Relativistic jet acceleration region in a black hole magnetosphere

Masaaki Takahashi∗

Department of Physics and Astronomy, Aichi University of Education, Kariya, Aichi 448-8542, Japan

Motoki Kino†

Kogakuin University of Technology & Engineering, Academic Support Center,2665-1 Nakano, Hachioji, Tokyo 192- 0015, Japan and

National Astronomical Observatory of Japan 2-21-1 Osawa, Mitaka, Tokyo, 181-8588, Japan

Hung-Yi Pu‡

National Taiwan Normal University (NTNU), No. 88,Sec. 4, Tingzhou Road, Taipei 116, Taiwan, R.O.C. andInstitute of Astronomy and Astrophysics, Academia Sinica,11F of Astronomy-Mathematics Building, AS/NTU No. 1,

Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan, R.O.C.

We discuss stationary and axisymmetric trans-magnetosonic outflows in the magnetosphere of arotating black hole (BH). Ejected plasma from the plasma source located near the BH is acceleratedfar away to form a relativistic jet. In this study, the plasma acceleration efficiency and conversion offluid energy from electromagnetic energy are considered by employing the trans-fast magnetosonicflow solution derived by Takahashi & Tomimatsu (2008). Considering the parameter dependenceof magnetohydrodynamical flows, we search for the parameters of the trans-magnetosonic outflowsolution to the recent M87 jet observations and obtain the angular velocity values of the magneticfield line and angular momentum of the outflow in the magnetized jet flow. Therefore, we estimatethe locations of the outer light surface, Alfven surface, and separation surface of the flow. We alsodiscuss the electromagnetic energy flux from the rotating BH (i.e., the Blandford–Znajek process),which suggests that the energy extraction mechanism is effective for the M87 relativistic jet.

I. INTRODUCTION

The system of a supermassive black hole (BH) withaccreting matter is widely believed to be the central en-gine of the galactic nuclei, and relativistic jets are of-ten observed. Recently, very-long baseline interferometry(VLBI) observations have revealed the configuration andvelocity distribution of the M87 jet near the BH [1, 2].The parsec-scale jet of the galaxy M87 is parabolic-like[3, 4], and it becomes conical-like near the BH [5]. The ra-dial profile of the jet velocity was observed by Park et al.[6] using the KVN and VERA Array (KaVA) (KVN: Ko-rean VLBI Network; VERA: VLBI Exploration of RadioAstronomy). The rapid increase in velocity is observedapproximately 100 to 1,000 times the BH’s radius. Thisarea is called the acceleration region and is considered toextend from slightly inside the outer light surface to thearea several times the fast-magnetosonic surface. Thus,detailed data for the region closer to the base of the jetwere obtained. Exploring the base of the jet would ad-vance our understanding of plasma near the BH.

A magnetic field is generated by accretion plasmaaround the BH, which is dominant around the axis of ro-tation. Such a region is called the BH magnetosphere. At

∗Electronic address: mtakahas@auecc.aichi-edu.ac.jp†Electronic address: motoki.kino@nao.ac.jp‡Electronic address: hypu@ntnu.edu.tw

the low- and mid-latitudes of this rotating system, a gastorus is formed by accretion gas. The outflow is ejectedfrom a plasma source located in the magnetosphere andaccelerated by the Lorentz force toward a distant region.The BH magnetosphere extends along the flow from theBH to the region where the magnetic field and the fluidenergies become comparable. We consider stationary andaxisymmetric magnetohydrodynamic (MHD) outflows inthe BH magnetosphere as relativistic jets. Figure 1 showstrans-fast magnetosonic outflow and inflow in a BH mag-netosphere. The flow region (a funnel) is confined by acorona and/or disk wind, and cold MHD flow is consid-ered. It is shown separately for the electromagnetic fieldcomponent (Poynting flux: blue arrows) and fluid compo-nent (red arrows) of the MHD flow’s energy flux. The jetpower derived from the rotational energy of the BH wouldbe explained via the Blandford–Znajek (BZ) process [7, 8](see also, [9, 10]). The rotation of the magnetospheregenerates a strong centrifugal force on plasma, therebycreating a region that produces an outward plasma flowalong a magnetic field line; moreover, there is a regionof accretion flow because of strong gravity near the BH.Therefore, there is a plasma supply between the inflowand outflow regions, which is a watershed for inflow andoutflow ejected from the plasma source with low veloc-ity and is located between the inner and outer light sur-faces (e.g., [11, 12]). Thus, the BH magnetosphere can beclassified into inner and outer regions; i.e., the inner andouter BH magnetospheres. The region where gravity andcentrifugal force act on the plasma along the streamline

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FIG. 1: Schematic of MHD flows driven by rotating BH(BZ power). The thin green curves are the global magneticfield lines in the magnetosphere. The blue and red arrowsindicate the direction of the Poynting and fluid componentfluxes, respectively. The ejected outflow and inflow from aplasma source, which may be near the separation surface, passthrough the Alfven surface (A), light surface (L), and fast-magnetosonic surface (F) in order, then accelerate toward adistant and the BH, respectively. The thin black curve aroundthe BH is the boundary of the ergosphere.

balance is called the “separation surface,” rsp(θ).For a stationary and axisymmetric ideal MHD flow so-

lution in the BH magnetosphere, in addition to the pa-rameters of BH spacetime and the function of the mag-netic field shape Ψ(r, θ), the total energy E(Ψ) and totalangular momentum L(Ψ) of the flow, the angular velocityof the magnetic field lines ΩF (Ψ), and the particle num-ber flux per magnetic flux tube η(Ψ) should be specified.These parameters are conserved along the magnetic fieldline (i.e., the flow’s streamline). By analyzing these pa-rameter values for the outflow solution with a solutionof magnetic field distribution, various jet morphologiesand behaviors would be elucidated. To apply the flowsolution to an observed relativistic jet, it is necessaryto associate the physical quantity obtained from the ob-servation with the model parameters as the field-alignedconserved quantities.

We examine the nature of the magnetized outflowand then the acceleration of the jet near its base re-

gion. Further, we compare the outflow solution withthe M87 observed data in the framework of general rel-ativity. We expect this attempt to impose restrictionson unknown plasma sources and inflow onto the BH.Generally, solving the equation of motion for magnetizedfluid involves analysis of critical conditions at the Alfvenpoint as well as fast and slow magnetosonic points un-der a given magnetic field line. This analysis is complexand tedious (see, e.g., [13, 14]). However, Takahashi &Tomimatsu [15] (hereinafter, TT08) analytically devisedtrans-magnetosonic flow solution without a complicatedregularity condition analysis at the magnetosonic sur-faces. In this study, we adopt the method of TT08 to ana-lyze trans-fast magnetosonic flow solutions. This methodfacilitates the analysis of stationary trans-magnetosonicflow solutions; i.e., we can easily solve the MHD flowsand discuss the distributions of plasma density, the flowvelocity, and the ratio of the magnetization of the flow.Hence, we apply the flow solution to the M87 jet.

The asymptotic structures of the magnetic field andrelativistic flow have been discussed (e.g, [16–19]). Re-cently, Huang, Pan & Yu [20] numerically discussed a self-consistent magnetosphere of a BH by solving the trans-field equation (the relativistic Grad–Shafranov equation)[21, 22] (see also [23]) with the poloidal equation and ap-plied it to the M87 jet to explain the velocity distribution.Kino et al. [24] discussed the flow acceleration of theM87 jet by employing an analytical model of Tomimatsu& Takahashi [25] (hereinafter, TT03), which solved theapproximated transfield equation outside the outer lightsurface.

Pu & Takahashi [26] discussed trans-fast magnetosonicjets in a BH magnetosphere by using TT08, where theinjection surface rinj(Ψ) of flows was set to the locationof the separation surface rsp(Ψ), and the injected flowstarts with a zero velocity, urinj = 0. They performed pa-rameter searches for ΩF , E, Ψ and the BH spin depen-dencies. Notably, the angular momentum L of outflowinjected from the separation surface with zero velocitycan be described by using other conserved quantities.

In this study, we release such an initial setting foroutflows by Pu & Takahashi [26] that restricted the Lparameter, and search the dependence of the values ofE, L and ΩF on the jet acceleration motivated by theM87 observations, where the configuration of the mag-netic field is considered to be of conical shape for someparameter searches. The parameter search of L is to al-low the flows to have a finite initial velocity (urinj > 0)at the separation surface, rinj = rsp, or an initial zerovelocity (urinj = 0) ejected from the radius of rinj ≥ rsp.

The specific location of rinj(θ) is still under debate inthe BH magnetosphere as the jet base. It could be dueto electron-positron pair-creation from the backgroundMeV photons [27–30] or modest acceleration within acharge-starved region at the null surface or the separa-tion surface [31–36]). Recently, the structure of a BHmagnetosphere in consideration of the pair-creation nearthe horizon was studied via general-relativistic particle-

3

in-cell simulations [37, 38]. Thus, the initial position andvelocity (rinj, u

rinj), which are associated with the angular

momentum L, are expected to shift from rinj = rsp withurinj = 0 to rinj 6= rsp with urinj 6= 0.

The rest of this manuscript is organized as follows. InSection II, we review the trans-magnetosonic flow solu-tions discussed by TT08. In Section III, we summarizethe properties of cold trans-fast magnetosonic flow solu-tions by analyzing the field-aligned flow parameters. InSection IV, we discuss the outflow solution in detail forcomparison with the jet observations. The application tothe M87 jet is presented in Section V, and we discuss it inSection VI. Using the physical parameters of the outflowsolution obtained by fitting with observed data, we esti-mate the values of E, ΩF , and L; i.e., the locations of theouter light surface, Alfven surface, and separation surfaceof the outflow. Further, the observed M87 jet power canbe explained by the BZ power. The stationary model ofmagnetically driven jets in a BH magnetosphere wouldexplain the observed properties of the M87 jet and simi-lar low luminosity AGN jets. Finally, we give concludingremarks in Section VII.

