Chapter 5 Thin Accretion Disks 1. Disk formation with oche lobe overflow In a non-rotating frame, the components of stream velocity at the L 1 point along and perpendicular to the instantaneous line of centers are 1/ 3 1/ 3 1/ 3 1 v ~ b1 ~ 100m1 (1 q) Pday kms, and v c s ~10 kms -1 for normal stellar envelope. 1
Implications: (1) The transferring material has high specific angular momentum, so that it cannot accrete directly onto the compact star. (2) The gas stream issuing through the L 1 point is supersonic, so that pressure forces can be neglected and the stream will follow a ballistic trajectory determined by the oche potential. 2
The initial trajectory would be an elliptical orbit lying in the binary plane. The presence of the secondary causes the orbit to precess slowly. The stream will therefore intersect itself, resulting in dissipation of energy. Meanwhile, the angular momentum is conserved. So the gas will tend to the orbit of lowest energy for a given angular moment, i.e. a circular orbit. 3
In most cases the total mass of gas in the disk is so small that we can neglect the self-gravity of the disk. The circular orbit is then Keplerian with angular velocity K ()=(GM 1 / 3 ) 1/2. The radius of this circular orbit is called the circularization radius circ, which follows from the relation 1/ 2 2 ( GM1 circ ) b1. Then circ /a = (1+q)(0.5-0.227log q) 4. 4
Within the ring of radius of circ, there will be dissipative processes, e.g. collisions, shocks, viscous dissipation, etc. These will convert some of the energy of the ordered bulk orbital motion into internal energy (heat). Eventually some of this energy is radiated and therefore lost from the gas. The gas has to sink deeper into the gravitational potential of the primary, orbiting it more closely. This in turn requires it to lose angular momentum. So most of the gas will spiral towards the primary through a series of approximately circular orbits. 5
The angular momentum is transferred outwards through the disk by viscous torques. The outer parts of the ring will gain angular momentum and will spiral outwards. The original ring of matter at = circ will spread to both smaller and larger radii by this process, to form an accretion disk. 6
To see it in detail, let s write the conservation equations for the mass and angular momentum transport in the disk due to radial drift motion. The mass of an annulus of (, +) is (2) 2, where H is the disk height and is the surface density. The angular momentum is then 2 2. 7
8 For the mass of the annulus, ) ( 2 ), ( ) ( )2, ( ), ( )2, ( ) (2 v t t v t t v t In the limit 0, we get the mass conservation equation 0 ) ( v t
The conservation equation of angular momentum is 2 2 (2 ) v (, t)2 (, t) t v t t 2 (, )2 ( ) (, )( ) ( ) G( ) G( ) 2 G 2 ( v ) where G(, t) 2 2 is the viscous torque exerted by the outer ring (here = d/d, and is the viscosity). 9
In the limit 0, we get ( t 2 ) ( v 2 ) 1 G 2. The equation can be further simplified by use of the mass conservation equation to be 1 G 1/ 2 1/ 2 v 3 ( ) 2 2 ( )' where we have used 3 1/ 2 ( ) K( ) ( GM / ). 10
We finally get the equation governing the time evolution of surface density in a Keplerian disk, t ( v ) 3 [ 1/ 2 ( 1/ 2 )]. The radial velocity is v 3 2 1/ 2 1/ ( ) ~ 11
As an example, the figure shows the spreading of a ring of matter with a Keplerian orbit at 0, under the action of viscous torques. The viscosity is assumed to be constant. The typical timescale of the ring s spreading is 2 ~ / v ~ / tvisc 12
Near the outer edge of the disk out ~ (0.70.8) L1 > circ, some other process must finally remove this angular momentum, and it is likely that angular momentum is fed back into the binary orbit through tides exerted by the secondary. 13
