------------------------------------------------------------------------ ms.tex ApJ, May 2005, submitted From: qianlei Reply-To: qianl@vega.bac.pku.edu.cn To: gcnews@aoc.nrao.edu Content-Type: text/plain Organization: Peking University Date: Fri, 01 Jun 2007 07:54:24 +0800 Message-Id: <1180655664.2704.8.camel@b84> Mime-Version: 1.0 X-Mailer: Evolution 2.2.2 (2.2.2-5) Content-Transfer-Encoding: 7bit X-MailScanner-Information: Please contact the postmaster@aoc.nrao.edu for more information X-MailScanner: Found to be clean X-MailScanner-SpamCheck: not spam, SpamAssassin (not cached, score=1.68, required 5, autolearn=disabled, OBSCURED_EMAIL 1.68) X-MailScanner-SpamScore: s X-MailScanner-From: qianl@vega.bac.pku.edu.cn X-Spam-Status: No %arXiv:0705.2792 \documentclass[12pt,preprint]{aastex} %% manuscript produces a one-column, double-spaced document: %% \documentclass[manuscript]{aastex} %% preprint2 produces a double-column, single-spaced document: %% \documentclass[preprint2]{aastex} %% Sometimes a paper's abstract is too long to fit on the %% title page in preprint2 mode. When that is the case, %% use the longabstract style option. %% \documentclass[preprint2,longabstract]{aastex} %% If you want to create your own macros, you can do so %% using \newcommand. Your macros should appear before %% the \begin{document} command. %% %% If you are submitting to a journal that translates manuscripts %% into SGML, you need to follow certain guidelines when preparing %% your macros. See the AASTeX v5.x Author Guide %% for information. %\newcommand{\vdag}{(v)^\dagger} \newcommand{\myemail}{liusm@lanl.gov} %% You can insert a short comment on the title page using the command below. %\slugcomment{Not to appear in Nonlearned J., 45.} %% If you wish, you may supply running head information, although %% this information may be modified by the editorial offices. %% The left head contains a list of authors, %% usually a maximum of three (otherwise use et al.). The right %% head is a modified title of up to roughly 44 characters. %% Running heads will not print in the manuscript style. \shorttitle{Polarized Emission in Sagittarius A*} %\shortauthors{Djorgovski et al.} %% This is the end of the preamble. Indicate the beginning of the %% paper itself with \begin{document}. \begin{document} %% LaTeX will automatically break titles if they run longer than %% one line. However, you may use \\ to force a line break if %% you desire. \title{The Nature of Linearly Polarized Millimeter and Sub-millimeter Emission in Sagittarius A*} %% Use \author, \affil, and the \and command to format %% author and affiliation information. %% Note that \email has replaced the old \authoremail command %% from AASTeX v4.0. You can use \email to mark an email address %% anywhere in the paper, not just in the front matter. %% As in the title, use \\ to force line breaks. \author{Siming Liu,\altaffilmark{1} Lei Qian,\altaffilmark{2} Xue-Bing Wu,\altaffilmark{2} %Fulvio Melia,\altaffilmark{3} } \altaffiltext{1}{Los Alamos National Laboratory, Los Alamos, NM 87545; liusm@lanl.gov, hli@lanl.gov} \altaffiltext{2}{Department of Astronomy, Peking University, Beijing 100871; qianl@vega.bac.pku.edu.cn} %\altaffiltext{2}{Center for Space Science and Astrophysics, Department of Physics and Applied Physics, Stanford %University, Stanford, CA 94305; vahe@astronomy.stanford.edu} %\altaffiltext{3}{Physics Department and Steward Observatory, The University of Arizona, %Tucson, AZ 85721; melia@physics.arizona.edu; Sir Thomas Lyle Fellow and Miegunyah Fellow.