II. BASIC EQUATIONS FOR TRANS-FASTMAGNETOSONIC FLOW

We assume stationary and axisymmetric ideal MHDflows in a BH magnetosphere in Kerr geometry. Thebackground metric is written by the Boyer–Lindquist co-ordinates with c = G = 1,

ds2 =

(1− 2mr

Σ

)dt2 +

4amr sin2 θ

Σdtdφ

−A sin2 θ

Σdφ2 − Σ

∆dr2 − Σdθ2 , (1)

where m and a denote the mass and angular momentumper unit mass of the BH, respectively, and ∆ ≡ r2 −2mr+a2, Σ ≡ r2 +a2 cos2 θ, A ≡ (r2 +a2)2−∆a2 sin2 θ.The particle number conservation is (nuµ);µ = 0, wheren is the number density of the plasma, and uµ is thefluid 4-velocity. The ideal MHD condition is uνFµν = 0,where Fµν is the electromagnetic tensor, and it is writ-ten as Fµν = Aν,µ − Aµ,ν using the vector potentialAµ. The relativistic polytropic relation is P = KρΓ

0 ,where Γ is the adiabatic index, K is related to the en-tropy, ρ0 = nmpart is the rest mass density, and mpart

is the mass of plasma particles. The equation of mo-tion is Tµν ;ν = 0. The energy–momentum tensor isgiven by Tµν = Tµνfluid + TµνEM, where the fluid part isTµνfluid = nµuµuν − Pgµν , the electromagnetic part is

TµνEM = (1/4π)[FµλFλν + (1/4)gµνF 2], µ ≡ (ρ + P )/n

is the relativistic enthalpy, ρ is the total energy density,and F 2 ≡ FµνFµν .

A. Relativistic Bernoulli Equation for MHD Flows

We define the magnetic and electric fields as fol-lows: Bµ ≡ ∗Fµνk

ν and Eµ ≡ Fµνkν , where ∗Fµν ≡

(1/2)√−gεµνσλFσλ, and kν = (1, 0, 0, 0) is the timelike

Killing vector. The poloidal component Bp of the mag-netic field seen by a lab-frame observer is given by

B2p ≡ −(BrBr +BθBθ)/G

2t

= −[grr(∂rΨ)2 + gθθ(∂θΨ)2

]/ρ2w , (2)

where Gt ≡ gtt+gtφΩF and ρ2w ≡ g2

tφ−gttgφφ = ∆ sin2 θ.

The function Ψ(r, θ) is the magnetic stream function (theφ component of the vector potential, Aφ(r, θ)). Theideal MHD flow stream along the magnetic field lines,Ψ(r, θ) = constant lines, and have five field-aligned flowparameters; the particle number flux per magnetic fluxη(Ψ) = nup/Bp, total energy E(Ψ) = µut−ΩFBφ/(4πη),total angular momentum L(Ψ) = −µuφ − Bφ/(4πη),entropy, and angular velocity of magnetic field linesΩF (Ψ) = −FtA/FφA [39, 40]. Thus, we can treat one-dimensional MHD flows along a Ψ(r, θ) = constant line.The physical variables of flows are denoted as a functionof r and Ψ with field-aligned flow parameters.

We define the poloidal velocity up by u2p ≡ −(urur +

uθuθ). The relativistic Alfven Mach number M is definedby

M2 ≡4πµnu2

p

B2p

=µupBp

, (3)

where the term Bp ≡ Bp/(4πµcη) is introduced to nondi-mensionalize and µc is the enthalpy for a cold flow; i.e.,µc = mpart. The nondimensional toroidal component ofthe magnetic field Bφ is defined as

Bφ ≡(

1

ρw

)Bφ

4πµcη=GφE +GtL

ρw(M2 − α), (4)

where E ≡ E/mpart, L ≡ L/mpart, Gφ ≡ gtφ + gφφΩF ,and α ≡ Gt + GφΩF . The toroidal magnetic fieldBφ = (∆/Σ)Fθr can be expressed in terms of the field-aligned flow parameters and Alfven Mach number. Thepoint of M2 = α with GφE + GtL = 0 on the flow so-lution is called the “Alfven point” (labeled as “A”); i.e.,the point (rA,M

2A; Ψ) in the r–M2 plane. At the Alfven

point, BφA has a nonzero finite value. The radius rA iscalled the “Alfven radius.” Notably, we may identify theradius given by GφE +GtL = 0 without M2 = α. Sucha radius is called an “anchor radius” for the magneticfield line considered (TT08), where Bφ = 0. Althoughthe magnetic field line in an axisymmetric magnetic sur-face (i.e., Ψ = constant surface) has a spiral shape in thetoroidal direction, the direction of winding is reversed atthe anchor radius if such a radius appears on the mag-netic surface. Both the Alfven and anchor radii can belocated between the inner and outer light surfaces (i.e.,

4

the α(r, θ) > 0 region), where the two light surfaces aregiven by the locations α(r,Ψ; ΩF , a) = 0.

The relativistic Bernoulli equation for MHD flows,which determines the poloidal velocity (or the AlfvenMach number) along a magnetic field line, can be ex-pressed as follows: [11, 15, 41]

e2 = µ2α+M4(αB2p + B2

φ) , (5)

where e ≡ E − ΩF L. The relativistic enthalpy µ is ex-pressed in terms of up and Bp

µ ≡ µ

mpart= 1+

(µinj

mpart

)(Bpup

)Γ−1

= 1+µhot

(Bpup

)Γ−1

,

(6)where µhot ≡ (µinj/mpart)(4πµcη)Γ−1, and the term µinj

is evaluated at the plasma injection point by

µinj ≡ΓK

Γ− 1(mpartη)Γ−1 =

Γ

Γ− 1

Pinj

ninjmpart

(uinjp

Binjp

)Γ−1

.

(7)Now, to solve the relativistic Bernoulli equation (5),

we introduce the function β(r; Ψ) as follows: [15]

β ≡ BφBp

, (8)

i.e., the pitch angle of a magnetic field line on a mag-netic flux surface. Moreover, we can introduce thepoloidal electric-to-toroidal magnetic field amplitude ra-tio ξ(r, θ) ≡ Ep/|BT |, seen by a zero angular momen-tum observer (ZAMO), where E2

p ≡ −(ErEr + EθE

θ)

and B2T ≡ −BφBφ. The electric and magnetic fields in

ZAMO are Eα ≡ Fαβhβ and Bα ≡ (1/2)ηαβγδh

βF γδ

with hβ = (Σ∆/A)−1/2(1, 0, 0, ω). The function ξ(r, θ) isrelated to the above function β(r, θ), as ξ2 = −gφφ(ΩF −ω)2/β2, where ω ≡ −gtφ/gφφ.

By assuming the functions of the magnetic field lineΨ = Ψ(r, θ) and β = β(r, θ) [or ξ = ξ(r, θ)], we canspecify the cross-section of the magnetic flux tube of theMHD flow. Hence, using Eq. (4), Eq. (5) can be ex-pressed in terms of M2 and β with the conserved quan-tities. Although the ratio β should be determined bysolving the transfield equation with the poloidal equationself-consistently, in this study, we only consider a mag-netic flux tube with a certain functional form of β(r, θ).Notably, the function β(r; Ψ) must satisfy certain restric-tions at several characteristic locations in a BH magne-tosphere (TT08).

B. Cold MHD Flow Solutions

The Alfven Mach number and poloidal velocity alongthe flow are obtained by solving Eq. (5) with Eq. (4),which is a higher-order equation for M2 (or up). How-ever, for a cold MHD flow (P = 0), we obtain the follow-ing quadratic equation for M2: [15]

AM4 − 2BM2 + C = 0 , (9)

where

A = −(k + 1)− 1

β2ρ2w

(GφE +GtL

)2

, (10)

B = e2 − α , (11)

C = α(e2 − α

), (12)

with k ≡ (gφφE2+2gtφEL+gttL

2)/ρ2w. The Alfven Mach

number of the flow is obtained using the following:

M2(±) =

B ±√B2 −ACA

, (13)

where M2(+) (> α) denotes the super-Alfvenic flow, and

M2(−) (< α) denotes the sub-Alfvenic flow. At the Alfven

point, we see the condition LΩF = YA with the defini-tions L ≡ L/E and Y (r, θ) ≡ −GφΩF /Gt; therefore, wehave (B2 − AC)A = 0, and we confirm that the AlfvenMach number becomes M2

A ≡ M2(+)A = M2

(−)A = αA.

At the light surface, we have M2(−)L = 0 and M2

(+)L =

2e2/[−(kL + 1)− e2/β2L ].

For the cold MHD flow, we define two characteristicAlfven Mach numbers related to the Alfven and fast-magnetosonic wave speeds, M2

AW(r, θ) ≡ α(r, θ), andM2

FW(r, θ) ≡ α(r, θ) + β2(r, θ), respectively [11, 15].At the light surface, we obtain M2

AW(rL; Ψ) = 0 andM2

FW(rL; Ψ) = β2, and at the event horizon rH = m +√m2 − a2, we have M2

AW(rH; Ψ) = gφφ(ΩF − ωH)2 < 0and M2

FW(rH; Ψ) = 0, where ωH = a/(2mrH) is theangular velocity of the BH. Hence, the inflow into theBH must be a trans-fast magnetosonic flow between theAlfven surface and the event horizon. The location of(α)′sp = 0, where the prime is a derivative along a stream-

line ( )′ ≡ ∂r + (Bθ/Br)∂θ, makes the separation surfacer = rsp(θ).

