# On the M87 jet structure near the central engine

V. S. Beskin1,2, T. I. Khalilov2,1, E. E. Nokhrina1,2, I. N. Pashchenko1 and
E. V. Kravchenko1,2

1P.N.Lebedev Physical Institute, Leninsky prospekt 53, Moscow 119991, Russia
2Moscow Institute of Physics and Technology, Institutsky per. 9, Dolgoprudny
141700, Russia

(Accepted. Received; in original form)

###### Abstract

At present, there is no doubt that relativistic jets observed in active
galactic nuclei pass from highly magnetized to weakly magnetized stage, which
is observed as a break in the dependence on their width $d_{\rm jet}(z)$ on
the distance $z$ to the central engine. In this paper, we discuss the
possibility of observing another break, which should be located at shorter
distances. The position of this discontinuity can be associated with the
region of formation of the dense central core near the rotation axis which was
predicted both analytically and numerically more than a decade ago, but has
not yet received sufficient attention. In this case, the observed jet width
should be determined by the dense core rather than the full transverse
dimension of the jet. We found the evidence of the predicted break in the jet
expansion profile using stacked 15 GHz VLBA image of M87 radio jet and
constrain the light cylinder radius.

###### keywords:

galaxies: active, galaxies: jets, Physical data and processes: MHD, Physical
data and processes: radiation mechanisms: non-thermal

††pagerange: On the M87 jet structure near the central
engine–Acknowledgements††pubyear: 2018

##  1 Introduction

A real breakthrough in the study of the internal structure of relativistic
jets emanating from active galactic nuclei (AGN), achieved over the past
decade thanks to the high angular resolution obtained using very long baseline
interferometry (VLBI) observations (Kovalev1; MOJAVE_V; MOJAVE_XII;
MOJAVE_XIII; Mertens) allows us to investigate directly their internal
structure. In particular, it give us the direct information about the
dependence of the jet width $d_{\rm jet}(z)$ on the distance $z$ to the
central engine. It finally became possible to compare the results of
observations with the predictions of a theory, the basic elements of which
have long been constructed (B76; L76; Ardavan; BZ-77; BBR84; Camenzind86;
HN89; Camenzind90; Tomimatsu; PP92; BP93; Heyvaert96).

We recall that the nature of relativistic jets from AGN is believed to be
associated with highly magnetized magneto-hydrodynamical (MHD) outflows
generated by fastly rotating supermassive black holes (Krolik; Camenzind07;
MHD; Meier). Within this approach, the key role is played by the poloidal
magnetic field generated in the accretion disc as within this model both
plasma outflow and the energy flux propagate along magnetic field lines from
the central engine to active regions. As was shown by BZ-77, a strongly
magnetized flow (in which the main role in energy transfer is played by the
electromagnetic field, i.e. the Poynting vector flux) is really capable of
taking the energy away from a rotating black hole and transmitting it to
infinity. As a result, thanks to numerous works devoted to both analytical and
numerical analysis of MHD equations (Chiueh91; Appl92; Eichler93; Bogovalov95;
Ustyugova95; Beskin97; LHAN99; BM00; VK03; McKinney06; BN06; Komissarov07;
BN09; Romanova; Lyu09; MHD; Tchekhovskoy_11; Porth_etal11; McKinney12; PC15;
BCKN-17), a consistent model was constructed, which is now generally accepted.

Let us note from the very beginning the following key points characterizing
the MHD model of relativistic jets; the assumptions of the model we use will
be described in detail in Section 2. First, it should be stressed the role of
the ambient pressure $P_{\rm ext}$ which actually determines the transverse
size of the jet $d(z)$ and, hence, their inner structure (Beskin97; Lyu09). In
particular, it is the pressure dependence $P_{\rm ext}(z)$ on the distance $z$
that is to determine the dependence of transverse size of the jet on the
distance from the central engine.

Second, as was found both analytycally (BN06; Lyu09) and numerically
(McKinney06; Komissarov07; Tchekhovskoy_11; Porth_etal11), for a sufficiently
high external pressure $P_{\rm ext}$ (i.e. not so far from the central
engine), the longitudinal poloidal magnetic field within the jet is to be
homogeneous. However, as the external pressure decreases, a denser core begins
to form in the very centre of the jet, while the poloidal magnetic field
$B_{\rm p}$ and the number density $n_{\rm e}$ of the outflowing plasma begin
to decrease significantly with distance $\varpi$ from the jet axis. But the
flow still remains highly magnetized. Recently this result was comfirmed by
zero for relativistic jets with zero velocity along the axis.

Third, as in the magnetically dominated flow the energy of particles increases
with the distance $\varpi$ from the jet axis, at large enough distances $z$
corresponding to lower external pressures (where the jet width $d_{\rm
jet}(z)$ also becomes large enough), inevitably, saturation should occur, when
almost all the electromagnetic energy flux (which, as was already stressed, is
dominated near the black hole) will be transferred to the energy of outflowing
plasma (see Sect. 2 for more detail).

Currently, there are direct indications that saturation occurs at the
distances of several parsecs ($10^{5}$–$10^{6}$ of gravitational radii $R_{\rm
g}=GM/c^{2}$) from the central engine (NKP20). Such a transition from highly
magnetized to weakly magnetized regime was predicted to be observed as a break
in the dependence on the width of the jet $d_{\rm jet}(z)$ on the distance $z$
(BCKN-17; NGBNAH19). This results in the observed parabolic streamline in the
upstream region, and the jet shape becomes conical downstream a few
$10^{5}~{}R_{\rm g}$. This structural transition of a jet shape was first
discovered in M87 (2012ApJ...745L..28A) and the similar transition was lately
found in a number nearby sources (2016ApJ...833..288T; 2018ApJ...860..141H;
2018ApJ...854..148N; 2018Galax...6...15A; Kovalev2020; 2020AJ....159...14N).

Remember that for M87 such a nature of the break has also been discussed by
Mertens. But they talked about a feature located at a distance about 10 mas,
i.e., hundreds of times smaller. Therefore, it is of interest to discuss the
question of whether the feature at the very base of the jet in the M87 is not
connected with the formation of the central core, i.e. with the transition
from the jet with homogeneous poloidal magnetic field to the jet having a
dense central core. An indication of such a break can be found in the work of
2016ApJ...817..131H. However, this issue has not yet been discussed in detail.

Thus, in this paper, we investigate the internal structure of relativistic
jets at small distances $z$ from the central engine. Having decided on the
model of the internal structure of electromagnetic fields (which, within the
framework of the MHD approach, allows us to determine the plasma number
density as well), we then find the observed width of the jet, the radiation of
which is associated with the synchrotron radiation of outflowing particles.
Our goal is to find out whether, in addition to external break in the
dependence of the width of the relativistic jet $d_{\rm jet}(z)$ on the
distance $z$, at which the transition from magnetically dominated to particle
dominated flow occurs, there is also an internal break in which the central
core near the jet axis begins to form.

