Evolution of finite viscous disks with time-independent viscosity
aa r X i v : . [ a s t r o - ph . H E ] M a y A strophysical J ournal Preprint typeset using L A TEX style emulateapj v. 01 / / EVOLUTION OF FINITE VISCOUS DISKS WITH TIME-INDEPENDENT VISCOSITY
G. V. L ipunova
Lomonosov Moscow State University, Sternberg Astronomical Inst., Universitetski pr. 13, Moscow 119991, Russia; [email protected] (Received February 8, 2015; Accepted March 30, 2015; Published May 5, 2015)
ABSTRACTWe find the Green’s functions for the accretion disk with the fixed outer radius and time-independent viscos-ity. With the Green’s functions, a viscous evolution of the disk with any initial conditions can be described.Two types of the inner boundary conditions are considered: the zero stress tensor and the zero accretion rate.The variable mass inflow at the outer radius can also be included. The well-known exponential decline of theaccretion rate is a part of the solution with the inner zero stress tensor. The solution with the zero central ac-cretion rate is applicable to the disks around stars with the magnetosphere’s boundary exceeding the corotationradius. Using the solution, the viscous evolution of disks in some binary systems can be studied. We apply thesolution with zero inner stress tensor to outbursts of short-period X-ray transients during the time around thepeak. It is found that for the Kramers’ regime of opacity and the initial surface density proportional to the ra-dius, the rise time to the peak is t rise ≈ . r /ν out and the e -folding time of the decay is t exp ≈ . r /ν out .Comparison to non-stationary α -disks shows that both models with the same value of viscosity at the outerradius produce similar behaviour on the viscous time-scale. For six bursts in X-ray novae, which exhibit fast-rise-exponential-decay and are fitted by the model, we find a way to restrict the turbulent parameter α . Subject headings: accretion, accretion disks – binaries: general – methods: analytical INTRODUCTIONThe accretion processes are recognized to power many as-trophysical sources, which are observable due to the release ofenergy when matter is spiralling down into the gravity well.The study of viscous accretion disks as underlying mechaismsfor the extraction of potential gravitational energy began withthe works of Lynden-Bell (1969); Shakura & Sunyaev (1973),Novikov & Thorne (1973, pp.343-450).Time-dependent problems for the accretion disks arisewhen describing a wide variety of outburst phenomena, ob-served in the accreting sources, i.e. binary systems with masstransfer and galactic nuclei, and systems with planet forma-tion. It is usually possible to consider separately the verticaland radial structure of an accretion disk due to significantlydi ff erent characteristic time-scales. In such a case the time-dependent radial structure of a disk can be described by asecond-order partial di ff erential equation, which is a conse-quence of the angular momentum and mass conservation. Theenergy conservation in a geometrically thin optically thickdisk is provided by the local balance of the viscous heatingand radiative cooling.The basic equation of the viscous evolution, Eq. (1) or (6)below, describes the viscous angular momentum flux alongthe disk radius. This is a di ff usion type equation with avariable ‘di ff usion’ coe ffi cient. Choice of an analytic ap-proach to solve the equation depends on the nature of theviscosity involved. The preferential way of describing vis-cosity in astrophysical disks, although not exclusive, is to use α -viscosity (Shakura 1973; Shakura & Sunyaev 1973), i.e.,the proportionality of the viscous stress tensor to the totalpressure in the disk. One can generally assume that the kine-matic viscosity ν is a product of two power functions: onefunction of radius and another of some local physical param-eter, the latter usually represented by the surface density.The boundary conditions imposed on the accretion disk areimportant. While the outer boundary can be a freely expand-ing surface due to the spreading of the matter corresponding to the outward viscous angular momentum flux, in some sit-uations certain conditions should be posed at some distancefrom the accreting object. Disks in the binary systems are themain focus for such models.In a binary system, the tidal torques, acting inside the Rochelobe of the accreting object, truncate the disk and provide asink for the angular momentum. Papaloizou & Pringle (1977)show that the e ff ective tidal radius is close to the Roche lobe: ∼ . ν ∝ r b ,which is time-independent. We obtain Green’s functions (19),(22), (31), and (32), which are the kernels in the integrals(20), (21), and (35) to compute the viscous angular momen-tum flux F and the accretion rate ˙ M . The method of Green’sfunctions allow one to compute Σ ( r , t ) and ˙ M ( r , t ) for arbitraryinitial surface density distributions. Di ff erent boundary con-ditions are considered: no stress or no accretion at the innerboundary, which is located at the zero radial coorinate, zeroor time-dependent mass inflow at the outer boundary.It is beleived that the mechanism of bursts in X-ray novaeinvolves the thermal instability connected with opacity varia-tions in the disks, as in dwarf novae. When a burst is ignited,and a su ffi cient portion of the disk is in the hot state, the vis- The couple g of Lynden-Bell & Pringle (1974). Lipunovacous di ff usion starts to govern the evolution, driving the char-acteristic rise and drop of the accretion rate onto the centralobject. It is a question, which probably allow no universalanswer, why X-ray novae light curves have di ff erent profilesand pecularities. If the disk is not very large, as it happensin short-period X-ray transients, its whole body is likely tobe engaged in the viscous evolution. It has been shown thatan episodic mass input at the outer radius causes the fast-rise-exponential-decay (FRED) profile due to the viscous evolu-tion (Wood et al. 2001). Using the Green’s function, one canexplore the situation with an arbitrary initial distribution ofthe matter. Actually, as we show in the present study afterobtaining the Green function, for realistic initial density dis-tributions and for the case with the zero inner viscous stress,the evolution is generally a FRED. It is interesting to comparethe results of the model with what happens in the α -disks, theprevalent model for the disks’ turbulence in the stellar bina-ries.The second case, when the viscous stress is non-zero at thecenter, applies to the disks with the central source of the an-gular momentum, e.g., rotating stars with magnetosphers. Ifthe magnetosphere’s radius is large enough, the matter hasa super-Keplerian angular momentum at its boundary. Di ff er-ent regimes for such disks have been proposed: propellers and‘dead’ disks. Using the Green founction, the disks of constantmass that evolve to dead disks can be described.In binary systems the mass transfer rate is a key factor. Atsome stages it can be considered negligible comparing to theaccretion rate in the disk. In other systems, mass transfer vari-ations may cause the activity in the disk. The case of time-dependent mass transfer, which includes the constant transferrate as a particular case, can be also studied analytically if ν ∝ r b . Using the Green’s functions, it is possible to express Σ ( r , t ) and ˙ M ( r , t ) for any outer boundary conditions (for any˙ M out ( t )) as well as for any initial conditions.In Section 2, we present the viscous evolution equation andreview its known solutions for accretion disks. In Section 3,the Green’s functions are found for di ff erent inner and outerboundary conditions. An analytic way to calculate the diskevolution with the variable mass transfer at the outer radius ispresented; the features of a radiating dead disk are considered.In Section 4, a stage of decaying accretion is dealt with anda comparison to the α -viscosity models is made. Applicationto the FRED light curves of short-period X-ray transients ispresented in Section 5. The last sections are dedicated to thediscussion and summary. VISCOUS ACCRETION DISK EQUATIONThe equation of the viscous evolution in the accretion diskis the di ff usion-type equation (e.g., Kato et al. 1998): ∂ Σ ∂ t = r ∂∂ r " ∂ ( ω r ) /∂ r ∂∂ r ( W r ϕ r ) , (1)where, among the standard designations of the time, radius,angular speed, and the surface density we have the compo-nent of the viscous stress tensor w r ϕ , integrated over the fullthickness of the disk W r ϕ ( r , t ) = z o Z w r ϕ d z , where z is the half-thickness of the disk at radius r , and thestress tensor is related to the di ff erential rotation of gas masses by the kinematic viscosity ν as follows: w r ϕ = − ρ ν r d ω d r . (2)For the Keplerian disks d ω/ d r = − (3 / ω k / r , and we rewrite W r ϕ ( r , t ) = ω k z o Z ν ρ d z . (3)With the formula for the surface density Σ ( r , t ) = z Z ρ ( r , z , t ) d z , (4)assuming that ν is independent on z , we have W r ϕ ( r , t ) = ω k ν Σ . (5)It is convenient to introduce a new independent variable h ( r ) = v ϕ ( r ) r = ω r – the specific angular momentum. Then,for the Keplerian disk, equation (1) can be rewritten as: ∂ Σ ∂ t =
34 (
