# On the consistency of the Lagrange multiplier method in classical mechanics

Nivaldo A. Lemos<EMAIL_ADDRESS>Marco Moriconi<EMAIL_ADDRESS>Instituto de Física, Universidade Federal Fluminense, Campus da Praia
Vermelha, Niterói, 24210-340, RJ, Brazil.

###### Abstract

Problems involving rolling without slipping or no sideways skidding, to name a
few, introduce velocity-dependent constraints that can be efficiently treated
by the method of Lagrange multipliers in the Lagrangian formulation of the
classical equations of motion. In doing so one finds, as a bonus, the
constraint forces, which must be independent of the solution of the equations
of motion, and can only depend on the generalized coordinates and velocities,
as well as time. In this paper we establish conditions the Lagrangian and the
constraints should obey in order to guarantee that the constraint forces can
be obtained consistently.

##  I Introduction

Sometimes the mathematical treatment of certain physical problems seems so
well established, so ingrained in the tradition of theoretical physics and
vindicated by so many examples, that it is widely believed to provide
physically correct answers in all instances. But even if a physical theory
appears to be definitely settled, a closer examination of its mathematical
foundations is not a waste of time because it may reveal overlooked conditions
for the theory to be free from physical inconsistencies. A case in point is
the application of the method of Lagrange multipliers in classical mechanics
to deal with a large class of constrained systems.

In classical mechanics one often encounters problems involving rolling without
slipping, such as that of a rolling coin on a slanting table Lemos  or a ball
on a rough surface of revolution English ; White , and also those entailing
the requirement of not skidding sideways, as in the case of a skate or a
bicycle Lemos ; Neimark , a two-wheeled cart driven by electrostatic forces
Gatland  or a snakeboard Janova . The restrictions that characterize the
allowed motions of these systems are velocity-dependent constraints, making
them a class of constraints worth our attention. The equations that express
these constraints are first-order ordinary differential equations that depend
linearly on the velocities associated with the coordinates chosen to describe
the configurations of the system. In most cases these constraints are
nonholonomic, and they are usually dealt with by the technique of Lagrange
multipliers. A bonus of this approach is that the constraint forces are also
found in the process of solving the equations of motion. Even holonomic
constraints can be advantageously tackled by this method when one is
interested in determining the constraint forces.

Forces are functions only of positions, velocities and time; they cannot
depend on accelerations or even higher derivatives of the positions Zangwill .
This imposes a physical consistency requirement on the method of Lagrange
multipliers that, as far as we know, is not discussed in the standard
textbooks. Here we state clearly what this consistency requirement is and find
conditions under which the technique of Lagrange multipliers is physically
consistent.

##  II The Lagrange multiplier method in classical mechanics

Consider an $n$-degree-of-freedom mechanical system described by coordinates
$q_{1},\ldots,q_{n}$ subject to a set of $p<n$ mutually independent
differential constraints that depend linearly on the velocities, namely

$\sum_{k=1}^{n}\,a_{lk}dq_{k}+\alpha_{l}dt=0\,,\,\,\,\,\,\,\,\,\,\,l=1,\ldots,p\,.$
(1)

The coefficients $a_{lk}$ and $\alpha_{l}$ do not depend on the velocities;
that is, they are functions of $q_{1},\ldots,q_{n},t$ alone:
$a_{lk}=a_{lk}(q,t),\,\alpha_{l}=\alpha_{l}(q,t)$. In order to ensure the
mutual independence of the above constraint equations we assume that the
$p\times n$ rectangular matrix

$\boldsymbol{\mathsf{A}}=\left(\begin{array}[]{ccccc}a_{11}&a_{12}&a_{13}&\cdots&a_{1n}\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\
a_{p1}&a_{p2}&a_{p3}&\cdots&a_{pn}\end{array}\right)$ (2)

has rank $p$ for all $q,t$. This means that, with a suitable numbering of the
coordinates,

$\det\left(\begin{array}[]{ccc}a_{11}&\cdots&a_{1p}\\\ \vdots&\vdots&\vdots\\\
a_{p1}&\cdots&a_{pp}\end{array}\right)\neq 0\,\,\,\,\,\mbox{for
all}\,\,\,\,\,q,t.$ (3)

By way of illustration, for an upright coin rolling without slipping on a
horizontal plane the constraints are Lemos

${\dot{x}}-R{\dot{\phi}}\cos\theta=0,\,\,\,\,\,\,\,\,\,\,{\dot{y}}-R{\dot{\phi}}\sin\,\theta=0.$
(4)

With $q_{1}=x,q_{2}=y,q_{3}=\phi,q_{4}=\theta$ and upon multiplication of
these constraint equations by $dt$, it follows that

$\boldsymbol{\mathsf{A}}=\left(\begin{array}[]{cccc}1&\,\,\,\,0&-R\cos\theta&\,\,\,\,0\\\
0&\,\,\,\,1&-R\sin\theta&\,\,\,\,0\end{array}\right).$ (5)

This is a coordinate-dependent matrix whose rank is $2$ for all $q,t$.

Equation (3), combined with the fact that transposition does not change the
determinant of a matrix, has the following important consequence: if
$\boldsymbol{\mathsf{s}}$ is a $p$-dimensional column vector and
$\boldsymbol{\mathsf{A}}^{T}\boldsymbol{\mathsf{s}}=0$ then
$\boldsymbol{\mathsf{s}}=0$. This fact will be used in the proof of our main
result in Section IV.

Let $L(q,{\dot{q}},t)$ be the Lagrangian for the system, written as if there
were no constraints. The Lagrange multiplier method to take into account the
constraints (1) in Hamilton’s principle Lemos  starts by noting that the
variations $\delta q_{k}$ that enter Hamilton’s variational principle are
virtual displacements that, because of (1), obey

$\sum_{k=1}^{n}\,a_{lk}\delta q_{k}=0\,,\,\,\,\,\,\,\,\,\,\,l=1,\ldots,p\,$
(6)

inasmuch as $dt=0$ for virtual displacements. Equation (3) implies that the
first $p$ variations $\delta q_{1},\dots,\delta q_{p}$ can be expressed in
terms of the remaining ones $\delta q_{p+1},\dots,\delta q_{n}$, which are
independent and arbitrary. The introduction of Lagrange multipliers to take
the subsidiary conditions (6) into account in Hamilton’s principle leads to
the equations of motion Lemos

$\frac{d}{dt}\Bigl{(}\frac{\partial
L}{\partial{\dot{q}_{k}}}\Bigr{)}-\frac{\partial L}{\partial
q_{k}}=\sum_{l=1}^{p}\,{\lambda}_{l}a_{lk}\,,\,\,\,\,\,\,\,\,\,\,k=1,\ldots,n\,,$
(7)

where $\lambda_{1},\ldots,\lambda_{p}$ are Lagrange multipliers.

Of course, the differential constraints (1) can be written in the equivalent
form of a set of $p$ first-order ordinary differential equations:

$\sum_{k=1}^{n}\,a_{lk}{\dot{q}}_{k}+\alpha_{l}=0\,,\,\,\,\,\,\,\,\,\,\,l=1,\ldots,p\,.$
(8)

Equations (7) and (8) comprise a set of $n+p$ equations for $n+p$ unknowns —
the $n$ coordinates $q_{1},\ldots,q_{n}$ and the $p$ Lagrange multipliers
$\lambda_{1},\ldots,\lambda_{p}$ — which uniquely determine the motion of the
system once initial conditions obeying the constraints are stipulated. In
terms of the Lagrange multipliers, which are determined in the process of
solving equations (7) and (8), the generalized constraint forces
$Q^{\prime}_{k}$ responsible for the enforcement of (8) are given by Lemos

$Q^{\prime}_{k}=\sum_{l=1}^{p}\,{\lambda}_{l}a_{lk}\,,\,\,\,\,\,\,\,\,\,\,k=1,\ldots,n\,.$
(9)

The technique of Lagrange multipliers can be extended to nonholonomic
constraints of the more general form

$g_{l}(q,{\dot{q}},t)=0\,,\,\,\,\,\,\,\,\,\,\,l=1,\ldots,p\,.$ (10)

In this case the accepted equations of motion are Rund ; Saletan

$\frac{d}{dt}\Bigl{(}\frac{\partial
L}{\partial{\dot{q}_{k}}}\Bigr{)}-\frac{\partial L}{\partial
q_{k}}=\sum_{l=1}^{p}\,{\lambda}_{l}\frac{\partial
g_{l}}{\partial{\dot{q}}_{k}}\,,\,\,\,\,\,\,\,\,\,\,k=1,\ldots,n\,.$ (11)

These equations of motion define the d’Alembert-Lagrange theory of
nonholonomic systems. It is clear that equations (11) reduce to the previous
equations of motion (7) when the functions $g_{l}$ depend linearly on the
velocities. All of our forthcoming results remain valid for this more general
case by just putting $a_{lk}=\partial g_{l}/\partial{\dot{q}}_{k}$, despite
the fact that the $a_{lk}$ will depend on the velocities, as long as it is
assumed that the matrix $\boldsymbol{\mathsf{A}}=(\partial
g_{l}/\partial{\dot{q}}_{k})$ has rank $p$ for all $q,{\dot{q}},t$.

It is also worth pointing out that the physical Lagrange multiplier method to
deal with nonholonomic constraints in classical mechanics Rund ; Saletan ;
Flannery1 ; Flannery2 ; Flannery3  leads to equations of motion that are
different from those derived in the treatment of the corresponding
mathematical problem of Lagrange in the calculus of variations Bliss . The
latter gives rise to the so-called vakonomic mechanics Ray ; Goedecke ; Kozlov
; Borisov , which features ${\dot{\lambda}_{l}}$ in the equations of motion in
addition to the Lagrange multipliers $\lambda_{l}$ themselves. The reader is
referred to the Appendix for a short overview of this controversial topic.

##  III A consistency requirement

In Newtonian mechanics the most general forces are functions of coordinates,
velocities and time, and this must be true of the constraint forces (9).
Therefore, without having to explicitly solve for the motion, it should be
possible to express the Lagrange multipliers solely in terms of
$q,{\dot{q}},t$. The accelerations cannot appear in the solution for the
Lagrange multipliers.

Particular examples Lemos ; Neimark  show that by combining the equations of
motion (7) with the constraint equations (8) one manages to express
$\lambda_{1},\ldots,\lambda_{p}$ in terms of $q,{\dot{q}},t$ only. Thus, for
consistency, one must prove that this can be achieved in general. To this end,
let us start by writing the equations of motion in the explicit form

$\sum_{j=1}^{n}\frac{\partial^{2}L}{\partial{\dot{q}}_{j}\partial{\dot{q}}_{k}}\,{\ddot{q}}_{j}+\sum_{j=1}^{n}\frac{\partial^{2}L}{\partial
q_{j}\partial{\dot{q}}_{k}}\,{\dot{q}}_{j}+\frac{\partial^{2}L}{\partial
t\partial{\dot{q}}_{k}}-\frac{\partial L}{\partial
q_{k}}=\sum_{l=1}^{p}\,{\lambda}_{l}a_{lk}\,,\,\,\,\,\,\,\,k=1,\ldots,n\,.$
(12)

We assume that we are dealing with a regular Lagrangian, which means that the
Hessian matrix $\boldsymbol{\mathsf{W}}=(W_{jk})$ with elements

$W_{kj}=W_{jk}=\frac{\partial^{2}L}{\partial{\dot{q}}_{j}\partial{\dot{q}}_{k}}\,$
(13)

is nonsingular:

$\det\boldsymbol{\mathsf{W}}\neq 0\,\,\,\,\,\,\,\,\,\,\mbox{for
all}\,\,\,\,\,\,\,\,\,\,(q,\dot{q},t)\in{\cal D},$ (14)

where ${\cal D}$ is the Lagrangian’s domain of definition. This condition is
sufficient to allow the usual Lagrange’s equations to be solved for the
accelerations and put in the standard form
${\ddot{q}}_{k}=f_{k}(q,{\dot{q}},t)$. Then, if (14) holds and the Lagrangian
is smooth, the existence and uniqueness theorems of the theory of ordinary
differential equations apply and establish that once we fix the coordinates
and velocities at some initial instant Lagrange’s equations uniquely determine
the motion at least locally in time, a physical requisite one cannot do
without. It is also worth noting that condition (14) is all that is required
in order for the Hamiltonian formalism associated with the Lagrangian $L$ to
be locally well defined Lemos . To ensure the global existence of the
Hamiltonian formalism a stronger condition is needed: the Lagrangian should be
hyperregular Marsden .

