MAGNETIZATION DEGREE AT THE JET BASE OF M87 DERIVED FROM THE EVENT HORIZON TELESCOPE DATA: TESTING THE MAGNETICALLY DRIVEN JET PARADIGM

Elucidating the formation mechanism of relativistic jets in active galactic nuclei (AGNs) is one of the longstanding challenges in astrophysics. Although the magnetically driven jet and wind models are widely discussed in the literature (e.g., Okamoto 1974; Blandford & Znajek 1977; Blandford & Payne 1982; Chiueh et al. 1991; Li et al. 1992; Uchida 1997; Okamoto 1999; Koide et al. 2002; Tomimatsu & Takahashi 2003; Vlahakis & Konigl 2003; McKinney & Gammie 2004; Krolik et al. 2005; McKinney 2006; Komissarov et al. 2007, 2009; Tchekhovskoy et al. 2011; McKinney et al. 2013; Nakamura & Asada 2013; Toma & Takahara 2013), the actual value of the strength of the magnetic field (B) at the base of the jet is still an open problem. In order to test the magnetic jet paradigm, it is most essential to clarify the energy density of the magnetic fields (${{U}_{B}}\equiv B_{{\rm tot}}^{2}/8\pi$) and that of the particles at the upstream end of the jet, where ${{B}_{{\rm tot}}}$ is the strength of the total magnetic fields.

Recently, short-millimeter radio observations at 1.3 mm (equivalent to the frequency 230 GHz) have been performed against the nearby giant radio galaxy, M87. M87 is located at a distance of ${{D}_{L}}=16.7\;{\rm Mpc}$ (Jordán et al. 2005; Blakeslee et al. 2009) and hosts one of the most massive supermassive black holes, ${{M}_{\bullet }}=(3{--}6)\times {{10}^{9}}\;{{M}_{\odot }}$ (e.g., Macchetto et al. 1997; Gebhardt & Thomas 2009; Walsh et al. 2013), and thus M87 is known as the best target for studying the upstream end of the jet (e.g., Junor et al. 1999; Hada et al. 2011). The Schwarzschild radius is ${{R}_{s}}\equiv 2G{{M}_{\bullet }}/{{c}^{2}}\approx 2\times {{10}^{15}}\;{\rm cm}$ for the central black hole with ${{M}_{\bullet }}=6\times {{10}^{9}}\;{{M}_{\odot }},$ where G is the gravitational constant and c is the speed of light. This corresponds to the angular size of $\sim 7\;\mu {\rm as}$. Hereafter, we set this mass as the fiducial one. The Event Horizon Telescope (EHT), composed of stations in Hawaii and the western United States, has detected a compact region at the base of the M87 jet at 230 GHz with its size 40 μas (Doeleman et al. 2012). Furthermore, Kuo et al. (2014) obtained the first constraint on the Faraday rotation measure (RM) for M87, using the submillimeter array (SMA) at 230 GHz.

Short-millimeter VLBI observations of EHT at 230 GHz (equivalent to 1.3 mm) are crucially beneficial in order to minimize the blending effect of substructures below the spatial resolutions of telescopes. Historically, single-dish observations of AGN jets at the centimeter waveband (with arcminutes spatial resolution) revealed that their spectra are flat at the centimeter waveband (Owen et al. 1978). Marscher (1977) suggested the importance of VLBI observations for distinguishing various possible explanations for the observed flatness. Cotton et al. (1980) conducted VLBI observations at the centimeter waveband and found that the flat spectrum results from a blending effect of substructures with the milliarcsecond (μas) scale. This was a significant forward step. However, subsequent VLBI observations have revealed that such μas-scale components still have substructures when observed at higher spatial resolution (i.e., shorter wavelength). This is a vicious circle between telescopes' spatial resolutions and the sizes of the substructures. In the case of M87, we finally start to overcome this problem because the spatial resolution of EHT almost reaches one of the fundamental scales, i.e., the Innermost Stable Circular Orbit (ISCO) scale (Doeleman et al. 2012). Hence, in this work, we will assume the ISCO radius (${{R}_{{\rm ISCO}}}$) to be the minimum size of the jet nozzle.

Motivated by the significant observational progresses by EHT, we explore the magnetization degree (${{U}_{\pm }}/{{U}_{B}}$) in the core of M87 seen at 230 GHz. We note that Doeleman et al. (2012) did not derive ${{B}_{{\rm tot}}}$ and ${{U}_{\pm }}/{{U}_{B}}$, and we will estimate them at the EHT region for the first time. In the theoretical point of view, we have developed the methodology for the estimation of ${{U}_{\pm }}/{{U}_{B}}$ and ${{B}_{{\rm tot}}}$ in Kino et al. (2014; hereafter K14), and it is also applicable to 230 GHz. In K14, we estimated ${{U}_{\pm }}/{{U}_{B}}$ and ${{B}_{{\rm tot}}}$ in the radio core at 43 GHz with ${{\theta }_{{\rm FWHM}}}=110\;\mu {\rm as}$ and 0.7 Jy. We obtained the tight constraint of field strength ($1\;{\rm G}\leqslant {{B}_{{\rm tot}}}\leqslant 15\;{\rm G}$), but the resultant energetics are consistent with either the ${{U}_{\pm }}$-dominated or U_B-dominated ($1\times {{10}^{-5}}\leqslant {{U}_{\pm }}/{{U}_{B}}\leqslant 6\times {{10}^{2}}$) cases. The radio core at 230 GHz with ${{\theta }_{{\rm FWHM}}}=40\;\mu {\rm as}$ directly corresponds to the upstream end of the M87 jet and would then give the tightest constraints for testing the magnetic jet paradigm. The goal of this work is applying the method of K14 to the EHT-detected region and exploring its properties.

Opacity of the EHT region against synchrotron self-absorption (SSA) is critically important. We should emphasize that it is not clear whether the EHT region is SSA-thick or not because short-millimeter VLBI observations are conducted only at 230 GHz, and therefore it is not possible to obtain spectral information at the moment. Intriguingly, Rioja & Dodson (2011) detect the core-shift between 43 and 86 GHz (in Figure 5 in their paper), which means that the radio core at 86 GHz contains the SSA-thick region. With the aid of interferometry observations, we can also infer the turnover frequency. The fluxes measured by IRAM at 89 GHz (Despringre et al. 1996) and by SMA at 230 GHz (Tan et al. 2008) also seem to indicate that the radio core is SSA-thick above 89 GHz, although the data are not obtained simultaneously (see also Abdo et al. 2009). The submillimeter spectrum obtained by ALMA also shows the spectral break above ∼100 GHz (Doi et al. 2013). Therefore, we may infer that the SSA turnover frequency for the EHT region is above ∼100 GHz. As a working hypothesis, we first assume that the EHT region includes the SSA-thick region and apply the method of K14. We will also discuss the fully SSA-thin case in Section 6.

The layout of this paper is as follows. In Section 2, we briefly review the method of K14. In Section 3, we apply the method to the EHT region. In Section 4, the resultant ${{U}_{\pm }}/{{U}_{B}}$ and ${{B}_{{\rm tot}}}$ are presented. In Section 5, we further discuss constraints on the proton component. In Section 6, we discuss the fully SSA-thin case. In Section 7, we summarize the results and give an account of important future work to be pursued. In this work, we define the radio spectral index α as ${{S}_{\nu }}\propto {{\nu }^{-\alpha }}$.

Following K14, here we briefly review the method for constraining the magnetic field and relativistic electrons in radio cores.

2.1. Basic Assumptions

2.2. General Consideration

Given the SSA turnover frequency (${{\nu }_{{\rm ssa}}}$) and the angular diameter size of the emission region at ${{\nu }_{{\rm ssa}}}$, one can uniquely determine ${{B}_{{\rm tot}}}$ and ${{K}_{\pm }},$ where ${{K}_{\pm }}$ is the normalization factor of relativistic (nonthermal) electrons and positrons (e.g., Kellermann & Pauliny-Toth 1969; Burbidge et al. 1974; Jones et al. 1974a, 1974b; Blandford & Rees 1978; Marscher 1987). Recently, K14 points out that the observing frequency is identical to ${{\nu }_{{\rm ssa}}}$ when we can identify the SSA-thick surface at the observing frequency.

