------------------------------------------------------------------------
From: Eliot Quataert eliot@ias.edu
To: Galactic Center Newsletter
Subject:astro-ph/0004286
MIME-Version: 1.0
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\begin{document}
\title{Constraining the Accretion Rate Onto Sagittarius A$^*$ Using
Linear Polarization}
\author{Eliot Quataert and Andrei Gruzinov}
\affil{Institute for Advanced Study, School of Natural
Sciences, Einstein Drive, Princeton, NJ 08540; eliot@ias.edu,
andrei@ias.edu}
\begin{abstract}
Two possible explanations for the low luminosity of the supermassive
black hole at the center of our galaxy are (1) an accretion rate of
order the canonical Bondi value, but a very low radiative efficiency
for the accreting gas or (2) an accretion rate much less than the
Bondi rate. Both models can explain the broad-band spectrum of the
Galactic Center. We show that they can be distinguished using the
linear polarization of synchrotron radiation. Accretion at the Bondi
rate predicts no linear polarization at any frequency due to Faraday
depolarization. Low accretion rate models, on the other hand, have
much lower gas densities and magnetic field strengths close to the
black hole; polarization is therefore observable at high frequencies.
If confirmed, a recent detection of linear polarization from Sgr A$^*$
at $\gsim 150$ GHz argues for an accretion rate $\sim 10^{-8} \ \mpy$,
much less than the Bondi rate. This test can be applied to other
low-luminosity galactic nuclei.
\end{abstract}
\section{Introduction}
In this paper we discuss the effect of Faraday depolarization on
synchrotron radiation in spherical accretion flow models of
low-luminosity galactic nuclei. We focus on the radio source Sgr
A$^*$ at the Galactic Center, but our results can also be applied to
other systems (see \S4). This paper was motivated by a possible
detection of linear polarization from Sgr A$^*$ (Aitken et al. 2000).
The Bondi accretion rate onto the supermassive black hole at the
center of our galaxy is estimated to be $\sim 10^{-4}-10^{-5} \ \mpy$
(e.g., Coker \& Melia 1997; Quataert, Narayan, \& Reid 1999), implying
a luminosity of $\sim 10^{41} \egs$ if the radiative efficiency is
$\sim 10 \%$. This is roughly 5 orders of magnitudes larger than the
observed luminosity (see Narayan et al. 1998 for a recent
compilation). Comparable discrepancies are obtained for massive
elliptical galaxies in nearby X-ray clusters (e.g., Fabian \& Rees
1995; Di Matteo et al. 1999).
One explanation for the low luminosity of nearby supermassive black
holes is that they accrete via an advection-dominated accretion flow
(ADAF), in which most of the dissipated turbulent energy is stored as
thermal energy rather than being radiated (e.g., Rees et al. 1982;
Narayan \& Yi 1994, 1995; Abramowicz et al. 1995). In such models the
accretion rate is of order the Bondi rate while the radiative
efficiency is extremely small ($\approx 10^{-6}$ for Sgr A$^*$).
Another explanation for very low luminosity accreting systems is that
the Bondi accretion rate estimate is inapplicable. Several physical
mechanisms for reducing the accretion rate have been suggested: (1)
Blandford \& Begelman (1999) proposed that most of the mass supplied
to non-radiating accretion flows is lost to an outflow/wind, rather
than being accreted onto the black hole, (2) Gruzinov (1999) argued
that thermal conduction can suppress the accretion rate by
``over-heating'' the outer parts of the accretion flow, (3) Numerical
simulations of non-radiating accretion flows with small values of the
dimensionless viscosity parameter $\alpha$ find that the gas density
scales with radius as $\rho \propto r^{-1/2}$ rather than the
canonical Bondi/ADAF scaling of $\rho \propto r^{-3/2}$ (Stone,
Pringle, \& Begelman 1999; Igumenshchev \& Abramowicz 1999, 2000;
Igumenshchev, Abramowicz, \& Narayan 2000). Narayan, Igumenshchev, \&
Abramowicz (2000) and Quataert \& Gruzinov (2000a) explained these
simulations in terms of a ``convection-dominated accretion flow''
(CDAF). In such a flow angular momentum is efficiently transported
inwards by radial convection, nearly canceling the outward transport
by magnetic fields.
In all three of the above models the accretion rate onto the black
hole is suppressed with respect to the Bondi rate for a fixed density
and temperature in the interstellar medium of the host galaxy. The
low accretion rate is not due to a dearth of material at large radii.
