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\begin{document} 
  
   \thesaurus  { 
04(02.01.2; 
02.08.1; 
10.03.1;
11.01.2;
11.09.1 M31;
11.14.1)
                                   } 
% 
\title{ Starving accretion disks in the center of galaxies:\break\hfill
           The examples of Sgr A* and M31} 
  \author{Heino Falcke\inst{1} and Olaf M. Heinrich\inst{2}
          } 
  
   \offprints{HFALCKE@mpifr-bonn.mpg.de}
  
   \institute{
Max-Planck-Institut f\"ur Radioastronomie, Auf dem H\"ugel 69, 
D--53121 Bonn, Germany
%\instn{1}
 \and
Institut f\"ur Theoretische 
              Astrophysik der Universit\"at Heidelberg, Im Neuenheimer 
              Feld 561, 
              D-69120 Heidelberg, Germany
%\instn{2}
             }
  
   \date{Received April 1994; accepted July 1994} 
  
\maketitle 
\markboth{Falcke \& Heinrich: Starving accretion disks in the center of galaxies: Sgr A* and M31}{Falcke \& Heinrich: Starving accretion disks in the center of galaxies: Sgr A* and M31}  

\begin{abstract} 
%______________________________________ Do not leave a blank line here! 
% 
%  14.Sep.'90: Demo-Vs. 
%_____________________________________ Do not leave a blank line here!
Observations of the Galactic Center source Sgr~A* suggest the presence
of a central hot accretion disk around a supermassive black hole with
an accretion rate a factor $10^{5}$ below the Eddington limit.  We
argue that purely spherical accretion models have problems with
obervational constraints.  The low accretion rate inferred for Sgr A*
requires a distinct type of accretion disk -- a starving disk.  We
investigate the structure of this type of disk within the framework of
thin Keplerian disk models taking into account relativistic effects,
detailed opacitiy tables and turbulent heat transport. It appears as
if starving disks are the perfect realization of geometrical thin,
optically thick disks with well behaved structure functions. Although
in general gas-pressure dominated, radiation pressure may become
important in the inner region of the disk.  All in all starving disks
are very stable structures with high surface density, high optical
depth and long viscous time scales.  However, S-curves in the
$\Sigma-\dot M$ diagram occur, indicating a thermal limit-cycle
behaviour observable in the IR-NIR with timescales ranging between
three years and some thousand years.  We propose that, at least in Sgr
A*, the main mass supply for the disk might be a wind from the
surrounding star cluster. As the disk is a very stable and slowly
evolving structure, it will inhibit spherical accretion and
incorporate the inflowing matter in the general accretion flow.  On
the other hand self-gravitation and fragmentation of the outermost
part of the disk could have resulted in starformation at $0.1$ pc and
hence even the wind of the star cluster can already have substantial
angular momentum.  The situation in M31 may be very similar but with a
more extreme set of parameters.
      \keywords{accretion disks  --
                hydrodynamics --
                Galaxy: center --
                galaxies: active --
	        galaxies: individual: M31--
                galaxies: nuclei 
}
   \end{abstract} 
% 
%  14.Sep.'90: Demo-Vs. 
%________________________________________________________________ 

%\twocolumn  
\section{Introduction} 
In recent years the interest in the physical processes happening in
the center of our galaxy has grown considerably, especially because of
possible similarities between the Galactic Center (GC) region and
the nuclei of active galaxies (AGN). It is believed that the very
center of the Galaxy hosts a supermassive black hole with mass
$M_\bullet\simeq10^6 M_{\sun}$ (e.g. Genzel \& Townes 1987). This is
strongly supported not only by mass estimates but also by the presence
of the compact flat spectrum radio source Sgr A* with size
$\la10^{14}$ cm $\simeq 1$ mas (Krichbaum et al. 1993) at the
dynamical center of the central starcluster (Eckart et al. 1993) which
could be interpreted as the core of a jet-like wind originating from
a central accretion disk (Falcke et al. 1993b).  The presence of an
accretion disk around a supermassive black hole is also consistent
with NIR measurements of this region (see Falcke et al. 1993a and
refererences therein).

Black hole, jet and disk could make a perfect active nucleus in the
Galaxy but the actually observed signs of activity in this region are
not very impressive. Generally this is attributed to a very low
central accretion rate of the order $\la10^{-7} M_{\sun}/$yr making
the GC, if at all, only an AGN on a starvation diet (Falcke \&
Biermann 1994a).  For a black hole the critical Eddington accretion
rate is a factor $\sim10^5$ higher than this value.  Usually nuclear
disks around black holes are studied only in a regime close to being
critical while the putative disk around Sgr A* has to be extremely
subcritical.

If we believe that our galaxy is not a very special one in the
universe one should think that the configuration of a starving disk
plus the remnant of a radio jet can be found also in the nuclei of
other ordinary spirals. Alas, the characteristic of starving disks --
its low accretion rate -- will make them almost invisible in more
distant galaxies. At least our neighbour M31 does show in its
nucleus a weak central radio source (Crane, Dickel \& Cowan 1992) and
indication for the presence of a central black holes (Dressler
\& Richstone 1988; Kormendy 1988). Therefore starving disks may be a
frequent phenomenon in the center of galaxies.

