Issue 
A&A
Volume 574, February 2015



Article Number  A48  
Number of page(s)  9  
Section  Astrophysical processes  
DOI  https://doi.org/10.1051/00046361/201424306  
Published online  22 January 2015 
A magnetized torus for modeling Sagittarius A^{∗} millimeter images and spectra
^{1}
Nicolaus Copernicus Astronomical Center, ul. Bartycka 18,
00716
Warszawa,
Poland
email:
fvincent@camk.edu.pl
^{2}
LUTH, Observatoire de Paris, CNRS, Université Paris
Diderot, 5 place Jules
Janssen, 92190
Meudon,
France
email:
odele.straub@obspm.fr
^{3}
Physics Department, Gothenburg University,
41296
Göteborg,
Sweden
email:
marek.abramowicz@physics.gu.se
^{4}
Institute of Physics, Faculty of Philosophy and Science, Silesian
University in Opava, Bezručovo nám.
13, 74601
Opava, Czech
Republic
Received: 30 May 2014
Accepted: 17 November 2014
Context. The supermassive black hole, Sagittarius (Sgr) A*, in the centre of our Galaxy has the largest angular size in the sky among all astrophysical black holes. Its shadow, assuming no rotation, spans ~50 μas. Resolving such dimensions has long been out of reach for astronomical instruments until a new generation of interferometers being operational during this decade. Of particular interest is the Event Horizon Telescope (EHT) with resolution ~20 μas in the millimeterwavelength range 0.87 mm–1.3 mm.
Aims. We investigate the ability of the fully general relativistic Komissarov (2006, MNRAS, 368, 993) analytical magnetized torus model to account for observable constraints at Sgr A* in the centimeter and millimeter domains. The impact of the magnetic field geometry on the observables is also studied.
Methods. We calculate raytraced centimeter and millimeterwavelength synchrotron spectra and images of a magnetized accretion torus surrounding the central black hole in Sgr A*. We assume stationarity, axial symmetry, constant specific angular momentum and polytropic equation of state. A hybrid population of thermal and nonthermal electrons is considered.
Results. We show that the torus model is capable of reproducing spectral constraints in the millimeter domain, and in particular in the observable domain of the EHT. However, the torus model is not yet able to fit the centimeter spectrum. 1.3 mm images at high inclinations are in agreement with observable constraints.
Conclusions. The ability of the torus model to account for observations of Sgr A* in the millimeter domain is interesting in the perspective of the future EHT. Such an analytical model allows very fast computations. It will thus be a suitable test bed for investigating large domains of physical parameters, as well as nonblackhole compact object candidates and alternative theories of gravity.
Key words: Galaxy: center / accretion, accretion disks / black hole physics / relativistic processes
© ESO, 2015
1. Introduction
A black hole is characterized by its event horizon, a boundary that causally separates the external universe from its interior and makes the black hole appear black. The reason why there are so far no direct observations of their immediate environment is that astrophysical black holes observed from Earth have a very small apparent angular size in the sky. The measurable apparent size of a black hole refers to the size of its shadow (we will use the term introduced by Falcke et al. 2000), the dark area on the observer’s sky cast by the black hole illuminated by nearby radiation. The edge of this shadow is a very thin ring of light produced by photons approaching the black hole as close as its photon orbit, i.e. the innermost circular orbit around the black hole. Photons visiting the region inside the photon orbit may either fall below the event horizon, or may still escape but with such high redshifts that the whole area inside the black hole shadow appears dark. The black hole with largest shadow of all is the supermassive black hole in the centre of our Galaxy, Sagittarius (Sgr) A*. Given its mass M = 4.3 × 10^{6} M_{⊙} and distance D = 8.3 kpc (Ghez et al. 2008; Gillessen et al. 2009a,b), the angular size of its shadow projected on sky, taking into account the magnification due to gravitational lensing, is only ≈50 μas for a nonrotating black hole. This angular size is a decreasing function of the black hole spin, reaching ≈40 μas for a maximally rotating black hole. Sgr A* is not the only black hole on sky with such an apparent size: the black hole at the center of the M 87 galaxy reaches an apparent size of 40 μas (assuming no rotation). However, this article is only dedicated to studying Sgr A*.
Sgr A* was first observed in the radio band (Balick & Brown 1974), but its observed emission ranges from radio to Xray energies. The most remarkable feature of Sgr A* is its complex variability at all observable wavelengths. The luminosity fluctuations increase with increasing energy, from a factor of a few at radio to a few orders of magnitude in the Xray band (see e.g. Genzel et al. 2010, for a review). The spectral peak lies in the millimeter radio band and brings forth a peak luminosity of ≲10^{35} erg s^{1} (or ≈10^{9}L_{Edd}). The accretion structure around Sgr A* is thus extremely dim given its enormous mass. Therefore, adequate disk models describe a radiatively inefficient emitter like an advection dominated accretion flow (ADAF, Narayan & Yi 1995, see also the recent review by Yuan & Narayan 2014). The term advection means here that a large part of the gravitationally liberated thermal energy is not converted into radiation but carried inward with the ionised, hot accretion flow.
