Vertical Convection in Turbulent Accretion Disks and Light Curves of the A0620-00 1975 Outburst
aa r X i v : . [ a s t r o - ph . H E ] N ov Vertical Convection in Turbulent Accretion Disks andLight Curves of the A0620–00 1975 Outburst
K. L. Malanchev , ∗ , N. I. Shakura Sternberg Astronomical Institute, M. V. Lomonosov Moscow State University, Moscow, Russia Faculty of Physics, M. V. Lomonosov Moscow State University, Moscow, Russia
We present a model of the non-stationary α -disk with account for the irradiation and the verticalconvection in the outer accretion disk where hydrogen is partially ionized. We include the viscousenergy generation in the mix-length convection equations in accretion disks. The optical and X-ray light curves of X-ray nova A0620–00 are investigated in terms of this model. The turbulentviscosity parameter of the accretion disk is estimated, α = 0 . ÷ . , which is necessary to explainthe luminosity decay rate on the descending branch of the X-ray light curve for the A0620–00 1975outburst. The secondary luminosity maximum on the light curves is explained by assuming anadditional injection of matter into the accretion disk from the optical companion. Keywords:
X-ray sources, accretion, black holes.
Astronomy Letters,
Vol. 41, No. 12, 2015.
DOI: ∗ E-mail: < [email protected] > K. L. Malanchev & N. I. Shakura
1. Introduction
X-ray novae are close binary systems with a relativistic object (a black hole or a neutronstar) and a low-mass Roche-lobe-filling star (see, e.g. Cherepashchuk (2000); Postnov & Yungelson(2014)). The time evolution of an accretion disk may be considered in terms of a linear problem,where the kinematic viscosity in the disk does not depend on its surface density Σ . An analyticsolution for the disk evolution in this case was obtained by Lipunova (2015). The standard diskaccretion model developed more than forty years ago (Shakura, 1972; Shakura & Sunyaev, 1973)implies that the kinematic viscosity depends on the surface density, and, hence, the accretion diskevolution problem is nonlinear. For an α -disk, analytic solutions were found for various stages ofaccretion disk evolution (Lyubarskij & Shakura, 1987; Lipunova & Shakura, 2000).Analytic solutions of the nonlinear disk accretion problem are possible only in the casewhere an analytic relation exists between the accretion disk viscosity and surface density (forexample, using the solution of the vertical disk structure equations; see Ketsaris & Shakura (1998);Lipunova & Shakura (2000)). In the remaining cases, the vertical structure equations and theviscous accretion disk evolution equations should be solved numerically. The numerical simulationsof light curves by taking into account the thermal instability, convection, and self-irradiation inthe case of dwarf novae, where matter is accreted onto a black hole (e.g. Hameury et al., 1998),and X-ray novae (e.g. Dubus et al., 2001; Wisniewicz et al., 2015) were considered in a number ofpapers. The main difference of the present work from the previous papers is a study of an impactof the turbulence on the vertical convection in the outer disk. We describe the outer disk verticalstructure using the mixing length theory in the Appendix.The X-ray nova A0620–00 1975 outburst is one of the brightest X-ray novae and has beenstudied thoroughly. The X-ray curve of this outburst rises on scales of several days and thendecays quasiexponentially on scales of tens of days. In this paper, we consider only the descendingbranches of the Xray and optical light curves for the X-ray nova A0620–00.The X-ray and optical light curves of X-ray nova A0620–00 within the first 50 days after itspeak were jointly modeled in Suleimanov et al. (2008), where α & . were obtained. In this paper,we model the light curves of this nova within the first 120 days after its peak. A characteristicfeature of the light curves for A0620–00 is the secondary brightness peak observed in both X-rayand optical bands approximately on the 55th day after the primary maximum. The nature of this ertical Convection in Turbulent Accretion Disks 3 secondary maximum remains unclear and can be explained by a host of various physical processes.One of the explanations can be the radiative heating of the hitherto cold outer disk (King & Ritter,1998; Ertan & Alpar, 2002). Another suggestion is the evaporation of matter in the central hotdisk regions and their subsequent filling with material (Cannizzo, 2000). Lipunova and Shakurain their 2001 and 2003 papers associate the appearance of a secondary peak with the onset ofconvection in the outer accretion disk, which can effectively increase the viscous friction. In thispaper, we use the assumption about an additional injection of mass into the disk from the opticalcompanion. Such an ejection can be explained both by the irradiation of the optical star by X-raysfrom the accretion disk and by internal instabilities in its convective envelope.