As a first step, we assume that the EHT region is a one-zone sphere with isotropic magnetic field (${{B}_{{\rm tot}}}$) and particle distributions in the present work. Locally, we denote (${{B}_{\bot ,{\rm local}}}={{B}_{{\rm tot}}}{\rm sin} \alpha$) as the magnetic-field strength perpendicular to the direction of electron motion (Ginzburg & Syrovatskii 1965 hereafter GS65), where α is the pitch angle between the vectors of electron velocity and the magnetic field (e.g., Rybicki & Lightman 1979). Then, we can obtain pitch-angle-averaged ${{B}_{\bot ,{\rm local}}},$ defined as ${{B}_{\bot }},$ as follows:

$$\begin{eqnarray}&&B_{{\rm tot}}^{2}=\frac{3}{2}B_{\bot }^{2}\end{eqnarray} \tag{ 1 }$$

because $B_{\bot }^{2}=(1/4\pi )\int B_{{\rm tot}}^{2}{{{\rm sin} }^{2}}\alpha d{\Omega}=2B_{{\rm tot}}^{2}/3$. (This is a slightly different definition of ${{B}_{\bot }}$ than given in K14. The corresponding slight changes of numerical factors are summarized in the Appendix.) Because we assume an isotropic field, hereafter we choose the ${{B}_{\bot }}$ direction to the line of sight (LOS).

The number density distribution of relativistic electrons and positrons ${{n}_{\pm }}({{\epsilon }_{\pm }})$ is defined as (e.g., Equation (3.26) in GS65)

$$\begin{eqnarray}&&{{n}_{\pm }}({{\epsilon }_{\pm }})d{{\epsilon }_{\pm }}={{K}_{\pm }}\epsilon _{\pm }^{-p}d{{\epsilon }_{\pm }}\quad ({{\epsilon }_{\pm ,{\rm min} }}\leqslant {{\epsilon }_{\pm }}\leqslant {{\epsilon }_{\pm ,{\rm max} }}),\end{eqnarray} \tag{ 2 }$$

where ${{\epsilon }_{\pm }}={{\gamma }_{\pm }}{{m}_{e}}{{c}^{2}}$, $p=2\alpha +1$, ${{\epsilon }_{\pm ,{\rm min} }}={{\gamma }_{\pm ,{\rm min} }}{{m}_{e}}{{c}^{2}}$, and ${{\epsilon }_{\pm ,{\rm max} }}={{\gamma }_{\pm ,{\rm max} }}{{m}_{e}}{{c}^{2}}$ are the electron energy, spectral index, minimum energy, and maximum energy of relativistic (nonthermal) electrons and positrons, respectively. Although electrons and positrons may have a different heating/acceleration process in ${{e}^{-}}/{{e}^{+}}/p$ mixed plasma (e.g., Hoshino & Arons 1991), here we assume that minimum energies of electrons and positrons are the same for simplicity. By evaluating the emission at the SSA frequency, we obtain

$$\begin{eqnarray}\begin{array}{rcl} {{B}_{\bot }} & = & b\left( p \right){{\left( \frac{{{\nu }_{{\rm ssa},{\rm obs}}}}{1\;{\rm GHz}} \right)}^{5}}{{\left( \frac{{{\theta }_{{\rm obs}}}}{1\;{\rm mas}} \right)}^{4}}{{\left( \frac{{{S}_{{{\nu }_{{\rm ssa}}},{\rm obs}}}}{1\;{\rm Jy}} \right)}^{-2}} \\ {} & {} & \times \left( \frac{\delta }{1+z} \right), \\ \end{array}\end{eqnarray} \tag{ 3 }$$

where $b(p)$ is tabulated in Marscher (1983), Hirotani (2005), and K14. The term ${{K}_{\pm }}$ is given by

$$\begin{eqnarray}\begin{array}{rcl} {{K}_{\pm }} & = & k\left( p \right){{\left( \frac{{{D}_{{\rm A}}}}{1\;{\rm Gpc}} \right)}^{-1}}{{\left( \frac{{{\nu }_{{\rm ssa},{\rm obs}}}}{1\;{\rm GHz}} \right)}^{-2p-3}}{{\left( \frac{{{\theta }_{{\rm obs}}}}{1\;{\rm mas}} \right)}^{-2p-5}} \\ {} & {} & \times {{\left( \frac{{{S}_{{{\nu }_{{\rm ssa}}},{\rm obs}}}}{1\;{\rm Jy}} \right)}^{p+2}}{{\left( \frac{\delta }{1+z} \right)}^{-p-3}}, \\ \end{array}\end{eqnarray} \tag{ 4 }$$

where $k(p)$ is tabulated in K14. The cgs units of ${{K}_{\pm }}$ and $k(p)$ depend on p: erg $^{p-1}$ cm⁻³. It is useful to show the explicit expression of the ratio ${{U}_{\pm }}/{{U}_{B}}$ as follows:

$$\begin{eqnarray}&&\begin{array}{l} \frac{{{U}_{\pm }}}{{{U}_{B}}}=\frac{16\pi }{3{{b}^{2}}(p)}\frac{k(p)\epsilon _{\pm ,{\rm min} }^{-p+2}}{\left( p-2 \right)}{{\left( \frac{{{D}_{{\rm A}}}}{1\;{\rm Gpc}} \right)}^{-1}}{{\left( \frac{{{\nu }_{{\rm ssa},{\rm obs}}}}{1\;{\rm GHz}} \right)}^{-2p-13}} \\ \qquad \times {{\left( \frac{{{\theta }_{{\rm obs}}}}{1\;{\rm mas}} \right)}^{-2p-13}}{{\left( \frac{{{S}_{{{\nu }_{{\rm ssa}}},{\rm obs}}}}{1\;{\rm Jy}} \right)}^{p+6}}{{\left( \frac{\delta }{1+z} \right)}^{-p-5}} \\ \qquad ({\rm for}\quad p\gt 2). \\ \end{array}\end{eqnarray} \tag{ 5 }$$

From this, we find that ${{\nu }_{{\rm ssa},{\rm obs}}}$ and ${{\theta }_{{\rm obs}}}$ have the same dependence on p. Using this relation, we can estimate ${{U}_{\pm }}/{{U}_{B}}$ without the minimum energy (equipartition B field) assumption. It is clear that the measurement of ${{\theta }_{{\rm obs}}}$ is crucial for determining ${{U}_{\pm }}/{{U}_{B}}$.

We further impose two general constraint conditions.

• 1.  
  Time-averaged total power of the jet (${{L}_{{\rm jet}}}$) estimated by jet dynamics at large scale should not be exceeded by the one at the jet base

  $$\begin{eqnarray}\begin{array}{rcl} {{L}_{{\rm jet}}} & \geqslant & {\rm max} \left[ {{L}_{{\rm poy}}},{{L}_{\pm }} \right], \\ {{L}_{\pm }} & = & \frac{4\pi }{3}{{{\Gamma}}^{2}}\beta {{R}^{2}}c{{U}_{\pm }}, \\ {{L}_{{\rm poy}}} & = & \frac{4\pi }{3}{{{\Gamma}}^{2}}\beta {{R}^{2}}c{{U}_{B}}, \\ \end{array}\end{eqnarray} \tag{ 6 }$$

  where ${{L}_{\pm }}$, ${{L}_{{\rm poy}}}$, ${\Gamma}$, and $\beta c$ are electron/positron kinetic power, Poynting power, bulk Lorentz factor, and bulk speed of the jet at the EHT region, respectively. Note that U_B, ${{U}_{\pm }}$, and R are directly constrained by the VLBI observations.
• 2.  
  The minimum Lorentz factor of the relativistic electrons and positrons (${{\gamma }_{\pm ,{\rm min} }}$) should be smaller than the ones radiating the observed synchrotron emission (${{\nu }_{{\rm syn},{\rm obs}}}$); for example, 230 GHz. Otherwise, we would not be able to observe the synchrotron emission at the corresponding frequency. This is generally given by

  $$\begin{eqnarray}&&{{\nu }_{{\rm syn},{\rm obs}}}\geqslant 1.2\times {{10}^{6}}{{B}_{\bot }}\gamma _{\pm ,{\rm min} }^{2}\frac{\delta }{1+z}.\end{eqnarray} \tag{ 7 }$$

  These relations significantly constrain the allowed values of ${{\gamma }_{\pm ,{\rm min} }}$ and ${{B}_{{\rm tot}}}$.

In the next section, we will add another constraint condition (i.e., minimum size limit).

Here we apply the method to the EHT region in M87.

3.1. On the Basic Physical Quantities

3.2. Difficulties for the SSA-thick One-zone Model

First, we estimate the magnetic-field strength in the EHT region by assuming that all of the EHT region with ${{\theta }_{{\rm FWHM}}}=40\;\mu {\rm as}$ is fully SSA-thick. The field strength of the EHT region is estimated as

$$\begin{eqnarray}&&\begin{array}{l} {{B}_{{\rm tot}}}=3.4\times {{10}^{2}}\;{\rm G}{{\left( \frac{{{\nu }_{{\rm ssa},{\rm obs}}}}{230\;{\rm GHz}} \right)}^{5}} \\ \quad \times {{\left( \frac{{{\theta }_{{\rm obs}}}}{72\;\mu {\rm as}} \right)}^{4}}{{\left( \frac{{{S}_{{{\nu }_{{\rm ssa}}},{\rm obs}}}}{1.0\;{\rm Jy}} \right)}^{-2}}\left( \frac{\delta }{1+z} \right). \\ \end{array}\end{eqnarray} \tag{ 10 }$$

Marscher (1983) pointed out VLBI-measured ${{\theta }_{{\rm FWHM}}}$ is connected with true angular size ${{\theta }_{{\rm obs}}}$ by the relation ${{\theta }_{{\rm obs}}}\approx 1.8{{\theta }_{{\rm FWHM}}}$ for partially resolved sources (see also Krichbaum et al. 2006; Loeb & Waxman 2007). Taking such deviation into account, we examine the case of $72\;\mu {\rm as}=1.8\times 40\;\mu {\rm as}$ for the estimate of B-field strength.

