The limb-brightened jet of M 87 down to 7 Schwarzschild radii scale

In Fig. 1 we show the $(u,v)$-coverage of the GMVA observations in 2015. The inclusion of the large and sensitive telescopes PV, PB, and GBT improved the data quality and gave robust fringe detections even on the longest baselines (up to $\sim 3$G$\lambda$). In Fig. 2 we show a representative plot of the radial dependence of the visibility amplitudes versus the $uv$-distance. The visibilities drop from $\sim 1$ Jy at short uv-spacings to $\sim 50-100$ mJy at the longest $uv$-distances (e.g., 3G$\lambda$, PV and PdB to MK).

Figure 3 shows the 86 GHz images of the inner jet region of M87 between 2004 and 2015 (forward in time from top to bottom). The images show the basic source structure (core-jet morphology with limb-brightening) is similar in all epochs (though the data and image quality differ between observations). In addition to the prominent east-west oriented jet, faint emission east of the bright VLBI core is visible. This feature may be the counter-jet which is also visible at longer wavelengths (Kovalev et al. 2007; Walker et al. 2016). The validity of our counter-jet detection at 86 GHz was tested by applying different phase and amplitude self-calibration schemes which do not allow removal of this feature from the final image. We also point out that fine-scale structure of the jet and the positions of several bright emission features change with time. We note that our sparse time-sampling does not allow us to robustly cross-identify these bright components across different epochs, which is not unexpected in view of the complex nature of the kinematics in the inner jet (Mertens et al. 2016). We note that for the two-days long experiment in 2009 no significant signature of flux variability or motion in the jet was detected over a timescale of two days. We therefore combined the two-days long experiment into a single visibility data set and made a single image (Figure 3c).

In order to measure the properties of the VLBI core, we fitted elliptical Gaussian components to the fully calibrated visibilities using the Modelfit task in the Difmap package. The model-fit provided us the core flux $S_{\rm core}$, the FWHM size $\psi_{\rm maj,min}$ along the major and minor axes of the ellipse, and the position angle of the major axis $\rm PA_{\rm core}$ for each epoch. To estimate uncertainties in the model-fit parameters we followed Schinzel et al. (2012) in which the authors accounted for the effects of the finite signal-to-noise and strong side-lobe interference. The ratio of the peak to the rms noise near the core is $\gtrsim 70$ in all epochs. For this signal-to-noise, the errors of our model-fit parameters are approximately $\sim$15% and $\sim 20$% for the flux density and the core size, respectively.

The model-fit parameters for the VLBI core are shown in Tab. 2. In all cases, the FWHM size of the core is larger than 64% and 23% of the beam along its minor and major axis, respectively. These sizes are larger than the empirical resolution limit of 1/5th of the beam. If we take the signal-to-noise of $\sim 70$ into account, model-fit components whose sizes are larger than $\sim$10% of the beam can be claimed to be spatially resolved (Lobanov 2005; Schinzel et al. 2012). Thus we conclude that the core of M 87 is spatially resolved (See also Baczko et al. 2016 for a more detailed discussion about determination of the resolution limit of a typical GMVA data set). The flux density of the VLBI core is in the range of $\sim(0.53-0.67)$ Jy for all observations except 2009, when a significantly higher (factor 2) core flux was seen. We checked the amplitude calibration using 3C 273, which was observed in alternate VLBI scans in the same experiment and found no evidence for a systematic amplitude mis-calibration. We therefore regard the elevated flux density as intrinsic to the source. Details regarding this particular epoch will be studied in a future publication.

Owing to the sparse time sampling of our observations, we focus on an approach to average all the images in time (image stacking), in order to increase the image sensitivity and study the time-averaged emission, which characterizes the overall shape of the jet launching region. Such stacking analysis has also been adopted in other studies, e.g., in the analysis of the jet structure of AGN at 15 GHz and other frequencies (e.g. Fromm et al. 2013; MacDonald et al. 2015; Boccardi et al. 2016b; Pushkarev et al. 2017).

For the stacking, the individual jet images (Fig. 3) were restored with a common beam. The restored maps were aligned by the positions of their intensity peaks. The assumption of a stable core position is supported by astrometric VLBA 43 GHz observations of M 87 over multiple epochs (Acciari et al. 2010), which revealed a stationary peak position on scales of $6\leavevmode\nobreak\ R_{\rm sch}$. We performed the stacking procedure with two different beam sizes: (i) with a larger beam size of $0.1\times 0.3$ mas in order to recover the faint jet on $1-4$ mas core separation, and (ii) with a smaller beam of 0.051$\times$0.123 mas, in order to reveal the fine scale structure on $<1$ mas. The latter beam corresponds to 50% super-resolution in N-S direction for uniformly weighted data.

In the top panel of Fig. 4 we show the stacked image with the larger beam (Fig. 4a) and below with the smaller beam (Fig. 4b). The 1$\sigma$ rms level in the higher-resolution stacked image is $\sim 0.1$ mJy/beam, making it the deepest and highest resolution view of the M87 jet structure to date.