Rather, it is a consequence of the dynamics of quasi-spherical
accretion (either because most of the gas is lost to an outflow/wind
or because convection/conduction strongly suppresses the inflow
velocity of the accreting gas at large radii).
Broad band spectra have thus far had difficulty distinguishing between
these explanations for the low luminosity of nearby supermassive black
holes. For example, Quataert \& Narayan (1999; hereafter QN) showed
that models with accretion rates much less than the Bondi rate could
produce spectra quite similar to ADAF models.
In this paper we show that the linear polarization of synchrotron
radiation can be markedly different in low efficiency and low
accretion rate models. This is because the gas density and magnetic
field strength near the black hole are orders of magnitude higher in
the former than in the latter; depolarization by Faraday rotation in
thus significantly more important.
In the next section (\S2) we present simple estimates of the physical
parameters of the accretion flow relevant for our analysis. We then
discuss Faraday depolarization in spherical accretion flow models of
Sgr A$^*$ (\S3). In \S4 we compare these predictions with
observational constraints on the linear polarization of Sgr A$^*$ and
summarize our results. We also generalize our analysis to other
low-luminosity galactic nuclei.
Throughout this paper, we focus on accretion models of Sgr A$^*$. An
unresolved jet or outflow may, however, contribute to the observed
emission (e.g., Falcke, Mannheim, \& Biermann 1993; Lo et al. 1998;
Falcke 1999); this is briefly discussed in \S4.
\section{Plasma Parameters for Sgr A$^*$}
Stellar kinematics show that there are $\approx 2.6 \times 10^6
M_\odot$ within $\approx 0.015$ pc of the Galactic Center (Eckart \&
Genzel 1997, Ghez et al. 1998), centered on the radio source Sgr A$^*$
(Menten et al. 1997). The most plausible explanation is that Sgr
A$^*$ is a $\approx 2.6 \times 10^6 M_\odot$ accreting black hole.
Sgr A$^*$ is believed to accrete the winds from nearby ($\sim 0.1$ pc)
massive stars (Krabbe et al. 1991). The Bondi accretion rate of these
winds onto the supermassive black hole is $\approx 10^{-4}-10^{-5} \
\mpy$ (e.g., Coker \& Melia 1997; Quataert, Narayan, \& Reid 1999).
Recent Chandra observations detect a point source coincident with the
radio source Sgr A$^*$ to within $\approx 0.5'' \approx 10^5 R_g$.
Its $0.1-10$ keV luminosity is $L_X \approx 4 \times 10^{33}$ ergs
s$^{-1}$ (Baganoff et al. 2000). This emission can be modeled as
thermal bremsstrahlung from large radii in a quasi-spherical accretion
flow (e.g, Quataert \& Gruzinov 2000b). The required density is $n
\approx 10^4 \cc$ at $r \approx 10^4$, where $r$ is the radius in the
flow in units of the Schwarzschild radius, $R_g$.\footnote{Regardless
of the interpretation of the Chandra detection, the density near $r
\approx 10^4$ cannot be much larger than $\approx 10^4 \cc$ or the
X-ray luminosity from the accretion flow would exceed the observed
value.}
If the accretion rate close to the black hole is of order the Bondi
value the gas density near $r \sim 1$ is $n \approx \ 10^9 - 10^{10}
\cc$ (since $v_r \approx c$ near the horizon); this is consistent with
the gas density normalized to the Chandra observations, and an $n
\propto r^{-3/2}$ scaling as in Bondi and ADAF models. The
corresponding magnetic field strength, assuming rough equipartition
with the nearly relativistic protons, is $B \approx 2 \times 10^3 \
$G. At such magnetic field strengths, relativistic electrons cooling
by synchrotron radiation would have a cooling time much less than the
inflow time of the gas. In order to not overproduce the observed
radio to sub-mm luminosity of Sgr A$^*$, the bulk of the electrons
must therefore be marginally relativistic, with $T_e \sim
10^9-10^{10}$ K. These plasma parameters ($n, B, T_e$) describe Bondi
and ADAF models of Sgr A$^*$ (e.g., Melia 1992, 1994; Narayan, Yi, \&
Mahadevan 1995; Narayan et al. 1998). In such models the electrons
are assumed to be adiabatically compressed from large radii in the
accretion flow, with virtually no additional turbulent heating.