In the millimeter radio range, i.e. at wavelengths corresponding to the spectral peak of Sgr A*, the Event Horizon Telescope (EHT, Doeleman et al. 2009), operational in 2015–2020, will be able to perform high resolution Very Long Baseline Interferometry (VLBI) observations. Like this, images of the close environment of the black hole will be obtained, in particular images of the accretion flow. Recently, the intrinsic emitting region around Sgr A* was constrained to merely 37 μas by millimeter VLBI (Doeleman et al. 2008). Since this is smaller than the shadow size of the presumed black hole, the observed emission should originate from the surrounding accretion flow, seen with a high inclination.
These new observational possibilities on Sgr A* have stimulated a lot of recent research dedicated to modeling the accretion flow surrounding this black hole (see e.g. Goldston et al. 2005; Noble et al. 2007; Chan et al. 2009; Mościbrodzka et al. 2009; Dexter et al. 2010; Shcherbakov et al. 2012; Mościbrodzka & Falcke 2013), as well as works specifically related to the future EHT (reviewed by Broderick et al. 2014). The hope is that a detailed knowledge of theoretically predicted observational appearance of the structure of the accretion flow in Sgr A* will provide powerful and reliable tools to test Einstein’s general relativity at its strongfield limit. While eventually sophisticated general relativistic magnetohydrodynamic (GRMHD) numerical models (see e.g. McKinney et al. 2014) of Sgr A* will be used to make a meaningful comparison between theory and observations, in the foreseeable future simple analytic models will be invaluable as a secure guide in the vast parameter space that needs to be explored. In the past 15 years, mainly two such analytic models have been proposed: the radiatively inefficient accretion flow (RIAF) model (Narayan et al. 1995; Özel et al. 2000; Yuan et al. 2003; Broderick et al. 2011) and the jet model (Falcke & Markoff 2000). We propose here a third analytic model, that we will call the torus model.
We have recently constructed an analytic optically thin torus (socalled Polish Doughnut) model of Sgr A* (Straub et al. 2012). It assumed that the magnetic field in Sgr A* had no global structure, but instead was chaotic i.e. locally isotropic. In this paper we make the next logical step by considering a model with a globally ordered (toroidal) magnetic field. For this, we explicitly calculate images and spectra of Sgr A* in the centimeter and millimeter domains, from 10 GHz to 10^{3} GHz. Hereafter, we will refer to the ≈10 GHz–100 GHz part of the spectrum as the centimeter spectrum, and the to the ≈100 GHz–10^{3} GHz as the millimeter spectrum. In particular, the millimeter spectrum is relevant for the future EHT observations at the wavelengths 0.87 mm and 1.3 mm (345 GHz and 230 GHz). We use the Komissarov (2006) analytic model of a magnetized optically thin Polish Doughnut and follow all its assumptions. In the Komissarov model, all general relativistic effects, and influence of the (toroidal) magnetic field are fully and exactly taken into account. They are calculated from the first principles with no approximation. The presence of a magnetic field is important in calculations of the synchrotron radiation emissivity, which we also derive following Wardziński & Zdziarski (2000). We consider here a torusshaped, barotropic, and stationary disk with axisymmetry and constant angular momentum around a Kerr black hole. The disk is fully ionized. These assumptions reflect the basic physics of the real object. We compute synchrotron spectra emitted by such tori in the range 10 GHz–10^{3} GHz and fit them to observed data. The spectral fits are compared to the predictions of the RIAF model. We compute images at 1.3 mm and compare their sizes to observable constraints. We also study the difference in the millimeter spectra and images implied by the magnetic field geometry (chaotic or ordered).
We summarize the basic features of the magnetized torus model and its synchrotron radiation in Sects. 2 and 3, respectively. Section 4 investigates the ability of the torus model to reproduce spectral observable constraints. Section 5 is dedicated to studying the millimeter images predicted by our model and Sect. 6 presents conclusions and perspectives.
2. Magnetized accretion torus
In this section we investigate the observable appearance of an accretion torus for two distinct magnetic field configurations: toroidal and isotropic (i.e. chaotic). We are also interested in the electron number density distribution, particularly as compared to the RIAF distribution.
2.1. Toroidal magnetic field (the Komissarov model)
We constructed a magnetized accretion torus at the Galactic centre using the model developed by Komissarov (2006), which describes analytically a polytropic accretion torus with toroidal magnetic field in the Kerr spacetime.
