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\Chapter[Radio and X-ray Emission]
{Radio and X-ray Emission from the Galactic Black Hole}{
Heino Falcke\\
Max-Planck-Institut f\"ur Radioastronomie, Bonn, Germany}
\label{Falcke}

\begin{center}
{\it To appear in: The Galactic Black Hole, Lectures on General Relativity
and Astrophysics, H. Falcke and F.W. Hehl (eds.), IOP Publishing,
Bristol (2002)}
\end{center}
\bigskip

{\bf Summary:} In this chapter the radio properties of the Galactic
Center black hole (Sgr~A*) are reviewed: variability, size and
position from VLBI (Very Long Baseline Interferometry), spectrum, and
polarization.  Radio and X-ray emission are discussed within the
framework of black hole plasma jet models and simple equations for the
emission are derived. It is also shown that the radio emission can be
used to actually image the event horizon of the black hole in the near
future.

\tableofcontents

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\section{Introduction}
\label{chap:intro}

We have already seen in Chapter~\ref{Eckart} that the evidence for the
presence of a dark mass in the Galactic Center is very strong. A
central point mass of about $3\cdot10^6M_\odot$ seems to coincide with
the compact radio source Sgr A*. The existence of Sgr A* has always
been considered a good sign for a black hole itself. In fact, based on
analogous detections of compact radio cores in the nuclei of active
galaxies, the existence of Sgr A* was predicted by
\citeN{Lynden-BellRees1971}. \citeN{BalickBrown1974} detected the source in one 
of the early VLBI (Very Long Baseline Interferometry) experiments and
a couple of years later named it Sgr A* (by simply adding an asterisk
to the name of the nuclear radio region Sgr A --- see
\citeNP{PalmerGoss1996} for an account of the history of naming
sources in the Galactic Center).

Ever since, Sgr A* has been the focus of great attention. From the
near-infrared speckle observations (Eckart, this volume) we know that
it is at the very center of the gravitational potential. However, until
today one is still unsure about the exact nature of the detected
emission in radio and X-ray bands. On the other hand, understanding
the radiation spectrum of Sgr A* will have important implications for
all other supermassive black holes, since most galaxies will have
black holes that are as inactive as the Galactic Center and we are
investigating the low-power end on the black hole activity scale.  In
fact, surveys have shown that a large number of nearby galaxies host
radio sources very similar to Sgr A* in our Milky Way
\cite{WrobelHeeschen1984,NagarFalckeWilson2000,FalckeNagarWilson2000}.
Hence, what we learn about Sgr A* is quite typical for the (rather
silent) majority of black holes in the universe.

In the following section we will attempt to summarize the main radio
properties of the ``Galactic Black Hole''. In the subsequent sections
we will try to explain the main emission characteristics of Sgr A* in
an almost back-on-the-envelope fashion. Finally we will look ahead on
how this source could finally make the theoretical concept of an event
horizon observable in the near future.

\begin{figure}
\begin{center}
\centerline{\psfig{figure=sgr.eps,width=0.75\textwidth}}
\end{center}
\caption[]{\label{hf-sgra}Sgr A West, a spiral-like pattern of thermal ionized gas that appears to be falling into the very center of the galaxy. Near its center is Sgr A*, a point-like radio source that many suspect is the nucleus of the Milky Way and indicates the presence of a black hole. Figure courtesy of Prof. K.Y. Lo}
\end{figure}

\section{Radio Properties of Sgr A*}
Most of the direct information about Sgr A* is available at radio
wavelengths. This includes the total intensity spectrum, the
variability, the polarization, and the source structure. In the
following we will go through these various issues step by step.

\subsection{Variability of Sgr A*}

\begin{figure}
\centerline{\psfig{figure=gbibw.jpg.ps,width=0.49\textwidth}\psfig{figure=tresmontosas2.jpg.ps,width=0.49\textwidth}}
\caption[]{The Green Bank Interferometer (GBI; left) which was instrumental in
detecting Sgr A* and the Very Large Array (VLA; right). Both
interferometers are operated by the National Radio Astronomy
Observatory (NRAO) in the United States and were extensively used to
monitor the variability of Sgr A*.\label{gbivla}}
\end{figure}

First evidence for a a very compact structure in a radio source often
comes from its variability.  And indeed soon after its discovery, Sgr
A* was established as a variable source \cite{BrownLo1982} and
extended campaigns were set up to monitor this variability. The most
extensive data sets were obtained using the Very Large Array (VLA) and
the Green Bank Interferometer (GBI; see Fig. \ref{gbivla}). The most
recent data from these instruments are presented in
\citeN{ZhaoBowerGoss2001} and
\citeN{Falcke1999a}. The amplitude of variability can reach up to
200\% for strong flares. The degree of variability seems to increase
with increasing frequency. Strong flares seem to occur on time scales
of 100 days (see Fig.~\ref{sgrvar}). \citeN{ZhaoBowerGoss2001} even
claim to have found a periodicity of 106 days on which such strong
flares occur regularly.  The shortest times scale of radio variability
was probably found by \citeN{BowerFalckeSault2002} at 15 GHz: 20\%
within one hour.

\begin{figure}
\centerline{\psfig{figure=sgrvar.ps,height=0.8\textwidth,bblly=0.6cm,bbllx=0.9cm,bburx=20cm,bbury=26.8cm}}
\caption[]{
Variability of Sgr A* as observed with the VLA at different wavelengths
(given in the top left corner of each frame). From Zhao, Bower, Goss
(2001)\label{sgrvar}.}
\end{figure}

Some of the radio variability may be due to scintillation due to an
foreground screen (see below), but at least the large-amplitude flares
at higher radio frequencies are most likely of an intrinsic
nature. The fastest variations are fundamentally limited by the source
size.  We can then convert the measured time scale $\tau$ and
amplitude $\Delta S/S$ of the variability into a limit on the
characteristic size $R$ of Sgr A*. For a given maximum signal speed
$v_{\rm max}\la c$ in the emitting plasma, one obtains

\begin{equation}
R < \left(\Delta S/S\right)^{-1}  \tau\cdot v_{\rm max}.
\end{equation}

For $(\Delta S/S)=20\%$ and $\tau=1$ hr we get $R<5\cdot10^{14}$ cm
$\cdot (v_{\rm max}/c)$, which is 36 Astronomical Units (AU) or 4.5
milli-arcseconds (mas) at the Galactic Center distance of 8 kpc. Based
on the variability time scale alone the radio emission therefore has
to come from a region smaller than a planetary system. In fact,as we
will learn in the next subsection, direct radio imaging shows that the
emission comes from an even smaller scale.

Alternatively, taking the much longer 100 day time scale we find a
characteristic size of $R<2.7\cdot 10^{17}$ cm $\cdot (v_{\rm max}/c)$
or 2.3 arcseconds. This is relatively large and does not appear to be
a useful limit for the size of Sgr A*. It is more likely that this
variability signals some other underlying physical process, e.g. a
process related to the accreting material powering the source. 

If the accreting gas is in orbit around the central point mass
$M_\bullet$ the characteristic velocity will be set by the the
Keplerian velocity\footnote{Note that the sound speed of infalling gas
in optically thin accretion models is typically of similar order and
hence the same numbers apply.} $v=\sqrt{G M_\bullet/R}$ and the
corresponding time scale is set by $\tau=2\pi R/v$, yielding

\begin{equation}
R\approx\left({\tau\over2\pi}\right)^{2/3}\left({G M_\bullet}\right)^{1/3}=0.9\cdot 10^{15}\,{\rm cm}\;\left({\tau\over\,106 {\rm days}}\right)^{2/3}\left({M_\bullet\over3\cdot10^6 M_\odot}\right)^{1/3}.
\end{equation}

This corresponds to about 1000 Scharzschild radii or 8 mas. The
accretion radius (Coker, this volume) of the hot X-ray gas in the
Galactic Center is somewhat further out, but it is not impossible that
instabilities in the capture process of surrounding gas by the black
hole are the ultimate source of the slow, large-amplitude
variability. If the flares are indeed periodic as claimed by
\citeN{ZhaoBowerGoss2001}, then one probably has to think of a star
orbiting Sgr A* at this distance which modulates the accretion flow.

On the other hand, in stellar mass black holes, quasi-periodic signals
are often related to beat and precession frequencies of accretion
disks close to the black hole. If true, this can be used to derive
informations about the black hole spin. Such a scenario has been
considered by \citeN{LiuMelia2002}. Based on such a model and the 100
day periodicity, they derive a black hole spin parameter (e.g.,
Sec.~\ref{imevent}) of Sgr A* of $a\simeq0.1$. It is difficult to
assess which of these interpretations is correct.

The fastest variations are expected at the shortest wavelengths, since
high-frequency emission -- in essentially all models for Sgr A* -- is
typically produced at the smallest scales. The problem with
high-frequency measurements of Sgr A* is that these observations are
extremely sensitive to weather and the low elevations of the source
encountered in typical observations from the northern
hemisphere. Because of the large confusing flux in the Galactic
Center, interferometers have to be used.  The flux density measured on
an interferometer baseline can be artificially reduced by rapid
changes in the atmospheric opacity and loss of coherence (of
instrumental or atmospheric origin) which are difficult to
track. Early observations
\cite{WrightBacker1993,ZylkaMezgerWard-Thompson1995} already indicated
possibly strong variations in the sub-millimeter wavelength
region. This was later strengthened by
\citeN{TsuboiMiyazakiTsutsumi1999} and very recently \citeN{ZhaoYoungMcGary2001} claim relatively 
strong sub-millimeter flares in Sgr A* from measurements with the
sub-millimeter array (SMA) in Hawaii. These results will certainly
become more and more important in the years to come since, as we will
see below, this emission most certainly comes directly from the
immediate environment of the event horizon.


