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Astron. Astrophys. 334, 427-438 (1998)

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3. Oblique shocks and non-collinearity of the radio structure of 3C 275.1

3C 275.1 (S(178 MHz)=11.4 Jy) exhibits significant non-collinearity of the radio structure (bending angle [FORMULA]) and belongs to the "dog-leg" QSOs. It has a straight steep spectrum between 0.02 and 15 GHz with [FORMULA] and no sign of a low frequency turnover (see Fig. 2). The flux densities designated by stars in Appendix A are taken from "A Catalogue of Radio Sources" by Kühr et al. (1979). They are on the absolute scale of Baars et al. (1977). The others are taken from papers quoted in Appendix A. The first radio observations with the Ryle 5 km radio telescope at 5 and 15 GHz (Jenkins et al. 1977, Riley and Pooley 1978) indicate that at 15 GHz 3C 275.1 consists of three components, namely a compact one, which coincides with the QSO, the slightly extended southern one at [FORMULA] and an unresolved component [FORMULA] north-west of the QSO, while at 5 GHz one component coincides with the north-west (NW) lobe and another one intermediate between the core and the south-east (SE) lobe.

[FIGURE] Fig. 2. Flux density - frequency relation

VLA observations at 6 and 20 cm by Stocke et al. (1985) show a disturbed "bridge" extended back from the edge brightened SE lobe toward a core as well as a hot spot midway through the NW lobe and off axis with respect to emission preceding and following it. There is less depolarization on the (NW) jet side (Garrington et al., 1988).

3C 275.1, the FRII-type quasar having LAS= [FORMULA] at 5GHz, is variable in radio and optical spectral regions. The SE lobe is brighter than NW one at low frequencies and fainter at high ones (Stocke et al., 1985), while the NW lobe is more distant from the core than the SE one.

3.1. Jet velocity and orientation

The ratio of the distances between the radio core and the brightest feature in each radio lobe is defined by: [FORMULA]. At 5 GHz it is equal to 1.50. We adopt a kinematical model of the radio source, in which two plasma streams are ejected from the active core with velocities [FORMULA] in two opposite directions at angle [FORMULA] to the line of sight. Then, the ratio Q is given by: [FORMULA]. The value of Q=1.5 implies [FORMULA]. The relation between jet velocity [FORMULA] and the angle to the line of sight [FORMULA] for the observed Q is shown in Fig. 3 (lower line).

[FIGURE] Fig. 3. The jet velocity [FORMULA] and the angle to line of sight [FORMULA] relations for [FORMULA] and [FORMULA] (lower line) and [FORMULA] (upper line)

It is worth noting that 3C 275.1 is observed at [FORMULA] for ejection velocities larger than 0.3 c. In general, the jet and counter-jet might be ejected with different velocities and at different angles to the line of sight, although the probability is quite low (Saikia, 1991). In this case, we have: [FORMULA]. If we assume that the non-collinearity of the radio structure of 3C 275.1 is due to different ejection angles, then the [FORMULA] - [FORMULA] relation will have a minimum for [FORMULA] at [FORMULA] (upper line in Fig. 3). However, it seems more probable that the deflection of the SE lobe (of about [FORMULA]) is due to local density or pressure enhancement than motion of the radio galaxy through the intergalactic or intra-cluster medium (see the Table - [FORMULA]) (Parma et al., 1993). The candidate for localized deflectors might be the gaseous halo and/or the interstellar medium of galaxies in the immediate vicinity of the QSO. If the non-collinearity of the radio structure of 3C 275.1 results from asymmetry of the density of ISM/ICM gas then [FORMULA] ; where [FORMULA] and [FORMULA] are the radio luminosities of radio lobes; [FORMULA] and [FORMULA] - the ISM/ICM gas densities in the vicinity of radio lobes. Putting [FORMULA] [FORMULA] and [FORMULA] [FORMULA] as given by Strom and Conway (1985) one estimates the ratio of the ISM/ICM gas densities equal to 11.5. We consider that the jet plasma ejected from the core toward the SE lobe exhibits an oblique shock at the interface with the halo of the neighbouring galaxy (galaxy "381" in Table 1) leading to the observed non-collinearity. Furthermore, the observed ratio R of the integrated flux densities S of jet and counter-jet is given by: [FORMULA], where [FORMULA] is the average spectral index of jet and counter-jet. The index "n" depends upon the ejection model and n=2 stands for continuous ejection. However, 3C 275.1 being a lobe-dominated quasar has only a weak jet ([FORMULA] Jy), with no sign of a counter-jet. Therefore, to derive the ratio R we have used lobes instead of jet and counter-jet. [FORMULA], changes from 0.44 to 1.30 at 0.408 and 5 GHz respectively. Accordingly, the spectral indices of lobes vary from [FORMULA] and 0.94 to [FORMULA] and 1.23 for the NW and SE lobes respectively. The SE lobe dominates at low frequencies, i.e. it contains many low energy electrons.