The fluid 4velocity is assumed to be (1)using BoyerLindquist coordinates. We assume a constant specific angular momentum (2)This quantity is expressed in terms of the dimensionless specific angular momentum (3)where ℓ_{ms} and ℓ_{mb} are the specific angular momentum at the marginally stable and bound orbits respectively. These assumptions fully determine the 4velocity.
The gas and magnetic pressures are assumed to follow the polytropic prescription (4)where p is the gas pressure, p_{m} is the magnetic pressure, κ and κ_{m} are polytropic constants, k is the polytropic index (assumed identical for gas and magnetic pressures), h = p + ρ is the particle enthalpy where ρ is the gas energy density, and where g_{μν} is the Kerr metric.
The conservation of stressenergy leads to (5)where the potential W = −ln  u_{t}  is used. We assume that the torus fills its Roche lobe, which fixes the central radius of the torus and its surface, thus the values of the potential at the centre, W_{c}, and at the surface, W_{s}, of the torus. This immediately gives (6)where h_{c} is the central enthalpy, ω = (W − W_{s})/(W_{c} − W_{s}) and ℒ_{c} is the value of ℒ at the center of the torus.
The polytropic constants κ and κ_{m} can be expressed according to (7)where β_{c} is the central magnetic pressure ratio, β_{c} ≡ p_{c}/p_{m,c}. The electron number density (8)is then known analytically, as well as the magnetic pressure.
The magnetic pressure has a different expression depending on the magnetic field geometry. For a chaotic, isotropic field it is 1/3 of the magnetic energy density, whereas it is equal to the magnetic energy density for a toroidal field (Leahy 1991). Thus, here (9)where B is the magnetic field 3vector magnitude in the fluid frame.
The magnetic field in the BoyerLindquist frame is assumed to be toroidal, b = (b^{t},0,0,b^{ϕ}), and to be orthogonal to the fluid 4velocity, b·u = 0. In the fluid frame, i.e. in an orthonormal tetrad (u,e_{i}), its components are b = (0,0,0,B). The norm of b is thus equal to the quantity B, the magnitude of the magnetic field 3vector in the fluid frame, which is known analytically. It is then easy to get (10)which is fully known analytically. Let us consider one synchrotron photon emitted at a given point of the torus. Let p be the 4vector tangent to the photon geodesic and l be its projection orthogonal to u. The angle ϑ between the magnetic field b and the direction of emission is given by l·b =   l     b   cosϑ. It is known analytically as well.
We now note that such an accretion torus cannot be made of a perfect gas. If it were, then pm_{u}/ (ρk_{B}) = T/μ_{e} where T is the electron temperature and k_{B} is the Boltzmann constant. However, it is easy to see that p/ρ is independent of the central value of the enthalpy h_{c}. Thus the temperature would be independent of h_{c} as well, and would be purely determined by the geometry of spacetime, which does not make sense. We will still assume that there exists a relation T = Cp/ρ where C is a constant, but does not take its perfectgas value. Rather, we choose T_{c} at the center of the torus and define the constant C by T_{c} = Cp_{c}/ρ_{c}. Then (11)depends on the choice of T_{c}, and no longer only on spacetime geometry.
2.2. Isotropic magnetic field
An accretion torus with isotropic (i.e. chaotic) magnetic field has already been studied around Sgr A* by Straub et al. (2012). This model is simply the limit of the Komissarov (2006) model with κ_{m} = 0. The section above thus directly applies to this simpler case. The magnetic field strength is obtained by assuming that the magnetic pressure is everywhere related to the gas pressure through (12)thus, the β parameter is valid in the whole torus, not only at its center. Then the magnetic field magnitude is known from its link to the magnetic pressure, which for a chaotic magnetic field is given by (Leahy 1991) (13)
2.3. Torus crosssection and density distribution
This section aims at giving an overview of how the accretion torus properties depend on its parameters, and in particular to compare its density distribution to the RIAF model.
The crosssection of the torus is dictated by the spin value and angular momentum λ parameter. Figure 1 shows the crosssection of a torus for λ = 0.9 and spin a = 0 and a = 0.95. The outer radius of the torus strongly decreases when increasing spin, from r_{out} = 120 r_{g} to r_{out} = 30 r_{g} in the case illustrated. The inner radius of the torus (the Roche lobe overflowing point) decreases when spin increases, from r_{in} = 4 r_{g} at spin 0 to r_{in} = 1.5 r_{g} at spin 0.95. An important point is that the central radius, where all physical quantities (density, temperature, magnetic field) reach their maximum, is very different from the geometrical center. In our example case, it goes from r_{center} = 10 r_{g} at spin 0 to r_{center} = 3 r_{g} at spin 0.95. In all cases, the central radius, i.e. the location of the maximum of emission, is at only a few r_{g}. In the optically thin part of the spectrum (i.e., the millimeter spectrum essentially), only the very central part of the torus will contribute to the flux, no matter what the value of λ is.
Fig. 1 Crosssection of the torus surface for a fixed angular momentum λ = 0.9 when varying the spin parameter from a = 0 (solid line) to a = 0.95 (dashed line). As the spin increases, the torus becomes smaller and its inner radius decreases. The axes are in units of the gravitational radius GM/c^{2}. 