Let us recall that the source of radiating leptons are currently unknown.
Since the seminal paper by BZ-77 it has been believed that they can be formed
as a result of the collision of thermal gamma rays emitted by the highly
heated internal regions of the accretion disk. Detailed calculations of this
process can be found in (Monika). In this case, as in most works devoted to
this issue (Porth_etal11; 2012MNRAS.423..756P; 2013MNRAS.429.1189P;
2019MNRAS.488..939P; 2019A&A...629A...4F), we assume a power-law spectrum of
radiating particles, without discussing the very nature of particle
acceleration. This assumption is consistent with the observational data
(2014AJ....147..143H; 2014MNRAS.437.3396K).

The paper is organized as follows. In Section 2 the basic relations of the MHD
theory are formulated, allowing a detailed description of the internal
structure of relativistic jets. Based on this theory, in Section 3 we
determine the transverse size of relativistic jets near their base, which
allows us to find the position of the internal break. Finally, in Section 4 we
employ the observational data on the break and discuss the imposed
constraints.

##  2 Internal structure of relativistic jets

First of all, let us dwell in detail on our model of the internal structure of
relativistic jets. It is based on the results obtained by BN06, according to
which, for strongly collimated jets, one can consider their internal structure
as a sequence of cylindrical flows depending on a jet radius $\varpi$ only.
For such cylindrical configurations, the Grad-Shafranov (GS) equations
describing internal structure of ideal MHD flows (see HN89; Camenzind90; PP92;
BP93; Heyvaert96 for more detail) can be easily integrated (Beskin97; BM00;
Lyu09) as it reduces to two ordinary differential equations of the first
order. As a result, by specifying four integrals of motion at the base of a
magnetically dominated flow (see below), it is possible to obtain transverse
profiles of electromagnetic fields (and, within the framework of this
approach, hydrodynamic speed and particle number density as well) at an
arbitrary distance from the central engine.

A detailed discussion of the model used can be found in our previous works
(BN09; NBKZ15; BCKN-17; PaperI). Here we recall only two significant
circumstances. First, as already noted, one of the important features of the
model is that at sufficiently large distances from the central engine, a
denser core begins to form in the very center of the jet, and the poloidal
magnetic field $B_{\rm p}$ and the number density $n_{\rm e}$ of the
outflowing plasma begin to decrease significantly with distance $\varpi$ from
the jet axis. This structure, shown in Figure 1, was also confirmed by
numerical simulation (Komissarov07; Tchekhovskoy_11; Porth_etal11).

Figure 1: The structure of the magnetic field in the conical model under
consideration. At a distances $z>z_{\rm cr}$ from the central engine, when the
transverse radius of the jet reaches the scale $\sim r_{\rm cr}$, in a conical
flow (in which the plasma density and the magnetic field weakly depend on the
distance from the axis), a denser central core begins to form. The light
cylinder $\varpi=c/\Omega_{0}$ is shown by a dashed line.

Second, despite the cumbersome form of the GS equation, rather simple
asymptotic solutions have been formulated, which will be enough for us to
state the main points of this paper. As was already stressed, the structure of
the poloidal magnetic field within relativistic jets outflowing from active
galactic nuclei substantially depends on the ambient pressure $P_{\rm ext}$
which, in turn, depends on the distance $z$ from the central engine.
Therefore, in what follows, we consider all quantities as functions of the
distance $z$.

Figure 2: Comparison of analytical approximation (1)–(2) (black dashed lines)
with the structure of a poloidal magnetic field (blue solid lines) obtained by
solving a system of two ordinary differential equations describing internal
structure of a cylindrical jet. In this figure, we terminated the solid lines
of numerical solution at smaller radii so that the analytical approximation
may be better seen. We plot three jet crosscuts for a conical jet with
intrinsic opening angle $10^{\circ}$.

As a result, with a good accuracy, poloidal magnetic field $B_{\rm p}$ can be
represented as

$\displaystyle B_{\rm p}(\varpi)\approx B_{\rm
p}(0),\qquad\qquad\qquad\quad\quad B_{\rm p}(0)>B_{\rm cr},$ (1)
$\displaystyle B_{\rm p}(\varpi)\approx
B_{0}(z)\left(1+\frac{\varpi^{2}}{r_{\rm
core}^{2}}\right)^{-\alpha(z)/2},B_{\rm p}(0)<B_{\rm cr}.$ (2)

Here

$r_{\rm core}=\gamma_{\rm in}R_{\rm L}$ (3)

is the core radius where the integration constant $\gamma_{\rm in}\sim 1$
corresponds to the Lorentz-factor of a flow along the axis, and $R_{\rm
L}=c/\Omega_{0}$ is the radius of the light cylinder (the constant
$\Omega_{0}$ will be defined below).

In other words, at small distances from the central engine, poloidal magnetic
field remains actually constant (so that $B_{0}^{2}/8\pi\approx P_{\rm
ext}$)111This relationship is valid in the case when the longitudinal current
is closed inside the jet itself, which we are considering here., while at
large distances it decreases with distance from the axis $\varpi$. As was
already stressed, this result has been repeatedly confirmed by numerical
calculations (Komissarov07; Tche08; Porth_etal11). Moreover, this structure is
realized for zero hydrodynamical velocity along the jet axis ($\gamma_{\rm
in}=1$) as well (zero). As shown in Figure 2, analytical approximation (1)–(2)
perfectly reproduces the structure of a poloidal magnetic field obtained by
solving a system of two ordinary differential equations describing internal
structure of cylindrical jet (see BN09; Lyu09; PaperI for more detail).

Wherein, the characteristic magnetic field $B_{\rm cr}$, at which the
formation of the central core begins, can be presented as

$B_{\rm cr}=\frac{B_{\rm L}}{\gamma_{\rm in}\sigma_{\rm M}},$ (4)

where

$\sigma_{\rm M}=\frac{\Omega_{0}^{2}\Psi_{\rm tot}}{8\pi^{2}\mu\eta c^{2}}$
(5)

is so-called Michel69 magnetization parameter. Here $\mu\approx m_{\rm
e}c^{2}$ is the relativistic enthalpy, and $\eta=nu_{\rm p}/B_{\rm p}$ is the
particle-to-magnetic flux ratio222Unfortunately, at present there is no
reliable data concerning the dependence of the integral of motion
$\eta=\eta(\Psi)$ on the magnetic flux $\Psi$. Therefore, in what follows we
assume $\eta=$ const.. Due to numerous evaluations (see, e.g. NBKZ15), for
most jets from active galactic nuclei the Michel magnetization parameter
$\sigma_{\rm M}\sim 10$–$50$. For M87 we assume $\sigma_{\rm M}\sim 20$. In
turn, the magnetic field $B_{\rm L}$ is determined from the relation

$\Psi_{\rm tot}=\pi R_{\rm L}^{2}B_{\rm L},$ (6)

where $\Psi_{\rm tot}$ is the total magnetic flux within the jet. In what
follows we assume $B_{\rm L}=100$ G, which corresponds to $R_{\rm L}=10R_{\rm
g}$.