} \altaffiltext{3}{Physics Department, The University of Arizona, Tucson, AZ 85721; fryer@lanl.gov} %% Mark off your abstract in the ``abstract'' environment. In the manuscript %% style, abstract will output a Received/Accepted line after the %% title and affiliation information. No date will appear since the author %% does not have this information. The dates will be filled in by the %% editorial office after submission. \begin{abstract} The linearly polarized millimeter and sub-millimeter emission in Sagittarius A* is produced within 10 Schwarzschild radii of the supermassive black hole at the Galactic Center. We show that the millimeter emission likely originates from a hot accretion disk, where electrons are heated efficiently by turbulent plasma waves. The observed flux density and polarization requires that the disk have an inclination angle of $\sim45^\circ$ and its rotation axis be aligned with the major axis of the intrinsic polarization. The disk also needs to be strongly magnetized with a magnetic field energy density comparable to the thermal energy density of the gas. The high flux density and hard spectrum of the sub-millimeter ($<1$ mm) emission, on the other hand, suggest that it is emitted from small emission regions and therefore associated with flare events occurring either in coronas of the disk or within the last stable orbit. Simultaneous spectrum and polarization measurements will be able to test the model. \end{abstract} %% Keywords should appear after the \end{abstract} command. The uncommented %% example has been keyed in ApJ style. See the instructions to authors %% for the journal to which you are submitting your paper to determine %% what keyword punctuation is appropriate. %% Authors who wish to have the most important objects in their paper %% linked in the electronic edition to a data center may do so in the %% subject header. Objects should be in the appropriate "individual" %% headers (e.g. quasars: individual, stars: individual, etc.) with the %% additional provision that the total number of headers, including each %% individual object, not exceed six. The \objectname{} macro, and its %% alias \object{}, is used to mark each object. The macro takes the object %% name as its primary argument. This name will appear in the paper %% and serve as the link's anchor in the electronic edition if the name %% is recognized by the data centers. The macro also takes an optional %% argument in parentheses in cases where the data center identification %% differs from what is to be printed in the paper. \keywords{acceleration of particles --- black hole physics --- Galaxy: center --- plasmas --- radiation mechanisms: thermal--- turbulence} %\object{NGC 6624}, \objectname[M 15]{NGC 7078}, %\object[Cl 1938-341]{Terzan 8})} %% From the front matter, we move on to the body of the paper. %% In the first two sections, notice the use of the natbib \citep %% and \citet commands to identify citations. The citations are %% tied to the reference list via symbolic KEYs. The KEY corresponds %% to the KEY in the \bibitem in the reference list below. We have %% chosen the first three characters of the first author's name plus %% the last two numeral of the year of publication as our KEY for %% each reference. \section{Introduction} Our understanding of physical processes in Sagittarius A*, the compact radio source associated with a supermassive [$M\simeq 3.7 \times 10^{6}M_\odot$] black hole at the Galactic Center (Ghez et al. 2005), has improved dramatically since the detection of linear polarization of its millimeter and sub-millimeter emission (Aitken et al. 2000). It is generally accepted that Sagittarius A* is powered by accretion of the black hole in stellar winds (Melia 1992; Narayan 1998; Rockefeller et al. 2004). The linear polarization reveals a synchrotron origin of the emission and sets strict constraints on the magnetic field, gas density, and electron temperature (Agol 2000; Melia et al. 2000; Quataert \& Gruzinov 2000). Recent VLBI imaging shows that the 3.5 mm emission is produced within $10\ r_S$ of the black hole, where $r_S \simeq 1.1\times 10^{12} [M/(3.7\times 10^6M_\odot)]$ cm is the Schwarzschild radius for a non-spinning black hole (Shen et al. 2005). The millimeter emission may originate from a hot magnetized accretion torus and simultaneous spectrum and polarization measurements can be used to determine its orientation and plasma properties (Bromley et al. 2001). High spatial and spectral resolution X-ray and near-infared (NIR) observations routinely detect flares from the direction of Sagittarius A* (Baganoff