S hw 2 v and hw 2 p D are the relativistic densities

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1 THE ASTROPHYSICAL JOURNAL, 448 : L105 L108, 1995 August The American Astronomical Society. All rights reserved. Printed in U.S.A. MORPHOLOGY AND DYNAMICS OF HIGHLY SUPERSONIC RELATIVISTIC JETS J. M a.martí 1 AND E. MÜLLER Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Strasse 1, Garching, Germany J. A. FONT Departamento de Física Teórica, Universidad de Valencia, Burjassot (Valencia), Spain AND J. M a.iba ÑEZ Departamento de Astronomía y Astrofísica, Universidad de Valencia, Burjassot (Valencia), Spain Received 1995 March 20; accepted 1995 May 17 ABSTRACT We present a simulation of a diffuse ( 0.01), high-beam Mach number (M b 6.0), relativistic (beam Lorentz factor 22.4) axisymmetric jet, and we discuss its morphology and dynamics. The jet exhibits a prominent structure of oblique shocks inside the beam and possesses an extensive cocoon. This result is qualitatively different from the findings of other recent simulations of low-beam Mach number relativistic jets, where both features are absent. We find that the jet propagates very efficiently through the ambient medium. Its mean velocity is 15 times larger than that expected from classical (i.e., nonrelativistic) simulations. The simulations are performed with a high-resolution shock-capturing scheme using a Riemann solver which is based on the spectral decomposition of the Jacobian matrices of relativistic hydrodynamics. Subject headings: galaxies: jets hydrodynamics relativity 1. INTRODUCTION In the widely accepted beaming model (Blandford & Königl 1979), a jet moving at relativistic speed at a small angle to the line of sight is thought to be the cause of the apparent superluminal motion on parsec scales found in VLBI components of 140 compact radio sources (see, e.g., Ghisellini et al. 1993). The observed superluminal motion extends out to more than 100 pc as, e.g., in the case of the quasar 3C 273 (see Davis, Unwin, & Muxlow 1991) and the inferred bulk velocities are as large as c, where c is the speed of light. Besides the evidence for relativistic bulk motion inferred from VLBI observations of superluminal motion, there are now additional independent indications of highly relativistic outflow on these small scales. If the observed intraday radio variability occurring in more than one-fourth of all compact radio sources (Krichbaum, Quirrenbach, & Witzel 1992; Witzel 1992) is intrinsic and results from incoherent synchrotron radiation, then the associated jets must have bulk Lorentz factors in the range W (Begelman, Rees, & Sikora 1994). At kiloparsec scales, indirect evidence for relativistic bulk motion is based on asymmetries in the radio flux emitted by the twin jets of powerful radio sources. In addition, direct measurements of the proper motion of knots in the jet of M87 obtained with the VLA (Biretta & Owen 1990; Biretta, Zhou, & Owen 1995) have proven the existence of apparent superluminal motions within the first kiloparsec to the central source. Numerical simulations of jets have become an important tool in understanding the morphology and dynamics of extragalactic jets ever since Norman et al. (1982) verified the jet model of Blandford & Rees (1974) by means of hydrodynamical simulations of axisymmetric supersonic flows. Until recently these simulations have been performed in the framework of Newtonian fluid dynamics. With the advent of 1 On leave from Departamento de Astronomía y Astrofísica, Universidad de Valencia, Spain. L105 hydrodynamical codes based on high-resolution shock-capturing techniques accurate descriptions of ultrarelativistic flows involving strong shocks have become feasible (see, e.g., Martí, Ibáñez, & Miralles 1991; Font et al. 1994; Eulderink & Mellema 1995). Applying these modern techniques simulations of unsteady relativistic jets have been performed (Martí, Müller, & Ibáñez 1994; Duncan & Hughes 1994; hereafter MMI94 and DH94, respectively). However, all previous simulations only considered jets with a small beam Mach number. In this Letter, we present first results concerning the structure and dynamics of a highly supersonic relativistic jet. Besides being highly supersonic, the flow in the beam of the jet also has a bulk Lorentz factor of 22.37, which is the largest of any models calculated until today. Further results obtained within a comprehensive study of the structure and dynamics of relativistic jets will be presented in a forthcoming publication (Martí et al. 1995). As in MMI94 and DH94, we have solved the equations of special relativistic hydrodynamics, for a perfect fluid, for the vector of conserved variables (D, S, ), where D W, S hw 2 v and hw 2 p D are the relativistic densities of rest