------------------------------------------------------------------------ From: "[gb2312] Huang Lei" muduri@shao.ac.cn To: gcnews Subject: [gb2312] submit mnshadow.tex MNRAS accepted %astro-ph/0703254 %http://www.arxiv.org/pdf/astro-ph/0703254 % mn2esample.tex % % v2.1 released 22nd May 2002 (G. Hutton) % % The mnsample.tex file has been amended to highlight % the proper use of LaTeX2e code with the class file % and using natbib cross-referencing. These changes % do not reflect the original paper by A. V. Raveendran. % % Previous versions of this sample document were % compatible with the LaTeX 2.09 style file mn.sty % v1.2 released 5th September 1994 (M. Reed) % v1.1 released 18th July 1994 % v1.0 released 28th January 1994 \documentclass[useAMS,usenatbib]{mn2e} \usepackage{graphicx} \usepackage{amsmath} % If your system does not have the AMS fonts version 2.0 installed, then % remove the useAMS option. % % useAMS allows you to obtain upright Greek characters. % e.g. \umu, \upi etc. See the section on "Upright Greek characters" in % this guide for further information. % % If you are using AMS 2.0 fonts, bold math letters/symbols are available % at a larger range of sizes for NFSS release 1 and 2 (using \boldmath or % preferably \bmath). % % The usenatbib command allows the use of Patrick Daly's natbib.sty for % cross-referencing. % % If you wish to typeset the paper in Times font (if you do not have the % PostScript Type 1 Computer Modern fonts you will need to do this to get % smoother fonts in a PDF file) then uncomment the next line % \usepackage{Times} %%%%% AUTHORS - PLACE YOUR OWN MACROS HERE %%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title[Black Hole Shadow Image and Visibility Analysis of Sgr~A*] {Black Hole Shadow Image and Visibility Analysis of Sagittarius~A*} \author[L. Huang et al.]{ Lei Huang$^{1,3}$, %\thanks{E-mail:email@address (AVR); otheremail@otheraddress (ANO)} Mike Cai$^{4}$, Zhi-Qiang Shen$^{1,2}$, and Feng Yuan$^{1,2}$\\ %\footnotemark[1] %\thanks{This file has %been amended to highlight the proper use of \LaTeXe\ code with the %class file. These changes are for illustrative purposes and do not %reflect the %original paper by A. V. Raveendran.}\\ $^{1}$Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China\\ $^{2}$Joint Institute for Galaxy and Cosmology (JOINGC) of ShAO and USTC, Shanghai 200030, China\\ $^{3}$Graduate School of the Chinese Academy of Sciences, Beijing 100039, China\\ $^{4}$Academia Sinica, Institute of Astronomy and Astrophysics, Taipei, China} \begin{document} %\date{Accepted 1988 December 15. Received 1988 December 14; in original form 1988 October 11} \pagerange{\pageref{firstpage}--\pageref{lastpage}} %\pubyear{2002} \maketitle \label{firstpage} \begin{abstract} The compact dark objects with very large masses residing at the centres of galaxies are believed to be black holes. Due to the gravitational lensing effect, they would cast a shadow larger than their horizon size over the background, whose shape and size can be calculated. For the supermassive black hole candidate Sgr A*, this shadow spans an angular size of about 50 micro~arc~second, which is under the resolution attainable with the current astronomical instruments. Such a shadow image of Sgr~A* will be observable at about 1 mm wavelength, considering the scatter broadening by the interstellar medium. By simulating the black hole shadow image of Sgr A* with the radiatively inefficient accretion flow model, we demonstrate that analyzing the properties of the visibility function can help us determine some parameters of the black hole configuration, which is instructive to the sub-millimeter VLBI observations of Sgr A* in the near future. \end{abstract} \begin{keywords} black hole physics -- relativity -- methods: numerical -- scattering -- Galaxy:centre -- sub-millimeter -- techniques:interferometric. \end{keywords} \section{Introduction} There is compelling evidence that a $~4\times 10^6 M_\odot$ black hole, associated with an extremely compact radio source Sagittarius A* (Sgr A*), resides at the Galactic Centre \citep*{Sd03, Gz05, Me01}. Recent