et al. 2001; Genzel et al. 2003). The characteristic variation time scale of a few tens of minutes, comparable to the dynamical time at the last stable orbit, indicates that they are also produced within a few $r_S$ of the black hole. The spectra, polarization, and variability, especially when considering results from simultaneous multi-wavelength observations, suggest that the NIR radiation is emitted through synchrotron processes and the X-rays are likely due to the synchrotron self-Componization (Eckart et al. 2006a, 2006b; Yusef-Zadeh et al. 2006a). The flares can be triggered by rapid releases of magnetic field energy near the black hole, which then heats electrons to a few tens of MeV producing the observed emission (Liu et al. 2004; Bittner et al. 2007). The long wavelength radio emission is less variable. The low quiescent-state X-ray flux and high centimeter radio flux density uncover a nonthermal origin of the emission and/or emission from an unbounded flow at large radii (Liu \& Melia 2001). Correlated flare activities in the X-ray and radio bands suggest that the radio emission is associated with an outflow (Zhao et al. 2004; Yusef-Zadeh et al. 2006b). Recent SMA and BIMA observations confirm the linear polarization of the millimeter and sub-millimeter emission (Bower et al. 2005 Marrone et al. 2006a, 2006b). These observations not only show that the flux density and polarization are variable on time scales as short as a few hours but also reveal a relatively high mean flux density and hard spectrum in the sub-millimeter band as compared with the millimeter spectral bump first observed by Falcke et al. (1998). The observed position angle of the polarization vector has been used to infer a Faraday rotation measure of $\sim -50$ rad cm$^{-2}$ (Macquart et al. 2006; Marrone et al. 2007) and a position angle of $\sim 170^\circ$ for the intrinsic polarization. In this {\it Letter}, we consider a Keplerian accretion flow in a pseudo-Newtonian potential. Instead of assuming that the electron and proton temperatures are the same (Melia et al. 2001), we describe the electron heating by turbulence with a single parameter and treat the cooling process more accurately. These leave five basic model parameters, namely the inclination angle of the disk, accretion rate, electron heating rate, and the ratios of the stress to the magnetic and gas pressures. \S\ \ref{eqs} gives the basic equations describing the disk structure. We then discuss the synchrotron emission from the disk and its polarization and apply the model to Sagittarius A* in \S\ \ref{app}. In \S\ \ref{dis}, we summarize the main predictions, discuss the model limitation, and draw conclusions. \section{Basic Equations for the Disk Structure} \label{eqs} %We consider a fully ionized hydrogen plasma. Then the gas pressure and the thermal energy density are given, respectively, by %\begin{eqnarray} %P&=& n k_B (T_p+T_e)\,, \\ %{\cal E}&=& n k_B (1.5 T_p + \alpha T_e)\,, %\end{eqnarray} %where the gas density, the proton and electron temperatures are denoted by $n$, $T_p$ and $T_e$, respectively, $k_B$ is the Boltzmann constant, and $\alpha = x[3K_3(x)+K_1(x)-4 K_2(x)]/4K_2(x)$ with $x=m_ec^2/k_B T_e $, where $m_e$ and c denote the electron mass and the speed of light, respectively, and $K_i$ refers to the $i$th order modified Bessel function and $K_i(x)\rightarrow2^{i-1} (i-1)!/x^i$ as $x\rightarrow 0$. %In the pseudo-Newtonian potential, the potential and Keplerian angular velocity are given, respectively, by %\begin{eqnarray} %\phi &=& -{GM\over r-r_S}\,,\\ %\Omega_K&=& \left[{GM\over r(r-r_S)^2}\right]^{1/2}\,, %\end{eqnarray} %where For a Keplerian accretion disk in a pseudo-Newtonian potential with a magnetic field dominated by the azimuthal component, the scale height of the disk is given by \begin{equation} H = \left[{r k_B(T_p+T_e)(1+2\beta_p)\over GMm_p}\right]^{1/2}(r-r_S)\,, \end{equation} where $G$, $k_{\rm B}$, $m_p$, $M$, $r$, $r_S\equiv 2GM/c^2$, $c$, $T_e$ and $T_p$ denote the gravitational constant, Boltzmann constant, proton mass, black hole