Equations (12) can be conveniently recast as

$\sum_{l=1}^{p}\,(-{\lambda}_{l})a_{lk}+\sum_{j=1}^{n}W_{kj}{\ddot{q}}_{j}=b_{k}\,,\,\,\,\,\,\,\,k=1,\ldots,n\,,$
(15)

where $b_{k}=b_{k}(q,{\dot{q}},t)$.

Differentiating the constraint equations (8) with respect to time we arrive at

$\sum_{k=1}^{n}\,a_{lk}{\ddot{q}}_{k}=c_{l}\,,\,\,\,\,\,\,\,\,\,\,l=1,\ldots,p\,,$
(16)

where $c_{l}=c_{l}(q,{\dot{q}},t)$. In the following, all that matters is the
fact that $b_{k}$ and $c_{l}$ depend only on coordinates and velocities, not
their specific functional form.

Let $\boldsymbol{\mathsf{B}}$ be the $(n+p)\times(n+p)$ matrix defined in
terms of the matrices $\boldsymbol{\mathsf{A}}$ and $\boldsymbol{\mathsf{W}}$
by

$\boldsymbol{\mathsf{B}}=\left(\begin{array}[]{lc}\boldsymbol{\mathsf{O}}&\boldsymbol{\mathsf{A}}\\\
\boldsymbol{\mathsf{A}}^{T}&\boldsymbol{\mathsf{W}}\\\ \end{array}\right)$
(17)

where $\boldsymbol{\mathsf{O}}$ is the $p\times p$ zero matrix. Let us also
introduce the $(n+p)$-dimensional column vectors

$\boldsymbol{\mathsf{v}}=\left(\begin{array}[]{c}-\lambda_{1}\\\ \vdots\\\
-\lambda_{p}\\\ {\ddot{q}}_{1}\\\ \vdots\\\
{\ddot{q}}_{n}\par\end{array}\right)\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\boldsymbol{\mathsf{e}}=\left(\begin{array}[]{c}c_{1}\\\
\vdots\\\ c_{p}\\\ b_{1}\\\ \vdots\\\ b_{n}\par\end{array}\right)\,.$ (18)

In terms of these column vectors and the matrix (17), the set of $n+p$
equations (15) and (16) can be summarized as

$\boldsymbol{\mathsf{B}}\boldsymbol{\mathsf{v}}=\boldsymbol{\mathsf{e}}\,.$
(19)

Note that both the matrix $\boldsymbol{\mathsf{B}}$ and the vector
$\boldsymbol{\mathsf{e}}$ depend only on $q,{\dot{q}},t$. Therefore, if
$\boldsymbol{\mathsf{B}}$ is a nonsingular matrix, the unique solution of
equation (19) for $\boldsymbol{\mathsf{v}}$ is
$\boldsymbol{\mathsf{v}}=\boldsymbol{\mathsf{B}}^{-1}\boldsymbol{\mathsf{e}}$,
which gives $\lambda_{1},\dots,\lambda_{p}$ as functions of $q,{\dot{q}},t$
alone. Incidentally, the accelerations are also determined in the form
${\ddot{q}}_{k}=f_{k}(q,{\dot{q}},t)$, which, if the Lagrangian as well as the
constraint coefficients $a_{lk}$ and $\alpha_{l}$ are smooth functions,
guarantees existence and uniqueness for the motion at least locally in time
once initial coordinates and velocities obeying the constraints are specified.

It follows that in order for the method of Lagrange multipliers in classical
mechanics to yield physically meaningful results one must demand that

$\det\boldsymbol{\mathsf{B}}\neq 0\,\,\,\,\,\,\,\,\,\,\mbox{for
all}\,\,\,\,\,\,\,\,\,\,(q,{\dot{q}},t)\in{\cal D}$ (20)

as an essential consistency requirement. To be precise, the condition
$\det\boldsymbol{\mathsf{B}}\neq 0$ ensures that the Lagrange multipliers are
physically sensible functions of $q,\dot{q},t$.

Note that Eq.(20) is only a sufficient condition: if it fails it may still be
possible to determine the Lagrange multipliers in terms of $q,{\dot{q}},t$
only. But even if this is the case, $\det\boldsymbol{\mathsf{B}}=0$ implies
that the $n+p$ equations (19) for the $\lambda$s and ${\ddot{q}}$s are not
mutually independent. This leads to the appearence of at least one equation
relating only coordinates and velocities, meaning that some initial conditions
cannot be satisfied by the general solution to the equations of motion. In
order to justify this assertion, let us start by noting that the matrix
$\boldsymbol{\mathsf{B}}$ defined by equation (17) is symmetric. Suppose
$\det\boldsymbol{\mathsf{B}}=0$ for all values of $q,{\dot{q}},t$ in some set
$\cal O$. Then there exists at least one $(n+p)$-dimensional nonzero vector
$\boldsymbol{\mathsf{u}}(q,{\dot{q}},t)$ such that
$\boldsymbol{\mathsf{B}}\boldsymbol{\mathsf{u}}=0$ on $\cal O$, which is
equivalent to

$\boldsymbol{\mathsf{u}}^{T}\boldsymbol{\mathsf{B}}=0\,\,\,\,\,\,\,\,\,\,\mbox{for
all}\,\,\,\,\,\,\,\,\,\,(q,{\dot{q}},t)\in{\cal O},$ (21)

where the symmetry of $\boldsymbol{\mathsf{B}}$ has been used. Combining the
above equation with
$\boldsymbol{\mathsf{B}}\boldsymbol{\mathsf{v}}=\boldsymbol{\mathsf{e}}$ we
are led to

$\boldsymbol{\mathsf{u}}^{T}\boldsymbol{\mathsf{e}}=\boldsymbol{\mathsf{u}}^{T}\boldsymbol{\mathsf{B}}\boldsymbol{\mathsf{v}}=0\,\,\,\,\,\,\,\,\,\,\mbox{for
all}\,\,\,\,\,\,\,\,\,\,(q,{\dot{q}},t)\in{\cal O}.$ (22)

In terms of components, the condition
$\boldsymbol{\mathsf{u}}^{T}\boldsymbol{\mathsf{e}}=0$ on $\cal O$ reads

$\sum_{r=1}^{n+p}u_{r}(q,{\dot{q}},t)e_{r}(q,{\dot{q}},t)=0\,\,\,\,\,\,\,\,\,\,\mbox{for
all}\,\,\,\,\,\,\,\,\,\,(q,{\dot{q}},t)\in{\cal O}.$ (23)

Unless it is identically satisfied owing to an accident, this equation
provides an additional constraint. A consequence of this additional constraint
is that some allowed initial conditions cannot be fulfilled, which is a
physical inconsistency. The ocurrence of this state of affairs will be shown
explicitly by means of Example 1 in Section V.

In short, even though it is possible to have physically consistent Lagrange
multipliers without (20) being satisfied, failure to satisfy (20) is almost
certain to bring about some inconsistency.

The Lagrangians associated with most mechanical systems are not only regular
but also smooth functions of $q,{\dot{q}},t$. This smoothness allied to the
regularity condition (14) is sufficient to make sure that Lagrange’s equations
in their usual form locally yield a unique motion under standard initial
conditions. It is also enough for the derivation of a Hamiltonian and
Hamilton’s equations from the given Lagrangian $L$, albeit only locally.
Therefore, it would be quite natural and reasonable to expect that it is also
sufficient for the technique of Lagrange multipliers to lead to physically
sensible results, that is, for the validity of (20). In the following we show
that this is not the case.

##  IV A consistency proof

The standard Lagrangian in classical mechanics is of the form $L=T-V$, where
$T$ is a positive quadratic form in the variables
${\dot{q}}_{1},\ldots,{\dot{q}}_{n}$ and $V$ does not depend on the
velocities. This implies that the associated Hessian matrix
$\boldsymbol{\mathsf{W}}$ is not only real, symmetric and nonsingular, but
also positive, since, on physical grounds, the kinetic energy is positive for
any set of velocities, that is,

$\boldsymbol{\mathsf{t}}^{T}\boldsymbol{\mathsf{W}}\boldsymbol{\mathsf{t}}>0\,\,\,\,\,\,\,\mbox{if}\,\,\,\,\,\boldsymbol{\mathsf{t}}\neq
0\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\boldsymbol{\mathsf{t}}^{T}\boldsymbol{\mathsf{W}}\boldsymbol{\mathsf{t}}=0\,\,\Longleftrightarrow\,\,\boldsymbol{\mathsf{t}}=0,$
(24)

where $\boldsymbol{\mathsf{t}}$ is an $n$-dimensional column vector. It is to
be noted that a real symmetric positive matrix is automatically nonsingular,
since its eigenvalues are all real and greater than zero. Moreover, its
inverse is also positive, since the eigenvalues of the inverse matrix are just
the reciprocals of the eigenvalues of the original positive matrix.

The previous remarks, and the significant fact that the method of Lagrange
multipliers is successful in all known physical examples, lead us to suspect
that one cannot prove in general that the matrix (17) is nonsingular unless
the Hessian matrix is assumed to be positive.

Proposition Let $p$ and $n$ be positive integers such that $p<n$. If the
$p\times n$ matrix $\boldsymbol{\mathsf{A}}$ has rank $p$ and the $n\times n$
matrix $\boldsymbol{\mathsf{W}}$ is positive then the $(n+p)\times(n+p)$
matrix $\boldsymbol{\mathsf{B}}$ defined by equation (17) is invertible.

Proof In order to prove that $\boldsymbol{\mathsf{B}}$ is invertible we must
show that $\det{\boldsymbol{\mathsf{B}}\neq 0}$, that is,
$\boldsymbol{\mathsf{B}}$ has no zero eigenvalue. We show this by proving that
$\boldsymbol{\mathsf{B}}\boldsymbol{\mathsf{v}}=0$ implies
$\boldsymbol{\mathsf{v}}=0$.

Let us write $\boldsymbol{\mathsf{v}}$ as

$\boldsymbol{\mathsf{v}}=\left(\begin{array}[]{c}\boldsymbol{\mathsf{s}}\\\
\boldsymbol{\mathsf{t}}\end{array}\right)$ (25)

where $\boldsymbol{\mathsf{s}}$ and $\boldsymbol{\mathsf{t}}$ are
$p$-dimensional and $n$-dimensional vectors, respectively. The equation
$\boldsymbol{\mathsf{B}}\boldsymbol{\mathsf{v}}=0$ becomes

$\displaystyle\boldsymbol{\mathsf{A}}\boldsymbol{\mathsf{t}}=0;$ (26)
$\displaystyle\boldsymbol{\mathsf{A}}^{T}\boldsymbol{\mathsf{s}}+\boldsymbol{\mathsf{W}}\boldsymbol{\mathsf{t}}=0.$
(27)

Since $\boldsymbol{\mathsf{W}}$ is invertible, applying
$\boldsymbol{\mathsf{s}}^{T}\boldsymbol{\mathsf{A}}\boldsymbol{\mathsf{W}}^{-1}$
on (27) and using (26), we obtain

$\boldsymbol{\mathsf{s}}^{T}\boldsymbol{\mathsf{A}}\boldsymbol{\mathsf{W}}^{-1}\boldsymbol{\mathsf{A}}^{T}\boldsymbol{\mathsf{s}}=0,$
(28)

which, together with the positivity of $\boldsymbol{\mathsf{W}}^{-1}$, implies
$\boldsymbol{\mathsf{A}}^{T}\boldsymbol{\mathsf{s}}=0$. As pointed out in
Section II, since $\boldsymbol{\mathsf{A}}$ has rank $p$, this implies
$\boldsymbol{\mathsf{s}}=0$ which, together with (27), implies
$\boldsymbol{\mathsf{t}}=0$. $\square$

Thus, if the conditions stated in this proposition are fulfilled, the Lagrange
multipliers are uniquely determined as functions only of $q,{\dot{q}},t$ and
so are the constraint forces (9).

