Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://hea-www.harvard.edu/QEDT/Papers/ewol.ps Äàòà èçìåíåíèÿ: Tue Aug 29 00:44:42 1995 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 23:01:07 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: extreme ultraviolet

EVOLUTION OF AN ACCRETION DISK IN AGN
Aneta Siemiginowska 1;2 , Bo—zena Czerny 2 & Vadim Kostyunin 3
1 Harvard -- Smithsonian Center for Astrophysics, 60 Garden St. M/S 4, Cambridge, MA 02138
2 Copernicus Astronomical Center,Bartycka 18, 00­716 Warsaw, Poland
3 Institute of Astronomy, Russian Academy of Sciences, 48 Pyatnitskaya St, Moscow 109017, Russia
August 28, 1995

Abstract
We present results of the modeling of thermal­viscous ionization instabilities
in accretion disks in AGN. This instability, analogous to the instability which drives
outbursts in cataclysmic variables (Smak 1984b) or X­ray transients (see Mineshige
& Kusunose 1993, for review), is expected also to occur in accretion disks around
supermassive black holes (Lin & Shields, 1986; Clarke & Shields 1989; Mineshige
& Shields 1990) unless self­gravity effects or irradiation prevent the existence of the
partial ionization zone (which causes the instability). We assume that viscosity scales
with the gas pressure (fi­viscosity). We discuss in detail evolution of the accretion
disk around a M bh =10 8 M fi black hole mass, assuming a range of accretion rates
corresponding to luminosities between 10 \Gamma5 \Gamma 1, in Eddington units.
We show that the ionization instability does operate in such disks. Depending
on the viscosity, the instability can develop in a very narrow unstable zone thus
causing a flickering, or can propagate over the entire disk resulting in the large
amplitude outbursts on the timescales of order 10 3 ­10 5 years. As a consequence:
(1) AGN are not always in their active state; and (2) the emitted optical­UV spectra
are quite different from the spectrum of a thin stationary accretion disk. External
irradiation is shown to have a large effect on the instability and on the emitted
optical­UV spectra. We briefly consider observational consequences.
Subject headings: accretion, accretion disks ­ galaxies: active ­ quasars
1

1. INTRODUCTION
The accretion process that is the leading model for powering quasars (e.g. Rees
1984) is frequently described by the model of a stationary thin disk (Lynden­Bell 1969,
Shakura & Sunyaev 1973). However, there are both observational and theoretical
arguments indicating that the time­dependent effects are of extreme importance.
Observationally, the evidence for global evolutionary effects is compelling in
a number of accretion disks around Galactic X­ray sources. Outbursts (by factors
? 10 4 ) of cataclysmic variables or X­ray novae last for weeks or months and happen
every few months to a number of years. The outbursts are essentially caused by the
disk thermal instability in the partial ionization zone (Meyer & Meyer­Hoffmeister
1982, Smak 1982, see also Cannizzo 1993b for recent review), although some secondary
effects due to variability in the outflow rate from the companion star are also observed
(e.g. the secondary maximum in X­ray novae outbursts, Chen, Livio & Gehrels 1993).
The conclusion from the study of transient Galactic sources is that neither the gas
supply rate (that determines the instantaneous accretion rate), nor a stationary disk
model can account for the observed disk properties.
There is a strong similarity between Galactic X­ray sources and AGN both in
spectral behavior and overall variability (Fiore & Elvis 1994, Tanaka & Lewin 1995)
which leads us to expect similar accretion disk behavior in AGN. However, as the
characteristic timescales scale roughly with the central mass the expected variability
takes thousands to millions of years in AGN. This is necessarily much longer than
the observed variability timescales, and so the time­dependent disk models have not
been widely applied to AGN. However, such models need to be investigated since
the emitted spectrum predicted by these models can differ strongly from that of
stationary disk models. Moreover, physical models for the observed evolution of
the quasar luminosity function (e.g. Cavaliere & Padovani, 1988) will be affected if
quasars undergo recurrent strong outbursts.
Theoretically, accretion disks around the massive black holes in AGN are
expected to have a partial ionization zone, as in Galactic binaries, and therefore
to be subject to the same instability. This zone forms at a distance of a few hundred
Schwarzschild radii (¸0.01 M 8 pc) from the center (Lin & Shields 1986, Clarke 1987)
unless self­gravity effects or irradiation of the outer parts prevent its existence.
Current models of time evolution of accretion disks in AGN have confirmed
the presence of disk eruptions, but they appeared to be of minor amplitude (Clarke &
Shields 1989). In these models the viscosity parameter ff (Shakura & Sunyaev 1973)
for hot and cold gas was assumed to be the same. However, stronger outbursts can
be produced if the viscosity of the cold gas is considerably lower than the viscosity of
the hot gas (Mineshige & Shields 1990), as seems to be the case for CV disks (Smak
1984a,b).
2

The ratio of the hot and cold state viscosity parameter (ff hot ; ff cold ) between
these two states depends on the source and may be as high as 5 (e.g. in SS Cyg,
Cannizzo 1993a). Therefore the use of a variable viscosity in AGN disk models seems
to be worth exploring.
The models of accretion disks in AGN studied by Mineshige & Shields (1990)
did not cover the range of luminosities appropriate for bright Seyfert galaxies and
quasars because of numerical difficulties (Mineshige 1990). The parameters of these
models were set for low luminosity objects with a small central black hole mass.
The case of a massive black hole (10 8 M fi ) was considered only for a very low
accretion rate (L/LEdd ¸ 10 \Gamma4 \Gamma 10 \Gamma5 ). Also the effects of irradiation were completely
neglected although irradiation strongly modifies the predictions. This can be seen
from stationary models, as well as from the time evolution of a single disk zone
(Tuchman, Mineshige & Wheeler 1990, Mineshige, Tuchman & Wheeler, 1990).
In this paper we use the approximate but very efficient method of Smak
(1984a) to calculate the evolution of disks for the relevant range of parameters.
We study a broad range of external accretion rates corresponding to luminosities
ranging from the value as low as 10 \Gamma5 up to 1, in Eddington units. The lowest
values are characteristic of starving disks, like the one postulated at the center of our
Galaxy (Falcke and Heinrich 1994), while higher values are appropriate for luminous
quasars. We constrain ourselves to fi disks, i.e. models in which the viscous torque
is proportional only to the gas pressure (as in Clarke & Shields, 1989 and Mineshige
& Shields, 1990). This avoids the Lightman­Eardley (1974) instability in the inner,
radiation dominated parts of ff disks (i.e. models in which the viscous torque is
proportional to the total, gas plus radiation, pressure). This instability has not been
observationally confirmed, nor is it theoretically understood. However, if it is present,
it certainly operates on much shorter timescales. Luminous AGN are known to vary
on short timescales, but there is no clear connection between this variability and
the Lightman­Eardley instability. In this paper we will consider only the ionization
instability which operates on timescales of ¸ 10 5 years.
We show that the ionization instability does operate in accretion disks sur­
rounding supermassive black holes. Depending on the viscosity, the instability can
develop in a very narrow unstable zone, or can propagate over the entire disk resulting
in (1) large amplitude outbursts (? 10 4 ); and as a consequence (2) accretion disks
around supermassive black holes at the center of AGN are not always in their active
state so many low luminosity AGN may contain an accretion disk in quiescence; and
(3) the emitted optical­UV spectra are quite different from the spectrum of a thin
stationary accretion disk. Finally external irradiation is shown to have a large effect
on the instability and on the emitted optical­UV spectra. We briefly consider some
further observational consequences.
We discuss the local disk structure and the scaling laws in Section 2, Section
3 contains the details of the method and in Section 4 we present our results. We
3

discuss irradiation effects in the Section 5. Section 6 contains a discussion of some
observational consequences.
4

