Hawley & Balbus: The Generic NRAF Model

4. The Generic NRAF Model

4.1 The Three Component Model

In this section we assemble into a simple model those features suggested by our simulation that we expect will prove to be enduring and representative of radiatively inefficient accretion flows more generally. This model has the virtue of limited flexibility, is falsifiable, and is grounded in uncontroversial dynamics. Whether it enjoys subsequent support or must ultimately be abandoned, it will have served a useful role. It is schematically illustrated in Figure 1.

The structure emerges as follows. Differentially rotating magnetized gas is subject to the MRI. The MRI produces a turbulent $R\phi$ Maxwell stress, and the angular momentum transport results in accretion. (A somewhat smaller $R\phi$ Reynolds stress component is also present.) The sizable outward transport of angular momentum rapidly changes the flow profile to near Keplerian, at which point the supporting pressure gradients are small. The global energy minimum state separates the matter in the disk's inner region from the angular momentum in its outer regions, as in the classical disk evolution envisioned by Lynden-Bell & Pringle (1974). The bulk of the gas is decidedly disk-like: whether it radiates or not, the main body of the accretion flow is nearly Keplerian. Absent radiation, the disk is simply hot, vertically thickened, and radial pressure gradients are minimized.

The disk temperature increases rapidly inward, with $T\propto R^{-3/2}$ . The vertical structure is dynamic, with gas lofting away from the equator, as pressure and centrifugal accelerations drive the gas out. This produces the hot, dynamic coronal envelope surrounding the disk. Disk-generated magnetic field also rises into the corona, where the resulting magnetic pressure significantly exceeds the gas pressure.

The disk remains vertically thick as it accretes, making an encounter with the centrifugal barrier inevitable, despite the loss of angular momentum. This is the centrifugal funnel wall. It is present because gravity weakens with increasing vertical distance from the central hole while the centrifugal force remains unchanged. Just outside the marginally stable orbit, a small hot torus of gas accumulates. The specific angular momentum in the torus is slightly greater that of the marginally stable orbit. Hot gas, pressed up against the funnel wall, accelerates up along this centrifugal barrier, and is held against it by the magnetic pressure of the surrounding corona. This is the magnetically-confined jet, but note: the magnetic confinement is from the outside medium! The jet apparently is stable.

The size of the inner torus varies depending on the jet flux, the accretion into the hole, and the rate at which matter is supplied from the Keplerian disk. It is highly variable. The final accretion into the black hole takes place only through the opening in the funnel wall at the equator, like threading a needle. A high resolution torus simulation (Hawley & Krolik 2001) found that the magnetic stress can remain large down to and beyond the marginally stable orbit. This effect might increase $\dot M$ into the hole, but the present simulation is not sufficiently well-resolved to address this.

To summarize: the combination of gravity, radiative inefficiency, angular momentum, and MHD turbulence found in black hole accretion leads to a three component flow structure--a hot Keplerian disk, an extended corona, and a jet-like central outflow.

4.2 Application to Sgr A*

An important application of ADAF-type systems has been to the source Sgr A* at the Galactic center. The properties of the black hole system at the Galactic center are reviewed by Melia & Falcke (2001). Compelling dynamical evidence suggests the presence of a massive black hole of $\approx 2.6\times 10^{6}\ M_\odot$ . Observations in X-ray and radio bands reveal a luminosity substantially below Eddington, making this system a prime candidate as an archetype low-radiative-efficiency accretion flow. Recent Chandra observations find a luminosity in 2-10 keV X-rays of $\approx 2\times 10^{33}$ erg s^-1, and also an X-ray flare rapidly rising to a level about 45 times as large, lasting for only $\sim 10^4$ s (Baganoff et al. 2001), indicating that the flare must originate near the black hole.