The proof of the above proposition shows only that the positivity hypothesis
on $\boldsymbol{\mathsf{W}}$ together with the full rank of the constraint
matrix $\boldsymbol{\mathsf{A}}$ guarantees that the Lagrange multipliers can
be expressed as functions of $q,{\dot{q}},t$ and nothing else. But is the
positivity of $\boldsymbol{\mathsf{W}}$ a condition one can go without? It
turns out that if the positivity of $\boldsymbol{\mathsf{W}}$ is relinquished
we run the risk of getting in trouble, as discussed in the next section.

##  V Status of the positivity of the Hessian matrix

In the typical examples of physical interest the Hessian matrix
$\boldsymbol{\mathsf{W}}$ is positive. Let us investigate what can happen when
$\boldsymbol{\mathsf{W}}$ is not positive by means of the study of two
specific cases. Example 1 illustrates the case alluded to before in which
$\det\boldsymbol{\mathsf{B}}=0$, but the multipliers can still be solved for
in terms of coordinates, velocities and time (but other inconsistencies
arise). Example 2 illustrates a case in which the Hessian matrix is non-
positive but physically consistent results for the Lagrange multipliers can
still be obtained.

Our first example shows that, as a rule, the positivity hypothesis on
$\boldsymbol{\mathsf{W}}$ cannot be dispensed with.

Example 1. Let the Lagrangian for a three-degree-of-freedom system be

$L=\frac{{\dot{q}}_{1}^{2}}{2}+\frac{{\dot{q}}_{2}^{2}}{2}+{\dot{q}}_{1}{\dot{q}}_{2}+{\dot{q}}_{2}{\dot{q}}_{3}-\frac{1}{2}(q_{1}^{2}+q_{2}^{2}+q_{3}^{2})\,.$
(29)

For the corresponding Hessian matrix we have

$\boldsymbol{\mathsf{W}}=\left(\begin{array}[]{ccc}1&1&0\\\ 1&1&1\\\
0&1&0\end{array}\right)\,.$ (30)

This matrix is invertible since its determinant equals $-1$, which means that
the Lagrangian (29) is regular. However, it is not positive. Indeed, since
$\boldsymbol{\mathsf{W}}$ is real and symmetric, its eigenvalues are all real,
and they cannot be all positive, for their product is $-1$.

Let us assume that the system is subject to the two constraints

${\dot{q}}_{1}+q_{3}=0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\dot{q}}_{2}-q_{3}=0,$
(31)

which can be readily shown to be nonholonomic by an appeal to the Frobenius
theorem Lemos . It follows that the constraint matrix is

$\boldsymbol{\mathsf{A}}=\left(\begin{array}[]{ccc}1&0&0\\\
0&1&0\end{array}\right)$ (32)

and it is clear that $\boldsymbol{\mathsf{A}}$ has rank $2$. The matrix
$\boldsymbol{\mathsf{B}}$ constructed from $\boldsymbol{\mathsf{A}}$ and
$\boldsymbol{\mathsf{W}}$ according to equation (17) is

$\boldsymbol{\mathsf{B}}=\left(\begin{array}[]{ccccc}0&0&1&0&0\\\ 0&0&0&1&0\\\
1&0&1&1&0\\\ 0&1&1&1&1\\\ 0&0&0&1&0\end{array}\right)$ (33)

It is easy to verify that $\det\boldsymbol{\mathsf{B}}=0$. However, the
Lagrange multipliers can be expressed in terms of coordinates and velocities
alone. In order to check this, we write the equations of motion (7) for the
system:

$\displaystyle{\ddot{q}}_{1}+{\ddot{q}}_{2}+q_{1}$ $\displaystyle=$
$\displaystyle\lambda_{1},$ (34)
$\displaystyle{\ddot{q}}_{1}+{\ddot{q}}_{2}+{\ddot{q}}_{3}+q_{2}$
$\displaystyle=$ $\displaystyle\lambda_{2}\,$ (35)
$\displaystyle{\ddot{q}}_{2}+q_{3}$ $\displaystyle=$ $\displaystyle 0.$ (36)

From the constraint equations (31) we find that
${\ddot{q}}_{1}+{\ddot{q}}_{2}=0$, which yields $\lambda_{1}=q_{1}$. We find,
then, that $\lambda_{2}={\ddot{q}}_{3}+q_{2}$. Differentiating the second of
constraint equations (31) with respect to time we have
${\ddot{q}}_{2}={\dot{q}}_{3}$, as consequence of which equation (36) becomes
${\dot{q}}_{3}+q_{3}=0$. Therefore, ${\ddot{q}}_{3}=-{\dot{q}}_{3}$ implying
$\lambda_{2}=-{\dot{q}}_{3}+q_{2}$. Both Lagrange multipliers have been
expressed solely in terms of coordinates and velocities. This shows that if
(20) fails it still may be possible to express the multipliers in terms of
coordinates and velocities only. However, the failure of (20) brings about a
different kind of predicament pointed out in Section III. We have just noticed
that the coordinate $q_{3}$ obeys ${\dot{q}}_{3}+q_{3}=0$, whose general
solution is $q_{3}(t)=Ce^{-t}$. Substituting this into the constraint
equations (31) and performing trivial integrations we arrive at the general
solution to the equations of motion:

$\displaystyle q_{1}(t)$ $\displaystyle=$ $\displaystyle A+Ce^{-t},$ (37)
$\displaystyle q_{2}(t)$ $\displaystyle=$ $\displaystyle B-Ce^{-t},$ (38)
$\displaystyle q_{3}(t)$ $\displaystyle=$ $\displaystyle Ce^{-t}.$ (39)

Note that the general solution to the equations of motion contains only three
arbitrary constants, $A,B,\,C$, but there are four initial conditions to be
satisfied, since, due to the constraints (31), the initial velocities
${\dot{q}}_{1}(0)$ and ${\dot{q}}_{2}(0)$ are given by $-q_{3}(0)$ and
$q_{3}(0)$, respectively. In particular, initial conditions such that
$q_{3}(0)=0$ and ${\dot{q}_{3}}(0)\neq 0$ cannot be satisfied.

In this example, the appearence of the additional constraint
${\dot{q}}_{3}+q_{3}=0$ illustrates the general discussion in Section III
about what can happen when $\det\boldsymbol{\mathsf{B}}=0$. With conceivable
but probably rare exceptions, the failure of condition (20) implies that the
method of Lagrange multipliers is not fully consistent.

The possibility remains that the constraints give rise to a nonsingular matrix
$\boldsymbol{\mathsf{B}}$ in spite of the non-positivity of
$\boldsymbol{\mathsf{W}}$. The next example illustrates exactly such a case.

Example 2. Consider the system described by the Lagrangian

$L=\frac{{\dot{q}}_{1}^{2}}{2}+\frac{{\dot{q}}_{2}^{2}}{2}-\frac{{\dot{q}}_{3}^{2}}{2}-\frac{1}{2}(q_{1}^{2}+q_{2}^{2}+q_{3}^{2})$
(40)

now under the single nonholonomic constraint

${\dot{q}}_{1}+q_{3}{\dot{q}}_{2}=0.$ (41)

The Hessian matrix is $\boldsymbol{\mathsf{W}}=\mbox{diag}\,(1,1,-1)$. The
constraint matrix is

$\boldsymbol{\mathsf{A}}=\left(\begin{array}[]{ccc}1&q_{3}&0\end{array}\right),$
(42)

and its rank is $1$ everywhere. Now we have

$\boldsymbol{\mathsf{B}}=\left(\begin{array}[]{cccc}0&1&q_{3}&0\\\ 1&1&0&0\\\
q_{3}&0&1&0\\\ 0&0&0&-1\end{array}\right)$ (43)

with $\det\boldsymbol{\mathsf{B}}=1+q_{3}^{2}$, which is everywhere
nonvanishing. Therefore, the Lagrange multiplier can be determined exclusively
in terms of the coordinates and velocities. Let us show how to do this
explicitly. In the present case the equations of motion (7) take the following
form:

$\displaystyle{\ddot{q}}_{1}+q_{1}$ $\displaystyle=$ $\displaystyle\lambda,$
(44) $\displaystyle{\ddot{q}}_{2}+q_{2}$ $\displaystyle=$
$\displaystyle\lambda q_{3},$ (45) $\displaystyle-{\ddot{q}}_{3}+q_{3}$
$\displaystyle=$ $\displaystyle 0.$ (46)

From the constraint equation (41) it follows that

${\ddot{q}}_{1}=-q_{3}{\ddot{q}}_{2}-{\dot{q}}_{2}{\dot{q}}_{3}.$ (47)

Now insert this into Eq.(44) and sum the resulting equation with Eq.(45)
previously multiplied by $q_{3}$ to obtain

$\lambda=\frac{q_{1}+q_{2}q_{3}-{\dot{q}}_{2}{\dot{q}}_{3}}{1+q_{3}^{2}}.$
(48)

From our main result in Section IV and these two examples one infers that the
positivity of the Hessian matrix $\boldsymbol{\mathsf{W}}$ together with the
full rank of the constraint matrix $\boldsymbol{\mathsf{A}}$ is a sufficient
condition that guarantees the consistency of the method of Lagrange
multipliers in classical mechanics, but it is not a necessary condition
because the method can be consistent even though $\boldsymbol{\mathsf{W}}$ is
not positive.

The previous examples are obviously contrived. Is there any physical theory in
which there appears a Lagrangian with a nonpositive Hessian matrix? In fact,
there is. In Weber’s electrodynamics, which has some adherents today, the
generalized potential (in Gaussian units) that yields the force on a moving
point charge $e$ exerted by a point charge $e^{\prime}$ fixed at the origin is
Whittaker

$U(r,{\dot{r}})=\frac{ee^{\prime}}{r}\Bigl{(}1+\frac{{\dot{r}}^{2}}{2c^{2}}\Bigr{)}.$
(49)

The Lagrangian in spherical coordinates is

$\displaystyle L$ $\displaystyle=$
$\displaystyle\frac{m}{2}({\dot{r}}^{2}+r^{2}{\dot{\theta}}^{2}+r^{2}\,{\dot{\phi}}^{2}\sin^{2}\theta\,)-U(r,{\dot{r}})$
(50) $\displaystyle=$
$\displaystyle\frac{m}{2}\bigg{(}1-\frac{ee^{\prime}}{mc^{2}r}\bigg{)}{\dot{r}}^{2}+\frac{m}{2}r^{2}{\dot{\theta}}^{2}+\frac{m}{2}r^{2}\,{\dot{\phi}}^{2}\sin^{2}\theta\,-\frac{ee^{\prime}}{r}.$

The Hessian matrix associated with this Lagrangian is not positive if

$r\leq\frac{ee^{\prime}}{mc^{2}}.$ (51)

For these values of $r$ this theory gives rise to physical inconsistencies
Whittaker .

As we have just seen, if $\boldsymbol{\mathsf{W}}$ is not positive the method
of Lagrange multipliers can be consistent, in the sense that the constraint
forces are uniquely determined as functions solely of $q,{\dot{q}},t$.
Nonetheless, a Lagrangian with a nonpositive Hessian matrix must be shunned,
for it is almost certainly bound to give rise to some other kind of physical
inconsistency

##  VI Conclusions

The interplay between mathematics and physics has long and often been the
cause of wonder and pondering for prominent mathematicians and theoretical
physicists. Wigner Wigner  justifiably marveled at the fact that “the
mathematical formulation of the physicist’s often crude experience leads in an
uncanny number of cases to an amazingly accurate description of a large class
of phenomena.” Dirac Dirac  ascribed the remarkable success of the method of
mathematical reasoning in physics to some mathematical quality in nature. To
him the principle of mathematical beauty plays a central role in the setting
up of physical theories. V. I. Arnold Arnold  viewed mathematics as a natural
science, and famously stated that “mathematics is the part of physics where
experiments are cheap.”