For a trans-Alfvenic outflow, we should select M2 =M2− (< M2

AW) in the sub-Alfvenic region of LΩF > Y ,where is the region from the injection point to the outerAlfven point, whereas M2 = M2

+ (> M2AW) is in the

super-Alfvenic region of LΩF < Y , where is the re-gion from the outer Alfven point to a distant region.The solution of M2 = M2

+ in the super-Alfvenic re-gion becomes trans-fast magnetosonic. Notably, at thefast-magnetosonic point (labeled as “F”), the solution ofthe above quadratic equation does not diverge providedβ(r; Ψ) is a smooth function such that we can obtain atrans-fast magnetosonic solution without the usual criti-cal analysis at the magnetosonic point. However, if thereis a location A = 0 along a distant region, the Mach num-ber M2 begins to diverge there, where the particle num-ber density and the magnetization parameter becomezero. To obtain a physical flow solution ejected fromthe plasma source, we require the condition A(r) > 0 inall areas of the super-Alfvenic flow.

Using the definition of the Alfven Mach number, thepoloidal velocity of the cold flow up is given by

u2p = B2

pM4 =

1

β2

(GφE +GtL)2

ρ2w(M2 − α)2

M4 , (14)

5

or using the poloidal Eq. (5), we obtain

u2p =

e2 − αα+ β2

. (15)

At the light surface, the poloidal velocity is up = e/β,where we should select M2(rL) = M2

(+)(rL). Notably,

M2(−)(rL) = 0 with Bp(rL) = 0; i.e., the solution M2(r) =

M2(−)(r) breaks at the light surface. At the Alfven and

fast-magnetosonic surfaces, we have u2p(rA) = u2

AW(rA)

and u2p(rF) = u2

FW(rF), respectively, where the 4-velocityof the Alfven and fast-magnetosonic waves are defined asu2

AW ≡ B2pM

4AW = (B2

φ/β2)α2, and u2

FW ≡ B2pM

4FW =

(B2φ/β

2) (α+ β2)2.From defining the Alfven Much number, the distribu-

tion of the particle number density is obtained as follows:

n =4πµcη

2

M2. (16)

The magnetization parameter σ, which is defined as theratio of the Poynting flux to the total mass-energy fluxseen by a ZAMO, is expressed as follows: [42, 43]

σ(r, θ) =BφGφ

4πµηutρ2w

= − e− αhe−M2h

, (17)

where h ≡ gtt(E − Lω). Thus, the energy conversionbetween the magnetic and fluid parts of the MHD flow’senergy E along the streamline is discussed depending onplasma acceleration.

III. PROPERTIES OF TRANS-FASTMAGNETOSONIC FLOWS

Before considering the trans-fast magnetosonic outflowsolution, the classification and constraints of the entiretrans-fast magnetosonic inflow and outflow solutions inthe BH magnetosphere is summarized.

For stationary and axisymmetric ideal MHD flows inthe cold limit, there are four conserved quantities alongthe flow. The flow properties are characterized by theseparameters with some constraints. In this section, weinvestigate the characteristics of the trans-fast magne-tosonic flow solution under the constraints.

A. Classification of the trans-Alfvenic Flows

The magnetic field lines in the magnetosphere aroundthe Kerr BH are dragged toward the rotation of theBH by the dragging effect of spacetime. In particu-lar, the toroidal component of the magnetic field ofBφ ∝ (ΩF − ω) and the angular momentum of the flowsL ∝ (ΩF − ωA) are strongly influenced by the drag ofspacetime. The value of the angular momentum of theflow can be specified by the location of the Alfven point

under given E and ΩF . Trans-Alfvenic MHD flows in aBH magnetosphere can be classified by LΩF , which givesthe constraint on the field-aligned flow parameters at theAlfven point by the relation LΩF = YA.

The conditions at the Alfven point can be typed by thespin of a BH a and the angular velocity of the magneticfield lines ΩF ; i.e., type I (ωH ≤ ΩF < Ωmax

F ), type II(0 < ΩF < ΩH), and type III (Ωmin

F < ΩF ≤ 0), where

Ωmin/maxF is the minimum/maximum values of ΩF for the

existence of the inner and outer light surfaces. For types Iand III, where (LΩF )min < LΩF ≤ 1 with E > 0 andL > 0 for type I and L < 0 for type III, both the inner andouter Alfven radii appear in the MHD flow solution. Fortype II, however, there is one Alfven radius (the inner orouter one) in the solution, and it can be further classified

into the following three cases: (a) type IIa: 0 < LΩF ≤ 1

with L > 0 and E > 0, (b) type IIb: LΩF ≤ 0 with L ≤ 0

and E ≥ 0 and (c) type IIc: 1 ≤ LΩF with L < 0 andE ≤ 0 (see also, TABLE I).

B. Constraints on Trans-fast MagnetosonicOutflows

For the relativistic outflow such as the M87 jet, wewill handle the flow through the Alfven surface locatedoutside the separation surface, whch is called the outerAlfven surface. Such a flow has a positive angular mo-mentum (L > 0). Hence, we will consider the types I andtype IIa cases. The steady magnetosonic flow in the BHmagnetosphere has several essential surfaces that charac-terize its behavior. They include the separation surface,Alfven surface, light surface, and fast-magnetosonic sur-face. The outflow ejected from the plasma source passesthrough these surfaces before reaching the far distant re-gion.

The locations of the Alfven surface are related to thefield-aligned parameters as LΩF = YA(Ψ), and as a con-dition that the trans-Alfvenic outflow reaches to a fardistant region, A(r, θ) > 0 should be required along theflow (rA < r <∞), as mentioned in the previous section.That is, at the outer Alfven surface r = rA, we find therestriction

A(rA; Ψ) = −kA − 1 > 0 . (18)

This imposes a limit on the field-aligned flow parameters.Thus, we have the condition

M2A = αA >

G2tA

E2, (19)

and we obtain the minimum energy for the trans-Alfvenicoutflows,

E2 >G2tA

αA=

GtA

1− LΩF≡ E2

min . (20)

6

TABLE I: Classification of the trans-Alfvenic flows and the relation between the flow’s total energy E and angular momentumL. The condition A(r; Ψ) > 0 in the super-Alfvenic region is considered.

ΩminF < ΩF ≤ 0 0 < ΩF < ωH ωH ≤ ΩF < Ωmax

F

type I · · · · · · (LΩF )min < LΩF ≤ 1 E > 0, L > 0 inflow or outflow

type IIa · · · 0 < LΩF ≤ 1 · · · E > 0, L > 0 outflow

type IIb · · · LΩF ≤ 0 · · · E ≥ 0, L ≤ 0 inflow

type IIc · · · 1 ≤ LΩF · · · E ≤ 0, L < 0 inflow

type III (LΩF )min < LΩF ≤ 1 · · · · · · E > 0, L < 0 inflow or outflow

Although the function A(r; Ψ) includes the function ξ2,the condition for the minimum energy (20) does not de-pend on the detail expression of ξ2(r, θ); i.e., the min-imum energy for jets does not limit the shape of themagnetic field as ξ2

A(r, θ). Notably, depending on thefunctional form of ξ2, there may be a situation whereA = 0 with M2 → ∞ in the radius to infinity. For atrans-Alvenic outflow solution, if A = 0 before reachingthe distant region, it becomes unphysical, unless MHDshock [42–44] or some type of MHD instability (e.g., [45])occurs in the flow. As discussed in TT03, the conditionξ2(r, θ) < 1 − (1/E2) ≡ ξ2

cr is required to reach a fardistant region. In the BH magnetosphere, the conditionthat the radius of A = 0 does not appear in the trans-fastmagnetosonic solution requires restrictions on LΩF andξ2. At the location of the A = 0 surface, we have thefollowing functions

(LΩF )±(r; Ψ) =Y

1 + Y +X

1 +X ±

[1 + (1− Y )X

−(1 + Y +X)Gt

E2

]1/2, (21)

where X ≡ gφφGt(1− ξ2)/ρ2w. For M2 not to diverge on

the way to the distant region, the condition, LΩF = YA <(LΩF )+

A < 1, must be satisfied at the Alfven surface.Here, the ratio of the toroidal to poloidal magnetic

fields is introduced as a model of the magnetic field lineshape by the function β2(r; Ψ). As a simple situationof the poloidal magnetic field, it may be assumed that amagnetic flux tube is distributed along one conical mag-netic flux surface Ψ(θ0) = constant, where θ0 is the angleof the magnetic surface to be considered, and it shows aspiral shape on this sheet. Notably, because the func-tional form of the magnetic surface Ψ(r, θ) in the entiremagnetosphere is not given, the cross-sectional area ofthe magnetic flux tube along the magnetic flux surfacemay differ from that of the monopole (or split-monopole)magnetosphere. In the conventional study of trans-magnetosonic flow solution, one trans-magnetosonic solu-tion was selected from the solution curve group generatedby combining field-aligned parameters and give the crit-ical condition at the magnetosonic point. On the otherhand, the handling introduced here has the merit that bygiving a regular function β2(r; Ψ) [ or ξ2(r; Ψ) ], a trans-magnetosonic flow can be solved without fine-tuning the

critical values of the field-aligned flow parameters at themagnetosonic point.

C. Constraints on Magnetic Field Lines

In the following, the function form of ξ2(r; Ψ) [insteadof β2(r, θ)] is assumed to be a regular function at themagnetosonic surface, and the magnetized plasma flowsdepending on the field-aligned conserved flow parame-ters are considered. Although the toroidal componentof magnetic field Bφ(r; Ψ) is specified by combining thefield-aligned parameters, the cross-sectional area changealong the flow is also specified accordingly. Moreover,the acceleration efficiency and the magnetization param-eter in the flow are determined depending on the functionξ2(r; Ψ) and the field-aligned parameters. As the func-tional form of ξ2(r, θ), the one considered in TT03 andTT08 is introduced.