G M ) h ∂ ( Σ ν h ) ∂ h , h ≡ h K . (6)The acknowledged approach is to solve the disk evolu-tion equation for the variable proportional to the full vis-cous torque acting between the adjacent rings in the disk, ω r ϕ · z π r · r . This way, the boundary conditions are easy toexpress. From the above expressions, the following relationbetween F = π W r ϕ r and the accretion rate can be obtained: − π Σ v r r ≡ ˙ M ( r , t ) = ∂ F ∂ h , (7)where the radial velocity v r is negative.One chooses a procedure to solve (6) provided the form of ν = ν ( r , Σ ) is known. This form is determined by the physicalconditions in the disk. For example, the vertical structure canbe solved for the α -disk, and the required relation between ν and Σ can be found. Let us give a perspective of some studies(mainly analytic) of the viscously evolving accretion disks.They di ff er in respect of the viscosity prescription and bound-ary conditions. The analytic solutions are the benchmarks formore sophisticated and involved numerical models.2.1. Viscosity ν ∝ r b For the kinematic viscosity in the form ν ∝ r b , the equationof the viscous evolution (6) becomes a linear partial di ff eren-tial equation. In 1952 L¨ust obtained particular solutions to theequation proposed by his teacher von Weizs¨acker (1948) anddescribed the principles to determine the general solution forthe infinite and finite problems.For the accretion disks that can extend infinitely ,Lynden-Bell & Pringle (1974) (hereafter LP74) find, for thetwo types of the inner boundary conditions, the Green’s func-tions that provide means to describe the whole developmentof the disk. The inner radius of the disk is zero in their exactsolution. The long-term self-similar evolution of the accretionrate goes as a power-law ˙ M ∝ t − ( l + with a parameter l < finite disk with the viscos-ity constant over the radius and in time and derive the expo-nential decay of the accretion rate in the disk.Zdziarski et al. (2009) study the mass flow rate through adisk resulting from a varying mass-supply rate, derive theGreen’s function for the accretion-rate and its Fourier trans-form. By means of numerical simulations, they investigatea finite disk, residing between non-zero r in and r out , withthe non-zero accretion rate at the outer radius. Tanaka et al.(2012) find general Green’s function for an explicit depen-dence of the inner boundary r in on time and a non-zero massacross r in .In the present work, we study analytically a finite accretiondisk with the zero r in and find an analytic solution describingits whole evolution, from the rise to decay. For the case of thezero stress tensor in the disk center and the mass depositionat the outer disk edge, the Green’s function of the problem isfound in Wood et al. (2001).2.2. Viscosity ν ∝ Σ a r b Similarity solutions of the first kind of the non-linear dif-ferential equation (Barenblatt 1996) have been found at the”decaying-accretion stage,” when the total angular momen-tum of an infinite disk is constant (Pringle 1974, 1981). Ac-cretion rate is found to decrease as ∝ t − / for the Thomsonopacity, and as ∝ t − / for the Kramers’ opacity (see alsoFilipov 1984; Cannizzo et al. 1990). Lyubarskij & Shakura(1987) obtain self-similar solutions that describe separatelythree stages of the disk evolution. The first two are the self-similar solutions of the second kind: the initial movement ofthe inner edge of a ring of matter toward the center and the riseof the accretion rate in the center ( ˙ M ∝ t . for the Kramers’opacity and ∝ t . for the Thomson opacity). The third stageis the decay of the accretion and corresponds to the solution ofPringle (1974). The first two stages of Lyubarskij & Shakura(1987) may be applicable to the finite disks when the condi-tions at the distant outer boundary do not a ff ect the behaviourof the disk near the center.Lin & Pringle (1987) derive an analytic self-similar solu-tion for a disk subject to the gravitational instability, whichtransfers mass and angular momentum in the disk, for which˙ M ∝ t − / , and a numerical solution for the whole evolution.Lin & Bodenheimer (1982) found a self-similar solution for ν ∝ Σ , if the viscosity is due to the action of convectivelydriven turbulent viscous stresses ( ˙ M ∝ t − / ).Accretion disks with a central source of angular momen-tum and no mass transfer through the inner boundary wereconsidered by Pringle (1991) who derived a self-similar so-lution (see also Ivanov et al. (1999)). Rafikov (2013) in theextensive study of the circumbinary disks around supermas-sive black holes obtained self-similar solutions for the diskswith a possible mass transfer across the orbit of the secondaryblack hole.For a finite disk, the radial and temporal solution is obtainedby Lipunova & Shakura (2000) yielding ˙ M ∝ t − / for theKramers’ opacity and ˙ M ∝ t − / for the Thomson opacity.In the context of the self-similarity solutions, we would liketo note that for an advection-dominated infinitely expandingaccretion flow, Ogilvie (1999) obtains by similarity methods a time-dependent solution with the conserved total angular mo-mentum. EVOLUTION OF THE FINITE DISK WITH STEADYVISCOSITYIf the kinematic viscosity ν is not a function of Σ and is afunction of the radius alone, we arrive at the linear di ff erentialequation (6). We write the kinematic viscosity as ν = ν r b . (8)Then Eq. (6) for the dependent variable F is as follows ∂ F ∂ t = ν h b − ( G M ) − b ∂ F ∂ h , (9)where F = π W r ϕ r = π h Σ ν r b . (10)We rewrite (9) in the form similar to that in LP74: ∂ F ∂ h = (cid:18) κ l (cid:19) h / l − ∂ F ∂ t , (11)where the constants are defined as follows:12 l = − b , κ = l ν ( G M ) / l . (12)There is a degeneracy of the index l at b =
2. But it is onlyapparent as l enters equation (11) in the denominators. Value b = ff erential equation, and is left out of thefollowing consideration. In the case of b = t vis ∝ r /ν o r b . We limitourselves to the finite positive values of l and the Kepleriandisks.In the astrophysical hot accretion disks, the turbulent vis-cosity is at work. According to the Prandtl concept (Prandtl1925), the kinematic turbulent viscosity is an averaged prod-uct of the two random values: ν t = l t υ t , where l t and υ t arethe path and speed of the turbulent motion. For the compo-nent of the viscous stress tensor we write w r ϕ = − ρ ν t r d ω/ d r .Following the Prandtl consideration, υ t = − l t r d ω/ d r , whichleads to w r ϕ = ρ υ . Shakura (1973) proposed the α -disks: w r ϕ = α ρ υ . From the hydrostatic balance one derives thesound speed υ s ≈ ω r ( z / r ) and obtains for a Keplerian disk ν t ≈ / α h ( z / r ) . If the relative half-thickness of the α -disk is invariable then b = / l = /
3. This value of b takes place for the ‘ β -viscosity disks’ with ν = β h , suggestedby Duschl et al. (2000) to act in the protoplanetary and galac-tic disks. The standard model of the α -disk implies that h / r varies with r and Σ .Let us find the Green’s function of the linear equation (11)with the specific boundary conditions at the boundaries. AGreen’s function or a di ff usion kernel is a reaction of the sys-tem to the delta impulse, in other words, the Green’s functionis the solution to the di ff erential equation with the initial con-dition F ( x , = δ ( x − x ), where δ ( x − x ) is the Dirac deltafunction. We search a solution by separating the variables, F ( h , t ) = f ( h out ξ ) exp( − s t ), where s is a constant and h out isthe outer disk radius. Substituting this into (11), we getd f ( ξ )d ξ + kl ! ξ / l − f ( ξ ) = , (13) Lipunovawhere k = κ h / l out s . This is a Lommel’s transformation ofthe Bessel equation, which can be derived by introducing anew independent variable x = ξ / l . A particular solution isrepresented as F k ( x , t ) = e − st ( kx ) l [ A J l ( kx ) + B J − l ( kx )](L¨ust 1952), where J l and J − l are the Bessel functions of anon-integer order. If l is an integer number, the Bessel func-tions of the second kind should be used instead of J − l . For b =
2, the solution is no more a Bessel function (see Ap-pendix I of Kato et al. 1998). The value b = r /ν is constantover the radius. We do not consider this case in the presentstudy.A general solution for an infinite problem is a superpositionof particular solutions and is expressed by LP74 as an integralover all positive values of k : F ( x , t ) = R ∞ F k ( x , t ) d k . For aproblem inside a finite interval, a general solution is a super-position of particular solutions and is expressed as an infinitesum (L¨ust 1952): F ( x , t ) = ∞ X i = e − t k i κ − h − / l out ( k i x ) l [ A i J l ( k i x ) + B i J − l ( k i x )] , (14)where constants k i , A i . and B i are to be defined from twoboundary conditions and one initial condition. The dimen-sionless coordinate x lies in the interval [0 , h (specific angular momentum) lies inside[0 , h out ]. The outer boundary condition, which sets the ac-cretion rate at the outer radius of the disk, can be written asfollows ∂ F ∂ h = ˙ M out ( t ) at h = h out . (15)In the simplest case, the homogeneous outer-boundary con-dition corresponds to the zero mass-inflow rate at the outerboundary. As this happens, the viscous angular momentumflux F takes some non-zero value, which implies withdrawalof the angular momentum at the outer boundary. This corre-sponds to the situation of the disk in the binary system, whenthe angular momentum is removed from the outer edge of thedisk by the tidal action of the secondary.At x =
0, one trivial inner boundary condition F ( x , t ) = ∂ F /∂ h = x =
0, describes the situation of no accretion through the innerdisk edge. This case is applicable for the stars with strongmagnetosphere, when the Alfv´en radius is greater than thecorotation radius, impeding the accretion of the matter. Weconsider these two cases separately.3.1.