mass, radius, Schwarzschild radius, speed of light, electron and proton temperatures, respectively. $\beta_p=B^2/8\pi nk_B(T_e+T_p)$ is the ratio of the magnetic field energy density $B^2/8 \pi$ to the gas pressure $nk_B(T_e+T_p)$, where $n$ gives the gas density. %In a non-radiative accretion flow, the accretion processes likely drive strong winds from the accretion disk (Blandford and Begelman 1999). There are currently no strong observational constraints to these processes. To simplify our model, we consider a radius independent accretion rate, $\dot{M} = - 4 \pi r H v_r n m_p={\rm constant}$, and assume that the stress is zero at the inner boundary $r_i=3r_S$. From the angular momentum conservation, one obtains the radial velocity %\begin{equation} %v_r = \nu {{\rm d}\ln \Omega_k\over {\rm d} r}\left[1-{r_i^2 \Omega_k(r_i)\over r^2 \Omega_k}\right]^{-1} %= -\nu \left({1\over 2r}+{1\over r-r_S}\right) \left[1-\left({r_i^3 (r-r_S)^2\over r^3(r_i-r_S)^2}\right)^{1/2}\right]^{-1} %\end{equation} %where the kinematic viscosity $\nu=2\beta_p\beta_\nu k_B (T_p +T_e)(r-r_S)^2r^{1/2}/m_p(3r-r_S)(GM)^{1/2}$, and in accord with Melia et al. (2001) $$ and $\beta_p= /8\pi P$. Then the radial velocity can be rewritten as $ v_r = -{\beta_\nu\beta_pk_B(T_p+T_e)(r-r_S)/ [fm_p(GMr)^{1/2}}]\,, $ where $f\equiv \left[1-{r_i^{3/2} (r-r_S)/r^{3/2}(r_i-r_S)}\right]$ and $\beta_\nu$ is the ratio of the total stress to $B^2/8\pi$ (Melia et al. 2001). So \begin{equation} n = {fGM\dot{M} m_p^{1/2}\over 4\pi \beta_\nu\beta_p [k_B(T_p +T_e)]^{3/2}(1+2\beta_p)^{1/2}r(r-r_S)^2}\,. \end{equation} The energy conservation equation is given by %\begin{equation} %{{\rm d} \over Hr{\rm d} r}\left\{Hrv_r[P(1+2\beta_p)+{\cal E}+nm_p[\phi+0.5(1-2f)v_K^2+0.5v_r^2]]\right\} = -\Lambda\,, %\end{equation} \begin{equation} {{\rm d} \over {\rm d} r}\left\{k_{\rm B}[T_e(\alpha+1+2\beta_p)+T_p(2.5 +2\beta_p)]+m_p\left[(1-2f){v_{\rm K}^2\over 2}+{v_r^2\over 2}-{GM\over r-r_S}\right]\right\} = -{\Lambda\over v_r n}\,, \label{energy} \end{equation} where $\Lambda$ is the radiative cooling rate and $v_{\rm K}= (GMr)^{1/2}/(r-r_S)$ is the Keplerian velocity. $\alpha = x[3K_3(x)+K_1(x)-4 K_2(x)]/4K_2(x)$, where $x=m_ec^2/k_B T_e$, $K_i$ refers to the $i$th order modified Bessel function and $m_e$ represents the electron mass. To obtain the disk structure, one also needs to specify the electron heating rate by the turbulent magnetic field (Liu et al. 2007; Sharma et al. 2007). The electron heating time $ \tau_{ac} = {3C_1 H \langle v_e\rangle /c_S^2}\,, $ where $C_1$ is a dimensionless constant, $\langle v_e \rangle={2c (1 +x)/x^2 K_2(x)\exp(x)}$ and $c_S = [k_B(T_i+T_e)(1+2 \beta_p)/m_p]^{1/2}$ are the mean speed of electrons and speed of fast mode waves, respectively. Then we have \begin{equation} {{\rm d}T_e\over {\rm d}r} = {T_e\over \tau_{ac} v_r}+{T_p-T_e \over \tau_{\rm Coul} v_r} -{\Lambda \over \alpha n k_Bv_r}\,, \label{energye} \end{equation} where $\tau_{\rm Coul} = {3\pi m_em_p\langle v_e\rangle ^3/ 256 n e^4\ln \lambda}$ is the electron-proton energy exchange time via Coulomb collisions and $\ln\lambda\simeq15$ (Spitzer 1962). The cooling is dominated by synchrotron, inverse Comptonization and bremmsstrahlung processes. In the optically thin regime, the synchrotron and bremmsstrahlung cooling rates are given, respectively, by \begin{eqnarray} \Lambda^0_{\rm syn} &=& {4e^4n\over 9 m_e^4 c^5} \langle p^2 \rangle B^2 \simeq 1.06\times 10^{-15}n B^2 {3x^2+12x+12\over x^3+x^2}\,, \label{syn0} \\ \Lambda_{\rm brem} &=& \left({2\pi k_{\rm B} T_e\over 3 m_e}\right)^{1/2} {32\pi e^6\over 3 h m_e c^3} n^2 g_{\rm B} =1.4\times 10^{-27} T_e^{1/2} n^2 g_{\rm B}\,, \end{eqnarray} where $\langle p^2\rangle $, $g_{\rm B}\simeq 1.2$, $e$, and $h$ are the mean momentum square of the electrons, Gaunt factor, elemental charge unit, and Planck constant, respectively, and all quantities here and in what follows are given in cgs units. Most of the thermal synchrotron emission is emitted at $\nu_E \simeq 20 \nu_c = {60eB(x+1)^2/ 4\pi m_e c x^2}=8.6\times 10^7 B{(x+1)^2/x^2} \, {\rm