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The electron number density distribution (Eq. (8)) is a rather nontrivial function of radius and angle. Figure 2 illustrates its profile in the equatorial plane as a function of radius for the same cases illustrated in Fig. 1, and compares it to the RIAF number density distribution. The RIAF distribution is taken from Broderick et al. (2011), Eq. (2): it is a simple power law with slope − 1.1. The torus model distribution is clearly not a power law, it is concentrated around the central radius of the torus, decreasing much more rapidly than the RIAF power law for bigger radii. As a consequence, the distant regions of the torus model will contribute less than the distant regions of the RIAF model. This point will be important later on.
3. Synchrotron radiation
In this section, we express the synchrotron emission and absorption coefficients for a directional magnetic field, and for either a thermal or nonthermal population of electrons. Radiative transfer for an angleaveraged magnetic field is simply obtained by angleaveraging the directional expressions.
3.1. Thermal directional synchrotron radiation
Wardziński & Zdziarski (2000) show that for a mildly relativistic Maxwellian electron distribution, (14)where θ_{e} = k_{B}T/ (m_{e}c^{2}), m_{e} being the electron mass, γ = (1 − v^{2}/c^{2})^{− 1/2} is the Lorenz factor and K_{2} is a modified Bessel function, the emission coefficient is (15)where ν_{0} ≡ eB/ (2πm_{e}c) is the cyclotron frequency. Then, (16)is the Lorenz factor of those thermal electrons that contribute most to the emission at ν, and (17)(18)The absorption coefficient is simply given by Kirchhoff’s law: where B_{ν} is the Planck blackbody function.
Fig. 2 Normalized distribution of the electron number density for the torus model with spin 0 and λ = 0.9 (solid blue) or spin 0.95 and λ = 0.9 (solid red). For comparison, the RIAF powerlaw distribution is plotted in dashed black (from Broderick et al. 2011). 

