arXiv:astro-ph/0601663v1 29 Jan 2006 o.Nt .Ato.Soc. Astron. R. Not. Mon. .Mayer M. therma cool, The component discs: accretion hole black of Variability 1Otbr2018 October 31 biu rgnfrflcutosi naceinds r turb are disc accretion an Th radius. in that fluctuations at ra timescale for viscous each origin the at as obvious variations long the (2001 as of least al. timescale at et Churazov be the by for o emphasised necessary In also is variations. as it rate work, accretion to the model of the origin (19 Lyubarskii physical by the noted as is model, emitting this x-ray with problem central major the The in release energy the modulate Lyubarskso di th of at in occurring model variations rate by the tion produced by is variability naturally the produced where th (1997) note be (2005) can al. relation et Uttley a and (2004) s Uttley explai the Both cannot by relation. flares, or produced shots, is occurring randomly curve of light mation the standard where the models, that mo emphasise shot-noise various (2005) – al. between et relation distinguish Uttley to rms-flux example diagnostic For the a – as 2001) used be McHardy can & (Uttley relati ampli flux linear the x-ray and strong variability the X-ray a the of of amplitude (2 existence the the al. between et that Uttley out and 2003; pointed (2004) have 1990; al. Uttley et However, Vaughan variabilit Hasinger this understood. of & 2003; well origin Belloni timescal The al. 2004). et of Remillard 1988; Markowitz & McClintock range 1994; ap McHardy broad Klis significant der example a van on display for (AGN) flickering, scales, (see galactic or binaries) the variability, (X-ray on odic both stellar sources, and x-ray powered Accretion INTRODUCTION 1 c nttt fAtooy aige od abig CB30HA, Cambridge 1 Road, Madingley Astronomy, of Institute -al [email protected] E-Mail: 05RAS 2005 1 .E Pringle E. J. & ff 000 rn ai hc rpgt nad,and inwards, propagate which radii erent –8(05 rne 1Otbr21 M L (MN 2018 October 31 Printed (2005) 1–18 , oe fKn ta.(04 saddt hs oes efidsim find we models, these to th added When is (2004) behaviour. before. al. limit-cycle et King find of also model impo and an plays structure, advection one- disc heat a radial develop that we agree authors, We structure. previous with common In proceeds. edta a epoue ylcldnm el nteds ol to itself. words: disc disc Key the the en within in the field cells dynamo limit var dynamo the local of to of by degree density need produced energy low we be the binaries, can with that hole consistent field black be of to component order disc in that find we nteKn ta.(04 oe eed estvl nterat the on sensitively depends model bl (2004) in disc. al. the variability et of for King properties (2004) the thermal in the al. et of King account proper of taking model the extend We ABSTRACT H / R ti motn oflo h iedpnec ftelcld local the of time-dependence the follow to important is it , eaenwbte lcdt u hsclcntanso model on constraints physical put to placed better now are We lc oepyis–glxe:jt -as binaries. X-Rays: – jets galaxies: – physics hole black UK uha such n accre- e drfor rder snot is y tsuch at region. ueof tude isto dius simple onship most e ulent dels. 005) um- 97), eri- es ), ii hs aepaeo h oa yaia iecl ( timescale dynamical local the on dynamo place disc the take with these associated perhaps hydro-magnetic, or Ω iecl yafco of factor a by timescale aaee,and parameter, es eod eas h oa icdnmswr sue to of assumed scale radial were a dynamos thic on disc disc independent of local spatially value the local be the because compute Second, to ness. need ign no be was could there equations cause structure disc local First, reasons. h ici htte sue htteds a osatthic constant a had disc the that assumed of ratio they ness that in disc the released. centr is the energy the at of timescales most dynamical where disc the the than are longer frequencies flickering much dominant cally the why for explanation yafco fodr2 order times of longer factor much a on by occurs chanc This by field. rise large-scale give annuli coherent disc neighbouring in small-sc processes when occur dom only can outflow th large-scale They that phenomeno timescale. tulate dynamical stochastic local small-scale the roughly a jet on or as operating (wind dynamo outflow the an of model form They the in loss a momentum angular can th ing disc that the suggest They in (2003). processes base al. dynamo (2004), et al. Livio by et forward King put by ideas proposed recently been has model, steaglrvlct) hc ssotrta h oa vis local the than shorter is which velocity), angular the is A T E ouint hspolm ntefr fa xlctphysica explicit an of form the in problem, this to solution A oee,Kn ta.(04 okavr daie oe for model idealised very a took (2004) al. et King However, tl l v2.2) file style X H / H R / hssmlfidtecmuain o w main two for computations the simplified This . R < R / h icoeigage(rnl 1981). (Pringle angle opening disc the 1 H ∼ Lvoe l 03.Ti lopoie an provides also This 2003). al. et (Livio α ( tn oei eemnn h inner the determining in role rtant H eas h ereo variability of degree the Because / tcatcmgei dynamo magnetic stochastic e R oo ictikest radius, to thickness disc of io ) oemdlfrtelcldisc local the for model zone s tutr stevariability the as structure isc s hnafwpreto the of percent few a than ess 2 c oeaceindss by discs, accretion hole ack rydniyo h poloidal the of density ergy ff where , lrvraiiybhvorto behaviour variability ilar aiiyse ntethermal the in seen iability c h crto aeb driv- by rate accretion the ect aaees nparticular, In parameters. α disc l ( ∆ < R )i h viscosity the is 1) ∼ ∼ H Ω hr was there , − rd be- ored, 1 where , l ran- ale npos- en local e cales, oa to e and , typi- cous on d of e k- k- n, ). l 2 M. Mayer & J. E. Pringle no need to vary the number of independent dynamos as the disc purely Newtonian gravity, around a point mass M, and truncate the 2 thickness varied with time. In this paper we take the first steps to- disc at an appropriate radius Rin = 6GM/c . Here again we take wards remedying this deficiency. For the time being we concern the view that the wealth of uncertainties inherent in the physical ourselves with a standard (optically thick, geometrically thin) ac- processes involved, and the complicated nature of the subsequent cretion disc, and thus any conclusions we have will be relevant behaviour, means that adding the complication of using the proper mainly to the thermally dominated (TD) state of the X-ray tran- space-time geometry is not warranted at this stage. Second, by tak- 3 sients (e.g. McClintock & Remillard 2004). We leave the extension ing the angular velocity of the disc material to be ΩK = GM/R to the more interesting, and more variable (low/hard) state, con- we have also neglected the effects of any radial pressure gradient in p taining both disc and corona, to future work. In Section 2 we ex- the disc. We shall show below that this approximation is justified plain how we generalise the work of King et al. (2004) by includ- for the disc models presented here. ing equations to compute the local disc structure, and also explain As is apparent from equation 3, angular momentum is trans- how we vary the grid spacings to follow the radial size of the local ported either by advection or viscous torques. For Keplerian rota- disc dynamo cells in such a way as to minimise unwanted numer- tion the viscous torque is given by ical mixing. In Sections 3 and 4 we investigate the models which G = 3πν ΣR2Ω , (4) result from our equations when there is a steady external accretion − t K rate, and in Section 5 compare our results with previous work in the where νt is the kinematic viscosity. field. In Section 6 we add the variability according to the stochastic Equations (3) and (2) can be combined (Pringle 1981) to give magnetic model of King et al. (2004). In Section 7, we discuss the observational data on the variability of the thermal disc component, ∂Σ 3 ∂ 1/2 ∂ 1/2 to which our models are relevant, and in Section 8 we show that for = R νtΣR . (5) ∂t R ∂R ∂R expected values of the model parameters our models are consistent " # with the data. We present brief conclusions in Section 9. We integrate this equation in time applying a zero-torque con- 2 dition at the inner boundary (Rin = 3RS, where RS = 2GM/c )
and thus set ΣRin=3RS = 0. At the outer boundary we feed the disc with a constant external accretion rate M˙ ext. For advection we use 2 METHODOLOGY a first-order donor cell upwind scheme. The integration conserves In this Section we present the input physics for our basic accre- mass up to machine accuracy, accounting for mass supply at Rout tion disc models. For a more detailed discussion, see for example and loss at Rin. Frank et al. (2002). The disc is treated as one-dimensional, in the sense that we re- 2.2 Radial force balance and hydrostatic equilibrium solve the disc in radial direction only, dividing the disc into annuli, and just use the one-zone approximation in the vertical direction. We assume throughout that the disc is in hydrostatic equilibrium Thus, all variables (e.g. mass density ρ, temperature T etc.) have in the vertical direction (i.e. perpendicular to the disc plane). In one value at each radius R, and we make no further assumptions the radial direction, the central gravitational pull is balanced by the about the vertical disc structure. Thus, for example the surface den- centrifugal force Σ sity is given in terms of density ρ and disc semi-thickness H by GM the relation Ω2 R = . (6) K R2 Σ= 2ρH (1) The vertical structure is treated in the one-zone- approximation. Thus we take the vertical component of the As we discuss in Section 5 it is possible to make more detailed gravitational force of the black hole to be balanced by the vertical assumptions about the vertical disc structure. However, in view of pressure gradient, i.e. the large number of uncertainties in the basic physical processes involved here, we regard such extra complications as unnecessary ∂P = ρg , (7) for our current purposes. ∂z − z
where gz is the vertical gravity. In terms of our one-zone model we may write 2.1 Surface density ∂P = P/H, (8) The viscous evolution of the disc is governed (Pringle 1981; ∂z Livio & Pringle 1992) by the equation of continuity and H Σ 2 ∂ 1 ∂ gz = ΩKR . (9) + (ΣRuR) = 0 (2) − R ∂t R ∂R Using this, together with Equation 1 we obtain the relationship ex- and by the equation of conservation of angular momentum pressing vertical hydrodynamic equilibrium in the form ∂ 1 ∂ 1 ∂G Σ 2Ω + Σ 2Ω = GM R K RuRR K . (3) P = Σ2. (10) ∂t R ∂R 2πR ∂R 4ρR3 Here Σ is the surface density, uR the radial velocity, M˙ = 2πΣRuR the accretion rate, Ω = GM/R3 the Keplerian rotation frequency K 2.3 Energy equation and G the viscous torque. p We should note two things here. First, although we are dis- We consider the heat content q of one half of an elemental disc cussing accretion discs around black holes in this paper, we use annulus of width ∆R at radius R, and height H. Then, from the
