------------------------------------------------------------------------
From: Eric Agol  agol@pha.jhu.edu 
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MIME-Version: 1.0
To: Galactic Center Newsletter <gcnews@aoc.nrao.edu>
Subject:Sgr A* Polarization: No ADAF ... etc.

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%\received{4 August 1988}
%\accepted{23 September 1988}
%\journalid{337}{15 January 1989}
%\articleid{11}{14}
%\slugcomment{Not to appear in Nonlearned J., 45.}
\lefthead{Agol}
\righthead{Sgr A* Polarization}
\begin{document}

\title{Sgr A* Polarization: No ADAF, Low Accretion Rate, and Non-Thermal 
Synchrotron Emission}

\author{Eric Agol}
\affil{Physics and Astronomy Department, Johns Hopkins University,
    Baltimore, MD 21218; agol@pha.jhu.edu}
\begin{abstract}
The recent detection of polarized radiation from Sgr A* requires a 
non-thermal electron distribution for the
emitting plasma. %, and rules out thermal and mono-energetic electron
%distributions.  
The Faraday rotation measure must be small, placing
strong limits on the density and magnetic field strength.  We show that 
these constraints rule out advection-dominated accretion flow models.
We construct a simple two-component model which can reproduce both the 
radio to mm spectrum and the polarization.  This model predicts that the 
polarization should rise to nearly 100\% at shorter wavelengths.  The 
first component, possibly a black-hole powered jet, is compact, low 
density, and self-absorbed near 1 mm with ordered magnetic field,
relativistic Alfv\'en speed, and a non-thermal electron distribution. 
The second component is poorly constrained, but may be a 
convection-dominated accretion flow with $\dot M \sim 10^{-9}M_\odot/$yr,
in which feedback from accretion onto the black hole suppresses the accretion 
rate at large radii.  The black hole shadow should be detectable with
sub-mm VLBI.
\end{abstract}

\keywords{accretion, accretion disks --- black hole physics --- polarization 
--- Galaxy: center }

\section{Introduction}

The nearest supermassive black hole candidate lies at the center of the
Milky Way galaxy, weighing in at $2.6\times10^6 M_\odot$, as inferred from
motions of stars near the galactic center (Ghez et al. 1998; Genzel et
al. 1997).  The low
luminosity of the point source associated with the center, $\la 10^{37}$ erg/s,
is a conundrum since accretion from stellar winds of neighboring
stars should create a luminosity of $\sim 10^{41}$erg/s.  One possibility
is that most of the energy is carried by the 
accreting matter into the black hole, as in the advection-dominated
accretion flow solution (ADAF, Narayan \& Yi 1994; Narayan, Yi, \& Mahadevan 
1995).  
Such a situation is achieved when most of the dissipated energy is channeled 
into
protons which cannot radiate efficiently.   Low efficiency also occurs
when gas accretes spherically and carries its energy in as kinetic
energy (Melia 1992).  Alternatively, the accretion rate may be overestimated,
and the emission may be due to a tenuous disk or jet.

The radio spectrum of Sgr A* can be described by a power law, $F_\nu \propto 
\nu^{1/3}$
from centimeter to millimeter wavelengths.  This is intriguingly close
to the spectrum of optically thin, mono-energetic electrons emitting
synchrotron radiation (Beckert \& Duschl 1997).  However, this explanation
is not unique:  a self-absorbed source which varies in size as a
function of frequency may produce a similar spectral slope (Melia 1992;
Narayan et al. 1995).  A possible technique to distinguish
these models is to measure the polarization of the emission:  Faraday
rotation and self-absorption can change the polarization magnitude
and wavelength dependence (Jones \& O'Dell 1977).

Only recently has linear polarization been detected
(Aitken et al. 2000: A00); previous searches showed only marginal detections 
or upper limits (Bower et al. 1999a,b).  After correcting for contamination by 
dust and
free-free emission, the inferred polarization is 10-20\%, implying
a synchrotron origin.  Remarkably, the polarization shows a change
in position angle of $\sim90^\circ$ around 1 mm, which A00 suggest might be
due to synchrotron self-absorption.  We first discuss the physics of 
synchrotron polarization (\S 2); we then apply it to various models in 
the literature (\S  3); next we discuss a model consistent with all of 
the observations (\S 4); and finally speculate on the physical implications 
of this model (\S 5).