mass, momentum, and energy. In the last expressions,, p, and v are, respectively, the proper rest-mass density, pressure, and fluid velocity. The specific enthalpy, h, is given by h 1 p/, being the specific internal energy. The flow is supposed to be axisymmetric, and hence the equations are discretized in two-dimensional cylindrical coordinates (r, z). An ideal gas equation of state with adiabatic index has been assumed. The hydrodynamical code is based on an approximate Riemann solver, which handles high Mach number flows extremely well (Marquina & Donat 1993). Contrary to the method used by DH94, this Riemann solver relies on the spectral decomposition of the Jacobian matrices of the multidimensional system of relativistic equations (Font et al. 1994). It guarantees that the coupling of the transverse velocities is

2 L106 MARTI ET AL. Vol. 448 taken into account properly in the calculation of the numerical fluxes, which is a particular and difficult feature of special relativistic numerical hydrodynamics. The spatial accuracy of the code is improved by means of a conservative monotonic piecewise-parabolic reconstruction of the pressure, proper rest-mass density and flow velocity. The reconstruction procedure is based on the work of Colella & Woodward (1984). The explicit formulae used in our relativistic version can be found in Martí & Müller (1995). Integration in time is done simultaneously in both spatial directions using a total variation diminishing (TVD) ge-kutta scheme of high order developed by Shu & Osher (1988). Except for the Riemann solver and the ge-kutta time discretization, our code is otherwise identical to the one of MMI94. A one-dimensional Newton-Raphson iteration allows to compute, at each time step, the primitive variables {,, p, v} from the conserved ones (see Martí &Müller 1995 for details). As stated above, previous simulations of relativistic jets focused on the study of low Mach number beams (see Table 1). The most interesting result of these simulations is the lack of structural features inside the beam and a remarkable reduction of Kelvin-Helmholtz instabilities near the contact surface for the fastest of these jets (see, e.g., MMI94 and simulation D of DH94), the latter being responsible for the presence of very stable lobes around the beam. Based on these results DH94 suggested that the morphological differences between BL Lac objects and quasars are due to the transition to more stable structures for the higher Lorentz factor quasars. However, the results presented in this Letter show that the beam Mach number plays an important role in the overall morphology of relativistic jets. Therefore, the conclusions obtained for low Mach number beams cannot be extended to highly supersonic jets. 2. RESULTS We present results obtained in the simulation of a relativistic (v b c), pressure-matched jet with a high (Newtonian) beam Mach number (M b 6.0). The full set of parameters defining the model are given in Table 1. We followed the propagation of the jet through a homogeneous ambient medium in a region extending 50 R b in z-direction and 7 R b in r-direction, where R b is the (initial) beam radius. The computational domain is covered by a uniform numerical grid of zones corresponding to a spatial resolution of 20 zones per beam radius. This resolution is the same as in Parameter TABLE 1 PARAMETERS OF RELATIVISTIC JET SIMULATIONS 1 MMI B DH94 C D This Paper v b M b b NOTE. v b is the bulk velocity of the fluid in the beam; M b is the beam Mach number, M v/c s, where c s is the proper sound speed; b is the beam proper Mach number, WM/W s, where W and W s are the Lorentz factors associated with the fluid bulk velocity and the proper sound speed, respectively; is the ratio of the beam proper rest-mass density to the ambient proper rest-mass density; and is the adiabatic exponent. MMI94 and only marginally smaller than the one quoted by DH94 in regions of highest grid refinement. The beam fluid is injected into the grid parallel to the symmetry axis (i.e., the z-axis) through a nozzle at the bottom (r 0) of the left boundary of the grid (z 0), which is 20 zones wide. The time evolution of our model is displayed in Figure 1 (Plate L4) by means of a sequence of snapshots of the logarithm of the proper rest-mass density, the unit of time being R b /c. The typical jet components, i.e., beam, cocoon, and bow shock, are clearly visible. In Figures 1a 1c (t 20.86) a terminal Mach disk of decreasing diameter is present at the end of the beam. This shock is able to decelerate the beam flow efficiently, and creates a stable cocoon around the central beam with a strong backflow starting at the head of the jet. At t (Fig. 1c), the terminal Mach disk has disappeared and the converging conical shock