In this paper we will first argue in Sec. 2 that `disk accretion',
i.e. implying the presence of an accretion disk with a large outer
radius and a radio jet, is a more likely scenario for Sgr A* than
recently proposed spherical accretion models where the radio emission
is produced during a mostly spherical infall. Nevertheless, a
reconciliation of disk and wind accretion seems possible by invoking a
pre-existing (fossil) disk. Moreover, we investigate the structure of
starving disks in more detail based on a code developed by Heinrich
(1994) using the basic Shakura \& Sunyaev (1973) $\alpha-$Ansatz with
relativistic corrections for the Kerr metric (Novikov \& Thorne 1973)
but with refined opacities for temperatures above $\sim2000$ K and
inclusion of turbulent heat transport. The basic equations are briefly
repeated in Section 3 and results for the example of Sgr A* are
presented in Sec.4. As an alternative parameter set we discuss in Sec.
5 shortly a situation applicable to M31.

\section{Spherical accretion vs. disk accretion}
\subsection{Arguments against purely spherical accretion in Sgr~A*}
One has to admit that there are alternative models to explain the
phenomena seen in Sgr A*. There are two main scenarios competing with
each other. One is the standard paradigm for AGN scaled down to the
Galactic Center with black hole, jet and accretion disk (Zylka et al.
1992; Falcke et al. 1993a\&b; Falcke 1994a\&b). Here it is assumed that
the matter accretes through a disk structure onto the black hole, part
of this matter is then expelled again by a jet formed in the very
inner parts of the disk close to the black hole.

The other model -- the spherical wind accretion scenario -- put
forward by Melia (1992a \& 1994), Melia et al. (1992) and Ozernoy
(1989) assumes that Sgr A* accretes in a Bondi-Hoyle type accretion
from the wind of the IRS 16 complex, a small subcluster of NIR bright
stars with possible temperatures of some $10,000$ K. Radio and X-ray
emission is then produced in onion-like shells around the central
black hole with the highest frequencies produced close to the black
hole. The spectral characteristics of this accretion process are then
used to obtain mass estimates for Sgr A*, but despite using similar
approaches both authors come to contrary conclusions. Melia, using a
more elaborate description of the magnetic bremsstrahlung, confirms a
supermassive balck hole of mass $\sim10^6M_{\sun}$ while Ozernoy
denies it and allows only $10^3 M_{\sun}$ for the central mass. In the
disk models a value of $10^6 M_{\sun}$ is favoured.

The main difference between the spherical and the disk scenario
therefore does not concern the central mass but the accretion rate.
Melia needs a very high accretion rate which is $10^{-4} M_{\sun}/$yr.
He argues that the spherical infall will still have a net angular
momentum which will lead to the formation of a small disk inside a
certain radius. Thus he is able to account for the NIR source at the
position of Sgr A*; to reduce the efficiency of the black hole in
producing the visible luminosity Melia considers only a Schwarzschild
hole, but still with the high accretion rate the total disk luminosity
would be well above the observed limits for the luminosity of the
central object. He therefore presents the spectrum of a highly
inclined Newtonian disk which brings down the apparent disk luminosity
by more than a factor 10.

Disregarding that relativistic effects (i.e. boosting and light
bending) would diminish this effect drastically the comparison of
apparent disk luminosity and observed luminosities is highly
misleading. The luminosity estimates from the central source are
obtained by making use of the fact that most of the nuclear luminosity
at optical wavelengths gets absorbed in the dust of the central region
and is reradiated in the IR, therefore one has to use {\em total} and
not {\em apparent} luminosity of the disk for the comparison.
Inclination effects will make no difference in this number.  Falcke et
al. (1993a) and Zylka et al. (1992) have shown that from the IR
luminosities and taking the stellar light contributions into account,
Sgr A* can not have a total luminosity $L_{\rm tot}$ larger than
$\sim7\cdot10^5 L_{\sun}$ which is also consistent with estimates for
the NIR source. If one relaxes all limits and postulates clumpiness
and a special geometry of the dust distribution Davidson et al.
(1992) get as a very strict upper limit $L_{\rm tot}< 10^7L_{\sun}$
for the nuclear luminosity which still is almost an order of magnitude
below the luminosity which would be produced by central disk in the
Melia model ($\sim10^8L_{\sun}$) and a rotating black hole would make
this even worse. Hence, in order to reproduce the radio and X-ray
emission of Sgr A* correct the spherical accretion model of Melia
(1992a \& 1994) is in serious conflict with observational constraints
in the energetically dominant NIR-UV regime. This discrepancy is
slightly reduced if the outer disk radius is very small, however, it
must be extremely low ($\la5R_{\rm s}$, $R_{\rm s}=2 GM/c^2$) such
that the disk is almost non-existing to resolve this problem
completely.


\subsection{Fossil disk and wind accretion}
Now, we do not deny that there are indications for a strong wind in
the Galactic Center (e.g. Yussef-Zadeh \& Melia 1992) but first of all
the inferred accretion rate of Sgr A* depends very sensitively on the
wind parameters at the postion of Sgr A*, the overall geometry of the
wind and the real distance between wind source and black hole.
Secondly, the question is if the wind has indeed the parameters
quoted by Melia will this wind affect the accretion rate in the
energetically important innermost region?

Imagine that there is a fossil disk in the center of the Galaxy -- a
remainder of past activity or even of galaxy formation. In how far does
it influence current accretion flows? If there is no significant mass
supply for the central region from larger scales those fossil disks
will appear as the starving disks discussed in this paper. As we will
show these disks are very stable and slowly evolving structures that
may last for a long time. The disks themselves still have a large
amount of angular momentum, mass and kinetic energy.  A spherically
infalling wind will sooner or later hit the disk which definitively
will lead to a strong distortion and even disruption of the spherical
infall.  Especially in the inner regions the wind will be incorporated
into the general disk accretion and mainly lead to a local enhancement
of the accretion rate. It will take a long viscous time scale until
this enhanced accretion rate reaches the black hole and becomes
``visible''.