It is not always appreciated that scrutinizing the mathematical foundations of
a physical theory may help identify conditions that must be imposed to add
physical content to the mathematical setting. In order to put a physical
theory on a rigorous mathematical basis it is not enough to explicitly state
conditions under which every problem admits a unique solution. Mathematical
existence does not by itself guarantee that the solution is physically
reasonable. Existence and uniqueness of a physical solution may require
conditions additional to those that ensure mathematical existence and
uniqueness. Rigorous mathematical formulations are helpful to clarify the
physics, and the physics assists in suggesting the most suitable mathematical
framework.

We have presented a sufficient condition, namely equation (20), that the
Lagrangian and the constraints must satisfy in order to guarantee that we can
solve for the Lagrange multipliers and obtain the physical constraint forces.
We have also cogently argued that the failure of (20) is nearly certain to
preclude the method of Lagrange multipliers from being fully consistent.
Moreover, through the examination of an important physical topic we hope to
have shown that, even though our physical intuition is fundamental for the
treatment of different classes of problems, there are cases in which a careful
mathematical analysis is needed to show the consistency, or lack thereof, of
our own intuition.

###### Acknowledgements.

We would like to thank the anonymous referees and the editor Todd Springer for
their careful reading and constructive criticism, which helped improve our
paper considerably.

## Appendix: Vakonomic mechanics

Vakonomic mechanics Ray ; Goedecke ; Kozlov ; Borisov  is an approach to the
dynamics of systems subject to velocity-dependent constraints of the general
form (10) that differs from the d’Alembert-Lagrange theory described in
Section II. Its inspiration comes from the mathematical problem of Lagrange in
the calculus of variations Bliss . The equations of motion of vakonomic
mechanics arise from the variational principle

$\delta\int_{t_{1}}^{t_{2}}\Bigl{[}L(q,{\dot{q}},t)+\sum_{l=1}^{p}\lambda_{l}(t)g_{l}(q,{\dot{q}},t)\Bigr{]}dt=0,$
(52)

in which the $q$s and $\lambda$s are varied independently, with the standard
boundary conditions $\delta q_{k}(t_{1})=\delta q_{k}(t_{2})=0$. Variation of
the Lagrange multipliers leads to the constraint equations (10), whereas
variation of the coordinates yields the equations of motion

$\frac{d}{dt}\Bigl{(}\frac{\partial
L}{\partial{\dot{q}_{k}}}\Bigr{)}-\frac{\partial L}{\partial
q_{k}}=\sum_{l=1}^{p}\,{\lambda}_{l}\bigg{[}\frac{\partial g_{l}}{\partial
q_{k}}-\frac{d}{dt}\Bigl{(}\frac{\partial
g_{l}}{\partial{\dot{q}_{k}}}\Bigr{)}\bigg{]}-\sum_{l=1}^{p}{\dot{\lambda}_{l}}\frac{\partial
g_{l}}{\partial{\dot{q}_{k}}}\,,\,\,\,\,\,\,\,\,\,\,k=1,\ldots,n\,.$ (53)

For holonomic constraints

$f_{l}(q,t)=0,\,\,\,\,\,\,\,\,\,\,l=1,\ldots,p,$ (54)

we have

$g_{l}(q,{\dot{q}},t)=\frac{d}{dt}f_{l}(q,t)=\sum_{j=1}^{n}\frac{\partial
f_{l}}{\partial q_{j}}{\dot{q}}_{j}+\frac{\partial f_{l}}{\partial t},$ (55)

and it is easy to check that

$\frac{\partial g_{l}}{\partial q_{k}}-\frac{d}{dt}\Bigl{(}\frac{\partial
g_{l}}{\partial{\dot{q}_{k}}}\Bigr{)}=0.$ (56)

This reduces the vakonomic equations of motion (53) to

$\frac{d}{dt}\Bigl{(}\frac{\partial
L}{\partial{\dot{q}_{k}}}\Bigr{)}-\frac{\partial L}{\partial
q_{k}}=-\sum_{l=1}^{p}{\dot{\lambda}_{l}}\frac{\partial
g_{l}}{\partial{\dot{q}_{k}}}\,,\,\,\,\,\,\,\,\,\,\,k=1,\ldots,n\,.$ (57)

Since only ${\dot{\lambda}_{l}}$ appear in these equations but not the
Lagrange multipliers $\lambda_{l}$ themselves, the mere renaming
${\dot{\lambda}_{l}}\to-\lambda_{l}$ shows that the equations of motion of
vakonomic mechanics coincide with (11) if the constraints are holonomic.

However, if the constraints are nonholonomic, equations (53) and (11) are
completely different. Although mathematical conditions are known Fernandez
under which equations (53) and (11) can be regarded as equivalent, the
presence of both $\lambda_{l}$ and ${\dot{\lambda}_{l}}$ in the equations of
motion of vakonomic mechanics means that in order to determine the motion
uniquely one has to specify $\lambda_{l}(0)$, the values of the Lagrange
multipliers at the initial instant $t=0$. This entails, given that the
Lagrange multipliers are directly connected to the constraint forces, that in
general, though not always, the initial accelerations would have to be
prescribed together with the initial positions and velocities, which is not
physically sound. Therefore, the physical significance of vakonomic mechanics,
if any, remains obscure Flannery2 ; Flannery3 . Moreover, experimental studies
Lewis ; Kai  favor (11) as the correct equations of motion for nonholonomic
systems.

This brief critical exposition of vakonomic mechanics is justified because
there is an ongoing controversy concerning its physical validity. Moreover, to
add to the confusion, equations (53) were sanctioned in the third edition of
Goldstein’s highly influential textbook Goldstein  but retracted without
explanation by the new co-authors almost as soon as it came to light
Goldstein2 .

## References

  * (1) N. A. Lemos, Analytical Mechanics (Cambridge University Press, Cambridge, 2018).
  * (2) L. Q. English and A. Mareno, “Trajectories of rolling marbles on various funnels,” Am. J. Phys. 80, 996-1000 (2012).
  * (3) G. D. White, “On trajectories of rolling marbles in cones and other funnels,” Am. J. Phys. 81, 890-898 (2013).
  * (4) Ju. I. Neĭmark and N. A. Fufaev, Dynamics of Nonholonomic Systems (American Mathematical Society, Providence, RI, 1972).
  * (5) I. R. Gatland, “Nonholonomic constraints: A test case,” Am. J. Phys. 72, 941-942 (2004).
  * (6) J. Janová and J. Musilová, “The streetboard rider: an appealing problem in non-holonomic mechanics,” Eur. J. Phys. 31, 333–345 (2010).
  * (7) Thus we are excluding self-interaction forces that arise in electromagnetism, such as the Abraham-Lorentz force. See, for example, A. Zangwill, Modern Electrodynamics (Cambridge University Press, Cambridge, 2013).
  * (8) H. Rund, The Hamilton-Jacobi Theory in the Calculus of Variations (Van Nostrand, London, 1966).
  * (9) E. J. Saletan and A. H. Cromer, “A variational principle for nonholonomic systems,” Am. J. Phys. 38, 892-897 (1970).
  * (10) M. R. Flannery, “The enigma of nonholonomic constraints,” Am. J. Phys. 73, 265-272 (2005).
  * (11) M. R. Flannery, “d’Alembert–Lagrange analytical dynamics for nonholonomic systems,” J. Math. Phys. 52, 032705 (2011).
  * (12) M. R. Flannery, “The elusive d’Alembert-Lagrange dynamics of nonholonomic systems,” Am. J. Phys. 79, 932-944 (2011).
  * (13) G. A. Bliss, “The problem of Lagrange in the calculus of variations,” Amer. J. Math. 52, 673-744 (1930).
  * (14) J. R. Ray, “Nonholonomic Constraints,” Am. J. Phys. 34, 406–408 (1966); Erratum 34, 1202–1203 (1966).
  * (15) G. H. Goedecke, “Undetermined multiplier treatments of the Lagrange problem,” Am. J. Phys. 34, 571–574 (1966).
  * (16) V. V. Kozlov, “Realization of nonintegrable constraints in classical mechanics,” Sov. Phys. Dokl. 28, 735-737 (1983).
  * (17) A. V. Borisov, I. S. Mamaev and I. A. Bizyaev, “Dynamical systems with non-integrable constraints, vakonomic mechanics, sub-Riemannian geometry, and non-holonomic mechanics,” Russ. Math. Surv. 72, 783-840 (2017).
  * (18) J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, 2nd ed. (Springer, New York, 1999).
  * (19) E. T. Whittaker, A History of the Theories of Aether and Electricity, Vol. I: The Classical Theories (Thomas Nelson and Sons, London, 1951), Chap. VII.
  * (20) E. Wigner, “The unreasonable effectiveness of mathematics in the natural sciences,” Commun. Pure Appl. Math. 13, 1-14 (1960).
  * (21) P. A. M. Dirac, “The Relation between Mathematics and Physics,” Proc. Royal Soc. (Edinburgh) 59, 122-129 (1938-39).
  * (22) V. I. Arnold, “On teaching mathematics,” address at the Palais de Découverte in Paris, March 7, 1997. Russian Math. Surveys 53, 229-236 (1998). https://www.uni-muenster.de/Physik.TP/$\%$7Emunsteg/arnold.html.
  * (23) O. E. Fernandez and A. M. Bloch, “Equivalence of the dynamics of nonholonomic and variational nonholonomic systems for certain initial data,” J. Phys. A: Math. Theor. 41, 344005 (2008).
  * (24) A. D. Lewis and R. M. Murray, “Variational principles for constrained systems: Theory and experiment,” Int. J. Non-Linear Mechanics 30, 793-815 (1995).
  * (25) T. Kai, “Experimental comparison between nonholonomic and vakonomic mechanics in nonlinear constrained systems,” Nonlinear Theory and Its Applications 4, 482-499 (2013).
  * (26) H. Goldstein, C. P. Poole, Jr. and J. L. Safko, Classical Mechanics, 3rd ed. (Addison Wesley, San Francisco, 2001).
  * (27) http://astro.physics.sc.edu/Goldstein/1-2-3To6.html. At this website, devoted to errata, corrections and comments on the third edition, co-author J. L. Safko states that Section 2.4 should be revised and cites, among others, reference Saletan  above.


# The Composition and Power of the Jet of the Broad-Line Radio Galaxy 3C 120

Andrzej A. Zdziarski Nicolaus Copernicus Astronomical Center, Polish Academy
of Sciences, Bartycka 18, PL-00-716 Warszawa, Poland<EMAIL_ADDRESS>Dakalo
G. Phuravhathu Centre for Space Research, North-West University, Potchefstroom
2520, South Africa  Marek Sikora Nicolaus Copernicus Astronomical Center,
Polish Academy of Sciences, Bartycka 18, PL-00-716 Warszawa, Poland;
<EMAIL_ADDRESS>Markus Böttcher Centre for Space Research, North-West
University, Potchefstroom 2520, South Africa  James O. Chibueze Centre for
Space Research, North-West University, Potchefstroom 2520, South Africa
Department of Physics and Astronomy, Faculty of Physical Sciences, University
of Nigeria, Carver Building, 1 University Road, Nsukka 410001, Nigeria

###### Abstract

We calculated the electron-positron pair production rate at the base of the
jet of 3C 120 by collisions of photons from the hot accretion flow using the
measurement of its average soft gamma-ray spectrum by Compton Gamma Ray
Observatory. We found that this rate approximately equals the flow rate of the
leptons emitting the observed synchrotron radio-to-IR spectrum of the jet
core, calculated using the extended jet model following Blandford & Königl.
This coincidence shows the jet composition is likely to be pair-dominated. We
then calculated the jet power in the bulk motion of ions, and found it greatly
exceeds that achievable by the magnetically arrested disk scenario for the
maximum black hole spin unless the jet contains mostly pairs. Next, we found
that the magnetic flux through the synchrotron-emitting jet equals the maximum
poloidal flux that can thread the black hole. Finally, we compared two
estimates of the magnetization parameter at the onset of the synchrotron
emission and found they are in agreement only if pairs dominate the jet
content.