\subsection{Size of Sgr A* -- VLBI Observations}
%1/Sqrt[10^12 Kelvin/(7.8 10^6 Kelvin)/1000 3^2] 2.5=0.073598 mas

A major issue for a long time has been the exact size and structure of
Sgr A*. In extragalactic sources the VLBI technique has allowed one to
probe deep into the hearts of active galactic nuclei and resolved the
radio structures into relativistically outflowing plasma jets (see,
e.g., \citeNP{Zensus1997}). In a VLBI experiment radio telescopes
distributed over a continent or the entire world are synchronized by
atomic clocks and observe jointly one source. The incoming radio waves
are digitized and stored (usually on magnetic tapes). Later, the
digitized waves are correlated in a specialized computer recreating a
virtual interferometer. Such a virtual interferometer will have a
spatial resolution similar to that of a giant telescope with the
diameter of roughly the separation of the individual telescopes
(called ``baselines''). This technique has yielded by far the highest
resolution images in astronomy (i.e., 50 $\mu$arcseconds at 86
GHz). The image quality improves with the number of participating
telescopes and the two major VLBI arrays today are the European VLBI
Network (EVN) and the Very Long Baseline Array (VLBA) in the United
States.

Of course, a prominent source such as Sgr A* has been a prime target
for VLBI experiments over the last 25 years. It is from these
observations that we have direct information on the size of the source
-- or at least have very tight upper limits.


\begin{figure}
\centerline{\psfig{figure=sgrsize2-eng.eps,width=0.9\textwidth}}
\caption[]{
The major source axis (filled circles) of Sgr\,A* and the minor source
axis (open diamonds) as measured by VLBI plotted versus wavelength
(adapted from Krichbaum et al.~1999.). The inclined lines show the
$\lambda^2$ scattering law and the horizontal line shows the size
scale expected for the visual imprint of the event horizon (Falcke,
Melia, Agol 2000).
\label{scattersize}}
\end{figure}
\nocite{KrichbaumWitzelZensus1999}
\nocite{FalckeMeliaAgol2000}

\begin{figure}
\centerline{\psfig{figure=sgrvlbmaps.ps,height=0.6\textheight}}
\caption[]{
Contour plots of VLBA images of Sgr A* at wavelengths of $\lambda=$
6.0, 3.6, 2.0, 1.35, \& 0.7 cm.  These images are smoothed to a
circular beam of FWHM = 2.62 $\lambda_{cm}^{1.5}$ mas as shown on the
left-bottom corner on each image.  The contours are 2 mJy
beam$^{-1}\times$ (-2, 2, 4, 8, 16, 32, 64, 128, 256). Figure from Lo
et al. (1999).
\label{sgrvlbmaps}}
\end{figure}
\nocite{LoShenZhao1999}

It was, however, quickly realized that, despite its relative
proximity, detecting the true structure of Sgr A* is unusually
difficult compared to other galactic nuclei. The reason is that that
interstellar material in the line line-of-sight towards the Galactic
Center scatter broadens the VLBI image. This produces a characteristic
$\lambda^2$-law (e.g. \citeNP{Scheuer1968}) for the size of Sgr A*
\cite{DaviesWalshBooth1976,vanLangeveldeFrailCordes1992,Yusef-ZadehCottonWardle1994,LoShenZhao1998}.
The scattering size apparently has not changed over a decade
\cite{MarcaideAlberdiLara1999}.
For that reason the {\it intrinsic} size and structure of Sgr A*
remains obscure until today
(Fig.~\ref{scattersize} \& \ref{sgrvlbmaps}).  The measured size of Sgr
A* is given by
\cite{LoShenZhao1998}
\begin{equation}\label{sizeeq}
\theta_{\rm minor}=0.76\,{\rm mas}\;(\lambda/{\rm cm})^2,\quad\theta_{\rm
major}=1.42\,{\rm mas}\;(\lambda/{\rm cm})^2.
\end{equation}


The front line of this research is currently at wavelengths of 7mm and
shorter. \citeN{LoShenZhao1998}
\citeN{KrichbaumZensusWitzel1993}, and \citeN{KrichbaumGrahamWitzel1998} claim to
have seen evidence for a deviation from the scattering law at these
wavelengths. Other experiments did not directly confirm this
\cite{RogersDoelemanWright1994,BowerBacker1998,DoelemanShenRogers2001}. The main problem is
that Sgr A* is never above $25^\circ$ elevation at the VLBA and
mm-VLBI is strongly affected by the variable atmosphere. Significant
distortions of the phase (of the radio waves) can happen on time
scales of 10 seconds at the short wavelengths. Therefore mm-VLBI
observations of Sgr A* are difficult to calibrate and are always
subject of intense scrutiny and nagging doubts.

On the other hand, the rapid variability of Sgr A* with time scales of
10 minutes (X-rays) to 3 hours (15 GHz radio) suggests that time-variable structure may exist
on similarly short time scales. We can estimate a scale for adiabatic
cooling of a plasma by using Eq.~\ref{sizeeq}, convert to a
linear size at the Galactic Center distance $D_{\rm GC}=8$ kpc, and
divide by the maximal sound speed of a relativistic plasma (photon
gas; see e.g. \citeNP{Konigl1980}) $c_{\rm s}=c/\sqrt{3}$. We get

%cnv[1.42 0.7^2 mas 8 kpc/(c/Sqrt[3])/hrs]

\begin{equation}\label{sgrcool}
t_{\rm cool}\ga\theta_{\rm major} D_{\rm GC}/c_{\rm s}=1.3\,{\rm hrs} \left({\lambda\over{\rm 7\,mm}}\right)^2.
\end{equation}
Hence, if for example, matter is ejected during one of the big X-ray
flares near the black hole it could in principle cool and fade away
within a few hours at most radio wavelengths\footnote{Note that
Eq.~\ref{sgrcool} is based solely on the {\it observed} upper limit on
the source size. Most models suggest that the intrinsic size of Sgr A*
grows less rapidly than $\lambda^2$ (e.g. linearly) and hence the
cooling time scale would grow accordingly slower.}. Consequently,
there is no reason why VLBI observations separated by one day always
have to look the same.

\subsection{Position of Sgr A*}

The exact location of Sgr A* is a very important factor for its
interpretation. This allows one to investigate whether the source is
indeed in the Galactic Center (and not behind) and to set a lower
limit on its mass. The position can be determined relatively
accurately with radio observations. 

This requires the use of so-called ``phase-referencing''
observations. In this type of experiments the telescopes of a
VLBI-array are switched rapidly from one source to another, where both
sources should be within one isoplanatic patch of the atmosphere
(typically 1$^\circ$ to 5$^\circ$). Within this patch the radiation
passes roughly through the same atmospheric irregularities.  The
switching also has to happen within the coherence time of the
atmosphere---at mm-wavelengths the telescopes switch sources every 10
seconds. The phase-difference of the incoming wave fronts between the
two observed sources can then be used to determine the relative
positions of the sources. By fixing a grid of phase-referenced sources
--- radio quasars at cosmological distances -- one can then establish
an absolute coordinate system, called the International Celestial
Reference System (ICRF; see
\citeNP{MaAriasEubanks1998}). Since 1997 the ICRF is the fundamental
reference system of astronomy as adopted by the International
Astronomical Union (IAU).

\begin{figure}
\centerline{\psfig{figure=sgra_propmotion.ps,height=0.5\textheight,bbllx=2.7cm,bblly=6.5cm,bburx=16.5cm,bbury=22.2cm}}
\caption[]{Position  of Sgr A* relative to a background quasar (J1745--283) on the
plane of the sky determined from VLBA observations.  North is to the
top and East to the left.  Each measurement is indicated with the date
of observation and $1-\sigma$ error bars.  The dashed line is the
best-fit proper motion, and the solid line gives the orientation of
the Galactic plane.  Figure from Reid et al. (1999).
\label{sgrapropmotion}}
\end{figure}
\nocite{ReidReadheadVermeulen1999}


Attempts have been made to relate the {\it absolute} position of Sgr
A* to bright background quasars. Averaging of VLA observations by
\citeN{Yusef-ZadehChoateCotton1999} yield a position for Sgr A* at the
epoch 1992.4 of

\begin{equation}
\alpha(1950)=17^h42^m29.3076^s \pm 0.0007s,\; \delta(1950)=-28^\circ59'18.484" \pm 0.014" 
\end{equation}
\begin{equation}
\alpha(2000)=17^h45^m40.0383s \pm 0.0007s,\; \delta(2000)=-29^\circ00'2^8.069"\pm0.014".
\end{equation}

A position using VLBA observations was also derived by
\citeN{RogersDoelemanWright1994}, which agrees with this position within
0.2''. In yet another experiment \citeN{MentenReidEckart1997} were
able to relate the position of Sgr A* to the position of near-infrared
stars (emitting a radio line) surrounding it. This allows one to
locate Sgr A* in a near-infrared image and try to find a
counterpart. 