2. Viscous torque How the orbital kinetic energy is converted into heat? The differential rotation of the gas means that elements on neighboring streamlines will slide past each other. Because of thermal and/or turbulent motions, viscous stresses are generated. The angular momentum is transported by shear viscosity. 14
We now consider local mechanisms of angular momentum transport. First consider a uniform gas whose streaming motion is in the x-direction with velocity u(z). Consider the x-momentum transport across an arbitrary plane z = z 0 due to the exchange of fluid elements (or molecules) between the levels z 0, with the typical turbulent scale (or mean free path) and speed v. 15
There is no net mass motion across z 0, so the average upward and downward mass fluxes are the same. During the motion there is no force acting on the turbulent elements, so the linear momentum (and angular momentum) are conserved. The upward (downward) moving elements carry with them the x-momentum from a level ~z 0 z 0. The net upward x-momentum flux density is vuz 0 uz 0 vu z 0 16
The viscous stress exerted by the gas below on the gas above is given by u Txz ~ vu '( z0) z, where u = du/dz and is the dynamical viscosity. The kinematic viscosity is defined as, so v. 17
In the case of molecular transport, and v are the mean free path and thermal speed of the molecules respectively. In the case of turbulent motions, is the characteristic spatial scale of the turbulence and v is the typical velocity of the eddies. 18
Now consider the similar process in a thin, differentially rotating disk in polar coordinates (, ). Here the problem is which should be conserved, the angular momentum or linear momentum, when gas elements are constantly exchanged across the surface = const? 19
If the elements are not interacting with the streaming fluid, and subject only to external forces (e.g. gravity), the appropriate assumption is that angular momentum is conserved. If the effects of surrounding fluid (e.g. pressure gradients) must be included, the external forces are canceled by bulk rotation and pressure gradients in a steady state, so stream-wise momentum is conserved. The example of the first case is the ring of Saturn, while the later situation may be applied to accretion disks. 20
Following the same method, the net upward -momentum flux density is ~ v~ [( / 2) v ( / 2) ( / 2) v ( / 2)] v~ '. -direction per unit area is T ( / / ) '~ ~ v v v', yielding a kinematic viscosity ~ v ~ See Hayashi and Matsuda (2001, astro-ph/0102484), Subramanian et al. (2004, astro-ph/0403362), and Hayashi et al. (2005, Progress of Theoretical Physics, 113, 1183) for detailed discussions on the angular momentum transfer. 21
3. The magnitude of viscosity The momentum transfer in the disk means that there is a force acting in the -direction on the volume element due to shear viscosity, 2 v v f ~ v ~ v 2 2. Comparing this with the inertia terms in the Euler equation leads to the eynolds number 2 inertia v / v e ~ 2 viscous vv / v. ( v ) v 22
If e1, viscous force dominates the flow; if e1, the viscosity is dynamically unimportant. In the case of molecular viscosity due to Coulomb collisions between charged particles in the ionized gas, given by d and v ~ c s, e 14 15-3 mol ~ 210 ( n/10 cm ) m 1/ 2 1 1/ 2 10 T 5/ 2 4, for a typical accretion disk in an X-ray binary. Hence molecular viscosity is far too weak to bring about the viscous dissipation and angular momentum required. 23
A number of hypotheses have been proposed to explain the much larger effective viscosity in accretion disks. The most important of these are: (1) A turbulent viscosity resulting from random small-scale turbulent fluid motions in the disk, generated by the strong shear in the differentially rotating disk. (2) A magnetic viscosity associated with the magnetic Lorentz force in a disk containing magnetic fields. (3) Nonlinear (spiral) waves or shocks in the disk. (4) Outflows from the disk 24