We measure the transverse width of the jet with cuts perpendicular to the mean jet axis. In Figure 4c we highlight the transverse intensity profiles in two cuts made at $\sim 0.6$ mas and $\sim 0.8$ mas core separations. The transverse jet profiles appear limb-brightened, with an additional signature of a fainter central emission component that is visible above the 5$\sigma$ image noise level. In order to characterize the brightness of this central component, we calculated the center-to-limb brightness ratio $\rho_{\rm CL}$. We integrated the fluxes in the central lane and the two limbs over $(0.45-0.95)$ mas core distance and we obtained $\sim 38$ mJy and $\sim 15$ mJy for each limb and the center, respectively. Accordingly, we find $\rho_{\rm CL}\sim 0.4\pm 0.1$. For this calculation we assumed 15% for the flux density uncertainty for the limb but 30% for the fainter central lane. As an illustration, we show the averaged transverse intensity profile in Fig. 5.

In order to analyze the jet base structure using the ridge lines, the following procedure was performed. We took the high-resolution jet image in Fig. 4.b and rotated the image by $21^{\circ}$ in the clockwise direction assuming that the overall jet position angle $PA_{\rm c}$ is $-69^{\circ}$ with respect to north. At each core distance $d$, we made slices transverse to the jet and fitted two Gaussians to the two humps in the transverse intensity profile. We were often forced to fit another Gaussian along the jet center axis to improve the fit quality. We began this fitting procedure at a core distance $d\sim 1$ mas and continued fitting the transverse profile down to $d\sim 0.06$ mas. We followed Mertens et al. (2016) in computing the uncertainties in the ridge line analysis. The diameter of the jet $W(d)$ has been determined by the distance between the positions of the Gaussian peaks at each $d$. Similarly, we also derived the apparent jet opening angle $\phi_{\rm app}(d)$ by computing the angle subtended by the two Gaussian peaks. The intrinsic opening angle $\phi_{\rm int}$ is then given by $\phi_{\rm int}=2\arctan(\sin\theta\times\tan(\phi_{\rm app}/2))$ (Pushkarev et al. 2017) under the assumption that the extended jet is azimuthally symmetric.

Figure 6 shows the results of the ridge line analysis. The jet diameter at $d<1$ mas apparently increases with core separation. The overall jet diameter at $d>0.3$ mas is $W\sim 0.3-0.7$ mas. At smaller core separations ($d<0.2\leavevmode\nobreak\$mas, 90$R_{\rm sch}$ projected), the collimation profile slightly flattens and the jet diameter does not sharply decrease. At the jet base ($d=0.06$ mas), the jet width is $0.29\pm 0.09$ mas ($40.8\pm 12.3\leavevmode\nobreak\ R_{\rm sch}$) with an apparent opening angle of $\phi_{\rm app}=127^{\circ}\pm 22^{\circ}$. Depending on the jet viewing angle, the intrinsic jet opening angle is $\phi_{\rm int}=63.6^{\circ}\pm 25.0^{\circ}$ ($90^{\circ}\pm 28^{\circ}$) for a jet viewing angle $\theta=18^{\circ}$ (30^∘), respectively.

Asymptotic structure in the measured jet width $W$ versus the distance from the central engine $z$ was fit with a power-law model $W(z)\propto z^{k}$ where $k$ is a dimensionless index which parameterizes the jet expansion and acceleration within theoretical models (e.g. Komissarov et al. 2007; Lyubarsky 2009). It is important to note that the core separation $d$ is not necessarily the same as the the distance from the central engine $z$ because of the jet opacity (e.g. Lobanov 1998). Hence, we associate the core separation $d$ to the distance from the central engine by $z=\epsilon+d$ where $\epsilon$ is the unknown offset between the BH and the 86 GHz core (see Fig. 7). We adopted $\epsilon\leq 41\leavevmode\nobreak\ \mu$as based on the results of Hada et al. (2011), where the authors performed astrometric VLBA observations toward M 87 at $2.3-43.2$ GHz and estimated the distance between the intensity peak and the jet apex at 43 GHz. Then we obtained $W$ as function of $z$ and the power-law model was fit to $W(z)$.

We find a jet expansion rate of $k=0.469\pm 0.019$ when we ignore the core-shift at 86 GHz (i.e., $\epsilon=0$). With a non-zero core-shift correction ($\epsilon\neq 0$), we find $k\sim 0.47-0.51$ with a mean value of $k=0.498\pm 0.025$ (the error represents uncertainties in both the core position and the statistical fitting). Both fits have a reduced chi square of $\chi^{2}_{\rm red}\sim 0.51$. This is in agreement with previous values of $k=0.56-0.60$ (Asada & Nakamura 2012; Hada et al. 2013; Mertens et al. 2016) within 3$\sigma$ uncertainty levels. To demonstrate the goodness of fit, we calculated the fractional difference between the model and the data, $\Delta W/W=(W_{\rm obs}/W_{\rm fit}-1)$ where $W_{\rm obs}$ and $W_{\rm fit}$ are the observed and model jet widths, respectively. The results are displayed in the bottom panel of Fig. 6. It can be seen that the fractional difference is nearly zero, but starts to grow within $d\leq 0.2$ mas.