Finally, one can also measure the {\it relative} position of Sgr A*
with respect to faint background quasars which are much closer than
the ICRF reference sources. Assuming that these sources are without
motion on the sky (because of their cosmological distances), 
\citeN{ReidReadheadVermeulen1999} (Fig.~\ref{sgrapropmotion}; see also \citeNP{BackerSramek1999a}) find a
proper motion for Sgr A* of $-3.33\pm0.1$ mas yr$^{-1}$ (E) and $-4.94 \pm 0.4$ mas
yr$^{-1 }$ (N), corresponding to $-5.90 �\pm 0.35$ mas yr$^{-1}$ and $+0.20 \pm 0.30$
mas yr$^{-1}$ in Galactic longitude and latitude. This agrees very
well with the apparent motion expected for a source at the Galactic
center due to the {\it Galactic rotation of the solar system} (220 km
s$^{-1}$). This implies that Sgr A* is indeed at the center of the
Galaxy and has very little motion of its own (${v_{\rm
Sgr\,A*}<15\,{\rm km s^{-1}}}$).  This is interesting, since in 
Chapter \ref{Eckart} it was shown that stars near Sgr A*
move in the deep potential well with velocities up to 1500 km
s$^{-1}$. The most likely reason for this is, of course, that Sgr A*
itself causes this potential well. One can use the assumption of
equipartition of momentum between the fastest stars ($m_* v_*$) and
Sgr~A* $(M_{\rm Sgr\,{A^*}}\,v_{\rm Sgr\,{A^*}})$ to infer a lower
limit on the mass of Sgr A* from the VLBI proper motion studies

\begin{equation}
M_{\rm Sgr\,A*}\ga1,000\, M_{\sun}
\left({m_*\over10\,M_{\sun}}\right)\left({v_*\over1,500\,{\rm
km/s}}\right)\left({v_{\rm Sgr\,A*}\over15\,{\rm km/s}}\right)^{-1}.
\end{equation}

Numerical modeling of n-body interactions suggest that under most
conditions this lower limit can be as high as $10^5 M_\odot$
\cite{ReidReadheadVermeulen1999}. This mass is way beyond those of any
stellar object and hence it is very reasonable to assume that
essentially all the dark mass of $2-3\cdot10^6 M_\odot$ is
concentrated inside Sgr A*---the radio source.

\subsection{Radio Spectrum of Sgr A*}
A major input for modeling the nature of the central black hole
candidate of our Galaxy is the emission spectrum. Typical luminous
black holes, e.g. those in quasars, emit over a broad range in
frequencies. In contrast, for Sgr A* only radio emission was reliably
detected over many years, with the late addition of X-ray
emission. One reason for this dimness is certainly the low accretion
rate and quite plausibly the presence of a radiatively deficient
accretion flow (Coker, this volume).

At frequencies below 1 GHz the spectrum is essentially
undetermined. First of all, the scattering size of Sgr A* becomes
larger than 1 arcsecond and the source starts to blend with its
surroundings. Secondly, the Sgr A complex becomes optically thick at
low frequencies \cite{PedlarAnantharamaiahEkers1989} and Sgr A* may be
obscured. It is also possible, but less likely, that the claimed
low-frequency turnover in the spectrum \cite{DaviesWalshBooth1976} has
an intrinsic nature.

At higher frequencies the radio spectrum of Sgr A* has been measured
with great accuracy in various campaigns up into the THz regime (e.g.,
\citeNP{WrightBacker1993,ZylkaMezgerWard-Thompson1995,SerabynCarlstromLay1997,FalckeGossMatsuo1998}). Since
the source is variable (see Fig.~\ref{sgrvar}), it is useful to
consider either simultaneous or time-averaged spectra. Such an average
spectrum is shown in Figure~\ref{sgrspec} compiled by
\citeN{MeliaFalcke2001}. The overall radio spectrum is slightly
inverted, i.e. has a positive spectral index ($\alpha\simeq0.2$,
$S_\nu\propto\nu^\alpha$) in the GHz regime. The average radio flux
density is around 1 Jy. At the highest radio frequency ($\ga100$ GHz),
the spectrum seems to become even more inverted until it abruptly cuts
off somewhere in the far-infrared. This upturn and subsequent cut-off
has been interpreted as an effect of the finite size of the central
object with a size scale as expected for a $3\cdot10^6 M_\odot$ black
hole (e.g.,
\citeNP{FalckeBiermann1994} and below). The up-turn in the spectrum 
at submm-wavelengths is often referred to as the ``submm-bump''.


\begin{figure}
\centerline{\psfig{figure=sgrspec.ps,width=\textwidth,bbllx=4.1cm,bburx=16cm,bblly=19.2cm,bbury=27.1cm}}
\caption[]{Time-averaged spectrum---flux density versus frequency---of
Sgr A* from radio to the near-infrared as compiled by Melia \& Falcke
(2001).  The error bars in the radio regime indicate variability (one
standard deviation).\label{sgrspec}}
\end{figure}

\subsection{Polarization of Sgr A*}
Finally, as a relatively recent development, the polarization
properties of Sgr A* are now relatively well established. For a long
time it was generally thought that, in marked contrast to more
luminous AGN, Sgr A* is unpolarized. This is true for linear
polarization in the GHz radio regime. \citeN{BowerBackerZhao1999}
found an upper limit of $\le0.1\%$ to the linear polarization at
cm-waves, however, they found plenty of circular polarization.

The results for linear polarization at low frequencies were obtained
with a rarely used technique in radio, called spectro-polarimetry.
This allows one to look for polarization in small frequency
bands. Usually in continuum observations one averages the polarization
of the radiation over the available bandwidth $\delta\nu$. However,
since Sgr A* may be embedded in a dense plasma, Faraday rotation in
the accretion flow or a foreground Galactic screen could lead to a
rotation of the linear polarization vector even within the small
bandwidth.

Faraday rotation is produced when radio waves pass through an ionized
and magnetized medium. Since left and right circularly polarized waves
have different refractive indices for a given magnetic field
orientation, a wavelength-dependent delay is induced that rotates the
position angle $\phi$ of the linear polarization vector by an amount

\begin{equation}
\Delta \phi = {\rm RM}\, \lambda^2.
\end{equation} 
The parameter RM is called the rotation measure and can be determined
by measuring the position angle of the linear polarization vector $\phi$
at different wavelengths. For a given frequency bandwidth $\delta\nu$,
significant de-polarization is obtained if $\Delta \phi\sim1$
rad. Hence, for a typical VLA bandwidth of $\Delta\nu=$ 50 MHz at 4.8
GHz a rotation measure of RM$=10^4$ rad m$^{-2}$ of any foreground
material would destroy any intrinsic polarization signal. Such a value
for RM is large compared to what is seen in other AGN but cannot be
excluded in the Galactic Center. By Fourier-transforming
spectro-polarimetric data (to look for periodic signals due to a fast
rotation of the polarization vector as a function of frequency),
\citeN{BowerBackerZhao1999} were able to set the 0.1\% limit and also 
exclude rotation measures below RM$\la 10^7$ rad m$^{-2}$ as the cause
for the low polarization.

Later, \citeN{AitkenGreavesChrysostomou2000} made linear polarization
observation with a single dish sub-millimeter wave telescopes and
found $\sim$10\% linear polarization above 150 GHz. This was confirmed
with an interferometer by \citeN{BowerWrightFalcke2002} and they also
found evidence for a large rotation measure $\la10^6$ rad m$^-2$ plus
some evidence for intrinsic depolarization towards lower
frequencies. At the moment of writing this is a strongly developing
field which promises many new insights in the future. For example, one
can use the measured RM to limit the accreting material engulfing
Sgr~A* \cite{Agol2000}.

As a big surprise, \citeN{BowerFalckeBacker1999} also found strong
circular polarization at the 0.3-1\% level. This is unusual because
typical AGN have more linear than circular polarization and it can be
used to constrain the electron content and distribution in Sgr A* (see
Sec.~\ref{hfcptheo}). Interestingly, the circular polarization turned
out to be variable itself. At higher frequencies the variability as
well as the fractional polarization increase
\cite{BowerFalckeSault2002}. Figure~\ref{sgrpol} shows a summary of
the currently known polarization properties of the Sgr A* spectrum.

\begin{figure}
\centerline{\psfig{figure=sgr_pol.drw.ps,width=0.75\textwidth,bbllx=1.4cm,bburx=20.5cm,bblly=5.2cm,bbury=23.6cm}}
\caption[]{The average fractional circular polarization of Sgr A* and upper limits
to the linear polarization from Bower et al.~(2002; and references
therein). In the top right corner we show the linear polarization
values given in Aitken et al.~(2000) from single dish values. The
error bars are $1\sigma$ errors.
\label{sgrpol}}
\end{figure}
\nocite{AitkenGreavesChrysostomou2000,BowerFalckeSault2002}



\section{Radio and X-ray Emission from a Black Hole Jet}
\label{sec:falcketheo}
A very common feature of active black holes in the radio regime is the
compact radio core with its characteristic, flat spectrum.  In
luminous quasars the cores are known for many years. Studying these
radio cores with VLBI has allowed us to make the most detailed images
of the physics and environment of black holes \cite{Zensus1997}. Such
flat-spectrum radio cores have also been found in many nearby galaxies
which show signs of nuclear activity
\cite{WrobelHeeschen1984,NagarFalckeWilson2000,FalckeNagarWilson2000}. A well studied example
is M81*, the compact core in the nearby galaxy M81, which shares many
characteristics with Sgr A*
\cite{ReuterLesch1996,BietenholzBartelRupen2000,BrunthalerBowerFalcke2001}. In essentially all cases these cores are
related to relativistic outflows or jets. For that reason, we start
with the most simple-minded assumption, namely that Sgr A* is not very
different either and we will discuss in the following how to obtain
the observed radio spectra within the context of jet physics.

\subsection{The Flat Radio Spectrum}
The fact, that flat radio spectra for radio cores are so ubiquitous,
suggests that this is a very robust feature that must arise
naturally. This is indeed the case for initially collimated, then
freely expanding, supersonic radio plasmas. Why is this so?