2. Viscous evolution of an accretion disk
We will consider a thin accretion disk symmetric relative to the vertical axis and the midplanein which the velocities of the accreting material depend only on its distance r to the disk center.In such a disk, the continuity equation integrated along the vertical axis is ∂ Σ ∂t = − r ∂∂r (Σ v r r ) , (1)where Σ = R ∞−∞ ρ d z is the surface density of the ring of material at radius r , and v r is the radialvelocity of the material in the disk.The angular momentum transport equation can be written as Σ v r ∂ ( ωr ) ∂r = 1 r ∂∂r ( W rϕ r ) , (2)where ω = p GM x /r is the angular velocity of the accreting material in the disk, G is thegravitation constant, M x is a black hole mass, and W rϕ = R ∞−∞ w rϕ d z is the rφ -component ofthe viscous stress tensor integrated along the vertical axis. Here, we neglect the tidal torques byassuming them to be negligible in the entire disk, except for its outermost ring (Ichikawa & Osaki,1994).Let us introduce the viscous torque acting between adjacent layers F : F = − πW rϕ r . (3) K. L. Malanchev & N. I. Shakura
Substituting (2) and (3) into (1) and using the specific angular momentum h = √ GM x r asa new radial coordinate, we will obtain a diffusion-type differential equation: ∂ Σ ∂t = 14 π ( GM x ) h ∂ F∂h . (4)The initial and boundary conditions should be specified to solve this equation. The material fromthe inner disk edge falls without viscosity, moving in the region of unstable circular orbits towardthe black-hole event horizon in the dynamical time. On this basis, we will set the following innerboundary condition on the viscous torque: F | r = r in = 0 , (5)where (Kato, Fukue & Mineshige, 2008): r in = GM x c (3 + Z − p (3 − Z )(3 + Z + 2 Z )) (6)is the radius of the innermost stable circular orbit ,Z = 1 + √ − a ( √ a + √ − a ) , (7) Z = q a + Z , (8) a is the Kerr parameter of the black hole , (9) c is the speed of light . (10)Note that π Σ ( r ) v r ( r ) r is equal to the accretion rate ˙ M ( r, t ) . Expressing the left-hand sideof Eq. (2) in terms of the accretion rate, we will obtain the relation between the accretion rate andthe viscous torque: ˙ M ( h ) = − ∂F∂h (11)At the outer disk edge, where there are no inflow and outflow of matter, ∂F∂h (cid:12)(cid:12)(cid:12)(cid:12) h out = 0 , (12)where h out = √ GM x r out is the specific angular momentum at the outer disk radius. The outerdisk radius is determined by the tidal radius, which we assume to be 80% of the effective Rochelobe radius for a black hole (Eggleton, 1983; Paczynski, 1977; Suleimanov et al., 2008): r out = 0 . a . M x /M opt ) / . M x /M opt ) / + ln (1 + ( M x /M opt ) / ) , (13) ertical Convection in Turbulent Accretion Disks 5 where ˜ a is the semimajor axis of the binary system, and M opt is the mass of the optical companion.We choose the initial condition in a form satisfying the boundary conditions: F ( h ) | t =0 = 2 π ( h out − h in ) ˙ M × sin (cid:18) π h − h in h out − h in (cid:19) , (14)where ˙ M ≡ ˙ M ( h in ) | t =0 is the accretion rate onto the black hole at the peak of X-ray luminosity.The relation between the two unknown function Σ ( h, t ) and F ( h, t ) should be establishedto solve the system of equations (5), (12) and (14); it can be found from the solution of thevertical structure equation for the accretion disk (see the next Section). This system was solvednumerically using the implicit method that guaranteed the stability of the solution. The solutionwas performed on a logarithmic spatial grid in h with a crowding near h out consisting of 400 gridpoints. The time step was constant and equal to 0.2 day. This exceeds the thermal time in theouter disk and allows us to disregard the effects associated with the fact that the radiation andmatter in the disk are nonequilibrium ones during its abrupt cooling.