##  1 Introduction

We consider possible mechanisms to produce electron-positron (e±) pairs at the
base of relativistic jets. There are a lot of indications that extragalactic
jets contain e± pairs dominating their content by number (e.g., Ghisellini
2012; Pjanka et al. 2017; Snios et al. 2018; Sikora et al. 2020; Liodakis et
al. 2022). However, it is not clear what the mechanism producing those pairs
is.

One possibility is pair production in the ergosphere of a rotating black hole
(BH; Blandford & Znajek 1977). If the charge density there is low enough, a
strong electric field forms a ‘spark gap’, where electrons are accelerated to
relativistic energies. These emit energetic $\gamma$-rays by Compton-
scattering ambient soft photons. The $\gamma$-rays in turn produce e± pairs in
collisions with ambient photons. However, the electric field will be screened
when the charge density in the ergosphere becomes higher than the Goldreich-
Julian (Goldreich & Julian, 1969) density, $n_{\rm GJ}=\Omega B/2\pi ec$,
where $\Omega$ is the BH angular velocity, $B$ is the strength of the magnetic
field of the magnetosphere and $e$ is the electron charge. The pairs are
produced above the gap (see fig. 1 in Levinson & Rieger 2011) and thus the
density there can exceed $n_{\rm GJ}$ by some factor, which Levinson & Rieger
(2011) estimate to be $\lesssim\\!10^{3}$. As found to follow from that
estimate, this mechanism can be be efficient in low-luminosity sources
(Levinson & Rieger, 2011; Mościbrodzka et al., 2011). On the other hand, the
electron density close to the BH in blazars has been estimated to be much
larger, $\sim\\!(10^{10}$–$10^{13})n_{\rm GJ}$ (Nokhrina et al., 2015). Here
we accounted for the synchrotron-emitting electrons only, without counting any
background electrons, which Nokhrina et al. (2015) also included. Even without
those electrons, it appears unlikely that such high electron densities can be
obtained by this process.

An alternative is pair production within the jet base via collisions of
energetic photons originating in the accretion flow (Henri & Pelletier, 1991;
Beloborodov, 1999; Levinson & Rieger, 2011; Aharonian et al., 2017; Sikora et
al., 2020). Figure 1 presents an illustration of the assumed geometry (see
also fig. 3 in Zdziarski et al. 2022, hereafter Z22). The rate of the pairs
produced at the base can be compared with the flow rate of non-thermal
relativistic electrons emitting synchrotron emission far downstream in the
jet. Such a comparison was done for the microquasar MAXI J1820+070, where
these two rates were found to be of the same order (Z22).

The ability of loading jets by pairs at their bases was also studied for AGNs
using blazar and/or core-shift models combined with data on their radio lobes
(Kang et al., 2014; Sikora, 2016; Pjanka et al., 2017). However, this can be
tested in some AGNs more directly, in a similar way as in the case of the
microquasar MAXI J1820+070, i.e., by using radio-IR spectra of compact jets
and hard X-ray/soft $\gamma$-ray spectra of their accretion disk coronae. This
is usually not possible in blazars, where the beamed jet emission dominates
over the X-rays from accretion. A suitable class of objects is broad-line
radio galaxies, where the jet is viewed from a side. We have found the broad-
line radio galaxy 3C 120 to be well suited for such calculations.

This is a well-studied, strong and prominent jet source (Marscher et al.,
2002), but at the same time its X-ray emission is dominated by its accretion
flow, as evidenced by the detection of strong reflection features (Zdziarski &
Grandi, 2001; Lohfink et al., 2013). The model of the broad-band jet spectrum
from 3C 120 by Janiak et al. (2016) also implies a weak jet contribution to
the X-rays. The source was observed about 10 times by the Oriented
Scintillation Spectroscopy Experiment (OSSE) detector (Johnson et al., 1993)
onboard of Compton Gamma Ray Observatory, whose average spectrum was published
by Woźniak et al. (1998), and compared to a number of X-ray observations at
lower energies in Zdziarski & Grandi (2001). Apart from a moderate difference
in the normalization, the OSSE spectrum (starting at 50 keV) was in a good
agreement with that from BeppoSAX (extending up to $\approx$100 keV), but
showed a power-law extension at high energies, with the detection up to
$\approx$400 keV. For the synchrotron spectrum, we use the spectral
measurements from radio to IR listed in Giommi et al. (2012).

The redshift to 3C 120 is $z_{\rm r}=0.033$, which implies the luminosity
distance of $D\approx 141$ Mpc (for $H_{0}=72$ km s-1 Mpc-1,
$\Omega_{\Lambda}=0.7$, $\Omega_{\rm M}=0.3$). The latest determination of its
BH mass is $M\approx 6.92^{+2.63}_{-2.45}\times 10^{7}{\rm M}_{\sun}$ (Grier
et al., 2017). It corresponds to the Eddington luminosity of $L_{\rm E}\approx
1.0\times 10^{46}$ erg s-1 assuming the H mass fraction of $X=0.7$, and to the
gravitational radius of $R_{\rm g}\equiv GM/c^{2}\approx 1.0\times 10^{13}$
cm. Its accretion luminosity was estimated by Ogle et al. (2005), Kataoka et
al. (2011), Zdziarski et al. (2015) and Janiak et al. (2016) as $L_{\rm
accr}\approx 1.5$, 0.8, 0.8 and $1.7\times 10^{45}$ erg s-1, respectively,
which correspond to the Eddington ratio of 0.08–0.17. Given this relatively
large ratio, the accretion efficiency, $\epsilon$, is implied to be large,
likely in the 0.1–0.3 range. The jet inclination was measured as $i\approx
20.5\pm 1.8\arcdeg$ (Jorstad et al., 2005), $i\approx 16\arcdeg$ (Agudo et
al., 2012), $i\approx 10.4\pm 2.3\arcdeg$ (Jorstad et al., 2017) and $i\approx
18.7\pm 0.5\arcdeg$ (Pushkarev et al., 2017). The average value of the jet
Lorentz factor was measured as $\Gamma\approx 5.3\pm 1.2$ (Jorstad et al.,
2005), $\Gamma\approx 10.7\pm 2.4$ (Jorstad et al., 2017) and $\Gamma\approx
7.9\pm 0.6$ (Pushkarev et al., 2017). The jet half-opening angle was estimated
as $\Theta\approx 3.8\pm 1.3\arcdeg$ (Jorstad et al., 2005), $\Theta\approx
1.2\pm 0.5\arcdeg$ (Jorstad et al., 2017) and $\Theta\approx 3.1\pm
0.1\arcdeg$ (Pushkarev et al., 2017). Based on the above estimates, we will
use in this work the following values,

$\displaystyle i\approx 15\pm 5\arcdeg,\quad\Gamma\approx 7.5\pm
2.5,\quad\Theta\approx 2\pm 1\arcdeg,$ (1) $\displaystyle L_{\rm accr}\approx
1.0\pm 0.2\times 10^{45}{\rm erg/s},\quad\epsilon\approx 0.2\pm 0.1.$ (2)

Figure 1: A sketch of of the pair-producing geometry based on fig. 2a of
Barkov & Komissarov (2008), which shows the result of their GRMHD simulation
for strongly magnetized accretion on a BH with the mass of $3{\rm M}_{\sun}$
and the spin parameter of $a_{*}=0.9$. The contours show the magnetic field
lines, the arrows show the velocity field, and the color contours show the
baryonic rest mass density, $\rho$, in a logarithmic scale in $\log_{10}\rho$
in units of g cm-3. The jet base, shown in blue, has a very low matter
density. We show e± pair production within the jet base with the arrows
representing pair-producing photons and the pairs.

##  2 The spectra and model

Figure 2 shows the radio to IR spectrum of the core of 3C 120, representing
the inner and steady part of the jet. While the shown measurements were not
simultaneous, we see relatively little scatter among them, indicating that the
core compact jet is indeed stable. Its radio to IR spectrum has a power-law
part approximately flat in $F_{\nu}$ (below $\sim\\!10^{12}$ Hz), followed by
a significantly steeper power law at higher frequencies. This implies the
presence of a break frequency in-between. The low-$\nu$ part is usually
interpreted as coming from jet synchrotron emission being partially self-
absorbed (Blandford & Königl, 1979; Königl, 1981), while the part above the
break frequency comes from synchrotron emission being optically thin in the
entire emitting part of the jet. There is a possible contribution in the IR
from the molecular torus. However, it appears to be negligibly small in 3C
120, which we have inferred based on the quasar template in Richards et al.
(2006).

The radio-to-submm part of the spectrum forms a hard power law, which we
fitted111Some of the fluxes we compiled from the literature had no measurement
errors, and for the $\chi^{2}$ fit (weighted by inverse squares of the errors)
we assumed them to equal 5% of the flux. as $F_{\nu}\propto\nu^{\alpha}$ with
$\alpha\approx-0.20\pm 0.04$ (where the uncertainty is $1\sigma$). This fits
well its entire range of 74 MHz–226 GHz. However, an extrapolation of this
power law to the IR band intersects the softer power law at a frequency a
factor of a few above its minimum measured energy. This shows the overall
spectrum is more complex than that implied by the model of Blandford & Königl
(1979). Namely, the $\alpha\approx-0.20$ radio-to-submm power law has to
change to a harder one in order to connect to the lowest IR point. Such
changes are related to the rates of the changes of the density of the
relativistic electrons and of the magnetic field strength, see, e.g., Königl
(1981), Zdziarski et al. (2019). However, our goal here is to estimate the
rate of the electron flow at the onset of the emitting region (with
$z\propto\nu^{-1}$) and we do not need to consider this complexity. Thus, we
approximate the overall spectrum as a once-broken power law, with the two
parts intersecting around the break frequency. The hard power law has
$\alpha_{\rm thick}=0$, and the power law fitted to the IR part of the
spectrum has $\alpha_{\rm thin}\approx-1.09\pm 0.02$. This is shown by the
dashed line in Figure 2, where we see it still provides a fair approximation
to the entire spectrum.

In our modelling, we follow the formulation of the partially synchrotron self-
absorbed conical jet model given in Z22, which we summarize in Appendix A
below. The break frequency, $\nu_{0}$, is defined as corresponding to a unit
optical depth at the base of the emitting part, $z_{0}$. The broken-power law
approximation is given by Equation (A1). For 3C 120, we find $\nu_{0}$ and
$F_{\nu}$ at $\nu_{0}$, denoted hereafter as $F_{\nu_{0}}$, as

$\nu_{0}\approx 3.15\times 10^{3}\,{\rm GHz},\quad F_{\nu_{0}}\approx
3.0\,{\rm Jy}.$ (3)

The hard index of $\alpha_{\rm thick}=0$ corresponds to the canonical
(Blandford & Königl, 1979) power law dependencies of the electron number
density and the magnetic field strength,
$n(z,\gamma)=n_{0}(z/z_{0})^{-a}\gamma^{-p}$ (where $\gamma$ is the electron
Lorentz factor), $B(z)=B_{0}(z/z_{0})^{-b}$, respectively, with $b=a/2=1$. The
soft index of $\alpha_{\rm thin}=-1.09$ corresponds to the power law index of
the synchrotron-emitting electrons of $p=1-2\alpha_{\rm thin}\approx 3.18$. We
also need to set the minimum and maximum values of $\gamma$, which we assume
to be $\gamma_{\rm min}=2$ (approximately the minimum for which the
relativistic formulae are valid) and $\gamma_{\rm max}=10^{4}$. These values
give a low average electron kinetic energy of $\approx\\!\\!2.7m_{\rm
e}c^{2}$, where $m_{\rm e}$ is the electron mass. Given the relative steepness
of the electron distribution, all derived quantities weakly depend on
$\gamma_{\rm max}$. Furthermore, we assume the ratio between the energy
densities of the matter (kinetic only) and magnetic field of $\beta_{\rm
eq}=1$, where we further assume the only contribution to the matter kinetic
energy density is from the relativistic electrons, see Equation (A5). We
stress, however, that a contribution from ions is possible.

Figure 2: The radio to IR spectrum of the core of 3C 120 from various
measurements compiled in Giommi et al. (2012). The red dashed line gives a fit
of the broken power law approximation to the partially synchrotron self-
absorbed jet spectrum, Equation (A1).

