears and propagates downthe disc at a speed of v front ≈ α h c s .In an irradiated disc, however, the transition between the hot and cold regions is set by T irr because a cold branchexists only for T irr (cid:46) K. In an irradiated disc a cooling front can propagate inwards only down to the radius atwhich T irr ≈ K, i.e. as far as there is a cold branch to fall onto. Thus the decay is still irradiation-controlled.The hot region remains close to steady-state but its size shrinks R hot ∼ ˙ M / (as can be seen in Eq. 92 with T irr ( R hot ) = const). Thermal decay
In the model shown in Fig. 8 irradiation is unimportant after t (cid:38) −
190 days because η becomes very small for˙ M in < g · s − when an ADAF forms. The cooling front thereafter propagates freely inwards, on a thermal timescale. In this particular case the decrease of irradiation is caused by the onset of evaporation at the inner edge whichlowers the efficiency. In general there is always a moment at which T irr becomes less than 10 K; evaporation justshortens the ‘linear’ decay phase.
D. Maximum accretion rate and decay timescale
Now we will see that there are two observable properties of X-ray transients that, when related one to to the other,provide informations and constraints on the physical properties of the outbursting system. The first is the maximumaccretion rate ˙ M max (Eq. 96). The second is the decay time of the X–ray flux: as we have seen, disk irradiationby the central X–rays traps the disk in the hot, high state, and only allows a decay of ˙ M on the hot–state viscoustimescale. This is t (cid:39) R ν (99)1which using Eq. (9) gives t (cid:39) ( GM R ) / αc s . (100)Taking the critical midplane temperature T +c ≈ t ≈ m / R / d, α − . days , (101)where α . = α/ .
2. Eliminating R between (96) and (101) gives the accretion rate through the disk at the start ofthe outburst as ˙ M = 5 . × m − . ( t α . ) . g s − , (102)with t = 30 t d. Assuming an efficiency of η of 10%, the corresponding luminosity is L = 5 . × η . m − . ( t α . ) . erg s − . (103) E. Comparison with observations
1. Sub-Eddington outbursts
The peak luminosities of most of the soft X-ray transients are sub-Eddington. Eq. (102) can be written using theEddington ratio m := ˙ M / ˙ M Edd as ˙ m = 0 . η . ( α . t ) . m − . . (104)This equation shows that the outburst peak will be sub–Eddington only if the outburst decay time is relatively shortor the accretor (black hole) mass is high, i.e. the observed decay timescale is t (cid:46) η − . . α − . m . d , (105)in good agreement with the compilation of X–ray transients outburst durations found in [59]. This shows that thestandard value of efficiency η . (cid:39)
1, and the value α . (cid:39) ≈ ≈
300 days). This equationalso implies that black hole transients should have longer decay timescales than neutron star transients, all else beingequal. Yan and Yu [59] find that outbursts last on average ≈ . × longer in black hole transients than in neutronstar transients thus confirming this conclusion.For sub–Eddington outbursts Eq. (103) gives a useful relationship between distance D , bolometric flux F andoutburst decay time t , D Mpc (cid:39) . m − . (cid:18) η . F (cid:19) / ( α . t ) . , (106)where D = D Mpc
Mpc and F = 10 − F erg s − cm − ; F = L/ πD and t = 50 t d.Eq. (106) shows that distant ( D >
2. Observational tests
Finally, one can test observationally if soft X-ray transients satisfy the necessary condition for instability ˙ M tr < ˙ M crit ( R d ), where ˙ M crit is the critical accretion rate for either non-irradiated or irradiated discs. In Fig. 9 the criticalaccretion rates (90) and (95) for respectively non-irradiated and irradiated disc around black holes are plotted as˙ M ( P orb ) relation. This relation was obtained from disc-radius – orbital-separation relation R d ( a ) [42], where (fromKepler’s law) the orbital separation a = 3 . × ( m + m ) / P / cm, where m i are the masses of the components2 Orbital period (hr) x x x x x x M a ss t r an s f e r r a t e ( g / s ) GRO J0422+32A0620-00GRS 1009-45XTE J1118+480GS 1124-683GS 1354-644U 1543-57XTE J1550-564XTE J1650-500GRO J1655-40MAXI J1659-352GX 339-44U 1705-250Swift J1753.5-0127GRS 1915+105GS 2000+25V 404 Cyg1E 1740.7-2942GRS 1758-258-------------------4U 1957+115Cyg X-1LMC X-1LMC X-3
Eddington limit (10 M ! ) D I M n o n i r r D I M i r r FIG. 9: Mass transfer rate as a function of the orbital period for SXTs with black holes. The transient and persistent sourceshave been marked with respectively filled and open symbols. The shaded grey areas indicated ‘DIM irr’ and ‘DIM non irr’represent the separation between persistent (above) and transient systems (below) according to the disc instability model when,respectively, irradiation is taken into account and when it is neglected. The horizontal dashed line indicates the Eddingtonaccretion rate for a 10M (cid:12) black hole. All the upper limits on the mass transfer rate are due to lower limits on the recurrencetime. The upper limits on the mass transfer rate of 4U 1957+115 and GS 1354-64 result from lower limits on the distanceto the sources. The three left closed arrows do not indicate actual upper limits on the orbital period of Cyg X-1, LMC X-1and LMC X-3. They emphasize that the radius of any accretion disk in these three high-mass XRBs is likely to be smallerthan the one derived from the orbital period since they likely transfer mass by a (possibly focused) stellar wind instead of fullydeveloped Roche lobe overflow. In the legend, the solid horizontal line separates transient and persistent systems. (The dashedhorizontal line stresses that the persistent nature of 1E 1740.7-2942 and GRS 1758-258 is unclear.) [From [13].] in solar units, and P hr the orbital period in hours. Against these two critical lines the actual positions of the observedsources are marked. The mass transfer rate being difficult to measure, a proxy in the form of the accumulation rate˙ M accum = ∆ Et rec ηc (107)has been used. ∆ E is the energy corresponding to the integrated X-ray luminosity from during an outburst and t rec the recurrence time of the outbursts. One can see that all low-mass-X-ray-binary (LMXB) transients are in theunstable part of the figure, as they should be if the model is correct. One can also see that all black hole LMXBsare transient. This is not true of neutron star LMXBs. Cyg X-1 in which the stellar companion of the black hole isa massive star is observed to be stable but according to Fig. 9 should be transient. This is not a problem becausein such a system matter from the high-mass companion is not transferred by Roche-lobe overflow as in LMXBs, butlost through a stellar wind. In this case the R d ( a ) relation used in the plot is not valid - the discs in such systemsare smaller which is marked by a left-directed arrow at the symbol marking the position of this and two other similarobjects (LMC X-1 and LMC X-3). Additional reading : References [15], [16], [21], [31], and [32].
