------------------------------------------------------------------------ totani-GC511.tex PASJ. July 2006. submitted From: Tomonori TOTANI X-Mailer: Mew version 2.3 on Emacs 20.7 / Mule 4.1 =?iso-2022-jp?B?KBskQjAqGyhCKQ==?= Mime-Version: 1.0 Content-Type: Text/Plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-MailScanner-Information: Please contact postmaster@aoc.nrao.edu for more information X-MailScanner: Found to be clean X-MailScanner-SpamCheck: not spam, SpamAssassin (score=0, required 5, autolearn=disabled) X-MailScanner-From: totani@kusastro.kyoto-u.ac.jp X-Spam-Status: No %astro-ph/0607414 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% PASJ LaTeX template for draft(body)<2001/02/08> %%% %%% IMPORTANT NOTICE FOR AUTHORS %%% 1. ``\draft'' creates single column and double spaces format. %%% 2. If you comment out ``\draft'', the output will be double column %%% and single space. %%% 3. 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Do NOT redefine commands provided by PASJ00.cls. %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass{pasj00} %\draft \begin{document} \SetRunningHead{TOTANI}{The Galactic Center Black Hole} %\Received{2000/12/31}%{yyyy/mm/dd} %\Accepted{2001/01/01}%{yyyy/mm/dd} \title{A RIAF Interpretation for the Past Higher Activity of the Galactic Center Black Hole and the 511 keV Annihilation Emission } %%% begin:list of authors \author{Tomonori \textsc{TOTANI}}% % \thanks{Example: Present Address is xxxxxxxxxx}} \affil{Department of Astronomy, Kyoto University, Sakyo-ku, Kyoto 606-8502} \email{totani@kusastro.kyoto-u.ac.jp} %% `\KeyWords{}' always has to be placed before `\maketitle'. \KeyWords{xxxx:xxxx ......} %Do NOT move this preamble from here! \maketitle \begin{abstract} There are several lines of evidence that the super-massive black hole at the Galactic center had higher activities in the past than directly observed at present. Here I show that these lines of evidence can quantitatively and consistently be explained if the mean accretion rate during the past $\sim 10^7$ yrs has been $\sim 10^{3-4}$ times higher than the current rate, by the picture of radiatively inefficient accretion flow (RIAF) and associated outflow that has been successfully applied to Sgr A$^*$. I argue that this increased rate and its duration are theoretically reasonable in the Galactic center environment, while the accretion rate suddenly dropped about 300 years ago most likely because of the shell passage of the supernova remnant Sgr A East. Then I show that a significant amount of positrons should have been created around the event horizon during the higher activity phase, and injected into interstellar medium by the outflow. The predicted positron production rate and propagation distance are close to those required to explain the observed 511 keV annihilation line emission from the Galactic bulge, giving a natural explanation for the large bulge-to-disk ratio of the emission. The expected injection energy is $\sim$ MeV, which is also favorable as an explanation of the 511 keV line emission. \end{abstract} \section{Introduction} A variety of active and high energy phenomena are seen in the direction towards the Galactic center over a broad range of wavelengths. It is well established that the center of our Galaxy, Sagittarius (Sgr) A$^*$, is a supermassive black hole (SMBH) with a mass of $M_\bullet \sim 3 \times 10^6 M_\odot$ (Genzel et al. 1997; Sch\"odel et al. 2003), and a number of models have been proposed to explain the radiation and its spectral energy distribution (SED) from Sgr A$^*$ by accretion of matter onto it (see Baganoff et al. 2003 for a review and comparison to X-ray observations). The concept of radiatively inefficient accretion flow (RIAF) is theoretically well-motivated based on the physics of accretion flows, and models based on the RIAF picture have successfully been applied to many low accretion rate systems (Narayan \& Quataert 2005 for a review). Sgr A$^*$ is one of the best studied examples, and the model of Yuan, Quataert, \& Narayan (2003, 2004, hereafter YQN03 and YQN04, respectively) can reproduce the observed SED of Sgr A$^*$ in a wide range of wavelengths from radio to X-ray bands. An important ingredient of this model is that the accretion rate decreases with decreasing radius from the SMBH, indicating a significant mass loss by a magnetically driven outflow or wind. This is necessary to make the model consistent with observations, and such wind activities are also supported by recent numerical simulations (YQN03 and references therein). Though this model can explain the SED of Sgr A$^*$ at present, there are several lines of evidence for higher activities in the Galactic center in the past, such as much higher X-ray luminosity and mass outflows (see \S \ref{section:past-activity} for a brief review). Hence, it is interesting to examine whether these lines of evidence can consistently be explained by changing the accretion rate within the framework of the RIAF model. The first aim of this paper is to show that it is indeed possible, and I also discuss whether such a high accretion rate is reasonable in the environment of the Galactic center, and the cause of the sudden decrease leading to the current low rate. The 511 keV electron-positron annihilation line emission in the Galaxy has been observed for a long time (Kn\"odlseder et al. 2005 and references therein), and the latest observation by the INTEGRAL gamma-ray observatory revealed that the all sky distribution is dominated by the extended bulge component ($\sim 8^\circ$ FWHM), with a weak evidence for the disk component (Kn\"odlseder et al. 2005). Although the disk component can be explained by positron emission from radioactive nuclei produced in supernovae, the origin of the bulge component is still a mystery. A positron production rate of $\sim 1.5 \times 10^{43} \ \rm s^{-1}$ is required to explain the bulge component. A number of candidates have been proposed for the origin of the bulge component, but the large bulge-to-disk ratio excludes many of them related to recent star formation activities, leaving type Ia supernovae (SNe Ia) and low-mass X-ray binaries (LMXBs) as the primary candidates (Kn\"odlseder et al. 2005). However, about one order of magnitude higher rate of bulge SNe Ia than the best estimate is required to explain the observed annihilation rate (Prantzos 2006). The positron production rate from LMXBs or microquasars may be sufficient (Guessoum et al. 2006), but the bulge-to-disk ratio of LMXBs is considerably lower than that of the 511 keV data. Most of the positrons produced in microquasars are probably lost by annihilation before injection into interstellar medium (ISM) (Guessoum et al. 2006), and direct annihilation gamma-ray emission from the sources would be inconsistent with the diffuse gamma-ray background around MeV band (see \S \ref{section:ejection}). Annihilation of dark matter particles with a mass of $\sim$ MeV is another possible solution, but such a particle is not naturally predicted by the particle physics theory, in contrast to the well-motivated candidates such as the supersymmetric particles that are much more massive (Bertone et al. 2004). Furthermore, the upper limit on 511 keV line flux from the Sagittarius dwarf galaxy excludes this scenario for almost all types of the halo density profile (Kn\"odlseder et al. 2005). The second aim of this paper is to show that, if the past accretion rate onto Sgr A$^*$ was in fact higher than now at a rate inferred from