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

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1. Introduction

Cosmic radio jets are believed to originate from the centers of active galaxies which harbor black holes (e.g., Chakrabarti 1996, hereafter C96). Even in so-called `micro-quasars', such as GRS 1915+105 which are believed to have stellar mass black holes (Mirabel & Rodriguez 1994), the outflows are common. The well collimated outflow in SS433 has been well known for almost two decades (Margon 1984). Similarly, systems with neutron stars also show outflows, as is believed to be the case in X-ray bursters (e.g. Titarchuk 1994).

There is a large number of articles in the literature which attempt to explain the origin of these outflows. These articles can be broadly divided into three sets. In one set, the jets are believed to come out due to hydrodynamic or magneto-hydrodynamic pressure effects and are treated separately from the disks (e.g., Fukue 1982; Chakrabarti 1986). In another set, efforts are made to correlate the disk structure with that of the outflow (e.g., Königl 1989; Chakrabarti & Bhaskaran 1992). In the third set, numerical simulations are carried out to actually see how matter is deflected from the equatorial plane towards the axis (e.g., Hawley 1984; Eggum et al. 1985; Molteni et al. 1994). Nevertheless, the definitive understanding of the formation of outflows is still lacking, and more importantly, it has always been difficult to estimate the outflow rate from first principles. In the first set, the outflow is not self-consistently derived from the inflow. In the second set, only self-similar steady solutions are found and in the third set, either a Keplerian disk or a constant angular momentum disk was studied, neither being the best possible assumption. On the other hand, the mass outflow rates of the normal stars are calculated very accurately from the stellar luminosity. The theory of radiatively driven winds seems to be very well understood (e.g. Castor et al. 1975). Given that the black holes and the neutron stars are much simpler celestial objects, and the flow around them is sufficiently hot to be generally ionized, it should have been simpler to compute the outflow rate from an inflow rate by using the methods employed in stellar physics.

Our approach to the mass outflow rate computation is somewhat different from that used in the literature so far. Though we consider simple-minded equations to make our points, such as those applicable to conical inflows and outflows, we add a fundamental ingredient to the system, whose importance is being revealed only very recently in the literature. This is the quasi-spherical centrifugally supported dense atmosphere with a typical size of a few tens of Schwarzschild radiusus around a black hole and a neutron star. Whether a shock actually forms or not, this dense region exists, as long as the angular momentum of the flow close to the compact object is roughly constant and is generally away from a Keplerian distribution as is the case in reality (C96). This centrifugally supported region (which basically forms the boundary layer of black holes and weakly magnetized neutron stars) successfully replaced the so called `Compton cloud' (Chakrabarti & Titarchuk 1995 [hereafter CT95]; Chakrabarti et al. 1996) in explaining hard and soft states of black hole candidates, and the converging flow property of this region successfully produced the power-law spectral slope in the soft states of black hole candidates (CT95). The oscillation of this region successfully explains the general properties of the quasi-periodic oscillation (Molteni et al. 1996, C96) of X-rays from black holes and neutron stars. It is therefore of interest to know if this region plays any major role in the formation of outflows.

Several authors have also mentioned denser regions forming due to different physical effects. Chang & Ostriker (1985) showed that pre-heating of the gas could produce standing shocks at a large distance. Kazanas & Ellison (1986) mentioned that pressure due to pair plasma could produce standing shocks at smaller distances around a black hole as well. Our computation is insensitive to the actual mechanism by which the boundary layer is produced. All we require is that the gas should be hot in the region where the compression takes place (i.e., the optical depth should be small). Thus, since Comptonization processes cool this region (CT95) for larger accretion rates ([FORMULA]) our process will produce outflows in hard states and low-luminosity objects consistent with current observations. Some workers talked about a so-called `cauldron' model of compact objects where jets were assumed to emerge from a dense mixture of matter and radiation through a de-Laval nozzle as in the `twin-exhaust' model (for a review of these models see Begelman et al. 1984). The difference between this model and the present one is that there a very high accretion rate was required ([FORMULA]) while we consider thermally driven outflows from smaller accretion rates. Second, the size of the `cauldron' was thousands of Schwarzschild radii (where gravity was so weak that the channel has to have the shape of a de-Laval nozzle), while we have a CENBOL of about [FORMULA] (where the gravity plays an active role in creating the transonic wind) in our mind. Third, in the present case, matter is assumed to pass through a sonic point using the pre-determined funnel where rotating pre-jet matter is accelerated (Chakrabarti 1984) and not through a `bored nozzle' even though symbolically a quasi-spherical CENBOL is considered for mathematical convenience. Fourth, for the first time we compute the outflow rate completely analytically starting from the inflow rate alone. To our knowledge such a calculation has not been done in the literature at all.

