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Deriving the Quasar Luminosity Function from Accretion Disk
Instabilities
Aneta Siemiginowska & Martin Elvis
Harvard­Smithsonian Center for Astrophysics
60 Garden Street, MS­70, Cambridge, MA 02138, USA
asiemiginowska@cfa.harvard.edu
March 14, 1997
ABSTRACT
We have derived the quasar luminosity function assuming that the quasar activity
is driven by a thermal­viscous unstable accretion disk around a supermassive black
hole. The instabilities produce large amplitude, long­term variability of a single source.
We take a light curve of a single source and calculate the luminosity function, from
the function of time it spends at each luminosity. Convolving this with an assumed
mass distribution we fit well the observed optical luminosity function of quasars at four
redshifts. As a result we obtain the evolution of the mass distribution between redshifts
2.5 and 0.5.
The main conclusions are following: 1) The quasar long­term variability due to the
disk thermal­viscous instabilities provides a natural explanation for the observed quasar
luminosity function. 2) The peak of the mass function evolves towards lower black hole
masses at lower redshifts by a factor ¸ 10. 3) High mass sources die subsequently when
redshift gets smaller. 4) The number of high mass sources declines rapidly, and so low
mass sources become dominant at lower redshift. 5) The periodic outbursts of activity
appear as long as the matter is supplied to the accretion disk. 6) Since the time­averaged
accretion rate is low, the remnant sources (or sources in the low activity phase) do not
grow to very massive black holes. 7) A continuous fuel supply at a relatively low accretion
rate ( ¸ 0:01 \Gamma 0:1 —
MEdd ) for each single source is required over the lifetime of the entire
quasar population.
Subject headings: accretion: accretion disks ­ cosmology:theory ­ quasars:general
to appear in Ap.J. Letters
1

1. INTRODUCTION
The quasar luminosity function has been studied
for the last 30 years and is observationally now quite
well determined as a function of redshift for z ! 4
(e.g. Boyle et al, 1991). However, there have been
few attempts to derive the luminosity function from
physical models of the quasar power engine. There are
three possible phenomenological scenarios (Cavaliere
& Padovani, 1988): long­lived objects, recurrent ob­
jects (possibly related to galaxy mergers) and a single
short event over the whole host galaxy life­time. Con­
tinuous models imply masses for the remnant black
holes that are too large, and accretion rates that are
too low (Cavaliere et al 1983; Cavaliere & Szalay
1986; Cavaliere & Padovani 1988, 1989; Caditz, Pet­
rosian & Wandel, 1991).
Short­lived models have been studied more re­
cently. Haehnelt & Rees (1993) assumed that new
quasars were born at successive epochs with a short
active phase followed by a rapid exponential fading
due to exhaustion of fuel. They used the Cold Dark
Matter formalism (Press & Schechter, 1974) to esti­
mate the number of newly forming dark matter ha­
los at different cosmic epochs. Small & Blandford
(1992) suggested a scenario involving a mixture of
continuous and recurrent activity. They assumed that
newly formed sources achieve the Eddington luminos­
ity quickly, such that the accretion rate is limited by
radiation pressure. The break in the luminosity func­
tion is related to the boundary between the continu­
ous and intermittent accretion phases originating in
the amount of fuel supply to the black hole.
However, none of these models relate directly to
the physical processes responsible for powering a
quasar. They simply invoke sources that emit at the
Eddington luminosity for a certain time and then fade
below an observational threshold. Here we describe a
scenario which for the first time derives the luminosity
function from a specific physical process.
The time evolution of an accretion disk around a
supermassive black hole (the main components of the
standard quasar paradigm), exhibits large variations
on long timescales due to thermal­viscous instabilities
(Siemiginowska, Czerny & Kostyunin 1996, hereafter
SCK96, Mineshige & Shields 1990). Depending on
the assumed disk model, variations of up to a factor
¸ 10 4 can be produced on timescales of 10 4 \Gamma 10 6
years. Here, we assume that all quasars are subject
to this variability. We then take the light curve of
a single source and calculate the luminosity function
of a population of identical sources from the fraction
of time it spends at each luminosity. Convolving this
with an assumed mass distribution we fit the observed
quasar luminosity function at four redshifts. As a
result we obtain the evolution of the mass distribution
between redshifts 2.5 and 0.5.
2. EVOLUTION OF AN ACCRETION DISK
Accretion onto a supermassive black hole is the
leading model for powering quasars (e.g. Rees 1984).
