# Jet Parameters in the Black-Hole X-Ray Binary MAXI J1820+070

Andrzej A. Zdziarski  Nicolaus Copernicus Astronomical Center, Polish Academy
of Sciences, Bartycka 18, PL-00-716 Warszawa, Poland<EMAIL_ADDRESS>Alexandra J. Tetarenko  NASA Einstein Fellow  East Asian Observatory, 660 N.
A’ohōkū Place, University Park, Hilo, Hawaii 96720, USA  Department of Physics
and Astronomy, Texas Tech University, Lubbock, Texas 79409-1051, USA  Marek
Sikora  Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences,
Bartycka 18, PL-00-716 Warszawa, Poland<EMAIL_ADDRESS>
###### Abstract

We study the jet in the hard state of the accreting black-hole binary MAXI
J1820+070. From the available radio-to-optical spectral and variability data,
we put strong constraints on the jet parameters. We find while it is not
possible to uniquely determine the jet Lorentz factor from the spectral and
variability properties alone, we can estimate the jet opening angle ($\approx
1.5\pm 1\arcdeg$), the distance at which the jet starts emitting synchrotron
radiation ($\sim$3$\times 10^{10}$ cm), the magnetic field strength there
($\sim$104 G), and the maximum Lorentz factor of the synchrotron-emitting
electrons ($\sim$110–150) with relatively low uncertainty, as they depend
weakly on the bulk Lorentz factor. We find the breaks in the variability power
spectra from radio to sub-mm are consistent with variability damping over the
time scale equal to the travel time along the jet at any Lorentz factor. This
factor can still be constrained by the electron-positron pair production rate
within the jet base, which we calculate based on the observed X-ray/soft
gamma-ray spectrum, and the jet power, required to be less than the accretion
power. The minimum ($\sim$1.5) and maximum ($\sim$4.5) Lorentz factors
correspond to the dominance of pairs and ions, and the minimum and maximum jet
power, respectively. We estimate the magnetic flux threading the black hole
and find the jet can be powered by the Blandford-Znajek mechanism in a
magnetically-arrested flow accretion flow. We point out the similarity of our
derived formalism to that of core shifts, observed in extragalactic radio
sources.

##  1 Introduction

Our knowledge of the structure of extragalactic radio jets is already quite
detailed, see, e.g., Blandford et al. (2019). While a number of aspects
remains to be determined, e.g., the jet lateral structure (Perlman et al.,
2019), radio maps provide us the projected structures of the jets, in
particular their opening angles. Magnetic fields can be determined via core
shifts, which are angular displacements of the position of the radio core
between two frequencies (e.g., Lobanov 1998; Zamaninasab et al. 2014;
Zdziarski et al. 2015, hereafter Z15). Superluminal motion allows us to
estimate the jet bulk Lorentz factors, $\Gamma$ (e.g., Jorstad et al. 2001;
Kellermann et al. 2004; Lister et al. 2019). They can also be independently
estimated from radiative models of blazars (Ghisellini & Tavecchio, 2015) and
from the radio core luminosity function (Yuan et al., 2018). The jet power can
be estimated from calorimetry of radio lobes (Willott et al., 1999; Shabala &
Godfrey, 2013) and core shifts (e.g., Pjanka et al. 2017). Finally, the e±
pair content can be obtained from comparison of the observed jet powers with
theoretical predictions (Sikora et al. 2020 and references therein).

On the other hand, our knowledge of jets in accreting black-hole (BH)
binaries, which are the main class of microquasars, is much more rudimentary.
From available radio maps, we can only set upper limits on the jet opening
angles (e.g., Stirling et al. 2001; Miller-Jones et al. 2006). If we know the
distance, we can constrain the Lorentz factors of ejected transient blobs,
which phenomenon is associated with transitions from the hard spectral state
to the soft one, e.g., Atri et al. (2020); Wood et al. (2021). The Lorentz
factors of steady compact jets commonly present in the hard spectral state are
even more difficult to constrain, with only rough estimates of $\Gamma\gtrsim
1.5$–2 (e.g., Stirling et al. 2001; Casella et al. 2010; Tetarenko et al.
2019). Therefore, an accurate determination of the jet parameters even for a
single source would be very important.

Here we study the jet in the transient BH X-ray binary MAXI J1820+070 during
its outburst in 2018. We use the observational data of outstanding quality for
that source presented by Tetarenko et al. (2021) (hereafter T21), which gives
us an opportunity of such an accurate parameter determination. We interpret
these data in terms of the classical model of Blandford & Königl (1979) and
Königl (1981). Here flat radio spectra are interpreted in terms of a
superposition of synchrotron self-absorbed and optically-thin spectra, spectra
above the break frequency are optically-thin synchrotron, and the electron
distribution and the magnetic field strength are parametrized by power laws.
The jet in the synchrotron-emitting part is assumed to be conical and of a
constant bulk-motion velocity. We provide an updated analysis of those data,
making corrections to the similar model used in T21. In particular, T21
followed the formulation of the model that suffered from some errors related
to the transformation from the comoving frame to that of the observer. Also,
we properly connect the break frequencies in the radio/mm spectra with the
propagation time along the jet, and we correct the expressions for jet power.
Furthermore, we link the dependencies of energy densities on the distance to
the observed hard inverted spectral index, as well as we use additional data
from Rodi et al. (2021). This allows us to obtain constraints based on the
full radio-through-optical spectrum.

MAXI J1820+070 was discovered during its outburst in 2018 (Tucker et al.,
2018; Kawamuro et al., 2018). The source is relatively nearby, with a distance
of $D\approx 2.96\pm 0.33$ kpc measured based on a radio parallax (Atri et
al., 2020). Then, Wood et al. (2021) determined $D\leq 3.11\pm 0.06$ kpc based
on the proper motion of the moving ejecta during the hard-to-soft state
transition. The inclination of the radio jet is $i\approx 64\arcdeg\pm
5\arcdeg$ (Wood et al., 2021), while the inclination of the binary is
constrained to $i_{\rm b}\approx 66\arcdeg$–$81\arcdeg$ (Torres et al., 2020).
The BH mass is given by $M\approx(5.95\pm 0.22){\rm M}_{\sun}/\sin^{3}i_{\rm
b}$ (Torres et al., 2020).

The data presented in T21 were obtained during a multiwavelength observational
campaign from radio to X-rays performed during a 7-h period during the hard
state on 2018 April 12 (MJD 58220). We also use the simultaneous IR and
optical data obtained by Rodi et al. (2021). During the campaign, the source
was in a part of the initial hard state which formed a plateau in the X-ray
hardness vs. flux diagram (Buisson et al., 2019).

Our theoretical model is presented in Section 2 and Appendix A. In Section 3,
we fit it to the data. In Section 4, we discuss various aspects of our
results, and show that our formalism based on time lags between different
frequencies of the flat spectrum is equivalent to the formalism based on core
shifts. We give our conclusions in Section 5.

##  2 Steady-state jets

###  2.1 Power-law dependencies

Following the theoretical interpretation in T21, we consider a continuous,
steady-state, jet in the range of its distance from the BH where it has
constant both the bulk Lorentz factor and the opening angle, i.e., it is
conical. We consider its synchrotron emission and self-absorption, and assume
that the hard, partially self-absorbed, part of the total spectrum results
from superposition of spectra from different distances with breaks
corresponding to unit self-absorption optical depth (Blandford & Königl,
1979). We use the formulation of the model of Königl (1981) (which is an
extension of the model of Blandford & Königl 1979 for cases with the self-
absorbed radio index different from zero) as developed in Zdziarski et al.
(2019), hereafter Z19. In this model, the jet is assumed to be laterally
uniform, which is a good approximation for $i\gg\Theta$, where $\Theta$ is the
jet (half) opening angle. We denote the observed and comoving-frame photon
frequencies as $\nu$ and $\nu^{\prime}$, respectively, and introduce the
dimensionless distance,

$\nu^{\prime}=\frac{\nu(1+z_{\rm
r})}{\delta},\quad\delta\equiv\frac{1}{\Gamma(1-\beta\cos
i)},\quad\xi\equiv\frac{z}{z_{0}},$ (1)

where $z_{\rm r}$ is the cosmological redshift (equal to null in our case),
$\Gamma$ and $\beta$ are the jet bulk Lorentz factor and the velocity in units
of the light speed, respectively, $z$ is the distance from the BH, and $z_{0}$
is the distance at which the jet becomes optically thin to self-absorption at
all considered frequencies. As in Königl (1981), we assume the electron
differential density distribution, $n(\gamma,\xi)$, and the magnetic field
strength, $B$, are parameterized by power-law dependencies,

$R(\xi)=z_{0}\xi\tan\Theta,\,n(\gamma,\xi)=n_{0}\xi^{-a}\gamma^{-p},\,B(\xi)=B_{0}\xi^{-b},$
(2)

where $R$ is the jet radius and $\gamma$ is the Lorentz factor of the emitting
electrons in the jet comoving frame, with $\gamma_{\rm
min}\leq\gamma\leq\gamma_{\rm max}$. The quantities $R$ and $z$ are measured
in the local observer’s frame, while $n$ and $B$ are given in the comoving
frame (for notational simplicity, we skip the primes). For a conserved
electron number along the jet and conserved magnetic energy flux dominated by
toroidal fields, we have $a=2$ and $b=1$, corresponding to the spectral index
of $\alpha=0$, independent of the value of $p$ (Blandford & Königl, 1979).
Here, we define $\alpha$ by the energy flux density of
$F_{\nu}\propto\nu^{\alpha}$. If either the electron or magnetic energy is
dissipated, $a>2$, $b>1$, respectively. Then, the emission weakens with the
distance and the synchrotron spectrum in the partially self-absorbed frequency
range becomes harder than in the conserved case, $\alpha>0$. The spectral
indices of partially self-absorbed and optically thin synchrotron emission are
given by

$\alpha=\frac{5a+3b+2(b-1)p-13}{2a-2+b(p+2)},\quad\alpha_{\rm
thin}=\frac{1-p}{2},$ (3)

respectively (Königl 1981; Z19). Using a delta-function approximation to the
single-electron synchrotron spectrum at $\gamma^{2}\gg 1$ (assumed hereafter),
the synchrotron frequency for a given $\gamma$ and $\xi$, and its range
emitted by the jet are

$\displaystyle\frac{h\nu^{\prime}}{m_{\rm e}c^{2}}=$
$\displaystyle\frac{B_{0}\xi^{-b}}{B_{\rm cr}}\gamma^{2},$ (4)
$\displaystyle\frac{h\nu^{\prime}_{\rm min}}{m_{\rm
e}c^{2}}=\frac{B_{0}\xi_{\rm M}^{-b}}{B_{\rm cr}}\gamma_{\rm min}^{2}$
$\displaystyle,\quad\frac{h\nu^{\prime}_{\rm max}}{m_{\rm
e}c^{2}}=\frac{B_{0}}{B_{\rm cr}}\gamma_{\rm max}^{2},$ (5)

respectively. Here $z_{0}\xi_{\rm M}$ is the distance at which the jet
terminates, $h$ is the Planck constant, $B_{\rm cr}={2\pi m_{\rm
e}^{2}c^{3}/(eh)}\approx 4.414\times 10^{13}$ G is the critical magnetic field
strength, and $m_{\rm e}$ and $e$ is the electron mass and charge,
respectively. The spectral density of the synchrotron emission for a single
jet parameterized by Equation (2) and for $\nu_{\rm min}\leq\nu\leq\nu_{\rm
max}$ is then given by (see equation A5 of Z19),

$\displaystyle F_{\nu}$ $\displaystyle\simeq
F_{0}\left(\frac{\nu}{\nu_{0}}\right)^{\frac{5}{2}}\int_{\xi_{\rm
min}}^{\xi_{\rm max}}{\rm
d}\xi\,\xi^{1+b/2}\left\\{1-\exp[-\tau(\frac{\nu}{\nu_{0}},\xi)]\right\\},$
(6) $\displaystyle F_{0}$ $\displaystyle\equiv{(1+z_{\rm
r})^{\frac{7}{2}}(m_{\rm e}h\delta)^{\frac{1}{2}}\pi
C_{1}(p)z_{0}^{2}\nu_{0}^{\frac{5}{2}}\tan\Theta\sin i\over
6cC_{2}(p)(B_{0}/B_{\rm cr})^{\frac{1}{2}}D^{2}}.$ (7)