We will discuss the first and second possibilities. Since little is known about turbulence, the most we can do is to place plausible limits on turb and v turb. First, the typical size of the largest eddies cannot exceed the disk thickness H, so turb H. Second, it is unlikely that the turnover velocity v turb is supersonic; otherwise the turbulent motions would be thermalized by shocks, so v turb c s. 25
Hence we can write the viscosity as = c s H with 1. This is the famous -prescription of Shakura and Sunyaev (1973, A&A, 24, 337). Note that with this semi-empirical approach all our ignorance about viscosity mechanism has been isolated in, which depends on other parameters and should not be taken as a constant. 26
The shear stress in a thin, Keplerian disk is 3 3 2 3 T ' cs HK cs P 2 2 2, so 2 T 3 P, which means that the viscous stress shouldn t exceed the gas pressure in the disk. 27
Another source of shear stress can be a magnetic field. The material in an accretion disk is usually ionized. This means that it can support electrical currents. These electrical currents generate a magnetic field, and the field and current together lead to a Lorentz force on the gas, f 1 1 L ( j B) [( B) B] TM c 4. The magnetic stress tensor is given by 2 1 B TM ( Î BB ) 4 2 28
The first term is the magnetic pressure, the second term B B/ is responsible for magnetic shear stress (magnetic tension). The idea of MHD turbulence was initially discussed by Velikhov (1959) and specifically developed by Balbus and Hawley (1991, ApJ, 376, 214; 376, 223). 29
Weakly magnetized accretion disks are subject to an axisymmetric shearing instability. The most important consequence of this instability is that the mechanism behind a generic means of transport in accretion disks has been elucidated. The underlying cause of turbulent structure in accretion disks stems from the tendency of a weak magnetic field to try to enforce corotation on displaced fluid elements, a behavior which results in excess centrifugal force at larger radii, and a deficiency at small radii. 30
Imagine that a magnetic field line initially connects two neighboring annuli in a radial direction, as shown by the dotted line. Because these two annuli have differing angular velocities, the field line will tend to become stretched as the shear proceeds (solid curve). The magnetic field will try to oppose the shear, and try to straighten out, which requires speeding up the outer annulus relative to the inner annulus, i.e. transferring angular momentum outward. 31
The image shows a cross section through a magnetized disk in which the magnetorotational instability has created turbulence. The blue indicates gas with less than Keplerian angular momentum; the red is gas with excess angularmomentum. (http://www.astro.virginia.edu/~jh8h/). However, recent MHD simulations of the magnetorotational instability by Fromang & Papaloizou (2007, A&A, 476, 1113) demonstrate that turbulent activity decreases as resolution increases. 32
3. Steady thin disks Assumptions (1) Steady disks: the external conditions (e.g. mass transfer rate) change on timescale much longer than the viscous timescale t visc 2 /, i.e. /t 0. (2) Thin disks: the disk height H() is much smaller than the radius. (3) Keplerian rotation: the angular velocity of disk material is Keplerian, i.e. (, z) K () (GM/ 3 ) 1/2. 33
Mass conservation t ( v ) 0 v constant M 2 ( v ) where M is the accretion rate (v ). 34
Angular momentum conservation ( t 2 v ) ( v 2 G(, t)=2 2 2 G const 2 2 ) 1 G 2 v const/(2 3 ) The constant is related to the boundary condition. 35
