Fast-spinning Black Holes Inferred from Symmetrically Limb-brightened Radio Jets

As shown below, our force-free model is essentially the same as in BL09. Although we employ a flat spacetime outside the BH, the magnetic field configuration is not much different from that with a general relativistic treatment even near the hole (McKinney & Narayan 2007). The resultant radio images will not be significantly changed as long as we focus on the limb-brightened features seen far from the central warped region. We put the detailed formulation in Appendix A and briefly explain the salient results below.

2.1.1. Electromagnetic Field

In a steady, axisymmetric force-free field, a stream function Ψ gives the electromagnetic field. Following BL09, we assume a parametrically controlled paraboloid-like jet instead of an exact solution of a force-free field. The stream function is given by

$$\begin{eqnarray}&&{\rm{\Psi }}={{Ar}}^{\nu }(1\mp \cos \theta ).\end{eqnarray} \tag{ 1 }$$

In the above expression, $(r,\theta ,\phi )$ denote the standard spherical coordinates, and the minus and plus signatures are for $z\geqslant 0$ and z < 0, respectively. Here, ν is the parameter to control the jet shape, where ν = 1 gives paraboloidal jets, and A is a constant that has the dimension of $[{r}^{2-\nu }B]$, with B being the magnetic field. The electromagnetic field is then given by

$$\begin{eqnarray}&&{{\boldsymbol{B}}}_{p}=\displaystyle \frac{1}{R}{\boldsymbol{\nabla }}{\rm{\Psi }}\times \hat{{\boldsymbol{\phi }}},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{B}_{\phi }=\mp \displaystyle \frac{2{{\rm{\Omega }}}_{{\rm{F}}}{\rm{\Psi }}}{{Rc}},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{\boldsymbol{E}}=-\displaystyle \frac{1}{c}{{\rm{\Omega }}}_{{\rm{F}}}{\boldsymbol{\nabla }}{\rm{\Psi }}=-\displaystyle \frac{R{{\rm{\Omega }}}_{{\rm{F}}}}{c}\hat{\phi }\times {\boldsymbol{B}},\end{eqnarray} \tag{ 4 }$$

where B and E denote the magnetic and electric fields, respectively. Here, $(R,\phi ,z)$ are the standard cylindrical coordinates, the subscript p is assigned for the poloidal component, $\hat{{\boldsymbol{\phi }}}$ is the azimuthal unit vector, and ${{\rm{\Omega }}}_{{\rm{F}}}={{\rm{\Omega }}}_{{\rm{F}}}({\rm{\Psi }})$ corresponds to the rotational frequency of magnetic fields. It should be noted that the magnetic field is wound up and toroidal dominant, $B\sim | {B}_{\phi }|$, for $R| {{\rm{\Omega }}}_{{\rm{F}}}| /c\gg 1$, while it is poloidal dominant, $B\sim | {{\boldsymbol{B}}}_{p}|$, for $R| {{\rm{\Omega }}}_{{\rm{F}}}| /c\ll 1$.

2.1.2. Fluid Velocity

The fluid velocity cannot be determined in the force-free limit in principle, since the inertia is totally neglected. In this paper, we use the following drift velocity, v, as the fluid velocity by following BL09:

$$\begin{eqnarray}&&{\boldsymbol{v}}=\displaystyle \frac{{\boldsymbol{E}}\times {\boldsymbol{B}}}{{B}^{2}}c=-R{{\rm{\Omega }}}_{{\rm{F}}}\displaystyle \frac{{B}_{\phi }}{{B}^{2}}{{\boldsymbol{B}}}_{p}+R{{\rm{\Omega }}}_{{\rm{F}}}\displaystyle \frac{{B}_{p}^{2}}{{B}^{2}}\hat{{\boldsymbol{\phi }}}.\end{eqnarray} \tag{ 5 }$$

The above velocity holds the following conditions for the electromagnetic field given by Equations (2)–(4): (1) the velocity does not exceed the speed of light in the entire region for $\nu \leqslant \sqrt{2}$, (2) the electric field vanishes in the fluid rest frame (the frozen-in condition), and (3) the velocity is asymptotically the same as in cold, ideal MHD when $R| {{\rm{\Omega }}}_{{\rm{F}}}| /c\gg 1$ (see Appendix A.3). Figure 1 sketches an example of the twisted magnetic and velocity field lines in a paraboloid-shaped jet (ν = 1). We note that the velocity is perpendicular to the magnetic field, while their poloidal components, ${{\boldsymbol{v}}}_{p}$ and B_p, are parallel to each other. The asymptotic relations of the velocity for $R| {{\rm{\Omega }}}_{{\rm{F}}}| /c\gg 1$ are given by

$$\begin{eqnarray}&&\beta \sim {\beta }_{p}\sim g(\theta ,\nu ),\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{\beta }_{\phi }\sim {\left(\displaystyle \frac{R| {{\rm{\Omega }}}_{{\rm{F}}}| }{c}\right)}^{-1}{[g(\theta ,\nu )]}^{2},\end{eqnarray} \tag{ 7 }$$

where β denotes the speed normalized by c, and $g(\theta ,\nu )$ is a factor that is an order of tenth and approaches unity toward the jet axis (θ = 0, π; see Figure 12). That is, the fluid velocity is dominated by the poloidal component and becomes relativistic around the jet axis if $R| {{\rm{\Omega }}}_{{\rm{F}}}| /c\gg 1$. For $R| {{\rm{\Omega }}}_{{\rm{F}}}| /c\ll 1$, on the other hand, the following relations are obtained:

$$\begin{eqnarray}&&\beta \sim {\beta }_{\phi }\sim \displaystyle \frac{R| {{\rm{\Omega }}}_{{\rm{F}}}| }{c},\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{\beta }_{p}\sim {\left(\displaystyle \frac{R{{\rm{\Omega }}}_{{\rm{F}}}}{c}\right)}^{2}\displaystyle \frac{1}{g(\theta ,\nu )}.\end{eqnarray} \tag{ 9 }$$

That is, the fluid velocity is not relativistic and is dominated by the toroidal component.

Zoom In Zoom Out Reset image size

2.1.3. Nonthermal Electrons

Motivated by the limb-brightened jets, we consider the case where the nonthermal electrons are distributed away from the jet axis, in contrast to BL09, who assumed the nonthermal electrons are clinging to the axis. Such a spatial distribution concentrated away from the axis could be realized for jets launched from an accretion disk and even for those launched from the BH. The particles are supplied from a disk at the jet foot point for the former case, while several options for the particle injection can be considered for the latter case. As shown in MHD simulations (McKinney & Gammie 2004; Komissarov 2005; Komissarov et al. 2007; Barkov & Komissarov 2008; McKinney & Blandford 2009; Tchekhovskoy et al. 2011; McKinney et al. 2014; Sadowski et al. 2014; Qian et al. 2018), the jets driven by the BZ mechanism are confined by the external pressure of the ambient matter, that is, a geometrically thick disk with an advection-dominated accretion flow (ADAF; Narayan & Yi 1994), or the disk wind. Thermal charged particles are prevented by the globally ordered magnetic field in the funnel region from diffusing into there from the ADAF, but high-energy hadrons can diffuse into there (Toma & Takahara 2012; Kimura et al. 2014, 2015), and high-energy photons can annihilate and supply ${e}^{-}{e}^{+}$ pairs there (Levinson & Rieger 2011; Mościbrodzka et al. 2011). The particles in a jet flow outward from the separation surface (the stagnation surface), which is much closer to the BH and the hottest part of the disk farther away from the jet axis (Takahashi et al. 1990; McKinney & Gammie 2004; Pu et al. 2015; Nakamura et al. 2018).⁷ Thus, the particle injection for the outflow can be dominated at the jet edge. A pair creation gap created around the separation surface could also be a particle supplier (Levinson & Rieger 2011; Broderick & Tchekhovskoy 2015). The fluid instability or magnetic reconnection at the layer between the jet and disk wind may also produce nonthermal particles (see Matsumoto & Masada 2013; Parfrey et al. 2015; Toma et al. 2017).

It is beyond the scope of this paper to discuss in detail the above injection and acceleration mechanisms (upcoming EHT data would shed light on those mechanisms). In this study, we simply assume that the spatial distribution of the nonthermal is described in a parametric way and the energy spectrum is given by a single power law. The spatial distribution is characterized by the cross sections at z = ±z₁ for simplicity, where the electrons are assumed to be in a ring shape and the number density is given by

$$\begin{eqnarray}&&n(R,\pm {z}_{1})={n}_{0}\exp \left[-\displaystyle \frac{{(R-{R}_{p})}^{2}}{2{{\rm{\Delta }}}^{2}}\right],\end{eqnarray} \tag{ 10 }$$

where R_p is the radius where n peaks on the plane, and Δ gives the width of the ring; n₀ is a normalization constant. Our prescription is identical to that in BL09 when R_p = 0 and ${\rm{\Delta }}={z}_{1}=5{r}_{g}$. We also set ${z}_{1}=5{r}_{g}$ hereafter while R_p and Δ remain as free parameters. In the vertical direction, n is assumed to obey the continuity equation (BL09), which is given as follows by using Equation (5):⁸

$$\begin{eqnarray}&&\displaystyle \frac{n}{{B}^{2}}=\mathrm{const}.\ \mathrm{along}\,{\rm{a}}\,\mathrm{magnetic}\,\mathrm{field}\,\mathrm{line}.\end{eqnarray} \tag{ 11 }$$

We assume that the nonthermal electrons are isotropic in the fluid rest frame and obey an energy distribution of a single power law given by an index p:

$$\begin{eqnarray}f(\gamma ^{\prime} )\propto \left\{\begin{array}{cc}{\gamma }^{{\prime} -p} & \mathrm{for}\ {\gamma }_{\min }^{{\prime} }\leqslant \gamma ^{\prime} \leqslant {\gamma }_{\max }^{{\prime} }\\ 0 & \mathrm{othewise}\end{array},\right.\end{eqnarray} \tag{ 12 }$$

where γ' is the Lorentz factor of an electron measured in the proper frame, which has lower and higher cutoffs at ${\gamma }_{\min }^{{\prime} }$ and ${\gamma }_{\max }^{{\prime} }$, respectively. The synchrotron emissivity does not depend on ${\gamma }_{\max }^{{\prime} }$ but only on ${\gamma }_{\min }^{{\prime} }$ provided ${\gamma }_{\min }^{{\prime} }$ and ${\gamma }_{\max }^{{\prime} }$ are sufficiently small and large, respectively (see Appendix A.5). We can, hence, set ${\gamma }_{\max }^{{\prime} }=\infty$ for a large higher cutoff, while we use ${\gamma }_{\min }^{{\prime} }=100$ for the lower cutoff by following BL09. As in BL09, the energy distribution is fixed to Equation (12) everywhere, which means that some energy supplier is assumed to replenish high-energy electrons to compensate for cooling processes such as synchrotron and adiabatic cooling.

