From rohan@kailash.harvard.edu Thu Jun 12 10:25:56 1997
Date: Thu, 12 Jun 1997 10:25:01 -0400
From: rohan@kailash.harvard.edu (Rohan Mahadevan)
To: gcnews@sgrastar.astro.umd.edu
Subject:  Sgr A*


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\begin{document}
{\centerline {\bf Submitted to {\em The Astrophysical Journal} on June 11, 1997.}}
\title{Advection-Dominated Accretion Model of Sagittarius A$^*$:
Evidence for a Black Hole at the Galactic Center}

\authoremail{authors@cfa.harvard.edu}
\author{Ramesh~Narayan, Rohan~Mahadevan, Jonathan~E.~Grindlay, 
Robert~G.~Popham, Charles~Gammie}

\affil{Harvard-Smithsonian Center for Astrophysics, 60 Garden St.,
Cambridge, MA 02138}

\begin{abstract}

Sgr A$^*$ at the Galactic Center is a puzzling source.  It has a mass
$M=(2.5\pm0.4)\times10^6M_\odot$ which makes it an excellent black
hole candidate.  Observations of stellar winds and other gas flows in
its vicinity suggest a mass accretion rate $\dot M\gsim {\rm
few}\times10^{-6}\msun\,{\rm yr^{-1}}$.  However, such an accretion
rate would imply a luminosity $> 10^{40}~{\rm erg\,s^{-1}}$ if the
radiative efficiency is the usual 10\%, whereas observations indicate
a bolometric luminosity $<10^{37}~{\rm erg\,s^{-1}}$.  The spectrum of
Sgr A$^*$ is unusual, with emission extending over many decades of
wavelength.  We present a model of Sgr A$^*$ which is based on a
two-temperature optically-thin advection-dominated accretion flow.
The model is consistent with the estimated $M$ and $\dot M$, and it
fits the observed fluxes in the radio and X-ray bands as well as
measured upper limits in the sub-millimeter and infrared bands.  The
model explains the very low luminosity of Sgr A$^*$ by invoking
advection.  Most of the viscously dissipated energy is carried into
the central star by the accreting gas, and therefore the radiative
efficiency is extremely low, $\sim5\times10^{-6}$.  A critical element
of the model is the presence of an event horizon at the center.  The
success of the model may thus be viewed as confirmation that Sgr A$^*$
is a black hole.

\end{abstract}

\keywords{accretion, accretion disks --- black holes --- galaxies: nuclei
--- Galaxy: center --- radiation mechanisms: bremsstrahlung, inverse
Compton, synchrotron --- radio sources: Sgr A$^*$}

\section{Introduction}

The enigmatic radio source, Sagittarius A$^*$ (Sgr A$^*$), has for
many years been a puzzle (Genzel \& Townes 1987; Genzel, Hollenbach \&
Townes 1994; Mezger, Duschl \& Zylka 1996).  The source is located at
the dynamical center of the Galaxy and is presumed to be associated
with a supermassive black hole.  However, observations provide
conflicting indications both for and against the black hole hypothesis
and there is currently no model that explains all the observations.
The paradoxical clues may be summarized briefly as follows:

\noindent
1. Dynamical measurements indicate a dark mass of $\sim(2.5\pm0.4)
\times10^6 M_\odot$ within the central 0.1 pc of the Galactic Center
(Eckart \& Genzel 1997).  This presumably represents the mass $M$ of
the putative supermassive black hole in Sgr A$^*$.

\noindent
2. Observations of stellar winds and gas flows near Sgr A$^*$, coupled
with the above estimate of the black hole mass, suggest that \sgr must
accrete gas from its surrounding at a mass accretion rate $\dot M
\gsim{\rm few}\times 10^{-6}M_\odot{\rm yr^{-1}}$ (Genzel et al. 1994).

\noindent
3. This $\dot M$ implies an accretion luminosity $L\sim0.1\dot Mc^2>
10^{40}~\ergs$, assuming a nominal radiative efficiency of 10\%.
However, \sgr is an unusually dim source with a total luminosity from
radio to $\gamma$-rays under $10^{37}~\ergs$.  The extremely low
luminosity has been used to argue against \sgr being an accreting
black hole (Goldwurm et al. 1994).

\noindent
4. For the above $M$ and $\dot M$ the peak emission from a standard
thin accretion disk will be in the near infrared (cf. Frank, King \&
Raine 1992).  In fact, this is a generic prediction of any model that
involves an optically thick flow radiating as a blackbody.  However,
Menten et al. (1997) have obtained a strong upper limit on the 2.2
micron flux of Sgr A$^*$ that effectively rules out such models.

\noindent
5. \sgr is brightest in the radio/mm band (see \S2 and Fig. 1), it is
weakly detected in X-rays (Predehl \& Trumper 1994), and it may have
been detected between 100 MeV and 2 GeV (Merck et al. 1996).  These
observations, combined with the Menten et al. infrared upper limit,
imply a spectral distribution completely unlike a blackbody.  The
observations suggest that \sgr is optically thin, and that many
different radiation processes are in operation.

Over the years a number of models have been proposed for Sgr A$^*$.
Some of these have been phenomenological approaches which aim to
explain the radio and infrared spectrum without including any detailed
dynamics (e.g. Falcke 1996, Duschl \& Lesch 1994).  
These models generally require mass accretion rates lower than
those indicated by the observations.  Other models (Melia 1992, 1994;
Mastichiadis \& Ozernoy 1994) have attempted to incorporate dynamics,
but with drastic simplifying assumptions such as ignoring the angular
momentum of the accreting gas.  One model of \sgr that has attempted a
self-consistent treatment of both viscous hydrodynamics and radiation
processes is the advection-dominated accretion flow (ADAF) model of
Narayan, Yi \& Mahadevan (1995).  (See Rees 1982 for some early ideas
along these lines.)  The present paper is based on this model.

An advection-dominated accretion flow is one in which most of the
energy released by viscous dissipation is stored in the gas and
advected to the center, and only a small fraction of the energy is
radiated (Narayan \& Yi 1994, 1995a, 1995b; Abramowicz et al. 1995;
Chen et al. 1995; see Narayan 1997 for a recent review).  Most current
work on ADAFs has been concerned with a branch of low $\dot M$
solutions which is present for mass accretion rates below a few
percent of the Eddington rate (Narayan 1997; Esin, McClintock \&
Narayan 1997).  These low-$\dot M$ solutions are based on a
two-temperature plasma (Shapiro, Lightman \& Eardley 1976; Rees et
al. 1982); by assumption, the bulk of the viscous energy in these
flows is deposited in the ions, and energy then flows from the ions to
the electrons through Coulomb collisions.  The ions achieve a nearly
virial temperature, $T_i\sim10^{12}~{\rm K}/r$, where $r$ is the
radius in Schwarzschild units, while the electron temperature
saturates at around $10^9-10^{10}$ K for $r\ \sles\ 100$.  The gas is
optically thin and the flow does not suffer from any serious thermal
or viscous instabilities (Narayan \& Yi 1995b; Abramowicz et al. 1995;
Kato, Abramowicz \& Chen 1996).

The low-$\dot M$ branch of ADAFs has several appealing features for
explaining the observations of \sgr and other low-luminosity accretion
systems.  First, since most of the energy is advected with the gas and
lost into the black hole, the model naturally explains the low
luminosity of Sgr A$^*$.  Second, being optically thin, the spectrum
is quite different from blackbody.  Indeed, the high electron and ion
temperatures allow a variety of radiation processes to operate:
synchrotron, bremsstrahlung and inverse Compton from the electrons
(Narayan \& Yi 1995b), and $\gamma$-ray emission from the ions via
pion production (Mahadevan, Narayan \& Krolik 1997).  Therefore, the
gas is likely to radiate over a wide range of wavelengths.

The Narayan et al. (1995) ADAF model of \sgr gave a reasonable fit to
the observations available at that time and provided a convincing
explanation of the low luminosity of the source.  However, the model
required a black hole mass of $7\times10^5M_\odot$ which differs from
the dynamical mass estimate of $2.5\times10^6M_\odot$ (Eckart \&
Genzel 1997).  This problem prompted us to take another look at the
model.

In addition, there have been important developments on the
observational and theoretical front which further motivate the present
study.  The recent Menten et al. (1997) limit on the infrared flux of
\sgr is well below previous ``detections'' (Rosa et al. 1991; Eckart et al. 1992)
which were used by Narayan et al. (1995) to fit the ADAF model.  
Similarly, ASCA observations by
Koyama et al. (1996) have shown that the X-ray emission from the
Galactic Center is dominated by an X-ray burster which lies within 1.3
arcminutes of \sgr.  This implies that most previous measurements of
the X-ray flux of \sgr (Pavlinskii, Grebenev, \& Sunyaev 1992, 1994; 
Skinner et al. 1987) are suspect since they were obtained with
inadequate spatial resolution.  Only the ROSAT PSPC detection by
Predehl \& Trumper (1994) survives as a reliable X-ray detection,
although the corresponding luminosity of the source in the ROSAT band
is uncertain due to uncertainties in the absorbing hydrogen column
(cf. \S2).

The modeling techniques have also advanced significantly during the
intervening two years.  The Narayan et al. (1995) model was a local
one in the sense that the dynamics were calculated using a local
self-similar solution (Narayan \& Yi 1994) and the radiation was
computed with a local approximation for Compton scattering (Dermer,
Liang \& Canfield 1991; Narayan \& Yi 1995b).  Consistent global
dynamical solutions have since been calculated, initially with a
pseudo-Newtonian potential (Narayan, Kato \& Honma 1997; Chen,
Abramowicz \& Lasota 1997) and more recently for the full relativistic
Kerr problem (Abramowicz et al. 1996; Peitz \& Appl 1997; Gammie \&
Popham 1997).  A global scheme for Comptonization has also been
developed (Narayan, Barret \& McClintock 1997), based on techniques
due to Poutanen \& Svensson (1996).  Finally, Nakamura et al. (1997)
have shown that in addition to energy advection by ions, which had
been considered in previous work, advection by electrons can be
important in some circumstances.  This effect is now included in the
calculations (Appendix A).  The present models are thus superior to
those used by Narayan et al. (1995).

In this paper we present improved ADAF models of Sgr A$^*$.  We make
use of the spectral data described in \S2 and employ the modeling
techniques outlined in \S3.  We present detailed results in \S4 and
conclude with a discussion in \S5.
 
\section{Spectral Data}

The radio to near infrared (NIR) spectrum of Sgr A$^*$ has been of
constant interest since the discovery of the source by Balick \& Brown
(1974), and observations have been carried out from 400 MHz (Davies et
al. 1976) to $\sim 10^{14}$ Hz (Menten et al. 1997).  Table 1 is a
compilation of the radio to NIR observations.  The data are also
plotted in Fig. 1.  The two general features in the table and figures
are 1) we have given only the maximum and minimum fluxes for
frequencies at which variability has been observed, and 2) open
circles correspond to low resolution measurements, which we treat as
upper limits because of possible contamination, while filled circles
correspond to high resolution measurements which most likely probe the
accretion flow in Sgr A$^*$.  This is discussed in detail below.