Here $F_{0}$ is a constant proportional to the bolometric flux,
$\tau(\nu/\nu_{0},\xi)$ is the synchrotron self-absorption optical depth,
$\nu_{0}$ is the break frequency, see Equation (13) below, and $C_{1}(p)$,
$C_{2}(p)$ are constants following from averaging the synchrotron emission and
absorption coefficients over the pitch angle,

$\displaystyle C_{1}(p)={3^{p+4\over 2}\Gamma_{\rm E}\left(3p-1\over
12\right)\Gamma_{\rm E}\left(3p+19\over 12\right)\Gamma_{\rm E}\left(p+1\over
4\right)\over 2^{5}\pi^{1\over 2}\Gamma\left(p+7\over 4\right)},$ (8)
$\displaystyle C_{2}(p)={3^{p+3\over 2}\Gamma_{\rm E}\left(3p+2\over
12\right)\Gamma_{\rm E}\left(3p+22\over 12\right)\Gamma_{\rm E}\left(p+6\over
4\right)\over 2^{4}\pi^{\frac{1}{2}}\Gamma_{\rm E}\left(p+8\over 4\right)}$
(9)

(cf. Jones et al. 1974; Zdziarski et al. 2012), where $\Gamma_{\rm E}$ is the
Euler Gamma function. In the extragalactic case, $D$ is the luminosity
distance. The lower and upper limits of the integral (6) are,

$\displaystyle\xi_{\rm min}(\nu)=\max\left[1,\left(\frac{B_{0}m_{\rm
e}c^{2}\gamma_{\rm min}^{2}}{h\nu^{\prime}B_{\rm
cr}}\right)^{\frac{1}{b}}\right],$ (10) $\displaystyle\xi_{\rm
max}(\nu)=\min\left[\left(\frac{\nu_{\rm
max}}{\nu}\right)^{\frac{1}{b}},\xi_{\rm M}\right],$ (11)

respectively. Figure 1 shows an example of the spatial dependencies of the
emission along the jet at different frequencies for $\gamma_{\rm min}=10$ and
$\nu_{\rm max}=10^{7}$ GHz. For $\gamma_{\rm min}\gtrsim 30$, the emission at
all frequencies in this case would be in the optically-thin regime only, cf.
Equation (16) below. We note that above we have assumed the single-electron
synchrotron emission is isotropic in the plasma frame, which is strictly valid
for a tangled magnetic field.

Figure 1: An example of the spatial structure of the jet emission at different
frequencies for $\nu_{0}=2\times 10^{4}$ GHz, $F_{0}=300$ mJy, $b=1.1$,
$a=2b$, $p=2$. The red dots, blue dashes, magenta dots, cyan dashes, and green
dots correspond to $\nu=5.25$, 25.9, 343.5, $1.4\times 10^{5}$ GHz, and
$5\times 10^{6}$ GHz, respectively. The black solid curve corresponds to
$\nu=\nu_{0}$. The three lowest and two highest frequency curves end at
$\xi_{\rm min}>1$ and at $\xi_{\rm max}$, respectively, beyond which there is
no emission at those $\nu$. These values of $\xi_{\rm min}$, $\xi_{\rm max}$
were calculated for $\gamma_{\rm min}=10$, $\nu_{\rm max}=10^{7}$ GHz,
$i=64\arcdeg$, $\Gamma=3$ and $B_{0}=10^{4}$ G (which correspond to
$\gamma_{\rm max}\approx 793$).

Figure 2: An example of the jet synchrotron spectrum for $\nu_{0}=2\times
10^{4}$ GHz, $F_{0}=300$ mJy, $b=1.1$, $a=2b$, $p=2$, $\nu_{\rm max}=10^{7}$
GHz, $i=64\arcdeg$. This spectrum is virtually independent of $\xi_{\rm
min}(\nu)$ in the shown range of $\nu$, as long as $\xi_{\rm
min}(\nu)\ll\xi_{\nu}$. The blue curve shows the accurate spectrum of Equation
(6), and the red dashes show the approximation of Equation (12). The gradual
high-energy cutoff of the accurate spectrum is due to $\xi_{\rm max}$
decreasing with increasing $\nu$ and reaching unity for $\nu_{\rm max}$.

Then, power-law dependencies assuming $\xi_{\rm min}=1$, $\xi_{\rm
max}=\infty$ in the optically-thick and optically thin cases are (cf. equation
A11 in Z19)

$F_{\nu}\simeq 2F_{0}\begin{cases}\displaystyle{\Gamma_{\rm
E}\left[\frac{2a-6+b(p+1)}{2a-2+b(p+2)}\right]}\frac{(\nu/\nu_{0})^{\alpha}}{4+b},&\nu\ll\nu_{0};\cr\displaystyle{\frac{(\nu/\nu_{0})^{\alpha_{\rm
thin}}}{2a-6+b(p+1)}},&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\nu_{0}\ll\nu\ll\nu_{\rm
max}.\cr\end{cases}$ (12)

Figure 2 shows an example comparison of the accurate spectrum of Equation (6)
with these power-law approximations. We see they are inaccurate around
$\nu_{0}$ as well as close to $\nu_{\rm max}$, where they fail to reproduce
the gradual cutoff of the accurate spectrum. While the power-law asymptotic
solutions intersect at a $\nu$ slightly different from $\nu_{0}$, that
frequency has no physical meaning since the actual spectrum in that range does
not follow the broken-power law form, see Figure 2. We can define a broken-
power law approximation by taking the minimum of the two branches in Equation
(12).

The optical depth along a line of sight crossing the jet spine can be written
as

$\tau(\nu/\nu_{0},\xi)=(\nu/\nu_{0})^{-(p+4)/2}\xi^{1-a-b(p+2)/2},$ (13)

where $\nu_{0}$ is defined by $\tau(\nu=\nu_{0},\xi=1)=1$. The place $\xi=1$,
or $z=z_{0}$, corresponds to the jet being optically thin for all
$\nu\geq\nu_{0}$. There is no synchrotron emission111We note that since the
partially optically-thick emission of a jet at $z>z_{0}$ would remain almost
unaffected if there were still emission following the scaling of Equation (2)
at $z<z_{0}$ (which would, however, decrease the actual value of $z_{0}$ and
increase $\nu_{0}$), it is also possible to formulate the structure of the
partially optically-thick part without invoking $z_{0}$ and $\nu_{0}$. Such a
formulation is presented in Equations (A1–A4) in Appendix A. at $z<z_{0}$, and
thus $z_{0}$ corresponds to the onset of the jet emission. The relationship of
$\nu_{0}$ to the jet parameters is given by equation (A8) of Z19. We express
it here as a formula for the normalization of the electron distribution,

$n_{0}=\left(\frac{B_{\rm
cr}}{B_{0}\delta}\right)^{1+\frac{p}{2}}\\!\\!\left[\frac{h\nu_{0}(1+z_{\rm
r})}{m_{\rm e}c^{2}}\right]^{2+\frac{p}{2}}\\!\\!\frac{\alpha_{\rm f}\sin
i}{C_{2}(p)\pi\sigma_{\rm T}z_{0}\tan\Theta},$ (14)

where $\alpha_{\rm f}$ is the fine-structure constant and $\sigma_{\rm T}$ is
the Thomson cross section. From Equation (13), the distance along the jet at
which $\tau(\nu,\xi_{\nu})=1$ at $\nu\lesssim\nu_{0}$ is

$\xi_{\nu}=\left(\frac{\nu}{\nu_{0}}\right)^{-q}\\!\\!,\,\,q\equiv\frac{p+4}{2a+bp+2b-2},\,\,z_{\nu}=z_{0}\xi_{\nu}.$
(15)

For $a=2$ and $b=1$, we have $q=1$ at any $p$. This distance is very close to
that at which most of the flux at a given $\nu$ is emitted, which can be
defined by the maximum of ${\rm d}F_{\nu}(\xi)/{\rm d}\,\ln\xi$, see Figure 1,
and can be calculated using Equation (6). For example, at $a=2.2$, $b=1.1$,
and $p=2$, that maximum is at $\xi\approx 1.19\xi_{\nu}$. The emission around
the peak has a broad spatial distribution; the 50% values of the maximum flux
are reached at $\xi=0.65\xi_{\nu}$ and $3.64\xi_{\nu}$.

Then, the Lorentz factor responsible for the bulk of emission at $\xi_{\nu}$
is

$\gamma_{\nu}=\left(\frac{B_{\rm cr}}{B_{0}}\frac{h\nu_{0}}{\delta m_{\rm
e}c^{2}}\right)^{1/2}\left(\frac{\nu}{\nu_{0}}\right)^{(1-bq)/2},$ (16)

which is usually weakly dependent on $\nu$. While the integral spectrum of
Equation (6) is valid for any $\gamma_{\rm min}$, the asymptotic power-laws of
Equation (12) require $\gamma_{\rm min}$ to be by a factor of at least a few
lower than $\gamma_{\nu}$ for values of $\nu$ of interest (in the range
$<\nu_{0}$) and $\gamma_{\rm max}$ is required to be a factor of a few larger
than $\gamma_{\nu}$. If a high-energy cutoff is observed, an additional
constraint follows from it, see Equation (4).

If we know $\alpha$ and $\alpha_{\rm thin}$, we still cannot determine the
values of $a$ and $b$ separately. However, a likely possibility is that the
ratio between the electron and magnetic energy densities remains constant,
i.e., maintaining the same degree of equipartition along the jet, in which
case $a=2b$. We define an equipartition parameter as the ratio of the energy
densities,

$\beta_{\rm eq}\equiv{u_{\rm p}\over B^{2}/8\pi}={n_{0}m_{\rm e}c^{2}(1+k_{\rm
i})(f_{E}-f_{N})\over B_{0}^{2}/8\pi},$ (17)

where

$f_{E}\equiv\begin{cases}{\gamma_{\rm max}^{2-p}-\gamma_{\rm min}^{2-p}\over
2-p},&p\neq 2;\cr\ln{\gamma_{\rm max}\over\gamma_{\rm
min}},&p=2,\cr\end{cases}\quad f_{N}\equiv\frac{\gamma_{\rm
min}^{1-p}-\gamma_{\rm max}^{1-p}}{p-1},$ (18)

the second equality in Equation (17) is at $z_{0}$, $u_{\rm p}$ is the
particle energy density, $k_{\rm i}$ accounts for the energy density in
particles other than the power-law electrons, in particular in ions (excluding
the rest energy), and $p>1$ has been assumed in the expression for $f_{N}$.
For $a=2b$, $\beta_{\rm eq}$ is constant along the jet (provided $k_{\rm i}$
is also constant) at $z\geq z_{0}$, which yields

$\alpha=\frac{(b-1)(13+2p)}{b(p+6)-2},\quad q=\frac{p+4}{b(p+6)-2}.$ (19)

Below, we use $\beta_{\rm eq}$ and $a=2b$ to constrain the jet parameters.

We note that the case of $a>2$ requires that either $\gamma_{\rm min}$
decreases or the electrons removed from their power-law distribution move to
some low energies below $\gamma_{\rm min}$ (with negligible emission). Since
we assume $\gamma_{\rm min}$ to be constant along the jet, the latter has to
be the case.

We next consider the difference between the arrival times of two photons. The
first photon, at $\nu_{1}$, is emitted toward the observer at $\xi_{\nu_{1}}$.
The second photon, with $\nu_{2}<\nu_{1}$, is emitted at $\xi_{\nu_{2}}$ by
the same comoving point of the jet after the time $\Delta t_{\rm e}$, which is
further downstream in the jet by $\beta c\Delta t_{\rm e}$. Since the jet
moves at an angle $i$ with respect to the line of sight, the distance of the
emitting point to the observer will become shorter during this time by $\beta
c\Delta t_{\rm e}\cos i$. For an observed difference in the arrival times of
$\Delta t_{\rm a}$, the intrinsic separation between the emission points
(measured in the local observer’s frame) will be

$z_{\nu_{2}}-z_{\nu_{1}}=z_{0}(\xi_{\nu_{2}}-\xi_{\nu_{1}})=\frac{\Delta
t_{\rm a}\beta c}{(1-\beta\cos i)(1+z_{\rm r})}.$ (20)

At frequencies $\leq\nu_{0}$, $\xi_{\nu}$ follows from Equation (15). Here, we
have also taken into account the redshift, making this expression correct also
for an extragalactic source. Given an observed $\Delta t_{\rm a}$, Equations
(15) and (20) imply

$\displaystyle z_{0}=\frac{\Delta t_{\rm a}\nu_{0}^{-q}\beta c}{(1-\beta\cos
i)\left(\nu_{2}^{-q}-\nu_{1}^{-q}\right)(1+z_{\rm
r})}=\frac{t_{0}c\beta\Gamma\delta}{1+z_{\rm r}},$ (21) $\displaystyle
t_{0}\equiv\frac{\Delta t_{\rm a}}{\Delta\xi}=\frac{\Delta t_{\rm
a}\nu_{0}^{-q}}{\nu_{2}^{-q}-\nu_{1}^{-q}},$ (22)

where $t_{0}$ can be obtained from time lag data if $\nu_{0}$ and $q$ are
known (from the spectrum).