Such a concept has several advantages, firstly, a fossil disk will
preserve the memory of the initial orientation of the Galactic plane.
This makes it more likely for an accretion disk like in Sgr A* to be
somehow aligned with the galactic plane, as indicated by the radio
observations (Krichbaum et al. 1993) although it is so small that the
disk potential of the Galaxy is irrelevant to the dynamics of this
region. Secondly, the problem of the mass supply in the central parsec
is no longer important. A few stars could provide enough fuel to keep
the central engine alive. First of all it supplies mass and secondly,
the lower angular momentum of the more or less spherically infalling
matter will counteract the tendency of the fossil accretion disk to
transport its original angular momentum outwards. Wind accretion will
simply lead to a shrinking of the fossil disk down to an equilibrium
radius -- the angular momentum of the disk is conserved, but the
matter is increased -- and a mass reservoir at the outer edges of
the disk will develop.  Hence, wind accretion can conserve angular
momentum and orientation of a fossil disk for a long time and refill
empty mass stores. Different epochs of star formation may then also
lead to different stages of activity in the center of galaxies.

For Sgr A* we can make some simple arguments. The parameters assumed
for a possible wind produced by nearby stars are a total mass loss
rate $\dot M_{\rm w*}=\dot m_{-2.5}\;3.5\cdot10^{-3} M_{\sun}/{\rm
yr}$, a wind velocity $V_{\rm w*}=v_{600} \;600$ km/sec and a distance
of the wind sources from Sgr A* of $D_*=d_{0.1} 0.1$ pc (Melia 1992a).
The accretion radius is $R_{\rm acc}=2 G M_{\bullet}/ V^2_{\rm w*}$
which is for a black hole of mass $M_{\bullet}=m_6\; 10^6 M_{\sun}$
approximately $R_{\rm acc}\approx7.3\cdot 10^{16} {\rm cm}\; m_6
/v_{600}^2$. The wind accretion rate onto the black hole is $\dot
M_{\rm w\bullet}=5\cdot10^{-5} M_{\sun}/$yr and the mass flow per area
through a disk with radius $R_{\rm acc}$ (disregarding inclination
effects) would be $\dot\Sigma_{\rm w\bullet}\approx1.8\cdot
10^{-13}\,{\rm g/sec/cm}^2 \;\dot m_{-2.5} d_{0.1}^{-2}$. We can
compare this with the radial accretion flow in the inner region $\dot
M_{\rm disk}/4\pi R_{\rm disk}$ and see that the wind accretion
becomes important at a scale $R_{\rm disk,out}\ga 3\cdot 10^{15}\,{\rm
cm}$ $d_{0.1} \dot m_{-2.5}$. This is much larger than the inner hot
disk which is smaller than $10^{14}$ cm with viscous timescales of the
order $>10^7$ years, thus larger than the expected lifetime of such a
wind.

Of course the whole argument rests on the assumption that a cold,
molecular fossil disk indeed extends to radii comparable to $R_{\rm
disk,out}$ which is able to catch the wind before it reaches the small
scales. The enclosed mass in the molecular disk can easily be of the
order of a few 100 solar masses and should be seen at infrared
wavelengths below $\lambda18\mu m$ (see Zylka et al. 1993 for
discussion of this topic). Another prediction of this concept would be
an increased activity in some $10^7$ years which, unfortunately, is
slightly longer than an astronomers lifetime.


% 
%  14.Sep.'90: Demo-Vs. 
%__________________________________________________________________ 
  
\section{The model} 
  
% 
   We now want to discuss the properties of starving disks like in
Sgr~A*. Here we concentrate only on the inner hot disk $(\ga 2000)$ K
with a length scale $\la 10^{14} {\rm cm}$. We, however, implicitly
postulate that a cold molecular disk extends further out which
incorporates the stellar wind flow onto the center as described in the
previous section and thus serves as a buffer between inner disk and
environmental impact.  This allows us to ignore the effects of time
dependence and vertical mass inflow for the hot disk. Distortion due
to the passage of stars is also not expected as this scale is much
smaller than the core radius of the star cluster (Eckart et al. 1993).
There is, however, one caveat that onw should bear in mind: to explain
the radio emission as emission from a jet requires that a large amount
of energy is transferred from the disk into the jet (Falcke et al.
1993b, Falcke et al. 1994) in the very inner part of the disk (Falcke
\& Biermann 1994b). This may considerably modify the inner boundary
conditions of the disk and in particular the effective temperature at
the disk surface. However, the problem of the backreaction of a strong
outflow on the disk itself is not solved and can not be calculated in
this paper.