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Variability of black hole accretion discs: The cool, thermal disc component 3 usual thermodynamic relations, and assuming hydrostatic equilib- 2.4.2 Viscosity rium (Equation 10), we find (see Appendix A) that We use the standard α-viscosity prescription put forward by dq ∂e ∂ (2πRHuR) Shakura & Sunyaev (1973) and take the the (rφ)-component of the = e˙ + uR + APH˙ + P . (11) dt ∂R ∂R stress tensor to be trφ = αP. Relating this to the rate-of-strain ten- − sor, through the viscosity ρν (e.g. Bisnovatyi-Kogan & Lovelace Here e is the internal energy of the semi-annulus, and the last two t 2001) so that t = 3 ν ρΩ we see that the kinematic viscosity is terms come from the PdV work done on the semi-annulus. A = rφ 2 t K ′ ′ related to α by the equation− 2πR∆R represents the surface area of a disc annulus of width ∆R at a given distance R from the black hole. The energy equation is then 2 P νt = α . (20) 3 ρΩK
dq + This viscosity prescription (eq. 20), using Eqns. (10), (1) = A Q Q− . (12) dt − and (18), can also be written in the form The source terms on the RHS of the energy equation (11) are 2 νt = α √γcs H. (21) given by the viscous dissipation of the disc per unit annulus area, 3 9 GM If we envisage the viscosity νt in physical terms as the product of a Q+ = ν Σ , (13) characteristic turbulent length and turbulent velocity, l and V , and 8 t R3 t t our disc is assumed to be geometrically thin, we can associate the and radiative losses, Q−, which we take to be of the form characteristic velocity and length scale with a fraction of the local scale height and of the local sound speed, respectively. In order 4σ 4 Q− = T , (14) that the turbulence remains subsonic we see that the factor 2/3α √γ 3τ needs to be smaller than, or of order, unity. where the optical depth τ is given in terms of the opacity κR by The viscous torque (4) using the viscosity prescription (20) can be written as 1 τ = Σκ (ρ, T). (15) 2 R G = 4παPHR2 . (22) − To ensure the consistency of the model we need disc to be optically thick in the vertical direction, i.e. τ 1. ≫ 2.4.3 Opacity We have written the numerical scheme so that internal energy e is conserved to machine accuracy, apart from losses on the inner We take the Rosseland mean opacity, κR, tabulated by the OPAL and outer boundary and local source terms (i.e. PdV). The bound- opacity project(Rogers & Iglesias 1992; Iglesias & Rogers 1996) ary condition at the outer disc edge for the energy equation is taken for the solar composition of Grevesse & Noels (1993). We use the to be divergence free, i.e. energy is being put in the last cell asit is X = 0.7 set of their opacity tables (X is the hydrogen mass fraction lost to the next innermost cell. At the inner boundary energy is lost, of the matter). Without further notice we fix the metallicity to be assumed captured by the black hole. Z = 0.02 (solar metallicity).
2.5 Numerical Grid and calculation procedure 2.4 Material functions We use a one-dimensional grid in radius with N points ranging from 2.4.1 Equation of state Rin = 3RS to Rout. We solve the time-dependent equations (3), (2) and (11) using the mass m and internal energy e of a disc semi- The total pressure P is given by the sum of gas and radiation pres- annulus as dependent variables. sure (since τ 1) ≫ In order to allow simple application of the concepts of vari- k T 4σ able dynamo-driven angular momentum presented by King et al. P = ρ B + T 4, (16) µmp 3c (2004), it is necessary for the widths (∆R) of the disc annuli to be comparable to the local disc thickness H. Within the context of the while the specific internal energy U of the mixture of monatomic one-zone approximation, this is a reasonable assumption, since in gas & radiation is this case it makes little physical sense to try to attribute physical reality to attempts at resolving structures significantly smaller than 3 kBT 4σ 4 U = + T . (17) the scale height H of the disc. 1 2 µmp cρ While Rin and Rout are kept constant, the number of grid-points The sound speed cs can be calculated using is time-dependent and the number of grid points is adjusted accord- ing to the following prescription (see Fig. 1). We remove/insert grid 2 P = γρcs , (18) points if the local radial extent of the annulus is larger/smaller than . where the adiabatic index γ is given by 2H and 0 5H, respectively. Grid points are inserted by conserv- ing disc mass, angular momentum and energy e. In typical runs, N 3 2 ∂ log P 16 12βP β ranges between a few dozens and a few hundred points. γ = = − − 2 P , (19) ∂ log ρ 12 21 β !S − 2 P 1 Hameury et al. (1998) demonstrate that formal mathematical conver- 4 5 and is a slowly varying function of βP, which ranges from 3 to 3 as gence of the numerical scheme may require a grid resolution much less βP ranges from 0 to 1, where βP is the ratio of gas pressure to total than the disc thickness. We note here that mathematical convergence does pressure. not necessarily imply more accurate modelling of physical reality. c 2005 RAS, MNRAS 000, 1–18
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-0.6 stabilising effect on the disc in that region. To illustrate this, we note that the terms responsible for the radial advection of energy can be -0.8 written in terms of the radial entropy gradient as if they correspond to a local additional heat loss term (e.g. Muchotrzeb & Paczynski -1 1982; Bisnovatyi-Kogan & Lovelace 2001), in the form MT˙ ∂S -1.2 Q = . (23) ad 2πR ∂R log H/R -1.4 The radial entropy gradient can be expanded in terms of pressure and radial density and temperature gradients. In doing so we obtain
-1.6 M˙ P Qad = χad, (24) -1.8 2πR2 ρ
where the dimensionless measure of the strength of the effect, χad, -2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 includes the radial derivatives. Usually, χad(R) is a slowly varying log R/(3 RS) function close to unity. However, although in the gas pressure dom- Figure 1. The refinement of the numerical grid. The Figure shows log H/R inated regime this value is indeed close to unity, in the radiation versus log R as it changes in a typical run. Every time the local scale height pressure dominated regime χad can be as large as 10, H exceeds(falls short of) twice(half) the local grid cell width, the resolution Using the expressions for the internal energy and pressure in is reduced(increased). Due do the binarity requirement of the grid (see text) Appendix A, we can express χad in terms of the radial derivatives the same grid points are restored after repeated refinement/coarsening. of ρ and T for a stationary disc 21 ∂ log T ∂ log ρ χ = 12 β + (4 3β ) . (25) The setup of the grid is implemented as follows. At the start of ad − − 2 P ∂ log R − P ∂ log R ! the calculation there are 2 points, at Rin and Rout. Then we calculate the scale height H(R ) and compare it with the interval R R . out out − in If Rout Rin > 2H(Rout), we put a new grid point at Rnew = √RoutRin 3 STATIONARY SOLUTIONS AND STABILITY − and calculate the scale height H(Rnew). This procedure is recur- sively applied on both intervals. The setup is complete if there is Before investigating the time dependent disc behaviour, and before no need to refinement any more. adding in the complications of stochastically driven accretion, we From a numerical point of view it is beneficial in preventing first investigate steady disc solutions, and the conditions for stabil- numerical mixing to use a scheme which retains the structure of the ity. grid cells fixed as far as possible. Thus, in order to maintain a binary structure of the grid (yielding the same coarse interval at the same 3.1 Stationary solutions place after several refinements), we assign each grid point with a ˙ number N, initially N(Rin) = 0 and N(Rout) = 1. As a refinement In the stationary case, with accretion rate M = const., we have occurs, the new grid point gets the mean number of the adjacent ∂/∂t = 0 and thus the conservation of angular momentum (Eqn. 3) grid points. If this number is not an integer, all N are multiplied by becomes 2. The condition of grid coarsening is accompanied by the condi- Mf˙ = 3πν Σ, (26) tion for keeping the grid binary, i.e. we only remove a point if the t N(R ) N(R ) = N(R ) N(R ) and N(R ) has to conservation of energy (11) becomes outer − remove remove − inner inner be an integer multiple of N(Router) N(Rinner). This binarity of the + − Q Q− Qad = 0, (27) grid ensures that, for example, the magnetic field is never mixed − − numerically across the entire grid, i.e. subsequent refinement and and vertical hydrostatic equilibrium (10) remains coarsening restores the same grid cell again. GM P = Σ2, (28) A typical coarsening/refinement can be seen in Fig. 1. The 4ρR3 ∆ ∆ = (log R) correlate very well with the H/R ratio since (log R) 1 + ∆ = + ∆ ∆ = where the function f (R) = 1 (R/Rin) 2 ensures compliance with log((R R)/R) log(1 R/R) R/R H/R as required. − Every ∆ log H/R = 0.3 dex, i.e. at log≈H/R 1, 1.3 and 1.6 the the inner boundary condition. For a given accretion rate, and at a ≈− − − resolution changes, as H de-/increases by a factor of 2. particular radius, these three equations can be written as two equa- tions, depending on pressure P, density ρ and temperature T. The energy equation yields 2.6 Advection ˙ 4 It is often the case (see, for example, the discussion in Pringle et al. M 3 2 2 P 16πσT αP Ω R f χad = 0, (29) 2πR2 4 K − ρ − 3κ ρΩ Mf˙ 1986) that the advective terms in the energy equation (i.e. those ! R K terms containing the radial velocity uR on the RHS of Equation 11) and hydrostatic equilibrium yields are omitted. However, it was realised early that even in the case of Ω4 M˙ 2 f 2ρ dwarf nova discs these terms can make a difference (Faulkner et al. P3 K = 0. (30) 1983). And more recently, Abramowicz et al. (1988) drew attention − 16π2α2 to the fact that the radial advection of heat can play an important Then for a given accretion rate, and at a particular radius, these two role in the local energy balance in the disc, especially at high ac- equations, together with the equation of state (Equation 16) can be cretion rates close to the black hole, and can, in particular, have a solved for P, ρ and T.