From the condition A(r, θ) > 0 for the trans-magnetosonic flow solution, we have

ξ2(r, θ) < (k+1)gφφ(ΩF − ω)2ρ2

w

(GφE +GtL)2=

−(k + 1)

gttE2(1− LΩF /Y )2.

(22)At the far distant region (r/m 1), this condition be-comes

ξ2(r, θ) < 1− 1

E2+

1

E2

2m

r+O(r−2) . (23)

(This corresponds to ζ0 < (1/E2)(2m/r), where ζ0 isintroduced in Eq. (A3) of TT08. ) Referencing TT03 andTT08, we consider the following ξ2 model for each type ofBH magnetosphere. For the outflows of type I/III/IIa, wecan apply the following as a functional form of ξ2(r; Ψ),

ξ2 = 1− ∆

Σ

1

E2+ ζ . (24)

The effect of ζ is related to the energy conversion effi-ciency of the outflow, and discussed in [26]. Assumingζ = 0 for simplicity, the radial velocity by Eq. (15) be-comes

(ur)2 =

[E2(1− LΩF )2 − α

] [E2 − (∆/Σ)

]E2 (Σ2/A)− α

. (25)

7

At a distant region (R ≡ r sin θ 1), the terminal veloc-ity of the trans-fast magnetosonic relativistic jet becomes(ur)∞ = γ∞ ∼ E. Thus, the total energy is primarily ki-netic in the distant region. On the way to the distantregion, the initially magnetically dominated MHD flowenergy is converted to the kinetic energy of the plasma.

When we discuss the inflows, certain restrictions on ξ2

should be considered (TT08). First, it must be ξ2 = 1 onthe event horizon. For example, for type I or III inflowcase, we can set up the following functional form:

ξ2 = 1− ∆

Σf(r, θ) , (26)

where the function f(r, θ) is a regular function. Whenthe anchor radius appears in the flow, where the toroidalmagnetic field becomes 0, it is necessary to satisfy thecondition ξ2 = ∞ (or β2 = 0). Therefore, we can con-sider the following as an example,

ξ2 =

(1− ∆

Σf(r, θ)

)(GφE +GtL)H

(GφE +GtL). (27)

For type II, the “corotation radius” defined by the radiusof ω(r; Ψ) = ΩF occurs in the BH magnetosphere. At thecorotation radius where Gφ(r, θ; ΩF , a) = 0, we obtainthe finite poloidal velocity [i.e., (ur)2 = finite], but weget M2 = 0 and n = ∞ where Bp = ∞ and β2 = 0because ξ2(r; Ψ) = finite 6= 0 is assumed. This indicatesthe break down of the flow solution there; therefore, werequire ξ2 = 0 at the corotation radius. Thus, we canintroduce the following functional form:

ξ2 =

(1− ∆

Σf(r, θ)

)(ω − ΩFωH − ΩF

)2

. (28)

For type IIa outflows (discussed in Section IV), theabove solution (15) can be applied in the area outside thecorotation radius, where Gφ > 0. Fortunately, becausethe separation surface is located outside the corotationradius, there should be no problem with the general out-flow, starting from the plasma source located near theseparation surface. However, to obtain physical inflowsolution (i.e., type IIb/IIc) starting around the separa-tion surface, we must set up ξ2 = 0 at the corotationradius, hence we should use Eq. (28) as a model of ξ2

rather than Eq. (26). In the following sections, we fo-cus only on outflows ejected from outside the corotationradius as the jet model.

D. Injection Surface and Initial Velocity

The outgoing and ingoing plasma flows are injectedinto the outer and inner BH magnetospheres from theplasma source, respectively. The plasma source must belocated between the inner and outer light surfaces so thatαinj > 0 is required. At the injection surfaces r = rinj(θ)with M2

inj 1, from Eq. (9), we have the relation of

αinj = (E−LΩF )2. Because the particle number flux permagnetic flux tube η(Ψ) is the conserved quantity along amagnetic field line, for an injected plasma from the injec-tion surface with a low velocity urinj 1, the number den-

sity of flow is very high ninj 1. If αinj > (E − LΩF )2,we have (urinj)

2 < 0; i.e., there is no physical solutionthat flows out from the injection surface with a finite ini-tial velocity. Specifically, for a highly magnetized flow ofLΩF ∼ E, the physical flow would start near the lightsurface, rinj . rL. Meanwhile, for a small value of the

angular momentum, |L| < (E − √αinj)/|ΩF |, the flowejected from the injection surface initially has a relativis-tic speed (i.e., urini & 1).

IV. PARAMETER DEPENDENCIES ON MHDOUTFLOW

We consider the parameter dependence of outflowin detail. For numerical plots of the radial velocityur(r; Ψ) about outflows, we set up the functional form ofξ2(r; Ψ). Hence, from Eq. (13), we obtain the distribu-tion M2(r; Ψ) along the flow line; using Eq. (4), we havethe toroidal component of the magnetic field Bφ(r; Ψ)and Bp(r; Ψ). As such, assuming the function ξ2(r, θ) isequivalent to assuming the poloidal magnetic field Bp.We discuss trans-fast magnetosonic flow solutions alonga single flux tube that is theoretically and observationallyvalid so that we can consider the qualitative understand-ing of the behavior of jet flow acceleration.

Certain physical parameters would be significantly af-fected by the general-relativistic effect; hence, we focusour research on this. The locations of the light and Alfvensurfaces, which are the typical scales of the magneto-sphere, primarily depend on the angular velocity ΩF ofthe magnetic field lines and the ratio of the rotationalenergy to the total energy LΩF of the plasma flow, re-spectively. In the following parameter search, the de-pendency of ΩF and L is particularly examined. Weassume ΩF (Ψ) = constant for the entire magnetosphere.Although an example of ΩF (Ψ) = non-constant is con-sidered by Pu & Takahashi [26], the basic properties aresimilar, whereas the distributions of the light surfacesdiffer slightly. Moreover, we assume η(Ψ) = constant forthe entire magnetosphere for simplicity. Notably, in theξ2 model, the parameter η is renormalized into the mag-netic field Bp and Bφ so that it does not directly appearin the expression of the MHD flow equations.

Figure 2 shows the velocity distribution ur(R,Z) whenEq. (24) is employed as the distribution of ξ2(R,Z) with

constant E and ΩF for a rapidly rotating BH (type IIa)case, where R = r sin θ and Z ≡ r cos θ. To understandthe angular momentum dependence on the flow, the casesof constant angular momentum L(θ) =constant are plot-

ted in Figures 2(a) and 2(b), and the case where L has

angular dependence of L(θ) = L0 sin2 θ with a constant

L0 is plotted in Figure 2(c). The outflow from the plasma

8

R/m

Z/m

0

2

4

6

8

10

0 2 4 6 8 10

(b)

Z/m

0

2

4

6

8

10

0 2 4 6 8 10

(c)

R/m

Z/m

0

2

4

6

8

10

0 2 4 6 8 10

(a)

R/m

P

QR

FIG. 2: The distribution of the radial velocity ur(R,Z) in the BH magnetosphere of a = 0.9m, ΩF = 0.5ωH; (a) the cases of

LΩF = 0.9, (b) LΩF = 0.93, and (c) L(Ψ) = L0 sin2 θ, where L0ΩF = 0.9. The broken magenta curves show the locations ofthe inner and outer light surfaces. The dotted green curve shows the location of the separation surface. The shaded area on theupper left of the dotted curve in (b) is the forbidden region, where u2

p < 0. The red lines (outside the corotation surface of thecyan dashed curve) are examples of the streamlines for outflow (Z = 2R, Z = 0.5R). The red dotted line in (b) is unacceptableas an outflow. The black region shows the BH. Points P, Q, and R are examples of the injection points.

source (e.g., points P, Q, and R in Fig. 2) flows out inthe distant region.

If L(θ) = constant in the magnetosphere, the higherthe latitude, the smaller the initial velocity. However,when the angular momentum is very large, the flow’s for-bidden region (i.e., u2

p < 0 region) appears in the high lat-itude region [see, the shaded region in Fig. 2(b)]. Alter-natively, the angular momentum at high latitudes shouldbe selected not too large for the trans-fast magnetosonicsolution to fill the magnetosphere in the jet region [see,Fig. 2(a)]. Moreover, we observe that the behavior of ac-celeration differs depending on whether the angular mo-mentum is constant throughout the magnetosphere orwhether it has a sin2 θ dependence [see, Fig. 2(c)]. Tofit the acceleration profile of the M87 jet (see Fig. 6),

LΩF = constant seems to be better in the entire regionof the magnetosphere. Afterward, we will consider thecase of L(Ψ) = constant.

Figure 2 also shows the velocity distribution within theseparation surface, where the parameter set used herecorresponds to the type IIa case; therefore, it fails asa physical solution on the corotation surface. Notably,from the area between the corotation and separation sur-faces (e.g., point P), an outgoing flow ejected with a suf-ficiently large initial velocity is possible.

Figures 3–5 show the dependences of the field-alignedparameters on the trans-fast magnetosonic outflow solu-tions (red curves) along a θ = constant magnetic field linefor the type IIa case. The label “Co” or orange dashedline indicates the location of the corotation surface. Thelabel “SP” or green dotted line indicates the location ofthe separation surface and the label “LS” or magenta bro-ken lines indicate the location of the light surfaces. Themark labeled “A” and the mark labeled “F” indicatethe Alfven and fast-magnetosonic points, respectively.