Suppose that the disk extends all the way to the surface of the central star, which is rotating at a rate < K (). The disk joins the star with a boundary layer with width b where the angular velocity of the disk material decreases from the Keplerian value K () to. There exists a radius b where = b and b K () with b for thin disks. 36
37 Then we have 2 1/ * * K 3 ) ( ) ( ) ( 2 const GM M v ] ) ( [1 3 2 1/ * M The viscous dissipation rate is independent of viscosity. ] ) ( [1 8 3 8 9 4 ' ) ( 2 1/ * 3 3 GMM GM G D
The luminosity produced by the disk between 1 and 2 is 2 3GMM 1 2 * 1/ 2 1 L( 1, 2 ) 2 D( )2d { [1 ( ) ] [1 2 3 1 1 2 2 ( 3 1 Let 1 and 2 we obtain the luminosity of the whole disk GMM 1 Ldisk Lacc 2 2 * * 2 ) 1/ 2 ]} 38
The vertical (z-direction) structure of the disk Assume that there is no motion in the z-direction, the hydrostatic equilibrium equation is 1 P GM [ ] 2 2 1/ 2 z z ( z ). For a thin disk (z) this becomes 1 P GMz 3 z, 39
2 cs GMH or 3 H for a typical scale-height H of the disk. H c v s K ( ) ( ) c s v K (the local Keplerian velocity is highly supersonic for a thin disk). 40
41 The radial velocity is highly subsonic s s 1 2 1/ * ~ ~ ] ) ( [1 2 3 2 c H c M v. Now consider the radial component of the Euler equation 0 1 2 2 GM P v v v 0 2 K 2 s 2 2 v c v v v c s v K vv K
The emitted spectrum Assume that energy transport is radiative, and the disk is optically thick, i.e. h (, Tc ) 1 (where is the osseland mean opacity), the radiation field is locally very close to the blackbody, the flux of radiant energy through the surface z constant is given by 3 16T T 4 4 F( z) ~ T ( z) 3 z 3 42
The energy balance equation is F Q z or F( H) F(0) 0 H Q ( z)dz D( ) 4 4 If Tc T ( H), it becomes approximately 4 4 Tc D( ) 3 43
Or the effective temperature of the disk is 4 T ( ) D( ) 3GMM T( ) { 8 For *, T T where T * 3/ 4 *( / * ) 3GMM ( ) 3 8 4.110 1.310 * 4 7 * 1/ 2 1/ 4 [1 ( ) ]} 3 1/ 4 M M 1/ 4 16 1/ 4 17 M M 1/ 4 1 1/ 4 1 3 / 4 9 3 / 4 6 K K 44
If we neglect the effect of the atmosphere of the disk, the spectrum emitted by each element of area of the disk is the blackbody with temperature T() I B 3 2h 1 [ T( )] 2 h / kt ( ) c e 1 The flux observed at a distance of D is 3 4 cos d h i c D e 1 out 2 out F I (2 d cos i / D ) * * 2 2 h / kt ( ) where i is the angle between the line of sight and the normal to the disk plane. 45
The spectrum is shown in the figure. If hkt( out ), B =2kT 2 /c 2 F 2 If hkt *, B =2kT 3 c -2 e h kt F 2 e h kt If kt( out )hkt *, F x e 5 / 3 1/ 3 0 x dx 1 1/ 3 46
Structure of the standard -disk The equations of the steady disk H H c 2 P c s kt P m T c 3 3/ 2 1/ 2 s /(GM) c p 3GMM 3 8 4 T 3c 4 c 4 4 1/ 2 * [1 ( ) ] (, ) T c M 3 csh * 1/ 2 [1 ( ) ] 47
The disk may be composed of a number of distinct regions (a) P r P g, T ff ( T 0.4 cm 2 g -1 ) 3 TM 4 H (cm) f 1.610 M16 f 8c 1 2 1/ 2 3/ 2 (gcm ) 23 M M f -3 2 16 1 8-1 2 1/ 2 5/ 2 v (cms ) 44 M M 16 6 1/ 4 T (K) 4.2 10 where f( * /) 1/2. 1 M 1/ 8 1 8 3/ 8 8 f 48
(b) P g P r, T ff H (cm) (gcm v T (cms (K) 5 1/10 8.0 10-3 -1 ) 1.9 10 4 5 ) 1.010 5 1/ 5 5.9 10 M 7 /10 4/ 5 M M 2 / 5 16 1/ 5 16 M 2/ 5 16 M M 2 / 5 16 M 3/10 1 7 / 20 1 M 1/ 5 1 11/ 20 1 9 /10 8 21/ 20 8 2/ 5 8 f f 1/ 5 33/ 20 8 f 2 / 5 3/ 5 f 2 / 5 The boundary between regions (a) and (b) lies on the radius 6 ab (cm) 2.510 2/ 21 M 16/ 21 16 M 7/ 21 1 f 16/ 21 49
(c) P g P r ff T ( = ff = 6.610 22 T -7/2 cm 2 g -1 ) 8 1/10 3/ 20 3/ 8 9 / 8 3/ 20 H (cm) 1.2710 M M f (gcm -3 ) 4.6 10 8 16 7 /10-1 4 4 / 5 v (cms ) 2.7 10 T (K) 4 1/ 5 1.4 10 M 3/10 16 M M M 1 11/ 20 16 3/10 16 1/ 4 1 M M 10 5/ 8 1 1/ 4 1 3/ 4 10 f 15/ 8 10 1/ 4 10 3/10 f f 11/ 20 7 /10 The boundary between regions (b) and (c) lies at 8 bc (cm) 2.910 M 2/3 16 M 1/ 3 1 f 2/ 3 50
51