created at the edge of the nozzle has reached the symmetry axis. Because the beam material is less efficiently decelerated in the conical shock than it was in the Mach disk, the working surface of the jet is accelerated and the backflow is reduced. Later on in the evolution (frame d, t 24.24), after the converging conical shock has been reflected off the axis, a new Mach disk is formed at the end of the beam. At this moment the pressure in the hot spot reaches its absolute maximum during the whole evolution. Figures 1e 1f (24.24 t 34.61) are characterized by the growth of vortices caused by Kelvin-Helmholtz instabilities at the interface between the cocoon and the shocked ambient medium. At least seven distinct vortices can be seen with a typical wavelength of 3.2 R b. Vortices closer to the nozzle are larger in size than those close to the head of the jet because they were emitted earlier and consequently had more time to evolve. The propagation of the jet proceeds with the emission of additional vortices at the head of the jet (see Figs. 1g [t 41.55] and Fig. 1h [t 51.83]). Finally, Figures 1i 1l (59.39 t 64.63) cover the process of formation and emission of a small vortex which starts to roll up in the last snapshot. The overall morphology of the jet is dominated by the presence of a broad cocoon mixing with the shocked ambient medium and a prominent internal structure inside the beam (in frame l two conical shocks are visible at 19 R b and 39 R b ). In Figure 2, which shows the flow pattern corresponding to the last snapshot (i.e., frame l), the turbulent character of the flow in the cocoon and the reconfining effect of the conical shock created at the nozzle on the beam are clearly seen. An estimate of the propagation speed of the jet through a uniform medium, V j, can be obtained by considering a onedimensional analog of the two-dimensional flow and assuming a stationary momentum transfer between beam and ambient medium. For a relativistic flow, this estimate (MMI94) leads to r V j v b, (1) 1 v b2 r where r (h b /h a ) (a and b correspond to the ambient medium and the beam, respectively). This expression has the well known nonrelativistic limit (see, e.g., Begelman, Blandford, & Rees 1984) but leads to much larger jet velocities in the case of ultrarelativistic, diffuse models. According to equation (1), the expected velocity for our model is V j 0.69 c, which is 18 times larger than the corresponding classical estimate for the same initial conditions. The mean propagation speed of the jet is v j 0.67 c at the time of the last snapshot when it has propagated a distance of 43.3 R b. This

3 No. 2, 1995 HIGHLY SUPERSONIC RELATIVISTIC JETS L107 FIG. 2. The flow pattern corresponding to frame l of Fig. 1 is shown. The largest velocities are found in the beam near the nozzle. The vortex structure of the cocoon is clearly visible. corresponds to a propagation efficiency, ( v j/v j ), of 97%, which is significantly larger than the propagation efficiency of nonrelativistic jets. For example, Norman, Winkler, & Smarr (1983) found beam efficiencies in the range 49% 90% for their comprehensive set of jets including diffuse and dense models as well as highly supersonic ones. In particular, for the Newtonian counterpart of our model they obtained an efficiency of only 51%. Extremely high propagation efficiencies (between 99% and 101%) were also obtained by MMI94 for their low-beam Mach number models. Figure 3 shows the distribution of the logarithm of the pressure at the final time t Contrary to models with a low-beam Mach number where the beam is in pressure equilibrium with the cocoon/ambient medium (see MMI94 and Martí et al. 1995), we find that the shocked ambient medium has a much larger pressure than the beam. This is more clearly seen in Figure 4, where the logarithm of the pressure normalized to the initial beam pressure is plotted as a function of position along several straight lines perpendicular to the symmetry axis of the jet at the end of the simulation. According to this figure, the originally pressure-matched jet is now embedded into a medium with an overpressure factor of 130. In Martí et al. (1995) it will be demonstrated that the overpressure factor is indeed an increasing function of the beam Lorentz factor in diffuse, high Mach number jets. Although the thermal pressure in the cocoon decreases with time trying to reach the equilibrium with the surrounding medium (Begelman & Cioffi 1989), our results suggest that highly relativistic jets could be confined by strongly overpressured envelopes which are produced by the jets themselves during the early stages of jet formation. The confinement effect of overpressured envelopes on nonrelativistic hypersonic jets was pointed out by Begelman & Ciofi (1989) and numerically explored by Loken et al. (1992). 