On the other hand, non-visibility of the counter-jet implies [FORMULA]. Putting [FORMULA] one derives slightly larger values of [FORMULA] for the given angle [FORMULA]. In general, our limitation upon [FORMULA] calculated from Q has been confirmed by the ratio R.

We have analysed if the drag force due to the motion of the host galaxy of 3C 275.1 with velocity of [FORMULA] through ICM could cause the bending of the radio jet. We derived the pressure gradient [FORMULA] as a sum of the buoyancy pressure gradient of the ICM gas [FORMULA] and the dynamic pressure gradient caused by slow galaxy motion [FORMULA], namely [FORMULA]. We have:

[FORMULA], [FORMULA], [FORMULA], [FORMULA] have been defined in Sect. 2. [FORMULA] is the velocity of host galaxy in respect to the mean velocity of the cluster. For [FORMULA] one might take the size of the host galaxy or the size of the tail or lobe. We have considered both these cases. In the first case we assumed [FORMULA], while in the second one [FORMULA], which corresponds to the size of the SE lobe. We discussed two following cases:
We obtained the buoyancy pressure gradients about an order of magnitude lower than those due to galaxy motion through ICM. The whole ram pressure gradient changes only slightly between both these cases. At the distance of SE lobe it is equal to about [FORMULA], while the ram pressure due to the motion of radio plasma in SE lobe is larger than [FORMULA] for the minimum pressure of the SE lobe, namely from a few [FORMULA] up to a few [FORMULA]. Therefore, it is higher than that due to the motion of the host galaxy through ICM and due to buoyancy effects at least by two orders of magnitude. Accordingly, the non-linear radio structure of the quasar cannot result from the galaxy motion. We suggest that the oblique shock of the plasma jet at the interface with the halo of the neighbouring galaxy or with the cloud of ionized gas in NELR might explain the non-collinearity of radio structure.