3. Vertical structure
We perform an independent calculation of the vertical disk structure at various radii. Thecalculation is performed in the approximation of hydrostatic equilibrium and the Eddington ap-proximation for radiative transfer. For the convenience of our calculations, the vertical coordinate ˜ z in the disk is measured from the photosphere toward the disk midplane. In view of the disksymmetry relative to the midplane, the vertical structure equations are solved only between thepoints with vertical coordinates ˜ z = 0 (optical disk photosphere) and ˜ z = z (disk midplane). Thevalue of z corresponds to the half-thickness of a standard accretion disk. The hydrostatic equilibrium equation is d P d˜ z = ρg z , (15)where P is the pressure, ρ is the density, g z = GM x /r × ( z − ˜ z ) /r is the vertical component ofgravity. We neglect the radiation pressure in our calculations. It is convenient to express g z in K. L. Malanchev & N. I. Shakura (15) in terms of the angular velocity of the accreting material ω = p GM x /r : d P d˜ z = ρω ( z − ˜ z ) . (16)Let us write the derivative of the pressure with respect to the optical depth τ measured along thevertical ˜ z axis: d P d τ = ω ( z − ˜ z ) κ , (17)where κ is the Rosseland opacity. We use the tabulated opacities from the OPAL project(Iglesias & Rogers, 1996) and Ferguson et al. (2005) for solar chemical composition (Asplund et al.,2009). The pressure at the level of the accretion disk photosphere where the optical depth τ = 2 / is estimated to be P | ˜ z =0 = 23 ω z κ . (18) Neglecting the surface density of the disk layers above the optical photosphere, we have ∂ Σ ∂ ˜ z = 2 ρ, (19)where Σ is twice the surface density of the disk layer between its photosphere and the currentvertical coordinate ˜ z . The boundary condition for Eq. (19) can be written as Σ | ˜ z =0 = 0 . (20)By integrating system (19) and (20), we will obtain the total surface density Σ : Σ = Σ | ˜ z = z . (21)Note that the latter relation is not a boundary condition, because Σ is assumed to be unknownwhen integrating the vertical structure equations. In the α -disk model, the generation of energy per unit time per unit volume by viscous forcesis w rϕ r d ω/ d r = αP ω (Shakura & Sunyaev, 1973). In the outer accretion disk regions, the rate ertical Convection in Turbulent Accretion Disks 7 z o z atm Black hole Xrays DiskAtmosphere θ Figure 1:
Schematic view of an accretion disk with an atmosphere and X-ray irradiation. of energy release through the thermalization of X-ray radiation ǫ x (˜ z ) arrived from its inner partsand incident on the surface is a significant source of energy: d Q d˜ z = − αP ω − ǫ x (˜ z ) . (22)Given (3), the total energy flux being released in the disk through turbulent viscosity Q vis is Q vis = Z + ∞−∞ w rϕ r d ω d r d z = − W rϕ ω = 38 π F ωr . (23)To calculate the flux of the X-ray radiation, we used a two-component disk model (see Fig.1). In this model, there is a hot atmosphere above the optical disk photosphere that scattersand absorbs some of the X-ray photons. The disk below the optical photosphere is assumed tobe cold relative to the X-ray radiation and the absorption in it dominates over the scattering.The disk under consideration is optically thick to the X-ray irradiation, and the energy releasenear its midplane is determined exclusively by viscosity. This allows the model of a semi-infiniteplane-parallel layer to be used in writing the X-ray radiation transfer equations.Let us write the X-ray flux Q x incident on the hot atmosphere by assuming its boundary tobe located at height z atm above the disk midplane (i. e. ˜ z = z − z atm ): Q x | ˜ z = z − z atm = L x πr Ψ( z atm /r ) sin θ, (24)where L x = ˜ η ˙ M in is the X-ray luminosity of the disk, ˜ η is the accretion efficiency, Ψ( z atm /r ) isthe transfer function of the inner disk for a remote observer, and θ is the angle between the X-ray K. L. Malanchev & N. I. Shakura photon propagation direction and the surface of the atmosphere. We used the transfer functionfrom Suleimanov et al. (2008). For a Schwarzschild black hole and a relation z atm /r = 0 . that istypical for the outer disk, the value of Ψ( z atm /r ) is about 0.35.The angle θ in the thin-disk approximation can be written as θ ≃ d z atm d r − z atm r . (25)According to numerical models for the atmospheres of accretion disks in X-ray binaries (e.g.Jimenez-Garate et al., 2002), to a first approximation, it can be assumed that z atm = k z z inthe outer disk, where the irradiation is significant. The expression for θ will then take the form θ = k z (cid:18) d z d r − z r (cid:19) = k z z r (cid:18) d ln z d ln r − (cid:19) . (26)Note that, formally, θ can take on negative values in the outer disk when convection sets in. In ourcalculations, we proceeded from the fact that the change in the shape of the photosphere at theonset of convection in the disk did not affect the shape of the surface of the atmosphere and useda constant value for d ln z / d ln r = 0 . , which we obtained for the outer disk at temperaturesabove 10 000 K.According to Mescheryakov et al. (2011a), the X-ray flux at the level of the optical diskphotosphere can be written as Q x | ˜ z =0 = λ atm e − k atm τ atmx Q x | ˜ z = z − z atm (27)where λ atm ≡ κ T / κ atmx is is the ratio of the Thomson opacity to the total opacity for X-ray photonsin the atmosphere, τ atmx is the optical depth of the atmosphere along the ˜ z axis for X-rays, and k atm ≡ p − λ atm ) .Let us designate the attenuation of the X-ray flux in the atmosphere as (1 − ˜ A ) : − ˜ A ≡ λ atm e − k atm τ atmx . (28)Using this designation, we will write the X-ray flux under the optical photosphere(Mescheryakov et al., 2011a) as Q x (˜ z ) | ˜ z ≥ = (1 − ˜ A ) L x πr Ψ( z atm /r ) sin θ e − k d κ x Σ(˜ z ) , (29)where κ x is the opacity for X-rays in a cold medium, was chosen to be 5.7 cm g − , correspondingto the opacity for photons with an energy of 3 keV in cold matter of solar