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
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
Subject headings: accretion, accretion disks — black hole physics — galaxies:
active — Galaxy: center — radiation mechanisms: non-thermal
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,
2Joint Institute for Galaxy and Cosmology (JOINGC) of SHAO and USTC
3Graduate School of Chinese Academy of Sciences, Beijing 100039, China
– 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
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-
– 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
– 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.
– 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