Further, the exponent $\alpha$ changes from 0 to 1 as the ambient pressure
decreases to values $P_{\rm eq}=B_{\rm eq}^{2}/8\pi$, where $B_{\rm
eq}=\sigma_{\rm M}^{-2}B_{\rm L}$. In this region, the magnetic field on the
axis $B_{0}$ weakly depends on $P_{\rm ext}$. In all this magnetically
dominated region, the asymptotic behaviour $\gamma(x)=x=\Omega\varpi/c$ is
fulfilled. At an even lower ambient pressure $P_{\rm ext}<B_{\rm
eq}^{2}/8\pi$, the flow becomes partically dominated. In this domain
$\alpha\approx 1$.

It should be emphasized that in order to determine the explicit form of
dependences $B_{0}(z)$ and $\alpha(z)$, it is necessary to specify either the
dependence of ambient pressure $P_{\rm ext}(z)$ on $z$ or the shape of the jet
boundary. However, since the ambient pressure is not known with sufficient
accuracy, below we use the second approach. It is based on the fact that the
parameter of the problem considered by BN06 was not the distance from the
central engine $z$, but the transverse radius of a jet $r_{\rm jet}$.
Therefore, by specifying the shape of the jet, one can also determine the
dependence of the parameters on the coordinate $z$. Below we use the conical
model (see Figure 1), which seems to be the most reasonable at the very base
of the jet. As we see below, this assumption does not contradict the observed
parabolic shape of the jets.

Using now the results of the work of BN06, we obtain the following
approximation for the functions $\alpha\left(z\right)$ and
$B_{0}\left(z\right)$ defining the poloidal magnetic field (1)–(2) in the case
of a conical structure for $z>z_{\rm cr}$

$\displaystyle\alpha\left(z\right)$ $\displaystyle\approx$ $\displaystyle
0.05\ln\left(z/z_{\rm cr}\right),$ (7) $\displaystyle B_{0}\left(z\right)$
$\displaystyle\approx$ $\displaystyle B_{\rm
cr}\frac{1-\alpha\left(z\right)/2}{\left(1+z^{2}/z_{\rm
cr}^{2}\right)^{1-\alpha\left(z\right)/2}-1}.$ (8)

In (8) $B_{0}\left(z\right)$ is determined from the conservation of the total
magnetic flux.

From here it can be seen that at short distances, before the formation of the
central core, we have $B_{0}\left(z\right)\,\propto\,z^{-2}$. It should be
noted that the approximate function (7) at distances $z\gg z_{\rm cr}$ becomes
more than one, which contradicts the condition in the region when the flow of
electromagnetic energy almost completely passes into the plasma energy flow.
Nevertheless, within the framework of this paper, such distances will not be
considered, and this question concerns an approximate function such that the
inequality $0<\alpha\left(z\right)<1$ is satisfied.

The advantage of the approach considered here (in contrast to works in which
it was necessary to solve a system of two ordinary differential equations) is
that the expression (2) allows one to write down the magnetic flux
$\Psi=2\pi\int B_{\rm p}\varpi\,{\rm d}\varpi$ explicitly

$\Psi(\varpi,z)=\frac{2\pi B_{0}(z)r_{\rm
core}^{2}}{2-\alpha(z)}\left[\left(1+\frac{\varpi^{2}}{r_{\rm
core}^{2}}\right)^{1-\alpha(z)/2}-1\right].$ (9)

In turn, it allows us to find all the integrals of motion in any point, as
they in standard GS approach (HN89; PP92) depend on magnetic flux $\Psi$ only.

Indeed, the key role in the MHD theory of strongly magnetized jets play the
integrals (MHD)

$\displaystyle E(\Psi)$ $\displaystyle=$ $\displaystyle\gamma\mu\eta
c^{2}+\frac{\Omega_{\rm F}I}{2\pi,}$ (10) $\displaystyle L(\Psi)$
$\displaystyle=$ $\displaystyle\varpi u_{\varphi}\mu\eta c+\frac{I}{2\pi},$
(11)

where $E(\Psi)$ (integral Bernoulli) corresponds to the energy flux and
$L(\Psi)$ to the angular momentum flux. Here $\Omega_{\rm F}(\Psi)$ is the
angular velocity which is also constant on magnetic surfaces (Ferraro’s
isorotation law), and $I$ is the total current inside the magnetic surface.
The value $\Omega_{0}$ we introduced earlier is equal to $\Omega_{\rm F}(0)$
by definition. Besides, in the black hole magnetosphere both the electric
current $I$ and the angular velocity $\Omega_{\rm F}$ are determined from the
critical conditions on the singular surfaces. Wherein, the current $I$
entering the relations (10)–(11) turns out to be close to the so-called
Goldreich-Julian current (appropriate current density $j_{\rm GJ}=\Omega_{\rm
F}B_{0}/2\pi$), and the angular velocity $\Omega_{\rm F}$ is close to half the
angular velocity of the black hole rotation $\Omega_{\rm H}$.

Thus, with good accuracy, the Bernoulli integral can be written in the form

$E(\Psi)\approx\Omega_{\rm F}(\Psi)L(\Psi)+\gamma_{\rm in}\mu\eta c^{2},$ (12)

and besides, in the region of strongly magnetized flow, the contribution of
particles can be neglected. On the other side, as we know, for both conical
(Michel73) and parabolic (B76) flows near the axis, we can set
$L(\Psi)=\Omega_{0}\Psi/(4\pi^{2})$. In what follows, however, we use the
integrals (NGBNAH19; PaperI)

$\displaystyle L(\Psi)$ $\displaystyle=$
$\displaystyle\frac{\Omega_{0}\Psi}{4\pi^{2}}\sqrt{1-\frac{\Psi}{\Psi_{\rm
tot}}},$ (13) $\displaystyle\Omega_{\rm F}(\Psi)$ $\displaystyle=$
$\displaystyle\Omega_{0}\sqrt{1-\frac{\Psi}{\Psi_{\rm tot}}},$ (14)

which allow us to include in consideration the reverse electric current
flowing within the jet.

As a result, knowing the potential $\Psi(\varpi,z)$ and explicit expressions
for the integrals of motion $\Omega_{\rm F}(\Psi)$, $E(\Psi)$, and $L(\Psi)$,
we can determine all components of electric and magnetic fields within the jet

$\displaystyle\mn@boldsymbol{B}_{\rm p}$ $\displaystyle=$
$\displaystyle\frac{\nabla\Psi\times\mn@boldsymbol{e}_{\varphi}}{2\pi\varpi},$
(15) $\displaystyle B_{\varphi}$ $\displaystyle=$
$\displaystyle-(1+\varepsilon)\frac{\Omega_{\rm F}(\Psi)}{2\pi
c}|\nabla\Psi|,$ (16) $\displaystyle\mn@boldsymbol{E}$ $\displaystyle=$
$\displaystyle-\frac{\Omega_{\rm F}(\Psi)}{2\pi c}\nabla\Psi.$ (17)

Here, however, we add the additional factor $(1+\varepsilon)$ into the
expression for the toroidal magnetic field, where
$\varepsilon=\varepsilon(\Psi)$ is a small parameter, $\varepsilon\ll 1$. This
increase of the toroidal magnetic field with respect to exact force-free
solution makes it possible to simulate the absence of particle acceleration at
large distances, when the electromagnetic energy flux has already been
transferred to the plasma flow. Moreover, such a correction does not violate
the basic relationship $\mn@boldsymbol{E}+(\mn@boldsymbol{\Omega}_{\rm
F}\times\mn@boldsymbol{r}/c)\times\mn@boldsymbol{B}=0$.