In determining the spectrum of the ADAF from Sgr A$^*$, special
attention has to made to the angular resolution of the observations,
which must be comparable to the size of the ADAF.  The angular size of
an ADAF at the Galactic Center is $\lsim 0.06\arcsec$ ($\sim 500$ AU),
where we have taken $M = 2.5\times10^{6} M_{\odot}$, and assumed a
linear size of $\sim 10^4$ Schwarzschild radii.

For frequencies $\lsim 40$ GHz, electron scattering in the ISM between
Sgr A$^*$ and the observer leads to source broadening.  The apparent
size of the source is larger than the intrinsic size, and is
proportional to the square of the observed wavelength (e.g. Davies et
al. 1976; Backer 1982).  Since the intrinsic source size ($\lsim
0.06{\arcsec}$) is smaller than the scattering size ($\sim 1{\arcsec}$
at low frequencies), the observed flux could be contaminated by
structures (e.g. outflows or jets) which exist on scales smaller than
the scattering size. Therefore, for frequencies $\lsim 40$ GHz, we
label the points in Fig. 1 with open circles, indicating that the flux
measurements should be considered as upper bounds rather than as firm
detections of the central engine.

For frequencies $\gsim 40$ GHz, the scattering size falls below the
intrinsic size.  High resolution observations can therefore determine
directly the emission from the central engine.  However, only a few
high resolution ($\lsim 1$ mas) observations have been made at these
wavelengths using VLBI or the VLBA.  These radio observations are
indicated by a $\star$ in Table 1, and are represented as filled
circles in Fig. 1.  We believe that these points best represent the
emission from the ADAF
\footnote{However since the scattering size approaches the intrinsic size at 
$\nu \sim 43$ GHz, this VLBI point might be slightly contaminated.}
and we take the point at 86 GHz to be the best determined radio flux
of \sgr for the purposes of modeling the source.  Apart from these
VLBI/VLBA points, all other radio points $\gsim 40$ GHz have been
represented by open circles since these observations were done with
poor resolution ($\gsim 4{\arcsec}$).

The sub--millimeter to NIR observations are all upper limits, and are
represented by open circles due to the low angular resolution of the
measurements.  However, the recent upper limit of Menten et al. (1997)
at 2.2$\mu$m, is a high resolution measurement since the authors were
able to determine accurately the relative position of the Galactic
Center in their speckle images by comparing with a radio map of nearby
stellar SiO masers. The effective resolution of the speckle image is
$0.15\arcsec$, which is comparable to the size of the ADAF. The flux
measured by Menten et al. is therefore an excellent upper limit for
the emission of \sgr at NIR wavelengths, and we plot this point in
Fig. 1 as a filled circle.

X--ray and hard X--ray as well as gamma-ray observations of Sgr A*
have long been limited by the relatively poor angular resolution
available at these energies and yet very crowded field of the Galactic
Center region. Even at hard X--ray energies (e.g. 20--100 keV) where
the density of sources is relatively low, the Sgr A$^*$ source region
contains more sources ($\sim$10) in the central 5$\degree$ of the
Galaxy than any other region (Goldwurm et al. 1994). 
At soft X--ray (e.g. 0.5--4 keV) and
low-medium X--ray energies (e.g. 2--10 keV), this same region is even
more crowded, with complex diffuse emission and at least 10 sources
within the central 1$\degree$ of Sgr A* (cf. Watson et al 1991 and
Koyama et al 1996). Thus the flux and spectral distribution of the
Galactic Center in the $\sim$1--100 keV band, where the ADAF
bremsstrahlung component peaks (cf. Fig. 1), is particularly critical
and angular resolution is paramount. At the soft X--ray end of this
band, the relatively uncertain absorbing column density, $N_H$,
towards the Sgr A$^*$ source is especially important since the value
chosen (which can in principle be determined by future observations
with AXAF) greatly affects the derived source flux and luminosity.

We have included in the X--ray/hard X--ray region of the
spectrum plotted in Figure 1 only 4 points or upper limits which all
are the highest angular resolution available in their respective
energy bands.  These data are listed in Table 2.  The one detection of
a point-like source with position fully consistent with Sgr A$^*$ is
the ROSAT detection (PSPC) with $\sim 20\arcsec$ resolution in the
0.8--2.5 keV band reported by Predehl and Trumper (1994). This
detection is plotted for an assumed $N_H$ = 6 $\times$ 10$^{22}$
cm$^{-2}$, which is the usual best estimate (cf. Watson et al 1981)
for the interstellar column density and which corresponds to the
usually quoted visual extinction of A$_V \sim$ 25--30 mag. The total integrated
luminosity in the 0.8--2.5 keV band is $L_X = 1.5 \times 10^{34}$ erg s$^{-1}$.
\footnote{The $L_X$ and $\nu L_{\nu}$ 
calculations were done for the ROSAT data using the
PIMMS program, supplied by the HEASARC at GSFC.}
The $N_H$ used here 
differs from the much higher value (1.5 $\times$ 10$^{23}$ cm$^{-2}$)
assumed by Predehl and Trumper.  These authors chose a higher $N_H$ in
order to make the soft X--ray luminosity more compatible with the
variable compact source within $\sim 1\arcmin$ of Sgr A$^*$ reported
by Skinner et al (1987) and Pavlinskii, Grebenev and Sunyaev (1992)
from 2--20 keV coded aperture imaging observations with modest (few
arcmin) resolution.  However, recent ASCA observations of the Galactic
center in the 2--10 keV band, with $\sim 1\arcmin$ angular resolution
but much higher spectral resolution, have been reported by Koyama et
al. (1996) and Maeda et al (1996). They show the Sgr A$^*$ region to be
complex, with diffuse emission over a $\sim$2 $\times 3 \arcmin$
region and a point source at $1.3 \arcmin $ from Sgr A$^*$ which is an
X--ray burster and therefore a neutron star in a (8.4h) binary
system. This burster has very likely dominated the coded aperture
imaging flux measurements. Koyama et al. therefore quote an upper limit
of 10$^{36}$ erg/s as the 2--10 keV luminosity for the entire Sgr
A$^*$ complex, with a possible actual value of $\sim$10$^{35}$ erg/s
for the point source alone. We have plotted this ASCA upper limit in
Fig. 1.

The ROSAT PSPC detection, although only with angular resolution of
$\sim 20\arcsec$, is a likely detection of Sgr A$^*$ with minimal
contamination from the surrounding diffuse source since it appears in
the ROSAT image to be consistent with a point source. However, higher
spatial resolution (e.g. AXAF) X--ray imaging, and/or evidence for
time variability, are needed to confirm this identification. In this
sense, even the ROSAT detection might be regarded as an upper limit,
although it is plotted as a solid symbol.  The flux level plotted in
Fig. 1 and listed in Table 2 corresponds to the likely $N_H$ value of
$6\times10^{22}~{\rm cm^{-2}}$ and was obtained by considering two
extreme values of a power law spectral index: photon index 1.0 and
2.0. These indices bracket the corresponding index ($\sim$1.4) for a
bremsstrahlung spectrum as predicted by our model. The detected flux
in the soft (0.8--2.5 keV) band is, however, still highly sensitive to
the assumed $N_H$ (as also pointed out by Predehl and Trumper): for
$N_H$ = 8 $\times$ 10$^{22}$ the flux increases by a factor of 2.8
over that shown; whereas for $N_H$ = 5 $\times$ 10$^{22}$, it
decreases by a factor of 0.6. Thus the fits to ADAF models described
in the next section, which are otherwise greatly constrained by this
ROSAT detection, should be regarded as uncertain in normalization by a
factor perhaps as large as 2 if uncertainties in the interstellar
$N_H$ are allowed for completely and perhaps as large as 3 if internal
self-absorption in the vicinity of the AGN is allowed for (as
suggested by Predehl and Trumper).  The box around the ROSAT point in
Fig. 1 includes both the uncertainty due to spectral index as well as
an additional factor of 2 uncertainty in normalizaton.  The uncertain
value of $N_H$ can, in principle, be directly measured or at least
greatly constrained by observations of Sgr A$^*$ with the ACIS
instrument on AXAF. This could observe the interstellar absorption
edge of Oxygen (at 0.8 keV) at the high angular resolution
(1$\arcsec$) needed to completely isolate the surrounding diffuse
emission.

At still higher energies, 20--100 keV, we plot the two upper limits
derived from the deep SIGMA observations of the Galactic Center
complex reported by Goldwurm et al. (1994). These observations are not
able to resolve the burster, and are likely affected by it (or,
rather, constrain the hard X--ray flux from the burster). More
sensitive observations in the hard X--ray band at energies above
60--100 keV, where the hard X--ray spectral component of neutron stars
in bursters usually is not detectable (e.g. Barret \& Grindlay 1995)
could provide a more stringent test of the ADAF spectrum as well as
the predicted spectral break.

The $\gamma$--ray spectrum shown in Fig. 1 corresponds to emission
from the EGRET source 2EG 1746--2852 which is coincident with the
Galactic Center (Merck et al. 1996).  The source appears to be
point--like to within the resolution of the instrument ($ \sim
1\degree$), and is a significant excess ($\sim$10$\sigma$) above the
local diffuse emission.  The source has a very hard spectrum with a
spectral slope of 1.7 which differs significantly from other
unidentified Galactic EGRET sources.  The source might therefore
represent emission from Sgr A*, or it might be unresolved emission
from a giant molecular cloud in the vicinity of the Galactic Center.
If the source is indeed Sgr A$^*$, the flux and spectrum may be
explained as $\gamma$-ray emission via pion production and decay in the
ADAF (Mahadevan et al. 1997).  If the source is instead associated
with a dense cloud of molecular hydrogen situated at the Galactic
Center, it will again show a pion decay spectrum but with a spectral
index (at energies above 100 MeV) given by the local cosmic ray proton
spectrum (i.e.  a power law with energy index $\sim-1.6$). Due to the
low resolution of EGRET and uncertainty in the nature of the source,
we plot the $\gamma$-ray data in Fig. 1 with open circles.  The EGRET
observations (as reported by Mayer-Hasselwander, 4th Compton Symp.)
indicate possible variability, which would of course suggest that the
source is indeed Sgr A$^*$, but the same EGRET data also indicate the
source may be slightly resolved, which would argue that the radiation
is significantly contaminated by surrounding diffuse emission.  Future
observations with higher sensitivity and better resolution would be
invaluable in resolving this issue.

\section{Modeling Techniques}

We consider a black hole of mass $M$ accreting through an ADAF at a
rate $\dot{m} \dot{M}_{\rm Edd}\msun\,{\rm yr^{-1}}$, where $\dot{M}_{\rm Edd}
= 10 L_{\rm Edd}/c^2 = 1.39 \times 10^{18}(M/\msun) ~{\rm g\,s^{-1}} =
2.21 \times 10^{-8}(M/\msun) ~\msun{\rm yr^{-1}}$.  The ADAF is in
some sense dynamically intermediate between a thin disk and a
spherical accretion flow: it is hot ($H/R \sim 1$), with rapid radial
inflow, yet centrifugal support still plays a significant role.  In
describing the ADAF we shall generally refer to the scaled radius $r
\equiv R/R_s$, where $R_s = 2 G M/c^2$ is the Schwarzschild radius.
We assume that the ADAF extends from the black hole horizon, $r=1$, to an outer
radius $r_{\rm out} = 10^5$.  
We take the mean angular momentum vector of the ADAF to be inclined at
an angle $i$ to the line of sight.