Many aspects of spectral models for Sgr A* follow from the simple scaling laws of black hole accretion and will be present in any model, regardless of the detailed dynamics. For example, Quataert & Narayan (1999) demonstrate the impressive range of spectra that may be generated with the adjustment of a few free parameters: the ratio of electron to ion temperature, the magnetic pressure, the run of density with radius, and the accretion rate. Our knowledge of the flow is not yet sufficient to tightly constrain these parameters. As the underlying dynamical models become more sophisticated, the spectral models should become more constrained as well.

Although our simulation lacks a formal treatment of the energetics necessary for a detailed application to Sgr A*, it is possible to look at some radiative properties of the computed flow and compare them with other, more detailed spectral predictions. The aim is to illustrate how the dynamical structures revealed in the simulation can be compared with current spectral models for Sgr A*.

We must first translate between computational and physical units, as discussed in §2.3. The black hole mass is $2.6\times 10^6 M_\odot$ , which gives a Schwarzschild radius of 7.8 x 10¹¹ cm. To keep things as general as possible, let the code value n=1 be equal to a physical value $n_o {\rm cm}^{-3}$ . With this parameterization, the initial torus mass is 6.8 x 10¹⁷n_o gm. The average accretion rate from the inner edge of the initial torus is roughly 2% of the torus mass per orbit (see fig. 6); in physical units this becomes 6 x 10¹⁰n_o gm s^-1, or $1.6\times 10^{-13} n_o \dot M_{Edd}$ . The average accretion rate into the central hole is about a factor of 10 smaller than this.

The accretion rate for Sgr A* is uncertain. Coker & Melia (1997) estimate a rate of 10²² gm s ${}^{-1} = 0.03 \dot M_{Edd}$ from Bondi-Hoyle accretion of winds from nearby stars. Reconciling this accretion rate estimate with the low X-ray luminosity is a problem, however. Quataert, Narayan, & Reid (1999) argue that the low luminosity requires that the accretion rate at large radius be substantially sub-Eddington. The best fit spectral model of Melia, Liu, & Coker (2001) has an accretion rate into the central hole of 10¹⁶ gm s^-1.

Figure 8 shows the run of temperatures along the equator in the model at the end time. This is a single fluid calculation with a simple equation of state, i.e., we assume that the electron and ion temperatures are the same. We estimate the total bremsstrahlung emissivity using an approximate form of the relativistic ion-electron bremsstrahlung formula (Svensson 1982)

$\begin{displaymath} \Lambda = 1.2\times 10^{-22} n^2 \Theta \ln\left(\Theta+1.5\right) {\rm ergs}\, {\rm s}^{-1}\, {\rm cm}^{-3} \end{displaymath}$

(14)

where $\Theta = kT/m_ec^2$ , and for simplicity we use Z=1 and n_e=n_i. The emission is calculated for each grid zone the total is obtained by summing up over the entire computational volume. The total bremsstrahlung luminosity on the grid at the end of the simulation is 3 x 10²¹ n_o² erg s^-1. The inner torus dominates the total emission. The total thermal energy is $\sim 10^{37}n_o$ ergs, so the cooling time is $\sim 3\times 10^{16} n_o^{-1}$ s. The simulation time is $\sim 10^6$ s, so the cooling time is substantially longer than the dynamical time for $n_o < 10^{10}{\rm cm}^{_3}$ , or $\dot M \le 10^{-3} \dot M_{Edd}$ . We note that the formal electron-ion equilibration time from Coulomb collisions (Spitzer 1962) is long compared to the flow time for $n_o \sim 10^8\,{\rm cm}^{-3}$ .