4. Steady disks: confrontation with observation Accretion disks: Inner regions: closely related to the compact star. Outer regions: T 10 6 K, radiating predominantly in UV, optical and I. To study the outer regions of the disks observationally, we require that the light in one or more of these parts of the spectrum is dominated by the disk contribution. 52
Galactic X-ray sources (1) HMXBs The donor stars are O or B giants or supergiants, L opt 37 38 ergs -1, much higher than the UV and optical luminosity of the disks. (2) LMXBs and CVs The donor stars are late type, low-mass, faint stars, the accreting compact stars are neutron stars (black holes) and white dwarfs respectively. The processes of mass transfer are similar in these two types of systems, but L opt,lmxb 100 L opt,cv 53
This means that re-absorption of X-rays in LMXB disks is very important. So CVs are the best candidates for testing the theory of steady thin disks. 54
Evidence of circular motion of accreting material from eclipse of a double-peaked emission line from the optically thin gas in a CV (i) and (iv) Outside eclipse the line appears double-peaked because of the nearly circular motion in the disk around the white dwarf. (ii) The advancing side of the disk is eclipsed first, leading to the disappearance of the blueward component of line. 55
(iii) As the eclipse proceeds this side of the disk re-emerges and the receeding side is eclipsed, leading to the loss of the redward component and the re-appearance of the blueward component. 56
Doppler tomography of binary accretion disks (from Steeghs et al. 2004, AN, 325, 185) 57
Comparing the predicted spectra from optically thick disks with observations Method: eclipsing mapping the surface brightness distribution in an accretion disk. Because of the temperature distribution in the disk, the light at short wavelengths is strongly concentrated towards the central disk regions, while for long wavelengths the brightness distribution is almost uniform outside the central regions. 58
Hence if we observe a CV with a sufficiently high orbital inclination that the companion star eclipses the central regions of the disk, there should be a deep and sharp eclipse at short wavelengths and a shallower broader one at long wavelengths. 59
The figure shows the effective temperature distribution given by maximum-entropy deconvolution, compared with the theoretical temperature distribution for various values of mass transfer rates. 60
5. Irradiation of accretion disks The disks in LMXBs are probably heated by X-ray irradiation by the central accretion source. If the central source with X-ray luminosity L x can be regarded as a point, the flux crossing the disk surface is F Lx (1 )cos 2 4 where is the albedo (~0.9, de Jong et al. 1996, A&A, 314, 484), and is the angle between the local disk normal and the direction of the incident radiation. 61
/ 2 where tan dh/d, and tanh/. Since dh/d and H/1 for thin disks, we have cos sin( ) tan tan dh /d H / The effective temperature T irr resulting from irradiation is T L (1 ) H dln H ( )[ 1] 4 dln 4 x irr 2 62
The effective temperature of the disk is a combination the irradiation temperature and the viscous temperature T T T 4 4 4 eff irr vis In the outer part of the disk where T irr >> T vis, the structure of the disk changes as (Fukue, 1992, PASJ, 44, 669) 2 1/ 7 1/ 7 3/ 7 1/ 7 2 / 7 H/ 1.2 10 (1 ) M M (gcm v (cms -3-1 ) ) 7.4 10 8 8.7 10 4 1/ 7 T (K) 1.2 10 (1 ) 3 16 (1 ) (1 ) M 2 / 7 16 1 3/ 7 2 / 7 M 1 M 1/ 7 1 *6 M 2 / 7 16 2 / 7 *6 4 / 7 16 M 10 M 3/ 7 10 11/14 1 5/14 1 3/ 7 *6 2 / 7 *6 33/14 10 1/14 10 Note that the disk height H changes from H 9/8 to H 9/7. 63
If the accretor is a black hole, the irradiating source is the inner region of the accretion disk, and there is an extra factor ~H/ to the irradiating flux, see Sanbuichi et al. (1993, PASJ, 45, 443) for details. 64