%\section{Optically-Thick Synchrotron Polarization}
\section{Synchrotron Theory Background}

In the synchrotron limit ($\gamma \gg 1$) for an isotropic electron velocity 
distribution, some analytic results have been 
derived, which we now summarize (Ginzburg \& Syrovatskii 1965 \& 1969: GS ).  
For a uniform slab of electrons 
with a power-law distribution, $d n_e/ d \gamma \propto \gamma^{-\xi}$
(with $\gamma_{min} \le \gamma \ll \gamma_{max}$ such that electrons with
$\gamma_{min}$ and $\gamma_{max}$ do not contribute to the frequency of 
interest), 
we can relate the magnetic field strength and electron density 
in the slab to the fluid-frame brightness temperature and
the spectral turnover due to self-absorption.
For $\xi=2$ and a uniform field $B_\bot$ (projected into the sky 
plane) we find $B_\bot \sim 2  T_{11}^{-2} \nu_{12}$  G and
$\tau_C \sim 3\times 10^{-2} T_{11}^4 \nu_{12}\gamma_{min}^{-1}$, 
where $T_{11}$ is the brightness temperature in units of $10^{11}$ K 
at the self-absorption frequency $\nu_t = 10^{12}  \nu_{12}$ Hz and 
$\tau_C$ is the Compton scattering optical depth of the emission region.
For $\nu < \nu_t$, the emission is self-absorbed so
$F_\nu \propto \nu^{5/2}$, while above this frequency the emission
is optically-thin and $F_\nu \propto \nu^{(1-\xi)/2}\exp(-\nu/\nu_{max})$
where $\nu_{max} = 3 B_\bot e \gamma_{max}^2/(4 \pi m_e c)$.

%For a mono-energetic electron distribution, the emitted
%radiation is 100\% polarized; however, when optically thick due
%to self-absorption, the net polarization is zero.
%The polarization scales as 
%optically-thin regime, and as $\Pi = -3 / (6 \xi + 13)$ in the 
%optically-thick regime for $\xi > 1/3$ (the minus sign indicates a switch
%in polarization angle by 90$^\circ$).  
In the optically-thin regime, the polarization plane is perpendicular to 
the magnetic field with polarization $\Pi = (\xi +1)/(\xi + 7/3)$, up to 100\% 
for $\xi \gg 1$. In the optically-thick regime,  $\Pi = -3 / (6 \xi + 13)$ (for
$\xi > 1/3$);  the radiation polarized perpendicular to the magnetic field 
is absorbed more strongly than the opposite polarization,
causing the radiation polarized along the magnetic field to dominate, 
switching the polarization angle by $90^\circ$, which changes the sign of $\Pi$.
Numerical calculations show that the optically-thick polarization peaks at 
$|\Pi| = 20$\% for $\xi=1/3$, but remains large for $0 < \xi < 2$.  

To compute the polarization near the self-absorption frequency
requires a knowledge of the polarized
opacity and emissivity, $\mu_{\bot,\|},\epsilon_{\bot,\|}$.  For 
$\xi=2$, these can be approximated as (GS):
$\mu_{\bot,\|} = r_s^{-1} (\nu/\nu_t)^{-3} (1\pm 3/4)$ and
$(\epsilon_{\bot,\|}/ \mu_{\bot,\|}) = 2S_t/9 (\nu/\nu_t)^{5/2}(13\pm9)/(4\pm1)$
where  $r_s$ is the size of the emission region, $\nu_t$ is the frequency for 
which the total source has an optical depth of unity (i.e. $\tau = \mu r_s = 
1/2(\mu_\bot+\mu_\|)r_s = 1$), $S_t$ is the source function near the 
frequency $\nu_t$, and the $+$ or $-$ signs go with the radiation emitted $\bot$ 
or 
$\|$ to the magnetic field, respectively.  GS then express the polarization and 
emission for a slab with uniform magnetic field strength and direction, constant 
density, and size $r_s$: $I_\bot = (\epsilon_\bot/\mu_\bot) (1-\exp(-\mu_\bot 
r_s))$,
$I_\|   = (\epsilon_\|/\mu_\|) (1-\exp(-\mu_\|   r_s))$, and
$\Pi = (I_\bot - I_\|)/(I_\bot + I_\|),$
where $I_\bot, I_\|$ are the intensities (erg/cm$^2$/s/Hz/sr) with polarization 
perpendicular and parallel to the projected direction of the magnetic field on 
the sky. %, $\Pi$ is the magnitude of the polarization ($\Pi$ is positive for
%polarization perpendicular to the magnetic field, and negative for 
%polarization parallel).
%The power-law assumption causes a flux which diverges
%if $\xi < 3$, so we have also added an upper frequency cutoff due to the
%%maximum energy of the electrons by fitting the data with $I = (I_\bot + I_\|) 
%\exp(-\nu/\nu_{max})$.  This upper frequency cutoff has a small effect
%on the polarization, which we have checked with numerical computations.