the observational evidence mentioned above, the RIAF model of Sgr A$^*$ gives a natural explanation for the bulge 511 keV line emission. There are models of positron production by accretion activity of Sgr A$^*$ (Titarchuk \& Chardonnet 2006; Cheng, Chernyshov, \& Dogiel 2006), but the model proposed here is considerably different from these and it gives a more natural explanation for the 511 keV line (\S\ref{section:comparison}). A variety of phenomena on various scales around the Galactic center will be discussed in this paper, and these are summarized in schematic diagrams in Fig. \ref{fig:schem} for the reader's convenience. Starting from a brief description about the RIAF model of Sgr A$^*$ (\S\ref{section:RIAF}), I review and summarize the lines of evidence for the past increased activity of Sgr A$^*$, and present RIAF interpretations for these, in \S\ref{section:past-activity}. We discuss the physical origin of the past high activity in the Galactic center environment in \S\ref{section:origin}. Then I calculate the positron production rate around the event horizon of Sgr A$^*$ in \S \ref{section:production}, and consider the ejection from Sgr A$^*$ (\S \ref{section:ejection}) and propagation in ISM (\S \ref{section:propagation}). In \S \ref{section:discussion}, I compare my model with the other models of the 511 keV emission, and discuss about future observational tests of this model. I apply 8 kpc as the distance to the Galactic center (Eisenhauer et al. 2003). \section{The RIAF Model of Sgr A$^*$ and Outflow} \label{section:RIAF} The accretion rate of the YQN03 model is normalized at the outer boundary corresponding to the Bondi radius ($r_B \sim 10^5 r_s = 0.029$ pc), which is inferred from the ambient temperature of X-ray emitting gas (Baganoff et al. 2003), where $r_s$ is the Schwartzshild radius of the SMBH. The size of the extended ($\sim 1''.4$ FWHM) X-ray emission of Sgr A$^*$ (Baganoff et al. 2003) nicely corresponds to this radius. The accretion rate at this radius is $\dot M_{\rm acc}(r_B) \sim \alpha_v \dot M_B \sim 10^{-6} M_\odot$/yr, where $\alpha_v \sim 0.1$ is the dimensionless viscousty parameter and $M_B$ is the Bondi accretion rate. The YQN03 model assumes a radially varying accretion rate, $\dot M_{\rm acc} \propto r^s$ with $s = 0.27$, based on the adiabatic inflow-outflow solutions (ADIOS, Blandford \& Begelman 1999) rather than the advection dominated accretion flow (ADAF, see Kato et al. 1998; Narayan et al. 1998 for reviews) that is the simplest RIAF solution with a radially constant mass accretion rate. This is necessary since the ADAF model predicts too large density and magnetic field strength in the innermost region, in contradiction with the observational upper bounds on rotation measure. The accretion flow is quasi-spherical and the density $\rho$ is determined by $\dot M_{\rm acc}(r) \sim 4 \pi \rho r^2 \upsilon_{\rm in}$, where $\upsilon_{\rm in}$ is the inflow velocity of the accretion flow. In the self-similar solution of RIAFs, $\upsilon_{\rm in}$ is close to and proportional to the free fall velocity $\upsilon_{\rm ff}$, and hence $\rho \propto r^{-3/2+s}$. In the YQN03 model, the particle density is $n_e \sim \rho/m_p \sim 3.9 \times 10^7 (r/r_s)^{-1.23} \rm \ cm^{-3}$, where $m_p$ is the proton mass, and hence $\upsilon_{\rm in} \sim 0.15 \upsilon_{\rm ff}$. This also means $\rho \propto \dot M_{\rm acc}$. It is reasonable to assume that the evaporated mass flow due to the non-conserving accretion rate is ejected as wind or outflow, whose velocity is comparable with the escape velocity $\upsilon_{\rm esc}$ at the region where the wind originates. Then the mass ejection rate per logarithmic interval of radius is $\dot M_w = r (d\dot M_{\rm acc}/dr) = s \dot M_{\rm acc}$, and the largest kinetic energy is produced from the innermost region where $\upsilon_{\rm esc}$ is close to the speed of light, $c$. We denote $r \sim r_*$ for the wind production region, and the outflow kinetic energy from this region becomes $L_{\rm kin} \sim \dot M_w \upsilon_{\rm esc}^2 / 2 \sim 1.5 \times 10^{38} r_3^{-0.73}$ erg/s, where $r_3 \equiv r_* / (3 r_s)$. The jet or outflow may also contribute to the Sgr A$^*$ SED, especially in the radio bands (see discussion of YQN03). The outflow kinetic energy derived here is in fact comparable with those assumed in the jet models of Sgr A$^*$ (Falcke \& Biermann 1999; Yuan, Markoff, \& Falcke 2002; Le \& Becker 2004), indicating that this estimate is a reasonable one. The outflow kinetic power is much larger than the X-ray luminosity of Sgr A$^*$. Such a trend of jet-energy dominance is indeed established for black holes in stellar X-ray binaries in the low/hard state, which is believed to be a state with an accretion rate much lower than the Eddington value like Sgr A$^*$ (Gallo et al. 2003, 2005). In fact, the Sgr A$^*$ X-ray luminosity ($\sim 10^{33}$ erg/s in the quiescent state and $\sim 10^{34-35}$ erg/s in the flaring state) and the above estimate of the kinetic luminosity are consistent with the relation inferred for the stellar black hole binaries: $L_{\rm kin} = f (L_X/0.02)^{0.5}$ with $f = 0.06$--1, where $L_{\rm kin}$ and $L_X$ are in units of the Eddington luminosity (Gallo et al. 2005). Though the mass scale of black holes is quite different, this can be understood if (1) the critical accretion rate between the standard thin disk and RIAF is independent of the black hole mass, (2) luminosity scales roughly as $\dot M_{\rm acc}^2$ in the RIAF regime, and (3) the mass outflow is proportional to $\dot M_{\rm acc}$. The first two are indeed the properties of the ADAF (Narayan et al. 1998), and the third is in accordance with the radially varying accretion rate assumed in YQN03. Some active galactic nuclei (AGNs) show the jet activity with much higher speed (the bulk Lorentz factor $\Gamma \sim 10$) than the above velocity estimate ($\sim \upsilon_{\rm esc}$, only mildly relativistic with $\Gamma \sim 1$). However, $\Gamma \sim 1$ is reasonable for Sgr A$^*$, since the same trend has been known for the outflows from stellar X-ray binaries in the low/hard state, which are not strongly beamed and not extremely relativistic ($\Gamma \lesssim 2$), in contrast to those of X-ray transients (Gallo et al. 2003). \section{Past Higher Activity and RIAF Interpretation} \label{section:past-activity} \subsection{Higher X-ray Luminosity} \label{section:past-X-ray-luminosity} There are a few independent lines of evidence that about 300 years ago Sgr A$^*$ was much more luminous than now in the X-ray band. Koyama et al. (1996) found fluorescent X-ray emission reflected from cold iron atoms in the giant molecular cloud Sgr B2 by ASCA observation (Fig. \ref{fig:schem}). Since there is no irradiation source to explain the iron line emission, they suggested a possibility that $\sim$300 yrs ago Sgr A$^*$ was much brighter than now. More recent studies by Murakami et al. (2000, 2001a, b) found a new X-ray reflection nebula associated with Sgr C, and estimated the increased past luminosity of Sgr A$^*$ as $L_X \sim 3 \times 10^{39}$ erg/s from Sgr B2 and C data. This claim was independently confirmed by an INTEGRAL observation covering higher energy band of 10--100 keV (Revnivtsev et al. 2004); the ASCA and INTEGRAL data of Sgr B2 can nicely be fit by a reflection of radiation from Sgr A$^*$ whose luminosity is $1.5 \times 10^{39}$ erg/s in 2--200 keV having a power-law spectrum with a photon index of $\beta = 1.8 \pm 0.2$ ($dN/dE \propto E^{-\beta}$). ``The