The accretion process 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 time­dependent effects in accretion process are
of extreme importance. Observationally, the evidence
for global evolutionary effects is compelling in accre­
tion 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 years. The outbursts are essentially caused
by the disk thermal instability in the partial ioniza­
tion zone (Meyer & Meyer­Hoffmeister 1982, Smak
1982, see also Cannizzo 1993 for review). There is
a strong similarity between Galactic X­ray sources
and AGN both in spectral behavior and overall vari­
ability (Fiore & Elvis 1994, Tanaka & Lewin 1995)
which leads us to expect similar accretion disk behav­
ior in AGN. However, as the characteristic timescales
are roughly proportional to the central mass, the ex­
pected variability takes thousands to millions of years
in AGN. Since these timescales are not directly ob­
servable, these changes have been considered little
more than a curiosity in AGN.
Theoretically, accretion disks around the massive
black holes in AGN are expected to have a partial
ionization zone, as in Galactic binaries, and there­
fore to be subject to the same instability (Lin &
Shields 1986, Clarke 1989, SCK96). Current models
of the time evolution of accretion disks in AGN have
confirmed the presence of disk eruptions (Clarke &
Shields 1989, Mineshige & Shields 1990, SCK96).
SCK96 considered a geometrically thin Keplerian
accretion disk around a supermassive black hole and
assumed that the viscosity scales with the gas pres­
sure (Ü rOE = ffP gas ). They found that, depending
on the viscosity, the instability can either develop
only in a narrow unstable zone, or can propagate
2

over the entire disk resulting in large amplitude opti­
cal/ultraviolet outbursts (¸ 10 4 ) (see Fig. 1a). The
calculation of these light curves is at present compu­
tationally demanding (SCK96).
3. FROM LUMINOSITY VARIATIONS TO
THE LUMINOSITY FUNCTION
3.1. A single mass, single accretion rate pop­
ulation.
The luminosity function of a population of quasars
with the same mass and accretion rate is given simply
by the product of a fraction of time one source spends
in each luminosity bin and their space density. In
Fig.1b we show the fraction of time a source emits at
each luminosity, relative to the Eddington luminosity,
for the light curve shown in Fig.1a. The luminosity
range is between 10 \Gamma4 LEdd and LEdd . The shape of
the function reflects the fact that the amplitude of
each outburst is not constant and the variability is not
precisely periodic. The details of each outburst and
the overall variability characteristics depend on the
physics of the accretion disk and the assumptions of
the model. These details average over many outbursts
(usually a few hundred over 10 8 \Gamma 10 9 years).
Fig. 1.--- a) Luminosity variations due to the disk
instabilities around a black hole of 10 8 M fi , when
the accretion rate is 0.1 M fi yr \Gamma1 and the viscos­
ity parameter is different in the high and low states:
ff hot = 0:1 and ff cold = 0:025. b) Fraction of the time
the source emits at a given luminosity for 10 8 M fi and
0.1 —
M fi yr \Gamma1 accretion rate. The luminosity is given in
the Eddington luminosity units.
A single source will spend about ¸ 75% of its life
in quiescence (L ! 0:001LEdd ) and about ¸ 25% in
an active state, with ¸ 10% in a high state (L ?
0:1LEdd ). Likewise in a population ¸10% of sources
will be in the high state, ¸25% will be active and
¸75% in quiescence at any given time.
There are two characteristic transition points in
the function shown in Fig.1b: a broad maximum at
¸ 0:1LEdd and a minimum at ¸ 0:001LEdd . When we
construct a luminosity function for a realistic popu­
lation these features will be modified by the distribu­
tion of accretion rates and masses. A range of accre­
tion rates affects the low luminosity part of the curve
by smoothing at the minimum. The maximum at
¸ 0:1LEdd is caused by the fraction of outburst am­
plitudes reaching close to the Eddington luminosities.
The maximum is thus smoothed by the distribution
of black hole masses.
3.2. Fit to the Observed Luminosity Func­
tion.
The luminosity in Fig.1b. is expressed in terms
of the Eddington luminosity, in order to make the
function independent of the central mass (see Fig.2a
``1­mass'' luminosity function). Thus for a given dis­
tribution of black hole masses we can calculate the
quasar luminosity function. The luminosity function
is defined as:
\Phi(L; z) =
Z
\Phi(L; z; M )N (M; z)dM
where \Phi(L; z; M ) describes which central mass con­
tributes to a given luminosity bin at a given redshift
and N (M; z) represents a number of sources with a
given central mass at a given redshift. The mass
density function (N (M; z)M ) can be derived, with
assumptions, from cosmological models and theoreti­
cal models on the formation of the structures in the
universe (Heahnelt & Rees 1993, Small & Bland­
ford 1992). Here we do not consider any particu­
lar model for the formation of the black holes, galax­
ies and quasars. Instead we assume, arbitrarily that
N (M; z) can be represented by a simple parabola. We
then fit the observed luminosity function from Boyle
et al (1991) at different epochs varying the peak, po­
sition and width of the parabola. The mass function
is convolved with the single mass luminosity function
(Fig.1b). We held —
M constant, and so did not change
the minimum in Fig.1b. However, this minimum is
not within the range covered by the Boyle et al (1991)
data.