Very Long Baseline Interferometry (VLBI) observations reveal that the intrinsic size of Sgr A* is only about 1 AU \citep{Shen05}. For such a high degree of mass concentration, the relaxation time scale is much shorter than the age of our Galaxy, and thus almost all the theoretical models predict the formation of a supermassive black hole via gravitational collapse \citep{Mz98}. However, to conclusively prove that Sgr A* is a supermassive black hole requires observations close to its event horizon, where relativistic effects can not be ignored. According to General Relativity, the photon trajectory will bend appreciably in a strong gravitational field \citep*[see, e.g., ][]{M73, Chdr83}. Because of this peculiar behavior, images of black holes and their immediate neighborhood we observe are highly distorted. The true physical structure in the very vicinity of the black hole must be decoded from the image via a ray tracing method \citep*[]{Ft97, Fc00, Schn04, Schn06}. In particular, if the impact parameter is smaller than a critical value, the photon is doomed to fall into the black hole \citep{Chdr83}. As seen by a distant observer, this effect allows the black hole to cast a shadow over the background source that is larger than its event horizon. The black hole candidate Sgr A*, at a distance of 8 kpc to us, spans an angular size of $\sim$ 20 micro arc second ($\mu$as) in diameter in the sky, which is beyond the resolution attainable with the current astronomical instruments. However, the shadow size is always of order $10GM/c^2$ in diameter, corresponding to an angular size of $50\mu$as for Sgr~A*. In principle, this resolution can be achieved using VLBI techniques at a baseline length $\sim4\times10^3\rm{km}$ if observed at 1 mm wavelength \citep*[e.g., ][]{Fc00}. In this paper, we adopt the ray-tracing method to map the Sgr A* black hole with the radiatively inefficient accretion flow (RIAF) model \citep*{Y03}. We then use our simulations to estimate the visibility function to be observed by a VLBI array operating at, if can be designed in the future, sub-millimeter wavelengths. We are on the verge of resolving the shadow of the presumed supermassive black hole at the Galactic Centre. The paper is organized as follows. Section $2$ briefly describes the formalism we use for the ray-tracing algorithm. We show the simulation results in Section $3$. Then we estimate the visibility functions in observations in Section $4$. Finally we give a discussion in Section $5$. Throughout this paper, we will work with geometric units where $c=G=1$. \section[]{Hamiltonian Geodesic Ray-tracing in Black Hole Spacetime} In a given spacetime metric, the trajectory of a photon is uniquely determined by the initial conditions. To efficiently simulate the image of a black hole and its environment, we opt to impose our initial conditions at infinity, where the observer is, and integrate backwards in affine parameter toward the black hole \citep[cf.][]{RB94, Ft97}. Starting from the standard kinetic Lagrangian, we compute the super-Hamiltonian \citep[cf.][]{M73, FQ03} in Kerr metric but with spin parameter $a=0$, since we discuss only the Schwarzschild black hole case in this work. Normal variational approach yields the Hamiltonian equations of motion in the phase space $(q^\mu,p_\mu)$, where $q^{\mu}$ and $p_\mu$ are Boyer-Linquist coordinate and the canonical momentum of the particle, respectively. If we choose the affine parameter to be proper time per unit rest mass (which has a finite limit even when the rest mass is zero in the photon case), then $p_\mu$ is interpreted as the covariant component of the four-momentum. Similar process can be seen in, e.g., \citet{Schn04, Br06}. As discovered by \citet{Bard72}, an infalling photon with its impact parameter \citep{Chdr83} less than $\sqrt{27}M$ will never escape the vicinity of the black hole. As seen by an outside observer, the black hole will cast a shadow of radius $r_\text{shadow} = \sqrt{27}M \approx 5.2M$ over the background light source. As reported by many numerical calculations (e.g., Falcke et al, 2000; Takahashi 