V. BLACK HOLES AND ADVECTION OF ENERGY
Until now, we have neglected advection terms in the energy and momentum equations for stationary accretionflows. There two regimes of parameters where this assumption is not valid, in both cases for the same reason: low3radiative efficiency when the time for radial motion towards the black hole is shorter than the radiative cooling time.Low density (low accretion rate), hot, optically thin accretion flows are poor coolers and they are one of the twoconfigurations were advection instead of radiation is the dominant evacuation-of-energy (“cooling”) mechanism. Suchoptically thin flows are called ADAFs, for Advection Dominated Accretion Flows. Also advection dominated arehigh-luminosity flows accreting at high rates but they are called “slim discs” to account for their property of notbeing thin but still being described as if this were not of much importance.We shall start with optically thin flows. • ADAFsAdvection Dominated Accretion Flows’ (ADAFs) is a term describing accretion of matter with angular momen-tum, in which radiation efficiency is very low. In their applications, ADAFs are supposed to describe inflowsonto compact bodies, such as black holes or neutron stars; but very hot, optically thin flows are bad radiatorsin general so that, in principle, ADAFs are possible in other contexts. Of course in the vicinity of black holesor neutron stars, the virial (gravitational) temperature is T vir ≈ × ( R S /R ) K, so that in optically thinplasmas, at such temperatures, both the coupling between ions and electrons and the efficiency of radiationprocesses are rather feeble. In such a situation, the thermal energy released in the flow by the viscosity, whichdrives accretion by removing angular momentum, is not going to be radiated away, but will be advected towardsthe compact body. If this compact body is a black hole, the heat will be lost forever, so that advection, in thiscase, acts as sort of a ‘global’ cooling mechanism. In the case of infall onto a neutron star, the accreting matterlands on the star’s surface and the (reprocessed) advected energy will be radiated away. There, advection mayact only as a ‘local’ cooling mechanism. (One should keep in mind that, in general, advection may also beresponsible for heating, depending on the sign of the temperature gradient – in some conditions, near the blackhole, advection heats up electrons in a two-temperature ADAF).In general the role of advection in an accretion flow depends on the radiation efficiency which in turns dependson the microscopic state of matter and on the absence or presence of a magnetic field. If, for a given accretionrate, radiative cooling is not efficient, advection is necessarily dominant, assuming that a stationary solution ispossible. • Slim discsAt high accretion rates, discs around black holes become dominated by radiation pressure in their inner regions,close to the black hole. At the same time the opacity is dominated by electron scattering. In such discs
H/R is no longer (cid:28)
1. But this means that terms involving the radial velocity are no longer negligible since v r ∼ αc s ( H/R ). In particular, the advective term in the energy conservation equation v r ∂S/∂R (see Eq. 39)becomes important and finally, at super-Eddington rates, dominant. When Q + = Q adv the accretion flow isadvection dominated and called a slim disc. A. Advection–dominated–accretion–flow toy models
One can illustrate the fundamental properties of ADAFs and slim discs with a simple toy model. The advection‘cooling’ (per unit surface) term in the energy equation can be written as Q adv = ˙ M πR c ξ a (108)(see Eq. 239).Using the (non-relativistic) hydrostatic equilibrium equation HR ≈ c s v K (109)one can write the advection term as Q adv = Υ κ es c R (cid:18) ˙ mη (cid:19) ξ a (cid:18) HR (cid:19) (110)whereas the viscous heating term can be written as Q + = Υ 38 κ es cR (cid:18) ˙ mη (cid:19) , (111)4where Υ = (cid:18) cR S κ es R (cid:19) . (112)Since ξ a ∼ Q adv ≈ Q + (cid:18) HR (cid:19) (113)and, as said before, for geometrically thin discs ( H/R (cid:28)
1) the advective term Q adv is negligible compared to theheating term Q + and in thermal equilibrium viscous heating must be compensated by radiative cooling. Things aredifferent at, very high temperatures, when ( H/R ) ∼
1. Then the advection term is comparable to the viscous termand cannot be neglected in the equation of thermal equilibrium. In some cases this term is larger than the radiativecooling term Q − and (most of) the heat released by viscosity is advected toward the accreting body instead of beinglocally radiated away as happens in geometrically thin discs.From Eq.(57) one can obtain a useful expression for the square of the relative disc height (or aspect ratio): (cid:18) HR (cid:19) = √ κ es (cid:18) ˙ mη (cid:19) ( α Σ) − (cid:18) R S R (cid:19) / . (114)Deriving Eq. (114) we used the viscosity prescription ν = (2 / αc / Ω K .Using this equation one can write for the advective cooling Q adv = ΥΩ K ξ a ( α Σ) − (cid:18) ˙ mη (cid:19) . (115)The thermal equilibrium (energy) equation is Q + = Q adv + Q − . (116)The form of the radiative cooling term depends on the state of the accreting matter, i.e. on its temperature, densityand chemical composition. Let us consider two cases of accretion flows: – optically thickand – optically thin.For the optically thick case we will use the diffusion approximation formula Q − = 83 σT c κ R Σ , (117)and assume κ R = κ es . With the help of Eq. (114) this can be brought to the form Q − thick = 8Υ (cid:18) κ es R S c (cid:19) / (cid:18) RR S (cid:19) Ω / K ( α Σ) − / (cid:18) ˙ mη (cid:19) / . (118)For the optical thin case of bremsstrahlung radiation we have Q − = 1 . × Hρ T / (119)which using Eq. (114) can be written as Q − thin = 3 . × − Υ (cid:18) RR S (cid:19) Ω K α − ( α Σ) . (120) • In the optically thick case we have therefore ξ a (cid:18) ˙ mη (cid:19) + 0 . (cid:18) RR S (cid:19) / ( α Σ) (cid:18) ˙ mη (cid:19) ++2 . (cid:18) RR S (cid:19) / ( α Σ) / (cid:18) ˙ mη (cid:19) / = 0 (121)5 FIG. 10: (a) Thermal equilibria for optically thick (The right solid S -shaped line) and optically thin (the left solid line) accretionflows. The upper branches represent advection-dominated solution (ADAFs). Flows above the dotted lines τ = 1 are opticallythin – τ is the effective optical depth calculated for radiation-pressure dominated (upper line) or gas-dominated (lower line)configurations. It is assumed that M BH = 10M (cid:12) , R = 5 R S , α = 0 . ξ a = 1. (b) The same for α = 0 . • In the optically thin case the energy equation has the form ξ a (cid:18) ˙ mη (cid:19) + 0 . (cid:18) RR S (cid:19) / ( α Σ) (cid:18) ˙ mη (cid:19) ++3 × − α − (cid:18) RR S (cid:19) ( α Σ) = 0 (122)There are two distinct types of advection dominated accretion flows: optically thin and optically thick. We willfirst deal with optically thin flows known as ADAFs .