Appendix A provides general solutions for $z_{\nu}$, $B$, $\Theta$ and $n$ to
the equations in this section assuming the validity of Equation (12) for
$\nu<\nu_{0}$ as functions of $b$ (assuming $a=2b$), $p$, $\nu_{0}$, $F_{0}$,
$t_{0}$, $i$ and $D$, as well as $\beta_{\rm eq}$, $\gamma_{\rm min}$ and
$\gamma_{\rm max}$.

###  2.2 The jet power

The jet power can be calculated using standard expressions. Note that it is
defined in terms of the proper enthalpy rather than the energy density, e.g.,
Levinson (2006), Zdziarski (2014). Then, the component of the jet power due to
both the relativistic electrons and magnetic fields (assuming they are
predominantly toroidal at $z_{0}$) including both the jet and counterjet for
$a=2b$ and $k_{\rm i}=0$ at $z\geq z_{0}$ is

$P_{B}+P_{\rm e}=\left(\frac{1}{2}+\frac{\beta_{\rm
eq}}{3}\right)\\!c\beta(B_{0}z_{0}\Gamma\tan\Theta)^{2}\xi^{2-2b}.$ (23)

The usable power in cold ions, at any $z$, but calculated at $z_{0}$, is

$P_{\rm i}=2\pi\mu_{\rm e}n_{0}f_{N}\\!\left(\\!1-\frac{2n_{+}}{n_{\rm
e}}\\!\right)\\!m_{\rm p}c^{3}\beta\Gamma(\Gamma-1)(z_{0}\tan\Theta)^{2},$
(24)

where $n_{\rm e}$ and $n_{+}$ is the density of both electrons and positrons
(which ratio is assumed to be constant along the jet), and positrons only,
respectively, $\mu_{\rm e}=2/(1+X)$ is the electron mean molecular weight, $X$
($\approx 0.7$ for the cosmic composition) is the H mass fraction, $m_{\rm p}$
is the proton mass, and $f_{N}$ is given by Equation (18). This is the power
in ions that has to be supplied to the jet, and then it can be dissipated,
hence the factor $(\Gamma-1)$. Equation (24) neglects the possible presence of
background electrons being piled up at $\gamma<\gamma_{\rm min}$ already at
$z_{0}$. On the other hand, $a>2$ at a constant $\gamma_{\rm min}$ requires
that leptons removed from $n(\gamma,\xi)$ at $z>z_{0}$ do appear at some low
energies below $\gamma_{\rm min}$ (with negligible emission).

We note that in a steady state, $2n_{+}/n_{\rm e}=2\dot{N}_{+}/\dot{N}_{\rm
e}$, where $2\dot{N}_{+}$ is the total rate of advection upstream of e± pairs
produced at the jet base, and $\dot{N}_{\rm e}$ is the total lepton flow rate,

$\dot{N}_{\rm e}\approx 2\pi n_{0}f_{N}c\beta\Gamma(z0\tan\Theta)^{2}.$ (25)

This implies

$P_{\rm i}=\mu_{\rm e}m_{\rm p}c^{2}(\Gamma-1)\left(\dot{N}_{\rm
e}-2\dot{N}_{+}\right)\geq 0.$ (26)

Since the pair production rate at the jet base, $\dot{N}_{+}$ (Section 2.3),
is independent of $\Gamma$, the condition of $P_{\rm i}\geq 0$ can give a
lower limit on $\Gamma$.

In any jet model, the total usable jet power is approximately constrained by
the accretion power,

$P_{\rm j}=P_{B}+P_{\rm e}+P_{\rm
i}\lesssim\dot{M}c^{2}=\frac{L}{\epsilon_{\rm eff}}$ (27)

where $\dot{M}$ is the mass accretion rate, $L$ is the bolometric luminosity
and $\epsilon_{\rm eff}\sim 0.1$ is the accretion efficiency. This then gives
an upper limit on $\Gamma$. The limit $\dot{M}c^{2}$ can be exceeded if the
rotation of the BH is tapped, but only by a factor $\lesssim$1.3 and for a
maximally rotating BH, see the references in Section 2.4 below.

Finally, we consider the power lost in synchrotron emission. It equals the
synchrotron luminosity emitted by both jets in all directions (which is
Lorentz invariant). Since $\nu_{0}$ depends on the direction and the partially
self-absorbed emission is not isotropic in the comoving frame, we neglect its
effect and assume the entire emission is optically thin and isotropic in that
frame, which is a good approximation for hard electron distributions with
$p\lesssim 2.5$ or so. This gives

$\displaystyle P_{\rm S}$
$\displaystyle\approx\frac{1}{3}(B_{0}\tan\Theta)^{2}\sigma_{\rm
T}cz_{0}^{3}n_{0}\Gamma f_{E2}f_{\xi},$ $\displaystyle f_{E2}$
$\displaystyle\equiv\begin{cases}{\gamma_{\rm max}^{3-p}-\gamma_{\rm
min}^{3-p}\over 3-p},&p\neq 3;\cr\ln\frac{\gamma_{\rm max}}{\gamma_{\rm
min}},&p=3,\cr\end{cases}$ (28) $\displaystyle f_{\xi}$
$\displaystyle\equiv\int_{1}^{\infty}{\rm d}\xi\xi^{2-2b-a}=\frac{1}{2b+a-3},$

where $2b+a>3$ is assumed. This $P_{\rm S}$ approximately equals the intrinsic
luminosity of both jets,

$L_{\rm jet}\approx 8\pi D^{2}\delta^{-3}\Gamma\int_{0}^{\nu_{\rm
max}(\delta)}{\rm d}\nu F_{\nu},$ (29)

where $F_{\nu}$ is for the approaching jet only, $\nu_{\rm max}$ is given by
Equation (4) and the transformation law is for a stationary jet emitting
isotropically in its comoving frame (Sikora et al., 1997). In addition to that
law, $\nu_{\rm max}$ depends on $\delta$. For self-consistency of our
equations, $P_{\rm S}\ll P_{\rm j}$ is required.

###  2.3 Pair production

As we see from Equation (26), the jet power in ions (given an observed
synchrotron spectrum) strongly depends on the abundance of e± pairs. In the
case of extragalactic jets, there are strong indications that they dominate by
number, though most of the rest mass is usually still in ions (Sikora et al.,
2020). In the case of jets in microquasars, this is uncertain. An important
issue for that is the origin of pairs. A likely mechanism is pair production
in photon-photon collisions by photons produced close to the BH.

Figure 3: A sketch of of the pair-producing geometry based on fig. 3.9 of
Tchekhovskoy (2015), which shows the result of his 3D GRMHD simulation for
magnetically-arrested accretion on a BH with the spin parameter of
$a_{*}=0.99$. In our case the disk is hot in its inner part (up to the radius
$R_{\rm hot}$) and surrounded by a cool outer disk. We consider e± pair
production within the jet base (shown in green), which is devoid of matter,
with the wavy arrows representing pair-producing photons. We denote the
characteristic jet radius of the pair-producing region as $R_{\rm jet}$. In
addition, pairs are produced within the hot disk, but it is magnetically
shielded from the jet base. The black solid curves show the poloidal magnetic
field.

Pairs can be produced within the hot flow, e.g, Svensson (1987). Since the
Larmor radius of either a proton or an electron is orders of magnitude lower
than $R_{\rm g}$ (where $R_{\rm g}=GM/c^{2}$ is the gravitational radius), the
magnetic base of the jet is shielded from the hot plasma and it is unlikely
that pairs produced in the accretion flow enter the jet. On the other hand,
pairs can also be produced within the magnetic base of the jet, outside the
hot plasma (Sikora et al., 2020). There, photon-photon collisions will create
e± pairs in an environment devoid of background matter, thus strongly reducing
the rate of pair annihilation. A possible geometry (see also Henri & Pelletier
1991; Ferreira et al. 2006) is shown in Figure 3. From an observed hard X-ray
spectrum and a radius, $R_{\rm hot}$, of the emitting hot plasma (inferred,
e.g., from X-ray spectroscopy; Bambi et al. 2021), we can estimate the average
photon density within the jet base, which then gives us the rate of pair
production per unit volume, $\propto R_{\rm hot}^{-4}$. We approximate the
pair-producing volume as two cylinders with the height $R_{\rm hot}$ and the
characteristic radius of the jet, $R_{\rm jet}$, i.e., $V=2\pi R_{\rm
jet}^{2}R_{\rm hot}$. We can then write the total lepton production rate as

$2\dot{N}_{+}=A_{\gamma\gamma}R_{\rm hot}^{-3}R_{\rm jet}^{2},$ (30)

where the factor $A_{\gamma\gamma}$ would follow from detailed calculations.
Depending on the equilibrium density of the pairs, some of them would
annihilate, and some would be advected to the BH, reducing the effective
$2\dot{N}_{+}$. We address this issue for the case of MAXI J1820+070 in
Section 3.1.

###  2.4 The Blandford-Znajek mechanism

We can also estimate the jet power in the framework of the model with
extraction of the rotational power of the BH (Blandford & Znajek, 1977), as
illustrated in Figure 3. The jet power in this case depends on the magnetic
flux, $\Phi_{\rm BH}$, threading the BH (on one hemisphere), which can be
written as

$\Phi_{\rm BH}=\phi_{\rm BH}(\dot{M}c)^{1/2}R_{\rm g},$ (31)

where $\phi_{\rm BH}$ is a dimensionless magnetic flux. Its maximum value of
$\approx$50 is obtained in magnetically arrested disks (MAD; Narayan et al.
2003), as it was found in GRMHD simulations of MAD accretion (Tchekhovskoy et
al. 2011; McKinney et al. 2012; see its more accurate value in Davis &
Tchekhovskoy 2020). Then it has been found that

$P_{\rm j}\approx 1.3\left(\frac{\phi_{\rm
BH}}{50}\right)^{2}h_{0.3}a_{*}^{2}\dot{M}c^{2},$ (32)

where $a_{*}$ is the BH spin parameter and $h_{0.3}$ is defined by the half-
thickness of the disk being $H_{\rm disk}=R_{\rm disk}0.3h_{\rm 0.3}$ (Davis &
Tchekhovskoy, 2020). This maximum differs from that of Equation (27) by the
factor $1.3h_{0.3}a_{*}^{2}$. In the spectrally hard state, the disk is most
likely hot, in which case $h_{0.3}\sim 1$.

Figure 4: MCMC fit results for the seven model-independent quantities, which
require only the assumptions of $a=2b$, see Section 3.1. Here and in Figure 8
below, the panels show the histograms of the one-dimensional posterior
distributions for the model parameters, and the two-parameter correlations,
with the best-fitting values of the parameters indicated by green
lines/squares. The best-fit results for fitted quantities are taken as the
medians of the resulting posterior distributions, and are shown by the middle
vertical dashed lines in the distribution panels. The surrounding vertical
dashed lines correspond approximately to a $1\sigma$ uncertainty.

