## 2. Ultracompact jetsA 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
(Daly & Marscher 1988) . A nozzle is formed at a distance
where the pressure ## 2.1. VLBI core and ultracompact jetIn the scheme described above, the protons become subrelativistic in the rest frame of the flow outside a characteristic radius , and the bulk Lorentz factor freezes at the value . 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 ) 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 . For given and , the corresponding is (e.g. Rybicki & Lightman 1979): Here is the electron mass and . The observed jet opening angle . is described in Blumenthal & Gould (1970), and . Setting to unity gives for the distance from the central engine to the core observed at frequency From (1), and . For
the equipartition value of , the choice of
, appears to be the most
reasonable. The corresponding in this case does
not depend on spectral index, and both and
decline (taking , for
instance, would result in , which is
unrealistic: would then remain nearly constant,
or even grow, with increasing If the apparent core positions measured at two frequencies differ by , one can introduce a measure of core position offset where is the luminosity distance to the source, and frequencies are given in Hz. can be used for assessing the quality of core position measurements. For ideal measurements, , for all frequency pairs. A dispersion in the values of derived from different frequency pairs reflects the inaccuracy of core position data. If has been determined inaccurately, or if it varies along the jet, then will exhibit a systematic trend. The variations of can occur when the jet crosses a region with rapidly changing absorption properties (e.g. the broad line region). In this case, can be used to estimate the change of . If two values (measured between frequencies and ) and (between and ) are different, the relation between the corresponding 's is: With a measured , equation (2) can be applied to determine the magnetic field in the jet at pc and the absolute distance of the core from the central engine: Formally, refers to the distance between the core and the sonic point, . However, since , in most cases, can be taken as a fair approximation for the distance to central engine. It should also be stressed that in (7) depends only on measured quantities ( and ) and on the jet viewing angle that can be determined from observations. Therefore can be used as a reliable estimate of core position, or at least of the projected distance from the central engine (if is poorly known). From (6) and (7), the magnetic field in the VLBI core observed at frequency is given by where . For a typical , we have . ## 2.2. Equipartition regimeWe now consider the equipartition between jet particle and magnetic field energy densities, with the magnetic field energy density given by (Blandford & Königl 1979) , where , . In this case, (with , ), and the core position offsets can also be used to determine the total (kinetic + magnetic field) power of the jet where . The can be further used to derive the magnetic field at 1 pc The magnetic field in the core can be obtained, similarly to derived in (8). The equipartition can also be used for predicting the core offset in a source with known synchrotron luminosity . In this case, the expected shift of the core position between two frequencies and is with , and assuming . Table 1 contains first order predictions of the core shift between 5 and 22 GHz, for several prominent radio sources.
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 , . The jet opening angle 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 . 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 gradientsIn the vicinity of an accretion disk, physical conditions in the jet become sensitive to the gradients of the pressure of the ambient medium. Since , to satisfy Bernoulli's equation, the jet Lorentz factor and opening angle vary. Consequently, the corresponding gradients in and 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 and spectral index 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 is , . The particle density distribution can be approximated by .
If the jet is confined by gaseous clouds supported by thermal
pressure from the central source and the stellar population with star
density , the ambient medium density can be
modelled by an exponential decrease , with the
characteristic size of the cloud system
(Blandford & Rees 1974) . Here is the
protom mass, and
## 2.4. Free-free absorptionForeground 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) where is the size of absorbing region. Assuming a pure hydrogen plasma with uniform density, we can take . The hydrogenic free-free Gaunt factor, , can be evaluated numerically (Hummer 1988). An analytical long-wave approximation of 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
cm For a spherical distribution of BLR clouds, we can take
( is the volume filling
factor of the cloud distribution). Then, from (12),
. A crude estimate can
be adopted (remembering that © European Southern Observatory (ESO) 1998 Online publication: January 8, 1998 |