Page 1

arXiv:astro-ph/0603683v1 24 Mar 2006

Testing RIAF model for Sgr A* using the size measurements

Feng Yuan1,2, Zhi-Qiang Shen1,2and Lei Huang1,3

ABSTRACT

Recent radio observations by the VLBA at 7 and 3.5 mm produced the

high-resolution images of the compact radio source located at the center of our

Galaxy—Sgr A*, and detected its wavelength-dependent intrinsic sizes at the two

wavelengths. This provides us with a good chance of testing previously-proposed

theoretical models for Sgr A*. In this Letter, we calculate the size based on the

radiatively inefficient accretion flow (RIAF) model proposed by Yuan, Quataert

& Narayan (2003). We find that the predicted sizes after taking into account

the scattering of the interstellar electrons are consistent with the observations.

We further predict an image of Sgr A* at 1.3 mm which can be tested by future

observations.

Subject headings: accretion, accretion disks — black hole physics — galaxies:

active — Galaxy: center — radiation mechanisms: non-thermal

1. Introduction

The compact radio source located at the center of our Galaxy, Sgr A*, is perhaps the

most intensively studied black hole source up to date (see review by Melia & Falcke 2001).

Substantial observational results put strict constraints on theoretical models. These models

include the spherical accretion model (Melia, Liu & Coker 2001; Liu & Melia 2002), the pure

jet model (Falcke et al. 1993; Falcke & Markoff 2000), the advection-dominated accretion

flow (ADAF) or radiatively inefficient accretion flow (RIAF) (Narayan et al. 1995; Narayan

et al. 1998; Yuan, Quataert & Narayan 2003, 2004), and the coupled ADAF-jet model

(Yuan, Markoff & Falcke 2002). In the present paper we concentrate on the RIAF model

proposed by Yuan, Quataert & Narayan (2003, hereafter YQN03).

1Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030,

China;fyuan,zshen,muduri@shao.ac.cn

2Joint Institute for Galaxy and Cosmology (JOINGC) of SHAO and USTC

3Graduate School of Chinese Academy of Sciences, Beijing 100039, China

Page 2

– 2 –

The YQN03 model explains most of the observations available at that time, including

the spectrum from radio to X-ray, the radio polarization, and the flares at both infrared

and X-ray wavebands (see YQN03 for detail). After the publication of YQN03, many new

observations are conducted. These include new spectral variability at millimeter wavelength

(Zhao et al. 2003; Miyazaki et al. 2004; Mauerhan et al. 2005; An et al. 2005), the high

angular resolution measurements of the linear polarization at submillimeter wavelengths and

its variability with SMA (Marrone et al. 2005), and very high energy emissions from the

direction of Sgr A* (INTEGRAL: B´ elanger et al. 2004; HESS: Aharonian et al. 2004; CAN-

GAROO: Tsuchiya et al. 2004; MAGIC: Albert et al. 2006). Several large multiwavelength

campaigns have been performed (e.g., Eckart et al. 2004, 2005; Yusef-Zadeh et al. 2005).

Some observations mentioned above confirm the YQN03 model, or they can be easily inter-

preted in the context of this model, while some observational results are not so obvious to be

understood and thus offer new challenges to the model. In the present Letter we will discuss

the size of Sgr A* at radio wavelengths, which has not been discussed in YQN03.

It has long been realized that due to the effect of scattering by the interstellar electrons,

the intrinsic size of Sgr A* is only detectable at short wavelength (Davis, Walsh & Booth

1976; Lo et al. 1985, 1998; Krichbaum et al. 1997; Bower & Backer 1998). This is because the

scattering theory shows that at long wavelength the observed image size will be dominated

by the scattering and scale quadratically as a function of wavelength (Narayan & Goodman

1989). At short wavelength, however, precise measurements of the size of Sgr A* are seriously

hampered by calibration uncertainties. Recently, great progress has been made in this aspect

due to the improvement of the model fitting procedure by means of the closure amplitude.

Using the VLBA, at 7 mm wavelength, Bower et al. (2004) successfully measured the size

of Sgr A* of 0.712+0.004

and 0.21+0.02

size, they obtained an intrinsic size of 0.237±0.02 mas (Bower et al. 2004) or 0.268 ±0.025

mas (Shen et al. 2005) at 7 mm and 0.126 ±0.017 mas at 3.5 mm (Shen et al. 2005). Since

this new constraint is independent of the other observations such as spectrum and variability,

it provides an independent test to investigate whether or not the RIAF model proposed by

YQN03 can account for the observed sizes.