We measured the integrated jet and counter-jet flux density at core separations of $0.2-0.5$ mas (projected distances of $\sim 28-140\leavevmode\nobreak\ R_{\rm sch}$) using the jet image in Fig. 4a. For the main jet, we placed a large box covering all three elements of the approaching jet (i.e., the northern limb, the southern limb, and the central emission). Another box of the same size was placed in the counter-jet region. The measured flux density of the jet and the counter jet were $\sim$95 mJy and $\sim 3.5$ mJy, respectively. Accordingly, we obtained the jet to counter-jet ratio $R=27.1\pm 9.1$ (assuming 15% and 30% of flux uncertainties for the approaching and the receding jets, respectively). We note observational evidence for limb-brightening in the counter-jet at 43 GHz (Walker et al. 2016). This is not seen in our data (possibly) due to sensitivity limitations and the higher frequency. Therefore, our counter-jet flux could be under-estimated. In order to correct for this, we have measured the integrated flux of only the southern limb of the approaching jet at the same core distances assuming that we see only the northern edge of the limb-brightened counter-jet. The integrated flux of the southern-limb of the approaching jet was reduced to $\sim 60$ mJy. This lowers the jet to counter-jet ratio to $R=17\pm 6$.

We also measured the jet to counter-jet ratio variation in the longitudinal direction using individual pixel values. For this, we cut the jet longitudinally through the counter-jet, the core, and the southern limb of the approaching jet. We took the jet image in Fig. 4a to obtain a smoother jet intensity gradient. For a more reliable measurement, we used only pixel values over the 7$\sigma$ level (0.77 mJy/beam). When calculating the jet to counter-jet ratio as function of the distance from the central engine, using the same jet apex to core offset $\epsilon$ may affect our measurements (see Fig. 7 for illustration). Therefore, we adopted the same $\epsilon\leq 41\,\mu$as in a direction east of the VLBI core (highlighted in gray in Fig. 7; see Sect. 3.5 for details) and calculated the brightness ratio at the corresponding jet distance. We accounted for thermal noise, systematic amplitude error, and the dispersion in the $R$ profile at each distance $z$ due to the uncertainty in the central engine position. We further excluded $R$ values measured within 0.1 mas of the central engine because of the relatively large beam size in the image presented in Fig. 4a. The result is shown in Fig. 8. The $R$ measured within $\sim 0.2$ mas from the central engine is largely affected by the positional uncertainty of the 86 GHz VLBI core, $\epsilon$, causing $R$ to vary by a factor $\sim 5$. Nevertheless, we find that a single constant value of the jet-to-counter jet ratio is not suitable in describing the relatively large variation of $R$ ($R=1-10$ near 0.1 mas and $R=10-25$ at larger distances).

The VLBI core brightness temperature parameterizes the physical conditions within compact energetic jet components (e.g., Kovalev et al. 2005; Lee et al. 2016). We calculate the intrinsic brightness temperature of the core in the source rest frame, $T_{\rm B}$, by

(Lee et al. 2016) where $\nu$ is the observing frequency in GHz, $S_{\rm core}$ is in Jy, $\psi_{\rm maj}$ and $\psi_{\rm min}$ are in mas, $z=0.00436$ is the redshift of M 87 (Smith et al. 2000), and $\delta$ is the Doppler factor. The observed apparent brightness temperature, $T_{\rm B,app}=T_{\rm B}\times\delta$, is shown in Tab. 2. We find that the $T_{\rm B,app}$ is generally quite low. Remarkably, the $T_{\rm B,app}$ is always lower than $T_{\rm eq}\sim(5\times 10^{10})\leavevmode\nobreak\ K$, which is often referred to as the brightness temperature of a plasma in which the energy density of the magnetic field $u_{\rm B}$ is equal to that of the particles $u_{\rm p}$ (i.e., the “equipartition brightness temperature”; Readhead 1994). The intrinsic brightness temperature will be even lower if Doppler boosting is accounted for.

In order to examine whether the equipartition brightness temperature for the non-thermal electrons in the jet of M 87 is higher or lower than $5\times 10^{10}$ K, we explicitly calculate the equipartition brightness temperature in the following manner: We follow the analysis presented by Singal (2009). Assuming equal energy density for particles $u_{p}$ and magnetic field $u_{B}$ in the jet frame (i.e. without Doppler boosting), we can rewrite Eq. 3 of Singal (2009) to express $T_{\rm B,eq}$ in the source frame by

