THE STRUCTURE OF THE M87 JET: A TRANSITION FROM PARABOLIC TO CONICAL STREAMLINES

We conducted the European VLBI Network (EVN) observations toward M87 on 2009 March 7 at a wavelength of 18 cm with the Cambridge (UK), Effelsberg (Germany), Jodrell Bank (UK), Knockin (UK), Medicina (Italy), Noto (Italy), Onsala (Sweden), Torun (Poland), and Westerbork (Netherlands) stations. Both left and right circular polarization data were recorded at each telescope using eight channels of 8 MHz bandwidth and 2 bit sampling. The data were correlated at the Joint Institute for VLBI in Europe correlator.

A priori amplitude calibration for each station was derived from a measurement of the system temperatures during each run and the antenna gain. Fringe fitting was performed using AIPS. After results of fringe fitting were applied, the data were averaged over 12 s in each intermediate frequency (IF) and self-calibrated using Difmap.

We conducted the MERLIN observations on 2007 March 9 at a wavelength of 18 cm with the Defford, Cambridge, Knockin, Darnhall, Jodrell Bank, and Tabley stations. Both left and right circular polarization data were transferred from each telescope with a 15 MHz bandwidth.

Initially, the data were pipelined, and a channel-by-channel flux scale derived from 3C 286 observation was applied. After this a priori amplitude calibration, we self-calibrated the data using Difmap.

We also analyzed archival VLBA data of the M87 jet (BK073). Those observations were carried out on 2000 January 22 at 15 GHz with all ten stations of the VLBA and one station of the VLA. Four IFs, each with an 8 MHz bandwidth, were recorded at each telescope. Data reduction was conducted in the same manner as for the EVN observations. We reproduced an M87 image, which is consistent with that published in Kovalev et al. (2007).

We consider a conical streamline (z∝r^a, a = 1) for supersonic jets. In an adiabatic jet, the internal pressure p_jet decreases with the axial distance z as z^−2Γ (Γ: the ratio of specific heats). So, we speculate that the constant expansion of the jet radius and a conical structure downstream of HST-1 to knot A in the M87 jet require the same axial gradient for the external interstellar medium (ISM) pressure, p_ism, as p_jet (Owen et al. 1989). If p_ism decreases slower than p_jet, then p_ism > p_jet at some distance so that a recollimation shock will be triggered (Sanders 1983). The self-similar solution of a conical streamline for the magnetized case (with a purely toroidal field component) requires p_ism∝z^−b, b = 4 (Zakamska et al. 2008). For a general (non-self-similar) case, b > 2 is allowed in analytical and numerical models (Tchekhovskoy et al. 2008; Lyubarsky 2009; Komissarov et al. 2009). X-ray observations reveal the ISM properties such as the Bondi radius r_B ∼ 250 pc and the King core radius r_c ≃ 1.4 kpc (see, e.g., Young et al. 2002; Di Matteo et al. 2003; Allen et al. 2006). Thus, the region of conical streamlines in the M87 jet lies between r_B and the marginal radius for the power-law decay beyond r_c in the King profile, indicating that the ISM distribution is essentially uniform. We thus rule out that the structure downstream of HST-1 is hydrostatically confined by p_ism in order to conform to a conical streamline.

In order to possess a conical streamline without any overcollimation between knots HST-1 and A, the condition p_jet ≳ p_ism should be maintained. Knots HST-1 to A do appear to be overpressured with respect to the external pressure (Owen et al. 1989). However, the p_jet of the inter-knot regions, as estimated by the minimum energy argument for the VLA data (Sparks et al. 1996), appears underpressured with respect to p_ism, as estimated by the recent X-ray observations (Young et al. 2002; Rafferty et al. 2006). One possibility is an underestimation of the magnetic field strength when the toroidal (azimuthal) components are not considered (cf. Owen et al. 1989). It has been further suggested that the magnetic field energy is at least in equipartition (or even larger with a factor of 1–2) with the energy of the radiating ultrarelativistic electrons (Stawarz et al. 2005).