Indeed, using the fundamental theoretical result, according to which, outside
the light cylinder, the hydrodynamic velocity ${\bf V}$ becomes almost equal
to the electrical drift velocity (Tche08; MHD)

$\mn@boldsymbol{U}_{\rm
dr}=c\frac{\mn@boldsymbol{E}\times\mn@boldsymbol{B}}{B^{2}},$ (18)

so the hydrodynamic Lorentz-factor of plasma can be written as
$\Gamma=\Gamma_{\rm dr}$ where

$\displaystyle\Gamma_{\rm dr}(x)=\frac{1}{\sqrt{1-U_{\rm
dr}^{2}/c^{2}}}=\left(1-\frac{E^{2}}{B_{\varphi}^{2}+B_{\rm
p}^{2}}\right)^{-1/2}$
$\displaystyle=\left[1-\frac{\omega^{2}}{(1+\varepsilon)^{2}\omega^{2}+1/x^{2}}\right]^{-1/2}\approx\left(2\varepsilon+\frac{1}{\omega^{2}x^{2}}\right)^{-1/2}.$
(19)

Here $x=\varpi/R_{\rm L}$ is the dimensionless distance to the jet axis, and
$\omega=\Omega_{\rm F}(x)/\Omega_{0}$. As a result, at small distances from
the central engine we get for $\omega=1$

$\Gamma_{\rm dr}\approx x,$ (20)

i.e., well-known asymptotic solution for collimated magnetized jets (Beskin97;
Lyu09; Tchekhovskoy_11). As already emphasized, this asymptotic behavior
remains valid in the entire region $B_{\rm ext}>B_{\rm eq}$. On the other
hand, at large distances, i.e. for $x>(2\varepsilon)^{-1/2}$, we have
$\Gamma_{\rm dr}\approx(2\varepsilon)^{-1/2}\approx$ const. This asymptotic
behavior models the saturation region, when the entire energy flow is already
concentrated in the hydrodynamic particle flow and Lorentz factor of the flow
reaches its maximal value $\Gamma_{\rm max}\approx(2\varepsilon)^{-1/2}$.

As a result, the function $\varepsilon(\Psi)$ can be easily found from
Bernoulli equation (10) which just determines the maximum Lorentz factor on a
given magnetic surface:

$\Gamma_{\max}(\Psi)=\frac{E(\Psi)}{\mu\eta c^{2}}.$ (21)

Using now relations (10)–(11), we obtain

$\varepsilon(\Psi)=\left(1-\frac{1}{\Gamma_{\rm
max}^{2}(\Psi)}\right)^{-1/2}-1\approx\frac{1}{2\Gamma_{\rm max}^{2}(\Psi)},$
(22)

where now due to (13)–(14)

$\Gamma_{\rm max}(\Psi)=\gamma_{\rm in}+2\sigma_{\rm M}\frac{\Psi}{\Psi_{\rm
tot}}\left(1-\frac{\Psi}{\Psi_{\rm tot}}\right),$ (23)

and the value of $\sigma_{\rm M}$ (5) has already been defined above. Below we
use the velocity field obtained to deternime the Doppler factor.

##  3 Observed jet transverse dimension

Let us now discuss the observed transverse dimension of a jet $d_{\rm
jet}(z)=2r_{\rm jet}(z)$ at an arbitrary distance $z$ from the central engine
(see Figure 1). Due to relation (4), the appearance of the central core occurs
at the moment when the transverse radius of the jet becomes of order

$r_{\rm cr}\sim(\gamma_{\rm in}\sigma_{\rm M})^{1/2}R_{\rm L}.$ (24)

Accordingly, the distance $z_{\rm cr}$ to the central engine can be evaluated
as

$z_{\rm cr}\sim\Theta_{\rm jet}^{-1}(\gamma_{\rm in}\sigma_{\rm
M})^{1/2}R_{\rm L}.$ (25)

For reference, we present here the corresponding estimates for parabolic flow
as well

$r_{\rm jet}\sim\left(\frac{z}{R_{\rm L}}\right)^{1/2}R_{\rm L}.$ (26) $z_{\rm
cr}\sim\gamma_{\rm in}\sigma_{\rm M}\,R_{\rm L}.$ (27)

As for the value $r_{\rm cr}$, it does not depend on the geometry of the jet.
This important result will be used significantly below.

It is clear that as long as the longitudinal magnetic field (and hence the
number density) remains uniform within the jet, observed transverse radius of
the conical jet $r_{\rm jet}(z)$ increase linearly with distance $z$ as the
total transverse radius

$r_{\rm jet}\approx\Theta_{\rm jet}z.$ (28)

Here $\Theta_{\rm jet}$ is the total angular half-width of the conical jet. On
the other hand, observed width of the jet $d_{\rm jet}(z)$, due to the
decrease in the integrals of motion (10)-(11) and the number density towards
its edge at large distances $z>z_{\rm cr}$, is to be smaller than
$2\Theta_{\rm jet}z$. Nevertheless, below we use relations (24)-(25) for the
estimate of the observed jet width for small distances up to $z=z_{\rm cr}$.

To more accurately determine the observed transverse size of the jet at any
distance from the central engine, we use standard relations (BIP03; PIB03;
LPB03; LPG), allowing to construct the brightness temperature maps. It is
based on the hypothesis of the power-law spectrum of radiating particles in
the comoving reference frame $n_{\gamma}^{\prime}(\gamma^{\prime})$

${\rm d}n_{\gamma}^{\prime}=K_{\rm e}\,(\gamma^{\prime})^{-p}\,{\rm
d}\gamma^{\prime}\,{\rm d}^{3}{r}^{\prime}\,{\rm d}\Omega,$ (29)

where $\gamma_{0}<\gamma^{\prime}<\gamma_{\rm max}$, $p$ is the spectrum
index, ${\rm d}\Omega$ is the solid angle, and $K_{\rm e}$ is the normalising
constant resulting from the condition $\int f(\gamma^{\prime}){\rm
d}\gamma^{\prime}=1$.