To find the dynamical structure of the ADAF, we use the fully
relativistic, self-consistent, steady-state global solutions in the
Kerr metric developed by Popham \& Gammie (1997).  Their model uses a
nearly standard viscosity prescription which is parameterized by the
efficiency of angular momentum transport, $\alpha$ (assumed constant,
independent of radius).  The viscosity prescription has been modified,
however, so as to preserve causality and to include relativistic
effects.  We also use a quasi-spherical prescription for the vertical
structure based on Abramowicz, Lanza, and Percival (1997).  The model
is uniquely specified by four parameters: $\alpha$, the viscosity
parameter; $\gamma$, the adiabatic index of the fluid, which is
assumed to be a mixture of gas and magnetic fields; $f$, the advection
parameter which gives the ratio of advected energy to the viscous heat
input; and $a$, the rotation parameter of the black hole.  The models
considered here are extremely advection-dominated and so $f$ is very
close to unity.  Also, we consider only the case a = 0, i.e. a
Schwarzschild black hole.  
Comparison of relativistic and nonrelativistic global models, and the
self-similar solution of Narayan \& Yi (1994) shows that they are
rather similar outside about $r = 5$.  Inside this radius the density
in our solution is very close to self-similar, but the temperature is
slightly lower.

The thermodynamic state of the flow is described by the ion
temperature $T_i$, the electron temperature $T_e$, and the magnetic
pressure $p_{\rm mag} \equiv (1 - \beta) p_{\rm tot}$.  The total pressure is
fixed by the dynamical model.  We assume that $\beta$ is a constant,
independent of radius; that implies $\gamma = (8 - 3\beta)/ (6 -
3\beta)$ (see Appendix A and Esin 1997).  Given $\beta$ we know the
run of gas pressure with radius.  Since the electrons can cool
efficiently and coupling between the ions and electrons is assumed to
be weak (provided solely by Coulomb collisions), we have $T_i \gg
T_e$.  The run of gas pressure then gives $T_i$ to a good
approximation.

The heart of the problem is now calculating the electron temperature,
which determines the spectral properties of the flow.  A detailed
description of our procedure is given in Esin et al. (1997).
Basically, the electron temperature is determined by solving an energy
balance equation:
\be
Q^{\rm e,adv} = Q^{\rm ie} + \delta\,Q^{\rm vis} - Q^{\rm rad}.  
\ee 
Here $Q^{\rm e,adv}$ is the rate at which energy in the electrons is
advected inward by the flow.  This term is discussed in greater detail
in Appendix A.  Our earlier work (with the exception of Esin et
al. 1997) ignored the advective term, but recently Nakamura et
al. (1997) (see also Mahadevan \& Quataert 1997) have shown that it
can be important, particularly at low mass accretion rates.  At
low $\mdot$ the electrons
are nearly adiabatic and the compressive heating as the gas flows in
becomes large.  The other terms in equation (1) are as follows:
$Q^{\rm ie}$ is the rate at which the electrons are heated by Coulomb
collisions, $Q^{\rm vis}$ is the total rate of viscous dissipation, and
$Q^{\rm rad}$ is the radiative cooling.  The parameter $\delta$ describes
the fraction of the viscous heating that goes into the electrons.  In
general we set $\delta = 10^{-3} \sim m_e/m_p$.

Because of Compton scattering, the radiative cooling term couples the
flow at different radii.  It is calculated using the iterative
scattering method described in detail in  Narayan, Barret \& McClintock (1997).  
The ADAF is represented
by a logarithmically spaced grid of nested shells.  Within each shell
all flow variables are assumed uniform for $\pi/2 - \theta_H < \theta <
\pi/2 + \theta_H$, where $\theta_H$ is an effective angular scale
height, calculated as in  Narayan, Barret \& McClintock (1997) Appendix A.  
We then guess a value of the
electron temperature and compute the synchrotron and bremsstrahlung
emission from each shell.  Then, using the probability matrix elements
$P_{ij}$, which give the probability that a photon emitted in shell $i$
of the ADAF is scattered by an electron in shell $j$, the rate of
cooling through Compton scattering is calculated (see  
Narayan, Barret \& McClintock (1997) for details).  
This procedure is iterated until convergence is achieved.
The iteration method ensures that multiple scattering within the ADAF
is properly taken into account.  Doppler shifts and ray deflections are
ignored.

Once the cooling term is calculated, we must get the photons out to the
observer.  We use essentially Newtonian photon transport, although we
have taken a step toward relativistic photon transport by including
gravitational redshift (this is also included in the Compton scattering
calculation).   We do not include Doppler shifts, which to lowest order
in $v/c$ broaden the spectrum.  Since all features in our spectra have
$\delta\nu/\nu$ of order unity, this should not greatly alter the gross
properties of the spectrum, except where it is very steep.  At next
higher order in $v/c$ gravitational redshift, higher order Doppler, and
various geometric effects all enter.  In our model we have included a
correction for the volume of the emitting region ($1/\sqrt{1 - 1/r}
\times$ the Euclidean value) and for gravitational redshift.  The sum
of these effects is that $L_\nu(\nu)$ observed at large radius is now
$L_\nu(\nu \sqrt{1 - 1/r})$, that is, $L_\nu$ is shifted redward by the
gravitational redshift factor.  Jaroszynski \& Kurpiewski (1997) have
recently considered the full effects of photon transport near a Kerr
black hole on the spectrum of an ADAF.

The above discussion is concerned with radiation from electrons.  We
also compute pion production by the hot protons in the ADAF and the
resulting $\gamma$-ray emission through pion decay.  The procedure we
follow for this calculation is described in Mahadevan et al. (1997)
and is based on earlier work by Dermer (1986).  The present
calculations differ in three respects from those in Mahadevan et al.:
1) the density profile of the protons is now given by the relativistic
global solution of Popham \& Gammie (1997) instead of the
pseudo-Newtonian solution of Narayan, Kato \& Honma (1997), 2) we
allow for the non--spherical geometry of the flow via the angle
$\theta_H$ mentioned above, and 3) we include the effect of
gravitational redshift.

The EGRET spectrum of the Galactic Center is consistent with a
power--law distribution of proton energies (Mahadevan et al. 1997).
In the present calculations we assume that the protons are nonthermal
with a power-law index $p=2.3$.  Note that it is consistent for us to
assume a nonthermal distribution of proton energies, since Mahadevan
\& Quataert (1997) have shown that the protons in an ADAF do not have
time to thermalize.

\section{Results}

In this section we combine the modeling techniques outlined in \S3
with the observational constraints described in \S2 to come up with
ADAF models of Sgr A$^*$.  Our standard parameters are as follows.
We assume that the ADAF extends from $r=1$ to 10$^{5}$. The outer radius is 
large enough that the results are insensitive to its value. 
For the black hole mass, we take the estimate of Eckart \& Genzel
(1997): $M=2.5\times10^6M_\odot$.  We assume exact equipartition
between gas and magnetic pressure in the accreting gas: $\beta=0.5$.
Following Hawley, Gammie \& Balbus (1996) we take the viscosity
parameter to be $\alpha\sim0.6(1-\beta)=0.3$.  We assume that
electrons receive only a small fraction $\delta\sim m_e/m_p$ of the
viscous dissipation as direct viscous heating: $\delta=0.001$.  We
assume a Schwarzschild black hole.  Since we do not know the
orientation of the angular momentum vector of the accreting gas, we
set the inclination of the system to a generic value: $i=60^o$.  The
models considered here are extremely optically thin and so the results
are virtually independent of $i$.

The only parameter that we consider fully adjustable is the mass
accretion rate $\mdot$.  In each model described below, we have
adjusted $\mdot$ so as to fit the ROSAT X-ray flux (but see the
discussion in \S2 for uncertainties in the X-ray flux due to the
hydrogen column).  Although we treat $\mdot$ as a free parameter,
there are in fact some constraints.  The infrared star IRS 16 appears
to be the primary supplier of gas to Sgr A$^*$.  Assuming that \sgr
captures a fraction of the wind of IRS 16 by Bondi accretion, Melia
(1992) estimated $\dot M\approx 2\times10^{-4}~M_\odot{\rm yr^{-1}}$
for a wind velocity of $600~{\rm km\,s^{-1}}$ while Genzel et
al. (1994) estimated a mass accretion rate of $\dot M\approx
6\times10^{-6}~M_\odot{\rm yr^{-1}}$ for a wind velocity of $1000~{\rm
km\,s^{-1}}$ .  We take these two estimates to be low and high
extremes.  Converting to Eddington units, $\mdot$ is thus likely to be
in the range $10^{-4}<\mdot<3\times10^{-3}$.

The solid line in Fig. 1 shows our baseline model where $M$, $\alpha$,
$\beta$, and $\delta$ have been set to their standard values
($2.5\times10^6M_{\odot}, ~0.3, ~0.5, ~0.001$ respectively) and $\mdot$
has been adjusted to fit the X-ray flux.  This model has a mass
accretion rate of $\mdot=1.3\times10^{-4}$, which lies within the
acceptable range of $\mdot$ discussed above.  The computed spectrum
has four well-defined peaks.  From the left, these correspond to
self-absorbed thermal synchrotron emission, singly Compton-scattered
synchrotron radiation, bremsstrahlung emission, and $\gamma$-rays from
neutral pion decay.  The calculated spectrum fits the radio and
infrared constraints quite well.  The model spectrum passes through
the VLBI radio flux measurement at 86 GHz (which we have identified as
the most reliable of the radio observations, cf. \S2) and turns over sharply
in the sub-millimeter band exactly as required by the sub-millimeter
and infrared upper limits.  The model also satisfies the X-ray upper
limits and detection.  Considering that only one parameter, $\mdot$,
has been adjusted, we consider the agreement with the data impressive.

Note that the infrared and X-ray fluxes have come down significantly
compared to the data shown in Narayan et al. (1995).  The present
model is compatible with both the new measurements.  Interestingly, we
cannot find any model that fits either the new infrared upper limit
with the old X-ray data, or the new X-ray flux with the old infrared
data.

The model does have a discrepancy in the $\gamma$-ray band; the
predicted flux is lower than the observations by approximately an
order of magnitude.  The pion decay model requires an accretion rate
of $\mdot=4.5\times10^{-4}$ to fit the observed flux whereas the fit
to the rest of the spectrum gives $\mdot=1.3\times10^{-4}$.  There is
thus a discrepancy of a factor $\sim3-4$ between the two values of
$\mdot$.  We have been unable to come up with a reasonable resolution
of this discrepancy; perhaps the EGRET source does not correspond to
Sgr A$^*$, but rather to a central concentration of molecular gas in
the Galaxy and resulting diffuse emission which appears unresolved to
the $1^o$ beam of EGRET.