We note that overpressured knots do appear to have trails of stationary recollimation shocks in purely hydrodynamic jets. Falle & Wilson (1985) performed hydrodynamic simulations to apply stationary recollimation shocks to the observed knots at VLA scales under the assumption of the shallow ISM gradient (p_ism∝z⁻¹). Stationary features, however, are in conflict with the observed large proper motions (Biretta et al. 1995, 1999), while the ISM also does not appear to have such a gradient. Therefore, we suggest that the highly magnetized nature of the jet may be responsible for the conical part of the M87 jet.

We next consider a parabolic streamline (1 < a ⩽ 2) for supersonic jets. It is shown that the magnetized jet can be parabolic in analytical and numerical models where the ISM pressure is decreasing as p_ism∝z^−b, b = 2 (Tchekhovskoy et al. 2008; Komissarov et al. 2009). A self-similar solution also exists for non-magnetized cases with a pure parabolic streamline (a = 2) under the same p_ism dependence with b = 2 (Zakamska et al. 2008). Their solution indicates p_jet ≲ p_ism with pressure increasing near the jet edge. So far, no analytical or numerical solutions, except a = 2, have been derived in the non-magnetized cases. We note that even in a hydrodynamic recollimation shock model, the jet boundary ("shocked zone" at the interface of the jet and the ISM) is expected to be parabolic. However, the ISM is required to be uniform (p_ism ∼ const.; Komissarov & Falle 1997; Nalewajko & Sikora 2009).

In order to have parabolic streamlines, b ∼ 2 is required in magnetized jets. It is important to examine the distributions of p_ism. A recent analytical model for a giant advection-dominated accretion flow (Narayan & Fabian 2011) shows b ∼ 2.3 from beyond r_B down to the SMBH, and it is universally derived for slow rotation of the ISM (≲30% of the Keplerian speed) with the wider range of the α-viscosity parameter (0.001–0.3; Shakura & Sunyaev 1973). A spherically symmetric accretion with MHD turbulence (Shcherbakov 2008) produces b ∼ 2.1 without a strong dependence on the parameters and the equation of state. However, the spatial resolution in current X-ray observations is not high enough to resolve this region.

The radius of HST-1 appears to be smaller than the size estimated by the two power-law streamlines. The maximum measured radius of the HST-1 region is 2200 r_s ± 300 r_s and the mean value is 1300 r_s ± 150 r_s, while the expected radii are 3000 r_s ± 400 r_s for the parabolic case and 4100 r_s ± 200 r_s for the conical case, respectively. One possibility is that the jet is subjected to overcollimation toward HST-1.

As is seen in Figure 2, HST-1 is located downstream of the Bondi radius (r_B), where the jet changes from a parabolic to a conical shape. This transition is presumably caused by the different profiles of the external pressure (changing from steep to shallow gradients). The distribution of the external gas follows the influence of gravity from the central SMBH inside of r_B, while it can be flattened outside of r_B. The field strength $|{\bm B}|$ at HST-1 is 0.5–20 mG as estimated by the X-ray variability (Perlman et al. 2003; Harris et al. 2009). Supposing $|{\bm B}| \sim$ 1 mG for our reference, the magnetic pressure is expected to be about 100 times larger (i.e., overpressured) than the ISM value p_ism ∼ 4.3 × 10⁻¹⁰ dyn cm⁻² (Allen et al. 2006; n_e = 0.17 cm⁻³ and kT = 0.8 KeV).

Another interesting objective is to understand the production of the superluminal components (Biretta et al. 1999; Cheung et al. 2007). The observed motions are modeled by the quad relativistic shocks in the MHD helical jet (Nakamura et al. 2010). Recent optical polarimetry requires a coherent helical field to explain the distribution of polarization vectors (perpendicular to the jet; Perlman et al. 2003) and variations of the radio electric vector positional angles as well as the fractional polarization degree (Chen et al. 2011), indicating either a helical distortion to the jet or a shock propagating through a helical jet (Perlman et al. 2011).