2004), rotating black holes will cast shadows of approximately the same size as well. In the following simulation, we specify initial conditions at a large distance (which we have chosen arbitrarily $r_0 = 200M$), and then integrate backward toward the black hole to obtain the full geodesic. \section{Results From Simulation} Fig. 1 is a sketch map, defining the inclination angle $i$ \citep{Chdr83} and position angle $\Theta$ used in our simulation. Three axes $(X,Y,Z)$ point east, north and the direction aligning with the line of sight for an observer, respectively. The thick blue arrow represents the direction of the black hole's equatorial plane (the spin axis for the Kerr black hole). The inclination angle $i\in [-\pi/2,\pi/2]$ is the angle between the spin axis and the observer's line of sight (Z axis), and the position angle $\Theta\in [-\pi,\pi]$ is the angle between the projection of spin axis on $X-Y$ plane and $Y$ axis, positive by east and negative by west. We limit $i\in [0,\pi/2]$ and $\Theta\in [-\pi/2,\pi/2]$ in the following discussion because of symmetry. \subsection{Black Hole Shadow of Sgr A* with the Radiatively Inefficient Accretion Flow Model} Even though there have been various theoretical models proposed to explain the observations of Sgr A* \citep*[see, e.g., ][]{Me01}, the precise structure of the dominant emitting region is still poorly known. In this paper, we only consider the RIAF model proposed by \citet{Y03}, which can satisfy most observational results including the observed spectrum of Sgr A* from radio to X-ray, the flaring activity at both infrared and X-ray bands, and its extremely faint luminosity. In the RIAF model, the emission of Sgr A* from millimeter to sub-millimeter is dominated by the synchrotron radiation of thermal electrons. Thus one can determine the frequency-dependent emissivity $j_\nu(r,0)$ in the equatorial plane. Assuming a Gaussian distribution of electron density in the vertical direction of a RIAF, the emissivity distribution function is then: \begin{eqnarray}\label{distr1} j_\nu(r,z) = j_\nu(r,0)\exp(-\frac{z^2}{H^2}), \end{eqnarray} where $z$ is the vertical distance from the equatorial plane and $H$ is the scale height determined by: \begin{eqnarray}\label{H} H = \frac{\alpha_s}{\Omega_\rmn{K}}, \end{eqnarray} where $\alpha_s$ is the thermal sound speed and $\Omega_\rmn{K}$ is the Keplerian angular velocity. The absorption coefficient $\kappa_\nu$ is then given by Kirchhoff's Law. In this work, exactly the same RIAF model parameters in \citet{Y03} and \citet*{Y06} are adopted to preserve the good fit to the spectrum of Sgr A* and other observations. To construct an observed emission structure, we then perform a full radiation transfer calculation using the ray-tracing method outlined in Section 2. Here, the radiative transfer equation takes the form: \begin{eqnarray}\label{radtrans} \frac{dI_{\nu}}{ds} = j_{\nu} - \kappa_{\nu}I_{\nu}. \end{eqnarray} Since the frequency is red-shifted along the geodesic and $I_\nu/\nu^3$ is a Lorentz invariant \citep[see, e.g., ][]{Ch06, Schn06, Nb07}, the observed specific intensity is related to the emitted intensity by \begin{eqnarray}\label{Iinv} I_{\nu_{\rm{obs}}} = \delta^3I_{\nu_{\rm{em}}}, \end{eqnarray} where the gravitational red-shift factor is calculated to be \begin{eqnarray}\label{redshf} \delta = \frac{\nu_{\rm{obs}}}{\nu_{\rm{em}}}={\sqrt{1-\frac{2M}{r}}} \end{eqnarray} for a non-spinning black hole. We neglect the re-emission process when solving the radiation transfer equation. Furthermore, light rays emitted from the RIAF bend to us due to strong gravitation of the black hole, suffer absorption and in addition, suffer interstellar scattering. That means, what we can observe is a scatter-broadened image. The two-dimensional scattering structure in the direction to the Galactic Centre is determined from fitting to the angular size measurements as a function of the observing wavelength \citep[cf.][]{Shen05}. It is a Gaussian ellipse with full-widths at half maximum (FWHM) of major axis and minor axis in milli-arcsecond (mas) to be $(1.39\pm0.02)\lambda^2$ and $(0.69\pm0.06)\lambda^2$ ($\lambda$ measured in $\rm{cm}$), respectively and position angle $\sim80^\circ$ \citep{Shen05}. We take the conversion that one gravitation radius corresponds to $~5\mu$as angular size for Sgr A* with $4\times10^6M_\odot$ mass at 8kpc distance. In Fig. 2, we show both the un-scattered (ray-tracing only) and scatter-broadened images at wavelengths of 1.3 mm (first two columns of panels) and 3.5 mm (last two columns), respectively, obtained from our simulations with parameters $\Theta=0^\circ$, $i=\pi/2,\pi/4,0$ (from top to bottom). For each two columns at two wavelengths, right one is obtained by convolving the left one (GR ray-tracing result) with the interstellar scattering. We use color to show the self-normalized specific intensity of the images. The abscissa axes point to west and the ordinal axes point to north. The non-scattered (intrinsic) images are different at different wavelengths. They exhibit predicted shadows as a result of relativistic effects. The scattered image becomes more obscured as the observing wavelength increases. It indicates that gradually with the availability of the high-resolution VLBI imaging at wavelengths of 1.3 mm or shorter, we will be at a very good position to unveil the shadow structure of the black hole. \subsection{Comparison with the VLBI Observations at 7 and 3.5 mm} Currently, VLBI observations can be steadily performed at millimeter wavelengths of 3.5 and 7 mm. Attempts to determine the Sgr A* structure with the VLBI observations, however, have suffered from the angular broadening caused by the diffractive scattering by the turbulent ionized interstellar medium, which dominates the resultant images with a $\lambda^2$-dependence apparent size. The development of the model fitting analysis by means of the amplitude closure relation along with the careful design of the observations at millimeter wavelengths has greatly improved the accuracy of the size measurements of the observed image \citep*[e.g.,][]{Shen03,Bo04,Shen05}. As a result, an intrinsic source size as compact as 1 AU was first detected at 3.5 mm \citep{Shen05}. However, for the reason discussed in \citet{Y06}, it is more meaningful to directly compare the observed apparent VLBI image with the scatter-broadened image obtained from the simulations. The apparent images of Sgr A* can be depicted quantitatively by an elliptical Gaussian distribution with the following parameters of the major and minor-axis sizes \citep*{Shen05}: 0.724$\pm$0.001 mas by 0.384$\pm$0.013 mas and 0.21$^{+0.02}_{-0.01}$ mas by 0.13$^{+0.05}_{-0.13}$ mas at wavelengths of 7 and 3.5 mm, respectively. The position angles at both wavelengths are about 80$^\circ$, consistent with the orientation of the scattering structure. While \citet{Y06} only considered a special configuration, i.e., the RIAF is face-on ($i=0^\circ$ and $\Theta=0^\circ$), here we will do a thorough investigation on the possible geometry of the RIAF with respect to the presumed supermassive black hole in the Galactic centre based on the available VLBI measurements at both 3.5 and 7 mm. For this purpose, we first made a series of simulated after-scattering images with different geometric configuration (different combination of $i$ and $\Theta$) between the black hole and the RIAF. Then, we fit the simulated images with an elliptical Gaussian distribution to estimate the FWHMs of the major and minor axes. In Fig. 3, the FWHMs of the major and minor axes of the simulated scattering-broadened images of Sgr A* are plotted as a function of the inclination angle ($i$) and the position angle ($\Theta$), respectively. Thick solid lines are the upper and lower limits of the FMHMs with the best-fit scattering angular size of 1.39$\lambda^2\times$0.69$\lambda^2$, and thick dashed lines are the upper and lower limits to account for the $\pm1\sigma$ ($0.02\lambda^2\times0.06\lambda^2$) in the scattering size. The measured angular sizes from high-resolution VLBA imaging \citep*{Shen05} are indicated in Fig. 3 by the thin solid lines with the $1\sigma$ uncertainties by the thin dotted