ionized halo'' surrounding Sgr A$^*$ with a density $n_{e, h} \sim 10^2$--$10^3 \ \rm cm^{-3}$ has been known from radio observations (Pedlar et al. 1989; Anantharamaiah et al. 1999), and it extends to a radius of $\sim 10$ pc (Fig. \ref{fig:schem}). Maeda et al. (2002) argued that currently no ionizing source is found for the ionized halo, and the past activity of the Sgr A$^*$ may be responsible for the ionization, requiring an X-ray luminosity of $L_X \sim 10^{40}$ erg/s, a similar flux to those inferred from the X-ray reflection nebulae. We can then estimate the boost factor of the accretion rate to achieve the X-ray luminosity of $L_X \sim 10^{39}$--$10^{40}$ erg/s in the RIAF model. It should be noted that this luminosity is $\sim 10^{-5}$ in the Eddington luminosity, and hence it is still well within the ADAF/RIAF regime (Narayan et al. 1998). According to Fig. 5 of YQN04, a boost factor $f_b \equiv \dot M_{\rm past} / \dot M_{\rm present} \sim 10^3$--$10^4$ (at a fixed radius $r$) is required to achieve the luminosity of $L_X \sim 10^{39}$--$10^{40}$ erg/s, assuming a constant value for $s$. The SED in YQN04 with $f_b \sim 10^3$ is roughly constant in $\nu L_\nu$ (luminosity per logarithmic frequency interval) in the X-ray band, consistent with the X-ray spectrum inferred from the X-ray reflection nebula. Then the outflow kinetic luminosity in such a higher activity phase should be $L_{\rm kin} \sim 4.7 \times 10^{41} f_{3.5} r_3^{-0.73}$ erg/s, where $f_{3.5} \equiv f_b / 10^{3.5}$. \subsection{Mass Outflows on Various Scales} There is evidence for powerful mass outflow from the Galactic center on scales from arcminutes to tens of degrees (Fig. \ref{fig:schem}). Bland-Hawthorn \& Cohen (2003) reported the mid-infrared emission from dust expanding in the Galactic center lobe (GCL) on a few degree scale, and estimated the total kinetic energy as $\sim 10^{55}$ erg with a velocity of $\sim$ 100 km/s and a dynamical time of $\sim 10^6$ yr. The size, energy, and time scales are similar to those of the expanding molecular ring (EMR) around the Galactic center (Kaifu et al. 1972; Scoville 1972). Bland-Hawthorn \& Cohen further argued that this result is consistent with the North Polar Spur (NPS) on an even larger scale (up to tens of degrees), which has been interpreted by Sofue (2000) to be an outflow with an energy scale of $\sim 10^{55-56}$ erg and a dynamical time scale of $\sim 10^7$ yr\footnote{There are other interpretations for NPS by closer objects on the Galactic disk like a supernova remnant, but see Bland-Hawthorn \& Cohen (2003) for arguments favoring the Galactic center interpretation.}. The kinetic luminosity inferred from these various observations is in a nice agreement with the estimate for the RIAF model with the boost factor of $f_b \sim 10^{3-4}$, and hence we can attribute these outflows to the past activity of Sgr A$^*$ that was responsible for the higher X-ray luminosity. This indicates the duration of $10^{6-7}$ yr for the higher activity, which is reasonable compared with various estimates for lifetimes of AGNs (Martini 2004). The origin of the outflow from the Galactic center may instead be starbursts in the nuclear region, as suggested by Bland-Hawthorn \& Cohen (2003). However, observations of gamma-rays from radioactive $^{26}$Al seem to disfavor the starburst scenario. The energy of $10^{55}$ erg in $10^6$ yr is equivalent to $\sim 1$ supernova per century, almost comparable with the Galaxy-wide core-collapse supernova rate of $1.9 \pm 1.1$ per century (Diehl et al. 2006) estimated from the flux of $^{26}$Al gamma-rays, whose spatial distribution is clearly associated along with the Galactic disk. Since the half-life of $^{26}$Al is $7.2 \times 10^5$ yr, $^{26}$Al cannot travel beyond the GCL region with the outflow velocity of $\sim 100$ km/s. Therefore, if the origin of the outflow is starbursts, we expect a strong $^{26}$Al gamma-ray emission concentrated within a few degree from the Galactic center, with a flux comparable with the total gamma-ray flux along with the disk. However such a strong concentration at the Galactic center is not found (Prantzos \& Diehl 1996; Kn\"odlseder et al. 1999), indicating that the accretion activity of Sgr A$^*$ is more plausible as the origin of the mass outflow. Muno et al. (2004) studied in detail the Chandra data of the diffuse X-ray emission within $\sim $ 20 pc of the Galactic center, and they concluded that the hard component plasma ($k T \sim 8$ keV, Koyama et al. 1989; Yamauchi et al. 1990) cannot be explained by unresolved point sources (but see also Revnivtsev et al. 2006). If it is truly diffuse, it cannot be gravitationally bound requiring a large energy input of $\sim 10^{40}$ erg/s to keep this hot plasma, assuming the escape simply by the sound velocity. This is too large to be explained by supernova explosions and the origin of this hot plasma is still a matter of debate. Later in this paper (\S \ref{section:propagation}), I will discuss in more detail about the consequences of the energy injection into ISM by the wind from Sgr A$^*$, and show that this hot plasma can be explained as a result of shock heating by the wind. \section{The Origin of the Past Higher Activity} \label{section:origin} If the activity of Sgr A$^*$ was much higher than now until 300 years ago for a time scale of $\sim 10^7$ yrs, questions are: (1) what is the source of the accreting matter during the high activity phase, and (2) what caused the sudden drop of accretion rate by a factor of $10^{3-4}$ on a time scale of just $\sim 10^{2-3}$ yrs. Here I give reasonable explanations for these. \subsection{The Role of Sgr A East and the Ionized Halo} Maeda et al. (2002) proposed that, based on their Chandra observation of the supernova remnant Sgr A East, the higher activity was induced by accretion from the dense supernova shell expanding into the ionized halo. The location of Sgr A$^*$ is in fact inside Sgr A East and close to its shell (Fig. \ref{fig:schem}). In this scenario the duration of such high accretion rate is only $\sim 10^3$ yr, as inferred from the shell thickness ($\sim$1/10 of the observed shell radius $r_{\rm sh} = $ 2.9 pc) and the shell expansion velocity ($\upsilon_{\rm sh} \sim$ 200 km/s estimated by a simple theoretical model of supernova remnants). For comparison, the age estimate of Sgr A East is $\sim 10^4$ yr. A much higher accretion rate than now is possible by the Bondi accretion with the shell density estimated from shock-compression of the gas of the ionized halo. However, according to this scenario, we expect a comparable or even higher accretion rate by accretion directly from the ionized halo before the passage of the Sgr A East shell, since the density enhancement by shock-compression is at most by a factor of 4 and the sound velocity of unshocked gas is probably much lower than the shell velocity. Suppose that the SMBH is embedded directly in the ionized halo. The Bondi radius in the halo, $r_{B, h} = 2 GM_\bullet/c_{s, h}^2$ is larger than 1 pc if the temperature $k T_h$ is lower than $\sim$0.11 keV, where $c_{s, h}$ is the sound velocity in the ionized halo. Then, a natural scale for the outer boundary of the accretion flow is likely determined by $r_{\rm grav} \sim 1$ pc, within which the gravity of the SMBH is dominant compared with stars around the SMBH (e.g., Sch\"odel et al. 2003). Then we can extrapolate the YQN03 model from $r_B = 0.029$ pc out to $r_{\rm grav}$ with the boost factor of $f_b \sim 10^{3-4}$ and $n_e \propto