For electron distributions which are highly peaked at a single
energy (such as mono-energetic or relativistic Maxwellian)  the
polarization for $\nu \la \nu_t$ is zero.
%For a 
%relativistic-Maxwellian distribution with $d n_e/d\gamma =
%n \gamma^2/(2T^3)\exp(-\gamma/T)$, optically thin radiation is again polarized
%perpendicular to the magnetic field at 100\% polarization.  However,
%when optically-thick, the radiation is in detailed balance, so the
%source functions for the perpendicular and parallel polarized radiation
%are identical, resulting in zero net polarization (Jones \& Hardee 1979).
%The relativistic-Maxwellian corresponds to $\xi=-2$ (with a high energy
%cutoff), which implies that there is some critical $-2 < \xi_c < 1/3$ such that
%the self-absorbed polarization is maximum (since it is zero for $\xi =-2$
%but increasing towards smaller $\xi$ at $\xi=1/3$).  

The Faraday effect rotates the polarization vector of photons emerging
from different optical depths by different amounts, causing a cancellation in 
polarization (Agol \& Blaes 1996).  The differential Faraday rotation angle 
within 
the source scales as $\Delta \theta = 3.6\times 10^{28} \tau_{phot} B \nu^{-2}
\gamma_{min}^{-2}$ (Jones \& O'Dell 1977), where $\tau_{phot}$ is the optical
depth of the photosphere.  When optically thin,  $\tau_{phot}\sim \tau_C$ is 
constant, so rotation is largest at the self-absorbed wavelength.  When 
self-absorbed, $\tau_{phot}$ of the photosphere scales as $\nu^{\xi/2+2}$, so 
the differential Faraday rotation angle $\propto \nu^{\xi/2}$ (for  $\xi > 
1/3$), 
again largest at the self-absorption wavelength.  The differential rotation
at $\nu_t$ is $\Delta \theta \sim 2\pi g(\xi) 
(\theta_b/\gamma_{min})^\xi/\gamma_{min}$,
where $\gamma_{min}$ is the minimum electron Doppler factor, $g(\xi)$ is a 
dimensionless
factor of order unity, and $\theta_b$ is
the brightness temperature in units of $m_e c^2/k_B$.

\section{Observational Constraints on Published Models}

The observations of polarization in Sgr A* provide the following
constraints on emission models:

1)  The Faraday rotation angle near the self-absorbed wavelength must
be $\ll \pi$.

2)  The electron distribution must be non-thermal since
thermal emission is unpolarized when self-absorbed.
If the beam correction by A00 is correct, 
then  $\Pi \sim$ 12\% at self-absorbed wavelengths, requiring 
$\xi \la 2$.

3)  The self-absorption frequency must lie near the change in
polarization angle, $\sim 1$mm.

4)  The component contributing at lower frequencies must have
zero linear polarization.

5)  The magnetic field must be ordered to prevent cancellation
of polarization.

These constraints rule out nearly several models proposed in the
literature, as will be discussed in turn.