lines. As already shown in \citet{Y06}, within the uncertainties of the measurements and calculations, the predicted sizes from the RIAF model with $i=0^\circ$ and $\Theta=0^\circ$ are in reasonable agreement with the observations at two wavelengths. With a more thorough consideration shown in Fig. 3, it seems to suggest a geometry of the RIAF in Sgr A* to have a large inclination angle $i$ and a small position angle $\Theta$. But it should be cautious since the results in the minor axis are not accurate, especially at 3.5 mm \citep*[see,][]{Shen05}. It is clear to us that in order to get a reliable estimate of the geometry ($i$ and $\Theta$) of the RIAF in Sgr A*, more accurate measurements are needed. \section{Visibility Functions of The Black Hole Shadow Images} As shown in Fig. 2 \citep*[also see Fig. 1 in][]{Y06}, the simulated apparent scatter-broadened images of Sgr A* at wavelengths of 1.3 mm and shorter cannot be well fitted by one elliptical Gaussian component, indicating that the interstellar medium scattering effects at 1.3 and 0.8 mm no longer dominate the observed morphology. So, it is very promising to image the shadow of the supermassive black hole Sgr A* with the high-resolution VLBI observations at sub-millimeter wavelengths. However, there is no such a sub-millimeter VLBI array available at present for the imaging observations. Therefore, we will not focus on the comparison of the images at wavelengths of 1 mm and shorter, but try the visibility analysis to show that some useful constraints on the emission of Sgr A* can be extracted from the limited visibility data without imaging. In the interferometric observation, the visibility function $V(u,v)$ measured on a baseline with coordinates $(u,v)$ is related, by the Fourier transform, to the sky brightness distribution, i.e., the shadow image in our simulation. To provide instructive information to the real observation, we first perform two-dimensional Fourier transform to obtain the visibility function of shadows like those in Fig. 2. Then we analyze the visibility distribution along four specific directions in the sky plane with position angles of (i) $80^\circ$ (E$^\prime$), i.e., along the major axis of the scattering structure; (ii) $-10^\circ$ (N$^\prime$), i.e., along the minor axis of the scattering structure; (iii) $35^\circ$ (NE$^\prime$); and (iv) $-55^\circ$ (NW$^\prime$). In Fig. 4, we show an example of such visibility slices with parameters, $i=78.75^\circ$ and $\Theta=-70^\circ$ at $\lambda=1.3\rm{mm}$. The abscissa is projected baseline length in the units of $10^9\lambda$ and the ordinate the normalized visibility amplitude. We plot slices in solid, long-dashed, short-dashed, and dotted line for E$^\prime$, N$^\prime$, NE$^\prime$, and NW$^\prime$ directions, respectively. To depict these visibility slices, we introduce two characteristic baseline lengths: $\Sigma$ (marked with open circle) to denote the baseline length at which a normalized visibility decreases to $0.5$, and $\sigma$ (marked with filled circle) to denote the baseline length at which the visibility reaches its first valley. Such a minimum in the visibility distribution is important because it implies that the image can no longer be described by a single Gaussian but must have an additional structure, which could be related to the black hole shadow. Visibilities at baselines longer than $\sigma$ have quite limited signal-to-noise ratio at millimeter wavelengths. The typical detection limit with the current VLBA (Very Long Baseline Array) at 3.5 mm is about $0.12$~mJy. It is obvious that different sets of geometrical parameters ($i$ and $\Theta$) will result in different visibility profiles at different wavelengths. For a given orientation of the RIAF model at a given wavelength, we can obtain two characteristic baseline lengths ($\Sigma$ and $\sigma$) along each of the four directions (E$^\prime$, N$^\prime$, NE$^\prime$, and NW$^\prime$), i.e. ($\Sigma_{E^\prime}$, $\sigma_{E^\prime}$), ($\Sigma_{N^\prime}$, $\sigma_{N^\prime}$), ($\Sigma_{NE^\prime}$, $\sigma_{NE^\prime}$), and ($\Sigma_{NW^\prime}$, $\sigma_{NW^\prime}$). We