where $\alpha$ is the spectral index ($S\propto\nu^{+\alpha}$; note that the sign convention is different from Singal 2009), $t(\alpha)$ is a numerical function of the spectral index tabulated in Singal (2009) ($t(\alpha)\sim 0.3-0.7$ for $\alpha\sim-0.3$ to $-1.5$), $s$ is the linear size of the emitting region, and $\nu_{\rm turn}$ is the synchrotron turn-over frequency. In the observer’s rest frame, $T_{\rm B,eq}$ is boosted by the Doppler factor $\delta$ (Readhead 1994). For $s$ we adopt the mean FWHM size of the VLBI core ⁴⁴4 Marscher (1983) suggested a correction factor of 1.8 for the Gaussian to spherical size conversion. This correction changes the $T_{\rm B,eq}$ by a small factor of $1.8^{1/8}\sim 1.08$. Thus we ignore this size correction. , $\leavevmode\nobreak\ 77\leavevmode\nobreak\ \mu$as ($\sim 11R_{\rm sch}\sim 0.006\leavevmode\nobreak\$pc). The exact values of $\nu_{\rm turn}$ and $\alpha$ on $\lesssim 11R_{\rm sch}$ scale are not well determined due to the lack of simultaneous multi-frequency VLBI observations at high resolution. Nevertheless, we compare the time averaged core flux at 86 GHz ($\sim 0.5$ Jy) and at 230 GHz ($\sim 1$ Jy; Doeleman et al. 2012; Akiyama et al. 2015) and find an indication of an inverted spectrum between 86 GHz and 230 GHz. Also, Broderick & Loeb (2009) suggested a turnover frequency $\nu_{\rm turn}\sim 230\leavevmode\nobreak\$GHz and a spectral index $\alpha\sim-1.0$ using other available total flux density measurements. Hence we adopt $\nu_{\rm turn}=230$ GHz and $\alpha=-1.0$ in our analysis. In addition, we note that Eq. 2 is insensitive to the exact value of $\nu_{\rm turn}$ and $\alpha$ due to the 1/8 power dependence. The Doppler factor is given by $\delta=1/\Gamma(1-\beta\cos\theta)$ where $\Gamma$ is the bulk Lorentz factor, $\beta$ is the intrinsic jet speed, and $\theta$ is the jet viewing angle. We adopt the apparent jet speed of $\sim 0.5c$ and the viewing angle of $\theta=18^{\circ}$ (Hada et al. 2016; Mertens et al. 2016), which gives $\delta\sim 2$. Computing Eq. 2, we obtain a value of $T_{\rm B,eq}$ in the observer’s frame of $\approx 7.7\times 10^{10}\leavevmode\nobreak\ K$. For consistency, we also calculate $T_{\rm B,eq}$ using Eq. 4b of Readhead (1994), which yields $T_{\rm B,eq}\approx 3.5\times 10^{11}\leavevmode\nobreak\ K$ in the same frame (which is within a factor of 2 of our current value).

Accordingly, the equipartition brightness temperature is larger than the observed values by nearly an order of magnitude. It is worth noting that the broadband spectral energy decomposition of the core of M 87 shows a large dominance of the jet emission. In particular, emission from non-thermal electrons in the jet dominate over the emission from thermal (and non-thermal) electrons in the surrounding accretion disk (Broderick & Loeb 2009; Prieto et al. 2016). We also recall that the observed brightness temperature is still higher than the effective temperature of an electron $m_{e}c^{2}/k_{B}\sim 6\times 10^{9}\leavevmode\nobreak\$K. Therefore, the core brightness temperature is a good representation of the microscopic energy of the non-thermal electrons in the jet. Hence, we conclude that in the VLBI core at 86 GHz and on spatial scales of $\sim 11R_{\rm sch}$, the magnetic field energy density in the jet is larger than that of the non-thermal particles.

The corresponding magnetic field strength $B$ in the jet can be estimated by $B=B_{\rm eq}(T_{\rm B,eq}/T_{\rm B})^{2}$ (Readhead 1994) where $B_{\rm eq}$ is the equipartition magnetic field strength. In Tab. 3 we list $T_{\rm B,eq}/T_{\rm B}$ and $B/B_{\rm eq}$, adopting an equipartition brightness temperature of $T_{\rm B,eq}=2\times 10^{11}\leavevmode\nobreak\ K$. If the characteristic equipartition magnetic field strength in the VLBI core region of M 87 would match the estimates for other AGN on larger scales (of the order of $\sim 1$ G, Pushkarev et al. 2012), the true magnetic field strength could lie in the range of 61-210 G at the 86 GHz jet base and even higher when approaching closer to the central engine. This agrees with an independent estimate from Kino et al. (2015), who also obtained $B\sim 100$ G. We note that such a strong magnetic field seems to be present in some other AGN-jet systems (Zamaninasab et al. 2014; Martí-Vidal et al. 2015; Baczko et al. 2016). Following Readhead (1994), the energy density ratio $u_{\rm p}/u_{\rm B}$ is obtained by

Using $T_{\rm B,eq}/T_{\rm B}$ tabulated in Tab. 2, we find $-6\lesssim\log(u_{\rm p}/u_{\rm B})\lesssim-4$. Thus, the inner jet of M 87 appears to be more magnetically dominated.