r^{-1.23}$, and the density at $r_{\rm grav}$ becomes $n_e \sim 1.1 \times 10^3 f_{3.5} \ \rm cm^{-3}$. This is consistent with the density of the ionized halo, $n_{e, h}$, and hence the RIAF at the increased accretion rate is naturally connected to the environment surrounding the SMBH. Since this is an extrapolation of the RIAF solution, the surrounding gas can accrete even if it has a significant angular momentum, because of the angular momentum loss by viscousty. This is in contrast to the simple picture of the spherical Bondi accretion. In this new scenario, the high accretion rate can last for a much longer time scale than $\sim 10^3$ yrs, and accretion from the dense supernova shell is no longer necessary. Still, Sgr A East must play an important role to explain the sudden drop of the accretion rate in $\sim$300 yrs, by the destruction of accretion flow when the dense shell passed through the SMBH. It should also be noted that there are the arm-like structures of Sgr A West and the circumnuclear disk surrounding it on the scale of $\sim$1 pc (e.g., Yusef-Zadeh, Melia, \& Wardle 2000; see also Fig. \ref{fig:schem}), which could be the remnants of the former accretion flow. \subsection{Destruction of the Accretion Flow by Sgr A East} We examine the destruction process more quantitatively as follows. Since the ADAF and ADIOS solutions have a positive Bernoulli parameter (Narayan et al. 1998; Blandford \& Begelman 1999), the flow is not gravitationally bound and change of flow velocity by external force would result in a destruction of the flow. Therefore we should compare the momentum of the flow and the supernova remnant to estimate the effect of the supernova shell passage. The momentum of the accretion flow is: \begin{eqnarray} P_{\rm acc} &\sim& \frac{4\pi}{3} r^3 \rho \ \upsilon_{\rm flow} \\ &\sim& 5.6 \times 10^{41} \left(\frac{r}{1 \ \rm pc}\right)^{1.27} f_{3.5} \ \rm g \ cm \ s^{-1}, \end{eqnarray} where we estimated the flow velocity $\upsilon_{\rm flow}$ by $\sim \upsilon_{\rm in}$, since the rotation velocity is negligible when the adiabatic index $\gamma_{\rm ad} \rightarrow 5/3$ in ADAFs (Narayan et al. 1998). The momentum of the supernova remnant given to the flow is: \begin{eqnarray} P_{\rm SN} &=& \frac{1}{4}\left(\frac{r}{r_{\rm sh}}\right)^2 \frac{2 E_{\rm SN}}{\upsilon_{\rm sh}} \\ &\sim& 3.0 \times 10^{42} \left(\frac{r}{1 \ \rm pc}\right)^2 \left(\frac{E_{\rm SN}}{10^{51} \ \rm erg}\right) \ \rm g \ cm \ s^{-1} \ , \end{eqnarray} where $E_{\rm SN}$ is the shell kinetic energy of the supernova remnant. Therefore the accretion flow could have been destroyed at $r \gtrsim 0.1$ pc from the SMBH. Destruction should have occurred with a time scale of the shell crossing ($\sim 10^3$ yrs), and the accretion time scale at this radius is also $r/\upsilon_{\rm in} \sim 1.3 \times 10^3$ yrs. These time scales are consistent with the required time scale of the accretion rate drop, $\sim 300$ yrs. The current low rate may be determined by either the diffuse gas in the supernova remnant, winds from nearby stars (e.g., Cuadra et al. 2006), or residual of the former accretion flow. \subsection{Comparison with Nearby Galaxies} We may ask how the suggested high activity of Sgr A$^*$ in the past compares with those found in nearby normal galaxies, because it would be statistically unlikely if our Galaxy had a much higher activity compared with nearby normal galaxies. As reviewed by Ho (2004), nuclear activity is quite commonly found in nearby galaxies. The YQN model with $f_b \gtrsim 10^3$ predicts a nuclear $B$-band luminosity of $L_B \sim 10^{39}$ erg/s, and the number density of such galaxies expected from the nuclear luminosity function is $\sim 10^{-2} (h/0.75)^3 \ \rm Mpc^{-3}$, which is similar to that of typical galaxies like our own, where $h = H_0/(\rm 100 \ km/s/Mpc)$ is the Hubble constant. Nuclear radio cores with flux of $10^{19}$--$10^{21} \ \rm W \ Hz^{-1}$ at 5 GHz are also commonly found, which is again a similar radio luminosity to that predicted by the YQN04 model with $f_b \sim 10^{3-4}$. Therefore, the increased activity of Sgr A* is not particularly rare compared with nearby normal galaxies, indicating that the characteristic time scale of the increased activity can be $\sim 10^7$ yr or even longer, possibly as long as the cosmological time scale. The accretion rate at the event horizon, i.e., the mass growth rate of the SMBH is $1.4 \times 10^{-4} f_{3.5} M_\odot \rm yr^{-1}$, and the mass growth in 10 Gyr is $1.4 \times 10^6 f_{3.5} M_\odot$, which is still less than $M_\bullet$. AGN activity is generally sporadic and showing strong variability, and hence it is naturally expected that the accretion rate was changing significantly always in the past $\sim 10^7$ yrs. In fact, the characteristic structures such as GCL, EMR, and NPS indicate such variability or modulation of the accretion and wind activity. However, conclusions of this paper are not seriously affected if the mean or characteristic accretion rate is given by the above value. %\section{Explaining the 511 keV Electron-Positron Annihilation Emission} \section{Positron Production around Sgr A$^*$} \label{section:production} \subsection{Physical Quantities around the Event Horizon} Now I consider the pair-production at the region where the wind originates, $r \sim r_*$. During the increased phase, the particle accretion rate is $\dot N_{\rm acc} = \dot M_{\rm acc}/m_p \sim 7.1 \times 10^{45} f_{3.5} r_3^{0.27} \ \rm s^{-1}$ and the particle outflow rate per logarithmic radius is $\dot N_w = s \dot N_{\rm acc}$. The particle density in the accretion flow is $n_e \sim 3.2 \times 10^{10} f_{3.5} r_3^{-1.23} \rm \ cm^{-3}$. The accretion time spent around this radius is $t_{\rm acc} \sim r_*/\upsilon_{\rm in}\sim 1.0 \times 10^3 r_3^{3/2}$ s. The electron temperature of the YQN03 model is $T_e \sim 8 \times 10^{10} r_3^{-1}$ K, and hence relative motion of electrons is sufficiently relativistic at the pair-production region if $r_* \sim 3 r_s$, with the mean electron Lorentz factor $\gamma_{e} \sim 3.151 k T_e / (m_e c^2) \sim 40 r_3^{-1}$ in the rest frame of the accretion flow. The temperature does not heavily depend on the enhancement factor $f_b \sim 10^{3.5}$, if the transfer efficiency of viscous heating energy from ions to electrons (the parameter $\delta$ in YQN03) does not change with the accretion rate, as assumed by YQN04. It may increase with increasing accretion rate because of higher density and hence more efficient interactions, but the value of $\delta$ assumed in YQN03 is already of order unity (=0.55), not leaving much room for the increase of $\delta$. \subsection{Pair Equilibrium Criterion} \label{section:equilibrium} The positron density produced in the accretion flow depends on whether the $e^\pm$ pair production process achieves the equilibrium with the $e^\pm$ pair annihilation, which can be evaluated by comparing the pair production rate density $\dot n_+$ and pair annihilation rate density $\dot n_{\pm, \rm ann} \equiv n_e n_+ \sigma_{\pm, \rm ann} c$, where $n_+$ is the produced positron density. The $e^\pm$ pair annihilation cross section is \begin{eqnarray} \sigma_{\pm, \rm ann} &=& \frac{\pi r_e^2}{\gamma_{\pm}} [\ln 2 \gamma_{\pm} - 1] \end{eqnarray} in the ultra-relativistic limit (Svensson 1982), where $r_e$ is the classical electron radius and $\gamma_\pm$ is the Lorentz factor in the rest frame of one particle. Hence $\gamma_\pm$ can be related as $\gamma_\pm \sim \gamma_+ \gamma_e$, where $\gamma_+$ is the positron Lorentz factor in the flow frame. Initially the positron density is small, and it increases with time as $n_+ \sim \dot n_+ t$ until $t \sim t_{\pm, \rm ann}$ when the equilibrium is achieved ($\dot n_+ = \dot n_{\pm, \rm ann}$), where the pair annihilation time scale is $t_{\pm, \rm ann} \equiv (n_e c \sigma_{\pm, \rm ann})^{-1}$. Since $t_{\pm, \rm ann}$ does not depend on $\dot n_+$, the equilibrium condition is the same for any pair production processes and it is determined by comparing $t_{\pm, \rm ann}$ to the accretion time scale, as: \begin{eqnarray} \frac{t_{\rm acc}}{t_{\pm, \rm ann}} &=& 4.2 \times 10^{-2} \gamma_+^{-1} f_{3.5} r_3^{1.27} \ , \end{eqnarray} where we estimated the logarithmic part of the cross section by $\gamma_+ \sim \gamma_e \sim 40$. For positrons produced by electron-electron scattering ($e^- e^- \rightarrow e^- e^- e^- e^+$), we expect $\gamma_+ \sim \gamma_e$, while for positrons produced by two photon annihilation ($\gamma \gamma \rightarrow e^- e^+$), a variety of $\gamma_+$ is possible (see the following subsections). The positron energy may also significantly change by interaction with the accreting material within the accretion time scale. However, for any value of $\gamma_+$, we find that $t_{\rm acc} \ll t_{\pm, \rm ann}$, and hence the equilibrium will not be achieved, meaning that we can estimate the pair density by $n_+ \sim \dot n_+ t_{\rm acc}$. \subsection{Spectral Energy Distribution of Sgr A$^*$} For pair production processes including photons, we must assume the form of the SED and the emission region of Sgr A$^*$ during the increased activity phase. Here I assume that the radiation is mainly coming from the region around the event horizon. It should be noted that the X-ray emission in the quiescent phase of the YQN03 model for present-day Sgr A$^*$ is dominated by thermal bremsstrahlung at large radii far from the SMBH, which is in agreement with the extended X-ray emission ($\sim 1''$). However, the synchrotron-self-Compton (SSC) component becomes dominant when the accretion rate is increased as $f_b \gtrsim 10^3$ (YQN04), and it is produced in the innermost region. Therefore the above assumption is reasonable for the higher activity phase. For calculations below, I assume a constant SED in $\nu L_\nu$. The predicted SED of the YQN04 model when $f_b \gtrsim 10^3$ is approximately flat in $\nu L_\nu$ from keV to MeV band. A flat $\nu L_\nu$ SED is also supported in 2--200 keV band by the observed spectrum of the X-ray reflection nebula (\S \ref{section:past-X-ray-luminosity}). Therefore, this assumption is well supported both by observation and theory in the photon energy band of keV--MeV. The SED beyond MeV is more uncertain, and here I simply examine the constraints on the current SED of Sgr A$^*$. The TeV gamma-ray emission detected by H.E.S.S. from the inner $10'$ of Sgr A$^*$ (Aharonian et al. 2004) is possibly coming from the region close to the event horizon, and its flux is similar to that extrapolated from X-ray bands with a constant $\nu L_\nu$ spectrum. Though the GeV flux of 3EG J1746$-$2851, which is the closest to Sgr A$^*$ among the EGRET sources, is considerably higher than the X-ray flux of Sgr A$^*$, the poor angular resolution in this band does not allow to establish a firm connection between the GeV flux and Sgr A$^*$ (Mayer-Hasselwander et al. 1998; Aharonian \& Neronov 2005)\footnote{A recent analysis of the EGRET data by Hooper \& Dingus (2004) excluded Sgr A$^*$ as the origin of 3EG J1746$-$2851 beyond the 99.9\% confidence level.}. Therefore, it is not unreasonable to assume a flat $\nu L_\nu$ SED at photon energy of $\gtrsim$ MeV, though uncertainty is large. On the other hand, the photon production region will become optically thick to $e^\pm$ pair production for very high energy photons ($\gtrsim$ GeV), and the constant $\nu L_\nu$ assumption will be no longer valid at such high energy band (see below). \subsection{Expected Pair Amount in the Wind} Now I estimate the expected amount of pairs produced by three processes of pair production, i.e., electron-electron scattering ($e^- e^- \rightarrow e^- e^- e^- e^+$), photon-electron collisions ($\gamma e^- \rightarrow e^- e^- e^+$), and two photon annihilation ($\gamma \gamma \rightarrow e^\pm$). The rates of corresponding proton processes (e.g., $ p \ e^- \rightarrow p \ e^- e^- e^+$) are about one order of magnitude smaller than these (Zdziarski 1982, 1985). \vspace{0.2cm} \subsubsection{Electron-Electron Scattering} I used the formula given in Svensson (1982) for the cross section in the ultra-relativistic limit, which is $\sigma_{ee} = 1.7 \times 10^{-28} \ \rm cm^2$ for $\gamma_e = 40$ and depends on $\gamma_e$ only logarithmically. Hence I ignore the dependence on $\gamma_e$. Then the density ratio of the produced positrons to electrons is given as: \begin{eqnarray} \frac{n_+}{n_e} &=& \frac{\dot n_+ t_{\rm acc}}{n_e} = c \sigma_{ee} n_e t_{\rm acc} \\ &=& 1.6 \times 10^{-4} f_{3.5} r_3^{0.27}\ , \end{eqnarray} and hence the total positron production rate as an outflow from Sgr A$^*$ is: \begin{eqnarray} \dot N_+ &\sim& \dot N_w \left(\frac{n_+}{n_e}\right) \\ &=& 3.2 \times 10^{41} f_{3.5}^2 r_3^{0.54} \ \rm s^{-1} \ . \end{eqnarray} This is a rate per $\ln r_*$, and integrating from $r_* = 3 r_s$ to $40 \times 3 r_s$, beyond which electrons become non-relativistic and hence the above formulations are no longer valid, the rate is increased by a factor of 11.7 leading to $\dot N_+ \sim 3.7 \times 10^{42} f_{3.5}^2 \ \rm s^{-1}$. This estimate is, within the model uncertainties, in nice agreement with the rate required for the bulge 511 keV emission, $1.5 \times 10^{43} \ \rm s^{-1}$ (Kn\"odlseder et al. 2005). \vspace{0.2cm} \subsubsection{Photon-Electron Collisions} The cross section for $e^-\gamma \rightarrow e^- e^- e^+$ depends on the photon frequency $\nu_{\rm er}$ measured in the electron's rest frame, which is given by (Svensson 1982)\footnote{The numerical factor $3 \sqrt{\pi} $ of the non-relativistic formula in Svensson (1982, eq. 31) should be corrected to $\pi \sqrt{3}$ as noted in Svensson (1984).}: \begin{eqnarray} \sigma_{e\gamma} = \frac{\pi \sqrt{3}}{972} \alpha r_e^2 \left( x_{\rm er} - 4 \right)^2 \end{eqnarray} in the non-relativistic limit ($x_{\rm er} - 4 \ll 4 $), and \begin{eqnarray} \sigma_{e\gamma} = \alpha r_e^2 \left[\frac{28}{9} \ln \left( 2 x_{\rm er} \right) - \frac{218}{27} \right] \end{eqnarray} in the ultra-relativistic limit ($x_{\rm er} - 4 \gg 4$), where $x_{\rm er} \equiv h\nu_{\rm er}/(m_e c^2)$, $\alpha$ is the fine structure constant, and the reaction threshold is $x_{\rm er, th} = 4$. Treating electrons as a single energy population with $\gamma_e \sim 40$, and estimating the photon number density per unit photon frequency ($\nu$) in the laboratory frame as $n_\nu \sim L_\nu / (4 \pi r_*^2 c h \nu)$, the positron production rate by this process can be written as: \begin{eqnarray} \dot n_+ &=& \int n_e n_\nu \sigma_{e\gamma} c d\nu \\ &=& \frac{\gamma_e n_e (\nu L_\nu)}{4 \pi r^2 m_e c^2} \int \frac{\sigma_{e\gamma}(x_{\rm er})}{x_{\rm er}^2} dx_{\rm er} \ , \end{eqnarray} where we have used $\nu_{\rm er} \sim \gamma_e \nu$. With the assumption of a flat $\nu L_\nu$ SED, the integration over $x_{\rm er}$ is mostly contributed from photons with $x_{\rm er} \sim 20$, i.e., $h \nu \sim 0.3$ MeV for $\gamma_e \sim 40$. This target photon energy is within the range where the Sgr A$^*$ luminosity during the increased activity can