The recent development of tomographic analysis of radio data shows the
presence of sheaths around several extragalactic jets. We have studied the
Kelvin-Helmholtz instability of such stratified jets, describing them as inner
relativistic beams embedded in slower plasma envelopes. We find that the para-
meters of the envelope generally determine the Kelvin-Helmholtz properties of
the jet which thus appears isolated from any ambient external medium.Comment: LaTeX, 5 pages, 1 figure and crckapb.sty, epsf.sty included, To
  appear in the proceedings of Girona Conference ``Blazars, Black Holes and
  Jets'', 9-12 September 1996, Girona, Spai

Hanasz, M.

Sol, H.

English

arXiv

The recent development of tomographic analysis of radio data shows the presence of sheaths around several extragalactic jets. We have studied the Kelvin-Helmholtz instability of such stratified jets, describing them as inner relativistic beams embedded in slower plasma envelopes. We find that the para- meters of the envelope generally determine the Kelvin-Helmholtz properties of the jet which thus appears isolated from any ambient external medium

Hanasz, Michał

Sol, Helene

Repository of Nicolaus Copernicus University

arXiv:astro-ph/9711240v1  20 Nov 1997HYDRODYNAMICAL INSTABILITY OF EXTRAGALACTICSTRATIFIED JETSM. HANASZInstitute of Astronomy, Nicolaus Copernicus University, Toruń,PolandANDH. SOLObservatoire de Paris, UPR 176 du CNRS, Meudon, France1. IntroductionA rather new view of extragalactic radio sources seems to come out fromrecent observational data. For instance, in the case of 3C273, comparisonof optical and radio maps suggested the presence of an inner optical jetembedded in a slow moving radio cocoon (Bahcall et al, 1995). Furthermorenew ways of combining radio data at different frequencies provided originalinformation on the sources. In particular sheaths of emission with specificspectrum and polarization have been found around the jets in Cygnus A,3C449, two wide-angle-tail and one CSS sources, suggesting that such kindof sheaths or envelopes are commonly present around extragalactic jets(Katz-Stone, Rudnick, 1994; Rudnick, Katz-Stone, 1996).The question of the origin of envelopes around extragalactic jets is stillopen. They can just emanate from the inner jets as the result of particlediffusion or correspond to some entrainment of the external gas by theinner jets. Another possibility, suggested a few years ago in the contextof two-component jet models (Smith, Raine, 1985; Baker et al, 1988; Solet al, 1989; Achatz et al, 1990), is that they have been directly ejectedfrom the central engines. Indeed such models proposed and analyzed thescenario where a fast beam (corresponding to the inner core of the jet) isejected from the vicinity of the central black hole and a slower collimatedouter wind (corresponding to the envelope) is emitted by all parts of theaccretion disk.2 M. HANASZ AND H. SOLAnyway, such envelopes likely modify the interaction of the jets with theambient media in which they propagate, and a question which immediatelyoccurs is that of the Kelvin-Helmholtz stability properties of such “core-envelope” or “stratified” jets.2. Kelvin-Helmholtz instability of stratified jetsWe have studied the linear stage of the Kelvin-Helmholtz instability ofsuch composite stratified jets, in the plane parallel approximation (Hanasz,Sol, 1996). Jets are assumed to be made of an inner core of relativisticgas with a relativistic bulk velocity surrounded on both sides by a sheet ofnon-relativistic gas with slower bulk velocity, and embedded in the externalambient medium. There are therefore two types of interfaces, the internalinterfaces between the core and the sheet, and the external ones betweenthe sheet and the ambient medium. Transition layers at all interfaces aredescribed in the vortex-sheet approximation. We perform a temporal sta-bility analysis and perturb the initial hydrodynamical equilibrium for thethree components (relativistic core and non-relativistic sheet and externalmedium). The dispersion relation is then obtained from the boundary con-ditions at the internal and external interfaces, namely equality of pressureand equality of transverse gas and interface velocities. It writes in compactform:ZsZcE = coth ikc⊥ for the symmetric solutionandZsZcE = th ikc⊥ for the antisymmetric solution.Here E is the term due to the effect of the envelope (or sheet),E =1 + Rsee2iks⊥(R−1)1 −Rsee2iks⊥(R−1).R = Rs/Rc is the ratio of the sheet radius to the core radius, kc⊥ and ks⊥the transverse wavenumbers in core and sheet, Zc, Zs and Ze the complexnormal acoustic impedances of core, sheet and external medium. Rse is thereflection coefficient at the external interface between sheet and ambientmedium, defined as a function of the acoustic