We note that the above calculations assume a negligible energy contribution from thermal particles in the accretion flow, which may be entrained in the sheath of the jet (e.g., Mościbrodzka et al. 2016). The brightness temperature of M 87 is also quite low compared to typical values found in other sources at this frequency ($T_{\rm B,app}\sim 10^{11}\leavevmode\nobreak\$K; Lee et al. 2016; Nair et al. 2018), perhaps indicating that the non-thermal particles should be at least mildly relativistic. In this situation, the non-thermal particles in the low energy tail of the particle density distribution cannot be easily distinguished from those in the thermal particle density distribution. For instance, we recall that near the 86 GHz the VLBI core the jet plasma is still opaque to synchrotron radiation (see also Kim et al. 2018). The self-absorbed synchrotron radiation may heat the low energy particles, which will in turn modify the spectral shape of the particle energy distribution at lower energies (Ghisellini et al. 1988).

With high-resolution 86 GHz VLBI jet images, it is possible to estimate the diameter of the jet base $D_{0}$ by using the width and collimation profile of the outflow. For this we assume a self-similar jet and that the observed power-law dependence can be used to back-extrapolate the jet width to its origin near the event horizon. We then determine the collimation profile to $z=1R_{\rm sch}$, keeping the unknown separation $\epsilon$ between the BH and the VLBI core at 86 GHz as a free parameter ($\epsilon\leq 41\mu$as; see Sect. 3.5 and Figure 7).

In Fig. 9 we show the estimated size of the jet base versus the displacement $\epsilon$ of the VLBI core from the BH. The width of the jet base is in the range of $\sim(4.0-5.5)R_{\rm sch}$, with a statistical uncertainty of $\sim 0.5R_{\rm sch}$. This small size is consistent with the upper limit given by the 86 GHz VLBI core FWHM size, which is $\sim 11R_{\rm sch}$. It is also consistent with a circular Gaussian modelfit size estimate of $R_{0}=(5.6\pm 0.4)R_{\rm sch}$ (or equivalently $40\pm$1.8 $\mu$as) obtained at 230 GHz with EHT observations (Doeleman et al. 2012), although the 230 GHz size measurements can be model-dependent due to the limited $(u,v)$-coverage in the early EHT experiments.

Physically, this jet base size is comparable to the diameter of the innermost stable circular orbit (ISCO) for a non-spinning BH $D_{\rm ISCO}\sim 6R_{\rm sch}$ and it is much larger than $D_{\rm ISCO}$ for a maximally spinning BH with prograde disk rotation ($D_{\rm ISCO}=1R_{\rm sch}$). On the other hand, the diameter of the ergosphere in a rotating BH ranges between $(1-2)R_{\rm sch}$ (Komissarov 2012). This suggests that the diameter of the jet base of M 87 matches the dimensions of the innermost portion of the accretion disk. It is interesting to note that for some other nearby AGN-jets (e.g., 3C 84, Giovannini et al. 2018; Cygnus A, Boccardi et al. 2016a) the jet base appears to be much wider (100s of $R_{\rm sch}$).

We also highlight a somewhat larger value of $D_{0}\simeq(9.6\pm 1.6)R_{\rm sch}$ which Mertens et al. (2016) determined for M 87 at 43 GHz via an independent analysis. The difference in values for the jet base at 86 and 43 GHz might be simply due to systematics, but several other physical explanations are also possible. One possibility is that multi-frequency VLBI observations see radio emission coming from different layers of a transversely stratified jet, i.e., with stronger emission from the outer layers at lower frequencies. For instance, we refer to VLBI observations of Cygnus A at 5 GHz and 86 GHz, which show significantly different jet transverse widths (Carilli et al. 1991; Boccardi et al. 2016a).

An alternate interpretation of these differing values at 86 and 43 GHz could also be attributed to the jet viewing angle. The jet of M 87 has a small viewing angle of $15^{\circ}-30^{\circ}$. A transverse velocity stratification of the layered jet sheath would lead to differential Doppler boosting (see Komissarov 1990). This effect depends on the viewing angle and could lead to different jet widths at different frequencies.

The accretion disk geometry is another factor. The accretion flow in the center of M 87 is believed to be geometrically thick and the scale height of the jet launching point could also be model-dependent (see Yuan & Narayan 2014 for a review).

It is important to note that the geometrically wide origin of the sheath does not exclude the existence of a more narrow relativistic spine which may be rooted in the BH ergosphere (Blandford & Znajek 1977). An apparently faint, Doppler-deboosted ultra-relativistic spine located at the center of the jet could explain the observed limb-brightening (e.g., Komissarov 1990). The high-energy emission observed from M 87 also favors the presence of a fast spine surrounded by the slower sheath (see discussions in Abramowski et al. 2012). For instance, more recent measurements by the EHT with longer baselines (Krichbaum et al. 2014) and theoretical modeling of the jet base region (Kino et al. 2015) provide hints of the existence of a compact emission region significantly smaller than originally inferred from EHT observations with less extended baseline coverage (Doeleman et al. 2012; Akiyama et al. 2015). We also point to complex transverse jet intensity profiles shown by recent deep imaging of M 87, which may be consistent with the idea of the multiple jet layers with different origins (Asada et al. 2016; Hada 2017).