The low efficiency of an ADAF implies a higher accretion rate and 
thus higher density than for a high efficiency flow of the same
luminosity and geometrical thickness.  For Sgr A*, an 
accretion rate of $\sim 10^{-(4-5)} M_\odot/$yr is inferred
due to capture of gas in the vicinity of the black hole
(Quataert, Narayan, \& Reid 1999; Coker \& Melia 1999), which is the
value assumed in ADAF models.  Assuming that the gas falls
in at near the free-fall speed, one infers 
an electron density $n_e = 10^{10}$ cm$^{-3} \dot m_{-5}
x^{-3/2}$ and a magnetic field strength of
$B= 10^3 {\rm G} \dot m_{-5}^{1/2} x^{-5/4} (v_A/0.1v_{ff})$,
where $x$ is the radius of the emission region in units of
$r_g=GM/c^2$, $\dot m_{-5}$ is the accretion rate in units
of $10^{-5} M_\odot$/yr, and $v_A/v_{ff}$ is the ratio of the
Alfv\'en speed to the free-fall speed.  These values imply a total
Faraday rotation angle at the self-absorption frequency $\nu_t$ of
$\Delta \theta \sim 10^{4} \dot m_{-5}^{3/2} \nu_{12}^{-2}
(v_A/0.1v_{ff})$.
% See p. 70 in my notes on Sgr A* polarization
%At frequencies $\nu < \nu_t$,
%the rotation angle should be scaled as $\nu/\nu_t$;
%at higher frequencies it scales as $(\nu/\nu_t)^{-2}$.
This value is so large that rotation of the emitted
radiation leads to zero net polarization, so ADAFs are in direct
conflict with the observed polarization.  Only significant
modifications of the model, such as a reduction in the accretion
rate by a factor of $10^{-3}$, can reduce the Faraday rotation
angle $\ll \pi$.  %However,
%an accretion rate of $10^{-9} M_\odot/$yr is much too low to
%explain the observed luminosity.
An accretion rate of $10^{-8} M_\odot/$yr is consistent with
the observed luminosity if the accretion flow has a higher efficiency
$\sim$2\%, no longer ``advection-dominated.''
In addition, ADAF models assume a Maxwellian electron distribution,
which cannot produce the observed switch in polarization angle\footnote{
Mahadevan (1999) and \"Ozel, Psaltis, \& Narayan (2000) have added a non-thermal
electron component to ADAF models which contributes to the flux at
wavelengths longer than 2 mm, not at the polarized wavelengths.}.
Finally, ADAFs predict a higher self-absorption frequency:
\"Ozel et al. (2000) find that $\nu_t \sim 5\times10^{12} 
\dot m_{-5}^{5/9}{\rm Hz}$, which implies $\dot M \sim 4\times10^{-7} 
M_\odot$/yr
to be consistent with the observed $\nu_t \sim 5\times 10^{11}$Hz.
The accretion rate might be reduced if there is significant gas lost
by a wind or jet (Begelman \& Blandford 1999; Quataert \& Narayan
1999) or if the Bondi rate is reduced by heating the infalling gas
with heat carried outwards by a convection-dominated accretion flow, 
or ``CDAF'' (Stone, Pringle, \& Begelman 1999, Quataert \& Gruzinov 
2000).  %To satisfy
%the observational constraints, these modifications must also include
%a non-thermal electron distribution which contributes strongly
%at the polarization wavelengths (Mahadevan 1999; \"Ozel et al. 2000
%add non-thermal electrons which contribute at cm and IR wavelengths,
%but not at mm wavelengths) and highly ordered magnetic field to be 
%consistent with the observed polarization.

%Quataert \& Narayan (1999) have modified the
%ADAF model for Sgr A* by adding an outflow following the prescription 
%of Begelman \& Blandford (1998), referred to as an Advection
%Dominated Inflow-Outflow Solution (ADIOS).  They showed that the
%ADIOS can only function for a very small outflow rate since the
%reduced density leads to underluminous radio emission for fixed
%X-ray luminosity.  The density is only reduced by an order of
%magnitude; not enough to get around the Faraday rotation or 
%self-absorption frequency constraints.

The model of Melia (1992) is rather similar to the ADAF model, and thus 
suffers the same problems: the high accretion rate implies high density 
which is inconsistent with the observed polarization.

%Recent simulations and analytic models indicate that ADAFs are
%convectively unstable, and thus should exhibit strong convective
%motions which change the run of density with radius (Stone, Blandford,
%\& Pringle 1999; Quataert \& Gruzinov 2000).  These models suffer the 
%same problem as ADAFs since a high accretion rate still requires a 
%high density near the horizon where the inflow speed is comparable to 
%the speed of light, so the density is fixed by the steady-state
%accretion rate.