chemical composition ertical Convection in Turbulent Accretion Disks 9 (Balucinska-Church & McCammon, 1992), λ d = 0 (the Thomson opacity is much smaller than theopacity for absorption), k d = p − λ d ) = √ .Having differentiated (29) with respect to ˜ z , we obtain the final expression for the rate ofenergy release per unit volume ǫ x through the thermalization of X-ray radiation: ǫ x (˜ z ) = k L x πr Ψ( z atm /r ) z r (cid:18) d ln z d ln r − (cid:19) k d κ x ρ e − k d κ x Σ(˜ z ) , (30)where k ≡ (1 − ˜ A ) k z is the unknown coefficient associated with the atmospheric structure, whichcan be determined by modeling the optical light curve of the X-ray nova.Using (23) и (27), we will write the total flux of thermal radiation through the disk photo-sphere as Q | ˜ z =0 = 38 π F ωr + k L x πr Ψ( z atm /r ) z r (cid:18) d ln z d ln r − (cid:19) . (31)In view of the disk symmetry relative to the midplane, the boundary condition for the energyflux in the plane of disk symmetry is Q | ˜ z = z = 0 . (32) The equation of energy transfer by radiation Q rad in the diffusion approximation can bewritten as c κ ρ d a r T d˜ z = Q rad , (33)where a r is the radiation constant, T is the temperature of the accreting matter.At a temperature below 10 000 K, hydrogen becomes partially ionized and the opacity increasesharply, which gives rise to a sharp increase in the vertical temperature gradient and convection.An expression for d T / d˜ z n this case was derived in the Appendix (A.37) in terms of themixing-length theory.Thus, the vertical temperature gradient will be defined by Eq. (A.37) when the criterion forthe onset of convection (A.8) is met and by Eq. (33) in the opposite case: d T d˜ z = κ ρ a r cT Q, ∇ r < ∇ ad ; g z ρTP ∇ , ∇ r ≥ ∇ ad . (34) The boundary condition for the temperature at the level of the disk photosphere is determinedby the effective temperature: T | ˜ z =0 = s Q | ˜ z =0 a r c . (35) Equations (16), (19), (22), and (34) together with the boundary conditions (18), (20), (31),(32), and (35) form a system of four linear differential equations with five boundary conditions(four are specified on the disk surface and one is specified in the plane of symmetry) and oneunknown, z . The disk halfthickness z is chosen in such a way that when this system of equationsis integrated from point ˜ z = 0 , at which the boundary conditions are specified for all unknownfunctions P (˜ z ) , Σ(˜ z ) , Q (˜ z ) , and T (˜ z ) , to point ˜ z = z , the boundary condition Q | z = z = 0 is met.As a result, for given radius, viscous torque, and X-ray flux incident on the disk photosphere,we can obtain the vertical distributions of all thermodynamic quantities and the value of the disksurface density. The derived surface density profile for a set of accretion disk rings can be used tosolve the viscous disk evolution Eq. (4). To optimize the time it takes to model the light curves,we used the table of integral disk parameters ( Σ , z , etc.) as a function of three parameters:the specific angular momentum h , the viscous torque F , and the ratio of the total flux from thedisk surface to the viscous flux f Q = Q | ˜ z =0 /Q vis . The table was compiled for the logarithms of h , F , and f Q and had the dimensions ( × × ). The table was filled with the values, asnecessary, when numerically solving Eg. (4); the final filling with the elements does not exceed 1%of the number of cells in the table.
4. Modeling the light curves of X-ray nova A0620–00
To test the described numerical model, we chose the well-known X-ray nova A0620–00. Theadvantage of this system is that the orbital parameters needed in our modeling are well known forit and there are both X-ray and optical observations of its 1975 outburst (Cherepashchuk, 2000;Chen et al., 1997).The observed light curves of this nova contain a secondary maximum on their exponentialbranches. To explain this phenomenon, we use the hypothesis about an additional injection of ertical Convection in Turbulent Accretion Disks 11 matter from the secondary component into the accretion disk (Lipunova, 2015). For this purpose,we instantaneously increase the surface density at radii r ≥ . r out on the 43th day after theprimary luminosity maximum. When the effective temperature of the outer accretion disk dropsbelow 10 000 K, it becomes important to take into account the thermal instability. The latterenables a rapid transition of the disk ring to the cold state, whereby the disk hydrogen becomesneutral (e.g. Hoshi, 1979; Meyer & Meyer-Hofmeister, 1981; Smak, 1982; Cannizzo et al., 1988).By the time of the transition to the cold state, the effective temperature is ÷ K andthe mass fraction of matter involved in convection at the radius under consideration is more than80%. During our numerical simulations, we assumed that the outer radius of the hot disk wasshifted inward together with the thermal instability zone and that the accretion in the outer colddisk stopped completely (Lipunova & Shakura, 2003).In our modeling, we used the following parameters of the binary system: the mass of theoptical companion M opt = 0 . M ⊙ , the orbital period P = 0 . day, and the Kerr parameter ofthe black hole a = 0 . (Chen et al., 1997; Cherepashchuk, 2000; Gou et al., 2010). We performedour modeling for two black hole masses: M ⊙ (Gelino et al., 2001) and . M ⊙ (Cantrell et al.,2010); orbital inclinations of ◦ and ◦ , respectively, at the system’s known mass function of . M ⊙ correspond to these parameters. The modeling results for the black hole mass of M ⊙ are presented in Figs. 2 and 3; the parameters of both models are given in Table 1. To achieveagreement between the model and observations, the values for two model parameters were chosen inthe segment of the X-ray light curve before the secondary peak: the turbulent viscosity parameter α and the accretion rate at maximum light ˙ M in units of ( L Edd /c ) , where L Edd is the Eddingtonluminosity for a black hole of the corresponding