1. Optically thin flows:
ADAFs
For prescribed values α and ξ a , Eq. (122) is a quadratic equation in ( ˙ m/η ) whose solutions in the form of ˙ m (Σ)describe thermal equilibria at a given value of R . Obviously, for a given Σ this equation has at most two solutions.The solutions form two branches on the ˙ m ( α Σ) – plane: • the ADAF branch ˙ m = 0 . κ es η (cid:18) RR S (cid:19) / ξ − a α Σ . (123)and • the radiatively–cooled branch ˙ m = 1 . × − η (cid:18) RR S (cid:19) / ξ − a α − ( α Σ) . (124)6From Eqs. (123) and (124) it is clear that there exists a maximum accretion rate for which only one solution of Eq.(122) exists. This implies the existence of a maximum accretion rate at˙ m max ≈ . × η α (cid:18) RR S (cid:19) / . (125)This is where the two branches formed by thermal equilibrium solutions on the ˙ m ( α Σ) – plane meet as seen on Figure10.The value of ˙ m max depends on the cooling mechanism in the accretion flow and free-free cooling is not a realisticdescription of the emission in the vicinity (( R/R S ) (cid:46) ) of a black hole. The flow there most probably forms atwo-temperature plasma. In such a case ˙ m max ≈ α with almost no dependence on radius. For larger radii ˙ m max decreases with radius.
2. Optically thick flows: slim discs
Since the first two terms in Eq. (121) are the same as in (Eq. 122), the high ˙ m , advection dominated solution isthe same as in the optically thin case but now represents the • Slim disc branch ˙ m = 0 . κ es η (cid:18) RR S (cid:19) / ξ − a α Σ . (123)Now, the full equation (121) is a cubic equation in ˙ m / and on the ˙ m ( α Σ) plane its solution forms the twoupper branches of the S -curve shown in Fig. 10. The uppermost branch corresponds to slim discs while thebranch with negative slope represents the Shakura-Sunayev solution in the regime a. (see Sect. III F), i.e. • a radiatively cooled, radiation-pressure dominated accretion disc˙ m = 160 κ − η (cid:18) RR S (cid:19) / ( α Σ) − (126)
3. Thermal instability of radiation–pressure dominated discs
Radiation–pressure dominated ( P = P rad ) accretion discs are thermally unstable when opacity is due to electronscattering on electrons. Indeed d ln T d ln T c = 4 (127)because κ R = κ es = const. , while in a radiation pressure dominated disc Q + ∼ ν Σ ∼ HT ∼ T / Σ so d ln Q + d ln T c = 8 > d ln T d ln T c (128)and the disc is thermally unstable. This solution is represented by the middle branch with negative slope (see Eq.126) in Fig. 10. The presence of this instability in the model is one of the unsolved problems of the accretion disctheory because it contradicts observations which do not show any unstable behaviour in the range of luminositieswhere discs should be in the radiative pressure and electron-scattering opacity domination regime.
4. Slim discs and super-Eddington accretion
From Eqs. (114) and (126) one obtains for the disc aspect ratio HR = 0 . (cid:18) ˙ mη (cid:19) R S R (129)7which shows that the height of a radiation dominated disc is constant with radius and proportional to the accretionrates.But this means that with increasing ˙ m advection becomes more and more important (see e.g. Eq. 113) and for˙ mη ≈ . RR S (130)advection will take over radiation as the dominant cooling mechanism and the solution will represent a slim disc.Equation (130) can be also interpreted as giving the transition radius between radiatively and advectively cooled discfor a given accretion rate ˙ m : R trans R S ≈ . η ˙ m (131)Another radius of interest is the trapping radius at which the photon diffusion (escape) time Hτ /c is equal to theviscous infall time
R/v r R trapp = Hτ v r c = Hκ Σ c ˙ M πR Σ = HR (cid:18) ˙ mη (cid:19) R S . (132)Notice that both R trans and R trapp are proportional to the accretion rate.In an advection dominated disc the aspect ration H/R is independent of the accretion rate: HR = 0 . ξ a (cid:18) RR S (cid:19) / , (133)therefore contrary to radiatively cooled discs, slim disc do not puff up with increasing accretion rate.Putting (133) into Eq. (132) one obtains R trapp R = 0 . ξ − / a (cid:18) RR S (cid:19) / (cid:18) ˙ mη (cid:19) . (134)Radiation inside the trapping radius is unable to stop accretion and since R trapp ∼ ˙ m there is no limit on the accretionrate onto a black hole.The luminosity of the toy-model slim disc can be calculated from Eqs. (118) and (123) giving Q − = σT = 0 . ξ a L Edd R , (135)which implies T eff ∼ /R / . The luminosity of the slim–disc part of the accretion flow is then L slim = 2 (cid:90) R trans R in σT πRdR = 0 . ξ a L Edd · ln R trans R in ≈ L Edd ln ˙ m, (136)where we used Eq. (131).Therefore the total disk luminosity L total = L thin + L slim = (137)4 π (cid:32)(cid:90) R trans R in σT RdR + (cid:90) R ∞ R trans σT RdR (cid:33) ≈ L Edd (1 + ln ˙ m ) , where L thin is the luminosity of the radiation-cooled disc for which Eq. (58) applies.It is easy to see that the same luminosity formula L ≈ L Edd (1 + ln ˙ m ) is obtained when one assumes mass–loss fromthe disc resulting in a variable (with radius) accretion rate: ˙ M ∼ R .At very high accretion rates the disc emission will be also strongly beamed by the flow geometry so that andobserver situated in the beam of the emitting system will infer a luminosity L sph = 1 b L Edd (1 + ln ˙ m ) , (138)where b is the beaming factor (see [27] for a derivation of b in the case of Ultra-Luminous X-ray sources).Numerical simulations do not seem to correspond to this analytical solutions (see e.g. [25], [51] and [52] but theyalso disagree between themselves. The reasons for these contradictions are worth investigating. Additional reading:
References [1], [3], [4], [33], [38], [49], [50], and [60].8
VI. ACCRETION DISCS IN KERR SPACETIME.
In this section we will present and discuss the set equations whose solutions represent α –accretion discs in the Kerrmetric. This section is based on references [5], [30] and [49] and to be understood requires some basic knowledge ofEinstein’s General Relativity. A. Kerr black holes
The components g ij of the metric tensor with respect to the coordinates ( t, x α ) are expressible in terms of the lapse N , the components β α of the shift vector and the components γ αβ of the spatial metric: g ij dx i dx j = − N dt + γ αβ ( dx α + β α dt )( dx β + β β dt ) , (139)which is a modern way of writing the metric. Remark 3.
In this section only I will use conventions different from those used in other parts of the Chapter.First, I will use the so-called geometrical units that are linked to the physical units for length, time and mass bylength in physical units = length in geometrical units , time in physical units = 1c length in geometrical units , mass in physical units = c G length in geometrical units . (140)Second, the radial coordinate will be called “ r ” and not “ R ”. This should not confuse the reader since R is usedonly in the non-relativistic context where it denotes a radial coordinate and a radial distance, while in the relativisticcontext it only a coordinate.