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We note that our raytracing code allows to naturally take into account synchrotron selfabsorption. The radiative transfer equation is simply solved along null geodesics until the torus becomes optically thick.
3.2. Nonthermal directional synchrotron radiation
Motivated by the RIAF model which is able to fit the whole radio (centimeter and millimeter) spectrum of Sgr A* using a population of nonthermal powerlaw electrons, we implemented such a population in our torus model.
We consider a population of electrons with a powerlaw energy spectrum (19)where is the powerlaw electron number density. Following Özel et al. (2000) we assume that the energy of the nonthermal electrons is equal a small fraction δ of the energy of the thermal electrons. This leads to (20)where a(θ_{e}) is an orderunity function of temperature given by Özel et al. (2000), Eq. (6).
The powerlaw emission and absorption coefficients are then given by Petrosian & McTiernan (1983)(21)We note here a typo in Özel et al. (2000) and Yuan et al. (2003): the frequency exponent of their absorption coefficient is not correct.
4. Torus model spectra
We use the opensource^{1} raytracing code GYOTO (Vincent et al. 2011) to compute images and spectra of the magnetized accretion torus. We have studied the impact of the models’ parameters on the observables in order to produce simulations that are in agreement with current constraints. Restricting ourselves to the centimeter and millimeter domains, we want our model to be able to reproduce the observed spectrum of Sgr A* (this section). The corresponding images should predict a correct size of the emitting zone which was constrained at 1σ to as at 1.3 mm (intrinsic size) or as (scatter broadened) by Doeleman et al. (2008, next section).
In order to allow a direct comparison with the angleaveraged RIAF model, all torus model simulations are performed with angleaveraging unless otherwise stated.
4.1. Fitting the millimeter spectrum
The RIAF model is able to fit equally well any values of spin and inclination: these two parameters are not constrained by spectral data only (Broderick et al. 2011). We obtain the same behavior with our torus model: spin and inclination are not constrained from the millimeter spectral data only.
Our torus model is parameterized by a set of 7 parameters listed in Table 1. Here, we have always kept constant the plasma β parameter, to β = 10, as well as the polytropic index k = 5/3. For a given pair of spin and inclination, there are thus 3 parameters to fit: the torus constant angular momentum λ, its central temperature T_{c} and central density n_{c}. We have performed a standard minimum χ^{2} analysis, minimizing the distance from model to data on a grid of parameters. We have been considering a grid of (λ,T_{c},n_{c}) with λ varying from 0.3 to 0.9, T_{c} from 7 × 10^{10} K to 3 × 10^{12} K, and n_{c} from 3 × 10^{6} cm^{3} to 10^{7} cm^{3}. We have considered 30 × 30 × 10 values of (λ,T_{c},n_{c}) within these boundaries. We have investigated only extreme values of the spin parameter, a = 0 and a = 0.95, considering three values of inclination for both cases (i = 5°, i = 45°, i = 85°). Inclination is defined from the axis of rotation of the black hole to the line of sight: i = 5° means nearly faceon view.
Torus model parameters.
Table 2 gives the bestfit parameters for these configurations. It shows that all fits lead to –0.4, thus to very good fits. Such low values of are obtained because of the large error of the 690 GHz data (see Fig. 3). This large error reflects the variability of the source rather than instrumental precision (see Marrone et al. 2006). All values of spin and inclination can thus be fitted with very good accuracy: the spectral millimeter data alone are not sufficient to constrain these parameters.
It is interesting to note that all bestfit physical values are rather close to each other, with r_{out} ≈ 15, n_{c} ≈ 5 × 10^{6} cm^{3}, T_{c} ≈ 5 × 10^{11} K. These values are in good agreement with other models of Sgr A* accretion flow, be they analytic or numeric (see e.g. Broderick & Loeb 2006; Mościbrodzka et al. 2009). A particularly striking point is the clear preference for very compact configurations, with the outer boundary of the torus being in all cases at 10–20 r_{g}, translating to approximately 70 μas from the black hole position. This makes a big difference with respect to the RIAF model, for which the distribution of electrons is extended to regions far away from the black hole.
Torus model bestfit parameters for 2 extreme values of spin and a range of inclinations.
Figure 3 shows the 6 spectra associated to the best fits in Table 2. It is interesting to note that the thermal bump of the spectrum is always obtained at rather high frequency (between 10^{12} Hz and 5 × 10^{13} Hz). For completeness, Fig. 3 also shows the nearinfrared upper limits: it is likely that some spin/inclination configuration might be excluded when fitting from radio to infrared data. However, here, the fit is only done on millimeter data, which means on the data points in Fig. 3 with frequency between ≈ 10^{11} Hz and 10^{12} Hz. The infrared upper limits are not taken into account in the fitting procedure.
Fig. 3 Bestfitting spectra obtained for the parameters listed in Table 2. Blue curves are for spin 0, black for spin 0.95. Solid line is for i = 5°, dashed for i = 45°, and dotted for i = 85°. Note that the fitting is done only on the millimeter error bars. The infrared upper limits are given for completeness, but are not taken into account in the fit. 