Detailed models for the vertical structure of the disk under the
assumption that the disk is geometrically thin and the fluid elements
move on nearly circular Keplerian orbits have been discussed by a
number of authors (e. g.  Pringle 1981, Meyer and Meyer Hofmeister
1982, Cannizzo 1992).  In the following we briefly list the structure
equations of the disk model used in this paper (see also Heinrich
1994).  The basic equations describe the mechanical equilibrium, heat
generation, energy transport and the viscosity law in the disk:
\hfill\break \begin{equation} {{dp_{\rm g} \over {dz}}}= -
\rho g_{\rm z} + c^{-1} \rho \kappa F_{\rm rad} , \end{equation}

   \begin{equation}
   {{dF} \over {dz}}= {9 \over 4} \rho \nu_{\rm turb} \omega^2  ,
   \end{equation}   

   \begin{equation}
    F = F_{\rm rad}+F_{\rm turb}  ,
   \end{equation}
  
   \begin{equation}
   F_{\rm rad}= - { {16 \sigma} \over {3 \kappa \rho} } T^3 { {dT} \over {dz} }.
   \end{equation}

   \begin{equation} F_{\rm turb}= - \rho \chi_{\rm turb} T {{ds} \over
{dz}} , \end{equation} where $g_{\rm z}$ is the vertical gravitational
acceleration, $\omega$ the angular velocity, $T$ the local
temperature, $\rho$ the mass density, $\kappa$ the Rosseland mean
opacity, $s$ the specific entropy and $F$ the total vertical energy
flux which is the sum of the radiation flux $F_{\rm rad}$ and the
turbulent heat flux $F_{\rm turb}$.  For the turbulent viscosity
$\nu_{\rm turb}$ we adopt the $\alpha$-parametrization of Shakura and
Sunyaev (1973) in the following form :

   \begin{equation}
   \nu_{\rm turb}= {2 \over 3} { {\alpha p_{\rm g}} \over {\rho \omega}}.
   \end{equation}

   The energy transport
   by turbulent eddies 
   is modelled 
   according to the approach of 
   Shakura et al. (1978) and
   R\"udiger et al. (1988, 1990).
   The ratio of turbulent heat
   conductivity $\chi_{\rm turb}$ to turbulent
   viscosity is given by the 
   dimensionless Prandtl 
   number
   \begin{equation}
   Pr = {\chi_{\rm turb} \over \nu_{\rm turb} }.
   \end{equation} 
   For the calculations in this paper
   we will assume a value $Pr=1$.
Rosseland mean opacities were calculated 
with a program of Hauschildt (1991)
based on absorption cross-sections of
Mathisen (1984) and Kurucz (1979).
It includes electron scattering, free-free absorption and the bound-free 
transitions of H I, $H^-$, He I, He II, 
C I-III, N I-III, O I-III, Na I, Mg I-II,
Al I, Si I-II, S I, K I, \nobreak Fe I-II; 
solar abundances are assumed.

\section{The Sgr A* disk}
\subsection{Structure of the disk}
   We have computed disk models for 
   four different values of the 
   viscosity parameter: 
   $ \log \alpha = -4, -3, -2, -1$.
   A black hole mass $M_{\bullet}=10^6 M_\odot$,
   an accretion rate $M= 10^{-7} M_{\odot} yr^{-1}$
   and a canonical Kerr-parameter
   a=0.9982 (Falcke et al. 1993a) is assumed. \hfill\break

   Figures 1\&2 show two typical
   results of a vertical structure
   calculation.
   Fig. 1 correspond to a low value of
   the surface temperature ( $T_{\rm s}= 2000K$).
   In the vertical profiles the influence of
   hydrogen ionization can be clearly seen.
   In the inner layers
   where hydrogen is partially ionized, 
   the eddy flux $F_{\rm turb}$
   is the
   dominant energy transport mechanism, 
    while
   the atmospheric and sub-atmospheric layers
   are almost purely radiative.
   The density profile (dash -- tripple dots) is rather flat in the
   interior, but has the usual exponential 
   behaviour in the outer layers.
   Figure 2 represent a vertical structure for
   a section with surface temperature $T_{\rm s}=4.1 \cdot 10^4 K$.
   In this case hydrogen is fully 
   ionized. The density profile
   is very smooth.The energy transport by
   radiation dominates although the turbulent
   heat flux still contributes 40 \% near the 
   midplane. \hfill\break

   In the thin Keplerian disk approach
   the knowledge of the vertical
   disk structure allows one to 
   compose a whole disk
   model from radial rings. 
   In this way we have displayed 
   in Figs. 3-8 the radial
   run of some important quantities. \hfill\break

   It is useful to compare the results for 
   starving disk models with those for
   disks with subcritical luminosity
   since the latter have been 
   studied more extensively in the
   context of AGN. 
   We have thus additionally
   calculated a disk model
   with the same black hole mass
   and $\log  \alpha = -1$ , but
   with an accretion rate
   $\dot M =1.97 \cdot 10^{-3} M_{\odot} yr^{-1}$
   corresponding to a luminosity
   $L/L_{\rm edd}=0.29$.
   For this model the condition
   of geometrical thinness
   is still fulfilled
   since one has
   $ (h/r)_{\rm max} \approx 0.1$
   (Laor \& Netzer 1989).
   The corresponding curves in Figs. 3-8
   are labelled with ``agn''. \hfill\break

   In Fig. 3 the variation of the disk halfthickness with radius is
shown. The condition of geometrical thinness is very accurately
fulfilled for starving disks, we found a maximal value
$(h/r)_{\max}=3.5 \cdot 10^{-3}$ for the model with $\alpha=10^{-4}$
while the absolute thickness of the disk is still substantial and
comparable with stellar radii.  The increase of $h$ with $r$ is nearly
linear, there is no sign of thickening near the inner edge as in the
AGN-case.
\hfill\break