We can estimate $\phi_{\rm BH}$ using the magnetic field strength measured far
from the BH by using the conservation of the magnetic flux. Specifically, we
use the expected equality between the poloidal and toroidal field components
at the Alfvén surface (in the observer’s frame), which radius, for strongly
magnetized jets, approaches the light cylinder radius, $R_{\rm LC}$
(Lyubarsky, 2010). This implies $\Gamma\langle
B^{\prime}_{\phi}\rangle\approx(R/R_{\rm LC})B_{\rm p}$, where $B_{\rm p}$ is
the poloidal field (which has the same value in the comoving and BH frames)
and $\langle B^{\prime}_{\phi}\rangle$ is the average toroidal field strength
in the comoving frame, denoted by $B$ in the remainder of this paper. Then,
the toroidal field dominates at $z$ satisfying $R(z)\gg\Gamma R_{\rm LC}$,
and, presumably, at $z\geq z_{0}$. The magnetic flux using this method was
determined for a sample of radio loud active galactic nuclei in Zamaninasab et
al. (2014) and Z15. We use the resulting formula as derived in Z15,

$\Phi_{\rm j}={2^{3/2}\pi R_{\rm H}sz_{0}B_{0}(1+\sigma)^{1/2}\over\ell
a_{*}},$ (33)

which allows us to estimate $\phi_{\rm BH}$ for a given $a_{*}$ by setting
$\Phi_{\rm j}=\Phi_{\rm BH}$. Here $R_{\rm H}=[1+(1-a_{*}^{2})^{1/2}]R_{\rm
g}$ is the BH horizon radius, $\ell\lesssim 0.5$ is the ratio of the field and
BH angular frequencies, and $s$ is the scaling factor relation between the jet
opening angle and the bulk Lorentz factor (Komissarov et al., 2009;
Tchekhovskoy et al., 2009), limited by causality to $\lesssim 1$,

$\Theta\approx s\sigma^{1/2}/\Gamma.$ (34)

Here, $\sigma$ is the magnetization parameter, which is defined as the ratio
of the proper magnetic enthalpy to that for particles including the rest
energy,

$\sigma\equiv{B^{2}/4\pi\over\eta u_{\rm p}+\rho c^{2}}=\frac{2}{\beta_{\rm
eq}}\left[\eta+\frac{\mu_{\rm e}m_{\rm p}(1-2n_{+}/n_{\rm e})f_{N}}{m_{\rm
e}(f_{E}-f_{N})(1+k_{\rm i})}\right]^{-1},$ (35)

where $\rho$ is the rest-mass density, and $4/3<\eta<5/3$ is the particle
adiabatic index. The second equality relates $\sigma$ to $\beta_{\rm eq}$
assuming that the only ions are those associated with the power law electrons
(i.e., neglecting the possible presence of ions associated with electrons with
$\gamma<\gamma_{\rm min}$, e.g., with a quasi-Maxwellian distribution). For
$p>2$ and a large enough $\gamma_{\rm max}$, $f_{E}/f_{N}\approx\gamma_{\rm
min}(p-1)/(p-2)$. Then, for $\beta_{\rm eq}(1-2n_{+}/n_{\rm e})/\gamma_{\rm
min}\gg m_{\rm e}/m_{\rm p}$, we have $\sigma\ll 1$.

##  3 Application to MAXI J1820+070

Here, we apply the model of Section 2 to the source. We use the VLA fluxes at
5.25, 7.45, 8.5, 11.0, 20.9, 25.9 and the ALMA flux at 343.5 (from table 1 in
T21). We also use the IR flux at $1.5\times 10^{5}$ from VLT/HAWK-I, and the
optical flux at $3.9\times 10^{5}$ GHz from NTT/ULTRACAM (T21), and the 13
fluxes between 1.37 and $7.00\times 10^{5}$ GHz from the VLT X-shooter and the
INTEGRAL/OMC flux at $5.66\times 10^{5}$ GHz (Rodi et al., 2021). All of the
IR/optical fluxes have been de-reddened with $E(B\\!-\\!V)=0.18$ (Tucker et
al., 2018), as assumed in T21. We use the time lags between 25.9 GHz and lower
frequencies, 11.0 GHz and lower frequencies, and between 343.5 GHz and lower
frequencies. The lags are given in tables 3 and 4 of T21. This gives us 23
spectral measurements and 14 time lags. We present analytical and numerical
estimates in Sections 3.1 and 3.2, respectively. We assume the ratio between
the electron and magnetic energy densities to be constant along the jet, i.e.,
$a=2b$.

In our fits below, we use the Markov-Chain Monte Carlo (hereafter MCMC)
technique with wide uniform priors, see T21 for details. We assume
$i=64\arcdeg\pm 5\arcdeg$ (Wood et al., 2021) with a Gaussian prior, and
$D=2.96\pm 0.33$ kpc (Atri et al., 2020) with a Gaussian prior, but truncated
at the upper limit of $D_{\rm max}=3.11$ kpc found by Wood et al. (2021).

We assume the observed spectrum is from the approaching jet only. At $i\approx
64\arcdeg$, this assumption is satisfied only roughly. The ratio of the jet-
to-counterjet fluxes in the optically-thick part of the spectrum is given by
$[(1+\beta\cos i)/(1-\beta\cos i)]^{(7+3p)/(4+p)}$ (which follows from
Zdziarski et al. 2012), which is $\approx$7 at the fitted $p\approx 2.21$ (see
below).

###  3.1 The initial fit and analytical estimates

Table 1: The basic parameters of the jet in MAXI J1820+070.  $b$ | $p$ | $\nu_{0}$ | $F_{0}$ | $\nu_{\rm max}$ | $\alpha_{\rm disk}$ | $F_{\rm disk}$ | $t_{0}$
---|---|---|---|---|---|---|---
|  |  $10^{4}$ GHz | mJy |  $10^{4}$ GHz |  | mJy | s
$1.10^{+0.01}_{-0.01}$ | $2.21^{+0.22}_{-0.19}$ | $2.32^{+0.65}_{-0.60}$ | $298^{+31}_{-38}$ | $32.5^{+5.0}_{-4.4}$ | $0.54^{+0.13}_{-0.12}$ | $30.4_{-5.9}^{+6.0}$ | $0.79^{+0.30}_{-0.19}$

Figure 5: The radio-to-optical spectrum from T21 (VLA, ALMA, VLT/HAWK-I,
NTT/ULTRACAM; red error bars), the 339 MHz measurement (VLITE, magenta error
bar; Polisensky et al. 2018), and from the VLT/X-shooter (blue error bars) and
the INTEGRAL/OMC (cyan error bar) as obtained by Rodi et al. (2021), but with
the de-reddening correction for $E(B\\!-\\!V)=0.18$ (Tucker et al., 2018). The
error bars for the radio and sub-mm measurements of T21 are the square roots
of the squares of their statistical and systematic errors, and 10% systematic
errors are assumed for the IR and optical measurements. The spectrum above 5
GHz is fitted by the jet model of Equation (6) using the best-fit parameters
shown in Figure 4 (cyan dashed curve) and a phenomenological power law
approximating the disk component with $\alpha_{\rm disk}=0.53$ (green solid
curve). The former is virtually independent of $\xi_{\rm min}$ within the
ranges obtained in the full fits (Equation 10; Section 3.2), so it can be
assumed to be unity. The sum is shown by the solid black curve. The
corresponding asymptotic optically thick and optically thin spectra of
Equation (12) are shown by the magenta dotted lines.

Figure 6: The time lags measured by T21 vs. the theoretically expected
distance in units of $z_{0}$ between the emission points for the partially-
self-absorbed part of the spectrum. The blue and red symbols correspond to the
lags between 343.5 GHz and radio frequencies (5.25–25.9 GHz), and within the
radio frequencies, respectively, where we assumed a constant ratio between the
electron and magnetic energy densities, $\xi_{\nu}=(\nu/\nu_{0})^{-q}$,
$q\approx 0.88$, and $\nu_{0}=2.32\times 10^{4}$ GHz. The diagonal line gives
the best-fit theoretical relationship between the two quantities corresponding
to $t_{0}=0.79$ s, see text.

We find we can solve for $b$, $p$, $\nu_{0}$, $F_{0}$, $\nu_{\rm max}$ and
$t_{0}$ by only assuming $a=2b$. From the measured fluxes, we obtain $b$, $p$,
$\nu_{0}$, $F_{0}$ and $\nu_{\rm max}$ using Equations (6–7). We find the
fitted spectrum is very insensitive to $\xi_{\rm min}(\nu)$, Equation (10), as
long as it is low enough. We can just use any low value of it, or just assume
$\xi_{\rm min}=1$, and check a posteriori the consistency of the choice.
Similar to Rodi et al. (2021), we find the presence of an additional hard
component beyond $\nu_{\rm max}$, apparently due to the emission of an
accretion disk. Given the limited range of the fitted frequencies, we fit it
phenomenologically as a power law, $F_{\nu,\rm disk}=F_{\rm
disk}(\nu/10^{5}{\rm GHz})^{\alpha_{\rm disk}}$. Then, using the obtained
values of $b$, $p$ and $\nu_{0}$, we can fit the time lags using Equations
(19) and (22). However, with the MCMC technique, we fit the flux and time-lag
measurements simultaneously. The fitted parameters and the correlations
between them are shown in Figure 4, and the parameters are listed in Table 1.
The best-fitting values are given as the median of the resulting posterior
distributions, and the lower and upper uncertainties are reported as the range
between the median and the 15th percentile, and the 85th percentile and the
median, respectively. These uncertainties correspond approximately to
$1\sigma$ errors. We use these best-fit values as well as the best-fit values
of $D$ and $i$ in our estimates in this subsection.

Figure 5 shows the observed average radio-to-optical spectrum fitted by the
above model. The best-fitting spectral indices in the optically thick and
optically thin regimes are then $\alpha\approx 0.25$ and $\alpha_{\rm
thin}\approx-0.61$, respectively. We show the theoretical spectrum calculated
by integrating Equation (6) and the asymptotic optically thick and thin
spectra of Equation (12) for this fit.

We then show the time lags in Figure 6, where we plot the values of the
measured $\Delta t_{\rm a}$ against the separation in the dimensionless units,
$\xi$, using Equations (15) and (22). At the best-fit values of $b$ and $p$,
$q\approx 0.883$, see Equation (19). The actual lags have to follow a single
dependence relating the physical separation between the emission points to
$\Delta t_{\rm a}$, which is shown by the diagonal line showing the linear
relationship between $t_{\rm a}$ and $\Delta\xi_{\nu}$ corresponding to the
best-fit value of $t_{0}$, see Equation 22. We see a certain offset between
the points corresponding to the lags between the sub-mm frequency of 343.5 GHz
and 6 radio frequencies (blue error bars), and the lags measured between the
radio frequencies (red error bars). This may be related to the different
methods used in T21 to determine those. On the other hand, the offset is
significantly reduced for $q=0.8$, which value of $q$, however, is not
compatible with $\alpha\approx 0.25$. This may indicate that the jet is more
complex than we assume, e.g., either $\Gamma$, $\Theta$ or $b$ are not
constant at $z\geq z_{0}$.

Our formalism assumes the lags correspond to propagation of perturbations
between different values of $z_{\nu}$ at the jet speed, $\beta$. With this
assumption, we obtain $z_{0}$ as a function of $\Gamma$, see Equation (21),

$z_{0}\approx(2.37\times 10^{10}\,{\rm cm})(t_{0}/0.79\,{\rm
s})\beta\Gamma\delta.$ (36)

We then use the solutions obtained in Appendix A assuming $\gamma_{\rm
min}=3$, $k_{\rm i}=0$. However, we only know $\nu_{\rm max}$ rather than
$\gamma_{\rm max}$, see Equation (5). Since the solutions depend on
$\gamma_{\rm max}$ relatively weakly, we assume here the best-fit values of
$B_{0}=10^{4}$ G and $\Gamma=2.2$ obtained in Section 3.2 for $\gamma_{\rm
min}=3$, which yield $\gamma_{\rm max}=125$, which we will use hereafter in
this subsection. Using Equation (A5), we obtain at the best fit

$\Theta\approx\frac{2.32\arcdeg}{\left(\beta\Gamma\right)^{1.89}\delta^{2.57}\beta_{\rm
eq}^{0.11}}.$ (37)

At $\beta\approx 1$ and $\beta_{\rm eq}=1$, $\Theta\approx
0.53\arcdeg\Gamma^{0.67}$. Next, Equations (A6–A7) give at the best fit,

$\displaystyle B_{0}\approx\frac{8.2\times 10^{3}\,{\rm
G}(\beta\Gamma)^{0.22}}{\delta^{0.13}\beta_{\rm eq}^{0.22}},$ (38)
$\displaystyle n_{0}\approx\frac{1.8\times 10^{12}\,{\rm
cm}^{-3}(\beta\Gamma)^{0.43}\beta_{\rm eq}^{0.57}}{\delta^{0.26}},$ (39)

with $B_{0}\propto\Gamma^{0.35}$ and $n_{0}\propto\Gamma^{0.70}$ at
$\beta\approx 1$. Equation (37) shows that we cannot determine both $\Theta$
and $\delta$ even assuming a value of $\beta_{\rm eq}$ (on which $\Theta$
depends very weakly). We can also calculate the Thomson scattering optical
depth along the jet radius at $z\geq z_{0}$, which equals

$\tau_{\rm T}(\xi)=\sigma_{\rm
T}n_{0}f_{N}z_{0}\tan\Theta\xi^{1-2b}\approx\frac{2.5\times 10^{-4}\beta_{\rm
eq}^{0.46}}{(\beta\Gamma)^{0.46}\delta^{1.82}\xi^{1.2}}.$ (40)

At $i=64\arcdeg$ and $\Gamma=2$, 3, 4, we have $\delta\approx 0.81$, 0.57,
0.43, and, at $\beta_{\rm eq}=1$, $\Theta\approx 1.4\arcdeg$, $1.4\arcdeg$,
$1.5\arcdeg$, $B_{0}\approx 1.0,\,1.1,\,1.2\times 10^{4}$ G, $z_{0}\approx
3.3,\,3.8,\,4.0\times 10^{10}$ cm, and $\tau_{\rm T}(\xi=1)\approx
2.9,\,4.3,\,6.1\times 10^{-4}$, respectively. The values of $z_{0}$ correspond
to $\approx(2.8$–$3.3)\times 10^{4}R_{\rm g}$ at an assumed $M=8{\rm
M}_{\sun}$. We find that $\Theta$, $B_{0}$ and $z_{0}$ depend relatively
weakly on $\Gamma$ for $1.5\lesssim\Gamma\lesssim 5$.