2. LOCAL STRUCTURE OF AN ACCRETION DISK AROUND
A MASSIVE BLACK HOLE
2.1. Scaling law for the \Sigma\GammaT eff S--curve.
We consider a geometrically thin Keplerian accretion disk around a super­
massive black hole. The local structure for such disks (assuming that viscosity is
proportional to gas pressure) is described by standard equations (e.g. Frank, King
& Raine 1992). We integrate these equations at a given radius using a procedure
described by Pojma'nski (1984, 1986). We included convection in our vertical structure
calculations however, we do not consider vertical advection (Kley, Papaloizou & Lin
1993). We assumed that the density at the surface of the disk is low (ae(H) = 10 \Gamma15 g
cm \Gamma3 ). Opacities were taken from Cox & Stewart (1969) for log T eff ? 4.0 and from
Alexander (1975) for lower temperatures.
There is a local equilibrium relationship between a steady--state accretion rate
—
M (or effective temperature, T eff ) and a surface density (\Sigma) for a given central black
hole mass -- M bh , viscosity parameter -- ff, and a distance r from the center. This
relation has a characteristic S--curve shape for an optically thick, geometrically thin
accretion disk. Three characteristic regions on the S--curve describe different physical
conditions in the disk 1 . The lower branch is thermally stable; cool and neutral
hydrogen dominates the chemical composition of the gas and molecules strongly
contribute to the opacities. A disk on the upper branch is also thermally stable
but is hot and the hydrogen is fully ionized; bound­free transitions in heavy metals as
well as free­free transition and electron scattering generate the opacity. The middle
branch corresponds to a partially ionized disk, which is thermally unstable due to the
rapid increase of opacity with temperature (Bath & Pringle 1982, Faulkner, Lin &
Papaloizou 1983). Small perturbations in the cooling--heating balance cause cooling
or heating of the disk until a stable condition on the upper or lower branch is achieved.
This thermally unstable situation has been analyzed by many authors for cataclysmic
variables, LMXRB, soft X­ray transients (eg. Huang & Wheeler, 1989, Mineshige &
Wheeler 1989), and in less detail also for AGN (Lin & Shields 1986, Clarke 1987,
Clarke & Shields 1989, Mineshige & Shields 1990).
We can calculate a local vertical structure of the disk at every radius and so
derive the S­curve for each location. The curves in Fig.1 show the local disk structure
(\Sigma vs. T eff ) at log r(cm) equal to 15:0; 15:5; 16:0 for a black hole mass of 10 8 M fi .
The dependence on the viscosity parameter ff is also shown.
1 for models with Ü rOE = ffP total there exists a fourth, thermally unstable, region where P rad AE P gas
5

Smak (1984a) parameterized disk solutions for dwarf novae using a standard S­
curve (\Sigma­T eff ) and the position of the point on the curve having the minimum surface
density (point A in Figure 2). Siemiginowska (1990) adopted this parameterization
for accretion disks around supermassive black holes:
logT r
eff = logT eff \Gamma 0:12logr 15 + 0:03logff 0:1 (1)
log\Sigma = f(logT r
eff ) + 1:05logr 15 \Gamma 0:69logff 0:1 ; (2)
where ff 0:1 = ff=0:1; r 15 = r=10 15 , T r
eff ­ effective temperature reduced to log r = 15:
and ff = 0:1. The coefficients were determined numerically.
We have estimated the accuracy of this parameterization by means of a
comparison of the surface density profile calculated from the vertical structure
equations with the profile calculated by scaling. An example is shown in Fig. 2
(for particular values of the black hole mass and accretion rate). As expected, the
parameterization works well ( ?
¸ 5%) in the middle parts of the disk where there is no
influence of the inner boundary condition and the dependence of all the parameters on
radius are monotonic and well approximated by power laws. Close to the marginally
stable orbit the representation of the disk structure by scaling laws is accurate only
to ?
¸ 10%. Departures from an exact scaling are also visible (at ¸ 20%) in the outer
parts of the disk, far from the radius used to define the S­curve (see Fig.2)
2.2. Radial Extent of the Instability Strip.
The thermally stable upper branch of the S­curve (``hot'' phase) is charac­
terized by an accretion rate higher than the external stationary value, while the
stable lower branch (``cold'' phase) has a lower than stationary accretion rate, and
so accumulates gas. For a given external accretion rate there is usually a range of
radii where the neither of these stationary solutions is stable, and in which the cyclic
instability is active.
We can estimate the size and the location of the unstable zone by finding inner
and outer radii, r A and r B , of the zone. Using the parameterization of the radius vs.
effective temperature relation (eqs. 1,2) we find the radius, r A , described by the local
values of \Sigma A and TA :
r A ¸ 1:6 \Theta 10 16 M 0:4
8
—
M 0:4 ff \Gamma0:05
0:1 cm; (3)
where M 8 = M bh =10 8 M fi , and —
M accretion rate in M fi yr \Gamma1 . The temperature TA on
the standard S­curve is log(T A ) = 3:64. The scaling relations then give us the radius
for any (stationary) effective temperature T:
log(r=r A ) ú \Gamma1:4 log(T=TA ): (4)
6

This expression gives the radius of the turning point B (Fig. 1., log(T B ) = 3:54) to
be r B ¸ 1:4r A , similar to the value given by Lin & Shields (1986).
However, the entire region between the points C and D on the S­curve is
affected by the limit cycle to some extent. A better estimate of the position and the
width of the instability strip is given by r D and r C (see Fig. 1). On the standard
curve (log(T C ) = 3:23; log(T D ) = 3:76) these are
r D ¸ 0:68r A and r C ¸ 5:5r D ¸ 3:8r A : (5)
The location of the instability zone depends on the accretion rate (Eq.3). The
unstable region is located close to the black hole for low accretion rates and far away
when the accretion rates are high. Since the bulk of the energy release in the disk
is at small radii we expect much stronger luminosity variations when the accretion
rates are low. The locations of the inner and outer edges of the instability strip for
the assumed model parameters are given in Table 1.
The position and the extent of this region is also sensitive to the adopted shape
of the standard S­curve (Mineshige & Shields, 1990). In particular if the viscosity
parameter for the cold phase ff cold = 0:025 (1/4 that for the hot phase) then the
temperatures TC and TD become more widely separated (Fig. 6., log(T D ) = 3:89,
log(T C ) = 3:01), and the coefficient in the equation (3) becomes equal to 0.41 close to
the inner radius of the disk (see Table 1). In such models the instability develops over
the entire disk, instead of being constrained to an instability belt, as for a constant
viscosity parameter and moderate accretion rates.
The characteristic temperatures also give a rough estimate of the ratio, Ü , of
the time spent by the disk in the high state t high relative to the time spent in the low
state (quiescent) t low . Ü is approximated by the ratio of the blackbody cooling times:
Ü = t high
t low
ú ( TB
TA ) 4 (6)
This gives Ü = 0:4 in the constant ff model (with ff = 0:1), but Ü = 0:03 in the model
with ff cold = 0:025. So only for a few percent of the time the disk remains in the
outburst phase when ff cold = 0:025 and ff hot = 0:1. This has significant observational
consequences which we discuss in Section 6. The accuracy of this estimate is good
only to a factor of a few, but shows a clear trend in the duty cycle and allows us to
estimate the basic properties of the solution.
2.3. Timescales.
The surface density and the local accretion rate evolve on the viscous timescale
(Lightman 1974) t visc ¸ r=v r (where v r -- radial velocity). For stationary ff--disk
models with parameters scaled to AGN values,
7