QN calculated spectral models of quasi-spherical accretion flows with
strong outflows. Such spectra are relevant for any model in which the
accretion rate close to the black hole is much less than the Bondi
value. They showed that accretion at much less than the Bondi rate
could produce the observed radio to sub-mm emission of Sgr A$^*$,
provided the electrons were much hotter than in standard ADAF models
(see their Table 2 and Fig. 8b).
A simple explanation for this result can be obtained by applying the
Burbidge (1958) estimate to Sgr A$^*$. We consider synchrotron
emission from a sphere of radius $R$ containing relativistic electrons
with a temperature $k T_e = \gamma m_e c^2$. We take the electron
heating rate to be comparable to the net turbulent (magnetic) heating
rate. As can be confirmed {\it a posteriori}, the synchrotron cooling
time is $\gg$ the inflow time of the gas. The electron energy density
is then similar to the magnetic energy density
\begin{equation}
n\gamma m_ec^2 \approx {B^2\over 8\pi }.
\end{equation}
The frequency of peak synchrotron emission and the synchrotron
luminosity are given by
\begin{equation}
\nu \approx 0.1\gamma ^2{eB\over m_ec}
\end{equation}
and
\begin{equation}
L \approx \sigma _T c B^2 \gamma ^2 R^3 n,
\end{equation}
where $\sigma _T$ is the Thomson cross section.
We express $n$, $\gamma$, and $B$ in terms of $R$, $\nu$, and $L$:
\begin{equation}
\gamma \approx 3.2 \left( {m_e\over c}~{\nu ^4 R^3\over L}\right) ^{1/7} \approx
\ 100, \label{gamma}
\end{equation}
\begin{equation}
n \approx {4\over \gamma ^5 \lambda ^2r_e} \approx 10^6~{\rm cm}^{-3}, \label{n}
\end{equation}
and \begin{equation} B \approx \sqrt{8\pi m_ec^2\gamma n} \ \approx \
45~{\rm G}, \label{B}
\end{equation}
where $\lambda =c/\nu$ is the wavelength and $r_e=e^2/(m_ec^2)$ is the
classical electron radius. For the numerical estimates in equations
(\ref{gamma})-(\ref{B}), we have used the observed values for Sgr A$^*$.
The peak synchrotron frequency is at $\nu \approx 10^3$ GHz with a
luminosity of $L \approx 10^{36} \egs$ (e.g., Serabyn et al. 1997).
In spherical accretion models, this high frequency emission arises
from very close to the black hole, so we have taken $R \approx R_g
\approx 10^{12}$ cm.
For a density close to the black hole of $n \sim 10^6 \cc$, the
implied accretion rate is $\sim 10^{-8} \mpy$, three to four orders of
magnitude smaller than the Bondi value (this estimate is consistent
with QN's detailed spectral calculations). Despite this much smaller
accretion rate, the radiative efficiency of the accreting gas is still
rather small, $\sim 10^{-3}$. This is because, even for $\gamma \sim
100$, the synchrotron cooling time is longer than the inflow time of
the gas at these low magnetic field strengths.
%Assuming that the proton temperature near the event horizon is close
%to the ``Bondi'' value, $k T_p=0.15m_pc^2$, the fraction of the
%available energy in the electrons and in the magnetic field is
%\begin{equation}
%\xi _e\equiv {\gamma m_ec^2 \over k T_p} = \xi _B \equiv {B^2\over
%8\pi n k T_p} \approx 0.25.
%\end{equation}
%As expected for this model, $\xi_e$ and $\xi_B$ are comparable to, but
%somewhat less than, unity.
The thermal blackbody emission at frequency $\nu$ from a sphere of
radius $R$ is \beq L_t = 2 \pi \nu^3 \gamma m_e 4 \pi R^2 \approx
10^{37} \egs, \eeq where the numerical estimate is for our fiducial
parameters. This comparison shows that the synchrotron emission in
our ``maximal heating'' model becomes optically thin below the peak
frequency, near $\nu \approx 300$ GHz. At lower frequencies the
emission is self-absorbed.
The above considerations show that both low ($\sim 10^{-8} \mpy$) and
high ($\sim 10^{-5}-10^{-4} \mpy$) accretion rate models can explain
the observed spectrum of Sgr A$^*$ (see QN). Both models are also
consistent with the Chandra X-ray detection coincident with Sgr A$^*$.
If we normalize the gas density to be $n \approx 10^4 \cc$ at $r \sim
10^4$, an $n \propto r^{-3/2}$ scaling predicts a density near the
black hole of $n \sim 10^{10} \cc$, consistent with Bondi and ADAF
models of the radio emission. Alternatively, if we use the CDAF
scaling of $n \propto r^{-1/2}$, the density close to the black hole
is $n \sim 10^6 \cc$, consistent with our ``maximal heating'' model.