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1 2 Z=0 χ=5 χ=20 0.001 0.02 0.5 1 χ 0.1 =0
0 0 ] ]
. Edd . Edd . -0.5 . -1 log M [M log M [M -1 (10 -2 -3 , 10 6 ) -3 , 10) -1.5 ) -3
6
,M)=(10 (α (1, 10) -2 (1, 10 -4 2 2.2 2.4 2.6 2.8 3 3.2 3.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 log αΣ [g cm-2] log Σ [g cm-2] 6 Figure 2. Local equilibrium solutions at R = 30RS for a range of accre- Figure 3. Local equilibrium solution at R = 30RS for M = 10 M , α = 1, ⊙ tion rates. We consider solutions for different values of viscosity parameter χad = 20 and different metallicities Z = 0, 0.001, 0.02(solar) and Z = 0.1. α, the black hole mass M and the strength of radial advection, χad. Parts The hydrogen content is fixed at X = 0.7. with negative M˙ Σ slope are thermally and viscously unstable. Note the − stabilising effect of advection (when χad , 0).
higher black hole mass M, higher α and lower χad the range of accretion rates we get instability becomes larger. We solve these equations with a 2D nested-intervals method It is well known (see, for example, Lasota 2001) that the insta- as described in Mayer & Duschl (2005). In general we consider the bility causes the disc to undergo limit-cycle behaviour. We investi- conditions at a particular radius R, for various values of the accre- gate this further below. tion rate M˙ . For illustrative purposes we also consider the solutions for three different particular values of the strength of the adiabatic heat flux χad, and for various values of the viscosity parameter α. 3.3 Dependency on black hole mass and metallicity We present our results in terms of the Eddington luminosity and the corresponding Eddington accretion rate. In terms of the For black hole masses typical of X-ray binaries (M 10 M ), ≈ ⊙ Eddington luminosity, LEdd = 4πGMmpc/σT we may define a cor- characteristic mid plane temperatures close to the black hole are of responding Eddington accretion rate by the order of 107 K. Since M˙ Ω2 T 4/τ (cf. eq. 27) in the absence of advection, ˙ 2 K ∝ R LEdd = ηK MEddc , (31) we get in scaled units (R in Schwarzschild Radii, M˙ in units of where η is the efficiency. For a typical value of η = 1/12 (see Eddington accretion rate) and roughly constant optical depth that K K 1/4 6 below) we find in numerical terms T M− . For a black hole mass of 10 M we then expect the mid∝ plane temperature of the disc to be of the⊙ order of a few times 5 7 M 1 10 K. M˙ Edd = 2.68 10− M yr− . (32) · · 10M ⊙ For 106 M the disc temperature reaches T 105.4 K for ⊙ ! ⊙ ≈ reasonable accretion rates. Then the disc shows an additional in- stability caused by the ”Z-Bump” (cf. Seaton et al. 1994) in the 3.2 S-Curves opacity for a metallicity larger than Z = 0.02. This so called Z- Bump is also responsible for pulsations in β Cep stars (Simon 1982; In Fig. 2 we plot the relationship between accretion rate M˙ and α Kiriakidis et al. 1992). We show a S-curve presenting this instabil- times disc surface density Σ for different values of radially constant ity in Fig. 3. It is evident that the disc for this instability locally can α and for different values of χ and of the central mass M. 4 3 1 ad jump from accretion rates of 10− ... 10− to 10− M˙ Edd. These plots give us direct information about the stability of the Generally, the disc becomes more unstable with increasing steady-state discs. There are two types of instability which these mass of the black hole, i.e. the lower turning point in the ”S-curve” discs can be subject to – viscous (e.g. Lightman & Eardley 1974) moves to lower accretion rates. and thermal (Pringle 1976). We show in Appendix B that the stabil- ity criterion in both cases is the same. The discs are unstable when 3.4 H/R in the presence of advection ∂ log M˙ < 0. (33) For the approximations used in this paper to be valid we require ∂ log Σ !P,T that the disc thickness be small compared to the radius. We note From Figure 2 we see that in the absence of heat advection the here that the addition of a local heat sink in the guise of advection discs become unstable (i.e. the M˙ (Σ) curves have negative gradient) implies that the disc is typically thinner than it would be if all the as the accretion rates approach the Eddington limit. But, as pointed energy generated locally were radiated at the same radius. In the out by Abramowicz et al. (1988), advection of heat (i.e. non-zero stationary case, when advection of heat strongly dominates radia- tive losses (so that Qad Q−), Equation (29) becomes χad) provides stability. For all combinations of α and M considered, ≫ we see that within the physical range for χad there are always values 3 P (Ω R)2 f χ = 0. (34) of the accretion rate for which the discs are unstable, whereas for 4 K − ρ ad c 2005 RAS, MNRAS 000, 1–18
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Then using hydrostatic equilibrium (Equation 10) we find that 10 H 3 f = . (35) R 4χad s stable
Thus, for example, taking χad = 10 as a representative value, = 1 and with f (corresponding to R 10RS) we find H/R 0.2. ] 2 ≈ ≈ 1 Indeed, we see that as long as χad > 4/3, H/R is smaller than unity. . Edd . M [M 3.5 Keplerian rotation unstable For similar reasons, if advective heat flow is important, the disc angular velocity stays close to Keplerian. If we include the radial 0.1 pressure gradient in the radial momentum balance equation, we find that the disc angular velocity Ω is given by 0.001 0.01 0.1 GM PR α Ω2 = 1 + ξ , (36) R3 GMρ P ! Figure 4. M˙ versus α showing the systems with limit cycles (to the lower where right of each line) and without (upper left) for a 10 M black hole. The ˙ ⊙ ∂ log P line is only a rough indication since the grid in M, α is very coarse. The = points, for which actually calculations have been done, are indicated by the ξP . (37) − ∂ log R crosses. This may be written as ∂ log ρ ∂ log T ξ = β + (4 3β ) . (38) P P ∂ log R − P ∂ log R The dominant term in the radiation pressure dominated do- and thus enabled us to carry out a parameter study concerning the global stability of these discs with respect to the instabilities dis- main (βP = 0) is, of course, the temperature gradient, and a repre- sentative value is ξ 2. cussed above. P ≈ In (36) we treat the extra term in the brackets as small devi- 2 ation. Thus we can replace GM/R by (ΩKR) and with the hydro- static equilibrium (10) we get H 2 Ω2 =Ω2 1 + ξ . (39) 4.1 General behaviour K R P ! Our time-dependent models depend on the mass of the black hole, When advective heat transport is strongly dominant we can use the external accretion rate and the value of the viscosity parameter Equation (35) to deduce that α. 3 f ξ As initial conditions we set up a stationary disc, assuming Ω2 =Ω2 + P K 1 . (40) Q+ = Q and neglecting advection. Depending on the accretion 4χad − ! rate and the viscosity parameter, the disc either continues to stay For small deviation (Ω Ω Ω ) this gives approximately − K ≪ K in this stationary state, or, if advection of heat is important, the in- Ω 3 f ξP ner disc re-adjusts itself to allow for the advection. As we see from 1 = . (41) ΩK − 8χad Figure 2 if the viscosity is high enough, we expect limit cycles to Thus, using representative values, the deviation from Keplerian ro- appear with the inner disc oscillating between a hot state with high tation when advective heat transport is strong never exceeds around accretion rate and a cool state with lower accretion rate (see Sec- 4 per cent. The assumption of Keplerian rotation is therefore a rea- tion 3.2). sonable one. If the inner disc is such that the mean accretion rate produces instability, the inner disc spends its time trying to jump between the two stable branches of the S-curve. While it is on the upper branch, with advection important, the disc is steadily depleted. The surface 4 TIME-DEPENDENCE - RESULTS density decreases while the accretion rate is higher than average. We have seen above that even in the absence of stochastic mag- As the inner disc is depleted, some matter is transported outward in netic phenomena our accretion disc models are expected to display a heating front. The inner disc is radiation pressure dominated and time-dependent behaviour for some ranges of the parameters. We the accretion rate there is radially constant. Matter then piles up have carried out a number of simulations for different values of α, ahead of this front and subsequently the inner disc begins to starve different external (or mean) accretion rates and different black hole and the accretion rate decreases. When the front reaches the stable, masses. gas pressure dominated region of the disc, the front slows down and In this section, for reasons of computational time, we use a the inner disc cools, becoming more gas pressure dominated, as the logarithmically equidistant grid with N = 500 grid points without supply of matter is interrupted. The dissipated energy is advected refinement and coarsening. We solve the equations with an implicit outwards and the accretion rate in the inner disc decreases, vary- solver. Comparison with the standard explicit solver and the refine- ing radially as M˙ R2. Then inner disc reheats, the accretion rate ∝ ment and coarsening as described in Sect. 2.5 does not show signifi- increases inwards, the pileup is accreted and eventually the cycle cant deviations. However the computing time decreased drastically restarts with the inner disc again becoming depleted.