t visc ¸ 2 \Theta 10 5 ff \Gamma0:8
0:1
—
M \Gamma0:3 M 0:25
8 r 1:25
16 f 1:2 yrs (7)
where r 16 = r=10 16 cm, f = [1 \Gamma (3r g =r) 0:5 ]. Thus the timescale on which the global
luminosity variations caused by the ``ionization instability'' can occur is ¸ 10 5 years
for a 10 8 M fi central black hole.
The local thermal timescale is described by t th ¸
(1=ff\Omega K ) and in our case it
is
t th ¸ 2:7 ff \Gamma1
0:1 M \Gamma0:5
8 r 1:5
16 yrs (8)
So
t visc =t th ¸ 10 5 ff 0:2
0:1
—
M \Gamma0:3 M 0:75
8 r \Gamma0:25
16 f 1:2 (9)
for standard accretion disk models (e.g. Frank, King & Raine 1992).
In CVs the ratio t visc =t th ¸ 10 2 , 10 3 times smaller than for AGN. This difference
has important consequences since changes in AGN disks which happen on the thermal
timescales will not then propagate over the entire disk as fast as in Galactic sources.
We can expect to observe variations on the thermal timescales in a quasar, while it
is impossible to observe global disk evolution on the viscous timescale. Instead the
global evolution of a nuclear disk must be studied through statistical departures from
stationary accretion in a population of quasars.
3. CONSTRUCTION OF TIME­DEPENDENT SOLUTIONS
3.1. Local Disk Structure ­ Time evolution under thermal instability.
When solutions for the local vertical structure are defined by points on the local
S­curve (Fig.1) the disk is locally in thermal equilibrium. We assume that solutions
which are not in thermal equilibrium are also described by the standard disk equations
(e.g. Frank, King & Raine 1992), but with an extra term (Smak 1984a):
dF 0
dz
= dF j
dz
+ dF th
dz ; (10)
where F 0 is the total radiation flux, F j is the standard energy flux generated by
viscous forces;
dF j
dz
= 3
2
ff\Omega KP gas (11)
and F th describes the extra flux due to vertical expansion or contraction of the disk.
In the simple case of a perfect gas and homologous expansion and contraction
the second term in eq. (10) can be transformed to:
dF th
dz
= \Gamma
` 3
2
dlnT e
dt
+ dlnH
dt
'
P gas ; (12)
8

where T e is the temperature in the equatorial plane and H is the disk thickness.
Therefore it can be represented as:
dF th
dz = 3
2 ff
th\Omega KP gas (13)
(Smak 1984a) where ff th describes thermal effects (ff th ! 0 in the heating case and
ff th ? 0 for cooling). The parameter ff th is a convenient parameterization of deviations
from thermal equilibrium such that models with ff th = 0 are in thermal equilibrium
and others are out of thermal equilibrium. This parameterization allows us to define
the quantity ff eff = ff + ff th . The total flux from the disk surface can then be defined
by:
F 0 = oeT 4
eff = F j + F th ; (14)
where
F j = ff
ff eff
F 0 (15)
F th = ff th
ff eff
F 0 (16)
3.2. The Temperature Equation.
The partial flux resulting from contraction+cooling or expansion+heating, can
be related to the total energy changes (including the internal energy and the potential
energy which depends on the height of the disk) using
\Gamma
2F th
\Sigma = dE
dt
= @E
@ \Sigma
d\Sigma
dt
+ @E
@T eff
dT eff
dt
(17)
where E = E (\Sigma; T eff ) = E(T r
eff ), where T r
eff (\Sigma; T eff ) is given by Eq.(1). Eq.17 can
be rewritten as:
@lnT eff
@t
= \Gamma
`
2F th
E + @lnE
@ln\Sigma
@ \Sigma
@t
'
`
\Sigma @lnE
@lnT eff
' \Gamma v r
@lnT eff
@r
: (18)
Cooling by advection (in which energy is carried inward with the accreting gas
as internal energy) was not included in our calculations, although it may be important
when the radial temperature gradient is very steep. The importance of the advection
was analyzed by Abramowicz et al. (1989) for high accretion rates (also recently
by Narayan & Yi (1994, 1995) for very low accretion rates) with viscosity scaled by
total pressure in an accretion disk around a supermassive black hole. We can expect
that at the boundary of the unstable region the temperature gradient will be quite
large, that a radial cooling flow may occur. This effect may help to smooth out the
discontinuity at the edge of the unstable region, and may affect the timescale on
which the instability travels through the disk.
9

3.3. The Surface Density Equation and the Boundary Conditions.
In the case of Keplerian motion the surface density evolution may be written
(Pringle 1981, Smak 1984a):
@ \Sigma
@t
= 3
r
@
@r
` p
r
@
@r
(š \Sigma p
r)
'
+ 1
2úr
@ —
M
@r
+ 1
úr
@
@r
`
( p
r \Gamma
p
r out ) @ —
M
@r
'
; (19)
where r out outer radius of the disk and š is the kinematic viscosity coefficient given
in ff disks by š \Sigma = 8F
j\Omega
\Gamma2
K =9.
The local accretion rate is given by
—
M (r) = 6úr 1=2 @
@r
(r 1=1 š \Sigma): (20)
Equation (19) describes a disk which is continuously supplied not only with the
matter but also with angular momentum. Only a fraction of the angular momentum
is transported to the central object (black hole) because the angular momentum of
the matter at the marginally stable orbit is approximately equal l 0 = (GMr in ) 1=2 .
The inner edge of the disk r in is situated at 3r g , and below this the matter is freely
falling onto the black hole.
In the case of binary systems, the excess of angular momentum is subtracted
from the disk by the tidal forces due to the presence of the companion. However,
in AGN accretion disks the supply mechanism is unknown. It seems most probable
that global dynamical instabilities are responsible for transporting the gas available
from distances of a few parsecs or kiloparsecs to the nucleus but there are no actual
predictions for the resulting conditions at the outer edge of the disk. Therefore we
simply postulate that the excess of the angular momentum is subtracted at the outer
radius in such a way as to produce stationarity of the disk outside the instability
strip. The amount of angular momentum `outflow' can be determined from the global
balance between the inflow rate ( —
M l out ) and the storage rate in the stationary case
( —
M l 0 ).
If we incorporate this stationarity requirement at the outer edge equation (19)
is changed into
@ \Sigma
@t
= 3
r
@
@r
` p
r
@
@r
(š \Sigma p
r)
'
+ 1
2úr
@ —
M
@r
+ 1
úr
@
@r
`
( p
r \Gamma
p
r in ) @ —
M
@r
'
: (21)
To solve the equations conveniently we use variables x = 2 p
r and S = x\Sigma
(Bath & Pringle 1981).
3.4. Method.
10

Two equations determine the time evolution of an accretion disk: eq (18) and
eq (21), give directly the surface density and the effective temperature. Scaling law
(eqs. 1­2) determines the new viscosity coefficient ff eff , which allows us to calculate
the viscous flux (eqs. 14­15) and the departure of the model from the thermal
equilibrium (eq. 16). The local accretion rate is given by eq. 20, and the radial
velocity results from the standard continuity equation.
Using these equations we have developed a numerical code based on that of
Smak (1984a) and have applied it to the case of accretion disks around supermassive
black holes. The disk was divided into N (45­100) annuli in the radial direction.
For each annulus the S­curve was defined by the parameterization of the standard
S­curve (eqs. 1­2). This parameterization allows the code to run on widely available
computers and still complete a disk structure calculation on a short time (usually
about one hour on IBM 58H or several hours on a SPARC IPX station). Mineshige &
Shields (1990) calculated their model using detailed vertical structure calculations at
each step separately, which is a more accurate method. However, this method requires
the use of supercomputers, takes more time, and for some parameters, especially large
central masses (10 8 M fi ), leads to numerical problems (Mineshige 1990).
4. RESULTS OF THE EVOLUTIONARY CALCULATIONS.
Two different kind of models can be distinguished for the assumed character
of the viscosity parameter. The first type has constant parameter ff, while the second
type has the viscosity dependent on the phase of the disk, and is different at the
hot and cold phases. We always assume that ff hot = 0:1 and vary only ff cold . All
computations were run for a central black hole mass of 10 8 M fi . Table 1 contains
the values of accretion rates and ff cold for sets of models we have calculated. The
boundaries of the unstable zone (r C ; r D ) are also shown for each model. We discuss
the two classes of models in the next sections.
4.1 Models with Constant Viscosity Parameter [ff=0.1].
This family of models show only relatively weak variations of the total
luminosity, as was noticed for low accretion rates by Mineshige & Shields (1990).
We discuss results of our calculations for the model C0. We also discuss other cases
briefly. We compare the two models with the lowest accretion in the Table 1 (model
A0, A1) directly with the Mineshige & Shields models.
The luminosity variations due to thermal and viscous instabilities developing
in the accretion disk (model C0: —
M = 0:1M fi yr \Gamma1 , and constant viscosity parameter
ff = 0:1) are shown in Fig. 3a. The luminosity variations are very small (\Sigma0:1 in log
L). They are not regular and show some clear substructure. Effectively the changes
11