Spectral models of Sgr A$^*$ can be distinguished by comparing the
observed brightness temperature and/or radio image as a function of
frequency with the theoretical predictions (see, e.g., \"Ozel,
Psaltis, \& Narayan 2000). This test has been difficult to implement
because interstellar scattering significantly broadens the image of
Sgr A$^*$.\footnote{Recent detections of Sgr A$^*$'s intrinsic size
(Lo et al. 1998; Krichbaum et al. 1998) still have sufficient
uncertainties that a range of theoretical models are allowed.} In the
next section we show that the linear polarization of Sgr A$^*$ at high
frequencies provides an additional discriminant.
\section{Faraday Depolarization}
The anisotropic index of refraction of a magnetized plasma leads to a
frequency-dependent rotation in the position angle, $\theta$, of
linearly polarized electromagnetic waves, \beq \theta = RM \lambda^2,
\eeq where $RM$ is the rotation measure. This can lead to significant
depolarization of intrinsically linearly polarized synchrotron
emission.
For a ``cold'' non-relativistic plasma, RM is given by (e.g., Rybicki
\& Lightman 1979) \begin{eqnarray} RM &=&{e^3\over 2\pi m_e^2c^4}\int
d{\bf l}\cdot {\bf B}n \nonumber \\ &=& 2.63\times 10^{-13}\times \int
d{\bf l}\cdot {\bf B}n~{{\rm rad}\over {\rm m}^2}, \label{cold1}
\end{eqnarray}
where $d {\bf l}$ is the differential path length from the observer to
the source. In the Appendix we show that the rotation measure for an
ultrarelativistic thermal plasma is given by
\begin{eqnarray}
RM_{\gamma }&=&{e^3\over 2\pi m_e^2c^4}\int d{\bf l}\cdot {\bf B}n {\log
\gamma \over 2\gamma ^2}\nonumber \\ &=& 2.63\times 10^{-13}\times \int d{\bf
l}\cdot {\bf B}n{\log \gamma \over 2\gamma ^2}~{{\rm rad}\over {\rm
m}^2},
\label{ultra1}
\end{eqnarray}
where $\gamma = kT_e/m_ec^2$. A comparable expression is obtained for
a power law distribution of relativistic electrons, with $\gamma$
replaced by $\gamma_{min}$, the minimum Lorentz factor of the
electrons (Jones \& O'Dell 1977). In what follows, we define $RM(r)$
to be the contribution to the net rotation measure from radii within
$dr \approx r$ of radius $r$ in the accretion flow.
\subsection{ADAF/Bondi Models}
In spherical accretion flow models, radio emission arises from close
to the black hole, where the electron temperature and magnetic field
strengths are the largest; this is also generally true for jet models
(e.g., Falcke 1999). \"Ozel et al. (2000) show that in ADAF models of
Sgr A$^*$ the synchrotron emission at frequency $\nu = 100 \ \nu_{100}$
GHz arises from a radius $r_\nu \approx 20 \ \nu_{100}^{-0.9}$ (see
their Fig. 5). This radius defines the $\tau = 1$ surface of the
synchrotron emission. For smaller radii the emission is self-absorbed
while for larger radii it is optically thin. Faraday rotation is only
important for $r \gsim r_\nu$, where the photons ``free stream'' out
of the accretion flow.
In fact, in Bondi/ADAF models Faraday rotation is so strong that the
synchrotron emission is completely depolarized in the vicinity of the
$\tau = 1$ surface where it is emitted. Taking $n \propto r^{-3/2}$
and $B \propto r^{-5/4}$; the rotation measure scales roughly as $RM
\propto r^{-7/4} \gamma^{-2}$. The relativistic suppression of the
rotation measure is small in all models which have an accretion rate
comparable to the Bondi rate, because the electrons must then be at
most marginally relativistic (\S2). The rotation measure as a
function of radius is thus given by \beq RM \approx 10^{13} r^{-7/4}
\rad. \label{ADAF} \eeq The normalization in equation (\ref{ADAF}) is
set by the Chandra observations ($n \approx 10^4 \cc$ at $r \approx
10^4$); we have also assumed that the magnetic field is in rough
equipartition with the gas pressure, is not finely tangled on scales
of $\sim r$, and has a significant component along the line of sight.