mass. To explain the secondary peak on the X-raylight curve, we chose the amount of matter added to the outer disk on the 43rd day after thelight-curve maximum δM in fractions of the then available mass in the disk. At the same time,the optical light curve (Fig. 2) exhibits an abrupt rise in the luminosity due to the assumptionabout an instantaneous increase in the mass.Once these parameters were established, we chose the parameter of the disk atmospherek to explain the B light curve. Just as in Suleimanov et al. (2008) we used parameter k z ≡ z atm /z = 2 in modeling A0620–00. Note that Mescheryakov et al. (2011b) derived the samerelation between the thickness of the atmosphere and the standard disk thickness at the outerradius from observations of the low-mass X-ray binary GS 1826–238. Using this value of k z , we Parameter M x = 6 . M ⊙ M x = 12 M ⊙ Distance to binary system, kpc 0.85 1.0Turbulent viscosity parameter α ˙ M , L edd /c δM on 43th day 0.3 0.25Atmospheric irradiation interception parameter k Parameters of the models for two different black hole masses obtained for the light curves ofthe 1975 outburst of A0620–00 M agn i t ude , B band Days after the peak Lloyd et al., BModel
Figure 2:
Optical light curve of X-ray nova A0620–00. The optical observations (Lloyd et al., 1977) areshown; the solid line indicates our model for the black hole mass of M ⊙ . can obtain ˜ A = 1 − k/k z , the fraction of the radiation reflected or absorbed in the disk atmosphere,which is 0.3 and 0.6 for the models with the black hole masses of . M ⊙ and M ⊙ , respectively.Assuming that the optical depth of the atmosphere for X-rays τ atmx is greater than unity, we canplace a constraint on the ratio of the scattering coefficient to the total absorption coefficient inthe atmosphere from (28): λ atm > . and λ atm > . for the black hole masses of . M ⊙ and M ⊙ , respectively. ertical Convection in Turbulent Accretion Disks 13 −9 −8 −7 −6
0 20 40 60 80 100 120 F l u x ( e r g / s / c m , − k e V ) Days after the peak Ariel 5, 3−6 keVModel
Figure 3:
Soft X-ray (3—6 keV) light curve of X-ray nova A0620–00. The Ariel 5 observations(Kaluzienski et al., 1977) are shown; the solid line indicates our model for the black hole mass of M ⊙ . The optical flux deficiency on the model light curve (Fig. 2) after the 100th day can beexplained by the appearance of an additional source of optical radiation. The cold parts of thedisk irradiated by scattered X-ray radiation in the atmosphere, whose area increases with timebut whose radiation is disregarded when constructing the light curve, can be such a source. TheX-ray-heated surface of the optical companion can be another possible source. As the radius ofthe hot disk decreases, the ratio z /r (Fig. 1) and, as a consequence, the size of the shadow castby the X-ray radiation from the outer disk decrease, increasing the area of the irradiated surfaceof the star.
5. Conclusions
The light curves of X-ray nova A0620–00 after its 1975 outburst were modeled in termsof the standard nonstationary α -disk model. The descending branches of the optical and X-ray light curves, including the secondary luminosity maximum, were investigated. We showedthat this secondary maximum could be explained by an additional injection of matter into theaccretion disk from the donor star. At the early outburst stage, the disk is in a hot state with its cmF = 1.15 x 10 dyn cm S = 36.4 g/cm f Q = 2.0 S / S (cid:209)(cid:209) ad (cid:209) r T / 10 K 0.1 1 10 0 0.2 0.4 0.6 0.8 1r = 1.21 x 10 cmF = 1.15 x 10 dyn cm S = 29.3 g/cm f Q =2.5 S / S (cid:209)(cid:209) ad (cid:209) r T / 10 K Figure 4:
Vertical distributions of the temperature and three logarithmic gradients, the mean ∇ , theadiabatic ∇ ad , and the fictitious radiative ∇ r . Convection takes place under the condition ∇ r > ∇ ad .The mass coordinate Σ / Σ , which is zero at the level of the optical photosphere and one in the plane ofsymmetry, is along the horizontal axis. The distributions are presented for the model with the black holemass of M ⊙ on the 80th day after the light-curve maximum. The upper and lower panels correspondto radii of . × cm and . × cm (the outer radius of the hot disk), respectively. outer radius determined according to Paczynski (1977). After the secondary luminosity maximum,approximately on the 60th day after the primary maximum, a zone with partially ionized hydrogenappears in the outer disk, which is shifted toward the disk center with time. A cold disk in whichhydrogen is completely neutral and the accretion is very slow is formed behind the front of partiallyionized hydrogen.Vertical convection and irradiation in the outer accretion disk affect significantly its vertical ertical Convection in Turbulent Accretion Disks 15 structure. Figure 4 shows the vertical distributions of the temperature and various logarithmictemperature gradients (see the Appendix) on the 80th day after the primary luminosity maximumfor the model with a black hole mass of M ⊙ . At this time, the outer radius of the hot disk is . × cm, while the radius of the cold disk is . × cm. The lower graph in Fig. 4 showsthe vertical structure at the outer radius of the hot disk, where hydrogen at the photosphericlevel is already neutral and convection affects ≃ of the disk mass fraction. The upper graphdemonstrates the structure at a smaller radius, where the disk temperature is considerably higherand convection affects only about a third of the matter in the upper disk layers. Nevertheless,there is no convection near the photosphere due to the decrease in the vertical temperature gradient d T / d˜ z associated with the thermalization of X-ray radiation incident on the photosphere. Acknowledgments
We thank G.V. Lipunova, A.V.Mescheryakov, and K.A. Postnov for fruitful discussions. Thiswork was supported by the Russian Science Foundation grant 14-12-00146. We used the equipmentfunded by M.V. Lomonosov Moscow State University Program of Development.