Evidence for X-ray irradiation in LMXB disks van Paradijs & McClintock (1994, A&A, 290, 133) show that there is a strong relation between the absolute magnitudes in optical of LMXB disks and the X-ray luminosities. This can be explained as follows. In the temperature range encountered in LMXB disks the visual surface brightness S v of a blackbody emitter approximately varies as S v T 2. So we have 1/ 2 1/ 2 2 / 3 Lv Lx a Lx Porb. 65
6. Time dependence and stability easons for studying time-dependent disks (1) To check that the steady disk models are stable against smaller perturbations; (2) To get information about disk viscosity from time-dependent disk behavior. 66
Typical timescales (1) Dynamical timescale, the timescale on which inhomogeneities on the disk surface rotate, or hydrostatic equilibrium in the vertical direction is established. t ~ / v ~ 1 K (2) Viscous timescale, the timescale on which matter diffuses through the disk under the effect of viscous torques. t ~ 2 / ~ / visc v (3) Thermal timescale, the timescale for re-adjustment to thermal equilibrium. t th ~ c 2 s / D( ) ~ ( H / ) 2 t visc 67
We have the following relation t ~ t ~ ( H / ) 2 th t visc Or numerically t ~ t ~ (100 s) M th 1/ 2 3/ 2 1 10 t ~ (3 10 s) M M visc 5 4/5 3/10 1/ 4 5/ 4 16 1 10., 68
Thermal instability The thermal equilibrium at a given radius in the disk is defined by the equation Q Q, where Q and Q are the heating and cooling rates per unit surface respectively, or T 4 eff 9 8 2 K. Since,,the thermal equilibrium equation can be represented as a T eff () relation. This relation forms an S curve on the (, T eff ) plane. 69
Each point on the (, T eff ) S curve represents an accretion disk s thermal equilibrium at a given radius. The middle branch of the S curve corresponds to thermally unstable equilibrium. A stable disk equilibrium can be represented only by a point on the lower cold or the upper hot branch of the S curve. 70
This means that the surface density in the cold state must be lower than the maximal value on the cold branch: 0.83 0.38 1.14 2 max 13.4 c M 1 10 gcm, whereas the surface density in the hot state must be larger than the minimum value on this branch: 0.77 0.37 1.11 2 min 8.3 h M 1 10 gcm. For thermal instability, since t visc t th, and t < t th, we can assume thatconstant during the growth time and the vertical structure of the disk can respond rapidly towards hydrostatic equilibrium. 71
The disk is thermally unstable on the middle branch because radiative cooling varies slower with temperature than viscous heating d lnt d lnt 4 eff c d ln F d lnt visc c, so that when the central temperature T c in an annulus of the disk initially in thermal equilibrium is increased by a small perturbation, T c will rise further because the cooling rate is inadequate. 72
For example, consider the regions of the disk where gas pressure dominates pressure. In general we can write the opacity as n T c so that 2 n ~ H Tc / H. Since 1/ 2 H c s T c, we get 4 4 9 / 2 2 / n T eff Tc Tc. 73
So the left hand side of the inequality d lnt d lnt 4 eff c is 9/2n. d ln F d lnt visc c From the-prescription we havec s H T c and F visc T c. So the inequality becomes 9/2n<1, or n7/2. 74
Further analysis gives T eff 132n 4(72n). This relation shows that T eff /<0, i.e. the disk is unstable in regions when 7/2 < n < 13/2. 75
In fact the values of n in the unstable range will always occur in hydrogen ionization zone, i.e. wherever T eff is close to the local hydrogen ionization temperature T H ~6500 K. Hydrogen is predominantly neutral when T<T H, and increases rapidly with temperature, i.e., n > 13/2. For T > T H, hydrogen is essentially fully ionized, and n takes the Krammers value 3.5. So the opacity changes abruptly when T~ T H. 76
(Figure from Menou, K. 2001, ApJ, 559, 1032) 77