1. General structure, Boyer-Lindquist coordinates
The Kerr metric in the Boyer-Lindquist (spherical) coordinates t, r, θ, ϕ corresponds to: N = ς √ A ∆ , β r = β θ = 0 , β ϕ = − ω, (141) g rr = ς ∆ , g θθ = ς , g ϕϕ = A ς sin θ (142)with ς = r + a cos θ, ∆ = r − M r + a , (143) A = (cid:0) r + a (cid:1) − ∆ a sin θ, ω = 2 JrA = 2
M arA , (144)where M is the mass and a = J/M is the angular momentum per unit mass. In applications one often uses thedimensionless “angular-momentum” parameter a ∗ = a/M .Therefore in BL coordinates the Kerr metric takes the form of ds = − ς ∆ A dt + A sin θς ( dϕ − ωdt ) + ς ∆ dr + ς dθ . (145)The time (stationarity) and axial symmetries of the metric are expressed by two Killing vectors η i = δ i ( t ) , ξ i = δ i ( ϕ ) , (146)where δ i ( k ) is the Kronecker delta.9 Remark 4.
Using Killing vectors (146) one can define some useful scalar functions: the angular velocity of thedragging of inertial frames ω , the gravitational potential Φ, and the gyration radius R , ω = − (cid:126)η · (cid:126)ξ(cid:126)ξ · (cid:126)ξ , e − = ω (cid:126)ξ · (cid:126)ξ − (cid:126)η · (cid:126)η, R = − (cid:126)ξ · (cid:126)ξ(cid:126)η · (cid:126)η . (147)In the Boyer-Lindquist coordinates the scalar products of the Killing vectors are simply given by the components ofthe metric, (cid:126)η · (cid:126)η = g tt , (cid:126)η · (cid:126)ξ = g tϕ , (cid:126)ξ · (cid:126)ξ = g ϕϕ , (148)and therefore quantities defined in Eq. (147) can be explicitly written down in terms of the Boyer-Lindquist coordinatesas: R = A r ∆ , e − = r ∆ A . (149) • The horizon
The black hole surface (event horizon) is at r H = M + (cid:112) M − a . (150)Therefore a horizon exists for a ∗ ≤ ω H = Ω H = a M r H , (151)where Ω H is the angular velocity of the horizon, i.e. the angular velocity of the horizon-forming light-rays with respectto infinity. The horizon rotates.The area of the horizon is given by S = 8 πM r H = 8 πM (cid:112) M − a . (152)The extreme (maximally rotating) black hole corresponds to a = M. (153)For a > M the Kerr solution represents a naked singularity . Such singularities would be a great embarrassmentnot only because of their visibility but also because the solution of Einstein equation in which they appear violatecausality by containing closed time-like lines. The conjecture that no naked singularity is formed through collapse ofreal bodies is called the cosmic censorship hypothesis (Roger Penrose). Remark 5.
Rotation of astrophysical bodies
Since this is a lecture in astrophysics let us leave for a moment the geometrical units. They are great for calculationsbut usually useless for comparing their results with observations. In the physical units r H = GMc + (cid:34)(cid:18) GMc (cid:19) − (cid:18) JM c (cid:19) (cid:35) / . (154)and therefore the maximum angular momentum of a black hole is J max = GM c = 8 . × (cid:18) M M (cid:12) (cid:19) g cm s − . (155)0This is slightly more than the angular momentum of the Sun ( J (cid:12) = 1 . × g cm s − , a (cid:12)∗ = 0 . ∼ . (cid:12) and radius ∼ J NS = I NS Ω S ≈ . × (cid:18) α ( x )0 . (cid:19) (cid:18) M NS . (cid:12) (cid:19) (cid:18) R NS
10 km (cid:19) (cid:18) P S (cid:19) − g cm (156)where I NS ≈ α ( x ) M NS R is the moment of inertia and x = ( M NS / M (cid:12) )(km / R NS ) the compactness parameter. Forthe most compact neutron star x ≤ .
24 and α ( x ) (cid:46) . a NS ∗ ≈ . (cid:18) α ( x )0 . (cid:19) (cid:18) M NS . (cid:12) (cid:19) − (cid:18) R NS
10 km (cid:19) (cid:18) P S (cid:19) − . (157)By definition • the specific (per unit rest-mass) energy is E := − (cid:126)η · (cid:126)u, (158) • the specific (per unit rest-mass) angular momentum L := (cid:126)ξ · (cid:126)u (159)and • the specific (per unit mass-energy) angular momentum (also called geometrical specific angular momentum) J := − LE = − (cid:126)ξ · (cid:126)u(cid:126)η · (cid:126)u (160) B. Privileged observers
Let us consider observers privileged by the symmetries of the Kerr spacetime. The results below apply to anyspacetime with the same symmetries, e.g. the spacetime of a stationary, rotating star. The four-velocity of a privilegedobserver is the linear combination of the two Killing vectors: (cid:126)u = Z (cid:16) (cid:126)η + Ω obs (cid:126)ξ (cid:17) (161)where the redshift factor Z is (from the normalization (cid:126)u · (cid:126)u = 1) Z − = (cid:126)η · (cid:126)η + 2Ω obs (cid:126)η · (cid:126)ξ + Ω (cid:126)ξ · (cid:126)ξ (162)Since for a (cid:54) = 0 the Kerr spacetime is stationary but not static, i.e; the timelike Killing vector η is not orthogonalto the space-like surfaces t =const. In such a spacetime ”non-rotation” is not uniquely defined. Stationary observers are immobile with respect to infinity; their four-velocities are defined as u i stat = ( ηη ) − / η i (163)but are locally rotating: L stat = ξ i u i stat (cid:54) = 0.The four-velocity of a locally non-rotating observer is a unit timelike vector orthogonal to the space-like surfaces t =const.: u i ZAMO = e Φ (cid:0) η i + ωξ i (cid:1) , (164)defines four-velocity of the local inertial observer or ZAMO, i.e. Zero Angular Momentum Observers since L ZAMO = ξ i u i ZAMO = 0.Finally, in presence of matter forming a stationary and axisymmetric configuration, there are privileged observers comoving with matter.1
TABLE I: Summary of properties of privileged observersObserver Four-velocity Angular velocity with respectto stationary observersStationary (cid:126)u = ( (cid:126)η · (cid:126)η ) − / (cid:126)η Ω stat = 0ZAMO (LNR) (cid:126)u = e − Φ (cid:16) (cid:126)η + ω(cid:126)ξ (cid:17) Ω ZAMO = ω Comoving (with matter) (cid:126)u = Z (cid:16) (cid:126)η + Ω (cid:126)ξ (cid:17) Ω com = Ω C. The ergosphere