Beckert \& Duschl (1997) considered several 1-zone, quasi-monoenergetic
and thermal emission models for the synchrotron emission.  As we
have pointed out, these electron distributions produce zero polarization
when self-absorbed, and so are ruled out.  Falcke, Mannheim, \& Biermann  (1993) 
present
a disk-plus-jet model which assumes a tangled magnetic field
topology which would erase any polarization.  The model is not
described in enough detail to easily ascertain whether it predicts the
correct self-absorption frequency or small Faraday rotation.

\section{A Phenomenological Model}

Now, we attempt to construct a model consistent with all of the
data, using uniform emission regions for simplicity.  
%We utilize the equations of Ginzburg \&
%Syrovatskii (1965, 1969) for the polarization of a synchrotron
%emitting slab to make qualitative estimates of conditions in
%the emitter.  
Typical optically-thin AGN spectra show $\xi \sim 2-3$; since
$\xi=2$ is consistent with the polarization from A00,  we fix
$\xi=2$ in our model fits.  The model parameters
for the polarized component are 
$S_t=6$ Jy, $\nu_t = 550 \Gamma^{-1}$ GHz (corresponding to $\lambda = 
0.55$ mm), and $\nu_{max} \sim 5000 \Gamma^{-1}$GHz, where
$\Gamma$ is the bulk Doppler boost parameter (Figure 1).

To explain the lack of polarization and spectral slope flatter than 5/2,
we require an additional component which is unpolarized 
and has a cutoff near 1 mm so that it doesn't dilute the polarization at 
shorter wavelengths.  Since Sgr A* has a spectral slope of 1/3 at mm 
wavelengths and appears to have a spectral turnover at 1 GHz, we model 
the spectrum as a monoenergetic electron distribution with energy
$\gamma$ and zero polarization (due to Faraday depolarization or
tangled magnetic field) which becomes self-absorbed at low frequency.
For the unpolarized component, we find 
$F_\nu=1.3(\nu/\nu_{max})^{1/3} \exp(-\nu/\nu_{max}){\rm Jy}$ with 
$\nu_{max} \sim 50$GHz, and $\nu_t \sim 1$GHz (Figure 1).  
%We describe the spectrum as
%\begin{equation}
%I={2I_c \over\left({\nu\over 
\nu_c}\right)^{-1/3}\exp\left({\nu\over\nu_c}\right)
%+\left({\nu_s\over \nu_c}\right)^{-1/3}
%\exp\left({\nu_s\over \nu_c}\right)\left({\nu\over\nu_s}\right)^{-5/2}
%}
%\end{equation}
%where $\nu_s$ is the self-absorption frequency, $I_c$ is
%the intensity near the cutoff frequency, and $\nu_c =
%3 B_\bot e \gamma_e^2/(4\pi m_e c)$ is the cutoff for the
%the electron energy of $\gamma_e$.  We do not require that the
%second component have the same source size or magnetic field strength
%as the polarized component.  The total number of model parameters
%for the two components is seven, while the data offers 
%approximately ten constraints, so there is not much freedom
%in the model choice.  

\vskip 2mm
\hbox{~}
\centerline{\psfig{file=fig1.ps,width=3.6in}} %FIGURE 2
\noindent{
\scriptsize \addtolength{\baselineskip}{-3pt}
\vskip 1mm
\begin{normalsize}
Fig.~1: Polarization and spectral energy distribution of Sgr A* 
compared to model.  The dashed line shows the polarized component,
the dotted line the unpolarized, mono-energetic component, and the 
solid line the sum of the two.  The dot-dash line shows the maximum 
CDAF model (assumed to be unpolarized;  the total polarization is similar 
if the CDAF replaces the monoenergetic component).  The diamonds are 
the data compiled by Narayan et al. (1995), while the asterisks are 
the data from Bower et al. (1999a,b) and A00.
\end{normalsize}
\vskip 3mm
\addtolength{\baselineskip}{3pt}
}