In the jet acceleration and collimation zone, the relationship between the jet opening angle and the jet speed is important. In particular, Hada et al. (2016) suggested that several reconfinement nodes may form or a jet breakout from a dense atmosphere might occur near the base of the jet in M 87. If this is the case, the expansion and acceleration of the jet will be significantly different from the collimation acceleration scenario (Komissarov et al. 2007; Lyubarsky 2009). For instance, for a sufficiently narrow jet of a given speed the jet-crossing time can be short enough for pressure disturbances from the ambient medium to propagate across the jet (i.e., causally connected). On the other hand, too wide an opening angle makes the jet less sensitive to the ambient pressure. The latter case could lead to a different jet acceleration mechanism such as the rarefaction acceleration (e.g., Tchekhovskoy et al. 2010; Komissarov et al. 2010).

We examine the relationship between the intrinsic opening angle $\phi_{\rm int}$ and the bulk Lorentz factor $\Gamma$ at the core separation $d=60\leavevmode\nobreak\ \mu$as ($27R_{\rm sch}$). According to Komissarov et al. (2009), a causally connected jet should satisfy

where the factor 2 accounts for the half opening-angle and the $\sigma$ is the level of jet magnetization defined by the ratio of the Poynting flux to kinetic energy (i.e., $\sigma=1$ for equipartition). For the jet viewing angle of $\theta=18^{\circ}-30^{\circ}$, the jet will be causally connected if $\Gamma\lesssim 2\sigma^{1/2}/\phi_{\rm int}=(1.3-1.8)\sigma^{1/2}$. If we presume $\sigma\sim 4$ at the core distance $d\sim 60\leavevmode\nobreak\ \mu$as (jet radius $\sim 0.15$ mas; see Fig. 18 of Mertens et al. 2016), we obtain upper limits on the apparent jet speed of $\sim(2.3-3.4)c$. Previous observations, in contrast, suggest apparent speeds of $\sim 0.5c$ in this region (Hada et al. 2016; Mertens et al. 2016) and corresponding $\Gamma\phi_{\rm int}/2\sim 0.7-0.9$, which satisfies Eq. 4.

Therefore, our $\Gamma\phi_{\rm int}/2\lesssim 1$ hints at a gradual collimation and acceleration of the jet base instead of a sudden acceleration accompanied by a jet breakout from a dense atmosphere, similar to GRB jet acceleration which has an order of magnitude higher $\Gamma\phi_{\rm int}/2$ (Panaitescu & Kumar 2002). It is also interesting to note that $\Gamma\phi_{\rm int}/2\sim 0.7-0.9$ in the M 87 jet base is significantly larger than $\Gamma\phi_{\rm int}/2\sim 0.1-0.2$ found in typical pc-scale jet systems (Jorstad et al. 2005; Pushkarev et al. 2009; Clausen-Brown et al. 2013). This implies $\Gamma\phi_{\rm int}$ is not constant for the whole AGN jet population.

It is also crucial to ascertain weather the wide expansion in the jet base can suppress different types of instabilities or not, in particular the current-driven kink instability (CDI). The CDI plays a critical role in determining the Poynting flux to kinetic energy conversion in high magnetization jet environments (e.g., Singh et al. 2016). In order for an instability to propagate across the jet, the jet expansion timescale should be longer than the timescale needed for an instability to propagate across the jet cross-section. This leads to the following criterion:

(Komissarov et al. 2009) where $t_{\rm dyn}=(\Gamma\phi_{\rm int}z/2)/(\beta_{s}c)$ is the jet crossing timescale for the instability, $\beta_{s}$ is the signal speed in units of $c$, $t_{\rm exp}=z/(\beta c)$ is the jet expansion timescale, and $\beta$ is the speed of the jet in units of $c$. If the magnetic field is dynamically important, $\beta_{s}$ is azimuthal component of the Alfvén speed, which is $\sim 1$ for a highly magnetized plasma (Giannios & Spruit 2006). By using numbers corresponding to the highly magnetized M 87 jet base, we find that the condition in Eq. 5 is satisfied for the jet base. This suggests that current-driven kink instabilities can survive the jet expansion and could be important in terms of the dynamics and energy conversion occurring within the jet.

The viewing angle and, more importantly, the velocity of the innermost M 87 jet are crucial for our understanding of the jet geometry and dynamics. Recent GRMHD simulations show that the filamentary jet structure becomes significantly complicated near the jet formation region (e.g., Mościbrodzka et al. 2016). This can potentially make it difficult to determine precisely the viewing angle and the jet velocity. Nevertheless, previous studies (e.g., Mertens et al. 2016; Walker et al. 2016) show that the two-dimensional kinematics of the jet in M 87 can be decomposed into longitudinal and transverse motions. The longitudinal motions can explain the majority of the relativistic boosting without additional jet curvature. In the light of this result, we can test two possible scenarios using the observed jet to counter-jet ratio. That is, the inner jet of M 87 could have (1) a stationary non-accelerating flow, with a constant jet speed and line of sight jet orientation (i.e., constant viewing angle) or (2) the viewing angle is constant but the jet is accelerating.