Here two important clarifications must be made. First, below we take into
account the evolution of the spectrum of radiating particles due to
conservation of the first adiabatic invariant (inv)

$I_{\perp}=\frac{(p_{\perp}^{\prime})^{2}}{h},$ (30)

where

$h=\sqrt{B^{2}-E^{2}}$ (31)

is a magnetic field in the comoving reference frame; for magnetically
dominated flow considered here one can put $h\approx B_{\rm p}$. Since the
condition $p_{\perp}^{\prime}\approx m_{\rm e}c\gamma^{\prime}$ is satisfied
for ultrarelativistic particles, we finally obtain

$\gamma^{\prime}=\frac{I_{\perp}^{1/2}}{m_{\rm e}c}\,h^{1/2}.$ (32)

Asumng now that $\gamma_{\rm max}^{\prime}\gg\gamma_{0}$ and $\gamma_{0}\gg 1$
we obtain for the normalising constant $K_{\rm e}$

$K_{\rm
e}=(p-1)n_{\gamma}^{\prime}({\mn@boldsymbol{r}}^{\prime})\,\gamma_{0}^{p-1}({\mn@boldsymbol{r}}^{\prime}).$
(33)

Thus, when moving away from the central engine, due to the preservation of the
first adiabatic invariant (30), the entire spectrum of radiating particles
will shift towards low energies without changing its shape. Accordingly the
normalising constant $K_{\rm e}$ will have the form

$K_{\rm e}\propto n_{\gamma}^{\prime}\,h^{(p-1)/2}.$ (34)

It is easy to check that expression (34) remains the same if the lower limit
corresponds to non-relativistic velocities ($\gamma_{0}=1$). This is due to a
decrease in the number of ultrarelativistic particles.

Second, despite the fact that the conservation of the first adiabatic
invariant formally leads to a noticeable angular anisotropy of the angular
distribution of radiating particles (inv), turbulence and plasma instabilities
should lead to isotropization of the distribution function. Therefore, in what
follows we do not take into account the effects of anisotropy of the angular
distribution of radiating particles.

On the other hand, below we are not limited to the approximation of an
optically thin plasma, although this approximation is also valid for
sufficiently high frequencies. The point is that we are interested in regions
close enough to the central engine, where the effects of synchrotron self-
absorption can play a significant role (FNP23).

Thus, to determine the brightness temperature, we use the standard radiative
transfer equation for the spectral radiance $I_{\nu}$

$\frac{{\rm d}I_{\nu}}{{\rm d}l}=a_{\nu}-\mu_{\nu}I_{\nu},$ (35)

where

$a_{\nu}=\frac{(p+7/3)}{(p+1)}\kappa(\nu)\frac{{\cal
D}^{2+(p-1)/2}}{\left(1+{\rm
z}\right)^{2+(p-1)/2}}|h\sin{\hat{\chi}}|^{(p+1)/2}$ (36)

is the synchrotron emissivity (Zhelez; LPB03), and

$\mu_{\nu}=\left(p+10/3\right)\zeta(\nu)\frac{{\cal
D}^{\left(p+4\right)/2-1}}{\left(1+{\rm
z}\right)^{\left(p+4\right)/2-1}}|h\sin{\hat{\chi}}|^{(p+2)/2}$ (37)

is the absorption opacity. Here

${\cal D}=\frac{1}{\Gamma(1-{\mn@boldsymbol{\beta}}{\mn@boldsymbol{n}})}$ (38)

is the Doppler-factor which we can now determine at any point, knowing the
spatial distribution of hydrodynamic velocities
${\mn@boldsymbol{V}}(\mn@boldsymbol{r})$ and the corresponding Lorentz factors
$\Gamma(\mn@boldsymbol{r})$,

$\displaystyle\kappa(\nu)=\frac{\sqrt{3}}{4}\Gamma\left(\frac{3p-1}{12}\right)\Gamma\left(\frac{3p+7}{12}\right)\,\frac{e^{3}}{m_{\rm
e}c^{2}}\,$ $\displaystyle\times\left(\frac{3e}{2\pi m_{\rm
e}c}\right)^{(p-1)/2}\nu^{-(p-1)/2}K_{\rm e},$ (39)

and

$\displaystyle\zeta(\nu)=\frac{\sqrt{3}}{4}\Gamma\left(\frac{3p+2}{12}\right)\Gamma\left(\frac{3p+10}{12}\right)$
$\displaystyle\times\frac{e^{3}}{2m_{\rm e}^{2}c^{2}}\left(\frac{3e}{2\pi
m_{\rm e}c}\right)^{p/2}\nu^{-\left(p+4\right)/2}K_{\rm e}.$ (40)

Using now relation $T_{\rm br}=I_{\nu}c^{2}/(2k_{\rm B}\nu^{2})$, we obtain

$\displaystyle T_{\rm br}=\frac{R(p)}{(1+z)^{2+(p-1)/2}}\frac{e^{3}}{m_{\rm
e}k_{\rm B}}\,\left(\frac{e}{m_{\rm e}c}\right)^{(p-1)/2}\nu^{-(p+3)/2}$
$\displaystyle\times\int_{0}^{\infty}{\cal
D}^{2+(p-1)/2}h^{(p+1)/2}n_{\gamma}({\bf r})\,\gamma_{0}^{p-1}({\bf
r})(\sin{\hat{\chi}})^{(p+1)/2}$ (41)
$\displaystyle\times\exp\left(-\int_{0}^{l}\mu_{\nu}(l^{\prime}){\rm
d}l^{\prime}\right){\rm d}l,$

where

$R(p)=\frac{3^{p/2}(p-1)(p+7/3)}{8(2\pi)^{(p-3)/2}(p+1)}\Gamma\left(\frac{3p-1}{12}\right)\Gamma\left(\frac{3p+7}{12}\right),$
(42)

and the integration goes from the point of the observer towards the source.
Here, in contrast to inv, we have added the cosmological factor $(1+z)$.

In what follows, it is convenient to write down the number density of high-
energy particles $n_{\gamma}(\mn@boldsymbol{r})$ entering in (41) (which now
corresponds to the laboratory reference frame) as

$n_{\gamma}=\lambda_{\gamma}n_{\rm GJ},$ (43)

where

$n_{\rm GJ}=\frac{|\mn@boldsymbol{\Omega B}|}{2\pi ce}$ (44)

is the so-called GJ number density (i.e., the lowest particle density required
to screen the longitudinal electric field), and $\lambda_{\gamma}$ is the
multiplicity factor for radiating particles. It is clear that the number
density of emitting particles $\lambda_{\gamma}n_{\rm GJ}$ should not exceed
the number density of the outflowing plasma $\lambda n_{\rm GJ}$, for which,
as estimates show (see, e.g. Monika; NBKZ15), $\lambda\sim 10^{11}$–$10^{13}$.
Therefore, below we assume that the number density of emitting particles is
about one percent of the density of the runaway plasma, so that
$\lambda_{\gamma}=10^{10}$. As a result, we finally get

$\displaystyle T_{\rm br}=\lambda_{\gamma}\frac{m_{\rm e}c^{2}}{k_{\rm
B}}\frac{R(p)}{2\pi}\left(\frac{eh_{0}}{m_{\rm
e}c}\right)^{(p+3)/2}\frac{\nu^{-(p+3)/2}}{(1+z)^{2+(p-1)/2}}$
$\displaystyle\int{\cal
D}^{2+(p-1)/2}\left(\frac{h}{h_{0}}\right)^{p}\,\frac{B_{0}(z)}{h_{0}}\,(\sin{\hat{\chi}})^{(p+1)/2}$
(45) $\displaystyle\frac{\Omega_{\rm
F}(\Psi)}{\Omega_{0}}\,\left(1+\frac{\varpi^{2}}{r_{\rm
core}^{2}}\right)^{-\alpha(z)/2}\,\exp\left(-\int_{0}^{l}\mu_{\nu}(l^{\prime}){\rm
d}l^{\prime}\right)\frac{{\rm d}l}{R_{\rm L}}.$

Here $h_{0}$ is the magnetic field at the light cylinder.