What happens with this field strength?

3.2.1. Too-large Poynting Power

A severe problem arises if ${{B}_{{\rm tot}}}\approx 3\times {{10}^{2}}\;{\rm G}$ is realized. Because we assume a nearly isotropic random field, which can be supported by the low linear polarization degree at 230 Ghz (Kuo et al. 2014), the corresponding Poynting power is given by

$$\begin{eqnarray}\begin{array}{rcl} {{L}_{{\rm poy}}} & = & 1.5\times {{10}^{47}}\;{\rm erg}\;{{{\rm s}}^{-1}} \\ {} & {} & \times {{\left( \frac{{{B}_{{\rm tot}}}}{300\;{\rm G}} \right)}^{2}}{{\left( \frac{2R}{1.8\times {{10}^{16}}\;{\rm cm}} \right)}^{2}}. \\ \end{array}\end{eqnarray} \tag{ 11 }$$

Here we adopt $2R=1.8\times {{10}^{16}}\;{\rm cm}=1.8\times 40\;\mu as\times 16.7\;{\rm Mpc}$. When the total power of the jet (i.e., the sum of the kinetic and Poynting ones) is conserved along the jet at a large scale, then this is too large compared with the jetʼs mean kinetic power inferred from its large-scale dynamics of a few $\times {{10}^{44}}\;{\rm erg}\;{{{\rm s}}^{-1}}$ (e.g., Rieger & Aharonian 2012 for review) We emphasize that a constraint on ${{B}_{{\rm tot}}}$ by ${{L}_{{\rm poy}}}$ is almost model-independent.

If we allow some kind of fast magnetic reconnection processes (e.g., Kirk & Skjæraasen 2003; Bessho & Bhattacharjee 2007; Bessho & Bhattacharjee 2012; Takamoto et al. 2012), in order to dissipate the magnetic fields at the EHT region, then fast and large variabilities would be naturally expected. However, there is no observational support for such variabilities. Therefore, it seems difficult to realize too-large ${{B}_{{\rm tot}}}$ at the EHT region.

3.2.2. Too-fast Synchrotron Cooling

Once we obtain a typical value of ${{B}_{{\rm tot}}}$, then we can estimate a typical synchrotron-cooling timescale. It significantly characterizes the observational behavior of the EHT region. The synchrotron-cooling timescale is correspondingly

$$\begin{eqnarray}&&{{t}_{\pm ,{\rm syn}}}\approx 1\times {{10}^{-2}}\;{\rm day}{{\left( \frac{{{B}_{{\rm tot}}}}{300\;{\rm G}} \right)}^{-2}}{{\left( \frac{{{\gamma }_{\pm }}}{10} \right)}^{-1}}.\end{eqnarray} \tag{ 12 }$$

This is much shorter than the day scale, although the flux at 230 GHz measured by the EHT remains constant during the subsequent 3 days (see the Supplementary Material of Doeleman et al. 2012). Then, a difficulty arises due to this short ${{t}_{\pm ,{\rm syn}}}$. The 230 GHz radio-emitting electrons are in the so-called fast cooling regime (Sari et al. 1998) in which injected electrons instantaneously cool down by synchrotron cooling. Hence, a slight change/fluctuation of B-field strength instantaneously (on timescale ${{t}_{\pm ,{\rm syn}}}$) is reflected on the synchrotron flux at the EHT region. Hence, for realizing the observed constant flux, a constant plasma supply of ${{B}_{{\rm tot}}}$ and ${{K}_{\pm }}$ with very small fluctuation is required to avoid rapid variability/decrease of the synchrotron flux. On the other hand, when the magnetic fields are not that large, ${{t}_{\pm ,{\rm syn}}}$ can become longer than the day scale. Then, we can avoid the rapid variability/decrease of the synchrotron flux without imposing a very small fluctuation of ${{B}_{{\rm tot}}}$ and ${{K}_{\pm }}$ in the bulk flow. Because some fine-tuning of  the ${{B}_{{\rm tot}}}$ and ${{K}_{\pm }}$ injection may be able to adjust the observed constant flux density at the EHT region, the too-fast cooling problem may be less severe than the aforementioned problem on too-large ${{L}_{{\rm poy}}}$. However, it is natural to suppose that smaller ${{B}_{{\rm tot}}}$ realizes in the EHT region to avoid fine-tuning of injection quantities.

3.3. Two-zone Model

3.3.1. Basic Idea

The difficulty of too-large ${{L}_{{\rm poy}}}$ can be resolved if the EHT region is composed of SSA-thick and SSA-thin regions, and the angular size of the SSA-thick region (${{\theta }_{{\rm thick}}}$) is more compact than ${{\theta }_{{\rm obs}}}$, i.e.,

$$\begin{eqnarray}&&{{\theta }_{{\rm obs}}}\gt {{\theta }_{{\rm thick}}}.\end{eqnarray} \tag{ 13 }$$

We show an illustration of our scenario in Figure 1. In this solution, most of the correlated flux density detected by EHT is attributed to the emission from the SSA-thin region. Because the ${{\nu }_{{\rm ssa}}}$ of the SSA-thin region is by definition smaller than 230 GHz, the magnetic field must be significantly smaller because the field strength is proportional to $\nu _{{\rm ssa}}^{5}$. For of this reason, we regard the SSA-thick region as the main carrier of the Poynting power.

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Here, we assume the ISCO radius for a nonrotating black hole (${{R}_{{\rm ISCO}}}=6G{{M}_{\bullet }}/{{c}^{2}}=3{{R}_{s}}\equiv {{\theta }_{{\rm ISCO}}}{{D}_{L}}$) as the minimum size of the SSA-thick region. This corresponds to the angular size, $21\;\mu {\rm as}$. Indeed, theoretical works (Broderick & Loeb 2009; Lu et al. 2014) comparing the EHT observations and jet models also indicate model images with short-millimeter bright region,⁵ with their size comparable to ISCO. Therefore, we examine the range of the SSA-thick region, ${{\theta }_{{\rm thick}}}\geqslant 21\;\mu {\rm as}$.

3.3.2. Gaussian Fitting with Two Components

In Figure 2, we estimate the correlated flux density of this SSA-thick region based on the EHT data. The observed flux data, plotted as a function of baseline length, are adopted from Doeleman et al. (2012). The black solid curve is the best-fit circular Gaussian model by Doeleman et al. (2012). The red solid curve is the best-fit model. The red dashed and dot-dashed curves represent the SSA-thick and the SSA-thin components, respectively.

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Below, we explain the details of the Gaussian fitting. To determine the correlated flux density for the compact SSA-thick region with its lower limit size, ${{\theta }_{{\rm FWHM}}}=21\;\mu {\rm as}/1.8=11.1\;\mu {\rm as},$ we conduct the two-component (SSA-thick and thin components) Gaussian fitting to the EHT data. First, we obtain the upper limit of the correlated flux density for the SSA-thick component as ${{S}_{\nu }}=0.27\;{\rm Jy}$. Next, we perform the the two-component (SSA-thick and thin components) Gaussian fitting by fixing ${{\theta }_{{\rm FWHM}}}=21\;\mu {\rm as}/1.8=11.1\;\mu {\rm as}$ and ${{S}_{\nu }}=0.27\;{\rm Jy}$. Then, we obtain the corresponding size and flux of the extended SSA-thin component, ${{S}_{\nu }}=0.75\;{\rm Jy}$ and ${{\theta }_{{\rm FWHM}}}=60\;\mu {\rm as}$.

Here, we limit on ${{B}_{{\rm tot}}}$, ${{\theta }_{{\rm thick}}}$, and ${{U}_{\pm }}/{{U}_{B}}$, in the EHT region without assuming plasma composition. The critical value, ${{\gamma }_{\pm ,{\rm min} }},$ is derived by the combination of the jet power limit (Equation (6)) and the synchrotron emission limit (Equation (7)). By eliminating ${{B}_{{\rm tot}}}$, we obtain

$$\begin{eqnarray}&&\begin{array}{l} {{\gamma }_{\pm ,{\rm min} }}\leqslant 1.2\times {{10}^{2}} \\ \quad \times {{\left( \frac{2R}{1.8\times {{10}^{16}}\;{\rm cm}} \right)}^{1/2}}{{\left( \frac{{{L}_{{\rm jet}}}}{5\times {{10}^{44}}\;{\rm erg}\;{{{\rm s}}^{-1}}} \right)}^{-1/4}}, \\ \end{array}\end{eqnarray} \tag{ 14 }$$

where ${{\nu }_{{\rm ssa}}}=230\;{\rm GHz}$ is used. Because ${{\gamma }_{\pm ,{\rm min} }}$ has R dependence, larger R allows slightly larger ${{\gamma }_{\pm ,{\rm min} }}$.