The case of zero viscous stress at the inner boundaryand zero accretion rate at r out
First, we consider the case of the zero viscous stress at theinner boundary. Approximately, this corresponds to the solu-tion F ( x , t ) = x = ffi cients at J l -termsin (14) remain non-zero. Thus, sum (14) at t = F ( x , = ∞ X i = ( k i x ) l A i J l ( k i x ) . (16)We assume the zero accretion rate at r out . Such boundarycondition approximates the situation of the low mass-transfer F ig . 1.— Green’s function for the finite accretion disk with the zero viscousstress in the center at five moments of time t = . t = . t = t ∞ max = / t = . t = .
3. Initial spike was at position x s = ( h / h out ) / l = . κ = l = / from the secondary in the binary system, for example, whenthe accretion rate in the disk is much higher than the masstransfer rate during an outburst. Dropping the time part of thesolution, we arrive at condition (15) required for every termin the sum (16), which can be rewritten as follows: l J l ( k i ) + k i J ′ l ( k i ) = . (17)Thus, the general solution for the case of zero viscous stresstensor at the center and zero accretion rate at the outer bound-ary is the sum (14), where B i = k i are the positive rootsof the transcendental equation (17). To define A i from theinitial condition, the finite Hankel transforms are used.The series ∞ P i = k li A i J l ( k i x ) with condition (17) are knownas the Dini’s series (see Watson 1944, § f ( x ) = F ( x , x − l can be expressed as the Dini’s expansionif the function satisfies the Dirichlet conditions in the inter-val and the coe ffi cients are defined as k li A i = f J ( k i ) J − l ( k i )(Watson 1944; Sneddon 1951) with the finite Hankel trans-form ¯ f J ( k i ) = Z x f ( x ) J l ( k i x ) d x , where f ( x ) = F ( x , x − l . We look for a solution for the δ -function as the initial condition: F ( x , = δ ( x − x ). Usingthe properties of the Dirac function, we obtain k li A i = x − l J l ( k i x ) J l ( k i ) . (18)Thus, for the specific boundary conditions, we derive theGreen’s function, which is the solution to (11) with δ -functionas the initial distribution G ( x , x , t ) = x l x − l X i e − t k i κ − h − / l out J l ( k i x ) J l ( k i x ) J l ( k i ) , (19)where k i are the positive roots of Eq. (17) and x = ( h / h out ) / l .The function is plotted in Fig. 1 for consecutive times. Thecurve for time t corresponds to the maximum accretion ratein the center. It is plotted at t = t ∞ max given by (25) below. F ig . 2.— Left: ratio of the time when the accretion rate peaks in the finite disk to that in the infinite disk versus the position of the initial thin ring. Right: ratioof the peak accretion rate in the finite disk to that in the infinite disk as a dependence on the initial ring position. Given the dimensional initial distribution F ( x ,
0) in the fi-nite accretion disk, the distribution of F at any t > F ( x , t ) = Z F ( x , G ( x , x , t ) d x . (20)The accretion rate can be found from˙ M ( x , t ) = Z F ( x , G ˙ M ( x , x , t ) d x . h out , (21)where the Green’s function for the accretion rate G ˙ M ( x , x , t ) ≡ ∂ G ( x , x , t ) ∂ x l == ( x x ) − l l X i e − t k i κ − h − / l out k i J l ( k i x ) J l − ( k i x ) J l ( k i ) . (22)Functions G and G ˙ M are found by Wood et al. (2001) for thecase x = t .One can express the initial F , proportional to the disk mass,from the initial distribution of the surface density, using (5)and (12): F ( x , = π l κ h / l r Σ ( r ) h (23)where r = h / GM and h = h out x l .3.1.1. Spike-like and power-law initial distribution of the viscousstresses
Let the initial mass be located at some radius r s where theKeplerian specific angular momentum is h s = x ls h out , and themass of the infinitely thin ring is M disk , h out is the maximumpossible specific angular momentum of the matter.The surface density of the δ -ring is Σ ( r ) = M disk π r s δ ( r − r s ) = M disk π l r s x s δ ( x − x s ) . We take into account that δ ( r − r s ) = δ ( x − x s )d x / d r and r = x l r out . Applying (23), we obtain F ( x , = l κ M disk h − / l out x l − s δ ( x − x s ) (24)and the accretion rate evolution is as follows˙ M ( x , t ) = M disk t vis x l − s G ˙ M ( x , x s , t ) , where we designate t vis = κ h / l out l = l r ν ( r out ) . The time t max of the maximum and its accretion rate in thedisk center ˙ M in , peak are of order of those for the infinite disks.Using the solution of LP74, we find the time of the accretionrate peak for the infinite disk t ∞ max = κ h / ls + l ) = l + l ) r ν ( r s ) (25)and value ˙ M ∞ in , peak (A1). The numerical factor in (25) is pre-sented in Table 1 for various, physically motivated values of l . In Fig. 2 the ratios of maximum times and accretion ratesfor the finite and infinite disk are shown for several cases of l .The ratios approach 1 for x s ≪ Σ ∝ r α Σ , we use the following expression inEq. (23): Σ ( r ) = M disk π r ( α Σ +
2) ( r / r ) α Σ + − ( r / r ) α Σ + , assuming the mass of the disk is enclosed between the radii r and r .In Fig. 3 the variation of the accretion rate at the disk centeris presented for three di ff erent initial distributions of F (seeFig. 4): two narrow top-hats ( α Σ = / l − /
2) with di ff erent r and r and the one corresponding to Σ ( r , ∝ r . All threeinitial distributions are constructed for the same initial massof the disk M disk =
1. The evolution of a thin ring located farfrom the outer disk edge has a characteristic power-law inter-val, when the disk behaves as if it was infinite. The interval Lipunova F ig . 3.— The inner accretion rate variation (the black curves with a peak) for three initial distributions of F ( h ) shown in Fig. 4. The initial mass of the disk isthe same in all cases. On the left panel, the cumulative accreted mass is shown by the monotonic (green in the electronic version) curves. The dashed curve has apower-law interval, which is clearly seen on the right panel. Constants M disk = h out = κ = l = / t vis = / lasts about t max [1 − ( h s / h out ) / l ]. If Σ ( r , ∝ r , the mass of thedisk at t = Disk luminosity
The viscous heating in the disk (its half) is found from F : Q vis = π F ω k r , (26)which can be equated to σ T ff assuming the local energy bal-ance.In the center of the disk, the quasi-stationary distributiondevelops soon and the accretion rate is constant over radii: F ( t ) = ˙ M ( t ) h . The zone of constant ˙ M expands over time.The viscous heating (26) with the quasi-stationary distribu-tion of F diverges at the zero coordinate. In the real astro-physical disks, the inner radius of the disk is a finite value r in ,
0. Note that integrating (26) over the disk surface from r in to r out also yields the incorrect result for the total power re-leased in the disk. (As a remark, the integration of (26) gives F ig . 4.— Three initial distributions of F ( h ) in the disk, whose inner accretionrate evolution is shown in Fig. 3. Top-hats are located at 0 . − . − .