Figure 3: Transverse brightness temperature profiles (spectrum index $p=2.5$,
magnetization parameter $\sigma_{\rm M}=20$, and the jet angular half-width
$\Theta_{\rm jet}=10^{\circ}$) at $z=320\,R_{\rm L}$ from the central engine
assuming the presence and the absence of a dense central core. Figure 4:
Dependence of the observed jet width $d_{\rm jet}(z)$ at the level of 10% of
the maximum brightness temperature for two half-widths $\Theta_{\rm jet}$ of
conical jets. Parabolic asymptotic behaviour $d_{\rm jet}\approx
0.06\,z^{0.57}$ pc (NGBNAH19) shown by the dashed line.

Figure 3 shows an example of predicted cross sections of the brightness
temperature $T_{\rm br}$ for characteristic jet values (magnetization
parameter $\sigma_{\rm M}=20$, jet angular half-width $\Theta_{\rm
jet}=10^{\circ}$, spectrum index $p=2.5$) for $z=320\,R_{\rm L}$ (which is
larger than $z_{\rm cr}\approx 30\,R_{\rm L}$) from the central engine
assuming the presence and the absence of a dense central core. Everywhere in
the calculations it is assumed $\nu=15$ GHz. As we see, the presence of a
central core does result in a significant reduction in the observed transverse
size of the jet. Also note that our chosen value of $\lambda_{\gamma}=10^{10}$
results in reasonable brightness temperature $T_{\rm br}$.

In addition, Figure 4 shows the dependence of the observed (at the level of
10% percent of the maximum brightness temperature) jet width $d_{\rm
jet}(z)=2\Theta_{\rm j}(z)z$ for two jet half-thicknesses $\Theta_{\rm jet}$;
the choice of the 10% level is discussed in detail in the next Section. As we
see, at a distance of $\sim 100R_{\rm L}$ from the origin, i.e. in full
agreement with our evaluation (25), the dependence $d=d(z)$ has a break. If at
small distances the observed jet width follows the conical expansion of the
flow $d\propto z$ (thin dashed lines), then at large distances it becomes
noticeably smaller. In this case, the width of the jet in the kink region
really does not depend on $\Theta_{\rm jet}$. It is also interesting to note
that the width of the jet approaches the parabolic asymptotic behaviour
$d_{\rm jet}\approx 0.06\,z^{0.57}$ pc (NGBNAH19), shown by the dashed line.
Thus, the formation of a central core should indeed lead to a significant
change in the dependence of the observed jet width $d_{\rm jet}(z)$ on the
distance $z$ to the central engine.

##  4 Observations and analysis

According to Section 3, the appearance of the core is accompanied by the break
in the jet geometrical profile. Jet in the radio galaxy M87 provides the best
opportunity to search for the predicted phenomenon due to its proximity. The
mass of the black hole in M87 $M_{\textrm{BH}}\approx 6.5\times
10^{9}M_{\odot}$ and distance is 16.8 Mpc (2019ApJ...875L...1E) provide the
angular to spatial scale ratio of $1~{}\mathrm{milliarcsecond~{}(mas)}\approx
0.08~{}\mathrm{pc}\approx 260~{}R_{\rm g}$. Thus, the expected position of the
break (24) corresponds to its apparent position from tens of $\mu$as for large
spins to few mas for moderate spin values $a\approx 0.1$.

###  4.1 Searching for the break

For the analysis of the internal jet structure, we considered publicly
available333https://www.cv.nrao.edu/MOJAVE/sourcepages/1228+126.shtml 15 GHz
VLBA observations obtained between 1995 July 28 and 2022 September 29 in frame
of the MOJAVE program (2019ApJ...874...43L), with inclusion of data from NRAO
archive. Single epoch images often poorly represent large-scale structure and
stacking of the single epoch images obtained within sufficiently large time
interval could reveal the whole jet cross-section (2017MNRAS.468.4992P). Jet
of M87 is known to be a hard target for the CLEAN (CLEAN) imaging procedure.
2023MNRAS.523.1247P showed that CLEAN produces a spurious central brightening
in the intrinsically edge-brightened jets that could be suppressed by
convolving CLEAN model with a larger beam. Also, M87 parsec-scale jet
appearance at 15 GHz is dominated by the helical threads of the Kelvin-
Helmholtz instability (2023MNRAS.526.5949N). These introduce significant
problems in the analysis of the jet structure. Thus, we convolved each single
epoch CLEAN model with a common circular beam of size 1.4 mas which
corresponds to typical size of the naturally weighted beam major axis. This
reduces CLEAN imaging artifacts, while decreases the resolution, thus, making
our results more conservative. We also filtered out 8 single epochs with a
significant image noise to increase the final stacking image fidelity
(2023MNRAS.520.6053P).

To extract the collimation profile we considered the distribution of the total
intensity in the image domain along the slices, taken transverse to the jet
direction, which was chosen to be at position angle $PA=17^{\circ}$ with
respect to the Right Ascension axis444This value was chosen by considering the
jet transverse slices from $z_{\rm obs}=0.5$ to 30 mas.. The slices were taken
starting from 0.5 mas from the core in steps of 0.1 mas, and for the each cut
we fitted two Gaussians to the brightness profile and taking the jet width $D$
as profile width at 10% of the maximal fitted profile value. For the complex
resolved asymmetric transverse structure of the M87 jet, where multiple
components can have significant the brightness difference, the usage of the
half maximum level (FWHM) can be misleading by measuring the width of one
component instead of the whole jet (2023MNRAS.526.5949N). As an additional
check, we employed simulations with a known parabolic jet model to ensure that
usage of 10% level does not bias the geometry estimate. The deconvolved jet
cross section therefore was computed as $d_{\rm
jet}=\sqrt{D^{2}-\theta_{10\%}^{2}}$, where $\theta_{10\%}$ is the full width
of the restoring beam at 10% of the maximum.

In the deriving the jet expansion profile from the images one has to consider
the dependence of the nearby pixels. This could affect the uncertainty
estimation if the data points are assumed independent and identically
distributed or even bias the parameters of the fit. To overcome this we
considered a correlated noise model, where the noise is modelled via Gaussian
Process (hereafter GP, RasmussenW06). We employed the Rational Quadratic
kernel that contains a mixture of scales.