in the luminosity happen on timescales of order of 10 4 years, shorter than estimated
by the eq. 9, although the major outbursts take rather 10 5 years, as predicted.
The instability does not propagate strongly beyond the unstable region. This
is illustrated in Fig. 4c. by variations in the local accretion rate. Major changes can
be seen inside instability strip and only small fluctuations occur outside that region.
This gives only small variability in the effective accretion rate and in the bolometric
luminosity.
The surface density and effective temperature for the hot and cold phases are
plotted for model C0 in Figs. 4a and 4b. The radial surface density distribution in
the high state generally evolves towards lower values thus approaching the limiting
values for the onset of the cooling front given by log \Sigma A (r) in the Fig. 4a. As the
distribution is not a monotonic function of a radius the region close to the minimum
becomes unstable first. The cooling front starts to propagate in both directions. In
the meantime mass accumulates in the disk and the surface density increases until it
reaches the limit given by \Sigma B (a straight line parallel to log \Sigma A (r) but shifted up).
The outbursts start at the radius at which the surface density is equal to the critical
local \Sigma B (see Fig. 1). This may happen before the cooling front completes its travel
through the entire unstable region. Therefore the disk never actually reaches the
quiescent state. This is reflected in the shape of the light curve, which shows the
flickering.
The disk eruptions start somewhere in the middle of the instability strip and
show significant irregularity. This is related to the fact that the value of \Sigma A is very
close to the \Sigma B value. As a results the two timescales ­ the viscous timescale (for
stationary accretion) at the outer edge of the instability strip and the viscous timescale
(in the cold state) at the inner edge of the strip ­ are almost equal. The first timescale
determines a supply of the mass and the second one the build up of the mass in the
disk. If either one is considerably shorter than the other then the outbursts always
start either at the inner or at the outer edge of the instability strip, while if the two
are similar the outburst starts in the middle of the instability strip.
The location of the unstable region for a given central mass depends on the
outer accretion rate (eq. 3). For very low accretion rates the unstable region is situated
closer to the center and influences the regions of strong gravitational potential and
significant energy generation. In Figs. 5a and 5b we show the luminosity and local
accretion rate variations for model A0 (which has an accretion rate ¸1/3000 that
of model C0). The inner edge of the instability strip is now situated very close to
the inner edge of the disk and the entire inner part of the disk is strongly affected
by the instability. This is reflected in the amplitude of variations (\Delta log L = 0:9).
This model corresponds to the model F0 of Mineshige & Shields. The amplitude of
the luminosity variations are of the same order and the timescales compare closely.
However, lightcurves do not follow each other closely since the scaling approach to
the \Sigma(r) distribution in our models predicts the flickering mode even close to the
12

marginally stable orbit. These models are not clearly applicable to quasars as the
average bolometric luminosity is at the extreme low end of the luminosity function
of observed AGN (see Tables 2 and 3). There is also a theoretically based argument
that such low accretion rates may produce instead an ion torus (e.g. Begelman 1985)
rather than a geometrically thin disk.
4.1.1. Spectra.
The amplitude of luminosity variations is not significant for constant viscosity
parameter models. However, as the changes in the structure of the outer cold parts
of a disk are considerable they are reflected in significant modifications of the disk
spectrum in the IR/optical band. We calculate these spectra assuming local black
body emission. The spectral energy distribution for the two states, hot and cold, for
model C0 is shown in Fig. 3b. In the hot state a bump in the spectrum can be seen
due to the additional emission from the regions in the high state. However, the UV
luminosity as well as the bolometric luminosity may be higher in the low state. The
inner regions which contribute the most to the UV emission, are not affected strongly
by the instability in the unstable zone except in model A0 (with the lowest accretion
rate, see Fig.5c). The changes in the local accretion rates in the inner regions are
seen a long time after the unstable zone has gone through the hot state (the viscous
timescale governs the evolution of the inner stable parts while in the unstable zone the
characteristic timescale is directly related to the instabilities continuously developing
there). Spectral changes in the optical and UV are therefore essentially independent
and uncorrelated.
A good representation of the output spectrum is the mean spectral index
(F š ¸ š ff ) in the frequency range 10 14:5 ­ 10 15:0 Hz (3000 ­ 9500 š A). This provides
a straightforward comparison with the available AGN samples (see Table 3). We
selected four independent AGN samples for which spectral slopes are available (Table
3). The spectral slopes, the range of the optical luminosities and the average
luminosity for each sample is given in the Table 3.
The mean index calculated from model C0 over the entire cycle is 0.11 +
\Gamma 0:18
which fits the observed spectral index of \Gamma0:2 +
\Gamma 0:4 in bright quasars (Neugebauer et
al. 1987; Elvis et al. 1994, Fiore et al. 1995) better than the 0.33 index given by
stationary models. Nevertheless it is not a good fit. Values for higher accretion rates
fare no better (Table 2). Because the instability zone moves further out as —
M is
increased the spectrum is determined mainly by the steady, hot inner regions and so
the index approaches the standard stationary disk model value. For the model with
the lowest accretion rate the spectral index is closer to the observed value. However,
as we have already mentioned, for such low accretion rates the luminosities are much
too low and the geometrically thin disk approximation may not be valid.
13

The range of the observed luminosities (43:5 ! log L ! 47:0) is higher than
the range of averaged luminosities covered by our studies (41:70 ! log L ! 46:20; see
Table 2). Our calculations have been carried out for only one central mass (10 8 M fi ),
so the average luminosity depends only on the accretion rate in out runs while, in
general the luminosity also depends on the black hole mass. Adopting a range of
masses, we may easily model the entire observed luminosity range.
The choice of mass also determines the location of the peak of the spectrum
for a given luminosity. If the spectrum peaks near or below the optical/UV band the
calculated UV slopes may become very steep, and would not apply to the observed
values. We can clearly see that our models brighter than log L ¸ 44 tend to have
spectra rising too strongly in the UV to agree with the data (Table 3). It is possible,
in principle to obtain more luminous models with a spectral peak in the UV, but only
by raising the mass by two orders of magnitude or more.
4.1.2. Summary.
We conclude that in constant viscosity parameter ff accretion disks around
supermassive black holes, instabilities develop inside a well defined unstable zone.
The instability results in periodic changes of the optical/IR tail of the disk spectrum
with negligible effect on the UV and on the bolometric luminosity. The amplitude
of bolometric luminosity variations is very small. The mean optical/UV spectra
generated from these models (assuming local black body emission) are significantly
steeper than the spectra predicted by stationary models, but are still not steep enough
(for all the models except the one with the lowest accretion rate) to match the
observed spectral distribution. The situation is very different for two­valued viscosity
disks.
4.2 Instability in accretion disks with a two­valued viscosity parameter.
The origin and the nature of the viscosity in accretion disks is unknown.
However, detailed studies of some Galactic systems show that the viscosity must
be different in the hot and cold states. For example in SS Cygni ff is up to 4 --5 times
smaller in the cold state than in the high state (Cannizzo 1993a). Similar variable
viscosities may well apply to accretion disks in AGN. This is at least consistent with
the limits on ff based on the thermal timescale in the radiation pressure dominated
region (Siemiginowska & Czerny 1989, Czerny & Czerny 1986). Two types of radial
relation for the viscosity parameter ff have been developed: (1) discrete values of ff
for the disk in hot (ff hot ) and cold (ff cold ) state (Smak 1984a, Cannizzo 1993a); (2)
a continuous change with the effective temperature ff ¸ ff 0 ( h
r
) n (Meyer & Meyer­
Hoffmeister 1984). Observations of binaries do not clearly discriminate between these
two cases. Mineshige & Shields (1990) used the second relation in their models.
Since h/r is smaller for accretion disks in AGN than in CV's there should be a
14