36 of r out (dashed line) and the wide distribution, correspondingto Σ ( r ) ∝ r , goes in (0 . − r out (dotted line). The disk mass is the same inall cases and equals 1. The outer specific angular momentum h out =
1. Thetop-hat distributions are constructed using α Σ = −
1. Parameter l = / the correct result for the case of the zero accretion rate at thecenter, see § F ( r in ) = r in leadsto a time-independent correction of F near r in . This cor-rection is found for the infinite disks by Tanaka (2011) andrepresents the classical factor (1 − √ r in / r ). Naturally, thecorrection holds for a finite disk. It follows that the bolo-metric luminosity can be calculated via the classical formula L bol = G M ˙ M / (2 r in ) at times t > t vis ( r in / r out ) − b with thefirst term in the correction of order of ( r in / r out ) ( t vis / t ) / (2 − b ) for earlier times. For the disks in the stellar binary systems,usually, ratio r in / r out <<
1, and we are safe to calculate L bol by the classical formula with ˙ M = ˙ M (0 , t ) found from (21).3.2. The case of the disk with no central accretion and zeroaccretion rate at r out
Condition ∂ F /∂ h = h = A i = t = x − l F ( x , = B + ∞ X i = k li B i J − l ( k i x ) , (27)where k i are the positive roots. In the previous section, thezero root contributed nothing to (16). Here the zero root pro-vides a term like ( k x ) l J − l ( k x ), which must contribute some-thing, because the Bessel function J − l behaves as ∝ ( k x ) − l inthe vicinity of zero.The method to find B i for i ≥ A i , since the Dini’s series are convergent for the or-der of the Bessel function > − . We consider only non-integer 0 < l < k li B i = x − l J − l ( k i x ) J − l ( k i ) . (28)It is known about the Dini’s series of the negative orderthat an additional first term is to be added to the sum, if theequation defining roots k i has particular coe ffi cients . Just this The Hankel integral transform was also proved for the Bessel orders > − ? Betancor & Stempak 2001). For the usual form of writing the boundary condition for the Dini’s se-ries,
H J ν ( z ) + zJ ′ ν ( z ) =
0, the the additional term appears if H + ν = F ig . 5.— Green’s function for the finite accretion disk with the zero accretion rate at the center at five moments of time t = . t = . t = . t = . t = . t = . t = . t = . t = . t = . x s = ( h / h out ) / l = κ = l = / t vis = / case occurs for the properly rewritten outer boundary condi-tion (15): l J − l ( k i ) + k i J ′− l ( k i ) = . (29)The additional term in the Dini’s series is expressed as B = − l ) x − l Z z − l F ( z , z − l d z , (30)where z is a free variable (Watson 1944). Thus, adopting F ( z ) = δ ( z − x ), from (14) we obtain the Green’s functionfor the equation of the non-stationary accretion with the zeroinner accretion rate and the finite outer radius: G ( x , x , t ) = − l ) x − l ++ xx ! l x ∞ X i = e − t k i κ − h − / l out J − l ( k i x ) J − l ( k i x ) J − l ( k i ) , (31)where k i are the positive roots of Eq. (29) and x = ( h / h out ) / l .Formulas (20) and (21) can be applied without changes.This Green function is illustrated in Fig. 5. At the latestages, the accretion disks forget all information about the ini-tial distribution of the viscous stresses. For the disks withoutaccretion on to the center, the uniform distribution of F devel-ops, with its magnitude being proportional to the mass storedin the disk.We take the derivative of (31) and obtain the Green functionfor the accretion rate: G ˙ M ( x , x , t ) = x − l l X i e − t k i κ − h − / l out k i J − l ( k i x ) J − l ( k i x ) J − l ( k i ) . (32)It approaches zero when x → t . This corre-sponds to our boundary condition at the center: ˙ M =
0. Thisdisk, without sink of mass and additional mass supply, emitsradiation even when it reaches the ‘end’ of its evolution, asthe viscous heating (26) does not stop when the flow of mat-ter ceases. If Fig. 6 its bolometric luminosity is plotted for thetwo initial positions of the ring (the details are given in thenext subsection). 3.2.1.
The dead disk
The stable configuration of the confined ‘dead’ disk (weadopt the name from the work by Syunyaev & Shakura 1977)has the following viscous angular momentum flux at each r : F d ≡ F | t →∞ = l (1 − l ) h − / l out M disk κ = − l ) h out M disk t vis , (33)where (24) is used and t vis = l r / l out (3 ν ) − as before.The total luminosity of the dead disk is a constant value F d ( ω in − ω out ). It can be obtained by the integration of (26)over the disk surface between r in and r out . Unlikely to thecase of the zero torque at r = r in , the late-time value of F does not depend on the location at which condition ˙ M = r in a ff ects the limunosity only by a change in the size of theemitting surface. There is still a time-dependent correctionobtained by Tanaka (2011) having an e ff ect at early times t < t vis ( r in / r out ) − b .The dead-disk state has to end sometime, as the disk can-not receive infinitely the angular momentum from the centralobject. If the central star has a magnetosphere, its radius r mag can be estimated from the equality of the viscous and mag-netic torque at the magnetosphere boundary (LP74): F ( x → = κ t µ ⋆ r , (34)where µ ⋆ is the magnetic dipole of the star, κ t is a dimen-sionless factor of order of unity (Lipunov 1992). Withinsome factor, this condition is equivalent to the equality ofthe gas and magnetic pressure at the inner disk bound-ary (Syunyaev & Shakura 1977). The accretion is inhibitedif the inner disk radius, r mag , is greater than the corotation ra-dius, because the drag exerted by the magnetic field is super-Keplerian: r cor = G M ⋆ ω ⋆ ! / = . × P / ⋆ M / . cm , where M ⋆ , ω ⋆ and P ⋆ are the mass, angular speed and periodof the revolution of the central star. In the last expression the Lipunova F ig . 6.— Variation of the luminosity of the disk with zero accretion ratethrough the inner radius normalized to value L dead ≡ F d ω in . The disk startsfrom the outer radius (the solid line) or from r = . l × r out (the dashed line),corresponding to the Green’s functions in Fig. 5. mass is normalized by 1 . ⊙ . Thus we obtain for a finalsteady configuration r mag , d r cor ≈ . (cid:18) κ t − l (cid:19) / µ / t / , P / ⋆ M / . M / , r / , ⊙ . The ratio can be rather close to unity. The normalizationsare 10 g for the disk mass, 10 [G cm ] for the magneticdipole, 100 days for the viscous time, R ⊙ for the outer disk ra-dius. Situation of r mag & r cor is studied by D’Angelo & Spruit(2010), who argue that no considerable expulsion of matterfrom the disk is expected in such regime, as contrasted to thepropeller scenario. Using the magnetosphere radius as the in-ner disk radius, we find the steady disk luminosity: L dead ≈ . × M / , M / . r / , ⊙ t − / , µ − κ − / erg s − for r in ≪ r out from the both sides, for typical l ∼ / L dead , max ≈ . × µ P − ⋆ m − . erg s − . The regime can be sustained for su ffi ciently strong magneticdipole and fast rotation of the neutron star: µ P − ⋆ & . p (1 − l ) /κ t m / . M / t − / , r / , ⊙ . The flux of the angular momentum F at small radii and theinner radius of the disk always manage to adjust themselvesto the outer variations that proceed on longer viscous times.The steady value F d ∝ ν r b − / could be low due to a degra-dation of ν in the outer disk with low luminosity. Then thenormalized viscous time in the formulae above may be greateraccordingly to t vis ∝ ν .Luminosity during the spreading of the ring (Fig. 5) is cal-culated by integrating expression (26) over both surfaces ofthe disk confined between the outer and inner radius; it is plot-ted in Fig. 6. The inner radius of the disk is varying accordingto Eq. (34) and depends on the star’s magnetic dipole. Its typ-ical value r mag , d = r cor ≈ . × − P / ⋆ M / . R ⊙ .The torque transfers the angular momentum of the centralstar to the disk and, via the disk, to the orbital motion of thebinary. The corotation radius increases gradually and, even-tually, accretion on the center begins (Syunyaev & Shakura1977; D’Angelo & Spruit 2011). The characteristic braketime is the angular momentum of the star 2 π I ⋆ / P ⋆ divided by the torque F d : t brake t vis ≈ I ⋆, M / . r / , ⊙ P ⋆ M disk , , where I ⋆, = I / g cm and a typical value of l is substi-tuted. If the mass of the disk grows due to the matter income,the braking time of the star shortens, while the magnetosphereradius decreases at the same time.3.3. The case of an arbitrary accretion rate at r out
For a variable mass inflow ˙ M out ( t ) at the outer disk edge r out the solution can be found by a procedure described in theAppendix B F ( x , t ) = Z F ( x ) − x l h out ˙ M out ( t ) G ( x , x , t ) d x ++ h out t " x l κ ! ˙ M out ( τ ) h / l out − ¨ M out ( τ ) x l G ( x , x , t − τ ) d x d τ ++ x l M out ( t ) h out , (35)where G ( x , x , t ) is given by (19) or (31). Substituting G by G ˙ M in the integrals above, and dividing the result by h out , wefind the accretion rate in the disk ˙ M ( x , t ). We have tried theformula obtained in the numerous tests. In particular, it suc-cessfully reproduces the results obtained by numerical simu-lations of Mineshige (1994). The further details, however, areout of the scope of the present work. THE ACCRETION DISK AT THE DECAY STAGEWe return now to the standard accretion disks with zerostress tensor at the center. For large t , the first term of thesum (22) dominates, and a single exponential dependence ontime prevails: G ˙ M (0 , x , t ) (cid:12)(cid:12)(cid:12)(cid:12) t > t vis = k l x − l l Γ ( l ) J l ( k x ) J l ( k ) exp − t k l t vis . This corresponds to the well-known exponential dependenceon time of the accretion rate in the disk with the time-independent viscosity ν . The characteristic time of the ex-ponential decay is t exp = h / l out κ k = l k r ν out , (36)where we use ν out = ν r b out . In Table 1 we give locations k of the first zeroes of equation (17) for several values of l . Thetable also lists the numerical factors in the expressions for therise and decay time, (25) and (36).At late times, the profile of F ( x , t ) is self-similar, because G ( x , x , t → ∞ ) → x l J l ( k , x ). The distribution Σ ( r ) is alsoself-similar. Actually, this profile develops already at thevery beginning of the exponential decay. This is illustratedin Fig. 7 by the fact that the ratio of the inner accretion rateto the disk mass, which is the integral of the surface den-sity, approaches a constant value that depends only on theform of the radial distribution. In Table 1, the values of the F ig . 7.— Ratio of the disk mass to the inner accretion rate vs. time normal-ized by the peak time. The constant value indicates a self-similar distributionin the disk has developed. The line styles and parameters are as in Fig. 3. parameter a = ˙ M in h out / F out can be found. Parameter a describes the self-similar profile of F ( h ) and was calculatedby Lipunova & Shakura (2000) for the non-stationary finite α -disks. For the stationary disks with non-zero accretion rate, a = σ T ff = (3 / π ) ˙ M in ω / a , i.e. the outer ring luminosity is byfactor a ≈ .