To search the possible jet break, we employed the following relations for the
jet width (Kovalev2020):

$\begin{split}d_{\rm jet}(z_{\rm obs})=a_{1}(z_{\rm
obs}+z_{0})^{k_{1}},~{}{\rm if}~{}z<z_{\rm break},\\\ d_{\rm jet}(z_{\rm
obs})=a_{2}(z_{\rm obs}+z_{1})^{k_{2}},~{}{\rm if}~{}z>z_{\rm
break},\end{split}$ (46)

Here $z_{\rm break}$ is the apparent distance of the break from the jet origin
and $z_{0}>0$ corresponds to a separation of the 15 GHz core component from
the true jet origin. That is necessary in order to better fit the jet shape
near the jet apex (Kovalev2020). The coefficient $a_{2}$ is chosen for two
power-laws to join each other at $z_{\rm break}$. We also forced the
transition region to be smooth by cubic interpolation model predictions on
both sides of the break at distance 0.5 mas. Both single and double slope
(i.e. with a break) dependencies were fitted to the data using the Diffusive
Nested Sampling algorithm (brewer2011) implemented in the DNest4 package
(JSSv086i07) for sampling the posterior distribution of the model parameters.
We made use the Bayesian factors for the model selection
(2008ConPh..49...71T).

Figure 5: Jet width versus apparent (projected) distance along the jet $z_{\rm
obs}$ fitted by a single (Upper) and broken (Lower) power-law models. Black
points show data points. Lines represents 500 samples from the posterior
distribution. Red colour corresponds to the power-law model, while blue colour
shows Gaussian Process component.

Accounting for the correlated noise led to an increase in the uncertainty
interval width of the break position by a factor of two, and for the errors of
the slope — by four to five times. The resulting fits are presented in Figure
5 where the red colour represents the power-law model (46) and the blue colour
corresponds to the GP component. The residual GP component significantly
deviates from zero, especially in $z_{\rm obs}>15$ mas region. This could be a
real effect or the result of the limited ($u$, $v$)-coverage. The break at
$z_{\rm obs}=5.80^{+0.65}_{-0.62}$ mas is significant with logarithm of the
Bayes factor $\Delta\log{Z}=14.9$, where $Z$ is the evidence or the marginal
likelihood of the model (2008ConPh..49...71T). The power-law before the break
is consistent with almost conical expansion, while it becomes significantly
flatter after the break. Although the break seems to be a robust feature (e.g.
it is clearly seen in independent high-sensitivity 15 GHz image of
2023MNRAS.526.5949N, see their Figure 4), the exact values of the exponents of
the expansion profile (46) should be treated with caution. 2019MNRAS.488..939P
showed that at least for a relativistic jet model of BlandfordKonigl79 the jet
shape measured from the stacked VLBI images could deviate systematically from
the true unobserved value.

We assessed the robustness of our break detection procedure with several
tests, including simulations with a known model brightness distribution. For
this we created artificial multi-epoch $uv$-data set using the edge-brightned
jet model without the break from 2023MNRAS.523.1247P and the same
$uv$-coverages and noise as for the real data from the MOJAVE archive. After
imaging the artificial data and stacking single epoch images we employed the
same procedure of the break estimation as described above for the real
observed data. Below we summarize our tests:

  * •

We fitted the synthetic data based on the smooth parabolic jet model with a
single power-law and different number of Gaussians (single, double and triple)
describing the transverse profile. The resulting fits are generally consistent
with the intrinsic parabolic expansion profile $k=0.5$ with single Gaussian
fit providing the worst estimate $k^{\rm 1G}=0.42_{-0.02}^{+0.02}$ and double
and triple Gaussian fit are consistent with the true value ($k^{\rm
2G}=0.46^{+0.03}_{-0.02}$ and $k^{\rm 3G}=0.47^{+0.01}_{-0.03}$).

  * •

We found that the residual oscillations of the jet width in the synthetic
stacked images of the intrinsically smooth model significantly deviate from
zero at $z_{\rm obs}<2$ mas and $z_{\rm obs}>15$ mas. Thus, imaging artifacts
could also contribute to the observed oscillations of the M87 jet width as
well as helical threads of Kelvin-Helmholtz instability observed in e.g.
2023MNRAS.526.5949N or oscillations of the expanding magnetized jet (Lyu09;
2015ComAC...2....9K; Mertens).

  * •

We also fitted the observed transverse profiles with a single and three
Gaussians instead of two. The position of the break is stable against
different number of Gaussians. The single Gaussian fit results in
insignificantly larger break distance $z^{\rm 1G}_{\rm
break}=6.24^{+0.39}_{-0.36}$ mas than a triple Gaussian fit $z^{\rm 3G}_{\rm
break}=5.81^{+0.56}_{-0.60}$ mas.

  * •

The fit of the synthetic data based on the smooth jet model revealed a
significant – with even larger Bayes factor $\Delta\log{Z}=22.4$ than for the
real data – break from the conical to the parabolic jet shape with $z^{\rm
2G}_{\rm break}=2.20^{+0.50}_{-0.49}$ mas. Interesting that this type of
imaging artefact is opposite to the systematics described in Kovalev2020 who
observed that jet profiles at $z_{\rm obs}<0.5$ mas could be artificially
flattened. The fits with single and triple Gaussians also showed a break:
$z_{\rm break}^{\rm 1G}=3.26^{+0.72}_{-0.58}$ with $\Delta\log{Z}=21.9$ and
much less significant $z_{\rm break}^{\rm 3G}=2.24^{+0.65}_{-0.77}$ mas with
modest $\Delta\log{Z}=2.6$.

  * •

We also checked more the influence of the convolving beam size on the break
appearance and employed circular beam size 0.85 mas, that is the typical
equivalent area circular beam size from the analysis of 2017MNRAS.468.4992P.
The real data fit showed a significant (with $\Delta\log{Z}=5.2$) break at
$z_{\rm break}^{\rm 2G}=4.44^{+1.77}_{-2.49}$ mas. The synthetic data revealed
$z_{\rm break}^{\rm 2G}=1.37^{+0.22}_{-0.38}$ mas with modest significance
$\Delta\log{Z}=2.7$.

We conclude that the detected change of the jet shape from conical to
parabolic could be affected by the imaging systematics. Thus, the constraints
from the break position in the MOJAVE 15 GHz data obtained in the following
section should be treated with caution, but could be regarded as a firm
limits.

###  4.2 Constrains from the break position

Adopting 2019ApJ...875L...1E estimation for M87 black hole mass
$M_{\textrm{BH}}=(6.5\pm 0.9)\times 10^{9}M_{\odot}$ and the distance
$(16.8\pm 0.8)$ Mpc, we can use the observed jet width at the observed break
position Equation 24 and its uncertainty to estimate the width of the allowed
region in ($\sigma_{\rm M}$, $R_{\rm L}$) space. As emphasized above, this
parameter does not depend on the jet geometry and therefore can be used for
both conical and parabolic flows.

Further constrains could be obtained by considering the observed apparent
motion at the position of HST-1 (1999ApJ...520..621B) and assuming that it
corresponds to the maximal Lorentz factor $\Gamma_{\rm max}$ in the jet
observed at the viewing angle $\Theta=17.2\pm 3.3^{\circ}$ (Mertens). Equation
B13 from 2022MNRAS.509.1899N connects $\Gamma_{\rm max}$ and the initial
magnetization $\sigma_{\rm M}$ for the considered MHD model. Using the
observed jet width $d_{\rm break}=3.8\pm 0.3$ mas at the position of the break
in the MOJAVE data estimated in subsection 4.1 and assuming $\gamma_{\rm
in}\approx 1$, we obtain the constraints presented in Figure 6. Here we also
plot the constrained region that corresponds to the end of the conical
expansion profile at $z_{\rm obs}\approx 0.6$ mas observed in
2016ApJ...817..131H.