−0.003mas, Shen et al. (2005) obtained averaged size of 0.724 ±0.001 mas

−0.01mas at 7 and 3.5 mm, respectively. By subtracting in quadrature the scattering

2. RIAF Model for Sgr A*

We first briefly review the RIAF model of YQN03, which can be considered as an up-

dated version of the original ADAF model for Sgr A* (Narayan et al. 1995; 1998). Compared

to the ADAF model, two main developments in the RIAF model are the inclusions of out-

Page 3

– 3 –

flow/convection and the possible existence of nonthermal electrons. The former is based

on the theoretical calculations and numerical simulations (e.g., Stone et al. 1999; Hawley

& Balbus 2002). The possible existence of nonthermal electrons is due to the acceleration

processes such as turbulent acceleration, reconnection, and weak shocks in accretion flow.

We characterize the nonthermal population by p [n(γ) ∝ γ−pwhere γ is the Lorentz factor],

and a parameter η, the ratio of the energy in the power-law electrons to that in the thermal

electrons. The dynamical quantities describing the accreting plasma, such as the density

and temperature, are obtained by globally solving a set of accretion equations including the

conservations of fluxes of mass, momentum, and energy. We assume that the accretion rate

is a function of radius, i.e.,˙M =˙M0(R/Rout)s(e.g., Blandford & Begelman 1999). Here Rout

is the outer radius of the flow, i.e., the Bondi radius, ˙M0is the accretion rate at Rout(the

Bondi accretion rate, fixed by Chandra observations of diffuse gas on ∼ 1′′scales; Baganoff

et al. 2003). The radiative processes we considered include synchrotron, bremsstrahlung

and their Comptonization by both thermal and nonthermal electrons. The sum of the self-

absorbed synchrotron radiation from the thermal electrons at different radii dominates the

radio emission of Sgr A* at ? 86 GHz, while the radio emission at ? 86 GHz is the sum of

the synchrotron emission of both thermal and nonthermal electrons. As we stated in YQN03,

there is no much freedom in the choice of parameter values in the RIAF model.

To calculate the intrinsic size of Sgr A* predicted by the RIAF model and compare

with observations, we need to adjust the mass of the black hole. The mass of the black

hole adopted in YQN03 is 2.5 × 106M⊙. Recent observations show that the mass should be

larger—M/M⊙= 3.7±1.5,3.3±0.6, and 4.1±0.6×106in Sch¨ odel et al. (2002, 2003), and

Ghez et al. (2003), respectively. We adopt M = 4 × 106M⊙. Thus the model parameters

need to be adjusted accordingly to ensure that the adjusted model can fit the spectrum of

Sgr A* equally well. The new parameters are:

the fraction of the turbulent energy directly heating electrons δ = 0.3. We note that the

values of ˙M0, s and η change little, but the value of δ decreases from 0.55 in YQN03 to the

present 0.3. This is because the electron temperature needs to decrease a bit to compensate

for the increase of flux due to the increase of the mass of the black hole.

˙M0≈ 10−6M⊙yr−1,s = 0.25, η = 0.4%, and

3. The size of Sgr A* predicted by the RIAF model

The observed radio morphology of Sgr A* is broadened by the interstellar scattering,

which is an elliptical Gaussian along a position angle of ∼ 80◦with the major and minor

axis sizes in mas of θmaj

et al. 2005). The observing wavelength λ is in cm. To get the intrinsic size of Sgr A*,

scat= (1.39 ± 0.02)λ2and θmin

scat= (0.69 ± 0.06)λ2, respectively (Shen

Page 4

– 4 –

observers have to subtract the scattering effect from the observed image. Here, all the sizes

estimated from observations are referred to as the FWHM (Full Width at Half Maximum)

of the Gaussian profile. This requires that not only the observed apparent image, but the

intrinsic intensity profile of the source can be well characterized by a Gaussian distribution.

However, this may not necessarily be the case. For Sgr A*, we will show that the intrinsic

intensity profile emitted by the RIAF can be quite different from the Gaussian distribution.

In this case, we are unclear to the definition of the “intrinsic size”, let alone the comparison

between the theoretically predicted size and the observationally derived one. Given this

situation, in the present paper we will not try to calculate the “intrinsic” size of Sgr A*.

Rather, we first calculate the intrinsic intensity profile from the RIAF model. Then we

take into account the scatter broadening toward the Galactic center to obtain the simulated

image. We will directly compare the simulated image with the observed one.

