4 Jet-ADAF model for SgrA^*

The idea of combining a jet and an ADAF was proposed by Falcke (1999) and Donea et al. (1999). Yuan (2000) first worked this out in detail and calculated the spectrum of the jet-ADAF system for SgrA^* and some nearby elliptical galaxies. There is only scant direct observational evidence for the existence of a jet in SgrA^*, from the near-simultaneous VLBA measurements by Lo et al. (1998). They found that the intrinsic source structure at 43 GHz is elongated along an essentially north-south direction, with an axial ratio of less than 0.3. However, it is interesting to note that the nearby spiral galaxy M81 has a very similar radio core and similar unusual polarization features as SgrA^* (Bietenholz et al. 2000; Brunthaler et al. 2001). In this source, a jet was clearly observed after many VLBI observations, with the length of the jet being only $\sim$400 AU at 43 GHz (Bietenholz et al. 2000). If we consider M81 to be a scaled-up version of SgrA^*, as suggested by their similarity, there could well exist a jet in SgrA^* as well. Of course, the jet in SgrA^* would be less powerful and hence smaller, making it difficult to detect because of the strong scattering of radio waves within the Galaxy. More generally, jets seem to be symbiotic with accretion disks (Falcke & Biermann 1995; Livio 1999) and they are found in basically all kinds of accretion powered systems. In this sense the model presented here may be quite general.

The picture of our jet-disk model presented here is as follows. The accretion disk is described by an ADAF. In the innermost region, r < r₀, where parameter r₀ is the jet location, a fraction $q_{\rm m}$ of the accretion flow is ejected out of the disk and forms a jet. Since in our model r₀ is very small ( $r_0 \approx 2~R_{\rm s}$, within the sonic point of the accretion disk), the radial velocity of the accretion flow is supersonic at this small radius (the Mach Number is $\sim$2-3). Therefore, when the supersonic accretion flow is transferred from the disk into the jet, which is normal to the disk, the plasma will be shocked before entering into the jet. The shocked gas passes through a nozzle where it becomes supersonic. Then it is accelerated along the jet axis through the gas pressure gradient force (the gravitational force is ignored since its effect is rather small in the supersonic regime far away from the black hole) and expands sideways with its initial sound speed. Given the initial physical states of the plasma at the sonic point (top of the nozzle), we can solve for all the quantities as a function of distance from the nozzle, and after calculating the density and the strength of the magnetic field, we can calculate the radiation of the jet (Falcke & Markoff 2000).

If, however, there exists a possibility that a substantial fraction of the accretion flow can be transferred into the jet directly without being shocked (e.g., the accretion flow outside of the sonic point also goes into the jet), we could also envisage a mixture of a relatively cold (un-shocked, $\sim$10¹⁰ K in the innermost region of ADAF) and hot (shocked, $\sim$10¹¹ K, see below for this value) electrons in the jet. Since the energy transfer between two species of particles is inefficient due to the infrequent collision between them, this kind of mixture could last for a long distance along the jet. For an emission model we can ignore this possibility, because the implied radiation from the cold unshocked plasma should be much less than the dashed line in Fig. 3 and its contribution to the overall spectrum can be neglected. On the other hand, such a mixture of hot and cold electrons may be needed when considering the circular polarization of SgrA^* (Beckert & Falcke 2002). This might increase the coupling constant between jet and disk.

The exact physics of the nozzle are difficult to model since we are at present unclear as to the physical mechanism of jet formation. In this paper we treat the nozzle only phenomenologically when calculating its spectrum. We simply assume that it consists of a series of cylinders with the same electron temperature but linearly decreasing density (increasing velocity) from bottom to top. The velocity of the gas at the base of the nozzle is assumed to be 1/5 of that at the top of the nozzle where it reaches sound speed. The emission is not very sensitive to the exact value of the initial nozzle speed. The main radiation mechanisms are synchrotron emission and its Comptonization. The parameters describing the jet include radius and height of the nozzle, r₀ and z₀, electron temperature $T_{\rm e}$, electron number density $n_{\rm e}$, the strength of the magnetic field B at the top of nozzle, and the angle between the jet axis and the line of sight $\theta$.

All above are free parameters in the original jet model (but most of them have obvious physical constraints to their range of values, see Falcke & Markoff 2000). But here in our coupled jet-disk system, more constraints are required so that the jet parameters are consistent with the underlying accretion disk. $T_{\rm e}$ is calculated self-consistently as follows. When some accretion gas passes through the shock and enters into the jet, the ordered kinetic energy in the pre-shock gas will be converted into thermal energy in the shock front. Neglecting the effects of the magnetic field on the shock jump condition (we will check the rationality of this approximation later), we calculate the electron temperature of the post-shock plasma by the following Rankine-Hugoniot relations, namely the conservation of flux of mass, momentum, and energy. Written in the conventional notation, they are

$$[\rho v]=0,$$ (1)

$$[P+\rho v^2]=0,$$ (2)

