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Astron. Astrophys. 330, 79-89 (1998)

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2. Ultracompact jets

A region of significant particle acceleration is assumed to exist at the very center of an AGN. The accelerated particles are confined by an ambient plasma with steep pressure and density gradients along the rotational axis of the central engine. A highly collimated relativistic outflow (jet) can be formed in result. The jet can contain both protons and electron-positron pairs (Sol et al. 1989). Interactions between the jet and the ambient medium are usually ignored, and the jet itself is assumed to have no transverse velocity gradient. The jet hydrodynamics is then given by a stationary adiabatic flow satisfying Bernoulli's equation [FORMULA] (Daly & Marscher 1988) . A nozzle is formed at a distance [FORMULA] where the pressure p drops below 4/9 of its initial value, and the flow becomes supersonic, with bulk Lorentz factor [FORMULA]. Beyond the nozzle, the flow is accelerated by the conversion of internal relativistic particle energy [FORMULA] to bulk kinetic energy [FORMULA], so that [FORMULA] ([FORMULA] const). The resulting particle energy distribution [FORMULA] can be approximated by a power law: [FORMULA] for [FORMULA], where s is the particle energy spectral index. The jet radiation is then described by the inhomogeneous synchrotron spectrum with spectral index [FORMULA]. Unless otherwise specified, the jet bulk Lorentz factor [FORMULA], and viewing angle [FORMULA] are assumed constant. The jet geometry can be approximated by a conical or paraboloidal expansion, with the transverse radius [FORMULA] ([FORMULA]), for [FORMULA], where [FORMULA] can be taken [FORMULA] - [FORMULA], for most of parsec-scale jets. The jet opening angle [FORMULA] is constant for [FORMULA]. The magnetic field and particle density decrease with r, and can be approximated as: [FORMULA] and [FORMULA], where [FORMULA], [FORMULA] are the magnetic field and the electron density at [FORMULA] pc. Königl (1981) shows that the combination [FORMULA], [FORMULA] can be used to describe the observed X-ray and synchrotron emission from the most compact regions of VLBI jets.

2.1. VLBI core and ultracompact jet

In the scheme described above, the protons become subrelativistic in the rest frame of the flow outside a characteristic radius [FORMULA], and the bulk Lorentz factor freezes at the value [FORMULA]. This location in the jet is most likely to be observed as the VLBI core (Marscher 1995) . It is also referred to as the "injection point" (at which the relativistic plasma is assumed to be injected into the jet at [FORMULA]) in most of the models dealing with parsec-scale relativistic jets.

At any given frequency, the VLBI core is observed at the location where the optical depth of synchrotron self-absorption is [FORMULA]. For given [FORMULA] and [FORMULA], the corresponding [FORMULA] is (e.g. Rybicki & Lightman 1979):

[EQUATION]

Here [FORMULA] is the electron mass and [FORMULA]. The observed jet opening angle [FORMULA]. [FORMULA] is described in Blumenthal & Gould (1970), and [FORMULA]. Setting [FORMULA] to unity gives for the distance from the central engine to the core observed at frequency [FORMULA]

[EQUATION]

with

[EQUATION]

From (1), [FORMULA] and [FORMULA]. For the equipartition value of [FORMULA], the choice of [FORMULA], [FORMULA] appears to be the most reasonable. The corresponding [FORMULA] in this case does not depend on spectral index, and both [FORMULA] and [FORMULA] decline (taking [FORMULA], for instance, would result in [FORMULA], which is unrealistic: [FORMULA] would then remain nearly constant, or even grow, with increasing r).