   The vertical optical
   thickness $\tau$ and the surface
   density $\Sigma$ as function of
   $r$ are shown in Figures 4\&5.
   For  
   starving disks 
   with low values 
   of the viscosity
   parameter $ \alpha $
   the values
   of $\tau$ and $\Sigma$ are 
   remarkably high.
   Lower values of $\alpha$ require
   higher values of $ \Sigma$ in
   order to maintain the required mass supply.
   The optical depth can even 
   exceed the values for an
   AGN-disk!
   The reason is the strong 
   contribution of the bound-free
   opacity in starving disks due to
   the low surface temperature.
   A vertically averaged value
   $\bar \chi$ for the opacity
   in ${\rm cm}^2 {\rm g}^{-1}$ is given
   as $\bar \chi \approx \tau / \Sigma$.
   It follows that $\bar \chi$ is several
   hundred times larger in starving disks
   than in AGN-disks.
   Due to hydrogen ionization
   the curves deviate from monotonic behaviour
   near the outer edge of the displayed
   radial range which is particularly
   obvious for the higher values of the
   viscosity parameter
   $\log  \alpha=-1, -2$. 
   The opacity changes rapidly over 
   several orders of magnitude.
   For radii larger than $ \log\, r > 13.5...14$
   the optical thickness is 
   strongly reduced 
   but the precise
   values depend on the 
   not satisfactorily known molecular
   and dust opacities.
   Figure 6 gives the temperatures in the
   midplane of the disk.
   The values of $T_{\rm c}$ are considerably
   high and exceed $10^6K$ for small
   radii. Near the outer edge the
   ratio $T_{\rm c}/T_{\rm s}$ decreases due to 
   the large decrease of the optical
   thickness. \hfill\break
   The high temperatures in the inner
   disk region raise the question about the
   role of radiation pressure.
   In Fig. 7 the ratio $p_{\rm r}/p_{\rm g}$ is shown.
   For low values of the viscosity parameter
   ($ \log  \alpha =-4, -3$) radiation pressure
   is dominant in the innermost part of the disk,
   while for larger values of $\alpha $ the radiation
   pressure does not exceed
   the gas pressure.
   Thus despite the low accretion rate the starving disks
   modelled here  
   are 
   at the verge of 
   becoming radiation pressure dominated in their
   inner region. \hfill\break

   In Fig. 8  characteristic timescales
   for the disk are shown.
   We have estimated the viscous 
   timescale according to $ t_{\rm visc} \approx r / v_{\rm r}$,
   where $v_{\rm r}$ is the radial accretion
   velocity due to viscous loss of angular
   momentum and
   the sound propagation time according to
   $ t_{\rm s} \approx  r/c_{\rm s}$ where
   $c_{\rm s}= (P_{\rm tot}/\rho)^{1/2}$
   is the local sound velocity.
   The timescales for starving disks are very long 
   compared with an AGN-disk.
   The sound propagation time is of the
   order of years, while the timescale
   for viscous evolution ranges in
   dependence of $\alpha$ from several
   thousand to some $3\cdot10^6$ years. \hfill\break
   The reason why
   despite of the geometrical thinness 
   the timescales for a
   starving disk  are considerably longer 
   than for the AGN-disk model
   is as follows.
   The equation of continuity requires
   $\dot M = 2\pi r \Sigma v_{\rm r}$.
   The ratio of accretion rates is
   $ \dot M_{\rm agn}/ \dot M_{\rm starv} \approx 5 \cdot 10^{-5}$.
   However, as can be seen from Fig. 5,
   the ratio $\Sigma_{\rm starv}/ \Sigma_{\rm agn}$
   is larger than this value even for 
   $\log \alpha = -1$.
   Thus one has $v_{\rm r,agn} / v_{\rm r,starv} \gg 1$,
   which means that in a starving disk a
   fluid element needs a much longer time for
   drifting inwards.
   In gas pressure dominated regions
   the velocity of sound scales roughly
   with $T_c^{1/2}$ and is thus
   also lower in a starving disk than
   in a subcritical disk.
   The difference is even bigger in the
   innermost region since for the subcritical
   disk model the high radiation pressure
   additionally increases the
   values of $c_s$. \hfill\break
   We can now complete the comparison
   of starving disks and agn disks.
   Somewhat surprisingly,
   a starving disk with low viscosity
   parameter $\alpha$ has values of
   $\tau$ and $\Sigma$ not very
   far from that of an AGN-disk
   with $\alpha=0.1$.The most important
   differences occur in the halfhickness
   $h$, the temperature and the viscous
   timescale.






   \begin{figure}[htbp]
%      \picplace{6.5cm}
\centerline{
\bild{f1.ps}{height=5.8cm,bbllx=2.5cm,bblly=12.4cm,bburx=20.3cm,bbury=24.5cm}
}
      \caption{Vertical structure for a region in the disk 
               with low surface temperature
               ($T_{\rm s}=2000 K$).
               Density, temperature and radiation flux
               as well as the ratio of turbulent heat flux 
               to total flux are plotted.
               Density and temperature are normalized by their
               midplane values $\rho_{\rm c}, T_{\rm c}$. The radiation
               flux is normalized by the flux $F_{\rm s}$ emerging
               from the disk surface. }
   \end{figure}
   \begin{figure}[htbp]
%      \picplace{6.5cm}
\centerline{
\bild{f2.ps}{height=5.8cm,bbllx=2.5cm,bblly=12.4cm,bburx=20.3cm,bbury=24.5cm}
}
      \caption{Vertical structure for a higher surface temperature.
                ($T_{\rm s}=4.1 \cdot 10^4 K$).}
   \end{figure}
   \begin{figure}[htbp]
%      \picplace{6.5cm}
\centerline{
\bild{f3.ps}{height=5.8cm,bbllx=2.5cm,bblly=12.4cm,bburx=20.3cm,bbury=24.5cm}
}
      \caption{The half thickness h of the 
               Sgr A* disk  models as a function of the radius.
               The curves are labelled by $\log \alpha$.
               Also shown is a subcritical disk model
               (agn) with $L/L_{edd}=0.29$ and $\alpha = 0.1$.}