We were able to obtain good fits (Table 1) to the
luminosity function at three of four redshifts as shown
in Fig. 2a. The black hole mass density function re­
3

Fig. 2.--- (a) Observed luminosity function (Boyle et
al 1991) at four redshifts indicated by points with er­
ror bars: 0:25 ! z ! 0:75 ­ open triangles, 0:75 ! z !
1:25 ­ filled squares, 1:25 ! z ! 2:00 ­ filled trian­
gles, 2:00 ! z ! 2:90 ­ empty squares. The predicted
single mass luminosity function is indicated with the
stars. The model luminosity function is plotted with
the solid line. (b) The predicted black hole mass den­
sity function at each epoch obtained from the fit to the
observed luminosity function: 0:25 ! z ! 0:75 ­ solid
line; 0:75 ! z ! 1:25 ­ dashed line 1:25 ! z ! 2:00
­ dashed­dot line 2:00 ! z ! 2:90 ­ dotted line. The
truncation at the mass required by the lowest lumi­
nosity data point is indicated by the straight lines at
the lowest mass point for each parabola.
quired by the fit is plotted in Fig. 2b and the param­
eters of the fit are given in the Table 1. The poor fit
at redshift ¸1 is due to the highest luminosity point
which seems to require a kink in the luminosity func­
tion. We fit the data excluding this point and obtain
a good fit (Table 1). The mass density peaks at lower
mass in this case, since the luminosity function ends
at lower luminosity.
Only active sources contribute to the black hole
mass density function. The maximum indicates which
central black hole mass dominates the population
at each redshift. This peak mass declines from ¸
2 \Theta 10 7 M fi at z ¸ 2:5, to ¸ 2 \Theta 10 6 M fi at z ¸ 0:5
with most of the change occuring between z ¸ 1:75
and z ¸ 1 (Table 1). The mass density of high mass
sources gets smaller rapidly (e.g. a factor ¸ 100 at
M = 10 9 M fi ) with lower redshift, while the peak
mass density declines by less than a factor 2. The rela­
tive constancy of this peak implies that the number of
sources at the peak remains constant between z = 2:5
and z = 1, and then decreases by a factor of ¸ 2 be­
tween redshift z = 1 and z = 0:5. The low mass end
of the distribution is not constrained by the data and
so we truncate the functions at the mass where the
Eddington limit gives the lowest observed luminosity.
This lowest luminosity point of the observed luminos­
ity function at z = 2:5 (LEdd ¸ 8 \Theta 10 44 ergs s \Gamma1 )
gives the limit of 6:3 \Theta 10 7 M fi . Even if less massive
sources are present in the population we cannot see
them.
4. DISCUSSION
We have shown that the thermal­viscous instability
provides a natural mechanism to generate the quasar
luminosity function. We were able to fit the observed
luminosity function and estimate the parameters of
the mass density function, independent of cosmolog­
ical models. The overall shape of the mass density
function and the evolution of the peak of the mass
distribution towards lower masses with lower redshift
are similar to the results obtained by previous stud­
ies (Haehnelt & Rees 1993, Small & Blandford 1992).
The 30% ``on'' fraction in these models is also compa­
rable with the fraction of active time input light curve
(Fig.1b). This is not too surprising because the same
observational luminosity function was used in all the
studies.
The problem of whether the low mass sources are
present at high redshift or are born subsequently re­
mains unsolved. In our scenario this question could
be answered by extending the observed luminosity
function at z ?
¸ 1 fainter by \Deltam ¸ 3. If the fitted
mass functions match the low z mass density func­
tions these would suggest that all quasars are born
at the same time and the high mass ones ``burn out''
much more quickly. This scenario, in which single
mass density function declines more rapidly at high
redshift Log M peak log N(M peak ) ü 2
z
2.0 ­ 2.9 8.1 ­5.30 2.78
1.25 ­ 2.0 7.7 ­5.33 4.98
0.7 ­ 1.25 7.3 ­5.34 12.79
6.9 ­ 5.38 3.48 a
0.3 ­ 0.7 6.9 ­5.52 1.18
a excluding the highest luminosity data point at redshifts 0:7 !
z ! 1:25
4

luminosities, is strikingly different from the conven­
tional ``pure luminosity evolution'' that is used to de­
scribe the apparent fading of the whole observed lu­
minosity function to lower z. It reminds us of the
warning by Green (1985) against interpreting phe­
nomenological descriptions as physically meaningful.