Figure 1 compare the model to the data.
%our fit to the data compiled in
%Narayan et al. (1995) as well as the polarization data from Aitken et al.
%(2000) and Bower et al. (1999a,b).  
To compare the polarization, we have plotted the Stokes' parameter that
lies at $83^\circ$.
%There is almost no flexibility in most of the model parameters.  
Remarkably, the polarization should rise to $\sim 100$\% at even shorter 
wavelengths.
%; for example,  $\nu_t$ is tightly constrained by the 
%wavelength of zero polarization, which ranges from 0.545-0.605 of the
%$\nu_t$ for $\xi=1/3-2$. 
%The only parameters with some some flexibility
%are $\xi$ and $\nu_c$ since smaller values of $\xi$ lead to somewhat higher 
%polarization, which can be offset by increasing the cutoff frequency leading 
%to more contribution of the unpolarized component at longer wavelengths.
%The upper energy cutoff of the polarized component, $\nu_{max}$, leads
%to a better fit for smaller values down to $\sim \nu_t$; however, the 
%infrared data limits $\nu_{max} < 9 \nu_t$ and smaller values of $\nu_{max}$
%lead to a decreased polarization in the optically-thick region.

\subsection{Physical conditions in Synchrotron emitting regions}

%The frequency of self-absorption along with the brightness
%temperature may be used to constrain the physical parameters of a
%synchrotron emission region (e.g. Krolik 1999).  
The brightness temperature of the polarized emission region is
somewhat uncertain due to the unknown source size.  Krichbaum et al. 
(1998) report a source radius of $55 \mu$as at 1.4 mm from VLBI 
observations; this corresponds to 19$r_g$.  We expect 
the radius of the emission region to be greater than the size of the 
event horizon of the black hole, which has an apparent size of $\sim 5r_g \sim 
15 \mu$as projected on the sky (including gravitational bending,
Bardeen 1973), so we use an intermediate size in further estimates.
The flux of the fitted model at the self-absorption frequency,
$\nu_t=550$ GHz, is $\sim 9$ Jy.  This implies a brightness 
temperature in the emission frame $T_b \sim 1.6\times10^{10} 
(r_s/10r_g)^{-2}\Gamma^{-1}$ 
K, where $r_s$ is the size of 
the source (we have assumed the area of the source is $\pi r_s^2$).  For a 
steeply falling electron number distribution, $k T_b \sim 4 \gamma m_e c^2$ 
(for $\xi=2$), where $\gamma m_e c^2$ is the energy of the emitting 
electrons, implying $\gamma \sim 10 (r_s/10r_g)^{-2}\Gamma^{-1}$ for the 
electrons at the self-absorption frequency.  Using the formulae from
\S 2,  we find: $B_\bot = 350  (r_s/10r_g)^4 \Gamma {\rm G}$, 
$\tau_C = 10^{-5}  (r_s/10r_g)^{-8} \Gamma^{-5}$, and 
$\gamma_{max} = 50 (r_s/10r_g)^{-2} \Gamma^{-1}$, implying
$n_e \sim 6\times 10^6 (r_s/10r_g)^{-9} \Gamma^{-5}$cm$^{-3}$.
The ratio of magnetic to rest-mass energy density is
$B^2/(8\pi n_e m_p c^2) \sim 1 (r_s/10r_g)^{17}
\Gamma^{9/2} $ for an electron-proton plasma, indicating a
relativistic Alfv\'en speed. %, or
The Faraday rotation angle at $\nu_t
=5.5\times10^{11}$ Hz is $\Delta \theta \sim 350
(r_s/10r_g)^{-4} \Gamma^{-2} \gamma_{min}^{-3}$, assuming $B_\| \sim B_\bot$.
%If the power-law electron distribution has a cutoff at some $\gamma_{min}$,
%then the Faraday rotation might be reduced by a factor of 
%$\sim \gamma_{min}^{-3}$.  
For $r_s \sim 10 r_g$, $\gamma_{min}$ can
be as large as 4, reducing $\Delta \theta$ to 5; for $r_s \sim 5 r_g$,
$\gamma_{min}$ can be as large as 20 reducing $\Delta \theta$ to $\sim 0.6$.
Alternatively, if the synchrotron emission is due to a pair plasma, 
Faraday rotation will be reduced by the ratio of the proton number
density to the pair number density.
The rotation angle is further reduced at the observed wavelengths by
a factor $\sim\nu/\nu_t$.  The high energy cutoff for the electron
distribution may be due to synchrotron cooling since 
$t_{cool} = 8\times10^8 \gamma_{max}^{-1} B^{-2}
\sim 6 (r_s/10r_g)^{-6}\Gamma^{-3}$ sec, similar to
the dynamical time, $t_D \sim 13 x^{-3/2}$ sec.