The model also does not agree with the radio flux at frequencies below
about 50 GHz.  We discuss this discrepancy at the end of the section.

Two features of the ADAF model in Fig. 1 should be highlighted.
First, the model fits the data using a reasonable mass for the black
hole, $M=2.5\times10^6M_\odot$.  This is an improvement over the model
described in Narayan et al. (1995) where the data could be fitted only
with a mass $M=7\times10^5M_\odot$.  The primary reason for the
improvement is the inclusion of electron energy advection (Nakamura et al. 1997), 
or compressive heating of the electrons, as we explain below.  Second,
the model is extremely advection-dominated.  The bolometric luminosity
$L_{\rm bol}$ integrated over all frequencies is only
$2.1\times10^{36}~\ergs$, which corresponds to a radiative efficiency,
$\epsilon=L_{\rm bol}/\dot Mc^2=5\times10^{-6}$.  It is this
extraordinarily low radiative efficiency that allows the model to fit
the observations with such a large mass accretion rate.

In contrast, a standard thin accretion disk model of \sgr is ruled out
quite comprehensively by the data.  The dotted line in Fig. 1 shows a
thin disk model with $\mdot=10^{-4}$, the lowest mass accretion rate
we consider reasonable.  The spectrum was calculated assuming that the
emission is blackbody at each radius.  The model predicts an infrared
flux which is many orders of magnitude above the measured upper
limits.  The short dashed line in Fig. 1 shows another thin disk model where
$\mdot$ has been reduced to $10^{-9}$.  This model does satisfy the IR
upper limits but does not fit any of the observations, either in the
radio or X-ray bands.  Further, the mass accretion rate is
unreasonably small.

A number of improvements have taken place in the modeling techniques
since the publication of our first model of \sgr (Narayan et
al. 1995).  It is interesting to investigate what effect each
improvement has had on the calculated spectrum.  Figure 2 shows a
sequence of models in which we start with the simplest version of the
model and progressively add features one by one.

The dotted line in Fig. 2 corresponds to the most primitive version of
the ADAF model, in which the flow is assumed to have a self-similar
form (Narayan \& Yi 1994) and neither compressive heating of electrons
nor gravitational redshift is included.  The optimized accretion rate
is $\mdot=6\times10^{-5}$.  The general shape of the spectrum is
similar to that of our standard model (the solid line), but this model
differs in three ways.  First, the emission is stronger in nearly all
bands compared to our standard model; in fact, the model is
inconsistent with the infrared limit.  Second, the mass accretion rate
is lower than in our standard model by a factor of 2.  This, combined
with the higher luminosity, means that this model is not as
advection-dominated as our standard model; we calculate a radiative
efficiency of $\epsilon=2.4\times10^{-4}$.  Third, the synchrotron
peak is well above the 86 GHz VLBI flux measurement.  The discrepancy
in the radio flux is serious since the flux predicted by the model is
more than an order of magnitude larger than that observed.  Within the
framework of the self-similar flow assumption, the only way to
eliminate this problem is by changing one or more of the model
parameters.  This was, in fact, the primary reason why the Narayan et
al. (1995) model required a black hole mass of $7\times10^5M_\odot$
instead of the measured mass of $2.5\times10^6M_\odot$ (see Fig. 3
below which shows how the spectrum is modified when the black hole
mass is reduced).

The short-dashed line in Fig. 2a shows the effect of replacing the
self-similar flow by a global flow based on a pseudo-Newtonian
potential (Narayan, Kato \& Honma 1997; Chen, Abramowicz \& Lasota
1997).  The changes are minimal.

The long-dashed line in Fig. 2a next shows the effect of including
compressive heating (Appendix A and Nakamura et al. 1997).  This
model, which has $\mdot=1.1\times10^{-4}$, has a very different
spectrum compared to the previous two models.  The overall emission is
significantly reduced, notably in the synchrotron peak, and the flow
is substantially more advection-dominated:
$\epsilon=2.1\times10^{-6}$.  This model fits the 86 GHz data point
very well without requiring any adjustment to the black hole mass.  We
thus confirm the result of Nakamura et al. (1997) that compressive
heating is an important effect.  Figure 2b shows the electron
temperature profiles of the various models.  The model with
compressive heating has a significantly different temperature
structure than the models without.  At very low $\mdot$, as in our
model of Sgr A$^*$, the dominant terms in the electron energy equation
are the two pieces of $Q^{\rm e,adv}$ written in the Appendix, viz. the
terms proportional to $dT_e/dR$ and $d\rho/dR$.  These two dominant
terms balance each other, while the rest of the terms in equation (1) are
small.  In other words, the electrons are essentially adiabatic.  The
adiabatic condition gives a lower electron temperature than in the
previous two models and this accounts for the change in the shape of
the synchrotron peak.

The dash-dot line in Fig. 2a shows next the effect of including a
fully relativistic global solution ($\mdot=1.3\times10^{-4}$), taken
from Popham \& Gammie (1997), instead of the pseudo-Newtonian global
solution employed in the previous two models.  Most of the features
are similar, but the overall emission is increased.  The primary
reason for this is that the relativistic solution has a lower radial
velocity close to the black hole compared to the pseudo-Newtonian
solution.  (This is because $v$ is constrained to be less than $c$
whereas the pseudo-Newtonian model gives $v>c$ close to the black
hole.)  Consequently, the density is higher and this leads to
increased emission.

Finally, the solid line in Fig. 2a shows our standard model, which
includes gravitational redshift.  As expected, this model has a lower
luminosity than the previous model, but is otherwise quite similar.

Figures 3--5 show the effect of varying the other parameters in the
model.  Figure 3a shows the effect of varying the mass. 
In addition to the baseline model, two other models are shown with 
$M = 1/3$ and $3$ times $M$ of the baseline model.  
As Mahadevan (1997) has shown, with increasing $M$,
the magnetic field strength decreases as $M^{-1/2}$ and this causes
the synchrotron peak to move to lower frequencies $\propto M^{-1/2}$.
Of the three models shown in Fig. 3a, the one with
$M=2.5\times10^6M_\odot$ gives by far the best fit to the VLBI radio
data.  Assuming our choices of $\alpha$ and $\beta$ are correct, this
independently confirms the black hole mass measured by Eckart \&
Genzel (1997).

Figure 3b shows the effect of varying the mass accretion rate.  In
addition to the baseline model, four other models are shown, with
$\mdot=1/2, ~1/\sqrt{2}, ~\sqrt{2}, $ and $~2$ times $\mdot$ of the baseline
model.  The radiative efficiency of ADAFs varies rapidly with $\mdot$:
$\epsilon\propto\mdot$ (Narayan \& Yi 1995b).  Therefore, the
luminosity varies as $\mdot^2$.  This can be seen in both the
synchrotron and bremsstrahlung peaks.  The Compton peak in the
infrared shows an even stronger dependence on $\mdot$.  This is
because the Comptonized flux is proportional to the product of the
synchrotron emission (which is $\propto \mdot^2$) and the optical
depth (which is $\propto\mdot$), so that the amplitude of this peak
varies approximately as $\mdot^3$.  The three intermediate models in
Fig. 3b are consistent with all the data, but the two extreme models
lie outside the X-ray error box and one of them also violates the
sub-millimeter and infrared limits.

Figure 4a shows the effect of varying $\alpha$.  For each $\alpha$, we 
have adjusted the accretion rate to fit the X-ray flux.  The spectrum is not
very sensitive to $\alpha$ (except in the optical/IR), though the
fitted mass accretion rates show modest variations:
$\mdot=6.3\times10^{-4}, ~1.0\times10^{-3}, ~1.3\times10^{-3},
~1.5\times10^{-3}$ for $\alpha=0.1,~0.2, ~0.3,~0.4$ respectively.

Figure 4b shows the effect of varying $\beta$. Again, for each $\beta$, we
have adjusted the accretion rate to fit the X-ray flux.  In this case, we see
quite substantial changes in the predicted spectrum.  An increase in
$\beta$ leads to a decrease in the magnetic field strength and one may
be tempted to think that this would cause the synchrotron peak to
reduce in amplitude.  In fact, the opposite behavior is seen.  The
reason can be traced to compressive heating.  Since the electrons are
effectively adiabatic, their temperature profile is determined by the
adiabatic index $\gamma_e$.  As Appendix A shows, because the gas is a
mixture of particles and magnetic field, the effective $\gamma_e$
depends on the parameter $\beta$.  As $\beta$ goes up, the magnetic
pressure goes down and $\gamma_e$ increases.  A larger $\gamma_e$
leads to hotter electrons, see Fig. 4c, which causes more synchrotron
emission.  Unfortunately, this means that the results are sensitive to
the details of how we model energy advection in the electrons.  In
this context we note that our equation of state for the electrons
differs somewhat from the one used by Nakamura et al. (1997).  We use
a relativistic equation of state which causes $\gamma_e$ to change as
the electron temperature approaches and crosses $T_e\sim m_ec^2/k$.
In addition, we modify the adiabatic index of the electrons to allow
for the equipartition magnetic field which is present in the gas and
is coupled to the electrons.  Nakamura et al. (1997) treat the
particles as a separate component and ignore the field in their
equation of state.

Figure 4d shows the effect of varying $\delta$.  For all $\delta\
\sles\ 0.01$ the spectrum is generally unaffected.  This result is
different from that found in Narayan, Barret \& McClintock (1997).
Once again, the reason is the inclusion of compressive heating.
Because the electrons are now essentially adiabatic, their temperature
profile is not affected by modest changes in the heating or cooling.
Only when $\delta$ is large, e.g. $\delta=0.03162$ (the dot-dashed
line in Fig. 4d), do the electrons experience significant additional
heating and only then does the spectrum show a noticeable change.

We turn now to the low frequency radio data, where the model deviates
substantially from the measurements.  In our model, different parts of
the synchrotron peak are produced at different radii in the flow, the
emission at $10^{12}$ Hz coming from close to the black hole and the
emission at lower frequencies coming from farther out.  Thus, one
simple way of improving the fit to the radio data is to modify the
electron temperature beyond a few tens of Schwarzschild radii.  Figure
5 is {\em only} an illustration.  The solid line in Fig. 5a corresponds to our
baseline model, while the dashed line is another model which is
identical in all respects except that we have arbitrarily set
$T_e=2\times10^9$ K over the radius range $r=20-1000$ (Fig. 5b).  This
model fits the data quite well.

Is there any reason to think that the electrons might have the profile
shown in Fig. 5b?  Yes, there are several radial transport mechanisms
which could drive the electrons to a nearly isothermal state.  First,
the long mean free path of the electrons can lead to fairly strong
radial heat conduction (parallel to field lines).  
Second, synchrotron radiation, which has a
thermalizing effect on the electrons (Mahadevan \& Quataert 1997), can
also cause significant energy diffusion.  Finally, the tangled
magnetic fields which we assume to be present in the flow may move
outwards as a result of buoyancy and may dissipate their energy and
heat the electrons at larger radii.  Some of these effects could well
be episodic; it is interesting in this connection that the radio flux
of \sgr is known to be quite variable (Zhao et al. 1989).  These effects need
further study.