Figure 3: The blue symbols show the average 1992–97 Compton Gamma Ray
Observatory OSSE spectrum of 3C 120 from Woźniak et al. (1998). The red line
gives the best power-law fit in the 50–500 keV range. For comparison, we show
the 2004–2013 average spectrum (Oh et al., 2018) from the Neil Gehrels Swift
BAT (fitted separately by a power law, not shown).

In order to estimate the uncertainty ranges of our derived quantities, we find
their extreme values by varying the values of $i$, $\Gamma$, $\Theta$ and
$L_{\rm accr}$ in the ranges given in Equations (1–2). Some of those ranges
are large, violating the assumptions behind the standard propagation of
errors, and thus the above procedure gives more realistic, as well as
conservative, error ranges.

Knowing $\nu_{0}$, $F_{\nu_{0}}$, $p$ (from the observed spectrum), and $D$,
$\Gamma$, $i$, $\Theta$ (from other observations, Section 1), as well as
assuming $\gamma_{\rm min}$, $\gamma_{\rm max}$, $\beta_{\rm eq}$, we can use
standard formulae for the synchrotron emission and absorption and the
relativistic transformations and determine the distance of the onset of the
emission, $z_{0}$, and the magnetic field strength there, $B_{0}$, see
Equations (A3) and (A4), respectively. For our estimated values, we find

$\displaystyle z_{0}\approx 2.0_{-0.5}^{+1.3}\\!\times\\!10^{16}\,{\rm
cm}\approx 1.9_{-0.4}^{+1.3}\\!\times\\!10^{3}R_{\rm g},$ (4) $\displaystyle
B_{0}\approx 47_{-17}^{+31}\,{\rm G}.$ (5)

The dependence of $z_{0}$ on $\beta_{\rm eq}$ is very weak, as
$\propto\beta_{\rm eq}^{-0.05}$, and also the dependence on $\gamma_{\rm min}$
is weak. Our estimate of $z_{0}$ is consistent with the existing estimates of
the location of blazar zones (where the bulk of the high-energy emission
originates), which are found at distances between those of the broad-line
regions and the molecular torii, $\sim\\!\\!10^{3}$–$10^{5}R_{\rm g}$ (e.g.,
Madejski & Sikora 2016).

We can compare the above estimate of $B_{0}(z_{0})$ with that using
measurements of the core shift, which is an angular displacement of the
position of the radio core between two frequencies (Lobanov, 1998). It appears
that the only such measurements for 3C 120 are those of the angular shift
between 2.3 and 8.6 GHz of $1.196\pm 0.018$ and $0.956\pm 0.016$ mas by
Kovalev et al. (2008). To estimate the magnetic field from them, we use eq.
(7) of Zdziarski et al. (2015). Using the range of the measured core shift and
the assumed $\Gamma$, $i$ and $\Theta$, we find $B_{0}(z_{0})\approx 27$ and
23 G, respectively, with the total uncertainty range of 18–34 G, i.e., about
50% of the values estimated from the break frequency and the flux. Since the
two sets of values are estimated based on frequencies separated by a factor of
$\sim\\!10^{2}$, we consider them to be in a relatively good agreement.

Given the values of $B_{0}$ and $\nu_{0}$, we can estimate the electron
Lorentz factor corresponding to the bulk of the emission in the partially
self-absorbed part, see eq. (16) of Z22,

$\gamma_{\rm bulk}\approx\left(\frac{B_{\rm cr}}{B_{0}}\frac{h\nu_{0}}{\delta
m_{\rm e}c^{2}}\right)^{1/2}\approx 90^{+10}_{-10},$ (6)

where $\delta$ is the Doppler factor, $B_{\rm cr}$ is the critical magnetic
field strength, and the numerical value is for our determined parameters. For
the validity of the assumed model, $\gamma_{\rm min}$ needs to be a factor of
at least a few lower than $\gamma_{\rm bulk}$. This shows $\gamma_{\rm min}$
needs to be relatively low, but it could still be larger than our assumed
$\gamma_{\rm min}=2$.

Next, we can calculate the rate of the flow of electrons through the jet,
$\dot{N}_{\rm e}$, which is given by the relativistic number flux, see
Equation (A7). Substituting our assumed values and obtained solutions, we find

$\dot{N}_{\rm e}\approx 2.7_{-2.3}^{+12}\times 10^{49}\,{\rm
s}^{-1}\propto\Theta^{\frac{8+2p}{13+2p}}\gamma_{\rm
min}^{\frac{7-12p}{13+2p}},$ (7)

which is an increasing function of $\Gamma$, $\Theta$ and $i$. The dependence
on $\gamma_{\rm min}$ is approximate, valid for $\gamma_{\rm min}\gg 1$.

We then compare the rate of Equation (7) to an estimate of the pair production
rate within the jet base by collisions of photons from the accretion flow. We
use the average OSSE spectrum obtained by Woźniak et al. (1998), shown on
Figure 3. For comparison, we also show the average
spectrum222https://swift.gsfc.nasa.gov/results/bs105mon/226 from Neil Gehrels
Swift Burst Alert Telescope (BAT; Oh et al. 2018). Apart from a slightly
higher normalization, its part at $\gtrsim$50 keV is compatible with the
average from the OSSE. We have fitted the OSSE spectrum by a power law, shown
by the red line. We find $\alpha_{\rm X}\approx-1.11^{+0.27}_{-0.26}$ and the
normalization of $EF_{E}(511\,{\rm keV})\approx 0.015^{+0.07}_{-0.05}$
keV/(cm2 s), where the uncertainties are $1\sigma$ (but $\alpha_{\rm X}$ and
$EF_{E}$ are strongly correlated). This gives us the density of photons within
the jet base, Equation (A10). We then calculate the rate of pair production by
photons with such a power law, which also depends on the assumed
characteristic sizes of the hot plasma, $R_{\rm hot}$, and the jet base,
$R_{\rm jet}$, see Equation (A11). From it, we obtain,

$2\dot{N}_{+}\approx 3.9^{+6.3}_{-2.8}\times 10^{49}\left(\frac{R_{\rm
hot}}{10R_{\rm g}}\right)^{-1}\left(\frac{R_{\rm jet}}{R_{\rm
hot}}\right)^{2}{\rm s}^{-1}.$ (8)

(The given uncertainty is for the $1\sigma$ joint $\alpha_{\rm
X}$–$EF_{E}(511\,{\rm keV})$ error contour.) This pair production rate is then
balanced by the sum of the rates of pair annihilation and pair advection. The
equilibrium Thomson optical depth of the pairs is of the order of unity.
Still, the advection downstream along the jet is likely to dominate since the
pairs are produced with a substantial net momentum along the jet axis
(Beloborodov, 1999).

We see that the two rates are similar. This is a remarkable coincidence since
the two numbers have been derived with very different physics. This shows that
pair production by accretion photons is fully capable to provide most of the
leptons responsible for the jet synchrotron emission. The same conclusion was
reached for the hard spectral state of the accreting X-ray binary MAXI
J1820+070.

Another argument for the presence of e± pairs in the source is given by a
calculation of the jet power, $P_{\rm j}$. The total jet power in the
framework of the theoretically predicted (Bisnovatyi-Kogan & Ruzmaikin, 1974,
1976; Narayan et al., 2003) and numerically confirmed (Punsly et al., 2009;
Tchekhovskoy et al., 2011; McKinney et al., 2012) magnetically arrested disk
(MAD) scenario is

$P_{\rm j}\approx 1.3\left(\phi_{\rm BH}/50\right)^{2}a_{*}^{2}\dot{M}c^{2},$
(9)

where $\dot{M}$ is the accretion rate, $a_{*}$ is the BH spin parameter, and
$\phi_{\rm BH}$ is a dimensionless magnetic flux threading the BH on one
hemisphere related to the dimensional flux by $\Phi_{\rm BH}=\phi_{\rm
BH}(\dot{M}c)^{1/2}R_{\rm g}$. GRMHD simulations have shown that $\phi_{\rm
BH}\lesssim 50h_{\rm 0.3}^{1/2}$, where $h_{0.3}$ is a dimensionless disk
half-thickness defined by $H_{\rm disk}=R_{\rm disk}0.3h_{\rm 0.3}$ (Davis &
Tchekhovskoy, 2020). Noting that $a_{*}<1$ and $h_{\rm 0.3}\lesssim 1$,
$P_{\rm j}\lesssim\dot{M}c^{2}\equiv L_{\rm accr}/\epsilon$, which is also a
general, model-independent, constraint. In 3C 120, we find

$P_{\rm j}\lesssim 5\pm 1\times 10^{45}\left(\epsilon/0.2\right)^{-1}{\rm
erg\,s}^{-1}.$ (10)

In our model, we can then calculate the jet power in the magnetic field and
electrons, $P_{B{\rm e}}$, using Equation (A9),

$P_{B{\rm e}}\approx 1.5_{-1.3}^{+10.2}(1/2+\beta_{\rm eq}/3)\times
10^{45}{\rm erg\,s}^{-1}.$ (11)

The power in cold ions associated with the electrons not in e± pairs is given
by Equation (A8). We obtain

$P_{\rm i}\approx 3.1^{+19.6}_{-2.8}\times 10^{47}\left(1-2n_{+}/n_{\rm
e}\right){\rm erg\,s}^{-1},$ (12)

where $n_{+}$ and $n_{\rm e}$ are the densities of the positrons and of all
leptons, respectively, and in steady state $n_{+}/n_{\rm
e}\approx\dot{N}_{+}/\dot{N}_{\rm e}$. Both $P_{B{\rm e}}$ and $P_{\rm i}$
increase with $\Gamma$, $i$ and $\Theta$, while $P_{B{\rm e}}/P_{\rm i}\approx
0.005/(1-2n_{+}/n_{\rm e})$ (for $\beta_{\rm eq}\approx 1$) remains
approximately constant. Thus, the power in the electrons and $B$ is negligible
compared to that in the ions unless the jet consists of almost pure pair
plasma. Then, $P_{\rm i}/P_{\rm j}\gtrsim 62^{+860}_{-60}(1-2n_{+}/n_{\rm
e})$. Thus, the presence of pairs is required by the physical requirement of
$P_{\rm i}\leq P_{\rm j}$, implying $1-2n_{+}/n_{\rm e}\ll 1$ (i.e., a pair-
dominated jet) in most of the parameter space. On the other hand, $P_{\rm i}$
is approximately $\propto\gamma_{\rm min}^{(7-12p)/(13+2p)}$, and thus the
pair abundance, $2n_{+}/n_{\rm e}$, can be close to null if $\gamma_{\rm
min}\gg 2$.

The above comparison also implies the jet power of 3C 120 can be near the
maximal possible power. In the case of the jet extracting the BH rotational
power, we can test this conjecture by comparing the magnetic flux threading
the BH with that estimated from the observed synchrotron emission downstream
in the jet at a $z\geq z_{0}$. The latter was determined for a sample of radio
loud AGNs in Zamaninasab et al. (2014) and Zdziarski et al. (2015). We use eq.
(21) in Zdziarski et al. (2015) for the quantities at $z_{0}$,

$\Phi_{\rm j}=2^{3/2}\pi R_{\rm H}sz_{0}B_{0}(1+\sigma_{0})^{1/2}/(\ell
a_{*}),$ (13)

where $R_{\rm H}=[1+(1-a_{*}^{2})^{1/2}]R_{\rm g}$ is the BH horizon radius,
$\ell\lesssim 0.5$ is the ratio of the angular frequencies of the magnetic
field lines dragged around the BH and the BH, $s\lesssim 1$ is defined by

$\sigma_{0}\equiv(\Theta\Gamma/s)^{2}$ (14)

(Komissarov et al., 2009; Tchekhovskoy et al., 2009), and $\sigma_{0}$ is the
magnetization parameter at $z_{0}$. In our estimates below, we adopt $a_{*}=1$
and $\ell=0.5$. We find

$\displaystyle\Phi_{\rm BH}\approx 2.1\pm 0.2(\phi_{\rm
BH}/50)\left(\epsilon/0.2\right)^{-1/2}10^{32}{\rm G\,cm}^{2},$ (15)
$\displaystyle\Phi_{\rm j}\approx
1.7_{-0.9}^{+3.0}s(1+\sigma_{0})^{1/2}10^{32}\,{\rm G\,cm}^{2}.$ (16)

We thus find here another remarkable coincidence (in agreement with that of
Zamaninasab et al. 2014 and Zdziarski et al. 2015), of the two values of the
magnetic flux fully compatible with being identical for $\phi_{\rm BH}\sim 50$
and $\sigma_{0}\lesssim 1$, expected at $z\gtrsim z_{0}$. Thus, the jet power
can be maximal, corresponding to a jet ejected from a system with a MAD
accretion flow and a maximally rotating BH.