In Figure 3, we show the value of ${\rm log} ({{U}_{\pm }}/{{U}_{B}})$ in the allowed ranges of ${{\gamma }_{\pm ,{\rm min} }}$ and ${{B}_{{\rm tot}}}$ with ${{L}_{{\rm jet}}}=5\times {{10}^{44}}\;{\rm erg}\;{{{\rm s}}^{-1}}$ and $p=3.0$. It is essential to note that the maximum value of ${{B}_{{\rm tot}}}$ is determined by the condition ${{L}_{{\rm poy}}}\leqslant {{L}_{{\rm jet}}}$ whereas the minimum value of ${{B}_{{\rm tot}}}$ is governed by the condition of ${{\theta }_{{\rm thick}}}\geqslant {{R}_{{\rm ISCO}}}/{{D}_{L}}\approx 21\;\mu {\rm as}$. The right side of the allowed region is determined by the ${{\nu }_{{\rm syn},{\rm obs}}}$ limit shown in Equation (7). Note that the maximum value of ${{\theta }_{{\rm thick}}}=26.3\;\mu {\rm as}$ is smaller than $40\;\mu {\rm as}$. This suggests that the EHT region has a more compact SSA-thick component in it. Interestingly, overall the SSA-thick region satisfies ${{U}_{B}}\gg {{U}_{\pm }}$. If protons do not contribute to jet energetics, then this result supports the magnetically driven jet scenario. In Table 1 we show the resultant allowed values. Summing up, we find that (1) the allowed ${{\theta }_{{\rm thick}}}$ satisfies $21\;\mu {\rm as}\leqslant {{\theta }_{{\rm thick}}}\leqslant 26.3\;\mu {\rm as}$ and that (2) the allowed field strength is $50\;{\rm G}\leqslant {{B}_{{\rm tot}}}\leqslant 124\;{\rm G}$.

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Table 1.  Results When the EHT Region Contains the SSA-thick Region

Lj	Allowed Btot	Allowed θthick	Allowed U±/UB
(erg s−1)	(G)	(μas)
5 × 1044	50 ≤ Btot ≤ 124	21 ≤ θthick ≤ 26.3	$7.9\times {{10}^{-7}}\leqslant \frac{{{U}_{\pm }}}{{{U}_{B}}}\leqslant 2.3\times {{10}^{-3}}$

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When we choose a smaller ${{L}_{j}}$, the upper limit of ${{\theta }_{{\rm thick}}}$ and ${{B}_{{\rm tot}}}$ becomes smaller, according to Equations (3) and (6). When ${{L}_{j}}=1\times {{10}^{44}}\;{\rm erg}\;{{{\rm s}}^{-1}}$, the allowed ${{B}_{{\rm tot}}}$ and ${{\theta }_{{\rm thick}}}$ are $50\;{\rm G}\leqslant {{B}_{{\rm tot}}}\leqslant 65\;{\rm G}$ and $21\;\mu {\rm as}\leqslant {{\theta }_{{\rm thick}}}\leqslant 22.4\;\mu {\rm as}$. In this case, the allowed regions of ${{B}_{{\rm tot}}}$ and ${{\theta }_{{\rm thick}}}$ are very narrow.

We add a short comment on the brightness temperature. The brightness temperature of the SSA-thick region can be estimated as

$$\begin{eqnarray}\begin{array}{rcl} {{T}_{b}} & = & \frac{{{S}_{\nu ,{\rm obs}}}{{c}^{2}}}{2\pi k\nu _{{\rm syn},{\rm obs}}^{2}{{\left( {{\theta }_{{\rm thick}}}/2 \right)}^{2}}} \\ {} & {} & \approx 2\times {{10}^{10}}\;K\left( \frac{{{S}_{\nu ,{\rm obs}}}}{0.27\;{\rm Jy}} \right){{\left( \frac{{{\theta }_{{\rm thick}}}}{21\;\mu {\rm as}} \right)}^{-2}}, \\ \end{array}\end{eqnarray} \tag{ 15 }$$

where ${{\nu }_{{\rm syn},{\rm obs}}}=230\;{\rm GHz}$. This value is comparable with the T_b at 86 GHz estimated by Lee (2013).

Last, it is worthwhile to add the following. According to the equation of state in the relativistic temperature regime (e.g., Chandrasekhar 1967; Kato et al. 1998), we obtain⁶

$$\begin{eqnarray}&&{{\gamma }_{\pm ,{\rm ssa}}}\approx 3\frac{k{{T}_{b}}}{{{m}_{e}}{{c}^{2}}}\approx 10{{\left( \frac{{{\theta }_{{\rm thick}}}}{21\;\mu {\rm as}} \right)}^{-2}},\end{eqnarray} \tag{ 16 }$$

where we use the fact that ${{\gamma }_{\pm ,{\rm min} }}{{m}_{e}}{{c}^{2}}$ can be identical to the average energy of the electrons and positrons because p is steeper than 2. The obtained ${{\gamma }_{\pm ,{\rm ssa}}}$ tends to be smaller than the minimum Lorentz factor obtained by Equation (14) by a factor of a few. While we may use ${{\gamma }_{\pm ,{\rm ssa}}}$ as ${{\gamma }_{\pm ,{\rm min} }}$, we conservatively use the condition Equation (14), taking some uncertainty of a numerical factor in Equation (14) into account.

In Section 5, we investigate constraint on the energy density of protons (U_p) by using Faraday RM measured by Kuo et al. (2014). From the measured RM, we will constrain the number density of the protons (n_p). Then, we examine U_p. The degree of proton contribution in the energetics has a significant influence over the relativistic jet formation (e.g., Begelman et al. 1984; Reynolds et al. 1996).

5.1. Further Assumptions

5.2. RM Limit

Here we introduce a new constraint using the RM observation data. This RM is important for estimating the kinetic power of the protons (L_p) because RM can constrain the proton number density. Generally speaking, an observed rotation measure (${\rm R}{{{\rm M}}_{{\rm obs}}}$) consists of two parts, i.e., RM by internal (jet) (${\rm R}{{{\rm M}}_{{\rm jet}}}$) and RM by external (foreground) matter (${\rm R}{{{\rm M}}_{{\rm ext}}}$). Therefore, the ${\rm R}{{{\rm M}}_{{\rm obs}}}$ can be decomposed into

$$\begin{eqnarray}&&{\rm R}{{{\rm M}}_{{\rm obs}}}={\rm R}{{{\rm M}}_{{\rm jet}}}+{\rm R}{{{\rm M}}_{{\rm ext}}}.\end{eqnarray} \tag{ 17 }$$

Basically, it is difficult to decouple ${\rm R}{{{\rm M}}_{{\rm jet}}}$ and ${\rm R}{{{\rm M}}_{{\rm ext}}}$ and obtain ${\rm R}{{{\rm M}}_{{\rm jet}}}$. However, it may be possible to discuss an upper limit of $|{\rm R}{{{\rm M}}_{{\rm jet}}}|$ with some reasonable assumptions. When the observed RM (${\rm R}{{{\rm M}}_{{\rm obs}}}$) is comparable to ${\rm R}{{{\rm M}}_{{\rm ext}}}$, then we obtain

$$\begin{eqnarray}&&{\rm R}{{{\rm M}}_{{\rm obs}}}\approx {\rm R}{{{\rm M}}_{{\rm ext}}},\quad |{\rm R}{{{\rm M}}_{{\rm jet}}}|\ll {\rm R}{{{\rm M}}_{{\rm obs}}}.\end{eqnarray} \tag{ 18 }$$