The quantities given by Equations (1)–(11) give the synchrotron emissivity at each location in a jet that is received by the observer at a frequency ω as follows (Rybicki & Lightman 1985; Shibata et al. 2003):

$$\begin{eqnarray}&&{j}_{\omega }({\boldsymbol{n}})=\displaystyle \frac{1}{{{\rm{\Gamma }}}^{2}{(1-\beta \mu )}^{3}}{j}_{\omega ^{\prime} }^{{\prime} }({\boldsymbol{n}}^{\prime} ),\end{eqnarray} \tag{ 13 }$$

where the quantities with prime symbols are evaluated in the fluid rest frame.⁹ Here, ${\boldsymbol{n}}$ is a unit vector that is directed to the observer at infinity, μ is the cosine of the angle between ${\boldsymbol{n}}$ and v, and ${\rm{\Gamma }}=1/\sqrt{1-{\beta }^{2}}$ is the Lorentz factor. The factor in the right-hand side is attributed to relativistic effects that are due to the bulk fluid motion. Note that ${j}_{\omega ^{\prime} }^{{\prime} }({\boldsymbol{n}}^{\prime} )$ is given by Equation (55).

In higher frequencies, radio jets in AGNs are optically thin to synchrotron emissions. The intensity of radio images observed on the sky is, then, calculated by integrating Equation (13) along the line of sight after fixing the viewing angle Θ:

$$\begin{eqnarray}&&{I}_{\omega }(X,Y)=\int {j}_{\omega }({\boldsymbol{n}},X,Y,Z){dZ},\end{eqnarray} \tag{ 14 }$$

where (X, Y) are the coordinates of the sky and ${dZ}$ is the line element parallel to the line of sight. In the following, the X axis is chosen to coincide with the x axis, and the Z axis is inclined toward the −y direction so that the angle between the z and Z axes becomes Θ (see Figure 1). A simulated VLBI image is obtained after the convolution with a beam kernel, which is introduced in the next subsection.

In our force-free model, 10 parameters remain: Ω_F, R_p, Θ, Δ, ν, p, M_BH, A, n₀, and ω. We systematically change them and investigate the effects on our synthetic radio images. The first two parameters (Ω_F and R_p) are especially important, since they can drastically change radio images, as shown in Section 3. The viewing angle Θ is found to be less important for limb-brightened features, while it can be important for the brightness ratio between the jet and counterjet (see Section 4.2). The choices of the other parameters do not qualitatively alter the synthetic images (see Appendix B).

We consider two patterns of Ω_F. The first choice of Ω_F is the same as in BL09, where the magnetic field is threaded through a razor-thin accretion disk at the equatorial plane. Since the field rotates with the disk, Ω_F is given by

$$\begin{eqnarray}{{\rm{\Omega }}}_{{\rm{F}}}=\left\{\begin{array}{ll}{{\rm{\Omega }}}_{\mathrm{Kep}}(\tilde{R}) & (\tilde{R}\gt {R}_{\mathrm{ISCO}})\\ {{\rm{\Omega }}}_{\mathrm{Kep}}({R}_{\mathrm{ISCO}}) & (\tilde{R}\leqslant {R}_{\mathrm{ISCO}})\end{array}\right.,\end{eqnarray} \tag{ 15 }$$

where $\tilde{R}$ is the foot point radius of a given magnetic field line measured on the equatorial plane, and R_ISCO is the radius of the innermost stable circular orbit (ISCO) for prograde rotations. Here, ${{\rm{\Omega }}}_{\mathrm{Kep}}$ is the Keplerian angular frequency given by the dimensionless Kerr parameter, a, as follows (Bardeen et al. 1972):

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{\mathrm{Kep}}=\displaystyle \frac{\sqrt{{{GM}}_{\mathrm{BH}}}}{\sqrt{{R}^{3}}+a\sqrt{{r}_{G}^{3}}},\end{eqnarray} \tag{ 16 }$$

where ${r}_{G}:= {r}_{g}/2={{GM}}_{\mathrm{BH}}/{c}^{2}$ is the gravitational radius.

The other choice of Ω_F is motivated by the BZ process, which was not considered in BL09. In the BZ process, Ω_F is nearly a constant given by

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{{\rm{F}}}=\displaystyle \frac{1}{2}{{\rm{\Omega }}}_{\mathrm{BH}}=\displaystyle \frac{{ac}}{4{r}_{+}},\end{eqnarray} \tag{ 17 }$$

where ${{\rm{\Omega }}}_{\mathrm{BH}}$ is the rotational frequency of the Kerr BH and ${r}_{+}=(1+\sqrt{1-{a}^{2}}){r}_{G}$ is the horizon radius (Blandford & Znajek 1977; McKinney & Gammie 2004). We assume that the shape of the magnetic field lines changes above the equator so that the field lines penetrate the event horizon, while the shape far away from the equator is given by Equation (1). Such a field configuration may be possible, depending on the profile of the external pressure of the disk wind or corona, which collimates the jet and is responsible for the global jet shape (McKinney 2006; Nakamura & Asada 2013). Since the jet structure near the central region is not important for the limb-brightened feature observed far from the central BH, we use Equation (1) in the entire region for simplicity.

The ring radius (R_p) is systematically changed from zero to some sufficiently large value. For comparison to BL09, the other parameters are fixed to the fiducial values in BL09: ${M}_{\mathrm{BH}}=3.4\times {10}^{9}{M}_{\odot }$, ${\rm{\Delta }}=5{r}_{g}$, ν = 1, p = 1.1, and Θ =25°, which were chosen for the M87 jet. Accordingly, we henceforth consider M87, which is an example of an AGN that shows a symmetrically limb-brightened jet with a dim counterjet.¹⁰ We calculate jet images on the scale of several milliarcseconds, where the limb-brightened feature of the jet is observed with VLBI (Ly et al. 2007; Walker et al. 2008; Hada et al. 2011, 2013, 2016; Mertens et al. 2016). For the fiducial mass of the BH, 1 mas corresponds to ∼0.08 pc ∼250 r_g for D = 16.7 Mpc (Blakeslee et al. 2009). We use the beam kernel for VLBA given in Walker et al. (2008) and assume that the M87 jet is inclined toward the northeast by 20° as measured from the east direction on the sky. We note that A and n₀ are related only to the normalization of the intensity. We adopt the following values throughout the paper: ${{Az}}_{1}^{2-\nu }=100$ G, which corresponds to the strength of the magnetic field at (R, z) = (0, ±z₁) (Kino et al. 2015); and n₀ = 1 cm${}^{-3}$, which produces a peak intensity that is roughly consistent with observations of M87 in order of magnitude for our best model described in Section 3.2.1.¹¹ A typical value of the magnetization factor $\sigma ={B}^{2}/(4\pi {\rm{\Gamma }}{{nm}}_{p}{c}^{2})$, where m_p is the proton mass, is given by

$$\begin{eqnarray}&&\sigma \sim 5.3\times {10}^{5}{{\rm{\Gamma }}}^{-1}{\left(\displaystyle \frac{B}{100{\rm{G}}}\right)}^{2}{\left(\displaystyle \frac{n}{1{\mathrm{cm}}^{-3}}\right)}^{-1}.\end{eqnarray} \tag{ 18 }$$

In fact, the force-free approximation σ ≫ 1 consistently holds for our jet models, as shown in the next section. We use the observed frequency of ω/(2π) = 44 GHz to synthesize the intensity maps, while the choice of ω does not affect the shape of radio contour maps under the optically thin assumption.¹² We summarize the model parameters in Table 1. In Appendix B, some of the above parameters are varied around our fiducial values to study the effects on radio images; they do not change our conclusions.