As for the case (1), if the approaching and receding jets are intrinsically of the same brightness and speed and there is no transverse velocity gradient in both jets, the jet to counter-jet brightness ratio $R$ can be expressed by

where $I_{\rm Jet}$ and $I_{\rm CJ}$ are the intensities of the jet and the counter-jet, $\beta$ is the intrinsic jet speed normalized by the speed of light $c$, $\alpha$ is the spectral index ($S\propto\nu^{+\alpha}$), and $\beta_{\rm app}$ is the observed apparent speed.

It is worth noting that Eq. 6 assumes no transverse velocity gradient across the jet. A faster jet speed closer to the central axis could make the apparent jet to counter-jet ratio $I_{\rm Jet}/I_{\rm CJ}$ lower because of different levels of Doppler boosting in the approaching and receding flows (see Fig.2 and 3 of Komissarov 1990). We note that kinematics studies of the inner jet of M 87 find the same intrinsic flow speeds in the boundary layers of the jet and counter-jet within observational uncertainties (Mertens et al. 2016). Hence, we do not include the transverse velocity gradient effect in the analysis of the jet to counter-jet ratio.

In order to find a range of $\theta$ and $\beta$ satisfying both Eq. 6 and 7, we assume $\beta_{\rm app}\sim 0.5$ and $\sim 0.1$ as the upper and the lower limits on the apparent speed, respectively (Hada et al. 2016; Mertens et al. 2016). This wide range represents the variety of the directly measured jet speeds. We also adopt a typical optically thin spectral index of the M 87 jet $\alpha=-1$ (Hovatta et al. 2014). The jet to counter-jet ratio value is taken from Sect. 3.6 ($17\pm 6$). In Fig. 10 we show the possible range of $\theta$ and $\beta$. The range of the possible jet viewing angle is wide ($6^{\circ}-38^{\circ}$), mainly because of the large scatter in the apparent jet speed. If the fast speed ($\sim 0.5c$) is directly associated with the true jet flow (Mertens et al. 2016), rather large jet viewing angles of $28^{\circ}-38^{\circ}$ are expected (c.f., Hada et al. 2016). On the other hand, recent observations found significant acceleration within the jet of M 87 (Mertens et al. 2016; Walker et al. 2016), which disfavor case (1). Therefore, we expand our analysis and put more emphasis on case (2). In this subsequent analysis we assume a viewing angle of $18^{\circ}$ (Mertens et al. 2016), which better explains fast superluminal motions in the outer jet. If the jet to counter-jet ratio $R$ is given as a function of the distance from the jet origin, the jet kinematic parameters including the intrinsic speed $\beta$, the apparent speed $\beta_{\rm app}$, the bulk Lorentz factor $\Gamma$, and the Doppler factor $\delta$ can be computed along the jet axis via:

In Fig. 11 we show the results of our calculations of the Lorenz factor $\Gamma$. In this model, the Lorentz factor mildly increases from $\sim 1.0-1.1$ at 0.1 mas to $\sim 1.1-1.2$ at $\sim 0.35$ mas, which suggests mild inner jet acceleration. In terms of the apparent speed, our fiducial model predicts $\beta_{\rm app}\approx(0.1-0.7)$ at 0.1 mas and $(0.6-1.0)$ at $>0.3$ mas. The latter is comparable to what has been directly measured from VLBA 43 GHz observations at similar and/or slightly larger distances (see Fig. 16 of Mertens et al. 2016). We also recall that no clear inter-day timescale structural variation was seen in the jet in the 2009 data. This implies an upper limit on the apparent jet speed of $0.2\times 50\leavevmode\nobreak\ \mu{\rm as}/1\leavevmode\nobreak\ {\rm day}\sim 3.65\leavevmode\nobreak\ {\rm mas}/{\rm yr}\sim 1\leavevmode\nobreak\ c$ (using the speed conversion factor $1\leavevmode\nobreak\ c=3.89\leavevmode\nobreak\ {\rm mas/yr}$ for M 87 and adopting 1/5th of the beam size as the resolution limit). The inferred range of the apparent jet velocity agrees well with this upper limit.

Hence, we conclude that a mildly accelerating inner jet model with a stationary viewing angle could explain the observed inner jet to counter-jet ratio.

The transverse intensity profiles presented in Sect. 3.4, Fig. 4c, and Fig. 5 suggest the existence of a central emission lane close to the jet base. Other deep imaging experiments also suggest similar complex structure at more distant regions along the jet (Mertens et al. 2016; Asada et al. 2016; Hada 2017; see also Sect. 4.2).