\section{Discussion}

The main result of this paper is that the ADAF model (Narayan et
al. 1995) provides a viable explanation of the spectral properties of
Sgr A$^*$.  Our basic model, shown by the solid line in Fig. 1, fits
all the high resolution measurements from the radio to the X-ray band,
including the stringent infrared limit of Menten et al. (1997) and the
significantly reduced X-ray flux discussed in \S2.

In contrast to the models presented in Narayan et al. (1995), where a
black hole mass of $7\times 10^5M_\odot$ was required to accommodate
the radio data, here we find that the model quite naturally fits the
data with the correct mass of $2.5\times10^6M_\odot$ (Eckart \& Genzel
1997).  This removes one of the main problems with the previous work.
The model also predicts a mass accretion rate $\mdot\sim
1.3\times10^{-4}$ (in Eddington units) which is within the range
considered likely by a direct estimate of the accretion rate (Genzel
et al. 1994).

We should emphasize that the model is fully consistent with a detailed
treatment of hydrodynamics, radiation processes, and thermal balance
of ions and electrons in the two-temperature plasma.  Most other
models in the literature are more primitive and/or ad hoc.  In
addition, the model is fairly robust to changes in the parameters
(Fig. 4).  The predicted spectrum is insensitive to large variations
in the viscosity parameter $\alpha$ and the electron heating parameter
$\delta$.  The results do, however, vary with the equipartition
parameter $\beta$ (Fig. 4b, 4c).

There are two relatively minor problems with the present model.
First, we under--predict the $\gamma$-ray flux in the EGRET band by an
order of magnitude.  This might indicate a residual uncertainty in the
relative normalization of the protons and electrons, but we have not
been able to come up with any specific proposal to eliminate the
discrepancy.  Perhaps the $\gamma$-ray source detected by EGRET with
its one degree beam is not Sgr A$^*$, but surrounding diffuse
emission.  Second, the model under--predicts the radio flux below about
50 GHz.  We can explain the observed emission by modifying the
electron temperature profile in the accretion flow (see Fig. 5), but
this is rather ad hoc.  Another possibility is that the radio emission
arises from a different component of the source, such as a jet, which
is outside the scope of the model.  (We note that Krichbaum et
al. 1993 claim to have seen evidence for a jet in their 43 GHz VLBI
image, though Backer et al. 1993 could not verify this.)  In any case,
there is no region of the spectrum where the model {\it over-predicts}
the flux and this is important because such a discrepancy would be
much harder to explain away.

A more serious problem is the somewhat low electron temperatures
predicted by the model.  As Fig. 4c shows, we obtain temperatures
below $10^{10}$ K at all radii, whereas VLBI observations at 43 GHz
and 86 GHz indicate brightness temperatures in excess of $10^{10}$ K
(Backer et al. 1993, Rogers et al. 1994).  Thus, although the model
fits the 86 GHz flux well, it seems to predict a larger angular size
for the source,  by a factor of $\sim 1.5$, than observations 
indicate.  For comparison with future observations, we list here 
linear source radii at various radio
frequencies according to the baseline model shown in Fig. 1:
$1.3\times10^{12}$ cm at $10^{12.5}$ Hz, $4.1\times10^{12}$ cm at
$10^{12}$ Hz, $1.0\times10^{13}$ cm at $10^{11.5}$ Hz,
$2.1\times10^{13}$ cm at $10^{11}$ Hz, $4.1\times10^{13}$ cm at
$10^{10.5}$ Hz, $8.3\times10^{13}$ cm at $10^{10}$ Hz.  Note that the
electron temperature depends on the value of the parameter $\beta$
(Fig. 4c).  Thus, one might be able to fit the observed brightness
temperatures by adjusting this parameter.

An outstanding feature of the ADAF model presented here is its
extraordinarily low radiative efficiency: $\epsilon=L_{\rm bol}/ \dot
Mc^2=5\times10^{-6}$.  The low efficiency permits the model to fit the
very low luminosity of \sgr with a fairly large $\mdot$.  Figure 1
shows two thin disk models with the standard radiative efficiency of
$\epsilon\sim0.1$.  A thin disk with the ``correct'' $\mdot$ (dotted
line) over--predicts the infrared flux by four or five orders of
magnitude.  A model with $\mdot$ reduced by a factor of $10^5$
accommodates the IR limits but does not fit any of the other data and
has a mass accretion rate which is wildly discrepant with independent
estimates of $\mdot$.  Some models in the literature which are based
on a thin disk attempt to solve the luminosity problem by hiding a
large fraction of the emission in the optical band where there is
substantial extinction by dust (e.g. Falcke et al. 1993a, Melia 1994).
However, the models invariably predict emission in the infrared at a
level well above the Menten et al. (1997) limit.  Other models in the
literature which are not based on a thin disk also have problems
fitting the low luminosity (e.g. Duschl \& Lesch 1994; Mastichiadis \&
Ozernoy 1994).  These models generally require values of $\mdot$
significantly below the lower limit of $10^{-4}$.

Can we save the thin disk model by assuming that the disk has no
viscosity at all, so that the gas in the disk does not accrete at the
present time?  Falcke \& Melia (1997) have recently shown that even
such an extreme model is inconsistent because the mere presence of the
disk, even though it does not accrete, will still lead to fairly
strong infrared emission.  The argument is that if there is inflow of
material towards the Galactic Center via a Bondi-Hoyle-like flow, when
the inflowing gas hits the disk and circularizes it will produce a
substantial amount of thermal radiation.  For the mass accretion rate
of $\mdot\ \sgreat\ 10^{-4}$ estimated in the case of \sgr (cf. \S4)
the predicted infrared flux is well above the Menten et al. (1997)
limit in nearly all the cases considered by Falcke \& Melia (1997).
The ADAF model circumvents this argument relatively easily.  Since the
accreting gas in the ADAF has a quasi-spherical shape (Narayan \& Yi
1995a), the outer Bondi-Hoyle flow will shock with the ADAF at a large
radius $\sim10^5R_s$.  The luminosity associated with this impact is
quite low.  Inside the impact radius, the accretion becomes
advection-dominated and the gas again does not radiate very much.

Several recent papers have made the point that the successful
application of the ADAF model to any observed system is direct
evidence that the accreting star is a black hole (Narayan, McClintock
\& Yi 1996; Narayan, Yi \& Mahadevan 1996; Narayan, Barret \&
McClintock 1997; Narayan, Garcia \& McClintock 1997).  The argument is
that if the object has an event horizon the advected energy in an
ADAF will disappear from sight, whereas if the central object has a
surface, then the stellar surface will be heated by the hot inflow
from the ADAF and the advected thermal energy will be emitted as
thermal radiation from the star.  Thus, for a standard star with a
surface, the radiative efficiency will be the canonical 10\% even if
the accretion occurs via an ADAF.  Only if the central star is a black
hole can the radiative efficiency be truly low.

\sgr is perhaps one of the best objects for this argument.
This source has two strong observational constraints: (1) The mass
accretion rate is estimated to be at least a few$\times10^{-6}
M_\odot\,{\rm yr^{-1}}$, and (2) the bolometric luminosity is no
greater than $10^{37}~\ergs$.  In our opinion the only plausible
resolution of these two conflicting pieces of evidence is to postulate
(1) that the accretion in \sgr occurs via an ADAF and (2) that the
central object is a supermassive black hole.  This ``proof'' of the
black hole nature of \sgr is qualitatively different from usual proofs
which rely on a measurement of the mass.  The usual argument is that
if an object is too massive to be a neutron star it must be a black
hole; it is a proof by elimination.  Our ``proof'' is somewhat more
direct and cuts to the essence of what constitutes a black hole,
namely the presence of an event horizon.  We argue that \sgr has an
enormous luminosity deficit for which the only reasonable explanation
is that the object has an event horizon; therefore, \sgr must be a
black hole.

One of the results to come out of this work is our confirmation that
compressive heating of electrons cannot be ignored and must be
included consistently in computations of ADAF spectra of low
luminosity systems like \sgr (see Fig. 2).  This point was made
recently in an important paper by Nakamura et al. (1997).  In view of
this result it would be useful to revisit other low-$\mdot$ systems to
which the ADAF model has been applied (e.g. soft X-ray transients,
Narayan, Barret \& McClintock 1997; Hameury et al. 1997; NGC 4258,
Lasota et al. 1996; low-luminosity nuclei of giant ellipticals, Fabian
\& Rees 1995; Mahadevan 1997; Reynolds et al. 1996) and redo the
analysis with the inclusion of compressive heating.  Interestingly,
Esin et al. (1997) have shown that compressive heating has a much
weaker effect on ADAFs at higher values of $\mdot$.  The reason is
that with increasing $\mdot$ the other terms in the electron energy
equation (1), notably the Coulomb collision term $Q^{\rm ie}$ and the
radiative cooling term $Q^{\rm rad}$, become more important, and
compressive heating no longer dominates.

In the model presented in this paper the emission in the infrared and
the radio (above about 50 GHz) comes from fairly small radii $\sim10$
Schwarzschild radii.  Since the gas in the ADAF is nearly in
free-fall, the characteristic time scales of the flow are quite short.
We may thus expect rapid variability in \sgr at these wavelengths.
The shortest likely time scale is the dynamical time,
$t_{\rm dyn}\sim(GM/R^3)^{-1/2}$, which gives $t_{\rm dyn}\sim5000$ s at
$R=10R_s$.  The viscous time is longer than this by a factor of
$1/\alpha \sim3$.  The longer wavelength radio emission is from larger
radii; for instance, the emission below 1 GHz comes from $R\gsim
10^3R_s$.  Variability at these wavelengths will be correspondingly
slower: $t_{\rm dyn}(10^3R_s)\sim5\times10^6$ s.  The bremsstrahlung
emission in X-rays is from a broad range of radii, but is dominated by
large radii.  Therefore, the X-ray flux should show much slower
variations (timescale $\gsim$ 1 year) than the radio, millimeter or infrared fluxes.

By assumption, the electrons in our model are fully thermal.
Mahadevan \& Quataert (1997) have, however, shown that at low mass
accretion rates similar to that in \sgr, electrons may not be
thermalized at larger radii.  The electron energy distribution at
these radii will then be truncated even more sharply than in a
Maxwellian, which will cause a reduction in the synchrotron emission.
This will act to increase the discrepancy between the model and the
data at low radio frequencies.  This is an area for further work.

Note, however, that a power-law distribution of electrons extending
over any reasonable range of energies is ruled out by the
observations.  For instance, since optically
thin nonthermal synchrotron emission usually has a spectral form
$L_\nu\sim\nu^{-0.7}$, this means that in a $\nu L_\nu$ plot the
optically thin emission would continue to rise as $\nu^{+0.3}$.
Such a rise beyond $\sim 10^{12} - 10^{13}$Hz is strongly ruled out by the observed upper
limits in the infrared.