Hz}$ (Liu et al. 2006). The optical depth at $\nu_E$ can be approximated as \begin{equation} \tau_E = H {3(x+1)\Lambda^0_{\rm syn} c^2[\exp(h\nu_E/k_B T)-1]\over 8(x +10) \pi h\nu_E^4} \simeq {3(x+1)H \Lambda^0_{\rm syn} c^2 \over 8(x +10)\pi k_{\rm B} T_e \nu_E^3}\,, \end{equation} where the last expression is true for $h\nu_E \ll k_{\rm B} T_e$ and the numerical factor is chosen such that the expression is accurate in the relativistic regime. When $\tau_E\ge 1$, the synchrotron cooling is suppressed due to self-absorption: \begin{equation} \Lambda^{\tau}_{\rm syn} \simeq {8(x+10)\pi k_{\rm B} T_e \nu_E^3\over 3(x+1) c^2 H}\,. \label{syntau} \end{equation} We consider the synchrotron self-Comptonization cooling, then the total cooling rate \begin{eqnarray} \Lambda &=& \Lambda_{\rm syn} + \Lambda_{\rm IC} + \Lambda_{\rm brem} = \Lambda_{\rm syn} [1+ 8\pi (\Lambda-\Lambda_{\rm brem})H/cB^2] +\Lambda_{\rm brem} \\ &=&\Lambda_{\rm syn}(1-8\pi\Lambda_{\rm syn}H/cB^2)^{-1} +\Lambda_{\rm brem} %\nonumber \\ %&=&{4e^4n\over 9 m_e^4 c^5} B^2(1-8\pi \Lambda_{\rm syn}H/cB^2)^{-1} +\left({2\pi k_{\rm B} Te\over 3 m_e}\right)^{1/2} {32 \pi e^6\over 3 h m_e c^3} n^2 g_{\rm B} \nonumber \\ %&=&1.06\times 10^{-15}n B^2 {3x^2+12x+12\over x^3+x^2} (1-8\pi \Lambda_{\rm syn}H/cB^2)^{-1} +1.4\times 10^{-27} T_e^{1/2} n^2 g_{\rm B} \,. \nonumber \end{eqnarray} where $\Lambda_{\rm syn}$ is given by equations (\ref{syn0}) and (\ref{syntau}) for $\tau_E<1$ and $\ge1$, respectively. For given $\dot{M}$, $C_1$, $\beta_\nu$, $\beta_p$, and $T_e$ and $T_p$ at an outer boundary, one can solve equations (\ref{energy}) and (\ref{energye}) numerically to obtain the disk structure. The thick lines in the left panel of Figure \ref{f1.eps} give profiles of the fiducial model, where $\dot{M}=1.0\times 10^{18}$g s$^{-1}$, $C_1=1.15$, $\beta_\nu=0.088$, $\beta_p=1.0$ and $T_e=T_p=GMm_p/5 k_{\rm B} r$ at the outer boundary $r=10^4 r_S$. For such a low accretion rate, cooling is not very efficient and the temperature profiles are determined by the ratio of the electron heating time to the viscous time, which is proportional to $C_1\beta_\nu\beta_e$. The thin solid and dashed lines give the temperature profiles for $C_1=1.1$ and $1.2$, respectively. The other model parameters remain the same. Due to the increase of the electron heating rate, the electron temperature is higher for the former, which has a slightly lower proton temperature due to energy conservation. The profiles of other quantities do not change significantly. %We also note that according to the model, the electron temperature can not be much higher than $\sim 10^{11}$K since the proton temperature will decrease to zero at small radii for very efficient electron heating as a result of the zero stress condition at the $3r_S$ and overall energy conservation. \section{Radiation Spectrum and Polarization} \label{app} The basic formulae for calculating synchrotron emission from the disk are given by Melia et al. (2001). %For the sake of completeness and to take into account the Faraday rotation effect in the emission region, we list the key expressions here. The flux density in the direction of the disk axis and its orthogonal component are given, respectively, by %\begin{eqnarray} %F_{1\nu_o} &=& {1\over D^2}\int (I^e_{\nu}\cos^2\phi^\prime +I^o_{\nu}\sin^2\phi^\prime)\left(1-{r_S\over r}\right)\cos i\ r\ {\rm d} r\ {\rm d} \phi\\ %F_{2\nu_o} &=& {1\over D^2}\int (I^e_{\nu}\sin^2\phi^\prime +I^o_{\nu}\cos^2\phi^\prime)\left(1-{r_S\over r}\right)\cos i\ r\ {\rm d} r\ {\rm d} \phi %\end{eqnarray} %where $i$ is the inclination angle of the disk and $D$ the distance to Sagittarius A*. The azimuthal coordinate of an emission element is indicated by $\phi$ and $\tan\phi^\prime = \cot\phi\ \cos i$. Then the observed radiation frequency $\nu_o$ is related to the frequency in the co-moving frame $\nu$ via %$\nu_o = \nu[(1-r_S/r)(1-v_{\rm K}^2/c^2)]^{1/2}/[1-(v_{\rm K}/c)\sin i \ \cos \phi]$. %The specific intensity of the extra-ordinary and ordinary emission components are given, respectively, by %\begin{eqnarray} %I^e_\nu &=& {h\nu^3[1-v_{\rm K}^2/c^2]^{3/2}[1-\exp{(-\tau_e)}]\over c^2[1-(v_{\rm K}/c)\sin i\ \cos \phi]^3[\exp{(h\nu/k_{\rm B}T_e)}-1]}\,, \\ %I^o_\nu &=& {h\nu^3[1-v_{\rm K}^2/c^2]^{3/2}[1-\exp{(-\tau_o)}]\over c^2[1-(v_{\rm K}/c)\sin i\ \cos \phi]^3[\exp{(h\nu/k_{\rm B}T_e)}-1]}\,, %\end{eqnarray} %where the opticaly depth $\tau$s are given by %\begin{eqnarray} %\tau_e &=& {2 H c^2[\exp{(h\nu/k_{\rm B}T_e)}-1][1-(v_{\rm K}/c)\sin i\ \cos \phi]\epsilon_e\over \cos i\ h %\nu^3[1-v_{\rm K}^2/c^2]^{1/2}}\,, \\ %\tau_o &=& {2 H c^2[\exp{(h\nu/k_{\rm B}T_e)}-1][1-(v_{\rm K}/c)\sin i\ \cos \phi]\epsilon_o\over \cos i\ h \nu^3[1-v_{\rm K}^2/c^2]^{1/2}}\,. %\end{eqnarray} To take into account the Faraday rotation effect, we introduce modified emission coefficients and Faraday rotation angle $\eta$ (Quataert \& Gruzinov 2000): \begin{eqnarray} \epsilon_e &=& [(1+\sin\eta/\eta)\epsilon^0_e+(1-\sin \eta/\eta)\epsilon^0_o]/2\,,\\ \epsilon_o &=& [(1+\sin\eta/\eta)\epsilon^0_o+(1-\sin \eta/\eta)\epsilon^0_e]/2\,, \\ \eta &=& {e^3HnB\sin i\ \cos \phi \over 2\pi m_e^2c^2\nu^2(k_{\rm B}T_e/m_ec^2+1)^2}\,, \label{eta} \end{eqnarray} where $\epsilon^0_e$ and $\epsilon^0_o$ are the emission coefficients given by Pacholczyk (1970) and $i$, $\phi$, and $\nu$ are the inclination angle of the disk, azimuthal coordinate, and co-moving frame radiation frequency of the emitting element in the disk, respectively. These modified emission coefficients should replace the emission coefficients given by Melia et al. (2001). We note that the Faraday rotation effect is weak at the near ($\phi\sim 270^\circ$) and far ($\phi\sim90^\circ$) sides of the disk and can be significant at the blue ($\phi\sim 180^\circ$) and red ($\phi\sim 0^\circ$) shifted sides, especially for low frequency radiation (Bromley et al. 2001). The right panel of Figure \ref{f1.eps} compares the model predicted spectrum and polarization with observations. The thick line in the top panel gives the spectrum of the fiducial model with $i=45^\circ$, which fits the millimeter spectral bump first observed by Falcke et al. (1998). Recent simultaneous multi-wavelength observations confirm such a spectral feature (An et al. 2005; Marrone et al. 2006b). The spectral index below $100$ GHz is $\sim 1.3$, which is determined by the structure of the disk. Emission from different radii peaks at different frequencies, above which the source becomes optically thin, and lower frequency emission is produced at relatively larger radii. Although most of the emission is produced in the optically thin regime, the strong frequency dependence of the Faraday rotation (see eq. [\ref{eta}]) suppresses the linear polarization at lower frequencies. The polarization is significant only in the millimeter range as shown in the middle panel, where the thick line corresponds to the fiducial model, and the two thin lines have $i=35^\circ$ and $45^\circ$. The polarization fraction in the sub-millimeter range increases dramatically with the increase of $i$. Our fitting to the observations gives an $i$ of $\sim 45^\circ\pm10^\circ$. More accurate polarization observations above $300$ GHz may give a better measurement of this angle. We note that for small values of $i$, the model predicts that the position angle of the polarization flips by $90^\circ$ in the sub-millimeter range due to the dominance of the emission by the blue shifted side (Bromley et al. 2001). This can be tested by future observations. The model also produces a low level of polarization at $150$ GHz, a potentially critical prediction awaiting future observations to check. The spectrum and polarization not only lead to a measurement of $i$ but also give very strict constraints on $\beta_p$, $C_1\beta_\nu\beta_p$, and $\dot{M}/\beta_\nu$. For $\beta_p<1$, a higher electron temperature is needed to produce the observed $220$ GHz flux density. Such a disk will emit a flux above the observed limit at $\sim 100$ GHz except that $i$ is large, which implies very strong ($>20\%$) polarization in the sub-millimeter range, in contradict with