Appendix: The mixing-length theory of convection
In this paper, the mixing-length theory of convection is used to calculate the convectivezones of an accretion disk. Here, we will present only the key points of this theory by consideringthe aspects of its application for accretion disks in more detail (for a thorough presentation ofthe mixing-length theory of convection, see, e.g., the book Weiss, Hillebrandt, Thomas & Ritter(2004)).Generally, the energy flux through a surface of unit area along the ˜ z axis is the sum of theradiative and convective fluxes: Q = Q rad + Q conv . (A.1)An expression for the energy flux transferred by radiation can be derived in the diffusionapproximation: Q rad = 4 a r cT κ ρ d T d˜ z , (A.2)where a r is the radiation constant. As long as the vertical temperature gradient d T / d˜ z is small, there is no convection and Q conv = 0 . According to the Schwarzschild criterion, convection sets in when the temperaturegradient (A.2) derived from Eq. (A.2) exceeds the adiabatic gradient: (cid:18) d T d˜ z (cid:19) rad > (cid:18) d T d˜ z (cid:19) ad . (A.3)For the convenience of our consideration of convection, we will introduce four logarithmic deriva-tives of the temperature with respect to the pressure, also called the logarithmic temperaturegradients: ∇ ≡ d ln T / d ln P , the logarithmic gradient calculated for the mean parameters of themedium at a given ˜ z ; ∇ ′ , the mean gradient for the rising convective cells (i.e., those movingin a direction away from the disk midplane toward its photosphere); ∇ r , the fictitious radiativegradient, i.e., the gradient that would be without convection; ∇ ad , the adiabatic gradient that isa function of the thermodynamic state of matter. Figure 4 shows the logarithmic gradients asfunctions of the mass coordinate Σ / Σ at two different accretion disk radii on the 80th day afterthe primary luminosity maximum.For a gas of pure hydrogen, the adiabatic gradient ∇ ad can be written as (Weiss et al., 2004) ∇ ad ≡ (cid:18) d ln T d ln P (cid:19) ad = 2 + i (1 − i ) (cid:0) + Ry kT (cid:1) i (1 − i ) (cid:0) + Ry kT (cid:1) , (A.4)where Ry = 2 . × − erg is the Rydberg constant, i is the degree of hydrogen ionization (theratio of the number of hydrogen ions to the total number of hydrogen ions and atoms) derivedfrom the Saha equation (Landau & Lifshitz, 1980): i = 1 √ P K P , (A.5) K P is the chemical equilibrium constant: K P ≡ (cid:18) π ~ m e (cid:19) / (cid:18) k B T (cid:19) / exp (cid:18) Ry k B T (cid:19) . (A.6)By the definition, the fictitious radiative gradient ∇ r can be expressed from (A.2) by substi-tuting the total flux Q for the radiative one Q rad : ∇ r ≡ κ ρP a r cT (cid:18) d P d˜ z (cid:19) − Q. (A.7)The Schwarzschild condition (A.3) can now be rewritten using the logarithmic gradients: ∇ r > ∇ ad . (A.8) ertical Convection in Turbulent Accretion Disks 17 We will consider the case where the Schwarzschild condition holds.Convective cells transfer heat due to a temperature difference between their contents andthe ambient medium. Having traversed some distance, a convective cell dissolves in the ambientmedium and transfers its heat to the latter. Therefore, let us write the heat flux transferred byconvection by assuming the pressure in the convective cells to coincide with the pressure in themedium: Q conv = 12 ρvC P Λ (cid:20) d T d˜ z − (cid:18) d T ′ d˜ z (cid:19)(cid:21) = 12 ρvC P T Λ d P d˜ z ( ∇ − ∇ ′ ) , (A.9)where T ′ is the temperature in the rising convective cell, Λ is the mixing length, i.e., the mean freepath of a convective element, v is the mean velocity of the convective elements, C P = R µ (cid:20)
52 + 12 i (1 − i ) (cid:18)
52 +
