Astron. Astrophys. 346, 713-720 (1999)

2. The 3D description of the fluid

2.1. The accretion flow

We investigate the stationary flow of matter and the propagation of light in the gravitational field of a rotating Kerr black hole using the Boyer-Lindquist coordinates t, , r, and the metric components as given by Bardeen (1973). We follow the signature convention. We use geometrical units, so the speed of light and the mass of the hole . The Kerr parameter a () gives the black hole angular momentum in geometrical units. We use the Einstein summation convention, where needed, and a semicolon for the covariant derivative. The normalization of four velocity with the chosen metric signature reads .

The system of equations we use follows in general ACGL. Since we are neglecting the cooling processes at this stage, and treat the accretion flow in the disk approximation, two components of velocity and the speed of sound given as functions of radius, fully describe the dynamics. We use the angular velocity and the physical radial velocity V as measured by locally nonrotating observers as main kinematic variables (compare ACGL). After the vertical averaging the velocity perpendicular to the equatorial plane is neglected () and other components of the four velocity are given as:

The above velocity components have to be known only at the equatorial plane and its close vicinity if one needs to obtain a system of equations describing an ADAF in the slim disk approximation (Abramowicz et al. 1988). To describe the matter distribution in space (also far from the equatorial plane) one has to introduce further specifications of the velocity and angular momentum distribution, since only the averaged values enter the equations. We assume that the poloidal velocity component () is absent and that the radial velocity component V depends on the radius only:

(The BL velocity component depends on through - compare Eq. 3)

The choice of angular momentum distribution is less obvious. Close to the horizon, where the velocities are highest and possible kinematic effects most important, the specific angular momentum is approximately constant (Paper I, ACGL, Peitz & Appl, 1997). We assume to be exactly constant on spheres:

Since the metric components do depend on the angular velocity is not constant on spheres:

The other kinematic quantities are given as:

We assume that the accreting plasma contains small scale, isotropically tangled magnetic field resulting from magnetohydrodynamical instability (Balbus et al. 1995). Hence we write the total pressure as

where is the gas pressure, - the magnetic pressure, B - the magnetic field, and is a constant parameter. The pressure, the rest mass density and the sound velocity are related:

In our calculations we use a two temperature plasma with a small amount of magnetic field to represent the matter properties. Thus the ion pressure dominates, the gas is non-relativistic and the specific enthalpy µ is given as:

where is the total (rest mass plus thermal) energy density.

In the ADAF set of equations only the vertically averaged sound speed is used. It is in spirit of our approximations to postulate:

which means that the gas is isothermal on spheres. The electron temperature is much lower than the ion temperature, so its influence on the equation of state and the structure of the flow can be neglected.

Our detailed (but approximate) description of the fluid kinematics makes it possible to find the density dependence on the angular coordinate . The set of equations describing ADAF contains viscosity terms, which are included in the energy equation, but neglected in the mechanical equilibrium equations (ACGL, Peitz & Appl 1997, Paper I). Thus it is sufficient to use the ideal fluid energy momentum tensor:

After some algebra we obtain:

Since V does not depend on , the LHS of the equation is a full gradient of a quantity which can be called a potential . This implies the solution for the density:

where the , denote the values measured at the equatorial plane.

The constant sound speed on the spheres may suggest that the exponential atmosphere never ends. For the rotating configuration there is, however, an infinite potential barrier close to the rotation axis, where and . Thus the vicinity of the rotation axis is empty and the density falls steeply down near this region. In this respect the ADAF solutions are similar to the so called thick accretion disks (Abramowicz et al. 1978). Both have empty funnels around their rotation axes.

The mass flow through the surface can be calculated as:

Combining the last equation and the formula for the dependence of the density we obtain the equatorial value of the density.

2.2. Two temperature plasma

While modeling the dynamics of ADAF we neglect the heat transfer and all the radiation processes, assuming that only a small part of the total energy generated by viscous processes can be affected by them. Now we are going to model the radiation processes.

We assume that the whole energy dissipated is transferred to the ions. The ions heat the electrons via Coulomb collisions and the electrons lose their energy by the synchrotron, bremsstrahlung and inverse Compton cooling processes. In the ADAFs thermalization time-scale greatly exceeds the dynamical time-scale and the plasma remains two temperature. We find the electron and ion temperatures, and , self-consistently using the equation of state

where and are effective molecular weights of the ions and electrons, and the condition of thermal equilibrium applied locally

where is the rate of Coulomb heating of electrons by ions (e.g. Mahadevan 1997), and are the synchrotron and bremsstrahlung cooling rates, and and are the Compton cooling rate of synchrotron and bremsstrahlung photons, respectively.