44 less for the viscously evolving disk ( l = / a ( r ) ≈ . r = . × r out .4.1. Comparing with the α -disks Let us now compare the finite-disk solution for the time-independent ν and that obtained for the turbulent viscosity inthe general form ν = ν Σ a ( t ) r b . For the latter form of viscos-ity, Eq. (6) acquires the following view: ∂ F ∂ t = D F m h n ∂ F ∂ h , (37)embracing a dimension constant D = a +
12 (
G M ) ν (2 π ) a ( G M ) b ! / ( a + , and dimensionless parameters m and n : m = aa + , n = a + − ba + . For a finite disk with such type of viscosity the inner accre-tion rate decays as a power law (Lipunova & Shakura 2000;Dubus et al. 2001):˙ M ( t ) = ˙ M (0) (1 + t / t ) − / m , (38)where ˙ M (0) is the accretion rate at the time t =
0, whichcan be attributed to any moment of the decay stage. Eq. (37)describes the viscous evolution of the α -disks.The characteristic time of the solution t = h n + / ( λ m D F m out (0)), where λ is a numerical constantthat can be calculated for specific a and b . Using theexpression for D and relation F out = π h out ν out Σ out , weobtain t = λ a r ν out ( t = . (39)For the α -disk in the Kramers’ regime of the opacity, parame-ters a = / b = /
14, and λ = .
137 (Lipunova & Shakura F ig . 8.— Relative variation of the accretion rate in two cases: ν = ν r b ( b = /
4, the solid line) and ν = ν Σ a r b (Kramers’ opacity, a = / b = /
14, the dashed line). Disk parameters r out and ν out are the same in theboth models. t de-pends on the choice of the zero time.In the Shakura–Sunyaev disks, the viscous stress is propor-tional to the total pressure in the disk, and this proportionalityis parametrized by the α -parameter (Shakura 1973): w r ϕ = α ρ v . Then the kinematic viscosity can be expressed using (5) as ν t = α v ω k (40)or ν t = α ω k r (cid:18) z r (cid:19) . (41)To approximate the viscous evolution of the α -disk bythe disk with steady viscosity, one has to estimate the ap-propriate parameter b for Eq. (8) and (9). This can be ef-fectively done using the relation ν t ∝ r / ( z / r ) ∝ r b .The solution for the stationary pressure-dominated station-ary regions (Shakura & Sunyaev 1973) gives the relative half-thickness z / r ∝ r / , consequently, b ≃ / l ≃ / α -disk with the Thomson opacity and the ‘dead’ disks mod-els (Syunyaev & Shakura 1977).In Fig. 8 the two laws are plotted: exponential ∝ exp( − t / t exp ) for l = / m = / ∼ t exp , the models cannot be easily discriminated (thiswas also pointed at by Dubus et al. 2001). This fact providesa possibility to estimate α using the e -folding time of the lightcurve. In the next Section, we show how it could be done us-ing the observations of the disks in X-ray novae, whose lightcurves in many cases show exponential decays. THE BURST LIGHT CURVES OF X-RAY NOVAEIn the X-ray novae, binary systems with a compact starand, typically, a less-than-solar-mass optical star, transientoutbursts are thought to originate due to a dwarf nova-typeinstability (e.g., Lasota 2001). The temperature rises in a por-tion of the ‘cold’ neutral disk, which becomes ionized, and theviscosity ν rises. Moreover, the turbulent parameter α changesfrom α cold to higher α hot . The heating front moves outwardfrom some ignition radius with the speed V front ∼ α hot v sound .0 Lipunova TABLE 1P arameters of the G reen ’ s functions for two types of inner boundary condition . b l k t ∞ max ( r /ν s ) − t exp ( r /ν out ) − a Comment(1) (2) (3) (4) (5) (6) (7)0 1 / /
15 0 .
298 1.267 constant ν / / / .
383 1.363 α − disks with constant h / r ; β − disks3 / /
14 1.2927 0 .
125 0 .
407 1.392 α − disks, Thomson opacity ( a = . )3 / / .
152 0 .
449 1.444 α − disks, Kramers’ opacity ( a = . )1 1 / / .
540 1.571 F ( h ) ∝ sin(( π/ h / h out )2 ∞ — — — — t visc does not depend on r / /
16 3.4045 0 .
189 — 0 dead α − disks, Thomson opacity3 / /
14 3.3425 0 .
265 — 0 dead α − disks, Kramers’ opacityN ote . — Columns are as follows: (1) The index of the radial dependence of ν ; (2) l defined by (12); (3) Thefirst zero of equations (17) and (29), in the upper and lower part, respectively; (4) The dimensionless factor in (25)(the upper part) and the one in a similar formula derived by LP74 for the time of maximum F in the disk withoutcentral accretion (the lower part of the Table); (5) The dimensionless factor in (36); (6) Parameter describing theself-similar radial profile, a = ˙ M in h out / F out ; (7) Note on the applicability. The opacity law is indicated for the α -disks. The applicability of parameters is approximate for the α -disk. Refs: Lipunova & Shakura (2000).