Indeed, the foreground Faraday screen in close vicinity of the jets seems to well explain the observed ${\rm R}{{{\rm M}}_{{\rm obs}}}$ for radio-loud AGNs (e.g., Zavala & Taylor 2004). The explicit form of ${\rm R}{{{\rm M}}_{{\rm jet}}},$ the rotation measure for relativistic plasma, is given as

$$\begin{eqnarray}\begin{array}{rcl} |{\rm R}{{{\rm M}}_{{\rm jet}}}| & = & \frac{{{e}^{3}}}{2\pi m_{e}^{2}{{c}^{4}}}\int dl{{B}_{{\rm LOS}}}{{n}_{p}}\frac{{\rm log} {{\gamma }_{\pm ,{\rm min} }}}{2\gamma _{\pm ,{\rm min} }^{2}} \\ {} & {} & \leqslant 5.36\times {{10}^{3}}{{B}_{{\rm tot}}}{{n}_{p}}\frac{{\rm log} {{\gamma }_{\pm ,{\rm min} }}}{2\gamma _{\pm ,{\rm min} }^{2}} \\ {} & {} & \times \left( \frac{R}{{{10}^{16}}\;{\rm cm}} \right)\;{\rm rad}\;{{{\rm m}}^{-2}}, \\ \end{array}\end{eqnarray} \tag{ 19 }$$

where we set $\int dl\approx 2R$ because the region is assumed as uniform. From this, we see that Faraday rotation is strongly suppressed in relativistic plasma (Jones & Odell 1977; Quataert & Gruzinov 2000; Broderick & McKinney 2010). Note that RM only includes the ionic plasma contribution and does not include the electron/positron-pair plasma. This is because the electron and positron have the same mass but have opposite (i.e., minus and plus) charges, and then the net Faraday rotation by them is canceled out. Qualitatively saying, the mixture of the ${{e}^{\pm }}$-pair plasma (i.e., $\eta \lt 1$) effectively reduces the value of ${\rm R}{{{\rm M}}_{{\rm jet}}}$.

Regarding the RM limit of M87, Kuo et al. (2014) have measured $|{\rm R}{{{\rm M}}_{{\rm obs}}}|\approx (3.4-7.5)\times {{10}^{5}}\;{\rm rad}\;{{{\rm m}}^{-2}},$ and they assume ${\rm R}{{{\rm M}}_{{\rm obs}}}\approx {\rm R}{{{\rm M}}_{{\rm ext}}}$. Following Kuo et al. (2014), we also assume ${\rm R}{{{\rm M}}_{{\rm obs}}}\approx {\rm R}{{{\rm M}}_{{\rm ext}}}$. Then, the RM limit can be written as

$$\begin{eqnarray}&&|{\rm R}{{{\rm M}}_{{\rm jet}}}|\leqslant 1\times {{10}^{5}}\;{\rm rad}\;{{{\rm m}}^{-2}}.\end{eqnarray} \tag{ 20 }$$

Note that the above constraint only gives the upper limit of n_p. Therefore, the finite value of $|{\rm R}{{{\rm M}}_{{\rm jet}}}|$ does not exclude the plasma composition of pure ${{e}^{\pm }}$ plasma.

In Section 5.4, we will constrain the proton contributions in the case of ${{B}_{{\rm tot}}}\approx {{B}_{{\rm LOS}}}$ in Equation (19). At the moment, this is the only case that we can deal with within this simple framework.

5.3. Plasma Composition and the ${{e}^{\pm }}/p$-coupling Rate

To further constrain physical properties at the jet base, here we introduce the basic plasma properties and define general notations. The number densities of protons (n_p), positrons (${{n}_{+}}$), and electrons (${{n}_{-}}$) are defined as follows, respectively:

$$\begin{eqnarray}\begin{array}{rcl} {{n}_{p}} & \equiv & \eta {{n}_{-}}, \\ {{n}_{+}} & = & (1-\eta ){{n}_{-}}\quad (0\leqslant \eta \leqslant 1), \\ {{n}_{p}} & = & {{n}_{{{e}^{-}}p}}\approx \frac{\eta }{2-\eta }\frac{1}{p-1}{{K}_{\pm }}\gamma _{\pm ,{\rm min} }^{-p+1}, \\ \end{array}\end{eqnarray} \tag{ 21 }$$

where η is a free parameter describing the proton-loading in the jet. Here we use the charge neutrality condition in the jet. It is convenient to define further quantities:

$$\begin{eqnarray}\begin{array}{rcl} {{n}_{\pm }} & \equiv & {{n}_{-}}+{{n}_{+}}=(2-\eta ){{n}_{-}}, \\ {{n}_{{{e}^{-}}p}} & \equiv & \eta {{n}_{-}}={{n}_{p}}, \\ \end{array}\end{eqnarray} \tag{ 22 }$$

where ${{n}_{\pm }}$ and ${{n}_{{{e}^{-}}p}}$ are the number density of the electrons and positrons and that of the proton-associated electrons, respectively. The case of $\eta =0$ corresponds to pure ${{e}^{\pm }}$ plasma, whereas $\eta =1$ corresponds to the pure ${{e}^{-}}/p$ plasma. Next, it is important to clarify the energy balances between the electrons and protons. It is useful to introduce the parameter ζ defining the average energy ratio between the protons and electrons as

$$\begin{eqnarray}&&{{\epsilon }_{\pm }}\equiv \zeta {{\epsilon }_{p}},\quad (\frac{{{m}_{e}}}{{{m}_{p}}}\leqslant \zeta \leqslant 1),\end{eqnarray} \tag{ 23 }$$

where ${{\epsilon }_{\pm }}$ is the average energy of the relativistic ${{e}^{\pm }}$. The case $\zeta =1$ can be realized for equipartition between the electrons, positrons, and protons via effective ${{e}^{\pm }}/p$ coupling, whereas $\zeta ={{m}_{e}}/{{m}_{p}}$ means inefficient ${{e}^{\pm }}/p$ coupling, for example, through randomization of bulk kinetic energy of the jet flow (e.g., Kino et al. 2012 and reference therein). Because we focus on the case of $p\gt 2$ suggested in M87 (Doi et al. 2013), relativistic electrons at minimum Lorentz factors characterize the total energetics. Here, ${{\epsilon }_{\pm }}\approx {{\gamma }_{\pm ,{\rm min} }}{{m}_{e}}{{c}^{2}}$ can be estimated as $0.5\;{\rm MeV}\leqslant {{\epsilon }_{\pm }}\leqslant 50\;{\rm MeV}$ together with $1\leqslant {{\gamma }_{\pm ,{\rm min} }}\leqslant 100$ based on the obtained ${{\gamma }_{\pm ,{\rm min} }}$. Then, the case $\zeta =1$ corresponds to that of nonrelativistic protons ($0.5\;{\rm MeV}\leqslant {{\epsilon }_{p}}\leqslant 50\;{\rm MeV}$), whereas the case $\zeta ={{m}_{e}}/{{m}_{p}}$ coincides with that of relativistic protons ($1\;{\rm GeV}\leqslant {{\epsilon }_{p}}\leqslant 100\;{\rm GeV}$).

In general, L is decomposed to

$$\begin{eqnarray}\begin{array}{rcl} {{L}_{{\rm jet}}} & = & {{L}_{\pm }}+{{L}_{p}}+{{L}_{{\rm poy}}}, \\ {{L}_{\pm }} & \equiv & {{L}_{-}}+{{L}_{+}}, \\ \end{array}\end{eqnarray} \tag{ 24 }$$

where ${{L}_{\pm }}$, ${{L}_{-}}$, ${{L}_{+}}$, L_p, and ${{L}_{{\rm poy}}}$ are, the powers of the sum of the electrons and positrons, electrons, positrons, protons, and magnetic fields. respectively. For convenience, we define ${{\eta }_{{\rm eq}}}$ for ${{L}_{p}}={{L}_{\pm }},$ and it is given by

$$\begin{eqnarray}\begin{array}{rcl} {{\eta }_{{\rm eq}}} & \equiv & \frac{2\zeta }{1+\zeta }, \\ {{L}_{p}} & \gt & {{L}_{\pm }}\quad {\rm for}\quad \eta \gt {{\eta }_{{\rm eq}}}, \\ {{L}_{p}} & = & {{L}_{\pm }}\quad {\rm for}\quad \eta ={{\eta }_{{\rm eq}}},{\rm and} \\ {{L}_{p}} & \lt & {{L}_{\pm }}\quad {\rm for}\quad \eta \lt {{\eta }_{{\rm eq}}}. \\ \end{array}\end{eqnarray} \tag{ 25 }$$

Because we set

$$\begin{eqnarray}&&{{U}_{\pm }}\approx {{\epsilon }_{\pm }}{{n}_{\pm }},\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&{{U}_{p}}\approx {{\epsilon }_{p}}{{n}_{p}}={{\epsilon }_{p}}{{n}_{{{e}^{-}}p}},\end{eqnarray} \tag{ 27 }$$

${{L}_{p}}/{{L}_{\pm }}={{U}_{p}}/{{U}_{\pm }}=\eta /[(2-\eta )\zeta ]$ holds. Finally, the time-averaged total power of the jet (${{L}_{{\rm jet}}}$) can be generalized as follows:

$$\begin{eqnarray}&&{{L}_{{\rm jet}}}\geqslant {\rm max} \left[ {{L}_{{\rm poy}}},\left( 1+\frac{\eta }{2-\eta }\frac{1}{\zeta } \right){{L}_{\pm }} \right].\end{eqnarray} \tag{ 28 }$$

Given the two model parameters, η and ζ, we obtain U_p.