reliably be assumed, and hence the uncertainty about the luminosity and SED is small. Then the produced positron density is given by: \begin{eqnarray} \frac{n_+}{n_e} &=& \frac{\dot n_+ t_{\rm acc}}{n_e} \sim 4.2 \times 10^{-4} L_{39.5} r_3^{-1.5} \ , \end{eqnarray} where $L_{39.5} = \nu L_\nu / (10^{39.5} \ \rm erg/s)$, and the total positron production rate in the outflow is: \begin{eqnarray} \dot N_+ &\sim& \dot N_w \left(\frac{n_+}{n_e}\right) \\ &=& 8.1 \times 10^{41} L_{39.5} f_{3.5} r_3^{-1.23}\ \rm s^{-1} \ . \end{eqnarray} Integration over $\ln r_*$ at $r_* > 3 r_s$ would slightly decrease the above number by a factor of 1/1.23. Though this number is smaller than the rate required to explain the 511 keV emission by about one order of magnitude, it may also be important taking into account the model uncertainties. Note that $L_\nu \propto \dot M_{\rm acc}^2$ in RIAFs, and hence $\dot N_+ \propto f_b^3$. \vspace{0.2cm} \subsubsection{Photon-Photon Annihilation} The pair-production by two photon annihilation most efficiently occurs with a cross section of $\sigma_{\gamma\gamma} \sim 1.7 \times 10^{-25} \ \rm cm^{-2}$, when the photon energy at the center-of-mass is about the electron mass, as $(h \nu_l h \nu_h)^{1/2} \sim 2 m_e c^2$, where $\nu_l$ and $\nu_h$ are the frequencies of two photons at the laboratory frame ($\nu_l < \nu_h$) (e.g., Salamon \& Stecker 1998). For photons meeting this condition, the pair-production rate density is given by \begin{eqnarray} \dot n_+ \sim \nu_h n_\nu(\nu_h) \ \nu_l n_\nu(\nu_l) \ \sigma_{\gamma \gamma} c \ , \end{eqnarray} which is constant against $\nu_h$ (and $\nu_l$) by the assumption of the constant $\nu L_\nu$ SED. It should be noted that this is valid only when the region is optically thin for high frequency photons, i.e., $\tau_{\gamma \gamma} \lesssim 1$, where \begin{eqnarray} \tau_{\gamma \gamma} &=& \nu_l n_\nu(\nu_l) \sigma_{\gamma \gamma} r_* \\ &=& 3.4 \times 10^{-4} L_{39.5} r_3^{-1} \left( \frac{h\nu_l}{\rm 1 \ MeV} \right)^{-1} \ . \end{eqnarray} The luminosity and pair production would then be suppressed for very high energy photons of $\gtrsim$ GeV.\footnote{Because of the increased luminosity compared with that of Sgr A$^*$ at present, this energy scale is much smaller than the estimate by Aharonian \& Neronov (2005) for the present-day Sgr A$^*$ ($\sim$ 10 TeV).} Therefore we expect that the pair production rate will mostly be contributed by photons in keV--GeV bands. Now the density of positrons produced is: \begin{eqnarray} \frac{n_+}{n_e} &=& \frac{\dot n_+ t_{\rm acc}}{n_e} \\ &=& 8.4 \times 10^{-5} L_{39.5}^2 f_{3.5}^{-1} r_3^{-1.27} \ , \end{eqnarray} and hence the total positron production rate from the Sgr A$^*$ is: \begin{eqnarray} \dot N_+ &=& \dot N_w \left(\frac{n_+}{n_e}\right) \\ &=& 1.6 \times 10^{41} L_{39.5}^2 r_3^{-1} \ \rm s^{-1} \ . \end{eqnarray} Note that this estimate is per unit logarithmic interval of $\nu_h$ (or $\nu_l$). Since the Sgr A$^*$ luminosity during the increased activity can reliably be modeled up to $\sim$ MeV, the uncertainty is small for photons of $h\nu_l \sim h\nu_h \sim m_e c^2$. Though it suffers larger uncertainty about the SED in MeV--GeV bands, integration over $h\nu_h \sim $ 1 MeV--1 GeV would increase the total rate by a factor of 6.9. The rate has a large dependence on the accretion rate as $\dot N_+ \propto L_\nu^2 \propto f_b^4$. Considering the model uncertainties, the two photon annihilation may also substantially contribute to the observed bulge 511 keV emission. \section{Positron Ejection from Sgr A$^*$} \label{section:ejection} Some of the positrons are lost by annihilation around the production site, producing direct annihilation gamma-ray emission from Sgr A$^*$. Its spectrum depends on the positron spectrum and gravitational redshift, and it is thermal with a temperature $T_e \sim 10$ MeV for the electron-electron scattering. However, the small value of $(t_{\rm acc}/t_{\pm, \rm ann}) \sim \dot n_{\pm, \rm ann}/\dot n_+ \sim 4.2 \times 10^{-2} \gamma_+^{-1}$ indicates that most of the produced positrons are conveyed into the SMBH or ejected by the wind. The direct annihilation gamma-ray luminosity from the pair production site, $L_{\pm, \rm ann} \sim (4\pi r_*^3/3) \ \dot n_{\pm, \rm ann}$, is then much smaller than the positron production and ejection rate by the wind, $(n_+/n_e) \dot N_w \sim 4 \pi s r_*^3 \dot n_+$. Here we check that the positrons trapped in the outflow are not lost by annihilation before escaping the SMBH gravity. We assume a constant wind speed as $\upsilon_{w} \sim \upsilon_{\rm esc}(r_*)$ and the density of the outflow is $n_{w} \sim \dot N_w / (4 \pi r^2 \upsilon_w)$ for a steady wind. Then the fraction of positrons that are lost by annihilation during wind propagation is: \begin{eqnarray} \eta_{\rm ann} &=& \int_{r_*}^\infty n_{w} \sigma_{\pm, \rm ann} \frac{\upsilon_{\pm}}{\upsilon_w} dr \ , \end{eqnarray} where $\upsilon_{\pm}$ is the relative velocity of electrons and positrons. Initially both electrons and positrons are relativistic, and hence $\upsilon_{\pm} \sim c$ and $\sigma_{\pm, \rm ann} \propto \gamma_\pm^{-1} \sim (\gamma_e \gamma_+)^{-1} \propto r^{4/3}$, since $\gamma_e$ and $\gamma_+$ scale as $\propto n_w^{1/3} \propto r^{-2/3}$ by adiabatic expansion. Either electrons or positrons become non-relativistic at $r_{\rm nr} = r_* \min(\gamma_{e*}, \gamma_{+*})^{3/2}$, and the scaling changes as $\sigma_{\pm, \rm ann} \propto r^{2/3}$ at $r > r_{\rm nr}$, where $\gamma_{e*}$ and $\gamma_{+*}$ are the Lorentz factor at the wind creation site ($r \sim r_*$). Then, $\sigma_{\pm, \rm ann} \upsilon_\pm$ becomes constant after both electrons and positrons become non-relativistic, since $\sigma_{\pm, \rm ann} = \pi r_e^2 / (\upsilon_\pm/c)$ in the non-relativistic limit (Svensson 1982). Therefore, the main contribution to the integration comes from $r \sim r_{\rm nr}$, and we obtain: \begin{eqnarray} \eta_{\rm ann} &\sim& 6 n_w(r_{\rm nr}) \sigma_{\pm, \rm ann} (r_{\rm nr}) \frac{c}{\upsilon_{\rm esc}(r_*)} \ r_{\rm nr} \\ &\sim& 6 n_w(r_*) \sigma_{\pm, \rm ann}(r_*) \frac{c}{\upsilon_{\rm esc}(r_*)} \ r_* \left(\frac{r_{\rm nr}} {r_*}\right)^{1/3}\\ &=& 1.5 \times 10^{-3} f_{3.5} \gamma_{+*}^{-1} r_3^{1.27} \ \left[\min(\gamma_{e*}, \gamma_{+*})\right]^{1/2} \ . \end{eqnarray} This is sufficiently small for any value of $\gamma_{+*}$, and hence we expect that almost all the positrons produced around the SMBH will escape from the SMBH gravity field once they are trapped in the outflowing material. Therefore the annihilation luminosity directly from Sgr A$^*$ is expected to be much smaller than the bulge 511 keV line emission, even in the phase of the past higher activity. The direct annihilation luminosity at present is even much smaller by the boost factor $f_b$, far below the detection limit of gamma-ray telescopes in the foreseeable future. This is important concerning the constraint from the diffuse gamma-ray background, whose flux is $E \ (dF/dE) \sim 10^{-4} \rm \ photons \ cm^{-2} s^{-1}$ at $\sim$1 MeV within 5$^\circ$ from the Galactic center (Beacom \& Y\"uksel 2006). If any source of the 511 keV emission directly emits annihilation gamma-rays with a broader spectrum before injection into ISM, the flux should not violate the observed MeV background, leading an upper bound on the annihilation rate as $\lesssim 3.9 \times 10^{41} \ \rm s^{-1}$, assuming no positronium formation for annihilation within the source. Hence the annihilation rate within the source