Figure 6: Constrains on the light cylinder $R_{\rm L}$ (in units of $r_{\rm
g}$) and the corresponding dimensionless spin $a$ of M87 SMBH, obtained from
the observed radius of the jet at the core appearance position, apparent
speeds in HST-1 region and accounting for the uncertainties in the black hole
mass, distance and viewing angle estimates. Shade of the color represents 1
and 2 $\sigma$ intervals. The blue color represents the $r_{\rm cr}$ observed
in this paper subsection 4.1. The green color corresponds to the end of the
conical expansion profile observed by (2016ApJ...817..131H, see their fig.9).

##  5 Discussion and conclusion

Thus, it was shown that the formation of a central core near the base of
relativistic jets, which was first predicted theoretically and then confirmed
by numerical modeling many years ago, should inevitably lead to a change in
the dependence of the observed jet width $d(z)$ on the distance $z$ to the
central engine. This is due to a sharp decrease in the poloidal magnetic field
$B_{\rm p}$ with distance $r$ from the jet axis at large enough distances
$z>z_{\rm cr}$ and, as a consequence, with an rapid decrease in the density of
radiating particles $n_{\rm e}\propto B_{\rm p}$ (43). As a result, the
observed width of the jet $d_{\rm jet}(z)$, which at small distances $z<z_{\rm
cr}$ can be estimated as $d_{\rm jet}\approx 2\,r_{\rm jet}$, inevitably
becomes noticeably smaller than the geometric size of the jet at larger
distances.

Thus, the theory predicts another important property of relativistic jets,
which in principle makes it possible to carry out additional diagnostics of
their physical parameters. The fact that this issue has not yet been discussed
in detail up to now is due to the obvious difficulty of studying the internal
structure of the jet at short distances $z\sim z_{\rm cr}$ from the central
engine. Note that despite the different literal expressions for conical (25)
and parabolic (27) flows, the numerical values of $z_{\rm cr}$ for reasonable
parameters ($\sigma_{\rm M}\sim 10$–$50$, $\gamma_{\rm in}\approx 1$,
$\theta_{\rm jet}\sim 10^{\circ}$) differ little from each other. As a result,
the internal break should be located at a distance of the order of 10–100
$R_{\rm L}$, i.e. only 100–1000 gravitational radii.

On the other hand, as was already stressed, some indications of the existence
of an internal break have already been noted for the closest source M87, for
which the observed position of the break corresponds to a fairly large angular
distance $\sim$ 1 mas. For example, Figure 15 given by Mertens shows the break
in the dependence of the hydrodynamic Lorentz factors on the distance $z$ to
the central machine. Since in the region of magnetically dominated flow we are
considering, the well-known connection $\Gamma=r/R_{\rm L}$ must be satisfied
(MHD), it can also be interpreted as a break in the dependence of the width of
the jet $d_{\rm jet}(z)$. Accordingly, the break in the dependence of the
observed jet width $\theta_{\rm jet}$ on distance $z$ is clearly visible in
Figure 9 presented by 2016ApJ...817..131H at $z_{\rm obs}=0.6$ mas.

We also searched for the predicted break in the jet geometry further
downstream the jet using stacked image of 37 single-epoch 15 GHz VLBA
observations of M87 radio jet from the MOJAVE archive. The transition from
almost conical shape at $z_{\rm obs}\approx 6$ mas was found. However, our
tests with a synthetic data revealed the presence of the imaging systematical
effects that could artificially steepen the expansion profile near the jet
origin.

Although the presence of instabilities is generally expected in axisymmetric
MHD jets, and the observations of M87 are consistent with development of
Kelvin-Helmholtz instability modes (Lobanov03; 2023MNRAS.526.5949N), they seem
not to disrupt the global jet structure. Porth15 have explored the stability
of a central core, which we connect to the existence of the inner break. It
turned out that we should expect the core stability for the ambient pressure
$P_{\mathrm{ext}}\propto z^{b}$ with $b>2$, which is consistent with our
results for the expected from the observations ambient medium pressure power
in M87 jet (see Table 2 in NGBNAH19).

We constrained the spin of the M87 black hole assuming the observed break in
the jet shape is due to the appearance of the core of the longitudinal
magnetic field. Employing values of the terminal Lorentz factor inferred from
the HST-1 speeds measurements and known viewing angle estimate we constrain
the dimensionless spin to be in range 0.05 - 0.1 if the break is located at
the $z_{\rm obs}\approx 6$ mas and 0.3 - 0.7 if the break is located upstream
at $z_{\rm obs}=0.6$ mas. The spin constrains dependent on the exact value of
the jet magnetization $\sigma_{\rm M}$ among those consistent with HST-1
speeds with higher $\sigma_{\rm M}$ favouring larger spins.

The estimated range for a black hole spin presented in Figure 6 is in
excellent agreement with the previous results, based on the outer break
(NGBNAH19; NKP20). The corresponding light cylinder radius coincides with the
estimate by Kino22, which was obtained by fitting the observed M87 kinematics.
Thus, there are three different methods to constrain the value of
$R_{\mathrm{L}}$: based on kinematics (Kino22), fitting the outer (NGBNAH19)
and inner break (this work). All three methods point to the same values of the
order of 0.02–0.04 pc (50–100 $R_{\mathrm{g}}$), which, if the field lines are
connected to the black hole ($\Omega_{\mathrm{F}}=\Omega_{\mathrm{H}}/2$
holds), corresponds to its rather small non-dimensional spin.

We should also note that the observed inner break in M87 jet requires the
ambient medium, collimating the jet, to have a single power-law pressure
dependence. This dependence must hold down to the scales of the order of
100-1000 gravitational radii.

## Data availability

The data underlying this work will be shared on reasonable request to the
corresponding author.

## Acknowledgements

This work was partially supported by the government of the Russian Federation
(agreement No. 05.Y09.21.0018). The analysis of the observational data to
estimate the inner break position is supported by the Russian Science
Foundation: project555Information about the project:
https://rscf.ru/en/project/20-72-10078/ 20-72-10078.

This research has made use of data from the MOJAVE database that is maintained
by the MOJAVE team (2019ApJ...874...43L). The MOJAVE program is supported
under NASA-Fermi grant 80NSSC19K1579. The National Radio Astronomy Observatory
is a facility of the National Science Foundation operated under cooperative
agreement by Associated Universities, Inc.

This research made use of Astropy, a community-developed core Python package
for Astronomy (2013A&A...558A..33A), Numpy (numpy), Scipy (scipy), FINUFFT
(2018arXiv180806736B). Matplotlib Python package (Hunter:2007) was used for
generating plots in this paper.