We can develop a qualitative sense of the bremsstrahlung spectrum by computing the nonrelativistic value $\propto n^2 T^{-1/2} e^{-h\nu/kT}$ for each grid zone. At the end of the simulation, the bremsstrahlung emission from the hot Keplerian disk inside of R=100r_g peaks at a few times 10¹⁹Hz. The inner torus contributes the bulk of the highest frequency emission, peaking at 10²¹Hz. The coronal gas outside of one scale height from the disk emits over a broad range of frequency, but because the density is lower, the total emissivity is about a factor of 10 below that of the Keplerian disk at lower frequencies, and 2.5 orders of magnitude below at the highest energies. Our model is constrained by the observed low X-ray flux in the same way as all accretion models: the net accretion rate must be low. We are aided in achieving this by the coronal backflow, which permits the escape of matter with little additional bremsstrahlung emission, while reducing the hot gas density near the hole.

In theoretical spectral models, the radio and submillimeter emission arises in the innermost regions of the flow from synchrotron emission and Compton scattering. Synchrotron emission is governed by the electron number and energy densities, and the strength of the magnetic field. In some analyses, the field strength $\beta$ and electron temperature are treated as free parameters. In the model of Melia et al. (2001) $\beta$ comes directly from the connection between the field strength and the magnetic stress that drives the accretion. This is closer in spirit to a direct numerical simulation in which $\beta$ emerges self-consistently.

We shall limit our analysis of the sub-mm excess to a calculation of the spatial distribution of the peak synchrotron frequency (Rybicki & Lightman 1979)

$\begin{displaymath} \nu_c = 10^{-20} (n_e T^5/\beta)^{1/2}\ {\rm Hz}, \end{displaymath}$

(15)

postponing a detailed spectral analysis to a future paper. Figure 9 shows that the region of high temperature flow in the inner torus can account for the observed sub-mm emission. The highest peak frequencies are $\sim 2.5\times 10^{11} (n_o/10^8)^{1/2}$ Hz, and they are found at the inner edge of the disk, where gas is compressed against the centrifugal barrier. The region immediately surrounding the inner disk torus is also an emitting region. The value of $\beta$ is $\sim 10$ at the inner edge of the torus and $\le 1$ in the surrounding area. A simple estimate for the synchrotron cooling time (Spitzer 1962) is $8\times 10^5 \beta (10^8/n_o)$ s, which is longer than the flow time.

Figure 9: Contour map of peak frequencies for synchrotron emission at the end of simulation F1. The highest frequencies emerge from the inner edge of the inner torus, The contours are equally spaced in log frequency, and the levels are indicated by the color bar.

It is interesting to compare this result with the spectral model of Melia et al. (2001). Their best-fit model is an accretion flow with $\dot M = 10^{16}$ gm s^-1, from which they compute the emission emerging from a region inside of 5r_g. The gas has a temperature $\sim 10^{11}$ K, number densities $\sim 10^7$ cm^-3 and $\beta \approx 30$ . In our simulation the temperature in the inner torus is 4 x 10¹⁰-10¹¹K, $\beta\sim 10$ , and the maximum number density along the equator lies between $n\sim n_o$ and $10\,n_o\,{\rm cm}^{-3}$ . The inner torus is time varying over timescales of tens of hours, with fluctuations in temperature of about 50%, and vertically-averaged number density $\langle n\rangle$ by about a factor of 5 (fig. 10). The dynamics are consistent with significant emission variability. This is encouraging, but given the very simple treatment of energy in our simulation, at present it is only suggestive.

Figure 10: Plot of time history of the height- and azimuthally-averaged temperature (top) and number density (bottom) in the inner torus at r=4r_g. The numerical values are scaled to the parameters of Sgr A*, with an assumed unit number density n_o=10⁸ cm^-3. Time is given in kiloseconds from the beginning of the simulation.

It is interesting to note that key features in our simulation have been suggested independently by others. For example, the jet outflow from the inner torus may also be a source of emission, as in the jet model of Sgr A* by Falcke & Markoff (2000). The observational evidence for coronal outflows in low luminosity black hole sources has recently emphasized out by Merloni & Fabian (2001). Combining the more sophisticated spectral treatments of these models with the dynamics observed here is an obvious next step.

3D NRAF Simulations | 5. Conclusions