systematic difference between the viscosity parameters in both cases. Mineshige &
Shields multiplied their equation (2.12) by a factor of 10 3 to avoid such differences.
We adopt the first prescription in our modeling and assume the ratio of ff hot =ff cold
from the CVs' studies. The effective viscosities should be the same in both cases.
We discuss models assuming that the viscosity on the lower branch is 0.025,
0.05 or 0.075 while keeping the viscosity coefficient on the upper branch always equal
to 0.1. The standard S­curves for these models are different from the ff cold = ff hot
case (Fig.6). The effective temperatures and surface densities of the unstable region
cover an increasing range as the ff hot =ff cold ratio increases, and the outer radii of the
unstable zone are located further out, relative to the case of constant ff.
A ratio between the two extreme values of the surface densities (\Sigma A =\Sigma B )
characterizes the outbursts. By decreasing the value of the viscosity in the low state
this ratio increases. Smak (1984a,b) noticed that the quiescent state was not reached
in any of his models when ff was the same in hot and cold states. The instability
was triggered again before the low state was reached by the disk after the previous
outbursts. As a result the heating and cooling transition fronts are always present
in such disk. However, a pure quiescent state after an outburst was achieved if the
ratio between the two critical surface densities was higher then 2­3 (Cannizzo 1993b).
Our models reach quiescence only if ff cold Ÿ 0:025 (then ff hot =ff cold –4), and only in
these models do we obtain strong outbursts and global changes in the disk structure
corresponding to strong and well separated outbursts.
In Fig.7 we compare the bolometric luminosity variations when the instability
develops in the disk around 10 8 M fi black hole and accretion rate 0.1 M fi yr \Gamma1 (models
C0, C2, C3) for the three considered values of ff cold . Only very small luminosity
fluctuations are present when ff cold =0.1. In the case of ff cold = 0:05 the modulations
become stronger ( +
\Gamma 0:55 in log L), but the low state of the disk is never reached. Only
in the last case, ff cold = 0:025, the large luminosity variations, well separated in time
are present.
The repeated total luminosity variations accompanying the developing insta­
bility are shown in Fig.8a (model B3). The large amplitude (\Delta log L ¸ 4) on a
timescale of about 10 6 years can be seen. The period is much longer than in the
previous model (x4.1) which is related to the wider unstable zone, and to the fact
that the instability travels through the whole disk.
The evolution of the effective temperatures and the surface densities for this
model (B3) are plotted in Figs. 9a,b and 10a,b. The mechanism of the developing
outbursts is exactly the same as in the case of the constant viscosity. However,
there are significant qualitative differences. In the cold, low state the surface density
increases with radius. The matter flows slowly through the disk and accumulates at
the inner parts. At certain point the critical value of surface density is reached at the
radius located inside the instability strip (\Sigma B (r); a straight line shifted up from \Sigma A (r),
15

see Fig. 9a) and the thermal instability starts. It propagates in both directions. Since
the surface density gradient is now steeper than in the constant viscosity case, infall
is favored and there is a high rate of matter transfer inwards. However, the peak of
the surface density also travels outwards and the rapid infall starts after it had passed
through a given radius (see Fig. 9a, 10a). The peak of the surface density propagates
up to the end of the instability strip, where there is no more matter supply and the
cooling starts. The transfer into the cold state happens more gradually (see Figs. 9b,
10b). The cooling starts at the outer edge and propagates into the inner regions
until the entire disk reaches the cold, quiescent state. Once in the cold state the disk
accumulates matter up to the point when the local conditions at a given radius put
it into the high state and the instability begins once again.
During the hot state the matter flows through the disk very quickly and the
local accretion rate is high. Variations of the local accretion rate are presented in
Fig.11a,b. The most dramatic changes happen inside the instability strip, where
a large discontinuity forms and locally matter can flow also outwards. However,
there is always infall beyond the instability strip and changes in the local accretion
rate propagate over the entire disk generating large luminosity variations. The
average luminosity and the dispersion for each model are given in Table 2. While
the dispersion is very small in all the models with ff hot =ff cold ! 4 it is significantly
larger in the models with a larger ratio. All models with ff cold = 0:025 show changes
of order a few in log L. The amplitude increases with the increase in the accretion
rate, since the size of the unstable region grows with the accretion rate too.
The outbursts discussed by Mineshige & Shields (1990) start always at the
outer edge of the instability strip and move inwards, opposite to the situation in most
of our models. The reason is that the dynamics of an outburst depends on the local
shape of the S­curve, and in particular on the mass supply timescale at the outer
edge and the accumulation (viscous) timescale in the low state at the inner edge,
as discussed in Sec. 4.1. When the viscous timescale is longer than the accretion
timescale the outburst starts inside the instability zone, while in the opposite case it
always starts at the outer edge. The definition of viscosity assumed by Mineshige &
Shields (1990) forces the second situation.
4.2.1. Spectra.
The large variations in the local accretion rates affect the emitted spectrum
significantly. The evolution of the spectral energy distribution for the outburst and
decay phases (model B3) is plotted in the Figs. 12a,b. In the low state the accretion
rate is very low and the peak of the emission moves into the IR part, while the
luminosity drops below the average values observed in bright AGN. The disk in the
hot state contributes to the optical­UV part of the spectrum and the spectral index
varies over the outburst and decay phases.
16

We compute the distribution of the spectral slopes in the optical/UV range.
However, in this case we do not include the entire cycle as the luminosity at the low
state is frequently below the level which makes the AGN signature detectable and
such objects would not be included in an AGN sample. Therefore we adopted an
arbitrary limit of log(šL š ) of 42.0 at log(š)=14.8. We checked that the results do
not depend significantly on this value. We also exclude the spectra for which the
luminosity peaks below 3000 š A since there is no data to support such spectral shape
for bright AGN. These values are quoted in the Table 2. The mean spectral index for
one cycle in the model B3 ( —
M = 0:01M fi yr \Gamma1 ) is equal to 0.13 +
\Gamma 0:54 and it is smaller
for models with higher accretion rates. The values of spectral index are between ­0.28
(model B3) and ­1.15 (model D3), for the disk in the high state, in which an outburst
extends over the entire instability strip (curve 7 in Figs. 9, 10, 12).
As the ratio of the two values of the viscosity coefficient increases the median
value for the spectral slope increases (e.g. from 0.02 to 0.31 for the B models). This
reflects the fact that during certain stages (see curves 8­10 in Figs. 9, 10, 12) the
innermost part of the disk dominates and spectrum is closer to a single­temperature
black body than a stationary disk. Such spectra are even more discrepant from the
observational data than single viscosity models. However, when the ratio of the two
viscosity parameters is 4 the outbursts are so strong that any luminosity limited
sample has to be biased towards high state models. Therefore we perhaps should be
looking at the high state spectral index only. If so, these models are quite promising,
particularly when compared with Neugebauer (1987) or Francis (1991) sample (see
Table 2 and 3).
4.2.3. Summary
For the models with ff hot =ff cold the instability develops over the entire disk
causing luminosity variations with the amplitude as large as \Delta log L ¸ 4 for the
models with the ff hot =ff cold =4. The light curves show outbursts clearly separated in
time. The disk remains in the hot state for only a few percent of the entire cycle.
The luminosity in the hot state is close to the Eddington value. The optical spectral
index varies a lot over the entire cycle. In the very high state the spectrum is even
harder than that of a stationary disk, close to that of a single black body, while in
the intermediate state it becomes close to the observed one. In the low state the disk
does not contribute to the optical­UV region but is relatively bright in the IR.
5. THE EFFECT OF CORONAL IRRADIATION
Observations show clearly that a considerable fraction of the energy of the
accreting gas is released close to the black hole in the form of X­ray emission (thermal
or non­thermal) and that substantial fraction of the X­ray emission suffers Compton
17