A. Energy E Dependence

In Figure 3, we show the E dependence on the trans-fast magnetosonic outflow solutions, where the values ofa, LΩF , and ΩF are fixed; therefore, the locations of theseparation surface, light surfaces, and Alfven surfaces arethe same. The Alfven Mach number M2(Z) rapidly in-creases from around the Alfven point and passes throughthe fast-magnetosonic point to a distant region. For largeE values, where the L value is also large, the locationof the fast-magnetosonic point appears on the outside,and the plot of the radial velocity ur(Z) shifts upwardover the entire area. In Figure 3(b), the injected plasmaaround the separation surface has a nonzero initial veloc-ity. For example, the outflow solution passing throughpoint S has an almost constant flow velocity betweenthe corotation and Alfven surfaces and then acceleratesacross the fast-magnetosonic surface. For an outflow so-lution passing through point T, the outflow deceleratesonce toward the separation surface, but after passing theseparation surface, it accelerates and passes through theAlfven and fast-magnetosonic surfaces in order. For alow energy flow, the plasma flows out at a very low speedfrom the vicinity of the Alfven surface (point U), whichis far outside the separation surface. Such outflow is sig-nificantly accelerated beyond the Alfven surface from theinjection surface.

The rescaled number density n ≡ n/(4πµcη2) = 1/M2

decreases in the distance, unlike the increase in theAlfven Mach number [see, Fig. 3(a)]. Figure 3(c) showsthe distribution of the magnetization parameter σ(Z).For a large energy outflow, the magnetic field energy isdominant at the beginning of the flow; however, as theoutgoing fluid accelerates, the fluid part of the energy in-

9

SP LSCo

A F

(a)

(b)

(c)

1.0

0.0

-1.0

-2.0

-3.0

-4.0

-1.0

-0.5

0.0

0.5

1.0

1.5

-6.0-4.0-2.00.02.04.06.08.010.0

S

T

U

0 1 2 3 4 5 6 7

A F

A F

FIG. 3: The dependences of total energy E on relativisticoutflows (Type IIa: E = 5.0, 8.0, 10.0, 20.0, 50.0 with a =

0.9m, ΩF = 0.05ωH = 0.0157/m, LΩF = 0.9, θ = 1/E) (a)the Alfven Mach number M2(Z), (b) the radial velocity ur(Z)and (c) the magnetization parameter σ(Z). The blue curvesinside the corotation surface (Co) are unphysical solutions asoutflows.

creases and becomes dominant at a far distance (σ 1).For a low energy outflow, which corresponds to a largeangular momentum outflow, the fluid part of the energyis dominant at the beginning of the flow when the injec-tion point is located just inside the outer light surface.

For a slowly rotating BH case of type I, both the outerlight surface and the outer Alfven surface shift outward;however, the asymptotic feature of outflows is the same.

A

Co SP LS

(a)

(b)

(c)0 1 2 3 4 5 6 7

1.0

0.0

-1.0

-2.0

-3.0

-4.0

-1.0

-0.5

0.0

0.5

1.0

1.5

-6.0-4.0-2.00.02.04.06.08.010.0

F

A F

AF

FIG. 4: The dependences of total angular momen-tum L on relativistic outflows (Type IIa: LΩF =0.70, 0.80, 0.90, 0.95, 0.99, a = 0.9m, ΩF = 0.05ωH =

0.0157/m, E = 10.0, θ = 1/E) (a) the Alfven Mach numberM2(Z), (b) the radial velocity ur(Z) and (c) the magnetiza-tion parameter σ(Z).

B. Angular Momentum LΩF (L) Dependence

Figure 4 shows the LΩF -dependence on the trans-fastmagnetosonic outflow solutions, where E and ΩF arefixed; i.e., the dependence of angular momentum L isinvestigated. The angular momentum L determines thebehavior of the initial stage of acceleration of the ejectedoutflow in the sub-Alfvenic region and near the light sur-face. For outflows with small angular momentum, the ve-locity distribution in the sub-Alfvenic region has a largevalue, i.e., a solution with nonzero velocity is obtainedaround the separation region. Meanwhile, for outflows

10

(c)

(b)

(a)

0 1 2 3 4 5 6 7

1.0

0.0

-1.0

-2.0

-3.0

-4.0

-1.0

-0.5

0.0

0.5

1.0

1.5

-6.0-4.0-2.00.02.04.06.08.010.0

AF

AF

AF

FIG. 5: The dependences of magnetosphere’s angularvelocity ΩF on relativistic outflows (Type IIa: mΩF =

0.0003, 0.003, 0.03, 0.10, 0.30, a = 0.9m, E = 10.0,

LΩF = 0.9, θ = 1/E) (a) the Alfven Mach number M2(Z),(b) the radial velocity ur(Z) and (c) the magnetization pa-rameter σ(Z).

with large angular momentum, LΩF ∼ 1, no physical so-lution region appears around the separation surface in thesub-Alfvenic region. Such a flow with large angular mo-mentum starts at a slower speed from slightly inside theouter light surface, where the toroidal motion of the flow

is dominated; i.e., vφini ≈ c and γini 1 (see also, [13]).

Thus, the value of L is related to the toroidal motion inthe sub-Alfvenic region. The closer the injection surfaceto the light surface, the more dominant toroidal motionof plasma (σini 1) is obtained in the initial stage ofthe MHD flow, whereas the poloidal motion is relativelysmall. Subsequently, it is converted to poloidal motion,

and eventually becomes magnetically dominated MHDoutflow (σ 1) in the next stage of the accelerationregion. Through the equipartition region (TT03), the ki-netic energy by the poloidal motion becomes dominantover the magnetic energy; i.e., σ∞ 1. The terminalvelocity (ur)∞ does not depend on angular momentum

L. When the angular momentum becomes large, the flowstarting near the separation surface is prohibited.

C. Angular Velocity ΩF Dependence

Figure 5 shows the ΩF -dependence on the trans-fastmagnetosonic outflow solutions, where E and LΩF arefixed; therefore, a large (or small) ΩF value corresponds

to a small (or large) L value. Each solution curve inFig. 5 is similar, although the location of the light sur-face depends on both ΩF and a. When the value of ΩFdecreases, the outer light surface shifts to the outside.Meanwhile, when the value of spin a increases, the ef-fect of the spacetime dragging on the magnetic field linesbecomes large and it shifts to the inside.

V. PLASMA ACCELERATION ON M87 JET

Now, we explain the observed data of the M87 jet byparameter searching using the stationary trans-fast mag-netosonic outflow model. The mass of the central BHin M87 is estimated to be mM87 = (6.5 ± 0.7) × 109m(e.g., [46, 47]). The inclination angle (viewing angle) ofthe M87 jet is estimated to be θinc ≈ 17 [48]. The jetpower in M87 is in a range of 1035–1037 J s−1, and thereare several theoretical calculations of jet power driven byelectromagnetic mechanisms represented by the BZ pro-cess [7] and/or Blandford–Payne process [49] (see also,e.g., [50]). The parsec-scale jet of the galaxy M87 isroughly parabolic (Z ∝ R1.7) in structure [3, 4], where Zis the deprojected distance along the jet, and R is the jetwidth, and the M87 jet becomes conical-like (Z ∝ R1.3)near the BH [5]. The radial profile of the M87 jet velocityis observed by [1, 48, 51–54], and recently detail data forthe region closer to the base of the jet has been obtainedby Park et al. [6] using the KaVA.

A. Acceleration Region of M87 Jet

In this study, we examined the general-relativistic ef-fects in the outer BH magnetosphere. In this section,the M87 jet data fitting is performed using the general-relativistic version of the ξ2 model by TT08 that incor-porates the effect of a rotating BH, where the plausi-ble magnetic field suggested by VLBI observations andgeneral-relativistic MHD (GRMHD) numerical simula-tions for the relativistic jet are adopted (e.g., [55–57]).To discuss the acceleration properties of the jet, we can

11

consider the angular momentum dependence on the out-flow in the acceleration region and the initial velocityaround the plasma source region.

In the following, we apply the trans-fast magnetosonicjet solution to the Park’s M87 data [6]. As the flow’sstreamlines in the BH magnetosphere, we apply the ap-proximated magnetic field configuration in the asymp-totic region obtained from Eq. (57) of TT03; i.e.,

θ0Z/m = (R/m)Ψ0/Ψ (29)

for θ 1, where Ψ→ 0 for θ → 0, although this solutionshould not extend inside the outer light surface becauseit is out of the range of the approximation of TT03. How-ever, it may be suitable for around the outer light surface,where current observation data are primarily obtained.We expect it to be applicable for qualitative understand-ing of the relativistic jets from the sub-Alfvenic regionof the outer BH magnetosphere. In Figure 6, we inter-polate the approximated magnetic field shape (29) intothe sub-Alfvenic region that includes the plasma source,where the boundary layer Ψ0 = Ψ(θ0) between the jet re-gion and middle/lower latitude accreting matter region

is given by the θ0 ∼ 1/E conical magnetic field line. No-tably, around or inside the outer light surface, Hada etal. [5] and TT03 model suggests a conical-like magneticfield shape. (The nature of the conical outflow has beendiscussed in Section IV.)

Figures 6(b) and 6(e) show the radial velocity ur(Z)of the outgoing jet from the BH magnetosphere. TheKaVA’s data is rescaled in the unit of mM87, where1 mas ≈ 260mM87 [6]. The correspondence between theapparent angle and actual scale significantly depends onthe inclination angle (viewing angle) and opening angle ofthe jet. The ejected outflow from the plasma source is ac-celerated around the light surface; after passing the fast-magnetosonic surface, the flow velocity reaches a nearlyconstant value that is large enough; i.e., the initiallymagnetically dominated outflow becomes the fluid’s ki-netic energy dominated flow. Although we assumed thata = 0.9m, the spin dependence on the outflow solutionsis weak at least outside the outer light surface. The effi-ciency of acceleration differs for each magnetic field line.The magnetic field lines close to the funnel wall (Ψ ∼ Ψ0)cause large acceleration; moreover, the acceleration effi-ciency is minute for the magnetic field lines near the axis(Ψ Ψ0). The plasma source is located near the sepa-ration surface of each curve for the magnetic flux surface.