3. DISCUSSION AND CONCLUSIONS We have demonstrated that both the morphology and dynamics of highly supersonic relativistic jets differs from that of low-beam Mach number models investigated by MMI94 and DH94. Contrary to these models, high-beam Mach number jets have an extended cocoon with distinct vortices and are subject to Kelvin-Helmholtz instabilities. Moreover, they exhibit a remarkable internal structure within the beam produced by conical shocks. The dependence of the morphology and dynamics of the jet on the beam Mach number is similar to the one found in Newtonian simulations; a larger beam Mach number gives rise to an extended cocoon and a distinct internal structure within the beam. Our simulations and the results quoted by MMI94 further show that relativistic jets propagate very fast and efficiently when compared to their Newtonian counterparts. For the jet presented here, the mean propagation velocity is 115 times larger than in the Newtonian case and the propagation efficiency is very close to 100%. This efficiency seems to be an intrinsic property of relativistic jets and reflects the resistance of relativistic beams to be perturbed in their motion. Finally, the dependence of the mean pressure in the cocoon and shocked ambient medium region surrounding the beam on the jet Lorentz factor suggests that ultrarelativistic jets could be confined in the early stages by strongly overpressured envelopes produced by the jets themselves. This result might be of importance for the confinement problem of extragalactic jets. FIG. 3. Contour plot of the logarithm of the thermal pressure corresponding to frame l in Fig. 1

4 L108 MARTI ET AL. Taken as a whole, our results imply the necessity of extending Newtonian studies of the morphology and dynamics of jets to the relativistic regime. This will be the subject of a subsequent paper. The authors are indebted to A. Marquina for providing us with a recent version of his Riemann solver and to P. A. Hughes for his interesting comments. One of the authors (J.M a.m.) would like to thank the Max-Planck-Institut für Astrophysik for the kind hospitality during his stay. This work has been supported by the Human Capital and Mobility Program of the Commission of the European Communities (contract No. ERBCHBICT930496) and by the Spanish DGI- CYT (ref. PB ). FIG. 4. Logarithm of the pressure normalized to the initial beam pressure as a function of position along several lines perpendicular to the jet axis at the end of the simulation (frame l in Fig. 1). Begelman, M. C., Blandford, R. G., & Rees, M. J. 1984, Rev. Mod. Phys., 56, 255 Begelman, M. C., & Ciofi, D. C. 1989, ApJ, 345, L21 Begelman, M. C., Rees, M. J., & Sikora, M. 1994, ApJ, 429, L57 Biretta, J. A., & Owen, F. N. 1990, in Parsec-Scale Radio Jets, ed. J. A. Zensus & T. J. Pearson (Cambridge: Cambridge Univ. Press), 125 Biretta, J. A., Zhou, F., & Owen, F. N. 1995, ApJ, in press Blandford, R. D., & Königl, A. 1979, ApJ, 232, 34 Blandford, R. D., & Rees, M. J. 1974, MNRAS, 169, 395 Colella, P., & Woodward, P. R. 1984, J. Comp. Phys., 54, 174 Davis, R. J., Unwin, S. C., & Muxlow, T. W. B. 1991, Nature, 354, 374 Duncan, G. C., & Hughes, P. A. 1994, ApJ, 436, L119 (DH) Eulderink, F., & Mellema, G. 1995, A&AS, 110, 587 Font, J. A., Ibáñez, J. M a., Marquina, A., & Martí, J. M a. 1994, A&A, 282, 304 Ghisellini, G., Padovani, P., Celotti, A., & Maraschi, L. 1993, ApJ, 407, 65 REFERENCES Krichbaum, T. P., Quirrenbach, A., & Witzel, A. 1992, in Variability of Blazars, ed. E. Valtaoja & M. Valtonen (Cambridge: Cambridge Univ. Press), 331 Loken, C., Burns, J. O., Clarke, D. A., & Norman, M. L. 1992, ApJ, 392, 54 Marquina, A., & Donat, R. 1995, J. Comp. Phys., submitted Martí, J. M a., et al. 1995, in preparation Martí, J. M a., Ibáñez, J. M a., Miralles, J. A. 1991, Phys. Rev. D, 43, 3794 Martí, J. M a.,&müller, E. 1995, J. Comp. Phys., in press Martí, J. M a., Müller, E., & Ibáñez, J. M a. 1994, A&A, 281, L9 (MMI94) Norman, M. L., Smarr, L., Winkler, K.-H. A., & Smith, M. D. 1982, A&A, 113, 285 Norman, M. L., Winkler, K.-H. A., & Smarr, L. 1983, in Astrophysical Jets, ed. A. Ferrari and A. G. Pacholzyk (Dordrecht: Reidel), 227 Shu, C., & Osher, S. 1988, J. Comp. Phys., 77, 439 Witzel, A. 1992, in Physics of Active Galactic Nuclei, ed. W. J. Duschl & S. J. Wagner (Berlin: Springer), 484

5 PLATE L4 FIG. 1. Twelve snapshots of the time evolution of a diffuse, highly supersonic, relativistic jet. The logarithm of the proper rest-mass density is plotted on a gray scale from black ( 2.91) to white (0.78). Note that because of the assumed axial symmetry only the upper half of each frame has been computed (for further explanations, see text). MARTI et al. (see 448, L106)