Let us consider a plasma jet ejected from the vicinity of the black
hole. Mechanisms for this collimated launching of jets have been
discussed in the literature (see, e.g., \citeNP{Ferrari1998} and
references therein) and are mostly magneto-hydrodynamic (MHD) in
origin.  Observationally, jets span an enormous range of spatial
scales --- from milliparsecs to Megaparsecs --- and maintain their
basic structure for long stretches (see, e.g.,
\citeNP{BridlePerley1984}, Fig.~2). Here, we consider the part, where
the jet has left the acceleration and collimation region and is
essentially in a free expansion. If the jet has not yet propagated and
expanded too far, it is usually a good assumption to assume that the
jet is highly over-pressured with respect to the external medium. We
use a cylindrical coordinate system where $z$ is along the jet axis
and $r$ is perpendicular to it.

Let us assume the jet plasma moves in the forward direction with a
relativistic and almost constant proper velocity (bulk speed)

\begin{equation}
v_{\rm z}=\gamma_{\rm j}\beta_{\rm j}c,
\end{equation}
along the jet axis. The sideways expansion will happen with the
respective sound velocity 

\begin{equation}
v_{\rm s}=\gamma_{\rm s}\beta_{\rm s}c,
\end{equation}
if we can ignore the external pressure and we are well beyond the
sonic point where we can neglect adiabatic losses.

Here we use the well known definition of the relativistic Lorentz
factor and the dimensionless velocity,

\begin{equation}
\gamma=\sqrt{1\over 1-\beta}\quad\mbox{and}\quad\beta={{v}\over c}.
\end{equation}

With longitudinal and lateral expansion being of constant velocity the
plasma will expand into a cone of half opening angle

\begin{equation}
\phi\simeq{1\over{\cal M}}\quad{\cal M}={\gamma_{\rm j}\beta_{\rm j}\over\gamma_{\rm s}\beta_{\rm s}},
\end{equation}
where ${\cal M}$ is the relativistic Mach number (see
\citeNP{Konigl1980}). The shape is given by
\begin{equation}\label{machnumber}
r={z\over{\cal M}}.
\end{equation}
This naturally resulting conical structure is the basis for the
self-similar structure of jets.

The scaling of the relevant parameters for calculating the synchrotron
emission, electron density $n_{\rm e}$, and magnetic field $B$, can be
obtained from simple conservation laws. First, we demand that the particle
number $N_{\rm e}$ is conserved along the flow and set the total
mass flux to

\begin{equation}
\dot M_{\rm j}={d(m_{\rm p}N_{\rm e})\over{dt}}=\rho
v_{\rm z} A={\rm const}.
\end{equation}

Note that the total mass flux is determined by the protons in the
fully ionized plasma and we assume charge balance between electrons
and protons ($N_{\rm e}=N_{\rm p}$); $\rho$ is the mass density and
$A=\pi r^2$ the cross section of the jet. The particle density $n_{\rm
e}$ is then given by

\begin{equation}\label{density}
n_{\rm e}(r)={\dot M_{\rm j}\over m_{\rm p}\cdot\gamma_{\rm j}\beta_{\rm j}c\cdot\pi r^2}\propto r^{-2}.
\end{equation}

We can use the same argument to get the scaling for the magnetic
field, by demanding that the comoving magnetic field energy in a
turbulent plasma is conserved:

\begin{equation}
\dot E_{\rm B}=L_{\rm B}=\rho_{\rm B}v_{\rm
z}A={\rm const}.
\end{equation}
Here we use $L_{\rm B}$ as a measure for the magnetic power fed into
 the jet. The energy density of the magnetic field is given by
\begin{equation}
\rho_{\rm B}=B_{\rm j}^2/8\pi
\end{equation}
and consequently we get

%cnv[Sqrt[8 Lb/(gammaj betaj c R^2)]/Gauss //. {Lb->L3 1000 Lsol,R->rs 2 3 10^6 mgc Msol G/c^2}]


\begin{equation}\label{bfield}
B_{\rm j}(r)=\sqrt{8L_{\rm B}\over\gamma_{\rm j}\beta_{\rm j}c\cdot r^2}=36\,{\rm G}\;\left(\gamma_{\rm j}\beta_{\rm j}\right)^{-1/2}\left({L_{\rm B}\over1000L_\odot}\right)^{1/2}\left({r\over R_{\rm s}}\right)^{-1}\propto r^{-1},
\end{equation}
where for the Galactic Center we have a Schwarzschild radius of 

\begin{equation}
R_{\rm s}={2 G M_\bullet\over c^2}=0.9\cdot10^{12}\,{\rm cm}\;\left({M_\bullet\over 3\cdot10^6M\odot}\right).
\end{equation}

Of course, this implies that the energy content in magnetic field and
relativistic particles remains a fixed ratio throughout the jet. One
therefore relates these two crucial parameters of a jet by an
``equipartition relation'' such that the total energy in particles is
a fraction $k$ of the total energy in the magnetic field. For
simplicity we assume that all electrons are of the same energy $E_{\rm
e}=\gamma_{\rm e}m_{\rm e}c^2$, with $\gamma_{\rm e}$ being the
electron Lorentz factor characterizing the internal energy or
temperature of the electrons (not to be confused with the bulk Lorentz
factor of the entire flow)\footnote{The results will not be very
different for a thermal distribution of electrons or a power-law
distribution with a low-energy cut-off around $\gamma_{\rm e}$}. We
can equate the energy densities,

\begin{equation}
n_{\rm e} \gamma m_{\rm e}c^2 =k{B^2\over8\pi},\mbox{\;and\;yield\;}
n_{\rm e}={k B^2\over8\pi\gamma m_{\rm e}c^2}.
\end{equation}

Here we only consider the internal energy of the jet. The total energy
of the jet will of course be still dominated by the kinetic energy of
the protons -- but not by a huge factor. A proper discussion of the
{\it total} energy budget requires solution of the relativistic
Bernoulli equation and is discussed in
\citeN{FalckeBiermann1995}.

To calculate the radio emission, we need to know the scaling of the
synchrotron emission and absorption coefficients. This can be
obtained, e.g. from \nocite{Pacholczyk1970} Pacholczyk (1970;
Eqs. 3.43 \& 3.44) for electrons with a pitch angle $\alpha_{\rm e}$
with respect to the magnetic field. The emission and absorption
coefficients are respectively

\begin{equation}
\epsilon_\nu= n_{\rm e} \cdot {\sqrt{3} e^3\over 4 \pi m_{\rm e} c^2} B \sin{\alpha_{\rm e}}\cdot F\left({\nu\over\nu_{\rm c}}\right),
\end{equation}
and

\begin{equation}
\alpha_\nu=n_{\rm e} \cdot c^2 \sqrt{3 e\over4 \pi m_{\rm e}^3 c^5} {\sqrt{3} e^3\over 4 \pi m_{\rm e} c^2}
  \left(B \sin{\alpha_{\rm e}}\right)^{3/2} \nu_{\rm c}^{-5/2} K_{5/3}\left({\nu\over\nu_{\rm c}}\right),
\end{equation}
with
\begin{equation}\label{nuc}
\nu_{\rm c}= {3 \gamma_{\rm e}^2 e\over 4\pi m_{\rm e} c} B \sin{\alpha_{\rm e}}.
\end{equation}
$F(x)$ is a function with asymptotic limit
$F(x)\sim{4\pi\over\sqrt{3}\Gamma(1/3)}\left({x\over2}\right)^{1/3}$ for
$x\ll1$  which has a maximum at 
\begin{equation}\label{numax}
\nu_{\rm max}=0.29 \nu_{\rm c}.
\end{equation}
 $K_{5/3}(x)$ is the Bessel K function which can be 
Taylor expanded into $K_{5/3}(x)=1.43 x^{-5/3}$ for $x<<1$. For the
pitch angle we can take an average value: 

\begin{equation}
\left<\alpha_{\rm e}\right>=\arcsin{\left({\int_0^{\pi/2}\sin{\alpha}\sin{\alpha}\;d{\alpha}\over\int_0^{\pi/2}\sin{\alpha}\;d{\alpha}}\right)}\simeq52^\circ.
\end{equation}

Using the asymptotic behavior and average pitch angle, we can
evaluate emission and absorption coefficients and obtain handy
approximate formulae:

%<<Math/synchrotron-mono.m
%cnv[n Chop[epsilonsyncelow[nu]/(erg/sec/Hz/cm^3)] //. {alphae->52 Degree,B->b Gauss, gammae->g100 100,n->k B^2/(8 Pi gammae me c^2),nu->nu9 GHz}]

\begin{equation}\label{emcoeff}
\epsilon_\nu=6.0\cdot10^{-20}\,{\rm erg\,s^{-1}\,Hz^{-1} cm^{-3}}\; k\left({B\over {\rm Gauss}}\right)^{8/3}\left({\gamma_{\rm e}\over100}\right)^{-5/3}\left({\nu\over{\rm GHz}}\right)^{1/3}
\end{equation}
and


%<<Math/synchrotron-mono.m
%cnv[n Chop[sigmasyncelow[nu]] //. {alphae->52 Degree,B->b Gauss, gammae->g100 100,n->k B^2/(8 Pi gammae me c^2),nu->nu9 GHz}]

\begin{equation}\label{abscoeff}
\alpha_\nu=3.5\,10^{-14}\,{\rm cm}^{-1}\; k\left({B\over {\rm Gauss}}\right)^{8/3}\left({\gamma_{\rm e}\over100}\right)^{-8/3}\left({\nu\over{\rm GHz}}\right)^{-5/3}.
\end{equation}


The synchrotron spectrum of a mono-energetic electron distribution will
have a characteristic shape consisting of three parts: a) an optically
thick spectrum with $S_\nu\propto\nu^2$ at frequencies $\nu\ll\nu_{\rm
ssa}$ below the self-absorption frequency, b) an optically thin
spectrum with $S_\nu\propto\nu^{1/3}$ at intermediate frequencies
$\nu_{\rm ssa}<\nu<\nu_{\rm c}$, and c) an exponential high-frequency
cut-off beyond $\nu\gg\nu_{\rm c}$. In most realistic cases for jets
the intermediate region will not assume the $\nu^{1/3}$-law, since
$\nu_{\rm c}$ and $\nu_{\rm ssa}$ are close together leading to a
curved spectrum.