$$\left[\frac{1}{2}v^2+\frac{2}{5}\frac{P}{\rho}\right]=0,$$ (3)

respectively. To obtain $T_{\rm e}$ immediately after the shock, we still need the ratio between the ion and electron temperatures, $\xi$. On the one hand, shock heating may favor the ions rather than electrons, as isotropization of the bulk flow velocities will give to each species a thermal energy proportional to its mass. On the other hand, Coulomb collisions, and maybe collisionless processes also, will bring about equilibration between ions and electrons temperatures. Determining the value of $\xi$ is a difficult task (see Laming 2000 for a recent review). In the context of supernova remnant shocks, Cargill & Papadopoulos (1988) predict $\xi=5$ from a numerical simulation, while Laming et al. (1996) derive $\xi=20$ from their fit to observations. We set $\xi=10$ in our model.

For a given shock location r₀, we first solve the radiation-hydrodynamical equations describing the underlying ADAF under selected parameters and outer boundary conditions to obtain the pre-shock physical quantities at r₀. We then substitute them in the above shock relations to get the post-shock values. Thus we obtain the electron temperature $T_{\rm e}$ in the nozzle. Therefore $T_{\rm e}$ is not a free parameter in our model, we can change its value only through changing the parameters and outer boundary conditions of the underlying accretion disk. We use the same "magnetic parameter'' $\beta$ as in the ADAF to describe the ratio between the gas pressure and total pressure in the nozzle to obtain the value of B if temperature and density are known. This means that B is no longer a free parameter, either. The density, $n_{\rm e}$, in the jet follows from the coupling constant $q_{\rm m}$ in the jet-disk symbiosis model (Falcke et al. 1993). This is defined as the ratio between mass loss in the jet and accretion rate outside r₀. For jets, $q_{\rm m}$is typically a few percent and we require $q_{\rm m} \ll 1$.

Figure 3: The jet-disk spectral model for SgrA*. The dotted line is for the ADAF contribution, the dashed line is for the jet emission, and the solid line shows their sum. See text for details.

Our best spectral fit is presented by the solid line in Fig. 3. The dashed line denotes the emergent spectrum from the jet, and the dotted line is from the underlying disk (ADAF). The solid line is their sum. For ADAFs, the parameters are $\dot{M}=8.8 \times 10^{-7}~M_{\odot}~{\rm yr^{-1}}$, $\alpha=0.1$, $\beta=0.95$ and $\delta =10^{-3}$. The outer boundary conditions are $T_{\rm i}\approx T_{\rm e}=8\times 10^6~{\rm K}$, $\Omega_{\rm out}=0.27~\Omega_{\rm Kepler}$ at $10^5~ R_{\rm s}$. The parameters for the jet are $r_0=1.7~R_{\rm s}$, z₀=3.5 r₀, $q_{\rm m}= 0.5\%$ and $\theta=35\hbox{$^\circ$ }$, the "calculated parameters'' are $B=23{\rm G}$, $T_{\rm e}=2.1 \times 10^{11}~{\rm K}$, and $n_{\rm e}=2.4 \times 10^6$. The mass loss rate in the jet is $\dot{M}_{\rm jet}= \pi r_0^2 c_{\rm s} n_{\rm e} m_{\rm p} =4.3 \times 10^{-9}~M_{\odot}\,{\rm yr}^{-1}$, i.e. 0.5% of the accretion rate. For $\beta=0.95$ and shock location $r_0=1.7~R_{\rm s}$, the Alfvén Mach number $M_A \equiv v/v_{A1} = (v/c_{\rm s})(4 \pi \gamma p/B^2)^{1/2} \approx 10 > 6$, here $v_{A1} \equiv B_1/(4\pi\rho_1)^{1/2}$ is the pre-shock Alfvén speed, the magnetic effects are weak in the shock transition condition, so our hydrodynamic approximation to the shock transition condition, Eqs. (1)-(3) above, is justified (Draine & McKee 1993).

This model fits the spectrum over the whole range of frequencies from radio to the X-ray quite well. The submm bump is slightly over-predicted, but it is acceptable considering the variability of the data in this band (Melia & Falcke 2001) and the uncertainty of the model. The low-frequency radio emission is mainly contributed by the jet outside the nozzle. The contribution from the ADAF is rather weak and can be neglected. The submm bump is the sum of the synchrotron radiation from both the ADAF and the nozzle of the jet. We note that the emission from the nozzle is much weaker than in the ADIOS with $\delta =1$ presented in the last section (solid line in Fig. 2) although the electron temperatures are both $10^{11}~{\rm K}$. Such a difference is not surprising considering the much smaller spatial scale of the nozzle, $r_0=1.7~R_{\rm s}$, while in that case, there is a larger radial range with high temperature. In this sense, an abrupt increase in the temperature profile is necessary to model the spectrum. This is naturally satisfied in our jet-disk model by the formation of a jet. If instead the nozzle in our model is replaced by a similar high-temperature component such as the inner region of a disk, since the temperature profile of the disk is in general smooth, the radial range of this high-temperature component would be considerable. In this case, we expect that the model would greatly over-predict the submm flux, and the low-frequency radio spectrum is still hard to explain, as indicated by our calculation for the ADIOS model with high $\delta$ in Sect. 3. This is the reason why in Melia et al. (2001)'s model the authors require an accretion disk as small as $\sim$ $5~R_{\rm s}$.