Now let’s calculate the specific intensity profile of the radiation from the RIAF. We

first assume that the black hole in Sgr A* is non-rotating and the RIAF is face-on. The

effects of the assumptions on the result will be discussed later. We first solve the global

solution to obtain the dynamical quantities of the RIAF as stated in Section 2. Because in

our calculation the Paczy´ nski & Wiita (1980) potential is used and the calculation is in the

frame of Newtonian mechanics rather than the exact general relativity (GR), the calculated

radial velocity of the accretion flow very close to the black hole is larger than the speed of

light thus not physical. As a result, at this region the density of the accretion flow is smaller

and correspondingly the electron temperature is also lower due to weaker compression work.

To correct this effect, for simplicity we compare the radial velocity obtained in our calculation

with that obtained by Popham & Gammie (1998) in the frame of GR. We found that our

radial velocity at r ? 30 should be divided by 0.93e2.13/rwhere r is the radius in unit of

Rg(≡ GM/c2). As for the electron temperature, following the result in Narayan et al. (1998),

a correction factor of 1.4r0.097is adopted. The above corrections are of course not precise,

but fortunately the result is not sensitive to them as we will discuss in Section 4.

The resulting intrinsic intensity profiles at 3.5 and 7 mm are shown by the red solid

lines in Fig. 1(b)&(f). Obviously, these two profiles can’t be well represented by a Gaussian

distribution. Before we incorporate the electron scattering, however, we take into account

the following additional relativistic effects, namely gravitational redshift, light bending, and

Doppler boosting (Jaroszynski & Kurpiewski 1997; Falcke et al. 2000). We implement these

effects using our GR ray-tracing code (Huang et al. in preparation). The dashed lines in Fig.

1(b)&(f) show the resultant intensity profiles after the above GR effects are considered. The

original peak of each solid line becomes lower because of the strong gravitational redshift

near the black hole. The outward movement of the peak location is due to light bending.

Page 5

– 5 –

Fig. 1(c)&(g) show the simulated image after the scattering has been included. The

scattering model mentioned at the beginning of this section is adopted. The images are

elliptical, consistent with observations. The open circles in Fig. 1(d)&(h) show the intensity

of the simulated image as a function of radius. The smoothness of the profile is because of

the scattering broadening. The solid lines in Fig. 1(d)&(h) are Gaussian fit to the open

circles. It can be seen that the intensity profile of the simulated image can be perfectly fitted

by a Gaussian, as we stated above. The FWHM of the simulated images at 7 mm and 3.5

mm are 0.729+0.01

in good agreement with the observed value by Shen et al. (2005) within the error bars but

slightly larger than the observed size by Bower et al. (2004); the size at 3.5 mm is a little

larger than the observation of Shen et al. (2005). Given that the size of the source may be

variable (Bower et al. 2004) and the uncertainties in our calculations that we will discuss in

§4, we conclude that the predictions of the YQN03 model are in reasonable agreement with

the size measurements.

−0.009mas and 0.248+0.001

−0.002mas, respectively. The simulated size at 7 mm is

In the above simulation, the “input” intensity profile for the scattering simulation is

the result of considering various effects or corrections. In the following we discuss the effects

of these corrections by considering various “input” intensity profiles. The first profile we

consider is the one without the GR effect, i.e., the red solid lines in Fig. 1(b)&(f). In

this case, the FWHM of the simulated image after considering electron scattering are 0.737

and 0.239 mas at 7 and 3.5 mm, respectively. So the GR effects make the size of Sgr A*

slightly larger at 3.5 mm. This is because the strong GR effects make the emission very

close to the black hole weaker, while the emission at large radii almost remain unchanged.

But at 7 mm, since the scattering effect is much stronger (4 times) than at 3.5 mm, the

emission at both the small and large radii in the scattered intensity profile becomes weaker

due to the GR effects. The total effect is that the size becomes smaller at 7 mm. We

have confirmed our interpretation by simulating the image at a longer wavelength—14 mm.

The second profile we consider is based on the last profiles (i.e., without considering GR

effects), with the only difference that we now only consider the emission of thermal electrons

in calculating the intrinsic intensity profiles. The FWHM values of the simulated image in

this case are 0.724 and 0.228 mas at 7 mm and 3.5 mm, respectively. So the inclusion of

the nonthermal electrons in the RIAF makes the size of Sgr A* at 7 mm and 3.5 mm larger.

This is because the intensity profile from the nonthermal electrons are flatter than that of

the thermal electrons. The last input intensity profile we consider is based on the second

profiles above (i.e., without considering nonthermal electrons) but with the difference that

the profiles of the density and electron temperature are directly obtained from the global

solution of RIAF and no relativistic corrections to the profiles of density and temperature

are adopted. In this case, the FWHM values of the simulated image are 0.727 mas and