The origin of peculiar jet-torus structure in the Crab nebula

Mon. Not. R. Astron. Soc. 344, L93 L96 (2003) The origin of peculiar jet-torus structure in the Crab nebula S. S. Komissarov 1 and Y. E. Lyubarsky 2 1 Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT 2 Physics Department, Ben-Gurion University, PO Box 653, Beer-Sheva 84105, Israel Accepted 2003 August 7. Received 2003 July 11; in original form 2003 June 6 ABSTRACT In this Letter, we present the results of two-dimensional axisymmetric relativistic magnetohydrodynamic (MHD) simulations of the interaction between anisotropic pulsar wind and slowly expanding supernova ejecta. They show that the jet-torus pattern recently discovered in the Crab nebula and other pulsar nebulae can be explained within the MHD approximation when the condition of spherical symmetry is no longer enforced. The incorporation of anisotropy consistent with modern theory of pulsar winds results in a highly non-spherical termination shock squeezed along the rotational axis of the pulsar. In fact, the simulations reveal a whole complex of shocks terminating the wind. Downstream of the complex, the outflow proceeds mainly in the equatorial direction, with a typical velocity of 0.6c until it reaches the supernova envelope and gets deflected back into the main body of the nebula. If the global magnetization parameter of the pulsar wind is about σ 0.01 or higher, the magnetic hoop stress halts the outflow in the surface layers of the equatorial disc and prompts formation of pressure-driven, magnetically collimated polar jets. The typical velocity of these transonic jets is about 0.5c. The simulated synchrotron X-ray images of the inner region of the nebulae are subject to strong Doppler beaming and resemble the Chandra data closely. Key words: MHD shock waves pulsars: general ISM: individual: Crab nebula ISM: jets and outflows supernova remnants. 1 INTRODUCTION The Crab nebula is a prototype compact synchrotron nebula continuously powered by ultrarelativistic, magnetized wind from a young, rapidly rotating pulsar. As this nebula is confined within a nonrelativistic surrounding, the wind must terminate at a shock wave. It is the wind plasma heated to relativistic temperatures at this shock that fills the nebula and produces the observed non-thermal electromagnetic emission from the radio to the gamma-ray band. Early spherically symmetric MHD models of the nebula involving weakly magnetized pulsar wind (Rees & Gunn 1974; Kennel & Coroniti 1984) seemed to describe its main properties perfectly well. In particular, an apparent central hole in the nebula brightness distribution (Scargle 1969) was identified with the termination shock in the theoretical models of the nebula. However, recent discoveries of the intriguing jet-torus structure in the Crab nebula and other pulsar nebulae (Hester et al. 1995, 2002; Weisskopf et al. 2000; Gaensler, Pivovaroff & Garmire 2001; Helfand, Gotthelf & Halpern 2001; Pavlov et al. 2001; Gaensler et al. 2002; Lu et al. 2002) prompted calls for re-examination of the theory. The most radical proposals involve abolishing the MHD approximation altogether and the E-mail: serguei@maths.leeds.ac.uk (SSK); lyub@bgumail.bgu.ac.il (YEL) development of purely electromagnetic models (Blandford 2002). However, such a dramatic turn seems to be a bit premature, as the condition of spherical symmetry utilized in the classical MHD models may be simply too restrictive, and the effects of the relaxation of this constraint still remain to be fully investigated. In fact, the available MHD models of the pulsar wind, although not totally self-consistent yet, predict the maximum of the energy flux to be in the equatorial direction (Michel 1973; Bogovalov 1999). Because the termination shock of such a wind is highly nonspherical, this may explain the origin of the X-ray tori (Bogovalov & Khangoulyan 2002; Lyubarsky 2002). The jets of the Crab nebula as well as jets found in other pulsar nebulae cannot be explained so straightforwardly within the pulsar wind model. These jets appear to originate from the pulsar (Weisskopf et al. 2000; Helfand et al. 2001; Pavlov et al. 2001, 2003; Gaensler et al. 2002; Hester et al. 2002), which seems to indicate that they may be formed within the pulsar wind via collimation by magnetic hoop stress. However, a closer look reveals a number of problems with this explanation. First of all, such a collimation is found to be extremely ineffective in ultrarelativistic flows (Beskin, Kuznetsova & Rafikov 1998; Chiueh, Li & Begelman 1998; Bogovalov & Tsinganos 1999; Lyubarsky & Eichler 2001). Moreover, the direct observations of proper motions in the jets of the Crab and Vela nebulae indicate rather moderate velocities of only C 2003 RAS