For ZAMOs Ω = ω but for stationary observers Ω − ω = − ω . Therefore ZAMOs rotate with respect to infinity (butare locally non-rotating). They may exist down to the black hole horizon, where they become null: (cid:126)u ZAMO · (cid:126)u ZAMO = 0.Stationary observers immobile with respect to infinity but rotating with angular velocity − ω with respect to ZAMOscan exist (their four-velocity must be timelike, (cid:126)η · (cid:126)η <
0) only outside the stationarity limit whose radius is definedby (cid:126)η · (cid:126)η = 0: r er ( θ ) = M + (cid:112) M − a cos θ. (165)The stationarity limit is called the ergosphere . D. Equatorial plane
We will discuss now orbits in the equatorial plane, where they have the axial symmetry. We are introducing thecylindrical vertical coordinate z = cos θ is defined very close to the equatorial plane, z = 0. The metric of the Kerrblack hole in the equatorial plane, accurate up to the ( z/r ) is ds = − r ∆ A dt + Ar ( dϕ − ωdt ) + r ∆ d r + dz , (166)where now ∆ = r − M r + a , A = (cid:0) r + a (cid:1) − ∆ a , ω = 2 M arA , (167)or simpler ds = − (cid:18) − Mr (cid:19) dt − ωdtdϕ + Ar dϕ + r ∆ dr + dz . (168)
1. Orbits in the equatorial plane
The four velocity of matter u i has components u t , u ϕ , u r , u i = u t δ i ( t ) + u ϕ δ i ( ϕ ) + u r δ i ( r ) . (169)The angular frequency Ω with respect to a stationary observer, and the angular frequency ˜Ω with respect to a localinertial observer are respectively defined by Ω = u ϕ u t , ˜Ω = Ω − ω, (170)2The angular frequencies of the corotating (+) and counterrotating (–) Keplerian orbits areΩ ± K = ± M / r / ± aM / , (171)the specific energy is E ± K = r − M r ± a ( M r ) / r (cid:0) r − M r ± a ( M r ) / (cid:1) / (172)and the specific angular momentum is given by L ± K = ± ( M r ) / (cid:0) r ∓ a ( M r ) / + a (cid:1) r (cid:0) r − M r ± a ( M r ) / (cid:1) / , (173)or J K = ± ( M r ) / (cid:0) r ∓ a ( M r ) / + a (cid:1) r − M r ± a ( M r ) / . (174)Both J and L have a minimum at the last stable orbit, more often called ISCO (Innermost Stable Circular Orbit).Because of the rotation of space there is no direct relation between angular momentum and angular frequency but J = (cid:126)η · (cid:126)ξ + Ω (cid:126)ξ · (cid:126)ξ(cid:126)η · (cid:126)η + Ω (cid:126)η · (cid:126)ξ = R Ω − ω Ω ω − . (175)For the Schwarzschild solution ( a = ω = 0) J K = R Ω K , (176)so a Newtonian-like relation (justifying the name “gyration radius” for R ) between angular frequency and angularmomentum exists for J . No such relation exists for L . ISCO
The minimum of the Keplerian angular momentum corresponding to the innermost stable circular orbit (ISCO) islocated at r ± ISCO = M { Z ∓ [(3 − Z )(3 + Z + 2 Z )] / } ,Z = 1 + (cid:0) − a /M (cid:1) / (cid:104) (1 + a/M ) / + (1 − a/M ) / (cid:105) ,Z = (cid:0) a /M + Z (cid:1) / . (177) Binding energy
The binding energy E bind = 1 − E K (178)at the ISCO is • − (cid:112) / ≈ .
06 for a = 0 • − (cid:112) / ≈ .
42 for a = 1.3 FIG. 11: Radii of characteristic orbits in the Kerr metric as a function of a ∗ = a/M . The innermost stable circular orbit: r ISCO , the marginally bound orbit r IBCO (marked r mb ), the photon orbit: r ph and the black hole horizon: r H (marked r h ).(Courtesy of A. S¸adowski.) This corresponds to the efficiencies of accretion in a geometrically thin (quasi-Keplerian) disc around a black hole.For a Schwarzschild black hole the frequency associated with the ISCO at r ISCO = 6 M is ν K ( r ISCO ) = 2197 (cid:18) M M (cid:12) (cid:19) − Hz . (179) IBCO
The binding energy of a Keplerian orbit 1 − E = 0 at the marginally bound orbit (or IBCO: Innermost BoundCircular Orbit) r ± IBCO = 2 M ∓ a + 2 (cid:112) M ∓ aM . (180)For a non-rotating black-hole r IBCO = 4 M and the frequency associated with the IBCO is ν K ( r IBCO ) = 4037 (cid:18) M M (cid:12) (cid:19) − Hz . (181) ICO (
Circular photon orbit ) The Innermost Circular Orbit (ICO), i.e. the circular photon orbit is at r ± ph = 2 M (cid:18) (cid:20)
23 cos − (cid:16) ∓ aM (cid:17)(cid:21)(cid:19) . (182)For a non-rotating black hole r ph = 3 M .
2. Epicyclic frequencies
We will consider now consider a perturbed orbital motion in, and slightly off the equatorial plane. In the Newtoniancase the angular frequency of such motions must be equal to the Keplerian frequency Ω K since in there is only onecharacteristic scale defined by the gravitational constant G . In General Relativity the presence of two constants G and c imply that the epicyclic frequency does not have to be equal to Ω K .The four–velocity for the perturbed circular motion can be written as u i = (cid:0) , ˜ u r , ˜ u θ , Ω K + ˜ u ϕ (cid:1) , (183)where ˜ u α are the velocity perturbations.4 • For perturbations in the equatorial plane the equation of motion is (cid:18) ∂ ∂t + κ (cid:19) (cid:32) ˜ u r ˜ u ϕ (cid:33) = 0 , (184)where κ = Ω K r − M r ± aM / r / − a r (185)is the (equatorial) epicyclic frequency . In the Schwarzschild case a = 0 this is κ = Ω K (1 − M/r ) and vanishesat ISCO. In the Newtonian limit the epicyclic frequency equal to the Keplerian frequency κ = Ω K . • For vertical perturbations the equation is (cid:18) ∂∂t + Ω K ∂∂ϕ (cid:19) ˜ u θ = − Ω ⊥ δθ, (186)where the vertical epicyclic (angular) frequency is given byΩ ⊥ = Ω K r − aM / r / − a M r (187)In the Schwarzschild case ( a = 0) the vertical epicyclic frequency is equal to the Keplerian angular frequencyΩ K , which is to be expected from the spherical symmetry of this solution.The angular velocity Ω ⊥ appears also in the equation of vertical equilibrium of a (quasi)Keplerian disc whichwill be discussed later (sect. VIII E). Here let us just notice that Eq.(222) can be written as ∂p∂z = − ρe Ω ⊥ z. (188)All these characteristic frequencies can be put into the formΩ = f ( x, a ∗ ) 1 M , (189)where x = r/M . For all relativistic frequencies x = x ( a ∗ ) and therefore they can be written asΩ = F ( a ∗ ) 1 M . (190)
Additional reading : Reference [2].