3.2. Non-relativistic oblique shock

The solid angle as seen from the core of 3C 275.1 covered by the companion galaxy with a gaseous halo of radius [FORMULA], which is located at a distance [FORMULA] from the core is: [FORMULA]. The radius of the gaseous halo of a galaxy depends on morphological type and evolutionary effects such as cannibalism and stripping. Hence, the magnitude of [FORMULA] is highly uncertain. Putting the radius of the R-image of galaxy "381" - [FORMULA] and [FORMULA] (see Table 1), we derive [FORMULA] sr. On the other hand, the solid angle occupied by the SE radio lobe is [FORMULA], where [FORMULA] is the radius of the SE radio lobe, i.e. [FORMULA] and [FORMULA] is the distance of the SE radio lobe from the compact component, namely [FORMULA]. Hence, [FORMULA] sr. Therefore, the size of the solid angle seen from the core over which deflection by the companion galaxy "381" can occur is [FORMULA] sr. Accordingly, the inelastic collision of the radio jet with the halo of neighbouring galaxy may occur. However, the true distances can differ significally from the projected ones. One might derive the luminosity distances, the comoving distances and the angular diameter distances. We used the last ones "R" to estimate the projection angles. For the projection angle [FORMULA], the real distance of the galaxy from the quasar core is [FORMULA], while the radius of the spherical galaxy does not change. Taking the distances "R" and " [FORMULA] " given in Table 1, we obtain that the projection angle might change from 50 to [FORMULA]. The accuracy of derived projection angle is about (50-60) %. Thereafter, for [FORMULA] one calculates [FORMULA], i.e. 38.59 arcsec or about [FORMULA] kpc and 34.55 arcsec or about [FORMULA] kpc for the distances of galaxy 381 and SE lobe respectively. For [FORMULA], we have [FORMULA], i.e. 10.45 arcsec (78 kpc) and 9.36 arcsec (70 kpc) respectively. Putting these values into above expressions we obtain smaller solid angles. For projection angle [FORMULA], we have [FORMULA] sr (for [FORMULA]), [FORMULA] sr, [FORMULA] sr and the probability of interaction of about 5 %.. For larger projection angles the solid angles are even smaller and the probability of the interaction of the galaxy and the plasma jet is very low. However, the projected positions of the discussed galaxies (see PA in Table 1) are contained within one hemisphere. If we take the impact parameter of about 0.5 Mpc, three galaxies might interact with the radio plasma and probability increases to about 15 %. The other possibility is that the knots emitting narrow-emission lines might be deflectors.

Then, the deflected SE lobe is seen closer to the core because its outward velocity is reduced by deflection. Moreover, its transverse radius is greater than that of the undeflected NW lobe due to internal heating from shock, as observed in 3C 275.1. Therefore, the plasma jet characterized by the jet velocity [FORMULA], density [FORMULA], Mach number [FORMULA] exhibits the oblique internal shock at the interface with the halo of the neighbouring galaxy or knots emitting narrow lines characterized by density [FORMULA] and pressure [FORMULA]. Taking the radius of galaxy "381" equal to 22 kpc ([FORMULA]) and mass [FORMULA], one derives the average density [FORMULA], while for knots the critical density calculated in previous section is about [FORMULA]. We shall consider classical and relativistic oblique internal shocks since the discussion of the relativistic bulk motion of the emitting plasma in the nuclei of about one hundred radio sources by Ghisellini et al. (1993) indicates that the lobe-dominated QSOs have small Doppler factors ([FORMULA]), Lorentz factors of the order of 10 and average viewing angles of [FORMULA] ([FORMULA]).

Firstly, we shall discuss the non-relativistic shock of the initially supersonic jet ([FORMULA]). The jet exhibits an oblique shock at the interface to the halo of the neighbouring galaxy " [FORMULA] and bends (see Fig. 4). Subscripts '1' and '2' denote the state of gas upstream of and following the shock (downstream) respectively.

[FIGURE] Fig. 4. The schematic picture of the oblique shock in 3C 275.1. The ellipse presents the R-image of the host galaxy of 3C 275.1, while the circle corresponds to the R-image of the neighboring galaxy "381"