Ryk B T (cid:19)(cid:21) (A.10)is the specific heat of the gas at constant pressure ,µ = 11 + i is the molar mass , (A.11) R is the universal gas constant. The factor / in (A.9) appears because of the assumption thatapproximately half of the matter moves in the convective cells upward and the second half movesdownward.We assume that the mixing length Λ = βz , where β is a constant in space and time, equalto . in our calculations.To find the mean velocity v , we will use the following considerations. Let us write the specificforce acting on the rising element of matter: ρ d δz d t = ρ d v d t = − g z ( ρ + ∆ ρ ) + d P d˜ z = g z ∆ ρ, (A.12)where δz is the distance traversed by the convective element after its formation, and ∆ ρ is thedifference between the density of the convective element and the mean density at a given ˜ z .Neglecting the change in gravity with height, let us write the specific work done by this forceto move the element through its free path ∆ z : W (∆ z ) = − Z ∆ z g z ∆ ρ ( δz ) dδz = − g z ∆ ρ (∆ z )∆ z. (A.13)Averaging over ∆ z gives the mean work done to move the element through the mean mixinglength Λ : W (Λ) = 14 W (Λ) = − g z ∆ ρ (Λ)Λ , (A.14) where the factor / is related to the variable cell velocity and variable vertical gravity g z (Weiss et al., 2004). Assuming that half of this work is converted into kinetic energy and thesecond half goes into heating due to the viscous friction of elements against one another, we willobtain ρv = 12 W (Λ) = − g z ∆ ρ (Λ)Λ . (A.15)The energy release through the viscous friction of convective elements is much smaller than theturbulent one; therefore, it does not contribute to the energy equation(22).Let us write the expression for ∆ ln ρ provided that the pressures in all convective elementsat the same ˜ z are equal: ∆ ln ρ = "(cid:18) ∂ ln ρ∂ ln µ (cid:19) P,T (cid:18) ∂ ln µ∂ ln T (cid:19) P + (cid:18) ∂ ln ρ∂ ln T (cid:19) µ,P ∆ ln T. (A.16)Let us introduce a quantity Q : Q ≡ − (cid:18) ∂ ln µ∂ ln T (cid:19) P = 1 + 12 i (1 − i ) (cid:18)
52 +
Ryk B T (cid:19) . (A.17)Assuming the role of the radiation pressure to be negligible, for an ideal gas of pure hydrogen wewill obtain ∆ ln ρ = − (cid:20) − (cid:18) ∂ ln µ∂ ln T (cid:19) P (cid:21) ∆ ln T = −Q ∆ ln T. (A.18)After the substitution of (A.18) into (A.15), we will obtain the final expression for the meanvelocity of the rising element v : v = Q g z ρ Λ P ( ∇ − ∇ ′ ) . (A.19)The latter expression is valid only under the condition of subsonic motion of the convective cells,which holds in our calculations.Substituting (A.19) into (A.9), we will obtain the final expression for the energy flux trans-ferred by convection Q conv in terms of the mean temperature gradient ∇ , the temperature gradientin the rising element of matter ∇ ′ , and the quantities that are functions of the thermodynamicstate of matter: Q conv = C P Q / ρ / g z T Λ √ P / ( ∇ − ∇ ′ ) / , (A.20) Q = Q rad + Q conv = 4 a r cT κ P ∇ + C P Q / ρ / g z T Λ √ P / ( ∇ − ∇ ′ ) / . (A.21) ertical Convection in Turbulent Accretion Disks 19 The three gradients ∇ , ∇ ′ , and ∇ r can now be related by using the definition of ∇ r (A.7): ∇ r = ∇ + 3 C P κ Q / ρ / g z Λ √ a r cP / T ( ∇ − ∇ ′ ) / . (A.22)To determine ∇ , it remains to find its relation to ∇ ′ . Considering the question of theconvection efficiency Γ , i.e., of what “excess” fraction of heat a convective element transfers in itslifetime with respect to the heat that it radiates in the same time, will be helpful for us. The lossof heat through radiation by a convective element from a unit area per unit time is a r cT κ ρ ∆ T Λ / , (A.23)where ∆ T is the temperature difference between the surface of the element (the mean ambienttemperature) and its center averaged over its lifetime, and Λ / corresponds to the distance atwhich this temperature changes. This formula is valid for an optically thick convective element,i.e., the condition κ ρ Λ / ≫ must hold. Multiplying (A.23) by the area of the element A and byits lifetime Λ /v , we will obtain the energy F Λ /v dissipated by the element in its lifetime: F Λ v = 4 a r c T κ ρ ∆ T Λ / v A . (A.24)The “excess” of heat transferred by the convective element is C P ρ ∆ T max V , where V is thevolume of the element, ∆ T max is the temperature difference