The self-absorption frequency can be calculated from
Eq.~{\ref{abscoeff}} and Eq.~\ref{bfield}, by requiring that the
optical depth through the jet, seen under an angle of $\theta\gg{\cal
M}^{-1}$ from the jet axis, is unity:
\begin{equation}
\tau\sin^{-1}{\theta}R\alpha_\nu=1.
\end{equation}
We find

%Solve[PowerExpand[cnv[R/Sin[theta] n Chop[sigmasyncelow[nu]] //. {alphae->52 Degree,B->b Gauss, gammae->g100 100,n->k B^2/(8 Pi gammae me c^2),nu->nu9 GHz,b->Sqrt[8 Lb/(gammaj betaj c R^2)]/Gauss,Lb->L3 1000 Lsol,R->rau AU}]^(3/5)]==1,nu9]

\begin{equation}
\nu_{\rm ssa}=2.3\,{\rm GHz}\; k^{3/5}\sin^{-3/5}\theta\left({\gamma_{\rm j}\beta_{\rm j}}\right)^{-4/5}\left({\gamma_{\rm e}\over100}\right)^{-8/5}\left({L_{\rm B}\over 1000 L_{\odot}}\right)^{4/5}\left({R\over{\rm AU}}\right)^{-1}.
\end{equation}

The maximum flux of synchrotron emission is found at a frequency of
$\nu_{\rm max}=0.29\nu_{\rm c}$ (Eq.~\ref{numax}; see also
\citeNP{RybickiLightman1979}, Fig.~6.6). Using Eqs.~\ref{nuc} \& \ref{bfield} we find

%cnv[0.29 nuc/GHz //. {alphae->52 Degree,B->b Gauss, gammae->g100 100,n->k B^2/(8 Pi gammae me c^2),nu->nu9 GHz,b->Sqrt[8 Lb/(gammaj betaj c R^2)]/Gauss,Lb->L3 10^3 Lsol,R->rau AU}]


\begin{equation}\label{nucofr}
\nu_{\rm max}=21\,{\rm GHz}\; \left({\gamma_{\rm j}\beta_{\rm j}}\right)^{-1/2}\left({\gamma_{\rm e}\over100}\right)^{2}\left({L_{\rm B}\over 1000 L_{\odot}}\right)^{1/2}\left({R\over{\rm AU}}\right)^{-1}.
\end{equation}

As one can see, and as hinted at above, both frequencies are within an
order of magnitude for near-equipartition situations and both scale
with $R^{-1}=\left(Z/{\cal M}\right)^{-1}$ (Eq.~\ref{machnumber}).


Since the synchrotron spectrum peaks near these frequencies one also
sees that for a given observing frequency the maximum of emission in
the spatial domain will be at one characteristic zone in the jet. A
different observing frequency will reveal a different maximum. This
effect produces a core shift that is well known in quasar
jets. Since the size of the emitting region, $\Delta R\simeq Z/{\cal
M}$ will be proportional to the distance one also expects a different
core-size for different frequencies:

\begin{equation}\label{zofnu}
Z_{\rm max}=21\,{\rm AU}\; {\cal M}\left({\gamma_{\rm j}\beta_{\rm j}}\right)^{-1/2}\left({\gamma_{\rm e}\over100}\right)^{2}\left({L_{\rm B}\over 1000 L_{\odot}}\right)^{1/2}\left({\nu\over{\rm GHz}}\right)^{-1}.
\end{equation}

The effect of a roughly $\nu^{-1}$ core size was also nicely
demonstrated by \citeN{BietenholzBartelRupen2000} for M81*. For Sgr
A* this effect is not visible at cm-waves due to the
frequency-dependence of the scatter broadening.

At the Galactic Center distance of $D=8$ kpc, 1 AU corresponds to 0.125
mas.  For comparison, the claimed size for Sgr A* at 43 GHz by
\citeN{LoShenZhao1998} was $\la$0.7 mas. Equation \ref{zofnu} predicts 
a shift of order 0.25~mas$\cdot\left({\cal M}/4\right)\left(L_{\rm
B}/1000L_\odot\right)$ at 43 GHz.

Another useful comparison is the maximum frequency of the entire
spectrum. For the smallest size of the system, one Schwarzschild radius
$R_{\rm s}=0.9\cdot10^{12}$ cm = 0.06 AU, we find a maximum frequency
of $\sim$350 GHz from Eq.~\ref{nucofr}. Hence it is immediately
understandable, why the Sgr A* spectrum has to turn over at higher
frequencies, beyond the submm-bump. In this respect the location of
the submm-bump is a rough indicator of the size of the black hole in
the Galactic Center. 

Finally, we can calculate the total spectrum. We know that each
frequency is dominated by a relatively small spatial region $Z(\nu)$
in the jet at the scale given by equation \ref{zofnu}. The volume of
this region can be approximated by a cylinder, $V=\pi R^2 Z$, where
$R$ is given by Eq.~\ref{machnumber}. This volume has to be multiplied
by the emission coefficient (Eq.~\ref{emcoeff}) with the magnetic
field (Eq.~\ref{bfield}) inserted. Divided by the surface of an
imaginary absorbing sphere at the observers distance $D$, we get the
flux density of the jet as a function of frequency:

%cnv[ (Pi R^2 Z n epsilonsyncelow[nuc])/(4 Pi (8 kpc)^2)/Jy //. {alphae->52 Degree,B->b Gauss, gammae->g100 100,n->k B^2/(8 Pi gammae me c^2),nu->nu9 GHz, Z->R Mach,Mach->4,b->Sqrt[8 Lb/(gammaj betaj c R^2)]/Gauss,Lb->L3 1000 Lsol,R->rau AU}]


\begin{eqnarray}
S_\nu&=&\epsilon_{\nu}(\nu_{\rm c}){\pi R^2 Z\over 4\pi D^2}\nonumber\\
&=&1.0 {\rm Jy}\,k{{\cal M}\over4} \left({\gamma_{\rm e}\over100}\right)^{-1}\left({\beta_{\rm j}\gamma_{\rm j}}\right)^{-3/2} \left({L_{\rm B}\over 1000 L_{\odot}}\right)^{3/2}.
\end{eqnarray}

As we can see, the frequency cancels out thus implying a perfectly
flat spectrum ($S\nu\propto\nu^0$). This is a fairly general result
that applies to essentially all flat-spectrum radio cores in black
hole jets. A schematic view of how the flat spectrum arises is also
shown below in Figure~\ref{schematic}.

The equations have all been normalized by a magnetic luminosity of
$L_{\rm B}\simeq1000 L_{\odot}$. Of course, the jet has also other
energetic components (e.g., turbulence or cold protons), however, for
a maximally efficient jet they will be of similar order
\cite{FalckeBiermann1995}.

The total energy content of the jet will therefore be a few times
higher, i.e.~of order $10^{37}$ erg sec$^{-1}$. Assuming the efficiency
for the jet production is of order of $0.1\dot M c^2$
(e.g., \citeNP{FalckeMalkanBiermann1995}) this would require an
accretion rate of at least $\ga 2\cdot10^{-9}M_\odot$ yr$^{-1}$ onto
Sgr A*.  This is quite in the range of -- and sometimes well below --
the accretion rates discussed for Sgr A*.

In our derivation we have so far completely ignored relativistic effects on
the emitted spectrum. For a continuous relativistic jet where
$\beta_{\rm j}\simeq1$ and $\gamma_{\rm j}\gg1$, relativistic beaming
will lead to a modification of the observed spectrum
\cite{LindBlandford1985} by

\begin{equation}
S^\prime_{\rm \nu}={\cal D}^2 S_{\rm \nu}\quad\mbox{\&}\quad\nu^\prime={\cal D}\nu,
\end{equation}
where one defines the relativistic Doppler factor as

\begin{equation}
{\cal D}={1\over\gamma_{\rm j}\left(1-\beta_{\rm j}\cos\theta\right)}.
\end{equation}

For moderate inclination angles and moderate jet velocities the effect
will be of order unity and we have neglected this for the sake of
clarity.  From this simple exercise we can conclude that the basic
properties of Sgr A*, spectrum, flux, and size, can be naturally
reproduced by a jet model.

For a more realistic model, one has to take several additional effects
into account. For example, we have here assumed a constant velocity of
the jet. This is somewhat inconsistent, since the jet as presented
here has a pressure gradient that will naturally lead to some mild
acceleration of the jet plasma along the jet axis. This effect
together with the relativistic corrections will lead to a slightly
inverted radio spectrum and a slightly smaller exponent for the
size-frequency relation
\cite{Falcke1996a}. 

In addition we have not yet yet dealt with the submm-bump in the
spectrum. From Eqs.~\ref{nucofr} \& \ref{zofnu} it is clear that this
emission has to come from the innermost region of the jet (the
``nozzle''; \citeNP{Falcke1996b}) or the inner edge of the accretion
flow \cite{MeliaLiuCoker2000,NarayanMahadevanGrindlay1998}. A simple descriptive calculation to estimate the parameters of this
region, which follows essentially the procedure outline here, is given
in (\citeNP{MeliaFalcke2001}, Sec.~5.2).