This $RM$ leads to a net rotation in the position angle of linearly
polarized waves of \beq \theta_\nu \approx 10^8 \nu_{100}^{-2}
r_\nu^{-7/4} \sim \ 10^6 \nu_{100}^{-0.43} \ {\rm rad}, \eeq where the
last estimate uses \"Ozel et al.'s (2000) fit to $r_\nu(\nu)$.
These rotation angles are so large that the synchrotron emission in
ADAF/Bondi models of Sgr A$^*$ is strongly depolarized by Faraday
rotation.\footnote{The $\tau = 1$ surface in ADAF models can be an
extremely thin shell, with a width $\ll r_\nu$. Depolarization may
therefore be best modeled as due to an ``external'' screen (the plasma
within $dr \sim r$ of $r_\nu$), rather than as due to ``internal''
depolarization within the $\tau = 1$ surface.} For example, in a
simple uniform source model, the observed polarization is $\propto
\theta_\nu^{-1}$ (Pacholczyk 1970). In general, the observed
polarization depends on the rotation measure power spectrum, but is
$\ll 1$ for $\theta_\nu \gg 1$ (e.g., Tribble 1991).\footnote{One way
of evading this conclusion is to posit that the magnetic field is
sufficiently tangled to decrease $RM$ to $\lsim 10^{6} \rad$. This
tangling would, however, also eliminate linear polarization. Note
also that detailed dynamical models of ADAFs normalized to the Chandra
observations can, depending on model parameters, predict RM's near the
black hole somewhat smaller than equation (\ref{ADAF}). At most, this
can reduce RM (and $\theta_\nu$) by a factor of $\approx 100$, and
thus does not change our conclusions.}
\subsection{$\dot M \ll \dot M_{\rm Bondi}$}
If the accretion rate onto Sgr A$^*$ is much less than the Bondi rate,
significant polarization may be observable at high frequencies. This
is because the gas density and magnetic field strength are smaller
than in ADAF/Bondi models, and because the electrons are relativistic;
both effects substantially reduce the rotation measure. For the $\dot
M \sim 10^{-8} \ \mpy$ model of \S2, e.g., the rotation measure
calculated using equation (\ref{ultra1}) is $RM \approx 10^3 \rad$
near $r \sim 1$. Moreover, if $n \propto r^{-1/2}$, as in CDAF
models, the magnetic field scales as $B \propto r^{-3/4}$ and \beq RM
\approx 10^3 r^{-1/4} \left(\gamma \over 100\right)^{-2}
\rad. \label{CDAF} \eeq The variation of the electron Lorentz factor
with radius is somewhat uncertain, but we expect roughly $\gamma
\propto r^{-1}$, so that the electrons become non-relativistic by $r
\sim 10^2$. Equation (\ref{CDAF}) then shows that RM has its maximal
value at large radii, $r \sim 10^2$, where $RM \sim 3 \times 10^6
\rad$.\footnote{Although we have used the CDAF scaling in equation
(\ref{CDAF}), the relativistic suppression of Faraday rotation ensures
that RM will be dominated by $r \sim 10^2-10^3$ in any model with an
accretion rate much less than the Bondi rate.}
Equation (\ref{CDAF}) demonstrates that there is no depolarization of
synchrotron emission at small radii in models with accretion rates
much less than the Bondi rate; $RM$ is negligible in the region where
the synchrotron emission is produced. Depolarization can still be
important, however, if observed photons experience different Faraday
rotation at large radii, $r \gsim 10^2$, on their way out of the
accretion flow (see Bower et al. 1999ab). Note that Bower et
al. (1999ab) have shown that depolarization in the interstellar medium
is unlikely to be important.
Spatial variation in the rotation measure will depolarize Sgr A$^*$ at
frequencies for which $\delta \theta = \lambda^2 \delta RM \gsim \pi$,
i.e., for \beq \nu \lsim 100 \left({\delta RM \over 10^6
\rad}\right)^{1/2} {\rm GHz}, \label{numax}\eeq where $\delta RM$ is
the difference in the rotation measure for photons of a given
frequency which travel through different parts of the accretion flow.