between the element and the ambientmedium at an instant before the element dissolution, which is assumed to be twice the meandifference ∆ T introduced above. Dividing this expression by (A.24) and substituting the meanvelocity v from (A.19), we will obtain an expression for the convection efficiency Γ : Γ = 316 √ a C P κ g z Q / ρ / Λ a r cT P / ( ∇ − ∇ ′ ) / , (A.25)where / (2 a ) ≡ V / ( A Λ) and is / for a spherical element and for a cubic one. We used a = 9 / for which V / A = 2 / (Weiss et al., 2004). Comparing the derived expression with (A.22), weobtain ∇ r = ∇ + a Γ( ∇ − ∇ ′ ) . (A.26)Let us express the logarithmic gradient ∇ ′ in terms of ∇ ad and the heat capacity in the risingconvective cell C ′ (Weiss et al., 2004): ∇ ′ = ∇ ad − C ′ /C P . (A.27) The heat capacity C ′ ≡ d q ′ / d T , where q ′ is the specific (per unit mass) quantity of heat. Therate of change of the specific quantity of heat d q ′ / d t is equal to the difference between the “excess”specific rate of energy release in the convective cell ∆ ε and the specific radiation power carriedaway from the cell surface F . C ′ C P = (cid:18) d q ′ d T ′ (cid:19) C P = (cid:18) d q ′ d t (cid:19) (cid:18) d T ′ dt (cid:19) − C P = (∆ ε − F ) / ( ρV ) C P ( dT ′ /dt ) == −F [1 − ∆ ε/ F ] / ( ρV ) C P [( dT ′ /dt ) − ( dT /dt )] + C P ( dT /dt ) . (A.28)Note that C P ρV [( dT ′ /dt ) − ( dT /dt )] / F = Γ , and, substituting the derived expression for C/C P into (A.27), we obtain Γ1 − η = ∇ − ∇ ′ ∇ ′ − ∇ ad , (A.29)where η ≡ ∆ ε/ F is the ratio between the “excess” rate of energy release in the element and itslosses through radiation.In our case, there are two sources of energy release in the medium, turbulent viscosity and thethermalization of X-ray emission. However, the thermalization of X-ray emission in cold matterdoes not depend on its thermodynamic state; hence, the “excess” energy release in the convectivecells with be related only to the rate of energy generation by turbulent viscosity ε vis : ε vis = − ρ d Q vis d˜ z = 32 αP ωρ . (A.30)The following amount of “excess” energy will be released in the convective element in its lifetime: ∆ ερV Λ v = ( ν ε − Q λ ε ) ln (∆ T ) ε vis ρV Λ v , (A.31)where for an ideal hydrogen gas λ ε ≡ (cid:18) ∂ ln ε vis ∂ ln ρ (cid:19) T = − i (1 − i )2 + i (1 − i ) , (A.32) ν ε ≡ (cid:18) ∂ ln ε vis ∂ ln T (cid:19) ρ = 2 + i (1 − i ) (cid:16) + Ryk B T (cid:17) i (1 − i ) . (A.33)We will obtain η by dividing (A.31) by (A.24): η = 316 a ( ν ε − Q λ ε ) ε vis κ ρ Λ a r cT . (A.34) ertical Convection in Turbulent Accretion Disks 21 For the convenience of our subsequent reasoning, let us introduce a quantity κ equal to theratio of the convective and radiative thermal conductivities of convective elements, which is afunction of the thermodynamic state of matter and does not depend on its logarithmic gradients: κ ≡ Γ( ∇ − ∇ ′ ) / = 316 √ a C P κ g Q / ρ / Λ a r cT P / . (A.35)Equations (A.26), (A.29) and identity (A.35) form a system of three algebraic equations withthree unknowns: ∇ , ∇ ′ , and Γ . Let us introduce a new variable ζ ≡ ( ∇ r − ∇ ) / ( ∇ r − ∇ ad ) . Aftersome transformations, the system will be reduced to an equation for this variable: (1 − η ) ζ / + Bζ / + a B ζ − a B = 0 , (A.36)where B ≡ [( κ /a )( ∇ r − ∇ ad )] / and Γ = Bζ / . This equation in the case of η ≥ underconsideration has exactly one nonnegative solution for ζ (for a detailed consideration of the case of η = 0 , see Weiss et al. (2004)). However, this solution at η > (1 + B ) will have a value greater thanone, which can correspond to a negative value of ∇ , but this case is not realized in our calculations.Thus, we have obtained the final expression for the sought-for vertical temperature gradientin the case where convection sets in, which is used to solve the vertical structure equations for anaccretion disk: d T d˜ z = ∇ d P d˜ z TP = g z ρTP [ ζ ∇ ad + (1 − ζ ) ∇ r ] . (A.37) References
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