Table 1.  Model Parameters

Quantity	Symbol	Fiducial value for Figures 2–10
Rotational frequency of the magnetic field	ΩF	Equation (15) (Keplerian) or Equation (17) (rigid)
Radius where n peaks on z = ±z1	Rp	Varied in $[0,100{r}_{g}]$
Viewing angle	Θ	25°
Width of the Gaussian ring	Δ	5rg
Jet shape	ν	1 (paraboloidal jet)
Energy spectral index of the nonthermal electrons	p	1.1
Mass of the BH	MBH	$3.4\times {10}^{9}{M}_{\odot }$
Strength of the magnetic field at (R,z) = (0,±z1)	${{Az}}_{1}^{2-\nu }$	100 G
Number density of the nonthermal electrons at $(R,z)=({R}_{p},\pm {z}_{1})$	n0	1 cm${}^{-3}$
Dimensionless Kerr parameter of the BH	a	Varied in $[0,0.998]$
Height of the plane where n is given in a ring shape by Equation (10)	z1	5rg
Minimal Lorentz factor of the nonthermal electrons	${\gamma }_{\min }^{{\prime} }$	100
Maximal Lorentz factor of the nonthermal electrons	${\gamma }_{\max }^{{\prime} }$	$\infty$
Observational frequency	ω/(2π)	44 GHz
Luminosity distance to the jet	D	16.7 Mpc
Inclination of the projected jet axis measured from the east direction		20° toward northeast
Beam kernel		Walker et al. (2008) (VLBA)

Download table as:  ASCIITypeset image

First, we examine the case of the disk-threaded model, in which the magnetic fields penetrate the Keplerian accretion disk and Ω_F is given by Equation (15). We here show the results for a = 0.998, which is the fiducial value in BL09, since those for smaller a are qualitatively the same. This model has ${R}_{\mathrm{ISCO}}\sim 1.2{r}_{+}\sim 0.62{r}_{g}$ and ${{\rm{\Omega }}}_{\mathrm{Kep}}({R}_{\mathrm{ISCO}})\sim 2.5\times {10}^{-5}$ s⁻¹.

The jet structure of this model is presented in Figure 2.¹³ The upper left panel shows the distribution of an important quantity, $R{{\rm{\Omega }}}_{{\rm{F}}}/c$ (see also Figure 18 for the close-up around the origin). As shown by the white lines, the so-called light "cylinder," where $R{{\rm{\Omega }}}_{{\rm{F}}}/c=1$ is satisfied, forms not only a vertical surface around the jet axis but also a curved one far from the jet axis (Blandford 1976). The former truncated cylinder is formed at $R={R}_{\mathrm{lc},1}:= c/{{\rm{\Omega }}}_{\mathrm{Kep}}({R}_{\mathrm{ISCO}})\sim 1.2{r}_{g}\sim 4.8\times {10}^{-3}$ mas due to the uniform rotation of the magnetic field passing through inside the ISCO. The latter curved surface ($z\propto {R}^{4/3}$ at $R\gg {r}_{G};$ see Appendix C) is, on the other hand, attributed to the differential rotation of the magnetic field lines anchored to the accretion disk. As a result, $R{{\rm{\Omega }}}_{{\rm{F}}}/c$ exceeds unity only in a limited region bound by these two surfaces. This means that the jet edge part is dominated by a poloidal magnetic field and is not efficiently accelerated to poloidal directions, as shown in the lower left and upper middle panels in Figure 2 (see Equations (6) and (8) for asymptotic relations between $| {{\boldsymbol{v}}}_{p}|$ and $R{{\rm{\Omega }}}_{{\rm{F}}}/c$). The highly relativistic poloidal speed is realized, on the other hand, only at $R\gtrsim {R}_{\mathrm{lc},1}$, which is near the jet axis. The jet rotational speed, v_ϕ, is shown in the upper right panel in Figure 2. Note that v_ϕ peaks ∼0.5c around the light cylinder and reduces away from it, as indicated in Equations (7) and (9). The lower middle panel in Figure 2 shows the Lorentz factor, which manifestly shows the jet is relativistic only near the jet axis, as explained above. The lower right panel shows the number density of the nonthermal electrons for R_p = 40r_g (one of the fiducial cases) in the logarithmic scale. As is evident, the nonthermal electrons are concentrated on the magnetic field lines that pass through $R=40{r}_{g}$ at z = 5r_g (${\rm{\Psi }}/A\sim 35.3{r}_{g};$ drawn as black lines in Figure 2), while the number density rapidly decreases away from the lines. Note that n is also reduced along a field line upward. We note that the magnetization factor σ given by Equation (18) is low at the dense region around the black lines but is much larger than unity in the displayed region (σ ≳ 8 × 10³), which ensures the use of the force-free approximation for this model.

Zoom In Zoom Out Reset image size

Figure 3 shows the beaming factor $\delta := 1/[{\rm{\Gamma }}(1-\beta \mu )]$ on some horizontal slices of the jet. The top and middle left panels are for the counterjet (z < 0), and the others are for the jet (z > 0) that is directed toward the observer with a viewing angle of Θ = 25°. The beaming effect becomes remarkable in the region with $R{{\rm{\Omega }}}_{{\rm{F}}}/c\gg 1$, where the poloidal speed becomes relativistic and the Lorentz factor is large. In the approaching jet side, the distribution of δ is highly asymmetric because of the jet rotation, which reaches ∼0.5c around the curved light "cylinder," as presented in the upper right panel in Figure 2. Through μ in Equation (13), jet rotations lead to the opposite effects of relativistic beaming in the left and right sides of the jet. The left side of the jet is coming toward the observer and, as a result, strongly beams light toward the observer, whereas the right side of the jet is going away from the observer and, hence, does not efficiently beam light to the observer. We note that the peaks of δ and Γ in each slice do not necessarily coincide, due to the misalignment of the observer and flow directions. In the counterjet side, on the other hand, δ is suppressed below unity almost in the entire region. The suppression is especially strong in the region with $R{{\rm{\Omega }}}_{{\rm{F}}}/c\gg 1$, and the asymmetry of δ due to the jet rotation is also seen as in the approaching jet side.

Zoom In Zoom Out Reset image size

The left panel in Figure 4 shows the calculated radio image for R_p = 0. This model is essentially the same as the standard (M0) model in BL09, where the nonthermal electrons cling to the jet axis. As expected, the jet axis is the brightest, due to the concentration of electrons, and any limb-brightened feature is not seen. The counterjet is not seen in the radio map, due to the relativistic beaming to the direction opposite to the observer. We also note that the radio intensity is larger in the left-hand side of the jet in the figure because of the asymmetric beaming effect shown in Figure 3.

Zoom In Zoom Out Reset image size

We here pick up the result for R_p = 40r_g, while the results for R_p > 0 are qualitatively the same, as mentioned later. In this example, the nonthermal electrons are nearly on the curved light-cylinder surface for $| z| \leqslant 10$ mas, as indicated by the magnetic field lines with ${\rm{\Psi }}/A\sim 35.3{r}_{g}$ (the black lines in Figures 2 and 3), which reasonably trace the dense region of the nonthermal electrons. The right panel in Figure 4 shows the synthesized radio image for R_p = 40r_g. One of the most striking features is the strongly asymmetric limb brightening: the left-hand side of the jet axis in the figure (i.e., the northern part on the sky) is more luminous than the counterpart in the right-hand side (the southern part). The limb brightening is understood just as a reflection of the assumed R_p. The large asymmetry is, on the other hand, due to the rotation of the jet. As seen in Figure 3, the asymmetry of the beaming factor in the approaching jet side is relatively large near the magnetic field line ${\rm{\Psi }}/A\sim 35.3{r}_{g}$, where v_ϕ reaches (∼0.5c), as plotted in the upper right panel in Figure 2. We note that the synchrotron emission is intrinsically asymmetric even in the fluid rest frame, since the pitch angle of the relativistic electrons that are directed toward the observer is different between the right and left sides of the jet because of winding magnetic field lines, which is included in Equation (55) through $\sin \psi ^{\prime} ({\boldsymbol{n}}^{\prime} )$. The intrinsic asymmetry is, however, found to be minor compared to the asymmetry induced by the relativistic beaming.

The luminous counterjet is another notable feature for R_p = 40r_g, as seen in the right panel in Figure 4. The counterjet becomes apparent, in contrast to the observations of the M87 jet (e.g., Hada et al. 2016), since the relativistic boost to poloidal directions is so weak. As shown in the upper middle panel in Figure 2, the poloidal speed on the magnetic field line ${\rm{\Psi }}/A\sim 35.3{r}_{g}$ is relatively small ($| {{\boldsymbol{v}}}_{p}| \lesssim \mathrm{0.7c}$). As a result of the decrease of $| {{\boldsymbol{v}}}_{p}|$ toward the jet edge, δ increases to unity toward the jet edge in the counterjet side, as shown in Figure 3. Note that δ is ∼0.5 on the magnetic field line ${\rm{\Psi }}/A\sim 35.3{r}_{g}$, which is not sufficient to darken the counterjet.

The results for other R_p > 0 are qualitatively the same. We confirmed that the radio images still keep the strong asymmetry for $0\lt {R}_{p}\lt 40{r}_{g}$, as indicated by the asymmetric candle-flame-like image with the brighter northern edge for R_p = 0 in the left panel in Figure 4. The counterjet for $0\lt {R}_{p}\lt 40{r}_{g}$ becomes less luminous than for R_p = 40r_g owing to larger $| {{\boldsymbol{v}}}_{p}|$. We also found for ${R}_{p}\gt 40{r}_{g}$ that the asymmetry of the limb brightening can be weaker thanks to the smaller asymmetry of δ between the right and left sides of the jet, which is due to smaller v_ϕ, but the counterjet becomes more luminous due to smaller $| {{\boldsymbol{v}}}_{p}|$.

We investigate the other case, where Ω_F is a constant given by Equation (17), motivated by the magnetic field lines penetrating the BH. The Kerr parameter is crucial in this case, since it directly controls Ω_F. We thus systematically study the dependence of the radio image on the Kerr parameter as well as the effect of R_p. We pick two extreme cases of a = 0.998 and a = 0.1 as the best examples.