observations. Higher values of $\beta_p$ have the opposite effects. So the millimeter spectrum and the level of polarization in the sub-millimeter range require $\beta_p\simeq 1.0$. The thin dotted lines in the top panel give the spectra for the two profiles with $C_1=1.1$ and $1.2$. The emission spectrum is very sensitive to the electron heating rate. Because the cooling processes and Coulomb collisions have very weak effects on the temperature profiles, which are dominated by the product $C_1\beta_\nu\beta_p$, our model fitting gives $C_1\beta_\nu\beta_p\simeq 0.1\pm0.005$. For given $i$, $\beta_p$, and $C_1\beta_\nu\beta_p$, the synchrotron emissivity, which is proportional to $n B^2$, scales as $(\dot{M}/\beta_\nu)^2$, and we have $\dot{M}/\beta_\nu \sim 10^{19}$g s$^{-1}$. However, our model gives very weak constraints on $\dot{M}$ and/or $\beta_\nu$, which determines the radial velocity. For $C_1=0.14$, $\beta_\nu=0.7$, and $\dot{M}=6\times 10^{18}$g s$^{-1}$ (with other parameters unchanged), the spectrum and polarization are very similar to that of the fiducial model. Because the cooling processes and Coulomb collisions become less important for higher values of $v_r$, the electron temperature increases slightly with $\beta_\nu$ and a lower value of $\dot{M}/\beta_\nu$ is needed to fit the data. Physical processes below $3\ r_S$ or a better understand of the viscosity described by $\beta_\nu$ (Pessah et al. 2006) can break this degeneracy. Observations show that the position angle of the intrinsic polarization is $\sim 170^\circ$ (Macquart et al. 2006; Marrone et al. 2007). Our model predicts that the emission is polarized along the axis of the disk below $700$ GHz. The axis of the disk therefore has a position angle of $\sim 170^\circ$. Future VLBI imaging may be able to test this prediction (Shen et al. 2005; Bromley et al. 2001; Huang et al. 2007). We also note that in the NIR band, the emission is dominated by the blue-shifted side. The corresponding position angle of the polarization is $\sim 80^\circ$, which is consistent with recent observations (Meyer et al. 2006) \section{Discussion and Conclusions} \label{dis} We have shown that the disk model can account for the millimeter spectrum and polarization. Our best-fitting model predicts a $\sim 5\%$ polarization at 150 GHz, a $90^\circ$ flip of the position of the intrinsic polarization near 700 GHz, a disk inclination angle of $45^ \circ$ with the axis aligning at $\sim 170^\circ$ position angle. Future polarimetric spectroscopy and imaging will be able to test the model and/or better constrain the parameters. Our disk model, however, has difficulties in reproducing the high sub-millimeter flux densities and hard spectrum obtained recently with SMA (Marrone et al. 2006b). We first note that the flux density at $1.3$ mm obtained with BIMA is 3 times lower than the SMA flux (Bower et al. 2005). There appears to be some problems with the flux calibration of at least one of these instruments. If the SMA observations are confirmed, they are likely produced by local events in coronas of the disk or within $3\ r_S$ of the black hole (Liu et al. 2004). The thin solid lines fit the SMA spectrum with uniform spheres with $B^2/8\pi = nk_{\rm B}T_e$. We show two models with radii of $r_S$ ($B=65$G, $x=1/33$, $n = 6.3\times 10^6$cm$^{-3}$) and $2\ r_S$ ($B=44$G, $x=1/44$, $n=1.4\times 10^6$cm$^{-3}$). For a given optically thin sub-millimeter spectrum, the smaller sphere becomes optically thin at a higher frequency. Even with a radius of $2\ r_S$, the model over-predicts the flux below $\sim 100$ GHz. The sub-millimeter emission is therefore likely associated with flares observed in the NIR and X-ray bands. However, to produce NIR and X-ray flares, very energetic electrons with temperature up to $\sim 100 $MeV need to be produced (Liu et al. 2006; Bittner et al. 2007). The sub-millimeter emission can be produced by mildly relativistic electrons with $T_e\ge 10^{11}$K, which, nevertheless, is difficult to achieve with our disk model (Liu et al. 2007; Sharma et al. 