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Fig. 4 distribution in the (n_{c},T_{c}) plane, for the a = 0, i = 85° fit. All values of bigger than 1 have been put to a high value and appear in white color in this plane to focus on the relevant configurations with . Left: using the error bar at 690 Hz used in the fitting routine. Right: reducing the 690 GHz error by a factor of 3: the region with immediately shrinks. We note that the bestfit temperature changes, while the bestfit density remains the same. 

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4.2. Robustness of the fit results
We have analyzed in more detail the distribution of χ^{2} for the two cases in Table 2 corresponding to i = 85°. This choice was dictated by the fact that higher inclination is favored by the imaging constraints (see next section).
The first important result is that the size of the torus is indeed firmly constrained to values of r_{out} ≲ 20 r_{g}. At spin 0, all are above 1 for λ> 0.63, and all 0.5 ≤ λ ≤ 0.63 lead to spectra distant by more than 1σ from the 690 GHz data, which has a very large error bar due to the source variability. We conclude that we can safely constrain the angular momentum to λ< 0.5, equivalent to r_{out}< 20. We obtain a comparable constraint from the a = 0.95 fit. The torus model is thus imposing a compact configuration.
Figure 4 shows the 2D distribution in the (n_{c},T_{c}) plane for the bestfitting value λ = 0.33 at spin 0 and inclination 85°. From the left panel, the central density is rather well constrained (up to a factor of 2), but the central temperature is allowed to vary by a factor of ≈10, always giving . This freedom is only due to the large error bar at 690 GHz. The right panel of Fig. 4 shows the same distribution, but this error has been reduced by a factor of 3. The central temperature is immediately much more constrained. However, the bestfit value of T_{c} with a reduced error bar at 690 GHz differs from its counterpart with the unmodified error bar (while the central density remains unchanged). The central temperature changes from T_{c} = 2.3 × 10^{11} K to T_{c} = 5.5 × 10^{11} K. A very similar analysis could be made with spin 0.95 fit.
We conclude from this analysis that the fit results presented in Table 2 are robust as far as the angular momentum (or torus outer radius) and central density are concerned. The central temperature is less robust due to the large error bar at 690 GHz.
4.3. Impact of magnetic field geometry on the millimeter spectrum
One of our interest in this article is to investigate possible observable differences due to the magnetic field geometry. Figure 5 shows one of the bestfit spectra (a = 0.95, i = 85°), obtained for an angleaveraged configuration, together with the bestfit spectrum associated to a directional magnetic field. This directional best fit is found for the parameters λ = 0.85,n_{c} = 6.3 × 10^{6} cm^{3},T_{c} = 3.6 × 10^{11} K. The small difference between these directional parameters and their angleaveraged counterparts is mainly due to the difference in the relation between magnetic pressure and magnetic field magnitude (see Eqs. (9) and (13)).
The difference between the bestfit spectra is weak and it is not very surprising to conclude that spectral data will not be helpfull to constrain the magnetic field geometry.
Fig. 5 Comparison of the bestfit a = 0.95, i = 85° spectra for an angleaveraged magnetic field (blue) and a directional toroidal magnetic field (black). Spectral data are not enough to put constraints on the magnetic field geometry. The infrared upper limits are given for completeness, but are not taken into account in the fit. 

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4.4. No fit for the centimeter part of the spectrum
So far, we have been focused only on the millimeter part of Sgr A* spectrum, which is likely to be accounted for by thermal synchrotron. Lower radio frequencies are not fitted by a pure thermal population as the lowfrequency slope of the thermal spectrum is too hard to fit the data. The RIAF model is able to fit this centimeter part of the spectrum by invoking a small proportion of nonthermal, powerlaw electrons (Özel et al. 2000). We have thus considered such a population in our model and tried to fit for the centimeter data.
Figure 6 shows the complete radio (centimeter to millimeter) spectrum associated to a = 0 and i = 5°, for both the torus model and the RIAF model. We have modeled a RIAF following Broderick et al. (2011). We consider a powerlaw distribution for electron density and temperature, following Eqs. (2) and (3) of Broderick et al. (2011). The nonthermal population is then constructed following the recipe given in Sect. 3.2. Figure 6 clearly shows that the RIAF model perfectly fits the whole radio spectrum, while the torus model is not able to reproduce the correct centimeter slope.
This difference between the two models is due to the very different electron distribution functions. It is well known that the νF_{ν} slope of powerlaw synchrotron equals 7/2. However, for an extended distribution of electrons, different parts of the accretion flow will emit different spectra, and the averaging of these spectra can end up in a much shallower slope. This is a well known effect in jets, and the same behavior is found in the RIAF model. However, the torus model electron distribution being much more compact, such an averaging is not obtained. We have checked that even for very big tori, the nonthermal spectrum slope is still too hard: indeed, as illustrated in Fig. 2, even for an extended torus, the electron distribution is clearly peaked towards small distances. We have also checked that decreasing the outer radius of the RIAF model leads to worse fits. For instance, in the simulation used in Fig. 6, the outer radius of the RIAF is at r_{out} = 300 r_{g}. Decreasing this value to r_{out} = 100 r_{g} leads to a very bad fit, the centimeter spectrum being too low (just as in the torus model), while the millimeter data are still perfectly fit.
Fig. 6 Comparison of the bestfitting a = 0, i = 5° spectra as predicted by the RIAF model (solid black) and the torus model (dotted blue). A nonthermal powerlaw population of electrons is taken into account. The torus model is not able to fit the slope of the centimeter data because of the compact distribution of electrons, very different from the extended distribution of the RIAF model. 