must be $\lesssim$10\% of the annihilation rate in ISM, taking into account that $\sim$24\% of the bulge 511 keV photons come from the region within 5$^\circ$. The model presented here well satisfies this constraint, but it puts a stringent constraint on another explanation for the 511 keV emission by accreting black holes, i.e., LMXBs or microquasars. Theoretical models of pair production in these objects predict that most ($\sim$90\%) of the produced pairs annihilate near the production site before injection into ISM (Guessoum et al. 2006), which is in serious conflict with the above constraint. \section{Positron Propagation in Interstellar Medium} \label{section:propagation} \subsection{Dynamics and Energetics of the Wind Injected into ISM} \label{section:wind-dynamics} When positrons escape from the gravitational potential well of the SMBH, the kinetic energy of the outflow is expected to be dominant compared with the thermal energy, and hence the positron energy is determined by the bulk Lorentz factor $\Gamma \sim 1$ of the wind, i.e., $\sim$ 1 MeV, as argued in \S \ref{section:RIAF}. This is important since too relativistic outflow would be inconsistent with the upper bound on the injection energy, $\lesssim$ 3 MeV, derived from the absence of the signature of in-flight annihilation in the Galactic gamma-ray background radiation in 1--10 MeV band (Beacom \& Y\"uksel 2005; but see also Sizun et al. 2006 who derived a less stringent constraint of $\lesssim$ 25--30 MeV). The wind will sweep up ISM and heat it up by shocks. The ram pressure of the wind will be balanced with ISM at $r_{p}$, which is determined as $\dot N_w m_p \upsilon_w / (4 \pi r_p^2) \sim P_{\rm ISM}$, assuming a quasi-isotropic wind. Assuming $P_{\rm ISM} = B^2/(8\pi)$ by interstellar magnetic field of $B \sim 10 \mu$G in the bulge region (LaRosa et al. 2005; Jean et al. 2006), we find $r_p \sim 3.4 \times 10^2 f_{3.5}^{1/2} r_3^{-0.12} B_{10}^{-1}$ pc, where $B_{10} \equiv B/(10\mu{\rm G})$. This radius is comparable with the size of GCL or EMR, and also with the FWHM of the spatial extent of the 511 keV emission. Beyond $r \sim r_p$, the shock-heated gas will expand by thermal pressure. Therefore, even if the wind originally has some anisotropy, it will not directly appear on a scale larger than $\sim r_p$. The observed expansion velocity of $\sim$100 km/s at the GCL/EMR region indicates an escape time of $t_{\rm esc} \sim 10^6$ yrs from this region. Then, the wind kinetic energy stored in the GCL region is \begin{eqnarray} E_{\rm GCL} &\sim& L_{\rm kin} t_{\rm esc} \\ &\sim& 1.5 \times 10^{55} f_{3.5} \ r_3^{-0.73} \left(\frac{t_{\rm esc}} {10^6 \ {\rm yrs}}\right) \ \rm erg \ . \end{eqnarray} This is interestingly similar to the energy of the hot gas ($kT \sim 8$ keV) in the Galactic center observed by X-rays (Koyama et al. 1989; Yamauchi et al. 1990; Muno et al. 2004), $E_{\rm hot} \sim 2.6 \times 10^{54} (r_p / \rm 300 \ pc)^{5/2}$ erg, where I obtained this value from surface energy density estimated by Muno et al. (2004) assuming the size and depth to be $\sim r_p$. In fact, the observed size of the hot gas ($\sim 1.8^\circ$ FWHM, Koyama et al. 1989; Yamauchi et al. 1990) is in good agreement with $r_p$. Therefore, the large amount of energy stored in the hard X-ray emitting gas can be explained by the wind activity. The typical cooling time of the hot gas is $\sim 10^8$ yr (Muno et al. 2004), which is much longer than the time scale of the large-scale mass outflow, and hence the expansion is adiabatic as argued by Sofue (2000) and Bland-Hawthorn \& Cohen (2003). The expansion velocity ($\sim 100$ km/s) is much lower than the sound velocity of the hot gas ($\sim 10^3$ km/s), but it is possible if the associated cold material work as a ballast, as inferred from the infrared emission from expanding dust (Bland-Hawthorn \& Cohen 2003). \subsection{Positron Propagation Distance} At $r \gtrsim r_p$, positrons are expected to interact with ISM and produce the 511 keV emission. In the hot phase of ISM, which is the dominant component in the volume filling factor in the bulge region (Jean et al. 2006), positrons are thermalized in a time scale of $\sim 3 \times 10^6$ yrs and then annihilate in a time scale of $\sim 10^7$ yrs (Guessoum et al. 2005; Jean et al. 2006). Positrons can maximally reach $\sim 1$ kpc by the large scale outflow ($\sim$ 100 km/s) within this time scale, being consistent with the observed maximum extent of the 511 keV emission ($\sim 20^\circ$). Most positrons must annihilate before traveling this distance, since the spectral analysis of the 511 keV line indicates that almost all positrons are annihilating in warm neutral or warm ionized phase of ISM, where the annihilation time scale is much shorter. Exact spatial profile of the 511 keV line emission is determined by the probability of positrons entering into the warm phase of ISM at $r \gtrsim r_p$. Diffusion in random magnetic field should also have significant effect. Jean et al. (2006) estimated the propagation length by quasilinear diffusion for MeV positrons in the bulge as $\sim 260$ pc for a time scale of $\sim 3 \times 10^6$ yrs, using a diffusion coefficient $D \sim 9.8 \times 10^{-4} \ \rm kpc^2 Myr^{-1}$ that was derived from $B = 10 \ \mu$G and the Kolmogorov turbulent spectrum. Though the Kolmogorov spectrum is not valid for cosmic ray propagation in the Galactic disk (Maurin et al. 2001), this is not significant because a similar diffusion coefficient of $4.1 \times 10^{-4} \ \rm kpc^2 Myr^{-1}$ is obtained by using the parameters derived by Maurin et al. (2001) to explain the cosmic ray data. A more detailed, quantitative prediction about the 511 keV line morphology is beyond the scope of this paper, but these considerations indicate that the model presented here is consistent with the observed morphology and spatial extent of the 511 keV line emission. \section{Discussion} \label{section:discussion} \subsection{Comparison with Other Models of the Positron Production from Sgr A$^*$} \label{section:comparison} Titarchuk \& Chardonnet (2006) proposed a scenario in which positrons are produced by annihilation of hard X-ray and $\sim$ 10 MeV photons around Sgr A$^*$. The hard X-ray photons are emitted from the accretion activity of the SMBH, while the $\sim 10$ MeV photons are produced by accretion onto hypothetical small mass black holes (SMMBHs) with a mass $\sim 10^{17}$ g, which is assumed to be distributed within $r \sim 10^{2-3} r_s$ of the SMBH. If they are accreting with the Eddington accretion rate and their density is comparable with the dark matter, the SMMBHs can supply enough 10 MeV photons for the required pair production. Such SMMBHs have not yet been excluded as a candidate of the dark matter, but there is few theoretical support in contrast to the well-motivated dark matter candidates such as neutralinos (e.g., Bertone et al. 2004). Furthermore, it is extremely difficult to supply the accreting material onto such SMMBHs at the Eddington rate by Bondi accretion; the Bondi accretion rate for a $10^{17}$ g SMMBH with typical parameters are: \begin{eqnarray} \dot M_B \sim 7.9 \times 10^{-23} \ \left(\frac{n_e}{\rm cm^{-3}}\right) \left(\frac{kT}{1 \ \rm eV}\right)^{-3/2} \ \rm g/s , \end{eqnarray} which should be compared with the Eddington accretion rate of $72$ g/s (assuming 10\% radiation efficiency). Therefore it seems quite unlikely that such SMMBHs have the Eddington accretion rate at any realistic astrophysical circumstances. Cheng, Chernyshov, \& Dogiel (2006) considered cosmic-ray production by jet or outflow from the SMBH, which is ejected when stars are captured by the SMBH. In their scenario, the cosmic rays produce pions by collisions with ISM, and then positrons are produced by pion decays. The most important difference of this model from that proposed in this paper is the high injection energy into ISM ($\gtrsim 30$ MeV) of pion-decay positrons. Such a high injection energy is inconsistent with the observational upper bound on the injection energy, $\lesssim 3$ MeV, as mentioned in \S \ref{section:wind-dynamics}. Another problem about positron production from pion decays is the observed large bulge-to-disk ratio of the 511 keV line emission. We know that the Galactic gamma-ray background in the GeV band is mainly composed of pion-decay gamma-rays produced by cosmic-ray interactions in ISM (Strong et al. 2000, 2004), i.e., the same process with the model of Cheng et al. The GeV background is clearly associated along with the Galactic disk, and we have a difficulty to explain why we do not see the strong disk component of the 511 keV emission if it is produced by the cosmic-ray interactions. \subsection{Predictions and Possible Tests by Future Observations} Although a quantitative prediction of the morphology of the 511 keV emission is beyond the scope of this paper, it could be more spherically asymmetric compared with other explanations such as SNe Ia or MeV-mass dark matter. Asymmetry is expected by the wind anisotropy in the region of $r \lesssim r_p \sim $ a few degree, or by matter distribution in the Galaxy at $r \gtrsim r_p$. Asymmetry has not yet been detected in the observed 511 keV morphology (Kn\"odlseder 2005), but it does not reject the model presented here since the degree of anisotropy is theoretically highly uncertain. Future observational studies on the 511 keV morphology with better angular resolution might, however, detect larger asymmetry than expected for the other explanations. Another prediction of the proposed scenario is that it is extremely difficult to detect 511 keV emission in regions other than the Galactic bulge from the source population that is responsible for the bulge component. SMBHs are generally found in galaxies having bulges, and the nearest SMBH is probably M31$^*$ in the Andromeda galaxy. The X-ray luminosity of M31$^*$ is $\sim 10^{36}$ erg/s (Garcia et al. 2005), which is about $10^3$ times larger than the quiescent phase of Sgr A$^*$, but $10^3$ times smaller than the past higher activity phase. If this X-ray luminosity reflects the typical activity averaged over a time scale of $\sim 10^7$ yr, we expect that the 511 keV luminosity of the M31 bulge is much fainter than that of the Galaxy. The 511 keV emission has been detected at $\sim 50 \sigma$ level (Kn\"odlseder et al. 2005), and considering the distance to M31 (770 kpc), a large improvement of the sensitivity is required. On the other hand, there is a better chance to detect 511 keV emission from regions other than the Galactic bulge, for some other models of the bulge 511 keV emission (e.g., Kn\"odlseder et al. 2005). For example, if the origin is SNe Ia, improved instruments in the near future will detect 511 keV line emission from nearby supernova remnants. In fact, an interesting limit on the positron escape fraction from SN 1006 has already been obtained (Kalemci et al. 2006). If the origin is the MeV-mass dark matter annihilation, we expect 511 keV line emission from nearby dwarf galaxies by a modest improvement of the sensitivity. If future negative results ruled out the other explanations of the 511 keV emission, it would strengthen the case for the scenario presented here. It should also be noted that the standard prediction for positron production from SNe Ia is only one order of magnitude short of that required to explain the bulge 511 keV emission (Prantzos 2006). Therefore, just a detection of 511 keV emission from a nearby SN Ia does not confirm SNe Ia as the origin of the bulge component, but a close examination of the positron production efficiency would be required. \section{Conclusion} In this paper, I have shown that the several independent lines of evidence for the past higher activity of the Galactic center (i.e., X-ray reflection nebulae, hot X-ray emitting gas, and large scale outflows) can quantitatively be explained by the RIAF model of Sgr A$^*$, in which energetic outflow plays an essential role. A single increased accretion rate from the current value explains both the past high X-ray luminosity and kinetic luminosity of outflow inferred from observations. The required boost factor of the accretion rate is about $10^{3-4}$ for a time scale of $\sim 10^7$ yrs in the past. I have shown that this accretion rate and its duration are naturally expected in the environment of the Galactic center. The current low accretion rate seems a rather rare situation for Sgr A$^*$ on a long time scale, caused by a sudden destruction of the accretion flow when the shell of the supernova remnant Sgr A East passed through the SMBH about 300 yrs ago. I then estimated the production rate of positrons during the high activity phase, which are created in the region around the event horizon and then ejected by the outflow. The rate is found to be comparable with that required to explain the 511 keV line emission toward the Galactic bulge. I considered three processes of $e^\pm$ pair-production via electron-electron scattering, photon-electron collision, and photon-photon annihilation, and interestingly all the three processes give a similar positron production rate. Therefore the model presented here gives a new and natural explanation for the 511 keV line emission toward the Galactic bulge. The favorable aspects of this model are: (1) the correct positron production rate, (2) the large bulge-to-disk ratio and correct spatial extent in the bulge, (3) (i) negligible annihilation near the positron production site before injection into ISM, and (ii) the injection energy of $\sim$ MeV, both of which are consistent with the constraint from the MeV gamma-ray background, and (4) no exotic assumptions or parameters. To my knowledge, the other explanations for the 511 keV emission proposed before do not satisfy all of these. Anisotropy of the morphology of the 511 keV emission larger than expected for other models would be a signature for the model proposed here, which might be revealed by future observations with better angular resolutions. Detection of 511 keV lines from centers of nearby galaxies by the same mechanism will be difficult even in the foreseeable future. In contrast, some other scenarios predict detectable 511 keV lines in directions other than the Galactic center by a modest improvement of the sensitivity. 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The upper-left diagram is the all sky map showing the scales of the 511 keV annihilation line emission and the outflow making the North Polar Spur (NPS). The upper-right diagram is for the region within $\sim 1^\circ$ of the Galactic center, showing the scale of the outflow observed as the Galactic center lobe (GCL) and the expanding molecular ring (EMR). Famous objects found in the radio image of this region are shown by greyscale, and the X-ray reflection nebulae are indicated by the black regions. The lower diagram is for the innermost region, showing the interaction of Sgr A East, West, and Sgr A$^*$ surrounded by the ionized halo. } \label{fig:schem} \end{figure} \end{document}