5.4. Limits on ${{B}_{{\rm tot}}}$, ${{\theta }_{{\rm thick}}}$, ${{U}_{\pm }}/{{U}_{B}}$, and ${{U}_{p}}/{{U}_{B}}$

Here, we give limits on ${{B}_{{\rm tot}}}$, ${{\theta }_{{\rm thick}}}$, ${{U}_{\pm }}/{{U}_{B}}$, and ${{U}_{p}}/{{U}_{B}}$ in the EHT region for ${{e}^{-}}/{{e}^{+}}/p$mixed plasma. As for the plasma properties, the following four cases with proton-loaded plasma can be considered, i.e., relativistic protons with ${{e}^{-}}/p$-dominated composition, relativistic protons with ${{e}^{\pm }}$-dominated composition, nonrelativistic protons with ${{e}^{-}}/p$-dominated composition, and nonrelativistic protons with ${{e}^{\pm }}$-dominated composition.

5.4.1. . Case for Relativistic Protons ($\zeta ={{m}_{e}}/{{m}_{p}}$)

Here we consider the case for relativistic protons ($\zeta ={{m}_{e}}/{{m}_{p}})$. In Figure 4, we show a typical example of the "${{e}^{-}}/p$-dominated" case with $\eta =0.99$. In this case, we obtain ${{\eta }_{{\rm eq}}}=1.09\times {{10}^{-3}}$. Because we consider the "${{e}^{-}}/p$-dominated" composition, the upper limit of RM significantly constrains smaller ${{\gamma }_{\pm ,{\rm min} }},$ according to Equation (19). In this case, ${{U}_{B}}\gg {{U}_{\pm }}$ still holds, as the smaller ${{\gamma }_{\pm ,{\rm min} }}$ region is excluded by the RM constraint. In Table 2, we summarize the resultant allowed physical quantities in this case. The maximum value of ${{B}_{{\rm tot}}}$ is determined by the condition ${{L}_{{\rm poy}}}\leqslant {{L}_{{\rm jet}}},$ whereas the minimum value of ${{B}_{{\rm tot}}}$ is governed by the condition that ${{\theta }_{{\rm thick}}}\geqslant {{R}_{{\rm ISCO}}}/{{D}_{L}}\approx 21\;\mu {\rm as}$. In the limit of inefficient ${{e}^{\pm }}/p$ coupling, the minimum energies of the electrons/positrons are smaller than that of the protons by a factor of ${{m}_{e}}/{{m}_{p}}$ (i.e., ${{\epsilon }_{\pm ,{\rm min} }}=({{m}_{e}}/{{m}_{p}}){{\epsilon }_{p,{\rm min} }}$). Therefore, ${{L}_{\pm }}$ decreases, and L_p tends to dominate over ${{L}_{\pm }}$. The energetics constraint in this case is given by ${{L}_{{\rm jet}}}\geqslant {\rm max} \left[ {{L}_{{\rm poy}}},\left( 1+\frac{\eta }{2-\eta }\frac{{{m}_{p}}}{{{m}_{e}}} \right){{L}_{\pm }} \right]$.

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Table 2.  Results for the Case of the e⁻/p-dominated Composition with Relativistic Protons

η	Allowed Btot	Allowed θthick	Allowed U±/UB	Allowed Up/UB
	(G)	(μas)
0.9	50 ≤ Btot ≤ 124	21 ≤ θthick ≤ 26.3	$7.9\times {{10}^{-7}}\leqslant \frac{{{U}_{\pm }}}{{{U}_{B}}}\leqslant 1.1\times {{10}^{-4}}$	$1.2\times {{10}^{-3}}\leqslant \frac{{{U}_{p}}}{{{U}_{B}}}\leqslant 0.17$
0.99	50 ≤ Btot ≤ 124	21 ≤ θthick ≤ 26.3	$7.9\times {{10}^{-7}}\leqslant \frac{{{U}_{\pm }}}{{{U}_{B}}}\leqslant 1.1\times {{10}^{-4}}$	$1.4\times {{10}^{-3}}\leqslant \frac{{{U}_{p}}}{{{U}_{B}}}\leqslant 0.20$
1	50 ≤ Btot ≤ 124	21 ≤ θthick ≤ 26.3	$7.9\times {{10}^{-7}}\leqslant \frac{{{U}_{\pm }}}{{{U}_{B}}}\leqslant 1.1\times {{10}^{-4}}$	$1.4\times {{10}^{-3}}\leqslant \frac{{{U}_{p}}}{{{U}_{B}}}\leqslant 0.21$

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In the case of the ${{e}^{\pm }}$-dominated composition with smaller η also leads to the same ${{B}_{{\rm tot}}}$ and ${{U}_{\pm }}/{{U}_{B}}$. In the same way as shown above, the maximum and minimum values of ${{B}_{{\rm tot}}}$ are determined by the jet power limit and minimum size limit at the EHT region. However, U_p is much smaller than U_B, simply because of the paucity of the relativistic proton component.

5.4.2. Cases for Nonrelativistic Protons ($\zeta =1$)

Next, let us consider the case of nonrelativistic protons ($\zeta =1$). When nonrelativistic protons are loaded, the corresponding energetic condition can be given by ${{L}_{{\rm jet}}}\geqslant {\rm max} \left[ {{L}_{{\rm poy}}},\left( 1+\frac{\eta }{2-\eta } \right){{L}_{\pm }} \right]$. Because the protons are nonrelativistic, the effect of proton-loading is quite small in terms of energetics. The coefficient resides in a narrow range, $1\lt (1+\eta /(2-\eta ))\lt 3/2$. Note that RM strongly depends on η, whereas RM is independent of ζ.

The "${{e}^{-}}/p$-dominated" case results in similar values of ${{B}_{{\rm tot}}}$ and ${{U}_{\pm }}/{{U}_{B}}$ to those shown in Table 2 because the maximum and minimum values of ${{B}_{{\rm tot}}}$ are also determined by the jet power limit and minimum size limit at the EHT region. The contributions of the protons are only ${{U}_{p}}={{U}_{\pm }}/2$. So, this does not give any significant effects on energetics.

Finally, we comment on the "${{e}^{\pm }}$-dominated" case. The main difference between the "${{e}^{-}}/p$-dominated" and "${{e}^{\pm }}$-dominated" cases is ${{n}_{{{e}^{-}}p}}$. Because the number density of the ${{e}^{\pm }}$-pairs does not contribute to RM, the constraint of RM becomes weaker when ${{n}_{{{e}^{-}}p}}$ becomes smaller. It leads to a wider allowed region for smaller ${{\gamma }_{\pm }}$ and a smaller B region. Therefore, the maximum value of the allowed ${{U}_{\pm }}/{{U}_{B}}$ for the "${{e}^{\pm }}$-dominated" case becomes larger than that for the "${{e}^{-}}/p$-dominated" case. However, this only changes the allowed ${{\gamma }_{\pm }}$ within a factor of ∼10, and it does not give a large impact on energetics.

It is worthwhile to examine a case of a fully SSA-thin model for the EHT region because the indication of ${{\nu }_{{\rm ssa}}}\gt 100\;{\rm GHz}$ by interferometry observations does not necessarily mean that ${{\nu }_{{\rm ssa}}}$ is larger than $230\;{\rm GHz}$. We can safely regard the SSA frequency as $43\;{\rm GHz}\lt {{\nu }_{{\rm ssa}}}\lt 230\;{\rm GHz}$ where the lower limit is warranted by the detection of the core-shift at 43 GHz in Hada et al. (2011).