3.2.1. Fast-spinning BH

First, we show the results for a = 0.998, for which Ω_F is $1.4\times {10}^{-5}$ s⁻¹. The upper left panel in Figure 5 shows the color map of $R{{\rm{\Omega }}}_{{\rm{F}}}/c$ in the jet. Since the magnetic field lines rigidly rotate, $R{{\rm{\Omega }}}_{{\rm{F}}}/c$ monotonically increases with R, and the light cylinder is given by $R={R}_{\mathrm{lc},2{\rm{f}}}:= {{\rm{\Omega }}}_{{\rm{F}}}/c\sim 2.1{r}_{g}\sim 8.6\,\times {10}^{-3}$ mas. We emphasize here that the jet structure is qualitatively different from those in the disk-threaded model, in which another curved surface of the light cylinder exists. The magnetic field is thus toroidally dominated in almost the entire region in the jet except for the inside of the thin, light cylinder, as depicted in the lower left panel in Figure 5, which is a sharp contrast to the previous case. As a result, the velocity field is also qualitatively different away from the jet axis: $| {{\boldsymbol{v}}}_{p}|$ becomes highly relativistic $(\sim c)$, and v_ϕ is suppressed to nonrelativistic speed $(\lesssim 0.1c)$, as presented in the upper middle and right panels in Figure 5. Around the jet axis ($R\lesssim {R}_{\mathrm{lc},2{\rm{f}}}$), on the other hand, the velocity field is not much different from that in the previous disk-threaded model, since the magnetic field lines near the jet axis rigidly rotate with comparable angular frequencies in these models (see ${{\rm{\Omega }}}_{{\rm{F}}}\sim 1.4\times {10}^{-5}$ s⁻¹ for this model and ${{\rm{\Omega }}}_{\mathrm{Kep}}({R}_{\mathrm{ISCO}})\sim 2.5\times {10}^{-5}$ s⁻¹ for the previous disk-threaded model). The Lorentz factor is shown in the lower middle panel in Figure 5. The lower right panel exhibits $\mathrm{log}n$ for R_p = 40r_g as an example. The nonthermal electrons are concentrated on the magnetic field lines ${\rm{\Psi }}/A\sim 35.3{r}_{g}$ (black lines), on which the magnetization factor σ is minimized to ∼8 × 10⁵ in the presented region but still holds a sufficiently large value for the force-free approximation.

Zoom In Zoom Out Reset image size

Figure 6 shows δ in the jet. As expected from the slow v_ϕ, the difference in δ is small between the right and left sides with respect to the observer. Due to the large $| {{\boldsymbol{v}}}_{p}|$, δ in the counterjet is suppressed and the radiation is strongly debeamed for the observer. In the approaching jet side, on the other hand, a part of the front side of the jet strongly beams the light to the observer (δ ∼ 5), whereas the back side does not, due to the misalignment of the highly relativistic velocity and the observer direction. We also note that the asymmetry that is due to the anisotropic synchrotron radiation in the fluid rest frame is again found to be negligible.

Zoom In Zoom Out Reset image size

The left panel in Figure 7 displays the synthesized radio map for R_p = 0. Neither a limb-brightened feature nor the counterjet is seen in the radio map as in Case 1 with R_p = 0. This result is again attributed to the strong beaming effect to polar directions that is due to the velocity field near the jet axis.

Zoom In Zoom Out Reset image size

The radio maps for R_p > 0 can successfully show a symmetrically limb-brightened jet without a luminous counterjet, as displayed in the right panel in Figure 7, where the result for R_p = 40r_g is shown for comparison to the counterpart in Figure 4. The symmetry of the jet image is recovered thanks to the small v_ϕ in the outer part of the jet away from the axis, which suppresses the beaming/debeaming asymmetry in the jet northern/southern sides, as displayed in Figure 6. The counterjet is less luminous due to the highly relativistic poloidal speed, which beams the emission to the direction opposite to the observer.

For larger ring radii (${R}_{p}\gt 40{r}_{g}$), the results are qualitatively the same as for R_p = 40r_g, while the width of the jet image becomes wider. For smaller R_p ($0\lt {R}_{p}\lt 40{r}_{g}$), the jet width becomes smaller while keeping the symmetrically limb-brightened feature and gradually approaches the result for R_p = 0.

3.2.2. Slowly Spinning BH

We here show the results for the slowly spinning BH with a = 0.1. The Kerr parameter results in ${{\rm{\Omega }}}_{{\rm{F}}}=7.5\times {10}^{-7}$ s⁻¹, which is $5.3\times {10}^{-2}$ times as large as that for a = 0.998 and shifts the light cylinder outward to the ∼19 times larger radius, $R={R}_{\mathrm{lc},2{\rm{s}}}\sim 40{r}_{g}\sim 0.16$ mas, as well as the other contour lines of $R{{\rm{\Omega }}}_{{\rm{F}}}/c$, as presented in the upper left panel in Figure 8. The change is also reflected in the distribution of the ratio of the toroidal to poloidal magnetic field strengths, as visible in the lower left panel in Figure 8. As a result, the region with slow poloidal speeds and fast azimuthal ones is extended from the jet axis to $R\lesssim {R}_{\mathrm{lc},2{\rm{s}}}$, as visible in the upper middle and right panels in Figure 8. In the outer part of the jet, $R\gg {R}_{\mathrm{lc},2{\rm{s}}}$, v_ϕ is increased by ∼19 times, compared to the case of a = 0.998 at the same radius, since v_ϕ is inversely proportional to $R{{\rm{\Omega }}}_{{\rm{F}}}/c$, as given by Equation (7). The poloidal speeds for $R\gg {R}_{\mathrm{lc},2{\rm{s}}}$ are not much different from those in the previous case of a = 0.998, on the other hand, since it is asymptotically determined by the angle from the jet axis, as given by Equation (6), where $R{{\rm{\Omega }}}_{{\rm{F}}}/c$ appears in the higher order corrections. The resultant Lorentz factor is displayed in the lower middle panel in Figure 8, which has the asymptotically same structure as for a = 0.998 in the jet edge part, due to the dominance of the poloidal speed. The lower right panel presents the density profile of the nonthermal electrons for R_p = 40r_g, which are concentrated on the magnetic field lines ${\rm{\Psi }}/A\sim 35.3{r}_{g}$ (black lines). The minimal value of σ ∼ 5 × 10³ in the displayed area is consistent with the force-free assumption.

Zoom In Zoom Out Reset image size

The plots of δ in Figure 9 clearly exhibit different patterns compared to the case for a = 0.998. In the approaching jet side, δ is more asymmetric between the right and left sides, due to the larger v_ϕ. In the counterjet side, δ is still suppressed almost in the jet edge region, due to highly relativistic $| {{\boldsymbol{v}}}_{p}|$, whereas δ is close to unity inside the light cylinder because of nonrelativistic $| {{\boldsymbol{v}}}_{p}|$.

Zoom In Zoom Out Reset image size

Figure 10 shows the produced radio images for R_p = 0 (left) and R_p = 40r_g (right). Most importantly, the radio image becomes highly asymmetric between the northern and southern parts because of the enhanced relativistic beaming by larger v_ϕ, which is incompatible with the M87 jet. We also note that the counterjet becomes more luminous, which is clearer in the case of R_p = 0 (see the left panels in Figures 7 and 10), due to smaller $| {{\boldsymbol{v}}}_{p}|$ around the jet axis, which relaxes the relativistic beaming to the antidirection to the observer.

Zoom In Zoom Out Reset image size

We also confirmed that the results for $0\lt {R}_{p}\lt 40{r}_{g}$ present extremely asymmetric jets, as inferred from the results for R_p = 0 and 40r_g. The large asymmetry is also maintained for larger R_p in our search up to ${R}_{p}=100{r}_{g}$, which produces a sufficiently wide jet image for M87.

In Section 3, it was found that the disk-threaded model would not produce the symmetrically limb-brightened jets in our model and seems to be inappropriate for the M87 jet. We here discuss whether this result changes or not if we give another velocity different from that given by Equation (5) as the jet velocity. This is worth considering, since the jet edge part in the disk-threaded model corresponds to the region with $\zeta \leqslant 1$, where the drift velocity, Equation (5), may not approach the velocity in cold, ideal MHD, which will be the next simplest approximation. Comparing the drift velocity with the velocity in cold, ideal MHD jets, we argue for expected changes of limb-brightened radio images in the MHD treatment through the modified beaming effects for the observer.

We first focus on the azimuthal speed, since it is critical to the asymmetry in radio images. The toroidal speed in cold, ideal MHD outflows under the steady and axisymmetric assumptions is given as follows (e.g., Toma & Takahara 2013):

$$\begin{eqnarray}&&\displaystyle \frac{{v}_{\phi }}{c}=\displaystyle \frac{1}{\zeta }\left[1-\displaystyle \frac{(1-{\tilde{\zeta }}^{2})\tilde{{\rm{\Gamma }}}}{{\rm{\Gamma }}}\right],\end{eqnarray} \tag{ 19 }$$

where $\zeta := R{{\rm{\Omega }}}_{{\rm{F}}}/c$, and the letters with tilde symbols denote the quantities at the inlet. Here, $\tilde{{\rm{\Gamma }}}\sim 1$ is the initial Lorentz factor at the inlet. The ratio to the azimuthal speed in our force-free model, the second term in Equation (5), is then given by

$$\begin{eqnarray}&&\displaystyle \frac{{v}_{\phi ,\mathrm{MHD}}}{{v}_{\phi ,\mathrm{FF}}}=\displaystyle \frac{{g}_{\mathrm{FF}}^{2}+{\zeta }_{\mathrm{FF}}^{2}}{{g}_{\mathrm{FF}}^{2}{\zeta }_{\mathrm{FF}}{\zeta }_{\mathrm{MHD}}}\left(1-\displaystyle \frac{1-{\tilde{\zeta }}_{\mathrm{MHD}}^{2}}{{{\rm{\Gamma }}}_{\mathrm{MHD}}}\right),\end{eqnarray} \tag{ 20 }$$

where the letters with MHD and FF are evaluated in a cold, ideal MHD model and our force-free one, respectively. As long as the force-free approximation is reasonable, the shapes of the poloidal magnetic fields are the same in ours and in MHD. This assumption of the same-shaped field yields ${\zeta }_{\mathrm{FF}}={\zeta }_{\mathrm{MHD}}=\zeta$. Since ${g}_{\mathrm{FF}}(\theta ,\nu )$ is also determined by the shape of B_p, we can omit the subscript, FF, hereafter: g_FF = g.