   \end{figure}
   \begin{figure}[htbp]
%      \picplace{6.5cm}
\centerline{
\bild{f4.ps}{height=5.8cm,bbllx=2.5cm,bblly=12.4cm,bburx=20.3cm,bbury=24.5cm}
}
      \caption{The variation of the vertical optical depth $\tau$ with radius.
               (Sgr A*)}
   \end{figure}
   \begin{figure}[htbp]
%      \picplace{6.5cm}
\centerline{
\bild{f5.ps}{height=5.8cm,bbllx=2.5cm,bblly=12.4cm,bburx=20.3cm,bbury=24.5cm}
}
      \caption{The surface density $\Sigma$ as a function of radius.(Sgr A*)}
   \end{figure}
   \begin{figure}[htbp]
%      \picplace{6.5cm}
\centerline{
\bild{f6.ps}{height=5.8cm,bbllx=2.5cm,bblly=12.4cm,bburx=20.3cm,bbury=24.5cm}
}
      \caption{The midplane temperature $T_{\rm c}$ as a function
               of radius. The run of the surface temperature
               $T_{\rm s}$ is plotted for comparison. (Sgr A*)}
   \end{figure}
   \begin{figure}[htbp]
%      \picplace{6.5cm}
\centerline{
\bild{f7.ps}{height=5.8cm,bbllx=2.5cm,bblly=12.4cm,bburx=20.3cm,bbury=24.5cm}
}
      \caption{The midplane ratio of radiation pressure $p_{\rm r}$
               to gas pressure $p_{\rm g}$ for the innermost disk region.
               The radius is given here in units of the
               gravitational radius $r_{\rm g}=2GM/c^2$. (Sgr A*)}
   \end{figure}
   \begin{figure}[htbp]
%      \picplace{6.5cm}
\centerline{
\bild{f8.ps}{height=5.8cm,bbllx=2.5cm,bblly=12.4cm,bburx=20.3cm,bbury=24.5cm}
}
      \caption{The viscous and sound propagation
               timescales for the Sgr A* disk models
               with $\log \alpha = -1, -4$ and the
               subcritical disk.  }
   \end{figure}
   \begin{figure}[htbp]
%      \picplace{6.5cm}
\centerline{
\bild{f9.ps}{height=5.8cm,bbllx=2.5cm,bblly=12.1cm,bburx=19.8cm,bbury=24.5cm}
}
      \caption{The relation between surface density $\Sigma$
               and mass flux rate $\dot M$ for $\log \alpha= -4$
               at different radii. The curves are labelled by r in
               units of $r_{\rm g}$. (Sgr A*)}
   \end{figure}
   \begin{figure}[htbp]
%      \picplace{6.5cm}
\centerline{
\bild{f10.ps}{height=5.8cm,bbllx=1.7cm,bblly=13.4cm,bburx=18.6cm,bbury=24.4cm}
}
      \caption{ $\Sigma - \dot M $ - diagrams for $\log \alpha = -2$. (Sgr A*)}
   \end{figure}
   \begin{figure}[htbp]
%      \picplace{6.5cm}
\centerline{
\bild{f11.ps}{height=5.8cm,bbllx=2.5cm,bblly=12.1cm,bburx=19.8cm,bbury=24.5cm}
}
      \caption{The thermal instability strips in the 
               $\log r - \log \dot M $ plane for 
               $\log \alpha = -2$ (dark regions) and 
               $\log \alpha=-4$ (hatched region). (Sgr A*)}
   \end{figure} 

\subsection{Gravitational instability}

We have seen that the disks under consideration here
are relatively massive but very thin. It is thus 
necessary to discuss the gravitational stability of
these disks.
Approximately, the gravitational stability of a thin
rotating disk is controlled by the dimensionless number
(Toomre 1964, Lin and Pringle 1987)

\begin{equation}
  Q= {h \over R} {M_{bh} \over M_d}
\end{equation}
where $M_{\rm d} (R)$ is the disk mass inside a radius R.  The disk is
gravitationally stable if $Q > 1$.

Here we have attempted to model the disk structure up to an outer
radius $ R_{\rm o} \approx 3 \cdot 10^{13} cm$ corresponding to
$T_{\rm s} \approx 2000K$. The disk mass is estimated as $M_{\rm d}
\approx
\bar\Sigma R^2$, where $\bar\Sigma$ is an averaged value for the
surface density. With $\bar\Sigma \approx 10^5 {\rm g\,cm}^{-2} $ this
gives $M_{\rm d}(R_{\rm o}) \approx 0.05M{_\odot}$ while one has $h/R
> 3 \cdot 10^{-3}$. Therefore we have $Q > 6
\cdot 10^4$ and the disk region modelled here is certainly
gravitationally stable.

The selfgravitation of the disk is expected to become important at a
radius at least 2 orders of magnitude larger than the above value of
$R_{\rm o}$ hence at $R\gg 10^{16} cm$.  At present it is not clear
what happens in a selfgravitating disk.  There are suggestions
(Paczy\'nski 1978, Koz\l owski et al.  1979, Lin and Pringle 1987)
that a stabilization could take place in the nonlinear regime due to a
gravitationally induced viscosity, roughly given by
\begin{equation}
 \nu_{grav} \approx Q^{-2} hc_s
\end{equation}
which always exceeds the $\alpha$-viscosity for
$Q < 1$. In this case the outward angular momentum transport
can take place at a reasonable rate in the unstable region.