Let [FORMULA] be the deflection angle or the angle by which the velocity vector turns after passage through the shock front and [FORMULA] the angle that the shock front makes with respect to the velocity [FORMULA] in the flow plane. Neglecting projection effects [FORMULA] is equal to about [FORMULA]. The pressure balance is maintained across the bend (shock) interface and the bend moves in the direction of the initial jet velocity with velocity W. In the rest frame of the bend (shock), the flow may be considered as steady. Then we have: [FORMULA]. The relations between the post-shock Mach number [FORMULA], the initial flow Mach number [FORMULA], the shock inclination angle [FORMULA] and the deflection angle [FORMULA] for a given value of [FORMULA] are, as follows: [FORMULA] [FORMULA]. Here [FORMULA] is the adiabatic index. These relations are fulfilled for values of [FORMULA] larger than Mach angles [FORMULA]. We have calculated the relation [FORMULA] for [FORMULA] and different values of initial Mach number [FORMULA]. We found the single values of [FORMULA], i.e. [FORMULA] for initial Mach numbers smaller than about 2.5. For larger initial Mach numbers and [FORMULA] two values of [FORMULA] are possible, namely [FORMULA] for supersonic downstream flow and [FORMULA] for sub-sonic downstream flow. On the other hand, the pressure, density and temperature before and after the shock are described by Rankine-Hugoniot relations derived directly from the Euler, energy conservation and state equations. Then, the pressure ratio is given by: [FORMULA]. Therefore, for the observed ratio [FORMULA], [FORMULA] and [FORMULA], one derives [FORMULA] and the limitation for the initial Mach number [FORMULA]. The observed initial Mach numbers [FORMULA] are larger than 8 (Strom & Conway, 1985). On the other hand, Wellman et al. (1997) determined the Mach numbers of lobe advance larger than 3. Thereafter, [FORMULA] seems to be supersonic and [FORMULA] might be larger than 8 and [FORMULA]. Stronger shocks are likely to be associated with more powerful jets and more compressive shocks would produce flatter spectra.

3.3. Relativistic oblique shock

We assumed the structure of the shock, as follows: the x-axis is normal to the shock and the y-axis is in the plane of the shock. The particle acceleration and field amplification occur at the shock. Hence, the emission is dominated by the post-shock region. In general, for the oblique shock the components of the magnetic field are not vanishing. Hence, we have considered the magnetic field in the momentum and energy conservation equations. We assume that the motion is planar and the velocity and the magnetic field are in plane (x,y) on either side of the discontinuity. The tangential components of velocity are continuous at the surface of discontinuity, i.e. [FORMULA]. Moreover, we take such a coordinate system that the components of the electric field vanish. We consider two cases:

i) the magnetic field is parallel to flow velocities in both upstream and downstream flows. We have: [FORMULA]


[FORMULA]. Such assumption gives the relations: [FORMULA] and [FORMULA]. Since the parallel components of shock velocity are preserved, we have [FORMULA]. It gives the relation: [FORMULA].

ii) the magnetic field of the upstream flow is parallel to the velocity flow, i.e. [FORMULA], while that of the downstream flow is perpendicular, namely [FORMULA]. Since the polarization observations of 3C 275.1 by Liu and Pooley (1990) indicate that the magnetic field of the SE lobe is most probably perpendicular or complex, this case seems to be more realistic one. Then one has [FORMULA]. In both above discussed cases, the number density of particle [FORMULA] is conserved. Hence, one has: [FORMULA] To derive the conservation equations we used the energy-momentum tensor given by: [FORMULA]. Here [FORMULA] is the relativistic enthalpy, e - internal energy and [FORMULA] - the pressure of the magnetic field. Hence, the conservation equations, namely the conservation of the energy and momentum, describe the boundary (jump) conditions. They are as follows: [FORMULA] [FORMULA] [FORMULA] [FORMULA]. For case ii), after the inclusion of perpendicularity of magnetic field condition, one has: [FORMULA] [FORMULA] The angles of upstream and downstream flow velocities to the shock normal are related by: [FORMULA] and [FORMULA] for cases i) and ii) respectively.

We have derived in Appendix B expressions for the velocity components of the gas in the observer's frame as well as for the angles of upstream and downstream flow velocities for the ultrarelativistic equation of state, i.e. [FORMULA].

Knowing the ratio "a" and "b" one might derive the values of other physical parameters for different shock [FORMULA] and different angles between fluid and shock normal.

