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\begin{document}
\title{The Shadow of the Black Hole at the Galactic Center\footnote{To
appear in: ``Cosmic Explosions!'', 10th Annual October Astrophysics
Conference in Maryland, Oct.~1999, eds. S.S. Holt and W.W. Zhang, AIP
Conf.~Proc.}}

\author{Heino Falcke$^*$, Fulvio Melia$^{\dagger\#}$, and Eric Agol$^+$}
\address{$^*$Max-Planck-Institut f\"ur Radioastronomie, Auf dem H\"ugel
69, D-53121, Bonn, Germany\\
$^{\dagger}$Physics Department and Steward
Observatory, The University of Arizona, Tucson, AZ 85721\\
$^+$ Physics and Astronomy Department, Johns Hopkins University,
Baltimore, MD 21218\\
$^\#$ Presidential Young Investigator and Sir Thomas Lyle Fellow\\}


\lefthead{Shadow of Black Hole}
\righthead{Falcke et al.}
\maketitle


\begin{abstract}
 We show that perhaps already with the next generation of
 long-baseline interferometers at submm-wavelengths we will able to
 image the shadow of the black hole in the Galactic Center. To a
 distant observer, the event horizon casts a relatively large
 ``shadow'' with an apparent diameter of $\sim 10$ gravitational radii
 due to bending of light by the black hole, nearly independent of the
 black hole spin or orientation. The predicted angular size for the
 Galactic Center black hole is $\sim 30\,\mu$arcseconds, a mere factor
 two smaller than the highest currently achieved resolution with VLBI
 techniques. Taking into account scatter-broadening of the image in
 the interstellar medium and the finite achievable telescope
 resolution, we show that the shadow of Sgr A* can be observed at
 suitably high frequencies. The main problems are possible optical
 depth effects for an ADAF model and Doppler boosting for a jet
 model. This has an influence on which dynamic range and which
 observing frequency is ultimately required to prove or disprove the
 existence of an event horizon.\end{abstract}

\section*{Introduction}
Many energetic events in the universe are considered to be connected
with black holes, be it powerful radio jets, UV, X-ray, and
$\gamma$-ray emission from quasars, stellar mass black hole
candidates, or even Gamma-ray bursts (e.g.,~Pugliese et al.~1999; see
also this volume). The best evidence for the existence of a
supermassive black hole is found in our Galaxy (Eckart \& Genzel 1996)
where the compact radio source Sgr~A* lies at the dynamical center of
the central stellar cluster (Ghez, et al.~1998; Reid, et al.~1999;
Backer \& Sramek 1999). Sgr~A* is surrounded by thermally radiating
gas streamers (Sgr~A~West) spiraling into the nucleus (Zhao \& Goss
1998) and a non-thermal radio shell, possibly a hypernova resulting
from the tidal disruption of a star by the central black hole
(Khokhlov \& Melia 1996). The nature of the radio emission in Sgr~A*
is not clear; the latter may be due either to an outflow/jet ({Falcke}
et al.~1993; {Falcke} \& {Biermann} 1999) from the black hole or an
inflow/accretion onto the black hole ({Melia} 1992,1994; {Narayan} et
al.~1995).

Very Long Baseline Interferometry (VLBI) observations have confirmed
that the radio emission of Sgr~A* is very compact up to the highest
frequencies (215 GHz; {Krichbaum} {et al.}  1998) and its radio
spectrum at mm- and submm-wavelengths indicates the presence of an
even more compact radio-emitting plasma component ({Falcke} {et al.}
1998; {Serabyn} {et al.}  1997). This is intriguing since the size of
Sgr~A* could be less than 17 Schwarzschild radii (0.11 mas at 215 GHz)
for a black hole mass of $2.6\times10^6M_\odot$ at a distance of 8
kpc. Of all the known black hole candidates, Sgr~A* is the source where
the angular size on the sky of its Schwarzschild radius is the
largest. In fact, the angular resolution of ground-based VLBI
experiments now comes interestingly close to the scale where
significant general relativistic effects are important. We here report
on calculations we have made with a general relativistic ray-tracing
program which aim at clarifying what general relativistic effects might
be realistically measurable in future VLBI experiments.