If the apparent core positions measured at two frequencies [FORMULA] differ by [FORMULA], one can introduce a measure of core position offset

[EQUATION]

where [FORMULA] is the luminosity distance to the source, and frequencies are given in Hz. [FORMULA] can be used for assessing the quality of core position measurements. For ideal measurements, [FORMULA], for all frequency pairs. A dispersion in the values of [FORMULA] derived from different frequency pairs reflects the inaccuracy of core position data. If [FORMULA] has been determined inaccurately, or if it varies along the jet, then [FORMULA] will exhibit a systematic trend. The variations of [FORMULA] can occur when the jet crosses a region with rapidly changing absorption properties (e.g. the broad line region). In this case, [FORMULA] can be used to estimate the change of [FORMULA]. If two values [FORMULA] (measured between frequencies [FORMULA] and [FORMULA]) and [FORMULA] (between [FORMULA] and [FORMULA]) are different, the relation between the corresponding [FORMULA] 's is:

[EQUATION]

With a measured [FORMULA], equation (2) can be applied to determine the magnetic field in the jet at [FORMULA] pc

[EQUATION]

and the absolute distance of the core from the central engine:

[EQUATION]

Formally, [FORMULA] refers to the distance between the core and the sonic point, [FORMULA]. However, since [FORMULA], in most cases, [FORMULA] can be taken as a fair approximation for the distance to central engine. It should also be stressed that [FORMULA] in (7) depends only on measured quantities ([FORMULA] and [FORMULA]) and on the jet viewing angle [FORMULA] that can be determined from observations. Therefore [FORMULA] can be used as a reliable estimate of core position, or at least of the projected distance from the central engine (if [FORMULA] is poorly known). From (6) and (7), the magnetic field in the VLBI core observed at frequency [FORMULA] is given by

[EQUATION]

where [FORMULA]. For a typical [FORMULA], we have [FORMULA].

2.2. Equipartition regime

We now consider the equipartition between jet particle and magnetic field energy densities, with the magnetic field energy density given by [FORMULA] (Blandford & Königl 1979) , where [FORMULA], [FORMULA]. In this case, [FORMULA] (with [FORMULA], [FORMULA]), and the core position offsets can also be used to determine the total (kinetic + magnetic field) power of the jet

[EQUATION]

where [FORMULA]. The [FORMULA] can be further used to derive the magnetic field at 1 pc

[EQUATION]

The magnetic field in the core can be obtained, similarly to [FORMULA] derived in (8). The equipartition can also be used for predicting the core offset in a source with known synchrotron luminosity [FORMULA]. In this case, the expected shift of the core position between two frequencies [FORMULA] and [FORMULA] is

[EQUATION]

with [FORMULA], and assuming [FORMULA]. Table 1 contains first order predictions of the core shift between 5 and 22 GHz, for several prominent radio sources.


[TABLE]

Table 1. Predicted core position shift between 5 and 22 GHz


The synchrotron luminosities in Table 1 are calculated from a database compiled by Ghisellini et al. (1992), using the model of Blandford & Königl (1979) with [FORMULA], [FORMULA]. The jet opening angle [FORMULA] was used for all sources. The synchrotron self-Compton Doppler factors from the database are used for estimating the jet viewing angles and Lorentz factors, assuming the minimum jet kinetic power condition [FORMULA]. This gives lower limits for synchrotron luminosities, and upper limits for the shifts. One can see, from Table 1, that the core shifts may be noticeably large, and can be measured from VLBI data.

2.3. External pressure gradients

In the vicinity of an accretion disk, physical conditions in the jet become sensitive to the gradients of the pressure [FORMULA] of the ambient medium. Since [FORMULA], to satisfy Bernoulli's equation, the jet Lorentz factor and opening angle vary. Consequently, the corresponding gradients in [FORMULA] and [FORMULA] should also change along the jet in this region (Georganopoulos & Marscher 1996) . The dependences of magnetic field and particle density distributions on pressure gradients are shown in Fig. 1. If [FORMULA] and spectral index [FORMULA] are measured, one can find, from Fig. 1, the particle density and magnetic field distribution satisfying the conditions of stationary adiabatic flow. The magnetic field at [FORMULA] is [FORMULA], [FORMULA]. The particle density distribution can be approximated by [FORMULA].

[FIGURE] Fig. 1. Physical conditions in the jet in the presence of strong pressure gradients. Power index a describes the pressure gradient ([FORMULA]) along the jet axis. The corresponding magnetic field and particle density gradients have power indices [FORMULA] (for [FORMULA]) and n, respectively. The n and [FORMULA] are given for different values of synchrotron spectral index [FORMULA].