All these effects including the X-ray spectrum are dealt with in more
sophisticated numerical calculation which are are discussed in the
next sections.

\subsection{The X-ray Spectrum}
After having calculated the radio spectrum, we can now make a rough
estimate of the expected X-ray spectrum from Sgr A*. For AGN, there
are typically four processes discussed to explain the observed X-ray
emission in various objects: a) synchrotron emission, b)
Bremsstrahlung, c) thermal Comptonization by a hot corona, d) inverse
Compton scattering of photon off the relativistic electrons in the jet
plasma. Possibility a) can be excluded here since the radio
synchrotron spectrum cuts off already in the mid-infrared, b) and c)
have been discussed by Coker (this volume). Thus, we will here
concentrate on the fourth possibility.

Since the only photons we see from Sgr A* outside the X-ray regime are
radio photons, we will here consider solely the synchrotron
self-Compton (SSC) process, which is absolutely unavoidable. The
relativistic electrons that produce synchrotron radiation also have a
finite probability to Compton up-scatter the very photons they have
produced in the first place. The frequency of the up-scattered photons
will be increased by a factor $\simeq\gamma^2$ with respect to the
target photons. Inverse-Compton is a scattering process, where the
probability of an interaction of an electron from a population with
particle density $n_{\rm e}$ with a photon of a population with photon
density $n_{\gamma}$ depends on $n_{\rm e}\cdot n_{\gamma}$. Since in
SSC the electrons are also responsible for the target photons, the
efficiency of SSC will go as $n_{\rm e}^2$. For the case of a jet,
where the density increases inwards with $R^{-2}$, while the volume
decreases inwards with $R^3$, the dominant contribution to the
up-scattered spectrum will be at the smallest scale in the system,
where $n_{\rm e}$ is maximal. Following the previous discussion, this
will be at a few Schwarzschild radii where the submm-bump in the
spectrum is produced.


One can show (\citeNP{RybickiLightman1979}, Sec. 7.2) that the
luminosity $L_{\rm SSC}$ of the inverse-Compton process is proportional
to the luminosity of the synchrotron emission $L_{\rm sync}$, with the
proportionality factor given by the ratio of the energy densities in
synchrotron photons 
\begin{equation}
U_{\rm ph}={L_{\rm sync}\over 4 \pi R^2 c}
\end{equation}
and magnetic field 
\begin{equation}
U_{\rm B}=B^2/8\pi,
\end{equation}
such that

\begin{equation}\label{ic}
L_{\rm SSC}= {U_{\rm ph}\over U_{\rm B}} L_{\rm sync}.
\end{equation}

From Fig.~\ref{sgrspec} we find that the maximum of emission in Sgr~A*
is about 3 Jy at $10^{12}$ Hz. Hence the synchrotron luminosity of Sgr
A* is

%cnv[(3 Jy 1000 GHz 4 Pi (d8 8 kpc)^2)/(erg/sec)]

\begin{equation}
L_{\rm sync}=2.3\cdot10^{35}\,{\rm {erg\over s}}\;\left({S_\nu\over3 {\rm Jy}}\right)\left({D\over 8{\rm kpc}}\right)^2 \left({\nu_{\rm max}\over10^{12}{\rm Hz}}\right)
\end{equation}
and from Eqs.~\ref{bfield} and \ref{ic} we get, independent of the radius, 

\begin{equation}
L_{\rm SSC}=3.4\cdot10^{33}\,{\rm {erg\over s}}\;\left({\beta_{\rm j}\gamma_{\rm j}}\right) \left({S_\nu\over3\, {\rm Jy}}\right)^2 \left({D\over 8{\rm kpc}}\right)^4\left({\nu_{\rm max}\over10^{12}{\rm Hz}}\right)^2 \left({L_{\rm B}\over 1000 L_{\odot}}\right)^{-1}.
\end{equation}

%cnv[(S3 3 Jy nu12 1000 GHz 4 Pi (d8 8 kpc)^2)^2/(4 Pi R^2 c)/((B)^2/8/Pi)/(erg/sec) //. {B->Sqrt[8 Lb/(gammaj betaj c R^2)], Lb->L3 1000 Lsol}]

As one can see the SSC emission is sensitive to the ratio between
synchrotron emission and magnetic field. Using a different
parameterization this gives one a dependence on the equipartition
factor $k$. In general one can also state that the SSC emission should
be more variable than the synchrotron emission since it depends with a
high power on flux density and peak synchrotron frequency.

The peak of the SSC emission itself will be roughly at
$\gamma^2\cdot\nu_{\rm max}$. For $\gamma_{\rm e}\simeq100$ and
$\nu_{\rm max}\simeq10^{12}$ Hz the peak will be above $10^{16}$ eV,
hence in the far ultraviolet and soft X-rays. 

All of this is quite consistent with the X-ray observations by
\citeN{BaganoffBautzBrandt2001a} who find a quiescent, soft X-ray
emission of a few times $10^{33}$ erg sec$^{-1}$ in Sgr A* which can
be rapidly variable at times.

A schematic view of how the spectrum of Sgr A* from radio to X-rays
can be composed from the various parts discussed here is shown in
Figure~\ref{schematic}.
\begin{figure}
\psfig{file=schematic.ps,width=\textwidth,bbllx=2.3cm,bburx=16.5cm,bbury=26.45cm,bblly=8.7cm}
\caption{\label{schematic} This is a schematic of the jet model. 
The accretion disk/flow is assumed to be radiatively unimportant. The
submm-bump and the X-ray emission are produced by the jet nozzle
region. The flat part of the radio spectrum is the sum of individual,
peaked synchrotron spectra from increasingly distant zones in the
jet. The peak frequency of each of these spectra shifts to lower
frequencies as one goes out, while the peak flux density stays
essentially at the same level or decreases only slowly. Figure
from S. Markoff, based on Falcke \& Markoff (2000).}
\end{figure}
\nocite{FalckeMarkoff2000}

\subsection{Numerical Results}
After we have verified, that the basic properties of Sgr A* can be
explained with a synchrotron+SSC jet model, we can consider a more
sophisticated numerical approach. This has been outlined in
\citeN{FalckeMarkoff2000}, \citeN{MarkoffFalckeYuan2001}, and
\citeN{YuanMarkoffFalcke2002}.

We start with the basic jet emission model
\cite{FalckeBiermann1999,FalckeMarkoff2000}, consisting of a conical jet with
pressure gradient, nozzle and relativistic effects. The parameters in
the nozzle for the quiescent state are determined from the underlying
accretion disk, assumed to be an ADAF, as described in
\citeN{YuanMarkoffFalcke2002}.  All quantities further out in the jet
are solved for using conservation of mass and energy, and the Euler
equation for the accelerating velocity field. The results are shown in
Figs.~\ref{sgradafspec} \& \ref{jetsize} and show that the model is
able to reproduce the observed spectrum and size in detail.


\begin{figure}
\psfig{file=sgr-jdafspec.ps,width=\textwidth,angle=270}
\caption{\label{sgradafspec}The jet-disk spectral model for Sgr A$^*$.
The dotted line is for the ADAF (optically thin, advection dominated
accretion flow) contribution, the dashed line is for the jet emission,
and the solid line shows their sum. For the most part, the emission is
dominated by the jet-spectrum. The submm-bump is produced by the jet
nozzle with a possible contribution from the accretion flow. The X-ray
emission is largely SSC emission from the nozzle with a slight
contribution from more extended thermal X-ray emission from the
accretion flow. We have here assumed an accretion rate of
$10^{-6}M_\odot$ yr$^{-1}$, where only 0.1\% of the power goes into
the jet. For a 10\% efficiency the required accretion rate is about
$10^{-8}M_\odot$ yr$^{-1}$ and the disk contribution would be
negligible. The model is discussed in more detail in Yuan et
al. (2002).}
\end{figure}
\nocite{YuanMarkoffFalcke2002}




\begin{figure}
\centerline{\psfig{figure=sgr-jetsize.ps,width=\textwidth,angle=-90}}
\caption[]{\label{jetsize}Projected size of the major and minor axis
of the jet in Sgr A* as a function of frequency. The filled dots mark
the size as measured by Lo et al.~(1998; 43 GHz) and Krichbaum et
al.~(1998; 86 \& 215 GHz). The lines represent the predictions of the
jet model. At frequencies above 30 GHz one obtains a two component
structure with an increasingly stronger core (nozzle, solid dashed
line) and a fainter jet component (dotted line).}
\end{figure}
\nocite{LoShenZhao1998,KrichbaumGrahamWitzel1998}


\subsection{The Circular Polarization}\label{hfcptheo}
Finally, for understanding the radio properties we also have to
consider the surprising results of the polarization observations,
where a relatively large circular polarization (CP) was found. The
following intuitive explanation is essentially a discussion of CP
based on the paper by \citeN{BeckertFalcke2002}, where one can find
more details. The main point is that linear polarization is naturally
obtained in synchrotron radiation (up to 70\% for homogeneous magnetic
fields), while CP is strongly suppressed in synchrotron plasmas with
$\gamma_{\rm e}\gg1$. The reason for this is that the narrow beaming
cone of the relativistic electron allows one to see only a small arc
along the gyration around the magnetic field. However, CP can be
obtained through radiation transfer, particularly through the fact
that a magnetic plasma will naturally be bi-refringent.

For simplicity let us now separate Faraday rotation from conversion
and only picture purely linearly or circularly polarized waves in a
homogeneous magnetic field.