further list these in the order of their lengths and denote them as $\Sigma_{n}$ and $\sigma_{n}$ with n=1 (the shortest) to 4 (the longest). Our simulations show that the scattering effect at 3.5~mm is still quite large (see Fig. 2), thus we cannot get enough information of any fine structure of the shadow. To minimize the scattering effect, we consider the observations to be performed at other two shorter wavelengths 1.3 and 0.8~mm. Shown in Fig. 5 are the two characteristic baseline lengths ($\Sigma$ and $\sigma$) as a function of the inclination angle $i$ at 1.3 (upper two panels) and 0.8~mm (lower two panels). With a specific $i$, each of $\Sigma_n$ and $\sigma_n$ (n=1$-$4) can vary within a region due to the different position angle $\Theta$. These allowed regions for $\Sigma_1$, $\Sigma_2$, $\Sigma_3$, and $\Sigma_4$ (or, $\sigma_1$, $\sigma_2$, $\sigma_3$, and $\sigma_4$) are shown as the darkest grey with long-dashed border lines, lightest grey with dotted border lines, second lighter grey with short-dashed border lines, and white with solid border lines, respectively. From these plots, we summarize some useful properties as follows: \begin{enumerate} \item Most of the characteristic baseline lengths are in a range of (1$-4)\times10^3~\rm{km}$, which should be achievable under current VLBI observation conditions. Therefore, VLBI observations of Sgr A* at sub-millimeter wavelengths are expected to provide very important results in the near future. Some extreme cases may require a project baseline length longer than 5$\times10^3~\rm{km}$ for $\sigma_4$. \item When the RIAF is nearly face-on ($i\approx0^\circ$), the four $\Sigma$s (or $\sigma$s) are converged to a single similar number, suggesting that the four visibility profiles are almost identical and thus not orientation-dependent. This is mainly due to the isotropy of the intrinsic shadow image without scattering. The roughly East-West (80$^\circ$ in position angle) elongated scattering structure is very compact, $\le 15 \mu$as along the major axis at about 1 mm wavelength, and thus will not affect the symmetry of the final scatter-broadened image. \item With a moderate inclination angle $i$, the differences between these projected baselines at different directions are quite significant. In fact, the four $\Sigma$s (or $\sigma$s) are always range in four separate regions (as shown in Fig. 5) because of the asymmetry of the image. The possible region, caused by the different position angle $\Theta$, for each $\Sigma$ and $\sigma$ becomes larger when the inclination angle $i$ increases. \end{enumerate} Therefore, in principle we would be able to constrain the inclination angle ($i$) of the RIAF in Sgr A* by comparing the above-mentioned characteristic baseline lengths determined from the future 1.3 and/or 0.8 mm VLBI experiments with the simulation demonstrated in this paper. It can be inferred from Fig. 5 that when the RIAF has a large inclination angle (in our simulation of Sgr A*, $i>\sim 30^\circ$), both $\Sigma$s and $\sigma$s become very sensitive to $\Theta$. No wonder, the position angle $\Theta$ should have effects on the characteristic baseline lengths too. For clarity, we define here another three normalized differences in the characteristic baseline lengths: \begin{eqnarray}\label{ss} S_1 &=& (\Sigma_{N^\prime}-\Sigma_{E^\prime})/(\Sigma_{4}-\Sigma_{1}), \nonumber\\ S_2 &=& (\Sigma_{NE^\prime}-\Sigma_{NW^\prime})/(\Sigma_{4}-\Sigma_{1}), \\ S_3 &=& (\Sigma_{E^\prime}-\Sigma_{NE^\prime})/(\Sigma_{4}-\Sigma_{1}).\nonumber \end{eqnarray} Note that $\Sigma_{4}$ and $\Sigma_{1}$ represent the longest and shortest characteristic baseline lengths, respectively. If the RIAF is nearly face-on ($i\approx0^\circ$), the scatter-broadened image of Sgr A* is roughly E-W elongated, resulting in the $\Sigma_{N^\prime}$ being the longest and $\Sigma_{E^\prime}$ the shortest (i.e., $\Sigma_{N^\prime}$=$\Sigma_{4}$ and $\Sigma_{E^\prime}$=$\Sigma_{1}$), and $\Sigma_{NE^\prime}$=$\Sigma_{NW^\prime}$ because of its symmetry. Therefore, it is always the case that $S_1=1$, $S_2=0$ and $-1