The calculation of the synchrotron cooling rate is somewhat complicated because the optical depth to absorption (Self Synchrotron Absorption) for majority of the synchrotron photons is high. In our approach we assume that a photon can either escape from the medium carrying out its energy and taking part in cooling process, or be absorbed very close to the point of its original emission and not contributing to the cooling. In reality all photons carry energy and some are absorbed far from the emission point transferring energy to distant portions of the fluid. Our assumption neglects the heat transport between different volume elements, but makes calculations doable.

To find the probability of a photon escape from given location we follow rays with directions randomly chosen at the frame comoving with the fluid. The optical depth along a ray measured at the frequency is

where is the synchrotron emissivity, is the Planck function corresponding to the electron temperature , and the integration is over the proper distance, corresponding to the point on the fluid boundary. Since the integrand in the above formula is the function of the electron temperature we must assume the approximate distribution of the electron temperature in the ADAF. As a first approximation we use our results from Paper I based on the approach of Narayan & Yi 1994. Averaging one gets the probability of escape:

Since we assume that cooling is provided only by the escaping photons, we have:

for the synchrotron cooling. We adopt the expressions for synchrotron emissivity from Pacholczyk (1970) and Mahadevan et al. (1996).

For the bremsstrahlung cooling the solution is straightforward. The absorption of low frequency bremsstrahlung photons has no practical meaning, so we adopt their frequency integrated emission as cooling rate

We take the expression for bremsstrahlung cooling rate from Stepney & Guilbert (1983).

Finally we take the cooling by Comptonization of both synchrotron and bremsstrahlung photons using the formulae of Esin et al. (1996). The mean optical depth to Compton scattering is calculated in the same manner as above. Solving the Eqs. 19 and 20 we get the electron temperature and the ion temperature.

2.3. Spectra neglecting comptonization

As a by-product of the calculations of the previous subsections one can obtain the spectrum of the model, which would be valid if the Comptonization were unimportant. For sufficiently low frequencies a photon is rather absorbed than scattered, so both approaches, neglecting and including Comptonization, should give similar results in this regime. This gives a chance of a self check of the simulations.

Calculation of many rays sent from a given fluid element can also be used to find the contribution of this element to the total luminosity of the configuration as seen by a distant observer. If the frequency and the direction of a photon in the fluid frame is known, its frequency in the Boyer-Lindquist coordinate frame can be calculated and this is the frequency that would be measured by a distant observer, unless the photon goes under the horizon. For photons going to infinity the redshift factor can be defined:

Distant observers measure photon energies divided by the factor . Also the time interval between the detection of two signals is the interval between their sending multiplied by . Thus the contribution to the total luminosity from the fluid element of the volume , measured by distant observers in their frequency interval is given as:

where for the i-th ray:

The summation is limited to rays which reach infinity. Integration over the volume of the fluid gives the total luminosity of the model.

The emission from the configuration is not isotropic, so the observers at different position angles measure a different flux of radiation. The luminosity calculated above is in fact an average of luminosities assigned to the disk by observers uniformly distributed on a sphere around the object. One can also find the average luminosity that would be measured by observers from a limited solid angle . To do so it is sufficient to neglect in the summation all the rays which do not enter the region of interest and multiply the result by the correction factor .

2.4. Spatial distribution of the photon emissivity

In the Monte Carlo simulations we follow individual photons as they travel through the fluid undergoing consecutive scatterings. We have to know what is the distribution of the points of emission of the photons. We divide the flow into several spherical layers. The radius in the middle of the layer numbered j is . Each layer is then subdivided into annuli of limited range in the polar angle between fluid boundaries and . The angular coordinate at the middle of the annulus numbered jk is . One can assume that all fluid parameters are almost uniform inside each annulus, and take their values at as representative. Using similar arguments as in the previous subsection we calculate the rate of photon emission from the region numbered jk of the volume :

We adopt the expressions for synchrotron cooling from Pacholczyk (1970) and Mahadevan et al. (1996), and for bremsstrahlung cooling from Svensson (1982) [and references therein]. The cooling rates are functions of the electron temperature, the number density of ions and electrons and the magnetic field density (synchrotron radiation) and hence they are functions of r and . We take . The expression under the integral is regular at low frequencies despite the presence of in the denominator because when . The redshift factor in the denominator takes care of the difference between clock rates in the fluid frame and at infinity.

© European Southern Observatory (ESO) 1999

Online publication: May 21, 1999