VII. ACCRETION FLOWS IN THE KERR SPACETIMEA. Kinematic relations
In the reference frame of the local inertial (non-rotating) observer the four velocity takes the form, u i = γ (cid:16) u i ZAMO + v ( ϕ ) τ i ( ϕ ) + v ( r ) τ i ( r ) (cid:17) . (191)The vectors τ i ( ϕ ) and τ i ( r ) are the unit vectors in the coordinate directions ϕ and r . The Lorentz gamma factor γ equals, γ = 1 (cid:113) − (cid:0) v ( ϕ ) (cid:1) − (cid:0) v ( r ) (cid:1) . (192)5The relation between the Boyer-Lindquist and the physical velocity component in the azimuthal direction is, v ( ϕ ) = ˜ R ˜Ω , (193)which justifies the name of ˜ R – gyration radius. It is convenient to use the (rescaled) radial velocity component V defined by the formula, V √ − V = γv ( r ) = u r g / rr . (194)The Lorentz gamma factor may then be written as, γ = (cid:18) − ˜Ω ˜ R (cid:19) (cid:18) − V (cid:19) , (195)which allows writing a simple expression for V in terms of the velocity components measured in the frame of the localinertial observer, V = v ( r ) (cid:113) − (cid:0) v ( ϕ ) (cid:1) = v ( r ) (cid:112) − ˜ R ˜Ω . (196)Thus, V is the radial velocity of the fluid as measured by an observer corotating with the fluid at fixed r .Although a different quantity could have been chosen as the definition of the “radial velocity”, only V has directlythree very convenient properties, all guaranteed by its definition: • (i) everywhere in the flow | V | ≤ • (ii) on the horizon | V | = 1, • (iii) at the sonic point | V | ≈ c s ,where c s is the local sound speed.To see that property (i) holds, let us define˜ V = u r u r = u r u r g rr ≥ . (197)Then, one has V = ˜ V / (1 + ˜ V ) ≤ . (198)Writing V = (cid:112) r u r u r / ( r u r u r + ∆) demonstrates property (ii) since | V | = 1 independent of the value of r u r u r .For the proof of property (iii) of V see [4].Other possible choices of the “radial velocity” such as u = | u r | are not that convenient. B. Description of accreting matter
The stress-energy tensor T ik of the matter in the disk is given by T ik = ( ε + p ) u i u k + p g ik + S ik + u k q i + u i q k , (199)where ε is the total energy density, p is the pressure, S ik = νρσ ik , (200)is the viscous stress tensor, ρ is the rest mass density and q i is the radiative energy flux. In the last equation ν is thekinematic viscosity coefficient and σ ik is the shear tensor of the velocity field. From the first law of thermodynamicit follows that dε = ε + pρ dρ + ρT dS, (201)6where T is the temperature and S is the entropy per unit mass. Note, that in the physical units ε = ρc + Π, whereΠ is the internal energy. For non-relativistic fluids, Π (cid:28) ρc and p (cid:28) ρc , and therefore ε + p ≈ c ρ. (202)We shall use this approximation (in geometrical units ε + p ≈ ρ ) in all our calculations. This approximation doesnot automatically ensure that the sound speed is below c , and one should check this a posteriori when models areconstructed. We write the first law of thermodynamics in the form: dU = − p d (cid:18) ρ (cid:19) + T dS, (203)where U = Π /ρ . VIII. SLIM DISC EQUATIONS IN KERR GEOMETRY
General-Relativistic effects play an important role in the physics of thin (
H/r (cid:28)
1) accretion discs close to theblack hole but they determine the properties of slim (
H/r (cid:46)
1) discs. We will derive the slim-disc equation and beforediscussing their properties we will say few words about thin discs.It is convenient to write the final form of all the slim disk equations at the equatorial plane, z = 0. Only theseequations which do not refer to the vertical structure could be derived directly from the quantities at the equatorialplane with no further approximations. All other equations are approximated — either by expansion in terms of therelative disk thickness H/r , or by vertical averaging.
A. Mass conservation equation
From general equation of mass conservation, ∇ i ( ρu i ) = 0 , (204)and definition of the surface density Σ, Σ = (cid:90) + H ( r ) − H ( r ) ρ ( r, z ) dz ≈ Hρ, (205)we derive the mass conservation equation, ˙ M = − π ∆ / Σ V √ − V . (206)In the Newtonian limit the mass conservation equation is:˙ M = − π Σ v r . (207) B. Equation of angular momentum conservation
From the general form of the angular momentum conservation, ∇ k (cid:0) T ki ξ i (cid:1) = 0 , (208)we derive, after some algebra, ˙ M πr d L dr + 1 r ddr (cid:18) Σ νA / ∆ / γ r d Ω dr (cid:19) − F − L = 0 , (209)where F − = 2 q z is the vertical flux of radiation, and L ≡ − ( uξ ) = − u ϕ = γ (cid:18) A / r ∆ / (cid:19) ˜Ω , (210)7is the specific (per unit mass) angular momentum. The term F − L represents angular momentum losses throughradiation. Although it was always fully recognized that angular momentum may be lost this way, it has been arguedthat this term must be very small. Rejection of this term enormously simplifies numerical calculations, because with F − L = 0 equation (209) can be trivially integrated,˙ M π ( L − L ) = − Σ νA / ∆ / γ r d Ω dr ≡ T π , (211)where L is the specific angular momentum of matter at the horizon (∆ = 0). In the numerical scheme for integratingthe slim Kerr equations (with F − L assumed to be zero) the quantity L plays an important role: it is the eigenvalueof the solutions that passes regularly through the sonic point. The rhs of Eq. (211) T represent the viscous torquetransporting angular momentum.In the Newtonian case, a geometrically thin disc is Keplerian Ω ≈ Ω K (see Eq. (214), L K = (cid:96) K = R Ω K , Eq. (211)takes the familiar form of ν Σ = ˙ M π (cid:20) − (cid:96) (cid:96) K (cid:21) , (see Eq.57). C. Equation of momentum conservation
From the r-component of the equation ∇ i T ik = 0 one derives V − V dVdr = A r − dPdr , (212)where P = 2 Hp is the vertically integrated pressure and A = − M Ar ∆Ω + k Ω − k (Ω − Ω + k )(Ω − Ω − k )1 − ˜Ω ˜ R . (213)Note that in Eq. (212) the viscous term has been neglected.The Newtonian limit of Eq.(212) is v r dv r dr − (cid:0) Ω − Ω K (cid:1) r + c s r = 0 (214)For a thin disc: H/r ≈ c s /r Ω K (cid:28) ≈ Ω K , i.e. a thin Newtonian disc is Keplerian. Thethickness of the disc depends on the efficiency of radiative processes: efficient radiative cooling implies a low speed ofsound. D. Equation of energy conservation