Now, we understand that the values of E and LΩF areestimated by fitting the terminal velocity ur∞ and theinitial velocity urini at the injection region (or the accel-eration region near the plasma source), and the locationof the light surface rL is specified by parameters ΩF anda. The trans-fast magnetosonic MHD outflow becomesa fluid-dominated outflow at the distant region. By ob-serving the terminal velocity ur∞ (or γ∞) of the M87 jet,

we can specify the value of E ≈ 10. Angular momentumL (or LΩF ) is largely related to urini and the velocitydistribution in the acceleration region around the outer

light surface. By comparing the theoretical solution withthe M87 observed data, we estimate LΩF ≈ 0.9 from theacceleration profile of the jet. If the angular momentumparameter of the flow reduces by some extent, the ejectedflow around the separation surface will have a faster ini-tial velocity. The initial flow acceleration obtained incertain GRMHD simulations (e.g., [2, 55, 56]) that showurini & 1 and do not explain the M87 observations wellwould be explained in a parameter range with a smallangular momentum.

The location of the outer light surface depends on boththe angular velocity of the magnetic field lines and theBH spin. In Figures 6(b) and 6(e), from the location ofthe acceleration region for the M87 jet, we estimate thatΩF ≈ 0.023/m for a = 0 case, and ΩF ≈ 0.022/m fora = 0.9m case. Thus, the outer light surface, RL ∼ c/ΩF ,will be located far enough from the central BH; i.e., theeffect of BH spin decreases at the outer light surface. Itcan be said that the behavior of the jet does not dependon the details of type I or type IIa.

B. Energy Conversion in M87 Jet

Figures 6(c) and 6(f) show the distribution of the mag-netization parameter σ(Z) in the jet. The efficiency ofenergy conversion from the magnetic energy to the fluid’skinetic energy differs for each magnetic field line. Al-though the jet has a large value of σ at the beginning,the value of σ decreases along the magnetic field linesof Ψ ∼ Ψ0; then, at a distant region, the jet becomesa kinetic energy-dominated outflow. However, along themagnetic field lines near the axis of rotation, Ψ Ψ0,where the centrifugal force on plasma does not work effec-tively, the acceleration efficiency is relatively small, andthe value of σ does not decrease significantly. Thus, themagnetic field lines close to the funnel wall cause effi-cient energy conversion, whereas the efficiency is minutefor the magnetic field lines near the axis.

The strength of the magnetic field at the footpoint ofthe jet is estimated from the Event Horizon Telescope(EHT) observation [58, 59]. The value of the magneti-zation parameter σ within the outer light surface is esti-mated by Kino et al. [24]; i.e., 1 × 10−5 ≤ σ ≤ 6 × 103

within the radio core with VLBA at 43 GHz and 5 ≤ σ ≤1 × 106 within the putative synchrotron self-absorption(SSA) thick region in the EHT emission region at 230GHz. Thus, we identify that the high values of the σprofile within the outer light surface agree the ξ2 modelfor the GRMHD jet.

Moreover, low-σ (σ ∼ 10−4) has been observed in themajor atmospheric gamma imaging Cerenkov telescope(MAGIC) TeV-gamma observation [60], although the ac-curacy of the radiation source position observation is un-certain. For example, in Figure 6(c), such a low-σ flowcan be achieved at a distance of Z/m > 106. Alterna-tively, the MAGIC observation region may be near theouter light surface, which is because the outflow solution

12

(b)

(a)

0 1 2 3 4 5 6 7

1.0

0.0

-1.0

-2.0

-3.0

-4.0

-1.0

-0.5

0.0

0.5

1.0

1.5

-6.0-4.0-2.00.02.04.06.08.010.0

(c)

AF

AF

AF

(d)

(e)

(f)

AF

AF

AF

0 1 2 3 4 5 6 7

1.0

0.0

-1.0

-2.0

-3.0

-4.0

-1.0

-0.5

0.0

0.5

1.0

1.5

-6.0-4.0-2.00.02.04.06.08.010.0

FIG. 6: The Ψ-dependence on the jet solution is shown (Ψ/Ψ0 = 1.0, 0.8, 0.6, 0.4, 0.2, 10−5, θ0 = 1/E). (a)–(c) type I:

a slowly rotating BH of a = 0 with E = 10.0, ΩF = ωH + 0.12(Ωmax − ωH) = 0.0231/m, LΩF = 0.9, and (d)–(f) type IIa: a

rapidly rotating BH of a = 0.9m with E = 10, ΩF = 0.07ωH = 0.0219/m, LΩF = 0.9 are presented. The measured data pointswith error bars are adopted from [6]. In panels (d)(e), the dotted blue curves inside the corotation surface show unphysicalbranches as the jet solution.

emitted from the region slightly inside the outer lightsurface is realized with very low-σini for LΩF ∼ 1 [see,Figs. 3(c), 4(c)]. Such an outflow initially rotates in thetoroidal direction at very high speed (γini 1), resultingin low-σ, and then it accelerates outward. Interestingly,the plasma moving at high speed in the toroidal directionjust inside the outer light surface generates an extremelylarge acceleration, which would be a region where very-high-energy gamma rays are generated. If the vicinityof the outer light surface is the source of high-energy γ-ray for some magnetic field lines in the jet, the γ-raysource may also be distributed along the jet (i.e., hol-low cylindrical-shape) because the outer light surface isdistributed in the region extending along the jet.

VI. DISCUSSION

A. Dependence on magnetic field line shapes

To discuss the parameter dependence of the ideal MHDfield-aligned quantities, we assume the magnetic surfacehas a conical shape. Further, it is easy to extend toother plausible magnetic field shapes; e.g., we can usethe model

θ0Z/m = (R/m)p(Ψ0/Ψ), (30)

where p is a parameter for a magnetic field configura-tion, and Ψ = Ψ0 is the streamline for the jet’s bound-ary wall. According to Hada’s observation [5], the M87

13

jet shape near the jet source region bends around thefast-magnetosonic surface, suggesting a conical-like shapeon the side near the central BH. This shape would sug-gest the magnetic field configuration in the BH magne-tosphere. When applied to the above model (30), it isabout p ≈ 1.3 for the boundary wall of the funnel on theinside of the bend and about p ≈ 1.7 on the outside.

(a)

AF

AF

AF

(c)

(b)

-1.0

-0.5

0.0

0.5

1.0

1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

0 1 2 3 4 5 6 7

-1.0

-0.5

0.0

0.5

1.0

1.5

FIG. 7: The radial 4-velocity ur of the flow with differ-ent magnetic field shapes, (a) p = 1.0, (b) p = 1.3, and (c)p = 1.7, overlapped with the observation jet velocity of M87.The trans-fast magnetosonic solutions along the magnetic fluxsurfaces of (Ψ/Ψ0) = 1.0, 0.8, 0.6, 0.4, 0.2, 10−5 are plot-

ted, where a = 0.9m, E = 10.0, LΩF = 0.9, θ0 = 1/E and(a) ΩF = 0.07ωH = 0.0219/m, (b) ΩF = 0.15ωH = 0.0470/m,(c) ΩF = 0.30ωH = 0.0940/m are assumed.

Figure 7 shows the dependence of the funnel wall con-figuration of p = 1.0, 1.3, and 1.7. Hence, one can un-derstand the dependence of the magnetic field shape onthe acceleration efficiency of jets. In Figure 7(a), we plot

the radial velocity ur for magnetic field lines within theconical boundary wall of θ = θ0. Figures 7(b) and 7(c)show the cases of parabolic boundary wall of p = 1.3and 1.7, respectively. When the jet’s boundary wall isconical (p = 1.0), it is estimated that ΩF ≈ 0.02/m,whereas when it is a parabolic shape, the value of ΩFis overestimated compared with the case of conical; i.e.,ΩF ≈ 0.05/m for p = 1.3, and ΩF ≈ 0.09/m for p = 1.7.By fitting with KaVA data within several times rF, avalue of about ΩF = (0.02 − 0.05)/m is obtained suchthat it should be slightly parabolic for a slowly rotatingBH magnetosphere according to Thoelecke et al. [62].Thus, the model of the magnetic line shape of Ψ(r, θ)and ξ2(r; Ψ) used in this study incorporates a plausiblesituation based on the observation results to some extent.

Although Hada et al. [5] observed bending of the jetshape, the physical reason for this bending was discussedin TT03. TT03 considered the flow of E = γ∞ 1, de-rived the approximated transfield equation for the outerregion of the outer light surface, and solved the self-consistent magnetic field configuration. The outflowejected from the plasma source was initially magneticallydominated. In the process of passing through the Alfvenand fast-magnetosonic surfaces and accelerating, the en-ergy conversion from magnetic field energy to fluid ki-netic energy occurs slowly but effectively. The bendingof the magnetic field lines occurs in the area where the en-ergy of both become equipartition. In other words, in theinner region where the magnetic field is dominant, it isconical-like or slightly parabolic, and when the plasma isaccelerated and the plasma inertia becomes effective, thetoroidal component Bφ of the magnetic field is enhanced.Therefore, the magnetic field lines are curved toward therotation axis. Although the shape of the magnetic fieldlines in the vicinity of the BH is still unexplored, theobservation results of the jet shape (i.e., magnetic fieldlines) of Hada et al.[5] in the region close to the footpointof the jet seem to roughly correspond to the magneticfield configuration suggested by TT03.

Notably, comparing the parabolic shape with the con-ical shape, for the same value of ΩF , the location of theouter light surface along the jet, ZL(Ψ0), shifts in the+Z-direction; however, the width of the outer light cylin-der, RL, is the same. Even if the ZL(Ψ0) is observablyestimated along the streamline of the jet, there is a de-pendence on both ΩF , p, and θ0. If the velocity distri-bution in the width direction of the jet is measured, thewidth RL (i.e., ΩF ) can be directly estimated.