Looking at the sources in the present epoch should
provide the information on the lowest luminosity end
of the distribution together with the contribution of
the massive sources. However, the luminosity of the
host galaxy becomes comparable to the nuclear lu­
minosity for low mass black hole and it is hard to
observed the nucleus of a normal galaxy even if it con­
tains an accretion disk in the active state (L ¸ LEdd ).
On the other hand Seyfert nuclei are found in ¸ 10%
of galaxies, consistent with the high state fraction
from SCK96. The problem is how can we see the
low mass sources in quiescence.
The high luminosity end of the luminosity function
accounts for all the high mass sources which are ac­
tive at each epoch. The number of these sources gets
smaller with redshift. This decrease is often supposed
to relate to the limited fuel supply and the mecha­
nisms of transfer of the matter into the disk. We note
though that the location and the size of the ionization
zone depends on the accretion rate onto the outer edge
of the disk (SCK96, Clarke & Shields 1989). For high
accretion rates this zone moves towards outer regions
of accretion disk. In the case of the high mass black
holes the ionization zone can be pushed out to the
self­gravitating regions of the disk and the instability
will not develop. The source then remains in the ac­
tive state until the fuel supply is exhausted, and then
dies. How the location of the ionization zone affects
the global evolution of the population requires further
study.
Another consideration that could lead to the more
rapid demise of high mass quasars is that massive
sources require more fuel than the low mass sources
to emit at a given L=LEdd . Only ¸ 0:027M fi yr \Gamma1 is
needed to power a 10 6 M fi black hole at 0.1 —
MEdd ,
while a 10 9 M fi black hole requires accretion rates
of order 2.7M fi yr \Gamma1 . Recent studies of quasar host
galaxies, (at z! 0:3), show that the most luminous
quasars reside in the most massive galaxies, while
lower luminosity quasars can be found in any type of
a galaxy (McLeod & Rieke 1995a, 1995b; Bahcall et
al 1996). Based on the HST observations of 61 ellip­
tical galaxies Faber et al (1996) conclude that about
¸ 1% of the galaxy mass is contained within a cen­
tral core of few parsecs. This means that for a typical
¸ 10 11 \Gamma 10 12 M fi galaxy, there is about 10 9 \Gamma 10 10 M fi
available to feed a black hole. While at 10 6 M fi it
would last for 10 11 \Gamma 10 12 years at 10 9 M fi it would
last for 10 9 years at 0.1 —
MEdd .
A third possibility is that the, unknown, mecha­
nism responsible for transferring the matter towards
the central potential well and into the outer parts of
accretion disks becomes s rapidly less efficient in mas­
sive systems, so they would systematically die young.
Recently Yi (1996) considered the cosmological
evolution of quasars assuming that advection becomes
important for accretion rates below 0.01LEdd accre­
tion rate. The theoretical and observational studies
of the X­ray transients suggest that advection is im­
portant in quiescence below a critical accretion rate
(Narayan & Yi 1994, Narayan et al 1996). Advection
has not been included in our accretion disk model. It
will modify the low luminosity part of the light curve
in Fig.1a and influence the quasar evolution. We shall
include the advection in our further studies, since the
quasar remains in quiescence for ¸ 75% of its life.
In previous studies the sources making up the lu­
minosity function were assumed to begin by emitting
at their Eddington luminosities and steadily becom­
ing fainter with time. This does not apply in our
model. The stationary accretion rate onto the outer
edge of the disk can be much lower than the Edding­
ton limit. This prevents accumulation of a large mass
in the center and removes the problem of creating
overly massive quasars remnants.
Small & Blandford (1992) suggested two phases
of the quasar activity, which are related to the accre­
tion rate. Just after a black hole is born the matter is
supplied at super­Eddington rates, but the actual ac­
cretion onto the black hole is limited by the radiation
pressure. The black hole accretes continuously at the
Eddington rate until the fuel supply gets lower and
then the accretion is intermittent. The intermittent
activity can be related to the active state of the disk
in our scenario.
The model we use to produce the quasar luminos­
ity function works for the optical/ultraviolet bands.
The radio and X­ray luminosity function show simi­
lar form and evolution (Maccacaro et al. 1992, Della
Ceca et al 1994). Physically the radio and X­ray lu­
minosities must then be a result of the accretion disk
state.
5

We thank Bo—zena Czerny, Andrzej Soltan, Tom
Aldcroft, Kim McLeod, Pepi Fabbiano and Avi Loeb
for valuable discussions. We also thank the anony­
mous referee for useful comments. This work was sup­
ported by NASA Contract NAS8­39073 (ASC) and
NASA grant NAG5­3066 (ADP).
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