Limit cycle behavior During an outburst a point representing a local accretion disk s state moves in the (, T eff ) plane as shown in the figure. A point out of the S curve is out of thermal equilibrium. In the region to the right of the S curve heating dominates cooling, so that the temperature increases and the system-point moves up towards the hot branch. 78
On the left to the S curve is the case opposite and the point moves down towards the cool branch. These upward and downward motions take place in thermal time since they correspond to the heating and cooling of a disk s ring. During decay from outburst and during the quiescent phase of the outburst cycle, the system-point moves along, respectively the upper and lower branches in viscous time. 79
Dwarf novae Dwarf novae are erupting cataclysmic variables (CVs). In these binary systems outbursts take place in the accretion disk, which is formed around the central white dwarf by matter transferred from low-mass, oche-lobe filling companion star. Dwarf novae include three tupes: U Gem, SU UMa, and Z Cam, named after their prototypes. 80
All three types of dwarf novae show normal outbursts and only SU UMa stars also show superoutbursts. Normal outbursts have amplitudes of 2-5 magnitudes and last 2-20 days. The recurrence times are typically from ~10 days to years. (figure from http://observe.arc.nasa.gov/nasa/space/stellardeath/stellardeath_ 4b.html) 81
Superoutbursts have amplitudes brighter by ~0.7 magnitude, lasting ~5 times longer, and their recurrence time is longer than that of normal outbursts. (http://vsnet.kusastro.kyoto-u.ac.jp/vsnet/dne/wxcet.html) 82
The disk instability model for dwarf novae uses the limit cycle behavior expected if the disk contains regions of partial ionization, i.e., the mass transfer rate is lower than the critical mass transfer rate given by T eff ( out ) = T H M cr 9 2 310 ( P/3 hr) M yr -1 In quiescence (, t) lies between min and to increases outwards. max at each and tends 83
An outburst is triggered once rises above max at some radius. The disk annulus at that point makes the transition to the hot state; the mass and heat diffuse rapidly into the adjacent annuli, stimulating them to make the same transition. This leads to the propagation of heating fronts both inwards and outwards from the initial instability. The inward moving front propagates at a velocity c s. 84
In fact if has a single constant value no large outburst results, because the resulting S curve is rather narrow, i.e., max / min 2. Consequently the heating front does not propagate very far through the disk before the cooling wave begins to sweep inwards, shutting off the outbursts before it develops fully. It is usually adopted that h on the upper (hot) branch of the S curve and c on the lower (cold) branch. 85
Soft X-ray transients (X-ray novae): effect of irradiation Accretion disks in low-mass X-ray binaries also subject to the thermal instability. However, the irradiation of the disk has enhanced the effective temperature of the disk, so that the required mass transfer rate is considerably lower than the rate given by the dwarf nova condition T eff ( out ) < T H. 86
For an LMXB disk irradiated by a point source the criterion is where C is defined in Stability limits and parameters of Low Mass X-ray Binaries. Filled circles represent steady (i.e. non-transient) LMXBs containing neutron stars. The two asterisks correspond to two neutron-star LMXBTs. Diamonds represent black-hole LMXBTs with known recurrence times and down-pointing triangles those where only the lower limits for the recurrence time are known. The up-pointing triangle corresponds to GO J1655-40 with the recurrence time between the 1994 and 1996 outbursts. From astro-ph/0102072. 87