Once the presence of our centrifugal-pressure -upported boundary layer (CENBOL) is accepted, the mechanism of the formation of the outflow becomes clearer. One basic criterion is that the outflowing winds should have a positive Bernoulli constant (C96) (although in the presence of radiative momentum deposition, a flow with negative initial energy could also escape as outflow, see Chattopadhyay & Chakrabarti 1999). Just as photons from the stellar surface deposit momentum on the outflowing wind and keep the flow roughly isothermal at least up to the sonic point, one may assume that the outflowing wind close to the black hole is kept isothermal due to deposition of momentum from hard photons. In the case of the sun, its luminosity is only [FORMULA] and the typical mass outflow rate from the solar surface is [FORMULA] year-1 (Priest 1982). Proportionately, for a star with an Eddington luminosity, the outflow rate would be [FORMULA] year-1. This is roughly half the Eddington rate for a stellar-mass star. Thus, if the flow is compressed and heated at the centrifugal barrier around a black hole, it would also radiate enough to keep the flow isothermal (at least up to the sonic point) if the efficiency were exactly identical. Physically, both requirements may be equally difficult to meet, but in reality with photons shining on outflows near a black hole with almost [FORMULA] solid angle (from the funnel wall) it is easier to maintain the isothermality in the slowly moving (subsonic) region in the present context. Another reason is this: the process of momentum deposition on electrons is more efficient near a black hole. The electron density [FORMULA] falls off as [FORMULA] while the photon density [FORMULA] falls off as [FORMULA].Thus the ratio [FORMULA] increases with the size of the region. Thus a compact object will have a lesser number of electrons per photon and the momentum transfer is more efficient. In a simpler-minded way, the physics is scale-invariant, though. In solar physics, it is customary to choose a momentum deposition term which keeps the flow isothermal to be of the form (Kopp & Holzer 1976; Chattopadhyay & Chakrabarti 1999),

[EQUATION]

where D is the momentum deposition (localized around [FORMULA]) factor with a typical spatial dependence,

[EQUATION]

Here, [FORMULA], [FORMULA] are constants and [FORMULA] is the location of the stellar surface. Since r and [FORMULA] appear in ratio, exactly the same physical consideration would be applicable to black hole physics, with the same result provided [FORMULA] is scaled with luminosity. (But, as we showed above, [FORMULA] increases for a compact object.) However, as CT95 showed, a high accretion rate ([FORMULA]) will reduce the temperature of the CENBOL catastrophically, and therefore our assumption of isothermality of the outflow would break down at these high rates. It is to be noted that in the context of stellar physics, it is shown (Pauldrach et al. 1986) that the temperature stratification in the outflowing wind has little effect on the mass loss rate.

Having thus been convinced that isothermality of the outflow, at least up to the sonic point, is easier to maintain near a black hole, we present in this paper a simple derivation of the ratio of the mass outflow rate and mass inflow rate assuming the flow is externally collimated. We find that the ratio is a function of the compression ratio of the gas at the boundary of the hot, dense, centrifugally supported region. We estimate that the outflow rate should generally be less than a few percent if the outflow is well collimated. In Sect. 3, we find some interesting effects when the region out to the sonic point of the outflow cools down periodically and causes periodic change of spectral states. Finally, in Sect. 4, we draw our conclusions.

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© European Southern Observatory (ESO) 1999

Online publication: November 2, 1999
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