The X-rays are mainly produced in the nozzle by SSC, although bremsstrahlung from the ADAF also contributes to some degree. The predicted X-ray spectrum is the sum of the very soft SSC from the nozzle and the relatively flatter bremsstrahlung spectrum from the ADAF, which is in very good agreement with the Chandra data, almost identical to the best fit of Baganoff et al. (2001b). The fit is also much better than that of the ADIOS with $\delta =1$ in the last section. In both cases, the X-ray emission is the sum of bremsstrahlung and SSC, but in the present case, bremsstrahlung produces a much flatter spectrum than in the case of an ADIOS due to the absence of a strong wind. Because of the contribution of SSC from the jet, the variability timescale of X-rays should be short, $t \approx r_0/v_{\rm jet} \approx 10$ min. This is consistent with the $\sim$1 hour variability timescale determined in the quiescent state and is in excellent agreement with the 600 s variability timescale detected in the flare state. We show that it is the variability of the flux from the nozzle that causes the huge-amplitude flare (Markoff et al. 2001b). On the other hand, since the bremsstrahlung radiation from the ADAF also contributes partly to the X-ray spectrum, this could explain the possibly detected extended source with $\sim$ $10^5~R_{\rm s} (\approx 1{\hbox{$^{\prime\prime}$ }})$, the 6.7 keV K$\alpha$ emission line, and steady X-ray flux on $\sim$one year timescale, as we stated in Sect. 2.

We note that the above nozzle parameters, temperature, spatial size and density, are very close to the "second component'' in the model of Beckert & Duschl (1997), which is also responsible for the submm bump of SgrA^*. These parameters seem to be the best ones to fit the submm bump. It is interesting that the nozzle with these parameters will "evolve'' naturally into a jet whose emission can well reproduce the low-frequency radio spectrum of SgrA^*, and the Comptonization of its synchrotron emission can produce a very soft X-ray spectrum which can fit the Chandra data excellently. In fact, to make the up-scattered submm bump extend to the Chandra band, an electron temperature as high as $10^{11}~{\rm K}$ is needed. The peak frequency of this bump is $\sim$10¹² Hz. To up-scatter it to the X-ray band, $\nu \sim 10^{16}$ Hz, the electron Lorentz factor must satisfy $4\gamma_{\rm e}^2 \approx 10^{16}/ 10^{12} \approx 10^4$. This corresponds to a temperature of $T \approx \frac{1}{k}\frac{\gamma_{\rm e}}{3.5}m_{\rm e}c^2 \approx 10^{11}$ K. This value is about 10 times higher than the highest temperature that a canonical ADAF can reach in its innermost region, but is naturally reached when some fraction of accretion matter is shocked. In addition to a high temperature, the spatial size of the dominant emission medium must be small, otherwise the model will over-predict the high-frequency radio flux as in ADIOS with high $\delta$ (the solid line in Fig. 2). This is also easily satisfied in the jet model by requiring a small r₀. In addition to the above parameters, a truncated (no hard tail) electron energy distribution is also required in the model, otherwise the synchrotron emission will extend above the observed IR flux upper limit. Beckert & Duschl (1997) simply assume a mono-energetic distribution. In our model, a relativistic thermal distribution, which is highly peaked at $\gamma \approx 3.5\frac{kT}{m_{\rm e}c^2}$, is a natural result of shock heating (e.g. Drury 1983) since the Mach number is not very large in our case, $\sim$2-3.

The mass accretion rate of the ADAF in our model, $8.8 \times 10^{-7}~M_{\odot}~{\rm yr^{-1}}$, is only marginally smaller than the lower limit of Baganoff et al. (2001a) estimate of $1 \times 10^{-6}~M_{\odot}~{\rm yr^{-1}}$. If we used a higher accretion rate, we would obviously slightly over-predict the flux at the submm bump band because of the higher flux of the synchrotron emission from the ADAF.

There are various ways to further evolve the model. One is to introduce global winds in the ADAF. The X-ray radiation from the disk would be almost unaffected but the radio emission from the disk would be greatly decreased because of the great decrease in density close to the black hole (Quataert & Narayan 1999). But the wind cannot be too strong, otherwise the X-ray spectrum would be too soft, as we argued in the case of ADIOS model with $\delta \sim 1$. Another modification is to assume that the accretion disk is radiatively truncated within r₀, the radius of the jet formation (Yuan 2000). The physical reason for the truncation is that to form a jet, some amount of energy is needed. If we assume this energy comes from the underlying disk, the disk will be left cold within r₀ because of the energy extraction (Blandford & Payne 1982). This will greatly suppress the synchrotron emission due to the very sensitive dependence of synchrotron radiation on the temperature.