L94 S. S. Komissarov and Y. E. Lyubarsky 0.3 0.7c (Hester et al. 2002; Pavlov et al. 2003). In order to overcome these problems, Lyubarsky (2002) proposed that the jets are formed downstream of the termination shock, where velocities are no longer ultrarelativistic and the magnetic collimation is much more effective. Because the termination shock is much closer to the pulsar along its rotational axis, this could give the impression of jets being produced by the pulsar itself. The interaction between a relativistic MHD wind from the pulsar and a dense, non-relativistic surrounding is a much more challenging problem in the case of anisotropic wind, and there is not much hope in its analytical solution being found. Fortunately, recent progress in numerical methods for relativistic gas dynamics and MHD (e.g. Marti & Muller 1999; Komissarov 1999) has made it possible to approach this problem numerically. The main results of the first attempt of such a study are described briefly in this Letter. Full details will be presented elsewhere. 2 THE MODEL OF PULSAR WIND IN THE FAR ZONE Although a self-consistent solution to the problem of pulsar wind remains to be found, it is commonly accepted that far away from the pulsar, the wind can be considered as an almost radial supermagnetosonic outflow with purely azimuthal magnetic field and anisotropic angular distribution of the energy flux (Michel 1982; Beskin et al. 1998; Chiueh, Li & Begelman 1998; Bogovalov & Tsinganos 1999). According to the simplified split-monopole models of pulsar magnetospheres (Michel 1973; Bogovalov 1999), the total energy flux density of the wind, f w, satisfies the following equation f w = f 0 r 2 ( sin 2 θ + 1/σ 0 ), (1) where r and θ are the spherical coordinates in a system whose polar axis is aligned with the rotation axis of the pulsar. The first term in the brackets represents the dominating contribution of the electromagnetic energy flux, whereas the second term accounts for the small initial contribution of particles, σ 0 1. The most intriguing result of the classical MHD models of the Crab nebula is a rather low magnetization parameter of the wind, σ, defined as the ratio of the Poynting flux to the kinetic energy flux, just upstream of the termination shock, σ few 10 3 (Rees & Gunn 1974; Kennel & Coroniti 1984; Emmering & Chevalier 1987). Gallant & Arons (1994) came to the same conclusion independently in their shock-structure model for the time-variable features observed in the shock region ( wisps ). This seems to indicate that most of the Poynting flux is somehow transferred to particles along the way to the termination shock (however, see Begelman 1998). Although the problem of energy conversion remains a subject of intensive debate (e.g. Melatos 2002), a number of mechanisms have been proposed in recent years. Because the pulsar magnetic axis is inclined with respect to its rotational axis, a significant fraction of the Poynting flux is carried out by the component of electromagnetic field oscillating with the rotational period of the pulsar. These small-scale waves can decay via various dissipation processes (Lyubarsky & Kirk 2001; Melatos 2002; Lyubarsky 2003a; Kirk & Skjæraasen 2003). Even if the dissipation time-scale is larger than the time of travel from the pulsar to the termination shock, these waves decay rapidly at the termination shock. In this case, the post-shock parameters are still the same as if the alternating fields had already annihilated in the upstream flow (Lyubarsky 2003b). For these reasons, we assume in our model of the pulsar wind that all these waves have already decayed and have transferred their energy to particles. The rest of the Poynting flux is transported by the large-scale azimuthal magnetic field. The exact latitudinal distribution of this field remains to be found, but it must vanish both along the rotational axis as any axisymmetric magnetic field does and in the equatorial plane, because the average magnetic field in the obliquely rotating magnetosphere is zero at the equator. We take the residual magnetic field in the wind in the form ( 4π f 0 ξ B = c r sin θ 1 2θ ). (2) π The free parameter ξ 1 controls the wind magnetization. From equations (1) and (2), it follows that the local value of σ depends on the polar angle, gradually increasing from σ = 0atθ = π/2 to σ ξ 2 at π/2 θ 1/ σ 0 and then quickly falling back to zero at θ = 0. The global value, defined as the ratio of total fluxes, turns out to be σ 0.1ξ 2. 3 NUMERICAL SIMULATIONS To carry out these two-dimensional axisymmetric simulations, we used the Godunov-type scheme for relativistic magnetohydrodynamics constructed by Komissarov (1999). We utilize a spherical grid with the symmetry boundary in the equatorial plane and the same angular size of cells, θ, throughout the computational domain. The radial cell size, r i = r i θ, grows like r. As the Currant time-step stability condition requires t < r i /c, we split the computational domain into a set of rings such that the outer radius of each ring is twice its inner radius, and advanced the solution separately for each kth ring with its own time-step, t k, such that t k+1 = 2 t k. The initial solution involves two distinctly different regions separated by r 0 = 2. For r > r 0, it