If the ratio given by Equation (20) exceeds unity (i.e., ${v}_{\phi ,\mathrm{MHD}}\geqslant {v}_{\phi ,\mathrm{FF}}$), the disk-threaded model (Case 1) will not be preferred even in cold, ideal MHD models, due to more asymmetric limb-brightened features (see Appendix D for the proof that faster rotational speeds always lead to more asymmetric images). From Equation (20), the inequality ${v}_{\phi ,\mathrm{MHD}}\geqslant {v}_{\phi ,\mathrm{FF}}$ holds for

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{\mathrm{MHD}}\geqslant {\left(1-\displaystyle \frac{{g}^{2}{\zeta }^{2}}{{g}^{2}+{\zeta }^{2}}\right)}^{-1}(1-{\tilde{\zeta }}^{2}).\end{eqnarray} \tag{ 21 }$$

We can put $1-{\tilde{\zeta }}^{2}\sim 1$, since we are now interested in the outer jet with $\zeta \lesssim 1$ in Case 1, which roughly corresponds to ${\rm{\Psi }}/A\gtrsim 35.3{r}_{g}$ for $| z| \lt 10$ mas (the black curve in Figure 2) and ${\tilde{\zeta }}^{2}\lesssim 0.01$. Since $g\sim 1$ at high latitudes where the limb brightens (see Figure 12), we can reduce Equation (21) to

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{\mathrm{MHD}}\gtrsim 1+{\zeta }^{2}.\end{eqnarray} \tag{ 22 }$$

That is, if the above condition holds, the simulated radio images for ${R}_{p}\geqslant 40{r}_{g}$ would be more asymmetric in the disk-threaded model with cold, ideal MHD treatment.

Although the actual value of ${{\rm{\Gamma }}}_{\mathrm{MHD}}$ in cold, ideal MHD treatment could be obtained with a detailed model, it is beyond the scope of this paper. Instead, we here consider whether the limb brightening of the M87 jet can emanate from the region with $\zeta \leqslant 1$ by assuming that the pattern speed observed in the M87 jet corresponds to ${{\rm{\Gamma }}}_{\mathrm{MHD}}$. Mertens et al. (2016) reports that the Lorentz factor of the fast component of the M87 jet exceeds ∼2 at $z\gtrsim 3$ mas, which means from Equation (22) that the azimuthal speed should be larger than that in our model provided the limb brightening originates from the region with $\zeta \leqslant 1$. A larger v_ϕ enhances the asymmetry of the radio images, which is not consistent with observations.

A possible change of the poloidal speed would always produce problematic jet images: the counterjet becomes more luminous for smaller $| {{\boldsymbol{v}}}_{p}|$, while the asymmetry of the emission from "the coming quadrisection" of the jet is enhanced for larger $| {{\boldsymbol{v}}}_{p}|$ (see Appendix D).

From the above discussions, the disk-threaded model would not be suitable for the M87 jet even in the cold, ideal MHD treatment, while MHD numerical simulations should be incorporated for more quantitative discussions. It is noted, on the other hand, that Mertens et al. (2016) conjectured a jet launched from a Keplerian accretion disk, based on analyses of observed pattern speeds in the M87 jet with cold, ideal MHD treatment. The reason for this discordance should be pursued, although it is beyond the scope of this paper.

The viewing angle Θ will be another important parameter, as well as Ω_F and R_p, for producing radio images, since it changes the line-of-sight speed, which strongly beams or debeams the synchrotron emission to the observer. While the viewing angle of the M87 jet is thought to be in the range of ∼10° to 45° based on optical observations of superluminal motion around the HST-1 (Biretta et al. 1999) and radio observations of proper motion and brightness ratio of the jet and counterjet (Ly et al. 2007; Hada et al. 2016; Mertens et al. 2016), it will be interesting to study whether the limb-brightened features are kept if the viewing angle is much larger or smaller than the above constraint. Below we set Θ = 5° and 75°, for example, while the other parameters are the same as in Case 2 with a = 0.998 and R_p = 40r_g.

The left panel in Figure 11 presents the result for Θ = 5°, which still shows a limb-brightening feature. As Θ decreases, the jet becomes more luminous, while the counterjet becomes dimmer because of stronger beaming and debeaming effects to the observer. At the same time, the jet image is expanded in the transverse direction by the projection effect.

Zoom In Zoom Out Reset image size

The right panel in Figure 11 displays the case for Θ = 75°. The large viewing angle softens the relativistic beaming to the observer. As a result, the jet becomes less luminous while the counterjet becomes more luminous. The limb-brightened feature is still visible in the jet side while it is also apparent in the counterjet.

As presented above, limb-brightened features are observed even for viewing angles much different from our fiducial value. This fact suggests that the limb-brightened features observed in other objects such as Mrk 501, which has a viewing angle of ${\rm{\Theta }}\sim 5-15^\circ$ (Giroletti et al. 2004, 2008), and Cyg A, which has ${\rm{\Theta }}\sim 75^\circ$ (Boccardi et al. 2016), are also attributed to the jet structure with magnetic field lines penetrating a fast-spinning BH and nonthermal electrons away from the jet axis. It is interesting, however, that Boccardi et al. (2017) came to another conclusion that the jet base of Cyg A is widely extended and appears to be anchored to the accretion disk. We may need more detailed models with parameters tuned for these objects to make credible conclusions, which will be studied in a forthcoming paper.

Steady, axisymmetric electromagnetic fields have been widely considered in the literature (Mestel 1961; Okamoto 1974; Bekenstein & Oron 1978; Camenzind 1986; Tomimatsu & Takahashi 2003; Vlahakis & Königl 2003; Beskin 2009). We review here such fields with the force-free approximation. The basic equations consist of the Maxwell equations and the conservation laws of a fluid coupled with an electromagnetic field.

We start from the relations that are derived only from the steady, axisymmetric condition before imposing the force-free approximation. Analogous to the two-dimensional incompressible flows, a stream function exists for the poloidal magnetic field, by which each magnetic surface is labeled, because of the divergence-free condition of the magnetic field in an axisymmetric geometry. The poloidal magnetic field, B_p, is then given as follows (Narayan et al. 2007; Broderick & Loeb 2009; Toma & Takahara 2013):

$$\begin{eqnarray}\begin{array}{rcl}{{\boldsymbol{B}}}_{p} & = & -\displaystyle \frac{1}{R}\displaystyle \frac{\partial {\rm{\Psi }}}{\partial z}\hat{{\boldsymbol{R}}}+\displaystyle \frac{1}{R}\displaystyle \frac{\partial {\rm{\Psi }}}{\partial R}\hat{{\boldsymbol{z}}}\\ & = & \displaystyle \frac{1}{{r}^{2}\sin \theta }\displaystyle \frac{\partial {\rm{\Psi }}}{\partial \theta }\hat{{\boldsymbol{r}}}-\displaystyle \frac{1}{r\sin \theta }\displaystyle \frac{\partial {\rm{\Psi }}}{\partial r}\hat{{\boldsymbol{\theta }}}=\displaystyle \frac{1}{R}{\boldsymbol{\nabla }}{\rm{\Psi }}\times \hat{{\boldsymbol{\phi }}},\end{array}\end{eqnarray} \tag{ 23 }$$

where ${\rm{\Psi }}:= {{RA}}_{\phi }$ is a stream function with A_ϕ being the toroidal component of the magnetic vector potential. The vectors with a hat are the unit coordinate bases. We note that the stream function Ψ(R,z) is essentially the total magnetic flux penetrating within radius R except for a factor of 2π: that is, Φ = 2πΨ is satisfied for any magnetic flux Φ (Narayan et al. 2007).

The steady, axisymmetric condition reduces the poloidal component of Faraday's law, ${\boldsymbol{\nabla }}\times {\boldsymbol{E}}=0$, to the relation ${E}_{\phi }\equiv 0$, where E_ϕ stands for the toroidal component of the electric field.

In the force-free approximation, the plasma inertia and thermal pressure are neglected in dynamics (Narayan et al. 2007; Broderick & Loeb 2009). In this prescription, the fluid contributes only as the charge and current sources. The equation of motion is, hence, reduced to

$$\begin{eqnarray}&&{\rho }_{e}{\boldsymbol{E}}+\displaystyle \frac{1}{c}{\boldsymbol{j}}\times {\boldsymbol{B}}=0,\end{eqnarray} \tag{ 24 }$$

where ρ_e and j are charge and current densities, respectively.