Another possibility which is worth to be considered in the context of
starving galactic disks has been outlined by Kolykhalov and Sunyaev
(1980). One can imagine a disk where the outer parts are fragmented
into stars and gas clouds forming a spinning ring around the
gravitationally stable inner disk region.  Material, e.g. from stellar
winds, could fall onto the disk over a broad range. While in the inner
region angular momentum is carried by viscous stresses and inflow of
matter takes place, in the outer parts the excess angular momentum is
carried mainly by a stream of matter to the fragmented torus.

Such a starforming disk picture was already suggested by Zylka et al.
(1993) for the Galactic Center. And in fact one finds a small
population of unsusual He\,I emission line stars (Krabbe et al. 1991)
at a distance of 0.1 pc away from Sgr A* part of which form the so
called IRS 16 complex.  Tamblyn \& Rieke (1993) explained IRS 16 as
result of a starburst some $10^7$ years ago but added a note that
recent observations imply that the IMF in this area must have been
very unusual -- if this are stars at all. One now is tempted to link
the existence of these stars to the fragmentation of the inner disk.
This would have the consequence that the stellar wind produced already
has a large angular momentum making purely spherical accretion even
more unlikely.


   \subsection{Thermal instability} The thermal behaviour of the disk
is determined by the relation between surface density $\Sigma$ and
mass flux rate $\dot M$ at a given radial location.  The concept of
thermal instabilities in accretion disks which arises due to a
hysteresis-like $ \dot M ( \Sigma) $-relation is now well established
in accretion theory (e. g. Meyer and Meyer-Hofmeister 1981, 1983 Bath
\& Pringle 1982, Smak 1984).  Thermal variations in low luminosity
AGN-disks have been discussed by Clarke and Shields (1989).
\hfill\break We have calculated the $\dot M (\Sigma)$-relation for
$\log \alpha = -2, -4$ at different radii (Figs. 9, 10).  While for
$\log \alpha= -4$ a simple S-shape results one has for $\log \alpha
=-2 $ a more complicated, double S-like relation.  The branches with
$\partial \dot M /\partial \Sigma < 0$ are thermally unstable.  The
occurence of double S-shapes is familiar from the study of AGN-disks
(Cannizzo 1992) and caused by the fact that both a peak in the opacity
mountain as well as the eddy heat transport can produce unstable
branches in the $\Sigma- \dot M$ diagram. \hfill\break Since starving
disks are cooler than AGN disks the thermal instability occurs much
nearer to the center.  The condition $\partial \dot M /\partial \Sigma
< 0$ defines instability strips in the $ \log r - \log \dot M$ plane
which are shown in Figure 11.  For a given mass flux rate which is
physically determined by the feeding mechanism at larger scales the
thermally unstable radial zones can be found from Figure 11.  While
for $\log \alpha =-4$ there is a single instability strip, two smaller
instability regions occur for $\log \alpha=-2$.\hfill\break According
to Meyer (1984), the thermal instability drives heating and cooling
fronts through the unstable disk region.  Since the front velocity is
$ v_f \approx \alpha c_s$, the crossing time for a front is $t_f
\approx t_s/\alpha$.  In the hydrogen ionization zone ($r \ge
10^{13}$cm) one has $t_s$ in the range of $10^7...10^8 s$ and thus
$t_f$ may range from 3 to $3 \cdot 10^4$ years.  As was pointed out by
Clarke and Shields(1989) the spectrum of a low- luminosity disk will
vary mainly in the infrared during a limit cycle.  Thus, if one has $
\alpha \ge 0.1$ for the Sgr A disk, an observational verification of
thermal variations is in principle possible, although still very
difficult in practice.


\section{The situation in M31}
In the introduction we suggested that starving disks may be common
objects in the center of ordinary galaxies but that the luminosities
of such disks are so low that it is really difficult to detect them.
The best chance to find another sytem like in Sgr A* in another galaxy
is our neighbour M31. It was discussed that in the center of this
galaxy also a massive black hole of some $10^7 M_{\sun}$ dominates the
dynamics of the inner region (see Bacon et al. 1994 and refs.).
Because of the presence of a weak radio source in the dynamical center
it is also argued that a similar process as in Sgr A* may be important
in the nucleus of M31 as well (Melia 1992b).  In the case of M31 we
are not blocked by absorption in the Galactic Plane so that direct HST
observations may provide reasonable constraints for the central
engine.

Crane et al. 1993 and Lauer et al. 1993 presented such observations of
the nucleus of M31. So far a clear pointsoure is not detected. The
brightness profile extends up to $13.7^{\pm0.1}$mag/sq.arcsec with a
pixel size of 0.044\arcsec. If we convert this into an absolute
magnitude for an M31 distance of 750 kpc, we obtain $-10.7$
mag/sq.arcsec and using the emiprical relation derived by Falcke et
al.  (1994) this converts into a disk luminosity of
$\la1.6\cdot10^{38} erg/sec$ when considering a point source within
the central $3\times3$ pixel box. For the maximaly rotating Kerr hole
we use here this translates into an accretion rate of $\la0.9 10^{-8}
M_{\sun}$/yr. This estimate should be considered as a crude upper
limit but as the radio flux of the point source in M31 is a factor of
5 lower than in Sgr A* a somewhat lower accretion rate in M31 compared
to Sgr A* is not unrealistic.