describes uniform cold dense unmagnetized plasma expanding radially with velocity of 5000 km s 1. This accounts for the supernova ejecta in the phase of free expansion. For r < r 0, the initial solution describes a radial ultrarelativistic wind with the energy flux density given by equation (1) with σ 0 = 100, and the magnetic field described by equation (2). Provided the wind is ultrarelativistic, the MHD solution is not sensitive to the exact value of the wind Lorentz factor, γ w, and the plasma composition. In these simulations, we set γ w = 10 (much higher Lorentz factors are rather problematic for our numerical technique) and utilize the polytropic equation of state, e = ρ + [γ/(γ 1)]p, with γ = 4/3. Gallant & Arons (1994) have shown that in the case of iondominated winds, the ion Larmor radius may not be small compared with the macroscopic length scale of this problem. Obviously, our purely MHD model is based on the opposite assumption and cannot capture the fine structure of the wind termination shock discussed in Gallant & Arons (1994). At present, we also ignore the line emitting filaments observed within the volume of the Crab nebula which may play an important role in the nebula dynamics. Standard dimensional analysis shows that when the typical size of the nebula becomes much larger than r 0, the solution should begin to evolve in a self-similar fashion (e.g. Emmering & Chevalier 1987). Fig. 1 (opposite p. L96) shows the overall structure of the flow for the model with ξ = 0.3, and Figs 2 and 3 (opposite p. L96) show its central part at time t = 190. (Notice that we use such units that the speed of light c = 1.) At this point, the typical radius of the nebula is about three times higher than r 0, and the solution seems to have approached the self-similar regime (by this time, the magnetosonic signal have travelled back and forward across the nebula many times). One can see that instead of a single termination shock, the numerical solution displays a whole complex of shocks. The

Jet-torus of Crab nebula L95 equatorial, weakly magnetized part of the pulsar wind terminates at an almost cylindrical shock crossing the equator, the Mach belt. At higher latitudes, the flow passes first through a highly oblique arch shock. Downstream of this shock, the flow remains supermagnetosonic with the typical velocity of 0.8 0.9c. Then it passes through another shock, the rim shock, that originates from the edge of the Mach belt. In contrast to the spherically symmetric expansion assumed in the current theories of pulsar nebulae, most of the downstream flow is confined to the equatorial plane. Its typical velocity, v 0.6c, agrees with the measurements of proper motions in the torus of the Crab nebula (Hester et al. 2002). This equatorial outflow is eventually pushed back by the pressure force and the magnetic hoop stress and forms a backflow. When this backflow reaches the central region of the nebula, it forms large-scale vortices at intermediate latitudes. Another backflow is launched from the surface of the equatorial disc as the result of strong magnetic braking. Immediately after the termination shock, the magnetic pressure is small compared to the gas pressure. However, as the shocked plasma moves away from the axis, the ratio of magnetic pressure to the gas pressure grows rapidly until it reaches the value of 2 3. At this point, the magnetic stress becomes an important dynamical factor. As the termination shock is highly non-spherical, this transition occurs much closer to the origin for the wind plasma passing through the shock at smaller polar angles and then forming the surface layers of the equatorial disc. Higher wind magnetization at smaller polar angles (see Section 2) amplifies this effect even further. In fact, for ξ 0.3 the magnetic hoop stress in the surface layers becomes large so soon that it halts the outflow and pushes it back towards the axis already within 1 2 equatorial radii of the termination shock. This leads to a significant increase of total pressure near the cusp region of the reconfinement shock that drives the jet-like magnetically collimated outflow along the polar axis. The typical velocity of these polar jets is v j 0.5c, which is in good agreement with observed velocities of the jets of the Crab and Vela nebulae (Hester et al. 2002; Pavlov et al. 2003). Similar jets are produced in the model with ξ = 0.5 but not found in the solution with ξ = 0.2. Thus, the polar jets are produced if the global σ 0.01, which is somewhat higher than the previous estimates for the Crab nebula based on simplified theoretical models (Kennel & Coroniti 1984; Emmering & Chevalier 1987; Begelman & Li 1992). Fig. 4 (opposite p. L96) shows the simulated synchrotron X-ray emission from the inner part of the nebula obtained for the same orientation of the wind relative to the observer as in the Crab nebula. In order to create this image, we assumed that synchrotron electrons (and positrons) with a power law energy spectrum, n(e) = n 0 E 2.2, are injected at the termination shock and then suffer synchrotron energy losses at a constant rate determined by the typical value of magnetic field in the numerical solution. Because the magnetic field in our simulations is purely azimuthal, it vanishes on the symmetry axis, and so does the synchrotron emissivity. In a real jet, the strong velocity shear would generate the poloidal component of the magnetic field, which does not have to vanish on the axis. To take this into account, we added, only inside the jet, the poloidal field aligned with the flow velocity at the level of 30 per cent of the gas pressure. 