Now we consider the value of the magnetization parameter, $\sigma_{0}$. By
definition,

$\sigma_{0}\equiv\frac{B_{0}^{2}/4\pi}{\eta u_{\rm
k,0}+\rho_{0}c^{2}}\approx\frac{B_{0}^{2}/4\pi}{n_{0}f_{N}\mu_{\rm e}m_{\rm
p}c^{2}(1-2n_{+}/n_{\rm e})},$ (17)

where $\eta$ is the particle adiabatic index, $u_{\rm k,0}$ is the kinetic
energy density of particles at $z_{0}$, $\rho_{0}c^{2}$ is the rest-energy
density at $z_{0}$, $f_{N}$ is given by Equation (A6), $\mu_{\rm e}=2/(1+X)$
is the mean electron molecular weight, and $m_{\rm p}$ is the proton mass. In
our case, $u_{\rm k,0}\ll\rho_{0}c^{2}$, e.g., 1/600 of the latter for our
default parameters and in the absence of pairs, which in turn leads to the
approximate equality on the right-hand side. For our assumed parameters, we
find using Equations (17) and (14),

$\displaystyle\sigma_{0}\approx\frac{2.5\times 10^{-3}}{\beta_{\rm
eq}(1-2n_{+}/n_{\rm e})},$ (18) $\displaystyle\sigma_{0}\approx
6.9^{+20.5}_{-6.1}s^{-2}\times 10^{-2},$ (19)

respectively, where the former is independent of $\Gamma$, $i$ and $\Theta$.
Setting the two values of $\sigma_{0}$ equal implies $(1-2n_{+}/n_{\rm
e})\beta_{\rm eq}\ll s^{2}$, i.e., a pair dominated plasma for most of the
parameter space. For our default values, $2n_{+}/n_{\rm e}\approx 0.96$, i.e.,
there are about 25 times more leptons in pairs than in the ionization
electrons. We also note that our estimated values of $\sigma_{0}$ are $\ll$1.

##  3 Summary and Discussion

We have found that the composition of the jet in 3C 120 is most likely
dominated by e± pairs. First, a sufficient amount of pairs can be produced by
collisions of hard X-rays/soft $\gamma$-rays emitted by the accretion flow, as
indicated by the estimated rate of pair production being approximately equal
to the rate of the flow of synchrotron-emitting electrons downstream in the
jet. Second, the high pair content is indicated by the calculation of the jet
power, which is dominated by that in the bulk motion of the ions. If the ion
density directly corresponds to that of the synchrotron-emitting electrons,
the jet power would greatly exceed its maximum possible value, $\dot{M}c^{2}$,
but it can be $\lesssim\dot{M}c^{2}$ if most of the emitting leptons are those
in e± pairs. Third, the estimates of the magnetization parameter from the flow
parameters and from the product of $\Theta$ and $\Gamma$ agree only at a high
pair content. However, apart from the implied result that the pair content,
$2n_{+}/n_{\rm e}$ is $\approx$1, we cannot determine precisely its value
because the jet parameters for this source are relatively loosely constrained.

In our calculations, we have assumed the jet is conical in the dissipation
zone, at $z\geq z_{0}$, and we have estimated $z_{0}$ to equal a few times
$10^{16}$ cm. On the other hand, Kovalev et al. (2020) found that the jet in
3C 120 becomes conical only at distances about 100 times larger. A departure
from the conical shape would, however, affect mostly the value of $z_{0}$. On
the other hand, the parameters of the flow are determined by the jet radius,
i.e., $\Theta z_{0}$, which is sensitive mostly to the local synchrotron
emission and absorption, and are only weakly affected by the jet shape.
Furthermore, we have found the radio-to-submm spectrum forms an almost pure
power law at 74 MHz–226 GHz, which indicates that the shape of the jet in the
emitting range, which starts relatively close to $z_{0}$ (since
$z\propto\nu^{-1}$ and 226 GHz is $\sim 0.1\nu_{0}$), is most likely conical
and cannot have a substantial curvature.

Another uncertainty in the assumptions regards the low-energy end of the
electron distribution. On one hand, the value of $\gamma_{\rm min}$ is
uncertain and can be higher than our assumed $\gamma_{\rm min}=2$. In this
case, it will be even easier for the pair production at the jet base to
produce enough leptons for the synchrotron emission, while it would reduce the
jet power. On the other hand, we expect the presence of a quasi-thermal
electron distribution below $\gamma_{\rm min}$ (regardless of its value),
which would increase the rate of the electron flow, $\dot{N}_{\rm e}$.
Depending on the relative fraction of the electrons in the quasi-thermal part,
the value of $\gamma_{\rm min}$ and other jet parameters, this may result in
the pair production rate being not sufficient to account for the full flow.
Such a quasi-thermal distribution is expected in the presence of shock
acceleration (e.g., Sironi & Spitkovsky 2009; Sironi et al. 2015; Crumley et
al. 2019). For acceleration via magnetic field reconnection, simulations
performed for $\sigma\geq 1$ show the quasi-thermal part to be modest, with
the lepton number still dominated by the power-law part (e.g., Comisso &
Sironi 2018, 2019; Petropoulou et al. 2019). Furthermore, the presence of
substantial excesses of cold electrons in jets of radio-loud AGNs (hence also
in 3C 120) is ruled out by the observed absence of soft X-ray excesses in
spectra of blazars. Such an excess would result from Comptonization of
external UV radiation by the cold electrons in a relativistic jet (Sikora et
al., 1997).

## Acknowledgements

We thank G. Henri, A. Levinson, P.-O. Petrucci and T. Savolainen for valuable
discussions, the referee for a valuable suggestion, M. Barkov for the
permission to use his figure in our illustration of the effect of pair
production, and P. Lubiński and N. Żywucka for help with software. We
acknowledge support from the Polish National Science Center under the grants
2015/18/A/ST9/00746 and 2019/35/B/ST9/03944. The work of M.B. was supported in
part by the National Research Foundation333Any opinion, finding and conclusion
or recommendation expressed in this material is that of the authors, and the
NRF does not accept any liability in this regard. of South Africa (DSI/NRF
SARChI Programme, Grant No. 64789).

##  Appendix A Formulae

In order to make this work self-contained, we present the formulae from Z22
and Zdziarski et al. (2021) that are used here. In Z22, they were given for
arbitrary power-law dependencies of $n(z)$ and $B(z)$, while here we give them
for the canonical case of $b=a/2=1$.

We rewrite eqs. (12–13) of Z22 as

$\displaystyle F_{\nu}\simeq
F_{\nu_{0}}\min\left[1,\frac{5(\nu/\nu_{0})^{\frac{1-p}{2}}}{(p-1)\Gamma\left(\frac{p-1}{p+4}\right)}\right],$
(A1)
$\displaystyle\tau(\nu,z)=\left[(\nu/\nu_{0})(z/z_{0})\right]^{-(p+4)/2},$
(A2)

which give a broken power-law aproximation to the synchrotron spectrum and the
self-absorption optical depth, respectively. Here $\nu_{0}$ is defined by
$\tau(\nu_{0},z_{0})=1$, i.e., the jet has $\tau<1$ at $\nu>\nu_{0}$ in the
entire synchrotron-emitting part, $z\geq z_{0}$, and the two power laws
intersect at a $\nu<\nu_{0}$. Note that
$F_{\nu_{0}}\equiv(2/5)\Gamma\left[(p-1)/(p+4)\right]F_{0}$, where $F_{0}$ is
defined by eq. (7) of Z22. We then use eqs. (A2) and (A4) of Z22, which give
the solutions for $z_{0}$ and $B_{0}$, respectively,

$\displaystyle
z_{0}=\frac{1}{\delta^{\frac{4+p}{13+2p}}\nu_{0}}\left[\frac{c(1+k_{\rm
i})(f_{E}-f_{N})}{2\pi\beta_{\rm eq}}\right]^{\frac{1}{13+2p}}\times$
$\displaystyle\left[\frac{C_{2}(p)}{\sin
i}\right]^{\frac{5+p}{13+2p}}\\!\\!\left[\frac{5F_{\nu_{0}}D^{2}}{m_{\rm
e}C_{1}(p)\Gamma\left(\frac{p-1}{p+4}\right)}\right]^{\frac{6+p}{13+2p}}\times$
(A3)
$\displaystyle\left(\frac{3}{\pi\tan\Theta}\right)^{\frac{7+p}{13+2p}}(1+z_{\rm
r})^{-\frac{19+3p}{13+2p}},$ $\displaystyle
B_{0}=\left[\frac{3C_{1}(p)\Gamma\left(\frac{p-1}{p+4}\right)(1+k_{\rm
i})^{2}(f_{E}-f_{N})^{2}\sin^{3}i}{5C_{2}(p)^{3}\beta_{\rm
eq}^{2}D^{2}F_{\nu_{0}}\tan\Theta}\right]^{\frac{2}{13+2p}}$
$\displaystyle\times\frac{\nu_{0}\pi^{\frac{7+2p}{13+2p}}2^{\frac{9+2p}{13+2p}}c^{\frac{17+2p}{13+2p}}m_{\rm
e}^{\frac{15+2p}{13+2p}}(1+z_{\rm
r})^{\frac{15+2p}{13+2p}}}{e\delta^{\frac{3+2p}{13+2p}}},$ (A4)

where $e$ is the electron charge, $C_{1,2}$ are constants defined in eqs. (8)
and (9) of Z22, respectively,

$\displaystyle\beta_{\rm eq}={n_{0}m_{\rm e}c^{2}(1+k_{\rm
i})(f_{E}-f_{N})\over B_{0}^{2}/8\pi},$ (A5) $\displaystyle f_{E}\equiv$
$\displaystyle\begin{cases}{\gamma_{\rm max}^{2-p}-\gamma_{\rm min}^{2-p}\over
2-p},&p\neq 2;\cr\ln{\gamma_{\rm max}\over\gamma_{\rm
min}},&p=2,\cr\end{cases}\quad f_{N}\equiv\frac{\gamma_{\rm
min}^{1-p}-\gamma_{\rm max}^{1-p}}{p-1},$ (A6)

and $k_{\rm i}$ accounts for the kinetic energy density in particles other
than the power-law electrons.

The total lepton flow rate and the usable power in ions are

$\displaystyle\dot{N}_{\rm e}$ $\displaystyle\approx 2\pi
n_{0}f_{N}c\beta\Gamma(z0\tan\Theta)^{2},$ (A7) $\displaystyle P_{\rm i}$
$\displaystyle=\mu_{\rm e}m_{\rm p}c^{2}(\Gamma-1)\left(\dot{N}_{\rm
e}-2\dot{N}_{+}\right),$ (A8)

respectively, where $\beta$ is the dimensionless bulk velocity, and both the
jet and counterjet are taken into account, see eqs. (25–26) of Z22. The power
in the relativistic electrons and magnetic fields is, see eq. (23) of Z22,

$P_{B{\rm e}}=\left(\frac{1}{2}+\frac{\beta_{\rm
eq}}{3}\right)\\!c\beta(B_{0}z_{0}\Gamma\tan\Theta)^{2}.$ (A9)

We then provide the formalism for calculating the pair production rate, based
on Zdziarski et al. (2021). We modify their eq. (1) to calculate the
differential photon density at 511 keV, $n_{1}$, above the hot disk,

$n_{1}\approx F_{E}(511\,{\rm keV})\frac{4\pi D^{2}}{2\pi R_{\rm hot}^{2}c},$
(A10)

where we divided the total rate of the photon emission by the source area
(including both sides), and both $F_{E}$ and $E$ have the same energy unit
(e.g., keV). We then use eq. (3) of Zdziarski et al. (2021), which uses the
rate derived by Svensson (1987), and approximate the pair-producing volume as
two cylinders with the height $R_{\rm hot}$ and the characteristic radius of
the jet, $R_{\rm jet}$ ($\leq R_{\rm hot}$), i.e., $V=2\pi R_{\rm
jet}^{2}R_{\rm hot}$. We then get the total pair production rate as

$\dot{N}_{+}\approx 2\pi R_{\rm jet}^{2}R_{\rm hot}n_{1}^{2}\sigma_{\rm
T}c\frac{\ln(E_{\rm c}/511\,{\rm keV})}{(1-\alpha_{\rm X})^{5/3}(2-\alpha_{\rm
X})},$ (A11)

where $\sigma_{\rm T}$ is the Thomson cross section, and $E_{\rm c}$ is the
upper cutoff of the photon power law, which we assume $\approx$2 MeV.