If the central lane appears fainter only due to the Doppler de-boosting, then the center-to-limb brightness ratio $\rho_{CL}$ measured at the core distances 0.5-1.0 mas can be related to the ratio of their Doppler factors by $\rho_{\rm CL}=(\delta_{\rm spine}/\delta_{\rm sheath})^{2-\alpha}$ . To calculate the expected range of the Spine Lorentz factor $\Gamma_{\rm spine}$, we assume $\alpha=-1$, a viewing angle of $18^{\circ}-30^{\circ}$, and an apparent speed of the sheath of $\sim 0.5c$ (the corresponding $\Gamma_{\rm sheath}\sim 1.18-1.29$ and $\delta_{\rm sheath}\sim 1.58-1.96$). The observed $\rho_{\rm CL}$ constrains $\delta_{\rm spine}$ to be $\sim 1.16-1.44$. In Fig. 12 we show the range of $\delta_{\rm spine}$ versus the spine Lorentz factor for small and relatively large viewing angles, respectively. It is apparent from the figure that a rather fast $\Gamma_{\rm spine}\sim 13-17$ ($\Gamma_{\rm spine}\sim 4.4-5.9$) is required ⁶⁶6 Here we exclude the apparent solution, i.e. $\Gamma_{\rm spine}\sim 1$. when the viewing angle is $18^{\circ}$ ($30^{\circ}$).

We note that the inferred $\Gamma_{\rm spine}$ is large considering the very small distance from the core ($\sim 0.5-1.0$ mas; $70-140R_{\rm sch}$ projected). For instance, $\Gamma_{\rm spine}\sim 13-17$ is comparable to the Lorentz factor of HST-1 determined at optical wavelengths by Biretta et al. (1999) ($\Gamma\sim 14$ for the viewing angle $18^{\circ}$). At such short wavelengths the observed radiation presumably traces relativistic plasma closer to the central axis of the jet (e.g., Perlman et al. 1999; Mertens et al. 2016).

GRMHD simulations show that accreting BH systems can form (i) a narrow, relativistic, and Poynting-dominated beam (i.e., spine) and (ii) broader, sub-relativistic, and mass-dominated outflow (i.e., sheath) as a natural consequence of the mass accretion and BH physics (e.g., Hawley & Krolik 2006; Sa̧dowski et al. 2013). The latter helps to main low density levels near the central axis of the jet. In such cases, the narrow beam propagates without significant mass loading and maintains its initial speed up to large distances.

On the other hand, Asada et al. (2016) showed that the central lane expands in a similar manner as the sheath. If the collimation and acceleration pattern of the spine is qualitatively similar to that of the sheath (Asada & Nakamura 2012; Hada et al. 2013; Asada et al. 2014; Mertens et al. 2016), the Lorentz factor of the spine may increase at larger distances. If this is the case, the smaller $\Gamma_{\rm spine}\sim 4-6$, inferred assuming a larger inner jet viewing angle ($30^{\circ}$), is perhaps a more plausible estimate of the intrinsic speed.

It should be noted, however, that the intrinsic synchrotron emissivity of the spine and the sheath are not necessarily the same, especially when the two different layers are launched from different origins (e.g., the spine from the BH while the sheath from the inner disk). Closer to the central engine, the intrinsic emissivity of the spine may decrease (Mertens et al. 2016). In addition, the central lane is comparably or even brighter than the limb at 1.6-5 GHz (Asada et al. 2016; see Fig. 3 and 5 therein). Such an intensity profile cannot solely be produced by a transverse velocity gradient. Therefore, the limb-brightened jet morphology close to the jet base may not be fully explained by the velocity stratification. Intrinsic differences in the jet plasma such as the jet composition and/or the magnetic field strength would need to be considered for the different jet emissivity.

The commonly invoked “differential collimation” process (see Sect. 4.3.2 of Komissarov 2012) requires significant magnetic hoop stress from toroidal magnetic fields. However, this condition can be satisfied only in the “far-zone” where the toroidal field is substantially stronger than the poloidal field (e.g., Komissarov et al. 2007, 2009; Lyubarsky 2009). Therefore, a magnetic hoop stress might not be valid near the jet launching region where the poloidal field components are much more dominant (e.g., Tchekhovskoy 2015). For this scenario, we can estimate the radial distance of such a critical point from the central engine. A substantial change in the B-field orientation occurs when the jet radius approaches the light cylinder radius $r_{\rm lc}=c/\Omega$ (Meier 2012) where $\Omega$ is the angular speed of the outflow. Mertens et al. (2016) recently determined the light cylinder $r_{\rm lc}$ in the M 87 jet to be $\sim 20R_{\rm sch}$. The observed jet radius becomes comparable to $r_{\rm lc}$ at $\sim 0.2$ mas ($28R_{\rm sch}$ projected) core separation. It is interesting to note that we detect a slight divergence between the data and the single power-law model at a similar core separation (see the bottom panel of Fig. 6). Future higher angular resolution imaging experiments with the EHT plus a phased Atacama Large Millimeter/submillimeter Array (ALMA) (Matthews et al. 2018) will be able to reveal the structure and propagation of the M 87 jet on such spatial scales.