Quantitative calculations of depolarization by differential Faraday
rotation are uncertain because $\delta RM$ depends on the size of the
emitting region as a function of frequency and on the poorly
understood density and magnetic field structure of the accretion flow
at $r \sim 100$. Two points are, however, clear: (1) At low
frequencies, $\ll 100$ GHz, Sgr A$^*$ is easily depolarized at $r
\gsim 10^2$. The required $\delta RM$ is $\ll 10^6 \rad$, orders of
magnitudes smaller than the values of $RM$ obtained at $r \sim
10^2-10^4$. Moreover, both the intrinsic and scatter broadened sizes
of Sgr A$^*$ increase at low frequencies. Large values of $\delta
RM/RM$ can thus be obtained. (2) Emission above $\sim 100$ GHz can
plausibly be linearly polarized if the accretion rate onto Sgr A$^*$
is much less than the Bondi rate. In particular, equations
(\ref{CDAF}) and (\ref{numax}) show that for $\dot M \ll \dot M_{\rm
Bondi}$, emission above $\approx 100$ GHz is not depolarized
propagating out of the accretion flow.
\section{Discussion}
ADAF/Bondi models assume that the accretion rate onto Sgr A$^*$ is of
order the Bondi rate ($\sim 10^{-4}-10^{-5} \mpy$) and that the radio
to infrared emission is produced by synchrotron emission from
marginally relativistic electrons ($T_e \approx 10^9-10^{10}$ K). In
such models the rotation measure is $\gsim 10^{10} \rad$ inside
$\approx 100$ Schwarzschild radii where the synchrotron emission is
produced. ADAF/Bondi models thus predict that Sgr A$^*$ should be
depolarized by Faraday rotation over the entire radio to infrared
spectrum, and should have nearly zero linear polarization.
The theoretical arguments summarized in \S1 propose that the accretion
rate onto Sgr A$^*$ is much less than the Bondi rate. We have
described one such model, in which the electron heating rate is of
order the rate of change of the magnetic energy density. For an
accretion rate $\sim 10^{3}$ times smaller than the Bondi rate, i.e.,
$\sim 10^{-8} \mpy$, and with relativistic electrons with $\gamma
\approx 100$, this model can explain the observed characteristics of
Sgr A$^*$ (see QN for spectral models). Moreover, it predicts that
the rotation measure in the accretion flow is much smaller than in
ADAF/Bondi models. This is because the gas density and magnetic field
strength close to the black hole are much smaller, and because the
electrons are relativistic ($RM \propto \gamma^{-2} \log\gamma$ for
$\gamma \gg 1$; see \S3 and the Appendix). The maximal contribution
to the rotation measure comes from $\sim 10^2-10^3$ Schwarzschild
radii, where $RM \sim 10^6 \rad$.
Rotation measures of $\sim 10^6 \rad$ can depolarize Sgr A$^*$ at $\nu
\ll 100$ GHz by differential Faraday rotation; photons of a given
frequency travel through different rotation measures on their way out
of the accretion flow. Following Bower et al. (1999ab), we believe
that this accounts for the $\lsim 0.2 \%$ linear polarization of Sgr
A$^*$ at low frequencies (from $\approx 4$ to $\approx 23$ GHz; see
Bower et al. 1999ab); it is less clear, however, that it can account
for Bower et al.'s (1999b) limit of $\lsim 1 \%$ linear polarization
at $86$ GHz (see below). In fact, rotation measures of $\approx 10^6
\rad$ are insufficient to depolarize emission above $\approx 100$ GHz.
As a result, in models with accretion rates much less than the Bondi
rate, $\gsim 100$ GHz emission is not depolarized propagating out of
the accretion flow; intrinsically polarized synchrotron emission may
therefore be observable at high frequencies.
The above considerations show that the linear polarization of Sgr
A$^*$ at high frequencies provides a means of distinguishing between
accretion at the Bondi rate, and accretion at a much smaller rate. In
fact, Aitken et al. (2000) report a possible detection of $\sim 10 \%$
linear polarization from Sgr A$^*$ between $150$ and $400$ GHz. These
observations cannot be readily explained by ADAF/Bondi models.
Instead, the most straightforward interpretation is that the accretion
rate onto Sgr A$^*$ is much less than the Bondi rate (our specific
model has $\dot M \sim 10^{-8} \mpy$).
One difficulty in interpreting Aitken et al's results is the large
beam ($\approx 20''$) of the SCUBA camera on the JCMT. This large
beam forced Aitken et al. to subtract out free-free and dust emission
in order to isolate the flux and polarization of Sgr A$^*$. Although
we believe that they have likely detected polarized flux from Sgr
A$^*$, future high resolution polarimetry at mm wavelengths is
necessary to further address this important issue.