## References

  * Agudo et al. (2012) Agudo, I., Gómez, J. L., Casadio, C., Cawthorne, T. V., & Roca-Sogorb, M. 2012, ApJ, 752, 92, doi: 10.1088/0004-637X/752/2/92
  * Aharonian et al. (2017) Aharonian, F. A., Barkov, M. V., & Khangulyan, D. 2017, ApJ, 841, 61, doi: 10.3847/1538-4357/aa7049
  * Barkov & Komissarov (2008) Barkov, M. V., & Komissarov, S. S. 2008, MNRAS, 385, L28, doi: 10.1111/j.1745-3933.2008.00427.x
  * Beloborodov (1999) Beloborodov, A. M. 1999, MNRAS, 305, 181, doi: 10.1046/j.1365-8711.1999.02384.x
  * Bisnovatyi-Kogan & Ruzmaikin (1974) Bisnovatyi-Kogan, G. S., & Ruzmaikin, A. A. 1974, Ap&SS, 28, 45, doi: 10.1007/BF00642237
  * Bisnovatyi-Kogan & Ruzmaikin (1976) —. 1976, Ap&SS, 42, 401, doi: 10.1007/BF01225967
  * Blandford & Königl (1979) Blandford, R. D., & Königl, A. 1979, ApJ, 232, 34, doi: 10.1086/157262
  * Blandford & Znajek (1977) Blandford, R. D., & Znajek, R. L. 1977, MNRAS, 179, 433, doi: 10.1093/mnras/179.3.433
  * Comisso & Sironi (2018) Comisso, L., & Sironi, L. 2018, Phys. Rev. Lett., 121, 255101, doi: 10.1103/PhysRevLett.121.255101
  * Comisso & Sironi (2019) —. 2019, ApJ, 886, 122, doi: 10.3847/1538-4357/ab4c33
  * Crumley et al. (2019) Crumley, P., Caprioli, D., Markoff, S., & Spitkovsky, A. 2019, MNRAS, 485, 5105, doi: 10.1093/mnras/stz232
  * Davis & Tchekhovskoy (2020) Davis, S. W., & Tchekhovskoy, A. 2020, ARA&A, 58, 407, doi: 10.1146/annurev-astro-081817-051905
  * Ghisellini (2012) Ghisellini, G. 2012, MNRAS, 424, L26, doi: 10.1111/j.1745-3933.2012.01280.x
  * Giommi et al. (2012) Giommi, P., Polenta, G., Lähteenmäki, A., et al. 2012, A&A, 541, A160, doi: 10.1051/0004-6361/201117825
  * Goldreich & Julian (1969) Goldreich, P., & Julian, W. H. 1969, ApJ, 157, 869, doi: 10.1086/150119
  * Grier et al. (2017) Grier, C. J., Pancoast, A., Barth, A. J., et al. 2017, ApJ, 849, 146, doi: 10.3847/1538-4357/aa901b
  * Henri & Pelletier (1991) Henri, G., & Pelletier, G. 1991, ApJ, 383, L7, doi: 10.1086/186228
  * Janiak et al. (2016) Janiak, M., Sikora, M., & Moderski, R. 2016, MNRAS, 458, 2360, doi: 10.1093/mnras/stw465
  * Johnson et al. (1993) Johnson, W. N., Kinzer, R. L., Kurfess, J. D., et al. 1993, ApJS, 86, 693, doi: 10.1086/191795
  * Jorstad et al. (2005) Jorstad, S. G., Marscher, A. P., Lister, M. L., et al. 2005, AJ, 130, 1418, doi: 10.1086/444593
  * Jorstad et al. (2017) Jorstad, S. G., Marscher, A. P., Morozova, D. A., et al. 2017, ApJ, 846, 98, doi: 10.3847/1538-4357/aa8407
  * Kang et al. (2014) Kang, S.-J., Chen, L., & Wu, Q. 2014, ApJS, 215, 5, doi: 10.1088/0067-0049/215/1/5
  * Kataoka et al. (2011) Kataoka, J., Stawarz, Ł., Takahashi, Y., et al. 2011, ApJ, 740, 29, doi: 10.1088/0004-637X/740/1/29
  * Komissarov et al. (2009) Komissarov, S. S., Vlahakis, N., Königl, A., & Barkov, M. V. 2009, MNRAS, 394, 1182, doi: 10.1111/j.1365-2966.2009.14410.x
  * Königl (1981) Königl, A. 1981, ApJ, 243, 700, doi: 10.1086/158638
  * Kovalev et al. (2008) Kovalev, Y. Y., Lobanov, A. P., Pushkarev, A. B., & Zensus, J. A. 2008, A&A, 483, 759, doi: 10.1051/0004-6361:20078679
  * Kovalev et al. (2020) Kovalev, Y. Y., Pushkarev, A. B., Nokhrina, E. E., et al. 2020, MNRAS, 495, 3576, doi: 10.1093/mnras/staa1121
  * Levinson & Rieger (2011) Levinson, A., & Rieger, F. 2011, ApJ, 730, 123, doi: 10.1088/0004-637X/730/2/123
  * Liodakis et al. (2022) Liodakis, I., Blinov, D., Potter, S. B., & Rieger, F. M. 2022, MNRAS, 509, L21, doi: 10.1093/mnrasl/slab118
  * Lobanov (1998) Lobanov, A. P. 1998, A&A, 330, 79.  https://arxiv.org/abs/astro-ph/9712132
  * Lohfink et al. (2013) Lohfink, A. M., Reynolds, C. S., Jorstad, S. G., et al. 2013, ApJ, 772, 83, doi: 10.1088/0004-637X/772/2/83
  * Madejski & Sikora (2016) Madejski, G. G., & Sikora, M. 2016, ARA&A, 54, 725, doi: 10.1146/annurev-astro-081913-040044
  * Marscher et al. (2002) Marscher, A. P., Jorstad, S. G., Gómez, J.-L., et al. 2002, Nature, 417, 625, doi: 10.1038/nature00772
  * McKinney et al. (2012) McKinney, J. C., Tchekhovskoy, A., & Blandford, R. D. 2012, MNRAS, 423, 3083, doi: 10.1111/j.1365-2966.2012.21074.x
  * Mościbrodzka et al. (2011) Mościbrodzka, M., Gammie, C. F., Dolence, J. C., & Shiokawa, H. 2011, ApJ, 735, 9, doi: 10.1088/0004-637X/735/1/9
  * Narayan et al. (2003) Narayan, R., Igumenshchev, I. V., & Abramowicz, M. A. 2003, PASJ, 55, L69, doi: 10.1093/pasj/55.6.L69
  * Nokhrina et al. (2015) Nokhrina, E. E., Beskin, V. S., Kovalev, Y. Y., & Zheltoukhov, A. A. 2015, MNRAS, 447, 2726, doi: 10.1093/mnras/stu2587
  * Ogle et al. (2005) Ogle, P. M., Davis, S. W., Antonucci, R. R. J., et al. 2005, ApJ, 618, 139, doi: 10.1086/425894
  * Oh et al. (2018) Oh, K., Koss, M., Markwardt, C. B., et al. 2018, ApJS, 235, 4, doi: 10.3847/1538-4365/aaa7fd
  * Petropoulou et al. (2019) Petropoulou, M., Sironi, L., Spitkovsky, A., & Giannios, D. 2019, ApJ, 880, 37, doi: 10.3847/1538-4357/ab287a
  * Pjanka et al. (2017) Pjanka, P., Zdziarski, A. A., & Sikora, M. 2017, MNRAS, 465, 3506, doi: 10.1093/mnras/stw2960
  * Punsly et al. (2009) Punsly, B., Igumenshchev, I. V., & Hirose, S. 2009, ApJ, 704, 1065, doi: 10.1088/0004-637X/704/2/1065
  * Pushkarev et al. (2017) Pushkarev, A. B., Kovalev, Y. Y., Lister, M. L., & Savolainen, T. 2017, MNRAS, 468, 4992, doi: 10.1093/mnras/stx854
  * Richards et al. (2006) Richards, G. T., Lacy, M., Storrie-Lombardi, L. J., et al. 2006, ApJS, 166, 470, doi: 10.1086/506525
  * Sikora (2016) Sikora, M. 2016, Galaxies, 4, 12, doi: 10.3390/galaxies4030012
  * Sikora et al. (1997) Sikora, M., Madejski, G., Moderski, R., & Poutanen, J. 1997, ApJ, 484, 108, doi: 10.1086/304305
  * Sikora et al. (2020) Sikora, M., Nalewajko, K., & Madejski, G. M. 2020, MNRAS, 499, 3749, doi: 10.1093/mnras/staa3128
  * Sironi et al. (2015) Sironi, L., Petropoulou, M., & Giannios, D. 2015, MNRAS, 450, 183, doi: 10.1093/mnras/stv641
  * Sironi & Spitkovsky (2009) Sironi, L., & Spitkovsky, A. 2009, ApJ, 698, 1523, doi: 10.1088/0004-637X/698/2/1523
  * Snios et al. (2018) Snios, B., Nulsen, P. E. J., Wise, M. W., et al. 2018, ApJ, 855, 71, doi: 10.3847/1538-4357/aaaf1a
  * Svensson (1987) Svensson, R. 1987, MNRAS, 227, 403
  * Tchekhovskoy et al. (2009) Tchekhovskoy, A., McKinney, J. C., & Narayan, R. 2009, ApJ, 699, 1789, doi: 10.1088/0004-637X/699/2/1789
  * Tchekhovskoy et al. (2011) Tchekhovskoy, A., Narayan, R., & McKinney, J. C. 2011, MNRAS, 418, L79, doi: 10.1111/j.1745-3933.2011.01147.x
  * Woźniak et al. (1998) Woźniak, P. R., Zdziarski, A. A., Smith, D., Madejski, G. M., & Johnson, W. N. 1998, MNRAS, 299, 449, doi: 10.1046/j.1365-8711.1998.01831.x
  * Zamaninasab et al. (2014) Zamaninasab, M., Clausen-Brown, E., Savolainen, T., & Tchekhovskoy, A. 2014, Nature, 510, 126, doi: 10.1038/nature13399
  * Zdziarski & Grandi (2001) Zdziarski, A. A., & Grandi, P. 2001, ApJ, 551, 186, doi: 10.1086/320064
  * Zdziarski et al. (2015) Zdziarski, A. A., Sikora, M., Pjanka, P., & Tchekhovskoy, A. 2015, MNRAS, 451, 927, doi: 10.1093/mnras/stv986
  * Zdziarski et al. (2019) Zdziarski, A. A., Stawarz, Ł., & Sikora, M. 2019, MNRAS, 485, 1210, doi: 10.1093/mnras/stz475
  * Zdziarski et al. (2022) Zdziarski, A. A., Tetarenko, A. J., & Sikora, M. 2022, ApJ, 925, 189, doi: 10.3847/1538-4357/ac38a9
  * Zdziarski et al. (2021) Zdziarski, A. A., Jourdain, E., Lubiński, P., et al. 2021, ApJ, 914, L5, doi: 10.3847/2041-8213/ac0147