The projection of both sides of the force-free condition, Equation (24), to the direction of B yields the condition that the magnetic and electric fields are orthogonal to each other: ${\boldsymbol{E}}\cdot {\boldsymbol{B}}=0$. Due to the absence of the toroidal electric field, the orthogonal condition gives the electric field as follows (Lyubarsky 2009; Toma & Takahara 2013):

$$\begin{eqnarray}&&{\boldsymbol{E}}=-\displaystyle \frac{1}{c}{{\rm{\Omega }}}_{{\rm{F}}}{\boldsymbol{\nabla }}{\rm{\Psi }}=-\displaystyle \frac{R{{\rm{\Omega }}}_{{\rm{F}}}}{c}\hat{\phi }\times {\boldsymbol{B}},\end{eqnarray} \tag{ 25 }$$

where ${{\rm{\Omega }}}_{{\rm{F}}}(R,z)$ is a scalar function. That is, the surface of Ψ =const. is also an equipotential surface. Substituting Equation (25) into Faraday's law, we obtain the following conservation law from the toroidal component:

$$\begin{eqnarray}&&{\boldsymbol{B}}\cdot {\boldsymbol{\nabla }}{{\rm{\Omega }}}_{{\rm{F}}}=0,\end{eqnarray} \tag{ 26 }$$

which means that Ω_F is conserved along a magnetic field line and, hence, is a function of Ψ: ${{\rm{\Omega }}}_{{\rm{F}}}={{\rm{\Omega }}}_{{\rm{F}}}({\rm{\Psi }})$ (Toma & Takahara 2013).

The other Maxwell equations recover the corresponding charge and current sources for a given electromagnetic field. The charge density is obtained by Gauss's law (Narayan et al. 2007):

$$\begin{eqnarray}&&{\rho }_{e}=\displaystyle \frac{1}{4\pi }{\boldsymbol{\nabla }}\cdot {\boldsymbol{E}}=-\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{F}}}}{4\pi c}{\rm{\Delta }}{\rm{\Psi }}-\displaystyle \frac{1}{4\pi c}\displaystyle \frac{d{{\rm{\Omega }}}_{{\rm{F}}}}{d{\rm{\Psi }}}| {\boldsymbol{\nabla }}{\rm{\Psi }}{| }^{2},\end{eqnarray} \tag{ 27 }$$

while the current density is given by Ampère's law as follows (Narayan et al. 2007):

$$\begin{eqnarray}&&{j}_{\phi }=\displaystyle \frac{c}{4\pi }{({\boldsymbol{\nabla }}\times {\boldsymbol{B}})}_{\phi }=-\displaystyle \frac{c}{4\pi R}\left(\displaystyle \frac{{\partial }^{2}{\rm{\Psi }}}{\partial {R}^{2}}-\displaystyle \frac{1}{R}\displaystyle \frac{\partial {\rm{\Psi }}}{\partial R}+\displaystyle \frac{{\partial }^{2}{\rm{\Psi }}}{\partial {z}^{2}}\right),\end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray}&&{{\boldsymbol{j}}}_{p}=\displaystyle \frac{c}{4\pi }{({\boldsymbol{\nabla }}\times {\boldsymbol{B}})}_{p}=-\displaystyle \frac{c}{4\pi }\displaystyle \frac{\partial {B}_{\phi }}{\partial z}\hat{{\boldsymbol{R}}}+\displaystyle \frac{c}{4\pi R}\displaystyle \frac{\partial ({{RB}}_{\phi })}{\partial R}\hat{{\boldsymbol{z}}}.\end{eqnarray} \tag{ 29 }$$

The toroidal component of the force-free condition, Equation (24), gives a conservation law for the total poloidal current passing through a toroidal loop of radius R, $I\propto {{RB}}_{\phi }$. In fact, the equation gives (${\boldsymbol{j}}\times {\boldsymbol{B}}{)}_{\phi }=0$ because ${E}_{\phi }\equiv 0$, which is satisfied only if j_p is parallel to B_p. Comparing these poloidal vectors given by Equations (23) and (29), one notices that RB_ϕ should be a function of Ψ (Narayan et al. 2007). That is, RB_ϕ is conserved along a magnetic field line:

$$\begin{eqnarray}&&{\boldsymbol{B}}\cdot {\boldsymbol{\nabla }}({{RB}}_{\phi })=0.\end{eqnarray} \tag{ 30 }$$

We already projected Equation (24) to the directions of B and $\hat{{\boldsymbol{\phi }}}$. Because of the orthogonal relations, ${\boldsymbol{E}}\cdot {\boldsymbol{B}}={\boldsymbol{E}}\cdot \hat{{\boldsymbol{\phi }}}=0$, the projection onto E gives a relation independent of the former ones. The last equation determines Ψ for a given Ω_F and B_ϕ as follows (Narayan et al. 2007):

$$\begin{eqnarray}&&\begin{array}{l}\left[1-{\left(\displaystyle \frac{R{{\rm{\Omega }}}_{{\rm{F}}}}{c}\right)}^{2}\right]\left(\displaystyle \frac{{\partial }^{2}{\rm{\Psi }}}{\partial {R}^{2}}+\displaystyle \frac{{\partial }^{2}{\rm{\Psi }}}{\partial {z}^{2}}\right)-\left[1+{\left(\displaystyle \frac{R{{\rm{\Omega }}}_{{\rm{F}}}}{c}\right)}^{2}\right]\\ \quad \times \,\displaystyle \frac{1}{R}\displaystyle \frac{\partial {\rm{\Psi }}}{\partial R}+\displaystyle \frac{1}{2}\displaystyle \frac{d{({{RB}}_{\phi })}^{2}}{d{\rm{\Psi }}}-\displaystyle \frac{{R}^{2}{{\rm{\Omega }}}_{{\rm{F}}}}{{c}^{2}}\displaystyle \frac{d{{\rm{\Omega }}}_{{\rm{F}}}}{d{\rm{\Psi }}}| {\boldsymbol{\nabla }}{\rm{\Psi }}{| }^{2}=0,\end{array}\end{eqnarray} \tag{ 31 }$$

which gives the shape of the magnetic field that satisfies the force balance in the transfield direction.

Equation (31) becomes singular at the critical surface RΩ_F/c =1, and a regular solution is found only for an appropriate choice of the functional form of B_ϕ for a given Ω_F. Otherwise, the solution cannot be continuous beyond the singular surface (Fendt 1997; Contopoulos et al. 1999; Beskin 2009; Takamori et al. 2014). Since it is generally a tough task to find such a fully consistent regular solution, we use a stream function that approximately describes the force-free numerical solution obtained by Tchekhovskoy et al. (2008), which gives a paraboloid-shaped jet and was also adopted in BL09. The stream function is given by

$$\begin{eqnarray}&&{\rm{\Psi }}={{Ar}}^{\nu }(1\mp \cos \theta ),\end{eqnarray} \tag{ 32 }$$

where A is a constant that has the dimension of $[{r}^{2-\nu }B]$, and ν is the parameter that determines the jet shape. The minus and plus signs are for z ≥ 0 and z < 0, respectively, and the function is symmetric with respect to the equatorial plane, z = 0. We note that Equation (32) is a good approximation to the exact solution of the steady, axisymmetric force-free field, as well as results in numerical simulations (Tchekhovskoy et al. 2008). As special cases, Equation (32) gives a split-monopole field for ν = 0 and a paraboloidal field for ν = 1. Since we are interested in collimated jets, we assume ν > 0 hereafter. The components of the poloidal magnetic field are given by

$$\begin{eqnarray}&&{B}_{r}=\displaystyle \frac{1}{{r}^{2}\sin \theta }\displaystyle \frac{\partial {\rm{\Psi }}}{\partial \theta }=\pm {{Ar}}^{-(2-\nu )}=\pm \displaystyle \frac{{\rm{\Psi }}}{{R}^{2}}(1\pm \cos \theta ),\end{eqnarray} \tag{ 33 }$$

$$\begin{eqnarray}\begin{array}{rcl}{B}_{\theta } & = & -\displaystyle \frac{1}{r\sin \theta }\displaystyle \frac{\partial {\rm{\Psi }}}{\partial r}\\ & = & -\nu {{Ar}}^{-(2-\nu )}\sqrt{\displaystyle \frac{1\mp \cos \theta }{1\pm \cos \theta }}=-\nu \displaystyle \frac{{\rm{\Psi }}}{{R}^{2}}\sin \theta ,\end{array}\end{eqnarray} \tag{ 34 }$$

which yield

$$\begin{eqnarray}&&{B}_{p}=\sqrt{{B}_{r}^{2}+{B}_{\theta }^{2}}=\displaystyle \frac{2{\rm{\Psi }}}{{R}^{2}}g(\theta ,\nu ),\end{eqnarray} \tag{ 35 }$$

where

$$\begin{eqnarray}&&g(\theta ,\nu )=\sqrt{\displaystyle \frac{1\pm \cos \theta }{2}\left[1-(1-{\nu }^{2})\displaystyle \frac{1\mp \cos \theta }{2}\right]}.\end{eqnarray} \tag{ 36 }$$

We henceforth assume $\nu \leqslant \sqrt{2}$ (for the drift speed less than c; see Appendix A.3). Then the function $g(\theta ,\nu )$ satisfies $\sqrt{1+{\nu }^{2}}/2\leqslant g(\theta ,\nu )\leqslant 1$, as shown in Figure 12. We note that $g(\theta ,\nu )$ is reduced to $\cos (\theta /2)$ and $\sin (\theta /2)$ for z ≥ 0 and z < 0, respectively, in the case of ν = 1.

Zoom In Zoom Out Reset image size

Corresponding to the given shape of the jet, Equation (32), B_ϕ is given by (Tchekhovskoy et al. 2008)

$$\begin{eqnarray}&&{B}_{\phi }=\mp \displaystyle \frac{2{{\rm{\Omega }}}_{{\rm{F}}}{\rm{\Psi }}}{{Rc}}=\mp \displaystyle \frac{2{\rm{\Psi }}}{{R}^{2}}\displaystyle \frac{R{{\rm{\Omega }}}_{{\rm{F}}}}{c}.\end{eqnarray} \tag{ 37 }$$

We note that BL09 also use the same prescription for B_ϕ.