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We conclude that the torus model is still unable to fit the centimeter spectrum of Sgr A*. Another possibility to fit the centimeter data is to introduce a jet (Falcke & Markoff 2000). It is well known that Polish doughnuts, the base model of the torus model, can naturally funnel jets (Abramowicz et al. 1978). We plan in the future to investigate whether a torus+jet analytic model will be able to fit the whole radio specrum. The recent numeric study of such a configuration by Mościbrodzka & Falcke (2013) allows to be rather confident. In this perspective, the recent searches for a putative jet at the Galactic center are particularly meaningful (YusefZadeh et al. 2012; Li et al. 2013).
To conclude this section, we highlight that our main result as far as spectral data are concerned is to demonstrate that the torus model is able to fit with perfect accuracy the millimeter spectrum of Sgr A*, and in particular the future EHT band. However, much work remains to be done in order to develop a complete synchrotron torus model of Sgr A*, from centimeter to nearinfrared wavelengths. Even though our model cannot yet be considered selfconsistent, we will investigate in the next section the image properties of the torus model at 1.3 mm. At this wavelength, the thermal synchrotron dominates, so we believe that our purely thermal model will give realistic predictions. We have checked that a pure thermal and a mixed thermal powerlaw image for the RIAF model are very similar at 1.3 mm.
5. Torus model images
5.1. 1.3 mm images for bestfit spectral models
Images of the bestfit models listed in Table 2 are readily computed at 1.3 mm, a wavelength at which a stringent limit on the size of the emitting region was imposed by Doeleman et al. (2008) at as (1σ, scatter broadened). We need to model the smearing effect due to scattering by interstellar electrons. Following Mościbrodzka & Falcke (2013) we use a Gaussian profile with FWHM 1.309 λ^{2} (Falcke et al. 2000; Bower et al. 2006). At 1.3 mm this boils down to smearing the image with a Gaussian of FWHM ≈ 20 μas, already altering the image quite a lot.
Figure 7 shows the 1.3 mm images corresponding to the bestfit parameters of Table 2. Only the two higher values of inclination were considered as the size limit imposed by VLBI measurements favor higher inclination (Broderick et al. 2011). These images compare the size constraint of Doeleman et al. (2008) to the size of the region emitting 50% of the total flux. The figures show that constraints on the inclination parameter are at hand: at low spin, the i = 45° case is clearly excluded, and marginally acceptable for high spin. At i = 85°, the whole spin range will give a perfect fit to the size constraint.
Fig. 7 Images (maps of specific intensity) at 1.3 mm of the bestfit models of Table 2. The dotted circle show the 1σ confidence domain from Doeleman et al. (2008). The thin solid curve encompass the region containing 50% of the total flux. The color bar is common to all panels, and graduated in cgs units (erg s^{1} cm^{2} str^{1} Hz^{1}). 

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A complete study of the parameter space, fitting for the size of the emitting region for a grid of spins and inclinations, goes beyond the scope of the current article. It will be done in the future, once our model will have been made self consistent over the whole radio to infrared domain. The results presented here still support the torus model as they show that the size predicted in highinclination configuration fits well the VLBI size constraint at 1.3 mm.
5.2. Impact of magnetic field geometry on the image
Figure 8 shows the comparison of the 1.3 mm images associated to the bestfit spectral model at a = 0.95,i = 85°, for the angleaveraged and directional models. It shows that the difference in the image is very weak, in the same way as for the spectral data. It is thus very unlikely that a constraint on the magnetic field geometry will be available by means of spectral and imaging data. The most natural way to get access to this information is to study the polarization properties of the radiation (see e.g. Zamaninasab et al. 2010).
Fig. 8 Comparison of the bestfitting a = 0.95, i = 85° images at 1.3 mm for an angleaveraged magnetic field (left) and a directional toroidal magnetic field (right). 