The unpolarized emission component dominates at $\sim 7$ mm,
where Lo et al. (1998) measure a source size of 
$\sim 5\times 10^{13}$ cm.  The self-absorption frequency
then requires $\gamma\sim 400$, 
$B \sim 0.1$ G, and $n_e \sim 4 \times 10^5$ cm$^{-3}$.  Though 
somewhat ad-hoc, this model reproduces the spectrum well.
The Faraday rotation parameter is rather small, so depolarization
requires field which is tangled on a scale $\sim 100$ times 
smaller than the size of the emission region.

\subsection{Accretion Component}

We have tried modeling the spectrum of the unpolarized component with a 
self-similar, self-absorbed accretion flow. 
%models with electrons and is self-absorbed.  
We used the cyclo-synchrotron emission formulae from Mahadevan, Narayan, \& Yi 
(1996) and 
we performed the radiation transfer in full general relativity (Kurpiewski
\& Jaroszy\'nski 1997).  %We found an
%adequate fit for $\dot M = 5\times10^{-8} M_\odot$/yr, $v_A=0.01 v_{ff}$,
%$n_e \propto r^{-1/2}$, and $T_e = 4\times10^{11} (r/r_g)^{-2/5}$;
%The flat temperature dependence with radius in this model is unphysical 
%however, the gas is unbound, so we reject this model.
% ; however, the flat spectral index is inconsistent 
%with the steep dependence of density and temperature in an ADAF ($n_e \propto
%r^{-3/2}$ and $T_e \propto r^{-3/5}$) or CDAF ( $n_e \propto r^{-1/2}$ and 
%$T_e \propto r^{-1}$, Quataert \& Gruzinov 1999) as both have $F_\nu \propto 
%\nu$.
%The ADAF model () has a spectral slope which is too steep 
%to be consistent with the 
%unpolarized component, so 
We can place an upper limit on the accretion rate of an ADAF component (using 
the model of \"Ozel et al. 2000) since its unpolarized flux must not dilute 
the polarized component:  we find $\dot M_{ADAF} \la 3\times 10^{-6} 
M_\odot/$yr.
If the ADAF surrounds the polarized emission region, then it will depolarize,
so the Faraday depolarization places a stronger upper limit (\S 2).  We can 
place a similar limit on the CDAF model (using the structure from Quataert \& 
Gruzinov, 2000, with equipartition $B$ field and $p_{gas} = 2n_e k_B T_e$):  we 
find $\dot M_{CDAF} \la 1.5\times 10^{-9}M_\odot/$yr;  this accretion rate can 
account for 
the unpolarized component at $\nu \ga 10$GHz (see Figure 1) and is consistent 
with the Faraday rotation constraint.  The CDAF luminosity is $2\times 
10^{34}$erg/s and the self-absorption frequency is $\sim 30$GHz, so the 
polarized 
component would be visible through it.  Finally, we can place a limit on
a standard thin disk from the infrared upper limits:  we find $\dot M_{thin}
\la 2\times 10^{-11}M_\odot/$yr; this upper limit can be increased to
a maximum of $3\times 10^{-7} M_\odot/$yr if the inner edge of the disk is 
truncated at $r=6000 r_g$.

\section{Conclusions}

The main success of advection-dominated accretion models for Sgr A*
is in explaining the high-frequency radio spectrum and skirting
below the upper limits at infrared frequencies. % (a small non-thermal
%is added in to explain the low-frequency radio spectrum, \"Ozel et al.
%2000).  
However, the ADAF model is unpolarized at the same high frequencies, 
inconsistent with the recent detection of linear polarization.  We have 
constructed a simple toy model for the millimeter polarization which 
predicts a rise towards shorter
wavelengths:  polarization of $\sim 70\%$ might be seen with
SCUBA at 350 $\micron$ if this model is correct.  The lack of polarization
and spectral slope of 1/3 at wavelengths longer than 2 mm indicates
that a different physical component may be contributing.  The presence
of two physical components can be confirmed by looking for a change
in variability amplitude and time-scale or source size and morphology 
around 2 mm.