We determine the typical Lorentz factors, $\gamma_{\nu}$, of relativistic
electrons giving rise to the emission at $\nu$, which in the partially self-
absorbed regime originates mostly from $z_{\nu}$, see Equation (16). We obtain

$\gamma_{\nu}\approx 32\beta_{\rm
eq}^{0.11}(\beta\Gamma)^{-0.11}\delta^{-0.43}(\nu/\nu_{0})^{0.014}.$ (41)

In order to obtain a power-law emission in that regime, we need $\gamma_{\rm
min}$ to be a factor of a few smaller. Thus, we require $\gamma_{\rm
min}\lesssim 10$ for the validity of the model. The maximum $\gamma$
corresponds to the fitted $\nu_{\rm max}$, Equation (5). From that, we obtain
$\gamma_{\rm max}$ ranging from $\approx$123 to 147 for $\Gamma$ increasing
from 2 to 4. Combining this with the values of $\tau_{\rm T}$ from Equation
(40), we find that the power in the synchrotron self-Compton component is
relatively similar to that in the synchrotron one, $P_{\rm
SSC}\lesssim\tau_{\rm T}\gamma_{\rm max}^{2}P_{\rm S}$.

Figure 7: The locations of the emission at the observed frequencies based on
the break in the power spectra as $z_{\rm b}=\beta c/f_{\rm break}$ for
$\Gamma=3$ and $i=64\arcdeg$, shown as their ratio to the locations based on
time lags and the slope of the partially self-absorbed spectrum,
$z_{\nu}\approx z_{0}(\nu/\nu_{0})^{-0.88}$.

We can then consider implications of the break frequencies, $f_{\rm b}$, in
the power spectra for different frequencies measured by T21. For those power
spectra, most of the variability power per $\ln f$ occurs at $f\leq f_{\rm
b}$, with the variability at higher frequencies strongly damped, see figs. 3
and 5 in T21. We define the distance, $z_{\rm b}$, as that covered by a jet
element moving with the jet velocity during the time222T21 assumed
$z_{\nu}=z_{\rm b}\equiv\beta c\delta/f_{\rm b}(\nu)$, which they used as the
final condition determining the jet parameters. Thus, they transformed the
observed variability frequency to the jet frame, $f_{\rm b}/\delta$, and
multiplied the resulting time scale, $\delta/f_{\rm b}$, by the jet velocity
in the observer’s frame, $\beta c$, which does not appear to be correct. We
note that in the present case we consider the light curve originating from a
fixed region of the jet around $z_{\nu}$. While the plasma in that region is
moving, two adjacent maxima in the observed light curve are emitted from the
same region in the frame connected to the BH, which is the same frame as the
observer’s one (in the absence of a redshift). Thus, a frequency inferred from
the variability power spectrum should not be transformed. $1/f_{\rm b}$,

$z_{\rm b}(\nu)\equiv\beta c/f_{\rm b}(\nu).$ (42)

We can compare it to the distance along the jet from the BH up to the location
of the peak emission at $\nu$, i.e., $z_{\nu}$ (Equations 15, 21). In our
model, $z_{\nu}\approx z_{0}(\nu/\nu_{0})^{-0.88}$ with
$z_{0}\propto\beta/(1-\beta\cos i)$, giving $z_{\rm b}/z_{\nu}\propto
1-\beta\cos i$. Then, this ratio depends only weakly on $\beta$ (or $\Gamma$);
at $i=64\arcdeg$, $1-\beta\cos i$ changes only from 1 at $\beta\ll 1$ to 0.56
at $\beta\approx 1$. This implies that this correlation cannot be used to
determine the actual bulk Lorentz factor of the jet.

Figure 7 shows $z_{\rm b}/z_{\nu}$ vs. $z_{\nu}$ for $\Gamma=3$. We see an
approximately constant ratio of $z_{\rm b}/z_{\nu}\approx 1.5$–2. Therefore,
$z_{\rm b}$ is proportional and close to the travel time along $z_{\nu}$ in
all of the cases. A possible explanation of the damping of the variability at
frequencies $>c/z_{\nu}$ appears to be a superposition of the contributions to
the emission from different parts of the region dominating at a given $\nu$,
which is $\propto z_{\nu}$, as shown in Figure 1. The peak of ${\rm
d}F_{\nu}/{\rm d}\,\ln z$ for $p=2.21$ is at $\approx 1.15z_{\nu}$ and its
width defined by ${\rm d}F_{\nu}/{\rm d}\,\ln z$ decreasing to the 50% of the
peak is $(0.65$–$3.16)z_{\nu}$. Thus, if different parts vary independently,
the variability will be damped at $f\gtrsim c/(2z_{\nu})$, as observed.

Alternatively, the observed radio/IR variability can be driven by the variable
power supplied from the vicinity of the BH with a wide range of frequencies
(Malzac, 2013, 2014) and then transferred upstream, the travel time can act as
a low-pass filter, removing most of the variability at frequencies $f>\beta
c/z_{\nu}$. This can happen due to damping of perturbations along the jet due
to some kind of internal viscosity, e.g., collisions between shells within the
jet moving with a range of velocities (Jamil et al., 2010). The process would
be then analogous to viscous damping in accretion disks, where modulations
with a period shorter than the signal travel time across the disk are strongly
damped (Zdziarski et al., 2009). This picture is also compatible with the
integrated fractional variability of the power spectra (RMS) decreasing with
the decreasing $\nu$ (as shown in fig. 5 of T21). This means increasing the
distance travelled along the jet leads to the increasing damping.

We note that the break frequencies in the power spectra of T21 have been
defined by choosing a specific, and not unique, algorithm, as well as the
obtained values of $f_{\rm b}$ are close to the minimum frequency at which the
power spectrum is measured for $f<10$ GHz, which limits the accuracy of the
determination of those $f_{\rm b}$. Also, while the damping of variability
above $\beta c/z_{\nu}$ clearly occurs, details of the physics behind it
remain uncertain, and the damping could start at $f\sim\beta c/(2z_{\nu})$
instead of exactly $\beta c/z_{\nu}$. Summarizing, our results are completely
compatible with the variability damping at time scales shorter than the
light/jet travel time across $z_{\nu}$. However, unlike our previous estimates
from the observed spectrum and time lags, which are based on a relatively
rigorous and well-understood model, the detailed cause of the connection
between the break frequencies and the distance along the jet remains
uncertain.

We can also consider the prediction of the location of the bulk of the 15 GHz
emission, $z_{\nu}\approx z_{0}(\nu/\nu_{0})^{-0.88}\approx 2.5\times 10^{13}$
cm (at $\Gamma=3$, but weakly dependent on it), with the jet angular size at
this frequency from the VLBA observation on 2018 March 16 (MJD 58193),
reported in T21 as $0.52\pm 0.02$ mas. the deprojected size is $(2.60\pm
0.10)\times 10^{13}$ cm. The total flux density at 15 GHz was measured as
$F_{\nu}\approx 20.0\pm 0.1$ mJy. However, the VLBA observation was 27 d
before the radio/sub-mm ones. On MJD 58220, our best-fit spectral model yields
$F_{\nu}\approx 56\pm 1$ mJy. Within the framework of the continuous conical
jet model, we have $z_{\nu}\propto F_{\nu}^{(p+6)/(2p+13)}$ (Equation A2;
Zdziarski et al. 2012). Thus, for $p=2.2$ we predict the size at 15 GHz on MJD
58220 being $(56/20)^{0.47}\approx 1.6$ times larger than that on MJD 58193,
namely $\sim 4\times 10^{13}$ cm. While somewhat larger than the above
$z_{\nu}$, this size appears consistent with it since the peak of ${\rm
d}F_{\nu}/{\rm d}\,\ln z$ for $p=2.21$ is at $\approx 1.15z_{\nu}\approx
3.0\times 10^{13}$ cm, and that spatial distribution is broad and skewed
toward higher distances, see Figure 1 and the discussion of it above.

We then estimate the rate of pair production. For MAXI J1820+070, pair
production within the hot plasma was calculated by Zdziarski et al. (2021)
based on the spectrum observed by INTEGRAL in the hard state. That spectrum
was measured up to $\sim$2 MeV, well above the pair production threshold of
511 keV, and modelled by Comptonization. It was found that an appreciable pair
abundance can be obtained only provided the hard X-ray source size is as small
as several $R_{\rm g}$, while the spectroscopy based on the relativistic
broadening of the fluorescent Fe K$\alpha$ line indicates a size of $\gtrsim
20R_{\rm g}$. Then, the pair abundance within the Comptonizing plasma is very
low.

However, as discussed in Section 2.3, pair production within the jet base can
be much more efficient. To calculate it, we adapt the results of Zdziarski et
al. (2021). We modify their equation (1) to calculate the photon density above
the hot disk, dividing the total rate of the photon emission by $2\pi R_{\rm
hot}^{2}$ (including both sides). We then use this photon density in equation
(3) of that paper for the spectral parameters of average spectrum (table 2 in
Zdziarski et al. 2021). This gives the pair production rate per unit volume.
With the assumptions as in Section 2.3, we have

$2\dot{N}_{+}\approx 4.65\times 10^{40}\,{\rm s}^{-1}\left(\frac{R_{\rm
hot}}{20R_{\rm g}}\right)^{-3}\left(\frac{R_{\rm jet}}{10R_{\rm
g}}\right)^{2}.$ (43)

This is then balanced by the sum of the rates of pair annihilation and pair
advection. Using formulae in Zdziarski et al. (2021), we have found that pair
annihilation can be neglected for the advection velocity of
$\beta_{\pm}\gtrsim 0.1$. It appears that such a velocity can be achieved due
to the net momentum component of the pair-producing photons along the $z$
axis, see Figure 3. Thus, while some of the produced pairs will annihilate
(and a small fraction will be advected to the BH), a major fraction of the
produced pairs will have a sufficient net bulk velocity to escape upstream.

Then, the lepton flow rate through the jet, Equation (25), for $\gamma_{\rm
min}=3$ is

$\dot{N}_{\rm e}\approx\frac{6.7\times 10^{40}{\rm s}^{-1}\beta_{\rm
eq}^{0.35}}{(\beta\Gamma)^{0.35}\delta^{3.39}}\propto\Gamma^{3.05},$ (44)

where Equations (36), (37), (39) have been used and the proportionality
assumes $\beta\approx 1$. Comparing with Equation (43), we find $\dot{N}_{\rm
e}>2\dot{N}_{+}$ at any $\Gamma$ for $R_{\rm hot}=20R_{\rm g}$, $R_{\rm
jet}=10R_{\rm g}$ and $\gamma_{\rm min}=3$. Thus, at these parameters the
synchrotron-emitting plasma is never composed of pure pairs. If we assume
either $R_{\rm jet}=15R_{\rm g}$ or $\gamma_{\rm min}=10$, we find
$\dot{N}_{\rm e}=2\dot{N}_{+}$ at $\Gamma\approx 2$, which thus represent the
minimum possible $\Gamma$ for these parameters. While the hot disk and jet
radii and $\gamma_{\rm min}$ are poorly constrained, we consider the fact that
the numbers in Equations (43) and (44), obtained with completely different
physical considerations, are of the same order of magnitude, to be highly
remarkable and indicating that indeed the two rates may be similar in this
source. Then, the jet can contain a large fractional abundance of pairs, and
they can dominate by number over the ions.