It stops at a radius, where the surface density of the diskis too low to sustain the stable (on the thermal time-scale)hot state (Meyer 1984; Menou et al. 2000; Dubus et al. 2001;Lasota 2001).In the dwarf novae, the hot state accretion is quenchedswiftly (comparing to the viscous time) by the matter return-ing to the cold state. It was suggested that in the X-ray no-vae the illumination from the center has a stabilizing e ff ecton the outer disk, keeping the temperatures above the hy-drogen ionization temperature (Meyer & Meyer-Hofmeister1984; Chen et al. 1993). King & Ritter (1998) propose thatin order for the disk to undergo the viscous evolution an in-tensive irradiation must be included.A canonical light curve of an X-ray nova is describedas a FRED (Chen et al. 1997). Theoretically, if the diskis truncated from outside, a steeply decaying light curve isproduced (King & Ritter 1998; Lipunova & Shakura 2000;Wood et al. 2001). The main property is that the disk does notexpand infinitely, which is a consequence of its being in a bi-nary system, and its radius does not change on a viscous timescale. Scenario by King & Ritter (1998) produces exponentialdecays due to the steadiness of the parameter ν in time. Morerealistic temporal dependence of ν leads to a steep power-lawdecay (Lipunova & Shakura 2000; Dubus et al. 2001), whichis close to an exponential one during few viscous times (seethe previous Section).The possibility to model the X-ray novae bursts, whichhave the FRED profile, by the viscously evolving disk withthe steady viscosity and after a descrete mass-transfer event,was shown by Wood et al. (2001) (see also the modelingin Sturner & Shrader 2005). If a burst is due to an instabilitya ff ecting a slab of matter extended along the radii, the Green’sfunction obtained in the present work should be used. E ff ec-tively, after a heating front has passed, the dwarf-nova typeinstability leads to some distribution of the hot-ionized matterwith high viscosity. In the most trivial case, when the bulkmass is concentrated near r out , the evolution of the disk ap-proaches the one found by Wood et al. (2001).Let us compare the typical time of the heating front prop-agation r / V front and the time t max it takes the inner accretionrate to peak, which is of order of t ∞ max (see (25) and Fig. 2). Relating α and ν by (40), we obtain: t front t max = l + l s r s v G M ∼ . s T r µ m x , where the gas temperature is normalized by 10 K, centralstar mass by a solar mass, and radius by 10 cm, µ is themolecular weight of the hot matter. Radius r s is the charac-teristic radius where the bulk mass resides at the beginning ofan outburst. This ratio indicates that we are safe to model anX-ray nova outburst at time ∼ t max using the Green’s functionobtained.If one assumes the commonly suggested surface densitydistribution in the quiescent state in the X-ray novae, Σ ∝ r (Lasota 2001), then, as it is shown in Fig. 3, after the peakof the burst, the only important parameters are the disk massand the outer radius. For such density distribution, the bulkmass of the disk resides close to the outer radius. As we showin Sect. 4, already at the maximum of an outburst the diskacquires the self-similar radial distribution, which is not de-pendent on the initial distribution of matter. The necessarycondition for this is that the bulk mass of the viscously evolv-ing disk is located near the outer radius. This is consistentwith the absence of a power-law interval before the exponen-tial decline on the the light curves of the X-ray novae thatshow the FRED behaviour.Eqs. (25) and (36) yield the ratio t exp / t max taking into ac-count that t max is very close to t ∞ max for r s ∼ r out . The value of t max / t ∞ max is depicted in Fig. 2, where t max is the peak timefor the finite disk. For l = / t exp / t max ≈
3. This provides a rough test whether the par-ticular light curve is produced by the viscously evolving diskmade of hot ionized matter and with approximately constantouter radius.The viscous-disk relation t rise ≈ . r /ν out for the hot-disks with the Kramers’ opacity in the outer parts leads to aconclusion that to obtain a secondary maximum on the X-raylight curve of an X-ray nova, the disk has to provide an extramass input at the radius that is determined by the time of theinput. The secondary maximuma are usually observed around(1 − t exp after the peak. If the extra mass input happens at thetime of the burst maximum then it should happen at least at aradius about 2 . r out (from t exp / t rise ∼ ν ∝ r / ), without1 F ig . 9.— The peak-normalized light curves of GRO J0422 +
32 in 1992, A 0620-00 in 1975, GS 1124-68 in 1991, GS 2000 +
25 in 1998 (as collected in Chen et al.1997) and 4U 1543-47 in 2002 and XTE J1753.5-0127 in 2005 (the quick-look results provided by the ASM / RXTE team). The energy bands of the X-ray dataare indicated in the plots. The model curves are the peak-normalized accretion rate, calculated by (21) using t exp indicated for each burst, and l = /
5. The initialsurface density distribution Σ ∝ r , and the inner radius of the initial hot zone is 0 . × r out . Note the di ff erent axis scale for A 0620-00, for which two di ff erentmodels are plotted, with the initial inner radius at 0 . × r out (solid line) and 0 . × r out (dotted line).F ig . 10.— Top panel: estimates for the turbulent parameter α times thesquare relative half-thickness of the disk at r out for the six bursts of X-raynovae. Lower panel: corresponding estimates of the viscosity parameter ν at r = R ⊙ . mass input from the intermediate radii. Another possibility isthat the secondary maximum is triggered later after the peakand then it can happen at a radius ∼ r out (as in the model ofErtan & Alpar 2002).If an X-ray nova has the viscously evolving accretion diskwith constant r out during the interval of the ‘exponential de-cay’ (usually, first ∼
50 days after the peak), we can infer the value of the turbulent parameter α , using the closeness of themodel with steady visosity and the α -disk. As the irradiation,which provides the steadiness of the hot zone outer radius,does not change the vertical structure of the disk (Dubus et al.1999), we can adopt l = / α -disks. Combining Eqs. (36) and (41), we get α ≈ . r out R ⊙ ! / z / r out . ! − M M ⊙ ! − / (cid:18) t exp d (cid:19) − , (42)which relates the α -parameter to the observed t exp , the e -folding time of the bolometric luminosity or the inner ac-cretion rate. The last formula resembles quite a few othersfound in the literature. Similarly, the relation of α to the half-thickness of the disk cannot be eluded. However, an advatageis that (42) contains the e-folding time which can be inferredquite accurately from observations.We have sampled six FRED light curves in di ff erent X-raynovae (Fig. 9) and fitted them with the steady- ν viscously-evolving disk model. For particular values of t exp , the modelcurves agree well with the peak-normalized count-rates be-fore the second maximum, which is observed in most of thecases. For A 0620-00, we use the value of t exp , estimated byChen et al. (1997) from the 3 − Σ ( r ), as can be seen in the panel for A 0620-00.In Fig. 10 we plot the estimates for α -parameter times thenormalized square of the relative half-thikness at the outer ra-dius of the disk. Theses estimates are obtained using (42)with the values t exp indicated in Fig. 9. In the lower panel,the viscosity ν is plotted, found accordingly to (36). As ν isnot constant over radius, we choose to present its value at thesame radius R ⊙ for all cases. The considered X-ray novaehave known orbital periods. The masses of the primary andsecondary components, mass ratios are from Charles & Coe2 Lipunova(2006). For black hole candidate XTE J1753.5-0127 with P orb = .
24 hr (Zurita et al. 2008) we set rather arbitrarily M = −
15 M ⊙ and M sec = . − . ⊙ . The Roche lobes R RL are calculated using the formula by Eggleton (1983), and r out / R RL = . ± .
02. The vertical barrs in Fig. 10 reflectthe errors related to the uncertainties of the binary parametersand do not take into account uncertainty of t exp .Suleimanov et al. (2007) give the half-thickness of the outerparts for the stationary α -disks when the hydrogen is com-pletely ionized: z / r ≈ .
05 ( M / M ⊙ ) − / ˙ M / r / α − / , with the parameters’ normalizations equal to 10 g s − and10 cm. It can be rewritten for the factor in Fig. 10 as follows z / r . ! ≈ × M
10 M ⊙ ! − / ˙ M . M cr ! / r out R ⊙ ! / α − / . , where α is normalized to 0.1 and ˙ M cr is the conventional rateof accretion producing the critical luminosity: 0 . M cr c = . × ( M / M ⊙ ) erg s − . For the black hole masses in therange 3 −
15 M ⊙ , accretion rates in (0 . − .
5) ˙ M cr , α in0 . − .
5, and r out in 1 − ⊙ , the above factor changes betweenextremes ∼ . ∼ .
8. For 4U 1543-47 with parameters˙ M / ˙ M cr ≈ .
25 (Wu et al. 2010), M = . ⊙ (Charles & Coe2006), r out = . ⊙ = . × R RL and α = . z / r / . ≈ .
8. It follows from Fig. 10 that α canbe less than 1 in this case, but this is still not a self-consistentvalue. Apparently, the disk in this longer-period X-ray novawas not entierly involved in the viscous evolution that pro-duced the FRED light curve in 2002. Levels of α of the otherfive bursts (Fig. 10, the upper panel) agree with the estimatesfor the fully ionized, time-dependent accretion disks obtainedby various approaches in X-ray and dwarf novae (King et al.2007; Suleimanov et al. 2008; Kotko & Lasota 2012), in thedecretion disks of Be stars (Carciofi et al. 2012).The values of t exp , indicated in Fig. 9, are to be consideredwith caution. Note that the observed light curves are limited toan energy band. Spectral modeling might be required in manycases to obtain accurate t exp . If an X-ray nova during the burstis in the high / soft state, when the disk emission dominates thespectrum, the spectral modeling can provide T in ( t ) ∝ ˙ M / ( t )in a straightforward way. We expect t exp not to alter by a factorlarger than 2 for the X-ray flux dominated by the disk (prelim-inary resluts for 2-10 keV).On the other hand, some X-ray novae bursts proceed en-tirely in the low / hard state , such as the burst of XTE J1753.5.-0127 (Miller & Ryko ff M ( t )as the latter is defined by the process of the viscous matterredistribution in the outer disk. DISCUSSIONEquation (6) is written without the tidal-stress term, fol-lowing the proposition that it should be negligible everywherein the disk, except in a thin ring near the tidal truncation ra-dius (Ichikawa & Osaki 1994). According to Pringle (1991),e ff ects near the tidal-torque radius can be approximated with an e ff ective boundary condition ˙ M =
0. Numerical modelsconfirm that most of the tidal torque is applied in a narrowregion at the edge of the disk, where perturbations becomenon-linear and strong spiral shocks appear (Pringle 1991;Ichikawa & Osaki 1994; Hameury & Lasota 2005, and refer-ences therein). As Smak (2000) notes, in the case of dwarf no-vae, during outbursts, the observed values of the outer radiusappear to be approximately consistent with the theoretical pre-dictions. Hameury & Lasota (2005) argue that the action ofthe tidal torques are important also well inside the tidal ra-dius. They notice at the same time that observations do not al-low discriminating between the rapidly growing and smoothertidal stresses, as the model light curves are not strongly af-fected.The tidal radius for all mass ratios can be approximated as0 . ± .