reflection from a `cold' medium (Mushotzky, Done & Pounds, 1993). (There is no
clear connection between this emission and an accretion disk, although a number of
models have been suggested). In the disk models this `reflector' is the disk, which is
then strongly irradiated.
Fully self­consistent calculations should include this irradiation when com­
puting the disk structure, as in the one zone model by Mineshige, Tuchman &
Wheeler (1990). The irradiation does not only modify the disk spectrum (Malkan
1991, Siemiginowska et al 1994) but influences its time evolution. We are unable to
follow such an evolution with our code because irradiation changes the shape of the
S­curve making the scaling technique is inapplicable. We can, however estimate the
importance of irradiation by calculating the increase in effective temperature due to
different levels of irradiation.
If a fraction of the X­ray photons originating close to the black hole hits the disk
surface these photons are either scattered or absorbed in an upper layer of Thomson
optical depth of order of a few. The scattered, or Compton­reflected component,
predicted by Lightman & White (1988) is observed in many Seyfert galaxies (e.g.
Pounds et al. 1990, Nandra & Pounds 1994) and contains 15 to 30% of the total
luminosity (e.g. —
Zycki et al. 1994) as expected if half of the primary X­rays is
reprocessed and the other half goes directly to an observer. The rest of the energy
goes into heating the absorber and eventually is re­emitted in the form of lines,
recombination continua and free­free emission (Dumont & Collin­Souffrin 1990 a,b;
Ross & Fabian 1993). As the disk is optically thick most of the re­emission forms a
thermal component (e.g. Hur`e et al. 1994).
For simplicity, we assume that the X­ray radiation flux intercepted by the disk
surface is thermalized with 100 % efficiency.
The irradiation may be either direct, due to the exposure of the disk surface
to the X­ray radiation, or indirect, due to the scattering of the X­rays by extended
fully ionized corona. In this paper we constrain ourselves to the second possibility. At
every radius r we assume that 10% of the total X­ray flux is intercepted by the disk
(LX =4úr 2 , where LX is the X­ray bolometric luminosity assumed to be connected
to the accretion rate at the inner disk radius through the relation LX = 0:1 —
Mc 2 ).
Such a description of the scattering of X­rays in the corona gives roughly the order of
magnitude for both the flux and its radial distribution. Accurate computations show
a strong dependence of the scattered flux on the ratio of luminosity to the bolometric
luminosity, as well as on the corona temperature, and give somewhat steeper decrease
of the flux with the radius (e.g. Kurpiewski 1994). This simple description allows
us to estimate the effect of irradiation on the evolving accretion disk spectra. An
example of the change of the effective temperature distribution and the resultant
spectra due to irradiation in the high state and in the low state in models C0 and B3
are shown in Fig. 13 and 14. The optical spectral indices for all models are given in
Table 2.
18

We can see that strong irradiation causes a profound change in the effective
temperature distribution (a factor of 2 in the instability strip) and results in a different
spectral shape. As the irradiating flux decreases more slowly with radius than the
flux generated in the disk, the enhancement of the disk radiation flux is stronger
in the outer parts leading to significant steepening of the spectra in low states and,
even more strongly, in high states. These models well fit the spectral range given
by the data (spectral index can be in order of ­0.7 as in the Cristiani sample, see
Table 3). Detailed conclusions, however, can be made only after full time­dependent
calculations of the irradiated disk are done.
19

6. DISCUSSION
6.1 Have Low State Disks been detected?
Since the amplitude of the bolometric luminosity variations can reach four
orders of magnitude it is interesting to ask whether the ``activity'' of AGN is connected
to its accretion disks being in the high state, while the ``inactivity'' of normal galaxies
or faint AGN is due to them having quasars with a disk in the low state.
There are just few normal, or weakly active, galaxies with detailed studies of
their nuclei. Fabbiano (1988) presented a spectral energy distribution (SED) from
radio to X­rays for the low luminosity nucleus of the spiral galaxy M81 which shows
UV, radio and X­ray activity. Relative to the rest of the SED the UV luminosity is
lower by an order of magnitude than in regular active AGN. Moreover, the stellar
contribution to this UV luminosity is rather high and the true non­stellar component
could be significantly lower. This abnormally weak UV bump strength may indicate
that we are observing an accretion disk in a low activity state in M81 (c.f. Fig. 12).
In the low state there is no contribution from the disk emission into the UV while
in the high state the ratio between the UV and IR fluxes is about ¸ 2. Scaling the
observed UV/IR emission of the M81 nucleus to that appropriate for a high state disk
values produces a UV bump of normal strength.
To determine whether the nuclei of low luminosity objects are indeed in the
low state we need to constrain L/LEdd , which requires estimation of the mass of the
central black hole. We cannot constrain the mass using the optical­UV spectra when
the disk instability is present, so other methods are needed. Surprisingly the best
method so far comes from recent observations of water masers in NGC 4258. These
water masers are located at distances only a fractions of a parsec away from the
central mass (Miyoshi etal. 1995). This gives a good measurement for the central
mass, and so central black hole mass, of 3.6\Theta10 7 M fi . The nuclear X­ray flux from
ROSAT HRI observations is of the order of ¸ 3\Theta10 39 erg s \Gamma1 (Cecil et al. 1995).
Combining these implies LX ¸5\Theta10 \Gamma7 LEdd . If we relate the X­ray emission to an
accretion rate at the inner edge of an accretion disk (as seems to be the case in the X­
ray transient sources, Huang & Wheeler, 1989; McClintock, Horne & Remillard 1995)
then this luminosity constrains the accretion rates to very low values ( —
M ¸ 10 \Gamma6 M fi
yr \Gamma1 ), suggesting that we see the disk in its low state. This galaxy is a known radio
source with an extraordinary ``braided'' jet (Cecil, Wilson & Tully 1992). Further
detailed study of the radio and X­ray structure may give more information about its
activity in the past.
The double­peaked line profiles observed in some radio­galaxies (Eracleous
& Halpern 1994) may originate in an accretion disk. For these radio­galaxies the
starlight contribution to the continuum emission near Hff is much higher than for
single­peaked radio­galaxies. About 9% of the total continuum emission in the objects
with the single peaked profiles comes from the starlight while in the objects with the
20

double­peaked profiles more than 50%. As a result the central optical­UV continuum
component is much weaker in the objects with double­peaked lines than in the other
objects, possibly indicating the presence of a disk in the low state.
6.2 Long Term Variability and Quasar Evolution.
Cavaliere & Padovani (1988) have studied luminosity functions for different
types of quasar evolution models. One possibility they consider is a recurrent model,
in which a quasar has a short active phase related to the outer fuel supply, for example
from mergers. Our modeling suggests another intrinsic mechanism controllinf the
matter supply into the innermost central disk region ­ the ionization instability in
ff cold ! ff hot disks. In our models the luminosity varies, over the cycle, between
10 \Gamma5 LEdd and LEdd and objects remain in the low state for a large fraction of the
time. Such objects would contribute to the low luminosity end of the luminosity
function while the rarer, high state objects would populate the high luminosity end.
Since low luminosity AGN are much more common than high luminosity AGN, this
model at least gives the correct sense of the quasar luminosity function. A detailed
analysis of the luminosity function generated by a population of objects with disk
instability driven activity is presented in Siemiginowska et al. (1995). We note only
that in such an evolutionary model the mass of the central black hole does not increase
significantly, since the average accretion rate is very small, even when the active state
luminosity reaches Eddington values. As a consequence central black holes can be
present in normal galaxies and, at the same time may not be very massive, even if
they accrete matter continuously over hundreds of millions of years.
Since the characteristic timescales for the thermal­viscous instabilities are of
the order of 10 4 ­10 5 years one method of studying the long timescale activity of
quasars can be to use the `proximity effect' (Kovner & Rees 1989, Dobrzycki &
Bechtold 1991). The characteristic recombination time for the Ly­ff forest clouds
is about 10 4 years. If there is any significant change in the UV luminosity of a quasar
it must be reflected in the Ly­ff forest clouds at different distances from the quasar.
We may be able to see the variations when observing the Ly­ff forest absorption lines
in the spectra of apparent pairs of quasars, which are at different redshifts but close
in angular separation (Bechtold 1995, Carswell 1995).
6.3. Location of the Unstable Zone vs. the Characteristic Radii.
There are some constraints on the location of the unstable zone we have to
consider. We can compare the inner radius of the instability strip r D (Eq. 5) with the
transition radius, r ab , from the gas dominated to the radiation pressure dominated
unstable region in a standard ff disk (Shakura & Sunyaev, 1973):
r ab = 1:8 \Theta 10 15 M 1=3
8
—
M 16=21 ff 2=21
0:1 cm
21