4 DISCUSSION The obtained image is excitingly similar to the X-ray image of the Crab nebula (Weisskopf et al. 2000). In addition to the polar jets, one can see a system of rings which makes an impression of a disc-like or even a toroidal structure. Well in agreement with the observations, these rings are brighter on the counter-jet side, which is entirely due to the relativistic beaming effect. The bright spot within the inner ring may correspond to the so-called sprite of the Crab nebula (Hester et al. 2002) or even to the mysterious knot 1 discovered by Hester et al. (1995) near the Crab pulsar. This emission originates in the high-velocity plasma flowing just above the arch shock, where its velocity vector points directly towards the observer. On the whole, our model captures the main properties of the Crab nebula remarkably well. There are, however, some qualitative and quantitative differences which demand further investigation. The jet of the Crab nebula is not so straight and well collimated and it cannot be traced that far away from the pulsar. It bends, spreads and eventually merges into the surrounding plasma (Weisskopf et al. 2000). The most likely reason for such a behaviour is the development of the kink instability (Begelman 1998), which is suppressed in our simulations by the condition of axisymmetry. Full three-dimensional simulations are needed to overcome this restriction. Some small-scale features of the Crab nebula, like its inner ring (Weisskopf et al. 2000) and wisps (Hester et al. 2002) are not seen in our images. It is possible that at least some of them are attributed to non-mhd effects, like the fine structure in an ion-dominated shock (Gallant & Arons 1994; Gaensler et al. 2002). On the other hand, we should stress that even within the scope of MHD approximation, the small-scale structure of the flow may be significantly affected by three-dimensional effects. The brightness contrast between the jet side and the counter-jet side of the rings in the simulated maps is too high. Since this asymmetry is entirely due to the relativistic beaming, it strongly depends on the velocity field very close to the termination shock and hardly depends on anything else. As the main factor determining this velocity field is the angular structure of the pulsar wind, the brightness asymmetry of the torus of the Crab nebula imposes strong observational constraints on the pulsar wind models. Future simulations should be able to determine the model parameters providing the best fit to the observational data. ACKNOWLEDGMENTS YEL acknowledges support from the Arnow Chair of Physics. REFERENCES Begelman M. C., 1998, ApJ, 493, 291 Begelman M. C., Li Z.-Y., 1992, ApJ, 397, 187 Beskin V. S., Kuznetsova I. V., Rafikov R. R., 1998, MNRAS, 299, 341 Blandford R. D., 2002, in Gilfanov M., Sunyaev R., Churazov E., Proc. MPA/ESO/MPE/USM Joint Astron. Conf., Lighthouses of the Universe: the Most Luminous Celestial Objects and Their Use for Cosmology. ESO, Garching Bogovalov S. 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Opposite p. L96, MNRAS, 344 Figure 1. The ratio of magnetic pressure to the gas pressure and velocity field for the model with the magnetization parameter ξ = 0.3. The equatorial outflow is seen as the region of a particularly high magnetic pressure. Figure 3. The velocity magnitude, represented in colour, and the flow direction, represented by arrows, in the central part of the solution shown in Fig. 1. Just above the equatorial outflow, a layer of backflow can be seen converging towards the symmetry axis. This backflow provides plasma for two transonic jets propagating in the vertical direction. Figure 2. The gas pressure distribution in the central part of the solution shown in Fig. 1. The white circle in the centre shows the inner boundary of the computational domain, whereas the adjacent region of dark blue colour shows the pulsar wind zone. The Mach belt shock runs across the equatorial plane at the distance of 0.8 from the symmetry axis; the arch shock is seen as the upper and the lower boundaries of the wind zone; and the rim shocks originate from the points of intersection of the Mach belt and the arch shock. Figure 4. Synchrotron X-ray image for the solution shown in Figs 1 3. The nebula is tilted to the plane of the sky at the angle of 30, just like the Crab nebula. The brightness distribution is shown in the logarithmic scale.