If the jet is confined by gaseous clouds supported by thermal pressure from the central source and the stellar population with star density [FORMULA], the ambient medium density can be modelled by an exponential decrease [FORMULA], with the characteristic size of the cloud system [FORMULA] (Blandford & Rees 1974) . Here [FORMULA] is the protom mass, and T is the ambient plasma temperature. Such a distribution reflects the likely conditions in the broad line region (BLR) surrounding the nucleus of an AGN. The jet properties in this case are shown in Fig. 2, for [FORMULA], [FORMULA], and [FORMULA]. The parameters are chosen so as to approach the equipartition at [FORMULA]. The jet opacity to synchrotron self-absorption increases significantly in the region of steepest pressure gradients. This implies that confinement effects may influence the observed frequency dependence of the VLBI core position in some radio sources. If [FORMULA] and T are known, the core position measurements can be used for estimating the size of nuclear cloud system.

[FIGURE] Fig. 2. Change of physical conditions along the jet axis, r. The pressure gradient a corresponds to gaseous clouds supported by thermal pressure and maintaining a mass distribution with spherically symmetrical gravitational potential. The cloud region extends up to 400 [FORMULA] ([FORMULA] refers to the distance at which the jet becomes supersonic). The equipartition regime is approached at the outer boundary of the cloud region, with [FORMULA], [FORMULA], [FORMULA]. Significant deviations from the equipartition are seen on smaller scales, resulting in stronger self-absorption in the inner parts of the jet ([FORMULA]).

2.4. Free-free absorption

Foreground free-free absorption in a hydrogen plasma is expected to affect the VLBI-scale radio emission propagating through a dense nuclear environment. The existence of absorbing, circumnuclear plasma has been suggested in 3C 84 (Vermeulen et al. 1994; Walker et al. 1994), Cen A (Jones et al. 1996), and Cyg A (Krichbaum et al. 1997). In all three sources, only the counter-jet emission is believed to be absorbed. One can also expect to find absorption in the jet-side emission, if the jet viewing angle is relatively large, and the absorbing medium extends sufficiently high above the accretion disk plane. One possibility would be the broad line region formed by hydrogen plasma clouds entrained from the disk (Cassidy & Raine 1993) . The optical depth of free-free absorption is given by (e.g. Levinson et al. 1995)

[EQUATION]

where [FORMULA] is the size of absorbing region. Assuming a pure hydrogen plasma with uniform density, we can take [FORMULA]. The hydrogenic free-free Gaunt factor, [FORMULA], can be evaluated numerically (Hummer 1988). An analytical long-wave approximation of [FORMULA] given by Scheuer (1960) can also be used.

The typical sizes of BLR inferred from reverberation mapping and modelling the optical broad line emission of AGN vary between 0.01 and 1 pc (e.g. Kaspi et al. 1996; Baldwin et al. 1996). The corresponding densities of hydrogen plasma are [FORMULA] cm-3. The disk-wind model (Cassidy & Raine 1993) predicts the BLR sizes of up to [FORMULA] pc, with [FORMULA] between [FORMULA] and [FORMULA] cm-3. There is evidence that weaker emission lines may originate from a very extended ([FORMULA] pc) thermal component with densities [FORMULA] cm-3 and electron temperatures [FORMULA] K (Ferguson et al. 1997) . Most of these plasmas can alter significantly the GHz-range emission from VLBI jets (for instance, [FORMULA] at 10 GHz, for [FORMULA] cm-3, [FORMULA] K, [FORMULA]). Udomprasert et al. (1997) report very high rotation measures, [FORMULA] rad m-2 in the VLBI core of the quasar OQ 172, which further supports the presence of high-density thermal medium around the ultracompact jets.

For a spherical distribution of BLR clouds, we can take [FORMULA] ([FORMULA] is the volume filling factor of the cloud distribution). Then, from (12), [FORMULA]. A crude estimate [FORMULA] can be adopted (remembering that n refers essentially to plasma density variations along the jet axis). Then [FORMULA], and the corresponding core shift is roughly 10 times smaller than that due to synchrotron self-absorption at typical VLBI observing frequencies.

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© European Southern Observatory (ESO) 1998

Online publication: January 8, 1998
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