¸ 0:11r AM
\Gamma1=15
8
—
M 38=105 ff 1=21
0:1 (23)
Thus both instabilities (Lightman­Eardley and ionization) will be present in
ff­disks in the same regions, and probably cannot be treated separately. The global
disk evolution we have described for fi­viscosity (Ü rOE ¸ P gas ) model may be invalid
if the doubly unstable ff­viscosity (Ü rOE ¸ P total ) is actually a better representation
of disk properties. Clearly, spectral predictions for the ultraviolet region would be
affected since this part of the spectrum comes from the innermost region undergoing
its own instabilities (in the ff­viscosity case). Luminous AGN are variable in the
optical­UV band on timescales of days to years. Therefore, regardless of the origin
of the observed variability we cannot expect the disk to be stable (at least in the
innermost regions) on short timescales either. There might be a connection between
this variability and the Lightman­Eardley instability, but making such connections
requires better theoretical understanding of this instability.
The second problem with the applicability of the model is related to the self­
gravity effects which are estimated to be important beyond the radius (Toomre 1964,
Clarke 1989):
r sg = 1:7 \Theta 10 16 M 1=3
8
—
M 4=9 ff 2=9
0:1 cm: (24)
A significant part of the instability strip lies within this region (see Fig. 4). This may
affect the appearance of the ionization instability and also change the overall evolution
of an accretion disk. Gravitational instabilities have been examined as a source of
the viscosity (Paczy'nski 1978). They can cause fragmentations of the outer parts of
the disk (Sakimoto & Coroniti 1981) and also lead to star formations (Shlosman &
Begelman 1989).
22

7. SUMMARY
The ionization instability operating in cataclysmic variables and also, most
probably responsible for X­ray transients, may well operate in AGN accretion disks
causing evolutionary changes on timescales of hundreds of thousands of years.
This variability has two basic consequences:
(i) the massive black hole does not always have to be in an active stage;
(ii) the spectrum of an accretion disk is not that of a stationary model.
The effects of the thermal­viscous instability depend on two parameters only:
an accretion rate and assumed viscosity prescription. The accretion rate determines
the location and dimension of the unstable region. If it is large then the hydrogen
ionization zone is far away from the center and the instability does not develop. On
the other hand for very low accretion rates the unstable region is situated close to
the inner radius of the disk, causing significant luminosity variations even in the case
of a constant viscosity parameter.
Viscosity plays an important role in the development of the instability in
the disk. For the constant viscosity parameter ff the instability propagates only
inside a very narrow unstable zone. Therefore the resulting luminosity variations
depend critically on the location of this zone, and so on the accretion rate. The main
contribution to the luminosity comes from the hottest, innermost regions which are
stable for high accretion rates and unstable for very low accretion rates.
In the case of low accretion rates the unstable zone moves closer to the center
and the infuence of the instability on the disk emission is clearly visible. However,
even in this case the amplitude is small ( +
\Gamma 0:4 in log L) and the average luminosity
is too low to be appropriate for AGN. There is no full quiescence of the disk reached
after the outburst which reflects in the shape of the light curve.
Models with a non­constant viscosity prescription have much broader unstable
zone covering a significant part of the disk and therefore they show much more
violent behavior. For the largest studied ratio between the viscosity in the high
and cold states the changes in the local accretion rates are very high. For these
models (ff cold = 0:25ff hot ) the variations are dramatic ( +
\Gamma 4:0 in log L). The hot and
cold phases of the disk are clearly separated in the light curve. After the outburst
the disk reaches quiescence and accumulates matter.
We have shown that effect of the disk evolution on the disk spectra also depends
on the description of the viscosity. However, the typical effect is just a slight softening
of the spectrum in constant viscosity case, not enough to make models compatible
with the observed spectral distribution, or even a hardening of the spectrum in the
case of non­constant viscosity.
23

A much stronger effect on disk spectra than the evolution itself is expected
due to the irradiation if some fraction of the accretion energy is converted into hard
X­rays, as observed, and the X­ray flux irradiates the disk surface via scattering in
the extended corona. Irradiated disk spectra compare well with the observational
samples. This approach is therefore worth pursuing. Future models should include
the effect of the irradiation on the disk structure (and not just on disk spectra) since
the feedback may well change the evolutionary pattern significantly.
We did not include advection in our evolutionary modeling. Since it may
affect the overall appearance of instability it should be considered in the future
modeling. Also self­gravity, which affects the outer regions of the disk, should
be included in overall studies of accretion disks in AGN. An interesting problem
comes from the possibility that there may exist two co­existing unstable zones in
supermassive accretion disks: one related to the ionization instability and the other
to the instability of the radiation pressure dominated regions. The presence of both
instabilities will result in different variability patterns which we cannot predict from
the present studies.
ACKNOWLEDGEMENTS
This work was supported in part by grant 2P30401004 of the Polish State
Committee for Scientific Research and NASA grant NAGW--2201. Part of the
computations have been done in Osservatorio Astronomico di Roma in Monteporzio
(Italy) during AS visit. The numerical code to calculate the disk evolution was
developed from the original dwarf novae code kindly provided by Prof.J.I.Smak. We
thank Jill Bechtold, John Cannizzo, Martin Elvis, Pepi Fabbiano and J.I.Smak for
lots of helpful discussions. We thank the anonymous referee for helpful comments.
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Figure Captions
Figure 1. S­curves (\Sigma \Gamma T eff ) at different radii (log r =15.0, 15.5, 16.0 cm) and ff=0.1
are plotted with solid lines. Short­dashed lines indicate the S­curves at log r = 15:5
and different ff (0.2, 0.1, 0.05). Central black hole mass M bh =10 8 M fi and Ü rOE = ffP gas .
The critical points A B are indicated for minimum and maximum surface densities.
Points C, D represent the maximum and minimum effective temperatures reached by
the transitions between two stable states when the surface denstity remains constant.
The long­dashed line shows the S--curve when Ü rOE = ffP total .
Figure 2. Surface densities in radial direction. Upper panel: the solid line represent
the surface densities calculated using the parameterization (eq. 1,2); filled triangles
indicate the surface density calculated directly from the standard vertical structure
equations. Lower panel: filled squares show differences in surface densities values
calculated using the parameterization and the direct vertical structure models.
Figure 3. (a) Luminosity variations for the accretion disk around a 10 8 M fi black
hole, —
M = 0:1M fi yr \Gamma1 , ff = 0:1 (ff cold = ff hot ). (b) The spectrum in the high (solid
line) and low state (dotted line) of the disk. A separation in time between the two
state is equal to 2\Theta10 4 years.
Figure 4. High and low state of an accretion disk around a black hole of M bh =10 8 M fi ,
accretion rate —
M = 0:1M fi yr \Gamma1 and parameter ff = 0:1 (ff cold = ff hot ). A separation
in time between the two state is equal to 2\Theta10 4 years. Panels show: (a) surface
density; (b) effective temperature and (c) evolution of a local accretion rate. The
major changes occur in the unstable zone. The arrows indicate the boundaries of the
instability strip.
Figure 5. (a) Luminosity variations for the accretion disk around a 10 8 M fi black
hole, —
M = 3 \Theta 10 \Gamma5 M fi yr \Gamma1 , ff = 0:1 (ff cold = ff hot ). (b) Evolution of a local accretion
rate (time separation is equal to 2\Theta10 3 years). (c) Spectrum in the high (solid line)
and low state (dotted line) of the disk.
Figure 6. Modified S--curves (log r = 15:0 cm): ff hot =0.1 and ff cold = 0.05, 0.025.
28