An interpretation for the origin of HST-1 as a hydrodynamic recollimation shock has been proposed by Stawarz et al. (2006, hereafter S06) and Bromberg & Levinson (2009, hereafter BL09). S06 do not explain the observed gradual collimation and are inconsistent with our observational result. They assume a freely expanding (i.e., a conical) jet which is already in the particle-dominated regime upstream of HST-1. S06 suggest a shallow external pressure profile p_ism∝z^−0.6 inside some critical radius of 3'' (≈234 pc), as determined by a stellar cusp (a break in the stellar surface brightness of the M87 host galaxy; Lauer et al. 1992). BL09 also model the recollimation shock at HST-1 with the same pressure profile.

In summary, we propose that a transition from parabolic to conical streamlines occurs via a strong compression due to the change in the slope of the axial profile of p_ism. This condition can be achieved by an imbalance between p_jet and p_ism in the radial direction due to the overcollimation. It is remarkable that recent general relativistic MHD simulations (McKinney 2006) reproduce the observed transition from parabolic (a ∼ 1.7) to conical (a ∼ 1.0) streamlines. However, the scale of the transition does not match with the observations. We suggest that the properties of MHD jets should be considered in a realistic ISM environment.

Recent VLBA observations indicate that the VLBI core at 43 GHz (core₄₃) is located at 20 r_s from the position of the SMBH for the viewing angle of 14° (Hada et al. 2011). With the VLBA image, the size of core₄₃ can be measured to be 17 ± 4 r_s at a position angle of 68°, which is approximately perpendicular to the jet axis. If we assume that the core is the innermost part of the jet emission, the radius of the jet is 9 ± 2 r_s. This is in good agreement with the extrapolated radius of the jet of 9 ± 2 r_s assuming parabolic streamlines in the inner region. If this is the case, the jet propagates with a single power-law streamline over more than four orders of magnitude of distance.

The jet opening angle is one of the key parameters in the MHD jet acceleration mechanism. Blandford & Payne (1982, hereafter BP82) suggest that the jet is initiated by the magneto-centrifugal force if the jet inclination angle between the penetrating magnetic line of force and the accretion disk is smaller than 60°, which corresponds to a jet opening angle larger than 60°. If we assume that core₄₃ is the innermost part of jet emission, the jet opening angle would be estimated to be 46° at 20 r_s. Therefore, core₄₃ would be located downstream of the Alfvén surface where the magnetic pressure gradient force plays a role. Thus, future submillimeter VLBI projects can access initial acceleration mechanisms by direct imaging and measure the deviation from parabolic streamlines upstream of core₄₃.

Furthermore, as we show in Figure 2, the parabolic jet can be extrapolated to the vicinity of the SMBH. At the "axial" distance of z = 3.4 r_s from the SMBH, it intersects the theoretically expected inner stable circular orbit (ISCO) of the accretion disk for a Schwarzschild black hole. If the jet radius departs from the extrapolation line well beyond 3.4 r_s, it will indicate that the jet originates in the accretion disk (BP82). If the streamline is as extrapolated down to 3.4 r_s, the jet must originate from the accretion disk (BP82) around a spinning black hole and/or the ergosphere of the spinning black hole (Blandford & Znajek 1977, hereafter BZ77). If we see no departure down to 0.5 r_s from the SMBH, it indicates that the jet originates from the ergosphere of the spinning black hole (BZ77). Note the above discussion for prograde spinning black holes, while the upper limit of the ISCO is 4.5 r_s for retrograde spinning black holes. Therefore, tracing back the ridge line and identifying the ejection point of the jet is one of the crucial tests to an evaluation of the formation mechanism of the jet (BP82 and/or BZ77). More fundamentally, it provides an important method for detection of the spin of the SMBH⁵ and evaluation of the general theory of relativity. Such conjectures will be explored by direct imaging with future submillimeter VLBI experiments.