\section*{Calculations and Results}
We consider an optically thin emission region with
frequency-independent emissivity around a black hole with arbitrary
spin. The intensity and structure of the emission region can be
arbitrary as well, but we choose a number of generic scenarios where
we have either a spherical distribution of the intensity scaling as a
power-law with radius $r$ or a jet-like distribution (hollow
cylinder). We also can allow for various velocity fields, e.g.,~with
rotation, inflow, or outflow.  The appearance of the emission region
for an observer at infinity, taking all general relatvistic effects
into account, is then calculated using standard formalism
(e.g.,~{Thorne} 1981; {Viergutz} 1993; {Jaroszynski} \& {Kurpiewski}
1997).

\begin{figure}[b!] % fig 1
\centerline{\epsfig{file=run7.1410.ps,width=0.25\textwidth}\hskip-0.5mm\epsfig{file=run7.1710.ps,width=0.25\textwidth}\hskip-0.5mm\epsfig{file=run8.0510.ps,width=0.2512\textwidth}}
\caption{Images of the shadow of a black hole for rotating and
non-rotating black holes and for spherical and jet-like emission
models.}
\label{fig1}
\end{figure}

To test whether general relatvistic effects would be visible we
convolved the resulting images from the ray-tracing calculations with
two Gaussian beams: one representing the scatter-broadening of the
image due to the interstellar material along our line-of-sight towards
the Galactic Center and one representing the finite resolution of VLBI
with 8000 kilometer baselines. The width of the former has a
$\nu^{-2}$ (e.g., Lo et al.~1998) and the latter a $\nu^{-1}$
dependence.

Regardless of the exact emission model we use, we find a
characteristic structure in all models: a bright ring of emission with
a pronounced deficit of emission inside of that (Fig.~1). We call the
deficit in the inner region the ``shadow'' of the black hole since it
is caused by the deficit of photons emitted near the black hole that
have disappeared into the event horizon or are bent away from our line
of sight. The circumference of the shadow is determined by the
`photon-orbit'---a theoretical orbit where photons can circle the
black hole an infinite number of times, but when perturbed may escape
to infinity (Bardeen 1973). Interestingly, the size of this shadow is
much larger than the event horizon---due to gravitational
lensing---and is always of the order 10 $R_{\rm g}$ ($R_{\rm
g}=GM_\bullet/c^2$) for rotating and non-rotating black holes.

The exact intensity distribution of the bright ring depends
significantly on the nature of the emission region, however. A
rotating inflow would produce a slightly asymmetric ring due to
Doppler-boosting of one side of the shells in Keplerian rotation. A
jet would look even more asymmetric since boosting due to rotation
plus fast outflow would enhance one quadrant of the ring
(Fig.~1).

The relatively large size of the shadow is of particular interest for
Sgr~A*, since at a wavelength of around 1.3 mm the black hole shadow,
the scattering disk, and the possible resolution of mm-VLBI become
comparable. This is illustrated in Figure~2. It is clear that at
wavelengths shortward of $\lambda$1.3 mm the shadow could actually be
imaged with ground-based telescopes.


\section*{Discussion}
The possibility of seeing the effect of an event horizon is
tantalizing. The shadow of the black hole in the Galactic Center is
expected to have a diameter of $\sim30\,\mu$as. The highest resolution
so far achieved with VLBI is $\sim50\,\mu$as. To achieve the
additional improvement of a factor of two to three in resolution would
require extending mm-VLBI to submm wavelengths. While this is
difficult because of atmospheric effects, it is not technically
impossible.  Quite a number of submm-telescopes and arrays are
currently under construction or consideration that could be used for
such an experiment.

\begin{figure}[b!] % fig 1
\centerline{\epsfig{file=fig2.ps,width=0.99\textwidth,bblly=16.4cm,bbury=21.5cm,bbllx=0.8cm,bburx=18.2cm,clip=}}
\caption{The expected shadow of Sgr~A*. a) the ray-tracing
simmulation; b) simmulated VLBI image at $\lambda0.6$mm; c)
simmulated VLBI image at $\lambda1.3$mm (see Falcke et al.~2000 for
details).}
\label{fig2}
\end{figure}

Another concern is whether the source itself could become an
obstacle. Clumpiness of the emission may not be a major source of
confusion because of the short rotation timescale of about 100
seconds. Optical depth effects could be a major problem. However,
currently available submm spectra of Sgr A* (e.g., Serabyn et al.~1997)
indicate a rather flat spectrum with a turnover towards the
infrared. Hence at some wavelength between mm-radio and IR the source
is bound to be optically thin. A second pitfall is anisotropic
beaming. For example, in the jet model a quadrant is amplified due
to relativistic beaming, making it more difficult to pick out the
entire faint ring with low resolution or low dynamic range
observations.

Without a better understanding of spectrum, structure, and nature of
the emission it will be difficult to predict exactly at what
wavelength the shadow will be unambiguously detectable and how much
technical development still has to be done. In any case there is no
reason to think that imaging the shadow is in principal impossible and
any upcoming VLBI experiment at 1.3 mm and shorter wavelengths
involving Sgr~A* from now on could already show the first signs of the
event horizon.



{\bf Acknowledgments} This work was supported in part by a Sir Thomas Lyle
Fellowship (FM), NASA grant NAG58239 (FM), DFG grants Fa 358/1-1\&2
(HF), and NSF grant AST-9616922 (EA). 

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\end{document}



                    Fulvio Melia
              The University of Arizona             
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