To highlight the principal differences in structure of the starving
disks in Sgr A* and the nucleus of M31 we now simply adopt the
parameters $M_\bullet=10^7 M_{\sun}$ and $\dot M=10^{-8} M_{\sun}/$yr
and perform the same calculations as for Sgr A*.  The variation of the
vertical optical depth along the disk is shown in Fig. 12. Since the
putative M31-disk is cooler by a factor of $10^{3/4}$ than the Sgr A*
disk, the hydrogen-recombination region is located nearer to the inner
edge when measured in units of $r_{\rm g}$.  The disk becomes
optically thin for $r \ge 20...40r_{\rm g}$. In this transition region
there is an increase of the surface density $\Sigma$ before it
decreases outward again (Fig. 13). Correspondingly, the viscous
timescale is prolonged to very large values (Fig. 14) and may approach
a Hubble time in the outer parts.  These values indicate that the
turbulent viscosity is probably not efficient enough in transporting
the angular momentum for radii $r\gg10^{14}{\rm cm}$.  This is
especially true in the regime where self-gravitation is not yet
important. In order to model this part of such a low-luminosity disk
one either has to abandon values of $\alpha$ as low as $10^{-4}$ or
invoke other angular momentum transport mechanisms for which magnetic
fields and tidal waves are good candidates.  On the other hand it
could be possible that the accretion ceases completely leaving a
rotating ring at the outer edge.
\begin{figure}[htbp]
%\picplace{6.5cm} 
\centerline{
\bild{f12.ps}{height=5.8cm,bbllx=2.5cm,bblly=12.4cm,bburx=20.3cm,bbury=24.5cm}
}
\caption{The variation of $\tau$ with radius for the
M31 disk models with $\log \alpha = -4, -2, -1$.} \end{figure} \begin{figure}[htbp] 
%\picplace{6.5cm}
\centerline{
\bild{f13.ps}{height=5.8cm,bbllx=2.5cm,bblly=12.4cm,bburx=20.3cm,bbury=24.5cm}
}
\caption{The surface density $\Sigma$ as a function of radius for the
M31 disk models.} \end{figure} \begin{figure}[htbp] 
%\picplace{6.5cm}
\centerline{
\bild{f14.ps}{height=5.8cm,bbllx=2.5cm,bblly=12.4cm,bburx=20.3cm,bbury=24.5cm}
}
\caption{The viscous and sound propagation timescales for the M31 disk
models} \end{figure}



\section{Summary}
In this paper we argue that starving disks arround black holes may be
a common phenomenon in the center of galaxies. These are disks around
supermassive black holes with accretion rates several orders of
magnitude below the Eddington limit. We suggest that they might be
fossil remains of a past and active disk accretion onto the center of
the galaxy and therefore keep the memory on the original orientation
of a larger disk. Even though the mass supply on larger scales down to
the inner parsec might have been interrupted, we propose that the wind
of stars in the central star cluster can prevent the disk from
complete dissolution. The more or less spherical wind can conserve the
angular momentum of the fossil disk and lead to increase of its mass
reservoir. The timescale on which the accreted wind reaches the
central black hole may be considerably longer than the freefall time
and be dominated by the long viscous timescale of the disk. Such a
scenario may explain why Sgr A* has such a low disk accretion rate
(Falcke et al. 1993a) in its inner region while the inferred wind
accretion rate should be much higher (Melia 1992a). Principally our
approach differs from the Melia (Bond-Hoyle) model by postulating a
separate inner boundary condition for the accretion flow due to a
fossil disk. Without such a fossil disk the infalling wind should most
likely produce a substantial central luminosity which is excluded
observationally. Another possibility would be that the accretion disk
has fragmented in its outer parts and formed stars.  The stellar winds
than would have substantial angular momentum themselves unless the
isotropization timescale of the star cluster is much smaller than the
(unknown) age of the present stars.

We have outlined the physical properties of starving disks. Such disks
are geometrically perfectly thin, but optically very thick in their
inner region.  Compared to AGN disks starving disks seem to be the perfect
realization of the Shakura \& Sunyaev (1973) accretion disk.
The timescale for the viscous evolution of such disks is found to be
very long (up to $10^{7}$ yr). The dominant source of opacity in these
disks are bound-free transitions. The radiation pressure can be
important, when the black hole mass is relatively small and the
viscosity is very low. In general there are thermally unstable radial
zones in which a hysteresis-like surface density-mass flux relation
occurs. The corresponding limit cycles can have periods in the range
from several years up to several thousand years and could be seen in
the IR-NIR regime. 

The same model as developed for Sgr A* may apply for the center of M31
as well. However, because of an even lower accretion rate onto a
probably more massive black hole the situation is more extreme. The
`hot' disk will be very small (some 20 $r_{\rm g}$) and the viscous
time scales may become even longer than in Sgr A*.

  
\acknowledgements 
We thank F. Meyer who outlined the concept of a fossil disk and
mentioned its relevance for the wind accretion scenario to us. The
referee, M. Rees, pointed out the possibility of self-gravitation and
star-formation. P.G.  Mezger, P.L. Biermann and W.J. Duschl stimulated
our interest in this field and contributed in various ways to this and
the preceding work. Interesting discussions with F. Melia helped to
clarify several points. The work was supported by the DFG (Bi 191/9 \&
SFB 328).
      
  
% 
%  14.Sep.'90: Demo-Vs. 
%_____________________________________________________________________ 
  
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