Aitken et al. find that the position angle of Sgr A$^*$ changes by
$\lsim 10^o$ between $\lambda = 0.135$ cm and $\lambda = 0.2$ cm; at
face value this implies $RM \lsim 10^5 \rad$, somewhat smaller than
the values of $\sim 10^6 \rad$ in our model. This assumes, however,
that the intrinsic position angle of Sgr A$^*$ is the same at $\lambda
= 0.135$ cm and $\lambda = 0.2$ cm, which need not be the case.
Moreover, our estimates of $RM$ are actually upper limits, since they
assume (1) equipartition magnetic fields aligned along the line of
sight and (2) that our line of sight passes through the equatorial
plane of the accretion flow.
One interesting feature of the polarization observations of Sgr A$^*$
is the $10^{+9}_{-4} \ \%$ polarization found by Aitken et al. (2000)
at $150$ GHz and the $\lsim 1 \%$ polarization found by Bower et
al. (1999b) at 86 GHz, an apparently abrupt change in polarization
over a factor of $\lsim 2$ in frequency. Note, however, that Aitken et
al's error bars are only $1 \sigma$, so the difference may not be so
large.
A rapid change in polarization with frequency could be due to the fact
that different frequencies probe different radii in the accretion
flow. In particular, the highest frequency emission from Sgr A$^*$
arises from very close to the black hole in our models, where one
might plausibly expect a change in the magnetic field structure,
electron distribution function, etc.; this possibility is also
suggested by the change in the spectrum of Sgr A$^*$ above $\approx
100$ GHz (e.g., Serabyn et al. 1997). Furthermore, if the electrons
are thermal, linear polarization is exponentially suppressed below the
self-absorption frequency; even power law electrons show a factor of
$\approx 10$ decrease in polarization from the optically thin to the
optically thick limit (e.g., Ginzburg \& Syrovatskii 1969). This
could contribute to a rapid decline in polarization at low
frequencies.
Our analysis of depolarization is likely applicable even if the radio
emission from Sgr A$^*$ is dominated by a jet/outflow, rather than the
accretion flow as we have assumed. In jet models, it is still natural
for the highest frequency emission to originate very close to the
black hole; in Falcke's model, e.g., the $\gsim 100$ GHz emission
arises from $\lsim 10 \ R_g$, in what is really a ``transition
region'' between the accretion flow and the jet (Falcke 1999). In
order for this emission to not be depolarized (either {\it in situ} or
propagating through the accretion flow), our constraints on the
rotation measure and the plasma conditions close to the black hole
still apply.
Two scenarios in which accretion at the Bondi rate could be consistent
with observed linear polarization at high frequencies are (1) if the
high frequency emission arises close to the black hole, but in a
nearly empty funnel pointed directly towards us (e.g., along the
rotation axis of an ADAF) or (2) if the high frequency emission from
Sgr A$^*$ is produced at very large distances from the black hole, $r
\gsim 10^3$. The former possibility requires a rather special
geometry and the latter is likely ruled out by the VLBI source size of
$\sim 10 \ R_g$ (Krichbaum et al. 1998) and the variability of Sgr
A$^*$ at $\approx 100$ GHz (Tsuboi, Miyazaki, \& Tsutsumi 1999).
\subsection{Application to Other Systems}
Although we have have focused our analysis on Sgr A$^*$ at the
Galactic Center, linear polarization of high frequency radio emission
can be used as a probe of the accretion physics in other
low-luminosity galactic nuclei. For a black hole of mass $M = m_9 10^9
M_\odot$ accreting (spherically) at a rate $\dot M = 10^{-4} \md \dot
M_{\rm edd} \approx 10^{23} \md m_9$ g s$^{-1}$, the density, magnetic
field strength, and rotation measure in ADAF models are \beq n \approx
3 \times 10^{6} \ \md \ m_9^{-1} \ r^{-3/2} \cc,
\label{n} \eeq \beq B \approx 100 \ \md^{1/2} \ m_9^{-1/2} \ r^{-5/4} \ {\rm
G}, \label{B2} \eeq and \beq RM \approx 3 \times 10^{10} \ \md^{3/2} \
m_9^{-1/2} \ r^{-7/4} \ \rad. \label{RMf} \eeq Equation (\ref{RMf})
shows that large rotation measures and the associated depolarization
of synchrotron emission by Faraday rotation are generic features of
ADAF models (unless $\md \ll 1$).
The absence of observed linear polarization in the radio spectrum of a
low-luminosity galactic nucleus would be consistent with ADAF models.
By contrast, detected linear polarization would argue against an ADAF
as the source of the observed radio emission. Although particular
systems should be analyzed on an individual basis, we expect that in
many cases both a jet and a CDAF-like model could account for observed
linear polarization. Distinguishing between these two possibilities
requires high resolution observations.