Figure 8: (a) The MCMC fit results for $\Gamma$, $\Theta$, $z_{0}$, $B_{0}$,
$P_{\rm j}$, $2n_{+}/n_{\rm e}$ and $\gamma_{\rm max}$ assuming $\gamma_{\rm
min}=3$ and $\epsilon_{\rm eff}=0.3$. The meaning of the panels and lines is
the same as in Figure 4. See Section 3.2 for details.

Figure 8: (b) The MCMC fit results for $\gamma_{\rm min}=10$ and $\epsilon_{\rm eff}=0.1$.  Table 2: The parameters of the jet in MAXI J1820+070 other than those given in Table 1.  $\gamma_{\rm min}$ | $\epsilon_{\rm eff}$ | $\Gamma$ | $\Theta$ | $\log_{10}z_{0}$ | $B_{0}$ | $\log_{10}P_{\rm j}$ | $\gamma_{\rm max}$
---|---|---|---|---|---|---|---
|  |  | $\arcdeg$ | cm |  $10^{4}$ G | erg s-1 |
3f | 0.3f | $2.20^{+0.69}_{-0.46}$ | $1.04_{-0.35}^{+0.48}$ | $10.63^{+0.09}_{-0.08}$ | $0.99^{+0.22}_{-0.18}$ | $38.31^{+0.32}_{-0.60}$ | $120^{+8}_{-11}$
10f | 0.1f | $3.10^{+1.03}_{-0.85}$ | $1.41_{-0.56}^{+0.47}$ | $10.57^{+0.10}_{-0.13}$ | $1.21^{+0.29}_{-0.22}$ | $38.66^{+0.37}_{-0.59}$ | $124^{+21}_{-14}$

Next, we calculate the jet power. The power in the relativistic electrons and
magnetic fields, Equation (23), becomes at $z_{0}$

$P_{B}+P_{\rm e}\approx 1.9\times 10^{36}{\rm erg\,s}^{-1}\frac{3+2\beta_{\rm
eq}}{6\beta_{\rm eq}^{0.65}}\frac{\Gamma^{0.65}}{\beta^{0.35}\delta^{3.39}}.$
(45)

which increases very fast with $\Gamma$, approximately as $\propto\Gamma^{4}$
at $\beta\approx 1$. At $\beta_{\rm eq}=1$, $\Gamma=3$, this power is $\approx
2.2\times 10^{37}$ erg s-1. We find from Equations (24), (26), the power
associated with the bulk motion of cold matter as

$\displaystyle P_{\rm i}\approx 1.2\times 10^{38}{\rm
erg\,s}^{-1}(\Gamma-1)\times$ (46) $\displaystyle\left[\frac{\beta_{\rm
eq}^{0.35}}{(\beta\Gamma)^{0.35}\delta^{3.39}}-0.7\left(\frac{R_{\rm
hot}}{20R_{\rm g}}\right)^{-3}\\!\\!\left(\frac{R_{\rm jet}}{10R_{\rm
g}}\right)^{2}\right]\\!.$

The first term is approximately $\propto\Gamma^{3}(\Gamma-1)$.

To constrain $P_{\rm j}$ by the accretion power, we use the estimate of the
hard-state bolometric flux of $F_{\rm bol}\approx 1.4\times 10^{-7}$ erg cm-2
s-1 (Shidatsu et al., 2019). This yields $L\approx 1.5(D/2.96\,{\rm
kpc})^{2}10^{38}$ erg s-1 and

$\dot{M}c^{2}\approx 1.5\times 10^{39}\left(D\over 2.96\,{\rm
kpc}\right)^{2}\left(\epsilon_{\rm eff}\over 0.1\right)^{-1}\,{\rm
erg\,s}^{-1}.$ (47)

For the default parameter values, $P_{\rm j}\lesssim\dot{M}c^{2}$ implies
$\Gamma\lesssim 3.3$. If pair production is efficient enough, we also have a
lower limit on $\Gamma$ from the requirement of $P_{\rm i}>0$. The allowed
range depends significantly on the assumed parameters, in particular
$\gamma_{\rm min}$, $R_{\rm hot}$ and $R_{\rm jet}$. E.g., at $\gamma_{\rm
min}=10$, $R_{\rm hot}=20R_{\rm g}$ and $R_{\rm jet}=10R_{\rm g}$,
$\Gamma\gtrsim 2.4$ is required.

We can then compare the total jet power, $P_{\rm j}$, with the synchrotron
power. At the low $\gamma_{\rm max}$ implied by the $\nu_{\rm max}$ fitted to
the spectrum, we find $P_{\rm S}\ll P_{\rm j}$ always. For example, $P_{\rm
S}\approx 0.009P_{\rm j}$ at the maximum allowed $\Gamma\approx 3.3$, and
$P_{\rm S}\approx 0.02P_{\rm j}$ at $\Gamma=2$. On the other hand, we have
found $P_{\rm S}\sim 0.5(P_{B}+P_{\rm e})(z_{0})$, weakly depending on either
$\Gamma$ or $\gamma_{\rm min}$. Thus, the synchrotron emission can be entirely
accounted for by the power in electrons and magnetic fields at $z_{0}$, and
most of the decline of $P_{B}+P_{\rm e}$ with the distance can be due to the
synchrotron losses. However, we may see that the decline of $(P_{B}+P{\rm e})$
with $\xi$ is slower than that of the synchrotron power. If the former would
be just to the synchrotron emission, we would have ${\rm d}(P_{B}+P{\rm
e})/{\rm d}\xi+{\rm d}P_{\rm S}/{\rm d}\xi=0$, while the former and the latter
terms are $\propto-\xi^{1-2b}$ and $\propto\xi^{2-4b}$. This implies either
some electron re-acceleration at $z>z_{0}$ at the expense of $P_{\rm i}$, or
more complexity of the actual physical situation, with the initial energy loss
in the flow being faster and followed by a slower one.

In the framework of models with the jet dissipation mechanism being the
differential collimation of poloidal magnetic surfaces, the obtained
$\Theta\Gamma\ll 1$ indicate the jet magnetization at $z\gtrsim z_{0}$ is low.
Using Equation (34), we have (at $\beta\approx 1$)

$\sigma=(\Theta\Gamma/s)^{2}\approx\frac{8.4\times
10^{-5}\Gamma^{3.35}}{\beta_{\rm eq}^{0.22}s^{2}}.$ (48)

At $\beta_{\rm eq}=1$ and assuming $s=0.6$ (as found as the average value for
a large sample of radio-loud AGNs by Pjanka et al. 2017), we obtain
$\sigma\approx 0.0093(\Gamma/3)^{3.35}$. This can be compared to $\sigma$ from
its definition, Equation (35), which equals,

$\sigma\approx\beta_{\rm eq}^{-1}\left[2/3+130(1-2n_{+}/n_{\rm
e})\right]^{-1}.$ (49)

In the absence of pairs, $\sigma\approx 0.0078/\beta_{\rm eq}$. Comparing the
two estimates of $\sigma$, we see it requires $\Gamma\gtrsim 3$ at $\beta_{\rm
eq}=1$. However, the actual value of $s$ is uncertain, there could be ions
associated with background electrons piled up at $\gamma<\gamma_{\rm min}$,
and, importantly, $\beta_{\rm eq}$ could be $\gg 1$. Still, the low
magnetization implied by Equation (48) disfavors the case of strong pair
dominance, $(1-2n_{+}/n_{\rm e})\ll 1$.

Using $\sigma\ll 1$, we can calculate the magnetic fluxes in the model with
extraction of the BH rotational power. The jet magnetic flux from Equation
(33) with $z_{0}B_{0}$ from Equations (36) and (38) is then

$\Phi_{\rm j}\approx(4.1\times 10^{21}\,{\rm
G\,cm}^{2})\frac{s[1+(1-a_{*}^{2})^{\frac{1}{2}}](\beta\Gamma)^{1.22}\delta^{0.87}}{(\ell/0.5)a_{*}\beta_{\rm
eq}^{0.22}},$ (50)

which is $\approx 5.3\times 10^{21}\,{\rm G\,cm}^{2}$ for $a_{*}=1$,
$\Gamma=3$, $\ell=0.5$, $s=0.6$, $\beta_{\rm eq}=1$. The flux threading the
BH, Equation (31) with $\dot{M}$ estimated as above from $L$, is

$\Phi_{\rm BH}\approx(1.3\times 10^{22}\,{\rm G\,cm}^{2})\frac{\phi_{\rm
BH}}{50}\frac{D}{3\,{\rm kpc}}\left(\frac{\epsilon_{\rm
eff}}{0.1}\right)^{-1/2},$ (51)

where $M=8{\rm M}_{\sun}$ was assumed for both ($\Phi\propto M$). At
$\phi_{\rm BH}=50$ and the assumed parameters, the two fluxes are
approximately equal for $a_{*}\approx 0.7$. We consider the close agreement of
the above two estimates to be very remarkable. They are based on completely
different physical considerations. Thus, our results are consistent with the
jet being powered by the BH rotation and the accretion flow being magnetically
arrested. In this case, the jet power is constrained by Equation (32). Then, a
low value of $a_{*}$ would constrain $\Gamma$ to values lower than those
implied by $P_{\rm j}\lesssim\dot{M}c^{2}$.

###  3.2 Numerical estimates

In order to solve directly for the physical jet parameters and their
uncertainties, we use again the MCMC method. In the fits shown in Figure 4, we
fitted $b$, $p$, $\nu_{0}$, $\nu_{\rm max}$, $F_{0}$, $t_{0}$, $F_{\rm disk}$
and $\alpha_{\rm disk}$ with the minimum assumption of $a=2b$, and, in
particular, without the need to specify the value of $\Gamma$. Now we fit for
all of the parameters. However, since the solution given in Appendix A is
given in terms of $\gamma_{\rm max}$ rather than $\nu_{\rm max}$, we fit for
the former (which yields $\nu_{\rm max}$ given the values of $\Gamma$, $i$ and
$B_{0}$, see Equation 5).

In particular, we determine $\Theta$ from Equation (A5), $z_{0}$ from Equation
(21) and $B_{0}$ from Equation (A6). That requires specifying $\Gamma$ (which
is then a free parameter) and $\gamma_{\rm min}$. We fix $\beta_{\rm eq}=1$
and $k_{\rm i}=0$. However, in order to be able to constrain $\Gamma$ rather
than have it entirely free, we include further constraints, using the pair
production rate of Equation (43) and requiring $2\dot{N}_{+}/\dot{N}_{\rm
e}\leq 1$ in Equation (26) and from the maximum possible jet power, $P_{\rm
j}\leq\dot{M}c^{2}$, Equations (23–27). These constraints require specifying
$R_{\rm hot}$ and $R_{\rm jet}$, the bolometric luminosity, $L$, and the
accretion efficiency, $\epsilon_{\rm eff}$. We then solve simultaneously for
all of the parameters, including $b$, $p$, $\nu_{0}$, $F_{0}$, $t_{0}$,
$F_{\rm disk}$ and $\alpha_{\rm disk}$. Those parameters have now values
similar to those shown in Figure 4, and we thus do not show them again.