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6. Conclusion and perspectives
We have constructed a millimeterwavelength synchrotron radiative model for Sgr A* based on the fully general relativistic, analytical magnetized torus model of Komissarov (2006). We have considered the original model with a toroidal magnetic field, as well as an angleaveraged, chaoticmagneticfield configuration. We show that such a model is able to account for the observable spectral constraints in the millimeter domain. However, the torus model is not yet able of reproducing the centimeter data. Constraints on the emitting region size at 1.3 mm by Doeleman et al. (2008) are satisfied by the torus model provided inclination is high enough. We also show that the magnetic field geometry (chaotic or toroidal) cannot be constrained by spectral and imaging data. This work shows that the torus model is a serious candidate to account for the properties of Sgr A* close environment in the millimeter domain, which in particular encompasses the EHT observation band.
Our torus model is the third analytic model proposed to account for Sgr A* properties. The jet model (Falcke & Markoff 2000) and the RIAF model (Narayan et al. 1995; Özel et al. 2000; Yuan et al. 2003; Broderick et al. 2011) are alternatives. We have compared in this article the spectral predictions of the torus and RIAF models. This comparison shows that the RIAF model is able to account for the centimeter spectral data of Sgr A* because of its extended distribution of nonthermal electrons. As the torus model is firmly constrained to be compact (within ≈15 r_{g} from the black hole), it cannot produce the same spectrum. Our goal is to develop in the near future a torus+jet model that would be able to account for both spectral and imaging constraints over the radio band.
The development of such a selfconsistent analytic model over the full synchrotron band is interesting in the perspective of the future EHT data. Such an analytic model allows very fast computations as opposed to GRMHD simulations. For example, one spectrum or one image at a relatively low resolution, sufficient to fit the data, requires a few minutes of computation on a standard laptop for our model. This allows us to investigate large domains of physical parameters, which is not doable with GRMHD simulations because of the computational time limitation. In this perspective, we believe that the torus model for Sgr A* as developed in this work will be of interest for the future data analysis linked with the EHT project. In particular, this model may be a suitable test bed for investigating the observational counterparts of compact objects alternative to the Kerr black hole.
Freely available at the URL http://gyoto.obspm.fr
Acknowledgments
The authors are particularly grateful to the referee, Heino Falcke, for his careful reading of their first manuscript and his very accurate and relevant comments that allowed to greatly improve the quality of this paper. F.H.V. acknowledges interesting discussions with Monika Mościbrodzka at the first Black Hole Cam workshop in Effelsberg. This work was supported by five Polish NCN grants: 2011/01/B/ST9/05439, 2012/04/M/ST9/00780, 2013/08/A/ST9/00795, 2013/09/B/ST9/00060, and 2013/10/M/ST9/00729, together with the European “Synergy” grant CZ.1.07/2.3.00/20.0071 aimed to support international collaboration at the Institute of Physics of the Silesian University in Opava. This work was conducted within the scope of the HECOLS International Associated Laboratory, supported in part by the Polish NCN grant DEC2013/08/M/ST9/00664. Computing was partly done using Division Informatique de l’Observatoire (DIO) HPC facilities from Observatoire de Paris (http://dio.obspm.fr/Calcul/).
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All Tables
Torus model bestfit parameters for 2 extreme values of spin and a range of inclinations.
All Figures
Fig. 1 Crosssection of the torus surface for a fixed angular momentum λ = 0.9 when varying the spin parameter from a = 0 (solid line) to a = 0.95 (dashed line). As the spin increases, the torus becomes smaller and its inner radius decreases. The axes are in units of the gravitational radius GM/c^{2}. 

Open with DEXTER  
In the text 
Fig. 2 Normalized distribution of the electron number density for the torus model with spin 0 and λ = 0.9 (solid blue) or spin 0.95 and λ = 0.9 (solid red). For comparison, the RIAF powerlaw distribution is plotted in dashed black (from Broderick et al. 2011). 

Open with DEXTER  
In the text 
Fig. 3 Bestfitting spectra obtained for the parameters listed in Table 2. Blue curves are for spin 0, black for spin 0.95. Solid line is for i = 5°, dashed for i = 45°, and dotted for i = 85°. Note that the fitting is done only on the millimeter error bars. The infrared upper limits are given for completeness, but are not taken into account in the fit. 

Open with DEXTER  
In the text 
Fig. 4 distribution in the (n_{c},T_{c}) plane, for the a = 0, i = 85° fit. All values of bigger than 1 have been put to a high value and appear in white color in this plane to focus on the relevant configurations with . Left: using the error bar at 690 Hz used in the fitting routine. Right: reducing the 690 GHz error by a factor of 3: the region with immediately shrinks. We note that the bestfit temperature changes, while the bestfit density remains the same. 

Open with DEXTER  
In the text 
Fig. 5 Comparison of the bestfit a = 0.95, i = 85° spectra for an angleaveraged magnetic field (blue) and a directional toroidal magnetic field (black). Spectral data are not enough to put constraints on the magnetic field geometry. The infrared upper limits are given for completeness, but are not taken into account in the fit. 

Open with DEXTER  
In the text 
Fig. 6 Comparison of the bestfitting a = 0, i = 5° spectra as predicted by the RIAF model (solid black) and the torus model (dotted blue). A nonthermal powerlaw population of electrons is taken into account. The torus model is not able to fit the slope of the centimeter data because of the compact distribution of electrons, very different from the extended distribution of the RIAF model. 

Open with DEXTER  
In the text 
Fig. 7 Images (maps of specific intensity) at 1.3 mm of the bestfit models of Table 2. The dotted circle show the 1σ confidence domain from Doeleman et al. (2008). The thin solid curve encompass the region containing 50% of the total flux. The color bar is common to all panels, and graduated in cgs units (erg s^{1} cm^{2} str^{1} Hz^{1}). 

Open with DEXTER  
In the text 
Fig. 8 Comparison of the bestfitting a = 0.95, i = 85° images at 1.3 mm for an angleaveraged magnetic field (left) and a directional toroidal magnetic field (right). 

Open with DEXTER  
In the text 
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