A particularly interesting class of systems for future polarimetry are
elliptical galaxies in nearby X-ray clusters (e.g., NGC 4649, 4472,
and 4636 in the Virgo cluster). As discussed by, e.g., Fabian \&
Canizares (1988), Fabian \& Rees (1995), and Di Matteo et al. (1999,
2000), many of these galaxies have extremely dim nuclei given the
inferred black hole masses ($\sim 10^9 M_\odot$) and Bondi accretion
rates. Linear polarization may shed important light on the physics of
these systems.
\acknowledgments We thank Don Backer, Roger Blandford, and Mark Reid
for useful correspondence, Bruce Draine for useful conversations, and
John Bahcall and Feryal \"Ozel for helpful comments on the paper. EQ
is supported by NASA through Chandra Fellowship PF9-10008, awarded by
the Chandra X--ray Center, which is operated by the Smithsonian
Astrophysical Observatory for NASA under contract NAS 8-39073. AG was
supported by the W. M. Keck Foundation and NSF PHY-9513835.
\newpage
\begin{appendix}
\section{Faraday rotation in an ultra-relativistic Maxwellian plasma}
Faraday rotation in a cold plasma is described by a change in
polarization angle given by
\begin{equation}
{d \theta \over dl}={k_{\parallel }\over 2}{\omega _p^2\omega _B\over
\omega ^3},
\end{equation}
where $\omega =ck$ is the frequency of the radio wave, $k_{\parallel
}$ is the projection of the wavenumber along the magnetic field,
$\omega _p^2=4\pi ne^2/m_e$ is the plasma frequency, and $\omega
_B=eB/(m_ec)$ is the cyclotron frequency. This corresponds to the
usual rotation measure
\begin{equation}
RM\equiv {\theta \over \lambda ^2}={e^3\over 2\pi m_e^2c^4}\int d{\bf
l}\cdot {\bf B}n ~=~2.63\times 10^{-13}\times \int d{\bf l}\cdot {\bf
B}n~{{\rm rad}\over {\rm m}^2}. \label{cold}
\end{equation}
Here we derive the rotation measure for an ultrarelativistic
Maxwellian plasma:
\begin{equation}
RM_{\gamma }={e^3\over 2\pi m_e^2c^4}\int d{\bf l}\cdot {\bf B}n {\log \gamma
\over 2\gamma ^2}~=~2.63\times 10^{-13}\times \int d{\bf l}\cdot {\bf B}n{\log
\gamma \over 2\gamma ^2}~{{\rm rad}\over {\rm m}^2},
\label{ultra}
\end{equation}
where we have defined $\gamma \equiv k T_e/(m_ec^2)$. The dominant
correction to the non-relativistic expression is the relativistic
mass: $m_e \rightarrow \gamma m_e$.
We use the Vlasov equations to calculate the plasma permittivity and
hence the dispersion relation for electromagnetic waves. For a
magnetic field and wavenumber along the z axis, the first-order (in
the unperturbed magnetic field) permittivity is given by
\begin{equation}
\epsilon ^{(1)}_{xy}~=~{-i\over 2\omega} {4\pi e^2\over m_e} {eB\over m_ec} \int
d^3p{1\over (\omega -kv_z)^2}{p_{\perp }^2\over p}{dF\over dp}{m_e^2c^2\over
p^2+m_e^2c^2},
\label{perm}
\end{equation}
where the unperturbed distribution function is normalized by $\int d^3p F(p)=n$,
and $p_\perp^2\equiv p_x^2+p_y^2$. For a cold plasma, equation (\ref{perm})
gives
\begin{equation}
\epsilon _{xy}={i\omega _p^2\omega _B\over \omega ^3}.
\end{equation}
Using standard arguments (e.g., Rybicki \& Lightman 1979) this leads
to the RM for a cold plasma given by equation (\ref{cold}). For an
ultra-relativistic plasma, equation (\ref{perm}) gives
\begin{equation}
\epsilon _{xy}={i\omega _p^2\omega _B\over \omega ^3}{\log \gamma
\over 2\gamma ^2},
\label{ultraperm}
\end{equation}
where we have not changed the definition of $\omega _p$ and $\omega
_B$ in the ultra-relativistic regime. Equation (\ref{ultraperm}) for
the permittivity gives the ultra-relativistic RM in equation
(\ref{ultra}).
\end{appendix}
\newpage
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