B. M87 Jet Power and BZ Power

The BZ power is obtained from estimating the mag-netic field strength near the rotating BH and the angularvelocity values (i.e., 0 < ΩF < ωH). It is difficult to esti-mate the value of the BH spin from the observation databy Park et al. [6]. However, the Ωoutflow

F value can beestimated from the observation and by assuming the ΩF

14

values match in the outflow and inflow across the plasmasource region without significant energy loss. Hence, itcan be considered whether BZ power can be applied asthe jet power. For example, we can assume the junc-tion condition at the plasma source as Ωinflow

F ≈ ΩoutflowF ,

and ErH ≈ (ηE)outflow, where Er ≡ T rαEMkα is the radialcomponent of the conserved electromagnetic energy flux(i.e., BZ power) and ηE is the energy flux of MHD flow.For the outflow where ηoutflow > 0 and Eoutflow > 0,(ηE)outflow indicates the jet power per magnetic flux.Note that the jet power (ηE)outflow comprises the mag-netic and fluid parts of the energy flux, whereas ErH ispurely magnetic energy flux from the BH.

We can also consider the case of ErH > (ηE)inflow ≈(ηE)outflow across the plasma source, where (ηE)inflow

can be obtained as a solution of type IIc negative energyMHD inflow; i.e., Einflow < 0, Linflow < 0, ηinflow < 0inflow [11, 61] (see also, [29] for gamma-ray bursts jets).Notably, ErH ≈ (ηE)inflow > 0 in the magnetically dom-inated limit (i.e., force-free case). For the solution oftype IIc inflow, the magnetic field geometry of Eq. (28)should be adopted to avoid the break at the corotaionsurface during accretion onto the event horizon. More-over, the boundary conditions at the plasma source thatagree with the outflow solution must be estimated withinthe physical parameter ranges such that A(r, θ) > 0 be-tween the plasma source and the event horizon.

When the magnetic field shape is conical near the eventhorizon, we have

LBZ = 2π

∫ θH

0

dθ ErHΣH sin θ

= 2πε0Ψ2eq

∫ θH

0

dθ2mrH

ΣHΩF (ωH − ΩF ) sin3 θ ,

(31)

where θH is the half opening angle of the jet at the eventhorizon. The magnetic flux function at the event horizonis assumed to Ψ(rH, θ) = Ψeq (1− cos θ) with a constantΨeq. The jet’s opening angle is narrow enough, θH 1,such that the BZ power is estimated as follows:

LBZ ≈ 3.3× 1038 ΛΩFωH

(1− ΩF

ωH

)( am

)2

×(BpH0.1T

)2(m

109m

)2

J s−1 , (32)

where Λ is the geometrical factor given by

Λ =

(2

3− 3

4cos θH +

1

12cos(3θH)

)=θ4

H

4+O(θ6

H) .

(33)

Although we estimate θ0 ∼ 1/E ≈ 0.1 rad at the distantregion from the BH, in the vicinity of the event horizon,the magnetic field lines for inflows would be parabolic[62, 63] such that θH > θ0 ∼ 1/E. So, when θH ≈ 0.3 radat the event horizon, we have Λ ≈ 0.002 as a rough value.Hence, for Λ = 0.002, BpH = 0.1T, m = mM87 = 6.5 ×

109m, a = 0.9m, and ΩF = 0.07ωH = 0.0219/m, wehave LBZ ≈ 1.5× 1036 J s−1, which agree with the valueestimated by the jet observation. In Figure 6, the value ofΩF was estimated for the conical boundary wall; however,for the parabola-like boundary shape, the estimated ΩFvalue is about a factor larger. The BZ power may be anorder of magnitude larger.

VII. CONCLUDING REMARKS

In this article, we discussed the parameter dependenceon the MHD outflows, and we realized that the solutionsof urinj > 0 at rinj = rsp or rsp < rinj with urinj = 0.That is, the initial velocity of the flow solution is not al-ways 0, and the plasma sources may be widely distributedbetween the inner and outer light surfaces. Thus, theplasma source will have some internal structure; there-fore, we should discuss some physical process for theplasma source. It may be some kind of plasma insta-bilities, nonideal MHD process, or particle pair-creationprocess [27–36]. By connecting the outflow solution ob-tained from the observation data to the plasma source, wecan consider the junction condition or restrictions there(i.e., [26, 64]). Further, we would discuss the inner BHmagnetosphere. These details are outside the scope ofthis study and are for future study.

In this study, we define the nondimentional magneticfield Bp and Bφ, and introduce the ratio β(r; Ψ) as aphysical model. Therefore, there is no discussion aboutthe particle number flux number η(Ψ) because this pa-rameter is used to make Bp and Bφ dimensionless, whichis related to the fact that the critical condition at themagnetosonic points is no longer necessary in this model.To estimate η value, it is necessary to measure notonly velocity distribution uα(r, θ) but also magnetic fieldstrength distribution Bp(r, θ) and density distributionn(r, θ). Therefore, it is paramount to understand themechanism of shock wave formation and the radiationmechanism in the jet region. Thus, by comparing obser-vational data, it is possible to quantitatively discuss theparticle number flux η(Ψ).

In summary, we applied the trans-fast magnetosonicoutflow solution in a BH magnetosphere to the M87 jet.Specifically, we discussed the flow velocity ur(Z; Ψ) andthe magnetization parameter σ(Z; Ψ). We evaluate the

values of field-aligned flow parameters; E ≈ 10, LΩF ≈0.9, ΩF ≈ (0.02 − 0.05)/m, i.e., ΩF ≈ (20 − 50) year−1.Using these values, the width of the outer light surface(light cylinder) is roughly RL ≈ (20 − 50)m, and thewidth of the Alfven surface is roughly RA ≈ 0.95RL. Asthe spatial resolution will be improved by future VLBIobservations and the spatial distributions of jet velocityur(R,Z), and the magnetic field strength are known, thedistribution and limits of E(Ψ), L(Ψ), ΩF (Ψ), and η(Ψ)will be revealed. Further, the future observations of theBH shadow by sub-millimeter wave and observations viaother wavelengths, such as X- and gamma-ray, will also

15

impose restrictions on the physical quantities of outflowand inflow. Thus, the quantitative understanding of BHmagnetosphere including the plasma sources; hence, theBH spacetime, will be deepened.

Acknowledgments

The authors thank Honoka Daikai for fruitful discus-sions. M.T. was supported in part by JSPS KAKENHI

Grant No. 17K05439. M.K. was supported in partby JSPS KAKENHI Grant Numbers JP18H03721 andJP21H01137. H.-Y. P. acknowledges the support of theMinistry of Education (MoE) Yushan Young Scholar Pro-gram, the Ministry of Science and Technology (MOST)under the grant 110-2112-M-003-007-MY2, and NationalTaiwan Normal University.

[1] F. Mertens, A. P. Lobanov, R. C. Walker, and P. E.Hardee, Kinematics of the jet in M87 on scales of 100–1000 Schwarzschild radii, Astron. Astrophys.595, A54(2016).

[2] K. Chatterjee, M. Liska, A. Tchekhovskoy, et al., Acceler-ating AGN jets to parsec scales using general relativisticMHD simulations, Mon. Not. R. Astron. Soc.490, 2200(2019).

[3] K. Asada and M. Nakamura, The Structure of the M87Jet: a Transition from Parabolic to Conical Streamlines,Astrophys. J. Lett. 745, L28 (2012).

[4] M. Nakamura and K. Asada, The Parabolic Jet Structurein M87 As a Magnetohydrodynamic Nozzle, Astrophys.J. 775, 118 (2013).

[5] K. Hada, et al., The Innermost Collimation Structure ofthe M87 Jet down to ∼10 Schwarzschild Radii, Astro-phys. J. 775, 70 (2013).

[6] J. Park, et al., Kinematics of the M87 Jet in the Colli-mation Zone: Gradual Acceleration and Velocity Strati-fication, Astrophys. J. 887, 147 (2019); arXiv:1911.02279

[7] R. D. Blandford and R. L. Znajek, Electromagnetic ex-traction of energy from Kerr black holes, Mon. Not. R.Astron. Soc.179, 433 (1977).

[8] R. L. Znajek, Black hole electrodynamics and the Cartertetrad, Mon. Not. R. Astron. Soc.179, 457 (1977).

[9] M. Livio, G. I. Ogilvie, and J. E. Pringle, Extractingenergy from black holes: The Relative importance ofthe Blandford–Znajek mechanism Astrophys. J. 512, 100(1999).

[10] V. S. Beskin and I.V. Kuznetsova, On the Blandford–Znajek mechanism of the energy loss of a rotating blackhole, Nuovo Cim.B 115, 795 (2000).

[11] M. Takahashi, S. Nitta, Y. Tatematsu, and A. Tomi-matsu, Magnetohydrodynamic Flows in Kerr Geomerty:Energy Extraction from Black Holes, Astrophys. J. 363,206 (1990).

[12] N. Globus and A. Levinson, Loaded magnetohydrody-namic flows in Kerr spacetime Phys. Rev. D , 88, 084046(2013).

[13] M. Takahashi and S. Shibata, Trans-Fast MHD Winds ina Pulsar Magnetosphere, Publ. Astron. Soc. Jpn.50, 271(1998).

[14] M. Takahashi, Transmagnetosonic Accretion in a BlackHole Magnetosphere, Astrophys. J. 570, 264 (2002).

[15] M. Takahashi and A. Tomimatsu, Constraints on the evo-lution of black hole spin due to magnetohydrodynamicaccretion, Phys. Rev. D 78, 023012 (2008) (TT08).

[16] Z.-Y Li, T. Chueh, and M. C. Begalman, Electromag-

netically Driven Relativistic Jets: a class of self-similarsolutions, Astrophys. J. , 394, 459 (1992).