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Variability of black hole accretion discs: The cool, thermal disc component 7
4.2 Influence of α and the black hole mass on the stability The parameters for the grid of models with constant α discs we ran are shown in Figure 4 for a 10 M black hole. Also shown in Fig- ⊙ 1 ure 4 are those parameter values for which limit-cycle behaviour is found. The higher the viscosity parameter, the more unstable the ] disc becomes since then advection can only stabilise for higher ac- . Edd cretion rates. > [M . obs
For higher black hole masses, there are several complica-
Rout cretion rate deviates from the actual M˙ . For a 10 M black hole ⊙ Lrad = 2π Q−RdR. (42) this relation is practically independent of α. This deviation may be Rin 4 Z related to the deviations from the Lrad T relationship recently re- ∝ in For an infinitely extended, steady standard disc (Rout , no ra- ported by Kubota & Makishima (2004) and Abe et al. (2005). They dial advection of heat), the energy dissipated in the→ disc ∞ by vis- find the so called ’apparently standard’ regime in observations of cous processes (using our inner ’black hole’ radius Rin = 3RS = XTE J1550-564 and 4U 1630-47, where the disc luminosity is pro- 2 2 6GM/c ) is portional to Tin and attribute this to effects of the radial advection ˙ of energy. 1 GMM 1 2 6 Ldiss = = Mc˙ . (43) For the 10 M black hole (Fig. 7). There are departures from 2 Rin 12 ⊙ < M˙ obs >= M˙ . While the disc is being fed with an accretion rate of Thus the efficiency ηK of our Newtonian, standard black hole ac- M˙ = 0.1M˙ Edd, < M˙ obs > is only 44 % of the actual accretion rate, cretion disc in Keplerian rotation extending from Rin = 3RS to in- while the time average for the low-M˙ state only is as low as 1.5 % 1 finity in converting rest mass energy into radiation is ηK = 12 . of the actual accretion rate. This is a result of the strong outburst In the absence of advective heat flow, for an assumed value of behaviour of these discs compared to the low-mass case. Most of the efficiency ηK a measurement of the observed radiative flux, i.e. the mass previously stored in the outer disc is pushed through the Lrad, yields an estimate of the accretion rate by setting Lrad = Ldiss. inner disc in the high-M˙ state. Then, however, most of the energy However, when advection of heat is important, we expect devia- created by viscous dissipation is advected into the black hole. Since tions from the standard Lrad M˙ relation. In particular, since some the outbursts are short compared to the complete limit cycle, the of the dissipated heat is now∝ advected directly into the black hole, efficiency is fairly low. c 2005 RAS, MNRAS 000, 1–18
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4.4 Power density spectra
We calculate lightcurves at equidistant time points. Suppose we 10 have a lightcurve covering the time T with N points. We calculate power density spectra using the canonical normalisation to get the 2 Power Pν in units of (rms/mean) /Hz. Integrating Pνdν gives the in- . 1
tegrated fractional rms. We compute the FFT of the lightcurve and (t)/M] then define Pν by (Lewin et al. 1988) . obs 2 2
2 aν T log [M Pν = | | (46) L¯ 2T N 0.1 where L¯ is the mean luminosity of the lightcurve and aν the non- normalised Fourier coefficient. The Pν’s are binned into logarithmi- cally equidistant points. For the FFT we use the routines from the 0.01 FFTW library2. 500 600 700 800 900 1000 The integrated fractional rms is given in percent t [yrs] Figure 7. Lightcurve for an accretion disc around a 106 M black hole ⊙ accreting at 0.1 M˙ Edd. α = 0.1. The metallicity is Z = 0.1, the number of r = 100% Pνdν (47) · s grid points has been set to N = 250. Note that the lightcurve resembles the Z instabilities present in Fig. 3. Using Parseval’s theorem, it can be shown, that r indeed cor- responds to the rms/mean of the luminosity fluctuations the evolution of the total energy. They apply correction factors to 1 N L 2 r = 100% i 1 (48) try to take account of details of the vertical disc structure. They N ¯ = = · tv i=1 L − mainly consider M 10M and α 0.1 but make use of a modified X ”α-P”-description by multiplying⊙ the rφ-component of the viscous where Li is the luminosity given at equidistant time intervals and L¯ q stress-tensor with a factor βP. (Here βP is the ratio of gas pressure the time-average of the total light curve. Unless otherwise stated, related to total pressure, so that q = 0 corresponds to what we use the last formula is used to calculate the integrated fractional rms. here, and q = 1 corresponds to the rφ-component being propor- tional to αPgas.) For q > 0.5 they find stable disc solutions while 4.5 Period of limit cycle for q < 0.5 they find limit-cycle solutions of increasing amplitude. Their q = 0 results are compatible with ours. A sample lightcurve and power spectrum for a 10 M black hole is In a series of papers Szuszkiewicz & Miller (1997, 1998, ⊙ shown in Fig. 6. We binned the FFT data in logarithmically equis- 2001) take a similar approach to Honma et al. (1991) except that paced bins of width ∆ν/ν = 0.005. The disc undergoes limit cycles they use different numerical factors to try to take account of the on a timescale of roughly 600 s. Note that besides the fundamental details of the vertical disc structure. For this reason, while their frequency and their higher harmonics there are more peaks present. results are in line with those of Honma et al. (1991) and those pre- These arise from the asymmetry of the lightcurve. sented here, they differ in some minor details. In particular, they ˙ ˙ 3 The luminosity of the disc is given in terms of Mobs/M, where find unstable solutions for a 10 M black hole and α = 10− for ˙ ˙ ⊙ M is the external/average accretion rate and Mobs the ”observed” 0.09 < LEdd < 1 which is in line with our models except the high-M˙ accretion rate. This ratio is equivalent to the ratio between the ac- end, where we find stability at 1 LEdd. They subsequently develop tual luminosity and the luminosity of a stationary disc where no a scheme where they take account of the possibility that the disc advection is taken into account. might become optically thin and there is non-Keplerian rotation. Nayakshin et al. (2000) present a model with the specific ap- plication to GRS1915+105. They additionally include thermal and 5 COMPARISON WITH PREVIOUS WORK radiation diffusion in radial direction. For the simulations presented in this paper, however, these terms never become important. They In this Section we briefly compare our models with previous and might become comparable to the terms containing the radial advec- recent work in the field. We restrict ourselves to optically thick disc tion of the internal energy, but this only will result in factors of two models. at most. In the context of working in the one-zone-approximation, There is no absolutely correct way of doing 1D accretion disc the omission of these terms in our simulations seems to be justi- models in the one-zone approximation, and in this paper we have fied. They investigate models with different radially varying values adopted the simplest approach, by ignoring all the details of the for the viscosity parameter α. They furthermore try to account for vertical structure. To do it properly clearly requires doing two- additional cooling in a corona and include some flickering. Their dimensional hydrodynamics. In Appendix A we derive the LHS of flickering is a random ad-hoc modulation of the efficiency of radi- the energy equation (11) explicitly from first principles. We work ation coming from the inner parts of the disk. Lightcurves similar in terms of the total internal energy and the mass of a disc annulus. to GRS1915+105 can be produced. No further diagnostics of the All other variables can be expressed in terms of mass and internal flickering process (i.e. power density spectrum) are tested there. energy of a disc annulus. Janiuk et al. (2002) consider a similar approach to the one pre- In early work on black hole accretion discs, Honma et al. sented in this paper. They write the change in specific heat in terms (1991) presented disc models with an energy equation containing of temperature and density (i.e. surface density and scale height, setting the radial advection of the scale height to zero). They apply 2 http://www.fftw.org/ yet another set of correction factors to try to take account of the
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2 2 1.8 1 1.6 ]
2 0 1.4
. 1.2 -1
(t)/M 1 -2 obs ) [(rms/mean)
. ν M 0.8 P
ν -3 0.6 log ( -4 0.4
0.2 -5
0 -6 1000 2000 3000 4000 5000 6000 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 t [s] log ν [Hz]
Figure 6. (left) Lightcurve (detail) and (right) Power density spectrum for an accretion disc around a 10 M black hole accreting at 0.5 M˙ Edd. α = 0.1. The FFT data are binned in logarithmically equispaced bins of width ∆ν/ν = 0.005. ⊙ vertical structure. Their results are in line with those obtained by acteristic radial width comparable to the local disc thickness. They Szuszkiewicz & Miller (1997, 1998, 2001). then propose that the dynamo mechanism also gives rise to a small In conclusion, these various approaches differ mainly in the poloidal field in each cell. Most of the time this local poloidal flux nature and actual numerical values of the correction factors (being is randomly aligned from cell to cell and so has no net effect. But in the range 0.2 – 16) applied when trying to take some account from time to time this poloidal field magnetic field is sufficiently of the vertical disc structure. There seems to be no unanimity on aligned from cell to cell that it gives rise to a large scale magnetic the values that should be applied. In this paper we have taken the field which is able to generate an outflow (wind or jet) which re- simple approach of setting all these factors to unity. moves angular momentum from the disc. Since the local field in Some of the work reviewed here (Nayakshin et al. 2000; each dynamo cell changes on about the local dynamical timescale, Janiuk et al. 2002) also include prescriptions for energy loss to a this overall alignment only happens on a large multiple of the dy- wind. While there is agreement that mass loss does not influence namical timescale and thus can be comparable or even longer than the results, energy loss (cooling) does influence the disk and sta- the viscous timescale. bilises it. Energy loss usually is only parametrised with a fraction We now apply these ideas to the thermal disc structure de- of the radiative losses taken away to the corona or the wind. This veloped above. We stress here the additional concepts required in fraction is either set constant or depends on the accretion rate (e.g. applying this model to a realistic disc. More details of the underly- Janiuk et al. 2002). Without the energy loss to a wind, our results ing ideas can be found in King et al. (2004). In the following sub- so far are in agreement with the results in literature. sections we describe the details of this model and the equations In the next section we shall introduce our model for the flick- involved. ering which contains a self-consistent coupling between the mass, energy and angular momentum loss of the disc to a wind coupled 6.2 Evolution of the poloidal magnetic field to the physical model for the flickering.