The magnitude of the magnetic field is given by Equations (35) and (37) as follows:

$$\begin{eqnarray}&&B=\displaystyle \frac{2{\rm{\Psi }}}{{R}^{2}}\sqrt{{[g(\theta ,\nu )]}^{2}+{\left(\displaystyle \frac{R{{\rm{\Omega }}}_{{\rm{F}}}}{c}\right)}^{2}},\end{eqnarray} \tag{ 38 }$$

which gives the following asymptotic relation:

$$\begin{eqnarray}B\sim \left\{\begin{array}{cc}| {B}_{\phi }| & \mathrm{for}\ \tfrac{R| {{\rm{\Omega }}}_{{\rm{F}}}| }{c}\gg 1\\ {B}_{p} & \mathrm{for}\ \tfrac{R| {{\rm{\Omega }}}_{{\rm{F}}}| }{c}\ll 1.\end{array}\right.\end{eqnarray} \tag{ 39 }$$

The force-free approximation does not give the fluid velocity, since the fluid inertia is totally neglected and, hence, the motion along a magnetic field cannot be determined. Following BL09, we use the so-called drift velocity as the fluid velocity (Narayan et al. 2007):

$$\begin{eqnarray}&&{\boldsymbol{v}}=\displaystyle \frac{{\boldsymbol{E}}\times {\boldsymbol{B}}}{{B}^{2}}c=-R{{\rm{\Omega }}}_{{\rm{F}}}\displaystyle \frac{{B}_{\phi }}{{B}^{2}}{{\boldsymbol{B}}}_{p}+R{{\rm{\Omega }}}_{{\rm{F}}}\displaystyle \frac{{B}_{p}^{2}}{{B}^{2}}\hat{{\boldsymbol{\phi }}}.\end{eqnarray} \tag{ 40 }$$

This prescription ensures that (1) the fluid speed does not exceed the speed of light for $\nu \leqslant \sqrt{2}$, (2) the electric field vanishes in the proper frame, which is consistent with the infinite conductivity, and (3) the velocity asymptotically approaches the fluid velocity in cold, ideal MHD as relativistically accelerated to poloidal directions.

The first and second statements are straightforwardly confirmed by calculation. In fact, the normalized speed of the fluid is given by

$$\begin{eqnarray}&&\beta := \displaystyle \frac{| {\boldsymbol{v}}| }{c}=\displaystyle \frac{E}{B}=\displaystyle \frac{R| {{\rm{\Omega }}}_{{\rm{F}}}| }{c}\displaystyle \frac{{B}_{p}}{B}=\sqrt{\displaystyle \frac{{[g(\theta ,\nu )]}^{2}{\left(\tfrac{R{{\rm{\Omega }}}_{{\rm{F}}}}{c}\right)}^{2}}{{[g(\theta ,\nu )]}^{2}+{\left(\tfrac{R{{\rm{\Omega }}}_{{\rm{F}}}}{c}\right)}^{2}}}\leqslant 1,\end{eqnarray} \tag{ 41 }$$

where the equality holds for $g(\theta ,\nu )=1$ and $R| {{\rm{\Omega }}}_{{\rm{F}}}| /c=\infty$. We also note that the azimuthal speed is bound by $c/2$, which can be shown in the same manner.

Equations (39) and (40) give the asymptotic relations of the fluid velocity for $R| {{\rm{\Omega }}}_{{\rm{F}}}| /c\ll 1$ as follows:

$$\begin{eqnarray}&&\beta \sim {\beta }_{\phi }\sim \displaystyle \frac{R| {{\rm{\Omega }}}_{{\rm{F}}}| }{c},\end{eqnarray} \tag{ 42 }$$

$$\begin{eqnarray}&&{\beta }_{p}\sim \displaystyle \frac{R| {{\rm{\Omega }}}_{{\rm{F}}}| }{c}\displaystyle \frac{| {B}_{\phi }| }{{B}_{p}}={\left(\displaystyle \frac{R{{\rm{\Omega }}}_{{\rm{F}}}}{c}\right)}^{2}\displaystyle \frac{1}{g(\theta ,\nu )},\end{eqnarray} \tag{ 43 }$$

$$\begin{eqnarray}&&{\rm{\Gamma }}:= \displaystyle \frac{1}{\sqrt{1-{\beta }^{2}}}\sim 1+\displaystyle \frac{1}{2}{\left(\displaystyle \frac{R{{\rm{\Omega }}}_{{\rm{F}}}}{c}\right)}^{2},\end{eqnarray} \tag{ 44 }$$

where ${\beta }_{p}:= | {{\boldsymbol{v}}}_{p}| /c$ and ${\beta }_{\phi }=| {v}_{\phi }| /c$ are the normalized poloidal and toroidal speeds, respectively. That is, the fluid velocity is nonrelativistic and dominated by the toroidal component. For $R| {{\rm{\Omega }}}_{{\rm{F}}}| /c\gg 1$, on the other hand, the following relations are obtained:

$$\begin{eqnarray}&&\beta \sim {\beta }_{p}\sim g(\theta ,\nu ),\end{eqnarray} \tag{ 45 }$$

$$\begin{eqnarray}&&{\beta }_{\phi }\sim \displaystyle \frac{R| {{\rm{\Omega }}}_{{\rm{F}}}| }{c}\displaystyle \frac{{B}_{p}^{2}}{{B}_{\phi }^{2}}={\left(\displaystyle \frac{R| {{\rm{\Omega }}}_{{\rm{F}}}| }{c}\right)}^{-1}{[g(\theta ,\nu )]}^{2},\end{eqnarray} \tag{ 46 }$$

$$\begin{eqnarray}&&{\rm{\Gamma }}\sim \displaystyle \frac{1}{\sqrt{1-{[g(\theta ,\nu )]}^{2}}}.\end{eqnarray} \tag{ 47 }$$

That is, the fluid velocity is dominated by the poloidal component, which becomes relativistic as $g(\theta ,\nu )$ approaches unity. We note here that, as $g(\theta ,\nu )\to 1$, the leading terms in Equations (45) and (46) approach those in the asymptotic relations in steady, axisymmetric cold outflows in ideal MHD (Toma & Takahara 2013):

$$\begin{eqnarray}&&{\beta }_{p}\sim 1-\displaystyle \frac{1}{{{\rm{\Gamma }}}^{2}}-{\left(\displaystyle \frac{R{{\rm{\Omega }}}_{{\rm{F}}}}{c}\right)}^{-2}\sim 1,\end{eqnarray} \tag{ 48 }$$

$$\begin{eqnarray}&&{\beta }_{\phi }={\left(\displaystyle \frac{R| {{\rm{\Omega }}}_{{\rm{F}}}| }{c}\right)}^{-1}\left[1-\left(1-\displaystyle \frac{{\tilde{R}}^{2}{{\rm{\Omega }}}_{{\rm{F}}}^{2}}{{c}^{2}}\right)\displaystyle \frac{\tilde{{\rm{\Gamma }}}}{{\rm{\Gamma }}}\right]\sim {\left(\displaystyle \frac{R| {{\rm{\Omega }}}_{{\rm{F}}}| }{c}\right)}^{-1},\end{eqnarray} \tag{ 49 }$$

which holds for $R| {{\rm{\Omega }}}_{{\rm{F}}}| /c\gg 1$ and ${\rm{\Gamma }}\gg \tilde{{\rm{\Gamma }}}\sim 1$, where the letters with tilde symbols denote quantities at the inlet.

The number density of the nonthermal electrons, n, is assumed to be given by the continuity equation for a fluid, ${\boldsymbol{\nabla }}\cdot (n{\boldsymbol{v}})=0$, by following BL09, although it is not so obvious whether the nonthermal electrons obey the equation. For ${{RB}}_{\phi }{{\rm{\Omega }}}_{{\rm{F}}}\ne 0$, the continuity equation is reduced to

$$\begin{eqnarray}&&{\boldsymbol{B}}\cdot {\boldsymbol{\nabla }}\left(\displaystyle \frac{n}{{B}^{2}}\right)=0,\end{eqnarray} \tag{ 50 }$$

which means that n scales with B² along a given magnetic field. We also note that the continuity equation also derives the conservations of the ratio of the mass flux to the magnetic flux as in ideal MHD: ${\boldsymbol{B}}\cdot {\boldsymbol{\nabla }}(n| {{\boldsymbol{v}}}_{p}| /{B}_{p})=0$ by using Equation (40). In this paper, we assume the following ring-shaped distribution of the nonthermal electrons on the planes z = ±z₁ (z₁ ≥ 0):

$$\begin{eqnarray}&&n(R,\pm {z}_{1})={n}_{0}\exp \left[-\displaystyle \frac{{(R-{R}_{p})}^{2}}{2{{\rm{\Delta }}}^{2}}\right],\end{eqnarray} \tag{ 51 }$$

where R_p is the radius where n has the peak on the plane and Δ gives the width of the ring, while n₀ is the number density at the peak. We note that BL09 considered only R_p = 0, where the nonthermal electrons are concentrated on the jet axis at z = ±z₁.

Equations (50) and (51) give the number density of the nonthermal electrons at a given point on a magnetic field labeled by ${\rm{\Psi }}^{\prime}$ as follows:

$$\begin{eqnarray}&&n(R,z)={n}_{0}\displaystyle \frac{{B}^{2}(R,z)}{{B}^{2}({R}_{1},{z}_{1})}\exp \left[-\displaystyle \frac{{({R}_{1}-{R}_{p})}^{2}}{2{{\rm{\Delta }}}^{2}}\right],\end{eqnarray} \tag{ 52 }$$

where ${R}_{1}({\rm{\Psi }}^{\prime} )$ denotes the radial coordinate of the intersections of Ψ = Ψ' and z = ±z₁. We omit an artificial factor of $(1-\exp [-{r}^{2}/{z}_{1}^{2}])$ in Equation (52) that was introduced in BL09 to reduce plasma in the innermost region, $r\lt {z}_{1}$. Our results are not qualitatively different, however, even if the factor is taken into account.