In this connection, we note that both Falcke (1996) and  
Duschl \& Lesch (1994) have proposed
a nonthermal model for \sgr in which most of the electrons have
Lorentz factors of around a few hundred.  Such a mono-energetic electron 
distribution is required in order to reproduce the sharp cutoff observed in 
the sub--millimeter band.  
Falcke assumes a mono-energetic distribution in the ``nozzle'' 
of his jet-disk model, while Duschl \& Lesch assume a homogeneous sphere of 
mono-energetic electrons. 
Such mono-energetic models are not ruled out by the argument of 
the previous paragraph and indeed Falcke and Duschl \& Lesch 
are able to fit the radio spectrum of \sgr reasonably
well.  We, however, find the idea of a nonthermal but mono-energetic
distribution somewhat artificial.  Our ADAF model fits the data in a
much more natural way by making use of a thermal distribution of
electrons.  Further, because of thermal synchrotron self-absorption, 
the cutoff of the spectrum in the sub-millimeter band is very sharp without
requiring any fine--tuning.


Further high spatial resolution observations of \sgr are highly
desirable.  The ADAF model can be tested and constrained with better
observations of the spectrum and variability, especially in the
millimeter, sub--millimeter, infrared and X-ray bands.  Better
$\gamma$-ray observations might also resolve the issue of whether the
source detected by EGRET is Sgr A$^*$.

\acknowledgments
This work was supported in part by NSF grant AST 9423209 and NASA
grant NAG 5-2837.  We thank Shoji Kato for sending a preprint of
Nakamura et al. (1997) prior to publication.  RM thanks Jun-Hui  Zhao and
 Mark Reid for useful discussions on the radio observations of the Galactic
Center.

\vfill\eject
\begin{appendix}
\section{Energy Advection by Electrons}

The advection term in equation (1) can be written per unit volume as
\be
Q^{\rm e,adv}=\rho T_e v \frac{d s_e}{d R},
\ee
where $s_e$ is the entropy of the electrons per unit total gas mass.
This term was ignored in much of the earlier work under the
assumption that the temperature of the electrons is determined
primarily by a balance between 
the Coulomb transfer term $Q^{\rm ie}$ and the radiative
cooling term $Q^{\rm rad}$ in equation (1).
However, Nakamura et al. (1997) have shown that
energy advection by electrons can be very important 
under some circumstances and may play a dominant role in determining
the electron temperature.  This is the case especially when the
mass accretion rate is low, as in Sgr A$^*$.

We begin with the relation
\be
T_e d s_e = d u_e + P_e d \left(\frac{1}{\rho}\right),
\ee
where $u_e$ is the internal energy of the electrons per unit mass and 
$P_e$ is the electron pressure.  We consider a mixture of gas and
magnetic fields.  If gas pressure contributes a constant fraction
$\beta$ to the total pressure $P_{tot}$, then
\be
P_{tot} = \frac{\rho k T_i}{\mu_i m_u} + \frac{\rho k T_e}{\mu_e m_u} +
\frac{B^2}{24 \pi} = \frac{\rho k T_i}{\beta \mu_i m_u} + 
\frac{\rho k T_e}{\beta \mu_e m_u}.
\ee
It seems natural to denote the two terms 
on the right as the effective ion and 
electron pressure, each including an appropriate fraction of
the magnetic pressure.  Therefore, we have
$P_e = \rho k T_e/(\beta \mu_e m_u)$.

The internal energy of the gas is a sum of the ion, electron and 
magnetic field internal energies:
\be
\label{inten}
u = \frac{3}{2} \frac{k T_i}{\mu_i m_u} + a(T_e) \frac{k T_e}{\mu_e m_u} + 
\frac{B^2}{8 \pi \rho} = \frac{6-3\beta}{2 \beta} \frac{k T_i}{\mu_i m_u} +
\left[\frac{3 (1-\beta)}{\beta} + a(T_e)\right] \frac{k T_e}{\mu_e m_u},
\ee
where the coefficient $a(T_e)$ varies from $3/2$ in the case of a
non-relativistic electron gas, to $3$ for fully relativistic electrons.  
The general expression for $a$ as a function of the dimensionless electron
temperature $\theta_e = k T_e/m_e c^2$ was derived by 
Chandrasekhar (1939, Chapter X, eq.[236]):
\be
a(\theta_e) = \frac{1}{\theta_e} \left(\frac{3 K_3 (1/\theta_e) + 
K_1 (1/\theta_e)}{4 K_2 (1/\theta_e)} - 1\right).
\ee
Note that the ions never become relativistic in these flows,
so that the corresponding coefficient for the ions is
always $\sim 3/2$.

As we have done for the pressure, the right hand side of 
Eq. (\ref{inten}) may be naturally divided into two terms, the ion 
and electron internal energies:
\be
u_i = \frac{6-3\beta}{2 \beta} \frac{k T_i}{\mu_i m_u},\ \ \ {\rm and}\ \ \ 
u_e= \left[\frac{3 (1-\beta)}{\beta}+ a(T_e)\right] \frac{k T_e}{\mu_e m_u}.
\ee
In this interpretation, $u_i$ and $u_e$ are again ``effective'' quantities, 
which include contributions from the particles as well as the associated 
magnetic field.  Note that the contribution 
of the magnetic field to the internal energy of each particle 
species is proportional to the contribution of these particles to the total 
pressure, a natural choice in our model where the ratio of the magnetic
to gas pressure is fixed.
Having defined $P_e$ and $u_e$, we can now write the energy
advection term for the electrons as
\be
Q^{\rm e,adv}=
\frac{\rho v k}{\mu_e m_u} \left[\frac{3 (1-\beta)}{\beta} + a(T_e) + 
T_e \frac{d a}{d T_e}\right] \frac{d T_e}{d R} - \frac{v k T_e}{\beta
\mu_e m_u} \frac{d \rho}{d R}.
\ee

From the pressure and internal energy of the electrons, we calculate the
effective adiabatic index of the electrons via the relation
$\gamma_e-1=P_e/u_e$.  This gives 
\be 
\gamma_e =\frac{4-3\beta+a\beta}{3-3\beta+a\beta}.  
\ee 
If the
particles are relativistic, then $a=3$ and $\gamma_e=4/3$ regardless
of $\beta$.  This is because both the particles and the tangled field
behave like radiation.  If the particles are non-relativistic, however,  
$a=3/2$ and in this case 
\be
\gamma_e=\frac{8-3\beta}{6-3\beta},
\ee
as shown by Esin (1997).  We see that $\gamma_e$
varies from 4/3 when $\beta=0$ to 5/3 when $\beta=1$.  Thus,
the adiabatic index depends fairly sensitively on $\beta$ so long
as the electrons are non-relativistic.
If the electrons in the accreting gas behave adiabatically, then
their temperature varies as $T_e\propto\rho^{\gamma_e-1}$ (technically
there is also the term proportional to $da/dT_e$ in eq A7 which becomes
important when the electrons are quasi-relativistic, but we
neglect this for simplicity).  Thus, for a given amount of compression,
the electrons are hotter when $\beta$ is large and cooler when
$\beta$ is small.  This explains the trend seen in Figs. 4b and 4c.

\end{appendix}
\newpage
\begin{references}

\reference{A} Abramowicz, M.A., Chen, X., Granath, M., \& Lasota, J.-P.
1996, ApJ, 471, 762

\reference{A} Abramowicz, M.A., Chen, X., Kato, S.,
Lasota, J.-P., \& Regev, O. 1995, ApJ,  438, L37 

\reference{A} Abramowicz, M.A., Lanza, A., \& Percival, M.J. 1997, 
ApJ 479, 179

\reference{A} Alberdi, A., et al., 1993, A\&A, 277, L1

\reference{A} Backer, D., C, 1982, in Proc. IAU Symposium, 
eds. D.S. Heeschen \& C. M. Wade,  97, 389
No. 97 

\reference{A} Balick, B., \& Brown, R., 1974, ApJ, 194,265

\reference{A} Barret, D., \& Grindlay, J. E., 1995, ApJ, 440, 841

\reference{A} Beckert, T., Duschl, W. J., Mezger, P. G., \& Zylka, R., 1996, 
A\&A, 307, 450

\reference{A} Brown, R. L, \& Lo, K. Y., 1982, ApJ, 253, 10

\reference{A} Chandrasekhar, S., 1939, in 
Introduction to the Study of Stellar Structure, (New York: Dover Publications)

\reference{A} Chen, X., Abramowicz, M. A., \& Lasota, J.-P. 1997,
ApJ, 476, 61

\reference{A} Chen, X., Abramowicz, M. A., Lasota, J.-P.,
Narayan, R., \& Yi, I. 1995, ApJ, 443, L61 

\reference{A} Davies, R. D., Walsh, D., \& Booth, R. S., 1976, MNRAS, 177, 319

\reference{A} Dent, W. R. F., Matthews, H. E., Wade, R., \& Duncan, W. D., 
1993, ApJ, 410, 650

\reference{A} Dermer, C. D. 1986, A\&A, 157, 223

\reference{A} Dermer, C. D., Liang, E. P., \& Canfield, E. 1991,
ApJ, 369, 410

\reference{A} Duschl, W. J., \& Lesch, H. 1994, A\&A, 286, 431

\reference{A} Eckart, A., et al. 1992, Nature, 355, 526

\reference{A} Eckart, A., \& Genzel, R. 1997, MNRAS, 284, 576

\reference{A} Esin, A. A. 1997, ApJ, 482, in press (astro-ph/9701039) 

\reference{A} Esin, A. A., McClintock, J. E., \& Narayan, R. 1997,
ApJ, submitted (astro-ph/9705237)

\reference{A} Fabian, A. C., \& Rees, M. J., 1995, MNRAS, 277, L55

\reference{A} Falcke, H., 1996, in IAU Symp. 169, Unsolved Problems of the 
Milky Way, eds. L. Blitz \& P. J. Teuben (Dordrecht: Kluwer), 163

\reference{A} Falcke, H., Biermann, P. L., Duschl, W. J., Mezger, P. G., 1993a,
A\&A, 270, 102

\reference{A} Falcke, H., Mannheim, K., \& Biermann, P. L. 1993b,
A\&A, 278, L1

\reference{A} Falcke, H., \& Melia, F., 1997, ApJ, 479, 740

\reference{A} Frank, J., King, A., \& Raine, D. 1992, Accretion Power in 
Astrophysics, (Cambridge: Cambridge Univ. Press)

\reference{A} Gammie, C., \& Popham, R. 1997, ApJ, submitted

\reference{A} Genzel, R., \& Townes, C. H. 1987, ARAA, 25, 377

\reference{A} Genzel, R., Hollenbach, D., \& Townes, C. H. 1994, Rep. Prog. Phys., 
57, 417

\reference{A} Gezari, D., Ozernoy, L., Varosi, F., McCreight, D., 
\& Joyce, R., 1994,
in The Nuclei of Normal Galaxies, eds. R. Genzel \& A. I. Harris (Dordrecht: Kluwer), 
427

\reference{A} Goldwurm, A. et al. 1994, Nature, 371, 589

\reference{A} Hameury, J. M., Lasota, J. P., McClintock, J. E., \& Narayan, R., 1997, ApJ, in press (astro-ph/9703095)

\reference{A} Hawley, J. F., Gammie, C. F., \& Balbus, S. A., 1996, ApJ, 464, 690

\reference{A} Jaroszynski, M., \& Kurpiewski, A. 1997 A\&A, 
submitted (astro-ph 9705044).