We consider a poloidal magnetic field Bz which evolves according to the induction equation 6 THE MODEL FOR THE FLICKERING ∂Bz 1 ∂ ∂Bz 1 ∂ = Rη∗ (RB U ) + S . (49) ∂t R ∂R ∂R − R ∂R z R B 6.1 The basic mechanism ! We allow the field to diffuse radially with an effective magnetic We now address the means of introducing a physical mechanism to diffusivity η and allow for radial advection at velocity U which produce the observed short-timescale variability in black hole ac- ∗ R is due to a magnetic wind torque. S stands for a source term for cretion discs. As we mentioned in the Introduction the main prob- B the poloidal field which is assumed to be generated by the local lem is to find a mechanism which gives the right amplitude and the magnetic dynamo. At the outer disc boundary we take B = 0, right timescale. z allowing magnetic flux to diffuse outwards, but preventing inward Lyubarskii (1997) showed that if the viscosity parameter α flux advection. changes in a spatially uncorrelated manner on the local viscous timescale at each radius, then the characteristic flickering spectrum and the long timescales can be produced, but was unable to suggest 6.3 The source of poloidal field, S B a physical mechanism which would produce this result. King et al. (2004), following Livio et al. (2003), put forward a solution. The The poloidal field Bz is assumed to be generated by the disc dy- main problem is that local variations in α are correlated with small namo which generates the main disc viscosity. The magnetic field in the disc, partitioned into disc annuli of radial extent ∆R H, is variations in the disc’s internal magnetic field which is generated ≈ by a local disc dynamo. This operates characteristically on about assumed to change independently in each dynamo cell in a stochas- the local dynamical timescale. They envisage the disc dynamo op- tic fashion on the characteristic dynamo time scale 1 erating as a set of essentially independent dynamo cells with char- τd = kdΩK− , (50) c 2005 RAS, MNRAS 000, 1–18
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smaller than the disc magnetic field. This implies β2 1, so that 5 S ≪ energetically the poloidal field is negligible. Thus this assumption 4 furthermore allows us to neglect the energy generation/loss due to our dynamo process. Tout & Pringle (1996) estimate this fraction 3 to be H/R. ∼ )
ν 2 P
ν 6.4 Magnetic torque and the wind 1 log ( We assume that angular momentum loss occurs due to a magnetic wind/jet when the poloidal field is of sufficiently large scale. This is 0 likely to occur when the poloidal fields generated by the individual -1 dynamo cells are, by chance, spatially correlated over a radial ex- tent of the order of R. To measure the degree of spatial correlation -2 in the simulation, we define the radial averages -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 R+∆+ log (ν τ ) d R ∆ BzRdR = − − Bz +∆+ , (55) Figure 8. The power spectrum is shown for a sample time-series of a h i R R R ∆ RdR Markoff chain series (see eq. 53) for α1 = 0.5. This series is used to mimic − − R+∆+ the effects of locally acting magnetic dynamoes. In this figure we show R B2RdR 2 R ∆ z = − − the power spectrum for a dynamo acting on a timescale τd. To account for Bz +∆+ . (56) R R timescales smaller than τd, we linearly interpolated the Markoff chain. The ∆ RdR D E R − frequency is normalised to τ . The peak at the dimensionless frequency − d We chooseR the total interval over which we measure the cor- log(ντd) = 0 is clearly visible including higher harmonics. Most of the relation as equal to R, that is we take power is released on slightly longer timescales (approximately 2.5τd) com- ing from the broad peak at log(ντd) 0.4. + ≈ − ∆− + ∆ = R. (57)
We then choose R to be the geometric mean of R ∆ and R + ∆+. which we take to be a multiple of the local dynamical time scale − − Thus we choose the dimensionless constant a so that R ∆− = R/a, − (we take kd = 10, cf. Tout & Pringle 1992; Stone et al. 1996) R + ∆+ = aR with a > 1. This then implies that a = (1 + √5)/2, the Each annulus of the disc is assumed to generate a suf- golden ratio. ficiently large internal tangled magnetic field of strength Bdisc Thus we obtain which generates the effective disc viscosity with α parameter R+∆+ (Shakura & Sunyaev 1973) BzRdR R ∆− Bz = = − , (58) 2 h i R2 a2 1/a2 Bdisc R − α = . (51) R+∆+ 4πP B2RdR 2 R ∆ z B = = − − , (59) Although we assume that each dynamo cell generates a local z R2 a2 1/a2 R − poloidal field Bz, we expect the magnitude of the poloidal com- D E where we note that a2 1/a2 = √5. ponent to be small. We define a parameter βs so that the maximum We now define the− quantity value of the poloidal field, Bz,max is given by 2 = Bz Bz,max βs Bdisc, (52) Q = h i , (60) B2 where B is given by Equation 51. We expect β 1. z disc s ≪ We then model the stochastic nature of the local poloidal field which represents the degree of spatialD E correlation of the poloidal in the following way. At each radius we define a set of times, tn = magnetic field in neighbouring disc annuli. We note that 0 6 Q 6 1, nτd, n = 1, 2,... as integral multiples of the local dynamo timescale where Q = 1 represents a fully coherent magnetic field, while for at which the field changes. At each of these times we generate a Q = 0 the magnetic field is fully randomly ordered. ff random number, u(tn), according the the Marko process The poloidal magnetic field Bz produces a torque Tmag on a disc annulus, width ∆R, of size (Livio & Pringle 1992) u(tn+1) = α1u(tn) + ǫ(tn) , (53) − 2 Bz where ǫ(tn) is a random variable of zero mean and unit variance, T = 4πR2 h i ∆R. (61) mag − 4π and we take α1 = 0.5. The new value of the magnetic field strength ! in the disc annulus then is given by Considering the change of angular momentum in the disc an- nulus, this can be combined to give Bz(tn+1) = u(tn+1)Bz,max. (54) ∂ We show a power spectrum of a sample time-series for u(tn) 2πR∆R ΣR2Ω + ∆ 2πRΣv R2Ω = ∆G + T , (62) ∂t K R K mag for a magnetic dynamo acting locally on the dynamo time scale τd in Fig. 8. For the purposes of the Figure, in order to resolve shorter where the terms on the LHS describe the local change of angular timescales than the dynamo timescale, we applied a linear inter- momentum and the radial advection of angular momentum from polation between the timesteps. We note that the power spectrum neighbouring annuli, while the terms on the RHS describe the vis- peaks at a period of a few times τd. cous and magnetic torques. Note that while angular momentum As mentioned above, in order to ensure the consistency of the transport due to radial advection and the viscous torque for the model, we need to make sure that the poloidal field Bz is always whole disc reduce to the values at the inner and outer boundary,
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Variability of black hole accretion discs: The cool, thermal disc component 11 the magnetic torque produces a local angular momentum loss term 6.5 Magneticdiffusivity across the disc. Following King et al. (2004), we take the effective magnetic diffu- Given this additional term (T ) in the angular momentum, mag sivity η in eq. 49 to be equation (3) now becomes ∗ η∗ = ν max (1, QR/H) , (70) ∂ 1 ∂ 1 ∂G B 2 t ΣR2Ω + RΣv R2Ω = 2R h zi . (63) ∂t K R ∂R R K 2πR ∂R − 4π where we assume that the Prandtl number is of the order unity (η = ! νt), but allow it to be enhanced by a factor R/H (van Ballegooijen In contrast to King et al. (2004), we now take explicit account 1989; Lubow et al. 1994) and include large scale effects by mul- of the fact that the wind/jet, which removes angular momentum tiplying it by the coherency factor Q. For further discussion see from the disc, may also remove a significant amount of mass. We King et al. (2004). assume that the magnetic outflow at radius R co-rotates with the disc along each field line, out to some Alfv´en radius RA(> R). Then the local mass loss rate, LΣ, is related to the torque by 6.6 The interplay between the timescales for the flickering 2 In this section we quantify the timescale argument for the appear- 2πR∆RLΣRAΩK = Tmag, (64) ance of flickering. As mentioned previously, fluctuations propagate which implies that through the disc, if the viscous timescale is shorter than the mag- 2 2 netic timescale, i.e, the timescale where we expect sufficient align- 2 (R/RA) Bz LΣ = h i . (65) ment of the poloidal magnetic field in neighbouring cells. − Ω R 4π K ! The viscous timescale is given by (Pringle 1981) The loss term leads to a modification of the continuity equa- 1 tion (2) in the form τvisc = τdyn , (71) α (H/R)2 ∂Σ 1 ∂ + (RΣvR) = LΣ. (66) where τdyn = 1/ΩK stands for the dynamical timescale. H/R is ∂t R ∂R the opening angle of the disc. The magnetic timescale, i.e. the Eqns. 66 and 63 now can be combined using the functional timescale, on which we expect neighbouring field lines to be suffi- form of the viscous torque (eq. 4) to give ciently aligned, is given by (Livio et al. 2003) ∂Σ 3 ∂ ∂ τ = 2R/H k τ . (72) = √R ν Σ √R mag d dyn ∂t R ∂R ∂R t ! Thus the condition for propagation is 2 5 2 4 ∂ Bz R R + h i 1 (67) rτ = τmag/τvisc > 1 , (73) R ∂R 4π GM − R ! r A ! which translates into 2 2 Bz R R 2 h i . r = 2R/H k α (H/R)2 > 1 . (74) − 4π R GM τ d ! A ! r We plot the ratio rτ in Fig. 9 for different values of α. The value For R = RA the enhancement of angular momentum due to the ex- tra torque T is balanced by the loss of angular momentum taken of kd is fixed to 10 (see Sect. 6.3). The ratio is smallest for mag = away by the wind since it is taken away with the same lever arm (H/R)min 0.35. For α 0.1 fluctuations formally propagate only &≈ . = (see eq. 64) and the poloidal magnetic field influences the evolu- for H/R 0.5 or H/R 0.25 only, while for α 1 fluctuations can = tion only through the mass loss. Note that King et al. (2004) es- propagate for all H/R. In contrast to that, for α 0.01 fluctuations only propagate for H/R . 