The high observed polarization implies a highly ordered magnetic field
lying near the sky plane.
This might be due to the poloidal field in a jet seen edge-on.
The non-thermal electron distribution might be produced by shock
acceleration, reconnection, or electric field acceleration near 
the event horizon of a spinning black hole (Blandford \& Znajek
1977).  The Blandford-Znajek mechanism can generate a maximum 
luminosity of $L_{BZ} \sim 10^{37} (B/600 G)^2$ erg/s
(Thorne, Price, \& MacDonald 1986), so the entire polarized luminosity 
of Sgr A* might be powered by black hole spin.

The dynamics of the emission region will be controlled by the ratio of
the magnetic field energy density to the matter energy density, 
$B^2/(8\pi\rho c^2)$; however, this ratio scales as $r_s^{17} \Gamma^{9/2}$, 
while $\Gamma$ and $r_s$ are unknown.  Doppler boosting decreases the 
brightness temperature, which reduces Faraday rotation but makes the 
electrons trans-relativistic.  Future sub-mm VLBI observations should 
accurately measure the $r_s$ as a function of frequency, and proper 
motions may constrain $\Gamma$.  Also uncertain 
are the pair fraction and minimum electron energy $\gamma_{min}$.
The pair number density can be constrained by measuring the circular
polarization; without pairs, the circular polarization may be as high as 
a few percent at optically-thin wavelengths (Jones \& O'Dell 1977), while 
pure pair emission should have no circular polarization.  The pair 
annihilation line should be looked for at higher spatial resolution; 
however, it will be strongly broadened by relativistic motions of the pairs.
Once the source size is known, $\gamma_{min}$ and the pair fraction will be 
constrained by the Faraday rotation limit.

%If the unpolarized component is due to a quasi-spherical accretion flow
%then the inferred accretion rate is $\sim 5\times 10^{-8} M_\odot/$yr,
%requiring an efficiency of about 0.5\% to account for the observed luminosity
%(including the polarized component).  Some of the accretion power may be lost
%as kinetic energy in the outflow/jet, which has energy $\ga 10^{38}$erg/s,
%or about 5\% of $\dot M c^2$.
An ADAF model must have a low accretion rate, $\la10^{-8} M_\odot/$yr, to
be consistent with the lack of Faraday rotation of the polarized emission.
Such a low inferred accretion rate disagrees with estimates 
of the Bondi accretion rate inferred from stellar winds near the region 
of the black hole.  If accretion is episodic due to outer-disk 
instabilities, then the current state might be one of low accretion 
rate in the inner disk.  Alternatively, the accretion rate might be 
reduced by depositing energy from the accretion flow in the 
surrounding gas (either through outflow or convection), thus
increasing the sound speed and decreasing the capture rate of gas
by the black hole.  The accretion flow must deposit energy 
$\dot M_A GM/r_A \sim 6\times 10^{35}$ erg/s, where $\dot M_A$ is the stellar 
mass loss rate which crosses the Bondi radius $r_A$ (Quataert et al. 1999).  
This can be supplied by accretion
which releases energy $\sim 5\times 10^{35}(\eta/0.01) \dot m_{-9} $ erg/s,
where $\eta$ is the efficiency with which accretion deposits energy at large
radius.  As remarked above, a 
convection-dominated accretion flow with $\dot M \sim 10^{-9}
M_\odot/$yr can explain part of the unpolarized component without diluting the 
polarized emission;  the associated convection can carry the required 
energy outward to suppress the Bondi accretion rate.

Since the self-absorption frequency occurs at $\sim 500\micron$, it will 
be possible to image shadow of a black hole from the ground using VLBI, 
providing a direct confirmation of the existence of an event horizon (Falcke, 
Melia, \& Agol 2000).  Future sub-mm polarimetric VLBI observations might 
show rotation of the polarization angle near the black hole, a general 
relativistic effect which becomes stronger for a spinning black hole 
(Connors, Stark, \& Piran 1980). 

Eliot Quataert \& Andrei Gruzinov have shown me work which reaches
similar conclusions about ADAFs, but attributes the polarization
swing to magnetic field geometry rather than self-absorption.

\acknowledgments

I acknowledge Ski Antonucci, Julian Krolik, Colin Norman, and
Eliot Quataert for ideas and corrections which greatly improved 
this letter.  This work was supported by NSF grant AST 96-16922.

%\appendix

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\end{document}


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