\reference{A} Lasota, J. P., Abramowicz, M. A., Chen, X., Krolik, J., Narayan, R., 
\& Yi, I., 1996, ApJ, 462, 142

\reference{A} Kato, S., Abramowicz, M. A., \& Chen, X. 1996, PASJ, 48, 67

\reference{A}  Krichbaum, T. P., et al., 1993, A\&A, 274, L37

\reference{A} Koyama, K. et al. 1996, PASJ, 48, 249

\reference{A} Maeda, Y. et al., 1996, PASJ, 48, 417

\reference{A} Mahadevan, R. 1997, ApJ, 477, 585

\reference{A} Mahadevan, R., \& Quataert, 1997, ApJ, in press (astro-ph/9705067)

\reference{A} Mahadevan, R, Narayan, R., \& Krolik, J. 1997, ApJ,  486, in press
(astro-ph/9704018)

\reference{A} Mastichiadis, A., \& Ozernoy, L. M., 1994, ApJ, 426, 599--603

\reference{A} Melia, F. 1992, ApJL, 387, L25

\reference{A} Melia, F. 1994, ApJ, 426, 577

\reference{A} Menten, K. M., Reid, M. J., Eckart, A., \& Genzel, R.
1997, ApJL, 475, L111

\reference{A} Merck, M., et al. 1996, A\&A Sup., 120, 465

\reference{A} Mezger, P. G., 1994, in The Nuclei of Normal Galaxies, eds. 
R. Genzel \& A. I. Harris (Dordrecht: Kluwer), 415

\reference{A} Mezger, P. G, Duschl, W. J., \& Zylka, R.,  1996, ARA\&A, 7, 289

\reference{A} Nakamura, K. E., Kusunose, M., Matsumoto, R.,
\& Kato, S. 1997, PASJ, in press

\reference{A} Narayan, R. 1997, Proc. IAU Colloq. 163, Accretion Phenomena
and Related Outflows, A.S.P. Conf. Series, eds. D. T. Wickramasinghe,
L. Ferrario, G. V. Bicknell, in press (astro-ph/9611113)

\reference{A} Narayan, R., Barret, D., \& McClintock, J. E. 1997, ApJ, 
482, 448

\reference{A} Narayan, R., Garcia, M. R., \& McClintock, J. E. 1997, ApJ, 478, 
L79

\reference{A} Narayan, R., Kato, S., \& Honma, F. 1997, ApJ, 476, 49

\reference{A} Narayan, R., McClintock,  J. E., \& Yi, I. 1996, ApJ, 457, 821

\reference{A} Narayan, R., \& Yi, I. 1994, ApJ, 428, L13 

\reference{A}  Narayan, R., \& Yi, I. 1995a, ApJ, 444, 231

\reference{A}  Narayan, R., \& Yi, I. 1995b, ApJ, 452, 710

\reference{A}  Narayan, R., Yi, I., \& Mahadevan, R. 1995, Nature, 374, 623

\reference{A}  Narayan, R., Yi, I., \& Mahadevan, R. 1996, A\&A Sup., 120, 287

\reference{A}  Pavlinsky, M. N., Grebenev, S. A., \& Sunyaev, R. A., 1992, Soviet 
Astr. Lett., 18, 116

\reference{A}  Pavlinsky, M. N., Grebenev, S. A., \& Sunyaev, R. A., 1994, ApJ,
425, 110

\reference{A} Peitz, J., \& Appl, S., 1997, MNRAS, 286, 681

\reference{A} Popham, R., \& Gammie, C. 1997, ApJ, submitted

\reference{A} Poutanen, J., \& Svensson, R. 1996, ApJ, 470, 249

\reference{A} Predehl, P., \& Trumper, J. 1994, A\&A, 290, L29

\reference{A} Rogers, A. E. E., et al., 1994, ApJ, 434, L59

\reference{A} Rosa, M. R., Zinnecker, H., Moneti, A., \& Melnick, J., 
A\&A, 257, 515

\reference{A} Rees, M. J. 1982, Nature, 333, 523

\reference{A} Rees, M. J., Begelman, M. C., Blandford, R. D.,
\& Phinney, E. S. 1982, Nature, 295, 17

\reference{A} Reynolds, C. S., Di Matteo, T., Fabian, A. C., 
Hwang, U., \& Canizares, C. R., 1996, MNRAS, 283, L111

\reference{A}  Serabyn, E., Carlstrom, J. E., \& Scoville, N. Z., 1992, ApJ, 
	401, L87

\reference{A} Shapiro, S.L., Lightman, A.P., \& Eardley, D.M. 1976,
ApJ, 204, 187

\reference{A} Skinner, G. K. et al., 1987, Nature, 330, 544

\reference{A} Telesco, C. M, Davidson, J. A., \& Werner, M. W., 1996, ApJ, 
456, 541

\reference{A} Watson, M. G., Willingale, R., Grindlay, J. E., Hertz, P., 1981, 
ApJ, 250, 142

\reference{A} Wright, M. C. H., Genzel, R., G\"usten, R., Jaffe, T., 1987, 
in The Galactic Center AIP conf. Proc, ed. D. C. Backer, 133

\reference{A} Yusef--Zadeh, F., Morris, M., \& Ekers, R. D., 1990, Nature,
348, 45

\reference{A} Zhao, J., Ekers, R. D., Goss, W. M., Lo, K. Y., \& Narayan, R., 1989, 
in The Center of the Galaxy, ed. M. Morris, 535 

\reference{A} Zhao, J., Goss, W. M., Lo, K. Y., \& Ekers, R. D., 1992, in 
Relationships  Between Active Galactic Nuclei and Starburst Galaxies, ed. 
A. Filippenko, 31

\reference{A} Zylka, R., \&  Mezger, 1988, A\&A, 190, L25

\reference{A} Zylka, R., Mezger, P. G., \& Lesch, H., 1992, A\&A, 261, 119

\reference{A} Zylka, R., Mezger, Ward--Thompson, D., Duschl, W. J., \& 
Lesch, H., 1995, A\&A, 297, 83

\end{references}

\newpage

\noindent{\bf Figure Captions}

\bigskip
\noindent
Figure~1.  The open and filled circles represent various flux
measurements and upper limits of Sgr A$^*$.  We consider the filled
circles to be more important as model constraints.  The box at 1 keV
represents the uncertainty in the X-ray flux.  The solid line is our
baseline ADAF model with the following parameters:
$M=2.5\times10^6M_\odot$, $\alpha=0.3$, $\beta=0.5$, $\delta=0.001$.
The Eddington-scaled mass accretion rate $\mdot$ has been adjusted to
fit the X-ray flux, giving $\mdot=1.3\times10^{-4}$.  The peak at the
left is due to synchrotron radiation, the next peak is due to Compton
scattering, the peak between 10--100 keV is due to bremsstrahlung and
the peak above 100 MeV is due to pion production.  The long--dashed
line is a model in which the pion peak has been artificially raised by
about an order of magnitude to fit the data.  The dotted line is the
spectrum corresponding to a standard thin accretion disk with
$\mdot=10^{-4}$ while the short--dashed line is a thin disk with
$\mdot=10^{-9}$.  Neither of these models is satisfactory.

\noindent 
Figure~2.  (a) Spectra corresponding to five models, showing the effect
of various approximations; see the text for details.  (b) Electron
temperature profiles corresponding to the same five models.  The
models represented with long-dashed and solid lines have nearly
identical temperatures.

\noindent
Figure ~3.  (a) The solid line is the baseline model shown in Fig. 1,
with a black hole mass of $M=2.5\times10^6M_\odot$.  The dotted line
corresponds to a model with $M=8.3\times10^5M_\odot$ and the dashed
line corresponds to $M=7.5\times10^6M_\odot$.  (b) The solid line is
the baseline model with a mass accretion rate
$\mdot=1.3\times10^{-4}$.  The dotted, short-dashed, long-dashed and
dash-dotted lines corresponds to models with
$\mdot=6.4\times10^{-5}, ~9.0\times10^{-5}, ~1.8\times10^{-4},
~\mdot=2.5\times10^{-4}$, respectively.

\noindent
Figure ~4.  (a) The solid line is the baseline model shown in Fig. 1,
with $\alpha=0.3$.  The dotted, short-dashed and long-dashed lines
correspond to models with $\alpha=0.1, ~0.2, ~0.4$ respectively.  (b)
The solid line is the baseline model shown in Fig. 1, with
$\beta=0.5$.  The dotted, short-dashed, long-dashed lines and
dot-dashed lines correspond to models with $\beta=0.3, ~0.4, ~0.6,
~0.7$ respectively.  (c) Electron temperature profiles corresponding
to the models shown in (b).  (d) The solid line is the baseline model
shown in Fig. 1, with $\delta=0.001$.  The dotted, short-dashed,
long-dashed and dot-dashed lines correspond to models with $\delta=0,
~0.003162, ~0.01, ~0.03162$ respectively.


\noindent
Figure ~5.  (a) The solid line is the baseline model shown in Fig. 1.
The dotted line corresponds to a model in which the temperature of the
electrons has been fixed at $T_e=2\times10^9$ K over the radius range
$r=20-1000$.  This model fits the low frequency radio data better.
(b) Electron temperature profiles corresponding to the two models
shown in (a).