0.1. sentially assumed that RA = , and thus that mass loss is not im- portant in the model presented∞ there. However we retain the ratio This is in line with findings of Churazov et al. (2001) who state that fluctuations close to the dynamical timescale are effec- RA/R as a parameter and allow for mass loss, since analytical esti- mates (Pudritz & Norman 1986) give R /R = 10 while numerical tively damped in geometrically thin discs but can survive in geo- A metrically thick discs since then the fluctuation timescale becomes simulations (Ouyed & Pudritz 1997) suggest RA/R 1.5. Unless ≈ comparable to the viscous timescale. In our model fluctuations for- otherwise stated we choose RA/R = 3 as the fiducial value. The mass lost also removes the internal energy of the gas at a mally survive for lower H/R ratios as well. Then, however, the rate magnetic timescale is so long that alignment becomes extremely rare. From the above analysis this probably puts a lower limit on 2 Qw = LΣU LΣcs , (68) the viscosity parameter α of α & 0.1 since most of the variability ≈ is expected to come from a optically thin, but geometrically thick and thus we also require a revised energy equation (see eq. 11) disc. which is in the form
dq ∂e ∂ (2πRHvR) = e˙ + vR + 2πR∆RPH˙ + P dt ∂R ∂R (69) 7 OBSERVATIONS OF THE FLICKERING IN THE + = 2πR∆R Q + Q Q− . THERMAL COMPONENT OF THE X-RAY SPECTRUM w − We now have three equations (eqns. 67, 49 and 69) describ- Before describing the numerical simulations we first consider the ing the evolution of the disc. These are integrated in time using likely observational scenario to which they are relevant. In gener- a one-step Euler scheme. Advection is treated in a first-order, up- alising the work of King et al. (2004) to more physically realistic wind donor cell procedure. We note that the main observational discs, we have initially confined ourselves to considering the stan- output from the disc is the radiation Lrad coming from the disc (see dard accretion disc configuration, with the addition of heat advec- eq. 42). tion. In general this is assumed to correspond in X-ray binaries to c 2005 RAS, MNRAS 000, 1–18
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no variability in LMC X-3 down to a level of about 0.8 per cent. They do detect variability in LMC X-1 of 7 per cent throughout α=0.01 α=0.1 the spectral range considered (0-9 keV), but since for LMC X-1 the α=1 100 power law component is much stronger that in LMC X-3, this could just be variability of the power law component. Observations of the transient 4U 1543-47 during its 2002 out- burst are presented in (Park et al. 2004). During most of the decay
visc 10 τ
/ the spectrum is soft and dominated by black body emission. The
mag PDS shape in this phase is roughly 1/ν and the variability is about τ = τ
r r = 1 per cent. However the fraction of black body emission never 1 exceeds 80 per cent, i.e. the power law always contributes more than 20 per cent. Comparison with the results of Miyamoto et al. (1994) shows, that for a black-body flux fraction of 20 per cent, if 0.1 the relation shown there is representative, the fractional rms must be r = 1 per cent. 0.1 1 Kalemci et al. (2004) report on the light curves of a number of H/R transients observed with RXTE. They again find strong variability in the low/hard state, while in the high/soft state they only can give Figure 9. Ratio r of the magnetic and viscous timescale in dependence τ upper limits, since they are counting noise dominated. Depending of H/R (see eq. 74) for different values of the viscosity parameter α and on the source, and the quality of the data, these limits are in the kd = 10. Only for rτ > 1 fluctuations can propagate. range 1 to 8 percent. To quantify the relation between the flux fraction and the the thermally dominant (TD) or high/soft state. However, from an rms variability, we have taken data from Kalemci et al. (2004) and observational point of view, the variability in the TD or high/soft Miyamoto et al. (1994) and plot the rms versus the black body flux state is much less than in the LH state. fraction in Fig. 10. The fluxes are calculated in the 3-25 keV band. Usually the main components in the spectrum of X-Ray bina- Since Miyamoto et al. (1994) only give the normalised power den- ries can be well fit by a black-body and a power-law component sity P(ν) at frequency ν = 0.3 Hz, we estimate the integrated rms r (cf. McClintock & Remillard (2004)). The black-body or thermal by using r = 2 √νPν. The factor 2 is introduced to account for the component for galactic X-Ray binaries appears at energies around dν/ν in the definition of the integrated rms (cf. eq. 47). The black- 1 keV, while the power-law component extends to higher energies. body fraction for the LMC X-1 and X-3 data of Nowak et al. (2001) The components are thought to represent the radiation from the op- has been calculated using their black-body + powerlaw fits. While tically thick disc and a optically thin corona, respectively. Thus, there is some scatter in the data, the global trend is that for black- since in the current paper, we only attempt to describe the time- body flux fractions lower than 5 per cent the rms variability is about dependent behaviour of the thermal disc component, i.e. the opti- 25 per cent, while it drops beyond detectable limits for larger val- cally thick disc, we need to restrict ourselves to observations which ues of the black body fraction. The transition is not very sharp, but explicitly concentrate on measuring the flickering in the thermal occurs around a black body fraction in the range 3 to 20 per cent. disc component of the X-Ray spectrum. The transition might be sharper than this but we are only using data Churazov et al. (2001) show a power spectrum density (PSD) in the 3-25 keV range where the black-body contribution is small of Cyg X-1 in the high-soft state for energies of 6-13 keV, and state, compared to the power-law component, and varies from source to that the amplitude in the softer bands is much smaller due to the source. In addition, we note that Zdziarski et al. (2005) carry out an influence of the black-body emission from the disc. They argue that energy-dependent analysis of the fractional rms for GRS 1915+105 their data is consistent with essentially all of the variability being and come to similar conclusions. associated with a power-law component and not with black-body To summarise, it is clear from the observations that the rms emission (i.e. disc component), but do not quote a formal limit. variability of the black-body component in the X-Ray spectra of Miyamoto et al. (1994) examine Ginga observations of Nova black hole sources is small. Indeed there are no clear detections, Mus 1991 (GS 1124-683). They disentangle the variability in the but only upper limits. As a representative value we take r = 1 per disc and power-law component by calculating the PSD for differ- cent as an upper limit for the variability of the thermal component. ent observations which have different fractional contributions from the power-law component to the total flux. For a power-law com- ponent fraction above approximately 10 per cent, the PSD at 0.3 8 RESULTS Hz is an increasing function of power law fraction. While for lower 5 1 values it is roughly constant (1.5 10− Hz− ). At these low values, To demonstrate the effect of the stochastic magnetic wind/jet on our · the slope of the PDS is about -0.7 but due to the error bars it still disc models model with the physical input so far, we ran some mod- could be fit with -0.5. Nowak et al. (1999) show that for counting els to assess the influence of the parameters on the variability. For noise dominated time-series the expected PDS slope is 0.5. Thus all models, we fix the mass of the black hole to be 10 M , the vis- − ⊙ we conclude the data are consistent with a non-detection of flicker- cosity parameter α = 0.1 (appropriate for standard optically thick ing in the disc component. By using the value of the PDS at 0.3 Hz discs, see Lewin et al. 1997) and take for the dynamo timescale (cf. only we estimate an upper limit to the rms variability of r < 0.2 per eq. 50) kd = 10. cent. First we consider a black hole accreting at half of the Edding- Nowak et al. (2001) consider LMC X-1 and LMC X-3. These ton rate, M˙ = 0.5M˙ Edd. We take the Alfv´en radius at each radius are persistent black hole sources (i.e. not transients), and generally to be a constant value of RA/R = 3. From Figure 4, in the absence show spectra dominated by a soft, thermal component. They detect of magnetic flickering (i.e. with βS = 0) the disc is unstable and
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Variability of black hole accretion discs: The cool, thermal disc component 13
Figure 11. Sample lightcurves for a 10 M mass black hole accreting at 0.5 M˙ Edd for different values of βS, the strength of the poloidal field compared to ⊙ the disc magnetic field. The ratio of the Alfv´en radius RA to the radial distance R is RA/R = 3, the viscosity parameter α = 0.1. The intervals indicate the time-segments shown in Fig. 12 used to calculate the power spectra shown in Fig. 13.