However, it is seen in Fig. 4, that the angle [FORMULA] is equal to [FORMULA] in the projection on the (x,y) plane. Here [FORMULA] corresponds to the deflection angle. Hence,



Therefore, from the above relation one derives the changes of the angle [FORMULA] with the ratios of the gas "a" and of magnetic field " [FORMULA] " pressures for values of [FORMULA] from [FORMULA] to [FORMULA] and of [FORMULA] from [FORMULA] to [FORMULA]. We realized that the values of [FORMULA] change only slightly with the value of upstream magnetic pressure and the ratios of magnetic pressures " [FORMULA] ". On the other hand, the angle [FORMULA] depends significantly upon the values of the ratio of gas pressures "a". It changes from about [FORMULA] up to [FORMULA] for "a" from 1.1 to 1.9 and for [FORMULA], [FORMULA] and [FORMULA]. For an example the changes of [FORMULA] with "a" and " [FORMULA] " are shown in Fig. 5.

[FIGURE] Fig. 5. The changes of [FORMULA] with the ratios of the gas "a" and of magnetic " [FORMULA] " pressures for [FORMULA] and [FORMULA]. [FORMULA] (solid line); [FORMULA] (dashed line).

There is a serious problem, how one might determine the ratios of the magnetic field pressures and of the gas (plasma) ones in upstream and downstream flows. There are a few possibilities for estimation of the gas pressure ratio. We used the following methods:

i) the minimum pressures, i.e. [FORMULA], calculated from radio observations of NW lobe and SE lobe, correspond to [FORMULA] and [FORMULA] respectively. We put [FORMULA] [FORMULA] and [FORMULA] [FORMULA] (Strom and Conway, 1985). Hence, [FORMULA].

ii) the gas pressure of intra-cluster medium (see Sect. 2).
At the distance of SE lobe we have: [FORMULA]. Recently, Wellman et al. (1997) have determined the density and pressures of the ambient gas in the vicinity of radio lobes assuming that radio lobes are ram-pressure confined by the ambient gas. We compared our ICM gas densities determined for " [FORMULA] " model with theirs and found our pressures higher and cooling times shorter. Moreover, ICM gas pressure at a distance of SE lobe is lower by about two orders of magnitudo than that derived in previous section for inhomogeneous NELR. Therefore, it seems that the ionized knots radiating in forbidden narrow-emission lines might also be responsible for the oblique shock of radio plasma. Thereafter, the ratio of gas pressures of upstream flow to downstream one might change from 1.6 to 3.3. Therefore, for case i) we have: [FORMULA]. On the other hand, for deflection of the upstream flow by the angle [FORMULA] the ratio "a" should be larger than about 4. For a=4, one derived [FORMULA].

There is no reliable estimations of the magnetic field in 3C 275.1. Basing upon the MERLIN observations of polarization at 18 cm Liu and Pooley (1990) have given the general discussion of the magnetic field. According to Strom and Conway (1985) the strengths of observed magnetic field are [FORMULA] and [FORMULA] for NW and SE lobes respectively. Hence, the magnetic pressures given by [FORMULA] are equal to [FORMULA] and [FORMULA] for [FORMULA] and [FORMULA] respectively. They consist only a few percent of gas pressures. However, they are stronger in the parsec scale jets. On the other hand, Liu and Riley (1992) have determined the strengths of magnetic fields as equal to [FORMULA] and [FORMULA]. Hence, we have [FORMULA]. For [FORMULA], [FORMULA] and [FORMULA] the angle [FORMULA] between the direction of upstream velocity and the surface normal to the shock changes from about [FORMULA] up to [FORMULA] for "a" from 1.7 to 1.1. Assuming [FORMULA] roughly [FORMULA] we calculated from the expression given in Appendix B the angle between observer and shock surface [FORMULA]. Putting the observed values of parameters, i.e. a=1.6, [FORMULA], [FORMULA] and [FORMULA], we obtain from the formula for velocities given in Appendix B, [FORMULA] and [FORMULA]. Thereafter, we have [FORMULA] and [FORMULA]. In summary, one should include the magnetic pressure into the considerations of oblique shock in spite that in kiloparsec jets it consists only a few percent of gas pressures.

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Online publication: May 15, 1998