We assume that the distribution of the nonthermal electrons is isotropic in the fluid rest frame and the energy distribution is described by a single power law with an index p:

$$\begin{eqnarray}f(\gamma ^{\prime} )=\left\{\begin{array}{cc}{Cn}^{\prime} {\gamma }^{{\prime} -p} & ({\gamma }_{\min }^{{\prime} }\leqslant \gamma ^{\prime} \leqslant {\gamma }_{\max }^{{\prime} })\\ 0 & \mathrm{otherwise}\end{array}\right.,\end{eqnarray} \tag{ 53 }$$

where and hereafter quantities with a prime are evaluated in the fluid rest frame. Here, γ' is the Lorentz factor of an electron, ${\gamma }_{\min }^{{\prime} }$ and ${\gamma }_{\max }^{{\prime} }$ are the minimal and maximal Lorentz factors, respectively, and C is a normalization constant, which is given for $p\ne 1$ by (Shibata et al. 2003)

$$\begin{eqnarray}&&C=\displaystyle \frac{(p-1){{\gamma }_{\min }^{{\prime} }}^{p-1}}{4\pi \left[1-{\left(\tfrac{{\gamma }_{\min }^{{\prime} }}{\gamma {{\prime} }_{\max }}\right)}^{p-1}\right]}.\end{eqnarray} \tag{ 54 }$$

We assume that the energy distribution is given by Equation (53) in the entire region; that is, we assume some energy supplier that compensates for the energy loss that is due to cooling processes such as the synchrotron cooling.

Since we consider highly relativistic electrons, the synchrotron emission is highly beamed in the direction of the electron motion. In this case, the synchrotron emissivity in the fluid rest frame, ${j}_{\omega ^{\prime} }^{{\prime} }({\boldsymbol{n}}^{\prime} )$, is given by (Rybicki & Lightman 1985; Shibata et al. 2003)

$$\begin{eqnarray}\begin{array}{rcl}{j}_{\omega ^{\prime} }^{{\prime} }({\boldsymbol{n}}^{\prime} ) & = & \displaystyle \frac{\sqrt{3}{e}^{3}{Cn}^{\prime} B^{\prime} \sin [\psi ^{\prime} ({\boldsymbol{n}}^{\prime} )]}{2\pi {m}_{e}{c}^{2}(p+1)}{\left(\displaystyle \frac{{m}_{e}c\omega ^{\prime} }{3{eB}^{\prime} \sin [\psi ^{\prime} ({\boldsymbol{n}}^{\prime} )]}\right)}^{-(p-1)/2}\\ & & \times \,\bar{{\rm{\Gamma }}}\left(\displaystyle \frac{p}{4}+\displaystyle \frac{19}{12}\right)\bar{{\rm{\Gamma }}}\left(\displaystyle \frac{p}{4}-\displaystyle \frac{1}{12}\right),\end{array}\end{eqnarray} \tag{ 55 }$$

where e, m_e, and $\bar{{\rm{\Gamma }}}(\ldots )$ are the elementary charge, the mass of an electron, and the gamma function, respectively. Here, $\psi ^{\prime} ({\boldsymbol{n}}^{\prime} )$ is the pitch angle of the electrons that are directed toward the observer, which are most responsible for producing radio images because of relativistic beaming effects (Shibata et al. 2003):

$$\begin{eqnarray}&&\cos [\psi ^{\prime} ({\boldsymbol{n}}^{\prime} )]=\displaystyle \frac{{\boldsymbol{n}}^{\prime} \cdot {\boldsymbol{B}}^{\prime} }{| {\boldsymbol{B}}^{\prime} | }=\displaystyle \frac{1}{{\rm{\Gamma }}(1-\beta \mu )}\displaystyle \frac{{\boldsymbol{n}}\cdot {\boldsymbol{B}}}{| {\boldsymbol{B}}| }.\end{eqnarray} \tag{ 56 }$$

In the derivation of Equation (55), we used the approximation that ${\gamma }_{\min }^{{\prime} }$ and ${\gamma }_{\max }^{{\prime} }$ are sufficiently small and large, respectively, to evaluate an energy integral (Rybicki & Lightman 1985). In this case, the energy cutoffs affect the synchrotron emissivity only through the normalization constant C.

We study the dependence of radio images on the parameters that were fixed in the main text. It is important to note that our conclusions in the main text are not changed even if these parameters are altered, whereas the radio images are slightly modified. We use the fast-spinning BH-threaded model (Case 2 with a = 0.998) with R_p = 40r_g shown in the right panel in Figure 7 as a fiducial model, since it resembles the observed images better than the other models. We change four parameters, Δ (ring width), ν (jet shape), p (power index of the energy distribution of electrons), and M_BH (BH mass) around the fiducial model as in Table 2 while fixing the other parameters such as Ω_F and a as well as R_p. Comparing the produced radio images, we discuss the effects of each parameter below.

The dependence on Δ is displayed in Figure 13. As naturally expected, the larger Δ makes radio images wider in the north–south direction, since the electrons are more distributed to the edge region, although the effect is rather limited within this range of Δ.

Zoom In Zoom Out Reset image size

The dependence on the jet shape is shown in Figure 14, where the jet is less (more) collimated in the left (right) panel. We note that the jet shape is expressed by $R\propto {z}^{\xi }$ far from the BH ($\theta \ll 1$), where ξ is defined by ν = 2–2ξ (Tchekhovskoy et al. 2008). That is, ν = 0.75, 1, and 1.25 (i.e., ξ = 0.625, 0.5, and 0.375) give the asymptotic jet shape of $z\propto {R}^{8/5}$, R², ${R}^{8/3}$, respectively. The jet shape is clearly reflected in the radio image as tightly collimated jets produce narrower radio images.

Zoom In Zoom Out Reset image size

Figure 15 manifests that the harder energy distribution of electrons leads to more compact radio images. That is, the contrast of intensity is enhanced for larger p, since the difference in the magnetic and velocity fields at different locations is enhanced by $(p-1)/2$, as given in Equation (55). The limb-brightened feature becomes discrete, as a result, for large p, while it is still discernible in Figure 15. We note that Hada et al. (2016) reported $p\sim 2.2\mbox{--}2.6$ for the M87 jet.

Zoom In Zoom Out Reset image size

Massive BHs produce "larger" radio images, as shown in Figure 16. One should be careful in interpreting this result, since the Schwarzschild radius changes as ${r}_{g}\propto {M}_{\mathrm{BH}}$, while we used ${R}_{p}=40{r}_{g}$ in both models. That is, the electrons are distributed more far away from the jet axis in model H with a more massive BH, which directly makes the radio image wider in the X direction. We also note that the BH mass changes Ω_F, which is proportional to ${M}_{\mathrm{BH}}^{-1}$ in the BH-threaded model for a fixed Kerr parameter. Thus, the increase in M_BH for a fixed a has effects similar to the decrease in a for a fixed M_BH (note: ${{\rm{\Omega }}}_{{\rm{F}}}\propto a/(1+\sqrt{1-{a}^{2}})$).

Zoom In Zoom Out Reset image size

We study here the parameter dependence of radio intensity maps of the disk-threaded model (Case 1). We focus on the disk rotation, which characterizes disk-threaded models, and consider sub-Keplerian motion. Introducing a factor q ($0\lt q\leqslant 1$), we modify Equation (15) as follows:

$$\begin{eqnarray}{{\rm{\Omega }}}_{{\rm{F}}}=\left\{\begin{array}{lc}q{{\rm{\Omega }}}_{\mathrm{Kep}}(\tilde{R}) & (\tilde{R}\gt {R}_{\mathrm{ISCO}})\\ q{{\rm{\Omega }}}_{\mathrm{Kep}}({R}_{\mathrm{ISCO}}) & (\tilde{R}\leqslant {R}_{\mathrm{ISCO}})\end{array}\right.,\end{eqnarray} \tag{ 57 }$$

where $0\lt q\lt 1$ gives a sub-Keplerian disk while q = 1 coincides with Case 1. We pick the cases with q = 0.1 and q = 0.5 for example, while keeping the other parameters the same as in Case 1. The former is an extreme case of slowly rotating disks, and the latter corresponds to ADAFs. Figure 17 shows the radio intensity maps for q = 0.1, 0.5, and 1 for reference. As the disk rotation slows down, the radio image recovers the symmetry. However, the limb feature becomes less prominent and the counterjet keeps the brightness, which are inconsistent with observations of M87.

Zoom In Zoom Out Reset image size

These changes in the radio image are explained as follows. As the disk rotational speed decreases, the light "cylinder" surfaces, both the curved and vertical ones, shrink inward. The shrinkage of the light cylinder decreases v_ϕ on ${\rm{\Psi }}/A\sim 35.3{r}_{g}$, where most of the emitting particles exist (black lines in Figure 2), since v_ϕ peaks around the light "cylinder" and decreases toward the jet edge part (see the upper right panel in Figure 2). This is why the asymmetry of radio images is weakened for smaller q. The shrinkage of the light cylinder, at the same time, slows down the poloidal speed, since $| {{\boldsymbol{v}}}_{p}|$ becomes smaller away from the curved light-cylinder surface. Thus, the light emanating from ${\rm{\Psi }}/A\sim 35.3{r}_{g}$ is less beamed as q decreases. As a result, the counterjet keeps the feature and the limb becomes less prominent with respect to the central core, which weakens the limb-brightening feature.