35 strength of the variability. In Fig. 13 we show the influence of the XTE J1650-500 XTE J1550-564 in 1998 parameter βS on the power density spectra (PDS), while the inte- XTE J1550-564 in 2000 30 4U 1630-47 in 1998 grated fractional rms r (see eq. 47) is shown in Fig. 14. 4U 1630-47 in 1999 The time segments used for the calculation of the PDS are 4U 1630-47 in 2001 25 XTE J1748-288 indicated in Fig. 11. Throughout this section, we only take seg- GRO J1655-40 GS 1124-683 ments from the lightcurve, where the disc was in the high-M˙ state. 20 LMC X-1 LMC X-3 Since for red-noise the lightcurve is only a stochastic realisation of the underlying process, the calculated PDS fluctuates randomly 15 around the ”true” spectrum (e.g. Uttley et al. 2002). To overcome
r (rms/mean) [%] this difficulty, we divided each segment in 10 equally spaced sub- 10 segments, calculated the PDS of each subsegments and took the av- erage of these. We further normalised each subsegment to remove 5 linear trends in the lightcurve. We see from Fig. 14 that the rms variability is r < 1 %, for 0 1 10 100 βS < 0.2 and that r strongly increases for larger βS and reaching 10- fraction of black-body emission [%] 20 % for βS = 1. The near quadratic increase of r with βS reflects Figure 10. Fractional rms variability r for different black-body flux frac- the fact, that the poloidal magnetic field influences the disc only 2 tions in the 3-25 keV band for different objects. All data from Kalemci et al. through the torque which is proportional to Bz which in turn is β2 β2 (2004) except the data for GS 1124-683, which have been taken from proportional to SP, i.e. a fraction S of theD viscousE torque (see Miyamoto et al. (1994) and LMC X-1 and X-3 from Nowak et al. (2001). eqns. 61, 52, 51 and 22). Given the observational constraints, discussed in Section 7, it is evident that we need to take βS to be smaller than around 0.1 - shows limit-cycle behaviour. Fig. 11 shows sample lightcurves for 0.2. Thus in this model we require the energy density in the poloidal different values of βS. The limit cycle is still present, i.e. the disc field component to be at most only a few percent of the energy oscillates between a high M˙ obs and low M˙ obs state. It is best evi- density in the magnetic field generated by MRI in the disc. dent from the βS = 1 case that while in the high M˙ obs state there Next we consider the influence of the accretion rate on the is some variability, in the low M˙ obs state there is hardly any sign variability for constant βS. Fig. 15 shows the PDS for two values of of variability. This is a result of the smaller H/R ratio in the in- βS for different accretion rates. While the integrated rms variability ner disc during the low M˙ obs state. Then the alignment timescale r does not depend on the accretion rate but on α, the shape of the (proportional to the factor 2R/H times the dynamical timescale, see PDS for different accretion rates is slightly different. Livio et al. 2003) is much longer than the viscous timescale (pro- We also briefly investigate the behaviour which occurs if βS, portional to (R/H)2 times the dynamical timescale): Fluctuations the ratio of poloidal and tangled magnetic field strengths, is as- formally could survive, but since alignment occurs on extremely sumed to vary with H/R. For example, in their simple dynamo long timescales only, they are unlikely to produce flickering (see model, Tout & Pringle (1996) suggest that βS might be proportional Section 6.6 and King et al. 2004). In the high-M˙ state, however, to H/R. If so, since H/R in the simulations described is at most both timescales are at least comparable and so fluctuations propa- H/R 1/5, we can put a constraint on the constant of proportion- gate to the inner disc without being smeared out by the viscosity. ality to∼ be smaller than unity. It is evident from Fig. 11, that a non-zero βS influences the In Figures 16 and 17 we show the results of simulations with c 2005 RAS, MNRAS 000, 1–18
14 M. Mayer & J. E. Pringle
βS = 1
-2 β S=1 -3
] β 2 -4 S=0.5
-5 β S=0.25 -6 ) [(rms/mean)
ν β S=0.125
P -7
βS = 0.5 ν -8 log (
-9
-10
-1 -0.5 0 0.5 1 1.5 2 2.5 log ν [Hz]
Figure 13. Power spectra for a 10 M model solar mass black hole, accret- ⊙ ing at 0.5 M˙ Edd for different values of βS for the lightcurve segments shown βS = 0.25 in Fig. 12. α = 0.1
10
βS = 0.125
1 r (rms/mean) [%]
0.1 0.1 1 β S Figure 12. Lightcurve segments for the cases shown in Fig. 11 for dif- ferent βS. These are used to calculate the PDS in Fig. 13. M˙ obs is given Figure 14. Integrated fractional rms r for a 10 M model solar mass black ⊙ in arbitrary units, but the amplitude still has got some meaning. Note that hole, accreting at 0.5 M˙ Edd in dependence of βS, the strength of the poloidal the overall shape of the lightcurve does not change considerably, while the field compared to the disc magnetic field (cf. Fig. 13). The viscosity param- range in M˙ obs changes significantly. The longterm trend in the data has been eter is α = 0.1. removed.
netic torque dominated regime. For the parameters chosen, the in- the same parameters as before but with βS = 0.5H/R and βS = H/R tegrated rms-variability r is increasing with RA/R and saturates for varying accretion rate. For the 1 M˙ Edd cases there is a significant for RA/R > 3. The power spectral shape then is indistinguishable decline of the PDS for small frequencies. This decline is a result of within the error bars. Thus the assumption of King et al. (2004) a declining H/R in the outer disc which shortens the amplitude of who set RA/R = is justified as long as the ”real” RA/R > 3. ∞ flares released from further out (i.e. longer timescales) through the H/R-dependence of βS. For the lower accretion rates this decline is far less obvious, but then the time the disc is spending in the high- 9 CONCLUSIONS M˙ state is becoming shorter and so does the time available for the FFT and averaging. The integrated fractional rms in for the same We have generalised the model of King et al. (2004) for variability accretion rate changes by a factor of about 4 when comparing the in black hole accretion discs by including proper consideration of cases of βS = 0.5H/R and βS = H/R consistent with the result of the local disc structure. We take the disc properties to depend only Fig. 14 where we show the quadratic dependence of r with βS. on radius R, and so, in common with other authors, make a local Finally we explored the influence of different values of RA/R one-zone approximation for the disc structure. on the variability. In Figure 18 we show the influence for a 10 In the absence of the addition of a stochastic magnetic dy- M black hole accreting at 0.5 M˙ Edd with βS = 0.25. The sys- namo, which drives the flickering, we mainly reproduce results ⊙ tem changes with RA/R increasing from a mass loss to a mag- similar theoretical work has shown. We agree that taking account
c 2005 RAS, MNRAS 000, 1–18
Variability of black hole accretion discs: The cool, thermal disc component 15
-5 0.4
] -6 2
0.3 -7 ) [(rms/mean) ν -8 P
ν 0.2 r (rms/mean) [%]
log ( -9 1 2 RA/R=3 -10 4 8 -1 -0.5 0 0.5 1 1.5 2 2.5 1 2 3 4 5 6 7 8
log ν [Hz] RA/R
Figure 18. (left) Power spectra for different values of RA/R, the ratio of the Alfv´en radius RA to the radial distance R for a 10 M black hole accreting at ⊙ 0.5 M˙ Edd with βS = 0.25, the ratio of the poloidal field compared to the disc magnetic field. (right) Integrated rms-variability r as a function of RA/R. For all RA/R > 3 the power spectra and the value of r is essentially indistinguishable.
