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Astron. Astrophys. 351, 185-191 (1999)

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2. Derivation of the outflow rate

Fig. 1a shows the schematic nature of the inflow and outflow that is understood to be taking place around a black hole. Rotational motion brakes the flow and forms a dense boundary layer (CENBOL) around a black hole. Matter compressed and heated (over and above the heating due to geometric compression) at the CENBOL comes out in between the centrifugal barrier and the funnel wall (Molteni et al. 1994, C96). For the present paper, in Fig. 1b, we choose a simplified description of this system, where the toroidal CENBOL is replaced by a quasi-spherical one. Matter is assumed to fall in a conical shape. The sub-Keplerian, hot and dense, quasi-spherical region may also form due to pair-plasma pressure or pre-heating effects, but the details are not essential. The outflowing wind is assumed to be also conical in shape for simplicity and is flowing out along the axis. It is assumed that the wind is collimated by an external pressure. Both the inflow and the outflow are assumed to be thin enough so that the velocity and density variations across the flow could be ignored.

[FIGURE] Fig. 1a and b. Schematic diagram of inflow and outflow around a compact object. The hot, dense region around the object either due to a centrifugal barrier or due to a plasma pressure effect or pre-heating, acts like a `stellar surface' from which the outflowing wind is developed. In a , the actual flow behavior is shown. In b , a simplified model is depicted where the toroidal CENBOL is replaced by a quasi-spherical CENBOL.

The accretion rate of the incoming accretion flow is given by,

[EQUATION]

Here, [FORMULA] is the solid angle subtended by the inflow, [FORMULA] and [FORMULA] are the density and velocity respectively, and r is the radial distance. For simplicity, we assume geometric units ([FORMULA]; G is the gravitational constant, [FORMULA] is the mass of the central black hole, and c is the velocity of light) to measure all the quantities. In these units, for a freely falling gas,

[EQUATION]

and

[EQUATION]

Here, [FORMULA] (with [FORMULA] assumed to be a constant) is the outward force due to radiation.

We assume that the boundary of the denser cloud (say, the shock) is at [FORMULA] (typically a few to few tens of Schwarzschild radii, see Chakrabarti 1990 [hereafter C90], C96) where the inflow gas is compressed. The compression could be abrupt due to a standing shock or gradual as in a shock-free flow with angular momentum. These details are irrelevant. At this barrier, then

[EQUATION]

and

[EQUATION]

where R is the compression ratio. The exact value of the compression ratio is a function of the flow parameters, such as the specific energy and the angular momentum (e.g. C90). Here, the subscripts - and + denote the pre-shock and post-shock quantities respectively. At the shock surface, the total pressure (thermal pressure plus ram pressure) is balanced.

[EQUATION]

Assuming that the thermal pressure of the pre-shock incoming flow is negligible compared to the ram pressure, using Eqs. (4a,b) we find,

[EQUATION]

The isothermal sound speed in the post-shock region is then,

[EQUATION]

where [FORMULA]. An outflow which is generated from this dense region with very low flow velocity along the axis is necessarily subsonic in this region, however, at a large distance, the outflow velocity is expected to be much higher compared to the sound speed, and therefore the flow must be supersonic. In the subsonic region of the outflow, the pressure and density are expected to be almost constant and thus it is customary to assume isothermality conditions up to the sonic point. As argued in the introduction, in the case of black hole accretion also, such an assumption may be justified. With the isothermality assumption or a given temperature distribution ([FORMULA] with [FORMULA] a constant) the result is derivable in analytical form. The sonic point conditions are obtained from the radial momentum equation,

[EQUATION]

and the continuity equation

[EQUATION]

in the usual way, i.e., by eliminating [FORMULA],

[EQUATION]

where

[EQUATION]

and

[EQUATION]

and putting [FORMULA] and [FORMULA]. These conditions yield, at the sonic point [FORMULA], for an isothermal flow,

[EQUATION]

and

[EQUATION]

where we have utilized Eq. (7) to substitute for [FORMULA].

Since the sonic point of a hotter outflow is located closer to the black hole, clearly, the condition of isothermality is best maintained if the temperature is high enough. However if the temperature is too high, so that [FORMULA], then the flow has to bore a hole through the cloud just as in the `cauldron' model, although it is a different situation - here the temperature is high, while in the `cauldron' model the temperature was low. In reality, a pre-defined funnel caused by a centrifugal barrier does not require the flow to bore any nozzle at all, but our simple quasi-spherical calculation fails to describe this case properly. This is done in detail in Das & Chakrabarti (1999).

The constancy of the integral of the radial momentum equation (Eq. (8)) in an isothermal flow gives:

[EQUATION]

where we have ignored the initial value of the outflowing radial velocity [FORMULA] at the dense region boundary, and used Eq. (11a). We have also put [FORMULA] and [FORMULA]. After simplification, we obtain,

[EQUATION]

where,

[EQUATION]

Thus, the outflow rate is given by,

[EQUATION]

where [FORMULA] is the solid angle subtended by the outflowing cone. Upon substitution, one obtains,

[EQUATION]

which explicitly depends only on the compression ratio:

[EQUATION]

apart from the geometric factors. Notice that this simple result does not depend on the location of the sonic point or the the size of the shock or the outward radiation force constant [FORMULA]. This is because the Newtonian potential was used throughout and the radiation force was also assumed to be very simple-minded ([FORMULA]). Also, effects of centrifugal force were ignored. Similarly, the ratio is independent of the mass accretion rate which should be valid only for low luminosity objects. For high luminosity flows, Comptonization would cool the dense region completely (CT95) and the mass loss will be negligible. Pair-plasma-supported quasi-spherical shocks form for low luminosity as well (Kazanas & Ellison 1986). In reality there would be a dependence on these quantities when full general relativistic considerations of the rotating flows are made.