02 of the average Roche lobe radius of a given com-ponent in circular binaries (Papaloizou & Pringle 1977). Thedisk can extend beyond this radius due to the high inclinationof the binary orbit (Okazaki 2007). E ff ects of the high eccen-tricity of the binary orbit on the disk truncation radius havebeen studied by various methods (e.g., Artymowicz & Lubow1994; Pichardo et al. 2005; Okazaki 2007).The gravitational influence of the secondary on the disk in abinary system leads to a number of interactions, non-resonant(or tidal) and resonant, any of those transferring angular mo-mentum from the disk to the binary (Artymowicz & Lubow1994). The tidal distortions generally lead to the truncationof the disk, whereas the resonances of di ff erent strength im-pact or do not the disk structure, depending on their growthrate and other conditions (Whitehurst & King 1991; Ogilvie2002). An example of a resonance action is the debated ex-planation of the superhamps in SU UMa stars (see, e.g. Lubow1991; Kornet & Rozyczka 2000; Hameury & Lasota 2005).In our mathematical set-up, we have presumed a fixed outerradius of the disk without going into details of its actual value.Thus the model is appplicable to the coplanar disks in low-eccentricity binaries. The outer radius of the disk can alsochange due to intrinsic processes. Reproducing the wholeevolutionary cycles of the dwarf novae and X-ray novae,Smak (2000), Dubus et al. (2001) should have taken into ac-count the variations of the outer disk radius, but these changesare significant during the time spans much greater than theviscous times. SUMMARYUnder some astrophysical conditions, viscously evolvingdisks formed in binary systems are e ff ectively truncated fromoutside. In the present work, we use the method proposed byL¨ust (1952) to find the Green’s functions for the linear viscousevolution equation for the disk with the finite outer radius.Green’s functions are obtained for the viscous angular mo-mentum flux F and the accretion rate ˙ M . Two inner boundaryconditions at the zero coordinate are considered. They cor-respond to the accreting disks and to the disks with no masssink at the center. The Green’s functions allow one to compute Σ ( r , t ) and ˙ M ( r , t ) for arbitrary initial surface density profilesthat develop into self-similar distributions on su ffi ciently longtime scale. The analytic formula to calculate the disk evolu-tion with the variable outer mass inflow is derived. A solution,which can be found with the use of the Green’s function, hasthe properties defined by the main equation of the viscous evo-lution and thus provides the basic description of the transientphenomena in the viscous disk.The Green’s functions are found in the form of quickly3converging series and can be easily reproduced with standardcomputer methods. The C-code written for the case of the ac-creting disc with the use of the GNU Scientific Library canbe downloaded .We present the relations between the rising time, e -foldingtime, and disk viscosity. For six bursts in the X-ray novae,which are of FRED type, we show that the model describeswell the peak-normalized light curves before the second max-imum. This favors the mainly viscous nature of their evolu-tion during this period and enables us to obtain an estimate ofviscosity ν , which depends on the outer disk radius.The models with time-independent viscosity are shown toapproximate well the evolution of α -disks during time inter-vals comparable to the viscous time. Consequently, estimatesof the α parameter may be derived for the disks in the bi-naries with known period and masses. These estimates relyon the relative half-thickness of the disk at the outer radius.The estimates of α for the five short-period X-ray novae out-bursts, GRO J0422 +
32 in 1992, A 0620-00 in 1975, GS 1124- 68 in 1991, GS 2000 +
25 in 1998, and XTE J1753.5-0127in 2005, are in line with the values estimated so far for thehot viscous disks (King et al. 2007; Suleimanov et al. 2008;Kotko & Lasota 2012).Another Green’s fucntion is found for the disk that has zeroaccretion rate at the inner radius and acquires angular mo-mentum from the central star. In the steady state, the mass ofthe disk cannot be very high and the disk has low luminosity.It radiates most of the rotational power transferred from thecentral star. This disk works as a gear transmitting the angu-lar momentum of the central star to the orbital motion. The‘dead’ stage is ended by an abrupt fall or trickle of the mat-ter on to the star after the centrifugal barrier has moved closeenough or beyond the magnetosphere’s radius.ACKNOWLEDGEMENTSThe author is grateful to N. I. Shakura, K. Stempak,J. L. Varona, M. Perez and the anonymous referee for use-ful comments. The work is supported by the Russian ScienceFoundation (grant 14-12-00146).APPENDIXINFINITE DISKS: GREEN’S FUNCTION AND BASIC RELATIONSGreen’s function for the infinite disk, obtained by LP74, can be written in the dimensionless form with our designations (c.f.formula (19)) as follows: G ∞ ( x , x , t ) = κ h / l c x l x − l t exp − x + x t κ h / l c I l (cid:18) x x t κ h / l c (cid:19) , where I l is the modified Bessel function of the first kind, x = ( h / h c ) / l , h c is some characteristic value of the specific angularmomentum, which we use instead of h out .To find the physical distribution of F ( x , t ) in the course of the evolution of the initial narrow ring of matter, initially located atthe coordinate x s , one takes the integral of F ( x ,
0) (24) with the kernel G ∞ on the interval x ∈ [0 , ∞ ]: F ( x , t ) = M disk h c l ( x x s ) l t exp − x s + x t κ h / l c ! I l (cid:18) x x s t κ h / l c (cid:19) , where M disk is the initial mass of the ring.The accretion rate at the center can be found from ˙ M in = ∂ F /∂ h | h → (see (21)). One finds˙ M in ( t ) = x − l l h c ∂ F ∂ x (cid:12)(cid:12)(cid:12)(cid:12) x → = M disk τ le Γ ( l ) e − τ e / t t + l , or ˙ M in ( t ) = ˙ M ∞ in , peak (cid:18) τ pl t (cid:19) + l e − τ e / t , where the typical times τ pl and τ e are introduced, τ e = κ h / l s = + le τ pl . We find that the accretion rate peaks at the time (25) t ∞ max = τ e / (1 + l ) with the value˙ M ∞ in , peak = M disk t ∞ max (1 + l ) l e + l Γ ( l ) . (A1)SOLUTION FOR THE CASE WITH VARIABLE ACCRETION RATE AT THE OUTER EDGE OF THE DISKFor infinite disks, see Metzger et al. (2012) and Shen & Matzner (2014), who have obtained the Green’s function solution tothe viscous evolution of a disk with mass sources / sinks, which are distributed in a fashion along the disk radius. http: // / software / gsl / http: // xray.sai.msu.ru / ∼ galja / lindisk.tgz ∂ F ∂ h = (cid:18) κ l (cid:19) h / l − ∂ F ∂ t ∂ F /∂ h | h = h out = ˙ M out ( t ) F | t = = F ( h ) . (B1)Let us substitute function F ( h ) by a function ˜ F ( h ) using the relation F ( h ) = ˜ F ( h ) + h h out ˙ M out ( t ) . (B2)This substitution is used, for example, to solve the problem of finding the temperature distribution in a cylinder, on a surface ofwhich a thermal flux is defined (Bogolyubov & Kravtsov 1998). For the new ˜ F ( h ) the problem is the following: ∂ ˜ F ∂ t = l κ ! h − / l ∂ ˜ F ∂ h + Φ ( h , t ) ,∂ ˜ F /∂ h | h = h out = , ˜ F | t = ≡ ˜ F ( h ) = F ( h ) − h h out ˙ M out ( t = , (B3)where Φ ( h , t ) ≡ l κ ! h − / l ˙ M out h out − h h out d ˙ M out d t . The corresponding Sturm–Liouville problem with a free variable ξ = h / h out is investigated in Sect. 3: it consists of Eq. (13) ∂ f i ( ξ ) ∂ξ + s i (cid:18) κ l (cid:19) h / l out ξ / l − f i ( ξ ) = , (B4)where s i or k i = s i κ h / l out plays a role of an eigenvalue, and the boundary condition ∂ f i ( ξ ) /∂ξ = ξ = ξ out . Changing to the freevariable x = ξ / l = ( h / h out ) / l , we can write for a non-integer lf i ( x ) = ( k i x ) l [ ˜ A i J l ( k i x ) + ˜ B i J − l ( k i x )] , where ˜ A i and ˜ B i depend on the inner-boundary condition. According to the Steklov theorem, the solution to (B3) can be obtainedin the form: ˜ F ( h , t ) = ∞ X n = u i ( t ) f i ( x ) , (B5)where coe ffi cients u i ( t ) are to be found from the inhomogeneous first-order di ff erential equation over t . Let us substitute (B5)into (B4) and (B3), multiply by f n ( x ) and integrate over x from 0 to 1, using the orthogonality property of the eigenfunctions. Wearrive at ∂ u n ( t ) ∂ t + s i u n ( t ) = Φ n ( t ) , u n | t = = φ n , (B6)with designations φ n = Z ˜ F ( x ) f n ( x )d x . k f n k ; Φ n ( t ) = Z Φ ( h ( x ) , t ) f n ( x )d x . k f n k , (B7)where k f n k is the norm of eigenfunction f n . Solution to (B6) is as follows u i ( t ) = φ i e − s i t + t Z Φ i ( τ ) e − s i ( t − τ ) d τ . Let us write down the solution to (B3) for the case ˙ M out =
0, that is, Φ ( h , t ) = F ( x , t ) = ∞ X i = φ i e − s i t f i ( x ) . ffi cients φ i = A i and B i for the Dirac delta function as theinitial condition and express F ( x , t ) using the Green’s function. In the analogy, the solution to the problem with Φ ( h , t ) , F ( x , t ) = Z ˜ F ( x ) G ( x , x , t ) d x + t " Φ ( x , τ ) G ( x , x , t − τ ) d x d τ , (B8)where G ( x , x , t ) is given by (19) or (31).Substituting (B8) into (B2) and using functions ˜ F and Φ from (B3), we finally arrive at (35). The coordinate integral in itssecond term can be taken analytically; it involves the Lommel functions and can be tabulated beforehand for the particular valuesof l and k i for the sake of the computational e ffi ciency. The second term of (35) for B i = h out x l X i J l ( k i x ) t Z exp − t − τ t vis k i l " l L ˙ M out ( τ ) t vis − L ¨ M out ( τ ) d τ , (B9)where L = Z x l − J l ( k i x ) d x . J l ( k i ); L = Z x l + J l ( k i x ) d x . J l ( k i ) . (B10) REFERENCESArtymowicz, P., & Lubow, S. H. 1994, ApJ, 421, 651Barenblatt, G. I. 1996, Scaling, Self-similarity, and IntermediateAsymptotics: Dimensional Analysis and Intermediate Asymptotics,Cambridge Texts in Applied Mathematics, Cambridge University PressBenedek, A., & Panzone, R. 1972, Revista de la Uni´on Matem´aticaArgentina, 26, 42Betancor, J. J., & Stempak, K. 2001, Tohoku Mathematical Journal, 53, 109Bogolyubov, A. N., & Kravtsov, V. V. 1998, Problems of MathematicalPhysics, ed. A. G. Sveshnikov, Moscow State University PressCannizzo, J. K., Lee, H. M., & Goodman, J. 1990, ApJ, 351, 38Carciofi, A. C., Bjorkman, J. E., Otero, S. A., Okazaki, A. T., ˇStefl, S.,Rivinius, T., Baade, D., & Haubois, X. 2012, ApJ, 744, L15Charles, P. A., & Coe, M. J. 2006, Optical, ultraviolet and infraredobservations of X-ray binaries, ed. W. H. G. Lewin & M. van der Klis,215–265, Cambridge University PressChen, W., Livio, M., & Gehrels, N. 1993, ApJ, 408, L5Chen, W., Shrader, C. R., & Livio, M. 1997, ApJ, 491, 312D’Angelo, C. R., & Spruit, H. C. 2010, MNRAS, 406, 1208—. 2011, MNRAS, 416, 893Dubus, G., Hameury, J.-M., & Lasota, J.-P. 2001, A&A, 373, 251Dubus, G., Lasota, J.-P., Hameury, J.-M., & Charles, P. 1999, MNRAS, 303,139Duschl, W. J., Strittmatter, P. A., & Biermann, P. L. 2000, A&A, 357, 1123Eggleton, P. P. 1983, ApJ, 268, 368Ertan, ¨U., & Alpar, M. A. 2002, A&A, 393, 205Filipov, L. G. 1984, Advances in Space Research, 3, 305Guadalupe, J. J., P´erez, M., Ruiz, F. J., & Varona, J. L. 1993, Journal ofMathematical Analysis and Applications, 173, 370Hameury, J.-M., & Lasota, J.-P. 2005, A&A, 443, 283Ichikawa, S., & Osaki, Y. 1994, PASJ, 46, 621Ivanov, P. B., Papaloizou, J. C. B., & Polnarev, A. G. 1999, MNRAS, 307,79Kato, S., Fukue, J., & Mineshige, S. 1998, Black-hole accretion disks, Vol.KyotKing, A. R., Pringle, J. E., & Livio, M. 2007, MNRAS, 376, 1740King, A. R., & Ritter, H. 1998, MNRAS, 293, L42Kornet, K., & Rozyczka, M. 2000, Acta Astronomica., 50, 163Kotko, I., & Lasota, J.-P. 2012, A&A, 545, A115Lasota, J.-P. 2001, New Astronomy Review, 45, 449Lin, D. N. C., & Bodenheimer, P. 1982, ApJ, 262, 768Lin, D. N. C., & Pringle, J. E. 1987, MNRAS, 225, 607Lipunov, V. M. 1992, Astrophysics of Neutron Stars (Springer-Verlag,Berlin)Lipunova, G. V., & Shakura, N. I. 2000, A&A, 356, 363Lubow, S. H. 1991, ApJ, 381, 259L¨ust, R. Z. 1952, Zeitschrift Naturforschung Teil A, 7, 87Lynden-Bell, D. 1969, Nature, 223, 690Lynden-Bell, D., & Pringle, J. E. 1974, MNRAS, 168, 603Lyubarskij, Y. E., & Shakura, N. I. 1987, Soviet Astronomy Letters, 13, 386 MacRobert, T. M. 1932, Proc. Roy. Soc. Edinburgh, 51, 116Menou, K., Hameury, J.-M., Lasota, J.-P., & Narayan, R. 2000, MNRAS,314, 498Metzger, B. D., Rafikov, R. R., & Bochkarev, K. V. 2012, MNRAS, 423, 505Meyer, F. 1984, A&A, 131, 303Meyer, F., & Meyer-Hofmeister, E. 1984, A&A, 140, L35Miller, J. M., & Ryko ff , E. 2007, The Astronomer’s Telegram, 1066, 1Mineshige, S. 1994, ApJ, 431, L99Novikov, I. D., & Thorne, K. S. 1973, in Black Holes (Les Astres Occlus),ed. C. Dewitt & B. S. Dewitt, 343–450, Gordon and Breach SciencePublishers, Inc.Ogilvie, G. I. 1999, MNRAS, 306, L9—. 2002, MNRAS, 331, 1053Okazaki, A. T. 2007, in Astronomical Society of the Pacific ConferenceSeries, Vol. 367, Massive Stars in Interactive Binaries, ed. N. St.-Louis &A. F. J. Mo ff at, 485, Astronomical Society of the Pacific ConferenceSeriesPaczynski, B. 1977, ApJ, 216, 822Papaloizou, J., & Pringle, J. E. 1977, MNRAS, 181, 441Pichardo, B., Sparke, L. S., & Aguilar, L. A. 2005, MNRAS, 359, 521Prandtl, L. 1925, Z. angew. Math. Mech., 5, 136Pringle, J. E. 1974, PhD thesis, , Univ. Cambridge, (1974)—. 1981, ARA&A, 19, 137—. 1991, MNRAS, 248, 754Rafikov, R. R. 2013, ApJ, 774, 144Shakura, N. I. 1973, Soviet Astronomy, 16, 756Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 337Shen, R.-F., & Matzner, C. D. 2014, ApJ, 784, 87Smak, J. 2000, New Astronomy Rev., 44, 171Sneddon, I. N. 1951, Fourier Transforms, International Series in Pure andApplied Mathematics (McGraw-Hill)Sturner, S. J., & Shrader, C. R. 2005, ApJ, 625, 923Suleimanov, V. F., Lipunova, G. V., & Shakura, N. I. 2007, AstronomyReports, 51, 549—. 2008, A&A, 491, 267Syunyaev, R. A., & Shakura, N. I. 1977, Soviet Astronomy Letters, 3, 138Tanaka, T. 2011, MNRAS, 410, 1007Tanaka, T., Menou, K., & Haiman, Z. 2012, MNRAS, 420, 705Watson, G. 1944, A Treatise on the Theory of Bessel Functions, CambridgeUniversity PressWeizs¨acker, C. F. V. 1948, Zeitschrift Naturforschung Teil A, 3, 524Whitehurst, R., & King, A. 1991, MNRAS, 249, 25Wood, K. S., Titarchuk, L., Ray, P. S., Wol ffff