2007). Any deviation from $B^2/8\pi = nk_{\rm B} T_e$ will require either a strong magnetic field or a high $T_e$, both of which can not be accommodated within the disk. We therefore conclude that emission at $\sim 300$ GHz and beyond is dominated by flare events with the less energetic flares occurring more frequently producing a quasi-steady state flux in the sub-millimeter range. The millimeter emission likely comes from the hot accretion disk beyond the last stable orbit. Compared with the previous models of a small hot accretion torus (Melia et al. 2001; Bromley et al. 2001), we make several significantly improvements. A more realistic treatment of electron heating by turbulence replaces the assumption of one temperature flow in the original model. This also makes the results almost independent of the outer boundary radius and temperature(s). With the pseudo-Newtonian potential, the inner radius is not a free parameter anymore and fixed at the last stable orbit. We also take into account the Faraday rotation effect in the disk, which turns out to be critical to explain the sharp increase of the polarization from 3.4 mm to 1.3 mm. In the previous models, this is achieved by choosing a small outer boundary radius. The major shortcomings of the model are the assumption of Keplerian motion and zero stress inner boundary condition. We also ignore the vertical structure of the disk and the effects of winds or outflows (Liu et al. 2007). A fully general relativistic treatment or global MHD simulations may better address these issues (Popham \& Gammie 1998; Goldston et al. 2005; Chan et al. 2007). \acknowledgments This work was funded in part under the auspices of the U.S.\ Dept.\ of Energy, and supported by its contract W-7405-ENG-36 to Los Alamos National Laboratory. %This research %was partially supported by NSF grant ATM-0312344, NASA grants NAG5-12111, NAG5 11918-1 (at Stanford), NSF %grant AST-0402502 (at Arizona), and NSF grant PHY99-07949 (at KITP at UCSB). %FM is very grateful to the University of Melbourne for its support (through a Miegunyah Fellowship). \newpage \begin{thebibliography}{} \bibitem[Agol et al. 2000]{A00} Agol, E. 2000, ApJ, 538, L121 \bibitem[Aitken et al. 2000]{Ai00} Aitken, D. K., et al. 2000, ApJ, 534, L173 \bibitem[An et al. (2005)]{An05} An, T., et al. 2005, ApJ, 634, L49 \bibitem[Baganoff et al. 2001] {Baganoff01} Baganoff, F. K., et al. 2001, Nature, 413, 45 \bibitem[Bittner et al. 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The thick solid line fits the millimeter bump observed by Falcke et al. (1998) with the fiducial model. The dotted lines are for the two profiles with $C_1=1.1$ (with a higher flux) and $1.2$. The thin solid lines fit the sub-millimeter spectrum (Marrone et al. 2006b) with uniform spheres with $B^2/8\pi = n k_{\rm B} T_e$ and radii of $r_S$ and $2r_S$. The sphere becomes optically thick at a higher frequency for the smaller radius. The {\it middle} panel compares the polarization fractions. The data are from Aitken et al. (2000), Macquart et al. (2006) and Marron et al. (2007). The thick line is for the fiducial model. The thin lines have $i=55^\circ$ (with a higher polarization at high frequencies) and $35^\circ$. %with the polarization fraction increasing in the sub-millimeter range with the increase of $i$. {\it Bottom}: the observed position angle of the linear polarization for the fiducial model. The rotation axis of the disk has a position angle of $168^\circ$. The solid and dashed lines assume an external Faraday rotation measure of 44 and 56 rad cm$^{-2}$, respectively (Macquart et al. 2006; Marrone et al. 2007). According to the model, the position angle of the polarization flips by $90^\circ$ in the sub-millimeter range at the frequency, where the polarization fraction approaches to 0. } \label{f1.eps} \end{figure} %\begin{figure}[bht] %\begin{center} %\includegraphics[height=8.4cm]{f2a.pdf} %\hspace{-0.6cm} %\includegraphics[height=8.4cm]{f2b.eps} %\end{center} %\caption{ %{\it Left:} %{\it Right:} %} %\label{fig2.ps} %\end{figure} \end{document}