In the solution, we sample $D$ and $i$ as described at the beginning of
Section 3. We assume $L=1.5\times 10^{38}$ erg s-1, $X=0.7$, $R_{\rm
hot}=20R_{\rm g}$, $R_{\rm jet}=10R_{\rm g}$ (for $M=8{\rm M}_{\sun}$). We
show the resulting posterior distributions for two cases with ($\gamma_{\rm
min}=3$, $\epsilon_{\rm eff}=0.3$), and with ($\gamma_{\rm min}=10$,
$\epsilon_{\rm eff}=0.1$), in Figures 8(a), (b), respectively, and list the
fitted parameters in Table 2. We see that the obtained ranges of $\Gamma$ and
$\Theta$ depend on those two sets of assumptions, being larger for for the
latter case. The allowed maximum jet power is $\propto\epsilon_{\rm
eff}^{-1}$, and then it is higher in case (b). On the other hand, the obtained
values of $z_{0}\approx 2$–$4\times 10^{10}$ cm and $B_{0}\approx 10^{4}$ G
depend relatively weakly on those assumptions. For the sake of brevity, we
have not shown the effect of changing the values of $R_{\rm hot}$ and $R_{\rm
jet}$. For example, for $R_{\rm hot}>20R_{\rm g}$, pair production will be
less efficient, which would in turn allow fewer leptons in the flow and lower
values of $\Gamma$, see Equations (43–44). Thus, we cannot conclusively rule
out values of $\Gamma\lesssim 1.5$. Then, values of $\Gamma$ higher than those
obtained above would be possible for $\epsilon_{\rm eff}<0.1$.

Figures 8(a–b) also show $\gamma_{\rm max}$ and the pair abundance,
$2n_{+}/n_{\rm e}$. The former ir relatively tightly constrained in the
$\approx 110$–150 range. The latter is strongly anticorrelated with the jet
power, being low at the maximum $P_{\rm j}$ and close to unity at the minimum
$P{\rm j}$, in agreement with our considerations in Section 3.1. We find the
synchrotron power, Equation (28), is typically $P_{\rm S}\sim 0.01P_{\rm j}$,
as in Section 3.1, and thus the jet radiative efficiency, $P_{\rm S}/P_{\rm
j}$, is low. In our fits, we have not used constraints from the break
frequencies in the power spectra and from the jet spatial extent measurement,
following our discussion in Section 3.1.

##  4 Discussion

###  4.1 Electron energy losses and re-acceleration

In our model, we parametrize the electron distribution as a power-law function
of the distance, and assume that distribution keeps a constant shape. Such a
situation requires the electron energy losses are moderate and satisfying
$\dot{\gamma}\propto\gamma$. We compare here the time scale for synchrotron
energy losses,

$t_{\rm syn}=\frac{6\pi m_{\rm e}c\xi^{2b}}{\sigma_{\rm T}B_{0}^{2}\gamma},$
(52)

with the adiabatic/advection time scale,

$t_{\rm ad}=\frac{3z_{0}\xi}{2\beta\Gamma c}$ (53)

(e.g., Z19). We consider the solution in Section 3.1 for $\Gamma=3$. At
$\gamma\approx 30$, which corresponds to the bulk of the partially self-
absorbed emission, $t_{\rm syn}$ is shorter than $t_{\rm ad}$ for $\xi\lesssim
3$, and it is $\approx$3 times shorter at $z_{0}$. This implies that electrons
responsible for the optically-thin part of the synchrotron emission have to be
re-accelerated above $z_{0}$.

Calculating the electron distribution self-consistently as a function of the
distance as well as accounting for the slope of the spectrum at $\nu<\nu_{0}$
is relatively complex, involving solving a kinetic equation with both losses
and spatial advection (e.g., Z19). This also requires taking into account
losses from Compton scattering of synchrotron photons as well as the reduction
of the electron energy loss rate due to self-absorption (Ghisellini et al.,
1988; Katarzyński et al., 2006). Such a model is beyond the scope of the
present work.

###  4.2 Comparison with other jet models of accreting black holes

The very long time lags found in T21 unambiguously show that the radio/sub-mm
emission originates at size scales several orders of magnitude higher than
$R_{\rm g}$. The time lags between $\nu_{1}$ and $\nu_{2}$ are found to be
approximately proportional to $\nu_{2}^{-1}-\nu_{1}^{-1}$. Knowing the break
frequency, $\nu_{0}$, above which the entire synchrotron emission is optically
thin, we can extrapolate this correlation and find the location corresponding
to $\nu_{0}$. This is found to be $z_{0}\sim 3\times 10^{4}R_{\rm g}$, with
the uncertainty of a factor of at most a few. This rules out jet models
predicting the onset of the synchrotron emission to in an immediate vicinity
of the BH, for example that described in Giannios (2005) (based on the model
of Reig et al. 2003).

The main independent study of the hard-state jet of MAXI J1820+070 is that by
Rodi et al. (2021). They had at their disposal only the spectral data. They
assumed $R/z=0.1$, corresponding to $\Theta=5.7\arcdeg$, which is much larger
than that found by us. They assumed $\Gamma=2.2$ following the result of
Bright et al. (2020) for the ejection during the hard-to-soft transition, but
that of the hard-state jet can be different. The jet model of Rodi et al.
(2021) is also different from ours, and considers an initial acceleration
event followed by synchrotron cooling assuming no adiabatic losses. Still,
they obtain relatively similar values of the distance of onset of electron
acceleration, $z_{0}\approx 2.8\times 10^{10}$ cm, and the magnetic field
strength at that distance, $B_{0}\approx 1.8\times 10^{4}$ G.

###  4.3 Other constraints and caveats

Our model is based on that of Blandford & Königl (1979) and Königl (1981),
which assumes uniform scaling of the emission regions, through the
coefficients $a$ and $b$. As we see in Figure 5, this model does not account
for the observed flux at 339 MHz, measured by Polisensky et al. (2018). This
hints for the decline of the energy content in the relativistic electrons and
magnetic field being initially faster (responsible for the emission closer to
$z_{0}$) and then slower (responsible for the emission farther away from
$z_{0}$). This would introduce more complexity in the modelling, and is beyond
the scope of this work. On the other hand, the flux at 339 MHz could be due to
another component, in particular a pair of radio lobes at the jet ends. An
assumption of our model is that the bulk of the emission at a given distance
in the partially self-absorbed part of the spectrum occurs at a $\nu$
corresponding to $\tau\approx 1$. As we have found out, this corresponds to
the synchrotron emission by electrons with $\gamma\sim 30$. If the minimum
Lorentz factor of the electron distribution were higher, $\gamma_{\rm
min}>30$, then the emission at a given distance in that part of the spectrum
would be dominated by the electrons at $\gamma_{\rm min}$ instead, with no
contribution from self-absorption.

We assumed the jet is already fully accelerated at $z_{0}$ and then does not
decelerate. This may be not the case, and the available data do not exclude
that. The jet model of Z19 allows for a variable $\Gamma$, and we could use
some parametrization of $\Gamma(z)$ and refit our data (as done in Zdziarski
2019 for another source). This would, however, introduce more free parameters,
and make the resulting fits less constrained than in the present case. We have
also considered the steady state, while variability has been observed.
However, the fractional variability was $\sim 0.3$ at the sub-mm range and
much less than that in the radio regime. Thus, the variability can be
considered as a small perturbation of the steady state.

We also use a $\delta$-function approximation to the synchrotron process,
which is a good approximation for power-law parts of the spectra, but becomes
less accurate at cutoffs, given the single-electron synchrotron spectrum is
quite broad (Ginzburg & Syrovatskii, 1965). We also assume the synchrotron
emission of a single electron is isotropic in the plasma frame, which is valid
for a tangled magnetic field, while we assume a toroidal field in some of our
equations. Furthermore, we assume a sharp cutoff in the electron distribution
at $\gamma_{\rm max}$. While this is not realistic, the actual form of the
cutoff depends on details of the acceleration process and is poorly
constrained. Thus, our determination of $\gamma_{\rm max}$ based on the
observed cutoff in the optical range is only approximate.

Then, we have used our self-consistent set of equations, in which the slope of
the partially self-absorbed part of the synchrotron spectrum is connected to
the rate of decline of the energy density along the jet. The latter determines
the relationship between the characteristic emitted frequency and the distance
(Equation 15), and thus the time-lag vs. frequency relation. A significant
discrepancy between the spectral slope and time lags vs. frequency was found
in Cyg X-1 (Tetarenko et al., 2019). In our case, the two are in an
approximate mutual agreement.

We have found that the break frequencies in the power spectra, $f_{\rm
b}(\nu)$, are compatible with the origin of the emission at $z_{\nu}$, which
are roughly equal to $\beta c/f_{\rm b}(\nu)$ for $\nu<\nu_{0}$. However, an
increasing $f_{\rm b}$ with increasing $\nu$ is also observed for the IR and
optical data (see fig. 5 of T21), for which $\nu>\nu_{0}$. In our jet model,
the emission at $\nu>\nu_{0}$ is the optically-thin synchrotron from the
entire part of the jet at $z>z_{0}$, which implies $z_{\nu>\nu_{0}}=z_{0}$.
Thus, we expect that the above scaling of $f_{\rm b}\propto z_{\nu}^{-1}$ no
longer holds at $\nu>\nu_{0}$. Then, the IR/optical variability at high
Fourier frequencies may be mostly due to electron energy losses and the re-
acceleration (see Section 4.1) rather than due to propagation of some
disturbances from $z<z_{0}$.

As shown in fig. 8 of T21, the optical and IR light curves are tightly
correlated, with no measurable lag ($-18^{+30}_{-50}$ ms), in spite of a
relatively large disk contribution in the optical range (at $3.7\times 10^{5}$
GHz), as shown in Figure 5. This shows the the disk contribution is constant
on the studied time scales, which is consistent with the rms variability in
the optical range reduced with respect to the IR one, see fig. 5 in T21. As
shown in T21, the upper limit on the lag is consistent with the synchrotron
energy losses at the magnetic field strength of $\sim 10^{4}$ G, which agrees
with our determination of $B_{0}$.

###  4.4 Relationship to core shifts

Time lags are closely related to core shifts, $\Delta\theta$, which are
angular displacements of the radio cores, observed at frequencies where the
synchrotron emission is partially self-absorbed. They are commonly found in
radio-loud active galactic nuclei (e.g., Pushkarev et al. 2012). The physical
cause of the physical displacement along the jet, $z_{\nu_{2}}-z_{\nu_{1}}$,
is the same for both the core shifts and time lags; only the methods to
determine it are different. Using equation (4) in Z15 and Equation (20), the
relationship of $\Delta\theta$ to $\Delta t_{\rm a}$ is

$\Delta\theta=\frac{\Delta t_{\rm a}\beta c(1+z_{\rm r})\sin i}{D(1-\beta\cos
i)}.$ (54)

We can then relate $\Delta t_{\rm a}$ to $z_{\nu}$, $z_{0}$ and $\nu_{0}$
using Equations (20–21) and (A2).

We can estimate $B_{0}$ using the core-shift method, but only assuming $a=2$,
$b=1$, which parameters have been assumed in published core-shift studies,
including Z15. Using equation (7) of Z15 and Equation (54), we obtain
$B_{0}\approx 1.0\times 10^{4}$ G at $p=2$, $\delta=1$ and $\beta_{\rm eq}=1$,
in a good agreement with our estimate of Equation (38). We can also obtain
$B_{0}$ from equation (8) in Z15 without specifying $\beta_{\rm eq}$.

##  5 Conclusions

We have based our study on the results of a multiwavelength campaign observing
MAXI J1820+070 when it was close to the peak of its luminous hard spectral
state, at $\sim$15% of the Eddington luminosity. We have used mostly the data
published in T21 as well as the IR/optical spectrum from Rodi et al. (2021).
Our main conclusions are as follows.

A major model-independent result of our study is the estimation of the
distances at which the jet emits below the observed break frequency, based on
the time lags between various frequencies measured by T21. These distances are
definitely several orders of magnitude higher than $R_{\rm g}$. By
extrapolating the observed approximate correlation of the time lags with the
differences between the photon wavelengths, the place where that emission
begins can be estimated to be at the distance of several tens of thousands of
$R_{\rm g}$ from the BH. This value of that distance also agrees with the
corresponding finding of Rodi et al. (2021), based on spectral modelling
alone.

We then use the classical model of Blandford & Königl (1979), as formulated in
detail in later works, to determine the parameters of the jet emitting in the
radio-to-optical range. The model assumes the hard inverted spectrum is due to
the superposition of locally-emitted spectra that are self-absorbed up to some
frequency and then are optically thin. Apart from some details, this is the
same model as that used by T21. The values of the jet parameters obtained by
us update those of T21, which suffered from some errors (see Section 1 and
Appendix A). Our analysis is also broader than that of T21, utilizing
constraints from the break frequency and the optically thin part of the
spectrum.