Figure 7. Luminosity variations due to instabilities in an accretion disk around
10 8 M fi black hole, —
M = 0:1M fi yr \Gamma1 , ff hot = 0:1 and ff cold is assumed to be equal to
(a) 0.1; (b) 0.05; (c) 0.025.
Figure 8. Variations of the luminosity for model with a black hole mass equal to
10 8 M fi , accretion rate 0.01 M fi yr \Gamma1 and viscosity parameter is different in the high
and low states: ff hot = 0:1 and ff cold = 0:025.
Figure 9. Evolution of the surface density of an accretion disk around a black hole
of 10 8 M fi , when accretion rate is equal to 0.01 M fi yr \Gamma1 and viscosity parameter is
different in the high and low states: ff hot = 0:1 and ff cold = 0:025. (a) outburst; (b)
decay: time between the curves in years: 1­2 2.82\Theta10 4 ; 2­3 700; 3­4 600; 4­5 600; 5­6
700; 6­7 600; 7­8 600; 8­9 1200; 9­10 700; 10­11 5200; 11­12 2600; 12­13 2600; 13­14
7800.
Figure 10 Evolution of the effective temperature of an accretion disk around a black
hole of 10 8 M fi , when accretion rate is equal to 0.01 M fi yr \Gamma1 and viscosity parameter
is different in the high and low states: ff hot = 0:1 and ff cold = 0:025. (a) outburst; (b)
decay: time between the lines in years: 1­2 2.82\Theta10 4 ; 2­3 700; 3­4 600; 4­5 600; 5­6
700; 6­7 600; 7­8 600; 8­9 1200; 9­10 700; 10­11 5200; 11­12 2600; 12­13 2600; 13­14
7800.
Figure 11. Evolution of a local accretion rate during (a) outburst and (b) decay
for the model with M bh =10 8 M fi and —
M =0.01 M fi yr \Gamma1 . and viscosity parameter is
different in the high and low states: ff hot = 0:1 and ff cold = 0:025. (a) outburst; (b)
decay: time between the curves in years: 1­2 400; 2­3 1100; 3­4 1000; 4­5 1100; 5­6
700; 6­7 1200; 7­8 1200; 8­9 1600.
Figure 12. Evolution of the spectrum emitted by an accretion disk around a black
hole of 10 8 M fi , when accretion rate is equal to 0.01 M fi yr \Gamma1 and viscosity parameter
is different in the high and low states: ff hot = 0:1 and ff cold = 0:025. We assumed
that the disk emits locally as a blackbody. (a) outburst; (b) decay: time between the
curves in years: 1­2 2.82\Theta10 4 ; 2­3 700; 3­4 600; 4­5 600; 5­6 700; 6­7 600; 7­8 600; 8­9
1200; 9­10 700; 10­11 5200; 11­12 2600; 12­13 2600; 13­14 7800.
Figure 13. (a) Effective temperature in the high and low state of the disk (10 8 M fi ,
—
M = 0.1 M fi yr \Gamma1 , ff cold = 0:1) which is irradiated by an external hard source (the
29

luminosity of the external source is assumed to be of the 10% of the total luminosity
generated in the disk). (b) Spectrum of the irradiated and non irradiated disk in the
high and low states.
Figure 14. (a) Effective temperature in the high and low state of the disk (10 8 M fi ,
—
M = 0.01 M fi yr \Gamma1 , ff cold = 0:025) which is irradiated by an external hard source (the
luminosity of the external source is assumed to be of the 10% of the total luminosity
generated in the disk). (b) Spectrum of the irradiated and non irradiated disk in the
high and low states.
30

TABLE 1: MODELS
model —
M [M fi yr \Gamma1 ] L=LEdd ff cold r D [r g ] r C [r g ]
A0 3 \Theta 10 \Gamma5 10 \Gamma5 0.1 6.06 61.7
A1 3 \Theta 10 \Gamma5 10 \Gamma5 0.075 4.39 106
B0 0.01 0.0034 0.1 90.9 665
B1 0.01 0.0034 0.075 81.1 1120
B2 0.01 0.0034 0.050 55.1 1120
B3 0.01 0.0034 0.025 59.5 1500
C0 0.1 0.034 0.1 234 1670
C1 0.1 0.034 0.075 209 2800
C2 0.1 0.034 0.050 143 2800
C3 0.1 0.034 0.025 155 3760
D0 1.0 0.34 0.1 385 2730
D1 1.0 0.34 0.075 344 4570
D3 1.0 0.34 0.025 255 6130
E0 3.4 1.0 0.1 973 6850
E1 3.4 1.0 0.075 870 7930
31

TABLE 2: Luminosity and Spectral Index
model ! L ? \DeltaL=L duty cycle a ff b
med ff mean oe ff high ff low ff irr
high ff irr
low
ff cold = 0:1
A0 41.70 0.41 55.5 ­1.05 ­1.41 1.15 ­1.33 -- ­1.25 --
B0 43.88 0.09 99.9 0.02 ­0.05 0.24 ­0.45 0.05 ­0.65 ­0.27
C0 44.78 0.07 99.9 0.15 0.11 0.18 ­0.08 0.23 ­0.57 ­0.34
D0 45.73 0.08 100 0.18 0.16 0.14 ­0.12 0.34 ­0.63 ­0.44
E0 46.2 0.08 100 0.23 0.22 0.13 0.01 0.43 ­0.66 ­0.55
ff cold = 0:075
A1 41.78 1.18 22.0 ­0.11 ­0.12 0.03 ­1.67 -- ­1.65 --
B1 44.11 0.80 30.1 0.23 0.14 0.26 0.08 0.16 ­0.44 0.01
C1 44.75 0.71 29.8 0.27 0.17 0.32 0.25 0.28 ­0.49 0.06
D1 45.28 0.81 27.0 0.31 0.15 0.42 0.21 0.33 ­0.57 0.01
E1 46.18 0.87 20.6 0.36 0.18 0.41 ­0.70 0.40 ­0.76 0.19
ff cold = 0:05
B2 44.15 0.63 18.1 0.14 ­0.06 0.49 ­0.14 0.20 ­0.53 ­0.08
C2 44.82 0.75 18.8 0.24 0.09 0.44 0.14 0.07 ­0.55 ­0.21
D2 45.19 1.01 11.0 0.28 0.19 0.29 ­0.12 0.11 ­0.64 ­0.21
ff cold = 0:025
B3 43.88 2.63 6.0 0.31 c 0.13 c 0.54 c ­0.28 -- ­0.58 --
C2 44.30 3.01 3.1 0.45 c 0.09 c 0.86 c ­0.15 -- ­0.63 --
D2 45.21 3.98 3.0 0.51 c 0.05 c 0.84 c ­1.15 -- ­0.7 --
a a fraction of time when the luminosity is larger than the half of the maximum value [in %]
b spectral index defined as F š ¸ š ff
c calculated adopting the lower limit of 42.00 for the luminosity log(šL š ) at log(š) = 14:8
32

TABLE 3: Observations
sample spectral range ff mean ff med \Deltaff log Lmin log Lmax Ref
Neugeb.87 14.5­15.0 ­0.2 0.8 43.6 47.1 1
Neugeb.87m 14.5­15.0 ­0.16 45.0 47.1 1
Francis 14.7 ­ 15.3 ­0.32 0.2 44.1 b 2
Francis 14.7 ­ 15.0 0.02 2
Cristiani 14.7­ 15.4 ­0.70 43.5 47.5 3
Elvis 14.5 ­15.2 ­0.1 0.3 44.8 47.0 4
a there have been 30 objects in the sample of 718 with log L between (39.9­44.1);
Samples used here are: (1) bright quasars of Neugebauer et al. (1987) (here:Neugeb.87), the same
sample but constrained to objects brighter than 11.5 in solar units (here: Neugeb.87m), (2) bright
quasars of Francis et al. (1991), (3) Cristiani & Vio (1990) (4) Elvis et al. 1994.
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47