In Figure 5, we show a schematic draw of the synchrotron spectrum when the EHT region is SSA-thin at 230 GHz (solid line). The upper limit of the flux density at 43 GHz of the 230 GHz core is estimated as $0.09\;{\rm Jy}=0.7\;{\rm Jy}\times {{\left( 40/110 \right)}^{2}}$ based on the Very Long Baseline Array (VLBA) measurements of the radio core flux and size given by Hada et al. (2013). The gray-colored scale shows the typical flux density obtained by SMA and CARMA. Interferometric observation shows some variability at 230 GHz (Akiyama et al. 2014). We define this as ${{F}_{{\rm upper}}},$ and we assume that ${{F}_{{\rm upper}}}$ is the upper limit of the flux density in the overall frequency range of $43\;{\rm GHz}\lt {{\nu }_{{\rm ssa}}}\lt 230\;{\rm GHz}$. First, from the EHT data, we can estimate a possible lower limit of ${{\nu }_{{\rm ssa}}}$ as

$$\begin{eqnarray}\begin{array}{rcl} {{\nu }_{{\rm ssa}}} & \geqslant & 230\;{\rm GHz}\times {{\left( \frac{{{F}_{{\rm upper}}}/2.3\;{\rm Jy}}{{{S}_{\nu }}/1\;{\rm Jy}} \right)}^{-1/\alpha }} \\ {} & \sim & 160\;{\rm GHz}\;({\rm for}\;\alpha =2.5). \\ \end{array}\end{eqnarray} \tag{ 29 }$$

Note that the choice of $\alpha =3.0$ leads to ${{\nu }_{{\rm ssa}}}\sim 170\;{\rm GHz}$. Second, from the VLBA data, we can estimate a possible upper limit of ${{\nu }_{{\rm ssa}}}$ as

$$\begin{eqnarray}\begin{array}{rcl} {{\nu }_{{\rm ssa}}} & \leqslant & 43\;{\rm GHz}\times {{\left( \frac{{{F}_{{\rm upper}}}/2.3\;{\rm Jy}}{{{S}_{\nu }}/0.09\;{\rm Jy}} \right)}^{2/5}} \\ {} & \sim & 160\;{\rm GHz}. \\ \end{array}\end{eqnarray} \tag{ 30 }$$

Allowing some flux measurement errors, somehow we can have a consistent case around ${{\nu }_{{\rm ssa}}}\sim 160\;{\rm GHz}$ with $\alpha \sim 2.5$.

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Then, let us discuss the physical quantities in this case. From Equation (5), ${{U}_{\pm }}/{{U}_{B}}\propto \nu _{{\rm ssa}}^{-2p-13}$. Therefore, in this case, the ratio would be typically larger by a factor of ${{\left( 160/230 \right)}^{-18}}\sim 6.9\times {{10}^{2}}$ (for $p=2.5$) than that for the SSA-thick case. However, this does not change the result of ${{U}_{\pm }}\ll {{U}_{B}}$ because ${{U}_{\pm }}\ll {{U}_{B}}\lt {{10}^{-4}}$ in any cases with the SSA-thick core existing. Hence, we can conclude that even for the fully SSA-thin EHT region case, ${{U}_{\pm }}\ll {{U}_{B}}$ holds in order not to overproduce fluxes between $43\;{\rm GHz}\lt {{\nu }_{{\rm ssa}}}\lt 230\;{\rm GHz}$.

However, a critical difference appears for the comparison between U_p and ${{U}_{B}}$. In the case of relativistic protons with "${{e}^{-}}/p$-dominated composition, ${{U}_{p}}\gt {{U}_{B}}$ can be realized for a certain range of ${{\nu }_{{\rm ssa}}}$. From Table 2 we know the values of ${{U}_{p}}/{{U}_{B}}$ when ${{\nu }_{{\rm ssa}}}=230\;{\rm GHz}$, by multiplying the factor of ∼200 − 400, and the maximum value reaches ${{U}_{p}}/{{U}_{B}}\gt 1$ at ${{\nu }_{{\rm ssa}}}\sim 160\;{\rm GHz}$.

We have explored the magnetization degree of the jet base of M87 based on the observational data of the EHT obtained by Doeleman et al. (2012). Following the method in K14, we estimate the energy densities of the magnetic fields (U_B) and electrons and positrons (${{U}_{\pm }}$) in the region detected by EHT (the EHT region) with its FWHM size, $40\;\mu {\rm as}$. Imposing the basic energetics of the M87 jet, the constraints from the EHT observational data, and the minimum size of the SSA-thick region as the ISCO radius, we find the following.

• 1.  
  First, we adopt the assumption that the EHT region contains an SSA-thick region. Then, the coexistence of the SSA-thick and SSA-thin regions is required for the EHT region not to overproduce ${{L}_{{\rm poy}}}$. The angular size of the SSA-thick region is limited as $21\;\mu {\rm as}\leqslant {{\theta }_{{\rm thick}}}\leqslant 25.5\;\mu {\rm as}$, whereas that of the SSA-thin region should be $40\;\mu {\rm as}$ to explain the EHT data. The derived flux density of the SSA-thick region is about 0.27 Jy. The allowed magnetic-field strength in the SSA-thick region is $58\;{\rm G}\leqslant {{B}_{{\rm tot}}}\leqslant 127\;{\rm G}$. In terms of the energetics, ${{U}_{B}}\gg {{U}_{\pm }}$ is realized at the overall SSA-thick region. If protons do not dominantly contribute to the jet energetics, then this result supports the magnetic-driven jet scenario at the SSA-thick region. We further examine the following four cases for the electron/positron/proton (${{e}^{-}}/{{e}^{+}}/p$) mixed plasma; nonrelativistic protons with ${{e}^{-}}/p$-dominated composition, nonrelativistic protons with ${{e}^{\pm }}$-dominated composition, relativistic protons with ${{e}^{-}}/p$-dominated composition, and relativistic protons with ${{e}^{\pm }}$-dominated composition, together with the assumption that RM detected by SMA (Kuo et al. 2014) gives an upper limit of RM of the EHT region. Although the RM limit can give tighter constraints on allowed ${{\gamma }_{\pm }}$, it does not change the results significantly. We find that ${{U}_{B}}\gg {{U}_{\pm }}$ always holds in any case.
• 2.  
  Second, the case of the completely SSA-thin (${{\nu }_{{\rm ssa}}}\lt 230\;{\rm GHz}$) EHT region is also discussed. Although lower ${{\nu }_{{\rm ssa}}}$ can increase the ratio ${{U}_{\pm }}/{{U}_{B}}$ by a factor of 200–400 than that for the SSA-thick case, this does not change the result of ${{U}_{\pm }}\ll {{U}_{B}}$ because ${{U}_{\pm }}/{{U}_{B}}\lt {{10}^{-3}}$. However, we also find that in the case of relativistic protons with "${{e}^{-}}/p$-dominated" composition, ${{U}_{p}}\gt {{U}_{B}}$ can be realized around ${{\nu }_{{\rm ssa}}}\sim 160\;{\rm GHz}$.

Future work and key questions are enumerated below.