Figs. 2a and b contain the basic results. Fig. 2a shows the ratio [FORMULA] as a function of the compression ratio R (plotted from 1 to 7). Fig. 2b shows the same quantity as a function of the polytropic constant [FORMULA] (drawn from [FORMULA] to 3), [FORMULA] being the adiabatic index. Fig. 2a is drawn for any generic compression ratio and Fig. 2b is drawn assuming the strong shock limit only: [FORMULA]. In both the curves, [FORMULA] has been assumed for simplicity. Note that if the compression (over and above the geometric compression) does not take place (namely, if the denser region does not form), then there is no outflow in this model. Indeed for, [FORMULA], the ratio [FORMULA] is zero as expected. Thus the driving force of the outflow is primarily coming from the hot and compressed region.

[FIGURE] Fig. 2a and b. Ratio [FORMULA] of the outflow rate and the inflow rate as a function of the compression ratio of the gas at the dense region boundary a . In b , variation of the ratio with the polytropic constant n in the strong shock limit is shown. Solid angles subtended by the inflow and the outflow are assumed to be comparable for simplicity.

In a relativistic inflow or for a radiation dominated inflow, [FORMULA] and [FORMULA]. In the strong shock limit, the compression ratio is [FORMULA] and the ratio of inflow and outflow rates becomes

[EQUATION]

For the inflow of a mono-atomic ionized gas [FORMULA] and [FORMULA]. The compression ratio is [FORMULA], and the ratio in this case becomes

[EQUATION]

Since [FORMULA] is smaller for a [FORMULA] case, the density at the sonic point in the outflow is much higher (due to the exponential dependence of density on [FORMULA], see Eq. (7)) which causes the higher outflow rate, even when the actual jump in density in the postshock region, the location of the sonic point and the velocity of the flow at the sonic point are much lower. It is to be noted that generally for [FORMULA] shocks are not expected (C90), but the centrifugal-barrier-supported dense region would still exist. As is clear, the entire behavior of the outflow depends only on the compression ratio, R and the collimating property of the outflow [FORMULA].

Outflows are usually concentrated near the axis, while the inflow is near the equatorial plane. Assuming a half angle of [FORMULA] in each case, we obtain

[EQUATION]

and

[EQUATION]

The ratios of the rates for [FORMULA] and [FORMULA] are then

[EQUATION]

and

[EQUATION]

respectively. Thus, in quasi-spherical systems, in the case of strong shock limit, the outflow rate is at most a few percent of the inflow. If this assumption is dropped, then for flow with a weaker shock the rate could be much higher (see Fig. 2a).

The angle [FORMULA] must be related to the collimation property of the ambient medium, as well as the strength of the angular momentum barrier, although it is doubtful if matter achieves the observed collimation right close to the black hole. The stronger the barrier is, the higher is [FORMULA] (Molteni et al. 1994) and therefore the higher the loss rate. If [FORMULA] is sufficiently high and [FORMULA] is low ([FORMULA]) the outflow may cause a complete evacuation of the disk causing a quiescence state of the black hole candidates. The peak area in Fig. 2a would have interesting effects on the relation between spectral states and quasi-periodic oscillations of spectra of black holes as will be discussed in Sect. 4.

It is to be noted that the above expression for the outflow rate is strictly valid if the flow could be kept isothermal at least up to the sonic point. If this assumption is dropped the expression for the outflow rate becomes dependent on several parameters. As an example, we consider a polytropic outflow of the same index [FORMULA] but of a different entropy function K (we assume the equation of state to be [FORMULA], with [FORMULA]) the expression (11b) would be replaced by

[EQUATION]

and Eq. (12) would be replaced by

[EQUATION]

where [FORMULA] is the polytropic constant of the flow and [FORMULA] and [FORMULA] are the adiabatic sound speeds at the starting point and the sonic point of the outflow. It is easily shown that a power law temperature falloff of the outflow ([FORMULA]) would yield

[EQUATION]

where [FORMULA] and [FORMULA] are the entropy functions of the inflow and the outflow. This derivation is strictly valid for a non-isothermal flow. Since [FORMULA], [FORMULA] and [FORMULA], for [FORMULA], [FORMULA] is guaranteed provided [FORMULA], i.e., if the temperature falls off sufficiently rapidly. For an isothermal flow [FORMULA] and the rate tends to be higher. Note that since [FORMULA] in this case, any small jump in entropy due to compression will balance the effect of the [FORMULA] factor. Thus [FORMULA] remains smaller than unity. The first factor decreases with the entropy jump while the second factor will have a minimum at [FORMULA] when [FORMULA]. Thus the solution is still expected to be similar to what is shown in Fig. 2a,b. Numerical results of the transonic flow using a non-isothermal equation of state are discussed elsewhere (Das & Chakrabarti 1999).

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© European Southern Observatory (ESO) 1999

Online publication: November 2, 1999
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