%------------------------------------------------------------------
\begin{figure}
    \centerline{\psfig{figure=h3363f1.eps,width=0.49\textwidth}\hfill\psfig{figure=h3363f2.eps,width=0.49\textwidth}}
    \caption{\label{CP}{\it Left:} A circularly polarized wave can be composed of two
    orthogonal linearly polarized modes shifted in phase. A phase
    shift would be produced by a plasma in a magnetic field
    perpendicular to the propagation direction of the waves (here
    along the $z$-direction). Without phase-shift the sum of the two
    modes would be a purely linearly polarized wave.  The accompanying
    movie shows the effect of how phase-shifts in a region
    will turn such a linearly polarized wave into a circularly
    polarized wave (conversion).\hfill\break
{\it Right:} A linearly polarized wave can be composed of two
    orthogonal circularly polarized modes shifted in phase. A phase
    shift would be produced by a plasma in a magnetic field along the
    propagation direction of the waves (here along the
    $y$-direction). The accompanying movie shows the effect of
    additional phase-shifts on the linear polarization, leading to
    Faraday rotation.\hfill\break
    The movies can be found at http://www.mpifr-bonn.mpg.de/staff/hfalcke/CP.
}

\end{figure}
%------------------------------------------------------------------

The two orthogonal normal modes for propagation perpendicular to the
magnetic field are linearly polarized and a purely circularly
polarized wave is split into the two normal modes with a relative
phase shift as shown in Fig.~\ref{CP} (left). Without a phase-shift the wave
will be purely linearly polarized.  If, for example, a locally
homogeneous magnetic field vertically pervades the box in
Fig.~\ref{CP} (left) along the $z$-direction, electrons or positrons
will be free to move along the field lines and resonate with the
vertical mode but hardly resonate with the horizontal mode along the
$x$-direction. This yields the bi-refringence discussed above. The
resonating electrons or positrons will themselves act as antennas and
emit a somewhat delayed wave that interferes with the incoming
vertical mode, leading to a slight phase-shift between vertical and
horizontal mode. The effect of this shift is shown in the accompanying
animation\footnote{http://www.mpifr-bonn.mpg.de/staff/hfalcke/CP},
where the resulting wave is circularly polarized and switches from
linear to circular polarization as a function of the shift.

Conversion acts also on initially only linearly polarized radiation.
The amount of this conversion will depend on the misalignment between
the incoming wave and the magnetic field direction since, obviously, a
phase-shift between two orthogonal modes will have little effect if
one mode is very small or non-existent. Moreover, a random
distribution of magnetic field lines on the plane of the sky will
reduce circular polarization from conversion in exactly the same way
as linear polarization would be reduced.


Analogous to the picture for conversion, one can view a linearly
polarized wave as composed of two circularly polarized normal modes when
propagating along the magnetic field. This is
sketched in Fig.~\ref{CP} (right), where we will assume a
longitudinal magnetic field, i.e. a field along the $y$-direction. The
circular modes will resonate with either electrons or positrons
gyrating around the magnetic fields. The latter will again emit a
circularly polarized wave, producing a phase-shift when interfering
with the incoming wave. The effect of the phase-shift in the circular
modes is shown in the accompanying animation of
Fig.~\ref{CP} (right), where one can see that the resulting
linearly polarized wave is simply (Faraday) rotated.

An important conclusion to remember therefore is, that conversion is
mainly produced by magnetic field components perpendicular to the
line-of-sight or photon direction, while Faraday rotation is produced
by magnetic field components along the line-of-sight. Moreover, one
can also see that conversion is insensitive to the electron/positron
ratio while Faraday rotation is not. In Fig.~\ref{CP} (left) an
electron and an positron are both free to move along the
$z$-axis. While they will respond in opposite directions to the
incoming wave, their respective emitted waves will also have opposite
signs because of opposite charges and hence be identical. In the case
of Faraday rotation, the incoming left- or right-handed circularly
polarized wave will only resonate with the particle that also has the
correct handedness in its gyration -- either electron or positron
depending on the magnetic field polarity. A pure pair plasma would
therefore produce exactly the same phase shift in left- and
right-handed modes and not produce any net Faraday rotation. In the
case of a charge-excess, the direction of Faraday rotation depends on
the sign of the charge-excess (presumably electrons) and the polarity
of the magnetic field. This will indirectly also affect the sign of
circular polarization, if Faraday rotation is the ultimate cause of
the misalignment between the plane of polarization and the magnetic
field direction.

One can include these effects on the polarization into a radiation
transfer code and try to reproduce the Sgr A* spectrum and
polarization with a jet/outflow model \cite{BeckertFalcke2002}. The
results are shown in Fig.~\ref{CPtheo} and nicely reproduce the
observed spectrum. Two major conclusion can be drawn from this
approach:

a) Since conversion is most effective for low-energy electrons one can
conclude that a larger number of these low-energy electrons
($1<\gamma_{\rm e}\ll 100$) are present in Sgr A*. In fact, in the
specific modeling mentioned here, one finds that up to a factor of 100
more low-energy electrons could be present then the 'hot' electrons
with $\gamma_{\rm e}\simeq100$ invoked above to explain the
spectrum. Hence, a large fraction of Sgr A*'s plasma could reside in a
rather inconspicuous, ``cold'' and non-radiating (hidden) plasma.

b) Since the conversion requires an asymmetric magnetic field
polarity, an outflow model with a helical magnetic field is strongly
favored. The stability of the sign of CP also suggests that the
polarity (the North pole of the black hole/jet) remains stable over
some 20 years --- a long time scale compared to the accretion time in
optically thin accretion flows. It is possible that this stable
polarity reflects the stable polarity of the Galactic magnetic field
pervading the central parsec of the Galaxy which is accreted onto
the black hole.

\begin{figure}
    \centerline{\psfig{figure=h3363f3.eps,width=\textwidth}}
    \caption{Outflow model for the radio spectrum of Sgr A$^*$ with
    polarization.  The result of model calculations for total flux $I$
    (solid line), linearly polarized $P$ (dense shaded area), and
    circularly polarized flux $V$ (sparse area) are shown for a
    distance of $8$ kpc.  Diamonds show the observed circular
    polarization and circles the observed linear polarization, the
    rest are observed total flux density values.  The numerical
    calculations are based on the model described in Beckert \& Falcke
    (2002). The shaded areas mark the expected variability due to
    turbulence. The global magnetic field structure is a spiral with
    $B_\phi/B_z = 1$.}  \label{CPtheo}
\end{figure}
%-------------------------------------------------------------------------
\nocite{BeckertFalcke2002}

The main uncertainty in the modeling of the polarization at present
reflects the uncertainty in what suppresses the linear polarization in
Sgr A*. Here, we have simply assumed that the de-polarization of the
linear polarization is due to intrinsic Faraday de-polarization in the
radio source itself. However, this could similarly be done in a
``foreground'' screen, which would most likely be associated with the
accretion flow. If that is the case, some of the ``hidden matter''
mentioned above would be in the actual accretion flow and not in the
outflow/jet. This can be used to constrain the accretion
rate. Estimates by \citeN{Agol2000} and
\citeN{QuataertGruzinov2000b} then yield a limit on the accretion flow
of $\dot M\la10^{-8}$ to $10^{-9} M_\odot$ yr$^{-1}$, given the
rotation measures inferred from the linear polarization observations
at $\lambda1$mm.

\subsection{Comparison with Other Supermassive Black Holes}
An important input factor to the model discussed so far is the power
of the jet (here we mainly considered the magnetic power $L_{\rm
B}$). A nice feature of the model is that it can be scaled over many
orders of magnitude by just changing the power input of the jet. This
is presumably done by a parallel change in the accretion disk
power. Doing so would change the power of the radio core but would
not change its spectral shape --- only the turnover frequency might
change. To a radio observer the jet would always appear as a
flat-spectrum core. This may be the reason why radio jets are expected
in almost every type of active black holes: from supermassive to
stellar, from powerful to faint.

Indeed, compact, flat-spectrum radio cores have been found in sources
like Quasars, Seyfert galaxies, Low-Luminosity AGN and LINERs, as well
as in X-ray binaries, confirming that the physics we have discussed
for Sgr A* are fairly universal. The exact nature of the cores and the
emission of these other engines is not the main focus of the book and
further discussion of this point can be found in
\citeN{Falcke2001}. However, the general point one can make is that
as the accretion power and the disk luminosity decreases, one expects
to see fainter radio cores. If there is a range in accretion rates
throughout the universe, we also expect a range of core
luminosities. This is demonstrated in Fig.~\ref{theplot}, where Sgr
A*-like radio cores of various different types are shown for a range
of luminosities. In such a plot, Sgr A* would come in a the bottom
left of the distribution for an accretion rate of
$\simeq10^{-8}M_\odot$ yr$^{-1}$ (as indicated by the upper black dot
with horizontal error bar; the other point indicates the estimated
position for M31* --- the core in the Andromeda galaxy). However,
since the accretion disk in Sgr A* is so faint and the accretion rate
so uncertain, we cannot actually derive an accretion disk luminosity
or accretion power and this should only be considered as a general
guideline. The bottom line is, however, that Sgr A* with its radio
properties is not alone in the universe but is at the bottom end of
the activity scale seen from supermassive black holes.

\begin{figure}
    \centerline{\psfig{figure=theplot-all.ps,width=\textwidth,bbllx=3.4cm,bblly=17cm,bburx=13.7cm,bbury=27cm,clip=}}
\caption{Correlation between thermal emission from the accretion
disk (with the exception of X-ray binaries this is basically
normalized to the narrow H$\alpha$ emission) and the monochromatic
luminosity of black hole radio cores. Open circles: Radio-loud
quasars; small open circles: FR\,I radio galaxies; open gray
circles: Blazars and radio-intermediate quasars (dark grey); black
dots: radio-quiet quasars and Seyferts; small dots: X-ray
binaries; small boxes: detected sources from the ``48 LINERs''
sample (Nagar et al.~2000). The latter apparently confirm the
basic prediction of Falcke \& Biermann (1996) and almost close the
gap between very low (on an absolute scale) accretion rate objects
and high accretion rate objects.  The shaded bands represent the
radio-loud and radio-quiet jet models as a function of accretion
as shown in Falcke \& Biermann (1996).}  \label{theplot}
\end{figure}
\nocite{FalckeBiermann1996,NagarFalckeWilson2000}

\section{Imaging the Event Horizon - An Outlook} \label{imevent}
One can easily see from the previous sections that the ever growing
interest in Sgr A* has already yielded a number of tantalizing
results, the most important being that Sgr A* is the best supermassive
black hole candidate we know. VLBI observations are already
approaching scales which are not far from the actual scale of the
black hole and the presence of the submm-bump indicates that even more
compact emission is present at yet smaller scales -- possibly as close
in as the event horizon of the black hole. It is therefore worth
exploring whether we have in principle a chance to actually approach
this scale with imaging techniques and to ask what we would expect to
see. This naturally will have to be done at the highest radio
frequencies where the resolution is the highest and the
scatter-broadening of Sgr A* is the lowest.