• 1.  
  An important future work is to confirm the existence of the SSA-thick region in the EHT region. If we confirm it, then we can exclude the case of ${{U}_{p}}/{{U}_{B}}\gt 1$. In the context of confirming the existence of the SSA-thick region, we also note the effectiveness of inclusions of longer baselines, even for a single-frequency VLBI observation. In Figure 2, it is clear that the visibility amplitude of the SSA-thin component is much smaller than that of the SSA-thick component above $\sim 3{\rm G}\lambda$ at the 1.3 mm wavelength. Therefore, inclusions of baselines with $\gt 3{\rm G}\lambda$ would be effective to distinguish the SSA-thick component. For example, phased ALMA plus SMT with an effective bandwidth of 4 GHz would be effective at $\sim 5{\rm G}\lambda$ (Figure 6 in Fish et al. 2013). In Figure 2, we show the corresponding baseline-length range (the blue-shaded region).
• 2.  
  Equally important future work is to observe the EHT region with the spatial resolution of $\sim 1\;{{R}_{s}}$ of M87. Currently, the EHT array with 20–30 μas resolution at 230 and 345 GHz (e.g., Lu et al. 2014) is not able to reach $\sim 1\;{{R}_{s}}$ of M87. Ground-based short-millimeter VLBI observations are very sensitive to weather conditions (e.g., Thompson et al. 2001). To confirm our assumption that the minimum ${{\theta }_{{\rm thick}}}{{D}_{L}}$ is comparable to $\sim {{R}_{{\rm ISCO}}}$ or even smaller, space VLBI observations would be required in the future. In the past missions and existing plan of space VLBI, it was not possible to reach the event horizon scale of M87 (e.g., Dodson et al. 2006; Asada et al. 2009; Takahashi & Mineshige 2011; Dodson et al. 2013) because the target wavelengths were not short enough. Thus, the atmospheric-free space (sub)millimeter VLBI observation would be indispensable to reach $\sim 1\;{{R}_{s}}$ of M87. The phased ALMA (e.g., Alef et al. 2012, Fish et al. 2013) will play a definitive role for such observations for obtaining visibilities between space and ground telescope baselines. Honma et al. (2014) have recently proposed a new technique of VLBI data analysis to obtain superresolution images with radio interferometry using sparse modeling. The usage of the sparse modeling enables us to obtain superresolution images in which structure finer than the standard beam size can be recognized. A test simulation for imaging of the jet base of M87 is actually demonstrated in Honma et al. (2014), and the technique works well. Therefore, this superresolution technique will become another important tool for obtaining better resolution images.
• 3.  
  The observational result of Doeleman et al. (2012) does not show flux variability at 230 GHz. However, the total epoch number of EHT observations is too scarce to confirm the absence of flux variability at 230 GHz all of the time. M87 might be in a quiescent state during the EHT observations in 2010 April by chance. We also emphasize that the derived field strength is still $\geqslant 58\;{\rm G},$ and ${{t}_{\pm ,{\rm syn}}}$ still tends to be smaller than the day scale. It is also intriguing that the same correlated flux densities in 2009 reported by Doeleman et al. (2012) are observed during another EHT observation performed in 2012 April (Akiyama et al. 2014). This result is quite different from the day-scale variability detected in Sagittarius A* by the EHT observations (Fish et al. 2011). To search for a possible flux variability of M87 in more detail, continuous monitoring by EHT would be essential.
• 4.  
  Based on the GRMHD model, well-ordered poloidal fields are dominant within the Alfvén point, and toroidal fields become dominant outside of the Alfvén point, whereas turbulence may not yet develop at the jet base (e.g., Spruit 2010 for review). In general, turbulent eddies which most probably generate turbulent fields are not expected before sufficient interactions with surrounding ambient matter (e.g., Mizuta et al. 2010 and reference therein). Therefore, a higher LP degree is likely to be expected. Conservatively saying, the reason of the low LP degree by Kuo et al. (2014) is most probably because of depolarization within the SMA beam. At the moment, we are not able to rule out a possible constitution of RIAF emission, which may also lead to low LP degree. If so, then studies of fundamental process for particle accelerations in RIAF (e.g., Hoshino 2013) and the effects on particle escape from RIAF (Le & Becker 2004; Toma & Takahara 2012; Kimura et al. 2014) would become more important.
• 5.  
  In terms of the brightness temperature of the 230 GHz radio core of M87, ${{T}_{b}}\sim 2\times {{10}^{10}}\;{\rm K}\left( \frac{{{S}_{\nu }}}{1\;{\rm Jy}} \right){{\left( \frac{{{\theta }_{{\rm FWHM}}}}{40\;\mu {\rm as}} \right)}^{-2}}$ seems slightly higher than the prediction of the hot electron temperature of $\sim {{10}^{9}}\;{\rm K}$ in RIAF flows (e.g., Manmoto et al. 1997). Hence, the jet emission seems to be preferred to explain the EHT emission in M87 (Dexter et al. 2012; see Ulvestad & Ho 2001 for similar arguments). However, it is not conclusive because the geometry near the ISCO regions is highly uncertain in the observational point of view. The scrutiny of the origin of the 230 GHz emission is still a noteworthy issue to explore.
• 6.  
  Further polarimetric observation would be required to examine the RM properties in more detail. Although we adopt the RM values of Kuo et al. (2014), it is found that the observed electric vector position angle trend does not show a sufficiently tight fit to the ${{\lambda }^{2}}$-law. This behavior may not be due to the consequence of blending of multiple cross-polarized substructures with different RM values, but simply rather due to the nonuniformity between the upper and lower side bands of the SMA. A polarimetric observation with ALMA is clearly one of the promising first steps to improve this point. Obviously, in the final stage, short-millimeter (and submillimeter) VLBI polarimetric observations are inevitable to avoid contamination from the extended region.
• 7.  
  The degree of the ${{e}^{\pm }}/p$ coupling is a critical factor for the results of the proton power. Theoretically, Hoshino & Arons (1991) found the energy transfer process from protons to positrons via the absorption of high harmonic ion cyclotron waves emitted by the protons. Amato & Arons (2006) indeed performed one-dimensional particle-in-cell (PIC) simulations for ${{e}^{-}}/{{e}^{+}}/p$ mixed plasma. However, there are several simplifications in PIC simulations, such as a smaller ${{m}_{p}}/{{m}_{e}}$ ratio, etc. More intensive investigations are awaited to clarify the degree of ${{e}^{\pm }}/p$ coupling at the base of the M87 jet.
• 8.  
  We make a brief comment on the effects of the magnetic-field topology and anisotropy of ${{e}^{-}}/{{e}^{+}}$ in terms of the energy distribution. If the ${{e}^{-}}/{{e}^{+}}$ energy distribution in the EHT region is isotropic, then the synchrotron absorption coefficient investigated by GS65 is applicable, and differences of field geometry would not have an impact on field strength estimation. For example, the difference of ${{B}_{{\rm tot}}}$ between the cases of the isotropic field (see Equation (1)) and the ordered field (${{B}_{\bot }}={{B}_{{\rm tot}}}$), which,directed toward LOS, is only a factor of $\sqrt{3/2}$.

However, if the ${{e}^{-}}/{{e}^{+}}$ energy distribution is highly anisotropic, then the well-known synchrotron emissivity and self-absorption coefficient are not applicable. Effects of the ${{e}^{-}}/{{e}^{+}}$ anisotropy on synchrotron radiation are not well-studied, and it is beyond the scope of this paper. Although we do not have any observational suggestions of such anisotropy of the ${{e}^{-}}/{{e}^{+}}$ energy distribution, it may be a new theoretical topic to be explored, if observational suggestions are found in the future.

We thank the anonymous referee for constructive comments. KH and KA are supported by the Japan Society for the Promotion of Science (JSPS) Research Fellowship Program for Young Scientists.

In order to obtain better accuracy calculation plus some modifications of the definition of ${{B}_{\bot }}$ and relevant corrections, the modified numerical coefficients of $b(p)$ and $k(p)$ are presented, although the corrections are small.

In K14, the magnetic-field strengths perpendicular to the local electron motions were not averaged over the pitch angle (in Equation (1) in K14). In this work, in Equation (1), we conduct the pitch-angle averaging for defining the averaged magnetic-field strengths perpendicular to the local electron motions.

The SSA coefficient measured in the comoving frame is given by (Equations 4.18 and 4.19 in GS65; Equation 6.53 in Rybicki & Lightman 1979)

$$\begin{eqnarray}&&\begin{array}{l} {{\alpha }_{\nu }}=\frac{\sqrt{3}{{e}^{3}}}{8\pi {{m}_{e}}}{{\left( \frac{3e}{2\pi m_{e}^{3}{{c}^{5}}} \right)}^{p/2}} \\ \quad {{c}_{1}}(p){{K}_{\pm }}B_{\bot }^{(p+2)/2}{{\nu }^{-(p+4)/2}}, \\ \end{array}\end{eqnarray} \tag{ A1 }$$

where the numerical coefficient ${{c}_{1}}(p)$ is expressed by using the gamma functions as follows; ${{c}_{1}}(p)={\Gamma}[(3p+2)/12]{\Gamma}[(3p+22)/12]$. For convenience, we define ${{\alpha }_{\nu }}={{X}_{1}}{{c}_{1}}(p)B_{\bot }^{(p+2)/2}{{K}_{\pm }}{{\nu }^{-(p+4)/2}}$.

Optically thin synchrotron emissivity per unit frequency ${{\epsilon }_{\nu }}$ from the uniform emitting region is given by (Equations (4.59) and (4.60) in BG70; see also Equations (3.28), (3.31) and (3.32) in GS65)

$$\begin{eqnarray}&&\begin{array}{l} {{\epsilon }_{\nu }}=4\pi \frac{\sqrt{3}{{e}^{3}}}{8\sqrt{\pi }{{m}_{e}}{{c}^{2}}}{{\left( \frac{3e}{2\pi m_{e}^{3}{{c}^{5}}} \right)}^{(p-1)/2}} \\ \quad {{c}_{2}}(p){{K}_{\pm }}B_{{\rm tot}}^{(p+1)/2}{{\nu }^{-(p-1)/2}}, \\ \end{array}\end{eqnarray} \tag{ A2 }$$

where the numerical coefficient is ${{c}_{2}}(p)={\Gamma}[(3p+19)/12]{\Gamma}[(3p-1)/12)]{\Gamma}[(p+5)/4)]/$ ${\Gamma}[(p+7)/4)]/(p+1)$. In K14, ${{B}_{{\rm tot}}}$ was wrongly written as ${{B}_{\bot }}$. Therefore, here we revise it, and it leads to larger $b(p)$ by the factor of $\sqrt{1.5}$. For convenience, we define ${{\epsilon }_{\nu }}\equiv 4\pi {{X}_{2}}{{c}_{2}}(p)B_{{\rm tot}}^{(p+1)/2}{{K}_{\pm }}{{\nu }^{-(p-1)/2}}$. The modified coefficient $b(p)$ is expressed as

$$\begin{eqnarray}&&b\left( p \right)={{\left( \frac{4\pi {{c}_{2}}{{X}_{2}}\times {{1.5}^{1/4}}}{6{{c}_{1}}{{X}_{1}}} \right)}^{2}}\times {{2}^{-4}}.\end{eqnarray} \tag{ A3 }$$

In K14, the index of the square bracket at the right-hand side of $b(k)$ should not be 2 but −2(typo). The expression of $k(p)\propto b{{\left( p \right)}^{(-p-2)/2}}$ does not change, but the value of $k(p)$ is changed. Although the modifications of b and k in Table 1 of K14 are straightforward, based on the above explanations, we provide Table A1 for convenience.