From the general form of the energy conservation ∇ i (cid:0) T ik η k (cid:1) = 0 , (215)and the first law of thermodynamics, T = 1 ρ (cid:18) ∂ε∂S (cid:19) ρ , p = ρ (cid:18) ∂ε∂ρ (cid:19) S − ε, (216)the energy equation can be written in general as Q adv = Q + − Q − , (217)where Q + = ν Σ A r γ (cid:18) d Ω dr (cid:19) (218)8is the surface viscous heat generation rate, Q − is the radiative cooling flux (both surfaces) which is discussed inSection 5, and Q adv is the advective cooling rate due to the radial motion of the gas. It is expressed as Q adv = Σ V √ − V ∆ / r T dSdr ≡ − ˙ M πr T dSdr . (219)In stationary accretion flows advection is important only in the inner regions close to the compact accretor. In therest of the flow the energy equation is just Q + = Q − . (220)In the newtonian limit of Eqs. (218) and (211) one obtains Q + = 38 π GM ˙ MR (cid:18) − (cid:96) (cid:96) (cid:19) . (58) E. Equation of vertical balance of forces
The equation of vertical balance is obtained by projecting the conservation equation onto the θ direction h iθ ∇ k T ki = 0 , where h iθ = δ iθ − u i u θ (221)and neglecting the terms O (cos θ ). For a non-relativistic fluid this leads to dPdz = − ρg z z = − ρ L − a (cid:0) E − (cid:1) r z . (222)In the newtonian limit Eq. (222) becomes dPdz = − ρ (cid:96) K r z (see Eq . . IX. THE SONIC POINT AND THE BOUNDARY CONDITIONSA. The “no-torque condition”
There have been a lot of discussion about the inner boundary condition in an accretion disc. The usual reasoningis that for a thin disc the inner boundary is at ISCO and since it is where circular orbits end the boundary conditionshould be simply that the “viscous” torque vanishes (there is no orbit below the ISCO to interact with). Severalauthors have challenged this conclusion but a very simple argument by Bohdan Paczy´nski [44] shows the fallacy ofthese challenges.Using Eq. (206), one obtains from Eq. (211) v ( r ) = ν A / γ r L − L . d Ω dr (223)Next, from the viscosity prescription ν ≈ αH Ω, and taking for simplicity the non-relativistic approximation (thisdoes not affect the validity of the argument but allows skipping irrelevant in this context multiplicative factors) onecan write v r ≈ α H (cid:96)(cid:96) − (cid:96) d Ω dr ≈ α H (cid:96)(cid:96) − (cid:96) Ω r ≈ α v ϕ (cid:18) HR (cid:19) (cid:96)(cid:96) − (cid:96) , (224)where v ϕ = R Ω Although we have dropped the GR terms, the equation (224) does not assume that the radial velocityis small, i.e.this equation holds within the disk as well as within the stream below the ISCO.9Far out in the disk, where (cid:96) (cid:29) (cid:96) , one obtains the standard formula (see Eq. 36) v r ≈ α v ϕ (cid:18) HR (cid:19) , R (cid:29) R in . (225)The flow crosses the black hole surface at the speed of light and since it is subsonic in the disc it must somewherebecome transonic, i.e. to go through a sonic point, close to disc’s inner edge.At the sonic point we have v r = c s ≈ ( H/R ) v ϕ , and the equation (224) becomes: v r c s = 1 ≈ α H in R in (cid:96) in (cid:96) in − (cid:96) , R = R in (226)If the disk is thin, i.e. H in /R in (cid:28)
1, and the viscosity is small, i.e. α (cid:28)
1, then Eq. (226) implies that ( (cid:96) in − (cid:96) ) /(cid:96) in (cid:28) T has to satisfy the equation of angular momentum conservation (211), which canbe written as T = ˙ M ( (cid:96) − (cid:96) ) , T in = ˙ M ( (cid:96) in − (cid:96) ) . (227)Thus it is clear that for a thin, low viscosity disk the ‘no torque inner boundary condition’ ( T in ≈
0) is an excellentapproximation following from angular momentum conservation .However, if the disk and the stream are thick, i.e.
H/r ∼
1, and the viscosity is high, i.e. α ∼
1, then the angularmomentum varies also in the stream in accordance with the simple reasoning presented above. However, the no–stresscondition at the disc inner edge might be not satisfied.
Additional reading : Reference [6].
Acknowledgements
I am grateful to Cosimo Bambi for having invited me to teach at the 2014 Fudan WinterSchool in Shanghai. Discussions with and advice of Marek Abramowicz, Tal Alexander, Omer Blaes and OlekS¸adowski were of great help. I thank the Nella and Leon Benoziyo Center for Astrophysics at the Weizmann Institutefor its hospitality in December 2014/January 2015 when parts of these lectures were written. This work has beensupported in part by the French Space Agency CNES and by the National Science Centre, Poland grants DEC-2012/04/A/ST9/00083, UMO-2013/08/A/ST9/00795 and UMO-2015/19/B/ST9/01099. [1] Abramowicz, M. A. 2005, in
Growing Black Holes: Accretion in a Cosmological Context , ESO astrophysics symposia,(Berlin: Springer), p.257[2] Abramowicz, M. A., & Klu´zniak, W. 2005, Ap&SS, 300, 127[3] Abramowicz, M. A., Czerny, B., Lasota, J. P., Szuszkiewicz E., 1988, ApJ, 332, 646[4] Abramowicz, M. A., Chen, X., Kato, S., Lasota, J.-P., Regev, O., 1995, ApJ, 438, L37[5] Abramowicz, M. A., Chen, X.-M., Granath, M., Lasota, J.-P., 1996, ApJ, 471, 762[6] Afshordi, N., Paczy´nski, B. 2003, ApJ, 592, 354[7] Balbus, S. A. 2011,
Physical Processes in Circumstellar Disks around Young Stars , ed. by P.J.V. Garcia, (Chicago, Uni-versity of Chicago Press), 237; arXiv:0906.0854[8] Balbus, S. A., Hawley, J. F., 1991, ApJ, 376, 214[9] Blaes, O. 2014, Space Science Reviews, 183, 21[10] Broderick, A. E., Johannsen, T., Loeb, A., & Psaltis, D. 2014, ApJ, 784, 7[11] Coleman, M. S. B., Kotko, I., Blaes, O., Lasota, J.-P. 2016, MNRAS, submitted[12] Collin, S., Zahn, J.-P., 2008, A&A, 477, 419[13] Coriat, M., Fender, R. P., & Dubus, G. 2012, MNRAS, 424, 1991[14] Cox, J.P., Giuli, R.T. 1968,