[17] M. C. Begelman, and Z.-Y. Li, Asymptotic Dominationof Cold relativistic MHD Winds by Kinetic Energy Flux,Astrophys. J. , 426, 269 (1994).

[18] S. Apple & M. Camenzind, Self-collimated jets beyondthe light cylinder, Astron. Astrophys., 270, 71 (1993).

[19] Y. Lyubarsky, Asymptotic structure of Poynting-Diminated jets, Astrophys. J. , 698, 1570 (2009).

[20] L. Huang, Z. Pan, and C.Yu, Toward a Full MHD JetModel of Spinning Black Holes. II. Kinematics and Ap-plication to the M87 Jet, Astrophys. J. 894, 45 (2020).

[21] M. Camenzind, Hydromagnetic flows from rapidly rotat-ing compact objects II. The relativistic axisymmetric jetequilibrium, Astron. Astrophys.184, 341 (1987).

[22] S. Nitta, M. Takahashi, and A. Tomimatsu, Effects ofmagnetohydrodynamic accretion flows on global struc-ture of a Kerr Black-Hole magnetosphere, Phys. Rev. D44, 2295 (1991).

[23] V. S. Beskin, Axisymmetric stationary flows in compactastrophysical objects, Phys. Usp. 40, 659 (1997).

[24] M. Kino, M.Takahashi, et al., (2021), in preparation.[25] A. Tomimatsu and M. Takahashi, Relativistic Accelera-

tion of Magnetically Driven Jets, Astrophys. J. 592, 321(2003) (TT03).

[26] H. -Y. Pu and M. Takahashi, Properties of Trans-fastMagnetosonic Jets in Black Hole Magnetospheres, As-trophys. J. 892, 32 (2020).

[27] M. Moscibrodzka, C. F. Gammie, J. C. Dolence, & H.Shiokawa, Pair Production in Low-luminosity GalacticNuclei, Astrophys. J. , 735, 9 (2011).

[28] G. N. Wong, B. R. Ryan, & C. F. Gammie, Pair Drizzlearound Sub-Eddington Supermassive Black Holes, Astro-phys. J. , 907, 73 (2021).

[29] N. Globus ,& A. Levinson, Jet Formation in GRBs: ASemi-analytic Model of MHD Flow in Kerr Geometrywith Relativistic Plasma Injection, Astrophys. J. , 796,26 (2014).

[30] N. Globus ,& A. Levinson, Plasma Injection and OutflowFormation in Kerr Black Holes, High Energy Phenomenain Relativistic Outflows (HEPRO IV) International Jour-nal of Modern Physics: Conference Series, 28, 1460164(2014).

[31] V. S. Beskin, Y. N. Istomin, and V. I. Parev, Fillingthe Magnetosphere of a Supermassive Black-Hole withPlasma, SvA 36, 642 (1992).

[32] K. Hirotani and I. Okamoto, Pair plasma production ina force-free magnetosphere around a supermassive black

16

hole, Astrophys. J. 497, 563 (1998).[33] K. Hirotani and H. -Y. Pu, Energrtic Gamma Radiation

from Rapidly Rotaing Black Holes, Astrophys. J. 818, 50(2016).

[34] A. Levinson and F. Rieger, Variable TeV Emission as aManifestation of Jet Formation in M87? Astrophys. J. ,730, 123 (2011).

[35] A. E. Broderick and A. Tchekhovskoy, Horizon-scale Lep-ton Acceleration in Jets: Explaining The Compact RadioEmission in M87, Astrophys. J. , 809, 97 (2015).

[36] S. Kisaka, A. Levinson, and K. Toma, ComprehensiveAnalysis of Magnetospheric Gaps around Kerr BlackHoles Using 1D GRPIC Simulations, Astrophys. J. 902,id.80 (2020).

[37] K. Parfrey, A. Phylippov and B. Cerutti, First-PrinciplesPlasma Simulations of Black–Hole Jet Launching, Phys.Rev. Lett., 122, 035101 (2019).

[38] B. Crinquand, B. Cerutti, A. Philippov, K. Parfrey,and G. Dubus, Multi-dimensional simulations of ergo-sphereric pair discharges around black holes, Phys. Rev.Lett., 124, 145101 (2020).

[39] Bekenstein, J. D., & Oron, E., New conservation laws ingeneral-relativistic magnetohydrodynamics, Phys. Rev.D 18, 1809 (1978).

[40] Camenzind, M., Centrifugally driven MHD-winds in ac-tive galactic nuclei, Astron. Astrophys.156, 137 (1986).

[41] M. Camenzind, Hydromagnetic flows from rapidly rotat-ing compact objects, Astron. Astrophys.162, 32 (1986).

[42] M. Takahashi, D. Rilett, K. Fukumura, and S. Tsuruta,Magnetohydrodynamic Shock Conditions for AccretingPlasma onto Kerr Black Holes. I., Astrophys. J. 572, 950(2002).

[43] M. Takahashi, J. Goto, K. Fukumura, D. Rilett, and S.Tsuruta, Standing Shocks in Transmagnetosonic Accre-tion Flows onto a Black Hole, Astrophys. J. 645, 1408(2006).

[44] M. Takahashi and R. Takahashi, Black Hole Aurora pow-ered by a Rotating Black Hole, Astrophys. J. Lett. 714,L176 (2010).

[45] A. Tomimatsu, T. Matsuoka, and M. Takahashi, Screwinstability in black hole magnetospheres and a stabiliz-ing effect of field-line rotation, Phys. Rev. D 64, 123003(2001).

[46] K. Gebhardt, et al., The black hole mass in M87 fromGEMINI/NIFS Adaptive Optics observations Astrophys.J. 729, 119 (2011).

[47] Event Horizon Telescope Collaboration (K. Akiyama, etal.), First M87 Event Horizon Telescope Results. I. TheShadow of the Supermassive Black Hole, Astrophys. J.Lett. 875, L1 (2019a).

[48] R. C. Walker, P. E. Hardee, F. B. Davies, C. Ly, and W.Junor, The Structure and Dynamics of the Subparsec Jetin M87 Based on 50 VLBA Observations over 17 Yearsat 43 GHz, Astrophys. J. 855,128 (2018).

[49] R. D. Blandford and D. G. Payne, Hydromagnetic flowsfrom accretion discs and the production of radio jets,Mon. Not. R. Astron. Soc.199, 883 (1982).

[50] Y. -R. Li et al., Spins of supermassive black holes in M87.II. Fully general relativistic calculations, Astrophys. J.699, 513 (2009).

[51] Y. Y. Kovalev, M. L. Lister, D. C. Homan, and K. I.Kellermann, The Inner Jet of the Radio Galaxy M87,Astrophys. J. Lett. 668, L27 (2007).

[52] K. Asada, M. Nakamura, A. Doi, H. Nagai, and M. Inoue,Discovery of sub- to superluminal motions in the M87jet: An implication of acceleration from sub-relativisticto relativistic speeds, Astrophys. J. Lett. 781, L2 (2013).

[53] K. Hada, et al., High-Sensitivity 86GHz (3.5mm) VLBIObservations of M87: Deep Imaging of the Jet Base at aResolution of 10 Schwarzschild-Radii Astrophys. J. 817,131 (2016).

[54] K. Hada, J. H. Park, M. Kino, et al., Pilot KaVA moni-toring on the M 87 jet: Confirming the inner jet structureand superluminal motions at sub-pc scales Publ. Astron.Soc. Jpn.69, 71 (2017).

[55] J. C. McKinney, General relativistic magnetohydrody-namic simulations of the jet formation and large-scalepropagation from black hole accretion systems, Mon.Not. R. Astron. Soc.368, 1561 (2006).

[56] M. Nakamura, K. Asada, K. Hada, et al., Parabolic Jetsfrom the Spinning Black Hole in M87, Astrophys. J. 868,146 (2018).

[57] J. F. Mahlmann, P. Cerda-Duran and M. A. Aloy, Nu-merically solving the relativistic Grad–Shafranov equa-tion in Kerr spacetimes: Numerical techniques, Mon.Not. R. Astron. Soc., 477, 3927 (2018).

[58] M. Kino, F. Takahara, K. Hada, and A. Doi, Relativisticelectrons and magnetic fields of the M87 jet on the ∼10Schwarzschild radii scale, Astrophys. J. 786, 5 (2014).

[59] M. Kino, F. Takahara, K. Hada, K. Akiyama, H. Nagai,and B. W. Sohn, Magnetization degree at the jet baseof M87 derived from the event horizon telescope data:Testing the magnetically driven jet paradigm, Astrophys.J. 803, 30 (2015).

[60] MAGIC Collaboration: V. A. Acciari, et al., Monitoringof the radio galaxy M 87 during a low-emission state from2012 to 2015 with MAGIC, Mon. Not. R. Astron. Soc.,492, 5354 (2020).

[61] S. S. Komisarov, General Relativistic Magnetohydrody-namic Simulations of Monopole Magnetosheres of BlackHoles, Mon. Not. R. Astron. Soc., 350, 1431 (2004).

[62] K. Thoelecke, S. Tsuruta, and M. Takahashi, Effects ofinner Alfven surface location on black hole energy ex-traction in the limit of a force-free magnetosphere, Phys.Rev. D 95, 063008 (2017).

[63] K. Thoelecke, M. Takahashi, and S. Tsuruta, The struc-ture of magnetically dominated energy-extracting blackhole magnetospheres: Dependencies on field line angularvelocity, PTEP 2019, 093E01 (2019).

[64] H. -Y. Pu, M. Nakamura, K. Hirotani, et al., Steadygeneral relativistic magnetohydrodynamic inflow/outflowsolution along large-scale magnetic fields that thread arotating black hole, Astrophys. J. 801, 56 (2015).