At submm wavelengths, the various models indeed predict that the
synchrotron emission of Sgr A* is not self-absorbed, allowing a view
into the region near the event horizon. The size of this event horizon
is $(1+\sqrt{1-a_*^2})R_g$, where $R_g\equiv GM/c^2$, $M$ is the mass
of the black hole, $G$ is Newton's constant, $c$ the speed of light,
$a_*\equiv Jc/(GM^2)$ is the dimensionless spin of the black hole in
the range 0 to 1, and $J$ is the angular momentum of the black hole.


The appearance of the emitting region around a black hole was
determined by \citeN{FalckeMeliaAgol2000} -- from which we take the
following discussion -- under the condition that it is optically thin.
For Sgr A* this might be the case for the submm-bump
\cite{FalckeGossMatsuo1998} indicated by the turnover in the spectrum,
and can always be achieved by going to a suitably high frequency.  For
the qualitative discussion the emissivity was assumed to be frequency
independent and to be either spatially uniform or to scale as
$r^{-2}$.  These two cases cover a large range of conditions expected
under several reasonable scenarios, be it a quasi-spherical infall, a
rotating thick disk, or the base of an outflow.  The calculations took
into account all the well-known relativistic effects, e.g., frame
dragging, gravitational redshift, light bending, and Doppler boosting.

For a planar emitting source behind a black hole, a closed curve on
the sky plane divides a region where geodesics intersect the horizon
from a region whose geodesics miss the horizon
\cite{Bardeen1973}\footnote{For the following discussion see also 
Chapter \ref{Frolov}}.  This curve, which is referred to as
the ``apparent boundary'' of the black hole, is a circle of radius
$\sqrt{27} R_g$ in the Schwarzschild case ($a_*=0$), but has a more
flattened shape of similar size for a Kerr black hole, slightly
dependent on inclination.  The size of the apparent boundary is much
larger than the event horizon due to strong bending of light by the
black hole.  When the emission occurs in an optically thin region {\em
surrounding} the black hole, the case of interest here, the apparent
boundary has the same exact shape since the properties of the
geodesics are independent of where the sources are located.  However,
photons on geodesics located within the apparent boundary that can
still escape to the observer experience strong gravitational redshift
and a shorter total path length, leading to a smaller integrated
emissivity, while photons just outside the apparent boundary can orbit
the black hole near the circular photon radius several times, adding
to the observed intensity
\cite{JaroszynskiKurpiewski1997}.  This produces a marked deficit of the
observed intensity inside the apparent boundary {}--{} the ``shadow'' of
the black hole.

We here consider a compact, optically-thin emitting region surrounding
a black hole with spin parameter $a_*=0$ (i.e., a Schwarzschild black
hole) and a maximally spinning Kerr hole with $a_*=0.998$.  In the set
of simulations shown in Fig.~\ref{bhimage},  the viewing angle $i$ was taken to be $45^\circ$ with respect to the spin axis (when it is
present) with two distributions of the gas velocity $v$. The
first has the plasma in free-fall, i.e.,
$v^r=-\sqrt{2r(a^2+r^2)}\Delta/A$ and $\Omega = 2ar/A$, where $v^r$ is
the Boyer-Lindquist radial velocity, $\Omega$ is the orbital
frequency, $\Delta\equiv r^2-2r+a^2$, and
$A\equiv(r^2+a^2)^2-a^2\Delta\sin^2{\theta}$. (We have set $G=M=c=1$
in this paragraph.)  The second has the plasma orbiting in rigidly
rotating shells with the equatorial Keplerian frequency $\Omega =
1/(r^{3/2}+a)$ for $r>r_{ms}$ with $v^r=0$, and infalling with
constant angular momentum inside $r<r_{ms}$
\cite{Cunningham1975a}, with $v^\theta=0$ for all $r$.


In order to display concrete examples of how realistic the proposed
measurements of these effects with VLBI will be, the expected images
were simulated for the massive black hole candidate Sgr A* at the
Galactic Center.

\begin{figure}[h]
\centerline{\psfig{figure=bhimage.cps,width=\textwidth,bblly=11.2cm,bbury=21.5cm,bbllx=0.8cm,bburx=18.2cm}}
\caption{\label{bhimage}
An image of an optically thin emission region surrounding a black hole
with the characteristics of Sgr A* at the Galactic Center.  The black
hole is here either maximally rotating ($a_* = 0.998 $, panels a-c) or
non-rotating ($a_*=0$, panels d-f). The emitting gas is assumed to be
in free fall with an emissivity $\propto r^{-2}$ (top) or on Keplerian
shells (bottom) with a uniform emissivity (viewing angle
$i=45^\circ$). Panels a\&d show the GR ray-tracing calculations,
panels b\&e are the images seen by an idealized VLBI array at 0.6 mm
wavelength taking interstellar scattering into account, and panels
c\&f are those for a wavelength of 1.3 mm. The intensity variations
along the $x$-axis (solid green curve) and the $y$-axis (dashed
purple/blue curve) are overlayed. The vertical axes show the intensity
of the curves in arbitrary units and the horizontal axes show the
distance from the black hole in units of $GM_\bullet/c^2$ ($1/2R_{\rm
s}$).}
\end{figure}

The results of the two different models with and without scattering at
two different observing wavelengths are shown in Fig.~\ref{bhimage}.
The two distinct features that are evident in the top panel for a
rotating black hole are (1) the clear depression in $I_\nu$ {}--{} the
shadow {}--{} produced near the black hole, which in this particular
example represents a modulation of up to 90\% in intensity from peak
to trough, and (2) the size of the shadow, which here is $9.2R_{\rm
g}$ in diameter.  This represents a projected size of 34
$\mu$arcseconds. Such a resolution has already been surpassed in some
$\lambda$2mm-VLBI experiments of other radio cores
\cite{KrichbaumGrahamAlef2002}.  The shadow is a generic feature of
various other models one can look at, including those with outflows,
cylindrical emissivity, and various inclinations or spins.

This black hole shadow is also visible in the second illustrated case
for a non-rotating black hole with a modulation in $I_\nu$ in the
range of 50-75\% from peak to trough, and with a diameter of roughly
$10.4\,R_g$.  In this case, the emission is asymmetric due to the
strong Doppler shifts associated with the emission by a rapidly moving
plasma along the line-of-sight (with velocity $v_\phi$).


The important conclusion is that the diameter of the shadow {}--{} in
marked contrast to the event horizon {}--{} is fairly independent of
the black hole spin and is always of order 5$R_{\rm s}$.  The
presence of a rotating hole viewed edge-on will lead to a shifting of
the apparent boundary (by as much as 2.5 $R_g$, or 9 $\mu$arcseconds)
with respect to the center of mass, or the centroid of the outer
emission region.  Other possible signature of general relativistic
effects may come from the polarization properties of the submm-wave
emission region. This has been calculated by
\citeN{BromleyMeliaLiu2001}.

The importance of the proposed imaging of Sgr A* at submm wavelengths
with VLBI cannot be overemphasized.  The submm-bump in the spectrum of
Sgr A* strongly suggests the presence of a compact component whose
proximity to the event horizon is predicted to result in a shadow of
measurable dimensions in the intensity map. Such a feature seems
unique and Sgr~A* seems to have all the right parameters to make it
observable.  The observation of this shadow would confirm the widely
held belief that most of the dark mass concentration in the nuclei of
galaxies such as ours is contained within a black hole, and it would
be the first direct evidence for the existence of an event horizon
largely independent of any modeling. {\it It would finally turn the
theoretical concept of an event horizon discussed in the beginning of
the book into an observable reality.}

A non-detection with sufficiently developed techniques, on the other
hand, might pose a major problem for the standard black hole
paradigm. Because of this fundamental importance, the experiment
proposed here should be a major motivation for intensifying the
current development of submm astronomy in general and mm- and
submm-VLBI in particular.

This result also shows the outstanding position Sgr A* has among known
radio cores and supermassive black hole candidates.  For other
supermassive black holes, with the exception perhaps of the very
massive black hole in M87, the shadow will be much smaller than in Sgr
A* because of the much larger distances.


\bibliography{../../../References/hfbib} %\bibliographystyle{unsrt}
\bibliographystyle{apj} 
\end{document}