Principles of Stellar Structure , (New York, Gordon & Breach)[15] Dubus, G., Hameury, J.-M., & Lasota, J.-P. 2001, A&A, 373, 251[16] Dubus, G., Lasota, J.-P., Hameury, J.-M., & Charles, P. 1999, MNRAS, 303, 139[17] Esin, A. A., Lasota, J.-P., & Hynes, R. I. 2000, A&A, 354, 987[18] Frank, J., King, A., & Raine, D. J. 2002,
Accretion Power in Astrophysics , (Cambridge University Press)[19] Gammie, C. F., 2001, ApJ, 553, 174[20] Goodman, J. 2003, MNRAS, 339, 937[21] Hameury, J.-M., Menou, K., Dubus, G., Lasota, J.-P., & Hure, J.-M. 1998, MNRAS, 298, 1048[22] Hirose, S., Blaes, O., Krolik, J. H., Coleman, M. S. B., Sano, T. 2014, ApJ, 787, 1 [23] Horne, K., Cook, M. C. 1985, MNRAS, 214, 307[24] Jiang, Y.-F., Stone, J. M., & Davis, S. W. 2014, ApJ, 778, 65[25] Jiang, Y.-F., Stone, J. M., & Davis, S. W. 2014, ApJ, 796, 106[26] Kato, S., Fukue, J., & Mineshige, S. 2008, Black-Hole Accretion Disks – Towards a New Paradigm , (Kyoto UniversityPress)[27] King, A. R. 2009, MNRAS, 393, L41[28] King, A., & Nixon, C. 2013, Classical and Quantum Gravity, 30, 244006[29] Kotko, I., Lasota, J.-P. 2012, A&A, 545, 115[30] Lasota, J.-P. 1994, in
Theory of Accretion Disks - 2 , ed. by Wolfgang J. Duschl et al. NATO Advanced Science Institutes(ASI) Series C, Volume 417 (Dordrecht, Kluwer) p.341[31] Lasota, J.-P. 2001, New Astronomy Reviews, 45, 449[32] Lasota, J.-P., Dubus, G., & Kruk, K. 2008, A&A, 486, 523[33] Lasota, J.-P., King, A. R., & Dubus, G., 2015, ApJL, 801, L4[34] Landau, L. D., Lifshitz, E. M. 1987,
Fluid Mechanics; Course of theoretical physics , (Oxford: Pergamon Press)[35] Lin, D. N. C., Pringle, J. E., 1987, MNRAS, 225, 607[36] Menou, K., Hameury, J.-M., & Stehle, R. 1999, MNRAS, 305, 79[37] Meyer, F., Meyer-Hofmeister, E. 1981, A&A, 104, L10[38] Narayan, R., Yi, I., 1994, ApJ, 428, L13[39] Novikov, I. D., Thorne, K. S., in
Black holes (Les astres occlus) , ed. C. DeWitt & B.S. DeWitt. ´Ecole de Houches 1972(Gordon & Breach), p. 343[40] Ogilvie, G. I. 1999, MNRAS, 304, 557[41] Paczy´nski, B. 1969, AcA, 19, 1[42] Paczy´nski, B. 1977, ApJ, 216, 822[43] Paczy´nski, B. 1978, AcA, 28, 91[44] Paczy´nski, B. 2000, arXiv:astro-ph/0004129[45] Papaloizou, J. C. B., & Pringle, J. E. 1983, MNRAS, 202, 118[46] Partnership ALMA; Brogan, C. L., Perez, L. M., et al. 2015, ApjL, 808, L3[47] Poutanen, J., Lipunova, G., Fabrika, S., Butkevich, A. G., & Abolmasov, P. 2007, MNRAS, 377, 1187[48] Prialnik, D. 2009,
An Introduction to the Theory of Stellar Structure and Evolution (Cambridge University Press)[49] S¸adowski, A. 2009, ApJS, 183, 171[50] S¸adowski, A. 2011, PhD Thesis (CAMK), arXiv:1108.0396[51] S¸adowski, A., & Narayan, R. 2015, MNRAS, 453, 3213[52] S¸adowski, A., Narayan, R., McKinney, J. C., & Tchekhovskoy, A. 2014, MNRAS, 439, 503[53] Shakura, N. I., Sunyaev, R. A. 1973, A&A, 24, 337[54] Smak, J. 1999, AcA, 49, 391[55] Spruit, H. C. 2010, arXiv:1005.5279[56] Toomre, A., 1964, ApJ, 139, 1217[57] Vincent, F. H., Paumard, T., Perrin, G., et al. 2011, The Galactic Center: a Window to the Nuclear Environment of DiskGalaxies, 439, 275[58] Wood, J., Horne, K., Berriman, G., et al. 1986, MNRAS, 219, 629[59] Yan, Z., Yu, W., 2015, ApJ, 805, 87[60] Yuan, F., Narayan, R., 2014, ARA&A, 52, 529
Appendix
Thermodynamical relations
The equation of state can be expressed in the form: P = P r + R µ i ρT i + R µ e ρT e + B π , (228)where P r is the radiation pressure, R is the gas constant, µ i and µ e the mean molecular weights of ions and elec-trons respectively, T i , and T e ion and electron temperatures, a the radiation constant (not to be confused with thedimensionless angular momentum a in the Kerr metric), and B the intensity of a isotropically tangled magnetic field,includes the radiation, gas and magnetic pressures. The radiation pressure P r , the gas pressure P g , and the magneticpressure P m correspond respectively to the first term, the second and third terms, and the last term in equation (228).The mean molecular weights of ions and electrons can be well approximated by: µ i ≈ X + Y , µ e ≈
21 +
X , (229)1where X is the relative mass abundance of hydrogen and Y that of helium. We may define a temperature as T = µ (cid:18) T i µ i + T e µ e (cid:19) , (230)where µ = (cid:18) µ i + 1 µ e (cid:19) − ≈
21 + 3 X + 1 / Y (231)is the mean molecular weight. In the case of a one-temperature gas ( T i = T e ), one has T = T i = T e . For an opticallythick gas, P r = (4 σ/ c ) T r .For the frozen-in magnetic field pressure P m ∼ B ∼ ρ / , therefore we may write the internal energy as U = 4 σρc T r + R Tµm u ( γ g −
1) + e o ρ / , (232)where e o is a constant ( P m = 1 / e o ρ / ) and γ g is the ratio of the specific heats of the gas. We define β = P g p , β m = P g P g + P m , β ∗ = 4 − β m β m β. (233)From equations (228) and (232) one obtains the following formulae (see e.g. “Cox & Giuli” 2004) for the specific heatat constant volume: c V = R µ ( γ g − (cid:20) − β/β m )( γ g −
1) + ββ (cid:21) = 4 − β ∗ Γ − PρT (234)and the adiabatic indices: Γ − − β ∗ )( γ g − − β/β m )( γ g −
1) + β (235)Γ = β ∗ + (4 − β ∗ )(Γ − . (236)The ratio of specific heats is γ = c p /c V = Γ /β . For β = β m we have Γ = γ g and Γ = (4 − β ) / β ( γ g − β = 0 .
5) one gets Γ = 1 . β = 0 .
95, Γ = 1 .
65 (here we have used γ g = 5 / β m ∼ . −
1. Since
T dSdR = c V (cid:20) d ln TdR − (Γ − (cid:18) d ln Σ dR − d ln HdR (cid:19)(cid:21) , (237)The advective flux is written in the form: Q adv = ˙ M πR Pρ ξ a (238)where ξ a = − (cid:20) − β ∗ Γ − d ln Td ln R + (4 − β ∗ ) d ln Σ d ln R (cid:21) . (239)The term ∝ d ln H/d ln R has been neglected. Since no rigorous vertical averaging procedure exists, the presenceor not of the d ln H/d ln R – type terms in this (and other) equation may be decided only by comparison with 2Dcalculations.The formulae derived in this section are valid for the optically thin case τ = 0 if one assumes β = β m . Reference : Weiss, A., Hillebrandt, W., Thomas, H.-C., & Ritter, H. 2004,