-3 -5.5 . 1 MEdd -4 -6 . β 0.5 MEdd S=0.5
] ] -6.5
2 -5 2
-7 . -6 0.2 MEdd -7.5 -7 ) [(rms/mean) ) [(rms/mean) ν ν -8 P P ν ν -8 β =0.125 S -8.5 log ( log ( . -9 1.0 M . Edd -9 0.5 M. Edd 0.2 MEdd -10 -9.5
-1 -0.5 0 0.5 1 1.5 2 2.5 -1 -0.5 0 0.5 1 1.5 2 2.5 log ν [Hz] log ν [Hz]
Figure 15. Power spectra for a 10 M model solar mass black hole, accret- Figure 16. Power spectra for a 10 M model solar mass black hole, accret- ⊙ ⊙ ing at different accretion rates for different βS = 0.125 and 0.5. α = 0.1. ing at different rates for βS = 0.5 (H/R). α = 0.1. to advective heat flow is of crucial importance in determining the the more highly variable low/soft state of the black hole binaries, structure of the inner regions, but find that the assumption of strictly as well as the X-rays from AGN (Uttley et al. 2005). Keplerian angular velocities, and the neglect of radial pressure gra- dients, are generally reasonable assumptions. When a stochastic dynamo is included we find behaviour sim- ACKNOWLEDGEMENTS ilar to that described by King et al. (2004). However, because the degree of variability of the thermal disc is observed to be small (less MM gratefully acknowledges financial support from PPARC. We than around 1 per cent), if it is detected at all, we can only draw also thank the anonymous referee for comments on the paper which limited conclusions. Our main finding is that for consistency we re- helped to improve it significantly. quire that the strength of the poloidal field generated by any radial disc dynamo cell must be at least an order of magnitude smaller than the field generated by the dynamo within the disc. The simula- REFERENCES tions reported here however are of help to understand the influence of the parameters on the variability. Abe Y., Fukazawa Y., Kubota A., Kasama D., Makishima K., , In this paper we have restricted ourselves to modelling just the 2005, Three Spectral States of the Disk X-Ray Emission of the thermal disc component, and so have had to restrict our attention to Black-Hole Candidate 4U 1630-47 the high/soft or thermally dominant state of black hole binaries. In Abramowicz M. A., Czerny B., Lasota J. P., Szuszkiewicz E., future work we shall begin to include consideration of a hot corona, 1988, ApJ, 332, 646 in addition to the cooler thermal disc, so that we can begin to model Belloni T., Hasinger G., 1990, A&A, 230, 103 c 2005 RAS, MNRAS 000, 1–18
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Variability of black hole accretion discs: The cool, thermal disc component 17 where dQ stands for the change in heat Q per unit mass, dE the internal energy E, we can express the variation in ρ in terms of change in internal energy E per unit mass, and Pd(1/ρ) the volume variations in the surface density Σ and specific internal energy E as work. P is the pressure and the specific volume is 1/ρ. The continuity equation in general form reads dρ 8 6η dΣ 4 3β dE = − − P . (A16) ρ 8 + β 7η Σ − 8 + β 7η E d (ρV) = ρdV + Vdρ = 0, (A2) P − P − With the combination of the last two equations we can eliminate which is equivalent to dρ/ρ to obtain dV dρ = . (A3) dH βP η dΣ 4 3βP dE V − ρ = − + − , (A17) H 8 + βP 7η Σ 8 + βP 7η E ff − − In total di erentials using the continuity equation, the energy 4βP 4 η dΣ 4 3βP de equation can be shown to be equivalent to = − − + − , (A18) 8 + βP 7η Σ 8 + βP 7η e 2 − −2 d (ρVQ) = d (ρVE) + PdV. (A4) 4β 11βP + 8 dΣ 3β + 10βP 8 de = P − + − P − , (A19) β2 + 13β 16 Σ β2 + 13β 16 e This relation implies that the change of total heat in a volume V is P P − P P − given by a change in the total internal energy and the expansion of (A20) the volume. where we use In cylindrical coordinates, vertically averaged, we have Egas η = , (A21) V = 2πRH∆R (A5) E Thus and Pgas d (πRΣ∆RQ) = d (πRΣ∆RE) + 2πR∆RPdH. (A6) β = , (A22) P P Replacing the specific heat Q by the total heat content q in a as the fractional contribution of the gas component to the specific semidisc annulus, internal energy and pressure, respectively. These two variables are q = πRΣ∆RQ, (A7) related by β and the specific internal energy E by the total internal energy e in a η = P . (A23) 2 β semidisc annulus, − P The specific internal energy and pressure are related by e = πRΣ∆RE , (A8) P 3 E = 3 β . (A24) we can write ρ − 2 P ! dq = de + Pd (2πRH∆R) . (A9) Equation A19, multiplied by H and taking the time-derivative leads to (using Equations (A23) and (1)) Time-dependently, the LHS of the energy equation is written 2 2 4β 11βP + 8 Σ˙ 3β + 10βP 8 e˙ dq de d (2πRH∆R) H˙ = P − + − P − (A25) = + P . (A10) β2 + 13β 16 2ρ β2 + 13β 16 2πRρ∆RE dt dt dt P P − P P − CΣ Ce Taking into account advection, with radial velocity uR, 1 The| coeffi{zcients }Ce and |CΣ are{z equal to} 2 for βP 0 and dq ∂e ∂ (2πRHuR) − → = e˙ + u + 2πR∆RPH˙ + P . (A11) β 1, while going through a minimum for β 0.75 at values dt R ∂R ∂R P P 0.38→ and 0.35, respectively. Then in both the limiting≈ cases of β P → In discredited form, 0 and β 1, we find H √e/Σ Σc2/Σ = c , recovering the P → ∝ ≈ s s dq hydrostatic equilibrium. = e˙ + ∆ (πRu ΣE) + 2πR∆RPH˙ + P∆ (2πRu H ) , (A12) p dt R R ad Using Equation (A25) in the energy equation (A12), we can write where Had is the advected value of H, and this term corresponds to the advected volume contributing to the ’PdV’ work. dq = e˙ + ∆ (πRuRΣE) + 2πR∆RPH˙ + P∆ (2πRuR Had) Now, given the specific internal energy E, where dt P P 3 kT 4σ = e˙ 1 + C + ∆ (πRu ΣE) + πR∆R C Σ+˙ 2πRP∆ (u H ) E = + T 4, (A13) e ρE R ρ Σ R ad 2 µmp cρ ! P 1 P 1 1 = e˙ 1 + C + ∆ ME˙ + πR∆R C Σ+˙ P∆ M˙ , and given hydrostatic equilibrium in the form e ρE 2 ad ρ Σ 2 ρ ! ! ad ! kT 4σ 4 M 2 ρ + = Σ , where we applied M˙ = 2πΣRuR in the last step. T G 3 (A14) µmp 3c 4ρR ρ and T can now be be computed, for given values of E and Σ. The variation of H in terms of E and Σ is given by using the definition APPENDIX B: LOCAL STABILITY ANALYSIS of the surface density (Equation 1) The thermal stability analysis (Pringle 1976), now including advec- dH dΣ dρ = . (A15) tion shows thermal instability for Σ H − ρ ∂ log Q+ ∂ log (Q + Q ) − ad > 0. (B1) Using both the hydrostatic equilibrium, and the definition of the ∂ log T − ∂ log T !P !P c 2005 RAS, MNRAS 000, 1–18
18 M. Mayer & J. E. Pringle
Thermal instability occurs, if the increase(decrease) of temper- where we used the variation of the energy equation and expressed ature compared to the equilibrium state leads to a stronger in- the density variation dρ/ρ in terms of temperature dT/T using the crease(decrease) of the heating compared to the cooling while variation of the hydrostatic equilibrium while keeping the surface keeping hydrostatic equilibrium. density Σ=const. + Using the equilibrium condition (Q = Q− + Qad) and We have viscous instability for
Qad CE BF AF CD ǫad = , (B2) (βP 1) − (4 3βP) − + 1 < 0 (B17) Qad + Q − − AE BD − − AE BD − − − we get the condition where we both expressed the density and temperature variations in terms of the surface density variations and use this expression in ∂ log Q+ ∂ log Q ∂ log Q ǫ ad (1 ǫ) − > 0. (B3) the angular momentum equation ∂ log T P − ∂ log T P − − ∂ log T P ! ! ! M˙ 3πν Σ= 0 (B18) Along similar lines, viscous instability (e.g. − t Lightman & Eardley 1974; Pringle 1981) only occurs as long to get the instability criterion. as In the radiation pressure (βP = 0) and Thomson scattering ∂ log M˙ (AR = BR = 0) dominated regime we get thermal instability only if < 0. (B4) ∂ log Σ !P,T 4 12ǫ > 0 (B19) The disc is viscously unstable, if an increase(decrease) of accretion − ad rate does lead to a lower/higher surface density. and viscous instability as long as For the calculation of the criterion we use the hydrostatic equi- 7 (ǫ 1) librium ad − < 0 (B20) 1 3ǫ GM − ad log P log Σ2 = f (ρ, T, Σ) = 0, (B5) − 4ρR3 1 Since the nominator of the LHS of the last expression is always negative (0 < ǫad < 1), the condition of thermal instability and and the stationary energy equation viscous instability are the same. + To conclude, thermal and viscous instability only occurs, if log Q log Q− + Qad = f2(ρ, T, Σ) = 0 . (B6) − 1 ǫad < 3 . A radiation pressure dominated disc can be thermally and With the contribution of gas and radiation to the total pressure viscously unstable as long as advection only contributes less than (see eq. 16) we can write the total variation of the hydrostatic equi- one third of the local energy loss. This clearly shows the stabilising librium and energy equation in terms of ρ, T and Σ, generalising the effect of energy advection on radiation pressure dominated accre- method of Mayer & Duschl (2005) for a non selfgravitating, opti- tion discs. cally thick accretion disc. We neglect changes in χad since χad is 5 For bound-free and free-free absorption (AR = 1, BR = 2 ), expected to vary only very little. it can be shown that in this case the disc is stable with respec−t to We take the variation of the hydrostatic equilibrium ( f ) 1 thermal and viscous instability for all βP. dρ dT dΣ It needs to be stated that these results are only based on a lo- A + B + C = 0 (B7) ρ T Σ cal stability analysis and it still needs to be shown by either time- dependent simulations or a global stability analysis when these un- and the stationary energy equation ( f ) 2 stable modes are operational. dρ dT dΣ D + E + F = 0 (B8) ρ T Σ with respect to ρ, T and Σ. The coefficients A ... E are ∂ f A = 1 = 1 + β (B9) ∂ log ρ P ∂ f B = 1 = 4 3β (B10) ∂ log T − P ∂ f C = 1 = 2 (B11) ∂ log Σ − ∂ f D = 2 = A (1 ǫ ) + (β 1)(1 2ǫ ) (B12) ∂ log ρ R − ad P − − ad ∂ f E = 2 = (B 4)(1 ǫ ) + (4 3β )(1 2ǫ ) (B13) ∂ log T R − − ad − P − ad ∂ f F = 2 = 2 (1 ǫ ) (B14) ∂ log Σ − ad with ∂ log κ (A , B ) = R . (B15) R R ∂ log (ρ, T) Thermal instability exists, if AE BD > 0 (B16) − c 2005 RAS, MNRAS 000, 1–18