\newpage
\begin{table}
\caption[tab 1]
{Sgr A$^*$: Radio and NIR Observations (Distance = 8.5 kpc)}
\vskip-3.0cm
\begin{center}
\begin{tabular}{lccccl}\hline 
& & & & &  \\
\rb{$\nu$} & \rb{$\lambda$}& \rb{$\theta$} & \rb{$S_{\nu}$} & \rb{$\nu L_{\nu}$} & \rb
{Ref.} \\
\rb{Hz} &\rb{$\mu$m}& \rb{$ \prime \prime$} &\rb{Jy}& \rb{ergs s$^{-1}$} & \\[-.5ex]
\hline \hline
4.08$\times 10^{8}$ & $735294$ & $4.3$ & $\le 0.05$& $\le 1.76 \times 10^{30}$& Davies et al. 76 \\
9.6$\times 10^{8}$ & $312500$ & $10$ & $0.29$& $ 2.41 \times 10^{31}$& Davies et al. 76 \\
9.6$\times 10^{8}$ & $312500$ & $10$ & $0.27$& $ 2.24 \times 10^{31}$& Davies et al. 76 \\
9.6$\times 10^{8}$ & $312500$ & $10$ & $0.30$& $ 2.49 \times 10^{31}$& Davies et al. 76 \\
1.66$\times 10^{9}$ & $180722$ & $2.5$ & 0.56& $8.04\times 10^{31}$& Davies et al. 76 \\
1.5 $\times 10^{9}$ & $200000$ & $\sim 1.2$ & 0.8& 1.04$\times 10^{32}$& Backer 82 \\
1.5 $\times 10^{9}$ & $200000$ & $\sim 1.2$ & 0.3& 3.89$\times 10^{31}$& Zhao et al. 89 \\
2.7 $\times 10^{9}$ & $110000$ & $\sim 0.65$  & .73& $ 1.7 \times 10^{32}$& Backer 82 \\
2.7 $\times 10^{9}$ & $110000$ & $\sim 0.65$  & .42& $ 9.78 \times 10^{31}$& Brown \& Lo  82\\
5.0 $\times 10^{9}$ & 60000 & $\sim 0.35$ & 1.13& 1.45$\times 10^{32}$& Zhao et al. 89 \\
5.0 $\times 10^{9}$ & 60000 & $\sim$ 0.35 & 0.55& 7.13$\times 10^{31}$& Zhao et al. 89 \\
8.1 $\times 10^{9}$ & 37000 & $\sim 0.22$ & 0.9& 6.23$\times 10^{32}$& Backer 82 \\
8.1 $\times 10^{9}$ & 37000 & $\sim 0.22$ & 0.58& $4.01 \times 10^{32}$& Brown \& Lo 82 \\
8.4 $\times 10^{9}$ & 35714 & $\sim 0.21$ & 1.07& $ 7.77 \times 10^{32}$& Zhao et al. 92 \\
8.4 $\times 10^{9}$ & 35714 & $\sim 0.21$ & 0.55& $3.99 \times 10^{32}$& Zhao et al. 92 \\
1.5 $\times 10^{10}$ & 20000 & $\sim 0.12$& 1.64& 2.1$\times 10^{33}$& Zhao et al. 92 \\
1.5 $\times 10^{10}$ & 20000 & $\sim 0.12$& .68& 8.7$\times 10^{32}$& Zhao et al. 92 \\
1.5 $\times 10^{10}$ & 20000 & 0.12$\times$0.24 & 1.15+0.01& (1.49+0.06)$\times 10^{33}$& Yusef--Zadeh et al. 90 \\
2.2 $\times 10^{10}$ & 13600 & $\sim 0.08$& 2.1& $3.99\times 10^{33}$& Zhao et al. 92 \\
2.2 $\times 10^{10}$ & 13600 & $\sim 0.08$& 0.8& 1.52$\times 10^{33}$& Zhao et al. 92 \\
\end{tabular}
\end{center}
\end{table}
\newpage
\addtocounter{table}{-1}
\begin{table}
\caption[]
{ Sgr A$^*$: Observations (continued).}
\vskip-3.0cm
\begin{center}
\begin{tabular}{lccccl}\hline 
& & & & &  \\
\rb{$\nu$} & \rb{$\lambda$}& \rb{$\theta$} & \rb{$S_{\nu}$} & \rb{$\nu L_{\nu}$} & \rb
{Ref.} \\
\rb{Hz} &\rb{$\mu$m}& \rb{$ \prime \prime$} &\rb{Jy}& \rb{ergs s$^{-1}$} & \\[-.5ex]
\hline \hline
\hspace{-.29cm}$\!\star$ 4.3 $\times 10^{10}$ & 7000 & 0.75$\times 10^{-3}$ & 1.4$\pm 0.1$& $(5.2\pm 0.4)\times 10^{33}$& Krichbaum et al. 94 \\
8.6 $\times 10^{10}$ & 3488 & 0.02 & 1.3& 9.67$\times 10^{33}$& Backer 82 \\
8.6 $\times 10^{10}$ & 3488 & 4$\times8$ & 1.05& 7.8$\times 10^{33}$& Wright et al. 87 \\
\hspace{-.29cm}$\!\star$ 8.6 $\times 10^{10}$ & 3488 & $0.16 \times 10^{-3}$ & 1.4$\pm0.2$&
$(1.04\pm 0.15)\times 10^{34}$& Rogers et al. 94 \\
2.2$\times 10^{11}$ &1350& 1.9$\times 4.3$& $2.4\pm 0.5$ &$(4.57\pm .95)\times 10^{34}$ & Serabyn et al. 92 \\
\hspace{-.29cm}$\!\star$ 2.2$\times 10^{11}$ &1350& 2.6$\times 10^{-3}$& $1.07\pm 0.15$ &$(2.04\pm 0.29)\times 10^{34}$ & Alberdi et al. 93 \\
2.3$\times 10^{11}$ &1300& 11& $2.5$ &$4.97\times 10^{34}$ & Zylka \& Mezger 88 \\
2.3$\times 10^{11}$ &1300& 11& $2.6\pm0.6$ &$(5.2\pm 1.2)\times 10^{34}$ & Zylka et al. 92 \\
3.5$\times 10^{11}$ &870& 8 & $4.8\pm 1.2$ &$(1.45\pm 0.36)\times 10^{35}$ & Zylka et al. 92 \\
3.75$\times 10^{11}$ &800& 13 & $3.5\pm 0.5$ &$(1.13\pm 0.16)\times 10^{35}$ & Zylka et al. 95\\
5.0$\times 10^{11}$ &600& 10 & $4.0\pm 1.2$ &$(5.19\pm 0.52)\times 10^{35}$ & Zylka et al. 95\\
6.7$\times 10^{11}$ &450& 7 & $\le 1.5$ &$ \le 8.69 \times 10^{34}$ & Dent et al. 93\\
6.7$\times 10^{11}$ &450& 8 & $3.0\pm 1.0$ &$(1.74\pm 0.58)\times 10^{35}$ & Zylka et al. 95\\
8.6$\times 10^{11}$ &350& 11 & $\le 10$ &$\le 7.4\times 10^{35}$ & Mezger 1994\\
8.6$\times 10^{11}$ &350& 30& $18.5\pm 9$ &$(1.38\pm 0.67)\times 10^{36}$ & Zylka et al. 92 \\
1.0$\times 10^{13}$ &30& 8 & $\le 120$ &$\le 1.04 \times  10^{38}$ & Zylka et al. 92 \\
1.0$\times 10^{13}$ &30& 4 & $\le 20$ &$\le 1.7 \times  10^{37}$ &  Telesco et al. 96 \\
1.5$\times 10^{13}$ &20& 1.6 & $\le 1$ &$ \le 1.3 \times 10^{36}$ & Gezari et al. 94\\
1.56$\times 10^{13}$ &19.2& 4 & $\le 1.4$ &$ \le 1.9 \times 10^{36}$ & Telesco et al. 96\\
1.7$\times 10^{13}$ &18& 2.3 $\times 1.3$ & $\le 0.3$ &$\le 4.4\times 10^{35}$ & Zylka et al. 92 \\
2.3--3.6$\times 10^{13}$ &13--8& 2.3 $\times 1.3$ & $\le 0.1$ &$\le 2.6\times 10^{35}$ & Zylka et al. 92 \\
\hspace{-.29cm}$\!\star$ 1.4$\times 10^{14}$ &2.2& 0.15 &$\le 9\times 10^{-3}$ & $\le 1.1\times 10^{35}$ & Menten et al. 97 \\
\end{tabular}
\end{center}
\end{table}
\begin{table}
\caption[]
{ Sgr A$^*$: X--Ray \& $\gamma$--Ray Observations (Distance = 8.5 kpc).}
%\vskip-3.0cm
\begin{center}
\begin{tabular}{lccccl}\hline 
& & & & &  \\
\rb{Energy} & \rb{Telescope/}& \rb{$\theta$}& 
\rb{$L_{\rm EB}$ \footnotemark[1]}
& \rb{$\nu L_{\nu}$} 
& \rb{Ref.} \\
\rb{Band (EB)}&\rb{Instrument} & &\rb{erg s$^{-1}$} & \rb{erg 
s$^{-1}$} &  \\[-.5ex]
\hline \hline
0.8 - 2.5 keV& ROSAT& $\sim 20 \arcsec$ & $ 1.55  \times 10^{34}$
& 1.6 $\times 10^{34}$ & Predehl \& Tr\"umper 94 \footnotemark[2] \\
2 - 10 keV& ASCA & $\sim 1\arcmin$ &$\le 6.4 \times 10^{35}$ & 
$\le 4.8 \times 10^{35}$ & Koyama et al. 96 \\ 
35 - 75 keV &SIGMA &$\sim 15 \arcmin$& $\le 3.5\times 10^{35}$ & 
$\le 4.8 \times 10^{35}$ &  Goldwrum et al. 94 \\
75 - 150 keV &SIGMA & $\sim 15 \arcmin$& $ \le 2.4 \times 10^{35}$&
$\le 3.6 \times 10^{35}$&  Goldwrum et al. 94 \\
30 - 50 MeV& EGRET & $\sim 1\degree$& & $\le 1.8 \times 10^{36}$ &Merck et al. 96\\
50 - 70 MeV& EGRET & $\sim 1\degree$& & $\le 1.1 \times 10^{36}$ &Merck et al. 96\\
70 - 100 MeV& EGRET & $\sim 1\degree$& & $\le 8.3 \times 10^{35}$ &Merck et al. 96\\
100 - 150 MeV& EGRET & $\sim 1\degree$& & $\le 6.2 \times 10^{35}$ &Merck et al. 96\\
150 - 300 MeV& EGRET & $\sim 1\degree$& & $(4.9^{+2.0}_{-2.1}) \times 10^{35}$&Merck et al. 96\\
300 - 500 MeV& EGRET & $\sim 1\degree$& & $(1.2^{+0.24}_{-0.26}) \times 10^{36}$ &Merck et al. 96\\
500 - 1000 MeV& EGRET & $\sim 1\degree$& & $(1.4^{+0.28}_{-0.28}) \times 10^{36}$ &Merck et al. 96\\
1 - 2 GeV& EGRET & $\sim 1\degree$& & $(1.7^{+0.28}_{0.28}) \times 10^{36}$ &Merck et al. 96\\
2 - 4 GeV& EGRET & $\sim 1\degree$& & $(2.2^{+0.82}_{-0.62}) \times 10^{36}$ &Merck et al. 96\\
4 - 10 GeV& EGRET & $\sim 1\degree$& & $(8.3^{+4.2}_{-4.7}) \times 10^{35}$ &Merck et al. 96\\
& &&&& \\
\multicolumn{6}{l}{
 \footnotesize{$^{1}L_{\rm EB}$ is the total luminosity integrated over the band.}} \\
\multicolumn{6}{l}{\footnotesize{$^2$ This flux is obtained using 
$N_H = 6\times 10^{22} $cm$^{-2}$ as opposed to the much higher }} \\
\multicolumn{6}{l}{\footnotesize{
column density used by Predehl \& Trumper (1994). This is discussed 
in \S 2.}}
\end{tabular}
\end{center}
\end{table}
\newpage
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\begin{figure}
\epsffile{adsgr5.ps}
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\end{document}


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