7. Tilted/warped accretion disks in XBs Observational clues of tilted accretion disks in XBs (1) The super-orbital variabilities observed in a number of X-ray binaries have long been interpreted as due to precession of a tilted accretion disk. (2) Jet precession SS433 (164 days) GO J1655-40 (3 days) CAL 83 (~69 days) 88
Driving mechanisms for disk tilt/warp (1) irradiation-driven wind (Schandl & Meyer 1994) (2) radiation pressure (Pringle 1996) (3) stellar magnetic field (Lai 1999, 2003) 89
Self-induced warping of accretion disks When an accretion disk is illuminated by a radiation source at its center, a twist or warp in the disk will be induced, because the surface of a warped disk is illuminated by a central radiation source in a non-uniform manner. Provided that the disk is optically thick, radiation received at a particular point on the disk surface is reemitted from that same point in the direction of the normal to the surface at that point, the back-reaction of the emitted radiation gives rise to an uneven distribution of forces on the disk surface. 90
The effect of these forces on a given annulus of the disk is to induce a torque on that annulus about the disk center. The effect of such a torque is to change the angular momentum of the annulus and so to change the twist of the disk at the radius. 91
According to Pringle (1996, MNAS, 281, 357), radiation driven warping occurs at radii S 2 8 ( ) 2 where S =2GM/c 2 is the Schwarzschild radius of the central star, 2 / where 2 is the vertical kinematic viscosity coefficient, and 2 L/ Mc. 92
It is very likely that disks in LMXBs are unstable to warping, but it is very unlikely to occur in white dwarf binaries. For example, for a 1 M accreting white dwarf with radius ~510 8 cm, and 1, warping occurs only for radii 510 13 cm, whereas for a 1 M neutron star with ~10 6 cm, warping occurs for radii 310 8 cm. 93
The figures show the numerically simulated results of disk warping by Wijers and Pringle (1998, MNAS, 308, 207). Left: the shape of a disk undergoing warping. ight: the behavior of the inclination of the outer disk with different dimensionless strength of radiation field. 94
8. Tides and resonances At the outer edge the disk experiences the tidal torque exerted by the companion star. The accretion disk is cut off at the tidal radius, tide 0.9 1, where 1 is the primary s oche lobe radius. Under certain circumstances (q0.25-0.33), tidal force causes the orbit of disk material to be eccentric, and precess on a period slighter longer than P orb. 95
esonance occurs in a disk when the frequency of radial motion of a particle in the disk is commensurate with the angular frequency of the secondary star as seen by the particle. This condition ensures that the particle will always receive a kick from the secondary at exactly the same phase of its radial motion, so allowing the cumulative effect of repeated kicks to build up and affect the motion significantly. 96
If the mean angular frequency in a given orbit is (measured in a non-rotating frame), and the orbit precesses at an apsidal precession frequency, the epicyclic frequency for the particle to return to the same radial distance is. The particle sees the orb. Thus the resonance occurs when k()j( orb ) where k and j are positive integers. 97
Assume that the orbit of the disk material is close to Keplerian, the radii jk of the resonant orbits near the j:k commensurability is j k 2 / 3 1/ 3 jk ( ) (1 q) a j This can be compared with the tidal radius tide shown in the figure. esonant orbits can only exist for sufficiently small mass ratio q; the j 3, k 2 resonances are responsible for superhumps in SU UMa systems only for q0.3. 98
eferences 1. Frank, J., King, A., and aine, D. 2002, Accretion power in astrophysics 2. Achterberg, A. 1996, Accretion in astrophysics 3. Lasota, J. P. 2001, astro-ph/0102072 99