By applying the model to the data, we find we cannot uniquely determine the
jet bulk Lorentz factor, $\Gamma$, which then needs to be specified as a free
parameter. However, it can be constrained from above by the requirement of the
jet power being less than the accretion power. It can also be constrained from
below by estimating the e± pair production rate in the base of the jet and
comparing it to the flux of e± required to account for the observed
synchrotron emission. We then use a Bayesian MCMC method to determine all of
the jet parameters, and find the most likely range of
$1.7\lesssim\Gamma\lesssim 4.1$. We find the jet half-opening angle, $\Theta$
constrained to $\approx$0.6–$2\arcdeg$. The onset of the emission is at
$z_{0}\approx 3$–$4\times 10^{10}$ cm, where the magnetic field strength is
$B_{0}\approx 0.8$–$1.4\times 10^{4}$ G. The total jet power is between
$P_{\rm j}\sim 10^{37}$ and $\sim\\!10^{39}$ erg s-1. The jet composition is
strongly correlated with $P_{\rm j}$, being mostly pairs at the lower limit
and mostly e- and ions at the upper limit. The optical spectral data imply a
rather low value of the maximum electron Lorentz factor, of $\gamma_{\rm
min}\approx 110$–150.

In order to explain the possible presence of e± pairs in the jet, we calculate
the rate of pair production in the jet base in immediate vicinity of the hot
accretion flow. We use the measurement of a power-law spectral component
extending at least to 2 MeV in the same state of MAXI J1820+070. This rate
depends on the geometry, see Figure 3, but we find it to be of the same order
as the rate of the electron flow through the synchrotron-emitting part of the
jet, both being $\sim 10^{40}$–$10^{41}$ s-1. We find this coincidence to be a
strong argument for the presence of pairs in the hard-state jet of MAXI
J1820+070.

We also consider the possibility of the jet power being limited by the power
from the rotation of the BH in the presence of magnetically arrested accretion
flow. To test it, we calculate the magnetic flux of the jet in the emitting
region and that threading the BH. We find them to be very similar to each
other, of $\sim\\!10^{21}$ G cm2, which remarkable coincidence argues for that
scenario. Then, the jet is initially magnetic, Poynting-flux dominated, slow,
and not emitting, and accelerates and dissipates its energy only at large
distances, in agreement with our finding of the emission being far from the
BH.

We find the synchrotron emitting power being only a small fraction,
$\sim\\!10^{-2}$, of the total jet power. On the other hand, the synchrotron
power is very similar to either the electron or magnetic powers at the onset
of the dissipation ($z_{0}$), showing that decline of those powers with the
distance necessary to explain the observations can be due to the synchrotron
emission.

Finally, we show the correspondence between the methods to determine the jet
parameters based on time lags and radio core shifts. We give a formula
relating the lags and the angular displacements of the radio cores.

## Acknowledgments

We thank B. De Marco, M. Böttcher and P.-O. Petrucci for valuable comments,
and A. Tchekhovskoy for permission to use his plot of GRMHD simulations. We
acknowledge support from the Polish National Science Centre under the grants
2015/18/A/ST9/00746 and 2019/35/B/ST9/03944, and from the International Space
Science Institute (Bern). Support for this work was provided by NASA through
the NASA Hubble Fellowship grant #HST-HF2-51494.001 awarded by the Space
Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Inc., for NASA, under contract
NAS5-26555. This paper makes use of the following ALMA data:
ADS/JAO.ALMA#2017.1.01103.T. ALMA is a partnership of ESO (representing its
member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST
and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the
Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and
NAOJ. The National Radio Astronomy Observatory is a facility of the National
Science Foundation operated under cooperative agreement by Associated
Universities, Inc.

##  Appendix A The general solution

We provide here the general solution to the equations providing the jet
structure, given in Section 2.1. We first give the solution parametrized by
the equipartition parameter, $\beta_{\rm eq}$, but without utilizing the time-
lag constraint. This solution is analogous to that given by equations (28–29)
in Zdziarski et al. (2012), which is valid for $a=2$, $b=1$ only. Here, we
assume that equipartition holds along the entire emitting jet, i.e., $a=2b$;
otherwise, it would be artificial to impose it only at $z_{0}$. We note that
the solutions below are functions of $\gamma_{\rm max}$ (through $f_{E}$ and
$f_{N}$), while observationally we determine $\nu_{\rm max}$. The relation
between the two involves $B_{0}$, $\Gamma$ and $i$, see Equations (1) and (5).
As a consequence, explicit solutions in terms of $\nu_{\rm max}$ would be
rather complicated, and we do not provide them.

We determine $B_{0}$ by setting $n_{0}$ from Equation (14) equal to that
following from Equation (17), and then use $z_{\nu}=z_{0}(\nu_{0}/\nu)^{q}$
from Equation (15), finding

$\displaystyle B(z_{\nu})=\frac{m_{\rm
e}c}{e}\left(\frac{\pi}{\delta}\right)^{\frac{2+p}{6+p}}\left[\frac{3c(1+k_{\rm
i})(f_{E}-f_{N})\sin i}{\beta_{\rm
eq}C_{2}(p)z_{\nu}\tan\Theta}\right]^{\frac{2}{6+p}}$
$\displaystyle\times\left[2\nu(1+z_{\rm r})\right]^{\frac{4+p}{6+p}}$ (A1)

for $z_{\nu}\geq z_{0}$, $\nu\leq\nu_{0}$. Then, $B_{0}=B(z_{0})$. We then
substitute $B(z_{\nu})$ of Equation (A1) in the formula for $F_{\nu}$ in the
optically-thick case, Equation (12), which yields for $z_{\nu}\geq z_{0}$

$\displaystyle
z_{\nu}=\frac{1}{\delta^{\frac{4+p}{13+2p}}\nu}\left[\frac{c(1+k_{\rm
i})(f_{E}-f_{N})}{2\pi\beta_{\rm eq}}\right]^{\frac{1}{13+2p}}\times$
$\displaystyle\left[\frac{C_{2}(p)}{\sin
i}\right]^{\frac{5+p}{13+2p}}\\!\\!\left[\frac{F_{\nu}D^{2}}{m_{\rm
e}C_{1}(p)g_{bp}}\right]^{\frac{6+p}{13+2p}}\times$ (A2)
$\displaystyle\left(\frac{3}{\pi\tan\Theta}\right)^{\frac{7+p}{13+2p}}(1+z_{\rm
r})^{-\frac{19+3p}{13+2p}},$

where $g_{bp}$ follows from Equation (12) for $a=2b$,

$g_{bp}\equiv\Gamma_{\rm E}\left[\frac{b(p+5)-6}{b(p+6)-2}\right]/(4+b).$ (A3)

Then, $z_{0}=z_{\nu_{0}}$, at which $F_{\nu_{0}}=2F_{0}g_{bp}$, see Equation
(12)333Equation (A2) also provides the correct form of equation (5) in Heinz
(2006) for $p=2$, $b=1$. His equation should be multiplied by $\delta^{1/2}$
factor, which is due to that factor missing in his equation (1), which should
have accounted for the frame transformation from $\nu^{\prime}$ to $\nu$. That
incorrect model formulation was used in T21..

We can then substitute the above $z_{\nu}\,(\geq z_{0})$ into Equation (A1),

$\displaystyle B(\nu)=\nu\left[\frac{3C_{1}(p)g_{bp}(1+k_{\rm
i})^{2}(f_{E}-f_{N})^{2}\sin^{3}i}{C_{2}(p)^{3}\beta_{\rm
eq}^{2}D^{2}F_{\nu}\tan\Theta}\right]^{\frac{2}{13+2p}}$
$\displaystyle\times\frac{\pi^{\frac{7+2p}{13+2p}}2^{\frac{9+2p}{13+2p}}c^{\frac{17+2p}{13+2p}}m_{\rm
e}^{\frac{15+2p}{13+2p}}(1+z_{\rm
r})^{\frac{15+2p}{13+2p}}}{e\delta^{\frac{3+2p}{13+2p}}},$ (A4)

and $B_{0}=B(\nu_{0})$. We see that the above solutions are obtained without
specifying either $\nu_{0}$ or $z_{0}$. Also, the spatial index $b$ enters
only in the factor $g_{bp}$, and does not modify the functional dependencies.
Equations (A2) and (A4) are equivalent to equations (28–29) in Zdziarski et
al. (2012), which are for $a=2$, $b=1$, and differ only in the definition of
$\beta_{\rm eq}$ and by factors of the order of unity due to a slightly
different way of integrating the emission along the jet.

Next, we can use the independent determination of $z_{\nu}$ from the time
lags, $\Delta t_{\rm a}$. A single measured lag between the frequencies
$\nu_{2}$ and $\nu_{1}$ determines, via Equation (20),
$z_{\nu_{2}}-z_{\nu_{1}}$. This can be compared to the prediction using
$z_{\nu}$ of Equation (A2), which yields a constraint between $\Theta$ and
$\Gamma$. However, a single measurement of $\Delta t_{\rm a}$ has typically a
large error. We can combine them by fitting the relationship between $\Delta
t_{\rm a}$ vs. $z_{\nu_{2}}-z_{\nu_{1}}$. This can be done even when the break
frequency, $\nu_{0}$, is unknown. However, here it is known, and we find it
convenient to define $t_{0}$ by $\Delta t_{\rm
a}=t_{0}(z_{\nu_{2}}-z_{\nu_{1}})/z_{0}$, fitted to a number of measured lags.
This then implies $z_{0}=ct_{0}\beta\Gamma\delta$. We can set it equal to that
implied by Equation (A2), and solve for $\tan\Theta$ as a function of
$\Gamma$,

$\displaystyle\tan\Theta=\frac{3\left(\beta\Gamma\nu_{0}t_{0}\right)^{-\frac{13+2p}{7+p}}}{\pi^{\frac{8+p}{7+p}}\delta^{\frac{17+3p}{7+p}}}\\!\left[\frac{(1+k_{\rm
i})(f_{E}-f_{N})}{\beta_{\rm eq}}\right]^{\frac{1}{7+p}}\times$
$\displaystyle\left[\frac{2C_{2}(p)}{\sin
i}\right]^{\frac{5+p}{7+p}}\left[\frac{F_{0}D^{2}}{m_{\rm
e}c^{2}C_{1}(p)}\right]^{\frac{6+p}{7+p}}(1+z_{\rm
r})^{-\frac{19+3p}{7+p}}\\!.$ (A5)

Note a relatively strong dependence of $\Theta$ on $t_{0}$,
$\Theta\mathrel{\vbox{ \offinterlineskip\halign{\hfil$#$\cr\propto\cr\kern
2.0pt\cr\sim\cr\kern-2.0pt\cr}}}t_{0}^{-2}$. We can then insert this
$\tan\Theta$ into Equation (A4) to obtain

$\displaystyle B_{0}=\frac{2^{\frac{3+p}{7+p}}\pi^{\frac{5+p}{7+p}}(m_{\rm
e}\nu_{0})^{\frac{9+p}{7+p}}c^{\frac{11+p}{7+p}}}{e\delta^{\frac{p-1}{7+p}}}\times$
(A6) $\displaystyle\left[\frac{\beta\Gamma t_{0}C_{1}(p)(1+k_{\rm
i})(f_{E}-f_{N})\sin^{2}i}{F_{0}\beta_{\rm
eq}C_{2}(p)^{2}D^{2}}\right]^{\frac{2}{7+p}}\\!\\!(1+z_{\rm
r})^{\frac{11+p}{7+p}}\\!.$

We determine $n_{0}$ using the above $B_{0}$ in Equation (17),

$\displaystyle n_{0}=\frac{\nu_{0}^{\frac{18+2p}{7+p}}m_{\rm
e}^{\frac{11+p}{7+p}}}{2^{\frac{15+p}{7+p}}\delta^{\frac{2p-2}{7+p}}e^{2}}\left[\frac{c^{2}\beta\Gamma
t_{0}C_{1}(p)\sin^{2}i}{F_{0}C_{2}(p)^{2}D^{2}}\right]^{\frac{4}{7+p}}$
$\displaystyle\times\left[\frac{\pi\beta_{\rm eq}}{(1+k_{\rm
i})(f_{E}-f_{N})}\right]^{\frac{3+p}{7+p}}(1+z_{\rm
r})^{\frac{22+2p}{7+p}}\\!.$ (A7)

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