impedances byRse =Ze − ZsZe + Zs.The dispersion relation reduces to the case of a single jet in different limits,(i) when there is no reflection at the external interface (Rse = 0, E = 1), (ii)when the sheet becomes very thick (R >> 1) since the factor of Rse thenHYDRODYNAMICAL INSTABILITY OF EXTRAGALACTIC STRATIFIED JETS3FM,  R = 20.01 0.10 1.00 10.000.0010.0100.100ωiFM,  R = 50.01 0.10 1.00 10.000.0010.0100.100FM,  R = 100.01 0.10 1.00 10.000.0010.0100.100FM, R = 20.01 0.10 1.00 10.000.0010.0100.100ωiFM, R = 50.01 0.10 1.00 10.000.0010.0100.100FM, R = 100.01 0.10 1.00 10.000.0010.0100.100Figure 1. Examples of temporal growth rate of the instability versus the longitudinalwavenumber, for the fundamental mode of the antisymmetric solution. The thickness ofthe sheet increases from first (R = 2) to second (R = 5) and third (R = 10) columns.Upper case corresponds to underdense core and sheet, with density ratios ρc/ρs = 0.01and ρs/ρe = 0.1 and lower case to a dense sheet with ρc/ρs = 0.01 and ρs/ρe = 10. Herethe relativistic bulk Lorentz factor of the core is 10 and the Mach number of the sheet iszero. Dotted lines correspond to the solution without sheet (R = 1) and dashed lines tothe solutions without external medium (R → ∞, infinite sheet).vanishes as exp[−2RIm(ks⊥)] and E tends to 1 again, (iii) when internaland external interfaces coincide (R = 1) and only the core and external gascomponents remain.3. Results and interpretationThe dispersion relation has been solved numerically and various sets ofparameters investigated. Figure 1 illustrates the temporal growth rate ob-tained in a few cases. The presence of the sheet generates an oscillatingpattern due to the interferences of acoustic waves reflected at the bound-aries of the sheet, namely at the internal and external interfaces.The oscillating pattern can be explained as follows. The acoustic wavesemitted at the internal interface are reflected at the external interface andcome back to the internal one. The reflection coefficient for acoustic waves4 M. HANASZ AND H. SOLat external interface |Rse| is smaller than 1, so there is no amplification ofthe wave amplitude there even if sheet is supersonic (this statement is validfor frequencies and wavenumbers typical for the core-sheet instability). Thespace between internal and external interfaces (i.e. the sheet itself) can beconsidered as the resonant cavity, which imposes the condition for resonanceof the type2L = nλsand antiresonance2L =(n +12)λswhere L is the distance between internal and external interfaces, and λs isthe wavelength of the acoustic waves in the sheet, both of them are mea-sured along the direction of propagation. The local maxima/minima of thevalue of growth rate correspond to the fulfillment of resonant/antiresonantcondition. This is worth noticing that the resonant/antiresonant conditionsare in general modified by an additional phase shift coming from the re-flection at interfaces. The phase shift is dependent mostly on the densitycontrasts. This is apparent in figure 1 where the upper and lower rows rep-resent two cases which differ only in the value of νs = ρs/ρe In the two casesthe positions of maxima and minima are interchanged due to the mentionedphase shift. In all cases with thick enough sheet (R > 1) the solutions ofthe dispersion relation rapidly converge to the case of infinite sheet. So asheet or envelope as thick as the inner core almost imposes the growth rate.Parameters of the envelope rather than those of the external medium ap-pear to determine the Kelvin-Helmholtz instability, and the inner beam issomewhat isolated from the ambient gas. Relatively to the solution withoutenvelope, the growth rate is increased for an envelope light compared to theexternal medium, and lowered for a dense envelope. So the presence of anheavy envelope can help to stabilize the jet configuration.ReferencesAchatz, U., Lesch, H. and Schlickeiser, R. (1990), AA, 233, p. 391.Bahcall, J.N., Kirhakos, S., Schneider, D.P., Davis, R.J., Muxlow, T.W.B., Garring-ton, S.T., Conway, R.G. and Unwin, S.C. (1995) ApJ, 452, p. L91.Baker, D.N., Borovsky, J.E., Benford, G. and Eilek, J.A. (1988) ApJ, 326, p. 110.Hanasz, M. and Sol, H. (1996) AA, 315, p. 355.Katz-Stone, D.M. and Rudnick, L. (1994) ApJ, 426, p. 116.Rudnick, L. and Katz-Stone, D.M. (1996) Energy transport in radio galaxies andquasars, ASP Conference Series, 100, P.E. Hardee, A.H. Bridle, J.A. Zensus, eds.p. 233.Smith M.D. and Raine, D.J. (1985) MNRAS, 212, p. 425.Sol, H., Pelletier, G. and Asséo, E. (1989) MNRAS, 237, p. 411.

Hydrodynamical instability of extragalactic stratified jets

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Hydrodynamical instability of extragalactic stratified jets

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